# Extensions of Dombi Aggregation Operators for Decision Making underm-Polar Fuzzy Information

Authors: Muhammad Akram; Naveed Yaqoob; Ghous Ali; Wathek Chammam
Journal: Journal of Mathematics
 (2020)

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## Abstract

Anm-polar fuzzy set is a powerful mathematical model to analyze multipolar, multiattribute, and multi-index data. The m-polar fuzzy sets have appeared as a useful tool to portray uncertainty in multiattribute decision making. The purpose of this article is to analyze the aggregation operators under the m-polar fuzzy environment with the help of Dombi norm operations. In this article, we develop some averaging and geometric aggregation operators using Dombi t-norm and t-conorm to handle uncertainty in m-polar fuzzy (mF, henceforth) information, which are mF Dombi weighted averaging (mFDWA) operator, mF Dombi ordered weighted averaging (mFDOWA) operator, mF Dombi hybrid averaging (mFDHA) operator, mF Dombi weighted geometric (mFDWG) operator, mF Dombi weighted ordered geometric operator, and mF Dombi hybrid geometric (mFDHG) operator. We investigate properties, namely, idempotency, monotonicity, and boundedness, for the proposed operators. Moreover, we give an algorithm to solve multicriteria decision-making issues which involve mF information with mFDWA and mFDWG operators. To prove the validity and feasibility of the proposed model, we solve two numerical examples with our proposed models and give comparison with mF-ELECTRE-I approach (Akram et al. 2019) and mF Hamacher aggregation operators (Waseem et al. 2019). Finally, we check the effectiveness of the developed operators by a validity test.


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## Body

## 1. Introduction

Multicriteria decision making (MCDM) is performing a vital role in different areas, including social, physical, medical, and environmental sciences. MCDM methods are not only used to determine a suitable object but also used to rank the objects in an appointed problem. To solve different uncertain problems for decision making, Atanassov [1] presented the concept of intuitionistic fuzzy set (IFS) which considers both membership and nonmembership parts, an extension of fuzzy set [2] in which simple membership part is characterized.Aggregation operators (AOs) perform an important role in order to combine data into a single form and solve MCDM problems. For example, Yager [3] introduced weighted AOs. Xu [4] proposed some new AOs under IFSs. Xu and Yager [5] developed certain new geometric AOs and solved some real-world MCDM problems. From the inspection of an object, it can be easily seen that there exist two properties of the object which are opposite to each other. With this perspective, Zhang [6] presented the idea of bipolar fuzzy set (BFS). BFSs provide generalized structure as compared to fuzzy sets [2] whose memberships belong to −1,0×0,1. Bipolarity plays an important role in different research areas and provides more flexibility as compared to the fuzzy methods. In the last decades, a lot of researchers, attracted by this efficient concept, applied it to aggregate bipolar information using different t-norms and their corresponding conorms, including Hamacher and Dombi t-norms and their corresponding conorms. For example, Wei et al. [7] developed some bipolar fuzzy Hamacher weighted averaging and geometric AOs. By combining Hamacher operations and prioritized AOs, Gao et al. [8] proposed dual hesitant bipolar fuzzy Hamacher prioritized weighted AOs and applied the proposed methodologies to an MCDM problem. Liu [9] utilized interval-valued intuitionistic fuzzy numbers with Hamacher AOs and developed multicriteria methods for group decision making. Jana et al. [10] applied weighted, ordered weighted, and hybrid average and geometric AOs for the aggregation of bipolar fuzzy information using Dombi t-conorm and t-norm. They also proposed bipolar fuzzy Dombi prioritized AOs in [11]. He [12] developed hesitant fuzzy Dombi AOs and investigated typhoon disaster assessment using proposed theory. Xu and Wei [13] introduced different dual hesitant bipolar fuzzy AOs to solve MCDM problems. Xu [14] proposed intuitionistic fuzzy power AOs for multiattribute group decision making. Xiao [15] constructed induced interval-valued intuitionistic fuzzy Hamacher AOs and discussed their application to MCDM. Chen and Ye [16] discussed MCDM problem under Dombi operations in single-valued neutrosophic situation. Garg [17] presented some generalized interactive AOs under Einstein operations in Pythagorean fuzzy environment and discussed a decision-making issue. Akram et al. [18] proposed different Pythagorean Dombi fuzzy AOs and studied their applications in MCDM. Shahzadi et al. [19] introduced Pythagorean fuzzy Yager AOs for decision making. Peng and Yang [20] investigated different basic properties of interval-valued Pythagorean fuzzy AOs. Wang et al. [21] introduced some new types of q-rung orthopair fuzzy Hamy mean AOs to handle MCDM situations. Arora and Garg [22] proposed robust AOs with an intuitionistic fuzzy soft environment. Wang and Li [23] developed Pythagorean fuzzy interaction power Bonferroni mean AOs and discussed their applications to MCDM. Chiclana et al. [24] introduced some ordered weighted geometric operators and solved a decision-making problem. Liang et al. [25] developed Pythagorean fuzzy Bonferroni mean AOs.Nowadays, experts believe that multipolarity performs a vital role in many practical situations. Due to the presence of multipolar data in different daily life problems of science and technology, Chen et al. [26] initiated the notion of mF set theory as generalization of fuzzy and bipolar fuzzy sets. Waseem et al. [27] studied recently MCDM problems based on mF Hamacher AOs. Khameneh and Kilicman [28] proposed mF soft weighted AOs and applied these AOs in decision making. In view of the fact that mF sets have an efficient strength to handle vague data which arise in several real-life problems, in this paper, we generalize Dombi AOs to aggregate the mF information. The study of AOs under Dombi operations is very popular. Thus, an efficient research topic is how to aggregate mF numbers with Dombi operations. To tackle this dilemma, in this article, we present some mF Dombi AOs on the ground of classical geometric, arithmetic, and Dombi operations. For more information and terminologies on AOs, the readers are referred to [29–45].AnmF model is more general than the fuzzy sets and BFSs due to the wider range of applicability over different complex problems. The mF sets can handle much more details about an element and can explain uncertainties concurrently more precisely than the other existing methods, like fuzzy set and BFS. The motivation of developed AOs is summarized as below.(1)
A very difficult MCDM problem is the estimation of the supreme option in anmF environment due to the involvement of several imprecise factors. Assessment of information in different MCDM techniques is simply depicted through fuzzy and bipolar fuzzy numbers which may not consider all the data in a real-world problem.(2)
As a general theory,mF numbers describe efficient execution in the assessment process about uncertain, imprecise, and vague multipolar information. Thus, mF theory provides an excellent approach for the assessment of objects under multinary data.(3)
In view of the fact that Dombi AOs are simple but provide a pioneering tool for solving MCDM problems when combined with other powerful mathematical tools, this article aims to develop Dombi AOs in anmF environment to handle complex problems.(4)
AnmF model is different from the mathematical tools like fuzzy sets and BFSs because the fuzzy set and BFS can only handle one-dimensional data and two-dimensional data, respectively, which may prompt a loss in data. Nevertheless, in many daily life problems, we handle the situations having higher dimension to sort out all the attributes and their subcharacteristics.(5)
The Dombi AOs employed in the construction ofmF Dombi AOs are more suitable than all other aggregation approaches to tackle the MCDM situations as developed AOs have ability to consider all the information within the aggregation procedure.(6)
Dombi AOs make the optimal outcomes more accurate and definite when utilized in practical MCDM problems undermF environment.(7)
The proposedmF Dombi operators handle the drawbacks of existing AOs, including bipolar fuzzy Dombi AOs [10].Therefore, somemF Dombi AOs are developed to choose the best option in different decision-making situations. The developed operators have some advantages over other approaches which are given as follows:(1)
Our proposed methods explain the problems more accurately which involve multiple attributes because they considermF numbers.(2)
The developed AOs are more precise and efficient with single attribute.(3)
To solve practical problems by using Dombi AOs withmF numbers is very significant.The rest of this article is structured as follows.Section 2 recalls some fundamental definitions and operations of the mF numbers (mFNs). Section 3 presents mFDWA, mFDOWA, mFDHA, mFDWG, mFDOWG, and mFDHG operators. Section 4 develops a methodology of these AOs to model mF MCDM problems. Section 5 discusses two applications: first for the selection of best agricultural land and second for the selection of best commercial bank. Section 6 provides comparative analysis of developed approaches with mF ELECTRE-I model [46] and mF Hamacher AOs [27]. Section 7 discusses the conclusions and future directions.


## 2. Preliminaries

Definition 1.
(see [26]). An mF set on a universal set U is a mapping ζ:U⟶0,1m. The membership of every object is described by ζu=p1∘ζu,p2∘ζu,…,pm∘ζu where pr∘ζ:0,1m⟶0,1 is the r-th projection mapping.
Letζ^=p1∘ζ,…,pm∘ζ be an mFN, where pr∘ζ∈0,1, ∀r=1,2,…,m. We define the score and accuracy functions of ζ^, respectively, as follows.Definition 2.
(see [27]). For an mFNζ^=p1∘ζ,…,pm∘ζ, we define a score function S as follows:(1)Sζ^=1m∑r=1mpr∘ζ,Sζ^∈0,1.Definition 3.
(see [27]). For an mFNζ^=p1∘ζ,…,pm∘ζ, an accuracy function H is defined as(2)Hζ^=1m∑r=1m−1r+1pr∘ζ−1,Hζ^∈−1,1.
From Definitions2 and 3, it can be readily seen that for any mFNζ^, Sζ^∈0,1 and Hζ^∈−1,1. Notice that Hζ^ represents the accuracy degree of ζ^. Thus, a higher value of Hζ^ represents a higher accuracy degree for mFNζ^.
Using Definitions2 and 3, we now give the following ordered relation criteria for any two mFNs.Definition 4.
(see [27]). Let ζ^1=p1∘ζ1,…,pm∘ζ1 and ζ^2=p1∘ζ2,…,pm∘ζ2 be two mFNs. Then,(1)
ζ^1<ζ^2,ifSζ^1<Sζ^2.(2)
ζ^1>ζ^2,ifSζ^1>Sζ^2.(3)
ζ^1=ζ^2,ifSζ^1=Sζ^2andHζ^1=Hζ^2.(4)
ζ^1<ζ^2,ifSζ^1=Sζ^2,butHζ^1<Hζ^2.(5)
ζ^1>ζ^2,ifSζ^1=Sζ^2,butHζ^1>Hζ^2.
Some basic operations formFNs are given by [27](1)
ζ^1⊞ζ^2=p1∘ζ1+p1∘ζ2−p1∘ζ1.p1∘ζ2,…,pm∘ζ1+pm∘ζ2−pm∘ζ1.pm∘ζ2.(2)
ζ^1⊠ζ^2=p1∘ζ1.p1∘ζ2,…,pm∘ζ1.pm∘ζ2.(3)
βζ^=1−1−p1∘ζ)β,…,1−1−pm∘ζβ,β>0.(4)
ζ^β=p1∘ζβ,…,pm∘ζβ,β>0.(5)
ζ^c=1−p1∘ζ,…,1−pm∘ζ.(6)
ζ^1⊆ζ^2, if and only if p1∘ζ1≤p1∘ζ2,…,pm∘ζ1≤pm∘ζ2.(7)
ζ^1∪ζ^2=maxp1∘ζ1,p1∘ζ2,…,maxpm∘ζ1,pm∘ζ2.(8)
ζ^1∩ζ^2=minp1∘ζ1,p1∘ζ2,…,minpm∘ζ1,pm∘ζ2.Theorem 1.
(see [27]). For two mFNsζ^1=p1∘ζ1,…,pm∘ζ1u and ζ^2=p1∘ζ2,…,pm∘ζ2 with β,β1,β2>0, we have(1)
ζ^1⊞ζ^2=ζ^2⊞ζ^1.(2)
ζ^1⊠ζ^2=ζ^2⊠ζ^1.(3)
βζ^1⊞ζ^2=βζ^1⊞βζ^2.(4)
ζ^1⊠ζ^2β=ζ^1β⊞ζ^2β.(5)
β1ζ^1⊞β2ζ^1=β1+β2ζ^1.(6)
ζ^1β1⊠ζ^2β2=ζ^1β1+β2.(7)
ζ^1β1β2=ζ^1β1β2.Dombi [47] proposed operations, namely, Dombi sum ⊕ and Dombi product ⊗, which are, respectively, t-conorm and t-norm given by(3)D∗a,b=a⊕b=1−11+a/1−ak+b/1−bk1/k,Da,b=a⊗b=11+1−a/ak+1−b/bk1/k,where k≥1 and a,b∈0,1.


