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OPTIMAL SEQUENCING OF MULTIPLE CROPPING SYSTEMS By YOU JENITSAI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1985 ACKNOWLEDGMENTS This work would not have been possible without the assistance and contribution of many people. I will always be grateful to my major professors, Drs. James W. Jones and J. Wayne 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 motherinlaw who had confidence in my abilities and gave me more than I can ever repay. TABLE OF CONTENTS ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES ABSTRACT . * . 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 * CHAPTERS I INTRODUCTION The Problem Scope of The Study Objectives .... II REVIEW OF THE LITERATURE Multiple Cropping . Optimization Models of Irrig Soil Water Balance Models Crop Response Models Crop Phenology Models Objective Functions Optimization Techniques III METHODOLOGY FOR OPTIMIZING MULTIPLE CROPPING SYSTEMS . Mathematical Model . Integer Programming Model Dynamic Programming Model Activity Network Model . CropSoil Simulation Model . Crop Phenology Model . Crop Yield Response Model Soil Water Balance Model . Model Implementation . Network Generation Procedures Network Optimization . Parameters and Variables . Page ii v vii 0 . 0 . 0 0 0 0 0 0 0 0 0 0 0 Page CHAPTER IV WHEAT EXPERIMENTS . Introduction . Experimental Procedures Experimental Design Modeling and Analysis Results and Discussion Field Experiment Results Model Calibration V APPLICATION OF THE MODEL oooeoooooooooooe ooooeeooooo oooooooooooo ooooooeeooeooooo oooooooooooooooo o o o o o o o o o o o o o o o o o o o oooooeoooooooooo * 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 SimulationOptimization Techniques Multiple Cropping Systems of a NonIrrigated Field in North Florida Effects of Irrigation on Multiple Cropping ..... Risk Analysis of NonIrrigated Multiple Cropping Sequences . . Applications to Other Types of Management . VI SUMMARY AND CONCLUSIONS Summary and Conclusions .. . Suggestions for Future Research ..... APPENDIX A GENERAL DESCRIPTIONS OF SUBROUTINES . APPENDIX B SOURCE CODE OF SUBROUTINES . . APPENDIX C INPUT FILE 'GROWS' APPENDIX D INPUT FILE 'FACTS' REFERENCES BIOGRAPHICAL SKETCH . .. 107 112 119 121 130 132 178 180 181 188 . . . 000000 0 0 O00 . . . . . LIST OF TABLES Table Page 1. Threshold values for physiological stages of growth of corn and peanut ............... 43 2. Coefficients of a multiplicative model for predicting wheat phenological stages ............... 44 3. Description and threshold values of phenological stages and phases for soybean cultivars ... 46 4. Values of the parameters for the nighttime accumulator function of the soybean phenology model . 50 5. Crop sensitivity factors, A. for use in the simulation 50 6. Observations of specific reproductive growth stages for winter wheat at Gainesville, Fla., in 19831984 77 7. Summary of results of winter wheat growth under various irrigation treatments, Gainesville, Fla., 19831984 80 8. Treatment effects on winter wheat yield, Gainesville, Fla., 19831984 .... ... 81 9. Seasonal and stagespecific ET for winter wheat grown in Gainesville, Fla., 19831984 . .. ... 85 10. Price, production cost and potential yield of different crops for typical north Florida farm .. 92 11. Simulation results of irrigated and nonirrigated 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 nonirrigated 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 nonirrigated '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 nonirrigated 'Wayne' soybean grown on different planting dates for 25 years of historical weather data for Gainesville, Fla. 98 15. Simulation results of irrigated and nonirrigated peanut grown on different planting dates for 25 years of historical weather data for Gainesville, Fla. 99 16. Simulation results of irrigated and nonirrigated 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 nonirrigated multiple cropping sequences to weather patterns ....... 113 19. Comparison of various multiple cropping systems under a nonirrigated 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 nonirrigated multiple cropping sequences in response to different weather patterns . 127 21. Analysis of net returns of nonirrigated 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 LIST OF FIGURES Figure Page 1. A system network of multiple cropping . 37 2. Rate of development of soybean as a function of temperature . . 47 3. Effects of night length on the rate of soybean development . . 47 4. Maximum yield factors that reduce yield of each crop below its maximum value as a function of planting day for wellirrigated conditions ........... 51 5. Crop rooting depth after planting under wellirrigated conditions .................. .. 53 6. Leaf area index for wellirrigated 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. 19831984 .. 