MADM Based on Generalized Interval Neutrosophic Schweizer-Sklar Prioritized Aggregation Operators

The interval neutrosophic set (INS) can make it easier to articulate incomplete, indeterminate, and inconsistent information, and the Schweizer-Sklar (Sh-Sk) t-norm (tm) and tconorm (tcm) can make the information aggregation process more flexible due to a variable parameter. To take full advantage of INS and Sh-Sk operations, in this article, we expanded the ShSk and to IN numbers (INNs) in which the variable parameter takes values from   ,0  , develop the Sh-Sk operational laws for INNs and discussed its desirable properties. After that, based on these newly developed operational laws, two types of generalized prioritized aggregation operators are established, the generalized IN Sh-Sk prioritized weighted averaging (INSh-SkPWA) operator and the generalized IN Sh-Sk prioritized weighted geometric (INSh-SkPWG) operator. Additionally, we swot a number of valuable characteristics of these intended aggregation operators (AGOs) and created two novel decision-making models to match with multiple-attribute decisionmaking (MADM) problems under IN information established on INSh-SkPWA and INSh-SkPRWG operators. Finally, an expressive example regarding evaluating the technological innovation capability for the high-tech enterprises is specified to confirm the efficacy of the intended models.


Introduction
The most important utility of multiple-attribute decision-making (MADM) problems is to go for the preeminent alternative from the set of finite alternatives as stated to the partiality values specified by decision makers (DMs) with admiration to the attributes.However, despite the complication of the decision situation, it is hard for DMs to convey the partiality values by a particular real number in realistic problems.To agree with such circumstances, an intuitionistic fuzzy set (IFS) initiated by Atanassov [1] is one of the most promising simplifications of the fuzzy set (FS) initiated by Zadeh [2] to articulate unsure and inaccurate information perfectly [3][4][5].Yet, in several circumstances, only a positive-membership degree (TMD) and negative-membership degree (FMD) cannot depict the incompatible information precisely.To agree with the corresponding circumstances, Smarandache [6] created a neutrosophic set (NS) which depicts the vague, inaccurate, and incompatible information by TMD, neutral-membership degree (IMD), and FMD.The values of the said functions are taken independently and are standard or nonstandard subsets of ]0 − , 1 + [.As the NS consists of the IMD, it can explain the vague information much better than FS and IFS, and in addition, it is more reliable when it comes to individual expected opinions and perceptions.However, NS is difficult to exploit in factual problems due to the included nonstandard subsets of ]0 − , 1 + [.As a result, to employ NS effortlessly in factual problems, Wang et al. [7,8] initiated the conceptions of single-valued NS and interval neutrosophic set (INS), which are subclasses of NS.
In factual decision making, we require aggregation operators (AGOs) to incorporate the specified information.In a neutrosophic environment, a lot of researchers have anticipated a number of AGOs.For example, the operational laws of single-valued neutrosophic numbers (SVNNs) was initially anticipated by Ye [9] and established the SVN weighted averaging (SVNWA) operator and SVN weighted geometric (SVNWG) operator.Afterwards, Peng et al. [10] located various drawbacks in the operational laws presented by Ye [9] and established enhanced operational rules for SVNNs and anticipated various SVN ordered weighted averages and SVN ordered weighted geometric operators.Ye [11] further presented a number of SVN hybrid averaging (SVNHA) and SVN hybrid geometric (ACNHG) operators and used these AGOs to solve MADM problems.Zhang et al. [12] initiated operational laws for IN numbers and established some IN weighted averaging and IN weighted geometric AGOs and applied these AGOs to solve MADM problems.Ye [13] initiated some IN ordered weighted averaging operators and a possibility ranking method and initiated an approach established based on these AGOs and a possibility ranking method to solve a MADM problem under an IN environment.Sun et al. [14] studied some Choquet integral AGOs for INNs.Garg and Nancy [15] initiated a nonlinear programming model established on TOPSIS to solve MAM problems.Wei et al. [16] initiated several generalized IN Bonferroni mean operators and applied them in the evaluation of high-tech technology enterprises.Tan et al. [17] established various exponential AGOs and specified their application in typhoon disaster evaluation.Wang et al. [18] established a MADM method with IN probability established based on regret theory.Khan et al. [19] initiated the