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

S2405-8440(24)12594-7
10.1016/j.heliyon.2024.e36563
e36563
Research Article
Fermatean fuzzy Linguistic term set based on linguistic scale function with Dombi aggregation operator and their application to multi criteria group decision –making problem
Barukab Omar a
Khan Asghar azhar4set@yahoo.com
b⁎
Khan Sher Afzal c
a Faculty of Computing and Information Technology, King Abdulaziz University, P.O. Box 411, 21911, Rabigh, Jeddah, Saudi Arabia
b Department of Mathematics, Abdul Wali Khan University, Mardan, 23200, KP, Pakistan
c Department of Computer Science, Abdul Wali Khan University, Mardan, 23200, KP, Pakistan
⁎ Corresponding author. azhar4set@yahoo.com
20 8 2024
15 9 2024
20 8 2024
10 17 e3656324 4 2024
17 8 2024
19 8 2024
© 2024 The Authors. Published by Elsevier Ltd.
2024

https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
The selection of an industrial location is a challenging multiple-criteria decision-making (MCDM) problem that depends on taking a variety of locations as well as incompatible and inconsistent criteria. This paper proposed a comprehensive framework for the strategic selection of industrial locations, considering both quantitative and qualitative aspects. Decision-makers (DMs) have to deal with ambiguous information throughout this process due to a complex decision environment or their insufficient knowledge. We present a new Fermatean Fuzzy (FF) Linguistic term set based on the Dombi aggregation operators (AOs). By combining the FF set with Linguistic variables, the FF Linguistic (FFL) set is an effective approach for thoroughly representing uncertain evaluation information. We establish a basic operational principles and certain aggregation operator under FFL information, such as the FF Linguistic Dombi weighted averaging (FFLDWA) operator FF Linguistic Dombi weighted geometric (FFLDWG) operator and some fundamental properties of these operators with appropriated elaboration. Based on these operators, a multi-criteria group decision-making technique is developed. Finally, we used a numerical example to compare the flexibility of the suggested technique with other existing methods. Thus, by knowing priorities industries, the best site can be selected.

Keywords

Linguistic Fermatean fuzzy set (LFFS)
Linguistic Fermatean fuzzy Dombi aggregation operators (LFFDAO)
Industrial site selection
Multi-criteria group decision-making (MCGDM)
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pmc1 Introduction

Determining an appropriate site for an industrial facility is a critical choice that can have a significant impact on operational performance. In Pakistan, a country with a wide range of physical, economic, and infrastructure characteristics, industrial location selection becomes a complicated MCDM challenge. MCDM includes analyzing and comparing possible locations based on a variety of factors to select the best site for industrial development. Raw materials, power, transportation networks, and government rules are all factors that contribute to the complexity of Pakistan's industrial best location selection process. Decision-making involves solving real-world problems by choosing the most favorable option from multiple alternatives. MCDM [[1], [2], [3]], a branch of operation research, examines alternatives over multiple criteria and has numerous applications in various fields. Real numbers often failed to fully characterize options in MCDM challenges because of incomplete information and unclear scenarios. Driven by the inadequate and ambiguous data pertaining to real world issues, Zadeh [4] introduced the fuzzy set (FS) Fˇ={⟨xj,ϑFˇ(xj)⟩|xj∈X}(ϑFˇ(xj)∈[0,1]) on X=(x1,x2,…,xn), where ϑFˇ(xj) is the membership degree (MD) of xj∈X. A crucial component of fuzzy sets is the degree of membership, which allows for a more adaptable and realistic portrayal of uncertainty and imprecision in a variety of applications. In contrast to several other approaches to decision-making, fuzzy logic is not standardized in terms of modelling or representation. Because of this, integrating or comparing fuzzy systems across several applications or domains may be difficult. But it only takes into account xj∈X degree of membership; it doesn't take into account xj∈X degree of non-membership. In order to improve the description of the pertinent data, Atanassov [5] introduced the intuitionistic FS (IFS) A‾={⟨xj,ϑA‾(xj),ΛA‾(xj)⟩|xj∈X} (ϑA‾(xj),ΛA‾(xj)∈[0,1]) on X=(x1,x2,…,xn) where ϑA‾(xj) and ΛA‾(xj) are the MD and NMD of xj∈X, respectively. In scenarios involving decision-making, intuitionistic fuzzy sets can provide a more comprehensive depiction of the criteria for making decisions, particularly when decision-makers must convey not just the known but also the unknown or uncertain. Since its proposal, the IFS has drawn a lot of interest from a variety of fields, including pattern recognition and medical diagnostics [[6], [7], [8]]. In certain real-world situations, decision-makers have strong opinions regarding the ranking or rating of an organization's plans, projects, or official statements. But there are circumstances in which the total MD and NMD in a problem involving decision-making exceeds 1, in which case IFS are no longer relevant. Yager [9,10], defined Pythagorean FS (PFS) an higher class of FS and IFS as P‾={⟨xj,ϑP‾(xj),ΛP‾(xj)⟩|xj∈X} (ϑP‾(xj),ΛP‾(xj)∈[0,1]) such that the (MD)2 + (NMD)2 is bounded by 1. Since its introduction, the PFS has been extensively used in a variety of sectors, including decision process for investment [11,12], decision process for quality service of local airline [13], and others [14]. Although PFS was designed to cope the deficient spaces of FS and IFS, but there are still some cases where PFS are failed to apply. For example, a group of experts was invited to evaluate the feasibility of a candidate in the recruitment process in a university, the panel was divided into two independent groups; the first group gave his evaluation for candidate feasibility as 0.8, while the other group considered the non-feasibility for candidate's recruitment as 0.78. By constraint conditions of IFS and PFS, 0.8 + 0.78 > 1, (0.8)2+(0.78)2>1. It is clearly seen that the expert's opinion didn't satisfies the basic conditions of IFS and PFS. Senapati and Yager [15], created the Fermatean FS (FFS), an enhanced form of FS, to address this shortcoming. An FFS is a structure of the o form Fˇ={⟨xj,ϑFˇ(xj),ΛFˇ(xj)⟩|xj∈X}(ϑFˇ(xj),ΛFˇ(xj)∈[0,1]), where ϑFˇ(xj), and ΛFˇ(xj) for every xj∈X, represent the MD and NMD, respectively such that 0 ≤ (ϑFˇ(xj))3 + (ΛFˇ(xj))3 ≤ 1 FFS has a major advantage over IFS and PFS in that it can express a greater number of uncertainties, which is useful in a variety of decision-making scenarios. There may be some current progress available in Refs. [16,17]. (see Table 8, Table 9, Fig. 3, Fig 4, Fig 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9)

