Group Title: Bulletin University of Florida. Agricultural Experiment Station
Title: Indian River citrus packinghouse location
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 Material Information
Title: Indian River citrus packinghouse location a dynamic programming approach
Series Title: Bulletin University of Florida. Agricultural Experiment Station
Physical Description: iv, 36 p. : map ; 23 cm.
Language: English
Creator: Kilmer, Richard L
Publisher: Agricultural Experiment Stations, Institute of Food and Agricultural Sciences, University of Florida
Place of Publication: Gainsville Fla
Publication Date: 1984
Copyright Date: 1984
 Subjects
Subject: Citrus fruit industry -- Florida   ( lcsh )
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Bibliography: Includes bibliographical references (p. 35-36).
Statement of Responsibility: by Richard L. Kilmer and Thomas Spreen.
General Note: "October 1984."
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Bibliographic ID: UF00026904
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: ltuf - ADA2739
oclc - 12495634
alephbibnum - 000575326

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Copyright 2005, Board of Trustees, University
of Florida






October 1984 Bulletin 848 (technical)



Indian River Citrus
Packinghouse Location:
A Dynamic Programming
Approach
Richard L. Kilmer and Thomas Spreen















Agricultural Experiment Stations
Institute of Food and Agricultural Sciences
University of Florida, Gainesville
F. A. Wood, Dean for Research














Indian River Citrus
Packinghouse Location:
A Dynamic Programming Approach


by
Richard L. Kilmer and Thomas Spreen













Richard L. Kilmer and Thomas Spreen are associate professors in the
Food and Resource Economics Department at the University of Florida.
The original publication of the research in this bulletin was reported by
Richard L. Kilmer, Thomas Spreen, and Daniel S. Tilley in "A Dynamic
Plant Location Model: The East Florida Fresh Citrus Packing Industry,"
American Journal of Agricultural Economics 65(1983):730-737.
L














TABLE OF CONTENTS
Page
LIST OF TABLES ............................................. iii
LIST OF FIGURES ............................................ iii
ABSTRACT .............. ........................... iv
INTRODUCTION ............................................. 1
LITERATURE REVIEW .................................................. 2
Static Plant M odels ............. ........................ 3
Extension to Dynamic Models ................................. 4
METHODOLOGY ..................... ................... 5
The Static Model ............................................. 5
Transition Costs ............... ....................... 6
Terminal Values and Discounting ....... ..... . ........... .. 7
Dynamic Model .............................................. 8
SUPPLY AND DEMAND ................ .. ................... 9
ASSEMBLY, DISTRIBUTION, AND PACKING COSTS ........... 10
RESULTS .......................................... 13
SENSITIVITY ANALYSIS .. .......... ...................... 17
Time Horizon .................................. ............ 17
Discount Interest Rate .......... .......................... 17
Transition Costs ............. ............................... 17
SUMMARY ............ ..................... ........... 20
CONCLUSIONS .................... .......................... 21
APPENDIX A: MIXED INTEGER PROGRAM .................... 23
APPENDIX B: DYNAMIC PROGRAM ALGORITHM ............. 25
The Solution of Dynamic Programming Problems .................. 25
Documentation ..................... ................... 25
Overview .... ............ ....................... 25
Variable Definitions ............... .................. 28
Specific Components of the Program .......................... 29
Output .................. .... .................. 30
REFERENCES .............. .................. ............ 35










ii














LIST OF TABLES
Table Page
1 Static and dynamic solutions to the packinghouse
location problem, 1979-80 through 1983-84 .................... 14
2 Packinghouse size configuration for the best dynamic
solution over five years ................................... 16
3 Packinghouse size configuration for the best dynamic
solution over nine years .................. .................. 18
4 Dynamic solutions ranked for each discount rate ............... 19
5 Dynamic solutions ranked for each level of existing
plant transition costs ............... ... ............... 20

LIST OF FIGURES
Figure
1 Indian River district grapefruit and orange production,
packinghouses, and ports for fresh fruits .................... 2
Al Integer program .............. ... ......... ............... 23
B1 Dynamic program .............. .......................... 26
B2 Dynamic program output .................................. 31

























iii













ABSTRACT
Static location models provide a long-run solution without the adjust-
ments required over time. Dynamic programming allows incorporation
of short- and long-run adjustments in the same plant location model. A
standard transshipment model with fixed quantities at supply and de-
mand points is integrated with a dynamic program.
Results indicate that new large packinghouses are not competitive with
existing large packinghouses, because existing plants can forego a return
on investment for a finite period. However, new large packinghouses are
competitive with existing small packinghouses. The southern movement
of citrus production necessitates the construction of a new packinghouse
in the southern part of the Indian River District. Existing Indian River
packinghouse capacity is located where needed. Existing capacity could
be rebuilt into larger packinghouses. Instead of building new plants in the
same cities where existing plants are located, old packinghouses could be
enlarged to take advantage of economies of size.


KEY WORDS: citrus, dynamic programming, integer programming, packing-
house, packing costs, plant location, transshipment, transition
costs






















iv













INTRODUCTION
The Indian River area is a legally defined market district on the east
coast of Florida (Figure 1) and one in which the spatial pattern of citrus
supplies is changing. Nearly two-thirds of its western border is separated
from the interior marketing district by swampland containing little or no
citrus. In the past 15 years, most new planting in the district has occurred
in its southern half. Existing packing plants are located near older groves.
As more citrus is grown in the southern area, increased assembly costs
will be incurred unless new packing plants are opened near the new
production areas.
The purpose of this bulletin is to determine if cost efficiencies are
present which could result in changes in the size, number, and location of
Indian River packinghouses over time when the volume and location of
production is changing. Specific objectives are (a) to develop static and
dynamic plant location models for the Indian River district, (b) to deter-
mine the potential dynamic adjustments in size, number, and location of
"Indian River packinghouses from 1979-80 through the 1983-84 season
when the volume and location of production is changing, and (c) to
determine the sensitivity of the dynamic adjustments to changes in the
time horizon, the discount interest rate, and transition costs.
The model in this bulletin analyzes only the cost side of the profit
equation. Model results when compared with the existing industry con-
figuration will indicate the potential for packinghouse size, number and
location changes. The demonstrated presence of structure-altering eco-
nomic forces (cost efficiencies) may not result in immediate real world
changes. This results from the facts that (a) all packinghouses in the
model are assumed to behave so as to minimize industry collection,
packing, and distribution costs, whereas in the real world there are
independent entrepreneurs making decisions, (b) firms generally maxi-
mize profits, rather than minimize costs, and (c) the cost efficiency of
management varies among packinghouses.
Structure-altering cost efficiencies become more important (a) during
recessions, (b) during periods of increased competition from new firms
entering the industry, (c) during periods of low output prices, (d) during
periods of inflation when costs are rising faster than output prices, and (e)
during periods when substitutes are entering the market. The Indian
River packinghouse industry has experienced several of these structure-
"altering forces in the recent past and will likely encounter them in the
future.

1












Jacksonville










Titusville E,N- Port Canaveral P
Cocoa EN

Tampa P Melbourne N

"Vero Beach E,N
Ft. Pierce E'N'P
S/" & Stuart N
Jupiter N
E Location of existing'J
packinghouses
N Potential location of
new packinghouses Port
P Ports Everglades P






Figure 1.--Indian River district grapefruit and orange production, packing-
houses, and ports for fresh fruit.


LITERATURE REVIEW
The solution to a dynamic plant location problem involves the simul-
taneous solution to four related sub-problems: (a) determine the optimal
flow of fruit from the groves to a given set of packing plants and from the
packing plants to the final demand points (transshipment problem); (b)
from a set of possible plant locations, determine that location subset
which minimizes the cost of moving fruit from the groves to final demand


2








points (plant location problem); (c) from a finite set of possible sizes,
determine the optimal size plants) at each location (plant size problem);
and (d) given that (a), (b), and (c) have been solved for periods 1, 2,...,
T, determine the adjustment that should occur from the current con-
figuration of plants. Solution of (a), (b), and (c) constitutes a static plant
location model. Solution (d) is an extension of the static model to a
dynamic framework. In the next section, we focus on the static model.

