FGP to Sugarcane Nutrient Management

Transcription

1 DECISION SCIENCES INSTITUTE Fuzzy Goal Programming and Decision Theoretic Approach to Sugarcane Nutrient Management (Full Paper Submission) R. K. Jana Indian Institute of Social Welfare Business Management Dinesh K. Sharma University of Maryland Eastern Shore Daniel Okunbor Fayetteville State University ABSTRACT The objective of this paper is to help sugarcane farmers to make better nutrient management decisions that would result in improved sugarcane yield at minimum cost. We propose a technique that combines fuzzy goal programming and decision theory for solving the multiobjective model, selecting the fertilizer combinations based on a given fertilizer expenditure. The effectiveness and applicability of proposed technique are demonstrated via a case example. KEYWORDS: Fuzzy Goal Programming, Decision Theory, Sugarcane, Nutrient Management INTRODUCTION Sugarcane farming requires proper use of nutrients for high productivity and large crop yields. Applications of fertilizer higher than the required level are likely to cause diverse effects on sugarcane yield. This could potentially be harmful to the environment. To maintain a sustainable production of sugarcane, a proper combination of nutrients must be applied. Imbalance in the use of fertilizers necessitated the need to formulate blended fertilizer for the sugarcane industry. Research experiments (Wiedenfeld, 1; Sundara et al., 00) have suggested that for producing higher cane and sugar yield on a sustainable basis, application of adequate amounts of Nitrogen (N), phosphorus (P), and potassium (K) is essential. In this paper, we present a nutrient management model for maximizing sugarcane yield in India by determining the optimum fertilizer combination. Due to vagueness in the available budget for fertilizer, yield of sugarcane, etc., the corresponding goals are defined in a fuzzy sense and a fuzzy goal programming (FGP) model has been developed. The yield goal is nonlinear due to the nonlinearity in the production function. The other goals are linear. All goals are converted to their deterministic equivalents based on specified tolerance values (Tiwari el al., 1) and a nonlinear programming problem is obtained. The model is solved under various cases and sensitivity analysis is done using Lingo 10. The maximin and maximax Decision Theory (DT) (Iyer, 00) is then applied and pessimistic and optimistic decisions on selecting appropriate fertilizer combinations are obtained.

2 The remainder of the paper is organized as follows: The second section presents a brief review of the literature and the third section presents the proposed mathematical model for sugarcane nutrient management. The fourth section presents the decision theoretic approach for selecting fertilizer combination. The fifth section demonstrates the model via a case example. The sixth section analyzes the results obtained from the case example and the final section presents concluding remarks. LITERATURE REVIEW Over the years, operations research techniques have been used in decision-making in agricultural sector. Linear programming is a widely used technique to maximize production of crops (Arnold & Bennet, 1), allocate land under cultivation (Glen, 1), and minimize cost (Barnard & Nix, 1). This technique is useful for solving single objective problems. Literature Review of Goal Programming The nutrient management problem for sugarcane typically involves more than one objective function. The solution of these problems can be obtained by applying goal programming (GP) technique. Table 1 provides a summary of literature supporting the application of GP techniques. Table 1: Summary of Literature Review of GP YEAR REFERENCES JOURNAL/BOOK 11 Charnes & Cooper John Wiley & Sons. Inc. 1 Ignizio Lexington Books 1 Minguez et al. JORS 1 Tamiz & Jones Annals of Operations Research 1 Sharma et al. Acta Ciencia Indica Literature Review of Fuzzy Goal Programming GP is an efficient solution technique for multi-objective problems provided the goals and constraints are accurately specified. However, in many situations it is not possible to define the goals precisely. For example, in the sugarcane nutrient management problem, precise definitions of expected yield and expenditure for fertilizer which depend on the economic condition of the farmer are not possible. GP can t provide solutions to such problems. Narasimhan (10) introduced the concept of fuzzy goal programming (FGP) to solve imprecise multi-objective models. The important literatures of FGP are presented in Table. Table : Summary of Literature Review of FGP YEAR REFERENCES JOURNAL/BOOK 1 Rubin & Narsimhan Fuzzy Sets and Systems 1, 1 Tiwari et al. Fuzzy Sets and Systems 1 Slowinski Fuzzy Sets and Systems 1 Sinha et al. Socio-Economic Planning Sciences 00 Sharma & Jana Int. Journal of Production Economics The use of FGP technique for nutrient management problem for sugarcane has received less attention. In this study, we consider nutrient management problem for sugarcane utilizing FGP

