
Citation 
 Permanent Link:
 https://ufdc.ufl.edu/UFE0021262/00001
Material Information
 Title:
 Models and Algorithms for Reliable Facility Location Problems and System Reliability Optimization
 Creator:
 Zhan, Lezhou
 Place of Publication:
 [Gainesville, Fla.]
 Publisher:
 University of Florida
 Publication Date:
 2007
 Language:
 english
 Physical Description:
 1 online resource (120 p.)
Thesis/Dissertation Information
 Degree:
 Doctorate ( Ph.D.)
 Degree Grantor:
 University of Florida
 Degree Disciplines:
 Industrial and Systems Engineering
 Committee Chair:
 Shen, ZuoJun
 Committee Members:
 Geunes, Joseph P.
Romeijn, Hilbrand E. Feng, Juan
 Graduation Date:
 8/11/2007
Subjects
 Subjects / Keywords:
 Algorithms ( jstor )
Approximation ( jstor ) Datasets ( jstor ) Forts ( jstor ) Heuristics ( jstor ) Linear programming ( jstor ) Objective functions ( jstor ) Optimal solutions ( jstor ) System reliability ( jstor ) Total costs ( jstor ) Industrial and Systems Engineering  Dissertations, Academic  UF algorithms, approximation, facility, fortify, global, heuristics, integer, location, models, monotonic, nonlinear, optimization, programming, reliability, reliable, stochastic, system, uncertainty, unreliable
 Genre:
 Electronic Thesis or Dissertation
borndigital ( sobekcm ) Industrial and Systems Engineering thesis, Ph.D.
Notes
 Abstract:
 Uncertainty is one of the elements that make this world so fascinating and dynamic. However, the existence of uncertainty also poses a great challenge to reliable system design. Our study uses various models and algorithms to address reliability issues in the context of (1) the uncapacitated facility location problem where facilities are vulnerable, and (2) the system reliability problem where components are subject to fail. We first study the uncapacitated reliable facility location problem in which the failure probabilities are sitespecific. The problem is formulated as a twostage stochastic program and then a nonlinear integer program. Several heuristics that can produce nearoptimal solutions are proposed for this computationally difficult problem. The effectiveness of the heuristics is tested through extensive computational studies. The computational results also lead to some managerial insights. For the special case where the probability that a facility fails is a constant (independent of the facility), we provide an approximation algorithm with a worstcase bound of 2.674. Another part of our research is related to the application of a monotonic branchreducebound algorithm, a powerful tool to obtain globally optimal solution to problems in which both the objective function and constraints possess monotonicity. We tailor the algorithm to solve a mixed integer nonlinear programming problem. Its convergence analysis and acceleration techniques are also discussed. The algorithm is then successfully applied to solve system reliability optimization problems in complex systems, including the redundancy allocation optimization problem and the reliabilityredundancy allocation optimization problem. Compared to the existing techniques, the monotonic branchreducebound algorithm is not only versatile but also very efficient in dealing with different types of problems in system reliability. We also develop several models that can be used to fortify the reliability of the existing facilities. They are the extensions to the models in the first part of the dissertation and offer insights on which facility to choose and to what extent it should be fortified. The properties and solution methodologies of the models are discussed. In particular, a monotonic branchreducebound algorithm is used to solve one of these models. The efficiency of the algorithm is demonstrated in the computational results. ( en )
 General Note:
 In the series University of Florida Digital Collections.
 General Note:
 Includes vita.
 Bibliography:
 Includes bibliographical references.
 Source of Description:
 Description based on online resource; title from PDF title page.
 Source of Description:
 This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
 Thesis:
 Thesis (Ph.D.)University of Florida, 2007.
 Local:
 Adviser: Shen, ZuoJun.
 Statement of Responsibility:
 by Lezhou Zhan.
Record Information
 Source Institution:
 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright Zhan, Lezhou. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 662596757 ( OCLC )
 Classification:
 LD1780 2007 ( lcc )

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Full Text 
CHAPTER 2
RELIABLE FACILITY LOCATION PROBLEM: MODELS AND HEURISTICS
2.1 Introduction
Facility location models have been extensively studied in the literature. Different
kinds of facilities have been modeled, such as routers or servers in a communication
network, warehouses or distribution centers in a supply chain, hospitals or airports in a
public service system. Facility location models typically try to determine where to locate
the facilities among a set of candidate sites, and how to assign 'customers' to the facilities,
so that the total cost can he minimized or the total profit can he maximized ([34], [8], and
[45]). Most models in the literature have treated facilities as if they would never fail; in
other words, they were completely reliable. In this chapter, we relax this assumption to
model a more realistic case.
The reliability issue we consider is under the framework of the socalled uncapacitated
facility location problem (ITFLP). In ITFLP, we are given a set of demand points, a set of
candidate sites, the cost of opening a facility at each location, and the cost of connecting
each demand point to any facility. The objective is to open a set of facilities from the
candidate sites and assign each demand point to an open facility so as to minimize the
total facility opening and connection costs.
ITFLP and its generalizations are NPhard, i.e., unless P = NVP they do not admit
polynomialtime algorithms to find an optimal solution. There is a vast literature on these
NPhard facility location problems and many solution approaches have been developed in
the last four decades, including integer programming, metaheuristics, and approximation
algorithms. One common assumption in this literature is that the input parameters of
the problems (costs, demands, facility capacities, etc.) are deterministic. However, such
assumptions may not he valid in many realistic situations since many input parameters in
the model are uncertain during the decisionmaking process.
APPENDIX B
DATASET USED IN CHAPTER 2
The meaning of each column in Table B1 is provided as follows: #i denotes the
facility name; (x, y) is the coordinates, di is the demand; fi is the fixed cost; ri is the
penalty cost; and pi is the failure probability.
Table B1. Dataset of URFLPSFP
#i x y di fi ri pi II#i x y di fi ri pi
1 10.82 0.18 957 938 5.32 0.81 1151 0.63 0.04 4863 971 13.14 0.63
2 0.54 0.7 202 6342 1.9 0.39 52 0.53 0.32 548 1023 0.31 0.94
3 0.911 0.72 186 1230 3.11 0.42 53 0.89 0.99 870 754 0.66 0.76
4 0.15 0.31 635 1008 1.83 0.36 54 0.02 0.19 335 734 0.22 0.97
5 0.74 0.16 737 1279 1.34 0.28 55 0.51 0.32 446 1249 3.93 0.91
6; 0.58 0.92 953 1431 2.3 0.83 56 0.53 0.06; 198 1371 3.77 0.31
7 0.6; 0.09 450 1187 7.96 0.98 57 0.81 0.86; 212 1489 9.92 0.58
8 0.37 0.19 188 1044 3.42 1 58 0.53 0.36 903 8633 4.06 0.29
9 0.7 0.52 206 1466 9.05 0.86 59 0.89 0.58 594 521 1.64 0.15
10 0.22 0.4 995 989 4.563 0.55 60 0.87 0.56 250 8635 5.11 0.21
11 0.5 0.45 429 948 9.87 0.58 61 0.91 0.16 472 1464 2.64 0.43
12 0.3 0.52 528 585 0.53 0.79 6;2 0.32 0.15 244 730 6.05 0.09
13 0.95 0.2 570 923 3.411 0.46 63 0.37 0.37 353 1034 4.27 0.78
14 0.65 0.07 938 758 8.98 0.15 64 0.38 0.73 183 585 6.18 0.23
15 0.53 0.11 726 552 3.53 0.48 65 0.96 0.34 749 782 9.13 0.6
16 0.95 0.95 533 1471 1.64 0.7 6;6 0.15 0.76 200 985 6;.6;3 0.42
17 0.15 0.13 565 930 1.36 0.98 67 0.15 0.48 321 6;62 7.02 0.1
18 0.31 0.4 322 1103 5.1 0.2 6;8 0.99 0 650 904 8.73 0.16
19 0.98 0.73 326; 586 1.22 0.49 6;9 0.47 0.28 946 1242 3.92 0.52
20 0.59 0.04 663 812 6.95 0.3 70 0.84 0.16 143 513 8.16 0.79
21 0.46 0.21 952 850 9.57 0.37 71 0.71 0.9 5635 1117 1.25 0.84
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS
LIST OF TABLES.
LIST OF FIGURES
ABSTRACT
CHAPTER
1 INTRODUCTION
2 RELIABLE FACILITY LOCATION PROBLEM: MODELS AND HEURISTICS
2.1 Introduction.
2.2 Literature Review.
2.3 Notations and Acronyms .. ........ ....
2.4 Uncapacitated Reliable Facility Location Problem: a ScenarioBased Model
2.5 Uncapacitated Reliable Facility Location Problem with a Singlelevel Failure
Probability
2.5.1 Nonlinear Integer Programming Model
2.5.2 Model Properties
2.5.3 A Special Case: Uniform Failure Probabilities
2.6 Uncapacitated Reliable Facility Location Problem
Probabilities.
2.7 Solution Methodologies.
2.7.1 Sample Average Approximation Heuristic
2.7.2 Greedy Methods.
2.7.3 Genetic Algorithm Based Heuristic
2.8 Computational Results.
2.8.1 Sample Average Approximation Heuristic
2.8.2 Greedy Methods: GADH and GADSH .
2.8.3 Genetic Algorithm Based Heuristic
2.8.4 Applying Heuristics to Solve ITRFLPSFP
2.8.5 ITRFLPMFP: GADSH vs. GA.
2.9 Conclusions
with Multilevel Failure
:3 ITNIFORM INCAPACITATED RELIABLE FACILITY LOCATION PROBLEM:
A 2.674APPROXIMATION ALGORITHM
:3.1 Introduction.
:3.2 Formulations
:3.3 Approximation Algforithms.
:3.4 Conclusions
1 5 3
Figure A2. Configurations based on state of subsystem 4 in Figure 45: A) Subsystem 4
works, B) Subsystem 4 fails
with R3, 73 With R6, 74 With R7, rs with Rs:
Pr(system works t b..4fis
(R1R2 6g R12R6 (3 7 R3R7 5 (1R2R3 6R7 1R2R3R67)( 5
(A7)
Substitution of equations A6 and A7 into equation A5 yields the reliability of the
fivecomponent bridge network depicted in Figure 45:
R, = (1 QQ6) 7 ,[ R3( 2a5 4 (1R2 + 6 1R2R6 3 7 3R7 5
+(R1R2 3 6 7 1 2 3 6 7) R5)( 4). (A8)
Equation A8 can be reformulated as the following one:
R, =R6R7 1R2R3 6 6Q7) R14R7Q 62 2Q3)
+R3R5R67 1 1Q2 R12R5R734Q 6
+R23R4R6Q15 7 1R3R4R5Q2Q67 (A9)
[31] X. J. Liu, T. Umegaki, Y. Yamamoto, Heuristic methods for linear multiplicative
programs, Journal of Global Optimization 15(4) (1999) 433447.
[32] M. Mahdian, Y. Ye, J. Zhang, Approximation algorithms for metric facility location
problems, SIAM Journal on Computing 36 (2006) 411432.
[33] A. Mc Masters, M. Thomas, Optimal interdiction of a suppy network, i.1 I1 Research
Logistics Quarterly 17 (1970) 261268.
[34] R. Mirchandani, R. Francis (eds.), Discrete Location Theory, Wiley, New York, 1990.
[35] K(. B. Misra, U. Sharma, An efficient algorithm to solve integerporgramming
problems arising in systemreliability design, IEEE Transactions on Reliability
40(1) (1991) 8191.
[36] Y. N I1: I, l.\ra, ? I1: I1!;lls I, A heuristic method for determining optimal reliability
allocations, IEEE Transactions on Reliability R26 (1977) 156161.
[37] Y. N I1: I, l.\ra, K(. N I1: I1!;lls I, Y. Hottori, Optimal reliability allocation by
branchandbound technique, IEEE Transactions on Reliability R27 (1978) 3138.
[38] S. H. Owen, M. S. Daskin, Strategic facility location: A review, European Journal of
Operational Research 111 (1998) 423447.
[39] C. R. Reeves (ed.), Modern Heuristic Techniques for Combinatorial Problems, Orient
Longman, Hyderabad, 1993.
[40] A. Rubinov, H. Tuy, H. Mays, An algorithm for monotonic global optimization
problems, Optimization 49 (2001) 205221.
[41] H. S. Ryoo, N. Sahinidis, Global optimization of multiplicative programs, Journal of
Global Optimization 26 (2003) 387418.
[42] T. Santoso, S. Ahmed, M. Goetshalckx, A. Shapriro, A stochastic programming
approach for supply chain network design under uncertainty, European Journal of
Operational Research 167 (2005) 96115.
[43] P. M. Scaparra, R. L. Cloth1~ lo An optimal approach for the interdiction median
problem with fortification, working paper, K~ent business School (2005).
[44] P. M. Scaparra, R. L. Cloth1~ lo A bilevel mixed integer program for critical
infrastructure protection pl1 ...Irr.r Computers and Operations ResearchTo appear.
[45] Z. J. Shen, A profit maximizing supply chain network design model, Operations
Research Letters 34 (2005) 673682.
[46] D. Shmoys, E. Tardos, K(. Aasdal, Approximation algortihms for facility locatoin
problems, in: Proceedings of 29th ACijl STOC, 1997.
BIOGRAPHICAL SKETCH
Lezhou Zhan was born in Zh.~ i; I>. China, in the year of the Horse. He is also
known as Roger whose pronunciation is similar to Lezhou in his hometown dialect.
Prior to college, he graduated from Yueqing Middle School in 1997. He received his
Bachelor of Science in Applied Mathematics and his Bachelor of Science in Business
Administration and Engineering Management from Clue~~~, 11png University in 2001.
Before he transferred to the University of Florida in the fall of 2002, he studied scientific
computation at the Hong K~ong University of Science and Technology in a Master of
Philosophy program. He served as SecretaryGeneral of FACSS, a Chinese student
association at ITF, from 2003 to 2004. He earned his Master of Science and Doctor of
Philosophy in Industrial and Systems Engineering from the University of Florida in
May, 2004 and August, 2007 respectively. His current research interests include reliable
supply chain design, auction mechanism design, operations research models in airline
applications, and system reliability optimization. His work has been presented in various
conferences, book chapters, and journals, including Proceedings of the 2005 Winter
Simulation C'onference, Proceedings of the 2005 IIE Research C'onference and Production
and Op~erations _Ifor..~ry.; n.;.1 He is a member of INFORMS, SIAM, and IIE.
GADH, GADSH and GA can he used to solve ITRFLPSFP and ITRFLP1\FP. Our
computational studies show that (1) GADSH is the best heuristics among all four in
terms of the solution quality and the efficiency, and (2) GA is also able to find the best
solution with a little more time than GADSH.
There are several interesting research directions. We note that the 1! in r~ limitation of
the current models is the assumption that the facilities are uncapacitated. Although the
assumption itself is very coninon in the facility location models, it may be unrealistic in
practice. In the capacitated case, customer of failed facilities can he assigned to the next
level backup facilities only if they have sufficient capacity to satisfy the additional demand.
This restriction may make the capacitated model very complex. It becomes a valuable
topic of future investigation. In addition, some new measurements of the reliability
concept in the facility location problem setting are worth pursuing.
max R, = R6 7 + 1 2 3 6 + 6Q 7 R1 4 7 6 2 + 2 3) (436)
+R3R5R67 1 1Q2 R12R5R734Q6
+R23R4R6Q157 1R34R5Q2Q67
ctf to gi = "2+ 0.5x I(1+i) x3 4 2xs+ll 03 exp (437)
g2 1 x + 2x2 + 1.2x3) 1n~ 2 l+ + 2x3) + 0.4x4 (38)
(0.02 0.01
0.5 < Ri < 0.99, Vi = 6, 7, (439)
subje
(440)
xi: integfer, Vi=1... 5,
where Ri = 1 (1 ri)"i,Vi = 1,...,5, Qi = 1 Ri,Vi = 1,...,7, and rl = 0.70, T2 =
0.90, T3 = 0.80, T4 = 0.65, rs = 0.70.
This problem was originally solved using convexification method coded in FORTRAN
in [51]. The comparison between the mBRB method and the convexification one is done
in Table 44, where both algorithms obtain the optimal objective value. Admittedly,
comparing CPU seconds directly does not reflect the absolute efficiency, since the
convexification method was implemented in an older computer system. However, the
table clearly shows relative efficiency of the mBRB algorithm: It takes only negligible
CPU seconds to get the optimal solution.
Table 44. Performance comparison of Example 4
Method (X, r) R, R,(UB) CPU (s)
mBRB (4, 1, 3, 4, 3, 0.9845, 0.9900) 0.99974 0.99974 0.01
Convexification (4, 1, 3, 4, 3, 0.9845, 0.9899) 0.99974 38.63"
" It is measured on a SUN SPARCstation 5.
4.4 Conclusions
A monotonic branchreducebound algforithm for mixed integfer nonlinear progframmingf
is presented in this chapter. Its convergence and acceleration techniques are also discussed.
14000
12500 H _SAAH
ia11000
9500
o
8000
6500
5000
10 15 20 25 30 40 50 60 70 80 90 100
Dataset #
Figure 25. Comparison of objective values from GADH, GADSH, GA, and SAAH
2.8.5 URFLPMFP: GADSH vs. GA
In this part, we apply GADSH and GA to solve ITRFLP1\FP, which is considerably
more difficult than ITRFLPSFP. The dataset is similar to the one used in the previous
sections but with each facility having :3 levels of failure probability. The full dataset is
presented in Table B2 of Appendix B. The first half of Table 21:3 reports the results
from GA, which has been run for 5 times with different random seeds. Eight datasets
have been generated for testing. The best open sites and their optimal levels are listed
in the second column. The best and worst results obtained in the 5 trials are listed in
the third and fourth columns respectively. The average time in seconds are reported in
the last column. The results of GADSH are shown in the bottom half of Table 21:3. In
only one case (the 80node problem) does the GA find a better solution than GADSH. In
that case, the objective function value is 11782.85 compared to 11859.40, which represents
only a 0.10' improvement. But GA takes more time than GADSH does to get this small
improvement. In all other cases, GADSH finds the same solution as GA but with much
less time. Overall, GADSH is more favorable than GA in solving model ITRFLP1\FP.
existing system. Such models are among the earliest fortification models in the
facility location literature. Several efficient algorithms are also provided.
The structure of our research is depicted in Figure 11. In C'!s Ilter 2, we present
several models for the uncapacitated reliable facility location problem in which some
facilities are subject to failure from time to time. These models are the foundation of
our research. Besides the general scenariobased model, they include the case in which
each facility has a sitespecific failure probability, and the case in which each facility has
multilevel failure probabilities. The properties and different formulations of the models
are thoroughly discussed. Several heuristics are presented along with the computational
results.
Models Algorithms
Uncapacitated Reliable Facility Location Model, ~ISample Average Approximation Heuristic
Chapter 2
Single Failure Probability, Greedy Add Heuristic
Multilevel Failure Probabilities,
Chapter 2 Genetic Algorithm
Uniform Failure Probability, Chapter 3 I 2.674Approximation Algorithm
Discrete/Continuous Facility Fortification Model,
Chapter 5
Monotonic BranchReduceBound
Algorithm
System Reliability Model, Chapter 4 
Figure 11. Research structure
In C'!s Ilter 3, we present a tighter approximation algorithm with a worstcase bound
of 2.674 for a special case of the uncapacitated reliable facility location problem, where all
failure probabilities are identical.
In C'!s Ilter 4, we present a monotonic branchreducebound algorithm for a
special case of the nonlinear mixed/pure integer programming problem where both
the difference of the total cost between current row and previous row. It indicates the
sensitivity of L on the objective value.
Table 51 and Figure 51 show how the system deteriorates (total cost increases) as
the resource to maintain the system reliability decreases. In addition, the last column
of Table 51 indicates that the curve in Figure 51 is piecewise linear. For example, the
difference, similarly the slope, in total cost is a constant when L changes from 1.0 to 2.0
in a step of 0.1. This is because of (1) the failure probabilities of all but one facility (20,
in this case) remain the same, the whole cost structure does not change, (2) CI lngs;th
failure probability in one facility will only affect all the service cost that related to this
particular facility.
4500
4000 
3500
a 3000
o 2500
2000
1500
1000
0 0.5 1 1.5 2 2.5 3 3.5 4
System (Un)reliability Level, L
Figure 51. Total cost at different system reliability level
There is a common pattern in Table 51: when L increases by 0.1, only one of the
facilities changes the failure probability accordingly. For example, when L increases
from 1.7 to 1.8, the failure probability of facility 20 jumps from 0.7 to 0.8, all the others
remaining the same. The failure probability at an individual facility usually changes by
either 0.1 or 0.0 as shown in most cases in Figure 52. However, there is one notable
25: update Zbest and fbest, if improved.
26: else
27: update Zbest and fbest, if improved.
28: 2 < 2U {Sk
29: end if
30: end for
31: goto Select and Branch.
Algorithm 2 Monotonic BranchReduceBound Algorithm
5.3.2 Computational Experiments
One advantage of using the monotonic branchreducebound algorithm to solve the
discrete facility fortification problem is that no closed form of Pi(yi), i E F, and IM(yi),
i E F are required as long as they are monotonic. In this section, we use the same dataset
as the one in section 5.2.2, which is listed in Appendix C. The dataset contains 20 demand
nodes and 5 open facilities with the following specification.
Table 52. Input data (IM(yi) and Pi(yi)) for the 3level model
Level 1 Level 2 Level 3
Open Facility i V P V P V P
2 0 0.55 79 0.39 6388 0.02
5 0 0.85 614 0.45 728 0.28
15 0 0.75 303 0.63 855 0.48
18 0 0.52 135 0.48 409 0.20
20 0 0.30 178 0.24 273 0.22
The monotonic branchreducebound algorithm is implemented in C++. The CPU
seconds are reported from a Dell Optiplex GX620 computer with a Pentium IV 3.6 GHz
processor and 1.0 GB RAM, running under the Windows XP operating system.
The computational results are reported in Table 53 that shows that the fortification
level at each open facility given the resource constraints. Table 54 shows the results that
the fortification level is limited to 2. That is, yi < 2, Vi E F. The CPU time reported in
both tables clearly show the efficiency of the monotonic branchreducebound algorithm
for this type of problem. Next, we start to analyze the computational results.
very efficient in finding the optimal or near optimal solutions. For the capacitated facility
fortification models, we believe that the nionotonic branchreducebound algorithm can
still be used to find the global solution.
It would be interesting to see if there exists any approximation algorithm with a
constant worstcase bound for the capacitated uniform reliable facility location problem.
Based on the research on approximation theory of facility location problem without
considering the reliability issue, we expect that some more delicate techniques are required
to develop such approximation algorithm.
Another interesting direction is to introduce different risk/reliablity measurements
into the current models depending on the needs in reality. By doing so, the objective
functions of current models will change accordingly. For example, one may have more
interests in the cost of the worstcase scenario instead of the expected cost in the current
models.
In suninary, the current work we present serves as a useful foundation for further
research of more complicated models and delicate algorithms.
(CFFM~)
F F+1
minimize~ dyey (1 ps) plP 1 51
j6D k=1 iEF 16F j6D k=1 16F
subject to gyp,., )<0 =1..m(52)
0 < pi < 1, Vi E F. (53
The objective function (51) is the sum of the expected failure cost and the expected
penalty cost. Constraints (52) denote various resource restrictions on the fortification
levels, pi,Vi E F. We assume gj (j = 1,..., m) are convex functions so that the solution
domain is convex. Constraints (53) are natural constraints on the failure probability.
5.2.1 Properties of the Continuous Facility Fortification Model
In this section, we show that CFFM is a special case of the following generalized
linear multiplicative programming (GLMP) problem (see [41] for details).
(GLM~P)
m~inimizej~ (:IcX x dz3) (54)
j= 1 i= 1
subject to x E X (55)
where cij E R"n, dij E R"n, j = 1, ..,t, i = 1, ..,pj, and x E R"n, X is a nonempty
convex set. Note that without the summation sign in the objective function 54, GLMP is
reduced to linear multiplicative programming, another active topic in global optimization.
We first show that the objective function (51) is the sum of linear multiplicative
terms with positive coefficients. In other words, the objective function (51) can be
reduced to the form of
j i= 1
where asj are nonnegative and pij are binary.
[61] B. Verweij, S. Ahmed, A. J. K~leywegt, G. Nemhauser, A. Shapriro, The sample
average approximation method applied to stochastic routing problems: a
computational study, Computational Optimization and Applications 24 (200:3)
2893:33.
[62] R. K(. Wood, Deterministic network interdiction, Mathematical and Computer
Modelling 17(2) (199:3) 118.
[6:3] T. Zeng, J. E. Ward, The stochastic locationassignment problem on a tree, Annals of
Operations Research 1:36 (2005) 8197.
point must be served by several facilities where the number is specified, and a weighted
linear combination is used to compute the connection costs. FTFLP has been motivated
by the reliability issue considered in this chapter, but the failure probabilities of facilities
are not explicitly modeled and penalty cost is not considered.
The remainder of this chapter is organized as follows. Several equivalent formulations
for UURFLP are proposed in Section 3.2, which lead us to develop a 2.674approximation
algorithm in Section 3.3. The chapter is concluded in Section 3.4.
3.2 Formulations
The notation of this chapter follows that in C'!s Ilter 2. Recall the formulation of
UJRFLPSFP in C'!s Ilter 2:
( URFLPSFP)
minimize fSI/ y ~diricly (1p)I pl
i6EF j6D k 1 iEF 16F
+ jeD +~1 nF16F; 1 dr~J
subjectto z Cz = 1, Vje D,k= 1...,F+1 (32)
i6EF t=1
x < i E F, jE D, k~ = 1, ...,  F (33)
xi5 i E F,j jED (34)
Consider a special case of URFLPSFP where all facilities have the same failure
probabilities, i.e., pi = p, Vi E F. This assumption simplifies formulation (URFLPSFP)
considerably based on the following observation. Because pi = p, Vi E F, it is
stilfrai ht o w ardi th atl I p~l= P1 Wh ich isind ep end ent of th e val ues, of xti
16F
This property is implicitly used in a multiobjective formulation proposed in [48].
Based on the above observation, we are able to reduce formulation URFLPSFP to a
linear integer program as follows.
2.8 Computational Results
In this section, we compare the computational performance of the four heuristics: the
sample average approximation heuristic (SAAH), the greedy adding heuristic (GADH),
the greedy adding and substitution heuristic (GADSH), and the genetic algorithm based
heuristic (GAH). In order to evaluate the performance of these four heuristics, we first
apply them to solve URFLPIP, where each facility has the identical failure probability.
The reason is that URFLPIP admits a linear integer programming formulation that
can be solved to optimality by using commercial solvers such as CPLEX, so that we can
compare the heuristic results with the exact solutions, which helps to better evaluate the
performance of each heuristic.
The test dataset is generated as follows. Coordinates of the sites were drawn from
U[0, 1] x U[0, 1], demand of each site was drawn from U[0, 1000] and rounded to the
nearest integer, fixed facility costs were drawn from U[500, 1500] and rounded to the
nearest integer, and penalty costs were drawn from U[0, 15]. Further, the transportation
cost cay is set to be the Euclidean distance between points i and j. The number of sites
varies from 10 to 100. The dataset is available in Appendix B.
All the algorithms were coded in C++ and tested on a Dell Optiplex GX620
computer running the Windows XP operating system with a Pentium IV 3.6 GHz
processor and 1.0 GB RAM.
2.8.1 Sample Average Approximation Heuristic
We first test how the sample size (NV) affects (1) the quality of the solution, and (2)
the efficiency of the program, for a 50node dataset with the uniform failure probabilities
varying from 0 to 1. Table 25 lists the objective values obtained from SAAH when
M~ = 1 and the sample size varies from 10 to 200.
It is clear from Table 25 that the solution quality can be improved by increasing the
sample size. The ratios of the objective value obtained from SAAH with sample sizes 10,
50, 100, 150, 200, and 250 to the optimal value are plotted in Figure 22.
Proposition 2.1. In r,.;, optimal solution to URFLPSFP, for r,.;, client j, ifx~
x' = 1, then cap < c,4.
Proof. We prove the proposition by contradiction. Suppose cej > cej, we will show that by
I1l1!1! the assignment of a and v, the objective function will strictly decrease.
In particular, if we set x =l 1 and Zk = 1 With the values of other variables
unchanged, we can compute the new objective value. The difference between the new
objective value and the original one is
< 0,
where
16F
The last inequality holds because pl > 0 and it is assumed that p, < 1, p, < 1, and
cmj < cy. This is clearly a contradiction to the optimality of the original solution.
Therefore, cej < ce. O
An implication of Proposition 2.1 is that if the set of open facilities is determined,
then it is trivial to solve the level assignment problem for each client: assigning levels
according to the relative distances of different facilities to the client. If at some level the
distance is beyond the penalty cost, then no facility will be assigned at this level (and
higher ones) and the demand node simply takes the (cheaper) penalty.
We would like to point out the relationship between formulation (URFLPSP) and
formulation (URFLPSFP). Since these two formulations are just two wwsi~ of modeling
the same problem, they should have the same minimum cost as long as the inputs to the
two models are consistent. In formulation (URFLPSFP), each facility i has independent
failure probability pi. This implies that there are 2 *I scenarios and the probability that
The algorithm is successfully applied to solve a wide range of the system reliability
optimization problems. Computational results have been reported to show the superior
efficiency of the algorithm over existing ones.
We expect that the nionotonic branchreducebound algorithm can he applied to
other classes of the problem, such as nonlinear nmultidintensional knapsack problems, and
generalized niultiplicative progranining problems. In the future research, we also would
like to compare the performance of the acceleration techniques mentioned in this chapter,
and analyze its worstcase performance theoretically and computationally.
continued from previous page
#i x y di ri #i x y di ri
0.87
0.69
0.24
0.84
0.49
0.56
0.38
0.43
0.61
0.48
0.81
0.13
0.41
0.72
0.3
0.02
0.62
0.8
0.1
0.15
0.48
0.83
0.82
0.67
0.03
0.26
0.59
0.79
0.15
0.28
0.73
0.24
0.77
0.05
0.22
0.73
0.88
0.75
0.71
0.34
0.14
0.28
1
0.9
0.06
0.98
0.13
0.44
0.22
0.39
0.53
0.73
0.39
0.56
292
48
581
659
986
486
915
282
310
980
134
20
151
615
369
875
73
776
342
929
445
6384
643
771
181
926
733
6.95
2.29
2.9
5.05
8.81
3
6.94
0.88
2.66
3.32
0.6;6
3.08
2.634
3.84
1.74
1.14
2.76
3.36
5.61
9.77
4.32
9.61
1.45
7.46
1.03
7.6
0.15
5.93
0.09
0.71
0.27
0.25
0.57
0.96
0.83
0.72
0.69
0.22
0.37
0.36
0.11
0.77
0.16
0.75
0.16
0.01
0.51
0.98
0.55
0.36
0.18
0.09
0.18
0.1
0.25
0.74
0.78
0.04
0.07
0.18
0.49
0.21
0.49
0.5
0.89
0.88
0.82
0.1
0.69
0.09
0.63
0.41
0.21
0.76
0.32
0.39
0.63
0.75
0.46
0.67
0.38
0.66
374
284
598
720
457
213
550
418
863
36;8
282
811
866 i
895
959
375
711
208
954
843
905
729
382
91
991
6344
539
2.635
1.45
9.63
3.72
2.38
7.09
8.3
2.55
1.53
3.5
7.15
9.07
2.463
3.19
5.52
5.28
7.34
3.99
4.06
7.77
0.08
8.47
6;.23
0.53
6;.6;2
3.13
0.5
0.22  0.66  326
0.68  0.49  294  7.27
The above discussion leads to a nonlinear integer programming formulation for
UJRFLPSFP.
i6EF j6D k=1 iEF 16F
F+1
j6D k=1 16F
subject to x2 z = 1c Vj e D, k = 1..., F + 1 (2 8)
i6EF t=1
x$ y /, Vi e F% je D, k = 1,...  F  (29)
x@ :5 1, Vi le F (2(10)
xfyzf ,ye {0,1}.(211)
The decision variables xfy, zf are defined earlier. The indicator variable yi = 1 if facility i
is open in the first stage, otherwise yi = 0. The objective function (27) is the summation
of the facility cost, the expected service cost, and the expected penalty cost. Constraints
(28) ensure that client j is either assigned to a facility or subject to a penalty at each
level k. Constraints (29) make sure that no client is assigned to an unopen facility.
Constraints (210) prohibit a client from being assigned to a specific facility at more than
one level. Note that constraints (29) and (210) can be tightened as
x@2 < ye Vi e F, je D. (2 12)
2.5.2 Model Properties
In formulation (URFLPSFP), we do not explicitly require that a closer open facility
be assigned as a lower level facility to a particular demand point. However, according to
the following proposition, it is true that the level assignments among the open facilities are
based on the relative distances between the demand point and the facilities regardless of
the failure probabilities.
As we can see, the GA heuristic performs quite well. It can produce the exact
solution in 2 trials. While there is no guarantee that the GA solutions are optimal, it
can easily find an optimal or near optimal one by running multiple times then taking the
best solution it finds. Considering the fast speed for each run, a nmultitrial GA is quite
attractive.
Figure 24 depicts the evolution of the nxininiun objective value, and the average
objective value in each generation when p = 0.5. The algorithm terminates at generation
176 after it finds the optimal solution at generation 76. In fact, in this example the GA
quickly converges to a close optimal solution after just 20 generations as shown in Figure
24.
50000
45000
40000
35000
30000
25000
20000
15000
10000
1 21 41 61 81 101 121 141 161
Generation
Figure 24. Evolution
of the solutions from GA
2.8.4 Applying Heuristics to Solve URFLPSFP
Dropping the uniformity assumption of facility failure probabilities introduces more
challenges to solve ITRFLP exactly, due to the nonlinearity in formulation (ITRFLPSFP).
monotone optimization problem. The algorithm recursively partitions the hyperrectangle
[X x"] into a smaller region that still contains the global maximizer. A variant of the
algorithm is able to handle the mixed integer version of SROP'.
In the description of Algorithm 1, S denotes a hyperrectangle partition; C is a list
of unfathomed hyperrectangles; e is a predefined optimality tolerance parameter; xbest
and fbest denote the current best solution and objective value respectively; UB(S) is the
upper bound of the objective function over S. Besides initialization, the 1!! r ~ steps are
described in detail as follows.
Initialization
2: if xU E g then
3: x" is the optimal solution, terminate;
4: else if xL Sf then
5: the problem is infeasible, terminate;
6: else
7: set S = [X x"], e > 0, C = {S}, Zbest = XL, best =(L)
8: end if
Select and Branch
9: if C = 0 then
10: output the current Zbest aS the solution, terminate;
11: else
12: select S = [s SU] eE such that UB(S) = maxsez{UB(S)} and < C\{S}
13: if (UB(S) fbest) < e then
14: output the current Zbest aS the solution, terminate;
15: else
16: select i such thaRt i = argm~axi,~ {sU a, bisect S into SI anld S2 along th~e edge i.
(case 1: i < q; case 2: i > q.)
17: end if
18: end if
Reduce and Bound
19: for k = 1, 2 do
20: reduce Sk, to S, [L" U x"] according to reduction rules
bounded by
:1 k1
~=~j_
k=ky+1
1
k=ky+1
1
kg
' dyrypky1 (
dyyky1 _
p/
This completes the proof.
Proof. Lemma 3.4.
The costs of interest depend on the value of the randomly generated P. Recall the
definition? of (i, y). For ecnch j D an id i e F, if
xfy, if k / mj; If mj k 1 then
.The latter is true only if 6 < ij. Therefore, in
23 =3 0 x if my
k
ky, then z2
both cases,
x < xx .
(313)
By the constraint of the linear programming relaxation of (37), we know that, for
each i E F and je D,
F 
Therefore ,

j~~~o ~ "" ~
Z 
k=1".
