Multi-objective non-linear programming problem in intuitionistic fuzzy environment: Optimistic and pessimistic view point

Highlights

• Multi-objective optimization problem under intuitionistic fuzzy environment has been considered.

• Degree of attainability and non-attainability is represented by non-linear functions.

• Optimistic and pessimistic aspect of the problem has been considered.

• The validity of the problem has been discussed.

• multiobjective problems arising in the transportation and manufacturing systems have been taken for demonstration.

Abstract

The objective of this manuscript is to present an algorithm for solving multi-objective optimization problem under the optimistic and pessimistic view point. The conflicting natures of the different objective have been handled by defining the membership functions corresponding to it in parabolic intuitionistic fuzzy set environment and thus the problem becomes parabolic multi-objective non-linear optimization programming problem (PMONLOPP). A linear and non-linear membership functions corresponding to each objective has been taken in account. An illustrative examples from transportation as well as in manufacturing systems are reported and compared with the typical approaches exist in the literature. As shown, the solutions obtained by the proposed approach are superior to those of existing best solutions reported in the literature. Further-more, experimental results indicate that the proposed approach may yield better solutions to these types of problems than those obtained by using current algorithms.

Introduction

In real life decision making process, the systems and their corresponding decisions are becoming complicated day by day and hence it will be difficult for the decision maker’s for getting an accurate decision within a reasonable time. Moreover, the system depends on so many factors that contains a lot of uncertainties during the collection of data phase. Thus, the analysis corresponding to these data are also uncertain, vague and imprecise. In order to handle these uncertainties and impreciseness in the data, Attanassov (1986) presented intuitionistic fuzzy sets (IFSs), which is an extension of fuzzy set (Zadeh, 1965), and are characterized by a membership degree, a non-membership degree and a hesitancy degree. Angelov (1997) gave an application of the IFSs to optimization problems. His technique is based on maximizing the degree of membership (satisfaction), minimizing the degree of non-membership (dissatisfaction) and the crisp model is formulated using the IF aggregation operator (Bellman & Zadeh, 1970). While addressing real world problems, a multi-objective model with fuzzy parameters is more realistic than the one with deterministic. A solution that optimizes all the objectives simultaneously is rarely possible. Hence, an aspiration level for each objective is decided depending upon the decision maker’s choice. Obtaining the exact aspiration level is not necessary always and one tries to find a solution as close as possible to the decided aspiration level. In most of the fuzzy multi-objective optimization literature, this is done by maximizing the degree of membership function for each of the objective. Furthermore, IFSs has been proven to be highly useful to deal with uncertainty and vagueness, and hence by applying this concept, it is possible to reformulate the optimization problem by using degree of rejection of the constraints and the value of the objective which are non-admissible. Pramanik and Roy (2005) solved a vector optimization problem using an intuitionistic fuzzy goal programming. Garg (2013) proposed a technique for analyzing the behavior of industrial systems in terms of various reliability parameters using vague set theory. Chakrabortty, Pal, and Nayak (2013) proposed a method to solve the multi-objective EPQ inventory model with fuzzy inventory costs and fuzzy demand rate and the problem is solved with IFO technique. Garg and Rani (2013) presented an efficient technique for computing the membership functions of various reliability parameters using PSO and IFS theory. Garg and Sharma (2013) presented a method for solving multi-objective optimization problem using fuzzy and particle swarm optimization techniques. Also, Garg (2015) presented a hybrid GA-GSA algorithm for finding the optimal solution by utilizing the uncertain and vague information. Apart from that a lot of work has been done to develop and enrich the IFS theory given in Bit, Biswal, and Alam (1993); Chakraborty, Jana, and Roy (2015a); 2015b); Chandra and Aggarwal (2014); Dey and Roy (2015); Dubey, Chandra, and Mehra (2012); Gani and Abbas (2014); Garg (2016a); 2016b); Garg and Rani (2013); Garg, Rani, and Sharma (2013); 2014); Garg, Rani, Sharma, and Vishwakarma (2014a); 2014b); Hadi-Vencheh, Hejazi, and Eslaminasab (2012); Huang, Gu, and Du (2006); Hussain and Kumar (2012); Islam and Roy (2006); Jana and Roy (2007); Kumar and Hussain (2014); 2015); Kundu, Kar, and Maiti (2013); Nehi and Hajmohamadi (2012); Pandian (2014); Singh, Abdullah, Mohamed, and Noorani (2015); Singh and Yadav (2014); 2015); Stanciulescu, Fortemps, Install, and Wertz (2003).

In particular, most of the research on transportation problems is limited to the single or multi-objective fuzzy linear programming. But, in today, most of the real-world decision-making problems in economic, technical and environmental ones are multi-dimensional and multi-objective. It is significant to realize that multiple objectives are often non-commensurable and conflict with each other in the optimization problem. An objective within exact target value is termed as fuzzy goal. So a multi-objective model with fuzzy objectives is more realistic than the one with deterministic. However, if the decision maker has some preference or biasness towards a particular objective, then a linear membership function may not serve the purpose. Thus, a conflicting nature between the objectives is resolved with the help of defining their non-linear membership functions. Also, there may exist situations when the problems can be modeled only as non-linear fuzzy optimization problems which may not be modeled and solved efficiently by using the traditional techniques. Thus, in such circumstances, there is a need to modify the approach by taking into the account of decision maker preferences towards the objective in both optimistic and pessimistic view.

Motivated by this idea, in this study an algorithm has been proposed for solving multi-objective problems in fuzzy environment. IFS theory has been used for resolving the conflict nature between the objectives where the degree of attainability and non-attainability of objectives are represented by nonlinear functions. Based on their corresponding membership functions, in the present work, the problem has been analyzed in two different ways - the optimistic and the pessimistic, and is explained in the subsequent sections. The model so obtained is solved to obtain the pareto optimal solution. The proposed technique has been illustrated through manufacturing and transportation problems. To the best of our knowledge, no one has applied the concept of considering membership as well as non-membership function for these problems. Moreover, the extension of this work over the others is to consider the input parameters as non-linear numbers instead of the fixed or linear ones.

Rest of the paper is organized as follows: The next section contains some preliminaries to be used later in the paper. Parabolic fuzzy number has been defined and their ordering is suggested using the preference function. The model for the crisp as well as fuzzy multi-objective non-linear programming problem (MONLPP) has been presented in Section 3. Section 4 explains the different views of the decision maker and summarizes the solution algorithm. The numerical illustration of the proposed technique is given in Section 5 by solving two different MONLPPs. The paper ends with conclusions and future scope in Section 6.