## 3.mF Dombi AOs

In this section, we first give Dombi operations formFNs via Dombi t-conorm and Dombi t-norm and then we present mF Dombi arithmetic and geometric AOs. Let ζ1^=p1∘ζ1,…,pm∘ζ1,ζ2^=p1∘ζ2,…,pm∘ζ2 and ζ^=p1∘ζ,…,pm∘ζ be mFNs. We give some fundamental Dombi operations of mFNs as follows:(4)ζ^1⊕ζ^2=1−11+p1∘ζ1/1−p1∘ζ1k+p1∘ζ2/1−p1∘ζ2k1/k,…,1−11+pm∘ζ1/1−pm∘ζ1k+pm∘ζ2/1−pm∘ζ2k1/k,ζ^1⊗ζ^2=11+1−p1∘ζ1/p1∘ζ1k+1−p1∘ζ2/p1∘ζ2k1/k,…,11+1−pm∘ζ1/pm∘ζ1k+1−pm∘ζ2/pm∘ζ2k1/k,βζ^=1−11+βp1∘ζ/1−p1∘ζk1/k,…,1−11+βpm∘ζ/1−pm∘ζk1/k,ζ^β=1−11+β1−p1∘ζ/p1∘ζk1/k,…,1−11+β1−pm∘ζ/pm∘ζk1/k,where k>0.


### 3.1.mF Dombi Arithmetic AOs

We presentmF Dombi arithmetic AOs as follows.Definition 5.
For a collection ofmFNsζ^j=p1∘ζj,…,pm∘ζj where j=1,2,…,n, a mapping from ζ^n to ζ^ is called an mFDWA operator, which is given by(5)mFDWAΘζ1^,ζ2^,…,ζn^=⊕j=1nΘjζ^j,where Θ=Θ1,Θ2,…,ΘnT denotes the weights of ζ^j, ∀j=1,…,n and Θj>0 with ∑j=1nΘj=1.
We give the following theorem, which is used to apply the Dombi operations onmFNs.Theorem 2.
For a collection ofmFNsζ^j=p1∘ζj,…,pm∘ζj where j=1,2,…,n, an accumulated value of these mFNs using the mFDWA operators is defined as(6)mFDWAΘζ1^,ζ2^,…,ζn^=⊕j=1nΘjζ^j,=1−11+∑j=1nΘjp1∘ζj/1−p1∘ζjk1/k,…,1−11+∑j=1nΘjpm∘ζj/1−pm∘ζjk1/k.Proof.
We utilize the induction approach to show it.
Case 1. For n=1, by equation (6), we obtain(7)mFDWAΘζ1^,ζ2^,…,ζn^=Θ1ζ^1=ζ^1,sinceΘ1=1=1−11+p1∘ζ1/1−p1∘ζ1k1/k,…,1−11+pm∘ζ1/1−pm∘ζ1k1/k.
Hence, equation (6) satisfies when n=1.
Case 2. Now, we presume that equation (6) satisfies for n=t; here t is an arbitrary natural number; then,(8)mFDWAΘζ1^,ζ2^,…,ζt^=⊕j=1tΘjζ^j,=1−11+∑j=1tΘjp1∘ζj/1−p1∘ζjk1/k,…,1−11+∑j=1tΘjpm∘ζj/1−pm∘ζjk1/k.
Forn=t+1,(9)mFDWAΘζ1^,ζ2^,…,ζt^,ζ^t+1=⊕j=1tΘjζ^j⊕Θt+1ζ^t+1,=1−11+∑j=1tΘjp1∘ζj/1−p1∘ζjk1/k,…,1−11+∑j=1tΘjpm∘ζj/1−pm∘ζjk1/k⊕1−11+Θt+1p1∘ζt+1/1−p1∘ζt+1k1/k,…,1−11+Θt+1pm∘ζt+1/1−pm∘ζt+1k1/k=1−11+∑j=1t+1Θjp1∘ζj/1−p1∘ζjk1/k,…,1−11+∑j=1t+1Θjpm∘ζj/1−pm∘ζjk1/k.
Therefore, equation (6) satisfies for n=t+1. Hence, we deduce that equation (6) satisfies for every natural number n.Example 1.
Letζ^1=0.4,0.3,0.8,ζ^2=0.3,0.5,0.1,ζ^3=0.7,0.2,0.4, and ζ^4=0.5,0.4,0.6 be 3FNs and Θ=0.2,0.3,0.1,0.4T be weights related to these 3FNs. Then, for k=3,(10)mFDWAΘζ1^,ζ2^,ζ3^=⊕j=13Θjζ^j=1−11+∑j=1nΘjp1∘ζj/1−p1∘ζjk1/k,…,1−11+∑j=1nΘjpm∘ζj/1−pm∘ζjk1/k=1−11+0.2×0.4/1−0.43+0.3×0.3/1−0.33+0.1×0.7/1−0.73+0.4×0.5/1−0.531/3,1−11+0.2×0.3/1−0.33+0.3×0.5/1−0.53+0.1×0.2/1−0.23+0.4×0.4/1−0.431/3,1−11+0.2×0.8/1−0.83+0.3×0.1/1−0.13+0.1×0.4/1−0.43+0.4×0.6/1−0.631/3,=0.5467,0.4312,0.7076.
We now explore some useful laws ofmFDWA operators as follows.Theorem 3.
(idempotent law). Letζ^j=p1∘ζj,…,pm∘ζj be a family of “n” mFNs, which are equal, i.e., ζ^j=ζ^; then,(11)mFDWAΘζ1^,ζ2^,…,ζn^=ζ^.Proof.
Sinceζ^j=p1∘ζj,…,pm∘ζj=ζ^, where j=1,…,n, then by equation (6),(12)mFDWAΘζ1^,ζ2^,…,ζn^=⊕j=1nΘjζ^j,=1−11+∑j=1nΘjp1∘ζj/1−p1°ζjk1/k,…,1−11+∑j=1nΘjpm∘ζj/1−pm°ζjk1/k,=1−11+p1∘ζ/1−p1∘ζk1/k,…,1−11+pm∘ζ/1−pm∘ζk1/k,=p1∘ζ,p2∘ζ,…,pm∘ζ,fork=1=ζ^.
Hence,mFDWAΘζ1^,ζ2^,…,ζn^=ζ^ holds if ζ^j=ζ^, for all “j” varies from 1 to n.Theorem 4.
(bounded law). Letζ^j=p1∘ζj,…,pm∘ζj be a collection of “n” mFNs, ζ^−=∩j=1nζj, and ζ^+=∪j=1nζj; then,(13)ζ^−≤mFDWAΘζ1^,ζ2^,…,ζn^≤ζ^+.Theorem 5.
(monotonic law). For two collections ofmFNsζ^j and ζj′^,j=1,2,…,n, if ζ^j≤ζj′^,(14)mFDWAΘζ1^,ζ2^,…,ζn^≤mFDWAΘζ1′^,ζ2′^,…,ζn′^.
Now, we presentmFDOWA operator.Definition 6.
For a collection ofmFNsζ^j=p1∘ζj,…,pm∘ζj, j=1,2,…,n, an mFDOWA operator is a function mFDOWA: ζ^n⟶ζ^, which is given by(15)mFDOWAwζ1^,ζ2^,…,ζn^=⊕j=1nwjζ^σj,where w=w1,w2,…,wnT denotes the weights and wj∈0,1 with ∑j=1nwj=1. σj,j=1,2,…,n represents the permutation, for which ζ^σj−1≥ζ^σj.Theorem 6.
For a collection ofmFNsζ^j=p1∘ζj,…,pm∘ζj where j=1,2,…,n, an accumulated value of these mFNs using the mFDOWA operators is defined as(16)mFDOWAwζ1^,ζ2^,…,ζn^=⊕j=1nwjζ^σj=1−11+∑j=1nwjp1∘ζσj/1−p1∘ζσjk1/k,…,1−11+∑j=1nwjpm∘ζσj/1−pm∘ζσjk1/k.Example 2.