(314)
To show that
i6EF
we consider the following two cases.
1 j Dk m
xty. From the constraint
*mj = k, 1:i In this case, Vj E D, k < mj, Of" = 0 and ify
31 &
xf = 1.i
i6EF
) :
The coninercial solvers, such as CPLEX, lack such ability to solve nonlinear integer
progranining problems. In addition, other generic global optimization solvers seems
not able to efficiently handle model ITRFLPSFP. For example, BARON has difficulty
in solving ITRFLPSFP with 10 demand nodes and 10 candidate facilities. The 1! in Un
difficulties come front the binary constraints on the decision variables and the large
number of nonlinear terms in model ITRFLPSFP. In this section, we apply heuristics:
SAAH, GADH, GADSH, and GA to solve such model and compare their performance.
For some small size problems, we also provide globally optimal solutions front a simple
enumeration method.
Several datasets are derived front the 100node dataset in Appendix B: for example,
dataset #10 is the first 10 lines from Table B1 of Appendix B; it has 10 demand nodes
and facility sites. The other datasets are derived in the same way. Table 211 lists the
objective values obtained front SAAH with different sample sizes. The colunin "Best
Obj .I :lists the nxininiun objective value among the different sample sizes. Time is
measured in seconds with results front the sample size of 100. "" in Table 211 means
that the program was out of nienory due to the surge in the problem size. "" in Table
212 means that the results of the enumeration method were not obtained due to the
exponentially increased computational time. Table 212 suninarizes the objective values of
the solutions obtained by GADH, GADSH, GA and the enumeration method, and their
corresponding computational time. The results of GA are obtained through a single run.
Comparing the results front heuristics with the globally optimal solutions in small
data sets (front #10, to #30), we can see that GADSH and GA find optimal solutions,
whereas SAAH and GADH find optimal or nearoptinmal solutions. To evaluate the
quality of the solutions found by these heuristics in all datasets, we plot the objective
values in Figure 25. The values front SAAH are the best ones available in Table 211
for each instance. Figure 25 shows that GADSH and GA can find the best solutions
in all datasets, whereas SAAH and GADH can find either the best known solutions or
Comparing these two "obj. I I i.; lines in Figure 54, we also see the benefit to have a
:$level option over a 2level one. When the available fortification resource is very limited,
both share the same objective values as there is no enough resource to fortify the facility
to level :3. But as the available fortification resource increases, the :$level model has more
flexibility so that it incurs much lower total cost.
Another phenomenon different from the continuous version is the fortification level
at individual facility. In the continuous version, the fortification level is more or less
monotonic as shown in Figure 52. However, this does not exist in the discrete version as
shown in Figure 55. As the fortification resource increases, the fortification level at an
individual facility does not necessarily increase as a consequence from this combinatorial
optimization. Figures D and E in Figure 55 clearly show this phenomenon, because
facility 18 and 20 do not admit any monotonicity pattern.
5.4 Conclusions
In this chapter, we present two optimization models (a continuous version and a
discrete one) on how to choose facilities to fortify and to what extent they should be
fortified given limited fortification resources. The objective of both models is to minimize
the sum of the expected service cost and failtoserve penalty cost.
In the continuous version of the model, the fortification effort is dividable. We show
that the model is a special case of the generalized linear multiplicative programming
problem. We solve an illustrative example by the vertex numeration method, which
is very effective in solving this type of problems. This example also demonstrates the
multiextremeness of the problem: several vertices achieve the minimum. 1\anagerially, the
example shows how to identify the key facilities to fortify in a system.
The discrete facility fortification model focuses on choosing the suitable fortification
level at each facility when the fortification effort is divided into different levels. The model
is shown to be a monotonicity optimization problem since the monotonicity property
is inherent in both its objective and constraints. This model is therefore solved by the
exception in this pattern. That is when L increases fron1:3.7 to :3.8, both facility 2 and
facility 18 change their failure probability in the optimal solution. It means that the
optimal vertex in L = :3.8 is dramatically changed from L = :3.7. This exception is
clearly signaled in Figure 52 if we look at the changes from L = :3.7 to L = :3.8 in
subfigures A to D: a 0.2 dip in A, a 1.0 surge in B, a 1.0 sink in C, and a 0.3 jump in D.
This phenomenon is front the niultiextrentalness of the CFFM problem that we discuss
in Section 5.2.1. Using a bisection search, we find out that at L = 0.749842, both vertex
(1, 0, 1, 0.749842, 1) and vertex (0.749842, 1, 0, 1, 1) produce the same objective value
of :3909.2476. It means that at L = 0.749842, the optimal vertex transient front (1, 0,
1, 0.749842, 1) to (0.749842, 1, 0, 1, 1). Then from L = 0.749842 to L = 0.8, the same
pattern still holds, facility 2 is the only one that changes its failure probability.
As we point out in the introduction that CFFM model can also help to find the
key facility to fortify. In this example, the frequency that a facility is completely open is
depicted in Figure 5:3. It shows that facility 5 is very critical in this system, because it is
chosen to be completely secured :38 times out of the total 41 cases in this example.
facility means that we close the facility. After the substitution process, another greedy
adding procedure is performed to further improve the solution. The whole process (a
greedy adding procedure followed by a greedy substitution procedure then followed by
another greedy adding procedure) is called the greedy adding and substitution heuristic
(GADSH).
2.7.3 Genetic Algorithm Based Heuristic
Genetic algorithm (GA) hased heuristics have been widely used to solve combinatorial
optimization problems. A GA imitates the mechanism of natural selection and natural
genetics. Generally, a GA starts with an initial set of solutions called a population. Each
member of the population is called a chromosome, representing an encoded version of a
solution to the optimization problem at hand. The goal of an encoding is to translate a
solution into a string of genes that make up the chromosome. There is a fitness function
that evaluates the quality of a chromosome at each iteration, called a generation. The
generation evolves through several operators: crossover, reproduction, immigration, and
mubstion. The crossover operator is a process to produce one or more offspring from the
current generation. Reproduction is simply a process that copies the best solutions from
the previous generation to the next. The immigration operator is to randomly create
certain new chromosome in each generation. The mutation operator is to randomly change
the genes in a chromosome to introduce randomness in each generation. The population
size through each generation is kept constant. After several generations, the best solutions
converge to an optimal or suboptimal solution to the problem. Several comprehensive
treatments of GAs are given in [11, 14, 39].
In principle, a GA can he applied to any optimization problem. But there is
no generic GA since it requires ]rn lw: design decisions, such as the encoding of the
chromosome, the selection of parents, the method of the crossover operator. In this
section, we describe a GA that is suitable for URFLP.
x. > 0,
O< <1,
and r,:;, as (0,1i], one can find an integer solution (x, y) ;aH itr;,i, the above inequalities
so that
i6EF j6D k=1 iEF j6D k=1 iEF
In Lemma 3.1, a~ is a parameter that we can choose to control the quality of
approximation. Indeed, the approximation ratios of the algorithms of Guha et al. are
functions of a~. One can then choose the best a~ to minimize the approximation ratio.
We are now ready to present our algorithm for UURFLP. We first solve a linear
programming relaxation of formulation (37).
F 00
minimize f eyi + dyey pkrlZZI11 _p 39
i6EF j6D k=1 iEF j6D k=1
sub~jct to xz + f= 1, Vj D, k = 1..., F (310)
i6EF
Zk > 0, Ok" > 0. (312)
Assume that (x, y, 8) is an optimal solution to this linear program. Our algorithm rounds
the fractional solution (x, y, 8) to an integer solution (x, y, 8) that is feasible to formulation
(37).
The algorithm is based on a property of the optimal fractional solution (x, y, 8), which
is formalized in the following lemma. This lemma enables us to utilize known algorithms
and analysis for the faulttolerant facility location problem.
Lemma 3.2. For each j e D, the following two statements are true.
the objective function and constraints possess monotonicity. Its convergence analysis and
acceleration techniques are also discussed. The algorithm then is applied to solve system
reliability optimization problems in complex systems, including the redundancy allocation
optimization problem and the reliabilityredundancy allocation optimization problem. The
efficiency of the algorithm is demonstrated via the computational results.
Based on the models in C'!s Ilter 2, we develop two models that are used to fortify
the reliability of the existing facilities. The properties and solution methodologies of the
models are discussed. In particular, the monotonic branchreducebound algorithm
presented in Chapter 4 is used to solve one of these models. The efficiency of the
algorithm is demonstrated through the computational results.
This dissertation is concluded in OsI Ilpter 6 with a discussion on future research
directions.
21: compute a suitable upper bound UB(Sk)
22: if UB(Sk) Ibest Or ZLr Sf 0 then
23: continue,
24: else if xU" E then
25: update Zbest and fbest, if improved.
26: else
27: update Zbest and fbest, if improved.
28: 2 < 2U {Sk
29: end if
30: end for
31: goto Select and Branch.
Algorithm 1 Monotonic BranchReduceBound Algorithm
4.2.1 Select and Branch
The hyperrectangle with the greatest upper bound is selected for branching, as shown
in line 12 in Algorithm 1. Then the subsequent branching is carried out along the longest
side i of S, with? i = argmaxj {s~ 8 If i < ql thenl S = [s s"] is pa~rtitionecd to SI anld
S2 on the discrete variable:
is partitioned to S1 and S2
St = s L, s" af2s e
22 SL 2 s
where ei is the ith unit vector in R". It is obvious that after partition the least and
greatest elements in S1 and S2 belongs to X ,. A branchingf process is said to be exhaustive
S = s C,S s" 2 afe
S2~~S SL 2 s+ e", s,
where ei is the ith unit vector in R". If i > q, then S = [s s"]
on the continuous variable:
where f() is a general expression of the system reliability.
Another way to increase the system reliability is to simply use more reliable
components, which certainly costs more in terms of various resources. This problem
is called a reliability allocation optimization problem ([28]). Suppose there are ui
discrete choices for component reliability at stage i for i = 1,...,k (< n) and the
choice for component reliability at stage k + 1,..., n is on a continuous scale. Let
re (1), re (2),. ., re (ui) denote the component reliability choices at stage i for i = 1,. ., k
(< n). Then the continuous/discrete reliability allocation optimization problem can be
formulated as follows:
max R=f(r(x)... kk k1** E)(45)
subject to gryx).. kk k1**E)
x4 e{1,,...4} i= 1... k,(47)
0
where ri and u" are the lower bound and upper bound respectively for the component
reliability at stage i. If k = 0, the above mixed integer nonlinear programming formulation
reduces to a pure nonlinear programming problem.
The systems we are interested in are not limited to series and parallel systems. They
can be complex (general) systems that are nonseries and nonparallel, such as the bridge
network in Figure 43. The system reliability of such system can be computed by the
conditional probability theory. For example, the system reliability of the fivecomponent
bridge network in Figure 43 can be computed based on whether component 5 is functional
or not. We refer readers to Appendix A for the details of the following expression.
R, = (rl + T3 r.1r.3 72 + 4 r.2r4 7.5 + 71r.2 +73r4 T1r2r3r4 75 r)
( URFLPM~FP1)
F
minimize CC s 1Jf,+Cc 2) +i~ EE ge,,1_ ivs) Jre))
i6EF t j6D k=1 iEF 16F
F+1
j6D k=1 16F
sub~jct to x z~r = ~ 1 Vk=1 .,+1
i6EF t=1
xi 5 yr, Vi E Fj jE D, k = 1..., F
yti < Vi =1,..., F
F
xi $3~ 1 Vi EF,j ED
(216)
(217)
(218)
where constraints (218) ensure at most one investment level is allowed at each facility
and all the other constraints are similar to the ones in URFLPSFP. Note that Proposition
2.1 still holds in this model.
The above formulation, URFLPMFP1, is a binary model. URFLPMFP can also be
modeled as a regular integer model by reinterpreting the definition of yi as the investment
level at facility i. Thus, yi is not binary any more; 0 < yi < Ui, where Ui is the highest
level at which facility i can be possibly built. The corresponding failure probability
and the fixed cost at facility i are denoted by functions Pi(yi) and Fi(yi) respectively.
Pi(yi), iE F, are nonincreasing functions of yi and Pi(0) = 1, whereas Fi(yi), iE F,
are nondecreasing functions of yi and Fi(0) = 0. Then URFLPMFP can be modeled as
follows.
The uncertainties can he generally classified into three categories: providerside
uncertainty, receiverside uncertainty, and inhetween uncertainty. The providerside
uncertainty may capture the randomness in facility capacity and the reliability of facilities,
etc.; the receiverside uncertainty can he the randomness in demands; and the inhetween
uncertainty may be represented by the random travel time, transportation cost, etc. 1\ost
stochastic facility location models focus on the receiverside and inhetween uncertainties
([47]). The common feature of the receiverside and inhetween uncertainties is that
the uncertainty does not change the topology of the providerreceiver network once the
facilities have been built. However, this is not the case if the built facilities are subject to
fail (providerside uncertainty). If a facility fails, customers originally assigned to it have
to be reassigned to other (operational) facilities, and thus the connection cost changes
(usually increases).
We focus on the reliability issue of providerside uncertainty in this chapter. The
uncertainty is modeled using two different approaches: 1) by a set of scenarios that
specify which subset of the facilities will become nonoperational; or 2) by an individual
and independent failure probability inherent in each facility. Although each demand
point needs to be served by one operational facility only, it should be assigned to a
group of facilities that are ordered by levels: in the event of the lowest level facility
becoming nonoperational, the demand can then he served hv the next level facility that is
operational; and so on. If all operational facilities are too far away from a demand point,
one may choose not to serve this demand point by p wiing a penalty cost. The objective is
thus to minimize the facility opening cost plus the expected connection and penalty costs.
This problem will be referred to as the uncapacitated reliable facility location problem
(ITRFLP).
In particular, two variants of ITRFLP are considered in this chapter in terms
of the characteristics of the failure probability at each facility. In the first one, we
assume that there is only one sitespecific failure probability at each facility. We
Similarly, when component 5 fails, the original network in Figure 43 is reduced to
Figure A1(B), which is a seriesparallel system with a reliability of
Pr(system works  component 5 fails) = (TIT2 r3r4 T1r2r3r4). (A3)
Substitution of equations A2 and A3 into equation A1 yields the reliability of the
fivecomponent bridge network depicted in Figure 43:
R, = (rl + T3 r1T3 7r 2 + 4 r2r4 75 +71r2 + 3r4 T1r2r3r4)( 5) (A4)
Reliability of the sevenlink ARPA network. Following the notation in Example
4, we assume that each block from blocks 1 to 5 in Figure 45 represents a subsystem.
Blocks 6 and 7 are individual components. Recall that Ri = 1 (1 riefi,Vi = 1,. .,5,
Qi = 1 Ri, Vi = 1, .., 7. Reliability of the network in Figure 45 can be written based on
whether I th ir. 11. 4 is functional or not.
R, = Pr(system works  I I ti1 r ii 4 works)R4 FT SyStem works subsystem 4 fails) (1 R4)
(A5)
When subsystem 4 works, the original network in Figure 45 is reduced to Figure A2(A),
whose reliability can be obtained by applying parallel and series reductions:
Pr(system works  subsystem 4 works) = (1 Qq6)1 &71 R31 &2 5)]} (A6)
When subsystem 4 fails, the original network in Figure 45 is reduced to Figure A2(B). A
series reduction on subsystems 1 and 2 produces a supercomponent, which helps to map
the topology in Figure A2(B) to the fivecomponent bridge network in Figure 43. After
the nor lpphlr we can directly use the result of equation A4 by replacing rl with R1R2, r2
CHAPTER 1
INTRODUCTION
Our study focuses on reliability issues arising in facility location design problems and
complex systems. In the former case, we consider mathematical models that minimize
the sum of facility opening costs and expected service and penalty costs when facilities
are subject to fail from time to time. These failures may come from disruptive events
(e.g. labor strikes, supplier business failures, terrorist attacks), or natural disasters (e.g.
hurricanes, earthquake). Facili Tp~ecific failure probabilities are explicitly considered in
our models. To the best of our knowledge, these appear to be the first such models in
the literature. These models help to make decisions in the system design phase. Several
heuristics and an approximation algorithm are proposed for solving these models.
If facilities have been built but are still subject to fail, we consider models to fortify
the reliability of the existing system given limited fortification resources. These models
can be reduced to a special class of global optimization problems, called monotonic
optimization, in which both the objective function and constraints possess monotonicity.
A specialized monotonic branchreducebound algorithm is developed to efficiently solve
these problems.
We also examine reliability issues in complex industrial and military systems.
The reliability of such a system is measured by the probability of successful operation.
We address the issue of allocating unreliable components in the system to achieve
the maximum probability of successful operation, a different objective from that
used in the facility location model. The problem is generally categorized as a system
reliability optimization problem, including the classes of redundancy allocation and
reliabilityredundancy allocation optimization problems. In redundancy allocation
optimization, one is given the option to allocate the appropriate levels of redundancy
to maximize reliability or minimize the cost of a system given the design constraints.
For example, if a component of reliability level at 0.9 is assigned in parallel to backup
the component failure rate As expressed by
where asi and pi are the inherent characteristics of component i. If component i follows
the negative exponential failure law, that is, ri = exp(Ast), then the component cost is
c4(ri) = as g where t is the duration for which component i is: requnired to operate.
The coefficients of Example 3 are given in Table 42.
Table 42. Coefficients in Example 3
i asc x 10s pi I, Pi P C W t
1 2.330 1 7 1.5
2 1.450 2 8 1.5
3 0.541 3 8 1.5 110 175 200 1000
4 8.050 4 6 1.5
5 1.950 2 9 1.5
This problem is considered difficult to solve in the literature. To the best of our
knowledge, there is no exact method being applied to solve this problem. We compare
the performance of mBRB with the THK( heuristic in [55], the GAG heuristic in [15], the
K(LXZ method in [26], the surrogateconstraints algorithm HNNN in [18], and the genetic
algorithm GA in [20]. The comparison results are summarized in Table 43, where the
CPU time listed in the last column is measured in seconds. The first column lists the
names of the methods. The solutions and the obtained system reliability are listed in the
second and the third column respectively. The fourth column, R,(UB), lists the upper
bound of the system reliability.
The numbers in brackets after "mBRB" in the first column is the value of e, the
predefined optimality tolerance. As one can see, the mBRB algorithm produces higher
quality solutions with known upper bound comparing to other algorithms, which are able
to output some feasible solutions but without upper bound guarantee. With additional
The chromosome or the solution of the URFLP model is represented as a bit stream
with one position for every candidate location. We will use chromosome and solution
interchangeably. A "1" in position k is interpreted as that candidate site k is located
to open, while a "O" indicates that it is not. Since it is optimal to assign demand
nodes to facilities based on the distance between the demand node and the facility as
indicated in Proposition 2.1, we do not need an explicit encoding of the demandtofacility
assignments. For model URFLPMFP, an additional element is encoded to represent the
level at which we invest in an open facility. Table 22 shows the encoding for a system
with 10 candidate sites for model URFLPMFP, for example, with open facilities at nodes
3, 4 and 8 at investment levels 2, 1 and 3 respectively.
Table 22. Sample chromosome for model URFLPMFP
Candidate site 1 2 3 4 5 6i 7 8 9 10
Open? 0 0 1 1 0 0 0 1 0 0
Investment level 0 0 2 1 0 0 0 3 0 0
ClsInin..~ nin 4~!~ are evaluated based on the value of objective function. A chromosome
with a smaller objective value is fitter than one with a larger objective value. The
followingf parameters are emploi II in our description of heuristic GAH.
Table 23. GAH parameters
Parameter Notation
Population size NVp
Maximum number of generations Noe
Maximum number of generations without improvement 1NMr
Number of reproduction NiR
Number of immigration NI
Mutation probability P
The initial population is randomly generated. For each solution of the population,
sites are selected randomly and the solution is checked against all other solutions in the
emerging population to ensure that the solution is unique. If it is, the solution is added to
the emerging population, if it is not, the solution is rejected and a new random solution is
generated. This process stops until NVp distinct solutions are populated.
Table 26. Runs from SAAH for the 50node dataset
M = 5 M = 10
P ijmzn a gap min"" to gap Exact
0.0 7197.27 7197.27 0.00 7197.27 7197.27 0.00 7197.27
0.1 7763.80 7687.03 1.00 7763.80 7760.14 0.05 7763.80
0.2 8425.99 8315.65 1.33 8425.99 8436.50 0.12 8425.99
0.3 9414.40 9054.28 6.79 9378.06 9112.31 3.32 9275.99
0.4 10872.60 9740.45 23.03 10479.80 9814.09 10.79 10253.90
0.5 11932.00 10457.00 25.47 11932 10497.60 13.66 11603.00
0.6; 17825.90 11377.60 57.73 17335.50 11475.90 55.33 13416.80
0.7 27157.40 12758.80 114.563 23227.40 12816.20 111.90 16157. 20
0.8 34912.50 14758.30 142.19 31284.20 14761.90 136.50 21500.70
0.9 54722.30 19703.50 177.73 5 !<.li 40 19428.60 181.66 35987. 70
1.0 128009.00 128009.00 0.00 128009.00 128009.00 0.00 128009.00
M~ = 15 M~ = 20
P Omin"" is gap Omin v20 gap Exact
0.0 7197.27 7197.27 0.00 7197.27 7197.27 0.00 7197.27
0.1 7763.80 7784.17 0.26; 7763.80 7768.5 0.06; 7763.80
0.2 8425.99 8484.24 0.6;9 8425.99 8453.66 0.33 8425.99
0.3 9378.06 9138.80 2.6;2 9275.99 9152.36 2.47 9275.99
0.4 10479.80 9826.75 6.65 10259.90 9842.41 6.48 10253.90
0.5 11932.00 10507.40 13.56 11932.00 10530.80 13.31 11603.00
0.6 17335.50 11523.50 50.44 17291.50 11563.00 49.92 13416.80
0.7 22894.20 12766.10 81.95 22894.20 12747. 70 79.59 16157. 20
0.8 31284.20 14736.80 112.29 31284.20 14746.00 112.15 21500.70
0.9 5 !<.2 40 19752.00 176.57 53343.90 19811.80 175.74 35987. 70
1.0 128009.00 128009.00 0.00 128009.00 128009.00 0.00 128009.00
much less time. The greedy adding algorithm seems to perform better when the facility
failure probability is high. It actually finds optimal solutions when the failure probability
exceeds 0.5. This is in contrast to the performance of the SAAH, which works better
when the failure probability is low.
As we pointed out in Section 2.7.2, the solution quality of GADHI can be further
improved by GADSHI. This is clearly demonstrated in the following computational results.
GADSHI actually finds the optimal solutions for all instances in Table 27 and the CPU
times are comparable with those reported by GADHI. The results are summarized in
Table 28.
In formulation (URFLPSFP), Pro~position 2.1 ensures that the level assignments can he
easily derived for a given set of open facilities. Therefore, one can concentrate on selecting
a set of open facilities without worrying too much on the decisions of level assignment. Let
t'(T) denote the objective function value given by the set of open facilities, T. Let Tt be
the set of open facilities at step t, and # he the empty set.
The Greedy Adding Heuristic
* Step 1: Initially the set of open facilities is empty. Set t = 0 and T' = 0.
* Step 2: Ch....~ a facility front the remaining candidates to open such that it can
reduce the total cost the most. Add this facility to the facility set. That is,
t= t+1,
T' = Ttl U {.t)}.
* Step 3: Repeat Step 2 until the current solution cannot he improved further.
In general, as we can see front the computational tests later, the greedy adding
heuristic is able to find a high quality solution very efficiently. The complexity of this
heuristic is O(n4 log n), where n = F. Given T'l, it takes O(n log n) to do the level
assignments for each node, mainly because it involves a sort process that is in complexity
of O(n log n). There is n such nodes, so it takes O(n2 log n) to evaluate the value of
t'(Ttl). In the worst case, it takes n such evaluations to get the most cost effective
facility, .), at step t. The greedy adding process iterates at most n times, which leads to
the complexity of O(n4 log n) for GADH.
After the greedy adding heuristic, we perform the following greedy substitution
heuristic to further improve the solution: at each iteration, a substitute facility is chosen
to replace the existing open facility if doing so reduces the total cost the most. This
procedure is repeated until no substitute facility can he found to further reduce the
total cost. The substitution can he a null facility. Replacing an open facility with a null
CHAPTER 6
CONCLUDING RE1\ARK(S
In this chapter, we summarize the various models and algorithms discussed
throughout the dissertation, and point out directions for future research.
We study the impact of uncertainty on the decisions of facility location and demand
assignment. The uncertainty is represented by the failure probability in each facility.
Several novel models have been presented to offer solutions for both the design of initial
supply chain systems and the improvement of the existing systems. We first investigate
the uncapacitated reliable facility location model, whose objective is to minimize the total
of opening cost, expected service cost, and expected failtoserve penalty cost when each
facility has a sitespecific failure probability. We also study a more general case that each
facility has multiple levels of failure probabilities that can he chosen. If the supply chain
system already exists, we propose two models for optimally allocating the fortification
resource to reduce the expected service and failtoserve penalty cost.
The algorithms presented in this dissertation include (1) four heuristics, the sample
average approximation heuristic, the greedy adding heuristic, the greedy adding and
substitution heuristic, and the genetic algorithm hased heuristic; (2) the approximation
algorithm with a worstcase bound of 2.674; (3) the monotonic branchreducebound
algorithm. An indepth theoretic treatment is provided for the approximation algorithm.
All other algorithms are thoroughly tested in the computational studies. The four
heuristics are used to solve the uncapacitated reliable facility location problem. The
monotonic branchreducebound algorithm is applied to solve the facility fortification
problem as well as the system reliability problem arising from industrial or military
appli cations.
One immediate extension is to study the capacitated version of the current reliable
facility location models. Although the capacitated constraints generally pose more
challenges on findings the efficient algorithms, we expect that some of the heuristics are still
Table 53. Solutions of the 3level model
Constraint Resource Fortification Level at CPUJ
R Used Objective 2 5 15 18 20 second
25000
2952
2857
2400
2305
2097
2002
1824
1710
16388
1593
1550
1415
1369
1274
1096
1095
1000
960
8635
822
687
665
487
408
391
351
256
213
78
2953
2858
2401
2306
2098
2003
1825
1711
1689
1594
1551
1416
1370
1275
1097
1096
1001
961
866
823
6388
666
488
409
392
352
257
214
79
0
2020.69
2030.01
2057.66
206;8.99
2087. 25
2100.18
2138.98
2276.06
2392.85
2408.41
2415.59
2455.11
2489.01
2516.42
2598.63
2797.01
2830.34
2841.01
2875.19
2930.35
2977.74
3308.47
3503.25
3894.44
4318.51
4331.16
4462.80
110; 43
4857. 74
5670.71
0.11
0.89
1.08
1.19
1.48
1.03
1.80
1.94
2.72
2.58
2.72
3.33
3.05
2.88
3.70
3.34
3.64
3.95
3.17
3.48
3.48
2.27
2.56
2.56
2.58
2.86;
2.09
2.09
1.97
1.64
In Figure 54, we plot the objective values in these two different settings (3level
constraint and 2level constraint) across the resource used. Unlike the result in the
continuous facility fortification model, as shown in Figure 51, there is no piecewise
linear property exhibited in this discrete version. Instead the curves are shown steeper
in the earlier stage. That is, the fortification efforts help to reduce a lot of total cost at
earlier stage. This indicates that reliability can be drastically improved without large
call it the uncapacitated reliable facility location problem with a singlelevel failure
probability(URFLPSFP). In the other variant, we assume that there are multiple levels
of failure probabilities that can he chosen at each facility. We call it the uncapacitated
reliable facility location problem with multilevel failure probabilities(URFLP1\FP).
Both of them can he modeled hv a scenariobased stochastic programming approach and a
nonlinear integer programming approach.
ITRFLP is clearly NPhard as it generalizes ITFLP. We propose several heuristics
to solve ITRFLP. They include the sample average approximation heuristic for the
scenariobased model, the greedy adding heuristics, the greedy adding and substitution
heuristics, and the genetic algorithm for the nonlinear integer programming model.
The rest of this chapter is organized as follows. In Section 2.2, we review the related
literature and provide some basic background for our models. The notation and acronyms
are introduced in Section 2.:3. In Section 2.4, a scenariobased model is proposed, which
is followed by the nonlinear integer model for ITRFLPSPF in Section 2.5. Section 2.6
contains the nonlinear integer model for ITRFLPSPF. The three heuristics are presented
in Section 2.7. In Section 2.8, we conduct computational studies on the performance of the
heuristics. In Section 2.9, we conclude the chapter by II r; h.; several future research
directions.
2.2 Literature Review
The importance of uncertainty in decision making has promoted a number of
researchers to address stochastic facility location models (e.g., [:38, 47]). However, as
we pointed out in the Introduction, a 1 in ~dl~y of the current literature mainly deals with
the receiverside and/or inhetween uncertainties. This includes [63], [10], [9], [7] and [42]
among others.
The following two papers, [48] and [5], are closely related to this chapter. In [48],
the authors assume that some facilities are perfectly reliable while others are subject to
failure with the same probability. On the contrary, we assume that the failure probability
as in parent 1; and value of positions 4 to 10 are from those in the same positions as in
parent 2.
Candidate site 1 2 3 4 5 6 7 8 9 10
Locate? 0 0O 1 1 0 0 0 O 1 0 0
Parent 1
Investment level 0 0 2 1 0 0 0 3 0 0
Candidate site 1 2 3 4 5 6 7 8 9 10
Crossover 4 Locate? 0 0 1 0 1 1 0 1 1 0 Child
Investment level 0 0 2 0 1 1 0 2 1 0
Candidate site 1 2 3 4 5 6 7 8 9 10
Locate? 1 0 0 0 1 1 0 1 1 0 Parent 2
Investment level 3 0 0 0 1 1 0 2 1 0
Figure 21. Example of crossover operation at position 3
After a child solution has been constructed in the manner outlined above, with
probability Pn/r, the solution is mutated. 1\utation is accomplished by randomly selecting
two candidate sites: one at which a facility opens and one at which a facility closes; then
swapping their states: from open ("1") to close ("O"), and from close ("O") to open ("1").
In the case of model URFLPSFP, a randomly selected investment level is associated with
the newly open facility site.
If the child solution generated in this manner differs from all other solutions in the
emerging population, it is added to the population; if it does not, the entire process (of
parent selection, crossover, and mutation) is repeated. We continue adding solutions to the
population until the population contains NVp total solutions. In other words, the size of
each generation is maintained to be the same.
The whole process is repeated until one of the following termination criteria is met:
(1) the algorithm reaches NoG generations, or (2) it fails to improve the bestknown
solution in 1Nz; generations.
Table 21:3. Computational results for ITRFLPMFP using GADSH and GA
GA, 5 Trials
Dataset Best Worst Average
# Best Sites (Levels) Result Result Time (s)
20 2(2) 7(2) 5214.55 5214.55 :3.1:3
:30 2(2) 20(1) 21(2) 6484.74 6484.74 :3.7:3
40 2(2) 20(1) :35(2) 7194.88 7194.88 4.52
50 2(2) 5(1) 20(1) :35(1) 8827.95 8827.95 5.71
6;0 2(2) 20(1) :35(2) 59(1) 9964.84 9964.84 7.20
70 2(2) 1:3(2) 20(1) :35(2) 108:37.65 108:37.65 9.06;
80 2(2) 20(1) :35(2) 59(1) 76(1) 79(1) 11782.85 11867.01 10.81
90 2(2) 20(1) :35(2) 76(1) 79(1) 88(1) 126321.95 12717.84 11.1:3
100 2(2) 1:3(2) 20(1) :35(2) 67(2) 76(1) 88(1) 1:371:3.42 1:3749.58 11.3:3
GADSH
Sites (Levels) Result Time (s)
20 2(2) 7(2) 5214.55 0.09
:30 2(2) 20(1) 21(2) 6484.74 0.14
40 2(2) 20(1) :35(2) 7194.88 0.18
50 2(2) 5(1) 20(1) :35(1) 8827.95 0.27
6;0 2(2) 20(1) :35(2) 59(1) 9964.84 0.3:3
70 2(2) 1:3(2) 20(1) :35(2) 108:37.65 0.39
80 2(2) 1:3(2) 20(1) :35(2) 76(1) 11859.40 0.57
90 2(2) 20(1) :35(2) 76(1) 79(1) 88(1) 12621.95 1.19
100 2(2) 1:3(2) 20(1) :35(2) 67(2) 76(1) 88(1) 1:371:3.42 1.1:3
2.9 Conclusions
In this chapter, we have proposed several novel facility location models to deal with
the uncertainty in facilities. The issue arises when a facility fails is that the customers
originally assigned to it have to be reassigned to other facilities that are operational. The
impact of such uncertainty is explicitly modeled in all of our models in order to build a
reliable facilitycustonler network. In particular, we use a popular scenariobased technique
to capture the uncertainty when the number of scenarios is relatively small. If the failure
probability at each facility is independent, we propose several nonlinear integer models,
ITRFLPSFP and ITRFLPMFP. These models greatly enrich the literature of facility
reliability.
Four heuristics, SAAH, GADH, GADSH and GA, have been proposed to solve
these problems. SAAH is a specialized heuristic for the scenariobased model, whereas
It only takes the mBRB algorithm 15 milliseconds to obtain the proved optimal
solution (3, 2, 2, 3, 3) with the corresponding system reliability at 0.9044673. It is superior
to many solution techniques collected in [28].
Example 2: Fourstage series system with 2outofn: G configuration.
In this example, stage 1 does not allow for component redundancy, but its reliability is
determined by choosing a component from a pool of six different components at that
stage. In other words, the reliability levels at stage 1 are discrete. The reliabilities of
stages 2 and 4 can only be enhanced by providing redundancy. However, stage 3 has a
special configuration called 2outof n: G configuration, which works (or is t;ood") if and
only if at least 2 of the a components work (or are good). Based on this definition, the
system reliability of stage 3 is shown in the expression of R3 X3
max Rs e4 (425)
i= 1
subject to gl =~ 10 exp +lx)~ 10Z2i + 3+154< 10. (426)
g2 xl = 10ex 4 exp(x2) + 2 3+ exp +6x< 0, 47
g3 =40x 6 exp~(Z2n 31. ex + x 70 (428)
xi: integer, Vi = 1, 2, 3, 4, (429)
where stage reliabilities are
Rl(xl) = 0.94, 0.95, 0.96, 0.965, 0.97, 0.975, for xi = 1, 2, .. ,6, respectively,
R2 2a) = 1 (1 0.75)""
R3 3) =(0.90)k (1 0.90)"3k"
R4 4q) = 1 (1 0.95)24
It takes the mBRB algorithm 27 iterations to produce and verify the optimal solution
(3, 3, 5, 3) with its objective value of 0.9444472 at no CPU time, or more precisely, less
Phase II: Solving FaultTolerant Problem
For ea~ch j eD and i e F, let = 1 and 2$ = 0 for all k: > mi, anld let = 0 for
all k < mj, where mj is defined in Phase I of the algorithm.
Use the algorithm(s) in [17] with a parameter as (0, 1] to round the solution
(x, y) to a feasible solution of a faulttolerant facility location problem, where a set
of facilities is open such that each client j is served by at least mj distinct open
facilities.
For each i E F, set yi = 1 if facility i is open, and set yi = 0 otherwise.
For ea~ch je D,~ if i: is the k"" closest open facility to client j, then let ify = 1, where
Output the solution (x, y, 8).
This twophase algorithm shall be referred to as Algorithm TP. It is obvious
that the solution (x, y, 8) is feasible to formulation (37). We now establish a worst
case approximation bound of our (randomized) algorithm, i.e., we shall show that the
(expected) total cost is no more than a constant factor times the optimal cost. We bound
the total penalty cost in Lemma 3.3, and bound the total facility and transportation costs
in Lemma 3.4.
