<%BANNER%>

Nonlinear Approximation Techniques to Solve Network Flow Problems with Nonlinear Arc Cost Functions

xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID E20110209_AAAACC INGEST_TIME 2011-02-09T12:38:18Z PACKAGE UFE0015623_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
FILE SIZE 8423998 DFID F20110209_AAAYFS ORIGIN DEPOSITOR PATH nahapetyan_a_Page_132.tif GLOBAL false PRESERVATION BIT MESSAGE_DIGEST ALGORITHM MD5
09c30d262180e33a4ae01ce71d17f0f3
SHA-1
cf225d4fa88279b0399115fcba55e3de8cead88e
5525 F20110209_AAAXZX nahapetyan_a_Page_044thm.jpg
f3a85c2908dbef64b9e679506a8ac436
cc42a8bfec9d2c4dced4c08248fb46f6bb090599
5018 F20110209_AAAYGG nahapetyan_a_Page_115thm.jpg
f9fcaa1269e81965b1b7715a42004378
a58c8d7ad9d4223eaaa118624a62406e1dbf68dc
794942 F20110209_AAAYFT nahapetyan_a_Page_137.jp2
660ffe6735367f4c5bfc7d130388c50d
fc9b1600d9a31720fc640958de95de3b0dce9a7a
52004 F20110209_AAAXZY nahapetyan_a_Page_136.jpg
36b53ead7481d8d40dcfc5685f631bdd
284348bcd0a4e62ccfc5a2f1283d8ed71a9dc4fd
1051982 F20110209_AAAYGH nahapetyan_a_Page_015.jp2
4c79d637d4fc5e5e49fe91d4b6ab60a4
a6b59ed80911e3b61837c39b90b64cb2c99150d1
35811 F20110209_AAAYFU nahapetyan_a_Page_004.jpg
c08e7d8bedb4d0ff1ef0dbf22a966313
d4cf431adc20d15c4aff229e134fa82e9a720e82
617736 F20110209_AAAXZZ nahapetyan_a_Page_010.jp2
fff0d786dd8c3a2002406693a376d639
c71325c7566b13d3f2c9dcef115e5189cd8f1e57
F20110209_AAAYGI nahapetyan_a_Page_128.tif
307b713a4e5217b9c494ea4ae502bd76
fe88bbfa7ac3fa8fc51aab4fd0d7d414e044bc89
F20110209_AAAYFV nahapetyan_a_Page_072.tif
6c683a84ee0d603eb7d476f234e5953a
1f456269a79faba76c4833b2b6fa80ef65a1a5d5
79751 F20110209_AAAYGJ nahapetyan_a_Page_126.jp2
cc1f2d82d2bb082452e49107dd277f26
4915e3325e7c2f895d9ad21607f99670227d8b2c
F20110209_AAAYFW nahapetyan_a_Page_092.tif
e00d09175681a32466ad1b19c77e48f9
8fe4b5de9721189e4dc5e1878d6939382f654b1c
48391 F20110209_AAAYGK nahapetyan_a_Page_027.pro
439304a4abd7fbc7eb97d926a7e00339
f0418eed952105bccbf33b7a2c6d45d271c2d8ec
4943 F20110209_AAAYFX nahapetyan_a_Page_117thm.jpg
09ca6ce768ced5d41ed0fd0521d004e0
cf6f402cdd3e8a5ad7b5bdd8cd6e9231ca077d42
69923 F20110209_AAAYGL nahapetyan_a_Page_079.jpg
85d1251020bbe28ec7f0641b59143c26
6f95b81da46955d3871329961e32121ca48f8658
25460 F20110209_AAAYFY nahapetyan_a_Page_108.QC.jpg
afcec21c6a72f0526123d6c6b878bfec
0dc6e2ae787181744f3cdbb177bbc911a41ee66b
1046715 F20110209_AAAYHA nahapetyan_a.pdf
48b063583a589cdde80186ceff2c8bc8
e98d79ceaa998d1308fa9bcab63493d32cfdad95
1051976 F20110209_AAAYGM nahapetyan_a_Page_098.jp2
022298ad0b606dd120b33137253ed138
cd2c54dfaf039ea5087d999eca2dd5e9b0823a1c
397159 F20110209_AAAYFZ nahapetyan_a_Page_135.jp2
4569799e37ee3968387997607908f149
c2b0c9beca0169ab32f789dec66d4e2e40a1df36
F20110209_AAAYHB nahapetyan_a_Page_113.tif
2fae1d189d000776d3ebfedf8aa57fb2
0d6f60c2a9235ed7a2098266c45140507c31b262
2239 F20110209_AAAYGN nahapetyan_a_Page_107.txt
11d355294f8a8f7ce59beb335f1fa4da
d5f6ae59f60c50e1f25ac616171b4510dbdacd52
F20110209_AAAYHC nahapetyan_a_Page_008.tif
d1b4d0d5ab792ada7fecc9ea1a80cf0d
4ed4a6bff97b132d545dd601ebb8f0226b4f579d
1847 F20110209_AAAYGO nahapetyan_a_Page_066.txt
4b43cb87fb105c0eae7e05bc281070eb
f0155562768b268fe8742a25868dac26eaa0dcf7
6450 F20110209_AAAYHD nahapetyan_a_Page_140.QC.jpg
cc56cdc588bac04330b1fc13ac4e62ee
0906b8c26386a21deebd534b93bfc7c9b8743106
1975 F20110209_AAAYGP nahapetyan_a_Page_112.txt
0e1ebc55a031bac4b762f85aba333cab
d7a9c635b27a8d913e79e02e2873c1258e2c0514
21938 F20110209_AAAYHE nahapetyan_a_Page_090.QC.jpg
67fd0bc98878a92d6fa2839bf14e4723
25224af41bfa5a1228339cb83fd68a276bcc3f9a
158 F20110209_AAAYGQ nahapetyan_a_Page_070.txt
efb0211e9c1c18193e8cfc3077bf6eba
5a7e4c15eb480816fccbeb6704c12c8c21a24f6c
1747 F20110209_AAAYHF nahapetyan_a_Page_050.txt
955c05850c0e4065613361309de46b44
35c9eac035eef06ceebf16b84f214d38dde47256
21552 F20110209_AAAYGR nahapetyan_a_Page_040.QC.jpg
2e71a3538992bec9d091abb589fd5b9c
f08099ef5574ba8cc4cb86d6baa387a90facae28
25890 F20110209_AAAYHG nahapetyan_a_Page_009.jpg
45a719da4d4fb0287f26e7fb6f07b89c
ee66c5a4b91a27925b17fab106bbb840f94d55df
1954 F20110209_AAAYGS nahapetyan_a_Page_041.txt
d52d23032ebcd3faf2a02c04f6f85d17
62bdba6ad8124af0b82f3e9599a68eaca5a6580e
837931 F20110209_AAAYHH nahapetyan_a_Page_091.jp2
fc8e2dc131cd271723e999b59c6d19f4
5f55d2ca8d6888b9ccc943c3db067e852212863c
73436 F20110209_AAAYGT nahapetyan_a_Page_046.jpg
8d8bbfe7df6190508fba73075a969321
675cc5e17b0132ff7927b48f4b780c182a2d5466
1972 F20110209_AAAYHI nahapetyan_a_Page_069.txt
374b4829aa5eb060c25ab5b330d07e2d
be25435361017ad0a348def1b3d0bbc20c61a327
68410 F20110209_AAAYGU nahapetyan_a_Page_042.jpg
4d7a4a657f3f668bf416600b5ebcf712
17e96d34fad27646ad96ec4370aaf85be57ffeea
F20110209_AAAYHJ nahapetyan_a_Page_021.tif
222a93724bfabccdfc282707a078033b
d06221f2fec95ac2f4d6fc60ca548f528aebba19
5235 F20110209_AAAYGV nahapetyan_a_Page_122thm.jpg
96bfdf0adb62b1f6b31123dc4a3e17ba
05c17f72fa414f0f16d18a6318ff91ad2ebeeb18
F20110209_AAAYHK nahapetyan_a_Page_004.tif
03726648332cfed5aceb657ce72fec83
9e33fc2309d5815efbd6353182c68bb7b1487f74
F20110209_AAAYGW nahapetyan_a_Page_126.tif
644f2dd921870322e0a5137219db4f13
36f5f9af4025723bc128caf05d191ef812034b49
86432 F20110209_AAAYHL nahapetyan_a_Page_107.jpg
8a6af5a45489b57cfd5f92a6e2822305
edcffb3f63435fd864271a0ccad14c86102a759a
78759 F20110209_AAAYGX nahapetyan_a_Page_096.jpg
c7aa0a316848255e1f4aa8ca15010320
58ab7b327117d069b00ea67ad60ae64c3588aa8a
746864 F20110209_AAAYIA nahapetyan_a_Page_134.jp2
ea2cb2fca5b2eac19b7ccfd10d1adc4b
ef6507c959037ba3db242e7cce601c6cb822af49
24464 F20110209_AAAYHM nahapetyan_a_Page_144.QC.jpg
8c45328906be12f1396a454d6e12e472
dcf0eef6d9cb2594c3391198a68ae20140b537e6
30233 F20110209_AAAYGY nahapetyan_a_Page_052.pro
1dc4f3698899e01353ef5d1ad322ab4c
26ff87452bc0b2c8fe8193e85d144a1c68c87903
2144 F20110209_AAAYIB nahapetyan_a_Page_032.txt
24778b2f76e12f72f031e85267d0e398
6f5b2160ffeb713e0579f08232be413d66b9c0c9
1786 F20110209_AAAYHN nahapetyan_a_Page_042.txt
b32caf0ba375c59105ea68e3a359da22
095e697b8f1a2a7fb03f4490eaaa67fc778a1a2e
1919 F20110209_AAAYGZ nahapetyan_a_Page_023.txt
d28b3889f26d2cef69aef518f2a3857b
5c3b949be61e9f0281a458c933cad1ed185053a4
88447 F20110209_AAAYIC nahapetyan_a_Page_014.jpg
805bdf9d224decb07ccaf0d577170386
6896597b5c01b38ab0ec6cca7a5276f38ca744f7
41594 F20110209_AAAYHO nahapetyan_a_Page_080.jpg
6c6b9c1d332509ac576ab879002b701f
ccc207f8f573d77d109b22920eaa7c21bda11e2e
1277 F20110209_AAAYID nahapetyan_a_Page_080.txt
ffdf6e8c2126b00999527ea7615f750d
2a06471badc7f57b29c33ac067e31e3ec0c67a54
23800 F20110209_AAAYHP nahapetyan_a_Page_043.QC.jpg
a90c77e9ea0b0edbcfc709b8c9b9a2aa
97a60566a6d8cf1603d24e108a199745e149f6b4
843 F20110209_AAAYIE nahapetyan_a_Page_012.txt
902dd6fa3b98d71e109d535806e348d2
8f3a8bd8670c94fc58471f5346dcf55d85029135
1051930 F20110209_AAAYHQ nahapetyan_a_Page_122.jp2
b1dd28a1b86d6b9704a855bed8b5ea0d
17abcbd6418d23f13f3bfe232befb52a9e3678e7
90376 F20110209_AAAYIF nahapetyan_a_Page_048.jpg
48f14b04e348d5bca8753d9c1a8c932e
6f89b1e2c021d6a9b71a363382754cf379f7b807
72597 F20110209_AAAYHR nahapetyan_a_Page_115.jpg
b06de477d9bb72086020cb29969a8d2b
655691334d3fb96b03945ea6a8db21dece9fe87f
85545 F20110209_AAAYIG nahapetyan_a_Page_088.jpg
9fe7d81f97681094f5ec458448c3d1dc
06cd836da0de8171b40dd9f09b34a99313ecee92
763 F20110209_AAAYHS nahapetyan_a_Page_126thm.jpg
ecbe5c3329f4b6713a02385987b19b90
c38b08472a57c5d84e557dfcc569ba58444af0e9
92315 F20110209_AAAYIH nahapetyan_a_Page_148.jpg
4964df161ffadda53d18e287d71e2352
f30127bd77d08bd3d9001b81988db26ea1b48a99
F20110209_AAAYHT nahapetyan_a_Page_071.tif
1f9ab8d6efde3dbf405efe25b1391aec
82bb622bee344651f174c3341369590ca14fa026
53685 F20110209_AAAYII nahapetyan_a_Page_097.pro
3a0a7794ac00a11ee19c8565765f770e
d00410d2d59d4e7dd057a4d558c31a6e4e64b422
F20110209_AAAYHU nahapetyan_a_Page_036.tif
a0368d0030383d0219469d2c77453e70
165169214b21bdcf44ccae98edb80a3d08ce22ab
39401 F20110209_AAAYIJ nahapetyan_a_Page_128.jpg
01a02e7249d50b602c100af92034d4c4
a3617f34b1695f4095099ca71cc741a69b8e77f6
63685 F20110209_AAAYHV nahapetyan_a_Page_077.jpg
48749de08f473ccc355b18a99caf3c32
ecaecb943c795faa7db643d653a9984a4f1cbc5b
22680 F20110209_AAAYIK nahapetyan_a_Page_054.QC.jpg
a6890e903928d76c38b37a370742374c
afe0f988fc0cbf79ff8eba9d88f1d091c4d57f04
1019767 F20110209_AAAYHW nahapetyan_a_Page_100.jp2
433253d3c98c245c8ff075e7761bbdb5
b8331b1bc820e89b31b728542a7e01d7f00305c4
2176 F20110209_AAAYIL nahapetyan_a_Page_110.txt
625ef0875ada978f61783b2e9785f9ce
9b0984a675250a6c1c41cc7236e932ca3ebdd38b
907737 F20110209_AAAYHX nahapetyan_a_Page_022.jp2
78084c814327c8a495507be71a364974
ed5bdb53a47877607d085d91b572d31074cc0251
5046 F20110209_AAAYJA nahapetyan_a_Page_034thm.jpg
f90fea18f31c5097c4b79623b0f104ef
dd902305711c91963ac2e862ae348024b0f2687e
80686 F20110209_AAAYIM nahapetyan_a_Page_125.jpg
c1b05a97ecbedac8ab0aa69637a7d816
4e4040546b6e175f455ca951c6e5b5b988216a3e
1312 F20110209_AAAYHY nahapetyan_a_Page_104.txt
fdf0c95b81e06416920162190d0b40f9
4e4f29eb47afa33b0f46b66e8690ae78ad62470f
3726 F20110209_AAAYIN nahapetyan_a_Page_134.txt
3e3f10c4b521e3105e2b3f157ef53e0e
1857092e36cc1d3ec84e3ba0b0c55a82b69740f6
F20110209_AAAYHZ nahapetyan_a_Page_143.tif
2a06986ef04c2546f252c40f1394c105
bceb2727c50db9be5c3f642057f4fdaa637c47ab
F20110209_AAAYJB nahapetyan_a_Page_059.tif
b27072c1e8825e3b30abdc90f980860e
aff12e8b1da05bd395c5ada07ef9cb1855561b34
40064 F20110209_AAAYIO nahapetyan_a_Page_042.pro
dd6d798edabaec8f512e11e1c6e42fe4
643be7bd5166bcf06713b740e7005ce10904dd2e
77176 F20110209_AAAYJC nahapetyan_a_Page_043.jpg
6bca6f4eba0ff295bad480f8f796f03c
129362160ec1f9aa35fd45d1a5526d4b4e3bf0a1
1968 F20110209_AAAYIP nahapetyan_a_Page_127.txt
dd85b4bb8da6d9d3fbe40019139599b6
b32ffa8b7b909488f911a4ded4f47da146fa45ca
5514 F20110209_AAAYJD nahapetyan_a_Page_144thm.jpg
f3d3e19d0d24f3a39e942ac64f29637a
dea1118b67fcb936daa9f30eebf05bed624c53ec
70461 F20110209_AAAYIQ nahapetyan_a_Page_066.jpg
556a70ad6a109ecb3502d5e34a3cf3f7
6d37766cdd368cd159dfbd3f84e3656e2347f736
2113 F20110209_AAAYJE nahapetyan_a_Page_097.txt
fb35ffba87ed7c75f71b2a2cb834436d
92410ce8fdd40dc80317cf28053ea852bfb31284
22317 F20110209_AAAYIR nahapetyan_a_Page_092.QC.jpg
830a1a4e15df7a8c3427f671e9198774
eec14412dc8731276a4d340baccc508acfd41ca4
2008 F20110209_AAAYJF nahapetyan_a_Page_120.txt
fdcd80cb3343bf4dcb831b57f812477c
f6c305eb993794f3cb6b9f3c2d1ba2629069c998
1751 F20110209_AAAYIS nahapetyan_a_Page_001thm.jpg
6eccf169b91a79bb49f14a3ca7f3b793
4dfad7d4570b662d2fcc319659f4cf71c65c955e
893743 F20110209_AAAYJG nahapetyan_a_Page_067.jp2
edc6d28071639038088536ac8353c454
1f0a2799843135e283ec029bf2340dfaaa0e0ea8
1051959 F20110209_AAAYIT nahapetyan_a_Page_049.jp2
cb32440ae64e12899e692595455df7d3
907bb2bd2f3a9d4bf8ecbb1660b47caa0225f0b7
4946 F20110209_AAAYJH nahapetyan_a_Page_079thm.jpg
e1d8a75c3b53c881141073a4b427c3c7
8eaab77133caddb21be801e11165d4fe1b30d923
F20110209_AAAYIU nahapetyan_a_Page_133.tif
afddb00804cf7e3d650ae7ad7f39911e
5fe14b352c7e28169e17149932a5a96de3e77d73
4239 F20110209_AAAYJI nahapetyan_a_Page_138thm.jpg
f8d039de84306081210ad9c3c7ae5757
92ebbcaff2ee2297578d006514908f2601746892
62493 F20110209_AAAYIV nahapetyan_a_Page_061.jpg
622db17028f99dc7729d895abe761788
3d67b0b5863a464e11716f5b22696f86fcd91baa
5531 F20110209_AAAYJJ nahapetyan_a_Page_043thm.jpg
1941291a0842df5d58564be46a299756
8ec733e8a9a42e7b50a896089752aa40c9253c38
36349 F20110209_AAAYIW nahapetyan_a_Page_012.jpg
5f8900e827b927f41a272cc3f80fb866
28d4e7d4327d41371507e209156db19cbcb496f6
2055 F20110209_AAAYJK nahapetyan_a_Page_056.txt
496854ac998ddcc43bf980144b385a49
68d05f7c325e2eee8b0a09535a5acfb9462d9c87
39496 F20110209_AAAYKA nahapetyan_a_Page_071.pro
ac7f292ea60f30c34e78ad00c2338902
b7dbddc3c1052ffd6bd11b1cc0cfff1f72db15dd
24026 F20110209_AAAYJL nahapetyan_a_Page_036.QC.jpg
cf11ee6897f41b1e1b83c731271e3b0d
456ae4a529fab4b1dedde1a6d1892b1e6525e1be
4877 F20110209_AAAYIX nahapetyan_a_Page_084thm.jpg
1e8710a2cec0516d1f036289e2c82ff9
893d64b6162af3f1d27cfb6cc86b823d7286d3f8
89354 F20110209_AAAYKB nahapetyan_a_Page_098.jpg
d2a0beaf5e8a0219f83d182a0e4eead8
d2f93da9263a10ddbefd23e29710d75ac1757e9f
17960 F20110209_AAAYJM nahapetyan_a_Page_118.QC.jpg
2635885a4f2e438d751eaa7bb3d31c14
7f23352b489d696af31fb432c4cde1edd0cd3c80
46955 F20110209_AAAYIY nahapetyan_a_Page_055.pro
885dc3dc00df30e3c6b60a9592f694f8
e23d0bec8d16230be71a2996d3a58495b267404d
F20110209_AAAYJN nahapetyan_a_Page_121.tif
5d16a67cfe9e504f575fe774edd0ac2d
b982fcd099a481ce44d072b4d6ad4e3f399ddd37
85182 F20110209_AAAYIZ nahapetyan_a_Page_110.jpg
7d2b68adfb8f847e1536b59e44daf644
ff9c6febcd847a81e7374fbaf5a8cc23c64ce0c3
2507 F20110209_AAAYKC nahapetyan_a_Page_004thm.jpg
2987096fd8b60ca34051c04784fdaddf
a0d7e48aa8ca502d28d4b82fd15e7f7eec4e6477
74175 F20110209_AAAYJO nahapetyan_a_Page_054.jpg
a9d042b982305e2183a1a4d9dd1f1e81
f42f22dfbadf9718eb3d1e3c98dc1c1147329116
5006 F20110209_AAAYKD nahapetyan_a_Page_077thm.jpg
53ac474ff022d47d63c73d0436e6ce4c
040eb67a2c0707d1e9e872362625326f2481a0af
17454 F20110209_AAAYJP nahapetyan_a_Page_116.QC.jpg
52129fbdd47520a9dc94ac25112961ec
8220a56a9b6f83593134dd2a1be930e5ceaa9195
61354 F20110209_AAAYKE nahapetyan_a_Page_119.jpg
4f817c2a718cd050275af3b4eec3a7a6
9036ed2d5e57f793cefc0d6ddab78ad7db880305
2156 F20110209_AAAYJQ nahapetyan_a_Page_064.txt
fcb68339977aa6413d1ffceffdbc4998
debdd5b4d3c4bf84cf4cff84a9b3333563c582d7
4521 F20110209_AAAYKF nahapetyan_a_Page_132.txt
834f30690f2440a6d98257065e834c7d
e3a3a5fdf54b751b42345a4e8680a24f5f3642e2
85691 F20110209_AAAYJR nahapetyan_a_Page_044.jpg
5114adc2db5456d5b045e29d6a690951
a41d30445f2a3c8d5ee44a704ac6bfc16c115b23
F20110209_AAAYKG nahapetyan_a_Page_131.tif
935da0477acc22a4d97e206f60d6dd45
14626f34ff03cf93cabe795ddc6ca3165d22476a
F20110209_AAAYJS nahapetyan_a_Page_085.tif
508bd597d07c45b54a5b3687fa7df55a
2bbffd36a81a94911461524d2672bb4f8bf9f128
72256 F20110209_AAAYKH nahapetyan_a_Page_093.jpg
73261513fd8c28c3b94ca8e0c1936362
6d586d77d58292b720d42d01bcecef388de7bf02
1846 F20110209_AAAYJT nahapetyan_a_Page_053.txt
c1420c092198cebae8d170c6b0e7a302
d71f980e98303b34dac42d73333be4c0ff43e2c7
80797 F20110209_AAAYKI nahapetyan_a_Page_006.jpg
e04e4ec35237db674f869810b865760d
5ebfae46f52adc40306a7c3578f1b30f186aa418
624859 F20110209_AAAYJU nahapetyan_a_Page_114.jp2
55097e762c50d2d8cbd1e8d45922f02a
90d1155326dbd6554fb75666169ae96f81c5a589
57176 F20110209_AAAYKJ nahapetyan_a_Page_137.pro
cbeb6a5c696f6c8c4e3d39bcd074ae68
5c0af897fcb78ea078742a0c5c6d663d7a73fc1d
42484 F20110209_AAAYJV nahapetyan_a_Page_023.pro
38d697c79b0e47f92526101e46ac1e80
928fc95129c279f209f519864132041d70fa5b78
F20110209_AAAYKK nahapetyan_a_Page_024.tif
9a6b2dea0a8873787f8427a7e3ac4a3e
da61d8c5a86f407db41bbf88ac86590afd6fa9b4
969891 F20110209_AAAYJW nahapetyan_a_Page_041.jp2
1617ab0deb20dfb45ae940693949dc50
3880fe333c1661865eb93bb00d4e327d63130a1c
57583 F20110209_AAAYKL nahapetyan_a_Page_048.pro
8967fd4707a8a0765700ee01522dcb36
540f48948161323f81db4fc0f8057fe60cc0148f
4640 F20110209_AAAYJX nahapetyan_a_Page_123thm.jpg
4761e0f7664069fa8f095e5ca2c10a73
cdf3a8754fdc15885d9c34e7d8f76da13615fb63
813669 F20110209_AAAYLA nahapetyan_a_Page_071.jp2
d6439960fea68654b33b40c6bc93b22e
e6d0985632dae1e7c32939303f126d7726823f75
4405 F20110209_AAAYKM nahapetyan_a_Page_019thm.jpg
565e79c0eb7a4b40e0f99dd4721cd7f6
d2ab26b3a76895731b784cff76cfd65d5006b2f6
66798 F20110209_AAAYJY nahapetyan_a_Page_022.jpg
458f704c90f6ed23870a1a4bf1b2ff9c
5ef7dd05251de6aaa4a1858b2cb05c929def9609
7915 F20110209_AAAYLB nahapetyan_a_Page_133.QC.jpg
3c5a628c72929f5e93aa72baa943e64c
5820afc50b1be99b5721616fafbc5bb0352ea214
2065 F20110209_AAAYKN nahapetyan_a_Page_076.txt
6a45d41f7f4c6e211ad312d6c1f9f1ad
70b01952eff0eb048bb5e5fc499584095b21198b
24490 F20110209_AAAYJZ nahapetyan_a_Page_097.QC.jpg
2ded5b4551099ec005296131c95062c2
802b627d5840b6c869441356760a7eef695ab2db
23202 F20110209_AAAYLC nahapetyan_a_Page_087.QC.jpg
3a5e7d8b074f326b0bd594ea57866cd1
b13f78c0d2dc987b643cdec3b57e912b13f2c5df
22178 F20110209_AAAYKO nahapetyan_a_Page_127.QC.jpg
9f56fc67b53b55f47bee93ce4da37a73
f1b98c45e27dfc42005d0ecb8b699552ddab49c4
5288 F20110209_AAAYKP nahapetyan_a_Page_085thm.jpg
81d337e9b4516aa4c1e798c581b3e3d3
2e788af0f8dc10ceff50fde976cde6d8342718e1
1735 F20110209_AAAYLD nahapetyan_a_Page_033.txt
69f630e5d784540e368dc7f1645a814c
6786bcfca4cb94c0825e4d14e80432d37257ba68
3336 F20110209_AAAYKQ nahapetyan_a_Page_003.pro
8454f0d2b0f55ee7c992f7ac66a2ae46
7914ae615b4d503ad500b9dfa02f6a238c772fbc
710 F20110209_AAAYLE nahapetyan_a_Page_009.txt
04c57ae968bf43dd588eb6bc89217224
d71169f40c4cdfcb656312c6a37909518ac81c9b
F20110209_AAAYKR nahapetyan_a_Page_119.tif
88d3d34e091f6f9ef84d0bf0b6493497
650f0ced1b913fdc4f83cb99978c961b444238fa
18189 F20110209_AAAYLF nahapetyan_a_Page_019.QC.jpg
63e10a3bef707c005e3f5ea8e01ba51d
cf8ec673fa7ac93ebab5c80bee86e2b2657ef46c
5366 F20110209_AAAYKS nahapetyan_a_Page_096thm.jpg
d246e67a66bccb4c3831054c343fadb9
edcfdbb43155184b35db94bd94e52a8128262142
19490 F20110209_AAAYLG nahapetyan_a_Page_051.QC.jpg
7bed9ab95eea95a838aaa229e5eb27bb
69ff90da968d2dff148b450f2da96f156565fc1c
1051807 F20110209_AAAYKT nahapetyan_a_Page_143.jp2
a8760f4c3ee65f93f3076f60aefbe213
9f56faeb45093d271f663cfea50f4ff1740a0819
200 F20110209_AAAYLH nahapetyan_a_Page_126.txt
0c27bedbc4370137750c2fda47c59b0a
bd821bf688e032f41ba9b07e369c7451135e7d10
F20110209_AAAYKU nahapetyan_a_Page_043.tif
ee23f5ae3821edd91dd14e550ff15827
4f0394cf632d1be00a410d422448310eafdb9952
1051956 F20110209_AAAYLI nahapetyan_a_Page_048.jp2
8724551565d3097d312e5e10694908ec
08dc729e5987bff2427cfb07a2f02fdeff6ecdcf
943 F20110209_AAAYKV nahapetyan_a_Page_142.txt
a2263eac66dabd7c9e1e5dfdb05b949d
a92606a0562604d48057e7168309dd46729bf0a2
1024 F20110209_AAAYLJ nahapetyan_a_Page_139.txt
6971f7e22bf8a0d2fd4906f48a48dc51
78e0327d5a0f28cbedf485e85baaeaa5cb31e3e9
21781 F20110209_AAAYKW nahapetyan_a_Page_013.QC.jpg
02de5c3918d769b7056698fdcb5821e4
63a79b70e08a0e150a578cd1d3bd69634684b108
19588 F20110209_AAAYLK nahapetyan_a_Page_071.QC.jpg
0ac5069b6f40ad46d4e0bb6db9004abe
3ec5a2bccc28beec5b39057f187c379bdc9ae481
3976 F20110209_AAAYKX nahapetyan_a_Page_113thm.jpg
3b1f838a0d9d5e3be5cd1eecd21fa85b
a8ca91509dd70855a78b9697f84f9ad1e9e96b19
1834 F20110209_AAAYMA nahapetyan_a_Page_119.txt
324008825642c0cc64737242734e6a6e
aff8aeedb6a7c5d8adad6b8b07dc1663789d7ec0
902156 F20110209_AAAYLL nahapetyan_a_Page_084.jp2
1ca79a838b8f9bdb8c3b6229eb88ec82
26aac2f182f026b7f9fdd9d8c5dd18e702ceb0e6
5247 F20110209_AAAYKY nahapetyan_a_Page_049thm.jpg
23ff944bb4e1f80b4a74ee72dd4ab3e8
aea6c4bd0735b23e5de179033b3a4a5fdc8b4065
23555 F20110209_AAAYMB nahapetyan_a_Page_096.QC.jpg
02c46399bcb9521c8c3f681b89ddd393
c85461a15c6c3d1b02fdf740aa1103d31c944431
5494 F20110209_AAAYLM nahapetyan_a_Page_078thm.jpg
e74cc1628d96fa9a5407a8d759687fa2
aff40e60031a2226f2f863e84e95a7b475f15571
1045503 F20110209_AAAYKZ nahapetyan_a_Page_073.jp2
67a2a2f87b5cd51850823d2162911070
518d08fed3ebfba75cd8c2fc92e518e171cd6fbb
25519 F20110209_AAAYMC nahapetyan_a_Page_044.QC.jpg
50ab95709eafe087c4b739ec55478599
bc688963f0d41facfd337bf1d57b1eab1a1c2154
42492 F20110209_AAAYLN nahapetyan_a_Page_066.pro
491be5bbd7146fba3fd277287b18063c
bd187d759c35c19c5f8fff6d8cb100d84d6f7bb7
5460 F20110209_AAAYMD nahapetyan_a_Page_064thm.jpg
ccbd9f4bf4f16efc4f82d7b211ca4087
291ae87712b0c489b51ddede957cce179bb39256
1574 F20110209_AAAYLO nahapetyan_a_Page_106.txt
8f25dd9cb4fe4c8e1ad555396b2ed016
dffb1ac23863d6559b15e8c56c782a5afd0d63a6
974907 F20110209_AAAYLP nahapetyan_a_Page_093.jp2
fb53eb85e5f33009704fd677eb48c675
1b59dd8c4b2d14ba227e75db7cdb843f54d0b0a2
22293 F20110209_AAAYME nahapetyan_a_Page_047.QC.jpg
6781530d47090f602ec3b50ec6e7adfb
17d5653e023201fe15953c42dd34bbc7fdc838c9
1832 F20110209_AAAYLQ nahapetyan_a_Page_046.txt
876a1a99daa47ee257cd9991ac2edb16
da580416a6a3f87ea1cad6bf4964d8cbd81ab4a7
F20110209_AAAYMF nahapetyan_a_Page_089.tif
55d9fcff1133730ec79cebb6a22ba055
78535d83904ce7a69bb65d208c40cc770faba234
F20110209_AAAYLR nahapetyan_a_Page_028.tif
0871753c52fbdc65b90ff1bff6832675
52d6baaa8bcb82e9fb38a2b37cc9348ff11c2932
999314 F20110209_AAAYMG nahapetyan_a_Page_039.jp2
164165c72bff75f97717f96389ecb23f
e3cc70eb76a8ea0b2619d388e80b386085176320
799485 F20110209_AAAYLS nahapetyan_a_Page_116.jp2
b4d365aebfbb704e72f594aa8a0c5ac7
c9b37753aedc2aaa6c8f39fbf4e4c2916dea90e9
6856 F20110209_AAAYMH nahapetyan_a_Page_129.QC.jpg
190ee68a092657b6d9b5747988650a78
c4be379a270628d9498ca171d36c740033047738
55308 F20110209_AAAYLT nahapetyan_a_Page_075.pro
0a5d51e99b72a093f1c3ecf700feaa09
6be9d9ddef8f61d0ad661aedfd2ec51c59741f60
F20110209_AAAYMI nahapetyan_a_Page_112.tif
a15ab62c6b0a33a21c2e4ad0d9520475
def51c5a40f645e7a9dfe9a5a3058627f022c6c5
F20110209_AAAYLU nahapetyan_a_Page_117.tif
0db2a4555f34b819eb13e4a4c1b8ef6f
ccad3688a64fc79f641b9c735f86eb185283bb0d
F20110209_AAAYMJ nahapetyan_a_Page_135.tif
9bb1d5b719730fd099add5e65971c85c
d048eed0ab34cae5e6dfd70a24df7c4b368f5c71
F20110209_AAAYLV nahapetyan_a_Page_054.tif
88acd8550cff4ca1b57578a1705c183d
016dd774bae990177a39c3259c99a4576e5f51eb
78003 F20110209_AAAYMK nahapetyan_a_Page_055.jpg
306331948ee04983c7f08921f1580c3b
bddfbc9de6935b7a380a0db6d663422d35a74d8f
42239 F20110209_AAAYLW nahapetyan_a_Page_053.pro
5cffcf0f72a1226bd01988d140965f58
be5b540c4ffae1d21a744464f10d3269ec31a844
5111 F20110209_AAAYNA nahapetyan_a_Page_062thm.jpg
eacee1edf09dcb641d4cd59bd7f0bba4
e1148e2e13019d93df713889fdb02b75a08a4f00
61662 F20110209_AAAYML nahapetyan_a_Page_068.jpg
56d86eb972b01734c1f85f7e7dbced76
f5af7926745f3171b6940f0f20752173f1e5f544
2068 F20110209_AAAYLX nahapetyan_a_Page_005.txt
70255db30b146cb9af0b00927163d58f
6d629b3d74e62de060adb002db439bad8e59d96f
12751 F20110209_AAAYNB nahapetyan_a_Page_089.pro
1648a29e867a045a7bfbf7b87be3f142
93abfe51318ae55b827a4518dbfd38c2a6e5e38e
F20110209_AAAYMM nahapetyan_a_Page_129.tif
e22e99b12da17281116f361a82cb96d5
e1504d8604a017430b7f5f9f4a073c690439d8ce
968509 F20110209_AAAYLY nahapetyan_a_Page_069.jp2
08a9ee72510f38bc25c28c790c2c65cf
092f6c4b5de103bd6f221c7608d305061d7753ea
85720 F20110209_AAAYNC nahapetyan_a_Page_018.jpg
a715cd602caa96423d826528084ebba1
598e5fdc684030c88b2556b54dbab99d974fba22
12144 F20110209_AAAYMN nahapetyan_a_Page_141.QC.jpg
a3b234ff243a24c05603b02817e23c8e
c64aa1f8280641d27f260380c28ad913b332ed87
49383 F20110209_AAAYLZ nahapetyan_a_Page_113.jpg
5a5a1dc4bd6a4d4fcb652c6d34fabefa
d1404e71ac019ce51d7b5cd0b85bfd617c8846a2
838606 F20110209_AAAYND nahapetyan_a_Page_082.jp2
404ca46f9818685fabfce5c52c3b7f8c
8904961a3761b7d73d8caa8ae66c9fae93f2385a
5417 F20110209_AAAYMO nahapetyan_a_Page_101thm.jpg
22caf83e15d42a4c2fc48e51df4f8c5c
d746d1eae9e5cd0b1ef4b73db773abdb88818717
42316 F20110209_AAAYNE nahapetyan_a_Page_022.pro
fb5af2725cf20da3995030e1aa8d7f66
42644ec416c190da1ccff246a38000a0ed252fed
1014309 F20110209_AAAYMP nahapetyan_a_Page_138.jp2
e16bffbd9964fd7d01807f475eef8acd
c34c1f14f8e4772223bf68934d44229299b50884
26564 F20110209_AAAYMQ nahapetyan_a_Page_103.jpg
f482d578ca83a32587a925244f8cc4fe
66a12945ccb0f1ba8d4d9f812ccdbdf75533bf0a
116 F20110209_AAAYNF nahapetyan_a_Page_002.txt
32e386c0495fefffa22d3bd162ac58fd
5c4396d71c3e54292ccba774a823bafce85fec87
4269 F20110209_AAAYMR nahapetyan_a_Page_029thm.jpg
e61d13745f1c4918e31f0594d2ef3a67
e1d74a9abec4bcf5a303317492d7238c3a797bd5
815182 F20110209_AAAYNG nahapetyan_a_Page_118.jp2
b992a3b6cf0a32a7ddbbb208e5bdb3bb
b43bc17f88f23dfbce62269706259419d58aee6c
25329 F20110209_AAAYMS nahapetyan_a_Page_075.QC.jpg
7c41b3e39bd709775b9d21c8ba314aaf
f5699f602b6a4f9fac6b5dd891406861a8467e7b
47989 F20110209_AAAYNH nahapetyan_a_Page_031.pro
ba2656e272c430efe385952387f04277
8e667a52a34cadcb6710d2e02252985442d0002a
21390 F20110209_AAAYMT nahapetyan_a_Page_023.QC.jpg
0e54a25135e976295afae8a251a28383
9dfbfa66217a9ddcf926300ea77c69e1584534c3
74504 F20110209_AAAYNI nahapetyan_a_Page_127.jpg
87ea79437a0ed67ea6d9a71daa028aa7
34c4f7b8d54089e99dc5508cc468be0a0d53bf51
F20110209_AAAYMU nahapetyan_a_Page_005.tif
0eb8baae3e2dd388c8e9ca97eabcee30
00b4fa69132f5e3dd8b0e90743adb7adbc6f0ed2
21699 F20110209_AAAYNJ nahapetyan_a_Page_069.QC.jpg
76026d289cdbb01a6aaa1b6173466365
b9c1cc575902d4f24d9b2c63653e169504c2e8d8
1860 F20110209_AAAYMV nahapetyan_a_Page_058.txt
a7f85f7904c7a43f5a8245fc65c3ac17
ab7bfb6fdd719f09fe1d6aa1a451a5954a474f38
3670 F20110209_AAAYNK nahapetyan_a_Page_130thm.jpg
b4513e8accabfc0a2ba47cf621f6c41a
5deffcceca4963900b3c48ac0c1103d4fca97dd5
F20110209_AAAYMW nahapetyan_a_Page_035.tif
b85e7403763a3952e36ce0d29db0302c
c8816717d18691a1ef8d95144d9c5d76997335a2
4845 F20110209_AAAYNL nahapetyan_a_Page_017thm.jpg
6e2d3ac67113a7b75f257b7be9e02115
a7304c8b033c5bbecbd71f54583ff5f84c560dcd
5718 F20110209_AAAYMX nahapetyan_a_Page_150thm.jpg
8f3266677c46987e68fa0e9dcb3cc6b5
e1ae5458cdf57ff32f08a2ef76e6258d1f06bf67
47523 F20110209_AAAYOA nahapetyan_a_Page_005.pro
42a31703346a9068fb4c8bf01bdea91c
dd5aaa0be6240be12cce44521a220896b403cdba
F20110209_AAAYNM nahapetyan_a_Page_091.tif
e52a1328c6ba7799ab2a09933db100fe
eb91ddd3fe542204223e3439c9c8095034eaa7d8
F20110209_AAAYMY nahapetyan_a_Page_114.tif
62e6688fd0202367791eae4e10c46d80
30a7f4eb6a885e0e02b4922ca231542e58a690ab
42196 F20110209_AAAYOB nahapetyan_a_Page_065.pro
b6db5231e9f88377340169f8bd2c44ee
41d1e79c59ad42705241fb23306a6ac3bee64ad9
4867 F20110209_AAAYNN nahapetyan_a_Page_127thm.jpg
5124a33e9b45470cc82c6134b3d589f5
4d0f8c5dfd6c6957cf0a0dd550442893ca465cea
1629 F20110209_AAAYMZ nahapetyan_a_Page_081.txt
2884bc1965eb32be5547738afe797e10
6d3c027985a7ec0660c9708925e593bce13dea89
21741 F20110209_AAAYOC nahapetyan_a_Page_017.QC.jpg
e18db5eaa06ab317960b8ce11006523a
2d7273f1abb9885ff0700ee7f768b10b98931fcb
F20110209_AAAYNO nahapetyan_a_Page_075.tif
34f0e765986ca7294f0685d0ff2d2681
b1d78ccabaa7c9a1eb60658965d399f5f56e488c
F20110209_AAAYOD nahapetyan_a_Page_013.tif
b45006277b9bba615a6cd2fdd6d1ddd9
9350137e112c4160a124e533866dd4d76c35ee95
5204 F20110209_AAAYNP nahapetyan_a_Page_092thm.jpg
8649f7e773663c7d16c5f691f4a76f8e
0b81cc4f767704a890e5c0865ecddf95ae8d9143
22662 F20110209_AAAYOE nahapetyan_a_Page_073.QC.jpg
c134f4b906e048f03cb1f63c08166842
ce741d8d37092e39cfcb3b0d0159ac4be4662c88
4016 F20110209_AAAYNQ nahapetyan_a_Page_138.txt
ae43d466a0b4230c6473a06a073677ee
c0cd5f075de2af91e1bc46637823cf55defe78ca
770015 F20110209_AAAYOF nahapetyan_a_Page_008.jp2
98e3e2733004e74555753425ca1704a2
ea1e9b7db495f86612fcc2d347562ee3643fc7c9
F20110209_AAAYNR nahapetyan_a_Page_152.tif
39245d77bc93f28d0d6fff8cf9130624
5f99aa73e3e651822689c3d121fbebf70c361863
2128 F20110209_AAAYNS nahapetyan_a_Page_018.txt
b17f9a387a7ad6f18a060700175bedc0
50cdad134dee95ca2d9d9da923f2670ba2c8404c
21590 F20110209_AAAYOG nahapetyan_a_Page_038.QC.jpg
cac1511200a3e3abae1f7447847a0bca
c32cd5a9f95222fe4726a82682392e22da592159
7247 F20110209_AAAYNT nahapetyan_a_Page_131.QC.jpg
f74180a509b2bf344ace48b917b56a20
4b6df65faeb6a8906248ae631d22fe3ed1120081
1750 F20110209_AAAYOH nahapetyan_a_Page_065.txt
9a94342d6af903c8b003bbfde73740d4
0a13d5513ced340e313956f15c014e1bf3b9568d
10733 F20110209_AAAYOI nahapetyan_a_Page_012.QC.jpg
88ec95873ccae4f07ae80113c2914115
e01368884d26430651788b1f839f8a303f7958bd
1051932 F20110209_AAAYNU nahapetyan_a_Page_075.jp2
7161e6b12d90e3e0c9bc79e2b335add7
79b52196d382a7d77aede3d62ceb770ec77d3347
1035780 F20110209_AAAYOJ nahapetyan_a_Page_058.jp2
8d9d0df46d9e59f74351241bb106ae82
11f7dc620281d129a9c6fab886dfdf287e9b524f
44634 F20110209_AAAYNV nahapetyan_a_Page_039.pro
c555b672d6da4db19e9454d0b56cf9ec
d777f7b5733a865731a548b7ad788a23496404b7
23001 F20110209_AAAYOK nahapetyan_a_Page_131.jpg
8b40429b33c5234655936bf0b9f55ec5
393c22c790f843f796c1fe1fb526dd0291886df8
5297 F20110209_AAAYNW nahapetyan_a_Page_030thm.jpg
ef32403e6371f1fab1fef3e5c761b18c
333caef8d5b5b15dd246ded7ecc47d44c61485c4
39200 F20110209_AAAYPA nahapetyan_a_Page_123.pro
3829721f63e2b3fd89aae12f27a490d1
93b31756930bbd95a2bda3ab8bd4064c77a1fdf1
65094 F20110209_AAAYOL nahapetyan_a_Page_008.jpg
fad73dc45bfebb1cd7764a5b5c794e87
5962d46384bc4ff16fe110e9ab9d90beae6abf63
5180 F20110209_AAAYNX nahapetyan_a_Page_053thm.jpg
5a77ba3563ea992879448cfc116f385d
9ace4a643e471fc87bf68ef4f35af2fd007cc3c5
F20110209_AAAYPB nahapetyan_a_Page_120.tif
f9fc550f1ed018422ddf43a89866b939
2ab86942b75bef46c775eba8c08fe74e0eb3e87e
54824 F20110209_AAAYOM nahapetyan_a_Page_098.pro
1652ddbfefc110fe0b341bd4a40fd7b3
f55a0d638671b97a2743cdcf21ffe187a9a2945e
63451 F20110209_AAAYNY nahapetyan_a_Page_091.jpg
514bd4989652b5bba8983ad1106efce7
23326bf0db62ab27b700a7d53c9ee6dd19002591
432793 F20110209_AAAYPC nahapetyan_a_Page_142.jp2
2d264d1e23eac12ec38f216a8c254234
bf252df90f80ad5e327afb4d1d53c14d18ef2e67
56873 F20110209_AAAYON nahapetyan_a_Page_121.jpg
59d98747119a5300e65e972a1064a3bd
671c1abdbc7edb4cbc1fca73d186c33f63edae88
2618 F20110209_AAAYNZ nahapetyan_a_Page_126.QC.jpg
4da3afaf07b343988651510abff6e070
9e5c057a0008a7b1b7ef3a8175d196b536cbb902
1748 F20110209_AAAYPD nahapetyan_a_Page_140thm.jpg
fe94dc31c10538c6d67328812ce9d2a1
a07b0172367ea9a73e307de86f8dddbe6a60fe3c
4967 F20110209_AAAYOO nahapetyan_a_Page_132thm.jpg
a573eeba800b27e9f767d6d390b210ec
3f3b29f5941e1dd70646095023ab84960728ba53
F20110209_AAAYPE nahapetyan_a_Page_142.tif
89fc7caf617887a590bf176119cf7976
c281cd05c1417c13333a2d4c93caaf7aa51430f7
48100 F20110209_AAAYOP nahapetyan_a_Page_087.pro
f3c1cbe8fd345fceb126119dd9899d12
061dd1f95d3e739164337344d325a0d92ec6aa19
20961 F20110209_AAAYPF nahapetyan_a_Page_089.jpg
d01c0fb44f4061daa61e69ff20ddbf94
501dbf9c155e9aa0812a02386a3aadea7db08578
4744 F20110209_AAAYOQ nahapetyan_a_Page_069thm.jpg
08535384fbd78873d12154c21ec2889a
3403d1d5f08efc4d0fbbe89e1d12407f4d0ff919
2033 F20110209_AAAYPG nahapetyan_a_Page_083.txt
be2a75eecb21d37c44c17abd0a271831
e5a83842a1d4f27196ffbf7a2f0f9f0868550cd1
F20110209_AAAYOR nahapetyan_a_Page_127.tif
4ae2b015a42182d6651af8491b443dfc
86fa70fe458a909607e91ba05d306e69f5e046cc
F20110209_AAAYOS nahapetyan_a_Page_084.tif
1702b02cd8397474b904f1674252954b
2ca4118f3e95ff82b51067e670d75a891a85dd32
2036 F20110209_AAAYPH nahapetyan_a_Page_028.txt
9e7659bc79a01a1b55d690e413fb6d2b
bf8bc232dac58ccc76d16b575415e7e7ec637774
76194 F20110209_AAAYOT nahapetyan_a_Page_047.jpg
19e02ade1ccc0b14e727cfb516c594e1
06bce7d6ae490b5100f1d9ce6645d57ebe12320e
2022 F20110209_AAAYPI nahapetyan_a_Page_022.txt
ae93f779b85769ebacb8090094cf9820
dc89743ca08293aba917b0a591c36c249ffd0782
1872 F20110209_AAAYOU nahapetyan_a_Page_030.txt
0049c29f8859d03e2d59a8bfd05d4978
207f3007fc9dba8c88f369502b29ad52cabe8454
740738 F20110209_AAAYPJ nahapetyan_a_Page_029.jp2
e661035fd7c78fe3e0289a70353c8590
9c930ab157ac5571c4887e97015aedaef08958da
77521 F20110209_AAAYOV nahapetyan_a_Page_087.jpg
dc875a80a8e5b4fd7a5537e10d49e174
94e77d0f29d48f69710e72700b938972ff95d5ab
F20110209_AAAYPK nahapetyan_a_Page_151.tif
3b4ab416263a71d9c6d6c24b1cde7fc0
f4b67cefca705a0dec68e8591df80cd4e4186b83
F20110209_AAAYOW nahapetyan_a_Page_093.txt
e7f343e3e5ab6d83d700e90c57aaee34
da86d044839993fe267ae147d04b1c8e84ca9471
17324 F20110209_AAAYPL nahapetyan_a_Page_009.pro
616d2b85d21971622176edd141b950f1
3fea5cb94deddc01f3ae8bdf29555751108bf374
20681 F20110209_AAAYOX nahapetyan_a_Page_016.QC.jpg
1083cfbb7ae6d16153b6ed9509f1d792
822080089e091e6b10792ba9796a0984990d2984
48352 F20110209_AAAYQA nahapetyan_a_Page_094.pro
3b2803a9bd476340c993b3aa359a3f2d
f79e632f842b83abdbd009194c2f292a73c3429c
2112 F20110209_AAAYPM nahapetyan_a_Page_125.txt
a8ba64ac1eaa6a327ce32659c7e346c9
52d78d510be01c20c4e7b88b6e7186eb67e6655f
53655 F20110209_AAAYOY nahapetyan_a_Page_007.jp2
d7e875c464e346a20043a2e3c73a19b5
00f344946fbfb3bfe53d063cf6c1a9616e9efde3
40455 F20110209_AAAYQB nahapetyan_a_Page_084.pro
d0ac6a7e32fbfce710a5315e6e54d41a
9bff4dd2152c629d6212b261b081b48cd33b0e44
57556 F20110209_AAAYPN nahapetyan_a_Page_147.pro
a31e6975a2849226582e0ee162b992a5
b94b565ccbf9485bfec3f9ea92dcee6f20680226
1051980 F20110209_AAAYOZ nahapetyan_a_Page_044.jp2
cd4c37a96aec5b4f00d10a06248bb459
42f4e2aa04c4267ee8b4bfccfa145733389fbbb0
14791 F20110209_AAAYQC nahapetyan_a_Page_124.QC.jpg
4982329c1665a16443b4a543c87d7786
0686da5a553853988960f1f2ab0fb5733e1e3b7a
22164 F20110209_AAAYPO nahapetyan_a_Page_094.QC.jpg
302fd64eb2fe9dda0f27fd951cc93c9a
41583c0490d2d5234dcb56adb61da0f80e2b6950
39421 F20110209_AAAYQD nahapetyan_a_Page_106.pro
4655219847f4866cbca825d9f71f9ec2
311dd1e6911ae7cb170ab5e7d74e2d6a8ecc695e
73990 F20110209_AAAYPP nahapetyan_a_Page_017.jpg
4f58da33c9f7de6fbc426bf1c0dbed1c
25c123891601962618f054665639c3dc29826a6b
25778 F20110209_AAAYQE nahapetyan_a_Page_135.pro
949794e6cfe956e33a6b6eba1345020a
7f337c1eb7b997b818a961a6c08b97f9a28dc606
1922 F20110209_AAAYPQ nahapetyan_a_Page_090.txt
88d0e7c34ca02db70e973a90c652a6c6
e22b7c90af127a629dbe2df25e28102ce56a8430
18321 F20110209_AAAYQF nahapetyan_a_Page_121.QC.jpg
433e9db230863b2d87d0b5ae7e8e1207
ef66eb0be231d09cda641ecce3f50f14470a7c1a
975809 F20110209_AAAYPR nahapetyan_a_Page_038.jp2
318d8ee02119c4d4d30e9bbaa99b0cd7
925ee12d0eb042c19e50828931bfc1b992e9f055
21752 F20110209_AAAYQG nahapetyan_a_Page_058.QC.jpg
fce0df364a144a9770b34f540a32a144
8656ce8f28945f4055992c8d6ac1412800d34615
27780 F20110209_AAAYPS nahapetyan_a_Page_133.jpg
04372d12925647509cd5da892183f58e
aa02c0904c72024579add089a8d1cce2af1f61eb
461 F20110209_AAAYQH nahapetyan_a_Page_001.txt
14ae9968787b5bccb30ff5187a9f9f15
c14636dd62a6761ee019cff225e5741fd9bde411
85066 F20110209_AAAYPT nahapetyan_a_Page_063.jpg
dd90fe8beea6c428749a3f05d72c2c96
ba7f34c64668a781a6937389d48a885f81e23aa2
82369 F20110209_AAAYPU nahapetyan_a_Page_109.jpg
135c058955a6b7f1393f3e29a9009efa
fd500b97e1a3fd89fca6aebd8f6db25f3cb47de1
54426 F20110209_AAAYQI nahapetyan_a_Page_006.pro
1bd943ea65ff5e11b648e327dc016096
60968f2aa8f9cc0b0344df6f6a445120dc6f1f1c
1050731 F20110209_AAAYPV nahapetyan_a_Page_034.jp2
ca4bc8b0e92f433766f035a1ab47a011
c5c78ef72670245908104fa2eeb6c862a2b85f01
5577 F20110209_AAAYQJ nahapetyan_a_Page_032thm.jpg
2e2eee5a218d4b21c49b3c57953df32e
1331e6eda4a4fc550d03e0841d998ce99d840427
37830 F20110209_AAAYPW nahapetyan_a_Page_019.pro
14212c419e25a3fa8ff2d624e74c88ab
bfd34788f00c13e5877989568e38df7aa057b2d8
11990 F20110209_AAAYQK nahapetyan_a_Page_140.pro
288040045068dd53ddefdbcb20eaf7d7
1be3f628151339464534308ca606beec0b8aa1f4
911322 F20110209_AAAYPX nahapetyan_a_Page_072.jp2
60a911ddd1ad2d47abe0b99000febbbf
88309e027102bc2f2af4bb4d952ea1b3de3ad8d5
3132 F20110209_AAAYRA nahapetyan_a_Page_007.pro
34e300292bb2c6bff6ceb8101ae3db9d
25312abfca265f817b55b4d90123affd03ac335e
20130 F20110209_AAAYQL nahapetyan_a_Page_086.QC.jpg
aa206749dd00402efc3e88ab660ebdd4
df1423c9e384d44240f8462ed038d6a2ef036c83
672508 F20110209_AAAYPY nahapetyan_a_Page_113.jp2
f7c1a28378b1919c9646f02c530bb1cc
b48f2c46c3b73e0078a5a1d37b7b62aef15bd4ad
51223 F20110209_AAAYRB nahapetyan_a_Page_028.pro
1308928db5c4c76a78cd56a020a468d5
10aebf2567383dedb7904c8ebaa5530d0fc789dc
2241 F20110209_AAAYQM nahapetyan_a_Page_006.txt
0a561cd5de749fd0af36e8f44f0316b4
13e0d23684ba6e943eb4c426de1319e9c51f3d5e
1767 F20110209_AAAYPZ nahapetyan_a_Page_129.txt
d53ea7d394176b257a4138ce588480a8
e78be18466fa749e3b02834bb949876339d2392e
4561 F20110209_AAAYRC nahapetyan_a_Page_016thm.jpg
390fe57ee39bf53ad3f8cfd5d2ae599a
e7f004184000e3938e8356283be5c4af37777c3c
19093 F20110209_AAAYQN nahapetyan_a_Page_081.QC.jpg
255b6144b4707d20bec8af10678e9308
ab3413fe39f4a7cb234817ed211bbf1c51c5ab63
F20110209_AAAYRD nahapetyan_a_Page_083.tif
6d069204e030269019a18fb8b19219dd
f37f4ebc80602e831eeadf37572d59334750659a
2027 F20110209_AAAYQO nahapetyan_a_Page_073.txt
53542b5434180c70a8954d844a33ae69
70a843ac70e926e45df1ba6446da60878ab4d26b
19697 F20110209_AAAYRE nahapetyan_a_Page_091.QC.jpg
3c49d9d3bdf0586e76e20f3c56c0c63a
20ccfd7eebafffa09baccee719af3ff2f395074e
20894 F20110209_AAAYQP nahapetyan_a_Page_004.pro
9de81ae849c1b33799707a72b7de0cb8
febcbc762d757eb9d26b2298b516f335e2e95bcb
55518 F20110209_AAAYRF nahapetyan_a_Page_111.pro
ad25a5c0c8a9daac813ce47b170492f7
03975ffc03aaa6756476f781b9570c79f54558bd
72524 F20110209_AAAYQQ nahapetyan_a_Page_132.jpg
ef36e2dd05ebdb24cc7af46ffc6af217
7bbf53e7c0ec7c12b704fbf748176b17174081c6
279217 F20110209_AAAYRG nahapetyan_a_Page_131.jp2
131e59a540232a35f34353cd93b5618a
07564c910199f889ef820874787962f22cf7e807
1042880 F20110209_AAAYQR nahapetyan_a_Page_013.jp2
dae81bbcd8e713e3b5ef700a7e5a86de
2ebc39a68eba46dcb71b08ef9d79c0e79315d8f3
2170 F20110209_AAAYRH nahapetyan_a_Page_063.txt
21e91f760e1e77937e74c0c7ce3dd6be
c94c36cd4d0ba5929c18815b66a879307db27003
24386 F20110209_AAAYQS nahapetyan_a_Page_028.QC.jpg
2f4ecb07f898aef8d79bec19bd8e7b08
c1d72be76aa648486f7d1e8d410f078c70dcb394
275762 F20110209_AAAYRI nahapetyan_a_Page_089.jp2
23d65a88a1f7074f7612d5af9c8823fe
06b9243bc71b208828821857e45371f1201343dd
41650 F20110209_AAAYQT nahapetyan_a_Page_033.pro
10ca2793c520df982b81fbbb68df22ad
c6ed0cf91b6138bf842ff9da9efa5c67ab38a22a
24606 F20110209_AAAYQU nahapetyan_a_Page_001.jpg
148f619088100b01be67cad20f5a8908
e3a348bbda1c39b7a4c6f90e6bf470e50ffe9d25
786 F20110209_AAAYRJ nahapetyan_a_Page_003thm.jpg
ad52b24755c953502fdf4eb10c997f62
3c9d5309e4d7aa3f6d048d0a01736f52cda7214c
4782 F20110209_AAAYQV nahapetyan_a_Page_042thm.jpg
f412b71be52e587d04559602742aec91
3e7e7915b0f7f6e8a3fa1e502e710877fd262e25
45565 F20110209_AAAYRK nahapetyan_a_Page_045.pro
819fd9cb08898ebc3f4de66899756bb6
21e4d613f98ff63e7abb8acd01ba16bc5cbccceb
5059 F20110209_AAAYQW nahapetyan_a_Page_112thm.jpg
f065850306ac554224a793e838ec2cfc
ba2ecf598245c0686bfe8947e690f2237c928c5f
18954 F20110209_AAAYSA nahapetyan_a_Page_138.QC.jpg
fde72a81090e90f943cb64b8325c09d5
5e6aac34e1ba5ffc79608cf61715aeb40313cb47
993996 F20110209_AAAYRL nahapetyan_a_Page_092.jp2
f02e91153e1ee5835c85d1e80591c093
843a4ab1bbfc7f5f867339dd0258fa7d0f7707c6
664599 F20110209_AAAYQX nahapetyan_a_Page_052.jp2
ca0c7b854563017e453e8d40a8d7f0eb
e2019038576657a3f25bfb0b8ee2657dcf227f07
29025 F20110209_AAAYSB nahapetyan_a_Page_114.pro
3a496698fa76b0c554bce45bc6873170
ed3651856f23e57af9a7927f5b9b807573116414
40973 F20110209_AAAYRM nahapetyan_a_Page_051.pro
c1d9483ed792060561149c273e2758ca
dba697cba92ce151a7d5f55a2b38e01eb7b3647c
F20110209_AAAYQY nahapetyan_a_Page_080.tif
7e7ae2c5e71a19bed5769bfd597dd5d5
cf571a3c047efa86bbb23115bde4ccc32df2af50
2783 F20110209_AAAYSC nahapetyan_a_Page_137.txt
336da8ce9d6590eb5443e2dfafa358cd
266eac7eedf7750464f095e71f7f767b09928eb4
F20110209_AAAYRN nahapetyan_a_Page_150.jp2
2d9b33c0bfa21ec35cbca31c9326f136
8e606267b2145e6b896ec0cce50da918af6ceb87
61683 F20110209_AAAYQZ nahapetyan_a_Page_071.jpg
a68a5d69a980ab51cc7c3c70241d6162
89da8621c56295bcca404f994e4aa98933754f58
70396 F20110209_AAAYSD nahapetyan_a_Page_138.jpg
269500dd7eb14e8a0b57516204b04533
af3dd4bcbbabb598b9d9273d64f5b0e0194cf6da
2181 F20110209_AAAYRO nahapetyan_a_Page_122.txt
e5bb28b2a3cf5e5b67628b0a7ee2e4a0
dcffe8d0dc9be5f424289db8f3025e684cf556ff
1971 F20110209_AAAYSE nahapetyan_a_Page_036.txt
f1fb3a64a44740e664d507843241880a
2bcb781c4f6805eef12bafd8ca09b08160a80056
14453 F20110209_AAAYRP nahapetyan_a_Page_114.QC.jpg
def582d5a86b7e7208720f7424ba309b
4ff3d029dd244b099dd23e15934350431c968f13
5312 F20110209_AAAYSF nahapetyan_a_Page_036thm.jpg
387d7141e008a10dd4bf12abd22f999d
984611d3537b612ee152a45d12d96f82633f8282
2187 F20110209_AAAYRQ nahapetyan_a_Page_111.txt
be43c423a09cc3b38710eca26cce9de1
48896fdeaa5755c495f6e3e7db34c59f0e094bbd
1769 F20110209_AAAYSG nahapetyan_a_Page_024.txt
345ccc570ecf7cec98f4718af00b5f55
3dc7f387ba2d7ea234a8d5c963da57831b4ee97c
1051979 F20110209_AAAYRR nahapetyan_a_Page_076.jp2
a8ee73011d286b42d69f24b6d1bbf6bf
09e03af4a2e4db9796b1fdff9f671be79a8c7cc6
22693 F20110209_AAAYSH nahapetyan_a_Page_129.jpg
a48c31bf88feaa7561cde79ab9f20e8a
1639b6237abaf3abcc57b77e84d5108ce8cf810b
520699 F20110209_AAAYRS nahapetyan_a_Page_080.jp2
cbf762ae04445040f8e2dfbb88ff06b8
67a4b30caf167844751758205ef2994370221380
47684 F20110209_AAAYSI nahapetyan_a_Page_069.pro
59e7083d3079cfa9cfe643eb1f399d8a
de4bbaf308d293b9688a51906bc1b3457c46294c
4739 F20110209_AAAYRT nahapetyan_a_Page_065thm.jpg
46871a93955d1b67562fa4a3f354f7e5
972e0af16cfc7dd5668e650b47cafd8b0485a26f
46684 F20110209_AAAYSJ nahapetyan_a_Page_090.pro
31e8eb126ebb5eabd08c8fdfeb330b95
d3822ae6e3de58e8ec43a27b563ee0032a60fb98
25336 F20110209_AAAYRU nahapetyan_a_Page_149.QC.jpg
2f1964d0e38dffc407f2bce74c307d49
82bcd0089d36d2482ccc98548c97280756207fd5
55197 F20110209_AAAYRV nahapetyan_a_Page_029.jpg
65e7ddfbdb43a51679f06ddb56741b30
fd14a90075177cb8e0816993ff4f316cf5e8424e
38185 F20110209_AAAYSK nahapetyan_a_Page_119.pro
f96189847096f4baf8a618974c27d90c
c7b5bc273f8a17f57d1bc853e5d014a5e12bfb78
354022 F20110209_AAAYRW nahapetyan_a_Page_133.jp2
fe0873c380d3f3dfd82668c32e81d3bb
f6615c6945010b653ebf9c4ef941cc687f907b26
25563 F20110209_AAAYSL nahapetyan_a_Page_110.QC.jpg
2a8e5c674a0180804e49a490aa53d5a9
bfc65767a2248d45461320c835db42b99595c5e5
55666 F20110209_AAAYRX nahapetyan_a_Page_110.pro
784422ba11c67d1a04388fb800ebc5ea
e635d6c735024fbba59f2979f4aa973ae47b5f44
2044 F20110209_AAAYTA nahapetyan_a_Page_045.txt
a6ddc20bff1e5102e3c018e0c62bf93c
c6dcd3c3afa8f418eb4edfab4764b36cac0655c4
79970 F20110209_AAAYSM nahapetyan_a_Page_036.jpg
6ee30a1524a9bd3e7eaaa1567687c160
255191699820d92ca0a60dcd953abefbe9021d09
1051936 F20110209_AAAYRY nahapetyan_a_Page_062.jp2
fdaff2508e6a1268db26f882bbfc4b19
81d32b0848f06abf85da15fa4ba65859427b0bd2
22319 F20110209_AAAYTB nahapetyan_a_Page_006.QC.jpg
c8fb0b29166e28b5dfe47a2fe603fc53
963efb206b3c1ecff5cf476d6351002a685c000d
3628 F20110209_AAAYSN nahapetyan_a_Page_005thm.jpg
6ec7f8eb5f05d9b75e0d4f40597be1f3
1df05cea836d3f2c4bdc1004f790c98f7f6d8480
5077 F20110209_AAAYRZ nahapetyan_a_Page_099thm.jpg
2ed00bd5de3d13c522c160f6d51cef98
ac437cf35c2f25259efe47fd7e84dcfe1bc1f3af
546 F20110209_AAAYTC nahapetyan_a_Page_089.txt
62e2af29dc0a4f32e902a887d1012634
8cada1a469d43aa61cd348774c94d758bde5e59e
1051922 F20110209_AAAYSO nahapetyan_a_Page_028.jp2
a91a123f7b50ecfb8e4028824c1e51fd
bd328809057e1fc63dd71f8c63e2d770b72fd661
19022 F20110209_AAAYTD nahapetyan_a_Page_068.QC.jpg
6f552d9aa9779948e067b974de1984dd
9ac7876b64db29e39cc6d0a99a51d86c088bf079
68864 F20110209_AAAYSP nahapetyan_a_Page_053.jpg
ceb8de03994e5f4d33f894c85e42aef7
65fcc040c1062f74bc702a2fb661050f94e014a1
54563 F20110209_AAAYTE nahapetyan_a_Page_134.jpg
811877fe6dd56f7e8cdc840573afb4ac
ac8a5320ed66f0b086ff5dda4fc71fe7cc523612
6983 F20110209_AAAYSQ nahapetyan_a_Page_007.jpg
41878cd818aaddca3df00718bba748e4
6d4b5907804ea9bc9ceb7d02ccfdad92946b4777
1051962 F20110209_AAAYTF nahapetyan_a_Page_083.jp2
0f051a8c121192398aecd7bb5dc922c7
0a57584a6ee71b098760a5a76d8374a9cd8cb0a2
5646 F20110209_AAAYTG nahapetyan_a_Page_109thm.jpg
bb411394603590fa3bf708a87aff06ec
3525cc54ca6d6a3480ccb46d494043ac8c8adde8
F20110209_AAAYSR nahapetyan_a_Page_107.tif
fa9d386248dd28106f269d36ce32555e
c81b70a085e730cc570a2b3cd0b6808c00fa3ec2
4975 F20110209_AAAYTH nahapetyan_a_Page_038thm.jpg
2397c32e40bb52c5b2d769ddd10fa8d5
e21e65b164efd3b3fe285f488c41a789fc032d79
41861 F20110209_AAAYSS nahapetyan_a_Page_091.pro
ed6cbf851b7c6d58a006370ea0681d1c
4452ff1d5df956b3ce94741c9809de978560c195
F20110209_AAAYTI nahapetyan_a_Page_068.tif
6717238f9d55eb0656d2f329a6a9f4c9
d0ffd446234457f6240c15b2f498f194b3a376a1
5423 F20110209_AAAYST nahapetyan_a_Page_063thm.jpg
aba3e18da4d8c4f6bfdb0e69605e8e0f
ed5a26f03ec60ecd71b1cd5aa65c24105935d50b
25692 F20110209_AAAYTJ nahapetyan_a_Page_111.QC.jpg
b44bf92d22dbf55664e5f41dffd42c86
7e9fcbe42907874f72af85935ff70bff57204bc0
52573 F20110209_AAAYSU nahapetyan_a_Page_096.pro
42445beb2c87c7e644877b2667b5d968
40773a82f83fd90b2fe4943a4fe53424c008b8c3
4704 F20110209_AAAYTK nahapetyan_a_Page_081thm.jpg
a23ae86f6b6d35f1074cf72c4c40d411
8717c1f32ca0168987ab79c720a524c6db6b5219
72190 F20110209_AAAYSV nahapetyan_a_Page_045.jpg
ccca8a99010998759a642bcf60300c76
dce14608c3f2e808dceb85b85ad82811a82f9655
5613 F20110209_AAAYSW nahapetyan_a_Page_107thm.jpg
10b9a3dffb56500e50279916b7b9b245
7d63876c4725cae4770beb73d0ab16650bd25239
F20110209_AAAYUA nahapetyan_a_Page_007.tif
07be01151e65ff36456335c0b88e488b
35fc174fa37b83f1994549cc474a647a0d65aa10
2244 F20110209_AAAYTL nahapetyan_a_Page_144.txt
10f2d7e77049d5a1d4a314011e6fd0df
ee0a5264d31122e111b80f9838ff7d7b15b7eec0
