An ELECTRE I-based multi-criteria group decision making method with interval type-2 fuzzy numbers and its application to supplier selection
Graphical abstract
Because of these abilities, IT2 FNs have been widely applied in decision support systems (DSS). In this paper, the elimination and choice translating reality method is extended using interval type-2 fuzzy numbers for more effective multi-criteria group decision-making under an uncertain environment. An α-based distance method is first proposed to measure the proximity between the interval type-2 fuzzy numbers. Then, an entropy measure for the IT2 FNs and an entropy weight model are developed to objectively determine the criteria weights without any weight information. By applying the α-based distance method, the concordance and discordance for each alternative are measured to determine the partial-preference outranking order. The process has been shown as follow.
Introduction
Since Zadeh [1] proposed the fuzzy numbers (FNs) in which the membership degree of an element to a set is represented by a real number between zero and one, there have been significant further theoretical developments. The most well-known classical FNs theories have been intuitionistic fuzzy set theory introduced by Atanassov [2] and interval valued fuzzy sets developed by Zadeh [3]. Type-1 fuzzy numbers (T1 FNs) represent the uncertainty in an entity using a crisp number in the range [0, 1]. Unfortunately, if the entity itself is uncertain,using crisp number may not always be the best approach [4]. Type-2 fuzzy numbers (T2 FNs), however, which are an extension of the T1 FNs [5], have a better uncertainty processing ability [6], [7] and are more capable of capturing the complexities in social environments as well as the inherent vagueness of people's preferences [8]; even so, due to its intensive computations to date T2 FNs have tended to have limited applications. Under these circumstances, IT2 FNs, a special category of T2 FNs, were developed to have constant secondary membership grades [9], [10] and a significantly simpler computation; therefore, they have been widely applied in decision support systems (DSS) [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21].
Multiple criteria group decision-making (MCGDM) problems are when a group of qualified decision makers seeks to resolve complex problems by comparing and ranking alternatives based on evaluations for multiple conflicting criteria [22]. In general, MCGDM involves two main steps: (1) determining the weights for the decision makers and the criteria; and (2) determining a suitable ranking order for the alternatives. Several methods have been proposed to determine the weights in an interval type-2 fuzzy environment. Chen and Lee [23] developed an interval type-2 fuzzy TOPSIS method to handle MCGDM problems with incomplete criteria weight information. Abdullah and Otheman [24] defined an entropy measure for IT2 FNs to deal with unknown MCGDM criteria weight problems. Qin and Liu [21] defined a combined weight measure and then used it to solve an interval type-2 fuzzy MCGDM. For the second step ranking procedure, many aggregation operators and decision making methods have been proposed: the trapezoidal interval type-2 fuzzy geometric Bonferroni mean (TIT2FGBM) operator and the trapezoidal interval type-2 fuzzy weighted geometric Bonferroni mean (TIT2FWGBM) operator [25]; the interval type-2 fuzzy TOPSIS method [19]; the interval type-2 fuzzy TODIM method [18]; the interval type-2 fuzzy DETAMEL method [26]; the interval type-2 fuzzy VIKOR method [27], [28]; and the interval type-2 fuzzy analytic hierarchal process [16].
However, existing weight-determination methods and interval type-2 fuzzy MCGDM methods have the following shortcomings: (1) significant information is lost when only the approximation between two IT2 FNs is calculated as theIT2 FNs’ pictorial properties can be neglected; (2) these methods are extremely complex and inflexible when only a partial order is required.
To overcome the above issues, in this paper, we propose an extended-ELECTRE method for multi-criteria group decision making in an interval type-2 fuzzy numbers environment. As uncertainty exists in real-world decision-making, the modelling of linguistic terms using mathematical notations has become increasingly important for MCGDM problems [29]. As IT2 FNs better deal with the expression of uncertainty, have better processing abilities and are simpler to compute [9], the linguistic terms can be appropriately represented using IT2 FNs to illustrate uncertainties in complex situations. This paper first develops an entropy measure for the IT2 FNs, and then establishes an entropy weight model to objectively determine the criteria weights without any weight information. As there are several different methods available to measure the distance between two IT2 FNs [21], [6], in this paper, the α-based distance method is employed to measure the proximity between the IT2 FNs, which allows for the concordance and discordance in each pair of alternatives to be identified for the partial-preference outranking order. As a full rank order may be required in some circumstances, a complementary analysis is presented to obtain the full rank order for all alternatives. The feasibility and applicability of the proposed method are illustrated in a numerical supplier selection example. This proposed extended-ELECTRE ranking method is flexible enough to solve other complex decision-making problems in which criteria weight information is difficult to find or determine, such as facilities location selection, strategic decision making and multi-criteria large-group decision making.