Lemma 3.3. In Algorithm TP, the expected total I.. ..rl;ll cost is bounded from above by
Lemma 3.4. For it.;;, as (0,1i), the expected f~r ..1.:1/ cost plus the expected transportation
cost is no more than
In T ~In ~ L 3
where FLP and CLP are the total f 7.:.7//// cost and the total transportation cost, ,* i'' 1.: I;,
corresponding to the solution (x, y, 0) in the linear; r,.ye~l,,I,,:nt:1 relaxation of formulation
(87).
Table 43. Performance comparison of Example 3
Method (x, r) R, R,(UB) CPU (s)
mBRB (0.01) (3, 2, 2, 3, 3, 0.77500, 0.87500, 0.930947 0.940913 1.38
I ;' 12 ".11, 0.71500, 0.79000)
mBRB (0.001) (3, 2, 2, 3, 3, 0.78250, 0.87500, 0.931541" 0.939004 4.76
0.89938, 0.70750, 0.79000)
THK( (3, 3, 2, 2, 3, 0.78438, 0.82500, 0.915363
0.90000, 0.77500, 0.77813)
GAG (3, 2, 2, 3, 3, 0.80000, 0.86250, 0.930289
0.90156, 0.70000, 0.80000)
K(LXZ (3, 3, 2, 3, 2, 0.77960, 0.80065, 0.929750
0.90227, 0.71044, O ;;' s 7)
HNNN (3, 2, 2, 3, 3, 0.77489, 0.87007, 0.931451
0.89855, 0.71652, 0.79137)
GA (3, 2, 2, 3, 3, 0.77943, 0.51.i 18, 0.931578b
0.90267, 0.71404, 0.78689)
Terminated at iteration 2000.
b Best solution obtained in multiple trials.
CPU time, the mBRB algorithm can produce a solution of
(3, 2, 2, 3, 3, 0.77781, 0.87187, 0.90281, 0.71313, 0.751.2',)
with the system reliability at 0.931669, and its upper bound of 0.933111. This result can
serve as a benchmark for this example.
Example 4: Sevenlink ARPA network. This problem is to maximize the
reliability of the following sevenlink ARPA network (see [35], [51]). The derivation of the
reliability of this network, i.e. the objective function, is provided in Appendix A.
Figure 45. Sevenlink ARPA network
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CHAPTER :3
ITNIFORM INCAPACITATED RELIABLE FACILITY LOCATION PROBLEM: A
2.674APPROXIMATION ALGORITHM
3.1 Introduction
In this chapter, we consider a special case of the uncapacitated reliable facility
location problem in which failure probabilities are not facili' p;lecific. We call it the
uniform uncapacitated reliable facility location problem (ITIRFLP). Because the failure
probabilities are the same across all facilities, ITRFLP is reduced to a linear integer
progranining problem. ITIRFLP is related to the model considered in [48], where the
authors assume that some facilities are perfectly reliable while others are subject to failure
with the same probability. They formulate their nmultiobjective problem as a linear
integer program and propose a Lagrangian relaxation solution method. However, we
consider penalty cost, a factor that is missed in [48].
ITIRFLP is clearly NPhard as it generalizes ITFLP. The focus of this chapter is
to propose and analyze an approximation algorithm with a constant worstcase bound
guarantee.
Designing approximation algorithms for the facility location problem and its
variations has recently received considerable attentions front the research coninunity.
However, to the best of our knowledge, this chapter presents the first approximation
algorithm for stochastic facility location problems with providerside uncertainty.
The vast ill I iG~~y of approximation algorithms for the facility location problem
mainly deal with deterministic problems, e.g. [17, 21, :32, 46]. Until very recently,
approximation algorithms for ITFLP with stochastic demand have been proposed; see the
survey by Shnioys and Swanly [52]. Another related paper [6] proposes an approximation
algorithm for a facility location problem with stochastic demand and inventory. Our
approximation algorithm makes use of the ideas from several papers [17, 21, :32, 46]. In
particular, this chapter is closely related to [17], which presents a 2.41approxiniation
algorithm for the socalled faulttolerant facility location problem (FTFLP): every demand
5.2 Continuous Facility Fortification Model
We first introduce some common notations that will be used throughout the chapter.
Let D denote the set of clients or demand points and F denote the set of facilities. Let fi
be the facility cost to open facility i, dj be the demand of client j, and csj be the service
cost if j is serviced by facility i. For each client j e D, if it is not served by any open and
operational facility, then a penalty cost rj will be incurred.
In the reliable facility location model setting, each client is assigned to a set of backup
facilities, which is differentiated by the levels: in case of a lower level facility fails, the next
level facility, if functional, will ba~ck it up?. x@ = 1 if facility i is the k~th? level ba~ckup
facility of demand node j and afy = otherwise, z" 1 if j has (k 1)th backup facility,
but has no kth backup facility so that j incurs a penalty cost at level k. Contrary to the
models in C'!s Ilter 2, xfy and (j are not decision variables anymore. They are used to
specify the existing network.
To compute the expected failure cost, we follow the logic in C'!s Ilter 2. First of
all, we need to compute the expected failure cost at level k served by facility i. Each
demand node j is served by its levelk facility if all the lower level facilities become
nonoperational. For any facility 1, if it is assigned to a lower level (i.e, less than k) for
dem~anld node j, thecn CE:r = 1, otherwise it, is zero. So th~e probabhility th~at a~ll lower
levels facilities fail is niev a And j is served at levelk by i, which has to be
operational. The probability is (1 p ). Therefore, the expected failure cost at level k
served. by facility i is dcqiif(1 i p lF~ ) zr '. Similarly, we can calculate the penalty
cost at level k, which is nev pt ir
We can now formulate the continuous facility fortification model as follows:
continued from previous page
#i x y di ri fl p1 f2 p2 f3 p3
57 0.811 0.86 212 9.92 12711 0.58 14463 0.4 1489 0.38
58 0.53 0.36 903 4.06 8633 0.75 1078 0.29 1354 0.05
59 0.89 0.58 594 1.64 521 0.64 1071 0.44 1100 0.15
6;0 0.87 0.56 250 5.11 865 0.95 1273 0.6 1348 0.21
472
244
353
183
749
200
321
650
946
143
565
11
374
284
598
720
457
213
550
418
863
3638
282
811
86;6
895
959
2.64
6.05
4.27
6.18
9.13
6;.6;3
7.02
8.73
3.92
8.16
1.25
4.44
2.65
1.45
9.63
3.72
2.38
7.09
8.3
2.55
1.53
3.5
7.15
9.07
2.46
3.19
5.52
974
689
745
585
782
561
569
880
1190
513
601
904
691
522
609
541
986
6;6;6
564
588
1147
510
598
674
568
665
980
0.43
0.58
0.78
0.78
0.6
0.42
0.5
0.26
0.58
0.48
0.84
0.32
0.33
0.6;8
0.22
0.14
0.43
0.6;6
0.43
0.57
0.77
0.52
0.73
0.4
0.37
0.44
0.68
1464
810
1034
1259
1392
1068
713
1424
1332
1355
1117
982
1291
1246
1026
1012
1493
861
1452
758
1310
1070
1180
1166
1192
1165
1412
0.91
0.32
0.37
0.38
0.96
0.15
0.15
0.99
0.47
0.84
0.71
0.46
0.09
0.71
0.27
0.25
0.57
0.96
0.83
0.72
0.69
0.22
0.37
0.36
0.11
0.77
0.16
0.16
0.15
0.37
0.73
0.34
0.76
0.48
0
0.28
0.16
0.9
0.86
0.74
0.78
0.04
0.07
0.18
0.49
0.21
0.49
0.5
0.89
0.88
0.82
0.1
0.6;9
0.09
0.92
0.88
0.89
0.79
0.84
0.93
0.68
0.79
0.64
0.79
1
0.53
0.48
0.94
0.85
0.25
0.75
0.77
0.74
0.78
0.93
0.65
0.95
0.58
0.63
0.93
0.92
1220
730
821
1026
1219
985
6;6;2
904
1242
993
945
928
1028
818
771
704
1307
736
796
619
1299
976
871
1045
1021
1015
1250
0.22
0.09
0.46
0.23
0.13
0.22
0.1
0.16
0.52
0.14
0.13
0.26
0.32
0.35
0.11
0.09
0.19
0.49
0.15
0.39
0.71
0.39
0.07
0.3
0.24
0.25
0.66
algforithm, if the algforithm produces a solution with cost no more than
RfF* + RcC* + R,P*,
where F*, C*, and P* are the optimal facility, transportation, and penalty cost, respectively.
We take advantage of several results for the faulttolerant version of UFLP, where
every demand point j must be served by kg distinct facilities, a concept close to our level
assignment. In [17], Guha et al. propose a couple of approximation algorithms for the
faulttolerant facility location problem using various rounding and greedy localsearch
techniques.
The faulttolerant facility location problem can be formulated as the following integer
program.
(FTFLP)
miniize eys dycyw x(3 8)
i6EF j6D k=1 iEF
subject to xfy~ > 1, Vj' e: D, < kg
i6EF
xfy, yi E {0, 1}.
Th~e notation? here follows that of model U!R~FLPIP with som~e subtle differece~cs. wf~ is the
weighted factor at level k for demand node j. For demand j, the corresponding weights
are assumed to be w] > 2>..>w" f denotes that demand j is assigned to facility i
and facility i is the kth closest open facility to j.
One of the key results on FTFLP from Guha et al. [17] is summarized below.
Lemma 3.1. For r,:; vector (x, y) //rlifying the following inequalities (the dimension of
(x, y) should be clear from the inequalities)
xE>1 VjDs
i6EF
2. Facility i" is assigned to it. And the corresponding service cost is
car,(1 p) pl (510)
16F
Adding terms 58 and 510, we have
16F 16F
From Proposition 2.1, we know that ci/ c ,j > 0, so the only negative term left is
yer.pl (512)
16F
This term can be absorbed in the next level assignment following a similar analysis at level
k. By doing this process recursively, demand j will be eventually subject to penalty cost
at certain level higher than k + 1 and only case 1 can happen. Therefore, all cost terms
related to demand node j have the form of n, np, This completes the proof. O
i= 1
As a direct result from Proposition 5.1, the following Corollary holds.
Corollary 5.1. The objective function 51 of the continuous f 7. .:1.:1; fori;:p,,,7;:.w model is
monotor...: a ll;i nondecreasing.
It is obvious that Function 56 is a special case of Function 54: If pij = 0 in
function 56, then set cij = 0 and dij = c4, If Pij = 1 then set cij = agje", e" being
the ith unit vector in RW" and dij = 0. Therefore, the continuous facility fortification
problem is a special form of the generalized linear multiplicative programming. The
latter is multiextremal and possesses several local minima [41]. The existing algorithms
for GLMP include outerapproximation methods [25], vertex enumeration methods [19],
heuristics methods [31], among others. Corollary 5.1 also allows us to apply the monotonic
branchreducebound algorithm presented in (I Ilpter 4 when the resource constraints
possess monotonicity as well.
To my parents,
and my brother, Lep~ing,
for their love and support
in complex systems, including the redundancy allocation optimization problem and
the reliabilityredundancy allocation optimization problem. Compared to the existing
techniques, the nionotonic branchreducebound algorithm is not only versatile but also
very efficient in dealing with different types of problems in system reliability.
We also develop several models that can he used to fortify the reliability of the
existing facilities. They are the extensions to the models in the first part of the dissertation
and offer insights on which facility to choose and to what extent it should be fortified. The
properties and solution methodologies of the models are discussed. In particular, a
nionotonic branchreducebound algorithm is used to solve one of these models. The
efficiency of the algorithm is demonstrated in the computational results.
:3.5 Proofs ......... ... . 56
4 SYSTEM RELIABILITY OPTIMIZATION AND MONOTONIC OPTIMIZATION 61
4.1 Introduction ....... ... ......
4.2 A Monotonic BranchReduceBound Algorithm ..........
4.2.1 Select and Branch ...........
4.2.2 Reduce and Bound .........
4.2.3 Convergence Analysis ..........
4.2.4 Acceleration Techniques ...... ... ....
4.3 Using Monotonic BranchReduceBound Algorithm to Solve System Reliabi
Optimization Problems ....... .....
4.3.1 Redundancy Allocation Optimization .........
4.3.2 ReliabilityRedundancy Allocation Optimization ..........
4.4 Conclusions .........
61
67
6;9
70
71
72
lity
7:3
7:3
76
79
5 FORTIFYING THE RELIABILITY OF EXISTING FACILITIES AND MONOTONIC
OPTIMIZATION ......... . .. .. 81
5.1 Introduction ......... . .. .. 81
5.2 Continuous Facility Fortification Model ... .. . .. 8:3
5.2.1 Properties of the Continuous Facility Fortification Model .. .. 84
5.2.2 An Example of the Continuous Facility Fortification Model .. .. 87
5.3 Discrete Facility Fortification Model ..... ... .. 92
5.3.1 Properties and Algorithms ...... ... . 9:3
5.3.2 Computational Experiments ...... ... .. 96
5.4 Conclusions ......... .. .. 99
6 CONCLUDING REMARK(S ......... .. .. 102
APPENDIX
A SYSTEM RELIABILITY COMPUTATION IN CHAPTER 4 .
104
107
B DATASET ITSED IN CHAPTER 2 .
C DATASET ITSED IN CHAPTER 5.
REFERENCES ........
BIOGRAPHICAL SKETCH ....
As in CFFM, the objective of DFFM is to minimize the total expected transport and
fa~iluretoserve cost. Nota~tion xi~ anld zf = 1 keep th~e samec meaning a~s in? CFFM': x@ = 1
if facility i is the kth level backup facility of demand node j and afy 0 otherwise,
zf = 1 if j h~as (k )th backup facility, but h~as n~o kth backup facility so that j in~curs a.
penalty cost at level k. The cost terms are calculated in parallel to CFFM as well, which
are shown in the following formulation.
(DFFM~)
minmie dces ( Pi(y ))n Pl(,,)C"
j6D k=1 iEF 16F
j6D k=1 16F
subject to y)
i6EF
1 < Yi < Ui, Vi E F (519)
yi: integer, Vi E F. (520)
The objective function (517) is the sum of the expected failure cost and the expected
penalty cost. Constraint (518) denotes the resource restrictions on the fortification.
Constraints (519) and (520) are integral constraints on the fortification levels.
5.3.1 Properties and Algorithms
In this section, we show that DFFM can be solved via the monotonic branchreducebound
algorithm. It is obvious that constraint (518) possesses the monotonicity, since I(yi),
i E F, are nondecreasing functions of yi. The objective function of the continuous facility
fortification model is shown to be monotonically nondecreasing in Corollary 5.1. A similar
result holds in the discrete case, becauseir Ps/ys), ie~ Faennncesn uctoso
From the fact that the composite of a nondecreasing function and a nonincreasing function
is noninl i. I; to we have the following corollary, which is parallel to Corollary 5.1.
Table 54. Solutions of the 2level model
Constraint Resource Fortification Level at CPUJ
R Used Objective 2 5 15 18 20 second
25000 1309 3315.22 2 2 2 2 2 0.11
1308 1174 3402.08 2 2 2 1 2 0.88
1173 1006 3461.27 2 2 1 2 2 1.03
1005 871 3557. 78 2 2 1 1 2 1.19
870 828 3666.45 2 2 1 2 1 1.02
827 693 3776.24 2 2 1 1 1 1.19
6;92 560 4193.94 2 1 2 1 2 1.50
559 392 4318.51 2 1 1 2 2 1.19
391 257 4462.80 2 1 1 1 2 1.34
2563 214 110# 43 2 1 1 2 1 1.33
213 79 4857. 74 2 1 1 1 1 1.19
78 0 5670.71 111 1 1 1 1 1.06
increases in fortification resource. After that, the whole system seems lacking the room for
improvement. In fact, when R is greater than 1500, the decrease in the total cost can not
justify the fortification resource. For example, when R increases from 1551 to 2003, the
total cost reduced from 2415.59 to 2100.18, a net loss of 136.59.
6000
5500
5000
4500
4000
3500
3000
2500
2000
1500
1000
0 500 1000 1500 2000 2500
Resource Used
3000
Figure 54. Tradeoff between objective and resource used
continued from previous page
#i x y di ri fl p1 f2 p2 f3 p3
0.84
0.49
0.56
0.38
0.43
0.61
0.48
0.81
0.13
0.41
0.72
0.3
0.02
0.62
0.8
0.1
0.15
0.48
0.83
0.82
0.67
0.03
0.26
0.59
0.22
0.63
0.53
0.89
0.02
0.51
0.53
0.73
0.24
0.77
0.05
0.22
0.73
0.88
0.75
0.71
0.34
0.14
0.28
1
0.9
0.06
0.98
0.13
0.44
0.22
0.39
0.53
0.73
0.39
0.56
0.66
0.04
0.32
0.99
0.19
0.32
0.06
659
986
486
915
282
310
980
134
20
151
615
36;9
875
73
776
342
929
445
6384
6343
771
181
926
733
326
486
548
870
335
446
198
5.05
8.81
3
6.94
0.88
2.6;6
3.32
0.66
3.08
2.64
3.84
1.74
1.14
2.76
3.36
5.61
9.77
4.32
9.61
1.45
7.46
1.03
7.6
0.15
5.93
3.14
0.31
0.66
0.22
3.93
3.77
764
910
928
1040
771
605
1106
1073
905
651
800
723
890
1104
510
536
1022
594
1130
518
892
1055
691
1001
699
971
853
754
611
1217
742
0.99
0.94
0.81
0.81
0.76
0.81
0.86
0.88
0.79
0.87
0.98
0.7
0.81
0.88
0.91
0.63
0.73
0.83
0.68
0.71
0.74
0.97
0.56
0.72
0.98
0.68
0.99
0.84
0.97
0.91
0.69
941
1005
1079
1065
1028
748
1304
1153
1107
652
1298
1351
1029
1289
981
1157
1064
1289
1192
677
1334
1156
865
1058
1244
1154
1023
787
734
1249
1025
0.2
0.25
0.72
0.4
0.6;8
0.41
0.21
0.34
0.62
0.25
0.57
0.51
0.04
0.42
0.69
0.06
0.67
0.76
0.28
0.38
0.72
0.6;3
0.26
0.61
0.42
0.63
0.94
0.76
0.83
0.88
0.49
1035
1058
1121
1108
1494
1381
1400
1315
1260
1066
1423
1479
1278
1485
1341
1351
1157
1386
1466
1072
1483
1445
1312
1450
1472
1493
1207
1271
1214
1367
1371
0.04
0.23
0.35
0.16
0.04
0.39
0.06
0.28
0.62
0.01
0.35
0.22
0
0.37
0.59
0
0.33
0.43
0.09
0.28
0.45
0.35
0.23
0.08
0.04
0.49
0.28
0.61
0.51
0.05
0.31
CHAPTER 5,
FORTIFYING THE RELIABILITY OF EXISTING FACILITIES AND MONOTONIC
OPTIMIZATION
5.1 Introduction
In C'!s Ilter 2, we consider the impact of unreliability front the facilities when the
system is initially designed. In that case, we are given the option to build facilities from
scratch. However, redesigning an entire system is not ahrl . an available option given
the potentially large expenses involved in closing existing facilities or opening new ones.
In many situations, methods for protecting existing infrastructure may be preferable
given limited resources available. In this chapter, we address the issues on fortifying the
reliability of existing facilities.
Am il I iG~~y of research on reliable supply chain design has been focused on the initial
system design, but not on how to improve the existing system. These works have been
well documented in [47] and surveyed in C'!s Ilter 2 as well. However, reinforcing the
components of an existing system may become more valuable and realistic considering
the increased potential disruptions and uncertainties. These disruptions and uncertainties
may evolve front the new challenges that were never faced when the initial systems were
designed. They can he niannmade disruptive events or natural disasters, for example, the
September 11, 2001 terrorist attack and Hurricane K~atrina in 2005.
Only a small strand of literature has been devoted to addressing the fortification of
existing facilities, which includes [4:3], [44], and [49]. These works typically focus on the
interdictionfortification framework hased upon the pniedian facility location problem.
The problems are generally formulated in the form of bilevel progranining. These models
can help to identify the critical facilities to protect under the events of disruption.
Another related strand of literature is on the network interdiction problems that are
mainly developed for military applications, e.g. [12], [:33], [62] and [:30]. These models
study the impact of losing one or more transportation links or network arcs based on the
(i) f therep exi;sts k such thatS 0 < Of < 1, then Of' = 0 anrd Of = 1 for k' < k and
k" > k.
Proof. The proof is intuitive and similar to the proof of theorem 3.1, which is thus omitted
here. O
We present our rounding procedure next. For each j E D, assume kj is the smallest
inlteger suchl that H," > 0.
The rounding procedure is carried out in two phases. The first phase rounds the
optimal fractional solution (x, y, 8) to another fractional solution (, y, 8), which is feasible
to a linear programming relaxation of an appropriately defined faulttolerant facility
location problem. In the second phase, we use an algorithm for the faulttolerant facility
location problem to round the fractional solution (i, y, 8) to an integer solution (x, y, 8),
which is feasible to formulation (37).
Phase I: Decomposition
For every j eD and i e F, let Of = Of and ifykj = xfy for all k > 1 except for k = ky.
C!. ..  a parameter 6 E (0, 1] whose value will be fixed later.
Randomly generate a variable P that is uniformly distributed in [0, 6]. For each
.j EZanld i EF, if B*'> 8,Ithensetl
0" = my = k 1, & f= 0
otherwise set
yi = max ~x, .
26D
MODELS AND ALGORITHMS FOR RELIABLE FACILITY LOCATION PROBLEMS
AND SYSTEM RELIABILITY OPTIMIZATION
By
ROGER LEZHOU ZHAN
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
2007
For instance, in the above series system, each component i can be enhanced to a
Ishirh II. i with xi identical components in parallel. In this chapter, we use subsystem
and stage interchangeably. The reliability of such subsystem i is denoted as Ri. And
because only one component is required to make the system perform properly and all
components are independent, Ri = 1 (1 ri)"i. Therefore the improved system reliability
R, is the following.
which is increasing in each xi. The improved system is a parallelseries one as shown in
Figure 42.
Stage 1
Figure 42. Parallelseries system
Stage i
The system reliability can be maximized by choosing the right combination of xi
under certain resource constraints, denoted by gj(), j = 1,..., m in this chapter. This
leads to a redundancy allocation optimization problem (RAOP) or a nonlinear integer
programming in general.
(RA OP): max R, = f(xl,..., x,)
subject to gyxx..,,
0 < le < Xi
xi: integfer, Vi=1..., ,
(41)
(42)
(43
(44)
Stage n
continued from previous page
#i :r y di fi ri pi II#i :r y di fi ri pi
22 0.77 0.44 919 561 1.49 0.38 72 0.46 0.86 11 928 4.44 0.26
23 0.87 0.79 292 750 6.95 0.6 73 0.09 0.74 374 1028 2.635 0.33
24 0.69 0.15 48 956 2.29 0.71 1174 0.71 0.78 284 522 1.45 0.35
25 0.24 0.28 581 1456 2.9 0.3 75 0.27 0.04 598 6;09 9.6;3 0.22
26; 0.84 0.73 659 764 5.05 0.99 76 0.25 0.07 720 541 3.72 0.25
27 0.49 0.24 986; 1005 8.81 0.23 77 0.57 0.18 457 1493 2.38 0.43
28 0.56 0.77 4863 1121 3 0.81 1178 0.96 0.49 213 736 7.09 0.49
29 0.38 0.05 915 1065 6.94 0.4 79 0.83 0.21 550 1452 8.3 0.43
30 0.43 0.22 282 1494 0.88 0.68 80 0.72 0.49 418 758 2.55 0.57
31 0.61 0.73 310 605 2.6;6 0.39 81 0.6;9 0.5 86;3 1147 1.53 0.93
32 0.48 0.88 980 1304 3.32 0.86 82 0.22 0.89 36;8 976 3.5 0.52
33 0.811 0.75 134 1073 0.66 0.28 83 0.37 0.88 282 1180 7.15 0.95
34 0.13 0.71 20 1107 3.08 0.62 84 0.36 0.82 8111 1045 9.07 0.3
35 0.411 0.34 151 651 2.634 0.01 1185 0.11 0.1 1866 1021 2.463 0.37
36; 0.72 0.14 615 800 3.84 0.35 86; 0.77 0.6;9 895 1165 3.19 0.44
37 0.3 0.28 36;9 1479 1.74 0.22 87 0.16 0.09 959 1250 5.52 0.6;6
38 0.02 1 875 1278 1.14 0 88 0.75 0.6;3 375 1406 5.28 0.06;
39 0.62 0.9 73 1289 2.76 0.42 89 0.16 0.41 711 515 7.34 0.46
40 0.8 0.06 776 510 3.363 0.69 90 0.01 0.21 208 923 3.99 0.87
41 0.1 0.98 342 5363 5.611 0 91 10.51 0.76 954 1378 4.06 0.31
42 0.15 0.13 929 1022 9.77 0.73 92 0.98 0.32 843 733 7.77 0.33
43 0.48 0.44 445 594 4.32 0.83 93 0.55 0.39 905 545 0.08 0.74
44 0.83 0.22 684 1466 9.61 0.28 94 0.36; 0.6;3 729 1047 8.47 0.84
45 0.82 0.39 6343 677 1.45 0.71 1195 0.18 0.75 382 13635 63.23 0.38
46 0.67 0.53 771 1334 7.46 0.74 96 0.09 0.46 91 513 0.53 0.16
47 0.03 0.73 181 1445 1.03 0.35 97 0.18 0.67 991 1338 6.62 0.92
48 0.26 0.39 926; 691 7.6 0.26 98 0.1 0.38 644 1341 3.13 0.11
49 0.59 0.56 733 1001 0.15 0.08 99 0.25 0.6;6 539 1256 0.5 0.41
50 0.22 0.6;6 326; 1244 5.93 0.98 100 0.6;8 0.49 294 1168 7.27 0.6;6
monotonic branchreducebound algorithm. The computational results of the illustrative
example show the efficiency of the algorithm. We also analyze the tradeoff between cost
deduction and fortification effort and empirically demonstrated that the tradeoff curve is
steeper in the earlier stage, indicating major cost deduction can be achieved without large
increases in fortification resource.
The main limitation of the current models is the assumption that the facilities are
uncapacitated. Although the assumption itself is very common in the facility location
models, it may be unrealistic in practice. In the capacitated case, a 'customer' of the failed
facilities can be assigned to the next level backup facilities only if they have sufficient
capacity to satisfy the additional demands. This may make the capacitated model very
complex. We expect that the monotonic branchreducebound algorithm will still be
applicable. We believe that this is a valuable topic worthy of future investigation.
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[14] D. Goldberg (ed.), Genetic Algorithms in Search, Optimization and Machine
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optimization problem, IEEE Transactions on Reliability R29 (1980) 3638.
S30
S15
S10
2 5 15 18 20
Facility
Figure 53. Frequency of completely open facility in Table 51
5.3 Discrete Facility Fortification Model
The underlying assumptions of the continuous facility fortification model in Section
5.2 are that (1) the limited fortification resource is uniformly distributed across all the
facilities, and (2) the distributed resource is dividable. However, in many cases, the
fortification resource can only be discretely distributed. That is, the fortification resource
at a facility is categorized into different levels, which are functions of available resources.
For example, 7000 units of resource may improve the failure probability of a facility to
p = 0.4, while 8000 units can improve it to p = 0.3. But there is no amount of resource
that can improve the failure probability to a number between p = 0.4 and p = 0.3. This
inspires us to consider a discrete facility fortification model (DFFM).
In DFFM, let yi be the fortification level at facility i. Naturally, we assume yi
is a positive integer and only one fortification level is allowed at each facility. The
corresponding failure probability and the amount of resource are denoted by functions
Fi~~~~~~vi)~\Yl an V4s repciey Py),i F, are nonincreasing functions of yi, whereas
IM(yi), i E F, are nondecreasing functions of yi. This is because more fortification resource
would make a facility more reliable, and consequently a smaller failure probability. Let the
upper bound of yi be Ui and the total resource be R.
reliability optimization. It appears to be the first of its kind in the literature on reliability
optimization.
The remainder of this chapter is organized as follows. The monotonic branchreducebound
algorithm is presented in Section 4.2 with its convergence analysis. Several convergence
acceleration techniques are also discussed. The algorithm is applied to solve both the
reliability allocation and the reliabilityredundancy allocation optimization problems in
Section 4.3 with a demonstration of its efficiency. The chapter is concluded in Section 4.4.
4.2 A Monotonic BranchReduceBound Algorithm
To facilitate the presentation of the monotonic branch and bound algorithm, we
recast the original SROP, i.e. (49), to a more general vector format by denoting x=
(x1,... Xy; r,... y), in other words, x,+i = ri, Vi = 1,..., p.
(SROP'): max R, = f (x) (414)
subject to x E 0 = {x gj(x) < cy j 1.m, (415)
x E[x xU] (416)
x E 3z = {x  x E 7?", xi: integer, Vi=1 (417)
where [X x"] denotes a hyperrectangle with lowest boundary xL and greatest boundary
x"; fu~nctinon f andl gs are nondec~reasing for j =, 1. .,m. To ensure 0 is closed, we
assume gj are semicontinuous for j = 1, .. ., m.
Since no concavity assumption has been made on the objective function, multiple
locally optimal solutions may exist. However, from the monotonicity of the functions f
and gi, the following proposition can be easily derived.
Proposition 4.1. [56] The 11g l..lrl maximum of f(x) over 0 n 31 n [xL ,x"], if it exists, is
attained on its bette,~lr,/;,
The property of Proposition 4.1 is used by Tuy to develop the original Polyblock
Algorithm for monotonic optimization in [56]. The branchreducebound implementation
of the polyblock algorithm was mentioned in [57] for the continuous version of the
the original set. Typically such algorithms are simple and easy to be incorporated in more
sophisticated algorithms, such as Tabu search in [60].
We also apply a genetic algorithm (GA) based heuristic to solve the models we
develop. GAs imitate the natural selection in biological evolution. As solution techniques,
they maintain a large number of solutions, called the population, and allow each member
of the population (called a chromosome) to evolve iteratively into good ones. Some good
descriptions of GAs are provided in [11, 14, 39].
GAs have been used to solve many combinatorial optimization problems with success,
including various facility location problems, e.g. [22] and [2]. In this chapter, we design a
specialized GA for the models in which we are interested. A computational study on all
three heuristics is also provided.
2.3 Notations and Acronyms
We first introduce some common notations that will be used throughout the chapter.
Let D denote the set of clients or demand points and F denote the set of facilities. F is
the number of the facilities. Let fi be the facility cost to open facility i, dj be the demand
of client j, and cij be the service cost if j is serviced by facility i. The service costs, cij,
are assumed to form a metric, i.e., they satisfy triangle inequalities. For each client j E D,
if it is not served by any open and operational facility, then a penalty cost rj will be
incurred.
The acronyms listed in Table 21 are frequently used in this chapter.
Table 21. Acronyms
Acronym Meaning
UJFLP U~ncapacitated facility location problem
URFLPSFP Uncapacitated reliable facility location problem with a singlelevel
failure probability
UJRFLPMFP Uncapacitated reliable facility location problem with multilevel
failure probabilities
GA Genetic algorithm
SAAH Sample average approximation heuristic
GADH Greedy adding heuristic
GADSH Greedy adding and substitution heuristic
We refer the reader to Section 3.5 for proofs of both lemmas. An immediate
consequence of these two lemmas is the following corollary.
Corollary 3.1. For Irl:;r nE (0, 1) and 8 t (0, 1), there is a ,n ~I ni
ap~prox~imation rll' .:thm for the problem with ;,,..T. rm probabilities.
By chloosinl au = 0.0497870517. = 0.2714781971 we derive thatl mraX( 1 16 3
6 1a' 6 1a 6
;: I.; :1., which leads to the main result of this section.
Theorem 3.2. The uniform case of URFLP admits a 3 1I :I ap~prox~imation rll'y..athm.
We use a technique called greedy improvement procedure (see [32] for details) to
further improve the approximation factor.
Phase III: Greedy Improvement
Apply Phase I and Phase II to an instance of the original problem where the facility
cost is scaled up by a given factor a > 1, and output a feasible solution.
Assume the costs of the current solution are (F, C, P). Pick a facility i with cost fi
such that the ratio
(C + P C Pi fi)/fi
is maximized, where Ci and Pi are the corresponding transportation cost and penalty
cost if facility i was added to the current solution. If the ratio is positive, open
facility i and repeat this step, otherwise, stop.
The greedy improvement procedure can improve the worst case bound of Algorithm
TP, as shown next. We omit the proof here as the analysis is very similar to those in [17]
and [32].
Lemma 3.5. For r: ; given (Rf, Rc,R,)ap~prox~imation rll' y..thm for (87), there is a
(R, + In(A), 1 + 3II, 1 + ~i)approximn atio rly,. :thm,.
By choosing a~ = 0.42539606, 6 = 0.17430753, and a = 2.82899675, we obtain the
followingf theorem.
Theorem 3.3. URFLP with ,,;..[. < I, failure probabilities admits a 2.674ap~proximation
Once we have a generation of solutions or chromosomes, we employ several operators
to create the next generation whose initial population size is zero. Reproduction carries
forward the best NVR solutions from the current generation to the next one. Naturally,
NVR < 1VP. Immigration (Teates NI solutions randomly, such that each new solution is
different from any of the solutions already in the emerging population. This immigration
process is identical to the process used for generating the initial solutions.
The main operator is crossover. Two solutions are selected at random from the
population at the current generation, with a hias toward the better solutions. The
probability of selecting the jt^ hest solution is given by z" /2.Ji~ Note rNp is the
number of solutions in the population. The denominator, NVp(N~p + 1)/2, is the sum from
1 to NVp. The numerator, NVp + 1 .), is the reverse order of j's fitness among all NVp
solutions. These values are listed in Table 24 as the weight in probability evaluation. As
we can see that the better a solution is, the higher weight it is assigned.
Table 24. Selection probabilities for a population with NVp solutions
Solution Rank Weight in Probability Evaluation Probability
1 NV
P ~Np(Np+1)/2
2 N ~ 1 >,l
P ~Np(Np+1)/2
.1 P Np(Np+1)/2
Np 1 Np(Np+1)/2
Total N~p(N~p + 1)/2 1
After two nonidentical parents are selected, a onepoint crossover position in the
list of genes (candidate sites) is randomly selected. A child solution is constructed using
the genes (candidate sites) to the left of the crossover position from parent 1 and to the
right of the crossover position from parent 2. This process is depicted in Figure 21 for two
solutions with the onepoint crossover position at 3. Encoding of the child solution in this
example is the following: values of positions 1 to 3 are from those in the same positions
where xi is the total number of parallel components at stage i, i = 1, .. ,q; rk is the
reliability level at stage k, k = 1,. ., p; q is not necessarily equal to p, because in some
stages, one may just have only one option to choose: allocating either redundancy or
reliability but not both. If only redundancy allocation is allowed at stage i, then xi is the
only decision variable at stage i; If only reliability allocation is allowed at stage i, then ri
is the only decision variable at stage i; If both redundancy and reliability allocations are
allowed at stage i, then both xi and ri are the decision variables at stage i. Without loss
of the generality, we assume q + p = n. f () is the function measuring the system reliability
that is nondecreasing in each of its variables; gj(x, r) is the consumption of resource j,
j = 1,. ., m. Naturally, gy (x, r) is assumed to be non decreasing in x and r.
The SROP model covers a vast 1 in 4 Gry of reliability optimization models discussed
in the literature. For example, model SROP reduces to model RAOP when p = 0, and
a continuous version of the reliability allocation optimization problem when q = 0. It
certainly can also model the general case of the reliability allocation optimization problem
by reinterpreting the definition of the variables, since objective functions 45 and 49 are
mathematically equivalent, so are constraints 46 and 410. In addition, model SROP
is obviously a reliabilityredundancy allocation model if p = q, where at each stage the
decisions are which component reliability to choose and how much redundancy as well.