Static Plant Models
The determination of optimal flow of fruit from the groves through
packing plants and to demand points is a transshipment problem. It has
two components. The flow of raw product from supply areas to a packing
plant is called the assembly problem. The second component involves the
flow of product from the plants to the consumption areas and is called the
distribution problem. The plants are viewed as transshipment points.
The extension of the transshipment model to a plant size and location
model is straightforward. First, a finite number of possible plant locations
are identified. Possible plant sites include existing plants and new loca-
tions (plant location problem). The identification of new sites may be
based on several factors such as the intersection of major roadways or
"railways or availability of land and/or labor. The plant location problem is
then solved by a constrained optimization technique such as mixed inte-
ger programming. Use of the simplex method to solve this problem does
not insure an integer solution of plant numbers.
Much of the research published on plant location problems prior to
1980 focused on solution of the mixed integer program. Mixed integer
solution algorithms were cumbersome and required extensive computer
time. Recently more efficient ways have been developed along with the
reduced cost of computer time. This has encouraged direct solution of the
plant location problem.
Stollsteimer is generally credited with first proposing an integrated
model for determining the optimal location of plants. His formulation
only included assembly of raw product. The Stollsteimer model has been
extended to include multiple products (Polopolus). King and Logan
extended the Stollsteimer model to include assembly and distribution in
the same model. Economies of size were incorporated by King and
Logan, Chern and Polopolus, and Kloth and Blakely.
The solution method employed in these studies involved first assuming
a fixed configuration of plants. Once plant location is determined, the
mathematical programming model is a transshipment model (i.e., plant
numbers are fixed and set to integer values) and can be solved via linear
programming techniques. The optimal value of the objective function is



3








the total system cost associated with that plant configuration. Next, a
different plant configuration is specified. The associated transshipment
model is optimized, and the system cost associated with the second plant
configuration is determined. The system costs associated with each plant
configuration are compared, and the plant configuration with the lower
cost is retained. In a similar manner, all feasible plant configurations are
examined and eventually the optimal plant configuration is determined.
A more recent study by Fuller et al. uses a similar conceptual approach
except they exploited the special structure of the transshipment model.
The transshipment model can be reformulated into a transportation
model (Hiller and Lieberman, Chapter 4) and as such is a member of a
class of mathematical problems called networks. Algorithms far more
efficient than the simplex method exist to solve network flow problems.
These algorithms are usually heuristic in nature. One such algorithm was
used by Fuller et al.
In the 1980s, the development of mixed integer programming algo-
rithms, utilizing either the branch-and-bround method (e.g. MIP) or
Bender's decomposition (Hilger et al.), has allowed direct solution of the
plant location problem. Faminow and Sarhan, and Hilger et al. are
studies which employ mixed integer programming.

Extension to Dynamic Models
French reviewed this literature, noting that a limitation of previous
studies was their static nature. In particular, static analysis assumes that
the period of observation of the model is a "snapshot" of a long-run
equilibrium. This assumption is not appropriate when (a) the spatial
pattern of supply and/or demand is changing and (b) the costs of closing
existing plants and opening new plants is a significant proportion of total
industry costs. Kilmer and Hahn relaxed the static assumption and pro-
jected dairy industry adjustment in size, number, and location of process-
ing plants over time; however, the cost associated with opening and
closing plants (transition costs) from one time period to another was not
considered.
This bulletin uses a methodology developed by Sweeney and Tatham
to handle dynamic plant location problems. A standard transshipment
model with fixed quantities at supply and demand points is integrated
with a dynamic programming model. A finite planning horizon is speci-
fied, say Tperiods, and the optimal size, number, and location of plants
in each year is solved via mixed integer programming. The best R,
solutions for each time period, denoted by R*, are rank-ordered, starting
with the least-cost solution. Each solution represents a different configu-
ration of plants. The optimal dynamic solution is the path which mini-


4









mizes the assembly, packing, and distribution costs over T periods, plus
transition costs incurred as plant configurations change. In this case,
packinghouses are viewed as transshipment points and specified as inte-
ger variables (Hilger et al.).


METHODOLOGY

The Static Model
For this problem, let Xijkl be the quantity of fruit type ( = 1 denotes
oranges and 1= 2 denotes grapefruit) shipped from supply area i to
demand point k through plant size-location j. Each possible plant size-
location can have multiple plants. For example, Vero Beach has three
plant size-locations. They are (a) one small existing packinghouse, (b)
seven large existing packinghouses, and (c) one large potential packing-
house. In this problem there are i = 1,... 13 supply areas with Sil fruit
available, k = 1, . 10 demand points with Dkl fruit required and = 1,
13 possible plant size-locations.' Let Cijkl be the assembly, elimina-
tion transportation,2 variable packing, and distribution costs associated
with Xijkl. An integer linear programming model which minimizes the
cost is
13 13 10 2 13
(1) MIN I I CijkeXijk + Yjf,.
i=1 j=1 k=l e=1 j=1
13 10
subject to: I Xijkt =Si (i= 1, . ,13;= 1,2)
j=1 k=1
13 13
"1 1 Xik1 = Dkt (k = 1, ...,10; = 1,2)
i=1 j=1
13 10 2
SXijkt Yj CAP (j= . 13)
i=1 k=l 1=1
Yj, Xijkt > O, Yj is integer.

Yj equals zero or a positive integer depending on the number of packing-
houses at a specific size-location that are open, fj is the fixed cost associ-
ated with a single plant at size-location j, and CAPj is the capacity of a
single plant at the jth size-location. Si (the supply side) varies over time.


'Plant size locations were based on the location of (a) citrus production, (b)
large cities adjacent to existing railroad and highway networks, and (c) existing
packinghouses.
2Packinghouse eliminations do not meet fresh market size. The cost of trans-
porting the eliminations to the nearest processing plant is added to the per box
cost for fresh citrus packed.


5








The demand points were assumed to maintain their relative shares.3
Suppose the model is solved and n plants are open. Define the set J,
where the n plants are located, as a subset of the 13 size-locations. To find
another feasible solution to (1), impose the constraint

(2) 1 y 0 n.

This constraint does not allow the optimal solution to be feasible. Ad i-
tional optimal solutions are identified by continually imposing a ne
constraint for every feasible solution identified and the model resolved.
This guarantees a complete enumeration only if a maximum of one plant
is allowed at each size-location (Y). Model (1), however, allows j to be
any non-negative integer value. Under this condition, the restriction
imposed by (2) excludes all solutions with the same zero and non-zero Yj's
and n plants.4 This does not allow a complete enumeration of feasible
solutions.
To avoid this problem, a series of conditional constraints can be
imposed on each non-zero j in the optimal solution so that only one
solution is excluded. This requires more integer variables, increasing the
computation time and is not used (Hillier and Lieberman, pp. 736, 737).
Identifying near optimal solutions is accomplished by utilizing the
MIP-MPSX search procedure (IBM) in combination with inequality (2).
First, the optimal solution is determined, and the search procedure then
identifies other feasible solutions. A constraint is imposed for every
feasible solution identified and the algorithm re-run.


Transition Costs
Transition costs are the costs of closing existing plants and opening new
ones. They may be public costs of communities or private costs. Public
costs are difficult to document and are generally subjective. Plant closings


'Other assumptions that could be used include a trend or a five year moving
average; however, we do not believe that our results would be affected by a
change. The reasons are twofold. First, East Coast Florida packinghouses are
geographically concentrated in a remote corner of the United States with all
domestic U.S. markets located to the north and west. Relatively speaking, all east
Florida packinghouses will be shipping approximately the same distance to inter-
state demand areas. Second, the distribution cost to Florida ports is insignificant
compared to packing costs as a percentage of total cost (14 and 80 percent). Thus,
the main influences on packinghouse location are the location of supply and
packing costs.
"For example, let Y1 = 2 and Y2 = 3 with other Yj's = 0. Impose restriction (2)
such that Y1 + Y2 + 5. This excludes Y1 = 3 and Y2 = 2 from the feasible set of
solutions.


6








and openings likely will yield both benefits and costs for communities
S involved. This issue has been discussed by researchers with little agree-
ment (Stoevener and Kraynick). Furthermore, private entrepreneurs
must make decisions that are best for the survivability of the firm. Public
costs, unlike private costs, do not affect the profits of a firm. Therefore,
public costs are not included in transition costs.
A change in plant configuration involves either closing an existing plant
and/or opening a new one. In this bulletin, all facility costs are included in
the static model. Thus, the private transition costs of opening a new plant
are start-up costs such as equipment break-in.5 The start-up costs are
assumed to be small and are ignored.
Existing plants do not remain in a long-run solution if new plants are
"more efficient. However, some costs are fixed, and do not enter the
short-run production decisions of the firm. Thus, a firm may remain in
operation in the short-run even though it is not earning a return on
investment or covering debt servicing costs.
Both long- and short-run decisions can be incorporated in a dynamic
model. Each static solution is a long-run solution. If investment does not
earn its opportunity cost, the plant should be liquidated and the capital
re-invested. In the short-run, however, existing plants may continue to
operate if cash costs are covered. Thus, the transition cost of a change
from plant configuration s to configuration r should equal the investment
S servicing cost (debt servicing plus return on net investment) of all existing
plants in configuration s which are closed in moving to configuration r.
The industry moves from configuration s in period t to configuration r in
t + 1 only if the total cost of configuration r is less than that of s in period
t + 1 minus the investment servicing cost of all existing plants that are
closed.

