Aggregation Operators for Interval-Valued Pythagorean Fuzzy Hypersoft Set with Their Application to Solve MCDM Problem

1  Introduction

MCGDM is deliberated as the most suitable method for a verdict on the adequate alternative from all possible choices, following criteria or attributes. Most decisions are taken when the intentions and confines are usually unspecified or unclear in real-life circumstances. Zadeh offered the idea of the fuzzy set (FS) [1] to overcome such vague and indeterminate facts. It is a fundamental tool to handle the insignificances and hesitations in decision-making (DM). The existing FS cannot deal with the scenarios when the experts consider a membership degree (MD) in intervals form during the DM procedure. Turksen [2] presented the interval-valued FS (IVFS) with fundamental operations. The prevailing FS and IVFS cannot deliver the information about any alternative’s non-membership degree (NMD). Atanassov [3] overcame the mentioned above limitations and developed the intuitionistic fuzzy set (IFS). Wang et al. [4] introduced several operations such as Einstein product, Einstein sum, etc., and AOs for IFS. Atanassov [5] prolonged the IFS to an interval-valued intuitionistic fuzzy set (IVIFS) with some basic operations and their properties. Garg et al. [6] protracted the idea of IFS and settled the cubic intuitionistic fuzzy set (CIFS).

The models mentioned above have been well-recognized by the specialists. Still, the existing IFS cannot handle the inappropriate and vague data because it envisions the linear inequality among the MD and NMD. For example, if decision-makers choose MD and NMD 0.6 and 0.7, respectively, then the IFS, as mentioned earlier, cannot deal with it because 0.6 + 0.7 ≥ 1. Yager [7] offered the Pythagorean fuzzy set (PFS) to resolve the inadequacy mentioned above by modifying the elementary state κ+δ≤1 to κ2+δ2≤1. He also established the score and accuracy functions to compute the ranking. Rahman et al. [8] planned Einstein weighted geometric operator for PFS and showed a multi-attribute group decision-making (MAGDM) technique using their planned operator. Zhang et al. [9] developed some basic operational laws and prolonged the approach for order of preference by similarity to ideal solution (TOPSIS) to resolve multi-criteria decision-making (MCDM) problems for PFS. Wei et al. [10] offered the Pythagorean fuzzy power AOs and discussed their important features. Using their presented operators, they also established a DM technique to resolve multi-attribute decision-making (MADM). Wang et al. [11] demonstrated the interaction operational laws for Pythagorean fuzzy numbers (PFNs) and settled power Bonferroni mean operators. IIbahar et al. [12] offered the Pythagorean fuzzy proportional risk assessment technique to assess the professional health risk. Zhang [13] proposed a novel DM approach based on similarity measures to resolve MCGDM problems for the PFS. Peng et al. [14] offered the AOs for interval-valued PFS (IVPFS) and established a DM technique using their planned methodology. Rahman et al. [15] prolonged the weighted geometric aggregation operator for IVPFS and offered a DM technique based on their developed operator.

All of the above techniques have broad applications, but these theories have some limitations on parametric chemistry due to their ineffectiveness. Molodtsov [16] introduced the soft sets (SS) theory and defined some basic operations with their features to handle the misperception and haziness. Maji et al. [17] extended the theory of SS and developed many basic and binary operations for SS. Maji et al. [18] developed the fuzzy soft set with some desirable properties by merging two existing notions, FS and SS. Maji et al. [19] protracted the intuitionistic fuzzy soft set (IFSS) and some important operations with their essential properties. Arora et al. [20] presented the AOs for IFSS and planned a DM technique based on their developed operators. Jiang et al. [21] introduced the interval-valued IFSS (IVIFSS) and discussed its basic properties. Zulqarnain et al. [22] planned the TOPSIS technique based on the correlation coefficient (CC) for IVIFSS to resolve MADM problems. Peng et al. [23] anticipated the Pythagorean fuzzy soft sets (PFSS) by merging two prevailing theories, PFS and SS. Zulqarnain et al. [24] presented some operational laws for PFSS and prolonged the AOs and interaction AOs for PFSS. Zulqarnain et al. [25] developed the operational interaction laws for PFSS and protracted the interaction AOs based on established operational laws. They also established the DM methodologies using their developed AOs and interaction AOs with their application in green supplier chain management. Zulqarnain et al. [26] prolonged the Einstein-ordered operational laws for PFSS and introduced the Einstein-ordered weighted ordered geometric AO for PFSS. They also established a MAGDM technique to solve complex real-life problems. Zulqarnain et al. [27] protracted the Einstein-ordered weighted ordered average AO for PFSS and offered a DM technique based on their developed operator. Zulqarnain et al. [28] settled the TOPSIS method for PFSS using correlation coefficient and developed the MADM approach to resolve DM obstacles. Zulqarnain et al. [29] prolonged the AOs for IVPFSS and presented a MAGDM approach to solving real-life difficulties.