The SROP model has received tremendous research attentions over decades and has
been extensively studied and solved using many different mathematical programming
techniques and heuristic approaches. K~uo et al. [28], along with [27], provide a detailed
introduction to the models and algorithms in the reliability optimization. SROP is often
characterized by a nonlinear objective function that is neither convex nor concave over
a nonconvex feasible region. Due to the extreme difficulty of such type of problem,
the solution methods in the literature are mainly heuristics, metaheuristics and
approximation algorithms. A comprehensive review on these methods can be found in
[27].
S2007 Roger Lezhou Zhan
Due to the longfestedgfe bisection branch rule,
lim? (Up~ L ) = Vj =1, 1. ..
k>oo ,
It implies that
lim L"" lim U"" = x*.
k>oo k>oo
We must also show that x* is an optimal solution. From the selection of greatest upper
bound hyperrectangle, we know that the upper bound of the partition in this sequence is
no less than any feasible solution. That is, f (U ") > f (x), Vx E g nx 3. Therefore,
lim f (L" ) = lim f (U" ) = f (x*) > f (x), Vx E g n 31z.
k>oo k>oo
It means that x* is an optimal solution. O
From Theorem 4.1, we can add the following condition after line 16 in Algorithm 1 to
terminate the algorithm in finite iterations.
if maxj {sy c a ,f}< then
output sjL as the solution, terminate,
end if
4.2.4 Acceleration Techniques
In Algorithm 1, besides the reduction process, there are several other acceleration
techniques worth mentioning.
Preprocess. The upper bound of xi can be tightened by a number that is derived
from its lower bound. Let
As = max{A : gj(Aei) + gj(x) < cj}, (419)
we have all feasible xi < xC + Agei. In other words, xi < min(x xC + Xiei), which provides
a tighter upper bound for xi.
network flow problem. These models can help to identify the critical arcs to protect under
the events of disruption.
In this chapter, we follow the theme of C'!s Ilter 2, but assume that the facilities have
been built and can he reinforced to be more reliable. We also assume that the facilities
are uncapacitated. We present two novel models whose objectives are to minimize the
expected connection (service) and penalty cost by allocating the limited fortification
resources to the open unreliable facilities. In the first model, the fortification efforts
are continuous, that is, the failure probability at each facility varies from 0 to 1 at the
fortification stage. We call it the Continuous Facility Fortification Model (CFFM). On
the contrary, in the second model the fortification efforts are subject to different level of
resources. The failure probability at each facility can only be chosen from a set of discrete
levels at the fortification stage. Accordingly we call it the Discrete Facility Fortification
Model (DFFM). Both models can help to identify the critical facilities to protect and
optimally determine how much resources should be allocated to achieve the objective. To
some extent, the models mathematically resemble the reliability allocation problem we
discussed in OsI I pter 4.
The remainder of this chapter is organized as follows. In Section 5.2, we present the
continuous facility fortification model and reveal its connection to the generalized linear
multiplicative programming and the inherent monotonicity. An example is presented in
Section 5.2.2 to illustrate the solution structure and properties. In Section 5.3, we present
the discrete facility fortification model and apply the monotonic branchreducealgorithm
to exploit the monotonicity properties inherent in the problem. The efficiency of the
algorithm is demonstrated in Section 5.3.2. Section 5.3.2 also contains an analysis of
the solution structure and tradeoff between cost deduction and fortification effort. We
conclude this chapter in Section 5.4.
increases. Another interesting pattern in Figure 23 is the tail effect of the CPU time
in terms of the failure probability. SAAH spends more time to obtain a solution when
the failure probability is around 0.5. One possible way to explain this phenomena is the
following: when the failure probability is around 0.5, the constraints .r; I ylusi among
different samples are quite different. As a result, the problem size increases, so does the
computational time.
Next, we examine the effect of the replication number (Af) on the solution quality by
fixing NV = 30. Table 26 provides the objective values obtained when At = 5, 10, 15, 20.
From the objective values obtained in different replication numbers, we can see that the
increase of the replication number has not affected the solution quality too much. The gap
in this table is defined as "'" x 10aI' The negative numbers in the y, Ip column
reveal that v" is not ak ls a lower bound for ~i,2i. However, it is a good indication of
the quality of the solution from SAAH. In this particular case, if the gap is within +1(1' .
the obtained objective value is close to the optimal value.
In general, SAAH is capable to produce a fairly good solution with a large sample
size for the uniform case. But it also requires a tremendous amount of time to do so and
may run out of memory due to the increase of problem size. We defer presenting the
computational results of ITRFLPSFP to Section 2.8.4.
2.8.2 Greedy Methods: GADH and GADSH
In this section, we report the computational results of GADH and GADSH on the
ITRFLPIP model.
Table 27 lists the computational results of a 50node dataset when the failure
probability varies from 0 to 1. The first column, P, is the failure probability at each
facility. The y, .p" column is defined as the percentage difference between the cost of the
solution obtained by GADH and the optimal cost.
As we can see from Table 27, GADH finds optimal or nearoptimal solutions in most
cases in less than 0.05 seconds. Compared to the exact method using CPLEX, it takes
#i x ydi ri fl p1 f2 92 p 3 p3
21 0.46 0.21 952 9.57 850 0.37 932 0.18 980 0.14
22 0.77 0.44 919 1.49 561 0.92 742 0.76 1290 0.38
23 0.87 0.79 292 6.95 750 0.6 1213 0.59 1311 0.24
24 0.69 0.15 48 2.29 6384 0.911 772 0.79 956 0.71
25 0.24 0.28 581 2.9 1040 0.62 1153 0.3 1456 0.23
The meaning of each column in Table B2 is provided as follows: #i denotes the
facility name; (x, y) is the coordinates, di is the demand; ri is the penalty cost; and fi, p
(i = 1, 2, 3) are the investment level and its corresponding failure probability.
Table B2. Dataset of URFLPMFP: 3level
0.82
0.54
0.91
0.15
0.74
0.58
0.6
0.37
0.7
0.22
0.5
0.3
0.95
0.65
0.53
0.95
0.15
0.31
0.98
0.59
0.18
0.7
0.72
0.31
0.16
0.92
0.09
0.19
0.52
0.4
0.45
0.52
0.2
0.07
0.11
0.95
0.13
0.4
0.73
0.04
957
202
186
6335
737
953
450
188
206
995
429
528
570
938
726
533
5635
322
326
6;6;3
5.32
1.9
3.11
1.83
1.34
2.3
7.96
3.42
9.05
4.56
9.87
0.53
3.41
8.98
3.53
1.64
1.36
5.1
1.22
6.95
938
642
1125
772
665
890
6;20
503
1231
989
948
551
6;82
758
552
791
930
694
586
634
0.81
0.55
0.63
1
0.85
0.9
0.98
1
0.86
1
0.87
0.84
0.46
0.78
0.75
0.7
0.98
0.52
0.95
0.3
954
721
1230
1008
1279
1034
1187
703
1278
1037
1082
585
923
987
855
1471
958
829
975
812
0.44
0.39
0.42
0.9
0.45
0.83
0.48
0.75
0.65
0.55
0.6
0.79
0.32
0.5
0.6;3
0.66
0.8
0.48
0.49
0.24
1260
1330
1355
1440
1393
1431
1394
1044
1466
1455
1422
1303
999
1497
1407
1488
1227
1103
1127
907
0.15
0.02
0.19
0.36
0.28
0.61
0.06
0.49
0.24
0.54
0.58
0.39
0.04
0.15
0.48
0.47
0.35
0.2
0.15
0.22
accommodate this pure nonlinear integer programming. In the description of Algorithm
2, S denotes a hyperrectangle partition; C is a list of unfathomed hyperrectangles; e is a
predefined optimality tolerance parameter; xbest and fbest denote the current best solution
and objective value respectively; UB(S) is the upper bound of the objective function over
S. Besides initialization, the 1!! ri ~ steps are described as follows.
Initialization
2: if xU E g then
3: x" is the optimal solution, terminate;
4: else if xL Sf then
5: the problem is infeasible, terminate;
6: else
7: set S = [X x"], e > 0, C = {S}, Zbest = XL, best =(L)
8: end if
Select and Branch
9: if C = 0 then
10: output the current Zbest aS the solution, terminate;
11: else
12: select S = [s SU] eE such that UB(S) = maxse~e{UB(S)} and < C\{S}
13: if (UB(S) fbest) < e then
14: output the current Zbest aS the solution, terminate;
15: else
16: select, i sulch that, i argmax,(~ {sU a bisect, S into SI and S2 atlongf the edgfe i.
17: end if
18: end if
Reduce and Bound
19: for k = 1, 2 do
20: reduce Sk, to S, [L" U x"] according to reduction rules
21: compute a suitable upper bound UB(Sk)
22: if UB(Sk) Ibest Or ZLr Sf 0 then
23: continue;
24: else if xU" E then
ACKENOWLED GMENTS
I would never he able to adequately thank Dr. Zuo.Jun Max Shen, my supervisor and
mentor, for helping me in developing my research topics. He allowed me to work freely,
but he was ah .14 there when I needed advice or guidance. I want to thank hint not only
for his tremendous guidance and encouragement throughout my study, but also his endless
trust and understanding.
Dr. Panos Pardalos deserves many sincere thanks. He recruited me to the University
of Florida and was ahrl . very supportive through my years there.
I thank Dr. .Joseph P. Geunes and Dr. Edwin Romeijn for their many good
II_0r~! 0.. in this research as well as their wonderful courses I have taken with them.
I also thank another nienter of my coninittee, Dr. .Juan F in for her time and guidance
in the last three years.
I am grateful to both of my coauthors Dr. .Jiawei Z1! Iu, and Dr. Mark Daskin
for their incisive insights. I also thank Prof. H. Tuy for introducing his nionotonic
optimization methodology to me while he visited Gainesville in August, 2003.
I acknowledge all the help and financial support front the Department of Industrial
and Systems Engineering. In particular, I thank Dr. Hearn for assigning me to teach
various courses at the University of Florida.
My deepest appreciation goes to my entire family, including my wife, Ngana, my
brother, Leping, my parents and inlaws. I could not have finished this research without
their support, love, and encouragement. Most specially, I want to thank my wife for being
with me in numerous late nights in Gainesville, Miami, and Greenwich, and helping to
proofread nly manuscript.
Last but not the least, I thank my friends, Leon, Lian, Shu, Gang, Bin, .Jean,
Chunhua, .Jun, Altannar, .Jie, .Jonathon, Cindy, Silu, .Junnmin, and many others who
made my experience at the University of Florida nienorable.
* Facility 2
0 500 1000 1500 2000 2500
Resource Used
A) Facility 2
3000
0 500 1000 1500 2000 2500 3000
Resource Used
B) Facility 5
0 500 1000 1500 2000 2500 3000
Resource Used
C) Facility 15
0 500 1000 1500 2000 2500 3000
Resource Used
D) Facility 18
0 500 1000 1500 2000 2500 3000
Resource Used
E) Facility 20
Figure 55. Fortification level at individual facility by resource used
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
MODELS AND ALGORITHMS FOR RELIABLE FACILITY LOCATION PROBLEMS
AND SYSTEM RELIABILITY OPTIMIZATION
By
Roger Lezhou Zhan
August 2007
Cl.! ny~: ZuoJun Alax Shen
Major: Industrial and Systems Engineering
Uncertainty is one of the elements that make this world so fascinating and dynamic.
However, the existence of uncertainty also poses a great challenge to reliable system
design. Our study uses various models and algorithms to address reliability issues in the
context of (1) the uncapacitated facility location problem where facilities are vulnerable,
and (2) the system reliability problem where components are subject to fail.
We first study the uncapacitated reliable facility location problem in which the failure
probabilities are sitespecific. The problem is formulated as a twostage stochastic program
and then a nonlinear integer program. Several heuristics that can produce nearoptimal
solutions are proposed for this computationally difficult problem. The effectiveness of the
heuristics is tested through extensive computational studies. The computational results
also lead to some managerial insights. For the special case where the failure probability
at each facility is a constant (independent of the facility), we provide an approximation
algorithm with a worstcase bound of 2.674.
Another part of our research is related to the development and application of a
monotonic branchreducebound algorithm, a powerful tool to obtain globally optimal
solution to problems in which both the objective function and constraints possess
monotonicity. We tailor the algorithm to solve a mixed integer nonlinear programming
problem. Its convergence analysis and acceleration techniques are also discussed. The
algorithm is then successfully applied to solve system reliability optimization problems
2.6 Uncapacitated Reliable Facility Location Problem with Multilevel
Failure Probabilities
In this section, we extend URFLPSFP to URFLPMFP so that each facility has
multilevel failure probabilities. In this model, the decision makers can make some key
facilities more sustainable than others by investing more if necessary. In URFLPMFP, we
model the failure probabilities as functions of the initial fixed investment. To do so, we
introduce t to devote different investment levels. In addition, fee is denoted as the fixed
cost at the facility i at the level t; yti, the decision binary variables for the facility i at the
level t. That is, yti = 1, if a level t investment is put at facility i; otherwise, ymi = 0. We
assume 0 < t < U.
Given different investment levels at facility i, the output of the failure probability at
facility i, ~Pf(yti), is determined by yes:
I~I(I1L) C~P(.fL.i)~lZ)
(214)
with foi = 0, P(0) = 1, and P() is a decreasing function. foi = 0 and P(0) = 1 imply that
if there is no investment at facility i, then it is completely nonfunctional.
After the failure probability at facility i is determined, URFLPMFP is essentially no
different from URFLPSFP. In the following formulation, pi in the objective function 27
of URFLPSFP is replaced by ~Pf(yti).
( URFLPM~FP2)
minimize F ,(yi) +CC dy~c,,x@(1 y)) (IU
i6EF j6D k=1 iEF 16F
F+1
j6D k=1 16F
sub~jct to xrk z~ = j ,.,+1
i6EF t=1
x EImin~yi,1} Vi EF, j ED,k~= 1,...,IF
k=1
xi, zfE ( {, 1}, y,: integer,
(222)
(223)
(224)
(225)
(226)
where constraints (223) ensure no client is assigned to an unopen facility. That is, when
yI = 0, xi = 0, Vi, j, k = 1,..., F. Because of th~e binary con~straints (226) on? xi, th~e
right hand side of constraints (223) can be replaced by yi without affecting the feasible
domain.
2.7 Solution Methodologies
2.7.1 Sample Average Approximation Heuristic
The Sample Average Approximation (SAA) method is widely used for solving
complicated stochastic discrete optimization problems ([24], [42], and [61]). The basic idea
of this method is to randomly generate samples, then use a sample average function to
estimate the true expected value function. By doing so, the original problem is reduced
to a relatively small problem that can be repeatedly solved. Such an approach has been
used by various authors over the years. We apply the following procedures to solve model
UJRFLPSP.
2.4 Uncapacitated Reliable Facility Location Problem: a ScenarioBased
Model
We first discuss a scenariobased approach to model URFLP. Given a finite set of
scenarios, where each scenario specifies the set of operational facilities, we can formulate
UJRFLP as a twostage stochastic program with recourse. The first stage decision is to
determine which facilities to open before knowing which facilities will be operational.
When the uncertainty is resolved, the clients (demand points) will be assigned to the
operational facilities. These are the second stage decisions. In this model, we are not
allowed to build new facilities in the second stage. In other words, no remedy can be made
to the first stage decision, except for optimally assigning the clients to the operational
facilities. The objective is to minimize the total expected cost which includes the first
stage cost and the expected second stage cost. The expected cost is the sum of the cost of
all scenarios times their specific probabilities.
Let S be the set of scenarios. For any A E S, let pA be the probability that scenario
A happens. Then URFLP can be formulated as the following twostage stochastic
program.
minimize fg~yi + plA9A y) Subhjct to ye, { 0, 1}, (21)
iLEF AES
where gA 9/) = mi Cdjli djz (22)
j6D iLEF j6D
s.t. xs + zf = 1, Vj EDn (23)
i6EF
x y, i t F,i jeD (2 4)
x ~ 1 5 AiV F, jeD (2 5)
x ,~ zf c{, 1}. (2 6)
In the above formulation, the binary variable yi indicates if facility i is opened in the first
stage. Parameter IA,i indicates if facility i is operational under scenario A, which is an
input regardless of th~e value of ys. Viariabhle x is th~e assignment variabhle which? indicates
APPENDIX A
SYSTEM RELIABILITY COMPUTATION IN CHAPTER 4
We use conditional probability to derive the expressions of reliabilities of the networks
in Figure 43 and Figulre 45.
Reliability of the fivecomponent bridge network. Reliability of the network in
Figure 43 can be written based on whether component 5 is functional or not.
R, = Pr(system works component 5 works)rs+Pr(system works component 5 fails) (1 rs).
(A1)
When component 5 works, the original network in Figure 43 is reduced to Figure A1(A),
which is a parallelseries system with a reliability of
Pr (system works component 5 works)
(rl + T.3 T1r3 7r2 + 74 r.2r4).
(A2)
B
Figure A1. Configurations based on state of component 5 in Figure 43: A) Component 5
works, B) Component 5 fails
S2h

1
3
4 "
*mj kj: when k < kj, from the proof of the first bullet, we know that Ci ify = 1.
when my, & f = 1. The last equality holds because of the
iEF 3
constraint 310, i.e. xr ~ 0i = 1.
i6EF
Thus & and y satisfy the following constraints:
iLEF
k=1> o
0<^ <1
Therefore, if we apply the algorithm in [17] with a parameter a~ to round the solution
(i, y), then by Lemma 3.1, we can construct a solution (x, y) such that the expected total
facility cost and total transportation cost is bounded above by
In I 0 m
iLEF j6D k=1 iEF
1 In 3
1P 1a~ 1a
i6EF j6D k=1 iLEF
i6EF j6D k=1 iEF
The first inequality holds because of inequalities 313 and 314, the second one holds
because of mj < kyj.
Finally notice that was uniformly distributed in (0, 5), thus
1 11 1 1
E [1 ] =1 1p dp = In 
This completes the proof of the Lemma.
CHAPTER 4
SYSTEM RELIABILITY OPTIMIZATION AND MONOTONIC OPTIMIZATION
4.1 Introduction
The performance reliability of a system is of utmost importance in many industrial
and military systems. System reliability is a measure of how well a system meets its design
objective, and is usually expressed in terms of the reliability (a probability of successful
operations) of the subsystems or components. For example, a series system works if and
only if every component works. Such a system fails whenever any component fails. The
reliability of the series system in Figure 41 is
i= 1
where R, is the system reliability and ri is the reliability of component i, which is the
probability that component i successfully operates during the intended period of time. In
this chapter, we assume all components operate independently.
~ 12 i  n
Figure 41. Series system
System reliability can be improved in various owsi~, such as physical enhancement of
component reliability, provision of redundant components in parallel, and allocation of
interchangeable components.
Unlike in a series system, in a parallel system, not all components are necessary for
the system to work successfully. Actually, only one component in such system needs to
work properly in order for the whole system to work properly. Including a components
when only one is essential is called redundancy. The other n 1 components are included
to increase the probability that there is at least one working component. Redundancy is a
widely used technique in engineering to enhance system reliability.
3.4 Conclusions
In this chapter, we employ various rounding and decomposition techniques to develop
a ;: I; :Ilapproxiniation algorithm, then improve this to a 2.674approxiniation algorithm
using greedy intprovenient procedures for the uniform uncapacitated reliable facility
location problem. To the best of our knowledge, these are the first type of approximation
algorithm for the facility location problem with uncertainties in the facility side. Whether
these bounds can he improved is an open problem for future research. Another interesting
topic is to develop an approximation algorithm for the uniform capacitated reliable facility
location problem.
3.5 Proofs
Proof. Theorem :3.1.
(1). Assume we have an optimal solution (.r, y, x) for problem (:36), then we can
construct a feasible solution (.r, y, 8) for problem (:37) with the same objective value.
L~et .r = .r anld y = y. For ealCh j D an ld k~ > 1, define~ Of = Ek=1 Z3.z Because for
ea~ch .), there is exactly on~e value of t woullld br equlln to 1, w W~Can conclude Of t {0, 1}.
And it is straightforward that (.r, y, 8) is a feasible solution to problem (:37). Now we
check the objective value corresponding to (.r, y, 8). In fact, we only need to consider
oox no in~k,F+1}
k= 1 k= 1 t= 1
F+1 xo
t=1 k=t
F+1
t=1
The first equality is due to the fact that Vt > F + 1,.) E D, a = 0.
(2). Now we prove the other direction; i.e., assume we have an optimal solution
(.r, y, 8) for problem (:37), then we can construct a feasible solution (.r, y, x) for problem
(:36) with the same objective value.
LIST OF TABLES
Table page
21 Acronyms ......... .... . 19
22 Sample chromosome for model URFLPMFP ..... .... 32
23 GAH parameters ......... . . 32
24 Selection probabilities for a population with NVp solutions .. .. .. 33
25 Objective values from SAAH for the 50node dataset ... .. .. .. 36
26 Runs from SAAH for the 50node dataset ...... .. 39
27 Fiftynode uniform case: greedy adding and exact solution .. .. .. .. 40
28 Fiftynode uniform case: GADSH and exact solution .. .. . .. 40
29 Values of the GAH parameters ......... ... .. 41
210 Fiftynode uniform case: GA and exact solution ... . .. 41
211 Objective values obtained from SAAH on URFLPSFP with different sample
sizes ........... ........ .... 44
212 Computational performance on URFLPSFP: GADH, GADSH, GA and the
enumeration method ......... .. .. 44
213 Computational results for URFLPMFP using GADSH and GA .. .. .. .. 46
41 Coefficients in Example 1 ......... . 74
42 Coefficients in Example 3 ......... . 77
43 Performance comparison of Example 3 . ..... 78
44 Performance comparison of Example 4 . ..... 79
51 Solution of a CFFM model ........ .. .. 89
52 Input data (Ve(yi) and Pi(yi)) for the 3level model ... ... .. 96
53 Solutions of the 3level model ........ ... .. 97
54 Solutions of the 2level model ........ ... .. 98
B1 Dataset of URFLPSFP ........ . .. 107
B2 Dataset of URFLPMFP: 3level ....... ... .. 109
C1 Dataset of DFFM ........ . .. 113
( URFLPIP)
F F+1
minimize.2~ f y yey 1 p)pk1 k1d
i6EF j6D k=1 iEF j6D k=1
sub~jectto xz z )~=1, VjeDI, k =1,...,F+1
i6EF t=1
_36)
In the next section, we shall present an approximation algorithm for URFLPIP.
We find that it is more convenient to deal with a slightly different formulation. In the
necw form~ula~tion, we introduce a. new set of decision? va~riables Of to replace zf. Define
Of = , Note that Of are n~ot decision? va~riables when? k > F, Of = 1 for a~ll
k > F + 1. We prove that the following integer program is equivalent to formulation
(URFLPIP), as stated in Theorem 3.1. We refer the reader to Section 3.5 for the proof.
i6EF j6D k=1 iEF j6D k=1
subject to x ~ + Of = 1, jeF 1,..., F
i6EF
P> (37)
xfy < yi., i; E Fj jED
Theorem 3.1. Formulation (86) and formulation (87) are equivalent.
3.3 Approximation Algorithms
In this section, we aim to propose a 2.674approximation algforithm for the special
case where the failure probabilities are uniform. We call it a (Rf, R,, R,)approximation
than 1 millisecond. The problem was solved by various specialized heuristics, such as
[36], [23], and [1], all of which spent more time to solve than the mBRB algorithm and
could not verify the optimality of their solutions, although they were able to produce the
same solution. As a correction to the literature, it is noted that [1] contains the wrong
expression of this example.
4.3.2 ReliabilityRedundancy Allocation Optimization
Reliabilityredundancy allocation optimization is a mixed integer nonlinear programming
problem that is the general form of SROP. In this setting, a design engineer can
improve the reliability of a system by increasing the component reliabilities or providing
redundancy at various stages. The following two examples are widely used in the
reliability literature.
Example 3: Fivestage series system with component reliability choice. This
example is a variant of Example 1.
max Rs e,4 (430)
i= 1
subject to gi = ) p,.x < P, (431)
i= 1
g2 ~i= In i x4 ex < C, (432)
g3= t'I x < W, (433)
i= 1
0
xi: integer, Vi=1 ,(435)
where the objective function and constraints are the same as those of Example 1 except
a more explicitly expression for the unit cost of component i, which is a function of the
component reliability rs. To drivew the expression of cs(ri) =i as g we follow the
method in [54] and assume that the unit cost of component i is a decreasing function of
Example 1: Fivestage series system. The following problem is to maximize the
system reliability of a fivestage series system which are subject to three nonlinear resource
constraints. This problem is widely used to demonstrate a number of optimization
techniques [28]. The problem was originally presented in [53].
max Rs = R (x4) (4 20)
i= 1
subject to gl = pg? < P. (4 21)
i= 1
g2=Ci Zi +exp(
i= 1
g3 =it I e(xp < W, (423)
i= 1
xi: integer, Vi=1 ,(424)
where Ri = 1 (1 ri)"i is the reliability of stage i. Constraint 421, gl, is imposed on
the combination of weight and volume: pi is the product of weight per unit and volume
per unit. Component reliability does not usually affect the weight nor the volume, hence
91 is not a function of ri ([54]). Constraint 422, g2, iS the cost constraint where cixi is the
cost of all components at stage i and ce exp (\ ) i thep adirt~ional cosnt for ;,tci~n~rtelrconctn
parallel components. Constraint 423, g3, iS the weight constraint where I, is the weight
of all componentst at stage i. The additional factor, exp (y), is added due to the hardware
required for interconnecting components ([54]). The weight constraint is not a function of
component reliability. The coefficients are given in Table 41.
Table 41. Coefficients in Example 1
i ri pi ci I, P C W
1 0.80 1 7 7
2 0.85 2 7 8
3 0.90 3 5 8 110 175 200
4 0.65 4 9 6
5 0.75 2 4 9
that fl + dllrlp', > dulrl, then this facility should not be open. In this example, when the
failure probability increases, fewer facilities should be open.
2.8.3 Genetic Algorithm Based Heuristic
In all the GAH tests, the values in Table 29 were used for the parameters in GAH
described in Section 2.7.3.
Table 29. Values of the GAH parameters
Parameter Value
Population size NVp 100
Maximum number of generations Noe 200
Maximum number of generations without improvement 1NM 100
Number of reproduction NsR 10
Number of immigration NI 10
Mutation probability PM 0.1
Table 210 lists the computational results of the 50node dataset when the failure
probability varies from 0 to 1. The first column, P, is the failure probability at each
facility. Because the GA heuristic is a probabilistic method, two trials are performed. We
report the minimum objective it obtained and the average CPU time (in seconds) in the
second and third column respectively. The ,Ip column is defined as the percentage
difference between the cost of the solution obtained by GA and the optimal cost.
Table 210. Fiftynode uniform case: GA and exact solution
GA Heuristic (2 Trials) Exact Algorithm
P Min Objective Time (s) Objective Time (s) gap( ~)
0.0 7197.27 5.41 7197.27 6.94 0.00
0.1 7763.80 5.23 7763.80 7.61 110.00
0.2 8425.99 5.20 8425.99 8.94 0.00
0.3 9275.99 5.11 9275.99 10.62 0.00
0.4 10253.90 5.34 10253.90 10.38 0.00
0.5 11603.00 5.66 11603.00 10.86 0.00
0.6; 13416.80 5.52 13416.80 12.84 0.00
0.7 16157.20 5.75 16157. 20 13.47 0.00
0.8 21500.70 7.95 21500.70 14.08 0.00
0.9 35987.70 9.83 35987. 70 14.27 0.00
1.0 128009.00 3.25 128009.00 9.27 0.00
Table 211. Objective values obtained front SAAH on ITRFLPSFP with different sample
s1Zes
Dateset I I Best Time (s)
# N50 N100 N150 N200 Objective (N100)
10 5850.47 5576.00 5128.24 5128.24 5128.24 0.4:3
15 6074.06 5:337.18 5:337.18 5:337. 18 5:337.18 2.11
20 8071.98 5761.79 5761.79 5761.79 5761.79 :3.09
25 6749.39 658:3.06 658:3.06 658:3.06 658:3.06 12.14
:30 7897.19 7622.22 7847.37 7847.37 7622.22 50.67
40 7474.92 7474.92 7474.92 7474.92 7474.92 54.20
50 8719.32 86341.28 8781.18 86341.28 86341.28 185.563
60 9:357.37 9:394.87 9:357.37 9:394.87 9:357.37 159.48
70 10:337.60 10:391.80 10:38:3.00 10:38:3.00 10:337.60 250.85
80 11054.30 11054.30 11054.30 11054.30 11054.30 :320.82
90 12405.50 12405.50 12448.70 1 12405.50 659.57
100 1:3977.50 14028.10 I 1 1:3977.50 2164.03
Dateset GADH GADSH GA Enumeration
# Objective T (s) Objective T (s) Objective T (s) Objective T (s)
10 5128.04 0.00 5128.04 0.00 5128.04 1.62 5128.04 0.02
15 5:305.04 0.00 5:305.04 0.00 5:305.04 1.81 5:305.04 14.75
20 5761.79 0.00 5761.79 0.00 5761.79 1.81 5761.79 206.51
25 64:39.87 0.00 64:39.87 0.00 64:39.87 1.82 64:39.87 1ml.1
:30 7420.86 0.00 7:382.04 0.00 7:382.04 2.22 7:382.04 188276.35
40 7474.92 0.00 7474.92 0.02 7474.92 4.09
50 876:3.75 0.00 86341.28 0.00 86341.28 4.634
60 9:357.37 0.00 9:357.37 0.02 9:357.37 5.14
70 10:337.60 0.00 10:337.60 0.03 10:337.60 5.84
80 11054.30 0.00 11054.29 0.05 11054.29 5.563
90 1:30:30.90 0.00 12405.50 0.06 12405.50 10.75
100 1446:3.40 0.02 1:3820.87 0.09 1:3820.87 11.2:3
close to the best known ones. In terms of computational efficiency, SAAH takes much
more time to achieve its solutions than all other three heuristics. Between GADSH and
GA, GADSH spends considerably less CPIT time than GA: GADSH takes less than
0.1 seconds to get the best result in each instance. Overall, these results II r that
GADSH is the best one among all four heuristics for model ITRFLPSFP in terms of both
solution quality and computational time.
Table 212. Computational performance on
enumeration method
ITRFLPSFP: GADH, GADSH, GA and the
is sitespecific, a much more general case. They formulate their problem as a linear integer
program and propose a Lagrangian relaxation solution method. Another related model
is proposed in [5], which is based on the pniedian problem rather than the framework of
ITFLP.
There is also a small strand of literature devoted to addressing the fortification of
reliability for existing facilities, which includes [4:3], [44], and [49]. These models typically
focus on the interdictionfortification framework hased upon the pniedian facility location
problem. They are generally formulated as bilevel progranining models. Their main focus
is to identify the existing critical facilities to protect under the events of disruption.
In our model, the failure probabilities are sitespecific, which significantly complicates
the problem when formulating it as a niathentatical program. The model is further
extended to ITRFLP1\FP, where each facility is allowed to have multiple levels of the
failure probabilities. We propose two different modeling approaches: a scenariobased
stochastic progranining approach and a nonlinear integer progranining approach. The
scenariobased model is attractive due to its structural simplicity and its ability to
model dependence among random parameters. But the model becomes computationally
expensive as the number of scenarios increases. If the number of scenarios is too large,
the nonlinear integer progranining approach provides an alternative way to tackle the
problem.
The sample average approximation method is widely used for solving complicated
stochastic discrete optimization problems, e.g., [24], [42], and [61]. The basic idea of this
method is to use a sample average function to estimate the expected value function. Thus
the original problem is transformed to the one that can he efficiently solved.
Other types of the coninonly used heuristics in optimization are the local search and
iterative intprovenient algorithms ([1:3], [16], [4] and [:3]). These heuristics start with an
empty set and repeatedly consider adding a potential facility into the solution. Or they
start with an nonenipty set and repeatedly consider deleting or substituting a facility in
Table 25. Objective values front SAAH for the 50node dataset
Failure Probability N10 N20 N:30 N40 N50
0 7197.27 7197.27 7197.27 7197.27 7197.27
0.1 7956.03 7956.03 776:3.80 776:3.80 776:3.80
0.2 9429.49 8770.31 8425.99 8425.99 8425.99
0.3 9908.52 10059.90 9669.49 10201.50 10095.20
0.4 11546.90 1175:3.50 11984.00 1:3887.60 119:37.60
0.5 18727.50 15128.1 1:3120.60 15052.80 1:3546.40
0.6; 27429.00 18161.70 17946.40 17946.40 17946.40
0.7 :32965.30 :331:39.20 27:374.80 2:3659.30 2:3690.40
0.8 62876.90 46:387.50 :3574:3.60 :3574:3.60 :3574:3.60
0.9 84406.10 69255.40 54722.30 54728.10 55647.40
1.0 128009.00 128009.00 128009.00 128009.00 128009.00
Failure Probability N100 N150 N200 N250 Exact
0 7197.27 7197.27 7197.27 7197.27 7197.27
0.1 776:3.80 776:3.80 776:3.80 776:3.80 776:3.80
0.2 8425.99 8425.99 8425.99 8425.99 8425.99
0.3 9:387.85 9:387.85 9275.99 9275.99 9275.99
0.4 10848.20 10599.90 10566.30 10762.00 1025:3.90
0.5 1:3157. 20 1:3041.50 1:3099.70 121:34.20 1160:3.00
0.6; 17762.80 15609.50 14559.10 14559.10 1:3416.80
0.7 20740.30 20671.60 19070.90 1'71;' .00 16157.20
0.8 29218.10 267:31.10 24066.80 2:3252.20 21500.70
0.9 46546.80 45061.00 42555.50 :38:316.90 :35987.70
1.0 128009.00 128009.00 128009.00 128009.00 128009.00
Figure 22 shows that SAAH obtains fairly good solutions with a small sample size
when the failure probability is low (p < 0.4). In contrast, a large sample size is needed
to achieve such quality of solutions for the cases with a higher failure probability. The
following could be a possible explanation. Note that a sample is a collection of different
scenarios. When the failure probability is low, the 1 in U. G~y of the facilities are candidate
sites for opening in each scenario, so the sets of candidate facilities are similar in different
scenarios. Thus, any individual sample can capture the characteristics of the system pretty
well, and the corresponding solution obtained front SAAH is close to optimal. For the
extreme case where p = 0, all facilities are available to open in each scenario, so the sets
of available facilities are the same in each scenario and the SAAH can produce the exact
solution in this case. In the case of the higher failure probability, the candidate sites for
continued from previous page
#i x y di ri fl p1 f2 p2 f3 p3
88 0.75 0.63 375 5.28 522 0.59 1206 0.06 1406 0.04
89 0.16 0.41 711 7. 34 515 0.63 707 0.48 1415 0.46
90 0.011 0.21 208 3.99 923 0.87 1049 0.84 12811 0.32
91 0.51 0.76 954 4.06 739 0.79 1186 0.31 1378 0.28
92 0.98 0.32 843 7. 77 733 0.52 834 0.33 1335 0.18
93 0.55 0.39 905 0.08 545 0.74 86;9 0.61 1178 0.61
94 0.36 0.63 729 8.47 1047 0.84 1098 0.66 1460 0.6
95 0.18 0.75 382 63.23 538 0.62 613 0.59 13635 0.38
96 0.09 0.46 91 10.53 513 0.67 521 10.26 1448 0.16
97 0.18 0.67 991 6;.6;2 723 0.92 1119 0.91 1338 0.41
98 0.1 0.38 644 3.13 6;28 0.54 822 0.46 1341 0.11
99 0.25 0.66 539 0.5 848 0.49 942 0.48 1256 0.41
100 0.68 0.49 294 7.27 1069 0.99 1108 0.88 1168 0.66
whether client j is assigned to facility i in scenario A or not. Finally, the vrariable zf
indicates whether client j receives service at all or is subject to a penalty. The objective
in the formulation, i.e. 21, is to minimize the sum of the fixed cost and the expected
second stage cost. The objective of the second stage, i.e. 22, is to minimize the service
and penalty cost. Constraints (23) ensure that client j is either assigned to a facility or
subject to a penalty at each level k in Scenario A. Constraints (24) and (25) make sure
that no client is assigned to an unopen facility or a nonfunctional facility respectively.
