APPLICATION OF GOAL PROGRAMMING IN FARM AGRICULTURAL PLANNING

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1 APPLICATION OF GOAL PROGRAMMING IN FARM AGRICULTURAL PLANNING Dr.P.K.VASHISTHA, Dean Acaemics, Vivekanan Institute of Technology & Science, Ghaziaba ABSTRACT In this paper we present an overview of the ifferent goal programming formulations, their assumptions, limitations an implications for agricultural ecision making. The concept of stanarize ual variables which provies a more meaningful interpretation of shaow prices in goal programming is introuce through a simple example of farm agricultural planning. Keywors: Goal Programming, Decision Making, Farm Planning. INTRODUCTION Goal programming technique has its great potential, particularly in ecision-making environments involving multiple objectives like farm agricultural planning an management. In many situations farmers are often face with several objectives simultaneously an no easy single choice. Examples of such objectives are: maximization of net revenue; minimization of capital borrowing an hire labour; an minimization of the risk associate with yiel an fiel ay variability. On a regional or national level, the agricultural ecision maker may be face not only with ecisions about economic growth, but also about population nutritional requirements, strategic planning, environmental an other institutional issues. In this paper, we present aspect of goal programming base on uality theory through an example of farm agricultural planning. GOAL PROGRAMMING MODEL The importance of more than one objective in agricultural planning, stuies is illustrate by the works of Ignizio et al., Barnett et al., Canler an Boehlje., Harper an Eastman., Hatch et al., an Willis an Perlack. Romero., an Schnieerjans provie excellent reviews of the evelopment in goal programming. A generic-type goal planning moel can be compactly written as: Minimize: Subject to: w w... () Gx g... () Ax < b... (3) International Journal of Research in Science An Technology

2 x,,... (4) where x is a (n ) vector of ecision variables, G is a (p n) matrix of goal contribution coefficients; g is a (p ) vector of esire goal levels; + an are (p ) vector representing, respectively, positive an negative eviations from goals; A is a (m n) matrix of technological coefficients; b is a (m ) vector of resource levels. In aition, w+ an w are ( p) vectors of weights which may or may not be preemptive. DUALITY INTERPRETATION OF GOAL PROGRAMMING When a goal programming is solve as a minimum sum of weighte eviations, the problem of choosing the weights has been the focus of a lot of work because of its complexity. The commonly use weighting proceure is to have the ecision maker associate the highest weights with the most important goals. The use of uality theory may help provie a new insight into the esign an interpretation of time weights. We focus on the interpretation of uality in goal programming an its usefulness in ecision making. Without loss of generality we restrict our attention to a primal problem with only goal constraints. In aition, the istinction between preemptive an non-preemptive formulations will be implicit in the istinction between a single ual or a sequence of prioritize uals. Consier the following primal goal programming an ual goal programming formulations in compact form: Primal : Minimize: Z = w w Subject to: Gx + Dual : Maximize: y = vt g Subject to: G t v x,, w v w where v is a vector of ual variables, y is the value of the ual objective function, the problem imensions are as in () (4), an t is use to inicate transpose. In goal programming, the primal may still be interprete in physical terms where an optimal prouct mix contributes to the achievement of a certain number of goal targets. However, the corresponing ual is a "pricing problem of the goal targets" in terms of the ecision maker's preferences. Hence, the ual shoul have preference utility interpretation with the ual variables representing absolute marginal utilities of the ifferent goals. We show below that a more useful interpretation of the ual variables is in relative terms, an we provie a numerical illustration. From primal-ual relationships between primal goal programming an ual goal programming, we have the following: g International Journal of Research in Science An Technology

3 y < z (for any pair of feasible solutions)... (5) y* = z* (at optimality)... (6) z g * i y * * i v * i w i... (7)... (8) The first two relations can be interprete together as follows: the total isutility of eviation from the goal targets (i.e., z) is always at least as large as the total utility of these goal targets (i.e., y), an equal to it at optimility. Relation (7) states that the ith optimal ual variable represents the marginal effect of the ith goal target of total isutility (or utility). Relation (8) is a restatement of the weights attache to eviations (either positive or negative) as marginal effects of these eviations. Finally, from (7) an (8) we also obtain the following relations: vi* ( gi) = wi ( i*) = z* = y*... (9) vi* / wi = i* / gi... () from (9) we see that the same marginal isutility / utility effect can be erive from either changing the goal targets or changing the corresponing eviations. But the most interesting result is given by () which efines a "marginal rate of substitution" between a goal target an the istance away from it as equal to the marginal utility of the goal ivie by its weight. Hence, we can use these stanarize ual variables (vi*/wi) as a measure of goal achievement across all goals. This provies a way of properly interpreting an using the ual variables in goal programming with or without preemption. NUMERICAL EXAMPLE The following simplifie example is use throughout to illustrate the analysis. We have chosen two variables an two constraints so that we may be able to have graphical interpretations for both the primal an the ual. Consier a farmer who can grow either Wheat or Gram. He has acres of lan an woul like to achieve at least Rs. 6,, of revenue while minimizing total prouction cost. Table contains the relevant information. Table : Cost / Revenue Data Wheat Gram Yiel cwt/acre 8 Price Rs/cwt 3 Cost Rs / Acre 8, 7 International Journal of Research in Science An Technology