concept of IN power Bonferroni mean operators and applied these to solve MADM problems under IN information.Zhou et al. [20] established several Frank IN weighted and geometric averaging operators.Rani and Garg [21] discovered various drawbacks in division and subtraction operations of INS and established modified division and subtraction operations for INS.Liu et al. [22] presented a MAGDM established on IN power Hamy mean operators.Yang et al. [23] initiated various new similarities and entropies for INS.Meng et al. [24] presented the concept of IN preference and its application in the selection of virtual enterprise partners.Kakati et al. [25] presented various IN hesitant Choquet integral AGOs established on Einstein operational laws and applied them to MADM.Liu et al. [26] presented a number of generalized Hamacher AGOs for NS and applied them to MAGDM.Liu et al. [27] introduced the generalized IN power averaging (GINPA) operator by combining power AGOs with INS to gain the full advantage of power AGOs under an IN environment.Yang et al. [28] established various IN linguistic power AGOs based on Einstein's operational laws.
All the above-presented AGOs are recognized on the anticipation that the input arguments to be aggregated are independent.These managing AGOs have not measured the condition where the attributes have a priority relationship between them.To resolve this difficulty, Yager [29] initiated the concept of PA operator.Wei et al. [30] presented the concept of generalized PA operators.These AGOs were further enlarged by several researchers, such as Wu et al. [31], who enlarged PA operators to the SVN environment by anticipating the notions of SVN prioritized WA (SVNPWA) and SVN prioritized WG (SVNPWG) operators and used them on MADM problems with SVN information.Additionally, Liu et al. [32] anticipated a number of prioritized ordered WA/geometric operators to agree with IN information.Ji et al. [33] fused PA operators with BM operators and presented a number of SVN Frank prioritized BM AGOs by exploiting Frank operations.Recently, Wei et al. [34] put forward a number of PA operators established on Dombi TN and TCN and used them on MADM problems with SVN information.Sahin [35] anticipated some generalized PA operators for normal NS and applied these aggregation operators to MADM.Liu and You [36] studied some IN Muirhead mean operators and applied them to solve MADM problems with IN information.Sarkar et al. [37] developed an Symmetry 2019, 11, 1187 3 of 28 optimization technique for a national income determination model with stability analysis of differential equations in discrete and continuous processes under uncertain environment.
From the mentioned AGOs, the majority of these AGOs for NS or INS are established on algebraic, Hamacher, Frank, and Dombi operational rules, which are particular cases of Archimedean tn (Atn) and tcn (Atcn).Atn and Atcn are definitely the expansions of numerous TNs and TCNs, which have a number of particular cases which are preferrable for articulating the union and intersection of SVNS [38].Sh-Sk operations [39] are the particular cases from Atn and Atcn, and they are with a changeable parameter, so they are additionally supple and better than the former operations.Still, the majority of research concerning Sh-Sk mostly determined the elementary theory and types of Sh-Sk TN (Sh-Sktn) and TCN (Sh-Sktcn) [40,41].Recently, Liu et al. [42] and Zhang [43] merged Sh-Sk operations with interval-valued IFS (IVIFS) and IFS and anticipated power WA/geometric AGOs and weighted power WA/geometric AGOs for IVIFSs and IFSs, respectively.Wang and Liu [44] anticipated a Maclaurin symmetric mean operator for IFS established on Sh-Sk operational laws.Liu et al. [45] further presented Sh-Sk operational laws for SVNS and presented some Sh-Sk prioritized AGOs and applied these AGOs to solve MADM problems.Later on, Zhang et al. [46] anticipated some Muirhead mean operators for SVNS established on Sh-Sk operational laws and applied them to MADM problems.Nagarajan et al. [47] presented some Sh-Sk operational laws for INNs by taking the values of the variable parameter from (0, +∞].They also anticipated some weighted averaging and geometric AOs based on these Sh-Sk operational laws for INNs.In recent years, INSs have gained much attention from the researchers and a great number of achievement have been made, such as VIKOR [48][49][50], cross entropy [51], MABAC, EDAS [52], out ranking approach [53], distance and similarity measures [54], TOPSIS [55].

Literature Review
In this section, the literature discussing IN MADM aggregation operators is reviewed.It has been noticed that research of IN MADM aggregation operators has been rapidly published since 2013.Table 1 provides the recent literature of IN MADM based on different types of aggregation operators.