It is straightforward to show that the formulation (21) is equivalent to the following
mathematical program.
(URFLPSP)
iLEF ACS \jeD iEF j6 / J;
s.t.i x + Z=1,V ,A ACS
iEF
x 5 elAi, i t F, je D A CS
yei, X ,3 zf 6 {, 1}.
One advantage of the scenariobased formulation is that it can easily capture the
dependence of the failure probabilities of different facilities by properly defining the
scenarios. If the number of scenarios is not too large, it is possible to solve URFLPSP
efficiently and effectively.
However, when the failure probabilities are independent, the possible number of
scenarios can be extremely large. Therefore, the number of variables and constraints
in URFLPSP is exponentially large accordingly, which makes it extremely difficult to
solve. Under this situation, we propose several alternative nonlinear integer programming
formulations and efficient solution algorithms. We discuss these alternative formulations in
the next two sections.

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IwouldneverbeabletoadequatelythankDr.ZuoJunMaxShen,mysupervisorandmentor,forhelpingmeindevelopingmyresearchtopics.Heallowedmetoworkfreely,buthewasalwaystherewhenIneededadviceorguidance.Iwanttothankhimnotonlyforhistremendousguidanceandencouragementthroughoutmystudy,butalsohisendlesstrustandunderstanding.Dr.PanosPardalosdeservesmanysincerethanks.HerecruitedmetotheUniversityofFloridaandwasalwaysverysupportivethroughmyyearsthere.IthankDr.JosephP.GeunesandDr.EdwinRomeijnfortheirmanygoodsuggestionsinthisresearchaswellastheirwonderfulcoursesIhavetakenwiththem.Ialsothankanothermemberofmycommittee,Dr.JuanFeng,forhertimeandguidanceinthelastthreeyears.IamgratefultobothofmycoauthorsDr.JiaweiZhang,andDr.MarkDaskinfortheirincisiveinsights.IalsothankProf.H.TuyforintroducinghismonotonicoptimizationmethodologytomewhilehevisitedGainesvilleinAugust,2003.IacknowledgeallthehelpandnancialsupportfromtheDepartmentofIndustrialandSystemsEngineering.Inparticular,IthankDr.HearnforassigningmetoteachvariouscoursesattheUniversityofFlorida.Mydeepestappreciationgoestomyentirefamily,includingmywife,Ngana,mybrother,Leping,myparentsandinlaws.Icouldnothavenishedthisresearchwithouttheirsupport,love,andencouragement.Mostspecially,IwanttothankmywifeforbeingwithmeinnumerouslatenightsinGainesville,Miami,andGreenwich,andhelpingtoproofreadmymanuscript.Lastbutnottheleast,Ithankmyfriends,Leon,Lian,Shu,Gang,Bin,Jean,Chunhua,Jun,Altannar,Jie,Jonathon,Cindy,Silu,Junmin,andmanyotherswhomademyexperienceattheUniversityofFloridamemorable. 4
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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 11 2RELIABLEFACILITYLOCATIONPROBLEM:MODELSANDHEURISTICS 15 2.1Introduction ................................... 15 2.2LiteratureReview ................................ 17 2.3NotationsandAcronyms ............................ 19 2.4UncapacitatedReliableFacilityLocationProblem:aScenarioBasedModel 20 2.5UncapacitatedReliableFacilityLocationProblemwithaSinglelevelFailureProbability ................................... 22 2.5.1NonlinearIntegerProgrammingModel ................ 22 2.5.2ModelProperties ............................ 23 2.5.3ASpecialCase:UniformFailureProbabilities ............ 25 2.6UncapacitatedReliableFacilityLocationProblemwithMultilevelFailureProbabilities ................................... 26 2.7SolutionMethodologies ............................. 28 2.7.1SampleAverageApproximationHeuristic ............... 28 2.7.2GreedyMethods ............................. 29 2.7.3GeneticAlgorithmBasedHeuristic .................. 31 2.8ComputationalResults ............................. 35 2.8.1SampleAverageApproximationHeuristic ............... 35 2.8.2GreedyMethods:GADHandGADSH ................ 38 2.8.3GeneticAlgorithmBasedHeuristic .................. 41 2.8.4ApplyingHeuristicstoSolveURFLPSFP .............. 42 2.8.5URFLPMFP:GADSHvs.GA .................... 45 2.9Conclusions ................................... 46 3UNIFORMUNCAPACITATEDRELIABLEFACILITYLOCATIONPROBLEM:A2.674APPROXIMATIONALGORITHM .................... 48 3.1Introduction ................................... 48 3.2Formulations .................................. 49 3.3ApproximationAlgorithms ........................... 50 3.4Conclusions ................................... 56 5
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...................................... 56 4SYSTEMRELIABILITYOPTIMIZATIONANDMONOTONICOPTIMIZATION 61 4.1Introduction ................................... 61 4.2AMonotonicBranchReduceBoundAlgorithm ............... 67 4.2.1SelectandBranch ............................ 69 4.2.2ReduceandBound ........................... 70 4.2.3ConvergenceAnalysis .......................... 71 4.2.4AccelerationTechniques ........................ 72 4.3UsingMonotonicBranchReduceBoundAlgorithmtoSolveSystemReliabilityOptimizationProblems ............................. 73 4.3.1RedundancyAllocationOptimization ................. 73 4.3.2ReliabilityRedundancyAllocationOptimization ........... 76 4.4Conclusions ................................... 79 5FORTIFYINGTHERELIABILITYOFEXISTINGFACILITIESANDMONOTONICOPTIMIZATION ................................... 81 5.1Introduction ................................... 81 5.2ContinuousFacilityForticationModel .................... 83 5.2.1PropertiesoftheContinuousFacilityForticationModel ...... 84 5.2.2AnExampleoftheContinuousFacilityForticationModel ..... 87 5.3DiscreteFacilityForticationModel ...................... 92 5.3.1PropertiesandAlgorithms ....................... 93 5.3.2ComputationalExperiments ...................... 96 5.4Conclusions ................................... 99 6CONCLUDINGREMARKS ............................. 102 APPENDIX ASYSTEMRELIABILITYCOMPUTATIONINCHAPTER4 .......... 104 BDATASETUSEDINCHAPTER2 ......................... 107 CDATASETUSEDINCHAPTER5 ......................... 113 REFERENCES ....................................... 115 BIOGRAPHICALSKETCH ................................ 120 6
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Table page 21Acronyms ....................................... 19 22SamplechromosomeformodelURFLPMFP .................... 32 23GAHparameters ................................... 32 24SelectionprobabilitiesforapopulationwithNPsolutions ............ 33 25ObjectivevaluesfromSAAHforthe50nodedataset ............... 36 26RunsfromSAAHforthe50nodedataset ..................... 39 27Fiftynodeuniformcase:greedyaddingandexactsolution ............ 40 28Fiftynodeuniformcase:GADSHandexactsolution ............... 40 29ValuesoftheGAHparameters ........................... 41 210Fiftynodeuniformcase:GAandexactsolution .................. 41 211ObjectivevaluesobtainedfromSAAHonURFLPSFPwithdierentsamplesizes .......................................... 44 212ComputationalperformanceonURFLPSFP:GADH,GADSH,GAandtheenumerationmethod ................................. 44 213ComputationalresultsforURFLPMFPusingGADSHandGA ......... 46 41CoecientsinExample1 .............................. 74 42CoecientsinExample3 .............................. 77 43PerformancecomparisonofExample3 ....................... 78 44PerformancecomparisonofExample4 ....................... 79 51SolutionofaCFFMmodel .............................. 89 52Inputdata(Vi(yi)andPi(yi))forthe3levelmodel ................ 96 53Solutionsofthe3levelmodel ............................ 97 54Solutionsofthe2levelmodel ............................ 98 B1DatasetofURFLPSFP ............................... 107 B2DatasetofURFLPMFP:3level .......................... 109 C1DatasetofDFFM ................................... 113 7
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Figure page 11Researchstructure .................................. 13 21Exampleofcrossoveroperationatposition3 .................... 34 22Objectiveratioacrossdierentsamplesizesforthe50nodedataset ....... 37 23CPUtimeacrossdierentsamplesizesforthe50nodedataset .......... 37 24EvolutionofthesolutionsfromGA ......................... 42 25ComparisonofobjectivevaluesfromGADH,GADSH,GA,andSAAH .... 45 41Seriessystem ..................................... 61 42Parallelseriessystem ................................. 62 43Fivecomponentbridgenetwork ........................... 64 44Reduceprocess .................................... 70 45SevenlinkARPAnetwork .............................. 78 51Totalcostatdierentsystemreliabilitylevel .................... 88 52FailureprobabilityatindividualfacilityvssystemreliabilityLevel ........ 91 53FrequencyofcompletelyopenfacilityinTable 51 ................. 92 54Tradeobetweenobjectiveandresourceused ................... 98 55Forticationlevelatindividualfacilitybyresourceused .............. 100 A1Congurationsbasedonstateofcomponent5inFigure 43 :A)Component5works;B)Component5fails ............................. 104 A2Congurationsbasedonstateofsubsystem4inFigure 45 :A)Subsystem4works;B)Subsystem4fails ............................. 106 8
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Uncertaintyisoneoftheelementsthatmakethisworldsofascinatinganddynamic.However,theexistenceofuncertaintyalsoposesagreatchallengetoreliablesystemdesign.Ourstudyusesvariousmodelsandalgorithmstoaddressreliabilityissuesinthecontextof(1)theuncapacitatedfacilitylocationproblemwherefacilitiesarevulnerable,and(2)thesystemreliabilityproblemwherecomponentsaresubjecttofail. Werststudytheuncapacitatedreliablefacilitylocationprobleminwhichthefailureprobabilitiesaresitespecic.Theproblemisformulatedasatwostagestochasticprogramandthenanonlinearintegerprogram.Severalheuristicsthatcanproducenearoptimalsolutionsareproposedforthiscomputationallydicultproblem.Theeectivenessoftheheuristicsistestedthroughextensivecomputationalstudies.Thecomputationalresultsalsoleadtosomemanagerialinsights.Forthespecialcasewherethefailureprobabilityateachfacilityisaconstant(independentofthefacility),weprovideanapproximationalgorithmwithaworstcaseboundof2.674. Anotherpartofourresearchisrelatedtothedevelopmentandapplicationofamonotonicbranchreduceboundalgorithm,apowerfultooltoobtaingloballyoptimalsolutiontoproblemsinwhichboththeobjectivefunctionandconstraintspossessmonotonicity.Wetailorthealgorithmtosolveamixedintegernonlinearprogrammingproblem.Itsconvergenceanalysisandaccelerationtechniquesarealsodiscussed.Thealgorithmisthensuccessfullyappliedtosolvesystemreliabilityoptimizationproblems 9
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Wealsodevelopseveralmodelsthatcanbeusedtofortifythereliabilityoftheexistingfacilities.Theyaretheextensionstothemodelsintherstpartofthedissertationandoerinsightsonwhichfacilitytochooseandtowhatextentitshouldbefortied.Thepropertiesandsolutionmethodologiesofthemodelsarediscussed.Inparticular,amonotonicbranchreduceboundalgorithmisusedtosolveoneofthesemodels.Theeciencyofthealgorithmisdemonstratedinthecomputationalresults. 10
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Ourstudyfocusesonreliabilityissuesarisinginfacilitylocationdesignproblemsandcomplexsystems.Intheformercase,weconsidermathematicalmodelsthatminimizethesumoffacilityopeningcostsandexpectedserviceandpenaltycostswhenfacilitiesaresubjecttofailfromtimetotime.Thesefailuresmaycomefromdisruptiveevents(e.g.laborstrikes,supplierbusinessfailures,terroristattacks),ornaturaldisasters(e.g.hurricanes,earthquake).Facilityspecicfailureprobabilitiesareexplicitlyconsideredinourmodels.Tothebestofourknowledge,theseappeartobetherstsuchmodelsintheliterature.Thesemodelshelptomakedecisionsinthesystemdesignphase.Severalheuristicsandanapproximationalgorithmareproposedforsolvingthesemodels. Iffacilitieshavebeenbuiltbutarestillsubjecttofail,weconsidermodelstofortifythereliabilityoftheexistingsystemgivenlimitedforticationresources.Thesemodelscanbereducedtoaspecialclassofglobaloptimizationproblems,calledmonotonicoptimization,inwhichboththeobjectivefunctionandconstraintspossessmonotonicity.Aspecializedmonotonicbranchreduceboundalgorithmisdevelopedtoecientlysolvetheseproblems. Wealsoexaminereliabilityissuesincomplexindustrialandmilitarysystems.Thereliabilityofsuchasystemismeasuredbytheprobabilityofsuccessfuloperation.Weaddresstheissueofallocatingunreliablecomponentsinthesystemtoachievethemaximumprobabilityofsuccessfuloperation,adierentobjectivefromthatusedinthefacilitylocationmodel.Theproblemisgenerallycategorizedasasystemreliabilityoptimizationproblem,includingtheclassesofredundancyallocationandreliabilityredundancyallocationoptimizationproblems.Inredundancyallocationoptimization,oneisgiventheoptiontoallocatetheappropriatelevelsofredundancytomaximizereliabilityorminimizethecostofasystemgiventhedesignconstraints.Forexample,ifacomponentofreliabilitylevelat0.9isassignedinparalleltobackup 11
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Thecontributionsofourstudyencompasstheoreticaldevelopments,computationalalgorithmsandpracticalapplications.Inparticular,wemakethefollowingcontributions: 12
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ThestructureofourresearchisdepictedinFigure 11 .InChapter 2 ,wepresentseveralmodelsfortheuncapacitatedreliablefacilitylocationprobleminwhichsomefacilitiesaresubjecttofailurefromtimetotime.Thesemodelsarethefoundationofourresearch.Besidesthegeneralscenariobasedmodel,theyincludethecaseinwhicheachfacilityhasasitespecicfailureprobability,andthecaseinwhicheachfacilityhasmultilevelfailureprobabilities.Thepropertiesanddierentformulationsofthemodelsarethoroughlydiscussed.Severalheuristicsarepresentedalongwiththecomputationalresults. Figure11. Researchstructure InChapter 3 ,wepresentatighterapproximationalgorithmwithaworstcaseboundof2.674foraspecialcaseoftheuncapacitatedreliablefacilitylocationproblem,whereallfailureprobabilitiesareidentical. InChapter 4 ,wepresentamonotonicbranchreduceboundalgorithmforaspecialcaseofthenonlinearmixed/pureintegerprogrammingproblemwhereboth 13
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BasedonthemodelsinChapter 2 ,wedeveloptwomodelsthatareusedtofortifythereliabilityoftheexistingfacilities.Thepropertiesandsolutionmethodologiesofthemodelsarediscussed.Inparticular,themonotonicbranchreduceboundalgorithmpresentedinChapter 4 isusedtosolveoneofthesemodels.Theeciencyofthealgorithmisdemonstratedthroughthecomputationalresults. ThisdissertationisconcludedinChapter 6 withadiscussiononfutureresearchdirections. 14
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34 ],[ 8 ],and[ 45 ]).Mostmodelsintheliteraturehavetreatedfacilitiesasiftheywouldneverfail;inotherwords,theywerecompletelyreliable.Inthischapter,werelaxthisassumptiontomodelamorerealisticcase. Thereliabilityissueweconsiderisundertheframeworkofthesocalleduncapacitatedfacilitylocationproblem(UFLP).InUFLP,wearegivenasetofdemandpoints,asetofcandidatesites,thecostofopeningafacilityateachlocation,andthecostofconnectingeachdemandpointtoanyfacility.Theobjectiveistoopenasetoffacilitiesfromthecandidatesitesandassigneachdemandpointtoanopenfacilitysoastominimizethetotalfacilityopeningandconnectioncosts. UFLPanditsgeneralizationsareNPhard,i.e.,unlessP=NPtheydonotadmitpolynomialtimealgorithmstondanoptimalsolution.ThereisavastliteratureontheseNPhardfacilitylocationproblemsandmanysolutionapproacheshavebeendevelopedinthelastfourdecades,includingintegerprogramming,metaheuristics,andapproximationalgorithms.Onecommonassumptioninthisliteratureisthattheinputparametersoftheproblems(costs,demands,facilitycapacities,etc.)aredeterministic.However,suchassumptionsmaynotbevalidinmanyrealisticsituationssincemanyinputparametersinthemodelareuncertainduringthedecisionmakingprocess. 15
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47 ]).Thecommonfeatureofthereceiversideandinbetweenuncertaintiesisthattheuncertaintydoesnotchangethetopologyoftheproviderreceivernetworkoncethefacilitieshavebeenbuilt.However,thisisnotthecaseifthebuiltfacilitiesaresubjecttofail(providersideuncertainty).Ifafacilityfails,customersoriginallyassignedtoithavetobereassignedtoother(operational)facilities,andthustheconnectioncostchanges(usuallyincreases). Wefocusonthereliabilityissueofprovidersideuncertaintyinthischapter.Theuncertaintyismodeledusingtwodierentapproaches:1)byasetofscenariosthatspecifywhichsubsetofthefacilitieswillbecomenonoperational;or2)byanindividualandindependentfailureprobabilityinherentineachfacility.Althougheachdemandpointneedstobeservedbyoneoperationalfacilityonly,itshouldbeassignedtoagroupoffacilitiesthatareorderedbylevels:intheeventofthelowestlevelfacilitybecomingnonoperational,thedemandcanthenbeservedbythenextlevelfacilitythatisoperational;andsoon.Ifalloperationalfacilitiesaretoofarawayfromademandpoint,onemaychoosenottoservethisdemandpointbypayingapenaltycost.Theobjectiveisthustominimizethefacilityopeningcostplustheexpectedconnectionandpenaltycosts.Thisproblemwillbereferredtoastheuncapacitatedreliablefacilitylocationproblem(URFLP). Inparticular,twovariantsofURFLPareconsideredinthischapterintermsofthecharacteristicsofthefailureprobabilityateachfacility.Intherstone,weassumethatthereisonlyonesitespecicfailureprobabilityateachfacility.We 16
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URFLPisclearlyNPhardasitgeneralizesUFLP.WeproposeseveralheuristicstosolveURFLP.Theyincludethesampleaverageapproximationheuristicforthescenariobasedmodel,thegreedyaddingheuristics,thegreedyaddingandsubstitutionheuristics,andthegeneticalgorithmforthenonlinearintegerprogrammingmodel. Therestofthischapterisorganizedasfollows.InSection 2.2 ,wereviewtherelatedliteratureandprovidesomebasicbackgroundforourmodels.ThenotationandacronymsareintroducedinSection 2.3 .InSection 2.4 ,ascenariobasedmodelisproposed,whichisfollowedbythenonlinearintegermodelforURFLPSPFinSection 2.5 .Section 2.6 containsthenonlinearintegermodelforURFLPSPF.ThethreeheuristicsarepresentedinSection 2.7 .InSection 2.8 ,weconductcomputationalstudiesontheperformanceoftheheuristics.InSection 2.9 ,weconcludethechapterbysuggestingseveralfutureresearchdirections. 38 47 ]).However,aswepointedoutintheIntroduction,amajorityofthecurrentliteraturemainlydealswiththereceiversideand/orinbetweenuncertainties.Thisincludes[ 63 ],[ 10 ],[ 9 ],[ 7 ]and[ 42 ]amongothers. Thefollowingtwopapers,[ 48 ]and[ 5 ],arecloselyrelatedtothischapter.In[ 48 ],theauthorsassumethatsomefacilitiesareperfectlyreliablewhileothersaresubjecttofailurewiththesameprobability.Onthecontrary,weassumethatthefailureprobability 17
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5 ],whichisbasedonthepmedianproblemratherthantheframeworkofUFLP. Thereisalsoasmallstrandofliteraturedevotedtoaddressingtheforticationofreliabilityforexistingfacilities,whichincludes[ 43 ],[ 44 ],and[ 49 ].Thesemodelstypicallyfocusontheinterdictionforticationframeworkbaseduponthepmedianfacilitylocationproblem.Theyaregenerallyformulatedasbilevelprogrammingmodels.Theirmainfocusistoidentifytheexistingcriticalfacilitiestoprotectundertheeventsofdisruption. Inourmodel,thefailureprobabilitiesaresitespecic,whichsignicantlycomplicatestheproblemwhenformulatingitasamathematicalprogram.ThemodelisfurtherextendedtoURFLPMFP,whereeachfacilityisallowedtohavemultiplelevelsofthefailureprobabilities.Weproposetwodierentmodelingapproaches:ascenariobasedstochasticprogrammingapproachandanonlinearintegerprogrammingapproach.Thescenariobasedmodelisattractiveduetoitsstructuralsimplicityanditsabilitytomodeldependenceamongrandomparameters.Butthemodelbecomescomputationallyexpensiveasthenumberofscenariosincreases.Ifthenumberofscenariosistoolarge,thenonlinearintegerprogrammingapproachprovidesanalternativewaytotackletheproblem. Thesampleaverageapproximationmethodiswidelyusedforsolvingcomplicatedstochasticdiscreteoptimizationproblems,e.g.,[ 24 ],[ 42 ],and[ 61 ].Thebasicideaofthismethodistouseasampleaveragefunctiontoestimatetheexpectedvaluefunction.Thustheoriginalproblemistransformedtotheonethatcanbeecientlysolved. Othertypesofthecommonlyusedheuristicsinoptimizationarethelocalsearchanditerativeimprovementalgorithms([ 13 ],[ 16 ],[ 4 ]and[ 3 ]).Theseheuristicsstartwithanemptysetandrepeatedlyconsideraddingapotentialfacilityintothesolution.Ortheystartwithannonemptysetandrepeatedlyconsiderdeletingorsubstitutingafacilityin 18
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60 ]. Wealsoapplyageneticalgorithm(GA)basedheuristictosolvethemodelswedevelop.GAsimitatethenaturalselectioninbiologicalevolution.Assolutiontechniques,theymaintainalargenumberofsolutions,calledthepopulation,andalloweachmemberofthepopulation(calledachromosome)toevolveiterativelyintogoodones.SomegooddescriptionsofGAsareprovidedin[ 11 14 39 ]. GAshavebeenusedtosolvemanycombinatorialoptimizationproblemswithsuccess,includingvariousfacilitylocationproblems,e.g.[ 22 ]and[ 2 ].Inthischapter,wedesignaspecializedGAforthemodelsinwhichweareinterested.Acomputationalstudyonallthreeheuristicsisalsoprovided. TheacronymslistedinTable 21 arefrequentlyusedinthischapter. Table21. Acronyms Acronym Meaning UFLP Uncapacitatedfacilitylocationproblem URFLPSFP Uncapacitatedreliablefacilitylocationproblemwithasinglelevelfailureprobability URFLPMFP Uncapacitatedreliablefacilitylocationproblemwithmultilevelfailureprobabilities GA Geneticalgorithm SAAH Sampleaverageapproximationheuristic GADH Greedyaddingheuristic GADSH Greedyaddingandsubstitutionheuristic 19
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LetSbethesetofscenarios.ForanyA2S,letpAbetheprobabilitythatscenarioAhappens.ThenURFLPcanbeformulatedasthefollowingtwostagestochasticprogram. minimizeXi2Ffiyi+XA2SpAgA(y)subjecttoyi2f0;1g; wheregA(y)=minXj2DXi2FdjcijxAij+Xj2DdjrjzAj s.t.Xi2FxAij+zAj=1;8j2D Intheaboveformulation,thebinaryvariableyiindicatesiffacilityiisopenedintherststage.ParameterIA;iindicatesiffacilityiisoperationalunderscenarioA,whichisaninputregardlessofthevalueofyi.VariablexAijistheassignmentvariablewhichindicates 20
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2{1 ,istominimizethesumofthexedcostandtheexpectedsecondstagecost.Theobjectiveofthesecondstage,i.e. 2{2 ,istominimizetheserviceandpenaltycost.Constraints( 2{3 )ensurethatclientjiseitherassignedtoafacilityorsubjecttoapenaltyateachlevelkinScenarioA.Constraints( 2{4 )and( 2{5 )makesurethatnoclientisassignedtoanunopenfacilityoranonfunctionalfacilityrespectively. Itisstraightforwardtoshowthattheformulation( 2{1 )isequivalenttothefollowingmathematicalprogram. (URFLPSP)minimizeXi2Ffiyi+XASpAXj2DXi2FdjcijxAij+Xj2DdjrjzAj!s.t.Xi2FxAij+zAj=1;8j2D;ASxAijyiIA;i;8i2F;j2D;ASyi;xAij;zAj2f0;1g: However,whenthefailureprobabilitiesareindependent,thepossiblenumberofscenarioscanbeextremelylarge.Therefore,thenumberofvariablesandconstraintsinURFLPSPisexponentiallylargeaccordingly,whichmakesitextremelydiculttosolve.Underthissituation,weproposeseveralalternativenonlinearintegerprogrammingformulationsandecientsolutionalgorithms.Wediscussthesealternativeformulationsinthenexttwosections. 21
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2.5.1NonlinearIntegerProgrammingModel 48 ]toamoregeneralsetting.Comparingto[ 48 ],URFLPSFPcanbeinterpretedslightlydierentlyasfollows.Eachclientshouldbeassignedtoasetoffacilitiesinitially.Thefacilitiesassignedtoanyclientcanbedierentiatedbythelevels:incasealowerlevelfacilityfails,thenextlevelfacility,ifoperational,willprovideserviceinstead. Mathematically,wedenetwotypesofnewbinaryvariablesxkij;zkjtocapturedierentleveloffacilitiesforaclientj.Inparticular,xkij=1iffacilityiisthekthlevelbackupfacilityofclientjandotherwise,xkij=0.zkj=1ifjhas(k1)thbackupfacility,buthasnokthbackupfacilitysothatjincursapenaltycostatlevelk. Giventhevariablesxkij;zkj,onecancomputetheexpectedtotalservicecostasfollows.Consideraclientjanditsexpectedservicecostatitslevelkfacility.Clientjisservedbyitslevelkfacilityonlyifallitsassignedfacilitiesatlowerlevelsbecomenonoperational.Ontheotherhand,foranyfacilityl,ifitisonthelowerlevel(i.e,lessthank)fordemandnodej,thenPk1s=1xslj=1,otherwisePk1s=1xslj=0.Itfollowsthatforclientj,theprobabilitythatallitslowerlevelfacilitiesfailisQl2FpPk1s=1xsljl.Ifjisseveredbyfacilityi,asj'slevelkbackupfacility,thenfacilityihastobeoperationalwhichoccurswithprobability(1pi).Therefore,theexpectedservicecostofclientjatlevelkisPi2Fdjcijxkij(1pi)Ql2FpPk1s=1xsljl.Similarly,wecancalculatetheexpectedpenaltycostofclientjatlevelk,whichisQl2FpPk1s=1xsljldjrjzkj. 22
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minimizeXi2Ffiyi+Xj2DjFjXk=1Xi2Fdjcijxkij(1pi)Yl2FpPk1s=1xsljl+Xj2DjFj+1Xk=1Yl2FpPk1s=1xsljldjrjzkj subjecttoXi2Fxkij+kXt=1ztj=1;8j2D;k=1;:::;jFj+1 (2{8) Thedecisionvariablesxkij;zkjaredenedearlier.Theindicatorvariableyi=1iffacilityiisopenintherststage;otherwiseyi=0.Theobjectivefunction( 2{7 )isthesummationofthefacilitycost,theexpectedservicecost,andtheexpectedpenaltycost.Constraints( 2{8 )ensurethatclientjiseitherassignedtoafacilityorsubjecttoapenaltyateachlevelk.Constraints( 2{9 )makesurethatnoclientisassignedtoanunopenfacility.Constraints( 2{10 )prohibitaclientfrombeingassignedtoaspecicfacilityatmorethanonelevel.Notethatconstraints( 2{9 )and( 2{10 )canbetightenedas 23
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Proof. Inparticular,ifwesetxk+1uj=1andxkvj=1withthevaluesofothervariablesunchanged,wecancomputethenewobjectivevalue.Thedierencebetweenthenewobjectivevalueandtheoriginaloneis AnimplicationofProposition 2.1 isthatifthesetofopenfacilitiesisdetermined,thenitistrivialtosolvethelevelassignmentproblemforeachclient:assigninglevelsaccordingtotherelativedistancesofdierentfacilitiestotheclient.Ifatsomelevelthedistanceisbeyondthepenaltycost,thennofacilitywillbeassignedatthislevel(andhigherones)andthedemandnodesimplytakesthe(cheaper)penalty. Wewouldliketopointouttherelationshipbetweenformulation(URFLPSP)andformulation(URFLPSFP).Sincethesetwoformulationsarejusttwowaysofmodelingthesameproblem,theyshouldhavethesameminimumcostaslongastheinputstothetwomodelsareconsistent.Informulation(URFLPSFP),eachfacilityihasindependentfailureprobabilitypi.Thisimpliesthatthereare2jFjscenariosandtheprobabilitythat 24
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48 ]. Basedontheaboveobservation,weareabletoreduceformulation(URFLPSFP)toalinearintegerprogramasfollows. (URFLPIP) minimizeXi2Ffiyi+Xj2DjFjXk=1Xi2Fdjcijxkij(1p)pk1+Xj2DjFj+1Xk=1pk1djrjzkj subjecttoXi2Fxkij+kXt=1ztj=1;8j2D;k=1;:::;jFj+1jFjXk=1xkijyi;8i2F;j2Dxkij;zkj;yi2f0;1g: 2.8 25
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Givendierentinvestmentlevelsatfacilityi,theoutputofthefailureprobabilityatfacilityi,P0i(yti),isdeterminedbyyti: (2{14) withf0i=0,P(0)=1,andP()isadecreasingfunction.f0i=0andP(0)=1implythatifthereisnoinvestmentatfacilityi,thenitiscompletelynonfunctional. Afterthefailureprobabilityatfacilityiisdetermined,URFLPMFPisessentiallynodierentfromURFLPSFP.Inthefollowingformulation,piintheobjectivefunction 2{7 ofURFLPSFPisreplacedbyP0i(yti). 26
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minimizeXi2FXt(ftiyti)+Xj2DjFjXk=1Xi2Fdjcijxkij(1P0i(yti))Yl2F(P0l(ytl))Pk1s=1xslj+Xj2DjFj+1Xk=1Yl2F(P0l(ytl))Pk1s=1xsljdjrjzkj subjecttoXi2Fxkij+kXt=1ztj=18j;k=1;:::;jFj+1 (2{16) whereconstraints( 2{18 )ensureatmostoneinvestmentlevelisallowedateachfacilityandalltheotherconstraintsaresimilartotheonesinURFLPSFP.NotethatProposition 2.1 stillholdsinthismodel. Theaboveformulation,URFLPMFP1,isabinarymodel.URFLPMFPcanalsobemodeledasaregularintegermodelbyreinterpretingthedenitionofyiastheinvestmentlevelatfacilityi.Thus,yiisnotbinaryanymore;0yiUi,whereUiisthehighestlevelatwhichfacilityicanbepossiblybuilt.ThecorrespondingfailureprobabilityandthexedcostatfacilityiaredenotedbyfunctionsPi(yi)andFi(yi)respectively.Pi(yi);i2F,arenonincreasingfunctionsofyiandPi(0)=1,whereasFi(yi);i2F,arenondecreasingfunctionsofyiandFi(0)=0.ThenURFLPMFPcanbemodeledasfollows. 27
