Fermatean Hesitant Fuzzy Multi-Attribute Decision-Making Method with Probabilistic Information and Its Application

Abstract

When information is incomplete or uncertain, Fermatean hesitant fuzzy sets (FHFSs) can provide more information to help decision-makers deal with more complex problems. Typically, determining attribute weights assumes that each attribute has a fixed influence. Introducing probability information can enable one to consider the stochastic nature of evaluation data and better quantify the importance of the attributes. To aggregate data by considering the location and importance degrees of each attribute, this paper develops a Fermatean hesitant fuzzy multi-attribute decision-making (MADM) method with probabilistic information and an ordered weighted averaging (OWA) method. The OWA method combines the concepts of weights and sorting to sort and weigh average property values based on those weights. Therefore, this novel approach assigns weights based on the decision-maker’s preferences and introduces probabilities to assess attribute importance under specific circumstances, thereby broadening the scope of information expression. Then, this paper presents four probabilistic aggregation operators under the Fermatean hesitant fuzzy environment, including the Fermatean hesitant fuzzy probabilistic ordered weighted averaging/geometric (FHFPOWA/FHFPOWG) operators and the generalized Fermatean hesitant fuzzy probabilistic ordered weighted averaging/geometric (GFHFPOWA/GFHFPOWG) operators. These new operators are designed to quantify the importance of attributes and characterize the attitudes of decision-makers using a probabilistic and weighted vector. Then, a MADM method based on these proposed operators is developed. Finally, an illustrative example of selecting the best new retail enterprise demonstrates the effectiveness and practicality of the method.

1. Introduction

The traditional retail industry is experiencing a bottleneck due to advancements in technology and increased consumer demand. The service economy and experience economy have been valued and developed. Adapting to digital consumption demand and developing relevant business strategies has become a critical problem [1]. A new retail system characterized by consumer-centered, data-driven, and intelligent services has emerged in this context. The new retail model is fast gaining popularity due to its simplicity and security, making it a significant player in the sector [2]. In today’s retail landscape, business model innovation is crucial for enterprise growth. Traditional e-commerce platforms have adapted to the development trend of new retail, combined with artificial intelligence and Internet of Things technology to explore new retail models. Numerous retail consumption scenarios have emerged in an endless stream. New retail reconstructs the value order of business entities. With physical stores, e-commerce, and mobile Internet as the core, community retail as the front end, Internet logistics as the middle end, and big data and financial innovation as the back end, business model innovation is realized through online and offline integration [3]. Typical representatives are Amazon, JD.com, Hema Fresh, and so on. Some scholars have studied these new retail platforms by building models or frameworks [4,5]. Evaluating new retail enterprises usually involves a variety of evaluation attributes, making it a multi-attribute decision-making (MADM) problem. The market environment faced by new retail enterprises is complex and changeable, and corporate data are often uncertain and ambiguous (market feedback, competitor behavior, etc.). As a result, it is impossible to evaluate them directly using typical precise decision-making approaches, and fuzzy set theory can mitigate the impact of single criteria and subjective preferences.

The rapid development of the new retail industry and the uncertainty of market changes require relevant enterprises to respond flexibly. As decision-making problems become increasingly complex, decision-makers continuously seek more effective methods to mitigate the influence of subjective conditions on the decision process. The presence of inaccurate and incomplete information, along with the limitations of evaluators’ cognition, often results in evaluation processes filled with various uncertainties. Traditionally, evaluation information is usually presented in quantitative form. However, as decision-making problems grow more complex and human thinking remains inherently fuzzy, precise mathematical models increasingly fall short of meeting practical decision-making needs. In this context, the development of fuzzy set theory provides a new idea and method for solving real decision-making problems. Since its initial proposal, the fuzzy set (FS) [6] has aroused the interest of numerous scholars as an effective method for handling uncertain information. Related research has provided effective tools for solving complex problems in uncertain environments. Meanwhile, as real decision-making problems rapidly evolve, FSs are continually promoted and applied in various fuzzy environments. The advancements in fuzzy set theory provide significant insights and tools for addressing the uncertainties in modern decision-making processes, especially in dynamic and rapidly evolving industries like the new retail sector.

Compared to FSs that only contain a membership function, the intuitionistic fuzzy set (IFS) proposed by Atanassov [7] considers membership, non-membership, and hesitation. This allows it to describe three different evaluator attitudes: support, neutrality, and opposition. Compared with classical fuzzy sets, the IFS introduces the non-membership degree to express the degree of opposition to the same object. Therefore, it can better capture the inconsistencies and incompleteness of decision-makers’ subjective judgments and describe ambiguous information in more detail. Numerous studies have explored the capability of IFSs to handle uncertainty and fuzziness [8,9,10,11,12], including aggregation operators [8,9], distance measures [10], correlation coefficients [11], and various extensions of IFSs [12]. In practical decision-making, if the membership and non-membership degrees of the attribute values of alternatives are given separately, the decision-makers may provide a solution where the sum of the two is greater than one. Then, Yager [13] proposed the Pythagorean fuzzy set (PFS) to extend the constraints of fuzzy sets and enhance the ability to describe uncertainty. The PFS relaxes the limitation on membership and non-membership degrees to the sum of their squares being less than or equal to 1. Thus, the range of information expression is expanded. Since its appearance, the PFS has been widely used to solve MADM problems. The current research on the PFS mainly focuses on aggregation operators [14,15,16], distance and similarity measures [17,18], and other extensions of the PFS [19,20]. Although the IFS and the PFS have proven to be effective tools for solving fuzzy problems, their shortcomings become more and more apparent as the complexity of issues increases. Thus, Senapati and Yager [21] proposed the concept of the Fermatean fuzzy set (FFS). The FFS incorporates some concepts from the Fermatean theorem, allowing the sum of the cubes of the membership and non-membership degrees to be less than or equal to 1. Therefore, the FFS is more flexible and practical in characterizing complexity and uncertainty. Compared to the IFS and the PFS, the FFS can describe fuzziness more accurately, helping decision-makers consider more uncertainty. Many research models and methods have proved the effectiveness of the FFS in fuzzy decision-making, pattern recognition, cluster analysis, and so on. The research on Fermatean fuzzy MADM method is constantly emerging, including information aggregation operators [22,23,24], information measures [25,26], and extensions of FFSs [27]. Seikh and Mandal [22] developed interval-valued Fermatean fuzzy Dombi weighted averaging operators to express uncertain and vague data. Barokab et al. [23] defined some Fermatean fuzzy Einstein prioritized arithmetic and geometric aggregation operators. Debbarma et al. [24] proposed a Fermatean fuzzy weighted power average operator to identify the optimal spent lithium-ion battery recycling technique. Ganie [25] proposed several distance measures and knowledge measures of FFSs. Deng and Wang [26] innovated the Fermatean fuzzy distance measure based on the Hellinger distance and triangular divergence. Attaullah et al. [27] developed a technique based on the Fermatean hesitant fuzzy rough set. Therefore, the research on FFSs is becoming more and more abundant.

The evaluation information in traditional fuzzy sets is commonly a single value. To solve the hesitation problem of decision-makers in the decision process, Torra [28] introduced the concept of hesitant fuzzy sets (HFSs). The membership degree of an element in a hesitant fuzzy set is an interval value, which can better represent the uncertainty range of the element. As an effective tool for representing decision-makers’ preferences in fuzzy environments, several extensions of HFSs have been introduced in recent literature. These include hesitant fuzzy linguistic sets [29], hesitant intuitionistic fuzzy sets [30], Pythagorean hesitant fuzzy sets [31], and hesitant Fermatean fuzzy sets [32,33]. If a Fermatean hesitant fuzzy number {{0.3,0.5,0.8},{0.4,0.7}} is used to describe the degree to which element x belongs to the fuzzy concept of “old”. Here, x has a membership degree of 0.3, 0.5, or 0.8 and a non-membership degree of 0.4 or 0.6. Compared with single fuzzy numbers, the Fermatean hesitant fuzzy number contains more evaluation information and reflects the hesitancy of the decision-maker. Hence, FHFSs can improve the accuracy of information representation and deal with more complex uncertainty problems. Since the appearance of the FFS, some scholars have combined it with HFS to portray the decision-maker’s hesitation information more comprehensively. Mishra et al. [32] developed a modified VIKOR MADM approach under the Fermatean hesitant fuzzy environment. Lai et al. [33] defined the hesitant Fermatean fuzzy sets (HFFSs) and proposed an improved CoCoSo based on the aggregation operators and information measures of HFFSs. Ruan et al. [34] introduced several Fermatean hesitant fuzzy Heronian mean aggregation operators based on the priority relationship between different attributes. Liu and Luo [35] applied the probabilistic hesitant Fermatean fuzzy set to express the comprehensive evaluation information. Wang et al. [36] defined a series of Bonferroni mean operators under the hesitant Fermatean fuzzy environment. Ruan et al. [37] discussed various novel distance measures under the Fermatean hesitant fuzzy environment. These developments show the continued development and implementation of FHFSs in dealing with complicated decision-making settings characterized by ambiguity and hesitation.

Scholars are dedicated to enriching the theoretical foundations of Fermatean fuzzy information aggregation operators and applying these operators to handle complex MADM problems. As a crucial research direction in MADM, information aggregation operators can transform input information into a unified evaluation value. Current research on information aggregation operators primarily focuses on two directions. One is the study of mutual independence between attributes, such as weighted averaging operators and ordered weighted geometric operators. Among them, the ordered weighted averaging (OWA) operator [38] is a commonly used evaluation tool in the field of information fusion. OWA operator generates a single output value by weighting and ordering the input information. The allocation of weights depends on the relative importance of information and the preferences of decision-makers so that it can reflect the influence or preferences of different members of the group through appropriate weight assignment. Jin [39] demonstrated some new properties and representation methods for OWA operators. Yager [40] defined the centered OWA operators that give preference to argument values lying in the middle between the largest and the smallest. Zeng et al. [41] presented the induced intuitionistic fuzzy ordered weighted averaging (I-IFOWA) operator and its application in DM. Xu et al. [42] proposed a Pythagorean fuzzy induced generalized ordered weighted averaging (PFIGOWA) operator. Liu et al. [43] presented the Fermatean fuzzy fairly weighted averaging and Fermatean fuzzy ordered weighted averaging operators. Additionally, several new types and generalizations of OWA operators have been discussed, including GOWA [44], IOWA [45], and linguistic OWA operators [46].

