
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12539-X
10.1016/j.heliyon.2024.e36508
e36508
Research Article
A novel algorithmic multi-attribute decision-making framework for solar panel selection using modified aggregations of cubic intuitionistic fuzzy hypersoft set
Sajid Muhammad a
Khan Khuram Ali a
Rahman Atiqe Ur aurkhb@gmail.com
b⁎
Bajri Sanaa A. c
Alburaikan Alhanouf d
Khalifa Hamiden Abd El-Wahed d
a Department of Mathematics, University of Sargodha, Sargodha, 40100, Pakistan
b Department of Mathematics, University of Management and Technology, Lahore, 54000, Pakistan
c Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh, 11671, Saudi Arabia
d Department of Mathematics, College of Science, Qassim University, Buraydah, 51452, Saudi Arabia
⁎ Corresponding author. aurkhb@gmail.com
22 8 2024
15 9 2024
22 8 2024
10 17 e3650828 3 2024
23 7 2024
16 8 2024
© 2024 The Author(s)
2024
https://creativecommons.org/licenses/by/4.0/ This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
To address the shortcomings of the cubic intuitionistic fuzzy sets (CIFSs) for the entitlement of multi-argument approximate function, the cubic intuitionistic fuzzy hypersoft set (Ω-set) is an emerging study area. This type of setting associates the sub-parametric tuples with the collection of CIFSs. Categorizing the evaluation of parameters into their corresponding sub-parametric values based on non-overlapping sets has significance in decision making and optimization related situations. Some operations of Ω-set are proposed in this study, along with certain practical features. We provide the complement, P-order, and R-order subsets, P-union (∪P), R-union (∪R), P-intersection (∩P) and R-intersection (∩R) of Ω-sets. The internal cubic intuitionistic fuzzy hypersoft set (ΩI-set) and the external cubic intuitionistic fuzzy hypersoft set (ΩE-set) are also proposed in this paper, which will aid researchers in applying this new theory to other areas of study. We show a few examples in this context and look into some more aspects of ∪P, ∪R, ∩P and ∩R of ΩI-sets and ΩE-sets. Arguments for a few significant theorems about ΩI-sets and ΩE-sets are also presented. Lastly, an algorithm is presented that assists decision-makers in evaluating appropriate solar panels to establish solar plants. The proposed algorithm uses the idea of ∪P and ∪R for two Ω-sets constructed based on expert opinions of decision makers.

Keywords

Fuzzy set
Cubic set
Cubic soft set
Solar energy
Optimization
Decision making
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pmc1 Introduction

One of the phenomena that we use the most in our daily lives is decision making (DM). To reach the ultimate conclusion, almost all decisions require several phases, some of which may be unclear. The decision-maker must incorporate the preferences for handling uncertainty in the analysis because if the assessment is carried out without addressing the uncertainties in the data, the related outcomes would be quite ambiguous. To tackle this problem, Zadeh [1] presented the idea of the fuzzy set (FS), which has subsequently gained widespread application. In a variety of real-world domains like social science, environmental science, engineering, and economics. Additional FS extensions have been developed and studied, including the intuitionistic fuzzy set (IFS) [2], the interval-valued intuitionistic fuzzy set (IVIFS) [3], linguistic IVIFS [4], complex IFS [5] and complex IVIFS [6]. A study has been done in FS, where decision-makers assess the object by merely expressing the degrees to which they are partial to it; they are unable to represent the non-preferences. Each alternative or object is represented as a pair of membership and non-membership in IFSs or IVIFSs, with the sum of their degrees always less than or equal to one.

In 1999, Molodtsov [7] presented the idea of soft sets (SS). Maji et al. [8], [9] subsequently presented numerous additional procedures on SSs. The study of hybrid models, which combine SS with other models, has been developed in the meantime. Examples of these include fuzzy SSs (FSSs) [10], [11], [12], [13], interval-valued FSSs (IVFSSs) [14], [15], intuitionistic FSSs (IFSSs) [16], [17], generalized FSSs [18], generalized IFSSs [19], etc.

In 2012, Jun et al. [20] introduced the idea of a cubic set (CS) by merging the theory of FS with the interval-valued fuzzy set (IVFS) theory. They looked at the complement of CSs, ∪P, ∪R, ∩P, ∩R, and other relevant properties of CSs. The field of CS theory has experienced a rapid increase in research recently. As an illustration, Muhiuddin and Al-Roqi [21] introduced the cubic soft set (CSS), a combination of CS and SS, and applied it to BCK/BCI algebras. The degree of rejection or non-membership plays an equal part throughout the performance analysis of any DM problem, even though CSs only considered the acceptance region. Therefore, in light of all of this, Jun [22] proposed the concept of cubic intuitionistic fuzzy set (CIFS), a hybrid set developed by fusing the elements of IVIFSs and IFSs and applying it in BCK/BCI algebras. The same author proposed the concept of a cubic interval-valued intuitionistic fuzzy set (CIVIFS) [23] and discussed its important applications in BCK/BCI algebra. As a generalization of the IFSs and IVIFSs, CIFS is a powerful and useful tool for describing imprecise information. Garg and Kaur [24] further define the ∩P, ∩R, ∪P, ∪R, and internal (external) CIFSs based on their fundamental property, and they also introduced several aggregation operators (AOs) for CIFSs [25]. The same authors examined many approaches, such as distance measures [26] and the TOPSIS method [27], for solving decision making problems (DMPs) in the CIFS environment. Recently, Faizi et al. [28] offered an application example that solves an MCDM problem to show the viability of the suggested operations on CIFSs. The idea of the SS was expanded to the hypersoft set (HSS) by Smarandache [29], who did this by converting the soft approximate function into a multi-argument approximate function. FSS environments cannot be used to tackle the problem if an attribute is more than one and is further split. Consequently, a novel setting, known as the fuzzy hypersoft set (FHSS) [30], was required to find a new method of solving such issues. They tackled a DM problem successfully implementing the FHSSs with Roy and Maji's technique. The hybrid form of interval and exact values may be a better way for the decision maker to express his preference in the complex DM dilemma. A cubic intuitionistic fuzzy soft set (CIFSS) was, therefore, developed by Saqlain et al. [31], concurrently defining two components: an IFS and an IVIFS. As a result, the CIFSS handles the alternative's truth value and falseness value over their corresponding intervals jointly. Rahman et al. [32] investigated the interval-valued FHSS (IVFHSS), which is an emerging field of study aimed at addressing the limitations of IVFSSs in handling multi-argument approximate functions (maaf). The IVFHSS has been studied by several scholars, but the work of investigators [33], [34], [35], [36] has been reported significantly as they have applied this concept in various DM scenarios.

Since solar energy is dependent on the sun, a plentiful and continuously regenerated resource, it is seen as an environmentally friendly form of energy. Unlike fossil fuels like coal and oil, we do not consume any limited supplies or release any toxic contaminants into the environment when we use solar energy to generate power. Using solar energy instead of fossil fuels can help us cut down on greenhouse gas emissions considerably. When solar power is generated, very little greenhouse gas is released into the atmosphere, mostly during the solar panel production procedure. Solar panels provide electricity without releasing contaminants after they are deployed. By reducing the rate of global warming, a decline in emissions contributes to the mitigation of climate change. In this effort, solar energy is essential since it offers a clean and sustainable substitute for fossil fuels, lowering the carbon footprint linked to the production of power. We can contribute to a more sustainable future for ourselves and future generations by switching to solar power and other alternative power providers.

There are several uncertainties when choosing solar panels, which may affect your choice. First, new panel types and features are always being introduced by market forces and technological breakthroughs, making it difficult to predict which technology will ultimately provide the highest level of value and performance. Furthermore, there are unknowns about how well solar panels will hold up in various circumstances and over time due to supplier differences in effectiveness, robustness, and dependability. The DM process is made more complex by elements including startup expenses maintenance needs, and integration with current infrastructure. To get the best results in terms of energy generation, cost-effectiveness, and long-term viability balancing these uncertainties necessitates thoughtful assessment of both short- and long-term repercussions.

Researchers have put forth significant endeavors to use various computational frameworks to analyze solar panel selection (SPS) to address these kinds of uncertainty. For example, Pythagorean fuzzy-based operators were employed by Rani et al. [37] to evaluate SPS performance. For the evaluation of SPS, Ihsan et al. [38] and Akram et al. [39] used the concepts of FHS and Fermatean FSS combined with multi-decisive settings. In their discussion of SPS, Raja et al. [40] utilized aggregation operators with generalized N-SS. To address the uncertainties associated with SPS, Tysüz and Kahraman [41], Ziemba and Szaja [42], and Arman and Kundakc [43] used various DM techniques. Jafar et al. [44] formulated trigonometric similarity measures for generalized hypersoft set and discussed the evaluation of renewable energy source selection using these formulations. Similarly, Saqlain et al. [45] integrated the similarity and distance measures with TOPSIS of the generalized hypersoft set to the evaluation of sustainable green security systems. Riaz et al. [46] employed new techniques for the evaluation of renewable energy sources based on distance and entropy measures, and Einstein averaging aggregation operators of bipolar cubic fuzzy sets. The development of hybrid structures that combine HSS and CIFS has yet to be previously investigated. By breaking each attribute down into its parts, this approach can lead to a deeper understanding of DM attributes and improved efficiency and effectiveness of DM processes. The cubic intuitionistic fuzzy hypersoft set (Ω-set), a hybrid fuzzy structure, was developed by Saeed et al. [47] in response to the need for a versatile analysis tool that could be fully evaluated at the sub-attribute level. A strong addition to FS theory, the CIFS combines two membership functions, i.e., membership and non-membership, with two fuzzy intervals, that is, membership and non-membership intervals that either contain or do not contain membership and non-membership functions. This makes it easier to represent ambiguity and uncertainty in DM processes. Conversely, the primary focus of HSS theory is a sub-attribute analysis of attribute-based data from SS theory. These two frameworks were combined to develop the Ω-set, which offers an accurate and adaptable method for decision analysis. Thus, it can be concluded that the development of Ω-set is meant to cope with the following challenges:1. The membership and non-membership grades in IFS and IFSS are typically real-valued, with the requirement that their sums fall between [0,1] to account for uncertainties and vagueness, respectively. This forces the decision-makers to rely on one another, but it is much more practical to give them precise ranges to choose from so they can make wise decisions.

2. For a consistent and dependable DM process, it is crucial to take into account all of the parameters and their sub-parametric values. Just taking into account a small number of parameters and ignoring their respective sub-parametric values can cast doubt on the process.

The versatile approximate function of the suggested Ω-set, with its multi-argument domain (HSS settings) for handling the second problem and its vast range (CIFS) for handling the first, allows it to readily manage these challenges. The salient contributions of the study are outlined as:1. Combining the concepts of CIFS and HSS, an adaptable theoretical structure called Ω-set is developed. The first one aims to give a more comprehensive understanding of membership and non-membership grades, as well as their corresponding interval-valued ranges. On the other hand, the latter offers an approximate function with a multi-argument domain to handle attribute-valued non-overlapping sets. Therefore, the incorporation of these relevant concepts increases the versatility of Ω-set in decision analysis.

2. The set operations ∪P, ∪R and the intersection between the Ω-set family have been introduced. The basic features of the internal (or external) Ω-sets have been shown by several results, which also demonstrate that their union or intersection need not be internal (or external) Ω-sets. Additionally, a few restrictions on the two internal (or external) Ω-sets are provided, according to which an internal or external Ω-set is the ∪P and ∪R or intersection of the two Ω-sets.

3. To solve a multi-attribute decision making (MADM) problem, an effective algorithm is presented using the suggested set operations of Ω-set. An analysis of a prototype case study that helps an agriculturist assess solar panels that are optimized helps to explain this algorithm.

4. The suggested strategy's effectiveness and adaptability are evaluated by employing a thorough comparison with a few pertinent specified approaches.

The following is how the study is set up: A few selected preliminary definitions that are necessary for introducing Ω-set are provided in Section 2. In Section 3, the concepts of Ω-set, ΩI-set, and ΩE-set are presented. Additionally, the ∪P and ∪R, ∩P and ∩R, and complement of Ω-sets are defined. Several significant characteristics of the proposed sets that describe the intrinsic behavior are also covered. The basic properties of ΩI-sets and ΩE-sets are discussed, along with the proofs necessary to support their use. In Section 4, the application is demonstrated through the development of a MADM algorithm utilizing the ∪P and ∪R of Ω-sets. The conclusion section then summarizes the paper's key findings and upcoming projects.