Qualitative evaluation of criteria is sometimes necessary due to the complexity of real-world decision-making scenarios and the ambiguity of human cognition, making precise values challenging. Linguistic evaluation is suitable for evaluating a company's performance when an expert deems it excellent, as it closely resembles human cognition. Because of these similarities to human thought processes, Zadeh [[18], [19], [20]], proposed the concept of a linguistic term set (LTS). For example the general seven LTS may be written as Sˆ= {s˘0: very poor, s˘1: poor, s˘2: slightly poor, s˘3: fair, s˘4: slightly good, s˘5: good, s˘6 = very good}. The expert s assessment of the company s performance can be written as {s˘6}, indicating that the company s performance is excellent and that its membership degree in {s˘6} is 1. The LTS struggles to appropriately measure the degree of appraisal of the linguistic term, which might occasionally differ from the actual condition. Experts have different opinions on the company's performance at membership and nonmembership levels. One group believes it's very good at membership level (0.75), while the other thinks it's good at nonmembership level (0.52). Liu et al. [21] developed a Fermatean (FFLTS) l‾={⟨xj,Φil‾(xj),ϑl‾(xj),Λl‾(xj)⟩|xj∈X} based on the FFS and LTS, where ϑl‾(xj) and Λl‾(xj), respectively represent the MD and NMD of xj to Φil‾(xj), to describe the decision information mentioned above. Further, Xu [22] presented the linguistic scale function (LSF), which imparts unique semantic value to LTS in a variety of choice scenarios. The linguistic scale function is commonly used for practical decision-making [23,24] because to its flexibility. FF sets (FFSs) and LTS are used in decision-making and computational intelligence to handle uncertainty and vagueness. Combining these concepts leads to FFLTSs, which provide a robust framework for dealing with complex decision-making problems involving subjective and imprecise information. Imagine a scenario where a company needs to evaluate multiple suppliers based on several criteria such as price, quality, delivery time, and service. The evaluation is done by a panel of experts who use linguistic terms to express their opinions. In many real-world scenarios, DM face uncertainty and vagueness, especially when dealing with subjective criteria such as quality, satisfaction, or reliability. FFLTSs provide a robust framework to capture and process this imprecise information effectively. Motivated by this, we will provide the FF Dombi operations in LTS (FFDLTS), which are built based on LTS and FFS, to represent such information, motivated by the LTS. The application of FF Linguistic Term Sets in this study is essential for effectively handling the inherent uncertainty and vagueness in expert judgments, enhancing the expressiveness and accuracy of evaluations, and ultimately leading to more robust and reliable decision-making outcomes.

2 Related work

2.1 Dombi aggregation operator

Dombi [25] developed the Dombi aggregation operator, which is commonly used to aggregate fuzzy information. It provides a framework for merging several fuzzy values while taking into account their degree of conflict. The operator's parameter, known as the aggregation parameter, allows you to change the amount of focus on contradictory data. Dombi norms are highly flexible due to the parameter which allows for adjusting the aggregation operator's behavior between t-norms (TN) and t-conorms (TCN). This flexibility makes them suitable for various decision-making scenarios. Dombi norms offer significant advantages in terms of flexibility, smooth transition, and generalization, they also come with limitations such as parameter sensitivity, computational complexity, and challenges in interpretability. Careful consideration and expert knowledge are essential for effectively leveraging Dombi norms in decision-making applications. Researchers investigated the Dombi aggregation operator's features and uses in a variety of decision-making contexts. For this advantage, Seikh et al. [26] Utilized Dombi operations on intuitionistic fuzzy sets to develop Intuitionistic Fuzzy Dombi Aggregation Operators and applied them to MADM. Dombi AOs for PFS were proposed by Akram et al. [27]. Aydemir et al. [28] Introduced the FF TOPSIS method utilizing Dombi AOs and applied it to MCDM. Shit et al. [29] Presented MADM using various types of Dombi AOs under Fermatean fuzzy (FF) information. Dombi AOs for Bipolar fuzzy were proposed by Jana et al. [30]. Jana et al. [31] Proposed Dombi Aggregation of Q-Rung Orthopair Fuzzy Numbers for MADM. Dombi AOs for Linguistic picture fuzzy were proposed by Qiyas et al. [32]. Inspired by the preceding discussion, we propose novel aggregation operators for Linguistic FF sets utilizing the Dombi TN and Dombi TCN.

2.2 TOPSIS method

This section presents an overview of the TOPSIS [33,34] technique, focusing on its concepts, stages, and popular applications in MCDM. The TOPSIS technique is a traditional approach to solving MADM issues. It belongs to the compensating MADM approaches. The method has been expanded to solve IFSs [35], interval-valued IFSs [36], PFSs [37] and FFSs [38]. However, the data structure of FFLTSs differs from the prior fuzzy information. Therefore, earlier TOPSIS approaches cannot be directly applied to FFLSs. To our knowledge, no study has been conducted on using the TOPSIS approach in LFFSs. A comparison study is carried out to determine the points of divergence and convergence between TOPSIS and the suggested linguistic FF Dombi method.

3 Motivation and main contributions

The primary inspiration behind this work is to develop a novel idea of FF linguistic Dombi aggregation operator (FFLDAO) which a comprehensive mathematical tool that combines linguistic knowledge and FF sets to handle difficult decision-making scenarios. This novel technique is especially well-suited for multi-criteria group decision-making challenges, in which different points of view and criteria must be reconciled in order to achieve a collective conclusion. The motivation for FFLDAO stems from its capacity to capture verbal expressions and ambiguity in decision-making processes. The LFFDAO improves its application to instances where conventional decision-making approaches fail by combining FF sets, which allow for a flexible representation of uncertainty and imprecision, with linguistic variables. In addition, we suggested a set of averaging and geometric AOs based on Dombi norms operators, which will aid in the selection of the optimum industrial site option. In this study, we provide various Dombi operational rules based on the linguistic scale function on FFLNs, as well as some new aggregation operators such as the FFLDWA operators and FFLDWG operators. Establishing a consensus in group decision-making situations can be difficult owing to various perspectives and judgements. FFLTSs contribute in unifying these various points of view into a logical decision framework, encouraging consensus among DMs.

4 Structure of the manuscript

The following describes the paper's structure. Section 2 explains several crucial concepts about LFFs in brief. In section 3, we define Linguistic FF set, score and accuracy function. In section 4 we define the novel operations of LFFN based on Dombi operator. In section 5 we define LFFD aggregation operator and prove their related properties. In section 6, we discuss algorithm of MCGDM problem used Dombi aggregation operator. In section 7 construct a case study about the selection of the best location for industry. Section 7 presents the article's conclusion and suggests potential future research directions.

5 Preliminaries

In this section, we will introduce the related concepts of LTS, IFS, PFS, and FFS. Throughout the paper, the set X=(x1,x2,…,xn) denotes the set of discourse, unless otherwise stated.Definition 2.1 [5] Let X be a fixed set; an IFS is a structure of the form:A‾={⟨xj,ϑA‾(xj),ΛA‾(xj)⟩|xj∈X}

where ϑA‾(xj),ΛA‾(xj)∈[0,1], are represent the MD and NMD of xj∈X, respectively, and ϑA‾(xj)+ΛA‾(xj)≤1. For xj∈X, the indeterminacy degree is defined by π‾A‾=1−ϑA‾(xj)−ΛA‾(xj).

Definition 2.2 [9] Let X be a fixed set; a PFS is a structure of the form:P‾={⟨xj,ϑP‾(xj),ΛP‾(xj)⟩|xj∈X}

where ϑA‾(xj),ΛA‾(xj)∈[0,1], are represent the MD and NMD of xj∈X, respectively, and (ϑP‾(xj))2+(ΛP‾(xj))2≤1. For xj∈X, the indeterminacy degree is defined by π‾P‾=1−(ϑP‾(xj))2−(ΛP‾(xj))2.