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 nonirrigated fullseason corn on different planting dates 102 13. Cumulative probability of profit for nonirrigated shortseason corn on different planting dates 102 14. Cumulative probability of profit for nonirrigated 'Bragg' soybean on different planting dates 103 Figure 15. Cumulative probability of profit for nonirrigated 'Wayne' soybean on different planting dates . 16. Cumulative probability of profit for nonirrigated peanut on different planting dates . 17. Cumulative probability of profit for nonirrigated wheat on different planting dates . 18. Sample output of optimal multiple cropping sequences for north Florida . . 19. Optimal multiple cropping sequences of a nonirrigated field with corn, soybean, peanut and wheat, allowing continuous cropping of peanut . . 20. Optimal multiple cropping sequences of a nonirrigated field with corn, soybean, peanut and wheat, not allowing continuous cropping of peanut . . 21. Optimal multiple cropping sequences of a nonirrigated 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 . . . viii Page 103 106 106 110 114 117 120 122 123 124 126 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy OPTIMAL SEQUENCING OF MULTIPLE CROPPING SYSTEMS By YOU JEN TSAI DECEMBER 1985 Chairman: Dr. J. W. Jones Cochairman: Dr. J. W. 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 nonirrigated 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 wheatcorn cropping would not be a profitable multiple cropping system. Instead, double cropping of cornsoybean 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. CHAPTER I INTRODUCTION Net farm income has been a major concern for farmers in commercial agriculture for a long time. Income has expanded through various ways including an increase in land area for production, fertilizer and pesticide applications, machinery and other capital expansions. However, these different methods of increasing net farm income usually increase the cost of production. A study (Ruhimbasa, 1983) showed that multiple cropping had the potential to reduce costs per unit of output and reduce production risks, and therefore could increase or at least stabilize net farm income where climatic and agronomic conditions allow its use. Multiple cropping may also be called sequential or succession cropping. Succession cropping is the growing of two or more crops in sequence on the same field during a year. The succeeding crop is planted after the preceding crop has been harvested. There is no intercrop competition. Only one crop occupies the field at one time; thus mechanization is possible. In summary, multiple cropping increases annual land use and productivity resulting in increased total food production per unit of land. It also allows more efficient use of solar radiation and nutrients by diversifying crop production. Thus, it reduces risk of total crop loss and helps stabilize net farm income. The Problem Multiple cropping is not without risk. The use of multiple cropping creates new management problems. It may create time conflicts for land and labor, may require new varieties or new crops for an area, may deplete soil resources, i.e. water and nutrient reserves, more rapidly, and may cause residuals from one crop that directly affect the next crop. For example, increasing the crop species grown on the same land makes herbicide selection more complex. Disease incidence may increase with an annual production of the same species on the same field each year. As a result, higher levels of management become more important in terms of operations needed. In designing optimal multiple cropping systems, managers need to take into consideration these effects. Of the above management areas, timing becomes dominant for successful multiple cropping, given substantial yield losses for each day of delay. As estimated by Phillips and Thomas (1984), if the losses of soybean yields after a given date are 62 75 kg/haday, the cash losses on a 200ha 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 withinseason management practices are the key to profitable multiple cropping. Soil water determines whether seeds will germinate and seedlings become established. With multiple cropping, seed zone water is even more critical because the second crop must be established rapidly to avoid possible yield reduction due to frost. Also, because of depletion 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 longterm average rainfall amount (148 cm per year) may be sufficient on the average for replenishing the soil water supply, yeartoyear 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, costineffective, and timeconsuming 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 Nyear 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 withinseason irrigation strategies that maximize net discounted return? This study at a fieldlevel needs to be differentiated from that of a farmlevel 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. Objectives The specific objectives of the study are 1. To develop a framework for optimal sequencing of crops in a multiple cropping production system and for determining optimal management of the crop land. 2. To apply the framework to study irrigation management in multiple cropping production. 