Letζ^1=0.4,0.5,0.3,0.8,ζ^2=0.3,0.4,0.1,0.7, and ζ^3=0.8,0.7,0.6,0.4 be 4FNs with weights w=0.3,0.1,0.6T. Then, for k=3, we compute the score values as(17)Sζ^1=0.4+0.5+0.3+0.84=0.5,Sζ^2=0.3+0.4+0.1+0.74=0.375,Sζ^3=0.8+0.7+0.6+0.44=0.625.
SinceSζ^3>Sζ^1>Sζ^2,(18)ζ^σ1=ζ^3=0.8,0.7,0.6,0.4,ζ^σ2=ζ^1=0.4,0.5,0.3,0.8,ζ^σ3=ζ^2=0.3,0.4,0.1,0.7.
Then, from Definition6,(19)mFDOWAwζ1^,ζ2^,ζ3^=⊕j=13wjζ^σj,=1−11+∑j=1nwjp1∘ζσj/1−p1∘ζσjk1/k,…,1−11+∑j=1nwjpm∘ζσj/1−pm∘ζσjk1/k,=1−11+0.3×0.8/1−0.83+0.1×0.4/1−0.43+0.6×0.3/1−0.331/3,1−11+0.3×0.7/1−0.73+0.1×0.5/1−0.53+0.6×0.4/1−0.431/3,1−11+0.3×0.6/1−0.63+0.1×0.3/1−0.33+0.6×0.1/1−0.131/3,1−11+0.3×0.4/1−0.43+0.1×0.8/1−0.83+0.6×0.7/1−0.731/3,=0.7284,0.6152,0.5017,0.7073.Remark 1.
Note thatmFDOWA operators satisfy properties, namely, idempotency, boundedness, and monotonicity, as described in Theorems 3, 4, and 5.Theorem 7.
(commutative law). For any two collections ofmFNsζ^j and ζj′^j=1,2,…,n, we get(20)mFDOWAwζ1^,ζ2^,…,ζn^=mFDOWζwζ1′^,ζ2′^,…,ζn′^,where ζj′^ is any permutation of ζj^.
We see thatmFDWA and mFDOWA operators aggregate weighted mFNs and their ordering, respectively. Now, we develop a novel operator called mFDHA operator, which obtains the properties of both mFDWA and mFDOWA operators.Definition 7.
For a family ofmFNsζ^j=p1∘ζj,p2∘ζj,…,pm∘ζj, j=1,2,…,n, an mFDHA operator is defined as(21)mFDHAw,Θζ1^,ζ2^,…,ζn^=⊕j=1nwjζ^˜σj,where w=w1,w2,…,wnT denotes the weights corresponding to the mFNsζj^ with the conditions wj∈0,1,∑j=1nwj=1, ζ^˜σj is the jth biggest mFN, ζ^˜σj=nΘjζj^,j=1,2,…,n, and Θ=Θ1,Θ2,…,Θn is a vector having weights, with Θj∈0,1,∑j=1nΘj=1.
Notice that whenw=1/n,1/n,…,1/n, mFDHA operator converts into mFDWA operator. If Θ=1/n,1/n,…,1/n, then mFDHA operator becomes mFDOWA operator. Thus, mFDHA operator is a generalization for both operators, mFDWA and mFDOWA, which describes the degrees and ordering of mFNs.
The following theorem can be readily showed by same steps as in Theorem2.Theorem 8.
For a collection ofmFNsζ^j=p1∘ζj,…,pm∘ζj where j=1,2,…,n, an accumulated score of these mFNs using the mFDHA operators is defined as(22)mFDHAw,Θζ1^,ζ2^,…,ζn^=⊕j=1nwjζ^˜σj,=1−11+∑j=1nwjp1∘ζσj/1−p1∘ζσjk1/k,…,1−11+∑j=1nwjpm∘ζσj/1−pm∘ζσjk1/k.Example 3.
Letζ^1=0.7,0.3,0.5,ζ^2=0.2,0.5,0.7, ζ^3=0.8,0.2,0.1, and ζ^4=0.6,0.7,0.9 be 3FNs with w=0.2,0.3,0.1,0.4T, a weight vector corresponding to given 3FNs, and a vector Θ=0.3,0.1,0.4,0.2T having weights. Then, by Definition 7, for k=3,(23)ζ^˜1=1−11+nΘ1p1∘ζ1/1−p1°ζ1k1/k,…,1−11+nΘ1p3∘ζ1/1−p3°ζ1k1/k,=1−11+4×0.3×0.7/1−0.731/3,1−11+4×0.3×0.3/1−0.331/3,1−11+4×0.3×0.5/1−0.531/3,=0.7126,0.3129,0.5152.
Similarly,(24)ζ^˜2=1−11+4×0.1×0.2/1−0.231/3,1−11+4×0.1×0.5/1−0.531/3,1−11+4×0.1×0.7/1−0.731/3,=0.1555,0.4242,0.6322,ζ^˜3=1−11+4×0.4×0.8/1−0.831/3,1−11+4×0.4×0.2/1−0.231/3,1−11+4×0.4×01/1−0.131/3,=0.8239,0.2262,0.1150,ζ^˜4=1−11+4×0.2×0.6/1−0.631/3,1−11+4×0.2×0.7/1−0.731/3,1−11+4×0.2×0.9/1−0.931/3,=0.5820,0.6842,0.8931.
Then, scores ofmFNs for k=3 are calculated as(25)Sζ^˜1=0.7126+0.3129+0.51523=0.5136,Sζ^˜2=0.1555+0.4242+0.63223=0.4040,Sζ^˜3=0.8239+0.2262+0.11503=0.3884,Sζ^˜4=0.5820+0.6842+0.89313=0.7198.
SinceSζ^˜4>Sζ^˜1>Sζ^˜2>Sζ^˜3,(26)ζ^˜σ1=ζ^4=0.5820,0.6842,0.8931,ζ^˜σ2=ζ^1=0.7126,0.3129,0.5152,ζ^˜σ3=ζ^2=0.1555,0.4242,0.6322,ζ^˜σ4=ζ^3=0.8239,0.2262,0.1150.
Then, from Theorem8,(27)mFDHAw,Θζ1^,ζ2^,ζ3^,ζ4^=⊕j=14wjζ^σj=1−11+∑j=14wjp1∘ζσj/1−p1∘ζσjk1/k,…,1−11+∑j=14wjp3∘ζσj/1−p3∘ζσjk1/k,=1−11+0.2×0.5820/1−0.58203+0.3×0.7126/1−0.71263+0.1×0.1555/1−0.15553+0.4×0.8239/1−0.823931/3,1−11+0.2×0.6842/1−0.68423+0.3×0.3129/1−0.31293+0.1×0.4242/1−0.42423+0.4×0.2262/1−0.226231/3,1−11+0.2×0.8931/1−0.89313+0.3×0.5152/1−0.51523+0.1×0.6322/1−0.63223+0.4×0.1150/1−0.115031/3,=0.7819,0.5620,0.8304.


