key: cord-0073666-ldn3q7on
authors: Tao, Xiwen; Jiang, Wenqi
title: Research on Two-Stage Hesitate Fuzzy Information Fusion Framework Incorporating Prospect Theory and Dichotomy Algorithm
date: 2022-01-15
journal: Int
DOI: 10.1007/s40815-021-01207-6
sha: 90e1666bea39be047e3b9e36662dfa39477ed1b7
doc_id: 73666
cord_uid: ldn3q7on

In order to control the systematic divergence among decision makers (DMs) and preserve the original decision preference, this paper proposes a novel decision information fusion framework under the hesitant fuzzy environment. First, a maximum compactness-based normalization method is presented to normalize hesitant fuzzy elements (HFEs) as pretreatment of decision data. Second, prospect theory is introduced to assign the optimal aggregation weights to maximize the efficiency of the preference aggregation process, in which the expected consensus threshold is viewed as a reference point estimated through statistic inference to distinguish DMs’ status. Third, an effective feedback mechanism is designed to improve group consensus, and the dichotomy algorithm is utilized to search optimal feedback weight to preserve original decision information. Finally, a case study and comparison analysis are illustrated to show the efficiency of the proposed hesitant fuzzy information fusion method.

Due to the expression and cognitive limitations of subjective decision making, fuzzy values such as interval fuzzy value [1] , intuitionistic fuzzy value [2] , Pythagorean fuzzy value [3] and et al., are usually used to express the DMs' subjective attitude. In recent years, hesitant fuzzy element (HFE) especially its application in MAGDM (seen in [4] [5] [6] ) have become hot topics. The diversity of group intelligence provides an essential reference for more reliable and reasonable group decision results. Meanwhile, individual diversity can also lead to systematic conflict within a group because of the difference in knowledge structure, cultural background, and risk attitude. Therefore, integrating diverse individual preferences with the required consensus level and preserving the diversity of individual preferences become an imperative topic. The literature mainly focuses on two aspects for efficient information fusion, including preference aggregation and consensus improvement.

In the individual preference aggregation process, aggregation weights assignment is the key issue. Some researchers preset aggregation weight vector subjectively (seen in [7, 8] ), as a static parameter through the whole decision-making process. Considering DMs' behavior and consensus requirement, several researchers propose objective weight determination methods. (1) Behavior patternbased weight assignment, behavioral features are the basis of DMs' importance. Liu and Xu et al. [9] aggregated individual preference relation using a self-confidence induced ordered the weighted average operator to give more importance for those self-confident experts. Xu and Cabrerizo et al. [10] used a consistency induced ordered weighted average to collect individual preference and calculate weight vector with quantifier function (seen in [11] ). (2) Optimization-based weight vector, objective weight is driven by consensus requirement. Zhang and Liang et al. [12] proposed a programming-based method to minimize maximum distance between individual and group to obtain optimal weight vector. Liu and He et al. [13] developed a maximum consensus model which maximizes & Xiwen Tao txwfight@163.com consensus degree by adjusting the experts' weights. Xiao and Wang et al. [14] devised a maximum consensus-based optimization model combining in-degree centrality, consistency, and similarity indexes. Zhang and Pedrycz [15] established a programming model to determine DMs' weights by minimizing the distance between individual and group. There are also some other novel methods to determine weight vectors. Mohammadi and Rezaei [16] proposed a half-quadratic-based method to determine optimal weights for each individual ranking. Mohammadi and Rezaei [17] introduced a Bayesian-based method to find aggregated final weights of a group of DMs. Wan and Zhong et al. [18] constructed reference matrices of DMs to measure their weight vector by relatively closeness degree Based on the TOPSIS method. Wu and Liu et al. [19] proposed an entropy-based method to determine DMs' weights. The purpose of the individual preference fusion process is to generate reliable a group opinion that can be acceptable to DMs as far as possible.

