Ii‘i;§2bu’%‘il“x~f -. .jIIJ 1 war MODELING YIELD AND WEATHER FOR SUGARCANE PRODUCTION SIMULATION Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY FRANCISCO YANTO PANOL 1 972 1’ III/ll/III/IIIIILIIII/III/IIZIIIII/II/I/I ‘I/ ‘ ~ A R Y M; 3 , 7404 Michigan State in) University _.v V ‘w‘ This is to certify that the thesis entitled MODELING YIELD AND WEATHER FOR SUGARCANE ‘PRODUCT‘ION SIMULATION presented by FRANC ISCO YANTO PANOL has been accepted towards fulfillment of the requirements for Ph.D degreein Agricultural Engineering ; Major professor Febru 22 , 1972 Date ary 0-7639 ,. amomq‘wmrs ' z p. 800K BINDER! ISNC LIBRARY BINDERS mum”. mm“: ABSTRACT MODELING YIELD AND WEATHER FOR SUGARCANE PRODUCTION SIMULATION by Francisco Yanto Panol The primary objective of this project was to de- velop sugarcane yield and weather models for simulation application. In addition, preliminary indications of al- ternative cropping cycles for the Victorias milling dis- trict were to be obtained by simulation, using the models developed. The yield models were formulated by multiple regression using the least-squares criterion. Separate models for tonnage and rendement were developed for the periods January to June and July to December. In the models, the climatic influence tends to be manifested in sequences of occurrence rather than the absolute value of the weather factors. The various area models indicate different controlling weather factors on growth and yield. Model verification using 1970 production data yielded a close agreement between the estimated and actual tonnage. However, there were slight discrepancies between the estimated and actual rendement. Possibly, this can be attributed to the effect of residual fertilizer from previous crops. Models for generating weather variables were Francisco Yanto Panol developed for the two areas of the district. Determination of rainfall occurrence in the first area is by a Monte Carlo technique using second—order Markov probabilities. The amount of rainfall was determined from a probability density function derived for the area. Sunlight and maxi- mum and minimum temperatures were generated from regres- sion equations in lagged values of the variables. The choice of the regression equation to use on a given day depends on the first order rain-no rain state in the area. For the second area, rainfall occurrence was also deter- mined by the Monte Carlo technique. Here, the probability of rain depends only on the rain-no rain state in the first area, for the same day. The models for sunlight and temperatures consist of regression equations with lagged values of the variables. The choice of the equation to use depends on the first order rain-no rain state in this area and the present rain-no rain state in the first area. Two simulations were made to obtain preliminary indications of alternative cropping cycles. One simula- tion used the historical weather records and another used stochastically generated weather factors. This also pro- vided a test of the performance of the stochastic weather generator in production simulation applications. Means and standard deviations of yields and revenues for each month and pair of months were calculated. Five annual sets of monthly prices were used in calculating revenue. The calculated mean yields and revenues with the two Francisco Yanto Panol simulations were in close agreement with each other. There are strong indications, based on yield and revenue, that the November-December period is not the best time to cease operations in the district. Conclusions derived from these results included: 1. The tonnage and rendement models developed are ade- quate for production simulation applications. 2. The weather simulator is adequate for production simulation applications. 3. There is an annual time trend of increasing ton- nage and decreasing rendement in the three areas of the district. Approved 1/) My Mfijbr Professor Approved g A Depaftment Chairman MODELING YIELD AND WEATHER FOR SUGARCANE PRODUCTION SIMULATION by Francisco Yanto Panol A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Engineering 1972 ACKNOWLEDGEMENTS The author wishes to express his appreciation and thanks to the following persons and institutions for the help which made this study possible. To Dr. J. B. Holtman, committee chairman, Dr. F. W. Snyder, Dr. R. Zemach and Prof. J. B. Kreer who served on the author's guidance committee. To the Victorias Milling Company for providing financial support and the data utilized in the study. To the Agricultural Engineering Department, Michi- gan State University for the opportunity granted the author to pursue this study. To his wife who willing exchanged the security of established home and community roles for the frustration often attendant to the completion of a graduate program. ii TABLE OF CONTENTS Page ACMOWLEDGEMENTS O O O O O O O O O 0 O 0 O O O 0 0 O 0 i i LIST OF TABLES . . . . . . . . . . . . . . . . . . . . v LIST OF FIGURES . . . . . . . . . . . . . . . . . . . Vi Chapter 1. INTRODUCTION . . . . . . . . . . . . . . . . . l 1.1 Description of the Victorias Area . . . . 3 1.2 Soils of the District . . . . . . . . . . 4 1.3 Climate of the District . . . . . . . . . 5 1.4 The Sugarcane Plant . . . . . . . . . . . 6 1.5 Influence of Climatic Factors . . . . . . 8 2. YIELD MODEL DEVELOPMENT . . . . . . . . . . . 12 2.1 Available Data . . . . . . . . . . . . . 12 2.1.1 Monthly Tonnage and Rendement . . 12 2.1.2 Weekly Rendement and Production . 14 2.1.3 Weather Data . . . . . . . . . . . 14 2.2 Model Formulation . . . . . . . . . . . . 15 2.2.1 Growing Period . . . . . . . . . . 15 2.2.2 Discrete Time Models . . . . . . . 16 2.2.3 Basic Assumptions . . . . . . . . 20 2.3 Parameter Estimation Procedure . . . . . 20 2.3.1 Stepwise Addition of Variables . . 21 2.3.2 Stepwise Deletion of Variables . . 23 Estimated Tonnage Models . . . . . . . . 24 Estimated Rendement Models . . . . . . . 31 Tests for Autocorrelation and Hetero- skedasticity . . . . . . . . 2.6.1 Autocorrelation . . . . . . . . . 3S NNN .0 O‘Ultb 2.6.2 Heteroskedasticity . . 2.7 Model Verification . . . . . . . . . . 39 3. WEATHER SIMULATOR . . . . . . . . . . . . . . 44 3.1 Victorias Weather Simulator . . . . . . . 44 iii Page Manapla Weather Simulator . . . . . . . 50 Stochastic Weather Simulation . . . . . . 56 3.3.1 Methodology . . . . . . . . . . . 56 3.3.2 Simulator Validation . . . . . . . 57 Wk) 4. SIMULATION STUDIES OF ALTERNATIVE CROPPING CYCLES . . . . . . . . . . . . . . . . . . . 66 4.1 Yield Simulation . . . . . . . . . . 66 4.2 Revenue Simulation . . . . . . . . . . 67 4.3 Results and Discussion . . . . . . . . 68 4.3.1 Simulated Yields . . . . . . . 68 4.3.2 Simulated Revenues . . . . . 76 5. SUMMARY AND CONCLUSIONS . . . . . . . . . . . 81 6 . RECOMMENDATIONS O O O O O O O O O O O C O O O 84 REFERENCES 0 O O O O O O O O O O O O O O O O O O I O O 85 APPENDICES Appendix A . . . . . . . . . . . . . . . . . . 90 Appendix B . . . . . . . . . . . . . . 92 Appendix C . . . . . . . . . . . . . . . . . 94 Appendix D . . . . . . . . . . . . . . . 108 Appendix E . . . . . . . . . 109 iv LIST OF TABLES Durbin-Watson test statistics Id.) . . . . . Bartlett's test statistics (Q/L) . . . . . . Statistics for the chi-square goodness-of-fit test 0 O O O O O O O O O O O I O O O O O O 0 Simulated weekly weather variables . . . . . Actual weekly weather variables . . . . . . . Page 36 38 51 65 65 Figure 1a. 1b. 7b. 7c. 8a. 8b. 8c. LIST OF FIGURES Actual and simulated values, Victorias IOWIand (1970) o o o o o o o o o o o o 0 Actual and simulated values, Victorias upland (1970) . . . . . . . . . . . . . Actual and simulated values, Manapla (1970) Flow chart for Victorias daily weather simulator Flow chart for Manapla daily weather simulator Actual and simulated values, Victorias lowland Actual and simulated values, Voctorias upland Actual and simulated values, Manapla . . Actual and simulated values, Manapla . . Monthly mean yield, Victorias lowland . . Monthly mean yield, Victorias upland . . Monthly mean yield, Manapla . . . . . . . Mean yield for paris of months, Victorias lOWland O O O O O O O O O O O O O O O O 0 Mean yield for pairs of months, Victorias upland O O O O O O O O I O O O O O O O 0 Mean yield for pairs of months, Manapla . Simulated revenue for five annual price series, Victorias lowland . . . . . . . . . . . . Simulated revenue for five annual price series, Victorias upland . . . . . . . . . . . . Simulated revenue for five annual price series, Manapla O I O O O O O O O Q C O O O O O 0 vi Page 40 41 42 58 59 60 61 62 63 69 70 71 72 73 73 77 78 79 1 . INTRODUCTION Successful sugarcane production requires an under- standing of the influence of and the interaction among the various factors affecting the growth and yield of the crop. Of primary importance among these factors are the soil productive capacity, climatic condition, sugarcane variety, cultural practice, labor supply, field machinery and trans- port equipment availability. Of particular interest in this study is the selection of an appropriate cropping cycle for a given region or farm district. The selection of an appropriate harvesting season or cropping cycle is dependent upon the interactive influences of the various production factors. In the Philippines, the harvesting season for sugarcane varies among milling districts. Some areas har- vest only for about six months each year, while others har— vest for as long as ten and one-half months. Basically, climatic conditions determine the length of the harvesting and planting season. In most areas, the crop is harvested only during the drier part of the year, a period of about five to seven months, depending upon the particular geographic location. Harvesting only in the drier period normally provides for relatively higher cane quality. It also facilitates field operations during the harvesting and planting. However, such short milling periods result in heavy seasonal labor and equipment demands both in the production and processing sectors. Thus, the longer milling season has positive social implications in terms of continuous employment opportunities. The Victorias milling district, which is the largest sugar producing district in the Philippines, harvests cane approximately ten and one-half months each year. Normally, milling in the district ceases in early or mid November and starts again in late December. This shutdown period of about six weeks is used for repairs and retooling of factory and farm equipment for the next milling season. The production and economic implications of this November- December period (or of the resultant cropping cycle) are not well established. Recommendations as to the best cropping cycle for the district have been made. Some were purely intuitive while others were based on conventional analyses given the existing data (VMC, 1967; VMC, 1968). Several researchers have applied computer simula- tion analysis as a tool in the study of the complex inter- play of the various factors in agricultural production systems (Holtman, et a1., 1970; Stapleton, 1967; Sowell, et a1., 1967; Morey, et a1., 1969). Results of such analyses would be useful in the search for an optimum cropping cycle in the Victorias milling district. The simulation analysis requires the development of representative stochastic models embodying the cause-and-effect relationships of the various production factors. It is the primary objective of this project to develop sugarcane yield and weather models for the Victorias milling district. Such models will pro- vide a fundamental step towards future simulation applica- tions and also other immediate benefits. Cane yield models, for instance, are useful in developing operational projec- tions for production, processing and marketing. Moreover, an understanding of the influence of various factors on cane growth andydeld will be valuable in assessing current cultural practices in the district. A secondary objective of this project is to obtain preliminary indications of alternative cropping cycles for the district. 1.1 Description of the Victorias Area The Victorias milling district covers a total area of about 40,000 hectares (1 hectare = 2.47 acres). It is situated on the northern part of Negros Island, Philippines. The area is bounded by the Visayan Sea on the east and by volcanic chains on the southwest and is characterized by an irregular coastline. The relief is level to undulat- ing to rolling. Of the total district area, about 30,000 hectares are currently planted in sugarcane. The re- mainder consists of second growth forest, coconut plan- tations and other crops such as rice, fruit trees, etc. The district is divided into three areas as follows: (1) Victorias lowland with 1,500 hectares under cane, (2) Victorias upland with 9,500 hectares of sugarcane, and (3) Manapla-Cadiz, with a crOpped area of 19,000 hectares. The Victorias upland and Manapla-Cadiz areas respectively have approximately 1,000 and 5,000 hectares more that could be put to sugarcane cultivation. However, these ad- ditional areas and even recent plantings are, in most cases, of marginal productivity. There are about 900 individual planters or growers in the district with farm units ranging in size from about 10 to as much as 1,000 hectares. The average farm size in the district is about 40 hectares. Sugarcane produced by the planters is processed into either brown or refined sugar by the Victorias Milling Company, which is located in the district. 