A Synergistic Multi-Attribute Decision-Making Method for Educational Institutions Evaluation Using Similarity Measures of Possibility Pythagorean Fuzzy Hypersoft Sets

1  Introduction

Educational institutions are essential in molding people and society. To cultivate the information, abilities, and values necessary for critical thinking, self-improvement, and well-informed decision-making, they offer an organized atmosphere. In addition to promoting academic excellence, they also help people develop their moral and intellectual capacities by promoting ethical behavior, social engagement, and cultural awareness. Educational institutions also stimulate research, innovation, and societal advancement, preparing the next generation for the difficulties of a world that is changing quickly [1,2]. The process of selecting an educational institution can be challenging because there are many different, interconnected factors involved. The relative significance of such features varies widely throughout individuals, making prioritization challenging. Furthermore, the decision-making process becomes more complex due to inherent uncertainties, which include future job prospects, the quality of education evolving, and individual adaptability to the institution’s environment. This results in a MADM problem with individualistic and improbable results [3,4]. Although numerous commendable attempts have been made to manage such uncertainties by developing theoretical frameworks, the Pythagorean fuzzy set (PyFS) [5] is unique in this sense. For membership function refinement, the fuzzy set (FS) [6] and intuitionistic fuzzy set (IFS) [7] are extended to develop the PyFS. The FS paved the way for more nuanced representations of vague and imprecise information by allowing elements to have partial memberships in sets. The IFS was projected to allow elements possessing dependent partial memberships and non-memberships to satisfy the condition that the sum must be within [0, 1]. The PyFS integrates the square sum of the membership and non-membership values within [0, 1], which offers more flexibility. Because PyFS now has more flexibility, it can better represent scenarios in which decision-makers are unsure or reluctant, which enhances modeling and decision-making accuracy in contexts marked by ambiguity and insufficient information. Garg [8] discussed decision-making applications based on geometric aggregation operators of generalized Pythagorean fuzzy set embedded with Einstein t-norm and t-conorm. Numerous scholars have investigated the representation and measurement of innate uncertainties in diverse fields, such as Qu et al. [9] presented intelligent and enhanced maximum expert consensus modeling embedded with dynamic feedback mechanisms to model inherent uncertainties. Sun et al. [10] previewed target signals using sliding mode control of discrete-time interval type-2 fuzzy Markov jump systems. Ge et al. [11] used a fuzzy neural network to model inherent uncertainties based on adaptive inventory control. Zhang et al. [12] developed observer-centered sliding mode control for fuzzy stochastic switching systems embedded with deception attacks. Xia et al. [13] discussed more results on fuzzy sampled-data stabilization of chaotic nonlinear systems. Gao et al. [14] investigated the representation and measurement of innate uncertainties in sliding mode control. These fields frequently deal with dynamic, complex environments where uncertainty results from various factors.