Table 1. Multiple-attribute decision-making (MADM) methods based on different aggregation operators.
Author the form of real numbers.So far, there are no such AGOs to handle MADM problems under IN information established on Sh-Sktn and Sh-Sktcn.In response to this limitation, we merged the ordinary generalized PA operator with Sh-Sk operations to handle MADM problems with the IN information.
Therefore, from the above research inspirations, the objectives and offerings of this article are revealed as follows.
(1) Anticipating a generalized IN Sh-Sk prioritized weighted averaging (GIN Sh-Sk PWA) operator and generalized IN Sh-Sk prioritized weighted geometric (GIN Sh-Sk PWG) operator.(2) Examining properties and precise cases of these anticipated AGOs.
(3) Put forward two novel MADM approaches based on the anticipated AGOs.(4) Confirming the efficacy and realism of the anticipated approaches.
To achieve these objectives, this article is structured as follows.In Section 3, we initiate some basic ideas of INSs and score and accuracy functions of PA operators.In Section 4, we examine a number of Schweizer-Sklar operational laws for INNs where the variable parameter takes values from [∞−, 0).In Section 5, we propose INSSPWA and INSSPWG operators and examine a number of properties and particular cases of the anticipated AGOs.In Section 6, we present two MADM approaches established on these AGOs.In Section 7, we resolve a numerical example to confirm the soundness and compensations of the anticipated approaches by contrasting with other existing approaches.Finally, a short conclusion is made in Section 8.

Preliminaries
In this part, some basic definitions about INSs, INN, the PA operator, Schweizer-Sklar TN and TCN, and their associated properties are argued.These definitions are given in Appendices A and B.

Sh-Sk Operations for INNs
The Sh-Sk (SS) operations contain Sh-Sk product and Sh-Sk sum, which are particular cases of Att.
Definition 1 [12].Assume that VN 1 = TS 1 , IS 1 , FS 1 and VN 2 = TS 2 , IS 2 , FS 2 are any two arbitrary INSs, then the generalized union and intersection are identified as follows: where Θ and Θ * respectively represent TN and TCN.
Based on the tn and tcn of Sh-Sk operations, we can provide the following definition for Sh-Sk operations for INNs.

Some Generalized Prioritized Aggregation Operators for INNs
In this part, we develop some generalized prioritized aggregation operators based on the developed operational laws for INNs.
Proof.(1).Since vn g = vn for all g = 1, 2, . . ., s, then:  , is . First, when M ∈ (0, ∞), then: Similarly, we have: , we have: , and: Similarly, the same process is for the falsity-membership function, that is: Thus, on the other hand, we have: then: , that is: then: From the above established analysis, we have: (2) If Υ = 1 and A = 0, then the GINSh-SkPWA operator reduces to an IN prioritized weighted average operator based on algebraic operation.That is:

Some Generalized Interval Neutrosophic Schweizer-Sklar Power Geometric Aggregation Operators
In this subsection, we develop a generalized interval neutrosophic Schweizer-Sklar prioritized weighted geometric (GINSh-SkPWG) operator and discuss its desirable properties and some particular cases.Definition 4. For a group of INNs vn g (g = 1, 2, . . ., s), the GIFSh-SkPWG operator is a mapping s → , where Υ ∈ (0, +∞), Proof.In the following, we first prove: by utilizing a mathematical induction on s.
From the operational rules defined for INNs in Definition ( 5), we have: , and: Similarly: Then, when s = m + 1, by the operational rules given in Definition (5), we have: That is, Equation ( 30) is true for g = m + 1.Thus, Equation (30), is true for all g.Then:

Models for Multiple-Attribute Decision Making Established on Proposed Aggregation Operators
In this section, we develop MADM methods based on GINSh-SkPWA and GINSh-SkPWGA operators to deal with interval neutrosophic information.The following assumptions or notions are employed to state the MADM problems.Let AE = AE 1 , AE 2 , . . ., AE s be the set of alternatives, and CA = CA 1 , CA 2 , . . ., CA t be the set of attributes with a prioritization along with the attributes signified by the linear-ordering CA 1 > CA 2 > . . . .> CA t−1 > CA t , then the specific attribute CA q has a superior priority than CA r if q < r.
The aim of this problem is to rank the alternatives.