3. To implement a computer model for North Florida soil and climate conditions, taking soybean, corn, peanut, and wheat as crops to be produced. 4. To perform field experiments designed to quantify the effect of water stress on wheat yield for Florida conditions, and to form a simplified wheat yield response model for use in the analysis. 5. To use the model as a decisionmaking tool to analyze multiple cropping practices in this region in order to increase net farm income. CHAPTER II LITERATURE REVIEW Multiple Cropping In the United States, sequential cropping systems are mostly found in southern states where a short cold season allows the planting of a second or a third crop on the same land. The use of notillage methods further enhances the success of sequential cropping systems in this region. A selected number of articles concerning the topics are reviewed. Multiple cropping in sequence has been criticized for being yield reducing. Crabtree and Rupp (1980) found that in Oklahoma wheat yield decreased from 2519 kg/ha in a monocropping system to 2200 kg/ha in a double cropping system. The following soybean yield decreased from 2000 kg/ha in 51cm rows and 1792 kg/ha in 76cm rows to 1603 and 1453 kg/ha, respectively. The use of notillage 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 notillage 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 notillage method. The potential of notillage 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 frostfree 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 croppingyear basis, we divide it into wintersummer double cropping, summersummer double cropping and wintersummersummer triple cropping. The major system of wintersummer double cropping is wheatsoybeans (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 summersummer systems, cornsoybeans 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 wintersummer sequence with a late fall planting of a coolseason vegetable. This type of system has the advantage of producing the vegetable crop when prices are relatively high, and still producing field crops competitively with the rest of the nation. Despite other attributes of multiple cropping, if it does not, over a period of time, provide more net income to the farmer, it will not be practiced. Economic analyses studied by a group of research scientists in the University of Georgia indicated that irrigated agronomic crops were generally profitable on a firstcrop basis, but the profitable agronomic secondcrop was limited to sorghum and soybeans (Anonymous, 1981). In 1980, the study also showed that most irrigated multiple cropping production was profitable on the welldrained, 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, doublecrop sweet corn and soybean production. They concluded that irrigated soybean as the second crop in a doublecrop 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, cornsoybean succession cropping can be successful in Florida" (page 4). Optimization Models of Irrigation In order to study irrigation policies to maintain favorable soil moisture conditions and thus avoid economic yield reduction, optimization techniques have been increasingly used for the last 15 years. Mathematical models are inherent in this methodology. Implicitly or explicitly a crop response model within the mathematical statement of the objective function is required. Furthermore, the soil water status, needed as a set of constraints in the optimization problem, is traditionally calculated in a soil water balance model. Then, various optimization techniques are applied for finding the best or optimal decisions in an organized and efficient manner. The role of models and simulation in irrigation optimization problems is reviewed herein. Soil Water Balance Water balance models for irrigation scheduling were developed as 'bookkeeping' approaches to estimate soil water availability in the root zone. Sn = Sn1 + 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 wellwatered 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 pressuretemperature 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 nonlimiting 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 max where 0a = average soil water content, and 0max = water content at field capacity. At water contents above 0.3 0 transpiration rates were max assumed to be controlled by climatic conditions only. Ritchie (1973) reported that this model predicted transpiration rates well for sorghum and corn. In summary, several models for predicting ET rates under both well watered and water stressed conditions are presented. The models