Multiattribute Decision Making Based on Entropy under Interval-Valued Intuitionistic Fuzzy Environment

Multiattribute decision making (MADM) is one of the central problems in artificial intelligence, specifically in management fields. In most cases, this problem arises from uncertainty both in the data derived from the decision maker and the actions performed in the environment. Fuzzy set and high-order fuzzy sets were proven to be effective approaches in solving decision-making problems with uncertainty. Therefore, in this paper, we investigate the MADM problem with completely unknown attribute weights in the framework of interval-valued intuitionistic fuzzy (IVIF) set (IVIFS). We first propose a new definition of IVIF entropy and some calculation methods for IVIF entropy. Furthermore, we propose an entropy-based decision-making method to solve IVIF MADM problems with completely unknown attribute weights. Particular emphasis is put on assessing the attribute weights based on IVIF entropy. Instead of the traditional methods, which use divergence among attributes or the probabilistic discrimination of attributes to obtain attribute weights, we utilize the IVIF entropy to assess the attribute weights based on the credibility of the decisionmaking matrix for solving the problem. Finally, a supplier selection example is given to demonstrate the feasibility and validity of the proposed MADMmethod.

In particular, IVIFS and IFS are effective in solving the fuzzy decision-making problems.In most fuzzy multi-attribute decision making (MADM) problems, the preference over alternatives provided by decision makers is usually not sufficient for the crisp membership and nonmembership degree values, because things are fuzzy, uncertain, and probably influenced by the subjectivity of the decision makers, or the knowledge and data about the problem domain are insufficient during the decision-making process [13][14][15][16][17]. Therefore, the preferences among alternatives with uncertainty may be denoted by IFS or IVIFS for decision-making problems [7,11,[13][14][15].Chen and Tan [18], Xu [11,19], Ye [20], Lakshmana Gomathi Nayagam et al. [21], and Wang et al. [22] proposed some methods to rank alternatives expressed with IVIFSs/IFSs and discussed their application in the MADM field.Atanassov [23] introduced some aggregation operators on IFS and IVIFS.After the pioneering work of Atanassov, some aggregation operators were proposed and utilized to tackle the MADM problems with uncertainty [24][25][26][27][28][29][30][31][32][33].Xu and Yager [34], Li et al. [35][36][37][38], and Dubey et al. [15] constructed IF and IVIF MADM models based on optimization theories, such as linear, nonlinear, and fractional programming.Meanwhile, decision-making methods based on some measures (such as distance, similarity degree, correlation coefficient, and entropy) were proposed in dealing with fuzzy IF and IVIF MADM problems [7,19,35,[39][40][41].In [13,17,22,34,37], Mathematical Problems in Engineering emphasis was given that not only the information of alternatives on attributes may be fuzzy but also the attribute weight information may be partially known or unknown in some situations.In fact, proper assessment of attribute weights plays an important and essential role in the decision-making process, because the variation of weight values may result in different final ranking order of alternatives [11,14,17].
Generally speaking, the attribute weights in MADM can be classified as subjective and objective attribute weights based on the information acquisition approach [14].The subjective attribute weights are determined by preference information on the attributes given by the decision maker, who provides subjective intuition or judgments on specific attributes.The objective attribute weights are determined by the decision-making matrix.Analytic hierarchy process (AHP) method [42] and Delphi method [16] are two classical methods for generating subjective attribute weights.In terms of determining objective attribute weights, one of the most famous approaches is the Shannon entropy method, which expresses the relative intensities of attribute importance to signify the average intrinsic information transmitted to the decision maker.So far, a lot of literature pertaining to MADM analysis under IF/IVIF environment has been published using subjective weights [11,[15][16][17].In the course of determining objective attribute weights under IF environment, Chen and Li [14] utilized IF entropy to assess the objective attribute weights in dealing with the IF MADM problem with completely unknown attribute weights.Despite existing research effort, solving the fuzzy MADM problems with completely unknown attribute weights in the framework of IVIFS remains an open problem [11,14,17,35,37,38].
In an attempt to address such problems, we propose a novel entropy-based decision-making method under IVIF environment.Our focus is on the assessment of attribute weights.Meanwhile, we propose a definition of entropy on IVIFS, as well as a method to calculate it.Instead of using traditional methods, which use divergence among attributes or the probabilistic discrimination of attributes to obtain attribute weights, IVIF entropy is utilized to assess the objective weights based on the credibility of the input data.Furthermore, we construct a MADM model based on the IVIF weighted averaging (IIFWA) operator and the ranking functions on IVIFS.
The rest of this paper is organized as follows: in Section 2, we briefly review the concepts of IFS, IVIFS and some of their operations.In Section 3, we propose a definition of IVIF entropy and some calculation methods on entropy.An entropy-based MADM method within the framework of IVIFS is proposed in Section 4. A numerical example is utilized to illustrate the applicability of the proposed method in Section 5. Finally, conclusion is given in Section 6.