4 This problem is formulate as the following linear programming problem: (LP) Minimize: z = 8,x + 7,x subject to: x + x <,x + 5,6x > 6,, x, x > where x an x represents acres of wheat an gram respectively. Clearly, the problem has no feasible solution, as shown in figure. To get aroun the infeasibility an still fin an acceptable "compromise" solution, the following simple goal programming formulation is use: (GP) Minimize: z = Subject to: x x 6,, =,x + 5,6x + = 6,, x, x,, > where an are weights attache to the eviation variables. In this form the objective function is expresse as a weighte sum of percent eviations from targets. For illustration purposes a particular set of weight is = = meaning equal importance, is given to both goals. We let w w ; an 6,, This formulation is easily interprete in light of Figure as one which seeks to minimize the total infeasibility in the constraints of the original linear programming problem. The resulting goal programming solution is x = (grow acres of gram) an + = Rs. 88, (revenue shortage) an correspons to P in Figure. For the farmer this solution will guarantee as Rs 5,, total return. International Journal of Research in Science An Technology

5 Figure : Solution of Farm Example Now, suppose that the farmer looks at this solution an requires that an absolute first priority is not violate the revenue constraint because it may be a bankruptcy level. Once this is achieve, a minimization of the infeasibility of the total lan constraint will be sought. This requirement is formulate as a lexicographic (or prioritize) goal programming with objective function {( ),( )} solve in two iterations: Iteration One: Minimize: z { } Subject to :,x + 5,6x + + = 6,, x, x,, with optimal solution: =,,x + 5,6x (Line segment PP3 in Figure ) =6,, Iteration Two : Minimize: z = { Subject to: x x } =,x + 5,6x + = 6,, International Journal of Research in Science An Technology

6 x, x,,, with optimal solution : x = 3.44, = 3.44 (P in Figure ) The case of an optimal but ominate solution, ue to alternative optima, can also be illustrate. Consier the case where the farmer wants to use up all the lan available ( acres). The corresponing goal programming will have objective function regular simplex coe gives the following solution: x =, { }. A =,6, (P4 in Figure ). But an alternate optimal solution is easily foun to be: x=, = 88, (P in Figure ). In terms of the original problem the two solutions are compare in Table, from which clearly the first solution is ominate. Table : Comparing Alternate GP Solutions Optimal Solution Alternate Optimal Solution x = ; x = x = ; x = Cost,6,,4, Revenue 4,4, 5,, Deficit,6, 88, To illustrate the uality result, consier the following goal programming moel corresponing to the original linear programming problem, which is inconsistent: (GP) Minimize : z = w w Subject to : x x,x 5, 6x 6,, x,, x, where w an w are chosen in a way that will make the objective function consistent an shoul reflect the importance of each goal. If we use the opportunity costs of incurre shortages as a measure of relative importance, then w =,7, is interprete as the cost of proviing an aitional acre of lan an w =. is the cost per unit of aitional revenue foregone. Now, we consier the ual problems of both linear programming an goal programming above: (DLP) Maximize: V = y 6,, y Subject to: y,y y 5,6y 8, 7, International Journal of Research in Science An Technology

7 y, y (DGP) Maximize: V = y 6,, y Subject to: y,y y 5,6y y o y,7, Figure, epicts both problems with the hache area corresponing to ual linear programming an the cross-hatche area corresponing to ual goal programming. Since the original linear programming problem was inconsistent, by uality theory, we know that its ual linear programming is unboune. The goal program which was formulate to resolve the inconsistency is a linear program which has feasible solutions. Its ual goal program has the finite optimal solution: y =5,6, y = as can be seen from Figure. This solution correspons to shaow prices of Rs. 5,6 for lan an Rs.. for total revenue, with the following interpretations, base on (5) () above: Figure : Dual Solutions for (DLP) an (DGP) () In absolute terms, the magnitue of the shaow prices can be misleaing in terms of the ranking of the constraints with respect to total goal achievement. That is, an aitional acre of lan will improve the total goal achievement by Rs. 5,6 even though it may cost at least Rs.,7, to acquire; on the other han, an aitional rupees of revenue forgone will only improve the total goal achievement by one rupee. The first part of Table 3 summerizes these effects. International Journal of Research in Science An Technology

