Assimilation of Matrix Operations with Picture Fuzzy Hypersoft Structures for Complex Decision Scenarios

Abstract

Matrices are a unifying concept that permeates diverse fields of study and provides a common framework for expressing and solving problems in all fields, including decision-making. In most decision-making scenarios, addressing uncertainty is paramount as real-world scenarios often involve incomplete information, ambiguous data, and unpredictable factors. The aptness of a picture fuzzy hypersoft set as a parameterization tool becomes evident when dealing with the complexities and challenges associated with managing imprecise data. In this article, we have initially developed the notions of the picture fuzzy hypersoft matrices (${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$) and their basic operations based on the enhanced framework of the picture fuzzy hypersoft set. We then proposed the ideas of fundamental operational principles for picture fuzzy hypersoft numbers based on the structure of picture fuzzy hypersoft matrices. Furthermore, the concepts for picture fuzzy hypersoft geometric aggregation operators have been presented. The application of picture fuzzy hypersoft geometric aggregation operators to energy policy design signifies a bridge between theoretical advancements and practical decision-making. The inclusion of a decision-making approach, explanatory example, and comparative analysis enhances the understanding of how the developed theory can be effectively used and demonstrates its potential contributions to the field of informed decision-making using human intuitionistic data.

1 Introduction

The concept of fuzzy sets (FS) [1], introduced by Lotfi Zadeh in 1965, was an innovative step in handling uncertainty and vagueness in different fields. The notion of FS permitted a more flexible and nuanced representation of information by introducing the notion of membership degrees. This opinion role laid the foundation for dealing with imprecise and uncertain data. Atanassov and Stoeva further extended the study in 1986 with the introduction of intuitionistic fuzzy sets (IFS) [2]. IFS was conceptualized as a native progression from FS theory, aiming to concentrate on some of the limitations coupled with it. The key peculiarity of IFS lies in the preamble of the non-membership degree, which quantifies the degree to which an element does not belong to a set. This accumulation complements the degree of membership, providing a more inclusive representation of uncertainty. The introduction of picture fuzzy sets (PFS) [3] by Cuong in 2013 represents a considerable improvement in set theory, particularly addressing restrictions encountered in handling inconsistent information, mainly in scenarios such as voting questions. PFS is bespoke to hold situations where individuals may express varying degrees of support or opposition to a particular option. The introduction of Soft Set (SS) [4] by the Russian researcher Molodtsov in 1999 made an important development in mathematical tools to model uncertainties. One of the important features of SS is that it does not impose conditions on the description of objects. Researchers have the freedom to choose any form of parameters they need. This lack of restriction improves the adaptability of the theory to diverse applications.

The conception of a fuzzy soft set (FSS) [5], proposed by Maji et al. [6], is a hybrid model that combines the concepts of FS and SS. This hybrid model aims to concentrate on the challenges posed by the inherent vagueness and imprecision in the data, generally in situations where attributes cannot be precisely characterized using only crisp values of 0 and 1. The theory of an intuitionistic fuzzy soft set (IFSS), given by Maji et al. in 2001, is also a hybrid model that combines the characteristics of IFS and SS. In this model, each element is characterized not only by its membership and non-membership degrees but also by a hesitation degree, similar to IFS. The concept of picture fuzzy soft set (PFSS) [7], as introduced by Yang et al. 2015, the hybrid model is designed to handle uncertainties in a picture fuzzy environment. The introduction of hypersoft sets (HSS) [8] by Smarandache in 2018 was motivated by the need to overcome limitations in existing theories when situations involve characteristics of a group of parameters with additional sub-attributes. The work by Abbas et al. [9] in 2020 appears to deal with the vital operations that can be performed on HSS. The study by Saeed et al. [10] in 2021 focuses on defining and exploring the concepts of HSS theory.

The idea of fuzzy hypersoft sets (FHSS) [11], introduced by Yolcu and Ozturk in 2021, is an extension of HSS to handle fuzzy and vague information. FHSS is a helpful tool in various domains where decision-making involves ambiguity. The theory of intuitionistic fuzzy hypersoft sets (IFHSS) [12] introduced by Yolcu et al. in 2021 is an expansion of the HSS to accommodate intuitionistic fuzzy information. IFHSS is advantageous to handle not only fuzzy and uncertain data but also incomplete and vague information. The concept of picture fuzzy hypersoft sets (PFHSS) [13] introduced by Saeed and Harl in 2023 is notably beneficial in handling situations where the data is not only uncertain but also exhibits inconsistencies. The concept of picture fuzzy hypersoft graphs (PFHSG) [14], introduced by Saeed et al. in 2023, represents a novel approach to risk analysis, predominantly from the perspective of product sales. The lattice-ordered picture fuzzy hypersoft (LOPFHSS) [15] introduced by Harl et al. in 2023, is a pioneering approach to handling complex decision-making situations where a partial order among parameters is needed. The idea of a bipolar picture fuzzy hypersoft set (BPFHSS) [16], introduced by Harl et al. in 2023, addresses the need to consider inconsistent, bipolar, and multiple sub-attribute information in decision-making processes.

For instance, several studies have proposed aggregation operators tailored for PFHSS to improve decision-making effectiveness in diverse contexts [17]. Similarly, divergence and similarity measures have been introduced to enrich the theoretical foundation and extend applications, such as in the symptomatic detection of COVID-19 [18], medical diagnosis [19], and pattern analysis [20].

Further advancements include the incorporation of T-spherical fuzzy hypersoft sets with specialized aggregation operators to expand their utility in soft computing [21], and the application of Schweizer–Sklar T-norm operators for PFHSS in sustainable technology and healthy social environments [22]. Moreover, novel evaluation frameworks have been developed by leveraging q-rung picture fuzzy hypersoft Schweizer–Sklar aggregation operators, thereby enhancing decision-making in intelligent transportation systems [23].

Matrices are extensively used in the fields of medicine, engineering, and information technology. Cagman et al. contributed to this area by introducing the concept of soft matrices (SM) [24] and applying them to decision-making problems. They further extended their work by introducing fuzzy soft matrices (FSM) [25] and discussing the fundamental operations associated with these matrices, as well as their advantageous properties. Saikia et al. extend the study of FSM by introducing the concept of generalized intuitionistic fuzzy soft matrices (IFSM) [26]. Broumi et al. explore the concept of an FSM, introducing concepts such as the fuzzy soft complement matrix (FSCM) [27] and a trace of the FSM. Additionally, they present a decision technique based on the FSM. Mondal et al. [28] a matrix that combines both fuzzy and intuitionistic fuzzy elements. This suggests an extension of the traditional soft matrix to incorporate not only fuzzy values but also intuitionistic fuzzy values. Picture fuzzy soft matrices (PFSM) [29] and their algebraic structures are examined by Arikrishnan and Sriram. In the era of sustainable development, the selection of renewable energy sources has become a critical decision-making challenge due to the involvement of multiple, conflicting, and uncertain factors. To address this complexity, methodologies have been developed utilizing picture fuzzy hypersoft information with choice and value matrices [30]

Picture fuzzy aggregation operators play a crucial role in combining information from different sources or attributes. Jana introduced Dombi aggregation operators [31] for picture fuzzy sets. Wei’s proposed picture fuzzy Hamacher aggregation Operators [32] and applied these operators to multi-attribute decision-making. Jana applied picture fuzzy Hamacher aggregation operators [33] to assess the best enterprise. Garg introduced picture fuzzy aggregation operators [34] based on a decreasing function that generates t-norm and t-conorm. Khan proposed picture fuzzy aggregation operators [35] using algebraic and Einstein t-norm. Luo introduced picture fuzzy geometric aggregation operators [36] based on a trapezoidal fuzzy number and applied them to multi-attribute decision-making. Ju proposed a picture fuzzy weighted interaction geometric operator [37] and applied it to the selection of addresses. Tian proposed weighted geometric aggregation operators [38] based on the Shapley fuzzy measure, fuzzy measure, and power aggregation operator. Wang proposed Muirhead mean operators [39] for picture fuzzy sets and applied them to the assessment of financial investment risk. Xu introduced a family of picture fuzzy Muirhead mean operators [40] and applied them to multi-attribute decision-making. Mahmood presents the basic principles of picture fuzzy soft power average aggregation operators and picture fuzzy soft power geometric aggregation operators [41].

1.1 Research Gap

Existing fuzzy, intuitionistic, picture fuzzy, and hypersoft matrices capture uncertainty only partially, either missing neutrality, sub-attributes, or a structured matrix calculus. Picture fuzzy models add neutrality but not sub-attributes, while hypersoft models refine attributes but omit neutrality/negativity. Our contribution introduces picture fuzzy hypersoft matrices, unifying both strengths with full algebraic operations and MCDM applications. Table 1 shows a comparison of existing models along with their limitations and our proposed contributions.

Table 1 Comparative summary of existing approaches and proposed PFHSM

1.2 Motivation

In real-world decision-making problems such as energy policy design, investment selection, medical diagnosis, and quality control, decision-makers often face environments with multiple attributes and sub-attributes. Existing models like PFSS and PFSM are useful for representing uncertainty but remain limited, as they cannot effectively capture hierarchical attribute (sub-attribute) structures or incorporate positive, neutral, and negative information simultaneously in complex environments. ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ extends these frameworks by providing a theoretical foundation for modeling higher order uncertainties; however, PFHSS are set-based and less suitable for computational efficiency, large-scale aggregation, and comparative analysis. To address these shortcomings, we propose the ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$, which reformulates PFHSS into a matrix structure, enabling faster computations, systematic aggregation, and improved handling of multiple alternatives and hierarchical parameters. The novelty of ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ lies in its ability to model complex decision environments more transparently and efficiently, offering greater flexibility and reliability than PFSS, PFSM, and PFHSS. The main contributions of this work include the formal development of PFHSM and its basic operations, the establishment of operational principles for ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ numbers, the introduction of geometric aggregation operators based on ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$, and their application to the design of energy policies as a practical demonstration. Furthermore, comparative analysis and sensitivity studies confirm that PFHSM produces more accurate, stable, and robust results than existing approaches, thereby offering a significant advancement in decision-making methodologies.

The article is organized in the following manner: In Sect. 2, we covered some basic definitions of the soft set, hypersoft set, picture fuzzy set, picture fuzzy hypersoft set, and power aggregation operators. Section 3 explores the basic concepts and operations of picture fuzzy hypersoft matrices. In Sect. 4 we covered the basic concepts of picture fuzzy hypersoft geometric aggregation operators. We developed the DM technique and provided an algorithm in the Sect. 5, along with a detailed example to show how to apply these newly developed concepts. In Sect. 6 the comparison of these conceptions with other existing notions is presented. The Sect. 7 contains concluding observations (Table 2).

2 Preliminaries

In this section, fundamental concepts that help in the development of picture fuzzy hypersoft matrices are briefly summarized.

Definition 1

(Picture Fuzzy Set) [3] A PFS on universe of discourse ${{\mathbb {R}^{\varrho }}}$ is defined as:

$$\begin{aligned} \textrm{S} = \{\langle \mu , \eta ,\nu \rangle | \ \ {\textbf{r}^{\varrho }}_{i} \in {\mathbb {R}^{\varrho }} \} \end{aligned}$$

where $\mu : {\mathbb {R}^{\varrho }} \longrightarrow [0, 1]$ denotes membership degree of ${\textbf{r}^{\varrho }}_{i}$ in $\textrm{S}$,

$\eta : {\mathbb {R}^{\varrho }} \longrightarrow [0, 1]$ is neutral degree of ${\textbf{r}^{\varrho }}_{i}$ in $\textrm{S}$, $\nu : {\mathbb {R}^{\varrho }} \longrightarrow [0, 1]$ is non-membership degree of ${\textbf{r}^{\varrho }}_{i}$ in $\textrm{S}$ satisfy $0 \le \mu + \eta + \nu \le 1$, $\rho = 1 - \mu - \eta -\nu$ represent the refusal degree of ${\textbf{r}^{\varrho }}_{i}$ in $\textrm{S}$, $\forall \ {\textbf{r}^{\varrho }}_{i} \in {\mathbb {R}^{\varrho }}$, $0 \le \rho \le 1$.

Definition 2

(Soft Set) [4] Let ${\mathbb {R}^{\varrho }}$ be a universe of discourse, $\mathcal {P}({\mathbb {R}^{\varrho }})$ be the power set of ${\mathbb {R}^{\varrho }}$ and ${\textbf{q}^{\varrho }}_{i}$ is a set of attributes. Then, the pair $(\mathbb {F}, {{{\textbf {h}}}_{i}^{\kappa }}),$ where $\mathbb {F}: {\textbf{h}^{\varrho }}_{i} \longrightarrow \mathcal {P}({\mathbb {R}^{\varrho }})$ is called a SS over ${\mathbb {R}^{\varrho }}$, also ${\textbf{h}^{\varrho }}_{i}$ is subset of ${\textbf{q}^{\varrho }}_{i}$.

Definition 3

(Picture Fuzzy Hyper Soft Set) [13] Let ${\mathbb {R}^{\varrho }}$ be a discourse universe, $\mathcal {P}(PF({\mathbb {R}^{\varrho }}))$ be the set of powers of PFS over ${\mathbb {R}^{\varrho }}$. Let ${{\textbf{q}^{\varrho }}_{i}} =\{ {{\textbf{q}^{\varrho }}_{1}}, {{\textbf{q}^{\varrho }}_{2}}, {{\textbf{q}^{\varrho }}_{3}}, {{\textbf{q}^{\varrho }}_{4}},\ldots ,{{\textbf{q}^{\varrho }}_{\beta }}\}$ be the set of $\beta$ disjoint parameters whose corresponding attribute values are ${{\textbf{Q}^{\varrho }}_{1}}, {{\textbf{Q}^{\varrho }}_{2}}, {{\textbf{Q}^{\varrho }}_{3}}, {{\textbf{Q}^{\varrho }}_{4}},\ldots ,{{\textbf{Q}^{\varrho }}_{\beta }}$. Suppose ${{\textbf{Q}^{\varrho }}}= {{\textbf{Q}^{\varrho }}_{1}} \times {{\textbf{Q}^{\varrho }}_{2}}\times {{\textbf{Q}^{\varrho }}_{3}}\times {{\textbf{Q}^{\varrho }}_{4}}\times \cdots \times {{\textbf{Q}^{\varrho }}_{\beta }}$ with ${{\textbf{Q}^{\varrho }}_{s}} \cap {{\textbf{Q}^{\varrho }}_{t}} = \emptyset$, $t \ne s$, and t, s $\in$ $\{1, 2,\ldots , \beta \}$. The pair $(\mathbb {F}, {{\textbf{Q}^{\varrho }}})$, where $\mathbb {F}$: ${{\textbf{Q}^{\varrho }}}$ $\rightarrow$ $\mathcal {P}(PF({\mathbb {R}^{\varrho }}))$ is called a PFHSS over ${\mathbb {R}^{\varrho }}$ and represented as follows; $(\mathbb {F}, {{\textbf{Q}^{\varrho }}}) = \mathbb {F}({{{\textbf {w}}}_{\varsigma }}_{s})=$ $\{\langle \mu , \eta ,\nu \rangle |\ \ {{\textbf{r}^{\varrho }}_{n}} \in {\mathbb {R}^{\varrho }} \}\ \forall \ \ {{\textbf{z}^{\varrho }}_{m}} \in {{\textbf{Q}^{\varrho }}}$

3 Picture Fuzzy Hypersoft Matrices

This section introduces the concept of picture fuzzy hypersoft matrices and its related operations, essentially laying the groundwork for a novel mathematical structural design.