3 framework. Due to vagueness in the available budget for fertilizer, yield of sugarcane, etc., the corresponding goals are defined in a fuzzy sense, resulting in the development of FGP model. The yield goal is nonlinear due to the nonlinearity in the production function. The other goals are linear. All goals are converted to their deterministic equivalents based on specified tolerance values (Tiwari el al., 1) to obtain a nonlinear programming problem. The model is solved using various cases, along with sensitivity analysis on Lingo 10. Also, the maximin and maximax Decision Theory (Iyer, 00) is applied and pessimistic and optimistic decisions on selecting appropriate fertilizer combinations are obtained. SUGARCANE NUTRIENT MANAGEMENT MODEL FORMULATION To formulate the FGP model of the problem, the following notations are defined. Notations index for the fertilizer {1,,,F}; index for nutrient {1,,,N}; : amount of fertilizer in the mixture; : unit cost for fertilizer in the mixture; amount of nutrient in fertilizer ; : lower limit of nutrient ; : upper limit of nutrient ; : estimated expenditure for fertilizers; : yield of sugarcane per ha; : elasticity of nutrient. Goals of the Model Goals of the model are to maximize the sugarcane yield, minimize the fertilizer expenditure, and minimize the upper limit nutrient requirements and they are defined in fuzzy sense: 1. Yield goal: The yield function for sugarcane in India was constructed, considering N, P and K elasticity (Rosas, 0) :. Expenditure goal: Total expenditure for fertilizers should not exceed the overall expected value of expenditure. This goal can be presented as: (1). (). Nutrient Requirement goal: Application of fertilizer of higher than the required level are likely to cause diverse effects on sugarcane yield. The goal is represented as:. () Constraints of the Model a. Minimum level of fertilizer requirement: Most of soils require some level of fertilizer to produce high sugarcane yield. The corresponding constraint can be written as:. ()

4 b. Utilization of nutrient used in previous year: From soil testing and practical experiences, it has been observed that the fertilizers used in previous year are not utilized fully. Let us assume that the primary nutrients are applied in the previous season through the use of (F-r) fertilizers. Then a certain percentage (s) of the primary nutrient requirements applied during the previous season is not required to apply in the next year. Therefore,. () DECISION THEORETIC APPROACH FOR SELECTING FERTILIZER COMBINATION Decision theoretic approach helps in making the best possible decision. Generally, the farmers will have two types of decisions. The first decision is based on the availability of budget and the second decision is based on their expectation on sugarcane yield for a particular season. Then the payoff matrix for the optimal decision on fertilizer combination can be constructed as follows: Table : Payoff Matrix for Yield and Budget Yield Y 1 Y Y j Y J Budget T 1 b b 1 b 1i b 1J T b 1 b b 1i b J T i b i1 b i b ij b ij T I b I1 b I b IJ where T 1, T,...,T i are costs of different fertilizer combinations, Y 1, Y,, Y j are yields of sugarcane and b ij are the amount of total nutrients (N-P-K) in the fertilizer mixture applied to per unit area of land. The decision making situation is analyzed using the maximin and maximax criteria to arrive to the optimal policy on fertilizer combination. CASE EAMPLE This case example is illustrated on the basis of data collected from India. The production of sugarcane in India has touched 0 million tons in 01-1 from.1 million ha of cultivated land with an average yield of about. ton/ha. For most soil, nitrogen (N), phosphorus (P), and potassium (K) are the three major nutrients required for sugarcane cropping. The N, P, and K rates used in India vary from region to region. As a common practice among agriculture experts, the minimum recommended rate of application in India is kg per ha of N, kg per ha of P, and kg per ha of K. The fertilizers & their nomenclature are presented in Table. Table summarizes the fertilizers according to N, P, and K percentages, and their costs as approved by the Fertilizer Corporation of India.

5 Table : Fertilizers & Their Nomenclature 1: Di Ammonium Phosphate : Muriate of Potash : Single Super Phosphate : Urea : Nitro Phosphate with Potash - 1 : Nitro Phosphate with Potash - : Nitro Phosphate with Potash - : Nitro Phosphate with Potash - : Nitro Phosphate with Potash - 10: Ammonium Nitro Phosphate - 1 : Nitro Phosphate with Potash - 1 : Ammonium Nitro Phosphate - Table : Fertilizers & Their Costs Fertilizer N (%) P (%) K (%) 01-1 Price (Rs/kg) RESULTS ANALYSIS In this study, the proposed technique is applied to nutrient management decision-making for sugarcane in India. The solution of the nonlinear FGP model was obtained using LINGO 10. We Table : Sugarcane yield, fertilizer application & budget Case Budget Yield Fertilizer I II III IV V VI VII VIII I * Table figures are rounded to one decimal place. Budget: Rs/ha, Yield: ton/ha, Fertilizer: kg/ha