On the aspect of group consensus improvement, linear sum method, programming-based method, and weight punishment method are applied here. (1) The first method updates individual preference by a linear sum of individual and group preference. Feedback weight is the key parameter to affect the improvement process. Considering the diversity of DMs' cooperation willingness, Cao and Wu et al. [20] introduced personal feedback weight to maximize harmony degree. Based on the matching of evaluation and decision environment, Perez and Cabrerizo et al. [21] proposed a dynamic feedback mechanism. With simulation-based consensus threshold, Tang and Zhou et al. [22] introduced a local linear sum method to adjust individual preference. (2) Programming-based method develops an optimization model to determine an optimal updated preference, which can often reach consensus within one interaction. Arieh and Easton [23] pointed that the adjustment of original opinion comes with a cost and proposed minimum adjustment cost model. If unit adjustment cost is constant, the minimum cost model can be transferred into the minimum adjustment distance model. Zhang and Dong et al. [24] investigated the efficiency difference of consensus improvement from the perspectives of minimum adjustment distance, elements, and DMs. Wu and Huang et al. [25] proposed a multi-stage optimization-based feedback mechanism. Xiao and Wang et al. [26] presents a two-stage consensus reaching model with minimum preference information loss. Zhang and Dong et al. [27] analyzed the equilibrium strategy to obtain optimal consensus cost of the minimum adjustment model. (3) Weight punishment method adjusts individual aggregation weights, which refuses to modify preference and to strengthen acceptability of group preference. Liu and Xu et al. [28] presented a dynamic weight adjustment strategy to improve group consensus. Du and Yu et al. [29] combined weight punishment and linear sum method to construct a mixed consensus improvement method. Guo and Xu et al. [30] developed a noncooperative behaviors management method based on weights modification.

The above literature improves efficiency on group information fusion. Original information preservation is a vital factor in the process. In decision practice, the consensus threshold, aggregation weights, and preference adjustment can influence the original preference preservation. However, the systematic influence is not comprehensively considered by the existing research:

(1) The consensus threshold is a crucial parameter in group decision making, which controls the start and termination of consensus improvement. Most research regards the threshold as an exogenous parameter preset before decision making, which cannot reflect the flexible consensus requirement. The exogenous threshold can lead to an overconsensus situation and force DMs to adjust opinion considerably. And it is meaningless to force DMs to adjust opinion passively to reach an excellent consensus target if a group has low-level compactness. Therefore, it is necessary to connect the consensus threshold to the preference distribution to set a reachable target for the consensus improvement process. (2) Within the aggregation weights optimization, the importance of excess and shortage parts to the threshold is different. Generally, the shortage part will affect the adjustment volume directly in further consensus improvement. Therefore, it is essential to highlight the shortage part optimization when calculating the aggregation weights. Moreover, the differences among attributes are often ignored, which can harm the performance of the group opinion and affect the adjustment volume. (3) Minimum adjustment distance strategy is commonly applied in the current consensus improvement. However, the existing methods may distort the preference features (individual preference direction and the order relationship of proximity), reducing the acceptance degree of adjustment advice. For example, if the original individual preference direction is positive, it is hard to force DM to accept the negative updated preference; if a DM is farthest from the group, it is impossible to force him/her to have the highest proximity degree after adjustment. Oriented to original preference preservation, the consensus improvement strategy should minimize the adjustment volume and avoid the distortion of preference features.

In order to tackle the above problems, a two-stage hesitant fuzzy information fusion framework is developed for a high-level group consensus. The main innovative points of the research are as follows: (1) The endogenous consensus threshold: The statistic inference method estimates the expected consensus threshold rather than preset subjectively; (2) Prospect-based aggregation weight assignment: The prospect theory is introduced to calculate the optimal weight matrix to maximize the closeness on each attribute with less potential adjustment volume; (3) The dichotomy-based consensus improvement method: With the proposed method, the individual acceptance limitation is identified to prevent preference direction distortion. Moreover, the optimal feedback weights are searched through a dichotomy algorithm to minimize the adjustment distance and preserve the proximity order relationship.

