Group Title: dynamic, mixed integer location model applied to Florida citrus packinghouses
Title: A dynamic, mixed integer location model applied to Florida citrus packinghouses
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Title: A dynamic, mixed integer location model applied to Florida citrus packinghouses
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Language: English
Creator: Machado, Virgílio Azuíl Páscoa, 1950-
Copyright Date: 1978
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Bibliographic ID: UF00102861
Volume ID: VID00001
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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

I-1. Problem Statement . . . . . . . . .. 1
I-2. Propositions and Objectives . . . . . . 4

II THE CHANGING FLORIDA CITRUS INDUSTRY . . . . . . 6

II-1. Florida Citrus Production . . . . . . . 6
11-2. Florida Grapefruit Production . . . . ... 16
11-3. Fresh Grapefruit Production . . . . . .. 22
11-4. Florida Citrus Packing. . . . . . . .. 28
11-5. Florida Citrus Processing . . . . . ... 34
11-6. Marketing Costs . . . . . . . . ... 34
11-7. Florida Citrus Markets. . . . . . . .. 39

III GENERAL PLANT LOCATION MODELS . . . . . . ... 40

IV SOLUTION PROCEDURES. . . . . . . . .... . 52

IV-1. Introduction. . . . . . . . .... . 52
IV-2. Linear Programming Procedures . . . . ... 53
IV-3. Transportation Model Procedures . . . . ... 55
IV-4. Combinatorial Procedures . . .. . . . 56
IV-5. Transshipment Procedures . . . . . . . 59
IV-6. Nonlinear Prograrning Procedures . . . ... 60
IV-7. Mixed-Integer Procedures. . . . . . . ... 61










TABLE OF CONTENTS--Continued


CHAPTER

V LOCATION MODEL FOR THE CITRUS INDUSTRY . . . . .. 65

V-I. Model Formulation . . . . . . . . . 65
V-2. Model Discussion . . . . .... . . . 70

V-2.1. Cost Minimization. . . . . .. . . 70
V-2.2. Size Limits and Economies of Scale. . . 71
V-2.3. Existing Packinghouses Utilization, Closing
and Expansion . . . . . . .. 71

V-3. Dynamic Analysis . . . . .. . . . . 72

VI ESTIMATION OF MODEL PARAMETERS AND COEFFICIENTS. . . . 78

VI-1. Production and Demand Constraints . . . . .. 78

VI-1.1. Number of Products . . . . . .. 78
VI-1.2. Fruit Origins . . . . . . . 79
VI-1.3. Fruit Supply . . . . . . . .. 80
VI-1.4. Shipments to Markets . . . . . .. 85
VI-1.5. Processing Plant Capacity.. . . . ... 90

VI-2. Capacity Constraints and Cost Coefficients. . ... 97

VI-2.1. Description of Assembly Operations. . 97
VI-2.2. Assembly Cost . . . . . . .. 98
VT-2.3. Eliminations Transport Cost. . . . ... 99
VI-2.4. Distribution Costs . . . . . .. 100
VI-2.5. Packinghouse Locations . . . . .. 100
VI-2.6. Capacity Constraints . . . . . .. 104
VI-2.7. Packing Cost . . . . . . . .. 105
VI-2.8. Cost Coefficients . . . . . .. 117

VII EMPIRICAL RESULTS. . . . . . . . . . ... 123

VII-1. Probationary Exercises . . . . . . .. 123

VII-1.1. Model Simplification . . . . .. 123
VII-1.2. First Auxiliary Model . . . . .. 126
VII-1.3. Second Auxiliary Model .. . ..... 136
VII-1.4. Third Auxiliary Model . . . . .. 137
VII-1.5. Fourth Auxiliary Model . . . . .. 137










TABLE OF CONTENTS--Continued


Page


CHAPTER

VII VII-2.
VII-3.
VII-4.


VII-5.

VIII SUMMARY

VIII-1.
VIII-2.
VIII-3.


Static Model Optimal Solution, 1981-82 Season .
Dynamic Model Solution, 1975-76 through 1881-82
Duality Analysis and Secondary Computer Runs. .