8 () In relative terms, when comparing the achievement of both goals, the ratios of the shaow prices to their corresponing primal weights (stanarize ual values) convey more meaningful information: y 5,6 y. an. w,7, w These ratios reveal that on a per unit basis total revenue contribute percent to goal achievement compare to percent for lan. This is can be seen from the fact that the Rs,7, opportunity cost of lan will yiel Rs.,7, improvement instea of Rs. 5,6 for lan (see table 3, secon part). This suggests that care shoul be taken in practically interpreting shaow prices in goal programming. Table 3: Shaow Price Effects Constraint (Acre) Constraint (Rs.) Total Goal Achievement V (Rs.) Change in V (Rs.) 6,, 88, 6,, 6,4 5,6 9 6,,,3,6 + 5,6 6,, 88, + 5,99,999 87,999-7,7,,5, +,7, 4,73, 39,,7, CONCLUSION The goal programming approach are two problems emboie in the following questions: (a) Priorities: How to associate the results of a given solution to the satisfaction of the ranking; (b) Weights: How to generate them an what they mean. We think that the uality interpretation of the weights can help both the analysts an the ecision makers in the esign an solutions of meaningful multicriteria ecision problems in farm agricultural planning an management. REFERENCES. Barnett D., Blake B., an McCarl B.A. "Goal Programming Via Multiimensional Scaling Applie to Senegalese Subsistence Farms." American Journal of Agricultural Economics. 64, 7-77, 98 International Journal of Research in Science An Technology

9 . Bazaraa, M.S., an Bouzaher, A. "A Linear Goal Programming Moel for Developing Economics with an Illustration from the Agricultural Sector in Egypt." Management Science. 7(4), , Canler, W., an Boehlje, M. "Use of Linear Programming in Capital Bugeting with Multiple Goals." American Journal of Agricultural Economics. 53, 35-33, Charnes, A., an Cooper, W.W. Management Moels an Inustrial Applications of Linear Programming. Vol. I, II John Wiley & Sons. New York Fahmy, D., an El-Shishiny, H., "A Goal Programming Moel for Desert Farm Planning". Avance Desert Ari Lan Technology Development. 5, 69-86, Ghosh, D., Pal, B.B., an Basu, M. "Determination of Optimal Solution for a MCDM Moel in Agricultural Planning: A Strategy". International Journal of Management an Systems. (), 67-83, Ghosh, D., Pal, B.B., an Basu, M. "Determination of Optimal Lan Allocation in Agricultural Planning Through Goal Programming with Penalty Functions". Opsearch. 3(), 5-34, Harper, W.M., an Eastman, C. "An Evaluation of Goal Hierarchies for Small Farm Operators." American Journal of Agricultural Economics. 6, , Ignizio, James P. Goal Programming an Extensions. Lexington Books Ijiri, Yuji. Management Goals an Accounting for Control. North-Hollan Publishing Company, Amesteram Lee, Sang M. Goal Programming for Decision Analysis. Auerbach Publishers, Inc. Philaephia Markowski, C.A., an Ignizio, J.P. "Duality an Transformations in Multiphase an Sequential Linear Goal Programming" Computer an Operations Research. (4), 3-333, Piech, B., an Rehman, T. "Application of Multiple Criteria Decision Making Metho to Farm Planning: A Case Stuy". Agricultural Systems. 4(3), 35-39, Romero, C., an Rehman, T. "Goal Programming an Multiple Criteria Decision- Making in Farm Planning: An Expository Analysis". Journal of Agricultural Economics. 35(), 984, International Journal of Research in Science An Technology

10 5. Romero, C., an Rehman, T. "Goal Programming an Multiple Criteria Decision- Making in Farm Planning: Some Extensions." Journal of Agricultural Economics. 36, 7-86, Schnieerjans, Marc J. Goal Programming: Methoology an Applications. Kluwer Acaemic Publishers Schnieerjans, M. J., an Kwak, N.K. "An Alternative Solution Metho for Goal Programming Problems: A Tutorial." Journal of the Operational Research Society. 33(3), 47-5, Willis, C.E., an Perlack, R.D. "A Comparison of Generating Technologies an Goal Programming for Public Investment, Multiple Objective Decision making". American Journal of Agricultural Economics. 6, 66-74, 98. International Journal of Research in Science An Technology