Traditional OWA methods assume that the weights and importance of attributes are fixed. This is often not the case in the real world. However, individual ambiguity and inadequate knowledge can negatively impact decision-making, especially when information is insufficient. Introducing probabilistic information into different fuzzy sets becomes an effective solution. Probabilistic information helps evaluate the likelihood of an event, while fuzzy sets handle the uncertainty and ambiguity of information. Combining these two approaches enables decision-makers to make more accurate and flexible decisions under uncertain conditions. For this reason, scholars developed the concepts of the probabilistic hesitant fuzzy set [47], the probabilistic interval-valued intuitionistic hesitant fuzzy set [48], the probabilistic Pythagorean fuzzy set [49], the probabilistic interval-valued Fermatean hesitant fuzzy set [50], and so on. Existing research has primarily focused on information aggregation methods for fuzzy sets and corresponding MADM methods. In these decision problems, it is typically assumed that each attribute’s influence on the decision result is fixed. This assumption may overlook the uncertainty and randomness of the evaluation data. To reflect the importance of attributes more accurately, scholars have introduced probabilistic information into aggregation operators. Merigo [51] unified the weights of probability and the OWA operator and proposed the probabilistic OWA operator (POWA). Wei and Merigo [52] developed some probabilistic weighted aggregation operators under the intuitionistic fuzzy environment. Zeng [53] presented a Pythagorean fuzzy probabilistic ordered weighted averaging (PFPOWA) operator that considers the probabilities and the OWA in the same formulation. Espinoza-Audelo et al. [54] proposed the Bonferroni probabilistic ordered weighted average (B-POWA) operator and used it to calculate the error between the price of an agricultural commodity. Casanovas et al. [55] investigated an induced ordered weighted average distance operator. In real-world decision-making, the importance of attributes may be adjusted as the environment or the decision-maker’s preferences change. Probabilistic information can capture the uncertainty and randomness of evaluation data. The combination of probability and OWA operators can comprehensively consider the importance of each attribute during the aggregation process and improve the accuracy of decision-making results. To assess the influence degree of each attribute, this paper introduces a probabilistic and weighted weighting vector into the Fermatean hesitant fuzzy environment and develops several weighted aggregation operators with probabilistic information.

Therefore, this paper aims to unify probabilistic information and the weighted vector to develop several new ordered weighted aggregation operators for FHFSs. These operators include the Fermatean hesitant fuzzy probabilistic ordered weighted averaging/geometric (FHFPOWA/FHFPOWG) operators and generalized Fermatean hesitant fuzzy probabilistic ordered weighted averaging/geometric (GFHFPOWA/GFHFPOWG) operators. Then, these new operators are used to aggregate evaluation information, and a new MADM method is proposed. This method quantifies the importance of each attribute more accurately and offers decision-makers more flexibility by assigning different values to the parameters. Therefore, the rationality and quality of the method based on FHFSs are further improved. To illustrate the application of the proposed method, a numerical example is provided for evaluating the business model innovation of new retail enterprises. The contributions of this paper are summarized as follows: (1) A Fermatean hesitant fuzzy MADM method with probabilistic information and ordered weighted averaging (OWA) method is developed. (2) New information aggregation operators for FHFSs that unify probabilistic information and weighted vectors are introduced and are named FHFPOWA, GFHFPOWA, FHFPOWG, and GFHFPOWG operators. (3) By assigning different values to the parameter, the new method can provide decision-makers with more options. Hence, this work provides a framework for improving decision-making processes in uncertain and complex environments.

The remainder of this paper is organized as follows. Some basic concepts and operational laws of FFSs are briefly reviewed in Section 2. In Section 3, the Fermatean hesitant fuzzy probabilistic ordered weighted average (FHFPOWA) and generalized Fermatean hesitant fuzzy probabilistic ordered weighted average (GFHFPOWA) operators are proposed. The Fermatean hesitant fuzzy probabilistic ordered weighted geometric (FHFOWG) and generalized Fermatean hesitant fuzzy probabilistic ordered weighted geometric (GFHFOWG) operators are defined in Section 4. Section 5 introduces a MADM process and presents an illustrative example. Section 6 concludes this paper with some comments.

2. Preliminaries

Definition 1 

([22]).  Let $X$ be a nonempty set. A Fermatean fuzzy set (FFS) $F$ on $X$ is defined as follows:

$$F=\{(x,{\xi }_F(x),{\phi }_F(x)):x\in X\}$$

(1)

where ${\xi }_F(x):X\to [0,1]$ and ${\phi }_F(x):X\to [0,1]$ are, respectively, the membership degree and non-membership degree of each element $x\in X$ to $F$ satisfying the condition that $0\le {\xi }_F^3(x)+{\phi }_F^3(x)\le 1$. The indeterminacy degree of each element $x$ of the set $F$ is ${\pi }_F(x)=\sqrt[3]{1-{\xi }_F^3(x)-{\phi }_F^3(x)}$. For convenience, $F=(\xi ,\phi )$ is called a Fermatran fuzzy number (FFN), and the set of all Fermatean numbers is denoted as $\Omega$.

Definition 2 

([22]).  Let $F=(\xi ,\phi )$, $F_1=({\xi }_1,{\phi }_1)$, and $F_2=({\xi }_2,{\phi }_2)$ be three Fermatean fuzzy numbers (FFNs), where $\xi$ and $\phi$ are the membership and non-membership degrees of each element $x\in X$ to an FFS, respectively. Then, some basic operations are defined as follows: 

(1)

$F_1\cup F_2=\left(\max \{{\xi }_1,{\xi }_2\},\min \{{\phi }_1,{\phi }_2\}\right)$;

(2)

$F_1\cup F_2=\left(\min \{{\xi }_1,{\xi }_2\},\max \{{\phi }_1,{\phi }_2\}\right)$;

(3)

$F^c=(\phi ,\xi )$, $c>0$;

(4)

$F_1\subseteq F_2$, if and only if ${\mu }_1>{\mu }_2,{\nu }_1<{\nu }_2$.

Definition 3 

([22]).  Let $F=(\xi ,\phi )$, $F_1=({\xi }_1,{\phi }_1)$, and $F_2=({\xi }_2,{\phi }_2)$ be three FFNs, where $\xi$ and $\phi$ are the membership and non-membership degrees of each element $x\in X$ of an FFS, respectively. Several basic operations between them are given as follows: 

(1)

$F_1\oplus F_2=\left(\sqrt[3]{{\xi }_1^3+{\xi }_2^3-{\xi }_1^3{\xi }_2^3},{\phi }_1{\phi }_2\right)$;

(2)

$F_1\otimes F_2=\left({\xi }_1{\xi }_2,\sqrt[3]{{\phi }_1^3+{\phi }_2^3-{\phi }_1^3{\phi }_2^3}\right)$;

(3)

$cF=\left(\sqrt[3]{1-{(1-{\xi }_3)}^c},{\phi }^3\right)$, $c>0$;

(4)

$F^c=\left({\xi }^3,\sqrt[3]{1-{(1-{\phi }^3)}^c}\right)$, $c>0$.

Zadeh [6] initially attempted to use the membership function to express uncertain information. The membership information in traditional fuzzy sets is given as a single value. To address the significant uncertainty in the decision-making process, particularly when the evaluated object has multiple evaluation indicators, Torra [28] proposed the concept of HFS. This concept allows the membership of each element belonging to a set to have multiple different values.

Definition 4 

([28]).  Let $X$ be a fixed set. A hesitant fuzzy set (HFS) $E$ on $X$ is a structure of the following form: 

$$E=\left\{\left.〈x,h_E(x)〉\right|x\in X\right\}$$

(2)

where $h_E(x)$ is the set of several values belonging to [0,1], denoting some possible membership degrees of the element $x\in X$ of the set $E$.

Definition 5 

([37]).  Let $X$ be a universe of discourse. A Fermatean hesitant fuzzy set (FHFS) $Z$ on $X$ can be defined as follows: 

$$Z=\left(\left.〈x,U_Z(x),V_Z(x)〉\right|x\in X\right)$$

(3)

where  $U_Z(x)$ and  $V_Z(x)$ are nonempty finite subsets on $[0,1]$ and represent the set of all possible membership and non-membership degrees of $x$ to the set $Z$, respectively. $\forall {\mu }_Z(x)\in U_Z(x)$ and $\forall {\nu }_Z(x)\in V_Z(x)$, the condition $0\le {\mu }_Z^3(x)+{\nu }_Z^3(x)\le 1$ holds for each element of  $Z$. For  $x\in X$,  $〈{\mu }_Z(x),{\nu }_Z(x)〉$ is called a Fermatean hesitant fuzzy number (FHFN). For convenience, this paper denotes it as  $\alpha =〈{\mu }_{\alpha },{\nu }_{\alpha }〉$ and denotes  $\Phi$ as all Fermatean hesitant fuzzy numbers.

Definition 6 

([37]).  Let $\alpha =〈{\mu }_{\alpha },{\nu }_{\alpha }〉(i=1,2,\dots ,n)$ be a FHFN. $\left|{\mu }_{\alpha }\right|$ and $\left|{\nu }_{\alpha }\right|$ are the cardinal numbers of the set ${\mu }_{\alpha }$ and ${\nu }_{\alpha }$. Then, the score function $S_{\alpha }$ and accuracy function $P_{\alpha }$ are defined as follows:

$$S_{\alpha }=\frac{1}{\left|{\mu }_{\alpha }\right|}{\displaystyle \sum _{\mu \in U_{\alpha }}{\mu }^3-}\frac{1}{\left|{\nu }_{\alpha }\right|}{\displaystyle \sum _{v\in V_{\alpha }}{v}^3}$$

(4)

$$P_{\alpha }=\frac{1}{\left|{\mu }_{\alpha }\right|}{\displaystyle \sum _{\mu \in U_{\alpha }}{\mu }^3+}\frac{1}{\left|{\nu }_{\alpha }\right|}{\displaystyle \sum _{v\in V_{\alpha }}{v}^3}$$

(5)

Theorem 1 

([37]).  Let $\alpha =〈{\mu }_{\alpha },{\nu }_{\alpha }〉$ and $\beta =〈{\mu }_{\beta },{\nu }_{\beta }〉$ be two different FHFNs, then

(1) 

If $S_{\alpha }>S_{\beta }$, then $\alpha \succ \beta$;

(2) 

If $S_{\alpha }<S_{\beta }$, then  $\alpha \prec \beta$;

(3) 

If $S_{\alpha }=S_{\beta }$, then

(i) 

If $P_{\alpha }>P_{\beta }$, then $\alpha \succ \beta$

(ii) 

If $P_{\alpha }<P_{\beta }$, then $\alpha \prec \beta$.

Definition 7 

([37]).  Let $\alpha =〈{\mu }_{\alpha },{\nu }_{\alpha }〉$, ${\alpha }_1=〈{\mu }_{{\alpha }_1},{\nu }_{{\alpha }_1}〉$, and ${\alpha }_2=〈{\mu }_{{\alpha }_2},{\nu }_{{\alpha }_2}〉$ be three different FHFNs, and let $\lambda >0$. Then, 

(1)

${\alpha }_1\oplus {\alpha }_2=〈{\displaystyle \underset{{\mu }_{{\alpha }_1}\in U_{{\alpha }_1},{\mu }_{{\alpha }_2}\in U_{{\alpha }_2}}{\cup }\left\{\sqrt[3]{{\mu }_{{\alpha }_1}^3+{\mu }_{{\alpha }_2}^3-{\mu }_{{\alpha }_1}^3{\mu }_{{\alpha }_2}^3}\right\},{\displaystyle \underset{v_{{\alpha }_1}\in V_{{\alpha }_1},{\nu }_{{\alpha }_2}\in V_{{\alpha }_2}}{\cup }\left\{{\nu }_{{\alpha }_1}{\nu }_{{\alpha }_2}\right\}}}〉$;

(2)

${\alpha }_1\otimes {\alpha }_2=〈{\displaystyle \underset{{\mu }_{{\alpha }_1}\in U_{{\alpha }_1},{\mu }_{{\alpha }_2}\in U_{{\alpha }_2}}{\cup }\left\{{\mu }_{{\alpha }_1}{\mu }_{{\alpha }_2}\right\},{\displaystyle \underset{v_{{\alpha }_1}\in V_{{\alpha }_1},{\nu }_{{\alpha }_2}\in V_{{\alpha }_2}}{\cup }\left\{\sqrt[3]{{\nu }_{{\alpha }_1}^3+{\nu }_{{\alpha }_2}^3-{\nu }_{{\alpha }_1}^3{\nu }_{{\alpha }_2}^3}\right\}}}〉$;

(3)

$\lambda \alpha =〈{\displaystyle \underset{{\mu }_{\alpha }\in U_{\alpha }}{\cup }\left\{\sqrt[3]{1-{(1-{\mu }_{\alpha }^3)}^{\lambda }}\right\},{\displaystyle \underset{{\nu }_{\alpha }\in V_{\alpha }}{\cup }\left\{{\nu }_{\alpha }^{\lambda }\right\}}}〉$;

(4)

${\alpha }^{\lambda }=〈{\displaystyle \underset{{\mu }_{\alpha }\in U_{\alpha }}{\cup }\left\{{\mu }_{\alpha }^{\lambda }\right\},{\displaystyle \underset{{\nu }_{\alpha }\in V_{\alpha }}{\cup }\left\{\sqrt[3]{1-{(1-{\nu }_{\alpha }^3)}^{\lambda }}\right\}}}〉$.

