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Optimal sequencing of multiple cropping systems

Material Information

Title:
Optimal sequencing of multiple cropping systems
Creator:
Tsai, You Jen, 1954-
Publication Date:
Language:
English
Physical Description:
x, 188 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Corn ( jstor )
Crops ( jstor )
Irrigation ( jstor )
Irrigation systems ( jstor )
Multiple cropping ( jstor )
Peanuts ( jstor )
Simulations ( jstor )
Soil water ( jstor )
Soybeans ( jstor )
Wheat ( jstor )
Agricultural Engineering thesis Ph. D
Cropping systems -- Mathematical models ( lcsh )
Dissertations, Academic -- Agricultural Engineering -- UF
Multiple cropping ( lcsh )
City of Gainesville ( local )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1985.
Bibliography:
Bibliography: leaves 181-187.
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by You Jen Tsai.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
029447721 ( ALEPH )
14399185 ( OCLC )
AEG6553 ( NOTIS )
AA00004972_00001 ( sobekcm )

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Full Text


OPTIMAL SEQUENCING OF MULTIPLE CROPPING SYSTEMS
By
YOU JEN TSAI
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1985


ACKNOWLEDGMENTS
This work would not have been possible without the assistance and
contribution of many people.
I will always be grateful to my major professors, Drs. James W.
Jones and J. Wayne Mi shoe. My intellectual development was actively
encouraged by Dr. Jones, who was enthusiastic and confident of my
work. Dr. Mishoe focused on my appreciation of systems analysis and
provided unending support and confidence. Drs. K. L. Campbell, D. H.
Hearn, C. Y. Lee and R. M. Peart critically reviewed this paper and
provided insightful suggestions.
I also wish to thank Paul Lane, whose dedication and persistence
made the field experimental study possible.
Finally, I cannot begin to express my appreciation to my wife,
Chin Mei, my boy, Hubert, and mother-in-law who had confidence in my
abilities and gave me more than I can ever repay.


TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ii
LIST OF TABLES v
LIST OF FIGURES vii
ABSTRACT
CHAPTERS
IINTRODUCTION 1
The Problem 2
Scope of The Study 5
Objectives 6
IIREVIEW OF THE LITERATURE 7
Multiple Cropping 7
Optimization Models of Irrigation 10
Soil Water Balance Models 11
Crop Response Models 14
Crop Phenology Models 18
Objective Functions 20
Optimization Techniques 23
IIIMETHODOLOGY FOR OPTIMIZING MULTIPLE CROPPING SYSTEMS ... 29
Mathematical Model 29
Integer Programming Model 29
Dynamic Programming Model 32
Activity Network Model 35
Crop-Soil Simulation Model 39
Crop Phenology Model 41
Crop Yield Response Model 48
Soil Water Balance Model 49
Model Implementation 59
Network Generation Procedures 61
Network Optimization 62
Parameters and Variables 67


Page
CHAPTER
IVWHEAT EXPERIMENTS 70
Introduction 70
Experimental Procedures 72
Experimental Design 72
Modeling and Analysis 75
Results and Discussion 76
Field Experiment Results 76
Model Calibration 84
VAPPLICATION OF THE MODEL 89
Introduction 89
Procedures for Analysis 90
Crop Production Systems 90
Crop Model Simulation 91
Optimization of Multiple Cropping Sequences 91
Risk Analysis 93
Results and Discussion 94
Crop Model Simulation 94
Evaluation of The Simulation-Optimization Techniques 107
Multiple Cropping Systems of a Non-Irrigated Field in
North Florida 112
Effects of Irrigation on Multiple Cropping 119
Risk Analysis of Non-Irrigated Multiple Cropping
Sequences 121
Applications to Other Types of Management 130
VISUMMARY AND CONCLUSIONS 132
Summary and Conclusions 132
Suggestions for Future Research 133
APPENDIX A GENERAL DESCRIPTIONS OF SUBROUTINES 135
APPENDIX B SOURCE CODE OF SUBROUTINES 143
APPENDIX C INPUT FILE GROWS1 178
APPENDIX D INPUT FILE 'FACTS' 180
REFERENCES 181
BIOGRAPHICAL SKETCH 188
iv


LIST OF TABLES
Table Page
1. Threshold values for physiological stages of
growth of corn and peanut 43
2. Coefficients of a multiplicative model for predicting
wheat phenological stages 44
3. Description and threshold values of phenological
stages and phases for soybean cultivars 46
4. Values of the parameters for the nighttime accumulator
function of the soybean phenology model 50
5. Crop sensitivity factors, \^ for use in the simulation 50
6. Observations of specific reproductive growth stages for
winter wheat at Gainesville, Fla., in 1983-1984 .... 77
7. Summary of results of winter wheat growth under various
irrigation treatments, Gainesville, Fla., 1983-1984 . 80
8. Treatment effects on winter wheat yield, Gainesville,
Fla., 1983-1984 81
9. Seasonal and stage-specific ET for winter wheat grown in
Gainesville, Fla., 1983-1984 85
10. Price, production cost and potential yield of different
crops for typical north Florida farm 92
11. Simulation results of irrigated and non-irrigated ful 1 -
season corn grown on different planting dates for 25
years of historical weather data for Gainesville, Fla. 95
12. Simulation results of irrigated and non-irrigated short-
season corn grown on different planting dates for 25
years of historical weather data for Gainesville, Fla. 96
13. Simulation results of irrigated and non-irrigated 'Bragg1
soybean grown on different planting dates for 25 years
of historical weather data for Gainesville, Fla 97
v


Tab! e Page
14. Simulation results of irrigated and non-irrigated 'Wayne'
soybean grown on different planting dates for 25 years
of historical weather data for Gainesville, Fla 98
15. Simulation results of irrigated and non-i rrigated peanut
grown on different planting dates for 25 years of
historical weather data for Gainesville, Fla 99
16. Simulation results of irrigated and non-irrigated wheat
grown on different planting dates for 25 years of
historical weather data for Gainesville, Fla 105
17. Summary of network characteristics and CPU time required
for various durations of planning horizon and two
irrigation conditions 108
18. Sensitivity analysis of non-i rri gated multiple
cropping sequences to weather patterns 113
19. Comparison of various multiple cropping systems under a
non-irrigated field. System I includes corn, soybean,
peanut, and wheat allowing continuous peanut cropping.
System II includes the same crops as system I, but not
allowing continuous peanut cropping. System III
excludes peanut from consideration. 118
20. Analysis of net returns of non-irrigated multiple
cropping sequences in response to different weather
patterns 127
21. Analysis of net returns of non-i rrigated multiple
cropping sequences under different crop pricing
schemes for weather pattern number 3 128
22. General descriptions of subroutines used in optimizing
multiple cropping systems 136
vi


LIST OF FIGURES
Figure Page
1. A system network of multiple cropping 37
2. Rate of development of soybean as a function of
temperature 47
3. Effects of night length on the rate of soybean
development 47
4. Maximum yield factors that reduce yield of each crop
below its maximum value as a function of planting day
for well-irrigated conditions 51
5. Crop rooting depth after planting under wel1-irrigated
conditions 53
6. Leaf area index for wel1-irrigated crops as a function
of time 55
7. A schematic diagram for optimal sequencing of multiple
cropping systems 60
8. Time intervals during which each crop can be planted . 63
9. Phenological observations, water stress treatments,
stage partitioning, and daily temperature in winter
wheat experiment, Gainesville, Fla. 1983-1984 78
10. The effect of water stress treatment on different yield
variables of wheat for each stress treatment (average of
3 replications), (a) Dry matter; (b) Number of heads;
(c) Head weight; (d) Grain weight 82
11. Plot of observed vs. predicted yield ratio for wheat . 87
12. Cumulative probability of profit for non-irrigated
full-season corn on different planting dates 102
13. Cumulative probability of profit for non-irrigated
short-season corn on different planting dates 102
14. Cumulative probability of profit for non-irrigated
'Bragg' soybean on different planting dates 103


Figure
Page
15. Cumulative probability of profit for non-irrigated
'Wayne1 soybean on different planting dates 103
16. Cumulative probability of profit for non-irrigated
peanut on different planting dates 106
17. Cumulative probability of profit for non-irrigated
wheat on different planting dates 106
18. Sample output of optimal multiple cropping sequences
for north Florida 110
19. Optimal multiple cropping sequences of a non-irrigated
field with corn, soybean, peanut and wheat, allowing
continuous cropping of peanut 114
20. Optimal multiple cropping sequences of a non-irrigated
field with corn, soybean, peanut and wheat, not allowing
continuous cropping of peanut 117
21. Optimal multiple cropping sequences of a non-irrigated
field considering corn, soybean and wheat, excluding
peanut 120
22. Optimal multiple cropping sequences of an irrigated
field considering corn, soybean, peanut and wheat,
allowing continuous cropping of peanut 122
23. Optimal multiple cropping sequences of an irrigated
field considering corn, soybean, peanut and wheat,
not allowing continuous cropping of peanut 123
24. Optimal multiple cropping sequences of an irrigated
field considering corn, soybean and wheat, excluding
peanut 124
25. A set of optimal multiple cropping sequences for a non-
irrigated field chosen from Figure 20 for additional
simulation study 126
vi i i


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
OPTIMAL SEQUENCING OF MULTIPLE CROPPING SYSTEMS
By
YOU JEN TSAI
DECEMBER 1985
Chairman: Dr. J. W. Jones
Cochairman: Dr. J. W. Mi shoe
Major Department: Agricultural Engineering
Multiple cropping is one of the means to increase or at least
stabilize net farm income where climatic and agronomic conditions allow
its use, such as in Florida. With several crops to be examined
simultaneously, the design of multiple cropping systems becomes
complex. Therefore, a systems approach is needed. The goal of this
study is to develop a mathematical method as a framework for optimizing
multiple cropping systems by selecting cropping sequences and their
management practices as affected by weather and cropping history.
Several alternative formulations of multiple cropping problems were
studied with regard to their practicality for solutions. A
deterministic activity network model that combined simulation and
optimization techniques has been developed to study this problem. In


particular, to study irrigation management in multiple cropping systems,
models of crop yield response, crop phenology, and soil water were used
to simulate the network. Then, the K longest paths algorithm was
applied to optimize cropping sequences.
Under a non-irrigated field in north Florida, winter wheat followed
by either soybean, corn, or peanut was found to be the most profitable
cropping system. Especially favorable was the cropping of wheat-
peanut. Another significant conclusion to be drawn concerned the effect
of irrigation management on multiple cropping sequences. Under
irrigated fields, peanuts were selected for production each year because
of their high net returns in comparison to the other crops. In a system
in which peanut was not considered as an option, inclusion of irrigated
wheat-corn cropping would not be a profitable multiple cropping
system. Instead, double cropping of corn-soybean was the dominant
optimal sequence under irrigation.
The importance of irrigation management in multiple cropping
systems was studied using the methodology developed. The methodology is
also capable of incorporating other aspects of farming (i.e. pest
management) into an integrated framework for determining optimal
cropping sequences.
x


CHAPTER I
INTRODUCTION
Net farm income has been a major concern for farmers in commercial
agriculture for a long time. Income has expanded through various ways
including an increase in land area for production, fertilizer and
pesticide applications, machinery and other capital expansions.
However, these different methods of increasing net farm income usually
increase the cost of production. A study (Ruhimbasa, 1983) showed that
multiple cropping had the potential to reduce costs per unit of output
and reduce production risks, and therefore could increase or at least
stabilize net farm income where climatic and agronomic conditions allow
its use.
Multiple cropping may also be called sequential or succession
cropping. Succession cropping is the growing of two or more crops in
sequence on the same field during a year. The succeeding crop is
planted after the preceding crop has been harvested. There is no
intercrop competition. Only one crop occupies the field at one time;
thus mechanization is possible.
In summary, multiple cropping increases annual land use and
productivity resulting in increased total food production per unit of
land. It also allows more efficient use of solar radiation and
nutrients by diversifying crop production. Thus, it reduces risk of
total crop loss and helps stabilize net farm income.
1


2
The Problem
Multiple cropping is not without risk. The use of multiple
cropping creates new management problems. It may create time conflicts
for land and labor, may require new varieties or new crops for an area,
may deplete soil resources, i.e. water and nutrient reserves, more
rapidly, and may cause residuals from one crop that directly affect the
next crop. For example, increasing the crop species grown on the same
land makes herbicide selection more complex. Disease incidence may
increase with an annual production of the same species on the same field
each year. As a result, higher levels of management become more
important in terms of operations needed. In designing optimal multiple
cropping systems, managers need to take into consideration these
effects.
Of the above management areas, timing becomes dominant for
successful multiple cropping, given substantial yield losses for each
day of delay. As estimated by Phillips and Thomas (1984), if the losses
of soybean yields after a given date are 62 75 kg/ha-day, the cash
losses on a 200-ha planting of soybeans would be as much as $4000 -
$5000/day. A delay of one week probably could make the difference
between profit and loss. Therefore, a timely planting and optimal
within-season management practices are the key to profitable multiple
cropping.
Soil water determines whether seeds will germinate and seedlings
become established. With multiple cropping, seed zone water is even
more critical because the second crop must be established rapidly to
avoid possible yield reduction due to frost. Also, because of depletion


3
by the preceding crop, soil water content at planting of subsequent
crops in multiple cropping systems may be low as compared to planting
following a fallow period. This is particularly true in areas of low
rainfall or where periodic droughts could result in a depleted soil
reservoir that would prevent successful planting and production of the
second crop. Hence, management practices that take advantage of soil
water storage should be beneficial in multiple cropping systems.
Plant growth is influenced by the process of evapotranspiration
(ET). During the time course of a seasonal crop, the crop system
changes from one in which ET is entirely soil evaporation to one in
which ET is mostly plant transpiration, and finally to one in which both
plant transpiration and soil evaporation are affected by crop
senescence. Plants store only a minor amount of the water they need for
transpiration; thus, the storage reservoir furnished by the soil and its
periodic recharge are essential in maintaining continuous growth. In
the event of relatively high ET demand coupled with depleted soil water
conditions, water deficits in plants occur as potential gradients
develop to move water against flow resistances in the transpiration
pathway. As plants become water stressed, their stomata close. The
resulting effects on transpiration and photosynthesis are essentially in
phase. This would represent the reduction of plant growth because of
less carbon dioxide uptake and reduced leaf and stem growth. Therefore,
soil water, undoubtedly more often than any other factors, determines
crop yield.
The soil water reservoir is supplied by rainfall. As evapo
transpi ration demand and supply of soil water are synchronized,
potential maximum yield is expected. Otherwise, irrigation may be


4
practiced to supplement rainfall supply of water to the soil and thus
avoid possible yield reductions. Hence, crop sequencing that shifts
crop demands for soil water according to weather patterns could be
beneficial in multiple cropping systems.
In Florida, where the cold season is short and the water supply
(precipitation or irrigation) is sufficient to grow two or more crops
per year on the same field, the potential of practicing multiple
cropping is high. However, water management is critical here. For
instance, although long-term average rainfall amount (148 cm per year)
may be sufficient on the average for replenishing the soil water supply,
year-to-year variability in rainfall amounts and the variability in
successive days without rain may result in one or more drought periods
during a growth season. On the other hand, irrigation development is
expensive. Inasmuch as benefits from irrigation may vary appreciably
from year to year, developing optimal multiple cropping systems is
intended to make maximum use of the expensive irrigated land.
As the number of crops, number of varieties, variability in soil,
and development of new integrated management systems (i.e., tillage,
irrigation, pest control, fertilization, weed control, etc.) increase,
planning of a multiple cropping production system becomes very complex
in terms of maximizing net farm income. However, actual experimentation
with the system may be infeasible, cost-ineffective, and time-consuming
due to the vast array of multiple cropping systems that possibly can be
grown. As a result, an alternate method for evaluating optimal multiple
cropping practices is needed.
At a field level, it is desirable to be able to select crops,
varieties, planting date, and to evaluate various management strategies


5
in a multiple cropping scheme. The overall goal of this study is to
develop a mathematical method as a framework for optimizing multiple
cropping systems by selecting cropping sequences and their management
practices as affected by weather pattern and cropping history. This
framework will be applied in particular to the study of irrigation
management in multiple cropping production.
Scope of the Study
Many efforts have contributed to developing irrigation programs
which would provide optimal return to growing a single crop during a
single season. Fewer studies have concentrated on investigating the
effect of irrigation management under multiple cropping systems. The
problem to be explored is as follows. A 'field1 is considered for
growing crops over an N-year production horizon. There are I number of
potential crops and each crop has J varieties to be considered. Only
one crop grows at one time and various idle periods are also considered
legitimate choices in a cropping sequence. Under the assumption that
other production practices are optimally followed, what are optimal
cropping sequences and associated within-season irrigation strategies
that maximize net discounted return?
This study at a field-level needs to be differentiated from that of
a farm-level system. A field can be defined as an unit area of uniform-
soil land or as an area constrained by the inherent operational
practicalities of the irrigation system used. For example, it may be
the area under a center pivot irrigation system. Applying systems
analysis methods, this study develops a mathematical model to optimize
multiple cropping systems.


6
Objecti ves
The specific objectives of the study are
1. To develop a framework for optimal sequencing of crops in
a multiple cropping production system and for determining
optimal management of the crop land.
2. To apply the framework to study irrigation management in
multiple cropping production.
3. To implement a computer model for North Florida soil and
climate conditions, taking soybean, corn, peanut, and wheat
as crops to be produced.
4. To perform field experiments designed to quantify the effect
of water stress on wheat yield for Florida conditions, and to
form a simplified wheat yield response model for use in the
analysis.
5. To use the model as a decision-making tool to analyze multiple
cropping practices in this region in order to increase net
farm income.


CHAPTER II
LITERATURE REVIEW
Multiple Cropping
In the United States, sequential cropping systems are mostly found
in southern states where a short cold season allows the planting of a
second or a third crop on the same land. The use of no-tillage methods
further enhances the success of sequential cropping systems in this
region. A selected number of articles concerning the topics are
reviewed.
Multiple cropping in sequence has been criticized for being yield
reducing. Crabtree and Rupp (1980) found that in Oklahoma wheat yield
decreased from 2519 kg/ha in a monocropping system to 2200 kg/ha in a
double cropping system. The following soybean yield decreased from 2000
kg/ha in 51-cm rows and 1792 kg/ha in 76-cm rows to 1603 and 1453 kg/ha,
respectively. The use of no-tillage practices increased soybean yield
to 1722 and 1543 kilogram per hectare in the double cropping system. In
fact, the long land preparation process in the conventional tillage
method led to a late planting for the second crop which resulted in
lower yields. The no-tillage method, allowing a direct planting of
crops into unprepared soil with standing crops or residues, had
significant impacts on reducing the risk of obtaining low yield due to
late planting in a multiple cropping system. Westberry and Gallaher
(1980) conducted two different studies on the influence of tillage
7


8
practices on yield which also led to a conclusion favoring a no-tillage
method.
The potential of no-tillage methods to reduce production costs when
associated with multiple cropping systems to increase land productivity
suggests that these two practices should be used together to increase
net farm income (Robertson et al., 1980). Other advantages of no
tillage systems become more apparent with multiple cropping, and these
include (1) elimination of moisture loss associated with conventional
tillage at planting time, ensuring stands of second and third crops
under restricted rainfall patterns; (2) further reduction of soil
erosion; and (3) maintenance of soil structure by elimination of plowing
and land preparation (Phillips and Thomas, 1984.)
It is obvious that multiple cropping for grain crops depends on a
reasonably long frost-free season. Guilarte (1974) and Smith (1981)
indicated that a double cropping system can be feasible during the 240
or more days of the warm growing season in north and west Florida.
Unfortunately, these long growing seasons are associated with elevated
temperatures, which may adversely depress the second crop yield as
witnessed by Widstrom and Young (1980). Their results showed that
double cropping of corn could be a viable option on the coastal plain of
the southeastern United States, when the second crop was taken as forage
rather than as grain.
To generalize types of multiple cropping on a cropping-year basis,
we divide it into winter-summer double cropping, summer-summer double
cropping and winter-summer-summer triple cropping. The major system of
winter-summer double cropping is wheat-soybeans (Gallaher and Westberry,
1980). The use of valuable irrigation water for a second crop of


9
sorghum or sunflower is not very practical except to produce favorable
emergence condition. Thus, soybean is favored as a second crop. Of
summer-summer systems, corn-soybeans appears to be most commercially
viable (Gallaher et al., 1980). Because soybeans bloom over a longer
period of time, their yields tend to be hurt less by short periods of
drought during flowering. Corn, on the other hand, requires excellent
soil water conditions during silking and tasseling, or else yields will
be low. The third multiple cropping system is adding a winter vegetable
crop to summer crops or following a winter-summer sequence with a late
fall planting of a cool-season vegetable. This type of system has the
advantage of producing the vegetable crop when prices are relatively
high, and still producing field crops competitively with the rest of the
nation.
Despite other attributes of multiple cropping, if it does not, over
a period of time, provide more net income to the farmer, it will not be
practiced. Economic analyses studied by a group of research scientists
in the University of Georgia indicated that irrigated agronomic crops
were generally profitable on a first-crop basis, but the profitable
agronomic second-crop was limited to sorghum and soybeans (Anonymous,
1981). In 1980, the study also showed that most irrigated multiple
cropping production was profitable on the wel1-drained, sandy soil.
Both irrigated and dryland peanut production were profitable; however,
irrigated peanuts were more profitable. Irrigated corn was also more
profitable than nonirrigated corn. Tew et al. (1980) further analyzed
costs and returns of irrigated, double-crop sweet corn and soybean
production. They concluded that irrigated soybean as the second crop in
a double-crop system was a questionable alternative since net returns


10
did not compare favorably with dryland production. However, irrigation
of soybean as the second crop was still justified because it reduced
income variance.
These results suggest that the economics of multiple cropping
systems differs significantly from that of a single, full season
monocrop. Knowledgeable management practices such as precise planting
dates, cultivars, and water management are essential. Gallaher et al.
(1980) strongly asserted that "if growers use management practices in
these studies, corn-soybean succession cropping can be successful in
Florida" (page 4).
Optimization Models of Irrigation
In order to study irrigation policies to maintain favorable soil
moisture conditions and thus avoid economic yield reduction,
optimization techniques have been increasingly used for the last 15
years. Mathematical models are inherent in this methodology.
Implicitly or explicitly a crop response model within the mathematical
statement of the objective function is required. Furthermore, the soil
water status, needed as a set of constraints in the optimization
problem, is traditionally calculated in a soil water balance model.
Then, various optimization techniques are applied for finding the best
or optimal decisions in an organized and efficient manner. The role of
models and simulation in irrigation optimization problems is reviewed
herein.


11
Soil Water Balance
Water balance models for irrigation scheduling were developed as
'bookkeeping' approaches to estimate soil water availability in the root
zone.
S = S .
n n-1
+ P + I + OR
- ET RO
n n
PC
(2.1)
where
S = soil water content on the end of day n,
Pn = total precipitation on day n,
I = total irrigation amount on day n,
DR = water added to root zone by root zone extension,
n J
ETn = actual evapotranspiration on day n,
RO = total runoff on day n, and
n
PCn = deep percolation on day n.
In general, a volume of soil water, defined in terms of the soil
water characteristics and the root zone of the crop being irrigated, is
assumed to be available for crop use. Depletions from this reservoir by
evapotranspiration (ET) are made on a daily basis. Soil water balance
models generally are classified into two categories: (a) those based on
the assumption that water is uniformly available for plant use between
the limits of field capacity and permanent wilting point, and (b) those
based on the assumption that transpiration rates were known functions of
soil water potential or water content (Jones and Smajstrla, 1979).
Uniformly available soil water. Models based on the assumption of
uniformly available soil moisture between field capacity and permanent
wilting point simulated water use based on climatic variables only.


12
Those simulation models for ET by various crops have been summarized by
Jensen (1973). For ET prediction, a technique used widely to calculate
potential ET is the modified Penman equation (Van Bavel 1966). The
Penman equation predicted reference ET (ETp), which is that of a well-
watered, vegetated surface. To predict actual rather than reference ET
for a well-watered crop, a crop coefficient, 1973) as
ET = K ET (2.2)
c p '
Crop coefficients for specific crops must be determined experimently.
They represent the expected relative rate of ET if water availability
does not limit crop growth. The magnitude of the crop coefficient is a
function of the crop growth stage. One of the major shortcomings of
this method is that they do not account for changes in ET rates due to
changing soil water levels.
Limiting soil water. To correct this shortcoming, a number of
researchers (Ritchie, 1972; Kanemasu et al., 1976) have developed models
to predict ET as functions of both climatic demands and soil water
availability. This resulted in a more complex model than the Penman
equation, which uses climatic indicators only. Ritchie's model
separated evaporation and transpiration components of water use.
Potential evaporation Ep from a wet soil surface under a row crop
(energy limiting) was defined as
T


13
where t = reduction factor due to crop cover, and a = proportional ity
constant due to crop and climate.
During the falling rate stage (soil limiting) evaporation rate E,
was defined as a function of time as
E = ct1/2 c ( t 1 )1/2 (2.4)
where c = coefficient dependent on soil properties, and t = time.
Transpiration rates were calculated separately from evaporation
rates. For plant cover of less than 50 percent, potential transpiration
rate, T was calculated as
Tp = av ( 1 T ) ( A / ( A + Y )) Rn (2.5)
where A = slope of the saturation vapor pressure-temperature curve, y =
psychrometric constant, Rr = net radiation, and ay = (a 0.5)/0.05.
For greater than 50 percent crop cover, was calculated as
Tp = (ot-t) (a / (a + y) ) Rp (2.6)
This formulation represented transpiration during non-limiting water
conditions only. To account for decreasing soil water potential with
water content, and effects on transpiration rate, a coefficient of
limiting soil water (K ) was defined by Kanemasu et al. (1976) as
(2.7)


14
where 0 = average soil water content, and e = water content at field
3 max
capacity. At water contents above 0.3 0^^^ transpiration rates were
assumed to be controlled by climatic conditions only. Ritchie (1973)
reported that this model predicted transpiration rates well for sorghum
and corn.
In summary, several models for predicting ET rates under both well
watered and water stressed conditions are presented. The models
presented are all simple approximations of complex dynamic systems.
Their simplicity has the advantage of requiring few data inputs, and
therefore, they can be applied with relatively few meteorological, soil,
or crop measurements taken. However, because of their simplicity,
several empirical coefficients are required in each model, and each must
be calibrated for specific crops, soil conditions and climatic
variables.
Crop Yield Response
Vast literature on this subject revealed yield relationships to
water use can range from linear to curvilinear (both concave and convex)
response functions (Stegman and Stewart, 1982). These variations are
influenced by the type of water parameter that is chosen, its
measurement or estimation accuracy, and the varied influences associated
with site and production conditions. The following is intended to
illustrate the more general relationships of crop yields with water when
they are expressed as transpiration, evapotranspi ration, or field water
supply.
Yield vs. transpiration or evapotranspiration. When yields are
transpiration limited, strong correlations usually occur between


15
cumulative seasonal dry matter and cumulative seasonal transpiration.
Hanks (1974) calculated relative yield as a function of relative
transpiration:
Y T
where Yp = potential yield when transpiration is equal to potential
transpiration and Yp = cumulative transpiration that occurs when soil
water does not limit transpiration. With the close correlation between
T and ET, dry matter yield vs cumulative ET also plotted as a straight
line relationship. Hanks' work demonstrated a physically oriented,
simple model to predict yield as a function of water use.
Based on the same idea, an approach which interprets ET or T
reduction below potential levels as integrators of the effects of
climatic conditions and soil water status on grain yield is used
frequently. Such an approach predicts grain yields from physically
based models which relate water stresses during various stages of crop
growth to final yield, accounting for increased sensitivity to water
stress at various stages of growth. Two basic mathematical approaches
were taken in the development of these models. One assumed that yield
reductions during each crop growth stage were independent. Thus
additive mathematical formulations were developed (Moore, 1961; Flinn
and Musgrave, 1967; Hiler and Clark, 1971). A second approach assumed
interactive effects between crop growth stages. These were formulated
as multiplicative models (Hall and Butcher, 1968; Jensen, 1968).
Additive models. The Stress Day Index model is an additive model
presented by Hiler and Clark (1971). The model is formulated as


16
Y
A n
1.0
2 (CS, SD.)
(2.9)
Y
P
vp i=1
where A = yield reduction per unit of stress day index, SO.. = stress day
factor for crop growth stage i, CS^ = crop susceptibility factor for
growth stage i. CS^ expresses the fractional yield reduction resulting
from a specific water deficit occurring at a specific growth stage.
SD^ expresses the degree of water deficit during a specific growth
period.
The stress day index model was utilized to schedule irrigations by
calculating the daily SDI value (daily SD daily CS) and irrigating
when it reached a predetermined critical level, SDI. This integrated
the effects of soil water deficit, atmospheric stress, rooting density
and distribution, and crop sensitivity into plant water stress factor.
Multiplicative models. Jensen (1968) developed the following model
Y
(2.10)
Y
P
physiological development, and A.. = crop sensitivity factor due to water
stress during the i-th growth stage.
Hill and Hanks (1975) modified the above equation by including
factors to account for decreased dry matter production due to planting
late season crops, and to account for decreased yields due to excess
water. Their equation is
Y
n T A.
Y
P
n ( ) 1 SYF LF
1=1 tp
(2.11)


17
where (T/T )^ = relative total transpiration for growth stage i when
soil water is not limiting, SYF = seasonal yield factor which approaches
1.0 for adequate dry matter production, and LF = lodging factor.
Because this model relates relative yield to relative transpiration, it
is also necessary to predict evaporation rates as a function of ETp in
order to maintain a soil water balance. This yield response model,
verified with Missouri soybean experiments, appeared to be an excellent
simulator of grain yields as affected by transpiration rates.
Minhas et al. (1974) proposed another multiplicative model
expressed as
Y n ET 9 A.
= n { 1.0 ( (1.0 T L } (2.12)
Y i =1 ET 1
P P
where all factors are as previously defined. Howell and Hiler (1975)
found that it described adequately the yield response of grain sorghum
to water stress.
Yields vs. field water supply. The field water supply (FWS) in
irrigated fields is derived from the available soil water at planting
(ASWP), the effective growth season rainfall (Re), and the total applied
irrigation depth (IRR). Stewart and Hagan (1973) demonstrated that crop
yields are related to seasonal ET and seasonal IRR. In a given season,
the ASWP and Re components of the seasonal FWS make possible a yield
level that is common to both functions. The ET component associated
with successive applications of irrigation defines the yield, Y vs ET
function above the dryland level, which rises to a Y ET level
when the seasonal crop water requirement is fully satisfied. The ET +
non-ET components of IRR define a Y vs IRR function of convex form.


18
That is, non-ET losses increase as water is applied to achieve ET
levels due to the inefficiencies of irrigation methods and the
inexactness of water scheduling. The amount of water not used in ET,
therefore, represents runoff, deep percolation, and/or residual
extractable water in the soil when the crop is harvested. The water
management implications of this type of yield function are discussed
further in the next sections.
In summary, considerable efforts have been directed toward
development of simple models for describing the yield response of crops
subjected to water stress conditions. The application of these models
to irrigation management appears to be tractable (Hill and Hanks, 1975).
Crop Phenology Model
As a plant goes through its life cycle, various changes occur.
Crop ontogeny is the development and course of development of various
vegetative and reproductive phases, whereas phenology is the timing of
the transition from one phase to the next phase as controlled by
environmental factors. To accurately simulate crop growth and yield
with biophysical models, crop phenology needs to be successfully
predicted (Mishoe et al., in press). Crop parameters needed for growth
simulation are closely related to the phenological stages of the
plant. These include the duration of leaf area expansion, stem and root
growth, as well as the onset and end of pod and seed growth. It is
therefore desirable to allow assimilate partitioning values in the model
to change as the plant progresses through its reproductive stages.
Currently, many of the practical yield response models have
coefficients that depend on crop growth stage (Ahmed et al., 1976;


19
Childs et al., 1977; Wilkerson et al., 1983; Meyer, 1985). However, in
some studies, the crop growth stages have been poorly defined. And most
applications of these models use only the mean development times and
assume that stochastic variation does not affect the performance of the
model. Hence, a systematic approach to define stages relative to
physiological development of the crop and to predict these stages under
various weather conditions is needed (Boote, 1982). This would lead to
more accurate application of yield response models. In the rest of this
section, several approaches to modeling phenology are described.
The wide range of controlling factors and crop responses makes
phenological modeling challenging. The effect of temperature as well as
photoperiod as controlling factors has long been recognized. The
concept of thermal time in the form of degree-days is used to account
for temperature effect. Degree-days are cumulative daily average air
temperature above the base temperature (Prine et al., 1975).
Most models are based on thermal time or photoperiod or a
combination of the two. Some models based on thermal time alone are
quantitative, based on the analysis of experimental evidence (Kiniry et
al., 1983; Tollenaar et al., 1979). Kiniry et al. found that the
photoperiod did not affect all of the cultivars of corn. Those that
were affected were still insensitive below a threshold photoperiod value
of between 10 and 13 hours. For wheat, a quadratic equation, based on
day and nighttime temperatures and photoperiod was applied by Robertson
(1968), and Doraiswamy and Thompson (1982) to predict the time between
phenological stages.
Other models are based on the hypothetical processes involved in
crop response (Mishoe et al., 1985; Schwabe and Wimble, 1976). Mishoe


20
et al. (1985) developed a phenological model based on physiological
processes of soybean. One important concept is that a critical period
of uninterrupted night length is needed to produce rapid flowering.
Also the promotional effect of night length is cumulative. An
accumulator (X) value needed to trigger an event is calculated from a
function of night length and nighttime temperature. When the cumulative
X becomes larger than a threshold level, it triggers the phenological
event such as flower initiation. These threshold values for different
stages are calibrated from experiments, and are variety dependent.
Incomplete knowledge of biochemical processes involved hampers the
development of process models. However, for production management,
models using thermal time and night length have successfully predicted
phenological events.
Objective Functions
An objective function is a quantitative representation of the
decision maker's goal. One may wish to maximize yield, net profit, or
water use efficiency. However, these objectives are not equivalent and
the use of different objectives may result in different solutions.
Maximizing yield per unit area. This objective may be economically
justified when water supplies are readily available and irrigation costs
are low. All production practices and inputs must be at yield
optimizing levels, and daily cycles of plant water potentials must be
managed within limits conducive to maximum seasonal net
photosynthesis. From an applied water management viewpoint, this
production objective is relatively easy to attain. Many applied
experiments (Salter and Goode, 1967) have shown that for many crops,


21
yields will be near their maximum values when root zone available water
is not depleted by more than 25 to 40 percent between irrigations.
Maximizing yield per unit water applied. As irrigation water
supplies become more limited or as water costs increase in an area, the
management objective may shift to optimizing production per unit of
applied water (Hall and Butcher, 1968; Stewart and Hagan, 1973; Howell
et al., 1975; Windsor and Chow, 1971). Hiler et al. (1974) have
demonstrated that significant improvements in water use efficiency are
possible by applying the Stress Day Index method. Stewart et al. (1975)
have more recently suggested a simplified management criterion by noting
that the maximum yield for a given seasonal ET deficit level tends to
occur when deficits are spread as evenly as possible over the growing
season. Thus, scheduling is based on the concept of high frequency
irrigation, i.e. applying small depths per irrigation at essentially
evenly timed intervals.
Maximizing net profit. Applying marginal value vs marginal cost
analysis to yield production functions, Stewart and Hagan (1973) were
able to determine optimum economic levels of production for maximum
water use efficiency, maximum profit under limited water supply, and
maximum profit under unlimited water supply, respectively. A problem
with this method is that it provides only general guidelines for water
management. These guidelines are most applicable to the average or
normal climatic conditions in a given region and, therefore, may not
apply to specific sites or specific years. In addition the guidelines
are seasonal in nature, i.e., they indicate only the seasonal irrigation
depth most likely to maximize net profit.


22
In recent years, numerous models (Dudley et al., 1971; Matanga and
Marino, 1979; Bras and Cordova, 1981; Huang et al., 1975) have been
developed to address the goal of profit maximization. Methodologies
such as dynamic programming are frequently utilized to illustrate how
optimal water scheduling or allocation strategies within the growing
season can be derived under stochastic conditions.
Risk analysis. Risk assessment of decision alternatives can be
approached in several ways. One of the more common approaches is an
expected value-variance (E-V) analysis where the decision maker is
assumed to maximize utility, where utility is a function of the expected
value and associated variance in returns. The specific functional form
of this relationship varies by individual depending upon each
individual's psychological aversion to risk. For example, the risk
averse individuals may be willing to trade a reduction in expected net
returns for a decrease in the variance of net returns.
Concerning within-season irrigation strategies, Boggess et al.
(1983) expressed the variance of net returns for a particular irrigation
strategy as
2 v2 2 d2 2 2 2 v2 2 0 to
\t Yi p + P Y. + T X. + Xi y 2oPY.,YX. (2'13)
K 1 1 1 1
where and ay are the mean and variance of yield associated with irrigation
1 2
strategy i, P and ap are the mean and variance of crop price, y and
are the mean and variance of irrigation pumping cost per unit of
o
water, X. and at are the mean and variance of irrigation water applied
1 *i
for irrigation strategy i, and Opy ^ is the covariance between PY^


23
and yXi Then the relative contribution of each component random
variable (price, yield, pumping cost, and irrigation water) to the
variance of it was analyzed by normalizing the above equation. Their
analysis indicated that irrigating soybeans increased the expected net
returns above variable costs and decreased the variability compared to
non-irrigated soybeans. Probability curve and convolution of risk
techniques were subsequently applied to quantify and interpret the risks
associated with alternative irrigation strategies.
Optimization Methods
Systems analysis basically is a problem-solving technique wherein
attempts are made to build a replica of a real world system or
situation, with the objective of experimenting with the replica to gain
some insight into the real world problem. It encompasses several
optimization techniques such as dynamic programming, linear programming
and simulation. Generally in dealing with irrigation management,
dynamic programming techniques are applied to models which are spatially
limited to a field of single crop and temporally to one growing season
(Hall and Butcher, 1968; Windsor and Chow, 1971; Dudley et al., 1971;
Howell et al., 1975; Bras and Cordova, 1981). Linear programming
algorithms on the other hand are utilized to analyze farm level cropping
patterns models (Windsor and Chow, 1971; Huang et al., 1975; Matanga and
Marino, 1979). Simulation is usually used to evaluate specific
irrigation policies (Ahmed et al., 1976; Jones and Smajstrla, 1979).
Dynamic programming models. Characteristically, dynamic
programming problems are decomposed into stages and decisions are
required at each stage. The decision at any stage transforms the system


24
States and increments the value of the objective function at a
particular stage. Changes in the system states may be described by a
probability distribution.
In the Howell et al. (1975) dynamic programming formulation, the
decision process consisted of whether to irrigate 0., 0.25, 0.5, 0.75,
or 1.0 times the potential ET during each of five crop growth stages for
grain sorghum. The states consisted of the remaining water to be
allocated at each stage and the soil water status, a stochastic state
variable. The stochastic state transitions were calculated by utilizing
simulation of a soil water balance model.
The solutions produced an optimal sequencing of water application
based on expected weather patterns and on differential crop
sensitivities to water deficits during each growth stage. The solutions
were tabulated. The table provided the stage-by-stage optimal policy.
As the season progressed, realizations of rainfall and ET caused the
soil water and the remaining water supply to vary from year to year.
Therefore, at each stage, the irrigator could update the optimal policy,
using the table to optimally allocate water during the remaining part of
the growing season.
Bras and Cordova (1981) attempted to solve the same problem by
using an analytical approach which included a physical model of a soil-
climate system and a stochastic decision-making algorithm. Expressions
for the soil water transition probabilities over a given time period and
the first two moments of associated actual evapotranspiration were
derived analytically. A stochastic dynamic programming algorithm was
then used to determine optimal control policies at each irrigation
decision point, conditional on the state of the system (soil water
content).


25
Dividing the irrigation season into N stages and taking irrigation
depth (In) at decision stage n as a decision variable, the objective
function (Bras and Cordova, 1981) can be formulated as:
*
B =
N I
Max E [ Z R n ] PC (2.14a)
Iett n=l n
where I =
1^ ^2 ** *
I
R n =
n
I I I
P Ynn 6 IDnn yCn (2.14b)

B =
maximum net return,
E[ ] =
expectation operator,
PC =
production costs different from irrigation costs,
TT -
feasible set of control policies,
\ -
type of control applied at decision stage n,
N =
R1"-
Kn
number of decision stages in the growing season,
net return by irrigating In at decision stage n,
P =
\n-
price per unit of crop yield,
contribution of irrigation decision i to actual yield,
3 =
!D!" -
unit cost of irrigation water,
depth of irrigation water associated with operation
policy In,
Y =
fixed cost of irrigation (labor cost), and
c1"
c"
= 0, when ID^ = 0;
n
= 1, otherwise.
1, otherwise


26
Since the production cost (excluding irrigation costs), PC, is a
constant value, the optimal control law that maximizes the above
function will be the same that
N rn
Max E [ l R n ] (2.15)
I sir n = l
The dynamic programming technique then decomposes this problem into
a sequence of simpler maximization problems which are solved over the
control space.
Linear programming models. If the objective is to select crops to
grow on a farm where water is limiting, linear programming techniques
may be applied. Windsor and Chow (1971) described a linear programming
model for selecting the area of land to allocate to each crop and the
irrigation intensity and type of irrigation system to select. As
defined, the set of decision variables, represented the number of
hectares of crop 1 to grow in field (or soil type) i, using irrigation
practice j, and irrigation system k. The solution would select X-j^ to
maximize net profit for the farmer. A required input was net profit
associated with X ^ C ^ which included a crop yield response to
various conditions. Windsor and Chow used dynamic programming to
estimate crop yield response for optimal unit area water allocation.
Their model is designed for decision analysis prior to planting.
Their model can also be modified to determine when to plant the crop to
take advantage of seasonal rainfall or water availabilities. The
within-season scheduling of irrigation on a farm basis (for multiple
fields) after crops are planted would require a different formulation.
Trava-Manzani11 a (1976) presented one example of such a problem.


27
In the study by Trava-Manzani11 a (1976), the objective was to
minimize irrigation labor costs in a multi-crop, multi-soil farm subject
to constraints on daily water availability, water requirement of the
crops and the irrigation method being used. The mathematical
formulation of the problem was of zero-one linear integer programming.
However, because of the nature of the problem formulation was then
transformed to a linear programming model. Two linear programming
techniques, Simplex procedure and the Dantzig-Wolfe decomposition
principle, were successfully used to resolve the solutions.
Simulation models. Simulation can be used to evaluate specific
irrigation policies in an enumerative search for the best policy among
those tested. For this approach, models of the soil water status and
crop yield responses are required (Ahmed et al., 1976; Jones and
Smajstrla, 1979). By defining several explicit, alternate policies and
simulating results for one or more crop seasons, crop yields or net
returns can be compared for the different policies and the best policy
can then be selected. This procedure will not necessarily produce an
optimal solution, but from a practical viewpoint, it can provide
valuable information to decision makers.
In many of the reported studies (Dudley et al., 1971; Yaron et al.,
1973; Minhas et al., 1974; Ahmed et al., 1976), the lack of suitable
crop response models was cited as a major limitation. It may not be
realistic to estimate crop yield response over a broad range of
conditions by empirical approach. Details are needed in the model.
Dynamic crop growth models were developed to predict growth and yield of
crops using more theoretical considerations and physiological detail
(Curry et al., 1975; Childs et al., 1977; Barfield et al., 1977;


28
Wilkerson et al., 1983). These models are attractive because crop
growth stresses, such as those caused by nutrition or pests, can be
included, in addition to those caused by water deficits, to provide a
more comprehensive tool for crop production management.
However, the crop growth models may have so much detail that they
may not be suitable for the problem of long-term production
management. Models at other levels of sophistication to describe crop
system responses to management practices, such as irrigation, are likely
to be more useful. Thus, a general framework for optimization of
multiple cropping systems using both optimization and simulation
concepts will be developed.


CHAPTER III
METHODOLOGY FOR OPTIMIZING MULTIPLE CROPPING SYSTEMS
Mathematical Model
Several alternative formulations of the multiple cropping problem
are studied with regard to their practicality for solutions. These are
reviewed, and the most suitable one is described in detail.
Integer Programming Model
Sequencing is concerned with determining the order in which a
number of jobs' are processed in a 'shop' so that a given objective
criterion is optimized (Taha, 1976). In the multiple cropping problem
the variable, t is defined and equal to one when crop i, variety
i j 111 ^
j, planted at t^ still grows in the field at time t^. Otherwise, it is
equal to zero. It is also assumed that the growth season for crop i,
variety j, planted at t^ is A^-t and the associated net return is
C.jj£ To properly describe the multiple cropping problem, two
constraints are considered: only one crop can occupy the field anytime,
and a growing season is continuous. Provided with the definition of
variables, X..
ijt1t2
, and constants A... and C,-,. the formulation of
ijtj ijt;
an objective function and constraint conditions is
29


30
Max 1 (Ci it } (Xi it t }
ij t x t g J 1 J 1 2
(3.1a)
s t # ^ i i T*
ijtlt2 ^ 2
X.. 1 for all tp ,
ijt1 1JV2
(3.1b)
(3.1c)
ti+Aijt
xiit t = 0 or
t2=t1 1Jtr2
11 "*A j t
1 x,'jtlt2 Autj- for a11 (3-ld)
where T2 is the total number of weeks of an N-year production horizon.
The first constraint (3.1b) simply says that a production horizon is of
T2 periods. The second constraint (3.1c) indicates that at any instant
of time t2 only one crop is scheduled to grow in the field. The
constraints represented by (3.Id) are imposed to ensure the continuity
of a growth season. However, these either-or constraints cannot be
implemented directly in a mathematical programming algorithm. To
overcome this difficulty, new variables, Y. are defined. When crop
i Jt
1
i, variety j, is scheduled for planting at t. then Y.. = 1.
1 1 Jtj
Otherwise, Y. .. = 0. This problem is then a zero-one integer
1 Jt^
programming model. The formulation is


31
Max
1 l (Cijt } (Yijt >
ijt1 1JC1
(3.2a)
s.t.
l Xi1t t T2
ijt1t2 1Jtlr2
(3.2b)
IX < 1 for all t
ijt1 1Jtlr2
(3.2c)
VAijt
tJt 1 (xiJtit2)(1 vijtl>= - for 311 '-i-H-
(3.2d)
tl+Aijt
tJt 'l (3.2e)
But several difficulties are associated with this formulation. It
is noted that the number of X variables in the formulation is equal to
(I J * T2) directly dependent on how often the decision needs
to be made. Assume that a decision is to be made every week. For a
4.5-year planning horizon, the total number of X variables is estimated
as 4 2 234 234 = 438,048. This cannot be solved economically by
the existing integer programming algorithm (Land and Powell, 1979).
Moreover, the nonlinear terms in the model should generally result in a
computationally difficult problem. Still, the need of constants,
C-.. and l\. requires the simulation of as many combinations of
ljtj
(i ,j ,tx). Because of all of these shortcomings, the integer programming
approach was not pursued further.


