Multiple Attribute Group Decision-Making Based on Intuitionistic Fuzzy Schweizer-Sklar Generalized Power Aggregation Operators

Abstract

An intuitionistic fuzzy set (IFS) is a valuable tool to execute uncertain and indeterminate information. IFSs are more suitable to identify decision-maker’s evaluation data in decision-making problems. Intuitionistic fuzzy aggregation operators (AOs) are of enormous consequences in multiple attribute group decision-making (MAGDM) problems with an intuitionistic fuzzy environment. Consequently, the main impacts of this article are: firstly, to instigate various new novel generalized power AOs based on Schweizer-Sklar operational rules for IFS. Secondly, the study aims to discuss characteristics and particular cases of AOs. The core edge of proposed AOs is that they can eradicate the influence of uncomfortable data which could be too high or too low, making them more admirable for efficiently solving more and more complex MAGDM problems. Thirdly, we instigate two new algorithms to deal with MAGDM established on the generalized Schweizer-Sklar power AOs. Lastly, we appertain the anticipated method and algorithms to health care waste treatment technology selection (HCW-TT) to show the competence of the anticipated method and algorithms. The prevailing novelties of these items are duplex. Firstly, new generalized AOs established on Schweizer-Sklar operational rules are initiated for IFNs. Secondly, two new approaches for IF MAGDM are initiated, one for known decision-makers (DMs) and attributes weights, while the other for unknown DMs and attributes weights.

1. Introduction

Nowadays, multiple attribute decision-making (MADM) or MAGDM problems are one of the most essential topics in decision theory. In MADM or MAGDM problems, the most favorable alternative is selected from a group of limited alternatives established on the inclusive data. Naturally, the information given about the alternatives is in the form of real numbers. Due to the complexity arising day by day in MADM and MAGDM problems, it is tough for DMs to provide assessment values about the objects in the form of crisp values owing to ambiguity and inadequate information. To some extent, it has been considered suitable that the valuations are specified by fuzzy set (FS) or its enlarged form. IFS [1, 2] is one of the strong extensions of FS [3] to pact with ambiguity by incorporating an equal falsity degree into the investigation. IFSs have excellent competence to clarify and clear DMs fuzzy decision information in MAGDM problems. Plenty of studies by disparate scholars were performed on IFS in various fields, such as operational rules for IF numbers (IFNs) [4], distance and similarity measures among IFNs [5–9], information entropies among IFNs [10–13], correlation measure among IFNs [14–16], etc. Furthermore, a huge number of methods have been presented and been applied to resolve MAGDM problems [17–24].

The aggregation operators (AOs) are very significant tools for handling MADM or MAGDM problems [25, 26], and these aggregation operators have additional advantages than various conventional approaches, such as TOPSIS [27, 28], VIKOR [29–31], TODIM [32, 33], and so on. Based on conventional aggregation operators [34–38], such as operators WA, OWA, OWG, etc., several expanded aggregation operators have been developed to sustain decision-making, such as IF AOs [39–41] and generalized AOs [42, 43]. Yet, most expanded AOs do not adequately take the supportive correlations between input arguments. Hence, Yager [44] proposed the concepts of power average (PA) operator and power ordered weighted average (POWA) operator, which have the capability of eradicating the bad impact of awful data offered by DMs and can also consider the attribute values to support each other. Moreover, Xu and Yager [45] introduced the concepts of power geometric (PG) operator and power ordered weighted geometric (POWG) operator. These AOs are further extended by several scholars under different environments, such as Xu [46] proposed the concept of intuitionistic fuzzy power AOs, Wan and Dong [47] presented the concept of power geometric AOs under trapezoidal intuitionistic fuzzy environment, Liu and Wang [48] proposed generalized power AOs for linguistic intuitionistic fuzzy numbers, and Zhang [49] presented generalized intuitionistic fuzzy power geometric operator.

For aggregating IFNs, several AOs are proposed utilizing different operational laws based on disparate T-norms (TN) and T-conorms (TCN), such as Wang and Liu [50, 51] proposed several AOs based on Einstein operational rules concerning IF information. Garg [52] developed some generalized geometric AOs established on Einstein TN and TCN and apply these AOs to deal with MADM problems under the IF environment. Huang [53] proposed some IF Hamacher AOs. Zhang et al. [54] presented a few Frank power AOs for IFNs and consider these AOs to concern with MAGDM problems under the IF environment. Some other AOs are proposed using Dombi operational laws such as, Khan et al. [55] proposed Dombi power Bonferroni mean for interval neutrosophic numbers and Li et al. [56] proposed IF Dombi Hamy mean operator and applied them to solve MAGDM problems. Generally, the Archimedean TN (ATN) and Archimedean TCN (ATCN) are the expansions of different TNs and TCNs, such as algebraic, Einstein, Hamacher, Frank, and Dombi TNs and TCNs. ATN and ATCN are the enlargements of numerous TNs and TCNs, which have several cases preferred to articulate the union and intersection of IFS [57]. Schweizer-Sklar (SS) operations [58] are the cases from ATN and ATCN; they are with a general parameter, so they are extra flexible and better than the former operations. However, most of the scholars mainly determine the elementary theory and types of Schweizer-Sklar TN (SSTN) and TCN (SSTCN) [59, 60]. Recently, Liu et al. [61, 62] and Zhang [63] merge SS operations with interval-valued IFS (IVIFS) and IFS, and established power average/geometric operators and weighted averaging operators, Maclaurin symmetric mean (MSM) operator for IVIFSs and IFSs, respectively. The MSM operator can take the correlation among multiple input arguments. Liu et al. [64] further proposed Schweizer-Sklar operational laws for single valued neutrosophic number (SVNN), proposed some prioritized AOs established on these operational laws and applied them to deal MADM problems with SVN information. Moreover, Zhang et al. [65] developed Schweizer-Sklar Muirhead mean operator and gave its application in MADM.