Definition 2.3 [15] Let X be a fixed set; a FFS is a structure of the form:F‾={⟨xj,ϑF‾(xj),ΛF‾(xj)⟩|xj∈X}

where ϑA‾(xj),ΛA‾(xj)∈[0,1], are represent the MD and NMD of xj∈X, respectively, and (ϑF‾(xj))3+(ΛF‾(xj))3≤1. For xj∈X, the indeterminacy degree is defined by π‾F‾=1−(ϑF‾(xj))3−(ΛF‾(xj))33.

A comparison of the grades of memberships of IFS, PFS and FFS has been shown in Fig. 1. Below (see Fig. 2).Fig. 1 Comparison of space of FFS, PFS and IFS.

Fig. 1

Fig. 2 The connections between LTS and their associated semantics in various contexts.

Fig. 2

Fig. 3 Flow chart of proposed procedure.

Fig. 3

Fig 4 Locational factors of industries.

Fig 4

Fig 5 Hierarchical Structure of suitable Location for Industries.

Fig 5

Fig. 6 Graphical representation of comparison.

Fig. 6

Fig. 7 Graphically representation of different LSFs.

Fig. 7

Fig. 8 Graphical representation of FFLDWG operator using different parameter.

Fig. 8

Fig. 9 Graphical representation of FFLDWA operator using different parameter.

Fig. 9

5.1 Linguistic term set (LTS)

Definition 1 [18] Let L═={Φi:i=1,2,…,2Υ} be a totally ordered, finite, discrete LTS, where Φi represents a possible value for a linguistic variable, and Υ is a positive integer.

The following criteria are satisfied by the LTS L═:(1) The set L═ is ordered set: (I) Φi≤Φj if i≤j; (II) max(Φi,Φj)=Φi if i≥j; (III) min(Φi,Φj)=Φi if i≤j.

(2) The negation operator Neg (Φi)=Φ2Υ−j, where i+j=2Υ.

Xu [39], expanded the discrete LTS into a continuous LTS L═={Φi:i∈[0,l]}(l>2Υ) to manage all the information, where Υ is a sufficiently large positive integer. For Φi,Φj∈L═,λ1,λ2∈[0,1], Xu, also introduced some properties of a continuous LTS as follows:1. Φi⊕Φj=Φi+j.

2. Φi⊖Φj=Φi−j.

3. (λ1+λ2)Φi=λ1Φi⊕λ1Φi.

4. λ1(Φi⊕Φj)=λ1Φi⊕λ1Φj.

5.2 Linguistic scale function

In various semantic contexts, linguistic terms convey decision information in multiple ways. Bao et al. [40] suggested that directly using the LTS subscript in computations could lead to information corruption. To resolve this problem, Xu [22] developed the linguistic scale function to handle linguistic information. In decision-making scenarios, decision-makers choose an appropriate linguistic scale function according to the linguistic decision environment, which can effectively translate the linguistic data.

Definition 1[22] Let L═={Φi:i∈[0,2Υ]} be a LTS and αi∈[0,1] then the LSF ψ is defined byψ:L═⟶αi

This is referred to as an LSF. It is evident that ψ is a strictly monotonically increasing function with respect to i, and ψ(Φi) falls within the range [0, 1].

Three commonly used LSFs are listed below:ψ1(Φi)=αi=i2Υ,i∈[0,2Υ]

ψ2(Φi)=αi={ρΥ−ρΥ−i2ρΥ−2,(i=0,1,…,Υ);ρΥ+ρi−Υ−22ρΥ−2,(i=Υ+1,Υ+2…,2Υ).

If the LTS is a set of seven terms, then Wang et al. has shown that ρ∈[1.36,1.4] [41].

In this paper, we assume that ρ=1.37.ψ3(Φi)=αi={Υη−(Υ−i)η2Υη,(i=0,1,…,Υ);Υμ−(i−Υ)μ2Υη,(i=Υ+1,Υ+2…,2Υ).

Where η,μ∈(0,1]. If the LTS is a seven terms set, then η=μ=0.8 [42]Example 1 Let L═={Φi:i∈[0,6]} represent a continuous LTS. We can then compute the inverse of the following LSF.1. If ψ¨1(Φi)=αi=i6(i=0,1,…,6), then ψ¨1−1(αi)=Φ6αi(αi∈[0,1]).

2. ψ¨2(Φi)=αi={ρ3−ρ3−i2ρ3−2,0≤i≤3ρ3+ρi−3−22ρ3−2,0<i≤3,thenψ¨2−1(αi)={Φ3−logρ[ρ3−(2ρ3−2)αi],αi∈[0,0.5],Φ3+logρ[(2ρ3−2)αi−ρ3+2],αi∈(0.5,1].

3. ψ¨3(Φi)=αi={3η−(3−i)η2×3η,0≤i≤33μ−(i−3)μ2×3μ,0≤i≤3,thenψ¨3−1(αi)={Φ3−[3η−2×3η×αi]1η,αi∈[0,0.5],Φ3+[2×3μ×αi−3μ]1μ,αi∈(0.5,1].

5.3 Fermatean fuzzy linguistic term sets

Definition 7 [21] Let X be a fixed set and Φil‾(xj)∈L═, then FFLTS l‾ on X is defined by:l‾={⟨xj,Φil‾(xj),ϑl‾(xj),Λl‾(xj)⟩|xj∈X}

where ϑl‾(xj) and Λl‾(xj), respectively represent the MD and NMD of xj to Φil‾(xj). The expression π‾l‾=1−(ϑl‾(xj))3−(Λl‾(xj))33 represent the indeterminacy degree of xj to Φil‾(xj). For xj∈X, 0≤ϑl‾(xj),Λl‾(xj)≤1, and 0≤(ϑl‾(xj))3+(Λl‾(xj))3≤1.

For ϑl‾(xj)=1 and Λl‾(xj)=0, the FFLTS is reduced to LTS. For singleton set X={x}, the FFLTS ⟨xj,Φil‾(xj),ϑl‾(xj),Λl‾(xj)⟩ is reduced to ⟨Φil‾(x),ϑl‾(x),Λl‾(x)⟩ and we call l‾=⟨Φil‾,ϑl‾,Λl‾⟩ the FF linguistic number (FFLN) and the corresponding indeterminacy is defined by π‾l‾=1−(ϑl‾(x))3−(Λl‾(x))33.

Definition 2 [21] Let l‾=⟨Φil‾,ϑl‾,Λl‾⟩, be a FFLN and ψ¨ be a LSF, then the score and accuracy functions are defined by:S(l‾)=ψ¨(Φil‾).ϑl‾3+1−Λl‾32

A(l‾)=ψ¨(Φil‾).(ϑl‾3+Λl‾3)

Definition 3.4 Dombi [25] Let X and Y be two real numbers. The Dombi t-norm and t-conorm are then defined as follows:T═(X,Y)=11+{(1−XX)C+(1−YY)C}1C

S═(X,Y)=1−11+{(X1−X)C+(Y1−Y)C}1C

Where C≥1 and (X,Y)∈[0,1]×[0,1].

5.4 Fermatean fuzzy linguistic Dombi aggregation operator

In this section, we will discuss Dombi operations in the context of FFLNs using LSF, specifically focusing on the Dombi TN and TCN. We will also develop the AOs FFLDWA and FFLDWG.