3. Generalized Fermatean Hesitant Fuzzy Probabilistic Ordered Weighted Averaging Operator

Definition 8 

([38]).  Let $R$ be a set of real numbers. An OWA operator of dimension $n$ is a mapping $OWA:R^n\to R$ such that the following holds:

$$OWA(a_1,a_2,\dots ,a_n)={\displaystyle \sum _{j=1}^n{\omega }_j{b}_j}$$

(6)

where $b_j$ is the j-th largest element of $a_i(i=1,2,\dots ,n)$ and $\omega$ is the associated weighting vector with ${\omega }_j\in \left[0,1\right]$ and ${\displaystyle \sum _{j=1}^n{\omega }_j=1}$.

The OWA operator processes a set of numbers in descending order. The weight is related to the number in the j-th position instead of the data. Zeng [53] further proposes the probabilistic ordered weighted averaging (POWA) operator, which provides a parameterized aggregation operator between the maximum and minimum operators.

Definition 9 

([53]).  Let $R$ be a set of real numbers. A POWA operator of dimension $n$ is a mapping $POWA:R^n\to R$ with an associated weighting vector $\omega$ satisfying ${\omega }_j\in \left[0,1\right]$ and ${\displaystyle \sum _{j=1}^n{\omega }_j=1}$ such that the following holds:

$$POWA(a_1,a_2,\dots ,a_n)={\displaystyle \sum _{j=1}^n{\overline{p}}_j{b}_j}$$

(7)

where $b_j$ is the j-th largest element of $a_i(i=1,2,\dots ,n)$ and each $a_i$ has an associated probability $p_i$ with ${\displaystyle \sum _{i=1}^n{p}_i}=1$ and $p_i\in \left[0,1\right]$. ${\overline{p}}_j=\sigma {\omega }_j+(1-\sigma )p_j$ with $\sigma \in \left[0,1\right]$, and $p_j$ is the ordered probability $p_i$ of $b_j$.

When $\sigma$ takes specific values, the POWA operator degenerates into different aggregation operators. Especially when $\sigma =1$, the POWA operator reduces to the OWA operator. When $\sigma =0$, the POWA operator reduces to the probabilistic aggregation.

Definition 10. 

Let ${\alpha }_i=〈{\mu }_{{\alpha }_i},{\nu }_{{\alpha }_i}〉(i=1,2,\dots ,n)$ be a set of FHFNs. A Fermatean hesitant fuzzy probabilistic ordered weighted averaging (FHFPOWA) operator for ${\alpha }_i$ is a mapping $FHFPOWA:{\Phi }^n\to \Phi$, such that

$$FHFPOWA({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)=\underset{j=1}{\overset{n}{\oplus }}{\overline{p}}_j{\beta }_j$$

(8)

where $\omega =({\omega }_1,{\omega }_2,\dots ,{\omega }_n)$ is the associated weight vector with ${\omega }_j\in \left[0,1\right]$ and ${\displaystyle \sum _{j=1}^n{\omega }_j=1}$ and ${\beta }_j$ is the j-th largest element of $a_i$. Each $a_i$ has an associated probability $p_i$ with ${\displaystyle \sum _{i=1}^n{p}_i}=1$ and $p_i\in \left[0,1\right]$. ${\overline{p}}_j=\sigma w_j+(1-\sigma )p_j$, where $p_j$ is the ordered probability corresponding to ${\beta }_j$ and $\sigma \in \left[0,1\right]$.

Theorem 2. 

The FHFPOWA operator has different forms when parameters $\sigma$ and $w$ take specific values.

(1)

When $\sigma =1$, the FHFPOWA operator reduces to the Fermatean hesitant fuzzy OWA (FHFOWA) operator.

(2)

When $\sigma =0$, the FHFPOWA operator reduces to the Fermatean hesitant fuzzy probabilistic averaging (FHFPA) operator.

(3)

When ${\omega }_1=1$ and ${\omega }_j=0(j\ne 1)$, the FHFPOWA operator denotes the Fermatean hesitant fuzzy probabilistic maximum (FHFMax) operator.

(4)

When ${\omega }_n=1$ and $w_j=0(j\ne n)$, the FHFPOWA operator denotes the Fermatean hesitant fuzzy probabilistic minimum (FHFMin) operator. 

Definition 11. 

Let ${\alpha }_i=〈{\mu }_{{\alpha }_i},{\nu }_{{\alpha }_i}〉(i=1,2,\dots ,n)$ be a set of FHFNs. A Fermatean hesitant fuzzy probabilistic ordered weighted averaging (FHFPOWA) operator for ${\alpha }_i$ is a mapping $FHFPOWA:{\Phi }^n\to \Phi$ , such that

$$FHFPOWA({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)=\underset{j=1}{\overset{n}{\oplus }}\sigma {\omega }_j{\beta }_j\oplus \underset{i=1}{\overset{n}{\oplus }}(1-\sigma )p_i{a}_i$$

(9)

where $\omega =({\omega }_1,{\omega }_2,\dots ,{\omega }_n)$ is the associated weight vector with ${\omega }_j\in \left[0,1\right]$ and ${\displaystyle \sum _{j=1}^n{\omega }_j=1}$ and $p_i=(p_1,p_2,\dots ,p_n)$ is the probabilistic vector with $p_i\in \left[0,1\right]$ and ${\displaystyle \sum _{i=1}^n{p}_i}=1$. ${\beta }_j$ is the j-th largest element of $a_i$. ${\overline{p}}_j=\sigma {\omega }_j+(1-\sigma )p_j$ with $\sigma \in \left[0,1\right]$ and $p_j$ is the associated probability of ${\beta }_j$.

Next, we will give a specific example to prove that Definitions 10 and 11 are identical.

Example 1. 

Consider three FHFNs: ${\alpha }_1=<\{0.6,0.8\},\{0.5,0.7\}>$, ${\alpha }_2=<\{0.5,0.8\},\left\{0.6\right\}>$, and ${\alpha }_3=<\{0.6,0.9\},\left\{0.5\right\}>$. The weighting vector is $\omega =(0.3,0.3,0.4)$ and the probabilistic vector is $p=(0.5,0.2,0.3)$, $\sigma =0.6$. The order of ${\alpha }_i$ is obtained according to Theorem 1: ${\alpha }_3\succ {\alpha }_1\succ {\alpha }_2$. Then, we can obtain ${\overline{p}}_1=0.6\times 0.3+0.4\times 0.3=0.30$, ${\overline{p}}_2=0.6\times 0.3+0.4\times 0.5=0.38$, and ${\overline{p}}_3=0.6\times 0.4+0.4\times 0.2=0.32$.

According to Definition 10, the aggregation process is given as follows:

$$\begin{array}{l}FHFPOWA({\alpha }_1,{\alpha }_2,{\alpha }_3)\\ =0.30\times \alpha <\{0.6,0.9\},\left\{0.5\right\}>+0.38\times \alpha <\{0.6,0.8\},\{0.5,0.7\}>+0.32\times \alpha <\{0.5,0.8\},\left\{0.6\right\}>\\ =\alpha <\left\{0.5728,0.6853,0.7426,0.7973,0.6885,0.7591,0.7991,0.8392\right\},\left\{0.5300,0.6023\right\}>\end{array}$$

According to Definition 11, the aggregation process is given as follows:

$$\begin{array}{l}FHFPOWA({\alpha }_1,{\alpha }_2,{\alpha }_3)\\ =0.60\times (0.30\times \alpha <\{0.6,0.9\},\{0.5\}>+0.30\times \alpha <\{0.6,0.8\},\{0.5,0.7\}>+0.40\times \alpha <\{0.5,0.8\},\{0.6\}>)+\\ 0.40\times (0.50\times \alpha <\{0.6,0.9\},\{0.5\}>+0.20\times \alpha <\{0.6,0.8\},\{0.5,0.7\}>+0.30\times \alpha <\{0.5,0.8\},\{0.6\}>)\\ =\alpha <\left\{0.5728,0.6853,0.7426,0.7973,0.6885,0.7591,0.7991,0.8392\right\},\left\{0.5300,0.6023\right\}>\end{array}$$

The result is the same using Definitions 10 and 11. Therefore, Definitions 10 and 11 are equivalent.

Theorem 3. 

Let ${\alpha }_i=〈{\mu }_{{\alpha }_i},{\nu }_{{\alpha }_i}〉(i=1,2,\dots ,n)$ be a set of FHFNs, then

$$FHFPOWA({\alpha }_1,{\alpha }_2,\cdot \cdot \cdot ,{\alpha }_n)=〈{\displaystyle \underset{{\mu }_{{\alpha }_i}\in U_{{\alpha }_i}}{\cup }\left\{\sqrt[3]{1-{\displaystyle \prod _{j=1}^n(1-{\mu }_{{\beta }_j}^3}{)}^{{\overline{p}}_j}}\right\},}{\displaystyle \underset{{\nu }_{{\alpha }_i}\in V_{{\alpha }_i}}{\cup }\left\{{\displaystyle \prod _{j=1}^n{\nu }_{{\beta }_j}^{{\overline{p}}_j}}\right\}}〉$$

(10)

where $\omega =({\omega }_1,{\omega }_2,\dots ,{\omega }_n)$ is the weighting vector with ${\omega }_j\in \left[0,1\right]$ and ${\displaystyle \sum _{j=1}^n{\omega }_j=1}$ and $p=(p_1,p_2,\dots ,p_n)$ is the probabilistic vector with $p_i\in \left[0,1\right]$ and ${\displaystyle \sum _{i=1}^n{p}_i}=1$. ${\beta }_j$ is the j-th largest element of $a_i$. ${\overline{p}}_j=\sigma {\omega }_j+(1-\sigma )p_j$ with $\sigma \in \left[0,1\right]$, and $p_j$ is the ordered probability $p_i$ related to ${\beta }_j$.

Proof. 