1.2 Soils of the District Soils of the Victorias milling district are gener- ally grouped into lowland and upland soils based on the relief of the area. Lowland soils in the district are formed from recent alluvial deposits and have generally level relief with poor to adequate natural drainage. The lowland soils are considered the most productive in the district. These soils are located mainly in the Victorias lowland section. Upland soils in the district are developed either from the weathered products of igneous rocks, from older alluvial deposits or from the weathered products of coralline limestone. They have generally sloping to rol- ling relief with excessive surface and poor to fair internal drainage, resulting in varying degrees of soil erosion. The majority of the upland soils have about the same pro- ductivity ratings (Locsin and Tabayoyong, 1953). The moisture equivalent of the soils in the district varies from about 20% to as much as 30%. Available moisture is estimated to range from 5 to 15 per cent. 1.3 Climate of the District Under the climatic classification in the Philip- pines, which is based on the distribution and amount of annual precipitation, the climate of the Victorias milling district is characterized as having no dry season and no pronounced maximum rain period. The average annual preci- pitation in the district is 101 inches. The monthly and daily means and standard deviations of rainfall and also the daily means and standard deviations of maximum tempera- ture, minimum temperature and sunlight for the Victorias and Manapla stations are tabulated in Appendix A, These values are based on 20-year weather records from both stations. Generally, February, March and April are drier months. These are also the months of higher daily sun- light. The period from July to December is normally the wettest part of the year. Likewise, this period has lower sunlight. Minimum temperature is lowest during the months of December, January and early February. Maximum tempera- ture is highest during the months of April and May. Under the climatic conditions in the district, sugarcane has been grown without irrigation. 1.4 The Sugarcane Plant Sugarcane (Saccharum officinarum, Linn.) belongs to the vast family of grasses. Being a tropical plant, sugar- cane thrives in hot, sunny areas. The cane plant is com- posed of four principal parts, the leaves, the stalk, the roots and the flower or arrow. The leaves contain the green chlorOphyll which makes possible the synthesis of sugar from water and carbon dioxide, which is absorbed from the air through the stomatal openings in the leaves. The cane stalk is the above-ground portion of the plant supporting the leaves and the flower. The small portion of it underground is known as the stubble or root stalk. The stalk, which is composed of a number of sec- tions or internodes, is almost cylindrical in cross-section and consists of three recognizable substances, the hard rind, a softer internal flesh and fibers. The softer tis- sues of the stalk surrounding the vascular bundles are made up largely of the cells which store the sweet sugary juices of the plant. Generally, a 12-month crop will have cane stalks varying in length from five to as much as 12 feet. Each cane stalk weighs from one-half to one and one— half kilograms when harvested. The root system of a sugarcane plant, as in many other crops, anchors the plant in the soil. More import- antly, it serves as a supplier of and vehicle for the plant nutrients and water absorbed from the soil. Depending on the soil horizon, cane roots may extend to a depth of more than six feet. Normally, however, the root mass concentra- tion is within the first two feet of soil. The depth to which the majority of the roots extend determines, to a great extent, the drought-resistance of the crop. Commercial sugarcane planting uses either the top portion of the harvested stalks or the stalks of young (about 7 to 10 months) cane grown in nurseries. These planting materials are cut to a length of 12 to 18 inches and covered in furrows of well-tilled fields. Depending on the climatic conditions and cultural practices of the area, the growing period varies from 10 to 36 months. In the Philippines, the average growing season is 12 months. When the cane is harvested, a regrowth will occur from the stubble left in the soil. This crop is called a ratoon crop. Several ratoon crops are being grown in some areas before a new crop is planted, at which time the field is tilled accordingly. In the Philippines, only one ratoon is normally grown. While a ratoon crop involves less production cost, since land preparation and planting are not required, ratoon yields are generally lower. 1.5 Influence of Climatic Factors Several workers have investigated the influence of various climatic conditions on sugarcane growth. The work of Burr and associates (1957) is of particular interest. Sugarcane was grown for several years in culture solutions under controlled conditions of temperature and light. Briefly, some of their findings were: 1. Below 70°F, root temperatures become strongly limiting to growth. At 50°F, there is no growth. An 80°F root temperature is optimum for both growth and nutrient uptake. 2. Cutting full Hawaiian sunlight one-half reduces growth one—half. 3. Using sugarcane grown under identical day conditions but different night temperatures—-one cooled to 57°F and the other warmed to 73°F--it was found after 20 weeks that the cool night group had a weight of approximately half that of the warm night group. A similar reduction was observed for the weight of the leaves and overnight translocation of sucrose from the leaves to the stalk. A close relationship of stalk length and diameter to temperature in Hawaii was earlier shown by Stender (1924). His measurements showed that winter growth of the primary stalks was reduced to one-third the summer growth. More- over, greenhouse studies in Hawaii showed that irrespective of air temperature, root temperatures of 62°F and below restrict nitrogen uptake, water consumption, translocation and growth (Anon., 1957). It was further reported that at root temperatures of 74°F and above, light becomes the dominant factor affecting growth. Das (1935) in studying the effect of climate in Hawaii, planted two cane varieties in pots containing the same soil. One group of pots (including both varieties) was grown at a location 40 feet above sea level with an annual rainfall of 30 inches. Another group was grown three miles away, but at 650 feet above sea level where the annual rainfall was 200 inches. The climate in the former is characterized by bright sunny weather. The latter area received a quantity of sunlight (hours) less than 50% of the former. Maximum temperatures are about 4°F higher in the former location and minimum temperatures are about equal. The result of the experiment was striking in that both varieties produced nearly three times as much cane in the former area than in the latter. The result of Borden's (1936) more elaborate studies gave further evidence of the dominant influence of climatic factors upon cane yield. Clements (1964) also reported a positive response of sugar- cane to increasing maximum and minimum temperatures as well as sunlight. Moisture shortages can exert a dominant influence on stalk elongation. Clements and Kubota (1942) for in- stance, reported a correlation coefficient of 0.756 between 10 the moisture content of the elongating cane and meristem and the rate of stalk elongation. Sun and Chow (1949) found a high positive correlation between rate of stalk elongation and rainfall in Taiwan. While it is generally agreed that extended periods of drought often depress growth, such periods may also have beneficial effects: the forced development of a deeper root system, the prevention of undesirable flowering and an increase of carbohydrate accumulation during the ripening stage (Clements, 1964). Moreover, excessive rainfall is not only ineffective but may cause reduced growth rates, particularly where drainage is impeded. A desired environ- ment provides a balance between transpiration and water absorption that is conducive to highest growth during the vegetative stage and to ripening during the maturity stage. Like growth, maturity of sugarcane is also influ- enced significantly by climatic factors. Humbert (1968) pointed out the dramatic effect of minimum temperatures on the maturity of field cane at Los Mochis, Mexico. It was noted that lower minimum temperatures about a month before harvest are favorable for ripening and hence contribute to higher juice quality. Johnson (1966), reported that sucrose percentages in cane are closely related to the diurnal range one month before harvesting. In studying the relationship of low atmospheric temperatures with juice quality, Panje, et a1. (1968) found that temperatures in the range of 2 to 12°C caused 11 a depression in juice quality. The cane was able to re- cover under normal growing temperatures, however. The work of Singh and Lal (1935), confirmed by the work of Hartt (1940), indicated that the optimum temperature for the synthesis of sucrose by excised blades of sugarcane is ap- proximately 30°C. Although sufficient moisture is required during the vegetative stage, moisture has a depressing influence on cane juice quality during the ripening stage. Escober (1961) related Victorias district yields to certain weather factors. There existed an inverse relation between juice quality and rainfall excesses over a calculated effective rainfall, preceding and during the harvesting months. In Hawaii, irrigation is terminated as cane approaches maturity in order to reduce the rate of vegetative growth, dehydrate the cane and force the conversion of reducing sugars to recoverable sucrose (Humbert, 1968). The influence of climatic factors on cane produc- tion is best summarized by Mangelsdorf (1950) in character- izing an ideal climate for the production of sugarcane: l. A long, warm summer growing season with adequate rainfall. 2. A fairly dry, sunny and cool, but frost free, ri- pening and harvesting season. 3. Freedom from typhoons and hurricanes. 2. YIELD MODEL DEVELOPMENT The amount of sugar obtained per unit area is the product of the quantity of cane produced and the quality of the juice extracted from it. The quantity of cane harvested is commonly referred to as cane tonnage and ex- pressed as tons cane per unit area. The juice quality is commonly referred to as rendement and expressed as weight of sugar obtained per ton cane processed. In the Philip- pines, rendement is expressed in piculs (1 picul = 63.25 kilograms) per ton cane. These expressions of cane yield and rendement are used throughout the report. Because of the observed differences in the pedo- 1ogic, physiographic and biotic complexes in each of the three areas of the district, it was advantageous to develop distinct yield models for each section. Likewise, because of the significant differences in average weather condi- tions during the periods January to June and July to Dec- ember, it was decided to construct yield models for each of these two periods. 2.1 Available Data 2.1.1 Monthly Tonnage and Rendement One of the two sets of yield data available is a 12 l3 record of monthly tonnage, rendement and hectarage harves- ted in each of the three areas for the period 1951 to 1970. The data for the year 1970 were reserved for model verifi- cation and were not utilized in the modeling process. The data were gathered by personnel of the Victorias Milling Company. During the harvesting season, field inspectors visit each farm every month to determine the actual area harvested. Using a record of the amount of cane that came from a given farm, the tonnage for the farm was calculated. Juice samples were obtained in the factory from each ship- ment of cane coming from a farm. They were used to deter- mine the rendement of the cane milled. At the end of each milling month, aVerage tonnage and rendement were computed for each area and reported with the total hectarage harves- ted. The percentages of ratoon crop were also reported at the end of each crop year. When a crOp was damaged by a typhoon, the reported tonnage did not include the tonnage lost. The tonnage data for the months affected by typhoons were adjusted, based upon the estimated losses due to typhoons (VMC, 1968). Appendix B gives the adjustments made for the tonnage af- fected by typhoons. Because of the regular shutdown period, the months of October, November and December have less tonnage data available than the rest of the months. In the 19-crop year sequence (1951 to 1969), there were two missing values for October, ten for November and four for 14 December. Moreover, some of the data reported for the months of November and December did not cover the entire month. 