32
Dynamic Programming Model
Because of the nature of dynamic programming techniques which solve
a problem by sequential decision-making, the constraint of appearance of
a single crop in the field anytime is implicitly coupled in the
formulation. In a sense, sequential decision-making provides an
interactive mode in the process of solution. When it is required, net
return associated with a specific crop candidate is generated and then
evaluated. It is very beneficial in terms of storage and computer
time requirements.
In a crop production system, management practices consist of
irrigation strategy, fertilizer application, pest and disease control,
crop rotation, etc. Discrete values assigned to each level of a
specific management practice represent the state of a system. For
example, percentage of available water in the soil profile (soil water
content), is a primary indicator for irrigation management. Under an
unlimited water supply situation, without losing generality, (C,W,N) are
chosen as state variables to identify state transition in the
optimization model, where C stands for the preceding crop, W for soil
water content, and N for soil nutrient level.
The inclusion of nutrient level (N) in the formulation is to
express the potential application to other areas of interest.
Nonetheless, irrigation policy is solely emphasized in the iterative
functional equation, because this framework is to be demonstrated with
the application to irrigation management.
The dynamic programming model of multiple cropping is formed as
follows. First, the optimal value function F(C,W,N,t) is defined as


33
F(C,W,N,t) = maximum return obtainable for the remainder t periods,
starting with the current state (C,W,N). (3.3)
In terms of these symbols, Bellman's principle of optimality gives the
recurrence relation,
F(C, ,W.,N.,t) = Max
C2 S(C1 ,t)
R*(C2,I*,t) + F(C2,Wf,Nf,t-a(C2))
(3.4)
where
W.
i
N.
l
S(Cj,t)
a(C2)

I
*
R
state of soil water at the beginning of the season,
state of soil water at the end of the season,
value of nutrient level at the start of the season,
value of nutrient level at the end of the season,
preceeding crop,
selected crop, decision variable,
proper subset of crop candidates dependent on
and season t, due to practical considerations of crop
production system,
growth season of crop C^,
optimal realization of irrigation policies, a vector
1c -k
(I^, I2, ... Ik) represents the depths of irrigation
water associated with individual operations,
maximum return obtained from growing crop C2 by

applying optimal irrigation policy I .
The state transition from the start of a season to the end of a
season is determined by the system equations:


34
wf = g (c2, i*, w.),
Nf = h (C2, I*, W., N.).
(3.5a)
(3.5b)
These functions are not explicitly expressible. It is not
realistic to represent the complicated soi1-piant-atmosphere continuum
in terms of simple functional relationships. Simulation models may be
employed to carry out state transitions.
In order to use the iterative functional equation, it is necessary
to specify a set of boundary conditions to initialize the computational
procedure. Because the functional equation expresses the optimal value
function at t in terms of the optimal value function at (t a(C2)),
the boundary conditions must be specified at the final stage t = 0.
Formally, the appropriate boundary conditions are
F(C,W,N,t) = 0, when t = 0
(3.6)
F(C,W,N,t) = -~, when t < 0
for every C,W,N.
In addition, an optimal policy function, the rule that associates
the best first decision with each subprogram, is needed to recover the
optimal decision for the original whole problem. The optimal policy
function in the problem is defined as
(3.7)
where = soil water status at end of a season, = nutrient level at
end of a season, C2 index of the selected crop, a(C2) = growing season
of C2 .


35
Starting with the boundary conditions, the iterative functional
equation is used to determine concurrently the optimal value and policy
functions backward. When the optimal value and decision are known for
the initial condition, the solution is complete and the best cropping
sequence can be traced out using the optimal policy function. Namely,
the optimal solution is F (CQ, W NQ, T), where T = the span of N-year
growing period, (Cq, Wq, Nq) is the initial condition in which
production plan is to be projected.
However, it is not very clear whether certain states (C, W, N, t)
are relevant to the possible optimal system. Total enumerations of
optimal value functions F(C, W, N, t) are required to resolve the

optimal solution F (CQ, WQ, NQ, T). In terms of computational
efficiency, this dynamic programming model is not very appealing.
Therefore, a more comprehensive, efficient model needs to be
investigated.
Activity Network Model
Selecting crop sequences to optimize multiple cropping systems can
be formulated as an activity network model. In a network, a node stands
for an event or a decision point. An activity, represented by an arc,
transfers one node to another. In this particular application to
irrigation management, nodes represent discrete soil water contents at
every decision period. Arcs, not necessarily connecting with adjacent
nodes, have lengths that denote net returns associated with the choice
of crop and irrigation strategy. The structure of the network is
demonstrated in Figure 1, where C. is crop variety i and S. is
J
irrigation stragegy j. The S and T nodes are dummy nodes, representing
the source and terminal nodes of the network, respectively.


36
As noted in Figure 1, all arcs point in one direction from left to
right. There is no cycle in this network. This feature will prove
advantageous in developing a simplified algorithm for network
optimization. While circles are all potential decision nodes, solid
line ones are actual decision nodes which are generated by system
simulations, and dashed-line circles are fictitious, not accessible to
other nodes. In the dynamic programming model, these inaccessible nodes
are not detectable so that efforts on computing optimal values for
dashed-line nodes are wasted. In contrast, the inaccessible nodes are
detectable in the activity network model and more efficient computation
is accomplished. Under different weather conditions, networks of a
multiple cropping system vary.
In designing multiple cropping systems, several principles verified
by field experiments should be considered. These are: an idle period
may be required to restore the soil water reservoir, or to alleviate
pest population or chemical residues; consecutive scheduling of the same
crop may require more intensive management; and genetic traits may
prohibit planting certain varieties in some season of a year. Some of
these system criteria can be incorporated into simulation to generate a
multiple cropping network. Other aspects of the system (i.e. improper
consecutive scheduling of the same crop) restricted by model
representation may be reconsidered by a post-optimization scheme. In
such a manner, a more realistic system network is considered for
obtaining optimal crop scheduling.
The objective of optimizing multiple cropping systems is defined to
maximize total net return over a specified long-term period. In network
analysis terminology, it is to seek the 'longest path' of a network.


Soil Water State
Decision Time (weeks)
CO
figure 1. A system network for multiple cropping.


38
Since devaluation of cash value needs to be taken into consideration in
a long-term production horizon, total discounted net return of future
profits is to be maximized in the study.
A longest path solution algorithm can be expanded to search for the
K longest paths from the start node to terminal node. Determining the K
longest paths provides useful information for system analysis. The
advantages are as follows: First, such information provides a means of
assessing the sensitivity of the optimal solution to possible suboptimal
decisions. Second, one may be interested in a class of solutions and
not just in a single solution. Third, the K longest paths provide a
measure of the robustness of the underlying model when the data are
approximate. Moreover, in case post optimization analyses are necessary
to impose additional constraints on good solution paths in a system
network, calculation of the K longest paths provides a means of
efficient computation.
As described, an arc length in a multiple cropping system network
represents the return resulting from an optimal, single crop production
season. This represents a second-level optimization problem, which is
referred to as within-season management, i.e. optimal irrigation
scheduling. The problem of temporal water allocation in an irrigated
field consists of deciding when and how much water to apply in order to
maximize net returns. This problem is complicated by the uncertainty of
weather and by the fact that many crops exhibit critical growth stages
during which the crop sensitivity to soil water stress is high.
Systems analysis techniques such as simulation and dynamic
programming have been used in the past to determine the optimal
operation policies in an irrigation system. The necessity of


39
implementation of more dynamic, detailed crop phenology and growth/yield
models makes mathematical programming impractical. Simulation therefore
is required to evaluate within-season management strategies. As a
result, the activity network model coupled with the simulation-
optimization techniques provides a framework for optimizing multiple
cropping systems by selecting crop sequence and determining optimal
within-season management practices.
Thus, methodology is developed and summarized as follows:
1. To provide base data, models for simulating crop growth and
yield are constructed.
2. Considering systems options and constraints, a realistic
multiple cropping network is generated.
3. Applying the longest path algorithm, the K longest paths are
solved to evaluate various cropping sequences.
Crop-Soil Simulation Model
The crop-soil simulation model serves two purposes in optimizing
multiple cropping systems. First, the simulation is necessary to define
the state transitions (i.e. soil water contents) in the previously
discussed mathematical model. Secondly, simulation is an approach to
study irrigation management strategies. The problem of optimally
distributing irrigation water over the growing season is difficult
primarily because of imperfect knowledge of rainfall distribution over
the season. In addition, uncertainty in the distribution of other
weather variables which affect crop yields complicates the optimization
problem.


40
In general, uncertainty in the time distribution of inputs or
resources to a process which is to be optimized can be treated using
some form of stochastic programming, the inputs as random variables, and
the objective function to be optimized as some fairly simple production
function of inputs. Unfortunately, the complex nature of crop
production lends itself to simple production functions only in a general
statistical sense. In order to investigate the effects of irrigation
decisions at different points within the growing season, a detailed
simulation model is useful.
Such a simulation model is intended to integrate the effects of
weather variables and irrigation schedules on crop growth. It simulates
the progress of a crop during the time in which it interacts with its
environment. As the crop grows from day to day and uses the water
stored in the root zone, water deficits develop and are counter-balanced
by irrigation or rainfall. This closed loop simulation describes the
frequency and duration of water deficits that affect evapotranspirati on
and crop yield. By imposing a series of alternate irrigation strategies
on the simulation model, one can evaluate the effect on yield of various
strategies. To find the optimal solution, ranking the estimated net
return gives the most efficient strategy for a given specific weather
pattern.
As discussed by Jones and Smajstrla (1979), simulation models at
different levels of sophistication have been developed to study the
problem. In this work, a crop yield response model is included with the
soil water balance model so that irrigation strategy for maximizing net
return can be studied. The soil water balance model is primarily used
to provide the necessary data (daily ET) for describing the yield


41
response of the crop by the yield model. In addition, a crop phenology
model is coupled to systematically predict growth stage relative to
physiological development of the crop. In so doing, different levels of
water use of the crop at various growth stages can be realistically
simulated, and more accurate estimation of yield is possible. These
models are described in detail below.
Crop Phenology Model
Corn and peanut phenology. For corn and peanut, heat units are
used to predict physiological development. In the model, the
physiological day approach, a modification of the degree-day method is
used. Because the units of degree-day are products of temperature and
time, it is convenient to express the accumulation in units of
physiological time. To accomplish this, the degree-day unit is
normalized with respect to a given temperature, 30 C? The
physiological days are calculated as follows.
PD = 0
for T < 7,
n T(At ) 7
PD = l 2 At_-
30 7
for 7 < T < 30,
n
PD = l
i =1
45 -T(At.)
45 30
At.
1
for 30 < T < 45,
(3.8)
PD = 0
for T > 45,


42
where PD = physiological clay, T(At.j) = temperature in the time
interval At. Physiological days accumulate until specific thresholds
are reached. Stages occur at the thresholds the stages are said to be
set. In this study, the crop season is divided into four stages. For
corn and peanut, stages of growth and threshold values of physiological
development are shown in Table 1.
Wheat phenology. For wheat, four stages, planting to late
tillering, late tillering to booting, heading to flowering, and grain
filling are used to characterize the wheat life cycle. Time between
phenological stages is predicted by using the Robertson model (1968).
The approach uses the multiplicative effects of temperature and
daylength to determine time between events. In the model, the average
daily rate AX of development is calculated as
AX = (a1(L-aQ) + a2(L-aQ)2) (b1(T1-bQ) + ^(T^)2 +
b3(Vbo} + b4(Vbo)2) (3*9)
where L = daily photoperiod,
Tj = daily maximum (daytime) temperature,
T? = daily minimum (nighttime) temperature.
And aQ, a^, a^, bQ, b^, etc. are characteristic coefficients of specific
stages. Values of these coefficients are shown in Table 2. A new
stage (S2) is initiated when the summation
S2
XM = l AX = 1 .
(3.10)


43
Table 1. Threshold values for physiological stages of growth of corn and
peanut.
Threshold Values of
Phenological Development
Crop
Stage of Growth
(Physiological Days)
Source
Full
Planting to silking
38.7
Bennett
season
Si 1 king to blister
45.1
(personal
corn
B1ister to early
66.3
communi-
soft dough
Early soft dough to
81.4
cation)
maturity
Short
Planting to silking
33.6
Agronomy
season
Si 1king to blister
40.7
Facts, 1983
corn
Blister to early
57.7
soft dough
Early soft dough to
70.4
maturity
Peanut
Planting to beginning
27.3
Boote,
flowering
Beginning flowering to
42.4
1982
a full pod set
A ful 1 pod set to
68.2
beginning maturity
Beginning maturity to
97.3
harvest maturity


Table 2. Coefficients of a multiplicative model for predicting wheat phenological stages
(Robertson, 1968).
Coefficients
Phenolog
ical Stage
Planting to
Emergence
Emergence to
Late Tillering
Late Tillering
to Booting
Heading to
FIowering
Grain
Fi 11ing
a0
*
8.413
10.93
10.94
24.38
al
*
1.005
0.9256
1.389
-1.140
a2
**
0.0
-0.06025
-0.08191
0.0
b0
44.37
43.64
42.65
42.18
37.67
bl
0.01086
0.003512
0.002958
0.0002458
0.00006733
b2
-0.000223
-0.00000503
0.0
0.0
0.0
b3
0.009732
0.0003666
0.003943
0.0003109
0.00003442
b4
-0.000227
-0.00000428
0.0
0.0
0.0
** In this early stage, growth is independent of daily photoperiod.


45
The summation (XM) is carried out daily from one phenological stage
to another S^.
Primarily, five growth stages and centigrade temperatures were used
in the Robertson model. Modification by combining stages 1 and 2 into a
single stage has been made to accommodate to the study.
Soybean phenology. The model of soybean phenology, developed by
Mishoe et al. (in press) and implemented by Wilkerson et al. (1985) is
complicated. A version of the model was adapted for the study. The
model uses cultivar specific parameters, night length, and temperatures
to generate physiological development. The development phases of
soybean are described in Table 3. Some phases of development are
dependent on night length and temperature whereas others are dependent
only on temperature.
Temperature effect on development is expressed as physiological
time. Physiological time is calculated as the cumulative sum of rates
of development, starting at the beginning of a phase. The end of a
phase occurs when the cumulative physiological time reaches the
threshold as indicated in Table 3.
A nighttime accumulator is used to represent photoperiod effects on
development. The nighttime accumulator of the model is represented as
fol1ows:
Xm = I TF NTA (3.11)
where X = the accumulator value to trigger an event,
TF = temperature factor computed using the function shown
in Figure 2,
NTA = night time accumulator function shown in Figure 3.


46
Table 3. Description and threshold values of phenological stages and
phases for soybean cultivars (Wilkerson et al., 1985).
Threshold
Growth
Stage
Description
Phase
'Bragg'
'Wayne'
I
Physiological time from planting
to emergence
1
6.522
6.522
Physiological time from planting
to uni foliate
2
10.87
10.87
Physiological time from unifoliate
to the end of juvenile phase
3
2.40
2.40
Photoperiod accumulator from the
end of juvenile phase to floral
induction
4
1.00
1.00
II
Physiological time from floral
induction to flower appearance
5
9.48
9.48
Photoperiod accumulator from
flowering to first pod set
6
0.14
0.20
Photoperiod accumulator from
flowering to R-4
7
3.0
6.0
III
Photoperiod accumulator from
flowering to the last V-stage
8
0.16
0.5
Photoperiod accumulator from
flowering to the last possible
flowering date
9
0.575
0.6
IV
Photoperiod accumulator from
flowering to R-7
10
20.35
14.5
Physiological time from R-7
to R-8
11
12.13
10.0


Physio. Days for Floral Induction ^ Physio. Day / Real Day
47
2. Rate of development of soybean as a function of temperature.
Figure 3. Effect of night length on the rate of soybean development.


43
Figure 2 is the normalized function to calculate physiological time. In
Figure 2, data used for minimum, optimal and maximum temperatures were
7, 30, 45C, respectively. Figure 3 shows the relationship between
night length and physiological days to development based on phase 4, the
floral induction phase. Because the threshold for development for phase
4 was defined to be a constant (1.0), the relationship varied with
cultivars in Figure 3. The values for this relationship of 'Bragg1 and
'Wayne' soybean shown in Table 4 were taken from Wilkerson et al.
(1985). Based on these calibrated curves, thresholds (Table 3) for
other photoperiod phases also vary among cultivars.
The amount of development during one night is calculated by
multiplying the average temperature for the nighttime by the inverse of
days to development at a given night length. The function (equation
3.10) is accumulated using a daily time step. When the prescribed
threshold is reached, the event is triggered and the crop passes into
the next stage.
Crop Yield Response Model
Crop growth is closely correlated to evapotranspi rati on (ET).
Based on this principle, yield response models which interpret ET
reduction below potential levels as integrators of the effects of
climatic conditions and soil water status on grain yield were
developed. To account for increased sensitivity to water stress at
various stages of growth, and the interactive effects between crop
growth stages, the Jensen (1968) multiplicative form
Â¥/vp :1 >3 (3.12)


49
is used, where potential yield, Y is varied as a function of planting
r
dates. Maximum yield factors that reduce yield of each crop below its
maximum value as a function of planting day for wel1-irrigated
conditions are shown in Figure 4. The length of each stage is predicted
by the use of crop phenology model.
To obtain crop sensitivity factors () to water stress, intensive
literature studies have been made. Boggess et al. (1981), based on many
simulations from SOYGRO were able to quantify these factors (shown in
Table 5) by statistical analysis. Smajstrla et al. (1982) also
estimated A., for soybean in a lysimeter study, and their estimates
of A., were similar to those found by Boggess et al. (1981). For corn
and peanut, attempts have been made without success to obtain the
factors from a series of experimental studies (Hammond, 1981). The
factors used in Table 5 were derived from FAO publication (Doorenbos,
1979). For wheat, no data were available for Florida conditions.
Therefore, an experiment on wheat to be described in the later chapter
was performed to obtain A^ and related crop response to irrigation
practices.
Soil Water Balance Model
In order to predict ET rates under well-watered and water stressed
conditions, a soil water balance model was developed to integrate
existing knowledge about crop water use, weather patterns, and soil
properties into a framework compatible with irrigation objectives. A
model previously described by Swaney et al. (1983) was adapted for this
study.


50
Table 4. Values of the parameters for the nighttime accumulator
function of the soybean phenology model (Wilkerson et al.,
1985).
Value of The
Parameters
Name of
Parameters
'Bragg'
'Wayne'
THVAR (day)
63.0
32.0
DHVAR (day)
2.0
2.0
TNLG1 (hour)
5.2
5.2
TNLGO (hour)
11.0
9.5
Table 5. Crop sensitivity factors, for use in the simulation.
Crop Sensitivity Factors
Crop
Stage 1
Stage 2
Stage 3
Stage 4
Source
Corn
0.371
2.021
1.992
0.475
Ooorenbos (1979)
Soybean
0.698
0.961
1.034
0.690
Boggess et al. (1981)
Peanut
0.578
1.032
1.531
0.772
Ooorenbos (1979)
Wheat
0.065
0.410
0.114
0.026
Personal observation


Yield Factors Yield Factors
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
Figure 4
Julian Day
Maximum yield factors that reduce yield of each crop below its maximum value
as a function of planting day for well-irrigated conditions.


52
The soil water balance model divides the soil into two zones: an
evaporation zone in the uppermost 10 cm of the soil, and a root zone of
variable depth underneath. This shallow evaporation zone is selected
for the sandy soil used in the model, and would not be sufficient for
heavier soils. Root zone depth is increased during the season by
simulating root growth. Under wel1-irrigated conditions, rooting depth
of the crop as a function of time is shown in Figure 5. The soil used
is characterized by its field capacity and permanent wilting point.
Evaporative water loss is removed from the evaporation zone and
transpiration water is lost from both zones depending on their
respective water contents. Due to the high infiltration rates of the
sandy soil, all rainfall is added to the profile until field capacity is
reached, and excess water is assumed to drain from the profile. When
the fraction of available soil water reaches a critical level of a pre
determined irrigation strategy, irrigation water is applied and treated
as rainfall. If both rainfall and irrigation occur on the same day, the
effect is additive.
The soil water balance model requires daily rainfall and potential
evapotranspiration (ETp), which is estimated by a modified version of
the Penman equation. The ETp is then used to calculate potential
transpiration (T ) using a function of leaf area index (Ritchie, 1972):
T = 0
P
T = ET (0.7 *
p p^
L < 0.1
ai
(3.13a)
0.1 < L < 3.0 (3.13b)
al
3.0 < Lai
(3.13c)


Rooting Depth (cm) Rooting Depth (cm)
Figure 5. Crop rooting depth after planting under well-irrigated conditions.


54
where L = leaf area index. For wel 1-i rrigated crops, leaf area index
31
functions as seasons progress are shown in Figure 6.
Values of actual evaporation (E) and transpiration (T) limited by
available water in the two soil zones are calculated from potential
values using time from the last rainfall in the case of E, and a soil
water stress threshold (0C) in the case of T. Calculation of
transpiration is as follows:
T = T ' > Q (3.14a)
p c
T = Tp (0'/0c) e' < (3.14b)
I
where 0 = ratio of soil water in root zone, as a fraction of field
I
capacity, 0 = (0p- 0rf) / (0fc- 0d),
I I
0 = critical value of 0 below which water stress occurs and
transpiration is reduced, various values are used for
different growth stages and crops,
0 = volumetric water content of root zone,
r
0^ = lower limit of volumetric water content for plant growth,
0^c = field capacity of the soil.
Two stages of evaporation from soil are implemented. In the
constant rate stage (immediately following rainfall event or
irrigation), the soil is sufficiently wet for the water to be evaporated
at a rate
E = Min E W )
^ p eJ
(3.15)


Leaf Area Index Leaf Area Index
Figure 6. Leaf area index for wel1-irrigated crops as a function of time.


56
evaporation below the canopy. In the falling rate stage (stage 2),
evaporation is more dependent on the hydraulic properties of soil and
less dependent on the available atmosphere energy. For each subsequent
day, the daily evaporation rate is obtained by (Ritchie, 1972)
E = Min {
(3.16)
where a is a constant dependent on soil hydraulic properties. For sandy
soil, a = 0.334 cm day" ^
For practical application, the Penman equation is considered the
most accurate method available for estimating daily ET. The Penman
formula for potential evapotranspi rati on is based on four major climatic
factors: net radiation, air temperature, wind speed, and vapor pressure
deficit. As summarized by Jones et al. (1984), the potential ET for
each day can be expressed as
ET
ARnA + yEa
(3.17)
P
A + y
where ETp = daily potential evapotranspirati on, mm/day
a = slope of saturated vapor pressure curve of air, mb/C
2
Rn = net radiation, cal/cm day
X = latent heat of vaporization of water, 59.59-0.055 T
avg
Ea = 0.263(ea ed) (0.5 + 0.0062u2)
ea = vapor pressure of air = (emax + emin) / 2, mb


57
ed = vapor pressure at dewpoint temperature Td
(for practical purposes = T ^ ),
u^ = wind speed at a height of 2 meters, Km/day
y = psychroinetric constant = 0.66 mb/C
emax= maximurn vapor pressure of air during a day, mb
em^n= minimum vapor pressure of air during a day, mb.
Saturated air vapor pressure as a function of air temperature,

e (T), and the slope of the saturated vapor pressure-temperature
function, A are computed as follows:
e*(T) = 33.8639{(.00738T + .8072)8 .000019(1.8T + 48) + .001316}
(3.18)
A = 33.8639{0.05904(0.00738T + 0.8072)7 0.0000342} (3.19)
In general, net radiation values are not available and must be
estimated from total incoming solar radiation, R$ and the outgoing
thermal long wave radiation, R^. Penman (1948) proposed a relationship
of the form
Rn = (1-ct) Rs Rb (3.20)
where Rn = net radiation in cal/cm day,
?
Rs = total incoming solar radiation, cal/cm day
R^ = net outgoing thermal long wave radiation,
a = albedo or reflectivity of surface for R^.


58
Albedo value a is calculated for a developing canopy on the basis of the
leaf area index, Lai, from an empirical equation (Ritchie, 1972),
a = a + 0.25 (a a ) L (3.21)
s v s' ai
where a$ is average albedo for bare soil and a for a full canopy is a.
And an estimate of Rb is found by the relationship:
R, = ctT4(0.56 0.08/e ,) (1.42R /R 0.42) (3.22)
b d s so
-ft ? n
where o = Stefan-Boltzmann constant (11.71*10 cal/cm day/ K),
T = average air temperature inK (C + 273),
R = total daily cloudless sky radiation,
so J J
Values of R are available from weather stations in Florida. Clear-sky
s
insolation (R ) at the surface of the earth though needs to be
estimated. The equation (3.16), along with the discussed procedures for
estimating variables, is then used to calculate potential ET from a
vegetated surface.
The calculation of potential evaporation below the canopy, is
essential to predict soil evaporation when the surface is freely
evaporating. Proposed by Ritchie (1972), Fp is calculated as follows:
Ep = (A/(A + Y))Rn (3.23)
where Rn is net radiation at soil surface.


59
Irrigation Strategy
In order to study irrigation decisions, irrigation options input by
the user are available to the simulation model. The irrigation
strategies take the following form. The grower will irrigate on any day
of the season, if the water content in the root zone of the soil is
depleted to the threshold value (70% of availability by volume)
specified by the strategy. If the condition is met, irrigation water is
applied in an amount specified by the user. Frequent irrigation
applying less water per application (1 cm) is used in the model. On the
other hand, the rain-fed strategy depends totally upon rainfall.
Model Implementation
In order to study multiple cropping systems as well as associated
management strategies, models are needed to summarize and operationalize
knowledge about plant growth, yield, weather patterns, soil properties
and economics into a framework compatible with system objectives.
Therefore, computer programs were written in FORTRAN 77 to evaluate the
methodology. Figure 7 shows a schematic diagram for the methodology.
As outlined in the previous section of mathematical model, in order to
optimize multiple cropping systems, three independent steps, system
description, generation of network, and network optimization are
essential. Detailed descriptions and source code of subroutines to
execute the methodology are given in appendix A. The purpose of this
section is to provide discussions on model implementation in general.


60
Figure 7
A schematic diagram for optimal sequencing of multiple
cropping systems.


61
Network Generation Procedures
As discussed, nodes of a system network are specified by their time
coordinate and their system states. In this particular application, it
is proper to use a weekly decision interval. For limited water
retaining capacity of sandy soil, soil water contents as state variable
are discretized by an 1% interval between field capacity (10%) and
permanent wilting point (5%). Hence, there are a total of 6 states of
the system.
Net profit is gross receipts from crop sale minus total variable
cost. The variable cost for crop production is calculated by the
collective cost of production excluding irrigation plus variable cost of
seasonal irrigation. In planning of longterm production, devaluation of
cash value needs to be taken into consideration. Assume current
depreciation rate is i (12%). Present value of a future sum (F) is
calculated as
P = F / (1 + i )n (3.24)
where n is the year when F occurs. When F will be the net return of
future crop production, P is then the discounted net return evaluated at
the planning time.
In order to have a multiple cropping system network, simulation
techniques are applied. The tasks of these simulations are to keep
track of soil water status daily in order to be compatible with
irrigation objectives, to project the next crop and its planting date
(new nodes), and to estimate returns (arc lengths) related to the
decisions.


62
In the simulation, a crop season includes a one-week period to
allow for land preparation, and one week to allow for the harvesting
operation. Once a crop and irrigation strategy are decided, phenology
and soil water balance models are used every day to simulate the states
of the system. After all simulations of one single season for different
crops and irrigation strategies are performed, several new nodes for the
next crop are generated and new arcs are extended. In simulations, the
limitations on planting seasons of specific crops are shown in Figure 8
(personal communication with extension agent, D.L. Wright). Yield also
depends on the time during each interval when planting occurs.
In the process of optimizing a network, it is advantageous to have
a network whose nodes are sequentially numbered from a source node to a
terminal node. Since a straightforward simulation procedure does not
result in such a sequentially ordered network. It is necessary to re
number a currently existing network when expanding the network by
extending arcs from the presently considered node to new nodes generated
by simulations. Therefore, a procedure composed of appending, inserting
and re-numbering nodes are required in order to have an ordered network.
At each node (present planting day), a combination process of
simulation and re-numbering is performed. The process continues to
expand a network until the end of a planning horizon. As a result, a
multiple cropping system network whose nodes are sequentially numbered
is generated and ready for optimization.
Network Optimization
The optimization algorithm to seek K longest, distinct path lengths
of a network of multiple cropping system is discussed herein. For


Cal endar
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Full-season corn
Short-season corn
I
45 130
!" I
45 148
Late-maturing soybean
(Bragg)
H
101
H
195
Early-maturing soybean ^
(Wayne) 70
H
220
Virginia-type peanut

82
Wheat (Florida 301)
i 1
281 365
Figure 8. Time intervals during which each crop can be planted.


64
computing the longest path, the 1abel-correcting method is a fundamental
algorithm. This algorithm requires that the network contains no self
loops and all circuits in the network are of positive lengths. The
algorithm, coded by Shier (1974) was actually used in this study.
Suppose that the K longest path lengths from source node (node 1)
to all nodes i of an n-node network are required. Then a typical label-
correcting algorithm proceeds according to the following three steps:
LC1. Start with an initial (lower bound) approximation to the
required K longest path lengths from the source node (node 1) to each
node i. That is, assign a K-vector XV (i) = (XV..^, XV..^, ,XViK) to
every node i, where the entries of XV(i) are listed in decreasing order.
LC2. Select a new arc and then 'process' the arc. By processing
an arc (l,i) whose length is A-^ this means that current K-vector for
node i will be improved if possible by means of a path to node i which
extends first to node 1 and which then uses the arc (l,i). More
precisely, if any of the quantities (XV-|m + A-^: m = 1, ... ,K) provides
a longer path length than any one of the tentative K longest path
lengths in XV(i), then the current K-vector XV(i) is updated by
inclusion of this longer path length. It is to be understood that all
such possible updatings of XV(i) using XV(1) are performed when
processing arc (1 ,i).
LC3. Check the termination criterion. If satisfied stop.
Otherwise, return to step LC2.
The method for processing the arcs of the network is in a fixed
order: namely, in increasing order by the ending node of each arc.
Thus, arcs incident to node 1 are processed before those incident to
node 2, and so forth. If at some stage a node contains the approximate


65
lower-bound label (-,->, ... then no improvements can result by
using such a label. It is useful to group the arcs by their ending
node. Accordingly, we shall examine nodes in the fixed order 1, 2, ...
,n and shall skip the examination of a node if its label is
(-,-, ... ,-). Here the examination of a node simply entails the
processing of all arcs incident to that node. Finally, the method will
terminate when after examining all nodes 1, 2, ... n, it is found that
none of the components of the current K-vectors have changed from their
previous levels.
The labeling algorithm starts with the root (source node) having
label zero and all other nodes having negative infinite label (INF).
Then it enters a loop to update the label for each node i.
At any step of the process, the K-vector (XV(i)), associated with
each node i will contain the l< longest path lengths found so far from
source node to the node. Moreover, these K path lengths are always
distinct (apart from negative infinite values) and are always arranged
in strictly decreasing order. Such an ordering allows the following two
computationally important observations to be made.
(1) If the value INF is encountered in some component of a K-
vector, then all subsequent components of the K-vector also contain INF
values. Therefore, when updating the K-vector for node i, the K-vector
for a node 1 incident to i need only be scanned as far as the first
occurrence of an INF value since an infinite value cannot possibly yield
an improved path length for node i.
(2) If (IXV), the sum of some current path length in the K-vector
for node 1 and the arc length A. _¡, is less than or equal to the minimum
element of the K-vector for node i, then no improvement in the latter K-


66
vector by use of the former can possibly be made. Therefore, it is
appropriate to keep track of the current minimum element (MIN) of the K-
vector for node i. If IXV is greater than MIN, then it is possible for
an improvement to be made, as long as the value IXV does not already
occur in the K-vector for node i (only distinct path lengths are
retained).
As compared to the use of some general sorting routines to find the
K longest elements in a list, the use of these two observations allows
for a subtantial reduction in the amount of computational effort
required to update the current path lengths. When all nodes have been
labeled, the K longest path lengths to each node i in the network are
found. From such path length information, the actual paths
corresponding to any of the K longest path lengths are determined by a
backward path tracing procedure.
The optimal paths joining various pairs of nodes can be
reconstructed if an optimal policy table (a table indicating the node
from which each permanently labeled node was labeled) is recorded.
Alternatively, no policy table needs to be constructed, since it can
always be determined from the final node labels by ascertaining which
nodes have labels that differ by exactly the length of the connecting
arc.
In essence, this latter path tracing procedure is based on the
following fact. Namely, if a t-th longest path tt of length 1 from node
i to node j passes through node r, then the subpath of it extending from
node i to node r is a q-th longest path for some q, 1 < q < t. This
fact can be used to determine the penultimate node r on a t-th longest
path of known length 1 from node i to node j. Indeed, any such node r


67
can be found by forming the quantity (1 1 ,) for all nodes r incident
1 J
to node j and determining if this quantity appears as a q-th longest
path length (q < t) for node r. If so, then there is a t-th longest
path of length 1 whose final arc is (r,j); otherwise, no such a path
exists. This idea is repeatedly used, in the manner of a backtrack
procedure, to produce all paths from i to j with the length 1, and
ultimately all the K longest paths from node i to j.
Parameters and Variables
The hypothetical farm is located at Gainesville, Florida. The
field is of an unit area (1 hectare) and of deep, well-drained sandy
soil which is characterized as having a field capacity at 10% by volume
and a wilting point at 5%. More specific information about the farm is
discussed as follows.
Data bases contain three separate files. Weather data files in
standard format contain historical, daily values of important weather
variables collected from an USDA class A weather station at the Agronomy
Farm, Gainesville, Florida. Available data are from the years 1954-1971
and 1978-1984. The daily weather information which is needed to run
simulations consisted of Julian day of year (JULIN), maximum temperature
in C (TMAX), minimum temperature (TMIN), sunrise, hour a.m. (SNUP),
sunset, hour p.m. (SNDN), total solar radiation, langleys (XLANG), wind,
miles/day (WIND), and rainfall, inches/day (RAIN).
Cultivar and crop parameters are given in the text. These data are
in the file named 'GROWS' and shown in Appendix D. Values for two
cultivars (Bragg MG VII and Wayne MG III) of soybean were obtained from
the model S0YGR0 V5.0 (Wilkerson et al., 1985). Data for use in this
study were the result of simulating a wel1-irrigated field in 1982.


68
Parameters for corn cultivars were based on experiments in 1980-
1982 in which corn hybrid response to water stresses were studied
(Bennett and Hammond, 1983; Loren, 1983; Hammond, 1981). Some of the
observations included were physiological and morphological development.
Data for peanut were obtained from a study by McGraw (1979). For wheat,
experimental results in this study were used. Leaf area index and
rooting depth of wheat, not available from the experiment were from
Hodges and Kanemasu (1977).
The other file 'FACTS' shown in Appendix E provides specific
information about model operation, crop production system and economical
consideration. To initiate model execution, the user first provides the
first decision day (IDDEC), initial soil water content (MOIST), number
of crop price schemes (MXRUN) and number of crop cultivars (MXCRP) to be
considered in multiple cropping system. Also required are source node
(NS) and number of optimal cropping sequences (KL) searched.
Variables contained in the rest of the file are mainly relevant to
system evaluation and design. Primary variables of a multiple cropping
system are concerned with within-season irrigation management. These
include irrigation system used (IRSYS), application rate by a strategy
(RATE) and energy costs (GASPC, DSLPC, WAGE). For this study, a low-
pressure center pivot system was selected. It was assumed that with a
return time of one day the system was technologically capable of
achieving an application rate as desired by the user. In addition to
irrigation, idle periods (LIDLE) between two-crop seasons are also
specified by users.
From a computerized crop budget generator (Melton, 1980), the
collective costs of production for various crops were obtained.


69
Equations of variable irrigation costs of different systems used i
study were obtained from D'Almeda (personal communication). By
regressing results which were obtained from the irrigation cost
simulator (D'Almeda et al., 1982), he developed the equations for
typical North Florida conditions. The other economical component
interest is crop price (PRICE), $/kg. Current market prices (May,
were provided as baseline data.
n the
of
1985)


CHAPTER IV
WHEAT EXPERIMENTS
Introduction
Wheat (Triticum aestivum L.) is an important crop in the multiple
cropping minimum tillage systems widely used in the Southeast USA. In
this system, wheat is usually planted in the fall after soybean
harvest. Despite the need for intensive management, wheat can be grown
sucessfully in Florida and can make a significant contribution to
Florida agriculture (Barnett and Luke, 1980).
In Florida, agriculture depends mostly upon rainfall for crop
production and irrigation is needed during relatively short but numerous
droughts. However, uneven rainfall distribution patterns coupled with
sandy soils which have limited water storage capacities and
characteristically restricted root zones thus create problems in the
scheduling of irrigation. Therefore, the need for new information on
timing, application intensity, method of application, and amounts of
water applied exists for the region to grow wheat.
Crop growth is influenced by the process of evapotranspiration.
Evapotranspiration (ET) is the combination of two processes: evaporation
and transpiration. Evaporation is the direct vaporization of water from
a free water surface, such as a lake or any wet or moist surface.
Transpiration is the flow of water vapor from the interior of the plant
to the atmosphere.
70


71
As water transpires from the leaves, the plant absorbs water from
the bulk soil through its root system and transports it to the leaves to
replace water transpired. Under well-watered conditions, the plants
usually absorb enough water through their root systems to maintain
transpiration rates at the potential rate, determined by the
environment. However, as the soil around the root system dries, the
ability of the soil to conduct water to the roots decreases and plants
can no longer supply water fast enough to maintain the potential rate.
In order to prevent leaf desiccation, the plant has a feedback control
system that causes stomatal closure, thereby decreasing actual
transpiration below the potential rate.
To study the problem of how to best allocate water over the crop
production season, it is essential to understand and quantify the crop
response to water stress throughout the irrigation seasons. Yield
relationships have long been investigated. Many researchers have shown
that crop dry matter production is directly related to water use by the
crop throughout its growth cycle (del-lit, 1958; Arkley, 1963; Hanks et
al., 1969). The results demonstrate the important fact that a reduction
in transpirational water use below the potential rate results in a
concomitant decrease in crop biomass yield. Tanner (1981), and Tanner
and Sinclair (1983) further concluded that diffusion of CO^ into the
stomata and loss of water vapor from the stomata was the coupling
mechanism between biomass yield (Y) and evapotranspiration. Hence,
knowledge of this ET-Y relationship is fundamental in evaluating
strategies of irrigation management (Bras and Cordova, 1981; Martin et
al., 1983.)


72
Because it is observed that interactive effects between crop growth
stages existed (i.e. reduced vegetative growth during early stages
caused a reduction in photosynthetic material for fruit production at
the later stage), it is necessary to investigate the critical stage
whose sensitivity factor to water stress is high. Peterson (1965)
defined important stages of the wheat life cycle as emergence,
tillering, stem extension, heading, spike development, grain setting,
and grain filling and ripening. Studies of the effects of accurately
defined levels of water stress on wheat growth at various stages of
development were conducted by Robins and Domingo (1962), Day and Intalap
(1970), Musick and Dusek (1980). Commonly, the three stages of plant
development selected for irrigation were late tillering to booting,
heading and flowering, and grain filling. Most of the researchers
agreed that the most critical period of grain wheat for adequate soil
water was from early heading through early grain filling.
The purpose of this study was to develop ET-Y functions to provide
base data for improving wheat water management practices in Florida.
The specific objectives of this work are: (1) to quantify the nature of
ET-Y relationship for wheat crop in Northern Florida, (2) to determine
the effects of timing and intensity of water deficits on wheat yield,
and (3) to parameterize the crop sensitivity factors to water stress.
Experimental Procedures
Experimental Design
This study was conducted in 24 lysimeters at the Irrigation Park,
University of Florida at Gainesville. The lysimeter installation was


73
described by Smajstrla et al. (1982). The lysimeters were cylindrical
steel tanks with 2.0 meter square surface areas and 1.85 meter depth
filled with an Arredondo fine sand soil taken from the site of the
lysimeters. Automatically movable rainfall shelters were provided to
eliminate the direct applications of rainfall on crops during the water
management studies. Preplanting preparation included cultivation with a
rotor-tiller and irrigation with sprinkler heads to prepare a semi
smooth surface and granulate subsurface soil.
Planting of "Florida 301" winter wheat in the lysimeters was on 29
November 1983 in 20-cm rows at a seeding rate of approximate 135
kg/ha. Seeds were manually drilled and covered lightly with soil.
Fertilizer was applied at a rate of about 90-18-18 kg/ha (Nitrogen-
Sulfur-Potash) in the lysimeters. One half of this amount was applied
by hand at planting and the other half in late January. Unusual
freezing weather on 26 December 1983 destroyed most of the seedlings in
the lysimeters. Transplanting of young plants from buffer areas on 11
January 1984 made the intended study continuous. Attempts were made to
maintain uniform plant densities in lysimeters, however in some cases
uniformity problems did exist.
The crop growth season was partitioned as emergence to late
tillering, late tillering to booting, heading and flowering, and grain
filling stages. The study involved 8 treatments (4 crop stages of
stress 2 levels of stress), and each treatment was replicated three
times in three lysimeters. Treatments were labeled as double-index
(S,L), where S indicated stress stage and L period (weeks) of stress.
In treatment (N,N), the control, the soil water at the top 50-cm depth
was maintained at field capacity (11 percent volumetric water content)


74
throughout the growth season. There were two treatments (N,N) to ensure
reliable maximum yield and potential ET during each stage. In treatment
(II,*), (III,*), (IV,*), soil water contents in the top 50-cm zone were
maintained at field capacity except during specific growth stages. Two
levels of water stress during each growth stage were induced by omitting
irrigations for 3 and 4 weeks, respectively.
A Tuesday-Friday schedule was employed to monitor soil water
contents in the lysimeters during the season. Soil water contents at
five depths (15, 30, 45, 75, 105 cm) of soil profile were measured with
a neutron soil moisture meter (TROXLER 3220 Series.) Additional work
which was performed on the same schedule included irrigation, collecting
of volumes of drainage water from the lysimeters, and monitoring of crop
phenology.
Irrigation decisions were made weekly immediately following neutron
probe readings. Amounts of application were computed as the volume of
water reduction below field capacity for the top 50-cm zone. A manually
operated, pre-calibrated drip irrigation grid was designed to irrigate
inside each lysimeter. A separate irrigation system was used to
irrigate the buffer crop area beneath the rain-out shelters but outside
of the lysimeters.
Plots were harvested on 9 May 1984. Samples of total dry matter
above the ground were obtained from lysimeters by manually cutting and
threshing. At the same time, plant heights were measured. Samples were
then oven-dried at 95C for 24 hours. For individual lysimeters, grain
weights and related yield variables were assembled and measured for
detailed analysis.