When $n=2$, we have ${\alpha }_1=〈{\mu }_1,{\nu }_1〉$ and ${\alpha }_2=〈{\mu }_2,{\nu }_2〉$. Then,

$${\overline{p}}_1{\alpha }_1=〈\sqrt[3]{1-{(1-{\mu }_1^3)}^{{\overline{p}}_1}},{\nu }_1^{{\overline{p}}_1}〉,{\overline{p}}_2{\alpha }_2=〈\sqrt[3]{1-{(1-{\mu }_2^3)}^{{\overline{p}}_2}},{\nu }_2^{{\overline{p}}_2}.〉$$

By the operational rules of FHFNs, we obtain

$$\begin{array}{ll}FHFPOWA({\alpha }_1,{\alpha }_2)& ={\overline{p}}_1{\beta }_1\oplus {\overline{p}}_2{\beta }_2\\ & =〈\sqrt[3]{1-{(1-{\mu }_1^3)}^{{\overline{p}}_1}},{\nu }_1^{{\overline{p}}_1}〉\oplus 〈\sqrt[3]{1-{(1-{\mu }_2^3)}^{{\overline{p}}_2}},{\nu }_2^{{\overline{p}}_2}〉\\ & =〈\sqrt[3]{1-{\displaystyle \prod _{j=1}^2{(1-{\mu }_{{\beta }_j}^3)}^{{\overline{p}}_j}}},{\displaystyle \prod _{j=1}^2{\nu }_{{\beta }_j}^{{\overline{p}}_j}}〉\end{array}$$

If Equation (10) holds for $n=k$, we have

$$\begin{array}{ll}FHFPOWA({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_k)& ={\overline{p}}_1{\beta }_1\oplus {\overline{p}}_2{\beta }_2\oplus \dots \oplus {\overline{p}}_k{\beta }_k\\ & =〈\sqrt[3]{1-{\displaystyle \prod _{j=1}^k{(1-{\mu }_{{\beta }_j}^3)}^{{\overline{p}}_j}}},{\displaystyle \prod _{j=1}^k{\nu }_{{\beta }_j}^{{\overline{p}}_j}}〉\end{array}$$

Now, for $n=k+1$, we can obtain

$$\begin{array}{ll}FHFPOWA({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{k+1})& =\left({\overline{p}}_1{\beta }_1\oplus {\overline{p}}_2{\beta }_2\oplus \dots \oplus {\overline{p}}_k{\beta }_k\right)\oplus {\overline{p}}_{k+1}{\beta }_{k+1}\\ & =〈\sqrt[3]{1-{\displaystyle \prod _{j=1}^k{(1-{\mu }_{{\beta }_j}^3)}^{{\overline{p}}_j}}},{\displaystyle \prod _{j=1}^k{\nu }_{{\beta }_j}^{{\overline{p}}_j}}〉\oplus 〈\sqrt[3]{1-{(1-{\mu }_{{\beta }_{k+1}}^3)}^{{\overline{p}}_{k+1}}},{\nu }_{{\beta }_{k+1}}^{{\overline{p}}_{k+1}}〉\\ & =〈\sqrt[3]{1-{\displaystyle \prod _{j=1}^{k+1}{(1-{\mu }_{{\beta }_j}^3)}^{{\overline{p}}_j}}},{\displaystyle \prod _{j=1}^{k+1}{\nu }_{{\beta }_j}^{{\overline{p}}_j}}〉\end{array}$$

Thus, Theorem 1 can be proved using mathematical induction. □

This paper provides an example to specify how to use the FHFPOWA operator to aggregate information.

Example 2. 

To enhance its operational management, a new retail enterprise has invited several decision-making experts to assess its marketing strategy capability (C₁), brand image maintenance (C₂), supply chain management capability (C₃), and data analysis capability (C₄). Among them, expert A gives evaluation information through investigation and analysis. The evaluation information under each evaluation indicator is represented as α_i = {α₁, α₂, α₃, α₄}. Specifically, ${\alpha }_1=<\{0.3,0.5,0.8\},\{0.6,0.7\}>$, ${\alpha }_2=<\{0.5,0.8\},\{0.3,0.4\}>$, ${\alpha }_3=<\{0.7,0.9\},\left\{0.6\right\}>$, and ${\alpha }_4=<\{0.4,0.7\},\{0.3,0.5,0.7\}>$. The weighting vector and probabilistic vector given by the experts are ω = (0.3, 0.2, 0.3, 0.2) and p = (0.4, 0.1, 0.2, 0.3), σ = 0.7. The order of α_i is obtained according to Theorem 1: ${\alpha }_3\succ {\alpha }_2\succ {\alpha }_4\succ {\alpha }_1$. Then, we can obtain the following results by ${\overline{p}}_j=\sigma {\omega }_j+(1-\sigma )p_j$:

$${\overline{p}}_1=0.7\times 0.3+0.3\times 0.2=0.27$$

$${\overline{p}}_2=0.7\times 0.2+0.3\times 0.1=0.17$$

$${\overline{p}}_3=0.7\times 0.3+0.3\times 0.3=0.30$$

$${\overline{p}}_4=0.7\times 0.2+0.3\times 0.4=0.26$$

Using the FHFPOWA operator to aggregate information, we can obtain the following:

$$h(\mu )=\left\{\begin{array}{l}0.1675,0.3013,0.3445,0.2450,0.2461,0.3673,\\ 0.4065,0.5018,0.1901,0.3203,0.3624,0.2656,\\ 0.2667,0.3845,0.4226,0.5154,0.3042,0.4160,\\ 0.4522,0.3690,0.3700,0.4712,0.5040,0.5837\end{array}\right\}$$

$$h(\nu )=\left\{\begin{array}{l}0.0813,0.1287,0.1743,0.0941,0.1491,0.2481,\\ 0.0917,0.1452,0.1965,0.1062,0.1681,0.2798\end{array}\right\}$$

Theorem 4 

(Monotonicity).  Let ${\alpha }_i=〈{\mu }_{{\alpha }_i},{\nu }_{{\alpha }_i}〉$ and ${\gamma }_i=〈{\mu }_{{\gamma }_i},{\nu }_{{\gamma }_i}〉(i=1,2,\dots ,n)$ be two sets of FHFNs, where $U_{{\alpha }_i}=\left\{{\mu }_{{\alpha }_{i1}},{\mu }_{{\alpha }_{i2}},\cdot \cdot \cdot ,{\mu }_{{\alpha }_{ik}}\right\}$, $V_{{\alpha }_i}=\left\{{\nu }_{{\alpha }_{i1}},{\nu }_{{\alpha }_{i2}},\cdot \cdot \cdot ,{\nu }_{{\alpha }_{is}}\right\}$, $U_{{\gamma }_i}=\left\{{\mu }_{{\gamma }_{i1}},{\mu }_{{\gamma }_{i2}},\cdot \cdot \cdot ,{\mu }_{{\gamma }_{ik}}\right\}$, $V_{{\gamma }_i}=\left\{{\nu }_{{\gamma }_{i1}},{\nu }_{{\gamma }_{i2}},\cdot \cdot \cdot ,{\nu }_{{\gamma }_{is}}\right\}$, $m=1,2,\cdot \cdot \cdot ,k$, and $n=1,2,\cdot \cdot \cdot ,s$. If ${\mu }_{{\alpha }_i}\ge {\mu }_{{\gamma }_i}$ and ${\nu }_{{\alpha }_i}\le {\nu }_{{\gamma }_i}$, then

$$FHFPOWA({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)\ge FHFPOWA({\gamma }_1,{\gamma }_2,\dots ,{\gamma }_n)$$

(11)

Proof. 

Based on Definition 3, we have the following:

$$FHFPOWA\left({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n\right)=\underset{j=1}{\overset{n}{\otimes }}{\beta }_j^{{\overline{p}}_j}=〈{\displaystyle \underset{{\mu }_{{\alpha }_i}\in U_{{\alpha }_i}}{\cup }\left\{\sqrt[3]{1-{\displaystyle \prod _{j=1}^n{\left(1-{\mu }_{{\beta }_j}^3\right)}^{{\overline{p}}_j}}}\right\}},{\displaystyle \underset{{\nu }_{{\alpha }_i}\in V_{{\alpha }_i}}{\cup }\left\{{\displaystyle \prod _{j=1}^n{\nu }_{{\beta }_j}^{{\overline{p}}_j}}\right\}}〉=\alpha$$

$$FHFPOWA\left({\gamma }_1,{\gamma }_2,\dots ,{\gamma }_n\right)=\underset{j=1}{\overset{n}{\otimes }}{\delta }_j^{{\overline{p}}_i}=〈{\displaystyle \underset{{\mu }_{{\gamma }_i}\in U_{{\gamma }_i}}{\cup }\left\{\sqrt[3]{1-{\displaystyle \prod _{j=1}^n{\left(1-{\mu }_{{\delta }_j}^3\right)}^{{\overline{p}}_j}}}\right\}},{\displaystyle \underset{{\nu }_{{\gamma }_i}\in V_{{\gamma }_i}}{\cup }\left\{{\displaystyle \prod _{j=1}^n{\nu }_{{\delta }_j}^{{\overline{p}}_j}}\right\}}〉=\gamma$$

where ${\beta }_j$ and ${\delta }_j$ denote, respectively, the j-th largest element of ${\alpha }_i$ and ${\gamma }_i$.

Suppose that ${\mu }_{{\alpha }_i}\ge {\mu }_{{\gamma }_i}$ and ${\nu }_{{\alpha }_i}\le {\nu }_{{\gamma }_i}$, then $S_{\alpha }>S_{\gamma }$.

Based on Theorem 1, we have

$$FHFPOWA({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)\ge FHFPOWA({\gamma }_1,{\gamma }_2,\dots ,{\gamma }_n)$$

□

Theorem 5 

(Boundedness).  Let ${\alpha }_i=〈{\mu }_{{\alpha }_i},{\nu }_{{\alpha }_i}〉(i=1,2,\dots ,n)$ be a set of FHFNs. Suppose that ${\alpha }^{+}=〈{\mu }^{+},{\nu }^{-}〉$ and ${\alpha }^{-}=〈{\mu }^{-},{\nu }^{+}〉$, where ${\mu }^{+}=\max \left({\mu }_1,{\mu }_2,\cdot \cdot \cdot ,{\mu }_n\left|\mu \in U_{\alpha }\right.\right)$, ${\mu }^{-}=\min \left({\mu }_1,{\mu }_2,\cdot \cdot \cdot ,{\mu }_n\left|\mu \in U_{\alpha }\right.\right)$, ${\nu }^{+}=\max \left({\nu }_1,{\nu }_2,\cdot \cdot \cdot ,{\nu }_n\left|\nu \in V_{\alpha }\right.\right)$, ${\nu }^{-}=\min \left({\nu }_1,{\nu }_2,\cdot \cdot \cdot ,{\nu }_n\left|\nu \in V_{\alpha }\right.\right)$, then

$${\alpha }^{-}\le FHFPOWA({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)\le {\alpha }^{+}$$

(12)

Proof. 

