A Goal Programming approach for solving Interval valued Multiobjective Fractional Programming problems using Genetic Algorithm

Abstract

In this article, the efficient use of a genetic algorithm (GA) to the goal programming (GP) formulation of interval valued multiobjective fractional programming problems (MOFPPs) is presented. In the proposed approach, first the interval arithmetic technique [1] is used to transform the fractional objectives with interval coefficients into the standard form of an interval programming problem with fractional criteria. Then, the redefined problem is converted into the conventional fractional goal objectives by using interval programming approach [2] and then introducing under-and over-deviational variables to each of the objectives. In the model formulation of the problem, both the aspects of GP methodologies, minsum GP and minimax GP [3] are taken into consideration to construct the interval function (achievement function) for accommodation within the ranges of the goal intervals specified in the decision situation where minimization of the regrets (deviations from the goal levels) to the extent possible within the decision environment is considered. In the solution process, instead of using conventional transformation approaches [4, 5, 6] to fractional programming, a GA approach is introduced directly into the GP framework of the proposed problem. In using the proposed GA, based on mechanism of natural selection and natural genetics, the conventional roulette wheel selection scheme and arithmetic crossover are used for achievement of the goal levels in the solution space specified in the decision environment. Here the chromosome representation of a candidate solution in the population of the GA method is encoded in binary form. Again, the interval function defined for the achievement of the fractional goal objectives is considered the fitness function in the reproduction process of the proposed GA. A numerical example is solved to illustrate the proposed approach and the model solution is compared with the solutions of the approaches [6, 7] studied previously.