Intuitionistic trapezoidal fuzzy multi-numbers and its application to multi-criteria decision-making problems

Intuitionistic trapezoidal fuzzy multi-numbers (ITFM-numbers) are a special intuitionistic fuzzy multiset on a real number set, which are very useful for decision makers to depict their intuitionistic fuzzy multi-preference information. In the ITFM-numbers, the occurrences are more than one with the possibility of the same or the different membership and non-membership functions. In this paper, we define ITFM-numbers based on multiple criteria decision-making problems in which the ratings of alternatives are expressed with ITFM-numbers. Firstly, some operational laws using t-norm and t-conorm are proposed. Then, some aggregation operators on ITFM-numbers are developed. Also, the ranking order of alternative is given according to the similarity of the alternative with respect to the positive ideal solution. Finally, a numerical example is given to verify the developed approach and to demonstrate its practicality and effectiveness.

"Many fields of modern mathematics have been emerged by violating a basic principle of a given theory only because useful structures could be defined this way. Set is a welldefined collection of distinct objects, that is, the elements of a set are pair wise different. If we relax this restriction and allow repeated occurrences of any element, then we can get a mathematical structure that is known as Multisets or Bags. For example, the prime factorization of an integer n > 0 is a Multiset whose elements are primes. The number 120 has the prime factorization 120 = 2 3 3 1 5 1 which gives the Multiset {2, 2, 2, 3, 5}" [37]. As a generalization of multiset, Yager [38] proposed fuzzy multiset which can occur more than once with possibly the same or different membership values. Then, Shinoj and John [37,39,40] proposed intuitionistic fuzzy multiset as a new research area. Many researchers studied intuitionistic fuzzy multisets. Ibrahim and Ejegwa [41] and Ejegwa [42] extended the idea of modal operators to intuitionistic fuzzy multisets. Rajarajeswari and Uma [43] developed normalized geometric and normalized hamming distance measures on intuitionistic fuzzy multisets. Ejegwa [44] gave a method to convert intuitionistic fuzzy multisets to fuzzy sets. Ejegwa and Awolola [45] proposed a application of intuitionistic fuzzy multisets in binomial dis-tributions. Deepa [46] examined some implication results and Das et al. [47] proposed a group decision-making method. Rajarajeswari and Uma [48][49][50][51][52] introduced some measure for intuitionistic fuzzy multisets. Also, Shinoj and Sunil [53] and Ejegwa and Awolola [54] gave algebraic structures of intuitionistic fuzzy multisets, called intuitionistic fuzzy multigroups, and its various properties were examined. Also, the same authors proposed the topological structures of the sets in [55].
From the existing research results, we cannot see any study on intuitionistic trapezoidal fuzzy multi-numbers (ITFMnumbers). The ITFM-numbers are a generalization of trapezoidal fuzzy numbers and intuitionistic trapezoidal fuzzy numbers which are commonly used in real decision problems, because the lack of information or imprecision of the available information in real situations is more serious. So the research of ranking ITFM-numbers is very necessary and the ranking problem is more difficult than ranking ITFM-numbers due to additional multi-membership functions and multi-non-membership functions. Therefore, the remainder of this article is organized as follows. In "Preliminary", some preliminary background on intuitionistic fuzzy multiset and intuitionistic fuzzy numbers is given. In "Intuitionistic trapezoidal fuzzy multi-number", ITFMnumbers and operations are proposed. In "Some aggregation operators on ITFM-numbers", some aggregation operators on ITFM-numbers by using algebraic sum and algebraic product is given in Definition 2.3. In "An approach to MADM problems with ITFM-numbers", we introduce a multi-criteria making method, called ITFM-numbers multicriteria decision-making method, by using the aggregation operator. In "Application", case studies are proposed to verify the developed approach and to demonstrate its practicality and effectiveness. In "Comparison analysis and discussion", some conclusions and directions for future work are initiated.
1. Drastic product: 2. Drastic sum: 3. Bounded product: 4. Bounded sum: 5. Einstein product: 6. Einstein sum: 7. Algebraic product: 8. Algebraic sum: 9. Hamacher product: 10. Hamacher sum: 11. Minumum: 12. Maximum: Definition 2.4 [58] Let X be a non-empty set. A multi-fuzzy set A on X is defined as: Definition 2.5 [1] Let X be a nonempty set. An intuitionistic fuzzy set (IFS) A is an object having the form 1] defines, respectively, the degree of membership and the degree of non-membership of the element x ∈ X to the set Definition 2.6 [39,40] Let X be a non-empty set. A intuitionistic fuzzy multiset IFM on X is defined as: . . , p} and x ∈ X . Also, the membership sequence is defined as a decreasingly ordered sequence of elements, that is, and the corresponding non-membership sequence will be denoted by ) such that neither decreasing nor increasing function x ∈ X and i = (1, 2, . . . , p) Definition 2.7 [7] Letα be an intuitionistic trapezoidal fuzzy number; its membership function and non-membership function are given, respectively, as

Definition 2.9 [59] Let
Then, the degree of similarity S(A, B) between the generalized trapezoidal fuzzy numbers P(A) and P(B) is calculated as follows: where S(A, B) ∈ [0, 1]; P(A) and P(B) are defined as follows:

P(A)
and P(B) denote the perimeters of the generalized trapezoidal fuzzy numbers A and B , respectively.

Intuitionistic trapezoidal fuzzy multi-number
is a special intuitionistic fuzzy multiset on the real number set R, whose membership functions and non-membership functions are defined as follows, respectively: otherwise.
Note that the set of all ITFM-numbers on R will be denoted by .