75
Modeling and Analysis
To account for increased sensitivity to water stress at various
stages of growth, and the interactive effects between crop growth
stages, a multiplicative model was selected. Jensen (1968) first
developed one such model which related water stresses during various
stages of crop growth to final yield. Using input of standard,
available climatological data, Rasmussen and Hanks (1978) used this
method sucessfully to simulate grain yields of spring wheat grown in
Utah under various irrigation regimes. To estimate grain and bean
production assuming that other factors, such as fertility levels, pest
or disease activity, and climatic parameters are not limiting, the
Jensen model is given as
Y N ET
= n ( ) (4.1)
Y i=1 ET i
P P
where
ET/ET
P
Xi
= the relative yield of a marketable product,
= the relative total ET during the given ith stage of
physiological development,
= the relative sensitivity of the crop to water stress
during the ith (i = 1, 2, ... ,N) stage of growth.
To model this ET Y relationship, daily ET of each lysimeter was
calculated based on soil water balance method
ET = IR + AS DR (4.2)
where IR = irrigation, AS = soil water depletion, and DR = drainage.
Daily ET's were summed to calculate stage ET according to phenological


76
observations in the field. Data from six lysimeters, the control
treatments, were used for estimation of potential grain yield and
potential ET in Equation 4.1. The NLIN regression procedure (SAS, 1982)
for least-squares estimates of parameters of nonlinear models (Equation
4.1) was used to calibrate crop sensitivity factors (x^). These values
were then compared to the results of other researchers.
Results and Discussion
Field Experiment Results
The crop growth stage observations have a range of variability. In
addition, the effect of water stress on crop phenology was apparent.
Therefore, a stage was said to be observed when at least 50 percent of
the plants that were wel1-irrigated were at that stage of development.
Wheat phenology data observed in 1983-1984 winter season were recorded
in Table 6 and shown in Figure 9, indicating a full season of 163 days.
At 53 days after planting, the first node of stem was visible. Booting,
when the sheath of the last leaf was completely grown out, occurred 91
days after planting. Signaled by the time first ears were just visible,
heading began on March 10, which is 103 days after planting. Following
the heading stage, white flowers were visible on March 22. At 141 days
after planting kernels reached full size. This observation is very
similar to one made at Quincy, Florida, 1977-1979, by Barnett and Luke
(1980). Heading dates at Quincy were March 23 and March 27 for the
1977-1978 and 1978-1979 seasons, respectively.
Also shown in Figure 9, is the initiation of stress treatments.
According to the phenological calendar, the start of water stress


77
Table 6. Observations of specific reproductive growth stages for winter
wheat at Gainesville, FL., in 1983-1984.
Stage Description
Date
Elapsed Time
after Planting
PI anting
Nov. 29
1
Emergence
Dec. 4
6
Node of stem visible
Jan. 20
53
Booting
Feb. 27
91
Heading
Mar. 10
103
FIowering
Mar. 22
115
Mi 1ky-ripe
Apr. 3
127
Kernel hard
Apr. 17
141
Harvest
May 9
163


Stage partitioning
b
emergence to ^late tillering^
late tillering and booting
^headincj grain-
flowering filling
CD
S-
CD
CD
~a
+->
-M
CL
s-
03
CD
co
CD
r
03
E
CD
CJ
c
S-
-C
C
C=
4 CD
CD
CD
r
1
+->
r~
CD
O r
c
C
s~
i
>>
i
CO
4->
CD
_Q
r~
r
O)
CD
S-
CD
CD
C
-
CD 'f
4->
~a
5
C
CD
C
>
03
CD
lu co
O
03
o
S-
4->
S*-
i
E
O -r-
O
CD
1
CD
03
CD
03
CL
CD
c >
_Q
sz
4-
_£Z
Phenological observations
Stress treatments
M k it i
1 T T
irt' i n m
Lit
CO
CO
CO
CO
CO
CD t
CD i
CD >
S-
S- t
+->
4-J
+->
CO
CO
to
Figure 9. Phenological observations, water stress treatments, stage partitioning, and daily temperatu
in the winter wheat experiment, Gainesville, FL. 1983-1984.


79
treatments were slightly delayed. Therefore, the intended stress
treatments during the grain filling stage were not completely
accomplished, which resulted in duplicating treatment (IV,2) as shown in
Table 7.
Detailed yield vs irrigation data are tabulated in Table 7.
Effects of stress treatments on winter wheat yield are demonstrated in
Table 8, and plotted in Figure 10. By observations, crop growth in
lysimeters 3, 16, and 19 did not seem normal after the hard freeze.
Also, difficulty had been experienced in water management in these
lysimeters. Without irrigation, lysimeter 16 always had high counts of
the neutron probe throughout the season. In lysimeter 3 and 19,
irrigation was applied, however, it seemed that most water was drained
out by suction cups at the bottom of lysimeters. Therefore, data from
these three lysimeters were considered subject to an uncontrolled
treatment (UC), and were excluded from the following yield analysis.
Two basically different yield levels were obtained from irrigation
management. The treatments that were well-watered (N,N) and the one
that experienced severe water stress during late booting stage (11,4)
yielded less; whereas the rest of treatments had significantly higher
yields. Comparisons of biomass yields and head numbers between
treatments of heading period stress (III,*) and those stressed during
grain filling stage (IV,*) show that there is no significant
difference. This may be because the duration of heading to flowering
stage lasts only a short period of time (Peterson, 1965; Doraiswamy and
Thompson, 1982). It thus requires precise initiation of treatments to
acquire differential results.


Table 7.
Summary of
Gainesvi11e
results of winter
, FI., 1983-1984.
wheat growth
under various irrigation
treatments,
Dry Mass
Plant Ht
. No. of
Head Wt.
Grain Wt.
Test Wt.
Treatment
Lysi meter
- gm -
- cm -
Heads
- gm -
- gm -
-lb/bu-
(N,N)
3
620
73
577
298.3
190.1
78.5
(N,N)
16
648
78
511
325.5
224.9
81.2
(N,N)
19
670
63
576
324.9
240.6
80.4
(N,N)
7
795
68
683
395.2
305.2
81.4
(N,N)
10
840
76
574
446.6
321.4
83.1
(M)
21
880
73
824
459.2
358.9
81.1
(11,2)
4
742
71
663
399.9
281.6
81.2
(11,2)
12
1051
73
577
527.4
402.2
81.3
(11,2)
17
796
74
663
421.9
323.9
82.8
(11,4)
6
670
54
502
264.1
178.9
80.9
(11,4)
11
698
69
544
349.0
261.5
80.0
(11,4)
23
698
65
620
328.5
241.3
80.7
(111,3)
1
704
69
649
322.4
237.7
79.2
(HI,3)
14
966
75
690
499.4
393.4
80.2
(III,3)
22
924
83
818
480.9
326.8
82.0
(III,4)
8
772
75
635
371.9
284.4
78.8
(III,4)
9
816
73
823
409.5
283.1
80.6
(III,4)
20
1291
84
710
657.0
511.5
80.4
(IV,2)
2
645
68
692
322.3
216.6
79.8
(IV,2)
13
876
72
659
467.6
369.1
84.0
(IV,2)
24
1108
86
669
559.7
419.6
80.5
(IV,2)
5
746
68
677
393.6
303.6
80.8
(IV,2)
15
811
63
504
425.8
329.6
82.1
(IV,2)
18
1128
85
776
582.5
423.9
79.5
TM) the
control, well irrigated; (II,
*) stressed
during late
tilling to booting;
(III,*) stressed during heading
and flowering; (IV,
*) stressed
during grain
fil1ing period,


81
Table 8. Treatment effects on winter wheat yield, Gainesville, FL.,
1983-1984
Treatment
Dry Mass
- gm -
No. of
Heads
Head Wt.
- gm -
Grain Wt.
- gm -
UC
646.4 b
555 b
316.2 b
218.5 c
(N,N)
838.3 ab
694 ab
433.7 ab
328.5 ab
(11,2)
863.0 ab
634 ab
449.8 a
335.9 a
(11,4)
688.7 b
555 b
313.9 b
227.2 be
(111,3)
864.7 ab
719 a
434.3 ab
319.3 abc
(111,4)
959.7 a
723 a
479.5 a
359.7 a
(IV, 2)
876.3 ab
673 ab
449.9 a
335.1 a
(IV,2)
895.0 ab
652 ab
467.3 a
352.4 a
C.V. (%)
16.2
11.4
15.8
18.1
Column means followed by the same letter are not significantly different
at the 5% level by Duncan's multiple range test.


Number of Heads Dry Matter (gm)
82
Figure 10.
The effect of water stress treatment on different yield
variables of wheat for each stress treatment (average of
3 replications), (a) Dry matter; (b) Number of heads;
(c) Head Weight; (d) Grain weight.


Figure 10. (continued)
Grain Weight (gm)
450
Stress treatment
Head Weight (gm)
00
GJ


34
Plant water stress limits leaf and tiller development during
vegetative growth and stress during the late tillering to booting stage
accelerates stem senescence and reduces spikelets per head (Musick and
Dusek, 1980). Consequently, for treatment (11,4), the effect of
extensive water stress during the late tillering stage significantly
reduced grain yield by 30 percent of the wel1-irrigated plants. This
agrees with results from Day and Intalap (1970) that water stress is
more critical during late tillering than during flowering or grain
filling stage.
An attempt was made to relate grain yield to seasonal irrigation
and seasonal ET. A regression analysis of the effects of seasonal
irrigation amounts on grain yield indicates that the linear relationship
is poor with an r value of 0.25. It implies that a linear model of
grain yield dependent upon total irrigation or upon seasonal ET is not
strongly recommended on the basis of this study. Therefore, the model
of Jensen (1968) was evaluated.
Model Calibration
Phenological development occurred over a range of time and caused a
large variation in the duration of various stages. The appropriate
scheme of partitioning the growth season into four stages was
illustrated in Figure 9. The periods of stage I, II, III, and IV were
53, 50, 24, and 36 days, respectively. Accordingly, stage ET and
seasonal ET were computed and tabulated in Table 9. As explained in the
last section, difficulty had been experienced in water management in
lysimeters 3, 16 and 19. For these three lysimeters, the seasonal and
stage-specific ET1s were very low.


Table 9. Seasonal and stage-specific ET for winter wheat grown in Gainesville, FL.,
1983-1984.
Treatment
Lysimeter
Evapotranspiration (cm)
Irrigation
Applied
(cm)
Grai n
Yield
(gm)
Stage I
Stage II
Stage III
Stage IV
Season
(M)
3
0.74
6.51
0.22
1.14
8.61
9.13
190.1
(N,N)
16
0.04
2.59
0.35
1.03
4.01
0.00
224.9
(N,N)
19
0.30
7.80
0.86
1.74
10.70
12.38
240.6
(N,N)
7
0.24
9.59
6.71
16.32
32.86
37.29
305.2
(N,N)
10
0.81
10.71
6.16
12.32
27.84
30.75
321.4
(N,N)
21
0.95
11.70
7.14
8.27
28.06
31.53
358.9
(11,2)
4
0.34
10.45
3.24
10.55
24.58
27.85
281.6
(11,2)
12
1.32
8.45
3.47
14.30
27.26
29.68
402.2
(11,2)
17
1.24
13.41
3.50
5.39
23.54
29.12
323.9
(11,4)
6
0.04
9.04
0.86
11.14
21.08
23.27
178.9
(11,4)
11
0.54
7.66
1.52
8.51
18.23
20.58
261.5
(11,4)
23
0.03
10.91
0.69
14.82
26.45
32.24
241.3
(HI,3)
1
0.73
10.14
4.95
6.08
21.90
24.35
237.7
(111,3)
14
0.63
11.05
6.08
6.85
24.61
25.43
393.4
(111,3)
22
0.42
10.49
5.78
6.49
23.18
23.97
326.8
(111,4)
8
0.52
10.15
6.00
5.13
21.80
23.69
284.4
(III,4)
9
0.04
10.11
4.67
3.63
18.45
19.79
283.1
(III,4)
20
0.19
9.36
7.59
4.93
22.07
24.37
511.5
(IV,2)
2
0.45
8.41
5.08
3.72
17.66
20.51
216.6
(IV,2)
13
0.80
9.49
3.98
4.88
19.15
18.33
369.1
(IV,2)
24
0.88
11.77
8.84
6.62
28.11
33.05
419.6
(IV,2)
5
0.11
10.52
7.39
11.49
29.51
31.27
303.6
(IV,2)
15
0.28
8.79
2.00
2.33
13.40
14.35
329.6
(IV,2)
18
1.62
11.11
8.28
10.00
31.01
35.50
423.9
Stage I: emergence to late tillering; Stage II: late tillering to booting;
Stage III: heading to flowering; Stage IV: grain filling.


86
Data from six lysimeters, the control units, would be used for
estimation of potential grain yield and potential ET in Equation 4.1.
As explained, the crop in lysimeters 3, 16, and 19 did not recover from
the freeze and grow normally. Therefore, average values of data from
lysimeters 7, 10, and 21 were calculated to define potential yield and
potential ET values. For model calibration, calculated potential stage
ET's for emergence to late tillering, late tillering to booting, heading
and flowering, and grain filling 0.67, 10.67 6.67, 12.30 cm,
respectively. For stage I of 53 days, potential ET of 0.67 cm was
low. That is because radiation was low in December and January and
irrigation was not initiated until January 13, 1984. Potential grain
yield was 328 gm for a 2 meter square area.
Using data from all 24-lysimeters, calibration of xs values
(Equation 4.1) was accomplished. Values of 0.065, 0.410, 0.114, 0.026
for all A's in Equation 4.1 gave the best fit. Predicted vs. observed
yields for all data from lysimeters were given in Figure 11. Because
the uncontrollable within-treatment errors and unexpected freezing
o
weather, the r = 0.42 does not seem high. However, the effect of
critical stages of growth has been quantified.
I
Values of published a s for wheat are inconsistent. The x. values
for booting, heading, soft dought, and maturity reported by Neghassi et
al. (1975) are -0.490, 2.71, -5.45, and 4.58, respectively. The
negative values do not have any physical relevance. Values of 0.25 for
all As were given by Rasmussen and Hanks (1978). By assigning the
relatively short grain-filling period a X of 0.25, Rasmussen and Hanks
argued that the grain filling stage was more important in irrigated
wheat production. The values obtained from this study illustrated that


Predicted Yield Ratio
87
Figure 11. Plot of observed vs. predicted yield ratio for wheat.


88
water stress during late booting, heading and flowering stages were
important. Robins and Domingo (1962), and Mogensen et al. (1985) had
the same conclusion that severe water deficits should be avoided from
the booting stage until the heads were filled.
In summary, the grain yield model developed in this study accounts
for variables of climate and irrigation. It has been shown that the
model has the capability to give very reasonable predictions of yield
reductions to water stress. Coupled with a soil water balance approach,
the grain yield model can be utilized effectively for water stress and
irrigation management applications. It should be of particular use to
economists and others concerned with the effects of drought or limited
irrigation. One type of applications in using this data set will be
demonstrated in the next chapter.


CHAPTER V
APPLICATION OF THE MODEL
Introduction
In Florida, where the cold season is relatively short and the water
supply (precipitation and irrigation) is sufficient to grow two or more
crops per year, the potential of practicing multiple cropping is high.
On the other hand, irrigation development is expensive. Inasmuch as
benifits from irrigation may vary appreciably from year to year,
developing optimal multiple cropping systems is intended to make maximum
use of the expensive irrigated land.
As the number of crops and development of new integrated management
systems (i.e. tillage, irrigation, pest, fertilization, weed, etc.)
increases, the problem of deciding multiple cropping sequences to be
followed becomes very complex. If it is to be analyzed properly, it
must be examined systematically.
An optimization simulation model composed of submodels to
integrate crops, soil water dynamics, weather, management, and
economic components has been developed to select optimal multiple
cropping sequences. However, decisions about optimal multiple cropping
systems are complicated by a number of factors including weather
uncertainty, the complex nature of the crop's response to management
strategies (i.e. irrigation), and uncertain crop prices. The
application of the model refers to its use as a tool for studying
various optimal cropping management decisions.
89


Full Text

OPTIMAL SEQUENCING OF MULTIPLE CROPPING SYSTEMS
By
YOU JEN TSAI
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1985

ACKNOWLEDGMENTS
This work would not have been possible without the assistance and
contribution of many people.
I will always be grateful to my major professors, Drs. James W.
Jones and J. Wayne Mi shoe. My intellectual development was actively
encouraged by Dr. Jones, who was enthusiastic and confident of my
work. Dr. Mishoe focused on my appreciation of systems analysis and
provided unending support and confidence. Drs. K. L. Campbell, D. H.
Hearn, C. Y. Lee and R. M. Peart critically reviewed this paper and
provided insightful suggestions.
I also wish to thank Paul Lane, whose dedication and persistence
made the field experimental study possible.
Finally, I cannot begin to express my appreciation to my wife,
Chin Mei, my boy, Hubert, and mother-in-law who had confidence in my
abilities and gave me more than I can ever repay.

TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ii
LIST OF TABLES v
LIST OF FIGURES vii
ABSTRACT
CHAPTERS
IINTRODUCTION 1
The Problem 2
Scope of The Study 5
Objectives 6
IIREVIEW OF THE LITERATURE 7
Multiple Cropping 7
Optimization Models of Irrigation 10
Soil Water Balance Models 11
Crop Response Models 14
Crop Phenology Models 18
Objective Functions 20
Optimization Techniques 23
IIIMETHODOLOGY FOR OPTIMIZING MULTIPLE CROPPING SYSTEMS ... 29
Mathematical Model 29
Integer Programming Model 29
Dynamic Programming Model 32
Activity Network Model 35
Crop-Soil Simulation Model 39
Crop Phenology Model 41
Crop Yield Response Model 48
Soil Water Balance Model 49
Model Implementation 59
Network Generation Procedures 61
Network Optimization 62
Parameters and Variables 67

Page
CHAPTER
IVWHEAT EXPERIMENTS 70
Introduction 70
Experimental Procedures 72
Experimental Design 72
Modeling and Analysis 75
Results and Discussion 76
Field Experiment Results 76
Model Calibration 84
VAPPLICATION OF THE MODEL 89
Introduction 89
Procedures for Analysis 90
Crop Production Systems 90
Crop Model Simulation 91
Optimization of Multiple Cropping Sequences 91
Risk Analysis 93
Results and Discussion 94
Crop Model Simulation 94
Evaluation of The Simulation-Optimization Techniques . 107
Multiple Cropping Systems of a Non-Irrigated Field in
North Florida 112
Effects of Irrigation on Multiple Cropping 119
Risk Analysis of Non-Irrigated Multiple Cropping
Sequences 121
Applications to Other Types of Management 130
VISUMMARY AND CONCLUSIONS 132
Summary and Conclusions 132
Suggestions for Future Research 133
APPENDIX A GENERAL DESCRIPTIONS OF SUBROUTINES 135
APPENDIX B SOURCE CODE OF SUBROUTINES 143
APPENDIX C INPUT FILE ‘GROWS1 178
APPENDIX D INPUT FILE 'FACTS' 180
REFERENCES 181
BIOGRAPHICAL SKETCH 188
iv

LIST OF TABLES
Table Page
1. Threshold values for physiological stages of
growth of corn and peanut 43
2. Coefficients of a multiplicative model for predicting
wheat phenological stages 44
3. Description and threshold values of phenological
stages and phases for soybean cultivars 46
4. Values of the parameters for the nighttime accumulator
function of the soybean phenology model 50
5. Crop sensitivity factors, , for use in the simulation 50
6. Observations of specific reproductive growth stages for
winter wheat at Gainesville, Fla., in 1983-1984 .... 77
7. Summary of results of winter wheat growth under various
irrigation treatments, Gainesville, Fla., 1983-1984 . . 80
8. Treatment effects on winter wheat yield, Gainesville,
Fla., 1983-1984 81
9. Seasonal and stage-specific ET for winter wheat grown in
Gainesville, Fla., 1983-1984 85
10. Price, production cost and potential yield of different
crops for typical north Florida farm 92
11. Simulation results of irrigated and non-irrigated ful 1 -
season corn grown on different planting dates for 25
years of historical weather data for Gainesville, Fla. 95
12. Simulation results of irrigated and non-irrigated short-
season corn grown on different planting dates for 25
years of historical weather data for Gainesville, Fla. 96
13. Simulation results of irrigated and non-irrigated 'Bragg1
soybean grown on different planting dates for 25 years
of historical weather data for Gainesville, Fla 97
v

Tab! e Page
14. Simulation results of irrigated and non-irrigated 'Wayne'
soybean grown on different planting dates for 25 years
of historical weather data for Gainesville, Fla 98
15. Simulation results of irrigated and non-i rrigated peanut
grown on different planting dates for 25 years of
historical weather data for Gainesville, Fla 99
16. Simulation results of irrigated and non-irrigated wheat
grown on different planting dates for 25 years of
historical weather data for Gainesville, Fla 105
17. Summary of network characteristics and CPU time required
for various durations of planning horizon and two
irrigation conditions 108
18. Sensitivity analysis of non-i rri gated multiple
cropping sequences to weather patterns 113
19. Comparison of various multiple cropping systems under a
non-irrigated field. System I includes corn, soybean,
peanut, and wheat allowing continuous peanut cropping.
System II includes the same crops as system I, but not
allowing continuous peanut cropping. System III
excludes peanut from consideration. 118
20. Analysis of net returns of non-irrigated multiple
cropping sequences in response to different weather
patterns 127
21. Analysis of net returns of non-i rrigated multiple
cropping sequences under different crop pricing
schemes for weather pattern number 3 128
22. General descriptions of subroutines used in optimizing
multiple cropping systems 136
VI

LIST OF FIGURES
Figure Page
1. A system network of multiple cropping 37
2. Rate of development of soybean as a function of
temperature 47
3. Effects of night length on the rate of soybean
development 47
4. Maximum yield factors that reduce yield of each crop
below its maximum value as a function of planting day
for well-irrigated conditions 51
5. Crop rooting depth after planting under wel1-irrigated
conditions 53
6. Leaf area index for wel1-irrigated crops as a function
of time 55
7. A schematic diagram for optimal sequencing of multiple
cropping systems 60
8. Time intervals during which each crop can be planted . . 63
9. Phenological observations, water stress treatments,
stage partitioning, and daily temperature in winter
wheat experiment, Gainesville, Fla. 1983-1984 78
10. The effect of water stress treatment on different yield
variables of wheat for each stress treatment (average of
3 replications), (a) Dry matter; (b) Number of heads;
(c) Head weight; (d) Grain weight 82
11. Plot of observed vs. predicted yield ratio for wheat . . 87
12. Cumulative probability of profit for non-irrigated
full-season corn on different planting dates 102
13. Cumulative probability of profit for non-irrigated
short-season corn on different planting dates 102
14. Cumulative probability of profit for non-irrigated
'Bragg' soybean on different planting dates 103

Figure
Page
15. Cumulative probability of profit for non-irrigated
'Wayne1 soybean on different planting dates 103
16. Cumulative probability of profit for non-irrigated
peanut on different planting dates 106
17. Cumulative probability of profit for non-irrigated
wheat on different planting dates 106
18. Sample output of optimal multiple cropping sequences
for north Florida 110
19. Optimal multiple cropping sequences of a non-irrigated
field with corn, soybean, peanut and wheat, allowing
continuous cropping of peanut 114
20. Optimal multiple cropping sequences of a non-irrigated
field with corn, soybean, peanut and wheat, not allowing
continuous cropping of peanut 117
21. Optimal multiple cropping sequences of a non-irrigated
field considering corn, soybean and wheat, excluding
peanut 120
22. Optimal multiple cropping sequences of an irrigated
field considering corn, soybean, peanut and wheat,
allowing continuous cropping of peanut 122
23. Optimal multiple cropping sequences of an irrigated
field considering corn, soybean, peanut and wheat,
not allowing continuous cropping of peanut 123
24. Optimal multiple cropping sequences of an irrigated
field considering corn, soybean and wheat, excluding
peanut 124
25. A set of optimal multiple cropping sequences for a non-
irrigated field chosen from Figure 20 for additional
simulation study 126
vi i i

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
OPTIMAL SEQUENCING OF MULTIPLE CROPPING SYSTEMS
By
YOU JEN TSAI
DECEMBER 1985
Chairman: Dr. J. W. Jones
Cochairman: Dr. J. W. Mi shoe
Major Department: Agricultural Engineering
Multiple cropping is one of the means to increase or at least
stabilize net farm income where climatic and agronomic conditions allow
its use, such as in Florida. With several crops to be examined
simultaneously, the design of multiple cropping systems becomes
complex. Therefore, a systems approach is needed. The goal of this
study is to develop a mathematical method as a framework for optimizing
multiple cropping systems by selecting cropping sequences and their
management practices as affected by weather and cropping history.
Several alternative formulations of multiple cropping problems were
studied with regard to their practicality for solutions. A
deterministic activity network model that combined simulation and
optimization techniques has been developed to study this problem. In

particular, to study irrigation management in multiple cropping systems,
models of crop yield response, crop phenology, and soil water were used
to simulate the network. Then, the K longest paths algorithm was
applied to optimize cropping sequences.
Under a non-irrigated field in north Florida, winter wheat followed
by either soybean, corn, or peanut was found to be the most profitable
cropping system. Especially favorable was the cropping of wheat-
peanut. Another significant conclusion to be drawn concerned the effect
of irrigation management on multiple cropping sequences. Under
irrigated fields, peanuts were selected for production each year because
of their high net returns in comparison to the other crops. In a system
in which peanut was not considered as an option, inclusion of irrigated
wheat-corn cropping would not be a profitable multiple cropping
system. Instead, double cropping of corn-soybean was the dominant
optimal sequence under irrigation.
The importance of irrigation management in multiple cropping
systems was studied using the methodology developed. The methodology is
also capable of incorporating other aspects of farming (i.e. pest
management) into an integrated framework for determining optimal
cropping sequences.
x

CHAPTER I
INTRODUCTION
Net farm income has been a major concern for farmers in commercial
agriculture for a long time. Income has expanded through various ways
including an increase in land area for production, fertilizer and
pesticide applications, machinery and other capital expansions.
However, these different methods of increasing net farm income usually
increase the cost of production. A study (Ruhimbasa, 1983) showed that
multiple cropping had the potential to reduce costs per unit of output
and reduce production risks, and therefore could increase or at least
stabilize net farm income where climatic and agronomic conditions allow
its use.
Multiple cropping may also be called sequential or succession
cropping. Succession cropping is the growing of two or more crops in
sequence on the same field during a year. The succeeding crop is
planted after the preceding crop has been harvested. There is no
intercrop competition. Only one crop occupies the field at one time;
thus mechanization is possible.
In summary, multiple cropping increases annual land use and
productivity resulting in increased total food production per unit of
land. It also allows more efficient use of solar radiation and
nutrients by diversifying crop production. Thus, it reduces risk of
total crop loss and helps stabilize net farm income.
1

2
The Problem
Multiple cropping is not without risk. The use of multiple
cropping creates new management problems. It may create time conflicts
for land and labor, may require new varieties or new crops for an area,
may deplete soil resources, i.e. water and nutrient reserves, more
rapidly, and may cause residuals from one crop that directly affect the
next crop. For example, increasing the crop species grown on the same
land makes herbicide selection more complex. Disease incidence may
increase with an annual production of the same species on the same field
each year. As a result, higher levels of management become more
important in terms of operations needed. In designing optimal multiple
cropping systems, managers need to take into consideration these
effects.
Of the above management areas, timing becomes dominant for
successful multiple cropping, given substantial yield losses for each
day of delay. As estimated by Phillips and Thomas (1984), if the losses
of soybean yields after a given date are 62 - 75 kg/ha-day, the cash
losses on a 200-ha planting of soybeans would be as much as $4000 -
$5000/day. A delay of one week probably could make the difference
between profit and loss. Therefore, a timely planting and optimal
within-season management practices are the key to profitable multiple
cropping.
Soil water determines whether seeds will germinate and seedlings
become established. With multiple cropping, seed zone water is even
more critical because the second crop must be established rapidly to
avoid possible yield reduction due to frost. Also, because of depletion

3
by the preceding crop, soil water content at planting of subsequent
crops in multiple cropping systems may be low as compared to planting
following a fallow period. This is particularly true in areas of low
rainfall or where periodic droughts could result in a depleted soil
reservoir that would prevent successful planting and production of the
second crop. Hence, management practices that take advantage of soil
water storage should be beneficial in multiple cropping systems.
Plant growth is influenced by the process of evapotranspiration
(ET). During the time course of a seasonal crop, the crop system
changes from one in which ET is entirely soil evaporation to one in
which ET is mostly plant transpiration, and finally to one in which both
plant transpiration and soil evaporation are affected by crop
senescence. Plants store only a minor amount of the water they need for
transpiration; thus, the storage reservoir furnished by the soil and its
periodic recharge are essential in maintaining continuous growth. In
the event of relatively high ET demand coupled with depleted soil water
conditions, water deficits in plants occur as potential gradients
develop to move water against flow resistances in the transpiration
pathway. As plants become water stressed, their stomata close. The
resulting effects on transpiration and photosynthesis are essentially in
phase. This would represent the reduction of plant growth because of
less carbon dioxide uptake and reduced leaf and stem growth. Therefore,
soil water, undoubtedly more often than any other factors, determines
crop yield.
The soil water reservoir is supplied by rainfall. As evapo¬
transpi ration demand and supply of soil water are synchronized,
potential maximum yield is expected. Otherwise, irrigation may be

4
practiced to supplement rainfall supply of water to the soil and thus
avoid possible yield reductions. Hence, crop sequencing that shifts
crop demands for soil water according to weather patterns could be
beneficial in multiple cropping systems.
In Florida, where the cold season is short and the water supply
(precipitation or irrigation) is sufficient to grow two or more crops
per year on the same field, the potential of practicing multiple
cropping is high. However, water management is critical here. For
instance, although long-term average rainfall amount (148 cm per year)
may be sufficient on the average for replenishing the soil water supply,
year-to-year variability in rainfall amounts and the variability in
successive days without rain may result in one or more drought periods
during a growth season. On the other hand, irrigation development is
expensive. Inasmuch as benefits from irrigation may vary appreciably
from year to year, developing optimal multiple cropping systems is
intended to make maximum use of the expensive irrigated land.
As the number of crops, number of varieties, variability in soil,
and development of new integrated management systems (i.e., tillage,
irrigation, pest control, fertilization, weed control, etc.) increase,
planning of a multiple cropping production system becomes very complex
in terms of maximizing net farm income. However, actual experimentation
with the system may be infeasible, cost-ineffective, and time-consuming
due to the vast array of multiple cropping systems that possibly can be
grown. As a result, an alternate method for evaluating optimal multiple
cropping practices is needed.
At a field level, it is desirable to be able to select crops,
varieties, planting date, and to evaluate various management strategies

5
in a multiple cropping scheme. The overall goal of this study is to
develop a mathematical method as a framework for optimizing multiple
cropping systems by selecting cropping sequences and their management
practices as affected by weather pattern and cropping history. This
framework will be applied in particular to the study of irrigation
management in multiple cropping production.
Scope of the Study
Many efforts have contributed to developing irrigation programs
which would provide optimal return to growing a single crop during a
single season. Fewer studies have concentrated on investigating the
effect of irrigation management under multiple cropping systems. The
problem to be explored is as follows. A 'field' is considered for
growing crops over an N-year production horizon. There are I number of
potential crops and each crop has J varieties to be considered. Only
one crop grows at one time and various idle periods are also considered
legitimate choices in a cropping sequence. Under the assumption that
other production practices are optimally followed, what are optimal
cropping sequences and associated within-season irrigation strategies
that maximize net discounted return?
This study at a field-level needs to be differentiated from that of
a farm-level system. A field can be defined as an unit area of uniform-
soil land or as an area constrained by the inherent operational
practicalities of the irrigation system used. For example, it may be
the area under a center pivot irrigation system. Applying systems
analysis methods, this study develops a mathematical model to optimize
multiple cropping systems.

6
Objecti ves
The specific objectives of the study are
1. To develop a framework for optimal sequencing of crops in
a multiple cropping production system and for determining
optimal management of the crop land.
2. To apply the framework to study irrigation management in
multiple cropping production.
3. To implement a computer model for North Florida soil and
climate conditions, taking soybean, corn, peanut, and wheat
as crops to be produced.
4. To perform field experiments designed to quantify the effect
of water stress on wheat yield for Florida conditions, and to
form a simplified wheat yield response model for use in the
analysis.
5. To use the model as a decision-making tool to analyze multiple
cropping practices in this region in order to increase net
farm income.

CHAPTER II
LITERATURE REVIEW
Multiple Cropping
In the United States, sequential cropping systems are mostly found
in southern states where a short cold season allows the planting of a
second or a third crop on the same land. The use of no-tillage methods
further enhances the success of sequential cropping systems in this
region. A selected number of articles concerning the topics are
reviewed.
Multiple cropping in sequence has been criticized for being yield
reducing. Crabtree and Rupp (1980) found that in Oklahoma wheat yield
decreased from 2519 kg/ha in a monocropping system to 2200 kg/ha in a
double cropping system. The following soybean yield decreased from 2000
kg/ha in 51-cm rows and 1792 kg/ha in 76-cm rows to 1603 and 1453 kg/ha,
respectively. The use of no-tillage practices increased soybean yield
to 1722 and 1543 kilogram per hectare in the double cropping system. In
fact, the long land preparation process in the conventional tillage
method led to a late planting for the second crop which resulted in
lower yields. The no-tillage method, allowing a direct planting of
crops into unprepared soil with standing crops or residues, had
significant impacts on reducing the risk of obtaining low yield due to
late planting in a multiple cropping system. Westberry and Gallaher
(1980) conducted two different studies on the influence of tillage
7

8
practices on yield which also led to a conclusion favoring a no-tillage
method.
The potential of no-tillage methods to reduce production costs when
associated with multiple cropping systems to increase land productivity
suggests that these two practices should be used together to increase
net farm income (Robertson et al., 1980). Other advantages of no¬
tillage systems become more apparent with multiple cropping, and these
include (1) elimination of moisture loss associated with conventional
tillage at planting time, ensuring stands of second and third crops
under restricted rainfall patterns; (2) further reduction of soil
erosion; and (3) maintenance of soil structure by elimination of plowing
and land preparation (Phillips and Thomas, 1984.)
It is obvious that multiple cropping for grain crops depends on a
reasonably long frost-free season. Guilarte (1974) and Smith (1981)
indicated that a double cropping system can be feasible during the 240
or more days of the warm growing season in north and west Florida.
Unfortunately, these long growing seasons are associated with elevated
temperatures, which may adversely depress the second crop yield as
witnessed by Widstrom and Young (1980). Their results showed that
double cropping of corn could be a viable option on the coastal plain of
the southeastern United States, when the second crop was taken as forage
rather than as grain.
To generalize types of multiple cropping on a cropping-year basis,
we divide it into winter-summer double cropping, summer-summer double
cropping and winter-summer-summer triple cropping. The major system of
winter-summer double cropping is wheat-soybeans (Gallaher and Westberry,
1980). The use of valuable irrigation water for a second crop of

9
sorghum or sunflower is not very practical except to produce favorable
emergence condition. Thus, soybean is favored as a second crop. Of
summer-summer systems, corn-soybeans appears to be most commercially
viable (Gallaher et al., 1980). Because soybeans bloom over a longer
period of time, their yields tend to be hurt less by short periods of
drought during flowering. Corn, on the other hand, requires excellent
soil water conditions during silking and tasseling, or else yields will
be low. The third multiple cropping system is adding a winter vegetable
crop to summer crops or following a winter-summer sequence with a late
fall planting of a cool-season vegetable. This type of system has the
advantage of producing the vegetable crop when prices are relatively
high, and still producing field crops competitively with the rest of the
nation.
Despite other attributes of multiple cropping, if it does not, over
a period of time, provide more net income to the farmer, it will not be
practiced. Economic analyses studied by a group of research scientists
in the University of Georgia indicated that irrigated agronomic crops
were generally profitable on a first-crop basis, but the profitable
agronomic second-crop was limited to sorghum and soybeans (Anonymous,
1981). In 1980, the study also showed that most irrigated multiple
cropping production was profitable on the wel1-drained, sandy soil.
Both irrigated and dryland peanut production were profitable; however,
irrigated peanuts were more profitable. Irrigated corn was also more
profitable than nonirrigated corn. Tew et al. (1980) further analyzed
costs and returns of irrigated, double-crop sweet corn and soybean
production. They concluded that irrigated soybean as the second crop in
a double-crop system was a questionable alternative since net returns

10
did not compare favorably with dryland production. However, irrigation
of soybean as the second crop was still justified because it reduced
income variance.
These results suggest that the economics of multiple cropping
systems differs significantly from that of a single, full season
monocrop. Knowledgeable management practices such as precise planting
dates, cultivars, and water management are essential. Gallaher et al.
(1980) strongly asserted that "if growers use management practices in
these studies, corn-soybean succession cropping can be successful in
Florida" (page 4).
Optimization Models of Irrigation
In order to study irrigation policies to maintain favorable soil
moisture conditions and thus avoid economic yield reduction,
optimization techniques have been increasingly used for the last 15
years. Mathematical models are inherent in this methodology.
Implicitly or explicitly a crop response model within the mathematical
statement of the objective function is required. Furthermore, the soil
water status, needed as a set of constraints in the optimization
problem, is traditionally calculated in a soil water balance model.
Then, various optimization techniques are applied for finding the best
or optimal decisions in an organized and efficient manner. The role of
models and simulation in irrigation optimization problems is reviewed
herein.

11
Soil Water Balance
Water balance models for irrigation scheduling were developed as
'bookkeeping' approaches to estimate soil water availability in the root
zone.
S = S . + P
n n-1 n
+ I + OR
- ET - RO
n n
PC
(2.1)
where
S = soil water content on the end of day n,
Pn = total precipitation on day n,
I = total irrigation amount on day n,
DR = water added to root zone by root zone extension,
n J
ETn = actual evapotranspiration on day n,
RO = total runoff on day n, and
n
PCn = deep percolation on day n.
In general, a volume of soil water, defined in terms of the soil
water characteristics and the root zone of the crop being irrigated, is
assumed to be available for crop use. Depletions from this reservoir by
evapotranspiration (ET) are made on a daily basis. Soil water balance
models generally are classified into two categories: (a) those based on
the assumption that water is uniformly available for plant use between
the limits of field capacity and permanent wilting point, and (b) those
based on the assumption that transpiration rates were known functions of
soil water potential or water content (Jones and Smajstrla, 1979).
Uniformly available soil water. Models based on the assumption of
uniformly available soil moisture between field capacity and permanent
wilting point simulated water use based on climatic variables only.

12
Those simulation models for ET by various crops have been summarized by
Jensen (1973). For ET prediction, a technique used widely to calculate
potential ET is the modified Penman equation (Van Bavel , 1966). The
Penman equation predicted reference ET (ETp), which is that of a well-
watered, vegetated surface. To predict actual rather than reference ET
for a well-watered crop, a crop coefficient, 1973) as
ET = K * ET (2.2)
c p '
Crop coefficients for specific crops must be determined experimently.
They represent the expected relative rate of ET if water availability
does not limit crop growth. The magnitude of the crop coefficient is a
function of the crop growth stage. One of the major shortcomings of
this method is that they do not account for changes in ET rates due to
changing soil water levels.
Limiting soil water. To correct this shortcoming, a number of
researchers (Ritchie, 1972; Kanemasu et al., 1976) have developed models
to predict ET as functions of both climatic demands and soil water
availability. This resulted in a more complex model than the Penman
equation, which uses climatic indicators only. Ritchie's model
separated evaporation and transpiration components of water use.
Potential evaporation Ep from a wet soil surface under a row crop
(energy limiting) was defined as
T

13
where t = reduction factor due to crop cover, and a = proportional ity
constant due to crop and climate.
During the falling rate stage (soil limiting) evaporation rate E,
was defined as a function of time as
E = ct1/2 - c ( t - 1 )1/2 (2.4)
where c = coefficient dependent on soil properties, and t = time.
Transpiration rates were calculated separately from evaporation
rates. For plant cover of less than 50 percent, potential transpiration
rate, T , was calculated as
Tp = av ( 1 - T ) ( A / ( A + Y )) Rn (2.5)
where A = slope of the saturation vapor pressure-temperature curve, y =
psychrometric constant, Rr = net radiation, and ay = (a - 0.5)/0.05.
For greater than 50 percent crop cover, was calculated as
Tp = (cx-t) (a / (a + y) ) Rp (2.6)
This formulation represented transpiration during non-limiting water
conditions only. To account for decreasing soil water potential with
water content, and effects on transpiration rate, a coefficient of
limiting soil water (K ) was defined by Kanemasu et al. (1976) as
(2.7)

14
where 0 = average soil water content, and e = water content at field
3 max
capacity. At water contents above 0.3 0^^^ , transpiration rates were
assumed to be controlled by climatic conditions only. Ritchie (1973)
reported that this model predicted transpiration rates well for sorghum
and corn.
In summary, several models for predicting ET rates under both well
watered and water stressed conditions are presented. The models
presented are all simple approximations of complex dynamic systems.
Their simplicity has the advantage of requiring few data inputs, and
therefore, they can be applied with relatively few meteorological, soil,
or crop measurements taken. However, because of their simplicity,
several empirical coefficients are required in each model, and each must
be calibrated for specific crops, soil conditions and climatic
variables.
Crop Yield Response
Vast literature on this subject revealed yield relationships to
water use can range from linear to curvilinear (both concave and convex)
response functions (Stegman and Stewart, 1982). These variations are
influenced by the type of water parameter that is chosen, its
measurement or estimation accuracy, and the varied influences associated
with site and production conditions. The following is intended to
illustrate the more general relationships of crop yields with water when
they are expressed as transpiration, evapotranspi ration, or field water
supply.
Yield vs. transpiration or evapotranspiration. When yields are
transpiration limited, strong correlations usually occur between

15
cumulative seasonal dry matter and cumulative seasonal transpiration.
Hanks (1974) calculated relative yield as a function of relative
transpiration:
Y T
where Yp = potential yield when transpiration is equal to potential
transpiration and Yp = cumulative transpiration that occurs when soil
water does not limit transpiration. With the close correlation between
T and ET, dry matter yield vs cumulative ET also plotted as a straight
line relationship. Hanks' work demonstrated a physically oriented,
simple model to predict yield as a function of water use.
Based on the same idea, an approach which interprets ET or T
reduction below potential levels as integrators of the effects of
climatic conditions and soil water status on grain yield is used
frequently. Such an approach predicts grain yields from physically
based models which relate water stresses during various stages of crop
growth to final yield, accounting for increased sensitivity to water
stress at various stages of growth. Two basic mathematical approaches
were taken in the development of these models. One assumed that yield
reductions during each crop growth stage were independent. Thus
additive mathematical formulations were developed (Moore, 1961; Flinn
and Musgrave, 1967; Hiler and Clark, 1971). A second approach assumed
interactive effects between crop growth stages. These were formulated
as multiplicative models (Hall and Butcher, 1968; Jensen, 1968).
Additive models. The Stress Day Index model is an additive model
presented by Hiler and Clark (1971). The model is formulated as

16
Y
A n
1.0
2 (CS, * SD.)
(2.9)
Y
P
vp i=1
where A = yield reduction per unit of stress day index, SO.. = stress day
factor for crop growth stage i, CS^. = crop susceptibility factor for
growth stage i. CS^ expresses the fractional yield reduction resulting
from a specific water deficit occurring at a specific growth stage.
SD^ expresses the degree of water deficit during a specific growth
period.
The stress day index model was utilized to schedule irrigations by
calculating the daily SDI value (daily SD * daily CS) and irrigating
when it reached a predetermined critical level, SDI. This integrated
the effects of soil water deficit, atmospheric stress, rooting density
and distribution, and crop sensitivity into plant water stress factor.
Multiplicative models. Jensen (1968) developed the following model
Y
(2.10)
Y
P
physiological development, and A.. = crop sensitivity factor due to water
stress during the i-th growth stage.
Hill and Hanks (1975) modified the above equation by including
factors to account for decreased dry matter production due to planting
late season crops, and to account for decreased yields due to excess
water. Their equation is
Y
n T A.
Y
P
n ( ) •1 * SYF * LF
1=1 tp
(2.11)

17
where (T/T )^ = relative total transpiration for growth stage i when
soil water is not limiting, SYF = seasonal yield factor which approaches
1.0 for adequate dry matter production, and LF = lodging factor.
Because this model relates relative yield to relative transpiration, it
is also necessary to predict evaporation rates as a function of ETp in
order to maintain a soil water balance. This yield response model,
verified with Missouri soybean experiments, appeared to be an excellent
simulator of grain yields as affected by transpiration rates.
Minhas et al. (1974) proposed another multiplicative model
expressed as
Y n ET 0
= n { 1.0 - ( (1.0 )¿ ) 1 } (2.12)
Y i =1 ET 1
P P
where all factors are as previously defined. Howell and Hiler (1975)
found that it described adequately the yield response of grain sorghum
to water stress.
Yields vs. field water supply. The field water supply (FWS) in
irrigated fields is derived from the available soil water at planting
(ASWP), the effective growth season rainfall (Re), and the total applied
irrigation depth (IRR). Stewart and Hagan (1973) demonstrated that crop
yields are related to seasonal ET and seasonal IRR. In a given season,
the ASWP and Re components of the seasonal FWS make possible a yield
level that is common to both functions. The ET component associated
with successive applications of irrigation defines the yield, Y vs ET
function above the dryland level, which rises to a Y - ET level
when the seasonal crop water requirement is fully satisfied. The ET +
non-ET components of IRR define a Y vs IRR function of convex form.