2.1.2 Weekly Rendement and Production Weekly average rendement and the amount of cane milled for the entire district were recorded by the fac- tory. Typhoon loss adjustments were also made on these rendement data; these adjustments are given in Appendix C. Data are missing for those periods corresponding to the periods without monthly tonnage. Because of the significant changes that took place in both the varieties planted in the district and the fac- tory extraction efficiency, it was decided to utilize only the weekly rendement data for the period 1960 to 1969. The data for the year 1970 were saved for model verifica- tion. The weekly rendement for each of the three areas were obtained by multiplying the district rendement data by a factor based on monthly yield data in the three areas. The formula for calculating the factor is given in Appen- dix C. 2.1.3 Weather Data Available weather records include daily rainfall, sunlight hours, and maximum and minimum temperatures for the Victorias and Manapla weather stations for the years 1949 to 1970. 15 2.2 Model Formulation 2.2.1 Growing Period A 12-month period from planting to harvesting was assumed. Actual harvesting age in the district varies from about 11 to 13 months. In extreme cases, some cane fields are harvested at the age of nine or 10 months while others are allowed to grow up to 14 or 15 months before they are harvested. A period of several weeks before planting is re- quired for field preparation. During this period, the amount of precipitation is of utmost importance. Exces- sive rainfall delays land preparation and may result in the deterioration of the planting materials. If the field is too dry, adequate tillage is hardly possible. This may result in poor germination and growth. It is, therefore, necessary to consider soil moisture conditions before planting. The highest vegetative growth (boom stage) of cane in the district occurs in the period five to nine months after planting. After the boom stage, the cane starts to ripen if climatic conditions are favorable. An eight-week period before planting and the 12- month (52 week) planting-to-harvesting period were consi- dered in developing the tonnage model. The 52-week period was divided into thirteen 4-week crop ages and the eight- week period before planting was divided into two 4-week l6 periods. The period beginning at the thirty-fourth week after planting and including the harvesting week (19 weeks) was considered for the rendement. This period was divided into six crop ages: four 4-week periods, one 2-week period before harvest and the harvesting week. 2.2.2 Discrete Time Models Models for simulation applications can be formula- ted either in the continuous time form (described as dif- ferential equations) or the discrete time form (described as difference equation). The factors that determine whether a continuous time model or a discrete time model should be selected are: (1) the level of detail necessary to answer relevant questions, (2) the frequency of events or the flow rate of objects relative to the minimum time interval of interest, and (3) cost of programming and operating the models (Manetsch, 1970). It was decided to develop the models in the discrete time form with one week as the time interval for rendement and a time interval of one month for tonnage. It was assumed that the following functional rela- tions exist: TC - F( [w 15 A RC YR) _ o I ‘I I i J ]j=l 1 TC: TC, =H([w]15,A, ,RC,YR,ZQ. 1) l7 6 TON T RENDn = 0‘ [wk] k=1' n' m’ YR) where: TC = tonnage per hectare for month i or i+6 i = 1, 2,...6, denoting the harvesting months January to June, i+6 denotes the harvesting months July to December 15 [W.] = set of weather factors occurring during the 3 j=1 period j A = area in hectares harvested during the month i or i+6 RC = percentage by area of ratoon crOp for the year RENDn = rendement for week n 6 [Wk] = set of weather factors occurring during the k=1 period k total amount of cane milled in the district during the week n TON T = tonnage per hectare during the month m which contains week n YR = cropping year (1951-1969 for tonnage and 1960- 1969 for rendement). It is necessary to identify an appropriate quanti- tative expression for each climatic factor. Twelve weather factors were considered for the tonnage and rendement models. These factors, computed weekly, are: (1) total rainfall, (2) sequence of days exceeding 25 days with rainfall less than 0.50 inches, (3) summation of daily heat units with 24°C as the base temperature, (4) summation of daily diurnal range, (5) sequence of days exceeding two days with minimum temperature less than 22°C, (6) square of the sequences in 5, (7) squared sequence of days exceeding one day with sunlight 18 less than one hour, (8) squared sequence of days exceeding three days with sunlight less than four hours, (9) sequence of days exceeding two days with maximum temperature greater than 33°C, (10) sequence of days exceedings two days with sunlight greater than 10.0 hours, (11) summation of sun- light hours in excess of 10.0 hours, and (12) summation of daily sunlight hours. For periods prior to harvest, the sequences are computed in the following manner. If the sequence had not ended, say, in week m, it is allowed to continue to week m+l. A non-zero value is then assigned only to the week in which the sequence ended. However, if week m is the end of the harvesting month in the case of tonnage or the harves- ting week for rendement, the sequence is terminated at the end of week m. The value is then assigned to week m. The use of sequences instead of absolute values of the climatic factors was based on the result of a prelim- inary regression analysis of factors affecting yield. A small amount of variety experiment data was used in this analysis. Monthly mean maximum and minimum temperatures did not show any significant influence. There was also no measurable influence on yield if the sequence of days with rainfall less than 0.50 inches was shorter than 25 days. Also, the length of the growing period (the age of the crop at harvest) did not show a significant influence on the tonnage or rendement. Only rainfall and sequences of days with rainfall 19 less than 0.50 inch were considered for the two 4-week periods before planting, giving four weather variables for this period. With thirteen 4-week crop ages and 12 wea- ther factors, there were a total of 156 weather variables for the period of planting to harvesting. Thus, a total of 160 weather variables were considered for the tonnage models. The area harvested in each month was included to account for possible yield inflation when a smaller area is planted and harvested. This may be caused by a shift from extensive to intensive production. The percentage of ratoon crop in the district was included to account for possible yield deflation due to the inherently lower ratoon yields. Based upon the observed relationship between the tonnage during the periods January to June and July to December, it was decided to include the average tonnage of the former period in the models for the latter period. The six crop ages and 12 weather factors considered for rendement yielded a total of 72 weather variables. The amount of cane milled for the week was included to consider the possible effect of the volume of cane milled on the factory extraction efficiency. Previous experience sug- gested the inclusion of tonnage per hectare for the area of interest. The cropping year, denoting a factor for time trend, was included in both the tonnage and rendement models to isolate the influence of technological changes. 20 2.2.3 Basic Assumptions Inherent in this modeling process was the assumption that the weather factors observed and recorded in each of the two weather stations sufficiently characterize the weather occurring throughout the respective area served by the station. The assumption was also made that the crop response to weather variation is uniform throughout the given area (i.e., the interactions between weather factors and the various agronomic variables are uniform throughout the particular area). 2.3 Parameter Estimation Procedure The parameters of both the tonnage and rendement models were estimated by multiple regression utilizing the method of least squares (Kmenta, 1971; Kane, 1968; Draper and Smith, 1966). A general linear hypothesis for k ex- planatory variables and N observations is: Y1 = H) + blxll + b2x21 + ... +bixil +... +bkxkl + 01 . = + + . +...+ X +...+bx +U ¥t H) b1x1t b2x2t bi it k kt t where: = observation t of the dependent variable Y Y X. = observation t of the explanatory varia- ble xi Ut = stochastic disturbance associated with observation t 21 and CODStantS: b0 , bl, b2 0 o o bi o o o bk. The method of least squares consists of determining estimates (bo,‘b1,‘82 ...‘bi ...‘bk) of the constants b bl' b 0' 2... b- ... bk' such that the sum of the squared residuals,‘U 1 18 t a minimum, i.e., %: ‘U£ is a minimum, where: A A A A A A A null hypothesis that the individual bi's equal zero is established and tested to obtain the regression equation. An important measure of how much of the variation in the dependent variable may be accounted for by the group of explanatory variables is the coefficient of multiple deter- mination (R2). R2 is the proportion of the sum of the squared deviation from the mean of the dependent variable accounted for by the explanatory variables (Kane, 1968). 2 The square root of R is the so-called coefficient of mul- tiple correlation. 2.3.1 Stepwise Addition of Variables In this work, the method of least squares with step- wise addition of variables (Rafter and Ruble, 1969; Draper and Smith, 1966) was first used because of the large number of variables involved and the possibility of singularity problems. High requirements of computer time and memory provide additional motivation to begin with stepwise addi- tion. The steps involved in the method of least squares 22 with stepwise addition of variables are summarized below. 1. A regression equation involving only the dependent variable and its mean is formed. 2. From among the explanatory variables not presently in the least squares equation, a candidate for in- clusion in the equation is selected. The candidate is that variable xi which will yield a maximum in- crease in R2. 3. An Fbi statistic is calculated for the variable and a significance probability* for this variable is determined. 4. If the significance probability is less than a pre- set value, the variable is added to the least squares equation. Then the procedure reverts to step 2 and the process is repeated. 5. When the significance probability is greater than the preset value, the candidate is not entered into the equation and the procedure is terminated. There is the inherent danger that a group of variables which individually account for little of the variation in the de- pendent variable, but as a group explain much of this varia- tion, may never be entered into the equation (Rafter and Ruble, 1969). Therefore, a relatively high preset signifi- cance probability level of 0.05 was used. *Significance probability is the maximum probability of rejecting the hypothesis: bi = 0, when bi = 0 (i.e., the probability of committing a type I error). 23 2.3.2 Stepwise Deletion of Variables A set of explanatory variables[:E] was established utilizing all of those variables obtained from stepwise ad- dition and some additional weather variables. The additional weather variables were formed from certain combinations of variables from [Wj115 j=1 rendement. For instance, one explanatory variable in the 6 for tonnage and from[Wkl]k=1 for tonnage model was total rainfall during the period beginning the twenty-first week and ending the twenty-fourth week (a 4-week period), R The rainfall in the preceding 4-week 5. period was added to R5 forming one new explanatory variable. Similarly, the rainfall in the following 4-week period was added to R5 yielding another explanatory variable. Multiple regression utilizing the method of least squares with step- wise deletion of variables was then applied toI:E]. The procedure of stepwise deletion is composed of the following steps: 1. A least squares equation is formed utilizing the elements of [E]. 2. The explanatory variable (having a significance probability greater than a preset level) which when deleted produces a minimum reduction in R2 is re- moved from [E]. 3. A new least squares equation utilizing the remaining elements ofIE]is formed. Then the process returns 1 to step 2. 24 Since the selection of a candidate variable for deletion is closely tied to the stopping criterion, the pre- set significance probability was set at 0.005. The resulting regression models were then scrutinized. Particular attention was given to the sign and magnitude of the coefficient of each explanatory variable, their standard errors of estimate as well as the magnitude of the coefficient of determina- tion (R2) and the overall standard error of estimate for the model. Three additional variables were also deleted from the models. Each of them had a marginal significance prob- ability (close to 0.005). Furthermore, the sign of their coefficients was inconsistent with results of previous in- vestigations. Least squares models were then estimated utilizing the remaining explanatory variables. 2.4 Estimated Tonnage Models The estimated models obtained for tonnage for each of the three areas are given below. J anuary-June Harves ting : TCli = 56.419307 - 0.539063 VLRl - 0.688571 VLR2 (7.27634) (0.09759) (0.08198) .