Information Fusion Model of Group Decision Making Based on a Combinatorial Ordered Weighted Average Operator

Aiming at the problem that the existing aggregation operators only consider the weight of the data itself or the weight of the data position when aggregating data, a group decision information fusion model based on the combinatorial ordered weighted average operator is proposed, which considers the incompleteness, uncertainty and plurality of the decision information, and effectively improves the accuracy of group consensus. In this paper, an interval intuitionistic fuzzy combinatorically weighted average operator is proposed, which embodies the weight of the set data itself and the weight of data position, and can adjust the importance of the two weights in the operator through parameters, and prove that the operator has the properties of boundedness, monotonicity, idempotency and permutation invariance. Secondly, a position weight solving model of interval intuitionistic fuzzy combinatorial ordered weighted average operator is established, and the position weight is solved with the help of cross-entropy and Orness measures. Finally, a group decision information fusion process based on IVIFCOWA operators is designed, and the decision maker weight information is obtained according to the decision matrix, and the comprehensive decision matrix is obtained by fusion of IVIFCOWA operators, and then the mean similarity is obtained. Through specific case analysis, it is proved that the proposed operator can make full use of the decision data, and can better reach group consensus after data fusion than the existing operator.


I. INTRODUCTION
Aggregation operators are a means to fuse decision information. Specifically, they can be used to fuse individual decision opinions into comprehensive decision results. Aggregation operators play an important role in solving multi-attribute group decision-making problems, for the same decision data, using different operators for data fusion may obtain different group fusion results, so it is particularly critical to choose reasonable and effective aggregation operators when making decisions. At present, according to the different weights used by aggregate operators when assembling data, they can be divided into two categories: the first consists of ordered The associate editor coordinating the review of this manuscript and approving it for publication was Francisco J. Garcia-Penalvo . operators, that sort the data to be assembled according to a certain rule, considering only the vector associated with the position of the assembled data, and not the weight information of the attribute itself. In the second category, unordered operators assemble data based only on the weight of the data itself, without sorting it.
Let us first consider unordered operators: this category can be divided into two types, depending on how they handle mutual influence between attributes. Among operators that do not take into account any relationship between attributes, the most classic is the interval-valued intuition fuzzy (IVIF) weighted average operators proposed by Xu [1], namely the IVIF weighted arithmetic mean and IVIF weighted geometric mean. Subsequent scholars have subsequently used and developed them [2], [3]. As to operators that take into consideration the mutual influence between attributes, for example in logistics management decision-making two different attributes are usually considered: distribution cost and distance. Since distribution cost is related to distribution distance, this kind of decision-making problem should use an operator that considers how each of these attributes affects the other. For this reason, Yu [4] proposed the interval-valued intuitionistic fuzzy Bonferroni mean operator to deal with multi-attribute decision-making problems in which attributes are correlated and mutually influenced. Xu [5] proposed the IVIF associative average operator and the IVIF associative geometry operator under the IVIF environment, according to the properties of the Choquet integral. Chen [6] proposed the IVIF power-weighted arithmetic mean and geometric mean operators in the IVIF environment, and further extended them to the dynamic decision-making context. In the setting of management decision cases, Liu [7] proposed a power-weighted average operator and used it in decision-making cases described in the IVIF language. Liu et al. [8] proposed two new interval intuitionistic fuzzy set settlers: interval intuitionistic fuzzy power McLaughlin symmetric mean operators and weighted interval intuitionistic fuzzy power McLaughlin symmetric mean operators, and used them in two decision cases in IVIF environments. Combining the Heronian mean operator with the Einstein algorithm, Song and Wang [9] proposed the interval-valued intuitionistic fuzzy Einstein Heronian mean operator, and used it in specific decision-making cases.
Let us now consider ordered operators. In the actual decision-making process, in order to deal with decisionmaking problems with unknown attributes or decision-maker weights, Yager [10] proposed the ordered weighted average operator (OWA). The data to be assembled is sorted according to a certain sorting rule, the sequence of the data at the time of assembly is then obtained, and the weight related to the data position is given; the data is then fused based on the position weight. Data fusion by this operator only focuses on the weight information of the data location, and has no relation with the weight of the data itself (the location weight information has nothing to do with the weight of the data itself). Due to the excellent characteristics of the OWA operator itself, it can also be used to fuse data when the weight is completely unknown. Therefore, it has numerous different applications. Most of the subsequent operators are based on expansions and improvements of the OWA operator. Xu and Da [11] proposed the ordered weighted geometric mean operator. Yager [12] first proposed a generalized ordered weighted average operator, and secondly, in order to determine the sorting position of the assembled data, Yager and Filev [13] proposes to induce an ordered weighted average operator. Compared with the OWA operator, this operator has more induced variables (given as certain sorting rules). On this basis, Xu and Da [14] proposed an induced ordered weighted geometric mean operator. The above are all extensions of the OWA operator, but in the presence of increasingly complex decision-making environments decision-making information can no longer be represented by simple real numbers. Therefore, based on these foundations, various scholars continue to extend the above-mentioned set operators to cope with fuzzy intuition and intervals. Wei [15] proposed the IVIF ordered weighted geometric mean operator, the IVIF induced ordered weighted mean operator, and additional improvements and applications under the IVIF environment [16], [17].
It can be seen from the study of the above literature that the above operators only consider one kind of weight information when assembling data: either the data itself or the location of the data. This is not sufficiently comprehensive; therefore, some important information will necessarily be lost when one of the existing aggregation operators is used for data set fusion, which will in turn influence the subsequent sorting process, and possibly even produce wrong sorting results. In order to solve this problem, this paper proposes a group decision information fusion model based on combinatorically ordered weighted average operator, which can consider the weight of the aggregated data itself and the weight of data position at the same time in the aggregated data, and make full use of the interval intuitive fuzzy decision information, so as to make the decision determination process more reasonable and provide better support for the subsequent ranking stage. Firstly, the position weight solving model of the operator is given, the properties of the operator are proved to verify its rationality, and finally the data fusion process based on the operator is given. Through specific case analysis and comparison, it is proved that the proposed operator can better reach group consensus than the existing classical operators in the data aggregation stage, and provide better support for the subsequent sorting stage. The rest of this article is organized below. The related basic concepts are described in section II. In the third section, the IVIFOWA operator weighting model is proposed. In the fourth section, the IVIFCOWA operator and its position weight weighting model are established. In the fifth section, the group decision information fusion process based on IVIFCOWA operator is given. An example analysis is given in Section VI. Conclusions are presented in section VII.