As the criteria or factors that direct the process, parameters are essential to decision-making. By offering a framework for comparison, these factors aid in the evaluation of various options and guarantee that choices are made under predetermined objectives or limitations. Parameters are important because they help make complex circumstances simpler by organizing and quantifying essential aspects. This makes it possible to make decisions that are more objective, consistent, and well-informed. Appropriately selected parameters can greatly improve the efficacy of decision-making by concentrating attention on the most important elements of the given problem. Molodtsov [15] introduced the idea of the soft set (SS) to equip FS, IFS, and PyFS with parameterization mode. Muthukumar et al. [16] discussed the problem of clinical diagnosis based on similarity measures of an intuitionistic fuzzy soft set (IFSS). Peng et al. [17] introduced Pythagorean fuzzy soft set (PyFSS) by integrating the theories of PyFS and SS. Kirişci et al. [18] discussed the clinical assessment-based algorithmic decisive method of PyFSS. Zulqarnain et al. [19] investigated supplier selection decisive issue based on aggregation operators of PyFSS. Using only generic parameters while making decisions can result in biased or incomplete results in several cases. Sub-parametric values are essential for an exhaustive examination since they offer more precise and in-depth insights into each parameter. By capturing the subtleties and complexity of the issue, these finer details make sure that all pertinent factors are taken into account. Decisions may be exaggerated or distorted as a result of essential elements being missed or misinterpreted when sub-parametric values are ignored. Consequently, it is crucial to incorporate both parameters and their sub-parametric values into the decision-making process to prevent dubious or biased results. To address this issue, Smarandache [20] introduced hypersoft set (HSS) by transforming the approximate mapping of SS to multi-argument approximate mapping, that considers the Cartesian product of sub-parametric valued based disjoint sets as domain and maps it to the collection of objects being evaluated. Researchers have recently shown a great deal of interest in the HSS because of its exceptional adaptability and versatility in handling scenarios involving sophisticated decision-making. Because of its versatility, HSS is especially useful in domains where a hierarchy of criteria influences decisions or where complicated, interconnected elements must be taken into account. The HSS is, therefore, emerging as a potent tool for handling a variety of issues in decision-making and related fields. Smarandache [21] introduced some new types of SS and HSS based on the indeterminate nature of parameters or sub-parametric values. Debnath [22] presented the notions of the fuzzy hypersoft set (FHSS) and its weightage operators. Musa et al. [23] discussed the evaluation of houses by proposing three algorithms based on aggregations of the N-hypersoft set (NHSS). Al-Quran et al. [24] discussed the evaluation of cars by proposing an algorithm based on the aggregations of bipolar fuzzy hypersoft set (BFHSS). Asaad et al. [25] discussed the recruitment of software engineers for a company by proposing a decisive framework based on aggregations of BFHSS. Sajid et al. [26] investigated the evaluation of solar panels based on the set-theoretic operations of the cubic intuitionistic fuzzy hypersoft set (CIFHSS). Hamid et al. [27] discussed mobile phone evaluation by proposing a decisive framework based on the integrated context of machine learning and FHSS. Saeed et al. [28] explored the similarity and entropy measures for FHSS and employed them for the evaluation of renewable energy resources. Rahman et al. [29] investigated operational risks based on supplier chain management based on set-theoretic operations of HSS mappings. Smarandache et al. [30] introduced the integrated context of HSS and game theory for decision-making systems. Siddique et al. [31] investigated the aggregation operators of a Pythagorean fuzzy hypersoft set (PyFHSS) for decision-making situations.

It is advisable to involve an impartial convener who can objectively evaluate the extent of partial acceptance of the approximations made by decision-makers to ensure neutrality and minimize bias in the decision-making process. By reducing the possible impact of individual preferences or prejudices that could distort the review process, this method promotes more impartial and equitable results. The convener increases the decision-making process’s legitimacy and dependability by offering an objective, outside viewpoint. Alkhazaleh et al. [32] introduced the possibility fuzzy soft set (pFSS) and they investigated the relevant operations and similarity measures of pFSS. They proposed a decisive framework for the diagnosis of dengue fever based on similarity measures of pFSS. Al-Sharqi et al. [33] explored the features of pFSS with interval-valued settings and investigated its operations and similarity measures. They proposed a decisive framework for the diagnosis of respiratory disease based on similarity measures of pFSS. Ponnalagu et al. [34] discussed cars evaluation problems using pFSS and expert set environments collectively. Ali [35] formulated similarity measures of bipolar pFSS by considering approximations of fuzzy parameters and applied the concepts in agricultural land selection scenarios. Bashir et al. [36] introduced the possibility of the intuitionistic fuzzy soft set (pIFSS) and investigated its set-theoretic operations, e.g., union, AND, intersection, and OR-operations. They designed a decision-assisted system for the evaluation of educational institutions. Selvachandran et al. [37] investigated its set-theoretic operations, e.g., union, AND, intersection, and OR-operations of pIFSS. They designed a decision-making framework for the recruitment process in the company. Garg et al. [38] investigated aggregation operators, information measures, and modified complex proportional assessment (COPRAS) under a pIFSS environment. Jia-Hua et al. [39] proposed the notions of possibility Pythagorean fuzzy soft set (pPyFSS) and investigated its operations like union, OR, intersection, and AND. They also proposed similarity measures for pPyFSS to apply in decision-making scenarios. Palanikumar et al. [40,41] discussed the notions of bipolar pPyFSS and pPyFSS with interval-valued settings. Rahman et al. [42] introduced the notions of several hybrids of HSS with possibility settings. They [43] employed the similarity measures of possibility fuzzy hypersoft set (pFHSS) in agri-automobile evaluation and HRM pattern recognition. Further, they [44] discussed decision-making scenarios based on the similarity measures of possibility intuitionistic fuzzy hypersoft set (pIFHSS). Sahoo et al. [45] and Bhatia et al. [46] applied different decision-making techniques to discuss the evaluation of educational institutions.