+ 0.142532 VLHU + 1.376277 YR (0.02191) (0.22355) R2 = 0.690 R = 0.830 S-Eo = 9.37046 25 TCZi = 52.233198 — 0.110380 VUSLS - 0.506847 VUTNl (1.31407) (0.02301) (0.08854) + 1.499525 YR (0.10947) R2 = 0.690 R = 0.830 S.E. = 5.68666 TC3. = 39.849315 - 0.349193 MR1 - 0.040779 DRM (9.40218) (0.06648) (0.00922) + 0.213806 MR2 + 0.083472 MHUl + 0.068853 MHU2 (0.05687) (0.02234) (0.01502) + 0.041533 SSM + 0.777629 YR (0.01075) (0.11319) R2 = 0.621 R = 0.788 S.E. = 5.60270 July-December Harvesting: TC1i+6 = 73.649783 + 0.054936 VLDRl - 0.069704 VLDRZ (10.5712) (0.00871) (0.01266) + 0.527413 ATONl (0.09344) R2 = 0.545 R = 0.738 S.E. = 9.57022 TC2 = 41.287846 - 0.283702 VURNl + 0.307148 VURNZ 1+6 (7.01971) (0.06328) (0.07271) + 0.289710 VURN3 + 0.056751 vunu - 0.005019 VUDR (0.07527 (0.01206) (0.00723) + 1.111755 ATON2 (0.06786) R2 = 0.808 R = 0.899 S.E. = 5.69818 TC3i+6 = 49.838779 — 0.221258 MR3 + 0.360850 MR4 (3.30415) (0.04702) (0.08822) - 0.064704 MSLSl - 0.269533 MSLSZ + 1.147577 ATONZ (0.02201) (0.06796) (0.09545) R2 = 0.768 R = 0.876 S.E. = 6.76793 where: TC1, TC2, TC3 VLRl VLR2 VLHU VUSLS VUTNl MR1 MR2 MHUl MHUZ DRM 26 l, 2, ... 6, denotes the harvesting months January to June respectively. tons cane per hectare for the Victorias lowland, Victorias upland and Manapla- Cadiz areas respectively. total rainfall in Victorias for the per- iod beginning the 5th week and including the 28th week after planting. total rainfall in Victorias during the 8-week period prior to planting. total heat units accumulated in Victorias for the period beginning the 29th week and including the 40th week after planting. sum of squared sequences of days exceed- ing one day with sunlight less than one hour in Victorias for the period begin- ning the let week and including the 36th week after planting. sum of sequences of days exceeding two days with minimum temperature less than 22°C for Victorias for the period begin- ning the 13th week and including the 40th week after planting. total rainfall in Manapla for the period beginning 8 weeks before planting and including the end of the planting month. total rainfall in Manapla for the period beginning the 41st week after planting and including the end of the harvesting month. total heat units accumulated in Manapla for the period beginning the 17th week and including the 28th week after plant- ing. total heat units accumulated in Manapla for the period beginning the 41st week and including the end of the harvesting month. sum of daily diurnal ranges in Manapla for the period beginning the 4lst week after planting and including the end of the harvesting month. SSM VLDR1 VLDR2 VURNl VURN2 VURN3 VUHU VUDR MR3 MR4 MSLSl 27 sum of squared sequences of days with sunlight less than four hours in Manapla for the period beginning the 41st week after planting and including the end of the harvesting month. sum of daily diurnal ranges in Victorias for the period beginning the 17th week and including the 28th week after plant- ing. sum of daily diurnal ranges in Victorias for the period beginning the 4lst week after planting and including the end of the harvesting month. total rainfall in Victorias for the per- iod beginning the 5th week and including the 16th week after planting. total rainfall in Victorias for the per- iod beginning the 29th week and including the 40th week after planting. total rainfall in Victorias for the per- iod beginning the 4lst week after plant- ing and including the harvesting month. total heat units accumulated in Victorias for the period beginning the 5th week and including the 16th week after plant— ing. sum of daily diurnal range in Victorias for the period beginning the 17th week and including the 28th week after planting. total rainfall in Manapla for the period beginning the lst week and including the 20th week after planting. total rainfall in Manapla for the period beginning the 29th week and including the 40th week after planting. sum of squared sequence of days exceed- ing one day with sunlight less than one hour in Manapla for the period beginning the 13th week and including the 28th week after planting. 28 MSLSZ sum of squared sequence of days exceed- ing one day with sunlight less than one hour in Manapla for the period beginning the 29th week and including the 44th week after planting. YR = harvesting year with 1951 taken as 1, 1952, 2, etc. ATON1,ATON2,ATON3 = average tonnage from January to June (minus their lowest average tonnage: 54.663, 44.973 and 44.950 respectively) for Victorias lowland, Victorias upland and Manapla areas respectively. R2, R, S.E. respectively the multiple coefficient of determination, multiple coefficient of correlation and standard error of estimate for the model. The numbers in parentheses are the standard errors of estimate of the corresponding coeffi- cients. The negative relation of tonnage with rainfall in the Victorias lowland model is due primarily to the poor internal and surface drainage conditions in the area. As emphasized by Humbert (1968), excessive rainfall is not only ineffec- tive but may cause reduced rates of growth, particularly where drainage is impeded. It is also possible that the phenomenon of limited uptake of potassium due to poor soil aeration, as shown by Lawton (1946) for corn plants, is occurring in this poorly drained area, particularly under excessive rainfall conditions. For the Manapla and Vic- torias upland areas, the negative relation of tonnage with rainfall, before planting, and during the earlier growth stage of the crOp may be attributed to the resulting poor land preparation, erosion of soil and erosion of applied fertilizer. Young sugarcane plants provide very little 29 protection against erosion. During the period of boom to ripening stage, tonnage had a positive relation with rainfall for the Manapla and Victorias upland areas. The effects of moisture in prolong- ing growth of sugarcane, i.e., preventing or delaying ripening, is well recognized (Clements, et a1., 1948; Humbert, 1968; Willey, 1955). Moreover, the relatively shallow soils, particularly in the Manapla area, require more frequent rainfall. This is particularly important at the full grown stage when the transpiration requirement is high. The positive relation of tonnage with decreasing sunlight during this stage of cane growth may be due to an effect similar to that of rainfall. Limited relative humi- dity records indicate that a long sequence of days with relatively lower sunlight is characterized by higher air humidity. Higher temperature levels as well as heavy dews, with more frequent very light showers are also typical. Humbert (1968) notes that light showers and heavy dews stimulate cane growth, since the cane plant is able to absorb moisture through its leaves and sheaths. The higher air humidity also reduces transpiration losses from the plant. Furthermore, it is possible that such a sunlight level is not sufficient to induce ripening, but is suffici- ent to allow further vegetative growth. The positive relation of tonnage with heat units is in agreement with the findings of Burr and associates 30 (1957), Stender (1924) and others cited earlier. The negative relation with low minimum temperatures, expressed in terms of sequences during the growth stage, in the Vic- torias upland area also agrees with these findings. More- over, the negative relation with very low sunlight occurring on successive days agrees with the result obtained by Das (1935). Martin and Eckart (1933) concluded that since photosynthesis is dependent upon sunlight as a source of energy, the role of light is of major importance in supply- ing the plant with the food materials necessary for its normal growth. Diurnal ranges occurring in the last 8 to 12 weeks prior to the end of the harvesting month have a negative relationship to tonnage. High diurnal ranges during the few weeks prior to harvest stimulate the synthesis of su- crose (Garza, 1968). Thus, further vegetative growth is inhibited. In general, there is an increasing yield-time trend for all three areas in the district. However, the annual rate of yield increase for the Manapla area is much lower than those of the Victorias areas. This is primarily due to the continuous addition of marginal productivity hec- tarages in the Manapla area. Agronomic and cultural improve- ment effects are attenuated due to the inclusion of these poorer fields. In the Victorias, however, the annual ton- nage increase with technological changes is higher, since significant additions of production areas has not occurred. 31 2.5 Estimated Rendement Models January-June Harvesting: RENle = 2.625883 - 0.000621 DRl - 0.001124 HU (0.05752) (0.00014) (0.00017) — 0.000382 SUN1 - 0.001582 SL52 (0.00011) (0.00043) - 0.001853 SLs3 - 0.052898 YR + 0.005397 YR2 (0.00063) (0.01103) (0.00103) - 0.002966 TC1- (0.00056) R2 = 0.533 R = 0.730 S.E. = 0.10072 REND2k = 2.457020 - 0.000451 DR - 0.001083 HU (0.05697) (0.000075) 1 (0.00012) 2 — 0.002379 SL82 - 0.002024 SL53 (0.00032) (0.00047) + 0.011206 H82 + 0.007420 H83 — 0.005493 TC2. (0.00129) (0.00183) (0.00072) R2 = 0.692 R = 0.832 S.E. = 0.07815 R8N03k = 2.483599 - 0.007146 RNlm - 0.004714 (RN2m+RN3m) (0.07717) (0.00087) — 0.000999 HU + 0.007141 (TN +TN ) (0.00020) 1m (0.00185) 2” 3m + 0.007046 TX - 0.000844 (HU +HU ) (0.00128) 3m (0.00012) 2m 3m - 0.020639 YR - 0.003451 TC3j (0.00275) (0.00088) R2 = 0.653 R = 0.808 S.E. = 0.08330 July-December Harvesting: RENle = 2.254159 - 0.004370 HU3 + 0.029783 TNl (0.07914) (0.00066) (0.00459) - 0.000721 SUNl + 0.001573 SUN3 - 0.001195 8481 (0.00009) (0.00031) (0.00028) 32 ‘ 0.024702 YR - 0.003653 TCl. (0.00282) (0.00078) R2 = 0.612 R = 0.783 S.E. = 0.10183 R8N02k = 2.337035 + 0.020440 DRY3 — 0.001686 HUl (0.06821) (0.00606) (0.00031) + 0.006921 Tx1 + 0.005143 (TX +TX3) (0.00140) (0.00108) 2 — 0.000541 SUNl + 0.034386 H53 — 0.002946 YR2 (0.00010) (0.00951) (0.00035) - 0.003200 TC2j (0.00061) R2 = 0.552 R = 0.743 5.8. = 0.10070 R8N03k = 1.997109 + 0.013711 DRY3m + 0.000979 Dle (0.06586) (0.00287) (0.00017) - 0.002488 HUlm + 0.026393 TN3 (0.00027) (0.00769) + 0.003983 TX2m - 0.012505 H51m - 0.036763 YR (0.00112) (0.00340) (0.00319) - 0.003540 TC3. (0.00044) R2 = 0.663 R = 0.814 3.8. = 0.08424 where: RENle, RENDZk, REND3k = rendement in piculs of sugar per ton cane for week k for Victorias lowland, Victorias upland and Manapla respectively. DR = sum of daily diurnal ranges. HU = sum of daily heat units. RN = sum of daily rainfalls. DRY = sum of sequences of days exceeding 25 days with rain less than 0.50 inch. SUN TN TX SLS S4S HS YR TC1- TCZj, TC3. 33 sum of daily sunlight hours. sum of sequences of days exceed- ing two days with minimum tempera- tures less than 22.0°C. sum of sequences of days exceeding two days with maximum temperatures greater than 33°C. sum of sequences of days exceeding one day with sunlight less than one hour. sum of sequences of days exceeding three days with sunlight less than four hours. sum of sequences of days exceeding two days with sunlight greater than or equal to 10.0 hours. The subscript l on the weather variables denotes the period begin- ning the 19th week and including the 11th week before harvesting. The subscript 2 denotes the period beginning the 10th week and includ- ing the fourth week before harvest- ing. The subscript 3 denotes the period beginning the third week before harvest and including the harvesting week for Victorias. The subscripts 1m, 2m, 3m indicate the corresponding periods described above, but for the Manapla area. harvesting year with 1960 taken as l, 1961, 2, etc. tonnage for Victorias lowland, Victorias upland and Manapla areas respectively for month j containing week k. The relationships of the various weather factors With rendement as manifested by the sign of their coeffi- Cients in the estimated models are in agreement with the Previously mentioned reports. There is a continuous annual 34 decline in the rendement in all models, except for the Victorias upland model. While these time trends are believed to be representative for the period considered, extensions into the future should be carefully considered. Technolo- gical develOpments will have a significant effect on the time trends. The decline in rendement is possibly due to the increased use of fertilizer, particularly nitrogen. The negative effect of nitrogen fertilizer on rendement of sugar- cane is well demonstrated (King, et a1., 1965; Humbert, 1968). A similar effect has been noted with sugarbeets (Snyder, 1968). There is a consistent negative influence of sun- light on rendement during the 34th to 4lst week period after planting. This is the latter part of the high vegeta- tive growth period and presumably, sufficient sunlight en- courages continuation of vegetative production. High heat units accumulations have a negative influence on the rende- ment. There is an indication that during the low rainfall months of January to June, the amount of rainfall variation has significant influence on rendement in Manapla. However, during the high rainfall months of July to December, rain- fall variation does not affect rendement. Escober's (1961) analysis has shown that during years of low total rainfall (76 and 66 inches), an increase in rainfall registers a corresponding decrease in rendement. Conversely, during years of higher total rainfall (103 and 88 inches), changes 35 in the amount of precipitation did not affect rendement. 2.6 Tests for Autocorrelation and Heteroskedasticity In least squares estimation,homoskedasticity and non-autocorrelation are generally assumed. Since these as- sumptions are of importance in regression problems, they were examined. 