5.5 The operational laws of FFLD operator based on LSF

Definition 3.5 Let L═ be a continuous LTS, l‾1=(Φi(l‾1),ϑl‾1,Λl‾1) and l‾2=(Φi(l‾2),ϑl‾2,Λl‾2) be any two FFLNs, where Φi(l‾v)∈L═(v=1,2), λ∈[0,1] and C≥1 be a real number. Let ψ¨ and ψ¨−1 be a LSF and its inverse function, respectively. Then, Dombi TN and TCN operations of FFLNs are defined as:1. l‾1⊕l‾2=(ψ¨−1(1−11+{(ψ¨(Φi(l‾1))1−ψ¨(Φi(l‾1)))C+(ψ¨(Φi(l‾2))1−ψ¨(Φi(l‾2)))C}1C),(1−11+{(ϑl‾131−ϑl‾13)C+(ϑl‾231−ϑl‾23)C}1C3),(11+{(1−Λl‾1Λl‾1)C+(1−Λl‾2Λl‾2)C}1C)).

2. l‾1⊗l‾2=(ψ¨−1(11+{(1−ψ¨(Φi(l‾1))ψ¨(Φi(l‾1)))C+(1−ψ¨(Φi(l‾2))ψ¨(Φi(l‾2)))C}1C),(11+{(1−ϑl‾1ϑl‾1)C+(1−ϑl‾2ϑl‾2)C}1C),(1−11+{(Λl‾131−Λl‾13)C+(Λl‾231−Λl‾23)C}1C3)).

3. λ(l‾1)=(ψ¨−1(1−11+{λ(ψ¨(Φi(l‾1))1−ψ¨(Φi(l‾1)))C}1C),(1−11+{λ(ϑl‾131−ϑl‾13)C}1C3),(11+{λ(1−Λl‾1Λl‾1)C}1C)).

4. (l‾1)λ=(ψ¨−1(11+{λ(1−ψ¨(Φi(l‾1))ψ¨(Φi(l‾1)))C}1C),(11+{λ(1−ϑl‾1ϑl‾1)C}1C),(1−11+{λ(Λl‾131−Λl‾13)C}1C3)).

Example 3.6 Let L═={Φi:i∈[0,6]} be a continuous LTS, l‾1=(Φ2,0.4,0.5) and l‾2=(Φ4,0.6,0.3) be two FFLNs. If ψ¨=ψ2(Φi)(ρ=1.38,Υ=3), C=3 and λ=0.8 thenl‾1⊕l‾2=(Φ3.3026,0.6008,0.2947),l‾1⊗l‾2=(Φ2.6974,0.3933,0.5004),0.8(l‾1)=(Φ2.6702,0.3908,0.5186),(l‾1)0.8=(Φ2.7457,0.4180,0.4892)

5.6 Fermatean fuzzy linguistic Dombi weighted average (FFLDWA) operator based on LSF

Definition 3.7 Let l‾p=(Φi(l‾p),ϑl‾p,Λl‾p)(p=1,2,…,n) be set of FFLNs. Then, the FF Linguistic Dombi weighted averaging (FFLDWA) operator is a function of the form LFFDWA:l‾n→l‾, such that:FFLDWA(l‾1,l‾2,…,l‾n)=W═1l‾1⊕W═2l‾2⊕…⊕W═nl‾n

Where W═=(W═1,W═2,…,W═n)T,W═p>0, and ∑p=1nW═p=1.

Theorem 3.8 Ifl‾p=(Φi(l‾p),ϑl‾p,Λl‾p)(p=1,2,…,n)is a collection of FFLNs, the aggregated value of them using the FFLDWA operation is also an FFLN.FFLDWA(l‾1,l‾2,…,l‾n)=W═1l‾1⊕W═2l‾2⊕…⊕W═nl‾n

(1) =(ψ¨−1(1−11+{∑p=1nW═p(ψ¨(Φi(l‾p))1−ψ¨(Φi(l‾p)))C}1C),(1−11+{∑p=1nW═p(ϑl‾p31−ϑl‾p3)C}1C3),(11+{∑p=1nW═p(1−Λl‾pΛl‾p)C}1C))

Where W═=(W═1,W═2,…,W═n)T represent the weight vector of l‾p(p=1,2,…,n) such that W═p>0, and ∑p=1nW═i=1.

The proof of the Theorem 3.8 is demonstrated in the Appendix.

Property 1 (Idempotency) If all FFLNs are same, i.e, l‾p=(Φi(l‾p),ϑl‾p,Λl‾p)=(Φi(l‾),ϑl‾,Λl‾) for all p, l‾p=l‾ then;FFLDWA(l‾1,l‾2,…,l‾n)=l‾=(Φi(l‾),ϑl‾,Λl‾)

Property 2 (Monotonicity) Let l‾p″=(Φi″(l‾p),ϑl‾p″,Λl‾p″)(p=1,2,…,n) be a set of FFLNs such that Φi(l‾p)≤Φi″(l‾p), ϑl‾≤ϑl‾p″ and Λl‾≥Λl‾p″ thenFFLDWA(l‾1,l‾2,…,l‾n)≤FFLDWA(l‾1″,l‾2″,…,l‾n″)

Property 3 (Boundedness) If l‾−=(minp(Φi(l‾p)),minp(ϑl‾p),maxp(Λl‾p)) and l‾+=(maxp(Φi(l‾p)),maxp(ϑl‾p),minp(Λl‾p)) be two FFLNs then we havel‾−≤FFLDWA(l‾1,l‾2,…,l‾n)≤l‾+

Fermatean fuzzy linguistic Dombi weighted geometric (FFLDWG) operator based on linguistic scale function

Definition Let l‾p=(Φi(l‾p),ϑl‾p,Λl‾p)(p=1,2,…,n) be set of FFLNs. Then, the FF Linguistic Dombi weighted geometric (FFLDWG) operator is a function LFFDWA:l‾n→l‾, such that:FFLDWG(l‾1,l‾2,…,l‾n)=(l‾1)W═1⊗(l‾2)W═2⊗…⊗(l‾n)W═n

Where W═=(W═1,W═2,…,W═n)T,W═p>0, and ∑p=1nW═p=1.Theorem 5.1 If l‾p=(Φi(l‾p),ϑl‾p,Λl‾p)(p=1,2,…,n) is a collection of FFLNs, the aggregated value of them using the FFLDWA operation is also an FFLN.FFLDWG(l‾1,l‾2,…,l‾n)=(l‾1)W═1⊗(l‾2)W═2⊗…⊗(l‾n)W═n

=(ψ¨−1(11+{∑p=1nW═p(ψ¨(Φi(l‾p))1−ψ¨(Φi(l‾p)))C}1C),(11+{∑p=1nW═p(ϑl‾p1−ϑl‾p)C}1C),(1−11+{∑p=1nW═p(1−Λl‾p3Λl‾p3)C}1C3))

Where W═=(W═1,W═2,…,W═n)T represent the weight vector of l‾p(p=1,2,…,n) such that W═p>0, and ∑p=1nW═i=1.

Like Theorem 5.2, we can also identify some properties of the LFFWG operator.