### 3.2.mF Dombi Geometric AOs

We now propose different types of Dombi geometric AOs withmFNs, namely, mFDWG operator, mFDOWG operator, and mFDHG operator.Definition 8.
For a family ofmFNsζ^j=p1∘ζj,p2∘ζj,…,pm∘ζj, j=1,2,…,n, a mapping mFDWG: ζ^n⟶ζ^ is called mFDWG operator, which is given by(28)mFDWGΘζ1^,ζ2^,…,ζn^=⊗j=1nζ^jΘj,where Θ=Θ1,Θ2,…,ΘnT represents the weights, with ∑j=1nΘj=1,Θj∈0,1.Theorem 9.
For a collection ofmFNsζ^j=p1∘ζj,…,pm∘ζj where j=1,2,…,n, an accumulated score of these mFNs using the mFDWG operators is defined by(29)mFDWGΘζ1^,ζ2^,…,ζn^=⊗j=1nζ^jΘj,1−11+∑j=1nΘj1−p1∘ζj/p1∘ζjk1/k,…,1−11+∑ΘjnΘj1−pm∘ζj/pm∘ζjk1/k.Proof.
Its proof is identical to Theorem2.Example 4.
Letζ^1=0.2,0.6,0.3,ζ^2=0.9,1.0,0.7,ζ^3=0.1,0.8,0.4, and ζ^4=0.4,0.7,0.3 be 3FNs with weights Θ=0.1,0.5,0.3,0.1T. Then, for k=3,(30)mFDWGΘζ1^,ζ2^,ζ3^=⊗j=13ζ^jΘj,=1−11+∑j=1nΘj1−p1∘ζj/p1∘ζjk1/k,…,1−11+∑j=1nΘj1−pm∘ζj/pm∘ζjk1/k,=1−11+0.1×1−0.2/0.23+0.5×1−0.9/0.93+0.3×1−0.1/0.13+0.1×1−0.4/0.431/3,1−11+0.1×1−0.6/0.63+0.5×1−0.1/0.13+0.3×1−0.8/0.83+0.1×1−0.7/0.731/3,1−11+0.1×1−0.3/0.33+0.5×1−0.7/0.73+0.3×1−0.4/0.43+0.1×1−0.3/0.331/3,=0.8589,0.2582,0.6052.
It can be readily shown that themFDWG operator holds the notions given below.Theorem 10.
(idempotent law). Letζ^j=p1∘ζj,…,pm∘ζj be a family of “n” mFNs, which are equal, i.e., ζ^j=ζ^; then,(31)mFDWGΘζ1^,ζ2^,…,ζn^=ζ^.Theorem 11.
(bounded law). Letζ^j=p1∘ζj,…,pm∘ζj be a collection of “n” mFNs, ζ^−=∩j=1nζj, and ζ^+=∪j=1nζj; then,(32)ζ^−≤mFDWGΘζ1^,ζ2^,…,ζn^≤ζ^+.Theorem 12.
(monotonic law). For two collections ofmFNsζ^j and ζj′^,j=1,2,…,n, if ζ^j≤ζj′^, then(33)mFDWGΘζ1^,ζ2^,…,ζn^≤mFDWGΘζ1′^,ζ2′^,…,ζn′^.
Now, we developmFDOWG operators.Definition 9.
For a family ofmFNsζ^j=p1∘ζj,p2∘ζj,…,pm∘ζj,j=1,2,…,n, an mFDOWG operator is a mapping mFDOWG: ζ^n⟶ζ^, which is given as(34)mFDOWGwζ1^,ζ2^,…,ζn^=⊗j=1nwjζ^σj,where w=w1,w2,…,wnT is the weight vector and wj∈0,1 with ∑j=1nwj=1. σj,j=1,2,…,n represents the permutation, such that ζ^σj−1≥ζ^σj.Theorem 13.
For a collection ofmFNsζ^j=p1∘ζj,…,pm∘ζj where j=1,2,…,n, an accumulated score of these mFNs using an mFDOWG operator is defined by(35)mFDOWGwζ1^,ζ2^,…,ζn^=⊗j=1nζ^σjwj,=1−11+∑j=1nwj1−p1∘ζσj/p1∘ζσjk1/k,…,1−11+∑j=1nwj1−pm∘ζσj/pm∘ζσjk1/k.Example 5.
Letζ^1=0.2,0.4,0.7,ζ^2=0.3,0.6,0.1,ζ^3=0.8,0.3,0.5, and ζ^4=0.6,0.4,0.7 be 3FNs and w=0.3,0.1,0.2,0.4T be a weight vector. Then, score values of mFNs for k=3 are calculated as(36)Sζ^1=0.2+0.4+0.73=0.4333,Sζ^2=0.3+0.6+0.13=0.3333,Sζ^3=0.8+0.3+0.53=0.5333,Sζ^4=0.6+0.4+0.73=0.5667.
SinceSζ^4>Sζ^3>Sζ^1>Sζ^2,(37)ζ^σ1=ζ^3=0.6,0.4,0.7,ζ^σ2=ζ^3=0.8,0.3,0.5,ζ^σ3=ζ^1=0.2,0.4,0.7,ζ^σ4=ζ^2=0.3,0.6,0.1.
Then, from Definition9,(38)mFDOWGwζ1^,ζ2^,ζ3^,ζ4^=⊗j=14ζ^σjwj,=1−11+∑j=14wj1−p1∘ζσj/p1∘ζσjk1/k,…,1−11+∑j=14wj1−p3∘ζσj/p3∘ζσjk1/k,=1−11+0.3×1−0.6/0.63+0.1×1−0.8/0.83+0.2×1−0.2/0.23+0.4×1−0.3/0.331/3,1−11+0.3×1−0.4/0.43+0.1×1−0.3/0.33+0.2×1−0.4/0.43+0.4×1−0.6/0.631/3,1−11+0.3×1−0.7/0.73+0.1×1−0.5/0.53+0.2×1−0.7/0.73+0.4×1−0.1/0.131/3,=0.7237,0.5926,0.8690.Remark 2.
Note thatmFDOWG operators satisfy properties, namely, idempotency, boundedness, and monotonicity, as described in Theorems 10, 11, and 12.Theorem 14.
(commutative law). For two arbitrary collections ofmFNsζ^j and ζj′^j=1,2,…,n, if ζ^j≤ζj′^, then(39)mFDOWGwζ1^,ζ2^,…,ζn^=mFDOWGwζ1′^,ζ2′^,…,ζn′^,where ζj′^ is any permutation of ζj^.In Definitions5 and 6, we see that mFDWG and mFDOWG operators aggregate weighted mFNs and their ordering, respectively. Now, we develop a new operator called mFDHG operator, which contains the properties of both mFDWG and mFDOWG operators.Definition 10.
For a family ofmFNsζ^j=p1∘ζj,p2∘ζj,…,pm∘ζj, j=1,2,…,n, an mFDHG operator is defined by(40)mFDHGw,Θζ1^,ζ2^,…,ζn^=⊗j=1nζ^˜σjwj,where w=w1,w2,…,wnT denotes the weights associated to the mFNsζj^, j=1,2,…,n, wj∈0,1,∑j=1nwj=1, ζ^˜σj is the j-th largest mFN, ζ^˜σj=nΘjζj^,j=1,2,…,n, and Θ=Θ1,Θ2,…,Θn is a vector having weights, with Θj∈0,1,∑j=1nΘj=1.
Notice that whenw=1/n,1/n,…,1/nT, mFDHG operator becomes mFDWG operator. When Θ=1/n,1/n,…,1/nT, then mFDHG operator converts into mFDOWG operator. Thus, mFDHG operator is a generalization of mFDWG and mFDOWG operators.
With the induction technique, one can readily show the next theorem.Theorem 15.
For a collection ofmFNsζ^j=p1∘ζj,…,pm∘ζj where j=1,2,…,n, an accumulated score of these mFNs using an mFDHG operator is defined as(41)mFDHGw,Θζ1^,ζ2^,…,ζn^=⊗j=1nζ^˜σjwj,=1−11+∑j=1nwj1−p1∘ζσj/p1∘ζσjk1/k,…,1−11+∑j=1nwj1−pm∘ζσj/pm∘ζσjk1/k.Example 6.
Letζ^1=0.4,0.6,0.3,ζ^2=0.3,0.2,0.9, ζ^3=0.6,0.3,0.5, and ζ^4=0.3,0.5,0.7 be 3FNs and w=0.4,0.1,0.3,0.2T be an associated weight vector and a vector Θ=0.5,0.2,0.1,0.2T be weights. By Definition 10, for k=3,(42)ζ^˜1=1−11+nΘ11−p1∘ζ1/p1∘ζ1k1/k,…,1−11+nΘ11−p3∘ζ1/p3∘ζ1k1/k,=1−11+4×0.5×1−0.4/0.431/3,1−11+4×0.5×1−0.6/0.631/3,1−11+4×0.5×1−0.3/0.331/3,=0.6540,0.4565,0.7462.
Similarly,(43)ζ^˜2=1−11+4×0.2×1−0.3/0.331/3,1−11+4×0.2×1−0.2/0.231/3,1−11+4×0.2×1−0.9/0.931/3,=0.6842,0.7878,0.0935,ζ^˜3=1−11+4×0.1×1−0.6/0.631/3,1−11+4×0.1×1−0.3/0.331/3,1−11+4×0.1×1−0.5/0.531/3,=0.3294,0.6322,0.4242,ζ^˜4=1−11+4×0.2×1−0.3/0.331/3,1−11+4×0.2×1−0.5/0.531/3,1−11+4×0.2×1−0.7/0.731/3,=0.6842,0.4814,0.2846.
Then, score values ofmFNs for k=3 are given as follows:(44)Sζ^˜1=0.6540+0.4565+0.74623=0.6189,Sζ^˜2=0.6842+0.7878+0.09353=0.5218,Sζ^˜3=0.3294+0.6322+0.42423=0.4620,Sζ^˜4=0.6842+0.4814+0.28463=0.4834.
SinceSζ^˜1>Sζ^˜2>Sζ^˜4>Sζ^˜3,(45)ζ^˜σ1=ζ^1=0.6540,0.4565,0.7462,ζ^˜σ2=ζ^2=0.6842,0.7878,0.0935,ζ^˜σ3=ζ^4=0.6842,0.4814,0.2846,ζ^˜σ4=ζ^3=0.3294,0.6322,0.4242.
Then, from Definition9,(46)mFDHζw,Θζ1^,ζ2^,ζ3^,ζ4^=⊗j=14ζ^σjwj,=1−11+∑j=14wj1−p1∘ζσj/p1∘ζσjk1/k,…,1−11+∑j=14wj1−p3∘ζσj/p3∘ζσjk1/k,=1−11+0.4×1−0.6540/0.65403+0.1×1−0.6842/0.68423+0.3×1−0.6842/0.68423+0.2×1−0.3294/0.329431/3,1−11+0.4×1−0.4565/0.45653+0.1×1−0.7878/0.78783+0.3×1−0.4814/0.48143+0.2×1−0.6322/0.632231/3,1−11+0.4×1−0.7462/0.74623+0.1×1−0.0935/0.09353+0.3×1−0.2846/0.28463+0.2×1−0.4242/0.424231/3,=0.5482,0.5073,0.8210.