Terminal Values and Discounting
SA five-year planning horizon (T = 5) is not uncommon in industry and
is assumed for this article. Sweeney and Tatham do not consider costs
beyond period T. If future (T + 1, T + 2 .. .) costs are not included in the
dynamic program, the future plant configurations may not differ appreci-
ably from the initial configuration. The initial plant configuration will
change only if the discounted cost savings (assembly, packing, and dis-
tribution costs over future years) from a plant configuration change is
greater than the one-time transition cost of changing the configuration.


"5Refurbishing and/or liquidation costs may indirectly influence a plant's transi-
tion by prolonging its existence. The costs of converting to a different enterprise
are assumed to be borne by the new enterprise.


7








Since static solutions for time periods T + 1, T + 2,... are not derived, it
is assumed that the conditions existing in period T will be infinitely
perpetuated. Thus the assembly, elimination transportation, packing,
and distribution costs associated with T are the assumed present value of
the costs for T, T+1, ....

Dynamic Model
The dynamic programming model which determines the optimal path
of adjustment is
(3) v* (s) = minimum [v,,, + Nt,sr + v +1 (r)]; t = 1, . ,T;
r
where
v*(s) = the cost of the best dynamic solution beginning with
configuration s in period (stage) t,

vts = the discounted assembly, packing, and distribution cost
of configuration s in period t,
N,,r = the discounted transition cost of moving from configura-
tion s in period t to configuration r in period t + 1,
v1 +1 (r) = the cost of the best dynamic solution beginning with
configuration r in period t + 1.
Sweeney and Tatham develop a method to determine the maximum
number of different configurations considered in each period. Suppose
the static model for each period is solved for the best R, solutions. Let vt
denote the cost of the best static configuration in period t. Then v1 = S vt
is the sum of the minimum cost configurations over the planning horizon.
Since transition costs are not included, v1 is a lower bound on the optimal
multiperiod solution.
Let v" be the total cost of any feasible solution to the multiperiod
problem, and let vA = v" v1. Sweeney and Tatham show that, in each
period, one only need consider those static solutions with cost vtc where
vtc vtl 2- va. In other words, if the difference between two static
solutions in t is less than the difference between two dynamic solutions
(v^), the largest static solution is included in the dynamic program. Thus,
it is necessary to consider only the best R* solutions where vt,R* Vtl VA
and vt, R* Vt> VA
If R7 is relatively large, considerable time and expense is expended to
generate the solutions. After R, solutions (R, < R*) are generated, one
may evaluate the need for additional static solutions, Sweeney and
Tatham show that the maximum improvement in the least-cost, dynamic



8








solution (v*) when the R* best solutions are not found is
(4) I= maximum [1,, . ., I,]
where
I, = maximum [vA Vt,Rt + v, 0], t = 1,. . T.
The size of I with respect to the sum of the best static solutions may be
small enough that additional static solutions are unnecessary.
The data requirements for the static and dynamic models are extensive.
The supply and demand data, the assembly, distribution and packing
costs, and the transition costs are presented next. For a detailed discus-
sion of this information, see Kilmer and Spreen (1983b).

SUPPLY AND DEMAND
Oranges and grapefruit represent 97 percent of the citrus packed in the
Indian River marketing district during the 1979-80 marketing season
"(Florida Department of Agriculture and Consumer Services, p. 37). To
project the future production of oranges and grapefruit by supply area,
tree data by age and variety (Florida Crop and Livestock Reporting
Service, 1980) are combined with yield information by tree age and
variety (Fairchild, pp. 24-32). The varieties are early and midseason
oranges, valencia oranges, temple oranges, seedy grapefruit, white seed-
less grapefruit, and pink seedless grapefruit. Grove owners in the Indian
River marketing district shipped 6.8 and 67.1 percent of their oranges and
grapefruit, respectively, to packinghouses in 1979-80 and sent the re-
sidual directly to processing plants (calculated from the Florida Crop and
Livestock Reporting Service, 1981, p. 28, and Florida Department of
Agriculture and Consumer Services, p. 37). Only 65.6 and 76.1 percent of
orange and grapefruit deliveries to packinghouses were actually packed
during the 1979-80 season (Hooks and Kilmer, 1981a, p. 4). The remain-
der was shipped to processing plants. After considering tree age, variety,
"yield, percentage of harvested citrus taken to a packinghouse, and per-
"centage actually packed, total boxes packed in the Indian River market-
ing district was projected for the 1979-80 through the 1983-84 marketing
season (one box equals one and three-fifths bushels).
The oranges and grapefruit packed are either exported to foreign
countries (1.7 and 40.0 percent) or shipped to another location in North
America (98.3 and 60.0 percent-Florida Department of Agriculture and
Consumer Services, pp. 33, 34). Five demand areas in North America
with central distribution points at New York City, Atlanta, Chicago, Los
Angeles, and Toronto, Canada are assumed. Each region is assumed to
maintain its 1979-80 market share for oranges and grapefruit through



9








1983-84 (Florida Department of Citrus).6 Fresh citrus is exported through
Ft. Pierce, Jacksonville, Port Canaveral, Port Everglades, and Tampa,
all in Florida. The 1979-80 sales share for each port is assumed to extend
through the 1983-84 marketing season (Florida Department of Agricul-
ture and Consumer Services, p. 35).7

ASSEMBLY, DISTRIBUTION, AND PACKING COSTS
The distribution costs from Indian River packinghouses to the five
North American cities are determined by averaging actual quoted rates
for oranges and grapefruit from November 1979 through May 1980 (U.S.
Department of Agriculture)." The distribution cost (DC) in dollars per
box from a packinghouse to a port is assumed to be a function of one way
mileage (OWM) (DC= 0.2049 + 0.00410WM: updated Machado,
p. 100, to 1979-80 dollars)? The bulk hauling cost for citrus from groves
to the packinghouses and the cost of hauling eliminations from packing-
houses to a processing plant is assumed to be $0.00727 per box mile, one
way (calculated from Hooks and Kilmer, 1981b, p. 7).10
The capacities of existing packinghouses are not available. In order to
establish the maximum annual packing capacity, the 1979-80 volume
packed by each packinghouse is used as a base. The 1979-80 base is used
because the annual boxes packed in Florida during the 1979-80 season
was 36,439,588 boxes which represents the second largest volume packed
during the 1957-58 through 1981-82 seasons (Florida Department of
Agriculture and Consumer Services, 1981-82, p. 17). Thus the 1979-80
volume is assumed to be an acceptable proxy for capacity.
Houses are permitted to increase the 1979-80 volume packed by
twenty percent to allow for use of unused capacity. It was found that
during the 1973-74 through the 1975-76 seasons, 29 Florida packing-
houses (packing 43 percent of the total fresh fruit shipments in 1975-76)
had an average capacity utilization of 50 percent (Kilmer and Tilley, p.



"6See footnote 3.
7See footnote 3, item b.
"Toronto, Canada, is estimated by taking the rate to New York times 1.19.
'Machado based his original equation on quoted rates for shipping fruit to a
port. Machado's results are updated to 1979-80 dollars by using an index based on
Florida retail diesel fuel prices.
"1The bulk hauling cost per box mile is determined by using a per box cost
estimate from Hooks and Kilmer (1981b, p. 7) and dividing by an estimate of 30
miles for the average length of haul from grove to packinghouse. Industry sources
indicate that 30 miles was realistic. Also, Bowman et al. (p. 42) found that hauling
distances varied from 5 to 55 miles, which contains the 30-mile estimate.


10









36, 37)." The 50 percent is based on an 11-month season because fresh
citrus harvesting starts in September, increases slowly to a peak in De-
cember and January, and declines through July. A capacity utilization of
100 percent is unattainable because of the seasonal nature of supply. An
increase from 50 to 60 (20 percent) seems realistic and is assumed in this
study.12
Existing plants are categorized as small (100,000 to 500,000 boxes)" or
large (500,001 to 850,000)." All new plants were assumed to be large.15
Variable costs for both existing and new packinghouses include labor
(less 30 percent of foreman labor which is assumed fixed), direct operat-
ing expenses less repairs and maintenance, 30 percent of administrative
expenses, and 50 percent of sales expense.'" Variable costs are estimated
to be $2.227 and $2.008 per box respectively for the small and large
packinghouses. Variable costs are computed from 1979-80 accounting
records of eight packinghouses in the Indian River district.
Fixed costs for existing plants are overhead and investment servicing
cost (debt servicing plus net return on investment). Overhead includes
repairs and maintenance, insurance, taxes and licenses, 30 percent of
foreman labor, 70 percent of administrative expense, and 50 percent of
sales expense." Overhead costs are estimated to be $0.655 and $0.467 per


"The 1979-80 season was typical of the 1973-74 through 1975-76 period. The
average annual boxes packed in Florida during the 1973-74 through the 1975-76
seasons was 35,420,964 boxes, which was 97.2 percent (36,439,588 boxes) of the
boxes packed in 1979-80 (Florida Department of Agriculture and Consumer
Services, 1981-82, p. 17).
During the 1979-80 season, 80.4 percent of the fruit was packed during 63.6
percent of the 11-month period (December 1979-June 1980-Florida Depart-
ment of Agriculture and Consumer Services, 1979-80, pp. 25-29). If a plant had
operated at 100 percent capacity from December through June and then closed
for the season, its capacity utilization for the 11-month period would have been
63.6 percent.
"Small plants are assumed to be larger than 100,000 boxes because 95.8 percent
of the Florida fresh citrus packed during the 1979-80 season was packed by houses
larger than 100,000 boxes (Florida Department of Agriculture and Consumer
Services, 1979-80, pp. 18-24).
"4The largest volume packed during the 1979-80 season in the Indian River
district was 826,026 boxes (Florida Department of Agriculture and Consumer
Services, 1979-80, pp. 18-24).
"Initial computer runs indicated that small new plants were not coming into
solution. Variable costs were 22 cents per box higher for small plants than large.
The small plants were eliminated to save computer time.
"Percentages were determined from packinghouse records and industry
sources.
""See footnote 16.