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minimizeXi2FFi(yi)+Xj2DjFjXk=1Xi2Fdjcijxkij(1Pi(yi))Yl2F(Pl(yl))Pk1s=1xslj+Xj2DjFj+1Xk=1Yl2F(Pl(yl))Pk1s=1xsljdjrjzkj subjecttoXi2Fxkij+kXt=1ztj=18j;k=1;:::;jFj+1 (2{22) 0yiUi;8i2F; whereconstraints( 2{23 )ensurenoclientisassignedtoanunopenfacility.Thatis,whenyi=0,xkij=0,8i;j;k=1;:::;jFj.Becauseofthebinaryconstraints( 2{26 )onxkij,therighthandsideofconstraints( 2{23 )canbereplacedbyyiwithoutaectingthefeasibledomain. 2.7.1SampleAverageApproximationHeuristic 24 ],[ 42 ],and[ 61 ]).Thebasicideaofthismethodistorandomlygeneratesamples,thenuseasampleaveragefunctiontoestimatethetrueexpectedvaluefunction.Bydoingso,theoriginalproblemisreducedtoarelativelysmallproblemthatcanberepeatedlysolved.Suchanapproachhasbeenusedbyvariousauthorsovertheyears.WeapplythefollowingprocedurestosolvemodelURFLPSP. 28
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minimizeXi2Ffiyi+NXs=11 RepeatthisstepMtimes.Foreachm=1;2;;M,letymandvmbethecorrespondingoptimalsolutionanditsoptimalobjectivevaluerespectively.InviewofProposition 2.1 ,thelevelassignmentdecision(thesecondstagedecision)canbesolelydeterminedbyym.Computethetrueobjectivevalue^vmusingtheformulationofURFLPSFPforeachym,m=1;2;;M. Tworemarksareinorder.First,inastandardSAAapproach([ 42 ]and[ 61 ]),oneadditionalindependentsampleisneededtoestimatethetrueexpectedvalue^vm.Butinourcase,ananalyticalformulaisreadyforestimatingthetrueexpectedvalue.Second,theaverageofthevmvalues,i.e.vM=PMm=1vm,doesnotprovideastatisticallowerboundfortheoptimalvalue.Thisisdierentfromtheresultin[ 42 ]wheretheuncertaintyonlycomesfromdemandside.Inthecurrentmodel,dierentsamplesmayleadtodierentsolutionspacesoftheproblem,duetotheprovidersideuncertainty.ThereforetheaveragevaluevMisnolongerunbiasedtothetrueexpectedvalue.Nonetheless,vMmaystillserveasagoodindicationofthequalityofthesolutionfromtheSAAapproach,aswewillillustrateinthecomputationaltests. 29
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2.1 ensuresthatthelevelassignmentscanbeeasilyderivedforagivensetofopenfacilities.Therefore,onecanconcentrateonselectingasetofopenfacilitieswithoutworryingtoomuchonthedecisionsoflevelassignment.Letv(T)denotetheobjectivefunctionvaluegivenbythesetofopenfacilities,T.LetTtbethesetofopenfacilitiesatstept,andbetheemptyset. Ingeneral,aswecanseefromthecomputationaltestslater,thegreedyaddingheuristicisabletondahighqualitysolutionveryeciently.ThecomplexityofthisheuristicisO(n4logn),wheren=jFj.GivenTt1,ittakesO(nlogn)todothelevelassignmentsforeachnode,mainlybecauseitinvolvesasortprocessthatisincomplexityofO(nlogn).Thereisnsuchnodes,soittakesO(n2logn)toevaluatethevalueofv(Tt1).Intheworstcase,ittakesnsuchevaluationstogetthemostcosteectivefacility,jtatstept.Thegreedyaddingprocessiteratesatmostntimes,whichleadstothecomplexityofO(n4logn)forGADH. Afterthegreedyaddingheuristic,weperformthefollowinggreedysubstitutionheuristictofurtherimprovethesolution:ateachiteration,asubstitutefacilityischosentoreplacetheexistingopenfacilityifdoingsoreducesthetotalcostthemost.Thisprocedureisrepeateduntilnosubstitutefacilitycanbefoundtofurtherreducethetotalcost.Thesubstitutioncanbeanullfacility.Replacinganopenfacilitywithanull 30
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11 14 39 ]. Inprinciple,aGAcanbeappliedtoanyoptimizationproblem.ButthereisnogenericGAsinceitrequiresmanydesigndecisions,suchastheencodingofthechromosome,theselectionofparents,themethodofthecrossoveroperator.Inthissection,wedescribeaGAthatissuitableforURFLP. 31
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2.1 ,wedonotneedanexplicitencodingofthedemandtofacilityassignments.FormodelURFLPMFP,anadditionalelementisencodedtorepresentthelevelatwhichweinvestinanopenfacility.Table 22 showstheencodingforasystemwith10candidatesitesformodelURFLPMFP,forexample,withopenfacilitiesatnodes3,4and8atinvestmentlevels2,1and3respectively. Table22. SamplechromosomeformodelURFLPMFP Candidatesite 1 2 3 4 5 6 7 8 9 10 Open? 0 0 1 1 0 0 0 1 0 0 Investmentlevel 0 0 2 1 0 0 0 3 0 0 Chromosomesareevaluatedbasedonthevalueofobjectivefunction.Achromosomewithasmallerobjectivevalueistterthanonewithalargerobjectivevalue.ThefollowingparametersareemployedinourdescriptionofheuristicGAH. Table23. GAHparameters Parameter Notation Populationsize 32
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Themainoperatoriscrossover.Twosolutionsareselectedatrandomfromthepopulationatthecurrentgeneration,withabiastowardthebettersolutions.TheprobabilityofselectingthejthbestsolutionisgivenbyNP+1j NP(NP+1)=2.NoteNPisthenumberofsolutionsinthepopulation.Thedenominator,NP(NP+1)=2,isthesumfrom1toNP.Thenumerator,NP+1j,isthereverseorderofj'stnessamongallNPsolutions.ThesevaluesarelistedinTable 24 astheweightinprobabilityevaluation.Aswecanseethatthebetterasolutionis,thehigherweightitisassigned. Table24. SelectionprobabilitiesforapopulationwithNPsolutions SolutionRank WeightinProbabilityEvaluation Probability 1 NP ... ... NP+1j NP(NP+1)=2 ... ... 21 fortwosolutionswiththeonepointcrossoverpositionat3.Encodingofthechildsolutioninthisexampleisthefollowing:valuesofpositions1to3arefromthoseinthesamepositions 33
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Figure21. Exampleofcrossoveroperationatposition3 Afterachildsolutionhasbeenconstructedinthemanneroutlinedabove,withprobabilityPM,thesolutionismutated.Mutationisaccomplishedbyrandomlyselectingtwocandidatesites:oneatwhichafacilityopensandoneatwhichafacilitycloses;thenswappingtheirstates:fromopen(\1")toclose(\0"),andfromclose(\0")toopen(\1").InthecaseofmodelURFLPSFP,arandomlyselectedinvestmentlevelisassociatedwiththenewlyopenfacilitysite. Ifthechildsolutiongeneratedinthismannerdiersfromallothersolutionsintheemergingpopulation,itisaddedtothepopulation;ifitdoesnot,theentireprocess(ofparentselection,crossover,andmutation)isrepeated.WecontinueaddingsolutionstothepopulationuntilthepopulationcontainsNPtotalsolutions.Inotherwords,thesizeofeachgenerationismaintainedtobethesame. Thewholeprocessisrepeateduntiloneofthefollowingterminationcriteriaismet:(1)thealgorithmreachesNGgenerations,or(2)itfailstoimprovethebestknownsolutioninNMgenerations. 34
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Thetestdatasetisgeneratedasfollows.CoordinatesofthesitesweredrawnfromU[0;1]U[0;1],demandofeachsitewasdrawnfromU[0;1000]androundedtothenearestinteger,xedfacilitycostsweredrawnfromU[500;1500]androundedtothenearestinteger,andpenaltycostsweredrawnfromU[0;15].Further,thetransportationcostcijissettobetheEuclideandistancebetweenpointsiandj.Thenumberofsitesvariesfrom10to100.ThedatasetisavailableinAppendix B AllthealgorithmswerecodedinC++andtestedonaDellOptiplexGX620computerrunningtheWindowsXPoperatingsystemwithaPentiumIV3.6GHzprocessorand1.0GBRAM. 25 liststheobjectivevaluesobtainedfromSAAHwhenM=1andthesamplesizevariesfrom10to200. ItisclearfromTable 25 thatthesolutionqualitycanbeimprovedbyincreasingthesamplesize.TheratiosoftheobjectivevalueobtainedfromSAAHwithsamplesizes10,50,100,150,200,and250totheoptimalvalueareplottedinFigure 22 35
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ObjectivevaluesfromSAAHforthe50nodedataset FailureProbability N10 N20 N30 N40 N50 0 7197.27 7197.27 7197.27 7197.27 7197.27 0.1 7956.03 7956.03 7763.80 7763.80 7763.80 0.2 9429.49 8770.31 8425.99 8425.99 8425.99 0.3 9908.52 10059.90 9669.49 10201.50 10095.20 0.4 11546.90 11753.50 11984.00 13887.60 11937.60 0.5 18727.50 15128.1 13120.60 15052.80 13546.40 0.6 27429.00 18161.70 17946.40 17946.40 17946.40 0.7 32965.30 33139.20 27374.80 23659.30 23690.40 0.8 62876.90 46387.50 35743.60 35743.60 35743.60 0.9 84406.10 69255.40 54722.30 54728.10 55647.40 1.0 128009.00 128009.00 128009.00 128009.00 128009.00 FailureProbability N100 N150 N200 N250 7197.27 7197.27 7197.27 7197.27 7197.27 0.1 7763.80 7763.80 7763.80 7763.80 7763.80 0.2 8425.99 8425.99 8425.99 8425.99 8425.99 0.3 9387.85 9387.85 9275.99 9275.99 9275.99 0.4 10848.20 10599.90 10566.30 10762.00 10253.90 0.5 13157.20 13041.50 13099.70 12134.20 11603.00 0.6 17762.80 15609.50 14559.10 14559.10 13416.80 0.7 20740.30 20671.60 19070.90 17289.00 16157.20 0.8 29218.10 26731.10 24066.80 23252.20 21500.70 0.9 46546.80 45061.00 42555.50 38316.90 35987.70 1.0 128009.00 128009.00 128009.00 128009.00 128009.00 Figure 22 showsthatSAAHobtainsfairlygoodsolutionswithasmallsamplesizewhenthefailureprobabilityislow(p0:4).Incontrast,alargesamplesizeisneededtoachievesuchqualityofsolutionsforthecaseswithahigherfailureprobability.Thefollowingcouldbeapossibleexplanation.Notethatasampleisacollectionofdierentscenarios.Whenthefailureprobabilityislow,themajorityofthefacilitiesarecandidatesitesforopeningineachscenario,sothesetsofcandidatefacilitiesaresimilarindierentscenarios.Thus,anyindividualsamplecancapturethecharacteristicsofthesystemprettywell,andthecorrespondingsolutionobtainedfromSAAHisclosetooptimal.Fortheextremecasewherep=0,allfacilitiesareavailabletoopenineachscenario,sothesetsofavailablefacilitiesarethesameineachscenarioandtheSAAHcanproducetheexactsolutioninthiscase.Inthecaseofthehigherfailureprobability,thecandidatesitesfor 36
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Objectiveratioacrossdierentsamplesizesforthe50nodedataset Figure23. CPUtimeacrossdierentsamplesizesforthe50nodedataset openingineachscenarioisrelativelysmallandascenariocanbequitedierentfromanotherone.Thusanincreasedsizeofsamplecanhelptocapturethecharacteristicsofthesystemuncertainty. Figure 23 depictsthecomputationtimefromtherunwithsamplesizes10,20,30,and40.ItshowsthatthecasewithN=10istheonlyonethatrequiresslightlylesstimethantheexactalgorithm,whereasothersrequiremoretimeasthesamplesize 37
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23 isthetaileectoftheCPUtimeintermsofthefailureprobability.SAAHspendsmoretimetoobtainasolutionwhenthefailureprobabilityisaround0.5.Onepossiblewaytoexplainthisphenomenaisthefollowing:whenthefailureprobabilityisaround0.5,theconstraintsxsijyiIAs;iamongdierentsamplesarequitedierent.Asaresult,theproblemsizeincreases,sodoesthecomputationaltime. Next,weexaminetheeectofthereplicationnumber(M)onthesolutionqualitybyxingN=30.Table 26 providestheobjectivevaluesobtainedwhenM=5;10;15;20.Fromtheobjectivevaluesobtainedindierentreplicationnumbers,wecanseethattheincreaseofthereplicationnumberhasnotaectedthesolutionqualitytoomuch.Thegapinthistableisdenedas^vminvM Ingeneral,SAAHiscapabletoproduceafairlygoodsolutionwithalargesamplesizefortheuniformcase.Butitalsorequiresatremendousamountoftimetodosoandmayrunoutofmemoryduetotheincreaseofproblemsize.WedeferpresentingthecomputationalresultsofURFLPSFPtoSection 2.8.4 Table 27 liststhecomputationalresultsofa50nodedatasetwhenthefailureprobabilityvariesfrom0to1.Therstcolumn,P,isthefailureprobabilityateachfacility.The\gap"columnisdenedasthepercentagedierencebetweenthecostofthesolutionobtainedbyGADHandtheoptimalcost. AswecanseefromTable 27 ,GADHndsoptimalornearoptimalsolutionsinmostcasesinlessthan0.05seconds.ComparedtotheexactmethodusingCPLEX,ittakes 38
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RunsfromSAAHforthe50nodedataset P ^vmin ^vmin Exact 0.0 7197.27 7197.27 0.00 7197.27 7197.27 0.00 7197.27 0.1 7763.80 7687.03 1.00 7763.80 7760.14 0.05 7763.80 0.2 8425.99 8315.65 1.33 8425.99 8436.50 0.12 8425.99 0.3 9414.40 9054.28 6.79 9378.06 9112.31 3.32 9275.99 0.4 10872.60 9740.45 23.03 10479.80 9814.09 10.79 10253.90 0.5 11932.00 10457.00 25.47 11932 10497.60 13.66 11603.00 0.6 17825.90 11377.60 57.73 17335.50 11475.90 55.33 13416.80 0.7 27157.40 12758.80 114.56 23227.40 12816.20 111.90 16157.20 0.8 34912.50 14758.30 142.19 31284.20 14761.90 136.50 21500.70 0.9 54722.30 19703.50 177.73 54628.80 19428.60 181.66 35987.70 1.0 128009.00 128009.00 0.00 128009.00 128009.00 0.00 128009.00 P ^vmin ^vmin Exact 0.0 7197.27 7197.27 0.00 7197.27 7197.27 0.00 7197.27 0.1 7763.80 7784.17 0.26 7763.80 7768.5 0.06 7763.80 0.2 8425.99 8484.24 0.69 8425.99 8453.66 0.33 8425.99 0.3 9378.06 9138.80 2.62 9275.99 9152.36 2.47 9275.99 0.4 10479.80 9826.75 6.65 10259.90 9842.41 6.48 10253.90 0.5 11932.00 10507.40 13.56 11932.00 10530.80 13.31 11603.00 0.6 17335.50 11523.50 50.44 17291.50 11563.00 49.92 13416.80 0.7 22894.20 12766.10 81.95 22894.20 12747.70 79.59 16157.20 0.8 31284.20 14736.80 112.29 31284.20 14746.00 112.15 21500.70 0.9 54628.80 19752.00 176.57 53343.90 19811.80 175.74 35987.70 1.0 128009.00 128009.00 0.00 128009.00 128009.00 0.00 128009.00 muchlesstime.Thegreedyaddingalgorithmseemstoperformbetterwhenthefacilityfailureprobabilityishigh.Itactuallyndsoptimalsolutionswhenthefailureprobabilityexceeds0.5.ThisisincontrasttotheperformanceoftheSAAH,whichworksbetterwhenthefailureprobabilityislow. AswepointedoutinSection 2.7.2 ,thesolutionqualityofGADHcanbefurtherimprovedbyGADSH.Thisisclearlydemonstratedinthefollowingcomputationalresults.GADSHactuallyndstheoptimalsolutionsforallinstancesinTable 27 andtheCPUtimesarecomparablewiththosereportedbyGADH.TheresultsaresummarizedinTable 28 39
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Fiftynodeuniformcase:greedyaddingandexactsolution GreedyAddingHeuristic ExactAlgorithm P Objective Time(s) Objective Time(s) gap(%) 0.0 7551.02 0.00 7197.27 6.94 4.92 0.1 8053.11 0.00 7763.80 7.61 3.73 0.2 8637.46 0.00 8425.99 8.94 2.51 0.3 9309.50 0.00 9275.99 10.62 0.36 0.4 10253.90 0.02 10253.90 10.38 0.00 0.5 11622.80 0.02 11603.00 10.86 0.17 0.6 13416.80 0.02 13416.80 12.84 0.00 0.7 16157.20 0.02 16157.20 13.47 0.00 0.8 21500.70 0.02 21500.70 14.08 0.00 0.9 35987.70 0.05 35987.70 14.27 0.00 1.0 128009.00 0.00 128009.00 9.27 0.00 Table28. Fiftynodeuniformcase:GADSHandexactsolution GADSH ExactAlgorithm Gap P Objective Time(s) Objective OpenFacilities Time(s) (%) 0.0 7197.27 0.02 7197.27 0.00 0.1 7763.80 0.02 7763.80 0.00 0.2 8425.99 0.03 8425.99 0.00 0.3 9275.99 0.02 9275.99 0.00 0.4 10253.90 0.02 10253.90 0.00 0.5 11603.00 0.03 11603.00 0.00 0.6 13416.80 0.06 13416.80 0.00 0.7 16157.20 0.09 16157.20 0.00 0.8 21500.70 0.20 21500.70 0.00 0.9 35987.70 0.48 35987.70 0.00 1.0 128009.00 0.00 128009.00 0.00 ItisinterestingtocomparethesetsofopenfacilitiesinTable 28 .Onemightconcludethatmorefacilitiesshouldbeopenasthefacilitiesgetmorevulnerable,thatis,whenthefailureprobabilityincreases.Althoughthisclaimisusuallyvalid,itisnotalwaystrue.Anextremecaseiswhenthefailureprobabilityis1sothatnofacilityshouldopen.Onecanalsoconsiderthefollowingcounterexamplewherethereisonlyonesinglefacilitytoopen.Iff1+d11r1p1
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29 wereusedfortheparametersinGAHdescribedinSection 2.7.3 Table29. ValuesoftheGAHparameters Parameter Value PopulationsizeNP MaximumnumberofgenerationsNG MaximumnumberofgenerationswithoutimprovementNM NumberofreproductionNR NumberofimmigrationNI MutationprobabilityPM Table 210 liststhecomputationalresultsofthe50nodedatasetwhenthefailureprobabilityvariesfrom0to1.Therstcolumn,P,isthefailureprobabilityateachfacility.BecausetheGAheuristicisaprobabilisticmethod,twotrialsareperformed.WereporttheminimumobjectiveitobtainedandtheaverageCPUtime(inseconds)inthesecondandthirdcolumnrespectively.The\gap"columnisdenedasthepercentagedierencebetweenthecostofthesolutionobtainedbyGAandtheoptimalcost. Table210. Fiftynodeuniformcase:GAandexactsolution GAHeuristic(2Trials) ExactAlgorithm P MinObjective Time(s) Objective Time(s) gap(%) 0.0 7197.27 5.41 7197.27 6.94 0.00 0.1 7763.80 5.23 7763.80 7.61 0.00 0.2 8425.99 5.20 8425.99 8.94 0.00 0.3 9275.99 5.11 9275.99 10.62 0.00 0.4 10253.90 5.34 10253.90 10.38 0.00 0.5 11603.00 5.66 11603.00 10.86 0.00 0.6 13416.80 5.52 13416.80 12.84 0.00 0.7 16157.20 5.75 16157.20 13.47 0.00 0.8 21500.70 7.95 21500.70 14.08 0.00 0.9 35987.70 9.83 35987.70 14.27 0.00 1.0 128009.00 3.25 128009.00 9.27 0.00 41
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Figure 24 depictstheevolutionoftheminimumobjectivevalue,andtheaverageobjectivevalueineachgenerationwhenp=0:5.Thealgorithmterminatesatgeneration176afteritndstheoptimalsolutionatgeneration76.Infact,inthisexampletheGAquicklyconvergestoacloseoptimalsolutionafterjust20generationsasshowninFigure 24 Figure24. EvolutionofthesolutionsfromGA 42
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Severaldatasetsarederivedfromthe100nodedatasetinAppendix B :forexample,dataset#10istherst10linesfromTable B1 ofAppendix B ;ithas10demandnodesandfacilitysites.Theotherdatasetsarederivedinthesameway.Table 211 liststheobjectivevaluesobtainedfromSAAHwithdierentsamplesizes.Thecolumn\BestObjective"liststheminimumobjectivevalueamongthedierentsamplesizes.Timeismeasuredinsecondswithresultsfromthesamplesizeof100.\"inTable 211 meansthattheprogramwasoutofmemoryduetothesurgeintheproblemsize.\"inTable 212 meansthattheresultsoftheenumerationmethodwerenotobtainedduetotheexponentiallyincreasedcomputationaltime.Table 212 summarizestheobjectivevaluesofthesolutionsobtainedbyGADH,GADSH,GAandtheenumerationmethod,andtheircorrespondingcomputationaltime.TheresultsofGAareobtainedthroughasinglerun. Comparingtheresultsfromheuristicswiththegloballyoptimalsolutionsinsmalldatasets(from#10,to#30),wecanseethatGADSHandGAndoptimalsolutions,whereasSAAHandGADHndoptimalornearoptimalsolutions.Toevaluatethequalityofthesolutionsfoundbytheseheuristicsinalldatasets,weplottheobjectivevaluesinFigure 25 .ThevaluesfromSAAHarethebestonesavailableinTable 211 foreachinstance.Figure 25 showsthatGADSHandGAcanndthebestsolutionsinalldatasets,whereasSAAHandGADHcanndeitherthebestknownsolutionsor 43
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ObjectivevaluesobtainedfromSAAHonURFLPSFPwithdierentsamplesizes Dateset Best Time(s) # N50 N100 N150 N200 Objective (N100) 10 5850.47 5576.00 5128.24 5128.24 5128.24 0.43 15 6074.06 5337.18 5337.18 5337.18 5337.18 2.11 20 8071.98 5761.79 5761.79 5761.79 5761.79 3.09 25 6749.39 6583.06 6583.06 6583.06 6583.06 12.14 30 7897.19 7622.22 7847.37 7847.37 7622.22 50.67 40 7474.92 7474.92 7474.92 7474.92 7474.92 54.20 50 8719.32 8641.28 8781.18 8641.28 8641.28 185.56 60 9357.37 9394.87 9357.37 9394.87 9357.37 159.48 70 10337.60 10391.80 10383.00 10383.00 10337.60 250.85 80 11054.30 11054.30 11054.30 11054.30 11054.30 320.82 90 12405.50 12405.50 12448.70 12405.50 659.57 100 13977.50 14028.10 13977.50 2164.03 Table212. ComputationalperformanceonURFLPSFP:GADH,GADSH,GAandtheenumerationmethod Dateset GADH GADSH GA Enumeration # Objective T(s) Objective T(s) Objective T(s) Objective T(s) 10 5128.04 0.00 5128.04 0.00 5128.04 1.62 5128.04 0.02 15 5305.04 0.00 5305.04 0.00 5305.04 1.81 5305.04 14.75 20 5761.79 0.00 5761.79 0.00 5761.79 1.81 5761.79 206.51 25 6439.87 0.00 6439.87 0.00 6439.87 1.82 6439.87 6806.57 30 7420.86 0.00 7382.04 0.00 7382.04 2.22 7382.04 188276.35 40 7474.92 0.00 7474.92 0.02 7474.92 4.09 50 8763.75 0.00 8641.28 0.00 8641.28 4.64 60 9357.37 0.00 9357.37 0.02 9357.37 5.14 70 10337.60 0.00 10337.60 0.03 10337.60 5.84 80 11054.30 0.00 11054.29 0.05 11054.29 5.56 90 13030.90 0.00 12405.50 0.06 12405.50 10.75 100 14463.40 0.02 13820.87 0.09 13820.87 11.23 closetothebestknownones.Intermsofcomputationaleciency,SAAHtakesmuchmoretimetoachieveitssolutionsthanallotherthreeheuristics.BetweenGADSHandGA,GADSHspendsconsiderablylessCPUtimethanGA:GADSHtakeslessthan0.1secondstogetthebestresultineachinstance.Overall,theseresultssuggestthatGADSHisthebestoneamongallfourheuristicsformodelURFLPSFPintermsofbothsolutionqualityandcomputationaltime. 44
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ComparisonofobjectivevaluesfromGADH,GADSH,GA,andSAAH B2 ofAppendix B .ThersthalfofTable 213 reportstheresultsfromGA,whichhasbeenrunfor5timeswithdierentrandomseeds.Eightdatasetshavebeengeneratedfortesting.Thebestopensitesandtheiroptimallevelsarelistedinthesecondcolumn.Thebestandworstresultsobtainedinthe5trialsarelistedinthethirdandfourthcolumnsrespectively.Theaveragetimeinsecondsarereportedinthelastcolumn.TheresultsofGADSHareshowninthebottomhalfofTable 213 .Inonlyonecase(the80nodeproblem)doestheGAndabettersolutionthanGADSH.Inthatcase,theobjectivefunctionvalueis11782.85comparedto11859.40,whichrepresentsonlya0.6%improvement.ButGAtakesmoretimethanGADSHdoestogetthissmallimprovement.Inallothercases,GADSHndsthesamesolutionasGAbutwithmuchlesstime.Overall,GADSHismorefavorablethanGAinsolvingmodelURFLPMFP. 45
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ComputationalresultsforURFLPMFPusingGADSHandGA GA,5Trials Dataset Best Worst Average # BestSites(Levels) Result Result Time(s) 20 2(2)7(2) 5214.55 5214.55 3.13 30 2(2)20(1)21(2) 6484.74 6484.74 3.73 40 2(2)20(1)35(2) 7194.88 7194.88 4.52 50 2(2)5(1)20(1)35(1) 8827.95 8827.95 5.71 60 2(2)20(1)35(2)59(1) 9964.84 9964.84 7.20 70 2(2)13(2)20(1)35(2) 10837.65 10837.65 9.06 80 2(2)20(1)35(2)59(1)76(1)79(1) 11782.85 11867.01 10.81 90 2(2)20(1)35(2)76(1)79(1)88(1) 12621.95 12717.84 11.13 100 2(2)13(2)20(1)35(2)67(2)76(1)88(1) 13713.42 13749.58 11.33 GADSH Sites(Levels) Result Time(s) 20 2(2)7(2) 5214.55 0.09 30 2(2)20(1)21(2) 6484.74 0.14 40 2(2)20(1)35(2) 7194.88 0.18 50 2(2)5(1)20(1)35(1) 8827.95 0.27 60 2(2)20(1)35(2)59(1) 9964.84 0.33 70 2(2)13(2)20(1)35(2) 10837.65 0.39 80 2(2)13(2)20(1)35(2)76(1) 11859.40 0.57 90 2(2)20(1)35(2)76(1)79(1)88(1) 12621.95 1.19 100 2(2)13(2)20(1)35(2)67(2)76(1)88(1) 13713.42 1.13 Fourheuristics,SAAH,GADH,GADSHandGA,havebeenproposedtosolvetheseproblems.SAAHisaspecializedheuristicforthescenariobasedmodel,whereas 46
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Thereareseveralinterestingresearchdirections.Wenotethatthemajorlimitationofthecurrentmodelsistheassumptionthatthefacilitiesareuncapacitated.Althoughtheassumptionitselfisverycommoninthefacilitylocationmodels,itmaybeunrealisticinpractice.Inthecapacitatedcase,customeroffailedfacilitiescanbeassignedtothenextlevelbackupfacilitiesonlyiftheyhavesucientcapacitytosatisfytheadditionaldemand.Thisrestrictionmaymakethecapacitatedmodelverycomplex.Itbecomesavaluabletopicoffutureinvestigation.Inaddition,somenewmeasurementsofthereliabilityconceptinthefacilitylocationproblemsettingareworthpursuing. 47
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48 ],wheretheauthorsassumethatsomefacilitiesareperfectlyreliablewhileothersaresubjecttofailurewiththesameprobability.TheyformulatetheirmultiobjectiveproblemasalinearintegerprogramandproposeaLagrangianrelaxationsolutionmethod.However,weconsiderpenaltycost,afactorthatismissedin[ 48 ]. UURFLPisclearlyNPhardasitgeneralizesUFLP.Thefocusofthischapteristoproposeandanalyzeanapproximationalgorithmwithaconstantworstcaseboundguarantee. Designingapproximationalgorithmsforthefacilitylocationproblemanditsvariationshasrecentlyreceivedconsiderableattentionsfromtheresearchcommunity.However,tothebestofourknowledge,thischapterpresentstherstapproximationalgorithmforstochasticfacilitylocationproblemswithprovidersideuncertainty. Thevastmajorityofapproximationalgorithmsforthefacilitylocationproblemmainlydealwithdeterministicproblems,e.g.[ 17 21 32 46 ].Untilveryrecently,approximationalgorithmsforUFLPwithstochasticdemandhavebeenproposed;seethesurveybyShmoysandSwamy[ 52 ].Anotherrelatedpaper[ 6 ]proposesanapproximationalgorithmforafacilitylocationproblemwithstochasticdemandandinventory.Ourapproximationalgorithmmakesuseoftheideasfromseveralpapers[ 17 21 32 46 ].Inparticular,thischapteriscloselyrelatedto[ 17 ],whichpresentsa2.41approximationalgorithmforthesocalledfaulttolerantfacilitylocationproblem(FTFLP):everydemand 48
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Theremainderofthischapterisorganizedasfollows.SeveralequivalentformulationsforUURFLPareproposedinSection 3.2 ,whichleadustodevelopa2.674approximationalgorithminSection 3.3 .ThechapterisconcludedinSection 3.4 2 .RecalltheformulationofURFLPSFPinChapter 2 : (URFLPSFP) minimizeXi2Ffiyi+Xj2DjFjXk=1Xi2Fdjcijxkij(1pi)Yl2FpPk1s=1xsljl+Pj2DPjFj+1k=1Ql2FpPk1s=1xsljldjrjzkj subjecttoXi2Fxkij+kXt=1ztj=1;8j2D;k=1;:::;jFj+1 (3{2) ConsideraspecialcaseofURFLPSFPwhereallfacilitieshavethesamefailureprobabilities,i.e.,pi=p;8i2F.Thisassumptionsimpliesformulation(URFLPSFP)considerablybasedonthefollowingobservation.Becausepi=p;8i2F,itisstraightforwardthatYl2FpPk1s=1xsljl=pk1,whichisindependentofthevaluesofxslj.Thispropertyisimplicitlyusedinamultiobjectiveformulationproposedin[ 48 ]. Basedontheaboveobservation,weareabletoreduceformulationURFLPSFPtoalinearintegerprogramasfollows. 49
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minimizeXi2Ffiyi+Xj2DjFjXk=1Xi2Fdjcijxkij(1p)pk1+Xj2DjFj+1Xk=1pk1djrjzkj subjecttoXi2Fxkij+kXt=1ztj=1;8j2D;k=1;:::;jFj+1jFjXk=1xkijyi;8i2F;j2Dxkij;zkj;yi2f0;1g: 3.1 .WereferthereadertoSection 3.5 fortheproof. minimizeXi2Ffiyi+Xj2DjFjXk=1Xi2Fdjcijxkijpk1(1p)+Xj2D1Xk=1djrjkjpk1(1p) (3{7) subjecttoXi2Fxkij+kj=1;8j2F;k=1;:::;jFjjFjXk=1xkijyi;8i2F;j2Dxkij;kj;yi2f0;1g: 3{6 )andformulation( 3{7 )areequivalent. 50
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WetakeadvantageofseveralresultsforthefaulttolerantversionofUFLP,whereeverydemandpointjmustbeservedbykjdistinctfacilities,aconceptclosetoourlevelassignment.In[ 17 ],Guhaetal.proposeacoupleofapproximationalgorithmsforthefaulttolerantfacilitylocationproblemusingvariousroundingandgreedylocalsearchtechniques. Thefaulttolerantfacilitylocationproblemcanbeformulatedasthefollowingintegerprogram. (FTFLP) minimizeXi2Ffiyi+Xj2DkjXk=1Xi2Fdjcijwkjxkij subjecttoXi2Fxkij1;8j2D;kkjkjXk=1xkijyi;8i2F;j2Dxkij;yi2f0;1g: OneofthekeyresultsonFTFLPfromGuhaetal.[ 17 ]issummarizedbelow.
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1Xj2DkjXk=1Xi2Fwkjdjcijxkij: 3.1 ,isaparameterthatwecanchoosetocontrolthequalityofapproximation.Indeed,theapproximationratiosofthealgorithmsofGuhaetal.arefunctionsof.Onecanthenchoosethebesttominimizetheapproximationratio. WearenowreadytopresentouralgorithmforUURFLP.Werstsolvealinearprogrammingrelaxationofformulation( 3{7 ). minimizeXi2Ffiyi+Xj2DjFjXk=1Xi2Fdjcijxkijpk1(1p)+Xj2D1Xk=1djrjkjpk1(1p) (3{9) subjecttoXi2Fxkij+kj=1;8j2D;k=1;:::;jFj Assumethat(x;y;)isanoptimalsolutiontothislinearprogram.Ouralgorithmroundsthefractionalsolution(x;y;)toanintegersolution(x;y;)thatisfeasibletoformulation( 3{7 ). Thealgorithmisbasedonapropertyoftheoptimalfractionalsolution(x;y;),whichisformalizedinthefollowinglemma.Thislemmaenablesustoutilizeknownalgorithmsandanalysisforthefaulttolerantfacilitylocationproblem.
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(ii).Ifthereexistsksuchthat0k. Proof. 3.1 ,whichisthusomittedhere. Wepresentourroundingprocedurenext.Foreachj2D,assumekjisthesmallestintegersuchthatkjj>0. Theroundingprocedureiscarriedoutintwophases.Therstphaseroundstheoptimalfractionalsolution(x;y;)toanotherfractionalsolution(^x;^y;^),whichisfeasibletoalinearprogrammingrelaxationofanappropriatelydenedfaulttolerantfacilitylocationproblem.Inthesecondphase,weuseanalgorithmforthefaulttolerantfacilitylocationproblemtoroundthefractionalsolution(^x;^y;^)toanintegersolution(x;y;),whichisfeasibletoformulation( 3{7 ).
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17 ]withaparameter2(0;1]toroundthesolution(^x;^y)toafeasiblesolutionofafaulttolerantfacilitylocationproblem,whereasetoffacilitiesisopensuchthateachclientjisservedbyatleastmjdistinctopenfacilities. ThistwophasealgorithmshallbereferredtoasAlgorithmTP.Itisobviousthatthesolution(x;y;)isfeasibletoformulation( 3{7 ).Wenowestablishaworstcaseapproximationboundofour(randomized)algorithm,i.e.,weshallshowthatthe(expected)totalcostisnomorethanaconstantfactortimestheoptimalcost.WeboundthetotalpenaltycostinLemma 3.3 ,andboundthetotalfacilityandtransportationcostsinLemma 3.4 1 1CLP; 3{7 ).
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3.5 forproofsofbothlemmas.Animmediateconsequenceofthesetwolemmasisthefollowingcorollary. 1 1 1;1 1 1 1;1 32 ]fordetails)tofurtherimprovetheapproximationfactor. ThegreedyimprovementprocedurecanimprovetheworstcaseboundofAlgorithmTP,asshownnext.Weomittheproofhereastheanalysisisverysimilartothosein[ 17 ]and[ 32 ]. 3{7 ),thereisa(Rf+ln();1+Rc1 ;1+Rp1 )approximationalgorithm.