presented are all simple approximations of complex dynamic systems. Their simplicity has the advantage of requiring few data inputs, and therefore, they can be applied with relatively few meteorological, soil, or crop measurements taken. However, because of their simplicity, several empirical coefficients are required in each model, and each must be calibrated for specific crops, soil conditions and climatic variables. Crop Yield Response Vast literature on this subject revealed yield relationships to water use can range from linear to curvilinear (both concave and convex) response functions (Stegman and Stewart, 1982). These variations are influenced by the type of water parameter that is chosen, its measurement or estimation accuracy, and the varied influences associated with site and production conditions. The following is intended to illustrate the more general relationships of crop yields with water when they are expressed as transpiration, evapotranspiration, or field water supply. 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 transpiration: Y T = (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 period. The stress day index model was utilized to schedule irrigations by calculating the daily SDI value (daily SD daily CS) and irrigating when it reached a predetermined critical level, SDI. This integrated the effects of soil water deficit, atmospheric stress, rooting density and distribution, and crop sensitivity into plant water stress factor. Multiplicative models. Jensen (1968) developed the following model Y n ET \. = n, ( ) (2.10) Yp i=1 ETp where ET/ET = relative evapotranspiration rate during the ith stage of physiological development, and Xi = crop sensitivity factor due to water stress during the ith 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 + nonET components of IRR define a Y vs IRR function of convex form. That is, nonET 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 degreedays is used to account for temperature effect. Degreedays 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 valuevariance (EV) 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 withinseason 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 nonirrigated 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 problemsolving 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 stagebystage 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 decisionmaking algorithm. Expressions for the soil water transition probabilities over a given time period and the first two moments of associated actual evapotranspiration were derived analytically. A stochastic dynamic programming algorithm was then used to determine optimal control policies at each irrigation decision point, conditional on the state of the system (soil water content). 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 (2.14a) I = 1I' 12, ... IN I I R n p= Y n n n B EI P I N I Rn P I Yn I IDn I _ ID n n I (2.14b) = 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 n otherwise. yield, = 0; where on Since the production cost (excluding irrigation costs), PC, is a constant value, the optimal control law that maximizes the above function will be the same that N I 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 withinseason scheduling of irrigation on a farm basis (for multiple fields) after crops are planted would require a different formulation. TravaManzanilla (1976) presented one example of such a problem. In the study by TravaManzanilla (1976), the objective was to minimize irrigation labor costs in a multicrop, multisoil 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 zeroone 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 DantzigWolfe 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 longterm production management. Models at other levels of sophistication to describe crop system responses to management practices, such as irrigation, are likely to be more useful. Thus, a general framework for optimization of multiple cropping systems using both optimization and simulation concepts will be developed. CHAPTER III METHODOLOGY FOR OPTIMIZING MULTIPLE CROPPING SYSTEMS Mathematical Model Several alternative formulations of the multiple cropping problem are studied with regard to their practicality for solutions. These are reviewed, and the most suitable one is described in detail. Integer Programming Model Sequencing is concerned with determining the order in which a number of 'jobs' are processed in a 'shop' so that a given objective criterion is optimized (Taha, 1976). In the multiple cropping problem the variable, X 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 Nyear 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 eitheror constraints cannot be implemented directly in a mathematical programming algorithm. To overcome this difficulty, new variables, Y ijt are defined. When crop ijt1 i, variety j, is scheduled for planting at tI then