A DYNAMIC, MIXED INTEGER LOCATION MODEL
APPLIED TO FLORIDA CITRUS PACKINGHOUSES
By
VIRGILIO AZUIL PASCOA MACHADG
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1978
Ii~
Copyright by
VIRGfLIO AZUfL PASCOA MACHADO
1978
TO MY BELOVED FATHER
ACKNOWLEDGEMENTS
The author wishes to thank the helpful assistance of Dr. D. S.
Tilley, his supervisor and friend, for all the guidance, encouragement
and support. Thanks also to Dr. L. Polopolus, whose critical revisions
much improved this dissertation; to Dr. R. W. Ward and Dr. R. S.
Leavenworth for their assistance.
Special thanks to Dr. B. E. Melton, whose expertise with the computer
packages opened doors to their effective utilization; to Nancy Melton
who assisted in the execution of all the computer work; and to the
Northeast Regional Data Center for providing the computational facilities.
Nancy Waters was kind enough to dedicate herself to the manuscript
typing. Molly Owen was superb in typing extensive early versions of
this document and was helped here and there by Vera Anderson and Debbie
Mixon. Susan Howard did the excellent art work.
The author's friends were always prodigal in dispensing their love
and attention and made "saudade" almost an unknown word.
Special recognition is given herein to those men of great vision
of the governments of the U.S.A. and Portugal who made the possibility
of this educational experience a reality. There is a certain pride in
having been found deserving of the financial support of the Florida
Department of Citrus.
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS . . . . . . . . . . . iv
LIST OF TABLES . . . . . . . . ... . . . .viii
LIST OF FIGURES. . . . . . . . . ... . . . xii
KEY TO SYMBOLS AND ABBREVIATIONS . . . . . . . ... .xiii
ABSTRACT . . . . . . . . . . . . . . xv
CHAPTER
I PROBLEM DEFINITION . . . . . . . . ... .. 1
I1. Problem Statement . . . . . . . . .. 1
I2. Propositions and Objectives . . . . . . 4
II THE CHANGING FLORIDA CITRUS INDUSTRY . . . . . . 6
II1. Florida Citrus Production . . . . . . . 6
112. Florida Grapefruit Production . . . . ... 16
113. Fresh Grapefruit Production . . . . . .. 22
114. Florida Citrus Packing. . . . . . . .. 28
115. Florida Citrus Processing . . . . . ... 34
116. Marketing Costs . . . . . . . . ... 34
117. Florida Citrus Markets. . . . . . . .. 39
III GENERAL PLANT LOCATION MODELS . . . . . . ... 40
IV SOLUTION PROCEDURES. . . . . . . . .... . 52
IV1. Introduction. . . . . . . . .... . 52
IV2. Linear Programming Procedures . . . . ... 53
IV3. Transportation Model Procedures . . . . ... 55
IV4. Combinatorial Procedures . . .. . . . 56
IV5. Transshipment Procedures . . . . . . . 59
IV6. Nonlinear Prograrning Procedures . . . ... 60
IV7. MixedInteger Procedures. . . . . . . ... 61
TABLE OF CONTENTSContinued
CHAPTER
V LOCATION MODEL FOR THE CITRUS INDUSTRY . . . . .. 65
VI. Model Formulation . . . . . . . . . 65
V2. Model Discussion . . . . .... . . . 70
V2.1. Cost Minimization. . . . . .. . . 70
V2.2. Size Limits and Economies of Scale. . . 71
V2.3. Existing Packinghouses Utilization, Closing
and Expansion . . . . . . .. 71
V3. Dynamic Analysis . . . . .. . . . . 72
VI ESTIMATION OF MODEL PARAMETERS AND COEFFICIENTS. . . . 78
VI1. Production and Demand Constraints . . . . .. 78
VI1.1. Number of Products . . . . . .. 78
VI1.2. Fruit Origins . . . . . . . 79
VI1.3. Fruit Supply . . . . . . . .. 80
VI1.4. Shipments to Markets . . . . . .. 85
VI1.5. Processing Plant Capacity.. . . . ... 90
VI2. Capacity Constraints and Cost Coefficients. . ... 97
VI2.1. Description of Assembly Operations. . 97
VI2.2. Assembly Cost . . . . . . .. 98
VT2.3. Eliminations Transport Cost. . . . ... 99
VI2.4. Distribution Costs . . . . . .. 100
VI2.5. Packinghouse Locations . . . . .. 100
VI2.6. Capacity Constraints . . . . . .. 104
VI2.7. Packing Cost . . . . . . . .. 105
VI2.8. Cost Coefficients . . . . . .. 117
VII EMPIRICAL RESULTS. . . . . . . . . . ... 123
VII1. Probationary Exercises . . . . . . .. 123
VII1.1. Model Simplification . . . . .. 123
VII1.2. First Auxiliary Model . . . . .. 126
VII1.3. Second Auxiliary Model .. . ..... 136
VII1.4. Third Auxiliary Model . . . . .. 137
VII1.5. Fourth Auxiliary Model . . . . .. 137
TABLE OF CONTENTSContinued
Page
CHAPTER
VII VII2.
VII3.
VII4.
VII5.
VIII SUMMARY
VIII1.
VIII2.
VIII3.
Static Model Optimal Solution, 198182 Season .
Dynamic Model Solution, 197576 through 188182
Duality Analysis and Secondary Computer Runs. .
VII4.1. Duality Analysis. . . . . .
VII4.2. "What If . .?" Analysis . . .
VII4.3. Sensitivity Analysis. . . . .
VII4.4. TradeOff Analysis. . . . . .
VII4.5. Priority Analysis . . . . .
Computational Experience. . .
AND CONCLUSIONS . . . . .
Summary . . . . . .
Conclusions . . . . .
Suggestions for Further Research
APPENDICES
APPENDIX A PRODUCTION AND SHIPMENTS DATA . . .
APPENDIX B COST DATA . . . . . . . .
APPENDIX C LOCATION MODEL REVIEW . . . . .
APPENDIX D MIP CONTROL PROGRAM TO OBTAIN SECONDBEST
LIST OF REFERENCES . . . . . . . . . .
SOLUTIONS
BIOGRAPHICAL SKETCH . . . . . . . . . . . .
vii
S139
143
S 150
S 150
S 151
. 155
S 161
S164
168
171
171
173
178
182
201
209
231
232
249
LIST OF TABLES
Table Page
1 Citrus Production for 197576 Crop Year . . . . . 7
2 Number of Carlots Received by 41 U.S. and Canadian Cities,
197576 Season. . . . . . . . . .. . 7
3 Nonbearing Citrus Acreage and Trees, and Trees Less than
Twenty Years Old, County Percentage in 1965 and 1975. . 9
4 Counties with Gains in Citrus Acreage Between 1973 and 1975 12
5 County Production Estimated Rates of Growth, 196364
through 197576 Seasons . . . . . . . ... 14
6 County Grapefruit Acreage in 1975, Percentage of Nonbearing
Grapefruit Acreage and Trees, and Percentage of Trees
Less than Twenty Years Old in 1965 and 1975 ...... 17
7 Counties with Gains in Grapefruit Acreage Between 1973
and 1975. . . . . . . . . ... ..... 18
8 County Grapefruit Production Estimated Rate of Growth . 20
9 Number of Florida Grapefruit Trees by Type and Year Set,
Percent of Trees Set by Year, January 1976 ...... 21
10 Average Box Yield Estimates and Percent Increase in Yields
for Selected Tree Ages of Florida Grapefruit. . . ... 23
11 Fresh Florida Citrus Production, 197071 through 197576
Seasons . . . . . . . . ... ...... 25
12 Total Shipments, Number of Packinghouses and Average Ship
ments per Packinghouse, by County, 195556 and 197576
Seasons . . . . . . . . ... . . . 29
13 County Shipments Estimated Rate of Growth . . . ... 31
14 County Grapefruit Shipments Estimated Rate of Growth . 33
viii
LIST OF TABLESContinued
Table Page
15 Agricultural Location Models . . . . . . ... 42
16 Main Features of Selected Location Models . . . ... 49
17 Box Yield Estimates by Age of Tree, Average of 196667
through 197576 Seasons. . . . . . . . .. 83
18 Percentage of Florida Citrus Used as Fresh Fruit, by
Variety, 197374 through 197576 Seasons . . . ... 84
19 Fruit Supply, Indian River Origins (1=13), 197576 through
198182 Seasons. . . . . . . . . ... ... 86
20 Population of U.S. Market Regions and Canada, 197576
through 198182 Seasons . . . . . . . ... 92
21 Indian River Fruit Demand, U.S., Canadian Markets and
Exports (K=8), 197576 through 198182 Seasons . . .. 93
22 Existing Citrus Cannery Locations in the East Coast Region
of Florida, 197576 . . . . . . . . ... 96
23 Monthly Truck Rates for Oranges and Grapefruit, 1977 . .. 101
24 Existing Packinghouse Locations (J=10) and Volume Handled,
197576 through 198182 Seasons . . . . . ... 103
25 Average Packing Cost Function Variable Definitions and
Hypothesized Relationships . . . . . . ... 106
26 Regression Results for Average Packing Cost Model . .. .108
27 Sample Average and Assumed Values of Selected Cost Function
Variables. . . . . . . . . ... . . .109
28 Per Unit (v.) and "Fixed" Cost (f.) of Existing and New
Packinghouses, and Operating Volume Range, per Packing
house . . . . . . . . .. . . . . . 116
29 Assembly Distances . . . . . . . . ... .. .119
30 Distribution Distances to Ports . . . . . ... 120
LIST OF TABLESContinued
Table Page
31 Distribution Distances to Processors . . . . . .. 121
32 Upper Bounds on xj. . . . ....... . ... . . 127
33 First Auxiliary Model Solution Costs, 1975076 Season . . 130
34 First Auxiliary Model Solution Flows, 197576 Season . . 131
35 First Auxiliary Model Transportation Costs, 197576
through 198182 Seasons . . . . . . .... 134
36 First Auxiliary Model, Routes Changed Between 197576 and
198182 Alternative Solutions . . . . . .... 135
37 Fourth Auxiliary Model Solution Summary, 198182 Season .140
38 Optimal Location, Number and Size of Citrus Packinghouses
and Optimal Fruit Flows, Indian River Region, 198182
Season . . . .. . . . . . . 141
39 Total Cost and Packinghouse Configuration of Minimum Cost
Solution to Dynamic Packinghouse Location Problem. . . 149
40 "Shadow" Prices of Fruit at the Origins, Indian River, 1981
82 Season . . . . . . . .... ..... 152
41 "Shadow Prices of Fruit at the Markets and Ports, Indian
River, 198182 Season. . . . . . . . . ... 153
42 Comparative Advantage of Packinghouse Locations and Sizes 154
43 Supply, Supply Range and "Shadow" Prices for the ShortRun
Optimum, 197576 Season . . . . . . . .. 157
44 Demand, Demand Range and "Shadow" Prices for the ShortRun
Optimum, 197576 Season . . . . . . . .. 159
45 Packinghouse Optimum ShortRun Operating Volume and Volume
Range Over Which "Shadow" Prices Apply, 197576 Season 160
46 Total Cost and Integer Variables Values of Six Solutions,
197677 Season . . . . . . . . .. . . 162
LIST OF TABLESContinued
Table Page
47 Fourth Auxiliary Model Solution Summary, 197576 through
197778 Seasons . . . . . . . . ... .. . 165
48 Second Auxiliary Model Capacity Constraint Dual Activities
197576 Season . . . . . . . . ... .. . 167
49 Number of Variables, Constraints and Solution Time for
Different Models . . . . . . . .... . 169
LIST OF FIGURES
Figure Page
1 State of Florida, Interior and Indian River Citrus
Marketing Districts . . . . . . . . .. 11
2 Grapefruit market shares, selected countries, 197576 26
3 World production of grapefruit . . . . . . 27
4 The ct best solutions to the static packinghouse
location problem by season . . . . . . . 74
5 The orange production origins for fresh fruit assembly
in Florida. . . . . . . . . ... . .81
6 Florida citrus U.S. market regions . . . . ... .91
7 Longrun packing cost functions for the Indian River . 111
8 Existing and optimum number, size and location of citrus
packinghouses, Indian River, 197576 and 198182
Seasons . . . . . . . . ... .. .. 142
9 The 107 best and current solutions to the static packing
house location problem by season. . . . . ... 144
10 Minimum cost solution to the dynamic packinghouse location
problem . . . . . . . . . . . .148
11 Total, variable and fixed costs of solutions 1 through 6
(Table 27), as a function of number of packinghouses
of any size, 197677 season . . . . . .. .163
KEY TO SYMBOLS AND ABBREVIATIONS
APC = Average packing cost
E = Packout percentage
F = Fixed cost
H = Number of products
I = Number of origins of products
J = Number of plant locations
K = Number of markets for final products
L = Number of processing plants + K
LTC = Longrun total cost
M2 = 1 if technology level 2,0 otherwise
M3 = 1 if technology level 3.0 otherwise
N = Number of plants
0 = Natural logarithm of the
proportion of oranges packed
to grapefruit packed
PL = Location, 1 for Indian River,
0 otherwise
PO = Natural logarithum of percent
of fruit delivered that is
acceptable for fresh shipment
xili
PK = Percent of fruit that is not
packed in standard 4/5 bushel
cartons
q = Quantity shipped and/or processed
Q = capacity or total quantity
Q = Large number
S = Mean weekly shipments divided
by the standard deviation of
weekly shipments
SL = Natural logarithm of number
of weeks in which packinghouse
packs fruit
STC = Shortrun total cost
T = Time
u = Average variable unit cost
v = Average variable unit cost
x = Integer number of plants open
at a site
y = Fraction of Q
y. = Year, 1 for 1974, 0 otherwise
4
y. = Year, 1 for 1975, 0 otherwise
Z = Objective function
8 = Constant, growth rate
xiv
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
A DYNAMIC, MIXED INTEGER LOCATION MODEL
APPLIED TO FLORIDA CITRUS PACKINGHOUSES
By
VIRGILIO AZUIL PASCOA MACHADO
December, 1978
Chairman: Dr. Leo Polopolus
Major Department: Food and Resource Economics
Southward movement in production location, the increasing importance
of grapefruit as a fresh variety, increases in size of packinghouses and
cost inflation have an effect on the efficiency of the Florida fresh
citrus industry, and, therefore, on producer returns and consumer prices.
