Accepted Manuscript

An analysis of fully fuzzy linear programming with fuzzy decision variables through logistics network design problem

Adil Baykasog˘lu , Kemal Subulan

PII: S0950-7051(15)00359-7

DOI: 10.1016/j.knosys.2015.09.020

Reference: KNOSYS 3281

To appear in: Knowledge-Based Systems

Received date: 24 December 2014

Revised date: 30 July 2015

Accepted date: 19 September 2015

Please cite this article as: Adil Baykasog˘lu , Kemal Subulan , An analysis of fully fuzzy linear programming with fuzzy decision variables through logistics network design problem, Knowledge-Based

Systems (2015), doi: 10.1016/j.knosys.2015.09.020

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An analysis of fully fuzzy linear programming with fuzzy decision variables through logistics network design problem

Adil Baykasoğlu1, Kemal Subulan

Dokuz Eylül University, Faculty of Engineering, Dept. of Industrial Engineering, Izmir, Turkey

A B S T R A C T

Recently, there is a growing attention by the researchers to solve and interpret the analysis of fully fuzzy linear programming problems in which all of the parameters as well as the decision variables are considered as fuzzy numbers. Under a fully uncertain environment where all of the data are stated as fuzzy, presenting the reasonable range of values for the decision variables may be comparatively better than the currently available crisp solutions so as to provide ranges of flexibility to decision makers. However, there is still a scarcity of solution methodologies on fuzzy mathematical programs with fuzzy decision variables. Based on this motivation, a new parametric method which is mainly based on α-cut representation of fuzzy intervals is proposed in this paper by incorporating the decision maker’s attitude toward risk. In order to illustrate validity and practicality of the proposed method, it is applied to a generic reverse logistics network design model including fuzzy decision variables. To the best of our knowledge, this is the first study in the literature which presents fuzzy efficient solutions and analysis for a fully fuzzy reverse logistics network design problem with fuzzy decision variables. The provided solutions by the proposed method are also compared to the available solution methodologies from the literature in terms of computational efficiency, solution quality and ease of use. By using the proposed method, the decision makers can be supported by yielding fuzzy efficient solutions under different uncertainty levels and risk attitudes. The computational results have also shown that more reliable and necessarily precise solutions can be generated by the proposed method for a risk-averse decision maker.

Keywords: Fully fuzzy linear programming, Reverse logistics network design, Fuzzy decision variables, Fuzzy mathematical programming, Risk attitude. 1. Introduction

In the literature, most of the fuzzy mathematical programming models deal with the fuzziness related to the aspiration levels of the objective(s) and some of the model parameters, i.e., the objective function(s)’ coefficients, technological coefficients and right hand side values of the constraints. Because, available information on these parameters may not be precise and/or the decision maker(s) generally may not precisely know the values of these parameters. On the other hand, fully fuzzy linear programming (FFLP) problems in which all of the parameters and decision variables are stated as fuzzy numbers has been an attractive topic for the researchers in recent years. However, the literature on fuzzy mathematical * Corresponding author, e-mail: adil.baykasoglu@deu.edu.tr

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T 2 programming with fuzzy decision variables is not rich as there are a few studies available on this research topic. In fact, decision variables of a fuzzy mathematical program where all of the parameters are stated as fuzzy should also be considered as fuzzy numbers. Because, the fuzzy characteristic of the decision may be partially lost and the decision making process is limited with the crisp solutions when the decision variables of this problem are crisp [5]. For this reason, instead of crisp solutions, obtaining fuzzy efficient solutions which provide ranges of flexibility to decision maker(s) seems more impressive in fully uncertain environments [15]. In other words, presenting the reasonable range of values for the decision variables can be comparatively better than the currently available crisp solutions [36]. In addition, fuzzy solutions may serve the regions containing potential satisfactory solutions around the optimal solutions to the decision maker(s). Thus, the final decisions can be made by the decision maker(s) as crisp ones. By taking the decision variables as fuzzy, the choice of the crisp decisions among the fuzzy solutions will also be supported [34]. Furthermore, it was emphasized by Xiaozhong et al. [40] that it is possible to encounter some optimization problems such as the cardinality of optimal solutions and the number of fairly superior solutions for a fuzzy linear program. Nevertheless, achieving the optimal solutions of a FFLP problem is a challenging task since there is no algorithm which determines the exact values of the optimal fuzzy decision variables [10].

In this study, a fully fuzzy reverse logistics (RL) network design problem whose parameters as well as the decision variables are taken as fuzzy numbers is discussed. Because, the RL network design problems are surrounded by many uncertainties on product returns, capacity of the recovery facilities, disposal and recovery rates. Actually, these are the main essential factors contributed to the uncertainty in RL environments [42]. In other words, uncertainty in timing, quality and quantity of product returns are important aspects of RL network design problems. The decision variables of a fully fuzzy RL network design problem should also be stated as fuzzy numbers because of the following reasons: There are two important sources of uncertainty regarding to collection process in a RL network design problem. The first one is the uncertainty related to the quantity of returned products from customer zones and the second one is the capacity of collection/inspection centres. Moreover, uncertain amounts of the collected products may be lost or perished because of the several reasons during the transportation process. Thus, collection quantities should be stated as fuzzy decision variables in a RL network design model. Similarly, quantities of recyclable and disposal products should also be defined as fuzzy decision variables due to the uncertainties regarding disposal rate, conformity/acceptance ratio, recycling and disposal capacities. Finally, the amounts of recycled materials/components should also be stated as fuzzy decision variables since including some sort of ambiguity on the recovery rates. Since the output of any stage in a RL network constitutes the inputs of the consecutive stages, fuzziness in one of the decision variables will also causes uncertainty into the other decision variables. For instance, if the collection quantities are defined as fuzzy, the amounts of recyclable and disposed products should also be stated as fuzzy. Because, the output of the collection stage will be used as inputs in the recovery and disposal stages.