After careful analysis of the above-discussed literature, it is clear that the following important features are ignored:

1.    Ignoring the biased tendency of the decision maker in decision-making is tantamount to questioning the validity of the decision. Therefore, the observations and opinions of the decision-makers must be evaluated with the help of an impartial expert.

2.    Short-term decisions strengthen the credibility of decision-making, while long-term decisions undermine its credibility. Approximation of alternatives by decision-makers considering each attribute individually is not only a time-consuming but difficult process. However, if the same process is repeated with multiple arguments in mind, it will be completed in less time and more reliably.

3.    Optimal membership and non-membership functions play a key role in controlling uncertainty based on incomplete and ambiguous information.

The proposed theoretical framework, the possibility Pythagorean fuzzy hypersoft set (pPyFHSS), collectively incorporates all of the above features. The idea of possibility degree is meant to deal with the first feature, the hypersoft setting is aimed at providing a multi-argument domain for the approximation of alternatives; the Pythagorean fuzzy setting provides the optimal and generalized membership and non-membership functions for controlling uncertainty. The pPyFHSS extends the notions of IFS, IFSS, FHSS, pFHSS, and pIFHSS. The extended range of pPyFHSS allows for a finer distinction in the degree of uncertainty. This is particularly useful in decision-making problems, where experts may need to express a more nuanced confidence or hesitation. In certain scenarios, the sum constraint of IFS, IFSS, and pIFHSS can be overly restrictive.

The salient contributions of the study are enlisted as:

1.    By integrating the ideas of possibility degree, hypersoft set, and Pythagorean fuzzy set, a novel theoretical framework called pPyFHSS, is introduced that considers the possibility degree within [0, 1] to evaluate the acceptance level of approximations made by decision-makers. It helps reduce the prejudice of the decision makers or minimize their inclined impact on the decisions.

2.    For the sake of the applicability of pPyFHSS in different fields of study, several fundamentals including axiomatic properties, aggregation operations, and types of pPyFHSS are investigated. Additionally, proofs of several essential theorems are also provided.

3.    The formulation for similarity measures between two pPyFHSSs is developed and the related theorems are investigated. Additionally, the proposed formulation is explained with an illustration.

4.    Based on the proposed similarity measures, a robust algorithm is proposed which is then validated by the prototype case study of educational institution evaluation.

The structure of this article is organized as in Section 2, and some fundamental definitions are discussed. The concept of pPyFHSS is introduced in Section 3, along with some of its intriguing properties. To address decision difficulties, a similarity measure for comparing two pPyFHSSs is proposed. Additionally, an algorithm for evaluating the optimum educational institution is provided in Section 4. At the end of this section, discussion is made on the comparison of the proposed model. Finally, Section 5 concludes with recommendations for future research.

2  Preliminaries

Some basic definitions are provided here, which will be useful in the remaining portion of the work.

Definition 1. Reference [17] A pair (F˘,E¨) is called a SS over U¨ and U¨ be an initial universal set, where F˘ is a mapping F˘:E¨→PU¨. For E¨ (parameters set), set of elements in the SS (F˘,E¨) that approximate η can be represented as F˘(η) and PU¨ indicates the power set of U¨.

Definition 2. Reference [20] Let x¨1,x¨2,...,x¨n for n⩾1, be n distinct attributes, whose corresponding sub attribute values are contained in the sets Ξ¨1,Ξ¨2,Ξ¨3,...,Ξ¨n respectively, with Ξ¨i ∩ Ξ¨j≠∅, for i≠j∈{1,2,3,...,n} such that Ξ¨=Ξ¨1×Ξ¨2×Ξ¨3×...×Ξ¨n. Then the pair (F˘,Ξ¨), where: F˘:Ξ¨→PU¨, is called HSS over U¨.

Definition 3. Reference [22] A HSS (Ω,Ξ¨) is referred to as a FHSS over U¨, where Ω:Ξ¨⟶FU¨, if Ω(ρ¨)={⟨ρ¨,Ω(ρ¨)(ℏ¨)⟩:ℏ¨∈U¨∧ρ¨∈Ξ¨=Ξ¨1×Ξ¨2×...×Ξ¨n}, where, FU¨ is the collection of all the fuzzy subsets on U¨.

