Optimal sequencing of multiple cropping systems


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Optimal sequencing of multiple cropping systems
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x, 188 leaves : ill. ; 28 cm.
Tsai, You Jen, 1954-
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Subjects / Keywords:
Multiple cropping   ( lcsh )
Cropping systems -- Mathematical models   ( lcsh )
Agricultural Engineering thesis Ph. D
Dissertations, Academic -- Agricultural Engineering -- UF
bibliography   ( marcgt )
non-fiction   ( marcgt )


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

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University of Florida
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oclc - 14399185
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Full Text








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 Mishoe. 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.






* . S S S

* S S S S S S S S S S S S S

* S S S S S S S S S S *



The Problem
Scope of The Study
Objectives ....


Multiple Cropping .
Optimization Models of Irrig
Soil Water Balance Models
Crop Response Models
Crop Phenology Models
Objective Functions
Optimization Techniques


Mathematical Model .
Integer Programming Model
Dynamic Programming Model
Activity Network Model .
Crop-Soil Simulation Model .
Crop Phenology Model .
Crop Yield Response Model
Soil Water Balance Model .
Model Implementation .
Network Generation Procedures
Network Optimization .
Parameters and Variables .





0 . 0 .

0 0 0 0 0 0 0 0 0 0 0




Introduction .
Experimental Procedures
Experimental Design
Modeling and Analysis
Results and Discussion
Field Experiment Results
Model Calibration



o o o o o o o o o
o o o o o o o o o o

* S S S S S S S

Introduction . .
Procedures for Analysis ...
Crop Production Systems . .
Crop Model Simulation . .
Optimization of Multiple Cropping Sequences .....
Risk Analysis . .
Results and Discussion . ...
Crop Model Simulation . ..
Evaluation of The Simulation-Optimization Techniques
Multiple Cropping Systems of a Non-Irrigated Field in
North Florida
Effects of Irrigation on Multiple Cropping .....
Risk Analysis of Non-Irrigated Multiple Cropping
Sequences . .
Applications to Other Types of Management .


Summary and Conclusions .. .
Suggestions for Future Research .....















. . .

0 0


. . . . .


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, A. 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 full-
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 'Bragg'
soybean grown on different planting dates for 25 years
of historical weather data for Gainesville, Fla. .. 97

Table 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-irrigated 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-irrigated 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-irrigated 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


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 well-irrigated
conditions .................. .. 53

6. Leaf area index for well-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


15. Cumulative probability of profit for non-irrigated
'Wayne' soybean on different planting dates .

16. Cumulative probability of profit for non-irrigated
peanut on different planting dates .

17. Cumulative probability of profit for non-irrigated
wheat on different planting dates .
18. Sample output of optimal multiple cropping sequences
for north Florida . .

19. Optimal multiple cropping sequences of a non-irrigated
field with corn, soybean, peanut and wheat, allowing
continuous cropping of peanut . .

20. Optimal multiple cropping sequences of a non-irrigated
field with corn, soybean, peanut and wheat, not allowing
continuous cropping of peanut . .

21. Optimal multiple cropping sequences of a non-irrigated
field considering corn, soybean and wheat, excluding
peanut . . .

22. Optimal multiple cropping sequences of an irrigated
field considering corn, soybean, peanut and wheat,
allowing continuous cropping of peanut .

23. Optimal multiple cropping sequences of an irrigated
field considering corn, soybean, peanut and wheat,
not allowing continuous cropping of peanut .

24. Optimal multiple cropping sequences of an irrigated
field considering corn, soybean and wheat, excluding
peanut . . ..

25. A set of optimal multiple cropping sequences for a non-
irrigated field chosen from Figure 20 for additional
simulation study . . .














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





Chairman: Dr. J. W. Jones
Cochairman: Dr. J. W. Mishoe
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.


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.

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


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


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

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-

transpiration demand and supply of soil water are synchronized,

potential maximum yield is expected. Otherwise, irrigation may be

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

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.


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


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.


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


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

practices on yield which also led to a conclusion favoring a no-tillage


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

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


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 well-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

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


Soil Water Balance

Water balance models for irrigation scheduling were developed as

'bookkeeping' approaches to estimate soil water availability in the root


Sn = Sn-1 + Pn + In + DRn ETn ROn PCn (2.1)

where Sn = soil water content on the end of day n,

P = total precipitation on day n,

I = total irrigation amount on day n,

DR = water added to root zone by root zone extension,

ET = actual evapotranspiration on day n,

RO = total runoff on day n, and

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.

