Abstract

Fuzzy goal programming (FGP) is applied to solve fuzzy multi-objective optimization problems. In FGP, the weights are associated with fuzzy goals for the preference among them. However, the hierarchy within the fuzzy goals depends on several uncertain criteria, decided by experts, so the preference relations are not always easy to associate with weight. Therefore, the preference relations are provided by the decision-makers in terms of linguistic relationships, i.e., goal A is slightly or moderately or significantly more important than goal B. Due to the vagueness and ambiguity associated with the linguistic preference relations, intuitionistic fuzzy sets (IFSs) are most efficient and suitable to handle them. Thus, in this paper, a new fuzzy goal programming with intuitionistic fuzzy preference relations (FGP-IFPR) approach is proposed. In the proposed FGP-IFPR model, an achievement function has been developed via the convex combination of the sum of individual grades of fuzzy objectives and amount of the score function of IFPRs among the fuzzy goals. As an extension, we presented the linear and non-linear, namely, exponential and hyperbolic functions for the intuitionistic fuzzy preference relations (IFPRs). A study has been made to compare and analyze the three FGP-IFPR models with intuitionistic fuzzy linear, exponential, and hyperbolic membership and non-membership functions. For solving all three FGP-IFPR models, the solution approach is developed that established the corresponding crisp formulations, and the optimal solution are obtained. The validations of the proposed FGP-IFPR models have been presented with an experimental investigation of a numerical problem and a banking financial statement problem. A newly developed distance measure is applied to compare the efficiency of proposed models. The minimum value of the distance function represents a better and efficient model. Finally, it has been found that for the first illustrative problem considered, the exponential FGP-IFPR model performs best, whereas for the second problem, the hyperbolic FGP-IFPR model performs best and the linear FGP-IFPR model shows worst in both cases.

Highlights

  • Multi-objective programming is a robust analytical method for the formulation of real-world problems in which two or more than two goals have to be optimized, simultaneously

  • Based on the above discussion, we have proposed fuzzy goal programming with intuitionistic fuzzy preference relation (FGP-intuitionistic fuzzy preference relations (IFPRs)) among fuzzy goals to model the problems with uncertainties and ambiguities more realistically

  • In the proposed FGP-IFPR model, we have presented the linear and non-linear membership function (MF), which are used in the representation of the intuitionistic fuzzy preference relations (IFPR)

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Summary

Introduction

Multi-objective programming is a robust analytical method for the formulation of real-world problems in which two or more than two goals have to be optimized, simultaneously. FGP models dealing the imprecise preference relations with maximization of the belongingness degree They do not consider the uncertainty causes by vagueness, incorrect data, and deliberate decisions. Based on the above discussion, we have proposed fuzzy goal programming with intuitionistic fuzzy preference relation (FGP-IFPR) among fuzzy goals to model the problems with uncertainties and ambiguities more realistically. In the proposed FGP-IFPR model, we have presented the linear and non-linear MFs, which are used in the representation of the intuitionistic fuzzy preference relations (IFPR). A novel fuzzy goal programming with intuitionistic fuzzy preference relation named as FGP-IFPR is proposed for multi-objective programming problems. For the extensive analysis and comparison, three different FGP-IFPR models are formulated corresponding to intuitionistic fuzzy linear, exponential and hyperbolic functions.

Preliminaries
Fuzzy Goal Programming with Intuitionistic Fuzzy Preference Relations
Basic Fuzzy Goal Programming Model
Fuzzy Preference Relations under Intuitionistic Fuzzy Enviourment
Linear Membership and Non-Membership Function
Exponential Membership and Non-Membership Function
Hyperbolic Membership and Non-Membership Function
Proposed FGP-IFPR Model
Solution Approach
Experimental Study
Example 1
Solution Steps for Example 1
Results and Discussion for Example 1
Example 2
Solution Steps for Example 2
Results and Discussion for Example 2
Efficiency Analysis
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