The following definition is the modified form of the definition presented in [31].

Definition 4. Let x¨1,x¨2,x¨3,...,x¨n be the n distinct attributes whose corresponding sub-attributes are contained in disjoint sets Ξ¨1,Ξ¨2,Ξ¨3,...,Ξ¨n. A set pair (Ω,Ξ¨) is referred to as a PyFHSS over U¨, if Ω:Ξ¨⟶PFU¨, where (Ω,Ξ¨)={⟨ρ¨,Ω(ρ¨)⟩:∀ρ¨∈Ξ¨=∏i=1nΞ¨i}, and Ω(ρ¨)={ℏ¨iΩ(ρ¨)(ℏ¨i):ℏ¨i∈U¨∧∀ρ¨∈Ξ¨=∏i=1nΞ¨i},Ω(ρ¨)(ℏ¨i)=⟨εΩ(ρ¨)(ℏ¨i),τΩ(ρ¨)(ℏ¨i)⟩⊆PFU¨ (collection of all the Pythagorean fuzzy subset on U¨), such that εΩ(ρ¨)(ℏ¨i),τΩ(ρ¨)(ℏ¨i)∈[0,1], and 0≤εΩ2(ρ¨)(ℏ¨i)+τΩ2(ρ¨)(ℏ¨i)≤1. Ω(ρ¨)(ℏ¨i) is the Pythagorean value of ℏ¨i∈U¨ in Ω(ρ¨).

Example 1. Let U¨={ℏ¨1,ℏ¨2} be the universe of discourse and suppose that x¨1=Teaching methodology, x¨2=Subjects and x¨3=Classes be the attributes and their corresponding non-overlapping attributes values are Teaching methodology=Ξ¨1={x¨11=project base,x¨12=class discussion}, Subjects=Ξ¨2={x¨21=Mathematics,x¨22=Computer Science,x¨23=Statistics} and Classes=Ξ¨3={x¨21=Master,x¨22=Bachelor}. If Ω:Ξ¨1×Ξ¨2×Ξ¨3⟶PFU¨ and the Cartesian product is Ξ¨=Ξ¨1×Ξ¨2×Ξ¨3 ={ρ1¨,ρ2¨,ρ3¨,ρ4¨,ρ5¨,ρ6¨,ρ7¨,ρ8¨,ρ9¨,ρ10¨,ρ11¨,ρ12¨} then Pythagorean fuzzy hypersoft set over U¨ is given as follows:

Ω(ρ1¨)={ℏ¨1(0.6,0.3),ℏ¨2(0.5,0.7)},Ω(ρ2¨)={ℏ¨1(0.6,0.7),ℏ¨2(0.7,0.5)},Ω(ρ3¨)={ℏ¨1(0.4,0.8),ℏ¨2(0.3,0.7)},Ω(ρ4¨)={ℏ¨1(0.6,0.5),ℏ¨2(0.5,0.6)},Ω(ρ5¨)={ℏ¨1(0.7,0.3),ℏ¨2(0.4,0.8)},Ω(ρ6¨)={ℏ¨1(0.5,0.4),ℏ¨2(0.6,0.5)},Ω(ρ7¨)={ℏ¨1(0.5,0.6),ℏ¨2(0.4,0.5)},Ω(ρ8¨)={ℏ¨1(0.2,0.5),ℏ¨2(0.3,0.9)},Ω(ρ9¨)={ℏ¨1(0.4,0.6),ℏ¨2(0.8,0.5)},Ω(ρ10¨)={ℏ¨1(0.7,0.4),ℏ¨2(0.7,0.2)},Ω(ρ11¨)={ℏ¨1(0.4,0.5),ℏ¨2(0.5,0.3)},Ω(ρ12¨)={ℏ¨1(0.5,0.7),ℏ¨2(0.4,0.7)},

and its parameterized family is:

(Ω,Ξ¨)={(ρ1¨,{⟨ℏ¨1(0.6,0.3)⟩,⟨ℏ¨2(0.5,0.7)⟩}),(ρ2¨,{⟨ℏ¨1(0.6,0.7)⟩,⟨ℏ¨20.7,0.5)⟩}),(ρ3¨,{⟨ℏ¨1(0.4,0.8)⟩,⟨ℏ¨2(0.3,0.7)⟩}),(ρ4¨,{⟨ℏ¨1(0.6,0.5)⟩,⟨ℏ¨2(0.5,0.6)⟩}),(ρ5¨,{⟨ℏ¨1(0.7,0.3)⟩,⟨ℏ¨2(0.4,0.8)⟩}),(ρ6¨,{⟨ℏ¨1(0.5,0.4)⟩,⟨ℏ¨2(0.6,0.5)⟩}),(ρ7¨,{⟨ℏ¨1(0.5,0.6)⟩,⟨ℏ¨2(0.4,0.5)⟩}),(ρ8¨,{⟨ℏ¨1(0.2,0.5)⟩,⟨ℏ¨2(0.3,0.9)⟩}),(ρ9¨,{⟨ℏ¨1(0.4,0.6)⟩,⟨ℏ¨2(0.8,0.5)⟩}),(ρ10¨,{⟨ℏ¨1(0.7,0.4)⟩,⟨ℏ¨2(0.7,0.2)⟩}),(ρ11¨,{⟨ℏ¨1(0.4,0.5)⟩,⟨ℏ¨2(0.5,0.3)⟩}),(ρ12¨,{⟨ℏ¨1(0.5,0.7)⟩,⟨ℏ¨2(0.4,0.7)⟩}).}

Definition 5. Reference [42] If U¨ be the universe of discourse and x¨1,x¨2,x¨3,...,x¨n be n distinct attributes whose corresponding n distinct sub-attributes values are contained in disjoint sets Ξ¨1,Ξ¨2,...,Ξ¨n. Then (U¨,Ξ¨=Ξ¨1×Ξ¨2×...Ξ¨n) will be a hypersoft universe. Let Ω:Ξ¨⟶FU¨ and μ be the fuzzy subset of Ξ¨, i.e., μ:Ξ¨⟶FU¨. Let Ωμ:Ξ¨⟶FU¨×FU¨ be the function defined as Ωμ(ρ¨)={⟨Ω(ρ¨)(ℏ¨i),μ(ρ¨)(ℏ¨i)⟩:∀ℏ¨i∈U¨∧ρ¨∈Ξ¨=∏i=1nΞ¨i}, where; Ω(ρ¨)(ℏ¨),μ(ρ¨)(ℏ¨)⊆FU¨. Then Ωμ is referred to as a pFHSS over the hypersoft universe (U¨,Ξ¨).

3  The Notions of pPyFHSSs

In this section, we introduce the concept of pPyFHSS as an extension of the pFHSS model. The fundamental characteristics of the pPyFHSS are intended to control the ambiguity and uncertainty that are frequently encountered in MADM situations, like assessing educational institutions. When handling ambiguous data, the possibility degree-based option offers a layer of flexibility by assisting in evaluating the likelihood of different outcomes. Through the depiction of multi-argument qualities, the hypersoft setting extends conventional fuzzy models and captures intricate interdependencies in the decision-making process. Finally, decision-makers can more precisely indicate hesitation through membership and non-membership values in the Pythagorean fuzzy setting, which offers a more sophisticated treatment of uncertainty. Together, these characteristics improve the model’s capacity to manage complexity and ambiguity.

Definition 6. Let x¨1,x¨2,x¨3,...,x¨n be a set of n distinct attributes and their corresponding sub-attributes values are contained in non-overlapping sets Ξ¨1,Ξ¨2,...,Ξ¨n and Ξ¨=∏i=1nΞ¨i. Suppose that Ω:Ξ¨→PFU¨ and m:Ξ¨⟶FU¨, where PFU¨ and FU¨ are the collection sets of all the Pythagorean fuzzy and fuzzy subsets on U¨, respectively. If Ωm:Ξ¨→PFU¨×FU¨ is a mapping defined as:

Ωm(ρ¨)={⟨ℏ¨Ω(ρ¨)(ℏ¨),m(ρ¨)(ℏ¨)⟩:ℏ¨∈U¨,∀ρ¨∈Ξ¨=∏i=1nΞ¨i},

then Ωm is said to be a pPyFHSS over the hypersoft universe (U¨,Ξ¨). For each parameter ρ¨, it can be written as:

