Trapezoidal Hesitant Intuitionistic Fuzzy Numbers and Their Applications to Multiple-Criteria Decision Making Problems

In this paper, we introduce an extension theory of the trapezoidal intuitionistic fuzzy numbers under intuitionistic hesitant fuzzy sets called trapezoidal hesitant intuitionistic fuzzy number (THIF-number). This new theory provides very eﬀectively to model uncertainties of some events by several diﬀerent trapezoidal intuitionistic fuzzy numbers based on the same support set in the set of real numbers R. Also, to demonstrate the application of this theory, a new multi-criteria decision-making(MCDM) method based on THIF-numbers is presented. To do this, we ﬁrst propose operations of THIF-numbers with properties. We second give score, standard deviation degree, deviation degree of THIF-numbers to compare THIF-numbers. We third develop geometric operators and arithmetic operators of THIF-number. Finally, a numerical example is presented to illustrate the application of the developed method in THIF-numbers.


Introduction
Fuzzy set theory introduced by Zadeh [32] with a membership function on [0, 1] to model uncertainty information which classical set with a membership function on {0, 1} is unable to handle. Then, intuitionistic fuzzy set theory proposed by Atanassov [3] with a membership function and a non-membership function on [0, 1] which has proven its usefulness over the years and able to solve many problems which classical set and fuzzy set is unable to handle. After the introduction of fuzzy set theory and intuitionistic fuzzy set theory, the theories have widely been applied by many researchers in [12,33]. Although an element of fuzzy set has only one membership values, some decision making problems may need more than one membership values. For this, Torra and Narukawa [23,22] introduced the theory of hesitant fuzzy sets. After the work of Torra and Narukawa [23,22], different studies on hesitant fuzzy set were carried out in [1,2,4,10,11,18,19,20,25,29].
By using intuitionistic fuzzy, Beg and Rashid [5] first proposed hesitant intuitionistic fuzzy sets. Then, they defined a distance measure and developed a TOPSIS method on hesitant intuitionistic fuzzy sets. Peng et al. [17] developed operations including the averaging operator under Archimedean t-norms and t-conorms. By using the cross-entropy, Peng et al. [16] developed two model by defining cross-entropy of intuitionistic hesitant fuzzy sets. After the pioneer work of Beg and Rashid [5], Zhang [28] generalized the theory to interval-valued intuitionistic fuzzy sets(IVIF-sets). Also, he proposed some aggregation operators of IVIF-sets and introduced a method for multiple attribute group decision-making on the IVIF-sets. Zhou et al [30] said that "Preference relations are a powerful quantitative decision approach that assists decision makers in expressing their preferences over alternatives." Therefore, Zhou et al [30] introduced a proposal for the hesitant intuitionistic fuzzy preference relation (HIFR) and presented a group decision-making by initiating operational laws and aggregation operators of HIFR. Yu and Wang [27] developed a group decision making method by studying on HIFR including improved fuzzy preference relation and consistency index. Nazra et al. [13]- [14] combined hesitant intuitionistic fuzzy sets and soft sets and gave some operations of the sets, such as; complement, union and intersection. Also, Zhou and Xu [31] defined extended intuitionistic fuzzy number as an alternative to hesitant intuitionistic fuzzy sets.
On reel number R, Deli and Karaaslan [7] defined concept of generalized hesitant trapezoidal fuzzy numbers(GHTF-numbers) based on hesitant fuzzy sets whose membership degrees are expressed by several possible generalized trapezoidal fuzzy numbers. Deli [6] developed an approach for multi criteria decision making problems based on TOPSIS method by introducing some novel distance measures. Moreover, Deli [8] presented a multiple attribute decision-making method with GHTFnumbers by defining two aggregation techniques under Bonferroni mean operator for aggregating the GHTF-information. As far as we know, there is no study on generalized trapezoidal hesitant intuitionistic fuzzy number in the literature. To fill this gap, the rest part of this paper is organized as follows: Section 2 first reviews some basic concepts of fuzzy sets, hesitant fuzzy sets, generalized trapezoidal hesitant fuzzy numbers, intuitionistic fuzzy sets and intuitionistic hesitant fuzzy sets. Section 3 extends the intuitionistic hesitant fuzzy sets to generalized trapezoidal hesitant intuitionistic fuzzy environments and propose the concept of generalized trapezoidal hesitant intuitionistic fuzzy number (THIF-number) according to same support set in the set of real numbers R. Also, the section contains some desired operational laws of THIF-numbers and some THIF aggregation operators called the THIF-number weighted geometric operator and THIF-number weighted arithmetic operator including some properties of them. Section 4 develops a multi-criteria decision-making(MCDM) problems based on THIF-number. Also the section offers a practical example to illustrate the application of the developed method in THIF-numbers. The conclusion is shown in the last section.