The remainder of this paper is organized as follows. In Section 2, we review previous research to position our research. In Section 3, we briefly review the basic concepts and arithmetic operational laws for IT2 FNs. In Section 4, the α-based distance method for IT2 FNs and its properties are introduced. In Section 5, a new entropy measure method for the IT2 FNs and the entropy weight method are proposed to weight the criteria. Then, in Section 6, the proposed extended-ELECTRE outranking method is presented for MCGDM problems with IT2 FNs. In Section 7, two practical examples and relative analyses are presented; one on supplier selection and the other on health-care waste (HCW) management to illustrate the advantages of the proposed methods. Conclusions and possible further extensions are given in Section 8.
Section snippets
Literature review
In classical decision-making problems, the decision makers’ preference values for each alternative are expressed using precise numbers [30]. However, as human preferences are by nature vague, practical decision-making problems are fuzzy and uncertain. Because the best way for decision-makers to clearly state their opinions is by using natural language [29], linguistic variables [3], the value of which are expressed using natural language words and sentences, have been widely used to indicate
Preliminaries
This section briefly reviews the basic concepts for the IT2 FNs. More flexible than T1 FNs which require a crisp membership function, T2 FNs are able to express uncertainty by providing a measure of dispersion to better capture inherent uncertainties, which is especially useful in problems when it is difficult to determine the exact membership function of a fuzzy set [58].
Definition 2.1 Let X be a universe of discourse, then a type-2 fuzzy number can be defined as follows:[58]
α-based distance
Initially proposed by Figueroa-García et al. [42], the α-based distance can be used to measure the similarities between IT2 FNs. However, the original α-cut based distance is only able to calculate the distance between IT2 FNs, which is in a α-representation, which limits the application of this method. Based on [42], the α-based distance was modified as follows.
Definition 3.1 Let and be non-negative IT2 FNs defined on X and X ≠ {0}. If x is N discrete numbers, the distance between them can be computed
Criteria weight determination
Given that IT2 FNs can better express uncertainty, have better processing abilities and are simpler to compute [9], to adequately illustrate the uncertainties in a complex situation, linguistic terms can be appropriately represented by IT2 FNs. It is assumed that there are t decision makers D1, D2 …, Dt, who express their preference regarding m alternatives A1, A2 …, Am under n criteria C1, C2 …, Cn. First, the decision makers provide their judgements; after which and a decision matrix
The extended-ELECTRE outranking choices method
The classical ELECTRE method compares alternatives in pairs to determine the concordance and discordance of the sets. The method constructs different matrices using the concordance and discordance of the sets and filters the alternatives using the threshold value [55]. Different from the interval-valued fuzzy ELECTRE methods [65] and the interval type-2 trapezoid fuzzy ELECTRE method [6], a IT2 FNs outranking choice method is proposed, which is an integration of the classical ELECTRE method and
Numerical examples
The proposed effective multi-criteria decision-making method can be applied to a variety of fields, such as energy selection, environmental evaluations, supplier selection, health-care waste treatment technology selection, and others. In this section, we apply the proposed decision-making method to two practical examples: an electronic materials supplier selection problem and a health-care waste treatment technologies selection problem. Assume that the experts provide their decision preferences
Conclusion
The ELECTRE method, a relationship model based on outranking relations, has been widely applied to MCGDM. However, the limitation of this method is that it requires precise preferences; however, in real-world situations, decision-making problems take place under complex conditions and involve uncertain data, imprecise information and vague concepts. As IT2 FNs are able to inherently handle intrinsic and extrinsic uncertainty, they can be used to express the vague preferences in the ELECTRE
Acknowledgements
The work is supported by the National Natural Science Foundation of China (Grant No. 71671118), the National Natural Science Foundation for Young Scholars of China (Grant No. 71301109), and Soft Science Program of Sichuan Province (Grant No. 2017ZR0154). Besides, this work is also supported by National Natural Science Foundation of China (71301110), the Humanities and Social Sciences Foundation of the Ministry of Education (3XJC630015), and Research Fund for the Doctoral Program of Higher
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