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3.1 (1).Assumewehaveanoptimalsolution(x;y;z)forproblem( 3{6 ),thenwecanconstructafeasiblesolution(x;y;)forproblem( 3{7 )withthesameobjectivevalue. Letx=xandy=y.Foreachj2Dandk1,denekj=Pkt=1ztj.Becauseforeachj,thereisexactlyonevalueofztjwouldbeequalto1,wecanconcludekj2f0;1g.Anditisstraightforwardthat(x;y;)isafeasiblesolutiontoproblem( 3{7 ).Nowwechecktheobjectivevaluecorrespondingto(x;y;).Infact,weonlyneedtoconsider1Xk=1pk1(1p)kj=1Xk=1pk1(1p)minfk;jFj+1jgXt=1ztj=jFj+1Xt=11Xk=tpk1(1p)ztj=jFj+1Xt=1pt1ztj 3{7 ),thenwecanconstructafeasiblesolution(x;y;z)forproblem( 3{6 )withthesameobjectivevalue. 56
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Nowwedeneanewsolution(~x;~y;~)forproblem( 3{7 )asfollows:~y=y;forj6=j0,~xkij=xkij,~kj=jforalli6=i0andk;fork6=k0;k0+1,~xkij0=xkij0,~kj0=j0foralli6=i0;xk0i0j0=1,k0j0=0,xk0+1i0j0=0andk0+1j0=1.Itiseasytoverifythat(~x;~y;~)isafeasiblesolutionforproblem( 3{7 ).Thedierencebetweentheobjectivevaluecorrespondingto(x;y;)andtheobjectivevaluecorrespondingto(~x;~y;~)isdj0ci0j0pk0(1p)+dj0rj0pk01(1p)dj0ci0j0pk01(1p)+dj0rj0pk0(1p) 3{7 ).Itfollowsthatrj0ci0j0: 3{7 )anditsatisestheconditionthatif^k0j0=1then^k0+1j0=1. Therefore,wecanassumethatforeachj2D,ifkj=1,thenk+1j=1.Nowwearereadytondafeasiblesolutionforproblem( 3{6 ).Letx=x;y=y.Andforeachj,ifkj=1andk1j=0,thenletzkj=1andletzk0j=0fork06=k;if1j=1,thenletz1j=1andzk0j=0fork02.Thenitisclearfromthedenitionthatkj=Pkt=1ztj.Thenwe 57
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3{6 ).Andfromtherstpartoftheproof,weknowtheobjectivevaluecorrespondingto(x;y;z)isthesameastheonecorrespondingto(x;y;).Thiscompletestheproof. 3.3 Weboundtheexpectedpenaltycostforeachclientj2D,whichdependsonthevalueofthatisrandomlygenerated.Weconsidertwocases. 58
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3.4 Thecostsofinterestdependonthevalueoftherandomlygenerated.Recallthedenitionof(^x;^y).Foreachj2Dandi2F,^xkij=xkij,ifk6=mj;Ifmj=kj1,then^xkjij=0xkjij;ifmj=kj,then^xkjij=xkjij ^xkij1 1xkij:(3{13) Bytheconstraintofthelinearprogrammingrelaxationof( 3{7 ),weknowthat,foreachi2Fandj2D,jFjXk=1xkijyi: ^yi=maxj2DkjXk=1^xkijmaxj2D1 1kjXk=1xkij1 1yi:(3{14) ToshowthatXi2F^xkij=18j2D;kmj; 3{10 ,Xi2F^xkij=Xi2Fxkij=1. 59
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3{10 ,i.e.Xi2Fxkij+kjj=1. Thus^xand^ysatisfythefollowingconstraints:Xi2F^xkij=18j2D;kmjmjXk=1^xkij^yi8i2F;j2D^xkij0;0^yi1 Therefore,ifweapplythealgorithmin[ 17 ]withaparametertoroundthesolution(^x;^y),thenbyLemma 3.1 ,wecanconstructasolution(x;y)suchthattheexpectedtotalfacilitycostandtotaltransportationcostisboundedabovebyln1 1Xj2DmjXk=1Xi2Fdjcij^xkijpk1(1p)1 1ln1 1Xj2DmjXk=1Xi2Fdjcijxkijpk1(1p)!1 10@ln1 1Xj2DkjXk=1Xi2Fdjcijxkijpk1(1p)1A: 3{13 and 3{14 ;thesecondoneholdsbecauseofmjkj. Finallynoticethatwasuniformlydistributedin(0;),thusE[1 1]=Z01 11 1: 60
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41 isRs=nYi=1ri; Figure41. Seriessystem Systemreliabilitycanbeimprovedinvariousways,suchasphysicalenhancementofcomponentreliability,provisionofredundantcomponentsinparallel,andallocationofinterchangeablecomponents. Unlikeinaseriessystem,inaparallelsystem,notallcomponentsarenecessaryforthesystemtoworksuccessfully.Actually,onlyonecomponentinsuchsystemneedstoworkproperlyinorderforthewholesystemtoworkproperly.Includingncomponentswhenonlyoneisessentialiscalledredundancy.Theothern1componentsareincludedtoincreasetheprobabilitythatthereisatleastoneworkingcomponent.Redundancyisawidelyusedtechniqueinengineeringtoenhancesystemreliability. 61
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42 Figure42. Parallelseriessystem Thesystemreliabilitycanbemaximizedbychoosingtherightcombinationofxiundercertainresourceconstraints,denotedbygj(),j=1;:::;minthischapter.Thisleadstoaredundancyallocationoptimizationproblem(RAOP)oranonlinearintegerprogrammingingeneral. (RAOP):maxRs=f(x1;:::;xn) (4{1) subjecttogj(x1;:::;xn)cj;8j=1;:::;m; 0lixiui;8i=1;:::;n; 62
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Anotherwaytoincreasethesystemreliabilityistosimplyusemorereliablecomponents,whichcertainlycostsmoreintermsofvariousresources.Thisproblemiscalledareliabilityallocationoptimizationproblem([ 28 ]).Supposethereareuidiscretechoicesforcomponentreliabilityatstageifori=1;:::;k(n)andthechoiceforcomponentreliabilityatstagek+1;:::;nisonacontinuousscale.Letri(1);ri(2);:::;ri(ui)denotethecomponentreliabilitychoicesatstageifori=1;:::;k(n).Thenthecontinuous/discretereliabilityallocationoptimizationproblemcanbeformulatedasfollows: maxRs=f(r1(x1);:::;rk(xk);rk+1;:::;rn) (4{5) subjecttogj(r1(x1);:::;rk(xk);rk+1;:::rn)cj;8j=1;:::;m; 0rliriuui;8i=k+1;:::;n; whererlianduuiarethelowerboundandupperboundrespectivelyforthecomponentreliabilityatstagei.Ifk=0,theabovemixedintegernonlinearprogrammingformulationreducestoapurenonlinearprogrammingproblem. Thesystemsweareinterestedinarenotlimitedtoseriesandparallelsystems.Theycanbecomplex(general)systemsthatarenonseriesandnonparallel,suchasthebridgenetworkinFigure 43 .Thesystemreliabilityofsuchsystemcanbecomputedbytheconditionalprobabilitytheory.Forexample,thesystemreliabilityofthevecomponentbridgenetworkinFigure 43 canbecomputedbasedonwhethercomponent5isfunctionalornot.WereferreaderstoAppendix A forthedetailsofthefollowingexpression.Rs=(r1+r3r1r3)(r2+r4r2r4)r5+(r1r2+r3r4r1r2r3r4)(1r5):
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Figure43. Fivecomponentbridgenetwork Constraintsofasystemcanbethetotalweight,thetotalcost,thetotalvolume,andsoon.Ingeneral,suchconstraintsareinnonlinearforms[ 28 ].Asthenumberofcomponentsineachsubsystem/stageincreases,moreconnectingequipmentisrequiredandthus,thecostandweightmayincreaseexponentially[ 53 ]. Next,wepresentamoregeneralformulationforthesystemreliabilityoptimizationproblem(SROP),whichincludestheredundancyallocationoptimizationandthereliabilityallocationoptimizationproblemsasspecialcases.Formally,SROPcanbeformulatedasanonlinearmixedintegerprogrammingproblem. (SROP):maxRs=f(x1;:::;xq;r1;:::;rp) (4{9) subjecttogj(x1;:::;xq;r1;:::;rp)cj;8j=1;:::;m; 0lixiui;8i=1;:::;q; 0rlkrkruk;8k=1;:::;p; 64
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TheSROPmodelcoversavastmajorityofreliabilityoptimizationmodelsdiscussedintheliterature.Forexample,modelSROPreducestomodelRAOPwhenp=0,andacontinuousversionofthereliabilityallocationoptimizationproblemwhenq=0.Itcertainlycanalsomodelthegeneralcaseofthereliabilityallocationoptimizationproblembyreinterpretingthedenitionofthevariables,sinceobjectivefunctions 4{5 and 4{9 aremathematicallyequivalent,soareconstraints 4{6 and 4{10 .Inaddition,modelSROPisobviouslyareliabilityredundancyallocationmodelifp=q,whereateachstagethedecisionsarewhichcomponentreliabilitytochooseandhowmuchredundancyaswell. TheSROPmodelhasreceivedtremendousresearchattentionsoverdecadesandhasbeenextensivelystudiedandsolvedusingmanydierentmathematicalprogrammingtechniquesandheuristicapproaches.Kuoetal.[ 28 ],alongwith[ 27 ],provideadetailedintroductiontothemodelsandalgorithmsinthereliabilityoptimization.SROPisoftencharacterizedbyanonlinearobjectivefunctionthatisneitherconvexnorconcaveoveranonconvexfeasibleregion.Duetotheextremedicultyofsuchtypeofproblem,thesolutionmethodsintheliteraturearemainlyheuristics,metaheuristicsandapproximationalgorithms.Acomprehensivereviewonthesemethodscanbefoundin[ 27 ]. 65
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37 ],[ 28 ],[ 51 ],[ 29 ]).Thebranchandboundalgorithmisapopularapproachforndingglobaloptimalsolutionsforthenonconvexproblems.Theydierin(i)methodsofselectingabranchingvariable,(ii)methodsofselectingabranchingnode,(iii)methodsofcalculatingupperandlowerbounds.Forexample,[ 28 ]providesawaytotransformtheintegervariablesintobinaryvariablesthatarebranchedonafterwards.[ 37 ]branchesonthevariablesinincreasingorderoftherange,thedierencebetweenupperboundandlowerbound,thenxestheminthatorder.[ 51 ]and[ 29 ]adoptaconvexicationmethodtotransformtheproblemintoaconvexmaximizationproblemduringtheprocessofthebranchandbound.Thebranchingisdoneonafractionalvariable.[ 50 ]followsasimilarthemeto[ 51 ]and[ 29 ]exceptthatthebranchisdoneontheselectedboundarypointsanditonlysolvesthecontinuousversionofthemonotoneoptimizationproblem. TheMonotonicBranchReduceBound(briey,mBRB)algorithmpresentedinthischapterfollowsthedevelopmentofaspecializedalgorithmformonotonicoptimizationproposedbyTuyandhiscollaboratorsinaseriesofpapers,[ 40 ],[ 56 ],[ 57 ].Theyuseaunionofhyperrectanglestocovertheboundaryofthefeasibleregion.Thehyperrectangleiscalledapolyblockintheirlanguagesothatthealgorithmiscalledapolyblockalgorithm.Abranchandboundimplementationofthepolyblockalgorithmwasmentionedin[ 57 ]forthecontinuousversionofthemonotoneoptimizationproblem.Theeciencyofthisapproachisreportedonvariousclassesofglobaloptimizationproblems,suchaspolynomialfractionalprogramming[ 59 ],anddiscretenonlinearprogramming[ 58 ]. Comparedtothepreviousbranchandboundalgorithms,theproposedmethodexploitsthemonotonicitypropertiesinherentinSROPwithoutrequiringanyconvexity,concavity,dierentiability,andseparability.Unlike[ 28 ],itdoesnotrequireconversionoftheoriginaldecisionvariablesintobinaryones.Themethodisecientandexibleenoughtosolvepure,mixedinteger,andintegernonlinearprogrammingproblemsarisingfrom 66
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Theremainderofthischapterisorganizedasfollows.ThemonotonicbranchreduceboundalgorithmispresentedinSection 4.2 withitsconvergenceanalysis.Severalconvergenceaccelerationtechniquesarealsodiscussed.ThealgorithmisappliedtosolveboththereliabilityallocationandthereliabilityredundancyallocationoptimizationproblemsinSection 4.3 withademonstrationofitseciency.ThechapterisconcludedinSection 4.4 4{9 ),toamoregeneralvectorformatbydenotingx=(x1;:::;xq;r1;:::;rp),inotherwords,xq+i=ri,8i=1;:::;p. (SROP'):maxRs=f(x) (4{14) subjecttox2G=fxjgj(x)cj;8j=1;:::;mg; where[xL;xU]denotesahyperrectanglewithlowestboundaryxLandgreatestboundaryxU;functionsfandgiarenondecreasingforj=1;:::;m.ToensureGisclosed,weassumegjaresemicontinuousforj=1;:::;m. Sincenoconcavityassumptionhasbeenmadeontheobjectivefunction,multiplelocallyoptimalsolutionsmayexist.However,fromthemonotonicityofthefunctionsfandgi,thefollowingpropositioncanbeeasilyderived. 56 ]Theglobalmaximumoff(x)overG\Xz\[xL;xU],ifitexists,isattainedonitsboundary. 4.1 isusedbyTuytodeveloptheoriginalPolyblockAlgorithmformonotonicoptimizationin[ 56 ].Thebranchreduceboundimplementationofthepolyblockalgorithmwasmentionedin[ 57 ]forthecontinuousversionofthe 67
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InthedescriptionofAlgorithm 1 ,Sdenotesahyperrectanglepartition;Lisalistofunfathomedhyperrectangles;isapredenedoptimalitytoleranceparameter;xbestandfbestdenotethecurrentbestsolutionandobjectivevaluerespectively;UB(S)istheupperboundoftheobjectivefunctionoverS.Besidesinitialization,themajorstepsaredescribedindetailasfollows. SelectandBranch ReduceandBound 68
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12 inAlgorithm 1 .ThenthesubsequentbranchingiscarriedoutalongthelongestsideiofS,withi=argmaxjfsUjsLjg.Ifiq,thenS=[sL;sU]ispartitionedtoS1andS2onthediscretevariable:S1=sL;sUsUisLi 69
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19 ].Thelongestedgebisectionruleisexhaustive,asindicatedintheproofofTheorem 4.1 Althoughthelongestedgebisectionruleisapopularchoiceforbranching,onecanuseotherrulessuchasthelongestedgetrisection,andthelargestincrementedgebisection.Inthelargestincrementaledgebisection,theedgeisselectedonwhichwiththemostincrementintheobjectivefunction.Thatisi=argmaxjff(sLnsLj[sUj)g,wheresLnsLj[sUjdenotesthevalueofsLexceptthatitsjthindexisreplacedbysUj. 44 showsS1andS2afterreduction. Figure44. Reduceprocess Inthecase,i>q,withS1=sL;sUsUisLi 70
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56 ].Inthispaper,wesimplyusef(sU).Apartitionisdiscardedifitsupperboundislessthanthecurrentbestsolution,orifitdoesnotcoveranyfeasiblesolution,asshowninline 22 inAlgorithm 1 1 .TheresultissummarizedinTheorem 4.1 1 eitherterminatesafterniteiterations,producinganoptimalsolutionordetectingitsinfeasibility;oritisinnitewithasequenceofhyperrectanglesSpk=[Lpk;Upk]suchthatlimk!1Lpk=limk!1Upk=x,wherexisanoptimalsolution. Proof. 1 terminateswithinniteiterations,itproducesanoptimalsolutionordetectsitsinfeasibility.IfAlgorithm 1 doesnotterminatewithinniteiterations,allthediscretevariablesmusthavebeenxed.Inaddition,amongtheinnitelygeneratedpartitions,thereexistsasequencesuchthat[Lpk+1;Upk+1][Lpk;Upk]. 71
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FromTheorem 4.1 ,wecanaddthefollowingconditionafterline 16 inAlgorithm 1 toterminatethealgorithminniteiterations. 4.2.4AccelerationTechniques 1 ,besidesthereductionprocess,thereareseveralotheraccelerationtechniquesworthmentioning. wehaveallfeasiblexixLi+iei.Inotherwords,ximin(xUi;xLi+iei),whichprovidesatighterupperboundforxi. 72
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1 choosesthepartitionwiththegreatestupperbound.Computationally,adelicateruletoperiodicallyselectthepartitionbetweenthegreatestupperboundandtheleastupperboundhelpstheconvergenceofthealgorithm. Algorithm 1 employsthelongestedgebisectionrule.Onecanuseotherrulessuchasthelongestedgetrisection,andthelargestincrementaledgebisectiontospeeduptheconvergenceinsomecases.Inthelargestincrementaledgebisection,theedgeisselectedonwhichwiththemostincrementontheobjectivefunction.Thatis,i=argmaxjff(sLnsLj[sUj)g,wheresLnsLj[sUjdenotesthevalueofsLexceptthatitsjthindexisreplacedbysUj. 1 )isappliedtosolvevariousproblemsarisingfromsystemreliabilityoptimization.Mostoftheseproblemsarethoroughlystudiedintheliteraturewithvarioussolutiontechniques,mainlyheuristics.Thelimitationofaheuristicisthatitmayndgoodsolutions,butwithoutguaranteedoptimality.Incontrasttotheheuristictechniques,thepresentedmBRBAlgorithm,asanexactmethod,canndanoptimalsolutioninanitenumberofsteps.ThealgorithmthatsolvesallthefollowinginstancesisimplementedinC++andtestedonaDellOptiplexGX620computerwithaPentiumIV3.6GHzprocessorand1.0GBRAM,runningundertheWindowsXPoperatingsystem. 73
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28 ].Theproblemwasoriginallypresentedin[ 53 ]. maxRs=5Yi=1Ri(xi) (4{20) subjecttog1=5Xi=1pix2iP; whereRi=1(1ri)xiisthereliabilityofstagei.Constraint 4{21 ,g1,isimposedonthecombinationofweightandvolume:piistheproductofweightperunitandvolumeperunit.Componentreliabilitydoesnotusuallyaecttheweightnorthevolume,henceg1isnotafunctionofri([ 54 ]).Constraint 4{22 ,g2,isthecostconstraintwherecixiisthecostofallcomponentsatstageiandciexpxi 4{23 ,g3,istheweightconstraintwherewixiistheweightofallcomponentsatstagei.Theadditionalfactor,expxi 54 ]).Theweightconstraintisnotafunctionofcomponentreliability.ThecoecientsaregiveninTable 41 Table41. CoecientsinExample1 74
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28 ]. maxRs=4Yi=1Ri(xi) (4{25) subjecttog1=10exp0:02 1R1(x1)+10x2+6x3+15x4150; wherestagereliabilitiesare 75
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36 ],[ 23 ],and[ 1 ],allofwhichspentmoretimetosolvethanthemBRBalgorithmandcouldnotverifytheoptimalityoftheirsolutions,althoughtheywereabletoproducethesamesolution.Asacorrectiontotheliterature,itisnotedthat[ 1 ]containsthewrongexpressionofthisexample. maxRs=5Yi=1Ri(xi) (4{30) subjecttog1=5Xi=1pix2iP; 0ri1;8i=1;:::;5; wheretheobjectivefunctionandconstraintsarethesameasthoseofExample1exceptamoreexplicitlyexpressionfortheunitcostofcomponenti,whichisafunctionofthecomponentreliabilityri.Toderivetheexpressionofci(ri)=it 54 ]andassumethattheunitcostofcomponentiisadecreasingfunctionof 76
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42 Table42. CoecientsinExample3 Thisproblemisconsidereddiculttosolveintheliterature.Tothebestofourknowledge,thereisnoexactmethodbeingappliedtosolvethisproblem.WecomparetheperformanceofmBRBwiththeTHKheuristicin[ 55 ],theGAGheuristicin[ 15 ],theKLXZmethodin[ 26 ],thesurrogateconstraintsalgorithmHNNNin[ 18 ],andthegeneticalgorithmGAin[ 20 ].ThecomparisonresultsaresummarizedinTable 43 ,wheretheCPUtimelistedinthelastcolumnismeasuredinseconds.Therstcolumnliststhenamesofthemethods.Thesolutionsandtheobtainedsystemreliabilityarelistedinthesecondandthethirdcolumnrespectively.Thefourthcolumn,Rs(UB),liststheupperboundofthesystemreliability. Thenumbersinbracketsafter\mBRB"intherstcolumnisthevalueof,thepredenedoptimalitytolerance.Asonecansee,themBRBalgorithmproduceshigherqualitysolutionswithknownupperboundcomparingtootheralgorithms,whichareabletooutputsomefeasiblesolutionsbutwithoutupperboundguarantee.Withadditional 77
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PerformancecomparisonofExample3 Method (x;r) CPU(s) mBRB(0.01) (3,2,2,3,3,0.77500,0.87500,0.89250,0.71500,0.79000) 0.930947 0.940913 1.38 mBRB(0.001) (3,2,2,3,3,0.78250,0.87500,0.89938,0.70750,0.79000) 0.931541 4.76 THK (3,3,2,2,3,0.78438,0.82500,0.90000,0.77500,0.77813) 0.915363 GAG (3,2,2,3,3,0.80000,0.86250,0.90156,0.70000,0.80000) 0.930289 KLXZ (3,3,2,3,2,0.77960,0.80065,0.90227,0.71044,0.85947) 0.929750 HNNN (3,2,2,3,3,0.77489,0.87007,0.89855,0.71652,0.79137) 0.931451 GA (3,2,2,3,3,0.77943,0.86848,0.90267,0.71404,0.78689) 0.931578 CPUtime,themBRBalgorithmcanproduceasolutionof(3;2;2;3;3;0:77781;0:87187;0:90281;0:71313;0:78625) withthesystemreliabilityat0.931669,anditsupperboundof0.933111.Thisresultcanserveasabenchmarkforthisexample. 35 ],[ 51 ]).Thederivationofthereliabilityofthisnetwork,i.e.theobjectivefunction,isprovidedinAppendix A Figure45. SevenlinkARPAnetwork 78
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(4{36) +R3R5R6Q7(Q1+R1Q2)+R1R2R5R7Q3Q4Q6+R2R3R4R6Q1Q5Q7+R1R3R4R5Q2Q6Q7subjecttog1=x1x2+0:5x1ln(1+x3)+x4+2x5+0:3exp0:02 1R6 +0:3exp0:01 1R727;g2=(x1+2x2+1:2x3)ln(1+x1+x2+2x3)+0:4x4 +0:2x5exp0:02 1R6+0:5exp0:01 1R729;0:5Ri0:99;8i=6;7; whereRi=1(1ri)xi;8i=1;:::;5;Qi=1Ri;8i=1;:::;7;andr1=0:70;r2=0:90;r3=0:80;r4=0:65;r5=0:70: 51 ].ThecomparisonbetweenthemBRBmethodandtheconvexicationoneisdoneinTable 44 ,wherebothalgorithmsobtaintheoptimalobjectivevalue.Admittedly,comparingCPUsecondsdirectlydoesnotreecttheabsoluteeciency,sincetheconvexicationmethodwasimplementedinanoldercomputersystem.However,thetableclearlyshowsrelativeeciencyofthemBRBalgorithm:IttakesonlynegligibleCPUsecondstogettheoptimalsolution. Table44. PerformancecomparisonofExample4 Method(x;r)RsRs(UB)CPU(s) mBRB(4,1,3,4,3,0.9845,0.9900)0.999740.999740.01Convexication(4,1,3,4,3,0.9845,0.9899)0.9997438.63 a 79
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Weexpectthatthemonotonicbranchreduceboundalgorithmcanbeappliedtootherclassesoftheproblem,suchasnonlinearmultidimensionalknapsackproblems,andgeneralizedmultiplicativeprogrammingproblems.Inthefutureresearch,wealsowouldliketocomparetheperformanceoftheaccelerationtechniquesmentionedinthischapter,andanalyzeitsworstcaseperformancetheoreticallyandcomputationally. 80
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2 ,weconsidertheimpactofunreliabilityfromthefacilitieswhenthesystemisinitiallydesigned.Inthatcase,wearegiventheoptiontobuildfacilitiesfromscratch.However,redesigninganentiresystemisnotalwaysanavailableoptiongiventhepotentiallylargeexpensesinvolvedinclosingexistingfacilitiesoropeningnewones.Inmanysituations,methodsforprotectingexistinginfrastructuremaybepreferablegivenlimitedresourcesavailable.Inthischapter,weaddresstheissuesonfortifyingthereliabilityofexistingfacilities. Amajorityofresearchonreliablesupplychaindesignhasbeenfocusedontheinitialsystemdesign,butnotonhowtoimprovetheexistingsystem.Theseworkshavebeenwelldocumentedin[ 47 ]andsurveyedinChapter 2 aswell.However,reinforcingthecomponentsofanexistingsystemmaybecomemorevaluableandrealisticconsideringtheincreasedpotentialdisruptionsanduncertainties.Thesedisruptionsanduncertaintiesmayevolvefromthenewchallengesthatwereneverfacedwhentheinitialsystemsweredesigned.Theycanbemanmadedisruptiveeventsornaturaldisasters,forexample,theSeptember11,2001terroristattackandHurricaneKatrinain2005. Onlyasmallstrandofliteraturehasbeendevotedtoaddressingtheforticationofexistingfacilities,whichincludes[ 43 ],[ 44 ],and[ 49 ].Theseworkstypicallyfocusontheinterdictionforticationframeworkbaseduponthepmedianfacilitylocationproblem.Theproblemsaregenerallyformulatedintheformofbilevelprogramming.Thesemodelscanhelptoidentifythecriticalfacilitiestoprotectundertheeventsofdisruption. Anotherrelatedstrandofliteratureisonthenetworkinterdictionproblemsthataremainlydevelopedformilitaryapplications,e.g.[ 12 ],[ 33 ],[ 62 ]and[ 30 ].Thesemodelsstudytheimpactoflosingoneormoretransportationlinksornetworkarcsbasedonthe 81
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Inthischapter,wefollowthethemeofChapter 2 ,butassumethatthefacilitieshavebeenbuiltandcanbereinforcedtobemorereliable.Wealsoassumethatthefacilitiesareuncapacitated.Wepresenttwonovelmodelswhoseobjectivesaretominimizetheexpectedconnection(service)andpenaltycostbyallocatingthelimitedforticationresourcestotheopenunreliablefacilities.Intherstmodel,theforticationeortsarecontinuous,thatis,thefailureprobabilityateachfacilityvariesfrom0to1attheforticationstage.WecallittheContinuousFacilityForticationModel(CFFM).Onthecontrary,inthesecondmodeltheforticationeortsaresubjecttodierentlevelofresources.Thefailureprobabilityateachfacilitycanonlybechosenfromasetofdiscretelevelsattheforticationstage.AccordinglywecallittheDiscreteFacilityForticationModel(DFFM).Bothmodelscanhelptoidentifythecriticalfacilitiestoprotectandoptimallydeterminehowmuchresourcesshouldbeallocatedtoachievetheobjective.Tosomeextent,themodelsmathematicallyresemblethereliabilityallocationproblemwediscussedinChapter 4 Theremainderofthischapterisorganizedasfollows.InSection 5.2 ,wepresentthecontinuousfacilityforticationmodelandrevealitsconnectiontothegeneralizedlinearmultiplicativeprogrammingandtheinherentmonotonicity.AnexampleispresentedinSection 5.2.2 toillustratethesolutionstructureandproperties.InSection 5.3 ,wepresentthediscretefacilityforticationmodelandapplythemonotonicbranchreducealgorithmtoexploitthemonotonicitypropertiesinherentintheproblem.TheeciencyofthealgorithmisdemonstratedinSection 5.3.2 .Section 5.3.2 alsocontainsananalysisofthesolutionstructureandtradeobetweencostdeductionandforticationeort.WeconcludethischapterinSection 5.4 82
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Inthereliablefacilitylocationmodelsetting,eachclientisassignedtoasetofbackupfacilities,whichisdierentiatedbythelevels:incaseofalowerlevelfacilityfails,thenextlevelfacility,iffunctional,willbackitup.xkij=1iffacilityiisthekthlevelbackupfacilityofdemandnodejandxkij=0otherwise;zkj=1ifjhas(k1)thbackupfacility,buthasnokthbackupfacilitysothatjincursapenaltycostatlevelk.ContrarytothemodelsinChapter 2 ,xkijandzkjarenotdecisionvariablesanymore.Theyareusedtospecifytheexistingnetwork. Tocomputetheexpectedfailurecost,wefollowthelogicinChapter 2 .Firstofall,weneedtocomputetheexpectedfailurecostatlevelkservedbyfacilityi.Eachdemandnodejisservedbyitslevelkfacilityifallthelowerlevelfacilitiesbecomenonoperational.Foranyfacilityl,ifitisassignedtoalowerlevel(i.e,lessthank)fordemandnodej,thenPk1s=1xslj=1,otherwiseitiszero.SotheprobabilitythatalllowerlevelsfacilitiesfailisQl2FpPk1s=1xsljl.Andjisservedatlevelkbyi,whichhastobeoperational.Theprobabilityis(1pi).Therefore,theexpectedfailurecostatlevelkservedbyfacilityiisdjcijxkij(1pi)Ql2FpPk1s=1xsljl.Similarly,wecancalculatethepenaltycostatlevelk,whichisQl2FpPk1s=1xsljldjrjzkj. Wecannowformulatethecontinuousfacilityforticationmodelasfollows: 83
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minimizeXj2DjFjXk=1Xi2Fdjcijxkij(1pi)Yl2FpPk1s=1xsljl+Xj2DjFj+1Xk=1Yl2FpPk1s=1xsljldjrjzkj subjecttogj(p1;:::;pjFj)0;j=1;:::;m 0pi1;8i2F: Theobjectivefunction( 5{1 )isthesumoftheexpectedfailurecostandtheexpectedpenaltycost.Constraints( 5{2 )denotevariousresourcerestrictionsontheforticationlevels,pi;8i2F.Weassumegj(j=1;:::;m)areconvexfunctionssothatthesolutiondomainisconvex.Constraints( 5{3 )arenaturalconstraintsonthefailureprobability. 41 ]fordetails). (GLMP) minimizetXj=1kjYi=1(cTijx+dij) (5{4) subjecttox2X wherecij2Rn,dij2Rn,j=1;:::;t;i=1;:::;pj,andx2Rn,Xisanonemptyconvexset.Notethatwithoutthesummationsignintheobjectivefunction 5{4 ,GLMPisreducedtolinearmultiplicativeprogramming,anotheractivetopicinglobaloptimization. Werstshowthattheobjectivefunction( 5{1 )isthesumoflinearmultiplicativetermswithpositivecoecients.Inotherwords,theobjectivefunction( 5{1 )canbereducedtotheformof wherejarenonnegativeandijarebinary. 84
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5{1 ,ofthecontinuousfacilityforticationmodelcanbereducedtotheformofXjjjFjYi=1pijiwherejarenonnegativeandijarebinary. Proof. Theonlynegativetermis Atlevelk+1,twocasescanhappen: 1. Nofacilityisassignedtodemandjatlevelk+1.Itissubjecttopenaltycost Becauserjci0jinthiscase,addingterms 5{8 and 5{9 leadstodj(rjci0j)pi0Yl2FpPk1s=1xsljl; 85
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Facilityi00isassignedtoit.Andthecorrespondingservicecostis Addingterms 5{8 and 5{10 ,wehave FromProposition 2.1 ,weknowthatci00jci0j0,sotheonlynegativetermleftis Thistermcanbeabsorbedinthenextlevelassignmentfollowingasimilaranalysisatlevelk.Bydoingthisprocessrecursively,demandjwillbeeventuallysubjecttopenaltycostatcertainlevelhigherthank+1andonlycase1canhappen.Therefore,allcosttermsrelatedtodemandnodejhavetheformofjjFjYi=1piji.Thiscompletestheproof. AsadirectresultfromProposition 5.1 ,thefollowingCorollaryholds. 5{1 ofthecontinuousfacilityforticationmodelismonotonicallynondecreasing. 5{6 isaspecialcaseofFunction 5{4 :Ifij=0infunction 5{6 ,thensetcij=0anddij=j;Ifij=1,thensetcij=jei,eibeingtheithunitvectorinRnanddij=0.Therefore,thecontinuousfacilityforticationproblemisaspecialformofthegeneralizedlinearmultiplicativeprogramming.Thelatterismultiextremalandpossessesseverallocalminima[ 41 ].TheexistingalgorithmsforGLMPincludeouterapproximationmethods[ 25 ],vertexenumerationmethods[ 19 ],heuristicsmethods[ 31 ],amongothers.Corollary 5.1 alsoallowsustoapplythemonotonicbranchreduceboundalgorithmpresentedinChapter 4 whentheresourceconstraintspossessmonotonicityaswell. 86
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whereLisnonnegativeandcanbeinterpretedastheindicationofthe(un)reliabilityofthewholesystem.Inotherwords,withtheavailableforticationresourcesthesystemcanbemaintainedatthe(un)reliabilitylevelofL.IfLispositive,noteveryfacilitycanoperateatpi=0.Ontheotherhand,L=0meansthatthesystemistotallyfunctional:Nofacilityhasapossibilitytofail.TheCFFMmodelisthenformulatedasfollows, minimizeXj2DjFjXk=1Xi2Fdjcijxkij(1pi)Yl2FpPk1s=1xsljl+Xj2DjFj+1Xk=1Yl2FpPk1s=1xsljldjrjzkj subjecttoXi2FpiL 0pi1;8i2F: Anetworkwith20demandnodesand5openfacilitiesisconsideredinthisexample.DemandsdjweredrawnfromU[0;1000]androundedtothenearestinteger,andxandycoordinatesweredrawnfromU[0;1].TransportationcostscijaresettobeequaltotheEuclideandistancebetweeniandj.ThepenaltycostrjweredrawnfromU[0;15].ThedatasetisavailableinAppendix C .Facilities2,5,15,18,and20areopeninthisexample.WeuseavertexenumerationmethodtosolvethisproblemandobtaintheresultsatvariouslevelofLwhichareshowninTable 51 andplottedinFigure 51 .TherstcolumnofTable 51 isthe(un)reliabilityofthesystem.Thesecondtosixthcolumnsreportthesolutionofthisexample:theproleofthefailureprobabilityateachopenfacility.Itisfollowedbyacolumnshowingthetotalcost.Thelastcolumncalculates 87
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Table 51 andFigure 51 showhowthesystemdeteriorates(totalcostincreases)astheresourcetomaintainthesystemreliabilitydecreases.Inaddition,thelastcolumnofTable 51 indicatesthatthecurveinFigure 51 ispiecewiselinear.Forexample,thedierence,similarlytheslope,intotalcostisaconstantwhenLchangesfrom1.0to2.0inastepof0.1.Thisisbecauseof(1)thefailureprobabilitiesofallbutonefacility(20,inthiscase)remainthesame,thewholecoststructuredoesnotchange;(2)Changingthefailureprobabilityinonefacilitywillonlyaectalltheservicecostthatrelatedtothisparticularfacility. Figure51. Totalcostatdierentsystemreliabilitylevel ThereisacommonpatterninTable 51 :whenLincreasesby0.1,onlyoneofthefacilitieschangesthefailureprobabilityaccordingly.Forexample,whenLincreasesfrom1.7to1.8,thefailureprobabilityoffacility20jumpsfrom0.7to0.8,alltheothersremainingthesame.Thefailureprobabilityatanindividualfacilityusuallychangesbyeither0.1or0.0asshowninmostcasesinFigure 52 .However,thereisonenotable 88