F20110209_AAAYSX nahapetyan_a_Page_111.tif
b956a704f64161ce558cd8810437b881
b6a0e7557fe98f6f17397563a4918c2561ec542e
44768 F20110209_AAAYUB nahapetyan_a_Page_038.pro
fbbe6b45a60c392b83ddee9462bc43f0
55d56cb05f62a260c0552688b9206836b07c4080
3299 F20110209_AAAYTM nahapetyan_a_Page_070thm.jpg
57e101bb99715299de08c6f2bf2d8f2c
8662241ba55feae68b5f808d544e9c6053da8fee
1717 F20110209_AAAYSY nahapetyan_a_Page_118.txt
35449af539c2acbeb47a55381b0a09f8
c8e084831598833a595ca3055b91045c4c3f8456
F20110209_AAAYUC nahapetyan_a_Page_118.tif
c4bcfbbf2699137971da9b0c8d7d5b3b
b580c28bb035f3ac58d7a6830c3fc212bbd9e44c
2270 F20110209_AAAYTN nahapetyan_a_Page_145.txt
4a45832cafab719f5e08f731a53ae6af
6a76d5d7478f3b92e5e9d41dfd75c5079228910c
1051986 F20110209_AAAYSZ nahapetyan_a_Page_060.jp2
84ccdf28b3dfb7d18adbe4ebc19d6214
872b901812bdef78ede44028e6099c3c3089b583
7788 F20110209_AAAYUD nahapetyan_a_Page_009.QC.jpg
aba4f8b4c8334a8387c98ffb93a070ca
624a4fb9574f54601d727f818c6fcd52ab185f9a
2005 F20110209_AAAYTO nahapetyan_a_Page_055.txt
6935cc38d9142b845f8da4ea854a64b1
8617ef0aac0b4c210d010ccbc92cc8de5632b5d8
1051919 F20110209_AAAYUE nahapetyan_a_Page_056.jp2
9c54678040fcaa1c8b6d44d6c7a0bd4d
6b3c1b17a60c020ebca7012901f28220c1c7afcd
128 F20110209_AAAYTP nahapetyan_a_Page_007.txt
180b212c07c1f02c0788c67442727d4b
e0a0aa6ebb5b033bf64aaa30b7b031a6b29dfb74
1965 F20110209_AAAYUF nahapetyan_a_Page_043.txt
c561eb3f489d437c6c27c626ddaf29a9
1761ec8e5a0b9f19b2817254c414eb59d4283430
5302 F20110209_AAAYTQ nahapetyan_a_Page_076thm.jpg
c2cac99f121b37698b62081e05351190
5da980f4b934f8d8471ca30f38455d329d229bbb
50321 F20110209_AAAZAA nahapetyan_a_Page_073.pro
96deee8e5e12c243326a7446bc208bf4
d804e1eb43862df19c3023bd7c8e8f66fadd67ef
53837 F20110209_AAAYUG nahapetyan_a_Page_078.pro
81c3d09fde30087cbdf96c45ad2cff95
9dccc89276f0a408e1f996676851066952892e2b
3712 F20110209_AAAYTR nahapetyan_a_Page_130.txt
638a132650ea432fe99ffd41b779e718
be59378f5f8cf18386b1877e029b7f1d5acb27a0
41471 F20110209_AAAZAB nahapetyan_a_Page_074.pro
e3c830b94ff115fd09b68b6f4a82d410
38e421ea4c7a2487b39124a27c37b4d52596973d
74394 F20110209_AAAYUH nahapetyan_a_Page_112.jpg
9f687014078a311f72c9b9551d04976e
1aa4002b60ef31ee655a62ea36af3d9c016dbe6b
5722 F20110209_AAAYTS nahapetyan_a_Page_146thm.jpg
4b9d607d26f43efd8128495d440f205f
aa45fa188c84b75ea32df9e8f89aaa14e6cebd7e
52275 F20110209_AAAZAC nahapetyan_a_Page_076.pro
ec3fddc65bbab64dbd63947817f31a51
6f04080e7c579663f5ebb5cc90309c98328d0f53
2152 F20110209_AAAYUI nahapetyan_a_Page_059.txt
4dd0d9d06b8fd1a2cb3768a4bb2a9238
16cf18a19d8df3b2b3bd4585270e8f7ec959ca95
F20110209_AAAYTT nahapetyan_a_Page_018.tif
cd056a5440cde761691e73626d861d5a
c21e6ca01d355f7a79848caa23bdbf5e56af3aff
43658 F20110209_AAAZAD nahapetyan_a_Page_077.pro
b82b06489d0ca3800e9299957c5e5ea0
c2dcb0b44938a1dd4c45a4002ca0dd448491f6eb
41564 F20110209_AAAYUJ nahapetyan_a_Page_067.pro
08d1a0622e2c2d80e206227d8f773618
ec77096670129e5d4bb36baf5ec57aeb01083d29
39673 F20110209_AAAYTU nahapetyan_a_Page_095.pro
7a9900bf618ea11a69f5a89bd6599658
42abb0d524717ad87af9cf0abad1bd626e804a6a
23494 F20110209_AAAZAE nahapetyan_a_Page_080.pro
10abf923b8c0a182365a2be8d0c85b9e
4b8bd06ddf20cfb4a7dd57ebf7b518d62d28ecee
1051978 F20110209_AAAYUK nahapetyan_a_Page_144.jp2
db02d5c86839d2c55047e207d4505ebe
e2b414d42ec9956e430110aaa3187988919634f0
F20110209_AAAYTV nahapetyan_a_Page_106.tif
32882c70d5c983351a4957f03ed1b561
cc95cf5c42f0ef510c01722f9425a8289608c037
37884 F20110209_AAAZAF nahapetyan_a_Page_081.pro
0a38c2bd4af3964f8b16d1e81488697d
7fbdfcb78c40a83e0edef7784ae6995dac2cee23
20457 F20110209_AAAYUL nahapetyan_a_Page_067.QC.jpg
fbaf743530292922b252f96e9a1b7b1b
459af029b8e9b39aabfd73773dc840ad18f123a1
442774 F20110209_AAAYTW nahapetyan_a_Page_037.jp2
30c00f8757046e362df572e042d657a0
84dd7eeafbb40d86cd1ed412782e4f90379e1ce3
50548 F20110209_AAAZAG nahapetyan_a_Page_083.pro
84ac70559640306e752c7a6e3c3d2260
6ecb340c9fd1b4a27f8d322d3b0efbfdc7a22880
55403 F20110209_AAAYTX nahapetyan_a_Page_108.pro
239939765f56c1121e9c5567de7b6a0a
7f46d7e83fa6b5cbe58026ff2ed2c758e671b9ef
F20110209_AAAYVA nahapetyan_a_Page_123.tif
efe2c5e9cccae71873db5c45e60de224
5a7fac9aecdbc2923503e39bb7543c86ea273a04
40038 F20110209_AAAZAH nahapetyan_a_Page_086.pro
cd6fde53b3752ef4b52eda3b22deb465
37366d7e087311061dc64038ed283bfb728548ca
F20110209_AAAYUM nahapetyan_a_Page_116.tif
fa69c63337cc06c5d1af8d7a33f1c8a5
2c3bc62e254e608750ccfa5eed00b61ff883ab15
F20110209_AAAYTY nahapetyan_a_Page_056.tif
7eceaff7f8bf9d2fcfbd2c1f41fe99b6
3e31059db2a7eed580e3c3b926d753e80f111398
F20110209_AAAYVB nahapetyan_a_Page_101.tif
e6a1fb83e420e010c20713d626e314be
e1327050a6c0709e8f1fcf6d3ae0f29ed5c5b18a
13924 F20110209_AAAZAI nahapetyan_a_Page_103.pro
c4c5152ae4c4d385e7057b0073720747
7c6d42c129ca44b6e7106e33712d8d65f3b2ecb1
66644 F20110209_AAAYUN nahapetyan_a_Page_072.jpg
ebc5dbabbe81e33a97dd9719b3d7dfb0
f317b662586625987a5de068d322b1257adbc674
43078 F20110209_AAAYTZ nahapetyan_a_Page_040.pro
a2a6ae3261ceb3aece028cf53e34ceed
9ac1466f605c0196c7a80444806fbfa6f58edce6
43568 F20110209_AAAYVC nahapetyan_a_Page_117.pro
d25d9b332ff0755181068d981f625bcd
a2a966a648200fad8c27ce97778257b42f97b361
28273 F20110209_AAAZAJ nahapetyan_a_Page_104.pro
286b7e71cfe242f6172f2600ad808c49
a8e403e9453273ac443b4b31c39f623eb709e913
852031 F20110209_AAAYUO nahapetyan_a_Page_123.jp2
83b639a7c7a15529c5b527e1d86a4242
6dc062f34baa30bbc0981aa97dcb1a8a609c8140
26202 F20110209_AAAYVD nahapetyan_a_Page_098.QC.jpg
a29e4aa8bc8aa9e67a7331305d0b8f23
643b29fc7b1b5af6210ee13ba1b11d4e8b39be58
53358 F20110209_AAAZAK nahapetyan_a_Page_109.pro
21b14643f72ec724314a2409cda28660
7ab4870d1c4492fbe971b022e731b955d3890b7d
1020026 F20110209_AAAYUP nahapetyan_a_Page_112.jp2
3379f86bd3323c80ca14798dbf5e8c1a
1bb05e6777a272cc5c3fbc6350823c672ed4fcab
64601 F20110209_AAAYVE nahapetyan_a_Page_065.jpg
795a4f361d01d713888d8d326ba3714c
e6e0c31c6e2ae015d1ccdc6bf9a04e034b88adda
47671 F20110209_AAAZAL nahapetyan_a_Page_112.pro
c7d57e356134d75505c7c5f22308a75e
7e642c9efc3da410506cd9a00da3c0a36e81981c
1798 F20110209_AAAYUQ nahapetyan_a_Page_072.txt
b7cbd8fb6ea1dc665afe148775f41061
809796a18aa5801012c6c0f071b285cd685bffc1
F20110209_AAAYVF nahapetyan_a_Page_079.tif
ee48856deacc84fc35551f68e17688f3
0637faabb1088f8ed1c12176121a079e34b46c72
6501 F20110209_AAAZBA nahapetyan_a_Page_001.QC.jpg
dc9d8c65d29b28fd519cf0041e718a4f
ee227b22a1137c963c7a5fbfc9142b2f3db66951
37281 F20110209_AAAZAM nahapetyan_a_Page_118.pro
a7eea45943e5a93b24a1286532e857a3
84ffdf3f80d60e0709f1f7d915040b0bcde3b9e7
F20110209_AAAYUR nahapetyan_a_Page_070.tif
629ca43973910ad1892120687743f4ae
91141826a1a1ec9fabaa343da731b8979ca6b00b
F20110209_AAAYVG nahapetyan_a_Page_103.tif
5645db6756a61ea98b4856c107e58b7d
ab70b54e0da15fc9034dbc681278b887e55ac377
15262 F20110209_AAAZBB nahapetyan_a_Page_010.QC.jpg
4df9081684616768b8056abbe9fc15e8
985b68b504c6ea00dee0c08503b8604854b60d5d
48296 F20110209_AAAZAN nahapetyan_a_Page_120.pro
da376a1aaba97e525c1223808f5f0bba
2652d9c8967df77e3d4625c89b6963fea49fb63e
1752 F20110209_AAAYUS nahapetyan_a_Page_103thm.jpg
8b0150e3c444d3edda35219e4cd9a200
89f795839621f2fd3ec6abb11489e0bb40e62440
24857 F20110209_AAAYVH nahapetyan_a_Page_032.QC.jpg
bc52b6832fef663c3c55760827d81c92
72c6a5f90fe7b43b640939bac21d7dd9ad6ffdf6
71574 F20110209_AAAZBC nahapetyan_a_Page_011.jpg
a6956c1642a8361dd9306683268cc3ef
82dfc410b27a0062c5ca700f59adb55d9e00eb6d
53807 F20110209_AAAZAO nahapetyan_a_Page_122.pro
8ae27bff0696fdd6de98b322cd10c7e6
3ee10240629c3b7c19f45ddf48e872ab4703aa94
5328 F20110209_AAAYUT nahapetyan_a_Page_087thm.jpg
1332f24774a92255baa0cbe826553fde
9bd13bc7eb251a21ccbf3246aeee8e90ab99fff9
4603 F20110209_AAAYVI nahapetyan_a_Page_095thm.jpg
ea52e53df7293c75db1478ce51b0de93
5189a4e5810daf752888a9594823364b8ce99e2c
73317 F20110209_AAAZBD nahapetyan_a_Page_013.jpg
d5a99077c253a733844d8e6968779f11
bf68985c031206c69b8ca1dc3deb24d75d0f20f8
64124 F20110209_AAAZAP nahapetyan_a_Page_130.pro
fbbb9461684eca6982fc4e5d25f88410
bce7d2fde5541bb25ee0a9cb2d4f787c9d861042
26431 F20110209_AAAYUU nahapetyan_a_Page_048.QC.jpg
9a09ac333ab7ea5c76149694c0897e8b
849bc204db8243307433ebc9da2bd2037a3a1554
825477 F20110209_AAAYVJ nahapetyan_a_Page_121.jp2
d44833eb326cfb404c7db26a214cd426
d4dcdfe4265621f30cb731a67c0141f0389bf59d
69622 F20110209_AAAZBE nahapetyan_a_Page_016.jpg
ee98819cd669d833decc61dd46d2492f
ae2295265d8a92547c079d8d6101bc87decd0449
20125 F20110209_AAAZAQ nahapetyan_a_Page_131.pro
9c8fb945c4abaf0539e84071328393ba
c3fe34474c85934ed865ddc6bde7ccd0a617a6ad
72556 F20110209_AAAYUV nahapetyan_a_Page_120.jpg
8a9dd50f42aa2443723a69c49ac7ed8e
89da7e3d12a7b034d885b3b325125cbf69a31c24
8204 F20110209_AAAYVK nahapetyan_a_Page_152.QC.jpg
ce389227273b8f7372cbd410e579525d
e44b7a2ca31b0dfec4c88d197aa90b9798ba95fb
25132 F20110209_AAAZBF nahapetyan_a_Page_018.QC.jpg
29469912efbecdf7a20389934f4ab964
82e98dddc0a2c13da2d40ee65e68f5e14989e599
73223 F20110209_AAAZAR nahapetyan_a_Page_138.pro
9a8e9bd54f0cb477ffed0bcbec0f481f
d8b60332ab6eb67ca52b8342938fe2fb9c91cfc7
F20110209_AAAYUW nahapetyan_a_Page_020.tif
f608b7860873553162e391ef1eec4fa0
6d0810e1eaae77cbab671ff1015aa7c6e873fe15
23913 F20110209_AAAYVL nahapetyan_a_Page_083.QC.jpg
c380cad04214d4db362ba57b6ad7e665
92b5cac811d3c8b83cc5c02e96b0e8d8997745c0
60790 F20110209_AAAZBG nahapetyan_a_Page_019.jpg
3f401bf795a7551f788dbae830aaad12
c1f8a8a21944a243d86516bacafda602ac36feee
22585 F20110209_AAAZAS nahapetyan_a_Page_139.pro
1870bf240d757c61cdca264781696d73
9d87fa13e6cd092806fe571bceea41f29d828093
1051981 F20110209_AAAYUX nahapetyan_a_Page_078.jp2
46811f5d0458094f3707a127dc779053
56d09d337a95bd935dc8ff6ffa2b21c00ef64ab8
F20110209_AAAYWA nahapetyan_a_Page_023.tif
eebb4157dfbb0e9657804a09394503b2
9bd3c92a4185095970806e2db65985232648467f
60892 F20110209_AAAYVM nahapetyan_a_Page_074.jpg
ee6e6e68595165883d97c39d3a1b2373
2005549f6b17b47c4b2c690fdd4aedaa8d90aa6a
56528 F20110209_AAAZBH nahapetyan_a_Page_020.jpg
7b71c528727adddd09dd555b7f93f23e
e9704ab82095acfec245c1e0c78e54f7068921bd
12242 F20110209_AAAZAT nahapetyan_a_Page_142.pro
8686da85801738a5920f971c39bcee63
a609ede7d2968132fb330f25df54135dd7d61194
1762 F20110209_AAAYUY nahapetyan_a_Page_084.txt
992efeef8b2f354a652c02afa3d73f81
4bc8bb3f4815d85efa8c778eab42ecbc39cf6aa6
F20110209_AAAYWB nahapetyan_a_Page_025.tif
fcfb9e90d4f33d2ec5bf79f03c79b462
c9ae541076b0fc8818529e9666878b045950d836
66796 F20110209_AAAZBI nahapetyan_a_Page_021.jpg
77b34a0b95c53fa86c9d1c29ab475003
fe330d0fa9deff6c8c955b93fd101089959c177b
51238 F20110209_AAAZAU nahapetyan_a_Page_143.pro
62349706a868e380759bcd6a32618f43
15b2541ed2bb89dd4ad438cdbae793e1199e7e85
886747 F20110209_AAAYUZ nahapetyan_a_Page_086.jp2
271532559bca17a4a92c6fd21bdb1409
8af5ab37fd4ac8b9d19372f852c40765e1c78920
23042 F20110209_AAAXTA nahapetyan_a_Page_055.QC.jpg
1c153752b5208672c0478b8b1f48d8dd
8e901d7e3d07b7f08028b89f12710d7466b7daf6
F20110209_AAAYWC nahapetyan_a_Page_027.tif
cecd95871f914838a89ecec14ec16755
7d7edd7ca0d519a395ba0c41dc72a6756cb851ae
F20110209_AAAYVN nahapetyan_a_Page_063.tif
cc87d05db0ab982ae2cde645f6fd6193
968513af3e77ae661be4e3b7908a815d11a88274
20366 F20110209_AAAZBJ nahapetyan_a_Page_021.QC.jpg
7ab5fb0608cb5800afef83a3c66a87e6
37cca54e82f34cf9e79b193f949d1e59a927ec6d
56091 F20110209_AAAZAV nahapetyan_a_Page_145.pro
83601af0ad4d9febf11ab8dc5dc72b65
fac582fb2e7e711331a789cf4175cf11666a3d90
2092 F20110209_AAAXTB nahapetyan_a_Page_129thm.jpg
f9a72bef9bd69a9f91ca66fb49052de3
9ef37618c486f84a00403dec3089a56cb342505f
F20110209_AAAYWD nahapetyan_a_Page_029.tif
c8d4180ce5c42ca0aa3ae2cf8d2e2505
47cba433bf1086a423aef3d2cbae8426e5c2496c
12958 F20110209_AAAXSM nahapetyan_a_Page_142.QC.jpg
30fa96f42959eda8c9c0d1f7fc16aec3
070813cd959b59aac39c3ef3109f778b1efa1895
22384 F20110209_AAAYVO nahapetyan_a_Page_049.QC.jpg
499b9407fb09d1c0eaed55c5b4a7a990
70a1c0057b35314ae7fccf9c36e77b454e608981
78754 F20110209_AAAZBK nahapetyan_a_Page_025.jpg
0e1915b6b83d77fc8ad4671837dc63a3
a796af3cf1ec189d579506402f444d51c3943045
58192 F20110209_AAAZAW nahapetyan_a_Page_148.pro
bab635738dbef41951d42728ce41014c
e950542d83179524ec25543354df2105412c22a8
2062 F20110209_AAAXTC nahapetyan_a_Page_054.txt
69e443f1a0c4dab2007e9912a890e029
79d377d529e527e5431a6c251cffb484ca5ed4ef
F20110209_AAAYWE nahapetyan_a_Page_030.tif
6546c65d2c5b5eae3065677866115c77
dec01d5867bc19e8beecca25a274718adcf1ba7d
68308 F20110209_AAAXSN nahapetyan_a_Page_086.jpg
af1047c0197b770f98e4cc53674afb71
b4f9914d7e12b407e70694a6c45174d7422fc578
F20110209_AAAYVP nahapetyan_a_Page_031.tif
c8ab116a6c53d57bfbfe020449dd14eb
2670a4a801ea6fc49f664f5b9e9cd29747c0ff03
23566 F20110209_AAAZBL nahapetyan_a_Page_025.QC.jpg
a62838810351270283ade9c121c529d2
9408293fdcc328be8c1199c1c891a6a1db5ae47f
23701 F20110209_AAAXTD nahapetyan_a_Page_128.pro
eb2c06b02572c92715a5d11f63e9293b
1d787ef352dbc7d16f2a8be474af32c669316fea
F20110209_AAAYWF nahapetyan_a_Page_033.tif
abf94200088d414352aa9f1f0dab0ec9
39aca2cae7d2dc10a12522d6dcf5c424fa8debdc
21851 F20110209_AAAXSO nahapetyan_a_Page_034.QC.jpg
17677494952ca52408cfefc93ff3b0bd
a6236567e412f3f97ff500e76cf05af5a4f89a36
18694 F20110209_AAAYVQ nahapetyan_a_Page_005.QC.jpg
5c6c95916167d221064fdf4c5e503ffd
052e87c81b50935c0e6d4d01c4e588f005c2297c
24389 F20110209_AAAZCA nahapetyan_a_Page_060.QC.jpg
9af7b81ddf67f53ae5a47829107ac842
3fbe5d1fd2d600350e667bc5af3170423584a2b8
66272 F20110209_AAAZBM nahapetyan_a_Page_026.jpg
32f789e42159df0ffa91da70b7d42b88
06bf363f51b7c31a7c772ce2aa9f48e6264ec21d
60137 F20110209_AAAZAX nahapetyan_a_Page_149.pro
7295238768c5035bcb6f9dcff5e30b6c
fab06daf1e5fffa5afc560b34ff250029c912a43
85843 F20110209_AAAXTE nahapetyan_a_Page_111.jpg
e65f9a16bf4454ad653cd95c7ce4cb56
37d9cef7f48da26da45d1b0f80df3c339fa97f4e
F20110209_AAAYWG nahapetyan_a_Page_034.tif
7d28cb3dce9349a3588a890d0c668cdb
eb88bb995d3ff2a3763cdfb2cbbc74b739a2c55f
13688 F20110209_AAAXSP nahapetyan_a_Page_151.pro
176e3c16067f53f887fbe8aeda250bd0
bdeed06a7d875dbde5d059626c017a92d9b62300
4810 F20110209_AAAYVR nahapetyan_a_Page_093thm.jpg
c496b668acea4fa89946da6f7509dd45
2282c3bf9bdddfe49bc52de05a369314c05919a7
18810 F20110209_AAAZCB nahapetyan_a_Page_061.QC.jpg
86b440b08bf62a884fbc224aeeff2d92
83da23317657c0c35ffd92a329965609031c5130
77941 F20110209_AAAZBN nahapetyan_a_Page_031.jpg
e163f80929368eaf8e0629582bd59671
810a772f60a7084a2eec727e4e00b10c0673a74e
58628 F20110209_AAAZAY nahapetyan_a_Page_150.pro
7d20e9ce804f0aa3b5ce231c2ec46496
0b9f611e6897cb3ec338f0f2cd5e25885ca16f16
1916 F20110209_AAAXTF nahapetyan_a_Page_025.txt
4852783827eac26b4c510b0ef25be762
28458357cd6cdc9ee267700456cef9c85a26c594
F20110209_AAAYWH nahapetyan_a_Page_039.tif
01115c5747929668a2b84861bd82aafc
4c8754d6af46ac02b0c3217daf9d75458af33592
2345 F20110209_AAAXSQ nahapetyan_a_Page_003.QC.jpg
bfc1e6dc2cc2165909a94a4ef1ec00ca
b9f9f67d533511de9d247cbab0b4462f482bcddc
249494 F20110209_AAAYVS UFE0015623_00001.xml FULL
520bebf2e5bd27421a25e0f542a7d346
683ee1e9f75a446a677af20ba80fa217ab08d37f
77476 F20110209_AAAZCC nahapetyan_a_Page_062.jpg
14c8ed067e603371105330aaf89ab8b5
d8c5edcb022438c3f3a369e152fc75af7100aa0c
22885 F20110209_AAAZBO nahapetyan_a_Page_031.QC.jpg
8ec324d3a9ed803285f7deab3a1870b1
8a32085c01ce4b19d5c316069ee97cd4c01885c3
15138 F20110209_AAAZAZ nahapetyan_a_Page_152.pro
7e53d8cc4239237f70df04ce7fe952ff
280c28a020808a157066dd9e582b432249fb88be
F20110209_AAAXTG nahapetyan_a_Page_065.tif
598cfa0c6a18fca8b1666215e7741daa
9c03d97bc48a8091f811b6c213556a60a4b39722
F20110209_AAAYWI nahapetyan_a_Page_040.tif
8a9740724bb0de40f4bf5e102da9bd0b
d67bbd19773c1fd71da72166d8bbb1725775e1d1
4423 F20110209_AAAXSR nahapetyan_a_Page_121thm.jpg
8b777a02f0cb9e864c20735d58fb9f13
f0858e3b799bf076d1b218b602c9435c5077b4e9
22303 F20110209_AAAZCD nahapetyan_a_Page_062.QC.jpg
617e549ce6f1e4615f36b2292f14e841
ca60e11085d7c0aa055428fd8532e70f79d0d9a1
83508 F20110209_AAAZBP nahapetyan_a_Page_032.jpg
c3fc020f8894710b9e83494941bfd0c8
66c4f9119f10f05da1953da0b54023fd164f8487
49255 F20110209_AAAXTH nahapetyan_a_Page_036.pro
1775992dfb504e68fb80500ed622c9b4
6b125d2e42645bc5282184177d87362517066232
F20110209_AAAYWJ nahapetyan_a_Page_041.tif
0dd6f2ffc04bc3711b5d004d25cb31c4
b026217a58a32fc75dc18a297805f15d9635ceef
15194 F20110209_AAAXSS nahapetyan_a_Page_113.QC.jpg
a167bbdb0f9335f54bc48371053f9cde
321c579829499cf656de6e9e1909925851df5c3f
25250 F20110209_AAAZCE nahapetyan_a_Page_063.QC.jpg
c3815de26c86fa7addd84f887004e8c5
e8a0c6a298bb2b941e7b0f62d7d99346cae8a05f
82581 F20110209_AAAZBQ nahapetyan_a_Page_035.jpg
0bce814d74fc06e9033057cc21e570d6
959261ce8bc83a8faba5d95db40f8370d463c296
83076 F20110209_AAAXTI nahapetyan_a_Page_059.jpg
4133c9963f6ccb2372b08c769c8da37a
ec07c500cab34f75e1e200bedac6366e3b712c20
F20110209_AAAYWK nahapetyan_a_Page_045.tif
0892badcc36694a7cca96b13145d45b3
f5cf13373733bee57b05da6be3dbe177624a85c9
F20110209_AAAXST nahapetyan_a_Page_131thm.jpg
ac81696e525debeb0868e70c073812ad
f03e2a06a5fe9a6dc2685b2955128d91c843fb99
F20110209_AAAYVV nahapetyan_a_Page_006.tif
6c7d25906d795f024970070e43be03ec
28e3a98a1ba209c57eb0124ac8b1f87f1449cdfe
25253 F20110209_AAAZCF nahapetyan_a_Page_064.QC.jpg
9efe131c19a9c37428f1bc80b1bb06e2
91ea0ec50aa587f1de779d659d3a74bde7781971
35358 F20110209_AAAZBR nahapetyan_a_Page_037.jpg
6eb6766a91c8a41d6403c107f50d7059
05e67830ce091db93090492652ffae4d25e18f07
71063 F20110209_AAAXTJ nahapetyan_a_Page_040.jpg
8c4576514d3554fbd9c79f3bd5080d19
5cf7adeba5f44f691dcee5a47534ea1da877e58a
F20110209_AAAYWL nahapetyan_a_Page_046.tif
43e2de86b8b0b5c072135fa8ac3dd938
3620e7618d773a155d084d2cbd2b85d40037726d
46824 F20110209_AAAXSU nahapetyan_a_Page_025.pro
16184dacc30ac629352347ee740f5185
e7718e01eadad6b2b5aa61cf88559c6e37016d46
F20110209_AAAYVW nahapetyan_a_Page_009.tif
cd836a6a611d0272f5c0138576661514
5e8953bc988c2961d35b7d35408da122f79ae704
12947 F20110209_AAAZCG nahapetyan_a_Page_070.QC.jpg
5237c66936ddc5fcfafe7ef0c144813f
9f1fad9fffa239820a3ea94e297cc4aeb68c7380
71550 F20110209_AAAZBS nahapetyan_a_Page_038.jpg
8cd128765b97b81d6aa00d9734eeeaf7
c9eddd855895c119023aeeb9980d96b418457d41
F20110209_AAAYXA nahapetyan_a_Page_087.tif
cb8cd8c2389c6847da1a855c865531cb
8a6b89f6315a946a8be6eef7316a7f5896092e90
82368 F20110209_AAAXTK nahapetyan_a_Page_078.jpg
043eb02278469ead5384ca18b47fd910
8a64e49c7525466f98a0970b3e85b47738aedafc
F20110209_AAAYWM nahapetyan_a_Page_047.tif
28811436627879be049a0704e7cfa73f
da1b1b3f6fdb1817a5380e881988742143badd64
1051964 F20110209_AAAXSV nahapetyan_a_Page_108.jp2
b765dd4d1cdd959e26d1e9979e81b67a
d05fec7c07d9c958725e824c86ff3fa8dda400a3
F20110209_AAAYVX nahapetyan_a_Page_014.tif
0e8da48b806d8c2cf5da7e325ec3ca4c
373b7e1176bce032bdbf9d54ac789be0f1b518f9
20228 F20110209_AAAZCH nahapetyan_a_Page_072.QC.jpg
3fd179b6511e310283469b919229a388
834dd5aee2b7f0cd802f46ccbf94d28ee9c411f0
75312 F20110209_AAAZBT nahapetyan_a_Page_039.jpg
691d8ca37b9f1d9c0ac4634af0455a9c
02990c0c19308971cfd56ce3b356d4059fbffe25
F20110209_AAAYXB nahapetyan_a_Page_090.tif
e05ba24c7b773672b56050dc17fb3fba
e821182cedabb0a6f6b0e871ed7e2a72a965aef6
811262 F20110209_AAAXTL nahapetyan_a_Page_019.jp2
a7f60bbb366b423d4b8dddba21bb0b3d
dc761461ea563be6b142f44e50b2a8c3e9dbfa5f
F20110209_AAAYWN nahapetyan_a_Page_049.tif
3edbb81dde4cd1aeb5f5b2880f304eab
9ca0555a46a95a76c29d0002704b02445c9302cd
44537 F20110209_AAAXSW nahapetyan_a_Page_041.pro
d0b038502c998db4a1b2003f9dc55956
05b82549e6bc88d39bd8ef9078fdf41d09e1e5a3
F20110209_AAAYVY nahapetyan_a_Page_017.tif
4833553034fcd3a4453fe5ca64b71fd7
0fb8eaebb60862ed37b3dad1dff36b7316fab964
76888 F20110209_AAAZCI nahapetyan_a_Page_073.jpg
0d83b4ac5008e0a96d44f9dba0ecdd93
61d2efe6c8f286d82db894e3bedb808d6c0f33e8
75368 F20110209_AAAZBU nahapetyan_a_Page_049.jpg
c096a772d66490e9734261c4aac4e587
032d560a8f13d97c4c2956934a8b3384f4042544
F20110209_AAAYXC nahapetyan_a_Page_094.tif
9fa45a49a9b0d6861e08d9bd5e27141d
f468dafdc75f2de54e36c74aa70d24ef83d9059c
47043 F20110209_AAAXUA nahapetyan_a_Page_030.pro
02b0bf2f98bf66ad92845db935ffd110
0fca4040efd623e7d5a91dd3dff963854ed3ab99
2252 F20110209_AAAXSX nahapetyan_a_Page_133thm.jpg
7d4a5caca0a6642d34b107f71d38a8eb
4bb9dc85e47f721782fda867878dc9b9a339f3d6
F20110209_AAAYVZ nahapetyan_a_Page_019.tif
430fb0e5aee5c9b0b3841a34dd35d7c4
914838a3eeacf574ff26bf2cd6263b7cc24cba8e
18376 F20110209_AAAZCJ nahapetyan_a_Page_074.QC.jpg
22d6b537574cbf3349ddf178569833ad
919f5ec66fa666ec71e080ecc2db86db52f2f3cd
16035 F20110209_AAAZBV nahapetyan_a_Page_052.QC.jpg
02f596ac587c3d7840ebc416cae4d170
89aa1c60a38aea1a8456d15d39b1004e0a319bb1
F20110209_AAAYXD nahapetyan_a_Page_095.tif
633e02ab307880cb176f2d6e77b0b874
37583113d80678c56e35abece7afa5199bf289cd
1620 F20110209_AAAXUB nahapetyan_a_Page_124.txt
800dc53af739b48dcfe041ea33f51a24
7ed7d3022c517d887182d66bff18e42637a866f1
F20110209_AAAXTM nahapetyan_a_Page_093.tif
10850a5f8e817b85d371b4a93a689537
8af604c0660bf97546bbe87c433a34e01d800e4e
F20110209_AAAYWO nahapetyan_a_Page_051.tif
346ce9f0e121379074494adc0d9eb273
e7daf19816572c6ef2c2762af214d124a0bea784
4500 F20110209_AAAXSY nahapetyan_a_Page_050thm.jpg
a0924e9db854cc8a3e40f47221f04a15
fad81bfe8d6897144222bc7aa23b03c81c32a812
85629 F20110209_AAAZCK nahapetyan_a_Page_075.jpg
f74861acf6444123a5452b774df1f2b2
ac625c1f595ae2f1047c3280b4b7f9c659a61f16
21293 F20110209_AAAZBW nahapetyan_a_Page_053.QC.jpg
2bd14490c567a9fb624dcdfcd8b8463e
ff0b61ed0c599955d9084b29f0cd30b6d33dae28
F20110209_AAAYXE nahapetyan_a_Page_097.tif
ef45c75e0836ef33bb807dd4fa526855
b96a82ba456da270a2adeb96468e40902be95981
F20110209_AAAXUC nahapetyan_a_Page_026.tif
344374f390493f30e8c3e94a954c5d3f
058e05f24205d8369d71065133475d91fd578daa
52844 F20110209_AAAXTN nahapetyan_a_Page_125.pro
4e5c9e23a6aced6de3ce3f573d99a47f
b6b92109c3f2c6382439e30ab1cd07fef5611705
F20110209_AAAYWP nahapetyan_a_Page_057.tif
3e6cde3c280642f2a9f59dde099989cc
e5b1cf9f66d2756295c922080f95eeaf5f299413
47463 F20110209_AAAXSZ nahapetyan_a_Page_114.jpg
c77009b1bc21cb497aa221d43abc1965
d82db99208d622b41e56f87c1b9fee7803048ce4
80753 F20110209_AAAZCL nahapetyan_a_Page_076.jpg
94891fd7b2b8d5e2908e499670f7cf61
772b0fa180c16af7abf452557e0293e463216569
23597 F20110209_AAAZBX nahapetyan_a_Page_056.QC.jpg
ff4a5aad6ed02bf0566ba226636e49ad
63e32cd659d15242c291c75365128ffb27b40c9d
F20110209_AAAYXF nahapetyan_a_Page_098.tif
516569e9c4e0f630800be8dadac35c39
4e4f9e5c93ea951ca137a17ff4f71b59641dca8d
22998 F20110209_AAAXUD nahapetyan_a_Page_057.QC.jpg
6d14c1ddd52464288e68c2d806f500d9
63544ecffb3a9e6926c212d34ec8ed033f3194b7
870324 F20110209_AAAXTO nahapetyan_a_Page_051.jp2
cf4ef01d281faaa855ec2e5cff37da41
d1b0f42cc254cfe707c2ed3f09b78377a8808be8
F20110209_AAAYWQ nahapetyan_a_Page_058.tif
6e5744aaee2b2358d76129ed92aa6a92
0eeacd2c969e6c45c5ce46ed4f0c92c34977c0a5
16083 F20110209_AAAZDA nahapetyan_a_Page_105.QC.jpg
0f0fe5323b7a60a7b52306e804b2c34f
ea2c3ea31f92214d4257903233f44d0fe24a8b4d
20055 F20110209_AAAZCM nahapetyan_a_Page_077.QC.jpg
32660201a9fbe011c9618d9739fe902d
9b6685524fdca7f405cafc14f5812a0a6a61089e
F20110209_AAAYXG nahapetyan_a_Page_099.tif
517ea24acef95facc8390c668e6908a3
d5e45de98f67fe1ed6f208e0ef3f44cfd9b19965
4237 F20110209_AAAXTP nahapetyan_a_Page_052thm.jpg
7c4461f334aa512b884951f6c944040b
e2c32c0478b9ff7f5bad995514ecdacae11f9d3a
F20110209_AAAYWR nahapetyan_a_Page_060.tif
591c6e21b38e94e3bb3f9d8075e88b5a
49bb31227066a21a68e401179b26670d51257d60
900642 F20110209_AAAXUE nahapetyan_a_Page_106.jp2
f9f953a9631d3d962b1732b086f491e4
47613ef1d1c232d542c5b0903d35d6bd373ea90f
21348 F20110209_AAAZDB nahapetyan_a_Page_106.QC.jpg
2e18c2ec3fdea973e01be9e07bc430cb
dd91e7bdefe375d25d82bf400e9ae24c636bb56d
60561 F20110209_AAAZCN nahapetyan_a_Page_082.jpg
5295e4c33c306b82fdbae567c54118b5
9b78ef105008d35ca13d626692d4b28999a99cea
75059 F20110209_AAAZBY nahapetyan_a_Page_058.jpg
2393b50978c4b10c2bf058d600094ef9
49b7db6630dfee5bc8ded5f3048b4d8c98a94e00
F20110209_AAAYXH nahapetyan_a_Page_110.tif
69752f04ead8d8b3d43f5e6d2bfd8af6
1d4bd8804da3406dee8fa4f2a4cf0e0efc06f6a4
F20110209_AAAYWS nahapetyan_a_Page_062.tif
31583eacffd3eb8d7f83cc3355b076b1
68aebe98e9ada57e19c7ab1273c89288ab057f2a
4903 F20110209_AAAXTQ nahapetyan_a_Page_046thm.jpg
6b54dbc903b6f19e191cee9cd7d7085a
d408928f6b7313364d4a537d95cae9ee33bdf59a
3385 F20110209_AAAXUF nahapetyan_a_Page_141thm.jpg
90ff89a16eca9236b61e868da51c394d
a2ea87be01092c1b63157b56842d5ec6a1780845
86170 F20110209_AAAZDC nahapetyan_a_Page_108.jpg
f5d1ca87c89cb4ceb97a8ea1b6c5a22c
8d6529dd653b24cbf8821a4644e2b6c8d927e8ef
80259 F20110209_AAAZCO nahapetyan_a_Page_083.jpg
ddde4aa7121e1347e9307001348f903b
2bac1887c8eb0c574865746448d5722257130bbf
24659 F20110209_AAAZBZ nahapetyan_a_Page_059.QC.jpg
16e22213459b87e6c4a30ecd7c206195
16bb98f3dc0fc6639e59178988d78c6e1c60ec6c
F20110209_AAAYXI nahapetyan_a_Page_122.tif
414f7026670978aea44101724489888e
78d95c54c07186d3caa75991fc69d92018f38f68
F20110209_AAAYWT nahapetyan_a_Page_064.tif
1bf4b03a02a8d12742e1697fc25f4e3c
37f72205fde002670da116397f1c4dab894a3a13
2324 F20110209_AAAXTR nahapetyan_a_Page_147.txt
acd92719a1c04e0434c30793a7fbb16b
3e7333bad1d97450cf76ffa8433af48be46a9b3c
F20110209_AAAYAA nahapetyan_a_Page_001.tif
3f9142f07ec817361cbde61fff46bed7
517b42283cc29a737dffa0e2be5f07d3f93147cd
70755 F20110209_AAAXUG nahapetyan_a_Page_005.jpg
9b6b569aebacd4c726ee98d02caabbd7
58eea925f9e6f3079f71a6bb15c8baeda58efd73
21883 F20110209_AAAZDD nahapetyan_a_Page_112.QC.jpg
901249d5e86961295614f88aac5a74f9
cf950fd95329e3c4da321183d3ff76264ef9261c
77754 F20110209_AAAZCP nahapetyan_a_Page_085.jpg
561c8526da9272e6e500c1f8152c375e
a5adaef5585af4c26fef648f02967b59d6677092
F20110209_AAAYXJ nahapetyan_a_Page_124.tif
68c32decd02e877683d75a9cba698de3
5a2f6082e428eee45bd3297965d48c6c67dc884a
F20110209_AAAYWU nahapetyan_a_Page_066.tif
da691693f21e3778c596f11b47af0827
bd50a65bcd898a862c46b46f40503f3cc7975d90
24079 F20110209_AAAXTS nahapetyan_a_Page_078.QC.jpg
3c02625c478d5673d60caa77323f3d74
266c4db05d3e2b596c1b5044bcd4766d9685c33c
971206 F20110209_AAAYAB nahapetyan_a_Page_040.jp2
59733b8e4ffbee09b9aecd8204c925b4
fbef0a735d8a7f5ca53c42c9b45c517263cf2022
56487 F20110209_AAAXUH nahapetyan_a_Page_116.jpg
abdd92a679d14f32ae845d29cef3d986
ff1676113733219abf1093a54c06b036414c0874
22105 F20110209_AAAZDE nahapetyan_a_Page_115.QC.jpg
e5f44947dda682546615384c015e16bf
cdce03c384bcc046cac9597afe46458d7289fe02
23535 F20110209_AAAZCQ nahapetyan_a_Page_085.QC.jpg
b0067dc5509cc896d815c8247b67730f
6a715c889364af23f9193572fdca59935fa60739
F20110209_AAAYXK nahapetyan_a_Page_125.tif
48528eba6ad67d9cb7797b99aa37ac59
812d2b8126419abb4ffb4fa3d0f0caaf1d4cb4a8
F20110209_AAAYWV nahapetyan_a_Page_069.tif
ddadc89811eda2e28bb5792f37c78429
4f3bf60a07ff5260de60ed3b80e753f69ddfebe8
74963 F20110209_AAAXTT nahapetyan_a_Page_027.jpg
579fbab29c337091fe3094fd3fac4db4
ded4334ccf9b5266f13df12a7fcada70d5d1d239
943534 F20110209_AAAYAC nahapetyan_a_Page_023.jp2
012c5ec65a9cb61707d4f8eb97546783
8f1e5acb2bd584356f363c2120a750c61d9e2fdc
1793 F20110209_AAAXUI nahapetyan_a_Page_123.txt
b0d5311b773c054e832ade32541eff8b
736d2df1e1a09dc2721ddbf66fa7cf7c74da39ba
57602 F20110209_AAAZDF nahapetyan_a_Page_118.jpg
bccb11a21d904cb27e948d6a4246d32f
586f9a8d179014bcd50e914936965bbc21965c37
6575 F20110209_AAAZCR nahapetyan_a_Page_089.QC.jpg
b997139a831461f467fbf0760dbfacae
5afda5bb8f5058ace5942659fe2e087c7a87649a
F20110209_AAAYXL nahapetyan_a_Page_130.tif
525888a29c76e7a949def774c5767914
bf007b8edabe413594098b92491341e2cb02a4ab
F20110209_AAAYWW nahapetyan_a_Page_073.tif
5cbfccde07e0f1330addd1c26db8a3c2
46b4a401a728557b1604e4c9b4e006daaafb5c82
69433 F20110209_AAAXTU nahapetyan_a_Page_084.jpg
6dc46e7a3fd2dabd4c25a778a87ed7d4
287fab26aea46716b8cfc7f5cf15917516be1575
21872 F20110209_AAAYAD nahapetyan_a_Page_046.QC.jpg
ca03672f36966f9a23f93caad0e60a1f
a06b1480741a1e05960f724e0b6efb6fe35a6b67
95896 F20110209_AAAXUJ nahapetyan_a_Page_149.jpg
f964abd2b2a79a09b277f208b0981931
6471d8a8754e9f031bdda99f82a0b179ec33a497
22248 F20110209_AAAZDG nahapetyan_a_Page_120.QC.jpg
2cb76fbdacc3c8380397591880c8853d
445292c80eef503be0245d8fbfe5c76f0ed1690e
61381 F20110209_AAAZCS nahapetyan_a_Page_095.jpg
a6e8c85a50bfef4c5ce5931deb1ef6da
8ebb436fa6755d7abd670310d122e40f5ea84210
2038 F20110209_AAAYYA nahapetyan_a_Page_031.txt
a35315040232b0e7da0fcb67eb72f850
bc5ecd1933341f2c36fd1a3ac6f209dbe14b8a16
F20110209_AAAYXM nahapetyan_a_Page_134.tif
983f251fc1ef004632a9737d07485b99
8a89221d0c25659d804b31610dc89a043d06c916
F20110209_AAAYWX nahapetyan_a_Page_074.tif
4671b8488d1caaa3f1f3fae20d7a218f
86815c3e01c43737b4624be24a006e6ffa649fad
F20110209_AAAXTV nahapetyan_a_Page_032.tif
2bb6978e9a5ea6b7c4a4c1a06bb3ecfe
abd3a8bb743f3112833cd37a4a4544024d407836
13276 F20110209_AAAYAE nahapetyan_a_Page_139.QC.jpg
57790b785607d9458c015442b7dff211
403e0a16677d27e768c4731b1a60297015f1e7dc
4945 F20110209_AAAXUK nahapetyan_a_Page_143thm.jpg
d85559ac2993700a2e989b7b4cef5352
ac107a9d434975f654a5c5c5a58123d07f525576
23717 F20110209_AAAZDH nahapetyan_a_Page_122.QC.jpg
f579ef1beaf1f72b29186ce082648c90
8ba73251dc820a6cedf2197e5f357824768366cb
19132 F20110209_AAAZCT nahapetyan_a_Page_095.QC.jpg
63aad1310a28ac102af8597d1ba85ddd
a5c74558a3b2ca17822ef1f6b14cb62ac369a918
1907 F20110209_AAAYYB nahapetyan_a_Page_034.txt
0f52fb813473f5c0b93cae1a565c759c
c64385e5326e65b08d0f3ebdf2a4274243eb8481
F20110209_AAAYXN nahapetyan_a_Page_136.tif
ce12b15bc5eeaf39698b207f63b82d60
973368eb13b781bfba2d87c5d942c0b587eaf91a
F20110209_AAAYWY nahapetyan_a_Page_078.tif
73ac8e5421709720f317c14453103805
c82ab14ef3aae9b48fa9e09eadcd853c5bd94cdf
15192 F20110209_AAAXTW nahapetyan_a_Page_136.QC.jpg
acd3c95166a533ce88b9b9d9f63b9224
db8bee03c868b25ebeb2d2c524c7e7da475db9f5
5683 F20110209_AAAYAF nahapetyan_a_Page_098thm.jpg
ffea0a2635e023e934c34d6a320e9347
e2f39fe2b062aa1b2ce9502e5c6af07d4980a751
5550 F20110209_AAAXUL nahapetyan_a_Page_059thm.jpg
d3721fe6a32a7a3b6ed41ea6bb049666
5ca499f7e1a5e64dedc591c6fc37868cf593fbb4
65990 F20110209_AAAZDI nahapetyan_a_Page_123.jpg
948593d83668229a70702399e70b1053
c59572be4da5f75bb5289c93137b6771b7495e1c
21974 F20110209_AAAZCU nahapetyan_a_Page_099.QC.jpg
574884a40b99783a60845ebd64dfd6df
02276750439b65b88960b795f99d8aa3e4d54d99
2056 F20110209_AAAYYC nahapetyan_a_Page_035.txt
7627bf3d393bef9335dd1f1a811fb86b
f9192b0f821d2c87dd2c47649d71ab17e6261093
F20110209_AAAYXO nahapetyan_a_Page_137.tif
f4e5c369125f3d324c431224b3b3525b
4bdb988deadc89427e35019b72a9ca1c55598a4f
F20110209_AAAYWZ nahapetyan_a_Page_081.tif
a37a67464fccd8c299ed0b24337fb19c
219b34259bab1adb92f155fc7a0616e771c91dcd
314163 F20110209_AAAXTX nahapetyan_a_Page_151.jp2
eff17246f6fb81c8c0d6739e60a60a0a
877e2299621d52a147f4407ae4648ccd5d484dae
F20110209_AAAYAG nahapetyan_a_Page_139.tif
5ecd30946c7209228776bf1284d9d69d
c05329b53c9eebd78a8861d093313117d45500e5
82028 F20110209_AAAXVA nahapetyan_a_Page_060.jpg
0ec4ff4993a23c1497df3865399458f0
e72cc8210ed13adc69ceb4575b27e29921a785e6
598856 F20110209_AAAXUM nahapetyan_a_Page_070.jp2
768894362c55b0331b7fae7d387c2970
dbf7a189ec1ff9d7e34a7dcd610e17c9631a0244
20063 F20110209_AAAZDJ nahapetyan_a_Page_123.QC.jpg
7751e10df9e42e5b4b1362bff0589097
7bc5716fbae6031d7ae0cfe6eee921d6c8282acd
22460 F20110209_AAAZCV nahapetyan_a_Page_100.QC.jpg
6d4070b947e46e63e0d1a59a9136ec2e
765ad69985d6b9fa3d5539b8d436ae3ff2b06e9c
1896 F20110209_AAAYYD nahapetyan_a_Page_038.txt
15bfbf18995ad3d3f2fbeabf3e75541c
1783c165fec39696934feec04ec59fdda8168400
F20110209_AAAXTY nahapetyan_a_Page_141.tif
32e7260bd1585287b371b581d91fd09b
bdf34dece15c54425e260f05af30fa056fabc189
25611 F20110209_AAAYAH nahapetyan_a_Page_146.QC.jpg
f9868beaa867b3b7858f9eae04319200
57af8f40eae8bb34325187c361554f3c15cac650
18945 F20110209_AAAXVB nahapetyan_a_Page_119.QC.jpg
e43d0f8221835898b53968642604c7e5
2c877d572da1f5be5e71d557bbbde18de2dd69a7
51294 F20110209_AAAZDK nahapetyan_a_Page_124.jpg
06d63c156d1000054562010126077227
a3648a8e4a7a2bac001e380fbb31e7768cd78fbf
24051 F20110209_AAAZCW nahapetyan_a_Page_101.QC.jpg
d425a00c240f6dd1327da43b43f72a9c
bb7726f72f1fcf5ea213ba245b39f0343324c9e1
1818 F20110209_AAAYYE nahapetyan_a_Page_039.txt
32b38e95058744edec89b120b4c204ec
70c8eed7478fa8d62cc2b7aa1d38b77efe8294a8
F20110209_AAAYXP nahapetyan_a_Page_138.tif
8286bd7885f16c0f2ab42f8b223455fd
7d45d5abacb3adfde5de24cb0e7eff44a508b76a
1753 F20110209_AAAXTZ nahapetyan_a_Page_074.txt
9440ba69f0ea8cf54924467f3a7bea70
ab3cd0400686cd08487942b22ec5d587a596abf2
913978 F20110209_AAAYAI nahapetyan_a_Page_021.jp2
b8c121b5afa4b79c5930800ed3232a86
875adcb08f86343a270a564cb47093fcc3aa822e
4770 F20110209_AAAXVC nahapetyan_a_Page_091thm.jpg
8293ee8377db4fd8b6747d7e5378018e
23f1b5c34b02cec9fffbb7d43daf613bd1280cbf
55234 F20110209_AAAXUN nahapetyan_a_Page_063.pro
59490be8afdb75fb963c8eb94fcd9325
d20d1b80fca6a7c18e0754ff42c3ad13c114e461
24277 F20110209_AAAZDL nahapetyan_a_Page_125.QC.jpg
0ec9d90f704d90add80369188777d103
1996d64db314570ed8d44ec7a4ea8f309402b482
80890 F20110209_AAAZCX nahapetyan_a_Page_102.jpg
c3ab370eaf45793ce5fc834cb7a2742f
143828e2db3d0cba9e829560a240b3945fece997
2123 F20110209_AAAYYF nahapetyan_a_Page_044.txt
c7936a8bb793b5fdbf4071e227cd6e8a
953e5bf9c21067ed87dfa2f06a0703caf8a207bd
F20110209_AAAYXQ nahapetyan_a_Page_145.tif
d70618c79806b972d1307d22dd4d679c
a34e9d324b1d0f4ce9fcdfaf7d1c62f71104a51a
4375 F20110209_AAAYAJ nahapetyan_a_Page_082thm.jpg
7cf77cf76e342ea92ec7276c2d7d4eb9
2b8534d868198593938a0dba4963565e6a6acf0e
F20110209_AAAXVD nahapetyan_a_Page_011.tif
f4f05a67833396c349fe591cb47bc8fe
57b241a8f008b4057e2807a69db2d33e4252c373
89512 F20110209_AAAXUO nahapetyan_a_Page_145.jpg
2c79ead4d61d57521c4f8e1c16405e0c
2fea8a83736f71ca04167cb8a84fb4cea5803979
24681 F20110209_AAAZEA nahapetyan_a_Page_148.QC.jpg
1eddb4e334e6e243169d936ebabbb340
1d830a314b92c77128b95a20774fff1cff61ca61
7784 F20110209_AAAZDM nahapetyan_a_Page_126.jpg
fb9458bfbe18d8a90a7b3cc47edfb527
aee9d5831f26b6a0a2cddda02d9e7b87784997c7
23807 F20110209_AAAZCY nahapetyan_a_Page_102.QC.jpg
cd6f931d4aa733d3b38389c648b3cd35
a6b6869e6db06beb93e3f0ce2b78b5de24c3c4e1
1927 F20110209_AAAYYG nahapetyan_a_Page_047.txt
daec452e7806c4f1a6baf31d847dc0fb
6efb8302ffb28b9979fb850e074558798d0df6f9
F20110209_AAAYXR nahapetyan_a_Page_147.tif
d56bdfc41c136011752f83de889e8879
a1c8a41756356e1813dda4e232acc4fd50d089b0
26637 F20110209_AAAYAK nahapetyan_a_Page_105.pro
16947ab6f598faa62e2fc1bd491e2767
424a56102c994cbbbfdc72cca6a58bcbefdd6e76
F20110209_AAAXVE nahapetyan_a_Page_100.tif
61a88bb7a8a1aa2ecfd0ed726db3a253
5dd04d1ea73a112441400771bb9643b636f37cca
1015310 F20110209_AAAXUP nahapetyan_a_Page_094.jp2
aab9190a012a996063fb1318009c363e
6e344bde428aa0aa7bc73ab2c34a7ad438ed6392
24980 F20110209_AAAZEB nahapetyan_a_Page_150.QC.jpg
f9933907f83a044bfdd15c727adfea61
b0bb773d39bc17896fc9a8499de9127c8b19c949
11985 F20110209_AAAZDN nahapetyan_a_Page_128.QC.jpg
13c43495487c977823708f34c437523d
eacd4be9dbf463d18221efe6912ff7c218cffd9d
2260 F20110209_AAAYYH nahapetyan_a_Page_048.txt
521d56cd2aca109d87f7c2dba4bded4c
1f0b108c006ff0b964723dad25d9fff818fdea93
F20110209_AAAYXS nahapetyan_a_Page_148.tif
13b760d71b69f65933d1bb86ff11225d
3a11cdc2ea379e9e695f2ac1230f0f33c1f6d4b8
F20110209_AAAYAL nahapetyan_a_Page_053.tif
1ee0f500a6c6e5d55c286b2f08c37021
edd148b1bfc962c0fc26a48a1c1b541e23f2f604
F20110209_AAAXVF nahapetyan_a_Page_050.tif
0861bb96191b8b5cf94148a745613f62
6b9b09d534561963d787b75538485fcae465528c
35349 F20110209_AAAXUQ nahapetyan_a_Page_133.pro
0ca8a056b3ca504c24cb0cb5af86e21e
dcd70b3af59a1c91320dc53d657e870b81b6457c
27103 F20110209_AAAZEC nahapetyan_a_Page_152.jpg
b960772e17fa1efa70d264438979e937
14074e9ae21c98382337e70298633eab34b1eb46
58070 F20110209_AAAZDO nahapetyan_a_Page_130.jpg
17642ca3e4fcb7a2f5fb55384a3b4c37
02d18ba5fcbfc6354a9ba73d299285cb5bcd7e10
7952 F20110209_AAAZCZ nahapetyan_a_Page_103.QC.jpg
6dd72c508be168aefcb3cb7bfe931d04
8d85d7e5eae923dc657314f2946261f0d8c3e017
F20110209_AAAYYI nahapetyan_a_Page_057.txt
bd30e83e44d3c53431864b0dda902db6
b913477f882e4b4650a55525861e55593b2311a6
F20110209_AAAYXT nahapetyan_a_Page_150.tif
c38ab2db3c740397f2c641c2103bc5e7
ab8c7f691c8e25ef599fc58a52653f135badea68
1602 F20110209_AAAYBA nahapetyan_a_Page_086.txt
4410b604f9eb4d23c78af8e98c6c840e
dcd72fb07596c15cf84c9aecd0452937daf6f079
78142 F20110209_AAAYAM nahapetyan_a_Page_030.jpg
4f6d952479f02c6bdfaeef316085d4b1
18f52e0edbb8dff9b9f481bcf9d481d0a2b90d1c
826330 F20110209_AAAXVG nahapetyan_a_Page_074.jp2
10ae662965497fd2cd648226d351e872
d8caece075ba2b2336df22af3c896d84f0a7829b
1051965 F20110209_AAAXUR nahapetyan_a_Page_031.jp2
88bb84be380db2c39c6f3a545d08c2cd
5a17678f931ea43b7ed754f44b95c415e2698811
270991 F20110209_AAAZED nahapetyan_a_Page_001.jp2
8dbd09ab37f7ceb938e37617bf665ca8
007b432eb236a826e77899340d58a8b268ffcd68
21218 F20110209_AAAZDP nahapetyan_a_Page_132.QC.jpg
1695ecd299b8da1dba40b13daa022bb2
6de5a3a6caeb450b572b71f30846231183eb0c04
2142 F20110209_AAAYYJ nahapetyan_a_Page_060.txt
477c35247029babc2dfc3e88c54be9bd
6f0af5eaddbb80f4c399a5912ba16651023c6e00
1952 F20110209_AAAYXU nahapetyan_a_Page_013.txt
263cba998c293fea82c339e817f57d8d
26843b9b2c186959638db4be4fde12610ba85d66
F20110209_AAAYAN nahapetyan_a_Page_014.jp2
2dc86aedffc77ed799ce3b53e9c3031a
9d7f554c49f94ef4d7cd8de7dd2fd6d6f26c806b
38233 F20110209_AAAXVH nahapetyan_a_Page_116.pro
1623ac2dfb333087278f653e74cecc94
f1aa38bd0777d61ea8795371871fa19057dbb9a1
F20110209_AAAXUS nahapetyan_a_Page_086.tif
6c34999a6ef253bb6db13ee2a95561fc
32b5a1fe8c3ae3d62bb778a5b1c3d32fb822d4b5
1007561 F20110209_AAAYBB nahapetyan_a_Page_046.jp2
7f554512dd74be3c62a6444185aa4524
74d932388f30b08a103eff44ce7dcc3a9d65de94
838119 F20110209_AAAZEE nahapetyan_a_Page_005.jp2
68f8957424da016e68cf531f8354bb0e
849edecaff322f7b5f1023a361c783807baccf57
10541 F20110209_AAAZDQ nahapetyan_a_Page_135.QC.jpg
1f773e861892596bd4a62a49d37bfc50
bc331cd239223a2490acdfc71f4a155a56dc60ae
1542 F20110209_AAAYYK nahapetyan_a_Page_068.txt
d8dd0057215e2061b19cf71f2bb9fc7d
fcafa5fdd02838f0c3b3cde92277cc63e3b8237b
2251 F20110209_AAAYXV nahapetyan_a_Page_014.txt
0c0959d7ae49984db3be5a38b6787bd5
8f06a1407411d5a6521a3b23d15205da4190e854
1845 F20110209_AAAYAO nahapetyan_a_Page_011.txt
79c44667f1098092508737f67a99468e
48ab9ce8b60ff853f3c0e51714d04cdd6b5bc5a8
872724 F20110209_AAAXVI nahapetyan_a_Page_061.jp2
3e2762777f4d1703cdfd6c64db22b4e0
b936dace1a80fc30475f73a192203a0554990913
90249 F20110209_AAAXUT nahapetyan_a_Page_144.jpg
63dba034529430868ae8a962d0143faf
b05d008b8be450dfeeb227f15ee8b805c1bb7b70
73445 F20110209_AAAYBC nahapetyan_a_Page_092.jpg
0ffbd64afb815b1508a1836f9c839d02
d092df12f58e6b9b3405d6cae461544b85b2432f
979473 F20110209_AAAZEF nahapetyan_a_Page_006.jp2
948bb38054a7f7b8b9260e83116aca4d
62c256496edc84a1824cf1647df6ebdc048c1f0b
14875 F20110209_AAAZDR nahapetyan_a_Page_137.QC.jpg
a3dd494448a8f9257f365d2ae892b112
3f88b024f1f98aab4165ce99628878312e7f07b5
2292 F20110209_AAAYYL nahapetyan_a_Page_075.txt
02e8a1f7735ca7abb63e947bf2bbefc8
0851908c57aa8bcbb8ec4fb631aa97f052ca57e2
1772 F20110209_AAAYXW nahapetyan_a_Page_016.txt
61af8545583718e559cec6fa4a88c080
2ed938c1a9b46adb57d23fffc8119891a925265f
1951 F20110209_AAAYAP nahapetyan_a_Page_095.txt
6424168ac03cd7cdf5277d8b1cbe12a3
c397624cc82df453da38be50b7162cd53289845f
F20110209_AAAXVJ nahapetyan_a_Page_003.tif
a5dd0123cb37327d45334155a55ee92e
f1ee1609831eca5d25a3c70cec01a419fc7224d4
43527 F20110209_AAAXUU nahapetyan_a_Page_079.pro
e281f1b2765dd75cb936b33c8cb37e7e
0284854c2d4b7a2acb4a8b1c7ab7f8a346bdedbd
F20110209_AAAYBD nahapetyan_a_Page_149.tif
f16052dee22fd9781d304d28ef102dbe
96fd0e1fa5f72b2115e95552ec36d2a0bb26f47a
289454 F20110209_AAAZEG nahapetyan_a_Page_009.jp2
8de823dad69aa54751c2ad220aa2476c
44fa06bd53f26ca85adca0a7ca48454bcf684179
23312 F20110209_AAAZDS nahapetyan_a_Page_140.jpg
35f6dc5f69f82389be1d5002b2bd5c82
bb1a34aadb144b333d635533d0b7ea539d48b0bf
948 F20110209_AAAYZA nahapetyan_a_Page_128.txt
3b98caed459d9d400a62a965bcabb879
c435c2ff8cb4a8397168c3b12b5a90b2c833b99d
1888 F20110209_AAAYYM nahapetyan_a_Page_077.txt
bbd2b115ce090e4e5f43ecc26a227a5d
7fbd6ec182f1bb5733b937d97b4e203d9d551ba1
1827 F20110209_AAAYXX nahapetyan_a_Page_020.txt
64ec81c34d09fef6d0b9e7942e11eb57
8600a4dde0ff7f30ba155ae6d148f202f8df1d9a
738970 F20110209_AAAYAQ nahapetyan_a_Page_024.jp2
3954b47597db155dfa80d2714d8369c4
4cb8550b3c206eb3217f8128a25205ad0c30e1d2
1417 F20110209_AAAXVK nahapetyan_a_Page_052.txt
a8143b22396950cc48c6614dd044f827
51099ab95a51fe6d4c8dfbf7fa7c0b94012bd723
5072 F20110209_AAAXUV nahapetyan_a_Page_045thm.jpg
8fbcef432393d27e8e4f97d47a35ba10
9a2dbb7c5522aeb7b8151d62715d45de54ee6a40
25730 F20110209_AAAYBE nahapetyan_a_Page_015.QC.jpg
4f013294131349e1f5791481f056fe91
97609e9ab36b8ec972597c45bd37107d8840792a
964983 F20110209_AAAZEH nahapetyan_a_Page_011.jp2
38b67d09ff391eff3187f2885caab0a3
979de99baa314ae18548f413887c822e36ef61da
40893 F20110209_AAAZDT nahapetyan_a_Page_141.jpg
184332f2a8066dbd206f39457c258521
2ce97bd7aaa5b6efe4976cfc94b839751da9a904
1616 F20110209_AAAYZB nahapetyan_a_Page_131.txt
a3d315ddb05aeff8e54c18d8591af8f6
526a8063bf2bef6aea64c94c1c3239cf410e9911
2154 F20110209_AAAYYN nahapetyan_a_Page_078.txt
ebb68c8caeab502de0a5e7415313fd6b
04b0987d4af9fbc10e94cb7c118b044c7e92fe1b
1854 F20110209_AAAYXY nahapetyan_a_Page_026.txt
47f99904b2742c6d83a4870d84713986
8124e60a864dd531e24eaf942bf4d1f5333e1600
203 F20110209_AAAYAR nahapetyan_a_Page_003.txt
f20ab9edc8a741f2f7c1e1b44178bab8
02fff2930d9a293c84e7d9f3b26e5663725b8a9c
F20110209_AAAXVL nahapetyan_a_Page_102.tif
26e38ca1f6f998378871ec11af3d5397
e182288c3cd97648b1c96af7eef5d2d6ccabb563
F20110209_AAAXUW nahapetyan_a_Page_140.tif
9674fdd5653e34285e37791d8e154e37
5d649dd0d270d200d8cf220349795bedcfcd91f6
1051971 F20110209_AAAYBF nahapetyan_a_Page_110.jp2
6d0c438606f8cc5154ddbe5bf9271c9c
c3bfef101afd1fc50d1630988bace3b29d870c90
467532 F20110209_AAAZEI nahapetyan_a_Page_012.jp2
96e9a5068e5003be22626981cdf5885c
0d3b69bacaa2fbd433fa9182241c1d65ad9cbf89
40627 F20110209_AAAZDU nahapetyan_a_Page_142.jpg
f375b2185df2258fb5edb956808842b9
5ef5b23c0ca437b8592ed90a86b91f34b5db048d
2037 F20110209_AAAYZC nahapetyan_a_Page_135.txt
6f119368e384cd298c101c51e746f1aa
4285f2c575bc47a2982209101a0edb547f3e9525
1777 F20110209_AAAYYO nahapetyan_a_Page_079.txt
b12463f407407f6c943cca267a2f7d1c
dccc45193ca18112528c2cb3a3433dd98e5e6f6d
1958 F20110209_AAAYXZ nahapetyan_a_Page_027.txt
c2381b9b8cb3df94fa10ab201215af89
a79bf20cc5ca85b3ae2473f3c95fc8a15fcbda15
1034199 F20110209_AAAXWA nahapetyan_a_Page_090.jp2
db5d6376d03b00e1701c472d818b06f8
d27bf4c807b8c1a313751592f09a873376a9f566
F20110209_AAAYAS nahapetyan_a_Page_055.tif
af942ed5c5f008b7c490733802367b2f
3d575e965a7391e677fd154eb74a4f1156fc6148
467708 F20110209_AAAXVM nahapetyan_a_Page_004.jp2
514c2478565690779819cbe86bb403b7
b7b3a2aa2ad5969cbe5a973895cc13d24d987e73
20400 F20110209_AAAXUX nahapetyan_a_Page_022.QC.jpg
4898e0a4dd4011b48d5bcb88520dc964
d8c6e9dc07646ad01dbcb72d33bda8fe19f3e367
F20110209_AAAYBG nahapetyan_a_Page_011.QC.jpg
d0b9a3b3c2a00cffbc3d4984b83ab584
cb00f4521e67a73ddf39558f72c5ba5187587f00
958669 F20110209_AAAZEJ nahapetyan_a_Page_016.jp2
2f9eb56da97664b0a85ac7c7aeec07d2
2a771cd8a89cf9977a2d5d2d1a71b94eaa69abfe
80747 F20110209_AAAZDV nahapetyan_a_Page_143.jpg
6c8828d0d349a2118673537fc7637cd5
d36c322dbd6300f71d6728369d98f1671665d9fc
3346 F20110209_AAAYZD nahapetyan_a_Page_136.txt
dacaceef264ee630f5119124ab03a749
72235fbfeb3b885f0b740707fee378103c10e286
2117 F20110209_AAAYYP nahapetyan_a_Page_087.txt
1bf3ef6a3492c1c11557f7409057fd08
11d8b9c2082111a7b1a724271d69422fadef4721
1973 F20110209_AAAXWB nahapetyan_a_Page_062.txt
1bea39733c016452227daaf18b5bd21e
f477cc467688a9311217e64707bef09df44f943c
5733 F20110209_AAAYAT nahapetyan_a_Page_147thm.jpg
f5b58db6393992e6f5ddd7bc05acc812
fe29523cf2389acc6915c8f51d6abd6d7c965150
5361 F20110209_AAAXVN nahapetyan_a_Page_097thm.jpg
fdd3b6f94ebce3b56c67287c1768af82
df769036895c23540779bdd3d66a1fbbc6d79e21
25639 F20110209_AAAXUY nahapetyan_a_Page_107.QC.jpg
d9e2d8727290ef648421e0c4dcdb1d57
c8eb8efbe18961fa360ea2dee0c0bc856b019b53
4487 F20110209_AAAYBH nahapetyan_a_Page_116thm.jpg
ea1060471c1abd515da4684e57e2becb
e0e46c001ba1dd2538108a574a0c69a9b3e1f991
1037903 F20110209_AAAZEK nahapetyan_a_Page_017.jp2
9d159429a9f0e496078794590c4acc9f
4309487b81dc1e516a638dbf3ca952366c076ad4
