Generalized cross-entropy based group decision making with unknown expert and attribute weights under interval-valued intuitionistic fuzzy environment

Highlights

• MAGDM with unknown attribute and expert weights with preference in IVIFSs is studied.

• Generalized cross-entropy for IVIFSs is given to cover extra DM’s characteristics.

• We use the cross-entropy measure to form several models to ensure unknown weights.

• For illustration purpose, application example is studied.

• Results show devised methods can use DM’s preference more completely and objectively.

Abstract

This paper focuses on the multiple attribute decision making problems widespread in industry engineering, typically the supplier selection problems, and investigates effective methods utilizing preference information objectively for multiple attribute group decision making (MAGDM) with unknown attribute weights and expert weights under interval-valued intuitionistic fuzzy environments (IVIFEs). Firstly, a novel generalized cross-entropy measure for interval-valued intuitionistic fuzzy sets (IVIFSs) is proposed to enable decision makers to express attitudinal characteristic information of hesitation degree rather than arbitrary equal assignment by other cross-entropy methods. Further, based on the generalized cross-entropy measure, simultaneously considering the divergence of attribute assessments from the most fuzzy number in IVIFSs and the deviation between attribute assessments, a maximizing optimization model is proposed for objectively obtaining unknown attribute weights, which is then extended to accommodate decision situations with incomplete attribute weighting information. And also based on the generalized cross-entropy measure, an integrated algorithm is developed for ensurement of unknown expert weights by fusing two optimization models: one is to maximize divergence of decision matrices from positive or negative ideal decision matrix, and the other is to minimize similarity degree between individual decision matrices. Then based on the aforeproposed models, an approach is constructed for MAGDM problems with unknown attribute and expert weights under IVIFEs. Finally, case study on a simplified but representative supplier selection problem is carried out, and comparative experiments indicate the practicality and effectiveness of proposed methods.

Introduction

To survive from today’s fierce competitive environment, companies are forced to cooperate with supply chain partners to improve the chain’s total performance, resulting in increased competitiveness, customer satisfaction and profitability (Sanayei, Mousavi, & Yazdankhah, 2010). Being the main process in the upstream chain and affecting all areas of an organization, the purchasing function is taking an increasing importance (Boran, Genc, Kurt, & Akay, 2009). As the most important activity of purchasing function, selecting appropriate suppliers can significantly decrease purchasing costs and increase the organization competition ability because costs of raw materials and parts for a large portion of final product price in most industries, as a result, supplier selection problem has received more and more attentions from both academia and industry (Roshandel, Miri-Nargesi, & Hatami-Shirkouhi, 2013).

Supplier selection problem is normally recognized as a kind of multiple criteria decision making (MCDM) problem, and several categories of existent approaches can be found in literatures for tackling this problem (Chai et al., 2013, Roshandel et al., 2013, Sanayei et al., 2010): multiple attribute decision making (MADM) methods, mathematical programming methods, statistics/probabilistic methods, intelligent methods, and hybrid ones. Especially, MADM methods are the most widely adopted approaches for the analysis of supplier selection problems and have been accumulatively studied under different decision environments for last decades (Chai et al., 2013, Sanayei et al., 2010). Because of the complex and unconstructed nature and context of many real world supplier selection problems (Boran et al., 2009), decision preference usually have to be expressed by use of uncertain modelling tools (Roshandel et al., 2013), such as traditional fuzzy set theories including triangular fuzzy set and trapezoidal fuzzy set. However, increasing complexity of socioeconomic environments brings about the increasing uncertainty in preference of decision makers (DMs) and the incompetence of single DM for considering all the relevant aspects of supplier selection problems. So recently, several researches have started to address novel MADM methods (Chai et al., 2012, Chen, 2011) and multiple attribute group decision making (MAGDM) methods (Boran et al., 2009, Khaleie and Fasanghari, 2012) for supplier selection problems of high uncertainty based on intuitionistic fuzzy sets (IFSs) theory (Atanassov, 1986, Chen and Yang, 2012), which can depict decision uncertainty multifacetedly by membership degree, nonmembership degree and hesitation degree. But there is only few researches can be found in the literatures concerning supplier selection by utilizing interval-valued intuitionistic fuzzy sets (IVIFSs) theory (Atanassov & Gargov, 1989), which can include the merits of IFSs and can handle decision uncertainty more flexibly by providing membership and non-membership function with interval-values rather than exact numbers in IFSs. Such as, Wang, Li, and Xu (2011) proposed a single-person MADM method for supplier selection in light of TOPSIS framework, Chen, Wang, and Lu (2011) presented a multicriteria group decision making approach based on a model for obtaining subjective criteria weights, and Ye (2013) proposed a multiple attribute group decision making approach integrating a simple traditional entropy based method for ensurement of both attribute and expert weights, but without considering the DMs’ preference on hesitation degrees. Therefore, there is a significantly important need to further investigate more effective and suitable MADM methods utilizing IVIFSs theory for better handling supplier selection problems of increasingly high ambiguity and uncertainty, especially in group settings.

Generally in the process of MAGDM, aggregation operators (Merigó and Casanovas, 2011a, Merigó and Casanovas, 2011b, Merigó and Gil-Lafuente, 2009), attribute weighting methods and expert weighting methods can be outlined as three key aspects (Wang et al., 2011). As for aggregation operators for MAGDM under IVIFEs, plenty achievements have gained in recent years (Xu & Cai, 2010b), among which several most intensively investigated types of aggregation operators can be outlined as the weighted/ordered weighted aggregation operators (Wu & Liu, 2013), the correlated aggregation operators (Wei and Zhao, 2012, Xu, 2010a), the induced aggregation operators (Su et al., 2012, Wei, 2010, Xu and Xia, 2011, Zhang and Qi, 2012), the generalized aggregation operators (Qi, Liang, & Zhang, 2013) and the hybrid aggregation operators (Liu, 2013, Zhang and Qi, 2012).

