Fuzzy mathematical programming applied to multi-level programming problems

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Abstract

An alternative multi-level programming technique based on fuzzy mathematical programming (FMP) is developed which gives better solutions than that proposed by Shih et al. (Comput. Oper. Res. 23(1) (1996) 73). The method uses the objectives in a sequential order of the hierarchy interactively and takes into account the desire of the DM at each stage. The technique is equally applicable to linear/nonlinear multi-level (say P levels) programming problems (MLPPs).

Scope and purpose

In this paper, we propose an alternative FMP method for a linear/nonlinear MLPP of maximization- and minimization-type objectives which gives better optimal solutions.

Introduction

Shih et al. [1] have proposed a methodology using fuzzy mathematical programming FMP to solve multi-level (say P levels) programming problems MLPPs, namely, BLPP, BLDPP and TLPP. In case of MLPPs, the iterative method starts by considering the top two levels (P=2) to find a satisfactory solution, then the successive lower level (P=3) is included into the system, for which another satisfactory solution is obtained and the process continues. At P=3, the lower bound (fpL,p=2,3) is assumed to be zero while the upper bound (fpU) is calculated at the satisfactory solution obtained. They represent the objective by linear membership functionμfp[fp(x̄)]=fp(x̄)−fpLfpU−fpL⩾λwherefpL⩽fp(x̄)⩽fpUand0⩽λ⩽1⇒fp(x̄)⩾fpUλ. Therefore, fp(x̄)⩾0 at λ=0 and fp(x̄)=fpU at λ=1 since fpU is the upper bound. Thus, the values of the objectives at P=3 will not change from that at P=2 and hence the point where the satisfactory solution is obtained will also not change. The value of the minimum acceptable level of satisfaction (λ) at P=3 is higher than that at P=2, which is not expected. Once λ=1, the inclusion of other levels into the system will not affect the solution and hence the hierarchical order becomes redundant.

In this paper we have proposed a methodology using FMP approach to solve MLPPs of maximization- and minimization-type objectives and demonstrate by solving TLPPs. The procedure used is equally applicable to multi-level decentralized programming problems (MLDPP) also. By choosing a nonlinear membership function, the linear MLPP reduces to a nonlinear crisp MLPP. The procedure followed is independent of the nature of the objectives or the constraints (whether linear or nonlinear) and hence is applicable to nonlinear problems as well. By following this approach, λ decreases as we move down the hierarchy and in general, the values of the objectives and the point where the satisfactory solution is attained may also change.

FMP has diverse applications as employed by Walle et al. [2]. They have analyzed three different cutting techniques for dismantling the thermal shield of the Belgian BR3 nuclear reactor within the framework of classical multi-criteria utility theory. In order to allow for indifference and incomparability in the decision process, the problem is redefined in terms of a newly developed fuzzy methodology. Sakawa et al. [3] focus on large-scale LPPs with block angular structure. They adopt a convex fuzzy decision for combining fuzzy goal for the objective and fuzzy constraints for coupling constraints. They show that under some appropriate conditions, the formulated problem can be reduced to a number of independent linear subproblems and the overall satisfying solution for the DM is directly obtained just by solving the subproblems. Chanas and Kuchta [4] define the optimal solution of the transportation problem with fuzzy cost coefficients as well as propose a solution algorithm. Carlsson and Fuller [5] have reviewed fuzzy MCDM and introduced a novel approach—interdependence in MCDM. Sakawa et al. [6] propose a fuzzy satisfying method for the solution of multi-objective linear continuous optimal control problems. They use fuzzy decision for combining the objectives into an LPP and obtain satisfying solution through the simplex method. Angelov [7] treats a new concept of the optimization problem under uncertainty which is an extension of fuzzy optimization in which the degrees of rejection of objective(s) and of constraints are considered together with the degrees of satisfaction. Ekel et al. [8] describe a general approach to solving a wide class of optimization problems containing fuzzy coefficients in objective functions and constraints. In 1999, Sakawa et al. [9] had proposed an interactive FMP for 0–1 MLPPs, in which the DMs are essentially cooperative, through genetic algorithms. In another paper [10] they use similar approach for MLPPs with fuzzy parameters.

Section snippets

Fuzzy mathematical programming approach to solve an MLPP

We consider a general linear maximization P-level hierarchical system and represent it as:maxx̄1f1(x̄)=c̄11x̄1+c̄12x̄2+⋯+c̄1Px̄P,maxx̄2f2(x̄)=c̄21x̄1+c̄22x̄2+⋯+c̄2Px̄P,maxx̄PfP(x̄)=c̄P1x̄1+c̄P2x̄2+⋯+c̄PPx̄P,s.t.Āi1x̄1+Āi2x̄2+⋯+ĀiPx̄P(⩽,⩾,=)bi,i=1,2,…,m,x̄1⩾0,x̄2⩾0,…,x̄P⩾0,

x̄1={x11,x12,…,x1n1}T: decision variables under the control of the Center,

x̄2={x21,x22,…,x2n2}T: decision variables under the control of DM on second level,

x̄P={xP1,xP2,…,xPnP}T: decision variables under the control of DM

Numerical examples

Example 1 Linear TLPP

maxx1,x2f1(x̄)=7x1+3x2−4x3+2x4,maxx3f2(x̄)=x2+3x3+4x4,maxx4f3(x̄)=2x1+x2+x3+x4s.t.x1+x2+x3+x4⩽5;x1+x2−x3−x4⩽2;x1+x2+x3⩾1−x1+x2+x3⩽1;x1−x2+x3+2x4⩽4;x1+2x3+3x4⩽3;x4⩽2;x̄⩾0

Example 2 Linear TLPP

maxx1,x2f1(x̄)=2x1+5x2−x3+x4+2x5,maxx3f2(x̄)=x2+3x3+x5,maxx4,x5f3(x̄)=4x1+2x3+2x4+2x5s.t.x1+x2+x3+x4⩽5;x1+x2−x3+x5⩽3;−x1+x2+x3⩽1x1−x3+x5⩽2;x1−x2+x3+2x4⩽4;x3+x4+x5⩽3;x̄⩾0

Example 1: We find that the optimal solution is f1=16.25 at (2.25,0,0,0.25); f2=5 at (0,1,0,1) and f3=5 at (1.33,1.5,0.83,0). This is not a satisfactory solution.

Conclusions

In the proposed algorithm the bounds of all the objective functions are so chosen that the satisfactory solution is forced towards the optimal solution. Therefore, we get better solution as compared to the procedure when one of the bound is taken to be 0. On taking the bound as 0, the satisfactory solution moves away from the optimal solution. The difficulty that arises is that the levels of the hierarchical system after the second level are not accounted for. Thus, in effect, the multi-level (P

Surabhi Sinha is a research scholar at Indian Institute of Technology, Kharagpur, India. She received her B.Sc., M.Sc. and Ph.D. degrees from I.I.T. Kharagpur. Her interests presently include multi-level, hierarchical, stochastic and possibilitic programming problems. Her papers have appeared in the Journal of Fuzzy Mathematics, OPSEARCH, Journal of Operational Research Society, and European Journal of Operations Research.

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Surabhi Sinha is a research scholar at Indian Institute of Technology, Kharagpur, India. She received her B.Sc., M.Sc. and Ph.D. degrees from I.I.T. Kharagpur. Her interests presently include multi-level, hierarchical, stochastic and possibilitic programming problems. Her papers have appeared in the Journal of Fuzzy Mathematics, OPSEARCH, Journal of Operational Research Society, and European Journal of Operations Research.

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