2.6.1 Autocorrelation When successive values of the stochastic disturbance term show some degree of dependence, autocorrelation is in- dicated. In ordinary least squares estimation, the presence of autocorrelation signals possible inadequacy of the regres- sion model formulation. Generally, autocorrelation does not destroy unbiasedness and consistency of the estimates of the coefficients, but rather of their variances (Kane, 1968). When positive autocorrelation is present, the variances of the coefficients are generally underestimated leading to more frequent rejection of the null hypothesis of hi equals zero. A well-known test for the existence of autocorrela- tion is the Durbin-Watson test (Durbin and Watson, 1951). A test statistic d for the null hypothesis of residual in- dependence is computed. This statistic is also called the Von Neuman ratio. It is the sum of squares of the first differences of the least squares estimated disturbances, divided by the sum of squares of the estimated disturbances, 36 i.e., W A A Z (U-U.)2 =t=2 t t1 N d 2 t=1 where: d= Durbin-Watson test statistic .A t= the least squares estimator of the disturbance for observation t. If there is no autocorrelation, d is equal to 2. Lower val- ues of d indicate positive correlation, while higher d values indicate negative correlation. Regions of acceptance and re- jection of the null hypothesis are tabulated for comparison with the computed d value (Kane, 1968; Durbin and Watson, 1951). Application of the Durbin-Watson test to the resi- duals of the models developed indicated a rather high degree of positive autocorrelation, particularly in the rendement models (Table 1). Actually, the d values should still be Table l. Durbin-Watson test statistics (d). JanuaryéJune July:December Area Tonnage Rendement Tonnage Rendement Victorias Lowland 1.526 0.468 1.553 0.649 Victorias Upland 0.911 0.536 1.191 0.448 Manapla 1.176 0.489 1.210 0.596 37 slightly lower than those shown in the table, since these were calculated treating the residuals as continuous series. This is not the case as there are separate models for the January to June and July to December periods. Methods are available to correct for the effect of autocorrelation (Durbin, 1960; Theil and Nagar, 1961). How- ever, because of the discontinuity of the data used for each period, these methods are not appropriate. The autocorrela- tions should not produce large error accumulations in a simulation, since alternate models are applied every six months. To reduce the possibility of type I error, a mini- mum significance probability of 0.005 was required. Further- more, each variable in the model was scrutinized to determine if its effect in the model agrees with known influences or is theoretically possible. This procedure, of course, does not yield strong assurances that none of the bi's in the regression is equal to zero. 2.6.2 Heteroskedasticity Heteroskedasticity or non—homogeneity of variance of stochastic disturbances does not result in bias or in- consistency, but rather in inefficient estimates. One commonly used procedure to determine the existence of heteroskedasticity is Bartlett's test (Kane, 1968). The observations of stochastic disturbances are divided into sets of q independent subsamples. An error variance is com- puted for each. Then the hypothesis that these subsamples 38 have been drawn from a single pOpulation is tested. The Bartlett's test statistic is given as the ratio Q/L where: q n. 2 q :E: 1. S ) - E: n -Log 8,2 ___. i i 1 i=1 i=1 q L=1+ 1 2: 1_1 Biq"l) i=1 ni N for which S error variance for each subsample i g i=1 and N Under the assumption that the error term is normally and independently distributed, the ratio Q/L has a chi-square distribution with q-l degrees of freedom (Anderson and Ban- croft, 1952). To test the yield models for homoskedasticity, the total number of observations was grouped into two sub— samples and Bartlett's formula was applied. The Q/L values for the models are given in Table 2. Table 2. Bartlett's test statistics (Q/L). .7 Area January-June July-December Tonnage Rendement Tonnage Rendement Victorias Lowland 0.006 0.006 1.687 0.442 Victorias Upland 0.003 1.466 0.027 0.742 Manapla 0.012 4-549 0.006 0.740 39 All the values are significant at the 0.01 level. There- fore, the hypothesis of homoskedasticity was accepted. 2.7 Model Verification Verification of the models was attempted using the weather records for the period 1968 to 1970. Monthly ton- nage and weekly rendement were estimated. The simulated and actual yields are shown in Figures la, lb, and 1c for the Victorias lowland, Victorias upland and Manapla areas respectively. The total estimated tonnage was in reasonable agree- ment with the actual value, with the exception of the ton- nage in Victorias lowland in the months of June and July. The January to June models for Victorias lowland and upland slightly overestimated the actual rendement, particularly 'in the months of April, May and June. However, the July to December model for Victorias upland underestimated the actual values. The Manapla models overestimated the June and July rendement. There seems to be no unusual weather conditions during the period that could affect the rendement, except for an unusually dry October in 1969. One possible dis- turbance that could have affected the model performance is the effect of residual fertilizer from the previous crop. The 1969 crop was a comparatively low yield due to a long drought that occurred during the growing period. Partic- ularly affected by this drought is the cane tonnage Sugar Production (Picul Sugar/Hectare) Tonnage (Tons Cane/Hectare) Rendement (Picul Sugar/Ton Cane) 160“ 140-N 120.: 100 _I 80.. 40 x Actual ‘ Simulated O \\/\/"./’ 90- / 80- 70- 50-. /\. x— J x-—-*'“‘5 * .———-X /x \\\‘:::§y/,X \\\ //’ 0\ .fr/ 0” N O 1 P‘ H a: (h l l 5...: O L } X/ J Figure la. 'M'ATM'J'JTATSTOTN'Ifr Months Actual and simulated values, Victorias lowland (1970) 41 Ia X Actual Yield q . .343 160 ‘1 ‘ Simulated Yield 23% 140 ... \ ~ 2 _. :\ P34 f/fi;#df ' E.” 120 at " ><"—_'"\ \ . a. 3‘ / X\ 0 x X/ um KKX—X\x/. (OH 100 q» '~0——— ms at) -H 91 80- 447 90 a. \ «IJ \ /\‘ 7‘ - .;7‘x x $03 70 -~m\\‘x czc - 88 a 60. m 6‘ E: 50 a ’53 2.0.. c: (U U a 1.81 ,/’. U 0 '°\ ——-". . ég l.6-/7x\x__—fi-x \ (D (U '0 o3 )9 m \ .‘O '3 102 -I ./ \0/. O .31 .5 1.0- J IF 1IM I A I M IJ I J I A I S I O I N I D‘r Months Figure lb. Actual and simulated values, Victorias upland (1970) 42 ...—f7 IFTM‘A'IvlrJ'JrR's'o'NIDr J d e C t \\ \\\~ 1 a 31 u u o o o x Col A S O O C x x . \ “”9 MAN 1. x. \ x W") I I v /V q 1 HI d d J a a q 1 d - d u d dI 0 0 O 0 0 o 0 0 0 O 0 6 4 2 o 6 4 2 o 8 9 8 7 6 S o o o o o l 1 1* ..l. 2 l l 1 1 Aoumuoom\ummsm asoflmv coauooooum.Hmmsm mmmcsoa Aoumuoom\mcmo mcosv Amado soe\nmmsm Hsowmv ucosoocom Months Figure 1c. Actual and simulated values, Manapla (1970) 43 harvested in the period February to July of 1969. It is highly probable that the poor crop left a significant amount of unused nutrients, particularly nitrogen, in the soil. It is hypothesized that this resulted in the reduc- tion of rendement in spite of favorable weather conditions. 3 . WEATHER SIMULATOR In crop production analyses where the weather fac- tors are normally taken as exogenous system inputs, weather time series can be obtained, either by using historical weather records, or by generating them according to the describing process. The former offers the advantage of providing an exact replication of historical occurrences. It has the serious disadvantage, however, of providing a series of limited length. To obtain a longer time series than available from historical records, a stochastic model was developed for generation of daily weather time series. Actually, models for simulating weather time series were developed for both the Victorias and Manapla areas. 3.1 Victorias Weather Simulator Preliminary analysis of the 20-year weather time- series (see section 2.1.3) indicated a high degree of cross- correlation and autocorrelation (one day lag) among the weather factors. The following development includes con- sideration of these relationships. The stochastic weather models follow the format: 1. The generation of quantity of rainfall is dependent upon the occurrence of rainfall on previous days. 44 45 2. The sunlight hours probability density function (pdf) parameters are dependent upon sunlight and temperatures (lagged one day) and current and one day lagged rainfall occurrences. 3. The maximum temperatures pdf parameters are depend- ent upon temperatures (one day lagged), current sunlight and current and one day lagged rainfall occurrences . 4. The minimum ent upon minimum temperature (one day lagged), cur- temperatures pdf parameters are depend- rent sunlight, current maximum temperature and current and one day lagged rainfall occurrences. Most pdf parameters were dependent upon the previous two days' rainfall occurrences.* no rain sequences for these two periods. The superscript ji is utilized in denoting the rain-no rain states: li——. rain on 21—- rain on 31—. no rain 4i ——— no rain The following pdf‘s the 20-year weather day i and rain on day i—l day i and no rain on day i-l on day i and rain on day i-l on day i and no rain on day i-l were hypothesized and estimated using records. *Rainfall was said to occur if recorded rainfall was greater than 0.01 inch. There are four possible rain- 46 Rainfall Probability Density Function: ji-l f(ri ) o, I 1.- ((l-Pil 1) o, 0.7 A. 1 0.3 Ai' 3.. P.1 1°(l- 1 quantity of rain occurring on no rain state j. ii- pdf of ri 3°- 90:11 1) I j-- P-l 1' 1 pii’l, 1 j1-1 —r. 1 A.)0Mr..€ 1 1. 1-1' 31-1 1 l r . l j'- r,1 L< 0 1 r?i_l 0 1 j. osril‘1< 0.01 3-_ 0.015ril s 0.05 day i given rain- = 1, 2, 3, 4 probability of rain on day i given rain-no rain state ji—l unit impuls estimate of Prob (.015 ri xii—E3 0.01 1 1 estimate of Prob (.Olsri J . 0.015ril'1 e function 5 0.10 ji-l :5 0.10) given that S 0.05) given that 47 j. .3 = estimate of Prob (0.05< ril'l).<_ 0.10) given ji-l that 0.0151:i s 0.10 f4r. = mean rainfall given rainfall greater than 0.10 on day i. Sunlight Probability Density Function: j. -. j. 2 f(S.l) = N h , («5.1) 1 1 where: ji . . . . S, = sunlight hours on day 1 for rain-no rain 1 state j. 1 j. j. f(S 1) = pdf of 3,1 1 1 N(M,g?) = Normal (Gaussian) probability density function with mean/(4, and varianceo’ j. J. j- ji-l j- j°—1 1 _ 1 , .1 , 1 X451 — asil + bsi i-l + csl ri j. j._ j. j-_ + ds 1 ' X_1 1 + es.l ' Y.1 1 i 1-1 1 1‘1 ji-l . 1'1 = maximum temperature on day 1-1 Y?i;1 = minimum temperature on day i-l 1- j. aSill b3 1! i ji . . dsi , = estimated parameters reflecting auto and cross-correlations. 48 Maximum Temperature Probability Density Function: ji ji ji 2 f(xi ) - NMXi I (O'Xi) where: Xi1 = maximum temperature on day i for rain-no rain state ji 3'- j. X l = .1 f( i ) pdf of X1 3- '. j. -. j. j. -. .. 44X.l = ax?1 + bx,1°x?l"l + cx.1-S.l + dx?lor?1-l l l l 1‘]. 1 l l 1 ji ji-1 ji ji-1 J 1 3 i - . . cx, , dx.1,= estimated parameters reflecting auto and 1 cross-correlations. J 1 Minimum Temperature Probability Density Function: j. j- j- 2 1 _ 1 1 f(Yi ) - N ,uyi I ( in ) where: Y.i = minimum temperature on day i for rain-no 1 rain state ji 3- j. 1 = 1 f(Yi ) pdf of Y1 ji ji j- j. j' ji ji j. 1 = + 1. 1‘1 + .1“ . +d ° .1'* ’uyi ayi byi Yi-l Cyl Sl yi r1 31 + eyiix 49 j. 3. any}, Ivy.1 1 1 I 51 cy_i, dyi ,= estimated parameters reflecting auto and 1 cross-correlations. eyji Monthly or bimonthly average estimates of all parameters are given in Appendix C. Sufficient data were not available to estimate daily values of the pdf parameters. The weather data were divided into bimonthly groups and bimonthly parameter averages were estimated by multiple regression utilizing the least squares technique. Because Ai and/“xi were not dependent upon previous rain-no rain states, it was possible to estimate monthly averages of them. A preliminary weather simulation utilized only the rain-no rain state on the previous day (first-order Markov assumption). The results indicated some inadequacies in capturing the persistency of rain-no rain sequences. Sub- stantial improvement was obtained when a second-order Markov process was assumed. For example, in the two-month period May-June, the probability of rain on day i given rain on day i-l (first-order Markov process) was estimated to be 0.588. However, in the second-order process, the parameter P11“; (the probability of rain on day 1 given rain on days i-l and i-2) was estimated to be 0.652. In this same two- month period, the probability of no rain on day i given no rain on day i-l was estimated to be 0.552. The estimate of 50 4-_ P11 1 was 0.600. While second-order estimates yielded sub- stantial improvement, third-order estimates were not con- sidered because of the limited amount of data. Several alternative stochastic models of rainfall quantity have been proposed (Jones, et a1., 1969; Sorensen, 1967). The more common assumption is that rainfall quantity (given rain) is distributed exponentially. A chi-square goodness-of-fit test (Larson, 1968) was applied to see if the rainfall data were distributed exponentially. The results of this test are given in Table 3. Since the test indicates that the hypothesis should be accepted in none of the twelve months, this hypothesis was rejected. The hypothesis that rainfall greater than 0.10 inch is dis- tributed exponentially was accepted in ten of the twelve months. Therefore an exponential distribution was assumed only for rainfall greater than 0.1 inches. The probability of rainfall less than or equal to 0.10 was determined for each month. Histograms suggested that rainfall was dis- tributed uniformly in the ranges (0.01 - 0.05) and (0.05 ~ 0.10) inch. 3.2 Manapla Weather Simulator There is a similarity between the weather condi- tions, particularly rainfall occurrences, in the Victorias and Manapla areas. An attempt was made to maintain Spatial correlation between the simulated weather in these two areas. This was done by relating the weather generation .b.H~ ma Eowmwum mo mmmummp mafia QUflB coausnfluumflp mumsqmlflno was mo maflucmonmm numm one « 51 hm.mm mm.