Preliminary
Definition 2.1. [3] Let X be a nonempty set. An Intuitionistic Fuzzy Set (IFS) A is an object having the form where the function µ A : X → [0.1],ν A : X → [0.1] define respectively the degree of membership and the degree of non membership of the element x ∈ X to the set Definition 2.2. [24] Letα is an intuitionistic trapezoidal fuzzy number, its membership function and non-membership function as given, respectively. [7] Let ξ 1 GT HF N and ξ 2 GT HF N be two GTHF-numbers, S(ξ 1 GT HF N ) and S(ξ 2 GT HF N ) the scores of ξ 1 GT HF N and ξ 2 GT HF N , respectively, and D(ξ 1 GT HF N ) and D(ξ 2 GT HF N ) the deviation degrees of ξ 1 GT HF N and ξ 2 GT HF N , respectively. Then, 1. If S(ξ 1 GT HF N ) < S(ξ 2 GT HF N ) then ξ 1 GT HF N < ξ 2 GT HF N Definition 2.8. [7] Let ξ j GT HF N , j ∈ I n be a collection of GTHF-numbers. Then, 1. GTHF-number weighted geometric operator is defined as;

Trapezoidal Hesitant Intuitionistic Fuzzy Number
Then, a trapezoidal hesitant intuitionistic fuzzy number(THIF-number) is defined as follows; is a special hesitant intuitionistic fuzzy set on the real number set R, whose membership functions and non-membership functions as given, respectively.
be three THIF-numbers and γ = 0 be any real number. Then, 2. standard deviation degree of , is denoted by D s ( ), is defined as Based on the score of , give the deviation degree of as; 3. deviation degree of , is denoted by D( ), is defined as Now, we give a method to compare the two THIF-numbers.
Note that if the number of elements of (α, β) is one, then the equations are provided. Definition 3.6. Let j = (a j , b j , c j , d j ); (α j , β j ) (j ∈ I n+1 ) be a collection of THIF-numbers. For then Ω G w is called THIF-number weighted geometric operator of dimension n, where w = (w 1 , w 2 , ..., w n ) T is the weight vector of j , j ∈ I n , with w j ∈ [0, 1] and n j=1 w j = 1.
Definition 3.7. Let j = (a j , b j , c j , d j ); (α j , β j ) (j ∈ I n+1 ) be a collection of THIF-numbers. For then Ω A w is called THIF-number weighted arithmetic operator of dimension n, where w = (w 1 , w 2 , ..., w n ) T is the weight vector of j , j ∈ I n , with w j ∈ [0, 1] and n j=1 w j = 1.

An approach to MCDM problems with THIF-numbers
In this section, we developed a method for THIF-numbers by using the proposed concepts in section 3.
Definition 4.1. Let K = {k 1 , k 2 , ..., k m } be a set of alternatives, L = {l 1 , l 2 , ..., l n } be the set of criteria. If A ij = (a ij , b ij , c ij , d ij ); ; (α ij , β ij ) be THIF-numbers then decision matrix is given as; Here A ij denotes evaluation of the alternative k i with respect to the criteria l j made by expert or decision maker. Example 6. Let's assume that an agricultural firm has a 40-50 year plan to maximize its profit by producing agricultural products. There is five alternatives is denoted by K = {k 1 = Walnut, k 2 =Banana, k 3 =Grape, k 4 =Apple, k 5 =annual vegetables such as tomatoes, peppers, eggplants} that it can produce. The five possible alternatives are to be evaluated under the four criteria is denoted by L = {l 1 = market, l 2 =support of government, l 3 =seasonal factors, l 4 =sustainability} by corresponding to linguistic values of THIF-numbers for linguistic terms as shown in Table 1. The weight vector of the criterions is w = (0.15, 0.40, 0.30, 0.15) T .  Step 1. The expert construct the decision matrix as follows: Step 2. The ρ G i = Ω G w (A i1 , A i2 , ..., A in )) for i ∈ I 5 are computed as; 1. The scores s(ρ G i ) (i ∈ I 5 of the THIF-numbers ρ A i (i ∈ I 5 are found as; s(ρ G 1 ) = 0.0427 s(ρ G 2 ) = 0.0346 s(ρ G 3 ) = 0.0123 s(ρ G 4 ) = 0.0247 s(ρ G 5 ) = 0.0275 It is obvious that s(ρ G 1 ) > s(ρ G 2 ) > s(ρ G 5 ) > s(ρ G 4 ) > s(ρ G 3 ) Therefore, the ranking order of the alternatives k i (i = 1, 2, 3, 4, 5) is generated as follows: The best supplier for the enterprise is k 1 .

Conclusion
In this paper, a new trapezoidal hesitant intuitionistic fuzzy number (THIF-number) theory has been proposed along with its associated properties, theorems and definitions. Also, to demonstrate the application of this theory, a new multi-criteria decision-making(MCDM) method based on THIFnumber is presented. The applicability of the proposed method, a numerical example is presented to illustrate the application of the developed method in THIF-numbers. Several future research directions are put forward: (1) other decision-making methods such as the LINMAP method, Topsis method and (2) new aggregation operators is another direction.