21555 F20110209_AAAZDW nahapetyan_a_Page_143.QC.jpg
540f235588e74853cbefa05cdf8c11b8
f72e6174edf210978cb1cccb660ad5620fe3f6b7
620 F20110209_AAAYZE nahapetyan_a_Page_140.txt
b4e6e8a965ae5ad3316a6299592608da
a8dbc7a8ea45030232e6a1216d6bc03890ab6313
47665 F20110209_AAAXWC nahapetyan_a_Page_013.pro
8a4f30c63a6cbc92f6bf39110ca0e798
cd728b79c1e23963408d159a2b4c7f77c3575fdd
F20110209_AAAYAU nahapetyan_a_Page_077.tif
2a32e956659fe4d5a6ff449b26c48aed
739fc00df620b1eaee083f795af9793d23144a0f
15724 F20110209_AAAXUZ nahapetyan_a_Page_134.QC.jpg
2ee019b53f8d2b13f880eda1a270ba41
89d76077419d48e1271f41d322af945173b448df
13355 F20110209_AAAYBI nahapetyan_a_Page_080.QC.jpg
ee2b8445106d8a5ba24755e674f62f85
eb8baa460e26bfb6ad589e8d1b28050ba398e3ac
1051969 F20110209_AAAZEL nahapetyan_a_Page_018.jp2
a2aa9a6abb0ce532872b1b7ff9e0fcff
7e1ae2a88b8792e49d8c59250a13393bf211bf0f
23812 F20110209_AAAZDX nahapetyan_a_Page_145.QC.jpg
ad350053ccf5a9a38208b6eb616bbc0d
17222a93400b006a18b4ef043b60e597b873fc33
875 F20110209_AAAYZF nahapetyan_a_Page_141.txt
86d26f3c5e66edee5987c02255dff81b
f5ce3c0da3df0540120063fcb84e472d87de4688
2327 F20110209_AAAYYQ nahapetyan_a_Page_088.txt
125f678e7035f5fa0ee285d951326ba7
1036227fd93eee6bd17c9c6c3e68abaf4270cfa2
33196 F20110209_AAAXWD nahapetyan_a_Page_135.jpg
9dfbdfe3170aa67db2c465aeb3613c64
a798ce12c2569e65d1e5a9cf1e48939917535ae4
F20110209_AAAYAV nahapetyan_a_Page_109.txt
740f47935a334378c5de95d394126d09
07eaa3f4e5e10fbbcf9424b3f7b38f2a5ecaa7a7
1609 F20110209_AAAXVO nahapetyan_a_Page_151thm.jpg
004dac2bad67bd50714fac8ba1a8f832
56344fd46795c64ddbb72da4b51b70fed787e56c
F20110209_AAAYBJ nahapetyan_a_Page_104.tif
5c4eb698e8ac4e585f6bcf3e769cdb27
dcf68fce073dca35d2c713deeb50b32dcddd4d69
F20110209_AAAZFA nahapetyan_a_Page_088.jp2
8ab11a40a68b26cfec8958452af77f15
821b7b5fc48c231e43d6c438fd48a1bdff9d2055
F20110209_AAAZEM nahapetyan_a_Page_025.jp2
6c7665ab983f8ec64d9b338d215122fc
078e0ce73147bec13e09569b200bff2d67724d4f
92250 F20110209_AAAZDY nahapetyan_a_Page_147.jpg
1cf9d0fb472b42ea5379a28b03b2c9f9
2b26f3723c79ffcaaa1cb486887c3067e5e34c08
2079 F20110209_AAAYZG nahapetyan_a_Page_143.txt
e10b083401fe8d5de5faeed54d100d21
9d2838db09ef9574f9401bddf81bb578c111ad8f
2059 F20110209_AAAYYR nahapetyan_a_Page_091.txt
59d4b9039c6f9e5ecaced7c9123c5576
9c4e5a9bc0ce5551ce56bfaa25c1e95632e6be46
50078 F20110209_AAAXWE nahapetyan_a_Page_136.pro
39d8fd7e110b24a762d0c73e3f24da19
a01990b325c94533b3ffe17a567c8d1bab7c35f2
35681 F20110209_AAAYAW nahapetyan_a_Page_050.pro
1af5b521a493cf653674da27512d2e67
eaafafd4c012fb9e6ccafbee838e72cfeda9d9ce
F20110209_AAAXVP nahapetyan_a_Page_076.tif
817087b57c98091e4218b0577870ce46
0fe209c1af43a8502195e93afe0b423fc031ea33
5351 F20110209_AAAYBK nahapetyan_a_Page_102thm.jpg
b60880df3cb49e30fabd75853ba4e939
7932a2f7ae61f127b62a677ca65511235b42dd60
843728 F20110209_AAAZFB nahapetyan_a_Page_095.jp2
a799019cd79f51b7b41e1f7f735c6d24
0093b7774dc5117d7a325f18d74c86d846e46e0c
1041613 F20110209_AAAZEN nahapetyan_a_Page_027.jp2
64ba61ed657aba7ee8d57474c8661e6a
7ff337251702f246f1014b11c96ffab16c0e2091
24850 F20110209_AAAZDZ nahapetyan_a_Page_147.QC.jpg
b37f24040d94332a5aa073ddc41e49a8
8e56d6e29a549b2ce63d7e09797d493b0bade786
2392 F20110209_AAAYZH nahapetyan_a_Page_150.txt
cb8fd75f1524f2aa8f4a8efdbefda511
125004d8bc106a4d4d49d4339799ca4316baa864
2161 F20110209_AAAYYS nahapetyan_a_Page_096.txt
349404647cac4130b493465a2f1341d7
d25e5922ecd936ba4bea874eb8a37a75cc323ee8
21231 F20110209_AAAXWF nahapetyan_a_Page_066.QC.jpg
724a2fb07e7ba84358b5ede59990df38
af3930b3e6214267792427fa8ed7e50d9ab1f6c3
17813 F20110209_AAAYAX nahapetyan_a_Page_141.pro
f9365cab15a8172e1a8acb191c3881a0
e824e57cfab69f3d010482dc49e7570bd47ccd03
51666 F20110209_AAAXVQ nahapetyan_a_Page_085.pro
49f5402a19a742b23f331ef40872132f
849c6a2194740d7074e96518cee410cc9f854c53
2259 F20110209_AAAYBL nahapetyan_a_Page_015.txt
214286bc232dff38ae629b472f3a2516
b47091c66292bc4ee36f2903096c08e6286e5de3
1051937 F20110209_AAAZFC nahapetyan_a_Page_097.jp2
edf3cbe0b291a772e08ed594831270ff
9613ce439090b5e87e9c9abf3d84e1682d776941
F20110209_AAAZEO nahapetyan_a_Page_032.jp2
91edb031f7b8c2052644095fb133d762
4fb99e6c3db61182ad93615907233e133f5c5b01
600 F20110209_AAAYZI nahapetyan_a_Page_151.txt
a5c00c219b0e04d874ebedf3c540c8e7
29d48211616ea8d0bc7d2b44c35addd4131438ab
2175 F20110209_AAAYYT nahapetyan_a_Page_098.txt
7fb827187fc7f00641a52aed5b8f0596
e170b598cef053424437615d249b9abcc5d37de5
5079 F20110209_AAAXWG nahapetyan_a_Page_022thm.jpg
3cea1a620dc5b04c8a4a1d696eb97d4a
b5f837d73307d65692f7b6c9436988676acccc76
F20110209_AAAXVR nahapetyan_a_Page_042.tif
18703bf425e83ae1bfa4e98d5a8e44b4
68afea1ed6c1d117da3d720cdf2a726ad3376cf5
F20110209_AAAYCA nahapetyan_a_Page_105.tif
7367804c33cbac90e853cf2b758da207
8a547c03f846850e0dcaa2391dd4f4b8824642f0
50320 F20110209_AAAYBM nahapetyan_a_Page_010.jpg
4cd40ef2ceafa3acc96ab4ec13aa4b63
95352b9180c4212d1a670d7093ee8392fa5055fa
991943 F20110209_AAAZFD nahapetyan_a_Page_099.jp2
9f3dcc1354e0e088de10cc7facc47127
498fe221862db78de9e51ef9c7cebb6828cdaea1
F20110209_AAAZEP nahapetyan_a_Page_035.jp2
1b18ed4e226c8ceb783dcc3e14e14d0e
f143f96f95019755b0c41979e1965a0f44251d5f
8884 F20110209_AAAYZJ nahapetyan_a_Page_001.pro
c22f3c4a7599ab16e6b347efa2b6e6f7
0aa4675e815291215ae475338099a0771349eeed
2042 F20110209_AAAYYU nahapetyan_a_Page_099.txt
56c0f900b91d53b906e4e83ccae394f3
e29f184c76863215404a804994064abc0188b7df
1967 F20110209_AAAXWH nahapetyan_a_Page_049.txt
15624bb4e67d9eb939a74117f6e9d833
11ba746dff9f83c137296a7d67d35d81a8b67183
85946 F20110209_AAAYAY nahapetyan_a_Page_132.pro
73924f947b552d321f10e6fc0bf1d1be
6bbc5a3cc4623a845ad1247da59f78d2a2aa81f9
4279 F20110209_AAAXVS nahapetyan_a_Page_061thm.jpg
510fb72d1d4e220d5a9091b657288da7
49400fa872e3e0c654192b9545531b7652b666e6
4596 F20110209_AAAYCB nahapetyan_a_Page_074thm.jpg
8234718ff97d0d79d53d9c855fc38c73
5927c8e61e78d928b03c4ff4dc93633fd845add9
1270 F20110209_AAAYBN nahapetyan_a_Page_105.txt
b60a8229d7921327ab816d01a06f06a9
8cfa463c80cf8c125fe42c3ac755bb12e0e34ea4
1051946 F20110209_AAAZFE nahapetyan_a_Page_101.jp2
07d0d52b63401860658610b0f04350f6
09ed8ea851c1fe7d21d6aa6ab38f7506545c006c
1051954 F20110209_AAAZEQ nahapetyan_a_Page_036.jp2
319bacd2bf8fceb325c6ee4ec331939c
fede67a763a1a3f79b2cd702a953bd74f9df7bf5
51075 F20110209_AAAYZK nahapetyan_a_Page_008.pro
e5c1d05f274bcdb4843a4fe0c572fe13
2810267ba36d549575691c5e79733d95cdcf982e
624 F20110209_AAAYYV nahapetyan_a_Page_103.txt
71f7c3cc274bbf313bd597247ac5338e
2a9b9b1665899ba93fe4e406de4d112a0c013a64
F20110209_AAAXWI nahapetyan_a_Page_111.jp2
742ef7156eafd46fe74b44fd917f65b2
b3b31033b6aadb4c110e4f88501ecad02029f6b2
35601 F20110209_AAAYAZ nahapetyan_a_Page_020.pro
ee4d1dcc4f9a1662d992cc7bd32d4e7e
dea155022a5c8cb6986312d58cec2669d05f59db
32375 F20110209_AAAXVT nahapetyan_a_Page_124.pro
b71550009631e05125ab07e08ee767fa
dd7d863fa3f23b6bb078c398a334c0f593dba2d3
4264 F20110209_AAAYCC nahapetyan_a_Page_002.jpg
03cc1ecf5b11eafbbbb1e40069b2e908
49a31cd170d1598da312ab2cfd43d8bd7bb225a2
54781 F20110209_AAAYBO nahapetyan_a_Page_064.pro
1a40dc8ab5d3fe935c5faedac5716f02
f47945a1f9be061bac9753ed9e7ce6d044ff7b80
F20110209_AAAZFF nahapetyan_a_Page_102.jp2
6114fed375f6d074d6cf14060380e096
13ffe38edf851b97bba54dc18fda4f4c89004a9a
1051938 F20110209_AAAZER nahapetyan_a_Page_043.jp2
33eb6041aecd5052c14704d34d6ae653
a2965a58695cf3f79a6673bb51cc3fe673fb6385
57489 F20110209_AAAYZL nahapetyan_a_Page_014.pro
24a2503ec483e8772580922aec017afc
b94d00983ab14d845ebf9c7cf157acdcb40f5680
1173 F20110209_AAAYYW nahapetyan_a_Page_113.txt
4d8a1781747b7a562c885f0c6ac04fa3
21874a6130aaca7f015116e224518c29368b2e18
5407 F20110209_AAAXWJ nahapetyan_a_Page_018thm.jpg
073c3a9478f22a1aa63e662f51e01153
e5e29ebb55881e0a1e2f8de270ccf2a6361a64b2
5371 F20110209_AAAXVU nahapetyan_a_Page_055thm.jpg
680824b6c66fad3ddbf71493d3f61094
7e2b0285a37c097a21b5f131fa4f8fb1efe9d7ea
67744 F20110209_AAAYCD nahapetyan_a_Page_117.jpg
b6d3faa700aa0eb38eeb52f0ef0d3ae6
3eb55534491b2fea7bd738c832a6b0aac097d940
F20110209_AAAYBP nahapetyan_a_Page_082.tif
e922631efeaaa3984b2a5adfbb62e847
30bf1f9d601cdf703c8ab0f66715764be8344072
697074 F20110209_AAAZFG nahapetyan_a_Page_104.jp2
ce147d0136d6a483f9cf2dc95f65d1e1
b1f28b099305f10749b019d9dfed010ff65bb368
1051912 F20110209_AAAZES nahapetyan_a_Page_047.jp2
ae68e39490c2e8da6b028c8c4ce8b491
f50b9e80131d3a5b5fa222bb970a0c4e42791863
44532 F20110209_AAAYZM nahapetyan_a_Page_016.pro
1898c289eb1bd9cbe8f7d66eb06acc8d
b12855b6d0903b9fb6e47e20beb98c91c4552c67
1636 F20110209_AAAYYX nahapetyan_a_Page_114.txt
0ccd1684ac7e9f178efd712f09f31a50
047694205cb7f53bb0490ddc05758e83df7e68b3
1878 F20110209_AAAXWK nahapetyan_a_Page_116.txt
27fdcb717583d0962b9585f90347dc63
ede34f7995719dbd136beb1b4904c1ef2d6817ac
1039736 F20110209_AAAXVV nahapetyan_a_Page_085.jp2
226ecce711cb090a55c591ef28723d28
7a6b44113de32ad882181f49072b5eb09cc17c3d
1051984 F20110209_AAAYCE nahapetyan_a_Page_030.jp2
4805862a90eeb295485928494de86ed6
c256d147d1c465ef58e60bf7494cca26e704c3fa
4501 F20110209_AAAYBQ nahapetyan_a_Page_068thm.jpg
89e5f79096939fae29d39b2fef08ce15
071e2f2ba1159e0afc92ef890f07842686c9c9b8
642112 F20110209_AAAZFH nahapetyan_a_Page_105.jp2
e539ca66e678e4e4a8b6a7bd1cebac63
d69dc3ce6d0b9a7050d1d123080ad6409cb06581
1051968 F20110209_AAAZET nahapetyan_a_Page_063.jp2
9904b49616fc1df0c8ea1ba25f70f9de
033f7d8a0656f8fe3677df5b5b2be88c768d6353
54135 F20110209_AAAYZN nahapetyan_a_Page_018.pro
b6b2ee88a5d05355567ac207f84c2bba
01527d70679a1a05c41b557d895297d2c05f524b
2090 F20110209_AAAYYY nahapetyan_a_Page_115.txt
4fbbaa6b9b199776cd62ec9bd44d636c
4c404209c0e94a84d20ecf346f1757d7a86f6b40
52434 F20110209_AAAXWL nahapetyan_a_Page_052.jpg
e7d640010712a5542a8d0c133d5fc5e8
311a31054c332a44260f9839996939665f080fd7
5603 F20110209_AAAXVW nahapetyan_a_Page_075thm.jpg
b86577dd5259d2e196a02744b2beef16
b42276275a1e2a67014d6d242c87dc93d87301fb
3952 F20110209_AAAYCF nahapetyan_a_Page_126.pro
d23b32ef6182b93ba23be441c4eec638
9653a798dc92b1ee69ceac4ca982bee26b0a12c8
19439 F20110209_AAAYBR nahapetyan_a_Page_026.QC.jpg
0015df5ca28ee539e53e132a49018560
edbeeb0cb9c59e1ebe6393ff5a1424269b442523
1051973 F20110209_AAAZFI nahapetyan_a_Page_109.jp2
7f897d8619d2d77748a49fdfc1f300cd
f937a936b5290534e14dbfc61bb0bf6221aa721c
1051948 F20110209_AAAZEU nahapetyan_a_Page_064.jp2
b433c5aedb801ee93d1eac92df438240
ad9b2a38a5b5b008386c9426692b5adb062b5bfc
42116 F20110209_AAAYZO nahapetyan_a_Page_021.pro
3b63ee5fcf54ec4a73d785be4c01dbcd
2c98b1bb2ccec0c3824b0d47a241ec8de81f5c67
1887 F20110209_AAAYYZ nahapetyan_a_Page_117.txt
ba35b265380d44ca58f0ff000438aabb
4708ab4f813fabaf22bf68cb745fb5ce8774bcb7
5735 F20110209_AAAXWM nahapetyan_a_Page_088thm.jpg
aced647e861d37f633cba9bdae18f6b4
3e362c432accc3c4bf155c1559ada9065fa3cd40
65719 F20110209_AAAXVX nahapetyan_a_Page_051.jpg
618c009d804b27a2fcaa20b52035b576
04e59fbe4484638fb001cf1e1e123b50b5793d09
51976 F20110209_AAAYCG nahapetyan_a_Page_060.pro
654066ab8e54a9accb62350e8dc610c6
34289a404ca2acbe85471aedafdbe27d1783c0c6
1051975 F20110209_AAAXXA nahapetyan_a_Page_096.jp2
cae1197e3ed519a64856512c484e4c64
b5680f121a4c8f5fc6f63471482adf6d8a9e7657
1231 F20110209_AAAYBS nahapetyan_a_Page_002.pro
f1214f2d2b3fcd672dbc87bb87ee5c39
96766263128bfa18822d2b4988d2b696da93d75a
997161 F20110209_AAAZFJ nahapetyan_a_Page_115.jp2
c336735f79b6c445d3e3228787277cd0
40c8f68cd4033e3fc723708144c52f6a54eaabe6
886959 F20110209_AAAZEV nahapetyan_a_Page_065.jp2
e894181b43b9368f673dd7c8c63cd613
036a9a5cf3796405f7ca8fd19d6ae4a5e1c225c1
33297 F20110209_AAAYZP nahapetyan_a_Page_024.pro
d354a4befd5cdf1fc8f67da9d5138b84
dc2b1920ed9f7c123c6808fde44a2dfc5d256edb
26525 F20110209_AAAXWN nahapetyan_a_Page_113.pro
a846dbc5f96e723f2ff84507cc6a44a4
b4d2fd0338acf70a29c8444fce86fa04671d2e35
942116 F20110209_AAAXVY nahapetyan_a_Page_053.jp2
afa66f09bf467cf2cafe22b302f3fd3f
4d8bb055da430f081dc8f687512ccc3e8680b6c5
74741 F20110209_AAAYCH nahapetyan_a_Page_094.jpg
8fbce61086da3b6fd83d34ec4a6ff6fa
05d762fd9e522a2a607c85f768c5c30e915b7c94
74059 F20110209_AAAXXB nahapetyan_a_Page_090.jpg
273121b500650a0f6ba0c46de87b9314
44aee0058f34d27eb791a47f9dce6f93c1c4ff74
4952 F20110209_AAAYBT nahapetyan_a_Page_106thm.jpg
e3c5080cb4aca41e57d3de53cd235d88
78f4e95c501804f9778245c003cc20779b741d16
931145 F20110209_AAAZFK nahapetyan_a_Page_117.jp2
3ece87b443baaaa6f91e4bfef040728c
484759da63aa3c93c79f635ddbd08b59a3ce8efd
923627 F20110209_AAAZEW nahapetyan_a_Page_066.jp2
ad87bdc7b9e0eadae16ec13370c6f19e
4944ec18d39532ebd945766333b1faf2db5737a7
34528 F20110209_AAAYZQ nahapetyan_a_Page_029.pro
43c1e2fb7ebabcfcbfb675ba405b76be
b14280aa73c708162db93d09de7a07ba0479c43e
4868 F20110209_AAAXWO nahapetyan_a_Page_066thm.jpg
aa7c1c96435c7830b0392595144f7d48
c40399ec432e8a8bac9954f41bb793d8b75744f5
1682 F20110209_AAAXVZ nahapetyan_a_Page_067.txt
c6cf18ebe02e0f5b45a4e358a67a82fb
379a1351a5608f0945e35b9070836863a2415473
47299 F20110209_AAAYCI nahapetyan_a_Page_054.pro
463097e1bf330d39b88370e32c67950f
0119ccaad6c98b2e76532dfea991e7c6fd3b3042
17275 F20110209_AAAXXC nahapetyan_a_Page_024.QC.jpg
006773e5c3e40424f6d32d335390005e
e1f7512b07c913f65dd3a0a52998d4c50718ccc8
F20110209_AAAYBU nahapetyan_a_Page_107.jp2
0de57c8a2d036c5898b9c2a6713bb52e
ef52495cbd578dbdd5b53c47bad2fec0fc68e860
809386 F20110209_AAAZFL nahapetyan_a_Page_119.jp2
6eb153899dd93ce6de480a3ab64bef35
c18dff4faa7ae5b6d2f5f2eef3d1d7172528afac
932497 F20110209_AAAZEX nahapetyan_a_Page_079.jp2
31b300f32719412c95552bb983684011
c28c77d259efe0558d2dd23d4e9aacdd865bf66a
18529 F20110209_AAAYCJ nahapetyan_a_Page_082.QC.jpg
f40343b44f73c10254dcc6296d9b5a3f
97b06ea4b9896416611dbe8f1a3c1d0c6a094272
20744 F20110209_AAAXXD nahapetyan_a_Page_117.QC.jpg
83833bc971816a8ef07bd388b019c145
9d0b5fd4cd38fd9a36afe24aee7efef3e6008635
53502 F20110209_AAAYBV nahapetyan_a_Page_104.jpg
bd0afc444238359e360e9544d3c52ce2
1e18a0d064769eb4a298b07de696dc85fa2833e6
4471 F20110209_AAAZGA nahapetyan_a_Page_011thm.jpg
199ec35126b0cd9927b4661db6954030
7150918609f82ba3fc1ba71742062d9eccf78444
687073 F20110209_AAAZFM nahapetyan_a_Page_124.jp2
ec4a493b5d646ddf5ecb3112cc3b4030
efd27e3b953c9defd19e579542b58c0abc83c9c0
829857 F20110209_AAAZEY nahapetyan_a_Page_081.jp2
c5b97a1e1524127a9766b8a4b1b9e41d
95d1200abe64ad02e248e24eb5628607da75eaba
52059 F20110209_AAAYZR nahapetyan_a_Page_032.pro
17c2b9e4f70905b5c9df4a2c7e3ba1dc
95f0197d867d84a7e20c31749f3dc78c58f6b680
47425 F20110209_AAAXWP nahapetyan_a_Page_057.pro
07019fe03f6ce516f9eb291368baaa38
1ce23160199e60d1c12053c5d69295c7094451b4
F20110209_AAAYCK nahapetyan_a_Page_038.tif
6302480743d540e1dd31e0ac69dc5b82
f3ed1cd72bd02a18dd88d0b3089f8dbcd558c783
4959 F20110209_AAAXXE nahapetyan_a_Page_094thm.jpg
9b302208f521454365ac661ea1ea8657
3315d5ccdb9c5470228a80dad2426b5cf6b3ae37
F20110209_AAAYBW nahapetyan_a_Page_109.tif
73df47b8c2289960cd5937e0134e1db4
08b80adaa66148e8c17be16309e0376e7e768404
4787 F20110209_AAAZGB nahapetyan_a_Page_013thm.jpg
53f5632a288153ef2c6742dcc931d5d0
2b08656231c8841a48d2478bb493e5f05f2b08d9
F20110209_AAAZFN nahapetyan_a_Page_125.jp2
a9461eb3c71dbbd5c81a2794406a2c41
dc43eb817b48944299abd95c49bdb54eb36bbdb6
1036704 F20110209_AAAZEZ nahapetyan_a_Page_087.jp2
1050b2819f586a424c7d96ec9b4e8e45
c0e837cd19ef4dcfe2eb3cb136c995cb795b4003
46521 F20110209_AAAYZS nahapetyan_a_Page_034.pro
1c7a69cc2680560afe8d4e968e8efa36
a28bd7687bd6ad97d5dc2eddf6235425e78493c0
52885 F20110209_AAAXWQ nahapetyan_a_Page_105.jpg
42ceaed0e607a96c1a1991b3bdc1e9f0
29d075e80834b488e989f866fb2efacdd87e6924
923600 F20110209_AAAYCL nahapetyan_a_Page_077.jp2
caf9e16a363bac9182e92059fd63bd7b
3022099cf567519c0058b0cee9da50b8f3bbf53e
39186 F20110209_AAAXXF nahapetyan_a_Page_121.pro
e0809ab9e8c9feacc568b9e126788976
171e9bf17acefcc42677f1fa2a93629f66e5aafe
2029 F20110209_AAAYBX nahapetyan_a_Page_102.txt
2e3335b5731c64a68bcf26e8d7d85936
26efbed89e093f8b9644fa914d78fec7e618dd9a
5598 F20110209_AAAZGC nahapetyan_a_Page_014thm.jpg
7fd0da30f29cecb000dc126b3db9dfa3
454d832b10b5232807a5f2453c2d79f8b0760947
662735 F20110209_AAAZFO nahapetyan_a_Page_136.jp2
fd32bcf298996cd1809ff540ce8fc03d
6acf87c125b02fe6215bee65cd57ca3f6e13186d
52147 F20110209_AAAYZT nahapetyan_a_Page_035.pro
39c741cf05aee0198a0810a66a6ddab6
21d0c71a9de2e778de9aa3b6ed228ba73a4eb01d
24590 F20110209_AAAXWR nahapetyan_a_Page_088.QC.jpg
7fddf497c21f7fb805ac197e5d9113c1
c7b828b0c51f98dffd66d2984738936aea1d3988
41825 F20110209_AAAYDA nahapetyan_a_Page_072.pro
dc324e20203e083f0ddc8c2360d0b836
01594ca5d14195de8e6440e76ae584708f6d0fdb
2346 F20110209_AAAYCM nahapetyan_a_Page_148.txt
7e466776dd5ccf562f14932d5fc89b5f
49aa4d5ce6281676ba17c223dda36c72ab334ddb
80603 F20110209_AAAXXG nahapetyan_a_Page_122.jpg
d0791b57e0c261037af2f351ec8de0ca
9025a9b0656bc92a3f8438547796a2783a0d86ed
1899 F20110209_AAAYBY nahapetyan_a_Page_082.txt
ea44f18ed708407639ba9fb001336ba3
481ca6b7857a4bdd1001c5593245cdb1642c776f
5544 F20110209_AAAZGD nahapetyan_a_Page_015thm.jpg
2bcc93acf88ed1eda5c22ee60c2672d2
3e7f2ce44cfcc48248cfc0dc754bc15881d0f0f1
600502 F20110209_AAAZFP nahapetyan_a_Page_139.jp2
82005527349177f2333040f09e4dff57
7df29936f34f716a7616d0a3df69d7b89f2bb91d
53861 F20110209_AAAYZU nahapetyan_a_Page_044.pro
55f37f737f682daa752e759bf6b0244b
a8e7c31a1a960e7d8d25d40a1270200658d2c592
F20110209_AAAXWS nahapetyan_a_Page_067.tif
6a2b7f2f95697723f20f1ae49985b7c3
ad2220f5ef88bdb681882a49c87843aff630af40
76617 F20110209_AAAYDB nahapetyan_a_Page_003.jp2
8e64d7e2a7abf33a98d1ffa36feef663
103d9b813bf575e55e90aa9cc7eef9d267294233
1773 F20110209_AAAYCN nahapetyan_a_Page_029.txt
756cef8d1447922da2ab89000a81a176
87036b92654661f5b29309eba717ac0248b4b019
3436 F20110209_AAAXXH nahapetyan_a_Page_139thm.jpg
4591a69e7f70c673c6b366c4a5060b44
6d01e103accde99b345d2659e44f8170c361d138
4133 F20110209_AAAZGE nahapetyan_a_Page_020thm.jpg
87cfc5e51d826b3eae030e128e643fa5
0039b16a76c42d461c53afba923ff89e46a727a1
485990 F20110209_AAAZFQ nahapetyan_a_Page_141.jp2
967fcb2e1fc6a5cf6d41c43d3032c6c1
8c541682374c7225493a60341594fdcc97022870
47103 F20110209_AAAYZV nahapetyan_a_Page_047.pro
2a768a19e91d3c78e36695d7af9d0548
2de2b9e55c723e7730ae619724c31fec8e0092af
11523 F20110209_AAAXWT nahapetyan_a_Page_037.QC.jpg
41bebda298573762eb3d493078248892
89408d61b33ee59faf947041c1009db820e3f276
2133 F20110209_AAAYDC nahapetyan_a_Page_101.txt
ac4a6a3a35ef0124d69b6a2c2e3ed79c
d8327b967267e7d743e854b4e7a7652980575d57
21201 F20110209_AAAYCO nahapetyan_a_Page_012.pro
1f75e9765d600f431b19f1c2679d6417
5c51e374298c68024fd6219554abe3d62a9c0242
21876 F20110209_AAAXXI nahapetyan_a_Page_045.QC.jpg
3af3d2e63718eead19090d7505bc593d
ad333b2eeb592a9e622dcda68d0b245122cd0e6d
3571 F20110209_AAAYBZ nahapetyan_a_Page_137thm.jpg
517beca3907a4c96f2ef260b07a20090
c06bde92d7fdc60a9f3350502963f2cbec726f2c
4925 F20110209_AAAZGF nahapetyan_a_Page_021thm.jpg
9fd01638ef36a6b82053981f33e92d9e
3ec29a6ceafaed474add76ee39f415bbbebb9e10
1051941 F20110209_AAAZFR nahapetyan_a_Page_146.jp2
3429806f2eca650830a916579ae94018
149c7a596bb05685ed8e79d3d091816af556249a
49117 F20110209_AAAYZW nahapetyan_a_Page_049.pro
9fb35778d8f8d30bd3ec082de14e8f4c
bc1b7e435812dbf4f239a01bcd3035713db855fd
57426 F20110209_AAAXWU nahapetyan_a_Page_137.jpg
cec122b458a04a8a00709eee7905f4ed
86182e1ef1ede28fdfd01585686b1aee3bfd5189
921166 F20110209_AAAYDD nahapetyan_a_Page_033.jp2
a646feaa542b0ed2a94816a808ac41a5
ceee1b90131f95102eee04bebcd3660af16f91c4
1219 F20110209_AAAYCP nahapetyan_a_Page_002.QC.jpg
0792e091718cca39f99de94e03958b70
202fa3cfe5a31def4347875c8fcaf85b19cfe0f6
F20110209_AAAXXJ nahapetyan_a_Page_035thm.jpg
08c6e95aa7273382f16bf2e7c55bcd06
015aced1d2afe4bd07a0e262ef77cc7db5fa6b09
4844 F20110209_AAAZGG nahapetyan_a_Page_023thm.jpg
c52bfb2dc462d890a1a70d5a5c3c7988
4a625770a618ff856b7516dff42f24e91a9a20e9
1051983 F20110209_AAAZFS nahapetyan_a_Page_147.jp2
28a51f2c33fb28a34db2afc74fe78b59
8d9f209a13eb64919d005c5e305c9324b3432697
54214 F20110209_AAAYZX nahapetyan_a_Page_059.pro
6b68022e1c9aa09b47bf4a38c7eb5ccd
b9017446886aebe0bfb8de1bf7bb9fbd4a500b0d
F20110209_AAAXWV nahapetyan_a_Page_015.tif
fa03eb61b62679018c131594995f0db4
47e964535af0bd77a909a7b42857d1b3a9733f84
F20110209_AAAYDE nahapetyan_a_Page_096.tif
bb72f28684a2ee83b60dd8fd56326242
97fe7dbcfd8c94e167e6e5c1133536ed60a1800d
67791 F20110209_AAAYCQ nahapetyan_a_Page_067.jpg
d00a39a6b600daf04c61dc76e3019b5c
1b8476e113728fe58858b80891cdf3e39aa6d675
F20110209_AAAXXK nahapetyan_a_Page_016.tif
67ee4c48c546c8d0ce1b79dd1d3eff90
407aa9483d1b9e42d33b878132dfcfcd0054f609
4278 F20110209_AAAZGH nahapetyan_a_Page_024thm.jpg
eb7f846564744f130e56d55a32a2cb47
7dfbbf52cbcc3a1d4dcefd004aa559f8c715fcca
F20110209_AAAZFT nahapetyan_a_Page_149.jp2
7c735da23247eddffab26b88bc50e322
d864bcba781c716c55d4760e49087f5b80317bc1
39395 F20110209_AAAYZY nahapetyan_a_Page_061.pro
3cfc0b69bda08cd769b9a8f814c7b74b
4a7f296a4f0a0d664e140c112ae7f62d2f5e285b
3759 F20110209_AAAXWW nahapetyan_a_Page_114thm.jpg
7b7c5523ca3ebf2017e18ecec24f8318
3669dd64a065d130580fda5edf8f6896aff6f7ed
44514 F20110209_AAAYDF nahapetyan_a_Page_092.pro
6285d46263b42a3ffcdfb17fbca8a097
34db70ff6fcb0fb5d6c4635f20c5dba996ae3706
1701 F20110209_AAAYCR nahapetyan_a_Page_071.txt
3e6bb605537de3a3a51a3db5a0a2bbde
70428e013526cc8337fd993c684f980ef1454efa
84144 F20110209_AAAXXL nahapetyan_a_Page_064.jpg
755fb192ff8b08fd1a8523417db0cc6d
a434aa07c234337721361b4ba1fcf51dc2d5722b
5440 F20110209_AAAZGI nahapetyan_a_Page_025thm.jpg
1bd2c186fd20afe183a8eade26f3a487
ec0ecbaa4f386fa9865726afb736b5a13582afe5
356288 F20110209_AAAZFU nahapetyan_a_Page_152.jp2
054a403c154a1fcc81c8ef1096a191ac
d1ae7a1c35d3d6b4e7c6807e404e36c01ccb6a1a
48363 F20110209_AAAYZZ nahapetyan_a_Page_062.pro
507eaff5716c0388e29b677a3408b60c
6014b320081e2d82991b60cfa375c79991ba5487
44154 F20110209_AAAXWX nahapetyan_a_Page_010.pro
ec2eb0479651715b926e356f3a3034dd
fef5e1afc2824a9fc2e2cf95f063fd9b6bb07036
69697 F20110209_AAAYDG nahapetyan_a_Page_023.jpg
73a6c77353803bd65b41cd69f35c9a9d
ea65f811f38c4ab5b9251979f894513f5aabeaad
1933 F20110209_AAAXYA nahapetyan_a_Page_021.txt
fdd5ae72d9d105f33e82a7e299e4aef7
7ee73a42ab4e2aef9df2ea4ca72892f06226eb7a
F20110209_AAAYCS nahapetyan_a_Page_108.tif
1151a9f9a22ba4e81aecb8cbabfdec23
f8af687a0648653db1108265716071259cbebebb
F20110209_AAAXXM nahapetyan_a_Page_048.tif
425470e83227756188783cbf5a519b2e
625285082004f649e07bbd3c031e9b773dc2443e
4570 F20110209_AAAZGJ nahapetyan_a_Page_026thm.jpg
9839959d0e8afed011221bb434459a4b
ce3d2cedcc47a507c7e073db884dae28fa4f9efe
501 F20110209_AAAZFV nahapetyan_a_Page_002thm.jpg
bcca14bfcab674269a1d027d8a4472f0
35d8165b3bcae6295a6b8de689c6960a587c229b
17890 F20110209_AAAXWY nahapetyan_a_Page_050.QC.jpg
dae73d33f053c582378db3969638e9df
747dc902bf970817e89752b62425bc542a557105
F20110209_AAAYDH nahapetyan_a_Page_059.jp2
a3030eafc083fb9d80b5c1af422ae33e
9f7069d16db046f961be3928e5505b26d9821f99
20152 F20110209_AAAXYB nahapetyan_a_Page_042.QC.jpg
d94b778994b8852c7465c4e410fa1145
62ab057f1987d060a1e7a39d3715493ed2bd0869
2009 F20110209_AAAYCT nahapetyan_a_Page_092.txt
ee7837743b445e74e61052577301e1b7
9686c91a84e88a60d04842b162551c7c71832a69
74378 F20110209_AAAXXN nahapetyan_a_Page_100.jpg
cce4365b0425cb10129328684576b58b
73539948abd8ee420df307142de007c79fcdbf36
5424 F20110209_AAAZGK nahapetyan_a_Page_027thm.jpg
d531fba9b9a65041ecac8f02a4abf883
2a485214747a40b0d1101fb4eff1a0524de48047
4157 F20110209_AAAZFW nahapetyan_a_Page_006thm.jpg
6894c855b45a05d38b5143919b72b868
6a72db38b49b3fc31b0e64bc5d97d598d6803b42
1924 F20110209_AAAXWZ nahapetyan_a_Page_152thm.jpg
094bc5ae09b62cef7a7a057b6645aef5
07492812c9f8cef0ff0f2111bae5dcdc54522aee
1051944 F20110209_AAAYDI nahapetyan_a_Page_057.jp2
b9ddf0f99854319a2435f5fd17e32897
cb5c7d6c08b02f871d4ce225ecdb64218360fb55
5453 F20110209_AAAXYC nahapetyan_a_Page_028thm.jpg
a2b1098aabe952058276773eba9f8243
3d6a4a299071cef2de44f52ca2bf506b2776ee08
22903 F20110209_AAAYCU nahapetyan_a_Page_027.QC.jpg
9021d4b7d5541145eb1712d80b73de9d
8d7002472e526f0209f74ea587112776ae29e3ec
517442 F20110209_AAAXXO nahapetyan_a_Page_128.jp2
c60ad0d4ac9f897c2bb6b0ba43fc52c5
fd46409a68f5814a0f142ca7dac8ebacb129ccf7
5249 F20110209_AAAZGL nahapetyan_a_Page_031thm.jpg
f10ab36d458b4a1c8a66c2568c2d9dfe
9fc411e7013bedf1b6d7a6947478c7e1c99f257b
3809 F20110209_AAAZFX nahapetyan_a_Page_008thm.jpg
427d23738e2edac54f71ec7043905716
fc16e35caaec8eaa902a1b4b025a31017bb18cee
28335 F20110209_AAAYDJ nahapetyan_a_Page_129.pro
eaab4a98c8943f3f6ad34736415755af
9131c64c83c5239ed99715f23a17b10a1defbba4
45212 F20110209_AAAXYD nahapetyan_a_Page_058.pro
363420b7dbb8059fa9965f885cc5a256
95422c833d0be7e1ca25166ab15e3c1d0722dc10
882533 F20110209_AAAYCV nahapetyan_a_Page_042.jp2
90c9e6e90253a02e1b6b2c4c7d57a8d7
dc51b710d81d8e96480969f27f24ca854c080e26
5325 F20110209_AAAXXP nahapetyan_a_Page_125thm.jpg
179ef91ca952d933a4c2b1c31bc0368b
bbc872113ead473e4db2d23b4b47339dd22ade77
5195 F20110209_AAAZHA nahapetyan_a_Page_073thm.jpg
b646722bdf29ab30afeb856cf15856f5
5d12e583b5f13e8cdd1783667800afda4ff39e17
4554 F20110209_AAAZGM nahapetyan_a_Page_033thm.jpg
caf32fba23cb0daf4618625e920ea552
a8d8e38248521e62837072da31824da57bae4cb8
1670 F20110209_AAAZFY nahapetyan_a_Page_009thm.jpg
7f3d3aa4a745bbddc36701b0af81f911
01994dedc81e5e019028dcbceb4bdd19a72350bc
93783 F20110209_AAAYDK nahapetyan_a_Page_150.jpg
de223c40dcb47f02d4eab32a232fe5e0
fc8bbc3130a71478e9016c73111bcec1be75a9fc
757227 F20110209_AAAXYE nahapetyan_a_Page_050.jp2
828ac258355985ee15efcd27eb14e686
8cb98ed3a438feb24ddc7bb5b63b5cc0c2895938
57639 F20110209_AAAYCW nahapetyan_a_Page_015.pro
f269961996456970750b64e461c2dcaf
2d7cc140639ce87021323fcbb8b1be1c535dbd3a
3662 F20110209_AAAZHB nahapetyan_a_Page_080thm.jpg
9bf825aef965d0c71df5146fe6ea6adf
cbae73f76c2be478bef7652ce0610c95696d0f3b
3106 F20110209_AAAZGN nahapetyan_a_Page_037thm.jpg
a53c1aeb7e26b684df4e77c0d3912b18
041b5428d5d9c34f6670d95a1182c0b14d4cbbb4
3160 F20110209_AAAZFZ nahapetyan_a_Page_010thm.jpg
f1f811af7de20370b525f0191d117c6d
8fc71599d32097437c74e1e24c358adfd104311c
1051985 F20110209_AAAYDL nahapetyan_a_Page_145.jp2
7b3232d62e3417c1a542c4918a83a2c7
d25653e9fdcbfc811b458c6623552a33b6947d6f
2108 F20110209_AAAXYF nahapetyan_a_Page_007.QC.jpg
b8f724ea6859aa4176b87215aebb74d3
aadf106da79cb302661b60afe8009e7e24e3da12
22300 F20110209_AAAYCX nahapetyan_a_Page_039.QC.jpg
d9801a4c55fa536172edd3d0c974e9bb
b68bba65002373ca4fc4fa4e68c52ab939cbf275
21098 F20110209_AAAXXQ nahapetyan_a_Page_093.QC.jpg
9390042ea1edda4d28f0052f02d5e38f
53fc0ea2bd97bf871a05501374bbd79d90f58768
5378 F20110209_AAAZHC nahapetyan_a_Page_083thm.jpg
f87e8c2c197e72ae6ba0b6bd3ff4278d
f1f1792a8b84bbdd4f263d36248f4823e510dfa1
5452 F20110209_AAAZGO nahapetyan_a_Page_039thm.jpg
2de7f7d7cfb065716dcd105f2f5684f9
37c8b1c43024e6ab6ba28ac96d9f1a6db5e33312
56477 F20110209_AAAYDM nahapetyan_a_Page_024.jpg
9930fb72cf2d8675d0c9a2ee2a5c7fb1
8d160c85a600c1c0b62a651a6e06f042a5ad8e0a
23792 F20110209_AAAXYG nahapetyan_a_Page_076.QC.jpg
45316aff27b3f5b2b8643dc6d6a38d24
e97ca4e52331e2a367cc0e9e2c12f55a44654834
1906 F20110209_AAAYCY nahapetyan_a_Page_017.txt
0deb243d087273c51a53370f8caabc1d
4d73d803e19ded0e025e3f94b6974bacafb3d435
318980 F20110209_AAAXXR nahapetyan_a_Page_103.jp2
571416ab942377d6bd74878798f973ea
35ebfa18405e93d092f2b427b012875bcfaec450
F20110209_AAAYEA nahapetyan_a_Page_144.tif
8819e65fa434a3a37a9de74a79114109
c4dde4a0d3d44fc9e35a8a3e8f07b8b332f9ba38
4659 F20110209_AAAZHD nahapetyan_a_Page_086thm.jpg
2fdb5b15bff29a1db56c65eb38b613f7
243bb0dea6a12f000162441451d87c0ad8880d69
4888 F20110209_AAAZGP nahapetyan_a_Page_040thm.jpg
2386ecd57f1b7ca30feee5ba2e9c174f
04a0ab816e9d953f70a6bde4d00eb7c6a293828e
5704 F20110209_AAAYDN nahapetyan_a_Page_149thm.jpg
ad8ab46878cb5e105620db7e9a0f8bb7
12adaabffc50427b0b18a1516521c6431085f503
874812 F20110209_AAAXYH nahapetyan_a_Page_026.jp2
06c6eed5841d48f09804d1e14f444113
b4d44130ffeb4c1dc232815462a1aff920aa1c7e
743418 F20110209_AAAYCZ nahapetyan_a_Page_020.jp2
5ad2ab12ad4d7cfde6103866f5c9b3b7
35826d8a1d7ffa95f1c0e714c12791bc8d55c0d5
56900 F20110209_AAAXXS nahapetyan_a_Page_134.pro
fef7c4f99ab603301b53195ce3a59a71
294f4ac1f426e2bd8939a299e081a3cc58549728
39632 F20110209_AAAYEB nahapetyan_a_Page_026.pro
8c093bbdf456c4ed60ab05e8b48bfcf0
07a53d279d67903d75e52ab18359ed18be57e172
5231 F20110209_AAAZHE nahapetyan_a_Page_100thm.jpg
6573d45b7b12f405536df001addd4157
dcd1399931baff4090e353ca694956e9a56eeaba
5060 F20110209_AAAZGQ nahapetyan_a_Page_041thm.jpg
a200df4a929ad87b7584ad425013da43
66581a90835c5ce69098a15454e7a560fdbf725c
17041 F20110209_AAAYDO nahapetyan_a_Page_020.QC.jpg
de47688667452ae003afef32f0ec2959
2f0e8b270dae5941781fc0b1d8619d6caaed4a21
F20110209_AAAXYI nahapetyan_a_Page_146.tif
2c411f5f70fcddacc9d1e265f78ecb95
d6b07193f109a1034a8cd72b1ca0d4ee6418b896
95456 F20110209_AAAXXT nahapetyan_a_Page_146.jpg
7b662487657ab47d0a6abc05f19b1464
8bfb419730a70e1aeb8ce05b56150aa656946208
21703 F20110209_AAAYEC nahapetyan_a_Page_041.QC.jpg
18a30dbbf44a590189d145bcae9c873f
8c7b0d1f853fe679eaf3304162ac776c892330d4
4207 F20110209_AAAZHF nahapetyan_a_Page_104thm.jpg
cbdfb729d3d36b8f0d31ff02f1f6ab6a
a0e6c01855d0f18a9abff7f5f54f924719c10e02
5034 F20110209_AAAZGR nahapetyan_a_Page_047thm.jpg
c66c5fd0ea088dde0c344f0a333e46e2
a4208d6d676b98bdbf60a624f55da75d8faef3ff
1817 F20110209_AAAYDP nahapetyan_a_Page_019.txt
507f8d1e06903079e4c7b5f10771b304
51fdd26707d8ae7d26b9d8ba8e8732020b4b6420
1894 F20110209_AAAXYJ nahapetyan_a_Page_121.txt
617a25bc6bf1bcb0794adb217623aa43
145e141f30d4d350c2fef6ffcb205d7e5fc91b1d
22767 F20110209_AAAXXU nahapetyan_a_Page_030.QC.jpg
d6e0289f6f1552dd6cdce57c5ad460f2
e018ce3b8aebcca6080226c2edb0201174a5913b
4956 F20110209_AAAYED nahapetyan_a_Page_090thm.jpg
ebf5ab4562608603b6ea78374fc26c38
ae256746359db64422835c49404b2ea3db188ccb
5497 F20110209_AAAZHG nahapetyan_a_Page_108thm.jpg
4352541cf90eb6e61e4ec3f8057f0a9b
329daf48347f2bfd0e6ee32da25ab9d26443f1c8
5709 F20110209_AAAZGS nahapetyan_a_Page_048thm.jpg
40c686aa7bde982a30f540fae0714335
3c339e0f353c93150491864b066cc65da278587f
F20110209_AAAYDQ nahapetyan_a_Page_037.tif
bdab7364f3e5aec00b80c2c4c9dc4eda
0c702d2f92486bf83eb75e0ced5f446d475a5811
F20110209_AAAXYK nahapetyan_a_Page_012.tif
4213ffbbee047ad12bf7ba41831211cf
91eded7e3ddb6f5167d9e155a4e5e242e09184a8
20974 F20110209_AAAXXV nahapetyan_a_Page_084.QC.jpg
d26e2b8fb1d555865ab50daf2463bb85
af082406925d5d3d4f93046cf306b9c243e2a360
1014870 F20110209_AAAYEE nahapetyan_a_Page_132.jp2
f71f3857bb9664b817330587620831ed
5e8bca3c24ced4f07c559438113924ab893811e7
5631 F20110209_AAAZHH nahapetyan_a_Page_111thm.jpg
96c42d4c2c60f9fc9620cbb233e6735e
4bbaa29312c45beaaaa6227fba3bd16d630d1705
F20110209_AAAZGT nahapetyan_a_Page_051thm.jpg
a56aff34a8e51e0bcbe0a3961739692b
bfdea6b99da15d2d711966521531265a65583214
20337 F20110209_AAAYDR nahapetyan_a_Page_033.QC.jpg
dfa6fa291e47e8e7bb2245715ec8d68c
01a4554a89187b55e5a3ac0cc6bf44c86271d4e4
5744 F20110209_AAAXYL nahapetyan_a_Page_110thm.jpg
9be3f7c9e9663445f6fe6a167f354c8a
fbc352520dc12376fb995b85799ddcdc7decc768
51354 F20110209_AAAXXW nahapetyan_a_Page_102.pro
a3f92e62daebeacdaeb4f71d9a1fb73f
8ac203d384ba184d464a0dc99ae60f4d99c3661d
38819 F20110209_AAAYEF nahapetyan_a_Page_082.pro
a109c5e77bb437e84e09ba1f992188a1
9881d6c09a7c78770212544619cb34e63cf569ef
4817 F20110209_AAAZHI nahapetyan_a_Page_118thm.jpg
aa7a7c64fcad8203960ec9a02671f14e
8e28e8e84eea64e3ee9711ce4ae7a1901ef4acd1
5320 F20110209_AAAZGU nahapetyan_a_Page_054thm.jpg
b47256d0a94aa90075aa25d0868878c5
ce02f7829076b72e02e7704fcd1dd1db6878535d
75333 F20110209_AAAXZA nahapetyan_a_Page_034.jpg
cccd1b54c2e052fa904b86654267ab25
3ca3dd0fa7e35d334b0ba311be492b02a2fc6d83
18222 F20110209_AAAYDS nahapetyan_a_Page_008.QC.jpg
239db083488e20da8cf0158d668203f3
44c094c9f7d70ca790764f1f0fdb0af501fc2fa5
F20110209_AAAXYM nahapetyan_a_Page_022.tif
328f2916780ffd387aae9b73df48f65f
e5eda40c590c1fb82f7b8678244b7be8f5816bad
19673 F20110209_AAAXXX nahapetyan_a_Page_065.QC.jpg
e1995afd68c9ab86518ef87b37d9afab
f83cb4bf8be75b47c25f701492b9c18220f04c8f
48184 F20110209_AAAYEG nahapetyan_a_Page_139.jpg
430d35cbcbfd893dd2ea7edfad90e538
70b13ca9651e5e4008cdd13f6bf0db98a934289b
4783 F20110209_AAAZHJ nahapetyan_a_Page_119thm.jpg
8c2cc4f787ee88da6ab0383125c44229
4afd5710a2f680f3626a03660dc14bfb9371e3ed
5458 F20110209_AAAZGV nahapetyan_a_Page_056thm.jpg
14193827e33f8a6c58f6c60579255b75
056d5d2b9a627fa9257b33a75796d690387f0fa4
46863 F20110209_AAAYDT nahapetyan_a_Page_093.pro
48a63f4c4fdc63b708a30abf7ca6d2ba
c02479c723a1d6a038b65ffbf2baf2194c5aaf9a
F20110209_AAAXYN nahapetyan_a_Page_127.jp2
eb95b9decf444642af522f3cfafb5dd4
887c1110a51f2d57cf54a741bc63248950371c9f
69772 F20110209_AAAXXY nahapetyan_a_Page_106.jpg
df95c8bd8e36166ac63ee8ea85bf165e
2accaf5ad30511beeb34c416dfd6d1de25ba4f2a
73226 F20110209_AAAYEH nahapetyan_a_Page_099.jpg
12244acbb707e62c5ff62c2c2ba497eb
9fbd5bd7738dbcde4952b1b5c2472029fed08753
48442 F20110209_AAAXZB nahapetyan_a_Page_127.pro
8358880adf9baed0382ead0672528afd
5bbfba65180b19f1177f68b1f4854d9914e5df81
5257 F20110209_AAAZHK nahapetyan_a_Page_120thm.jpg
fb2fbcf62ca1f05873ce677640dab1e8
d3eae63564a5eb89a05e6c0d69e38b3af60ea955
F20110209_AAAZGW nahapetyan_a_Page_057thm.jpg
e6be4a7e537a3a68928a1d86d9dec8d8
a8a19d075040a7a6390941dd9c503c6d59ba25ad
1862 F20110209_AAAYDU nahapetyan_a_Page_010.txt
d0995a7a869a10ac5d043fbd832dbfac
9f42d3261aac40254009cf1d122ed6648b4eb7f0
48811 F20110209_AAAXYO nahapetyan_a_Page_056.pro
343ec09c431360bdfd71b0a5c6b7c19d
48771461703e88dfbb6a0de7faf56f291aff714d
269484 F20110209_AAAXXZ nahapetyan_a_Page_140.jp2
95d7a4b212039dab6e551faecfbc378a
ed083fb21ea66039b2c1f98ea0cc7a8730700b00
45450 F20110209_AAAYEI nahapetyan_a_Page_046.pro
e792f5d45b70b384724ce9dbe1c47ff3
0f479d19ca2e9e4ac55bbe349e9d7c71a2acafe7
892 F20110209_AAAXZC nahapetyan_a_Page_004.txt
702f6b0da9c33ea65d1afafb22525258
caa23b828a04bb4132a9e603e36f888ecd47fadf
2781 F20110209_AAAZHL nahapetyan_a_Page_128thm.jpg
27296e522c3ffdf45b59be71a878e3a2
2d3f2c2860074380df9bb7263a4f43ac45f76204
5085 F20110209_AAAZGX nahapetyan_a_Page_058thm.jpg
48aade4fa5c8efa471e4c8f67478db7f
2d8a1bcad4fc261245b6b324a91479304350a936
F20110209_AAAYDV nahapetyan_a_Page_002.tif
c0784fe7108b3bb61e28b69605d041bf
e9735e3b25a7fdb043ba0957baa703a26a9e080a
52447 F20110209_AAAXYP nahapetyan_a_Page_101.pro
c48b6a1f852878e943784a111142d9c8
f6d3f3a0d6d3d2e98f4bd7876f04a2d7585a628f
4504 F20110209_AAAYEJ nahapetyan_a_Page_105thm.jpg
23ac2571a489e1dec83774ed61bda6ca
42384fca1507249ec921e1a221fd31cc9e6861dc
F20110209_AAAXZD nahapetyan_a_Page_115.tif
126561a7f020b6f7073a36cd777046db
f4d979f752cbc6f624ddb109a138b5b326abaff6
3614 F20110209_AAAZHM nahapetyan_a_Page_134thm.jpg
03ce6b20c5efb05102e4668bd0aaaa47
5f6596ee9faa8a3bda209f39c4b0ece9e926e84a
4819 F20110209_AAAZGY nahapetyan_a_Page_067thm.jpg
7d1df0a706e3c855bbe5de0d46258f8a
249d44c1442476be150102380291c2e40fa39083
1051960 F20110209_AAAYDW nahapetyan_a_Page_148.jp2
5fc217288475ee42872aef407ba8a775
8890b48e91c0b94da6d9b24cb67389a9ebfd385d
73560 F20110209_AAAXYQ nahapetyan_a_Page_041.jpg
69bb07c25b2103b45a0ef6ea1e3a53eb
c1eef795e7cc4713bbae689e33d193c4e3887279
1013553 F20110209_AAAYEK nahapetyan_a_Page_054.jp2
9fbd9774f7d12a6570a544f1a6b31a77
1aa4113089e2d5a34d18adcc421c139472b9a794
F20110209_AAAXZE nahapetyan_a_Page_044.tif
dc792188202d4067467d9d079747b00d
722f8d9e9669dd853254a3691589eeaf39d4a0d0
4041 F20110209_AAAZHN nahapetyan_a_Page_142thm.jpg
df8c5cfe22b07fa3521ea8206a362227
18852e3877725426e940828e6f6ddd6d18cde46c
4860 F20110209_AAAZGZ nahapetyan_a_Page_071thm.jpg
63e4f6efbfc984b773642ece391a736c
b022bfa742f895a713dc9f5fda52c2466d9c2189
245052 F20110209_AAAYDX nahapetyan_a_Page_129.jp2
892809408f2daeb8995cdf65fdf77717
bad1b8afcd0f76bf4fd463645c3c2aeabed7e040
F20110209_AAAYEL nahapetyan_a_Page_061.tif
45b9171d83795968d2fa59978d899ad1
a9f1f448c450c33651bafd0dc3ec0fb9ec20be42
42955 F20110209_AAAXZF nahapetyan_a_Page_011.pro
03aceaa85bcb16f29e696c2cefd7030e
e3d4f3cb928f2a666bb2c7d5960aa5ff31518d0e
181914 F20110209_AAAZHO UFE0015623_00001.mets
23180b575dcd2f8e6c305dbf7a5a2418
4dfc0164018736178d38292005ad57f95e28d7bc
15000 F20110209_AAAYDY nahapetyan_a_Page_130.QC.jpg
ee3c9c05115fa31130cbcf1b9c7f5e8a
cff1f148ec75aa969693be73820a0ec3c8535a84
60169 F20110209_AAAXYR nahapetyan_a_Page_146.pro
c5cf363a64c12af6258dab731fc676e1
58007be20d3e2570c0a29989bf79f9c773ee6be9
2141 F20110209_AAAYFA nahapetyan_a_Page_008.txt
860b7687c0cbc8eaad03ef39fb85238f
7d8057f6b7434a5d0ad89cce76e168e3e7833b94
608 F20110209_AAAYEM nahapetyan_a_Page_007thm.jpg
11d36a7b5b81edbd8a0eed6f4fa0519e
5ee78c7fb8cba569ee4372c6b33748ebb363481c
79441 F20110209_AAAXZG nahapetyan_a_Page_056.jpg
00d6fcd25ed6e68f6b5dd49126c864dd
835cd7c1d4c14a98630bbd27f6ec9ed7e89e24fa
2624 F20110209_AAAYDZ nahapetyan_a_Page_135thm.jpg
bbab72c6b7f59dac6698a5e4b5431169
bcfcf40b580c1374c9cabc9c42cb222bc74b582d
F20110209_AAAXYS nahapetyan_a_Page_060thm.jpg
635240f82e3d871d2381ed7e4d043646
b86f6890b094e4e35f81cd154548468b61abecda
1649 F20110209_AAAYFB nahapetyan_a_Page_089thm.jpg
b2ea111252ff28e59c3713bf96dca3dc
6034fe4bcfa34f6e0922308ec67f6a33ea628553
87100 F20110209_AAAYEN nahapetyan_a_Page_015.jpg
fd975d664f9ce117b3019dc15e42d2b2
a3729bb9a2d5cf9e0424c4249b601f4771eeba45
2490 F20110209_AAAXZH nahapetyan_a_Page_012thm.jpg
4b769fc47dc5900fcaad9723f8fcd13c
fe1a73adc6074ee14fda41fc28f8fa34b0a4cb85
56006 F20110209_AAAXYT nahapetyan_a_Page_050.jpg
269e4b794266d31cfb4607136306fca4
0a031f4add5474fd42c649227e49e291cfebd58d
53735 F20110209_AAAYFC nahapetyan_a_Page_088.pro
7f1deb0773dd8d9f46c8f5d2da38d7da
0e5330e2cde048cd182cf7567e3dab6d77a99887
F20110209_AAAYEO nahapetyan_a_Page_108.txt
254220a05b8f139d4116fe2c072b8612
1d0b8344dabe0c8371f960b1835bd838bcd77504
791811 F20110209_AAAXZI nahapetyan_a_Page_130.jp2
d81673b446e22295ff68de471975ecec
6fc12ff406a32aafe52bb107a02a09fa6fb9766c
2099 F20110209_AAAXYU nahapetyan_a_Page_094.txt
d7cc7e2ad7823c44c9681c9584bf45cd
d5d383e5b9d27caa540e9580760b40435d8ac262
48146 F20110209_AAAYFD nahapetyan_a_Page_043.pro
4507d1f57d60266e9e39bb97d6e46922
1e231df22a6577684bee7c5eeff46d5d83251810
2066 F20110209_AAAYEP nahapetyan_a_Page_100.txt
cd8f7c0f39be35071877309ef750abcb
377a3071f15b2bd1270dc9653fbb1a6827b34612
2155 F20110209_AAAXZJ nahapetyan_a_Page_085.txt
b2625f39f38c24e96f9f4f497a442727
c3414d8280693ed8bb67a1acdae315446cdd1602
24836 F20110209_AAAXYV nahapetyan_a_Page_035.QC.jpg
1ec5a1655332d759b55189ce618acc76
4f297a22b7c181d6aa2de24f09c335c956210fef
1779 F20110209_AAAYFE nahapetyan_a_Page_070.pro
2a8cc7341fe9a73b69f3bddde93e43d0
daedb7b4651926a06082c72adcba5193a75373fe
1161 F20110209_AAAYEQ nahapetyan_a_Page_037.txt
5a71091cd542370a878ad9ea89722859
fec5b45f2aebcbdbe74e4cc807b034cd15ddc34c
19070 F20110209_AAAXZK nahapetyan_a_Page_037.pro
48d2b5a7b71f0850bb05d8a2534d43c1
a92c36f26bdc205ec189d28a5a93b29cb52acbe8
1029824 F20110209_AAAXYW nahapetyan_a_Page_120.jp2
13bea3b71435c59347c541697a0f464a
2aaa2ff4ce9c895594f43c31812de289942b6b21
48464 F20110209_AAAYFF nahapetyan_a_Page_100.pro
6d02bbe2a61c84cc48f0cd9d1f1730f2
d707899f91a922d76643837f6ef810b1dbfeb487
F20110209_AAAYER nahapetyan_a_Page_052.tif
b0a6a441b4a7934bbcb402a4b6d1ed5b
c67113818e7bfe89811711857fa7b7895343f011
26073 F20110209_AAAXZL nahapetyan_a_Page_014.QC.jpg
ec7fad13e5a7846db5727736b8e0fded
76e7d02343f70e43c9a458490a495f2854646240
7019 F20110209_AAAXYX nahapetyan_a_Page_151.QC.jpg
e0ac534d27950b2fc38986ec55eb3063
7aae5786961e9e028c281c955355f7e8e20fceb7
46443 F20110209_AAAYFG nahapetyan_a_Page_115.pro
b9b67a1284ab800f65ca306e4a754484
fcefe2bcd1320e4a51166c5bad0c2a877567a6d5
46209 F20110209_AAAYES nahapetyan_a_Page_017.pro
2bbbe2f3613d924538316d4906e06057
edda4a78c09a7c85db583c9c7488c3c9c6b0b969
647 F20110209_AAAXZM nahapetyan_a_Page_152.txt
1a0489dc90e0511c8dd3aabaf42fabc4
d35521ec4f3a9f37f74769845d72f6ccadf3e53e
78081 F20110209_AAAXYY nahapetyan_a_Page_057.jpg
6cd37c29477eca3b28c9300420dfb2f2
aff0c027f6ff3440c3fc9447a1736aff6f6e67c9
45744 F20110209_AAAYFH nahapetyan_a_Page_070.jpg
0aa2591d56edddfb6ccca31464a74b8a
b2dc03c3cc323caedce930ac1d13a2834318d8ec
55713 F20110209_AAAYET nahapetyan_a_Page_144.pro
7742404489e7eb9e5bc5743992d00d01
109419795c5e02a862d9d6d4ddec5aa9816b9449
83462 F20110209_AAAXZN nahapetyan_a_Page_097.jpg
2f144f1641bd4b7fbb643960e2b849bf
a2b28670b119ae29c0b09b4c32023b13749153fc
62657 F20110209_AAAXYZ nahapetyan_a_Page_081.jpg
9bf52c9ed64cca68e36109e405c95cfb
c008906dfe1a2456475eadc8b88148b75d3afc2b
7723 F20110209_AAAYFI nahapetyan_a_Page_003.jpg
4f43341e54244fe079b7fcab1bfe914f
d5b4723c74cbc67087b4817162f2e840b9787068
71921 F20110209_AAAYEU nahapetyan_a_Page_069.jpg
facfed1c6b2c966239940881c7633fd9
a6ec59a0957acb20a0d7a08af3679208a154780c
56987 F20110209_AAAXZO nahapetyan_a_Page_107.pro
e5c57c6cf14e722257f163aaa1e42abf
8cb6ca70a50b2b32cbfd38d639b060e898184cea
F20110209_AAAYFJ nahapetyan_a_Page_010.tif
9bf5c66a2d71d0490df43263f4b9806e
a3130fe51788e4e2e78ba5e1e44acd0cf5d564f0
16399 F20110209_AAAYEV nahapetyan_a_Page_104.QC.jpg
31baecdc775e737083c476790e7f9fa4
c34809e6ddcb6b220f9edfdc473618fe20658aa3
20676 F20110209_AAAXZP nahapetyan_a_Page_079.QC.jpg
54b38ee828c9abc9b7d7b5858addf6c0
4089647d611aca436a35cbd5d00a34207ec1dee8
82472 F20110209_AAAYFK nahapetyan_a_Page_101.jpg
6dc7bdb33e3a5fb5b18b49b91dd37c8b
8eef766c18b0fe8af73f56a570ed53df1ab53465
80415 F20110209_AAAYEW nahapetyan_a_Page_028.jpg
0597eb9d9c1c0d5b85259cc1912f8ee3
37dd26c94c423e602daa757f136b32f2c5eaa6b8
1727 F20110209_AAAXZQ nahapetyan_a_Page_040.txt
dfef305aa0e42bc1ec0e5c5ad7a4dc55
6eba9445c4fd4286b3e8226fe0187ded4d834110
795291 F20110209_AAAYFL nahapetyan_a_Page_068.jp2
788651e3a1c9aa0b92cca493d35d6e6d
cca79e42a572be4fbed55e2157921cde0f4e8454
5579 F20110209_AAAYEX nahapetyan_a_Page_145thm.jpg
0ba6d8562a60a49dfcfd18a10b87312c
e85cc6aec6d87116cb03834b38825abdf4531ee0
1016253 F20110209_AAAXZR nahapetyan_a_Page_045.jp2
a0c6806421c4173046f28db2a4089e16
80d8effb779bed8f79c85905350d426fbd08f2ad
67040 F20110209_AAAYFM nahapetyan_a_Page_033.jpg
04c6425d14ea9ee0c73816376ce9848e
583b812e488580878efb421ff1b272b79519adc1
25852 F20110209_AAAYEY nahapetyan_a_Page_151.jpg
f7f65d37b95d7b1c88bef83ba7b11316
304f9c2541eeb5efba444ce1d5ed22bffd979e99
5758 F20110209_AAAYGA nahapetyan_a_Page_148thm.jpg
f5758bb109ba11dbc1866de1cf006580
9148257813dab867975722469502c8bccca5eec4
10818 F20110209_AAAYFN nahapetyan_a_Page_004.QC.jpg
365795c18034b55c1f18237e7e1301e9
eb3466ed75fd7779254986ad41156a1044ede3e4
F20110209_AAAYEZ nahapetyan_a_Page_051.txt
7d4008721425691592f046fa300ddc51
e710f62a2672b66fc09d0dfc4bb3491f3fd6cdee
2421 F20110209_AAAXZS nahapetyan_a_Page_149.txt
1580bd41a5aafb624750c152c7f270d0
f8cde1a2d5791c9928796438a57b656ca5881663
F20110209_AAAYGB nahapetyan_a_Page_133.txt
f7e4ed72bdd51a6946b893de634ee6dd
aa58664150667f7ccf9689a5d828617809ced59f
24807 F20110209_AAAYFO nahapetyan_a_Page_109.QC.jpg
16bc0fc04fe58f8d465a6b47ea776968
bffbc1fc52d0c51ab2b6bb4f455220fa1d7dbfc4
36362 F20110209_AAAXZT nahapetyan_a_Page_068.pro
51cd345aa70d8f270f12360e0ba7f7be
989e36df0a21cf8fb9589af81b0672e29dbbeffb
3766 F20110209_AAAYGC nahapetyan_a_Page_124thm.jpg
7524d05a673b1202d98c39ed45b3a7b2
8eb3d6385bfed20ce887e361d1058dfcb4e1d144
1028884 F20110209_AAAYFP nahapetyan_a_Page_055.jp2
fa3e5d2525806bda7c0f314fd6c3541b
62c559715143ebdd8750f8a7f43b42ae12091b59
1596 F20110209_AAAXZU nahapetyan_a_Page_061.txt
456e82c33184a8e8ecd12e1bf2526117
b5ad470df80fdf37919c5d11ffd2dcaa297f9f7b
F20110209_AAAYGD nahapetyan_a_Page_072thm.jpg
76c93f2be2c92e3a7921c045c6e72865
a7909876d5f7a6a5eda3619b1ee4d89680903c82
17389 F20110209_AAAYFQ nahapetyan_a_Page_029.QC.jpg
f9cbd2428e8582c2ba181db6e6f675fc
56963d1435723761c94cd38c34cf476713e7cf8b
3781 F20110209_AAAXZV nahapetyan_a_Page_136thm.jpg
858347e3b03f3f805929677eeb5d2fe5
d2f66c638c3bd3e12b9f64e0fd58c72b85ce6fca
48223 F20110209_AAAYGE nahapetyan_a_Page_099.pro
06e971eda2fc601021134ed1d1c3c892
8d73eaf5855155e963e0c968ac4fadc62c151ca3
30127 F20110209_AAAYFR nahapetyan_a_Page_002.jp2
074797c97588b1b61fa1ce93d93cbf64
7b6b2b2d9494466353ebd66ade62c1fa57540fa1
2418 F20110209_AAAXZW nahapetyan_a_Page_146.txt
230057f574c3979c72a4d0592c95154c
47f3bf1da5a8803411c2803c07e68d09b4c28959
F20110209_AAAYGF nahapetyan_a_Page_088.tif
255aaf22916e7a99f995c394b9a7cbfc
da6c5e0f5d201078274493be33749805a3da3d71