An improvement in the industry's productive efficiency is achieved by
optimizing the number, location, size, supply and market areas of the
Indian River citrus packinghouses.
Two models were developed and used. One is a locationallocation
mixedinteger model with fixed costs, demand, supply and capacity con
straints. The integer variables are the number of packinghouses of the
same size at each location. The other model used dynamic programming to
find the minimum cost path through a matrix of static solutions, taking
into consideration the cost of moving from one system configuration to
another. The static solutions are obtained with the mixedinteger model
for each season, and ranked by cost.
Included in the model are oranges and grapefruit. Townships in
the Indian River are aggregated into 13 origins of fruit that supply fruit
to the packinghouses. Supply is determined by the number and age of
trees and yield per tree, and estimated for the seasons of 197576
through 198182. Export quantities being shipped out of the major
Florida ports are estimated together with quantities being shipped to
domestic and Canadian markets. Assembly, packing and distribution costs
are estimated in order to compute the model coefficients. Locations with
existing packinghouses are Cocoa, Ft. Pierce, Titusville and Vero Beach.
Economies of scale are included by considering packinghouses of different
sizes on those locations. Potential packinghouse locations are Jupiter,
Melbourne and Stuart. Capacity constraints are then defined as the
capacity percentage that will insure packing all fruit during the peak
weeks and a minimum degree of operating efficiency. A longrun packing
cost function is statistically estimated by using a variance component
model on cross sectional, timeseries accounting data.
Solutions to the model are obtained using IBM's MPSX (1972) and MIP
(1973) computer programs. The results indicate that there should be fewer
and larger packinghouses in the region. The solution Ls not sensitive
to use of nonoptimal assembly and distribution routes, and to changes
in transportation cost. To adjust to the proposed future efficient
xvi
marketing organization, the industry should phase out small plants
operating at high unit costs. Large plants are desirable if they can
be operated at low unit costs.
xvii
CHAPTER I
PROBLEM DEFINITION
I1. Problem Statement
Changes in the location of production, product mix, technology and
costs have an adverse effect on the efficiency of the Florida citrus
industry, and, consequently, on producer profits and consumer prices.
Structural adjustments to these changes are needed. An improvement in
productive efficiency can be achieved by optimizing the number, location,
size, supply and market areas of citrus packinghouses in Florida. This
will contribute to reduced distribution costs,accomnodating future out
put expansions without deterioration of market prices and grower returns.
The adjustment of the present industry configuration and product flow
to a lower cost level can be facilitated and encouraged by providing
guidelines consistent with the minimization of industry costs over a
planning horizon.
The Florida citrus industry is quite large at the state, national
and international level, in terms of production and market shares
(Florida Crop and Livestock Reporting Service, 1976a). It employs con
siderable manpower and capital (Florida Department of Agriculture and
Consumer Services, 1977). Successive commercial citrus tree inventories
show that removal of citrus acreage and new plantings have been shifting
production within the state (Florida Crop and Livestock Reporting Service,
1976a). Citrus production has been increasing and more than doubled 
in the past twentyfive years, but production growth is substantially
higher in south Florida counties (Florida Crop and Livestock Reporting
Service, 1967 through 1976a). All counties with high growth rates have
percentages of trees less than twenty years old. Counties with signif
icant citrus commercial acreage and high percentages of young trees are
all south of Lakeland. This was not the case ten years earlier, in
1965, when more counties had high percentages of young trees. In the
same ten years the state percentage of nonbearing trees had decreased.
Counties with high percentages of nonbearing trees are also now limited
to some south of Lakeland. Bearing acreage in Florida has been decreasing
since 197071, although bearing acreage of seedless grapefruit continues
to increase.
Despite its small percentage of total crop utilization, the fresh
citrus market is nevertheless an important source of potential profit.
For grapefruit, tangelos and tangerines most of a grower's profit comes
from fruit marketed as fresh (Growers Administrative Committee, 1976).
Grapefruit is the most important of these. Expansion of foreign markets
for fresh grapefruit has caused its fresh shipments to increase faster
than production which is also expected to increase rapidly in the next
five years. World production of grapefruit is dominated by the U.S. but
other countries have been progressively increasing their shares in the
international market (Myers, 1977).
Southern counties in Florida have experienced high grapefruit produc
tion growth, increasing their share of the state output (Florida Crop and
Livestock Reporting Service, 1967 through 1976a). Counties with high
growth rates are expected to continue growing faster due to maturity
of their large percentage of young grapefruit trees (Fairchild, 1977).
Fresh shipments of grapefruit have been increasing faster in the East
Coast and North Interior regions, presenting a marketing challenge
which must be met (Florida Department of Agriculture and Consumer
Services, 1964 through 1976).
Large fresh fruit shipments were made in 1976 from packinghouses in
the same counties where the average shipment per packinghouse is high
and there is a large number of packers (Florida Department of Agricul
ture and Consumer Services, 1976). Not all these counties have been
experiencing a comparable rate of growth in their shipments.
Over the past twenty years the average size of packinghouses has
increased. Volume of shipments over the season varies widely from as
little as 30 boxes to more than 3 million 4/5 bushel boxes. Moreover,
volume of shipments per packinghouse per county, varies from 8 to 800
thousand boxes packed. Capacity utilization of packinghouses has been
reportedly low (Kilmer and Tilley, 1978).
Costs have been increasing in all phases of the industry (Hooks and
Spurlock, 1977 and Hooks and Kilmer, 1977 a and b) with reductions in
returns to growers expected to continue under present marketing conditions
(Myers, 1977). Increasing capacity utilization is an apparent shortrun
key to cost reductions. Furthermore, packinghouses that are capable of
coordinating grove production and harvesting such that they receive a
steady volume of high quality fruit are likely to have lower costs (Kilmer
and Tilley, 1978).
12. Propositions and Objectives
For the above problem the following propositions are formulated:
a) An improved Florida citrus fruit assembly, packing and
distribution system can be proposed which, if adopted, would minimize
costs.
b) The optimal locations and sizes of new packinghouses to accommo
date future output expansion, costs and other expected changes can be
determined.
c) If shipping routes and plants excluded from the efficient
solution are not used, then substantial savings will be made by the
industry.
d) Knowledge of the most minimizing marketing network will
help future industry decision making and provide a firm management tool.
a) A leastcost plan for the adjustment of the industry to an
optimal organization can be provided to management.
The primary objective of this study is to determine a marketing
network that will improve the productive efficiency of the Florida
fresh citrus industry. Specifically the objectives are to:
a) Determine the cost minimizing number, location, size, supply
and market areas of citrus packinghouses in Florida, and assembly and
distribution patterns.
b) Determine the sensitivity of a minimum cost solution to changes
in supply and other parameters and costs.
5
c) Present the citrus industry with a criterion and technique
that can be used to provide guidelines for the industry or a plan of
action for citrus firms to adjust towards a more efficient industry
marketing organization.
CHAPTER II
THE CHANGING FLORIDA CITRUS INDUSTRY
II1. Florida Citrus Production
World grapefruit production is dominated by the United States,
which is also the largest producer of oranges and tangerines. Florida
is a major factor in domestic citrus production and markets (Tables 1
and 2). In the 197576 crop year, 852,369 acres containing 70.547
million trees bore fruit in Florida (Florida Crop and Livestock Reporting
Service, 1976b).
The citrus industry accounted for 27 percent of Florida farm cash
receipts in 1975. Approximately 54,000 persons are employed annually
by the industry: 33,000 in field operations, 9,000 in processing plants
and 12,000 in packing plants (Florida Department of Agriculture and
Consumer Services, 1977). Fresh citrus was shipped by 170 packinghouses
and processed in 52 registered citrus canning or concentrate plants
during the 197576 season (Florida Department of Agriculture and Con
sumer Services, 1976).
In 1975, commercial citrus acreage had decreased 0.7 percent since
the first commercial citrus inventory was conducted in 1965. The new
survey shows a removal of 40,518 acres between 1973 and 1975 surveys,
more than offsetting the 28,789 acres of new trees set in 1974 and 1975,
a net decline of 1.4 percent from the 864,098 acres in December, 1973
(Florida Crop and Livestock Reporting Service, 1976a).
6
Table 1 .Citrus Production for 197576 Crop Year
Florida United States World
1,000 Percent of 1,000 Percent of 1,000 Percent of
Metric World Metric World Metric World
Tons Production Tons Production Tons Production
Grapefruit 1,803 50.0 2,574 71.0 3,624 100.0
Oranges and
Tangerines 7,402 23.7 9,908 31.7 31,267 100.0
Source: Florida Crop and Livestock Reporting Service, 1976a.
Table 2.Number of Carlots Received by 41 U.S. and Canadian Cities,
197576 Season
Florida
Grapefruit:
Oranges:
Tangerines:
U.S.
Canada
U.S.
Canada
U.S.
Canada
Cars
16,401
2,451
11,712
1,307
3,894
499
Percent
68.7
79.0
31.2
19.2
73.2
80.6
Cars
23,866
3,102
37,533
6,792
5,319
619
Total
Percent
100.0
100.0
100.0
100.0
100.0
100.0
Source: Florida Crop and Livestock
Reporting Service, 1976a.
Half of the acreage that was planted in the calendar years 1974
and 1975 was reset planting. Land that had never before been used for
citrus accounted for 25 percent of the new acreage. Of the acreage that
was removed from commercial production, approximately 12.5 percent was
taken out for commercial development.
The percentage of nonbearing acres and trees, and the percentage
of trees less than twenty years old, all decreased from 1965 to 1975
(Table 3).
In 1975, Polk and Lake counties had the largest commercial citrus
acreage (Figure 1). Acreage gains between the surveys of 1973 and
1975 were recorded in a few counties, i.e., Hendry, Indian River and
St. Lucie (Table 4).
All types of citrus showed net declines in commercial acreage be
tween 1973 and 1975, except seedless grapefruit. White seedless grape
fruit bearing acreage increased by 2,047 acres and pink seedless by
1,110 acres. Nonbearing acreage, however, increased by 1,641 and 2,718
acres bring the gains in total acreage to 3,688 and 3,828 acres for
white and pink seedless grapefruit, respectively. Indian River and St.
Luice are the counties with the largest acreage and number of seedless
grapefruit trees (Florida Crop and Livestock Reporting Service, 1976b).
St. Lucie and part of Indian River counties are in the Indian River
District (Figure 1). This region is one of two Marketing Agreement
Regulation Areas: Interior and Indian River (Code of Fegeral Regulations,
1975). The Marketing Agreements regulate size, grades, containers,
shipping holidays, and prorated shipments. Under Marketing Agreement
Table 3.Nonbearing Citrus Acreage and Trees, and Trees Less than
Twenty Years Old, County Percentage in 1965 and 1975
Nonbearing Nonbearing Trees Less than
Acres Trees Twenty Years Old
1965 1975 1965 1975 1965 1975
     Percent      
Indian River:b
Brevard 28.5 5.87 34.9 6.05 67.0 60.9
Indian River 45.5 14.9 48.4 16.7 74.9 82.2
Martin 70.6 10.8 73.1 10.2 97.6 97.4
Palm Beach 75.4 1.61 80.9 1.64 89.6 97.8
St. Lucie 44.4 8.79 49.7 9.40 80.7 78.4
Volusia 14.9 2.07 15.0 2.48 45.8 35.2
Interior:
Alachua 21.1 0 25.0 0 66.7 45.6
Broward 16.7 0.482 21.9 0.105 60.6 33.7
Charlotte 68.5 7.32 71.1 7.62 93.6 94.9
Citrus 27.8 0.508 28.6 0.518 75.0 50.4
Collier 75.8 5.10 79.7 4.77 99.2 99.8
Dade 21.2 16.5 26.9 16.6 89.5 89.3
De Soto 40.4 25.2 43.8 31.1 81.2 89.9
Flagler 17.2 13.0 26.3 17.4 94.7 100.0
Glades 74.9 2.91 72.1 2.61 100.0 100.0
Hardee 31.5 6.55 34.4 7.28 85.6 76.3
Hendry 90.8 20.5 93.0 21.0 98.3 99.2
Hernando 31.6 1.89 32.0 1.92 83.2 68.0
Highlands 23.0 5.25 28.1 5.67 69.6 59.0
Hillsborough 31.4 4.73 34.2 5.15 79.7 63.8
Lake 23.8 2.52 26.7 2.75 75.5 50.7
Lee 20.8 8.81 21.1 10.1 63.3 92.3
Manatee 34.6 4.33 35.0 4.72 83.4 75.6
Marion 21.8 1.02 23.9 1.20 59.7 45.5
Okeechobee 72.3 11.8 70.1 12.3 95.7 97.0
Orange 14.3 2.61 15.6 2.62 51.1 35.3
Osceola 28.4 2.97 30.4 3.02 81.7 61.9
Pasco 28.2 2.59 29.2 2.52 76.8 60.3
Pinellas 11.8 0.849 12.4 0.814 29.2 24.6
Polk 14.5 3.20 16.8 3.51 48.1 42.6
Putnam 25.8 1.12 24.9 1.15 49.9 35.0
St. Johns 13.6 0 16.7 0 50.0 25.0
Sarasota 27.5 16.0 32.1 15.3 56.6 65.5
Table 3.Continued
Nonbearing Nonbearing Trees Less than
Acres Trees Twenty Years Old
1965 1975 1965 1975 1965 1975
     Percent      
Seminole 18.5 1.46 18.4 1.39 62.9 46.3
Sumter 21.9 6.99 22.3 6.14 81.2 75.0
State 29.2 6.59 33.1 7.80 71.0 66.0
aSet in 1945 and 1955 or later, respectively.
with the exception of St. Lucie these countries have some townships
in the Interior Region (Figure 1).