From the above literature, we can notice that all the aggregation operators proposed on Schweizer-Sklar operational laws for aggregating different types of fuzzy information can consider interrelationship among input arguments or can consider priority among the attributes. Yet, there is no research to combine Schweizer-Sklar operational laws and generalized power AOs developed by Yager. So, there is a need to initiate some generalized power aggregation operators based on Schweizer-Sklar operational laws. The main advantages of these aggregation operators are that they consist of generalized parameters in which one parameter can take values and can remove the effect of awkward data from the final ranking results. Therefore, to achieve the following, the main contributions of our article are:(1)To develop some generalized power aggregation operators based on Schweizer-Sklar operational laws for IFNs.(2)To discuss some properties and some special cases of the developed aggregation operators.(3)To develop two MAGDM algorithms based on these newly developed aggregation operators. The develop aggregation operators have some advantages:(i)These newly developed AOs can have the facility of eradicating the influence of awful data.(ii)They consist of two parameters, both taking values from infinite set, which makes the decision-making process additionally supple.(4)To initiate two MAGDM models based on these aggregation operators to deal with MAGDM problems under the IF environment.

To do so, this article is planned as follows. In Section 2, several fundamental definitions concerning IFSs, Schweizer-Sklar TN and TCN, PA operators, and correlated properties are investigated. In Section 3, based on Schweizer-Sklar operational rules, we initiated several generalized intuitionistic fuzzy Schweizer-Sklar power AOs, with associated properties and particular cases being conferred. In Section 4, two novel MAGDM models are presented and established on these newly initiated AOs. In Section 5, a numerical example is specified to confirm the efficacy of the initiated MAGDM approaches and comparison of the initiated approaches with several approaches being given. In the last part, conclusion, future work, and references are provided.

2. Preliminaries

In this section, several essential notions of IFSs, operational laws of IFNs, the PA operator and associated properties are investigated.

2.1. IFS and Their Operations

Definition 1 (see [1]). Let be the reference set with common element in represented by . An IFS in the reference set is expressed mathematically as follows:where and symbolize, respectively, the MS and the NM degrees of an element to the IFS , respectively, , with the restriction The hesitancy degree of an element belonging to IFS is described by
The pair in the IFS is said to be an IFN. To be easily understood, we shall represent IFN by , where and

Definition 2 (see [39]). Let and be any two arbitrary IFNs, . Subsequently, the operational rules for IFNs are explained as follows:

Definition 3 (see [17]). Let be an IFN. The score function of the IFN is explained as follows:where the score values of IFN lies among the closed interval . The higher the score value , the better the IFN .

Definition 4 (see [18]). Let be an IFN. The accuracy function of the IFN is explained as follows:where the accuracy values of IFN lies among the closed interval . The bigger the accuracy value , the bigger the IFN .

Definition 5 (see [41]). Let and be any two arbitrary IFNs. Then, for comparing two IFNs, Xu [41] classified the judgment rules, which are explained as follows: 1. If then  2. If then(a)If afterward (b)If afterward

Definition 6 (see [9]). Let and be any two IFNs. Then, the similarity measure between and is classified as follows.where and are, respectively, the knowledge measures of and and are described as follows:

Definition 7 (see [5]). Let the two IFNs be and . Then, the Hamming distance measure amongst and is classified as follows.

2.2. The PA Operator

PA operator initiated by Yager [44] is one of the imperative AOs. The PA operator reduces several unconstructive influences of unreasonably high or unreasonably low arguments given by DMs. The conservative PA operator can contract with real numbers, and is identified as follows.

Definition 8 (see [44]). Let be a faction of positive real numbers. A PA operator is classified as follows:where and is the support degree (SPD) for from satisfying the following axioms:
(1) (2) (3) if .

Definition 9 (see [45]). Let be a faction of positive real numbers. A PG operator is described as follows:where and is the SPD for from satisfying the above axioms.