PAGE 4

Iwouldliketothankmychairandcochair,Prof.SiriphongLawphongpanichandProf.DonaldW.Hearn,fortheirvaluableadvice,supportandguidanceduringmystudies.Ourmeetingsanddiscussionswerealwaysveryhelpful.AlsoIwouldliketoexpressmysinceregratitudetothecommitteemembersProf.PanosPardalos,Prof.WilliamHager,andProf.RavindraAhujafortheirencouragement.Especially,IamgratefultoProf.PanosPardalosforhisvaluablesuggestionsandadviceonthesupplychainproblemsIhaveworkedon.Thetremendoussupportfrommyparentsisinvaluable,andtherearenowordstoexpressmyappreciationforthat.Finally,Iwouldliketothankallmyfriendsandcollaboratorswhomademystudiesenjoyableandproductive. iv

PAGE 5

page ACKNOWLEDGMENTS ............................. iv LISTOFTABLES ................................. viii LISTOFFIGURES ................................ x ABSTRACT .................................... xi CHAPTER 1INTRODUCTION .............................. 1 2ABILINEARREDUCTIONBASEDALGORITHMFORCONCAVEPIECEWISELINEARNETWORKFLOWPROBLEMS ......... 5 2.1IntroductiontotheChapter ...................... 5 2.2ABilinearReductionTechniquefortheConcavePiecewiseLinearNetworkFlowProblem ......................... 7 2.3ConcavePiecewiseLinearProblemswithSeparableObjectiveFunctions ................................ 11 2.4DynamicCostUpdatingProcedure .................. 13 2.5OntheDynamicSlopeScalingProcedure ............... 16 2.6NumericalExperiments ......................... 19 2.7ConcludingRemarks .......................... 21 3ADAPTIVEDYNAMICCOSTUPDATINGPROCEDUREFORSOLVINGFIXEDCHARGENETWORKFLOWPROBLEMS ........... 22 3.1IntroductiontotheChapter ...................... 22 3.2ApproximationoftheFixedChargeNetworkFlowProblembyaTwo-PieceLinearConcaveNetworkFlowProblem .......... 24 3.3AdaptiveDynamicCostUpdatingProcedure ............. 27 3.4OntheDynamicSlopeScalingProcedure ............... 30 3.5NumericalExperiments ......................... 31 3.6ConcludingRemarks .......................... 34 4ABILINEARREDUCTIONBASEDALGORITHMFORSOLVINGCAPACITATEDMULTI-ITEMDYNAMICPRICINGPROBLEMS ... 35 4.1IntroductiontotheChapter ...................... 35 4.2ProblemDescription .......................... 37 v

PAGE 6

43 4.4NumericalExperiments ......................... 46 4.5ConcludingRemarks .......................... 49 5DISCRETE-TIMEDYNAMICTRAFFICASSIGNMENTMODELSWITHPERIODICPLANNINGHORIZON:SYSTEMOPTIMUM ... 50 5.1IntroductiontotheChapter ...................... 50 5.2PeriodicPlanningHorizon ....................... 53 5.3Discrete-TimeDynamicTracAssignmentProblemwithPeriodicTimeHorizon .............................. 54 5.4BoundsfortheDTDTAProblem ................... 67 5.5NumericalExperiments ......................... 71 5.6ConcludingRemarks .......................... 76 6ANONLINEARAPPROXIMATIONBASEDHEURISTICALGORITHMFORTHEUPPER-BOUNDPROBLEM .................. 78 6.1IntroductiontotheChapter ...................... 78 6.2NonlinearRelaxationofDTDTA-UProblem ............. 83 6.3NonlinearRelaxationBasedHeuristicAlgorithm ........... 86 6.4NumericalExperiments ......................... 89 6.5ConcludingRemarks .......................... 91 7ADYNAMICTOLLPRICINGFRAMEWORKFORDISCRETE-TIMEDYNAMICTRAFFICASSIGNMENTMODELS ............. 92 7.1IntroductiontotheChapter ...................... 92 7.2TheReducedTime-ExpandedNetworkandUESolution ...... 97 7.3TheDynamicTollSet ......................... 103 7.4DynamicTollPricingProblems .................... 108 7.5IllustrativeExamples .......................... 110 7.6ConcludingRemarks .......................... 113 8DIRECTIONSOFFUTURERESEARCH ................. 115 APPENDIX ACOMPUTATIONALRESULTSFORCHAPTER 2 ............ 117 BCOMPUTATIONALRESULTSFORCHAPTER 3 ............ 121 CCOMPUTATIONALRESULTSFORCHAPTER 4 ............ 125 DCOMPUTATIONALRESULTSFORCHAPTER 6 ............ 127 ECOMPUTATIONALRESULTSFORCHAPTER 7 ............ 129 vi

PAGE 7

................................... 131 BIOGRAPHICALSKETCH ............................ 140 vii

PAGE 8

Table page 5{1Demandpatterns ............................... 72 5{2Optimalsolutionstothetwo-arcproblem. ................. 73 5{3Solutionsfromthelowerandupper-boundproblems:lineartravelcostfunction. .................................... 75 5{4Solutionsfromthelowerandupper-boundproblems:quadratictravelcostfunction. ................................. 75 5{5Qualityofrenedupperandlower-boundsolutions:lineartravelcostfunction. .................................... 76 5{6Qualityofrenedupperandlower-boundsolutions:quadratictravelcostfunction. .................................... 76 6{1Equivalentobjectivefunctions ........................ 88 6{2Distributionsofparametersofrandomlygeneratedtraveltimefunctions 89 7{1Additionalconstraints ............................ 110 7{2Distributionsofparametersofrandomlygeneratedtraveltimefunctions 111 A{1Setofproblems. ................................ 117 A{2Computationalresultsofsets1-18:qualityofthesolutionandtheCPUtimes. ..................................... 118 A{3Computationalresultsofsets1-18:DSSPvs.DCUP. ........... 119 A{4Computationalresultsforsets19-30. .................... 120 A{5Computationalresultsforthecombinedmode. ............... 120 B{1Setofproblems. ................................ 121 B{2ComputationalresultsofgroupsG1andG2:qualityofthesolutionsandtheCPUtimes. ................................ 122 B{3ComputationalresultsofgroupsG1andG2:thepercentageofproblemwhereoneofthealgorithmsndsabettersolutionthananotherone. .. 123 viii

PAGE 9

................ 124 C{1Thequalityofthesolution:Procedure 6 .................. 125 C{2Thequalityofthesolution:Procedure 5 .................. 126 C{3TheCPUtimeoftheprocedures. ...................... 126 D{1Computationalresultsoftheexperiments. ................. 127 D{2Computationalresultsofthecombinedmode. ............... 128 E{1Thetotalcollectedtollandthetotalcostforeachproblemandparameter". ........................................ 129 E{2Thenumberoftollcollectingcentersforeachproblemandparameter". 129 ix

PAGE 10

Figure page 3{1Approximationoffunctionfa(xa). ...................... 25 3{2"aa(xa)and"kaa(xa)functions. ........................ 28 4{1Thepriceandtherevenuefunctions. .................... 38 5{1Linearversuscircularintervals. ....................... 53 5{2Eventsoccurringintwoconsecutiveplanninghorizons. .......... 54 5{3Three-nodenetwork. ............................. 55 5{4Timeexpansionofarc(1;2)att=1. .................... 56 5{5Time-expansionofthethree-nodenetwork. ................. 58 5{6a(xa(t))2(s;s]versusxa(t)2(1a(s);1a()]. .......... 68 5{7Two-arcnetwork. ............................... 72 5{8Four-nodenetwork. .............................. 74 6{1Twofeasiblesolutions. ............................ 80 7{14-Nodenetworkandtracdemand. ..................... 92 7{2Userequilibriumowsandtraveltimes. .................. 93 7{3Systemoptimumowsandtraveltimes. .................. 93 7{4Tolleduserequilibriumowsandtraveltimes. ............... 94 7{5Thevalueoft. ................................ 111 D{1TwoNetworks. ................................ 127 E{19-nodenetwork. ................................ 129 E{2Thetollvectorfordierentvaluesof"intheMinRev(")problem. ... 130 E{3Thetollvectorfordierentvaluesof"intheMinCost(")problem. ... 130 x

PAGE 11

Inthisdissertationweinvestigatenetworkowproblemswithnonlineararccostfunctions.Therstgroupofproblemsconsistsofconcavepiecewiselinearnetworkow,xedchargenetworkow,anddynamicpricingproblemsthatariseintheareasofsupplychainmanagementandlogistics.BasedontheMIPformulation,weconstructbilinearreductionproblems,inwhichtheglobalsolutionofthelatterisasolutionoftheinitialformulation.Tosolvethereductionproblem,weproposesomeheuristicalgorithms.Intheexperiments,wecomparethesolutionprovidedbyouralgorithmwithanexactsolutionaswellasasolutionprovidedbyotherheuristicalgorithmsintheliterature.Numericalexperimentsonrandomlygeneratedinstancesconrmthequalityofthealgorithms. Thesecondgroupofproblemsisrelatedtothedynamictracassignmentproblem.Inparticular,weconsideraperiodicdiscretetimedynamictracassignmentproblem(DTDTA),inwhichthetraveltimeisafunctionofthenumberofcarsontheroad,andtheplanninghorizoniscircular.Themathematicalformulationbelongstotheclassofnonlinearmixedintegerproblems.Toobtainan xi

PAGE 12

xii

PAGE 13

Networkowproblemsareminimization/maximizationproblemswithunderlyingnetworkstructure.Althoughtherearedierentrepresentationsofthenetwork,perhapsthemostpopularoneisbasedonowconservationconstraintsviaanode-arcincidencematrix.Apartfromtheowconservationconstraints,manyproblemshaveadditionalrestrictionsonthevariables,e.g.,non-negativityandlower/upperboundaryconstraints.Basedontheobjectivefunctionandotheradditionalconstraintstheproblemscanbeclassiedaslinearornonlinear,wherethelattercanbefurtherdecomposedintoconvex,concave,orotherproblems. Thelinearproblemsassumethattheconstraintsaswellastheobjectivefunctionarelinear.Polynomialtimealgorithmsforsolvingtheproblemsarewellknown.Someclassicalexamplesofthenetworkowproblemswithlinearconstraintsincludeshortestpath,minimumspanningtree,minimumcut,maximumow,minimumcostnetworkow,andotherproblems.Basedontheoptimalityconditionsandotherpropertiesoftheproblem,severalalgorithmshavebeenproposedtosolvetheproblems.Fordetailsonthelinearnetworkowproblems,seeAhujaetal.[ 2 ]. Despitethenicetheoreticalresultsdevelopedforthelinearproblems,mostofthepracticalproblemsarenotlinear,i.e.;theobjectivefunctionand/orsomeoftheconstraintsarenonlinear.Ifitisaconvexminimization(concavemaximization)problemwithaconvexfeasibleregionthenanylocalminimum(maximum)isaglobalsolutionoftheproblem,andappropriatealgorithmssuchthattheFrank-Wolfalgorithm,gradientbasedanddirectionndingmethods,canbeusedtosolvetheproblem.Alargevarietyofalgorithmsforsolvingconvexminimization 1

PAGE 14

(concavemaximization)problemscanbefoundinBazaraaetal.[ 5 ].Whentheobjectivefunctionisnotconvex(concave)and/orthefeasibleregionisnotconvex,thesealgorithmsdonotnecessarilyconvergetoaglobaloptimalsolution.Findingaglobalsolutionisahardtask,andglobaloptimizationtechniquesarerequiredtosolvetheproblem(see,e.g.,Horstetal.[ 54 ]andHorstandTuy[ 55 ]). Inthisdissertation,weconsidertwogroupsofnon-convexnetworkowproblems.Therstgroupincludespiecewiselinearnetworkow,xedchargenetworkow,anddynamicpricingproblems(seeChapters 2 3 ,and 4 )thathavealargevarietyofapplicationsintheproductionplanning,scheduling,investmentdecision,networkdesign,locationofplantsanddistributioncenters,pricingpolicy,andmanyotherpracticalproblemsthatariseinsupplychains,logistics,transportationscience,andcomputernetworks.ItiswellknownthattheproblemsintheirgeneralformareNP-Hard;therefore,therearenopolynomialtimealgorithmstosolvetheproblemsunlessP=NP.Althoughthemathematicalformulationbelongstotheclassoflinearmixedintegerproblems,solvinglargeproblemsrequiresalargeamountofCPUtimeandmemory.Ontheotherhand,onecanconsiderapproximationtechniquesthatareabletoprovideagood-qualitysolutionusinglesscomputerresources.Manyofthesetechniquesemployalinearrelaxationoftheproblem.Unlikethoseintheliterature,weproposenonlinearreductiontechniquestosolvetheproblems.Inparticular,inallthreeproblemswedevelopamethodtoreducetheproblemtoabilinearoneandproposeaheuristicalgorithmtosolvetheresultingproblem.Inthenumericalexperimentswecomparetheresultswithanexactsolution(orthebestfeasiblesolution)providedbyMIPsolvers,aswellaswithDynamicSlopeScalingProcedure(DSSP),sinceitisknowntobeoneofthebestheuristicalgorithmstosolvesuchproblems.Numericalexperimentsonrandomlygeneratedproblemsconrmthequalityofthesolutionsprovidedbyouralgorithms.Inparticular,itoutperformstheDSSPin

PAGE 15

thequalityofthesolutionaswellasinthecomputationaltime.Inaddition,wetransformtheproblemsintoalternativecontinuousnetworkowproblemswithowdependentcostfunctionandprovethataglobalsolutionoftheresultingproblemisasolutionoftheinitialMIPformulation.Despiteanunusualstructureofthecostfunction,themathematicalformulationoftheproblemsissimilartothesystemoptimumproblemsarisinginthetracassignmentmodelling.Usingthesamecostfunction,wealsoconstructavariationalinequalityproblemsimilartothoseinthetransportationliteratureandprovethattheDSSPconvergestoasolutionoftheresultingproblem;i.e.,itprovidesanequilibriumsolution.However,theproblemrequiresndingasystemoptimumsolution,andthealgorithmsweproposendsanapproximatesolutiontotheproblem. Thesecondgroupofproblemsisrelatedtothedynamictracassignmentproblem.Unlikethestaticcase,wherethetraveltimeisafunctionofthearcow,thedynamicmodelsinvolvethreevariables:inowrate,outowrate,anddensity,andthetraveltimecanbeafunctionofallthreevariables.Intheliteratureseveralcontinuousanddiscretetimemodelshavebeenproposedfordierenttraveltimefunctions.Themodelinthisdissertationassumesthatthetraveltimeisafunctionofthedensity,andallcarsthatenteranarcatthesamepointoftimeexperiencethesametracconditions;therefore,theyleavethearcatthesametime.Inaddition,themodelsintheliteratureassumethatthenetworkisemptyatthebeginningandtheendofaplanninghorizon.Inthecasewhensomecarsarepresentinthenetwork,thetimetoenterthenetworkforthosecarsisunknown,anditishardtomodelthepropagationofthecarsinthenetwork.Unlikeothermodelsintheliterature,weconsideraperiodicplanninghorizonandassumethattheprocessesrepeatthemselvesfromoneperiodtoanother(seeChapter 5 ).Themathematicalformulationoftheproblemminimizesthetotaldelayandbelongstotheclassofnonlinearmixedintegerproblems,ahardproblemtosolve.

PAGE 16

Bylinearizingtheobjectivefunctionandtheconstraints,weconstructlinearmixedintegerproblemsthatprovideupperandlowerbounds.Thesolutionoftheboundingproblemscanbemadearbitrarilyclosetoasolutionoftheinitialformulationbydecreasingthediscretizationparameter.However,theboundingproblemsinvolvebinaryvariables,anditishardtosolvelargeproblemsusingMIPsolvers.InChapter 6 wediscussaheuristicalgorithmbasedonanonlinearrelaxationoftheproblem.Inparticular,weconstructacontinuousbilinearproblem,whichprovidesatighterlowerboundthantheLPrelaxation.Usingthebilinearrelaxation,theheuristicalgorithmaimstondanintegersolution,whichhasanobjectivefunctionvalueclosetotheoneprovidedbytherelaxationproblem. Anotherproblemofinterestisthetollpricingframeworkforthedynamictracassignmentproblem(seeChapter 7 ).Similartothestaticcase,weconstructasetofvalidtollvectorssuchthatasystemoptimumsolutionisasolutionofthetolleduserequilibriumproblem.Thelatterisauserequilibriumproblemwherethearccostfunctionsincludetollsinadditiontothetraveltimes.Akeycomponentinthedevelopmentofsuchtechniqueisthereducedtime-expanded(RTE)networkconstructedbasedonafeasiblevector.Usingthenetwork,weshowthatafeasiblevectorisauserequilibriumsolutionifandonlyifitisasolutionofalinearproblemwithanunderlyingRTEnetworkstructure.Thelatterallowstheconstructionofasetofvalidtollsandformulationofatollpricingproblemwithasecondaryobjective,andweprovideseveralexamplesofsuchproblems.

PAGE 17

45 ]andGeunesandPardalos[ 41 ]).Weconsidertheconcavepiecewiselinearnetworkowproblem(CPLNF),whichhasdiverseapplicationsinsupplychainmanagement,logistics,transportationscience,andtelecommunicationnetworks.Inaddition,theCPLNFproblemcanbeusedtondanapproximatesolutionfornetworkowproblemswithacontinuousconcavecostfunction.ItiswellknownthattheseproblemsareNP-hard(seeGuisewiteandPardalos[ 45 ]). ThischapterdealswithanonlinearreductiontechniqueforthelinearmixedintegerformulationoftheCPLNFproblem.Inparticular,theproblemisreducedtoacontinuousonewithlinearconstraintsandabilinearobjectivefunction.ThereductionhasaneconomicalinterpretationanditssolutionisproventobethesolutionoftheCPLNFproblem.Basedonthereduction,weproposeanalgorithmforndingalocalminimumoftheproblem,whichwerefertoasthedynamiccostupdatingprocedure(DCUP).Inthechapter,weshowthatDCUPconvergesinanitenumberofiterations. Thetheoreticalresultspresentedinthischaptercanbeextendedtoamoregeneralconcaveminimizationproblemwithaseparablepiecewiselinearobjectivefunctionandlinear/nonlinearconstraints.ItshouldbeemphasizedthatHorst 5

PAGE 18

etal.[ 54 ](seealsoHorstandTuy[ 55 ])discussabilinearprogramwithdisjointfeasibleregionsandprovethattheproblemisequivalenttoasubclassofpiecewiselinearconcaveminimizationproblems.Theresultsinthischaptershowthatanyconcaveminimizationproblemwithaseparableconcavepiecewiselinearobjectivefunctionisequivalenttoabilinearprogram.Itiswellknownthatanoptimalsolutionofageneraljointlyconstrainedbilinearprogrambelongstotheboundaryofthefeasibleregionandisnotnecessarilyavertex(seeHorstetal.[ 54 ]).However,thereductiontechniquepresentedinthischapterhasajointlyconstrainedfeasibleregionwithaspecialstructureanditisstillequivalenttoaconcavepiecewiselinearprogram.Fromthelatteritfollowsthattwopartsofasolutionoftheproblemareverticesoftwodierentpolytopesthatare\joined"byasetofconstraints.Inthatsense,thesetypesofproblemsareweaklyjoinedbilinearprograms. TheCPLNFproblemcanbetransformedintoanequivalentnetworkowproblemwithowdependentcostsfunction(NFPwFDCF).UsingNFPwFDCF,itcanbeshownthatthedynamicslopescalingprocedure(DSSP)(seeKimandPardalos[ 61 ]and[ 62 ])convergestoanequilibriumsolutionofNFPwFDCF.AlthoughDSSPprovidesasolution,whichcanbequiteclosetothesystemsolution,itiswellknownthattheequilibriumandthesystemsolutionsingeneralarenotthesame.Ontheotherhand,DCUPconvergestoalocalminimumoftheproblem.Inthenumericalexperiments,wesolvedierentproblemsusingDCUPandDSSPandcomparethequalityofthesolutionaswellastherunningtime.ComputationalresultsshowthatDCUPoftenprovidesabettersolutionthanDSSPandusesfeweriterationsandlessCPUtime.SinceDCUPstartsfromafeasiblevectorandconvergestoalocalminimum,oneconsidersrstsolvingDSSPandthenimprovingthesolutionusingDCUP.Thenumericalexperimentsusingthiscombinedmodeareprovidedaswell.

PAGE 19

Fortheremainder,Section 2.2 discussesthenonlinearreductiontechniquefortheCPLNFproblem.Section 2.3 generalizestheresultsfromSection 2.2 foraconcavepiecewiselinearproblemwithaseparableobjectivefunction.Section 2.4 describesDCUPandtheoreticalresultsontheconvergenceandthesolutionoftheprocedure.InSection 2.5 weprovethatthesolutionoftheDSSPisanequilibriumsolutionofanetworkowproblemwithowdependentcostfunctions.TheresultsofnumericalexperimentsonDCUPandDSSPareprovidedinSection 2.6 ,andnally,Section 2.7 concludesthechapter. s.t.Bx=b(2{1) whereBisthenode-arcincidentmatrixofthenetworkG,andfa(xa)arepiecewiselinearconcavefunctions,i.e.,fa(xa)=8>>>><>>>>:c1axa+s1a(=f1a(xa))xa2[0a;1a)cnaaxa+snaa(=fnaa(xa))xa2[na1a;naa];

PAGE 20

Usingbinaryvariables,yka,k2Ka,onecanformulatetheCPLNFproblemasthefollowinglinearmixedintegerprogram(CPLNF-IP).minxXa2AXk2Kackaxka+Xa2AXk2Kaskayka whereMisasucientlylargenumber. Intheaboveformulation,equality( 2{6 )makessurethat8a2A,thereisonlyone2Kasuchthatya=1andyka=0,8k2Ka,k6=.Thecorrectchoiceofdependsonthevalueofxaandhastosatisfyconstraint( 2{5 ).Inparticular,ifxa2[1a;a]thenfromconstraints( 2{5 )and( 2{6 ),itfollowsthatya=1.Asfortherestoftheconstraints,inequality( 2{7 )ensuresthatxka=0ifyka=0,andequalities( 2{3 )and( 2{4 )makesurethatthedemandissatisedandthesumofxkaoverallindicesk2Kaisequaltotheowonarca.Inaddition,itiseasytoshowthattheobjectiveoftheproblemisequivalenttotheobjectiveofCPLNFandoneconcludesthattheCPLNFandtheCPLNF-IPproblemsareequivalent. ConsiderarelaxationoftheCPLNF-IPproblemwhereconstraint( 2{7 )andtheintegralityofykaarereplacedby

PAGE 21

andyka0,respectively.Observethatintheresultingproblemconstraint( 2{4 )isredundantandfollowsfrom( 2{6 )and( 2{9 );therefore,itcanberemovedfromtheformulation.Inaddition,noticethatonecanremovethevariablexkafromtheformulationaswellbysubstituting( 2{9 )intotheobjectivefunction.ThemathematicaldescriptionoftheresultingproblemisprovidedbelowandwerefertotherelaxationproblemasCPLNF-R.minx;yg(x;y)=Xa2A"Xk2Kackayka#xa+Xa2AXk2Kaskayka=Xa2AXk2Kafka(xa)yka 2{10 )-( 2{13 )arepresentintheCPLNF-IPproblem.Therefore,anyfeasiblevectoroftheCPLNF-IPproblemsatisesconstraints( 2{10 )-( 2{13 ).

PAGE 22

thefollowingformminfykajk2Kag"Xk2Kackayka#xa+Xk2Kaskayka=Xk2Ka[ckaxa+ska]yka Letxa2[k1a;ka].Aswehavementionedbefore,fa(xa)=mink2Kaffka(xa)g;thereforefa(xa)=mink2Kaffka(xa)g=mink2Kafckaxa+skag=ckaxa+ska: 2{14 )becausexa2[k1a;ka]and(ii)^ya=argminfPk2Ka[ckaxa+ska]ykajPk2Kayka=1;yka0g.Basedontheabove,oneconcludesthat^yaisanoptimalsolutionoftheproblem.Ifxa2(k1a;ka)then^yistheuniquesolutionoftheproblembecauseckaxa+ska>ckaxa+ska,8k2Ka,k6=k;therefore,y=^y.Ifxa=k1aorxa=ka,thereareexactlytwobinarysolutionsoftheproblem,andbothhavethesameobjectivefunctionvalue.Asaresult,eitheronecanbeusedtoconstructabinarysolution^y.Asimilarresultholdsforallarcsa2A.Regardingvariablexka,given(x;y),theonlyfeasibleoneisxka=xaandxka=0,8k2Ka,k6=k. 2.2.2 ,itfollowsthataglobaloptimumofCPLNF-RiseitherfeasibletoCPLNF-IPorcanbeusedtoconstructafeasiblesolutionwiththesameobjectivefunctionvalue.SinceallfeasiblevectorsofCPLNF-IParefeasibleto

PAGE 23

CPLNF-R(seeLemma 2.2.1 ),oneconcludesthataglobalsolutionofCPLNF-RleadstoasolutionofCPLNF-IP. 2.2.1 ,itfollowsthatsolvingtheCPLNF-IPproblemisequivalenttondingaglobaloptimumofthebilinearproblemCPLNF-R.IfthesolutionofCPLNF-RisnotfeasibletoCPLNF-IPthentheproofofLemma 2.2.2 providesaneasywaytoconstructafeasiblesolutionwiththesameobjectivefunctionvalue.OtherpropertiesoflocalminimaoftheCPLNF-RproblemarediscussedinSection 2.4 ItisnoticedthattheCPLNF-Rproblemhasthefollowingeconomicinterpretation.Observethatbecauseofequality( 2{12 ),yka2[0;1],andonecaninterpretthevariablesykaasweights.Underthisassumption,onecanviewtheobjectivefunctionasthesumoftheweightedaveragesofthevariablecostsmultipliedbytheow,Pk2Kackaykaxa,andthexedcosts,Pk2Kaskayka.Inotherwords,theobjectivefunctionconsistsoftheweightedaveragesoffunctionsfka(xa).However,theweightshavetosatisfyconstraint( 2{11 ),wheretheow,xa,isboundedbytheweightedaveragesoftheleftandtherightendsoftheintervals[k1a;ka],k2Ka.AccordingtoLemma 2.2.2 ,alocal(global)optimumleadstoasolutionwheretheweightsareeitherequal0or1. 2{1 )isreplacedbyarequirementx2XRn. wherethefi(xi)arepiecewiselinearconcavefunctions,i.e.,

PAGE 24

2{10 )byx2X,i.e., 2.2.1 and 2.2.2 ,andTheorem 2.2.1 ,arestillvalid.Asaresult,oneconcludesthattheCPLPwSOFandtheCPLPwSOF-RproblemsareequivalentinthesensethatasolutionoftheCPLPwSOFproblemcanbeeasilyconstructedfromaglobalsolutionoftheCPLPwSOF-Rproblem. IfthesetXisapolytopethenCPLPwSOF-Risabilinearprogramwithajointlyconstrainedlinearfeasibleregion.LetY=fyjPk2Kiyki=1;yki0gandX+=fxjx2X;xi2[0i;nii]g.DenotebyV(X+)andV(Y)thesetsofverticesofthepolytopesX+andY,respectively.NoticethatthesetsX+andYare\joined"bytheconstraintsPk2Kik1iykixiPk2Kikiyki.Itiswellknownthatanoptimalsolutionofageneralbilinearprogramwithjointly

PAGE 25

constrainedfeasibleregionoccursattheboundaryofthefeasibleregionandisnotnecessarilyavertex(seeSection3.2.2,Horstetal.[ 54 ]andtherelatedproblemset).However,CPLPwSOF-RisequivalenttoCPLPwSOF.Inparticular,if(x;y)isaglobalsolutionofCPLPwSOF-RthenfromTheorem 2.2.1 ,itfollowsthatxisasolutionofCPLPwSOF.Thelatterisaconcaveminimizationproblemwherethefeasibleregionisapolytope.Itiswellknownthatthesolutionofsuchaconcaveminimizationproblemisoneoftheverticesofthepolytope;thereforex2V(X+).Inaddition,fromthetheoremitfollowsthatthereexists^y2V(Y)suchthat(x;^y)isaglobalsolutionofCPLPwSOF-Rproblem.Inthatsense,CPLPwSOF-Risaweaklyjoinedbilinearprogram.Theabovediscussionissummarizedinthefollowingtwotheorems. ConsiderthefollowingtwolinearproblemswhichwerefertoasLP(y)andLP(x),wherey(x)denotetheparameteroftheproblemLP(y)(LP(x)),i.e.,xedtoaparticularvalue.

PAGE 26

s.t.Bx=bxa2[0;naa] IntheLP(y)problemweassumethatvariablesykaaregiven,andthexaaretheonlydecisionvariables.Similarly,intheLP(x)problem,thexaaregiven,andtheykaarethedecisionvariables.AswehaveshownintheproofofLemma 2.2.2 ,problemLP(x)canbedecomposedintojAjproblemsandthesolutionsofthedecomposedproblemsarebinaryvectors,whichsatisfyconstraint( 2{14 ).Therefore,givenvectorx,asolutionoftheLP(x)problemcanbefoundbyasimplesearchtechniquewhereyka=1ifxa2[k1a;ka]. Weproposeadynamiccostupdatingprocedure(DCUP),whereoneconsiderssolvingtheproblemsLP(x)andLP(y)iteratively,usingthesolutionofoneproblemasaparameterfortheother(seeProcedure 1 ).Althoughintheproceduretheinitialvectory0,issuchthaty10a=1andyk0a=0,8k2Ka,k6=1,onecanchooseanyotherbinaryvector,thatsatisesconstraint( 2{12 ).Itisnoticedthatasimilariterativeprocedurehasbeenusedforsolvingabilinearprogramwithadisjointfeasibleregion(see,e.g.,Horstetal.[ 54 ]andHorstandTuy[ 55 ]).In

PAGE 27

theDCUP,LP(y)doesnotincludeconstraint( 2{11 ).Inotherwords,L(x)istheCPLNF-Rproblemwithxedvariableyandrelaxedconstraint( 2{11 ).Thelatterallowsusingtheiterativeproceduretosolvebilinearprogramswithaweaklyjoinedfeasibleregion.LetVrepresentthefeasibleregionofCPLNF-Rand(x;y)bethesolutionoftheDCUP. 54 ]. 2.2.2 wehaveshownthatyisnotuniqueifandonlyifoneofthecomponentsofvectorx,xa,isequaltothevalueofoneofthebreakpointska.However,observethatx=argminfLP(y)g,andthefeasibleregionofproblemLP(y)doesnotinvolvebreakpoints.Asaresult,inpracticeitisunlikelythatxaisequaltooneofthebreakpoints. 2{12 ),DCUPconvergesinanitenumberofiterations.

PAGE 28

Observethatineachiterationtheprocedurechangesthebinaryvectory.Ifym=ym1thentheprocedurestopsandg(xm;ym1)=g(xm;ym)=g(xm+1;ym).Ifthereexistm1andm2suchthatm11>m2andym1=ym2,thenxm1+1=argminfLP(ym1)g=argminfLP(ym2)g=xm2+1,i.e.,g(xm2+1;ym2)=g(xm2+1;ym1)=g(xm1+1;ym1).Fromthenon-increasingpropertyofthesequenceitfollowsthatg(xm2+1;ym2)=g(xm2+1;ym2+1);therefore,ym2=ym2+1andthealgorithmmuststoponiterationm2.Fromthelatteritfollowsthatallvectorsym,constructedbytheprocedurebeforeitstops,aredierent.Sincethesetofbinaryvectorsyisnite,oneconcludesthattheprocedureconvergesinanitenumberofiterations. 61 ]and[ 62 ]). AlthoughintheCPLNFproblemtherearenorestrictionsonthevaluesofparameterss1aand0a,bysubtractings1afromfunctionfa(xa)andreplacingthevariablexaby^xa=xa1a,onecantransformtheproblemintoanequivalentone

PAGE 29

wheres1a=0and0a=0.Therefore,withoutlossofgenerality,weassumethats1a=0and0a=0. ToinvestigateDSSP,letFa(xa)=8><>:fa(xa) whereMisasucientlylargenumber.ConsiderthefollowingnetworkowproblemwithowdependentcostfunctionsFa(xa)(NFPwFDCF).minxFT(x)x whereF(x)isthevectoroffunctionsFa(xa). 2 ).Thenititerativelyupdates

PAGE 30

thevalueofthecostvectorusingthefunctionFa(xa),Fma=Fa(xm1a),andsolvestheresultingNFPwFDCF(Fm)problem.Inthecostupdatingprocedure,dierentvariationsofthealgorithmusedierentvaluesforM.Inparticular,onemayconsiderreplacingMbyFm1aormaxn
PAGE 31

Thesetofproblemsisdividedintovegroupsthatcorrespondtothenetworkswithdierentsizesandnumbersofsupply/demandnodes.Foreachgroupwerandomlygeneratethreetypesofdemand;U[10;20],U[20;30],orU[30;40],andconsider5or10linearpieces(seeTable A{1 ,Appendix A ).InKimandPardalos[ 62 ]theauthorsconsiderincreasingconcavepiecewiselinearcostfunctionsforexperiments.AlthoughthebilinearreductiontechniqueaswellasDCUParevalidforanyconcavepiecewiselinearfunction,toremainimpartialforcomparisonwegeneratesimilarincreasingcostfunctions.Doingso,rstforeacharcwerandomlygenerateaconcavequadraticfunctionoftheformg(x)=x2+x.Noticethatthemaximumofthefunctionisreachedatthepointx= A Sets1-18havearelativelysmallnetworksize,anditispossibletosolvethemexactlyusingCPLEX(seeTable A{2 ).TherelativeerrorsforthosesetsarecomputedusingthefollowingformulasREDCUP(%)=fDCUPfexact

PAGE 32

A{3 ,columnsB,C,andDdescribethepercentageofproblemswhereDCUPisbetterthanDSSP,DSSPisbetterthanDCUP,andtheyarethesame,respectively.ThenumbersincolumnAaretheaverages(maximumvalues)ofthenumbersREDSSPREDCUP,givenREDSSPREDCUP>0.AccordingtothenumericalexperimentsDCUPprovidesabettersolutionthanDSSPinabout41%oftheproblemsandthesamesolutionin36%ofproblems.AlsonoticethatDCUPrequiresfeweriterationstoconvergeandconsumeslessCPUtime.RegardingCPLEX,thecomputationaltimevariesfromseveralsecondsinthesets1-6toseveralthousandsofsecondsinthesets13-18. Inthecaseoftheproblemsets19-30,CPLEXisnotabletondanexactsolutionwithin10,000secondsofCPUtime,andthebestfoundsolutionisnotbetterthantheoneprovidedbytheheuristics;therefore,wecomparetheresultsofDCUPversusDSSP.InTable A{4 ,columnsBandDdescribethepercentageofproblemswhereDCUPisbetterthanDSSP,andDSSPisbetterthanDCUP,respectively.ThenumbersincolumnsAandCarecomputedbasedontheformulafDSSPfDCUP Intheabovenumericalexperiments,wehaveusedthevectory0(8a2A,y10a=1andyk0a=0,8k2Ka,k6=1)asaninitialbinaryvector.However,DCUPcanstartfromanyotherbinaryvectorthatsatisesconstraint( 2{12 ).Inparticular,onecanconsidersthesolutionofDSSPasaninitialvectoranduseDCUPtoimprovethesolution.Table A{5 comparestheresultsofDCUPversus

PAGE 33

DSSPwherecolumnAissimilartotheoneinTable A{4 (i.e.thenumbersinthecolumnarecomputedbasedontheformulafDSSPfDCUP Basedonthetheoreticalresults,wehavedevelopedaniteconvergentalgorithmtondalocalminimumofthebilinearrelaxation.ThecomputationalresultsshowthatthedynamiccostupdatingprocedureisabletondanearoptimumoranexactsolutionoftheproblemusinglessofCPUtimethanCPLEX.Inaddition,wecomparethequalityofthesolutionandtherunningtimewiththedynamicslopescalingprocedure.SinceDCUPisfast,onecanaimtondtheglobalminimumbyrandomlygeneratingtheinitialbinaryvectorandrunningDCUP.Inaddition,DCUPcanbeusedincuttingplanealgorithmsforndinganexactsolution.

PAGE 34

41 ]). TheFCNFproblemiswellknowntobeNP-Hardandbelongstotheclassofconcaveminimizationproblems.Theproblemcanbemodeledasa0-1mixedintegerlinearprogram(seeHirschandDantzig[ 50 ])andmostsolutionapproachesutilizebranch-and-boundtechniquestondanexactsolution(seeBarretal.[ 4 ],CabotandErenguc[ 12 ],Gray[ 44 ],KenningtonandUnger[ 59 ],andPalekaretal.[ 83 ]).Sincetheconcaveminimizationproblemattainsasolutionatoneoftheverticesofthefeasibleregion,Murty[ 76 ]proposedavertexrankingproceduretosolvetheproblem.However,ndinganexactsolutioniscomputationallyexpensiveanditisnotpracticalforsolvinglargeproblems.SomeheuristicproceduresarediscussedinCooperandDrebes[ 27 ],Diaby[ 31 ],KhangandFujiwara[ 60 ],andKuhnandBaumol[ 63 ].RecentlyKimandPardalos[ 61 ](seealsoKimandPardalos[ 62 ])proposedaheuristicalgorithm,DynamicSlopeScalingProcedure(DSSP),tosolvethexedchargenetworkowproblem.Theproceduresolvesasequenceoflinearproblems,wheretheslopeofthecostfunctionisupdatedbased 22

PAGE 35

onthesolutionofthepreviousiteration.ThealgorithmisknowntobeoneofthebestheuristicprocedurestosoleFCNFproblems. Notethatallapproachestosolvetheproblemarebasedonlinearapproximationtechniques.Instead,weapproximateFCNFbyaconcavepiecewiselinearnetworkowproblem(CPLNF),wherethecostfunctionshavetwolinearpieces.AproperchoiceoftheapproximationparameterensurestheequivalencebetweenFCNFandtheresultingCPLNFproblem.However,ndingtheproperparameteriscomputationallyexpensive;therefore,weproposeanalgorithmthatsolvesasequenceofCPLNFproblemsbygraduallydecreasingtheparameteroftheproblem.WeprovethatthestoppingcriteriaofthealgorithmisconsistentinthesensethatasolutionofthelastCPLNFprobleminthesequenceisasolutionoftheFCNFproblem. Despitetheabovementionedtheoreticalresults,thealgorithmrequiresndingexactsolutionsoftheCPLNFproblems,whichareNP-Hard(seeGuisewiteandPardalos[ 45 ]).InChapter 2 (seealsoNahapetyanandPardalos[ 79 ]),wehaveshownthattheCPLNFproblemisequivalenttoabilinearprogram.Inaddition,wehaveproposedaniteconvergentdynamiccostupdatingprocedure(DCUP)tondalocalminimumoftheresultingbilinearprogram.TosolvetheFCNFproblem,inthealgorithmonetransformstheCPLNFproblemsintoequivalentbilinearprogramsandusestheDCUPtosolvetheresultingproblems.Werefertothecombinedalgorithmastheadaptivedynamiccostupdatingprocedure(ADCUP). SimilartotheresultpresentedintheChapter 2 ,weprovethatthesolutionprovidedbyDSSPisasolutiontoavariationalinequalityproblem,whichisformulatedbasedonthefeasibleregionoftheFCNFproblem.Althoughingeneralanequilibriumsolutionandasystemsolutionarenotthesame,thedierencebetweentheobjectivefunctionvaluesofthesolutionscanbefairlysmall.On

PAGE 36

theotherhand,ADCUPisaheuristicprocedureforndingasystemoptimumsolution.Tocomparethesetwoprocedures,weconductnumericalexperimentson36problemssetsfordierentnetworksandchoicesofcostfunctions.Thereare30randomlygeneratedproblemsforeachproblemset.Intheexperiments,wecompareADCUPversusDSSPintermsofthequalityofthesolutionaswellasCPUtime.Inaddition,forsmallnetworkswendanexactsolutionoftheproblemsusingMIPsolversofCPLEXandcomputerelativeerrors.ThecomputationalresultsshowthatADCUPprovidesanearoptimumsolutionusinganegligibleamountofCPUtime.Inaddition,theprocedureoutperformsDSSPinthequalityofthesolutionaswellasCPUtime.Thedierencebetweensolutionsismorenoticeableinthecasesofsmallgeneralslopesandlargexedcosts. Fortheremainder,Section 3.2 discussestheapproximationtechniqueandestablishestheequivalencebetweentheFCNFproblemandaCPLNFproblemwithaspecialstructure.AsolutionalgorithmforsolvingtheFCNFproblemisprovidedinSection 3.3 .SomepropertiesoftheDSSPareintroducedinSection 3.4 .TheresultsofnumericalexperimentsonADCUParesummarizedinSection 3.5 ,andnally,Section 3.6 concludesthechapter. ConsiderageneralxedchargenetworkowproblemconstructedonanetworkG(N;A),whereNandAdenotethesetsofnodesandarcs,respectively.Letfa(xa)denotethecostfunctionofarca2A,and

PAGE 37

Figure3{1. Approximationoffunctionfa(xa). FCNF:minxf(x)=Xa2Afa(xa)s.t.Bx=b;xa2[0;a];8a2A; Observethatthecostfunctionisdiscontinuousattheoriginandlinearontheinterval(0;a].Althoughweassumethattheowsonthearcsareboundedbya,theboundscanbereplacedbyasucientlylargeM,andtheproblemtransformsintoanunboundedone. Let"a2(0;a],and"aa(xa)=8><>:caxa+saxa2["a;a]c"aaxaxa2[0;"a)

PAGE 38

wherec"aa=ca+sa="a.Observethat"aa(xa)=fa(xa),8xa2f0gS["a;a]and"aa(xa)
PAGE 39

3.2.2 makessurethatbychoosingasucientlysmall">0(e.g.,"a=,8a2A),bothproblemshavethesamesolution;therefore,FCNFisequivalenttoaconcavepiecewiselinearnetworkowproblem. ConsiderProcedure 3 .InStep1,theprocedureassignsinitialvaluesfor"a.Step2solvestheresultingCPLNFproblem.NoticethatCPLNF("1)isindeedalinearproblem,because["1a;a]=fag.If9a2Asuchthatxma2(0;"ma),wedecreasethevalueof"ato"a,whereisaconstantfromtheopeninterval(0;1).Assumethattheprocedurestopsatiterationkandletxk=argminfCPLNF("k)g.

PAGE 40

Figure3{2. thedenitionof"-approximationitfollowsthat"aa(xa)"kaa(xa),8a2Aandxa2[0;a](seeFigure 3{2 ),andthesecondinequalityfollows. Observethatbecauseofthestoppingcriteria,xka=0orxka2["ka;a].Since"a<"ka,"aa(xa)="kaa(xa),8a2Aandxa2f0gS["ka;a];therefore,"(xk)="k(xk).Thelattertogetherwith( 3{1 )insuresthat"(x")="k(xk).Sincebothproblems,CPLNF(")andCPLNF("k),havethesameobjectivefunctionvalueatx"andxk,oneconcludesthattheyareequivalent. 3.3.1 itfollowsthat8"suchthat0<"a<"ka,8a2A,theCPLNF(")andtheCPLNF("k)problemshavethesamesetofsolutions.Ontheotherhand,bychoosing0<"a
PAGE 41

minimumoftheproblemandcanbeusedinStep2ofProcedure 3 tondasolutionoftheCPLNF("m)problem.TheresultingalgorithmissummarizedinProcedure 4 ,whichwerefertoasadaptivedynamiccostupdatingprocedure(ADCUP).Belowweprovidethemathematicalformulationofthebilinearproblem,whichisequivalenttoCPLNF("m).Fordetailsontheformulationoftheproblem,niteconvergenceandotherpropertiesoftheDCUPwerefertoChapter 2 ProblemCPLNF-R("m)isdenedby:minx;yXa2Ac"maay"maa+cayaxa+Xa2Asayas.t.Bx=b;"mayaxa"may"maa+aya;8a2A;y"maa+ya=1;8a2A;xa0;y"maa0;andya0;8a2A: 3.5 ).

PAGE 42

61 ](seealsoKimandPardalos[ 62 ]).Inthepaper,theauthorsdiscussfourvariationsofDSSPbasedonthechoiceoftheinitialvectorandtheslopeupdatingscheme.However,regardlessoftheinitialvectorandtheupdatingscheme,DSSPprovidesanequilibriumtypeofsolution.Toprovethestatement,wersttransformFCNFintoanalternativeproblemthenprovethatthesolutionprovidedbyDSSPisasolutionofavariationalinequalityproblemconstructedbasedonthenewformulation.ThetheoreticalresultsprovidedbelowareverysimilartothoseinChapter 2 ,wherewehaveshownthatthepropertyholdsfortheconcavepiecewiselinearnetworkowproblem. LetFa(xa)=8><>:fa(xa) whereMisasucientlylargenumber.ConsiderthefollowingnetworkowproblemwithowdependentcostfunctionsFa(xa). NFPwFDCF:minxFT(x)x whereF(x)isthevectoroffunctionsFa(xa). 3.4.1 ,Chapter 2 )

PAGE 43

61 ],theauthorsproposedierentupdatingschemes,wheretheyreplaceMbyasmallervalue.However,thenexttheoremprovesthatregardlessoftheinitialvectorF0andtheupdatingscheme,thenalsolutionprovidedbyDSSPisasolutionofavariationalinequalityproblem. ndxfeasibleto( 3{2 )and( 3{3 )suchthatFT(x)(xx)0,8xfeasibleto( 3{2 )and( 3{3 ) 3.4.2 itfollowsthatthesolutionofDSSP,x,satisestheinequalityFT(x)xFT(x)x,8xfeasibleto( 3{2 )and( 3{3 ),i.e.,xisanequilibriumsolution.However,sinceNFPwFDCFisequivalenttotheFCNF(seeTheorem 3.4.1 ),oneisinterestedinndingafeasible^xsuchthatFT(x)xFT(^x)^x,8xfeasibleto( 3{2 )and( 3{3 ),i.e.,^xisasystemoptimumsolution.Noticethattheequilibriumandthesystemoptimumsolutionsmaynotbethesame,unlessFa(xa)isconstant.

PAGE 44

KimandPardalos[ 61 ]).Tocomparetheresultsoftheheuristicprocedures,inthecaseofsmallproblemswendanexactsolutionusingCPLEXMIPsolverandcomputerelativeerrors.Inthecaseoflargeproblems,CPLEXisnotabletondanexactsolutionwithin5,000secondsofCPUtime;therefore,wecomparetheresultsofDCUPversusDSSP. Intheexperiments,wesolveproblemsusingallfourvariationsofDSSPandchoosethebestsolutiontocomparewiththesolutionprovidedbytheADCUP.Inadditiontothenalsolution(thesolutionthatthealgorithmreturnswhenitstops),duringtheiterativeprocessDSSPaswellasADCUPconstructfeasiblevectorsthatmighthaveabetterobjectivefunctionvalue.Intheprocedures,werecordthosevectorsandchoosethebestone.Thecomparisonbetweenthebestsolutionsofbothalgorithmsisalsoprovided.Withregardtothecomputationaltime,wecomparetheCPUtimeofADCUPversusthebestCPUtimeamongfourvariationsofDSSP. Therearefourgroupsoftestproblemsbasedonthesizeofthenetworkandthenumberofsupply/demandnodes(seeTable B{1 ,Appendix B ).Foreachgroup,weconstructdierenttypesoffunctionsfa(xa),wheretheslopeandthexedcostaregeneratedrandomlyaccordingtothespecieddistributions.Intotal,thereareninesetsofproblems(ninetypesoffunctionfa(xa))foreachgroup,i.e.,onesetofproblemsforeachchoiceofdistributionfortheslopeandthexedcost.Thereare30randomlygeneratedproblemsforeachproblemset.Thecomponentsofthesupply/demandvectoraregenerateduniformlybetween30and50units.ThemodelisconstructedusingtheGAMSenvironmentandsolvedbyCPLEX9.0.ComputationsweremadeonaUnixmachinewithdualPentium43.2Ghzprocessorsand6GBofmemory.AllresultsaretabulatedintheAppendix B Tables B{2 and B{3 illustratethecomputationalresultsforgroupsG1andG2.Sincethesizeofthoseproblemsisnotbig,wehavesolvedtheproblemsexactly

PAGE 45

usingtheCPLEXMIPsolver.TherelativeerrorsarecomputedbasedonthefollowingformulasREADCUP(%)=fADCUPfexact InthecaseofgroupsG3andG4,wecompareADCUPversusDSSPusingthefollowingformulaDSSPADCUP(%)=fDSSPfADCUP B{4 .Similartotheprevioustwogroups,oneobservesthatonaverageADCUPprovidesabettersolutionthanDSSP.NoticethatDSSPconsumesmuchmoreCPUtimebeforeterminationthanADCUP.Inaddition,thepercentageofproblemswhereADCUPprovidesabettersolutionthanDSSPishigherthaninthepreviouscases.SimilartogroupsG1andG2,thedierencebetweenthesolutionsprovidedby

PAGE 46

bothalgorithmsissmallfortheproblemsetswithalargerslopeandsmallerxedcost.Whentheslopedecreases(orthexedcostincreases),ADCUPprovidesaperceptiblybettersolutionthanDSSP. Inthenumericalexperiments,wehaveshownthattherelativeerrorofthesolutionsofbothproceduresincreasesinthecasesofsmallslopesandlargexedcosts.Toexplainthisphenomena,observethatbydecreasingthevalueoftheslopetheanglebetweenfunctionfa(xa)andtherstlinearpieceoffunction"aa(xa)increases(seeFigure 3{1 ).Asaresult,"aa(xa)doesnotapproximatethefunctionfa(xa)aswellasinthecaseoflargevariablecosts.Thesamediscussionappliestothecaseofalargeslope.

PAGE 47

43 ],FlorianandKlein[ 36 ],vanHoeselandWagelmans[ 53 ],Loparicetal.[ 70 ],andLoparicetal.[ 71 ]).Itiswellknownthatuncapacitatedproblemscanbereducedtoashortestpathproblem.FlorianandKlein[ 36 ]studiedcapacitatedsingle-itemproblems,wheretheycharacterizedtheoptimalsolutionandproposedasimpledynamicprogrammingalgorithmforproblemsinwhichthecapacitiesarethesameineveryperiod.Thesingle-itemproblemswithvaryingcapacitiesareknowntobeNP-hard.AclassicationofdierentproblemsandasurveyonexistenceofapolynomialalgorithmforsolvingproblemsfordierentclassescanbefoundinWolsey[ 100 ]andPochetandWolsey[ 88 ].TightformulationsforpolynomiallysolvableproblemsarediscussedinMillerandWolsey[ 75 ]andPochetandWolsey[ 88 ]. Almostallpracticalproblemsinvolvemultipleitems,machinesand/orlevels,andpolynomialresultsforthoseproblemsarelimited.Usingbinaryvariables,onecanconstructamixedintegerlinearprogramming(MIP)formulationoftheproblemwithanimbeddednetworkstructure.Tosolvetheproblem,branch-and-boundandcuttingplanealgorithmshavebeenused(see,e.g.,Barretal.[ 4 ],CabotandErenguc[ 12 ],Gray[ 44 ],KenningtonandUnger[ 59 ],Palekar 35

PAGE 48

etal.[ 83 ],Marchandetal.[ 72 ],andWolsey[ 100 ]).Inaddition,severalheuristicalgorithmshavebeenproposed(see,e.g.,CooperandDrebes[ 27 ],Diaby[ 31 ],KhangandFujiwara[ 60 ],KuhnandBaumol[ 63 ],vanHoeselandWagelmans[ 52 ],KimandPardalos[ 61 ]and[ 62 ],NahapetyanandPardalos[ 79 ]and[ 80 ]). Inthischapterwediscussacapacitatedmulti-itemdynamicpricing(CMDP)problemwhereonemaximizestheprotbychoosingaproperproductionlevelaswellaspricingpolicyforeachproduct.Intheproblem,thedemandisadecisionvariable,andinordertosatisfyahigherdemandoneneedstoreducethepriceoftheproduct.Ontheotherhand,reducingthepricecandecreasetherevenue,whichistheproductofthedemandandtheprice.Inaddition,theproblemincludesaninventoryandproductioncostforeachproduct,wherethelatterinvolvesasetupcost.Theobjectiveoftheproblemistondanoptimalproductionstrategy,whichmaximizestheprotsubjecttoproductioncapacitiesthatare\shared"bytheproducts.Dierentvariationsofasingle-itemuncapacitatedproblemwithdeterministicdemandsarediscussedbyGilbert[ 43 ],Loparicetal.[ 71 ],andThomas[ 96 ].Acapacitatedsingle-itemproblemwithtimeinvariantcapacitiesisdiscussedinGeunesetal.[ 42 ].Thepolynomialalgorithmsproposedbytheauthorsarebasedonthecorrespondingresultsforthelot-sizingproblems. InChapters 2 and 3 (seealsoNahapetyanandPardalos[ 79 ]and[ 80 ])wehaveproposedabilinearreductiontechnique,whichcanbeusedtondanapproximatesolutionofconcavepiecewiselinearandxedchargenetworkowproblems.AsimilartechniqueisproposedtosolvetheCMDPproblem.Inparticular,weconsiderabilinearreductiontechniqueoftheproblemandprovethatsolvingtheCMDPproblemisequivalenttondingaglobalmaximumofthebilinearproblem.Thelatterbelongstotheclassofbilinearproblemswithdisjointfeasibleregion,andoneconsidersaheuristicalgorithmtondasolutionoftheproblem.Theheuristicalgorithmemploysawellknowniterativeprocedureforndingalocal

PAGE 49

maximumoftheproblem(see,e.g.,Horstetal.[ 54 ]andHorstandTuy[ 55 ]).Numericalexperimentsonrandomlygeneratedproblemsconrmtheeciencyofthealgorithm. Fortheremainder,Section 4.2 providesalinearmixedintegerformulationoftheproblemanddiscussesabilinearreductionoftheproblem.WeprovethatsolvingtheCMDPproblemisequivalenttondingaglobalmaximumofthebilinearreduction.InSection 4.3 weproposeaheuristicalgorithmforsolvingthebilinearproblem.NumericalexperimentsonthealgorithmareprovidedinSection 4.4 ,andnally,Section 4.5 concludesthechapter. LetPandrepresentthesetofproductsanddiscretetimes,respectively.Inaddition,letf(p;j)(d)denotethepriceofproductpattimejasafunctionofthedemandd,andg(p;j)(d)=f(p;j)(d)d,i.e.,g(p;j)(d)representstherevenueobtainedfromsellingdamountofproductpattimej.Intheproblem,weassumethatf(p;j)(d)andg(p;j)(d)arenonincreasingandconcavefunctions,respectively(seeFigures 4{1 ).Iff(p;j)(d)isaconcavefunction,thenitiseasytoshowthatconcavityofg(p;j)(d)follows. Letx(p;i;j)denoteanamountofproductpthatisproducedattimeitosatisfythedemandattimej,andy(p;i)representabinaryvariable,whichequalsoneifproductpisproducedattimeiandzerootherwise.Assumethatinventorycosts,cin(p;i;j),productioncosts,cpr(p;i),andsetupcosts,cst(p;i),aswellasproductioncapacities,Ci,aregiven,wherep,i,andjrepresenttheproduct,producingtime,

PAGE 50

Figure4{1. Thepriceandtherevenuefunctions. andsellingtime,respectively.ThefollowingisthemathematicalformulationoftheCMDPproblem. Althoughtheaboveformulationbelongstotheclassofnonlinearmixedintegerprograms,usingstandardtechniquesonecanapproximatetherevenuefunctionbyapiecewiselinearoneandlinearizetheobjectivefunction.Doingso,observethatfromtheconcavityoftherevenuefunctionitfollowsthatthereexistsapoint,~d(p;j),wherethefunctionreachesitsmaximum(seeFigure 4{1 ).Asaresult,producing

PAGE 51

andsellingmorethan~d(p;j)isnotprotable,andatoptimalityPi2jijx(p;i;j)~d(p;j).Tolinearizetherevenuefunction,divideh0;~d(p;j)iintoNintervalsofequallength.Letdk(p;j)denotetheendpointsoftheintervals,i.e.,dk(p;j)=k~d(p;j)=N,8k2f1;:::;NgSf0g=KSf0g,andgk(p;j)representsthevalueoftherevenuefunctionatthepointdk(p;j),i.e.,gk(p;j)=g(p;j)(dk(p;j))=f(p;j)(dk(p;j))dk(p;j).Usingthoseparameters,constructthefunction~g(p;j)((p;j))=NXk=0gk(p;j)k(p;j)=NXk=1gk(p;j)k(p;j); 5 ]).Thefollowingisthemathematicalformulationoftheapproximationproblem:maxx;y;Xp2PXj2Xk2Kgk(p;j)k(p;j)Xp2PXi;j2jijhcin(p;i;j)+cpr(p;i)ix(p;i;j)Xp2pXi2cst(p;i)y(p;i);s.t.Xp2PXj2jijx(p;i;j)Ci;8i2;Xj2jijx(p;i;j)Ciy(p;i);8p2Pandi2;Xi2jijx(p;i;j)=Xk2Kdk(p;j)k(p;j);8p2Pandj2;

PAGE 53

whereqk(p;i;j)=fk(p;j)cin(p;i;j)cpr(p;i).Observe,thatatoptimalityxk(p;i;j)=0forallindicessuchthatqk(p;i;j)0,andthosevariablescanberemovedfromtheformulation.Therefore,withoutlostofgenerality,intheanalysisbelow,weassumethatqk(p;i;j)>0. DeneX=fxjx0andxk(p;i;j)arefeasibleto( 4{1 )and( 4{3 )g,andY=[0;1]jPjjj.Considerthefollowingbilinearprogram: 4{1 ),

PAGE 54

( 4{3 )and( 4{4 ).If(x;^y)violatesconstraint( 4{2 )then9p2Pandi2suchthatPj2jijPk2Kxk(p;i;j)>0and^y(p;i)=0.Fromthelocaloptimalityof(x;^y)itfollowsthatPj2jijPk2Kqk(p;i;j)xk(p;i;j)cst(p;i)0,anditisnotprotabletoproduceproductpattimei.Furthermore,byassigningxk(p;i;j)=0,8j2andk2K,theobjectivefunctionvalueoftheACMDP-Bproblemremainsthesame.Let^xdenotetheresultingvector,i.e., ^xk(p;i;j)=8><>:0ifPj2jijPk2Kqk(p;i;j)xk(p;i;j)cst(p;i)0xk(p;i;j)ifPj2jijPk2Kqk(p;i;j)xk(p;i;j)cst(p;i)>0(4{5) Thevector(^x;^y)isfeasibletotheACMDPaswellastheACMDP-Bproblemandhasthesameobjectivefunctionvalueas(x;y). 4{2 )andrelaxingtheintegralityofthevariabley(p;i).Inotherwords,theACMDP-BproblemisarelaxationoftheACMDPproblem.FromTheorem 4.2.1 itfollowsthataglobalsolutionofACMDP-Bisasolution(orleadstoasolution)oftheACMDPproblem.

PAGE 55

solvetheACMDP-BproblemandifthesolutionisnotfeasibletotheACMDPproblem,thenusethemethoddescribedintheproofofTheorem 4.2.1 toconstructafeasibleonewiththesameobjectivefunctionvalue. Observethattheproblembelongstotheclassofbilinearprograms.Byxingvectorxorytoaparticularvalue,theproblemcanbereducedtoalinearone.LetLP(x)andLP(y)denotethecorrespondinglinearprograms,i.e.,LP(x):maxy2YPp2PPi2hPj2jijPk2Kqk(p;i;j)xk(p;i;j)cst(p;i)iy(p;i),andLP(y):maxx2XPp2PPi2Pj2jijPk2Khqk(p;i;j)y(p;i)ixk(p;i;j). NoticethatthesolutionoftheLP(x)iseasytoobtain.Inparticular,y(p;i)=8><>:0ifPj2jijPk2Kqk(p;i;j)xk(p;i;j)cst(p;i)01ifPj2jijPk2Kqk(p;i;j)xk(p;i;j)cst(p;i)>0 isanoptimalsolutionoftheproblem.TheProcedure 5 describesawellknownalgorithm,whichstartsfromaninitialbinaryvectorandconvergestoalocalmaximumoftheACMDP-Bprobleminanitenumberofiterations(seeHorstandTuy[ 55 ]orHorstetal.[ 54 ]). However,theprocedurehasthefollowingdisadvantage.Let(xm;ym)representthesolutionobtainedoniterationm,andassumethat9p2Pandi2suchthatym(p;i)=0.Asaresult,intheLP(ym)problemqk(p;i;j)ym(p;i)=0,8j2,ij,and

PAGE 56

Let'1(p;i)(x(p;i))=Pj2jijPk2Kqk(p;i;j)xk(p;i;j)cst(p;i)and'2(p;i)(x(p;i))="(p;i)

PAGE 57

5 tondalocalmaximumoftheproblem,whereymisaninitialbinaryvector.Let(xm+1;ym+1)denotethelocalmaximum. Asbefore,thesolutionoftheLP"(x)problemiseasytoobtainbyassigningy(p;i)=8><>:0ifPj2jijPk2Kqk(p;i;j)xk(p;i;j)cst(p;i)"(p;i)1ifPj2jijPk2Kqk(p;i;j)xk(p;i;j)cst(p;i)>"(p;i)=8><>:0if'1(p;i)(x(p;i))'2(p;i)(x(p;i))1if'1(p;i)(x(p;i))>'2(p;i)(x(p;i)): 6 )startswithasucientlylarge"andndsalocalmaximumoftheresulting"-approximationproblem.IfthestoppingcriteriaisnotsatisedinStep3thenitdecreasesthevalueofthevector"to",whereisaconstantfromtheopeninterval(0;1),andtheprocesscontinuesusinganew"-approximationproblem.ObservethatProcedure 6 usesvectorymfromthepreviousiterationasaninitialvector. Theproceduredependsontwoparameters:theinitialvector"andthevalueof.Thevalueof"(p;i)dependsonparametersoftheproblem,andonecanconsiderthemequaltothemaximumprot,whichcanbeobtainedbyproducingonlyproductpattimei.AlthoughsuchmaximizationproblemiseasytosolveusingstandardLPsolvers,forlargejPjandjjonendscomputationallyexpensivesolvingtheproblemforallpairs(p;i)2P.Instead,wepropose

PAGE 58

analgorithmforndingthevaluesof"(p;i)(seeProcedure 7 ).Observethatxk(p;i;j)dk(p;j),andthemaximumadditionalprotthatcanbeobtainedusingthevariablexk(p;i;j)isqk(p;i;j)dk(p;j).Usingthisproperty,forallpairs(p;i)theprocedureiterativelyndsthemaximumamongqk(p;i;j)dk(p;j)andassignsthedemand(ortheremanningofthecapacity)tothecorrespondingvariablexk(p;i;j).Thevalueof"(p;i)iscomputedbasedontheformula"(p;i)=Pj2jijPk2Kqk(p;i;j)xk(p;i;j)cst(p;i).Asfortheparameter,itslargervalueincreasesthecomputationaltimeoftheprocedure,anditislikelytoprovideabettersolution. 5 aswellasProcedure 6 usingdierentvaluesfortheparameter.ThelatterprocedureemploysProcedure 7 tondaninitialvalueforthevector".Inaddition,wesolvetheproblemsbytheMIPsolverofCPLEXusingtheACMDPformulation.InthecaseswheretheMIPsolverisnotabletosolvelargeproblemswithinpostedCPUandmemorylimitations,wecomparethesolutionsoftheprocedureswiththebestsolutionsfoundbyCPLEX.Themainpurposeofthecomputationsistheperformanceoftheproceduresfordierentcapacities. Assignxk(p;i;j)=0,8p2P,i;j2,andk2K Assignxkmax(p;i;jmax)=minf^C;dkmax(p;jmax)g,^C=^Cxkmax(p;i;jmax),^qk(p;i;jmax)=0,8k2K,andqmax=maxf^qk(p;i;j)dk(p;j)jk2K;j2;jig

PAGE 59

Inthenumericalexperimentsweconsiderproblemsetswithdierentnumbersofproducts,jPj=5,10,or20,andtimehorizons,jj=12or52.Foreachproblemsetwerandomlygeneratecapacitiesforalli2usingtheformulaCi=jPjU,whereUisarandomnumberuniformlygeneratedfrominterval[10;100],[50;150],[100;200],or[150;250].Notethatallintervalsallowgeneratingcapacitiesthataretightatoptimalitywithrespecttotherevenuefunctiondiscussedbelow.Inaddition,usingtermjPjonegeneratescapacitiesthatdependonthenumberofproducts.Thelatterallowscomparingofresultsacrossdierentnumbersofproducts.Asforthecosts,wegeneratetheproductioncostscpr(p;i)andtheinventorycostscin(p;i;j)accordingtotheuniformdistributionsU[20;40]andU[4;8],respectively.Observethatonaveragetheinventorycostisequalto20%oftheproductioncost.Finallythesetupcostcst(p;i)isgenerateduniformlyfrominterval[600;1000]. Intheexperimentswerestrictourselfbyconsideringonlylinearpricefunctionsoftheformf(p;j)(d)=fmax(p;j)fmax(p;j)=dmax(p;j)d.Toavoidgeneratingfunctionsthatatoptimalityresultinunrealisticallylargeprots,weintroduceanindex,where=Pp2PPi2hPj2jijPk2Kfk(p;j)cin(p;i;j)cpr(p;i)xk(p;i;j)cst(p;i)y(p;i)i Pp2PPi2hPj2jijPk2K(cin(p;i;j)+cpr(p;i))xk(p;i;j)+cst(p;i)y(p;i)i: 4{1 ).Inaddition,theproposedpricefunctionanddistributionsofthecostsandcapacitiesallowgeneratingproblemsthathaveanoptimalobjectivefunctionvaluerangingfromhundredsofthousandstoseveral

PAGE 60

millions.Finally,intheconstructionofthepiecewiselinearapproximationoftherevenuefunctionweuseN=10. ThemodelisconstructedusingtheGAMSenvironmentandsolvedbyCPLEX9.0withaCPUrestrictionof2000secandamemoryrestrictionof2Gb,wherethelatteristhememorythatisrequiredtostorethetreeinthebranch-and-boundalgorithm.ComputationsaremadeonaUnixmachinewithdualPentium43.2Ghzprocessorsand6GBofmemory.TheresultsaretabulatedintheAppendix C Intheexperimentswesolve10randomlygeneratedproblemsforeachproblemsetandcapacity.Tables C{1 and C{2 comparetheresultsprovidedbyCPLEXwiththesolutionsprovidedbybothprocedures.TherelativeerroriscomputedusingtheformulaRE(%)=ObjCPLEXObjProc: ( 5 ) C{1 ,columnAindicatesthenumberofproblemswheretheheuristicprocedurendsabettersolutionthanCPLEX.NotethatCPLEXisabletoprovideanexactsolutionforallcapacitiesfromtheproblemset5-12.Inallothercases,thesolverstopsafterreachingtheCPUlimitorthememorylimitandreturnsthebestfoundsolution.Althoughtherelativeoptimalitygapofthenalsolutionsofthoseproblemsetsvariesfrom2%to5%,webelievethatthesolutionisanoptimalorclosetoanoptimalone,andthelargeoptimalitygapisduetoimperfectlowerbounds.Thefactthattheheuristicproceduresprovideaslightlybettersolutioninthecasewithjj=52thanjj=12partiallyconrmsourassumptions. TherelativeerrorsintheTable C{1 conrmstheeectivenessoftheheuristicprocedure.Inparticular,inthemajorityoftheproblemstheheuristicalgorithmisabletoprovideasolutionwithin1%fromtheoptimaloneorthebestoneprovidedbyCPLEX.Observethatthelargervalueofprovidesabettersolutionandthenumberofproblemswheretheheuristicprocedurendsabettersolutionthan

PAGE 61

CPLEXisincreasingwiththesizeoftheproblem.BycomparingwiththesolutionsprovidedbyProcedure 5 (seeTable C{2 )onenoticesthatProcedure 6 outperformstheProcedure 5 ,anditismorestablewithchangesinthecapacities.AsfortheCPUtime(seeTable C{3 ),theheuristicproceduresrequirefewerresourcesthanCPLEX.Inaddition,unlikeCPLEXtheheuristicproceduresdonotrequiregigabytesofmemorytostorethetree. 4.3 ,intheverybeginningoftheiterativeprocesstheprocedureeliminatessomeproductsfromthefurtherconsideration.Thelatterworsenthequalityofthesolutionreturnedbytheprocedure.Inthesecondprocedureweconstructapproximateproblemsandgraduallydecreaseparametersoftheproblems.Asaresult,duringtheiterativeprocessthecostsoftheeliminatedproductsremainpositiveandtheprocedureconsidersthemagainifneedbe.AlthoughthesecondprocedurerequiresmoreCPUtimetostopthantherstone,itprovidesahigher-qualitysolution.