Preliminaries
In this section, we briefly review some basic concepts, for example, IFS, IVIFS, and their relevant operations.
Notation. = { 1 ,  2 , . . .,   } is the universal set (it is clear that  = Cardinal()).FS (), IFS (), and IVIFS () denote all the fuzzy sets on , all the IFSs on , and all the IVIFSs on , respectively.Definition 1 (see [1]).Let  be a set.An IFS  in  is defined as where are two maps satisfying () and ]  () denote the membership and nonmembership degrees of  to , respectively.For each IFS  in , we designate an intuitionistic index of  in .
The following expressions are defined in [1,11] for all  and  belonging to IFS (): Definition 2 (see [2,23]).Let  be a set and Int [0, 1] the set of all closed subintervals of [0, 1].An IVIFS  in  is defined as where for which The intervals   () and ]  () denote the membership and nonmembership degrees of  to , respectively.Let where and ]   () = ]   (), then IVIFS  reduces to an IFS.
For each IVIFS , we designate an intuitionistic interval of  in , where The following expressions are defined for all  and  belonging to IVIFS () [11]: Ranking the alternatives expressed with IVIFSs is necessary to deal with the MADM problem within the framework of IVIFS.We introduce the score function and accuracy function of IVIFS proposed by Xu [11].
Based on the score function and the accuracy function, Xu further introduced an algorithm of ranking alternatives expressed with IVIFSs.

Entropy on IVIFSs
The definition of entropy on IFSs/IVIFSs has a great importance in the theoretical research of IFSs/IVIFSs.In 1996, Burillo and Bustince [43] introduced the IF entropy for the first time, and, in 2001, Szmidt and Kacprzyk [44] proposed a non-probabilistic-type entropy measure for IFSs.Later, Hung [45] and Zhang et al. [46] constructed the IF entropy based on the distance measures among IFSs.Vlachos and Sergiadis [9] introduced the De Luca-Termini nonprobabilistic entropy for IFSs.Zeng and Su [32] introduced the IF entropy based on the similarity among IFSs.Ye [47] constructed two IF entropies based on the fuzzy entropy established by Zadeh [48].Chen and Li [14] presented different entropies on IFSs.So far, a great deal of literature on entropy on IFSs is available, but, unfortunately, to our knowledge, only a few works on entropy on IVIFSs exist.Two published papers are found in [49] which extended De luca-Termini axioms and proposed an IVIF entropy, and in [41] which proposed an IVIF entropy in dealing with multiple-attribute decision-making problem.

Entropy on IFSs.
Based on the definition of fuzzy entropy [48], Burillo and Bustince [43] defined the IF entropy as follows.
Definition 6 (see [43]).A real function  : IFS () →  + is called an entropy if  has the following properties: (1) () = 0 if and only if  ∈ FS (), According to Definition 6 and function Φ, Bustince and Burillo proposed a theorem to construct the entropy on IFSs.

Entropy on IVIFSs.
Based on Definition 6 and some properties on IFS [50,51], we define the entropy on IVIFSs as follows.

MADM Method within the Framework of IVIFS
In this section, we present a novel method to solve the decision-making problems with unknown attribute weights based on the ranking functions and IVIF entropy.Let  = { 1 ,  2 , . . .,   } be a set of alternatives and  = { 1 ,  2 , . . .,   } be a set of attributes.The IVIF decision matrix  of  on  is where   ( = 1, 2, . . ., ;  = 1, 2, . . ., ) denotes the IVIF numbers.In the following, we introduce our method.

Experimental Analysis.
Similarly, we adopt  Φ ,exp and  Φ ,sin ( = 1, 10, 100;  = 0.1, 0.5, 0.9) to implement the decision-making process for this supplier selection problem.All the ranking results are the same as the ranking order of  Φ ,cost ( = 1, 10, 100;  = 0.1, 0.5, 0.9), that is,  4 ≻  5 ≻  3 ≻  2 ≻  1 .By applying the methods in [11,41] to Example 13, the ranking order is  4 ≻  5 ≻  3 ≻  2 ≻  1 , and the most desirable one is  4 which is identical to ours.From the aforementioned results, the proposed decisionmaking method in this paper can be suitably utilized to solve the IVIF MADM problem with completely unknown attribute weights.Different IVIF entropies clearly offer various iuchoices for assessing the attribute weights.

Conclusion
In this paper, we have focused on solving the IVIF MADM problem with completely unknown attribute weights.We first introduced a new definition of IVIF entropy and some calculation methods of different IVIF entropies.Subsequently, we proposed an entropy-based MADM method in the framework of IVIFS.The proposed method can be utilized to solve fuzzy and uncertain decision-making problems derived from supplier selection, public risk, medical diagnosis, and other problems in any aspects.