2 Preliminaries

This section consists of some elementary definitions that are necessary for understanding the concept of this paper. Let X be an initial universe set and let E be a set of parameters and G⊆E. A FS A on X is characterized by mapping μA:X⟶[0,1]. For each x∈X, the value μA(x) is called the membership grade of x to μA. A function μ˜A:X⟶P([0,1]) is called the IVFS on X, where P([0,1]) is the collection of sub-intervals of [0,1]. For every x∈X, the value μ˜A(x)=[μ˜AL(x),μ˜AU(x)] is the membership grade of x to μ˜A, where μ˜AL:X⟶[0,1] and μ˜AU:X⟶[0,1] are both FSs known as lower and upper FS on X respectively. Combination of IVFS and FS on X in the form {〈x,μ˜A(x),μA(x)〉:x∈X} known as CS on X. CX represents the collection of all CSs on X. Definition 2.1 [2]

An IFS B on X, is given byB={〈x,σB(x),ςB(x)〉:x∈X}

where σB:X⟶[0,1] and ςB:X⟶[0,1] with condition 0≤σB(x)+ςB(x)≤1. Also, σB(x)∈[0,1] and ςB(x)∈[0,1] represent the degrees of membership and non-membership of x∈X respectively. The value πB(x)=1−σB(x)−ςB(x) represents the level of hesitancy for membership of element x∈X. The IFS can simply be written as (σB,ςB). The collection of all IFSs over X will be denoted by F(X).

Definition 2.2 [2]

Let B=(σB,ςB),andD=(σD,ςD) be two IFSs. The following expressions are thus defined as1. B⊆D if σB(x)≤σD(x) and ςB(x)≥ςD(x) for all x∈X;

2. B=D if and only if B⊆D and D⊆B;

3. Bc={〈x,ςB(x),σB(x)〉:x∈X};

4. B∪D=(max(σB(x),σD(x)),min(ςB(x),ςD(x)))=B∨D;

5. B∩D=(min(σB(x),σD(x)),max(ςB(x),ςD(x)))=B∧D.

The membership functions in IFSs are depicted as pointed numbers. But if a decision must be made using interval numbers, Atanassov and Gargov [3] expanded the idea of IFSs to develop an IVIFS, which can be defined as follows: Definition 2.3 [3]

An IVIFS, A˜ on X is given asA˜={〈x,σ˜A˜(x),ς˜A˜(x)〉:x∈X}

where σ˜A˜:X⟶P([0,1]) and ς˜A˜:X⟶P([0,1]) such that σ˜A˜(x)=[σA˜L(x),σA˜U(x)] and ς˜A˜(x)=[ςA˜L(x),ςA˜U(x)] represent the membership and non-membership grade of x∈X respectively, with 0≤σA˜L(x)≤σA˜U(x)≤1,0≤ςA˜L(x)≤ςA˜U(x)≤1 and σA˜U(x)+ςA˜U≤1. The IVIFS can simply be written as A˜=(σ˜A˜,ς˜A˜). The collection of all IVIFSs over X will be denoted by F˜(X).

We define the join ∨ and meet ∧ operations for two IVIFSs, in X. For A˜,B˜∈F˜(X), we have:1. (A˜∨B˜)=([sup(σA˜L,σB˜L),sup(σA˜U,σB˜U)],[inf(ςA˜L,ςB˜L),inf(ςA˜U,ςB˜U)]),

2. (A˜∧B˜)=([inf(σA˜L,σB˜L),inf(σA˜U,σB˜U)],[sup(ςA˜L,ςB˜L),sup(ςA˜U,ςB˜U)]).

Definition 2.4 [7]

Let P(X) be the power set of universe set X. A SS FE over X, is defined by a function fE representing a mapping:fE:E⟶P(X)such thatfE(x)=ϕifx∉E.

In other words, FE is parameterized family of subsets of X. Here, for each x∈E, the set fE(x) is called the value set of x in FE. Thus, FE over X can be represented by the set of ordered pairsFE={(x,fE(x)):x∈E,fE(x)∈P(X)}.

Definition 2.5 [16]

An IFSS over X is a pair (ζG,G), where ζG is an approximate mapping given asζG:E⟶F(X)such thatζG(ϵ)=ϕifϵ∉G,

G⊆E and F(X) is the collection of all IFSs over X. For each ϵ∈E,ζG(ϵ) represents the ϵ-element of the (ζG,G). Where ζG(ϵ) can be written as:ζG(ϵ)={〈x,μζG(ϵ)(x),νζG(ϵ)(x)〉:x∈X}

where μζG(ϵ)(x) and νζG(ϵ)(x) are the membership and non-membership degrees of x in the F(X), satisfying 0≤μζG(ϵ)(x)+νζG(ϵ)(x)≤1 for all x∈X. Thus(ζG,G)={(ϵ,ζG(ϵ)):ϵ∈E}.

Definition 2.6 [20]

A CS, U on X≠ϕ is a structure U={〈x,A(x),λ(x)〉:x∈X} in which A is an interval-valued fuzzy set (IVF) on X and λ is a fuzzy set on X. It is denoted by U=〈A,λ〉. The collection of all CSs is denoted by CX.

Definition 2.7 [22]

A CIFS, U˜ define over X≠ϕ has a structure given asU˜={〈x,A˜(x),λ˜(x)〉:x∈X,A˜(x)∈F˜(X),λ˜(x)∈F(X)}

we denote CIFS by pairs as U˜=(A˜,λ˜), where A˜=(σ˜A˜=[σA˜L,σA˜U],ς˜A˜=[ςA˜L,ςA˜U]) and λ˜=(σA˜,ςA˜) are IVIFS and IFS. The family of all CIFSs over X is denoted by C˜X.

Definition 2.8 [31]

A CIFSS, (JG,G) has an approximate mapping JG given as JG:E⟶C˜Xsuch thatJG(ϵ)=ϕifϵ∉G⊆E For each ϵ∈E,JG(ϵ) denotes the ϵ-element of the CIFS and expressed asJG(ϵ)={〈x,G˜(x),λ˜(x)〉:x∈X,G˜(x)∈F˜(X),λ˜(x)∈F(X)}.

Thus a CIFSS, (JG,G) over X can be expressed by the set of ordered pairs(JG,G)={(ϵ,JG(ϵ)):ϵ∈G⊆E}.

Definition 2.9 [29]

Let Z1,Z2,Z3,...,Zn be n sets of parameter with no common elements having the sub-parametric values z1,z2,z3,...,zn of the parameters respectively. The HSS ΨA on universe set X can be written in the form of pairs asΨA={(ϑj,Ψ(ϑj)):ϑj∈Z=∏i=1nZi},

where A⊆ZandΨ:Z⟶P(X) such that Ψ(ϑj)=ϕifϑj∉A.

3 The notions of Ω-set and its properties

The definition of Ω-set and basic concepts associated with it are covered in this section. The following defintion of Ω-set is modified version of Ω-set discussed by Saeed et al. [47]. Definition 3.1 If F˜(X) and F(X) are the collections of all IVIFSs and IFSs on X respectively. Also zi are sub parametric values contained in the sets of parameter Zi for 1≤i≤n respectively, with Zi∩Zj=ϕ for i≠j,1≤i,j≤n. Now Ω-set, (Γ,A) on the universe set X can be expressed as:(Γ,A)={(ϑj,Γ(ϑj)):ϑj∈Z=∏i=1nZi},

where Γ:Z⟶C˜X such that Γ(ϑj)=ϕifϑj∉A⊆Z, and ϕ is cubic intuitionistic fuzzy empty set. Γ is called the approximate function of Ω-set, (Γ,A). For each ϑj∈A,Γ(ϑj) is the set of ϑj-approximate element of the CIFS. It can be written as:Γ(ϑj)={〈x,A˜Γ(ϑj)(x),λ˜Γ(ϑj)(x)〉:x∈X,A˜Γ(ϑj)(x)∈F˜(X),λ˜Γ(ϑj)(x)∈F(X)}orΓ(ϑj)={〈x,[σΓ(ϑj)L(x),σΓ(ϑj)U(x)]σΓ(ϑj)(x),[ςΓ(ϑj)L(x),ςΓ(ϑj)U(x)]ςΓ(ϑj)(x)〉:x∈X}.

If πΓ(ϑj)(x)=1−σΓ(ϑj)(x)−ςΓ(ϑj)(x). Hence, πΓ(ϑj)(x) represents the level of hesitancy for membership of element x∈X. The collection of all Ω-sets over X is denoted by Ω(Γ,A).

Remark 3.2 Some special cases of Ω-set for zero level of hesitancy are summarized as follows:1. A Ω-set, (Γ,A)={(ϑj,{x,〈[0,0]1,[1,1]0〉}):∀ϑj∈Z,x∈X} is denoted by 0¨.

2. A Ω-set, (Γ,A)={(ϑj,{x,〈[1,1]0,[0,0]1〉}):∀ϑj∈Z,x∈X} is denoted by 1¨.

3. A Ω-set, (Γ,A)={(ϑj,{x,〈[0,0]0,[1,1]1〉}):∀ϑj∈Z,x∈X} is denoted by 0ˆ.

4. A Ω-set, (Γ,A)={(ϑj,{x,〈[1,1]1,[0,0]0〉}):∀ϑj∈Z,x∈X} is denoted by 1ˆ.

Remark 3.3 For any X≠ϕ, let 1(x)=1,0(x)=0. Then,(Γ,A)={(ϑj,{x,〈σ˜Γ(ϑj)(x)1,ς˜Γ(ϑj)(x)0〉}):∀ϑj∈A⊆Z,x∈X},(Γ,B)={(ϑj,{x,〈σ˜Γ(ϑj)(x)0,ς˜Γ(ϑj)(x)1〉}):∀ϑj∈B⊆Z,x∈X},(Γ,C)={(ϑj,{x,〈σ˜Γ(ϑj)(x)12(σΓ(ϑj)L+σΓ(ϑj)U),ς˜Γ(ϑj)(x)12(ςΓ(ϑj)L+ςΓ(ϑj)U)〉}):∀ϑj∈C⊆Z,x∈X},

are all Ω-sets on X.

Definition 3.4 For any Ω-set, (Γ,A)∈Ω(Γ,A), the score value of (Γ,A) is defined asSc[(Γ,A)]=13[(σΓ(ϑj)L+σΓ(ϑj)U+σΓ(ϑj))−(ςΓ(ϑj)L+ςΓ(ϑj)U+ςΓ(ϑj))],

where Sc[(Γ,A)]∈[−1,1]. Score values of Ω-sets, 0¨,0ˆ,1¨, and 1ˆ given in Remark 3.2 are Sc(0¨)=−0.333, Sc(0ˆ)=−1, Sc(1¨)=0.333, and Sc(1ˆ)=1 respectively.

Definition 3.5 For (Γ,A)and(Θ,B)∈Ω(Γ,A), where A and B are any two subsets of Z. We define the following as:1. Equality: (Γ,A)=(Θ,B)⇔A=B and σ˜Γ(ϑj)(x)=σ˜Θ(ϑj)(x);ς˜Γ(ϑj)(x)=ς˜Θ(ϑj)(x)⇔σΓ(ϑj)L(x)=σΘ(ϑj)L(x) and σΓ(ϑj)U(x)=σΘ(ϑj)U(x);ςΓ(ϑj)L(x)=ςΘ(ϑj)L(x) and ςΓ(ϑj)U(x)=ςΘ(ϑj)U(x)∀ϑj∈Z,x∈X.

2. P-order: (Γ,A)⊆P(Θ,B)⇔A⊆B and σ˜Γ(ϑj)(x)⊆σ˜Θ(ϑj)(x),σΓ(ϑj)(x)≤σΘ(ϑj)(x);ς˜Γ(ϑj)(x)⊇ς˜Θ(ϑj)(x),ςΓ(ϑj)(x)≥ςΘ(ϑj)(x)∀ϑj∈Z,x∈X.

3. R-order: (Γ,A)⊆R(Θ,B)⇔A⊆B and σ˜Γ(ϑj)(x)⊆σ˜Θ(ϑj)(x),σΓ(ϑj)(x)≥σΘ(ϑj)(x);ς˜Γ(ϑj)(x)⊇ς˜Θ(ϑj)(x),ςΓ(ϑj)(x)≤ςΘ(ϑj)(x)∀ϑj∈Z,x∈X.

