A New Multi Attribute Decision Making Method Based on the T-Spherical Hesitant Fuzzy Sets

In this paper, we first introduce the concept of T-spherical hesitant fuzzy set (T-SHFS) on the basis of the combination of T-spherical fuzzy sets (T-SFSs) and hesitant fuzzy sets (HFSs). This model can provide more accuracy in expressing fuzzy and indeterminate data. Then, we develop the basic operational laws of T-SHFSs and study their properties. Also, we propose the T-spherical hesitant fuzzy weighted averaging (T-SHFWA) operator and the T-spherical hesitant fuzzy weighted geometric (T-SHFWG) operator and investigate their properties. Furthermore, two newly approaches to multi attribute decision making within the framework of T-SHFS are developed on the basis of the proposed aggregation operators. An illustrative example is given to demonstrate the effectiveness of the developed approaches in dealing with uncertain and indeterminate decision making problems. Finally, the proposed models’ performance is evaluated to the existing models. Results from this comparison analysis reveal a great similarity and consistency while using other models.


I. INTRODUCTION
Decision theory has been a chief field of investigation in many scientific disciplines were the area of choice under uncertainty represents the heart of decision theory. The process of decision-making, in the majority of cases, consists of the evaluation of alternatives and the choice of the most preferable from them [1]- [3]. Multiple attribute decision making (MADM) -firstly proposed by Churchman et al. [4] -refers to rank alternatives or select the best choice based on multiple attribute evaluation values of the different alternatives. In traditional MADM problems, decision-makers usually provide deterministic measurements to express their preference. On account of the time limitations, decision-makers' capability, and the increasing uncertainty of problems and the complexity of human's cognitive information, decision-makers frequently find it difficult to provide deterministic measurements when attempting to solve MADM problems. To tackle this issue, Zadeh [5] originated fuzzy set (FS) to process the fuzzy information, and then it was used to easily characterized the attribute values in uncertain MADM problems. However, The associate editor coordinating the review of this manuscript and approving it for publication was Giovanni Pau . in some cases, fuzzy set alone cannot precisely describe vague, ambiguous, incomplete and indeterminate information in MADM problems. To effectively deal with these cases, many generalizations and variations of fuzzy sets have been generated. Among them, we underline, for their relevance in this paper, hesitant fuzzy sets (HFS) [6], intutionistic fuzzy set (IFS) [7], Pythagorean fuzzy set (PYFS) [8] and picture fuzzy set (PFS) [9].
PFS is a generalization of FS and IFS which is characterized by a truth membership function (T), indeterminacy membership function (I), and falsity membership function(F), each of which lies in the standard interval of [0, 1] and their summation is greater than or equal 0 and less than or equal 1, i.e., 0 ≤ T + I + F ≤ 1. Basically, PFS has been originated to express the indeterminacy, which cannot be accurately expressed in the traditional FS and IFS. PFS has further been developed to picture hesitant fuzzy set(PHFS) [10] and spherical fuzzy set (SFS) [11]. In SFS memberships grades are gratifying the condition 0 ≤ T 2 + I 2 + F 2 ≤ 1 instead of 0 ≤ T + I + F ≤ 1 as is in PFS. SFS has further been extended to spherical hesitant fuzzy set (SHFS) [12] which is a generalization of PHFS. Comparing with PHFS to model uncertain information, SHFS has wider feasible region, which can handle more room of uncertainty.
In addition, IFS and PYFS has been extended to q-rung orthopair fuzzy set (q-ROFS) [13] which is more flexible than IFS and PYFS since the sum of the qth power of the truth membership and falsity membership less than 1. In other words, q-ROFSs relax the constraint of IFS and PYFSs.
In order to enhance the quality of q-ROFS, so it can deal with vagueness and impreciseness, Li et al. [14] proposed q-rung picture fuzzy set (q-RPFS), which takes the advantages of both q-ROFS and PFS. To effectively aggregate q-RPF data, some aggregation operators have been developed [14]- [19]. In [14], the authors proposed the q-rung picture linguistic weighted Heronian mean (q-RPLWHM) and the q-rung picture linguistic geometric Heronian mean (q-RPLGHM) operators. He et al., developed some q-RPF Dombi Hamy mean (q-RPFDHM) operators in [15]. Akram et al. [16], proposed Einstein operational laws for q-rung picture fuzzy numbers and introduced some q-rung picture fuzzy Einstein weighted averaging operators. Specific types of q-RPF Yager average and geometric aggregation operators have been studied in [17]. Yang et al., [18] presented the interval q-RPF Heronian mean (IVq-RPtFHM) operators based on the new operational laws of the IVq-RPtF numbers. Pinar and Boran [19] introduced a novel distance measure for q-RPFS. The proposed distance measure is used in q-RPF ELECTRE integrated with TOPSIS method.