18
That is, non-ET losses increase as water is applied to achieve ET
levels due to the inefficiencies of irrigation methods and the
inexactness of water scheduling. The amount of water not used in ET,
therefore, represents runoff, deep percolation, and/or residual
extractable water in the soil when the crop is harvested. The water
management implications of this type of yield function are discussed
further in the next sections.
In summary, considerable efforts have been directed toward
development of simple models for describing the yield response of crops
subjected to water stress conditions. The application of these models
to irrigation management appears to be tractable (Hill and Hanks, 1975).
Crop Phenology Model
As a plant goes through its life cycle, various changes occur.
Crop ontogeny is the development and course of development of various
vegetative and reproductive phases, whereas phenology is the timing of
the transition from one phase to the next phase as controlled by
environmental factors. To accurately simulate crop growth and yield
with biophysical models, crop phenology needs to be successfully
predicted (Mishoe et al., in press). Crop parameters needed for growth
simulation are closely related to the phenological stages of the
plant. These include the duration of leaf area expansion, stem and root
growth, as well as the onset and end of pod and seed growth. It is
therefore desirable to allow assimilate partitioning values in the model
to change as the plant progresses through its reproductive stages.
Currently, many of the practical yield response models have
coefficients that depend on crop growth stage (Ahmed et al., 1976;

19
Childs et al., 1977; Wilkerson et al., 1983; Meyer, 1985). However, in
some studies, the crop growth stages have been poorly defined. And most
applications of these models use only the mean development times and
assume that stochastic variation does not affect the performance of the
model. Hence, a systematic approach to define stages relative to
physiological development of the crop and to predict these stages under
various weather conditions is needed (Boote, 1982). This would lead to
more accurate application of yield response models. In the rest of this
section, several approaches to modeling phenology are described.
The wide range of controlling factors and crop responses makes
phenological modeling challenging. The effect of temperature as well as
photoperiod as controlling factors has long been recognized. The
concept of thermal time in the form of degree-days is used to account
for temperature effect. Degree-days are cumulative daily average air
temperature above the base temperature (Prine et al., 1975).
Most models are based on thermal time or photoperiod or a
combination of the two. Some models based on thermal time alone are
quantitative, based on the analysis of experimental evidence (Kiniry et
al., 1983; Tollenaar et al., 1979). Kiniry et al. found that the
photoperiod did not affect all of the cultivars of corn. Those that
were affected were still insensitive below a threshold photoperiod value
of between 10 and 13 hours. For wheat, a quadratic equation, based on
day and nighttime temperatures and photoperiod was applied by Robertson
(1968), and Doraiswamy and Thompson (1982) to predict the time between
phenological stages.
Other models are based on the hypothetical processes involved in
crop response (Mishoe et al., 1985; Schwabe and Wimble, 1976). Mishoe

20
et al. (1985) developed a phenological model based on physiological
processes of soybean. One important concept is that a critical period
of uninterrupted night length is needed to produce rapid flowering.
Also the promotional effect of night length is cumulative. An
accumulator (X) value needed to trigger an event is calculated from a
function of night length and nighttime temperature. When the cumulative
X becomes larger than a threshold level, it triggers the phenological
event such as flower initiation. These threshold values for different
stages are calibrated from experiments, and are variety dependent.
Incomplete knowledge of biochemical processes involved hampers the
development of process models. However, for production management,
models using thermal time and night length have successfully predicted
phenological events.
Objective Functions
An objective function is a quantitative representation of the
decision maker's goal. One may wish to maximize yield, net profit, or
water use efficiency. However, these objectives are not equivalent and
the use of different objectives may result in different solutions.
Maximizing yield per unit area. This objective may be economically
justified when water supplies are readily available and irrigation costs
are low. All production practices and inputs must be at yield
optimizing levels, and daily cycles of plant water potentials must be
managed within limits conducive to maximum seasonal net
photosynthesis. From an applied water management viewpoint, this
production objective is relatively easy to attain. Many applied
experiments (Salter and Goode, 1967) have shown that for many crops,

21
yields will be near their maximum values when root zone available water
is not depleted by more than 25 to 40 percent between irrigations.
Maximizing yield per unit water applied. As irrigation water
supplies become more limited or as water costs increase in an area, the
management objective may shift to optimizing production per unit of
applied water (Hall and Butcher, 1968; Stewart and Hagan, 1973; Howell
et al., 1975; Windsor and Chow, 1971). Hiler et al. (1974) have
demonstrated that significant improvements in water use efficiency are
possible by applying the Stress Day Index method. Stewart et al. (1975)
have more recently suggested a simplified management criterion by noting
that the maximum yield for a given seasonal ET deficit level tends to
occur when deficits are spread as evenly as possible over the growing
season. Thus, scheduling is based on the concept of high frequency
irrigation, i.e. applying small depths per irrigation at essentially
evenly timed intervals.
Maximizing net profit. Applying marginal value vs marginal cost
analysis to yield production functions, Stewart and Hagan (1973) were
able to determine optimum economic levels of production for maximum
water use efficiency, maximum profit under limited water supply, and
maximum profit under unlimited water supply, respectively. A problem
with this method is that it provides only general guidelines for water
management. These guidelines are most applicable to the average or
normal climatic conditions in a given region and, therefore, may not
apply to specific sites or specific years. In addition the guidelines
are seasonal in nature, i.e., they indicate only the seasonal irrigation
depth most likely to maximize net profit.

22
In recent years, numerous models (Dudley et al., 1971; Matanga and
Marino, 1979; Bras and Cordova, 1981; Huang et al., 1975) have been
developed to address the goal of profit maximization. Methodologies
such as dynamic programming are frequently utilized to illustrate how
optimal water scheduling or allocation strategies within the growing
season can be derived under stochastic conditions.
Risk analysis. Risk assessment of decision alternatives can be
approached in several ways. One of the more common approaches is an
expected value-variance (E-V) analysis where the decision maker is
assumed to maximize utility, where utility is a function of the expected
value and associated variance in returns. The specific functional form
of this relationship varies by individual depending upon each
individual's psychological aversion to risk. For example, the risk
averse individuals may be willing to trade a reduction in expected net
returns for a decrease in the variance of net returns.
Concerning within-season irrigation strategies, Boggess et al.
(1983) expressed the variance of net returns for a particular irrigation
strategy as
2 v2 2 . d2 2 . 2 2 . v2 2 0 ro n\
°,i - Yi °p + p °y. + T °x. ui',- 2opy.,yx. (2'13)
K 1 1 1 ’ 1
where and ay are the mean and variance of yield associated with irrigation
1 " 2
strategy i, P and ap are the mean and variance of crop price, y and
are the mean and variance of irrigation pumping cost per unit of
o
water, X. and at are the mean and variance of irrigation water applied
1 *i
for irrigation strategy i, and Opy ^ is the covariance between PY^

23
and yXi . Then the relative contribution of each component random
variable (price, yield, pumping cost, and irrigation water) to the
variance of it was analyzed by normalizing the above equation. Their
analysis indicated that irrigating soybeans increased the expected net
returns above variable costs and decreased the variability compared to
non-irrigated soybeans. Probability curve and convolution of risk
techniques were subsequently applied to quantify and interpret the risks
associated with alternative irrigation strategies.
Optimization Methods
Systems analysis basically is a problem-solving technique wherein
attempts are made to build a replica of a real world system or
situation, with the objective of experimenting with the replica to gain
some insight into the real world problem. It encompasses several
optimization techniques such as dynamic programming, linear programming
and simulation. Generally in dealing with irrigation management,
dynamic programming techniques are applied to models which are spatially
limited to a field of single crop and temporally to one growing season
(Hall and Butcher, 1968; Windsor and Chow, 1971; Dudley et al., 1971;
Howell et al., 1975; Bras and Cordova, 1981). Linear programming
algorithms on the other hand are utilized to analyze farm level cropping
patterns models (Windsor and Chow, 1971; Huang et al., 1975; Matanga and
Marino, 1979). Simulation is usually used to evaluate specific
irrigation policies (Ahmed et al., 1976; Jones and Smajstrla, 1979).
Dynamic programming models. Characteristically, dynamic
programming problems are decomposed into stages and decisions are
required at each stage. The decision at any stage transforms the system

24
States and increments the value of the objective function at a
particular stage. Changes in the system states may be described by a
probability distribution.
In the Howell et al. (1975) dynamic programming formulation, the
decision process consisted of whether to irrigate 0., 0.25, 0.5, 0.75,
or 1.0 times the potential ET during each of five crop growth stages for
grain sorghum. The states consisted of the remaining water to be
allocated at each stage and the soil water status, a stochastic state
variable. The stochastic state transitions were calculated by utilizing
simulation of a soil water balance model.
The solutions produced an optimal sequencing of water application
based on expected weather patterns and on differential crop
sensitivities to water deficits during each growth stage. The solutions
were tabulated. The table provided the stage-by-stage optimal policy.
As the season progressed, realizations of rainfall and ET caused the
soil water and the remaining water supply to vary from year to year.
Therefore, at each stage, the irrigator could update the optimal policy,
using the table to optimally allocate water during the remaining part of
the growing season.
Bras and Cordova (1981) attempted to solve the same problem by
using an analytical approach which included a physical model of a soil-
climate system and a stochastic decision-making algorithm. Expressions
for the soil water transition probabilities over a given time period and
the first two moments of associated actual evapotranspiration were
derived analytically. A stochastic dynamic programming algorithm was
then used to determine optimal control policies at each irrigation
decision point, conditional on the state of the system (soil water
content).

25
Dividing the irrigation season into N stages and taking irrigation
depth (In) at decision stage n as a decision variable, the objective
function (Bras and Cordova, 1981) can be formulated as:
*
B =
N I
Max E [ z R n ] - PC (2.14a)
Iett n = l n
where I =
^2’ *’* *
I
R n =
n
I I I
P Ynn - 6 IDnn - yCn (2.14b)
★
B =
maximum net return,
E[ ] =
expectation operator,
PC =
production costs different from irrigation costs,
TT =
feasible set of control policies,
Tn =
type of control applied at decision stage n,
N =
R1"-
Kn
number of decision stages in the growing season,
net return by irrigating In at decision stage n,
P =
price per unit of crop yield,
contribution of irrigation decision i to actual yield,
3 =
!D!" -
unit cost of irrigation water,
depth of irrigation water associated with operation
policy In,
Y =
fixed cost of irrigation (labor cost), and
c1"
c'"
= 0, when ID*" = 0;
n
= 1, otherwise.
1, otherwise.

26
Since the production cost (excluding irrigation costs), PC, is a
constant value, the optimal control law that maximizes the above
function will be the same that
N rn
Max E [ l R n ] (2.15)
I sir n = l
The dynamic programming technique then decomposes this problem into
a sequence of simpler maximization problems which are solved over the
control space.
Linear programming models. If the objective is to select crops to
grow on a farm where water is limiting, linear programming techniques
may be applied. Windsor and Chow (1971) described a linear programming
model for selecting the area of land to allocate to each crop and the
irrigation intensity and type of irrigation system to select. As
defined, the set of decision variables, represented the number of
hectares of crop 1 to grow in field (or soil type) i, using irrigation
practice j, and irrigation system k. The solution would select X-j^ to
maximize net profit for the farmer. A required input was net profit
associated with X — ^ , C — ^ which included a crop yield response to
various conditions. Windsor and Chow used dynamic programming to
estimate crop yield response for optimal unit area water allocation.
Their model is designed for decision analysis prior to planting.
Their model can also be modified to determine when to plant the crop to
take advantage of seasonal rainfall or water availabilities. The
within-season scheduling of irrigation on a farm basis (for multiple
fields) after crops are planted would require a different formulation.
Trava-Manzani11 a (1976) presented one example of such a problem.

27
In the study by Trava-Manzani11 a (1976), the objective was to
minimize irrigation labor costs in a multi-crop, multi-soil farm subject
to constraints on daily water availability, water requirement of the
crops and the irrigation method being used. The mathematical
formulation of the problem was of zero-one linear integer programming.
However, because of the nature of the problem formulation was then
transformed to a linear programming model. Two linear programming
techniques, Simplex procedure and the Dantzig-Wolfe decomposition
principle, were successfully used to resolve the solutions.
Simulation models. Simulation can be used to evaluate specific
irrigation policies in an enumerative search for the best policy among
those tested. For this approach, models of the soil water status and
crop yield responses are required (Ahmed et al., 1976; Jones and
Smajstrla, 1979). By defining several explicit, alternate policies and
simulating results for one or more crop seasons, crop yields or net
returns can be compared for the different policies and the best policy
can then be selected. This procedure will not necessarily produce an
optimal solution, but from a practical viewpoint, it can provide
valuable information to decision makers.
In many of the reported studies (Dudley et al., 1971; Yaron et al.,
1973; Minhas et al., 1974; Ahmed et al., 1976), the lack of suitable
crop response models was cited as a major limitation. It may not be
realistic to estimate crop yield response over a broad range of
conditions by empirical approach. Details are needed in the model.
Dynamic crop growth models were developed to predict growth and yield of
crops using more theoretical considerations and physiological detail
(Curry et al., 1975; Childs et al., 1977; Barfield et al., 1977;

28
Wilkerson et al., 1983). These models are attractive because crop
growth stresses, such as those caused by nutrition or pests, can be
included, in addition to those caused by water deficits, to provide a
more comprehensive tool for crop production management.
However, the crop growth models may have so much detail that they
may not be suitable for the problem of long-term production
management. Models at other levels of sophistication to describe crop
system responses to management practices, such as irrigation, are likely
to be more useful. Thus, a general framework for optimization of
multiple cropping systems using both optimization and simulation
concepts will be developed.

CHAPTER III
METHODOLOGY FOR OPTIMIZING MULTIPLE CROPPING SYSTEMS
Mathematical Model
Several alternative formulations of the multiple cropping problem
are studied with regard to their practicality for solutions. These are
reviewed, and the most suitable one is described in detail.
Integer Programming Model
Sequencing is concerned with determining the order in which a
number of ‘jobs' are processed in a 'shop' so that a given objective
criterion is optimized (Taha, 1976). In the multiple cropping problem
the variable, X-^ t , is defined and equal to one when crop i, variety
• j 111 ^
j, planted at t^ still grows in the field at time t^. Otherwise, it is
equal to zero. It is also assumed that the growth season for crop i,
variety j, planted at t^ is A^-t and the associated net return is
C— £ . To properly describe the multiple cropping problem, two
constraints are considered: only one crop can occupy the field anytime,
and a growing season is continuous. Provided with the definition of
variables, X..
, and constants A... and C,-,. , the formulation of
ijtj ijt;
an objective function and constraint conditions is
29

30
Max 1 * (Ci it } (Xi it t }
i j t x t g J 1 J 1 2
(3.1a)
s • t ^ • i i T*
1jtlt2 1JV2 2
X.. 1 , for all t„ ,
ijt1 1JV2 ¿
(3.1b)
(3.1c)
VAtjt,
xiit t = 0 or
t2=t1 1Jtr2
11 "*”A - j t
t 1 x,'jtlt2 * Autj- for a11 <3-ld)
where T2 is the total number of weeks of an N-year production horizon.
The first constraint (3.1b) simply says that a production horizon is of
T2 periods. The second constraint (3.1c) indicates that at any instant
of time t2 only one crop is scheduled to grow in the field. The
constraints represented by (3.Id) are imposed to ensure the continuity
of a growth season. However, these either-or constraints cannot be
implemented directly in a mathematical programming algorithm. To
overcome this difficulty, new variables, Y. . are defined. When crop
i Jt
1
i, variety j, is scheduled for planting at t. then Y.. = 1.
1 1 Jtj
Otherwise, Y. .. = 0. This problem is then a zero-one integer
1 Jt^
programming model. The formulation is

31
Max
1 - l (Cijt } (Yijt >
ijt1 1JC1
(3.2a)
s.t.
l Xi1t t * T2 ’
ijt1t2 1Jtlr2
(3.2b)
l X < 1 , for all t
ijt1 1Jtlr2 ¿
(3.2c)
VA1jt
tit 1 (xijt1t2,(1 - Yijt1)* °- for 411 ’-J-V
(3.2d)
tl+Aijt
tJt ' 'l (3.2e)
But several difficulties are associated with this formulation. It
is noted that the number of X variables in the formulation is equal to
(I * J * * T2) , directly dependent on how often the decision needs
to be made. Assume that a decision is to be made every week. For a
4.5-year planning horizon, the total number of X variables is estimated
as 4 * 2 * 234 * 234 = 438,048. This cannot be solved economically by
the existing integer programming algorithm (Land and Powell, 1979).
Moreover, the nonlinear terms in the model should generally result in a
computationally difficult problem. Still, the need of constants,
C. .. and A- .. , requires the simulation of as many combinations of
l J t ^
(i»j*t^). Because of all of these shortcomings, the integer programming
approach was not pursued further.

32
Dynamic Programming Model
Because of the nature of dynamic programming techniques which solve
a problem by sequential decision-making, the constraint of appearance of
a single crop in the field anytime is implicitly coupled in the
formulation. In a sense, sequential decision-making provides an
interactive mode in the process of solution. When it is required, net
return associated with a specific crop candidate is generated and then
evaluated. It is very beneficial in terms of storage and computer
time requirements.
In a crop production system, management practices consist of
irrigation strategy, fertilizer application, pest and disease control,
crop rotation, etc. Discrete values assigned to each level of a
specific management practice represent the state of a system. For
example, percentage of available water in the soil profile (soil water
content), is a primary indicator for irrigation management. Under an
unlimited water supply situation, without losing generality, (C,W,N) are
chosen as state variables to identify state transition in the
optimization model, where C stands for the preceding crop, W for soil
water content, and N for soil nutrient level.
The inclusion of nutrient level (N) in the formulation is to
express the potential application to other areas of interest.
Nonetheless, irrigation policy is solely emphasized in the iterative
functional equation, because this framework is to be demonstrated with
the application to irrigation management.
The dynamic programming model of multiple cropping is formed as
follows. First, the optimal value function F(C,W,N,t) is defined as

33
F(C,W,N,t) = maximum return obtainable for the remainder t periods,
starting with the current state (C,W,N). (3.3)
In terms of these symbols, Bellman's principle of optimality gives the
recurrence relation,
F(C, ,W.,N.,t) = Max
C2 S(C1 ,t)
R*(C2,I*,t) + F(C2,Wf,Nf,t-a(C2))
(3.4)
where
W.
i
N.
l
S(Cj,t)
a(C2)
•k
I
*
R
state of soil water at the beginning of the season,
state of soil water at the end of the season,
value of nutrient level at the start of the season,
value of nutrient level at the end of the season,
preceeding crop,
selected crop, decision variable,
proper subset of crop candidates dependent on
and season t, due to practical considerations of crop
production system,
growth season of crop C^,
optimal realization of irrigation policies, a vector
•k ie k
(I^, I2, ... Ik) represents the depths of irrigation
water associated with individual operations,
maximum return obtained from growing crop C2 by
★
applying optimal irrigation policy I .
The state transition from the start of a season to the end of a
season is determined by the system equations:

34
wf = g (c2, i*, w.),
Nf = h (C2, I*, W., N.).
(3.5a)
(3.5b)
These functions are not explicitly expressible. It is not
realistic to represent the complicated soi1-piant-atmosphere continuum
in terms of simple functional relationships. Simulation models may be
employed to carry out state transitions.
In order to use the iterative functional equation, it is necessary
to specify a set of boundary conditions to initialize the computational
procedure. Because the functional equation expresses the optimal value
function at t in terms of the optimal value function at (t - a(C2)),
the boundary conditions must be specified at the final stage t = 0.
Formally, the appropriate boundary conditions are
F(C,W,N,t) = 0, when t = 0
(3.6)
F(C,W,N,t) = -~, when t < 0
for every C,W,N.
In addition, an optimal policy function, the rule that associates
the best first decision with each subprogram, is needed to recover the
optimal decision for the original whole problem. The optimal policy
function in the problem is defined as
(3.7)
where = soil water status at end of a season, = nutrient level at
end of a season, C2 = index of the selected crop, a(C2) = growing season
of C2 .

35
Starting with the boundary conditions, the iterative functional
equation is used to determine concurrently the optimal value and policy
functions backward. When the optimal value and decision are known for
the initial condition, the solution is complete and the best cropping
sequence can be traced out using the optimal policy function. Namely,
the optimal solution is F (CQ, WQ, NQ, T), where T = the span of N-year
growing period, (Cq, Wq, Nq) is the initial condition in which
production plan is to be projected.
However, it is not very clear whether certain states (C, W, N, t)
are relevant to the possible optimal system. Total enumerations of
optimal value functions F(C, W, N, t) are required to resolve the
★
optimal solution F (CQ, WQ, NQ, T). In terms of computational
efficiency, this dynamic programming model is not very appealing.
Therefore, a more comprehensive, efficient model needs to be
investigated.
Activity Network Model
Selecting crop sequences to optimize multiple cropping systems can
be formulated as an activity network model. In a network, a node stands
for an event or a decision point. An activity, represented by an arc,
transfers one node to another. In this particular application to
irrigation management, nodes represent discrete soil water contents at
every decision period. Arcs, not necessarily connecting with adjacent
nodes, have lengths that denote net returns associated with the choice
of crop and irrigation strategy. The structure of the network is
demonstrated in Figure 1, where C. is crop variety i and S. is
«1
irrigation stragegy j. The S and T nodes are dummy nodes, representing
the source and terminal nodes of the network, respectively.

36
As noted in Figure 1, all arcs point in one direction from left to
right. There is no cycle in this network. This feature will prove
advantageous in developing a simplified algorithm for network
optimization. While circles are all potential decision nodes, solid¬
line ones are actual decision nodes which are generated by system
simulations, and dashed-line circles are fictitious, not accessible to
other nodes. In the dynamic programming model, these inaccessible nodes
are not detectable so that efforts on computing optimal values for
dashed-line nodes are wasted. In contrast, the inaccessible nodes are
detectable in the activity network model and more efficient computation
is accomplished. Under different weather conditions, networks of a
multiple cropping system vary.
In designing multiple cropping systems, several principles verified
by field experiments should be considered. These are: an idle period
may be required to restore the soil water reservoir, or to alleviate
pest population or chemical residues; consecutive scheduling of the same
crop may require more intensive management; and genetic traits may
prohibit planting certain varieties in some season of a year. Some of
these system criteria can be incorporated into simulation to generate a
multiple cropping network. Other aspects of the system (i.e. improper
consecutive scheduling of the same crop) restricted by model
representation may be reconsidered by a post-optimization scheme. In
such a manner, a more realistic system network is considered for
obtaining optimal crop scheduling.
The objective of optimizing multiple cropping systems is defined to
maximize total net return over a specified long-term period. In network
analysis terminology, it is to seek the 'longest path' of a network.

Soil Water State
Decision Time (weeks)
GJ
Mgure 1. A system network for multiple cropping.

38
Since devaluation of cash value needs to be taken into consideration in
a long-term production horizon, total discounted net return of future
profits is to be maximized in the study.
A longest path solution algorithm can be expanded to search for the
K longest paths from the start node to terminal node. Determining the K
longest paths provides useful information for system analysis. The
advantages are as follows: First, such information provides a means of
assessing the sensitivity of the optimal solution to possible suboptimal
decisions. Second, one may be interested in a class of solutions and
not just in a single solution. Third, the K longest paths provide a
measure of the robustness of the underlying model when the data are
approximate. Moreover, in case post optimization analyses are necessary
to impose additional constraints on good solution paths in a system
network, calculation of the K longest paths provides a means of
efficient computation.
As described, an arc length in a multiple cropping system network
represents the return resulting from an optimal, single crop production
season. This represents a second-level optimization problem, which is
referred to as within-season management, i.e. optimal irrigation
scheduling. The problem of temporal water allocation in an irrigated
field consists of deciding when and how much water to apply in order to
maximize net returns. This problem is complicated by the uncertainty of
weather and by the fact that many crops exhibit critical growth stages
during which the crop sensitivity to soil water stress is high.
Systems analysis techniques such as simulation and dynamic
programming have been used in the past to determine the optimal
operation policies in an irrigation system. The necessity of

39
implementation of more dynamic, detailed crop phenology and growth/yield
models makes mathematical programming impractical. Simulation therefore
is required to evaluate within-season management strategies. As a
result, the activity network model coupled with the simulation-
optimization techniques provides a framework for optimizing multiple
cropping systems by selecting crop sequence and determining optimal
within-season management practices.
Thus, methodology is developed and summarized as follows:
1. To provide base data, models for simulating crop growth and
yield are constructed.
2. Considering systems options and constraints, a realistic
multiple cropping network is generated.
3. Applying the longest path algorithm, the K longest paths are
solved to evaluate various cropping sequences.
Crop-Soil Simulation Model
The crop-soil simulation model serves two purposes in optimizing
multiple cropping systems. First, the simulation is necessary to define
the state transitions (i.e. soil water contents) in the previously
discussed mathematical model. Secondly, simulation is an approach to
study irrigation management strategies. The problem of optimally
distributing irrigation water over the growing season is difficult
primarily because of imperfect knowledge of rainfall distribution over
the season. In addition, uncertainty in the distribution of other
weather variables which affect crop yields complicates the optimization
problem.

40
In general, uncertainty in the time distribution of inputs or
resources to a process which is to be optimized can be treated using
some form of stochastic programming, the inputs as random variables, and
the objective function to be optimized as some fairly simple production
function of inputs. Unfortunately, the complex nature of crop
production lends itself to simple production functions only in a general
statistical sense. In order to investigate the effects of irrigation
decisions at different points within the growing season, a detailed
simulation model is useful.
Such a simulation model is intended to integrate the effects of
weather variables and irrigation schedules on crop growth. It simulates
the progress of a crop during the time in which it interacts with its
environment. As the crop grows from day to day and uses the water
stored in the root zone, water deficits develop and are counter-balanced
by irrigation or rainfall. This closed loop simulation describes the
frequency and duration of water deficits that affect evapotranspirati on
and crop yield. By imposing a series of alternate irrigation strategies
on the simulation model, one can evaluate the effect on yield of various
strategies. To find the optimal solution, ranking the estimated net
return gives the most efficient strategy for a given specific weather
pattern.
As discussed by Jones and Smajstrla (1979), simulation models at
different levels of sophistication have been developed to study the
problem. In this work, a crop yield response model is included with the
soil water balance model so that irrigation strategy for maximizing net
return can be studied. The soil water balance model is primarily used
to provide the necessary data (daily ET) for describing the yield

41
response of the crop by the yield model. In addition, a crop phenology
model is coupled to systematically predict growth stage relative to
physiological development of the crop. In so doing, different levels of
water use of the crop at various growth stages can be realistically
simulated, and more accurate estimation of yield is possible. These
models are described in detail below.
Crop Phenology Model
Corn and peanut phenology. For corn and peanut, heat units are
used to predict physiological development. In the model, the
physiological day approach, a modification of the degree-day method is
used. Because the units of degree-day are products of temperature and
time, it is convenient to express the accumulation in units of
physiological time. To accomplish this, the degree-day unit is
normalized with respect to a given temperature, 30 C? The
physiological days are calculated as follows.
PD = 0
for T < 7,
n
PD = l
i =1
T (Ati) - 7
30 - 7
At â– 
for 7 < T < 30,
n
PD = l
i =1
45 -T(At.)
45 - 30
At.
1
for 30 < T < 45,
(3.8)
PD = 0
for T > 45,

42
where PD = physiological clay, T(At.j) = temperature in the time
interval At. . Physiological days accumulate until specific thresholds
are reached. Stages occur at the thresholds the stages are said to be
set. In this study, the crop season is divided into four stages. For
corn and peanut, stages of growth and threshold values of physiological
development are shown in Table 1.
Wheat phenology. For wheat, four stages, planting to late
tillering, late tillering to booting, heading to flowering, and grain
filling are used to characterize the wheat life cycle. Time between
phenological stages is predicted by using the Robertson model (1968).
The approach uses the multiplicative effects of temperature and
daylength to determine time between events. In the model, the average
daily rate AX of development is calculated as
AX = (a1(L-aQ) + a2(L-aQ)2) (b1(T1-bQ) + ^(T^)2 +
b3(Vbo} + b4(Vbo)2) (3‘9)
where L = daily photoperiod,
Tj = daily maximum (daytime) temperature,
T? = daily minimum (nighttime) temperature.
And aQ, a^, a^, bQ, b^, etc. are characteristic coefficients of specific
stages. Values of these coefficients are shown in Table 2. A new
stage (S2) is initiated when the summation
S2
XM = l AX = 1 .
(3.10)

43
Table 1. Threshold values for physiological stages of growth of corn and
peanut.
Threshold Values of
Phenological Development
Crop
Stage of Growth
(Physiological Days)
Source
Full
Planting to silking
38.7
Bennett
season
Si 1 king to blister
45.1
(personal
corn
B1ister to early
66.3
communi-
soft dough
Early soft dough to
81.4
cation)
maturity
Short
Planting to silking
33.6
Agronomy
season
Si 1king to blister
40.7
Facts, 1983
corn
Blister to early
57.7
soft dough
Early soft dough to
70.4
maturity
Peanut
Planting to beginning
27.3
Boote,
flowering
Beginning flowering to
42.4
1982
a full pod set
A ful 1 pod set to
68.2
beginning maturity
Beginning maturity to
97.3
harvest maturity

Table 2. Coefficients of a multiplicative model for predicting wheat phenological stages
(Robertson, 1968).
Coefficients
Phenolog
ical Stage
Planting to
Emergence
Emergence to
Late Tillering
Late Tillering
to Booting
Heading to
FIowering
Grain
Fi 11ing
a0
★ *
8.413
10.93
10.94
24.38
al
★ *
1.005
0.9256
1.389
-1.140
a2
**
0.0
-0.06025
-0.08191
0.0
b0
44.37
43.64
42.65
42.18
37.67
bl
0.01086
0.003512
0.002958
0.0002458
0.00006733
b2
-0.000223
-0.00000503
0.0
0.0
0.0
b3
0.009732
0.0003666
0.003943
0.0003109
0.00003442
b4
-0.000227
-0.00000428
0.0
0.0
0.0
** In this early stage, growth is independent of daily photoperiod.

45
The summation (XM) is carried out daily from one phenological stage
to another S^.
Primarily, five growth stages and centigrade temperatures were used
in the Robertson model. Modification by combining stages 1 and 2 into a
single stage has been made to accommodate to the study.
Soybean phenology. The model of soybean phenology, developed by
Mishoe et al. (in press) and implemented by Wilkerson et al. (1985) is
complicated. A version of the model was adapted for the study. The
model uses cultivar specific parameters, night length, and temperatures
to generate physiological development. The development phases of
soybean are described in Table 3. Some phases of development are
dependent on night length and temperature whereas others are dependent
only on temperature.
Temperature effect on development is expressed as physiological
time. Physiological time is calculated as the cumulative sum of rates
of development, starting at the beginning of a phase. The end of a
phase occurs when the cumulative physiological time reaches the
threshold as indicated in Table 3.
A nighttime accumulator is used to represent photoperiod effects on
development. The nighttime accumulator of the model is represented as
fol1ows:
Xm = I TF * NTA (3.11)
where X = the accumulator value to trigger an event,
TF = temperature factor computed using the function shown
in Figure 2,
NTA = night time accumulator function shown in Figure 3.

46
Table 3. Description and threshold values of phenological stages and
phases for soybean cultivars (Wilkerson et al., 1985).
Threshold
Growth
Stage
Description
Phase
'Bragg'
'Wayne'
I
Physiological time from planting
to emergence
1
6.522
6.522
Physiological time from planting
to uni foliate
2
10.87
10.87
Physiological time from unifoliate
to the end of juvenile phase
3
2.40
2.40
Photoperiod accumulator from the
end of juvenile phase to floral
induction
4
1.00
1.00
II
Physiological time from floral
induction to flower appearance
5
9.48
9.48
Photoperiod accumulator from
flowering to first pod set
6
0.14
0.20
Photoperiod accumulator from
flowering to R-4
7
3.0
6.0
III
Photoperiod accumulator from
flowering to the last V-stage
8
0.16
0.5
Photoperiod accumulator from
flowering to the last possible
flowering date
9
0.575
0.6
IV
Photoperiod accumulator from
flowering to R-7
10
20.35
14.5
Physiological time from R-7
to R-8
11
12.13
10.0

Physio. Days for Floral Induction ^ Physio. Day / Real Day
47
2. Rate of development of soybean as a function of temperature.
Figure 3. Effect of night length on the rate of soybean development.

43
Figure 2 is the normalized function to calculate physiological time. In
Figure 2, data used for minimum, optimal and maximum temperatures were
7, 30, 45°C, respectively. Figure 3 shows the relationship between
night length and physiological days to development based on phase 4, the
floral induction phase. Because the threshold for development for phase
4 was defined to be a constant (1.0), the relationship varied with
cultivars in Figure 3. The values for this relationship of 'Bragg1 and
'Wayne' soybean shown in Table 4 were taken from Wilkerson et al.
(1985). Based on these calibrated curves, thresholds (Table 3) for
other photoperiod phases also vary among cultivars.
The amount of development during one night is calculated by
multiplying the average temperature for the nighttime by the inverse of
days to development at a given night length. The function (equation
3.10) is accumulated using a daily time step. When the prescribed
threshold is reached, the event is triggered and the crop passes into
the next stage.
Crop Yield Response Model
Crop growth is closely correlated to evapotranspi rati on (ET).
Based on this principle, yield response models which interpret ET
reduction below potential levels as integrators of the effects of
climatic conditions and soil water status on grain yield were
developed. To account for increased sensitivity to water stress at
various stages of growth, and the interactive effects between crop
growth stages, the Jensen (1968) multiplicative form
Â¥/vpâ–  :1 X3 (3.12)

49
is used, where potential yield, Yp, is varied as a function of planting
dates. Maximum yield factors that reduce yield of each crop below its
maximum value as a function of planting day for wel1-irrigated
conditions are shown in Figure 4. The length of each stage is predicted
by the use of crop phenology model.
To obtain crop sensitivity factors () to water stress, intensive
literature studies have been made. Boggess et al. (1981), based on many
simulations from SOYGRO were able to quantify these factors (shown in
Table 5) by statistical analysis. Smajstrla et al. (1982) also
estimated A., for soybean in a lysimeter study, and their estimates
of A., were similar to those found by Boggess et al. (1981). For corn
and peanut, attempts have been made without success to obtain the
factors from a series of experimental studies (Hammond, 1981). The
factors used in Table 5 were derived from FAO publication (Doorenbos,
1979). For wheat, no data were available for Florida conditions.
Therefore, an experiment on wheat to be described in the later chapter
was performed to obtain A^ and related crop response to irrigation
practices.
Soil Water Balance Model
In order to predict ET rates under well-watered and water stressed
conditions, a soil water balance model was developed to integrate
existing knowledge about crop water use, weather patterns, and soil
properties into a framework compatible with irrigation objectives. A
model previously described by Swaney et al. (1983) was adapted for this
study.

50
Table 4. Values of the parameters for the nighttime accumulator
function of the soybean phenology model (Wilkerson et al.,
1985).
Value of The
Parameters
Name of
Parameters
'Bragg'
'Wayne'
THVAR (day)
63.0
32.0
DHVAR (day)
2.0
2.0
TNLG1 (hour)
5.2
5.2
TNLGO (hour)
11.0
9.5
Table 5. Crop sensitivity factors, , for use in the simulation.
Crop Sensitivity Factors
Crop
Stage 1
Stage 2
Stage 3
Stage 4
Source
Corn
0.371
2.021
1.992
0.475
Ooorenbos (1979)
Soybean
0.698
0.961
1.034
0.690
Boggess et al. (1981)
Peanut
0.578
1.032
1.531
0.772
Ooorenbos (1979)
Wheat
0.065
0.410
0.114
0.026
Personal observation

Yield Factors Yield Factors
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
Figure 4
Julian Day
Maximum yield factors that reduce yield of each crop below its maximum value
as a function of planting day for well-irrigated conditions.

52
The soil water balance model divides the soil into two zones: an
evaporation zone in the uppermost 10 cm of the soil, and a root zone of
variable depth underneath. This shallow evaporation zone is selected
for the sandy soil used in the model, and would not be sufficient for
heavier soils. Root zone depth is increased during the season by
simulating root growth. Under wel1-irrigated conditions, rooting depth
of the crop as a function of time is shown in Figure 5. The soil used
is characterized by its field capacity and permanent wilting point.
Evaporative water loss is removed from the evaporation zone and
transpiration water is lost from both zones depending on their
respective water contents. Due to the high infiltration rates of the
sandy soil, all rainfall is added to the profile until field capacity is
reached, and excess water is assumed to drain from the profile. When
the fraction of available soil water reaches a critical level of a pre¬
determined irrigation strategy, irrigation water is applied and treated
as rainfall. If both rainfall and irrigation occur on the same day, the
effect is additive.
The soil water balance model requires daily rainfall and potential
evapotranspiration (ETp), which is estimated by a modified version of
the Penman equation. The ETp is then used to calculate potential
transpiration (T ) using a function of leaf area index (Ritchie, 1972):
T = 0
P
T = ET (0.7 *
p p^
L . < 0.1
ai
(3.13a)
0.1 < L . < 3.0 (3.13b)
al
3.0 < Lai
(3.13c)

Rooting Depth (cm) Rooting Depth (cm)
Figure 5. Crop rooting depth after planting under well-irrigated conditions.

54
where L â–  = leaf area index. For wel 1 -i rrigated crops, leaf area index
31
functions as seasons progress are shown in Figure 6.
Values of actual evaporation (E) and transpiration (T) limited by
available water in the two soil zones are calculated from potential
values using time from the last rainfall in the case of E, and a soil
water stress threshold (0C) in the case of T. Calculation of
transpiration is as follows:
T = T ©' > Q (3.14a)
p c
T = Tp * (0'/0c) e' < ©’ (3.14b)
I
where 0 = ratio of soil water in root zone, as a fraction of field
I
capacity, 0 = (0p- 0d) / (0fc- 0d),
I I
0 = critical value of 0 below which water stress occurs and
transpiration is reduced, various values are used for
different growth stages and crops,
0 = volumetric water content of root zone,
r
0^ = lower limit of volumetric water content for plant growth,
0^c = field capacity of the soil.
Two stages of evaporation from soil are implemented. In the
constant rate stage (immediately following rainfall event or
irrigation), the soil is sufficiently wet for the water to be evaporated
at a rate
E = Min (E , W )
^ p eJ
(3.15)

Leaf Area Index Leaf Area Index
Figure 6. Leaf area index for wel1-irrigated crops as a function of time.

56
evaporation below the canopy. In the falling rate stage (stage 2),
evaporation is more dependent on the hydraulic properties of soil and
less dependent on the available atmosphere energy. For each subsequent
day, the daily evaporation rate is obtained by (Ritchie, 1972)
E = Min {
(3.16)
where a is a constant dependent on soil hydraulic properties. For sandy
soil, a = 0.334 cm day" ^
For practical application, the Penman equation is considered the
most accurate method available for estimating daily ET. The Penman
formula for potential evapotranspi rati on is based on four major climatic
factors: net radiation, air temperature, wind speed, and vapor pressure
deficit. As summarized by Jones et al. (1984), the potential ET for
each day can be expressed as
ET
ARnA + yEa
(3.17)
P
A + y
where ETp = daily potential evapotranspirati on, mm/day
a = slope of saturated vapor pressure curve of air, mb/°C
2
Rn = net radiation, cal/cm day
X = latent heat of vaporization of water, 59.59-0.055 T
avg
Ea = 0.263(ea - ed) (0.5 + 0.0062u2)
ea = vapor pressure of air = (emax + emin) / 2, mb

57
ed = vapor pressure at dewpoint temperature Td
(for practical purposes Td = T ^ ),
u^ = wind speed at a height of 2 meters, Km/day
y = psychroinetric constant = 0.66 mb/°C
emax= maximurn vapor pressure of air during a day, mb
em^n= minimum vapor pressure of air during a day, mb.
Saturated air vapor pressure as a function of air temperature,
★
e (T), and the slope of the saturated vapor pressure-temperature
function, A are computed as follows:
e*(T) = 33.8639{(.00738T + .8072)8 - .000019(1.8T + 48) + .001316}
(3.18)
A = 33.8639{0.05904(0.00738T + 0.8072)7 - 0.0000342} (3.19)
In general, net radiation values are not available and must be
estimated from total incoming solar radiation, R$ , and the outgoing
thermal long wave radiation, R^. Penman (1948) proposed a relationship
of the form
Rn = (1-ct) Rs - Rb (3.20)
where Rn = net radiation in cal/cm day,
?
Rs = total incoming solar radiation, cal/cm day
R^ = net outgoing thermal long wave radiation,
a = albedo or reflectivity of surface for R^.

58
Albedo value a is calculated for a developing canopy on the basis of the
leaf area index, Lai, from an empirical equation (Ritchie, 1972),
a = a + 0.25 (a - a ) L . (3.21)
s v s' ai
where a$ is average albedo for bare soil and a for a full canopy is a.
And an estimate of Rp is found by the relationship:
R, = ctT4(0.56 - 0.08/e ,) (1.42R /R - 0.42) (3.22)
b d s so
-ft ? n
where o = Stefan-Boltzmann constant (11.71*10" cal/cm day/ K),
T = average air temperature in°K (°C + 273),
R = total daily cloudless sky radiation,
so J J
Values of R are available from weather stations in Florida. Clear-sky
s
insolation (R ) at the surface of the earth though needs to be
estimated. The equation (3.16), along with the discussed procedures for
estimating variables, is then used to calculate potential ET from a
vegetated surface.
The calculation of potential evaporation below the canopy, Ep, is
essential to predict soil evaporation when the surface is freely
evaporating. Proposed by Ritchie (1972), Ep is calculated as follows:
Ep = (A/(A + Y))Rn (3.23)
where Rn is net radiation at soil surface.

59
Irrigation Strategy
In order to study irrigation decisions, irrigation options input by
the user are available to the simulation model. The irrigation
strategies take the following form. The grower will irrigate on any day
of the season, if the water content in the root zone of the soil is
depleted to the threshold value (70% of availability by volume)
specified by the strategy. If the condition is met, irrigation water is
applied in an amount specified by the user. Frequent irrigation
applying less water per application (1 cm) is used in the model. On the
other hand, the rain-fed strategy depends totally upon rainfall.
Model Implementation
In order to study multiple cropping systems as well as associated
management strategies, models are needed to summarize and operationalize
knowledge about plant growth, yield, weather patterns, soil properties
and economics into a framework compatible with system objectives.
Therefore, computer programs were written in FORTRAN 77 to evaluate the
methodology. Figure 7 shows a schematic diagram for the methodology.
As outlined in the previous section of mathematical model, in order to
optimize multiple cropping systems, three independent steps, system
description, generation of network, and network optimization are
essential. Detailed descriptions and source code of subroutines to
execute the methodology are given in appendix A. The purpose of this
section is to provide discussions on model implementation in general.

60
Figure 7
A schematic diagram for optimal sequencing of multiple
cropping systems.

61
Network Generation Procedures
As discussed, nodes of a system network are specified by their time
coordinate and their system states. In this particular application, it
is proper to use a weekly decision interval. For limited water
retaining capacity of sandy soil, soil water contents as state variable
are discretized by an 1% interval between field capacity (10%) and
permanent wilting point (5%). Hence, there are a total of 6 states of
the system.
Net profit is gross receipts from crop sale minus total variable
cost. The variable cost for crop production is calculated by the
collective cost of production excluding irrigation plus variable cost of
seasonal irrigation. In planning of longterm production, devaluation of
cash value needs to be taken into consideration. Assume current
depreciation rate is i (12%). Present value of a future sum (F) is
calculated as
P = F / (1 + i )n (3.24)
where n is the year when F occurs. When F will be the net return of
future crop production, P is then the discounted net return evaluated at
the planning time.
In order to have a multiple cropping system network, simulation
techniques are applied. The tasks of these simulations are to keep
track of soil water status daily in order to be compatible with
irrigation objectives, to project the next crop and its planting date
(new nodes), and to estimate returns (arc lengths) related to the
decisions.

62
In the simulation, a crop season includes a one-week period to
allow for land preparation, and one week to allow for the harvesting
operation. Once a crop and irrigation strategy are decided, phenology
and soil water balance models are used every day to simulate the states
of the system. After all simulations of one single season for different
crops and irrigation strategies are performed, several new nodes for the
next crop are generated and new arcs are extended. In simulations, the
limitations on planting seasons of specific crops are shown in Figure 8
(personal communication with extension agent, D.L. Wright). Yield also
depends on the time during each interval when planting occurs.
In the process of optimizing a network, it is advantageous to have
a network whose nodes are sequentially numbered from a source node to a
terminal node. Since a straightforward simulation procedure does not
result in such a sequentially ordered network. It is necessary to re¬
number a currently existing network when expanding the network by
extending arcs from the presently considered node to new nodes generated
by simulations. Therefore, a procedure composed of appending, inserting
and re-numbering nodes are required in order to have an ordered network.
At each node (present planting day), a combination process of
simulation and re-numbering is performed. The process continues to
expand a network until the end of a planning horizon. As a result, a
multiple cropping system network whose nodes are sequentially numbered
is generated and ready for optimization.
Network Optimization
The optimization algorithm to seek K longest, distinct path lengths
of a network of multiple cropping system is discussed herein. For

Cal endar
Jan
Feb
Mar
Apr
May
Jun
1 Jul
Aug
Sep
Oct
Nov
Dec
Full-season corn
Short-season corn
t—— I
45 130
!"â–  â– â– â–  I
45 148
Late-maturing soybean
(Bragg)
H
101
H
195
Early-maturi ng soybean )â– 
(Wayne) 70
H
220
Virginia-type peanut
â–º
82
Wheat (Florida 301)
i 1
281 365
Figure 8. Time intervals during which each crop can be planted.