Definition 4

Suppose the pair $(\mathbb {F}^{\varrho }, {\textbf{Q}^{\varrho }})$, is PFHSS over ${\mathbb {R}^{\varrho }}$. Then a subset of ${\mathbb {R}^{\varrho }} \times {\textbf{Q}^{\varrho }}$ is defined by ${\textbf{N}}^{{\textbf{Q}^{\varrho }}} = \{({\textbf{r}^{\varrho }}_{n}, {\textbf{z}^{\varrho }}_{m}): {\textbf{z}^{\varrho }}_{m} \in {\textbf{Q}^{\varrho }}, \ {\textbf{r}^{\varrho }}_{n} \in \mathbb {F}({\textbf{z}^{\varrho }}_{m}) \}$ is relation of $(\mathbb {F}^{\varrho }, {\textbf{Q}^{\varrho }})$. The characteristic function of ${\textbf{N}}^{{\textbf{Q}^{\varrho }}}$ is defined as $\complement _{f}({\textbf{r}^{\varrho }}_{n}, {\textbf{z}^{\varrho }}_{m}) = \{\langle \mu _{mn}, \eta _{mn}, \nu _{mn}\rangle \}$

If $\textbf{e}_{nm} = \complement _{f}({\textbf{r}^{\varrho }}_{n}, {\textbf{z}^{\varrho }}_{m})$ then the matrix (Matrix 1):

$$\begin{aligned} {\textbf{M}}_{H}^{\texttt {P}} = [ \textbf{e}_{nm}]_{\alpha \times \beta } = \left[ \begin{array}{cccc} \textbf{e}_{11} & \textbf{e}_{12} & ...... & \textbf{e}_{1\beta } \\ \textbf{e}_{21} & \textbf{e}_{22} & ...... & \textbf{e}_{2\beta } \\ \textbf{e}_{31} & \textbf{e}_{32} & ...... & \textbf{e}_{3\beta } \\ . & . & ...... & . \\ . & . & ...... & . \\ \textbf{e}_{\alpha 1} & \textbf{e}_{\alpha 2} & ...... & \textbf{e}_{\alpha \beta } \\ \end{array} \right] \end{aligned}$$

(1)

is called picture fuzzy hypersoft matrix (${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$) of the order $\alpha \times \beta$.

Example 1

: Suppose ${\mathbb {R}^{\varrho }} = \{{\textbf{r}^{\varrho }}_{1}, {\textbf{r}^{\varrho }}_{2}, {\textbf{r}^{\varrho }}_{3}\}$ be universe of discourse. Let ${\textbf{q}^{\varrho }}_1, {\textbf{q}^{\varrho }}_2,{\textbf{q}^{\varrho }}_{3}$ be the sets of attributes and their corresponding attributive values are, respectively, the set ${\textbf{Q}^{\varrho }}_1, {\textbf{Q}^{\varrho }}_2, {\textbf{Q}^{\varrho }}_3$. Where ${\textbf{Q}^{\varrho }}_1 = \{ b_{11}$, $b_{12}$, $b_{13}$, $b_{14} \}$${\textbf{Q}^{\varrho }}_2 = \{ b_{21}$, $b_{22}$, $b_{23}$, $b_{24}\}$${\textbf{Q}^{\varrho }}_3 = \{ b_{31}$, $b_{32}$, $b_{33}$, $b_{34}\}$.

Now ${{\textbf {Q}}}^{\varrho } = {\textbf{Q}^{\varrho }}_1 \times {\textbf{Q}^{\varrho }}_2 \times {\textbf{Q}^{\varrho }}_3$, there are sixty four possible outcomes but for simplicity we take only three.

$$\begin{aligned} {{\textbf {Q}}}^{\varrho }= & \left\{ \begin{array}{l}\ {\textbf{z}^{\varrho }}_1 = ( b_{11}, b_{21}, b_{32}, b_{44}),\\ {\textbf{z}^{\varrho }}_2 = (b_{14}, b_{24}, b_{33}, b_{43}),\\ {\textbf{z}^{\varrho }}_3 = ( b_{12}, b_{23}, b_{31}, b_{41}).\ \ \ \end{array}\right\} \\ (\mathbb {F}^{\varrho })({\textbf{z}^{\varrho }}_{1})= & \left\{ \begin{array}{l}\ \langle {\textbf{r}^{\varrho }}_1, (0.5, 0.1, 0.3)\rangle , \langle {\textbf{r}^{\varrho }}_2, (0.2, 0.2, 0.3)\rangle , \\ \langle {\textbf{r}^{\varrho }}_3, (0.4, 0.1, 0.3)\rangle \end{array}\right\} \\ (\mathbb {F}^{\varrho })({\textbf{z}^{\varrho }}_{2})= & \left\{ \begin{array}{l}\ \langle {\textbf{r}^{\varrho }}_1, (0.6, 0.2, 0.2)\rangle , \langle {\textbf{r}^{\varrho }}_2, (0.3, 0.4, 0.2)\rangle , \\ \langle {\textbf{r}^{\varrho }}_3, (0.5, 0.2, 0.2)\rangle \end{array}\right\} \\ (\mathbb {F}^{\varrho })({\textbf{z}^{\varrho }}_{3})= & \left\{ \begin{array}{l}\ \langle {\textbf{r}^{\varrho }}_1, (0.7, 0.3, 0.0)\rangle , \langle {\textbf{r}^{\varrho }}_2, (0.4, 0.5, 0.1)\rangle , \\ \langle {\textbf{r}^{\varrho }}_3, (0.6, 0.3, 0.1)\rangle \end{array}\right\} \end{aligned}$$

Then the matrix:

${\textbf{M}}_{H}^{\texttt {P}} = [ \textbf{e}_{nm}]_{3 \times 3}= \left[ \begin{array}{cccc} & {\textbf{z}^{\varrho }}_{1}& {\textbf{z}^{\varrho }}_{2}& {\textbf{z}^{\varrho }}_{3} \\ {\textbf{r}^{\varrho }}_{1}& (0.5, 0.1, 0.3) & (0.6, 0.2, 0.2) & (0.7, 0.3, 0.0) \\ {\textbf{r}^{\varrho }}_{2}& (0.2, 0.2, 0.3) & (0.3, 0.4, 0.2) & (0.4, 0.5, 0.1) \\ {\textbf{r}^{\varrho }}_{3}& (0.4, 0.1, 0.3) & (0.5, 0.2, 0.2) & (0.6, 0.3, 0.1) \\ \end{array} \right]$ is ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ of order $3 \times 3$.

A diagrammatic representation of PFHSM formation is provided in Fig. 1, using Example 1 for clarity.

Fig. 1

Diagrammatic representation of picture fuzzy hypersoft matrix formation

Definition 5

The picture fuzzy hypersoft row matrices serve to capture the sub-attribute values across multiple attributes for a single decision entity. They are essential for entity-wise evaluation, enabling structured comparisons and ranking of alternatives in applications such as employee performance evaluation or investment selection.

A ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ of order $1 \times \beta$ is called picture fuzzy hypersoft row matrix (${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}^{ro}}$), defined as:

$$\begin{aligned} {\textbf{M}}_{H}^{\texttt {P}} = [ \textbf{e}_{nm}]_{1 \times \beta } = \left[ \begin{array}{cccc} \textbf{e}_{11} & \textbf{e}_{12} & ...... & \textbf{e}_{1\beta } \\ \end{array} \right] \end{aligned}$$

For example: ${\textbf{M}}_{H}^{\texttt {P}} = [ \textbf{e}_{nm}]_{1 \times 4} = \left[ \begin{array}{ccccc} (0.5, 0.2, 0.1)& (0.5, 0.3, 0.1)& (0.3, 0.5, 0.1)& (0.4, 0.3, 0.2)\\ \end{array} \right]$ is ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}^{ro}}$ of order $1 \times \beta$.

Definition 6

The picture fuzzy hypersoft column matrices are used to represent the set of sub-attributes corresponding to a single attribute. They facilitate attribute-wise aggregation and simplify computations when combining information across different sub-attributes, which is particularly useful in MCDM problems.

A ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ of order $\alpha \times 1$ then ${\textbf{M}}_{H}^{\texttt {P}}$ is called picture fuzzy hypersoft column matrix(${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}^{co}}$ ), defined as (Matrix 2):

$$\begin{aligned} {\textbf{M}}_{H}^{\texttt {P}} = [ \textbf{e}_{nm}]_{\alpha \times 1} = \left[ \begin{array}{c} \textbf{e}_{11} \\ \textbf{e}_{21} \\ \textbf{e}_{31}\\ .\\ .\\ .\\ .\\ \textbf{e}_{\alpha 1} \\ \end{array} \right] \end{aligned}$$

(2)

For example (Matrix 3),

$$\begin{aligned} {\textbf{M}}_{H}^{\texttt {P}} = [ \textbf{e}_{nm}]_{4 \times 1} = \left[ \begin{array}{c} (0.5, 0.2, 0.1)\\ (0.5, 0.3, 0.1)\\ (0.3, 0.5, 0.1)\\ (0.4, 0.3, 0.2)\\ \end{array} \right] \end{aligned}$$

(3)

is ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}^{co}}$ of order $\alpha \times 1$.

Definition 7

The picture fuzzy square matrices are employed when the number of attributes(sub-attributes ) equals the number of alternatives, allowing full pairwise interactions. They are particularly suitable for computing similarity or distance measures in PFHSM-based TOPSIS or VIKOR methods.

If numbers of rows is equal to numbers of columns in a ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ of order $\alpha \times \beta$ then ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ is called picture fuzzy hypersoft square matrix(${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}^{sq}}$). It means $\alpha = \beta$.

For example ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}^{sq}}$ of order $4 \times 4$ is given as (Matrix 4):

$$\begin{aligned} {\textbf{M}}_{H}^{\texttt {P}} = [ \textbf{e}_{nm}]_{4 \times 4} = \left[ \begin{array}{ccccc} (0.5, 0.2, 0.1)& (0.5, 0.3, 0.1)& (0.3, 0.5, 0.1)& (0.4, 0.3, 0.2)\\ (0.7, 0.1, 0.1)& (0.2, 0.5, 0.2)& (0.6, 0.1, 0.2)& (0.4, 0.1, 0.3) \\ (0.5, 0.2, 0.2)& (0.6, 0.2, 0.1)& (0.3, 0.4, 0.1)& (0.5, 0.2, 0.2)\\ (0.4, 0.4, 0.1)& (0.6, 0.2, 0.1)& (0.5, 0.3, 0.2)& (0.5, 0.3, 0.1)\\ \end{array} \right] \end{aligned}$$

(4)

Definition 8

The picture fuzzy rectangular matrices arise when the number of attributes and alternatives differ. These matrices enable modeling heterogeneous attribute-sub-attribute structures, which is common in real-world scenarios like energy policy design, where different factors may have varying numbers of sub-factors.

If numbers of rows does not equal to numbers of columns in a ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ of order $\alpha \times \beta$ then ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ is called picture fuzzy hypersoft rectangular matrix (${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}^{rt}}$). It means $\alpha \ne \beta$.

For example ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}^{rt}}$ of order $3 \times 4$ is as (Matrix 5):

$$\begin{aligned} {\textbf{M}}_{H}^{\texttt {P}} = [ \textbf{e}_{nm}]_{3 \times 4} = \left[ \begin{array}{ccccc} (0.5, 0.2, 0.1)& (0.5, 0.3, 0.1)& (0.3, 0.5, 0.1)& (0.4, 0.3, 0.2)\\ (0.7, 0.1, 0.1)& (0.2, 0.5, 0.2)& (0.6, 0.1, 0.2)& (0.4, 0.1, 0.3) \\ (0.5, 0.2, 0.2)& (0.6, 0.2, 0.1)& (0.3, 0.4, 0.1)& (0.5, 0.2, 0.2)\\ \end{array} \right] \end{aligned}$$

(5)

Definition 9

Suppose ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ of order $\alpha \times \beta$ then the transpose of ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ is denoted by $[{\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}]^{t }$ and is defined as: $[{\textbf{M}}_{H}^{\texttt {P}}]^{t }$ = $[ \textbf{e}_{nm}]_{\alpha \times \beta }^{t }$ = $[ \textbf{e}_{mn}]_{\beta \times \alpha }$. In other words rows and columns of ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ are interchanged (Matrix 6).

$$\begin{aligned} {[}{\textbf{M}}_{H}^{\texttt {P}}]^{t } = [[ \textbf{e}_{nm}]_{\alpha \times \beta }]^{t } = \left[ \begin{array}{cccc} \textbf{e}_{11} & \textbf{e}_{12} & ...... & \textbf{e}_{1\beta } \\ \textbf{e}_{21} & \textbf{e}_{22} & ...... & \textbf{e}_{2\beta } \\ \textbf{e}_{31} & \textbf{e}_{32} & ...... & \textbf{e}_{3\beta } \\ . & . & ...... & . \\ . & . & ...... & . \\ \textbf{e}_{\alpha 1} & \textbf{e}_{\alpha 2} & ...... & \textbf{e}_{\alpha \beta } \\ \end{array} \right] ^{t } = \left[ \begin{array}{cccc} \textbf{e}_{11} & \textbf{e}_{21} & ...... & \textbf{e}_{\alpha 1} \\ \textbf{e}_{12} & \textbf{e}_{22} & ...... & \textbf{e}_{\alpha 2} \\ \textbf{e}_{13} & \textbf{e}_{23} & ...... & \textbf{e}_{\alpha 3} \\ . & . & ...... & . \\ . & . & ...... & . \\ \textbf{e}_{1\beta } & \textbf{e}_{2\beta } & ...... & \textbf{e}_{\alpha \beta } \\ \end{array} \right] \end{aligned}$$

(6)

For example, (Matrix 7):

$$\begin{aligned} {[}{\textbf{M}}_{H}^{\texttt {P}}]^{t }&= [[ \textbf{e}_{nm}]_{4 \times 4}]^{t } = \left[ \begin{array}{ccccc} (0.5, 0.2, 0.1)& (0.5, 0.3, 0.1)& (0.3, 0.5, 0.1)& (0.4, 0.3, 0.2)\\ (0.7, 0.1, 0.1)& (0.2, 0.5, 0.2)& (0.6, 0.1, 0.2)& (0.4, 0.1, 0.3) \\ (0.5, 0.2, 0.2)& (0.6, 0.2, 0.1)& (0.3, 0.4, 0.1)& (0.5, 0.2, 0.2)\\ (0.4, 0.4, 0.1)& (0.6, 0.2, 0.1)& (0.5, 0.3, 0.2)& (0.5, 0.3, 0.1)\\ \end{array} \right] ^{t } \nonumber \\&= \left[ \begin{array}{ccccc} (0.5, 0.2, 0.1)& (0.7, 0.1, 0.1)& (0.5, 0.2, 0.2)& (0.4, 0.4, 0.1)\\ (0.5, 0.3, 0.1)& (0.2, 0.5, 0.2)& (0.6, 0.2, 0.1)& (0.6, 0.2, 0.1) \\ (0.3, 0.5, 0.1)& (0.6, 0.1, 0.2)& (0.3, 0.4, 0.1)& (0.5, 0.3, 0.2)\\ (0.4, 0.3, 0.2) & (0.4, 0.1, 0.3)& (0.5, 0.2, 0.2)& (0.5, 0.3, 0.1)\\ \end{array} \right] \end{aligned}$$

(7)

Definition 10

Suppose ${\textbf{M}}_{H}^{\texttt {P}}$ is ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}^{sq}}$ of order $\alpha \times \beta$ then ${\textbf{M}}_{H}^{\texttt {P}}$ is symmetric if $[{\textbf{M}}_{H}^{\texttt {P}}]^{t }$ = ${\textbf{M}}_{H}^{\texttt {P}}$.