A. ABBREVIATIONS AND ACRONYMS
Atanassov [18] proposed the notion of interval-valued intuitionistic fuzzy set (IVIFS), as given below: Definition (1): Let X be a non-empty set, then the IVIFSÃ on X is written as: (1)  [20]: be an interval-valued intuitionistic fuzzy number, then its score function and exact function are respectively defined as follows: Definition (3) [20]: Letã i ,ã j be two interval-valued intuitionistic fuzzy numbers, then their sorting method is given as follows: Their operation rules are given as follows: Definition (5) [22]: Their similarity degree Sãb is defined as : . . n be a set of IVIFS and w = (w 1 , w 2 , · · · , w n ) T its weight vector, with w j ∈ [0, 1] such that n j=1 w j = 1. Then the IVIF weighted arithmetic means operator and the IVIF weighted geometric mean operator can be defined respectively as follows: Definition (7) [22]: . . n be a set of IVIFS. Then the IVIF ordered weighted arithmetic mean operator and the IVIF ordered weighted geometric mean operator can be defined respectively as follows: where w = (w 1 , w 2 , · · · , w n ) T is the n-dimension weight vector associated with the IVIFOWA operator with w j ∈ [0, 1] and n j=1 w j = 1. The data a i is not related to w i ; in fact the latter weight w i is only associated with the position in the assembly process. Among them,b j is the j-th largest element after sorting is assembled based on the given weight vector. The weighting model of the position weight vector is given below.

III. IVIFOWA OPERATOR WEIGHTING MODEL
The IVIFOWA operator aggregates data by position weight, so it is essential to correctly specify position weights. Most of the existing methods for determining the weight vector of IVIFOWA operators are inseparable from the following two pieces of information: the Orness measure and the maximum entropy measure. The Orness measure is defined as representing the level of optimism of the decision maker. The maximum entropy measure is defined as disp(w) = − n i=1 w i ln w i , which represents the discrete degree of w. The classic IVIFOWA empowerment methods mainly include the following: HAGAN [23] proposed the following maximum entropy model: FULLÉR [24] proposed a variance-based weight optimization model in order to minimize variance, as follows: Wang [25] proposed the following model based on a min-max inconsistency method: Finally, EMROUZNEJAD [26] proposed the following model: The common point of the above models is that the calculated position weights have the smallest degree of dispersion.

IV. IVIFCOWA OPERATOR AND ITS POSITION WEIGHTING MODEL
A. IVIFCOWA OPERATOR Definition (8): 1, 2, . . . , n) be a set of IVIFS; then the IVIF combined ordered weighted arithmetic mean operator is given by: This operator consists of two parts. In the first part  (20) given below. In the first part,b j is the value ofã i corresponding to the j-th largest order-induced variable u i in the pair ⟨µ i ,ã i ⟩. β is a parameter whose value can be changed to reflect the degree of importance between the weight of the data itself and the weight of the location in the operator. By deforming both parts of the IVIFCOWA operator into the geometric mean form, the IVIFCOWG operator can be obtained: A set of examples will now be given to allow us to analyze and compare the results obtained with the operator proposed in this paper and with the IVIFWA and . From these results it can be seen that the three operators obviously produce different results after data fusion.