PAGE 62

73 ]and[ 74 ])rstproposedtheirmodelin1978,therehavebeenanumberofpapers(see,e.g.,CareyandSubrahmanian[ 23 ],Carey[ 15 ],Carey[ 13 ],Carey[ 14 ],Friesz[ 37 ],Friesz[ 38 ],Wieetal.[ 98 ],ChenandHsueh[ 25 ],Janson[ 57 ],Ho[ 51 ],Ziliaskopoulos[ 104 ],Drissi-KaitouniandHameda-Benchekroun[ 32 ],Lietal.[ 66 ],Kaufmanetal.[ 58 ],Boyceetal.[ 11 ],RanandBoyce[ 90 ],Ranetal.[ 92 ],andWieetal.[ 99 ])discussingvariationalinequalityormathematicalprogrammingformulationsforthedynamictracassignmentproblemwiththeassumptionthattheplanninghorizonisasetofdiscretepointsinsteadofacontinuousinterval.Manyofthesepapersuseadynamicortime-expandednetwork(see,e.g.,Ahujaetal.[ 2 ])tosimultaneouslycapturethetopologyofthetransportationnetworkandtheevolutionoftracovertime.Implicitlyorotherwise,thesepaperstypicallyassumethatthereisnotracatthebeginningoftheplanninghorizon(orattimezero)andthatalltripsmustexitthenetworkpriortotheend.Whentherearecarsatthetimezero,thetimesatwhichthesecarsenterthenetworkmustbeknowninordertodeterminewhentheywillexitthearcsonwhichtheyweretravelling.Inpractice,datawithsuchdetailsdonotgenerallyexist. Therearetwomainfactorsthatdistinguishthemodelsinpapersreferencedabove.First,some(e.g.,MerchantandNemhauser[ 73 ],CareyandSubrahmanian[ 23 ],Ho[ 51 ],Carey[ 15 ],Ziliaskopoulos[ 104 ],Kaufmanetal.[ 58 ],Garciaetal.[ 40 ])seekasystemoptimalsolutionandothers(e.g.,Janson[ 57 ],Wieet 50

PAGE 63

al.[ 98 ],ChenandHsueh[ 25 ],andDrissi-KaitouniandHameda-Benchekroun[ 32 ])computeauserequilibriuminstead.Theotherfactoristhetravelcostfunctionusedbythesemodels.Amongotherparameters(physicalorotherwise),atraveltimeorcostfunctionmaydependonthenumberofcarsonthelinkandtheinputandoutputrates.Many(e.g.,CareyandGe[ 20 ],CareyandMcCartney[ 18 ],Carey[ 16 ],Carey[ 17 ],LinandLo[ 69 ],HanandHeydecker[ 46 ],Daganzo[ 28 ])haveanalyzedtheeectsoftravelcostfunctionsonvariousmodels.Some(e.g.,LinandLo[ 69 ]andHanandHeydecker[ 46 ])haveshownthatsometravelcostfunctionsarenotconsistentwiththemodelsthatusethem. SimilartoCareyandSrinivasan[ 21 ],CareyandSubrahmanian[ 23 ],Carey[ 15 ],ChenandHsueh[ 25 ]andKaufmanetal.[ 58 ],themodelinthischapterisbasedonthetime-expandednetwork.However,insteadofassumingthatthenetworkisemptyatthebeginningorattheend,wetreattheplanninghorizonasacircularintervalinsteadoflinear.Forexample,considertheinterval[0,24],i.e.a24-hourplanninghorizon.Whenviewedinalinearfashion,itistypicallyassumedthatthereisnocarinthenetworkattimes0and24.Inturn,thisimpliesthereisnotraveldemandaftertimek<24.Otherwise,carsthatenterthenetworkaftertimekcannotreachtheirdestinationsbytime24,therebyleavingcarsinthenetworkattheendofthehorizon.Ontheotherhand,ifthereisacarenteringastreetat23:55h(11:55PM)andexitingat24:06h(12:06AM,thenextday)inacircularplanninghorizon,theexittimeofthiscarwouldbetreatedas00:06hinstead.Whenaccountedforinthismanner,itispossibletodeterminetheexittimeforeverycarthatisinthenetworkattimezerowithoutrequiringanyadditionaldata.Additionally,modelsthatviewtheplanninghorizoninacircularfashionaremoregeneralinthattheyincludethosewithalinearplanninghorizon.Bysettingthetraveldemandsandothervariablesduringanappropriatetime

PAGE 64

intervaltozero,modelswithacircularplanninghorizoneectivelyreducetooneswithalinearhorizon. ItisoftenarguedthatthenumberofcarsatthebeginningandtheendofthehorizonaresmallandsolutionstoDTAarenotdrasticallyaectedbysettingthemtozero.Whenthepathsthatthesecarsusedonotoverlap,theargumentisvalid.However,whenthesecarshavetotraversethesamearcinreachingtheirdestinations,thenumberofcarsonthearcmaybesignicantandignoringitmayleadtoasolutionsignicantlydierentfromtheonethataccountsforallcars. Thischaptermakestwomainassumptions.Onerequiresthelinktraveltimeattimettobeafunctionofonlythenumberofcarsonthelinkatthattime.CareyandGe[ 20 ]showthatthesolutionsofmodelsusingfunctionsofthistypeconvergetothesolutionoftheLighthill-Whitham-Richardsmodel(seeLighthillandWhitham[ 68 ]andRichards[ 93 ])asthediscretizationoflinksintosmallersegmentsisrened.Becauseminimizingthetotaltraveltimeordelaymitigatesitsoccurrence,modelsdiscussedhereindonotexplicitlyaddressspillback.Ontheotherhand,themodelscanbeextendedtohandlespillbackusingatechniquesimilartotheoneinLieberman[ 67 ]oranalternativetraveltimefunctionthatincludestheeectofspillback(see,e.g.,PerakisandRoels[ 86 ]).However,asindicatedinthereference,usingsuchafunctionmaynotleadtoamodelwithasolution. Fortheremainder,Section 5.2 denestheconceptofperiodicplanninghorizon.Section 5.3 formulatesthesystemversionofthediscrete-timedynamictracassignmentproblemwithperiodicplanninghorizonorDTDTAandprovethatafeasiblesolutionexistsunderarelativelymildcondition.Toourknowledge,thereareonlyfourpapers(Brotcorne[ 10 ],Smith[ 95 ],Wieetal.[ 98 ],andZhuandMarcotte[ 103 ])thataddresstheexistenceissueandsome(see,e.g.,Smith[ 95 ]andZhuandMarcotte[ 103 ])considerthissmallnumbertobelacking.Allfourdeal

PAGE 65

Figure5{1. Linearversuscircularintervals. withuserequilibriumproblemsinsteadofsystemoptimal.Section 5.4 describestwolinearintegerprogramsthatprovideboundsforDTDTA.Section 5.5 presentsnumericalresultsforsmalltestproblemsand,nally,Section 5.6 concludesthechapter. 5{1 .Indoingso,time0andTarethesameinstant.Forexample,time0:00hand24:00h(ormidnight)arethesameinstantina24-hourday.Forthisreason,Tisexcludedandtheplanninghorizonishalf-open.Tomakethediscussionhereinmoreintuitive,weoftenrefertotheplanninghorizonasa24-hourday,i.e.,T=24.Intheory,theplanninghorizoncanbeofanylengthaslongaseventsoccurinaperiodicfashion.Ifanevent(e.g.,vecarsenterastreet)occursattimet,thenthesameeventalsooccursattimet+kT,forallintegerk1. Becausetheplanninghorizoniscircular,eventsoccurringtomorrowareassumedtooccurinthesameintervalthatrepresentstoday.Forexample,consideracarthatentersastreetatt1=23:00h(or11PM)todayandtraversesthestreetuntilitleavesatt2=01:00h(or1AM)tomorrow.(SeeFigure 5{2 .)Inacircularplanninghorizon,thesetwoevents,acarenteringandleavingastreet,occurat

PAGE 66

Figure5{2. Eventsoccurringintwoconsecutiveplanninghorizons. time23:00hand01:00hinthesameinterval[0,24).Ingeneral,ifacarentersastreetattimet1
PAGE 67

Figure5{3. Three-nodenetwork. pairsandthetraveldemandforODpairkduringthetimeinterval[t;t+],t2,ishkt. Thereisalsoatraveltimefunctionassociatedwitheacharcinthenetwork.Intheliterature(see,e.g.,Wuetal.[ 101 ],RanandBoyce[ 89 ]andCareyetal.[ 19 ]),thesefunctionscandependonanumberoffactorssuchasin-owandout-owratesandtracdensities.Weassumeinthisformulationthata,thetraveltimeassociatedwitharca,dependsonlyonthenumberofcarsonthearc.Furthermore,aiscontinuous,non-decreasingandboundedbyT,i.e.,0
PAGE 68

Figure5{4. Timeexpansionofarc(1;2)att=1. isa=fs:s=da(w) ToincorporatethetimecomponentintheTEnetwork,everynodeinthestaticnetwork(orstaticnode)is`expanded'orreplicatedonceforeacht2.Forthethree-nodenetwork,staticnode1istransformedintoveTEnodes,oneforeacht2,intheTEnetwork.Forexample,node1isexpandedintonodes10;11;12;13;and14intheTEnetwork.(SeeFigure 5{4 .)Similarly,eacharc(i;j)inthestaticnetwork(orstaticarc)isreplicatedonceforeachpairof(t;s),wheret2ands2(i;j).Considerarc(1;2)inthethree-nodenetwork.Carsthatenterthisarcattime1cantake2,3,or4unitsoftimetotraversedepending(asassumedearlier)onthenumberofcarsonthearcatt=1.Toallowallpossibilities,arc(1;2)isexpandedintothreeTEarcs(11;23),(11;24),and(11;20).Thelatterrepresentsacarthatentersarc(1,2)attime1,takes4unitsoftimetotraverse,andleavesthearcattime5ortime0(ormod(1+4;5))ofthefollowingday.Similarexpansionappliestoeacht2.Ingeneral,eachstaticarc(i;j)expandsintojjj(i;j)jTEarcsoftheform(it;jmod(t+s;T));8t2;s2(i;j).

PAGE 69

Figure 5{5 displaysthecompletetimeexpansionofthethree-nodenetwork.Inadditiontothetime-expandednodesandarcs,thegurealsodisplaysthetraveldemandattheoriginTEnodes(i.e.,node1t;8t2)anddecisionvariablesgkd(k)trepresentingnumberofcarsarrivingatthedestinationnoded(k)ofODpairkattimet,i.e.,atnode3t;8t2. ToreferenceowsonTEarcs,letyka(t;s)denotetheamountofowforcommoditykthatentersstaticarcaattimet2,takess2aunitsoftimetotraverseit,andthenexitsthearcattimemodft+s;Tg.Inparticular,ifa=(i;j),thenthesubscripta(t;s)referstoTEarcsoftheform(it;jmod(t+s;T));8t2;s2(i;j).Inaddition,Ya(t;s)=Pk2Cyka(t;s)representsthetotalowonarca(t;s). Tocomputethetimetotraverseastaticarcattimet,let a(t)=f(;s):=[t1]T;[t2]T;;[ts]T;s2ag: Inwords,a(t)containspairsofentrance,,andtraveltimes,s,forstaticarcasuchthat,ifacarentersstaticarcaattimeandtakesstimeunitstotraverseit,thecarwillstillbeonthearcattimet.Forexample,ift=11:00handthetimetotraversearcaisvehoursforthepreviousveconsecutivetimeperiods,thencarsenteringarcaattime=10:00h,9:00h,8:00h,7:00h,and6:00hwillbeonthearcat11:00h.(Weassumeherethatcarsenteringarcaat,e.g.,6:00harestillonthearcat11:00heventhoughitisscheduledorexpectedtoleaveat11:00h.)Whentisrelativelynearthebeginningoftheplanninghorizon,thenotation[]Taccountsforcarsonthearcattimetthatenteritfromthepreviousday.Continuingwiththeforegoingexample,lett=3:00hinstead.Then,carsenteringarcaattime=

PAGE 70

Figure5{5. Time-expansionofthethree-nodenetwork.

PAGE 71

2:00h,1:00h,0:00h,23:00h,and22:00harestillonthearcat3:00h.Usingtheseta(t),thetotalamountofowonstaticarcaattimetorxa(t)isP(;s)2a(t)Ya(;s). Therearetwoadditionalsetsofdecisionvariables.Onesetconsistsofza(t;s),abinaryvariablethatequalsoneifittakesbetween(s)andsunitsoftimetotraversearcaattimet.Intheformulationbelow,thevalueofza(t;s)dependsonxa(t)and,foreacht,za(t;s)=1foronlyones2a.Theothersetconsistsofgk,avectorwithacomponentforeachnodeintheTEnetwork.ComponentitofgkissettozeroifiisnotthedestinationnodeofODpairk.Otherwise,gkd(k)t,whered(k)denotesthedestinationnodeofODpairk,isadecisionvariablethatrepresentstheamountofowforcommoditykthatreachesitsdestination,d(k),attimet. Belowisamathematicalformulationofthediscrete-timedynamictracassignmentproblemwithperiodicplanninghorizon(DTDTA).min(x;y;z;g)Xt2Xa2A"a(xa(t))Xs2aYa(t;s)#

PAGE 72

Intheobjectivefunction,Ps2aYa(t;s)representsthenumberofcarsthatenterarcaattimetand,basedonourassumption,thesecarsexperiencethesametraveltime,a(xa(t)).Thus,thegoalofthisproblemistominimizethetotaltraveltimeordelay.Usingconstraint( 7{10 ),theobjectivefunctioncanbeequivalentlywrittenasmin(x;y;z;g)Xt2Xa2A24a(X(;s)2a(t)Ya(;s))Xs2aYa(t;s)35 (a(P(;s)2a(t)Ya(;s)))whosecomponentsaredenedsothattheirinnerproductisconsistentwiththesummations. Constraint( 7{8 )ensuresthatowsarebalancedateachnodeintheTEnetwork.Inthisconstraint,Bdenotesthenode-arcincidencematrixoftheTEnetworkandbkisaconstantvectorwithacomponentforeachTEnodeanddenedasfollows:bkit=8><>:0ifi6=o(k)hktifi=o(k) whereo(k)denotestheoriginnodeofODpairk.Constraint( 7{9 )guaranteesthatthenumberofcarsarrivingatthedestinationnoded(k)equalsthetotaltraveldemandofODpairkduringtheplanninghorizon.Then,constraint( 7{10 )computesthetotalowoneachTEarcand( 5{4 )determinesthenumberofcarsthatarestillonstaticarcaattimet. Incombination,thenextthreeconstraints,i.e.,constraints( 5{5 )-( 5{7 ),computethetraveltimeforthecarsthatenterarcaattimetandonlyallowowstotraversethecorrespondingarcintheTEnetwork.Inparticular,constraint( 5{5 ),inconjunctionwith( 5{6 ),choosesone(discretized)traveltimes2a

PAGE 73

thatbestapproximatesa(xa(t)),i.e.,a(xa(t))2(s;s].Whenarepresentsarc(i;j),constraint( 5{7 )onlyallowsarc(it;imod(t+s;T))tohaveapositiveow.Otherwise,(7)forcesowsonarc(it;imod(t+;T)),for2aand6=s,tobezero.Finally,constraint( 7{11 )makessurethatappropriatedecisionvariablesareeithernonnegativeorbinary. Asformulatedabove,thetraveltimeassociatedwithza(t;s)inequation( 5{6 )canonlytakeondiscretevaluesfromthesetawhilethetraveltimeintheobjectivefunctionvariescontinuously.Althoughitmaybemoreconsistenttousediscretevaluesoftraveltimesintheobjectivefunction,theabovemodelwouldprovideabettersolutionbecausethetruetraveltimeisusedtocalculatethetotaldelay.ThemodelalsohasinterestingpropertiesdiscussedinSection 5.4 .Inaddition,thetreatmentsoftraveltimesinboththeobjectivefunctionandconstraintscanbemadeconsistentbysolvingthe(approximation)renementproblemalsodiscussedinthesamesection. Underarelativelymildsucientcondition,weshowbelowthatDTDTAhasasolutionbyconstructingafeasiblesolution.Infact,thesolutionweconstructbelowisgenerallyfarfrombeingoptimal.However,itsucesforthepurposeofprovingexistence.LetRa(t)beasetofdiscretetimesatwhichacarentersarcaandstillremainsonthearcattimet.Below,werefertoRa(t)astheenter-remainset.Givenxa(t),Ra(t)isaunionoftwosets,i.e.,Ra(t)=fw2:w(t1);w+(xa(w)) 5{4 )-( 5{7 )andrelevantconditionsin( 7{11 )existswhenMaissucientlylarge.

PAGE 74

5{4 )to( 5{7 )andtherelevantconditionsin( 7{11 ). Form=1,let Asdenedabove,R1tistheenter-remainsetbasedonthetraveltimes1,avectorofs1!;8!2.Form2,let Sequencesfsmtg,fxma(t)gandfRmtgconstructedabovearemonotonicallynon-decreasing.Considerthesequencefsmtg.Observethats2ts1t;8t2becausex1a(t)0;8t2,andasassumedearliera()isnon-decreasing.Itfollowsthat,foranyt2,!+s2!!+s1!tand!+s2!T!+s1!Tt.Thus,!2R1t

PAGE 75

impliesthat!2R2t,i.e.,R1tR2tforallt2.Thelatter,andthefactthatua(t)isnonnegative,implythatx2a(t)=P!2R2tY2a(!;s2!)P!2R1tY1a(!;s1!)=x1a(t);8t2. Assumethattheclaimistrueuptosomexedm.Forallt2,sm+1t=da(xma(t))=eda(xm1a(t))=e=smt;becausexma(t)xm1a(t)anda()isnon-decreasing.Usinganargumentsimilartoabove,RmtRm+1tandxm+1a(t)xma(t).Thus,thethreesequencesaremonotonicallynon-decreasing.Inaddition,allthreesequencesarebounded,i.e.,smt
PAGE 76

become(i;j0)and(j;i0).LetpkdenoteapathinG(N;A)connectingtheODpairk,i.e.,pk2Pk.Thesetofthesepaths,=fpk:k2Cg,inducesasubgraphG(bN;bA),wherebNNandbAAdenotethesetsofnodesandarcs,respectively,belongingtothepathsin.Foreachi2bN,dene[i+]=f(i;j):(i;j)2bAgand[i]=f(j;i):(j;i)2bAg.Inwords,[i+]and[i]arethesetsofarcsinG(bN;bA)thatemanatefromandterminateatnodei,respectively.Also,letorder(i)denoteatopologicalorderofnodei(seeAhujaetal.[ 2 ]).If(i;j)2bAandG(bN;bA)canbetopologicallyordered,thenorder(i)
PAGE 77

Toconstructthevariablesxa(t),Ya(t;s),za(t;s),andyka(t;s)forarcsemanatingfromnodesofhigherorder,assumethatthedecisionvariablesforarcsemanatingfromnodeswithordermorlesshavebeenconstructedandletnodeibeoforder(m+1). 7{8 )and( 7{9 ). Consideranarca2[i+].Foreachba2[i],deneQ(ba;a)=fk:ba2pk;a2pk;k2Cgand,foreachk2Q(ba;a),letuka(t)denotetheowintoarcaattimetforODpairk.Then,uka(t)=Xfbt:bt+s1ba(bt)=tgykba(bt;s1ba(bt))+Xfbt:bt+s1ba(bt)T=tgykba(bt;s1ba(bt)); 5.3.1 ensuresthatxa(t),Ya(t;s),yka(t;s)andza(t;s)feasibletorelevantconstraintsexist. Thus,whencarriedoutinthetopologicalorderforeveryarcinbA,theaboveprocessmustproduceafeasiblesolutiontoDTDTA.

PAGE 78

eachODpairtotraverseoverseveralpathsaslongastheydonotinducecyclesinG(bN;bA).Withmorecumbersomenotation,theaboveargumentcanbeextendedtothecasewithmultiplepathsperODpairaswell. WhenappliedtotheaboveexampleinwhichtheODpairs(i;j)and(j;i)become(i;j0)and(j;i0),theacyclicsubgraphassumptionimpliesthatthepathsfromitoj0andfromjtoi0cannotformacycle.Intuitively,thismeansthattheremustexisttworoutesbetweentheoriginalnodesi(e.g.,home)andj(e.g.,work)withnoroadincommon.Theseroutesneednotbeoptimalandthereisnorequirementinourformulationoralgorithmstousethem.TheyareusedonlytoestablishedtheexistenceinTheorem 5.3.1 TheFirst-In-First-Out(FIFO)conditionrequiresthatcarsenteringanarcattimetmustleavethearcbeforethoseenteringaftertimet.Intheliterature,many(see,e.g.,Ranetal.[ 91 ],ZhuandMarcotte[ 103 ],andParakisandRoels[ 86 ])assumethatthetravelcostfunctionsatisedcertainconditionstoensureFIFO.Toavoidmakingadditionalassumptions,weensureFIFObyaddingthefollowingconstraintstoDTDTAinstead.Doingsomaymaketheproblemhardertosolvebecauseoftheadditionalconstraints.t+Ps2asza(t;s) t2:(t+) t+Ps2asza( t2:(t+)

PAGE 79

today)as(t+T)(e.g.,as08:00hofyesterdayplusT)andforcescarsenteringthearcatthistimetodepartafterthosethatenteratyesterday'stime Exceptforconstraint( 5{6 ),theconstraintsforDTDTAarelinear.Todevelopalinearversionof( 5{6 ),assumethatthetraveltimefunction,a,isinvertibleforalla2A.Forexample,ifaisacontinuousandincreasingfunction,then1aexistsontheinterval[a(0);a(Ma)].(SeeFigure 5{6 .)Underthisassumption,a(xa(t))2(s;s]ifandonlyifxa(t)2(1a(s);1a()].Thus,therequirement(s)za(t;s)
PAGE 80

Figure5{6. 5{6 ),thefollowingholdforanyfeasiblesolutiontoDTDTA:Xt2Xa2A"Xs2a(s)za(t;s)#"Xs2aYa(t;s)#(Y)TY; 5{7 )impliesthatYa(t;s)>0onlyifza(t;s)=1.Inaddition,constraint( 5{5 )ensuresthat,foreachpair(a;t),za(t; s)=1forsome s)0andYa(t;s)=0;8s2a;s6= s)=Xs2asYa(t;s):

PAGE 81

Asimilarresultholdsfortherstsetofsummations.Thus,theaboveinequalitiesreducetothefollowingXt2Xa2AXs2a(s)Ya(t;s)(Y)TYXt2Xa2AXs2asYa(t;s): 7{8 )-( 5{5 ),( 5{9 ),( 5{7 ),and( 7{11 )and,forconvenient,(Y;Z)representsanelementinS().Inaddition,let(Yl;Zl),(Y;Z),and(Yu;Zu)besolutionstothelower-boundproblem(orminfqTlY:(Y;Z)2S()g),theoriginalproblem(orminf(Y)TY:(Y;Z)2S()g),andtheupper-boundproblem(orminfqTuY:(Y;Z)2S()g),respectively.Then,thefollowinglemmaholds. 5.4.1 :qTlYlqTlY(Y)TY: Finally,thelastinequalityholdsbecauseYlisnotnecessarilyoptimaltominfqTuY:(Y;Z)2S()g.

PAGE 82

Inviewoftheabovelemma,thesolutionstotheupperandlower-boundproblemsareapproximationsofthesolutiontotheoriginalproblem.Thetheorembelowstatesthattheapproximationcanbemadearbitrarilyclosetotheoriginalproblembychoosingasucientlysmall. Then,thefollowingsequencemusthold:0(quql)TYl=eTYl=Pa2APk2CPt2Ps2ayka(t;s)=Pa2APk2CPt2yka(t;sa(t))Pa2APk2CHk=HPa2A1=HjAj 5.4.2 .Then,theaboverelationshipbetweenquandqlandlettingPk2Cyka(t;s)denoteindividualscomponentsofYlyieldthersttwoequalities.Thethirdequalityfollowsfromthedenitionofsa(t).Followingthis,thesecondinequalityholdsbecausethetotalamountofowon(static)arcaforODpairkduringtheentireplanninghorizoncannotexceedHk.ThesumofthelatterisH,aconstantthatcanbefactoredoutofthesummationoverA.Thisvalidatesthepenultimateequality.Finally,thelast

PAGE 83

equalityfollowsfromthefactthatPa2A1simplydenotesthenumberofelementsinthesetA.Choosing=" HjAjguaranteesthatqTuYlqTlYl.WhencombinedwiththeresultsinLemma2,thelatterimpliesthatqTuYu(Y)TYand(Y)TYqTlYl: 5{5 )and( 5{7 )fromtheproblem.InDTDTA,weusexa(t),thenumberofcarsonarcaattimet,tocomputethetraveltimeonarcaand,subsequently,toselectwhichTEarctouseorwhichza(t;s)tosettoone.Thus,whenZisgiven,xa(t)becomesunnecessary.Additionally,lets(t)besuchthatzua(t;s(t))=1foreacht2.Then,constraint( 5{9 ),originally( 5{6 ),reducestorequiringP(;s)2a(t)Ya(;s)tobeintheinterval(s(t);s(t)].Inotherwords,theoriginalproblemwithZ=Zuisanonlinearmulti-commodityowproblemwiththelatterassideconstraints. Let^Yubeanoptimalsolutiontominf(Y)TY:(Y;Zu)2S()g.Then,thefollowingcorollaryshowsthat^YubetterapproximatesY. 5.4.1

PAGE 84

Table5{1. Demandpatterns Time TracIntensity 0123456789 Total Low 20253035404035302520 300 Medium 30354045505045403530 400 High 40455055606055504540 500 Inallproblems,theplanninghorizonis[0,10)andthetravelcostfunctionsareeitherlinear,i.e.,(w)=1:5+2:5(w 5{1 .Inallthreepatterns,traveldemandsatdiscretepointsincreasesgraduallyuntiltime4,levelsobriey,andthendecreasesgraduallyaftertime5.Theindividualdemandsinthethreepatternsaredierentandrepresentthreetracintensities:low,medium,andhigh.WeusedGAMS[ 39 ]toimplementandsolveallproblemsusingNEOSServerofOptimization[ 82 ].Inparticular,weusedSBB[ 94 ]tosolveournonlinearintegerprogrammingproblem,i.e.,DTDTA,XPress-XP[ 102 ]tosolveourlinearintegerprograms,i.e.,thelowerandupper-boundproblems,andCONOPT[ 26 ]tosolveourlinearlyconstrainedoptimizationproblems,i.e.,theapproximationrenementproblems.AllCPUtimesreportedhereinarefromtheNEOSserver. ToempiricallyverifythatDTDTAproblemisnotconvex,werstconsiderthetwo-arcnetworkinFigure 5{7 thathasoneODpair.Welet=1.Thus,thediscrete-timeplanninghorizonis=f0;1;;9g.Weusetheabovequadratic Figure5{7. Two-arcnetwork.

PAGE 85

Table5{2. Optimalsolutionstothetwo-arcproblem. Solution1 Solution2 Inow Traveltime Inow Traveltime Time 020 1.6001.500 200 1.5001.600 1 250 1.5001.600 025 1.6001.500 2 030 1.6561.500 300 1.5001.656 3 350 1.5001.725 035 1.7251.500 4 040 1.8061.500 400 1.5001.806 5 400 1.5001.900 040 1.9001.500 6 035 1.9001.500 350 1.5001.900 7 300 1.5001.806 030 1.8061.500 8 025 1.7251.500 250 1.5001.725 9 200 1.5001.656 020 1.6561.500 traveltimefunctionforbotharcsandthefunctionyieldstraveltimesintheinterval[1:5;4:0].Because=1,thesetofdiscretetraveltimesis=f2;3;4g.UsingthelowtracintensitydemandpatterninTable 5{1 ,wesolvedDTDTAusingSBBandterminateditwhentherelativeoptimalitygapislessthan0.005(or0.5%).Therearetwooptimalsolutions(seeTable 5{2 )tothetwo-arcproblemwithanoptimaltotaldelayof450. Considertherstsolution,labelled`Solution1',intheTable 5{2 .Attime0,thereare20carstotravelfromnode1tonode2.Atthistime,therearealso20carsalreadyonarca1.Thesecarsenterthearcattime9andhavenotreachedtheirdestinationattime0.BecauseDTDTAassumesthatthetimetotraversearca1dependsonthenumberofcarsonthearcattheentrancetime,thetraveltimeforarca1attime0is1:5+2:5(20 100)2=1:6.Ontheotherhand,thereisnocarona2attime0.Carsthatenterthearcattime8alreadyleftthearcbytime0.Thus,thetraveltimefora2attime0is1.5,thefree-owtraveltime.Tominimizethetraveltime,all20carsenteringthenetworkattime0musttravelona2.Infact,everycarinSolution1travelsatthefree-owtraveltimeof1.5.Thus,therecannotbeanysolutionwithlesstotaldelayandSolution1mustbeoptimal.

PAGE 86

Figure5{8. Four-nodenetwork. Becauseofthesymmetryinthedata,switchingtheowsbetweenthetwoarcsinthenetworkyieldsSolution2,anotheroptimalsolution.Furthermore,itiseasytoverifythateveryconvexcombinationofthesetwosolutionsisfeasibletoDTDTAandyields,ontheotherhand,alargertotaldelay,therebyconrmingempiricallythattheobjectivefunctionisnotconvex. Additionally,the\extreme"travelbehaviordisplayedinTable 5{2 maynotbeintuitive.Thisisduetotheassumptionthatthesystemoperatorisextremelysensitivetothedierenceintraveltimesandiswillingtoswitchroutesinordertosaveaminuteamountoftraveltime. Whenthenetworkislarge,itwouldbetootime-consumingtosolveDTDTAoptimallyorotherwise.Inourexperiments,weconsiderfourapproximatesolutionstoDTDTA:(Yl;Zl),(YU;ZU),(^Yl;Zl),and(^YU;ZU),wherethelasttwoarerenementsofthersttwo.Toevaluatethequalityandthecomputationtimesofthesesolutions,weconsiderthefour-nodenetworkinFigure 5{8 withtwoODpairs,(1,4)and(2,4).Inourexperiments,bothODpairshavethesamedemandpatternandallarcshavethesametravelcostfunction,linearorquadratic,asspeciedabove. First,wesolvedthelowerandupper-boundproblemswithusingtwolevelsofdiscretization,=1and=0:5.Asbefore,when=1,thediscrete-timeplanninghorizonis=f0;1;;9g.Ontheotherhand,when=0:5,becomes

PAGE 87

Table5{3. Solutionsfromthelowerandupper-boundproblems:lineartravelcostfunction. Trac Intensity low 820.01580.0760.0 1187.51560.0372.5 medium 1200.02230.01030.0 1705.02230.0525.0 high 1500.02875.01375.0 2187.52870.0682.5 5{3 and 5{4 ),weassumethat,when=0:5,thereisnodemandatfractionaltimes(e.g.,at0.5,1.5,2.5,etc.)andthedemandsatintegraltimes(i.e.,1,2,3,etc.)areasshowninTable 5{1 Forbothtypesoftravelcostfunctions,thesizeoftheoptimalitygap(i.e.,qTuYuqTlYl)decreasesbyapproximately50%asdecreasesfrom1to0.5.However,theresultsinTables 5{3 and 5{4 suggestthatthereductioninthegapisduemainlytotheimprovementinthesolution,Yl,ofthelower-boundproblem.TheapproximatetraveldelaysasestimatedbyYuchangerelativelylittleforthetwovaluesof. Tables 5{5 and 5{6 comparethesolutionsfromDTDTA,(Y;Z),againsttwoapproximations,(^Yu;Zu)and(^Yl;Zl).Asinthetwo-nodeproblem,wesolveDTDTAusingSBBtoobtaina(integer)solution(Y;Z)withlessthan0:5%relativeoptimalitygap.Toobtain(^Yu;Zu),werstsolvetheupper-boundproblemusingXPress-MPtoobtain(Yu;Zu),a(integer)solutionwithlessthan0:5%optimalitygap,and,then,solvetheapproximationrenementproblem(with Table5{4. Solutionsfromthelowerandupper-boundproblems:quadratictravelcostfunction. Trac Intensity Low 600.01200.0600.0 900.01200.0300.0 Medium 822.21644.5822.2 1233.31644.5411.1 High 1124.52248.91124.5 1686.72248.9562.2

PAGE 88

Table5{5. Qualityofrenedupperandlower-boundsolutions:lineartravelcostfunction. (^Yu;Zu) (^Yl;Zl) Rel. cpu Trac cpu Ratio Intensity Delay(sec) Delay(sec) Delay(sec) (%) cpu 1337.5027.42 1385.002.57 1392.502.38 3.55 10.711.5 Medium 1800.0015.92 1866.302.66 1815.302.90 0.85 6.05.5 High 2290.0095.02 2327.504.07 2315.001.25 1.09 23.376.0 Forbothlinearandquadratictraveltimefunctions,thetwoapproximationschemesprovidesolutionswithrelativelysmallerrorsusingmuchlessCPUtimerequiredtosolveDTDTA(seetheratiosofthecputimesinTables 5{5 and 5{6 ).Forquadratictraveltimefunctions,theapproximatesolutionsareidenticaltoDTDTAsolutions,exceptforthehightracintensitycasewhentheapproximatesolutionsareslightlybetter(by0:06%). Table5{6. Qualityofrenedupperandlower-boundsolutions:quadratictravelcostfunction. (^Yu;Zu) (^Yl;Zl) Rel. cpu Trac cpu Ratio Intensity Delay(sec) Delay(sec) Delay(sec) (%) cpu 1054.500.88 1054.500.09 1054.500.08 0.00 9.811.0 Medium 1543.806.62 1543.800.14 1543.800.34 0.00 47.319.5 High 2129.80501.17 2128.600.10 2128.600.13 -0.06 5011.73855.2

PAGE 89

andhighways.Theresultingproblemisanonlinearprogramwithbinaryvariables,adicultclassofproblemstosolve.Alternatively,twolinearintegerprogramsareconstructedtoobtainapproximatesolutionsandboundsonthetotaltraveldelay.ItisshownthatsolutionsfromthelattercanbemadearbitrarilyclosetosolutionsofDTDTA.Furthermore,numericalresultsfromsmalltestproblemssuggestthatsolvingthelinearintegerprogramismoreecient.

PAGE 90

5 wehavediscussedaperiodicdiscretetimedynamictracassignmentproblem,whichisconstructedbasedontwoassumptions:(i)allcarsthatenterthearcduringthesametimeintervalexperiencethesametraveltimeandleavethearcduringthesametimeinterval,and(ii)thetraveltimeisafunctionofthenumberofcarsontheroad(seealsoNahapetyanandLawphongpanich[ 78 ]).Aswehaveseen,theinitialmathematicalformulationofsuchmodelleadstoamixedintegerproblemwithlinearconstraintsandanonlinearobjectivefunction.Bylinearizingtheobjective,onecanconstructanupperandalowerboundproblems.Althoughthesolutionofsuchproblemscanbemadearbitrarilyclosetothesolutionoftheinitialproblembydecreasingthediscretizationparameter,observethattheboundingproblemsbelongtotheclassoflinearmixedintegerprogram,whicharecomputationallyexpensivetosolve.Inthecaseoftheboundingproblems,thetaskbecomesmorechallengingbecauseofthespecialstructureofthefeasibleregion. Observethatthemodelisconstructedbasedonatime-expandednetworkandtherearebinaryvariables,za(t;s),associatedwiththearcsofthenetwork.Bydecreasingtheparameter,thenumberofthebinaryvariablesincreases.Forexample,givenatracnetworkG(N;A)andasetofpossiblediscretetraveltimesa(),thetotalnumberofthebinaryvariablesintheDTDTA-Uproblemisj()jPa2Aja()j.Ifreducesto=2,thenja(=2)j=2ja()j,j(=2)j=2j()j,andthetotalnumberofthebinaryvariablesinthenewproblemis 78

PAGE 91

22j()jPa2Aja()j.Becauseofresourcelimitations,MIPsolverscannotsolvelargeproblems. Inthischapterweconsideraheuristicalgorithmtosolvetheboundingproblems.Althoughthesametechniquecanbeappliedtosolvebothboundingproblems,wemainlyfocussontheupperboundproblemDTDTA-U.Forconvenienceofreference,werestatetheproblembelow.min(x;y;z;g)qTuY Inthewell-knownheuristicalgorithmssuchasneighborhoodsearch,greedyalgorithmortabusearch,itisrequiredtomovefromonefeasiblesolutiontoanother.However,ndingafeasiblesolutiontotheDTDTA-Uproblemisnoteasy.Todemonstrate,consideraone-arc-network,a=(1;2),alineartraveltimefunctiona(xa(t))=0:3+0:05xa(t),andasetofpossible(discrete)traveltimesa=f1;2;3g.Inaddition,assumethatthetimehorizon[0;5)isdividedintoveintervals,and10carsenterintothearcateachdiscretetimet2=f0;1;2;3;4g.Byassigningthosecarstothearcsa(t;1)oftheTEnetwork,oneconcludesthatxa(t)=10,

PAGE 92

6{5 )and( 6{6 )(seetheleftnetworkinFigure 6{1 ).ItcanbeshownthatthisisanoptimalsolutiontotheDTDTA-Uproblem.However,thereisanotherfeasiblesolution,where10carsareassignedtothearcsa(t;2)(seetherightnetworkinFigure 6{1 ).Usingthosesettings,xa(t)=20,a(xa(t))=1:3,andza(t;2)=1,8t2,whichagainsatisesconstraints( 6{5 )and( 6{6 ).Assumethatthesecondsolutionisknownandonedecidestoimprovethesolutionandmovetoanotherfeasiblesolutionbyassigningthecarsattimet=0toarca(0;1),i.e.,za(0;1)=1andza(0;2)=0.Asaresult,xa(1)=10anda(xa(1))=0:8,whichviolatesinequality( 6{5 ).Thesamefollowsfromthechangeofotherarcs;thus,weconcludethatthesecondsolutionisisolatedinthesensethattheadjacentsolutions,i.e.,changingonlyonearc,areinfeasible.FindinganadjacentfeasiblesolutionbecomesmorecomplicatedwhenthestaticnetworkG(N;A)islargerbecause,withrespecttoagivenpath,thechangesonupstreamarcshaveaninuenceontheowsofdownstreamarcs.Becauseheuristicalgorithmssimilartotheneighborhoodsearch,greedyalgorithms,andtabusearchrequiremovingfromonefeasiblesolutiontoaneighboringfeasiblesolution,thedicultiesofndinganeighboringsolutionmakesinappropriatetheuseofsuchtechniques. Figure6{1. Twofeasiblesolutions.

PAGE 93

AnotherapproachtosolvetheDTDTA-Uproblemistherelaxationoftheintegralityofthevariableza(t;s)andconstructinganequivalentformulationwithcontinuousvariables.Todoso,replacetheconstraintsza(t;s)2f0;1gbyinequalities0za(t;s)1,i.e.z2[0;1]n,n=j()jPa2Aja()j,andza(t;s)(1za(t;s))0.Thelattercanbeincludedintotheobjectivefunctionwithapenalty.Asaresult,theproblemreducestoacontinuousconcaveminimizationoneandaglobalsolutionoftheresultingproblemisasolutionoftheDTDTA-Uproblem(see,e.g.,Horstetal.[ 54 ]orHorstandTuy[ 55 ]).Althoughitisknownthatanoptimalvectorofthebinaryvariables,z,representsoneoftheverticesofthendimensionalunitcube(eachvertexcorrespondstoanintegersolution),becauseofconstraints( 6{1 )-( 6{8 )mostofthemareinfeasibleanditishardtondanoptimalone. ObservethattheLPrelaxationoftheDTDTA-Uproblemprovidesalowerbound,whichisfarfromanoptimalone.Toillustrate,considertheDTDTA-Uproblem,wheretheconstraintsza(t;s)2f0;1garereplacedbytheinequalities0za(t;s)1.Tondoptimalvaluesofvariablesyka(t;s)andgkd(k)tintheresultingproblem,itissucienttosolvethefollowinglinearproblem.min(y;g)qTuYs.t.Byk+gk=bk8k2CXt2gkd(k)t=Xt2hkt8k2Cyka(t;s)0;gkd(k)t0;8t2;a2A;s2aandk2C

PAGE 94

6{3 )and( 6{4 ).Theoptimalvaluesofthevariablesza(t;s)canbeobtainedbysolvingthefollowingsystemofequations. Noticethatthelastinequalityin( 6{9 )issatised8za(t;s)2[0;1],s6=1,becauseYa(t;s)=0,8s2a,s6=1.Inthecaseofs=1,asucientlylargevalueofMareducestheinequalitytoza(t;1)>0andmakessurethatarcsa(t;1)areallowedtohavepositiveowsgivenanypositivevaluesofthevariableza(t;1)(seeinequality( 6{7 )).Otherequationsin( 6{9 )areeasytosatisfyanditcanbeshownthatforanyvalueofxa(t)thesetofsolutionstothesystemisnotemptyandnotunique.However,becauseofthecongestionitishighlyunlikelythatatoptimalityoftheDTDTA-UproblemalldriversexperiencethefreeowtraveltimeandoneconcludesthatasolutionoftheLPrelaxationoftheproblemisnotrealistic,andtheoptimalobjectivefunctionvalueoftherelaxationproblemisfarfromtheoptimalvalueoftheobjectivefunctionofDTDTA-U. Despiteallcomplicationsdescribedabove,theDTDTA-Uproblemhasthefollowingusefulproperty:ifatoptimalitythetotalinowintoarcaattimetiszero,i.e.,Ps2aYa(t;s)=0,thenconstraint( 6{7 )issatisedforanyvalueofza(t;s);therefore,correspondingconstraints( 6{5 )-( 6{7 )canberemovedfromtheformulationandthesolutionoftheresultingproblemremainsthesame.Unknownvaluesofthevariablesza(t;s)canberestoredbysolvingthesystemofequations( 6{5 )-( 6{6 )usingthevaluesofxa(t;s). TheaboveanalysismotivatesdevelopingalowerboundproblemfortheDTDTA-U,whichis(i)tighterthantheLPrelaxation,(ii)easiertosolve

PAGE 95

thantheconcaveminimizationproblemdiscussedabove,and(iii)preservestheabovementionedproperty.Inparticular,inthischapterweconsideranonlinearrelaxationoftheproblemwithbilinearconstraints.Usingtherelaxationtechnique,weproposeaheuristicalgorithmtosolvetheDTDTA-Uproblem. Fortheremainder,Sections 6.2 discussesthenonlinearrelaxationoftheDTDTA-Uproblem.Usingtherelaxation,inSection 6.3 weproposeaheuristicalgorithmtosolvetheDTDTA-Uproblem.NumericalexperimentsonthealgorithmareprovidedinSection 6.4 ,andnally,Section 6.5 concludesthechapter. yka(t;s)0;gkd(k)t0;xa(t)08t2;a2A;s2aandk2C(6{15) Observethat(i)intheDTDTA-Rproblemconstraints( 6{10 )-( 6{13 )arethesameascorrespondingconstraintsoftheDTDTA-Uproblem,(ii)theDTDTA-Rproblemdoesnotincludethebinaryvariablesandconstraints( 6{5 )and( 6{7 ),and(iii)theconstraints( 6{6 )arereplacedbybilinearconstraints( 6{14 ).

PAGE 96

Case1:Pr2aYa(t;r)6=0.Fromequality( 6{16 )itfollowsthatza(t;s)=Ya(t;s) 6{5 )and( 6{7 )givenasucientlylargeMa;thus,theconstraintscanberemovedfromtheformulation.Inaddition,observethatafterappropriatesubstitutionsofthevariablesza(t;s)theconstraint( 6{6 )transformsintotheconstraint( 6{14 ),andthevariablesza(t;s)canberemovedfromtheformulation. Case2:Pr2aYa(t;r)=0.Observethatequation( 6{16 )andconstraint( 6{7 )aresatisedforanyvalueofthevariableza(t;s),s2a.Becauseconstraint( 6{7 )isredundant,removethevariablesza(t;s),s2a,fromtheformulationaswellascorrespondingconstraints( 6{5 )-( 6{7 ).Inaddition,noticethatconstraint( 6{14 )issatisedandcanbeaddedtotheformulationwithoutchangingthefeasibleregion. Basedontheaboveanalysis,oneconcludesthatbothproblemshavethesameoptimalobjectivefunctionvalueandthesameoptimalvaluesforthevariablesyka(t;s)andxa(t).Inaddition,theoptimalvaluesofthevariablesza(t;s)canbeobtainedusingthevaluesofyka(t;s)andxa(t).Inparticular,ifPr2aYa(t;r)6=0thenza(t;s)=Ya(t;s) 6{5 )-( 6{6 ). 6{16 )becausefromconstraints( 6{5 )-( 6{7 )itfollowsthateitherbothsidesof( 6{16 )arezero,i.e.,za(t;s)=Ya(t;s)=0,or

PAGE 97

6{16 )totheDTDTA-Uproblemdoesnotchangethefeasibleregion.Next,observethataccordingtotheTheorem 6.2.1 FromTheorem 6.2.1 6{16 ).Nextweshowthattheconstraintsarenotredundantunlessatoptimalityalldriversexperiencethefreeowtraveltime.Inparticular,theconstraint( 6{16 )requiressendingaportionofthetotalow,i.e.,za(t;s)Pr2aYa(t;r),alongthearca(t;s)ifPr2aYa(t;r)6=0.Ontheotherhand,recallthatintheLPrelaxationoftheDTDTA-Uproblem,atoptimalityonlyarcsthatcorrespondtothefreeowtraveltimehaveapositiveow,i.e.,yka(t;s)=0,8s2a,s6=1,andthevariablesza(t;s)mayhavepositivevaluesforalls2a,s6=1,aslongastheysolvethesystem( 6{9 )(seeSection 6.1 ).Itcanbeshownthatthesolutionsatisesequation( 6{16 )onlyifza(t;1)=1solvesthesystem( 6{9 ).Thelatterholdsonlyintrivialproblemswithnocongestion.FromtheaboveanalysisitfollowsthattheDTDTA-RproblemhasatighterfeasibleregionthantheLPrelaxationoftheDTDTA-Uproblem. 6{14 )isneitherconvexnorconcave.However,itiscomputationallymoreattractivethantheconcaveminimizationproblemdiscussedinSection 6.1 .Inparticular,theDTDTA-Rproblemhasfewerconstraintsandvariables,anddoesnothaveapenaltytermintheobjectivefunction.Inaddition,theDTDTA-Rproblempreservestheabovementionedproperty;ifattimetanarc,a,hasnoinow,i.e.,Ps2aYa(t;s)=0,thenconstraint( 6{14 )issatisedforanyvalueofthevariablexa(t).

PAGE 98

6{14 )andsolvingtheLPrelaxationoftheDTDTA-Uproblem.(NoticethatthelatterisequivalenttothendingtheshortestpathintheTEnetwork.)Asaresult,theoptimalvaluesofthevariablesxa(t)betterapproximatetheoptimalvaluesofthecorrespondingvariablesoftheDTDTA-Uproblem.However,thesolutionisunlikelytobefeasibletotheoriginalproblem;thusitisessentialtondanintegersolutionthathastheobjectivefunctionvalueascloseaspossibletotheoneprovidedbytheDTDTA-Rproblem. Intheheuristicalgorithm(seeProcedure 8 ),rstwesolvetheLPrelaxationoftheDTDTA-Uproblem,whichprovidesaninitialsolutionforDTDTA-Rproblem.NexttheproceduresolvesDTDTA-Rproblem.InStep3,ndingthevaluesof^za(t;s)iseasyandcanbeaccomplishedbyperformingasimplesearchtechnique.ByxingthebinaryvariablesoftheDTDTA-Uproblemtothevaluesof^za(t;s),i.e.,za(t;s)^za(t;s),theproblemreducestoalinearone.IftheresultingLPisfeasiblethenthealgorithmstopsandreturnsthesolution.Otherwise,thealgorithmrunstheUpSetprocedure(seeProcedure 9 )thengoestoStep2. constraints( 6{5 )and( 6{6 ).

PAGE 99

forall8s2ado else forall8s2ado endif endfor

PAGE 100

numericalexperiments,wefounditmoreusefultotakethevaluesforparameterfromtheinterval[0:2;0:5].Asimilarprocedureappliestothecasewhenonlyonearcisused.Finally,ifthetotalinowintothearciszero,i.e.,Ps2aYa(t;s)=0,thenwerelaxrestictionsonthevariablesYa(t;s),8s2a.AlthoughtheDTDTA-Rproblemisnon-convexandrequiresndingaglobaloptimum,theUpSetprocedurepotentiallyeliminatesthecurrentsolutionfromfurtherconsiderationbynarrowingthefeasibleregion. TheabovedescribedheuristicalgorithmmaynotconvergeandtheperformanceofthealgorithmisdiscussedinSection 6.4 .However,ifthealgorithmdoesnotconverge,analternativeobjectivefunction(seeTable 6{1 )canbeusedtosolvetheproblem.Toconstructthealternativeobjectivefunction,observethatthecostvectorquconsistsofdiscretetraveltimess2a.Ontheotherhand,thevariablexa(t)iscomputedbasedonthesetofindicesa(t)(seeequation( 6{13 )).Inparticular,eacharca(t;s),s2a,isincludedintoatleastoneofthesetsa(t),forsomet2.Noticethatthearcmaybeincludedintomorethanoneset,andthetotalnumberofsuchsetsisequaltos=.(Notethats=isanintegerbecausesisamultipleof.)Asaresult,qTuY=Xa2AXt2Xs2as Ya(t;s)=Xa2AXt224Xr2a(t)Ya(t;r)35=Xa2AXt2xa(t)=eTx; Table6{1. Equivalentobjectivefunctions ObjectiveFunction1 ObjectiveFunction2 min(x;y;z;g)qTuY

PAGE 101

D{1 ,Appendix D ),wherethe4-nodenetworkhastwoorigin-destinationpairs,(1;4)and(3;4),andthe9-Nodesnetworkhasfourofthem;(1;8),(1;9),(2;8)and(2;9).Inaddition,weconsidertwoplanninghorizons,[0;10)and[0;30),traveltimefunctionsoftheforma(xa(t))=a+a(xa(t)=ca)a,wherea,a,caanda,aregeneratedrandomlyaccordingtothedistributionsspeciedinTable 6{2 .Thediscretizationparameterissettoone. ThedemandforeachODpairateachtimet2isgenerateduniformlybetween20and100cars.Thedistributionofthedemandischosentobeconsistentwiththeparametersofthetraveltimefunction.Inparticular,theaveragedemandisabout1=4oftheaveragecapacityca.Asaresult,onaverageupto4arcswith(average)arcowsof60carscanbeincludedintothecomputationofxa(t).Inotherwords,seta(t)consistsofupto4indices,whichcorrespondtothearcswith(average)owsof60cars.Thelatterisconsistentwithparametersandbecauseonaverage4<5=jaj.Thespecieddistributionsallowgeneratinglargevarietyofeasyandhardsolvableinstances. Foreachchoiceofthenetwork,timehorizonandobjectivefunction(seeTable 6{1 ,Appendix D ),50problemsarerandomlygeneratedandsolvedbytheheuristicalgorithmaswellastheCPLEXMIPsolver.Intheheuristicalgorithm,welimitthenumberofiterationsto500.Ifthealgorithmreachesthelimit,westoptheprocessandreportthattheheuristicalgorithmhasfailedtosolvetheinstance.TheaverageperformanceissummarizedinTable D{1 ,Appendix D .Becausetheheuristicalgorithmcouldnotsolvesomeoftheinstances,theaverageCPU Table6{2. Distributionsofparametersofrandomlygeneratedtraveltimefunctions uniform(2,8) uniform(200,300) random(1or2)

PAGE 102

timeandthequalityofthesolutionarecomputedbasedonthesetofthesolvedproblems.TheratiobetweentheCPUtimesoftheheuristicalgorithmandtheCPLEXsolveriscomputedforeachinstanceandthetablepresentstheaverageofthosenumbers. Onaveragetheheuristicalgorithmisabletosolveabout90%ofthesmallproblems(seethersttworowsinTable D{1 ,Appendix D )usingalmostthesameCPUtimeastheCPLEXsolver.However,inthecaseofthetimehorizon[0;30),theaverageCPUtimeofthelatterincreasesrapidly.Ontheotherhand,theCPUtimeoftheheuristicalgorithmremainsfairlysmallandthealgorithmisabletosolveabout80%oftheproblems.Inthecaseofthe9-nodenetworkandthetimehorizon[0;10),noticethattheperformanceofthealgorithmisdierentfordierentobjectivefunctions.Inparticular,thealgorithmrequiresmuchfewerresourcesandprovidesahigherqualitysolutioninthecaseofthesecondobjectivefunction.Inthelasttwoexperiments,theTEnetworkislarge,andwepostalimitof5000secondsofCPUtimeontheCPLEX.Inmostofthecases,theCPLEXterminatesduetotheCPUrestrictionandreturnsasolution,whichhasalowerqualitythantheoneprovidedbytheheuristicalgorithm.Inparticular,onaveragetheheuristicalgorithmprovidesasolution,whichisabout34%betterthanoneprovidedbytheCPLEXandusesabout1000-1500secondsoftheCPUtime. Althoughtheheuristicalgorithmcouldnotsolvesomeinstancesusingtherstorthesecondobjectivefunctions,onemayconsidersolvingtheproblemsonparallelmachinesusingbothobjectivefunctions,wherethealgorithmstopsifoneofthemsolvestheproblem.Table D{2 inAppendix D providestheresultsforthecombinedmode.ObservethatthepercentageoftheunsolvedproblemsistwicelessthanintheTable D{1 andthequalityofthesolutionremainsthesame.

PAGE 104

7{1 ,whichhastwoorigin-destinationpars,(1;4)and(2;4),andthetracdemandforeachODpairistwocars.Inaddition,assumethatthetraveltimesonarcs(1;3),(1;4),(2;3)and(2;4)areconstantandittakes2,15,4and16unitsof Figure7{1. 4-Nodenetworkandtracdemand. 92

PAGE 105

Figure7{2. Userequilibriumowsandtraveltimes. timetotraversethem,respectively.Ontheotherhand,assumethatthetraveltimeonarc(3;4)isafunctionofow 7{1 ). ItiseasytoverifythatthearcowsinFigure 7{2 satisfytheuserequilibriumcondition.Inparticular,twocarsofODpair(1;4)areassignedtopath1-3-4,whichisshorterthanpath1-4.InthecaseoftheODpair(2;4),paths2-3-4and Figure7{3. Systemoptimumowsandtraveltimes.

PAGE 106

Figure7{4. Tolleduserequilibriumowsandtraveltimes. 2-4havethesametraveltimeandonecarisassignedtoeachpath.Thetotaltraveltimeofallusersinthenetworkis60. NextconsiderthearcowsinFigure 7{3 .Thetotaltraveltimeofallcarsinthenetworkis47:67<60.However,observethatthesolutiondoesnotsatisfytheuserequilibriumcondition.Inparticular,paths1-3-4and2-3-4areshorterthanpaths1-4and2-4,respectively.ItiseasytoverifythattheowvectorinFigure 7{3 isasystemoptimumsolution. Thetollpricingproblemimposesadditionalcostsontheroadstotransformthesystemoptimumsolutionintoasolutiontoauserequilibriumproblemwithtolls(see,e.g.,Beckmann[ 6 ]andDafermosandSparrow[ 30 ]).Intheliteratureatollvectorthatallowsthistransformationisoftencalledavalidtollvector.Intheaboveexample,onemayconsideradding8.10unitsofdelaytothetraveltimeonarc(3;4),i.e.,(3;4)(x(3;4))=x(3;4)+x2(3;4)+8:10.Asaresult,thetraveltimeonpaths1-3-4and1-4arethesameandequalto15unitsoftime(seeFigure 7{4 ).Ontheotherhand,path2-3-4ismoreexpensivethanthealternativeone,i.e.,path2-4;therefore,thereisnoowonit,andweconcludethatthesolutionsatisestheuserequilibriumcondition.Thevalueoftimefordierentuserclassesandits

PAGE 107

inuenceonthesystemoptimumanduserequilibriumproblemsinthepresenceofroadpricingisdiscussedinEngelsonandLindberg[ 33 ]. Inthestaticnetworkowmodels,wherethedemandisknownandtimeinvariant,andthetraveltimeisafunctionofthearcow,thetollpricingframeworkscanbeclassiedasa\rst-best"anda\second-best"problems.Theformerassumesthateveryroadinthenetworkcanbetolled,anditiswellknownthatatollvectorderivedfromthemarginalsocialcostsisavalidtollvector.Bergendoretal.[ 8 ],andHearnandRamana[ 47 ]showthatitisnottheonlyvalidtollvectorandmathematicallydescribethesetofvalidtollvectors.Intheidealcase,whenthetraveltimefunctionisstrictlymonotone,andthesystemoptimumproblemisconvex,thesetofvalidtollvectorsassumesaverysimpleformofaconvexcone.Thetollsetallowsconstructingdierenttollpricingproblemswithasecondaryobjective(e.g.,minimizingthetotalrevenue,maximumtoll,ornumberoftollbooths).YildirimandHearn[ 106 ]extendedtheresultsfortheelasticdemandcase.ComputationalmethodsforsuchmodelsarediscussedinHearnetal.[ 48 ],Baietal.[ 3 ],andYildrimandHearn[ 106 ].Thesecond-bestmodelassumesthatonlyasubsetoftheroadscanbetolled.Thisandothertypesofrestrictionscanyieldtoanemptyvalidtollset.Inotherwords,thereisnoatollvectorthattransformsasystemoptimumsolutionintoasolutionofthetolleduserequilibriumproblem.Mathematicalformulationsofthesecond-bestproblemareeitherbileveloptimizationproblemsormathematicalprogramswithequilibriumconstraints(see,e.g.,Belleietal.[ 7 ],ChenandBernstein[ 24 ],Henderson[ 49 ],Labbeetal.[ 64 ],Ferrari[ 34 ],Brotcorneetal.[ 9 ],YangandLam[ 105 ],PatrikssonandRockafellar[ 84 ],andLawphongpanichandHearn[ 65 ]). Dynamicmodelsassumethatthedemandvariesduringaplanningtimehorizon,andthesetofdecisionvariablesinvolvesinowrates,outowrates,anddensities.Complexrelationshipbetweenthevariablesmakesthemathematical

PAGE 108

descriptionofthesystemoptimumaswellasuserequilibriumproblemsmorecomplicated.Despiteavarietyofmodelsforbothproblem,fewpapersaddressthedynamictollpricingframework.Agnew[ 1 ]presentsadynamicmodelforahighwayandusingoptimalcontroltheoryanalyzesmarginalcostsandassociatedtolls.CareyandSrinivasan[ 22 ]derivesystemmarginalcosts,userperceivedcostsanduserexternalitycosts.Theauthorsconsiderasetoftollsthatdependnotonlyonthelevelofcongestionbutalsotherateofchangeofcongestion.Usingoptimalcontroltheory,HuangandYang[ 56 ]proposeacongestionprisingproblemonanetworkofparallelrouteswithelasticdemand.AnotheroptimalcontrolformulationoftheproblemaswellasamarginalcostbasedtollvectorisdiscussedbyWieandTobin[ 97 ]. Inthischapter,weconsideradynamictollpricingframeworksimilartotheonedevelopedinBergendor[ 8 ],andHearnandRamana[ 47 ]forthestaticcase.Inparticular,theframeworkconsistsofthefollowingsteps:(i)solveasystemoptimumproblem,e.g.,theDTDTAproblemfromChapter 5 ,(ii)derivethesetofvalidtollvectorsforasolutionoftheproblemand(iii)solveatollpricingproblemwithasecondaryobjective.However,theDTDTAproblemaswellasmostofothersystemoptimumproblemswithdynamicsettingsarenon-convexandcanhavemultiplesolutions.Furthermore,ndinganexactsolutionofalargeproblemiscomputationallyexpensive,andheuristicalgorithmsprovideanapproximatesolutionoftheproblem.Inthischapter,weslightlychangethedenitionofthesetofvalidtollvectorsanddenethesetwithrespecttoanapproximateorafeasiblesolutionoftheSOproblem.Inparticular,givenanapproximate(feasible)solutionoftheSOproblematollvectoriscalledvalidwithrespecttothesolutionifittransformsthesolutionintoasolutionofthetolleduserequilibriumproblem.TheabovedenitionallowsdevelopingatollpricingframeworkbasedonanapproximatesolutionoftheSOproblem.

PAGE 109

Thekeycomponentofthetollpricingframeworkinthischapterisareducedtime-expandednetwork(RTE),whichisconstructedbasedonafeasiblesolutiontotheDTDTAproblem(seeChapter 5 ).Usingthenetwork,weprovethatthefeasiblesolutionisauserequilibriumsolutionifandonlyifitisasolutionofalinearminimizationproblemwiththeunderlyingRTEnetwork.ByincludingthetollcomponentintotheobjectiveoftheminimizationproblemandapplyingthedualitytheoryforLP,themathematicalexpressionofthesetofvalidtollvectorsconsistsoflinearequations.Inthecaseofasystemoptimumsolution,weshowthatthereisavalidtollvector,whichhasasimilarstructureasoneobtainedfromthemarginalcostsinthestaticcase. Fortheremainder,Sections 7.2 and 7.3 discussthereducedtime-expandednetworkandthesetofvalidtollvectors,respectively.ThetollpricingproblemswithdierentobjectivesandconstraintsarepresentedinSection 7.4 .SomeillustrativeexamplesareprovidedintheSection 7.5 andnally,Section 7.6 concludesthechapter. Let(^y;^g;^z)denoteafeasiblesolutionoftheDTDTAproblemand^a(t)=a(t)(^y)=a(P(;s)2a(t)Pk2Cyka(;s)).Foreachpair(a;t),a2A,t2,let^sa(t)denotetheelementfromthesetasuchthat^za(t;^sa(t))=1.Observethat^za(t;s)=0,foralls2a,s6=^sa(t).Asaresult,^yka(t;s)=0,8s2a;s6=^sa(t),andintheTEnetworkarcsa(t;s),s6=^sa(t),donothaveows.Byremovingthosearcs,i.e.,all

PAGE 110

arcsa(t;s)suchthat^za(t;s)=0,werefertothesimpliedTEnetworkasreducedTEnetworkanddenotebyRTE(^z).NoticethatthestructureofthereducedTEnetworkdependsonvector^z,andthenetworkconsistsofarcsa(t;^sa(t)).IntheRTE(^z)network,therearejjcopiesoforiginanddestinationnodes,i.e.,onecopyforeacht2.ForeachODpairk,letOktrepresentthecorrespondingcopyoftheoriginnodeattimetandPk(t)denotethesetofpathsintheRTE(^z)networkthatemanatefromtheOktnodeandterminateatthedestinationnodeDk^t,forsome^t2.Inaddition,letPkrepresentthesetofpathsinthestaticnetworkG(N;A)thatemanatefromtheoriginnodeandterminateatthedestinationnodeoftheODpairk.

PAGE 111

i.e.pk(t0)=fa1(t0;^sa1(t0));:::;an(tn1;^san(tn1))g.Thisshowsthat(pk;t0)correspondstopk(t0).Conversely,itiseasytoseethatpk(t0)alsocorrespondsto(pk;t0).Thus,thereisaone-to-onecorrespondencebetweenpair(pk;t)andpathpk(t).Furthermore,because^ai(ti1)representsthetraveltimeonarcaatthetimetoenterthearcti1,thetraveltimeonpathpkatthetimetoenterthearctisequaltothetraveltimeonpathpk(t)oftheRTE(^z)network. 7.2.1 itfollowsthatgivenafeasiblesolution,(^y;^g;^z),onecanconstructthereducedTEnetworkandanalyzethesolutionusingpathsintheRTE(^z)network.AlthoughRTE(^z)isasubgraphoftheTEnetwork,tosomeextentitcanbeviewedasastaticnetwork.Thismakesiteasiertoprovesomepropertiesofthefeasiblesolution.Becauseoftheone-to-onecorrespondence,belowweusenotationp(t)todescribepathp2Sk2CPkattimet. TheDTDTAproblemdiscussedinChapter 5 isasystemoptimumproblem,wheretheobjectiveistominimizethetotaldelayofalldriversduringthetimehorizon[0;T).Intheuserequilibriumproblem,theobjectiveistondafeasiblevectortotheDTDTAproblem,(^y;^g;^z),suchthatalluserschooseoneoftheshortestpathsamongallalternativepathsavailableatthetimetoenterthenetwork.Givenafeasiblevector(^y;^g;^z),^yka(t;^sa(t))representstheowonthearca(t;^sa(t))oftheRTE(^z)networkthatbelongstotheODpairkaswellasthenumberofcarsthatenterthearcaofthestaticnetworkduringthetimeinterval[t;t+).Let^up(t)denotethenumberofcarsenteringthepathp2Sk2CPkduringthetimeinterval[t;t+).Observethat^up(t)dependsonthevector^y.Similarly,letp(t)(^y)representthetraveltimeonpathpattimet.Usingthesenotations,formallytheconceptoftheuserequilibriumsolutioncanbedenedasfollows:

PAGE 112

7{1 )issatised.Thetheorembelowprovidesanalternativewaytocheckifafeasiblevectorisauserequilibriumsolution.Theprocedureinvolvessolvingthefollowingminimizationproblem,wheretheunderlyingnetworkisRTE(^z). where^anduarethevectorsofp(t)(^y)andup(t),respectively.ObservethattheproblemcanbedecomposedintojCjjjproblems,i.e.,oneproblemforeachpair(k;t)2C,wheretisa(discrete)timetoenterthenetwork.Inaddition,notethateachdecomposedproblemisequivalenttondingashortestpathfromnodeOkttooneofthecopiesofthedestinationnodeofthesameODpairk.AlthoughtheaboveproblemisformulatedbasedontheRTE(^z)network,SP(^y;^z)isanordinaryshortestpathproblemwithpathcostsp(t)(^y). 7{1 )ifandonlyifitisanoptimalsolutionofthecorrespondingSP(^y;^z)problem.

PAGE 113

7.2.1 ).Noticethat^up(t)isfeasibletotheSP(^y;^z)problem.Letup(t)denoteafeasiblesolutionofSP(^y;^z).Observethatup(t)[p(t)(^y)kt]0,8t2,k2Candp(t)2Pk(t).Because^up(t)[p(t)(^y)kt]=0,Xt2Xk2CXp(t)2Pk(t)[up(t)^up(t)][p(t)(^y)kt]0:+Xt2Xk2CXp(t)2Pk(t)p(t)(^y)up(t)^up(t)Xt2Xk2Ckt24Xp(t)2Pk(t)up(t)Xp(t)2Pk(t)^up(t)350+Xt2Xk2CXp(t)2Pk(t)p(t)(^y)up(t)^up(t)0+^Tu^T^u:

PAGE 114

whereB^zisthenode-arcincidentmatrixofRTE(^z),and^andykarethevectorsofa(t)(^y)andyka(t;^sa(t)),respectively.Usingthearcbasedformulation,Theorem 7.2.1 canberestatedasfollows.

PAGE 115

7.2 allowdevelopingatollpricingframeworksimilartothestaticcase(seeHearnandRamana[ 47 ]andBergendoretal.[ 8 ]). Let(^y;^g;^z)and^denoteafeasiblesolutionoftheDTDTAproblemandatollvector,respectively.Werefertothevector^asavalidtollwithrespectto(^y;^g;^z)if(^y;^g;^z)isasolutionofthetolleduserequilibriumproblem.Considerthefollowingproblem:

PAGE 116

wherethefeasibleregionisthesameasintheproblemSP(^y;^z)-Aandtheobjectivefunctionincludesthetollvector^inadditiontothetraveltime.ThefollowingcorollaryisadirectconsequenceofCorollary 7.2.1 ^kd(k)t^k8k2Candt2(7{7) wherekrepresentsthevectorofdualvariablesofcorrespondingnodeconservationconstrains( 7{2 ),andkarethedualvariablesoftheconstraints( 7{3 ).FromCorollary 7.3.1 itfollowsthatvector^isavalidtollifandonlyif(^y;^g)isasolutionoftheSP(^y;^z;^)-Aproblem.Observethat(^y;^g)isfeasibletotheproblem.Usingthedualitytheory,vector(^y;^g)isanoptimalsolutionofSP(^y;^z;^)-Aifandonlyiftherearevectors^kand^feasibletoequations( 7{5 )-( 7{7 ).