Yijtl 1. Otherwise, Y. = 0. This problem is then a zeroone 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 tl+Ai 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 4.5year 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 decisionmaking, the constraint of appearance of a single crop in the field anytime is implicitly coupled in the formulation. In a sense, sequential decisionmaking 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,ta(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 soilplantatmosphere 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 Nyear 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 investigated. Activity Network Model Selecting crop sequences to optimize multiple cropping systems can be formulated as an activity network model. In a network, a node stands for an event or a decision point. An activity, represented by an arc, transfers one node to another. In this particular application to irrigation management, nodes represent discrete soil water contents at every decision period. Arcs, not necessarily connecting with adjacent nodes, have lengths that denote net returns associated with the choice of crop and irrigation strategy. The structure of the network is demonstrated in Figure 1, where 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 dashedline 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 dashedline 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 postoptimization 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 longterm 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 Dc S  4 5 z, Since devaluation of cash value needs to be taken into consideration in a longterm 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 secondlevel optimization problem, which is referred to as withinseason 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 withinseason 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 withinseason 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. CropSoil Simulation Model The cropsoil simulation model serves two purposes in optimizing multiple cropping systems. First, the simulation is necessary to define the state transitions (i.e. soil water contents) in the previously discussed mathematical model. Secondly, simulation is an approach to study irrigation management strategies. The problem of optimally distributing irrigation water over the growing season is difficult primarily because of imperfect knowledge of rainfall distribution over the season. In addition, uncertainty in the distribution of other weather variables which affect crop yields complicates the optimization problem. 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 counterbalanced 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 pattern. As discussed by Jones and Smajstrla (1979), simulation models at different levels of sophistication have been developed to study the problem. In this work, a crop yield response model is included with the soil water balance model so that irrigation strategy for maximizing net return can be studied. The soil water balance model is primarily used to provide the necessary data (daily ET) for describing the yield 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 degreeday method is used. Because the units of degreeday are products of temperature and time, it is convenient to express the accumulation in units of physiological time. To accomplish this, the degreeday 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(Lao) + a2(Lao)2) (bl(Tlbo) + b2(Tlbo)2 + b3(T2b0) + b4(T2b0)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 S2 XM = C AX = 1 (3.10) S1 Table 1. Threshold values for physiological stages of growth of corn and peanut. Threshold Values of Phenological Development Crop Stage of Growth (Physiological Days) Source Planting to silking Silking to blister Blister to early soft dough Early soft dough to maturity Planting to silking Silking to blister Blister to early soft dough Early soft dough to maturity Planting to beginning flowering Beginning flowering to a full pod set A full pod set to beginning maturity Beginning maturity to harvest maturity 38.7 45.1 66.3 81.4 33.6 40.7 57.7 70.4 27.3 42.4 68.2 97.3 Bennett (personal communi cation) Agronomy Facts, 1983 Boote, 1982 Full season corn Short season corn Peanut 44 kO cn UC rC. m Dc o o o C co mr~ o C C) mU . C C CM C Co) * L , C\J I Cn 0 , CO 0 0 0 L 0 Ct+ C0 0'C: CD cu c c CM Co .c: .,c 0"i 1 0 o ". C L of CO 00 C0 0 C m  0 m CD (4 0 C 0 CD *i c o o o rf o o o o O 0 C '4 C CM C C C C o1) "U ) I s3 0 ern ( C0 C: Q. S*i co m' 0 oL Ln :r 4) 4u C .. Ci ma > *U cu m C to l CD CD o C c r 0 C) LA CM D 0 C C V) 4) 0 C 8 0 (O O C 8 I a) co+ 4 1 pc o )0  0 ' U L C C C 0 S) 4 C C C C C m :r C) CC C C C C C L a)+ co CD m CD C C) C Eo I I c E 4 ) V o c o 3 c 1U4' C0) 0 0 I0 cc (v Coo o.) m Co Q L. 4t.0 0'U) u 0 * c'.i 0'" 00'. CC C C) 0m C 0.r Cl C) C) 4C cK . *i E 4) a L oC C)sUC 0m t 4) 0) 4 C  I *4 * L ) >ccu Ir 4O .0 t CUY 0 (0I 0v Q; V) O^ Iv> C V) OU *ir * I *< ~ o~ o . 