Intuitionistic Trapezoidal Fuzzy Group Decision-Making Based on Prospect Choquet Integral Operator and Grey Projection Pursuit Dynamic Cluster

In consideration of the interaction among attributes and the influence of decision makers’ risk attitude, this paper proposes an intuitionistic trapezoidal fuzzy aggregation operator based onChoquet integral andprospect theory.With respect to amultiattribute group decision-making problem, the prospect value functions of intuitionistic trapezoidal fuzzy numbers are aggregated by the proposed operator; then a grey relation-projection pursuit dynamic cluster method is developed to obtain the ranking of alternatives; the firefly algorithm is used to optimize the objective function of projection for obtaining the best projection direction of grey correlation projection values, and the grey correlation projection values are evaluated, which are applied to classify, rank, and prefer the alternatives. Finally, an illustrative example is taken in the present study to make the proposed method comprehensible.


Introduction
Multiattribute group decision-making (MAGDM) provides a systematic quantitative approach for decision-making problems involving multiple attributes and actions and can assist decision makers (DMs) in rationally considering all the important objective and subjective attributes of a problem [1,2].With the complexity of social environment and boundedness of human cognition, it is difficult to use crisp values to characterize the evaluation information.Fuzzy sets (FS) are generalized to model the vagueness in decision evaluation.But, a fuzzy set has limitation on presenting a broad description of all information related to the specific decision problem under an uncertain environment [3].In 1965, Atanassov put forward the idea of intuitionistic fuzzy sets (IFS) which contained membership function and nonmembership function that represented the decision maker's supporting degree and opposition degree.Owed to the capacity of expressing the fuzziness and hesitancy originating from imprecise knowledge or information, scholars have made many achievements in MAGDM problems with IFS, such as traditional decision-making methods with new definitions, new hybrid decision-making methods, extension of distance measure, and aggregation operators.As the combination of traditional fuzzy numbers and IFS, the intuitionistic fuzzy number (IFN) based on real numbers is a good match for describing an ill-known quantity.In recent years, the studies focused on three kinds of IFNs: interval-valued intuitionistic fuzzy number, intuitionistic triangular fuzzy number, and intuitionistic trapezoidal fuzzy number (ITFN).Chen et al. (2016) proposed four aggregation operators based on interval-valued intuitionistic fuzzy numbers and made the experimental results to show the drawbacks of existing methods [4].As the extension of triangular intuitionistic fuzzy numbers, ITFNs have the better capability to contain rich fuzzy information.Yuan and Li (2016) defined intuitionistic trapezoidal fuzzy random variable based on ITFNs and proposed decision-making method based on Mahalanobis-Taguchi Gram-Schmidt and evidence theory [5].Lakshmana et al. (2016) proposed a linear ordering class method to rank the alternatives in MADM with ITFNs [6].Kumar et al. (2015) utilized the shortest path as a classical network optimization method to classify alternatives by finding the shortest path and shortest distance [7].Wu and Cao (2013) 2 Mathematical Problems in Engineering presented some cases of aggregation operators with intuitionistic trapezoidal fuzzy numbers and then applied the ITFWG and the ITFHG operators to MADM problem [8].Based on Wu and Cao's research, Wan and Dong (2015) investigated four kinds of power geometric operators of ITFNs and developed four MADM methods for different cases [9].Motivated by the aggregation ability of operators, Yuan et al. (2016) proposed a new aggregation method for interval-valued intuitionistic hesitant fuzzy information based on Choquet integral [10].Chen and Chang (2016) established transformation techniques between intuitionistic fuzzy values and right-angled triangular fuzzy numbers and proposed new geometric averaging operators to deal with ITFNs [11].
To sum up, aggregation operators are effective tools to aggregate ITFNs, but the existing research has some drawbacks.
(1) Some decision methods use subjective weight determination methods.This approach would add the fuzziness and incongruence in decision information.(2) Most aggregation operators do not consider the interdependency characteristics of attributes, but the assumption of independency of attribute is too strong to be satisfied in many MADM problems.(3) The researches premises are formed commonly based on the condition that the decision makers are absolutely reasonable, whereas actually the subjective psychological factors and attitude towards risk in decision leaders directly influence the risk evaluation results.(4) In group decision-making, the construction of decision maker evaluation system often needs to meet the requirement of completeness, representativeness, and independence.However, it is slightly harsh to meet the requirement of independence in the actual decision-making problems.Preferences of decision makers are often affected by status, prestige, knowledge, and other factors.
In view of studying the existing literatures, this paper proposes an intuitionistic trapezoidal fuzzy aggregation operator based on Choquet integral in consideration of the validity of fuzzy measure.Combined with prospect theory, the ITFN prospect value function and the intuitionistic trapezoidal fuzzy Choquet integral prospect operator are proposed.Aiming at the MAGDM problem with ITFNs, this paper presents grey relation-projection pursuit dynamic cluster method based on intuitionistic trapezoidal fuzzy Choquet integral prospect operator.Through history information, the fuzzy measures of decision makers are obtained.The intuitionistic trapezoidal fuzzy numbers are aggregated by the proposed operator, and then the alternatives ranking is obtained by grey relation with projection pursuit dynamic cluster method.The remainder of this paper unfolds as follows: Section 2 introduces some basic definitions of intuitionistic trapezoidal fuzzy number.The aggregation operator based on Choquet integral and prospect theory is developed in Section 3.
Section 4 shows the decision-making method.In Section 5, an example and its comparison analysis are given.Section 6 ends the paper with some concluding remarks.
The main contributions of this paper are as follows.
(1) An intuitionistic trapezoidal fuzzy aggregation Choquet integral operator is proposed to eliminate the interdependence between attributes.(2) The prospect value function for ITFNs is defined in consideration of different risk attitude.(3) In order to avoid subjective weights and information loss, a grey relation-projection pursuit dynamic cluster method based on firefly algorithm is developed to rank alternatives.
Step 8. Construct the optimization model to obtain the best projection vector   . max Step 9. Calculate the grey correlation projection value   .Draw the experience from [14]; the cosine vector of the projection angle between the alternative   and the normalized ideal reference vector is Obviously, 0 ≤   ≤ 1, the bigger the better.The grey correlation projection value   between the alternative   and the normalized ideal reference vector is where (  ) = ∑  =1 [    ] 2 is the moduli of   .And   is the grey correlation projection vector, and it can be calculated by According to the grey correlation projection value   , we can rank the alternatives.