The remaining parts of this paper are organized as follows: Sect. 2 introduces the concept of hesitant fuzzy MAGDM and prospect theory. Section 3 focuses on the preference aggregation process. To standardize hesitant fuzzy decision information, maximize compactness-based normalization is proposed. And then, proximity-induced hesitant fuzzy-ordered weighted averaging (P-IHFOWA) operator and its quantifier function are designed to aggregate individual preference and estimate group consensus threshold. In Sect. 4, the acceptance limit of each DM is identified firstly. Moreover, a dichotomy-based linear sum algorithm is given to improve consensus with minimum adjustment distance. Section 5 gives a numerical example and comparison analysis to verify the proposed approach and highlight its efficiency and rationality. Finally, in Sect. 6, we draw some conclusions and discuss future research possibilities.

In this section, the concepts of hesitant fuzzy MAGDM and prospect theory are reviewed for further discussion.

Let A ¼ fa i ji ¼ 1; 2; :::; mg, C ¼ fc j j ¼ 1; 2; :::; ng and D ¼ fd i jq ¼ 1; 2; :::; tg be the set of alternatives, attributes and DMs, respectively. h q ij ¼ fc ql ij l ¼ 1; 2; :::; #h q ij g denotes the evaluation from DM e q ðq 2 f1; 2; :::; tg ¼ TÞ on alternative a i ði 2 f1; 2; :::; mg ¼ MÞ over attribute c j ðj 2 f1; 2; :::; ng ¼ NÞ. H q ¼ ðh q ij Þ mÂn is the evaluation matrix of DM e q ðq 2 TÞ. Definition 1 [31, 32] . Let X be a fixed set, a HFS on X is in terms of a function h E that returns a subset of ½0; 1, which can be defined as follows: E ¼ f\x; h E ðxÞ [ jx 2 Xg. h E ðxÞ denotes the possible membership degrees of the element x 2 X to the set E, which is a set of values in ½0; 1½0; 1. For convenience, h E ðxÞ is called a HFE denoted as follows: h E ðxÞ ¼ fc l l ¼ 1; 2; :::; #h E ðxÞg, where #h E ðxÞ denotes the number of elements. Generally, elements in h E ðxÞ are sorting in ascending order.

Remark 1 Considering the correspondence of linguistic term s t ðt ¼ Às; :::; À1; 0; 1; :::; sÞ with membership cðc 2 ½0; 1Þ [33] , Às s l  and 0 c:5 share the same negative attitude. On the contrary, 0\s l s and 0:5

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proposed an optimization-based weight determination method. Group consensus index GCI ij of each position is compared with subjective weight [7] [7] and optimization-based weight [13] . (2.1) To compare with subjective method, 1000 weight vectors are generated randomly to simulate subjective weight and the normalization parameter is n ¼ 0:5. The comparison results regarding the group consensus index that are used in Sect. 5.2 are offered in Fig. 4 . (2. 2) According to optimization-based method [13] , the optimal weight vectors are calculated under different normalization parameters, which minimize distance between individual and group:The comparison results are offered in Fig. 5 . From the above two figures, the proposed aggregation method performs better than the two existing methods almost in each position. Due to better acceptability of group preference, less adjustment range will be produced and more original information will be preserved. (3) Adjustment volume comparison From consensus improvement, the proposed dichotomybased automatic feedback method is compared with the traditional linear sum method and minimum adjustment method. [7] Global and local feedback methods is used to improve consensus, respectively. Traditional feedback mechanism has not provided adjustment recommendation, which may lead to too much loss of information. To control adjustment distance, [18] and [40] constructed programming model to enhance consensus. However, the original attitude of DMs may be changed during modification. To unify the comparison standard, supposing consensus threshold 0.95 and all DMs are the same important.(3.1) With traditional feedback method, the feedback weight is usually set by DMs. Local feedback method [10] is applied to improve consensus. Supposing l ¼ 0:6. The updated individual preferences are