Definition 3.6 A Ω-set, (Γ,A) is said to be an internal cubic intuitionistic fuzzy hypersoft set (ΩI-set) or external cubic intuitionistic fuzzy hypersoft set (ΩE-set), accordinglyσΓ(ϑj)L(x)≤σΓ(ϑj)(x)≤σΓ(ϑj)U(x)and ςΓ(ϑj)L(x)≤ςΓ(ϑj)(x)≤ςΓ(ϑj)U(x)σΓ(ϑj)(x)∉(σΓ(ϑj)L(x),σΓ(ϑj)U(x))and ςΓ(ϑj)(x)∉(ςΓ(ϑj)L(x),ςΓ(ϑj)U(x)).

∀ϑj∈A⊆Z,x∈X.

Example 3.7 Suppose the Example 4.2, if (Γ,A)={(ϑj,{x,〈σ˜Γ(ϑj)(x)σΓ(ϑj)(x),ς˜Γ(ϑj)(x)ςΓ(ϑj)(x)〉})} and (Θ,B)={(ϑj,{x,〈σ˜Θ(ϑj)(x)σΘ(ϑj)(x),ς˜Θ(ϑj)(x)ςΘ(ϑj)(x)〉})}∈Ω(Γ,A) are two Ω-sets on the universal set X.1. Let σ˜Γ(ϑj)(x)=[0.2,0.6], σΓ(ϑj)(x)=0.3 and ς˜Γ(ϑj)(x)=[0.1,0.3], ςΓ(ϑj)(x)=0.15, ∀ϑj∈A⊆Z,x∈X. Then (Γ,A) is ΩI-set on X.

2. Let σ˜Θ(ϑj)(x)=[0.3,0.5], σΘ(ϑj)(x)=0.2 and ς˜Θ(ϑj)(x)=[0.2,0.4], ςΘ(ϑj)(x)=0.45, ∀ϑj∈B⊆Z,x∈X. Then (Θ,B) is ΩE-set on X.

3.1 Set theoretic operations

In this section, we define the basic set theoretic operations, namely the complement, P(R)-union, and P(R)-intersection of Ω-sets. Definition 3.8 The complement of a Ω-set,(Γ,A)={(ϑj,{x,〈σ˜Γ(ϑj)(x)σΓ(ϑj)(x),ς˜Γ(ϑj)(x)ςΓ(ϑj)(x)〉}):ϑj∈A⊆Z,x∈X}

is denoted by (Γ,A)c and is defined as(Γ,A)c={(ϑj,{x,〈ς˜Γ(ϑj)(x)ςΓ(ϑj)(x),σ˜Γ(ϑj)(x)σΓ(ϑj)(x)〉}):ϑj∈A⊆Z,x∈X}.

Obviously ((Γ,A)c)c=(Γ,A). Example 3.9 The complement (Γ,A)c of the Ω-set (Γ,A) defined in Table 1 is given in Table 2.Table 1 Tabular form of Cubic Intuitionistic Fuzzy Hypersoft set (Γ,A)c.

Table 1(Γ,A)c	ϑ1	ϑ2	ϑ3	ϑ4	
C1	〈[0.4,0.5]0.65,[0.3,0.45]0.3〉	〈[0.2,0.4]0.55,[0.3,0.55]0.2〉	〈[0.1,0.3]0.35,[0.2,0.55]0.4〉	〈[0.3,0.4]0.35,[0.25,0.4]0.6〉	
C2	〈[0.25,0.4]0.3,[0.1,0.5]0.2〉	〈[0.3,0.5]0.45,[0.2,0.45]0.35〉	〈[0.2,0.3]0.1,[0.35,0.6]0.5〉	〈[0.2,0.35]0.5,[0.2,0.6]0.3〉	
C3	〈[0.1,0.3]0.25,[0.25,0.5]0.6〉	〈[0.2,0.4]0.3,[0.15,0.5]0.4〉	〈[0.3,0.55]0.5,[0.2,0.4]0.25〉	〈[0.2,0.3]0.25,[0.3,0.65]0.6〉	
C4	〈[0.2,0.4]0.1,[0.3,0.55]0.7〉	〈[0.3,0.5]0.25,[0.25,0.4]0.2〉	〈[0.35,0.4]0.3,[0.3,0.5]0.45〉	〈[0.1,0.4]0.45,[0.4,0.55]0.5〉 .	

Table 2 Tabular form of Ω-sets (Γ1,A),(Γ2,B),(Γ1,A)⁎ and (Γ2,B)⁎.

Table 2X	(Γ1,A)	(Γ2,B)	(Γ1,A)⁎	(Γ2,B)⁎	
C1	〈[0.45,0.6]0.65,[0.1,0.25]0.3〉	〈[0.1,0.35]0.4,[0.4,0.55]0.35〉	〈[0.45,0.6]0.4,[0.1,0.25]0.35〉	〈[0.1,0.35]0.65,[0.4,0.55]0.3〉	
C2	〈[0.2,0.35]0.15,[0.4,0.65]0.7〉	〈[0.3,0.45]0.5,[0.15,0.3]0.1〉	〈[0.2,0.35]0.5,[0.4,0.65]0.1〉	〈[0.3,0.45]0.15,[0.15,0.3]0.7〉	

Definition 3.10 If (Γ1,A)and(Γ2,B)∈Ω(Γ,Z), are any two Ω-sets on X. We define ∪P (Γ1,A)∪P(Γ2,B)=(ϒ,C), where A and B are any two subsets of Z and C=A∪B⊆Z andϒ(ϑj)={Γ1(ϑj)Γ2(ϑj)Γ1(ϑj)∨PΓ2(ϑj)ifϑj∈A−B⊆Zifϑj∈B−A⊆Zifϑj∈A∩B⊆Z,

whereas Γ1(ϑj)∨PΓ2(ϑj) is defined asΓ1(ϑj)∨PΓ2(ϑj)={〈x,rmax{σ˜Γ1(ϑj)(x),σ˜Γ2(ϑj)(x)}max{σΓ1(ϑj)(x),σΓ2(ϑj)(x)},rmin{ς˜Γ1(ϑj)(x),ς˜Γ2(ϑj)(x)}min{ςΓ1(ϑj)(x),ςΓ2(ϑj)(x)}〉:x∈X}.

Definition 3.11 If (Γ1,A)and(Γ2,B)∈Ω(Γ,Z), are any two Ω-sets on X. We define ∩P (Γ1,A)∩P(Γ2,B)=(Θ,C), where A and B are any two subsets of Z and C=A∪B⊆Z andΘ(ϑj)={Γ1(ϑj)Γ2(ϑj)Γ1(ϑj)∧PΓ2(ϑj)ifϑj∈A−B⊆Zifϑj∈B−A⊆Zifϑj∈A∩B⊆Z,

whereas Γ1(ϑj)∧PΓ2(ϑj) is defined asΓ1(ϑj)∧PΓ2(ϑj)={〈x,rmin{σ˜Γ1(ϑj)(x),σ˜Γ2(ϑj)(x)}min{σΓ1(ϑj)(x),σΓ2(ϑj)(x)},rmax{ς˜Γ1(ϑj)(x),ς˜Γ2(ϑj)(x)}max{ςΓ1(ϑj)(x),ςΓ2(ϑj)(x)}〉:x∈X}.

Definition 3.12 If (Γ1,A)and(Γ2,B)∈Ω(Γ,Z), are any two Ω-sets on X. We define ∪R (Γ1,A)∪R(Γ2,B)=(ϒ˜,C), where A and B are any two subsets of Z and C=A∪B⊆Z andϒ˜(ϑj)={Γ1(ϑj)Γ2(ϑj)Γ1(ϑj)∨RΓ2(ϑj)ifϑj∈A−B⊆Zifϑj∈B−A⊆Zifϑj∈A∩B⊆Z,

whereas Γ1(ϑj)∨RΓ2(ϑj) is defined asΓ1(ϑj)∨RΓ2(ϑj)={〈x,rmax{σ˜Γ1(ϑj)(x),σ˜Γ2(ϑj)(x)}min{σΓ1(ϑj)(x),σΓ2(ϑj)(x)},rmin{ς˜Γ1(ϑj)(x),ς˜Γ2(ϑj)(x)}max{ςΓ1(ϑj)(x),ςΓ2(ϑj)(x)}〉:x∈X}.

Definition 3.13 If (Γ1,A)and(Γ2,B)∈Ω(Γ,Z), are any two Ω-sets on X. We define ∩R (Γ1,A)∩R(Γ2,B)=(Θ˜,C), where A and B are any two subsets of Z and C=A∪B⊆Z andΘ˜(ϑj)={Γ1(ϑj)Γ2(ϑj)Γ1(ϑj)∧RΓ2(ϑj)ifϑj∈A−B⊆Zifϑj∈B−A⊆Zifϑj∈A∩B⊆Z,

whereas Γ1(ϑj)∧RΓ2(ϑj) is defined asΓ1(ϑj)∧RΓ2(ϑj)={〈x,rmin{σ˜Γ1(ϑj)(x),σ˜Γ2(ϑj)(x)}max{σΓ1(ϑj)(x),σΓ2(ϑj)(x)},rmax{ς˜Γ1(ϑj)(x),ς˜Γ2(ϑj)(x)}min{ςΓ1(ϑj)(x),ςΓ2(ϑj)(x)}〉:x∈X}.

Theorem 3.14 Let(Γ,A)=〈σ˜ΓσΓ,ς˜ΓςΓ〉be Ω-set on X which is not an ΩI-set. Then there exist an x∈X such that σΓ(x)∉(σΓ(ϑj)L(x),σΓ(ϑj)U(x)) and ςΓ(x)∉(ςΓ(ϑj)L(x),ςΓ(ϑj)U(x)).

Proof Straightforward. □

This theorem tells us that if (Γ,A) is a Ω-set which is not ΩI-set then some x∈X must exist that satisfy the condition of ΩE-set. See definitions of ΩI-set and ΩE-set in Definition 3.6. Theorem 3.15 Let(Γ,A)=〈σ˜ΓσΓ,ς˜ΓςΓ〉be both anΩI-set andΩE-set∀ϑj∈A⊆Zandx∈X, thenσΓ(ϑj)(x)∈Uσ∪LσandςΓ(ϑj)(x)∈Uς∪Lς, whereUσ(x)={σΓ(ϑj)U(x):x∈X},Lσ(x)={σΓ(ϑj)L(x):x∈X}andUς(x)={ςΓ(ϑj)U(x):x∈X},Lς(x)={ςΓ(ϑj)L(x):x∈X}.

Proof Since (Γ,A) is both ΩI-set and ΩE-set. So, we haveσΓ(ϑj)L(x)≤σΓ(ϑj)(x)≤σΓ(ϑj)U(x)and σΓ(ϑj)(x)∉(σΓ(ϑj)L(x),σΓ(ϑj)U(x))ςΓ(ϑj)L(x)≤ςΓ(ϑj)(x)≤ςΓ(ϑj)U(x)and ςΓ(ϑj)(x)∉(ςΓ(ϑj)L(x),ςΓ(ϑj)U(x)).Thus,σΓ(ϑj)(x)=σΓ(ϑj)L(x)orσΓ(ϑj)U(x)andςΓ(ϑj)(x)=ςΓ(ϑj)L(x)orςΓ(ϑj)U(x).Hence,σΓ(ϑj)(x)∈Uσ∪Lσ and ςΓ(ϑj)(x)∈Uς∪Lς.

 □

This theorem depicts that if (Γ,A) is a Ω-set which both ΩI-set and ΩE-set then membership degree σΓ(ϑ)(x) of all elements of the universe set is the union of Uς and Lς (i.e. membership degree of all elements of the universe set either upper limit or lower limit of membership interval. Theorem 3.16 Let(Γ,A)be a Ω-set in X. If (Γ,A) is ΩI-set (respectively, ΩE-set), then (Γ,A)c is also ΩI-set (respectively, ΩE-set).

Proof Since (Γ,A)=〈σ˜ΓσΓ,ς˜ΓςΓ〉 is ΩI-set (respectively, ΩE-set). So ∀ϑj∈A⊆Z,x∈X, we have:σΓ(ϑj)L(x)≤σΓ(ϑj)(x)≤σΓ(ϑj)U(x)and ςΓ(ϑj)L(x)≤ςΓ(ϑj)(x)≤ςΓ(ϑj)U(x)(respectively,σΓ(ϑj)(x)∉(σΓ(ϑj)L(x),σΓ(ϑj)U(x))and ςΓ(ϑj)(x)∉(ςΓ(ϑj)L(x),ςΓ(ϑj)U(x))).Hence(Γ,A)c=〈ς˜ΓςΓ,σ˜ΓσΓ〉is also ΩI-set (respectively, ΩE-set).