Source: Florida Crop and Livestock Reporting Service, 1966 and 1976b.
Indian River
Marketing District
164,120 Acres
All Citrus
Interior
Marketing District
688,249
All Citrus
Figure 1.
State of Florida,
Interior and Indian
Citrus Marketing
Districts
/ ;
r 'salt ,
Source: Florida Crop and Livestock Reporting Service, 1976b.
Table 4 .Counties with Gains in Citrus Acreage Between 1973 and 1975
County 1973 1975 Gain
  Acres  
Indian River:a
Indian River 52,261 56,206 3,945
St. Lucie 73,036 73,812 876
Interior:
Dade 4,340 4,536 196
Flagler 182 192 10
Hendry 24,225 25,944 1,719
Marion 11,223 11,327 104
Okeechobee 4,087 4,162 75
St. Johns 125 127 2
Sarasota 1,449 1,661 212
Sumter 1,677 1,760 83
aIndian River county has some townships in the Interior Region (Figure 1).
Source: Florida Crop and Livestock Reporting Service, 1974 and 1976b.
regulations, grower committees make recommendations for marketing policy
at the beginning of the season and for changes in regulations during
the season.
Florida citrus production has more than doubled in the past twenty
five years. Counties with the largest production are in the Interior
and East Coast of Florida (Appendix Table Ai). Eight counties produced
a total of almost 175 million boxes or 70.7 percent of the state total.
The highest rates of production growth since the freeze of 1962 occurred
in the southern Florida counties of Collier, Glades, Hendry, Martin and
Palm Beach (Table 5). Data for the 197677 season are available from
the Florida Crop and Livestock Reporting Service but are not used in this
research because the freeze of January of 1977 rendered these data in
appropriate.
Collier, Glades, Hendry, Martin and Palm Beach counties had an
estimated production in the 196566 season of only 1,553 thousand boxes
of fruit or 1.1 percent of the state total. Those same counties ten
years later or in the 197576 season produced an estimated 21,009
thousand boxes of fruit or 8.5 percent of the state's estimated citrus
production (Appendix Table Al).
Production growth is determined fitting an exponential model such as
Bt
Q = Qoe ,
to the data using ordinary least squares, and estimating 8. After
differentation with respect to time and division by Q one obtains,
dQ/dt = e
Q Q0eBt
the percentage growth rate.
Table 5.County Production Estimated Rates of Growth, 196364 through
197576 seasons
^a b
a r
Percent
Indian River: c
Brevard 3.82 0.84
Indian River 8.52 0.95
Martin 25.4 0.98
Palm Beach 24.3 0.95
St. Lucie 8.13 0.95
Volusia 3.88 0.66
Interior:
Broward 0.0821 0.03
Charlotte 18.0 0.97
Citrus 12.4 0.74
Collier 39.9 0.96
De Soto 10.3 0.91
Glades 25.1 0.96
Hardee 9.45 0.86
Hendry 33.9 0.96
Hernando 11.9 0.76
Highlands 4.98 0.88
Hillsborough 9.91 0.67
Lake 6.75 0.84
Lee 12.5 0.92
Manatee 11.7 0.78
Marion 5.44 0.72
Okeechobee 19.0 0.96
Orange 3.96 0.66
Osceola 7.98 0.86
Pasco 12.4 0.78
Pinellas 2.63 0.24
Polk 3.60 0.68
Putnam 1.64 0.29
Sarasota 7.87 0.67
Seminole 2.85 0.49
Sumter 6.89 0.73
State 6.83 0.90
aEstimate of 8, defined in footnote, page 13.
correlation coefficient = T Et InQ Et InQg
T Et2 (Zt)2 / T E(InQ 2 (ZnQt
CWith the exception of St. Lucie these counties have some townships
in the Interior Region (Figure 1).
Other counties have seen their share of the state citrus output
increased. Counties with an estimated citrus production growing at a
higher rate than the state growth rate had their share of the estimated
production increased from 29.4 percent to 44.6 percent between the
196566 and 197576 seasons (Appendix Table Ai).
Brevard and Volusia counties in the northern half of the Indian
River have much lower growth rates than the counties located in the
southern half of the Indian River.
There are indications that the shift of citrus production cowards
southern and coastal counties is going to continue. All the counties
with the highest rate of growth had more than 90 percent of the trees
twenty years old or younger, as of January, 1976 (Table 3). Counties
with a growth rate greater than average had more than 70 percent of the
immature trees, with somewhat lower figures for Citrus, Pasco, and
Sarasota counties, all of them, however, with more than 50 percent of
immature trees (Table 3). Dade county supplies only limes and lemons,
which are not considered in this study although some of this fruit is
packed in the summer offseason in packinghouses that pack oranges and
grapefruit.
A higher percentage of nonbearing trees also predominates in the
southern half of Florida. Counties with significant citrus commercial
acreage and more than the statewide percentage of nonbearing trees as
of January, 1976,were DeSoto, Hendry, Indian River, Lee, Martin,
Okeechobee, St. Lucie and Sarasota, all south of Lakeland (Table 3).
Those counties had 3,751.7 thousand nonbearing trees or 68 percent of
the state total. Counties in 1975 that had a higher percentage of
young trees than in 1965 are also all south of Lakeland (Table 3).
One finds that in 1965 high percentages of young were not
as prevalent in southern Florida, but extended throughout the West
Coast and Interior regions. Note that all citrus producing counties
in 1965 had a percentage of nonbearing acres and trees above the 1975
state average (Table 3).
112. Florida Grapefruit Production
Grapefruit represents the most important fresh fruit category
and the one whose production growth has been very high in the Indian
River region.
Grapefruit acreage in Florida amounted to 137,909 acres or 16
percent of total citrus acreage in 197576. There was a slight
decrease in nonbearing acreage from 16.9 to 14.5 percent and in non
bearing trees from 19.5 to 17.3 percent between 1965 and 1975. The
percentage of trees less than twenty years old, however, increased
from 46.8 to 60.5 percent in the same period (Table 6). This is in
sharp contrast with the situation for all citrus acreage and trees
(Table 3).
In 1965, Indian River, Lake, Polk and St. Lucie counties had
the largest grapefruit acreage. Acreage gains between the surveys
of 1973 and 1975 were recorded for most counties, particular Indian
River and St. Lucie (Table 7).
Grapefruit production has increased only 50 percent in Florida
in the past twentyfive years. Production for the five year period
Table 6.County Grapefruit Acreage in 1975, Percentage of Nonbearing
Grapefruit Acreage and Trees, and Percentage of Trees Less
than Twenty Years Old in 1965 and 1975
Grapefruit Nonbearing Nonbearing Trees Less than
Acreage Acres Trees Twenty Years Old
1975 1965 1975 1965 1975 1965 1975
Acres    Percent      
Indian River:a
Brevard 3,442 9.13 12.3 8.10 12.6 50.0 34.8
Indian River 30,477 29.7 23.4 30.4 27.4 64.8 79.0
Martin 5,682 57.8 10.1 62.9 9.20 94.3 98.2
Palm Beach 3,405 65.9 0.294 75.9 0.318 83.9 98.8
St. Lucie 20,050 35.8 16.5 39.7 19.0 76.0 75.4
Volusia 681 13.8 9.69 13.2 11.2 26.4 32.0
Interior:
Broward 389 3.15 0 2.70 0 81.1 21.8
Charlotte 250 29.5 8.40 38.5 9.52 84.6 74.4
Citrus 38 5.88 5.26 0 8.70 100.0 26.1
Collier 529 76.0 36.7 100.0 42.2 100.0 99.6
De Soto 1,400 9.00 26.2 9.52 26.4 41.3 64.8
Glades 75 0 0 0 0 0 100.0
Hardee 960 6.35 26.4 4.88 30.9 48.8 55.6
Hendry 4,404 95.9 38.8 96.9 40.3 98.5 99.4
Hernando 152 0 4.60 0 5.66 16.7 23.6
Highlands 4,771 4.49 11.5 4.51 13.7 16.2 37.4
Hillsborough 2,593 6.40 15.7 7.11 18.8 38.4 38.9
Lake 12,564 5.39 5.83 5.69 7.70 42.8 25.0
Lee 430 0.373 1.16 0 0.771 31.6 77.9
Manatee 2,149 10.8 16.4 10.8 17.6 41.6 50.3
Marion 407 3.85 2.21 3.13 3.34 18.8 12.7
Okeechobee 978 62.0 29.0 61.9 32.3 95.2 98.3
Orange 2,445 4.42 5.32 4.15 5.16 33.7 22.2
Osceola 1,078 6.67 22.4 7.14 21.4 53.6 45.6
Pasco 1,515 7.86 7.59 8.08 8.32 39.4 32.8
Pinellas 1,176 4.04 1.70 4.42 1.51 16.6 11.7
Polk 25,282 2,28 5.58 2.48 6.41 13.9 17.5
Putnam 39 2.15 0 0 0 0 6.45
Sarasota 323 16.9 17.3 16.0 17.9 40.0 52.0
Seminole 216 0.850 4.17 0 4.64 24.0 9.27
Sumter 9 0 0 0 0 0 0
State 137,909 16.9 14.5 19.5 17.3 46.8 60.5
With the exception of St. Lucie these counties have some townships in
the Interior Region (Figure 1).
Source: Florida Crop and Livestock Reporting Service, 1966 and 1976b.
Table 7 .Counties with Gains in Grapefruit Acreage Between 1973 and
1975
County 1973 1975 Gain
   Acres    
Indian River:a
Brevard 3,207 3,442 235
Indian River 26,536 30,477 3,941
Martin 5,441 5,682 241
St. Lucie 27,652 30,050 2,398
Interior:
Citrus 32 38 6
Collier 528 529 1
Hardee 858 960 102
Hendry 3,607 4,404 797
Hillsborough 2,587 2,593 6
Lake 12,486 12,564 78
Manatee 1,910 2,149 239
Marion 398 407 9
Okeechobee 954 978 24
Osceola 895 1,078 183
Pasco 1,443 1,515 72
Sarasota 227 323 46
Sumter 7 9 2
aWith the exception of St. Lucie these counties have some townships
in the Interior Region (Figure 1).
Source: Florida Crop and Livestock Reporting Service, 1974 and 1976b.
19
197172 through 197576 oscillated around an average of 46.8 million
boxes per season. Counties with the largest production are in the
Interior and East Coast of Florida. Five counties in these two
regions produced a total of almost 37 million boxes or 74.7 percent
of the state total (Appendix Table A2). Grapefruit production is
expected to increase rapidly between the 197677 and 198182 seasons
or at a rate of 2 million boxes per year. Collier, Glades, Hendry,
Martin and Palm Beach counties in South Florida also have high rates
of grapefruit production growth (Table 8). These counties had an
estimated grapefruit production in the 196566 season of only 224
thousand boxes or 0.64 percent of the state total. Ten years later
or in the 197576 season these South Florida counties produced an
estimated 4,659 thousand boxes or 9.5 percent of the state estimated
grapefruit production (Appendix Table A2).
A high percentage of nonbearing grapefruit trees predominates
in the southern half of Florida. Counties with more than the state
wide percentage of nonbearing grapefruit trees as of January, 1976
were in three separate areas in Florida (Table 6), Indian River,
Okeechobee, Osceola and St. Lucie counties in the East Coast,
Collier and Hendry counties in the South, and De Soto, Hardee, Hills
borough, Manatee and Sarasota counties in the West Coast. These
countries had 1,477.2 thousand nonbearing grapefruit trees or 82 per
cent of the state total.
The age distribution of trees has an important impact on long
range production expectations. As can be seen in Table 9, 37 percent
Table 8.County Grapefruit Production Estimated Rate of Growth
8a rb
83 r
Percent
Indian River:c
Brevard 0.232 0.08
Indian River 7.26 0.91
Martin 32.7 0.95
Palm Beach 24.9 0.93
St. Lucie 7.78 0.92
Volusia 3.43 0.51
Interior:
Broward 1.72 0.42
Charlotte 7.02 0.83
Citrus 45.9 0.92
Collier 45.9 0.92
De Soto 3.48 0.58
Glades 159.0 0.77
Hardee 1.89 0.36
Hendry 38.3 0.94
Hernando 3.78 0.36
Highlands 1.40 0.34
Hillsborough 1.68 0.18
Lake 3.61 0.57
Lee 3.02 0.64
Manatee 5.73 0.48
Marion 2.59 0.40
Okeechobee 16.7 0.90
Orange 2.10 0.34
Osceola 4.33 0.58
Pasco 4.10 0.44
Pinellas 1.54 0.14
Polk 0.580 0.14
Putnam 4.88 0.52
Sarasota 4.43 0.47
Seminole 0.726 0.10
State 4.18 0.86
a
Estimate of S, defined in footnote, p. 13.
bT Zt InQt Zt *.1nQt
bCorrelation coefficient = ____Et _nQt t ZlnQt
/ T Et2 (Zt) / T E(nQ,) (ZlnQt)2
With the exception of St. Lucie these counties have some townships
in the Interior Region (Figure 1).