Let ${\alpha }^{+}=〈{\mu }^{+},{\nu }^{-}〉=A$ and ${\alpha }^{-}=〈{\mu }^{-},{\nu }^{+}〉=B$, we have

$$FHFPOWA({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)=\underset{j=1}{\overset{n}{\oplus }}{\overline{p}}_j{\beta }_j\le \underset{j=1}{\overset{n}{\oplus }}{\overline{p}}_{j}A=A\underset{j=1}{\overset{n}{\oplus }}{\overline{p}}_j,$$

$$FHFPOWA({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)=\underset{j=1}{\overset{n}{\oplus }}{\overline{p}}_j{\beta }_j\ge \underset{j=1}{\overset{n}{\oplus }}{\overline{p}}_{j}B=B\underset{j=1}{\overset{n}{\oplus }}{\overline{p}}_j.$$

Since $\underset{j=1}{\overset{n}{\oplus }}{\overline{p}}_j=1$, then

$$FHFPOWA({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)\le A,FHFPOWA({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)\ge B$$

Therefore, we can obtain

$${\alpha }^{-}\le FHFPOWA({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)\le {\alpha }^{+}$$

□

Theorem 6 

(Idempotency).  Let ${\alpha }_i=〈U_{{\alpha }_i},V_{{\alpha }_i}〉$ be a set of FHFNs, $i=1,2,\cdot \cdot \cdot ,n$. If ${\alpha }_i=\alpha =〈U_{{\alpha }_i},V_{{\alpha }_i}〉$ for all $i$, then

$$FHFPOWA({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)=\alpha$$

(13)

Proof. 

Since ${\alpha }_i=\alpha =〈U_{{\alpha }_i},V_{{\alpha }_i}〉$ for all $i$, we have

$$FHFPOWA({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)=\underset{j=1}{\overset{n}{\oplus }}{\overline{p}}_j{\alpha }_j=\underset{j=1}{\overset{n}{\oplus }}{\overline{p}}_j\alpha =\alpha \underset{j=1}{\overset{n}{\oplus }}{\overline{p}}_j$$

Given that ${\displaystyle \sum _{j=1}^n{\overline{p}}_j}={\displaystyle \sum _{j=1}^n\left(\sigma {\omega }_j+(1-\sigma )p_j\right)=}\sigma {\displaystyle \sum _{j=1}^n{\omega }_j}+(1-\sigma ){\displaystyle \sum _{j=1}^n{p}_j}=1$, then

$$FHFPOWA({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)=\alpha$$

□

Theorem 7 

(Permutation invariance).  Let ${\alpha }_i=〈{\mu }_{{\alpha }_i},{\nu }_{{\alpha }_i}〉(i=1,2,\dots ,n)$ be a set of FHFNs. If ${{\alpha }_i}^{\prime }=({{\alpha }_1}^{\prime },{{\alpha }_2}^{\prime },\dots ,{{\alpha }_n}^{\prime })$ is any permutation of ${\alpha }_i=({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)$, then

$$FHFPOWA({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)=FHFPOWA({{\alpha }_1}^{\prime },{{\alpha }_2}^{\prime },\dots ,{{\alpha }_n}^{\prime })$$

(14)

Proof. 

Since ${{\alpha }_i}^{\prime }=({{\alpha }_1}^{\prime },{{\alpha }_2}^{\prime },\dots ,{{\alpha }_n}^{\prime })$ is a permutation of ${\alpha }_i=({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)$, it follows that

$$\begin{array}{l}FHFPOWA({{\alpha }_1}^{\prime },{\alpha }_2\prime ,\cdot \cdot \cdot ,{{\alpha }_n}^{\prime })\\ =\underset{j=1}{\overset{n}{\oplus }}\sigma {\omega }_j{\beta }_j\oplus \underset{i=1}{\overset{n}{\oplus }}(1-\sigma )p_i{a_i}^{\prime }=〈{\displaystyle \underset{{\mu }_{{{\alpha }_i}^{\prime }}\in U_{{{\alpha }_i}^{\prime }}}{\cup }\left\{\sqrt[3]{1-{\displaystyle \prod _{j=1}^n(1-{\mu }_{{\beta }_j}^3}{)}^{{\overline{p}}_j}}\right\},}{\displaystyle \underset{{\nu }_{{{\alpha }_i}^{\prime }}\in V_{{{\alpha }_i}^{\prime }}}{\cup }\left\{{\displaystyle \prod _{j=1}^n{\nu }_{{\beta }_j}^{{\overline{p}}_j}}\right\}}〉\\ =FHFPOWA({\alpha }_1,{\alpha }_2,\cdot \cdot \cdot ,{\alpha }_n)\end{array}$$

where $\omega =({\omega }_1,{\omega }_2,\dots ,{\omega }_n)$ is the weighting vector with ${\omega }_j\in \left[0,1\right]$ and ${\displaystyle \sum _{j=1}^n{\omega }_j=1}$ and $p=(p_1,p_2,\dots ,p_n)$ is the probabilistic vector with $p_i\in \left[0,1\right]$ and ${\displaystyle \sum _{i=1}^n{p}_i}=1$. ${\beta }_j$ is the j-th largest element of $a_i$. ${\overline{p}}_j=\sigma {\omega }_j+(1-\sigma )p_j$ with $\sigma \in \left[0,1\right]$, and $p_j$ is the ordered probability $p_i$ corresponding to ${\beta }_j$. □

Definition 12. 

Let ${\alpha }_i=〈{\mu }_{{\alpha }_i},{\nu }_{{\alpha }_i}〉(i=1,2,\dots ,n)$ be a set of FHFNs. A generalized Fermatean hesitant fuzzy probabilistic ordered weighted averaging (GFHFPOWA) operator for ${\alpha }_i$ is a mapping $GFHFPOWA:{\Phi }^n\to \Phi$ such that

$$GFHFPOWA({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)={\left(\underset{j=1}{\overset{n}{\oplus }}{\overline{p}}_j{\beta }_j^{\lambda }\right)}^{\frac{1}{\lambda }}$$

(15)

where $\omega =({\omega }_1,{\omega }_2,\dots ,{\omega }_n)$ is the associated weighting vector with ${\omega }_j\in \left[0,1\right]$ and ${\displaystyle \sum _{j=1}^n{\omega }_j=1}$ and ${\beta }_j$ is the j-th largest element of $a_i$, $\lambda >0$. Each $a_i$ has an associated probability $p_i$ with ${\displaystyle \sum _{i=1}^n{p}_i}=1$ and $p_i\in \left[0,1\right]$. ${\overline{p}}_j=\sigma {\omega }_j+(1-\sigma )p_j$ and $p_j$ is the ordered probability $p_i$ according to ${\beta }_j$, $\sigma \in \left[0,1\right]$.

Similarly, when $\lambda =1$, the GFHFPOWA operator reduces to the FHFPOWA operator.

Theorem 8. 

Let ${\alpha }_i=〈{\mu }_{{\alpha }_i},{\nu }_{{\alpha }_i}〉(i=1,2,\dots ,n)$ be a set of FHFNs, and let $\lambda >0$. Then,

$$\begin{array}{l}GFHFPOWA({\alpha }_1,{\alpha }_2,\cdot \cdot \cdot ,{\alpha }_n)\\ =〈{\displaystyle \underset{{\mu }_{{\alpha }_i}\in U_{{\alpha }_i}}{\cup }\left\{{\left(\sqrt[3]{1-{\displaystyle \prod _{j=1}^n(1-{\mu }_{{\beta }_j}^{3\lambda }}{)}^{{\overline{p}}_j}}\right)}^{\frac{1}{\lambda }}\right\},}{\displaystyle \underset{{\nu }_{{\alpha }_i}\in V_{{\alpha }_i}}{\cup }\left\{\sqrt[3]{1-{\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{(1-{\nu }_{{\beta }_j}^3)}^{\lambda }\right)}^{{\overline{p}}_j}}\right)}^{\frac{1}{\lambda }}}\right\}}〉\end{array}$$

(16)

where $\omega =({\omega }_1,{\omega }_2,\dots ,{\omega }_n)$ is the weighting vector with ${\omega }_j\in \left[0,1\right]$ and ${\displaystyle \sum _{j=1}^n{\omega }_j=1}$ and $p=(p_1,p_2,\dots ,p_n)$ is the probabilistic vector with $p_i\in \left[0,1\right]$ and ${\displaystyle \sum _{i=1}^n{p}_i}=1$. ${\beta }_j$ is the j-th largest element of $a_i$. ${\overline{p}}_j=\sigma {\omega }_j+(1-\sigma )p_j$ with $\sigma \in \left[0,1\right]$, and $p_j$ is the ordered probability $p_i$ related to ${\beta }_j$.

Similarly, the GFHFPOWA operator has several excellent properties:

Theorem 9 

(Monotonicity).  Let ${\alpha }_i=〈{\mu }_{{\alpha }_i},{\nu }_{{\alpha }_i}〉$ and ${\gamma }_i=〈{\mu }_{{\gamma }_i},{\nu }_{{\gamma }_i}〉(i=1,2,\dots ,n)$ be two sets of FHFNs, where $U_{{\alpha }_i}=\left\{{\mu }_{{\alpha }_{i1}},{\mu }_{{\alpha }_{i2}},\cdot \cdot \cdot ,{\mu }_{{\alpha }_{ik}}\right\}$, $V_{{\alpha }_i}=\left\{{\nu }_{{\alpha }_{i1}},{\nu }_{{\alpha }_{i2}},\cdot \cdot \cdot ,{\nu }_{{\alpha }_{is}}\right\}$,  $U_{{\gamma }_i}=\left\{{\mu }_{{\gamma }_{i1}},{\mu }_{{\gamma }_{i2}},\cdot \cdot \cdot ,{\mu }_{{\gamma }_{ik}}\right\}$, $V_{{\gamma }_i}=\left\{{\nu }_{{\gamma }_{i1}},{\nu }_{{\gamma }_{i2}},\cdot \cdot \cdot ,{\nu }_{{\gamma }_{is}}\right\}$, $m=1,2,\cdot \cdot \cdot ,k$, and $n=1,2,\cdot \cdot \cdot ,s$. If ${\mu }_{{\alpha }_i}\ge {\mu }_{{\gamma }_i}$ and ${\nu }_{{\alpha }_i}\le {\nu }_{{\gamma }_i}$, then

$$GFHFPOWA({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)\ge GFHFPOWA({\gamma }_1,{\gamma }_2,\dots ,{\gamma }_n)$$

(17)

Theorem 10 

(Boundedness).   Let ${\alpha }_i=〈{\mu }_{{\alpha }_i},{\nu }_{{\alpha }_i}〉(i=1,2,\dots ,n)$ be a set of FHFNs. Suppose that ${\alpha }^{+}=〈{\mu }^{+},{\nu }^{-}〉$ and ${\alpha }^{-}=〈{\mu }^{-},{\nu }^{+}〉$, where ${\mu }^{+}=\max \left({\mu }_1,{\mu }_2,\cdot \cdot \cdot ,{\mu }_n\left|\mu \in U_{\alpha }\right.\right)$, ${\mu }^{-}=\min \left({\mu }_1,{\mu }_2,\cdot \cdot \cdot ,{\mu }_n\left|\mu \in U_{\alpha }\right.\right)$, ${\nu }^{+}=\max \left({\nu }_1,{\nu }_2,\cdot \cdot \cdot ,{\nu }_n\left|\nu \in V_{\alpha }\right.\right)$, and ${\nu }^{-}=\min \left({\nu }_1,{\nu }_2,\cdot \cdot \cdot ,{\nu }_n\left|\nu \in V_{\alpha }\right.\right)$, then

$${\alpha }^{-}\le GFHFPOWA({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)\le {\alpha }^{+}$$

(18)

Theorem 11 

(Idempotency).  Let ${\alpha }_i=〈U_{{\alpha }_i},V_{{\alpha }_i}〉$ be a set of FHFNs, $i=1,2,\cdot \cdot \cdot ,n$. If ${\alpha }_i=\alpha =〈U_{{\alpha }_i},V_{{\alpha }_i}〉$ holds for all $i$, then

$$GFHFPOWA({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)=\alpha$$

(19)

Theorem 12 

(Permutation invariance).  Let ${\alpha }_i=〈{\mu }_{{\alpha }_i},{\nu }_{{\alpha }_i}〉(i=1,2,\dots ,n)$ be a set of FHFNs. If ${{\alpha }_i}^{\prime }=({{\alpha }_1}^{\prime },{{\alpha }_2}^{\prime },\dots ,{{\alpha }_n}^{\prime })$ is any permutation of ${\alpha }_i=({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)$, then

$$GFHFPOWA({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)=GFHFPOWA({{\alpha }_1}^{\prime },{{\alpha }_2}^{\prime },\dots ,{{\alpha }_n}^{\prime })$$

(20)

The proof process refers to Theorems 4–7, which is omitted here.