11








box for a small and large plant (from the 1979-80 accounting records of
eight packinghouses in the district). The investment servicing cost is
$0.125 per box (Hooks and Kilmer, 1981a, and Florida Department of
Agriculture and Consumer Services, p. 37).18
The overhead estimate for existing plants also is used for new plants
($0.467 per box). The investment servicing cost is different, however.
Total estimated facility costs for a new large plant in 1980 including land,
building, offices, and equipment are $1.7 million (Kmetz). It is assumed
that a 20 percent downpayment of $340,000 would be required, the
remainder financed at 16 percent for 20 years. The annual debt servicing
costs are $.27 per box. The yearly net return on investment is figured on
the downpayment ($340,000) plus the average annual principle paid back
during a five-year period ($11,943.92). Since all costs are in constant 1979
dollars, a three percent real rate of return on net investment is assumed
($0.10 per box). The total investment servicing cost (debt servicing plus
net return on investment) is $0.28 per box for a new large packinghouse.
The downpayment is considered a transition cost, since it is incurred
only in the year the plant opened. The annual packinghouse cost of using
the downpayment is included in the investment servicing cost (net return
on investment). The construction of a new packinghouse requires an
initial investment ($340,000) unlike the annual opening of an old plant.
Once a new plant is opened, the model does not allow it to close. It is
assumed that the venture is viable or it would not have been undertaken.
An existing plant which covers cash costs but not all investment servic-
ing costs is closed after three years. A rational entrepreneur may remain
in business for several years without having received a return on invest-
ment. At some point, however, he should have considered the opportu-
nity cost of capital. It is assumed that after three years of no return on
investment, the packinghouse is liquidated or transformed to a new
endeavor. The three-year time period could have differed among firms
depending on tax considerations and/or the rate of property value appre-
ciation.
If an existing plant is closed for less than three years, it reopens at zero
start-up cost. Most packinghouses close for a few months during each
year. The fixed costs associated with start-up after being closed for two
years should not be appreciably different than they are after being closed
for a few months. The fixed cost associated with start-up are included in
the model.



"The $0.125 figure is taken from accounting records where it is called deprecia-
tion and rent. Actual debt servicing and net return on investment was not
available.


12








RESULTS
The model includes oranges and grapefruit produced in 13 locations in
the Indian River district, 35 existing packinghouses at six plant size-
locations (four geographic locations, Figure 1), potential new packing-
houses at seven plant size-locations (Figure 1), five consumption regions
in the United States and Canada, and five export points.
The static mixed integer solutions are obtained for 1979-80 through the
1983-84 seasons. Costs associated with the best R, solutions are shown in
Table 1. They are discounted to 1979 with a 3 percent real discount rate.
The costs in 1983-84 are adjusted to reflect the present value of packing
costs for 1983-84 and beyond. Using estimated discounted transition
costs and static solutions from the mixed integer programming model,
dynamic solutions to the packinghouse location problem are obtained.
"Two such solutions are shown in Table 1.
It is found, from the estimate of va, that the R* best static solutions are
not obtained. Thus, the optimality of the least cost dynamic solution
shown in Table 1 is not assured. The benefit from a better dynamic
solution is offset by additional computer cost and the potential size of the
dynamic model. Using Equation 4, the maximum improvement in the
dynamic solution is 0.16 percent ($4,172,000) of v1 (the sum of the best
integer solution for each year plus zero transition costs).
The best computed solution calls for the immediate closing of 24
existing plants with 11 remaining open. Six new large plants are built for a
total of 17 plants in 1979-80 (Table 2). By the 1983-84 season, nine
existing plants are still operating. One of the new packinghouses is
located at Jupiter in the southern part of the region (Figure 1), where no
packinghouses are now located.
In the model, most of the existing small packinghouses close in the
1979-80 season. This is feasible. From 1964-65 to 1965-1966, packing-
house numbers increased from 160 to 225 (State of Florida total-Florida
Department of Agriculture and Consumer Services). By the 1968-69
season, the number of packinghouses declined to 169. A similar decrease
occurred from 1969-70 until 1971-72, when packinghouse numbers de-
clined from 211 to 164.
A notable result is that the best dynamic solution did not include the
best static solution for the 1979-80, 1980-81, and 1981-82 seasons (Table
1). The fourth best dynamic solution did not include the best solution in
1983-84. A static analysis based on assembly, packing, and distribution
costs for any particular season, without taking into consideration transi-
tion costs, selects the best static solution in that year. Industry transition
costs increase the number of terminal solutions from one to many.




13







Table 1. Static and dynamic solutions to the packinghouse location problem, 1979-80 through 1983-84.a

Static Solutions for Seasons
Rank Dynamic
Ordered Program 1983-84
Solutions Solution 1979-80 1980-81 1981-82 1982-83 (through infinity)b

thousand $
1 2,548,660 59,083 60,762 62,799 64,829 66,901
(Best) (2,296,922)

2 2,548,982 59,083 60,782 62,807 64,832 66,911
(Fourth Best) (2,297,276)
3 59,094 60,838 62,821 64,863 66,914
(2,297,396)
4 59,101 60,852 62,826 64,865 66,925
(2,297,750)
5 59,109 60,862 62,900 64,882 67,001
(2,300,387)
6 59,118 60,877 62,907 64,884 67,012
(2,300,741)
7 59,139 60,884 62,919 64,904 67,015
(2,300,831)
8 59,146 60,922 62,920 64,924 67,025
(2,301,186)
9 59,152 60,939 62,941 64,926 67,025
(2,301,186)









Static Solutions for Seasons
Rank Dynamic
Ordered Program 1983-84
Solutions Solution 1979-80 1980-81 1981-82 1982-83 (through infinity)b

thousand $
10 59,176 60,943 64,977 67,046
(2,301,896)
Initial
Configuration 2,605,366 62,350 63,687 65,167 66,779 68,371
(2,347,383)
Transition cost
(Best) 2903c 365 320 329 302
Transition cost
"(Fourth Best) 2903c 365 320 329 302
Transition Cost
(Initial Conf.) Oc 0 0 0 0
"aAll costs are in 1979 dollars.
bPresent value of collection, packing, and distribution cost from 1983-84 to infinity, assuming plant configuration, supply, and demand remain
unchanged.
"Transition cost to initial configuration.











Table 2. Packinghouse size configuration for the best dynamic solution over five years.

Initial Packinghouse Number for Seasons
Capacity Config-
Location (1-3/5 bu. box) uration 1979-80 1980-81 1981-82 1982-83 1983-84

-- 1,000's -
North
Titusville 100-500 2* -- -
501-850 1 1 1 1 1
Cocoa 100-500 1* 1* 1* 1* -
501-850 -
Melbournea 501-850 -
South
Vero Beach 100-500 11* 1*
501-850 7* 7*,1 7*,2 7*,3 7*,3 7*,4
Ft. Pierce 100-500 12* -
501-850 2* 2*,3 2*,3 2*,3 2*,4 2*,4
Stuarta 501-850 -
Jupiter" 501-850 1 1 1 1 1

*Denotes existing plants. 7*,1 means seven existing plants operating and 1 new plant operating in that year.
"aNew location.






SENSITIVITY ANALYSIS
The results of the dynamic programming analysis depend directly on
the exogenous data: namely, projected production and consumption;
collection, elimination transportation, packing, and distribution costs;
and transition costs. The results may also be dependent on the length of
the time horizon and the discount rate. Because of the high computer cost
of generating multiple static solutions, sensitivity analysis of the results
are limited to the time horizon, the discount rate, and the transition costs.