Table 9.Number of Florida Grapefruit Trees by Type and Year Set,
Percent of Trees Set by Year, January 1976
Year Set Seedy White Pink Total Percent
Seedless Seedless of Total
   1,000 trees
1951 or earlier
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
Total
1233.1
2.5
0.7
1.7
0.7
1.0
1.2
3.3
2.0
9.5
6.1
10.6
8.0
7.9
43.4
36.5
66.6
20.3
15.6
14.7
10.2
31.1
10.1
23.4
9.7
1569.9
1553.7
13.6
11.6
17.7
14.2
7.9
6.5
69.9
49.8
96.2
99.2
63.3
128.5
232.9
650.0
609.9
528.9
220.7
131.4
98.2
174.7
151.3
216.7
261.9
198.0
5607.6
1086.5
68.2
73.9
45.6
37.1
6.7
5.8
24.9
7.4
16.5
4.8
12.6
12.6
32.0
155.5
156.1
170.0
123.5
109.1
68.5
107.4
169.9
298.8
286.6
141.3
3221.3
3873.3
84.3
86.2
65.0
52.0
15.6
13.5
98.1
59.2
122.2
110.1
86.5
149.1
272.8
849.8
802.5
765.5
364.5
256.1
181.4
292.3
352.3
525.6
571.9
349.0
10,398.8
Percent
37.2
0.8
0.8
0.6
0.5
0.2
0.1
0.9
0.6
1.2
1.1
0.8
1.4
2.6
8.2
7.7
7.4
3.5
2.5
1.7
2.8
3.4
5.1
5.5
3.4
100.0
Source: Fairchild, 1977.
22
of the existing grapefruit trees have attained full maturity. Only 9
percent of the grapefruit trees are in the 1324 year ranges and thus
will reach full maturity during the next 12 years. Furthermore, over
onehalf (53.8 percent) of all existing grapefruit trees are less than
13 years of age, of which 20 percent were less than five years old in
1976 (Fairchild, 1977).
As emphasized by Fairchild (1977), the major implication of age
distribution for future grapefruit production is that the full growth
effect of the heavy plantings of the midsixties will be felt over
the next 1218 years.
The effect of tree age distribution on citrus production is ex
plained by Fairchild (1977) using the data provided in Table 10. The
average box yields per tree for five, ten, fifteen, twenty, and twenty
five year old trees illustrates the influence of the age distribution
on total production. The bottom part of Table 10 compares the yields
for trees of various ages. For example, an average 25 year old Pink
Seedless tree yields 282 percent more fruit than an average 5 year old
Pink Seedless tree. The effect of age distribution is even more dramatic
when plantings vary considerably from year to year, as is the case with
Florida citrus.
113. Fresh Grapefruit Competition
Florida grapefruit is expected to face increasing competition from
both domestic and foreign sources. This competition will increase the
need to work toward a more efficient fresh fruit packing industry.
23
Table 10.Average Box Yield Estimates and Percent Increase in Yields
for Selected Tree Ages of Florida Grapefruit
Grapefruit
Tree Age
White Pink Seedy
Seedless Seedless
   Boxes/tree    
5 yr. 2.00 1.81 1.52
10 yr. 3.71 3.61 2.85
15 yr. 5.14 5.29 3.97
20 yr. 6.53 6.72 5.07
25 yr. 8.21 6.92 6.78

 Percent increase in yield  
25 yr. versus 5 yr. 310 282 346
25 yr. versus 10 yr. 121 92 138
25 yr. versus 15 yr. 60 31 71
25 yr. versus 20 yr. 26 3 34
20 yr. versus 15 yr. 27 27 28
15 yr. versus 10 yr. 39 47 39
10 yr. versus 5 yr. 86 99 88
aCalculated as Older Age Yield Younger Age Yield
Younger Age Yield
Source: Fairchild, 1977.
24
As noted earlier, Florida accounts for a large share of domestic,
Canadian and world markets, and seedless grapefruit was the only type
of citrus that showed gains in acreage between 1973 and 1975, accounting
for 50 percent of Florida citrus production for fresh use (Table 11).
Fresh shipments of grapefruit have been increasing at a rate faster than
the general growth rate in production. According to Myers (1977), this
trend results mainly from the expansion of foreign markets for fresh
fruit. Fresh sales of Florida grapefruit increased 23 percent during
197172 through 197576 while the total Florida crop increased by only
4 percent. While increased sales occurred in the domestic and Canadian
markets, the major factor affecting the fresh growth rate was the
development of European and Japanese markets. Japan has played the
primary role since 1972, accounting for 73 percent of all offshore
Florida grapefruit exports in 197576. Florida grapefruit market shares
were 32 percent in France and 12 percent in the Netherlands in 197576
(Figure 2).
In the domestic market, Florida's share of total U.S. grapefruit
production has declined due to a stronger growth rate in Texas and
CaliforniaArizona. By 198485, it is projected that Florida will pro
duce 70 percent of the grapefruit produced in the U.S., down from 76
percent in 197172 and 73 percent in 197576 (Myers, 1977).
Israel, Argentina and South Africa are the only three other coun
tries currently producing significant quantities of grapefruit (Figure
3). Israel, the largest producing country outside the U.S., increased
production 36 percent between 197172 and 197576. Argentina increased
0
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production by 78 percent over the same 5 year period. Total world sup
plies increased 12 percent from 197172 through 197576. Myers (1977)
also reports that grapefruit production in countries other than the
U.S. is projected by the FAO to expand rapidly by 198182. Israel
and Cuba are expected to be then the major world suppliers of grape
fruit other than the U.S. Other important countries will be Argentina,
South Africa and Central America. Overall, world supplies of grape
fruit are expected to increase approximately 34 percent between 197576
and 198182.
Given the increasing levels of domestic and foreign competition,
the Florida citrus industry will perhaps face increased pressure to improve
packing industry efficiency.
114. Florida Citrus Packing
Size and concentration of packinghouses throughout the state vary
widely. Shipments from packinghouses ranged from 30 boxes to more than
three million 4/5 bushel boxes in the 197576 season (Appendix Table A3).
Packinghouses in six counties in the Interior and East Coast of the
stateHighlands, Indian River, Lake, Orange, Polk and St. Lucie
shipped large amounts of fresh fruit, exceeding one million 4/5 bushel
boxes each. Average quantity of fruit shipped per packinghouse per
county ranged between 6 and 800 thousand boxes (Table 12). All the
above counties, plus De Soto, Duval and Hillsborough had an acreage of
fruit shipped per packinghouse above the state average of 446 thousand
boxes per packinghouse in 197576 (Table 12).
Table 12.Total Shipments, Number of Packinghouses
per Packinghouse, by County, 195556 and
and Average Shipments
197576 Seasons
Total Fresh Number of Average Shipments
Citrus Shipped Packinghouses per Packinghouses
5556 7576 5556 7576 5556 7576
 /5 hb boxes   A/I h"i bxvc
Indian River:a
Brevard
Indian River
Martin
Palm Beach
St. Lucie
Volusia
Interior:
Alachua
Broward
Citrus
Dade
De Soto
Duval
Hardee
Hernando
Highlands
Hillsborough
Lake
Lee
Manatee
Marion
Orange
Osceola
Pasco
Pinellas
Polk
Putnam
Sarasota
Seminole
Sumter
Other
2,303,048
5,028,016
m
60,748
4,610,648
990,258
266
48,886
48,918
5,966
1,221,310
1,002,298
30,776
562,746
3,459,962
3,603,150
7,615,730
1,210
63,618
1,939,910
15,848,806
434,648
730,214
3,484,288
26,067,280
449,006
80,784
3,403,310
82,698
2,330,044
16,102,331
16,147
90,484
9,882,085
669,307
50,117
28
447,499
585,610
6,984
432,770
2,102,618
1,912,549
9,196,442
28,432
505,451
8,124,932
177,153
470,195
21,761,642
77,088
745,736
42,492
83,239,498 75,758,196
241
170
345,392 445,636
aWith the exception of St. Lucie these counties have
the Interior Region (Figure 1).
some townships in
Source: Florida Department of Agriculture and Consumer Service, 1956 and
1976.
255,894
419,001
60,748
461,065
76,174
266
12,472
24,459
5,966
610,655
501,149
10,259
187,582
494,280
360,315
217,592
1,210
63,618
149,224
480,267
144,883
243,405
316,753
637,251
44,901
80,784
378,146
338,341
596,383
16,147
18,097
449,136
95,615
m
7,168
28
447,449
585,610
6,984
432,770
700,978
478,137
540,967

7,108
63,181
507,808
88,576
156,732
805,987
38,544
372,868
21,246
State
.I  
.rl u urs
30
The variability in fruit shipped per packinghouse also shows itself
in the capacity utilization as measured by Kilmer and Tilley (1978).
From a sample of 23 Florida citrus packinghouses for three seasons
(197374 to 197576), a maximum annual capacity was defined as the
maximum monthly output through a packinghouse during three seasons multi
plied by 11. Capacity utilization is then equal to the volume packed
in a season as a percentage of maximum annual capacity. The minimum
value found for the firms in the sample was 19 percent, and the maximum
90 percent. The average was 50 with a standard deviation of 14 percent.
Using the same growth model as before, and data shown in Appendix
Table A4, the shipment growth rates vary from negative values to
almost 20 percent (Table 13). Some counties, e.g., Charlotte, Collier,
Dade, Osceola and Sarasota, are not now shipping citrus although
they did in the past (Appendix Table A4). High growth rates can
be found mostly in the southern East Coast in Broward, Indian River,
Martin, Palm Beach and St. Lucie counties. Other counties with high
growth rates include Hardee, Hernando, Hillsborough, Lake and Sumter
(Table 13). These counties shipped approximately 37 million boxes
of fruit in 197576 or 50 percent of the state's shipments. The four
fastest growing counties in terms of fresh citrus shipments had a
very small share of the state total in 197576 with only 546 thousand
boxes or 0.7 percent.
Over time there has been a clear trend towards fewer and larger
packinghouses, using quantity shipped as a measure of packinghouse
size. While in 195556 there were 241 facilities shipping an average
of about 345 thousand boxes, in 197576 only 170 plants were operating.
Table 13.County Shipments Estimated Rate of Growth
a rb
Percent
Indian River:c
Brevard 0.019 0.34
Indian River 7.00 0.95
Martin 19.5 0.35
Palm Beach 19.4 0.76
St. Lucie 5.25 0.83
Volusia 3.70 0.77
Interior:
Broward 9.75 0.77
Citrus 30.8 0.62
De Soto 1.06 0.18
Duval 0.212 0.036
Hardee 16.4 0.17
Hernando 17.2 0.64
Highlands 2.40 0.57
Hillsborough 3.75 0.62
Lake 4.95 0.71
Manatee 07.48 0.75
Marion 1.49 0.16
Orange 1.46 0.36
Osceola 19.4 0.87d
Pasco 8.08 0.62
Pinellas 2.77 0.37
Polk 1.06 0.37
Putnam 3.09 0.33
Sarasota 60.0 0.77d
Seminole 2.20 0.26
Sumter 8.12 0.23
State 2.46 0.73
aEstimate of 8, defined in footnote, page 13.
Correlation coefficient = T Et InQt t lnQt
ST Zt2 (Zt)2 / T E(lnQt)2 (lnQt)2
CWith the exception of St. Lucie these counties have some townships
in the Interior Region (Figure 1).
dNo shipments in 197576.
This is a reduction of 29 percent in the number of packinghouses. In
197576, however, total shipments had decreased by 9 percent below
the 195556 level, but the average shipments per packinghouse were
at more than 445 thousand boxes, an increase of 29 percent since 195556
(Table 12). Furthermore, packinghouses with an output exceeding 100
thousand boxes handled 95 percent of the fruit packed in the 197576
season, an increase from the 90 percent packed in 195556.
Despite the state trend of declining number of packinghouses the
following counties have more packinghouses in 197576 than before:
Broward, Indian River, Manatee, Martin, Palm Beach and Sumter. With
the exception of Manatee and Sumter, all counties are in the Southern
East Coast region. Four counties (Hardee, Hernando, Hillsborough and
Lake) have experienced reductions in the number of packinghouses
although shipments have been increasing over the past ten years (Table
13).
The grapefruit shipment growth rates vary from negative values
to 30 percent per year (Appendix Table A5 and 14). Charlotte, Collier,
Hardee, Osceola and Sarasota counties no longer ship grapefruit (Appen
dix Table A5). Two regions in Florida have high grapefruit shipments
growth rates: The East Coast with Broward, Indian River, Martin, Palm
Beach and St. Lucie counties; Central Florida with Lake, Seminole and
Sumter counties (Table 14). These counties shipped almost 26 million
boxes of grapefruit in 197576 or 67 percent of the state grapefruit
shipments. The three fastest growing counties in terms of fresh grape
fruit shipments had a very small share of the state total in 197576,
with slightly over 50 thousand boxes or 0.13 percent.
Table 14.County Grapefruit Shipments Estimated Rate of Growth
ea r b
Percent
Indian River:c
Brevard 0.108 0.02
Indian River 6.93 0.94
Martin 30.7 0.64
Palm Beach 18.6 0.67
St. Lucie 6.78 0.86
Volusia 0.181 0.03
Interior:
Broward 9.32 0.40
Citrus 16.3 0.26
Duval 13.7 0.82
Hernando 5.61 0.25
Highlands 8.17 0.75
Hillsborough 0.809 0.18
Lake 4.96 0.55
Manatee 11.6 0.69
Marion 7.16 0.48
Orange 0.848 0.22
Osceola 12.4 0.63
Pasco 14.5 0.68
Pinellas 14.6 0.89
Polk 0.608 0.28
Putnam 16.7 0.80
Sarasota 66.9 0.77
Seminole 3.82 0.17
Sumter 22.7 0.38
State 3.05 0.79
aEstimate of defined in footnote, page 13.
bCorrelation coefficient = T .* t InQt Et J nQt
T t2 (Et)2 / T Z(lnQt)2 (ZlnQt)2
cWith the exception of St. Lucie these counties have some townships
in the Interior Region (Figure 1).