PAGE 117

Theorem 7.3.1 providesamathematicaldescriptionofthevalidtollset.Inparticular,givenafeasiblevector(^y;^g;^z),equations( 7{5 )-( 7{7 )describethesetofvalidtollsassociatedwithtriplet(^y;^g;^z).Usingtheset,secondarytollpricingproblemscanbeformulated.Observethatforallfeasiblevectors(^y;^g;^z),^=^0isavalidtollvector,i.e.,thevalidtollsetisnotempty. Let(y;g;z)andRTE(z)denotethesystemoptimumsolution,i.e.,thesolutionoftheDTDTAproblem,andthecorrespondingreducedtime-expandednetwork,respectively.Denea(t)=n(;s):(;s)2a(t);za(;s)=1o.Inwords,a(t)isasubsetofa(t)suchthatarca(;s)remainsintheRTE(z)network.BecauseRTE(z)consistsofarcsoftheforma(t;sa(t)),a(t)=(;sa()):(;sa())2a(t).Inaddition,givenzcomputethelowerandtheupperboundsintheequation( 5{9 ),i.e.,La(t)=Ps2a1a(s)za(t;s)andUa(t)=Ps2a1a(s)za(t;s).ConsiderthefollowingoptimizationproblemwiththeunderlyingRTE(z)networkstructure. whereBzdenotesthenode-arcincidentmatrixoftheRTE(z)networkandthevectorykconsistsofcomponentsyka(t;sa(t)).Inwords,problemDTDTA(z)istheDTDTAproblem,wherethevectorzisxedtothevalueofthevectorz.The

PAGE 118

theorembelowprovestheexistenceofanothervalidtollvectorwithrespecttothesystemoptimumsolution. 7{10 ),anda(t)a(t)=0,8a2Aandt2. wherevectors,,,andarethedualvariablesofcorrespondingconstraints( 7{8 ),( 7{10 ),and( 7{11 ),andiandjdenotethetailandtheheadnodesofthearca(t;sa(t)),respectively.Asaresult,ifyka(t;sa(t))>0thenka(t;sa(t))=0anda(t)(y)+Xr2j(t;sa(t))2a(r)"ryka(t;sa(t))a(r)(y)Xk2Cyka(r;sa(r))+a(r)a(r)#=jkik

PAGE 119

LetMSCa(t;sa(t))=Pr2j(t;sa(t))2a(r)hryka(t;sa(t))a(r)(y)Pk2Cyka(r;sa(r))+a(r)a(r)iandMSCdenotethevectorofMSCa(t;sa(t)).ByviewingthedualmultipliesikaspotentialsofthenodesintheRTE(z)network,itiseasytoshowthatisavalidtollvectorwithrespecttothesystemoptimumsolution(y;g;z);i.e.,(y;g)isanoptimumsolutionofSP(y;z;MSC)-A.Inaddition,ifr2issuchthat(t;sa(t))2a(r)thenryk1a(t;sa(t))a(r)(y)=ryk2a(t;sa(t))a(r)(y)=rxa(r)a(r)xa(t),8k1andk22C,k16=k2. FromtheKKTconditionitalsofollowsthata(t)0@La(t)X(;sa())2a(t)Xk2Cyka(;sa())1A=0; 7.3.1 itfollowsthattherearevectorsandsuchthatthesystemXk2C"(bk)Tk+kXt2hkt#=(+MSC)TXk2CykBTzk+MSC8k2Ckd(k)tk8k2Candt2 issatised.Ontheotherhand,theabovesystemfollowsdirectlyfromtheKKTconditionsoftheDTDTA(z)problem,andcorrespondingdualvariablesand

PAGE 120

ThetollvectorinTheorem 7.3.2 issimilartooneobtainedfromthemarginalsocialcostpriceinthestaticcase.Inparticular,therstcomponentofthetollvector,i.e.,Xr2j(t;sa(t))2a(r)"rxa(r)a(r)(xa(r))Xk2Cyka(r;sa(r))#; 7{10 ),andtheyrepresentthechangesintheobjectivefunctionvalueinthepresenceofminorperturbationsofthevaluesofUa(t)andLa(t),respectively.However,tochangethevaluesofUa(t)orLa(t),thebinaryvariableza(t;s)shouldbechangedaswell,andtheresultingvectormaynotbefeasibleoroptimal.Inthatsense,thesecondcomponentofthetollvector,i.e,Xr2j(t;sa(t))2a(r)a(r)a(r); 7{5 )-( 7{7 ).Intheprevioussection,wehaveshownthat(^;0;0)2=(^y;^g;^z)forallvectors(^y;^g;^z)feasibletotheDTDTA.Inthecaseofthesystemsolution,i.e.,(^y;^g;^z)=(y;g;z),Theorem 7.3.2 providesanothervalidtollvector;therefore,thereareatleasttwovectorsintheset=(y;g;z).Thelattersuggestsdevelopingtollpricingproblemswithasecondaryobjective,wheretheunderlyingfeasibleregionisconstructedbasedontheset=(y;g;z).Forinstance,atracmanagementmightbeinterestedin

PAGE 121

minimizingthetotaloperatingcost,costsofconstructingaproperinfrastructure(e.g.,constructingtollboothes),orminimizing/maximizingthetotalrevenue.Inaddition,thesetofvalidtollscanbenarrowedbyaddingconstraintstotheset=(y;g;z).Thelatterallowsustoconsidermorerealistictollsets.Althoughtheproblemsbelowareconstructedbasedonasystemsolution(y;g;z),asimilarframeworkappliestoanapproximatesolutionorafeasiblevectortotheDTDTAproblem. Letopa(t)andinfadenotebinaryvariablessuchthatopa(t)=8><>:1ifja(t;sa(t))j>00ifja(t;sa(t))j=0;andinfa=8>><>>:1ifPt2ja(t;sa(t))j>00ifPt2ja(t;sa(t))j=0: wherecopandcinfarethevectorsofoperatingandinfrastructurecosts,respectively,andMisasucientlylargenumber.Theoptimizationproblembelowminimizesthetotalrevenue.min(;;)2=(y;g;z)^yT 7{1 .Inparticular,addingthenonnegativityand/ormaximumtoll

PAGE 122

constraintspreventfromcharginganegativeand/oranunrealisticallylargetoll,respectively.Ifthetollsareallowedtobenegative;i.e.,itisallowedtosubsidizethedriversonparticulararcs,thenthemaximumsubsidyconstraintpreventspayingalargeamountofmoneytothedrivers.Theroadandtimerestrictionconstraintspreventtollingcertainroadsatcertaintimesofthetimehorizon,e.g.,from1:00amto5:00am.Finally,thevariabilityconstraintrestrictsthevariationofthetollduringtheplanninghorizon.Itisnoticedthataddingthoseconstraintsmaybetoorestrictiveandviolatethefeasibilityofthetollset.Insuchcasesthesecondbestpricingproblemshouldbeconsidered. Table7{1. Additionalconstraints Constraint 5 .Inthissection,theapproximate

PAGE 123

Table7{2. Distributionsofparametersofrandomlygeneratedtraveltimefunctions U[2,8] U[200,300] random(1or2) solutionisusedtoconstructavalidtollsetandconsiderseveralexamplesofthetollpricingproblemsbasedonthesetofvalidtollsandsomeoftheconstraintsfromTable 7{1 Allexperimentsareconductedonthe9-nodenetwork(seeFigure E{1 ,Appendix E ),whichhasfourODpars:(1,8),(1,9),(2,8),and(2,9).Intheproblems,thetimehorizonis[0;30),and=1,i.e.,=f0;1;:::;29g.Inaddition,thetraveltimefunctionsaregeneratedaccordingtotheformulaa(xa(t))=Aa+Ba(xa(t)=Ca)Da,whereparametersAa,Ba,CaandDa,arerandomnumbers(seeTable 7{2 ). Toimitatethecongestiononthearcs,foreachODpairademandisgeneratedaccordingtotheformulahkt=1:1t,wherethevalueoftdependsontimet(seeFigure 7{5 ),andisanumberuniformlygeneratedfromtheinterval[20;30].Asaresult,thedemandgraduallyincreasesduringthetimeinterval[0;9],remainsonthesamehighlevelduringthenext10unitsoftime,andthendecreasestotheinitiallevelattheendoftheplanninghorizon.Notethataccordingtotheformulathedemandvariesfromto2:36.Asitwasmentionedabove,wendanapproximatesolutionoftheproblembysolvingtheupperboundproblemandthe Figure7{5. Thevalueoft.

PAGE 124

renementproblem.Let(y;g;z)and=(y;g;z)denotetheobtainedsolutionandcorrespondingsetofvalidtolls,respectively. Oneofthetollpricingproblemsminimizesthetotalcollectedtoll,subjecttothesetofvalidtollvectors=(y;g;z)andadditionalnonnegativityandvariabilityconstraints.Belowisthemathematicalformulationoftheresultingproblem,whichwerefertoasMinRev(").min(;;)yTs.t.(;;)2=(y;g;z)"a(a;t;sa(t))(a;t+;sa(t+))"a8a2Aandt2[0;T]0 Inthenumericalexperimentswesolvetheproblemwithoutthevariabilityconstraintsaswellaswithvariabilityconstraints,where"=1,0.5,or0.1. Similarlyconsiderminimizingthetotalcostsubjecttothesamenonnegativityandvariabilityconstraints.Intheexperiments,thecopandcinfcostsarerandomlygeneratedfromtheintervals[5;10]and[100;150],respectively.WerefertotheresultingproblemasMinCost(").min(;;;op;inf)(cop)Top+(cinf)Tinfs.t.(;;)2=(y;g;z)"a(a;t;sa(t))(a;t+;sa(t+))"a8a2Aandt2[0;T]a(t;sa(t))Mopa(t)8a2Aandt2opa(t)infa8a2Aandt20;opa(t);infa2f0;1g;8a2A;andt2

PAGE 125

Inbothproblems,atoptimalitythetotalcollectedtollaswellasthetotalcostiscomputed(seeTable E{1 ,Appendix E ).Observethatinbothproblemstheobjectivefunctionvalueincreaseswiththedecreaseofthevalueof".Inthecaseof"=0:1,thevariabilityconstraintsaretoorestrictive,andthereisnononnegativevalidtollvectorthatsatisestheconstraints.Table E{2 inAppendix E describesthenumberoftollcollectingcentersthatarenecessarytobuild.Figures E{2 and E{3 (seeAppendix E )illustratetheinuenceofthevariabilityconstraintontheoptimaltollvector.Inparticular,theguresillustratethechangesinthetollonthearc(6;9)duringthetimeinterval[19;28].Observethatinthecaseof"a=0:5thetollvectorissmootherthaninothertwocases. IdeallyonewouldliketoconstructthesetofvalidtollsandthecorrespondingtollpricingproblemswithrespecttoanoptimalsolutionoftheDTDTAproblem.However,thelatterbelongstotheclassofnonlinearmixedintegerprograms,anditishardtondanexactsolutionoftheproblem.Section 7.5 discussesseveralexamplesoftollpricingproblemsbasedonanapproximatesolution,whichisobtainedusingthetechniquediscussedinChapter 5 .Forotherapproximatesolutions,thetollpricingframeworkissimilar.Inparticular,givenanapproximatesolution(y;g;z),constructthecorrespondingRTEnetwork,derivethesetof

PAGE 126

validtollsusingequations( 7{5 )-( 7{7 ),i.e.,theset=(y;g;z),andsolveatollpricingproblembasedontheset=(y;g;z).

PAGE 127

Inthisdissertationwehavediscussedbilinearreductionapproachestosolvethepiecewiselinearnetworkow,xedchargenetworkow,anddynamicpricingproblems.InparticularwehaveshownthataglobalsolutionofthebilinearproblemsisasolutionorleadstoasolutiontotheinitialMIPformulation.Theproposedheuristicalgorithmsallowndinganapproximatesolutiontothereductionproblem.Althoughtheproceduresoftenconvergetoaglobalsolution,theycanonlyguaranteetheconvergencetoalocalminimumoftheproblembecausethebilinearreductionproblemisnotconvex.Tondaglobalsolutiontotheproblem,onemayndusefulincorporatingthecuttingplanemethodinwhichthemasterproblemgeneratescuttingplanestoeliminatethecurrentlocalminimumfromthefeasibleregion,andtheheuristicprocedureisusedtondanewlocalminimumoftheresultingproblem.Becauseoftheeectivenessoftheheuristicprocedure,overallperformanceofthecuttingplanealgorithmisexpectedtobeeectiveaswell. Thedynamictracassignmentprobleminthedissertationisconstructedbasedontheassumptionthatthetraveltimeisafunctionofthedensity.Inaddition,weassumethatthedemandisxedandtheeventsoccurinaperiodicfashion.Themathematicalformulationoftheproblembelongstotheclassofnonlinearmixedintegerproblemsandiscomputationallyhardtosolve.However,thetheoreticalresultssuggestsolvingthelinearmixedintegerboundingproblemsinstead.Inparticular,wehaveshownthatbydecreasingthediscretizationparameterasolutionoftheboundingproblemcanbemadearbitrarilyclosetoasolutionoftheinitialproblem.Tosolvetheboundingproblem,wehaveproposed 115

PAGE 128

aheuristicalgorithm,whichisbasedonabilinearrelaxationofthefeasibleregion.Asaresult,ineachiterationitisrequiredtondalocalminimumofthebilinearproblem.AvailablecommercialsolversspendabouthalfofthetotalCPUtimeonndingafeasiblesolutionduringtherstiterationoftheheuristicprocedure;therefore,afastprocedureforndingafeasiblesolutionisdesirable.Inaddition,wewouldliketoexploreotheralgorithmsthatcanbeusedtondalocalminimumoftheproblem. InChapter 7 wehaveusedCPLEXLPandMIPsolverstosolveMinRev(")andMinCost(")tollpricingproblems,respectively.TosolvelargeMinRev(")problemsadecompositiontechniquesimilartheoneinBaietal.[ 3 ]canbeutilized.InthecaseofMinCost("),theprobleminvolvesbinaryvariables,andtosolvelargeproblemsaheuristicalgorithmsshouldbedeveloped.

PAGE 129

2 TableA{1. Setofproblems. SetNo 123522U[10,20]5 2 10 3 U[20,30]5 4 10 5 U[30,40]5 6 10 7 2010033U[10,20]5 8 10 9 U[20,30]5 10 10 11 U[30,40]5 12 10 13 4030044U[10,20]5 14 10 15 U[20,30]5 16 10 17 U[30,40]5 18 10 19 10020002020U[10,20]5 20 10 21 U[20,30]5 22 10 23 U[30,40]5 24 10 25 20050005050U[10,20]5 26 10 27 U[20,30]5 28 10 29 U[30,40]5 30 10

PAGE 130

TableA{2. Computationalresultsofsets1-18:qualityofthesolutionandtheCPUtimes. CPUTime Iterations DCUP DSSP DCUPDSSP DCUPDSSP RE(%) RE(%) Aver.Aver. Aver.Aver. SetNo (min,max) (min,max) (min,max)(min,max) (min,max)(min,max) 1 1.39 1.45 0.020.05 2.235.30 (0.00,8.47) (0.00,9.19) (0.01,0.04)(0.02,0.10) (2,3)(3,9) 2 1.43 1.55 0.050.14 2.205.47 (0.00,9.45) (0.00,9.45) (0.03,0.09)(0.05,0.22) (2,3)(3,9) 3 0.79 0.73 0.020.06 2.175.77 (0.00,3.48) (0.00,5.04) (0.01,0.04)(0.02,0.09) (2,3)(3,8) 4 0.81 0.91 0.050.14 2.205.67 (0.00,3.47) (0.00,10.35) (0.03,0.09)(0.07,0.23) (2,3)(3,8) 5 0.80 0.89 0.070.17 2.376.27 (0.00,4.22) (0.00,6.03) (0.05,0.10)(0.06,0.26) (2,4)(3,9) 6 0.83 0.88 0.060.16 2.506.43 (0.00,4.42) (0.00,6.11) (0.04,0.11)(0.08,0.23) (2,4)(3,9) 7 1.05 1.25 0.070.23 2.478.20 (0.00,6.24) (0.00,5.63) (0.04,0.16)(0.12,0.38) (2,5)(5,13) 8 1.13 1.28 0.060.23 2.478.73 (0.00,6.12) (0.00,5.59) (0.03,0.12)(0.12,0.39) (2,4)(5,14) 9 0.73 1.02 0.080.23 2.738.33 (0.00,3.48) (0.00,5.38) (0.05,0.12)(0.13,0.31) (2,4)(5,11) 10 0.91 1.20 0.030.09 2.477.63 (0.00,5.59) (0.00,5.34) (0.01,0.05)(0.03,0.16) (2,4)(3,14) 11 0.92 0.69 0.070.23 2.508.80 (0.00,4.31) (0.00,4.13) (0.03,0.13)(0.03,0.13) (2,4)(5,13) 12 0.96 0.60 0.070.23 2.879.57 (0.00,4.55) (0.00,3.93) (0.04,0.13)(0.13,0.33) (2,5)(6,14) 13 0.71 1.25 0.070.27 2.5310.00 (0.00,6.27) (0.00,6.40) (0.04,0.10)(0.16,0.41) (2,4)(6,17) 14 0.92 1.46 0.070.30 2.6310.77 (0.00,6.06) (0.00,6.42) (0.03,0.11)(0.13,0.54) (2,3)(6,17) 15 0.99 1.34 0.100.34 3.0010.47 (0.00,3.03) (0.11,3.36) (0.04,0.21)(0.21,0.62) (2,6)(6,18) 16 1.32 1.28 0.070.28 2.8011.30 (0.00,4.85) (0.00,3.43) (0.02,0.14)(0.09,0.56) (2,5)(8,18) 17 1.03 0.96 0.090.36 2.7711.73 (0.00,4.26) (0.00,2.83) (0.06,0.17)(0.06,0.17) (2,6)(8,17) 18 1.03 1.00 0.070.28 3.2712.60 (0.00,4.18) (0.00,2.91) (0.02,0.22)(0.13,0.52) (2,7)(10,17)

PAGE 131

TableA{3. Computationalresultsofsets1-18:DSSPvs.DCUP. SetNo (%)(%)(%)(%) 1 0.2120773 (3.42) 2 0.2523374 (3.43) 3 0.30271756 (4.37) 4 0.52232057 (9.54) 5 0.43403030 (1.80) 6 0.36303060 (1.75) 7 0.71403030 (4.91) 8 0.75333730 (5.14) 9 0.62502030 (5.13) 10 0.80502723 (4.92) 11 0.17473023 (1.09) 12 0.11373726 (0.54) 13 0.67631027 (3.40) 14 0.77501337 (4.33) 15 0.58602020 (2.38) 16 0.58374023 (2.61) 17 0.45504010 (1.28) 18 0.43603010 (1.37) A-aver.(max)improvement;B-DCUPisbetterthanDSSP; C-DSSPisbetterthanDCUP;D-botharethesame.

PAGE 132

TableA{4. Computationalresultsforsets19-30. CPUTime Iterations DCUPvs.DSSP DCUPDSSP DCUPDSSP Set ABCD Aver.Aver. Aver.Aver. No (%)(%)(%)(%) (min,max)(min,max) (min,max)(min,max) 19 0.87570.5843 0.401.69 4.7019.59 (2.02)(1.28) (0.18,0.78)(1.04,2.90) (2,9)(13,34) 20 0.56430.3757 0.492.03 5.1720.90 (1.74)(1.06) (0.27,0.83)(1.11,3.89) (3,10)(13,36) 21 0.63640.4233 0.441.76 5.1020.03 (2.09)(1.26) (0.21,0.91)(1.14,2.19) (3,10)(15,24) 22 0.66700.3030 0.511.96 6.0322.57 (1.74)(0.62) (0.24,1.12)(1.12,2.55) (3,13)(14,32) 23 0.54770.4223 0.462.03 5.3323.13 (1.04)(0.95) (0.81,0.22)(1.32,2.61) (3,9)(16,30) 24 0.50570.4643 0.492.06 5.7023.70 (1.27)(1.52) (0.23,0.88)(1.19,2.94) (3,9)(15,33) 25 0.58330.2767 1.345.31 6.0322.90 (1.38)(0.79) (0.65,2.41)(4.55,6.95) (3,11)(19,29) 26 0.72370.4663 1.716.50 7.8027.83 (1.63)(1.39) (0.60,2.94)(4.63,8.54) (3,14)(21,36) 27 0.56630.2537 1.635.96 6.9025.03 (1.27)(0.71) (0.51,3.59)(4.17,9.95) (3,14)(19,40) 28 0.48700.2830 1.976.83 8.8029.03 (1.14)(1.29) (1.02,3.21)(4.89,9.03) (5,15)(23,42) 29 0.34700.1430 1.836.28 7.9726.27 (0.77)(0.36) (0.84,3.39)(4.78,8.71) (4,14)(21,37) 30 0.39670.2933 2.367.21 10.1330.60 (0.89)(0.69) (1.06,4.11)(5.29,10.75) (4,19)(24,43) A-aver.(max)offDSSPfDCUP C-aver.(max)offDCUPfDSSP Computationalresultsforthecombinedmode. A(%)B(%) A(%)B(%) Aver. Aver. Aver. SetNo (min,max) SetNo (min,max) SetNo (min,max) 1 0.4610 11 0.1947 21 0.17100 (0.02,0.70) (0.01,0.53) (0.01,0.58) 2 0.437 12 0.1643 22 0.1293 (0.01,0.85) (0.01,0.54) (0.01,0.57) 3 0.5920 13 0.6150 23 0.1597 (0.04,1.57) (0.01,3.40) (0.01,0.47) 4 0.6120 14 0.4437 24 0.0997 (0.04,1.87) (0.04,1.40) (0.01,0.39) 5 0.3043 15 0.2057 25 0.17100 (0.07,1.02) (0.03,0.74) (0.01,0.49) 6 0.4030 16 0.1443 26 0.13100 (0.02,1.09) (0.02,0.68) (0.01,0.61) 7 0.3140 17 0.1957 27 0.16100 (0.01,1.14) (0.01,0.90) (0.02,0.44) 8 0.2640 18 0.3060 28 0.14100 (0.02,1.35) (0.01,1.62) (0.02,0.38) 9 0.3347 19 0.2793 29 0.12100 (0.03,0.97) (0.01,1.32) (0.02,0.32) 10 0.5940 20 0.1893 30 0.08100 (0.01,3.48) (0.01,1.08) (0.02,0.29) A-percentageofimprovement;B-percentageofproblemsthatareimproved.

PAGE 133

3 TableB{1. Setofproblems. Set Spp/DemVar.Fixed No No G1 1 20/1003/3U[1,5]U[50,100] 2 U[100,200] 3 U[200,400] 4 U[10,20]U[50,100] 5 U[100,200] 6 U[200,400] 7 U[30,40]U[50,100] 8 U[100,200] 9 U[200,400] G2 1 40/3004/4U[1,5]U[50,100] 2 U[100,200] 3 U[200,400] 4 U[10,20]U[50,100] 5 U[100,200] 6 U[200,400] 7 U[30,40]U[50,100] 8 U[100,200] 9 U[200,400] G3 1 100/100010/10U[1,5]U[50,100] 2 U[100,200] 3 U[200,400] 4 U[10,20]U[50,100] 5 U[100,200] 6 U[200,400] 7 U[30,40]U[50,100] 8 U[100,200] 9 U[200,400] G4 1 150/300015/15U[1,5]U[50,100] 2 U[100,200] 3 U[200,400] 4 U[10,20]U[50,100] 5 U[100,200] 6 U[200,400] 7 U[30,40]U[50,100] 8 U[100,200] 9 U[200,400]

PAGE 134

TableB{2. ComputationalresultsofgroupsG1andG2:qualityofthesolutionsandtheCPUtimes. Average Var.Fixed FinalSol.* BestSol.** CPUTime costcost ADCUPDSSP ADCUPDSSP ADCUPDSSP G1U[1,5]U[50,100] 2.417.5 1.67.8 0.891.47 (0.0,13.3)(0.0,53.0) (0.0,13.3)(0.0,21.8) U[100,200] 3.39.4 2.69.4 0.911.60 (0.0,16.9)(0.0,33.4) (0.0,16.9)(0.0,33.4) U[200,400] 3.513.4 2.710.3 0.901.66 (0.0,14.9)(0.0,34.3) (0.0,14.9)(0.0,34.2) U[10,20]U[50,100] 0.61.1 0.20.2 0.430.45 (0.0,3.7)(0.0,7.3) (0.0,1.5)(0.0,2.3) U[100,200] 0.84.3 0.51.3 0.460.61 (0.0,2.8)(0.0,24.2) (0.0,2.8)(0.0,10.0) U[200,400] 1.612.9 0.95.0 0.450.72 (0.0,8.7)(0.0,43.1) (0.0,4.3)(0.0,16.2) U[30,40]U[50,100] 0.20.5 0.10.1 0.330.20 (0.0,1.7)(0.0,4.8) (0.0,0.9)(0.0,1.5) U[100,200] 0.51.3 0.20.5 0.360.27 (0.0,1.9)(0.0,6.1) (0.0,1.9)(0.0,2.9) U[200,400] 0.91.8 0.40.9 0.390.48 (0.0,5.3)(0.0,10.1) (0.0,2.5)(0.0,6.0) G2U[1,5]U[50,100] 3.417.2 2.511.5 1.113.16 (0.0,15.3)(0.0,41.1) (0.0,11.1)(0.0,34.1) U[100,200] 4.117.0 3.715.1 1.093.32 (0.0,15.3)(0.0,37.8) (0.0,14.4)(0.0,31.2) U[200,400] 5.413.8 5.113.4 1.053.77 (0.0,13.3)(0.0,36.7) (0.0,12.2)(0.0,31.4) U[10,20]U[50,100] 0.52.4 0.11.2 0.541.20 (0.0,2.2)(0.0,17.8) (0.0,4.4)(0.0,4.4) U[100,200] 1.08.9 0.54.1 0.531.46 (0.0,6.1)(0.0,28.1) (0.0,4.3)(0.0,15.2) U[200,400] 2.511.5 1.66.3 0.491.39 (0.0,6.0)(0.0,29.0) (0.0,4.7)(0.0,15.0) U[30,40]U[50,100] 0.20.7 0.10.4 0.560.76 (0.0,1.4)(0.0,2.4) (0.0,1.6)(0.0,1.6) U[100,200] 0.82.0 0.21.1 0.571.15 (0.0,5.1)(0.0,6.5) (0.0,1.8)(0.0,4.9) U[200,400] 1.44.5 0.92.7 0.531.30 (0.0,4.9)(0.0,18.8) (0.0,4.6)(0.0,7.5) A-ADCUPisbetterthanDSSP;B-DSSPisbetterthanADCUP. *Thenalsolutionreturnedbythealgorithmwhenitstops.**Thebestsolutionfoundduringtheiterativeprocess.

PAGE 135

TableB{3. ComputationalresultsofgroupsG1andG2:thepercentageofproblemwhereoneofthealgorithmsndsabettersolutionthananotherone. Var.Fixed FinalSol.* BestSol.** costcost AB AB G1U[1,5]U[50,100] 7710 770 U[100,200] 7013 6010 U[200,400] 870 703 U[10,20]U[50,100] 237 1310 U[100,200] 4320 307 U[200,400] 6310 5313 U[30,40]U[50,100] 2310 1313 U[100,200] 3017 3010 U[200,400] 3020 2020 G2U[1,5]U[50,100] 9010 937 U[100,200] 903 900 U[200,400] 903 873 U[10,20]U[50,100] 5713 6310 U[100,200] 873 807 U[200,400] 807 803 U[30,40]U[50,100] 4017 407 U[100,200] 6713 537 U[200,400] 7310 7010 A-ADCUPisbetterthanDSSP;B-DSSPisbetterthanADCUP. *Thenalsolutionreturnedbythealgorithmwhenitstops.**Thebestsolutionfoundduringtheiterativeprocess.

PAGE 136

TableB{4. ComputationalresultsofgroupsG3andG4. PercentageofProblems Average Var.Fixed Aver.(min,max) FinalSol.* BestSol.** CPUTime costcost FinalSol.* BestSol.** AB AB ADCUPDSSP G3U[1,5]U[50,100] 16.8 13.8 973 1000 2.1012.86 (-0.4,39.5) (0.4,23.6) U[100,200] 17.0 16.9 973 973 1.9214.16 (-2.4,29.6) (-0.7,28.7) U[200,400] 16.0 15.3 1000 1000 1.9614.34 (3.6,29.8) (3.6,29.8) U[10,20]U[50,100] 3.3 1.2 937 903 2.1513.08 (-0.4,7.4) (-0.2,2.9) U[100,200] 7.7 3.9 973 970 2.2213.25 (-0.8,16.4) (0.0,7.6) U[200,400] 15.7 8.1 1000 1000 2.2412.97 (0.7,29.9) (0.2,13.8) U[30,40]U[50,100] 1.3 0.5 907 837 2.1111.38 (-0.3,3.7) (-0.1,1.6) U[100,200] 1.9 1.2 8713 930 2.1613.04 (-1.2,5.5) (0.0,3.5) U[200,400] 4.7 3.2 1000 1000 2.2613.56 (0.5,14.4) (0.3,7.5) G4U[1,5]U[50,100] 21.6 17.1 1000 1000 3.7261.31 (8.2,32.6) (7.8,26.4) U[100,200] 21.2 20.7 1000 1000 3.7267.87 (13.5,32.5) (14.1,31.5) U[200,400] 18.4 18.4 1000 1000 3.5967.57 (10.7,31.8) (10.7,31.5) U[10,20]U[50,100] 5.0 1.1 1000 900 4.4558.87 (0.1,10.6) (0.0,3.6) U[100,200] 12.3 2.9 1000 970 4.3963.07 (4.2,23.0) (0.0,6.6) U[200,400] 18.2 6.0 1000 1000 4.2369.47 (8.9,29.8) (1.1,11.8) U[30,40]U[50,100] 1.8 0.3 1000 900 4.2458.46 (0.0,3.9) (0.0,1.0) U[100,200] 4.0 0.7 1000 870 4.3057.12 (0.6,9.4) (0.0,2.0) U[200,400] 8.8 1.3 1000 7723 4.4859.79 (3.2,15.5) (-1.5,3.8) A-ADCUPisbetterthanDSSP;B-DSSPisbetterthanADCUP. *Thenalsolutionreturnedbythealgorithmwhenitstops.**Thebestsolutionfoundduringtheiterativeprocess.

PAGE 137

4 TableC{1. Thequalityofthesolution:Procedure 6 Procedure 6 usingProcedure 7 tondvector" =1=2 ObjRE%A ObjRE%A ObjRE%A 5-12[10,100] 132,803 131,2061.19131,3121.11131,6120.88[50,150] 216,831 215,5620.58215,9560.40215,9920.39[100,200] 283,494 282,4180.38282,7520.26282,9130.20[150,250] 320,602 319,3250.39319,5010.34319,5180.345-52[10,100] 543,397 537,5881.07538,1820.97539,2640.76[50,150] 910,778 906,2110.50907,3040.381 909,1560.183 [100,200] 1,209,347 1,206,8400.211,208,1550.102 1,208,4610.072 [150,250] 1,378,794 1,375,4310.251,376,0590.201,377,2270.111 10-12[10,100] 246,410 243,2901.29244,2680.87244,7540.70[50,150] 420,000 417,0320.70417,0340.70418,3370.40[100,200] 563,297 562,0490.221 562,2860.18562,6650.112 [150,250] 648,362 646,2400.33646,1870.331 646,8750.232 10-52[10,100] 1,142,132 1,132,2920.871,134,6300.651,136,3690.51[50,150] 1,878,849 1,872,7710.321,873,8330.271,877,1280.091 [100,200] 2,470,661 2,466,9890.152,469,4740.053 2,470,966-0.017 [150,250] 2,799,250 2,796,3960.101 2,798,0490.043 2,798,8080.025 20-12[10,100] 542,549 537,3460.95539,2100.61539,7540.51[50,150] 881,231 875,9630.59877,1760.46878,8360.27[100,200] 1,152,475 1,149,7010.241,150,2680.191,150,9670.13[150,250] 1,305,170 1,301,5870.281,302,1340.241 1,302,9630.173 20-52[10,100] 2,303,214 2,285,5080.772,286,7160.712,292,5450.46[50,150] 3,791,838 3,776,7680.403,780,3100.313,787,1920.121 [100,200] 4,993,056 4,983,5990.194,987,7460.114,992,3680.013 [150,250] 5,671,140 5,661,3380.175,666,8160.085,669,5740.033 A-NumberofproblemswheretheheuristicprocedureprovidesabettersolutionthanCPLEX.

PAGE 138

TableC{2. Thequalityofthesolution:Procedure 5 Procedure 5 ObjRE% 5-12[10,100] 1.02 132,803 123,7476.84 [50,150] 0.95 216,831 208,9763.65 [100,200] 0.86 283,494 278,3421.82 [150,250] 0.76 320,602 315,9691.45 5-52[10,100] 1.03 543,397 498,5038.27 [50,150] 0.96 910,778 873,7494.08 [100,200] 0.88 1,209,347 1,183,3192.16 [150,250] 0.78 1,378,794 1,357,7181.53 10-12[10,100] 1.13 246,410 226,5388.11 [50,150] 0.98 420,000 403,3603.96 [100,200] 0.98 563,297 553,2781.78 [150,250] 0.84 648,362 641,3891.08 10-52[10,100] 1.06 1,142,132 1,059,6957.24 [50,150] 0.98 1,878,849 1,811,7663.58 [100,200] 0.98 2,470,661 2,425,8021.82 [150,250] 0.77 2,799,250 2,765,5601.20 20-12[10,100] 1.08 542,549 504,7716.97 [50,150] 1.00 881,231 850,3083.51 [100,200] 1.00 1,152,475 1,133,5551.64 [150,250] 0.79 1,305,170 1,292,0681.01 20-52[10,100] 1.08 2,303,214 2,139,8377.11 [50,150] 1.00 3,791,838 3,658,2023.53 [100,200] 1.00 4,993,056 4,900,8791.85 [150,250] 0.79 5,671,140 5,601,9321.22 TheCPUtimeoftheprocedures. Procedure 6 Procedure 5 CPUA CPUA CPUA CPUA 5-12[10,100] 335 0.52641 0.78432 1.78188 0.191,811 [50,150] 1,113 0.711,570 0.831,341 2.29486 0.205,621 [100,200] 1,580 0.602,656 0.841,886 2.55617 0.266,148 [150,250] 559 0.74752 0.96583 3.10180 0.212,651 5-52[10,100] 821 10.9675 14.9855 40.9720 3.77218 [50,150] 348 12.7327 18.4219 52.067 4.8272 [100,200] 391 15.8625 19.0620 56.017 5.7967 [150,250] 596 12.7147 28.5121 60.4310 3.23185 10-12[10,100] 820 1.19688 1.64500 4.27192 0.392,096 [50,150] 748 1.24602 1.87401 5.48137 0.421,802 [100,200] 802 1.35594 2.05392 5.77139 0.441,814 [150,250] 660 1.47449 2.16306 5.30124 0.431,542 10-52[10,100] 376 24.5415 35.4211 103.824 8.3045 [50,150] 324 25.6913 39.518 115.943 10.1932 [100,200] 408 27.1915 39.9810 118.083 11.8934 [150,250] 327 26.3412 37.679 107.773 7.1646 20-12[10,100] 857 2.51341 3.69232 10.8279 0.851,005 [50,150] 657 2.72242 4.21156 12.7252 0.92717 [100,200] 633 3.07206 4.00158 12.0553 0.93684 [150,250] 675 2.75246 4.04167 10.7263 0.77873 20-52[10,100] 890 52.0217 76.7412 237.574 17.3851 [50,150] 750 55.2714 86.829 256.993 28.6426 [100,200] 756 58.0413 84.789 248.123 33.3123 [150,250] 785 57.7014 77.9810 236.523 14.1156

PAGE 139

6 FigureD{1. TwoNetworks. TableD{1. Computationalresultsoftheexperiments. HeuristicApproach Cplex Rel. ACPUIter. CPU Err. B (%)(sec.) (sec.) (%) (ratio) 4-Nodes,[0,10),OBJ1 12%0.8110.23 0.84 4.09% 1.74 4-Nodes,[0,10),OBJ2 8%1.0310.35 1.00 4.13% 1.41 4-Nodes,[0,30),OBJ1 20%19.7634.47 376.58 3.92% 22.62 4-Nodes,[0,30),OBJ2 22%27.0538.64 396.51 3.32% 9.55 9-Nodes,[0,10),OBJ1 28%633.0040.58 1,515.44 4.16% 7.80 9-Nodes,[0,10),OBJ2 10%111.929.98 1,758.65 2.09% 38.91 9-Nodes,[0,30),OBJ1 30%1338.377.56 5,000.30 -3.33% 12.65 9-Nodes,[0,30),OBJ2 27%1029.794.53 4,945.14 -4.22% 14.28 A-percentageoftheproblemsnotsolvedbytheheuristic B-CPUCplex/CPUheur:

PAGE 140

TableD{2. Computationalresultsofthecombinedmode. HeuristicApproach Cplex Rel. ACPU CPU Err. B (%)(sec.) (sec.) (%) (ratio) 4-Nodes,[0,10) 4%0.59 0.97 3.63% 1.85 4-Nodes,[0,30) 10%19.91 543.00 3.70% 24.22 9-Nodes,[0,10) 6%166.86 1,359.16 2.50% 30.27 9-Nodes,[0,30) 14%706.03 4,952.60 -3.62% 16.97 A-percentageoftheproblemsnotsolvedbytheheuristic B-CPUCplex/CPUheur:

PAGE 141

7 FigureE{1. 9-nodenetwork. TableE{1. Thetotalcollectedtollandthetotalcostforeachproblemandparameter". TotalCost A"a=1"a=0:5"a=0:1 A"a=1"a=0:5"a=0:1 385440155157NoSol. 267032983870NoSol. 543355636978NoSol. 124916242418NoSol. TableE{2. Thenumberoftollcollectingcentersforeachproblemandparameter". NumberofTollCollectingCenters A"a=1"a=0:5"a=0:1 171617N/A 8912N/A A-NoVariabilityConstraint 129

PAGE 142

FigureE{2. Thetollvectorfordierentvaluesof"intheMinRev(")problem. FigureE{3. Thetollvectorfordierentvaluesof"intheMinCost(")problem.

PAGE 143

[1] C.Agnew,\TheTheoryofCongestionTolls,"JournalofRegionalScience,vol.17(3),pp.381-393,1977. [2] R.Ahuja,T.Magnanti,andJ.Orlin,NetworkFlows,PrenticeHall,UpperSaddleRiver,1993. [3] L.Bai,D.Hearn,andS.Lawphongpanich,\DecompositionTechniquesfortheMinimumTollRevenueProblem,"Networks,vol.44(2),pp.142-150,2004. [4] R.Barr,F.Glover,andD.Klingman,\ANewOptimizationMethodforLargeScaleFixedChargeTransportationProblems,"OperationsResearch,vol.29(3),pp.448-463,1981. [5] M.Bazaraa,H.Sherali,andC.Shetty,NonlinearProgramming:TheoryandAlgorithms,Wiley,2ndEdition,NewYork,1993. [6] M.Beckmann,\OnOptimalTollsforHighways,TunnelsandBridges,"In:VehicularTracScience,AmericanElsevier,NewYork,1965. [7] G.Bellei,G.Gentile,andN.Papola,\NetworkPricingOptimizationinMulti-UserandMultimodalContextwithElasticDemand,"TransportationResearchPartB:Methodological,vol.36(9),pp.779-798,2002. [8] P.Bergendor,D.Hearn,andM.Ramana,\CongestionTollPricingofTracNetworks,"NetworkOptimization,P.Pardalos,D.Hearn,andW.Hager(Eds),LectureNotesinEconomicsandMathematicalSystems450,Springer-Verlag,pp.51-71,1997. [9] L.Brotcorne,M.Labbe,P.Marcotte,andG.Savard,\ABilevelModelforTollOptimizationonaMulticommodityTransportationNetwork,"TransportationScience,vol.35(4),pp.345-358,2001. [10] L.Brotcorne,D.DeWolf,M.Gendreau,andM.Labbe,\ADynamicUserEquilibriumModelforTracAssignmentinUrbanAreas,"TransportationandNetworkAnalysis:CurrentTrends,M.GendreauandP.Marcotte(Eds.),Kluwer,pp.49-69,Boston,2002. [11] D.Boyce,B.Ran,andL.LeBlanc,\SolvinganInstantaneousDynamicUser-OptimalRouteChoiceModel,"TransportationScience,vol.29(2),pp.128-142,1995. 131

PAGE 144

[12] A.CabotandS.Erenguc,\SomeBranch-and-BoundProceduresforFixed-CostTransportationProblems,"NavalResearchLogisticsQuarterly,vol.31,pp.145-154,1984. [13] M.Carey,\AConstraintQualicationforaDynamicTracAssignmentModel,"TransportationScience,vol.20(1),pp.55-58,1986. [14] M.Carey,\OptimalTime-VaryingFlowonCongestedNetwork,"OperationsResearch,vol.35(1),pp.58-69,1987. [15] M.Carey,\DynamicTracAssignmentwithMoreFlexibleModellingwithinLinks,"NetworkandSpatialEconomics,vol.1,pp.349-375,2001. [16] M.Carey,\LinkTravelTimesI:DesirableProperties,"NetworkandSpatialEconomics,vol.4,pp.257-268,2004. [17] M.Carey,\EcientDiscretisationforLinkTravelTimeModels,"NetworkandSpatialEconomics,vol.4,pp.269-290,2004. [18] M.CareyandM.McCartney,\BehaviourofaWhole-LinkTravelTimeModelUsedinDynamicTracAssignment,"TransportationResearchB,vol.36,pp.83-95,2002. [19] M.Carey,Y.Ge,andM.McCartney,\AWhole-LinkTravel-TimeModelwithDesirableProperties,"TransportationScience,vol.37(1),pp.83-96,2003. [20] M.CareyandY.Ge,\ConvergenceofaDiscretisedTravel-TimeModel,"TransportationScience,vol.39(1),pp.25-38,2005. [21] M.CareyandA.Srinivasan,\SolvingaClassofNetworkModelsforDynamicFlowControl,"EuropeanJournalofOperationalResearch,vol.75,pp.151-170,1994. [22] M.CareyandA.Srinivasan,\Externalities,AverageandMarginalCosts,andTollsonCongestedNetworkswithTime-VaryingFlows,"OperationsResearch,vol.41(1),pp.217-231,1993. [23] M.CareyandE.Subrahmanian,\AnApproachtoModellingTime-VaringFlowsonCongestedNetworks,"TransportationResearchB,vol.34(3),pp.157-183,2000. [24] M.ChenandD.Bernstein,\SolvingtheTollDesignProblemwithMultipleUserGroups,"TransportationResearchB,vol.38(1),pp.61-79,2004. [25] H.ChenandC.Hsueh,\AModelandanAlgorithmfortheDynamicUser-OptimalRouteChoiceProblem,"TransportationResearchB,vol.32(3),pp.219-234,1998.

PAGE 145

[26] [27] L.CooperandC.Drebes,\AnApproximateSolutionMethodfortheFixedChargeProblem,"NavalResearchLogisticsQuarterly,vol.14,pp.101-113,1967. [28] C.Daganzo,\PropertiesofLinkTravelTimeFunctionUnderDynamicLoading,"TransportationResearchB,vol.29(2),pp.95-98,1995. [29] S.DafermosandF.Sparrow,\TheTracAssignmentProblemforaGeneralNetwork,"JournalofResearchoftheNationalBureauofStandards,vol.73B,pp.91-118,1969. [30] S.DafermosandF.Sparrow,\OptimalResourceAllocationandTollPatternsinUser-OptimizedTransportationNetwork,"JournalofTransportationEconomicsandPolicy,vol.5,pp.198-200,1971. [31] M.Diaby,\SuccessiveLinearApproximationProcedureforGeneralizedFixed-ChargeTransportationProblem,"JournaloftheOperationsResearchSociety,vol.42,pp.991-1001,1991. [32] O.Drissi-KaitouniandA.Hameda-Benchekroun,\ADynamicTracAssignmentModelandaSolutionAlgorithm,"TransportationScience,vol.26(2),pp.119-128,1992. [33] L.EngelsonandP.Lindberg,\CongestionPricingofRoadNetworkwithUserHavingDierentTimeValue,"MathematicalandComputationalModelsforCongestionCharging,S.Lawphongpanich,D.HearnandM.Smith(Eds),Springer,pp.81-104,NewYork,2006. [34] P.Ferrari,\RoadNetworkTollPricingandSocialWelfare,"TransportationResearchB,vol.36(5),pp.471-483,2002. [35] M.FlorianandD.Hearn,\NetworkEquilibriumandPricing,"HandbookofTransportationScience,2ndEdintion,KluwerAcademicPublishers,Norwell,2003. [36] M.FlorianandM.Klein,"DeterministicProductionPlanningwithConcaveCostsandCapacityConstraints,"ManagementScience,vol.18(1),pp.12-20,1971. [37] T.Friesz,L.Leque,R.Tobin,andB.Wie,\DynamicNetworkTracAssignmentConsideredasaContinuousTimeOptimalControlProblem,"OperationsResearch,vol.37(6),pp.893-901,1989.

PAGE 146

[38] T.Friesz,D.Bernstein,T.Smith,R.Tobin,andB.Wie,\AVariationalInequalityFormulationoftheDynamicNetworkUserEquilibriumProblem,"OperationsResearch,vol.41(1),pp.178-191,1993. [39] [40] A.Garcia,D.Reaume,andR.Smith,\FictitiousPlayforFindingSystemOptimalRoutinginDynamicTracNetworks,"TransportationResearchB,vol.34(2),pp.147-156,2000. [41] J.GeunesandP.Pardalos,SupplyChainOptimization,Springer,NewYork,2005. [42] J.Geunes,E.Romeijn,andK.Taae,\RequirementsPlanningwithPricingandOrderSelectionFlexibility,"OperationsResearch,vol.54(2),pp.394-401,2006. [43] S.Gilbert,\CoordinationofPricingandMulti-PeriodProductionforConstantPricedGoods,"EuropeanJournalofOperationalResearch,vol.114,pp.330-337,1999. [44] P.Gray,\ExactSolutionfortheFixed-ChargeTransportationProblem,"OperationsResearch,vol.19(6),pp.1529-1538,1971. [45] G.GuisewiteandP.Pardalos,\MinimumConcave-CostNetworkFlowProblems:Applications,Complexity,andAlgorithms,"AnnalsofOperationsResearch,vol.25,pp.75-100,1990. [46] S.HanandB.Heydecker,\ConsistentObjectiveandSolutionofDynamicUserEquilibriumModels,"TransportationResearchB,vol.40(1),pp.16-34,2006. [47] D.HearnandM.Ramana,\SolvingCongestionTollPricingModels,"EquilibriumandAdvancedTransportationModeling,P.MarcotteandS.Nguyen(Eds),KluwerAcademicPublishers,pp.109-124,Boston,1998. [48] D.Hearn,M.Yildirim,M.RomanaandL.Bai,\ComputationalMehtodsforCongestionCongestionTollPricingModels,"ProceedingsofIEEEConferenceonItelligentTransportationSystems,pp.257-262,Berkeley,2001. [49] J.V.Henderson,\RoadCongestion:aReconsiderationofPricingTheory,"JournalofUrbanEconomics,vol.1,pp.346-365,1974. [50] W.HirschandG.Dantzig,\TheFixedChargeProblem,"NavalResearchLogisticsQuarterly,vol.15,pp.413-424,1968. [51] J.Ho,\ASuccessiveLinearOptimizationApproachtotheDynamicTracAssignmentProblem,"TransportationScience,vol.14(4),pp.295-305,1980.

PAGE 147

[52] C.vanHoeselandA.Wagelmans,\FullyPolynomialApproximationSchemesforSingle-ItemCapacitatedEconomicLot-SizingProblems,"MathematicsofOperationsResearch,vol.26(2),pp.339-357,2001. [53] C.vanHoeselandA.Wagelmans,\AnO(T3)AlgorithmfortheEconomicLot-SizingProblemwithConstantCapacities,"ManagementScience,vol.42(1),pp.142-150,1996. [54] R.Horst,P.Pardalos,andN.Thoai,IntroductiontoGlobalOptimization,Springer,2ndEdition,Boston,2000. [55] R.HorstandH.Tuy,GlobalOptimization,Springer,3rdEdition,1996. [56] H.J.HuangandH.Yang,\OptimalVariableRoad-UsePricingonaCongestedNetworkofParallelRouteswithElasticDemand,"Proceed-ingsofthe13thInternationalSymposiumontheTheoryofTrafcFlowandTransportation,pp.479-500,1996. [57] B.Janson,\DynamicTracAssignmentforUrbanRoadNetwork,"Trans-portationResearchB,vol.25(2/3),pp.143-161,1991. [58] D.Kaufman,J.Nonis,andR.Smith,\AMixedIntegerLinearProgrammingModelforDynamicRouteGuidance,"TransportationsResearchB,vol.32(6),pp.431-440,1998. [59] J.KenningtonandV.Unger,\ANewBranch-and-BoundAlgorithmfortheFixedChargeTransportationProblem,"ManagementScience,vol.22,pp.1116-1126,1976. [60] D.KhangandO.Fujiwara,\ApproximateSolutionofCapacitatedFixed-ChargeMinimumCostNetworkFlowProblems,"Networks,vol.21,pp.689-704,1991. [61] D.KimandP.Pardalos,\ASolutionApproachtotheFixedChargeNetworkFlowProblemUsingaDynamicSlopeScalingProcedure,"OperationsResearchLetters,vol.24(4),pp.195-203,1999. [62] D.KimandP.Pardalos,\DynamicSlopeScalingandTrustIntervalTechniquesforSolvingConcavePiecewiseLinearNetworkFlowProblems,"Networks,vol.35,pp.216-222,2000. [63] H.KuhnandW.Baumol,\AnApproximateAlgorithmfortheFixedChargeTransportationProblem,"NavalResearchLogisticsQuarterly,vol.9,pp.1-15,1962. [64] M.Labbe,P.Marcotte,andG.Savard,\ABilevelModelofTaxationanditsApplicationtoOptimalHighwayPricing,"ManagementScience,vol.44,pp.16081622,1998.

PAGE 148

[65] S.LawphongpanichandD.Hearn,\AnMPECApproachtoSecondBestTollPricing,"MathematicalProgramming,vol.101(1),pp.33-55,2004. [66] Y.Li,S.WallerandT.Ziliaskopoulos,\ADecompositionSchemeforSystemOptimalDynamicTracAssignmentModels,"NetworksandSpatialEconomics,vol.3,pp.441-455,2003. [67] E.Lieberman,\AnAdvanceApproachtoMeetingSaturatedFlowRequirements,"The72ndAnnualTransportationResearchBoardMeet-ing,Washington,D.C.,1993. [68] M.LighthillandG.Whitham,\OnKinematicWavesI:FlowMovementinLongRivers"and\OnKinematicWavesII:ATheoryofTracFlowonLongCrowdedRoads,"ProceedingsoftheRoyalSocietyA,vol.229,pp.281-345,1955. [69] W.LinandH.Lo,\AretheObjectiveandSolutionsofDynamicUser-EquilibriumModelsAlwaysConsistent?"TransportationResearchA,vol.34(2),pp.137-144,2000. [70] M.Loparic,H.Marchand,andL.Wolsey,\DynamicKnapsackSetsandCapacitatedLot-Sizing,"MathematicalProgramming,vol.95B,pp.53-69,2003. [71] M.Loparic,Y.Pochet,andL.Wolsey,\TheUncapacitatedLot-SizingProblemwithSalesandSafetyStocks,"MathematicalProgramming,vol.89A,pp.487-504,2001. [72] H.Marchand,A.Martin,R.Weismantel,andL.Wolsey,\CuttingPlanesinIntegerandMixedIntegerProgramming,"DiscreteAppliedMathematics,vol.123,pp.397-446,2002. [73] D.MerchantandG.Nemhauser,\AModelandanAlgorithmfortheDynamicTracAssignmentProblems,"TransportationScience,vol.12(3),pp.183-199,1978. [74] D.MerchantandG.Nemhauser,\OptimalityConditionsforDynamicTracAssigmentProblem,"TransportationScience,vol.12(3),pp.200-207,1978. [75] A.MillerandL.Wolsey,\TightFormulationsforSomeSimpleMixedIntegerProgramsandConvexObjectiveIntegerPrograms,"MathematicalProgramming,vol.98B,pp.73-88,2003. [76] K.Murty,\SolvingtheFixedChargeProblembyRankingtheExtremePoints,"OperationsResearch,vol.16(2),pp.268-279,1968. [77] A.Nagurney,NetworkEconomics:AVariationalInequalityApproach,KluwerAcademicPublishers,2ndEdition,Boston,1999.

PAGE 149

[78] A.NahapetyanandS.Lawphongpanich,\Discrete-TimeDynamicTracAssignmentModelwithPeriodicPlanningHorizon:SystemOptimum,"toappearinJournalofGlobalOptimization. [79] A.NahapetyanandP.Pardalos,\ABilinearRelaxationBasedAlgorithmforConcavePiecewiseLinearNetworkFlowProblems,"toappearinJournalofIndustrialandManagementOptimization. [80] A.NahapetyanandP.Pardalos,\AdaptiveDynamicCostUpdatingProcedureforSolvingFixedChargeNetworkFlowProblems,"toappearinComputationalOptimizationandApplications. [81] A.NahapetyanandP.Pardalos,\ABilinearReductionBasedAlgorithmforSolvingCapacitatedMulti-ItemDynamicPricingProblems,"submittedtoComputersandOperationsResearch. [82] [83] U.Palekar,M.Karwan,andS.Zionts,\ABranch-and-BoundMethodforFixedChargeTransportationProblem,"ManagementScience,vol.36,pp.1092-1105,1990. [84] M.PatrikssonandR.T.Rockafellar,\AMathematicalModelandDescentAlgorithmforBilevelTracManagement,"TransportationScience,vol.36(3),pp.271291,2002. [85] S.PeetaandA.Ziliaskopoulos,\FoundationsofDynamicTracAssignment:ThePast,thePresentandtheFuture,"NetworksandSpecialEconomics,vol.1,pp.233-265,2001. [86] G.PerakisandG.Roels,\AnAnalyticalModelforTracDelaysandtheDynamicUserEquilibriumModels,"TechnicalReport,OR-368-04,OperationsResearchCenter,MassachusettsInstituteofTechnology,Cambridge,2004. [87] A.Pigou,TheEconomicsofWelfare,MacMillanandCo.,London,1920. [88] Y.PochetandL.Wolsey,AlgorithmsandReformulationsforLot-SizingProb-lems,CombinatorialOptimization,DIMACSSeriesinDiscreteMathematicsandTheoreticalComputerScience,vol.20,Providence,1995. [89] B.RanandD.Boyce,ModelingDynamicTransportationNetworks,Springer-Verleg,Berlin,1996. [90] B.RanandD.Boyce,\ALink-BasedVariationalInequalityFormulationforIdealDynamicUser-OptimalRouteChoiceProblem,"TransportationResearchC,vol.4(1),pp.1-12,1996.

PAGE 150

[91] B.Ran,D.Boyce,andL.Leblanc,\ANewClassofInstantaneousDynamicUser-OptimalTracAssignmentModels,"OperationsResearch,vol.41(1),pp.192-202,1993. [92] B.Ran,R.Hall,andD.Boyce,\ALink-BasedVariationalInequalityModelforDynamicDepartureTime/RouteChoice,"TransportationResearchB,vol.30(1),pp.31-46,1996. [93] P.Richards,\ShockWavesontheHighway,"OperationsResearch,vol.4(1),pp.42-51,1956. [94] [95] M.Smith,\ANewDynamicTracModelandtheExistenceandCalculationofDynamicUserEquilibriaonCongestedCapacity-ConstrainedRoadNetwork,"TransportationResearchB,vol.27(1),pp.49-63,1993. [96] J.Thomas,\Price-ProductionDecisionswithDeterministicDemand,"ManagementScience,vol.16,pp.747-750,1970. [97] B.WieandR.Tobin,\DynamicCongestionPricingModelsforGeneralTracNetworks,"TransportationResearchB,vol.32(5),pp.313-327,1997. [98] B.Wie,R.Tobin,andM.Carey,\TheExistence,UniquenessandComputationofanArcBasedDynamicNetworkUserEquilibriumFormulation,"TransportationResearchB,vol.36(10),pp.897-928,2002. [99] B.Wie,R.Tobin,T.Friesz,andD.Bernstein,\ADiscreteTime,NestedCostOperatorApproachtotheDynamicNetworkUserEquilibriumProblem,"TransportationScience,vol.29(1),pp.79-92,1995. [100] L.Wolsey,\SolvingMulti-ItemLot-SizingProblemswithanMIPSolverUsingClassicationandReformulation,"ManagementScience,vol.48,pp.1587-1602,2002. [101] J.Wu,Y.Chen,andM.Florian,\TheContinuousDynamicNetworkLoadingProblem:aMathematicalFromulationandSolutionsMethods,"TransportationResearchB,vol.32(3),pp.173-187,1997. [102] [103] D.ZhuandP.Marcotte,\OntheExistenceofSolutionstotheDynamicUserEquilibriumProblem,"TransportationScience,vol.34(4),pp.402-414,2000.

PAGE 151

[104] A.Ziliaskopoulos,\ALinearProgrammingModelforaSingleDestinationSystemOptimumDynamicTracAssignmentProblem,"TransportationScience,vol.34(1),pp.37-49,2000. [105] H.YangandW.Lam,\OptimalRoadTollsUnderConditionsofQueuingandCongestion,"TransportationResearchA,vol.30(5),pp.319332,1996. [106] M.YildirimandD.Hearn,\AFirstBestTollPricingFrameworkforVariableDemandTracAssignmentProblems,"TransportationResearchB,vol.39(8),pp.659-678,2005.

PAGE 152

ArtyomNahapetyanwasbornonApril15,1974,inGiumri,Armenia.HereceivedhisbachelordegreeinmathematicsfromYerevanStateUniversityin1996,MSdegreeinmathematicsfromArmenianStateEngineeringUniversityin1998,andMSdegreeinindustrialandsystemsengineeringfromAmericanUniversityofArmeniain2001.In2002heenteredthegraduateprograminindustrialandsystemsengineeringattheUniversityofFlorida.HereceivedhisPh.D.degreeinindustrialandsystemsengineeringfromtheUniversityofFloridainAugust2006. 140


Permanent Link: http://ufdc.ufl.edu/UFE0015623/00001

Material Information

Title: Nonlinear Approximation Techniques to Solve Network Flow Problems with Nonlinear Arc Cost Functions
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0015623:00001

Permanent Link: http://ufdc.ufl.edu/UFE0015623/00001

Material Information

Title: Nonlinear Approximation Techniques to Solve Network Flow Problems with Nonlinear Arc Cost Functions
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0015623:00001


This item has the following downloads:


Full Text











NONLINEAR APPROXIMATION TECHNIQUES TO SOLVE NETWORK
FLOW PROBLEMS WITH NONLINEAR ARC COST FUNCTIONS
















By

ARTYOM NAHAPETYAN


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2006

































Copyright 2006

by

Artyom Nahapetyan


































I dedicate this work to my parents, Nina Hovhanni- i, and Garush Nahapetyan,

who alv--x supported me in my studies.















ACKNOWLEDGMENTS

I would like to thank my chair and cochair, Prof. Siriphong L ,i.--1 i,.! i ich

and Prof. Donald W. Hearn, for their valuable advice, support and guidance during

my studies. Our meetings and discussions were ahv--l- very helpful.

Also I would like to express my sincere gratitude to the committee members

Prof. Panos Pardalos, Prof. William Hager, and Prof. Ravindra Al!n I, for their

encouragement. Especially, I am grateful to Prof. Panos Pardalos for his valuable

si.-.- -I ir 1 and advice on the supply chain problems I have worked on.

The tremendous support from my parents is invaluable, and there are no words

to express my appreciation for that.

Finally, I would like to thank all my friends and collaborators who made my

studies enjov- 1'l1' and productive.















TABLE OF CONTENTS
page

ACKNOWLEDGMENTS ................... ...... iv

LIST OF TABLES ................... .......... viii

LIST OF FIGURES ................................ x

ABSTRACT ....................... ........... xi

CHAPTER

1 INTRODUCTION .................... ....... 1

2 A BILINEAR REDUCTION BASED ALGORITHM FOR CONCAVE
PIECEWISE LINEAR NETWORK FLOW PROBLEMS ........ 5

2.1 Introduction to the C!i ipter ................... ... 5
2.2 A Bilinear Reduction Technique for the Concave Piecewise Linear
Network Flow Problem ........... ..... ....... .. 7
2.3 Concave Piecewise Linear Problems with Separable Objective
Functions .............................. 11
2.4 Dynamic Cost Updating Procedure ................ 13
2.5 On the Dynamic Slope Scaling Procedure ....... ....... 16
2.6 Numerical Experiments ......... ................ 19
2.7 Concluding Remarks ............... ....... .. 21

3 ADAPTIVE DYNAMIC COST UPDATING PROCEDURE FOR SOLVING
FIXED CHARGE NETWORK FLOW PROBLEMS ........ 22

3.1 Introduction to the C!i lpter ................... ... 22
3.2 Approximation of the Fixed CI irge Network Flow Problem by a
Two-Piece Linear Concave Network Flow Problem . ... 24
3.3 Adaptive Dynamic Cost Updating Procedure . . 27
3.4 On the Dynamic Slope Scaling Procedure . . ..... 30
3.5 Numerical Experiments .................. ..... .. 31
3.6 Concluding Remarks .................. ....... .. 34

4 A BILINEAR REDUCTION BASED ALGORITHM FOR SOLVING
CAPACITATED MULTI-ITEM DYNAMIC PRICING PROBLEMS... 35

4.1 Introduction to the C!i lpter .................. .... 35
4.2 Problem Description .................. ....... .. 37









4.3 A Bilinear Reduction Based Algorithm for Solving AC'\ Il)P Problem 43
4.4 Numerical Experiments .................. ... .. 46
4.5 Concluding Remarks .................. ....... .. 49

5 DISCRETE-TIME DYNAMIC TRAFFIC ASSIGNMENT MODELS
WITH PERIODIC PLANNING HORIZON: SYSTEM OPTIMUM 50

5.1 Introduction to the C'!i lter .................. .... 50
5.2 Periodic Planning Horizon ......... .... .... ....... 53
5.3 Discrete-Time Dynamic Traffic Assignment Problem with Periodic
Time Horizon ................... ....... 54
5.4 Bounds for the DTDTA Problem .................. .. 67
5.5 Numerical Experiments .................. ...... .. 71
5.6 Concluding Remarks .................. ....... .. 76

6 A NONLINEAR APPROXIMATION BASED HEURISTIC ALGORITHM
FOR THE UPPER-BOUND PROBLEM ....... ......... 78

6.1 Introduction to the C!i lpter ................... ... 78
6.2 Nonlinear Relaxation of DTDTA-U Problem ............. 83
6.3 Nonlinear Relaxation Based Heuristic Algorithm . .... 86
6.4 Numerical Experiments .................. ...... .. 89
6.5 Concluding Remarks .................. ....... .. 91

7 A DYNAMIC TOLL PRICING FRAMEWORK FOR DISCRETE-TIME
DYNAMIC TRAFFIC ASSIGNMENT MODELS ............. 92

7.1 Introduction to the C!i lpter ................... ... 92
7.2 The Reduced Time-Expanded Network and UE Solution . 97
7.3 The Dynamic Toll Set .................. ..... .. 103
7.4 Dynamic Toll Pricing Problems .............. .. .. 108
7.5 Illustrative Examples .................. .. .... .. .. 110
7.6 Concluding Remarks .................. ..... .. 113

8 DIRECTIONS OF FUTURE RESEARCH . . 115

APPENDIX

A COMPUTATIONAL RESULTS FOR CHAPTER 2 . . .... 117

B COMPUTATIONAL RESULTS FOR CHAPTER 3 . . .... 121

C COMPUTATIONAL RESULTS FOR CHAPTER 4 ............ 125

D COMPUTATIONAL RESULTS FOR CHAPTER 6 ............ 127

E COMPUTATIONAL RESULTS FOR CHAPTER 7 ............ 129









REFERENCES .. .... ............................ 131

BIOGRAPHICAL SKETCH ........ ........ ............ 140















LIST OF TABLES
Table page

5-1 Demand patterns .................. ............ .. 72

5-2 Optimal solutions to the two-arc problem. ............... 73

5-3 Solutions from the lower and upper-bound problems: linear travel cost
function .................. ................. .. 75

5-4 Solutions from the lower and upper-bound problems: quadratic travel
cost function. .................. .............. .. 75

5-5 Quality of refined upper and lower-bound solutions: linear travel cost
function .................. ................. .. 76

5-6 Quality of refined upper and lower-bound solutions: quadratic travel cost
function. .................. ................. .. 76

6-1 Equivalent objective functions .................. .... .. 88

6-2 Distributions of parameters of randomly generated travel time functions 89

7-1 Additional constraints .................. ....... .. .. 110

7-2 Distributions of parameters of randomly generated travel time functions 111

A-1 Set of problems. .................. ............. .. 117

A-2 Computational results of sets 1-18: quality of the solution and the CPU
times .................. .................. .. 118

A-3 Computational results of sets 1-18: DSSP vs. DCUP. . .... 119

A-4 Computational results for sets 19-30. .............. .. 120

A-5 Computational results for the combined mode. . . ...... 120

B-1 Set of problems. .................. ............. .. 121

B-2 Computational results of groups G1 and G2: quality of the solutions and
the CPU times. .................. ............ 122

B-3 Computational results of groups G1 and G2: the percentage of problem
where one of the algorithms finds a better solution than another one. 123









B-4 Computational results of groups G3 and G4 ............... ..124

C-1 The quality of the solution: Procedure 6. ................ 125

C-2 The quality of the solution: Procedure 5. ................ 126

C-3 The CPU time of the procedures. ................ ..... 126

D-1 Computational results of the experiments. ............... 127

D-2 Computational results of the combined mode. ............. ..128

E-1 The total collected toll and the total cost for each problem and parameter
S.. ........................................ .. 129

E-2 The number of toll collecting centers for each problem and parameter e.. 129















LIST OF FIGURES
Figure page

3-1 Approximation of function fa(xa). ............ . .. 25

3-2 ,a (xa) and 0a (X,) functions. .................. ..... 28

4-1 The price and the revenue functions. . ......... 38

5-1 Linear versus circular intervals. .................. .... 53

5-2 Events occurring in two consecutive planning horizons. . .... 54

5-3 Three-node network. ............... ..... .... 55

5-4 Time expansion of arc (1, 2) at t = ................ .. 56

5-5 Time-expansion of the three-node network. ................ 58

5-6 Oa(Xa(t)) C ( 6, s] versus Xa(t) E (1( 6), 1(6)] . . 68

5-7 Two-arc network ................ ........... .. 72

5-8 Four-node network. ............... .......... 74

6-1 Two feasible solutions. ............... ...... 80

7-1 4-Node network and traffic demand. ............. .. 92

7-2 User equilibrium flows and travel times. ................ 93

7-3 System optimum flows and travel times. ................ 93

7-4 Tolled user equilibrium flows and travel times. .. . ..... 94

7-5 The value of at. ............... ............ .. 111

D-1 Two Networks. ............... ............ 127

E-1 9-node network. .................. .. ...... ...... 129

E-2 The toll vector for different values of E in the MinRev(E) problem. 130

E-3 The toll vector for different values of E in the MinCost(E) problem. 130















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

NONLINEAR APPROXIMATION TECHNIQUES TO SOLVE NETWORK
FLOW PROBLEMS WITH NONLINEAR ARC COST FUNCTIONS

By

Artyom Nahapetyan

August 2006

C('! ,': Siriphong L i.v !. wi! )panich
Cochair: Donald W. Hearn
Major Department: Industrial and Systems Engineering

In this dissertation we investigate network flow problems with nonlinear arc

cost functions. The first group of problems consists of concave piecewise linear

network flow, fixed charge network flow, and dynamic pricing problems that

arise in the areas of supply chain management and logistics. Based on the MIP

formulation, we construct bilinear reduction problems, in which the global solution

of the latter is a solution of the initial formulation. To solve the reduction problem,

we propose some heuristic algorithms. In the experiments, we compare the solution

provided by our algorithm with an exact solution as well as a solution provided by

other heuristic algorithms in the literature. Numerical experiments on randomly

generated instances confirm the quality of the algorithms.

The second group of problems is related to the dynamic traffic assignment

problem. In particular, we consider a periodic discrete time dynamic traffic

assignment problem (DTDTA), in which the travel time is a function of the number

of cars on the road, and the planning horizon is circular. The mathematical

formulation belongs to the class of nonlinear mixed integer problems. To obtain an









appropriate solution to the problem, we construct a linear mixed integer problem

for providing an upper bound and discuss an approximation scheme based on the

bounding problem. However, the bounding problem involves binary variables, and

when the problem is large, it is hard to solve. To overcome these difficulties, we

propose a heuristic algorithm based on a bilinear relaxation of the problem. Using

an approximate solution, in the dissertation we develop a toll pricing framework for

the dynamic case. In particular, based on a feasible vector of DTDTA we describe

a set of valid tolls and discuss several toll pricing problems. By constructing

an appropriate time-expanded network, one may consider a similar toll pricing

framework for other solutions obtained, for example, from a simulation.















CHAPTER 1
INTRODUCTION

Network flow problems are minimization/maximization problems with

underlying network structure. Although there are different representations of the

network, perhaps the most popular one is based on flow conservation constraints

via a node-arc incidence matrix. Apart from the flow conservation constraints,

many problems have additional restrictions on the variables, e.g., non-negativity

and lower/upper boundary constraints. Based on the objective function and other

additional constraints the problems can be classified as linear or nonlinear, where

the latter can be further decomposed into convex, concave, or other problems.

The linear problems assume that the constraints as well as the objective

function are linear. Polynomial time algorithms for solving the problems are

well known. Some classical examples of the network flow problems with linear

constraints include shortest path, minimum spanning tree, minimum cut, maximum

flow, minimum cost network flow, and other problems. Based on the optimality

conditions and other properties of the problem, several algorithms have been

proposed to solve the problems. For details on the linear network flow problems,

see Al!,i et al. [2].

Despite the nice theoretical results developed for the linear problems, most of

the practical problems are not linear, i.e.; the objective function and/or some of the

constraints are nonlinear. If it is a convex minimization (concave maximization)

problem with a convex feasible region then any local minimum (maximum)

is a global solution of the problem, and appropriate algorithms such that the

Frank-Wolf algorithm, gradient based and direction finding methods, can be used

to solve the problem. A large variety of algorithms for solving convex minimization









(concave maximization) problems can be found in Bazaraa et al. [5]. When the
objective function is not convex (concave) and/or the feasible region is not convex,

these algorithms do not necessarily converge to a global optimal solution. Finding

a global solution is a hard task, and global optimization techniques are required to

solve the problem (see, e.g., Horst et al. [54] and Horst and Tuy [55]).

