Three-way Group Decisions with Interval-valued Intuitionistic Fuzzy Information Based on Optimization Method

LIU Jiubing,Jü Hengrong,LI Huaxiong,HUANG Bing,BU Xiangzhi

(1.School of Business, Shantou University, Shantou 515063, China;2.School of Information Science and Technology, Nantong University, Nantong 226019, China;3.School of Management and Engineering, Nanjing University, Nanjing 210093, China;4.School of Information Engineering, Nanjing Audit University, Nanjing 211815, China)

Abstract：A new three-way group decision method is proposed based on three representative interval-valued intuitionistic fuzzy (IVIF) aggregation operators and optimization methodology. First, we elaborate on the three-way group decision problem with IVIF information, use the IVIFWA, GIVIFWA and IVIFWG operators to aggregate these information provided by experts and obtain collective evaluation results of IVIF collection loss functions, and some relations among collective loss functions are discussed. Second, based on the aggregated loss functions, we construct an optimization model pair for capturing the thresholds and three-way decisions with the aid of optimization methodology. Third, we develop a three-way group decision model based on these three aggregation operators and optimization methodology. Finally, an illustrative example and its related comparison are given to verify the rationality and advantage of our model.

Key words：Three-way group decisions； interval-valued intuitionistic fuzzy sets； optimization models

0 Introduction

Three-way decisions (3WDs) with acceptance, further investigation and rejection provide a powerful tool for dealing with the uncertainty and risk. The theory of 3WDs was originally proposed by Yao in 2010 based on rough sets and the Baye-sian decision procedure[1]. Later, the studies on this aspect have been gradually received attention and many related achievements have been obtained[2-5], especially in some fields such as three-way recommendation[6], three-way clustering[7], three-way concept learning[8-9], three-way classification[10-11], group decision-making[12-13]and face recognition[14], etc.

As one of key elements of 3WDs, the determination for the thresholdsαandβis of great study topic. Usually, this threshold pair is determined by loss functions and these loss functions are traditionally given/evaluated by experts as crisp values. However, experts are sometimes difficult to give an exact evaluation in some cases, and may be more inclined to a fuzzy or uncertain evaluation due to the increasing complexity and uncertainty of random factors[5]. The interval-valued intuitionistic fuzzy sets (IVIFSs), which were proposed by Atanassov and Gargov in 1989[15], were regarded as one of important fuzzy or uncertain evaluation forms with applications to multiple attribute decision-making[16]and group decision-making[17]. In view of the effectiveness of IVIFSs in uncertain evaluations, the IVIFSs are here introduced into loss functions and naturally how to determine the threshold pair under interval-valued intuitionistic fuzzy environments becomes a key research problem. Nevertheless, there are few studies in this area. Inspired by our previous work[5], this paper will present an optimization-based 3WD framework with IVIFSs for determining the thresholds.

In the classical model of 3WDs, loss functions are often evaluated by single expert. It is necessary to explore multiple experts because of the limit of personal domain knowledge and cognition[18]. To achieve a final solution from the evaluation information given by multiple experts. Group decision making, which can aggregate wisdom of experts from different domains, provides an effective method to handle the uncertain evaluation in decision making[19]. Clearly, aggregation of experts’ information in group decision making becomes an essential step. 3WDs with group decision semantics are achieved by aggregating loss functions of multiple experts. Liangetal.[18]utilized the principle of justifiable granularity to form majority suggestions of experts in group decision making for obtaining interval-valued loss values, and thus developed three-way group decisions with decision-theoretic rough sets. Based on that, Zhangetal.[19]investigated three-way group decisions with interval-valued decision-theoretic rough sets based on aggregating inclusion measures. Sunetal.[20]proposed a novel approach to linguistic information-based three-way group decisions. In 2018, Yeetal.[21]first introduced IVIFSs into 3WDs, and designed a simple and straightforward algorithm to induce 3WD rules in group decision making. From the above, there are few researches on three-way decisions with IVIFSs under group decision making and it is necessary to explore this topic in depth, which can further enrich these theories of three-way decisions and interval-valued intuitionistic fuzzy sets. To do so, in this paper we propose an optimization-based three-way group decision method with IVIFSs and group semantics. The main contributions of this paper are summarized as follows: 1) we study the relationships between the aggregated collective loss functions, thereby establishing the optimization models for determining the thresholds; 2) we develop an optimization based three-way group decision approach under interval-valued intuitionistic fuzzy environments.