34
II5. Florida Citrus Processing
About 84 percent of Florida citrus production was processed in
197576, including eliminations from packinghouses. The percentage of
citrus processed has been practically at the same level over the past
decade, after increasing from between 60 and 70 percent twenty years
ago (Florida Crop and Livestock Reporting Service, 1976a).
Approximately 200 million boxes were processed in 197576 by 52
plants. The number of plants has been quite stable over the past twenty
years. In 195556, 57 plants processed about 84 million boxes. Thus,
the increase in output of canned and concentrated products has been
accompanied by a substantial increase in average number of boxes pro
cessed per plant or from 1.5 million in 195556 to 4 million in 197576
(Florida Department of Agriculture and Consumer Services, 1956 through
1976).
Since 195556 several processing plants have closed while new plants
opened in the same or different counties. While there are no packing
houses in Dade, Hendry and Highlands counties, there are processing
plants. The opposite is true in Broward, Citrus, Duval, Marion, Putnam
and Sumter counties.
II6. Marketing Costs
Costs have been rising in all stages of the citrus industry, as
reported by Hooks and Spurlock (1977) and Hooks and Kilmer (1977 a and
b).
35
Marketing cost increases imply a need to improve citrus industry
efficiency in order to remain competitive in world and domestic markets.
Orange picking and roadsiding costs in the 197576 season were
85 cents per box, an increase of about 1 percent from 197475. Grape
fruit and tangerine picking and roadsiding were estimated to be 69
and 158 cents per box, respectively.
Hauling costs for all types of fruit were estimated to be 15.4 cents
per box, an increase of about 1 percent from 197475 levels.
Total citrus picking and hauling costs increased slightly in 197576
and were higher than in any preceding season except 197374. Total
picking and hauling cost estimates for oranges and grapefruit were higher
by 1.2 and 1.4 percent, respectively, from 197475 season estimates.
Tangerines remained virtually unchanged. Some of the seasontoseason
variation in cost for each service is due to changes in the firms in
the sample. The longterm trend shows that costs have increased almost
steadily over the past 19 seasons before 197576 (Appendix Table B1).
Orange picking and hauling costs decreased relative to the delivered
in value of oranges for 197576. Appendix Table B2 shows the relation
ship between picking and hauling cost and the perbox deliveredin price
of oranges reported by the Florida Canners Association. During the first
five seasons as reported in Appendix Table B2, picking and hauling
costs averaged 16.4 percent of the perbox deliveredin price of oranges.
During the last five year period, picking and hauling costs averaged
36.8 percent of the deliveredin price for a box of oranges used in con
centrate. In the 197576 season, picking and hauling costs represented
32 percent of the deliveredin value of fruit.
36
Appendix Table B3 shows how total picking and hauling costs
have changed compared with the average reported figure for the 196061
through 196465 seasonsthe base period. With few exceptions, the
increases for all types of fruit have been very steady. For the 197576
season, orange and grapefruit picking and hauling costs are about twice
the levels during the base period, while tangerine costs are estimated
to be 1.7 times the average level reported during the base period.
Hooks and Spurlock (1977, p. 10) show how the components of total
orange picking and hauling costs have changed in absolute magnitude and
relative to their base period levels. Picking labor, the largest
component, has shown the greatest increase relative to the base period.
While Appendix Table B3 shows that total picking and hauling costs for
oranges in 197576 were 198 percent of the base period level, picking
labor has increased to 227 percent of the average in the base period.
From the base period, (196061 through 196465) total picking and hauling
costs for oranges are estimated to have increased 49.77 cents and picking
labor costs have increased 28.28 cents; that is, 57 percent of the in
crease is accounted for by picking labor cost increases. Labor costs
have increased relatively more than any other component and also account
for a major share of the total absolute increase.
Packing costs for two 4/5 bushel cartons of Florida oranges and
grapefruit are estimated at $2.22 and $2.27, respectively, for the
197576 season. The estimates are 1.8 and 2.9 percent above 197475
season estimates. Tangerine packing costs for two 4/5 bushel cartons
are estimated at $2.59, up 1.6 percent from previous season estimates.
The cost to pack two 4/5 bushel cartons of tangelos was $2.35.
37
Packing costs have increased steadily over the past 16 seasons with
the greatest increases in direct and indirect operating expenses, as
reported in Hooks and Kilmer (1977a, p. 15). Specifically, material
costs increased 5 and 9 percent for oranges and grapefruit while labor
costs increased one percent for oranges and decreased three percent
for grapefruit. Other direct operating costs were unchanged for oranges
but decreased 10 percent for grapefruit. Finally, administrative
costs decreased 10 percent for oranges but increased 21 percent for grape
fruit.
Trends in orange and grapefruit packing costs for the past 17
seasons are reported in Appendix Table B4. In general, packing costs
have increased over time; however, the rate of cost increase slowed in
197576 when compared with the change between the 197374 and 197475
seasons.
The relationship between average packing costs and the average
FOB value of packed fruit for 4/5 bushel cartons of oranges and grape
fruit in the Interior and Indian River regions for the past nine seasons
is shown in Appendix Table B5. The changes in the ratio have been
cyclical. The value of fruit has shown greater variability than costs
which have tended to increase over time (Hooks and Kilmer, 1977).
Processing, warehousing and selling costs remained reasonably stable
until the 197172 season. As shown in Hooks and Kilmer (1977b, p. 16),
costs for processing, warehousing and selling 48 six ounce cans of con
centrate in cases were at only 107 percent of the 196568 average unit
the 197172 season. Since the 197172 season the index has risen to 153
percent in 197475 and declined slightly to 145 for 197576 season.
38
While 197576 processing costs were estimated at 145 percent of 1965
1968 levels, 197576 prices were at only 131 percent of 19651968 levels.
Changes in the cost categories that make up the total processing,
warehousing and selling expenses for 48 sixounce cans of concentrate
are shown in Hooks and Kilmer (1977, p. 17). Materials, other pro
essing expenses and warehousing expenses are the items that increased
the most relative to 19651968 levels while materials, selling and
other expenses have shown relatively small increases. Hooks and
Kilmer (1977, pp. 1819) also show how costs of producing other products
have changed since the 196162 seasons.
Kilmer and Tilley (1978) estimated that approximately 10 cents
in the cost of packing a box could be saved with a volume increase
relative to capacity such that capacity utilization will increase by
10 percent. Also found was that firms with a large capacity and the
ability to procure sufficient volume at a steady rate throughout the
season in which they are open would tend to have lower costs than firms
that have greater difficulty achieving favorable levels of the supply
related variables.
Soule (1974) discusses how careful handling can increase the per
centage of fruit harvested that reaches the consumer's table. Kilmer
and Tilley (1978) report that packout percentage among a sample of pack
inghouses ranges from 30 percent to 88 percent. The cost differential
between these two extremes is 28 cents per box or 11.4 percent of an
average firm with a packout percentage of 88, which is a substantial
potential cost reduction. Packout percentage is the proportion of fruit
delivered to the packinghouse that meets quality standards for fresh fruit
shipments.
39
117. Florida Citrus Markets
The major markets for Florida citrus are the large metropolitan
areas in the Northeast of the U.S.: Baltimore, Boston, Chicago, New
York and Philadelphia. All these markets received more than two
thousand truck carlots in 197576. In Canada the major market is
Toronto. Other major export markets are Japan, France and Germany,
each with more than 1 million 4/5 bushel boxes. This fruit is shipped
mainly through the Tampa, Fort Everglades and Jacksonville ports
(Florida Crop and Livestock Reporting Service, 1976a).
CHAPTER III
GENERAL PLANT LOCATION MODELS
Selection of the optimal location, number and size of facilities
in a region has been the focal point for a variety of economic studies.
These include the location of educational and health service facilities
as well as agricultural and other types of processing plants. In this
chapter several classes of mathematical models which represent alterna
tive approaches to making such selections are cited.
The earliest developments in location theory are those by Thunen
(1826) and Laundardt (1872) and continued into this century by Weber
(1909) and Losch (1940) forming what is known as classical location
theory.
The assumptions of those models differ to a certain extent from
those which can be made with respect to the problem of improving the
efficiency of the citrus industry marketing (Appendix C). Particularly,
the set of possible locations is the plane, that is, the geographical
area where the production and markets are localized. These models
provide benchmark or ideal solutions which do not account for limita
tions on the solution such as a finite and discrete set of potential
and existing locations. Thus one has to go beyond the models of
classical location theory.
41
The need to introduce a discrete space as the set of location
possibilities led first to the use of the linear programming techniques
developed by Dantzig in 1947 (Schrader, 1961), and the transportation
models (Judge, Havlicek and Rizek, 1964), discussed in Appendix C.
Simultaneously, in order to include economies of scale in plant
operation, Stollsteimer (1961) developed a model which evaluates all
possible combinations of facilities and locations to determine the
optimal minimum cost solution, and King and Logan (1964) adapted the
transshipment model developed by Orden (1956) to plant location analyses
(Appendix C).
Several nonlinear programming models have also been used as is
pointed out in Appendix C. Separable programming introduced by Miller
in 1963 is used in a locationallocation context by Crowder (1967).
Candler, Snyder and Faught (1972) formulate the problem as one of con
cave programming. Graves (1972) uses quadratic programming for which
solutions had been provided earlier by Beale (1955) and Wolfe (1959).
Beale also studied mixedinteger problems in 1958 and such a formula
tion is used by Ballinski (1964) for a locationallocation model. Mixed
integer models are reviewed in Appendix C.
Many other contributions have been made to each one of these types
of models as shown in Table 15. The model classifications used in
Table 15 and throughout this chapter are not exclusive. All models that
follow a classical location theory approach are grouped under that
designation. Linear programming models which do not have or do not
exploit the special characteristics of a transportation problem are
42
Table 15.Agricultural Location Models
Model Author(s) Application
Classical Location Theory
Olson (1959)
Williamson (1962)
Miller and Henning (1966)
Araji and Walsh (1969)
Oppen and Hill (1970)
Ballestero Pareja (1971)
Oppen (1972 and 1974) and
Oppen and Scott (1976)
Lee (1975 and 1977)
Linear Programming
Schrader (1961), Schrader and
King (1962) and King and
Schrader (1963)
Egbert and Heady (1963)
Martin (1964)
Langemeier (1968) and
Langemeier and Finley (1971)
Leunis (1968)
Lee (1968) and Lee and
Snyder (1970)
Freeman (1969)
Shumway (1969) and Shumway
et al. (1970)
Free (1970 and 1971)
Milk Processing
Meat Processing
Livestock Markets
Grain Distribution
Grain Distribution
Sugar Beet Scales
Soybean Processing
Wastewater Treatment
Cattle Feeding
Field Crops
Poultry Industry
Cattle Feeding
Soybean Processing
Feed Industry
Snap Beans Processing
Field Crops and Vegetables
Soybean Processing
Table 15.Continued
Model Author(s) Application
Alexander and Ashley (1970)
Dietrich (1971)
Goodwin and Crow (1973)
Byrkett (1974) and Byrkett
et al. (1976)
Thomas (1975)
Fedeler and Heady (1976)
Milk Processing
Cattle Feeding
Meat Processing
Cattle Feeding
Milk Processing
Grain Distribution
Transportation
Pherson and Firch (1960)
Judge et al. (1964 and 1965),
Havlicek et al. (1964) and
Rizek et al. (1965)
Carley (1966)
Fraase and Anderson (1970)
Weindlmaier (1971)
Hopkins et al. (1971)
Hardy (1972) and Hardy et
al. (1973)
Rao (1973)
Fraase et al. (1974)
Doeksen and Oehrtman (1976)
Meat Processing
Meat Processing
Milk Processing
Malt Plants
Meat Processing
Feed Industry
Medical Clinics
Grain Distribution
Pasta Processing
Fire Fighting
Combinatorial
Stollsteimer (1961 and 1963),
Sammet and Courtney (1975)
and Stollsteimer et al.
(1975)
Pear Packing
Table 15.Continued
Model Author(s) Application
Mathia (1962 and 1963) and
Mathia and King (1962)
Peeler (1963) and Peeler and
King (1964)
Siebert (1964)
Polopolus (1965 and 1968),
Popolus and Strebeck (1965)
and Lopez (1966)
Sanders and Fletcher (1966)
Warrack (1967) and Warrack
and Fletcher (1970 a and b)
Courtney (1968)
O'Dwyer (1968 a and b)
Halvorson (1968 and Ladd and
Halvorson (1970)
Chern (1969) and Chern and
Polopolus (1970)
Hicks (1970) and Hicks and
Badenhop (1971)
Huie (1970)
Dawson and Warrack (1971)
Eddleman (1972)
Moore (1972) and Moore and
Courtney (1973)
Kedar (1972)
Tilley (1973)
Kuehn (1973)
Sweet Potato Processing
Egg Packing
Citrus Industry
Vegetable Processing
Egg Packing
Feed Industry
Feed Industry
Milk Processing
Turkey Processing
Citrus Industry
Livestock Markets
Meat Processing
Hog Markets
Hospitals
Cotton Ginning
Citrus Industry
Hog Markets
Livestock Markets
Table 15. Continued
Model Author(s) Application
Tosterud (1973) and
Tyrchniewicz and Tosterud
(1973)
Baumel et al. (1973), Lifferth
(1974) and Ladd and Lifferth
(1975)
Fuller and Washburn (1974)
Fuller (1975a)
King and Logan (1964) and
Logan and King (1964)
Hinton (1964)
Hurt and Tramel (1965)
Leath and Martin (1966 and
1967) and Leath (1970)
Bobst (1966) and Bobst and
Waananen (1968)
Cassidy (1968), Toft et al.
(1970) and Cassidy et al.