VII-4.1. Duality Analysis. . . . . .
VII-4.2. "What If . .?" Analysis . . .
VII-4.3. Sensitivity Analysis. . . . .
VII-4.4. Trade-Off Analysis. . . . . .
VII-4.5. 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 SECOND-BEST

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 1975-76 Crop Year . . . . . 7

2 Number of Carlots Received by 41 U.S. and Canadian Cities,
1975-76 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, 1963-64
through 1975-76 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, 1970-71 through 1975-76
Seasons . . . . . . . . ... ...... 25

12 Total Shipments, Number of Packinghouses and Average Ship-
ments per Packinghouse, by County, 1955-56 and 1975-76
Seasons . . . . . . . . ... . . . 29

13 County Shipments Estimated Rate of Growth . . . ... 31

14 County Grapefruit Shipments Estimated Rate of Growth . 33


viii










LIST OF TABLES--Continued


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 1966-67
through 1975-76 Seasons. . . . . . . . .. 83

18 Percentage of Florida Citrus Used as Fresh Fruit, by
Variety, 1973-74 through 1975-76 Seasons . . . ... 84

19 Fruit Supply, Indian River Origins (1=13), 1975-76 through
1981-82 Seasons. . . . . . . . . ... ... 86

20 Population of U.S. Market Regions and Canada, 1975-76
through 1981-82 Seasons . . . . . . . ... 92

21 Indian River Fruit Demand, U.S., Canadian Markets and
Exports (K=8), 1975-76 through 1981-82 Seasons . . .. 93

22 Existing Citrus Cannery Locations in the East Coast Region
of Florida, 1975-76 . . . . . . . . ... 96

23 Monthly Truck Rates for Oranges and Grapefruit, 1977 . .. 101

24 Existing Packinghouse Locations (J=10) and Volume Handled,
1975-76 through 1981-82 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 TABLES--Continued


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, 1975-76 Season . . 131

35 First Auxiliary Model Transportation Costs, 1975-76
through 1981-82 Seasons . . . . . . .... 134

36 First Auxiliary Model, Routes Changed Between 1975-76 and
1981-82 Alternative Solutions . . . . . .... 135

37 Fourth Auxiliary Model Solution Summary, 1981-82 Season .140

38 Optimal Location, Number and Size of Citrus Packinghouses
and Optimal Fruit Flows, Indian River Region, 1981-82
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, 1981-82 Season. . . . . . . . . ... 153

42 Comparative Advantage of Packinghouse Locations and Sizes 154

43 Supply, Supply Range and "Shadow" Prices for the Short-Run
Optimum, 1975-76 Season . . . . . . . .. 157

44 Demand, Demand Range and "Shadow" Prices for the Short-Run
Optimum, 1975-76 Season . . . . . . . .. 159

45 Packinghouse Optimum Short-Run Operating Volume and Volume
Range Over Which "Shadow" Prices Apply, 1975-76 Season 160

46 Total Cost and Integer Variables Values of Six Solutions,
1976-77 Season . . . . . . . . .. . . 162










LIST OF TABLES--Continued


Table Page


47 Fourth Auxiliary Model Solution Summary, 1975-76 through
1977-78 Seasons . . . . . . . . ... .. . 165

48 Second Auxiliary Model Capacity Constraint Dual Activities
1975-76 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, 1975-76 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 Long-run packing cost functions for the Indian River . 111

8 Existing and optimum number, size and location of citrus
packinghouses, Indian River, 1975-76 and 1981-82
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, 1976-77 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 = Long-run 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 = Short-run 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 location-allocation

mixed-integer 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 mixed-integer 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 1975-76

through 1981-82. 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 long-run packing

cost function is statistically estimated by using a variance component

model on cross sectional, time-series 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


I-1. 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 twenty-five 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 non-bearing trees had decreased.

Counties with high percentages of non-bearing trees are also now limited

to some south of Lakeland. Bearing acreage in Florida has been decreasing

since 1970-71, 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 short-run

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).








1-2. 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 least-cost 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


II-1. 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 1975-76 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 1975-76 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 1975-76 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,
1975-76 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 A-i). 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 1976-77 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 1965-66 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 1975-76 season produced an estimated 21,009

thousand boxes of fruit or 8.5 percent of the state's estimated citrus

production (Appendix Table A-l).