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 (ET p), which is that of a well-

watered, vegetated surface. To predict actual rather than reference ET

for a well-watered crop, a crop coefficient, Kc, was introduced (Jensen,

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

Ep ETp (2.3)
p a

where T = reduction factor due to crop cover, and a = proportionality

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, Tp, was calculated as

Tp a= v ( 1 T ) ( A / ( A + y )) Rn (2.5)

where A = slope of the saturation vapor pressure-temperature curve, y =

psychrometric constant, Rn = net radiation, and av = (a 0.5)/0.05.

For greater than 50 percent crop cover, T was calculated as

Tp = ( ) ( A / ( A + y ) ) Rn (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 s) was defined by Kanemasu et al. (1976) as

KS = ---- (2.7)
0.3 m

where 0a = average soil water content, and 0max = water content at field

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


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, evapotranspiration, or field water


Yield vs. transpiration or evapotranspiration. When yields are

transpiration limited, strong correlations usually occur between

cumulative seasonal dry matter and cumulative seasonal transpiration.

Hanks (1974) calculated relative yield as a function of relative


= (2.8)
Yp Tp

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

Y A n
= 1.0 E (CSi SDi) (2.9)
Y YV i=1 1
p P

where A = yield reduction per unit of stress day index, SD. = stress day

factor for crop growth stage i, CS. = crop susceptibility factor for

growth stage i. CSi expresses the fractional yield reduction resulting

from a specific water deficit occurring at a specific growth stage.

SDi expresses the degree of water deficit during a specific growth


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 n ET \.
= n, ( ) (2.10)
Yp i=1 ETp

where ET/ET = relative evapotranspiration rate during the i-th stage of

physiological development, and Xi = 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 x.
= n ( ).1 SYF LF (2.11)
Yp i=l Tp

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 2
= { 1.0- (1.0 )i } (2.12)
Yp i=1 ETp

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 Ymax ETmax 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.

That is, non-ET losses increase as water is applied to achieve ETmax

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;

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

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,

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.

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 y2 +2 2 2 + 2 a2 2 C2 -2 (2.13)
i i p Y i + X 1 y 2PYi,YXi

where o2i is the variance in net returns for irrigation strategy i, Yi

and a2 are the mean and variance of yield associated with irrigation
1 2
strategy i, P and aO are the mean and variance of crop price, y and

o2 are the mean and variance of irrigation pumping cost per unit of
Y 2
water, Xi and ao2 are the mean and variance of irrigation water applied

for irrigation strategy i, and aPYi'YXi is the covariance between PYi
1, Y" 1

and yXi Then the relative contribution of each component random

variable (price, yield, pumping cost, and irrigation water) to the

variance of t 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

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


Dividing the irrigation season into N stages and taking irrigation

depth (I ) at decision stage n as a decision variable, the objective

function (Bras and Cordova, 1981) can be formulated as:

N In
= Max E [ Z R ] PC
I e n=l n


I = 1I' 12, ... IN

R n p= Y n
n n







_ ID n



= maximum net return,

] = expectation operator,

'C = production costs different from irrigation costs,

t = feasible set of control policies,

= type of control applied at decision stage n,

= number of decision stages in the growing season,
n = net return by irrigating In at decision stage n,

= price per unit of crop yield,
n = contribution of irrigation decision In to actual

B = unit cost of irrigation water,

= depth of irrigation water associated with operati

policy In,

y = fixed cost of irrigation (labor cost), and

= 0,

= 1,

when ID n


= 0;



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

Max E [ I Rnn ] (2.15)
I eI 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, Xijkl 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 Xijkl to

maximize net profit for the farmer. A required input was net profit

associated with Xijkl, Cijkl 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-Manzanilla (1976) presented one example of such a problem.

In the study by Trava-Manzanilla (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;

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.


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 is defined and equal to one when crop i, variety

j, planted at t1 still grows in the field at time t2. Otherwise, it is

equal to zero. It is also assumed that the growth season for crop i,

variety j, planted at t1 is Aijtl and the associated net return is

Cijt 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, Xijt lt and constants Aijt and Cijt the formulation of
an objec tive function and constraint conditions is
an objective function and constraint conditions is

Max Z = (Ci ) (X ) (3.la)
ijtlt2 Jtl iJtt2

s.t. X T2 (3.1b)
ijtlt2 X lt

Xitlt2 1 for all t2 (3.1c)
tlAijt l '
ijt 12

1 t t t = 0 or
t2=tl ijtt2

t +Aijt
SX ijt = A ijt, for all i,j,t1, (3.1d)

t2=t12 1ij

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 ijt are defined. When crop
i, variety j, is scheduled for planting at tI then Yijtl 1.