 □

This theorem shows that if (Γ,A) is a Ω-set which satisfies the conditions of ΩI-set (or ΩE-set) Then its complement also satisfies the conditions of ΩI-set (ΩE-set).

3.2 Significant results of ΩI-set and ΩE-set

Theorem 3.17 Since(Γi,Ai)=〈σ˜ΓiσΓi,ς˜ΓiςΓi〉i∈Λbe a family ofΩI-set inX. Then∪Pand∩Pof(Γi,Ai)are alsoΩI-set inX.

Proof Since (Γi,Ai)=〈σ˜ΓiσΓi,ς˜ΓiςΓi〉 is ΩI-set. So for each i∈Λ we have:σΓiL≤σΓi≤σΓiUand ςΓiL≤ςΓi≤ςΓiU⟹(⋁i∈ΛσΓi)L≤(⋁i∈ΛσΓi)≤(⋁i∈ΛσΓi)Uand (⋁i∈ΛςΓi)L≤(⋁i∈ΛςΓi)≤(⋁i∈ΛςΓi)U(⋀i∈ΛσΓi)L≤(⋀i∈ΛσΓi)≤(⋀i∈ΛσΓi)Uand (⋀i∈ΛςΓi)L≤(⋀i∈ΛςΓi)≤(⋀i∈ΛςΓi)U⟹(⋁i∈ΛσΓi)∈(⋃Pi∈Λσ˜Γi)and (⋁i∈ΛςΓi)∈(⋃Pi∈Λς˜Γi)(⋀i∈ΛσΓi)∈(⋂Pi∈Λσ˜Γi)and (⋀i∈ΛςΓi)∈(⋂Pi∈Λς˜Γi).

Hence ⋃P(Γi,Ai) and ⋂P(Γi,Ai) is also ΩI-set in X. □

This theorem provides important information about the special type of union and intersection of Ω-sets that are the ∪P and ∩P of the collection of ΩI-sets are also ΩI-sets. The following example will demonstrate that ∪P and ∩P of ΩE-sets are not always ΩE-sets. Example 3.18 Let (Γ1,A)=〈σ˜Γ1σΓ1,ς˜Γ1ςΓ1〉 and (Γ2,B)=〈σ˜Γ2σΓ2,ς˜Γ2ςΓ2〉∈Ω(Γ,Z) be two ΩE-sets on the universal set X with σ˜Γ1=[0.1,0.4],σΓ1=0.45;ς˜Γ1=[0.15,0.3],ςΓ1=0.4 and σ˜Γ2=[0.2,0.5],σΓ2=0.15;ς˜Γ2=[0.3,0.45],ςΓ2=0.2. Now1. Let (Γ1,A)∪P(Γ2,B)=(Γ3,C) with σ˜Γ3=[0.2,0.5],σΓ3=0.45;ς˜Γ3=[0.3,0.45],ςΓ3=0.4 which is not ΩE-set on X.

2. Let (Γ1,A)∩P(Γ2,B)=(Γ4,C) with σ˜Γ4=[0.1,0.4],σΓ4=0.15;ς˜Γ4=[0.15,0.3],ςΓ4=0.2 which is not ΩE-set on X.

In the example below, we will demonstrate that ∪R and ∩R of ΩI-sets are not always ΩI-sets. Example 3.19 Let (Γ1,A)=〈σ˜Γ1σΓ1,ς˜Γ1ςΓ1〉 and (Γ2,B)=〈σ˜Γ2σΓ2,ς˜Γ2ςΓ2〉∈Ω(Γ,Z) be two ΩI-sets on the universal set X with σ˜Γ1=[0.1,0.25],σΓ1=0.2;ς˜Γ1=[0.3,0.6],ςΓ1=0.4 and σ˜Γ2=[0.35,0.5],σΓ2=0.45;ς˜Γ2=[0.15,0.3],ςΓ2=0.2. Now1. Let (Γ1,A)∪R(Γ2,B)=(Γ3,C) with σ˜Γ3=[0.35,0.5],σΓ3=0.2;ς˜Γ3=[0.15,0.3],ςΓ3=0.4 which is not ΩI-set on X.

2. Let (Γ1,A)∩R(Γ2,B)=(Γ4,C) with σ˜Γ4=[0.1,0.25],σΓ4=0.45;ς˜Γ4=[0.3,0.6],ςΓ4=0.2 which is not ΩI-set on X.

The ∪R and ∩R of ΩE-sets may not be ΩE-sets, as we will demonstrate in the following example. Example 3.20 Let (Γ1,A)=〈σ˜Γ1σΓ1,ς˜Γ1ςΓ1〉 and (Γ2,B)=〈σ˜Γ2σΓ2,ς˜Γ2ςΓ2〉∈Ω(Γ,Z) be two ΩE-sets on the universal set X with σ˜Γ1=[0.15,0.30],σΓ1=0.55;ς˜Γ1=[0.4,0.6],ςΓ1=0.25 and σ˜Γ2=[0.5,0.65],σΓ2=0.75;ς˜Γ2=[0.2,0.3],ςΓ2=0.1. Now1. Let (Γ1,A)∪R(Γ2,B)=(Γ3,C) with σ˜Γ3=[0.5,0.65],σΓ3=0.55;ς˜Γ3=[0.2,0.3],ςΓ3=0.25 which is not ΩE-set on X.

2. Let (Γ1,A)∩R(Γ2,B)=(Γ4,C) with σ˜Γ4=[0.15,0.3],σΓ4=0.75;ς˜Γ4=[0.4,0.6],ςΓ4=0.1 which is not ΩE-set on X.

We have shown in Example 3.19 that the ∪R of two ΩI-sets is not necessarily an ΩI-set. The following theorem is meant to present a condition for the ∪R of two ΩI-sets to be an ΩI-set. Theorem 3.21 Let(Γ1,A)=〈σ˜Γ1σΓ1,ς˜Γ1ςΓ1〉and(Γ2,B)=〈σ˜Γ2σΓ2,ς˜Γ2ςΓ2〉be any twoΩI-sets on the universal setXsuch thatmax{σΓ1L,σΓ2L}≤min{σΓ1,σΓ2}andmin{ςΓ1U,ςΓ2U}≥max{ςΓ1,ςΓ2}, then∪Rof(Γ1,A)and(Γ2,B)isΩI-set onX.

Proof Since (Γ1,A) and (Γ2,B) are ΩI-sets. So,σΓ1L≤σΓ1≤σΓ1Uand ςΓ1L≤ςΓ1≤ςΓ1UσΓ2L≤σΓ2≤σΓ2Uand ςΓ2L≤ςΓ2≤ςΓ2U⇒min{σΓ1,σΓ2}≤max{σΓ1U,σΓ2U}and max{ςΓ1,ςΓ2}≥min{ςΓ1L,ςΓ2L}.

By given conditions it follows thatmax{σΓ1L,σΓ2L}≤min{σΓ1,σΓ2}and min{σΓ1U,σΓ2U}≥max{ςΓ1,ςΓ2}≤max{σΓ1U,σΓ2U}and ≥min{ςΓ1L,ςΓ2L}⇒min{σΓ1,σΓ2}∈rmax{σ˜Γ1,σ˜Γ2}and max{ςΓ1,ςΓ2}∈rmin{ς˜Γ1,ς˜Γ2}.

By Definition 3.12 ∪R of (Γ1,A) and (Γ2,B) is ΩI-set on X. □

In this theorem, ∪R of two ΩI-sets is again a ΩI-set under the following two constraints:1. The maximum value of lower limits of the membership interval of both sets is less than or equal to the minimum membership value of both sets.

2. The minimum value of the upper limits of the non-membership interval of both sets is greater than or equal to the maximum non-membership value of both sets.

Theorem 3.22 Let(Γ1,A)=〈σ˜Γ1σΓ1,ς˜Γ1ςΓ1〉and(Γ2,B)=〈σ˜Γ2σΓ2,ς˜Γ2ςΓ2〉be any twoΩI-sets on the universal setXsuch thatmin{σΓ1U,σΓ2U}≥max{σΓ1,σΓ2}andmax{ςΓ1L,ςΓ2L}≤min{ςΓ1,ςΓ2}, then∩Rof(Γ1,A)and(Γ2,B)isΩI-set onX.

Proof Since (Γ1,A) and (Γ2,B) are ΩI-sets. So,σΓ1L≤σΓ1≤σΓ1Uand ςΓ1L≤ςΓ1≤ςΓ1UσΓ2L≤σΓ2≤σΓ2Uand ςΓ2L≤ςΓ2≤ςΓ2U⇒max{σΓ1,σΓ2}≥min{σΓ1L,σΓ2L}and min{ςΓ1,ςΓ2}≤max{ςΓ1U,ςΓ2U}.

By given conditions it follows thatmin{σΓ1U,σΓ2U}≥max{σΓ1,σΓ2}and max{σΓ1L,σΓ2L}≤min{ςΓ1,ςΓ2}≥min{σΓ1L,σΓ2L}and ≤max{ςΓ1U,ςΓ2U}⇒max{σΓ1,σΓ2}∈rmin{σ˜Γ1,σ˜Γ2}and min{ςΓ1,ςΓ2}∈rmax{ς˜Γ1,ς˜Γ2}.

By Definition 3.13 ∩R of (Γ1,A) and (Γ2,B) is ΩI-set on X. □

In this theorem, ∩R of two ΩI-sets is again a ΩI-set under the following two constraints:1. The minimum value of the upper limits of the membership interval of both sets is greater than or equal to the maximum membership value of both sets.

2. The maximum value of the lower limits of the non-membership interval of both sets is less than or equal to the minimum non-membership value of both sets.

If we swap out σΓ1 for σΓ2 and ςΓ1 for ςΓ2 for two Ω-sets (Γ1,A)=〈σ˜Γ1σΓ1,ς˜Γ1ςΓ1〉 and (Γ2,B)=〈σ˜Γ2σΓ2,ς˜Γ2ςΓ2〉 on the universal set X. These Ω-sets on X are denoted by (Γ1,A)⁎=〈σ˜Γ1σΓ2,ς˜Γ1ςΓ2〉 and (Γ2,B)⁎=〈σ˜Γ2σΓ1,ς˜Γ2ςΓ1〉 respectively.

The following example shows that (Γ1,A)⁎ and (Γ2,B)⁎ do not have to be ΩI-sets on X for any two ΩE-sets (Γ1,A) and (Γ2,B) on X.

Example 3.23 Let (Γ1,A)=〈σ˜Γ1σΓ1,ς˜Γ1ςΓ1〉 and (Γ2,B)=〈σ˜Γ2σΓ2,ς˜Γ2ςΓ2〉 be any two ΩE-sets on the universal set X={C1,C2} defined in Table 3 for all ϑj∈A,B. Clearly (Γ1,A)⁎ and (Γ2,B)⁎ are not ΩE-sets on X.Table 3 Tabular form of (Γ1,A)∪P(Γ2,B) and (Γ1,A)∩P(Γ2,B)∀ϑj∈A∩B.

Table 3X	(Γ1,A)	(Γ2,B)	(Γ1,A)∪P(Γ2,B)	(Γ1,A)∩P(Γ2,B)	
C1	〈[0.45,0.6]0.65,[0.1,0.25]0.3〉	〈[0.1,0.35]0.4,[0.4,0.55]0.35〉	〈[0.45,0.6]0.65,[0.1,0.25]0.3〉	〈[0.1,0.35]0.4,[0.4,0.55]0.35〉	
C2	〈[0.2,0.35]0.15,[0.4,0.65]0.7〉	〈[0.3,0.45]0.5,[0.15,0.3]0.1〉	〈[0.3,0.45]0.5,[0.15,0.3]0.1〉	〈[0.2,0.35]0.15,[0.4,0.65]0.7〉	

We are going to demonstrate that the ∪P of two ΩE-set in X may not be an ΩI-set with the example that follows as. Example 3.24 Examine two ΩE-set (Γ1,A) and (Γ2,B) in X, given in Table 3. In this case (Γ1,A)∪P(Γ2,B) and (Γ1,A)∩P(Γ2,B) are not ΩI-sets on the universal set X as shown in Table 4.Table 4 Tabular form of (Γ1,A)∪R(Γ2,B),∀ϑj∈A∩B.