Time Horizon
To determine the impact on the model of alternate specifications of the
time horizon, an analysis was performed using a nine-year horizon. The
length of planning horizon did not influence the dynamic adjustments
required in the first five years, as the five and nine year model results were
identical (Tables 2 and 3). The nine-year model did show the number of
packinghouses in the south increasing from 15 in 1978-80 to 20 in 1987-88
(Table 3). This may be somewhat understated because the nine year
model did not include the production from trees planted after the 1979-
80 season. The 1980-81 plantings would have started bearing fruit in
1984-85, thus increasing the packinghouse capacity required in 1984-85
and beyond. Plantings beyond 1980-81 would produce similar results.

Discount Interest Rate
In the earlier analysis, a 3 percent discount rate was used as the
long-run real discount rate. Rates from 0.1 to 25 percent were used to test
the sensitivity of the dynamic model (Table 4). As the discount rate
increased, the present value of the dynamic paths with lower costs in
1982-83 and 1983-84 (paths 4, 5, 8, 11-Table 4) became relatively
higher than the present value of the dynamic paths with lower costs in
1980-81 and 1981-82 (paths 2, 3, 6, 7, 9, 10-Table 4). This was expected
because the present value of the dollar saved now increases relative to a
dollar saved later as the discount rate increases. With high real interest
rate values, firms will tend to postpone cost saving changes with high
initial cost. Over a reasonable range of real interest rates (0.0 to 0.06),
one, two, and three best dynamic paths retain their relative ranking;
therefore, the results are relatively insensitive to changes in interest rates.

Transition Costs
Setting different limits for industry transition costs for existing and new
plants alters existing plant numbers in the best dynamic solution. If


17










Table 3. Packinghouse size configuration for the best dynamic solution over nine years.

Initial Packinghouse Number for Seasons
Capacity Config-
(1 3/5 bu. box) uration 1979-80 1980-81 1981-82 1982-83 1983-84 1984-85 1985-86 1986-87 1987-88

-- 1,000's-
North
Titusville 100-500 2* -
501-850 1 1 1 1 1 1 1 1 1
Cocoa 100-500 1* 1* 1* 1* -
501-850 -
Melbournea 501-850 1 1 1 1
00
South
Vero Beach 100-500 11* 1* -
501-850 7* 7*,1 7*,2 7*,3 7*,3 7*,4 7*,4 7*,4 7*,5 7*,5
Ft. Pierce 100-500 12* -
501-850 2* 2*,3 2*,3 2*,3 2*,4 2*,4 2*,4 2*,5 2*,5 1*,6
Stuarta 501-850 - -
Jupitera 501-850 1 1 1 1 1 1 1 1 1

*Denotes existing plants. 7*,1 means seven existing plants operating and 1 new plant operating in that year.
"aNew location.














Table 4. Dynamic solutions ranked for each discount rate.

Static Solutions in Dynamic Path Rank Ordered Dynamic Solutions by Discount Rate
Dynamic
Path 1979-80 1980-811981-821982-831983-84 0.001 0.003 0.005 0.009 0.01 0.03 0.06 0.12 0.25

No. ----------------- Ranka ----- ---- ------------------------------------- Rankb -------
2 2 1 1 2 2 8 8 7 7 7 7 7 8 7
3 3 1 1 2 2 5 5 4 4 4 4 4 2 1
6 6 1 1 2 2 6 6 5 5 5 5 5 4 3
7 7 1 1 2 2 7 7 6 6 6 6 6 5 4
9 9 3 1 2 2 10 10 9 9 9 9 9 9 9
10 10 1 1 2 2 9 9 8 8 8 8 8 6 6
1 Infeasible Infeasible
4 4 2 2 1 1 1 1 1 1 1 1 1 1 2
5 5 2 2 1 1 3 3 3 3 3 3 3 7 8
8 8 2 2 1 1 2 2 2 2 2 2 2 3 5
11 11 2 2 1 1 4 4 10 10 10 10 10 10 10

"Ranking represents ordering of static solutions within each year; lowest cost equals one.
bRanking represents ordering among the dynamic paths for each discount rate; lowest cost equals one.








transition costs are reduced from $4,219,000 (Table 1) to zero (v1), the
number of existing packinghouses decreases from 11 to 10 in 1979-80.
The remaining seasons are unchanged. If transition costs increase from
$4,219,000 to $60,925,000, the number of existing packinghouses in the
best dynamic solution increases from 11 to 35 (initial configuration, Table
2) in 1979-80 and remains at 35 through 1983-84.
The transition costs for existing plants (debt servicing plus net return
on investment) are varied from zero to $4.75 per 1/5 bushel box. The
present value of the dynamic paths with lower initial transition costs
become relatively lower than the present value of the dynamic paths with
higher initial transition costs (Table 5). However, the best dynamic
solution retains its best ranking over all levels of transition costs.


Table 5. Dynamic solutions ranked for each level of existing plant transition
costs.a

Dynamic Initial Rank Ordered Dynamic Solutions by
Path Transition Costsb Level of Transition Costs'

No. $1,000s 0 $0.125 $.30 $4.75
Ranked
3 2905.9 5 4 4 3
6 2905.9 6 5 5 4
7 2905.0 7 6 6 5
8 2905.9 3 2 2 2
10 2889.9 8 8 7 6e
11 0 10 10 10 7
1 Infeasible
4 2905.9 1 1 1 1
2 3363.4 4 7 8 9
5 3363.4 2 3 3 8
9 3363.4 9 9 9 Infeasible

"aThe discount rate was 0.03 and the time horizon was 5 years. The transition costs
were debt servicing plus net return on investment. The down payment for a new plant
($340,000) was not excluded.
bTransition costs from initial configuration to 1979-80.
"Cost per 1 3/5 bushel box.
dRanking represents ordering among the dynamic paths for each transition cost, lowest
present value of dynamic paths equals one.
"Packinghouse configuration changed.


SUMMARY

The southern movement of citrus production necessitates the construc-
tion of a new packinghouse in Jupiter, Florida, located in the south of the


20








Indian River district. However, existing Indian River packinghouse
capacity is located where needed. Existing capacity could be rebuilt into
larger packinghouses. Instead of building new plants in the same cities
where existing plants are located, old packinghouses could be enlarged to
take advantage of economies of size.

CONCLUSIONS
Information has been disseminated to the industry documenting the
presence of structure-altering economic forces (Kilmer and Spreen,
1983a). A large packinghouse is efficient enough to be competitive with
existing small packinghouses even though existing plants forego a return
on investment for several years in order to stay in business. Total collec-
tion, packing, and distribution costs can be reduced (mostly packing cost)
by an estimated $0.086 per box in 1983-84 (3.0 and 3.5 percent of small
and large plant packing cost) if the industry adjusts. Given the risk and
uncertainty in the actual operation of the industry, we conclude that the
industry is operating in a cost efficient manner.
The model used in this analysis approximates a cost-minimizing model
that assumes that packinghouses are operated by one firm. The industry,
on the other hand, is composed of many individual profit-seeking entre-
preneurs. They tend to maximize profits and not minimize costs. The
results of this model may differ from the real world results because of this
discrepancy. In the long run, however, the industry will encounter times
when cost minimization will become more of an objective, such as during
a recession. The industry would be expected to trend in the direction of
the model results during periods when one of the few ways to enhance
profits is to take a more critical look at reducing cost.



















21















APPENDIX A
MIXED INTEGER PROGRAM

The mixed integer programming algorithm from IBM was used to
obtain the static solutions for the plant location problem. The model
consumed a large block of processing time and produced a large volume
of output. Steps taken to overcome these problems are presented.
To decrease the processing time, XMXDROP (lines 41-43) was em-
ployed to decrease the number of potential solutions to the problem
(Figure Al). XMXDROP eliminated all potential problem solutions that
had an objective function value greater than a user-specified amount.
This was initially determined by adding an arbitrary amount to the
objective function value of a noninteger solution.
The output was decreased by using an integer solution counter (CT,
lines 54, 55), XMXFNLOG (lines 38-40), and "Active" (line 51) (Figure
Al). The CT counter put a maximum limit on the number of integer
solutions (lines 56, 57). The XMXFNLOG decreased the output gener-
ated as the MIP searched for an integer solution. Finally, "Active"
limited the output to those variables that were in solution.


Figure Al. Integer program.