Otherwise, Y. = 0. This problem is then a zero-one integer

programming model. The formulation is

Max Z = I (C.. ) (Yijt ) (3.2a)
ijtI 1jt 1

s.t. ijt T2 (3.2b)
ijtlt 2 t2

I X ijt < 1 for all t2 (3.2c)
ijtI rjt t2

1+A ijt
S (X ijt )(1 Yijtl)= 0, for all i,j,tI,

1 = (3.2d)

t +A ijt
(Xijt Aijt )(Yi ) = 0, for all i,j,t1,
t2=1 (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 T1 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
ijt1 ijt 1
(ijtl). Because of all of these shortcomings, the integer programming
approach was not pursued further.

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

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(C1,Wi,Ni,t) = Max R (C2,I ,t) + F(C2,Wf,Nf,t-a(C2)) (3.4)
C2 S(CIt)

where W. = state of soil water at the beginning of the season,

Wf = state of soil water at the end of the season,

N. = value of nutrient level at the start of the season,

Nf = value of nutrient level at the end of the season,

C1 = proceeding crop,

C2 = selected crop, decision variable,

S(C1,t) = proper subset of crop candidates dependent on C1

and season t, due to practical considerations of crop

production system,

a(C2) = growth season of crop C2,

I = optimal realization of irrigation policies, a vector

(I1, 12, ... Ik) represents the depths of irrigation

water associated with individual operations,

R = 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:

Wf = g (C2, I Wi), (3.5a)

Nf = h (C2, I Wi, Ni). (3.5b)

These functions are not explicitly expressible. It is not

realistic to represent the complicated soil-plant-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

P(C1, Wi, Ni, t) = (C2, Wf, Nf, a(C2)) (3.7)

where Wf = soil water status at end of a season, Nf = nutrient level at

end of a season, C2 = index of the selected crop, a(C2) = growing season

of C2

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 (Co, W0, No, T), where T = the span of N-year

growing period, (C W N ) 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 (CO, Wo, No, T). In terms of computational

efficiency, this dynamic programming model is not very appealing.

Therefore, a more comprehensive, efficient model needs to be


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 Ci is crop variety i and S. is

irrigation strategy j. The S and T nodes are dummy nodes, representing

the source and terminal nodes of the network, respectively.

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.

" I

Lfl /\-

w o'
Y -f I I'

E /
\- A L I

0A l jn

,;i J3

S -


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

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


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 evapotranspiration

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


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

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(Ati) 7
PD = Ati for 7 < T < 30, (3.8)
i=l 30 7

n 45 -T(At.)
PD = At for 30 < T < 45,
i=1 45 30

for T > 45,

PD = 0

where PD = physiological day, T(Ati) = temperature in the time

interval Ati 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 = (al(L-ao) + a2(L-ao)2) (bl(Tl-bo) + b2(Tl-bo)2 +

b3(T2-b0) + b4(T2-b0)2) (3.9)

where L = daily photoperiod,

T = daily maximum (daytime) temperature,

T2 = daily minimum (nighttime) temperature.

And a al, a2, b0, b1, 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

XM = C AX = 1 (3.10)

Table 1. Threshold values for physiological stages of growth of corn and

Threshold Values of
Phenological Development
Crop Stage of Growth (Physiological Days) Source

Planting to silking
Silking to blister
Blister to early
soft dough
Early soft dough to

Planting to silking
Silking to blister
Blister to early
soft dough
Early soft dough to

Planting to beginning
Beginning flowering to
a full pod set
A full pod set to
beginning maturity
Beginning maturity to
harvest maturity










Facts, 1983






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ern (

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I- a) co+ -4 1 pc o

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m :r C) CC C C C C C

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The summation (XM) is carried out daily from one phenological stage S1

to another S2.

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


Xm = 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.

Table 3. Description and threshold values of phenological stages and
phases for soybean cultivars (Wilkerson et al., 1985).