Meanwhile, to consider human's hesitance, q-ROFS has been extended to q-rung orthopair hesitant fuzzy set (q-ROHFS) by Liu et al. [20]. Authors proposed the distance measures between q-ROHFSs and developed TOPSIS method into the proposed measures. Hussain et al. [21] also presented a new concept of hesitant q-rung orthopair fuzzy set (H q ROFS) and proposed the H q ROF weighted averaging (H q ROFWA) and H q ROF weighted geometric (H q ROFWG)operators. Wang et al. [22] defined the distance, similarity measures and the entropy of q-ROHFSs. Based on these concepts, they constructed a TOPSIS model under the q-ROHF environment. Wang et al. [23] presented the dual hesitant q-rung orthopair fuzzy set (DHq-ROFS) for handling real MADM problems. To aggregate the information in DHq-ROFSs some Muirhead mean operators are defined. Afterward, the defined operators are used to solve the MADM with DHq-ROF numbers. In the same year Wang et al. [24] proposed some power Heronian mean operators under the q-ROHF environment to deal with green supplier selection in supply chain management. Yang and Pang [25] developed a new MADM based on the q-ROHFS, where the linear programming technique for multidimensional analysis of preference (LINMAP) and TOPSIS have been extended to q-ROHF environment in order to handle MADM problems.
Mahmood et al. [11] proposed T-spherical fuzzy set (T-SFS) as a generalization of q-ROFS and q-RPFS. Some aggregation operators have been developed on T-SFS. [11], [26]- [31]. In [11], the authors proposed the T-spherical fuzzy weighted geometric operator. Garg et al., developed some weighted ordered weighted and hybrid geometric aggregation operators in [26]. Garg et al. [27], define several weighted averaging and geometric power aggregation operators. The stated operators named as T-spherical fuzzy weighted, ordered weighted, hybrid averaging and geometric operators for the collection of the T-SFSs. Zeng et al. [28] proposed some new Einstein operations for TSF numbers on which they define some T-spherical Einstein interactive averaging and geometric operators. In [29], a generalized parameter is defined for TSFSs and based on the proposed parameter, a group generalized TSF geometric operators are proposed. In the complex space, Ali et al. [30] proposed two aggregation operators to the complex TSF numbers, including weighted geometric and weighted averaging operators. In addition, Karaaslan et al. [31], introduced some complex TSF Dombi weighted aggregation operators on the basis of the Dombi t-norm and t-conorm.
Summarizing above discussion, we combine the advantages of both T-SFS and HFS by proposing the notion of T-spherical hesitant fuzzy set(T-SHFS) which is a more comprehensive model to cope with the most real complicated situations that can not be handled by the existing models. In the T-SHFS, for each element of the reference set, the grades of truth, indeterminacy and falsity memberships consist of the set of several values in the interval [0, 1] rather than a single value. Furthermore, the sum of the qth power of the truth membership, the qth power of the indeterminacy membership and the qth power of the falsity membership is not more than 1, which makes it more influential and resilient than other current models. Thus, the contributions of this paper are stated as follows: (1) the novel notion of T-SHFS with its operations are proposed; (2) to effectively aggregate T-SHF data, we propose the aggregation operators, called T-SHFWA and T-SHFWG operators; (3) MADM problem is addressed based on T-SHFNs by utilizing the T-SHFWA and T-SHFWG operators. To examine the efficiency and validity of the proposed models, we solve an example by using the proposed operators;(4) the proposed operators are compared with existing operators with an MADM example.
The remaining portions of the paper is set out as follows. In section 2, we present some essential concepts related to PHFS, SHFS, q-ROFS, T-SFS and q-ROHFS. In Section 3, we provide the formal concept of the T-SHFS and define the score and accuracy function of the T-SHFNs. The operational rules of the T-SHFNs along with their properties are also introduced in this section. We propose the T-SHFWA operator and the T-SHFWG operator in Section 4. Properties of these operators are also investigated in this section. On the basis of the T-SHFWA operator and the T-SHFWG operator, two new methods for MADM are proposed in Sction 5. To verify the application of the suggested models, Section 6 provides real example about the ranking of mobile phone products. The comparison analysis with other existing models has been conducted in Section 7. Finally, Section 8, outlines the main results in this research. VOLUME 9, 2021