64
computing the longest path, the 1abel-correcting method is a fundamental
algorithm. This algorithm requires that the network contains no self¬
loops and all circuits in the network are of positive lengths. The
algorithm, coded by Shier (1974) was actually used in this study.
Suppose that the K longest path lengths from source node (node 1)
to all nodes i of an n-node network are required. Then a typical label-
correcting algorithm proceeds according to the following three steps:
LC1. Start with an initial (lower bound) approximation to the
required K longest path lengths from the source node (node 1) to each
node i. That is, assign a K-vector XV (i) = (XV.^, XV..^, ••• .XV.^) to
every node i, where the entries of XV(i) are listed in decreasing order.
LC2. Select a new arc and then 'process' the arc. By processing
an arc (l,i) whose length is A-^ , this means that current K-vector for
node i will be improved if possible by means of a path to node i which
extends first to node 1 and which then uses the arc (l,i). More
precisely, if any of the quantities (XV-|m + A-^: m = 1, ... ,K) provides
a longer path length than any one of the tentative K longest path
lengths in XV(i), then the current K-vector XV(i) is updated by
inclusion of this longer path length. It is to be understood that all
such possible updatings of XV(i) using XV(1) are performed when
processing arc (1 ,i).
LC3. Check the termination criterion. If satisfied stop.
Otherwise, return to step LC2.
The method for processing the arcs of the network is in a fixed
order: namely, in increasing order by the ending node of each arc.
Thus, arcs incident to node 1 are processed before those incident to
node 2, and so forth. If at some stage a node contains the approximate

65
lower-bound label (-«,-«>, ... then no improvements can result by
using such a label. It is useful to group the arcs by their ending
node. Accordingly, we shall examine nodes in the fixed order 1, 2, ...
,n and shall skip the examination of a node if its label is
(-°°,-°°, ... ,-°°). Here the examination of a node simply entails the
processing of all arcs incident to that node. Finally, the method will
terminate when after examining all nodes 1, 2, ... , n, it is found that
none of the components of the current K-vectors have changed from their
previous levels.
The labeling algorithm starts with the root (source node) having
label zero and all other nodes having negative infinite label (INF).
Then it enters a loop to update the label for each node i.
At any step of the process, the K-vector (XV(i)), associated with
each node i will contain the l< longest path lengths found so far from
source node to the node. Moreover, these K path lengths are always
distinct (apart from negative infinite values) and are always arranged
in strictly decreasing order. Such an ordering allows the following two
computationally important observations to be made.
(1) If the value INF is encountered in some component of a K-
vector, then all subsequent components of the K-vector also contain INF
values. Therefore, when updating the K-vector for node i, the K-vector
for a node 1 incident to i need only be scanned as far as the first
occurrence of an INF value since an infinite value cannot possibly yield
an improved path length for node i.
(2) If (IXV), the sum of some current path length in the K-vector
for node 1 and the arc length A. _¡, is less than or equal to the minimum
element of the K-vector for node i, then no improvement in the latter K-

66
vector by use of the former can possibly be made. Therefore, it is
appropriate to keep track of the current minimum element (MIN) of the K-
vector for node i. If IXV is greater than MIN, then it is possible for
an improvement to be made, as long as the value IXV does not already
occur in the K-vector for node i (only distinct path lengths are
retained).
As compared to the use of some general sorting routines to find the
K longest elements in a list, the use of these two observations allows
for a subtantial reduction in the amount of computational effort
required to update the current path lengths. When all nodes have been
labeled, the K longest path lengths to each node i in the network are
found. From such path length information, the actual paths
corresponding to any of the K longest path lengths are determined by a
backward path tracing procedure.
The optimal paths joining various pairs of nodes can be
reconstructed if an optimal policy table (a table indicating the node
from which each permanently labeled node was labeled) is recorded.
Alternatively, no policy table needs to be constructed, since it can
always be determined from the final node labels by ascertaining which
nodes have labels that differ by exactly the length of the connecting
arc.
In essence, this latter path tracing procedure is based on the
following fact. Namely, if a t-th longest path tt of length 1 from node
i to node j passes through node r, then the subpath of tt extending from
node i to node r is a q-th longest path for some q, 1 < q < t. This
fact can be used to determine the penultimate node r on a t-th longest
path of known length 1 from node i to node j. Indeed, any such node r

67
can be found by forming the quantity (1 - 1 ,) for all nodes r incident
1 J
to node j and determining if this quantity appears as a q-th longest
path length (q < t) for node r. If so, then there is a t-th longest
path of length 1 whose final arc is (r,j); otherwise, no such a path
exists. This idea is repeatedly used, in the manner of a backtrack
procedure, to produce all paths from i to j with the length 1, and
ultimately all the K longest paths from node i to j.
Parameters and Variables
The hypothetical farm is located at Gainesville, Florida. The
field is of an unit area (1 hectare) and of deep, well-drained sandy
soil which is characterized as having a field capacity at 10% by volume
and a wilting point at 5%. More specific information about the farm is
discussed as follows.
Data bases contain three separate files. Weather data files in
standard format contain historical, daily values of important weather
variables collected from an USDA class A weather station at the Agronomy
Farm, Gainesville, Florida. Available data are from the years 1954-1971
and 1978-1984. The daily weather information which is needed to run
simulations consisted of Julian day of year (JULIN), maximum temperature
in °C (TMAX), minimum temperature (TMIN), sunrise, hour a.m. (SNUP),
sunset, hour p.m. (SNDN), total solar radiation, langleys (XLANG), wind,
miles/day (WIND), and rainfall, inches/day (RAIN).
Cultivar and crop parameters are given in the text. These data are
in the file named 'GROWS' and shown in Appendix D. Values for two
cultivars (Bragg MG VII and Wayne MG III) of soybean were obtained from
the model S0YGR0 V5.0 (Wilkerson et al., 1985). Data for use in this
study were the result of simulating a wel1-irrigated field in 1982.

68
Parameters for corn cultivars were based on experiments in 1980-
1982 in which corn hybrid response to water stresses were studied
(Bennett and Hammond, 1983; Loren, 1983; Hammond, 1981). Some of the
observations included were physiological and morphological development.
Data for peanut were obtained from a study by McGraw (1979). For wheat,
experimental results in this study were used. Leaf area index and
rooting depth of wheat, not available from the experiment were from
Hodges and Kanemasu (1977).
The other file 'FACTS' shown in Appendix E provides specific
information about model operation, crop production system and economical
consideration. To initiate model execution, the user first provides the
first decision day (IDDEC), initial soil water content (MOIST), number
of crop price schemes (MXRUN) and number of crop cultivars (MXCRP) to be
considered in multiple cropping system. Also required are source node
(NS) and number of optimal cropping sequences (KL) searched.
Variables contained in the rest of the file are mainly relevant to
system evaluation and design. Primary variables of a multiple cropping
system are concerned with within-season irrigation management. These
include irrigation system used (IRSYS), application rate by a strategy
(RATE) and energy costs (GASPC, DSLPC, WAGE). For this study, a low-
pressure center pivot system was selected. It was assumed that with a
return time of one day the system was technologically capable of
achieving an application rate as desired by the user. In addition to
irrigation, idle periods (LIDLE) between two-crop seasons are also
specified by users.
From a computerized crop budget generator (Melton, 1980), the
collective costs of production for various crops were obtained.

69
Equations of variable irrigation costs of different systems used in the
study were obtained from D'Almeda (personal communication). By
regressing results which were obtained from the irrigation cost
simulator (D'Almeda et al., 1982), he developed the equations for
typical North Florida conditions. The other economical component of
interest is crop price (PRICE), $/kg. Current market prices (May, 1985)
were provided as baseline data.

CHAPTER IV
WHEAT EXPERIMENTS
Introduction
Wheat (Triticum aestivum L.) is an important crop in the multiple
cropping minimum tillage systems widely used in the Southeast USA. In
this system, wheat is usually planted in the fall after soybean
harvest. Despite the need for intensive management, wheat can be grown
sucessfully in Florida and can make a significant contribution to
Florida agriculture (Barnett and Luke, 1980).
In Florida, agriculture depends mostly upon rainfall for crop
production and irrigation is needed during relatively short but numerous
droughts. However, uneven rainfall distribution patterns coupled with
sandy soils which have limited water storage capacities and
characteristically restricted root zones thus create problems in the
scheduling of irrigation. Therefore, the need for new information on
timing, application intensity, method of application, and amounts of
water applied exists for the region to grow wheat.
Crop growth is influenced by the process of evapotranspiration.
Evapotranspiration (ET) is the combination of two processes: evaporation
and transpiration. Evaporation is the direct vaporization of water from
a free water surface, such as a lake or any wet or moist surface.
Transpiration is the flow of water vapor from the interior of the plant
to the atmosphere.
70

71
As water transpires from the leaves, the plant absorbs water from
the bulk soil through its root system and transports it to the leaves to
replace water transpired. Under well-watered conditions, the plants
usually absorb enough water through their root systems to maintain
transpiration rates at the potential rate, determined by the
environment. However, as the soil around the root system dries, the
ability of the soil to conduct water to the roots decreases and plants
can no longer supply water fast enough to maintain the potential rate.
In order to prevent leaf desiccation, the plant has a feedback control
system that causes stomatal closure, thereby decreasing actual
transpiration below the potential rate.
To study the problem of how to best allocate water over the crop
production season, it is essential to understand and quantify the crop
response to water stress throughout the irrigation seasons. Yield
relationships have long been investigated. Many researchers have shown
that crop dry matter production is directly related to water use by the
crop throughout its growth cycle (del-lit, 1958; Arkley, 1963; Hanks et
al., 1969). The results demonstrate the important fact that a reduction
in transpirational water use below the potential rate results in a
concomitant decrease in crop biomass yield. Tanner (1981), and Tanner
and Sinclair (1983) further concluded that diffusion of CO^ into the
stomata and loss of water vapor from the stomata was the coupling
mechanism between biomass yield (Y) and evapotranspiration. Hence,
knowledge of this ET-Y relationship is fundamental in evaluating
strategies of irrigation management (Bras and Cordova, 1981; Martin et
al., 1983.)

72
Because it is observed that interactive effects between crop growth
stages existed (i.e. reduced vegetative growth during early stages
caused a reduction in photosynthetic material for fruit production at
the later stage), it is necessary to investigate the critical stage
whose sensitivity factor to water stress is high. Peterson (1965)
defined important stages of the wheat life cycle as emergence,
tillering, stem extension, heading, spike development, grain setting,
and grain filling and ripening. Studies of the effects of accurately
defined levels of water stress on wheat growth at various stages of
development were conducted by Robins and Domingo (1962), Day and Intalap
(1970), Musick and Dusek (1980). Commonly, the three stages of plant
development selected for irrigation were late tillering to booting,
heading and flowering, and grain filling. Most of the researchers
agreed that the most critical period of grain wheat for adequate soil
water was from early heading through early grain filling.
The purpose of this study was to develop ET-Y functions to provide
base data for improving wheat water management practices in Florida.
The specific objectives of this work are: (1) to quantify the nature of
ET-Y relationship for wheat crop in Northern Florida, (2) to determine
the effects of timing and intensity of water deficits on wheat yield,
and (3) to parameterize the crop sensitivity factors to water stress.
Experimental Procedures
Experimental Design
This study was conducted in 24 lysimeters at the Irrigation Park,
University of Florida at Gainesville. The lysimeter installation was

73
described by Smajstrla et al. (1982). The lysimeters were cylindrical
steel tanks with 2.0 meter square surface areas and 1.85 meter depth
filled with an Arredondo fine sand soil taken from the site of the
lysimeters. Automatically movable rainfall shelters were provided to
eliminate the direct applications of rainfall on crops during the water
management studies. Preplanting preparation included cultivation with a
rotor-tiller and irrigation with sprinkler heads to prepare a semi¬
smooth surface and granulate subsurface soil.
Planting of "Florida 301" winter wheat in the lysimeters was on 29
November 1983 in 20-cm rows at a seeding rate of approximate 135
kg/ha. Seeds were manually drilled and covered lightly with soil.
Fertilizer was applied at a rate of about 90-18-18 kg/ha (Nitrogen-
Sulfur-Potash) in the lysimeters. One half of this amount was applied
by hand at planting and the other half in late January. Unusual
freezing weather on 26 December 1983 destroyed most of the seedlings in
the lysimeters. Transplanting of young plants from buffer areas on 11
January 1984 made the intended study continuous. Attempts were made to
maintain uniform plant densities in lysimeters, however in some cases
uniformity problems did exist.
The crop growth season was partitioned as emergence to late
tillering, late tillering to booting, heading and flowering, and grain
filling stages. The study involved 8 treatments (4 crop stages of
stress * 2 levels of stress), and each treatment was replicated three
times in three lysimeters. Treatments were labeled as double-index
(S,L), where S indicated stress stage and L period (weeks) of stress.
In treatment (N,N), the control, the soil water at the top 50-cm depth
was maintained at field capacity (11 percent volumetric water content)

74
throughout the growth season. There were two treatments (N,N) to ensure
reliable maximum yield and potential ET during each stage. In treatment
(II,*), (III,*), (IV,*), soil water contents in the top 50-cm zone were
maintained at field capacity except during specific growth stages. Two
levels of water stress during each growth stage were induced by omitting
irrigations for 3 and 4 weeks, respectively.
A Tuesday-Friday schedule was employed to monitor soil water
contents in the lysimeters during the season. Soil water contents at
five depths (15, 30, 45, 75, 105 cm) of soil profile were measured with
a neutron soil moisture meter (TROXLER 3220 Series.) Additional work
which was performed on the same schedule included irrigation, collecting
of volumes of drainage water from the lysimeters, and monitoring of crop
phenology.
Irrigation decisions were made weekly immediately following neutron
probe readings. Amounts of application were computed as the volume of
water reduction below field capacity for the top 50-cm zone. A manually
operated, pre-calibrated drip irrigation grid was designed to irrigate
inside each lysimeter. A separate irrigation system was used to
irrigate the buffer crop area beneath the rain-out shelters but outside
of the lysimeters.
Plots were harvested on 9 May 1984. Samples of total dry matter
above the ground were obtained from lysimeters by manually cutting and
threshing. At the same time, plant heights were measured. Samples were
then oven-dried at 95°C for 24 hours. For individual lysimeters, grain
weights and related yield variables were assembled and measured for
detailed analysis.

75
Modeling and Analysis
To account for increased sensitivity to water stress at various
stages of growth, and the interactive effects between crop growth
stages, a multiplicative model was selected. Jensen (1968) first
developed one such model which related water stresses during various
stages of crop growth to final yield. Using input of standard,
available climatological data, Rasmussen and Hanks (1978) used this
method sucessfully to simulate grain yields of spring wheat grown in
Utah under various irrigation regimes. To estimate grain and bean
production assuming that other factors, such as fertility levels, pest
or disease activity, and climatic parameters are not limiting, the
Jensen model is given as
Y N ET
— = n ( ) (4.1)
Y„ i=1 ET i
P P
where
ET/ET
P
Xi
= the relative yield of a marketable product,
= the relative total ET during the given ith stage of
physiological development,
= the relative sensitivity of the crop to water stress
during the ith (i = 1, 2, ... ,N) stage of growth.
To model this ET - Y relationship, daily ET of each lysimeter was
calculated based on soil water balance method
ET = IR + AS - DR (4.2)
where IR = irrigation, AS = soil water depletion, and DR = drainage.
Daily ET's were summed to calculate stage ET according to phenological

76
observations in the field. Data from six lysimeters, the control
treatments, were used for estimation of potential grain yield and
potential ET in Equation 4.1. The NLIN regression procedure (SAS, 1982)
for least-squares estimates of parameters of nonlinear models (Equation
4.1) was used to calibrate crop sensitivity factors (x^). These values
were then compared to the results of other researchers.
Results and Discussion
Field Experiment Results
The crop growth stage observations have a range of variability. In
addition, the effect of water stress on crop phenology was apparent.
Therefore, a stage was said to be observed when at least 50 percent of
the plants that were wel1-irrigated were at that stage of development.
Wheat phenology data observed in 1983-1984 winter season were recorded
in Table 6 and shown in Figure 9, indicating a full season of 163 days.
At 53 days after planting, the first node of stem was visible. Booting,
when the sheath of the last leaf was completely grown out, occurred 91
days after planting. Signaled by the time first ears were just visible,
heading began on March 10, which is 103 days after planting. Following
the heading stage, white flowers were visible on March 22. At 141 days
after planting kernels reached full size. This observation is very
similar to one made at Quincy, Florida, 1977-1979, by Barnett and Luke
(1980). Heading dates at Quincy were March 23 and March 27 for the
1977-1978 and 1978-1979 seasons, respectively.
Also shown in Figure 9, is the initiation of stress treatments.
According to the phenological calendar, the start of water stress

77
Table 6. Observations of specific reproductive growth stages for winter
wheat at Gainesville, FL., in 1983-1984.
Stage Description
Date
Elapsed Time
after Planting
PI anting
Nov. 29
1
Emergence
Dec. 4
6
Node of stem visible
Jan. 20
53
Booting
Feb. 27
91
Heading
Mar. 10
103
FIowering
Mar. 22
115
Mi 1ky-ripe
Apr. 3
127
Kernel hard
Apr. 17
141
Harvest
May 9
163

Stage partitioning
b
emergence to ^late tillering^
late tillering anc| booting
^headincj grain-
flowering filling
CD
S-
CD
CD
-o
+->
-M
CL
S-
03
CD
co
CD
•r—
03
E
CD
CJ
c
S-
sz
C
C=
4— CD
CD
CD
•r—
1
+-)
•r—
CD
o •—
c
c
s~
i—
>)
1—
co
4->
CD
-Q
• r~
• r—
O)
CD
S-
CD
CD
C
Í-
CD *r-
4->
~a
5
C
CD
c
>
03
CD
"O co
o
03
o
S-
4->
s*.
S-
i—
E
O -r-
o
CD
1—
CD
03
CD
03
CL
CD
c >
_Q
sz
4-
_£Z
Phenological observations
Stress treatments
M j it J Á * i l
1 T T
irt' i n m
CO1
CO
to
CO
CO •—«
CO
CD •—!
CD »—1
CD >
S- ►—«
S- »—t
+->
4-J
+->
CO
CO
to
Figure 9. Phenological observations, water stress treatments, stage partitioning, and daily temperatu
in the winter wheat experiment, Gainesville, FL. 1983-1984.

79
treatments were slightly delayed. Therefore, the intended stress
treatments during the grain filling stage were not completely
accomplished, which resulted in duplicating treatment (IV,2) as shown in
Table 7.
Detailed yield vs irrigation data are tabulated in Table 7.
Effects of stress treatments on winter wheat yield are demonstrated in
Table 8, and plotted in Figure 10. By observations, crop growth in
lysimeters 3, 16, and 19 did not seem normal after the hard freeze.
Also, difficulty had been experienced in water management in these
lysimeters. Without irrigation, lysimeter 16 always had high counts of
the neutron probe throughout the season. In lysimeter 3 and 19,
irrigation was applied, however, it seemed that most water was drained
out by suction cups at the bottom of lysimeters. Therefore, data from
these three lysimeters were considered subject to an uncontrolled
treatment (UC), and were excluded from the following yield analysis.
Two basically different yield levels were obtained from irrigation
management. The treatments that were well-watered (N,N) and the one
that experienced severe water stress during late booting stage (11,4)
yielded less; whereas the rest of treatments had significantly higher
yields. Comparisons of biomass yields and head numbers between
treatments of heading period stress (III,*) and those stressed during
grain filling stage (IV,*) show that there is no significant
difference. This may be because the duration of heading to flowering
stage lasts only a short period of time (Peterson, 1965; Doraiswamy and
Thompson, 1982). It thus requires precise initiation of treatments to
acquire differential results.

Table 7.
Summary of
Gainesvi11e
results of winter
, FI., 1983-1984.
wheat growth
under various irrigation
treatments,
Dry Mass
Plant Ht
. No. of
Head Wt.
Grain Wt.
Test Wt.
Treatment
Lysi meter
- gm -
- cm -
Heads
- gm -
- gm -
-lb/bu-
(N,N)
3
620
73
577
298.3
190.1
78.5
(N,N)
16
648
78
511
325.5
224.9
81.2
(N,N)
19
670
63
576
324.9
240.6
80.4
(N,N)
7
795
68
683
395.2
305.2
81.4
(N,N)
10
840
76
574
446.6
321.4
83.1
(M)
21
880
73
824
459.2
358.9
81.1
(11,2)
4
742
71
663
399.9
281.6
81.2
(11,2)
12
1051
73
577
527.4
402.2
81.3
(11,2)
17
796
74
663
421.9
323.9
82.8
(11,4)
6
670
54
502
264.1
178.9
80.9
(11,4)
11
698
69
544
349.0
261.5
80.0
(11,4)
23
698
65
620
328.5
241.3
80.7
(111,3)
1
704
69
649
322.4
237.7
79.2
(HI,3)
14
966
75
690
499.4
393.4
80.2
(HI,3)
22
924
83
818
480.9
326.8
82.0
(HI,4)
8
772
75
635
371.9
284.4
78.8
(III,4)
9
816
73
823
409.5
283.1
80.6
(III,4)
20
1291
84
710
657.0
511.5
80.4
(IV,2)
2
645
68
692
322.3
216.6
79.8
(IV,2)
13
876
72
659
467.6
369.1
84.0
(IV,2)
24
1108
86
669
559.7
419.6
80.5
(IV,2)
5
746
68
677
393.6
303.6
80.8
(IV,2)
15
811
63
504
425.8
329.6
82.1
(IV,2)
18
1128
85
776
582.5
423.9
79.5
TM) the
control, well irrigated; (II,
*) stressed
during late
tilling to booting;
(III,*) stressed during heading
and flowering; (IV,
*) stressed
during grain
fil1ing period,

81
Table 8. Treatment effects on winter wheat yield, Gainesville, FL.,
1983-1984
Treatment
Dry Mass
- gm -
No. of
Heads
Head Wt.
- gm -
Grain Wt.
- gm -
UC
646.4 b
555 b
316.2 b
218.5 c
(N,N)
838.3 ab
694 ab
433.7 ab
328.5 ab
(11,2)
863.0 ab
634 ab
449.8 a
335.9 a
(11,4)
688.7 b
555 b
313.9 b
227.2 be
(111,3)
864.7 ab
719 a
434.3 ab
319.3 abc
(111,4)
959.7 a
723 a
479.5 a
359.7 a
(IV, 2)
876.3 ab
673 ab
449.9 a
335.1 a
(IV,2)
895.0 ab
652 ab
467.3 a
352.4 a
C.V. (%)
16.2
11.4
15.8
18.1
Column means followed by the same letter are not significantly different
at the 5% level by Duncan's multiple range test.

Number of Heads Dry Matter (gm)
82
Figure 10.
The effect of water stress treatment on different yield
variables of wheat for each stress treatment (average of
3 replications), (a) Dry matter; (b) Number of heads;
(c) Head Weight; (d) Grain weight.

Figure 10. (continued)
Grain Weight (gm)
450
Stress treatment
Head Weight (gm)
oo
GJ

34
Plant water stress limits leaf and tiller development during
vegetative growth and stress during the late tillering to booting stage
accelerates stem senescence and reduces spikelets per head (Musick and
Dusek, 1980). Consequently, for treatment (11,4), the effect of
extensive water stress during the late tillering stage significantly
reduced grain yield by 30 percent of the wel1-irrigated plants. This
agrees with results from Day and Intalap (1970) that water stress is
more critical during late tillering than during flowering or grain
filling stage.
An attempt was made to relate grain yield to seasonal irrigation
and seasonal ET. A regression analysis of the effects of seasonal
irrigation amounts on grain yield indicates that the linear relationship
is poor with an r value of 0.25. It implies that a linear model of
grain yield dependent upon total irrigation or upon seasonal ET is not
strongly recommended on the basis of this study. Therefore, the model
of Jensen (1968) was evaluated.
Model Calibration
Phenological development occurred over a range of time and caused a
large variation in the duration of various stages. The appropriate
scheme of partitioning the growth season into four stages was
illustrated in Figure 9. The periods of stage I, II, III, and IV were
53, 50, 24, and 36 days, respectively. Accordingly, stage ET and
seasonal ET were computed and tabulated in Table 9. As explained in the
last section, difficulty had been experienced in water management in
lysimeters 3, 16 and 19. For these three lysimeters, the seasonal and
stage-specific ET1s were very low.

Table 9. Seasonal and stage-specific ET for winter wheat grown in Gainesville, FL.,
1983-1984.
Treatment
Lysimeter
Evapotranspiration (cm)
Irrigation
Applied
(cm)
Grai n
Yield
(gm)
Stage I
Stage II
Stage III
Stage IV
Season
(M)
3
0.74
6.51
0.22
1.14
8.61
9.13
190.1
(M)
16
0.04
2.59
0.35
1.03
4.01
0.00
224.9
(N,N)
19
0.30
7.80
0.86
1.74
10.70
12.38
240.6
(N,N)
7
0.24
9.59
6.71
16.32
32.86
37.29
305.2
(N,N)
10
0.81
10.71
6.16
12.32
27.84
30.75
321.4
(N,N)
21
0.95
11.70
7.14
8.27
28.06
31.53
358.9
(11,2)
4
0.34
10.45
3.24
10.55
24.58
27.85
281.6
(11,2)
12
1.32
8.45
3.47
14.30
27.26
29.68
402.2
(11,2)
17
1.24
13.41
3.50
5.39
23.54
29.12
323.9
(11,4)
6
0.04
9.04
0.86
11.14
21.08
23.27
178.9
(11,4)
11
0.54
7.66
1.52
8.51
18.23
20.58
261.5
(11,4)
23
0.03
10.91
0.69
14.82
26.45
32.24
241.3
(III,3)
1
0.73
10.14
4.95
6.08
21.90
24.35
237.7
(111,3)
14
0.63
11.05
6.08
6.85
24.61
25.43
393.4
(111,3)
22
0.42
10.49
5.78
6.49
23.18
23.97
326.8
(111,4)
8
0.52
10.15
6.00
5.13
21.80
23.69
284.4
(III,4)
9
0.04
10.11
4.67
3.63
18.45
19.79
283.1
(III,4)
20
0.19
9.36
7.59
4.93
22.07
24.37
511.5
(IV,2)
2
0.45
8.41
5.08
3.72
17.66
20.51
216.6
(IV,2)
13
0.80
9.49
3.98
4.88
19.15
18.33
369.1
(IV,2)
24
0.88
11.77
8.84
6.62
28.11
33.05
419.6
(IV,2)
5
0.11
10.52
7.39
11.49
29.51
31.27
303.6
(IV,2)
15
0.28
8.79
2.00
2.33
13.40
14.35
329.6
(IV,2)
18
1.62
11.11
8.28
10.00
31.01
35.50
423.9
Stage I: emergence to late tillering; Stage II: late tillering to booting;
Stage III: heading to flowering; Stage IV: grain filling.

86
Data from six lysimeters, the control units, would be used for
estimation of potential grain yield and potential ET in Equation 4.1.
As explained, the crop in lysimeters 3, 16, and 19 did not recover from
the freeze and grow normally. Therefore, average values of data from
lysimeters 7, 10, and 21 were calculated to define potential yield and
potential ET values. For model calibration, calculated potential stage
ET's for emergence to late tillering, late tillering to booting, heading
and flowering, and grain filling 0.67, 10.67 6.67, 12.30 cm,
respectively. For stage I of 53 days, potential ET of 0.67 cm was
low. That is because radiation was low in December and January and
irrigation was not initiated until January 13, 1984. Potential grain
yield was 328 gm for a 2 meter square area.
Using data from all 24-lysimeters, calibration of x’s values
(Equation 4.1) was accomplished. Values of 0.065, 0.410, 0.114, 0.026
for all A's in Equation 4.1 gave the best fit. Predicted vs. observed
yields for all data from lysimeters were given in Figure 11. Because
the uncontrollable within-treatment errors and unexpected freezing
o
weather, the r = 0.42 does not seem high. However, the effect of
critical stages of growth has been quantified.
I
Values of published a s for wheat are inconsistent. The x. values
for booting, heading, soft dought, and maturity reported by Neghassi et
al. (1975) are -0.490, 2.71, -5.45, and 4.58, respectively. The
negative values do not have any physical relevance. Values of 0.25 for
all A’s were given by Rasmussen and Hanks (1978). By assigning the
relatively short grain-filling period a X of 0.25, Rasmussen and Hanks
argued that the grain filling stage was more important in irrigated
wheat production. The values obtained from this study illustrated that

Predicted Yield Ratio
87
Figure 11. Plot of observed vs. predicted yield ratio for wheat.

88
water stress during late booting, heading and flowering stages were
important. Robins and Domingo (1962), and Mogensen et al. (1985) had
the same conclusion that severe water deficits should be avoided from
the booting stage until the heads were filled.
In summary, the grain yield model developed in this study accounts
for variables of climate and irrigation. It has been shown that the
model has the capability to give very reasonable predictions of yield
reductions to water stress. Coupled with a soil water balance approach,
the grain yield model can be utilized effectively for water stress and
irrigation management applications. It should be of particular use to
economists and others concerned with the effects of drought or limited
irrigation. One type of applications in using this data set will be
demonstrated in the next chapter.

CHAPTER V
APPLICATION OF THE MODEL
Introduction
In Florida, where the cold season is relatively short and the water
supply (precipitation and irrigation) is sufficient to grow two or more
crops per year, the potential of practicing multiple cropping is high.
On the other hand, irrigation development is expensive. Inasmuch as
benifits from irrigation may vary appreciably from year to year,
developing optimal multiple cropping systems is intended to make maximum
use of the expensive irrigated land.
As the number of crops and development of new integrated management
systems (i.e. tillage, irrigation, pest, fertilization, weed, etc.)
increases, the problem of deciding multiple cropping sequences to be
followed becomes very complex. If it is to be analyzed properly, it
must be examined systematically.
An optimization - simulation model composed of submodels to
integrate crops, soil - water dynamics, weather, management, and
economic components has been developed to select optimal multiple
cropping sequences. However, decisions about optimal multiple cropping
systems are complicated by a number of factors including weather
uncertainty, the complex nature of the crop's response to management
strategies (i.e. irrigation), and uncertain crop prices. The
application of the model refers to its use as a tool for studying
various optimal cropping management decisions.
89

90
In this chapter, efforts are made to evaluate the combined
simulation - optimization method for studying crop management decisions
under multiple cropping; and to apply the concept to study the impact of
irrigation management on the decision of crop sequencing. Specific
objectives include (1) determine the efficiency and utility of the
combined optimization - simulation technique as related to the multiple
crop problem; (2) apply the model using north Florida as an example to
study optimal multiple cropping sequences under a non-irrigated field
with corn, soybean, peanut and wheat; (3) determine the effect of
irrigation on optimal cropping sequences with the same crops considered;
(4) evaluate the risk of various optimal cropping sequences with respect
to variations in weather and crop prices; (5) suggest better cropping
sequences for north Florida.
Procedures for Analyses
Crop Production Systems
Using north Florida as an example region, the model was applied to
study optimal multiple cropping sequences under an irrigated or non-
irrigated field. In the study, the first day of planting was set on
March 16 (Julian day 75), and a 4.5-year production schedule was
projected. Three crop production systems were investigated. Under
system I, crops to be considered in sequential cropping were full-season
corn (F.S.Corn), short-season corn (S.S.Corn), early-maturing soybean
(Wayne), late-maturing soybean (Bragg), peanut, and winter wheat (Wheat
301). System II had the same crops considered, but did not allow for
repeating peanut seasons in sequence. System III was studied, which

91
excluded peanut from cropping production. Each of these three cropping
systems were studied under no-irrigation and irrigation production
practices.
Constant crop prices (May 1985) through the whole 4.5-year planning
period were used to project the results. Table 10 shows the crop price
along with production costs and potential yields under a typical north
Florida farm.
Crop Model Simulations
As described in chapter III, crop phenology, soil water balance,
and crop models were developed to respond to weather and management
practices for day-to-day management decisions. Therefore, weather input
was an essential part of the system. Several schemes of implementation
of weather data were available. They were the use of historical weather
records, stochastically generated weather based on probability function,
and the use of average weather data for each day. In this study,
historical weather records were used to drive crop simulation models.
To study the production management decisions and crop responses in
a single season, simulations on the effects of different planting dates
and types of irrigation management were performed using weather records
from 25 different seasons. The results were then averaged to evaluate
crop response to various management strategies over the long term. In
particular, the effects of management strategies on the feasibility of
multiple cropping were examined.
Optimization of Multiple Cropping Sequences
Multiple cropping sequences of all three production systems for an

Table 10. Price,
north
production cost
Florida farm.
and potential
yield of
different crops for
a typical
SI Unit
Farmer Unit
Crop
Potential
Yield
(kg/ha)
Price
($/kg)
Production
Cost
($/ha)
Potential
Yield
(bu/ac)
Price
(S/bu)
Production
Cost
($/ac)
Full-Season Corn
9800
0.103
346.2
200
2.58
141.3
Short-Season Corn
8600
0.103
346.2
175
2.58
141.3
'Wayne1 soybean
4010
0.238
308.2
59
6.50
125.8
'Bragg' soybean
4680
0.238
308.2
70
6.50
125.8
Peanut
3180
0.473
481.2
60
13.00
196.4
Wheat
3680
0.137
248.9
70
3.75
101.6
Source: Agronomy Facts. Fla. Coop. Ext. Serv., Insti. of Food and Agri. Sci., IJniv. of Fla

93
irrigated or non-irrigated field were to be explored by applying the
activity network model. For the study of multiple cropping, different
weather sequences were needed. There were only five 5-year-sequences
weather patterns obtainable from the available 25-year weather data.
Therefore, Monte Carlo simulation techniques were applied. By using a
random number generator, six 1-year weather data records were randomly
drawn from historical records to compose a 6-year-sequence weather
file. By so doing, 20 synthetic weather patterns were generated for
use. Given a weather pattern, a network of multiple cropping systems
was obtained by simulations. The K longest path algorithm was then used
to optimize the K optimal multiple cropping sequences of the simulated
network.
Risk Analysis
Multiple cropping sequences are especially susceptible to the
unpredictable influences of weather and market prices. Future
uncertainties are so great that a clear answer cannot exist.
Alternatively, the techniques involved in risk assessment provide
assistance in quantifying these uncertainties and aid in decision-making
processes.
Simulation techniques were utilized to assess the relative risk of
non-irrigated multiple cropping sequences with respect to variations in
weather and crop marketing. There were 10 cropping sequences used for
comparison. These 10 candidates represented unique sequences chosen
from those obtained from the K longest path optimization program with
the best production potential.

94
In various optimal multiple cropping sequences, optimal planting
dates of every cultivar were frequently observed. Hence, in the
simulations, whenever the cultivars were scheduled after an idle period,
planting would begin on observed, fixed dates. This procedure allowed
an adjustable idle period which was very practical and intuitive for
real production conditions.
To analyze the risk of cropping sequences under the influence of
crop marketing, simulations driven by a single set of 6-year-sequence
weather data were performed. While other crops price were kept
constant, the increase or decrease of a crop price by 10 or 20 percent
each time resulted in a total of 17 pricing schemes for use.
The results suggested by these analyses can not be extrapolated to
other locations, crops, or soil types. The current version of the model
is restricted to a deep, well drained, sandy soil typical of Florida
conditions. However, the approach used in the analyses could be applied
to other areas, soil types, and crops.
Results and Discussion
Crop Model Simulation
The growth models were used to determine crop development and
yield, as well as to examine the effects of production management on
final profit. Under north Florida conditions the crops were assumed to
grow on the sandy soil. Tables 11 through 16 show the average results
of different crops that would be considered in a multiple cropping farm.
The first aspect studied was the effect of planting date on length
of growth season and final yield. For the spring/summer crop (corn,

95
Table 11. Simulation results^ of irrigated and non-irrigated full-season
corn grown on different planting dates for 25 Years of
historical weather data for Gainesville, FL.
Non-Irrigated
Feb.
15
1
Mar.
PI anting
15
Date -
Apr.
15
May :
15
Season (days)
141.2
(4.4)
124.4
(3.4)
112.3
(1.8)
105.5
(1.5)
Yield (kg/ha)
2133.0
(1759)
3556.0
(2244)
4015.0
(1624)
3771.0
(792)
Profit ($/ha)
-126.2
(180.8)
20.1
(230.6)
67.3
(166.8)
42.0
(81.4)
Act. ET (cm)
35.9
(3.5)
36.1
(2.8)
34.7
(2.2)
34.5
(2.3)
Ref. ET (cm)
49.2
(1.8)
49.5
(1.5)
48.5
(1.3)
46.2
(1.1)
Rainfall (cm)
50.8
(15.3)
48.2
(11.7)
50.9
(12.3)
59.0
(14.8)
Irrigation (cm)
0.0
0.0
0.0
0.0
Irrigated
Season (days)
141.2
(4.4)
124.4
(3.4)
112.3
(1.8)
105.5
(1.5)
Yield (kg/ha)
7589.0
(533)
8252.0
(156)
6363.0
(199)
4913.0
(274)
Profit ($/ha)
283.8
(56.5)
349.0
(30.2)
177.3
(27.3)
61.4
(41.0)
Act. ET (cm)
44.5
(1.7)
44.8
(1.4)
42.8
(1.2)
40.7
(1.1)
Ref. ET (cm)
49.2
(1.8)
49.5
(1.5)
48.5
(1.3)
46.2
(1.1)
Rainfall (cm)
50.8
(15.3)
48.2
(11.7)
50.9
(12.3)
59.0
(14.8)
O
Irrigation' (cm)
19.9
(4.2)
20.3
(3.9)
17.3
(2.8)
12.9
(3.8)
1. mean value followed by standard deviation in parenthesis.
2. water required to avoid most water stress.

96
Table 12. Simulation results1 of irrigated and non-irrigated short-
season corn grown on different planting dates for 25 Years of
historical weather data for Gainesville, FL.
Non-Irrigated
Feb.
15
- - - Planting
Mar. 15
Date -
Apr.
15
May 15
Season (days)
127.2
(4.5)
110.5
(3.2)
98.5
(1.9)
91.7
(1.4)
Yield (kg/ha)
1225.0
(1105)
2685.0
(1946)
3653.0
(1675)
3733.0
(1230)
Profit ($/ha)
-219.6
(113.9)
-59.5
(200.0)
29.9
(172.1)
38.2
(126.3)
Act. ET (cm)
29.6
(3.2)
30.4
(2.7)
29.7
(1.9)
29.9
(2.2)
Ref. ET (cm)
42.7
(1.8)
42.6
(1.4)
41.9
(1.2)
39.9
(1.1)
Rainfall (cm)
43.5
(14.7)
40.3
(12.7)
41.1
(9.8)
51.7
(14.0)
Irrigation (cm)
0.0
0.0
0.0
0.0
Irrigated
Season (days)
127.2
(4.5)
110.5
(3.2)
98.5
(1.9)
91.7
(1.4)
Yield (kg/ha)
4540.0
(894)
7767.0
(289)
6675.0
(61)
5238.0
(78)
Profit ($/ha)
-14.8
(86.3)
295.0
(36.1)
206.4
(22.0)
92.6
(28.4)
Act. ET (cm)
36.9
(1.7)
39.5
(1.2)
38.3
(1.2)
36.1
(1.0)
Ref. ET (cm)
42.7
(1.8)
42.6
(1.4)
41.9
(1.2)
39.9
(1.1)
Rainfall (cm)
43.5
(14.7)
40.3
(12.7)
41.1
(9.8)
51.7
(14.0)
O
Irrigation" (cm)
18.0
(4.1)
20.8
(3.8)
17.7
(2.7)
13.2
(3.3)
1. mean value followed by standard deviation in parenthesis.
2. water required to avoid most water stress.

97
Table 13. Simulation results^ of irrigated and non-irrigated 'Bragg'
soybean grown on different planting dates for 25 years of
historical weather data for Gainesville, FL.
Non-Irrigated
---------- Planting Data ---------
April 10 May 10 June 10 July 10
Season (days)
Yield (kg/ha)
Profit ($/ha)
Act. ET (cm)
Ref. ET (cm)
Rainfall (cm)
Irrigation (cm)
Season (days)
Yield (kg/ha)
Profit ($/ha)
Act. ET (cm)
Ref. ET (cm)
Rainfall (cm)
186.5(2.3)
576.0 (215)
-170.6 (51.3)
48.8 (4.2)
76.0 (1.7)
84.8 (19.2)
186.5
(2.3)
1294.0
(338)
-138.1
(92.3)
58.4
(2.8)
76.0
(1.7)
84.8
(19.2)
18.2
(3.3)
161.4(2.7)
934.0 (365)
-85.4 (86.6)
45.1(3.9)
65.0 (1.4)
78.8(18.1)
161.4(2.7)
1848.0 (340)
247.0 (88.6)
52.5(2.0)
65.0 (1.4)
78.8(18.1)
14.1(4.8)
137.4 (3.2)
1622.0 (519)
77.8 (123.3)
39.5 (3.5)
52.4 (1.3)
68.2 (17.7)
137.4(3.2)
2634.0 (213)
222.3 (65.7)
44.9(1.6)
52.4(1.3)
68.2(17.7)
12.6(4.5)
117.5 (4.7)
1655.0 (631)
85.4(149.6)
31.2(3.0)
40.1 (1.4)
53.7(15.2)
117.5 (4.7)
2723.0 (304)
251.8 (88.6)
36.0 (1.4)
40.1 (1.4)
53.7(15.2)
11.6 (3.6)
Irrigation11 (cm)
1. mean value followed by standard deviation in parenthesis.
2. water required to avoid most water stress.
0.0
0.0
0.0
0.0
Irrigated

98
Table 14. Simulation results* of irrigated and non irrigated 'Wayne'
soybean grown on different planting dates for 25 years of
historical weather data for Gainesville, FL.
Non-Irrigated
- - - Planting
Date -
March 10
May
2
June
15
Aug
. 2
Season (days)
108.2 (3.9)
87.1
(1.9)
80.5
(1.2)
85.7
(3.4)
Yield (kg/ha)
496.0 (251)
788.0
(305)
1400.0
(466)
1031.0
(475)
Profit ($/ha)
-189.6 (60.0)
-120.1
(72.4)
25.2
(110.4)
-62.6
(112.7)
Act. ET (cm)
23.2 (3.1)
25.8
(2.2)
27.1
(2.6)
21.4
(2.7)
Ref. ET (cm)
42.2 (1.6)
39.1
(l.D
35.0
(1.0)
29.5
(l.D
Rainfall (cm)
37.4 (13.0)
40.4
(10.3)
48.7
(13.6)
39.1
(13.8)
Irrigation (cm)
0.0
0.0
0.0
0.0
Irrigated
Season (days)
108.2 (3.9)
87.1
(1.9)
80.5
(1.2)
85.7
(3.4)
Yield (kg/ha)
1738.0 (212)
2484.0
(133)
2372.0
(142)
2018.0
(163)
Profit ($/ha)
30.8 (59.0)
169.5
(51.0)
177.8
(51.0)
93.3
(52.7)
Act. ET (cm)
33.0 (1.6)
33.9
(1.0)
31.5
(l.D
26.1
(1.0)
Ref. ET (cm)
42.2 (1.6)
39.1
(1.1)
35.0
(1.0)
29.5
(l.D
Rainfall (cm)
37.4 (13.0)
40.4
(10.3)
48.5
(13.6)
39.1
(13.8)
O
Irrigation (cm)
18.0 (3.2)
14.9
(2.5)
10.3
(3.6)
10.3
(3.5)
1. mean value followed by standard deviation in parenthesis.
2. water required to avoid most water stress.

99
Table 15. Simulation results^ of irrigated and non-irrigated peanut
grown on different planting dates for 25 years of historical
weather data for Gainesville, FL.
Non-Irrigated
- - - Planting
Date -
Apri 1
1
May
1
June
1
July
1
Season (days)
137.0
(2.9)
128.2
(1.4)
124.8
(1.7)
133.1
(4.0)
Yield (kg/ha)
1088.0
(346)
1328.0
(353)
1333.0
(318)
1069.0
(345)
Profit ($/ha)
33.7
(163.4)
146.6
(166.8)
149.0
(149.9)
24.4
(162.6)
Act. ET (cm)
42.2
(2.7)
41.6
(2.7)
40.2
(2.9)
36.3
(3.1)
Ref. ET (cm)
56.4
(1.5)
54.3
(1.2)
50.2
(1.3)
44.6
(1.3)
Rainfall (cm)
62.9
(13.5)
65.6
(14.5)
67.8
(18.2)
60.6
(15.4)
Irrigation (cm)
0.0
0.0
0.0
0.0
Irrigated
Season (days)
137.0
(2.9)
128.2
(1.4)
124.8
(1.7)
133.1
(4.6)
Yield (kg/ha)
2446.0
(78)
2393.0
(85)
1945.0
(105)
1580.0
(96)
Profit ($/ha)
516.4
(50.4)
527.0
(64.5)
337.6
(67.9)
171.6
(62.4)
Act. ET (cm)
52.1
(1.5)
49.6
(1.1)
45.8
(1.3)
41.2
(1.2)
Ref. ET (cm)
56.4
(1.5)
54.3
(1.2)
50.2
(1.3)
44.6
(1.3)
Rainfall (cm)
62.9
(13.5)
65.6
(14.5)
67.8
(18.2)
60.6
(15.4)
Irrigation' (cm)
21.0
(3.4)
16.2
(4.4)
13.3
(4.1)
12.4
(3.8)
1. mean value followed by standard deviation in parenthesis.
2. water required to avoid most water stress.

100
soybean, peanut), early spring planting lengthened the growing season.
For example, there was a 36-day difference in growth seasons between
early planting and late planting of corn. A difference of 69 days was
even more prominent between the April and July plantings of the late-
maturing 'Bragg1 soybean. The effect is less significant for early-
maturing 'Wayne' soybean, which has the longest season of 109 days and
the shortest season of 81 days to harvest maturity.
On the average, full-season corn took 2 more weeks than short-
season corn to mature. Planted on March 15, full-season corn took 125
days to grow before harvesting on June 20. So in the multiple cropping
system under Florida conditions, soybean may be a good second crop to
immediately follow the corn harvest. Soybean cultivars are very
sensitive to photoperiod especially the late maturity groups. When
'Bragg' soybean was early planted in late spring, it required a full 6
months to grow before harvesting. If planted late, it took only 4
months to mature. On the other hand, day length had less influence on
the early-maturing soybean cultivar. Hence, 'Wayne' soybean could grow
109 days before harvesting when it was planted as early as March 10.
Due to these genetic traits, agronomists have advocated a soybean-
soybean double cropping practice that would adopt this cultivar as the
first crop. For peanuts, the range of predominate planting seasons is
relatively narrow. Having an average of 130 days, the growth season of
peanuts did not vary significantly.
Under unlimited soil water, yield potential of a crop is a function
of photosynthetically active radiation. The length of a crop growth
season generally correlates with higher yield. However, in Florida
rainfall is scarce and less frequent in the spring and early summer.