For example, (Matrix 8):

$$\begin{aligned} {[}{\textbf{M}}_{H}^{\texttt {P}}] = \left[ \begin{array}{ccccc} (0.5, 0.2, 0.1)& (0.7, 0.1, 0.1)& (0.5, 0.2, 0.2)& (0.4, 0.4, 0.1)\\ (0.7, 0.1, 0.1)& (0.2, 0.5, 0.2)& (0.6, 0.2, 0.1)& (0.6, 0.2, 0.1) \\ (0.5, 0.2, 0.2)& (0.6, 0.2, 0.1)& (0.3, 0.4, 0.1)& (0.5, 0.3, 0.2)\\ (0.4, 0.4, 0.1) & (0.6, 0.2, 0.1)& (0.5, 0.3, 0.2)& (0.5, 0.3, 0.1)\\ \end{array} \right] \end{aligned}$$

(8)

is symmetric.

Definition 11

Scalar multiplication or scaling represents adjusting the weight or importance of an attribute in decision-making. For example, in investment selection, a high-priority factor such as return on investment can be scaled to reflect its greater influence on the final evaluation.

Suppose ${\textbf{M}}_{H}^{\texttt {P}}$ is ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ of order $\alpha \times \beta$ and $\Re$ be any scalar such that $0 \le \Re \le 1$, then the matrix $\Re {\textbf{M}}_{H}^{\texttt {P}}$ is formed by multiplying each element of matrix ${\textbf{M}}_{H}^{\texttt {P}}$ by $\Re$.

$\Re [{\textbf{M}}_{H}^{\texttt {P}}]$ = $[\Re [\textbf{e}_{mn}]_{4 \times 4}]$. Suppose $\Re = 0.5$ and ${\textbf{M}}_{H}^{\texttt {P}}$ is given as (Matrix 9):

$$\begin{aligned} { [}{\textbf{M}}_{H}^{\texttt {P}}] = \left[ \begin{array}{ccccc} (0.5, 0.2, 0.1)& (0.7, 0.1, 0.1)& (0.5, 0.2, 0.2)& (0.4, 0.4, 0.1)\\ (0.5, 0.3, 0.1)& (0.2, 0.5, 0.2)& (0.6, 0.2, 0.1)& (0.6, 0.2, 0.1) \\ (0.3, 0.5, 0.1)& (0.6, 0.1, 0.2)& (0.3, 0.4, 0.1)& (0.5, 0.3, 0.2)\\ (0.4, 0.3, 0.2) & (0.4, 0.1, 0.3)& (0.5, 0.2, 0.2)& (0.5, 0.3, 0.1)\\ \end{array} \right] \end{aligned}$$

(9)

then $\Re [{\textbf{M}}_{H}^{\texttt {P}}]$ is (Matrix 10):

$$\begin{aligned} \Re [{\textbf{M}}_{H}^{\texttt {P}}] = \left[ \begin{array}{ccccc} (0.25, 0.1, 0.05)& (0.35, 0.1, 0.1)& (0.5, 0.2, 0.2)& (0.4, 0.4, 0.1)\\ (0.5, 0.3, 0.1)& (0.2, 0.5, 0.2)& (0.6, 0.2, 0.1)& (0.6, 0.2, 0.1) \\ (0.3, 0.5, 0.1)& (0.6, 0.1, 0.2)& (0.3, 0.4, 0.1)& (0.5, 0.3, 0.2)\\ (0.4, 0.3, 0.2) & (0.4, 0.1, 0.3)& (0.5, 0.2, 0.2)& (0.5, 0.3, 0.1)\\ \end{array} \right] \end{aligned}$$

(10)

Definition 12

Suppose ${\textbf{M}}_{H}^{\texttt {P}} = [\textbf{e}_{nm}] = \langle {\mu _{mn}}_{1}, {\eta _{mn}}_{1}, {\nu _{mn}}_{1}\rangle$ of order $\alpha \times \beta$ is ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ then compliment of $[{\textbf{M}}_{H}^{\texttt {P}}]^{c}= [\textbf{e}_{nm}]^{c} = \langle {\nu _{mn}}_{1}, {\eta _{mn}}_{1}, {\mu _{mn}}_{1}\rangle$.

Definition 13

Suppose ${\textbf{M}}_{H}^{\texttt {P}}$ is ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ of order $\alpha \times \beta$ is called picture fuzzy hypersoft null matrix(${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}^{nl}}$) if $\textbf{e}_{mn} = [0.0, 0.0, 1.0]$ for all m and n.

For example, ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}^{nl}}$ of order $\alpha \times \beta$ (Matrix 11).

$$\begin{aligned} {[}{\textbf{M}}_{H}^{\texttt {P}}] = \left[ \begin{array}{ccccc} (0.0, 0.0, 1.0)& (0.0, 0.0, 1.0)& (0.0, 0.0, 1.0)& (0.0, 0.0, 1.0)\\ (0.0, 0.0, 1.0)& (0.0, 0.0, 1.0)& (0.0, 0.0, 1.0)& (0.0, 0.0, 1.0) \\ (0.0, 0.0, 1.0)& (0.0, 0.0, 1.0)& (0.0, 0.0, 1.0)& (0.0, 0.0, 1.0)\\ (0.0, 0.0, 1.0) & (0.0, 0.0, 1.0)& (0.0, 0.0, 1.0)& (0.0, 0.0, 1.0)\\ \end{array} \right] \end{aligned}$$

(11)

Definition 14

Suppose ${\textbf{M}}_{H}^{\texttt {P}}$ is ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}^{sq}}$ then the elements $[\textbf{e}_{mn}]$ s.t $m = n$ is called a principal diagonal.

$$\begin{aligned} {\textbf{M}}_{H}^{\texttt {P}} = [ \textbf{e}_{nm}]_{\alpha \times \beta } = \left[ \begin{array}{cccc} \textbf{e}_{11} & \textbf{e}_{12} & ...... & \textbf{e}_{1\beta } \\ \textbf{e}_{21} & \textbf{e}_{22} & ...... & \textbf{e}_{2\beta } \\ \textbf{e}_{31} & \textbf{e}_{32} & ...... & \textbf{e}_{3\beta } \\ . & . & ...... & . \\ . & . & ...... & . \\ \textbf{e}_{\alpha 1} & \textbf{e}_{\alpha 2} & ...... & \textbf{e}_{\alpha \beta } \\ \end{array} \right] \end{aligned}$$

The elements $\textbf{e}_{11}, \textbf{e}_{22}, \textbf{e}_{33},................\textbf{e}_{\alpha \alpha }$ are principal diagonal.

Definition 15

The picture fuzzy diagonal matrices isolate the main attribute–alternatives contributions by placing non-zero elements along the diagonal. This is useful for weighted aggregation and computational simplification, allowing focus on the most significant components without losing structural integrity.

Suppose ${\textbf{M}}_{H}^{\texttt {P}}$ is ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}^{sq}}$ of order $\alpha \times \beta$ is called picture fuzzy hypersoft diagonal matrix (${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}^{dg}}$ ) if any one the element $\textbf{e}_{nm}$ s.t $n= m$ is non zero and all $\textbf{e}_{nm}$ are zero when $n \ne m$. For example, ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}^{sq}}$ is given as (Matrix 12):

$$\begin{aligned} {[}{\textbf{M}}_{H}^{\texttt {P}}] = \left[ \begin{array}{ccccc} (0.5, 0.2, 0.1)& (0.0, 0.0, 1.0)& (0.0, 0.0, 1.0)& (0.0, 0.0, 1.0)\\ (0.0, 0.0, 1.0)& (0.2, 0.5, 0.2)& (0.0, 0.0, 1.0)& (0.0, 0.0, 1.0)) \\ (0.0, 0.0, 1.0)& (0.0, 0.0, 1.0)& (0.3, 0.4, 0.1)& (0.0, 0.0, 1.0)\\ (0.0, 0.0, 1.0) & (0.0, 0.0, 1.0)& (0.0, 0.0, 1.0)& (0.5, 0.2, 0.2)\\ \end{array} \right] \end{aligned}$$

(12)

is ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}^{dg}}$.

Definition 16

The picture fuzzy scalar matrices, containing a single repeated element along the diagonal, facilitate uniform scaling or normalization operations across all attributes and alternatives. They are helpful in standardizing ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ entries before aggregation or distance computation.

Suppose ${\textbf{M}}_{H}^{\texttt {P}}$ is ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}^{sq}}$ of order $\alpha \times \beta$ is called picture fuzzy hypersoft scalar matrix (${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}^{sc}}$ ) if $\textbf{e}_{nm} = [\mu _{mn}, \eta _{mn}, \nu _{mn}]$ s.t $n = m$ and all $\textbf{e}_{nm} = [0.0, 0.0, 1.0]$ $m \ne n$.

For example, (Matrix 13):

$$\begin{aligned} {[}{\textbf{M}}_{H}^{\texttt {P}}] =\left[ \begin{array}{ccccc} (0.5, 0.2, 0.2)& (0.0, 0.0, 1.0)& (0.0, 0.0, 1.0)& (0.0, 0.0, 1.0)\\ (0.0, 0.0, 1.0)& (0.5, 0.2, 0.2)& (0.0, 0.0, 1.0)& (0.0, 0.0, 1.0)) \\ (0.0, 0.0, 1.0)& (0.0, 0.0, 1.0)& (0.5, 0.2, 0.2)& (0.0, 0.0, 1.0)\\ (0.0, 0.0, 1.0) & (0.0, 0.0, 1.0)& (0.0, 0.0, 1.0)& (0.5, 0.2, 0.2)\\ \end{array} \right] \end{aligned}$$

(13)

is ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}^{sc}}$.

Definition 17

The picture fuzzy identity matrices serve as neutral elements in matrix operations, particularly in addition and multiplication. They ensure that matrix operations preserve the original PFHSM information when combining multiple matrices, which is crucial for iterative decision algorithms and preserving consistency.

Suppose ${\textbf{M}}_{H}^{\texttt {P}}$ is ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}^{sq}}$ of order $\alpha \times \beta$ is called picture fuzzy hypersoft identity matrix (${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}^{id}}$ ) if $\textbf{e}_{nm} = (1.0, 0.0, 0.0)$ s.t $n = m$ and all $\textbf{e}_{nm} = (0.0, 0.0, 1.0)$ are zero when $n \ne m$.

For example, (Matrix 14):

$$\begin{aligned} {[}{\textbf{M}}_{H}^{\texttt {P}}] = \left[ \begin{array}{ccccc} (1.0, 0.0, 0.0)& (0.0, 0.0, 1.0)& (0.0, 0.0, 1.0)& (0.0, 0.0, 1.0)\\ (0.0, 0.0, 1.0)& (1.0, 0.0, 0.0)& (0.0, 0.0, 1.0)& (0.0, 0.0, 1.0)) \\ (0.0, 0.0, 1.0)& (0.0, 0.0, 1.0)& (1.0, 0.0, 0.0)& (0.0, 0.0, 1.0)\\ (0.0, 0.0, 1.0) & (0.0, 0.0, 1.0)& (0.0, 0.0, 1.0)& (1.0, 0.0, 0.0)\\ \end{array} \right] \end{aligned}$$

(14)

is ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}^{id}}$ of order $4 \times 4$.

Definition 18

Suppose ${\textbf{M}}_{H_{1}}^{\texttt {P}} = [\textbf{e}_{nm}] = \langle {\mu _{mn}}_{1}, {\eta _{mn}}_{1}, {\nu _{mn}}_{1}\rangle$ of order $\alpha \times \beta$ is picture fuzzy hypersoft sub matrix (${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}^{sub}}$ ) of ${\textbf{M}}_{H_{2}}^{\texttt {P}}= [\textbf{f}_{nm}] = \langle {\mu _{mn}}_{2}, {\eta _{mn}}_{2}, {\nu _{mn}}_{2}\rangle$ of order $\alpha \times \beta$ if

$$\begin{aligned} & {\mu _{mn}}_{1} \le {\mu _{mn}}_{2}, \ \ {\eta _{mn}}_{1} \le {\eta _{mn}}_{2}, \ \ {\nu _{mn}}_{1} \ge {\nu _{mn}}_{2}\nonumber \\ & {\textbf{M}}_{H_{1}}^{\texttt {P}} = \left[ \begin{array}{ccccc} (0.5, 0.1, 0.4)& (0.4, 0.2, 0.4)& (0.6, 0.1, 0.3)& (0.3, 0.3, 0.2)\\ (0.4, 0.2, 0.3)& (0.3, 0.3, 0.2)& (0.4, 0.1, 0.3)& (0.5, 0.2, 0.3)\\ (0.6, 0.1, 0.3)& (0.8, 0.1, 0.1)& (0.2, 0.5, 0.3))& (0.4, 0.3, 0.3)\\ (0.4, 0.1, 0.3)& (0.4, 0.2, 0.4)& (0.5, 0.1, 0.3)& (0.3, 0.3, 0.3) \\ \end{array} \right] \end{aligned}$$

(15)

$$\begin{aligned} & {\textbf{M}}_{H_{2}}^{\texttt {P}} = \left[ \begin{array}{ccccc} (0.6, 0.2, 0.2)& (0.5, 0.3, 0.1)& (0.7, 0.3, 0.1)& (0.5, 0.4, 0.1)\\ (0.5, 0.3, 0.2)& (0.5, 0.4, 0.1)& (0.5, 0.2, 0.2)& (0.6, 0.3, 0.2) \\ (0.7, 0.2, 0.2)& (0.8, 0.1, 0.1)& (0.3, 0.6, 0.2)& (0.5, 0.4, 0.1)\\ (0.5, 0.2, 0.2)& (0.5, 0.3, 0.1)& (0.6, 0.2, 0.2)& (0.4, 0.4, 0.2)\\ \end{array} \right] \end{aligned}$$

(16)

Hence ${\textbf{M}}_{H_{1}}^{\texttt {P}}$ (Matrix 15) is ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}^{sub}}$ of ${\textbf{M}}_{H_{2}}^{\texttt {P}}$ (Matrix 16).

Definition 19

Addition of ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ entries can be interpreted as aggregating multiple expert opinions on the same attribute–alternatives pair. For instance, when evaluating employee performance, each supervisor provides a positive, neutral, and negative assessment. Addition combines these inputs to produce an overall evaluation reflecting collective judgment.