The Model Established on GINSh-SkPWA Operator
Subsequently, a procedure for positioning and picking of the mainly superior alternative(s) is offered as follows.
Step 1. Equalize the decision matrix.
First, the decision-making information vn gh in matrix VN = vn gh s×t must be equalized.Accordingly, the criteria can be categorized into the benefit and cost types.For the criterion of benefit type, the evaluation information does not need to be distorted, but for the criterion of cost type, it should be customized with the complement set.Thus, the decision matrix can be equalized by exploiting the formula: f or cos t type attribute CA gh (30) Step 2. Find out the values of T gh (g = 1, 2, . . ., s; h = 1, 2, . . ., t) by exploiting the following formula: where T g1 = 1 for g = 1, 2, . . ., s.
Step 3. Exploit the decision information from decision matrix VN = vn gh s×t and the INSh-SkPWA operator specified in Equation ( 18): to acquire the overall INN vn g (g = 1, 2, . . ., s).
Step 5. Rank all the alternatives AE g (g = 1, 2, . . ., s) and pick the preeminent one exploiting Definition A3 given in Appendix A.

The Model Established on GINSh-SkPWGA Operator
Steps 1 and 2 are same as those given for the GINSh-SkPWA operator.
Step 3. Exploit the decision information from decision matrix VN = vn gh s×t and the INSh-SkPWG operator specified in Equation ( 26): to acquire the overall INN vn g (g = 1, 2, . . ., s).
Step 5. Rank all the alternatives AE g (g = 1, 2, . . ., s) and pick the preeminent one exploiting Definition A3 given in Appendix A.

Numerical Example
In this part, a numerical example is modified from [16] about assessing technological innovation competency for high-tech enterprises with INNs.
Assume that AE g (g = 1, . . ., 5) signifies the promising five high-tech enterprises (alternatives), which are to be evaluated.The experts exploit the following four attributes to evaluate the promising five high-tech enterprises: (1) the innovative culture signified by CA 1 ; (2) the infrastructure and funding for the enlargement of industry signified by CA 2 ; (3) the organizational learning and knowledge Symmetry 2019, 11, 1187 20 of 28 management signified by CA 3 ; and (4) support for technological innovation signified by CA 4 .The following priority relationship CA 1 > CA 2 > CA 3 > CA 4 between the four attributes is considered.The experts evaluated the promising five high-tech enterprises AE g (g = 1, . . ., 5) with respect to the above four attributes and provide the evaluation information in the form of INNs listed in Table 2. Now, we exploit the developed approach to solve the MADM problem.
Step 1.Since all the attributes are of same type, there is no need to equalize decision matrix VN = vn gh 5×4 .
Steps 1 and 2 are the same.
Step 3. Exploit the decision information from decision matrix VN = vn gh 5×4 and the INSSPWA operator specified in Formula (36), and we have: Step 5.According to the values, the ranking order of the alternatives is AE 2 > AE 4 > AE 1 > AE 3 > AE 5 .Hence, the best high-tech enterprise is AE 2 , while the worst one is AE 5 .

Effect of the Parameters Υ and A on Decision Result
Firstly, we fix the value of the parameter A and give different values to parameter Υ.The effect of parameter Υ on the decision results exploiting the GINSh-SkPWA operator and GINSh-SkPWG operator is revealed in Figures 1 and 2, respectively.Secondly, we fix the value of parameter Υ and give different values to parameter A. The effect of parameter Υ on the decision results exploiting the GINSh-SkPWA operator and GINSh-SkPWG operator is revealed in Figures 3 and 4 Firstly, we fix the value of the parameter A and give different values to parameter ϒ .The effect of parameter ϒ on the decision results exploiting the GINSh-SkPWA operator and GINSh- SkPWG operator is revealed in Figures 1 and 2, respectively.Secondly, we fix the value of parameter ϒ and give different values to parameter A .The effect of parameter ϒ on the decision results exploiting the GINSh-SkPWA operator and GINSh-SkPWG operator is revealed in Figures 3 and 4, correspondingly.
From Figures 1 and 2, one can perceive that although the ranking order for dissimilar values of parameter ϒ may be diverse, the best and worst alternatives stay the same.From Figures 1 and 2, we can also perceive that, when the values of parameter ϒ go up, exploiting the INSh-SkPWA operator, the score values of the alternatives are reduced, while when exploiting the INSh-SkPWG operator, the score values of the alternatives raises.
Similarly, from Figures 3 and 4, one can perceive that the ranking order for dissimilar values of the parameter A may be different, but the best and worst alternatives remain the same.From Figures 3 and 4

Comparison with Existing Approaches
In the following, we compare our developed approach established on these innovative developed AGOs with some existing approaches, such as the approaches developed by Zhang et al. [12], Nagarajan et al. [47], TOPSIS [55], and MABAC [52].The score values and ranking orders are specified in Table 3. INSSWA operator [47] ( 2).