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SolutionofaCFFMmodel FailureProbabilityat 5 15 18 20 TotalCost Dierence 0 0 0 0 0 0 1582.93 n/a 0.1 0 0 0.1 0 0 1590.37 7.44 0.2 0 0 0.2 0 0 1597.80 7.43 0.3 0 0 0.3 0 0 1605.24 7.44 0.4 0 0 0.4 0 0 1612.67 7.43 0.5 0 0 0.5 0 0 1620.11 7.44 0.6 0 0 0.6 0 0 1627.55 7.44 0.7 0 0 0.7 0 0 1634.98 7.43 0.8 0 0 0.8 0 0 1642.42 7.44 0.9 0 0 0.9 0 0 1649.86 7.44 1.0 0 0 1 0 0 1657.29 7.43 1.1 0 0 1 0 0.1 1689.40 32.11 1.2 0 0 1 0 0.2 1721.51 32.11 1.3 0 0 1 0 0.3 1753.62 32.11 1.4 0 0 1 0 0.4 1785.73 32.11 1.5 0 0 1 0 0.5 1817.84 32.11 1.6 0 0 1 0 0.6 1849.95 32.11 1.7 0 0 1 0 0.7 1882.06 32.11 1.8 0 0 1 0 0.8 1914.17 32.11 1.9 0 0 1 0 0.9 1946.28 32.11 2.0 0 0 1 0 1 1978.39 32.11 2.1 0.1 0 1 0 1 2050.81 72.42 2.2 0.2 0 1 0 1 2123.23 72.42 2.3 0.3 0 1 0 1 2195.65 72.42 2.4 0.4 0 1 0 1 2268.06 72.41 2.5 0.5 0 1 0 1 2340.48 72.42 2.6 0.6 0 1 0 1 2412.90 72.42 2.7 0.7 0 1 0 1 2485.31 72.41 2.8 0.8 0 1 0 1 2557.73 72.42 2.9 0.9 0 1 0 1 2630.15 72.42 3.0 1 0 1 0 1 2702.56 72.41 3.1 1 0 1 0.1 1 2863.49 160.93 3.2 1 0 1 0.2 1 3024.41 160.92 3.3 1 0 1 0.3 1 3185.34 160.93 3.4 1 0 1 0.4 1 3346.26 160.92 3.5 1 0 1 0.5 1 3507.19 160.93 3.6 1 0 1 0.6 1 3668.11 160.92 3.7 1 0 1 0.7 1 3829.04 160.93 3.8 0.8 1 0 1 1 3972.66 143.62 3.9 0.9 1 0 1 1 4099.09 126.43 4.0 1 1 0 1 1 4225.52 126.43 89
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52 ifwelookatthechangesfromL=3:7toL=3:8insubguresAtoD:a0.2dipinA,a1.0surgeinB,a1.0sinkinC,anda0.3jumpinD.ThisphenomenonisfromthemultiextremalnessoftheCFFMproblemthatwediscussinSection 5.2.1 .Usingabisectionsearch,wendoutthatatL=0:749842,bothvertex(1,0,1,0.749842,1)andvertex(0.749842,1,0,1,1)producethesameobjectivevalueof3909.2476.ItmeansthatatL=0:749842,theoptimalvertextransientfrom(1,0,1,0.749842,1)to(0.749842,1,0,1,1).ThenfromL=0:749842toL=0:8,thesamepatternstillholds,facility2istheonlyonethatchangesitsfailureprobability. AswepointoutintheintroductionthatCFFMmodelcanalsohelptondthekeyfacilitytofortify.Inthisexample,thefrequencythatafacilityiscompletelyopenisdepictedinFigure 53 .Itshowsthatfacility5isverycriticalinthissystem,becauseitischosentobecompletelysecured38timesoutofthetotal41casesinthisexample. 90
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FailureprobabilityatindividualfacilityvssystemreliabilityLevel 91
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FrequencyofcompletelyopenfacilityinTable 51 5.2 arethat(1)thelimitedforticationresourceisuniformlydistributedacrossallthefacilities;and(2)thedistributedresourceisdividable.However,inmanycases,theforticationresourcecanonlybediscretelydistributed.Thatis,theforticationresourceatafacilityiscategorizedintodierentlevels,whicharefunctionsofavailableresources.Forexample,7000unitsofresourcemayimprovethefailureprobabilityofafacilitytop=0:4,while8000unitscanimproveittop=0:3.Butthereisnoamountofresourcethatcanimprovethefailureprobabilitytoanumberbetweenp=0:4andp=0:3.Thisinspiresustoconsideradiscretefacilityforticationmodel(DFFM). InDFFM,letyibetheforticationlevelatfacilityi.Naturally,weassumeyiisapositiveintegerandonlyoneforticationlevelisallowedateachfacility.ThecorrespondingfailureprobabilityandtheamountofresourcearedenotedbyfunctionsPi(yi)andVi(yi)respectively.Pi(yi),i2F,arenonincreasingfunctionsofyi,whereasVi(yi),i2F,arenondecreasingfunctionsofyi.Thisisbecausemoreforticationresourcewouldmakeafacilitymorereliable,andconsequentlyasmallerfailureprobability.LettheupperboundofyibeUiandthetotalresourcebeR. 92
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(DFFM) minimizeXj2DjFjXk=1Xi2Fdjcijxkij(1Pi(yi))Yl2FPl(yl)Pk1s=1xslj+Xj2DjFj+1Xk=1Yl2FPl(yl)Pk1s=1xsljdjrjzkj subjecttoXi2FVi(yi)R; 1yiUi;8i2F Theobjectivefunction( 5{17 )isthesumoftheexpectedfailurecostandtheexpectedpenaltycost.Constraint( 5{18 )denotestheresourcerestrictionsonthefortication.Constraints( 5{19 )and( 5{20 )areintegralconstraintsontheforticationlevels. 5{18 )possessesthemonotonicity,sinceVi(yi),i2F,arenondecreasingfunctionsofyi.TheobjectivefunctionofthecontinuousfacilityforticationmodelisshowntobemonotonicallynondecreasinginCorollary 5.1 .Asimilarresultholdsinthediscretecase,becausePi(yi),i2F,arenonincreasingfunctionsofyi.Fromthefactthatthecompositeofanondecreasingfunctionandanonincreasingfunctionisnonincreasing,wehavethefollowingcorollary,whichisparalleltoCorollary 5.1 93
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5{17 ofthediscretefacilityforticationmodelismonotonicallynonincreasing. 4 .Tobeconsistentwiththegeneralformatofthemonotonicoptimizationinthisthesis,wetransformDFFMmodeltoamaximizationproblem. (DFFMmax) maxmizeXj2DjFjXk=1Xi2Fdjcijxkij(1Pi(yi))Yl2FPl(yl)Pk1s=1xsljXj2DjFj+1Xk=1Yl2FPl(yl)Pk1s=1xsljdjrjzkj subjecttoXi2FVi(yi)R; 1yiUi;8i2F AmoregeneralformatofDFFMmaxcanbewrittenasthefollowingbyreplacingthedecisionvariablesyiwithxi. (MO):maxRs=f(x) (5{25) subjecttox2G=fxjgi(x)ci;8i=1;:::;mg; where[xL;xU]denotesahyperrectanglewithlowestboundaryxLandgreatestboundaryxU,functionsfandgiarenondecreasingfori=1;:::;m.ToensureGisclosed,weassumegiaresemicontinuousfori=1;:::;m. Forthecompletenessofexposition,wealsoincludethedescriptionofthemonotonicbranchreduceboundalgorithmwithslightdierencesthanthatinChapter 4 to 94
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2 ,Sdenotesahyperrectanglepartition;Lisalistofunfathomedhyperrectangles;isapredenedoptimalitytoleranceparameter;xbestandfbestdenotethecurrentbestsolutionandobjectivevaluerespectively;UB(S)istheupperboundoftheobjectivefunctionoverS.Besidesinitialization,themajorstepsaredescribedasfollows. SelectandBranch ReduceandBound
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5.2.2 ,whichislistedinAppendix C .Thedatasetcontains20demandnodesand5openfacilitieswiththefollowingspecication. Table52. Inputdata(Vi(yi)andPi(yi))forthe3levelmodel Level1 Level2 Level3 OpenFacilityi V P V P V P 0 0.55 79 0.39 688 0.02 5 0 0.85 614 0.45 728 0.28 15 0 0.75 303 0.63 855 0.48 18 0 0.52 135 0.48 409 0.20 20 0 0.30 178 0.24 273 0.22 ThemonotonicbranchreduceboundalgorithmisimplementedinC++.TheCPUsecondsarereportedfromaDellOptiplexGX620computerwithaPentiumIV3.6GHzprocessorand1.0GBRAM,runningundertheWindowsXPoperatingsystem. ThecomputationalresultsarereportedinTable 53 thatshowsthattheforticationlevelateachopenfacilitygiventheresourceconstraints.Table 54 showstheresultsthattheforticationlevelislimitedto2.Thatis,yi2;8i2F.TheCPUtimereportedinbothtablesclearlyshowtheeciencyofthemonotonicbranchreduceboundalgorithmforthistypeofproblem.Next,westarttoanalyzethecomputationalresults. 96
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Solutionsofthe3levelmodel Constraint Resource ForticationLevelat CPU Objective 5 15 18 20 25000 2953 2020.69 3 3 3 3 3 0.11 2952 2858 2030.01 3 3 3 3 2 0.89 2857 2401 2057.66 3 3 2 3 3 1.08 2400 2306 2068.99 3 3 2 3 2 1.19 2305 2098 2087.25 3 3 1 3 3 1.48 2097 2003 2100.18 3 3 1 3 2 1.03 2002 1825 2138.98 3 3 1 3 1 1.80 1824 1711 2276.06 3 2 1 3 1 1.94 1710 1689 2392.85 3 3 1 1 3 2.72 1688 1594 2408.41 3 3 1 1 2 2.58 1593 1551 2415.59 3 3 1 2 1 2.72 1550 1416 2455.11 3 3 1 1 1 3.33 1415 1370 2489.01 3 1 1 3 3 3.05 1369 1275 2516.42 3 1 1 3 2 2.88 1274 1097 2598.63 3 1 1 3 1 3.70 1096 1096 2797.01 3 1 1 2 3 3.34 1095 1001 2830.34 3 1 1 2 2 3.64 1000 961 2841.01 3 1 1 1 3 3.95 960 866 2875.19 3 1 1 1 2 3.17 865 823 2930.35 3 1 1 2 1 3.48 822 688 2977.74 3 1 1 1 1 3.48 687 666 3308.47 2 1 1 3 2 2.27 665 488 3503.25 2 1 1 3 1 2.56 487 409 3894.44 1 1 1 3 1 2.56 408 392 4318.51 2 1 1 2 2 2.58 391 352 4331.16 2 1 1 1 3 2.86 351 257 4462.80 2 1 1 1 2 2.09 256 214 4688.43 2 1 1 2 1 2.09 213 79 4857.74 2 1 1 1 1 1.97 78 0 5670.71 1 1 1 1 1 1.64 InFigure 54 ,weplottheobjectivevaluesinthesetwodierentsettings(3levelconstraintand2levelconstraint)acrosstheresourceused.Unliketheresultinthecontinuousfacilityforticationmodel,asshowninFigure 51 ,thereisnopiecewiselinearpropertyexhibitedinthisdiscreteversion.Insteadthecurvesareshownsteeperintheearlierstage.Thatis,theforticationeortshelptoreducealotoftotalcostatearlierstage.Thisindicatesthatrealiabilitycanbedrasticallyimprovedwithoutlarge 97
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Solutionsofthe2levelmodel Constraint Resource ForticationLevelat CPU Objective 5 15 18 20 25000 1309 3315.22 2 2 2 2 2 0.11 1308 1174 3402.08 2 2 2 1 2 0.88 1173 1006 3461.27 2 2 1 2 2 1.03 1005 871 3557.78 2 2 1 1 2 1.19 870 828 3666.45 2 2 1 2 1 1.02 827 693 3776.24 2 2 1 1 1 1.19 692 560 4193.94 2 1 2 1 2 1.50 559 392 4318.51 2 1 1 2 2 1.19 391 257 4462.80 2 1 1 1 2 1.34 256 214 4688.43 2 1 1 2 1 1.33 213 79 4857.74 2 1 1 1 1 1.19 78 0 5670.71 1 1 1 1 1 1.06 increasesinforticationresource.Afterthat,thewholesystemseemslackingtheroomforimprovement.Infact,whenRisgreaterthan1500,thedecreaseinthetotalcostcannotjustifytheforticationresource.Forexample,whenRincreasesfrom1551to2003,thetotalcostreducedfrom2415.59to2100.18,anetlossof136.59. Figure54. Tradeobetweenobjectiveandresourceused 98
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54 ,wealsoseethebenettohavea3leveloptionovera2levelone.Whentheavailableforticationresourceisverylimited,bothsharethesameobjectivevaluesasthereisnoenoughresourcetofortifythefacilitytolevel3.Butastheavailableforticationresourceincreases,the3levelmodelhasmoreexibilitysothatitincursmuchlowertotalcost. Anotherphenomenondierentfromthecontinuousversionistheforticationlevelatindividualfacility.Inthecontinuousversion,theforticationlevelismoreorlessmonotonicasshowninFigure 52 .However,thisdoesnotexistinthediscreteversionasshowninFigure 55 .Astheforticationresourceincreases,theforticationlevelatanindividualfacilitydoesnotnecessarilyincreaseasaconsequencefromthiscombinatorialoptimization.FiguresDandEinFigure 55 clearlyshowthisphenomenon,becausefacility18and20donotadmitanymonotonicitypattern. Inthecontinuousversionofthemodel,theforticationeortisdividable.Weshowthatthemodelisaspecialcaseofthegeneralizedlinearmultiplicativeprogrammingproblem.Wesolveanillustrativeexamplebythevertexnumerationmethod,whichisveryeectiveinsolvingthistypeofproblems.Thisexamplealsodemonstratesthemultiextremenessoftheproblem:severalverticesachievetheminimum.Managerially,theexampleshowshowtoidentifythekeyfacilitiestofortifyinasystem. Thediscretefacilityforticationmodelfocusesonchoosingthesuitableforticationlevelateachfacilitywhentheforticationeortisdividedintodierentlevels.Themodelisshowntobeamonotonicityoptimizationproblemsincethemonotonicitypropertyisinherentinbothitsobjectiveandconstraints.Thismodelisthereforesolvedbythe 99
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Forticationlevelatindividualfacilitybyresourceused 100
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Themainlimitationofthecurrentmodelsistheassumptionthatthefacilitiesareuncapacitated.Althoughtheassumptionitselfisverycommoninthefacilitylocationmodels,itmaybeunrealisticinpractice.Inthecapacitatedcase,a`customer'ofthefailedfacilitiescanbeassignedtothenextlevelbackupfacilitiesonlyiftheyhavesucientcapacitytosatisfytheadditionaldemands.Thismaymakethecapacitatedmodelverycomplex.Weexpectthatthemonotonicbranchreduceboundalgorithmwillstillbeapplicable.Webelievethatthisisavaluabletopicworthyoffutureinvestigation. 101
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Inthischapter,wesummarizethevariousmodelsandalgorithmsdiscussedthroughoutthedissertation,andpointoutdirectionsforfutureresearch. Westudytheimpactofuncertaintyonthedecisionsoffacilitylocationanddemandassignment.Theuncertaintyisrepresentedbythefailureprobabilityineachfacility.Severalnovelmodelshavebeenpresentedtooersolutionsforboththedesignofinitialsupplychainsystemsandtheimprovementoftheexistingsystems.Werstinvestigatetheuncapacitatedreliablefacilitylocationmodel,whoseobjectiveistominimizethetotalofopeningcost,expectedservicecost,andexpectedfailtoservepenaltycostwheneachfacilityhasasitespecicfailureprobability.Wealsostudyamoregeneralcasethateachfacilityhasmultiplelevelsoffailureprobabilitiesthatcanbechosen.Ifthesupplychainsystemalreadyexists,weproposetwomodelsforoptimallyallocatingtheforticationresourcetoreducetheexpectedserviceandfailtoservepenaltycost. Thealgorithmspresentedinthisdissertationinclude(1)fourheuristics,thesampleaverageapproximationheuristic,thegreedyaddingheuristic,thegreedyaddingandsubstitutionheuristic,andthegeneticalgorithmbasedheuristic;(2)theapproximationalgorithmwithaworstcaseboundof2.674;(3)themonotonicbranchreduceboundalgorithm.Anindepththeoretictreatmentisprovidedfortheapproximationalgorithm.Allotheralgorithmsarethoroughlytestedinthecomputationalstudies.Thefourheuristicsareusedtosolvetheuncapacitatedreliablefacilitylocationproblem.Themonotonicbranchreduceboundalgorithmisappliedtosolvethefacilityforticationproblemaswellasthesystemreliabilityproblemarisingfromindustrialormilitaryapplications. Oneimmediateextensionistostudythecapacitatedversionofthecurrentreliablefacilitylocationmodels.Althoughthecapacitatedconstraintsgenerallyposemorechallengesonndingtheecientalgorithms,weexpectthatsomeoftheheuristicsarestill 102
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Itwouldbeinterestingtoseeifthereexistsanyapproximationalgorithmwithaconstantworstcaseboundforthecapacitateduniformreliablefacilitylocationproblem.Basedontheresearchonapproximationtheoryoffacilitylocationproblemwithoutconsideringthereliabilityissue,weexpectthatsomemoredelicatetechniquesarerequiredtodevelopsuchapproximationalgorithm. Anotherinterestingdirectionistointroducedierentrisk/reliablitymeasurementsintothecurrentmodelsdependingontheneedsinreality.Bydoingso,theobjectivefunctionsofcurrentmodelswillchangeaccordingly.Forexample,onemayhavemoreinterestsinthecostoftheworstcasescenarioinsteadoftheexpectedcostinthecurrentmodels. Insummary,thecurrentworkwepresentservesasausefulfoundationforfurtherresearchofmorecomplicatedmodelsanddelicatealgorithms. 103
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WeuseconditionalprobabilitytoderivetheexpressionsofreliabilitiesofthenetworksinFigure 43 andFigure 45 43 canbewrittenbasedonwhethercomponent5isfunctionalornot. Whencomponent5works,theoriginalnetworkinFigure 43 isreducedtoFigure A1 (A),whichisaparallelseriessystemwithareliabilityof FigureA1. Congurationsbasedonstateofcomponent5inFigure 43 :A)Component5works;B)Component5fails 104
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43 isreducedtoFigure A1 (B),whichisaseriesparallelsystemwithareliabilityof Substitutionofequations A{2 and A{3 intoequation A{1 yieldsthereliabilityofthevecomponentbridgenetworkdepictedinFigure 43 : (A{4) 45 representsasubsystem.Blocks6and7areindividualcomponents.RecallthatRi=1(1ri)xi;8i=1;:::;5;Qi=1Ri;8i=1;:::;7:ReliabilityofthenetworkinFigure 45 canbewrittenbasedonwhethersubsystem4isfunctionalornot. Whensubsystem4works,theoriginalnetworkinFigure 45 isreducedtoFigure A2 (A),whosereliabilitycanbeobtainedbyapplyingparallelandseriesreductions: 45 isreducedtoFigure A2 (B).Aseriesreductiononsubsystems1and2producesasupercomponent,whichhelpstomapthetopologyinFigure A2 (B)tothevecomponentbridgenetworkinFigure 43 .Afterthemapping,wecandirectlyusetheresultofequation A{4 byreplacingr1withR1R2,r2
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Congurationsbasedonstateofsubsystem4inFigure 45 :A)Subsystem4works;B)Subsystem4fails withR3,r3withR6,r4withR7,r5withR5: (A{7) Substitutionofequations A{6 and A{7 intoequation A{5 yieldsthereliabilityofthevecomponentbridgenetworkdepictedinFigure 45 : Equation A{8 canbereformulatedasthefollowingone: 106
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ThemeaningofeachcolumninTable B1 isprovidedasfollows:#idenotesthefacilityname;(x;y)isthecoordinates,diisthedemand;fiisthexedcost;riisthepenaltycost;andpiisthefailureprobability. TableB1.DatasetofURFLPSFP #i x y di x y di 0.82 0.18 957 938 5.32 0.81 51 0.63 0.04 486 971 3.14 0.63 2 0.54 0.7 202 642 1.9 0.39 52 0.53 0.32 548 1023 0.31 0.94 3 0.91 0.72 186 1230 3.11 0.42 53 0.89 0.99 870 754 0.66 0.76 4 0.15 0.31 635 1008 1.83 0.36 54 0.02 0.19 335 734 0.22 0.97 5 0.74 0.16 737 1279 1.34 0.28 55 0.51 0.32 446 1249 3.93 0.91 6 0.58 0.92 953 1431 2.3 0.83 56 0.53 0.06 198 1371 3.77 0.31 7 0.6 0.09 450 1187 7.96 0.98 57 0.81 0.86 212 1489 9.92 0.58 8 0.37 0.19 188 1044 3.42 1 58 0.53 0.36 903 863 4.06 0.29 9 0.7 0.52 206 1466 9.05 0.86 59 0.89 0.58 594 521 1.64 0.15 10 0.22 0.4 995 989 4.56 0.55 60 0.87 0.56 250 865 5.11 0.21 11 0.5 0.45 429 948 9.87 0.58 61 0.91 0.16 472 1464 2.64 0.43 12 0.3 0.52 528 585 0.53 0.79 62 0.32 0.15 244 730 6.05 0.09 13 0.95 0.2 570 923 3.41 0.46 63 0.37 0.37 353 1034 4.27 0.78 14 0.65 0.07 938 758 8.98 0.15 64 0.38 0.73 183 585 6.18 0.23 15 0.53 0.11 726 552 3.53 0.48 65 0.96 0.34 749 782 9.13 0.6 16 0.95 0.95 533 1471 1.64 0.7 66 0.15 0.76 200 985 6.63 0.42 17 0.15 0.13 565 930 1.36 0.98 67 0.15 0.48 321 662 7.02 0.1 18 0.31 0.4 322 1103 5.1 0.2 68 0.99 0 650 904 8.73 0.16 19 0.98 0.73 326 586 1.22 0.49 69 0.47 0.28 946 1242 3.92 0.52 20 0.59 0.04 663 812 6.95 0.3 70 0.84 0.16 143 513 8.16 0.79 21 0.46 0.21 952 850 9.57 0.37 71 0.71 0.9 565 1117 1.25 0.84 107
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x y di x y di 0.77 0.44 919 561 1.49 0.38 72 0.46 0.86 11 928 4.44 0.26 23 0.87 0.79 292 750 6.95 0.6 73 0.09 0.74 374 1028 2.65 0.33 24 0.69 0.15 48 956 2.29 0.71 74 0.71 0.78 284 522 1.45 0.35 25 0.24 0.28 581 1456 2.9 0.3 75 0.27 0.04 598 609 9.63 0.22 26 0.84 0.73 659 764 5.05 0.99 76 0.25 0.07 720 541 3.72 0.25 27 0.49 0.24 986 1005 8.81 0.23 77 0.57 0.18 457 1493 2.38 0.43 28 0.56 0.77 486 1121 3 0.81 78 0.96 0.49 213 736 7.09 0.49 29 0.38 0.05 915 1065 6.94 0.4 79 0.83 0.21 550 1452 8.3 0.43 30 0.43 0.22 282 1494 0.88 0.68 80 0.72 0.49 418 758 2.55 0.57 31 0.61 0.73 310 605 2.66 0.39 81 0.69 0.5 863 1147 1.53 0.93 32 0.48 0.88 980 1304 3.32 0.86 82 0.22 0.89 368 976 3.5 0.52 33 0.81 0.75 134 1073 0.66 0.28 83 0.37 0.88 282 1180 7.15 0.95 34 0.13 0.71 20 1107 3.08 0.62 84 0.36 0.82 811 1045 9.07 0.3 35 0.41 0.34 151 651 2.64 0.01 85 0.11 0.1 866 1021 2.46 0.37 36 0.72 0.14 615 800 3.84 0.35 86 0.77 0.69 895 1165 3.19 0.44 37 0.3 0.28 369 1479 1.74 0.22 87 0.16 0.09 959 1250 5.52 0.66 38 0.02 1 875 1278 1.14 0 88 0.75 0.63 375 1406 5.28 0.06 39 0.62 0.9 73 1289 2.76 0.42 89 0.16 0.41 711 515 7.34 0.46 40 0.8 0.06 776 510 3.36 0.69 90 0.01 0.21 208 923 3.99 0.87 41 0.1 0.98 342 536 5.61 0 91 0.51 0.76 954 1378 4.06 0.31 42 0.15 0.13 929 1022 9.77 0.73 92 0.98 0.32 843 733 7.77 0.33 43 0.48 0.44 445 594 4.32 0.83 93 0.55 0.39 905 545 0.08 0.74 44 0.83 0.22 684 1466 9.61 0.28 94 0.36 0.63 729 1047 8.47 0.84 45 0.82 0.39 643 677 1.45 0.71 95 0.18 0.75 382 1365 6.23 0.38 46 0.67 0.53 771 1334 7.46 0.74 96 0.09 0.46 91 513 0.53 0.16 47 0.03 0.73 181 1445 1.03 0.35 97 0.18 0.67 991 1338 6.62 0.92 48 0.26 0.39 926 691 7.6 0.26 98 0.1 0.38 644 1341 3.13 0.11 49 0.59 0.56 733 1001 0.15 0.08 99 0.25 0.66 539 1256 0.5 0.41 50 0.22 0.66 326 1244 5.93 0.98 100 0.68 0.49 294 1168 7.27 0.66 108
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B2 isprovidedasfollows:#idenotesthefacilityname;(x;y)isthecoordinates,diisthedemand;riisthepenaltycost;andfi;pi(i=1;2;3)aretheinvestmentlevelanditscorrespondingfailureprobability. TableB2.DatasetofURFLPMFP:3level #i x y di 0.82 0.18 957 5.32 938 0.81 954 0.44 1260 0.15 2 0.54 0.7 202 1.9 642 0.55 721 0.39 1330 0.02 3 0.91 0.72 186 3.11 1125 0.63 1230 0.42 1355 0.19 4 0.15 0.31 635 1.83 772 1 1008 0.9 1440 0.36 5 0.74 0.16 737 1.34 665 0.85 1279 0.45 1393 0.28 6 0.58 0.92 953 2.3 890 0.9 1034 0.83 1431 0.61 7 0.6 0.09 450 7.96 620 0.98 1187 0.48 1394 0.06 8 0.37 0.19 188 3.42 503 1 703 0.75 1044 0.49 9 0.7 0.52 206 9.05 1231 0.86 1278 0.65 1466 0.24 10 0.22 0.4 995 4.56 989 1 1037 0.55 1455 0.54 11 0.5 0.45 429 9.87 948 0.87 1082 0.6 1422 0.58 12 0.3 0.52 528 0.53 551 0.84 585 0.79 1303 0.39 13 0.95 0.2 570 3.41 682 0.46 923 0.32 999 0.04 14 0.65 0.07 938 8.98 758 0.78 987 0.5 1497 0.15 15 0.53 0.11 726 3.53 552 0.75 855 0.63 1407 0.48 16 0.95 0.95 533 1.64 791 0.7 1471 0.66 1488 0.47 17 0.15 0.13 565 1.36 930 0.98 958 0.8 1227 0.35 18 0.31 0.4 322 5.1 694 0.52 829 0.48 1103 0.2 19 0.98 0.73 326 1.22 586 0.95 975 0.49 1127 0.15 20 0.59 0.04 663 6.95 634 0.3 812 0.24 907 0.22 21 0.46 0.21 952 9.57 850 0.37 932 0.18 980 0.14 22 0.77 0.44 919 1.49 561 0.92 742 0.76 1290 0.38 23 0.87 0.79 292 6.95 750 0.6 1213 0.59 1311 0.24 24 0.69 0.15 48 2.29 684 0.91 772 0.79 956 0.71 25 0.24 0.28 581 2.9 1040 0.62 1153 0.3 1456 0.23 109
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x y di 0.84 0.73 659 5.05 764 0.99 941 0.2 1035 0.04 27 0.49 0.24 986 8.81 910 0.94 1005 0.25 1058 0.23 28 0.56 0.77 486 3 928 0.81 1079 0.72 1121 0.35 29 0.38 0.05 915 6.94 1040 0.81 1065 0.4 1108 0.16 30 0.43 0.22 282 0.88 771 0.76 1028 0.68 1494 0.04 31 0.61 0.73 310 2.66 605 0.81 748 0.41 1381 0.39 32 0.48 0.88 980 3.32 1106 0.86 1304 0.21 1400 0.06 33 0.81 0.75 134 0.66 1073 0.88 1153 0.34 1315 0.28 34 0.13 0.71 20 3.08 905 0.79 1107 0.62 1260 0.62 35 0.41 0.34 151 2.64 651 0.87 652 0.25 1066 0.01 36 0.72 0.14 615 3.84 800 0.98 1298 0.57 1423 0.35 37 0.3 0.28 369 1.74 723 0.7 1351 0.51 1479 0.22 38 0.02 1 875 1.14 890 0.81 1029 0.04 1278 0 39 0.62 0.9 73 2.76 1104 0.88 1289 0.42 1485 0.37 40 0.8 0.06 776 3.36 510 0.91 981 0.69 1341 0.59 41 0.1 0.98 342 5.61 536 0.63 1157 0.06 1351 0 42 0.15 0.13 929 9.77 1022 0.73 1064 0.67 1157 0.33 43 0.48 0.44 445 4.32 594 0.83 1289 0.76 1386 0.43 44 0.83 0.22 684 9.61 1130 0.68 1192 0.28 1466 0.09 45 0.82 0.39 643 1.45 518 0.71 677 0.38 1072 0.28 46 0.67 0.53 771 7.46 892 0.74 1334 0.72 1483 0.45 47 0.03 0.73 181 1.03 1055 0.97 1156 0.63 1445 0.35 48 0.26 0.39 926 7.6 691 0.56 865 0.26 1312 0.23 49 0.59 0.56 733 0.15 1001 0.72 1058 0.61 1450 0.08 50 0.22 0.66 326 5.93 699 0.98 1244 0.42 1472 0.04 51 0.63 0.04 486 3.14 971 0.68 1154 0.63 1493 0.49 52 0.53 0.32 548 0.31 853 0.99 1023 0.94 1207 0.28 53 0.89 0.99 870 0.66 754 0.84 787 0.76 1271 0.61 54 0.02 0.19 335 0.22 611 0.97 734 0.83 1214 0.51 55 0.51 0.32 446 3.93 1217 0.91 1249 0.88 1367 0.05 56 0.53 0.06 198 3.77 742 0.69 1025 0.49 1371 0.31 110
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x y di 0.81 0.86 212 9.92 1271 0.58 1446 0.4 1489 0.38 58 0.53 0.36 903 4.06 863 0.75 1078 0.29 1354 0.05 59 0.89 0.58 594 1.64 521 0.64 1071 0.44 1100 0.15 60 0.87 0.56 250 5.11 865 0.95 1273 0.6 1348 0.21 61 0.91 0.16 472 2.64 974 0.92 1220 0.43 1464 0.22 62 0.32 0.15 244 6.05 689 0.88 730 0.58 810 0.09 63 0.37 0.37 353 4.27 745 0.89 821 0.78 1034 0.46 64 0.38 0.73 183 6.18 585 0.79 1026 0.78 1259 0.23 65 0.96 0.34 749 9.13 782 0.84 1219 0.6 1392 0.13 66 0.15 0.76 200 6.63 561 0.93 985 0.42 1068 0.22 67 0.15 0.48 321 7.02 569 0.68 662 0.5 713 0.1 68 0.99 0 650 8.73 880 0.79 904 0.26 1424 0.16 69 0.47 0.28 946 3.92 1190 0.64 1242 0.58 1332 0.52 70 0.84 0.16 143 8.16 513 0.79 993 0.48 1355 0.14 71 0.71 0.9 565 1.25 601 1 945 0.84 1117 0.13 72 0.46 0.86 11 4.44 904 0.53 928 0.32 982 0.26 73 0.09 0.74 374 2.65 691 0.48 1028 0.33 1291 0.32 74 0.71 0.78 284 1.45 522 0.94 818 0.68 1246 0.35 75 0.27 0.04 598 9.63 609 0.85 771 0.22 1026 0.11 76 0.25 0.07 720 3.72 541 0.25 704 0.14 1012 0.09 77 0.57 0.18 457 2.38 986 0.75 1307 0.43 1493 0.19 78 0.96 0.49 213 7.09 666 0.77 736 0.66 861 0.49 79 0.83 0.21 550 8.3 564 0.74 796 0.43 1452 0.15 80 0.72 0.49 418 2.55 588 0.78 619 0.57 758 0.39 81 0.69 0.5 863 1.53 1147 0.93 1299 0.77 1310 0.71 82 0.22 0.89 368 3.5 510 0.65 976 0.52 1070 0.39 83 0.37 0.88 282 7.15 598 0.95 871 0.73 1180 0.07 84 0.36 0.82 811 9.07 674 0.58 1045 0.4 1166 0.3 85 0.11 0.1 866 2.46 568 0.63 1021 0.37 1192 0.24 86 0.77 0.69 895 3.19 665 0.93 1015 0.44 1165 0.25 87 0.16 0.09 959 5.52 980 0.92 1250 0.68 1412 0.66 111
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x y di 0.75 0.63 375 5.28 522 0.59 1206 0.06 1406 0.04 89 0.16 0.41 711 7.34 515 0.63 707 0.48 1415 0.46 90 0.01 0.21 208 3.99 923 0.87 1049 0.84 1281 0.32 91 0.51 0.76 954 4.06 739 0.79 1186 0.31 1378 0.28 92 0.98 0.32 843 7.77 733 0.52 834 0.33 1335 0.18 93 0.55 0.39 905 0.08 545 0.74 869 0.61 1178 0.61 94 0.36 0.63 729 8.47 1047 0.84 1098 0.66 1460 0.6 95 0.18 0.75 382 6.23 538 0.62 613 0.59 1365 0.38 96 0.09 0.46 91 0.53 513 0.67 521 0.26 1448 0.16 97 0.18 0.67 991 6.62 723 0.92 1119 0.91 1338 0.41 98 0.1 0.38 644 3.13 628 0.54 822 0.46 1341 0.11 99 0.25 0.66 539 0.5 848 0.49 942 0.48 1256 0.41 100 0.68 0.49 294 7.27 1069 0.99 1108 0.88 1168 0.66 112
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ThemeaningofeachcolumninTable C1 isprovidedasfollows:#idenotesthefacilityname;(x;y)isthecoordinates,diisthedemand;fiisthexedcost;riisthepenaltycost. TableC1.DatasetofDFFM #i x y di x y di 2,5,15,18,20 1 0.82 0.18 957 5.32 51 0.63 0.04 486 3.14 2 0.54 0.7 202 1.9 52 0.53 0.32 548 0.31 3 0.91 0.72 186 3.11 53 0.89 0.99 870 0.66 4 0.15 0.31 635 1.83 54 0.02 0.19 335 0.22 5 0.74 0.16 737 1.34 55 0.51 0.32 446 3.93 6 0.58 0.92 953 2.3 56 0.53 0.06 198 3.77 7 0.6 0.09 450 7.96 57 0.81 0.86 212 9.92 8 0.37 0.19 188 3.42 58 0.53 0.36 903 4.06 9 0.7 0.52 206 9.05 59 0.89 0.58 594 1.64 10 0.22 0.4 995 4.56 60 0.87 0.56 250 5.11 11 0.5 0.45 429 9.87 61 0.91 0.16 472 2.64 12 0.3 0.52 528 0.53 62 0.32 0.15 244 6.05 13 0.95 0.2 570 3.41 63 0.37 0.37 353 4.27 14 0.65 0.07 938 8.98 64 0.38 0.73 183 6.18 15 0.53 0.11 726 3.53 65 0.96 0.34 749 9.13 16 0.95 0.95 533 1.64 66 0.15 0.76 200 6.63 17 0.15 0.13 565 1.36 67 0.15 0.48 321 7.02 18 0.31 0.4 322 5.1 68 0.99 0 650 8.73 19 0.98 0.73 326 1.22 69 0.47 0.28 946 3.92 20 0.59 0.04 663 6.95 70 0.84 0.16 143 8.16 21 0.46 0.21 952 9.57 71 0.71 0.9 565 1.25 22 0.77 0.44 919 1.49 72 0.46 0.86 11 4.44 113
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x y di x y di 0.87 0.79 292 6.95 73 0.09 0.74 374 2.65 24 0.69 0.15 48 2.29 74 0.71 0.78 284 1.45 25 0.24 0.28 581 2.9 75 0.27 0.04 598 9.63 26 0.84 0.73 659 5.05 76 0.25 0.07 720 3.72 27 0.49 0.24 986 8.81 77 0.57 0.18 457 2.38 28 0.56 0.77 486 3 78 0.96 0.49 213 7.09 29 0.38 0.05 915 6.94 79 0.83 0.21 550 8.3 30 0.43 0.22 282 0.88 80 0.72 0.49 418 2.55 31 0.61 0.73 310 2.66 81 0.69 0.5 863 1.53 32 0.48 0.88 980 3.32 82 0.22 0.89 368 3.5 33 0.81 0.75 134 0.66 83 0.37 0.88 282 7.15 34 0.13 0.71 20 3.08 84 0.36 0.82 811 9.07 35 0.41 0.34 151 2.64 85 0.11 0.1 866 2.46 36 0.72 0.14 615 3.84 86 0.77 0.69 895 3.19 37 0.3 0.28 369 1.74 87 0.16 0.09 959 5.52 38 0.02 1 875 1.14 88 0.75 0.63 375 5.28 39 0.62 0.9 73 2.76 89 0.16 0.41 711 7.34 40 0.8 0.06 776 3.36 90 0.01 0.21 208 3.99 41 0.1 0.98 342 5.61 91 0.51 0.76 954 4.06 42 0.15 0.13 929 9.77 92 0.98 0.32 843 7.77 43 0.48 0.44 445 4.32 93 0.55 0.39 905 0.08 44 0.83 0.22 684 9.61 94 0.36 0.63 729 8.47 45 0.82 0.39 643 1.45 95 0.18 0.75 382 6.23 46 0.67 0.53 771 7.46 96 0.09 0.46 91 0.53 47 0.03 0.73 181 1.03 97 0.18 0.67 991 6.62 48 0.26 0.39 926 7.6 98 0.1 0.38 644 3.13 49 0.59 0.56 733 0.15 99 0.25 0.66 539 0.5 50 0.22 0.66 326 5.93 100 0.68 0.49 294 7.27 114
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LezhouZhanwasborninZhejiang,China,intheyearoftheHorse.HeisalsoknownasRogerwhosepronunciationissimilartoLezhouinhishometowndialect.Priortocollege,hegraduatedfromYueqingMiddleSchoolin1997.HereceivedhisBachelorofScienceinAppliedMathematicsandhisBachelorofScienceinBusinessAdministrationandEngineeringManagementfromChongqingUniversityin2001.BeforehetransferredtotheUniversityofFloridainthefallof2002,hestudiedscienticcomputationattheHongKongUniversityofScienceandTechnologyinaMasterofPhilosophyprogram.HeservedasSecretaryGeneralofFACSS,aChinesestudentassociationatUF,from2003to2004.HeearnedhisMasterofScienceandDoctorofPhilosophyinIndustrialandSystemsEngineeringfromtheUniversityofFloridainMay,2004andAugust,2007respectively.Hiscurrentresearchinterestsincludereliablesupplychaindesign,auctionmechanismdesign,operationsresearchmodelsinairlineapplications,andsystemreliabilityoptimization.Hisworkhasbeenpresentedinvariousconferences,bookchapters,andjournals,includingProceedingsofthe2005WinterSimulationConference,Proceedingsofthe2005IIEResearchConferenceandProductionandOperationsManagement.HeisamemberofINFORMS,SIAM,andIIE. 120