~m mm.m mo.m mn.m on.ma oa.o cflmm mm.mm hm.mm mm.mm ma.mm mv.mm mo.ooa Ho.o :flmm nonsmowo quEm>oz Honouoo quEmumom umsms< NHSh mm.mH mo.om cv.ma mm.HH mh.ma mo.oa oa.o cflmm mm.m> ma.mm HH.MHH mm.om mv.vaa mm.mm Ho.o seam mash Nmz kumm comm: Nmmsunmm NHMSGMh mango: . Hammcflmm «.ummu uflMIHOImmmspoom mumsvmlfino map “Om mowumwumum .m GHQMB 52 process in Manapla with the rain-no rain states in Victorias. The format of the stochastic weather models for this area is the same as that in the Victorias with the following exceptions: l. The generation of quantity of rainfall is dependent only upon the current occurrence of rainfall in the Victorias. 2. The generation of sunlight and temperatures depends upon current rainfall occurrence in the Victorias as well as current and one day lagged occurrences in Manapla. The superscript ki is utilized in denoting the rain-no rain states given in the second exception: li—p rain on day i and rain on day i-l in Manapla and rain in Victorias on day i 2 —4- rain on day i and no rain on day i-l in Manapla and rain in Victorias on day i 31——p no rain on day i and rain on day i-l in Manapla and rain in Victorias on day i 4i-—c-no rain on day i and no rain on day i-l in Manapla and rain in Victorias on day i 5 6 7i,Bi-—-the same as l-, 21 , 3., 4i respecti- i ' O I O vely, but no rain in Victorias. i! The following probability density functions were hypothesi- zed and estimated for the Manapla weather variables: 53 Rain Probability Density Function: If rain occurred in Victorias on day i 0 r.<:0 1 f (1 - Pi) - €(ri) 00.10 \ 1 Mi 1 f o r. < o 1 f(ri) = J (l—Qi) ' SYri) 0<:ri<:0.01 1 'ri . Qi 'wi '€ wi rig 0.01 where: ri = quantity of rainfall on day i in Manapla f(ri) = pdf of ri £3.01, estimated parameters. [Li'wdi 54 Sunlight Probability Density Function: ki ki ki 2 where: 1‘1 S. = sunlight hours in Manapla on day i for rain- 1 . no ra1n state k-, k- = l, 2, ... 8 1 1 ki ki f(S. ) = pdf of S. 1 1 k. k. k. -_ k- k- k. lfls,l = as.1 + bs.l - S 111 + cs.1'r. + ds.J‘°X.]"'l 1 1 1 1 1 l 1 1-1 . k. + es.1'Y,l"l 1 1-1 k1-1 i-l X. ,Y. = respectively maximum and minimum temperatures 1-1 1-1 . . 1n Manapla on day 1-1 k k. asoi, bs 1, 1 k. k. cs.1, ds.l, = estimated parameters reflecting auto and l * 1 cross correlations. k. es,1 1 Maximum Temperature Probability Density Function: k. k. k. 2 £(x.1)= u(,qx_1, (dx,1) 1 1 1 where: X.i = maximum temperature in Manapla on day i for l rain-no rain state ki k0 k' 1 1 f(Xi ) — pdf of Xi k k. k _ k k- k /ux 1 = ax 1 + bx 1'X l l + ex 1.8 l + dx 1°r 1 1-1 1 k k. + ex,l°Y.l-l 1 1-1 k k. ax.i, bx.l, 1 1 1‘1 k1 cxi , dxi , = estimated parameters reflecting auto and cross-correlations. k. ex,1 1 Minimum Temperature Probability Density Function: ki ki ki 2 f y. = N , ( 1 ) Myi (qyi ) where: k1 Y. = minimum temperature in Manapla on day i for l rain-no rain state ki ki ki f(Yi )= pdf of Yi ki k. ki ki 1 ki k, ki = 1 + - . ‘ + ° .1 + . . xLyi ayi byi Y1-l cyi 51 dy1 ri - k l i + X eyi i ki ki ayi I byi I ki k. cyi , dy,1, = estimated parameters reflecting auto and 1 cross-correlations. k. 1 eyi The formulation of the weather models for Manapla follows the lines similar to the ideas expressed regarding the Victorias weather models. The parameters of the pdf's were estimated by multiple regression using the least 56 squares criterion. Pi' Qi' Bi',ui andIJi were estimated on a monthly basis. All other parameters were estimated bi- monthly. 3.3 Stochastic Weather Simulation 3.3.1 Methodology Daily values of all weather model pdf parameters were obtained via straight-line interpolation in time on the monthly or bimonthly estimates. Variates of the daily weather variables were generated recursively in the order in which the pdf's were formulated, i.e., rainfall in Vic- torias on day i is the first, sunlight hours in Victorias, second, ... concluding a one day simulation with the gen- eration of minimum temperature in Manapla on day i. The necessary initial conditions were selected randomly from the historical records. The necessary random numbers are transforms of uni- form (0, 1) random numbers generated via a multiplicative congruential technique (Hillier and Lieberman, 1968). Ex- ponentially distributed random variables (3) were generated by the inverse transform technique (Naylor, et a1., 1968): 3 = —E(%)Log RN where: RN = uniform (0, 1) random number E(B) = expected value of;.. The generation of normally distributed random variates (wk) 57 with mean.Ac and standard deviation cr are generated via the application of the central limit theorem (Naylor, et a1., 1968): 12 “r =M+ 04 2 RN, ~6) i=1 1 where: RN = uniform (0, 1) random number. Figures 2 and 3 depict the flow diagrams for genera- tion of respectively, Victorias and Manapla weather. 3.3.2 Simulator Validation To determine how well the generated weather factors compare with the actual data, a BOO-year simulation run was made. The means and standard deviations of monthly and daily rainfall, daily sunlight and maximum and minimum tem- peratures were calculated. (Figures 4a and 4b give the simulated and actual means of rainfall, sunlight and maximum and minimum temperatures for the Victorias. The means for Manapla are given in Figures 5a and 5b.) In comparing the simulated means of the four weather factors with the corresponding means based on the actual record, slight descrepancies in the mean are noted. Some months have lower simulated values while others have higher. These discrepancies are not entirely unexpected. The pri- mary reason is the nature of the interpolation method used. Using bimonthly pdf parameters, interpolation between two 58 Ftart Yes r?i'2 0 Yes 1-1 No No 21- Y No R P. es -5 1 Yes No .0 4 Generate sun, R P i- max. & min. i temp. 31 = 4 Yes L etermine quan- tity of rain— H———J fall Generate sun, max. & min. temp. 3i = 2 Determine i"quantity of wt rainfall Generate sun , a max. & min. 1 temp. ii = 1 Generate sun, x. m' . ¥._ma & 1n temp. 31 = 3 Figure 2. Flow chart for Victorias daily weather simulator Rain Yes Yes Yes Victorias Day i No \0 0 Generate Ye: Yes sun, max. & - r. 0 . 1—1 min. temp No No Generate sun, max. & 0 No min. temp ri-l T Generate su11 max. & min. Yes temp. k1 = 4 Determine rainfall quan tity imean = Generate su i ax. & min temp= enerate sun, max. & in. temp , ki = 1 Generate sun, max. min. temp. a o ki = 7 - . 474 Generate eterm1ne ra1 No sun, max. uantity & min.temp ean = i k. = 6 1 es Generate sun, max. & min. temp ki = 5 Figure 3. Flow chart for Manapla da ily weather simulator 0.64 0.5‘ 0.3‘ Daily Mean Rain Given Rain- Inches 0.2. 60 /‘ X Actual Value 0 Simulated X / i / 14" 12' 10‘ Monthly Mean Rainfall-Inches m 1 Figure 4a. F M A M J J A Months Actual and simulated values, Victorias Temperature °C Sunlight HOUI’S Minimum Maximum 61 x Actual 0 4 . ‘ 0 Simulated O 32 - xfé" 31 ~ /. \. 3o - \. .. . \ 28 ‘ F 1 r 1 T JFFIMIAIMIJ J'A's o N D Months Figure 4b. Actual and simulated values, Victorias Daily Mean Rain Given Rain-Inches Monthly Mean Rainfall-Inches 006 -‘ 0.5-4 0.2 ‘ 62 . ::::i x Actual Value \ 0 Simulated 14'- 12" Figure I T I rfi ASOND J'F'M‘A'M J J Months 5a. Actual and simulated values, Manapla as o l Sunlight Hours U1 2: 1 b O l 63 v‘ x Actual Value o\\\* ' Simulated / \' ° x ’/’3 '\\\ .;74. F" 7‘ .\'/"X \ ...—— x— 24 q 234%.?! E s E -a c -a z 0 ° 22'# m u s 4) m 3‘. 9‘ 32“ 5 B 31.- .5 30- 6 z 29'- 28-— / fi>fi\‘\. 7/ \"‘§ I I JTF'MIATM'J'JjAIS o'N'D Months Figure 5b. Actual and simulated values, Manapla 64 bimonthly values would expectedly yield some errors. High actual values are underestimated and low actual values are overestimated. In spite of these drawbacks, interpolation was thought necessary since only by doing so could the dynamic behavior of the weather system be adequately simulated. It was considered more important to capture the system dynamics than to zero in on the mean values of the weather factors. To use the weather models for crop production simulation, it is necessary to remedy the discrepancies in the mean values. This was accomplished by adding the differences between the simulated means and the actual means, for the particular month, to the generated daily weather factors. These added differences were subtracted from the lagged values of these factors for the following daily simulation cycle in order to preserve the dynamics of the weather system. Utilizing this procedure, the BOO—year simulation was repeated. Weekly values of each of the 10 weather vari- ables utilized in the yield models were calcuated. Their means, standard deviations and maximum values are given in Table 4. The corresponding values based on the actual weather data are given in Table 5. 65 .m.~ coauomm an nmcflmmn ma . mm.HH oo.HH oo.om oo.sm~ oo.ooa oo.m mm.~m mo.m¢ oo.mm mv.o~ msam> .xmz mo.~ ms.o mo.~ mm.¢a mo.o mm.o om.oa om.m es.¢ ma.m .>mo .oum mm.m ma.o ov. mmm.~ mv.H oa.o H~.om om.Hm mo.H mm.a cam: mmmmmmm m¢.HH oo.HH oo.s~ oo.moa oo.mm oo.m oo.Hm oo.~v oo.mn am.ma msam> .xmz wH.~ mo.H m~.~ mm.HH Hm.m mm.o nm.HH Hm.» ma.o ma.~ .>mo .num sm.m m~.o vm.o m~.~ mm.H mm.o mo.¢m mm.ma mm.a nm.a cams mmfluouofl> zam mm xa mvm mum 29 mo on who chum «.mmHQMHHm> Hmcummz maxmmB Hmsuod .m wands -.~H oo.oa oo.~H co.a~a oo.am oo.ma mm.maa Hm.mv oo.mm mm.mH moam> .xmz sm.a mm.o H5.o mo.m Hm.~ mm.o m¢.m mm.m ov.~ mm.a .>mo .6um mm.m oa.o mH.o mo.H mm.o oa.o on.ms mo.H~ om.o nm.a cams Mamas: sq.ma oo.~a oo.- oo.m- oo.¢m oo.m~ sm.am sn.mv oo.mm o¢.mH msam> .xmz Ho.~ mn.o nm.H so.s mo.m mm.o mo.aa Hm.> mn.~ sm.a .>mo .cum mm.m ma.o m¢.o hm.H m>.o om. Ha.qm oa.o~ ov.o hm.a cams mnemouow> zom mm xe mew mam 29 mo as use cflmm «.mmanmflumb Hmnummz mdxmms meMHSEwm . v wanna 4. SIMULATION STUDIES OF ALTERNATIVE CROPPING CYCLES There are several important factors that may be included in a simulation analysis of alternative cropping cycles. They include yield, sugar price, market demand, inventory policy, production lag, labor supply and field Operational requirement. In this preliminary simulation study, only yield and sugar price were considered. 4.1 Yield Simulation Two simulations were made; one used the 20-year‘ weather records and the other used the stochastic weather simulator to obtain the weather variables used in the yield models. Using historical weather records, only 18 produc- tion years can be considered. Utilizing the stochastic weather simulator, a 300-production year simulation was made. The yield models develOped for each area were used in the studies. To account for the uneXplained variations in each of the tonnage and rendement models, random varia- tion was generated and introduced. The random variation was computed as the product of the standard error of esti- mate (of the tonnage or rendement model) and a normal 66 67 variate of mean zero and variance one. New random varia- tions were computed and added to each simulated tonnage and rendement value. Monthly tonnages and weekly rendements were compu- ted. The weekly rendements were then converted to monthly values and the monthly sugar production was determined. Because the normal shutdown period varies from one to two months, summary statistics were computed monthly and in pairs of months (e.g., June—July, July-August) for each area. 4.2 Revenue Simulation There are available monthly data on domestic and export prices from 1953 to 1969. Of main interest in simulation studies of alternative cropping cycles is the seasonal variation in sugar prices, if it exists. An at- tempt was made to develop a model embodying price season- ality. However, neither domestic nor export prices exhi- bited (in a linear or quadratic sense), to an acceptable level of significance (0.005), any price seasonality. Therefore, it was decided for this preliminary analysis to use the monthly domestic and export prices for the five- year period (1965 to 1969), computing the gross revenue for each year. Monthly domestic, export and total revenues were calculated for each area. In calculating domestic and export revenues, it was assumed that 67 per cent of the sugar produced is exported and 33 per cent sold in the 68 domestic market. Means and standard deviations of domestic, export and total gross revenues for each month and pair of months were calculated for the three areas 0 4.3 Results and Discussion 4.3.1 Simulated Yields Monthly mean simulated yields based on the weather records and generated weather values are plotted in Fig- ures 6a, 6b and 6c for Victorias lowland, Victorias upland and Manapla respectively. Figures 7a, 7b and 7c give corresponding values for each pair of months for the three areas. The monthly yields for the two simulations are in close agreement with each other. There are however, slight differences in the standard deviations obtained from each simulation. The variation among months of the variance of yield was less using stochastically generated weather than with the historical weather records. The standard deviation for tonnage, rendement, and sugar production for the three areas are tabulated in Appendix D. Based on sugar production, it is obvious that the months of June to December comprise the possible region for cessation of harvesting and planting operations. Since Sugar Production (Picul Sugar/Hectare) Tonnage (Tons Cane/Hectare) Rendement (Picul Sugar/Ton Cane) Figure 6a. 160- 1404 120‘ 100- 801 69 0 Used the weather records 0 Used simulated weather 90 1 80 q 70 a 60 - 50 J Jl FTMjA'MjJrJ‘ATSIO'NrDI Months Monthly mean yield, Victorias lowland Sugar Production Tonnage Rendement 70 F‘ 0 Used the weather records m 8160 - a Used simulated weather ‘6 0 33140 -‘ 0 \. 