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 follows: 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). Threshold Growth 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 induction Physiological time from floral induction to flower appearance Photoperiod accumulator from flowering to first pod set Photoperiod accumulator from flowering to R4 Photoperiod accumulator from flowering to the last Vstage Photoperiod flowering flowering Photoperiod flowering accumulator from to the last possible date accumulator from to R7 Physiological time from R7 to R8 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 6.522 10.87 2.40 1.00 9.48 0.20 6.0 0.5 0.6 14.5 10.0 TPHMIN TOPT TPHMAX Temperature (C) Rate of development of soybean as a function of temperature. TNLG1 TNLGO Night Length (hr.) Effect of night length on the rate of soybean development. Figure 2. o U S 0 0 4 LL 1/1 0 THVAR DHVAR 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 wellirrigated 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 practices. Soil Water Balance Model In order to predict ET rates under wellwatered and water stressed conditions, a soil water balance model was developed to integrate existing knowledge about crop water use, weather patterns, and soil properties into a framework compatible with irrigation objectives. A model previously described by Swaney et al. (1983) was adapted for this study. Table 4. Values of the parameters for the nighttime accumulator function of the soybean phenology model (Wilkerson et al., 1985). Name of Parameters 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 Ij pLaLA C) cN 0 rq CO 4 Q r CM C) (.0 CD m OCO C C CM C0 sJoI.o PLta.L o <4 CD 0 4 CD  0 CC k C0 CO CJ C SJO;OPJ plaLA rD 0 0 0 0 s1C= CP j CP l C. SAoIPJ PPL~A C o C >, o c CMi L) E r00 x E r o o a) 4) 00 (1) u 0 LU U 4+ (U e S, ra C c 4 L S ra a) u c 0 S4 L, 04 4)0 U e C  o z  E3 L CM o r4 CO 0 14 U, C >, CM 0r 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 wellirrigated 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 LO CD Lon C LO 1 '4 C.' C'\ (w3) qjdaa 6u14oo0 C\J 0  0 4, 0( 0) 04 C C 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 wellirrigated 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) (3.15) 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, 59.590.055 T avg 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 pressuretemperature function, A are computed as follows: e (T) = 33.8639{(.00738T + .8072)8 .000019(1.8T + 48) + .001316} (3.18) A = 33.8639{0.05904(0.00738T + 0.8072)7 0.0000342} (3.19) In general, net radiation values are not available and must be estimated from total incoming solar radiation, Rs and the outgoing thermal long wave radiation, Rb. Penman (1948) proposed a relationship of the form Rn = (1a) 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 = StefanBoltzmann constant (11.71*108 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. Clearsky 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 rainfed strategy depends totally upon rainfall. Model Implementation In order to study multiple cropping systems as well as associated management strategies, models are needed to summarize and operationalize knowledge about plant growth, yield, weather patterns, soil properties and economics into a framework compatible with system objectives. Therefore, computer programs were written in FORTRAN 77 to evaluate the methodology. Figure 7 shows a schematic diagram for the methodology. As outlined in the previous section of mathematical model, in order to optimize multiple cropping systems, three independent steps, system description, generation of network, and network optimization are essential. Detailed descriptions and source code of subroutines to execute the methodology are given in appendix A. The purpose of this section is to provide discussions on model implementation in general. 60 START D 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  I Sort newly created nodes by  increasing order of decision da E Organize the expanding 0 multicroppingsystem network C 1 potential^ odes considered   yes Format arc list in increasing order 4 'J by arc ending node number 4w E I I + .4L Longest Path Algorithm o Summarize Results TOP 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 decisions. In the simulation, a crop season includes a oneweek 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 renumbering nodes are required in order to have an ordered network. At each node (present planting day), a combination process of simulation and renumbering 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: 0) >s 0 ., r c  rc^ I ECDL i en ,,) n3, .._j ., . w o >, 0 V) ro E ro L ' rCI S f computing the longest path, the labelcorrecting 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 nnode 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 