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, 1963-64 through
1975-76 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

1965-66 and 1975-76 seasons (Appendix Table A-i).

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 off-season 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).


11-2. 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 1975-76. 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 twenty-five 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

1971-72 through 1975-76 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 A-2). Grapefruit production is

expected to increase rapidly between the 1976-77 and 1981-82 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 1965-66 season of only 224

thousand boxes or 0.64 percent of the state total. Ten years later

or in the 1975-76 season these South Florida counties produced an

estimated 4,659 thousand boxes or 9.5 percent of the state estimated

grapefruit production (Appendix Table A-2).

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 Z-lnQt
/ 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 13-24 year ranges and thus

will reach full maturity during the next 12 years. Furthermore, over

one-half (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 mid-sixties will be felt over

the next 12-18 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.


11-3. 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

1971-72 through 1975-76 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 1975-76. Florida grapefruit market shares

were 32 percent in France and 12 percent in the Netherlands in 1975-76

(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

California-Arizona. By 1984-85, it is projected that Florida will pro-

duce 70 percent of the grapefruit produced in the U.S., down from 76

percent in 1971-72 and 73 percent in 1975-76 (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 1971-72 and 1975-76. Argentina increased











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production by 78 percent over the same 5 year period. Total world sup-

plies increased 12 percent from 1971-72 through 1975-76. Myers (1977)

also reports that grapefruit production in countries other than the

U.S. is projected by the FAO to expand rapidly by 1981-82. 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 1975-76

and 1981-82.

Given the increasing levels of domestic and foreign competition,

the Florida citrus industry will perhaps face increased pressure to improve

packing industry efficiency.


11-4. 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 1975-76 season (Appendix Table A-3).

Packinghouses in six counties in the Interior and East Coast of the

state--Highlands, 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 1975-76 (Table 12).










Table 12.--Total Shipments, Number of Packinghouses
per Packinghouse, by County, 1955-56 and


and Average Shipments
1975-76 Seasons


Total Fresh Number of Average Shipments
Citrus Shipped Packinghouses per Packinghouses
55-56 75-76 55-56 75-76 55-56 75-76

- /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

(1973-74 to 1975-76), 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 A-4, 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 A-4). 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 1975-76 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 1975-76 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 1955-56 there were 241 facilities shipping an average

of about 345 thousand boxes, in 1975-76 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 1975-76.








This is a reduction of 29 percent in the number of packinghouses. In

1975-76, however, total shipments had decreased by 9 percent below

the 1955-56 level, but the average shipments per packinghouse were

at more than 445 thousand boxes, an increase of 29 percent since 1955-56

(Table 12). Furthermore, packinghouses with an output exceeding 100

thousand boxes handled 95 percent of the fruit packed in the 1975-76

season, an increase from the 90 percent packed in 1955-56.

Despite the state trend of declining number of packinghouses the

following counties have more packinghouses in 1975-76 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 A-5 and 14). Charlotte, Collier,

Hardee, Osceola and Sarasota counties no longer ship grapefruit (Appen-

dix Table A-5). 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 1975-76 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 1975-76,

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

II-5. Florida Citrus Processing


About 84 percent of Florida citrus production was processed in

1975-76, 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 1975-76 by 52

plants. The number of plants has been quite stable over the past twenty

years. In 1955-56, 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 1955-56 to 4 million in 1975-76

(Florida Department of Agriculture and Consumer Services, 1956 through

1976).

Since 1955-56 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.


II-6. 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 1975-76 season were

85 cents per box, an increase of about 1 percent from 1974-75. 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 1974-75 levels.

Total citrus picking and hauling costs increased slightly in 1975-76

and were higher than in any preceding season except 1973-74. Total

picking and hauling cost estimates for oranges and grapefruit were higher

by 1.2 and 1.4 percent, respectively, from 1974-75 season estimates.

Tangerines remained virtually unchanged. Some of the season-to-season

variation in cost for each service is due to changes in the firms in

the sample. The long-term trend shows that costs have increased almost

steadily over the past 19 seasons before 1975-76 (Appendix Table B-1).