Ωm={(ρ¨,{⟨ℏ¨Ω(ρ¨)(ℏ¨),m(ρ¨)(ℏ¨)⟩:ℏ¨∈U¨}),∀ρ¨∈Ξ¨=∏i=1nΞ¨i},

where Ω(ρ¨)(ℏ¨)=⟨εΩ(ρ¨)(ℏ¨),τΩ(ρ¨)(ℏ¨)⟩⊆PFU¨ and 0≤εΩ(ρ¨)2(ℏ¨)+τΩ(ρ¨)2(ℏ¨)≤1 with εΩ(ρ¨)(ℏ¨),τΩ(ρ¨)(ℏ¨)∈[0,1], such that Ω(ρ¨)(ℏ¨) is the Pythagorean fuzzy number of ℏ¨∈U¨ in Ωm(ρ¨). Also m(ρ¨)(ℏ¨)=μm(ρ¨)(ℏ¨)⊆FU¨ with μm(ρ¨)(ℏ¨)∈[0,1], such that m(ρ¨)(ℏ¨) is the possibility membership grade of ℏ¨∈U¨ in Ωm(ρ¨).

Note: For convenience, it can be expressed as:

Ωm(ρ¨)={⟨(εΩ(ρ¨)(ℏ¨),τΨ(ρ¨)(ℏ¨)),μm(ρ¨)(ℏ¨)⟩:ℏ¨∈U¨,∀ρ¨∈Ξ¨=∏i=1nΞ¨i},

and collection of all pPyFHSS is symbolized by Θ^λ^.

Note: The difference between the above proposed mathematical framework, pPyFHSS, and the existing relevant frameworks like PyFHSS and pFHSS, is given as:

1.    In PyFHSS, the possibility degree-based setting has been ignored, therefore, it is insufficient to reduce the prejudice of the decision makers or minimize their inclined impact on the decisions.

2.    In pFHSS, the generalized membership and non-membership functions are ignored, therefore, it does not support the decision-makers to indicate hesitation more precisely through membership and non-membership values in the Pythagorean fuzzy setting, which offers a more sophisticated treatment of uncertainty.

Example 2. Let U¨={ℏ¨1,ℏ¨2} be the set of two vehicles in auto showroom and x¨1= luxury cars, x¨2= sports cars, be the attributes and corresponding attributes values are luxury cars =Ξ¨1={x¨11=beautiful,x¨12=high safety}, sports cars =Ξ¨2={x¨21=expensive,x¨22=less space}. Their Cartesian product is Ξ¨=∏i=1nΞ¨i=Ξ¨1×Ξ¨2={ρ1¨,ρ2¨,ρ3¨,ρ4¨} is Ωm:Ξ¨→PFU¨×FU¨ then the pPyFHSS can be constructed as:

Ωm(ρ1¨)={⟨ℏ¨1(0.9,0.4),0.51⟩,⟨ℏ¨2(0.7,0.4),0.61⟩},Ωm(ρ2¨)={⟨ℏ¨1(0.8,0.5),0.52⟩,⟨ℏ¨2(0.6,0.5),0.62⟩},

Ωm(ρ3¨)={⟨ℏ¨1(0.7,0.5),0.53⟩,⟨ℏ¨2(0.5,0.4),0.63⟩},Ωm(ρ4¨)={⟨ℏ¨1(0.6,0.4),0.54⟩,⟨ℏ¨2(0.4,0.3),0.64⟩},

Ωm={(ρ1¨,{⟨ℏ¨1(0.9,0.4),0.51⟩,⟨ℏ¨2(0.7,0.4),0.61⟩}),(ρ2¨,{⟨ℏ¨1(0.8,0.5),0.52⟩,⟨ℏ¨2(0.6,0.5),0.62⟩}),(ρ3¨,{⟨ℏ¨1(0.7,0.5),0.53⟩,⟨ℏ¨2(0.5,0.4),0.63⟩}),(ρ4¨,{⟨ℏ¨1(0.6,0.4),0.54⟩,⟨ℏ¨2(0.4,0.3),0.64⟩}).}(1)

In the matrix notion, data can be represented as follows:

Ωm=(⟨(0.9,0.4),0.51⟩⟨(0.7,0.4),0.61⟩⟨(0.8,0.5),0.52⟩⟨(0.6,0.5),0.62⟩⟨(0.7,0.5),0.53⟩⟨(0.5,0.4),0.63⟩⟨(0.6,0.4),0.54⟩⟨(0.4,0.3),0.64⟩).

