Multiattribute Group Decision Making with Unknown Decision Expert Weights Information in the Framework of Interval Intuitionistic Trapezoidal Fuzzy Numbers

The aim of this paper is to investigate an approach to multiattribute group decision making with interval intuitionistic trapezoidal fuzzy numbers, in which the decision expert weights are unknown. First, we introduce a distance measure between two interval intuitionistic trapezoidal fuzzy matrixes, and based on the distance between individual matrix and extreme matrix, as well as the average matrix, we obtain the decision expert weights. Second, we utilize the interval intuitionistic trapezoidal fuzzy weighted geometric (IITFWG) operator and the interval intuitionistic trapezoidal fuzzy ordered weighted geometric (IITFOWG) operator to aggregate all individual interval intuitionistic trapezoidal fuzzy decision matrices into a collective interval intuitionistic trapezoidal fuzzy decision matrix and then derive the group overall evaluation values of the given alternatives. Finally, an illustrative example of emergency alternatives selection is given to demonstrate the effectiveness and superiority of the proposed method.

Wan and Dong [19] gave the definitions of ITFN about expectation and expectant score, hybrid aggregation operator, and ordered weighted aggregation operator.Ye [20,21] presented an intuitionistic trapezoidal fuzzy multicriteria decision making method based on expected values and multicriteria group decision making method based on the similarity measures between ITFNs, respectively.Wan et al. [22] proposed a possibility mean-variance based method for intuitionistic trapezoidal fuzzy group decision making.Zhang et al. [23] investigated a grey relational projection method for multiattribute decision making based on intuitionistic trapezoidal fuzzy numbers.
Wan [24,25] studied the IITFN in depth.He not only discussed the operational laws and properties of IITFN, giving a series of definitions of it, such as the score function, weighted arithmetic average operator, weighted geometric average operator, and the Hamming and Euclidean distances for interval-valued trapezoidal intuitionistic fuzzy numbers, but also established a multiattribute decision making model based on IITFN.Wei et al. [26] put forward some new aggregation operators of IITFN anddeveloped a multiple attribute group decision making approach with known decision expert weights information.

Mathematical Problems in Engineering
In the existing literature, few studies have been carried out on IITFN.For the membership function value and nonmembership value of IITFN depending on different interval number, IITFN can better reflect decision maker's preference and express a different dimension of decision making information.Thus, it provides a new idea to solve the multiattribute decision making problems effectively.
To reflect the decision making more scientifically and democratically, in some large or important decision making problems, it often requires multiple decision makers to participate.In the process of group decision making, decision expert weights play an important role to aggregate all individual decisions into a collective decision because the decision makers may have different abilities, interests, hierarchical ranks, and so forth.
Due to the limited knowledge of the authors, no research has been carried out to handle multiattribute group decision making problems with unknown decision maker weights in the framework of IITFN.So this paper proposed a new group decision making methodology based on distance measure to derive the weights of experts, in which the attribute values take the form of interval intuitionistic trapezoidal fuzzy numbers.
To do that, the structure of this paper is organized as follows.A brief introduction of the IITFN theory is given in Section 2. In Section 3, we develop a new multiattribute group decision making method with incompletely known decision expert weights information under IITFN.In Section 4, we illustrate our proposed method with an example.Final conclusions are shown in Section 5.

Preliminaries
In the following, we will introduce some basic concepts related to interval intuitionistic trapezoidal fuzzy numbers.
In the following, we introduce an order relation between any pair of IITFNs based on (3) and (4).

Multiattribute Group Decision Making Method Based on Interval Intuitionistic Trapezoidal Fuzzy Numbers
In this section, we propose a new multiattribute interval intuitionistic trapezoidal fuzzy group decision making methodology based on distance measure, where the decision makers' weights are unknown.
In the following, we apply distance measure, IITFWG operator, and IITFOWG operator to multiattribute group decision making based on interval intuitionistic trapezoidal fuzzy information.The proposed method involves the following steps.
Step 1. Determine the weights of the decision makers.

𝑚×𝑛
, where ẽ() ).And the normalized values for benefit-related criteria () and cost-related criteria () are calculated as follows: Second, determine the extreme matrixes Ẽ+ = (ẽ +  ) × , Ẽ− = (ẽ −  ) × and the average matrix Ẽ * = (ẽ *  ) × , shown as follows: where  = 1, 2, . . ., ,  = 1, 2, . . ., , and  is the total number of decision makers.Third, calculate the distance measure of individual matrix Ẽ to extreme matrixes and average matrix, respectively, shown as follows: The average matrix reflects the group opinion, and the extreme matrixes reflect the extreme views of the experts.It is clear that the closer a decision matrix Ẽ is to the average matrix Ẽ * , the better the decision Ẽ of th ( = 1, 2, . . ., ) decision maker is, so the heavier weight should be given.At the same time, the closer a decision matrix Ẽ is to the extreme matrixes, the worse the decision of th expert is, so the smaller weight should be given.Based on the above analysis, the weight of the th decision maker is defined as follows: Step 2. Calculate the weights of attributes.
Step 5. Rank the preference order of all alternatives according to the score function and accuracy function.

Illustrative Example
In this section, we present an illustrative example of emergency alternatives selection to demonstrate the potential application of the proposed method.
To implement the rescue action of a certain urban fire happened in a Cotton Mill, an emergency department is desired to select the most suitable emergency alternatives from three preevaluation emergency alternatives: { 1 ,  2 ,  3 }.Three criteria are considered as follows: cost of consumption ( 1 ), negative impact of the rescue ( 2 ), and disposal time ( 3 ).In order to select the best alternative, a committee composed of three experts   ( = 1, 2, 3) has been found.The three alternatives are to be evaluated using the IITFNs by the three experts under the above three criteria, as listed in Table 1.
In the following, we utilize the proposed method to select the most desirable emergency alternatives.
Step 1. Calculate the decision expert weights vector.
First, by applying (9), the fuzzy decision matrix is normalized, which is shown in Table 2.
Step 2. Calculate the weights vector of the attributes.
Step 3. Determine the individual overall IITFNs.
Step 4. Determine the collective overall IITFNs and rank the alternatives.
By using the IITFWOG operator, we can obtain the collective overall IITFNS ẽ ( = 1, 2, 3), and based on (3) and (4), we can calculate the score degree of the overall IITFNs, which is shown in Table 3.
The final group ranking of the three alternatives can thus be obtained as  3 ≻  1 ≻  2 , and the most desirable alternative is  3 .

Concluding Remarks
In this paper, concerning the multiattribute group decision making problems in which the preference values are presented in the form of interval intuitionistic trapezoidal fuzzy numbers and the decision expert weights are unknown, a new analyzing method of decision making is proposed.By employing distance measure between individual matrix and extreme matrix, as well as the average matrix, the decision makers' weights have been measured.The proposed method can better reflect and describe the essential characteristic of the objective world and provide a more accurate, effective, and systematic decision support tool.In addition, an illustrative example of emergency alternative selection problem is provided to illustrate the practicality and effectiveness of the proposed methodology.Further research is to carry out sensitivity analyses and may continue extending the developed method to other domains.

Table 1 :
The decision matrix with IITFNs for each decision maker.

Table 3 :
Overall IITFNs, score degrees, and ranking of alternatives.