Stage Description Phase 'Bragg' 'Wayne'

Physiological time from planting
to emergence

Physiological time from planting
to unifoliate

Physiological time from unifoliate
to the end of juvenile phase

Photoperiod accumulator from the
end of juvenile phase to floral

Physiological time from floral
induction to flower appearance

Photoperiod accumulator from
flowering to first pod set

Photoperiod accumulator from
flowering to R-4

Photoperiod accumulator from
flowering to the last V-stage



accumulator from
to the last possible

accumulator from
to R-7

Physiological time from R-7
to R-8

1 6.522

2 10.87

3 2.40

4 1.00

5 9.48

6 0.14

7 3.0

8 0.16

9 0.575

10 20.35

11 12.13













Temperature (C)

Rate of development of soybean as a function of temperature.



Night Length (hr.)

Effect of night length on the rate of soybean development.

Figure 2.









Figure 3.

Figure 2 is the normalized function to calculate physiological time. In

Figure 2, data used for minimum, optimal and maximum temperatures were

7, 30, 450C, 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 'Bragg' 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 evapotranspiration (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

(1 2 3 4
Y/Yp = (ET1/ETpl) (ET2/ETp2) (ET 3 /ETp3) (ET4/ETp4) (3.12)

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 well-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 (Ai) 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 X. for soybean in a lysimeter study, and their estimates

of X. 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 Xi and related crop response to irrigation


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


Table 4. Values of the parameters for the nighttime accumulator
function of the soybean phenology model (Wilkerson et al.,

Name of

Value of The 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, x., 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 Doorenbos (1979)

Soybean 0.698 0.961 1.034 0.690 Boggess et al. (1981)

Peanut 0.578 1.032 1.531 0.772 Doorenbos (1979)

Wheat 0.065 0.410 0.114 0.026 Personal observation

C 00 I'D a CM C0
C c C; C; c ;

s o I-j pLaLA












sJoI.o PLta.L








rD 0 0 0 0
s1C= CP j CP l C.

C >,

o c





o o
a) 4-)

(1) u


U 4+-

(U e-
-C c

4- L
S- ra






-e C
-- o

z -







C >,


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 well-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 (ET p), which is estimated by a modified version of

the Penman equation. The ET is then used to calculate potential

transpiration (T p) using a function of leaf area index (Ritchie, 1972):

Tp = 0 Lai < 0.1 (3.13a)

Tp = ET p(0.7* (Lai)/2- 0.21) 0.1 < Lai < 3.0 (3.13b)

Tp = ETp 3.0 < Lai (3.13c)

C 0 0 0 C C
--1 '-4 C.' C'\

(w3) qjdaa 6u14oo0

0 -

0 4-,



C 0 0 0 0 C
LO 0 LO 0:) LO
-4 -4 C\J C\J

(wUo) qjdag 6uj40oy

0 0 0 C0 C0 0 0 00 0 0 0
in o in 0o LO im CD m Ln

(uwo) q;daa 6ut4oyo (wo) q;dao 6upLooy

where Lai = leaf area index. For well-irrigated crops, leaf area index

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 e> 9 (3.14a)
P c

T=T *(9/9) 0 < 0c (3.14b)

where 9 = ratio of soil water in root zone, as a fraction of field

capacity, 0 = (r- d) / (Ofcd- ),

0c = 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,

"d = lower limit of volumetric water content for plant growth,
0fc = 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 We)


xapuI Paiv J4pa

xapuI Pe i Jea1

xapuI vaiv 412@

XapuI eaJV .4ea

where We = volume of water in the evaporation zone (cm), Ep = potential

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 { (at1/2- (t- 1)1/2), Wel (3.16)

where a is a constant dependent on soil hydraulic properties. For sandy

soil, a = 0.334 cm day- /2.

For practical application, the Penman equation is considered the

most accurate method available for estimating daily ET. The Penman

formula for potential evapotranspiration 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

ARn/X + yEa
ET = n a (3.17)
P A + y

where ET = daily potential evapotranspiration, mm/day
A = slope of saturated vapor pressure curve of air, mb/C

Rn = net radiation, cal/cm2 day

X = latent heat of vaporization of water, 59.59-0.055 T
cal/cm2 mm or about 58 cal/cm2 mm at 29 C

Ea = 0.263(ea ed) (0.5 + 0.0062u2)
ea = vapor pressure of air = (emax + emin) / 2, mb

ed = vapor pressure at dewpoint temperature Td

(for practical purposes Td = Tmin), mb

U2 = wind speed at a height of 2 meters, Km/day

y = psychrometric constant = 0.66 mb/C

emax= maximum vapor pressure of air during a day, mb

e min= 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}

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, Rs and the outgoing

thermal long wave radiation, Rb. Penman (1948) proposed a relationship

of the form

Rn = (1-a) Rs Rb (3.20)

where Rn = net radiation in cal/cm2 day,

Rs = total incoming solar radiation, cal/cm2 day

Rb = net outgoing thermal long wave radiation,
a = albedo or reflectivity of surface for R .