0000 //KSTPR090 JOB (7000,1009,180,10,0),'KILMER',CLASS-1
0001 /*PASSWORD
0002 /*ROUTE PRINT LOCAL
0003 // EXEC MPS
0004 //CONTROL.SYSIN DD *
0005 PROGRAM
0006 INITIAL
0007 TITLE('CITRUS PACKING 1979-80')
0008 MOVE(XDATA,'SDCCX')
0009 MOVE(XPBNAME,'KST')
0010 CONVERT('FILE'r'FORT','CMASKS', 'Q**01**''OQ***02**' 'Q'**03*', 'O***01
0011 5*'r,'0t* 07**','QS*09*', 'a***11**l''G**t13**' 'G***15**''Q*17**',2
0012 '0*1ttl9t*'' O*t21*','G*=S23**','XJ01 ''XJ02','XJ03','XJ05','XJ07','XJ3
0013 09'>'XJll'>'XJ13','XJ15','XJ17','XJ19','XJ21','XJ23' 'ALPHA','OMEGA',' 4
0014 ')
0015 MOVE(XDATA 'SMALL')
0016 MOVE(XPBNAME,'KST')
0017 MOVE(XOLDNAME 'KST')
0018 REVISE('FILE,''NEWB')
0019 *SETUP 'NODES' EXPANDS THE STORAGE ALLOWED FOR THE NODES TABLE
0020 SETUP('BOUND'r'XRELAX','NODES',1000)
0021 MOVE(XRHS,'SDC79-80')
0022 MOVE(XOBJ,'TC')
0023 MOVE(XDATA 'SDCCX')
0024 INSERT
0025 PRIMAL
0026 *SOLUTION PRINTS ONLY THOSE VARIABLES THAT ARE ACTIVE IN SOLUTION
0027 SOLUTION('ACTIVE')
0028 *INIMIX SETS PARAMETERS TO STANDARD VALUES AND ESTABLISHES STANDARD
0029 *PROCESSING FOR DEMANDS IN THE MIXED INTEGER PROGRAM
0030 INIMIX
0031 tMIXSTART INITIATES THE SEARCH FOR INTEGER SOLUTIONS
0032 MIXSTART('COST')

(Continued)


23











Figure Al. (Continued)


0033 $MVADR ALLOWS CONTROL OF WHAT IS ACCOMPLISHED AFTER AN INTEGER SOLUTION
0034 *IS FOUND. FOR EXAMPLE,ABC(FOUND BELOW)IS A SUBROUTINE IN WHICH THE
0035 *'ACTIVE' VARIABLES ARE PRINTED AS SOLUTION OUTPUT. THEN THE
0036 *XMXDROP IS RESET AND SEARCH RESUMES FOR INTEGER SOLUTIONS
0037 MVADR(XDOPRINTiABC)
0038 *XMXFNLOG: LOG LINES ARE ONLY PRINTED AT EACH NODE WHOSE NUMBER
0039 *IS A MULTIPLE OF THE VALUE ARBITRARILY ASSIGNED BY USER
0040 XMXFNLOG=1000
0041 *XMXDROP DROPS ALL POTENTIAL PROBLEM SOLUTIONS THAT HAVE A FUNCTIONAL
0042 *VALUE > XMXDROP. THIS DECREASES PROCESSING TIME
0043 XMXDROP=.60964E08
0044 CT=0
0045 *MIXFLOW PERFORMS THE SEARCH FOR INTEGER SOLUTIONS
0046 MIXFLOW
0047 tMIXSTATS PRINTS A SURVEY OF THE CURRENT STATUS OF THE SEARCH
0048 STOP MIXSTATS
0049 EXIT
0050 *ABC IS A SUBROUTINE INITIATED BY MVADR(XDOPRINTABC)
0051 ABC SOLUTION('ACTIVE')
0052 *XMXDROP IS REDEFINED
0053 XMXDROP=.60964E08
0054 *CT MAINTAINS COUNT OF NUMBER OF INTEGER SOLUTIONS
0055 CT=CT + 1
0056 *TERMINATES PROGRAM AND SENDS PROGRAM TO MIXSTATS FOR OUTPUT
0057 IF(CT.EG.10,STOP)
0058 CONTINUE
0059 CT DC(0)
0060 PEND
0061 //PROBLEM.SYSIN DD DSN=UF.D0011272.S5.MPSBASISDISP=SHR,
0062 // DCB=(RECFM=FBDLRECL=80,BLKSIZE=400)
0063 //PROBLEM.FORT DD DSN=UF.D0011272.S5.MPSFORTDISP=SHR,
0064 // DCB=(RECFM=FBLRECL80sBLKSIZE=6400)
0065 //PROBLEM.NEWS DD *
0066 NAME SMALL
0067 ROWS
0068 AFTER
0069 L CON1
0070 L CON2
0071 COLUMNS
0072 MODIFY
0073 XJ01 CON2 1.
0074 XJ02 CON1 1.
0075 XJ02 CON2 1.
0076 XJ05 CON1 1.
0077 XJ07 CON1 1.
0078 XJ07 CON2 1,
0079 XJ11 CON1 1.
0080 XJ13 CON1 1.
0081 XJ13 CON2 1.
0082 XJ15 CON2 1.
0083 XJ17 CON1 1.
0084 XJ17 CON2 1.
0085 XJ23 CON1 1.
0086 XJ23 CON2 1.
0087 RHS
0088 MODIFY
0089 SDC79-80 CON1 16.
0090 SDC79-80 CON2 16.
0091 ENDATA
0092 /1
















24











APPENDIX B
DYNAMIC PROGRAM ALGORITHM
The purpose of this appendix is to describe in more detail the solution
of the dynamic programming problem given in Table 1. First, a general
discussion of the solution of dynamic programming models is presented
followed by documentation of a computer program developed to spe-
cifically solve the dynamic plant location problem (Figure Bi).

The Solution of Dynamic Programming Problems
Unlike linear programming problems, which can always be solved
using a specific algorithm, the solution of a dynamic programming prob-
lem is generally specific to the problem itself. No widely distributed
software packages are available. The researcher must develop software
for each problem encountered.
Dynamic programming refers to a problem solving approach which
involves a sequence of interrelated decisions (Hillier and Lieberman, p.
266). The dynamic programming problem in this study is deterministic for
which the computational solution is straightforward. A computer pro-
gram is developed to allow quick solution to alternative specifications of
the model.

Documentation
The computer program is written in FORTRAN (Figure B1). Com-
ments are placed in the program to provide explanation of the specific
steps required.

Overview
The number of periods to be considered (T), discount factor (interest
"rate), initial plant configurations and the transition costs of opening new
plants and closing old plants are given as data to the program. The best R,
solutions for each period t, t = 1, . T, have been previously deter-
mined. The program starts with the last two time periods and successively
works backwards in time to determine the best path from each static
solution in period 1 to period T The transition cost of moving from the
initial configuration to each of the period 1 static solutions is then calcu-
lated.
The output of the program includes the optimal path starting from each
"of the first period solutions and the total cost of that path. The user can
compare these paths to find the best overall path.


25











Figure Bl. Dynamic program.


0000 //DYNAMIC JOB (4001,1272,11,0O),'SPREEN',CLASS=A
0001 /*PASSWORD
0002 // EXEC FORTGCG
0003 //FORT.SYSIN DD *
0004 C
0005 C DYNAMIC PROGRAMING ALGORITHM FOR PLANT LOCATION MODELS
0006 C
0007 INTEGER IEXIST(6)
0008 REAL XPLCST(6),NEWPCT
0009 C THE NUMBER OF EXISTING PLANTS IS STORED IN IEXIST(.)
0010 DATA IEXIST/1,2t122,7,11/
0011 C THE TRANSITION COST OF CLOSING AN EXISTING PLANT IS STORED
0012 C IN XPLCST(.)
0013 DATA XPLCST/19.53,68.55,35.64,39.6,82.25,35.57/
0014 C THE TRANSITION COST OF OPENING A NEW PLANT IS NEWPCT,
0015 DATA NEWPCT/340./
0016 DIMENSION IPATH(10,20),ICON(10,20,13),NSOL(10),TRANS(20)
0017 DIMENSION XSCST(10,20),TRAN(22020),COSTC(10,20),XCT(20),TCOST(10)
0018 100 FORMAT(I2,FB.3,213)
0019 101 FORMAT(40I2)
0020 102 FORMAT(1312,FI4.0)
0021 103 FORMAT(2F10.2)
0022 106 FORMAT(1H/,'OPTIMAL PATH',IX,12)
0023 107 FORMAT(6X,12)
0024 108 FORMAT(' INFEASIBLE PATH')
0025 109 FORMAT(' TOTAL COST OF PATH'riX,F12.2)
0026 112 FORMAT(1X,'TRANSITION COST FROM EXISTING')
0027 113 FORMAT(1X,'TRANSITION COSTS FROM STAGE'1r3,' TO STAGE',13)
0028 C
0029 C THE PROGRAM READS NUMBER OF PERIODS (NYR), THE DISCOUNT FACTOR
0030 C (DISC), THE NUMBER OF OLD PLANT LOCATIONS (NOPL), THE NUMBER OF
0031 C NEW PLANT LOCATIONS (NNPL), AND THE NUMBER OF STATIC SOLUTIONS
0032 C TO BE CONSIDERED EACH PERIOD (NSOL(,)).
0033 C
0034 READ(5,100) NYR,DISC,NOPL,NNPL
0035 READ(5,101)(NSOL(L),L=1,NYR)
0036 NL=NOPL+NNPL
0037 DO I L=I,NYR
0038 LL=NSOL(L)
0039 DO 1 I=1,LL
0040 1 READ(5,102)(ICON(LI,J),J=1,NL),XSCST(L,I)
0041 C
0042 C FOR EACH PARTICULAR STATIC SOLUTION, THE PLANT CONFIGURATION IS
0043 C ICON(,,.,.) AND THE COST OF THAT SOLUTION IS XSCST(,,.),
0044 C
0045 C EACH STATIC SOLUTION COST IS DISCOUNTED BACK TO THE INITIAL TIME PERIOD
0046 C
0047 DO 2 L=1,NYR
0048 FAC=1./(1.+DISC)**L
0049 LL=NSOL(L)
0050 DO 2 I=ILL
0051 2 XSCST(L,I)=XSCST(L,I)*FAC
0052 LYR=NSOL(NYR)
0053 ADJ=(1.+DISC)/DISC
0054 DO 4 I=1,LYR
0055 4 COSTC(NYR,I)=XSCST(NYR,I)ADJ
0056 C
0057 C THROUGH LINE IS A LOOP ON TIME PERIOD WHICH CONSTITUTES THE
0058 C COMPUTATIONAL ASPECTS OF THE PROGRAM, THE STEPS ARE KEYED TO
0059 C ACCOMPANYING TEXT,
0060 C
0061 L=NYR
0062 22 L1=L-l
0063 LL=NSOL(L)
0064 LL1=NSOL(L1)
0065 DO 3 I=1,20
0066 DO 3 J-1,20
0067 3 TRAN(IJ)=0O
0068 DO 9 M 1,LL
0069 10 9 K 1.,LL1
0070 C STEP 1