Comparison with Existing Approaches
In the following, we compare our developed approach established on these innovative developed AGOs with some existing approaches, such as the approaches developed by Zhang et al. [12], Nagarajan et al. [47], TOPSIS [55], and MABAC [52].The score values and ranking orders are specified in Table 3. INSSWA operator [47] ( 2).From Figures 1 and 2, one can perceive that although the ranking order dissimilar values of parameter Υ may be diverse, the best and worst alternatives stay the same.From Figures 1 and 2, we can also perceive that, when the values of parameter Υ go up, exploiting the INSh-SkPWA operator, the score values of the alternatives are reduced, while when exploiting the INSh-SkPWG operator, the score values of the alternatives raises.
Similarly, from Figures 3 and 4, one can perceive that the ranking order for dissimilar values of the parameter A may be different, but the best and worst alternatives remain the same.From Figures 3  and 4, we can also perceive that, when the values of parameter A reduce, exploiting the INSh-SkPWA operator, the score values of the alternatives go up, while when exploiting the INSh-SkPWG operator, the score values of the alternatives are reduced.

Comparison with Existing Approaches
In the following, we compare our developed approach established on these innovative developed AGOs with some existing approaches, such as the approaches developed by Zhang et al. [12], Nagarajan et al. [47], TOPSIS [55], and MABAC [52].The score values and ranking orders are specified in Table 3. From Table 3, we can perceive that the ordering acquired from the INWA [16], INSh-SkWA, and INSh-SkWG operators [47] and TOPSIS is the same.While solving the same problem utilizing the MABAC method [51], the ranking order is different.The best alternative remains the same, and only the worst alternative is changed.In the above methods, the weight vector of the criteria is calculated using PA operators.However, the existing AGOs do not judge the priority relationship among attributes.Thus, the proposed aggregation operators have some advantages over these aggregation operators.Firstly, they developed Sh-Sk operational rules that consist of variable parameters, which makes the decision process suppler.Secondly, the anticipated AGOs can judge the priority relationship along with attributes.Therefore, the enlarged AGOs in this article are more realistic and supple to be employed in the decision-making procedure.

Conclusions
In practical decision making, accessible information is frequently imperfect and incompatible, and the INS is a superior tool to signify such types of information.In this article, initially, a number of Schweizer-Sklar operational rules for INNs were developed.Secondly, we created two new AGOs, an INSh-SkPWA operator and an INSh-SkPWG operator, and discussed their desirable properties.The leading qualities of these enlarged Schweizer-Sklar AGOs are that they can consider the priority relationship among attributes and are more flexible due to variable parameters.Moreover, based on these Schweizer-Sklar prioritized aggregation operators, two novel MADM approaches we are instituted.The novelty of the proposed method is compared to other methods.It is shown in Table A1 of Appendix C. The proposed method is illustrated by a numerical example.This example is specified to confirm the realism and efficacy of the proposed approach, and a comparison with the presented approaches is also given.In future research, we will apply the proposed approach to some new applications, such as evaluation of traffic control management [46], tourism recourses [47], enterprise green technology innovation behavior [48], mobile robot navigation [56,57], and so on, or extend the proposed method to some more extended form of INSs.
, we can also perceive that, when the values of parameter A reduce, exploiting the INSh-SkPWA operator, the score values of the alternatives go up, while when exploiting the INSh-SkPWG operator, the score values of the alternatives are reduced.

Figure 1 . 32 Figure 1 .
Figure 1.Score values of the alternatives for different values of parameterΥ .

Figure 2 .Figure 2 .
Figure 2. Score values of the alternatives for different values of parameter ϒ .

Figure 2 .
Figure 2. Score values of the alternatives for different values of parameter ϒ .

Figure 3 .
Figure 3. Score values of the alternatives for different values of parameter A .

Figure 4 .
Figure 4. Score values of the alternatives for different values of parameter A .

Figure 3 .
Figure 3. Score values of the alternatives for different values of parameter A .

Figure 2 .
Figure 2. Score values of the alternatives for different values of parameter ϒ .

Figure 3 .
Figure 3. Score values of the alternatives for different values of parameter A .

Figure 4 .
Figure 4. Score values of the alternatives for different values of parameter A .

Figure 4 .
Figure 4. Score values of the alternatives for different values of parameter A .
Theorem 6.Let vn g (g = 1, 2, . . ., s) be a group of INNs, then the value aggregated employing Definition (4) is still INN, and we have: Assume that VN = vn gh s×t

Table 3 .
Comparison with different approaches.

Table 3 .
Comparison with different approaches.