Depending on the information acquisition in attribute weighting methods for MAGDM with IVIF information, attribute weights can be categorized as subjective and objective weights (Hwang and Lin, 1987, Ma et al., 1999). Subjective weights are obtained from preference relations information in DM’s subjective comparative judgments matrices on specific attributes (Szmidt and Kacprzyk, 2003, Szmidt and Kacprzyk, 2005), while objective weights are derived from the information in decision matrices through mathematical models (Chen & Li, 2011). Most current related researches are conducted to study MAGDM in an IVIF environment integrating subjective weighting methods based on certain preference relations information, such as additive and multiplicative consistent intuitionistic relations (Gong et al., 2011, Gong et al., 2009, Xu, 2007a, Xu, 2012), consistent interval-valued intuitionistic preference relations (Chen et al., 2011, Wang et al., 2009), incomplete IVIF preference relations (Wang & Li, 2012). Comparatively, there are still only few researches on the extensions of objective weighting methods in traditional MAGDM to intuitionistic fuzzy environments, such as deviation functions (Wang, 1997, Wei, 2008; Xu, 2010b, Xu and Cai, 2010a) and fuzzy entropies (Burillo and Bustince, 1996, Hung and Yang, 2006, Szmidt and Kacprzyk, 2001, Vlachos and Sergiadis, 2007, Zeng and Li, 2006). And therein, especially due to the intrinsical suitability for measurement of fuzziness, fuzzy entropies have been increasingly employed and studied to comprehensively consider the decision preference for allocating attribute weights more reasonably. Ye (2010a) firstly designed a straight method in light of traditional fuzzy entropy to obtain attribute weights in an IF environment. Then some novel optimization models (Chen and Li, 2010, Chen and Li, 2011, Xia and Xu, 2012) were proposed to ensure objective attribute weights by modified intuitionistic fuzzy entropies and cross-entropies to additionally accommodate hesitation degree in an IF number rather than only consider the information of membership and non-membership degree. In respect of IVIF environments, only Ye, 2010b, Ye, 2013 proposed an attribute weighting method based on traditional IVIF entropies, but did not consider the hesitation information. So it is still a blank but vital to investigate suitable approaches for deriving objective attribute weights utilizing entropies or cross-entropies to comprehensively accommodate information of membership, non-membership and hesitation degree in decision preference under IVIF environments.

There is another strong necessity in the process of MAGDM to indicate the insufficient competence of DMs for all aspects of decision problems or to indicate the different influential saliencies of DMs (Yue, 2011b), so expert weighting is also significantly and fundamentally important to MAGDM under IVIF environments. However, just only few works have been carried out and all unanimously emphasized the utilization of objective information in IVIF decision matrices to derive appropriate expert weights. Such as, the nonlinear optimization models by minimizing the divergence between individual opinions and group opinions (Xu & Cai, 2010a), the method for deducing expert weights according to the distances between individual matrices and mean-value group decision matrices (Chen & Yang, 2011), the method depending on divergence between individual matrices and ideal group decision matrices (Yue, 2011a), and the models based on traditional IVIF entropies (Ye, 2013) without considering preference on hesitation degrees, but there is still no research till now that investigates approaches for obtaining expert weights objectively through cross-entropy measure to utilize preference information in IVIF decision evaluations more comprehensively.

The aim of this paper is to investigate effective MAGDM methods for the complex decision making problems like supplier selection problems in group settings where DM’s preference is denoted with IVIF information and attribute/expert weights are both very difficult to obtain beforehand. We propose a novel generalized IVIF cross-entropy measure that can comprehensively accommodate all the objective information of membership, nonmembership and hesitation degrees in IVIF evaluations, and also can flexibly reflect the DMs’ attitudinal characteristics on how hesitation degrees support membership and nonmembership functions. Based on the generalized IVIF cross-entropy measure, two programming models are developed to derive unknown attribute weights by simultaneously considering the credibility of IVIF evaluations and the deviation between attribute assessments, also another algorithm is devised for deriving unknown expert weights by combining two optimization models to concurrently consider the maximum divergence of individual decision matrices from ideal decision matrix and the minimum similarity between individual decision matrices. Then an effective MAGDM approach is constructed by integrating the aforepresented methods. Additionally, it is important to note that this paper presents a feasible way to MAGDM modelling through generalized IVIF cross-entropies and that the decision making methods investigated can also be applied to solve other complex group decision making problems of high uncertainty such as strategic decision making, emergency solution evaluation, exploitation investment evaluation.

The remainder of this work is organized as follows. In Section 2, some basic concepts and operations of IVIFSs are briefly reviewed. In Section 3, a generalized cross-entropy measure for IVIFSs is proposed to flexibly reflect DM’s attitudinal characteristics on how hesitation degree π supports membership and nonmembership degrees in IVIF evaluations. Based on the generalized cross-entropy measure, in Section 4, some programming models are established for obtaining unknown attribute weights and expert weights objectively. Then an approach integrating aforeproposed methods for MAGDM in IVIF environments is developed in Section 5. Furthermore, case study on simplified practical supplier selection problem is conducted in Section 6 to demonstrate effectiveness and practicality of presented methods. Finally, this work is concluded in Section 7.