Table 4X	(Γ1,A)	(Γ2,B)	(Γ1,A)∪R(Γ2,B)	
C1	〈[0.2,0.3]0.5,[0.4,0.5]0.25〉	〈[0.4,0.6]0.7,[0.2,0.4]0.2〉	〈[0.4,0.6]0.5,[0.2,0.4]0.25〉	
C2	〈[0.1,0.3]0.6,[0.5,0.7]0.3〉	〈[0.4,0.6]0.5,[0.2,0.4]0.1〉	〈[0.4,0.6]0.5,[0.2,0.4]0.3〉	

A criterion for the ∪P of two ΩE-sets to be an ΩI-set is identified in the following result. Theorem 3.25 For twoΩE-sets(Γ1,A)=〈σ˜Γ1σΓ1,ς˜Γ1ςΓ1〉and(Γ2,B)=〈σ˜Γ2σΓ2,ς˜Γ2ςΓ2〉on the universal setX. If(Γ1,A)⁎=〈σ˜Γ1σΓ2,ς˜Γ1ςΓ2〉and(Γ2,B)⁎=〈σ˜Γ2σΓ1,ς˜Γ2ςΓ1〉areΩI-sets onX. Then(Γ1,A)∪P(Γ2,B)and(Γ1,A)∩P(Γ2,B)areΩI-sets onX.

Proof Since (Γ1,A) and (Γ2,B) are ΩE-sets on X. So,σΓ1∉(σΓ1L,σΓ1U)and ςΓ1∉(ςΓ1L,ςΓ1U),σΓ2∉(σΓ2L,σΓ2U)and ςΓ2∉(ςΓ2L,ςΓ2U),

for all x∈X,ϑj∈Z. Also (Γ1,A)⁎ and (Γ2,B)⁎ are ΩI-sets on X. So,σΓ1L≤σΓ2≤σΓ1Uand ςΓ1L≤ςΓ2≤ςΓ1U,σΓ2L≤σΓ1≤σΓ2Uand ςΓ2L≤ςΓ1≤ςΓ2U,

for all x∈X,ϑj∈Z. We have the following cases for any x∈XCase 1σΓ1≤σΓ1L≤σΓ2≤σΓ1U;ςΓ1≤ςΓ1L≤ςΓ2≤ςΓ1UσΓ2≤σΓ2L≤σΓ1≤σΓ2Uand ςΓ2≤ςΓ2L≤ςΓ1≤ςΓ2U.Case 2σΓ1L≤σΓ2≤σΓ1U≤σΓ1;ςΓ1L≤ςΓ2≤ςΓ1U≤ςΓ1σΓ2L≤σΓ1≤σΓ2U≤σΓ2and ςΓ2L≤ςΓ1≤ςΓ2U≤ςΓ2.Case 3σΓ1≤σΓ1L≤σΓ2≤σΓ1U;ςΓ1≤ςΓ1L≤ςΓ2≤ςΓ1UσΓ2L≤σΓ1≤σΓ2U≤σΓ2and ςΓ2L≤ςΓ1≤ςΓ2U≤ςΓ2.Case 4σΓ1L≤σΓ2≤σΓ1U≤σΓ1;ςΓ1L≤ςΓ2≤ςΓ1U≤ςΓ1σΓ2≤σΓ2L≤σΓ1≤σΓ2Uand ςΓ2≤ςΓ2L≤ςΓ1≤ςΓ2U.

Since the explanations in every case are similar, we only take the first one into consideration. We have σΓ1=σΓ1L=σΓ2=σΓ2L and ςΓ1=ςΓ1L=ςΓ2=ςΓ2L. Also (Γ1,A)⁎ and (Γ2,B)⁎ are ΩI-sets. So, σΓ2≤σΓ1U; ςΓ1L≤ςΓ2 and σΓ1≤σΓ2U; ςΓ2L≤ςΓ1. It follows that:(σΓ1∪σΓ2)L=max{σΓ1L,σΓ2L}=max{σΓ1,σΓ2}=σΓ1∨σΓ2≤max{σΓ1U,σΓ2U}=(σΓ1∪σΓ2)U⇒max{σΓ1,σΓ2}∈rmax{σ˜Γ1,σ˜Γ2},and(ςΓ1∩ςΓ2)U=min{ςΓ1U,ςΓ2U}=min{ςΓ1,ςΓ2}=ςΓ1∧ςΓ2≥min{ςΓ1L,ςΓ2L}=(ςΓ1∩ςΓ2)L⇒min{ςΓ1,ςΓ2}∈rmin{ς˜Γ1,ς˜Γ2}.

Hence (Γ1,A)∪P(Γ2,B) is an ΩI-set on X. Similarly, (Γ1,A)∩P(Γ2,B) can be performed. □

From the Example 3.24 it is clear that ∪P and ∩P of ΩE-sets are not necessarily ΩE-set on X. Next condition shows that the ∪P of two ΩE-sets is ΩE-set on X. Theorem 3.26 Let(Γ1,A)=〈σ˜Γ1σΓ1,ς˜Γ1ςΓ1〉and(Γ2,B)=〈σ˜Γ2σΓ2,ς˜Γ2ςΓ2〉be twoΩE-sets on the universal setX. If(Γ1,A)⁎=〈σ˜Γ1σΓ2,ς˜Γ1ςΓ2〉and(Γ2,B)⁎=〈σ˜Γ2σΓ1,ς˜Γ2ςΓ1〉areΩE-sets onX. Then(Γ1,A)∪P(Γ2,B)is alsoΩE-set onX.

Proof Since (Γ1,A),(Γ2,B),(Γ1,A)⁎ and (Γ2,B)⁎ are ΩE-sets on X. So,σΓ1∉(σΓ1L,σΓ1U)and ςΓ1∉(ςΓ1L,ςΓ1U).σΓ2∉(σΓ2L,σΓ2U)and ςΓ2∉(ςΓ2L,ςΓ2U).σΓ2∉(σΓ1L,σΓ1U)and ςΓ2∉(ςΓ1L,ςΓ1U).σΓ1∉(σΓ2L,σΓ2U)and ςΓ1∉(ςΓ2L,ςΓ2U).HenceσΓ1∨σΓ2∉(max{σΓ1L,σΓ2L},max{σΓ1U,σΓ2U})⇒max{σΓ1,σΓ2}∉rmax{σ˜Γ1,σ˜Γ2}andσΓ1∧σΓ2∉(min{σΓ1L,σΓ2L},min{σΓ1U,σΓ2U})⇒min{σΓ1,σΓ2}∉rmin{σ˜Γ1,σ˜Γ2}

Therefore (Γ1,A)∪P(Γ2,B) is ΩE-set on X. □

Note that ∩P of two ΩE-sets may not be an ΩE-set. Now, here we provide a condition for the ∩P of two ΩE-sets to be an ΩE-set on X. Theorem 3.27 Let(Γ1,A)=〈σ˜Γ1σΓ1,ς˜Γ1ςΓ1〉and(Γ2,B)=〈σ˜Γ2σΓ2,ς˜Γ2ςΓ2〉be twoΩE-sets on the universal setXsuch thatmin{max{σΓ1L,σΓ2U},max{σΓ1U,σΓ2L}}≥(σΓ1∧σΓ2)>max{min{σΓ1L,σΓ2U},min{σΓ1U,σΓ2L}}andmin{max{ςΓ1L,ςΓ2U},max{ςΓ1U,ςΓ2L}}>(ςΓ1∨ςΓ2)≥max{min{ςΓ1L,ςΓ2U},min{ςΓ1U,ςΓ2L}}

then(Γ1,A)∩P(Γ2,B)is anΩE-set onX.

Proof For each x∈X, we takeαx=min{max{σΓ1L,σΓ2U},max{σΓ1U,σΓ2L}},βx=max{min{σΓ1L,σΓ2U},min{σΓ1U,σΓ2L}},αx⁎=min{max{ςΓ1L,ςΓ2U},max{ςΓ1U,ςΓ2L}},βx⁎=max{min{ςΓ1L,ςΓ2U},min{ςΓ1U,ςΓ2L}}.

Now αx is one of the σΓ1L,σΓ1U,σΓ2L,σΓ2U and αx⁎ is one of the ςΓ1L,ςΓ1U,ςΓ2L,ςΓ2U. With out loss of generality we will consider only when αx=σΓ1L and αx⁎=ςΓ1L or αx=σΓ1U and αx⁎=ςΓ1U. For the all remaining cases, the arguments are similar to these cases.

If αx=σΓ1L and αx⁎=ςΓ1L, then σΓ2L≤σΓ2U≤σΓ1L≤σΓ1U, ςΓ2L≤ςΓ2U≤ςΓ1L≤ςΓ1U. So, βx=σΓ2U and βx⁎=ςΓ2U. ThusσΓ2L=min{σΓ1L,σΓ2L}≤min{σΓ1U,σΓ2U}=ςΓ2U=βx⁎<(σΓ1∧σΓ2)ςΓ1L=max{ςΓ1L,ςΓ2L}=αx⁎>(ςΓ1∨ςΓ2)So,(σΓ1∧σΓ2)∉(min{σΓ1L,σΓ2L},min{σΓ1U,σΓ2U})=rmin{σ˜Γ1,σ˜Γ2}and(ςΓ1∨ςΓ2)∉(max{ςΓ1L,ςΓ2L},max{ςΓ1U,ςΓ2U})=rmax{ς˜Γ1,ς˜Γ2}.

Thus, in this case (Γ1,A)∩P(Γ2,B) is an ΩE-set on X.

If we take αx=σΓ1U and αx⁎=ςΓ1U, thenσΓ2L≤σΓ1U≤σΓ2UandςΓ2L≤ςΓ1U≤ςΓ2Uso,βx=max{σΓ1L,σΓ2L}andβx⁎=max{ςΓ1L,ςΓ2L}.

Consider that βx=σΓ1L and βx⁎=ςΓ1L, thenσΓ2L≤σΓ1L=βx<(σΓ1∧σΓ2)≤σΓ1U≤σΓ2UandςΓ2L≤ςΓ1L=βx⁎≤(ςΓ1∨ςΓ2)<ςΓ1U≤ςΓ2U.

Above inequalities arise following two cases.Case 1σΓ2L≤σΓ1L=βx<(σΓ1∧σΓ2)<σΓ1U≤σΓ2UandςΓ2L≤ςΓ1L=βx⁎<(ςΓ1∨ςΓ2)<ςΓ1U≤ςΓ2UCase 2σΓ2L≤σΓ1L=βx<(σΓ1∧σΓ2)=σΓ1U≤σΓ2UandςΓ2L≤ςΓ1L=βx⁎=(ςΓ1∨ςΓ2)<ςΓ1U≤ςΓ2U.

Since (Γ1,A) and (Γ2,B) are ΩE-sets on X contradict the case 1. Case 2 implies that(σΓ1∧σΓ2)∉(min{σΓ1L,σΓ2L},min{σΓ1U,σΓ2U})∵min{σΓ1U,σΓ2U}=σΓ1U=(σΓ1∧σΓ2)and(ςΓ1∨ςΓ2)∉(max{ςΓ1L,ςΓ2L},max{ςΓ1U,ςΓ2U})∵max{ςΓ1L,ςΓ2L}=ςΓ1L=(ςΓ1∨ςΓ2).

Assume that βx=σΓ2L and βx⁎=ςΓ2L, thenσΓ1L≤σΓ2L=βx<(σΓ1∧σΓ2)≤σΓ1U≤σΓ2UandςΓ1L≤ςΓ2L=βx⁎≤(ςΓ1∨ςΓ2)<ςΓ1U≤ςΓ2U.

We have following two cases.Case 1σΓ1L≤σΓ2L=βx<(σΓ1∧σΓ2)<σΓ1U≤σΓ2UandςΓ1L≤ςΓ2L=βx⁎<(ςΓ1∨ςΓ2)<ςΓ1U≤ςΓ2UCase 2σΓ1L≤σΓ2L=βx<(σΓ1∧σΓ2)=σΓ1U≤σΓ2UandςΓ1L≤ςΓ2L=βx⁎=(ςΓ1∨ςΓ2)<ςΓ1U≤ςΓ2U.

