Solving Multi-Objective Linear Fractional Programming Problem - First Order Taylor's Series Approximation Approach
In this paper, a method is proposed for solving multi-objective linear fractional programming (MOLFP) problem. Here, the MOLFP problem is transformed into an equivalent multi-objective linear programming (MOLP) problem. Using the first-order Taylor's seri
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stract In this paper, a method is proposed for solving multi-objective linear fractional programming (MOLFP) problem. Here, the MOLFP problem is transformed into an equivalent multi-objective linear programming (MOLP) problem. Using the first-order Taylor’s series approximation, the MOLFP problem is reduced to single-objective linear programming (LP) problem. Finally, the solution of MOLFP problem is obtained by solving the resultant LP problem. The proposed procedure is verified with the existing methods through the numerical examples. Keywords Linear programming problem problem Taylor’s series
Multi-objective linear programming
1 Introduction The decision makers, in the sectors such as financial and corporate planning, production planning, marketing and media selection, university planning and student admissions, health care and hospital planning, often face problems to take decisions that optimize department/equity ratio, profit/cost, inventory/sales, actual cost/standard cost, output/employee, student/cost, nurse/patient ratio, etc. The above problems can be solved efficiently through linear fractional programming (LFP) problems.
C. Veeramani (&) M. Sumathi Department of Applied Mathematics and Computational Sciences, PSG College of Technology, Coimbatore 641004, India e-mail: [email protected] M. Sumathi e-mail: [email protected]
G. S. S. Krishnan et al. (eds.), Computational Intelligence, Cyber Security and Computational Models, Advances in Intelligent Systems and Computing 246, DOI: 10.1007/978-81-322-1680-3_38, Springer India 2014
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C. Veeramani and M. Sumathi
A general LFP problem can be formulated as follows: 8 T > Max ZðxÞ ¼ dc T xþp > xþq < subject to > Ax b; > : x0
ð1Þ
where x; cT ; dT 2 Rn , A 2 Rmn , b 2 Rm , and p; q 2 R. Charnes and Cooper [2] showed that if the denominator is constant in sign on the feasible region, the LFP problem can be optimized by solving a linear programming problem. Many authors (see [4, 5, 7, 14]) have presented methods for solving LFP problems. However, in many applications, there are two or more conflicting objective functions, which are relevant, and some compromise must be sought between them. For example, a management problem may require the profit/cost, quality/ cost, and other ratios to be maximized and these conflict. Such types of problems can be solved efficiently through multi-objective linear fractional programming (MOLFP) problems. The general form of MOLFP problem can be formulated as follows: 8 cT xþp > Max Zi ðxÞ ¼ diT xþqii > < i subject to ð2Þ > Ax b; > : x0 where i ¼ 1; 2. . .k, A 2 Rmn , x 2 Rn , b 2 Rm , c; d 2 Rn , and p; q 2 R. There exist several methodologies to solve MOLFP problem. Kornbluth and Steuer [9] have proposed an algorithm for solving the MOLFP problem for all w-efficient vertices of the feasible region. The solution set concepts of w-efficient subsumes that of s-efficient. Kornbluth and Steuer [8] have discussed a generalized approach for solving a goal program with linear fractional criteria.
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