II. BASIC CONCEPTS
We shortly review the basic concepts of PHFS, SHFS, q-ROFS, T-SFS and q-ROHFS which be utilized in the subsequent research.

B. SPHERICAL HESTANT FUZZY SET (SHFS)
Definition 2 [12]: Consider the ground set M = φ. A SHFS F on M can be defined as follows.

C. Q-ROFS
Definition 4 [13]: If M is a finite set. Then the q-ROFS H on M is defined as where T H (µ) and F H (µ) represent degree of positive membership and the degree of negative membership respectively, where T H (µ) and F H (µ) belong to the unit interval [0, 1] and Mahmood et al. [10] define the T-SFS by taking the neutral membership degree into account in q-ROFSs.

D. T-SPHERICAL FUZZY SET (T-SFS)
Definition 5 [10]: If M is a fixed set. A T-SFS H on M is defined as follows.
Based on the q-ROFS and HFS, Liu et al. [20] define the q-ROHFS as follows. Definition 6 [20]: where D(m i ) and E(m i ) are two sets of values in [0, 1], denoting all the possible qth rung memberships and the qth rung nonmemberships of the elements m i ∈ M , and they satisfy: The qth rung orthopair hesitancy membership degree ξ can be defined as ξ = (1 − α q − β q ) 1/q . The basic operational rules of q-ROHFNs are defined as follows.
Definition 7 [20]: Then the operational laws of q-ROHFNs are represented as: The following is the definition of the score and accuracy functions of the q-ROHFN.
Definition 8 [20]: IfQ = DQ, EQ q is a q-ROHFN, such that γ (DQ) and γ (EQ) equal the number of objects in sets DQ and EQ, respectively, then a score function (Q) is: Accuracy function of q-ROHFN λ(Q) can be stated as Hussain et al. [21] provide the basic notions of related aggregation operators of the q-ROHFN as follows.
Based on the operational rules of the q-ROHFNs, the aggregation result for the q-ROHFNs is given as in Theorem 1.
Definition 10 [21]: The aggregation result for q-ROHFNs through operation rule is given as in Theorem 2.
Theorem 2 [21]: IfQ k = DQ k , EQ k q (k = 1, 2, . . . , n) is a set of q-ROHFNs, then the aggregated value by utilizing the q-ROHFWG operator is still q-ROHFN and In the following, we propose the notion of T-SHFS as an extension of q-ROHFS by adding the hesitant neutral membership function to the structure of the q-ROHFS.