26












0071 DO 5 I=1,NL
0072 ID=ICON(LM,I)-ICON(L1,K,I)
0073 IF(I.G1.NOPL) GO TO 6
0074 IF (ID) 7,5,8
0075 8 CONTINUE
0076 C AN OLD PLANT IS REOPENED, TRANSITION COST IS ZERO
0077 00 TO 5
0078 7 ID=-ID
0079 C OLD PLANTS CLOSE, TRANSITION COST INCREASES
0080 TRAN(KM)=TRAN(K,M)+XPLCST([)*FLOAT(ID)
0081 GO TO 5
0082 6 IF (ID) 11,5,32
0083 C NEW PLANTS OPEN, TRANSITION COST INCREASES
0084 32 TRAN(KM)=TRAN(K,M)+NEWPCT*FLOAT(ID)
0085 GO TO 5
0086 C NEW PLANTS CLOSE, PATH IS INFEASIBLE
0087 C TRANSITION COST IS SET EOUAL TO -1
0088 11 TRAN(K,M)=-1.
0089 GO TO 9
0090 5 CONTINUE
0091 9 CONTINUE
0092 C TRANSITION COSTS ARE DISCOUNTED AND STORED IN 1RAN(K,M)
0093 10 FAC=l./(1.+DISC)**Ll
0094 DO 13 I=1,LL1
0095 DO 13 J=1,LL
0096 IF(TRAN(IJ).EQ--1.) GO TO 13
0097 TRAN(IJ)=TRAN(I.J)*FAC
0098 13 CONTINUE
0099 WRITE(6,113)L1,L
0100 DO 34 I=1,LL1
0101 34 WRITE(6,110)(TRAN(I.J),J=1,LL)
0102 110 FORMAT(1XO10F7.0)
0103 K=O
0104 C STEP 2
0105 20 DO 16 I=1,LL
0106 16 XCT(I)=0.
0107 M=I
0108 K=K+1
0109 IF(K.GT.LL1) GO TO 15
0110 XMIN=100000000000.
0111 MMIN=99
0112 19 IF(TRAN(R1M).EQ.--I.) GO TO 17
0113 XCT(M)=XSCST(Ll,K)+TRAN(KM)+COSTC(L,M)
0114 IF(XCT(M).GF.XMIN) GO TO 17
0115 XMIN=XCT(M)
0116 MMIN=M
0117 17 M=M+l
0118 IF(M.GT.LL) GO TO 18
0119 0O TO 19
0120 18 COSTC(L1,K)=XMIN
0121 IPATH(L,1K)=MMIN
0122 GO TO 20
0123 15 L-L-1
0124 IF(L.LE.1) GO TO 21
0125 GO TO 22
0126 C STEP 3
0127 21 L1L=NSOL(1)
0128 DO 35 I=1,20
0129 35 TRANS(I)=0.
0130 DO 27 K=I,LIL
0131 DO 23 I=1,NOPL
0132 ID=ICON(1,K,I)-IEXIST(I)
0133 IF (ID) 25,23,23
0134 25 ID=-ID
0135 TRANS(K)=TRANS(K)+XPLCST(I)*FLOAT(ID)
0136 23 CONTINUE
0137 II=0
0138 NN=NOPLtl
0139 DO 24 I=NNNL
0140 24 II=II+ICON(1,KI)
0141 TRANS(K)=TRANS(K)+NEWPCT*FLOAT(II)
0142 27 TCOST(K)CCOSTC(1,K)+TRANS(K)

(Continued)




27









Figure Bl. (Continued)

0143 WRITE(6,112)
0144 WRITE(6,110)(TRANS(K) ,K=1,L11 )
0145 C STEP 4
0146 DO 28 K=1,L1L
0147 KA=K
0148 WRITE(6,106) K
0149 L=1
0150 WRITE(6,107)KA
0151 29 KI=IPATH(LKA)
0152 IF(K1.EQ.99) GO TO 30
0153 WRITE(6,107)K1
0154 L=L+1
0155 MNYR=NYR 1
0156 IF(L.GT.MNYR) GO TO 31
0157 KA=K1
0158 0O TO 29
0159 30 WRITE(6,108)
0160 GO TO 28
0161 31 WRITE(6,109)TCOST(K)
0162 28 CONTINUE
0163 STOP
0164 END
0165 //GO.SYSIN DD *
0166 /*INCLUDE JAN5
0167 /*



Variable Definitions

NOPL the number of existing (old) plant locations.
NNPL the number of new plant locations.
NL total number of possible plant locations (NL =
NOPL + NNPL).

IEXIST(NOPL) contains the number of existing plants at each
location.

XPLCST(NOPL) the transition cost of closing an existing plant at
each location.

NEWPCT the transition cost of opening a new plant. The
cost is assumed invariant with respect to location.

NYR the number of time periods.
NSOL(L) the number of static solutions to be considered
each period (L = 1, . NYR).

ICON(L,I,J) the number plants at each location J in static
solution I for time period L (J = 1, .. NL;
I = 1,... ,NSOL(L); L = 1,... ,NYR).

XSCST(L,I) the corresponding cost of the static solutions in
ICON.

DISC the discount factor (interest) expressed as a dec-
imal.





28








L index for time periods.
L1 defined to be L 1, thus L and L1 are adjacent
time periods.
LL the number of static solutions in L.
LL1 the number of static solutions in period L1.
These variables are determined within the program.
TRAN(I,J) the transition cost of moving from static solution
I in period L1 to static solution J in period L. If that
path is infeasible, TRAN(I,J) is set to -1.
XCT(M) total cost of a path starting with static solution I
in period L1 through static solution M in period L
and follows the best path thereafter.
COSTC(L1,K) the total cost of the best path starting from static
solution Kin period L1. (Note that COSTC(L1,K)
= Minimum [XCT(M)].)
IPA TH(L1,K) the path of the best dynamic solution from static
solution K in period L1. (That is, IPATH(L1,K) is
the value of M for which XCT(M) is minimum).
TRANS(K) the transition cost of moving from the initial
configuration to static solution K in period 1.
TCOST(K) the total cost of best overall dynamic solution
that passes through static solution K in period 1.

Specific Components of the Program
The initial section of the program (Figure Bl) specifies the number and
closing cost of old plants at each location as well as the cost of opening a
new plant. Arrays are dimensioned. The current version assumes NOPL
= 6 and NNPL = 7. If these values are increased, the arrays IEXIST,
XPLCST, and ICON must be correspondingly increased. Currently, the
program can handle up to 10 time periods (NYR = 10), and 20 static
solutions each time period (NSOL(L)).
The FORTRAN statements from line 34 to line 40 read all exogenously
specified variables. Lines 47 to 55 discount all static costs back to the
initial time period. The program assumes that the discount rate (DISC) is
in the same time units as the time periods.
Beginning with line 62 and continuing through line 125 is a loop which,
for each pair of adjacent time periods, finds the best path from each static
solution in the earlier time period to the later time period. The comment
statements included in the listing explain the specific calculations. Start-



29








ing with Step 1 (line 70), the transition cost of moving from a static
solution in period L -1 to a static solution in period L is computed. If
certain paths are infeasible, the transition cost is set to -1 to indicate the
infeasibility. At Step 2 (line 104), the program computes the total cost of
a path beginning with a particular static solution in period L 1,
successively through each static solution in period L and using the best
path thereafter. In this manner it can find the best path. It stores in
IPA TH(L1,K) the static solution in period L to which static solution K in
period L 1 should move, and in COSTC(L1,K) stores the total cost of
that path to the last time period NYR.
After circling through Steps 1 and 2 NYR 1 times, the program has
found the best path from each static solution in period 1 to period NYR.
At Step 3 (line 126), the transition costs from the initial plant configura-
tion to the first period static solutions are computed. The overall cost of
each path is stored in TCOST(K), K = 1, . NSOL(1).