Suppose ${\textbf{M}}_{H_{1}}^{\texttt {P}} = [\textbf{e}_{nm}] = \langle {\mu _{mn}}_{1}, {\eta _{mn}}_{1}, {\nu _{mn}}_{1}\rangle$ and ${\textbf{M}}_{H_{2}}^{\texttt {P}}= [\textbf{f}_{nm}] = \langle {\mu _{mn}}_{2}, {\eta _{mn}}_{2}, {\nu _{mn}}_{2}\rangle$ be two ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ of order $\alpha \times \beta$ then addition of ${\textbf{M}}_{H_{1}}^{\texttt {P}}$ and ${\textbf{M}}_{H_{2}}^{\texttt {P}}$ is denoted by ${\textbf{M}}_{H_{3}}^{\texttt {P}} = [\textbf{g}_{nm}] = \langle {\mu _{mn}}_{3}, {\eta _{mn}}_{3}, {\nu _{mn}}_{3}\rangle$ of order $\alpha \times \beta$ and defined as ${\textbf{M}}_{H_{3}}^{\texttt {P}}$ = ${\textbf{M}}_{H_{1}}^{\texttt {P}}$ + ${\textbf{M}}_{H_{2}}^{\texttt {P}}$

$[\textbf{g}_{nm}] = [\textbf{e}_{nm}] + [\textbf{f}_{nm}]$ = (max $[{\mu _{mn}}_{1}, {\mu _{mn}}_{2}]$, min $[{\eta _{mn}}_{1}, {\eta _{mn}}_{2}]$, min $[{\nu _{mn}}_{1}, {\nu _{mn}}_{2}])$

For example, ${\textbf{M}}_{H_{1}}^{\texttt {P}}$ (Matrix 17) and ${\textbf{M}}_{H_{2}}^{\texttt {P}}$ (Matrix 18) be two ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ then addition is given in ${\textbf{M}}_{H_{3}}^{\texttt {P}}$ (Matrix 19).

$$\begin{aligned} & {\textbf{M}}_{H_{1}}^{\texttt {P}} = \left[ \begin{array}{ccccc} (0.9, 0.0, 0.1) & (0.2, 0.1, 0.7) & (0.6, 0.3, 0.1)& (0.7, 0.1, 0.1)\\ (0.3, 0.3, 0.3) & (0.4, 0.2, 0.4)& (0.5, 0.3, 0.1) & (0.1, 0.1, 0.8) \\ (0.6, 0.1, 0.1)& (0.7, 0.1, 0.2)& (0.5, 0.2, 0.1)& (0.4, 0.1, 0.4) \\ (0.1, 0.4, 0.2)& (0.2, 0.2, 0.2)& (0.5, 0.1, 0.1)& (0.4, 0.1, 0.2)\\ \end{array} \right] \end{aligned}$$

(17)

$$\begin{aligned} & {\textbf{M}}_{H_{2}}^{\texttt {P}} = \left[ \begin{array}{ccccc} (0.3, 0.3, 0.4)& (0.6, 0.2, 0.1)& (0.6, 0.1, 0.3)& (0.3, 0.5, 0.2)\\ (0.7, 0.2, 0.1)& (0.3, 0.3, 0.2)& (0.4, 0.1, 0.3)& (0.5, 0.3, 0.1)\\ (0.6, 0.2, 0.2)& (0.8, 0.1, 0.1)& (0.2, 0.6, 0.2))& (0.2, 0.3, 0.4)\\ (0.4, 0.1, 0.2)& (0.4, 0.1, 0.4)& (0.5, 0.1, 0.2)& (0.3, 0.3, 0.3) \\ \end{array} \right] \end{aligned}$$

(18)

$$\begin{aligned} & {\textbf{M}}_{H_{3}}^{\texttt {P}} = \left[ \begin{array}{ccccc} (0.9, 0.0, 0.1)& (0.6, 0.1, 0.1)& (0.6, 0.1, 0.1)& (0.7, 0.1, 0.1)\\ (0.7, 0.2, 0.1)& (0.4, 0.2, 0.2)& (0.5, 0.1, 0.1)& (0.5, 0.1, 0.1) \\ (0.6, 0.1, 0.1)& (0.8, 0.1, 0.1)& (0.5, 0.2, 0.1)& (0.4, 0.1, 0.4)\\ (0.4, 0.1, 0.2)& (0.4, 0.1, 0.2)& (0.5, 0.1, 0.1)& (0.4, 0.1, 0.2)\\ \end{array} \right] \end{aligned}$$

(19)

Definition 20

Multiplication models the compounding effect of interdependent criteria. In investment analysis, if two financial indicators influence each other (e.g., market risk and liquidity), multiplication captures how their combined effect impacts the final decision.

Suppose ${\textbf{M}}_{H_{1}}^{\texttt {P}} = [\textbf{e}_{nm}] = \langle {\mu _{mn}}_{1}, {\eta _{mn}}_{1}, {\nu _{mn}}_{1}\rangle$ of order $\alpha \times \beta$ and ${\textbf{M}}_{H_{2}}^{\texttt {P}}= [\textbf{f}_{mk}] = \langle {\mu _{mn}}_{2}, {\eta _{mn}}_{2}, {\nu _{mn}}_{2}\rangle$ of order $\beta \times \gamma$ be two ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ then multiplication of ${\textbf{M}}_{H_{1}}^{\texttt {P}}$ and ${\textbf{M}}_{H_{2}}^{\texttt {P}}$ is denoted by ${\textbf{M}}_{H_{3}}^{\texttt {P}} = [\textbf{g}_{nk}] = \langle {\mu _{mn}}_{3}, {\eta _{mn}}_{3}, {\nu _{mn}}_{3}\rangle$ of order $\alpha \times \gamma$ and defined as ${\textbf{M}}_{H_{3}}^{\texttt {P}}$ = ${\textbf{M}}_{H_{1}}^{\texttt {P}} \circledast {\textbf{M}}_{H_{2}}^{\texttt {P}}$ $[\textbf{g}_{nk}] = [\textbf{e}_{nm}] \circledast [\textbf{f}_{mk}]$ = $\{$max$_{m}$ [ min$_{m}$ $({\mu _{mn}}_{1}, {\mu _{mn}}_{2})$], min$_{m}$[max$_{m}$ $[{\eta _{mn}}_{1}, {\eta _{mn}}_{2}]$], min$_{m}$[max$_{m}$ $[{\nu _{mn}}_{1}, {\nu _{mn}}_{2}]]\}$. For example, ${\textbf{M}}_{H_{1}}^{\texttt {P}}$ (Matrix 20) and ${\textbf{M}}_{H_{2}}^{\texttt {P}}$ (Matrix 21) be two ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ then multiplication is given in ${\textbf{M}}_{H_{3}}^{\texttt {P}}$ (Matrix 22).

$$\begin{aligned} & {\textbf{M}}_{H_{1}}^{\texttt {P}} = \left[ \begin{array}{ccccc} (0.9, 0.0, 0.1) & (0.2, 0.1, 0.7) & (0.6, 0.3, 0.1)& (0.7, 0.1, 0.1)\\ (0.3, 0.3, 0.3) & (0.4, 0.2, 0.4)& (0.5, 0.3, 0.1) & (0.1, 0.1, 0.8) \\ (0.6, 0.1, 0.1)& (0.7, 0.1, 0.2)& (0.5, 0.2, 0.1)& (0.4, 0.1, 0.4) \\ (0.1, 0.4, 0.2)& (0.2, 0.2, 0.2)& (0.5, 0.1, 0.1)& (0.4, 0.1, 0.2)\\ \end{array} \right] \end{aligned}$$

(20)

$$\begin{aligned} & {\textbf{M}}_{H_{2}}^{\texttt {P}} = \left[ \begin{array}{ccccc} (0.3, 0.3, 0.4)& (0.6, 0.2, 0.1)& (0.6, 0.1, 0.3)& (0.3, 0.5, 0.2)\\ (0.7, 0.2, 0.1)& (0.3, 0.3, 0.2)& (0.4, 0.1, 0.3)& (0.5, 0.3, 0.1)\\ (0.6, 0.2, 0.2)& (0.8, 0.1, 0.1)& (0.2, 0.6, 0.2))& (0.2, 0.3, 0.4)\\ (0.4, 0.1, 0.2)& (0.4, 0.1, 0.4)& (0.5, 0.1, 0.2)& (0.3, 0.3, 0.3) \\ \end{array} \right] \end{aligned}$$

(21)

$$\begin{aligned} & {\textbf{M}}_{H_{3}}^{\texttt {P}} = \left[ \begin{array}{ccccc} (0.6, 0.1, 0.2)& (0.6, 0.1, 0.1)& (0.6, 0.1, 0.1)& (0.3, 0.3, 0.2)\\ (0.5, 0.1, 0.2)& (0.5, 0.1, 0.1)& (0.4, 0.1, 0.2)& (0.4, 0.3, 0.3) \\ (0.7, 0.1, 0.2)& (0.6, 0.1, 0.1)& (0.6, 0.1, 0.2)& (0.5, 0.3, 0.2)\\ (0.5, 0.1, 0.2)& (0.5, 0.1, 0.1)& (0.4, 0.1, 0.2)& (0.3, 0.3, 0.2)\\ \end{array} \right] \end{aligned}$$

(22)

Definition 21

The union operation represents optimistic or inclusive aggregation, taking the best-case evaluation among alternatives. For example, in energy policy design, union can model the scenario where a site satisfies at least one favorable environmental sub-criterion, highlighting potential locations for further consideration.

Suppose ${\textbf{M}}_{H_{1}}^{\texttt {P}} = [\textbf{e}_{nm}] = \langle {\mu _{mn}}_{1}, {\eta _{mn}}_{1}, {\nu _{mn}}_{1}\rangle$ of order $\alpha \times \beta$ and ${\textbf{M}}_{H_{2}}^{\texttt {P}}= [\textbf{f}_{nm}] = \langle {\mu _{mn}}_{2}, {\eta _{mn}}_{2}, {\nu _{mn}}_{2}\rangle$ of order $\alpha \times \beta$ be two ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ then union of ${\textbf{M}}_{H_{1}}^{\texttt {P}}$ and ${\textbf{M}}_{H_{2}}^{\texttt {P}}$ is denoted by ${\textbf{M}}_{H_{3}}^{\texttt {P}} = [\textbf{g}_{nk}] = \langle {\mu _{mn}}_{3}, {\eta _{mn}}_{3}, {\nu _{mn}}_{3}\rangle$ of order $\alpha \times \beta$ and defined as ${\textbf{M}}_{H_{3}}^{\texttt {P}} = {\textbf{M}}_{H_{1}}^{\texttt {P}} \cup {\textbf{M}}_{H_{2}}^{\texttt {P}}$

$[\textbf{g}_{nm}] = [\textbf{e}_{nm}] \cup [\textbf{f}_{nm}]$ = $\{$max$[({\mu _{mn}}_{1}, {\mu _{mn}}_{2})$], min$[{\eta _{mn}}_{1}, {\eta _{mn}}_{2}]$, min$[{\nu _{mn}}_{1}, {\nu _{mn}}_{2}]\}$. For example, ${\textbf{M}}_{H_{1}}^{\texttt {P}}$ (Matrix 23) and ${\textbf{M}}_{H_{2}}^{\texttt {P}}$ (Matrix 24) be two ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$, then union is given in ${\textbf{M}}_{H_{3}}^{\texttt {P}}$ (Matrix 25).

$$\begin{aligned} & {\textbf{M}}_{H_{1}}^{\texttt {P}} = \left[ \begin{array}{ccccc} (0.9, 0.0, 0.1) & (0.2, 0.1, 0.7) & (0.6, 0.3, 0.1)& (0.7, 0.1, 0.1)\\ (0.3, 0.3, 0.3) & (0.4, 0.2, 0.4)& (0.5, 0.3, 0.1) & (0.1, 0.1, 0.8) \\ (0.6, 0.1, 0.1)& (0.7, 0.1, 0.2)& (0.5, 0.2, 0.1)& (0.4, 0.1, 0.4) \\ (0.1, 0.4, 0.2)& (0.2, 0.2, 0.2)& (0.5, 0.1, 0.1)& (0.4, 0.1, 0.2)\\ \end{array} \right] \end{aligned}$$

(23)

$$\begin{aligned} & {\textbf{M}}_{H_{2}}^{\texttt {P}} = \left[ \begin{array}{ccccc} (0.3, 0.3, 0.4)& (0.6, 0.2, 0.1)& (0.6, 0.1, 0.3)& (0.3, 0.5, 0.2)\\ (0.7, 0.2, 0.1)& (0.3, 0.3, 0.2)& (0.4, 0.1, 0.3)& (0.5, 0.3, 0.1)\\ (0.6, 0.2, 0.2)& (0.8, 0.1, 0.1)& (0.2, 0.6, 0.2))& (0.2, 0.3, 0.4)\\ (0.4, 0.1, 0.2)& (0.4, 0.1, 0.4)& (0.5, 0.1, 0.2)& (0.3, 0.3, 0.3) \\ \end{array} \right] \end{aligned}$$

(24)

$$\begin{aligned} & {\textbf{M}}_{H_{3}}^{\texttt {P}} = \left[ \begin{array}{ccccc} (0.9, 0.0, 0.1)& (0.6, 0.1, 0.1)& (0.6, 0.1, 0.1)& (0.7, 0.1, 0.1) \\ (0.3, 0.2, 0.3)& (0.3, 0.2, 0.4)& (0.4, 0.1, 0.3)& (0.1, 0.1, 0.8)\\ (0.6, 0.1, 0.2)& (0.7, 0.1, 0.1)& (0.2, 0.2, 0.2)& (0.2, 0.1, 0.4)\\ (0.1, 0.1, 0.2)& (0.2, 0.1, 0.4)& (0.5, 0.1, 0.2)& (0.3, 0.1, 0.3)\\ \end{array} \right] \end{aligned}$$

(25)

Definition 22

Intersection reflects conservative or consensus-based aggregation, focusing on the common strength across criteria. For employee appraisal, it ensures that only sub-attributes where all evaluators agree positively are highlighted, emphasizing reliability in critical assessments.

Suppose ${\textbf{M}}_{H_{1}}^{\texttt {P}} = [\textbf{e}_{nm}] = \langle {\mu _{mn}}_{1}, {\eta _{mn}}_{1}, {\nu _{mn}}_{1}\rangle$ of order $\alpha \times \beta$ and ${\textbf{M}}_{H_{2}}^{\texttt {P}}= [\textbf{f}_{nm}] = \langle {\mu _{mn}}_{2}, {\eta _{mn}}_{2}, {\nu _{mn}}_{2}\rangle$ of order $\alpha \times \beta$ be two ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ then intersection of ${\textbf{M}}_{H_{1}}^{\texttt {P}}$ and ${\textbf{M}}_{H_{2}}^{\texttt {P}}$ is denoted by ${\textbf{M}}_{H_{3}}^{\texttt {P}} = [\textbf{g}_{nm}] = \langle {\mu _{mn}}_{2}, {\eta _{mn}}_{2}, {\nu _{mn}}_{2}\rangle$ of order $\alpha \times \gamma$ and defined as ${\textbf{M}}_{H_{3}}^{\texttt {P}}$ = ${\textbf{M}}_{H_{1}}^{\texttt {P}} \cap {\textbf{M}}_{H_{2}}^{\texttt {P}}$

$[\textbf{g}_{nm}] = [\textbf{e}_{nm}] \cap [\textbf{f}_{nm}]$ = $\{$min$[({\mu _{mn}}_{1}, {\mu _{mn}}_{2})$], min$[{\eta _{mn}}_{1}, {\eta _{mn}}_{2}]$, max$[{\nu _{mn}}_{1}, {\nu _{mn}}_{2}]\}$. For example, ${\textbf{M}}_{H_{1}}^{\texttt {P}}$ (Matrix 26) and ${\textbf{M}}_{H_{2}}^{\texttt {P}}$ (Matrix 27) be two ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ then intersection is given in ${\textbf{M}}_{H_{3}}^{\texttt {P}}$ (Matrix 28).

$$\begin{aligned} & {\textbf{M}}_{H_{1}}^{\texttt {P}} = \left[ \begin{array}{ccccc} (0.9, 0.0, 0.1) & (0.2, 0.1, 0.7) & (0.6, 0.3, 0.1)& (0.7, 0.1, 0.1)\\ (0.3, 0.3, 0.3) & (0.4, 0.2, 0.4)& (0.5, 0.3, 0.1) & (0.1, 0.1, 0.8) \\ (0.6, 0.1, 0.1)& (0.7, 0.1, 0.2)& (0.5, 0.2, 0.1)& (0.4, 0.1, 0.4) \\ (0.1, 0.4, 0.2)& (0.2, 0.2, 0.2)& (0.5, 0.1, 0.1)& (0.4, 0.1, 0.2)\\ \end{array} \right] \end{aligned}$$

(26)

$$\begin{aligned} & {\textbf{M}}_{H_{2}}^{\texttt {P}} = \left[ \begin{array}{ccccc} (0.3, 0.3, 0.4)& (0.6, 0.2, 0.1)& (0.6, 0.1, 0.3)& (0.3, 0.5, 0.2)\\ (0.7, 0.2, 0.1)& (0.3, 0.3, 0.2)& (0.4, 0.1, 0.3)& (0.5, 0.3, 0.1)\\ (0.6, 0.2, 0.2)& (0.8, 0.1, 0.1)& (0.2, 0.6, 0.2))& (0.2, 0.3, 0.4)\\ (0.4, 0.1, 0.2)& (0.4, 0.1, 0.4)& (0.5, 0.1, 0.2)& (0.3, 0.3, 0.3) \\ \end{array} \right] \end{aligned}$$