Case 1 contradicts the fact that (Γ1,A) and (Γ2,B) are ΩE-sets on X. Case 2 implies that(σΓ1∧σΓ2)∉(min{σΓ1L,σΓ2L},min{σΓ1U,σΓ2U})∵min{σΓ1U,σΓ2U}=σΓ1U=(σΓ1∧σΓ2)and(ςΓ1∨ςΓ2)∉(max{ςΓ1L,ςΓ2L},max{ςΓ1U,ςΓ2U})∵max{ςΓ1L,ςΓ2L}=ςΓ2L=(ςΓ1∨ςΓ2).

We can obtain the similar result if we assume that βx=σΓ2L,βx⁎=ςΓ1Lorβx=σΓ1L,βx⁎=ςΓ2Landβx=σΓ2L,βx⁎=ςΓ2L. Hence, (Γ1,A)∩P(Γ2,B) is an ΩE-set on X. □

Example 3.28 Consider two ΩE-sets (Γ1,A) and (Γ2,B) in X={C1,C2}, defined in Table 5, Table 6 respectively. In this example (Γ1,A)∪R(Γ2,B) and (Γ1,A)∩R(Γ2,B) are not ΩE-sets on the universal set X as shown in Table 5, Table 6 respectively.Table 5 Tabular form of (Γ1,A)∩R(Γ2,B)∀ϑj∈A∩B.

Table 5X	(Γ1,A)	(Γ2,B)	(Γ1,A)∩R(Γ2,B)	
C1	〈[0.2,0.3]0.1,[0.4,0.5]0.55〉	〈[0.4,0.6]0.25,[0.25,0.4]0.45〉	〈[0.2,0.3]0.25,[0.4,0.5]0.45〉	
C2	〈[0.15,0.4]0.1,[0.25,0.55]0.35〉	〈[0.35,0.6]0.25,[0.15,0.35]0.55〉	〈[0.15,0.4]0.25,[0.25,0.55]0.35〉	

Table 6 Expert E1 provides a decision matrix M1.

Table 6X	ϑ1	ϑ2	ϑ3	ϑ4	
C1	〈[0.3,0.45]0.3,[0.4,0.5]0.65〉	〈[0.3,0.55]0.2,[0.2,0.4]0.55〉	〈[0.3,0.55]0.4,[0.1,0.3]0.35〉	〈[0.25,0.4]0.6,[0.3,0.4]0.25〉	
C2	〈[0.1,0.5]0.2,[0.25,0.4]0.3〉	〈[0.2,0.45]0.35,[0.3,0.5]0.45〉	〈[0.35,0.7]0.5,[0.2,0.3]0.1〉	〈[0.2,0.6]0.3,[0.2,0.35]0.5〉	
C3	〈[0.2,0.5]0.55,[0.1,0.4]0.25〉	〈[0.15,0.5]0.4,[0.2,0.4]0.3〉	〈[0.2,0.4]0.25,[0.3,0.55]0.5〉	〈[0.3,0.65]0.6,[0.2,0.35]0.4〉	
C4	〈[0.3,0.55]0.7,[0.2,0.4]0.1〉,	〈[0.25,0.4]0.2,[0.3,0.5]0.25〉,	〈[0.3,0.5]0.45,[0.35,0.4]0.25〉,	〈[0.3,0.55]0.5,[0.1,0.4]0.45〉	

It can be easily observed in Example 3.28 that ∪R and ∩R of two ΩE-sets may not be ΩE-set on X. In next result, we will show the criterion for ∪R and ∩R of two ΩE-sets is an ΩE-set on X. Theorem 3.29 Let(Γ1,A)=〈σ˜Γ1σΓ1,ς˜Γ1ςΓ1〉and(Γ2,B)=〈σ˜Γ2σΓ2,ς˜Γ2ςΓ2〉be twoΩE-sets on the universal setXsuch thatmin{max{σΓ1L,σΓ2U},max{σΓ1U,σΓ2L}}>(σΓ1∧σΓ2)≥max{min{σΓ1L,σΓ2U},min{σΓ1U,σΓ2L}}andmin{max{ςΓ1L,ςΓ2U},max{ςΓ1U,ςΓ2L}}≥(ςΓ1∨ςΓ2)>max{min{ςΓ1L,ςΓ2U},min{ςΓ1U,ςΓ2L}}

then(Γ1,A)∪R(Γ2,B)is anΩE-set onX.

Proof For each x∈X, we takeαx=min{max{σΓ1L,σΓ2U},max{σΓ1U,σΓ2L}},βx=max{min{σΓ1L,σΓ2U},min{σΓ1U,σΓ2L}},αx⁎=min{max{ςΓ1L,ςΓ2U},max{ςΓ1U,ςΓ2L}},βx⁎=max{min{ςΓ1L,ςΓ2U},min{ςΓ1U,ςΓ2L}}.

Now αx is one of the σΓ1L,σΓ1U,σΓ2L,σΓ2U and αx⁎ is one of the ςΓ1L,ςΓ1U,ςΓ2L,ςΓ2U. With out loss of generality we will consider only when αx=σΓ2L and αx⁎=ςΓ2L or αx=σΓ2U and αx⁎=ςΓ2U. For the all remaining cases, the arguments are similar to these cases.

If αx=σΓ2L and αx⁎=ςΓ2L, thenσΓ1L≤σΓ1U≤σΓ2L≤σΓ2U,ςΓ1L≤ςΓ1U≤ςΓ2L≤ςΓ2U.

So, βx=σΓ1U and βx⁎=ςΓ1U. Thusmax{σΓ1L,σΓ2L}=σΓ2L=αx>(σΓ1∧σΓ2)andmin{ςΓ1U,ςΓ2U}=ςΓ1U=βx⁎<(ςΓ1∨ςΓ2).So,(σΓ1∧σΓ2)∉(max{σΓ1L,σΓ2L},max{σΓ1U,σΓ2U})=rmax{σ˜Γ1,σ˜Γ2}and(ςΓ1∨ςΓ2)∉(min{ςΓ1L,ςΓ2L},min{ςΓ1U,ςΓ2U})=rmin{ς˜Γ1,ς˜Γ2}.

Thus, in this case (Γ1,A)∪R(Γ2,B) is an ΩE-set on X.

If we take αx=σΓ2U and αx⁎=ςΓ2U, thenσΓ1L≤σΓ2U≤σΓ1UandςΓ1L≤ςΓ2U≤ςΓ1Uso,βx=max{σΓ1L,σΓ2L}andβx⁎=max{ςΓ1L,ςΓ2L}.

Consider that βx=σΓ1L and βx⁎=ςΓ1L, thenσΓ2L≤σΓ1L=βx≤(σΓ1∧σΓ2)<αx=σΓ2U≤σΓ1UandςΓ2L≤ςΓ1L=βx⁎<(ςΓ1∨ςΓ2)≤αx⁎=ςΓ2U≤ςΓ1U.

We have following two cases.Case 1σΓ2L≤σΓ1L=βx<(σΓ1∧σΓ2)<αx=σΓ2U≤σΓ1UandςΓ2L≤ςΓ1L=βx⁎<(ςΓ1∨ςΓ2)<αx⁎=ςΓ2U≤ςΓ1UCase 2σΓ2L≤σΓ1L=βx=(σΓ1∧σΓ2)<αx=σΓ2U≤σΓ1UandςΓ2L≤ςΓ1L=βx⁎<(ςΓ1∨ςΓ2)=αx⁎=ςΓ2U≤ςΓ1U.

Case 1 contradicts the fact that (Γ1,A) and (Γ2,B) are ΩE-sets on X. Case 2 implies that(σΓ1∧σΓ2)∉(max{σΓ1L,σΓ2L},max{σΓ1U,σΓ2U})∵max{σΓ1L,σΓ2L}=σΓ1L=(σΓ1∧σΓ2)and(ςΓ1∨ςΓ2)∉(min{ςΓ1L,ςΓ2L},min{ςΓ1U,ςΓ2U})∵min{ςΓ1U,ςΓ2U}=ςΓ2U=(ςΓ1∨ςΓ2).

We can obtain the similar result if we assume that βx=σΓ1L,βx⁎=ςΓ2Lorβx=σΓ2L,βx⁎=ςΓ1Landβx=σΓ1L,βx⁎=ςΓ2L. Hence, (Γ1,A)∪R(Γ2,B) is an ΩE-set on X. □

The following theorem can be easily verified and proved; therefore, we omit the details. Theorem 3.30 Let(Γ1,A)=〈σ˜Γ1σΓ1,ς˜Γ1ςΓ1〉and(Γ2,B)=〈σ˜Γ2σΓ2,ς˜Γ2ςΓ2〉be twoΩE-sets on the universal setXsuch thatmin{max{σΓ1L,σΓ2U},max{σΓ1U,σΓ2L}}≥(σΓ1∨σΓ2)>max{min{σΓ1L,σΓ2U},min{σΓ1U,σΓ2L}}andmin{max{ςΓ1L,ςΓ2U},max{ςΓ1U,ςΓ2L}}>(ςΓ1∧ςΓ2)≥max{min{ςΓ1L,ςΓ2U},min{ςΓ1U,ςΓ2L}}

then(Γ1,A)∩R(Γ2,B)is anΩE-set onX.

Proof The proof is similar to Theorem 3.29; therefore, we omit the details. □

Theorem 3.31 Let(Γ1,A)=〈σ˜Γ1σΓ1,ς˜Γ1ςΓ1〉and(Γ2,B)=〈σ˜Γ2σΓ2,ς˜Γ2ςΓ2〉be any twoΩI-sets on the universal setXsuch thatmin{σΓ1,σΓ2}≤max{σΓ1L,σΓ2L}andmax{σΓ1,σΓ2}≥min{ςΓ1U,ςΓ2U}then(Γ1,A)∪R(Γ2,B)isΩE-set onX.

Theorem 3.32 Let(Γ1,A)=〈σ˜Γ1σΓ1,ς˜Γ1ςΓ1〉and(Γ2,B)=〈σ˜Γ2σΓ2,ς˜Γ2ςΓ2〉be any twoΩI-sets on the universal setXsuch thatmax{σΓ1,σΓ2}≥min{σΓ1U,σΓ2U}andmin{σΓ1,σΓ2}≤max{ςΓ1L,ςΓ2L}then(Γ1,A)∩R(Γ2,B)isΩE-set onX.

4 Application of Ω-set in MADM

This section describes the set-theoretic aggregation operations of a Ω-set that were applied in the design of a MADM recommendation model.

4.1 Problem description

Although thermal and hydroelectric energy sources are more expensive than other energy sources, they are used to produce electricity in some developing nations, such as Pakistan. Since alternative energy sources require slightly investment in the beginning, all types of customers choose to utilize them. Since solar panels are the most important kind of alternative energy in this context, their use in Pakistan has risen dramatically. Many clients from various sectors have made the switch to solar power systems. Homes, workplaces, farms, and industries all use it. These environmentally friendly solar systems ensure energy conservation, making them an intelligent option for consumers. Pakistan offers abundant sunlight throughout the year, making solar energy a sensible option for clean and renewable energy. On the market, there are multiple kinds of solar panels. Customers are very careful and concerned about their purchases because low-quality and inadequate solar panels are easily accessible. The process of choosing high-quality, long-lasting solar panels is difficult and involves several factors. Under the circumstances of an unclear algebraic setting, the MADM technique greatly assists decision-makers in choosing an appropriate solar panel product by taking into account relevant parameters. With the aid of some decision-makers, an algorithm (Algorithm 4.1) is suggested in the subsection that follows to help consumers buy a suitable solar panel product.

Algorithm 4.1 Let X={C1,C2,...,Cm} be the set of choices, Z={Z1,Z2,...,Zn} be the set of attributes which has following sub attributes values Zi={ei1,ei2,...,eix} for each 1≤i≤n,x∈{1,2,3,....} and E={E1,E2,...,Ek} be the set of experts. Suppose that ϑ={ϑ1,ϑ2,...,ϑj′} is the set of n-tuples of G=∏i=1nZi. Assume that each alternative Ci′(1≤i′≤n) is observed by experts Ek(1≤k≤K) with respect to each ϑj∈G using the Ω-sets. The suggested MADM algorithm's step-by-step growth is shown below.1. Based on the evaluated values of the experts Ek(1≤k≤K) in the form of Ω-sets aijk, construct the decision matrices Mk=(aijk)m×n.

2. Apply the suggested operations provided in Definition 3.11, Definition 3.13 to calculate the aggregated decision matrix M=(aij)m×n where aij=⋃P1≤k≤Kaijk or aij=⋃R1≤k≤Kaijk.

