Continuous Optimization
A goal programming procedure for solving problems with multiple fuzzy goals using dynamic programming

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Abstract

This paper describes how the preemptive priority based goal programming (GP) can be used to solve a class of fuzzy programming (FP) problems with the characteristics of dynamic programming (DP).

In the proposed approach, the membership functions of the objective goals of a problem with fuzzy aspiration levels are defined first. Then, under the framework of preemptive priority based GP a multi-stage DP model of the problem for achievement of the highest degree (unity) of each of the membership functions is developed. In the decision process, the goal satisficing philosophy of GP is used recursively to arrive at the most satisfactory solution.

Two numerical examples are provided to illustrate the approach.

Introduction

The concept of goal programming (GP) was first introduced by Charnes and Cooper in 1961 [5] as a tool to resolve infeasible linear programming (LP) problems. Thereafter, the significant methodological development of GP was made by Ijiri [30], Lee [34] and Ignizio [20] and others. As a promising tool for solving problems involving multiple conflicting objectives, GP has been studied extensively in [9], [21], [24], [27], [32], [43], [44], and widely circulated in the literature [7], [19], [28], [59], [71]. Since 1970s, GP has been successively implemented to several real-world multiple objective decision making (MODM) problems in [22], [33], [37], [46], [49], [55], [56], [61]. The methodological extensions of GP and its application to real-life problems have been surveyed by Kornbluth [31], Lin [37] and Romero [50].

In the field of GP, the preemptive priority based GP, introduced by Ijiri in 1965 [30], is actually the most prominent tool for solving MODM problems. Ignizio [23], [26], [27], [28], [29] has become a pioneer and main contributor in this area. In preemptive priority based GP, the goals are rank-ordered according to their priorities of achieving the aspiration levels assigned to them in the decision making context.

The primary advantage of a GP approach is that it leads to arrive at an acceptable compromise solution directly. However, the main weakness of GP is that the aspiration levels of the goals need be specified precisely in making a decision. But, in such a situation, decision maker (DM) is always faced with the problem of assigning the definite aspiration levels to the goals, because they are imprecise in nature for the DM.

To overcome the above difficulty, the concept of the fuzzy set theory, initially proposed by Zadeh in 1965 [70], has been introduced in the field of conventional MODM problems, where aspiration levels of the goals are assigned in an imprecise manner. A fuzzy goal considered here is a goal with an imprecise aspiration level.

The concept of fuzzy mathematical programming in a general level was first proposed by Tanaka et al. [60] in the framework of fuzzy decision of Bellman and Zadeh [3]. Thereafter, FP approach to LP with several objectives was introduced by Zimmermann in 1978 [72]. During the last two decades, various aspects of MODM problems using FP have been investigated in [4], [10], [11], [12], [13], [14], [16], [35], [38], [51], [53], [54], [57], [68], [69], [73], [74], [75].

The use of fuzzy set theory in GP was first studied by Narasimhan [40] and further developed by Hannan [17], Narasimhan [41], and Ignizio [25]. During the last two decades, fuzzy goal programming (FGP) has been deeply investigated in [6], [48], [52], [62]. However, in the approaches, MODM problems are transformed into the equivalent LP problems by using the max–min operator in [3]. The major drawback of such an approach is that the actual solution for achievement of the aspired levels of the goals according to their relative importance in the decision making situation cannot be obtained, because the goals are often conflicting in nature in most of the real-world MODM problems, and (single-objective) LP approach to such a problem is not appropriate in actual practice.

To overcome the above drawback, Hannan [18] has proposed an additive model for solving FGP problems. In his approach, the membership functions for the fuzzy goals are expressed as achievable goals by means of assigning elicited membership values (not unity) as their aspiration levels. Then the deviational variables of the membership goals are minimized by using conventional GP approach to achieve the aspired levels of the goals. But, in actual practice, it is difficult to assign such aspiration levels to the membership functions while the aspiration levels of the corresponding goals themselves are fuzzy. Also, in his model, the membership functions are not freely allowed to achieve their highest degree.

Tiwari et al. [63] have proposed an alternative additive model for maximizing the membership functions directly. But in their approach, the problem for achievement of each of the membership functions to its highest value by minimizing the deviational variable of the corresponding membership goal has not been considered.

In the recent past, Mohamed [39] has investigated some new FP forms by using the concept of conventional GP approach, and further studied by Pal and Moitra [47]. In the present investigation, the problem of stage-wise achievement for the highest aspiration level (unity) of each of the membership goals in the framework of preemptive priority based GP is considered.

During the last three decades, the methodology known as DP, introduced by Bellman [2], has been used for modelling and solving single-objective as well as multiple-objective multi-stage decision problems in [8], [64], [65], [67]. The use of DP for solving GP problems has been investigated by Levary [36] and Pal and Basu [45].

The DP formulation of fuzzy decision problems was first investigated by Bellman and Zadeh in 1970 [3]. Thereafter, fuzzy decision policies for multi-stage decision problems have been studied in [1], [15], [58], [66]. However, works in this area are yet limited.

This paper presents a preemptive priority based GP procedure to solve a class of FGP problems having a set of linear and/or non-linear fuzzy goal objectives with the characteristics of DP and a set of linear system constraints. In model formulation, instead of optimizing the membership functions directly, the problem of achieving the highest aspiration level of each of the membership goals by minimizing the deviational variable of it in the goal achievement function is taken into consideration. In the solution process, the recursive relations in DP incorporated to make a stage-wise decision on the basis of the priorities of the goals in the decision making context.

Section snippets

Formulation of dynamic FGP problem

In FGP, aspiration levels of the objective goals are always considered as fuzzy, whereas the right-hand side values of the system constraints may be precisely or imprecisely defined that depend on the fuzziness of the decision environment. To formulate the multi-stage FGP model of the problem, the system constraints are treated here as defining a conventional (crisp) feasible solution set over which achievement of the fuzzy goals to their aspired levels is determined.

The generic FGP formulation

Multi-stage FGP approach

To formulate the multi-stage decision making model of the problem in (2.5), the following definitions are introduced.

Let XjS designate the set of all feasible solutions of xj. Then from the system constraints in (2.1), the equivalent set of state constraints can be obtained asyj−1,i=tji(yji,xj),j=1,2,…,n,i=1,2,…,m,where yji and tji are the jth state variable and jth state transformation, respectively, which correspond to the ith system constraint in (2.1).

Then for the defined state constraints

Illustrative examples

Example 1

To illustrate the potential use of the proposed solution approach, a modified version of the problem presented by Nemhauser [42] is considered.


FindX(x1,x2)so as to satisfyG1:5x1+2x1255,G2:4x2+3x2241,G3:7.5x2+x2345,G4:x14,G5:x23,subject tox1+2x2⩽10,x1⩽10,x2⩽5,x1,x2⩾0andintegers.Let the tolerance ranges of the five fuzzy goals be (20, 8, 15, 2, 1), respectively. Then the lower tolerance limit of the first goal (G1) will be 35. The lower and upper tolerance limits of the second goal (G2) will be

Conclusion

This paper illustrates the potential use of preemptive priority based GP for solving a class of multi-objective FP problems with the characteristics of DP. The use of existing FP approaches to such a problem frequently involves computational complexity due to the typical non-linear nature of objectives in most of the fuzzy decision situations. Such a difficulty does not arise with the proposed approach for using the goal satisficing philosophy of GP in the process of making decision.

The primary

Acknowledgements

The authors would like to thank the Editor and the Referees for their helpful suggestions and comments which have led to an improvement in both the quality and clarity of this paper.

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