III. T-SPHERICAL HESITANT FUZZY SET
We will propose the definition of T-SHFS as follows: and F(m i ) are three sets of values in [0, 1], denoting, respectively all the possible truth membership degrees, indeterminacy membership degrees and falsity membership degrees of the elements m i ∈ M to the set Q and they satisfy the following conditions: The degree of refusal membership can be defined The accuracy function is defined as: According to the definition of score and accuracy functions above, the method for comparing two T-SHFNs can be defined as follows.
3) According to (1) and (3) in Definition 14, we can get for the left side of the Equation For the right side of the Equation, we can get: Thus, we get λ(Q 1 ⊕Q 2 ) = λQ 1 ⊕ λQ 2 . 4) The proof is similar to that of (3).

IV. T-SHF AGGREGATION OPERATORS
On the basis of the operational rules of T-SHFNs in Section 3, we will propose the T-SHFWA operator and the T-SHFWG operator and investigate their properties. Proof: We will use the method of mathematical induction to prove this theorem.
Obviously Equation (1) holds for n = 2. 2) If Equation (1) holds for k = n, then When k = n + 1, based on the operational rules of the T-SHFNs, we have  (1)and (2), we can get Equation (1) holds for any k.
In the following, we present the properties of the T-SHFWA operator along with their proofs. Proof: According to Theorem 4, sinceQ k = Q = TQ, IQ, FQ q , we have where, Similarly, as z − ≤ zQ and, where J − , K − , and L − are the number of values in the sets TQ, IQ and FQ, respectively, such that TQ, IQ and FQ belong to (Q) − . J + , K + , and L + are the number of values in the sets TQ, IQ and FQ, respectively, which belong to (Q) + .
Note that J − = K − = L − = J + = K + = L + = n, where n represents the number of the T-SHFNs.

B. THE T-SHFWG OPERATOR
The definition of T-SHFWG operator can be stated as follows.

V. DECISION-MAKING METHOD BASED ON THE T-SHFWA AND T-SHFWG OPERATORS
In this section, based on the T-SHFWA and T-SHFWG operators, we will come up with two new approaches for MADM with T-SHF information. Suppose M = {M 1 , M 2 , . . . , M n } is a collection of n alternatives which are assessed according to k attributes F = {F 1 , F 2 , . . . , F k }. If the weight vector of the attributes is w = {w 1 , w 2 , . . . , w k }, where k j=1 w j = 1 w j ≥ 0. Suppose that D = Q ij n×k represents the decision matrix, whereQ ij = T ij , I ij , F ij q is the assessment value of alternative M i , for attribute F j , which is represented by T-SHFN. Then the algorithm for solving the MADM problem based on the T-SHFWA operator or T-SHFWG operator is explained below.
Algorithm 1: Step 1: Normalize the decision matrix D by converting all cost attributes to benefit attributes. The converted values can be calculated by the following formulã Q ij = T ij , I ij , F ij q for benefit attributes F ij , I ij , T ij q for cost attributes (3) Step 2: Aggregate all attributes valuesQ ij (j = 1, 2, . . . , k) to obtain the overall valueS i by T-SHFWA operator such Step 3: RankS i (i = 1, 2, . . . , n) based on the score function (S i ) and accuracy function λ(S i ) by Definition 12.
Step 4: Rank the corresponding alternatives according to their scores to get the best one.

VI. ILLUSTRATIVE EXAMPLE
To reveal the application of the suggested models, the following example about the ranking of mobile phone products is adapted from [32]. The proposed methods are utilized to rank these phones according to the following features.  Table 1.
Then we can rank the alternative mobile phones using Algorithm 1.
Step 1: Normalize D = Q ij n×k : Because the only cost attribute in this example is battery draining F 5 , we convert it to benefit attribute using Equation 3, and then we obtain D * as in Table 2.
Step 2: Utilizing the T-SHFWA operator, the overall evaluation values of the alternatives is obtained as:   Step 4: We get the rank of the four mobile phones as.  Obviously, these two approaches have the same ranking result.