## 3.1.mF Dombi Arithmetic AOs

We presentmF Dombi arithmetic AOs as follows.Definition 5.
For a collection ofmFNsζ^j=p1∘ζj,…,pm∘ζj where j=1,2,…,n, a mapping from ζ^n to ζ^ is called an mFDWA operator, which is given by(5)mFDWAΘζ1^,ζ2^,…,ζn^=⊕j=1nΘjζ^j,where Θ=Θ1,Θ2,…,ΘnT denotes the weights of ζ^j, ∀j=1,…,n and Θj>0 with ∑j=1nΘj=1.
We give the following theorem, which is used to apply the Dombi operations onmFNs.Theorem 2.
For a collection ofmFNsζ^j=p1∘ζj,…,pm∘ζj where j=1,2,…,n, an accumulated value of these mFNs using the mFDWA operators is defined as(6)mFDWAΘζ1^,ζ2^,…,ζn^=⊕j=1nΘjζ^j,=1−11+∑j=1nΘjp1∘ζj/1−p1∘ζjk1/k,…,1−11+∑j=1nΘjpm∘ζj/1−pm∘ζjk1/k.Proof.
We utilize the induction approach to show it.
Case 1. For n=1, by equation (6), we obtain(7)mFDWAΘζ1^,ζ2^,…,ζn^=Θ1ζ^1=ζ^1,sinceΘ1=1=1−11+p1∘ζ1/1−p1∘ζ1k1/k,…,1−11+pm∘ζ1/1−pm∘ζ1k1/k.
Hence, equation (6) satisfies when n=1.
Case 2. Now, we presume that equation (6) satisfies for n=t; here t is an arbitrary natural number; then,(8)mFDWAΘζ1^,ζ2^,…,ζt^=⊕j=1tΘjζ^j,=1−11+∑j=1tΘjp1∘ζj/1−p1∘ζjk1/k,…,1−11+∑j=1tΘjpm∘ζj/1−pm∘ζjk1/k.
Forn=t+1,(9)mFDWAΘζ1^,ζ2^,…,ζt^,ζ^t+1=⊕j=1tΘjζ^j⊕Θt+1ζ^t+1,=1−11+∑j=1tΘjp1∘ζj/1−p1∘ζjk1/k,…,1−11+∑j=1tΘjpm∘ζj/1−pm∘ζjk1/k⊕1−11+Θt+1p1∘ζt+1/1−p1∘ζt+1k1/k,…,1−11+Θt+1pm∘ζt+1/1−pm∘ζt+1k1/k=1−11+∑j=1t+1Θjp1∘ζj/1−p1∘ζjk1/k,…,1−11+∑j=1t+1Θjpm∘ζj/1−pm∘ζjk1/k.
Therefore, equation (6) satisfies for n=t+1. Hence, we deduce that equation (6) satisfies for every natural number n.Example 1.
Letζ^1=0.4,0.3,0.8,ζ^2=0.3,0.5,0.1,ζ^3=0.7,0.2,0.4, and ζ^4=0.5,0.4,0.6 be 3FNs and Θ=0.2,0.3,0.1,0.4T be weights related to these 3FNs. Then, for k=3,(10)mFDWAΘζ1^,ζ2^,ζ3^=⊕j=13Θjζ^j=1−11+∑j=1nΘjp1∘ζj/1−p1∘ζjk1/k,…,1−11+∑j=1nΘjpm∘ζj/1−pm∘ζjk1/k=1−11+0.2×0.4/1−0.43+0.3×0.3/1−0.33+0.1×0.7/1−0.73+0.4×0.5/1−0.531/3,1−11+0.2×0.3/1−0.33+0.3×0.5/1−0.53+0.1×0.2/1−0.23+0.4×0.4/1−0.431/3,1−11+0.2×0.8/1−0.83+0.3×0.1/1−0.13+0.1×0.4/1−0.43+0.4×0.6/1−0.631/3,=0.5467,0.4312,0.7076.
We now explore some useful laws ofmFDWA operators as follows.Theorem 3.
(idempotent law). Letζ^j=p1∘ζj,…,pm∘ζj be a family of “n” mFNs, which are equal, i.e., ζ^j=ζ^; then,(11)mFDWAΘζ1^,ζ2^,…,ζn^=ζ^.Proof.
Sinceζ^j=p1∘ζj,…,pm∘ζj=ζ^, where j=1,…,n, then by equation (6),(12)mFDWAΘζ1^,ζ2^,…,ζn^=⊕j=1nΘjζ^j,=1−11+∑j=1nΘjp1∘ζj/1−p1°ζjk1/k,…,1−11+∑j=1nΘjpm∘ζj/1−pm°ζjk1/k,=1−11+p1∘ζ/1−p1∘ζk1/k,…,1−11+pm∘ζ/1−pm∘ζk1/k,=p1∘ζ,p2∘ζ,…,pm∘ζ,fork=1=ζ^.
Hence,mFDWAΘζ1^,ζ2^,…,ζn^=ζ^ holds if ζ^j=ζ^, for all “j” varies from 1 to n.Theorem 4.
(bounded law). Letζ^j=p1∘ζj,…,pm∘ζj be a collection of “n” mFNs, ζ^−=∩j=1nζj, and ζ^+=∪j=1nζj; then,(13)ζ^−≤mFDWAΘζ1^,ζ2^,…,ζn^≤ζ^+.Theorem 5.
(monotonic law). For two collections ofmFNsζ^j and ζj′^,j=1,2,…,n, if ζ^j≤ζj′^,(14)mFDWAΘζ1^,ζ2^,…,ζn^≤mFDWAΘζ1′^,ζ2′^,…,ζn′^.
Now, we presentmFDOWA operator.Definition 6.
For a collection ofmFNsζ^j=p1∘ζj,…,pm∘ζj, j=1,2,…,n, an mFDOWA operator is a function mFDOWA: ζ^n⟶ζ^, which is given by(15)mFDOWAwζ1^,ζ2^,…,ζn^=⊕j=1nwjζ^σj,where w=w1,w2,…,wnT denotes the weights and wj∈0,1 with ∑j=1nwj=1. σj,j=1,2,…,n represents the permutation, for which ζ^σj−1≥ζ^σj.Theorem 6.
For a collection ofmFNsζ^j=p1∘ζj,…,pm∘ζj where j=1,2,…,n, an accumulated value of these mFNs using the mFDOWA operators is defined as(16)mFDOWAwζ1^,ζ2^,…,ζn^=⊕j=1nwjζ^σj=1−11+∑j=1nwjp1∘ζσj/1−p1∘ζσjk1/k,…,1−11+∑j=1nwjpm∘ζσj/1−pm∘ζσjk1/k.Example 2.
Letζ^1=0.4,0.5,0.3,0.8,ζ^2=0.3,0.4,0.1,0.7, and ζ^3=0.8,0.7,0.6,0.4 be 4FNs with weights w=0.3,0.1,0.6T. Then, for k=3, we compute the score values as(17)Sζ^1=0.4+0.5+0.3+0.84=0.5,Sζ^2=0.3+0.4+0.1+0.74=0.375,Sζ^3=0.8+0.7+0.6+0.44=0.625.
SinceSζ^3>Sζ^1>Sζ^2,(18)ζ^σ1=ζ^3=0.8,0.7,0.6,0.4,ζ^σ2=ζ^1=0.4,0.5,0.3,0.8,ζ^σ3=ζ^2=0.3,0.4,0.1,0.7.
Then, from Definition6,(19)mFDOWAwζ1^,ζ2^,ζ3^=⊕j=13wjζ^σj,=1−11+∑j=1nwjp1∘ζσj/1−p1∘ζσjk1/k,…,1−11+∑j=1nwjpm∘ζσj/1−pm∘ζσjk1/k,=1−11+0.3×0.8/1−0.83+0.1×0.4/1−0.43+0.6×0.3/1−0.331/3,1−11+0.3×0.7/1−0.73+0.1×0.5/1−0.53+0.6×0.4/1−0.431/3,1−11+0.3×0.6/1−0.63+0.1×0.3/1−0.33+0.6×0.1/1−0.131/3,1−11+0.3×0.4/1−0.43+0.1×0.8/1−0.83+0.6×0.7/1−0.731/3,=0.7284,0.6152,0.5017,0.7073.Remark 1.
Note thatmFDOWA operators satisfy properties, namely, idempotency, boundedness, and monotonicity, as described in Theorems 3, 4, and 5.Theorem 7.
(commutative law). For any two collections ofmFNsζ^j and ζj′^j=1,2,…,n, we get(20)mFDOWAwζ1^,ζ2^,…,ζn^=mFDOWζwζ1′^,ζ2′^,…,ζn′^,where ζj′^ is any permutation of ζj^.
We see thatmFDWA and mFDOWA operators aggregate weighted mFNs and their ordering, respectively. Now, we develop a novel operator called mFDHA operator, which obtains the properties of both mFDWA and mFDOWA operators.Definition 7.
For a family ofmFNsζ^j=p1∘ζj,p2∘ζj,…,pm∘ζj, j=1,2,…,n, an mFDHA operator is defined as(21)mFDHAw,Θζ1^,ζ2^,…,ζn^=⊕j=1nwjζ^˜σj,where w=w1,w2,…,wnT denotes the weights corresponding to the mFNsζj^ with the conditions wj∈0,1,∑j=1nwj=1, ζ^˜σj is the jth biggest mFN, ζ^˜σj=nΘjζj^,j=1,2,…,n, and Θ=Θ1,Θ2,…,Θn is a vector having weights, with Θj∈0,1,∑j=1nΘj=1.
Notice that whenw=1/n,1/n,…,1/n, mFDHA operator converts into mFDWA operator. If Θ=1/n,1/n,…,1/n, then mFDHA operator becomes mFDOWA operator. Thus, mFDHA operator is a generalization for both operators, mFDWA and mFDOWA, which describes the degrees and ordering of mFNs.
The following theorem can be readily showed by same steps as in Theorem2.Theorem 8.
For a collection ofmFNsζ^j=p1∘ζj,…,pm∘ζj where j=1,2,…,n, an accumulated score of these mFNs using the mFDHA operators is defined as(22)mFDHAw,Θζ1^,ζ2^,…,ζn^=⊕j=1nwjζ^˜σj,=1−11+∑j=1nwjp1∘ζσj/1−p1∘ζσjk1/k,…,1−11+∑j=1nwjpm∘ζσj/1−pm∘ζσjk1/k.Example 3.
Letζ^1=0.7,0.3,0.5,ζ^2=0.2,0.5,0.7, ζ^3=0.8,0.2,0.1, and ζ^4=0.6,0.7,0.9 be 3FNs with w=0.2,0.3,0.1,0.4T, a weight vector corresponding to given 3FNs, and a vector Θ=0.3,0.1,0.4,0.2T having weights. Then, by Definition 7, for k=3,(23)ζ^˜1=1−11+nΘ1p1∘ζ1/1−p1°ζ1k1/k,…,1−11+nΘ1p3∘ζ1/1−p3°ζ1k1/k,=1−11+4×0.3×0.7/1−0.731/3,1−11+4×0.3×0.3/1−0.331/3,1−11+4×0.3×0.5/1−0.531/3,=0.7126,0.3129,0.5152.
Similarly,(24)ζ^˜2=1−11+4×0.1×0.2/1−0.231/3,1−11+4×0.1×0.5/1−0.531/3,1−11+4×0.1×0.7/1−0.731/3,=0.1555,0.4242,0.6322,ζ^˜3=1−11+4×0.4×0.8/1−0.831/3,1−11+4×0.4×0.2/1−0.231/3,1−11+4×0.4×01/1−0.131/3,=0.8239,0.2262,0.1150,ζ^˜4=1−11+4×0.2×0.6/1−0.631/3,1−11+4×0.2×0.7/1−0.731/3,1−11+4×0.2×0.9/1−0.931/3,=0.5820,0.6842,0.8931.
Then, scores ofmFNs for k=3 are calculated as(25)Sζ^˜1=0.7126+0.3129+0.51523=0.5136,Sζ^˜2=0.1555+0.4242+0.63223=0.4040,Sζ^˜3=0.8239+0.2262+0.11503=0.3884,Sζ^˜4=0.5820+0.6842+0.89313=0.7198.
SinceSζ^˜4>Sζ^˜1>Sζ^˜2>Sζ^˜3,(26)ζ^˜σ1=ζ^4=0.5820,0.6842,0.8931,ζ^˜σ2=ζ^1=0.7126,0.3129,0.5152,ζ^˜σ3=ζ^2=0.1555,0.4242,0.6322,ζ^˜σ4=ζ^3=0.8239,0.2262,0.1150.
Then, from Theorem8,(27)mFDHAw,Θζ1^,ζ2^,ζ3^,ζ4^=⊕j=14wjζ^σj=1−11+∑j=14wjp1∘ζσj/1−p1∘ζσjk1/k,…,1−11+∑j=14wjp3∘ζσj/1−p3∘ζσjk1/k,=1−11+0.2×0.5820/1−0.58203+0.3×0.7126/1−0.71263+0.1×0.1555/1−0.15553+0.4×0.8239/1−0.823931/3,1−11+0.2×0.6842/1−0.68423+0.3×0.3129/1−0.31293+0.1×0.4242/1−0.42423+0.4×0.2262/1−0.226231/3,1−11+0.2×0.8931/1−0.89313+0.3×0.5152/1−0.51523+0.1×0.6322/1−0.63223+0.4×0.1150/1−0.115031/3,=0.7819,0.5620,0.8304.