0““ 5§§3 H ¢::3:7 §‘12° "/0 \ c/ m _" 4;:- H100 J °\9__q ....o 5’: .E.80 — g 90.. 13 U 30 42:5 ~<9::=O==.Q -——0u G’sfifir“ :1: .‘.\ E 70 \°==9/ ‘97 Q '— m U 60 _ m c 8 so .. 'g2.0- {1.6 -/ :3 / 011.4 -—I O / C?) O\VM=O=O Hl.2 .— s o are 1 . Figure 6b. J'F'MTATM'J'J‘AISIOINIW Months Monthly mean yield, Victorias upland Sugar Production (Picul Sugar/Ton Cane Tonnage (Tons Cane/Hectare) Rendement (Picul Sugar/Ton Cane) 160‘ 140- 120- 100 804 +6>—0==°“o# 71 I Used the weather records 0 Used simulated weather ——45§§§GK‘§ ==flaczfi2=v °W°fo 90‘ 801 70~ 60« SO- ..vogjipzb Nw§how._. JTFIMFA'M'JrJIAIS'OTNrDT Months Figure 6c. Monthly mean yield, Manapla ...: O: O ...: b O 120‘ Sugar Production H O O 1 (Picul Sugar/Hectare) 80- 72 0 Used the weather records 0 Used simulated weather 90~ 804 70d Tonnage (Tons Cane/Hectare) 60% 504 N H on p.» m H Rendement b (Picul Sugar/Ton Cane) H Figure 7a. TIT—Tr F-M' M~N A—M M~JJ=J 1J--1UA:«SIS--OTO--‘-N'N-«D‘ D-UF Months in pairs Mean yield for pairs of months, Victorias lowland Tonnage (Ton Cane/Hectare) Rendement (Picul Sugar/Ton Cane) Sugar Production (Picul Sugar/Hectare) 73 Used the weather records 160-1 0 Used simulated weather 140- 1201 O 100.. \o 80- 9o- 80.. 70.1 60. 50+ J-F 'F—M 'M—A lA-MTM-J IJ-JIJ-Al A-s‘ s--ol o—NTN-DTD—J' Months in pairs Figure 7b. Mean yield for pairs of months, Victorias upland Sugar Production (Picul Sugar/Hectare) Tonnage (Tons Cane/Hectare) Rendement (Picul Sugar/Ton Cane) 160‘ 74 0 Used the weather records (3 Used simulated weather 90— 80-1 701 60- J-F'F-MIM-AT A-M'M—J‘J—J'J-ATA—s' s-o' o—fi N-d D—J‘ Months in pairs Figure 7c. Mean yield for pairs of months, Manapla 75 the present shutdown period in the district is during the months of November and December, comparison of yields and revenue will use a base of the values for these months. In the following discussion, if the difference in the means between two periods is significant at the five per cent level, the difference will be referred to simply as signi- ficant. If the mean difference is significant at the one per cent level, it will be referred to as highly signifi- cant. Determination of the significance between the means assumes that the differences are normally distributed (Spiegel, 1961). In the Victorias lowland area, lowest sugar produc- tion occurs during the July—August period. The mean dif- ference of 35 piculs sugar between this and the November- December period is highly significant (Figure 7a). Even the mean yield differences between the August-September and November*December period are highly significant, as are the differences between September—October or June-July and the November-December periods. The low sugar yield during the June to October period is due primarily to low rendement (Figure 6a). Sugar yield in the Victorias upland area is lowest during the August-September period (Figure 7b). The mean yield difference of 10.0 piculs between this and the November-December period is significant. The low sugar yield is attributed both to low tonnage and low rendement during this period. As in the lowland area, tonnage is 76 'lowest during the May—June period. Although the differ- ences in tonnage for the area are not striking, the dr0p in rendement during the August to December period (Figure 6b) brought the yield down. In the Manapla area, the lowest sugar production occurs during the July-August and August-September periods with a difference of 13.02 and 13.0 piculs respectively compared with the November-December yield (Figure 7c). Again, these differences are highly significant. Here the low rendement occurring in August and September is respon- sible for low sugar yield. Tonnage during this period is relatively high. 4.3.2 Simulated Revenues Since the simulated sugar yields using actual and generated weather variables are in close agreement with each other, only the gross revenues obtained with the lat- ter are plotted in Figures 8a, 8b and 8c. Total gross revenue for the Victorias lowland are lowest during the months of July and August (Figure 8a) for all five sugar price series used. The differences in revenue between the July-August and November-December per- iods for the five sugar price series are all highly sig- nificant. In the Victorias upland area, lowest revenue occurs during the months of August and September (Figure 8b). The differences in mean revenue of these periods as compared Total Gross Revenue (Pesos Per Hectare) 77 Months in pairs ‘-F -M -A -M -JlJ-JlJ-AlA-SIS-OIO—N|N-DLD—JJ n/ 5000 -/ \ 4000- 30004 . \\./,/ / .\\\q::::EE;;¢§5;//////// 2000— \'\./ / //’”\\\_ 5000— /§\ / I \ / 2000.1 \ /- \. J r1a'M'A'i‘JTJ'A's'orN‘13‘ Months Figure 8a. Simulated reVenue for five annual price series (1965—1969), Victorias lowland Total Gross Revenue (Pesos Per Hectare) 5000‘ 4006‘ 20004 5000‘ Months in pairs ~F .F-M 1M-A lA-M .M-J .J-J lJcAlAcSi s—OLo—N; anLn—Jl /. /. ___....__ ././ °/ \ \ ./,/:: .-—-. \ .a”fl.’v”-—‘”:::§ ’,/”” // ./—o\:§°\ ./ //’//””fl' “\»:§:§f\~M___u,/” .///”:: " \\ /.’/,.—:/ ~———__.__1..§5-—3.§/J‘¢' J'F'M' A' M' J‘J7A'STO'NTDT Months Figure 8b. Simulated revenue for five annual price series (1965-1969), Victorias upland Total Gross Revenue (Pesos Per Hectare) 79 Months in pairs 'J-F F-M y-AJA-M lM-JiJ-J lJ-AlA-SlS-OlO-NlN-Dl D-JL 50004 4000‘ 30001-"’” 2000‘ 5000‘ 40004 3000-. 20004 Figure 8c. J'F‘M'ATM'J'J’A's'o‘N'DT Months Simulated revenue for five annual price series (1965—1969), Manapla 80 with the November—December period are all significant. The July-August and August-September periods have the lowest revenue in the Manapla area. The difference of the mean of these periods compared with the November-December mean are all highly significant for all five sugar price series. On a monthly basis, the lowest revenue occurs in Manapla during the month of August (Figure 8c). 5. SUMMARY AND CONCLUSIONS Models for estimating sugarcane yields for the three areas of the Victorias milling district were devel- oped using multiple regression with least-squares criterion. Separate models were formulated for monthly tonnage and weekly rendement for the periods January to June and July to December. In the models, the climatic influence tends to be manifested in sequences of occurrence rather than the absolute value of the weather factors. The various area models indicate different controlling weather factors on growth and yield. Model verification using 1970 production data yielded a close agreement between the estimated and actual tonnage. However, there were slight discrepancies between the estimated and actual rendement. Possibly, this can be attributed to the effect of residual fertilizer from pre- vious crops. Models for generating weather factors were developed for the two areas of the district. Determination of rain- fall occurrence in the Victorias area was by a Monte Carlo technique using second-order Markov probabilities. The amount of rainfall was determined from a probability den- _ sity function derived for the area. Sunlight and 81 82 maximum and minimum temperatures were generated from re- gression equations in lagged values of the variables. The choice of the regression equation to use on a given day depends on the first-order rain-no rain state in the area. For the second area, rainfall occurrence was also deter- mined by the Monte Carlo technique. Here, the probability of rain depends only on the rain-no rain state in the Vic- torias area for the same day. The models for sunlight and temperatures consist of regression equations with lagged values of the variables. The choice of the equation to use depends on the first-order rain-no rain state in this area and the present rain-no rain state in the Victorias. Two simulations were made to obtain preliminary indications of alternative cropping cycles. One simula- tion used the historical weather records and another used stochastically generated weather factors. This also pro- vided a test of the performance of the stochastic weather generator in production simulation applications. Summary statistics on yields and revenues for each month and pair of months were calculated. Five annual sets of monthly prices were used in calculating revenue. The calculated mean yields and revenues with the two simulations were in close agreement with each other. There are strong indications, based on yield and revenue, that the November—December period is not the best time to cease operations in the district. Conclusions derived from these results included: 83 The tonnage and rendement models developed are adequate for production simulation applications. The weather simulator is adequate for production simulation applications. There is an annual time trend of increasing tonnage and decreasing rendement in the three areas of the district. 6 . RECOMMENDAT IONS The results of this project suggest the need for further work in simulation analysis of alternative cropping cycles for the Victorias milling district. Future studies can utilize the yield and stochastic weather models devel- oped here. Additional factors that should be considered in future simulation studies are: l. Shift in market orientation from export towards domestic markets. 2. Field operational requirements based on tracta- bility. 3. Farm and factory labor supplies 4. Marketing lag times 5. Inventory costs. The trend of decreasing rendement in the district suggests the need to appraise current cultural practices. Particular attention should be given to fertility programs in the area. 84 REFERENCES REFERENCES Anderson, R. L. and T. A. Bancroft 1952 Statistical Theory in Research. McGraw-Hill, New York, 399 pp. Anonymous 1957 Climate and cane growth. Hawaiian Planters' Record, Vol. 57, No. 1, p. 58. Borden, R. J. 1936 Cane growth studies--The dominating effect of climate. Hawaiian Planters' Record, Vol. 40, pp. 143-156. Burr, G. O., C. E. Hartt, H. W. Brodie, T. Tanimoto, H. P. 1957 Kortschak, D. TakaHashi, F. M. Ashton, and R. E. Coleman The sugarcane plant. Annual Review of Plant Physiology, Vol. 8, pp. 275-308. Clements, H. F. 1964 Interaction of factors affecting yield. Annual Review of Plant Physiology, Vol. 15, pp. 409- 442. Clements, H. F., G. Shigeura and E. K. Akamine 1948 Ripening sugarcane. Hawaii Agr. Expt. Sta. Rept., 1946-48, pp. 120-124. Clements, H. F. and T. Kubota 1942 Internal moisture relations of sugarcane-- The selection of a moisture index. Hawaiian Planters' Record, Vol. 46, p. 17. Das, U. K. 1935 A pot experiment with cane grown in the same soil but under different climatic conditions. Hawaiian Planters' Record, Vol. 39, pp. 26-29. Draper, N. R. and H. Smith 1966 Applied Regression Analysis. John Wiley & Sons, Inc., New York, 407 pp. Durbin , J . 1960 Estimation of parameters in time series regres- sion models. Journal of the Royal Statistical Society, Vol. 22, series B, pp. 139-153. 85 86 Durbin, J. and G. S. Watson 1951 Testing for serial correlation in least squares regression. Biometrika, Vol. 38, pp. 159-177. Escober, T. R. 1961 The influence of weather factors on rendement and tonnage of sugarcane. Proc. 9th Ann. Con- vention, Philippine Sugarcane Technologist, pp. 175-181. Garza, A. M. 1968 Aspects of climate on sugarcane. Proc. 13th Cong. Intern. Soc. Sug. Tech., Taiwan, pp. 882-836. Hartt, C. E. 1940 The synthesis of sucrose by excised blades of sugarcane. Time and temperature. Hawaiian Planters' Record, Vol. 44, pp. 89-116. Hillier, F. S. and G. L. Lieberman 1968 Introduction of Operations Research. Holden Day, Inc., San Francisco, 639 pp. Holtman, J. B., L. K. Pickett, D. L. Armstrong and L. J. Connor 1970 Modeling a corn production system--A new ap- proach. ASAE Paper 70-125. Humbert, R. P. 1968. The Growing of Sugarcane. Elsevier Publishing Company, New York, 779 pp. Johnson, C. A. 1966 Sucrose per cent cane and diurnal temperature range. Intern. Sugar Journal, Vol. 68, pp. 808-809. Jones, J. W., R. F. Colwick and E. D. Threadgill 1970 A simulated environmental model of temperature, rainfall, evaporation and soil moisture. ASAE Paper 70-404. Kmenta. J- 1971 Elements of Econometrics. MacMillan, New York, 750 pp. lKane, E. J. 1968 Economic Statistics and Econometrics. Harper & Row Publishers, New York, 437 pp. Ring, N. J. 1965 Manual of Cane Growing. Elsevier Publishing Co., New York, 375 pp. 87 Larson, H. J. 1969 Introduction to Probability Theory and Statis- tical Inference. John Wiley & Sons, New York, 387 pp. Lawton, K. 1946 The influence of soil aeration on the growth and absorption of nutrients by corn roots. Soil Science Soc. Amer. Proc. Vol. 10, pp. 263-268. Locsin, C. L. and F. T. Tabayoyong 1953 Soils of the Victorias Milling District--A Handbook for the Planters of the District. Victorias Milling Company, Victorias, Negros Occidental, Philippines, 86 pp. Manetsch, T. J. 1971 Associate Professor of Electrical Engineering and Systems Science, Michigan State University, East Lansing, Michigan. Unpublished class notes prepared for Systems Sci. 812. Mangelsdorf, A. J. 1950 Sugarcane--as seen from Hawaii. Econ. Botany, Vol. 4, pp. 150-176. Martin, J. P. and R. C. Eckart 1933 The effect of various intensities of light on the growth of the H109 variety of sugarcane. Hawaiian Planters' Records, Vol. 37, pp. 53-56. Morey, R. V., R. M. Peart and D. L. Deason 1969 A corn growth harvesting and handling simula— tor. ASAE Paper 69-674. Naylor, T. H., J. L. Balintfy, D. S. Burdick and K. Chu 1968 Computer Simulat1on Techniques. John Wiley & Sons, New York, 352 pp. Panje, R. R., B. Singh and S. K. Saxena 1968 Relationship of medium low atmospheric tempera- ture with the juice quality of standing crops of sugarcane. Proc. 13th Cong. Intern. Soc. Sug. Tech., Taiwan, pp. 859-866. Rafter, M. E. and W. L. Ruble 1969 Stepwise Addition of Variables. Agr. Expt. Sta. Stat. Series Description No. 9. Michigan State University, East Lansing, Michigan. 