American Journal of Operational Research 2018, 8(1): 10-13 DOI: 10.5923/j.ajor.20180801.02 Sen's Multi-Objective Programming Method and Its Comparison with Other Techniques Chandra Sen Department of Agricultural Economics, Institute of Agricultural Sciences, Banaras Hindu University, Varanasi, India Abstract The present paper evaluates the Sen's Multi-Objective Programming (MOP) method for solving multi-objective optimization problems. The method has been successfully used in formulating suitable farm plans for achieving several objectives of maximizing income, maximizing employment, minimizing fertilizer use, minimizing irrigation and plant protection chemicals etc. Few studies have reported several alternative techniques of multi-objective optimization and concluded their superiority over Sen's MOP method with illogical interpretations. The examples used to demonstrate the solutions of these MOP techniques were also not appropriate. Keywords Linear Programming, MOP, Mean, Median and Optimal average techniques 1. Introduction Linear programming has been extensively used to optimize (maximize or minimize) single objective function subject to certain constraints. It is a little difficult to optimize two or more objectives at a time and becomes more difficult if the objectives are conflicting in nature. It was realized to explore the possibilities of generating the compromising solution that achieves all the objectives simultaneously. Several methods have been developed for solving multi-objective optimization problems. In the constraint method, the most preferred objective is optimized keeping other objectives as constraints. The weighted sum method scalarizes the set of objective functions into the single objective function. Most of the new methods are weighted sum methods proposed during past decade. These methods have been evaluated with respect to the formulation of multi-objective function, suitability of the numerical examples solved and the interpretations of the solution. 2. Multi-Objective Programming Methods 2.1. Sen's Multi-Objective Programming Method Sen [1] proposed a method of Multi-Objective * Corresponding author: chandra.sen@bhu.ac.in (Chandra Sen) Published online at http://journal.sapub.org/ajor Copyright © 2018 The Author(s). Published by Scientific & Academic Publishing This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Programming for achieving several conflicting objectives simultaneously. A Multi-Objective Function is formulated and optimized under common constraints. The mathematical form of MOP is described as: Optimize Z= [Max. Z 1 , Max. Z 2 ......Max. Z r Min. Z r+1 .......Min. Z s ] Subject to: AX = b and X≥ 0 The individual optima are obtained for each objective separately as: Z optima = [W 1, W 2 ..........W s ] The Multi-Objective Function is formulated as: Maximize,               Subject to: AX = b and X≥ 0 W j ≠ 0 for J=1, 2..........s. W j = Optimum value of jth objective function The combined objective function was formulated by weighting each objective function by inverse of its optima which make the objective function dimension free. Therefore the combined objective function is constructed without any problem with the objective functions of different dimensions. The method has been successfully used by many research scholars/ scientists [3-11, 13] to formulate an alternative cropping plan for the farmers for achieving two to six objectives simultaneously. The objectives were the maximization of income and employment and minimization of fertilizer use, irrigation water, CO 2 emissions, Plant protection chemicals, etc. The results of all the studies were satisfactory.