4. Generalized Fermatean Hesitant Fuzzy Probabilistic Ordered Weighted Geometric Operator

Definition 13. 

Let ${\alpha }_i=〈{\mu }_{{\alpha }_i},{\nu }_{{\alpha }_i}〉(i=1,2,\dots ,n)$ be a collection of FHFNs. A Fermatean hesitant fuzzy probabilistic ordered weighted geometric (FHFPOWG) operator for ${\alpha }_i$ is a mapping $FHFPOWG:{\Phi }^n\to \Phi$ such that

$$FHFPOWG\left({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n\right)=\underset{j=1}{\overset{n}{\otimes }}{\beta }_j^{{\overline{p}}_j}$$

(21)

where $\omega =({\omega }_1,{\omega }_2,\dots ,{\omega }_n)$ is the weighting vector with ${\omega }_j\in \left[0,1\right]$ and ${\displaystyle \sum _{j=1}^n{\omega }_j=1}$ and ${\beta }_j$ is the j-th largest element of $a_i$. Each $a_i$ has an associated probability $p_i$ with ${\displaystyle \sum _{i=1}^n{p}_i}=1$ and $p_i\in \left[0,1\right]$. ${\overline{p}}_j=\sigma {\omega }_j+(1-\sigma )p_j$, where $p_j$ is the ordered probability $p_i$ corresponding to ${\beta }_j$ and $\sigma \in \left[0,1\right]$.

Theorem 13. 

Let ${\alpha }_i=〈{\mu }_{{\alpha }_i},{\nu }_{{\alpha }_i}〉(i=1,2,\dots ,n)$ be a set of FHFNs, then

$$\mathrm{FHFPOWG}({\alpha }_1,{\alpha }_2,\cdot \cdot \cdot ,{\alpha }_n)=〈{\displaystyle \underset{{\mu }_{{\alpha }_i}\in U_{{\alpha }_i}}{\cup }\left\{{\displaystyle \prod _{j=1}^n{\mu }_{{\beta }_j}^{{\overline{p}}_j}}\right\}},{\displaystyle \underset{{\nu }_{{\alpha }_i}\in V_{{\alpha }_i}}{\cup }\left\{\sqrt[3]{1-{\displaystyle \prod _{j=1}^n(1-{\nu }_{{\beta }_j}^3}{)}^{{\overline{p}}_j}}\right\}}〉$$

(22)

where $\omega =({\omega }_1,{\omega }_2,\dots ,{\omega }_n)$ is the weighting vector with ${\omega }_j\in \left[0,1\right]$ and ${\displaystyle \sum _{j=1}^n{\omega }_j=1}$ and $p=(p_1,p_2,\dots ,p_n)$ is the probabilistic vector with $p_i\in \left[0,1\right]$ and ${\displaystyle \sum _{i=1}^n{p}_i}=1$, and ${\beta }_j$ is the j-th largest element of $a_i$. ${\overline{p}}_j=\sigma {\omega }_j+(1-\sigma )p_j$ with $\sigma \in \left[0,1\right]$ and $p_j$ is the ordered probability $p_i$ related to ${\beta }_j$.

When $\sigma =1$, the FHFPOWG operator reduces to the Fermatean hesitant fuzzy probabilistic ordered weighted geometric (FHFOWG) operator. When $\sigma =0$, the FHFPOWG operator reduces to the Fermatean hesitant fuzzy probabilistic geometric (FHFPG) operator.

Example 3. 

Here, we still use the evaluation information given in Example 2 for calculation. The weighting vector and probabilistic vector given by the experts are $\omega =(0.3,0.2,0.3,0.2)$ and $p=(0.4,0.1,0.2,0.3)$, and $\sigma =0.7$. The ordering of ${\alpha }_i$ is obtained according to Theorem 1: ${\alpha }_3\succ {\alpha }_2\succ {\alpha }_4\succ {\alpha }_1$. Then, we can obtain the following results by ${\overline{p}}_j=\sigma {\omega }_j+(1-\sigma )p_j$:

$${\overline{p}}_1=0.7\times 0.3+0.3\times 0.2=0.27$$

$${\overline{p}}_2=0.7\times 0.2+0.3\times 0.1=0.17$$

$${\overline{p}}_3=0.7\times 0.3+0.3\times 0.3=0.30$$

$${\overline{p}}_4=0.7\times 0.2+0.3\times 0.4=0.26$$

Using the FHFPOWG operator to aggregate information, we can obtain the following:

$$h(\mu )=\left\{\begin{array}{l}0.1102,0.1682,0.1351,0.1281,0.1401,0.2138,\\ 0.1717,0.2621,0.1642,0.2506,0.2012,0.1908,\\ 0.2086,0.3185,0.2557,0.3904,0.2369,0.3616,\\ 0.2903,0.2753,0.3010,0.4595,0.3690,0.5633\end{array}\right\}$$

$$h(\nu )=\left\{\begin{array}{l}0.1322,0.1594,0.2287,0.1379,0.1650,0.2565,\\ 0.1712,0.1972,0.2633,0.1767,0.2025,0.2899\end{array}\right\}$$

Theorem 14 

(Monotonicity).  Let ${\alpha }_i=〈{\mu }_{{\alpha }_i},{\nu }_{{\alpha }_i}〉$ and ${\gamma }_i=〈{\mu }_{{\gamma }_i},{\nu }_{{\gamma }_i}〉(i=1,2,\dots ,n)$ be two sets of FHFNs where $U_{{\alpha }_i}=\left\{{\mu }_{{\alpha }_{i1}},{\mu }_{{\alpha }_{i2}},\cdot \cdot \cdot ,{\mu }_{{\alpha }_{ik}}\right\}$, $V_{{\alpha }_i}=\left\{{\nu }_{{\alpha }_{i1}},{\nu }_{{\alpha }_{i2}},\cdot \cdot \cdot ,{\nu }_{{\alpha }_{is}}\right\}$, $U_{{\gamma }_i}=\left\{{\mu }_{{\gamma }_{i1}},{\mu }_{{\gamma }_{i2}},\cdot \cdot \cdot ,{\mu }_{{\gamma }_{ik}}\right\}$, $V_{{\gamma }_i}=\left\{{\nu }_{{\gamma }_{i1}},{\nu }_{{\gamma }_{i2}},\cdot \cdot \cdot ,{\nu }_{{\gamma }_{is}}\right\}$, $m=1,2,\cdot \cdot \cdot ,k$, and $n=1,2,\cdot \cdot \cdot ,s$. If ${\mu }_{{\alpha }_i}\ge {\mu }_{{\gamma }_i}$, ${\nu }_{{\alpha }_i}\le {\nu }_{{\gamma }_i}$, then

$$FHFPOWG({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)\ge FHFPOWG({\gamma }_1,{\gamma }_2,\dots ,{\gamma }_n)$$

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Theorem 15 

(Boundedness).   Let ${\alpha }_i=〈{\mu }_{{\alpha }_i},{\nu }_{{\alpha }_i}〉(i=1,2,\dots ,n)$ be a set of FHFNs. Suppose that ${\alpha }^{+}=〈{\mu }^{+},{\nu }^{-}〉$ and ${\alpha }^{-}=〈{\mu }^{-},{\nu }^{+}〉$, where ${\mu }^{+}=\max \left({\mu }_1,{\mu }_2,\cdot \cdot \cdot ,{\mu }_n\left|\mu \in U_{\alpha }\right.\right)$, ${\mu }^{-}=\min \left({\mu }_1,{\mu }_2,\cdot \cdot \cdot ,{\mu }_n\left|\mu \in U_{\alpha }\right.\right)$, ${\nu }^{+}=\max \left({\nu }_1,{\nu }_2,\cdot \cdot \cdot ,{\nu }_n\left|\nu \in V_{\alpha }\right.\right)$, and ${\nu }^{-}=\min \left({\nu }_1,{\nu }_2,\cdot \cdot \cdot ,{\nu }_n\left|\nu \in V_{\alpha }\right.\right)$, then

$${\alpha }^{-}\le FHFPOWG({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)\le {\alpha }^{+}$$

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Theorem 16 

(Permutation invariance).   Let ${\alpha }_i=〈{\mu }_{{\alpha }_i},{\nu }_{{\alpha }_i}〉(i=1,2,\dots ,n)$ t be a set of FHFNs. If ${{\alpha }_i}^{\prime }=({{\alpha }_1}^{\prime },{{\alpha }_2}^{\prime },\dots ,{{\alpha }_n}^{\prime })$ is any permutation of ${\alpha }_i=({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)$, then

$$FHFPOWG({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)=FHFPOWG({{\alpha }_1}^{\prime },{{\alpha }_2}^{\prime },\dots ,{{\alpha }_n}^{\prime })$$

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Theorem 17 

(Idempotency).  Let ${\alpha }_i=〈U_{{\alpha }_i},V_{{\alpha }_i}〉$ be a set of FHFNs, $i=1,2,\cdot \cdot \cdot ,n$ . If ${\alpha }_i=\alpha =〈U_{{\alpha }_i},V_{{\alpha }_i}〉$ for all $i$, then

$$FHFPOWG({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)=\alpha$$

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Definition 14. 

Let ${\alpha }_i=〈{\mu }_{{\alpha }_i},{\nu }_{{\alpha }_i}〉(i=1,2,\dots ,n)$ be a set of FHFNs. A generalized Fermatean hesitant fuzzy probabilistic ordered weighted geometric (GFHFPOWG) operator for ${\alpha }_i$ is a mapping $GFHFPOWG:{\Phi }^n\to \Phi$ such that

$$GFHFPOWG({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)=\frac{1}{\lambda }\left(\underset{j=1}{\overset{n}{\otimes }}\lambda {\overline{p}}_j{\beta }_j\right)$$

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where $\omega =({\omega }_1,{\omega }_2,\dots ,{\omega }_n)$ is the associated weight vector with ${\omega }_j\in \left[0,1\right]$ and ${\displaystyle \sum _{j=1}^n{\omega }_j=1}$ and ${\beta }_j$ is the j-th largest element of $a_i$, $\lambda >0$. Each $a_i$ has an associated probability $p_i$ with ${\displaystyle \sum _{i=1}^n{p}_i}=1$ and $p_i\in \left[0,1\right]$. ${\overline{p}}_j=\sigma {\omega }_j+(1-\sigma )p_j$, where $p_j$ is the ordered probability corresponding to ${\beta }_j$ and $\sigma \in \left[0,1\right]$.

Theorem 18. 