## 3.2.mF Dombi Geometric AOs

We now propose different types of Dombi geometric AOs withmFNs, namely, mFDWG operator, mFDOWG operator, and mFDHG operator.Definition 8.
For a family ofmFNsζ^j=p1∘ζj,p2∘ζj,…,pm∘ζj, j=1,2,…,n, a mapping mFDWG: ζ^n⟶ζ^ is called mFDWG operator, which is given by(28)mFDWGΘζ1^,ζ2^,…,ζn^=⊗j=1nζ^jΘj,where Θ=Θ1,Θ2,…,ΘnT represents the weights, with ∑j=1nΘj=1,Θj∈0,1.Theorem 9.
For a collection ofmFNsζ^j=p1∘ζj,…,pm∘ζj where j=1,2,…,n, an accumulated score of these mFNs using the mFDWG operators is defined by(29)mFDWGΘζ1^,ζ2^,…,ζn^=⊗j=1nζ^jΘj,1−11+∑j=1nΘj1−p1∘ζj/p1∘ζjk1/k,…,1−11+∑ΘjnΘj1−pm∘ζj/pm∘ζjk1/k.Proof.
Its proof is identical to Theorem2.Example 4.
Letζ^1=0.2,0.6,0.3,ζ^2=0.9,1.0,0.7,ζ^3=0.1,0.8,0.4, and ζ^4=0.4,0.7,0.3 be 3FNs with weights Θ=0.1,0.5,0.3,0.1T. Then, for k=3,(30)mFDWGΘζ1^,ζ2^,ζ3^=⊗j=13ζ^jΘj,=1−11+∑j=1nΘj1−p1∘ζj/p1∘ζjk1/k,…,1−11+∑j=1nΘj1−pm∘ζj/pm∘ζjk1/k,=1−11+0.1×1−0.2/0.23+0.5×1−0.9/0.93+0.3×1−0.1/0.13+0.1×1−0.4/0.431/3,1−11+0.1×1−0.6/0.63+0.5×1−0.1/0.13+0.3×1−0.8/0.83+0.1×1−0.7/0.731/3,1−11+0.1×1−0.3/0.33+0.5×1−0.7/0.73+0.3×1−0.4/0.43+0.1×1−0.3/0.331/3,=0.8589,0.2582,0.6052.
It can be readily shown that themFDWG operator holds the notions given below.Theorem 10.
(idempotent law). Letζ^j=p1∘ζj,…,pm∘ζj be a family of “n” mFNs, which are equal, i.e., ζ^j=ζ^; then,(31)mFDWGΘζ1^,ζ2^,…,ζn^=ζ^.Theorem 11.
(bounded law). Letζ^j=p1∘ζj,…,pm∘ζj be a collection of “n” mFNs, ζ^−=∩j=1nζj, and ζ^+=∪j=1nζj; then,(32)ζ^−≤mFDWGΘζ1^,ζ2^,…,ζn^≤ζ^+.Theorem 12.
(monotonic law). For two collections ofmFNsζ^j and ζj′^,j=1,2,…,n, if ζ^j≤ζj′^, then(33)mFDWGΘζ1^,ζ2^,…,ζn^≤mFDWGΘζ1′^,ζ2′^,…,ζn′^.
Now, we developmFDOWG operators.Definition 9.
For a family ofmFNsζ^j=p1∘ζj,p2∘ζj,…,pm∘ζj,j=1,2,…,n, an mFDOWG operator is a mapping mFDOWG: ζ^n⟶ζ^, which is given as(34)mFDOWGwζ1^,ζ2^,…,ζn^=⊗j=1nwjζ^σj,where w=w1,w2,…,wnT is the weight vector and wj∈0,1 with ∑j=1nwj=1. σj,j=1,2,…,n represents the permutation, such that ζ^σj−1≥ζ^σj.Theorem 13.
For a collection ofmFNsζ^j=p1∘ζj,…,pm∘ζj where j=1,2,…,n, an accumulated score of these mFNs using an mFDOWG operator is defined by(35)mFDOWGwζ1^,ζ2^,…,ζn^=⊗j=1nζ^σjwj,=1−11+∑j=1nwj1−p1∘ζσj/p1∘ζσjk1/k,…,1−11+∑j=1nwj1−pm∘ζσj/pm∘ζσjk1/k.Example 5.
Letζ^1=0.2,0.4,0.7,ζ^2=0.3,0.6,0.1,ζ^3=0.8,0.3,0.5, and ζ^4=0.6,0.4,0.7 be 3FNs and w=0.3,0.1,0.2,0.4T be a weight vector. Then, score values of mFNs for k=3 are calculated as(36)Sζ^1=0.2+0.4+0.73=0.4333,Sζ^2=0.3+0.6+0.13=0.3333,Sζ^3=0.8+0.3+0.53=0.5333,Sζ^4=0.6+0.4+0.73=0.5667.
SinceSζ^4>Sζ^3>Sζ^1>Sζ^2,(37)ζ^σ1=ζ^3=0.6,0.4,0.7,ζ^σ2=ζ^3=0.8,0.3,0.5,ζ^σ3=ζ^1=0.2,0.4,0.7,ζ^σ4=ζ^2=0.3,0.6,0.1.
Then, from Definition9,(38)mFDOWGwζ1^,ζ2^,ζ3^,ζ4^=⊗j=14ζ^σjwj,=1−11+∑j=14wj1−p1∘ζσj/p1∘ζσjk1/k,…,1−11+∑j=14wj1−p3∘ζσj/p3∘ζσjk1/k,=1−11+0.3×1−0.6/0.63+0.1×1−0.8/0.83+0.2×1−0.2/0.23+0.4×1−0.3/0.331/3,1−11+0.3×1−0.4/0.43+0.1×1−0.3/0.33+0.2×1−0.4/0.43+0.4×1−0.6/0.631/3,1−11+0.3×1−0.7/0.73+0.1×1−0.5/0.53+0.2×1−0.7/0.73+0.4×1−0.1/0.131/3,=0.7237,0.5926,0.8690.Remark 2.
Note thatmFDOWG operators satisfy properties, namely, idempotency, boundedness, and monotonicity, as described in Theorems 10, 11, and 12.Theorem 14.
(commutative law). For two arbitrary collections ofmFNsζ^j and ζj′^j=1,2,…,n, if ζ^j≤ζj′^, then(39)mFDOWGwζ1^,ζ2^,…,ζn^=mFDOWGwζ1′^,ζ2′^,…,ζn′^,where ζj′^ is any permutation of ζj^.In Definitions5 and 6, we see that mFDWG and mFDOWG operators aggregate weighted mFNs and their ordering, respectively. Now, we develop a new operator called mFDHG operator, which contains the properties of both mFDWG and mFDOWG operators.Definition 10.
For a family ofmFNsζ^j=p1∘ζj,p2∘ζj,…,pm∘ζj, j=1,2,…,n, an mFDHG operator is defined by(40)mFDHGw,Θζ1^,ζ2^,…,ζn^=⊗j=1nζ^˜σjwj,where w=w1,w2,…,wnT denotes the weights associated to the mFNsζj^, j=1,2,…,n, wj∈0,1,∑j=1nwj=1, ζ^˜σj is the j-th largest mFN, ζ^˜σj=nΘjζj^,j=1,2,…,n, and Θ=Θ1,Θ2,…,Θn is a vector having weights, with Θj∈0,1,∑j=1nΘj=1.
Notice that whenw=1/n,1/n,…,1/nT, mFDHG operator becomes mFDWG operator. When Θ=1/n,1/n,…,1/nT, then mFDHG operator converts into mFDOWG operator. Thus, mFDHG operator is a generalization of mFDWG and mFDOWG operators.
With the induction technique, one can readily show the next theorem.Theorem 15.
For a collection ofmFNsζ^j=p1∘ζj,…,pm∘ζj where j=1,2,…,n, an accumulated score of these mFNs using an mFDHG operator is defined as(41)mFDHGw,Θζ1^,ζ2^,…,ζn^=⊗j=1nζ^˜σjwj,=1−11+∑j=1nwj1−p1∘ζσj/p1∘ζσjk1/k,…,1−11+∑j=1nwj1−pm∘ζσj/pm∘ζσjk1/k.Example 6.
Letζ^1=0.4,0.6,0.3,ζ^2=0.3,0.2,0.9, ζ^3=0.6,0.3,0.5, and ζ^4=0.3,0.5,0.7 be 3FNs and w=0.4,0.1,0.3,0.2T be an associated weight vector and a vector Θ=0.5,0.2,0.1,0.2T be weights. By Definition 10, for k=3,(42)ζ^˜1=1−11+nΘ11−p1∘ζ1/p1∘ζ1k1/k,…,1−11+nΘ11−p3∘ζ1/p3∘ζ1k1/k,=1−11+4×0.5×1−0.4/0.431/3,1−11+4×0.5×1−0.6/0.631/3,1−11+4×0.5×1−0.3/0.331/3,=0.6540,0.4565,0.7462.
Similarly,(43)ζ^˜2=1−11+4×0.2×1−0.3/0.331/3,1−11+4×0.2×1−0.2/0.231/3,1−11+4×0.2×1−0.9/0.931/3,=0.6842,0.7878,0.0935,ζ^˜3=1−11+4×0.1×1−0.6/0.631/3,1−11+4×0.1×1−0.3/0.331/3,1−11+4×0.1×1−0.5/0.531/3,=0.3294,0.6322,0.4242,ζ^˜4=1−11+4×0.2×1−0.3/0.331/3,1−11+4×0.2×1−0.5/0.531/3,1−11+4×0.2×1−0.7/0.731/3,=0.6842,0.4814,0.2846.
Then, score values ofmFNs for k=3 are given as follows:(44)Sζ^˜1=0.6540+0.4565+0.74623=0.6189,Sζ^˜2=0.6842+0.7878+0.09353=0.5218,Sζ^˜3=0.3294+0.6322+0.42423=0.4620,Sζ^˜4=0.6842+0.4814+0.28463=0.4834.
SinceSζ^˜1>Sζ^˜2>Sζ^˜4>Sζ^˜3,(45)ζ^˜σ1=ζ^1=0.6540,0.4565,0.7462,ζ^˜σ2=ζ^2=0.6842,0.7878,0.0935,ζ^˜σ3=ζ^4=0.6842,0.4814,0.2846,ζ^˜σ4=ζ^3=0.3294,0.6322,0.4242.
Then, from Definition9,(46)mFDHζw,Θζ1^,ζ2^,ζ3^,ζ4^=⊗j=14ζ^σjwj,=1−11+∑j=14wj1−p1∘ζσj/p1∘ζσjk1/k,…,1−11+∑j=14wj1−p3∘ζσj/p3∘ζσjk1/k,=1−11+0.4×1−0.6540/0.65403+0.1×1−0.6842/0.68423+0.3×1−0.6842/0.68423+0.2×1−0.3294/0.329431/3,1−11+0.4×1−0.4565/0.45653+0.1×1−0.7878/0.78783+0.3×1−0.4814/0.48143+0.2×1−0.6322/0.632231/3,1−11+0.4×1−0.7462/0.74623+0.1×1−0.0935/0.09353+0.3×1−0.2846/0.28463+0.2×1−0.4242/0.424231/3,=0.5482,0.5073,0.8210.


## 4. Mathematical Method for MCDM withmF Data

To solve MCDM problems containingmF data, we apply mF Dombi AOs. The following notions are utilized to tackle the MCDM situations having mF information. Suppose that Y1,Y2,…,Yk is a universal set and S1,S2,…,Sn is the universe of attributes. Assume Θ=Θ1,Θ2,…,Θn is a weight vector with ∑j=1nΘj=1,Θj∈0,1, for all j=1,…,n. Consider S^=s^ijk×n=p1∘ζij,p2∘ζij,…,pm∘ζijk×n is an mF decision matrix, which represents the membership values evaluated by the experts.We construct an algorithmic method to handle MCDM problems by anmFDWA (or mFDWG) operator.