101
Frequent, short droughts have detrimental effects on crop yields. As
studied by other researchers, growth of corn, soybean or peanut is very
sensitive to soil water availability during the transition period from
vegetative growth to reproductive stage. Shown in Table 11, for a non-
irrigated field, average yield of full-season corn planted too early
(February 15) would be as much as 53 percent of that planted on its
optimal date. This is also true for short-season corn, 'Bragg' soybean,
'Wayne' soybean, and peanut. The ratios are 0.33, 0.35, 0.35, 0.82,
respectively. As such, factors other than solar radiation should be
considered to determine optimal planting dates for various crops.
Irrigation is one of the production managements which may be used to
affect planting dates of crops.
The other aspect studied is the use of irrigation to improve crop
production systems. As shown in Tables 11 through 14, early planting of
corn and soybean using no irrigation made no profit in north Florida.
The later planted crops increased profitability by increasing yield.
Nonetheless, profits were not very high. To show the low profits and
large variances, Figures 12 through 17 plot profit against cumulative
probability. The examination of these figures shows that risk is high
to grow corn and soybean early under non-irrigated conditions. For
example, planting soybeans in June would result in losses 1 year out of
3 for 'Bragg', and 2 out of 5 for 'Wayne' approximately. Thus, non-
irrigated corn and soybean are not relatively profitable. It is even
more obvious for early-maturing 'Wayne' soybean. When 'Wayne' was
planted on June 15, it was profitable, but only $25 per hectare per
season. On the contrary, with or without irrigation, production of
peanut is profitable under the current high market value. For peanuts

102
Figure 12. Cumulative probability of profit for non-irrigated full-
season corn planted on Feb. 15 (——), March 15
Apri 1 15 (—), May 15 ( ).
_Q
03
_Q
O
S-
CL
O)
>
13
C_J
Figure 13. Cumulative probability of profit for non-irrigated short-
season corn planted on Feb. 15 ( ), March 15
Apri 1 15 (—), May 15 ( ).

103
Figure 14. Cumulative probability of profit for non-irrigated 'Bragg'
soybean planted on April 10 ( ), May 10 June 10
( —), July 10
Figure 15. Cumulative probability of profit for non-irrigated 'Wayne'
soybean planted on March 10 ( ), May 2 June 15
(---), August 2 ( ).

104
of normal-season planting (May 1), the yield of a non-irrigated field
was 45 percent less than an irrigated field. Even so, a profit of $147
per hectare was attainable. Irrigation then offered more of a return at
$527 per hectare.
Irrigation improves net farm income. In addition, it enhances the
production system by allowing early planting of crops. Simulation
results (Table 11 through 14) showed that profits were increased by
irrigation when corn was planted as early as February 15, and soybeans
to May 10. The results also revealed that the planting date of the
highest return under an irrigated field was earlier than that of a non-
irrigated field. As an example, Table 11 shows that it was better to
plant non-irrigated corn on April 15 and irrigated corn on March 15 for
maximum profit. Essentially, the optimal planting date occurred earlier
in the season. The importance of these results was that irrigated corn
and irrigated soybean made viable options for multiple cropping systems.
Wheat is drought-resistant even on the prevailing sandy soil in
north Florida. As shown in Figure 17, dry land production of winter
wheat usually results in positive incomes. These simulation results
(Table 16) indicate, except for very late planting (on December 20),
non-irrigated winter wheat production is appreciable. The use of
irrigation increases yield, ranging from 7 to 20 percent above non-
irrigated production. However, the increases of revenue accordingly
could not compensate for the expensive irrigation cost. Therefore,
under current low wheat price and high irrigation cost, irrigated
production of wheat does not seem to be an economic practice in Florida.
In summary, application of irrigation to corn and soybean is
essential for successful production. Irrigation makes peanut production

105
Table 16. Simulation results1 of irrigated and non-irrigated wheat
grown on different planting dates for 25 years of historical
weather data for Gainesville, FL.
Non-Irrigated
- - - Planting
Date â– 
Oct.
10
Nov.
1
Nov.
25
Dec I
?0
Season (days)
217.3
(3.9)
192.5
(3.9)
169.0
(4.2)
147.1
(4.0)
Yield (kg/ha)
2831.0
(151)
3262.0
(193)
2856.0
(352)
2186.0
(390)
Profit ($/ha)
123.6
(18.5)
176.2
(23.6)
126.7
(42.8)
44.8
(47.5)
Act. ET (cm)
23.4
(2.1)
23.9
(2.6)
23.7
(3.3)
21.4
(3.4)
Ref. ET (cm)
41.2
(1.9)
35.4
(1.8)
32.5
(1.8)
32.5
(1.2)
Rainfall (cm)
57.4
(17.4)
52.9
(18.3)
48.4
(17.4)
43.4
(15.8)
Irrigation (cm)
0.0
0.0
0.0
0.0
Irrigated
Season (days)
217.3
(3.9)
192.5
(3.9)
169.0
(4.2)
147.1
(4.0)
Yield (kg/ha)
3044.0
(72)
3484.0
(43)
3188.0
(56)
2630.0
(69)
Profit ($/ha)
43.3
(27.2)
96.8
(34.7)
57.1
(32.5)
-21.2
(31.0)
Act. ET (cm)
27.6
(2.1)
28.5
(1.7)
30.3
(1.5)
29.7
(1.3)
Ref. ET (cm)
41.2
(1.9)
35.4
(1.8)
32.5
(1.6)
32.5
(1.2)
Rainfall (cm)
57.4
(17.4)
52.9
(18.3)
48.4
(17.4)
43.4
(15.8)
O
Irrigation (cm)
15.7
(3.8)
15.7
(5.0)
16.3 (4.3)
17.9
(3.8)
1. mean value followed by standard deviation in parenthesis.
2. water required to avoid most water stress.

Cumulative probability Í71 Cumulative probability
106
Cumulative probability of profit for non-irrigated peanut
planted on April 1 ( ), May 1 June 1 (—),
July 1 ( ).
Figure 17. Cumulative probability of profit for non-irrigated wheat
planted on Oct. 10 ( ), Nov. 1 Nov. 25 (—),
Dec. 20 (- -).

107
more profitable than when it is not irrigated. Winter wheat is less
sensitive to water stress. Thus, the profitability of irrigating wheat
is doubtful. In addition, irrigation makes early planting of corn,
soybean, and peanut profitable production systems. It therefore makes
multiple cropping composed of these crops more adaptable to the north
Florida region.
Evaluation of the Simulation-Optimization Techniques
To evaluate the efficiency of the simulation-optimization
technique, the activity network model was applied to project optimal
cropping sequences of both irrigated and non-irrigated fields for varied
planning horizons. Using 20 randomly generated weather patterns to
simulate activity networks, results from the model are summarized in
Table 17. These data show that the networks were relatively small.
Computational requirements of the model is reasonably low. For
instance, it took only 21 minutes of CPU time on a mini-computer (PRIME
550) to optimize a non-irrigated multiple cropping field for a 4.5-year
planning horizon. The network contained 278 nodes and 710 arcs.
The networks did not expand significantly for each additional year
of planning. Accordingly, an activity network of multiple cropping
systems is a good method for system representation. In addition, the
longest path algorithm implemented in the model is very efficient and
the time required to select the K optimal multiple cropping sequences
was minor. This is suggested by two observations. First, referring to
Table 17, under different irrigation options it took almost equal time
to solve two networks that were different in size. Secondly, it took an
additional 5 minutes of CPU time to resolve the networks that were

108
Table 17. Summary of network characteristics and CPU time required for
various durations of planning horizon and two irrigation
conditions.
Non-Irrigated Field Irrigated Field
2.5
3.5
PIanning
4.5
Horizon
5.5
(years) - -
4.5
5.5
Nodes
147
203
278
352
325
414
Arcs
359
544
710
896
993
1283
CPU Time (min)
11
16
21
26
21
25

109
increased in size by 69 nodes on the average. Therefore, in the process
of obtaining optimal multiple cropping sequences, the simulation of the
network is more critical than optimization in terms of computational
requirements. In future work, efforts should be made to design
algorithms for simulation studies.
To illustrate the utility of the model, an output of the model is
shown in Figure 18. A cropping sequence is a series of crops.
Production of each crop is characterized by its attributes: day to
plant, crop cultivar, and associated seasonal management. Also,
reported in the output are soil water condition at the begining of a
season (system states), length of season, and the resultant discounted
net return of a crop in the sequence. The output as well lists various
sequences in order of total net discounted return. In this particular
run, results shown that under a non-irrigated field, peanut was a
favorable summer crop and winter wheat was profitable every year. It
also suggested no crop in the summer of the 4-th year. In terms of a
real year, it was 1954. This year was dry. During the period from May
to September, there was only 52.1 cm of rainfall compared to a 25 years
average of 76.2 cm. Such a result provides the decision-makers with
detailed information of the planning of cropping system under known
weather conditions.
To aid in decision-making, summarized results under the influence
of stochastic inputs, (i.e.: different weather patterns), are essential.
Since categorizing weather input involves voluminous work, determining
the effects of these factors on decisions about crop sequencing becomes
exhaustive. An alternative is the statistical analysis on maximum net
returns which result from various optimal cropping sequences. The

no
RUN # 6 WEATHER FILE:
CROP PRICE ($/KG)=0.103 0.103 0.238 0.238 0.137 0.473
SEQUENCE 1 HAVING TOTAL NET DISCOUNTED RETURN $1375
DECISION INITIAL SEASON IRRIGATION DISCOUNT
DATE
S.W.
CULTIVAR
(DAYS)
STRATEGY
RETURN
MAR-16-1
10%
IDLE
42
****
0
APR-27-1
5%
IDLE
42
kkkk
0
JUN- 8-1
8%
PEANUT
130
RAIN-FED
125
OCT-19-1
8%
WHEAT301
206
RAIN-FED
176
MAY-17-2
9%
PEANUT
129
RAIN-FED
314
SEP-27-2
7%
IDLE
42
kkkk
0
NOV- 8-2
8%
WHEAT301
187
RAIN-FED
162
MAY-16-3
9%
IDLE
42
kkkk
0
J UN-27—3
8%
BRAGG
132
RAIN-FED
221
NOV- 7-3
8%
WHEAT301
186
RAIN-FED
132
MAY-15-4
7%
IDLE
42
kkkk
0
JUN-26-4
6%
IDLE
42
kkkk
0
AUG- 7-4
7%
IDLE
91
kkkk
0
NOV- 6-4
10%
WHEAT301
188
RAIN-FED
125
MAY-14-5
8%
PEANUT
129
RAIN-FED
119
SEP-24-5
10%
kkkk
â– kick
kkkk
***

SEQUENCE 2 HAVING TOTAL NET DISCOUNTED RETURN $1343
DECISION INITIAL SEASON IRRIGATION DISCOUNT
DATE
S.W.
CULTIVAR
(DAYS)
STRATEGY
RETURN
MAR-16-1
10%
IDLE
42
kkkk
0
APR-27-1
5%
IDLE
42
kkkk
0
JUN- 8-1
8%
PEANUT
130
RAIN-FED
125
OCT-19-1
8%
WHEAT301
206
RAIN-FED
176
MAY-17-2
9%
PEANUT
129
RAIN-FED
314
SEP-27-2
7%
IDLE
42
kkkk
0
NOV- 8-2
8%
WHEAT301
187
RAIN-FED
162
MAY-16-3
9%
IDLE
42
kkkk
0
JUN-27-3
8%
BRAGG
132
RAIN-FED
221
NOV- 7-3
8%
WHEAT301
186
RAIN-FED
132
MAY-15-4
7%
IDLE
91
****
0
AUG-14-4
7%
IDLE
91
kkkk
0
NOV-13-4
8%
WHEAT301
181
RAIN-FED
114
MAY-14-5
6%
PEANUT
129
RAIN-FED
99
SEP-24-5
10%
kkkk
***
kkkk
kkk
Figure 18. Sample output of optimal multiple cropping sequences

Ill
SEQUENCE 3 HAVING TOTAL NET DISCOUNTED RETURN $1335
DECISION INITIAL SEASON IRRIGATION DISCOUNT
DATE
S.W.
CULTIVAR
(DAYS)
STRATEGY
RETURN
MAR-16-1
10%
IDLE
42
0
APR-27-1
5%
IDLE
42
kkkk
0
JUN- 8-1
8%
PEANUT
130
RAIN-FED
125
OCT-19-1
8%
WHEAT301
206
RAIN-FED
176
MAY-17-2
9%
PEANUT
129
RAIN-FED
314
SEP-27-2
7%
IDLE
42
kkkk
0
NOV- 8-2
8%
WHEAT301
187
RAIN-FED
162
MAY-16-3
9%
IDLE
42
kkkk
0
JUN-27-3
8%
BRAGG
132
RAIN-FED
221
NOV- 7-3
8%
WHEAT301
186
RAIN-FED
132
MAY-15-4
7%
IDLE
42
****
0
JUN-26-4
6%
IDLE
42
****
0
AUG- 7-4
7%
IDLE
42
kkkk
0
SEP-18-4
8%
IDLE
91
kkkk
0
DEC-18-4
10%
IDLE
91
kkkk
0
MAR-19-5
8%
F.S.CORN
125
RAIN-FED
205
JUL-23-5
10%
****
***
****
kkk

SEQUENCE 4 HAVING TOTAL NET DISCOUNTED RETURN $1334
DECISION INITIAL SEASON IRRIGATION DISCOUNT
DATE
S.W.
CULTIVAR
(DAYS)
STRATEGY
RETURN
MAR-16-1
10%
IDLE
42
****
0
APR-27-1
5%
IDLE
42
kkkk
0
JUN- 8-1
8%
PEANUT
130
RAIN-FED
125
OCT-19-1
8%
WHEAT301
206
RAIN-FED
176
MAY-17-2
9%
PEANUT
129
RAIN-FED
314
SEP-27-2
7%
IDLE
42
****
0
NOV- 8-2
8%
WHEAT301
187
RAIN-FED
162
MAY-16-3
9%
IDLE
42
★ ***
0
JUN-27-3
8%
BRAGG
132
RAIN-FED
221
NOV- 7-3
8%
WHEAT301
186
RAIN-FED
132
MAY-15-4
7%
IDLE
42
kkk*
0
JUN-26-4
6%
IDLE
91
kkkk
0
SEP-25-4
10%
IDLE
91
kkkk
0
DEC-25-4
10%
IDLE
91
kkkk
0
MAR-26-5
10%
F.S.CORN
124
RAIN-FED
204
JUL-30-5
10%
irk-k-k
kkk
kkkk
***
Figure 18. (Continued)

112
analysis (Table 18) shows that coefficients of variation are large. In
the case of a 4.5-year planning horizon, the coefficient of variation is
16.5%. It suggests that predictions of cropping sequences are sensitive
to different weather inputs. In addition, coefficients of variation
decrease as planning horizons increase. Hence, adoption of various
cropping sequences is essential to maximize the longterm profit of
cropping systems.
Based on the above discussion on evaluation and utilization of the
activity network model related to decision-making on multiple cropping
systems, the use of this model as a system approach to the problem is
promising. In the following sections the model is applied to the study
of multiple cropping systems in the region of north Florida by selecting
optimal cropping sequences.
Multiple Cropping Systems of a Non-Irrigated Field in North Florida
The network model of multiple cropping systems was applied to study
optimal cropping sequences under a non-irrigated field in north Florida.
In order to maximize profit in the long run, a study of various cropping
sequences in response to different weather patterns is necessary. Figure
19 shows the simulated optimal cropping sequences for 20 synthetic
weather patterns in the region.
Under a non-irrigated farm system, winter wheat followed by peanut
was such a profitable double cropping pattern that these component crops
were used most frequently in every optimal 4.5-year cropping sequence.
This is because in north Florida conditions both peanut and wheat do not
need irrigation to make a profitable income and they do not compete for
land at the same time. As studied, production of non-irrigated corn and

113
Table 18. Sensitivity analysis of non-irrigated multiple cropping
sequences to weather patterns.
Weather
Patterns
Maximum Total Net Return, ($/ha)
2.5
â–  - Planning Horizon (years) --------
3.5 4.5 5.5
1
625
956
1053
1331
2
820
1159
1421
1628
3
966
1210
1337
1546
4
659
938
1195
1436
5
353
610
858
1156
6
998
1130
1375
1626
7
802
1028
1253
1407
8
938
1336
1583
1823
9
1121
1338
1642
1929
10
1104
1451
1555
1806
11
768
898
1247
1513
12
573
855
1112
1232
13
973
1306
1548
1671
14
1239
1373
1479
1601
15
683
1030
1287
1479
16
1105
1508
1648
1951
17
1086
1249
1507
1806
18
949
1136
1247
1492
19
637
908
1063
1305
20
652
907
1154
1543
Mean
853.6
1116.3
1328.2
1564.0
Std. Oev.
229.7
232.3
219.3
222.8
C.V. (%)
26.9
20.8
16.5
14.2
Average per year
341.4
318.9
295.2
284.4

114
Weather
Pattern .--yearl —. —year2—. —year3 —. —year4—.—year5 —.
1
. BG .
•
FS. .
WH
.PE . .
WH .
PE ..
WH
•
2
. BG .
•
FS. .
WH
.FS ..
WH .
PE ..
WH
. .WN.
3
. BG .
WH
. PE.
•
WH
.PE ..
WH .
PE ..
WH
. .WN.
4
. FS . .
WH
•
WH
.PE . .
WH .
PE .
•
FS .
5
•
WH
•
WH
.SS. .
WH .
PE ..
WH
. PE .
6
.PE .
WH
.PE .
•
WH
. . BG .
WH .
•
WH
. PE.
7
.PE .
WH
.FS.
•
FS . .
WH .
PE. .
WH
. PE.
8
.PE .
WH
•
WH
.. BG .
WH .
.BG .
WH
. PE.
9
.PE .
WH
. .BG
•
•
FS . .
WH .
PE .
. WH
. PE.
10
.FS . .
WH
•
WH
. PE.
.SS
.BG .
WH
•
11
.PE .
WH
•
WH
. PE. .
WH .
•
WH
.PE .
12
. FS . .
WH
.PE .
•
WH
• •
WH .
PE .
FS .
13
. PE .
•
FS .BG
•
•
FS . .
WH .
. BG .
WH
.PE .
14
.SS .BG .
WH
. .BG
•
WH
.PE . .
WH .
•
WH
•
15
•
m
•
WH
.. BG
•
WH
.PE .
.SS
.BG .
WH
. .WN.
16
.FS . .
WH
.PE .
•
WH
•. BG .
WH .
.BG.
WH
. PE .
17
• PE .
WH
.PE .
•
WH
.PE .
. FS . .
WH
. .WN.
18
.PE .
WH
• PE ..
WH
.PE .
. WH
.PE .
. WH
•
19
.FS . .
WH
.PE .
•
WH
.SS. .
WH
.PE .
. WH
.SS.
20
. BG
. WH
•
WH
.PE . .
WH
.PE .
. WH
. .WN
Figure 19. Optimal multiple cropping sequences of a non-irrigated field
with full-season corn (FS), short-season corn (SS), soybean
'Bragg' (BG), soybean 'Wayne' (WN), peanut (PE) and wheat
(WH), allowing continuous cropping of peanut.

115
soybean is risky on a seasonal basis, though, it was feasible to
incorporate monocropping of soybean or corn and double cropping of
wheat-corn, wheat-soybean and corn-soybean into cropping sequences as
shown in Figure 19. Consequently, currently practiced, annual cropping
systems in north Florida can be integrated into a longterm cropping
sequence which will take into account of the variation in future weather
conditions.
In contrast to real-time decision models, the model developed is
for use in preseason planning. Provided that expected conditions are
known, the simulation-optimization model would solve the optimal
sequence to the multiple cropping system. However, future weather
variations and fluctuations of crop marketing prices are not known.
Hence, results of multiple simulation runs such as Figure 19 are
necessary in order to make rationale decisions.
One possible use of this figure is to help answer the following
question. Which crop should be planted now (March 16 for the results
presented here) in order to maximize total profit over a 4.5-year
period? Because future weather is unknown, this can be answered by
examining the frequency of the crop as the first crop in all possible
optimal cropping sequences. These optimal cropping sequences are the
simulated results under to various weather patterns. As a result, a
decision can be made to plant a crop according to its frequency. This
process could be repeated after each crop is harvested to provide
updated evaluations of crop selection.
From Figure 19, the frequency of the first selected crop was 9 for
peanut, 5 for full-season corn and 4 for 'Bragg' soybean out of 20
possible choices. There was a possibility for double cropping of short-

116
season corn followed by 'Bragg' soybean. However, this occurred only 1
time out of 20. As a result, growing peanut was a reasonable choice to
start the multiple cropping sequence. In addition, the farmer would
expect to plant winter wheat after the first summer season according to
these results.
From the agronomist's point of view, there may be potential
problems, especially diseases and nemotodes in continuous peanut
cropping. So, the optimal multiple cropping sequences suggested in
Figure 19 are not without risk. Consequently, a posterior optimization
technique was applied to seek alternate systems and to determine the
effect of various sequences. The procedure was the same in the
simulation phase of the network system. When optimizing the network,
paths which have continuous peanut cropping were then discarded from
optimal sequences. The modified optimal cropping sequences are
presented in Figure 20. The effects of changes are then demonstrated in
Table 19.
Indicated by rank, these new improved sequences did not deviate
much from the first optimal (Table 19) except a few of them. These few
exceptions include more than 3 peanut seasons. Peanut is the most
profitable of the crops under consideration. Therefore, dropping a
peanut season would mean a big loss to the farmer. However, on the
average, percentage of profit reduction was small (1.9%). Without
sacrificing much cumulative profit ($8 annually), the farmer could use a
cropping system which emcompasses more management techniques. These
modified cropping sequences are then the strategies which, in the long
run, not only maximize return but also maintain the balance of natural
crop environment. The posterior optimization technique proved to be a
valuable tool in this application.

117
Weather
Pattern
.--yearl—
.—year2—
-year3—
year4--
-year5—.
1
. BG .
.FS . .
WH
• SS. .
WH
. PE .
. WH
•
2
. BG .
.FS . .
WH
.FS ..
WH
. PE .
. WH
. .WN.
3
. BG .
WH . PE ..
WH
. .BG .
WH
.PE .
. FS .
4
. FS . .
WH .
WH
.. BG .
WH
. PE .
.FS .
5
•
WH .
WH
.SS.
•
FS . .
WH
. PE .
6
.BG .
WH . PE. .
WH
.. BG .
WH
•
. WH
. PE.
7
.PE .
WH .FS.
•
FS . .
WH
.PE .
. FS.
8
.PE .
WH .
WH
.. BG .
WH
. .BG
. WH
.PE .
9
.PE .
WH . .BG .
.FS . .
WH
. .BG
. WH
.PE .
10
.FS . .
WH . PE ..
WH
• .BG .
•
FS .BG
. WH
•
11
.PE .
WH .
WH
.. BG .
WH
• •
WH
• PE .
12
. FS . .
WH .
WH
• •
WH
. PE .
.FS .
13
. PE .
.FS .BG .
FS . .
WH
.. BG
. WH
.PE .
14
.SS .BG .
WH . .BG .
WH
.PE . .
WH
•
. WH
•
15
.PE .
WH .. BG .
WH
.PE .
•
SS .BG
. WH
. .WN.
16
.FS . .
WH .PE . .
WH
.. BG .
WH
. .BG.
WH
. PE .
17
**
18
.PE .
WH .SS. .
WH
. PE ..
WH
.WN..
WH
•
19
.FS . .
WH .PE .
.WH
.SS. .
WH
.PE .
. WH
.SS.
20
. BG .
WH .
WH
.PE . .
WH
.. BG
. WH
. .WN.
Figure 20. Optimal multiple cropping sequences of a non-irrigated field
with full-season corn (FS), short-season corn (SS), 'Bragg1
soybean (BG), 'Wayne' soybean (WN), peanut (PE), and wheat
(WH), not allowing continuous cropping of peanut. (** In the
post optimal analysis, search is beyond the first 100 optimal
sequences.)

118
Table 19. Comparison of various multiple cropping systems under a non-
irrigated field . System I includes corn, soybean, peanut,
and wheat allowing continuous peanut croppipng. System II
includes same crops as system I, but not allowing continuous
peanut cropping. Systems III excludes peanut from
consideration.
Weather
Pattern
Max.Net Profit
% of Profit Reduction
I
II
III
II
III
1
1053
1049 (2)1
966
0.4
8.3
2
1421
1421 (1)
1316
0.0
7.4
3
1337
1275 (24)
1110
4.6
17.0
4
1195
1159 (5)
1079
3.0
9.7
5
858
843 (3)
788
1.7
8.2
6
1375
1320 (5)
1068
4.0
21.0
7
1253
1233 (5)
1204
1.6
3.9
8
1583
1583 (1)
1428
0.0
9.8
9
1642
1618 (9)
1475
1.5
10.2
10
1555
1535 (3)
1436
1.3
7.6
11
1247
1180 (35)
998
5.4
20.0
12
1112
1091 (4)
1011
1.9
9.1
13
1548
1548 (1)
1330
0.0
14.1
14
1479
1479 (1)
1425
0.0
3.6
15
1287
1287 (1)
1191
0.0
7.4
16
1648
1648 (1)
1475
0.0
10.5
17
1507
1242
**
17.6
18
1247
1150 (18)
1005
7.8
19.4
19
1063
1063 (1)
968
0.0
8.9
20
1154
1114 (7)
1060
3.5
8.1
Mean
1328.2
1294.5
1179.6
1.9
11.1
Std. Dev.
219.3
227.8
202.8
2.2
5.2
C.V. (%)
16.5
17.6
17.2
118
47.0
1. The rank of the optimal cropping sequence in the system I.
2. Search is beyond the first 100 optimal sequences in the post
optimal analysis.

119
In another case, a farmer may decide to exclude peanut as an option
in his cropping system because of the complexity of production, a lack
of knowledge about peanut or lack of specialized equipment required for
peanut. Thus, optimal cropping sequences were studied when only three
crops, corn, soybean and wheat, were considered. Figure 21 shows the
optimal crop sequencing for this system. The significant difference
from the previous system was that wheat-soybean and wheat-corn double
cropping were commomly mixed in the sequences, plus one or two seasons
of monocropped corn, soybean and wheat. In this system, there were
fewer crops scheduled in the sequence. Thereby, lower returns (Table
19) were obtained from the system. On the average, there was 11.1%
reduction of maximum return compared to the system with peanut included.
From Figure 21, the frequency of the first selected crop was 11 for
'Bragg* soybean, 1 for 'Wayne' soybean and 6 for full-season corn out of
20 possible choices. As a result, growing 'Bragg' was a reasonable
choice to start the multiple cropping system which excluded peanut for
consideration. Soon after the first summer season, the following winter
wheat would be planted.
Effects of Irrigation on Multiple Cropping
The multiple cropping system under a non-irrigated field has been
studied. The results showed a majority of wheat-peanut double cropping
as a major profitable sequence. A balance of other crops in the system
was maintained though, and other practiced cropping systems, (i.e.:
wheat-soybean, wheat-corn, and corn-soybean) appeared to be relatively
competitive on a long-term basis. So cropping sequences under a non-
irrigated field were very diversified. Under an irrigated farm,

120
weather
Pattern
.—yearl-
year2--
-.—year3—
.—year4 —
.—year5—.
1
. BG
• •
FS . .
WH . SS . .
WH . .BG .
WH .
2
. BG
• •
FS . .
WH .FS ..
WH . .BG .
WH . .WN.
3
. BG
. WH
. .BG
. WH ..BG .
WH .. BG .
WH .
4
. FS . .
WH
. WH .. BG .
WH .
.FS .
5
•
WH
. WH .SS.
. FS . .
WH .WN.
6
.BG .
WH
.SS.
. WH .. BG .
WH .
.FS .
7
.WN.
WH
.FS.
. FS .
WH . .BG .
.FS .
8
. BG
. WH
. WH .. BG .
WH . .BG .
.BG .
9
• BG .
WH
. .BG
. .FS .
WH . .BG .
WH ..WN.
10
.FS . .
WH
. BG .
WH ..BG .
.FS .BG .
WH .
11
. BG
. WH
. WH .. BG .
WH .
WH .SS.
12
. FS .
. WH
. WH .
WH .
.FS .
13
. FS .
•
FS .BG
. . FS . .
WH .. BG .
.BG .
14
.SS .BG
. WH
. .BG
. WH .. BG .
WH .
WH .
15
.BG .
WH
.. BG
.FS .
.FS .BG .
WH . .WN.
16
.FS . .
WH
.. BG
. WH .. BG .
WH . .BG.
WH .
17
. BG
. WH
.. BG
. WH ..BG .
WH .
WH . .WN.
18
.BG .
WH
.SS.
. WH .SS. .
WH . .BG .
WH .
19
.FS.
WH
.SS.
. WH .SS. .
WH .SS. .
WH .SS.
20
. BG
. WH
. WH .. BG .
WH .. BG .
WH . .WN.
Figure 21. Optimal multiple cropping sequences of a non-irrigated field
considering full-season corn (FS), short-season corn (SS),
'Bragg' soybean (BG), 'Wayne' soybean (WN), and wheat (WH),
excluding peanut.

121
production of peanut became even more prominent and profitable (Figure
22). Given the current market values of the crops, corn-soybean was the
only competitive cropping system which occurred with some regularity.
Therefore, yearly production of peanut was the dominant sequence in
multiple cropping system under irrigation.
As shown in Figure 23, when peanut was not allowed to be
continuously cropped, the farmer could plant peanut one year and grow
corn-soybean double cropping the other year.
Under an irrigated field in north Florida, a multiple cropping
system including only corn, soybean and wheat was also studied. It was
found that corn-soybean double cropping was repeated each year (Figure
24). This sequence was the most profitable cropping system under
irrigation. Wheat-soybean became the substitute cropping practice in
dry years. Interestingly, wheat-corn cropping sequence was not a good
choice under irrigation. That was because wheat usually grows to
maturity in May. The second crop, corn, could not be planted on its
optimal date. Consequently, yield and profit would be reduced. Owing
to this, such a practice did not compare to the more prevailing wheat-
soybean system. In summary, combination of corn-soybean and wheat-
soybean appeared to be the optimal cropping sequences for the farmer who
uses irrigation to produce corn, soybean and wheat in north Florida.
Risk Analysis of Non-Irrigated Multiple Cropping Sequences
One of the risk assessment methods uses mean and variance values.
A risk averse individual faced with two strategies of equal expected
value would likely choose the one with the smaller variance. Likewise,
the choice between two strategies of equal variance will be based on the

122
Weather
Pattern
.—yearl-
-.—year2 —
-year3 —.
—year4—.
—year5—
1
. PE .
.PE .
• PE .
. PE .
. PE .
2
. PE .
.PE .
. WH
.PE .
. PE .
.SS .BG .
3
. PE .
.PE .
•
FS .BG .
. PE .
. PE .
4
. PE .
. PE .
. PE .
.PE .
.PE .
5
. PE .
. FS .BG
•
. PE .
. PE .
. PE .
6
. PE .
. PE .
•
FS .BG .
.PE .
. PE .
7
.SS .BG
. WH .PE .
. PE .
. PE .
. PE .
8
. PE .
. PE .
•
FS .BG .
.PE .
. PE .
9
.SS .BG
.SS .BG
. WH
.PE .
. PE .
. PE .
10
. PE .
. PE .
•
FS .BG .
. PE .
. PE .
11
. PE .
. PE .
•
FS .BG .
. PE .
. PE .
12
. PE .
. PE .
. PE .
.PE .
• PE .
13
. PE .
.PE .
. PE .
. FS .BG .
. PE .
14
. PE .
. PE .
. PE .
. PE .
. PE .
15
. PE .
. PE .
. PE .
. FS .BG .
.SS .BG .
16
. PE .
.FS .BG
•
.SS .BG .
.SS .BG .
. PE .
17
. PE .
. PE .
•
FS .BG .
. PE .
.SS .BG .
18
. PE .
. PE .
. PE .
. PE .
. PE .
19
. PE .
.PE .
.PE .
.PE .
. PE .
20
. PE .
.PE .
.PE .
.PE .
. FS .WN.
Figure 22. Optimal multiple cropping sequences of an irrigated field
with full-season corn (FS), short-season corn (SS), 'Bragg'
soybean (BG), 'Wayne' soybean (WN), peanut (PE), and
wheat (WH), allowing continous cropping of peanut.

123
Weather
Pattern
.—yearl--.
—year2 —.
—year3—.
—year4—
.—year5 —.
1
. PE .
.FS .WN.
. PE .
. FS .WN.
. PE .
2
kk
3
★ ★
4
. PE .
.FS .WN.
. PE .
. FS .WN.
. PE .
5
. PE .
.FS .BG .
. PE .
.FS .WN.
. PE .
6
. PE .
.FS .BG .
. PE .
. FS .WN.
. PE .
7
.SS .BG .
. PE .
. FS .WN.
. FS .BG .
. PE .
8
kk
9
.SS .BG .
.SS .BG .
WH .PE .
•SS .BG .
WH .PE .
10
**
11
kk
12
★ ★
13
. PE .
. FS .BG .
.PE .
. FS .WN.
. PE .
14
kk
15
•kle
16
. PE .
.FS .BG .
.SS .BG .
•SS .BG .
. PE .
17
kk
18
kk
19
kk
20
kk
Figure 23. Optimal multiple cropping sequences of an irrigated field
with full-season corn (FS), short-season corn (SS), 'Bragg'
soybean (BG), 'Wayne' soybean (WN), peanut (PE), and wheat
(WH), not allowing continuous cropping of peanut. (** In the
post optimal analysis, search is beyond the first 100 optimal
sequences.)

124
Weather
Pattern .—yearl —. —year2 —. —year3 —. —year4—.—year 5—.
1
.FS
.WN .
.FS
.WN.
•
FS
.BG .
WH
•
.BG .
.SS
• WN.
2
.FS
.WN .
.SS
.BG .
WH
• •
BG .
WH
•
.BG .
.SS
.WN.
3
.SS
.BG .
•
FS
.BG .
•
FS
.BG .
.SS
.BG .
WH .
BG .
4
.SS
.WN.
•
FS
.WN.
•
FS
.BG .
.FS
.WN.
.FS
.WN.
5
.SS
.BG .
.SS
.BG .
WH
• •
BG .
.FS
.WN.
. FS
.WN.
6
.SS
.BG .
.FS
.WN.
•
FS
.BG .
.SS
.BG .
.SS
.WN.
7
.SS
.BG .
WH
• •
BG .
.SS
.BG .
.SS
.BG .
.SS
• WN.
8
.SS
.BG .
WH
•
.BG .
WH
• •
BG .
.SS
.BG .
.SS
.WN.
9
.SS
.BG .
.SS
.BG .
.FS
.WN.
•
FS
.WN.
. FS
.WN.
10
.FS
.WN.
WH
•
BG .
•
FS
.BG .
.SS
.BG .
.SS
.WN.
11
.SS
.BG .
WH
• •
BG .
.SS
.BG .
.FS
.WN.
. FS
.WN.
12
. FS
.WN.
WH
• •
BG .
.FS
.WN.
.FS
.WN.
.FS
.WN.
13
. FS
.WN.
•
FS
.BG .
.SS
.WN.
•
FS
.BG .
.SS
.WN.
14
. SS
.BG.
WH
•
.BG .
.FS
.WN.
•
FS
.WN.
. FS
.WN.
15
. SS
.BG.
WH
• •
BG .
WH
• •
BG .
.FS
.BG .
.SS
.WN.
16
.FS
.WN.
.FS
.BG .
.SS
.BG .
.SS
.BG .
.SS
.WN.
17
.SS
.BG .
.FS
.WN.
•
FS
.BG .
WH
•
• BG .
.SS
.WN.
18
.FS
.WN.
•
FS
.BG .
•
FS
.BG .
.SS
.BG .
WH .WN.
19
.FS
.WN.
.FS
.BG .
.SS
.BG .
.SS
.BG .
WH .WN.
20
.SS
.BG .
•
FS
.WN.
•
FS
.BG .
• FS
.WN.
. FS
.WN.
Figure 24. Optimal multiple cropping sequences of an irrigated field
considering full-season corn (FS), short-season corn (SS),
'Bragg' soybean (BG), 'Wayne' soybean (WN), and wheat (WH),
excluding peanut.

125
higher expected value. Therefore, a mean-variance efficiency method was
applied to evaluate various multiple cropping systems.
The 10 cropping sequences shown in Figure 25 were used for
comparison. These 10 candidates were taken from Figure 20 to represent
unique near-optimal sequences for a crop production system considering
corn, soybean, peanut and wheat in a non-irrigated field. In the
simulation, the fixed optimal planting date used for full-season corn
was March 10; short-season corn was March 17; 'Bragg' soybean was June
8; 'Wayne' soybean was July 2; peanut was May 22; winter wheat was
November 6.
Analysis of net returns of non-irrigated multiple cropping
sequences in response to different weather and crop marketing are shown
in Table 20 and 21, respectively. Analysis of variance indicated that
the effects of changing weather and crop prices on net returns were
significant at the 5% level. Thus, for the future use of the model as a
real-time decision tool, inputs of weather data and crop prices deserve
special attention.
Statistical analysis of the results (Table 20) showed that
coefficients of variation of sequence 3, 5, 6, were higher than those of
others. This demonstrated that instability of these sequences would
occur from variations in weather. This is because frequent corn
monocropping and/or corn-soybean double cropping occurred in these
sequences. Therefore, in a long run these sequences should not be
considered favorable multiple cropping practices under non-irrigated
conditions.
All 10 cropping sequences were further analyzed on the basis of
mean-variance efficiency criteria. A plot of mean value against

126
Cropping
Sequence .
--yearl—
-year2—
.—year3 —
.—year4—
.—year5—.
1
. BG .
.FS . .
WH .FS ..
WH . PE ..
WH
. .WN.
2
. BG .
WH
. PE ..
WH ..BG .
WH .PE .
•
FS .
3
.PE .
WH
.FS.
. FS . .
WH .PE .
•
FS.
4
.PE .
WH
• •
WH .. BG .
WH . .BG .
WH
.PE .
5
.FS . .
WH
. PE ..
WH ..BG .
.FS .BG .
WH
•
6
. PE .
.FS .BG .
. FS . .
WH .. BG .
WH
• PE .
7
.SS .BG .
WH
• .BG .
WH .PE . .
WH .
WH
•
8
.PE .
WH
.. BG .
WH .PE .
.SS .BG .
WH
. .WN.
9
.FS . .
WH
.PE . .
WH .. BG .
WH . .BG.
WH
. PE .
10
.FS . .
WH
.PE .
.WH .SS. .
WH .PE . .
WH
.SS.
Figure 25.
A set of
optimal multiple cropping sequences
for
a non-
irrigated field chosen from Figure 20 for additional
simulation studies; FS: full-season corn, SS: short-season
corn, BG: 'Bragg' soybean, WN: 'Wayne' soybean, PE: peanut,
WH: wheat.

Table 20. Analysis of net returns of non-irrigated multiple cropping sequences in response
to different weather patterns.
Maximum Net Return ($/ha)
Weather
Pattern
SQ1
SQ2
S03
SQ4
SQ5
SQ6
SQ7
SQ8
SQ9
SQ10
Average
1
854
385
128
418
-40
350
216
389
222
342
326.4
2
1299
591
269
722
260
579
239
897
383
474
571.3
3
397
1077
514
926
623
220
658
866
747
670
669.8
4
13
699
184
435
613
-704
562
31
748
711
329.2
5
38
140
161
324
75
-389
190
83
215
362
119.9
6
488
818
125
805
410
245
614
247
879
475
510.6
7
823
1128
1107
1054
228
685
968
392
701
470
755.6
8
972
801
305
1471
836
692
548
980
1286
1032
892.3
9
652
1095
1157
1083
449
1241
912
670
863
656
877.8
10
743
804
86
594
1380
210
1028
1039
1113
1226
822.3
11
387
538
310
1128
219
674
590
538
727
355
546.6
12
69
731
160
322
600
-203
395
-127
740
554
324.1
13
862
807
693
1016
722
1354
903
691
1317
1209
957.4
14
233
671
106
515
691
-95
1531
441
838
853
578.4
15
996
541
628
615
645
456
618
1084
521
709
681.3
16
279
731
27
1182
1263
344
1050
733
1692
1140
844.1
17
588
806
133
751
527
-78
895
868
625
629
574.4
18
628
688
367
854
74
29
484
499
339
273
423.5
19
486
442
251
398
679
-164
473
156
659
892
427.2
20
610
551
-56
744
-12
-66
289
580
378
353
337.1
Statistical
Significance
abc
abc
de
a
cd
e
abc
be
ab
abc
Mean
570.8
702.2
332.8
767.8
512.1
269.0
658.1
552.8
749.6
669.2
Standard
Deviation
345.3
241.3
332.9
321.0
382.6
513.1
341.7
352.2
380.5
301.7
C.V. (%)
60.5
34.4
100.0
41.8
74.7
190.7
51.9
63.7
50.7
45.1

Table 21.
, Analysis of net returns of
pricing schemes for weather
non-irrigated multiple
pattern number 3.
cropping
sequences
under
different
crop
Crop
Maximum Net Return
($/ha)
Pricing
Schemes
Price
Changed
SQ1
SQ2
S03
SQ4
SQ5
SQ6
SQ7
SQ8
SQ9
SQ10
1
baseline
479
1077
514
926
623
220
658
866
747
670
2
corn(-20%)
293
1012
350
926
535
165
654
782
740
550
3
corn(-10%)
344
1045
432
926
578
192
656
824
743
609
4
corn(+10%)
449
1110
597
926
668
247
660
908
751
732
5
corn(+20%)
502
1143
680
926
713
275
662
950
755
792
6
soybean(-20%)
314
929
514
779
505
85
513
745
600
670
7
soybean(-10%)
355
1002
514
853
564
152
586
806
674
670
8
soybean(+10%)
438
1151
514
1000
683
288
732
927
821
670
9
soybean(+20%)
478
1223
514
1071
740
353
802
983
892
670
10
wheat(-20%)
208
869
378
663
421
102
394
666
481
404
11
wheat(-10%)
299
969
444
790
518
159
521
763
609
532
12
wheat(+10%)
496
1187
586
1064
728
282
797
970
886
809
13
wheat(+20%)
587
1287
652
1191
826
339
923
1068
1014
937
14
peanut(-20%)
307
859
314
753
500
47
529
625
561
457
15
peanut(-10%)
352
969
414
840
562
134
594
746
655
564
16
peanut(+10%)
442
1184
613
1012
683
306
722
984
839
775
17
peanut(+20%)
488
1295
715
1100
746
394
787
1106
934
884
Statistical
Significance
h
a
g
b
f
i
e
c
d
e
Mean
401.8
1077.1
514.4
926.2
623.1
220.0
658.2
865.8
747.2
670.3
Standard
Deviation
100.7
135.6
116.5
138.1
110.7
101.0
130.4
136.8
141.5
143.8
C.V. (%)
25.1
12.6
22.6
14.9
17.8
45.9
19.8
15.8
18.9
21.4
128

129
variance is helpful for the analysis. In the context of mean-variance
efficiency, pairwise comparisons eliminated alternative sequences 1, 3,
5, 6, 8 from an efficient set. For the remaining efficient sequences
(2, 4, 7, 9, 10), their use would depend on the farmer's preferences and
local conditions. Among them, sequences 4 and 2 yielded more income
with less variability. Sequence 2 and 4 continued cropping with
prominent component crops of wheat-peanut and wheat-soybean, alternating
each year.
The second type of variation was introduced to cropping sequences
by crop marketing. In Table 20, averages of net returns of different
cropping sequences were shown under various weather conditions. The
average net return for weather pattern 3 was the median of all 20
weather patterns. So weather pattern 3 was chosen to represent the
average climatic condition in the simulations. By simulating various
pricing schemes, variability of different sequences were calculated and
shown in Table 21. Of particular interest was that cropping sequences
1, 3, 6, 10 again had large coefficients of variation, while those of
sequence 2, 4, 8 and were relatively small. Such results are very
similar to those on the basis of risk to weather variability.
Therefore, statistically speaking, sequences 4 and 2 were preferred to
sequence 3 and 6.
In summary, for a higher net return with a smaller variation under
non-irrigated conditions, farmers could adopt either sequence 4 or 2,
which include mostly peanut, wheat, and 'Bragg' soybean in summer-winter
cropping each year.

130
Applications to Other Types of Management
Farming is much more than irrigation management. To farm
profitably for any significant period of time, it is also necessary to
understand the maintenance or improvement of organic matter, soil
structure and fertility, as well as the control of insects, weeds,
diseases, and erosion. Because of these many areas of management the
problem of deciding multiple cropping sequences to be followed become
more complex, and it must be examined systematically if it is to be
analyzed properly. The framework developed in the study is readily
applicable to these other areas of cropping management.
Application of the model to the specific area of pest and disease
management will be considered as an example of implementation
procedure. In multiple cropping systems, pests are of concern
throughout the entire cropping period. In order to minimize pest damage
to multiple cropping systems, a model of pest and disease balance is
needed for pest management just as a soil water balance model was
produced earlier for irrigation management. By representing the state
of the system by pest population, such as nematodes, and using a pest
model to simulate the state, a network model thus can be constructed for
studying optimal cropping sequences with particular application to pest
control and management.
To study more refined multiple cropping sequences, both irrigation
and pest management could be integrated. In such a case, levels of soil
water content and pest population would then be used to represent the
state of the system. There is no evidence that pest problems would
affect water conditions in the soil. Accordingly, soil water balance
model may not need to be modified. However, incidences of pests depend

131
on weather and the condition of soil. Therefore, more sophisticated
pest population models that are capable of responding to irrigation
strategy and pest management are required.
With these models at hand, the methodology developed is ready to
study the effects of irrigation and pest management on multiple cropping
system. Furthermore, provided other areas of management have been
investigated, more detailed, complicated multiple cropping systems can
be explored.