In this dissertation, we consider two groups of non-convex network flow

problems. The first group includes piecewise linear network flow, fixed charge

network flow, and dynamic pricing problems (see C'! lpters 2, 3, and 4) that have

a large variety of applications in the production 1p1 ,liiiir.- scheduling, investment

decision, network design, location of plants and distribution centers, pricing

policy, and many other practical problems that arise in supply chains, logistics,

transportation science, and computer networks. It is well known that the problems

in their general form are NP-Hard; therefore, there are no polynomial time

algorithms to solve the problems unless P = NP. Although the mathematical

formulation belongs to the class of linear mixed integer problems, solving large

problems requires a large amount of CPU time and memory. On the other hand,

one can consider approximation techniques that are able to provide a good-quality

solution using less computer resources. Many of these techniques employ a linear

relaxation of the problem. Unlike those in the literature, we propose nonlinear

reduction techniques to solve the problems. In particular, in all three problems

we develop a method to reduce the problem to a bilinear one and propose a

heuristic algorithm to solve the resulting problem. In the numerical experiments we

compare the results with an exact solution (or the best feasible solution) provided

by MIP solvers, as well as with Dynamic Slope Scaling Procedure (DSSP), since

it is known to be one of the best heuristic algorithms to solve such problems.

Numerical experiments on randomly generated problems confirm the quality of the

solutions provided by our algorithms. In particular, it outperforms the DSSP in









the quality of the solution as well as in the computational time. In addition, we

transform the problems into alternative continuous network flow problems with flow

dependent cost function and prove that a global solution of the resulting problem is

a solution of the initial MIP formulation. Despite an unusual structure of the cost

function, the mathematical formulation of the problems is similar to the system

optimum problems arising in the traffic assignment modelling. Using the same cost

function, we also construct a variational inequality problem similar to those in the

transportation literature and prove that the DSSP converges to a solution of the

resulting problem; i.e., it provides an equilibrium solution. However, the problem

requires finding a system optimum solution, and the algorithms we propose finds an

approximate solution to the problem.

The second group of problems is related to the dynamic traffic assignment

problem. Unlike the static case, where the travel time is a function of the arc flow,

the dynamic models involve three variables: inflow rate, outflow rate, and density,

and the travel time can be a function of all three variables. In the literature several

continuous and discrete time models have been proposed for different travel time

functions. The model in this dissertation assumes that the travel time is a function

of the density, and all cars that enter an arc at the same point of time experience

the same traffic conditions; therefore, they leave the arc at the same time. In

addition, the models in the literature assume that the network is empty at the

beginning and the end of a planning horizon. In the case when some cars are

present in the network, the time to enter the network for those cars is unknown,

and it is hard to model the propagation of the cars in the network. Unlike other

models in the literature, we consider a periodic planning horizon and assume

that the processes repeat themselves from one period to another (see C'! plter

5). The mathematical formulation of the problem minimizes the total delay and

belongs to the class of nonlinear mixed integer problems, a hard problem to solve.









By linearizing the objective function and the constraints, we construct linear

mixed integer problems that provide upper and lower bounds. The solution of

the bounding problems can be made arbitrarily close to a solution of the initial

formulation by decreasing the discretization parameter. However, the bounding

problems involve binary variables, and it is hard to solve large problems using

MIP solvers. In ('!i lpter 6 we discuss a heuristic algorithm based on a nonlinear

relaxation of the problem. In particular, we construct a continuous bilinear

problem, which provides a tighter lower bound than the LP relaxation. Using the

bilinear relaxation, the heuristic algorithm aims to find an integer solution, which

has an objective function value close to the one provided by the relaxation problem.

Another problem of interest is the toll pricing framework for the dynamic

traffic assignment problem (see ('!i lpter 7). Similar to the static case, we construct

a set of valid toll vectors such that a system optimum solution is a solution of

the tolled user equilibrium problem. The latter is a user equilibrium problem

where the arc cost functions include tolls in addition to the travel times. A key

component in the development of such technique is the reduced time-expanded

(RTE) network constructed based on a feasible vector. Using the network, we show

that a feasible vector is a user equilibrium solution if and only if it is a solution of

a linear problem with an underlying RTE network structure. The latter allows the

construction of a set of valid tolls and formulation of a toll pricing problem with a

secondary objective, and we provide several examples of such problems.















CHAPTER 2
A BILINEAR REDUCTION BASED ALGORITHM FOR CONCAVE
PIECEWISE LINEAR NETWORK FLOW PROBLEMS

2.1 Introduction to the Chapter

The cost functions in most of the network flow problems considered in the

literature are assumed to be linear or convex. However, this assumption might not

hold in practical real-world problems. In fact, the costs often have the structure of

a concave or piecewise linear concave function (see Guisewite and Pardalos [45] and

Geunes and Pardalos [41]). We consider the concave piecewise linear network flow

problem (CPLNF), which has diverse applications in supply chain management,

logistics, transportation science, and telecommunication networks. In addition,

the CPLNF problem can be used to find an approximate solution for network flow

problems with a continuous concave cost function. It is well known that these

problems are NP-hard (see Guisewite and Pardalos [45]).

This chapter deals with a nonlinear reduction technique for the linear mixed

integer formulation of the CPLNF problem. In particular, the problem is reduced

to a continuous one with linear constraints and a bilinear objective function. The

reduction has an economical interpretation and its solution is proven to be the

solution of the CPLNF problem. Based on the reduction, we propose an algorithm

for finding a local minimum of the problem, which we refer to as the dynamic cost

updating procedure (DCUP). In the chapter, we show that DCUP converges in a

finite number of iterations.

The theoretical results presented in this chapter can be extended to a more

general concave minimization problem with a separable piecewise linear objective

function and linear/nonlinear constraints. It should be emphasized that Horst









et al. [54] (see also Horst and Tuy [55]) discuss a bilinear program with disjoint

feasible regions and prove that the problem is equivalent to a subclass of piecewise

linear concave minimization problems. The results in this chapter show that

any concave minimization problem with a separable concave piecewise linear

objective function is equivalent to a bilinear program. It is well known that an

optimal solution of a general jointly constrained bilinear program belongs to

the boundary of the feasible region and is not necessarily a vertex (see Horst et

al. [54]). However, the reduction technique presented in this chapter has a jointly

constrained feasible region with a special structure and it is still equivalent to a

concave piecewise linear program. From the latter it follows that two parts of a

solution of the problem are vertices of two different polytopes that are "joined"

by a set of constraints. In that sense, these types of problems are n, .'/ 1.; joined

bilinear programs.

The CPLNF problem can be transformed into an equivalent network flow

problem with flow dependent costs function (NFPwFDCF). Using NFPwFDCF,

it can be shown that the dynamic slope scaling procedure (DSSP) (see Kim and

Pardalos [61] and [62]) converges to an equilibrium solution of NFPwFDCF.

Although DSSP provides a solution, which can be quite close to the system

solution, it is well known that the equilibrium and the system solutions in general

are not the same. On the other hand, DCUP converges to a local minimum of the

problem. In the numerical experiments, we solve different problems using DCUP

and DSSP and compare the quality of the solution as well as the running time.

Computational results show that DCUP often provides a better solution than

DSSP and uses fewer iterations and less CPU time. Since DCUP starts from a

feasible vector and converges to a local minimum, one considers first solving DSSP

and then improving the solution using DCUP. The numerical experiments using

this combined mode are provided as well.









For the remainder, Section 2.2 discusses the nonlinear reduction technique

for the CPLNF problem. Section 2.3 generalizes the results from Section 2.2 for a

concave piecewise linear problem with a separable objective function. Section 2.4

describes DCUP and theoretical results on the convergence and the solution of the

procedure. In Section 2.5 we prove that the solution of the DSSP is an equilibrium

solution of a network flow problem with flow dependent cost functions. The results

of numerical experiments on DCUP and DSSP are provided in Section 2.6, and

finally, Section 2.7 concludes the chapter.

2.2 A Bilinear Reduction Technique for the Concave Piecewise Linear
Network Flow Problem

Let G(N, A) represent a network where N and A are the sets of nodes and

arcs, respectively. The following is the mathematical formulation of the concave

piecewise linear network flow problem (CPLNF)

min fa(x)
aEA

s.t. Bx = b (2-1)

xa e [A, A"'] Va EA (2-2)

where B is the node-arc incident matrix of the network G, and fa(xa) are piecewise

linear concave functions, i.e.,

fa(X + xa ( f/(Xa)) Xa e [A2, A')

fa(xa) = ... ...

cjaXa + aS( ffa(xa)) xa G [A/a 1, al]

with ca > c > > c. Let Ka = {1,2,..., na}. Notice that because of the

concavity of fa(Xa), the function can be written in the following alternative form


fa(Xa) = min{f(xa)} = min{c xa + s}.
kEKa kEKa









Using binary variables, y, k e Ka, one can formulate the CPLNF problem as

the following linear mixed integer program (CPLNF-IP).


min a ac 5 E kak
aEA kEKa aEA EkEKa

s.t. Bxr b (2-3)

Sx= X Va e A (2-4)
kEKa
a A <1Y Xa< < A Va c A (2-5)
kEKa kEKa

S 1 Va A (2-6)
kEKa
xk < My Va E A (2-7)

S> 0, y {0,1} Va e A and k E Ka (2-8)

where M is a sufficiently large number.

In the above formulation, equality (2-6) makes sure that Va E A, there is only

one E Ka such that y = 1 and y = 0, Vk e Ko, k / The correct choice of

( depends on the value of Xa and has to satisfy constraint (2-5). In particular, if

Xa E [A-1, A4] then from constraints (2-5) and (2-6), it follows that y 1. As
for the rest of the constraints, inequality (2-7) ensures that xr = 0 if y = 0, and

qualities (2-3) and (2-4) make sure that the demand is satisfied and the sum of xk

over all indices k E Ka is equal to the flow on arc a. In addition, it is easy to show

that the objective of the problem is equivalent to the objective of CPLNF and one

concludes that the CPLNF and the CPLNF-IP problems are equivalent.

Consider a relaxation of the CPLNF-IP problem where constraint (2-7) and

the integrality of yj are replaced by


rk k= Xa
a ata


(2-9)









and y > 0, respectively. Observe that in the resulting problem constraint (2-4)

is redundant and follows from (2-6) and (2-9); therefore, it can be removed from

the formulation. In addition, notice that one can remove the variable xk from

the formulation as well by substituting (2-9) into the objective function. The

mathematical description of the resulting problem is provided below and we refer to

the relaxation problem as CPLNF-R.


min g(x, y) [: Zc xa > y f as
r,y
aEA LkEKa aEA kEKa aEA kEKa

s.t. Bx = b (2-10)

a y-1a < Xa< < A Va A (2-11)
kEKa kEKa

S 1 Va A (2-12)
kEKa
x_ > 0, y_ >0 Va A and k Ka (2-13)

Lemma 2.2.1. Any feasible vector of the CPLNF-IP problem is feasible to the

CPLNF-R.

Proof: Observe that constraints (2-10)-(2-13) are present in the CPLNF-IP

problem. Therefore, any feasible vector of the CPLNF-IP problem satisfies

constraints (2-10)-(2-13). U

Lemma 2.2.2. Any local optimum of the CPLNF-R problem is either feasible to

the CPLNF-IP or or ca be used to construct a feasible vector of CPLNF-IP with the

same objective function value.

Proof: Let (x*, y*) denote a local minimum of the CPLNF-R problem. From the

local optimality it follows that g(x*, y*) < g(x*, y) in the c-neighborhood of y*.

However, observe that by fixing the value of the vector x to x* in CPLNF-R, the

problem reduces to a linear one; therefore y* is a global minimum of the resulting

LP. In addition, notice that the problem can be decomposed into IA| problems of









the following form

min c x X + s k [c X + s]y
aya a aya a a a* a
keKa kEKa kEKa
s.t. A k-1 y < x A ky (2-14)

kEKa kEKa

y =a 1 (2-15)
kEKa
y >0 Vk c K (2-16)

Let x e [A*- Ak*]. As we have mentioned before, fa(x,) = minkEKafa(xa)};

therefore

f(x*) min{fk(x*)} min {c + s}.
EKKa keKa
Observe that by assigning y = 1 and yk = 0, Vk e Ka, k / k*, the resulting

vector ya (i) satisfies constraint (2-14) because xa e [A '*-, A*] and (ii) ya

argmin{ [cKa[ + s kKk = 1,ac > 0}. Based on the above, one

concludes that ya is an optimal solution of the problem. If x* C (A -1, A *) then

y is the unique solution of the problem because cxk + s > c *x + s Vk e Ka,

k / k*; therefore, y* y. If x A 1 or x A, there are exactly two binary

solutions of the problem, and both have the same objective function value. As a

result, either one can be used to construct a binary solution y. A similar result

holds for all arcs a E A. Regarding variable xa, given (x*, y*), the only feasible one

is x a = x* and x = 0, Vkc Ka, k / k*. U

The following theorem is a direct consequence of the above two lemmas.

Theorem 2.2.1. A 1 ,l.',l optimum of the CPLNF-R problem is a solution or ca be

used to construct a solution of the CPLNF-IP .

Proof: From Lemma 2.2.2, it follows that a global optimum of CPLNF-R is either

feasible to CPLNF-IP or can be used to construct a feasible solution with the same

objective function value. Since all feasible vectors of CPLNF-IP are feasible to









CPLNF-R (see Lemma 2.2.1), one concludes that a global solution of CPLNF-R

leads to a solution of CPLNF-IP. U

From Theorem 2.2.1, it follows that solving the CPLNF-IP problem is

equivalent to finding a global optimum of the bilinear problem CPLNF-R. If

the solution of CPLNF-R is not feasible to CPLNF-IP then the proof of Lemma

2.2.2 provides an easy way to construct a feasible solution with the same objective

function value. Other properties of local minima of the CPLNF-R problem are

discussed in Section 2.4.

It is noticed that the CPLNF-R problem has the following economic

interpretation. Observe that because of equality (2-12), yJ E [0, 1], and one

can interpret the variables yS as weights. Under this assumption, one can view

the objective function as the sum of the weighted averages of the variable costs

multiplied by the flow, [LkEK, c>ya] Xa, and the fixed costs, Ek:K sak In other

words, the objective function consists of the weighted averages of functions fa(xa).

However, the weights have to satisfy constraint (2-11), where the flow, Xa, is

bounded by the weighted averages of the left and the right ends of the intervals

[Ak-1, A ], k E Ka. According to Lemma 2.2.2, a local (global) optimum leads to a

solution where the weights are either equal 0 or 1.

2.3 Concave Piecewise Linear Problems with Separable Objective
Functions

Consider the following generalization of the CPLNF problem where constraint

(2-1) is replaced by a requirement x E X C R".

CPLPwSOF:
n
min f(xi)
i=1
s.t. xEX

Xi E [A, An1


where the fi(xi) are piecewise linear concave functions, i.e.,











ci Xi s+slj=f (x)) x [A',\1)


fi(xi) =

Cii "nn i ni 1
ci xi + s( f-'(xi)) ax E [An 1 ,An']

with ci > c > > c'. Let Ki = {1, 2,..., ni}. Although the theoretical results

in the previous section are derived for the concave piecewise linear network flow

problem, one can replace constraint (2-10) by x e X, i.e.,

CPLPwSOF-R:
nm
x,y
i=1 kEKi
s.t. xEX

E k-1 k< xi < k k
Ai Yi A
kEKi kEKi

1
kEK,
xi > 0, yk > 0,

and the bilinear reduction technique used before, as well as Lemmas 2.2.1 and

2.2.2, and Theorem 2.2.1, are still valid. As a result, one concludes that the

CPLPwSOF and the CPLPwSOF-R problems are equivalent in the sense that

a solution of the CPLPwSOF problem can be easily constructed from a global

solution of the CPLPwSOF-R problem.

If the set X is a polytope then CPLPwSOF-R is a bilinear program with a

jointly constrained linear feasible region. Let Y = {y EkeK y 1, y? > 0}

and X+ = {xlx e X,xi e [A, Ai ]}. Denote by V(X+) and V(Y) the sets

of vertices of the polytopes X+ and Y, respectively. Notice that the sets X+

and Y are "j.- 1I 1 by the constraints >kEKi 1i Y < i _< kEK, A-" It is

well known that an optimal solution of a general bilinear program with jointly









constrained feasible region occurs at the boundary of the feasible region and is not

necessarily a vertex (see Section 3.2.2, Horst et al. [54] and the related problem

set). However, CPLPwSOF-R is equivalent to CPLPwSOF. In particular, if (x*, y*)

is a global solution of CPLPwSOF-R then from Theorem 2.2.1, it follows that x* is

a solution of CPLPwSOF. The latter is a concave minimization problem where the

feasible region is a polytope. It is well known that the solution of such a concave

minimization problem is one of the vertices of the polytope; therefore x* e V(X+).

In addition, from the theorem it follows that there exists y E V(Y) such that

(x*, ) is a global solution of CPLPwSOF-R problem. In that sense, CPLPwSOF-R
is a ;, ., l./; joined bilinear program. The above discussion is summarized in the

following two theorems.

Theorem 2.3.1. A concave minimization problem with a separable piecewise linear

objective function, CPLPwSOF, is equivalent to a bilinear 1i. '",'r CPLPwSOF-R.

Theorem 2.3.2. Let (x*, y*) be a solution of problem CPLPwSOF-R. If the

set X+ is a ..1;,/. pe, then (i) x* E V(X+), and (ii) 3 y E V(Y) such that

Zi=fl kKi J (iI'l :i"l EkeKi Jik(i' .
2.4 Dynamic Cost Updating Procedure

In this section, we discuss an algorithm for solving the CPLNF-R problem. In

general, bilinear programs similar to CPLNF-R are not separable in the sense that

the feasible sets of variables x and y are joined by common constraints. By fixing

one of the variables to a particular value, the resulting problem transforms into a

linear one.

Consider the following two linear problems which we refer to as LP(y) and

LP(x), where y (x) denote the parameter of the problem LP(y) (LP(x)), i.e., fixed

to a particular value.

LP(y)


aEA EK, ICYa









s.t. Bx b

Xa C [0, \a]

LP(x)

min [caxa + sa
aEA kEKa

S.t. A k-1 < ky k A
kEKa kEKa


kEKa
S> 0

In the LP(y) problem we assume that variables y4 are given, and the Xa are

the only decision variables. Similarly, in the LP(x) problem, the Xa are given,

and the y are the decision variables. As we have shown in the proof of Lemma

2.2.2, problem LP(x) can be decomposed into IA| problems and the solutions

of the decomposed problems are binary vectors, which satisfy constraint (2-14).

Therefore, given vector x, a solution of the LP(x) problem can be found by a

simple search technique where y = 1 if Xa E [a -1, A].

We propose a dynamic cost updating procedure (DCUP), where one considers

solving the problems LP(x) and LP(y) iteratively, using the solution of one

problem as a parameter for the other (see Procedure 1). Although in the procedure

the initial vector yo, is such that yl0 = 1 and y0 = 0, Vk E Ka, k / 1, one can

choose any other binary vector, that satisfies constraint (2-12). It is noticed that

a similar iterative procedure has been used for solving a bilinear program with

a disjoint feasible region (see, e.g., Horst et al. [54] and Horst and Tuy [55]). In

Procedure 1 : Dynamic Cost Updating Procedure
Step 1: Let yo denote the initial vector of 0o, where y0 = 1 and o = 0, Vk e
Ka, k / 1. m -- 1.
Step 2: Let xm = argmin{LP(ym-l)}, and y = argmin{LP(x')}.
Step 3: If y"' y"-1 then stop. Otherwise, m -- m + 1 and go to Step 2.









the DCUP, LP(y) does not include constraint (2-11). In other words, L(x) is the

CPLNF-R problem with fixed variable y and relaxed constraint (2-11). The latter

allows using the iterative procedure to solve bilinear programs with a ; .,../;/l; joined

feasible region. Let V represent the feasible region of CPLNF-R and (x*, y*) be the

solution of the DCUP.

Theorem 2.4.1. The solution of the DCUP -.,I.f the following .! ,il.:l;'


min g(x*,y) g(x*,y*) min g(x,y*).
(x*,y)EV (x,y*)EV

Proof: Because of the stopping criteria, x* = argmin{LP(y*)} and y*

argmin{LP(x*)}. From the latter it follows that x* = argmin(x,y*)evg(x, y*) and

y* = argmin(X*,y)Evg(x*,y).

Theorem 2.4.2. If y* is a unique solution of the LP(x*) problem then (x*,y*) is a

local minimum of CPLNF-R.

Proof: See the proof of Proposition 3.3, Horst et al. [54]. 0

In the proof of Lemma 2.2.2 we have shown that y* is not unique if and

only if one of the components of vector x*, x*, is equal to the value of one of the

breakpoints A However, observe that x* = argmin{LP(y*)}, and the feasible

region of problem LP(y*) does not involve breakpoints. As a result, in practice it is

unlikely that x* is equal to one of the breakpoints.

Theorem 2.4.3. Given ,i.; initial binary vector yO that .,/I.:.- constraint (2-12),

DCUP converges in a finite number of iterations.

Proof: Let yo denote the initial binary vector, x1 = argmin{LP(yo)} and

y = argmin{LP(x1)}. According to the assumption of the theorem, Va E A,

there exists only one ( e Ka such that yo = 1 and y0 = 0, Vk e Ka, k / $ .

If x cE [A--1, A ] then the corresponding components of the vector yo do not

change their values in the next iteration, i.e. yl = 1 and yl = 0, Vk E Ka,

k $ IHowever, if x1 e [A-1A ], then y = 1 and yl = 0,
/ 0 0. Hoevr if0'EO









Vk E Ka, k / (. In addition, notice that c4xk + s > cix + s. As a result,

g(xl, yo) > g(xl, y') > g(x2, y) and one concludes that the objective function

value of the CPLNF-R problem does not increase in the next iteration. To

prove this by induction, assume that the objective function does not increase

until iteration m. Similar to the above, one can show that g(xm+l, y') >

g(xm+l, ym+l) > g(xm+2, ym+1). The constructed non-increasing sequence, i.e.,

{g(xO yo), g(xl, y'), g (x2, y),..., g(xm+, ym) g(xm+l, m+), g(xm+2, ym+1)...},
is bounded from below by the optimal objective function value, and one concludes

that the algorithm converges.

Observe that in each iteration the procedure changes the binary vector

y. If ym = t-1 then the procedure stops and g(xm, y- 1) g(xm, y")
g(xm+l, y'). If there exist mT and m2 such that mi 1 > m2 and y"l yp2, then

xm1+1 = argminLP(yp )} = argminLP(y 2)} x= +1, i.e., g(xm2+1, m2)

g(xm2+l, yml) = g(xml+, ym). From the non-increasing property of the sequence
it follows that g(x"2+1, y-2) = g(x2+l, y-2+1); therefore, yp2 = y2+1 and the

algorithm must stop on iteration m2. From the latter it follows that all vectors y",

constructed by the procedure before it stops, are different. Since the set of binary

vectors y is finite, one concludes that the procedure converges in a finite number of

iterations. U

2.5 On the Dynamic Slope Scaling Procedure

In this section, we discuss another equivalent formulation of the CPLNF

problem with a slightly different objective function. Using the new formulation, we

prove some properties regarding the solution of Dynamic Slop Scaling Procedure

(DSSP) (see Kim and Pardalos [61] and [62]).

Although in the CPLNF problem there are no restrictions on the values of

parameters sa and Ao, by subtracting s' from function fa(Xa) and replacing the
variable Xa by Xa = x A', one can transform the problem into an equivalent one









where s, = 0 and A = 0. Therefore, without loss of generality, we assume that
s = 0 and A 0.
a a
To investigate DSSP, let

1 Xa (0, A1]

Ca X E (A,2
(^ x,) > 0
F (x,) -
Fa(Xa) X > 0 ... ...
M Xa = 0
na a (A n-1 na]

M a 0

where M is a sufficiently large number. Consider the following network flow

problem with flow dependent cost functions Fa(xa) (NFPwFDCF).


min FT(x)x
x

s.t. Bx = b (2-17)

Xa E [0, A] (2-18)

where F(x) is the vector of functions Fa(Xa).

Theorem 2.5.1. The NFPwFDCF problem is equivalent to the CPLNF problem.

Proof: Observe that both problems have the same feasible region. Let x be a

feasible vector. If Xa > 0 then Fa(xa)xa = f Xaz fa(za). On the other

hand, if Xa = 0 then Fa(xa)xa = 0 = fa(xa). From the above it follows that

FT(x)x = aEA fa(Xa), and one concludes that NFPwFDCF is equivalent to

CPLNF. U

Let NFPwFDCF(F) denote the NFPwFDCF problem where the cost

vector function, F(x), is fixed to the value of the vector F. Notice that the

NFPwFDCF(F) problem is a minimum network flow problem with arc costs Fa

and flow upper bounds A=. DSSP starts with initial costs F = ca + s" /Aa and

solves the NFPwFDCF(Fo) problem (see Procedure 2). Then it iteratively updates









Procedure 2 : Dynamic Slope Scaling Procedure
Step 1: Let F = c"a + s~a/A~a be the initial arc costs and
x1 = argmin{NFPwFDCF(F0)}. m <- 2
Step 2: Update the arc costs, F <-- Fa(x'1), and let
xm+l argmMinNFPwFDCF(Fm-1)}.
Step 3: If x"+l xm then stop. Otherwise, m <- m + 1 and go to Step 2.


the value of the cost vector using the function Fa(xa), F' = Fa(x''1), and solves

the resulting NFPwFDCF(F") problem. In the cost updating procedure, different

variations of the algorithm use different values for M. In particular, one may

consider replacing M by F~-1 or maxn
Theorem 2.5.2. The solution of the DSSP is a user equilibrium solution of the

network flow problem with the flow dependent cost functions Fa(xa).

Proof: Assume that DSSP stops on the m*-th iteration, and let x* denote

the solution of the procedure. From the stopping criteria it follows that x*

argmin{(Fm)rx Bx = b, x E [0, A"]}. If xr / 0 then F7* = Fa(x*). On the

other hand, if x = 0 then one can replace Fa* by a sufficiently large M without

changing the optimality of x*. As a result, x* = argmin{FT(x*)xBx = b,Xa E

[0, An] }. From the latter it follows that x* is a solution of the following variational
inequality problem


find feasible x* such that FT(x*)(x x*) > 0, VXa E [0, A-], Bx = b,


and one concludes that x* is an equilibrium solution of the network flow problem

with arc cost functions Fa(xa). U

If the arc costs are not constant, it is well known that the equilibrium and

system optimum solutions may not be the same (see e.g., Pigou [87], Dafermos

and Sparrow [29], and N 1,;, n.:y [77]). However, in the NFPwFDCF problem we

are interested in the system optimum solution, x*, where FT(x*)x* < FT(x)x,

VXa C [0, AX"], Bx b.









2.6 Numerical Experiments

In this section, we provide computational results for the dynamic cost

updating procedure and compare the solution of the DCUP with solutions provided

by DSSP and CPLEX.

The set of problems is divided into five groups that correspond to the networks

with different sizes and numbers of supply/demand nodes. For each group we

randomly generate three types of demand; U[10, 20], U[20, 30], or U[30, 40],

and consider 5 or 10 linear pieces (see Table A-1, Appendix A). In Kim and

Pardalos [62] the authors consider increasing concave piecewise linear cost functions

for experiments. Although the bilinear reduction technique as well as DCUP are

valid for any concave piecewise linear function, to remain impartial for comparison

we generate similar increasing cost functions. Doing so, first for each arc we

randomly generate a concave quadratic function of the form g(x) = -ax2 + 3x.

Notice that the maximum of the function is reached at the point x = %. Next

we divide the interval [0, 2] into na equal pieces, i.e. [A -1, A ] [(k-i) 2, ]

k E Ka {1, 2,..., n}. Finally, we construct the function fa(xa) by approximating

the function g(x) in the breakpoints A\, i.e. fa() -(A) + 3\A. There

are 30 problems generated for each choice of the group, the demand distribution

and the number of linear pieces. We use the GAMS environment to construct the

model and CPLEX 9.0 to solve the problems. Computations are made on a Unix

machine with dual Pentium4 3.2Ghz processors and 6GB of memory. All results are

tabulated in the Appendix A.

Sets 1-18 have a relatively small network size, and it is possible to solve

them exactly using CPLEX (see Table A-2). The relative errors for those sets are

computed using the following formulas

RED p fDCUP exact
REDCUP ) xact
Jexact









REDSSP fDSSP fexact
REDSSP ) -
fexact
In addition to the relative errors, we compare the results of DCUP versus DSSP.

In Table A-3, columns B, C, and D describe the percentage of problems where

DCUP is better than DSSP, DSSP is better than DCUP, and they are the same,

respectively. The numbers in column A are the averages (maximum values) of the

numbers REDSSP REDCUP, given REDSSP REDCUP > 0. According to the

numerical experiments DCUP provides a better solution than DSSP in about 41.

of the problems and the same solution in 3:'.. of problems. Also notice that DCUP

requires fewer iterations to converge and consumes less CPU time. Regarding

CPLEX, the computational time varies from several seconds in the sets 1-6 to

several thousands of seconds in the sets 13-18.

In the case of the problem sets 19-30, CPLEX is not able to find an exact

solution within 10,000 seconds of CPU time, and the best found solution is not

better than the one provided by the heuristics; therefore, we compare the results

of DCUP versus DSSP. In Table A-4, columns B and D describe the percentage

of problems where DCUP is better than DSSP, and DSSP is better than DCUP,

respectively. The numbers in columns A and C are computed based on the formula

fDSSP-DCUP given fDP fDCUP > 0, and DCU-fDSSP, given fDCUP fDss > 0,
respectively. Observe that the percentage of problems where DCUP is better

than DSSP is higher in the problems with large demands. On average, in 59' of

problems DCUP finds a better solution than DSSP using fewer iterations and less

CPU time.

In the above numerical experiments, we have used the vector yo (Va E A,

0o = 1 and y0 = 0, Vk E Ka, k / 1) as an initial binary vector. However,

DCUP can start from any other binary vector that satisfies constraint (2-12). In

particular, one can considers the solution of DSSP as an initial vector and use

DCUP to improve the solution. Table A-5 compares the results of DCUP versus









DSSP where column A is similar to the one in Table A-4 (i.e. the numbers in the

column are computed based on the formula fDSSP-fDCUP given fDSSP-fDCUP > 0),
fDSSP
and column B is the percentage of problems that have been improved. Observe

that the percentage of problems where DCUP improves the solution of the DSSP

increases with the size of the network and the demand. Overall, the DCUP

improves the solution of the DSSP in about i'- of the problems.

2.7 Concluding Remarks

In this chapter, we have shown that the concave piecewise linear network flow

problem is equivalent to a bilinear program. Because of the special structure of

the feasible region of the reduced problem, we are able to prove that the optimum

is attained on a vertex of the di-i- iiil parts of the feasible region. In addition, we

have shown that the results are valid for a general concave minimization problem

with a piecewise linear separable objective function.

Based on the theoretical results, we have developed a finite convergent

algorithm to find a local minimum of the bilinear relaxation. The computational

results show that the dynamic cost updating procedure is able to find a near

optimum or an exact solution of the problem using less of CPU time than CPLEX.

In addition, we compare the quality of the solution and the running time with

the dynamic slope scaling procedure. Since DCUP is fast, one can aim to find the

global minimum by randomly generating the initial binary vector and running

DCUP. In addition, DCUP can be used in cutting plane algorithms for finding an

exact solution.














CHAPTER 3
ADAPTIVE DYNAMIC COST UPDATING PROCEDURE FOR SOLVING
FIXED CHARGE NETWORK FLOW PROBLEMS

3.1 Introduction to the Chapter

During the twentieth century due to a variety of applications many researchers

focused their attention on the fixed charge network flow problem (FCNF). In

particular, production p11 iii:"i' scheduling, investment decision, network design,

location of plants and distribution centers, pricing policy, and many other practical

problems that arise in the supply chain, logistics, transportation science, and

computer networks can be modeled as a FCNF problem (see, e.g., Geunes and

Pardalos [41]).

The FCNF problem is well known to be NP-Hard and belongs to the class

of concave minimization problems. The problem can be modeled as a 0-1 mixed

integer linear program (see Hirsch and Dantzig [50]) and most solution approaches

utilize branch-and-bound techniques to find an exact solution (see Barr et al. [4],

Cabot and Erenguc [12], Gray [44], Kennington and Unger [59], and Palekar et

al. [83]). Since the concave minimization problem attains a solution at one of the

vertices of the feasible region, Murty [76] proposed a vertex ranking procedure to

solve the problem. However, finding an exact solution is computationally expensive

and it is not practical for solving large problems. Some heuristic procedures

are discussed in Cooper and Drebes [27], Diaby [31], Khang and Fujiwara [60],

and Kuhn and Baumol [63]. Recently Kim and Pardalos [61] (see also Kim and

Pardalos [62]) proposed a heuristic algorithm, Dynamic Slope Scaling Procedure

(DSSP), to solve the fixed charge network flow problem. The procedure solves a

sequence of linear problems, where the slope of the cost function is updated based









on the solution of the previous iteration. The algorithm is known to be one of the

best heuristic procedures to sole FCNF problems.

Note that all approaches to solve the problem are based on linear approximation

techniques. Instead, we approximate FCNF by a concave piecewise linear network

flow problem (CPLNF), where the cost functions have two linear pieces. A proper

choice of the approximation parameter ensures the equivalence between FCNF

and the resulting CPLNF problem. However, finding the proper parameter is

computationally expensive; therefore, we propose an algorithm that solves a

sequence of CPLNF problems by gradually decreasing the parameter of the

problem. We prove that the stopping criteria of the algorithm is consistent in the

sense that a solution of the last CPLNF problem in the sequence is a solution of

the FCNF problem.

Despite the above mentioned theoretical results, the algorithm requires finding

exact solutions of the CPLNF problems, which are NP-Hard (see Guisewite and

Pardalos [45]). In C!i plter 2 (see also Nahapetyan and Pardalos [79]), we have

shown that the CPLNF problem is equivalent to a bilinear program. In addition,

we have proposed a finite convergent dynamic cost updating procedure (DCUP)

to find a local minimum of the resulting bilinear program. To solve the FCNF

problem, in the algorithm one transforms the CPLNF problems into equivalent

bilinear programs and uses the DCUP to solve the resulting problems. We refer

to the combined algorithm as the adaptive dynamic cost updating procedure

(ADCUP).

Similar to the result presented in the C(i plter 2, we prove that the solution

provided by DSSP is a solution to a variational inequality problem, which is

formulated based on the feasible region of the FCNF problem. Although in general

an equilibrium solution and a system solution are not the same, the difference

between the objective function values of the solutions can be fairly small. On









the other hand, ADCUP is a heuristic procedure for finding a system optimum

solution. To compare these two procedures, we conduct numerical experiments

on 36 problems sets for different networks and choices of cost functions. There

are 30 randomly generated problems for each problem set. In the experiments,

we compare ADCUP versus DSSP in terms of the quality of the solution as

well as CPU time. In addition, for small networks we find an exact solution of

the problems using MIP solvers of CPLEX and compute relative errors. The

computational results show that ADCUP provides a near optimum solution using

a negligible amount of CPU time. In addition, the procedure outperforms DSSP in

the quality of the solution as well as CPU time. The difference between solutions is

more noticeable in the cases of small general slopes and large fixed costs.

For the remainder, Section 3.2 discusses the approximation technique and

establishes the equivalence between the FCNF problem and a CPLNF problem

with a special structure. A solution algorithm for solving the FCNF problem is

provided in Section 3.3. Some properties of the DSSP are introduced in Section 3.4.

The results of numerical experiments on ADCUP are summarized in Section 3.5,

and finally, Section 3.6 concludes the chapter.

3.2 Approximation of the Fixed Charge Network Flow Problem by a
Two-Piece Linear Concave Network Flow Problem

This section discusses an approximation of FCNF by concave piecewise linear

network flow problems (CPLNF). In particular, by choosing a sufficiently small

approximation parameter one can guarantee the equivalence between the FCNF

and the CPLNF problems.

Consider a general fixed charge network flow problem constructed on a

network G(N, A), where N and A denote the sets of nodes and arcs, respectively.

Let f,(a() denote the cost function of arc a E A, and
















S I





Figure 3-1. Approximation of function fa,(.).




fa (Xa) CaXa + Sa Xa E (0, a,]
0 Xa= 0

where Aa is the capacity of the arc. The fixed charge network flow problem can be

stated as follows.

FCNF:

min f(x) fa(/a)
x
aEA
s.t. Bx b,

XaC [0, Aa, Va A,

where B is the node-arc incident matrix of network G, and b is a supply/demand

vector.

Observe that the cost function is discontinuous at the origin and linear on the

interval (0, Aa]. Although we assume that the flows on the arcs are bounded by Aa,

the bounds can be replaced by a sufficiently large M, and the problem transforms

into an unbounded one.

Let aE E (0, Ao], and


X) CaXa + Sa Xa C [Fa, Aal
a[(X)=a)
Caa Xa Xa C [0, a)









where ca = Ca + Sa/Ea Observe that ,' (xa) = fa(Xa), Vx, E {0} U[Fa, A] and

' (Xa) < fa(xa), VXa E (0, E) (see Figure 3-1). Consider the following concave
piecewise linear network flow problem

CPLNF(E):

min 0(x) = (Xa)
aEA
s.t. Bx b,

XaG [0, Aa, Va e A,

where E denotes the vector of Ea. The function 0((x) as well as problem CPLNF(E)
depend on the vector E. In the discussion below, we refer to 0"(x) and CPLNF(E)

as E-approximations of the function f(x) and the FCNF problem, respectively.

Denote by x* and x" the optimal solutions of the FCNF and the CPLNF(E)

problems, respectively. Let V represent the set of vertices of the feasible region,

and 6 minf{xal x" E V, a c A, x > 0}. That is, 6 is the minimum among all
positive components of all vectors x" e V; therefore, 6 > 0.

Theorem 3.2.1. For all E such that Ea e (0, al] for all a e A, 0~(x") < f(x*).
Proof: Notice that (x*) < fa(x), Va c A; therefore, "(x*) < f(x*). Since

x' = argmin{CPLNF(E)}, the statement of the theorem follows. U

Theorem 3.2.2. For all E such that Ea e (0,6] for all a c A, O~(x") = f(x*).

Proof: To prove the theorem by contradiction, assume that 0"(x") < f(x*).

Observe that CPLNF(E) is a concave minimization problem; therefore, solutions

of the problem attain on a vertice of the feasible region. From the latter it follows

that xa > 6 > E, or a = 0, Va c A. As a result, (xa) =fa,(x), Va E A,

Q ^(x ) f(xE) < f(x*). The latter contradicts the optimality of x*, and one
concludes that "'(x") = f(x*). U

From Theorem 3.2.1 it follows that for all E such that Ea e (0, Al] for all a c A,

CPLNF(E) provides a lower bound for the FCNF problem. In addition, Theorem









3.2.2 makes sure that by choosing a sufficiently small e > 0 (e.g., Fa = 6, Va E A),

both problems have the same solution; therefore, FCNF is equivalent to a concave

piecewise linear network flow problem.

3.3 Adaptive Dynamic Cost Updating Procedure

In this section we discuss an algorithm for finding a solution of the fixed

charge network flow problem. As we have shown in the previous section, the

problem is equivalent to CPLNF(E), where E is such that aE E (0, 6], for all a E A.

However, it is computationally expensive to find the value of 6. Instead, we propose

solving a sequence of CPLNF(E) problems by gradually decreasing the values of Ea.

Consider Procedure 3. In Step 1, the procedure assigns initial values for

E,. Step 2 solves the resulting CPLNF problem. Notice that CPLNF(e1) is

indeed a linear problem, because [ a] = {Aa}. If 3a E A such that x" e

(0, F7), we decrease the value of aE to a where a is a constant from the open
interval (0, 1). Assume that the procedure stops at iteration k and let xk

argmin{CPLNF(ek)}.

Lemma 3.3.1. For all e such that 0 < F < Va E A, problems CPLNF(E) and

CPLNF(k) have the same set of optimal solutions.

Proof: Let x" = argmin{CPLNF()e}. Consider the following sequence of

inequalities

_(xk) > 0X>) > kX) > O{X (31)

The first and the third inequalities follow from the optimality of x" and xk in the

CPLNF(E) and the CPLNF(Ek) problems, respectively. Since ,a < Ek, Va e A, from

Procedure 3
Step 1: Let e <-- Aa. m -- 1.
Step 2: Solve problem CPLNF(E") and let x" = argmin{CPLNF(')}.
Step 3: If 3a e A such that x~ e (0, E-) then Fe+1 <-- a e, m m + 1, and go
to step 2. Otherwise, stop.

















Figure 32. (xa) and (x) functions.

Figure 3 2. 't (xa) and aa (Xa) functions.


the definition of E-approximation it follows that (xa) > (Xa), Va E A and

Xa e [0, Aa] (see Figure 3-2), and the second inequality follows.
Observe that because of the stopping criteria, x = 0 or x E [e a].
k 'k
Since aE < Fa, (Xa) a (Xa:r), Va e A and Xa {0}U[ aAa]; therefore,

Q (xk) = O(xk). The latter together with (3-1) insures that Q(xE) = 0(k).
Since both problems, CPLNF(E) and CPLNF(ek), have the same objective function

value at xr and xk, one concludes that they are equivalent. U

Theorem 3.3.1. A solution of the CPLNF(Ek) is a solution of the FCNF problem.

Proof: From Lemma 3.3.1 it follows that VE such that 0 < aE < KE, Va E A, the

CPLNF(E) and the CPLNF(Ek) problems have the same set of solutions. On the

other hand, by choosing 0 < aE < min{E, )}, CPLNF(E) is equivalent to the FCNF

problem (see Theorem 3.2.2), and the statement of the theorem follows. U

From Theorem 3.3.1 it follows the consistency of the stopping criteria in

the sense that an optimal solution of the resulting problem, CPLNF(Ek), is an

optimal solution of the FCNF problem. Observe that Procedure 3 requires solving

a sequence of concave piecewise linear network flow problems, which are NP-Hard.

To overcome this difficulty, one considers a bilinear relaxation technique to solve

the CPLNF problems (see C'! lpter 2). In Section 2.2, we have shown that the

CPLNF problem is equivalent to a bilinear program. To solve the bilinear problem,

we propose the dynamic cost updating procedure (DCUP), which finds a local









minimum of the problem and can be used in Step 2 of Procedure 3 to find a

solution of the CPLNF(E") problem. The resulting algorithm is summarized in

Procedure 4, which we refer to as adaptive dynamic cost updating procedure

(ADCUP). Below we provide the mathematical formulation of the bilinear problem,

which is equivalent to CPLNF( '"). For details on the formulation of the problem,

finite convergence and other properties of the DCUP we refer to C'! Ilpter 2.

Problem CPLNF-R( ') is defined by:


minm [cI -y + cat.] a + s~ g
x,y
aEA aEA

s.t. Bx b,

?Ya < Xa < ?'I a" + AaYa, Va E A,

Y a + Ya = 1, Va e A,

Xa > 0, Y > > 0, and ga > 0, Va e A.

The ADCUP is a heuristic algorithm to find a solution to the FCNF problem.

Note that the choice of a has a direct influence on the CPU time of the procedure

as well as the quality of the solution. In particular if the value of the a is close

to one, the value of the F decreases slowly and the procedure requires a large

number of iterations to stop. On the other hand, small values of the parameter can

worsen the quality of the solution. In the numerical experiments, we use a = 0.5

because in our test problems the procedure with that parameter provides a fairly

high-quality solution using 1-4 seconds of CPU time (see Section 3.5).

Procedure 4 : Adaptive Dynamic Cost Updating Procedure (ADCUP)
Step 1: Let \ <-- Aa, 0 = 0, y 0, andy 1. m -- 1.
Step 2: Run DCUP to solve the CPLNF-R("') problem with initial vector
(4r1n- 1). Let (,- ,) be the solution that is returned by DCUP.
Step 3: If 3a e A such that e (0, F-) then eF+1 -- 7a', m -- n + 1, and go
to step 2. Otherwise, stop.









3.4 On the Dynamic Slope Scaling Procedure

This section discusses some properties of the dynamic slope scaling procedure

proposed by Kim and Pardalos [61] (see also Kim and Pardalos [62]). In the paper,

the authors discuss four variations of DSSP based on the choice of the initial vector

and the slope updating scheme. However, regardless of the initial vector and the

updating scheme, DSSP provides an equilibrium type of solution. To prove the

statement, we first transform FCNF into an alternative problem then prove that

the solution provided by DSSP is a solution of a variational inequality problem

constructed based on the new formulation. The theoretical results provided below

are very similar to those in C'! lpter 2, where we have shown that the property

holds for the concave piecewise linear network flow problem.

Let
'X)- Xa > 0aiC a +a s,, c (0, Aa,
M X"- 0 M X= 0

where M is a sufficiently large number. Consider the following network flow

problem with flow dependent cost functions F,(xa).

NFPwFDCF:

min FT (x)

s.t. Bx = b, (3-2)

Xa[0,X"], VacA, (3-3)

where F(x) is the vector of functions F,(Xa).

Theorem 3.4.1. The NFPwFDCF problem is equivalent to the FCNF problem.

Proof: (See proof of Theorem 3.4.1, ('! Ilpter 2) U

Let NFPwFDCF(F) denote the NFPwFDCF problem, where the vector

function F(x) is fixed to the value of the vector F. In the first step, DSSP

solves NFPwFDCF(F) problem with an initial vector F = F0. Let xm









argmin{NFPwFDCF(Fm)}. Next, DSSP iteratively updates the value of the

vector F using the solution x", i.e., F"+1 = F,(xj), and solves the resulting

NFPwFDCF(Fm+1) problem. The procedure stops if xm+l = xm. In Kim and

Pardalos [61], the authors propose different updating schemes, where they replace

M by a smaller value. However, the next theorem proves that regardless of the

initial vector F0 and the updating scheme, the final solution provided by DSSP is a

solution of a variational inequality problem.

Theorem 3.4.2. The solution of DSSP is the solution of the following variational

.,' .,. ;l,.;i problem

find x* feasible to (3 2) and (3 3) such that FT(x*)(x x*) > 0, Vx feasible to

(3-2) and (3-3)

Proof: Assume that DSSP stops on iteration k, and let x* = argmin{(Fk)TxlBx

b, xa E [0, OA ]}. From the stopping criteria it follows that x* = xk. If x* > 0

then Ff = Fa(x*). On the other hand, if x* 0 then one can replace Ff

by a sufficiently large M without changing the optimality of x*. As a result,

x* = argmin{FT(x*)x Bx = b,Xa E [0, A'-]}. From the latter it follows that

FT(x*)(x x*) > 0, for all feasible x. U

From Theorem 3.4.2 it follows that the solution of DSSP, x*, satisfies the

inequality FT(x*)x > FT(x*)x*, Vx feasible to (3-2) and (3-3), i.e., x* is an

equilibrium solution. However, since NFPwFDCF is equivalent to the FCNF

(see Theorem 3.4.1), one is interested in finding a feasible x such that FT(x)x >
FT(x)x, Vx feasible to (3-2) and (3-3), i.e., x is a system optimum solution. Notice

that the equilibrium and the system optimum solutions may not be the same,

unless Fa(xa) is constant.

3.5 Numerical Experiments

This section discusses numerical experiments on the adaptive dynamic cost

updating procedure. We solve all problem sets using ADCUP as well as DSSP (see









Kim and Pardalos [61]). To compare the results of the heuristic procedures, in the

case of small problems we find an exact solution using CPLEX MIP solver and

compute relative errors. In the case of large problems, CPLEX is not able to find

an exact solution within 5,000 seconds of CPU time; therefore, we compare the

results of DCUP versus DSSP.

In the experiments, we solve problems using all four variations of DSSP and

choose the best solution to compare with the solution provided by the ADCUP.

In addition to the final solution (the solution that the algorithm returns when it

stops), during the iterative process DSSP as well as ADCUP construct feasible

vectors that might have a better objective function value. In the procedures, we

record those vectors and choose the best one. The comparison between the best

solutions of both algorithms is also provided. With regard to the computational

time, we compare the CPU time of ADCUP versus the best CPU time among four

variations of DSSP.

There are four groups of test problems based on the size of the network and

the number of supply/demand nodes (see Table B 1, Appendix B). For each group,

we construct different types of functions fa(x,), where the slope and the fixed cost

are generated randomly according to the specified distributions. In total, there

are nine sets of problems (nine types of function f,(xa)) for each group, i.e., one

set of problems for each choice of distribution for the slope and the fixed cost.

There are 30 randomly generated problems for each problem set. The components

of the supply/demand vector are generated uniformly between 30 and 50 units.

The model is constructed using the GAMS environment and solved by CPLEX

9.0. Computations were made on a Unix machine with dual Pentium 4 3.2Ghz

processors and 6GB of memory. All results are tabulated in the Appendix B.

Tables B-2 and B-3 illustrate the computational results for groups G1 and G2.

Since the size of those problems is not big, we have solved the problems exactly









using the CPLEX MIP solver. The relative errors are computed based on the

following formulas
READCUP (i fADCUP exact
READCUP\ *)
fexact
REDSSP fDSSP fexact
REDSSPy *)-
fexact
From the results it follows that on average ADCUP provides a better solution

than DSSP using less CPU time. Notice that ADCUP outperforms DSSP in the

comparison of the final solutions as well as the best solutions. Although there

are some problems where DSSP provides a better solution than ADCUP, the

percentage of those problems is fairly small and decreases with the increase of the

size of the network. In addition, observe that the quality of the solutions provided

by both algorithms changes with the choice of the distributions for the slopes and

the fixed costs. In particular, both algorithms provide a near optimum solution

for the problems with a larger slope and smaller fixed cost. Although the relative

error of both algorithms increases with the decrease of the slope and the increase of

the fixed cost, observe that the quality of the solutions provided by DSSP changes

more than those provided by ADCUP.

In the case of groups G3 and G4, we compare ADCUP versus DSSP using the

following formula

DSSP ADCUPfDSSP fADCUP
min{fDssp, fADCUP}

The computational results on those groups are summarized in Table B-4. Similar

to the previous two groups, one observes that on average ADCUP provides a

better solution than DSSP. Notice that DSSP consumes much more CPU time

before termination than ADCUP. In addition, the percentage of problems where

ADCUP provides a better solution than DSSP is higher than in the previous cases.

Similar to groups G1 and G2, the difference between the solutions provided by









both algorithms is small for the problem sets with a larger slope and smaller fixed

cost. When the slope decreases (or the fixed cost increases), ADCUP provides a

perceptibly better solution than DSSP.

3.6 Concluding Remarks

Unlike other models in the literature, we consider concave piecewise linear

network flow problems to solve fixed charge network flow problems. A proper

choice of parameter a guarantees the equivalence between the CPLNF(E) and

the FCNF problems. Based on the theoretical results, we propose a solution

algorithm for the FCNF problem, where it is required to solve a sequence of

CPLNF(E) problems. To find a solution of CPLNF(E), the algorithm employs the

dynamic cost updating procedure. The computational results show that ADCUP

outperforms DSSP in the quality of the solution as well as CPU time. Although

in the computations we choose a = 0.5, one can use a higher value in an attempt

to improve the quality of the solution. Note that a large value of the parameter

increases the CPU time of the procedure. Although ADCUP often provides an

exact solution, it is not guaranteed because DCUP converges to a local minimum of

the bilinear relaxation problem, CPLNF-R(F').

In the numerical experiments, we have shown that the relative error of the

solutions of both procedures increases in the cases of small slopes and large fixed

costs. To explain this phenomena, observe that by decreasing the value of the slope

the angle between function fa(xa) and the first linear piece of function ,a' (xa)

increases (see Figure 3-1). As a result, ,' (x,) does not approximate the function

f,(ax) as well as in the case of large variable costs. The same discussion applies to
the case of a large slope.














CHAPTER 4
A BILINEAR REDUCTION BASED ALGORITHM FOR SOLVING
CAPACITATED MULTI-ITEM DYNAMIC PRICING PROBLEMS

4.1 Introduction to the Chapter

Supply chain problems with fixed costs and production planning problems

involving lot-sizing have been active research topics during resent decades. Many

research papers have addressed single-item problems with additional important

features such as backlogging, constant and varying capacities, and different

cost functions (see, e.g., Gilbert [43], Florian and Klein [36], van Hoesel and

Wagelmans [53], Loparic et al. [70], and Loparic et al. [71]). It is well known that

incapacitated problems can be reduced to a shortest path problem. Florian and

Klein [36] studied capacitated single-item problems, where they characterized

the optimal solution and proposed a simple dynamic programming algorithm for

problems in which the capacities are the same in every period. The single-item

problems with varying capacities are known to be NP-hard. A classification

of different problems and a survey on existence of a polynomial algorithm for

solving problems for different classes can be found in Wolsey [100] and Pochet and

Wolsey [88]. Tight formulations for polynomially solvable problems are discussed in

Miller and Wolsey [75] and Pochet and Wolsey [88].

Almost all practical problems involve multiple items, machines and/or levels,

and polynomial results for those problems are limited. Using binary variables,

one can construct a mixed integer linear programming (\!IP) formulation

of the problem with an imbedded network structure. To solve the problem,

branch-and-bound and cutting plane algorithms have been used (see, e.g., Barr

et al. [4], Cabot and Erenguc [12], Gray [44], Kennington and Unger [59], Palekar









et al. [83], Marchand et al. [72], and Wolsey [100]). In addition, several heuristic

algorithms have been proposed (see, e.g., Cooper and Drebes [27], Diaby [31],

Khang and Fujiwara [60], Kuhn and Baumol [63], van Hoesel and Wagelmans [52],

Kim and Pardalos [61] and [62], Nahapetyan and Pardalos [79] and [80]).

In this chapter we discuss a capacitated multi-item dynamic pricing (C' \I)P)

problem where one maximizes the profit by choosing a proper production level as

well as pricing policy for each product. In the problem, the demand is a decision

variable, and in order to satisfy a higher demand one needs to reduce the price of

the product. On the other hand, reducing the price can decrease the revenue, which

is the product of the demand and the price. In addition, the problem includes

an inventory and production cost for each product, where the latter involves a

setup cost. The objective of the problem is to find an optimal production strategy,

which maximizes the profit subject to production capacities that are -!I ied"

by the products. Different variations of a single-item uncapacitated problem

with deterministic demands are discussed by Gilbert [43], Loparic et al. [71], and

Thomas [96]. A capacitated single-item problem with time invariant capacities is

discussed in Geunes et al. [42]. The polynomial algorithms proposed by the authors

are based on the corresponding results for the lot-sizing problems.

In C'!i lpters 2 and 3 (see also Nahapetyan and Pardalos [79] and [80]) we have

proposed a bilinear reduction technique, which can be used to find an approximate

solution of concave piecewise linear and fixed charge network flow problems. A

similar technique is proposed to solve the C' I )P problem. In particular, we

consider a bilinear reduction technique of the problem and prove that solving the

C'\ I)P problem is equivalent to finding a global maximum of the bilinear problem.

The latter belongs to the class of bilinear problems with disjoint feasible region,

and one considers a heuristic algorithm to find a solution of the problem. The

heuristic algorithm employs a well known iterative procedure for finding a local









maximum of the problem (see, e.g., Horst et al. [54] and Horst and Tuy [55]).
Numerical experiments on randomly generated problems confirm the efficiency of

the algorithm.

For the remainder, Section 4.2 provides a linear mixed integer formulation

of the problem and discusses a bilinear reduction of the problem. We prove that

solving the C'\ 1 )P problem is equivalent to finding a global maximum of the

bilinear reduction. In Section 4.3 we propose a heuristic algorithm for solving the

bilinear problem. Numerical experiments on the algorithm are provided in Section

4.4, and finally, Section 4.5 concludes the chapter.

4.2 Problem Description

In this section we provide a nonlinear mixed integer formulation of the

problem. Using some standard linearization techniques, the problem can be

simplified. To solve the problem, we propose a bilinear reduction technique and

prove some properties of the bilinear problem.

Let P and A represent the set of products and discrete times, respectively.

In addition, let f(p,j)(d) denote the price of product p at time j as a function

of the demand d, and g(pj)(d) = f(p,j)(d)d, i.e., g(pj)(d) represents the revenue

obtained from selling d amount of product p at time j. In the problem, we assume

that f(pj)(d) and g(p,j)(d) are nonincreasing and concave functions, respectively

(see Figures 4-1). If f(p,j)(d) is a concave function, then it is easy to show that
concavity of g(p,j)(d) follows.
Let x(p,i,j) denote an amount of product p that is produced at time i to satisfy

the demand at time j, and y(p,i) represent a binary variable, which equals one

if product p is produced at time i and zero otherwise. Assume that inventory

costs, c(, production costs, c ,, and setup costs, c as well as production

capacities, Ci, are given, where p, i, and j represent the product, producing time,










(,j)( g(pj)(d)
/fmax
(pj)
A quadratic function

A linear function



dmax d 7 dmax d
d(PJ) d(pj)

Figure 4-1. The price and the revenue functions.


and selling time, respectively. The following is the mathematical formulation of the

C'\ I )P problem.

CMDP :


max .P ,. l (p,i) Y(13i)
pEP jE'e ieAi|ji pEP idjEA|ij pep iEAz


s.t. X(p) pEP jeAzli
SX(p,i,j) < CiY(p,i), Vp C P and i c A,
jEz'li
X(p,i,j) > 0, Y(p,i) E {0, 1}, Vp P and i,j e A.

The objective function of the problem maximizes the profit, which is the

difference between the revenue and the costs. The latter includes the inventory

as well as the production costs. The first constraint ensures the satisfaction of

the capacity restrictions, and the second one makes sure that y(p,i) equals one if

jeA|jE' 0.

Although the above formulation belongs to the class of nonlinear mixed integer

programs, using standard techniques one can approximate the revenue function by

a piecewise linear one and linearize the objective function. Doing so, observe that

from the concavity of the revenue function it follows that there exists a point, d(pj),

where the function reaches its maximum (see Figure 4-1). As a result, producing










and selling more than d(pj) is not profitable, and at optimality xiEAi
d(p,). To linearize the revenue function, divide 0, d(pj) into N intervals of equal

length. Let 4k i) denote the end points of the intervals, i.e., 4, i = kd(pj)/N,

Vk c {1,..., N} U{0} = K U{0}, and g Pj) represents the value of the revenue

function at the point 4k ), i.e., g = gP, )) ) ,' Using those

parameters, construct the function


N

k=0


N
A (pj),
k-1


where it is required that


N N
(P'') k k d k
k 0 k 1 (pj)'
iE^\i

Vp C P and j E A,


N
A k 1, A ) >0, Vp P and je A,
k-0
and Ak(p,) / 0 for at most two consecutive indices k. Observe that g(p,j)(d) is a

concave nondecreasing function on the interval 0, d(pj) ; therefore, its piecewise

linear approximation, i.e., g(p,j)(A(p,j)), preserves the same property. From the

latter, it follows that in the maximization problem the requirement on A) being

positive for at most two consecutive indices k can be removed from the formulation,

and it is satisfied at optimality (for details see C'! plter 11, Bazaraa et al. [5]). The

following is the mathematical formulation of the approximation problem:


max A3P) > >1 KcU + .] Cr, X(pYi ~ ji
x,y,A p ptE
pEP jEA kEK pEP i,jEAli
s.t. 3 Y X(ZPij) < C' V1 e A,
pEP jEAJi
.(p,ij) < Cy(pi), Vp c P and i c A,
jEAJi
X (pij) dk k,) VpcPand j A,
iEAl_

9(pj) (A(p,j))









N
A(py) = 1,
k-0
x(p,i,j) > O, A j > 0, Y(p,i) e {0, 1},


Vp E P and j e A,


Vp E P, i,j E A and k e KU {0}.


Next, we simplify the formulation using nonnegative variables x kj), k c K,

instead of x(p,ij), where x (j)k represents the amount of product p that is produced

at time i and sold at time j using unit price g kj)/ i) fk,) (pj)(

Doing so, the third constraint in the above formulation can be replaced by the

following one:


k1 k X dk Ak
xP(y) d(PJ) (pj),
tEAli

Vp c P,j c A and k E K.


The latter can be used to remove the variable Ak from the formulation. In

particular, the fourth constraint is replaced by inequalities


k
S_< 1,
kEK iEAli

Vp P and jE cA,


and in the objective


g()) A (3) ~ (3 x( ,,) Vp j c A and k K,
iEdli<_j

where fj) f1,=)k (k, i)). The following is the resulting alternative formulation of

the approximation problem, which we refer to as AC\ i )P:

ACMDP:


max Y (P j)X(P l)
pEP iEA jeA|i<_j kEK

s.t. X < ci,
pEP jEAti<_j kEK

x kj) < Cy(p,), Vp
jEA|ije kEK


st
c(pi) (p,i)


Vi eA,


- P and i E A,


(4-1)


(4-2)








k
S< 1, Vp P andj A, (4-3)
kEKiA\li
X _k) > 0, ,i) {0,1}, Vp e P, i,j A and k e K, (4-4)

where qg, j) = () c" ,. Observe, that at optimality x(,j) 0 for

all indices such that qk < 0, and those variables can be removed from the

formulation. Therefore, without lost of generality, in the analysis below, we assume

that kq,) > 0.

Define X = {xlx > 0 and x(pi,j) are feasible to (4-1) and (4-3)}, and

Y = [0, 1]IPlHl. Consider the following bilinear program:

ACMDP B:


max i'j)X(pi) -(pi) Y(pi) (Xy)
pEP izA jEAli<_j kEK

s.t. x EX and y E Y.

Theorem 4.2.1. Any local maximum of the A('CI)P-B problem is feasible or can

provide a feasible solution of the AC 'II)P problem with the same objective function

value.

Proof: Let (x*, y*) denote a local maximum of the AC I 1)P-B problem. Observe

that by fixing x to the value of the vector x*, the AC' I I)P-B problem reduces to

a linear one, and y* is an optimal solution of the resulting problem. Assume that

Ep E P and i E A such that y*i) E (0, 1), i.e., y*, is a fractional number. If

yEI<3 ZkEK q(p,ij) pi,) (p,) < 0 or 2Ejij< ZkEK q( ip,)Xi,) ,) (p,) > 0
then it is possible to improve the objective function value by assigning y*, = 0

or y*,i 1, respectively. The latter contradicts the optimality of (x*,y*). On the

other hand, if jElij -kEkK q pij)(P,) i ,. = 0 then by changing the value of

the variable y*,) to zero the objective function value remains the same. Construct

a vector y, where y(p,) = [YP,* Note that (x*, y) is feasible to constraints (4-1),









(4-3) and (4-4). If (x*, y) violates constraint (4-2) then 3p E P and i c A such

that E |<7 zkEKKij) > 0 and y(p,) = 0. From the local optimality of (x*, )

it follows that CeAli
produce product p at time i. Furthermore, by assigning x*,) = 0, Vj E A and

k e K, the objective function value of the AC\ I I)P-B problem remains the same.

Let x denote the resulting vector, i.e.,

k 0 if ECa|< ECkEK q(pj)xk() "- cs 0 -5
(*, i f k *k k- CSt > 0
( i(pj) .,
( Xj) if ZjAzJiZkEK q Cij) ,i,) >

The vector (x, y) is feasible to the AC'\ I)P as well as the AC' \I)P-B problem and

has the same objective function value as (x*, y*). *

Theorem 4.2.2. A 11..1l maximum of the AC('iP-B problem is a solution or can

be used to construct a solution of the AC('DP problem.

Proof: Observe that any feasible solution of the AC' I 1)P is feasible to the

AC'i\ )P-B problem. Furthermore, if (x, y) is feasible to the AC'i\I)P problem then


j) (p i' j) ~ (p, )yP(p, )= z z ^(Pij) (P ,) -- c(p,i) Y(p,i)
jEAli
From the latter it follows that the AC\ lI)P-B problem can be obtained from

AC\ I)P by replacing the objective function by

a q ( ) -- Y(p,i),
xpP iEz jEAli<_j kEK K

removing constraint (4-2) and relaxing the integrality of the variable y(p,i). In

other words, the AC\ i )P-B problem is a relaxation of the AC'\ I )P problem. From

Theorem 4.2.1 it follows that a global solution of AC'\ I )P-B is a solution (or leads

to a solution) of the AC' \ 1)P problem. U

The above two theorems prove that solving the AC'i\I)P problem is equivalent

to finding a global maximum of the AC' I 1)P-B problem. In particular, one can









solve the AC'\ i I)P-B problem and if the solution is not feasible to the AC'\ iI )P

problem, then use the method described in the proof of Theorem 4.2.1 to construct

a feasible one with the same objective function value.

4.3 A Bilinear Reduction Based Algorithm for Solving ACMDP
Problem

In this section we discuss a heuristic algorithm for solving the ACi\ I)P-B

problem, which employs a well known iterative procedure for finding a local

maximum of the AC'\ i I)P-B problem.

Observe that the problem belongs to the class of bilinear programs. By fixing

vector x or y to a particular value, the problem can be reduced to a linear one. Let

LP(x) and LP(y) denote the corresponding linear programs, i.e.,
LP(x) : ma ZpeP EieA [LEjA|i-< kEK C ,i p j st ] y(p,), and

LP(y) : maX pEP iEA jEA|i KCeK ['pi'j) (ip) ,i,j)'
Notice that the solution of the LP(x) is easy to obtain. In particular,

k k -st < 0
{ O if Zeli V(Ip,i) -
1 ) if YjEAlJi 0

is an optimal solution of the problem. The Procedure 5 describes a well known

algorithm, which starts from an initial binary vector and converges to a local

maximum of the AC'\ I)P-B problem in a finite number of iterations (see Horst and

Tuy [55] or Horst et al. [54]).

However, the procedure has the following disadvantage. Let (x", y') represent

the solution obtained on iteration m, and assume that 3p E P and i E A such that

y =,) 0. As a result, in the LP(ym) problem q(P,
Procedure 5
Step 1: Let yo denote an initial binary vector of y(p,i). m -- 1.
Step 2: Let xm ar,,,i., {LP(y -1)}, and ym = ,I.nt.,, {LP(xm)}.
Step 3: If y" y"-1 then stop. Otherwise, m <-- m + 1 and go to Step 2.









k E K, and perturbations of the values of the corresponding variables xzk ) do not

change the objective function value. Furthermore, because the products -I! ire"

the capacity and other products can have a positive cost in the LP(ym) problem, it

is likely that the value of the variable ximk decreases in the next iteration. From

the latter it follows that y'()1 = 0. To summarize, if y"(,i) 0 then it is likely that

(i) y( =,) 0, Vn > m, and (ii) the final solution is far from being a global one.
To overcome those difficulties, next we propose an approximation problem, which

avoids having zero costs in the objective of the LP(y) problem.

Let )(x(pL)) = E ECK pij)pi) and


(p ,) ( ( ,j) (pij),
1! jEAjti
where (p,) > 0 and x(p,,) is the vector of xk .). It is easy to show that both

functions have the same value, (pi), on the hyperplane CEAl
E(p,t) +c ,t Furthermore, p (i((p)) > (xi) ,) if ZE A|iz kEK 1i p,) >

E(p,i) + and p(p,)(X(p,)) < (pX)(X(p,) if EAi (pi) + cst Define


S(Xy) 0 >1 > LYp,)(x(p,t))u(p,t) + (#5pt)(X(p,t))(1 1(p,i))]
pEP iEA

where E denotes the vector of E(p,t). The function op(x, y) depends on the value

of the vector E, and "(x, y) > p(x, y), for all E > 0 and (x, y) feasible to

the AC\Ii)P-B problem. Observe that if E 0 then Y(ti)(x(p,i)) 0, and

P(x, y) p(x, y). In that sense, p"(x, y) is an E-approximation of p(x, y), and
it approximates the function from above. By replacing the objective function of

the AC\ I )P-B problem by the function (x, y), we refer to the resulting problem

as E-approximation of the AC\ I )P-B problem and denote by AC\ I )P-B(E).

Construct the corresponding LP"(x) and LP"(y) linear problems by fixing vectors

x and y in the ACi\I)P-B(E) problem to a particular value, i.e.,









Procedure 6 :
Step 1: Let (p,i) be a sufficiently large number, and yo be such that y = 1,
VpE P and i E A. m 0.
Step 2: Construct the E-approximation problem ACi\ I)P-B(E) and run
Procedure 5 to find a local maximum of the problem, where y" is an initial
binary vector. Let (x"m +l1) denote the local maximum.
Step 3: If 3p E P and i e A such that EY:Ei K p, and EjEA|i 0 then <-- aE, m <-- m + 1 and go to Step 2.
Otherwise, stop.


LP(x) : max pEp Ei (, p,)(x(p,i))y(p,i) + () (x((p,))(1 y(p,,)), and
yEY
LP"(y) : maX YpP,i, EkEK,jEAj i) (p) ,+ 1 Yp))i X))
As before, the solution of the LP"(x) problem is easy to obtain by assigning

0 jf Zke K V k k cst <
i(p,i) --
0 if jE zjiA| j Y kEfiK q(pi,j) (p,ij) > '(pl)


{ 0 if j )(X(P,)) < ) (X(,))

Sif ) X(p)) > X ))

The heuristic procedure (see Procedure 6) starts with a sufficiently large E and

finds a local maximum of the resulting E-approximation problem. If the stopping

criteria is not satisfied in Step 3 then it decreases the value of the vector E to aE,

where a is a constant from the open interval (0, 1), and the process continues using

a new E-approximation problem. Observe that Procedure 6 uses vector y' from the

previous iteration as an initial vector.