Property 1: (Idempotency) If all FFLNs are same, i.e, l‾p=(Φi(l‾p),ϑl‾p,Λl‾p)=(Φi(l‾),ϑl‾,Λl‾) for all p, l‾p=l‾ then;FFLDWA(l‾1,l‾2,…,l‾n)=l‾=(Φi(l‾),ϑl‾,Λl‾)

Property 2: (Monotonicity) Let l‾p″=(Φi″(l‾p),ϑl‾p″,Λl‾p″)(p=1,2,…,n) be a set of FFLNs such that Φi(l‾p)≤Φi″(l‾p), ϑl‾≤ϑl‾p″ and Λl‾≥Λl‾p″ thenFFLDWA(l‾1,l‾2,…,l‾n)≤FFLDWA(l‾1″,l‾2″,…,l‾n″)

Property 3: (Boundedness) If l‾−=(minp(Φi(l‾p)),minp(ϑl‾p),maxp(Λl‾p)) and l‾+=(maxp(Φi(l‾p)),maxp(ϑl‾p),minp(Λl‾p)) be two FFLNs then we havel‾−≤FFLDWA(l‾1,l‾2,…,l‾n)≤l‾+

The proofs of the above properties are presented in the Appendix.

6 MCGDM method with linguistic Fermatean fuzzy information

We construct the FF Linguistic model, uses FFLTSs to provide the evaluation values. The following are the computation stages for our suggested model. Assume there are m alternatives {H˜1,H˜2,H˜3,…,H˜m}, n attributes {R¨1,R¨2,R¨3,…,R¨n} and g experts {A˙1,A˙2,A˙3,…,A˙g}, let {W═1,W═2,…,W═n} and {ℏ´1,ℏ´2,…,ℏ´k} be the attributes weighting vector and experts weighting vector which satisfy W═p∈[0,1], ℏ´g ∈ [0, 1] and ∑q=1nW═j=1,∑g=1kℏ´g=1 such that each decision-maker has assessed the available alternatives based on various attributes and provided their preference values using FFLNs l‾pqg=(Φi(l‾pq)g,ϑl‾pqg,Λl‾pqg), where l‾pqg∈L═={Φi:i∈[0,2Υ]}. All the information from each decision maker is compiled into a decision matrix B=(l‾pqg)m×n. The following are the computational steps:

Step 1: Construct the Linguistic FF decision matrix B=(l‾pqg)m×n , where p=1,2,…,m,q=1,2,…,n and g=1,2,…,k, It can be represented as follows:B=(l‾pqg)m×n

=((Φi(l‾11)g,ϑl‾11g,Λl‾11g)(Φi(l‾12)g,ϑl‾12g,Λl‾12g)⋯(Φi(l‾1n)g,ϑl‾1ng,Λl‾1ng)(Φi(l‾21)g,ϑl‾21g,Λl‾21g)(Φi(l‾22)g,ϑl‾22g,Λl‾22g)⋯(Φi(l‾2n)g,ϑl‾2ng,Λl‾2ng)⋮⋮⋱⋮(Φi(l‾m1)g,ϑl‾m1g,Λl‾m1g)(Φi(l‾m2)g,ϑl‾m2g,Λl‾m2g)…(Φi(l‾mn)g,ϑl‾mng,Λl‾mng))

Step 2: In accordance with the decision-making matrix B=(l‾pqg)m×n and expert's weighting vector {ℏ´1,ℏ´2,…,ℏ´k}, we can derive the overall l‾pqg to l‾pq by applying the FFLDWA or FFLDWG operator, the computation results can be written as mentioned below.B=(l‾pq)m×n

=((Φi(l‾11),ϑl‾11,Λl‾11)(Φi(l‾12),ϑl‾12,Λl‾12)⋯(Φi(l‾1n),ϑl‾1n,Λl‾1n)(Φi(l‾21),ϑl‾21,Λl‾21)(Φi(l‾22),ϑl‾22,Λl‾22)⋯(Φi(l‾2n),ϑl‾2n,Λl‾2n)⋮⋮⋱⋮(Φi(l‾m1),ϑl‾m1,Λl‾m1)(Φi(l‾m2),ϑl‾m2,Λl‾m2)…(Φi(l‾mn),ϑl‾mn,Λl‾mn))

Step 3: Aggregate all rˆpq=(q=1,2,…,n) for each alternative H˜p=(p=1,2,…,m) apply the FFLDWA or FFLDWG operatorrˆpq=(Φi(l‾pq),ϑl‾pq,Λl‾pq)

rˆp=FFLDWA(rˆp1,rˆp2,…,rˆpn)

=(ψ¨−1(1−11+{∑p=1nW═p(ψ¨(Φi(l‾p))1−ψ¨(Φi(l‾p)))C}1C),(1−11+{∑p=1nW═p(ϑl‾p31−ϑl‾p3)C}1C3),(11+{∑p=1nW═p(1−Λl‾pΛl‾p)C}1C))

Or use FFLDWG operatorrˆpq=(Φi(l‾pq),ϑl‾pq,Λl‾pq)

rˆp=FFLDWG(rˆp1,rˆp2,…,rˆpn)

=(ψ¨−1(11+{∑p=1nW═p(ψ¨(Φi(l‾p))1−ψ¨(Φi(l‾p)))C}1C),(11+{∑p=1nW═p(ϑl‾p1−ϑl‾p)C}1C),(1−11+{∑p=1nW═p(1−Λl‾p3Λl‾p3)C}1C3))

Step 4: Rank all the alternatives H˜p based on the score function rˆp(p=1,2,…,m).

7 Case study about the selection of the best location for industry

The industry is a collection of methods, processes, and/or actions carried out on raw materials with the goal of producing completed goods. The evaluation that finds the most convenient location for a manufacturing installation, which gives the maximum profitability of operations in relation to its investment, or where it precisely meets the company's objectives, whether economic or social, is referred to as industrial site selection. No matter the size of your business, the location of the industrial estate is crucial to consider. The best site is one that will help to optimize costs and get the task done at the lowest possible cost. Industrial locations are not simple, instead of include a complicated operational structure. After the initial survey, five locations, such as H˜1= Karachi, H˜2= Lahore, H˜3= Faisalabad, H˜4= Multan and H˜5= Peshawar has been selected for further evaluation. A panel of three DMs has been appointed to thoroughly assess all sites based on the following criteria: R¨1= Raw material, R¨2= Power, R¨3= Transport infrastructure and R¨4= Government policies/regulation.➢ Raw material: Raw materials are available from natural resources.

➢ Power: Any industrial organization requires conventional (coal, mineral oil, or hydroelectricity) or non-conventional energy.

➢ Transport infrastructure: The location of an industry is consistently shaped by the availability of basic transportation infrastructure. Consequently, intersections of waterways, highways, and railways often become bustling hubs of industrial activity.

➢ Government policies/regulation: Another aspect influencing industrial location is government policies. The government implements a range of restrictions on land distribution for companies to reduce regional disparities, control excessive pollution, and prevent overcrowding of industries in major cities.

The four criteria' weight vector is W═=(0.2738,0.1967,0.1711,0.3584)T. The three experts A˙1, A˙2, and A˙3 of Table 1, Table 2, Table 3 are assigned to evaluate each alternative on each criteria using the linguistic word set L═= {Φ0 = very poor, Φ1 = poor, Φ2 = slightly poor, Φ3 = fair, Φ4 = slightly good, Φ5 = good, Φ6 = very good}. Then, as follows, we use the above-mentioned operators to obtain the most desirable alternative(s):Table 1 Fermatean fuzzy Linguistic decision matrix A˙1.