(27)

$$\begin{aligned} & {\textbf{M}}_{H_{3}}^{\texttt {P}} = \left[ \begin{array}{ccccc} (0.3, 0.0, 0.4)& (0.2, 0.1, 0.7)& (0.6, 0.1, 0.3)& (0.3, 0.1, 0.2)\\ (0.5, 0.1, 0.2)& (0.5, 0.1, 0.1)& (0.4, 0.1, 0.2)& (0.4, 0.3, 0.3) \\ (0.7, 0.1, 0.2)& (0.6, 0.1, 0.1)& (0.6, 0.1, 0.2)& (0.5, 0.3, 0.2)\\ (0.5, 0.1, 0.2)& (0.5, 0.1, 0.1)& (0.4, 0.1, 0.2)& (0.3, 0.3, 0.2)\\ \end{array} \right] \end{aligned}$$

(28)

Definition 23

Suppose ${\textbf{M}}_{H_{1}}^{\texttt {P}} = [\textbf{e}_{nm}] = \langle {\mu _{mn}}_{1}, {\eta _{mn}}_{1}, {\nu _{mn}}_{1}\rangle$ (Matrix 29) of order $\alpha \times \beta$ and ${\textbf{M}}_{H_{2}}^{\texttt {P}}= [\textbf{f}_{no}] = \langle {\mu _{mn}}_{3}, {\eta _{mn}}_{3}, {\nu _{mn}}_{3}\rangle$ (Matrix 30) of order $\alpha \times \beta$ be two ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ then AND product of ${\textbf{M}}_{H_{1}}^{\texttt {P}}$ and ${\textbf{M}}_{H_{2}}^{\texttt {P}}$ is denoted by ${\textbf{M}}_{H_{3}}^{\texttt {P}} = [\textbf{g}_{np}] = \langle {\mu _{mn}}_{3}, {\eta _{mn}}_{3}, {\nu _{mn}}_{3}\rangle$ s.t ${\textbf{M}}_{H_{3}}^{\texttt {P}}$ = ${\textbf{M}}_{H_{1}}^{\texttt {P}}$ $\wedge$ ${\textbf{M}}_{H_{2}}^{\texttt {P}}$ and define as $\wedge : [{\textbf{M}}_{H_{1}}^{\texttt {P}}]_{\alpha \times \beta } \times [{\textbf{M}}_{H_{2}}^{\texttt {P}}]_{\alpha \times \beta } \longrightarrow [{\textbf{M}}_{H_{3}}^{\texttt {P}}]_{\alpha \times {\beta }^{2}}$ and where $p = \beta (m - 1) + o$ $[\textbf{g}_{np}] = [\textbf{e}_{nm}] \wedge [\textbf{f}_{no}]$ = $\{$min$[({\mu _{mn}}_{1}, {\mu _{mn}}_{3})$], min$[{\eta _{mn}}_{1}, {\eta _{mn}}_{3}]$, max$[{\nu _{mn}}_{1}, {\nu _{mn}}_{3}]\}$

$$\begin{aligned} & {[}{\textbf{M}}_{H_{1}}^{\texttt {P}}]_{2 \times 2} = \left[ \begin{array}{cc} (0.6, 0.3, 0.1)& (0.7, 0.1, 0.1)\\ (0.5, 0.3, 0.1) & (0.1, 0.1, 0.8) \\ (0.5, 0.2, 0.1)& (0.4, 0.1, 0.4) \\ (0.5, 0.1, 0.1)& (0.4, 0.1, 0.2)\\ \end{array} \right] \end{aligned}$$

(29)

$$\begin{aligned} & {[}{\textbf{M}}_{H_{2}}^{\texttt {P}}]_{2 \times 2} =\left[ \begin{array}{cc} (0.6, 0.1, 0.3)& (0.3, 0.5, 0.2)\\ (0.4, 0.1, 0.3)& (0.5, 0.3, 0.1)\\ (0.2, 0.6, 0.2))& (0.2, 0.3, 0.4)\\ (0.5, 0.1, 0.2)& (0.3, 0.3, 0.3) \\ \end{array} \right] \end{aligned}$$

(30)

then AND product is given in $[{\textbf{M}}_{H_{3}}^{\texttt {P}}]$ (Matrix 31).

$$\begin{aligned} {[}{\textbf{M}}_{H_{3}}^{\texttt {P}}] =\left[ \begin{array}{ccccc} (0.6, 0.1, 0.3)& (0.3, 0.3, 0.2)& (0.6, 0.1, 0.3)& (0.3, 0.1, 0.2)\\ (0.4, 0.1, 0.3)& (0.5, 0.3, 0.2)& (0.1, 0.1, 0.8)& (0.1, 0.1, 0.8) \\ (0.2, 0.2, 0.2)& (0.2, 0.2, 0.4)& (0.2, 0.1, 0.4)& (0.2, 0.1, 0.4)\\ (0.5, 0.1, 0.2)& (0.3, 0.1, 0.3)& (0.4, 0.1, 0.2)& (0.3, 0.1, 0.3)\\ \end{array} \right] \end{aligned}$$

(31)

Definition 24

Suppose ${\textbf{M}}_{H_{1}}^{\texttt {P}} = [\textbf{e}_{nm}] = \langle {\mu _{mn}}_{1}, {\eta _{mn}}_{1}, {\nu _{mn}}_{1}\rangle$ (Matrix 32) of order $\alpha \times \beta$ and ${\textbf{M}}_{H_{2}}^{\texttt {P}}= [\textbf{f}_{no}] = \langle {\mu _{mn}}_{3}, {\eta _{mn}}_{3}, {\nu _{mn}}_{3}\rangle$ (Matrix 33) of order $\alpha \times \beta$ be two ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ then OR product of ${\textbf{M}}_{H_{1}}^{\texttt {P}}$ and ${\textbf{M}}_{H_{2}}^{\texttt {P}}$ is denoted by ${\textbf{M}}_{H_{3}}^{\texttt {P}} = [\textbf{g}_{np}] = \langle {\mu _{mn}}_{3}, {\eta _{mn}}_{3}, {\nu _{mn}}_{3}\rangle$ s.t ${\textbf{M}}_{H_{3}}^{\texttt {P}}$ = ${\textbf{M}}_{H_{1}}^{\texttt {P}}$ $\vee$ ${\textbf{M}}_{H_{2}}^{\texttt {P}}$ and define as $\vee : [{\textbf{M}}_{H_{1}}^{\texttt {P}}]_{\alpha \times \beta } \times [{\textbf{M}}_{H_{2}}^{\texttt {P}}]_{\alpha \times \beta } \longrightarrow [{\textbf{M}}_{H_{3}}^{\texttt {P}}]_{\alpha \times {\beta }^{2}}$ and where $p = \beta (m - 1) + o$

$[\textbf{g}_{np}] = [\textbf{e}_{nm}] \vee [\textbf{f}_{no}]$ = $\{$max$[({\mu _{mn}}_{1}, {\mu _{mn}}_{3})$], min$[{\eta _{mn}}_{1}, {\eta _{mn}}_{3}]$, min$[{\nu _{mn}}_{1}, {\nu _{mn}}_{3}]\}$

$$\begin{aligned} & {[}{\textbf{M}}_{H_{1}}^{\texttt {P}}]_{2 \times 2} = \left[ \begin{array}{cc} (0.6, 0.3, 0.1)& (0.7, 0.1, 0.1)\\ (0.5, 0.3, 0.1) & (0.1, 0.1, 0.8) \\ (0.5, 0.2, 0.1)& (0.4, 0.1, 0.4) \\ (0.5, 0.1, 0.1)& (0.4, 0.1, 0.2)\\ \end{array} \right] \end{aligned}$$

(32)

$$\begin{aligned} & {[}{\textbf{M}}_{H_{2}}^{\texttt {P}}]_{2 \times 2} =\left[ \begin{array}{cc} (0.6, 0.1, 0.3)& (0.3, 0.5, 0.2)\\ (0.4, 0.1, 0.3)& (0.5, 0.3, 0.1)\\ (0.2, 0.6, 0.2))& (0.2, 0.3, 0.4)\\ (0.5, 0.1, 0.2)& (0.3, 0.3, 0.3) \\ \end{array} \right] \end{aligned}$$

(33)

then OR product is given in $[{\textbf{M}}_{H_{3}}^{\texttt {P}}]$ (Matrix 34).

$$\begin{aligned} {[}{\textbf{M}}_{H_{3}}^{\texttt {P}}] =\left[ \begin{array}{ccccc} (0.6, 0.1, 0.1)& (0.6, 0.3, 0.1)& (0.7, 0.1, 0.1)& (0.7, 0.1, 0.1)\\ (0.5, 0.1, 0.1)& (0.5, 0.3, 0.1)& (0.4, 0.1, 0.3)& (0.5, 0.1, 0.1) \\ (0.5, 0.2, 0.1)& (0.5, 0.2, 0.1)& (0.4, 0.1, 0.2)& (0.4, 0.1, 0.4)\\ (0.5, 0.1, 0.1)& (0.5, 0.1, 0.1)& (0.5, 0.1, 0.2)& (0.4, 0.1, 0.2)\\ \end{array} \right] \end{aligned}$$

(34)

Definition 25

Suppose ${\textbf{M}}_{H}^{\texttt {P}} = [\textbf{e}_{nm}] = \langle {\mu _{mn}}_{1}, {\eta _{mn}}_{1}, {\nu _{mn}}_{1}\rangle$ of order $\alpha \times \beta$ is ${\mathcal {P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ then trace of ${\textbf{M}}_{H}^{\texttt {P}}$ is denoted by ${\textbf{M}}_{H}^{\texttt {P}^{tr}}$ and is define as

${\textbf{M}}_{H_{1}}^{\texttt {P}^{tr}} = \sum [{\mu _{mn}}_{1}- {\eta _{mn}}_{1}- {\nu _{mn}}_{1}]$ where m = n

For example, ${\textbf{M}}_{H}^{\texttt {P}}$ is ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ of order $4 \times 4$ is given as (Matrix 35):

$$\begin{aligned} {[}{\textbf{M}}_{H}^{\texttt {P}}] =\left[ \begin{array}{ccccc} (0.6, 0.1, 0.3)& (0.3, 0.3, 0.2)& (0.6, 0.1, 0.3)& (0.3, 0.1, 0.2)\\ (0.4, 0.1, 0.3)& (0.5, 0.3, 0.2)& (0.1, 0.1, 0.8)& (0.1, 0.1, 0.8) \\ (0.2, 0.2, 0.2)& (0.2, 0.2, 0.4)& (0.2, 0.1, 0.4)& (0.2, 0.1, 0.4)\\ (0.5, 0.1, 0.2)& (0.3, 0.1, 0.3)& (0.4, 0.1, 0.2)& (0.3, 0.1, 0.3)\\ \end{array} \right] \end{aligned}$$

(35)

then

${\textbf{M}}_{H}^{\texttt {P}^{tr}}$ = [0.6 -0.1 -0.3]+ [0.5 -0.3 -0.2]+ [0.2 -0.1 - 0.4]+ [0.3 - 0.1 - 0.3] = 0.2

Proposition 1

• (i) Suppose ${\textbf{M}}_{H_{1}}^{\texttt {P}} = [\textbf{e}_{nm}] = \langle {\mu _{mn}}_{1}, {\eta _{mn}}_{1}, {\nu _{mn}}_{1}\rangle$ be ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ then $[\textbf{e}_{nm}] \cup [\textbf{e}_{nm}] = [\textbf{e}_{nm}]$ and $[\textbf{e}_{nm}] \cap [\textbf{e}_{nm}] = [\textbf{e}_{nm}]$

• (ii) Suppose ${\textbf{M}}_{H}^{\texttt {P}} = [\textbf{e}^{null}_{nm}] = \langle {\mu _{mn}}^{{null\textbf{z}^{\varrho }}_{m}}_{{\textbf{r}^{\varrho }}_{n}}, {\eta _{mn}}^{{nl\textbf{z}^{\varrho }}_{m}}_{{\textbf{r}^{\varrho }}_{n}}, {\nu _{mn}}^{{null\textbf{z}^{\varrho }}_{m}}_{{\textbf{r}^{\varrho }}_{n}}\rangle$ be ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}^{null}}$ then $[\textbf{e}_{nm}] \cup [\textbf{e}^{null}_{nm}] = [\textbf{e}_{nm}]$ and $[\textbf{e}_{nm}] \cap [\textbf{e}^{null}_{nm}] = [\textbf{e}^{null}_{nm}]$

• (iii) Suppose ${\textbf{M}}_{H}^{\texttt {P}} = [\textbf{e}^{abst}_{nm}] = \langle {\mu _{mn}}^{{abst\textbf{z}^{\varrho }}_{m}}_{{\textbf{r}^{\varrho }}_{n}}, {\eta _{mn}}^{{abst\textbf{z}^{\varrho }}_{m}}_{{\textbf{r}^{\varrho }}_{n}}, {\nu _{mn}}^{{abst\textbf{z}^{\varrho }}_{m}}_{{\textbf{r}^{\varrho }}_{n}}\rangle$ be ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}^{abst}}$ then $[\textbf{e}_{nm}] \cup [\textbf{e}^{abst}_{nm}] = [\textbf{e}_{nm}]$ and $[\textbf{e}_{nm}] \cap [\textbf{e}^{abst}_{nm}] = [\textbf{e}^{abst}_{nm}]$

• (iv) Suppose ${\textbf{M}}_{H_{1}}^{\texttt {P}} = [\textbf{e}_{nm}] = \langle {\mu _{mn}}_{1}, {\eta _{mn}}_{1}, {\nu _{mn}}_{1}\rangle$ and ${\textbf{M}}_{H_{2}}^{\texttt {P}}= [\textbf{f}_{nm}] = \langle {\mu _{mn}}_{2}, {\eta _{mn}}_{2}, {\nu _{mn}}_{2}\rangle$ be two ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}s}$ then $[\textbf{e}_{nm}] \ \cup [\textbf{f}_{nm}] = [\textbf{f}_{nm}] \ \cup [\textbf{e}_{nm}]$ and $[\textbf{e}_{nm}] \ \cap [\textbf{f}_{nm}] = [\textbf{f}_{nm}] \ \cap [\textbf{e}_{nm}]$

• (v) Suppose ${\textbf{M}}_{H_{1}}^{\texttt {P}} = [\textbf{e}_{nm}] = \langle {\mu _{mn}}_{1}, {\eta _{mn}}_{1}, {\nu _{mn}}_{1}\rangle$, ${\textbf{M}}_{H_{2}}^{\texttt {P}}= [\textbf{f}_{nm}] = \langle {\mu _{mn}}_{2}, {\eta _{mn}}_{2}, {\nu _{mn}}_{2}\rangle$ and ${\textbf{M}}_{H_{3}}^{\texttt {P}} = [\textbf{g}_{nm}] = \langle {\mu _{mn}}_{3}, {\eta _{mn}}_{3}, {\nu _{mn}}_{3}\rangle$ be three ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}s}$ then $([\textbf{e}_{nm}] \ \cup [\textbf{f}_{nm}]) \ \cup [\textbf{g}_{nm}] = [\textbf{f}_{nm}] \ \cup ([\textbf{e}_{nm}] \ \cup [\textbf{g}_{nm}])$ and $([\textbf{e}_{nm}] \ \cap [\textbf{f}_{nm}]) \ \cap [\textbf{g}_{nm}] = [\textbf{f}_{nm}] \ \cap ([\textbf{e}_{nm}] \ \cap [\textbf{g}_{nm}])$

• (vi) Suppose ${\textbf{M}}_{H_{1}}^{\texttt {P}} = [\textbf{e}_{nm}] = \langle {\mu _{mn}}_{1}, {\eta _{mn}}_{1}, {\nu _{mn}}_{1}\rangle$, ${\textbf{M}}_{H_{2}}^{\texttt {P}}= [\textbf{f}_{nm}] = \langle {\mu _{mn}}_{2}, {\eta _{mn}}_{2}, {\nu _{mn}}_{2}\rangle$ and ${\textbf{M}}_{H_{3}}^{\texttt {P}} = [\textbf{g}_{nm}] = \langle {\mu _{mn}}_{3}, {\eta _{mn}}_{3}, {\nu _{mn}}_{3}\rangle$ be three ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}s}$ then $[\textbf{e}_{nm}] \ \cup ([\textbf{f}_{nm}]) \ \cap [\textbf{g}_{nm}]) = ([\textbf{f}_{nm}] \ \cup [\textbf{e}_{nm}]) \ \cap ([\textbf{f}_{nm}] \ \cup [\textbf{g}_{nm}])$ and $[\textbf{e}_{nm}] \ \cap ([\textbf{f}_{nm}] \ \cup [\textbf{g}_{nm}]) = ([\textbf{f}_{nm}] \ \cap [\textbf{e}_{nm}]) \ \cup ([\textbf{f}_{nm}] \ \cap [\textbf{g}_{nm}])$

Proof

The proofs are straightforward.