An Example and Analysis Process.
Jinzhong power supply company in ShanXi province plans to select one of the four equipment suppliers  = { 1 ,  2 ,  3 ,  4 } for an engineering project cooperation.There are four attributes in this case, including the product quality  1 , technology capability  2 , pollution control  3 , and environment management  4 , which are all benefit oriented attributes.The company invites three experts to form a group of decision makers.DM 1 is from the production department; DM 2 is from the operation and maintenance department; DM 3 is from the quality inspection department.
Step 1.Three experts {DM 1 , DM 2 , DM 3 } are invited to select the best alternative.Tables 1-3 show the experts' evaluation information with intuitionistic trapezoidal fuzzy numbers with respect to attributes under alternatives.
Step 2. Three experts provide reference points according to the four attributes, which are shown in Table 4.
Step 3. Set  = 0.89,  = 0.92,  = 2.25 [15] and calculate the prospect value function by ( 19).Then we transform the intuitionistic trapezoidal fuzzy decision matrixes to intuitionistic trapezoidal fuzzy prospect matrixes.Due to space limitation, the calculation process is omitted.According to (20), the comprehensive prospect decision matrix is obtained shown in Table 5.
Step 5. Normalize the comprehensive prospect decision matrix based on (23).The result is shown in Table 6.
Step 8. Set the parameters of the optimization model,  = 4 and  = 4, and the cluster is divided into two categories.Then the best projection vector is   = (0.5745, 0.2647, 0.6622, 0.4017).

Comparison
Analysis with Other Methods.The goal of decision-making in the real world is typically to find the optimal alternative from a group of alternatives.Lots of methods have been applied to MADM to make the result  more accurate or less biased.Some scholars consider several appropriate ways to select the optimal alternative based on the majority principle.But this suggestion assumes that all methods are appropriate for the decision-making problem.
Table 8 shows the comparison with other methods.Wu and Cao use the ITFN weighted geometric operator and hybrid geometric operator to obtain the collective overall values and calculate the distances between collective overall values and positive solution.Finally, the ranking result is  3 ≻  1 ≻  4 ≻  2 .Wu and Cao's method is only suitable for the case where the weights of attributes and decision makers are known.Therefore, they cannot solve the MAGDM problems with unknown weights, whereas the proposed method aggregates decision makers' data on bias of attribute independence and decision makers' risk preference.This implies that the proposed method has wider real application range than the method in [8].It shows from Tables 1-3 that there are too large ([6, 7, 8, 9]; 0.8, 0.1) for  3 on  4 in Table 1 and too small ([1, 2, 3, 4]; 0.4, 0.2) for  2 on  4 in Table 3, which results in the ranking order  3 ≻  1 ≻  4 ≻  2 by Wu and Cao's method.The two operators developed by [8] cannot reflect the influence of unfair data (too large or too small ITFNs), whereas, the proposed method can effectively relieve the influence of unfair data by assigning low weights to those unfair data.

Conclusion
Intuitionistic trapezoidal fuzzy number is suitable for expressing the fuzziness and uncertainty of the decision information in complex MAGDM problems.In the light of the faults in ITFN's operation rules, this paper redefines some basic operation rules of ITFN.In order to eliminate the interdependency in attributes, the intuitionistic trapezoidal fuzzy Choquet integral is defined.In consideration of decision makers' bounded rationality, the intuitionistic trapezoidal fuzzy Choquet integral prospect (ITFCIP) operator is defined to aggregate the ITFNs.Aiming at MAGDM problem with trapezoidal intuitionistic fuzzy numbers under unknown attribute weights, a grey relation-projection pursuit dynamic cluster method based on ITFCIP operator is presented.This method defines the prospect value function based on ITFNs, obtains the fuzzy measure according to the history sample information, and then constructs the comprehensive prospect decision matrix based on ITFCIP operator.In addition, a grey relation-projection pursuit dynamic cluster method is proposed to select the best alternative.This combination improves grey correlation projection method under the comprehensive consideration of preferential membership and grey correlation.The firefly algorithm is used to optimize the objective function of projection for obtaining the best projection direction of grey correlation projection values, and the grey correlation projection values are evaluated, which are applied to classify, rank, and prefer the alternatives.

Table 1 :
The evaluation information from DM 1 .

Table 2 :
The evaluation information from DM 2 .

Table 3 :
The evaluation information from DM 3 .

Table 6 :
The normalized comprehensive prospect decision matrix.

Table 7 :
The distance and grey correlation coefficients.