Table 1B	R¨1	R¨2	R¨3	R¨4	
H˜1	(Φ5,0.52,0.31)	(Φ4,0.83,0.24)	(Φ1,0.90,0.45)	(Φ5,0.76,0.33)	
H˜2	(Φ3,0.45,0.42)	(Φ3,0.65,0.51)	(Φ5,0.76,0.47)	(Φ2,0.55,0.48)	
H˜3	(Φ4,0.83,0.65)	(Φ4,0.66,0.47)	(Φ2,0.32,0.26)	(Φ3,0.39,0.37)	
H˜4	(Φ1,0.92,0.60)	(Φ5,0.77,0.54)	(Φ3,0.73,0.41)	(Φ1,0.64,0.55)	
H˜5	(Φ2,0.75,0.56)	(Φ3,0.89,0.44)	(Φ1,0.45,0.34)	(Φ1,0.34,0.19)	

Table 2 Fermatean fuzzy Linguistic decision matrix A˙2.

Table 2B	R¨1	R¨2	R¨3	R¨4	
H˜1	(Φ2,0.63,0.35)	(Φ1,0.76,0.64)	(Φ3,0.50,0.41)	(Φ3,0.86,0.49)	
H˜2	(Φ1,0.68,0.60)	(Φ2,0.35,0.34)	(Φ5,0.86,0.49)	(Φ1,0.59,0.50)	
H˜3	(Φ5,0.76,0.54)	(Φ3,0.56,0.49)	(Φ1,0.57,0.53)	(Φ2,0.77,0.57)	
H˜4	(Φ2,0.83,0.69)	(Φ3,0.67,0.55)	(Φ4,0.77,0.68)	(Φ4,0.84,0.59)	
H˜5	(Φ4,0.78,0.66)	(Φ1,0.69,0.65)	(Φ3,0.65,0.39)	(Φ5,0.94,0.26)	

Table 3 Fermatean fuzzy Linguistic decision matrix A˙3.

Table 3B	R¨1	R¨2	R¨3	R¨4	
H˜1	(Φ4,0.75,0.49)	(Φ3,0.94,0.53)	(Φ1,0.91,0.48)	(Φ2,0.87,0.40)	
H˜2	(Φ5,0.86,0.58)	(Φ1,0.85,0.59)	(Φ5,0.78,0.23)	(Φ3,0.58,0.52)	
H˜3	(Φ1,0.90,0.64)	(Φ5,0.69,0.59)	(Φ2,0.56,0.29)	(Φ5,0.45,0.30)	
H˜4	(Φ1,0.91,0.58)	(Φ3,0.73,0.69)	(Φ4,0.93,0.49)	(Φ4,0.59,0.36)	
H˜5	(Φ4,0.88,0.65)	(Φ2,0.63,0.35)	(Φ4,0.65,0.39)	(Φ1,0.95,0.37)	

Step 1: Construct the FF Linguistic term set decision matrix B=(l‾pqg)5×4 , where p=1,2,…,5,q=1,2,…,4,andg=1,2,3.

Step 2: We aggregate the overall l‾pqg, to single matrix l‾pq by utilizing the FFLDWG aggregation operator and we have taken the linguistic scaling function ψ¨=ψ2(Φi)(ρ=1.38) where C=3 and expert's weighting vector Φ=(0.33,0.29,0.38). The calculating results are shown in Table 4.Table 4 Aggregated values by using LFFDWG operator.

Table 4B	R¨1	R¨2	R¨3	R¨4	
H˜1	(Φ2.3844,0.5908,0.4467)	(Φ2.3250,0.8134,0.5828)	(Φ2.3165,0.6015,0.4572)	(Φ2.3797,0.8100,0.4410)	
H˜2	(Φ2.2952,0.5378,0.5696)	(Φ2.2633,0.4462,0.5488)	(Φ2.6252,0.7860,0.4580)	(Φ2.2860,0.5713,0.5037)	
H˜3	(Φ2.4569,0.8117,0.6297)	(Φ2.3578,0.6235,0.5488)	(Φ2.2461,0.3977,0.4702)	(Φ2.2780,0.4440,0.5084)	
H˜4	(Φ2.3487,0.8730,0.6416)	(Φ2.3184,0.7157,0.6446)	(Φ2.4119,0.7747,0.6172)	(Φ2.3795,0.6357,0.5482)	
H˜5	(Φ2.4189,0.7874,0.6376)	(Φ2.3440,0.6802,0.5862)	(Φ2.2936,0.5288,0.3790)	(Φ2.2602,0.4271,0.3349)	

Step 3: Aggregate all rˆpq=(q=1,2,…,n) for each alternative H˜p=(p=1,2,…,m) and the criteria weighting vector W═=(0.2738,0.1967,0.1711,0.3584), apply the FFLDWG operator. Table 5 provides a summary of the corresponding results.Table 5 Apply FFLDWG operator.

Table 5H˜1	H˜2	H˜3	H˜4	H˜5	
(Φ2.8120,0.6542,0.5118)	(Φ2.8032,0.5292,0.5362)	(Φ2.8039,0.4753,0.5758)	(Φ2.8135,0.6947,0.6189)	(Φ2.8012,0.4990,0.5825)	

Step 4: Rank all the alternatives H˜p by utilizing score function rˆp(p=1,2,…,m) given in Table 6.Table 6 Overall the score value of FFLDWG operator.

Table 6	S(H˜1)	S(H˜2)	S(H˜3)	S(H˜4)	S(H˜5)	
FFLDWG	0.2755	0.2385	0.2200	0.2641	0.2222	

Table 7 Comparative study and ranking of the alternatives.

Table 7	S(H˜1)	S(H˜2)	S(H˜3)	S(H˜4)	S(H˜5)	Ranking	
FFLDWG	0.2755	0.2385	0.2200	0.2641	0.2222	H˜1>H˜4>H˜2>H˜5>H˜3	
FFLDWA	0.3337	0.2912	0.3082	0.3272	0.3187	H˜1>H˜4>H˜5>H˜3>H˜2	
TOPSIS [21] method	0.4280	0.3259	0.3628	0.5859	0.4117	H˜4>H˜1>H˜5>H˜3>H˜2	

Table 8 The score values for different Linguistic Scale Functions (LSFs).

Table 8LSFsψ¨=ψ1(Φi)	S(H˜1)	S(H˜2)	S(H˜3)	S(H˜4)	S(H˜5)	Ranking	
FFLDWG	0.2728	0.2350	0.2492	0.2680	0.2569	H˜1>H˜4>H˜5>H˜3>H˜2	
FFLDWA	0.2730	0.2384	0.2507	0.2682	0.2579	H˜1>H˜4>H˜5>H˜3>H˜2	
LSFsψ¨=ψ3(Φi)
η=μ=0.8	
FFLDWG	0.2460	0.2112	0.2241	0.2418	0.2309	H˜1>H˜4>H˜5>H˜3>H˜2	
FFLDWA	0.2462	0.2158	0.2260	0.2421	0.2321	H˜1>H˜4>H˜5>H˜3>H˜2	

Table 9 Decision results of FFLDWG operator using different parameter.