Let ${\alpha }_i=〈{\mu }_{{\alpha }_i},{\nu }_{{\alpha }_i}〉(i=1,2,\dots ,n)$ be a set of FHFNs, and let $\lambda >0$. Then

$$\begin{array}{l}GFHFPOWG({\alpha }_1,{\alpha }_2,\cdot \cdot \cdot ,{\alpha }_n)\\ =〈{\displaystyle \underset{{\mu }_{{\alpha }_i}\in U_{{\alpha }_i}}{\cup }\left\{\sqrt[3]{1-{\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{(1-{\mu }_{{\beta }_j}^3)}^{\lambda }\right)}^{{\overline{p}}_j}}\right)}^{\frac{1}{\lambda }}}\right\}},{\displaystyle \underset{{\nu }_{{\alpha }_i}\in V_{{\alpha }_i}}{\cup }\left\{{\left(\sqrt[3]{1-{\displaystyle \prod _{j=1}^n(1-{\nu }_{{\beta }_j}^{3\lambda }}{)}^{{\overline{p}}_j}}\right)}^{\frac{1}{\lambda }}\right\}}〉\end{array}$$

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where $\omega =({\omega }_1,{\omega }_2,\dots ,{\omega }_n)$ is the associated weight vector with ${\omega }_j\in \left[0,1\right]$ and ${\displaystyle \sum _{j=1}^n{\omega }_j=1}$ and $p=(p_1,p_2,\dots ,p_n)$ is the probabilistic vector with $p_i\in \left[0,1\right]$ and ${\displaystyle \sum _{i=1}^n{p}_i}=1$. ${\beta }_j$ is the j-th largest element of $a_i$. ${\overline{p}}_j=\sigma {\omega }_j+(1-\sigma )p_j$ with $\sigma \in \left[0,1\right]$, where $p_j$ is the ordered probability $p_i$ according to ${\beta }_j$.

The GFHFPOWG operator has excellent properties such as monotonicity, boundedness, permutation invariance, and idempotency. The proofs are provided in Theorems 4–7, which are omitted here.

5. Illustrative Example

This section verifies the effectiveness and practicability of this method through an illustrative example showing the analysis of new retail enterprises.

5.1. Decision-Making Process

Suppose that the expert group is currently having a specific discussion on a MADM issue. Let $Y_i=\{Y_1,Y_2,\cdot \cdot \cdot ,Y_m\}$ be a set of alternatives and $C_i=\{C_1,C_2,\cdot \cdot \cdot ,C_n\}$ be a collection of attributes. $\omega ={\{{\omega }_1,{\omega }_2,\cdot \cdot \cdot ,{\omega }_n\}}^T$ is the weighting vector corresponding to each attribute with ${\omega }_j\ge 0$ and ${\displaystyle \sum _{j=1}^n{\omega }_j=1}$. ${\alpha }_i=〈{\mu }_{{\alpha }_i},{\nu }_{{\alpha }_i}〉\left(i=1,2,\dots ,n\right)$ is a set of FHFNs where ${\alpha }_{ij}$ denotes the evaluation value of alternative $Y_i$ with respect to attribute $C_j$. Then, the Fermatean hesitant fuzzy decision matrix $D={({\alpha }_{ij})}_{m\times n}$ is obtained. To solve this issue, this paper presents a Fermatean hesitant fuzzy MADM method based on the proposed operators. The specific steps are as follows:

Step 1. Relevant experts are invited to give evaluation information for each alternative, and then a generalized Fermatean hesitant fuzzy decision matrix $D={({\alpha }_{ij})}_{m\times n}$ is established.

Step 2. The value of probability ${\overline{p}}_j$ is calculated based on the weighting vector ${\omega }_i$, probabilistic vector $p_i$, and the given parameter $\sigma$, where $p_i\in [0,1]$, ${\displaystyle \sum _{i=1}^n{p}_i=1}$, and $\sigma \in [0,1]$.

$${\overline{p}}_j=\sigma {\omega }_j+(1-\sigma )p_j$$

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Step 3. Use the proposed operators to aggregate the alternatives and calculate the comprehensive attribute values of each alternative.

Step 4. The score function and accuracy function are used to sort the comprehensive attribute values. Then, the best solution is selected based on the sorting results.

5.2. Numerical Example

With the popularity of the Internet and mobile devices, consumers are increasingly inclined to use digital platforms for shopping, driving the growth of online shopping. However, consumers’ expectations for the shopping experience are also rising, and more than traditional online or offline shopping methods are needed to satisfy these demands fully. In this context, a new business model known as New retail is rapidly emerging and developing. New retail is a new model based on Internet technology and advanced technologies such as big data analysis and artificial intelligence. It breaks traditional brick-and-mortar retail and e-commerce boundaries, creating entirely new consumer scenarios. New retail changes the shopping habits of consumers and promotes the transformation of the entire retail industry. By 2025, the global new retail market is expected to reach more than 70 billion, with a compound annual growth rate of more than 30%. In China, E-business has recently been developing at full speed far beyond people’s imagination. Electronic commerce development in China has entered the stage of new retail, and new retail has become a driving force for economic growth [3]. Major e-commerce platforms such as Alibaba, JD.com, and Hema Fresh are actively deploying new retail strategies, and offline physical stores are actively transforming and exploring the path of new retail.

In recent years, China’s retail market has been booming, and technological progress superimposes demand changes, driving the business model innovation of new retail. Suppose that an existing logistics enterprise intends to cooperate with a new retail enterprise to expand its business scale. After an initial screening, the company selects four new retail enterprises $Y_i=\{Y_1,Y_2,Y_3,Y_4\}$ as alternatives. After clarifying the evaluation standards and indicators for the business model of new retail enterprises, this logistics enterprise selects nine experts with relevant professional backgrounds and experience. These experts come from academia and industry and have extensive experience in e-commerce, technological innovation, and supply chain management. After providing detailed explanations of the evaluation purpose, process, and development status of the candidate enterprises, the expert group engaged in thorough discussions and exchanged opinions. Finally, the consolidated evaluation opinions from each expert are used to form the initial evaluation matrix.

$C_1$: Customer equity.

As a new business model in the retail industry, new retail should regulate the behavior of enterprises and protect the legitimate rights and interests of consumers when improving their shopping experience. Yoon and Oh [56] proposed a modified model of retail customer equity to understand how the key drivers of customer equity affect store loyalty. Customer equity includes experience equity, brand equity, and relationship equity. The study found that experiential equity and brand equity significantly affected customer loyalty, while relationship equity did not.

$C_2$: Availability of technological innovations.

The phenomenon of digitalization is one of the most important transformations that is currently characterizing the retail sector [57]. Technological innovation has dramatically changed business opportunities, business models, production processes, and so on. The availability of innovative technologies and their diffusion in the retail industry can help related retailers build technological advantages and enhance competitiveness.

$C_3$: Social media platforms.

The expansion of social media platforms and the global distribution of users create new opportunities and challenges for online marketers [58]. For retail enterprises, social media platforms are important in accelerating product information dissemination and promoting brand building. Several studies have pointed out that retail brands should strive to understand the specific forms of value they can co-create in their social media brand communities, as well as their relative influence on consumers [59].

$C_4$: Logistics efficiency.

For new retail enterprises, the role of logistics is becoming increasingly important. Logistics has become an important bridge for online and offline integration. Logistics efficiency affects transportation costs, customer satisfaction, the environmental impact of enterprises, and so on. Jiang et al. [60] believe that it is of great significance for enterprises to evaluate logistics efficiency from two aspects: reducing logistics cost and improving logistics efficiency. Therefore, they developed a new logistics efficiency evaluation method.

The associated weighting vector, in this case, is given directly after a full discussion by the expert group: $\omega =(0.2,0.3,0.3,0.2)$. To ensure the scientificity and rationality of the decision-making results, experts should provide their evaluation information anonymously, and the repeated membership and non-membership degrees only take the values once. For example, when evaluating the performance of alternative $Y_1$ in terms of customer equity ($C_1$), the expert group still hesitates between several evaluation values after consultation. Among them, the group agreed that alternative $Y_1$ performed well in customer equity ($C_1$) to the extent of 0.3, 0.6, and 0.8 and disagreed that alternative $Y_1$ performed well in customer equity ($C_1$) to the extent of 0.1 and 0.2. Therefore, the final evaluation information given for $Y_1$ with respect to $C_1$ is {{0.3,0.6,0.8},{0.1,0.2}}, which is denoted as ${\alpha }_{11}$. Then, the generalized Fermatean hesitant fuzzy decision matrix $D={({\alpha }_{ij})}_{4\times 4}$ is established, where ${\alpha }_{ij}$ denotes a Fermatean hesitant fuzzy number. According to the established matrix, the four new retail enterprises are evaluated, and the best partner is selected based on the evaluation results. Figure 1 shows the complete decision-making process.

Step 1. The Fermatean hesitant fuzzy decision matrix is established based on the evaluation information provided by the decision-making group, which is shown in

Table 1. The Fermatean hesitant fuzzy decision matrix.

	${\mathit{C}}_1$	${\mathit{C}}_2$	${\mathit{C}}_3$	${\mathit{C}}_4$
$Y_1$	{{0.3,0.6,0.8},{0.1,0.2}}	{{0.7,0.8},{0.1,0.2}}	{{0.5, 0.7},{0.4}}	{{0.3,0.5},{0.6,0.7,0.9}}
$Y_2$	{{0.7,0.8},{0.6}}	{{0.8,0.9},{0.4}}	{{0.2,0.3,0.4},{0.6}}	{{0.3},{0.2,0.4,0.5,0.6}}
$Y_3$	{{0.5,0.6},{0.1,0.2,0.3}}	{{0.4,0.5},{0.2,0.3}}	{{0.9},{0.6}}	{{0.5,0.6},{0.1,0.2,0.3,0.4}}
$Y_4$	{{0.7,0.8},{0.4,0.6}}	{{0.5,0.6,0.7},{0.2,0.3}}	{{0.5,0.6},{0.3}}	{{0.5, 0.7},{0.1,0.3}}

.

Step 2. After full discussion, the experts give the weighting vector as $\omega =(0.2,0.3,0.3,0.2)$ and the probabilistic vector as $p=(0.1,0.2,0.4,0.3)$; $\sigma =0.6$. The score values of ${\alpha }_{1j}(j=1,2,3,4)$ in Table 1 are calculated and sorted as ${\alpha }_{12}\succ {\alpha }_{11}\succ {\alpha }_{13}\succ {\alpha }_{14}$. Then, calculate the value of the probability ${\overline{p}}_j$ according to the ordering of ${\alpha }_{1j}$:

$${\overline{p}}_1=0.6\times 0.2+0.4\times 0.2=0.20$$

$${\overline{p}}_2=0.6\times 0.3+0.4\times 0.1=0.22$$

$${\overline{p}}_3=0.6\times 0.3+0.4\times 0.4=0.34$$

$${\overline{p}}_4=0.6\times 0.2+0.4\times 0.3=0.24$$

Step 3. Use the suggested GFHFPOWA operator to aggregate the alternatives and calculate the comprehensive attribute values of each alternative (when $\lambda =0.3$). The aggregation results are given in

Table 2. The aggregation ranking results of each alternative when $\lambda =0.3$.