VII. THE COMPARISON ANALYSIS WITH THE OTHER METHODS
Apparently, T-SHFS is improved from SHFS which is the basis of the suggested approach. Also, the models H q ROFS and PHFS are the closest in structure to T-SHFS. Hence, to further ensure the validity of the proposed methods and explore their advantages, we compare the suggested operators with some SHF operators, some H q ROF operators and some PHF operators, including SHF weighted averaging (SHFWA) operator [12], SHF weighted geometric (SHFWG) operator [12], H q ROF weighted averaging ( H q ROFWA) operator [21], H q ROF weighted geometric (H q ROFWG)operator [21], PHF weighted averaging (PHFWA) operator [10] and PHF weighted geometric(PHFWG) operator [10]. In order to conduct the comparison, we solve the same decision making problem above using these operators. It is worth mentioning that the H q ROFSs only have two membership degrees, a truth and a falsity membership degrees, while the T-SHFS is constructed by three membership degrees, which are truth, indeterminacy and falsity membership degrees. Thus, the H q ROFS is a special case of T-SHFS and can be easily represented in the form of T-SHFS. In other words, the H q ROFS is a T-SHFS with indeterminacy term equal zero, so it cannot handle the T-SHFNs. Thus, to compare the suggested method with these in [22], we put the degree of indeterminacy to 0 in the suggested operator. For PHFS and SHFS they are also considered as special cases of T-SHFS with q = 1 and q = 2, respectively. In particular, we set q = 1 in the proposed operators when we compare the PHF operators with the proposed operators and q = 2, while comparing SHF operators with the proposed operators.
The comparison results are shown as Table 3. From Table 3, it is evident that the ranking results produced by the proposed methods are almost the same as those produced by other methods. However, their optimal selections are the same, which can prove the effectiveness of our methods very well. As mentioned above, H q ROFNs do not have the membership function which represents the neutrality, which will lead to the lack of some information, while the proposed T-SHFNs include truth, neutral and false membership degrees and give decision makers a more flexible environment to avoid information loss in the decision making process.
Wang and Li's [10] method is based on PHFW operators, which need simple calculations, but its area of applications very limited. It can only deal with the problem that parameters are represented as PHFNs, and PHFNs cannot fully represent the actual decision information, as the summation of its potential positive, neutral, and negative membership degrees must be between 0 and 1, i.e, (0 ≤ x + + y + + Z + ≤ 1). Thus, it will easily cause the bias of the data.
Khan et al.'s [12] method is based on the SHF operators which use the SHFNs. This makes the SHF operators more flexible than PHFW operators as the structure of SHFNs is with a constraint that the sum of the characteristic functions may exceeded from the unit interval but their square must belong to the unit interval, i.e, 0 ≤ (x + ) 2 +(y + ) 2 +(Z + ) 2 ≤ 1. However, the proposed method is based on the T-SHFW operators, which is expressed based on the T-SHFN. The T-SHFN is more flexible than the PHFN and SHFN because it needs the summation of potential truth, neutral and falsity membership degrees must meet 0 ≤ (x + ) q + (y + ) q + (Z + ) q ≤ 1 (q ≥ 1). It can represent the attributes more comprehensively. For the proposed T-SHFWA and T-SHFWG operators, we can find that they are best ways to express the neutral data by the T-SHFNs because they make the decision making process easier and changeable by a parameter q. By increasing q, the scope of the evaluated information will be wider, thus avoiding the bias of information.

VIII. CONCLUSION
This paper proposed the T-SHFSs, provided the operational laws of the T-SHFNs and discussed their properties. The proposed T-SHFSs inherent the advantages and superiorities of the SHFS, q-ROHFS and q-RPFS so that it is more flexible and powerful. Based on the operational laws of the T-SHFNs, the T-SHFWA operator and T-SHFWG opertor are defined. The properties of the proposed operators such as the idempotency, boundedness and monotonicity are investigated. Afterwards, we proposed two new methods for MADM, where the attributes values is in the form of T-SHFNs. We applied the newly developed methods to rank mobile phone products based on online reviews. The comparative study is also provided and the results show a great similarity and consistency while using other ranking methods such as SHFWA operator, SHFWG opertor, H q ROFWA operator, H q ROFWG operator, GPHFWA operator and GPHFWG operator.