Orange picking and hauling costs decreased relative to the delivered-

in value of oranges for 1975-76. Appendix Table B-2 shows the relation-

ship between picking and hauling cost and the per-box delivered-in price

of oranges reported by the Florida Canners Association. During the first

five seasons as reported in Appendix Table B-2, picking and hauling

costs averaged 16.4 percent of the per-box delivered-in price of oranges.

During the last five year period, picking and hauling costs averaged

36.8 percent of the delivered-in price for a box of oranges used in con-

centrate. In the 1975-76 season, picking and hauling costs represented

32 percent of the delivered-in value of fruit.






36

Appendix Table B-3 shows how total picking and hauling costs

have changed compared with the average reported figure for the 1960-61

through 1964-65 seasons--the base period. With few exceptions, the

increases for all types of fruit have been very steady. For the 1975-76

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 B-3 shows that total picking and hauling costs for

oranges in 1975-76 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, (1960-61 through 1964-65) 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

1975-76 season. The estimates are 1.8 and 2.9 percent above 1974-75

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 B-4. In general, packing costs

have increased over time; however, the rate of cost increase slowed in

1975-76 when compared with the change between the 1973-74 and 1974-75

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 B-5. 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 1971-72 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 1965-68 average unit

the 1971-72 season. Since the 1971-72 season the index has risen to 153

percent in 1974-75 and declined slightly to 145 for 1975-76 season.






38

While 1975-76 processing costs were estimated at 145 percent of 1965-

1968 levels, 1975-76 prices were at only 131 percent of 1965-1968 levels.

Changes in the cost categories that make up the total processing,

warehousing and selling expenses for 48 six-ounce 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 1965-1968 levels while materials, selling and

other expenses have shown relatively small increases. Hooks and

Kilmer (1977, pp. 18-19) also show how costs of producing other products

have changed since the 1961-62 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

11-7. 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 1975-76. 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 location-allocation 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 mixed-integer problems in 1958 and such a formula-

tion is used by Ballinski (1964) for a location-allocation 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


Mixed-Integer

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. Mixed-integer 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 mixed-integer 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) n-i n Yes
Leath and Martin (1966( n-i 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 mixed-integer location model with a

dynamic programming procedure for finding the optimal sequence of

configurations over multiple periods.

A mixed-integer 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


IV-1. 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) mixed-integer 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 mixed-integer 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.


IV-2. 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.










IV-3. 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.









IV-4. 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

IV-5. 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.


IV-6. 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.


IV-7. Mixed-Integer Procedures


To solve the mixed-integer 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 mixed-integer 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 zero-one 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 50-plant, 200-customer 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 zero-one 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 mixed-integer programming feature for the Mathematical Pro-

gramming System-Extended (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


V-1. 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 V-2. The model in

Section V-3 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 supply-demand 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 100-ehi .hij, i


h = ., H,
L = 1, ,
j = 1, . ., J,


supply-demand 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 positive-valued 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.


V-2. Model Discussion


V-2.1. 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

location-allocation 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).


V-2.2. 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

piecewise-linear 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.


V-2.3. 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.


V-3. 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 V-2 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 VII-6.5) 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 long-run 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 short-run problems

a MIP search-limiting 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

V-2, parameters and coefficients of the model need to be computed.

In this chapter the number of products (Section VI-1.1), origins

(Section VI-1.2), supply (Section VI-1.3), market demand (Section VI-

1.4), and processing capacity (Section VI-1.5) are established.

Following a description of the assembly operations (Section VI-

2.1), transfer costs are estimated (Sections VI-2.2 through VI-2.4).

Present and potential packinghouse locations are defined in Section

VI-2.5 and packing costs are discussed in Section VI-2.6. Capacity

constraints for existing and new packinghouses are defined in Section

VI-2.7. The computation of the model's cost coefficients is discussed

in Section VI-2.8.


VI-1. Production and Demand Constraints


VI-1.1. 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 11-3. 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 V-2) 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.


VI-1.2. 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 two-thirds 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 A-I).






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.


VI-1.3. 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 1975-76 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 1975-76 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 1966-67 through
1975-76 Seasons


Age Oranges Grapefruit
of
Tree Early and White Pink
Mid Season Valencias Seedy Seedless Seedless


- - - - - - 1 3/5 bu. boxes/tree - - - - -

1-3 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.




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