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 s) Lai (3.21)

where as is average albedo for bare soil and a for a full canopy is a.

And an estimate of Rb is found by the relationship:

Rb = aT4(0.56 O.08/ed) (1.42R /Rso 0.42) (3.22)

where a = Stefan-Boltzmann constant (11.71*10-8 cal/cm2 day/ K),

T = average air temperature inK (C + 273),

Rso= total daily cloudless sky radiation.

Values of Rs are available from weather stations in Florida. Clear-sky

insolation (R so) 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.

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.



Data Input :
1. the first decision day
2. crop potential yield
3. crop price g
4. irrigation system & operations .
5. crop phenology information
6. crop sensitivity factors 0

Initialize variables I o

Simulations to create
new nodes & arcs -
Sort newly created nodes by |
increasing order of decision da
Organize the expanding 0
multicropping-system network C

1 potential^
odes considered -- --

Format arc list in increasing order 4 'J
by arc ending node number
4-w E
I I +
Longest Path Algorithm o

Summarize Results


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

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


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




c -

i en
,,) n3,
.._j ., .-





L '
S- f

computing the longest path, the label-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) = (XVil, XV i2, ... ,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 Ali 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 (XVlm + Ali: 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 (l,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

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

(-Co,-g) ... ,-). 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 K 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 Ali, is less than or equal to the minimum

element of the K-vector for node i, then no improvement in the latter K-

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


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 substantial 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


In essence, this latter path tracing procedure is based on the

following fact. Namely, if a t-th longest path ir of length 1 from node

i to node j passes through node r, then the subpath of r 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

can be found by forming the quantity (1 rj) for all nodes r incident

to node j and determining if this quantity appears as a q-th longest

path length (q 4 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 SOYGRO V5.0 (Wilkerson et al., 1985). Data for use in this

study were the result of simulating a well-irrigated field in 1982.

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.


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.



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

successfully 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.

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 (deWit, 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 CO2 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.)

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

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)

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


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 950C for 24 hours. For individual lysimeters, grain

weights and related yield variables were assembled and measured for

detailed analysis.

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 successfully 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

= n ( ) (4.1)
Yp i=1 ETp i

where Y/Yp = the relative yield of a marketable product,

ET/ET = the relative total ET during the given ith stage of

physiological development,

Xi = 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

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 (Ai). 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 well-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

Table 6. Observations of specific reproductive growth stages for winter
wheat at Gainesville, FL., in 1983-1984.

Elapsed Time
Stage Description Date after Planting

Planting Nov. 29 1
Emergence Dec. 4 6
Node of stem visible Jan. 20 53
Booting Feb. 27 91
Heading Mar. 10 103
Flowering Mar. 22 115
Milky-ripe Apr. 3 127
Kernel hard Apr. 17 141
Harvest May 9 163

ajnqew ISEAJeq

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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.


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Table 8. Treatment effects on winter wheat yield, Gainesville, FL.,

Dry Mass No. of Head Wt. Grain Wt.
Treatment gm Heads gm 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 bc

(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.

UC (N,N) (11,2) (11,4) (III,3)(III,4) (IV,2) (IV,2)

Stress Treatment

UC (N,N) (11,2) (11,4) (III,3)(III,4) (IV,2) (IV,2)

Stress Treatment

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.



















UC (N,N) (11,2) (11,4) (III,3)(III,4) (IV,2) (IV,2)

Stress treatment

450. -F

400. 1

350. -

300. -

250. -



Stress Treatment

Figure 10. (continued)

7 I I I I

UC (N,N) (11,2) (11,4) (III,3)(III,4)

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 well-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 r2 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 ET's were very low.

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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 A'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

weather, the r2 = 0.42 does not seem high. However, the effect of

critical stages of growth has been quantified.

Values of published A s for wheat are inconsistent. The Ai 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




r= 0.42




0.0 0.5 1.0 1.5 2.0

Observed Yield Ratio

Figure 11. Plot of observed vs. predicted yield ratio for wheat.

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.



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

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 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.

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