3. Using Definition 3.4, determine the score value for each of the aij in the aggregated decision matrix M.

4. Determine the preferred value for every alternative Ci(1≤i≤m) where P(Ci)=∑i=1m∑j=1naij

5. Create the ranking order of the alternatives using the preference value's non-increasing order.

Example 4.2 Mr. Smith is a renowned landlord with sizable farmland. Electricity and gas farming have grown more challenging due to increased bills. He has chosen to install a solar panel system instead of such utility systems because it is more cost-effective as well as beneficial. However, he is worried about the number of inferior local brands available, so he has hired a few professionals to help him choose a decent brand based on several qualities. So, he and the experts examined the technical details of several models available on the market. Consider the universe of discourse, X={C1,C2,C3,C4} consisting of four different solar panel models manufactured by various companies. For this purchase, the set of experts is represented by E={E1,E2,E3}. After careful analysis of the literature [37], [48], [49], [50], [51], [52] and with the mutual understanding of the experts, attributes like rated power (pmax), cell size, wafer type, cell technology, and efficiency have been adopted for this evaluation. To make a sound decision, the respective sub-attributes of these attributes are Z1={e11=744W,e12=730W,e13=700W,e14=625W}, Z2={e21=210mm,e22=182mm}, Z3={e31=N−typer,e32=P−typer}, Z4={e41=Hetero junction (HJT),e42=Tunnel-OxidePassivatingContact (TOPCon ),e43=Mono PERC+(Passivated emitter&rear cell) }, and Z5={e51=23.96%,e52=23.5%,e53=22.5%}. To construct an Ω-set, the Cartesian product Z=Z1×Z2×Z3×Z4×Z5 of disjoint attributive valued sets is determined, which consists of 144 sub-attributes valued tuples, however, for the sake of avoiding computational complexity and getting reliability, four tuples (e11,e21,e31,e41,e51), (e12,e22,e32,e42,e53), (e13,e21,e32,e43,e52) and (e14,e22,e31,e41,e51) have been chosen for further evaluation. These tuples are given the names as ϑ1, ϑ2, ϑ3 and ϑ4 respectively. Assume the expert Ek(k=1,2,3) used Ω-set to evaluate each option Ci(i=1,2,3,4) based on the criteria ϑj(j=1,2,3,4). Each expert collects the information in the form of Ω-set, in which IVIFS is the opinion of the dealer and the corresponding IFS is agreement as well as disagreement with IVIFS. We will proceed with the following steps:Step 1 The expertise of experts is used to generate the decision matrices M1,M2,M3, as shown in Table 6, Table 7 and Table 8.Table 7 Expert E2 provides a decision matrix M2.

Table 7X	ϑ1	ϑ2	ϑ3	ϑ4	
C1	〈[0.2,0.6]0.65,[0.1,0.25]0.3〉	〈[0.45,0.6]0.4,[0.4,0.55]0.35〉	〈[0.2,0.4]0.15,[0.25,0.5]0.4〉	〈[0.3,0.45]0.15,[0.2,0.45]0.6〉	
C2	〈[0.1,0.35]0.4,[0.2,0.45]0.5〉	〈[0.1,0.35]0.4,[0.4,0.55]0.35〉	〈[0.45,0.6]0.65,[0.1,0.3]0.3〉	〈[0.1,0.35]0.4,[0.4,0.55]0.35〉	
C3	〈[0.2,0.35]0.5,[0.4,0.65]0.1〉	〈[0.3,0.45]0.5,[0.1,0.5]0.2〉	〈[0.3,0.6]0.5,[0.1,0.3]0.25〉	〈[0.2,0.35]0.15,[0.4,0.65]0.7〉	
C4	〈[0.25,0.6]0.7,[0.1,0.35]0.25〉	〈[0.3,0.45]0.5,[0.15,0.3]0.4〉	〈[0.45,0.6]0.4,[0.1,0.3]0.35〉	〈[0.1,0.35]0.65,[0.4,0.55]0.3〉	

Table 8 Expert E3 provides a decision matrix M3.

Table 8X	ϑ1	ϑ2	ϑ3	ϑ4	
C1	〈[0.3,0.45]0.4,[0.4,0.5]0.45〉	〈[0.2,0.3]0.5,[0.4,0.55]0.25〉	〈[0.2,0.6]0.5,[0.2,0.4]0.3〉	〈[0.25,0.6]0.5,[0.2,0.4]0.35〉	
C2	〈[0.1,0.35]0.6,[0.4,0.6]0.3〉	〈[0.4,0.6]0.3,[0.15,0.4]0.5〉	〈[0.3,0.6]0.5,[0.2,0.35]0.4〉	〈[0.1,0.4]0.3,[0.35,0.6]0.7〉	
C3	〈[0.25,0.5]0.5,[0.1,0.3]0.25〉	〈[0.15,0.4]0.1,[0.2,0.5]0.35〉	〈[0.3,0.6]0.7,[0.2,0.35]0.2〉	〈[0.15,0.4]0.2,[0.2,0.5]0.25〉	
C4	〈[0.2,0.4]0.3,[0.25,0.5]0.5〉	〈[0.3,0.5]0.25,[0.15,0.4]0.55〉	〈[0.15,0.4]0.25,[0.3,0.5]0.55〉	〈[0.4,0.6]0.5,[0.1,0.35]0.45〉	

Step 2 As mentioned in Definition 3.10, the suggested ∪P operation is used to build the aggregated decision matrix M=(aij)4×4 where aij=⋃P1≤k≤Kaijk. Table 9 displays the aggregated decision matrix M.Table 9 Aggregated decision matrix M by using ∪P operation.

Table 9X	ϑ1	ϑ2	ϑ3	ϑ4	
C1	〈[0.3,0.6]0.65,[0.1,0.25]0.3〉	〈[0.45,0.6]0.5,[0.2,0.4]0.25〉	〈[0.3,0.6]0.5,[0.1,0.3]0.3〉	〈[0.3,0.6]0.6,[0.2,0.4]0.25〉	
C2	〈[0.1,0.5]0.6,[0.2,0.4]0.3〉	〈[0.4,0.6]0.4,[0.15,0.4]0.35〉	〈[0.3,0.7]0.65,[0.1,0.3]0.1〉	〈[0.2,0.6]0.4,[0.2,0.35]0.35〉	
C3	〈[0.25,0.5]0.55,[0.1,0.3]0.1〉	〈[0.3,0.5]0.5,[0.1,0.4]0.2〉	〈[0.3,0.7]0.6,[0.1,0.3]0.2〉	〈[0.15,0.65]0.6,[0.2,0.35]0.25〉	
C4	〈[0.3,0.6]0.7,[0.1,0.35]0.1〉	〈[0.3,0.5]0.5,[0.15,0.3]0.25〉	〈[0.45,0.6]0.45,[0.1,0.3]0.25〉	〈[0.4,0.6]0.65,[0.1,0.35]0.3〉	

Step 3 The score value of each aij in the aggregated decision matrix M will be determined by applying Definition 3.4. Table 10 displays the matrix of score values for all the elements of M.Table 10 Score values of M.

Table 10X	ϑ1	ϑ2	ϑ3	ϑ4	
C1	0.3	0.2333	0.2333	0.2167	
C2	0.1	0.1667	0.3833	0.1	
C3	0.2667	0.2	0.3333	0.2	
C4	0.35	0.2	0.2833	0.3	

Step 4, 5 Calculate the preferred value of each 5-tuple sub parametric attributes ϑj(j=1,2,3,4) where P(ϑj)=maxi=14⁡aij corresponding to each alternatives by using the ∪P operation are given below:P(ϑ1)=0.35,P(ϑ2)=0.2333,P(ϑ3)=0.3833 and P(ϑ4)=0.3.

Based on the non-increasing order of their preference values of each 5-tuple sub-parametric attribute is ranked in the following order:P(ϑ3)=0.3833≥P(ϑ1)=0.35≥P(ϑ4)=0.3≥P(ϑ2)=0.2333.

Similarly, the preferred value of each 5-tuple sub-parametric attributes ϑj(j=1,2,3,4) where P(ϑj)=maxi=14⁡aij concerning each alternative by using the ∪R operation, is calculated and given in ranking order as follows:P(ϑ3)=0.2333≥P(ϑ1)=0.2≥P(ϑ4)=0.2≥P(ϑ2)=0.0833.

The robustness of the suggested method can be seen by the observation that the ranking order of sub-parametric attributes acquired with the aid of the ∪P operation and the ∪R operation is identical. Using the ∩P and ∩R processes described in Definition 3.11, Definition 3.13, we can readily observe that the sub-parametric attribute ranking order will result in the reverse order of the ranking orders acquired in the ∪P and ∪R operations, respectively (Fig. 1).Figure 1 Ranking based on ∪P and ∪R.

Figure 1

4.2 Comparison

It has not yet been explored how combinations of structures incorporating CIFS and HSS can be developed. This method can help increase the effectiveness and productivity of DM processes by dissecting each attribute into its parts, leading to a greater understanding of DM qualities. The Ω-set), a hybrid fuzzy structure, has been developed in recognition of a requirement for an adaptable analytical instrument that could be completely analyzed at the sub-attribute level. The combination of two membership functions: membership and non-membership, with two fuzzy intervals-membership and non-membership intervals that either include or lack both membership and non-membership functions, makes the CIFS a powerful contribution to FS theory. As a result, representing ambiguity and uncertainty in DM processes is made simpler. On the other hand, a sub-attribute analysis of attribute-based data from SS theory is the main emphasis of HSS theory. The Ω-set has been developed by combining these two frameworks to provide a precise and flexible approach to decision analysis. It is more adaptable as compared to existing ones as it generalizes them. To emphasize the advantages, Table 11 provides a comparison based on the structure in which the drawbacks of some relevant prior research are considered about the proposed framework.Table 11 Structure based comparison.

Table 11Literature	Case study	Approach	Shortcomings	
Rani et al. [37]	SPS	Pythagorean fuzzy DM	Two dimensional arrangement of IVIFS and IFS are missing.	
Ihsan et al. [38]	SPS	FHSS with multi decisive settings	Two dimensional arrangement of IVIFS and IFS are missing.	
Akram et al. [39]	SPS	Fermatean FSS with multi decisive settings	Two dimensional arrangement of IVIFS and IFS are missing.	
Riaz et al. [46]	Renewable Energy Resources	Cubic bipolar fuzzy DM	Two dimensional arrangement of IVIFS and IFS are missing.	
Suggested framework	SPS	Ω-set	A proper formulation has been provided to manage uncertainties effectively using two dimensional arrangement of IVIFS and IFS are missing.	

Additionally, some notable advantages of this study are:1. By promoting collaboration, the suggested context, the Ω-set, resolves the limitations of IFS and IFSS about the reliability of decision-makers opinions. At the same time, giving them specific ranges for these values improves their capacity to make well-informed and useful judgments.

2. In comparison to SS and IFSS, the approximate function of Ω-set is more flexible due to its multi-argument domain that takes into account the Cartesian product of attribute-valued non-overlapping sets. Its flexible range is cubic, allowing it to manage IFSS-related issues and facilitate well-informed DM.

3. The ability of Ω-set to capture expert judgments with both interval-valued and single-valued membership and non-membership grades improves flexibility and accuracy. This multifaceted approach enables more complex and in-depth evaluations, which makes it particularly useful in scenarios where precise analysis and DM require complex and multilayered data.

5 Conclusion

In this research effort, we presented a new hybrid structure combining a hypersoft set and a cubic intuitionistic fuzzy set called the Cubic intuitionistic fuzzy hypersoft set (Ω-set). We talked about some of its relevant features. We additionally discussed the ΩI-set and ΩE-sets, two more kinds of Ω-sets. Along with the relevant examples, the P-order, R-order, ∪P, ∪R, ∩P, ∩R, and several other helpful properties were also discussed. Furthermore, we established that ΩI-sets are also ∪P, and ∪R of ΩI-sets. Further, this study established some conditions under which the ∪P, ∪R, ∩P, and ∩R of two ΩE-sets are ΩI-sets. A few requirements for ∪P, ∪R, ∩P, and ∩R of two ΩE-sets to be ΩE-sets were also given. A case study on the optimized evaluation of solar panels is presented, which validates a proposed MADM-based algorithm. In the case study, four different solar panel models are evaluated according to 14 sub-parameters, including rated power, cell size, wafer type, efficiency, and cell technology. The proposed framework is insufficient for situations where decision-makers want to provide expert opinions regarding other mathematical structures like picture fuzzy, spherical fuzzy, neutrosophic, or plithogenic cubic hypersoft settings.

