Expected Values of Aggregation Operators on Cubic Trapezoidal Fuzzy Number and its Application to Multi-Criteria Decision Making Problems

In this paper, we define trapezoidal cubic fuzzy numbers and their operational laws. Started on these operational laws, each collection operators, with trapezoidal cubic fuzzy weighted arithmetic averaging operator and weighted geometric averaging operator are purposed. Expected values, score function, and accuracy function of trapezoidal cubic fuzzy numbers are defined. Overcoming on these, mindful of trapezoidal cubic fuzzy multi-criteria decision making program is proposed. A delineation illustration example is given to exhibit the sound judgment and openness of the procedure.


Introduction
Their get at a considerable lot of multi-criteria decision-making (MCDM) issues in indicating sociology. Different a period past the point of no return the fuzzy set institutionalization was offered and passed down to illuminate MCDM issues by Zadeh [14]. Therefor in [1], Atanassov presented the concept of intuitionistic fuzzy set (IFS) and discussed the degree of membership as well as the degree of non-membership function. Li reachable by theories and uses of fuzzy multi-criteria decision-making [9]. Wang displayed reading on multi-criteria decision-making drawing near with divided undoubting data [11]. There are differentiating preparing on the instrument of multi-criteria decision-making issues, in which the measures' weight coefficients are obvious and the criteria's principles are changed or are fuzzy numbers in [5,7,12], and here are likewise efficient readings on multi-criteria decision making or multi-criteria group decision making in [10,13], in which the weight sizes are tight and the standards' morals are fuzzy numbers.
Cubic set appeared by Jun in [8]. Cubic sets are the speculations of fuzzy sets and intuitionistic fuzzy sets, in which there are two portrayals, one is utilized for the degree of membership and other is utilized for the degree of non-membership. The membership function is hold as interim while non-membership is inside and out seen as the constant fuzzy set.
Aliya et al., [4] defined the triangular cubic fuzzy number and operational laws. We developed the triangular cubic fuzzy hybrid aggregation (TCFHA) administrator to total all individual fuzzy choice structure provide by the decision makers into the aggregate cubic fuzzy decision matrix. Aliya et al., [3] proposed the cubic TOPSIS method and cubic gray relation analysis (GRA) method. Finally, the proposed method is used for selection in sol-gel synthesis of titanium carbide Nano powders. Aliya et al., [2] defined weighted average operator of triangular cubic fuzzy numbers and hamming distance of the TCFN. We develop an MCDM method approach based on an extended VIKOR method using triangular cubic fuzzy numbers (TCFNS) and multi-criteria decision-making (MCDM) method using triangular cubic fuzzy numbers (TCFNs) are developed. Aliya et al., [5] defined the generalized triangular cubic linguistic hesitant fuzzy weighted geometric (GTCHFWG) operator, generalized triangular cubic linguistic hesitant fuzzy ordered weighted average (GTCLHFOWA) operator, generalized triangular cubic linguistic hesitant fuzzy ordered weighted geometric (GTCLHFOWG) operator, generalized triangular cubic linguistic hesitant fuzzy hybrid averag-ing (GTCLHFHA) operator and generalized triangular cubic linguistic hesitant fuzzy hybrid geometric (GTCLHFHG) operator. Aliya et al., [6] developed Trapezoidal linguistic cubic hesitant fuzzy TOPSIS method to solve the MCDM method based on trapezoidal linguistic cubic hesitant fuzzy TOPSIS method.
Thus, it is very necessary to introduce a new extension of cubic set to address this issue. The aim of this paper is to present the notion of Trapezoidal cubic fuzzy set, which extends the cubic set to Trapezoidal cubic fuzzy environments and permits the membership of an element to be a set of several possible Trapezoidal cubic fuzzy numbers. Thus, Trapezoidal cubic fuzzy set is a very useful tool to deal with the situations in which the experts hesitate between several possible Trapezoidal cubic fuzzy numbers to assess the degree to which an alternative satisfies an attribute. In the current example, the degree to which the alternative satisfies the attribute can be represented by the Trapezoidal cubic fuzzy set. Moreover, in many multiple attribute group decision-making (MAGDM) problems, considering that the estimations of the attribute values are Trapezoidal cubic fuzzy sets, it therefore is very necessary to give some aggregation techniques to aggregate the Trapezoidal cubic fuzzy information. However, we are aware that the present aggregation techniques have difficulty in coping with group decision-making problems with Trapezoidal cubic fuzzy information. Therefore, we in the current paper propose a series of aggregation operators for aggregating the Trapezoidal cubic fuzzy information and investigate some properties of these operators. Then, based on these aggregation operators, we develop an approach to MAGDM with Trapezoidal cubic fuzzy information. Moreover, we use a numerical example to show the application of the developed approach.
The rest parts of this paper are organized as follows: Section 2, we define the definotion of fuzzy set and cubic set. Section 3, we exhibit trapezoidal cubic fuzzy set and operational laws. Section 4, we exhibit Aggregation operators on trapezoidal cubic fuzzy numbers. Section 5, we define Expected values of trapezoidal cubic fuzzy numbers and comparison between them. Section 6, we define Multi-criteria decision making method based on trapezoidal cubic fuzzy numbers. Section 7, the application of the developed approach in group decision-making problems is shown by an illustrative example. Results and discussion are given in section 8. Finally, we give the conclusions in Section 9.

Preliminaries
Definition.2.1. [14] Give p a chance to be a nature of talk. The possibility of fuzzy set was speak to by Zadeh, and characterized as taking after: .

Definition.3.1. Let  be trapezoidal cubic fuzzy number in the set of real numbers, its membership function is defined as
Its non-membership function is defined as 11 11 11 ( ) The trapezoidal cubic fuzzy number is denoted as [ , ], . Generally from fuzzy numbers, trapezoidal cubic fuzzy numbers have another parameter: non-membership function, which is utilized to unequivocal the admeasurements to which the decision making that the component does not have a place with (( , , , ); ) the cubic fuzzy number is called trapezoidal cubic fuzzy number.

Definition.3.2. Let
: :         , and and the choice data is given as Table 1 by chiefs, extreme to impact positioning of the 4 options.

Expected Values of Trapezoidal Cubic Fuzzy Numbers and Comparison between them
Steps applying the ways and means in this unit are as continue from (1) homogenize material in Table 1;

Conclusion
In this paper, we define trapezoidal cubic fuzzy numbers and their operational laws. Started on these operational laws, each collection operators, with trapezoidal cubic fuzzyy weighted arithmetic averaging operator and weighted geometric averaging operator are purposed. Expected values, score function, and accuracy function of trapezoidal cubic fuzzy numbers are defined.