The rest of this paper is organized as follows: In Section 1, we describe a three-way group decision problem with IVIF information, and three representative operators are used to aggregate experts’ evaluations. Following it, we obtain collective loss functions and discuss some relations among them. A three-way group decision approach is further proposed based on these three aggregation operators and optimization methodologies. Section 2 presents an illustrative example and Section 3 summarizes the conclusion.

1 Tree-way group decisions with interval-valued intuitionistic fuzzy information (IVIFI)

1.1 Problem on three-way group decisions with IVIFI

We consider a three-way group decision problem described: assume there are two statesΩ={C,C}{P,N} and three kinds of decision actionsA={aP,aB,aN}. In this way, the actiona●of each expert under each stateowill lead to a corresponding loss denoted byλ●○, where●=P,B,Nand○=P,N. In the process of realistic evaluations,n(n≥2) experts are invited to evaluate these loss functions of multiple objectsU={o1,o2,…,om}. LetE={e1,e2,…,en} be a set ofnexperts andW=(w1,w2,…,wn)Tbe a weight vector of experts, whereandwk≥0. To derive three-way classified results of each objectoi, these experts need to provide the evaluated values of loss functions. In our real-life, experts may not be able to express, to some extent, his/her own preferences accurately,i.e. he/she may not be certain about risk preferences and that it would be partially sure and skeptical. It is therefore suitable and convenient to evaluate the loss function in the format of interval-valued intuitionistic fuzzy numbers (IVIFNs) given by expertekasi=1,2,…,m;k=1,2,…,n;●=P,B,N;○=P,N, using the interval membership degree, interval non-membership degree and interval hesitancy degree respectively, of objectoion the loss functionλ●○for the fuzzy concept of excellence. Thus, the evaluation results of objectoion six lossesλ●○are generated, which are shown in Table 1.

Note that， in Table 1, the values of loss functions satisfy a reasonable case with the following relations for each objectoi(1≤i≤m).

(1)

(2)

(3)

(4)

Table 1 The evaluation results of objects on six loss functions given by expert ek

Based on the evaluation results given in Table 1, how to obtain three-way classified rules of each objectoibecomes a key research problem. To solve it, we design two decision procedures: Deriving collective loss functions by the interval-valued intuitionistic fuzzy weight average (IVIFWA) operator, the generalized interval-valued intuitionistic fuzzy weighted average (GIVIFWA) operator and the interval-valued intuitionistic fuzzy weighted geometric (IVIFWG) operator, and establishing optimization models to determine the thresholds based on collective loss functions. We first consider the former in next section.

1.2 Deriving collective loss functions by using these three representative operators

In this subsection, the IVIFWA, GIVIFWA and IVIFWG operators are respectively considered to aggregate experts’ interval-valued intuitionistic fuzzy evaluation information, and we can calculate collective loss functions of each objectoi(i=1,2,…,m), which are shown in Table 2. Also, it is noted that wherei=1,2,…,m;●=P,B,N;○=P,N;r>0.

According to the aggregated results in Table 2, some conclusions on collective loss functions are drawn after we give Propositions 1 and 2.

Table 2 Collective evaluation results of objects on six loss functions

(5)

The results reported in [22] tell us that 0≤s(a)≤1 and the largers(a) is, the biggerais.

Based on Propositions 1 and 2, the relationships among collective loss functions are explored by the following proposition.

Proof. For anyi=1,2,…,m,●=P,B,Nand○=P,N, one has:

Based on (1)-(4), any of the IVIFWA, GIVIFWA and IVIFWG operators is used to aggregate IVIFI and the following relations obviously follow.

and thus

Namely,

On the other hand,

that is,

1.3 Optimization models for determining the thresholds based on collective loss functions

In light of the Bayesian decision theory, the following decision rules are given:

In reality, rules (P1)-(N1) can be further transformed as the following rules based on the similarity function-based ranking method for IVIFNs.