Proposition 2

Suppose ${\textbf{M}}_{H_{1}}^{\texttt {P}} = [\textbf{e}_{nm}] = \langle {\mu _{mn}}_{1}, {\eta _{mn}}_{1}, {\nu _{mn}}_{1}\rangle$ and ${\textbf{M}}_{H_{2}}^{\texttt {P}}= [\textbf{f}_{nm}] = \langle {\mu _{mn}}_{2}, {\eta _{mn}}_{2}, {\nu _{mn}}_{2}\rangle$ be two ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}s}$ then

• (i) $([\textbf{e}_{nm}] \ \cup [\textbf{f}_{nm}])^{c} = [\textbf{f}_{nm}]^{c} \ \cap [\textbf{e}_{nm}]^{c}$

• (ii) $([\textbf{e}_{nm}] \ \cap [\textbf{f}_{nm}])^{c} = [\textbf{f}_{nm}]^{c} \ \cup [\textbf{e}_{nm}]^{c}$

Proposition 3

Suppose ${\textbf{M}}_{H_{1}}^{\texttt {P}} = [\textbf{e}_{nm}] = \langle {\mu _{mn}}_{1}, {\eta _{mn}}_{1}, {\nu _{mn}}_{1}\rangle$, ${\textbf{M}}_{H_{2}}^{\texttt {P}}= [\textbf{f}_{nm}] = \langle {\mu _{mn}}_{2}, {\eta _{mn}}_{2}, {\nu _{mn}}_{2}\rangle$ and ${\textbf{M}}_{H_{3}}^{\texttt {P}}= [\textbf{h}_{nm}] = \langle {\mu _{mn}}_{3}, {\eta _{mn}}_{3}, {\nu _{mn}}_{3}\rangle$ be three ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}s}$ then

• (i) ${\textbf{M}}_{H_{1}}^{\texttt {P}} + {\textbf{M}}_{H_{2}}^{\texttt {P}}$ = ${\textbf{M}}_{H_{2}}^{\texttt {P}} + {\textbf{M}}_{H_{1}}^{\texttt {P}}$

• (ii) ${\textbf{M}}_{H_{1}}^{\texttt {P}} \circledast {\textbf{M}}_{H_{2}}^{\texttt {P}}$ = ${\textbf{M}}_{H_{2}}^{\texttt {P}} \circledast {\textbf{M}}_{H_{1}}^{\texttt {P}}$

• (iii) ${\textbf{M}}_{H_{1}}^{\texttt {P}} + [{\textbf{M}}_{H_{2}}^{\texttt {P}} + {\textbf{M}}_{H_{3}}^{\texttt {P}}]$ = $[{\textbf{M}}_{H_{1}}^{\texttt {P}} + {\textbf{M}}_{H_{2}}^{\texttt {P}}] + {\textbf{M}}_{H_{3}}^{\texttt {P}}$

• (iv) ${\textbf{M}}_{H_{1}}^{\texttt {P}} \circledast [{\textbf{M}}_{H_{2}}^{\texttt {P}} \circledast {\textbf{M}}_{H_{3}}^{\texttt {P}}]$ = $[{\textbf{M}}_{H_{1}}^{\texttt {P}} \circledast {\textbf{M}}_{H_{2}}^{\texttt {P}}] \circledast {\textbf{M}}_{H_{3}}^{\texttt {P}}$

• (v) ${\textbf{M}}_{H_{1}}^{\texttt {P}} \cup {\textbf{M}}_{H_{2}}^{\texttt {P}}$ = ${\textbf{M}}_{H_{2}}^{\texttt {P}} \cup {\textbf{M}}_{H_{1}}^{\texttt {P}}$

• (vi) ${\textbf{M}}_{H_{1}}^{\texttt {P}} \cap {\textbf{M}}_{H_{2}}^{\texttt {P}}$ = ${\textbf{M}}_{H_{2}}^{\texttt {P}} \cap {\textbf{M}}_{H_{1}}^{\texttt {P}}$

• (vii) ${\textbf{M}}_{H_{1}}^{\texttt {P}} \cup [{\textbf{M}}_{H_{2}}^{\texttt {P}} \cup {\textbf{M}}_{H_{3}}^{\texttt {P}}]$ = $[{\textbf{M}}_{H_{1}}^{\texttt {P}} \cup {\textbf{M}}_{H_{2}}^{\texttt {P}}] \cup {\textbf{M}}_{H_{3}}^{\texttt {P}}$

• (viii) ${\textbf{M}}_{H_{1}}^{\texttt {P}} \cap [{\textbf{M}}_{H_{2}}^{\texttt {P}} \cap {\textbf{M}}_{H_{3}}^{\texttt {P}}]$ = $[{\textbf{M}}_{H_{1}}^{\texttt {P}} \cap {\textbf{M}}_{H_{2}}^{\texttt {P}}] \cap {\textbf{M}}_{H_{3}}^{\texttt {P}}$

Proof

All results (i–viii) are straightforward because the union +, ’$\circledast$’ and the intersection are defined component-wise using $\max$ and $\min$, and both $\max$ and $\min$ are commutative and associative on [0, 1].

4 Picture Fuzzy Hypersoft Geometric Aggregation Operators

Definition 26

The basic laws for PFHSMs are defined as follows:

Let $({\textbf{r}^{\varrho }}_{b}, {\textbf{z}^{\varrho }}_{a})= \left\{ \begin{array}{l} ({\mu _{mn}}^{a}_{b}) , ({\eta _{mn}}^{a}_{b}) ,({\nu _{mn}}^{a}_{b}) \end{array}\right\} \$ and $({\textbf{r}^{\varrho }}_{d}, {\textbf{z}^{\varrho }}_{c})= \left\{ \begin{array}{l} ({\mu _{mn}}^{c}_{d}) , ({\eta _{mn}}^{c}_{d}) ,({\nu _{mn}}^{c}_{d}) \end{array}\right\}$ be two PFHSNs.

• (i) $({\textbf{r}^{\varrho }}_{b}, {\textbf{z}^{\varrho }}_{a}) \otimes ({\textbf{r}^{\varrho }}_{d}, {\textbf{z}^{\varrho }}_{c}) = \left\{ \begin{array}{l} \{[({\mu _{mn}}^{a}_{b})]\}\{[({\mu _{mn}}^{c}_{d})]\},\\ \{1- [1 - ({\eta _{mn}}^{a}_{b})]\}\{[1 - ({\eta _{mn}}^{c}_{d})]\}, \\ \{1- [1 - ({\nu _{mn}}^{a}_{b})]\}\{[1 - ({\nu _{mn}}^{c}_{d})]\} \end{array}\right\}$

• (ii) $({\textbf{r}^{\varrho }}_{b}, {\textbf{z}^{\varrho }}_{a}) \oplus ({\textbf{r}^{\varrho }}_{d}, {\textbf{z}^{\varrho }}_{c}) = \left\{ \begin{array}{l} \{1- [1 - ({\mu _{mn}}^{a}_{b})]\}\{[1 - ({\mu _{mn}}^{c}_{d})]\},\\ \{[({\eta _{mn}}^{a}_{b})]\}\{[({\eta _{mn}}^{c}_{d})]\}, \\ \{[({\nu _{mn}}^{a}_{b})]\}\{[({\nu _{mn}}^{c}_{d})]\} \end{array}\right\}$

• (iii) $\Re ({\textbf{r}^{\varrho }}_{b}, {\textbf{z}^{\varrho }}_{a}) = \left\{ \begin{array}{l} 1- [1 - ({\mu _{mn}}^{a}_{b})]^{\Re }, [({\eta _{mn}}^{a}_{b})]^{\Re }, [({\nu _{mn}}^{a}_{b})]^{\Re } \end{array}\right\}$

• (iv) $({\textbf{r}^{\varrho }}_{b}, {\textbf{z}^{\varrho }}_{a})^{\Re } = \left\{ \begin{array}{l} [({\mu _{mn}}^{a}_{b})]^{\Re }, 1- [1 - ({\eta _{mn}}^{a}_{b})]^{\Re }, 1- [1 - ({\nu _{mn}}^{a}_{b})]^{\Re }\end{array}\right\}$

• (v) $({\textbf{r}^{\varrho }}_{b}, {\textbf{z}^{\varrho }}_{a}) \oplus ({\textbf{r}^{\varrho }}_{d}, {\textbf{z}^{\varrho }}_{c}) = ({\textbf{r}^{\varrho }}_{d}, {\textbf{z}^{\varrho }}_{c}) \oplus ({\textbf{r}^{\varrho }}_{b}, {\textbf{z}^{\varrho }}_{a})$

• (vi) $\Re [({\textbf{r}^{\varrho }}_{b}, {\textbf{z}^{\varrho }}_{a}) \oplus ({\textbf{r}^{\varrho }}_{d}, {\textbf{z}^{\varrho }}_{c})]$ = $\Re ({\textbf{r}^{\varrho }}_{b}, {\textbf{z}^{\varrho }}_{a}) \oplus \Re ({\textbf{r}^{\varrho }}_{d}, {\textbf{z}^{\varrho }}_{c})$

Definition 27

Let ${N}^{P}_{H(nm)} = ({\textbf{r}^{\varrho }}_{n}, {\textbf{z}^{\varrho }}_{m}) = \langle \mu _{mn}, \eta _{mn}, \nu _{mn}\rangle$ where $n = \{1, 2, 3,\ldots , \alpha \}$ and $m = \{1, 2, 3,\ldots , \beta \}$, be a PFHSN. Then the score function $\Psi ({\textbf{r}^{\varrho }}_{n}, {\textbf{z}^{\varrho }}_{m})$ and accuracy function $\Omega ({\textbf{r}^{\varrho }}_{n}, {\textbf{z}^{\varrho }}_{m})$ of PFHSNs are defined as follows:

• (i) $\Psi ({\textbf{r}^{\varrho }}_{n}, {\textbf{z}^{\varrho }}_{m})$ = $(\mu _{mn} - \eta _{mn} - \nu _{mn})$

• (ii) $\Omega ({\textbf{r}^{\varrho }}_{n}, {\textbf{z}^{\varrho }}_{m})$ = $(\mu _{mn} + \eta _{mn} + \nu _{mn})$

Also the range of score function $\Psi ({\textbf{r}^{\varrho }}_{n}, {\textbf{z}^{\varrho }}_{m})$ is [−1,1] and range of accuracy function $\Omega ({\textbf{r}^{\varrho }}_{n}, {\textbf{z}^{\varrho }}_{m})$ is [0,1].

Definition 28

Let $({\textbf{r}^{\varrho }}_{b}, {\textbf{z}^{\varrho }}_{a})$ and $({\textbf{r}^{\varrho }}_{d}, {\textbf{z}^{\varrho }}_{c})$ be two PFHSSNs. The method to compare the two sets is defined as follows:

• (i) If $\Psi [({\textbf{r}^{\varrho }}_{b}, {\textbf{z}^{\varrho }}_{a})] > \Psi [({\textbf{r}^{\varrho }}_{d}, {\textbf{z}^{\varrho }}_{c})]$ then $({\textbf{r}^{\varrho }}_{b}, {\textbf{z}^{\varrho }}_{a}) > ({\textbf{r}^{\varrho }}_{d}, {\textbf{z}^{\varrho }}_{c})$

• (ii) If $\Psi [({\textbf{r}^{\varrho }}_{b}, {\textbf{z}^{\varrho }}_{a})] = \Psi [({\textbf{r}^{\varrho }}_{d}, {\textbf{z}^{\varrho }}_{c})]$ and $\Omega [({\textbf{r}^{\varrho }}_{b}, {\textbf{z}^{\varrho }}_{a})] > \Omega [({\textbf{r}^{\varrho }}_{d}, {\textbf{z}^{\varrho }}_{c})]$ then $({\textbf{r}^{\varrho }}_{b}, {\textbf{z}^{\varrho }}_{a}) > ({\textbf{r}^{\varrho }}_{d}, {\textbf{z}^{\varrho }}_{c})$

Definition 29

Suppose $[{N}^{P}_{H(nm)}] = ({\textbf{r}^{\varrho }}_{n}, {\textbf{z}^{\varrho }}_{m}) = \langle \mu _{mn}, \eta _{mn}, \nu _{mn}\rangle$ where $n = \{1, 2, 3,\ldots , \alpha \}$, $m = \{1, 2, 3,\ldots , \beta \}$ be a PFHSNs, A function from PFHSWG: $\xi _{p} \rightarrow \xi$ is called a picture fuzzy hypersoft weighted geometric (PFHSWG) operator.

$$\begin{aligned} & PFHSWG \{({\textbf{r}^{\varrho }}_{1}, {\textbf{z}^{\varrho }}_{1}), ({\textbf{r}^{\varrho }}_{1}, {\textbf{z}^{\varrho }}_{2}), ({\textbf{r}^{\varrho }}_{1}, {\textbf{z}^{\varrho }}_{3}),\ldots ,({\textbf{r}^{\varrho }}_{\alpha }, {\textbf{z}^{\varrho }}_{\beta })\}\\ & \quad = [({\textbf{r}^{\varrho }}_{1}, {\textbf{z}^{\varrho }}_{1})]^{T_{(1, 1)}} \otimes [({\textbf{r}^{\varrho }}_{1}, {\textbf{z}^{\varrho }}_{2})]^{T_{(1, 2)}} \otimes [({\textbf{r}^{\varrho }}_{1}, {\textbf{z}^{\varrho }}_{3})]^{T_{(1, 3)}} \otimes \cdots \otimes [({\textbf{r}^{\varrho }}_{\alpha }, {\textbf{z}^{\varrho }}_{\beta })]^{T_{(\alpha , \beta )}} \\ \end{aligned}$$

where $T_{(n, m)}$ is the weight vector matrix (WVM), $T_{(n, m)} \in [0,1]$ and $\sum _{(n, m)=(1, 1)}^{(\alpha , \beta )}T_{(n, m)} = 1.$,

Theorem 1

Suppose $[{N}^{P}_{H(nm)}] = ({\textbf{r}^{\varrho }}_{n}, {\textbf{z}^{\varrho }}_{m}) = \langle \mu _{mn}, \eta _{mn}, \nu _{mn}\rangle$ where $n = \{1, 2, 3,\ldots , \alpha \}$, $m = \{1, 2, 3,\ldots , \beta \}$ be a PFHSNs, then

$PFHSWG \{({\textbf{r}^{\varrho }}_{1}, {\textbf{z}^{\varrho }}_{1}), ({\textbf{r}^{\varrho }}_{1}, {\textbf{z}^{\varrho }}_{2}), ({\textbf{r}^{\varrho }}_{1}, {\textbf{z}^{\varrho }}_{3}),\ldots ,({\textbf{r}^{\varrho }}_{\alpha }, {\textbf{z}^{\varrho }}_{\beta })\}$ = $\left\{ \begin{array}{l} \prod _{{{(n, m)=(1, 1)}}}^{(\alpha , \beta )}[(\mu _{mn})]^{T_{(\alpha , \beta )}} ,\\ 1- \prod _{{{(n, m)=(1, 1)}}}^{(\alpha , \beta )}[1 - (\eta _{mn})]^{T_{(\alpha , \beta )}}, \\ 1- \prod _{{{(n, m)=(1, 1)}}}^{(\alpha , \beta )}[1 - (\nu _{mn})]^{T_{(\alpha , \beta )}} \end{array}\right\}$

Proof

This theorem is demonstrated through mathematical induction.