Output
The dynamic program output is shown as Figure B2. For debugging
purposes, the transition cost matrix is printed (lines 00-81). A label
indicating the time periods (stages) is printed, and the columns of the
matrix represent static solutions in period L and the rows are static
solutions in period L 1. In the sample output four transition cost
matrices are printed because the sample has five (NYR = 5) time periods.
An array with one row (lines 80, 81) appears next, whose elements
contain the transition costs from the initial configuration to each static
solution in period 1.
The optimal paths starting with the initial configuration through each
static solution in period 1 are listed next, along with the total discounted
cost of each path (lines 82-154). In the sample output, the path through
static solution 1 in period 1 is infeasible and is indicated as such. To
interpret a feasible path, consider "OPTIMAL PATH 4". It begins with
the initial configuration to static solution 4 in period 1 and continues to
static solution 2 in period 2 to static solution 2 in period 3 to static solution
1 in period 4 to static solution 1 in period 5. Total discounted cost of this
dynamic solution is $2,548,658,000. By comparing the total cost of the
dynamic solutions, an optimal dynamic solution can be identified. In the
sample output, this is OPTIMAL PATH 4.









30











Figure B2. Dynamic program output.


0000 TRANSITION COSTS FROM STAGE 4 TO STAGE 5
0001 302. -1. 302, -1. -1. -1. -1. -1. 302. -1.
0002 -1.
0003 -1. 302. -1. 302. -1. -1. -1. -1. -1. -1.
0004 -1.
0005 -1. -1. 302. -1. -1. -1 -1 -1. -1. -1.
0006 -1.
0007 -1. -1. -1. 302. -1. -1. -1. -1. -1. -1,
0008 -1.
0009 302.1. -1. -1 1, 1. .- -1-1. 302,
0010 -1.
0011 -1. 302. -1. -1. -1. -1. -1. -1. -1. 302,
0012 -1.
0013 319, -1. 319. -1. -1. -1. -1. -1. 319. -1.
0014 -1.
0015 -1. -1. -1. -1. 302. -1. 302. -1. 302. -1.
0016 -1.
0017 -1. -1. -1. -1. -1. 302. -1. 302. -1. -1.
0018 -1.
0019 -1. -1. -1. -1. 302. -1. -1. -1. -1. -1.
0020 -1.
0021 3836. 3836. 3836. 3836. 3836. 3836. 3836. 3836. 3836. 3836.
0022 0.
0023 TRANSITION COSTS FROM STAGE 3 TO STAGE 4
0024 -1. 329. -1. -1. -1. 329. -1. -1, -1. -1.
0025 -1.
0026 329. -1. -1. -1. 329. -1. 311. -1. -1. -1.
0027 -1.
0028 329. -1. 329. -1. -1. -1. 311. -1 -1. --1.
0029 -1.
0030 -1. 329. -1. 329. -1, -1. -1. -1. -1. -1.
0031 -1.
0032 344. -1. -1. -1. 344. -1. 344. -1. -1. -1.
0033 -1.
0034 -1. 344. -1. -1. -1. 344. -1. -1. -1. -1.
0035 -1.
0036 -1. -1. -1. -1. -1, 0. -1 1. -1. -1.
0037 -1.
0038 -1. 344. -1. -1. -1. 344. -1. -1. -1. -1.
0039 -1.
0040 -1. -1. -1. 344. -1. -1. -1. -1. -1, -1.
0041 -1.
0042 3640. 3640. 3640, 3640. 3640. 3640. 3622. 3640. 3640, 3640.
0043 0.
0044 TRANSITION COSTS FROM STAGE 2 TO STAGE 3
0045 320. -1. -1. 320. -1. 339. 737. 339. 339. -1.
0046 -1. 320. 320. -1. 339. -1. -1, -1. -1. --1,
0047 0. -1. -1. -1. -1. 0. 320. 0. -1. -1.
0048 354. -1. -1. 354. -1. 354. 752. 320. 354, -1.
0049 354. -1. -1. 354. -1. 320. 752. 354. 320. -1.
0050 -1. --1. -1, 320. -1. -1. -1. -1. -1. -1.
0051 -1. -1. -1. 0. --1. -1. -1. -1. --1. -1.
0052 358. -1. -1. 358. -1. 358. 756. 358. 3S8. -1.
0053 -1. -1. -1. 0. --1. -1. -1 -1. -1 --1.
0054 -1. 358. 358. -1. 358. -1 -1. -1. -1. -1.
0055 3410. 3410. 3410. 3410. 3395. 3395. 3827. 3395. 3395. 0,
0056 TRANSITION COSTS FROM STAGE 1 TO STAGE 2
0057 -1. --1. -1. -1. -1. -1. -1. -1. -1. -1.
0058 -1.
0059 0. -1. 349. 1 19 19. -1. 416. 19. 349. -1.
0060 -1.
0061 365, -1. 794. 349, 384. 365. 780, 384. 794. -1.
0062 -1.
0063 -1. 365. -1. -1. -1. -1. -1. -1. 384.
0064 --1.
0065 -1. 0. -1. --1, -1. -1. -1. 19.
0066 --1.
0067 365. -1. 794. 384. 349. -1. 780. 384. 794. -1.
0068 -1.
0069 365. -1. 794. 384. 349. 365. 780. 384. 794. -1.
0070 --1.

(Continued)




31











Figure B2. (Continued)

0071 -1 365. -1. -1. -1. -1. -1. -1. -1. 384.
0072 -1.
0073 -1. --1. 349, -1, -1. -1. -1. --1. -1. -.1.
00/4 -1.
0075 399, --1. 809, 365. 399. 399. 796. 399. 809. -1.
0076 -1,
0077 3183. 3183. 3612, 3167, 3167. 3183, 3598, 3163. 3612. 3163,
0078 0.
0079 TRANSITION COST FROM EXISTING
0080 2918, 3360. 2903, 2903, 3360. 2903. 2903, 2903, 3360, 2887.
0081 0.
0082 OPTIMAL PATH 1
0083 1
0084 INFEASIBLE PAIH
0085 OPTIMAL PATH 2
0086 2
0087 1
0008 1
0089
0090
0091 TOTAL COST (IF PATH 2549062.00
0092 OPTIMAL PA H 3
0093 3
0094 I
0095 1
0096 2
0097 2
0098 TOTAL COST OF PATH 2548979.00
0099 OPTIMAL PATH 4
0100 4
0101 2
0102 2
0103 1
0104 1
0105 TOTAL COST OF PATH 2548658.00
0106 OPTIMAL PATH ;
0107
0108 2
0109 2
0110 I
0111 1
0112 TOTAL COST OF PATH 2548759,00
0113 OPTIMAL PATH 6
0114 6
0115 1
0116 1
0111
0118 2
0119 TOTAL COST OF PATH 2549004.00
0120 OPTIMAL PATH 7
0121 7
0122 J
0123 1
0124 2
0125
0126 TOTAL COST OF PATH 2549024.00
0127 OPTIMAL PATH U
0128 U
0129
0130 2
0131 1
0132 1
0133 TOTAL COST OF PATH 2548703.00
0134 OPTIMAL PATH 9
0135 9
0136 3
0137 1
0138 2
0139 2
0140 TOTAL COST OF PATH 2549236,00
0141 OPTIMAL PATH 10
0142 10




32












0143 1
0144 1
0145 2
0146
0147 TOTAL COST OF PATH 2549079.00
0148 OPTIMAL PATH 11
0149 11
0150 2
0151 2
0152 1
0153 1
0154 TOTAL COST OF PATH 2551823.00






































































33














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60(1978): 385-392.





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Kilmer, Richard L., and Thomas H. Spreen. "Location of Indian River Fresh
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--- "Indian River Citrus Packinghouses and the Southward Movement of
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36






















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Agricultural Experiment Stations are open to all persons regardless of race,
color, national origin, age, sex, or handicap.




This public document was promulgated at an annual cost of $1900
or a cost of $1.58 per copy to present information on potential
dynamic adjustments in size, number, and location of Indian River
citrus packinghouses.



ISSN 0096-607X





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