(1970)
Strawn (1969) C
Ferguson and McCarthy (1970)
Ferguson and Pemberton
(1970), McCarthy et al.
(1971) and Ferguson et al.
(1972)
Blair et al. (1970) E
Stennis (1970) and Stennis
et al. (1971)
Jesse (1970) and Hudson and
Jesse (1972) C
Grain Distribution
Grain Distribution
Cotton Ginning
:otton Ginning
leat Processing
4ilk Processing
leat Processing
;rain Distribution
ilk Processing
teat Processing
orn Distribition
lool Processing
gg Packing
lilk Processing
:otton Ginning
Table 15 .Continued
Model Author(s)
Stammer (1971)
Holroyd (1972) and Holroyd
and Lessley (1973)
Haas (1972 and 1974) and Via
and Haas (1976)
King et al. and King (1973)
King and Seaver (1974)
Stennis and Hurt (1974 and
1975)
Boehm and Conner (1976 a and
b and 1977)
Nonlinear Programming
Crowder (1967)
Kloth (1970) and Kloth and
Blakley (1971)
Stennis (1970) and Stennis
et al. (1971)
Holder et al. (1971) and
Candler et al. (1972)
Graves (1972)
Howard (1975)
Kilmer (1975) and Kilmer
and Hahn (1978)
Baritelle and Holland (1975)
Holland and Baritelle
(1975)
Schools
Application
Feed Industry
Honey Packing
Hog Markets
Potato Packing
Apple Packing
Meat Processing
Milk Processing
Milk Processing
Milk Processing
Milk Processing
Rice Milling
Farm Inputs
Farm Inputs
Milk Processing
Table 15.Continued
Model Author(s) Application
MixedInteger
Barton (1974) Meat Processing
Fuller (1975b) and Fuller
et al. (1976) Cotton Ginning
Cleveland (1976) and Cleveland
and Blakley (1976) Cotton Ginning
Daberkow (1976) and Daberkow
and King (1977) Emergency Medical
Facilities
Hilger et al. (1977) Grain Distribution
48
listed separately. Models formulated on the basis of an optimization
of an objective function over all possible combinations of plant
numbers, sizes and locational patterns are classified as combinatorial,
although for each combination they might be a transportation or trans
shipment model. Transshipment implies shipment between an origin and
a destination through an intermediary point where some sort of product
transformation takes place. Nonlinear programming models are all those
whose objective function is nonlinear, mostly due to a nonlinear pro
cessing cost function. Mixedinteger models are those which include
continuous and integer decision variables. These integer variables
are associated with the operation or not of a specific plant. The
major features of the combinatorial, transshipment, nonlinear and mixed
integer models are listed and summarized in Table 16. Among these
developments, the works of Siebert (1964), Chern and Polopolus (1970)
and Kedar (1972) have special interest for not only extending the
existing models but also applying them directly to the citrus industry.
These models are discussed in Appendix C.
The only model and solution procedure with all of the desirable
characteristics shown in Table 16 is the mixedinteger model developed
by Barton (1974) for a meat processing firm. He could not solve and
analyze the firm's actual problem because all the essential data were
not provided, so an illustrative example was used. The model has
characteristics specific to meat marketing which do not apply to the
citrus industry.
None of the agricultural location models addresses the issue of
the optimal dynamic adjustment of the existing structure of the optimal
49
Table 16.Main Features of Selected Location Models
Type of Number of Stages Multiple
Model Author(s) Transportation Products
(assembly and Processing
distribution)
Combinatorial
Stollsteimer (1961, 1963) 1 1 No
Siebert (1964) 1 1 No
Polopolus (1965 1 1 Yes
Chern and Polopolus (1970) 1 1 No
Kedar (1972) 2 1 Yes
Tyrchniewicz and Tosterud
(1973) 2 1 No
Fuller (1975a) 1 1 No
Ladd and Lifferth (1975) 3 2 Yes
Transshipment
King and Logan (1964) 2 1 No
Hurt and Tramel (1965) ni n Yes
Leath and Martin (1966( ni n Yes
Bobst and Waananen (1968) 2 1 No
Stennis and Hurt (1974,
1975) 1 1 No
Separable Programming
Kloth and Blakley (1971) 2 1 No
Baritelle and Holland (1975) 2 1 Yes
Concave Programming
Candler et al. (1972) 2 1 Yes
Quadratic Programming
Graves (1972) 2 1 Yes
Mixed Integer
Balinski (1964) 1 1 No
Efroymson and Ray (1966) 1 1 No
Barton (1974) 8 3 Yes
Fuller et al. (1976) 1 2 No
Hilger et al. (1977) 2 1 No
Table 16.Extended
Capacity Considering Exact Efficient
Constraints Existing Solution Search Other
Facility Procedure
No
Yes
No
Yes
Yes
No
Yes
Yes
No
No
No
No
No
Yes
No
Yes
No
No
Yes
Yes
Yes
Yes
Yes
Yes
No
NA
No
No
No
Yes
No
Yes
Yes
No
Yes
Yes
No
No
Yes
Yes
No
No
Yes
No
Yes Yes
Yes
NA
Yes
Yes
Yes
Yes
Yes
No Piecewise linear cost function
NA Discounted costs
No Institutional constraints
No Piecewise linear cost function
No Multiperiod and piecewise
linear cost function
No
No
No
No Institutional constraints
No
No
No Inventory costs
No
NA
Yes
Yes
Yes
Yes Multiperiod and piecewise
linear cost function
Yes Multiperiod
 
proposed for the end of the planning horizon. Sweeney and Tatham
(1976) present a synthesis of a mixedinteger location model with a
dynamic programming procedure for finding the optimal sequence of
configurations over multiple periods.
A mixedinteger programming formulation is selected in this study
because it is the only one that is truly optimizing and that allows
multiple secondary optimization runs at relatively low computer cost.
These secondary runs allow the use of the procedure developed by Sweeney
and Tatham (1976) to solve the dynamic model.
A solution procedure is classified as exact when it provides the
user with the global optimum solution. An efficient procedure avoids
evaluating every possible combination of flows and locations. The
solution procedures associated with the models listed in Table 16 are
classified according to these criteria. They are analyzed in detail
in Chapter IV.
CHAPTER IV
SOLUTION PROCEDURES
IV1. Introduction
This chapter contains a presentation and discussion of the pro
cedures used to obtain solutions to the location models presented in
Chapter III.
A solution procedure to the problem of improving the efficiency
of an assembly and distribution system has the following major steps:
a) determine the origins and supply of initial product and
market demand of final products;
b) select potential plant locations;
c) estimate transfer and plant cost;
d) determine the optimum plant number, size and location;
e) study the solution behavior under different industry situations.
The main approaches used to determine the optimum number, size and
location of plants can be classified as follows:
a) iterative actualization of plant cost;
b) evaluation of all possible combinations;
c) analysis of cost effects of successively opening or closing
plants;
d) approximation of concave costs by a polygonal cost curve;
e) analysis of local optima;
f) mixedinteger algorithms.
53
Linear programming, transportation, transshipment or network
algorithms have been used to obtain solutions to the continuous distri
bution subproblems.
The combinatorial procedure gives an optimal solution at the
expense of lengthy computations for reasonably complex cases. Concave
programming allows the determination of the global optimum if all local
optima are considered. Most mixedinteger algorithms converge to the
optimal solution. All other procedures do not assure that an optimum
is obtained.
Most procedures can be stopped when a solution is obtained that is
close enough to a lower bound. Both concave programming and mixed
integer algorithms update the current lower bounds during the computa
tional procedure. The analysis of solution behavior under different
situations is generally done using sensitivity analysis and/or multiple
runs of the model with the data changed to correspond to the new condi
tions.
IV2. Linear Programming Procedures
When using linear programming as a location model (King and
Schrader, 1963) the primal provides the following information:
a) the location of plants and input usage;
b) shipment patterns of products and inputs;
c) equilibrium prices and consumption of final product.
54
The dual solution provides the following information:
a) the input prices of factors consistent with equilibrium flow; and
b) the cost associated with introducing activities not in the
optimum solution.
To analyze an industry, sets of prices, consumption and transfer
cost information are divided into specific sets of "situations" and the
cost for the industry is minimized for each of these situations (Martin,
1964).
To minimize total assembly and processing costs, an iterative pro
cedure is followed. First, each plant is assigned the minimum value
of processing costs, usually associated with a large plant operating
at capacity. A solution is then obtained to determine the volume of
processing under these conditions in each plant. Second, processing
costs used in the first step are compared for consistency with the
actual cost of processing the volume of product indicated by the first
solution. Third, processing costs are revised and another solution
obtained as in step one. The steps are repeated until no further
adjustments in processing costs need to be made. It is recognized that
this procedure may not yield an optimum answer.
The final step in this analysis consists of varying the situations
assumed in the analysis, and determining the effect on the optimum
number, size and location of plants.
IV3. Transportation Model Procedures
The transportation model is a special class of linear programming
models for which computing routines have been developed to obtain solu
tions more efficiently.
The transportation model was initially conceived to give a minimum
transport cost in satisfying a given set of needs from a given set of
sources. The need of each location and the capacity of each source
are predetermined.
An analytical procedure, similar to the one described for the
linear programming model, can be used, incorporating the transportation
model in the total assembly and processing cost minimization step. The
same iterative method is used, with a solution for each iteration being
obtained faster and more efficiently, but still with no guarantee that
an optimum answer will be obtained.
Another heuristic procedure devised by Rao (1973) consists of
solving the transportation model to get a set of potential locations.
Then, the location pattern obtained is modified in such a way that all
the facilities in the new location pattern have full capacity utilization
to the extent possible. The criterion which justifies the transfer of
volume is that the profit by this transfer should more thar. offset the
loss. In this process some locations will get eliminated and the facil
ities in the remaining locations will have full capacity utilization.
The process is repeated for each facility size. That particular size
for which the total cost is a minimum is then selected.
IV4. Combinatorial Procedures
The procedure for cost minimization varies with the presence or
absence of economies of scale in plant operations and the way in which
processing costs are influenced by plant location. For a model where
it is assumed that there are economies of scale in plant operations
with plant costs independent of plant locations, the problem of mini
mizing the sum of processing and transfer costs is accomplished in
two steps (Stollsteimer, 1963).
The first step is to obtain a transfer cost function that has been
minimized with respect to plant locations with varying number of plants.
N
There are E ( ) possible combinations of locations. The second step
n=l
is to add to the minimized total transfer costs, the processing costs
with varying numbers of plants, yielding a total transfer and processing
cost function minimized with respect to plant locations for varying
numbers of plants. With constant marginal processing costs in any
given plant, and a positive intercept in the plant cost function, the
total cost of processing a fixed quantity of material, Q, will increase
by an amount equal to the intercept value of the plant cost function
with each increase in plant number (Chern and Polopolus, 1970).
Stollsteimer (1963) noted that the first difference of total trans
fer costs with respect to plant numbers is negative or zero, and the
second difference is positive or zero with empirical applications yielding
positive second differences. This gives a total transfer cost function
of the number of plants that is an envelope to a set of total transfer
57
cost points. This envelope is decreasing at a decreasing rate. When
added to a constant increasing total plant cost function, a minimum or
extreme point is obtained, whenever the decrease in transfer costs
equals the increase in total processing costs.
The minimization procedure proposed by Stollsteimer (1963) has
been classified as a set of linear programming problems, automated to
select the least cost among repeated solutions with both plant numbers
and locations permitted to vary (Stollsteimer et al., 1975).
Further work by Hoch (1965), however, shows that it is possible to
obtain a negative second difference, even though the empirical evidence
indicates that the probability of this occurring may be low. As Hoch
(1965) puts it, transfer costs as a function of number of plants is a
decreasing monotonic concave function only for some special cases. In
other cases (apparently rare in occurrence), the function may not be
concave. This implies that local minima are possible for total costs,
where total costs equal transfer costs plus processing costs. This
has the practical implication that, to find the minimum minimorum
(or minimum of all possible cases), one cannot terminate computations
when a local minimum has been obtained but must work through all possible
cases. This, in fact, has already been indicated by Heady and Candler
(1958, p. 371) when a transportation problem is solved taking fixed costs
into account. A numerical example exhibiting negative second differences
of transfer costs is also given by Hoch (1965).
Although the correction involved in Hoch's note (1965) does not
affect Stollsteimer's major conclusions (1963), it contains a major rule
for any combinatorial model: all possible cases must be evaluated.
The complete enumeration rule for a combinatorial model is, of
course, applicable to the generalizations of the Stollsteimer (1963)
model which consider multiple products (Polopolus, 1965), discontinuous
plant cost (Chern and Polopolus, 1970), collection and distribution
costs (Tyrchniewicz and Tosterud, 1973), multiperiods, capacity expan
sions and economies of scale in transportation (Ladd and Lifferth,
1975). Failure to recognize all possible combinations has led to sub
optimal solutions as in Chern and Polopolus (1970), (Fuller, 1975), and
Chern, (1969, p. 35), (Machado, 1975). Thus, the combinatorial procedure
yields an optimal result at the expense of lengthy computations
(Warrack and Fletcher, 1970b).
All citrus industry location models (Siebert, 1964) Chern (1969),
Kedar (1972) developed until the present are extensions of the
Stollsteimer model (1961).
As it stands, the problem of optimizing the distribution system
of citrus has not been solved. None of the procedures proposed before
dealt effectively with it. The major drawback encountered is that
previous studies relied upon complete enumeration which when not pursued
to its fullest extent, may lead to suboptimal results. Several sub
optimization procedures have been suggested and used by Warrack and
Fletcher (1976), Kuehn and Hamburger (196), Shannon and Iqnizio (1970),
Hardy (1972), Hardy et al. (1973), Fuller (1975a), Sielken (1973) and
Fuller and Sielken (1978). None of the procedures necessarily obtain
the optimal solution.