In 2019, an optimization-based model pair was developed to determine the thresholds and 3WDs[5]. Motivated by this work, we extend the model pair to an interval-valued intuitionistic fuzzy group decision environment and establish its extended model pair based on (P2)-(N2), which is presented as follows:

Based on these two models, one induces the following proposition.

Proof. It is not difficult to prove Proposition 4 by the results reported in [5].

1.4 Three-way group decision approach based on collective loss functions and extended models

Step 1:nexperts use the IVIFNs to evaluate six loss functions of objectoi(i=1,2,…,m) and we obtain the interval-valued intuitionistic fuzzy evaluation matrix, which is shown in Table 1.

Step 2: Select a suitable operator from the IVIFWA, GIVIFWA and IVIFWG operators to aggregate these evaluation information given by experts and calculate collective evaluation results of loss functions,i.e. Table 2.

2 An illustrative example

To show the effectiveness of the proposed approach, an example is given[23]: in the development and production of the new energy vehicles (NEVs), the industry needs many capital, manpower and material resources to carry out the scientific research. Meanwhile, there is a huge risk so that the individual company can not indepen-dently undertake the research of the key generic technology. Therefore, the government decided to subsidize some companies of key technologies for NEVs and encourage them to conduct more effective research in key generic technologies. Three local car manufacturers were subsidized to conduct NEVs strategic research. But the government is still uncertain how to allocate these subsidies. Currently, the government’s decision is based largely on whether the companies have the capabilities to conduct and match the subsidy. So the subsidized companies have two states: have the capability to conduct key technology research(C), and do not have the capability to conduct it (C). The government has mainly three actions: subsidizing (P), not subsidizing (N) and delaying the decision making (B). Obviously, the risk lossesλ●○(●=P,B,N;○=P,N) of three actions of each companyoi(i=1,2,3) under two states are respectively evaluated by three experts as IVIFNs and the weight vector of expertsW=(0.3,0.5,0.2)T, and thus, the evaluation results are shown in Tables 3-5. For simplification, we assume Pr(C|[o1])=0.79, Pr(C|[o2])=0.38 and Pr(C|[o3])=0.44.

The proposed approach in this paper is employed to make a decision and the specified steps are given as follows:

Step 1: The GIVIFWA operator (assumer=2) is selected to aggregate three experts’ evaluation information and we obtain collective evaluation results of loss functions, which are listed in Table 6.

Table 3 Evaluation results of companies on loss functions given by expert e1

Table 4 Evaluation results of companies on loss functions given by expert e2

Table 5 Evaluation results of companies on loss functions given by expert e3

Table 6 Collective aggregation results of companies on loss functions

Table 7 Three-way decision thresholds for three objects

In the existing methods [18] and [21], they respectively establish intuitionistic fuzzy decision-theoretic rough sets and interval-valued intuitionistic fuzzy decision-theoretic rough sets to induce the rules of three-way decisions in group decision making, which were based on the model of DTRSs proposed by Yao. However, these existing methods does not determine the pair of thresholds under the corresponding fuzzy semantics. In order to show the rationality and advantage of our approach, we compare the proposed method with the existing method[21]as follows:

3 Discussion and conclusion

Considering the three-way group decision problem with interval-valued intuitionistic fuzzy loss functions, this paper develops a three-way group decision method based on three aggregation operators. This study gives a novel idea and methodology to capture the thresholds and 3WDs under interval-valued intuitionistic fuzzy environments, which enriches the theory of interval-valued intuitionistic fuzzy sets and three-way decisions. Future studies will focus on the generalization of the proposed method in group decision making with group consensus and attribute reduction.

Table 8 Ranking function values of each object under three actions

Fig.1 Three-way decision solutions for each object in three parameters r.

Fig.2 Three-way decision solutions for each object underthree parameters rand changed conditional probabilities