Definition 7. If Ωm and Ψq be two pPyFHSSs defined over (U¨,Ξ¨), then Ψq is considered a subset of Ωm in terms of pPyFHSS if and only if q(ρ¨)(ℏ¨)⊆m(ρ¨)(ℏ¨) if μq(ρ¨)(ℏ¨)≤μm(ρ¨)(ℏ¨),Ψ(ρ¨)(ℏ¨)⊆Ω(ρ¨)(ℏ¨) if εΨ(ρ¨)(ℏ¨)≤εΩ(ρ¨)(ℏ¨) and τΨ(ρ¨)(ℏ¨)≥τΩ(ρ¨)(ℏ¨), ∀ℏ¨∈U¨ and ∀ρ¨∈Ξ¨. This relationship is denoted as Ψq⊆Ωm.

Example 3. Consider that Ωm∈Θ^λ^ over (U¨,Ξ¨) given in Example 2. Let Ψq be another pPyFHSS over (U¨,Ξ¨). Suppose that Ψq:Ξ¨→PFU¨×FU¨ and defined as follows:

Ψq(ρ1¨)={⟨ℏ¨1(0.8,0.6),0.31⟩,⟨ℏ¨2(0.5,0.6),0.41⟩},Ψq(ρ2¨)={⟨ℏ¨1(0.7,0.8),0.32⟩,⟨ℏ¨2(0.4,0.8),0.42⟩},Ψq(ρ3¨)={⟨ℏ¨1(0.5,0.6),0.33⟩,⟨ℏ¨2(0.3,0.6),0.43⟩},Ψq(ρ4¨)={⟨ℏ¨1(0.4,0.8),0.34⟩,⟨ℏ¨2(0.2,0.7),0.44⟩}.

Ψq={(ρ1¨,{⟨ℏ¨1⟨0.8,0.6⟩,0.31⟩,⟨ℏ¨2⟨0.5,0.6⟩,0.41⟩}),(ρ2¨,{⟨ℏ¨1⟨0.7,0.8⟩,0.32⟩,⟨ℏ¨2⟨0.4,0.8⟩,0.42⟩}),(ρ3¨,{⟨ℏ¨1⟨0.5,0.6⟩,0.33⟩,⟨ℏ¨2⟨0.3,0.6⟩,0.43⟩}),(ρ4¨,{⟨ℏ¨1⟨0.4,0.8⟩,0.34⟩,⟨ℏ¨2⟨0.2,0.7⟩,0.44⟩}).}(2)

By comparing (1) and (2), we get Ψq⊆Ωm.

Definition 8. Suppose Ωm, Ψn∈Θ^λ^, then Ωm and Ψq are said to be equal pPyFHSS if and only if Ωm⊆Ψq, and Ωm⊇Ψq, which can be denoted by Ωm=Ψq.

Definition 9. If U¨ be the set of universe of discourse and Ξ¨i, for i∈{1,2,...,n} be the sub attributes values of attributes set x¨i. Let Ωm be the pPyFHSS over (U¨,Ξ¨). Complement of Ωm can be be denoted as Ωmc and it is defined as Ωmc=(Ωc(ρ¨)(ℏ¨),mc(ρ¨)(ℏ¨)), where Ωc(ρ¨)(ℏ¨)=⟨τΩ(ρ¨)(ℏ¨),εΩ(ρ¨)(ℏ¨)⟩, and mc(ρ¨)(ℏ¨)=μmc(ρ¨)(ℏ¨)=μm(ρ¨)c(ℏ¨)=1−μm(ρ¨)(ℏ¨).

Note: From Definition 9, we observe that (Ωmc)c=Ωm.

Example 4. From Example 2, compliment of pPyFHSS Ωmc is written as:

Ωmc(ρ1¨)={⟨ℏ¨1(0.4,0.9),0.49⟩,⟨ℏ¨2(0.4,0.7),0.39⟩},Ωmc(ρ2¨)={⟨ℏ¨1(0.5,0.8),0.48⟩,⟨ℏ¨2(0.5,0.6),0.38⟩},

Ωmc(ρ3¨)={⟨ℏ¨1(0.5,0.7),0.47⟩,⟨ℏ¨2⟨0.4,0.5⟩,0.37⟩},Ωmc(ρ4¨)={⟨ℏ¨1(0.4,0.6),0.46⟩,⟨ℏ¨2(0.3,0.4),0.36⟩}.