CHAPTER VI
SUMMARY AND CONCLUSIONS
Summary and Conclusions
Multiple cropping is one of the means to increase and help
stabilize net farm income where climatic and agronomic conditions allow
its use, such as in Florida. With several crops to be examined
simultaneously, the design of multiple cropping sequences becomes
complex. Therefore, a systems approach is needed. This study has
successfully developed a framework for optimizing the multiple cropping
system by selecting cropping sequences and their management practices.
By combining simulation and optimization techniques, the
deterministic activity network model for the multiple cropping system
was the best choice from those that were investigated in terms of system
representation and computational requirements. With particular
application to irrigation management, models of crop yield response,
crop phenology, and soil water balance were required for system
simulation. The level of sophistication of models has been determined
and component models were developed and implemented. Afterward, the
longest path algorithm was utilized to seek the K optimal multiple
cropping sequences for the production system considered.
By applying the methodology to study a farm in north Florida,
optimal multiple cropping sequences were identified. Under a non-
irrigated farm, winter wheat followed by either soybean, corn or peanut
132

133
forms the annual profitable cropping component in a multiple cropping
sequence. Especially favorable is the cropping of wheat-peanut.
Another significant conclusion to be drawn concerned the effect of
irrigation management on optimal multiple cropping sequences. After the
investment in an irrigation system, the multiple cropping system coupled
with irrigation was shown to almost double a farm net income in 4.5
years. An irrigated peanut crop was found to be prominent and was
scheduled each year assumming that the high value of peanut will
continue. In a system in which peanut was not considered an option,
inclusion of irrigated wheat-corn cropping could not be recommended as a
profitable multiple cropping system. Instead, double cropping of corn-
soybean was the main scheme under irrigation with the possible
substitution of wheat-soybean.
Suggestions for Further Research
The conclusions made with regard to multiple cropping systems in
north Florida were under the assumption that farming management other
than irrigation were optimally practiced. So the importance of
irrigation management on implementation of multiple cropping systems can
be stressed. In the meantime, a great effort has been devoted to
developing the methodology. The methodology developed is capable of
incorporating other aspects of farming into an integrated approach for
studying multiple cropping. However, basic research to quantify the
effect of farm management on crops and the development of component
models to describe each management area would be a noteworthy
contribution to the complete system by using this methodology.

134
Eventually, the model could be used in an expert system by the
farmer to help configure their cropping enterprise. The implementation
of input data of weather and crop price has been found to have great
influence on decision-making. The exploration of decisions based on the
logistic of inputing these parameters may be a fruitful exercise in
future research efforts.

APPENDIX A
GENERAL DESCRIPTIONS OF SUBROUTINES
Main Program
The main program, as an executive, takes control of the process for
optimizing multiple cropping systems. It calls various subroutines to
accomplish every procedure, system description, generation of network,
and network optimization. General descriptions of these subroutines are
given in Table 22. The flow diagram has been previously presented in
the section of model implementation.
Simulation proceeds node by node to create a network. At a node,
new nodes are generated (NNEW * 0) and subroutine ORDER is called to
expand a network. Simulation then advances to the next node
(NEXT=NEXT+1). Toward the end of a planning horizon, the crop growth
season is too long to be in time for harvest (T0SHT=1), and the
simulation is terminated. When the network generation is completed, the
main program moves on to optimize the network created and then to output
results.
Subroutine PATIN
Subroutine DATIN is first called from the main program to depict a
multicropping production system by inputing user-defined variables and
parameters of a system. This information may be categorized by
historical weather, crop growth, and production facts. These data for
DATIN are stored in input data files, 1WFILE‘, 'GROWS1, and 'FACTS',
135

136
Table 22. General descriptions of subroutines used in optimizing
multiple cropping systems.
Subroutine
General Descriptions
System Description
OATIN
called to input information concerning production system.
Generation of Network
FIELD
A simulation model in turn calls phenology, soil water
balance, crop response submodels for generating nodes and
arcs.
PHENO
crop phenology model.
SWBAL
soil water balance model.
PROFT
calculates net discounted profit by assessing crop yield.
WCALC
calculates hourly temperature from daily maximum and
minimum temperatures.
PENMAN
Penman formula for estimating potential evapotranspiration.
RADCL
equations for calculating daily insolation.
SORT
used to order newly created nodes by increasing order of
time argument.
ORDER
organizes the expanding network by numbering nodes in
increasing order by time argument.
Network Optimization
LISTN
called to list arcs in increasing order by arc ending node.
KPATH
K longest paths algorithm.
TRACE
produces all paths having any of the K longest path
lengths.
DCODE
decode and summarize cropping sequences.

137
respectively. Free format is used to read in data. As soon as values
are available, parameters derived from these input values are
calculated. Subroutine DATIN therefore solely completes the phase of
system description in the whole process of optimization.
Subroutine FIELD
In order to have a multiple cropping system network, simulation
techniques are applied to generate new nodes and new arcs. Subroutine
FIELD therefore is used as a planning manager to monitor field
activities and to organize simulation proceedings. Procedures to
simulate crop growth and state transition (soil water contents) have
been detailed in model implementation section of chapter III.
Subroutine FIELD is composed of two independent sections. The
first section is for considering main crop production. The second
section primarily simulates bare soil evaporation during cropping idle
time. After simulating the soil water status under various cropping
systems, results are stored in temporary arrays for future network
expansion. The number of new nodes generated in a simulation cycle,
NNEW, is used as one of criteria for terminating network generation
phase. Namely, when NNEW * 0, simulations are continued.
Subroutine ORDER(NEXT,NNEW)
In the process of optimizing a network, it is advantageous to have
a network whose nodes are sequentially numbered from a source node to a
terminal node. The network will be sequentially ordered according to
its first coordinate of a node. Subroutine ORDER exactly accomplishes
the objective.

138
Each node (NODES) is identified by two coordinates. The first
coordinate is decision date (IP(NODES)). The second one is soil water
content (IW(NODES)). An arc is specified by its starting node
(FROM(ARC)) and ending node (TO(ARC)). Data related to a specific arc
are crop choice, length of growing season, management strategy, and net
return, and are stored in the arrays JC, JG, JS, and JR, respectively.
A total of NNEW nodes are to be added to expand a network by
extending arcs from node NEXT to each new node. In terms of decision
date, when the first node of the new list of nodes is later than the
last node of the existing network, direct appending is only required and
expansion is complete. Otherwise, it is necessary to insert new nodes
and re-number an old network.
Insertion of a new node in the existing network is made by
examining the latest node first. As an insertion is located, a further
test on whether the node has been numbered will be done. If the node is
an existing one, new arcs are added to a network, and bookkeeping is
executed. Otherwise, re-numbering of nodes and subsequent update of arc
data need be completed, before a correct insertion can be made
possible. Accordingly, subroutine ORDER would output a sequentially
numbered network.
Subroutine SORT(NNEW)
For easily numbering nodes, new nodes need to be ordered before
being appended to the existent network. When there are a total of NNEW
nodes generated by simulations at a presently considered decision node,
subroutine SORT is used to order new nodes by increasing order of the
next decision date (IH(I)). INDEX is an array whose I-th element gives

139
the number of simulation runs that results in the node (X(I),Y(I)). It
is then used in subroutine ORDER as an index to search out relevent data
of an arc connecting the current decision node to newly appended nodes.
Subroutine LISTN
The K longest paths algorithm requires that a network
representation is such that an arc list is a non-decreasing sequence of
arc ending nodes. To serve the purpose, subroutine LISTN prepares a
description of a given network to be read in from subroutine KPATH. The
network description is achieved by specifying for each arc of a network
a record containing its starting node, its ending node and its length.
In this application, an arc length is the net profit associated with a
cropping decision. The essential operation in LISTN is sorting the
records in increasing order by arc ending node (TO(ARC)).
Subroutine KPATH
Subroutine KPATH implements an optimization algorithm to seek all K
longest, distinct path lengths of a network. It uses the label-
correcting method as discussed in the text. First of all, a description
of the given network is read in. The records are assumed to be stored
in increasing order by arc ending node. Moreover, the nodes are assumed
to be numbered consecutively from 1 to NODES. As the network is
entered, the variables and arrays needed by labeling procedure and TRACE
subroutine are created. The labeling algorithm starts with the root
(source node) having label zero and all other nodes having infinite
label (INF). Then it enters a loop to update the label for each node 1.

140
Subroutine TRACE
Subroutine TRACE will produce all paths (JJ) from source node (NS)
to terminal node (NF), having any of the K longest path lengths (LL).
The algorithm to reconstruct optimal paths from the final node labels
has been presented in chapter III.
Subroutine WCALC
This subroutine calculates hourly temperatures and a temperature
factor for use in phenological stage prediction. Provided with daily
maximum (TMAX(N)) and minimum (TMIN(N)) temperatures as well as time of
sunrise (SNUP(N)) and sundown (SNDN(N)), WCALC is able to calculate
hourly temperatures (THR(IXX)).
It assumes that hourly temperatures between 2-hour after daylight
and sundown are sinusoid-like. After sundown temperature cools off and
presumably decreases linearly to its minimum just at 2-hour after
sunrise of the next day. Accordingly, the first part of WCALC is coded.
The remainder of subroutine WCALC enables computation of
temperature factors on development (PHTFCT(IXX)) for use in phenological
calculations. Its calculation is based on a hypothetical curve which is
well defined by three variables, the optimal (TOPT), minimum (TPHMIN)
and maximum temperature (TPHMAX). The hypothesis is: Rate of
development, which is the inverse of the duration of a phase, is
linearly related to temperature if temperature is below an optimal
value. The relationship was given in Figure 2, where the development
rate is normalized to the rate at the optimum temperature for
development. Above the optimum temperature, development rate decreases
linearly to zero. The variables TPHMIN, TOPT, and TPHMAX are read into

141
the model in subroutine DATIN. In DATIN, the high and low temperature
slopes (PHC0N3, PHC0N5) of the relationship and intercepts (PHC0N4,
PHC0N6) are also calculated for use in this subroutine.
Subroutine PHENO
This subroutine is called each day to compute the phase of the crop
development from one phenological stage to the next. All crop
development phases depend on temperature. In PHENO, this temperature
effect is expressed as physiological time (PHTFCT(IXX)) which is
calculated hourly in subroutine WCALC. By cumulating the factors, the
rate of development (OTX) during any particular day as a function of
temperature is assessed. Subroutine PHENO is divided into sections.
Each section independently supports phenological modeling of a specific
crop.
Subroutine SWBAL
With the discussed procedures in the soil water balance model,
subroutine SWBAL estimates actual evaporation and transpiration as
affected by rainfall and irrigataion use. Simultaneously, soil water
status is updated properly. For each stage, actual ET and potential ET
are accumulated.
Subroutine PENMAN
The Penman formula is coded in this subroutine for calculation of
potential evapotranspiration at a given leaf area index.

142
Subroutine PROFT
Provided with stage ET's and season depth of irrigation water, this
subroutime returns a net discounted income from production of a specific
crop.

O O O O OOO O O O -G* G» ooooo
APPENDIX B
SOURCE CODE OF SUBROUTINES
** OPTIMAL A SYSTEM ANALYSIS OF MULTICROPPING SYSTEMS
** Y.J.TSAI VERSION?.1 05/15/85 AG.EN. U. OF FLORIDA
MAIN PROGRAM
INTEGER ARC ,FR0M,T0 ,MKMND,SW1,IND(6)
LOGICAL TOSHT
INSERT NAMCM1
INSERT NAMCM2
CALL DELETE ('SERIES',6,IC)
10(1)=4
10(2)=5
10(3)=6
10(4)=7
10(5)=8
10(6)=9
IIN2=14
IIN3=15
I0U1=16
I0U2=17
I0U3=18
I0U4=19
11 WE A=0
50 IIWEA=IIWEA+1
*** DATA INPUT PHASE
DO 80 1=1,6
IND(I)=M0D(INT(RND(0)*100),25)+1
80 CONTINUE
CALL DATIN(IND)
IIRUN=0
100 IIRUN=IIRUN+1
CALL DELETE ('NETWOK',6,IC)
CALL DELETE ('CROPIN1 ,6,IC)
INITIAL IAT ION PHASE
DO 200 M=1,2100
143

ooo ooo ooo o o ooo
144
150
200
300
***
400
C
C
DO 150 N=10,5,-l
N0DE(M,N)=0
CONTINUE
CONTINUE
ARC=0
N0DES=1
NEXT=0
IP(1)=IDDEC
IW(1)=M0IST
N0DE(IP(1),IW(1))=1
NNMAX=IDDEC+1640
PERFORM SIMULATIONS TO CREATE NEW NODES h ARCS
NEXT=NEXT+1
T0SHT=.FALSE.
MKMND=IP(NEXT)
SW1=IW(NEXT)
NNEW=0
CALL FIELD(T0SHT,NNEW)
IF (NNEW .NE. 0) THEN
ARRANGE NEW NODES IN INCREASING ORDER OF DECISION DATES
CALL SORT(NNEW)
EXPAND NETWORK, NUMBER NODES IN INCREAING ORDER OF DECISION DATES
CALL ORDER(NEXT,NNEW)
GOTO 300
ELSE
IF (.NOT. TOSHT) THEN
GOTO 300
END IF
END IF
NO MORE SIMULATION NEED, CONNECT DUMMY ARCS TO SINK NODE (NF)
NOMORE=NEXT
NF=N0DES+1
DO 400 I=N0M0RE,NODES
ARC=ARC+1
FR0M(ARC)=I
T0(ARC)=NF
JC(I,NF)=0
JG(I,NF)=0
JS(I,NF)=0
JR(I,NF)=0
CONTINUE
N0DES=N0DES+1
FORMAT ARC LIST IN INCREASING ORDER BY ARC ENDING NODE NUMBER

145
C
CALL LISTN
C
C *** APPLY LONGEST PATH OPTIMIZATION ALGORITHM
C
CALL KPATH
C
C *** PRINT OUT SUMMARY RESULTS
C
CALL DCODE
C
IF (IIRUN .LT. MXRUN) THEN
GOTO 100
END IF
IF (IIWEA .LT. MXYER) THEN
GOTO 50
END IF
C
STOP
END
Q •k-k-k-k-kic-k-k-k'kic-kic-k^C'k-k'k-k-k'k-k'k'k^ck^'kJc-kJcieick-k-k-k-k-k-k-k'kic'k-k-k-k-k-k-k-k-k-k-k-kJc-k'k'k-kic'k kick*
C
SUBROUTINE DATIN(IND)
C
0 ******************************************************************
$ INSERT NAMCM1
$ INSERT NAMCM2
DIMENSION ZZ(5),THRVAR(3),VRFRC(3),WEATR(25),IND(10)
DATA
WEATR/1G54W',1G55W1,
1G56W1
,1G57W1
9
'G58W1,1G59W'
,1G60W' ,
$
1G61W1 ,1G62W',
1G63W1
,1G64W'
9
1G65W1,'G66W1
,1G67W1 ,
$
'G68W',1G69W' ,
'G70W'
,1G71W'
9
1G78W' ,' G79W
,'G80W' ,
$
1G81W1 ,1G82W1,
1G83W'
,'G84W1
/
C
C
DO 100 J=1,6
WFILE(1)=WEATR(IND(J))
C
CALL OPENF(I0(J),WFILE(1),4,40,0,1,2,3,IC)
C
C *** WEATHER DATA, VARIABLE FILENAMES
C
DO 90 K=1,365
I=365*(J-l)+K
READ(IO(J),5) JULN(I),TMAX(I),TMIN(I),SNUP(I),SNDN(I),XLANG(I),
$ WIND(I),RAIN(I)
05 FORMAT(3X,I3,2(1X,F6.2),2(1X,F7.2),9X,F5.1,11X,F4.0,1X,F4.2)
SNDN(I)=12.+SNDN(I)
RAIN(I)=2.54*RAIN(I)
WIND(I)=1.61*WIND(I)
90 CONTINUE
C
CALL CLOSE(IO(J),IC)
100 CONTINUE
C

o o o o o o o o ooo oo ooo
146
IF (IIWEA .LE. 1) THEN
C
CALL 0PENF(IIN2,'FACTS',5,40,0,1,2,3,IC)
CALL 0PENF(IIN3,'GROWS',5,40,0,1,2,3,IC)
*** DATA CONCERNED WITH PROFIT ESTIMATE, FILE 'FACTS'
READ (IIN2,*) NS,KL,MXRUN,MXYER
READ (IIN2,*) IDDEC,MOIST,MXCRP
READ (IIN2,*) (LIDLE(J),J=1,3)
READ (IIN2,*) IRSYS,(RATE(J),J = 1,3)
READ (IIN2,*) (PDCST(J),J=1,MXCRP)
READ (IIN2 ,*) GASPC,DSLPC,WAGE,DEPRE
DO 150 1=1,MXRUN
READ (IIN2,*) (STD(1,J),J = 1,MXCRP)
150 CONTINUE
DO 200 1 = 1 ,MXRUN
READ (IIN2,*) (PRICE(I,J),J=1,MXCRP)
200 CONTINUE
IRRIGATION COST PER APPLICATION (IN-ACRE) FOR VARIOUS SYSTEMS
DO 240 N=1,4
UIRCS(N,1)=0.0
240 CONTINUE
DO 250 1=2,3
UIRCS(1,I)=6.96-(.25*RATE(I))+(.04*RATE(I)*RATE(I))+3.5*(DSLPC
$ -1.2) + (.0275+.065/RATE(I))*(WAGE-4.0)
UIRCS(2,1)=5.08-(,265*RATE(I)) + (.045*RATE(I)*RATE(I))+2.1*(DSLPC
$ -1,2)+(.0275+.065/RATE(I))*(WAGE-4.0)
UIRCS(3,I)=8.91-(1.01*RATE(I))+(.17*RATE(I)*RATE(I))+4.2*(DSLPC
$ -1.2) + (.055+.25/RATE(I))*(WAGE-4.0)
UIRCS(4,1 )=8.94-(.24*RATE(I)) + (.04*RATE(I)*RATE(I))+5.3*(DSLPC
$ -1.2)+(.055+.0625/RATE(I))*(WAGE-4.0)
250 CONTINUE
*** DATA FOR PHENOLOGY AND MULTIPLICATIVE YIELD MODELS, FILE 'GROWS'
HEAT DEGREE DAYS FOR CORN AND PEANUT
READ (IIN3,*) (HDGE(1,J),J=1,4)
READ (IIN3 ,*) (HDGE(2,J),J=1,4)
READ (IIN3,*) (HDGE(6,J),J=1,4)
COEFFICIENTS FOR WHEAT PHENOLOGY
READ (IIN3,*)
READ (IIN3 ,*)
READ (IIN3,*)
READ (IIN3,*)
READ (IIN3,*)
READ (IIN3,*)
READ (IIN3,*)
READ (IIN3 ,*)
(A0(J),J=1,5)
A1J ,0=1,5
(A2(J),J=1,5)
(B0(J),J=1,5)
(B1(J), J = 1,5)
(B2(J),J = 1,5)
(B3(J),J=1,5)
(B4(J),J=1,5)

ooo ooo o o o ooo
147
VALUES FOR SOYBEAN PHENOLOGY AND THEIR CALCULATION
READ (IIN3,*) TOPT,TPHMIN,TPHMAX
PHC0N3=1.0/(TOPT-TPHMIN)
PHC0N4=-PHC0N3*TPHMIN
PHC0N5=1.0/(TOPT-TPHMAX)
PHC0N6=-PHC0N5*TPHMAX
READ (IIN3,*) (ZZ(J),J = 1,5)
DO 330 1=1,2
DO 320 J=1,5
PHTHRS(I,J)=ZZ(J)
320 CONTINUE
330 CONTINUE
DO 350 1=1,2
READ (IIN3 ,*) Y1,Y2,Y3,Y4
TNLG1(I)=Y1
TNLG0(I)=Y2
THVAR(I)=Y3
DHVAR(I)=Y4
PHC0N2(I)=(THVAR(I)-DHVAR(I))/(TNLG1(I)-TNLGO(I))
PHC0N1(I)=DHVAR(I)-PHC0N2(I)*TNLG0(I)
READ (IIN3,*) (THRVAR(J),J=1,3)
READ (IIN3,*) (VRFRC(J),J=1,3)
PHTHRS(I,7)=THRVAR(1)
PHTHRS(I,10)=THRVAR(2)
PHTHRS(1,11)=THRVAR (3)
PHTHRS(I,6)=VRFRC(1)*PHTHRS(1,10)
PHTHRS(I,8)=VRFRC(2)*PHTHRS(I,10)
PHTHRS(I,9)=VRFRC(3)*PHTHRS(1,10)
350 CONTINUE
CROP POTENTIAL YIELD AS A FUNCTION OF PLANTING DATES
READ (IIN3,*)
READ (IIN3 ,*)
READ (IIN3,*)
READ (11N3,*)
READ (IIN3,*)
READ (IIN3,*)
READ (IIN3,*)
READ (11N3,*)
READ (IIN3,*)
READ (IIN3 ,*)
READ (IIN3,*)
READ (IIN3,*)
(XXS0W(1,J),J=1,8)
(YYILD(1,J),J=1,8)
(XXS0W(2,0),J=1,8)
(YYILD(2, J),J = 1,8)
(XXS0W(3,J),J=1,8)
(YYILD(3,J),J = 1,8)
(XXS0W(4,J),J=1,8)
(YYILD(4,J),J=1,8)
(XXS0W(5,J),J=1,8)
(YYILD(5,J),J=1,8)
(XXS0W(6,J),J=1,8)
(YYILD(6,J),J=1,8)
CROP LAI DATA
READ (IIN3,*) (XXLAI(1,J),J=1,11)
READ (IIN3 ,*) (YYLAI(1,J), J=1,11)

o o o o o o
148
READ
READ
READ
READ
READ
READ
READ
READ
READ
READ
(IIN3,*) (XXLAI(2,J),J=1,11)
(IIN3,*) (YYLAI(2,J),J«1,11)
(IIN3,*) (XXLAI(3,J), J = 1,11)
(IIN3,*) (YYLAI(3,J),J = 1,11)
(IIN3,*) (XXLAI(4,J),J=1,11)
(IIN3,*) (YYLAI(4,J),J=1,11)
(IIN3,*) (XXLAI(5,J),J = 1,11)
(IIN3,*) (YYLAI(5,J),J=1,11)
(IIN3,*) (XXLAI(6,J),J=1,11)
(IIN3,*) (YYLAI(6, J),J=1,11)
ROOT GROWTH DATA
READ (IIN3,*) (XXROT(1,J) ,J=1,11)
READ (IIN3,*) (YYROT(l,J),J=1,11)
DO 380 J=1,11
XXROT(2, J)=XXROT(1, J)
YYROT(2, J)=YYROT(1, J)
380 CONTINUE
READ (IIN3,*) (XXR0T(3,J),J=1,11)
READ (IIN3,*) (YYROT(3,J),J = 1,11)
DO 400 J=1,11
XXR0T(4,J)=XXR0T(3,J)
YYR0T(4,J)=YYR0T(3,J)
400 CONTINUE
READ (IIN3,*) (XXROT(5,J),J = 1,11)
READ (IIN3,*) (YYROT(5,J),J=1,11)
READ (IIN3,*) (XXROT(6,J),J = 1,11)
READ (IIN3,*) (YYROT(6,J),J = 1,11)
CROP SENSITIVITY FACTORS
DO 500 1=1,MXCRP
READ (11N3,*) (CS(I,J),J = 1,4)
500 CONTINUE
C
CALL CLOSE(IIN2,IC)
CALL CLOSE(IIN3,IC)
C
END IF
RETURN
END
Q ******************************************************************
C
SUBROUTINE FIELD(TOSHT,NNEW)
C
0 ******************************************************************
$ INSERT NAMCM1
$ INSERT NAMCM2
INTEGER CLEND.MKMND.SWl,SW2,TODAY,SEASN
LOGICAL FIRST,MATUR.TOSHT
C
CLEND=M0D(MKMND,365)
C

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149
0 ***
c
★ ★★
CONSIDER CROP(S) ONLY FOR THE CURRENT SEASON
DO 500 KCR0P=1,MXCRP
IF (KCROP .EQ. 1) THEN
IF (CLEND.LT.45 .OR. CLEND.GT.130) THEN
GOTO 500
END IF
ELSE
IF (KCROP .EQ. 2) THEN
IF (CLEND.LT.45 .OR. CLEND.GT.148) THEN
GOTO 500
END IF
ELSE
IF (KCROP .EQ. 3) THEN
IF (CLEND.LT.101 .OR. CLEND.GT.195) THEN
GOTO 500
END IF
ELSE
IF (KCROP .EQ. 4) THEN
IF (CLEND.LT.70 .OR. CLEND.GT.220) THEN
GOTO 500
END IF
ELSE
IF (KCROP .EQ. 5) THEN
IF (CLEND.LT.281 .OR. CLEND.GT.355) THEN
GOTO 500
END IF
ELSE
IF (CLEND.LT.91 .OR. CLEND.GT.175) THEN
GOTO 500
END IF
END IF
END IF
END IF
END IF
END IF
LPP=M0D(INT(RND(0)*100),3)+1
IHP=MOD(INT(RND(0)*100),3)+l
LPP=2
IHP=3
WITHIN-SEASON IRRIGATION STRATEGIES CONSIDERED
DO 400 IRSGY=1,2
IRSGY=1
IF (KCROP .NE. 5) THEN
IRSGY=2
ELSE
IRSGY=1
END IF
TODAY=MKMND
NNDAY=0
I

oooo oooo ooo o ooo o ooo
ISTG=1
MATUR=.FALSE
FIRST=.TRUE.
150
LAND PREPARATION PERIOD
DO 50 IL=1,LPP
TODAY=MKMND+NNDAY
NNDAY=NNDAY+1
CALL ETBARE
50 CONTINUE
FIRST=.TRUE.
SW2=ISCRE(THETA)
CROP GROWTH PERIOD
100 CALL PHENO(KCROP)
T0DAY=MKMND+NNDAY
NNDAY=NNDAY+1
CALL SWBAL(KCR0P,IRSGY)
IF (TODAY.GE.NNMAX .AND. ISTG.LT.4) THEN
T0SHT=.TRUE.
GOTO 500
END IF
IF (.NOT. MATUR) THEN
GOTO 100
END IF
HARVESTING PERIOD
DO 150 IV=1,IHP
T0DAY=MKMND+NNDAY
NNDAY=NNDAY+1
CALL SWBAL(KCR0P,IRSGY)
150 CONTINUE
SEASN=NNDAY
MAKE A PRODUCTION SEASON AS UNITS OF A WEEK
200 IF (M0D(NNDAY,7) .NE. 0) THEN
TODAY=MKMND+NNDAY
NNDAY=NNDAY+1
CALL SWBAL(KCROP,IRSGY)
GOTO 200
END IF
CALL PR0FT(KCR0P,IRSGY,MONEY)
DISCARD AN ARC OF NEGATIVE RETURN
IF (MONEY .GE. 0) THEN

o o o o o o o o o o o o o o o o o o
151
DISCRETE SOIL WATER CONTENT
SW2=ISCRE(THETA)
WETHER AN ARC IS REPLACED WITH LARGER RETURN ?
T0DA1=T0DAY+1
IF (NNEW .GT. 0) THEN
DO 250 1=1,NNEW
IF (TODAl.EQ.IH(I) .AND. SW2.EQ.ISW(I)) THEN
NN=I
IF (MONEY .LE. IR(NN)) THEN
GOTO 400
ELSE
GOTO 300
END IF
END IF
250 CONTINUE
END IF
STORAGE TEMPORARY INFORMATION OF ARCS NODES
NNEW=NNEW+1
NN=NNEW
300 IH(NN)=T0DA1
ISW(NN)=SW2
IC(NN)=KCR0P
IG(NN)=SEASN
IS(NN)=IRSGY
IR(NN)=MONEY
NEW(IH(NN),ISW(NN))=NN
END IF
400 CONTINUE
500 CONTINUE
IF (.NOT. TOSHT) THEN
*** CONSIDER IDLE PRACTICE AS OPTIONS OF CROPING SEQUENCE
DO 700 IDLE=1,2
TODAY=MKMND
NNDAY=0
FIRST=.TRUE.
IDUR=LIDLE(IDLE)
IF ((MKMND+IDUR) .LE. NNMAX) THEN
*** SIMULATE ET OF BARE SOIL
550 IF (NNDAY .LT. IDUR) THEN
TODAY=MKMND+NNDAY
NNDAY=NNDAY+1

o o o
152
CALL ETBARE
GOTO 550
END IF
C
SW2=ISCRE(THETA)
C
M0NEY=0
C
T0DA1=T0DAY+1
IF (NNEW .GT. 0) THEN
DO 600 1=1,NNEW
IF (TODAl.EQ.IH(I) .AND. SW2.EQ.ISW(I)) THEN
NN=I
IF (MONEY .LE. IR(NN)) THEN
GOTO 700
ELSE
GOTO 650
END IF
END IF
600 CONTINUE
END IF
C
NNEW=NNEW+1
NN=NNEW
C
650 IH(NN)=T0DA1
ISW(NN)=SW2
IC(NN)=MXCRP+1
IG(NN)=LIDLE(IDLE)
IS(NN)=4
IR(NN)=MONEY
NEW(IH(NN),ISW(NN))=NN
END IF
700 CONTINUE
END IF
C
RETURN
END
0 ******************************************************************
c
SUBROUTINE PHENO(CROP)
C
******************************************************************
C *** THIS SUBROUTINE IS CALLED EACH DAY TO COMPUTE THE PHASE OF THE
C *** CROP FROM ONE PHENOLOGICAL PHASE TO THE NEXT.
C
DIMENSION PHTFCT(24),PHTFCY(24)
INTEGER CROP,TODAY
LOGICAL FIRST,MATUR
$ INSERT NAMCM1
*** CALCULATE PHYSIOLOGICAL DAYS ACCUMULATED TODAY (DTX)
N=TODAY

o o o o o o o o o o o
153
DTX=0.
CALL WCALC(PHTFCT.PHTFCY)
DO 10 IXX=1,24
DTX=DTX+PHTFCT(IXX)/24.
10 CONTINUE
C
GOTO (100,100,300,300,200,100), CROP
*** FOR CORN, PEANUT PHENOLOGY IS A HEATING DEGREE DAY FUNCTION
100 CONTINUE
IF (FIRST) CUMDT=0.
CUMDT-CUMDT+DTX
IF (CUMDT .GE. HDGE(CROP,ISTG)) GOTO 150
RETURN
150 ISTG=ISTG+1
IF (ISTG .GT. 4) MATUR=.TRUE.
RETURN
*** FOR WHEAT, A MULTIPLICATIVE MODEL (ROBERTSON, 1968)
200 CONTINUE
IF (FIRST) JJ=1
IF (ISTG .GT. 1) GOTO 220
IF (JJ .NE. 1) GOTO 220
Vl=l.
GOTO 230
220 JJ = ISTG+1
DLEN=SNDN(N)-SNUP(N)
X=DLEN-AO(JJ)
V1=A1(JJ)*X+A2(JJ)*X*X
C
230 TEMPMX=TMAX(N)*1.8+32.
TEMPMN=TMIN(N)*1.8+32.
Y=TEMPMX-BO(JJ)
V2=B1(JJ)*Y+B2(JJ)*Y*Y
C
Z=TEMPMN-BO(JJ)
V3=B3(JJ)*Z+B4(JJ)*Z*Z
C
DELX=V1*(V2+V3)
IF (DELX .LE. 0.) 0ELX=0.
XM=XM+DELX
IF (XM .GE. 1.) GOTO 250
RETURN
C
250 IF (JJ .NE. 1) GOTO 260
JJ=2
XM=0.0

o o o o o o o o o o o o o
154
RETURN
c
260 ISTG=ISTG+1
XM=0.
IF (ISTG .GT. 4) MATUR=.TRUE.
RETURN
*** FOR SOYBEAN, A MODEL OF PHYSIOLOGICAL DAY AND NIGHT ACCUMULATOR
300 CONTINUE
IF (CROP .EQ. 3) JJ = 1
IF (CROP .EQ. 4) JJ=2
IF (FIRST) NPHEN=1
310 GO TO (320,320,320,350,320,350,350,350,350,350,320),NPHEN
320 CONTINUE
PHZDAY = PHZDAY + DTX
IF (PHZDAY .LT. PHTHRS(JJ,NPHEN)) GO TO 460
IF (NPHEN .NE. 1) PHZDAY = 0.0
GO TO 440
350 CONTINUE
XNT = 0.0
TNTFAC = 0.0
COMPUTE CHANGE IN NIGHTTIME ACCUMULATOR DURING THE PREVIOUS
NIGHT (USING TEMPERATURES AFTER SUNSET YESTERDAY AND BEFORE
SUNRISE TODAY IN THE CALCULATIONS)
DO 360 IXX = 12,24
XTMP = IXX
IF (XTMP .LT. SNDN(N-l)) GO TO 360
XNT = XNT + 1.0
TNTFAC = TNTFAC + PHTFCY(IXX)
360 CONTINUE
DO 370 IXX = 1,12
XTMP = IXX
IF (XTMP .GT. SNUP(N)) GO TO 380
XNT = XNT + 1.0
TNTFAC = TNTFAC + PHTFCT(IXX)
370 CONTINUE
380 CONTINUE
TNTFAC = TNTFAC / XNT
DURNIT = 24. - SNDN(N-l) + SNUP(N)
IF (DURNIT .LE. TNLGl(JJ)) DNIT = THVAR(JJ)
IF (DURNIT .GE. TNLGO(JJ)) DNIT = DHVAR(JJ)
IF (DURNIT.LT.TNLGO(JJ) .AND. DURNIT.GT.TNLG1(JJ))
+ DNIT = PHC0N1(JJ) + PHC0N2(JJ) * DURNIT
TDUMX = TNTFAC * (1.0 / DNIT)
ACCNIT = ACCNIT + TDUMX

c~> o o o w o o o o o o o o o
155
c
IF (ACCNIT .LT. PHTHRS(JJ,NPHEN)) GO TO 460
IF (NPHEN .LT. 6) ACCNIT = 0.0
440 CONTINUE
NPHEN = NPHEN + 1
IF (NPHEN .EQ. 5) ISTG=2
IF (NPHEN .EQ. 8) ISTG=3
IF (NPHEN .EQ. 10) ISTG=4
IF (NPHEN .GT. 11) MATUR=.TRUE.
IF GOING FROM NIGHT TIME ACCUMULATOR TO PHYSIOLOGICAL DAY
ACCUMULATOR, START ACCUMULATING IMMEDIATELY RATHER THAN
WAITING UNTIL THE NEXT DAY
IF (NPHEN .EQ. 5 .OR. NPHEN .EQ. 11) GO TO 310
460 CONTINUE
RETURN
END
* ***************** k kkkkkkk-kk-kk-k-k'kk-k-k'kkk'k'k-k-k'k'k'kk-k-kkk kkkkkkkkkkkkkkk
SUBROUTINE SWBAL(CROP,IRSG)
kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk
INTEGER CROP,TODAY,SW2
LOGICAL FIRST,MATUR
DIMENSION DEPLE(3),SWTHRS(7,4),XXR(11),YYR(11),XXL(11),YYL(11)
NSERT NAMCM1
DATA DEV/10./,ALFA/O.234/
DATA THETWP/4.4894E-02/ ,THETFC/10.00E-02/
DATA EPS/O.00001/,RNMAX/250./,DEPLE/.00,.60,.40/
DATA SWTHRS/.20,.20,.20,.20,.20,.15,.15,
$ .50, .50,.55,.55,.65,.60,.40,
$ .70,.70,.75,.75,.80,.80,.70,
$ .45, .45,.50,.50,.50,.50,.30/
IF (ISTG .GT. 4) ISTG=4
IF (FIRST) THEN
*** INITIALIZE PARAMETERS FOR CURRENT SEASON
DO 20 1=1,11
XXR(I)=XXROT(CROP, I)
YYR(I)=YYROT(CROP, I)
XXL(I)=XXLAI(CROP,I)
YYL(I)=YYLAI(CROP, I)
20 CONTINUE
DO 30 1=1,4
ETPO(I)=0.0
ETAC(I)=0.0
30 CONTINUE
DORY = 0
0RISW=FL0AT(SW2)/100.
APPLY=2.54*RATE(IRSG)

ooo ooo ooo ooo ooo
156
RANFL=0.0
DPIRR=0.
RZDEP=0.0
RZDP=0.0
WCMAX=THETFC-THETWP
WCAVL=ORISW-THETWP
WEP = OEV * WCMAX
WE = DEV * WCAVL
WTP = 0.00001
WT = 0.00001
RATI0=WE/WEP
END IF
FIRST=.FALSE.
N=T0DAY
RANFL=RANFL+RAIN(N)
IF (.NOT. MATUR) THEN
IMPLEMENT IRRIGATION STRATEGY
IF (RATIO.LT.DEPLE(IRSG) .AND. RAIN(N).LT.0.2) THEN
FXIN=RAIN(N)+APPLY
DPIRR=DPIRR+APPLY
ELSE
FXIN=RAIN(N)
DPIRR=DPIRR
END IF
NO IRRIGATION NEED WHILE HARVESTING
ELSE
FXIN=RAIN(N)
END IF
*** CALCULATE ROOT DEPTH AND UPDATE AVAILABLE SOIL WATER
RZDP=RZDEP
RN=FLOAT(NNDAY)
RZDEP=TABEX(YYR,XXR,RN,11)
IF (RZDEP .GT. DEV) THEN
RZINC = RZDEP-RZDP
WT = WT + WCAVL * RZINC
WTP = WTP + WCMAX * RZINC
END IF
WTZ = WEP+WTP
*** CALCULATE POTENTIAL ET (PET) A POTENTIAL SOIL EVAPO. (EP)
XLAI = TABEX(YYL,XXL,RN,11)
CALL PENMAN(XLAI,PET,EP)
*** CALCULATE POTENTIAL TRANSPIRATION RATE (RITCHIE, 1972)
IF (XLAI .LT. 0.1) TP=0.00001

o o o o o o o o o o o o o o o oooooo
157
***
c
IF (XLAI.GE.0.1 .AND. XLAI.IE.3.0) TP=PET*(0.7*SQRT(XLAI)-0.21)
IF (XLAI .GT. 3.0) TP=PET
THE FOLLOWING ADDED FOR SOIL FLUX TERM
IF (WT/WTP .LT. 0.2) WT = WT+0.05
CALCULATE TRANSPIRATION RATE, T
WET1 = WE + WT
THETA = THETWP+WET1/RZDEP
RATIO = WET1 / WTZ
THETAC = THETWP+SWTHRS(CROP,ISTG)*WCMAX
IF (THETA .GE. THETAC) THEN
T = TP
ELSE
T = TP * (THETA-THETWP)/(THETAC-THETWP)
END IF
CALCULATE EVAPORATION RATE , E
EP = AMIN1(AMAX1(0.00001,(PET-T)),EP)
COUNT NUMBER OF DAYS WITHOUT RAIN OR IRRIGATION
TWO-STAGE EVAPORATION PROCESS IMPLEMENTED
IF (FXIN .GE. EPS) THEN
DDRY = 0
E = EP
ELSE
DDRY = DDRY + 1
E = ALFA*(SQRT(DDRY)-SQRT(DDRY-1))
IF (E .GT. EP) THEN
E = EP
END IF
END IF
UPDATE SOIL WATER STATUS
RAINFALL
IF (FXIN .GE. EPS) THEN
WE = WE + FXIN
IF (WE .GE. WEP) THEN
WT = WT + WE - WEP
WE = WEP
END IF
END IF
EVAPORATION ZONE
IF (E .GT. WE) E = WE
WE = WE - E

o o o v* o o on o o o o o o o o o o o o o
158
C TRANSPIRATION ZONE
C
WET = WE + WT
IF (WET .IT. EPS) THEN
DO NOT LET SOIL WATER CONTENTS DECREASE BELOW PWP
(IF SOIL WATER DROPS BELOW ZERO, DRAW ON TRANSIENT WATER,
AND RESET THE WATER CONTENTS TO ZERO.)
EXCESS = EXCESS + WE + WT
WE = 0.00
WT = 0.00
ELSE
WE = WE - T * (WE/WET)
WT = WT - T * (WT/WET)
WET = WE + WT
IF (WET .LT. EPS) THEN
WE = 0.00
WT = 0.00
END IF
END IF
DRAIN WATER ABOVE FIELD CAPACITY FROM THE TRANSPIRATION ZONE
(AFTER TRANSPIRATION).
IF (WT .GE. WTP) THEN
EX2 = WT-WTP
WT = WTP
WE = WE+EX2
IF (WE .GE. WEP) THEN
WE = WEP
END IF
END IF
*** CUMULATE STAGE ET
ETPO(ISTG)=ETPO(ISTG)+PET
ETAC(ISTG)=ETAC(ISTG)+T+E
RETURN
END
******************************************************************
SUBROUTINE PROFT(CROP,IRSG,MONEY)
******************************************************************
INTEGER CROP,MKMND,TODAY,YEARTH
DIMENSION XXS(10),YYC(10)
NSERT NAMCM1
*** ESTIMATE YIELD AND GROSS RETURN
DO 50 J = 1,6
XXS(J)=XXSOW(CROP,J)
YYC(J)=YYILD(CROP,J)

ooo ooo ooo
159
50 CONTINUE
C
FRACT=1.0
DO 100 1=1,4
FRACT=FRACT*(ETAC(I)/ETP0(I))**CS(CR0P,I)
100 CONTINUE
C
DAY=FL0AT(JULN(MKMND))
PYDFC=TABEX(YYC,XXS,DAY,6)
YIELD=STD(1,CR0P)*PYDFC*FRACT
REVEN=YIELD*PRICE(11RUN,CROP)
*** IRRIGATION COST DEPENDENT ON IRRIGATION SYSTEM AND DEPTH
DPIRR=DPIRR/2.54
CSTIR=UIRCS(IRSYS,IRSG)*DPIRR*2.46
*** NET RETURN
TLCST=PDCST(CROP)+CSTIR
PROFT=REVEN-TLCST
*** CALCULATE NET DISCOUNT RETURN
YEARTH=INT(T0DAY/365)
M0NEY=PR0FT/((1.+DEPRE)**YEARTH)
C WRITE (1,30) CROP,FRACT,REVEN,PDCST(CROP),CSTIR,DPIRR,MONEY
C 30 FORMAT(I2,5(1X,F8.3),I5)
RETURN
END
Q ******************************************************************
C
SUBROUTINE SORT(NNEW)
C
Q •k'k-k'k-k-k-k-k-k-kic-k-k-k-k-k-k-k-k-k-k-k-k-k-k-k-k-k-kjcJcle-k-k'k'k-k'k'k-k'k'k-k-k-k'k'k-k-k-kJc'k-klrk-k-kle-k'k-klc-klc-k*
INTEGER X(40),Y(40),TEMP,FROM,TO
$ INSERT NAMCM2
C
DO 50 1=1,NNEW
X(I)=IH(I)
Y(I)=ISW(I)
50 CONTINUE
C
NM1=NNEW-1
DO 200 1=1,NM1
IP1=I+1
DO 100 J=IP1,NNEW
IF (X(I) .LE. X(J)) GOTO 100
TEMP=X(I)
X(I)=X(J)
X(J)=TEMP
TEMP=Y(I)
Y( I )=Y(J)
Y(J)=TEMP

oooo oooo*/*oooo
160
100 CONTINUE
200 CONTINUE
00 300 1=1,NNEW
INDEX(I)=NEW(X(I),Y(I))
300 CONTINUE
RETURN
END
•k'k-k'k-k'k'k^c'k'k'k^cie^c-k-k-kic^c-k-k-k'k'k-k-kicie'k-k'kie-k'k'k-k-k-k'k-k-k'kie-k-kk-k-k-k-k-k-k'k'k-k-k-k-k-k-k-k-k-k'k'kic
SUBROUTINE 0RDER(NEXT,NNEW)
******************************************************************
*** SEQUENTIAL NUMBERING SCHEME FOR NETWORK NODES
INTEGER ARC,FROM,TO
NSERT NAMCM2
IF (IH(INDEX(1)) - IR(NODES)) 300,300,100
*** APPEND NEW NODES TO THE LIST
100 INI = 1
GOTO 200
150 INI=I
200 DO 250 1=INI,NNEW
ARC=ARC+1
N0DES=N0DES+1
11 = INDEX(I)
IP(NODES)=1H(11)
IW(NODES)=ISW(II)
NODE(IP(NODES), IW(NODES))=N0DES
JC(NEXT,NODES) = IC(11)
JG(NEXT,NODES) = IG(11)
JS(NEXT,NODES) = IS(11)
JR(NEXT,NODES) = IR(11)
FROM(ARC)=NEXT
TO(ARC)=NODES
250 CONTINUE
RETURN
*** INSERT NEW NODES TO EXISTENT NODE SEQUENCE
*** SEARCH FOR CORRECT INSERTION FROM THE BOTTOM OF THE LIST
300 CONTINUE
DO 600 1=1,NNEW
11 = INDE X(I)
DO 350 J=N0DES,1 ,-l
IF (IH(II) .GE. IP(J)) GOTO 400
350 CONTINUE
400 IF (NODE(IH(II),ISW(II)) .NE. 0) GOTO 550
IF (J .GE. NODES) GOTO 150
C *** NODE NOT YET BEEN NUMBERED

o o o o o o
161
C *** FIRST, UPDATE INFORMATION OF OLD NODES AND ARCS
C
LOCATE=J+l
DO 500 K=NODES,LOCATE,-1
K1=K+1
IP(K1)=IP(K)
IW(K1)=IW(K)
NODE(IP(K1),IW(K1))=K1
DO 450 L=ARC,1 ,-l
NNALT=0
IF (TO(L) .NE. K) GOTO 450
M=FROM(L)
JC(M,K1)=JC(M,K)
JG(M,K1)=JG(M,K)
JS(M,K1)=JS(M,K)
JR(M,K1)=JR(M,K)
T0(L)=K1
C NNALT=NNALT+1
C IF (NNALT .GT. 120) GOTO 500
450 CONTINUE
500 CONTINUE
*** SECOND, NEW NODE IS INSERTED
N0DES=N0DES+1
IP(LOCATE)=IH(II)
IW(LOCATE)=ISW(II)
NODE(IP(LOCATE), IW(LOCATE))=LOCATE
JC(NEXT,LOCATE) = IC(11)
JG(NEXT,LOCATE)=IG(11)
JS(NEXT,LOCATE) = IS(11)
JR(NEXT,LOCATE) = IR(11)
ARC=ARC+1
FROM(ARC)=NEXT
T0(ARC)=LOCATE
GOTO 600
*** NODE HAS BEEN NUMBERED
550 NUMBR=NODE(IH(II) ,1SW(II))
JC(NEXT,NUMBR) = IC(11)
JG(NEXT,NUMBR)=IG(II)
JS(NEXT,NUMBR)=IS(II)
JR(NEXT,NUMBR) = IR(11)
ARC=ARC+1
FROM(ARC)=NEXT
TO(ARC)=NUMBR
600 CONTINUE
RETURN
END

ooooooooooooo
162
Q •k-k-k-k-k'k-k'k'k'kic-k'k-k'k'k'klc'k-k-k'k-k-k-k-k-k-kie'k-k'k-k'kie-k-k-k'k-k-k-kX-k-klc-k-kie-k'kic-k-kic'k-k-klc'k'k-k'k'k-k-k
C
SUBROUTINE LISTN
C
0 ******************************************************************
INTEGER ARC,FROM,TO,TEMP
$ INSERT NAMCM1
$ INSERT NAMCM2
C
CALL 0PENF(I0U1,1NETWOK',6,40,0,1,2,3,IC)
C
NM1=ARC-1
DO 100 1 = 1,NM1
IP1=I+1
DO 50 J=IP1,ARC
IF (TO(I) .LE. T0(J)) GOTO 50
TEMP=T0(I)
TO(I)=T0(J)
T0(J)=TEMP
TEMP=FR0M(I)
FR0M(I)=FR0M(J)
FR0M(J)=TEMP
50 CONTINUE
100 CONTINUE
C
DO 200 1=1,ARC
M=FR0M(I)
N=T0(I)
WRITE(I0U1,900) M,N,JR(M,N),IP(M),IW(M),JC(M,N),JG(M,N),
$ JS(M,N)
200 CONTINUE
C
CALL CLOSE(I0U1,IC)
RETURN
900 FORMAT(12X,815)
C
END
0 ******************************************************************
c
SUBROUTINE KPATH
******************************************************************
A DESCRIPTION OF THE NETWORK IS READ IN. THE NETWORK MUST BE
SORTED IN INCREASING ORDER BY ARC ENDING NODE NUMBER. MOREOVER,
IT IS ASSUMED THAT THE NODES ARE NUMBERED CONSECUTIVELY FROM 1
TO N. ALSO THE NETWORK SHOULD CONTAIN NO SELF-LOOPS AND ALL
CIRCUITS IN THE NETWORK ARE REQUIRED TO HAVE POSITIVE LENGTHS.
(APAPTED FROM D. R. SHIER, 1974)
THE VARIABLES AND ARRAYS IN COMMON ARE
NODES = THE NUMBER OF NODES IN THE NETWORK.
START = AN ARRAY WHOSE J-TH ELEMENT INDICATEOS THE FIRST