For (n, m)=(1, 1) and (n, m)=(2, 1)

$$\begin{aligned} & {[}({\textbf{r}^{\varrho }}_{1}, {\textbf{z}^{\varrho }}_{1})]^{T_{(1,1)}}= \left\{ \begin{array}{l} [({\mu _{mn}}^{1}_{1})]^{T_{(1,1)}}, [({\eta _{mn}}^{1}_{1})]^{T_{(1,1)}},[({\nu _{mn}}^{1}_{1})]^{T_{(1,1)}} \end{array}\right\} \end{aligned}$$

and

$$\begin{aligned} {[}({\textbf{r}^{\varrho }}_{2}, {\textbf{z}^{\varrho }}_{1})]^{T_{(2,1)}}= \left\{ \begin{array}{l} [({\mu _{mn}}^{{\textbf{z}^{\varrho }}_1}_{{\textbf{r}^{\varrho }}_2})]^{T_{(2, 1)}}, [({\eta _{mn}}^{{\textbf{z}^{\varrho }}_1}_{{\textbf{r}^{\varrho }}_2})]^{T_{(2, 1)}},[({\nu _{mn}}^{{\textbf{z}^{\varrho }}_1}_{{\textbf{r}^{\varrho }}_2})]^{T_{(2, 1)}} \end{array}\right\} \ \end{aligned}$$

Then, it follows that

$$\begin{aligned} [({\textbf{r}^{\varrho }}_{1}, {\textbf{z}^{\varrho }}_{1})]^{T_{(1, 1)}} \otimes [({\textbf{r}^{\varrho }}_{2}, {\textbf{z}^{\varrho }}_{1})]^{T_{(2, 1)}}= & \left\{ \begin{array}{l} \{[({\mu _{mn}}^{1}_{1})]^{T_{(1, 1)}}\}\{[({\mu _{mn}}^{1}_{2})]^{T_{(2, 1)}}\}\\ \{1- [1 - ({\eta _{mn}}^{1}_{1})]^{T_{(1, 1)}}\}\{[1 - ({\eta _{mn}}^{1}_{2})]^{T_{(2, 1)}}\}, \\ \{1- [1 - ({\nu _{mn}}^{1}_{1})]^{T_{(1, 1)}}\}\{[1 - ({\nu _{mn}}^{1}_{2})]^{T_{(2, 1)}}\} \end{array}\right\} \\= & \left\{ \begin{array}{l} \prod _{{{(n, m)=(1, 1)}}}^{(2, 1)}[(\mu _{mn})]^{T_{(n, m)}} , \\ 1- \prod _{{{(n, m)=(1, 1)}}}^{(2, 1)}[1 - (\eta _{mn})]^{T_{(n, m)}} , \\ 1- \prod _{{{(n, m)=(1, 1)}}}^{(2, 1)}[1 - (\nu _{mn})]^{T_{(n, m)}} \end{array}\right\} \end{aligned}$$

Suppose it holds for $(n, m) = (k, k)$,

$\{[({\textbf{r}^{\varrho }}_{1}, {\textbf{z}^{\varrho }}_{1})]^{T_{(1, 1)}} \otimes [({\textbf{r}^{\varrho }}_{1}, {\textbf{z}^{\varrho }}_{2})]^{T_{(1, 2)}} \otimes [({\textbf{r}^{\varrho }}_{1}, {\textbf{z}^{\varrho }}_{3})]^{T_{(1, 3)}} \otimes \cdots \otimes [({\textbf{r}^{\varrho }}_{k}, {\textbf{z}^{\varrho }}_{k})]^{T_{(k, k)}}\}$ $= \left\{ \begin{array}{l} \prod _{{{(n, m)=(1, 1)}}}^{(k, k)}[(\mu _{mn})]^{T_{(n, m)}} , \\ 1- \prod _{{{(n, m)=(1, 1)}}}^{(k, k)}[1 - (\eta _{mn})]^{T_{(n, m)}} , \\ 1- \prod _{{{(n, m)=(1, 1)}}}^{(k, k)}[1 - (\nu _{mn})]^{T_{(n, m)}} \end{array}\right\}$

Now we want to prove it is true $(n, m) = (k+1, k+1)$, by operational laws of PFHSNs, we have:

$$\begin{aligned} & \{[({\textbf{r}^{\varrho }}_{1}, {\textbf{z}^{\varrho }}_{1})]^{T_{(1, 1)}} \otimes [({\textbf{r}^{\varrho }}_{1}, {\textbf{z}^{\varrho }}_{2})]^{T_{(1, 2)}} \otimes [({\textbf{r}^{\varrho }}_{1}, {\textbf{z}^{\varrho }}_{3})]^{T_{(1, 3)}} \otimes \cdots \otimes [({\textbf{r}^{\varrho }}_{k+1}, {\textbf{z}^{\varrho }}_{k+1})]^{T_{(k+1, k+1)}}\}\\ & \quad = \left\{ \begin{array}{l} \prod _{{{(n, m)=(1, 1)}}}^{(k, k)}[(\mu _{mn})]^{T_{(n, m)}} ,\\ 1- \prod _{{{(n, m)=(1, 1)}}}^{(k, k)}[1 - (\eta _{mn})]^{T_{(n, m)}} , \\ 1- \prod _{{{(n, m)=(1, 1)}}}^{(k, k)}[1 - (\nu _{mn})]^{T_{(n, m)}} \end{array}\right\} \\ & \qquad \otimes \{ [({\mu _{mn}}^{{\textbf{z}^{\varrho }}_{a}}_{{\textbf{r}^{\varrho }}_{a}})]^{T_{(a, a)}} , [({\eta _{mn}}^{{\textbf{z}^{\varrho }}_{a}}_{{\textbf{r}^{\varrho }}_{a}})]^{T_{(a, a)}} ,[({\nu _{mn}}^{{\textbf{z}^{\varrho }}_{a}}_{{\textbf{r}^{\varrho }}_{a}})]^{T_{(a, a)}}\}\\ & \quad = \left\{ \begin{array}{l} \prod _{{{(n, m)=(1, 1)}}}^{(k+1, k+1)}[(\mu _{mn})]^{T_{(n, m)}} ,\\ 1- \prod _{{{(n, m)=(1, 1)}}}^{(k+1, k+1)}[1 - (\eta _{mn})]^{T_{(n, m)}} , \\ 1- \prod _{{{(n, m)=(1, 1)}}}^{(k+1, k+1)}[1 - (\nu _{mn})]^{T_{(n, m)}} \end{array}\right\} \end{aligned}$$

The above expression proves that $(n, m) = (k+1, k+1)$ meaning that theorem holds by the principle of mathematical induction for all values of $(\alpha , \beta )$.

$$PFHSWG \{({\textbf{r}^{\varrho }}_{1}, {\textbf{z}^{\varrho }}_{1}), ({\textbf{r}^{\varrho }}_{1}, {\textbf{z}^{\varrho }}_{2}), ({\textbf{r}^{\varrho }}_{1}, {\textbf{z}^{\varrho }}_{3}),\ldots ,({\textbf{r}^{\varrho }}_{\alpha }, {\textbf{z}^{\varrho }}_{\beta })\} $$

=$\left\{ \begin{array}{l} \prod _{{{(n, m)=(1, 1)}}}^{(\alpha , \beta )}[(\mu _{mn})]^{T_{(\alpha , \beta )}} , \\ 1- \prod _{{{(n, m)=(1, 1)}}}^{(\alpha , \beta )}[1 - (\eta _{mn})]^{T_{(\alpha , \beta )}} , \\ 1- \prod _{{{(n, m)=(1, 1)}}}^{(\alpha , \beta )}[1 - (\nu _{mn})]^{T_{(\alpha , \beta )}} \end{array}\right\}$

5 Design of the Decision-Making Algorithm

${\textbf {Algorithmic Design}}$: Let ${\mathbb {R}^{\varrho }} = \{{\textbf{r}^{\varrho }}_{1}, {\textbf{r}^{\varrho }}_{2}, {\textbf{r}^{\varrho }}_{3},\ldots ,{\textbf{r}^{\varrho }}_{\gamma }\}$ be the set of alternative, ${\textbf{v}^{\varrho }}_1, {\textbf{v}^{\varrho }}_2,\ldots ,{\textbf{v}^{\varrho }}_{\alpha }$ be set of experts and ${\textbf{q}^{\varrho }}_1, {\textbf{q}^{\varrho }}_2,\ldots ,{\textbf{q}^{\varrho }}_{\beta }$ be the sets of attributes and their corresponding attributive values are respectively the set $\textbf{Q}_{1}^{\varrho }, \textbf{Q}_{2}^{\varrho }, \textbf{Q}_{3}^{\varrho },\ldots ,\textbf{Q}_{\beta }^{\varrho }$. Suppose $\textbf{Q}^{\varrho }= \textbf{Q}_{1}^{\varrho } \times \textbf{Q}_{2}^{\varrho } \times \textbf{Q}_{3}^{\varrho } \times \cdots \times \textbf{Q}_{\beta }^{\varrho }$, with ${\textbf{Q}^{\varrho }}_{\theta } \cap {\textbf{Q}^{\varrho }}_{\phi } = \emptyset$, $\theta \ne \phi$, and $\theta , \varrho \ \in$ $\{1, 2,\ldots , n\}$. Where $T_{(n, m)}$ is the weight vector matrix (WVM), $T_{(n, m)} \in [0,1]$ and $\sum _{(n, m)=(1, 1)}^{(\alpha , \beta )}T_{(n, m)} = 1.$, The following step-by-step algorithm is provided for selecting the ideal alternative from the available ones.

${\textbf {Step 1}}:$ Acquire information about every alternative experts evaluate according to attributes in the shape of PFHSMs, provided by:

${\mathbb {R}^{\varrho }}_{i} = [ \textbf{e}_{nm}]_{\alpha \times \beta } =\langle {\mu _{mn}}^{m}_{n}, {\eta _{mn}}^{m}_{n}, {\nu _{mn}}^{m}_{n}\rangle = \left[ \begin{array}{ccccc} \textbf{e}_{11} & \textbf{e}_{12} & \textbf{e}_{13} & ...... & \textbf{e}_{1\beta } \\ \textbf{e}_{21} & \textbf{e}_{22} & \textbf{e}_{23} & ...... & \textbf{e}_{2\beta } \\ \textbf{e}_{31} & \textbf{e}_{32} & \textbf{e}_{33} & ...... & \textbf{e}_{3\beta } \\ . & . & ...... & . \\ . & . & ...... & . \\ \textbf{e}_{\alpha 1} & \textbf{e}_{\alpha 2} & \textbf{e}_{\alpha 3} & ...... & \textbf{e}_{\alpha \beta } \\ \end{array}\right]$

${\textbf {Step: 2}}$ Expert and parameter-corresponding weighted matrix for all alternatives is a crucial step in multi-criteria decision analysis to ensure that the values from different attributes or criteria are comparable and have equal importance.

${\textbf {Step: 3}}$ Calculate the aggregated value for each alternative, using the assigned weight matrix and the picture fuzzy hypersoft matrices for each criterion. The formula for the aggregated value is typically given by:

$$\begin{aligned} {\textbf{r}^{{\text{\AA }}}}_{i}= & \langle {\mu _{mn}^{{\text{\AA }}}}_{i}, {\eta _{mn}^{{\text{\AA }}}}_{i}, {\nu _{mn}^{{\text{\AA }}}}_{i}\rangle = PFHSWG \{({\textbf{v}^{\varrho }}_{1}, {\textbf{z}^{\varrho }}_{1}), ({\textbf{v}^{\varrho }}_{2}, {\textbf{z}^{\varrho }}_{2}), ({\textbf{v}^{\varrho }}_{3}, {\textbf{z}^{\varrho }}_{3}),\ldots ,({\textbf{v}^{\varrho }}_{\alpha }, {\textbf{z}^{\varrho }}_{\beta })\}\\= & \left\{ \begin{array}{l} \prod _{{{(n, m)=(1, 1)}}}^{(\alpha , \beta )}[({\mu _{mn}}^{m}_{n})]^{T_{(\alpha , \beta )}} , \\ 1- \prod _{{{(n, m)=(1, 1)}}}^{(\alpha , \beta )}[1 - ({\eta _{mn}}^{m}_{n})]^{T_{(\alpha , \beta )}} , \\ 1- \prod _{{{(n, m)=(1, 1)}}}^{(\alpha , \beta )}[1 - ({\nu _{mn}}^{m}_{n})]^{T_{(\alpha , \beta )}} \end{array}\right\} \end{aligned}$$

${\textbf {Step: 4}}$ To determine the score value of each alternative use the formula.

$$\begin{aligned} \Psi ({\textbf{r}^{{\text{\AA }}}}_{i}) = \langle {\mu _{mn}^{{\text{\AA }}}}_{i} - {\eta _{mn}^{{\text{\AA }}}}_{i} - {\nu _{mn}^{{\text{\AA }}}}_{i}\rangle \end{aligned}$$

${\textbf {Step: 5}}$ The score values for all alternatives, rank them in descending order. Higher score values indicate better alternatives.

The graphical representation of the algorithm is given below (Fig. 2):

Fig. 2

Overview of algorithmic design

Here is the pseudocode for the designed algorithm:

Algorithm 1

PFHSM-based Decision-Making Algorithm

5.1 Case Study

In essence, ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ offers a versatile and practical framework for decision-makers facing the challenges of inconsistent information and complex decision scenarios. By providing methods to manage inconsistencies, incorporate multiple perspectives, and enhance transparency, ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ contributes to more informed and robust decision-making processes in various domains, ultimately leading to better results and solutions.

During the planning stages of an energy project, it is essential that every step is meticulous and well thought out to allow the design of efficient and sustainable systems for the future. In this process, a lot has to be relied on expert opinions making the data entirely based on intuition leaving room for uncertainty. As each step involves the interpretation of numerous attributes and sub-attributes, it makes sense to use a defined system to interpret the human intuitionistic data for improved analysis and decision-making scenarios. During complicated decision-making scenarios, ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ becomes a powerful and adaptable tool. The following facts show that ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ is predominantly helpful in the circumstance of effective decision-making when it comes to the design of efficient energy systems:

• When analyzing factors like environmental impact, cost analysis, and social impacts of projects, a lot of factors under consideration are based on human intuition revealing great deal of complexity due to circumstances and conflicting opinions. Here, ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ can be used in a systemized way to analyze those opinions for effective decision-making.

• EMPTY ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ also has the ability to address data up to a sub-attribute level making it highly versatile when analyzing large datasets relating to geography and greenhouse gas emissions emitted by the energy projects.