59
IV5. Transshipment Procedures
Basically, the cost minimization procedure for the transshipment
model modifies the transportation problem by specifying that the product
is shipped twice. Each production and processing area is designated
as a possible shipping point. Then, the cost of shipping directly from
a production area to a market is given a very high cost thus forcing
the product to be shipped into a processing plant (or warehouse) before
it is shipped to the distribution center.
The computational advantage of the transshipment formulation over
the alternative linear program is pointed out by King and Logan (1964).
For a single product, 30 origins, 20 potential locations and 30 markets
case, a 60 x 60 matrix is required. This compares with an alternative
formulation requiring 90 equations and 1,800 activities.
Market demands are either known or estimated in the first steps
of the procedure. To consider economies of scale in plant operations,
an iterative procedure identical to the one described above for the
linear programming procedure is proposed by King and Logan (1964). This
procedure is used in the cost minimization step and may not provide an
optimum answer, when economies of scale exist in processing.
An investigation of the transshipment model solution sensitivity
to the processing cost function is performed by Toft et al. (1970).
Using the sensitivity analysis common in the transportation model it
is possible to determine:
a) the change in processing costs necessary to alter the actual
location of processing operations;
60
b) the effects on the solution caused by localized variations
in transportation costs;
c) a mapping of optimum solutions for continuous variations in
either the position or shape of the processing cost curves.
The iterative procedure used by Blair et al. (1970) to arrive at
the minimum cost solution is similar to the linear programming procedure
suggested by Martin (1964) and the iterative eliminations approach sug
gested by Warrack and Fletcher (1970).
For the King and Logan (1964) iterative technique to approximate
an optimum plant location, Stennis and Hurt (1975) listed three techni
ques which can be utilized to save computer time and expense. These
techniques involve eliminating in an iterative fashion some potential
plant locations from the admissable set in subsequent solutions.
IV6. Nonlinear Programming Procedures
Kloth and Blakely (1971) used separable programming in the cost
minimization step. The implementation of separable programming is based
on the fact that each nonlinear, separable function can be approximated
by a piecewise linear function, known as polygonal approximation. The
solution is obtained for the problem defined in terms of a polygonal
approximation of each separable function; thus the solution reached is
an approximation of the true solution. Each polygonal approximation is
represented by linear equations together with certain logical restric
tions on the variables in the equations.
The plant location problem with fixed supplies and demands can be
regarded as an example of concave programming. Candler et al. (1972)
61
discussed an algorithm capable of eventually reaching a global optimum
for concave programming problems. The most significant consequence of
a concave objective function is that there may be and usually will be,
many local optima. That is to say, there will be many solutions each
of which is better than any other solution in its immediate neighborhood.
The algorithm, developed utilizing Tui's ideas (1964), does an orderly
examination of the local optima. Given that an exhaustive search of
the solution space is possible, determining how many local optima should
be examined is an economic problem of balancing potential gains against
known additional computing expenditures.
IV7. MixedInteger Procedures
To solve the mixedinteger plant location model, Balinski (1954)
used a partitioning theorem of Benders (1962) to reduce the problem to
one of solving a sequence of increasingly more complex integer programs
whose only variables are x's. At each step, the integer x's produced
by the integer program are used to determine an optimal solution to a
continuous dual problem, which is in turn used to produce a new integer
program. It can be shown that this method will converge to the optimal
solution of the problem in a finite number of steps. The technique of
Benders decomposition is also used by Hilger et al. (1977).
A branch and bound algorithm is used by Efroymson and Ray (1966).
Branch and bound is a finite general purpose mixedinteger programming
technique developed by Land and Doig (1960). The basic idea is to solve
a sequence of linear programming problems (not necessarily meeting the
integer restrictions) that give progressively improved lower bounds1
on the value of the solution to the mixed integer problem. To illustrate
the technique one assumes a mixed integer formulation with objective
function Z, and zeroone integer variables x.
The problem is first solved as a linear program (without the integer
requirements on the x's) giving a value of Z0. If all the x's are integer,
then the problem is solved. If some x. is fractional,then it is first
fixed at zero and the linear program again solved producing Z1, and
2
then fixed at one and the linear program solved producing Z2.
It is clear that:
(4.1) Z = man (Zl, Z2),
is a new lower bound on the value of the solution.
What is happening is that a tree is being constructed whose nodes
are represented by the Z's and the corresponding values of the fixed x's.
So far two nodes have been constructed (Z1, xj, = 1) and (Z2, xj = 0).
One now "branches" on the node determined by Z by fixing some other
fractional x., first at zero, then at one to determine two nodes Z3,
Z4. One continues by branching on
(4.2) Z = min [(Z3', Z4' max (Z, Z2)],
'Here one assumes that the problem is a minimization one. The technique
produces upper bounds for maximization.
2The general technique can handle cases where the integer variables can
assume more than two values.
63
which is a new, improved lower bound. Of course, one needs only keep
track of "terminal" nodes, and if any node is infeasible, no branches
can emanate from it. The process terminates when a node is reached where
all the x's are iteger and its value is less than or equal to that of
any other terminal node.
A variety of node selection rules and branching rules can be used
to achieve computational efficiencies in solving the plant location prob
lem. Khumawala (1972) discussed several rules which perform well in
solving this problem. Francis and White (1974) illustrated the procedure
with a plant location problem involving five customers and three plant
sites.
As Efroymson and Ray (1966) mention, the chief difficulty with
branch and bound is computational. If a large number of linear programs
have to be solved and the computing time for each linear program is high,
the method could become prohibitively expensive. Thus, to use branch
and bound on plant location, the problem must be formulated in such a
way that the linear programming problem without the integer restrictions
can be efficiently solved.
In reporting their computational experience, Efroymson and Ray
(1966) indicate that a number of 50plant, 200customer problems were
solved. The average solution time is reported to be about 10 minutes on
a IBM 7094 computer. Khumawala (1972) solved 16 test problems of size
25 by 50 and reports average solutions times of approximately 10 seconds
on a CDC 6500 computer.
A zeroone mixed integer programming computer code is available
from McCarl et al. (1973) and an adaptation from Tyrrell and Barton
64
(1976), which can be used to solve small problems, up to 150 rows and
450 columns.
Among the packages commercially available, IBM (1973) has an
optional mixedinteger programming feature for the Mathematical Pro
gramming SystemExtended (MPSX), for problems with up to 4095 integer
variables.
Fuller et al. (1976) recognized that the transportation cost
minimization of the location problem could be formulated as a network
problem. The efficiency of this solution approach depends on the solu
tion speed of the network code and the number of subproblems that
actually have to be solved. The total number of subproblems is 2 1.
In order to avoid explicitly solving all subproblems, a branch and bound
implicit enumeration procedure is employed (Fuller et al., 1976).
CHAPTER V
LOCATION MODEL FOR THE CITRUS INDUSTRY
V1. Model Formulation
A general location model of the citrus industry is presented in
this chapter. The static model formulation for a single season is
given. A discussion of the flexibility of the static model to accomo
date various assumptions is made in Section V2. The model in
Section V3 provides a plan for adjustment toward a more efficient
marketing organization over time.
There are H citrus fruits produced at several groves, I, with
given variable yields. The quantity, Qhi, of fruit,h, supplied at
each origin, i, is sent to a packinghouse at one of the possible
locations, J, where quantity qhijk is packed and shipped to one Df
the markets, K, and quantity qhijl is eliminated and shipped to one
of the processors, L. There is an estimated consumer demand, Qhk' for
each fruit, h, at each market, k. This demand is satisfied by shipping
qhijk from groves, I, to markets, K, via potential packinghouse loca
tions, J. The quantity, Qhl. of each fruit, h, that each processor, 1,
is capable or willing to receive from packers cannot be exceeded by the
sum of the quantities, qhijl' shipped from all origins, I, through all
possible packinghouse locations, J, and sent to processing. There are
66
lower, Qmin' as well as upper bounds, max' on the allowable total
annual throughput of each packinghouse site. If a packinghouse is
open at site j, variable x. takes the value 1, and is 0 otherwise. The
products Qminxj and Qmaxxj either equal zero when x. equals zero or
Qminand Qma, when xj equals one. The sum of the quantities, qhijk'
going through location j is either zero when x. equals zero or is
between Qmin and Qmax The percentage of each fruit, h, from each
origin, i, going through a packinghouse location, j, that is actually
packed is the packout, ahi, and the percentage eliminated is 100 = ehi.
The proportional quantities packed and eliminated are qhijk and qhijl'
which can be sent to any of the,K, markets, and, L, processing centers.
Supply of each fruit, h, from all origins, I, that is actually packed,
ehi/100Qhi' will satisfy demand, Qhk, at all markets, K. Quantities
shipped, qhijk and qhijl, are always nonnegative.
Possible locations for packinghouses are selected beforehand, but
the particular sites to be used are to be selected so as to result in
the least total assembly and distribution costs. The objective function
has three components. The first is total variable cost, for all
types of fruit H, origins I and location J, of assembling qhijk units
of fruit h from origin i, packing them at location j and then shipping
to market k at a cost per unit of vhijk, for all markets K. The second
is the total variable cost of assembling a certain quantity qhijl of
fruit h, from origin i, which is eliminated at j, and shipped to a
processing plant at 1, at a cost per unit of u The third is the
hijl
sum of all fixed costs f. incurred when fruit is packed at location j,
J
in which case x. is 1. Packing costs are expressed as fixed charges
(imposed for the sites actually used) plus a linear per unit charge
constant between the bounds Qmin and Qmax. Transportation costs are
taken to be linear with respect to distance.
There are two types of decision variables in the model: continuous
and discrete. These last ones are the integer variables x that can
only take the values 0 and 1. There are HIJL continuous variables q
and J integer variables x.
The model contains HI supply constraints, HL demand and processor
constraints, 2J capacity constraints and HIJ packout constraints, and
H supplydemand balance equations, bring the total to H(I (1 + J) +
L + 1) + 2J.
The number of possible solution vectors x is 2 1.
The model will determine which packinghouse sites to use, what
size and how many packinghouses to have at each selected site, which
markets and processing plants should be served by each packinghouse,
and what the pattern of transportation flows should be for all fruit.
This is to be done so as to meet the estimated demand at minimum total
assembling costs subject to the fruit supply and packinghouse throughput
constraints.
The model can be written as the following mixed integer linear
program:
H I J
(5.1) minimize Z = Z Z E
h=l i=l j=l
h=l i=l j=l
subject to,
supply constraints,
J K
(5.2) Z Z qhijk
j=l k=l
J
+ j
j=!
K
Vhijkhijk +
k=l
L J
S hijk hiji+ J'
I=KI j
L
S 1 <
Q=K= hijl  qhi'
I=K=I
h = . H,
k = K,
demand constraints,
J K
(5.3) hijk q hk
j=1 k=l h 
processing constraints,
I J
(5.4) Z q < 0
i=l j=l hij hl
capacity constraints,
H I K
(5.5) E Z k hijk > Qmnx ,
h=l i=l k=l
h = 1, . H,
1 = K+l, .., L,
j = 1, J,
H
(5.6) Z
h=l
I K
i l kq < hij axx
i=l k=l a=
h = l, ., H,
k = l, ., K
j = 1 .. J,
packout constraints,
K
(5.7) 7qhijk
k= l
e hi
1 eh i hij
1=K+I 100ehi .hij, i
h = ., H,
L = 1, ,
j = 1, . ., J,
supplydemand balance equations,
I K
(5.8) Z e hi > 
i=tivi00t k= con
nonnegativity constraints,
h = 1, . ., H,
for all h, i, j, k, 1,
0, 1 constraints,
1, if plant is open at site j,
0, if otherwise,
j = 1, . ., J.
J
The objective function (5.1) gives the cost when xj packinghouses
j=l
are to be located at those sites corresponding to positivevalued xj.
Constraints (5.2) indicate that the quantities shipped to market k and
to processor 1 cannot exceed the quantities supplied. By (5.3) and (5.4)
all demands for market k must be met by some combination of plants, and
cannot exceed processors' capacity. Constraints (5.5) and (5.6) keep
the total throughput at site j either equal to zero when x. equals zero
or between Qmin and Qmax. Constraints (5.7) relate the capacity of
each fruit packed at location j with the volume of eliminations going
to processing. Constraint (5.8) is included to make sure that
(5.9) qhijk' qhijl > O'
70
total demand does not exceed total supply. Nonnegativity and integer
restrictions on the decision variables qhijk' qhijl and xj are given by
(5.9) and (5.10), respectively.
V2. Model Discussion
V2.1. Cost Minimization
Cost minimization is the most widely used objective in the design
of physical distribution systems. The obvious reason is the relative
ease with which costs can be calculated. Revenue response is not
explicitly considered. Given the conditions of perfect competition,
the equilibrium solution for cost minimization is equivalent to the
solution for profit maximization, that is, cost minimization is equiva
lent to profit maximization only if revenues are independent of the
locationallocation design. Such independence would mean revenues are
not influenced by the level of customer service measured by the speed
and reliability with which customer's orders are filled. Since this
is very unlikely, cost minimization and profit maximization are seldom
equivalent.
Sales (and therefore revenue) may be considered in the form of
minimum sales constraints. Market service need not be totally ignored.
Network configuration restrictions and assumed transportation, order
processing, and inventory policies can be imposed which provide satis
factory market service levels. By use of "what if" sensitivity analysis,
1
For an extensive discussion of location model features, see Geofrion
(1975).
71
the effects of various sales and customer service level policies can
be explored.
Measuring the revenue response of various design alternatives is
extremely difficult and so is the application of sales, customer service
and profit maximization objectives to physical distribution system
design (Barton, 1974).