OOO OOO O 0000 0 0 163
POSITION ON INC WHERE NODES INCIDENT TO NODE J ARE LISTED.
INC = AN ARRAY CONTAINING NODES I WHICH ARE INCIDENT TO NODE
J, LISTED IN ORDER OF INCREASING J.
VAL = AN ARRAY CONTAINING THE ARC LENGTH VALUES CORRESPONDING
TO ARCS IN INC.
XV = A TWO DIMENSIONAL ARRAY CONTAINING THE KTH LONGEST LABEL
OF THE NODE I.
VARIABLES WHOSE VALUES MUST BE SPECIFIED BY THE USER ARE
KL = THE NUMBER OF DISTINCT PATH LENGTHS REQUIRED.
NS,NF = THE INITIAL AND FINAL NODES OF ALL K LONGEST PATHS TO
BE GENERATED.
INTEGER START,VAL,INC,XV,A(300)
INSERT NAMCM1
INSERT NAMCM2
CALL OPENF(IOU1,1NETWOK1,6,40,0,1,2,3,IC)
INF=-999
AS THE NETWORK IS READ IN, THE VARIABLES AND ARRAYS NEEDED
BY LABELING PROCEDURE AND TRACE SUBROUTINE ARE CREATED.
J=0
N0W=1
10 J=J+1
READ (I0U1,15,END=30) NB,NA,LEN
15 F0RMAT(12X,3I5)
IF (NA .GT. NODES) NODES=NA
IF (NB .GT. NODES) NODES=NB
NUW=NA
IF (NUW .EQ. NOW) GOTO 20
START(NA)=J
20 INC(J)=NB
VAL(J)=LEN
NOW=NA
GOTO 10
30 START(NODES+1)=J
INITIALIZATION PHASE
DO 40 1 = 1,NODES
DO 35 J=1,KL
XV(I,J)=INF
35 CONTINUE
40 CONTINUE
XV(NS,1)=0
I=NS
START ALGORITHM

o o o o o o ooo ooo ooo ooo o o o o
164
50 1=1+1
INITIALIZE A TO THE CURRENT K LONGEST PATH LENGTHS FOR NODE I,
IN STRICTLY DECREASING ORDER.
DO 60 J=1,KL
A(J)=XV( I ,J)
60 CONTINUE
MIN=A(KL)
EACH NODE OF INC INCIDENT TO NODE I IS EXAMINED.
11S=START(I)
IFIN=START(1+1)-1
DO 200 L=IIS, IF IN
11 = INC(L)
IV=VAL(L)
TEST TO SEE WHETHER IXV IS TOO SMALL TO BE INSERTED INTO A
DO 180 M=1,KL
IX=XV(11,M)
IF (IX .LE. INF) GOTO 200
IXV=IX+IV
IF (IXV .LE. MIN) GOTO 200
IDENTIFY THE POSITION INTO WHICH IXV CAN BE INSERTED
00 110 J=KL,2,-l
IF (IXV-A(J-l)) 120,180,110
110 CONTINUE
J = 1
120 JJ=KL
150 IF (JJ .LE. J) GOTO 160
A(JJ)=A(JJ-1)
JJ=JJ-1
GOTO 150
160 A(J)=IXV
MIN=A(KL)
180 CONTINUE
200 CONTINUE
UPDATE THE K LONGEST PATH LENGTHS TO NODE I.
DO 250 J=1,KL
XV(I,J)=A(J)
250 CONTINUE
HAVE ALL NODES BEEN LABELLED ?
IF (I .NE. NODES) GOTO 50
THE K LONGEST PATH LENGTHS FROM NODE NS TO NODE NF ARE DETERMINED.

165
C
C
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
NF.
c
c
c
c
c
THE
C
c
CALL TRACE
CALL CLOSE(IOU1,IC)
RETURN
END
•k'k-k-k-k'k-k-k-k-kic'k'k'k'k-k'k-k-k'k-k-k-kic-k-k-k'k-k-kick'k-k-k'k-k-k-k-k-k-kic'kieic^e-k-k'k-k-kic-k-k-kie-k'k-kieickic^-k
SUBROUTINE TRACE
★ ★★**★*★** k-k-kic-k'k'kic-k-k-k-k'k-k-k-k'k'k-k-k'k-k-k'k'k'k-k-kic-k'k-k'k'k'k'k'kic-k'k'k-k'k'kicic'k-k-k'k'k'k'k-k-k'k
THIS SUBROUTINE WILL TRACE OUT THE PATHS CORRESPONDING TO THE K
DISTINCT LONGEST PATH LENGTHS FROM NODE NS TO NODE NF. AT MOST PMAX
SUCH PATHS WILL BE GENERATED. IT IS ASSUMED THAT ALL CITCUITS IN
THE NETWORK HAVE POSITIVE LENGTH. MOREOVER ONLY PATHS HAVING AT
MOST 50 ARCS WILL BE PRODUCED.
VARIABLES DEFINED
NS,NF = THE INITIAL AND FINAL NODES OF ALL K LONGEST PATHS BEING
GENERATED.
PMAX = THE MAXIMUM NUMBER OF PATHS TO BE GENERATED BETWEEN NODE
NS AND NF.
JJ = INDEX OF THE PATH LENGTH FROM NS TO NF BEING EXPLORED.
JJ CAN TAKE ON VALUES FROM 1 TO K.
NP = THE NUMBER OF PATHS FROM NS TO NF FOUND.
KK = CURRENT POSITION OF LIST P.
P = AN ARRAY CONTAINING NODES ON A POSSIBLE PATH FROM NS TO
Q = AN ARRAY WHOSE I-TH ELEMENT GIVES THE POSITION, RELATIVE
TO START, OF NODE P(I) ON THE INC LIST FOR P(I-l).
PV = AN ARRAY WHOSE I-TH ELEMENT IS THE ARC LENGTH EXTENDING
FROM NODE P(I) TO NODE P(I-l).
H = AN ARRAY WHOSE I-TH ELEMENT IS TOTAL NUMBER OF NODES OF
I-TH BEST PATH.
INTEGER P(50),Q(50),PV(50),PMAX,H,CO,C
INTEGER START,VAL,INC,XV
$ INSERT NAMCM1
$ INSERT NAMCM2
C
CALL OPENF(I0U2,1 CROP IN1,6,66,0,1,2,3,IC)
C
C INITIALIZATION PHASE
C
PMAX=20
INF=-999
DO 10 1=1,50
P(I)=0
Q(i)=o
PV(I)=0
10 CONTINUE
C

o o o c~> o o ooo o o o o o ooo o o o o o o
166
JJ=1
IF (NS .EQ. NF) JJ=2
NP=0
IF (XV(NF,JJ) .GT. INF) GOTO 15
WRITE (1002,909) NS,NF
909 F0RMAT(1H1,'THERE ARE NO PATHS FROM NODE',14,' TO NODE1,14)
GOTO 200
15 WRITE (I0U2,901) NS,NF
901 F0RMAT(1H ,1 THE K LONGEST PATHS FROM NODE1,14,' TO NODE1,14//
$ 1H ,1 PATH LENGTH NODE SEQUENCE'//)
THE JJ-TH DISTINCT PATH LENGTH IS BEING EXPLORED.
20 KK=1
LAB=XV(NF,JJ)
IF (LAB .LE. INF) GOTO 200
LL=LAB
P(1)=NF
C0=0
30 LAST=0
NODES INCIDENT TO NODE P(KK) ARE SCANNED.
40 NT=P(KK)
IIS=START(NT)
DO 45 ND=NT,NODES
IF (START(ND+1) .NE. 0) GOTO 48
45 CONTINUE
48 IIF=START(ND+1)-1
11 = IIS+LAST
50 IF (II .GT. I IF) GOTO 90
50 NI = INC(11)
NV=VAL(11)
LT=LAB-NV
TEST MADE, SEE IF THE CURRENT PATH CAN BE EXTENDED BACK TO NODE NI
DO 60 J=1,KL
IF (XV(NI,J)-LT) 70,80,60
60 CONTINUE
70 11=11+1
GOTO 50
80 KK=KK+1
THIS PORTION ADDED TO EXCLUDE SEQUENCES OF PEANUT REPETITION IN
SUMMER. 5/29/85 TSAI
C=J C(NI,NT)
C WRITE (1,801) C,NI,NT
C 801 FORMAT(315)

OOOOOOOOOOOOOOO OOO OOO C-> o o o
167
IF (C.LE.MXCRP .AND. C.NE.5) THEN
IF (C .EQ. 6) THEN
IF (CO .EQ. 6) THEN
GOTO 160
ELSE
C0=C
END IF
ELSE
C0=C
END IF
END IF
WRITE (1,802) C,C0,MXCRP
802 FORMAT(3110)
IF (KK .GT. 50) GOTO 190
P(KK)=NI
Q(KK) = 11 — IIS+1
PV(KK)=NV
LA8=LT
TESTS MADE TO SEE IF THE CURRENT PATH CAN BE EXTENDED FURTHER.
IF (LAB .NE. 0) GOTO 30
IF (NI .NE. NS) GOTO 30
COMPLETE PATH FROM NS TO NF HAS BEEN GENERATED
NP=NP+1
H(JJ)=KK
WRITE (I0U2,902) JJ ,LL,(P(J),J=KK,1,-l)
902 FORMAT(13,15,3014)
THIS PORTION IS COMMENTED OUT IN ORDER NOT TO DUPLICATE PATHS
OF THE SAME LENGTH. 5/15/85 TSAI
IF (NP .GE. PMAX) GOTO 200
90 LAST=Q(KK)
P(KK)=0
LAB=LAB+PV(KK)
KK=KK-1
IF (KK .GT. 0) GOTO 40
EXPLORATION OF THE CURRENT JJ-TH DISTINCT PATH LENGTH IS ENDED.
160 JJ=JJ+1
IF (JJ .GT. KL) GOTO 200
GOTO 20
C
190 WRITE (I0U2,903)
903 F0RMAT(1H0,'NUMBER OF ARCS IN PATH EXCEEDS 50')

168
200 CONTINUE
CALL CLOSE(I0U2,IC)
C
RETURN
END
Q •k-k'kick-k'k-k-k'kic-k-k'k'kic-k'k-k-k'k'k'k-k'kic-kie'k-k'k'k'k-kie-k-k'k-k'k-kie'k-k-k-k'k-k-kie-k-k-k'k-k'k-k'k'k'k'k-k-k-k'k'k
C
SUBROUTINE DCODE
C
0 ******************************************************************
$ INSERT NAMCM1
$ INSERT NAMCM2
C
INTEGER P(50),H,C,G,S,R,YEAR,AR,ARC
INTEGER RM0N,VAR(7,2),SGY(4,3), SENS(21)
C
DATA VAR/'F.S.','S.S.',1 BRAG','WAYN1,1WHEA',1 PEAN1,'IDLE' ,
$ ' CORN' , * CORN' ,' G ','E VT301VUT '/
DATA SGY/'RAIN1,'FREQ','INFR','****'-FED1,'UENT',‘EQUE',
$ ' V V 1 ,1 NT V 7
DATA SENS/'D.Ol','D.02','D.03','D.04','0.05','D.06‘,'D.07‘,
$ 1D.08' ,'0.091 ,'D.IO' ,'D.ll' ,'D. 121 ,'D.13' ,'0.14' ,
$ 'D.15‘,'0.16','D.17‘,'D.18‘,'D.19‘,'D.20‘,'D.21'/
C
NFILE=MXRUN*(IIWEA-1)+IIRUN
CALL OPENF( I0U2,1 CROPIN1,6,66,0,1,2,3,IC)
CALL OPENF(I0U4,SENS(NFILE),4,40,0,1,2,3,IC)
C
IF (NFILE .LE. 1) THEN
CALL 0PENF(I0U3,‘SERIES' ,6,40,0,1,2,3,IC)
END IF
C
WRITE (1,90) NFILE,NODES,ARC
WRITE (I0U3,900)
WRITE (I0U3,910) NFILE,(WFILE(I),1=1,2),
$ (PRICE(IIRUN,J),J=1,MXCRP)
C
JC0UN=0
100 READ (I0U2,110,END=300) NP,LL,(P(J),J=1,H(NP))
C
JC0UN=JC0UN+1
IF (JC0UN.LE.5 .AND. NP.LE.KL) THEN
C IF (M0D(NP,5) .EQ. 1) THEN
WRITE (I0U3,930) NP,LL
WRITE (I0U3,940)
JJ=0
150 JJ=JJ+1
IF (P(JJ) .NE. NF) THEN
M=P(JJ)
YEAR=INT(IP(M)/365)+l
JULD=MOD(IP(M),365)
CALL NAILUJ(JULD,RMON,NDAY)
N=P(JJ+1)
C=JC(M,N)

169
G=JG(M,N)
S=JS(M,N)
R=JR(M,N)
IF (C .NE. 0) THEN
WRITE (101)3,200) RM0N,NDAY,YEAR, IW(M), (VAR(C,J), J=1,2) ,G,
$ (SGY(S,J),J=1,3),R
200 F0RMAT(11X,A3,' -' ,12,'-' ,I1,3X,I2,'%‘,3X,2A4,3X, I3,2X,3A4,4X,
$ 14)
GOTO 150
ELSE
WRITE (I0U3.250) RMON,NDAY,YEAR,IW(M)
250 F ORMAT(11X,A3,'-',12,'-',11,3X,12,'%',3X,4(1H*),7X,3(1H*),2X,
$ 4(1H*),13X,'***')
WRITE (I0U3,920)
END IF
END IF
C END IF
GOTO 100
END IF
C
300 IF (IIWEA.LT.nixyer .or. iirun.lt.mxrun) then
CALL CLOSE(I0U2,IC)
CALL CLOSE(I0U4,IC)
ELSE
CALL CL0SE(I0U2,IC)
CALL CLOSE(I0U3,IC)
CALL CLOSE(I0U4,IC)
END IF
RETURN
C
90 FORMAT(1NFILE= ’,12,' N0DES= ',14,' ARCS*',15)
110 FORMAT(13,15,3014)
900 FORMAT(1H1,///,11X,'*** OPTIMAL SEQUENCING OF MULTICROPPING SYSTE
***1 j)
910 FORMAT(1íX,1 RUN 12,26X,'WEATHER FILE: ',2A4,/
$11X, 'CROP PRICE ($/KG) =1,F5.3,5F6.3/)
920 FORMAT(1H1,//,llX,11/)
930 FORMAT(11X,1 SEQUENCE',14,1 HAVING TOTAL NET DISCOUNTED RETURN $',
$14,/)
940 F0RMAT(11X,'DECISION INITIAL',10X,1 SEASON IRRIGATION DISCOUNT',/
S13X,'DATE S.W. CULTIVAR (DAYS) STRATEGY RETURN',/
$1IX,8(1H ),1X,7(1H_),1X,8(1H_),1X,6(1H_),1X,10(1H_),2X,
$8(1H ) ,/T
END
Q ******************************************************************
C
SUBROUTINE ETBARE
C
Q 'k-k-k-k-k'k-k-k-k-kieic-k-k-k'k'k-k-k'k'k-k'k-k'k'k'k-k-k'kie-kic'k-k-k'k'k-k'k'k'k-k-k'k-k-k'k'k-k'k-k'k-kick ic-k-k-k
INTEGER TODAY,SW1
LOGICAL FIRST
$ INSERT NAMCM1
DATA DEV/10./,DRZ/30./,ALFA/O.234/,EPS/O.00001/

o o o o c~> o oooooooooooo o o o o ooo
170
c
***
***
***
***
***
DATA THETWP/4.4894E-02/,THETFC/10.00E-02/
IF (FIRST) THEN
INITIALIZE PARAMETERS FOR CURRENT SEASON
0RISW=FL0AT(SW1)/100.
DDRY = 0.
WCMAX=THETFC-THETWP
WCAVL=0RISW-THETWP
THETAC=THETWP+0.5*WCMAX
WTP = DRZ * WCMAX
WT = DRZ * WCAVL
END IF
FIRST=.FALSE.
N=T0DAY
CALCULATE POTENTIAL ET (PET) USING PENMAN EQUATION
XLAI = 0.25
CALL PENMAN(XLAI,PET,EP)
CALCULATE POTENTIAL TRANSPIRATION RATE (RITCHIE, 1972)
IF (XLAI .LT. 0.1) TP=0.00001
IF (XLAI.GE.0.1 .AND. XLAI.LE.3.0) TP=PET*(0.7*SQRT(XLAI)-0.21)
IF (XLAI .GT. 3.0) TP=PET
TP=PET*(0.7*SQRT(XLAI)-0.21)
THE FOLLOWING ADDED FOR SOIL FLUX TERM
IF (WT/WTP .LT. 0.2) WT = WT+0.05
CALCULATE TRANSPIRATION RATE, T
WET1 = WE + WT
THETA = THETWP+WET1/(DEV+DRZ)
IF (THETA .GE. THETAC) THEN
T = TP
ELSE
T = TP * (THETA-THETWP)/(THETAC-THETWP)
END IF
CALCULATE EVAPORATION RATE , E
EP = AMIN1(AMAX1(0.00001,(PET-T)),EP)
TWO-STAGE EVAPORATION PROCESS
IF (RAIN(N) .GE. EPS) THEN
DDRY = 0.

oooo o o o o o o o o o o o o o o o o
171
irkk
E = EP
ELSE
DORY = DORY +1
E = ALFA*(SQRT(DDRY)-SQRT(DDRY-1))
IF ( E .GT. EP) THEN
E = EP
END IF
EN DIF
UPDATE SOIL WATER STATUS
RAINFALL
IF (RAIN(N) .GE. EPS) THEN
WE = WE + RAIN(N)
IF (WE .GE. WEP) THEN
WT = WT + WE - WEP
WE = WEP
END IF
END IF
EVAPORATION ZONE
IF (E .GT. WE) E = WE
WE = WE - E
TRANSPIRATION ZONE
WET = WE + WT
IF (WET .LT. EPS) THEN
DO NOT LET SOIL WATER CONTENTS DECREASE BELOW PWP
(IF SOIL WATER DROPS BELOW ZERO, DRAW ON TRANSIENT WATER,
AND RESET THE WATER CONTENTS TO ZERO.)
WE = 0.00
WT = 0.00
ELSE
WE = WE - T * (WE/WET)
WT = WT - T * (WT/WET)
WET = WE + WT
IF (WET .LT. EPS) THEN
WE = 0.00
WT = 0.00
END IF
END IF
DRAIN WATER ABOVE FIELD CAPACITY FROM THE TRANSPIRATION ZONE
(AFTER TRANSPIRATION).
IF (WT .GE. WTP) THEN
EX2 = WT-WTP
WT = WTP
WE = WE+EX2

OOO OOO OOO OOO 172
IF (WE .GE. WEP) THEN
WE = WEP
END IF
END IF
C WRITE (1,991) DDRY,WE,WT,PET,EP,TP,E,T,THETA
991 F0RMAT(F3.0,10(IX,F6.3))
C
RETURN
END
r ******************************************************************
SUBROUTINE WCALC(PHTFCT.PHTFCY)
******************************************************************
THIS SUBROUTINE CALCULATES HOURLY TEMPERATURES AND TEMPERATURE
FACTOR FOR USE IN PHENOLOGICAL STAGE CALCULATIONS
INTEGER TODAY
DIMENSION THR(24),PHTFCT(24),PHTFCY(24)
NSERT NAMCM1
N=T0DAY
IF (TODAY .EQ. 1) GOTO 200
DO 100 IXX=1,24
PHTFCY(IXX)=PHTFCT(IXX)
100 CONTINUE
200 CONTINUE
THIS SECTION CALCULATES HOURLY TEMPERATURES FOR THE DAY
DO 600 IXX = 1,24
X = IXX
IF (X .LT. SNUP(N) + 2.0) GO TO 400
IF (X .GT. SNDN(N)) GO TO 300
SINE CURVE
TAU = 3.1417 * (X-SNUP(N)-2.)/(SNDN(N)-SNUP(N))
THR(IXX) = TMIN(N) + ((TMAX(N)-TMIN(N)) * SIN(TAU))
GO TO 600
AFTER SUNSET BEFORE MIDNIGHT
300 TAU = 3.1417 * (SNDN(N)-SNUP(N)-2.)/(SNDN(N)-SNUP(N))
TLIN = TMIN(N) + ((TMAX(N)-TMIN(N)) * SIN(TAU))
HDARK = 24. - SNDN(N) + SNUP(N+1) + 2.
SLOPE = (TLIN - TMIN(N+1)) / HDARK
THR(IXX) = TLIN - (SLOPE * (X - SNDN(N)))
GO TO 600
BETWEEN MIDNIGHT AND SUNRISE + 2 HRS.
400 CONTINUE
IF (N .EQ. 1) GO TO 500

ooooooooooooooo o o oooo oooooo
173
TAU = 3.1417 * (SNDN(N—1)-SNUP(N—1)-2.)/(SNDN(N-l)-SNUP(N-l))
TLIN = TMIN(N-l) + ((TMAX(N-l) - TMIN(N-l)) * SIN(TAU))
HDARK = 24. - SNDN(N-l) + SNUP(N) + 2.
SLOPE = (TLIN - TMIN(N)) / HDARK
THR(IXX) = TLIN - SLOPE * (X + 24. - SNDN(N-l))
GO TO 600
500 CONTINUE
IF THIS IS DAY ONE OF SIMULATION THEN AVERAGE OF MAX AND MIN
TEMPERATURE FOR THE DAY IS USED IN COMPUTING THE HOURLY TEMPS
IN ORDER TO AVOID THE PROBLEM OF N-l BEING ZERO IN THE
ABOVE CALCULATIONS.
THR(IXX) = (TMAX(N) + TMIN(N)) / 2.0
600 CONTINUE
COMPUTE TEMPERATURE FACTORS FOR EACH HOUR FOR USE IN PHENOLOGICAL
CALCULATIONS
DO 700 IXX = 1,24
IF (THR(IXX) .LE. TOPT) PHTFCT(IXX) = PHC0N3 * THR(IXX) +
+ PHC0N4
IF (THR(IXX) .GT. TOPT) PHTFCT(IXX) = PHC0N5 * THR(IXX) +
+ PHC0N6
PHTFCT(IXX) = AMAX1(0.0,PHTFCT(IXX))
700 CONTINUE
RETURN
END
******************************************************************
SUBROUTINE PENMAN(XLAI,PET,EP)
******************************************************************
*** CALCULATE POTENTIAL ET (PET) & POTENTIAL SOIL EVAPORATION (EP)
*** USING PENMAN EQUATION (IFAS ET BULLETIN, 1981)
XLAI = LEAF AREA INDEX OF THE DAY (NNDAY)
ESUBD = DEWPOINT VAPOR PRESSURE (MILLIBARS)
VPD = VAPOR PRESSURE DEFICIT (MILLIBARS)
DELTA = SLOPE OF SATU. VAPOR PRESSURE CURVE AT MEAN AIR TEMP.
GAMMA = CONSTANT OF THE WET AND DRY BULB PSYCHROMETER EQN.
GAMMA = .0006595 * BARAMETRIC PRESSURE (MILLIBARS)
XLAMDA = LATENT HEAT OF VAPORIZATION OF H20 (CAL CM-2 MM)
RSO = CLEAR SKY RADIATION (LANGLEY/DAY)
RN = NET RADIATION (LANGLEY/DAY)
INTEGER TODAY
$ INSERT NAMCM1
DATA AS/.15/,ACROP/.25/,GAMMA/.66/,XLAMDA/58.4/,XLAT/0.61987/
C
N=T0DAY
ALPHA2 = AS+.25*(ACR0P-AS)*XLAI
ALPHA2 = AMIN1(ALPHA2,ACROP)
TC = (TMAX(N)+TMIN(N))/2.0

OOOOOOOOOOOOOOOOO O O ¡DOO o o o o o
174
TK = TC+273.
c
ESUBD = 33.8639*((.00738*TMIN(N)+.8072)**8-.000019*(1.8*TMIN(N)+
$ 48.)+0.001316)
C
DELTA = 33.8639*((.05904*(.00738*TC+.8072)**7)-.0000342)
C
VPD = 16.932*((,00738*TC+.8072)**8-(,00738*TMIN(N)+.8072)**8-
$ .000019*(1.8*(TC-TMIN(N))))
C
RS0 = RADCL(JULN(N),XLAT)
RN = (1.-ALPHA2)*XLANG(N)-(XLANG(N)/RS0*1.42-.42)*(.56-.08*SQRT
$ (ESUBD))*11.71E-08*TK**4
*** CHANGE FROM LANGLEYS TO MM H20 EVAPORATED.
RNO = AMAX1(R N/X LAMDA,0.)
EO = (DELTA/(DELTA+GAMMA))*RN0+.263*VPD*(.5+.0062*WIN0(N))*(GAMMA/
$ (GAMMA+DELTA))
EP = (DELTA/(DELTA+GAMMA))*RN0
*** CONVERT FROM MM TO CM.
PET = 0.1*E0
EP = 0.1*EP
PET = AMAX1(PET,0.00001)
EP = AMAX1(EP,0.00001)
RETURN
END
****************************************************************
FUNCTION RADCL(JDAY.XLAT)
****************************************************************
THIS FUNCTION ESTIMATES CLEAR-SKY INSOLATION AT THE SURFACE OF
THE EARTH BY CALCULATING EXTRATERRESTRIAL SOLAR RADIATION AS
A FUNCTION OF LATITUDE AND JULIAN DATE, AND REDUCING THIS VALUE
BY 20% TO ACCOUNT FOR AVERAGE CLEAR-SKY ATTENUATION. ATTENUATION
ESTIMATES ARE TO BE IMPROVED IN FUTURE VERSIONS.
(SEE ASCE REPORT "Consumptive Use of Water and Irrigation Water
Requirements, M. E. Jensen, Ed., 1973, & HANDBOOK OF
METEOROLOGY, F. A. Berry, E. Bol lay, and N. R. Beers, eds.,
McGraw Hill , 1945)
XLAT=LATITUDE OF USER'S LOCATION (RADIANS, NORTHERN HEMISPHERE
JDAY=JULIAN DATE
SC=SOLAR CONSTANT (CAL CM-2 HR-1, ASSUMING 1.94 CAL CM-2 MIN-1
RADCL=CLEAR-SKY RADIATION, CAL CM-2 DAY-1
PI = 3.141593
ATTFAC = .304
SC = 116.40

o o o o o o o o
175
DFAC=VARIATION IN RADIATION DUE TO VARIATION IN ORBITAL RADIUS
RM = 1.
RADIUS = RM*(l.-(0.01673*C0S(2.0*PI*JDAY/365.)))
DFAC = (RM/RADIUS)**2
DECLIN = PI*(23.47/180.)*SIN(2.0*PI*(284+JDAY)/365.)
COSZA = (COS(DECLIN+XLAT)-COS(XLAT-DECLIN))/(COS(XLAT+DECLIN)+COS
$ (XLAT-DECLIN))
CALCULATE ARCCOS OF THE COSINE OF ZENITH ANGLE
(TO OBTAIN HOUR ANGLE OF SUNRISE & SUNSET)
IF (ABS(COSZA) .LE. 0.00001) GO TO 100
ARG = ABS(1.0-C0SZA**2)
A = SQRT(ARG)
HASUN = ATAN(A/COSZA)
IF (COSZA .GT. 0.0) GO TO 200
HASUN = PI+HASUN
GO TO 200
100 HASUN = PI/2.0
200 CONTINUE
ACRIT = PI/2.
IF (XLAT-DECLIN .GE. ACRIT) GO TO 300
RADCL = ATTFAC*DFAC*(24./PI)*SC*(HASUN*SIN(DECLIN)*SIN(XLAT)+SQRT
$ (ABS(COS(DECLIN+XLAT))*COS(DECLIN-XLAT)))
RETURN
300 RADCL = 0.
RETURN
END
•k-k-k-k-k-k'k-k-k-k-k-k-k-k-k-k-k-k-k-k-k-k-k-k-k-k-k'k-k-k^e-k-k'k-kie-k-k-kic'k-kick'k-k-k-kie-k-k-k'k-k-k'kic-kic-k-k-k-k-k-kic
SUBROUTINE NAILUJ(JULD,RMON,NDAY)
C
Q ★★★★**★★★*★*********★*★*★★**★*★****★**★★★★*★**★**■★*-*■***★★*•*★**★★*★
INTEGER MON(12)
L0GICAL*4 RNAME(12),RMON
DATA MON/31,28,31,30,31,30,31,31,30,31,30,31/
DATA RNAME/1 JAN1,1 FEB1,1 MAR1,1 APR',1 MAY1,1JUN1,'JUL','AUG1,
$ 'SEP1,1 OCT1,1 NOV1,1 DEC1/
C
NSUM=0
DO 100 JC0UNT=1,12
NDIF=JULD-NSUM
IF (NDIF .LE. MON(JCOUNT)) GOTO 200
NSUM=NSUM+MON(JCOUNT)
100 CONTINUE
GOTO 300
200 NDAY=NDIF

176
RMON=RNAME(JCOUNT)
C
300 RETURN
END
Q •k'k-k'k-k-k-k-k-k^c'k'k'k'k'k'k'k'k-k'k^c-k'k-k'k-k-k-k'k-k-k'k'k-k-k-k-k-k'k-k-k^c'k-k-k'k-k'k'k-k'k'k-k'k'k'k-k'k-k-k-k'k-k-k'k-k
C
FUNCTION TABEX(VAL,ARG,DUMMY,K)
C
Q 'k-k-k-k-k'k'k'k-k-k'k-k-kie'k-k'k-k-k-k-k-k-k-kic-k'k-kickic-k-k'k-k-k'k-k'k-k'k'k-kic'k-k-k-k'k'k-k-k-k-k-k-kick-k-k-k-k-k'k'k'k
DIMENSION VAL(20),ARG(20)
DO 101 J=2 ,K
IF (DUMMY .GT. ARG(J)) GOTO 101
GOTO 102
101 CONTINUE
J=K
102 TABEX=(DUMMY-ARG(J-l))*(VAL(J)-VAL(J-l))/(ARG(J)-ARG(J-l))+
$ VAL(J-l)
RETURN
END
Q ******************************************************************
c
FUNCTION ISCRE(XX)
C
C ******************************************************************
XX=XX*100.
IF (AMOD(XX,1.) .LE. 0.5) THEN
ISCRE = I NT(XX)
ELSE
ISCRE=INT(XX)+1
END IF
RETURN
END

177
Q •k-k'k'k'k'k'k-k'k-k-k'k'k'k^cie'kie-k-kic-k'k'k'k-kic-kic-k-k'k'k-kic-k-kic'kie-k'k^c'k-k-k'kie'k'k-k^-k-kie-k-k-k-k-k-kie^-kicic
C
COMMON BLOCK
C
Q •k'k-k-k'k'k'k-k'k'k'k'k-k^e'k'k'k'k-k'k'k'k-k'k-k'k-k'k-k-k-k-k-k'k^cirk-k-k-k-k-k'k-k-k-k-k'k-k'kic-k-k'k-k-k-k'k'k'k'kie-k-k-kic
COMMON/SIMUL/ NNDAY,TODAY,MATUR,ISTG,ETP0(4),ETAC(4),THETA,
$ DPIRR,NNMAX,MKMND,SW1,SW2,FIRST,WFILE(2),IIRUN,I INI, IIN2,
$ IIN3,1IN4,10U1,10U2,IOU3,10U4,ICASH,10(10)
COMMON/WEATR/ JULN(2190),TMAX(2190),TMIN(2190),SNUP(2190),
$ SNDN(2190),XLANG(2190),WIND(2190),RAIN(2190)
COMMON/FACTS/ IDDEC,MOIST,STD(10,8),PRICE(98,8),GASPC,DSLPC,
$ WAGE, DEPRE,IRSYS,RATE(3),PDCST(8),UIRCS(4,3),LIDLE(3),MXCRP,
$ MXRUN,MXYER,IIWEA
COMMON/GROWS/ A0(5),A1(5),A2(5),80(5),B1(5),B2(5),B3(5),B4(5),
$ TNLG1(2),TNLG0(2),THVAR(2),DHVAR(2),PHC0N1(2),PHC0N2(2),TOPT,
$ TPHMIN,TPHMAX,PHC0N3,PHC0N4,PHC0N5,PHC0N6,PHTHRS(2,11),
$ XXLA1(8,11),YYLA1(8,11),XXROT(8,11),YYROT(8,11),HDGE(6,4),
$ CS(8,4),XXS0W(8,10),YYILD(8,10),
COMMON/NETWK/ IH(40),ISW(40),IC(40),IS(40),IR(40),INDEX(40),
$ IG(40),JC(450,450),JG(450,450),JS(450,450),JR(450,450),
$ (450) ,IW(450),NODE(2190,10),NEW(2190,10),FR0M(2000),
$ T0(2000)
COMMON/LONGS/ NS,NF,KL,START(451),INC(2000),VAL(2000),
$ XV(450,100),NODES,ARC,H(100)

APPENDIX C
INPUT FILE 'GROWS
38.7 45.1 66.3 81.4
33.6 40.7 57.7 70.4
27.3 42.4 68.2 97.3
0.0 8.41 10.93 10.94 24.38
0.0 1.005 0.925 1.389 -1.14
0.0 0.0 -.06025 -.08191 0.0
44.37 43.64 42.65 42.18 37.67
.01086 .0003512 .002958 .0002458
-.000223 -.00000503 0.0 0.0 0.0
.009732 .0003666 .003943 .0003109
-.000227 -.00000428 0.0 0.0 0.0
30.0 7.0 45.0
6.522 10.87 2.4 1.0 9.48
5.2 11.0 63.0 2.0
3.0 20.35 12.13
0.14 0.16 0.575
5.2 9.5 32.0 2.0
6.0 14.5 10.0
0.2 0.5 0.6
40. 48. 58. 77.
0.87 0.96 1.00 0.91
40. 48. 58. 77.
0.78 0.87 0.95 1.00
75. 106. 136. 152.
0.80 0.89 0.94 1.00
61. 91. 106. 121.
0.82 0.93 1.00 0.95
260. 275. 285. 300.
0.78 0.92 0.95 1.00
70. 85. 100. 112.
0.76 0.91 0.95 1.00
0 34 48 60 69 76
0 .6 1.1 2.4 3.2
0 24 39 46 53
0 .4 2.4 3.9 4.1
0 20 34 45 64
0 .2 0.7
0 15 28 33 47
0 .2 0.8 1.4 2.8
0 32 67 105 128
0 .5 0.8 1.4 1.9
0 35 49 63 70
0 .8 1.6 3.5 4.7
F.S.CORN (BENNETT 1982)
S.S.CORN
PEANUT (800TE, 1981)
AO (ROBERTSON,1969)
A1
A2
BO
00006733 B1
B2
.00003442 B3
B4
TOPT,TPHMIN,TPHMAX
PHTHRS(J),J=1,5
TNLG1 ,TNLGO,THVAR,DHVAR
THRVAR(J),J=1,3
VRFCR(J),J=1,3
TNLG1 TNLGO,THVAR,DHVAR
THRVAR(J),J=1,3
VRFCR(J),J=1,3
XXS0W(1,J) (AGRO.FKS 117)
YYILD(l.J)
XXS0W(2,J) (AGRO.FKS 117)
YYILD(2, J)
XXS0W(3,J) (S0YGR05.0)
YYILD(3,J)
XXS0W(4,J) (S0YGR05.0)
YYILD(4,J)
XXSOW(5,J)
YYILO(5,J)
XXS0W(6,J)
YYILD(6,J)
XXLAI,F.S.CORN (BENNETT)
YYLAI,
XXLAI,S.S.CORN (LOREN)
YYLAI,
XXLAI,BRAGG (S0YGR05.0)
YYLAI
XXLAI,WAYNE (S0YGR05.0)
YYLAI
XXLAI,WHEAT (HODGES ET AL)
YYLAI
XXLAI,PEANUT (MCGRAW)
YYLAI
90.
0.80
90.
0.92
166.
0.92
136.
0.90
315.
0.98
125.
0.90
90
3.7 4.0
65 74
3.8 3.7
90 108
1.8 3.2 3.1 2.1
56 72
3.3 2.4
135 142
3.6 4.0
77 84
4.4 4.6
102.
115
•
140.
0.74
0.67
0.
59
102.
125
•
150.
0.85
0.76
0.
62
182.
197
•
228.
0.87
0.81
0.
73
166.
197
•
228.
0.83
0.78
0.
70
330.
345
•
366.
0.90
0.82
0.
73
140.
160
i.
190.
0.82
0.76
0.
.70
110
118
133
180
2.8
1.9
0
.7
0.0
88
103
115
160
3.3
1.4
0
.4
0.0
112
123
155
200
1.0
0.5
0
.2
0.0
75
82
100
150
1.2
0.6
0
.2
0.0
149
161
178
250
3.8
2.3
0
.3
0.0
91
105
126
160
5.0
4.1
2
.5
0.0
178

179
0
11
30
37
44
57
62
78
89
0
18
28
40
55
103
128
155
165
0
10
28
46
50
53
83
128
156
0
13
32
67
80
95
152
176
182
0
12
40
70
118
132
170
182
195
0
7
18
42
60
80
130
150
180
0
10
28
46
50
53
83
128
156
0
13
32
67
80
95
152
176
182
0.
,371
2.
021 1
.992
0.
475
0.
.371
2.
021 1
.992
0.
475
0.
,698
0.
961 1
.034
0.
690
0.
,698
0.
961 1
.034
0.
690
0.
,065
0.
410 0
.114
0.
026
0.
,578
1.
032 1
.531
0.
627
111
250
XXROT,CORN (HAMMOND,1981)
169
175
YYROT
200
250
XXROT,SOYBEAN
184
187
YYROT
205
250
XXROT,WHEAT (TEARS)
192
213
YYROT
200
250
XXROT,PEANUT
184
187
YYROT
CS(I).F.S.CORN
CS(I),S.S.CORN
CS(I),BRAGG
CSC I),WAYNE
CS(I),WHEAT
CS(I).peanut

APPENDIX D
INPUT FILE 'FACTS'
1 05 01 20
NS,KL
,MXRUN
,MXYER
075 10 6
IDDEC
,MOIST
,MXCRP
42 91
175
LIDLE(3)
1 0.0
0.4 0,
.6
IRSYS
i,RATE(I) (IN.)
346.2
346.2
308.2
308.2
248.9
481.2
PRODUCTION
COST ($/HA)
1.30 I
..45 4,
.5 0.12
GASPC
:,DSLPC
,WAGE,DEPRE
9800.
8600.
4680.
4010.
3680.
3180.
POTENTIAL YIELD (KG/HA)
0.103
0.103
0.238
0.238
0.137
0.473
CROP
PRICE
($/KG)
0.083
0.083
0.238
0.238
0.137
0.473
CROP
PRICE
($/KG)
0.093
0.093
0.238
0.238
0.137
0.473
CROP
PRICE
($/KG)
0.113
0.113
0.238
0.238
0.137
0.473
CROP
PRICE
($/KG)
0.123
0.123
0.238
0.238
0.137
0.473
CROP
PRICE
(S/KG)
0.103
0.103
0.190
0.190
0.137
0.473
CROP
PRICE
(S/KG)
0.103
0.103
0.214
0.214
0.137
0.473
CROP
PRICE
(S/KG)
0.103
0.103
0.262
0.262
0.137
0.473
CROP
PRICE
(S/KG)
0.103
0.103
0.285
0.285
0.137
0.473
CROP
PRICE
(S/KG)
0.103
0.103
0.238
0.238
0.110
0.473
CROP
PRICE
(S/KG)
0.103
0.103
0.238
0.238
0.123
0.473
CROP
PRICE
(S/KG)
0.103
0.103
0.238
0.238
0.151
0.473
CROP
PRICE
(S/KG)
0.103
0.103
0.238
0.238
0.164
0.473
CROP
PRICE
(S/KG)
0.103
0.103
0.238
0.238
0.137
0.378
CROP
PRICE
(S/KG)
0.103
0.103
0.238
0.238
0.137
0.426
CROP
PRICE
(S/KG)
0.103
0.103
0.238
0.238
0.137
0.520
CROP
PRICE
(S/KG)
0.103
0.103
0.238
0.238
0.137
0.568
CROP
PRICE
(S/KG)
180

REFERENCES
Agronomy Facts, No. 142. 1983. Corn hybrids for Florida in 1983. Fla.
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1154.

BIOGRAPHICAL SKETCH
You Jen Tsai was born January, 1954, in Kaoshung, Taiwan, Republic
of China. He attended and was graduated from Kaoshung High School in
1973. He enrolled at National Chung-Hsing University, Taichung, Taiwan,
and received his Bachelor of Science degree in agricultural engineering
in 1978. He continued his education at Clemson University, Clemson,
South Carolina, and received his Master of Science degree in
agricultural engineering in 1981. He enrolled at The University of
Florida, Gainesville, in January 1982. You Jen is married to Chin Mei
Wu, and they have one child, Hubert Jeng Yowe.
188

I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Agricultural Engineering
I certify that I
conforms to acceptable
adequate, in scope and
Doctor of Philosophy.
have read this study and that in my opinion it
standards of scholarly presentation and is fully
quality, as a dissertation for the degree of
jy.mJus
J./ W. Mi shoe, Cocnai rman
Professor of Agricultural
Engineering
j III J ^ U I I'
Doctor of Philosophy.
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
K. 1. Camp belly'
Associate Professor of Agricultural
Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy. ^.
D. W. Hearn
Professor of Industrial and Systems
Engineering
I certify that I
conforms to acceptable
adequate, in scope and
Doctor of Philosophy.
have read this study and that in my opinion it
standards of scholarly presentation and is fully
quality, as a dissertation for the degree of
: v 1
C. Y. Lee
Assistant Professor of
Systems Engineering
Industrial and

This dissertation was submitted to the Graduate Faculty of the College
of Engineering and to the Graduate School and was accepted as partial
fulfillment of the requirements for t\ie degree of Doctor of Philosophy.
December 1985
Dean, Graduate School

UNIVERSITY OF FLORIDA
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