A multi-level non-linear multi-objective decision-making under fuzziness
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 xi∈Rni (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, .
Let F1:Rn1×Rn2×Rn3→RN1 be the first level objective functions and F2:Rn1×Rn2×Rn3→RN2 be the second level objective functions, and F3:Rn1×Rn2×Rn3→RN3 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]where x2, x3 solves
[2nd level]where x3 solves
[3rd level]where
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
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