## 5. Applications

### 5.1. Agriculture Land Selection

Agriculture is an essential part of Pakistan’s economic system. This area directly supports the population of the country and accounts for 26% of gross domestic product (GDP). The leading agricultural crops include sugarcane, wheat, rice, cotton, vegetables, and fruits. A business man wants to invest in agriculture sector and is searching for an appropriate land. The options in his brain areY1,Y2,…,Y5. He consults an expert to get his suggestion about the alternatives based on the following desired parameters:S1 denotes the “Location”S2 denotes the “Climate”S3 denotes the “Fertility”S4 denotes the “Price”Each parameter has been characterized into three parts to construct a 3FN.(i)
“Location” includes near to market, near to water channel, and transport availability.(ii)
“Climate” includes temperature, pollution level, and humidity level.(iii)
“Fertility” includes soil PH, level of nutrients, and water retention capacity of land.(iv)
“Price” includes low, medium, and high.The 3F decision matrix is shown in Table1.Table 1
3F decision matrix.
S1S2S3S4Y1(0.7, 0.6, 0.3)(0.5, 0.7, 0.2)(0.6, 0.3, 0.7)(0.9, 0.5, 0.4)Y2(0.9, 0.6, 0.5)(0.8, 0.5, 0.3)(0.4, 0.5, 0.8)(0.5, 0.8, 0.5)Y3(0.4, 0.7, 0.3)(0.5, 0.4, 0.3)(0.6, 0.8, 0.4)(0.6, 0.3, 0.8)Y4(0.5, 0.6, 0.3)(0.8, 0.6, 0.5)(0.7, 0.8, 0.2)(0.9, 0.3, 0.7)Y5(0.9, 0.7, 0.6)(0.8, 0.4, 0.3)(0.6, 0.5, 0.7)(0.4, 0.6, 0.5)According to the businessman, the expert assigns weights to parameters as follows:(47)Θ1=0.35,Θ2=0.25,Θ3=0.30,Θ4=0.10.Clearly,∑j=13Θj=1. To compute the most suitable land regarding agriculture, we use the two operators, namely, mFDWA and mFDWG, respectively:(1)
Fork=3, by applying the mFDWA operator, we calculate the values s^i of the lands Yi,i=1,2,…,5 regarding agriculture.(48)s^1=0.8107,0.6224,0.6109,s^2=0.8662,0.6679,0.7297,s^3=0.5443,0.7418,0.6515,s^4=0.8192,0.7347,0.5366,s^5=0.8663,0.6334,0.6327.(2)
Find the score valuesSs^i of 3FNs s^i,i=1,2,…,5 of the lands Yi:(49)Ss^1=0.6814,Ss^2=0.7546,Ss^3=0.6459,Ss^4=0.6968,Ss^5=0.7108.(3)
Rank the lands using scoresSsi,i=1,2,…,5 obtained from the preference values in the form of 3FNs: Y2>Y5>Y4>Y1>Y3.(4)
Y2 has a high score value, so it is the best land for agriculture.In a similar way, apply anmFDWG operator to find an appropriate land.(1)
Takek=3. Apply an mFDWG operator to determine the values s^i of the lands Yi.(50)s^1=0.4171,0.6142,0.7334,s^2=0.5092,0.4648,0.6058,s^3=0.5364,0.5633,0.6723,s^4=0.4196,0.5311,0.7423,s^5=0.4303,0.5153,0.6007.(2)
Determine the score valuesSs^i of 3FNs s^i of the lands Yi:(51)Ss^1=0.5882,Ss^2=0.5266,Ss^3=0.5907,Ss^4=0.5643,Ss^5=0.5154.(3)
Rank the lands using scoresSs^i,i=1,2,…,5 obtained from the preference values in the form of 3FNs: Y3>Y1>Y4>Y2>Y5.(4)
Y3 has high score, so it is the best land for agriculture.


### 5.2. Performance Evaluation of Commercial Banks

Commercial bank is one of the largest essential economic institutions. It can pull in money related streams, offering credit and different monetary administrations. These activities vitally affect national monetary improvements. Hence, commercial banks ought to be assessed by the modern and reliable procedures to rank commercial banks in the financial framework. This research establishes a MCDM model that usesmFDWA, mFDWG, and mF ELECTRE-I methods under a set of criteria and rank commercial banks. The board of specialists will assess each bank under chosen criteria. After a primer evaluation, six banks B1,B2,B3,B4,B5,B6 are assessed and ranked to pick the best bank. The banks are evaluated on the basis of four parameters.S1 denotes the “Net Income”S2 denotes the “Customer Service”S3 denotes the “Nonfinancial Performance”S4 denotes the “Potential Attractiveness”Each parameter has been characterized into four parts to form a 4FN.(i)
“Net Income” includes total equity, operating income, total assets, and net interest income.(ii)
“Customer Service” includes accessibility for customers, the evaluation of Internet page, the number of new services, and the number of new products.(iii)
“Nonfinancial Performance” includes support from main stake holders, bank management, employee stability, and ownership structure.(iv)
“Potential Attractiveness” includes location, involving environment, strategic dimension, and external and internal characteristics.The 4F decision matrix is represented by Table2.Table 2
4F decision matrix.
S1S2S3S4B1(0.7, 0.6, 0.7, 0.7)(0.5, 0.7, 0.6, 0.7)(0.5, 0.3, 0.5, 0.5)(0.5, 0.4, 0.4, 0.6)B2(0.9, 0.7, 0.8, 0.7)(0.8, 0.9, 0.8, 0.7)(0.8, 0.7, 0.7, 0.6)(0.7, 0.7, 0.8, 0.7)B3(0.4, 0.8, 0.5, 0.6)(0.5, 0.6, 0.6, 0.4)(0.5, 0.6, 0.5 0.4)(0.6, 0.6, 0.5, 0.3)B4(0.8, 0.7, 0.6, 0.5)(0.7, 0.5, 0.4, 0.6)(0.5, 0.5, 0.6, 0.5)(0.4, 0.5, 0.6, 0.5)B5(0.7, 0.7, 0.6, 0.7)(0.5, 0.6, 0.4, 0.4)(0.7, 0.5, 0.5, 0.4)(0.5, 0.5, 0.4, 0.5)B6(0.8, 0.8, 0.7, 0.6)(0.7, 0.6, 0.6, 0.5)(0.8, 0.5, 0.5, 0.4)(0.6, 0.6, 0.7, 0.6)The expert assigns weights to parameters as follows:(52)Θ1=0.28,Θ2=0.34,Θ3=0.22,Θ4=0.16.Clearly,∑j=14Θj=1. To select the most efficient bank, we use the two operators, namely, mFDWA and mFDWG, respectively:(1)
Fork=3, utilize the mFDWA operator to compute the values s^i for the banks Bi,i=1,2,…,6.(53)s^1=0.6188,0.5392,0.6306,0.6723,s^2=0.8617,0.8639,0.7895,0.6875,s^3=0.5139,0.7319,0.5492,0.5097,s^4=0.7385,0.6188,0.5699,0.5492,s^5=0.6551,0.6323,0.5227,0.6112,s^6=0.7689,0.7302,0.6562,0.6228.(2)
Calculate the score valuesSs^i of 4FNs s^i,i=1,2,…,6 for the banks Bi.(54)Ss^1=0.6152,Ss^2=0.8007,Ss^3=0.5762,Ss^4=0.6191,Ss^5=0.6053,Ss^6=0.6945.(3)
Now, rank the banks using scoresSsi,i=1,2,…,6 obtained from the preference values in the form of 4F numbers: B2>B6>B4>B1>B5>B3.(4)
B2 has a high score value, so it is the best bank.In a similar way, apply themFDWG operator to determine the most efficient bank.(1)
Takek=3. We employ the mFDWG operator to compute the values s^i for the banks Bi.(55)s^1=0.4752,0.6016,0.4896,0.4052,s^2=0.2182,0.2723,0.2361,0.3343,s^3=0.5366,0.3756,0.4772,0.6136,s^4=0.4006,0.5206,0.5246,0.4772,s^5=0.4487,0.4429,0.4952,0.5604,s^6=0.3029,0.4185,0.4146,0.5118.(2)
Find the score valuesSs^i of 4FNs s^i for the banks Bi.(56)Ss^1=0.4929,Ss^2=0.2652,Ss^3=0.5008,Ss^4=0.4808,Ss^5=0.4868,Ss^6=0.4269.(3)
Rank the banks with scoresSs^i,i=1,2,…,6 obtained from the preference values in the form of 4F numbers: B3>B1>B5>B4>B6>B2.(4)
B3 has high score, so it is the best bank.The methodology utilized in the applications to find the best alternative is shown in Figure1.Figure 1
Flowchart of selecting the best option.


### 5.3.mF-ELECTRE-I Method

In this section, we applymF-ELECTRE-I approach [27] to the problem (performance evaluation of commercial banks, Section 5.2) (Tables 3–5).(1)
Table3 describes the 4F decision matrix.(2)
Tables4 and 5, respectively, describe the 4F concordance and discordance values.(3)
The 4F concordance matrix is calculated by(57)F=−00.840.620.620.341−11110.160−0.160.3400.3801−0.560.220.6600.660.44−00.66010.781−.(4)
The 4F concordance levelf¯=0.498.(5)
The 4F discordance matrix is computed by(58)G=−10.5220.2890.748410−000011−111110.413−10.8303110.70101−10.950010−.(6)
The 4F discordance levelg¯=0.65.(7)
The 4F concordance and discordance matrices are given by(59)H=−001101−111100−000001−101010−010111−,L=−011000−111100−000001−000000−001101−.(8)
The 4F aggregated dominance matrix is constructed as(60)M=−001001−111100−000001−000000−000101−.(9)
Figure2 shows the preference relations between the banks.Table 3
4F weighted decision matrix.
S1S2S3S4B1(0.196, 0.168, 0.196, 0.196)(0.17, 0.238, 0.204, 0.238)(0.11, 0.066, 0.11, 0.11)(0.08, 0.064, 0.064, 0.096)B2(0.252, 0.196, 0.224, 0.196)(0.272, 0.306, 0.272, 0.238)(0.176, 0.154, 0.154, 0.132)(0.112, 0.112, 0.128, 0.112)B3(0.112, 0.224, 0.14, 0.168)(0.17, 0.204, 0.204, 0.136)(0.11, 0.132, 0.11, 0.038)(0.096, 0.096, 0.08, 0.048)B4(0.224, 0.196, 0.168, 0.14)(0.238, 0.17, 0.136, 0.204)(0.11, 0.11, 0.132, 0.11)(0.064, 0.08, 0.096, 0.08)B5(0.196, 0.196, 0.168, 0.196)(0.17, 0.204, 0.136, 0.136)(0.154, 0.11, 0.11, 0.038)(0.08, 0.08, 0.096, 0.08)B6(0.224, 0.224, 0.196, 0.168)(0.238, 0.204, 0.204, 0.17)(0.176, 0.11, 0.11, 0.038)(0.096, 0.096, 0.112, 0.096)Table 4
4F concordance set.
j123456F1j—{}{1, 2, 3}{1, 2}{1, 2}{2}F2j{1, 2, 3, 4}—{1, 2, 3, 4}{1, 2, 3, 4}{1, 2, 3, 4}{1, 2, 3, 4}F3j{4}{}—{4}{2}{}F4j{3, 4}{}{1, 2, 3, 4}—{2, 3}{3}F5j{1, 3, 4}{}{1, 3, 4}{1, 4}—{}F6j{1, 3, 4}{}{1, 2, 3, 4}{1, 2, 4}{1, 2, 3, 4}—Table 5
4F discordance set.
j123456G1j—{1, 2, 3, 4}{4}{3, 4}{1, 3, 4}{1, 3, 4}G2j{}—{}{}{}{}G3j{1, 2, 3}{1, 2, 3, 4}—{1, 2, 3, 4}{1, 3, 4}{1, 2, 3, 4}G4j{1, 2}{1, 2, 3, 4}{4}—{1, 4}{1, 2, 4}G5j{1, 2}{1, 2, 3, 4}{2}{2, 3}—{1, 2, 3, 4}G6j{2}{}{}{3}{}—From Figure2, it is clear that B2 is the best option (Tables 6 and 7).Figure 2
Outranking relation of banks.Figure 3
Comparison of first application inSection 5.1.Figure 4
Comparison of second application inSection 5.2.Table 6
Comparison ofmF Dombi AOs with mF Hamacher AOs [27] in agriculture land selection.
AOSs^1Ss^2Ss^3Ss^4Ss^5Ranking orderProposedmFDWA0.68140.75460.64590.69680.7108Y2>Y5>Y4>Y1>Y3ProposedmFDWG0.58820.52660.59070.56430.5154Y3>Y1>Y4>Y2>Y5mFHWA [27]0.54030.62870.51510.57250.6327Y5>Y2>Y4>Y1>Y3mFHWG [27]0.50840.58810.49080.54450.5909Y5>Y2>Y4>Y1>Y3Table 7
Comparison ofmF Dombi AOs with mF Hamacher AOs [27] in bank selection.
AOSs^1Ss^2Ss^3Ss^4Ss^5Ss^6Ranking orderProposedmFDWA0.61520.80070.57620.61910.60530.6945B2>B6>B4>B1>B5>B3ProposedmFDWG0.49290.26520.50080.48080.48680.4269B3>B1>B5>B4>B6>B2mFHWA [27]0.58510.76810.53420.57270.54900.6355B2>B6>B1>B4>B5>B3mFHWG [27]0.57110.75920.52670.56110.53810.6262B2>B6>B1>B4>B5>B3