• In energy policy design data from different sources may conflict, leading to uncertainty. ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ excels in situating such inconsistencies by providing tools to reconcile conflicting data and make informed decisions.

• Energy policies frequently engage multiple stakeholders, each with their own viewpoints and priorities. ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ facilitates the incorporation of these assorted viewpoints into the decision-making process, promoting inclusivity and consensus.

• Transparency is crucial in energy policy design to assemble public trust and support effective policy implementation. ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ based decision models augment transparency by visualizing complex relationships between attributes and sub-attributes, making it easier for stakeholders to understand and contribute to the decision process.

• EMPTY ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ allows for adaptive modeling, where the significance and relevance of attributes and sub-attributes can be adjusted based on changing.

For the practical implementation of the designed ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ structure, a case study is presented. In this case study, an organization is tasked with accessing environmental factors if a wind farm is installed at 3 different locations. As mentioned above, it is essential to develop some estimation analysis for improved decision-making and sustainability. For this purpose, 5 experts ${\textbf{v}^{\varrho }}_i = \{{\textbf{v}^{\varrho }}_1, {\textbf{v}^{\varrho }}_2, {\textbf{v}^{\varrho }}_3, {\textbf{v}^{\varrho }}_4, {\textbf{v}^{\varrho }}_5\}$ from the field are assigned to provide their expert opinion by analyzing the data for each of the attributes and sub-attributes. The attributes that are considered for the analysis for the experts are: ${\textbf{q}^{\varrho }}_{i} =$ $\{ {\textbf{q}^{\varrho }}_1$ = Factors relating to wind speed and consistency, ${\textbf{q}^{\varrho }}_2$= Land Topography, ${\textbf{q}^{\varrho }}_3$ = Proximity to water bodies, ${\textbf{q}^{\varrho }}_4$ = Climatic Conditions$,$ ${\textbf{q}^{\varrho }}_5$ = Impact on Wildlife$\}$. For improved analysis, each of the above-mentioned attributes is divided into sub-attributes allowing for a more detailed analysis. So, the following sub-attributes are selected for the study: The sets of sub-attributes are given below:

${\textbf{Q}^{\varrho }}_1$ = $\{ g_{11}$ = Extreme wind events, $g_{12}$ = Wind resource data assessment (Historical Data), $g_{13}$ = Wind Patterns, $g_{14}$ = Wind Consistency, $g_{15}$ = Wind Speed (Average)$\}$

${\textbf{Q}^{\varrho }}_2 = \{ g_{21}$ = Stability of Terrain, $g_{22}$ = Installing Elevation, $g_{23}$ = Distance from Existing Infrastructure, $g_{24}$ = Land use compatibility$, g_{25}$ = Slope $\}$

${\textbf{Q}^{\varrho }}_3 = \{ g_{31}$ = Effects of wind patterns, $g_{32}$ = Micro-climatic variations, $g_{33}$ = Environmental Constraints, $g_{34}$ = Water body proximity constraints$, g_{35}$ = Impact on local climatic conditions$\}$

${\textbf{Q}^{\varrho }}_4 = \{ g_{41}$ = Temperature Ranges, $g_{42}$ = Humidity, $g_{43}$ = Frequency of Extreme Weather Events, $g_{44}$ = Seasonal Variation in Climate$, g_{45}$ = Short Term Climate Stability$\}$,

${\textbf{Q}^{\varrho }}_5 = \{ g_{51}$ = Bird and Bat Mortality, $g_{52}$ = Habitat Disruption, $g_{53}$ = Noise Disturbance, $g_{54}$ = Electromagnetic Interference, $, g_{55}$ = Light Pollution $\}$

Based on the attributes mentioned above, ${\textbf{Q}^{\varrho }}_{i} = \textbf{Q}_{1}^{\varrho } \times \textbf{Q}_{2}^{\varrho } \times \textbf{Q}_{3}^{\varrho } \times \textbf{Q}_{4}^{\varrho } \times \textbf{Q}_{5}^{\varrho }$, there are 3125 possible outcomes but for but due to computational barriers and sake of brevity, five of the possible outcomes are addressed below:

$${\textbf{Q}^{\varrho }} = \left\{ \begin{array}{l}\ {\textbf{z}^{\varrho }}_1 = ( g_{11}, g_{21}, g_{32}, g_{44}, g_{55}),\\ {\textbf{z}^{\varrho }}_2 = (g_{14}, g_{24}, g_{33}, g_{43}, g_{52}),\\ {\textbf{z}^{\varrho }}_3 = ( g_{12}, g_{23}, g_{31}, g_{41}, g_{53}),\\ {\textbf{z}^{\varrho }}_4 = (g_{13}, g_{22}, g_{32}, g_{42}, g_{51}),\\ {\textbf{z}^{\varrho }}_5 = (g_{11}, g_{23}, g_{34}, g_{44}, g_{55} ) \end{array}\right\} $$

In the context of decision-making in investment problems, observer (decision-makers) plays a supervisory role in the decision-making process. Observers are not directly involved in making investment decisions but may have an interest in or influence over the decisions being made. Their involvement can contribute to more responsible and well-informed investment decision-making. These observers in this scenario and market experts and financial guru’s that have studied the economy and are well-aware of the developments taking place around the glove. Now, the use of picture fuzzy hypersoft matrices $\mathbb{P}\mathbb{F}_{\mathbb {HSM}}$ in investment decision-making suggests a more advanced and sophisticated approach to gathering and presenting opinions from these expert decision-makers. The structure allows for the incorporation of these opinions in the decision-making process. These opinions are then organized into $\mathbb{P}\mathbb{F}_{\mathbb {HSS}s}$. In this matrix

• (i) Rows represent the experts opinion.

• (ii) Columns represent attributes being evaluated.

• (iii) Each cell of the matrix contains a picture fuzzy hypersoft number, which represents the degree of preference or assessment of the corresponding decision-maker for a specific alternative.

Step: 01

Assume that five experts evaluate every alternative according to attributes that are suggested in the form of Picture fuzzy hypersoft matrices are given as (Matrix 36,37,38):

$$\begin{aligned} & {\textbf{r}^{\varrho }}_{1} = \left[ \begin{array}{cccccc} & {\textbf{z}^{\varrho }}_1 & {\textbf{z}^{\varrho }}_2 & {\textbf{z}^{\varrho }}_3 & {\textbf{z}^{\varrho }}_4 & {\textbf{z}^{\varrho }}_5\\ {\textbf{v}^{\varrho }}_1 & ( 0.4, 0.1, 0.5)& (0.6, 0.2, 0.1)& ( 0.3, 0.3, 0.4)& (0.3, 0.1, 0.4)& (0.8, 0.1, 0.1)\\ {\textbf{v}^{\varrho }}_2& ( 0.5, 0.2, 0.1)& (0.5, 0.3, 0.1)& (0.7, 0.1, 0.2)& (0.6, 0.2, 0.1)& (0.4, 0.3, 0.2)\\ {\textbf{v}^{\varrho }}_3& ( 0.8, 0.1, 0.1)& (0.4, 0.3, 0.2)& (0.2, 0.2, 0.4)& ( 0.4, 0.1, 0.2)& (0.7, 0.2, 0.1)\\ {\textbf{v}^{\varrho }}_4 & ( 0.6, 0.1, 0.3)& ( 0.4, 0.1, 0.3)& (0.5, 0.2, 0.2)& ( 0.6, 0.2, 0.1)& (0.4, 0.1, 0.4)\\ {\textbf{v}^{\varrho }}_5 & ( 0.7, 0.2, 0.1 )& (0.4, 0.2, 0.2)& (0.8, 0.1, 0.1)& (0.9, 0.1, 0.0)& (0.5, 0.2, 0.3)\\ \end{array}\right] \end{aligned}$$

(36)

$$\begin{aligned} & {\textbf{r}^{\varrho }}_{2} = \left[ \begin{array}{cccccc} & {\textbf{z}^{\varrho }}_1 & {\textbf{z}^{\varrho }}_2 & {\textbf{z}^{\varrho }}_3 & {\textbf{z}^{\varrho }}_4 & {\textbf{z}^{\varrho }}_5 \\ {\textbf{v}^{\varrho }}_1 & (0.8, 0.1, 0.1)& (0.4, 0.1, 0.2)& (0.4, 0.4, 0.1)& (0.5, 0.2, 0.1)& (0.4, 0.2, 0.2)\\ {\textbf{v}^{\varrho }}_2 & (0.5, 0.3, 0.1)& (0.4, 0.2, 0.1)& (0.6, 0.2, 0.1)& (0.4, 0.3, 0.3)& (0.5, 0.2, 0.3)\\ {\textbf{v}^{\varrho }}_3 & (0.5, 0.3, 0.1)& (0.5, 0.2, 0.1)& (0.5, 0.3, 0.2)& (0.5, 0.2, 0.2)& (0.5, 0.1, 0.3)\\ {\textbf{v}^{\varrho }}_4 & (0.3, 0.4, 0.2)& (0.5, 0.2, 0.2)& (0.5, 0.3, 0.1)& (0.7, 0.2, 0.1)& (0.6, 0.1, 0.2)\\ {\textbf{v}^{\varrho }}_5 & 0.6, 0.2, 0.1)& (0.4, 0.3, 0.1)& (0.4, 0.2, 0.4)& (0.6, 0.1, 0.3)& (0.3, 0.3, 0.3)\\ \end{array}\right] \end{aligned}$$

(37)

$$\begin{aligned} & {\textbf{r}^{\varrho }}_{3} = \left[ \begin{array}{cccccc} & {\textbf{z}^{\varrho }}_1 & {\textbf{z}^{\varrho }}_2 & {\textbf{z}^{\varrho }}_3 & {\textbf{z}^{\varrho }}_4 & {\textbf{z}^{\varrho }}_5 \\ {\textbf{v}^{\varrho }}_1 & (0.4, 0.1, 0.5)& (0.4, 0.1, 0.2)& (0.3, 0.3, 0.4)& (0.3, 0.1, 0.4)& (0.4, 0.1, 0.2)\\ {\textbf{v}^{\varrho }}_2 & 0.5, 0.2, 0.1)& (0.4, 0.2, 0.1)& (0.6, 0.1, 0.2)& (0.4, 0.2, 0.3)& (0.4, 0.2, 0.3)\\ {\textbf{v}^{\varrho }}_3 & (0.5, 0.1, 0.1)& (0.4, 0.2, 0.2)& (0.2, 0.2, 0.4)& (0.4, 0.1, 0.2)& (0.5, 0.1, 0.3)\\ {\textbf{v}^{\varrho }}_4 & (0.3, 0.1, 0.3)& (0.4, 0.1, 0.3)& (0.5, 0.2, 0.2)& (0.6, 0.2, 0.1)& (0.4, 0.1, 0.4)\\ {\textbf{v}^{\varrho }}_5 & (0.6, 0.2, 0.1)& (0.4, 0.2, 0.2)& (0.4, 0.1, 0.4)& (0.6, 0.1, 0.3)& (0.3, 0.2, 0.3)\\ \end{array}\right] \end{aligned}$$

(38)

Step: 02

Expert and parameter-corresponding weighted matrix ${T_{(n, m)}}$ for all alternatives is given below:

$$ {T_{(n, m)}} = \left[ \begin{array}{ccccc} 0.09& 0.01& 0.03& 0.04& 0.01\\ 0.02& 0.03& 0.07& 0.01& 0.05\\ 0.03& 0.02& 0.01& 0.07& 0.02\\ 0.08& 0.03& 0.01& 0.06& 0.05\\ 0.04& 0.06& 0.03& 0.04& 0.09\\ \end{array}\right] $$

Step: 03

We now aggregate the preferences of several alternatives using picture fuzzy hypersoft geometric aggregation operators.

${\textbf{r}^{{\text{\AA }}}}_{1} = (0.5148819061, 0.1594837374, 0.2360134381)$

${\textbf{r}^{{\text{\AA }}}}_{2} = (0.4894337849, 0.2270999550, 0.1794359199)$

${\textbf{r}^{{\text{\AA }}}}_{3} = (0.4224103592, 0.1451697325, 0.2681704837)$

Step: 04

To determine each alternative’s score value, use the formula,

$$\Psi ({\textbf{r}^{{\text{\AA }}}}_{i}) = \langle {\mu _{mn}^{{\text{\AA }}}}_{i} - {\eta _{mn}^{{\text{\AA }}}}_{i} - {\nu _{mn}^{{\text{\AA }}}}_{i}\rangle $$

$\Psi ({\textbf{r}^{{\text{\AA }}}}_{1}) = 0.11193847306$, $\Psi ({\textbf{r}^{{\text{\AA }}}}_{2}) = 0.0828979100$, $\Psi ({\textbf{r}^{{\text{\AA }}}}_{3}) = 0.0090701430$

Step: 05

An alternative with the highest rank is the best option.

${\textbf{r}^{\varrho }}_{3} \le {\textbf{r}^{\varrho }}_{2} \le {\textbf{r}^{\varrho }}_{1}$

Hence ${\textbf{r}^{\varrho }}_{1}$ is the best alternative.

6 Sensitivity Analysis

The primary goal is to demonstrate that the newly invented ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ theory and its geometric aggregate operators by subjecting them to a rigorous comparative analysis against various existing theories, providing a basis for the assertion that the newly introduced operators are more beneficial and realistic.

• The method discussed in a study by Mehmood et al. [42] involves the use of intuitionistic fuzzy soft information using membership grades, non-membership grades, and parameterization, a deficiency exists possibly due to the proposed work containing information in the form of truth, abstinence, and falsity grades with parameters and sub-parameters.

• In a study by Garg and Arora [41], the aggregation operators for PFSSs are discussed, and a deficiency is observed in the existing operator. The deficiency appears to stem from the existing operator having three grades (truth, abstinence, and falsity) with parameters in a one-dimensional form. The issue raised is that this format may not be suitable for the proposed work, which seems to involve information in the shape of truth, abstinence, and falsity grades with parameters that are further bifurcated.

• A study by Zulqarnain et al. [43] contains information on Robust Aggregation Operators for Intuitionistic Fuzzy Hypersoft Set. The deficiency highlighted in the existing operator is related to its suitability, specifically in handling information presented in the form of truth, abstinence, and falsity grades along with parameters in the proposed work.

The table 3 presents both the characteristics of the models and the sensitivity analysis of the alternatives.

Table 2 Computational complexity or decision accuracy comparison

Table 3 Table of symbols and explanations

7 Conclusion

This paper presents the concept of picture fuzzy hypersoft matrix theory and introduces major operations associated with it. We also introduced geometric aggregate operators that are more useful for conducting theoretical research in the picture fuzzy hypersoft domain. These evolved aggregation operators have the advantage of being reducible to their most basic form. ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ offers a versatile and robust framework for addressing the intricacies of energy policy design. They enable policymakers and energy experts to navigate complex attributes and sub-attributes, manage inconsistent information, and foster transparent and inclusive decision-making processes. As informed decision-making continues to evolve to meet the challenges of sustainability, security, and affordability, PFHS emerges as a valuable tool to support informed and effective decision-making. A comparative analysis has been conducted on the introduced work to demonstrate its reliability, and this structure is considered a more advanced structure. Thus, in the future, this idea can be extended to the spherical picture fuzzy hypersoft environment. Future research directions include extending ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ to more advanced environments such as spherical picture fuzzy hypersoft sets, bipolar complex ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$, and T-spherical ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$. Beyond these theoretical expansions, ${\textbf{P}^{\mathbb{F}\mathbb{H}}}_{\mathbb{S}\mathbb{M}}$ concepts can also be integrated with machine learning models for predictive analysis, applied in optimization problems for resource allocation, and employed in large-scale decision systems across sectors such as energy, digital manufacturing, and sustainable development.