A multi-level non-linear multi-objective decision-making under fuzziness

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Abstract

This paper studies a three-level non-linear multi-objective decision-making (TLN-MODM) problem with linear (or non-linear) constraints, and in which the objective function at every level are non-linear functions which are to be maximized. This paper makes an extension work of Abo-Sinna [J. Operat. Res. Soc. India (OPSEARCH) 38 (5) (2001) 484–495] which deal with a bi-level non-linear multi-objective decision-making problem under Fuzziness.

The three level programming (TLP) problem, whether from the stand point of the three-planner Stackelberg behavior or from the interactive organizational behavior, is a very practical problem and encountered frequently in actual practice.

This paper proposes a three-planner multi-objective decision-making model and solution method for solving this problem. This method uses the concepts of tolerance membership function and multi-objective optimization at each level to develop a fuzzy Max–Min decision model for generating Pareto optimal (satisfactory) solution for TLN-MODM problem; the first level decision maker (FLDM) specifies his/her objective functions and decisions with possible tolerances which are described by membership functions of fuzzy set theory. Then, the second level decision-maker (SLDM) specifies his/her objective functions and decisions, in the view of the FLDM, with possible tolerances which are described by membership functions of fuzzy set theory. Finally, the third level decision-maker (TLDM) uses the preference information for the FLDM and SLDM to solves his/her problem subject to the two upper level decision-makers restrictions. An illustrative numerical example is given to demonstrate the obtained results.

Introduction

Multi-level programming (MLP) is computationally more complex and expensive that conventional mathematical programming. This approach can be a powerful analytical tool once solutions difficulties are overcome.

Three-level programming (TLP) is a class of MLP problem in which there are three independent decision-makers (DMs) (see [3], [8], [13], [15]). Each DM attempts to optimize its objective function and is affected by the actions of the others DMs. The characteristics of TLP problem are summarized as follows (see [4], [5], [6], [7]):

  • (1)

    The system has interacting decision-making units with a hierarchal structure.

  • (2)

    The third level decision-maker (TLDM) executes its policies after, and in view of the decision of the second level decision maker (SLDM); who do the same action with the first level decision maker (FLDM).

  • (3)

    For every level optimizes its own objective independently of other levels, but is affected by the actions and reactions of other levels.

  • (4)

    The effect of the FLDM on the lower level problem is reflected in both its objective function and the set of feasible decision.


Several three level programming problems and their solution method have been presented, such as, the hybrid extreme-point search algorithm [7], [11], mixed-integer problem with complementary slackness [11], and the penalty-function approach [2], [11].

These approaches have been used widely in searching for the optimal solutions (see [11]). Apart from these approaches, fuzzy sets has been employed to formulate and solve three-level non-linear multi-objective decision-making (TLN-MODM).

In this paper, we use the concepts of membership functions as well as multi-objective optimization, at each level, to develop a fuzzy Max–Min decision model (see [1], [10]) for solving the TLN-MODM problem. We extend Abo-Sinna work [1] to TLN-MODM problem and propose that the FLDM defines his/her objective functions and decisions with possible tolerances, which are described by linear membership functions of fuzzy set theory and fuzzy decision. This information is delivered to the SLDM who defines his/her objective functions and decisions with possible tolerances in view of the FLDM. Finally, the TLDM solves his/her problem under restrictions of the FLDMs and SLDMs requirements. Then, the TLDM presents his/her solution to the FLDM. If the FLDMs rejects this proposal, the FLDM must update and change former goals and decisions as well as their corresponding tolerances also the SLDM must do the same until a satisfactory solution is reached.

Section snippets

Problem formulation and solution concept

Let xiRni (i=1,2,3) be a vector variables indicating the first decision level's choice, the second decision level's choice, and the third decision level's choice, ni⩾1(i=1,2,3).

Let F1:Rn1×Rn2×Rn3RN1 be the first level objective functions and F2:Rn1×Rn2×Rn3RN2 be the second level objective functions, and F3:Rn1×Rn2×Rn3RN3 be the third level objective functions, N1, N2, N3⩾3.

Let the FLDM, SLDM and TLDM have N1, N2 and N3 objective functions, respectively. Let G be the set of feasible choices

Fuzzy decision models for TLN-MODM problem

To solve the TLN-MODM by adopting, the three-planner Stakelberg (see [10], [11]), and the well-known fuzzy decision model of Sakawa [12]. One first gets the satisfactory solution that is acceptable to FLDM, and then give the FLDM decision variables and goals with some leeway to the SLDM for him/her to seek the satisfactory solution. Then the SLDM give the decision variables and goals with some leeway to the TLDM for him/her to seek the satisfactory solution, and to arrive at the solution which

Numerical example for TLN-MODM problem

To demonstrate the solution method for TLN-MODM, let us consider the following example.

  • [1st level]Maxx1F1(x1,x2,x3)=Maxx1[(x1−3)2+x22+x32,(x1−2)2+x22+x32],where x2, x3 solves

  • [2nd level]Maxx2F2(x1,x2,x3)=Maxx2[(x1−2)2+(x2+1)2+(x3−1)2,(x1−1)2+(x2+2)2+x32],where x3 solves

  • [3rd level]Maxx3F3(x1,x2,x3)=Maxx3[(x1−3)2+(x2−1)2+(x3+1)2,(x1−2)2+(x2+2)2+(x3+1)2]s.t.(x1,x2,x3)∈G={(x1,x2,x3)|2x1+x2+x3⩽8,x1+2x2+x3⩽6,x1,x2,x3⩾0},wheref11=(x1−3)2+x22+x32,f12=(x1−2)2+x22+x32,f21=(x1−2)2+(x2+1)2+(x3−1)2,f22=(x1−1)

Summary and concluding remarks

This paper has proposed a three-planners non-linear multi-objective decision-making TLN-MODM model and a solution method for solving this problem. This solution method uses the concepts of tolerance membership functions and multi objective optimization at every level to develop a fuzzy Max–Min decision model for generating Pareto optimal (satisfactory) solution for (TLN-MODM).

Based on Abo-Sinna [1] satisfactory solution concepts, the proposed solution method precede form FLDM to the TLDM

References (15)

  • G. Anandalingam et al.

    Multi-level programming and conflict resolution

    European Journal of Operational Research

    (1991)
  • M.A. Abo-Sinna

    A bi-level non-linear multi-objective decision-making under Fuzziness

    Journal of Operational Research Society (OPSEARCH)

    (2001)
  • G. Anandolingam

    A mathematical programming model of decentralized multi-level system

    Journal of Operational Research Society

    (1988)
  • J.F. Bard

    An efficient point algorithm for a linear two-stage optimization problem

    Operations Research

    (1983)
  • J.F. Bard

    Optimality conditions for the bi-level programming problem

    Naval Research Logistics Quarterly

    (1984)
  • W.F. Bialas et al.

    On two-level optimization

    IEEE Transactions on Automatic control

    (1982)
  • W.F. Bialas et al.

    Two-level linear programming

    Management Science

    (1984)
There are more references available in the full text version of this article.

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