6 studied different situations (Table ) to find the relationship between fertilizer expenditure and sugarcane yield (Figure 1 Figure ). Figure 1 shows that there exists a nonlinear relationship between sugarcane yield and expenditure for fertilizers. Therefore, spending more on fertilizer doesn t necessarily guarantee more sugarcane yield. More yield may be obtained by proper selection of fertilizer combination. Figure shows a fairly linear trend between fertilizer expenditure and fertilizer used. This is true as expenditure on fertilizer increases with increase in quantity. Figure shows that there exists a nonlinear relationship with sugarcane yield and amount of fertilizer used. This supports the facts reflected by Figure 1 and Figure. Figure 1: Fertilizer Cost v Sugarcane Yield Figure : Fertilizer Expenditure v Fertilizer Used

7 Figure : Fertilizer Used v Sugarcane Yield Let us assume that a farmer has the capacity to spent any amount between Rs 00/ha to Rs 00/ha for the fertilizer expenditure. Then a sensitivity analysis is performed with differential fertilizer expenditure and expected sugarcane yields and results obtained are presented in Table and are shown in Figure. Figure : Sugarcane Yield v Fertilizer Used for Different Fertilizer Expenditure Figure shows that sugarcane yields can be increased using less amount of fertilizer with optimum combination. This is one of the major finding of this research.

8 Table : Sensitivity Analysis for Budget and Sugarcane Yields Budget Yield Budget: Rs/ha, Yield: ton/ha Now the pessimistic and optimistic decisions can be obtained by applying maximin and maximax decision theoretic rules as follows: Table : Minimum and Maximum Values Alternative on Yield (ton/ha) Minimum Amount of Fertilizer (kg/ha) Maximum Amount of Fertilizer (kg/ha) Maximum {Minimum Amount of Fertilizer} = 1.1 kg/ha. Therefore, according to the maximin decision theoretic rule, to achieve a yield of.0 ton/ha a farmer has to spend Rs 00/ha as the fertilizer expenditure which is a pessimistic decision. The corresponding fertilizer combination is obtained as follows: Table : Fertilizer Combination, N, P and K by Maximin Rule N P K * Table figures are rounded to one decimal place. Fertilizer, N, P, K: kg/ha Maximum {Maximum Amount of Fertilizer} =. kg/ha. Therefore, according to the maximax decision theoretic rule, to achieve a yield of.1 ton/ha a farmer has to spend Rs 00/ha as the fertilizer expenditure. The corresponding fertilizer combination is given as follows: Table 10: Fertilizer Combination, N, P and K by Maximax Rule N P K * Table figures are rounded to one decimal place. Fertilizer, N, P, K: kg/ha

9 CONCLUSIONS This paper presents a method for obtaining optimum fertilizer combination for sugarcane nutrient management using FGP and Decision Theory. The traditional linear programming based models are incapable of modeling uncertain factors in complicated decision making environment. The FGP approach overcomes this drawback and helps to capture uncertain factors which are obvious in such situations. The goals and constraints of the model can be modified so that the proposed methodology will be suitable for decision-making irrespective of regions or Countries. REFERENCES Arnold, G.W., & Bennet, D. (1). The problem of finding an optimal solution in: Study of agricultural systems, in: G.E. Dalton (ed.), Applied Science Publishers, London, 1-1. Barnard, C.S., & Nix, J.S. (1). Farm Planning and Control. Cambridge: Cambridge University Press. Glen, J. (1). Mathematical models in farm planning: A survey. Operations Research, (), 1-. Rosas, F. (0). World Fertilizer Model The World NPK Model. CARD Working Paper Series: -WP 0. Center for Agricultural and Rural Development, Iowa State University. Sharma, Dinesh K., & Jana, R.K. (00). Fuzzy goal programming based genetic algorithm approach to nutrient management for rice crop planning. International Journal of Production Economics, 11, -. Sundara, B., Natarajan, V., & Hari, K. (00). Influence of phosphorus solubilizing bacteria on the changes in soil available phosphorus and sugarcane and sugar yields. Field Crops Research,, -. Wiedenfeld, R.P. (1). Effects of irrigation and N fertilizer application on sugarcane yield and quality. Field Crops Research,, (A complete list of references is available upon request)