Definition 10 (see [44]). A POWA operator of dimension is a function , representing by the following formula:whereHere, represents the largest arguments among ; is a fundamental BUM mapping [38] which must hold the following axioms: if

Definition 11 (see [45]). A POWG operator of dimension is a function , represented by the following formula:

Definition 12 (see [36]). Let be a faction of positive real numbers. A WGPA operator is described as follows:Where and is the SPD for from satisfying the following axioms:

Definition 13 (see [36]). Let be a group of positive real numbers. A WGPOWA operator is depicted as follows:

2.3. Schweizer-Sklar Operational Laws for IFNs

The Schweizer-Sklar (SS) operational rules for IFNs consist of SS product and SS sum, which are some particular cases of ATT, respectively.

Definition 14 (see [58]). Assume that and be any two arbitrary IFSs, then the generalized union and intersection are identified as follows:where and , respectively, represent TN and TCN.
The Schweizer-Sklar T-norm (SSTN) and T-conorm (TCN) are explained as follows:where .
Moreover, when , SSTN and SSTCN perverted into algebraic TN and TCN.
On the established TN and TCN of SS operations, we can offer the following Definition 10 about SS operations for IFNs.

Definition 15 (see [62]). Let and be any two arbitrary IFNs, then based on SS operations, the following generalized union and generalized intersection are developed:Based on Definitions 1 and 2, the SS operational laws for IFNs are initiated as follows :

Theorem 1 (see [62]). Let and be any three IFNs. Then,

3. Some Generalized Power Aggregation Operators for IFNs

In this segment, we exploit a variety of generalized power AOs launched on the initiated SS operational rules for IFNs.

3.1. Weighted Generalized Intuitionistic Fuzzy Schweizer-Sklar Power Aggregation Operator

In this subpart, we advanced intuitionistic fuzzy Schweizer-Sklar generalized power average (IFSSGPA) operator, weighted intuitionistic fuzzy Schweizer-Sklar generalized power average (WIFSSGPA) operator, intuitionistic fuzzy Schweizer-Sklar generalized power ordered weighted averaging (GIFSSPOWA) operator, and conferred their privileged properties and various cases.

Definition 16. For a faction of IFNs IFSSGPA operator is a function where , parameters and are the SPD for from with the following constraint:
(1) (2) (3) if , where is the distance measure among two IFNs.
To compose (20) in unsophisticated mode, we haveSo, equations (21) and (20) become

Theorem 2. Let be a faction of IFNs, then the value aggregated utilizing Definition 16 is still IFN, and we have

Proof. In the succeeding, firstly, we can prove the subsequent:By utilizing mathematical induction on
For .
From the operational rules stated for IFNs in Definition 15, we haveandSimilarly,Then,If equation (24) secures for Then, while by the operational rules provided in Definition 15, we haveThat is, equation (24) is true for So, equation (24) is true for all Then,Therefore,which completes the proof of the Theorem 2.

Theorem 3. The IFSSGPA operator has the following properties:(1)Commutativity: Let be any permutation of , then(2)Idempotency: If all then(3)Bonudedness: The IFSSGPA operator lies among the min and max operators

Proof. (1)Suppose that is any permutation of , then for each there exists one and only one such that and vice versa. And, Then, based on Theorem (2), we have(2)Since for all thus(3)Let , where for all we haveAnd, meanwhile,So, we haveThen, according to Definition 3, Definition 5, and Theorem 2, we haveFurthermore, we shall discuss a few cases of the initiated AO with respect to the parameter , which is listed as follows:(1)If , then the IFSSGPA operator reduces into IFSS power average (PA) operator:(2)If and , then the IFSSGPA operator reduces into IF power average operator based on algebraic operation. That is,(3)If , ( a constant) and , then the IFSSGPA operator reduces into IF average operator based on algebraic operation. That is,

Definition 17. For a group of IFNs a weighted intuitionistic fuzzy Schweizer-Sklar generalized power average (WIFSSGPA) operator is a function where is the weight vector for such that and parameter , and is the support for from with the following constraint:
(1) (2) (3) if , where is the distance measure among two IFNs.
To write (43) in an uncomplicated way, we haveSo, from equations (46) and (45) ,we have

Theorem 4. Let be a group of IFNs, then the value aggregated utilizing Definition 17 is still IFN, and we have

Proof. Proof of the Theorem is the same as Theorem 2.

Theorem 5. For a faction of IFNs , is a parameter and . is the weighting vector for , and If then WIFSSGPA degenerates into the IFSSGPA operator:where is the power weight vector. Furthermore, we shall discuss a few cases of the initiated AO with respect to the parameters and , which are listed below:(1)If , then the WIFSSGPA operator reduces into the IFSS power average (PA) operator:(2)If , and , then the WIFSSGPA operator reduces into the IFPA operator based on algebraic operation. That is,(3)If , , ( a constant) and , then the WIFSSGPA operator reduces into IF average operator based on algebraic operation. That is,