Declaration of generative AI and AI-assisted technologies in the writing process

During the preparation of this work the author(s) used chatGPT. After using this tool/service, the author(s) reviewed and edited the content as needed and take(s) full responsibility for the content of the publication.

CRediT authorship contribution statement

Muhammad Sajid: Writing – original draft, Visualization, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. Khuram Ali Khan: Writing – review & editing, Supervision, Methodology, Investigation, Formal analysis, Conceptualization. Atiqe Ur Rahman: Writing – original draft, Visualization, Methodology, Data curation, Conceptualization. Sanaa A. Bajri: Writing – review & editing, Software, Project administration, Funding acquisition, Data curation, Conceptualization. Alhanouf Alburaikan: Writing – review & editing, Software, Project administration, Funding acquisition, Data curation, Conceptualization. Hamiden Abd El-Wahed Khalifa: Writing – review & editing, Software, Project administration, Funding acquisition, Data curation, Conceptualization.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

Data will be made available on request.

Acknowledgement

This research is funded by 10.13039/501100004242 Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R527 ), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
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References

1 Zadeh L.A. Fuzzy sets Inf. Control 8 3 1965 338 353 10.1016/S0019-9958(65)90241-X
2 Atanassov K.T. Intuitionistic fuzzy sets Fuzzy Sets Syst. 20 1 1986 87 96 10.1016/S0165-0114(86)80034-3
3 Atanassov K.T. Gargov G. Interval-valued intuitionistic fuzzy sets Fuzzy Sets Syst. 31 3 1989 343 349 10.1016/0165-0114(89)90205-4
4 Garg H. Kumar K. Linguistic interval-valued Atanassov intuitionistic fuzzy sets and their applications to group decision making problems IEEE Trans. Fuzzy Syst. 27 12 2019 2302 2311 10.1109/TFUZZ.2019.2897961
5 Alkouri A.M.D.J.S. Salleh A.R. Complex intuitionistic fuzzy sets AIP Conf. Proc. 1482 1 2012, September 464 470 10.1063/1.4757515
6 Garg H. Rani D. Complex interval-valued intuitionistic fuzzy sets and their aggregation operators Fundam. Inform. 164 1 2019 61 101 10.3233/FI-2019-1755
7 Molodtsov D. Soft set theory—first results Comput. Math. Appl. 37 4–5 1999 19 31 10.1016/S0898-1221(99)00056-5
8 Maji P.K. Roy A.R. Biswas R. An application of soft sets in a decision making problem Comput. Math. Appl. 44 8–9 2002 1077 1083 10.1016/S0898-1221(02)00216-X
9 Maji P.K. Biswas R. Roy A.R. Soft set theory Comput. Math. Appl. 45 4–5 2003 555 562 10.1016/S0898-1221(03)00016-6
10 Maji P.K. Biswas R. Roy A.R. Fuzzy soft sets J. Fuzzy Math. 9 3 2001 589 602
11 Roy A.R. Maji P.K. A fuzzy soft set theoretic approach to decision making problems J. Comput. Appl. Math. 203 2 2007 412 418 10.1016/j.cam.2006.04.008
12 Çağman N. Çıtak F. Enginoğlu S. Fuzzy parameterized fuzzy soft set theory and its applications Turk. J. Fuzzy Syst. 1 1 2010 21 35
13 Çağman N. Enginoğlu S. Çıtak F. Fuzzy soft set theory and its applications Iran. J. Fuzzy Syst. 8 3 2011 137 147 10.22111/IJFS.2011.292
14 Yang X. Lin T.Y. Yang J. Li Y. Yu D. Combination of interval-valued fuzzy set and soft set Comput. Math. Appl. 58 3 2009 521 527 10.1016/j.camwa.2009.04.019
15 Feng F. Li Y. Leoreanu-Fotea V. Application of level soft sets in decision making based on interval-valued fuzzy soft sets Comput. Math. Appl. 60 6 2010 1756 1767 10.1016/j.camwa.2010.07.006
16 Maji P.K. Biswas R. Roy A.R. Intuitionistic fuzzy soft sets J. Fuzzy Math. 9 3 2001 677 692
17 Deli I. Çağman N. Intuitionistic fuzzy parameterized soft set theory and its decision making Appl. Soft Comput. 28 2015 109 113 10.1016/j.asoc.2014.11.053
18 Majumdar P. Samanta S.K. Generalised fuzzy soft sets Comput. Math. Appl. 59 4 2010 1425 1432 10.1016/j.camwa.2009.12.006
19 Agarwal M. Biswas K.K. Hanmandlu M. Generalized intuitionistic fuzzy soft sets with applications in decision-making Appl. Soft Comput. 13 8 2013 3552 3566 10.1016/j.asoc.2013.03.015
20 Jun Y.B. Kim C.S. Yang K.O. Cubic sets Ann. Fuzzy Math. Inform. 4 1 2012 83 98
21 Muhiuddin G. Al-roqi A.M. Cubic soft sets with applications in BCK/BCI-algebras Ann. Fuzzy Math. Inform. 8 2 2014 291 304
22 Jun Y.B. A novel extension of cubic sets and its applications in BCK/BCI-algebras Ann. Fuzzy Math. Inform. 14 5 2017 475 486
23 Jun Y.B. Song S.Z. Kim S.J. Cubic interval-valued intuitionistic fuzzy sets and their application in BCK/BCI-algebras Axioms 7 1 2018 7 10.3390/axioms7010007
24 Garg H. Kaur G. Cubic intuitionistic fuzzy sets and its fundamental properties J. Mult.-Valued Log. Soft Comput. 33 6 2019 507
25 Kaur G. Garg H. Cubic intuitionistic fuzzy aggregation operators Int. J. Uncertain. Quantificat. 8 5 2018 405 427 10.1615/Int.J.UncertaintyQuantification.2018020471
26 Garg H. Kaur G. Novel distance measures for cubic intuitionistic fuzzy sets and their applications to pattern recognitions and medical diagnosis Granul. Comput. 5 2 2020 169 184 10.1007/s41066-018-0140-3
27 Garg H. Kaur G. Extended TOPSIS method for multi-criteria group decision-making problems under cubic intuitionistic fuzzy environment Sci. Iran. 27 1 2020 396 410 10.24200/SCI.2018.5307.1194
28 Faizi S. Svitenko H. Rashid T. Zafar S. Sałabun W. Some operations and properties of the cubic intuitionistic set with application in multi-criteria decision-making Mathematics 11 5 2023 1190 10.3390/math11051190
29 Smarandache F. Extension of soft set to hypersoft set, and then to plithogenic hypersoft set Neutrosophic Sets Syst. 22 2018 168 170 10.5281/zenodo.2159755
30 Debnath S. Fuzzy hypersoft sets and its weightage operator for decision making J. Fuzzy Ext. Appl. 2 2 2021 163 170 10.22105/jfea.2021.275132.1083
31 Saqlain M. Imran R. Hassan S. Cubic intuitionistic fuzzy soft set and its distance measures Sci. Inq. Rev. 6 2 2022 59 75 10.32350/sir.62.04
32 Rahman A.U. Saeed M. Khan K.A. Nosheen A. Mabela R.M. An algebraic approach to modular inequalities based on interval-valued fuzzy hypersoft sets via hypersoft set-inclusions J. Funct. Spaces 2022 2022 1 15 10.1155/2022/1384541
33 Arshad M. Saeed M. Rahman A.U. Mohammed M.A. Abdulkareem K.H. Nedoma J. Martinek R. Deveci M. A robust framework for the selection of optimal COVID-19 mask based on aggregations of interval-valued multi-fuzzy hypersoft sets Expert Syst. Appl. 238 2024 121944 10.1016/j.eswa.2023.121944
34 Saeed M. Smarandache F. Arshad M. Rahman A.U. An inclusive study on the fundamentals of interval-valued fuzzy hypersoft set Int. J. Neutrosophic Sci. 20 2 2023 135 161 10.54216/IJNS.200209
35 Arshad M. Saeed M. Rahman A.U. Mohammed M.A. Abdulkareem K.H. Alghawli A.S. Al-Qaness M.A. A robust algorithmic cum integrated approach of interval-valued fuzzy hypersoft set and OOPCS for real estate pursuit PeerJ Comput. Sci. 9 2023 e1423 10.7717/peerj-cs.1423
36 Arshad M. Saeed M. Rahman A.U. Zebari D.A. Mohammed M.A. Al-Waisy A.S. Albahar M. Thanoon M. The assessment of medication effects in omicron patients through madm approach based on distance measures of interval-valued fuzzy hypersoft set Bioengineering 9 11 2022 706 10.3390/bioengineering9110706 36421107
37 Rani P. Mishra A.R. Mardani A. Cavallaro F. Štreimikienė D. Khan S.A.R. Pythagorean fuzzy SWARA-VIKOR framework for performance evaluation of solar panel selection Sustainability 12 10 2020 4278 10.3390/su12104278
38 Ihsan M. Saeed M. Rahman A.U. Kamacı H. Ali N. An MADM-based fuzzy parameterized framework for solar panels evaluation in a fuzzy hypersoft expert set environment AIMS Math. 8 2 2022 3403 3427 10.3934/math.2023175
39 Akram M. Ali G. Alcantud J.C.R. Riaz A. Group decision-making with Fermatean fuzzy soft expert knowledge Artif. Intell. Rev. 55 7 2022 5349 5389 10.1007/s10462-021-10119-8 35035018
40 Raja M.S. Hayat K. Munshi A. Mahmood T. Sheraz R. Matloob I. Aggregation operators on group-based generalized q-rung orthopair fuzzy N-soft sets and applications in solar panel evaluation Heliyon 10 15 2024 e27323 10.1016/j.heliyon.2024.e27323
41 Tüysüz N. Kahraman C. An integrated picture fuzzy Z-AHP & TOPSIS methodology: application to solar panel selection Appl. Soft Comput. 149 2023 110951 10.1016/j.asoc.2023.110951
42 Ziemba P. Szaja M. Fuzzy decision-making model for solar photovoltaic panel evaluation Energies 16 13 2023 5161 10.3390/en16135161
43 Arman K. Kundakcı N. A fuzzy best worst method based prioritization of solar panel selection criteria Rezaei J. Brunelli M. Mohammadi M. Advances in Best-Worst Method BWM 2022 Lecture Notes in Operations Research 2023 Springer Cham 10.1007/978-3-031-24816-0_9
44 Jafar M.N. Saeed M. Saqlain M. Yang M.S. Trigonometric similarity measures for neutrosophic hypersoft sets with application to renewable energy source selection IEEE Access 9 2021 129178 129187 10.1109/ACCESS.2021.3112721
45 Saqlain M. Riaz M. Saleem M.A. Yang M.S. Distance and similarity measures for neutrosophic hypersoft set (NHSS) with construction of NHSS-TOPSIS and applications IEEE Access 9 2021 30803 30816 10.1109/ACCESS.2021.3059712
46 Riaz M. Habib A. Saqlain M. Yang M.S. Cubic bipolar fuzzy-VIKOR method using new distance and entropy measures and Einstein averaging aggregation operators with application to renewable energy Int. J. Fuzzy Syst. 25 2 2023 510 543 10.1007/s40815-022-01383-z
47 Saeed M. Saeed M.H. Khalid M. Mekawy I. Development of Hamming and Hausdorff distance metrics for cubic intuitionistic fuzzy hypersoft set in cement storage quality control: development and evaluation PLoS ONE 18 9 2023 e0291817 10.1371/journal.pone.0291817
48 Balo F. Şağbanşua L. The selection of the best solar panel for the photovoltaic system design by using AHP Energy Proc. 100 2016 50 53 10.1016/j.egypro.2016.10.151
49 Kozlov V. Sałabun W. Challenges in reliable solar panel selection using MCDA methods Proc. Comput. Sci. 192 2021 4913 4923 10.1016/j.procs.2021.09.269
50 El-Bayeh C.Z. Alzaareer K. Brahmi B. Zellagui M. Eicker U. An original multi-criteria decision-making algorithm for solar panels selection in buildings Energy 217 2021 119396 10.1016/j.energy.2020.119396
51 Brian What are heterojunction technology (HJT) solar panels: advantages & applications Online available at https://www.maysunsolar.com/blog-what-are-heterojunction-technology-hjt-solar-panels-advantages-applications/ 2023
52 PV Manufacturing https://pv-manufacturing.org/tunnel-oxide-passivated-contact-topcon-solar-cells/