Kvector 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 Kvector 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 Kvector 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 lowerbound 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 Kvectors 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 Kvector (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 Kvector also contain INF values. Therefore, when updating the Kvector for node i, the Kvector 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 Kvector for node 1 and the arc length Ali, is less than or equal to the minimum element of the Kvector 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 Kvector for node i (only distinct path lengths are retained). As compared to the use of some general sorting routines to find the K longest elements in a list, the use of these two observations allows for a 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 arc. In essence, this latter path tracing procedure is based on the following fact. Namely, if a tth 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 qth longest path for some q, 1 < q < t. This fact can be used to determine the penultimate node r on a tth 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 qth longest path length (q 4 t) for node r. If so, then there is a tth 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, welldrained 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 19541971 and 19781984. 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 wellirrigated 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 withinseason 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 twocrop seasons are also specified by users. From a computerized crop budget generator (Melton, 1980), the collective costs of production for various crops were obtained. 69 Equations of variable irrigation costs of different systems used in the study were obtained from D'Almeda (personal communication). By regressing results which were obtained from the irrigation cost simulator (D'Almeda et al., 1982), he developed the equations for typical North Florida conditions. The other economical component of interest is crop price (PRICE), $/kg. Current market prices (May, 1985) were provided as baseline data. CHAPTER IV WHEAT EXPERIMENTS Introduction Wheat (Triticum aestivum L.) is an important crop in the multiple cropping minimum tillage systems widely used in the Southeast USA. In this system, wheat is usually planted in the fall after soybean harvest. Despite the need for intensive management, wheat can be grown 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 wellwatered 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 ETY 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 ETY 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 ETY 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 rotortiller 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 20cm 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 901818 kg/ha (Nitrogen SulfurPotash) 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 doubleindex (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 50cm 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 50cm 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 TuesdayFriday schedule was employed to monitor soil water contents in the lysimeters during the season. Soil water contents at five depths (15, 30, 45, 75, 105 cm) of soil profile were measured with a neutron soil moisture meter (TROXLER 3220 Series.) Additional work which was performed on the same schedule included irrigation, collecting of volumes of drainage water from the lysimeters, and monitoring of crop phenology. Irrigation decisions were made weekly immediately following neutron probe readings. Amounts of application were computed as the volume of water reduction below field capacity for the top 50cm zone. A manually operated, precalibrated drip irrigation grid was designed to irrigate inside each lysimeter. A separate irrigation system was used to irrigate the buffer crop area beneath the rainout 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 ovendried 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 Y N ET A. = 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 leastsquares 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 wellirrigated were at that stage of development. Wheat phenology data observed in 19831984 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, 19771979, by Barnett and Luke (1980). Heading dates at Quincy were March 23 and March 27 for the 19771978 and 19781979 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 19831984. 