Samarandche [30] proposed the idea of hypersoft set (HSS), which penetrates multiple sub-attributes in the parameter function f, which is a characteristic of the cartesian product with the n attribute. Samarandche HSS is the most suitable theory compared to SS and other existing concepts because it handles the multiple sub-attributes of the considered parameters. Several HSS extensions and their DM methods have been proposed. Rahman et al. [31] developed the DM techniques based on similarity measures for the possibility IFHSS. Zulqarnain et al. [32] extended the notion of IFHSS to PFHSS with fundamental operations and their properties. Rahman et al. [33] established a DM methodology for neutrosophic HSS. Saeed et al. [34] utilized the neutrosophic hypersoft mapping to diagnose the brain tumor. Zulqarnain et al. [35] extended the TOPSIS method based on the correlation coefficient for IFHSS and used it to resolve MADM complications. Zulqarnain et al. [36] expanded the AOs under the IFHSS environment and developed a DM approach based on their presented AOs. Zulqarnain et al. [37] developed the correlation-based TOPSIS approach for PFHSS and utilized their established technique to select the most appropriate face mask. PFHSS is a hybrid intellectual structure of PFSS. An enhanced sorting process fascinates investigators to crack baffling and inadequate information. Rendering to the investigation outcomes, PFHSS plays a vital role in decision-making by collecting numerous sources into a single value. The existing AOs for PFHSS cannot cope with the situation when the information of any multi-sub attribute is given in the form of intervals. To overcome the shortcomings mentioned above, we merged the IVPFS and hypersoft set (HSS) and introduced IVPFHSS, a novel hybrid structure to cope with uncertain problems. Therefore, to inspire the current research of IVPFHSS, we will state AOs based on rough data. The core objectives of the present study are given as follows:

•   The IVPFHSS capably contracts the multifaceted concerns seeing the multi sub-attributes of the deliberated parameters in the DM procedure. To preserve this benefit in concentration, we prolong the PFHSS to IVPFHSS and establish the AOs for IVPFHSS.

•   The AOs for IVPFHSS are well-known attractive estimate AOs. It has been observed that the prevalent AOs aspect is unresponsive to scratch the precise finding over the DM process in some situations. To overcome these specific complications, these AOs necessary to be revised. We determine innovative operational laws for interval-valued Pythagorean fuzzy hypersoft numbers (IVPFHSNs).

•   Interval-valued Pythagorean fuzzy hypersoft weighted average and geometric operators have been introduced with their necessary properties using developed operational laws.

•   A novel algorithm based on the planned operators to resolve the DM problem is established to resolve MCGDM issues under the IVPFHSS scenario.

•   Material selection is an imperative feature of manufacturing as it realizes the concrete conditions for all ingredients. MS is an arduous but significant step in professional development. The manufacturer’s efficiency, productivity, and eccentric will suffer due to the absence of material selections.

•   A comparative analysis of advanced MCGDM technique and current methods has been presented to consider utility and superiority.

The organization of this paper is assumed as follows: the second section of this paper involves some basic notions that support us in developing the structure of the subsequent study. In Section 3, some novel operational laws for IVPFHSN have been projected. Also, in the same section, IVPFHSWA and IVPFHSWG operators have been introduced based on our developed operators’ basic properties. In Section 4, an MCGDM approach has been constructed based on the proposed AOs. In the same section, a numerical example has been discussed to confirm the pragmatism of the established technique for material selection in the manufacturing industry. Furthermore, a brief comparative analysis has been delivered to confirm the competency of the developed approach in Section 5.

2  Preliminaries

This section contains some basic definitions that will structure the following work.

Definition 2.1. [16] Let U and N be the universe of discourse and set of attributes, respectively. Let P(U) be the power set of U and A⊆N. A pair (Ω,A) is called a SS over U, and its mapping is expressed as follows:

Ω:A→P(U)

Also, it can be defined as follows:

(Ω,A)={Ω(t)∈P(U):t∈N,Ω(t)=∅ if t∉A}

Definition 2.2. [30] Let U be a universe of discourse and P(U) be a power set of U and t = {t1, t2, t3,…,tn}, (n ≥ 1) and Ti represented the set of attributes and their corresponding sub-attributes, such as Ti ∩ Tj = φ, where