88 Singh, B. N. and K. M. Lal 1935 Limitations of Blackman's law of limiting fac- tor and Harder's concept of relative minimum as applied to photosynthesis. Plant Physiol- ogy, Vol. 10, pp. 245-268. Snyder, F. W. 1968 How do your sugar beets grow? Sugar Beet Journal, Vol. 31, No. 2, pp. 4-5. Sorensen, E. E. 1967 Regional rain model with space-and time cor- related structure. Management Science, Vol. 14, No. 3, pp. 239-249. Sowell, R. S., T. Liang and D. A. Link 1967 Simulation of expected crop returns. ASAE Paper 67-614. Spiegel, M. R. 1961 Theory and Problems of Statistics. McGraw-Hill, New York, 359 pp. Stapleton, H. N. 1968 Crop production system simulation. ASAE Paper 67-562. Stender, H. K. 1924 Some sugarcane growth measurements. Hawaiian Planters' Record, Vol. 28, pp. 472-495. Sun, V. G. and N. P. Chow 1949 The effect of climatic factors on the yield of cane in Tainan, Taiwan. Taiwan Sug. Expt. Sta. Rept. No. 4, Part 2, pp. 1-40. Theil, H. and A. L. Nagar 1961 Testing the independence of regression distur- bances. Journal Amer. Stat. Assoc., Vol. 56, pp. 793-806. V M C 1967 Unpublished report to the Victorias Milling Company, Inc. Victorias, Negros Occidental, Philippines. V M C 1968 Analysis of the shift in the mill shutdown period. Unpublished report to the Victorias Milling Company, Inc. 89 V M C 1968 Estimated loss in sugar due to typhoons Seniang and Reming. Unpublished report to the Victorias Milling Co., Inc. Willey, K. S. 1955 Ripening the crop. Proc. 14th Meeting Hawaiian Sugarcane Technologist, pp. 12-14. APPENDIX A ACTUAL AND SIMULATED WEATHER VALUES 90 mm.~ mo.a mw.a nm.o Hm.m av.m mm.mm mm.m~ mm.o mo.HH .omo om.~ mm.o ww.H mn.o mm.v mm.m om.m~ om.m~ no.0 ¢¢.ma .>oz Hm.m om.o mm.a m>.o mm.m vm.m mm.mm mm.om mm.o mv.~a poo vm.m mm.o Hm.H Hm.o ma.m h~.m hm.N~ mH.Hm om.o mm.m .ummm Hm.m vm.o mm.H Hw.o hm.m mo.m mo.m~ mm.Hm om.o mm.m .m:< mn.m mm.o ~m.a mm.o H¢.m no.m om.m~ om.am mm.o om.m .Hsb mm.m mm.o mn.H mm.o va.m mm.m mm.m~ mo.~m hv.o vm.n mash mm.~ hm.o mh.a ww.o mH.m mm.m mm.mm mo.mm mv.o ~v.n hm: Hm.~ mo.a mh.H mm.o mm.m oa.h mm.mm vm.am mm.o mH.m .HQfl mm.~ mH.H mm.H mv.o wo.m NH.h mm.m~ m~.om mm.o oa.v .Hmz mo.m 5H.H mv.a vm.o hv.a oa.m mo.mm Ha.mm hm.o mo.¢ .nmm wo.m ma.H ov.a mv.o mm.m mm.m mw.mm mm.mm mm.o mm.w .smn unmfis .cflz .xmz .Naamo msgucoz snags .cnz .xmz Magma wasucoz Icsm musumummfima cflmm Icsm OHDDmHmmEmB COHu8w>mo pummcmum smmz.i moauou0H>llmmsHm> Hmnumm3 umumasaflm NH.m mo.H oo.H nm.o HH.m mm.¢ ow.mm om.m~ mm.o ~m.aa .omo Hm.m mo.a mm.a mo.a vm.m mo.m oo.mm ma.om mo.o mm.va .>oz «m.~ em.o nm.a Hm.o mm.m mm.m mm.m~ mH.Hm mm.o om.ma .uoo Hm.m Hm.o on.a Hw.o om.m mh.v mm.- H¢.Hm mv.o mv.m .ummm eo.m vm.o em.a no.0 m>.~ om.¢ mo.mm v~.Hm om.o mh.m mam mm.~ hH.H Hm.H Hm.o mm.v mm.¢ va.mm H¢.Hm mm.o hm.m Hub vm.m mm.o om.a mm.o oa.m «H.m o~.mm NH.~m ov.o mH.w mash mo.m mm.o mn.a vm.o mm.m mo.h mm.m~ mm.mm mm.o mm.h >82 Hm.m NH.H hq.a mm.o on.~ mm.m mm.mm mm.Hm hm.o hm.m .Hma v~.m vH.H mm.H vv.o hm.~ vo.h Hh.- ma.om Hm.o ma.q .Hmz mv.m mo.H Hm.H mm.o mm.a mH.w mm.- om.m~ v~.o H>.m .nmm m~.m mN.H mm.a mh.o om.v mm.m Hm.mm Hm.m~ mm.o om.m .swn Dawns .cflz .xmz saamo manage: snags .afiz .xmz manna assumes scam masumuwmfims cflmm Icsm wusumummfima :flmm soflu8w>ma pumpsmum com: mafiuouofl>nlmmsam> Monummk Hmsuom 91 00.0 5H.H 00.0 05.0 05.0 00.0 00.00 00.00 00.0 00.00 .000 00.0 00.0 00.0 55.0 0H.v 00.0 50.00 50.00 00.0 00.00 .>oz 00.0 00.0 00.0 05.0 05.0 05.0 00.00 05.00 00.0 H0.0H .uoo 00.0 00.0 50.0 00.0 00.0 05.0 00.00 00.00 00.0 00.0 .ummm 00.0 00.0 0v.H 00.0 00.0 00.0 00.00 00.H0 00.0 05.0 .0:< 00.0 00.0 H0.H 00.0 00.0 H0.0 00.00 00.00 00.0 00.0 .Hsh 00.0 00.0 50.0 50.0 00.0 00.0 05.00 00.00 00.0 00.0 wash 55.0 00.0 00.0 00.0 00.0 05.0 00.00 00.00 00.0 00.5 002 05.0 00.0 00.0 50.0 00.0 00.5 55.00 00.00 00.0 00.0 .Hmfi 00.0 00.0 00.0 00.0 00.H 00.5 50.00 00.00 00.0 0H.¢ .mm: 50.0 05.0 00.0 00.0 05.0 00.0 50.00 00.00 00.0 00.0 .200 00.0 00.0 «0.0 00.0 50.0 00.0 50.00 05.00 50.0 05.0 .cm0 unmfla .00: .x02 maflmo manage: unmfla .cfiz .xmz 00000 manage: Icsm musumummfima samm Issm musumummame soflu00>mo oumvcmum cam: mammcmzllmmsam> umnummz 000009800 00.0 00.0 00.0 HH.H 05.5 00.0 00.00 00.00 50.0 00.00 .000 00.0 0H.H 00.0 50.0 00.0 00.0 00.00 HH.00 00.0 00.00 .>oz 00.0 00.0 00.0 00.0 50.0 00.0 v0.00 00.00 00.0 00.00 .uoo 00.0 00.0 00.0 55.0 00.0 00.0 00.00 00.00 00.0 05.0 .ummm 00.0 00.0 00.0 00.0 00.0 00.0 00.00 00.00 00.0 00.0 .090 00.0 00.0 00.0 05.0 H0.v 00.0 00.00 00.00 00.0 00.0 .050 55.0 H0.0 00.0 00.0 00.0 00.0 50.00 50.00 00.0 00.0 0:50 00.0 00.H 00.0 05.0 05.0 00.5 v0.00 00.00 00.0 «0.5 mm: 00.0 00.0 00.0 00.0 00.0 00.0 00.00 05.00 00.0 50.0 .uma 00.0 00.0 50.0 00.0 00.0 00.5 00.00 00.00 00.0 00.0 .Hmz 00.0 00.0 00.0 50.0 00.0 00.0 00.00 00.00 00.0 00.0 .nmm 00.0 00.0 00.0 00.0 00.0 00.0 00.00 50.00 00.0 00.5 .smn usmfla .cfiz .xmz maamorwanucoz unmfla .cHz .xmz magma manages scam wusumnmmfime c000 Icsm musumnmmEmB swam somwmw>mo oumocmum sum: n mammsmzllmmsam> Hmnumm3 Hmsuo< APPENDIX B TYPHOON LOSS ADJUSTMENT AND RENDEMENT CONVERSION FACTOR 92 .msam> vmumsmwm 000 on msam> 00000000 0:» £003 wmfiamfluanz 0 000.0 000.0 00000000 HmnEm>oz umnouoo 003800000 umsmsd >050 0c50 000.0 000.0 002 000.0 000.0 00000 000.0 000.0 00002 000.0 000.0 00000000 000.0 000.0 0000000 I00001-(00®0.|I 0000 0000 .4 in, 000002 ucmEmvcmm||0000000 ucmEumsnpm mmoH coosmwa 000.0 000.0 000.0 000.0 000.0 00000000 000.0 000.0 000.0 000.0 000E0>oz 000.0 000.0 000.0 000.0 0000000 000.0 000.0 000.0 000.0 000000000 000.0 000.0 000.0 000.0 000000 000.0 000.0 000.0 000.0 000.0 0000 000.0 000.0 000.0 000.0 000.0 000s 000.0 000.0 000.0 000.0 000.0 002 000.0 000.0 000.0 000.0 000.0 00004 000.0 000.0 000.0 000.0 000.0 00002 000.0 000.0 000.0 000.0 000.0 00000000 000.0 000.0 000.0 000.0 000.0 0000000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 000002 000::08I10000000 pameumsnpm mmoa coonmme 93 Factor needed to convert district rendement into rendement for each of the three areas: FACTOR.. 1 where: FACTOR TON SUGAR factor to be multiplied with the district weekly rendement to get weekly rendement for area i, during month j. If the week extends to the next month, it is considered part of the month having the most number of days in the particular week. amount of cane produced in area i, during month j. This is equal to the product of the tonnage and area harvested. amount of sugar produced in area i during month j. This is equal to the product of the cane produced and rendement for area i during month 3. l, 2, 3 referring to Victorias lowland, Vic- torias upland and Manapla respectively. 1, 2, . . . . 12 months. APPENDIX C WEATHER MODEL PARAMETERS V ICTORIAS WEATHER PARAMETERS 94 Jo- Bimonthly Transition Probabilities (Pi1 1) 31-1 Periods 1 2 3 *47 Jan-Feb 0.702 .696 0.416 0.406 Mar-Apr .600 0.526 0.322 0.338 May-Jun 0.652 0.528 0.476 0.400 Jul-Aug 0.716 0.667 0.496 0.464 Sept-Oct 0.714 0.716 0.505 0.496 Nov-Dec 0.795 0.672 0.553 0.512 Monthly Ai and.ari Jan Feb Mar Apr Max June Ai 0.429 0.493 0.456 0.474 0.383 0.364 auri 0.591 0.446 0.545 0.665 0.824 0.698 Jul Aug Sept Oct Nov Dec Ai 0.369 0.324 0.331 0.305 0.284 0.336 .ar 0.801 0.728 0.700 0.898 0.942 0.815 95 11 f(Si ) 1. 1, - 1- l- l. as.1 bsfL 05.1 ds.1 es.l 05.1 l 1 l l 1 1 Jan-Feb 2.640 .372 -.747 0.000 0.000 2.843 Mar-Apr 3.492 .400 -1.453 0.000 0.000 2.794 May-Jun 2.076 .513 -.432 0.000 0.000 2.405 Jul-Aug 13.634 .366 -.887 0.000 -.477 2.418 Sept-Oct 2.246 .515 -.419 0.000 0.000 2.589 Nov-Dec 2.318 .484 “.368 0.000 0.000 2.720 1. 1 f(Xi ) i 1i i d i 11 a, i axi bxi cxi xi exi xi Jan-Feb 15.855 .395 .259 -.l91 0.000 1.007 Mar-Apr 18.000 .332 .307 0.000 0.000 1.083 May-Jun 20.093 .307 .359 0.000 0.000 1.507 Jul-Aug 25.405 .255 .359 -.260 -.172 1.097 Sept-Oct 20.090 .301 .285 ".247 0.000 1.264 Nov-Dec 16.323 .395 .261 0.000 0.000 1.167 11 f(Yi ) 1' 1' l 1' 10 1' 1 1 ' 1 1 1 ayi byi cyil dyjL eyi dyi Jan-Feb 8.287 .517 0.000 0.000 .094 .851 Mar-Apr 13.948 .256 0.000 0.000 .110 .834 May—Jun 17.582 .231 .075 -.l94 0.000 .725 Jul-Aug 16.205 .287 .042 -.144 0.000 .754 Sept-Oct 15.190 .327 .057 -.113 0.000 .692 Nov-Dec 14.406 .379 0.000 -.140 0.000 .790 96 21 HS. ) 1 . 2« 2. 2. . . 1 1 1 1 1 1 asi bsi csi dsi esi dsi Jan-Feb 2.586 .481 0.000 0.000 0.000 2.820 Mar-Apr 2.314 .630 0.000 0.000 0.000 2.427 May-Jun 2.612 .456 0.000 0.000 0.000 2.471 Jul-Aug 2.959 .328 0.000 0.000 0.000 2.518 Sept-Oct 16.541 .383 0.000 0.000 -.585 2.536 Nov-Dec 2.540 .488 0.000 0.000 0.000 2.554 21 f(xi ) . 2. 2- 2- 2- 2- 1 1 1 1 1 axi bxi cxi dxi exi «xi Jan-Feb 8.041 .686 .158 0.000 0.000 .806 Mar-Apr 9.176 .667 .143 0.000 0.000 .790 May—Jun 15.216 .497 .194 0.000 0.000 1.190 Jul-Aug 20.184 .323 .259 0.000 0.000 1.019 Sept-Oct 15.302 .486 .196 0.000 0.000 .997 Nov-Dec 8.539 .687 .127 0.000 0.000 .908 21 f(Yi ) ayi1 byil cyi dyi eyi «syi Jan-Feb 12.676 .446 0.000 0.000 0.000 .875 Mar-Apr 12.498 .450 .060 0.000 0.000 .802 May-Jun 15.478 .340 0.000 -.240 0.000 .835 Jul-Aug 9.957 .548 .065 0.000 0.000 .795 Sept-Oct 14.718 .348 .067 -.225 0.000 .714 Nov-Dec 12.464 .464 0.000 -.243 0.000 .804 97 3. 1 f(Si ) 3- 3- 3- 3. 3. 3. as.1 bs,l cs.1 ds.1 es.1 ‘s.1 1 1 l 1 l l Jan-Feb 5.255 .267 0.000 0.000 0.000 2.736 Mar-Apr 5.393 .400 0.000 0.000 0.000 2.548 May-Jun 4.584 .409 0.000 0.000 0.000 2.420 Jul-Aug 4.999 .203 0.000 0.000 0.000 2.790 Sept-Oct 4.581 .352 0.000 0.000 0.000 2.508 Nov-Dec 3.997 .454 0.000 0.000 0.000 2.561 3. f (x. l) 1 3. 3. . 3i . . axil bxi1 cxi1 dxi exil «ix 1 Jan-Feb 12.809 .383 .168 0.000 .179 .932 Mar-Apr 14.321 .501 .142 0.000 0.000 .963 May-Jun 20.226 .356 .123 0.000 0.000 1.055 Jul-Aug 24.946 .169 .238 0.000 0.000 .930 Sept-Oct 24.218 .204 .154 0.000 0.000 .992 Nov-Dec 14.615 .501 .086 0.000 0.000 1.154 3. 1 f (Y1 ) 3- 3- 3- 3- 3- 3- 1 1 1 1 1 1 ayi byi cyjL dyi eYi dyi Jan-Feb 5.936 .728 0.000 0.000 0.000 1.082 Mar-Apr 9.687 .577 0.000 0.000 0.000 .906 May-Jun 12.017 .495 0.000 0.000 0.000 .901 Jul-Aug 10.339 .564 0.000 0.000 0.000 1.073 Sept-Oct 12.141 .470 0.000 0.000 0.000 .875 Nov-Dec 10.304 .545 0.000 0.000 0.000 .983 4. 1 f (Si ) . 4. . 4. 4. . as.l bs 1 cs 1 ds-1 es.1 «3,1 1 i 1 1 1 Jan-Feb 3.377 .567 0.000 0.000 0.000 2.340 Mar-Apr 4.323 .526 0.000 0.000 0.000 1.768 May-Jun 4.241 .475 0.000 0.000 0.000 2.187 Jul-Aug 4.009 .410 0.000 0.000 0.000 2.412 Sept-Oct 4.330 .380 0.000 0.000 0.000 2.357 Nov-Dec 4.025 .481 0.000 0.000 0.000 2.400 4. 1 f(Xi ) 4i b 4i 4i a i i i Jan-Feb 8.878 .678 .097 0.000 0.000 .815 Mar-Apr 7.132 .781 0.000 0.000 0.000 .905 May-Jun 14.862 .534 .099 0.000 0.000 1.146 Jul-Aug 20.448 .463 .162 0.000 -.169 .826 Sept-Oct 13.498 .567 0.000 0.000 0.000 .960 Nov-Dec 5.943 .812 0.000 0.000 0.000 .987 4i f(Yi ) 4- 4- 4- 4- 4. 4- 1 1 1 1 1 1 aYi in CYi in 9Y1 “Y1 Jan-Feb 10.718 .518 0.000 0.000 0.000 1.176 Mar-Apr 10.080 .561 0.000 0.000 0.000 1.142 May-Jun 15.891 .329 0.000 0.000 0.000 .972 Jul-Aug 9.586 .596 0.000 0.000 0.000 .894 Sept-Oct 13.886 .397 0.000 0.000 0.000 .831 Nov-Dec 9.546 .574 0.000 0.000 0.000 1.168 99 MANAPLA WEATHER PARAMETERS * Monthly Pi' Qi'.ui’1ui £21._ F_eI_D__ Mar 52;. ”.921!— 3&2. Pi 0.838 0.825 0.773 0.742 0.749 0.830 Qi 0.826 0.850 0.862 0.813 0.759 0.776 Aai 0.618 0.450 0.509 0.601 0.813 0.741 'wi 0.198 0.231 0.130 0.205 0.203 0.240 Jul 521.. §EEE_. Oct Nov Dec Pi 0.797 0.815 0.820 0.842 0.865 0.869 Qi 0.800 0.876 0.868 0.691 0.747 0.765 ’“i 0.761 0.692 0.723 0.902 0.907 0.878 1ui 0.240 0.231 0.327 0.226 0.179 0.102 * Pi = probability of rain in Manapla given rain in Victorias. Qi = probability of no rain in Manapla given no rain in V1ctor1as. ALi = daily mean rain given rain greater than 0.10 in Manapla given ra1n 1n V1ctor1as. 1Ui = daily mean rain given rain in Manapla given no rain in Victorias. 100 1. f(S.l) 1 1 l 1 1 1 l asi bsi csi dsi esi Gs Jan-Feb 2.292 .364 -.654 0.000 0.000 2.755 Mar-Apr 2.681 .403 0.000 0.000 0.000 3.048 May-Jun 2.030 .451 0.000 0.000 0.000 2.515 Jul-Aug 2.254 .348 0.000 0.000 0.000 2.672 Sept-Oct 2.311 .457 0.000 0.000 0.000 2.755 Nov-Dec 4.015 0.000 0.000 0.000 0.000 3.137 f X1i ( i ) 11 11 i l. 11 1. 1 1 axi bxi cxi dxi exi