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 Milkyripe Apr. 3 127 Kernel hard Apr. 17 141 Harvest May 9 163 ajnqew ISEAJeq pJieq [auial adLjRla@4PM [auJa>l 6u i.aMo [1 6uLpeaq 6uL4ooq as 0 apou Wais 1o apou a6uaaluewa f6u Lu Ld  A ssaiq4s III ssaJs II 4) a) 4) 0) S t, '/, 4) L/) a) S. 4) 43 S Q4, 0)4 > CF 4 r 01 U) 4)I EUr c, Ln Cl) 4 ), 0 C)a C S  u  L a ,, o > > (U U CE C C S. 1 to C1J C j (30) ainqeaadwai 'L.Ge 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 wellwatered (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. :3n U, ~ a,i I U, C: a E c. a, L 4 c 01 c o 0 01 L c, '" VI 0 l rcr L (0 Oi L a, *0 = 0 L 0, 01 a, La a, . Et r CO 0 COi 4 '3 ,LL ar, ca a, 4 I 0 ^ *r LU, oao 310 rN a, .0 fI LD CM *^1 < 4 1 00 1 0 C4 c . r oo 00 0o o0 00 N ICOC CM C C 4 oM k CJ 1d Ch Cl C t 0 CM9 LA fi C4M CM~ co) nY CY C") L) 0m CMI L0 C" 0o LA d ;n L9 0'c 0m. CMJ CM 0m ^td LA NCM .4 A C 3 N^ .4 N CO N CJ i^~ i r~ 00 i' CMj LO Ln LA LA LA OD m oo mo cc (.0 CY r^ i (.0 t.0 r~. r^ C:) 00 C") CO C) C) LALALAN CO CO o oo ko o 00 C"0 LA 10' N 00 00 f Im I CM Z2;2Z=Z 2;2; z z z z z z C. C%) CO 0C 0 N 4 4 CJ C :00 C0 OCC00 00 00 to CJ mn n LO C) : CM C) Co : : COO CDm NLA C\M < CV) . C\ M m' t 0 m 40C: LA C01 N! 4 C0' CO 0'. CMJ CMJ LA . CM CY) LA d C\J CV) c") C) N Cm C" d 0 to P LA 0 C 1 CM Lo LA LA LA LA LA 4 c md :* 0h LAn Nl r rN LA LA LA "CM 4 LA 0 CO CO a LA 0M r 0m 0ON NI 0 rN LA LA u 4 mt CMj N LA o CM 4 :d 4r Cqr CM CM CMC C 4 L4 14  4 4 4  C% C% C0 CO L t CO'CUC"JCOO rCOCOCC000 _ o CO o 4 L Nc") LA m C") .4 cz. C") c;) _: (3; LA: CJ oo0' Co N C) LO (Y) Cd :d CV) tt L ) 0CD CO LAM C)0 L0 LACO A C co CM L coo L CM) e ( N CO 1, N t0 LA CoJ N 40i'. CM tn coi 1, o all qn Cr oD CM N 10' 00. N CO (".3 ^0 1~ oi> ^o 04 44 i4<44o e 44r4M M coc c a* ^ 4 I E (0 3: 4 * (0 0 a, ==I Co 0 Ln 0o r4 Ln NCOCOCO)CON* mA qzI CA LA LA 0'. fLA 00 C)O CMn 10 .4 o t.0 Jo e 10 c; eN coL CO c m 1.0 ri ko 0M CM) c"L LA 0'I. Coo CM r: qn co Ln CM C"M %0 LO Cn CMj LA co Od 0'. N d LC 0'. LA LA N 0 N0 A LA LA rLA LA N CO U). LA CO C:) LA LA No Co Lo LA CO k 00 CM o oo co i LO flo oo 110 10 oo0 LA LA COD) L CO ko o 4 r4 o0 4 0i m cdO i)0 mC CO .l CO ino e'JC") LA LA C i4 CMJ 4 i CM CMJ CM. (''CM CM CM >>>>44 0 L 04 0 0 C 0.. 4 L .0 0 L I a r C L3 (V tc 0o a, * .p0, C * L< Q) *r 3 t "0 '0  t < z' ">1n Table 8. Treatment effects on winter wheat yield, Gainesville, FL., 19831984 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. 1000. 900. 800. 700. 600. 500. 750. 700. 650. 600. 550. 500. 500. 450. 400. 350. 300. 250. UC (N,N) (11,2) (11,4) (III,3)(III,4) (IV,2) (IV,2) Stress treatment 450. F 400. 1 350.  300.  250.  200. (IV,2)(IV,2) 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 wellirrigated 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 stagespecific ET's were very low. c (v ol ** e 0 0 . W. 4)~ E C  c t_ L iJ LL a) VI 0) 3, C L. 0 4 L U3 c 4 a) CL C) 4) L C: + 4 "3 0 1 Li U X: (A U m0 0) V) 0 r s 4 0 4 I 0 cU 00) 01 c c 10 ,0) o e cV o I U5CO enr o o o In t4 o m~ c" 0t CDJ Lc 4 CMJ CMJ co' C) cM c) 0c oo i O) M A o cr) C tr LO 0r 0 CM r. o _ 4 Cr) Cr) Cr) 4 0: kD cd '. oo oCD rC oC CDi CO M C 'j c, *a Ci CM r~ ~0 Cr) Cr e i 4 o o co.. 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CMrlC oo Zn o^ uo co co cO 00 00 CD) f) C) l0 cD r C) toC C)o o in Co) . in cc oo 0 i 0r> 0)0 04. gr InC)  '.o 1 in '.0 .si r^ ^l in 0) In '4410 c o C;" ii ( co o ii 00 4 C) co co CM CM Y n F4 000000 ,4 (Moo '4 Cr) C CM) CM ~~ co coc s d 44I IO4 '.0 '.0 4.C 4.00) _q kn kn co mnc . 4.0 40 (CM CM C' C) C:) "r oj m4 co) In i In 0 n Mr 0 ('4 Cr) In 0m c D CM co In c; o; cv .i 4 Cr) '.0 In m4 10 m t.04 _qO C r i r cc ^i o r.: c;> oo an co r II CM CM 4 cM VC'J c (N0P r) 00 CQ a) m C) ri0 c '.o t m o COr) '0 <4 CM 0 o4 4no c oCM ^l *=3u i~inr~  r4 4 cno oo oi o= 0M 0 m C 0 0 Cl in cr4 c; r: cZ. cc , co r In iN 4 ~nccat~Nocc; In M c cc0 00oC0004 4 CO4 c 4 004. %er. . CMi CM (M CM CM CM >>>>>>* 44 4 44 C, 0 4) c S4^ 0 HO 0l [434) 0) c: L cu [0) 04l CC L 0) pC? \4, Li) 0) 00 U,) Lt * CO Ot ^c? ; u u0 0) 0) ~UC  c c*i, 0)0) 01 0 4'4 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 24lysimeters, 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 withintreatment 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 grainfilling 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 87 2.0 1.5 0 r= 0.42 rr= S1.0 0, 0.5 0.0ii 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. CHAPTER V APPLICATION OF THE MODEL Introduction In Florida, where the cold season is relatively short and the water supply (precipitation and irrigation) is sufficient to grow two or more crops per year, the potential of practicing multiple cropping is high. On the other hand, irrigation development is expensive. Inasmuch as 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 nonirrigated 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 4.5year production schedule was projected. Three crop production systems were investigated. Under system I, crops to be considered in sequential cropping were fullseason corn (F.S.Corn), shortseason corn (S.S.Corn), earlymaturing soybean (Wayne), latematuring 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 