Example 3.2
The ITFM-numbers function

Definition 3.3 Let
. . , ϑ P A ) ∈ and γ = 0 be any real number. Then, In the following example, we use the Einstein sum and Einstein product is given in Definition 2.3.

Definition 3.5 Let
A is called neither positive nor negative ITFM-numbers if a > 0 and d < 0.

Theorem 3.8 Let
Proof In the following proof, we use the Einstein sum and Einstein product is given in Definition 2.3.

Some aggregation operators on ITFM-numbers
In the section, we use the algebraic sum and algebraic product is given in Definition 2.3. From now on, we use I n = {1, 2, . . . , n} and I m = {1, 2, . . . , m} as an index set for n ∈ N and m ∈ N, respectively.

An approach to MADM problems with ITFM-numbers
In this section, we define a multi-criteria making method, called ITFM-numbers multi-criteria decision-making method, by using the ITFMWG and (ITFMWA) operators.
is called an ITFM-number multi-criteria decision matrix of the decision maker. Now, we can give algorithm of the ITFM-numbers multicriteria decision-making method as follows:
Note that if r i for all i ∈ I m is not normalized ITFMnumbers, then we compute the normalized ITFMnumbers according to Definition 3.9.
Step 3 Calculate the distances between collective overall values Step 4 Rank all the alternatives A i (i = 1, 2, 3, . . . , m) and select the best one(s) in accordance with S(r i , r + ).
The bigger the distance S(r i , r + i ), the better are the alternatives A i , i ∈ I m .
Note that if r i for all i ∈ I m is not normalized ITFMnumbers, then we compute the normalized ITFMnumbers according to Definition 3.9.
Step 3 Calculate the distances between collective overall values Step 4 Rank all the alternatives A i (i = 1, 2, 3, . . . , m) and select the best one(s) in accordance with S(r i , r + ). The bigger the distance S(r i , r + i ), the better is the alternatives A i , i ∈ I m .

Application
The anonymous review of the doctoral dissertation in Turkey universities.
In many Turkey universities, doctoral dissertation will be reviewed by three experts anonymously and they have same importance in this review process. They will review dissertation according to five criteria, including topic selection and literature review, innovation, theory basis and special knowledge, capacity of scientific research and theses writing. Different weights are given to different criteria and the standards for those principles are as follows. After thorough investigation, four universities (alternatives) are taken into consideration, i.e., {x 1 , x 2 , x 3 , x 4 }. There are many factors that affect the review process and five factors are considered based on the experience of the department personnel, including u 1 ; topic selection and literature review (such as belonging to the leading edge of the subject or the hot research point has important theoretic significance and applied value; familiar with the research status and process for subject.), u 2 ; innovation (such as have theoretical breakthrough; have positive influence and impact on the development of social economy and culture; creativity points, u 3 ; theory basis and special knowledge (such as solid and broad theoretical foundation, also have spe-cialized knowledge for the subject and related area) and u 4 ; capacity of scientific research (such as independently scientific research ability; informative citing information; subject to be explored in depth) u 5 ; theses writing(such as clear concept and logistics, smooth sentences, format specification, good school ethos) · (whose weighted vector ω = (0.1, 0.3, 0.2, 0.3, 0.1)) Our solution is to examine the university at different time intervals (four times a year: autumn, spring, winter, summer), which in turn gives rise to different membership functions for each university.
Step 1 Construct the decision-making matrix A = (a i j ) m×n for decision as: Step 2 Applying the ITFMWG operator to derive the collective overall preference intuitionistic trapezoidal fuzzy multiset r i : Step 3 Calculate the distances between collective overall values r i and intuitionistic trapezoidal fuzzy positive ideal solution r + . Step 4 Rank all the alternatives A i (i = 1, 2, 3, 4) in accordance with the ascending order of S(r i , r + ): Step 5 End Step 1 Construct the decision-making matrix A = (a i j ) m×n , for decision as: Step 4 Rank all the alternatives A i (i = 1, 2, 3, 4) in accordance with the ascending order of S(r i , r + ): A 1 < A 3 < A 4 < A 2 , thus the most desirable alternative is A 2 .
Step 5 End

Comparison analysis and discussion
To verify the feasibility and effectiveness of the proposed decision-making approach, a comparison analysis with TFM-numbers multi-criteria decision-making method, used by Ulucay et al. [49], is given, based on the same illustrative example. Clearly, the ranking order results are consistent with the result obtained in [49] (Table 1).

Conclusion
In this study, we have defined ITFM-numbers and operational laws, which are mainly based on t norm and t conorm. The ITFM-numbers are a generalization of trapezoidal fuzzy numbers, and intuitionistic trapezoidal fuzzy numbers which are commonly used in real decision problems with the lack of information or imprecision of the available information in real situations is more serious. So the research of ranking ITFM-numbers is very necessary and the ranking problem is more difficult than ranking ITFM-numbers due Ulucay et al. [49] The proposed method to additional multi-membership functions and multi-nonmembership functions. So, some aggregation operators on ITFM-numbers by using algebraic sum and algebraic product is given in Definition 2.3. Based on the aggregation operators, we developed a multi-criteria making method, called ITFMnumbers multi-criteria decision-making method, by using the ITFMWG operator. Finally, we have proposed a practical example to discuss the applicability of ITFM-numbers multi-criteria decision-making method. In future work, we shall develop some new method and apply our theory to other fields, such as medical diagnosis, game theory, investment decision making, military system efficiency evaluation, and so on.