.	${\mathbf{h}}_{\mathbf{Y}}\left(\mathsf{\mu }\right)$	${\mathbf{h}}_{\mathbf{Y}}\left(\mathsf{\nu }\right)$
$Y_1$	{0.4713,0.5329,0.5982,0.5115,0.5687,0.6291,0.5548,0.6071,0.6624,0.6217,0.6664,0.7135,0.4713,0.5329,0.5982,0.5501,0.6029,0.6587,0.5901,0.6384,0.6894,0.6217,0.6664,0.7135}	{0.2481,0.2888,0.2848,0.3313,0.2586,0.3009,0.2968,0.3452,0.2802,0.3259,0.3214,0.3736 }
$Y_2$	{0.5265,0.5662,0.5864,0.6213,0.5447,0.5830,0.6025,0.6360,0.5645,0.6012,0.6198,0.6520}	{0.4011,0.4914,0.5249,0.5545}
$Y_3$	{0.6674,0.6835,0.6796,0.6952,0.6892,0.7043,0.7006,0.7152}	{0.1991,0.2225,0.2433,0.2351,0.2737,0.2991,0.2655,0.3089,0.3376,0.2897,0.3369,0.3680,0.2073,0.2413,0.2638,0.2550,0.2967,0.3243,0.2879,0.3349,0.3658,0.3140,0.3651,0.3986}
$Y_4$	{0.5389,0.5674,0.5649,0.5918,0.5961,0.6211,0.5668,0.5937,0.5913,0.6167,0.6207,0.6443,0.6042,0.6288,0.6267,0.6499,0.6536,0.6752,0.6283,0.6515,0.6495,0.6713,0.6748,0.6951}	{0.2036,0.2179,0.2829,0.3026,0.2263,0.2422,0.3142,0.3361}

.

Step 4. Use the score function to calculate the comprehensive attribute values.

$$S_{\alpha }=\frac{1}{\left|{\mu }_{\alpha }\right|}{\displaystyle \sum _{\mu \in U_{\alpha }}{\mu }^3-}\frac{1}{\left|{\nu }_{\alpha }\right|}{\displaystyle \sum _{v\in V_{\alpha }}{v}^3}$$

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When $\lambda =0.3$, then $S(Y_1)=0.1975$, $S(Y_2)=0.0852$, $S(Y_3)=0.3040$, $S(Y_4)=0.2228$. We can have $S(Y_3)>S(Y_4)>S(Y_1)>S(Y_2)$ according to the final scores. Therefore, the enterprises are ranked as $Y_3\succ Y_4\succ Y_1\succ Y_2$, and the best solution is selected.

5.3. Parameter Analysis

To observe the influence of the parameter $\lambda$ on the ranking results, this paper further selects different values of 0.01, 0.3, 0.5, 1, 2, 3.5, 5, 7.6, 8, 10, and 20 for a comparative analysis. Similarly, the GFHFPOWA operator is used to calculate the scoring values of each alternative, and the results are listed in

Table 3. The ranking of alternatives when different values of $\lambda$ are selected.

$\mathit{\lambda }$	$\mathit{S}({\mathit{Y}}_1)$	$\mathit{S}({\mathit{Y}}_2)$	$\mathit{S}({\mathit{Y}}_3)$	$\mathit{S}({\mathit{Y}}_4)$	Ranking
0.01	0.1865	0.0527	0.2847	0.2188	$Y_3\succ Y_4\succ Y_1\succ Y_2$
0.3	0.1975	0.0852	0.3040	0.2228	$Y_3\succ Y_4\succ Y_1\succ Y_2$
0.5	0.2064	0.1092	0.3183	0.2258	$Y_3\succ Y_4\succ Y_1\succ Y_2$
1	0.2279	0.1668	0.3561	0.2344	$Y_3\succ Y_4\succ Y_1\succ Y_2$
2	0.2664	0.2503	0.4274	0.2544	$Y_3\succ Y_1\succ Y_4\succ Y_2$
3.5	0.3079	0.3198	0.5034	0.2837	$Y_3\succ Y_2\succ Y_1\succ Y_4$
5	0.3350	0.3607	0.5498	0.3068	$Y_3\succ Y_2\succ Y_1\succ Y_4$
7.6	0.3638	0.4043	0.5958	0.3336	$Y_3\succ Y_2\succ Y_1\succ Y_4$
8	0.3670	0.4093	0.6007	0.3367	$Y_3\succ Y_2\succ Y_1\succ Y_4$
10	0.3800	0.4300	0.6204	0.3492	$Y_3\succ Y_2\succ Y_1\succ Y_4$
20	0.4101	0.4840	0.6646	0.3783	$Y_3\succ Y_2\succ Y_1\succ Y_4$

.

The data in Table 3 show that when the parameter $\lambda$ is set to 0.01, 0.3, 0.5, 1, 2, 3.5, 5, 7.6, 8, 10, and 20, the overall ranking changes, but the optimal solution remains as $Y_3$. This indicates that the parameter does not have a significant impact on the choice of the best solution. Therefore, decision-makers can choose appropriate values according to personal preferences when evaluating. Figure 2 shows the trends in the final scores for each alternative. By changing the value of the parameter $\lambda$ from 0.01 to 20, the score value of each alternative increases while maintaining the best option ranking. However, the growth rates of different solutions are different. $S(Y_2)$ and $S(Y_3)$ show a faster growth rate, while $S(Y_1)$ and $S(Y_4)$ grow more slowly, indicating that Alternatives 2 and 3 are more significantly affected by changes in the parameter. Additionally, when $\lambda \le 1$, the overall ranking result is $Y_3\succ Y_4\succ Y_1\succ Y_2$. When $\lambda \ge 3.5$ the ranking result is $Y_3\succ Y_2\succ Y_1\succ Y_4$. When the parameter value is between 1 and 3.5, the ranking of solutions shows significant fluctuations. Therefore, to ensure the stability of decision-making results, decision-makers are suggested to set the parameter to $\lambda \le 1$ or $\lambda \ge 3.5$.

Figure 3 illustrates the influence of parameter changes on the scores of each alternative. It can be seen that the parameter changes have the most significant impact on $S(Y_3)$, followed by $S(Y_2)$. The scores of $S(Y_1)$ and $S(Y_4)$ do not change significantly with increasing parameter values. Therefore, Alternatives 2 and 3 are more significantly affected by changes in the parameter.

5.4. Comparative Analysis

To further verify the validity of the GFHFPOWG operator, we consider the FHFPOWA, the FHFPOWG, the GFHFPOWG, the FHFOWA, the FHFPA, the FHFOWG, and the FHFPG operators. The results are shown in

Table 4. The ranking of different operators.

Operator	$\mathit{S}({\mathit{Y}}_1)$	$\mathit{S}({\mathit{Y}}_2)$	$\mathit{S}({\mathit{Y}}_3)$	$\mathit{S}({\mathit{Y}}_4)$	Ranking
GFHFPOWA	0.1975	0.0852	0.3040	0.2228	$Y_3\succ Y_4\succ Y_1\succ Y_2$
FHFPOWA	0.2282	0.1668	0.3561	0.2344	$Y_3\succ Y_4\succ Y_1\succ Y_2$
FHFOWA	0.2434	0.1963	0.3699	0.2469	$Y_3\succ Y_4\succ Y_1\succ Y_2$
FHFPA	0.1990	0.1201	0.4203	0.2150	$Y_3\succ Y_4\succ Y_1\succ Y_2$
GFHFPOWG	0.0868	−0.0430	0.1875	0.1973	$Y_4\succ Y_3\succ Y_1\succ Y_2$
FHFPOWG	0.0123	−0.0632	0.1437	0.1823	$Y_4\succ Y_3\succ Y_1\succ Y_2$
FHFOWG	0.0425	−0.0410	0.1474	0.1887	$Y_4\succ Y_3\succ Y_1\succ Y_2$
FHFPG	−0.0278	−0.0880	0.1617	0.1738	$Y_4\succ Y_3\succ Y_1\succ Y_2$

.

From Table 4, it can be observed that, regardless of the operator used for information integration, there are only two results for the best solution in the obtained ranking: $Y_3$ and $Y_4$. Most of the obtained optimal solution is $Y_3$, which is consistent with the results in the parameter analysis. In addition, the sorting results obtained using OWA and OWG methods are somewhat different, reflecting the choice of optimal scheme and sub-optimal scheme. The reason might be that the two methods handle input values differently. The OWA operator focuses on aggregation through arithmetic means, making it more suitable for conventional weighted decision-making problems. In contrast, the OWG operator aggregates information based on geometric means, which is appropriate for situations that require an emphasis on product or proportional effect.

In summary, compared to intuitionistic fuzzy operators, Pythagorean fuzzy operators, and Pythagorean hesitant fuzzy operators, the proposed Fermatean hesitant fuzzy probabilistic aggregation operators in this paper have a wider range of applications. The specific illustration is as follows:

(1)

The proposed operators combine the probability and order weighting methods. They can quantify the importance of each attribute and provide more choices for decision-makers by assigning different values to the parameters. By using probabilities, decision-makers can better represent the inherent uncertainty and variability in attribute weights, leading to more realistic and flexible decision models.

(2)

The proposed operators effectively combine the advantages of FFSs and HFSs in information representation and can consider the hesitancy of the evaluator. They are more flexible and practical because decision-makers can choose different parameters according to personal preferences when making decisions.

(3)

The proposed operators expand the limits of the membership and non-membership degrees. They can describe more uncertainty and deal with stronger ambiguity, making them more suitable for dealing with complex decision problems.

6. Conclusions

This paper proposes a new MADM method that combines probabilistic information, the OWA operator, and the Fermatean hesitant fuzzy set. Firstly, this paper introduces several new Fermatean hesitant fuzzy probabilistic aggregation operators, including the FHFPOWA, GFHFPOWA, FHFPOWG, and GFHFPOWG operators. Among them, Theorems 3–12 and Theorems 13–18 are our original contributions for exploring the properties of the operators proposed in this paper. Theorems 3–12 provide the calculation formulas for the FHFPOWA and GFHFPOWA operators and present their unique properties. Theorems 13–18 define formulas for the FHFPOWG and GFHFPOWG operators and show their excellent properties. The OWA operator can reflect the influence or preferences of different members within a group through appropriate weight distribution, while probabilistic information can capture the uncertainty and randomness associated with attribute weights. By combining probabilistic weights, the OWA method enhances accuracy in decision-making by accommodating attribute variability and uncertainty more effectively. Therefore, compared with traditional aggregation operators, the newly proposed operators broaden the dimension of information expression by incorporating corresponding probabilistic information and appropriate weight distribution. They provide a comprehensive consideration of diverse expert evaluations. Then, the paper develops a MADM method based on these operators. Finally, this paper applies the method to the business partner decision-making problem. In this problem, we study how a large logistics company evaluates the business model innovation of new retail enterprises and then finds the best partner. The effectiveness and practicability of the proposed MADM method are verified by a numerical example, parameter analysis, and comparative analysis.

In conclusion, this paper further improves the MADM method under Fermatean hesitant fuzzy environments and enriches the theoretical system of Fermatean hesitant fuzzy aggregation operators. This work serves as a significant supplement to the current research on FHFSs. Future studies could explore combining probabilistic information with Fermatean hesitant fuzzy distance measures or similarity measures to expand the applications of Fermatean fuzzy set research. Additionally, we will study the different applications of the proposed method in other practical decision-making problems such as bank investment decisions, green supplier selection, merchant service quality evaluation, and so on.