V2.2. Size Limits and Economies of Scale
If the maximum capacity of a packinghouse is never exceeded at any
location, there is no need to consider opening more than one plant at
each possible site.
Stipulation of lower and upper bounds on capacity permits a
piecewiselinear representation of economies of scale and other non
linearities (or even discontinuities) in packinghouse costs as a func
tion of throughput. Alternative packinghouses are simply introduced
at a given site with different size ranges controlled by in and Q ,
mm max
with fj and v i specialized accordingly. For instance, a piecewise
3 hijk
linear packinghouse cost function with three segments would require three
alternative packinghouses (small, medium and large) each with f. and
vhijk dictated by the corresponding segment of the packinghouse cost
function. More than one packinghouse of the same size may be open at
each site.
V2.3. Existing Packinghouse Utilization, Closing and Expansion
To consider expansion or closing of existing packinghouses existing
facilities are included among the J locations in the model.
Expansion of existing packinghouses can be considered using the
technique described in the previous section to permit a piecewise linear
representation of economies of scale. Expanded packinghouses at a given
site are introduced in the model with size controlled by Qmin and Qmax'
with f. and vhijk specialized accordingly.
V3. Dynamic Analysis
It is understood that once an optimal solution to (5.1 5.10) is
provided for some time in the future, one of the first questions asked
is: How should the industry move from its present structure to the
optimum in the future?
If the optimal solution does not differ much from the present
industry configuration, the analysis described in Section V2 above
provides helpful suggestions. Consider for instance the case where only
some locations need to be abandoned. Then by closing those plants
over the period considered, the final optimum stage is attained. Pri
ority analysis (Section VII6.5) establishes those plants that should
preferably be closed first, producing the largest savings. If some
existing facilities are to be expanded, then again priority analysis
shows those that will bring the highest reduction in costs. A combina
tion of both strategies can be outlined to open and close packinghouses
simultaneously. The results of the priority analysis relative to new
locations are a guide to further industry relocation.
Another possible approach is the use of a multiperiod model. If the
relocation of the industry from period to period can only be made at a
certain cost, a new situation might arise where some configurations
might result for certain years whose costs are only compensated by
savings realized over the entire period. For example, a configuration
which yielded the second best static solution in each planning season
could quite possibly yield the longrun optimal location, since no
relocation costs would be necessary over the planning horizon.
Consider now a matrix (Figure 4) of static solutions, Zt for each
season, t, ranked by cost. The optimal solution is found by using
dynamic programming to find the minimum cost path through the matrix,
taking into consideration, at each stage, the cost of moving from one
system configuration to any other (Sweeney and Tatham, 1976).
The computational feasibility of this procedure is dependent on
the number of stages and the number of alternative solutions that
must be considered at each stage. The computational time required
by any dynamic programming procedure increases linearly with the number
of stages and exponentially with the number of states. Therefore,
the number of configurations to be considered at each stage is the
most critical factor from a computational point of view.
Let Ztl denote the cost of the best static configuration in period
t and Ztc any other solution. Then Z1 = ZZt is the sum of the minimum
t
cost configurations for the entire planning horizon. Since no reloca
tion expenses are considered, Z1 is a lower bound on the value of the
optimal multiperiod solution.
Let Zu be an upper bound corresponding to any feasible solution to
the multiperiod problem. Then it can be shown (Sweeney and Tatham, 1976)
Rank Ordered
Solutions
(Low to High
cost)
Season
t
Z1l
Z12
13
Z21
Z22
z23
Z2c*
Z31
"32
z33
ZT2
ZT3
ZTc*
Z3c*
z2c
Z3c3
ZTc
z
Icl
Figure 4. The c best solutions to the static packinghouse location
problem by season.
ZIc*
75
A u 1
that if Z = Z Z is the maximum possible improvement that can be
u
made over the solution corresponding to Z then in period t, no static
solution with value Ztc such that c>c*, with Z t Z < Z and
t tc*t tl 
Ztc*t+ Ztl > Z may become part of an optimal multiperiod solu
tion. Thus it is only necessary to consider the c* best static
solutions in each period for possible inclusion in an optimal multi
period solution.
Since the number of static solutions ranked in each period depends
upon Z it is desirable to have a good upper bound, Zu available.
Any feasible solution, such as maintaining the current configuration
over the entire planning period, can be used to determine an initial
value for Zu. Better upper bounds can be generated as the solution
procedure progresses. The following approach is recommended by Sweeney
and Tatham (1976). Using the initial value for Z rank order the ct
best solutions in each period where c < c*. This may be some pre
t t
determined number of solutions, or one may rank order solutions until
Z Z > constant. In the former case ct will be fixed across
tct tl t
periods and in the latter c will vary. A new (and hopefully better)
upper bound can then be generated by using dynamic programming to find
the multiperiod solution considering as alternatives in each period
the c best static solutions. This new upper bound is clearly a
feasible solution,and if it consists of a sequence of optimal static
u
solutions, Ztl, it is the optimum solution. Using the new value for Z
one can recompute Z = Zu Z If now Zt Ztl > Z for all periods
76
one is finished, and Zu is the value of the optimal multiperiod solution.
If not, more solutions must be rank ordered in those periods where Ztt
tct
Z < Z Since the maximum possible improvement for further ranking
tl
is given by
(5.11) 2i = maxA I tct+ Ztl, 0 1
t
one might chose to terminate if Z1 is sufficient small. The value of
Z is a measure of the maximum possible opportunity loss associated with
implementing the solution associated with Zu. Computation cost also
limit the number of solutions to be ranked.
To obtain the next ct best static solution for the shortrun problems
a MIP searchlimiting process is used (IBM, 1973). This is preferred
to the procedure of adding additional constraints, and followed by Sweeney
and Tatham (1976), because of its computational simplicity.
The dynamic programming model that is used to find the multiperiod
optimum defines
Z (s) = minimum policy cost when in configuration s with T
T
more stages to go to final stage,
r (s) = a decision yielding Z (s),
and computes
(5.12) ZT(s) = minimum [ Zs + Z (r) ] for T = 1, 2, .,
(sr T
(sr)
where Zsr is the cost of configuration s plus the cost of changing from
configuration s to r. The recursion (5.12) is equivalent to the method
of finding a shortest route in a network.
The reason why this dynamic model is preferred over a multiperiod
model derives mostly from the fact that solutions for each season are
obtained independently from each other. Data inaccuracies for future
periods do not affect results obtained for other seasons. Thus for
each season one is able to rank order a reasonable number of alternatives
which is often of interest. This procedure of obtaining solutions for
each season also follows Geoffrion's recommendation (1975) for solving
the same model with different requirement scenarios.
The computational difficulty of a multiperiod model when compared
to this dynamic model should also be considered. The number of integer
variables and row constraints, both limiting factors in computational
feasibility, are multiplied by the number of seasons considered and the
number of constraints is increased by the between periods constraints
that have to be added to the model. The dynamic programming model that
gives the minimum cost solution over the seasons considered can be
solved by hand for a problem of reasonable size.
Most important of all, however, is the fact that the dynamic model
presented and used in this study includes the costs of moving from one
plant configuration to another. Obviously, these costs influence the
sequence of configurations that the dynamic programming procedure will
select as optimal for the multiperiod problem.
CHAPTER VI
ESTIMATION OF MODEL PARAMETERS AND COEFFICIENTS
To determine the cost minimizing number, location, size, supply
and market areas of citrus packinghouse, using the model of Section
V2, parameters and coefficients of the model need to be computed.
In this chapter the number of products (Section VI1.1), origins
(Section VI1.2), supply (Section VI1.3), market demand (Section VI
1.4), and processing capacity (Section VI1.5) are established.
Following a description of the assembly operations (Section VI
2.1), transfer costs are estimated (Sections VI2.2 through VI2.4).
Present and potential packinghouse locations are defined in Section
VI2.5 and packing costs are discussed in Section VI2.6. Capacity
constraints for existing and new packinghouses are defined in Section
VI2.7. The computation of the model's cost coefficients is discussed
in Section VI2.8.
VI1. Production and Demand Constraints
VI1.1. Number of Products
The major Florida citrus fruits are oranges, grapefruit, tangerines,
Temples and tangelos. Orange and grapefruit represent 79 percent of
Florida fresh citrus shipments (Table 11). Fresh grapefruit offer a
remarkable marketing challenge as detailed in Section 113. The
79
inclusion of tangerines, Temples and tangelos in the number of products
in the model would bring to 96 the percentage of fresh citrus production
under analysis. Thus, the model could include any number of products
between two and five depending on level of aggregation. Since model
size (Section V2) is proportional to the number of products considered
the study is made for oranges (h=l) and grapefruit (h=2). This is
justified by the fact that other varieties mature between the peak
orange and grapefruit packing seasons and can be packed without adding
capacity. Furthermore, production of specialty fruit is not concen
trated in any one set of origins and thus its inclusion in the model
is not likely to affect the locations of packinghouses.
VI1.2. Fruit Origins
Practically all Indian River fruit is packed in the area. Some
packers outside the region pack Indian River fruit for marketing reasons.
While the Indian River and Interior production regions are technically
geographically adjacent to one another, there is little fruit production
in the area near the southern twothirds of the boundary. Furthermore,
given the characteristics of the Indian River region, as discussed in
Chapter II, it is selected for application of the model.
Data on citrus production are available by section and land grant
in the State of Florida. This is the lowest level of aggregation by
which fruit origins can be represented. Sections can be grouped by
township and range, and these by county. There are 34 orange and/or
grapefruit producing counties in Florida with 23 counties accounting for
97.4 percent of the total production (Appendix Table AI).
80
Since Chern (1969) considered the county as too aggregative for
the purpose of his study, he aggregated townships into larger units of
less than county size. For convenience and accuracy in measuring the
distance from production origins to plant sites, an attempt was made
to define a production origin as consisting of approximately nine town
ships. However, due to the geographic restrictions and the uneven
dispersion of producing townships, some production origins were specified
to have more or less than nine townships. In this study the Indian
River region is divided in 13 fruit origins as in Chern (1969), and
indexed, i, according to Figure 5.
Although, in reality, shipments flow from groves to packinghouses,
to markets, it is assumed that those flows take place between central
reference points. This assumption implies that costs of shipments be
tween all sets of groves in region i, packinghouse in j, and markets
in k, or processing plants in 1, do not differe from the cost of shipment
between the centers of those regions. A town or city, as large and as
close as possible to the center of the origin, is selected to represent
the assembly point in each production origin.
VI1.3. Fruit Supply
For each origin, i, and each type of fruit, h, the supply to the
packinghouses is calculated as:
Qhi = Number of crees (age, variety, location) x yield (age,
variety) 9 average fraction packed fresh (variety) x
(100/packout percentage (type of fruit, region).
Figure 5.
The orange produc
tion origins for
fresh fruit assembly
in Florida, in Chern
(1969).
Source: Chern, 1969, p. 47.
82
The number, age, variety and location of trees as of January of
1976 were obtained from the Florida Crop and Livestock Reporting Service
(1976b). Data were accumulated for oranges and grapefruit at each origin.
Trees designated as "unidentified" by the Florida Crop and Livestock
Reporting Service in the round orange and grapefruit sections of the
report are prorated to the identified varieties according to the dis
tribution of total identified trees. Trees set in 1975 are considered
one year old in the 197576 season.
Yield data are published by Fairchild (1977) for early and mid
season oranges, Valencia oranges, seedy grapefruit, white seedless grape
fruit, and pink seedless grapefruit (Table 17).
For the 197576 season, the number of trees of each age and variety,
at each origin, multiplied by the appropriate yield gives total produc
tion. Only a percentage of this total production is packed as fresh
fruit, and certain varieties are more predominantly packed fresh (Table
18). Thus, total production of each variety is adjusted for each origin
by the average of the percentage used fresh in the past three years
(Table 18). These more recent data are assumed to better reflect the
present consumer preferences for the different types of fresh fruit.
Furthermore, fruit shipped from origins to packinghouses exceeds
fruit used fresh and shipped from the packinghouses. Packout, or the
percentage of fruit going through the house that is actually packed,
is 64.8 percent for oranges and 71.9 percent for grapefruit in the East
Coast (Hooks and Kilmer, 1977a). Dividing total quantity used fresh by
the packout percentage of each kind of fruit one obtains an estimate of
the total supply of fruit from each origin, Qhi'
Table 17.Box Yield Estimates by Age of Tree, Average of 196667 through
197576 Seasons
Age Oranges Grapefruit
of
Tree Early and White Pink
Mid Season Valencias Seedy Seedless Seedless
      1 3/5 bu. boxes/tree     
13 0. 0. 0. 0. 0.
4 0.81 0.66 1.22 1.59 1.48
5 1.03 0.85 1.52 2.00 1.81
6 1.25 1.05 1.82 2.41 2.14
7 1.48 1.26 2.09 2.78 2.49
8 1.72 1.47 2.34 3.09 2.86
9 1.95 1.69 2.60 3.39 3.24
10 2.19 1.90 2.85 3.71 3.61
11 2.43 2.12 3.11 4.01 3.99
12 2.66 2.33 3.36 4.33 4.36
13 2.86 2.48 3.57 4.59 4.67
14 3.01 2.63 3.77 4.87 4.98
15 3.28 2.78 3.97 5.14 5.29
16 3.48 2.93 4.18 5.42 5.60
17 3.69 3.08 4.38 5.69 5.92
18 3.90 3.23 4.58 5.96 6.23
19 4.10 3.38 4.79 6.23 6.54
20 4.32 3.57 5.07 6.53 6.72
21 4.56 3.78 5.41 6.86 6.76
22 4.79 3.99 5.76 7.21 6.79
23 5.02 4.21 6.09 7.54 6.84
24 5.25 4.42 6.44 7.88 6.88
25 & over 5.48 4.63 6.78 8.21 6.92
Source: Fairchild, 1977.