Table 9C	S(H˜1)	S(H˜2)	S(H˜3)	S(H˜4)	S(H˜5)	Ranking	
1	0.2952	0.2464	0.2349	0.2788	0.2463	H˜1>H˜4>H˜2>H˜5>H˜3	
3	0.2755	0.2385	0.2200	0.2641	0.2222	H˜1>H˜4>H˜2>H˜5>H˜3	
5	0.2647	0.2336	0.2128	0.2566	0.2134	H˜1>H˜4>H˜2>H˜5>H˜3	
10	0.2543	0.2267	0.2052	0.2485	0.2054	H˜1>H˜4>H˜2>H˜5>H˜3	
20	0.2487	0.2218	0.2002	0.2435	0.2008	H˜1>H˜4>H˜2>H˜5>H˜3	

According to the score function we get the ranking of alternatives H˜1>H˜4>H˜2>H˜5>H˜3.

8 Comparison with existing literature

To demonstrate the superiority and efficacy of the investigated technique, a detailed evaluation and comparison of various frameworks (see Refs. [[40], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52]] for additional information) developed by numerous investigators has been carried out (see Table 10) (see Table 7). The comparison demonstrated that the approach under consideration outperforms current methods and is significantly more effective for multi-parameter decision-making challenges. A practical example of MADM was displayed for selecting the optimal site for a small hydropower plant (SHPP) cited from Muneeza et al. [53] For this purpose, an architectural company based in Pakistan has commenced construction on a nationwide undertaking comprising four places for the construction of SHPP. The company is going to be evaluated further in order to determine which power facility meets the requirements of the organization most effectively. An architecture firm contracted three experts to assess the optimal site for SHPP. The experts evaluated all four SHPPs with the weight vector according to the following five criteria: purchase, feed-in tariff, accessibility, economic environment, constructability, and technical feasibility. The results of a comparative analysis of our established models and those that exist in the open literature (for additional details, see Refs. [[40], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52]]) are presented in Table 11. From Tables 11 and it could be observed that the IF imprecise values are employed and cannot be resolved using various aggregation operators, such as IF-GRA schemes, IF-VIKOR schemes, IF-TOPSIS schemes, or IF-EDAS methods. The failure of conventional approaches to properly compile the information leads to their inability to resolve and evaluate decision-making problems, as displayed in Table 11. Therefore, the innovative technique outperforms the classical model in terms of capabilities and efficiency.Table 10 Decision results of FFLDWA operator using different parameter.

Table 10C	S(H˜1)	S(H˜2)	S(H˜3)	S(H˜4)	S(H˜5)	Ranking	
1	0.3192	0.2638	0.2743	0.3042	0.2898	H˜1>H˜4>H˜5>H˜3>H˜2	
3	0.3337	0.2912	0.3082	0.3273	0.3187	H˜1>H˜4>H˜5>H˜3>H˜2	
5	0.3390	0.3051	0.3199	0.3379	0.3294	H˜1>H˜4>H˜5>H˜3>H˜2	
10	0.3433	0.3183	0.3303	0.3484	0.3386	H˜4>H˜1>H˜5>H˜3>H˜2	
20	0.3456	0.3274	0.3369	0.3547	0.3436	H˜4>H˜1>H˜5>H˜3>H˜2	

Table 11 Comparative analysis of the demonstrated approach and traditional techniques.

Table 11Techniques	Score values of alternatives	Ranking	
H˜1	H˜2	H˜3	H˜4	
IFWA [40]	Inaccessible	X	
IFDWA [43]	Inaccessible	X	
IFWG [44]	Inaccessible	X	
IFHWA [45]	Inaccessible	X	
IFR [[46], [47], [48]]	Inaccessible	X	
IFRSS [49]	Inaccessible	X	
IF-GRA scheme [50]	Inaccessible	X	
IF-TOPSIS scheme [51]	Inaccessible	X	
IF-VIKOR scheme [52]	Inaccessible	X	
IF-EDAS scheme [54]	Inaccessible	X	
FFLDWG (proposed)	0.6769 > 0.6710 > 0.6637 > 0.6484	H˜1>H˜2>H˜3>H˜4
	
FFLDWA (proposed)	0.6580 > 0.6454 > 0.6087 > 0.6031	H˜1>H˜2>H˜3>H˜4	

9 Conclusion

To explain ambiguous information from a qualitative point of view, FFLS is an extension of FF. We developed a multi-criteria group decision-making technique to overcome decision-making problem in this paper. To begin, we establish the operational laws for FFLNs using the Dombi operation. Following that, we proposed several aggregation operators for combining the diverse evaluation data of alternatives, including Fermatean Linguistic fuzzy Dombi weighted averaging (FFDFWA) operator and Fermatean Linguistic fuzzy Dombi weighted geometric (FFLDWG) operator. Some of the essential characteristics of these aggregating operators are comprehensively discussed. Furthermore, we used these aggregation operations to build a system of decision-making for solving real-life issues of choosing the best location for industry from cities in Pakistan that are Karachi, Lahore, Faisalabad, Multan and Peshawar using FFL data.On the basis of the available information, we found that Karachi is the best location for industry. To determine the validity and implementation of the proposed technique, illustration examples are used. To demonstrate the superiority of the proposed approach, a comparison study with existing methods is performed.

Future work could extend the results of this paper to interval-valued contexts FFS [55,56], complex FF set [57,58] and Fermatean Cubic fuzzy set [59,60], probabilistic linguistic q-rung orthopair fuzzy [61], q-rung probabilistic dual hesitant fuzzy environment [62], interval-valued intuitionistic fuzzy hypersoft set [63], q-rung orthopair fuzzy WASPAS method based on softmax function and Frank operations [64]. We will further extend the develop notion to cosine similarity under double hierarchy hesitant fuzzy linguistic environment [65]., ORESTE method with linguistic preference orderings, probabilistic double hierarchy linguistic term set [66], score function based on concentration degree for probabilistic linguistic term sets with TOPSIS and VIKOR schemes [67], q-rung orthophair hesitant fuzyy rough sets [[68], [69], [70], [71]].

Data availability statement

All data generated during this study are reported in this article.

CRediT authorship contribution statement

Omar Barukab: Funding acquisition. Asghar Khan: Supervision, Conceptualization. Sher Afzal Khan: Data curation.

Declaration of competing interest

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

Appendix Proof of Theorem 3.8. Use mathematical induction for proof.

For n = 2FFLDWA(l‾1,l‾2)=W═1l‾1⊕W═2l‾2

=(ψ¨−1(1−11+{W═1(ψ¨(Φi(l‾1))1−ψ¨(Φi(l‾1)))C+W═2(ψ¨(Φi(l‾2))1−ψ¨(Φi(l‾2)))C}1C),(1−11+{W═1(ϑl‾131−ϑl‾13)C+W═2(ϑl‾231−ϑl‾23)C}1C3),(11+{W═1(1−Λl‾1Λl‾1)C+W═2(1−Λl‾2Λl‾2)C}1C))

=(ψ¨−1(1−11+{∑p=12W═p(ψ¨(Φi(l‾p))1−ψ¨(Φi(l‾p)))C}1C),(1−11+{∑p=12W═p(ϑl‾p31−ϑl‾p3)C}1C3),(11+{∑p=12W═p(1−Λl‾pΛl‾p)C}1C))