B. IVIFCOWA POSITION WEIGHTING MODEL BASED ON CROSS ENTROPY
According to the analysis of the IVIFOWA weight optimization model of formulas (13)(14)(15)(16), the objective function of the above solution minimizes the dispersion degree of the solved weights as much as possible. Based on the above idea, this paper uses cross entropy in order to solve the position weight of the second part of the IVIFCOWA operator. The crossentropy ϕ (t) is defined as follows: Definition (9) [27]: Let p and q be two probability distributions of a discrete random variable x, then the cross-entropy of p and q is defined as: Cross-entropy is an important concept in Shannon's information theory. It is mainly used to measure the difference in known and unknown information between two probability distributions. The smaller the cross-entropy between two probability distributions, the more similar they are. Since the weight vector belongs to the probability distribution, p can be used to represent the weight vector to be solved, and q can represent the known weight vector. Because the degree of dispersion of p to be solved is low, q can be expressed as (1/n, 1/n, . . . , 1/n), where n is the vector dimension. Therefore, according to the above formula combined with the Orness measure, we propose in this paper the following weight optimization model: We set the value of α to 0.7, which means it is more inclined to the top-ranked data. The position weight vector ϕ (t) can be obtained by solving the above equation with the help of the Lingo software.

C. PROPERTIES OF THE IVIFCOWA OPERATOR
IVIFCOWA enjoys the following main properties: commutative, monotonic, bounded, idempotent. The proof is given below.
In addition, the IVIFCOWG operator also has the above properties; the proofs are similar to those shown above for IVIFCOWA.

V. GROUP DECISION INFORMATION FUSION PROCESS BASED ON THE IVIFCOWA OPERATOR
In the case where the decision-maker's decision-making opinion is an interval-valued intuitionistic fuzzy number, we set the target set as A = {A 1 , A 2 , . . . , A m }, the attribute set as G = {G 1 , G 2 , . . . , G n }, and the decision maker set as D = {D 1 , D 2 , . . . , D t }. Suppose decision maker D k evaluates attribute G j as (µ k ij , v k ij ) under scheme A i . We describe in detail below the steps of the information fusion interval intuition fuzzy group decision-making process based on the IVIFCOWA operator: Step 1: is constructed for each decision maker.
Step 2: The decision maker weights are calculated.
Step 3: The weight vector associated with IVIFCOWA operator, namely the weight of data positions, is calculated through the weight optimization model based on cross entropy as given by Equation (20).
Step 4: Using the weight information obtained in Step 2 and Step 3, the evaluation information of each attribute under different schemes is fused by the IVIFCOWA operator to obtain a comprehensive decision matrix. Among them, the induced variable u in the IVIFCOWA operator is obtained according to the similarity method in the literature [17].
Step 5: The similarity between the comprehensive decision matrix and the matrix of each decision maker under each scheme A i of attribute c i is at last calculated by formula (13), and the average value is taken to obtain the average similarity degree.