Ωmc={(ρ1¨,{⟨ℏ¨1(0.4,0.9),0.49⟩,⟨ℏ¨2(0.4,0.7),0.39⟩}),(ρ2¨,{⟨ℏ¨1(0.5,0.8),0.48⟩,⟨ℏ¨2(0.5,0.6),0.38⟩}),(ρ3¨,{⟨ℏ¨1(0.5,0.7),0.47⟩,⟨ℏ¨2(0.4,0.5),0.37⟩}),(ρ4¨,{⟨ℏ¨1(0.4,0.6),0.46⟩,⟨ℏ¨2(0.3,0.4),0.36⟩}).}.

In the matrix notion, data can be represented as:

(Ωmc,Ξ¨)=(⟨(0.4,0.9),0.49⟩⟨(0.4,0.7),0.39⟩⟨(0.5,0.8),0.48⟩⟨(0.5,0.6),0.38⟩⟨(0.5,0.7),0.47⟩⟨(0.4,0.5),0.37⟩⟨(0.4,0.6),0.46⟩⟨(0.3,0.4),0.36⟩).

Definition 10. Suppose Ωm and Ψq be two pPyFHSSs over (U¨,Ξ¨). Then union and intersection operations on Ωm and Ψq are denoted as Ωm∪Ψq and Ωm ∩ Ψq respectively, and are defined by two mappings ℜr:Ξ¨→PFU¨×FU¨, and ζz:Ξ¨→PFU¨×FU¨. These two mappings give the same meanings such that

{ℜr(ρ¨)=(ℜ(ρ¨)(ℏ¨),r(ρ¨)(ℏ¨)):ℏ¨∈U¨,∀ρ¨∈Ξ¨=∏i=1nΞ¨i},

{ζz(ρ¨)=(ζ(ρ¨)(ℏ¨),z(ρ¨)(ℏ¨)):ℏ¨∈U¨,∀ρ¨∈Ξ¨=∏i=1nΞ¨i},

where ℜ(ρ¨)(ℏ¨)=Ω(ρ¨)(ℏ¨)∪Ψ(ρ¨)(ℏ¨) and r(ρ¨)(ℏ¨)=m(ρ¨)(ℏ¨)∪q(ρ¨)(ℏ¨). Also, ζ(ρ¨)(ℏ¨)=Ω(ρ¨)(ℏ¨) ∩ Ψ(ρ¨)(ℏ¨) and z(ρ¨)(ℏ¨)=m(ρ¨)(ℏ¨) ∩ q(ρ¨)(ℏ¨).

Example 5. Lets consider that Ωm, Ψq∈Θ^λ^ over (U¨,Ξ¨), where they are explained as follows:

Ωm(ρ1¨)={⟨ℏ¨1(0.9,0.4),0.75⟩,⟨ℏ¨2(0.7,0.4),0.95⟩},

Ωm(ρ2¨)={⟨ℏ¨1(0.8,0.5),0.77⟩,⟨ℏ¨2(0.6,0.5),0.96⟩},

Ωm(ρ3¨)={⟨ℏ¨1(0.7,0.5),0.78⟩,⟨ℏ¨2(0.5,0.4),0.98⟩}.

Ψq(ρ1¨)={⟨ℏ¨1⟨0.8,0.6⟩,0.31⟩,⟨ℏ¨2(0.5,0.6),0.51⟩},

Ψq(ρ2¨)={⟨ℏ¨1(0.7,0.3),0.32⟩,⟨ℏ¨2(0.4,0.8),0.52⟩},

Ψq(ρ3¨)={⟨ℏ¨1(0.5,0.6),0.33⟩,⟨ℏ¨2(0.3,0.0.4),0.53⟩}.

By using Definition 10, union of these two functions is:

ℜ(ρ1¨)=Ω(ρ1¨)∪Ψ(ρ1¨)={⟨ℏ¨1(0.9,0.4),0.75⟩,⟨ℏ¨2(0.7,0.4),0.95⟩},

ℜ(ρ2¨)=Ω(ρ2¨)∪Ψ(ρ2¨)={⟨ℏ¨1(0.8,0.3),0.77⟩,⟨ℏ¨2(0.6,0.5),0.96⟩},