Hence equation (1) is true for n = 2

We assume that equation (13) is hold for n = kW═1l‾1⊕W═2l‾2⊕…⊕W═kl‾k

=(ψ¨−1(1−11+{∑p=1kW═p(ψ¨(Φi(l‾p))1−ψ¨(Φi(l‾p)))C}1C),(1−11+{∑p=1kW═p(ϑl‾p31−ϑl‾p3)C}1C3),(11+{∑p=1kW═p(1−Λl‾pΛl‾p)C}1C))

Now, for n = k+1FFLDWA(l‾1,l‾2,…,l‾k,l‾k+1)=⊕p=1k+1(W═pl‾p)

=(ψ¨−1(1−11+{∑p=1kW═p(ψ¨(Φi(l‾p))1−ψ¨(Φi(l‾p)))C}1C),(1−11+{∑p=1kW═p(ϑl‾p31−ϑl‾p3)C}1C3),(11+{∑p=1kW═p(1−Λl‾pΛl‾p)C}1C))

⊕(ψ¨−1(1−11+{W═k+1(ψ¨(Φi(l‾k+1))1−ψ¨(Φi(l‾k+1)))C}1C),(1−11+{W═k+1(ϑl‾k+131−ϑl‾k+13)C}1C3),(11+{W═k+1(1−Λl‾k+1Λl‾k+1)C}1C))

=(ψ¨−1(1−11+{∑p=1k+1W═p(ψ¨(Φi(l‾p))1−ψ¨(Φi(l‾p)))C}1C),(1−11+{∑p=1k+1W═p(ϑl‾p31−ϑl‾p3)C}1C3),(11+{∑p=1k+1W═p(1−Λl‾pΛl‾p)C}1C))

Hence equation (1) is true for n = k+1

FF Linguistic Dombi weighted averaging (FFLDWA) operator has some important properties which are listed below.

Proof of property 1: We haveFFLDWA(l‾1,l‾2,…,l‾n)=⊕i=1n(W═pl‾p)

=(ψ¨−1(1−11+{∑p=1nW═p(ψ¨(Φi(l‾p))1−ψ¨(Φi(l‾p)))C}1C),(1−11+{∑p=1nW═p(ϑl‾p31−ϑl‾p3)C}1C3),(11+{∑p=1nW═p(1−Λl‾pΛl‾p)C}1C))

=(ψ¨−1(1−11+{(ψ¨(Φi(l‾))1−ψ¨(Φi(l‾)))C}1C),(1−11+{(ϑl‾31−ϑl‾3)C}1C3),(11+{(1−Λl‾Λl‾)C}1C))

l‾=(Φi(l‾),ϑl‾,Λl‾)

Proof ofproperty 2: Since l‾1≤l‾p″, then Φi(l‾p)≤Φi″(l‾p), ϑl‾p≤ϑl‾p″ and Λl‾p≥Λl‾p″ for p=1,2,…,n thereforeΦi(l‾p)≤Φi″(l‾p)⇒ψ¨(Φi(l‾p))≤ψ¨(Φi″(l‾p))

⇒ψ¨(Φi(l‾p))1−ψ¨(Φi(l‾p))≤ψ¨(Φi″(l‾p))1−ψ¨(Φi″(l‾p))

⇒{∑p=1nW═p(ψ¨(Φi(l‾p))1−ψ¨(Φi(l‾p)))C}1C≤{∑p=1nW═p(ψ¨(Φi″(l‾p))1−ψ¨(Φi″(l‾p)))C}1C

⇒1+{∑p=1nW═p(ψ¨(Φi(l‾p))1−ψ¨(Φi(l‾p)))C}1C≤1+{∑p=1nW═p(ψ¨(Φi″(l‾p))1−ψ¨(Φi″(l‾p)))C}1C

⇒11+{∑p=1nW═p(ψ¨(Φi(l‾p))1−ψ¨(Φi(l‾p)))C}1C≥11+{∑p=1nW═p(ψ¨(Φi″(l‾p))1−ψ¨(Φi″(l‾p)))C}1C

⇒1−11+{∑p=1nW═p(ψ¨(Φi(l‾p))1−ψ¨(Φi(l‾p)))C}1C≤1−11+{∑p=1nW═p(ψ¨(Φi″(l‾p))1−ψ¨(Φi″(l‾p)))C}1C

⇒ψ¨−1(1−11+{∑p=1nW═p(ψ¨(Φi(l‾p))1−ψ¨(Φi(l‾p)))C}1C)

≤ψ¨−1(1−11+{∑p=1nW═p(ψ¨(Φi″(l‾p))1−ψ¨(Φi″(l‾p)))C}1C)

Hence Φi(l‾p)≤Φi″(l‾p). In the similar manner, we can show that ϑl‾p≤ϑl‾p″ and Λl‾p≥Λl‾p″. The prove of ϑl‾p≤ϑl‾p″ and Λl‾p≥Λl‾p″ is similar as Φi(l‾p)≤Φi″(l‾p).(ψ¨−1(1−11+{∑p=1nW═p(ψ¨(Φi(l‾p))1−ψ¨(Φi(l‾p)))C}1C),(1−11+{∑p=1nW═p(ϑl‾p31−ϑl‾p3)C}1C3),(11+{∑p=1nW═p(1−Λl‾pΛl‾p)C}1C))

≤(ψ¨−1(1−11+{∑p=1nW═p(ψ¨(Φi″(l‾p))1−ψ¨(Φi″(l‾p)))C}1C),(1−11+{∑p=1nW═p((ϑl‾p″)31−(ϑl‾p″)3)C}1C3),(11+{∑p=1nW═p(1−Λl‾p″Λl‾p″)C}1C))

FFLDWA(l‾1,l‾2,…,l‾n)≤FFLDWA(l‾1″,l‾2″,…,l‾n″)

Proof ofproperty 3:l‾p=(Φi(l‾p),ϑl‾p,Λl‾p)(p=1,2,…,n) be a set of FFLNs.

Let l‾−=min(l‾1,l‾2,…,l‾n)=(Φi−(l‾p),ϑl‾p−,Λl‾p−),

and l‾+=max(l‾1,l‾2,…,l‾n)=(Φi+(l‾p),ϑl‾p+,Λl‾p+) whereΦi−(l‾p)=minp(Φi(l‾p)),ϑl‾p−=minp(ϑl‾p),Λl‾p−=maxp(Λl‾p)

Φi+(l‾p)=maxp(Φi(l‾p)),ϑl‾p+=maxp(ϑl‾p),Λl‾p+=minp(Λl‾p).

Based on the monotonicity of FFLDWA operator, one hasFFLDWA(l‾1,l‾2,…,l‾n)≤FFLDWA(l‾1+,l‾2+,…,l‾n+);

FFLDWA(l‾1,l‾2,…,l‾n)≥FFLDWA(l‾1−,l‾2−,…,l‾n−).

Furthermore, based on the idempotency of LFFDWA operator, one has(l‾1+,l‾2+,…,l‾n+)=l‾+,(l‾1−,l‾2−,…,l‾n−)=l‾−.

Accordingly, l‾−≤FFLDWA(l‾1,l‾2,…,l‾n)≤l‾+.

This completes the proof.

Acknowledgements

This research work was funded by Institutional Fund Projects under grant no. (IFPRP: 227-830-1442). Therefore, authors gratefully acknowledge technical and financial support from the Ministry of Education and 10.13039/501100004054 King Abdulaziz University , DSR, Jeddah, Saudi Arabia.
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