VI. EXAMPLE ANALYSIS
We use data found in the literature [17] in order to illustrate the rationale behind the method proposed in this paper, as well as comparing it with other methods.
A company is looking for the best supplier, among 4 suppliers corresponding to A i i ∈ (1, 2, 3, 4). Each supplier has the following attributes: cost c1, quality c2, service level c3; the attribute weight vector is w = (0.34, 0.5, 0.16). The decision is made by 4 decision makers d k . The evaluation values of each decision maker for all suppliers are interval intuition fuzzy numbers.
Step 1: Build a decision matrixX k for each decision maker as shown in Table 1.
Step 2: Calculate the weight of the decision maker following the weighted bidirectional projection-based decision maker weight determination method described in the literature [2]. The weight of the decision maker is (0.26, 0.21, 0.26, 0.27).
Step 4: Using the weight information obtained in steps 2 and 3, fuse the evaluation information of the different decision makers for different attributes c i under different schemes A i using the IVIFCOWA operator of formula (18), to obtain a comprehensive decision matrix as shown in Table 2. In this VOLUME 11, 2023   paper we set β to the value 0.5, which means that the two weights have the same degree of importance.
Step 5: Calculate the similarity between the comprehensive decision matrix and the individual decision matrix of each scheme A i under the attribute c i by formula (8), and take the average to obtain the average similarity. The results are shown in Table 3.
In order to demonstrate the effectiveness of the proposed method, we compare the similarity mean calculated by the proposed operator IVIFCOWA with the results obtained with the existing classical aggregation operators IVIFOWA, IVIFOWA, and the IIFIOWA operator proposed in [17]. Secondly, the implementation with the IVIFCOWG, IVIFWG, IVIFOWG, and the I-IVIFOWG operator proposed in this paper, as well as of the IIFIOWG operator proposed in literature [17] are also compared. The calculation results are shown in Table 4.
It can be seen from Table 4 that each scheme A i obtained by the IVIFCOWA / IVIFCOWG operator proposed in this paper displays better similarity between the comprehensive decision matrix and the individual decision matrix under attribute c i than obtained by schemes using other set operators. The important purpose of group opinion fusion is that it should fully take into account the decision-making opinions of each individual decision maker, that is, the result of fusion should attain the largest possible group consensus and produce the smallest differences. Reasonable group opinion fusion results should not only reflect consistency between groups as a whole, but also cover conflicts between individuals.
Compared with the IVIFWA / IVIFWG operator, the operator IVIFCOWA / IVIFCOWG proposed in this paper increases the position weight information when aggregating data. Taking into account the weights both of the data itself and of the location information, this algorithm respects the importance of both the data itself and of its location, and therefore the degree of similarity obtained between group  opinions and the opinions of all the individual decision makers is higher than with the IVIFWA / IVIFWG operator. The above results confirm the advantages of the proposed method in data aggregation, resulting in fact in the largest group consensus in the data aggregation stage.
When the IVIFCOWA operator proposed in this paper is used to fuse data, the value of is chosen to be 0.5, which means that the two weights have the same degree of importance. The average similarity between the comprehensive matrix of scheme A i under attribute c i and each decision maker matrix, obtained when the value of β ranges from 0.1 to 0.9, can then be calculated. In the following figures we draw change graphs representing the variation of the mean similarity of scheme A i under attribute c i with the value of β.
It can be seen from figures 1-3 that the mean similarity decreases as the value of β increases from 0.1 to 0.9, when scheme A 1 is under attribute c 1 , A 4 is under attribute c 1 , A 2 is under attribute c 3 , and A 4 is under attribute c 3 . This shows that a smaller β value should be selected when merging the decision makers' evaluation information under the corresponding attributes of the above schemes, that is, when   the data position weight information is preferred. On the other hand, the similarity of scheme A 1 increases with the increase of β under attribute c 3 , which indicates that a larger β value should be selected when merging the decision-maker evaluation information of scheme A 1 under attribute c 3 , that is, when the weight information of the data itself is preferred. In addition to the above two cases, the similarity of other schemes does not change significantly with β under the corresponding attributes, so the value of β can be appropriately selected.
To sum up, when using the IVIFCOWA operator to fuse the evaluation information of different attributes of different decision makers under different schemes, the β value should be dynamically adjusted according to actual needs. As an example, when four decision makers fuse the evaluation information of attribute c1 under scheme A 1 , the value of β can be set to 0.1; if on the other hand the same decision makers fuse the evaluation information of attribute c 3 under scheme A 1 , β can be set to 0.9.
The average similarity between the comprehensive matrix of the scheme A i under the attribute c i and the matrix of each decision maker after dynamically adjusting the β value through sensitivity analysis is given in Table 5. The maximum value of the mean similarity of scheme A in attribute C, as shown in figures 1-3, is compared with a fixed β value of 0.5, in terms of different criteria. It can be seen from Table 5 that dynamically adjusting the value of β can lead to better fusion results, and further improve the group consensus.

VII. CONCLUSION
Aggregation operators play an important role in solving multi-attribute group decision-making problems, for the same individual decision-maker decision data, different groups of fusion results will be obtained when using different operators for data fusion, so it is particularly critical to choose reasonable and effective aggregation operators when making decisions. However, most of the existing aggregation operators only consider the weight information of the data itself or the weight information of the position of the data when aggregating data, so this paper proposes a combination of ordered weighted average operators, which can consider the two weight information of the aggregated data at the same time, namely the weight information or the weight of the location of the data, and make full use of the interval intuitive fuzzy decision-making information to obtain a more certain and reasonable and scientific decision-making process. Firstly, the position weight solving model of the operator is given, and secondly, the commutative, monotonicity, boundedness and idempotency of the operator are proved, and the mathematical rationality of the operator is verified, and finally the data aggregation process based on the operator is given, and the results are compared and analyzed by the example of a company looking for the best supplier, which proves that the proposed group decision information fusion model can reach group consensus better than the existing classical model in the data aggregation stage, and provide better support for the subsequent ranking stage.