The procedure depends on two parameters: the initial vector E and the value

of a. The value of E(p,t) depends on parameters of the problem, and one can

consider them equal to the maximum profit, which can be obtained by producing

only product p at time i. Although such maximization problem is easy to solve

using standard LP solvers, for large IPI and |A| one finds computationally

expensive solving the problem for all pairs (p, i) c P x A. Instead, we propose









an algorithm for finding the values of (p,i) (see Procedure 7). Observe that

x, : j*) -< i ), and the maximum additional profit that can be obtained using the
variable x j)is q(i .k i). Using this property, for all pairs (p, i) the procedure

iteratively finds the maximum among q ,dk i), and assigns the demand (or the

remaining of the capacity) to the corresponding variable xij).k The value of (p,i)

is computed based on the formula E(p,) = jeA|z, < k eK C ,ij)Xpij) cst .. As for

the parameter a, its larger value increases the computational time of the procedure,

and it is likely to provide a better solution.

4.4 Numerical Experiments

In this section we discuss numerical experiments conducted on randomly

generated problems. The problems are solved by Procedure 5 as well as Procedure

6 using different values for the parameter a. The latter procedure employs

Procedure 7 to find an initial value for the vector E. In addition, we solve the

problems by the MIP solver of CPLEX using the ACl\ I)P formulation. In the

cases where the MIP solver is not able to solve large problems within posted CPU

and memory limitations, we compare the solutions of the procedures with the

best solutions found by CPLEX. The main purpose of the computations is the

performance of the procedures for different capacities.

Procedure 7
Assign xzi) = 0, Vp e P, i,j e A, and k e K
for all p e P and i c A do
C C ,j) q~,) max = max{q(~p, i)k c K, j A,j > i}
while C / 0 and qmax / 0 do
Let jmax and kmax are such that qmax = q.m~ma,.m
Assign xl, ,max, = min{C',., ), Ca C X jmax p,ijmax = 0, Vk
K, and qmax max{q 4k'j), i),k e K,j A,j > i}
end while
(pi) jEa|J end for









In the numerical experiments we consider problem sets with different numbers
of products, |P| = 5, 10, or 20, and time horizons, |A| = 12 or 52. For each
problem set we randomly generate capacities for all i E A using the formula C

PI U, where U is a random number uniformly generated from interval [10,100],
[50,150], [100, 200], or [150,250]. Note that all intervals allow generating capacities
that are tight at optimality with respect to the revenue function discussed below.
In addition, using term IPI one generates capacities that depend on the number
of products. The latter allows comparing of results across different numbers
of products. As for the costs, we generate the production costs cp' and the
inventory costs c" i) according to the uniform distributions U[20, 40] and U[4, 8],
respectively. Observe that on average the inventory cost is equal to 211' i of the
production cost. Finally the setup cost c, i)is generated uniformly from interval
[600, 1000].
In the experiments we restrict ourself by considering only linear price functions
of the form f(p,)(d) = fa f_ /d, I4 d. To avoid generating functions that at
optimality result in unrealistically large profits, we introduce an index /, where
Vk _^ LJ < YV (if() cPr \) X$,J) cst il* 1i)

__^ V^ fv^ in + (,Pr ,,*k + st *
EpeP Ez [ 'e j YkK C i) L Z,)(p,i,j) + .+ t (p,i)j

That is, the index measures the amount of the profit per unit of investment and
it is computed based on the optimal or the best solution provided by CPLEX.
By generating fma' and nC i according to the uniform distributions U[70, 90]
and U[500, 1000], respectively, at optimality (i) / E [0.7, 1.3], (ii) all capacities
considered above are tight, and (iii) in most of the cases the satisfied demand
is less than d(pj) (see Figure 4-1). In addition, the proposed price function and
distributions of the costs and capacities allow generating problems that have an
optimal objective function value ranging from hundreds of thousands to several









millions. Finally, in the construction of the piecewise linear approximation of the

revenue function we use N = 10.

The model is constructed using the GAMS environment and solved by CPLEX

9.0 with a CPU restriction of 2000 sec and a memory restriction of 2Gb, where the

latter is the memory that is required to store the tree in the branch-and-bound

algorithm. Computations are made on a Unix machine with dual Pentium 4 3.2Ghz

processors and 6GB of memory. The results are tabulated in the Appendix C.

In the experiments we solve 10 randomly generated problems for each problem

set and capacity. Tables C 1 and C-2 compare the results provided by CPLEX

with the solutions provided by both procedures. The relative error is computed

using the formula
ObjCPLEX ObjProc.6(5)
RE( ,)=
ObjCPLEX

In the Table C 1, column A indicates the number of problems where the heuristic

procedure finds a better solution than CPLEX. Note that CPLEX is able to

provide an exact solution for all capacities from the problem set 5-12. In all other

cases, the solver stops after reaching the CPU limit or the memory limit and

returns the best found solution. Although the relative optimality gap of the final

solutions of those problem sets varies from _' to 5'. we believe that the solution

is an optimal or close to an optimal one, and the large optimality gap is due to

imperfect lower bounds. The fact that the heuristic procedures provide a slightly

better solution in the case with |A| = 52 than |A| = 12 partially confirms our

assumptions.

The relative errors in the Table C 1 confirms the effectiveness of the heuristic

procedure. In particular, in the 1i I ii iiy of the problems the heuristic algorithm is

able to provide a solution within 1 from the optimal one or the best one provided

by CPLEX. Observe that the larger value of a provides a better solution and the

number of problems where the heuristic procedure finds a better solution than









CPLEX is increasing with the size of the problem. By comparing with the solutions

provided by Procedure 5 (see Table C-2) one notices that Procedure 6 outperforms

the Procedure 5, and it is more stable with changes in the capacities. As for

the CPU time (see Table C-3), the heuristic procedures require fewer resources

than CPLEX. In addition, unlike CPLEX the heuristic procedures do not require

gigabytes of memory to store the tree.

4.5 Concluding Remarks

We have discussed a bilinear reduction scheme for the capacitated multi-item

dynamic pricing problem, where solving the latter is equivalent to finding a global

solution of the former. Based on theoretical results of the reduction problem, two

procedures have been proposed to find a global maximum of the problem. The

first one is a well known technique and has been intensively used to solve other

bilinear problems. Because of the reasons discussed in Section 4.3, in the very

beginning of the iterative process the procedure eliminates some products from the

further consideration. The latter worsen the quality of the solution returned by

the procedure. In the second procedure we construct approximate problems and

gradually decrease parameters of the problems. As a result, during the iterative

process the costs of the eliminated products remain positive and the procedure

considers them again if need be. Although the second procedure requires more

CPU time to stop than the first one, it provides a higher-quality solution.














CHAPTER 5
DISCRETE-TIME DYNAMIC TRAFFIC ASSIGNMENT MODELS WITH
PERIODIC PLANNING HORIZON: SYSTEM OPTIMUM

5.1 Introduction to the Chapter

Since Merchant and Nemhauser (see, Merchant and Nemhauser [73] and

[74]) first proposed their model in 1978, there have been a number of papers

(see, e.g., Carey and Subrahmanian [23], Carey [15], Carey [13], Carey [14],
Friesz [37], Friesz [38], Wie et al. [98], C('. ,1 and Hsueh [25], Janson [57], Ho [51],

Ziliaskopoulos [104], Drissi-Kaitouni and Hameda-Benchekroun [32], Li et al. [66],

Kaufman et al. [58], Boyce et al. [11], Ran and Boyce [90], Ran et al. [92], and

Wie et al. [99]) discussing variational inequality or mathematical programming

formulations for the dynamic traffic assignment problem with the assumption that

the planning horizon is a set of discrete points instead of a continuous interval.

_M iw, of these papers use a dynamic or time-expanded network (see, e.g., Al!mi et

al. [2]) to simultaneously capture the topology of the transportation network and

the evolution of traffic over time. Implicitly or otherwise, these papers typically

assume that there is no traffic at the beginning of the planning horizon (or at time

zero) and that all trips must exit the network prior to the end. When there are cars

at the time zero, the times at which these cars enter the network must be known in

order to determine when they will exit the arcs on which they were travelling. In

practice, data with such details do not generally exist.

There are two main factors that distinguish the models in papers referenced

above. First, some (e.g., Merchant and Nemhauser [73], Carey and Subrahmanian

[23], Ho [51], Carey [15], Ziliaskopoulos [104], Kaufman et al. [58], Garcia et
al. [40]) seek a system optimal solution and others (e.g., Janson [57], Wie et









al. [98], C'!, i, and Hsueh [25], and Drissi-Kaitouni and Hameda-Benchekroun [32])

compute a user equilibrium instead. The other factor is the travel cost function

used by these models. Among other parameters (physical or otherwise), a travel

time or cost function may depend on the number of cars on the link and the input

and output rates. 1T ,tn (e.g., Carey and Ge [20], Carey and McCartney [18],

Carey [16], Carey [17], Lin and Lo [69], Han and Heydecker [46], Daganzo [28])

have analyzed the effects of travel cost functions on various models. Some (e.g.,

Lin and Lo [69] and Han and Heydecker [46]) have shown that some travel cost

functions are not consistent with the models that use them.

Similar to Carey and Srinivasan [21], Carey and Subrahmanian [23], Carey

[15], C('! i and Hsueh [25] and Kaufman et al. [58], the model in this chapter

is based on the time-expanded network. However, instead of assuming that the

network is empty at the beginning or at the end, we treat the planning horizon

as a circular interval instead of linear. For example, consider the interval [0, 24],

i.e. a 24-hour planning horizon. When viewed in a linear fashion, it is typically

assumed that there is no car in the network at times 0 and 24. In turn, this implies

there is no travel demand after time k < 24. Otherwise, cars that enter the

network after time k cannot reach their destinations by time 24, thereby leaving

cars in the network at the end of the horizon. On the other hand, if there is a car

entering a street at 23:55h (11:55 PM) and exiting at 24:06h (12:06 AM, the next

day) in a circular planning horizon, the exit time of this car would be treated as

00:06h instead. When accounted for in this manner, it is possible to determine the

exit time for every car that is in the network at time zero without requiring any

additional data. Additionally, models that view the planning horizon in a circular

fashion are more general in that they include those with a linear planning horizon.

By setting the travel demands and other variables during an appropriate time









interval to zero, models with a circular planning horizon effectively reduce to ones

with a linear horizon.

It is often argued that the number of cars at the beginning and the end of

the horizon are small and solutions to DTA are not drastically affected by setting

them to zero. When the paths that these cars use do not overlap, the argument

is valid. However, when these cars have to traverse the same arc in reaching their

destinations, the number of cars on the arc may be significant and ignoring it may

lead to a solution significantly different from the one that accounts for all cars.

This chapter makes two main assumptions. One requires the link travel time

at time t to be a function of only the number of cars on the link at that time.

Carey and Ge [20] show that the solutions of models using functions of this type

converge to the solution of the Lighthill-Whitham-Richards model (see Lighthill

and Whitham [68] and Richards [93]) as the discretization of links into smaller

segments is refined. Because minimizing the total travel time or delay mitigates

its occurrence, models discussed herein do not explicitly address spillback. On

the other hand, the models can be extended to handle spillback using a technique

similar to the one in Lieberman [67] or an alternative travel time function that

includes the effect of spillback (see, e.g., Perakis and Roels [86]). However, as

indicated in the reference, using such a function may not lead to a model with a

solution.

For the remainder, Section 5.2 defines the concept of periodic planning

horizon. Section 5.3 formulates the system version of the discrete-time dynamic

traffic assignment problem with periodic planning horizon or DTDTA and prove

that a feasible solution exists under a relatively mild condition. To our knowledge,

there are only four papers (Brotcorne [10], Smith [95], Wie et al. [98], and Zhu and

Marcotte [103]) that address the existence issue and some (see, e.g., Smith [95] and

Zhu and Marcotte [103]) consider this small number to be lacking. All four deal













I I vs
0 T




Figure 5-1. Linear versus circular intervals.


with user equilibrium problems instead of system optimal. Section 5.4 describes

two linear integer programs that provide bounds for DTDTA. Section 5.5 presents

numerical results for small test problems and, finally, Section 5.6 concludes the

chapter.

5.2 Periodic Planning Horizon

The models in this chapter assume that the planning horizon is a half-open

interval of length T, i.e., [0, T). Instead of viewing this interval in a linear fashion,

the interval is treated in a circular manner as shown in Figure 5-1. In doing

so, time 0 and T are the same instant. For example, time 0:00h and 24:00h (or

midnight) are the same instant in a 24-hour d-iv. For this reason, T is excluded and

the planning horizon is half-open. To make the discussion herein more intuitive, we

often refer to the planning horizon as a 24-hour d-4v, i.e., T = 24. In theory, the

planning horizon can be of any length as long as events occur in a periodic fashion.

If an event (e.g., five cars enter a street) occurs at time t, then the same event also

occurs at time t + kT, for all integer k > 1.

Because the planning horizon is circular, events occurring tomorrow are

assumed to occur in the same interval that represents tod-iv. For example, consider

a car that enters a street at tl = 23:00h (or 11 PM) todci- and traverses the street

until it leaves at t2 = 01:00h (or 1 AM) tomorrow. (See Figure 5-2.) In a circular

planning horizon, these two events, a car entering and leaving a street, occur at










t O/T


vs
0 ti T t2




Figure 5-2. Events occurring in two consecutive planning horizons.


time 23:00h and 01:00h in the same interval [0, 24). In general, if a car enters a

street at time t1 < T and takes r < T units of time to traverse, then the two events

are assumed to occur at t1 and mod{ti + 7, T} on the interval [0, T).

5.3 Discrete-Time Dynamic Traffic Assignment Problem with Periodic
Time Horizon

Although, it is possible to formulate the dynamic traffic assignment problem

with a periodic time horizon as an optimal control problem, solving it is typically

troublesome (see, e.g., Peeta and Ziliaskopoulos [85]). This section presents a

discrete-time version of the problem in which the interval [0, T) is represented as

a set of discrete points, i.e., A {0, 6, 26,. T 6}, where 6 = and N is

a positive integer. (In general, the subdivision of the planning horizon need not

be uniform. For example, the subdivision during the period between 22:00h to

06:00h may be coarser than the one for the period between 06:00h to 22:00h.) In

order to avoid using fractional numbers in the set of indices and to simplify our

presentation, we typically assume that 6 = 1.

To formulate the problem, let G(N, A) represent the underlying transportation

network where N and A denote the set of nodes and arcs in the network,

respectively. It is convenient to refer to elements of A either as a single index a

or a pair of indices (i,j). The latter is used when it is necessary to reference the

two ends of an arc explicitly. Furthermore, C is a set of origin-destination (OD)


















Figure 5-3. Three-node network.


pairs and the travel demand for OD pair k during the time interval [t, t + 6], t E A,

is ht.

There is also a travel time function associated with each arc in the network. In

the literature (see, e.g., Wu et al. [101], Ran and Boyce [89] and Carey et al. [19]),

these functions can depend on a number of factors such as in-flow and out-flow

rates and traffic densities. We assume in this formulation that 0a, the travel time

associated with arc a, depends only on the number of cars on the arc. Furthermore,

Oa is continuous, non-decreasing and bounded by T, i.e., 0 < a((w) < T,

Vw E [0, .1,], where .i, is a sufficiently large upper bound for the range of Oa(w)

and there is no feasible solution whose flow on arc a can exceed .i,. In particular,

a(0) represents the free-flow travel time on arc a.
We use the dynamic or time-expanded (TE) network (see, e.g., Section 19.6

in A!,li et al. [2]) to determine the state of vehicular traffic in the system at

each time t c A. To illustrate the concept of time expansion, consider the static

network with three nodes shown in Figure 5-3 or the three-node network. In

this network, all arcs have the same upper bound value, ., = M, and there

is only one OD pair, (1,3). Let the planning horizon be the interval [0, 5) and

6 = 1. Thus, A {0, 1, 2, 3, 4}. The travel time function for every arc is Q and

1.5 < O(w) < 4,Vw c [0, M]. To construct the TE network, the travel time also

needs to be discretized. In general, the set of possible discrete travel times for arc a











1 (2
00





1 2




Figure 5-4. Time expansion of arc (1, 2) at t = 1.

is F, { { s : 0 = ], 0 < w < 1, }. For our example, the set of possible discrete

travel times for each arc is F, {2, 3, 4},Va.

To incorporate the time component in the TE network, every node in the

static network (or static node) is 'expanded' or replicated once for each t E A.
For the three-node network, static node 1 is transformed into five TE nodes, one

for each t E A, in the TE network. For example, node 1 is expanded into nodes

lo, 11, 12, 13, and 14 in the TE network. (See Figure 5-4.) Similarly, each arc (i,j)
in the static network (or static arc) is replicated once for each pair of (t, s), where

t E A and s E F(cj). Consider arc (1, 2) in the three-node network. Cars that
enter this arc at time 1 can take 2, 3, or 4 units of time to traverse depending

(as assumed earlier) on the number of cars on the arc at t = 1. To allow all
possibilities, arc (1,2) is expanded into three TE arcs (11, 23), (1, 24), and (11,20).
The latter represents a car that enters arc (1,2) at time 1, takes 4 units of time to

traverse, and leaves the arc at time 5 or time 0 (or mod(1 + 4, 5)) of the following

day. Similar expansion applies to each t E A. In general, each static arc (i,j)

expands into |A| x IP(yj) T TE arcs of the form (it, jmod (t+s,T)),V t E A, s cE (j).









Figure 5-5 di-pl--'i the complete time expansion of the three-node network. In

addition to the time-expanded nodes and arcs, the figure also di pl!-- the travel

demand at the origin TE nodes (i.e., node 1t, Vt c A) and decision variables g(k)t

representing number of cars arriving at the destination node d(k) of OD pair k at

time t, i.e., at node 3t,Vt E A.

To reference flows on TE arcs, let y>(t,s) denote the amount of flow for

commodity k that enters static arc a at time t E A, takes s E Fa units of time to

traverse it, and then exits the arc at time mod{t + s, T}. In particular, if a = (i,j),

then the subscript a(t, s) refers to TE arcs of the form (it,j mod (t+s,T)),V t E A, s E

F(i,). In addition, Ya(t,s) EkEC (t,s) represents the total flow on arc a(t, s).
To compute the time to traverse a static arc at time t, let



a(t,) = {(, S) : 7 = [t 1]r, [t 2]r, [t S]r, S c Fa .

where
[ q if q > 0

T+q ifq < 0

In words, Qa(t) contains pairs of entrance, T, and travel times, s, for static arc a

such that, if a car enters static arc a at time T and takes s time units to traverse it,

the car will still be on the arc at time t. For example, if t = ll:00h and the time to

traverse arc a is five hours for the previous five consecutive time periods, then cars

entering arc a at time T 10:00h, 9:00h, 8:00h, 7:00h, and 6:00h will be on the arc

at ll:00h. (We assume here that cars entering arc a at, e.g., 6:00h are still on the

arc at ll:00h even though it is scheduled or expected to leave at ll:00h.) When t

is relatively near the beginning of the planning horizon, the notation [-]T accounts

for cars on the arc at time t that enter it from the previous d4 i. Continuing with

the foregoing example, let t = 3:00h instead. Then, cars entering arc a at time -





















































Figure 5-5. Time-expansion of the three-node network.









2:00h, 1:00h, 0:00h, 23:00h, and 22:00h are still on the arc at 3:00h. Using the set

Qo(t), the total amount of flow on static arc a at time t or Xa(t) is (T,s)Ea(t) Ya(r,s).
There are two additional sets of decision variables. One set consists of za(t,s),

a binary variable that equals one if it takes between (s 6) and s units of time

to traverse arc a at time t. In the formulation below, the value of Za(t,s) depends

on Xa(t) and, for each t, Za(t,s) = 1 for only one s E Fa. The other set consists of

g9, a vector with a component for each node in the TE network. Component it

of gk is set to zero if i is not the destination node of OD pair k. Otherwise, g(k)t

where d(k) denotes the destination node of OD pair k, is a decision variable that

represents the amount of flow for commodity k that reaches its destination, d(k), at

time t.

Below is a mathematical formulation of the discrete-time dynamic traffic

assignment problem with periodic planning horizon (DTDTA).


mm in a(a(t)) Yat
(x,y,z,g) t aA I sEFa

s.t. Byk +gk bk Vk C

gd(k)t ht Vk G C
tEA tEA

a(t,s) E (t Vt e A, a c A and se c
kEC


sEVa


Xa(t) Y Ya(r,s) Vt
(Ts)EQa(t)

YE Za(t,s) 1 Vt E z
sEra

)oa(t,s) < oa(Xa(t)) < Y. SZa(t,s)
SEra


Ya(t,s) < 1 ,,.,)


E A and a c A


N and a c A


Vt c A and a E A


Vt c A,a E A and a c Fa


(5-3)


(5-4)


(5-5)

(5-6)

(5-7)


a(t,s), 9g(k)t X(t) > 0, za(t,s) E {0, 1}


Vt e A,a E A, s Fa and k C (5-8)









In the objective function, serP Ya(t,s) represents the number of cars that enter arc

a at time t and, based on our assumption, these cars experience the same travel

time, Qa(xa(t)). Thus, the goal of this problem is to minimize the total travel time

or delay. Using constraint (7-10), the objective function can be equivalently written

as

min MYY)
( tEA aEA (T,s)EQ(t) SEFa
or, more compactly, as

min () fY,

where Y and K(Y) are vectors of arc flows (Ya(t,s)) and travel times

(a((,Ts)Ean( Ya(r,s))) whose components are defined so that their inner product is
consistent with the summations.

Constraint (7-8) ensures that flows are balanced at each node in the TE

network. In this constraint, B denotes the node-arc incidence matrix of the TE

network and bk is a constant vector with a component for each TE node and

defined as follows:

k 0 if i o(k)
hk if i = o(k)

where o(k) denotes the origin node of OD pair k. Constraint (7-9) guarantees

that the number of cars arriving at the destination node d(k) equals the total

travel demand of OD pair k during the planning horizon. Then, constraint (7-10)

computes the total flow on each TE arc and (5-4) determines the number of cars

that are still on static arc a at time t.

In combination, the next three constraints, i.e., constraints (5-5) (5-7),

compute the travel time for the cars that enter arc a at time t and only allow flows

to traverse the corresponding arc in the TE network. In particular, constraint

(5-5), in conjunction with (5-6), chooses one (discretized) travel time s E Fa









that best approximates ,a(Xa(t)), i.e., a(x,(t)) E (s 6, s]. When a represents

arc (i,j), constraint (5-7) only allows arc (it, imod(t+s,T)) to have a positive flow.

Otherwise, (7) forces flows on arc (it, imod(t+7,T)), for T E Fa and T / s, to be zero.

Finally, constraint (7-11) makes sure that appropriate decision variables are either

nonnegative or binary.

As formulated above, the travel time associated with Za(t,s) in equation (5-6)

can only take on discrete values from the set Fa while the travel time in the

objective function varies continuously. Although it may be more consistent to use

discrete values of travel times in the objective function, the above model would

provide a better solution because the true travel time is used to calculate the

total delay. The model also has interesting properties discussed in Section 5.4.

In addition, the treatments of travel times in both the objective function and

constraints can be made consistent by solving the (approximation) refinement

problem also discussed in the same section.

Under a relatively mild sufficient condition, we show below that DTDTA has

a solution by constructing a feasible solution. In fact, the solution we construct

below is generally far from being optimal. However, it suffices for the purpose of

proving existence. Let Ra(t) be a set of discrete times at which a car enters arc a

and still remains on the arc at time t. Below, we refer to Ra(t) as the enter-remain

set. Given xa(t), Ra(t) C A is a union of two sets, i.e.,


Ra(t) ={w A: w < (t 1),w+ L(xi(w)) > t}U


{w : w > (t + 1),w+ ((xw) >t}.

In addition, let Ua(t) denote the total flow into arc a at time t. When Ua(t) is

given for each t E A, the lemma below shows that a set of Xa(t), Ya(t,s), and Za(t,s)

consistent with constraints (5-4)-(5-7) and relevant conditions in (7-11) exists

when if is sufficiently large.









Lemma 5.3.1. Assume that Ua(t) is known for a given a E A and all t E A. If 31,

is suff:.' i.l i o then there exists a set of Xa(t), Ya(t,s) and Za(t,s) that -.,l:fi.

constraints (5-4) to (5-7) and the relevant conditions in (7 11).

Proof: (Because a is given, we discard the subscript a in places where there

is no confusion in order to simplify our notation.) Below, we construct sequences

{sm}, {zm(,S)}, {(Y~t,)}, {x(t)} and {R~} whose limits yield the set of decision
variables feasible to constraints stated above.

For m = 1, let

s = [(a(0)/6l, i.e., s' is the discretized free flow travel time for arc a,
z(t,) 1 and (ts) 0, Vs e A, s / s1
1 andY 0t
Y1(t,s) a() at)and Y(t,) 0, Vs A, s / s}
a t )at ,
Ri e :w < (t -),+s >t}U{ A : > (t+1),w+s T T>t},

and
Si I
(t)- ZWERa "oiy)
As defined above, R1 is the enter-remain set based on the travel time s1, a vector of

1,Vw E A. For m > 2, let


zat,s-) tand -- a (s A, t
s( /

n ,) 1 and z ,) 0, Vs e A, s / sm

(,)) d() a (nd Y = Vs A,
Rm {w A : < (t ),w+s > t}U{ A :w > (t+1),w+s -T>t},

and

(t) ZwERYa a(,s)

Sequences {st}, {x t)} and {R-} constructed above are monotonically

non-decreasing. Consider the sequence {s-}. Observe that s2 > sl,V t c A because

Xi(t) > 0, V t A, and as assumed earlier O a() is non-decreasing. It follows that,

for any t A, + s2 > > t and + s- T > + s1- T>t. Thus, uw R
for~ WntA W3s ss- s _









implies that w E R2, i.e., R1 C R2 for all t E A. The latter, and the fact that Ua(t)

is nonnegative, imply that xa(t) -= Yw ERY (,) > ERt a(w,s) X t) t E A.

Assume that the claim is true up to some fixed m. For all t E A, s'l

[ra(x t))/1 > [a(X1t) )/] s, because ) > x-1 and a(.) is
non-decreasing. Using an argument similar to above, R C C R+1 and x"+1 > X "
a(t) a (t)
Thus, the three sequences are monotonically non-decreasing. In addition, all three

sequences are bounded, i.e., s' < T, R' C A, and xZ < tea U) and, therefore,

convergent. Let s', R', and xt) be their limits. Based on our construction,
s = [Oa(x(t))/6], ats) = 1 and z"t,s) 0, Vs E A, s / s'. In combination,

these ensure that constraints (5-5) and (5-6) are satisfied. Our construction also

implies that Y'sO) U a() and Yo -(t, 0, Vs e A, s / s". Because if, is

sufficiently large, Y~t,,) satisfies constraint (5-7).

In the limit, R' = {w E A : u < (t 1), u + s > t} U { EA :

w > (t + 1),w + s- T > t}. Thus, R' is consistent with s' and x't)
yoo because Y'
EER aR w,s) (T xt) satisfies (5-4). Furthermore, xz and Y ) are both nonnegative and z"t,)is
(t a(t) (t,)nonnegative a is

binary. Thus, the proof is complete. U

In the above proof, if, for some m, s' is larger than the maximum travel time

for arc a, i.e., max{s :s E Fa} (or, equivalently, xZ > .1,), then Ua(t) is infeasible

or not compatible with the upper bound i. ,.

To establish the existence of a feasible solution to DTDTA, recall that

G(N, A) denotes the (static) transportation network. For the theorem below,

assume without loss of generality that each node in N can be either an origin or

destination, but not both. If node i is both an origin and a destination, then we

create a dummy node i' and use node i as the origin node and i' as a destination.

For example, consider OD pairs (i,j) and (j, i). In this case, i and j are both

origins and destinations. When the dummy nodes are added, the two OD pairs









become (i,j') and (j, i'). Let pk denote a path in G(N, A) connecting the OD pair

k, i.e., pk E pk. The set of these paths, {pk : k E C}, induces a subgraph

G(N, A), where N C N and A C A denote the sets of nodes and arcs, respectively,

belonging to the paths in T. For each i E N, define [i+] = {(i,j) : (i,j) A} and

[i-] = {(j,i) : (j, i) c A}. In words, [i+] and [i-] are the sets of arcs in G(N,A)
that emanate from and terminate at node i, respectively. Also, let order(i) denote

a topological order of node i (see Al!ni et al. [2]). If (i,j) A and G(N,A) can be

topologically ordered, then order(i) < order(j).

Theorem 5.3.1. Assume that if, is suff. .ii:l, '1,n'/ for all a E A and a node

can be either an origin or a destination, but not both. Then, DTDTA has a feasible

solution, if there exists a path pk for each k E C such that the -.l'.q'rl, they induce
is i. ,. 1.:.

Proof: Let T be a set of paths, one per OD pair, such that the subgraph,

G(N, A), it induces has no cycle. Thus, N can be ordered topologically. (See

Al!li et al. [2].) Below, we construct a feasible solution one arc at a time and in

a topological order using Lemma 5.3.1 and only the paths in T. The latter implies
k
that Ya(t,s) = (t,s) =x(t) 0 for all a A.

Let node i E N be of topologicall) order 1 and, for each arc a in [i+], define

Q(a) to be the set of paths in T that contain or use arc a, i.e., Q(a) = {k :
a E pk, k E C}. (It is not necessary to index Q(a) with i because each arc a
can belong to only one [i+].) For each k E Q(a), arc a must be the first arc in

path pk because node i is of order 1. Let Ua(t) = ckQ(a) hk. Because .1, is

sufficiently large, Lemma 5.3.1 ensures that there exist Xa(t), Ya(t,s), and Za(t,s)

feasible to (5-4) (5-7) and the relevant conditions in (7 11). Let y(t, ()) = h
and /(ts) = 0,V s c A, s / s o(t). So constructed, these y(t,s)'s are consistent with

Ya(t,s) and satisfy the flow balance equation (7-8) for node i.









To construct the variables Xa(t), Ya(ts), Za(t,s) and y(ts) for arcs emanating

from nodes of higher order, assume that the decision variables for arcs emanating

from nodes with order m or less have been constructed and let node i be of order

(m + 1).

Case 1: The set [i+] is empty. Then, i must be a destination node for some

commodity k, i.e., i = d(k). For a c [i-], k c Q(a) and t c A, set


(ta )a(t)
{*: t+-", -t} {f: t+-", -T-t}

For each k E Q(a), every demand hk uses arc a. Thus, rk, as constructed must

satisfy the appropriate constraints in (7-8) and (7-9).

Case 2: The set [i+] is not empty. Let a E [i-], also a nonempty set. Assume

that a (q, i). Then, order(q) < order(i) and, by the above assumption, xa(t),

Ya(t,s), Za(t,s), and k(t,s) are available.

Consider an arc a c [i+]. For each a c [i-], define Q(a, a) = {k : a E pk a

pk, k e C} and, for each k E Q(a, a), let U(,k) denote the flow into arc a at time t
for OD pair k. Then,

> tj")= Ec)+ nE > : sc
e(t) )(t)
ti t\' ," t}{ t+ )-T=t}

and the total flow into arc a at time t is Ua(t) = keQ(,a) U(t)I Because I, is

sufficiently large, Lemma 5.3.1 ensures that Xa(t), Ya(t,s), ya(t,s) and za(t,s) feasible to

relevant constraints exist.

Thus, when carried out in the topological order for every arc in A, the above

process must produce a feasible solution to DTDTA. U

The theorem above assumes that each .[ is sufficiently large so that it is

feasible to send the entire flow for each OD pair along a single path. Although this

assumption appears to be stringent, it can be made less so by allowing the flow for









each OD pair to traverse over several paths as long as they do not induce cycles in

G(N, A). With more cumbersome notation, the above argument can be extended to

the case with multiple paths per OD pair as well.

When applied to the above example in which the OD pairs (i,j) and (j, i)

become (i, j') and (j, i'), the ... i- lic subgraph assumption implies that the paths

from i to j' and from j to i' cannot form a cycle. Intuitively, this means that

there must exist two routes between the original nodes i (e.g., home) and j (e.g.,

work) with no road in common. These routes need not be optimal and there is no

requirement in our formulation or algorithms to use them. They are used only to

established the existence in Theorem 5.3.1.

The First-In-First-Out (FIFO) condition requires that cars entering an arc at

time t must leave the arc before those entering after time t. In the literature, many

(see, e.g., Ran et al. [91], Zhu and Marcotte [103], and Parakis and Roels [86])

assume that the travel cost function satisfied certain conditions to ensure FIFO.

To avoid making additional assumptions, we ensure FIFO by adding the following

constraints to DTDTA instead. Doing so may make the problem harder to solve

because of the additional constraints.

t + E sza(t,s) < t + E sza(t,s), Va Al and t,t E A : (t + 6) < t
SECa SECa
t+ SZa(t,s) < (t + T) + Y SZa(t,s), Va c A1 and t,t E A: (t + J) < t
sEPa SECa

When t and t represent two instances of time on the same d4i-, the first inequality

ensures that cars entering arc a at time t leave the arc before those that enter at

time t > t. On the other hand, t and t may refer to times on consecutive d-iv- e.g.,

t = 08:00h todci- and t = 09:00h yesterday. Because of our periodic assumption,

these two times are on the same interval [00:00h, 24:00h) and t (incorrectly)

appears to be an earlier time than t. To distinguish times on consecutive di-4

and preserve FIFO, the second equation represents tod i-'s time t (e.g., 08:00h of









tod(vI) as (t + T) (e.g., as 08:00h of yesterday plus T) and forces cars entering the

arc at this time to depart after those that enter at yesterday's time t (e.g., 09:00h

yesterday).

5.4 Bounds for the DTDTA Problem

As formulated in the previous section, DTDTA is a nonlinear optimization

problem with binary decision variables, a difficult class of problems to solve. This

section describes mixed integer programs for obtaining an approximate solution to

DTDTA as well as bounds for the optimal delay.

Except for constraint (5-6), the constraints for DTDTA are linear. To develop

a linear version of (5-6), assume that the travel time function, Q0, is invertible for

all a E A. For example, if Oa is a continuous and increasing function, then Oa1

exists on the interval [,(0), ,(a ,)]. (See Figure 5-6.) Under this assumption,

Qa(Xa(t)) C (s 6, s] if and only if Xa(t) C (a 1(s 6), oa1()]. Thus, the requirement
(s 6)Za(t,s) < Oa(Xa(t)) < sza(t,s) is equivalent to 01(s 6)Za(t,s) < Xa(t) <

al(s)z(t,,). Recall that Fa {s :s ['='], 0 < w < 11,}. Let si].
Then, (si 6) g [a (0), O,(a(,)] and al'(sl ) is not well defined. (In Figure 5-6,

a1(si 6) = Oa(1) is not well defined.) In this thesis, we set a1'(si 6) = 0.
Using this convention, constraint (5-6) can be replaced by the following linear

equivalence:


Sa-1s -I )Za(t,s) < Xa(t) Q-l(s)za(t,s) Vt c A and a c A (5 9)
sEPa SECa


The following lemma implies that there exist linear functions that approximate

the objective function of DTDTA.

Lemma 5.4.1. There exist vectors ql and q, such that qY < (Y)TY < qfY for

all Y feasible to DTDTA.














Oa a(Xt0
2 -----------



S1 --


1 (2) a1 (3) a( (4)

Xa(t)

Figure 5-6. Oa(xa(t)) E (s s] versus Xa(t) E ((-1(S 6), a-1()].


Proof: As defined earlier, )(Y)TY = Y ~(Xa(t)) Y Ya(t,s) From constraint
tE aEA Lsea
(5-6), the following hold for any feasible solution to DTDTA:

y ( 6)za(t)] [ Y(ts) )j ( te aEA sEFera sera

and

N(Y)TY< S Sa(t) E(a(ts)
ted EA LseF, seFa

The summand of the last set of summations (i.e., K sZa(t,s) s8 Ya(t,s) ) can
LsrFa I LsEFa J
be simplified. Constraint (5-7) implies that Ya(t,s) > 0 only if Za(t,s) = 1. In

addition, constraint (5-5) ensures that, for each pair (a, t), Za(t,-) = 1 for some

s E Fa and za(t,s) = 0,V s E Fa,s / s. This implies that Ya(t,s) > 0 and

Ya(t,s) 0, Vs F,, / s and


Y SZa(ts) Ya(t, s) SYa(t,s) sYa(t,s)-
-sera sera I sera









A similar result holds for the first set of summations. Thus, the above inequalities
reduce to the following


E SE (s -J)Y(6,s)< tEz aEA sEFa tEz aEA sEFa
Let ql and q, be two constant vectors with a component for each arc in the
TE network such that [q]a(t,s) = (s 6) and [q,]a(t,s) = s, respectively, for all

a A, tE A, and sE Fa. Then,

q1T 555 (s 6)ya(ts) < qjy)Ty < 55 SYa(t,) qY. U
tEz aEA sEFa tEz aEA sEFa

Let S(6) denote the feasible region defined by linear constraints (7-8) (5-5),

(5-9), (5-7), and (7-11) and, for convenient, (Y, Z) represents an element in S(6).
In addition, let (Y', ZI), (Y*, Z*), and (Y", Z") be solutions to the lower-bound
problem (or min{qTy : (Y, Z) E S()}), the original problem (or min{j(Yy)T

(Y, Z) c S(6)}), and the upper-bound problem (or min{qY (Y, Z) e S(6)}),
respectively. Then, the following lemma holds.
Lemma 5.4.2. For i,., 6 > 0, qTY1 < (Y*)T* < qTU < qTy.

Proof: In following sequence of inequalities, the first one holds because Y* is
feasible to the lower-bound problem and the second follows from Lemma 5.4.1:

qYlV < qlY* < F(Y*)TY*.

Similarly, the following sequence also holds

q(Y*)TY* < (Yu)Ty" < qY".

Combining the above two sequences yield the first two inequalities in the lemma.

Finally, the last inequality holds because yl is not necessarily optimal to
min{qTY: (Y, Z) e S()}. U









In view of the above lemma, the solutions to the upper and lower-bound

problems are approximations of the solution to the original problem. The theorem

below states that the approximation can be made arbitrarily close to the original

problem by choosing a sufficiently small 6.

Theorem 5.4.1. Given c > 0, there exists 6 > 0 such that qTV" {Y*)Y* < e

and (Y*)Ty* (Y-
Proof: By construction, qu = qi + 6e, where e is (1, 1, 1)T. Let Hk denote the

travel demand for OD pair k during the entire planning horizon, i.e., Hk = teA ht

and set H = Ekec Hk. For each t c A, sa(t) is such that sa(t) e Fa and

z(ts ) 1. In words, Sa(t) is the approximate travel time for (static) arc a at

time t in the optimal solution (Y', ZI).

Then, the following sequence must hold:

0 < (q- qi)Tyl

6eTYl

E E E y y s)
aEA kEC tEA sEra

aEA kEC tEA
< 6 EE Hk
aEA kEC
-6H E 1
aEA
S6H IA

The first inequality follows from Lemma 5.4.2. Then, the above relationship

between q1 and q, and letting EckEC Y(t,s) denote individuals components of YI

yield the first two qualities. The third equality follows from the definition of

Sa(t). Following this, the second inequality holds because the total amount of

flow on (static) arc a for OD pair k during the entire planning horizon cannot

exceed Hk. The sum of the latter is H, a constant that can be factored out of

the summation over A. This validates the penultimate equality. Finally, the last









equality follows from the fact that ZaEA 1 simply denotes the number of elements

in the set A. Choosing 6 = HA guarantees that qY qfY' < e. When combined

with the results in Lemma 2, the latter implies that qTY" N(Y*)TY* < c and

_(y*)Ty* q Ty
The approximate solution Yu can be improved by solving an additional

nonlinear program. In particular, consider the approximation refinement problem

min{t(Y)TY : (Y,Z") E S(6)}, i.e., this is the original problem with Z = Z".

Doing so makes it possible to remove TE arcs corresponding to z (t,) 0 from the

TE network and discard decision variables Xa(t) and constraints (5-5) and (5-7)

from the problem. In DTDTA, we use Xa(t), the number of cars on arc a at time t,

to compute the travel time on arc a and, subsequently, to select which TE arc to

use or which Za(t,s) to set to one. Thus, when Z is given, Xa(t) becomes unnecessary.

Additionally, let s(t) be such that z"ts()) = 1 for each t E A. Then, constraint

(5-9), originally (5-6), reduces to requiring E(rs)EQ Ya(T,s) to be in the interval

(s(t) 6, s(t)]. In other words, the original problem with Z = Zu is a nonlinear
multi-commodity flow problem with the latter as side constraints.

Let Y" be an optimal solution to min{)(Y)TY : (Y, Z") e S(6)}. Then, the

following corollary shows that Y" better approximates Y*.

Corollary 5.4.1. (Y*)TY* < +(Y")TY" < ((yu)TY" < qTY"

Proof: In the above sequence of inequalities, the first one follows because Y* is

optimal to the original problem and Yu is only feasible. The second holds because

Y" is feasible to min{)(Y)TY : (Y, Z") e S(6)}. Finally, the last is due to Lemma

5.4.1. U

5.5 Numerical Experiments

We conducted numerical experiments using small test networks to empirically

verify our understanding of DTDTA as well as to evaluate the efficiency and

effectiveness of the approximation schemes discussed in previous sections.









Table 5-1. Demand patterns

Time
Traffic Intensity 0 1 2 3 4 5 6 7 8 9 Total
Low 20 25 30 35 40 40 35 30 25 20 300
Medium 30 35 40 45 50 50 45 40 35 30 400
High 40 45 50 55 60 60 55 50 45 40 500


In all problems, the planning horizon is [0, 10) and the travel cost functions

are either linear, i.e., O(w) = 1.5 + 2.5(1-'), or quadratic, i.e. O(w) = 1.5 + 2.5(1o )2,

where w is the number of cars on the arc. We consider the three different demand

patterns di-p'1l ,i'1 in Table 5-1. In all three patterns, travel demands at discrete

points increases gradually until time 4, levels off briefly, and then decreases

gradually after time 5. The individual demands in the three patterns are different

and represent three traffic intensities: low, medium, and high. We used GAMS [39]

to implement and solve all problems using NEOS Server of Optimization [82]. In

particular, we used SBB [94] to solve our nonlinear integer programming problem,

i.e., DTDTA, XPress-XP [102] to solve our linear integer programs, i.e., the lower

and upper-bound problems, and CONOPT [26] to solve our linearly constrained

optimization problems, i.e., the approximation refinement problems. All CPU times

reported herein are from the NEOS server.

To empirically verify that DTDTA problem is not convex, we first consider

the two-arc network in Figure 5-7 that has one OD pair. We let 6 = 1. Thus, the

discrete-time planning horizon is A = {0, 1, ,9}. We use the above quadratic


Figure 5-7. Two-arc network.









Table 5-2. Optimal solutions to the two-arc problem.

Solution 1 Solution 2
Inflow Travel time Inflow Travel time
Time al a2 al a2 al a2 al a2
0 0 20 1.600 1.500 20 0 1.500 1.600
1 25 0 1.500 1.600 0 25 1.600 1.500
2 0 30 1.656 1.500 30 0 1.500 1.656
3 35 0 1.500 1.725 0 35 1.725 1.500
4 0 40 1.806 1.500 40 0 1.500 1.806
5 40 0 1.500 1.900 0 40 1.900 1.500
6 0 35 1.900 1.500 35 0 1.500 1.900
7 30 0 1.500 1.806 0 30 1.806 1.500
8 0 25 1.725 1.500 25 0 1.500 1.725
9 20 0 1.500 1.656 0 20 1.656 1.500


travel time function for both arcs and the function yields travel times in the

interval [1.5, 4.0]. Because 6 = 1, the set of discrete travel times is F {2, 3, 4}.

Using the low traffic intensity demand pattern in Table 5-1, we solved DTDTA

using SBB and terminated it when the relative optimality gap is less than 0.005 (or

0.5'.-). There are two optimal solutions (see Table 5-2) to the two-arc problem with

an optimal total d. 1 iv of 450.

Consider the first solution, labelled 'Solution 1', in the Table 5-2. At time 0,

there are 20 cars to travel from node 1 to node 2. At this time, there are also 20

cars already on arc al. These cars enter the arc at time 9 and have not reached

their destination at time 0. Because DTDTA assumes that the time to traverse

arc al depends on the number of cars on the arc at the entrance time, the travel

time for arc al at time 0 is 1.5 + 2.5( 0)2 1.6. On the other hand, there is

no car on a2 at time 0. Cars that enter the arc at time 8 already left the arc by

time 0. Thus, the travel time for a2 at time 0 is 1.5, the free-flow travel time. To

minimize the travel time, all 20 cars entering the network at time 0 must travel on

a2. In fact, every car in Solution 1 travels at the free-flow travel time of 1.5. Thus,

there cannot be any solution with less total d, 1 iv and Solution 1 must be optimal.




















Figure 5-8. Four-node network.


Because of the symmetry in the data, switching the flows between the two arcs in

the network yields Solution 2, another optimal solution. Furthermore, it is easy to

verify that every convex combination of these two solutions is feasible to DTDTA

and yields, on the other hand, a larger total delay, thereby confirming empirically

that the objective function is not convex.

Additionally, the "extreme" travel behavior di -p i 1 in Table 5-2 may not

be intuitive. This is due to the assumption that the system operator is extremely

sensitive to the difference in travel times and is willing to switch routes in order to

save a minute amount of travel time.

When the network is large, it would be too time-consuming to solve DTDTA

optimally or otherwise. In our experiments, we consider four approximate solutions

to DTDTA: (Yl, Z), (Y, ZU), (Yl, Z), and (YU, Z), where the last two are

refinements of the first two. To evaluate the quality and the computation times

of these solutions, we consider the four-node network in Figure 5-8 with two OD

pairs, (1, 4) and (2, 4). In our experiments, both OD pairs have the same demand

pattern and all arcs have the same travel cost function, linear or quadratic, as

specified above.

First, we solved the lower and upper-bound problems with using two levels of

discretization, 6 = 1 and 6 = 0.5. As before, when 6 = 1, the discrete-time planning

horizon is A = {0, 1, ,9}. On the other hand, when 6 = 0.5, A becomes









Table 5-3. Solutions from the lower and upper-bound problems: linear travel cost
function.

Traffic = 1 6 = 0.5
Intensity ql Y q, Y" Gap ql Y q Y'" Gap
low 820.0 1580.0 760.0 1187.5 1560.0 372.5
medium 1200.0 2230.0 1030.0 1705.0 2230.0 525.0
high 1500.0 2875.0 1375.0 2187.5 2870.0 682.5


{0, 0.5, 1, 1.5, ... ,9, 9.5}. For the comparison below (see Tables 5-3 and 5-4), we

assume that, when 6 = 0.5, there is no demand at fractional times ( e.g., at 0.5,

1.5, 2.5, etc.) and the demands at integral times (i.e., 1, 2, 3, etc.) are as shown in

Table 5-1.

For both types of travel cost functions, the size of the optimality gap (i.e.,

quY" q1Yy) decreases by approximately 5('. as 6 decreases from 1 to 0.5.

However, the results in Tables 5-3 and 5-4 sl--'-, -1 that the reduction in the gap

is due mainly to the improvement in the solution, yl, of the lower-bound problem.

The approximate travel d-1i, as estimated by Yu change relatively little for the

two values of 6.

Tables 5-5 and 5-6 compare the solutions from DTDTA, (Y*, Z*), against

two approximations, (Y", Z") and (Yi, ZI). As in the two-node problem, we solve

DTDTA using SBB to obtain a (integer) solution (Y*, Z*) with less than 0.5'.

relative optimality gap. To obtain (Y", Z"), we first solve the upper-bound problem

using XPress-MP to obtain (Y", Z"), a (integer) solution with less than 0.5'.

optimality gap, and, then, solve the approximation refinement problem (with

Table 5-4. Solutions from the lower and upper-bound problems: quadratic travel
cost function.

Traffic = 1 6 = 0.5
Intensity q1 Y qu Y" Gap qf Y q Y" Gap
Low 600.0 1200.0 600.0 900.0 1200.0 300.0
Medium 822.2 1644.5 822.2 1233.3 1644.5 411.1
High 1124.5 2248.9 1124.5 1686.7 2248.9 562.2









Table 5-5. Quality of refined upper and lower-bound solutions: linear travel cost
function.

(Y*, Z*) (y", ZU) (Yl Zl) Rel. cpu
Traffic cpu* (1i cpu' Err Ratio
Intensity Delay (sec) Delay (sec) Delay (sec) ( .) pu cpu*
Low 1337.50 27.42 1385.00 2.57 1392.50 2.38 3.55 10.7 11.5
Medium 1800.00 15.92 1-.., ; 2.66 1815.30 2.90 0.85 6.0 5.5
High 2290.00 95.02 2327.50 4.07 2315.00 1.25 1.09 23.3 76.0


Z = Z") using CONOPT to obtain (Y", Z"). The solution (Y', Z') are obtained in

the same manner. In the two tables, the CPU times for the two approximations are

times for solving both bounding and refinement problems.

For both linear and quadratic travel time functions, the two approximation

schemes provide solutions with relatively small errors using much less CPU time

required to solve DTDTA (see the ratios of the cpu times in Tables 5-5 and 5-6).

For quadratic travel time functions, the approximate solutions are identical to

DTDTA solutions, except for the high traffic intensity case when the approximate

solutions are slightly better (by 0.0 .'.).

Table 5-6. Quality of refined upper and lower-bound solutions: quadratic travel
cost function.

(Y*, Z*) (y", Z) (Y' Z') Rel. cpu
Traffic cpu* < cpu Err Ratio
Intensity Delay (sec) Delay (sec) Delay (sec) ( ) cpu
Low 1054.50 0.88 1054.50 0.09 1054.50 0.08 0.00 9.8 11.0
Medium 1. I ; SO 6.62 1. ; SO 0.14 1'. ; SO 0.34 0.00 47.3 19.5
High 2129.80 501.17 2128.60 0.10 2128.60 0.13 -0.06 5011.7 .-. 2


5.6 Concluding Remarks

This chapter formulates a discrete-time dynamic traffic assignment problem

(DTDTA) in which the planning horizon is treated in a circular fashion and events

occur periodically. Doing so allows positive flows on the network both at the

beginning and at the end of the planning horizon. The structure underlying the

formulation is the time-expansion of the (static) network representation of streets






77


and highi--,-. The resulting problem is a nonlinear program with binary variables,

a difficult class of problems to solve. Alternatively, two linear integer programs are

constructed to obtain approximate solutions and bounds on the total travel delay.

It is shown that solutions from the latter can be made arbitrarily close to solutions

of DTDTA. Furthermore, numerical results from small test problems si,--l. -1 that

solving the linear integer program is more efficient.














CHAPTER 6
A NONLINEAR APPROXIMATION BASED HEURISTIC ALGORITHM FOR
THE UPPER-BOUND PROBLEM

6.1 Introduction to the Chapter

In C'! ipter 5 we have discussed a periodic discrete time dynamic traffic

assignment problem, which is constructed based on two assumptions: (i) all

cars that enter the arc during the same time interval experience the same

travel time and leave the arc during the same time interval, and (ii) the travel

time is a function of the number of cars on the road (see also Nahapetyan and

L i.|- !..ii I-)anich [78]). As we have seen, the initial mathematical formulation

of such model leads to a mixed integer problem with linear constraints and a

nonlinear objective function. By linearizing the objective, one can construct an

upper and a lower bound problems. Although the solution of such problems can

be made arbitrarily close to the solution of the initial problem by decreasing the

discretization parameter 6, observe that the bounding problems belong to the class

of linear mixed integer program, which are computationally expensive to solve. In

the case of the bounding problems, the task becomes more challenging because of

the special structure of the feasible region.

Observe that the model is constructed based on a time-expanded network

and there are binary variables, a(ts), associated with the arcs of the network.

By decreasing the parameter 6, the number of the binary variables increases. For

example, given a traffic network G(N, A) and a set of possible discrete travel

times FP(6), the total number of the binary variables in the DTDTA-U problem

is IA(65) EaEA Ia()l If 6 reduces to 6/2, then IF,(6/2)| = 21|,(6)1|,A(6/2)| =

21A(6)|, and the total number of the binary variables in the new problem is









221A()l aEA I a() 1. Because of resource limitations, MIP solvers cannot solve

large problems.

In this chapter we consider a heuristic algorithm to solve the bounding

problems. Although the same technique can be applied to solve both bounding

problems, we mainly focuss on the upper bound problem DTDTA-U. For

convenience of reference, we restate the problem below.


min qTY
(x,y,z,g)

s.t. Byk+ k bIk Vkc C (6-1)

Sgd(k)t h Vk C (6-2)
tEA tEA

Ya(t,s) Y (t,s) Vt c A, a E A and s E F, (6 3)
kEC

Xa(t) > Ya( ,s) Vt e and a c A (6-4)
(T,s)Ena(t)

Za(t,) 1 Vt A and a c A (6 5)
sEPa

SOa'-( 6)Za(ts) < Xr(t) "< E a'(s)Za(ts) Vt e A and a e A (6-6)
sEPa sEPa
a(t,s) < ,,, ., Vt e A,a E A and a E Fo (6-7)

k k
Ya (t,s) 9d(k)t X(t) > 0, a(t,s) {0, 1} Vt A, a e A, s E Fa and k C C (6-8)

In the well-known heuristic algorithms such as neighborhood search, greedy

algorithm or tabu search, it is required to move from one feasible solution to

another. However, finding a feasible solution to the DTDTA-U problem is not easy.

To demonstrate, consider a one-arc-network, a = (1, 2), a linear travel time function

o(Ka(t)) = 0.3 + 0.05Xa(t), and a set of possible (discrete) travel times Fa {1, 2, 3}.

In addition, assume that the time horizon [0, 5) is divided into five intervals, and

10 cars enter into the arc at each discrete time t E A {0, 1, 2, 3, 4}. By assigning

those cars to the arcs a(t, 1) of the TE network, one concludes that Xa(t) = 10,









a(xa(t)) = 0.8, and Za(t,1) = 1, Vt A, which satisfies constraints (6-5) and
(6-6) (see the left network in Figure 6-1). It can be shown that this is an optimal

solution to the DTDTA-U problem. However, there is another feasible solution,

where 10 cars are assigned to the arcs a(t, 2) (see the right network in Figure 6-1).

Using those settings, Xa(t) = 20, Oa(xa(t)) = 1.3, and Za(t,2) = 1, Vt A, which again

satisfies constraints (6-5) and (6-6). Assume that the second solution is known

and one decides to improve the solution and move to another feasible solution by

assigning the cars at time t = 0 to arc a(0, 1), i.e., Za(0,1) = 1 and Za(0,2) = 0. As

a result, Xa(l) = 10 and Qa(xa(l)) = 0.8, which violates inequality (6-5). The same

follows from the change of other arcs; thus, we conclude that the second solution

is isolated in the sense that the .,1i ,i:ent solutions, i.e., changing only one arc,

are infeasible. Finding an .,ii ,i:ent feasible solution becomes more complicated

when the static network G(N, A) is larger because, with respect to a given path,

the changes on upstream arcs have an influence on the flows of downstream arcs.

Because heuristic algorithms similar to the neighborhood search, greedy algorithms,

and tabu search require moving from one feasible solution to a neighboring feasible

solution, the difficulties of finding a neighboring solution makes inappropriate the

use of such techniques.

10 o 10 10
10 10 1
10 10


12 u 22 1 2 s22
10 /10 /
13 /23 23
10 10
1424 24


Figure 6-1. Two feasible solutions.









Another approach to solve the DTDTA-U problem is the relaxation of the

integrality of the variable Za(t,s) and constructing an equivalent formulation with

continuous variables. To do so, replace the constraints Za(t,s) E {0, 1} by inequalities

0 < Za(t,s) < 1, i.e. z e [0, 1], n = A(5) EaeA 1a(5)1, and Za(t,s)(1 Za(t,s)) < 0.

The latter can be included into the objective function with a penalty. As a

result, the problem reduces to a continuous concave minimization one and a

global solution of the resulting problem is a solution of the DTDTA-U problem

(see, e.g., Horst et al. [54] or Horst and Tuy [55]). Although it is known that an

optimal vector of the binary variables, z*, represents one of the vertices of the n

dimensional unit cube (each vertex corresponds to an integer solution), because of

constraints (6-1)-(6-8) most of them are infeasible and it is hard to find an optimal

one.

Observe that the LP relaxation of the DTDTA-U problem provides a lower

bound, which is far from an optimal one. To illustrate, consider the DTDTA-U

problem, where the constraints Za(t,s) E {0, 1} are replaced by the inequalities

0 < Za(t,s) < 1. To find optimal values of variables y(t,) and g(k)t in the resulting

problem, it is sufficient to solve the following linear problem.


min qTY
(y,9)

s.t. Byk + gk bk Vk C

t k14 Vk c C
tEA tEA

a(t,s) > 0, g(k), > 0, Vt e A, a A, s e F and k C

Because the arcs a(t, 1) have a lower cost in the objective than the arcs a(t, s),

s / 1, one concludes that at optimality only the arcs that corresponds to the free
flow travel time, i.e., the arcs a(t, 1), Va E A and t E A, have positive flows. Using

the optimal values of those variables it is easy to compute values of the variables









X,(t) through equations (6-3) and (6-4). The optimal values of the variables Za(t,s)

can be obtained by solving the following system of equations.

ZsEra Za(t,s) = 1

a )Za(to) < a(t)

e a(s)Z(ts) > x (6 9)

Za(t,s) C [0, 1]
Y*
Za(t,s) > Ma()

Notice that the last inequality in (6-9) is satisfied VZa(t,s) E [0,1, s / 1, because

Y(t,s) = 0, Vs c Fa, s / 1. In the case of s 1, a sufficiently large value of .3,
reduces the inequality to Za(t,1) > 0 and makes sure that arcs a(t, 1) are allowed to

have positive flows given any positive values of the variable Za(t,1) (see inequality

(6-7)). Other equations in (6-9) are easy to satisfy and it can be shown that for

any value of x*(t) the set of solutions to the system is not empty and not unique.

However, because of the congestion it is highly unlikely that at optimality of

the DTDTA-U problem all drivers experience the free flow travel time and one

concludes that a solution of the LP relaxation of the problem is not realistic, and

the optimal objective function value of the relaxation problem is far from the

optimal value of the objective function of DTDTA-U.

Despite all complications described above, the DTDTA-U problem has the

following useful property: if at optimality the total inflow into arc a at time t

is zero, i.e., C r Y(ts) = 0, then constraint (6-7) is satisfied for any value of

Za(t,s); therefore, corresponding constraints (6-5)-(6-7) can be removed from the
formulation and the solution of the resulting problem remains the same. Unknown

values of the variables Za(t,s) can be restored by solving the system of equations

(6-5)-(6-6) using the values of xa(t,s).

The above analysis motivates developing a lower bound problem for the

DTDTA-U, which is (i) tighter than the LP relaxation, (ii) easier to solve









than the concave minimization problem discussed above, and (iii) preserves the

above mentioned property. In particular, in this chapter we consider a nonlinear

relaxation of the problem with bilinear constraints. Using the relaxation technique,

we propose a heuristic algorithm to solve the DTDTA-U problem.

For the remainder, Sections 6.2 discusses the nonlinear relaxation of the

DTDTA-U problem. Using the relaxation, in Section 6.3 we propose a heuristic

algorithm to solve the DTDTA-U problem. Numerical experiments on the

algorithm are provided in Section 6.4, and finally, Section 6.5 concludes the

chapter.

6.2 Nonlinear Relaxation of DTDTA-U Problem

Consider the following continuous nonlinear minimization problem, which we

refer to as DTDTA-R.

min qY
(x,y,z,g)
s.t. Byk +gk bk VE C (6-10)

9d(k), = hi Vk cC (6-11)
tEA tEA

Ya(t,) = (t,s) Vt e A,a c A and s e (6-12)
kEC

a(t) = Ya(r,s Vt c A and a A (6-13)
(T,s)ena(t)

S-I(s 6y)a(t,s) < Xa(t) Ya(tr) < E .la(S)Y(ts) (6-14)
SECa rEFa SECa
Vt e A and a e A

y(t,s) > 0, ,, > 0, ,Xa() > 0 Vt e A,a c A,s c Fa and k C (6-15)

Observe that (i) in the DTDTA-R problem constraints (6-10)-(6-13) are the

same as corresponding constraints of the DTDTA-U problem, (ii) the DTDTA-R

problem does not include the binary variables and constraints (6-5) and (6-7), and

(iii) the constraints (6-6) are replaced by bilinear constraints (6-14).









Theorem 6.2.1. The DTDTA-R problem is equivalent to the LP relaxation of the

DTDTA-U problem with additional constraints of the form


Za(t,s) Y. Ya(t,r)= Ya(t,), (6 16)
rECF

Proof: Consider the following two case:

Case 1: rEr. Ya(t,r) / 0. From equality (6-16) it follows that Za(t,s)
Ya(t ,) e [0, 1, Vs E F,. The latter satisfies constraints (6-5) and (6-7) given
E-Cra Ya(t,r)
a sufficiently large [,. thus, the constraints can be removed from the formulation.

In addition, observe that after appropriate substitutions of the variables Za(t,s) the

constraint (6-6) transforms into the constraint (6-14), and the variables Za(t,s) can

be removed from the formulation.

Case 2: ErErF Ya(t,r) = 0. Observe that equation (6-16) and constraint (6-7)

are satisfied for any value of the variable Za(t,s), s E Fa. Because constraint (6-7)

is redundant, remove the variables Za(t,s), s E Fa, from the formulation as well as

corresponding constraints (6-5)-(6-7). In addition, notice that constraint (6-14) is

satisfied and can be added to the formulation without changing the feasible region.

Based on the above ,n i i- one concludes that both problems have the same

optimal objective function value and the same optimal values for the variables

Ya(t,s) and Xa(t). In addition, the optimal values of the variables Za(t,s) can be
obtained using the values of y*,s) and x*(). In particular, if rEFr Y*(t,r) / 0 then

z*, *t Otherwise, given the vector x*, to find the optimal value of the
Z (ts) ErEa (t r)
variables Za(t,s) it is sufficient to solve the system of equations (6-5)-(6-6). U

Theorem 6.2.2. The DTDTA-R problem provides a tighter lower bound solution

for the DTDTA-U problem than the LP relaxation.

Proof: Consider a feasible solution, (y, x, z), to the DTDTA-U problem.

Observe that it satisfies equation (6-16) because from constraints (6-5)-(6-7)

it follows that either both sides of (6-16) are zero, i.e., Za(t,s) = Ya(t,s) = 0, or









Za(t,s) = 1 and Zrer, Ya(t,r) Ya(t,s). As a result, adding the equality (6-16) to
the DTDTA-U problem does not change the feasible region. Next, observe that

according to the Theorem 6.2.1 the LP relaxation of the resulting problem is

equivalent to the DTDTA-R problem; therefore, the solution of the DTDTA-R

problem is a lower bound of the DTDTA-U problem.

From Theorem 6.2.1 it also follows that the DTDTA-R problem is equivalent

to the LP relaxation of the DTDTA-U problem with additional constraints of

the form (6-16). Next we show that the constraints are not redundant unless

at optimality all drivers experience the free flow travel time. In particular, the

constraint (6-16) requires sending a portion of the total flow, i.e., Za(t,s) YrEr Ya(t,r),

along the arc a(t, s) if ErEr, Ya(t,r) / 0. On the other hand, recall that in the LP

relaxation of the DTDTA-U problem, at optimality only arcs that correspond to

the free flow travel time have a positive flow, i.e., y,) = 0, Vs E Fa, s / 1, and

the variables Za(t,s) may have positive values for all s E Fa, s / 1, as long as they

solve the system (6-9) (see Section 6.1). It can be shown that the solution satisfies

equation (6-16) only if za(t,i) = 1 solves the system (6-9). The latter holds only

in trivial problems with no congestion. From the above analysis it follows that

the DTDTA-R problem has a tighter feasible region than the LP relaxation of the

DTDTA-U problem. U

The DTDTA-R belongs to the class of global optimization problems because

constraint (6-14) is neither convex nor concave. However, it is computationally

more attractive than the concave minimization problem discussed in Section 6.1.

In particular, the DTDTA-R problem has fewer constraints and variables, and

does not have a penalty term in the objective function. In addition, the DTDTA-R

problem preserves the above mentioned property; if at time t an arc, a, has no

inflow, i.e., Eser, Ya(t,s) = 0, then constraint (6-14) is satisfied for any value of the

variable Xa(t).









6.3 Nonlinear Relaxation Based Heuristic Algorithm

The nonlinear relaxation problem DTDTA-R can be very useful to develop

a heuristic algorithm for solving the DTDTA-U problem because solving the

DTDTA-R problem is a trade off between satisfying constraint (6-14) and solving

the LP relaxation of the DTDTA-U problem. (Notice that the latter is equivalent

to the finding the shortest path in the TE network.) As a result, the optimal values

of the variables Xa(t) better approximate the optimal values of the corresponding

variables of the DTDTA-U problem. However, the solution is unlikely to be feasible

to the original problem; thus it is essential to find an integer solution that has the

objective function value as close as possible to the one provided by the DTDTA-R

problem.

In the heuristic algorithm (see Procedure 8), first we solve the LP relaxation of

the DTDTA-U problem, which provides an initial solution for DTDTA-R problem.

Next the procedure solves DTDTA-R problem. In Step 3, finding the values of

Za(t,s) is easy and can be accomplished by performing a simple search technique. By

fixing the binary variables of the DTDTA-U problem to the values of Za(t,s), i.e.,

Za(t,s) Za(t,s) the problem reduces to a linear one. If the resulting LP is feasible
then the algorithm stops and returns the solution. Otherwise, the algorithm runs

the UpSet procedure (see Procedure 9) then goes to Step 2.

Procedure 8 : Heuristic Algorithm for Solving DTDTA-U Problem
Step 1: Solve the LP relaxation of the DTDTA-U problem, and let the solution
be an initial solution for solving the DTDTA-R problem in the next step.
Step 2: Solve the DTDTA-R problem, and let (yR, xR) denote the solution of
the problem
Step 3: Let Xa(t,s) R(t,) and find binary variables Za(t,s) that satisfy
constraints (6-5) and (6-6).
Step 4: In the DTDTA-U problem, fix the binary variables to the values of
Za(t,s) and solve the resulting LP problem.
Step 5: If the LP problem is feasible, stop and return the solution. Otherwise
run the UpSet procedure and go to Step 2.









Procedure 9 : (UpSet) Setting the upper bounds on variables Ya(t,s)
for all a c A, t c A do
Compute values of smax and smin
nmax max{sls E Fa, Ya(ts) / 0} ser Y(t's) 0
m 0 Otherwise
m min{s|s E FC Y(ts) / 0} Eer Y(t) / 0
S 0 Otherwise
if smax smiTn 0 then
relax all bounds on variables Ya(t,s), Vs E Fa
else if max Smin m < 1 and Y("'t"). > a then
2rer Ya(t,r)
for all Vs E F, do
if s > smax + 1 then set the upper bound of Ya(t,s) to zero. Otherwise, relax.
end for
else
for all Vs c Fa do
if s > smax 1 then set the upper bound of Ya(t,s) to zero. Otherwise, relax.
end for
end if
end for


In the DTDTA-R problem, at optimality variables Ya(t,s) may have positive

values for different indices s C F,. However, in the DTDTA-U problem only one

of them is allowed to be positive. Procedure UpSet restricts the flow on the arcs

of the DTDTA-R problem in order to avoid using a large variety of indices from

the set Fa and, at the same time, remain close to the optimal objective function

value. For example, consider an arc, a, and assume that Fa {2,3,..., 8}. If in

the TE network arcs a(t, 2) and a(t, 8) are used then the procedure sets the upper

bound on the variable Ya(t,s) to zero. As a result, in the next iteration the solution

is required to use indices 2, 3,..., 7 that are more compact in the sense that the

smallest and the largest indices are closer to each other. However, if the set of

used arcs consists of only two i,, _!lhor" indices and the arc with a larger index,

e.g., (, carries a significant portion of the total inflow, e.g., the flow on the arc

a(t,() is greater than a (Ya(t,(c-) + Ya(t,c)), where a c [0, 1], then one may consider

the settings too restrictive and allow a positive flow on the arc a(t, ( + 1). In the









numerical experiments, we found it more useful to take the values for parameter a

from the interval [0.2,0.5]. A similar procedure applies to the case when only one

arc is used. Finally, if the total inflow into the arc is zero, i.e., SEa Y4(t,s) = 0,

then we relax restrictions on the variables Ya(t,s), Vs E F,. Although the DTDTA-R

problem is non-convex and requires finding a global optimum, the UpSet procedure

potentially eliminates the current solution from further consideration by narrowing

the feasible region.

The above described heuristic algorithm may not converge and the performance

of the algorithm is discussed in Section 6.4. However, if the algorithm does not

converge, an alternative objective function (see Table 6-1) can be used to solve

the problem. To construct the alternative objective function, observe that the

cost vector q, consists of discrete travel times s E Fa. On the other hand, the

variable Xa(t) is computed based on the set of indices a,(t) (see equation (6-13)). In

particular, each arc a(t, s), s E Fa, is included into at least one of the sets ao(t), for

some t E A. Notice that the arc may be included into more than one set, and the

total number of such sets is equal to s/6. (Note that s/6 is an integer because s is

a multiple of 6.) As a result,

TY E E -6 a(s) [ Ycjx j 5 5 X 6JT,

aa aEA tEA rEQa() aEA tEA

where e = (1, 1,..., 1). Although the second objective function includes parameter

6, it is not necessarily decreasing with the value of 6. In fact, by decreasing 6,

the number of variables Xa(t,s) and the total number of sets Qa(t) that includes arc

a(t, s) increase, thereby increasing eTx.

Table 6-1. Equivalent objective functions

Objective Function 1 Objective Function 2
min qY min 6eTx
(x,y,z,g) (x,y,z,g)