## 5.1. Agriculture Land Selection

Agriculture is an essential part of Pakistan’s economic system. This area directly supports the population of the country and accounts for 26% of gross domestic product (GDP). The leading agricultural crops include sugarcane, wheat, rice, cotton, vegetables, and fruits. A business man wants to invest in agriculture sector and is searching for an appropriate land. The options in his brain areY1,Y2,…,Y5. He consults an expert to get his suggestion about the alternatives based on the following desired parameters:S1 denotes the “Location”S2 denotes the “Climate”S3 denotes the “Fertility”S4 denotes the “Price”Each parameter has been characterized into three parts to construct a 3FN.(i)
“Location” includes near to market, near to water channel, and transport availability.(ii)
“Climate” includes temperature, pollution level, and humidity level.(iii)
“Fertility” includes soil PH, level of nutrients, and water retention capacity of land.(iv)
“Price” includes low, medium, and high.The 3F decision matrix is shown in Table1.Table 1
3F decision matrix.
S1S2S3S4Y1(0.7, 0.6, 0.3)(0.5, 0.7, 0.2)(0.6, 0.3, 0.7)(0.9, 0.5, 0.4)Y2(0.9, 0.6, 0.5)(0.8, 0.5, 0.3)(0.4, 0.5, 0.8)(0.5, 0.8, 0.5)Y3(0.4, 0.7, 0.3)(0.5, 0.4, 0.3)(0.6, 0.8, 0.4)(0.6, 0.3, 0.8)Y4(0.5, 0.6, 0.3)(0.8, 0.6, 0.5)(0.7, 0.8, 0.2)(0.9, 0.3, 0.7)Y5(0.9, 0.7, 0.6)(0.8, 0.4, 0.3)(0.6, 0.5, 0.7)(0.4, 0.6, 0.5)According to the businessman, the expert assigns weights to parameters as follows:(47)Θ1=0.35,Θ2=0.25,Θ3=0.30,Θ4=0.10.Clearly,∑j=13Θj=1. To compute the most suitable land regarding agriculture, we use the two operators, namely, mFDWA and mFDWG, respectively:(1)
Fork=3, by applying the mFDWA operator, we calculate the values s^i of the lands Yi,i=1,2,…,5 regarding agriculture.(48)s^1=0.8107,0.6224,0.6109,s^2=0.8662,0.6679,0.7297,s^3=0.5443,0.7418,0.6515,s^4=0.8192,0.7347,0.5366,s^5=0.8663,0.6334,0.6327.(2)
Find the score valuesSs^i of 3FNs s^i,i=1,2,…,5 of the lands Yi:(49)Ss^1=0.6814,Ss^2=0.7546,Ss^3=0.6459,Ss^4=0.6968,Ss^5=0.7108.(3)
Rank the lands using scoresSsi,i=1,2,…,5 obtained from the preference values in the form of 3FNs: Y2>Y5>Y4>Y1>Y3.(4)
Y2 has a high score value, so it is the best land for agriculture.In a similar way, apply anmFDWG operator to find an appropriate land.(1)
Takek=3. Apply an mFDWG operator to determine the values s^i of the lands Yi.(50)s^1=0.4171,0.6142,0.7334,s^2=0.5092,0.4648,0.6058,s^3=0.5364,0.5633,0.6723,s^4=0.4196,0.5311,0.7423,s^5=0.4303,0.5153,0.6007.(2)
Determine the score valuesSs^i of 3FNs s^i of the lands Yi:(51)Ss^1=0.5882,Ss^2=0.5266,Ss^3=0.5907,Ss^4=0.5643,Ss^5=0.5154.(3)
Rank the lands using scoresSs^i,i=1,2,…,5 obtained from the preference values in the form of 3FNs: Y3>Y1>Y4>Y2>Y5.(4)
Y3 has high score, so it is the best land for agriculture.


## 5.2. Performance Evaluation of Commercial Banks

Commercial bank is one of the largest essential economic institutions. It can pull in money related streams, offering credit and different monetary administrations. These activities vitally affect national monetary improvements. Hence, commercial banks ought to be assessed by the modern and reliable procedures to rank commercial banks in the financial framework. This research establishes a MCDM model that usesmFDWA, mFDWG, and mF ELECTRE-I methods under a set of criteria and rank commercial banks. The board of specialists will assess each bank under chosen criteria. After a primer evaluation, six banks B1,B2,B3,B4,B5,B6 are assessed and ranked to pick the best bank. The banks are evaluated on the basis of four parameters.S1 denotes the “Net Income”S2 denotes the “Customer Service”S3 denotes the “Nonfinancial Performance”S4 denotes the “Potential Attractiveness”Each parameter has been characterized into four parts to form a 4FN.(i)
“Net Income” includes total equity, operating income, total assets, and net interest income.(ii)
“Customer Service” includes accessibility for customers, the evaluation of Internet page, the number of new services, and the number of new products.(iii)
“Nonfinancial Performance” includes support from main stake holders, bank management, employee stability, and ownership structure.(iv)
“Potential Attractiveness” includes location, involving environment, strategic dimension, and external and internal characteristics.The 4F decision matrix is represented by Table2.Table 2
4F decision matrix.
S1S2S3S4B1(0.7, 0.6, 0.7, 0.7)(0.5, 0.7, 0.6, 0.7)(0.5, 0.3, 0.5, 0.5)(0.5, 0.4, 0.4, 0.6)B2(0.9, 0.7, 0.8, 0.7)(0.8, 0.9, 0.8, 0.7)(0.8, 0.7, 0.7, 0.6)(0.7, 0.7, 0.8, 0.7)B3(0.4, 0.8, 0.5, 0.6)(0.5, 0.6, 0.6, 0.4)(0.5, 0.6, 0.5 0.4)(0.6, 0.6, 0.5, 0.3)B4(0.8, 0.7, 0.6, 0.5)(0.7, 0.5, 0.4, 0.6)(0.5, 0.5, 0.6, 0.5)(0.4, 0.5, 0.6, 0.5)B5(0.7, 0.7, 0.6, 0.7)(0.5, 0.6, 0.4, 0.4)(0.7, 0.5, 0.5, 0.4)(0.5, 0.5, 0.4, 0.5)B6(0.8, 0.8, 0.7, 0.6)(0.7, 0.6, 0.6, 0.5)(0.8, 0.5, 0.5, 0.4)(0.6, 0.6, 0.7, 0.6)The expert assigns weights to parameters as follows:(52)Θ1=0.28,Θ2=0.34,Θ3=0.22,Θ4=0.16.Clearly,∑j=14Θj=1. To select the most efficient bank, we use the two operators, namely, mFDWA and mFDWG, respectively:(1)
Fork=3, utilize the mFDWA operator to compute the values s^i for the banks Bi,i=1,2,…,6.(53)s^1=0.6188,0.5392,0.6306,0.6723,s^2=0.8617,0.8639,0.7895,0.6875,s^3=0.5139,0.7319,0.5492,0.5097,s^4=0.7385,0.6188,0.5699,0.5492,s^5=0.6551,0.6323,0.5227,0.6112,s^6=0.7689,0.7302,0.6562,0.6228.(2)
Calculate the score valuesSs^i of 4FNs s^i,i=1,2,…,6 for the banks Bi.(54)Ss^1=0.6152,Ss^2=0.8007,Ss^3=0.5762,Ss^4=0.6191,Ss^5=0.6053,Ss^6=0.6945.(3)
Now, rank the banks using scoresSsi,i=1,2,…,6 obtained from the preference values in the form of 4F numbers: B2>B6>B4>B1>B5>B3.(4)
B2 has a high score value, so it is the best bank.In a similar way, apply themFDWG operator to determine the most efficient bank.(1)
Takek=3. We employ the mFDWG operator to compute the values s^i for the banks Bi.(55)s^1=0.4752,0.6016,0.4896,0.4052,s^2=0.2182,0.2723,0.2361,0.3343,s^3=0.5366,0.3756,0.4772,0.6136,s^4=0.4006,0.5206,0.5246,0.4772,s^5=0.4487,0.4429,0.4952,0.5604,s^6=0.3029,0.4185,0.4146,0.5118.(2)
Find the score valuesSs^i of 4FNs s^i for the banks Bi.(56)Ss^1=0.4929,Ss^2=0.2652,Ss^3=0.5008,Ss^4=0.4808,Ss^5=0.4868,Ss^6=0.4269.(3)
Rank the banks with scoresSs^i,i=1,2,…,6 obtained from the preference values in the form of 4F numbers: B3>B1>B5>B4>B6>B2.(4)
B3 has high score, so it is the best bank.The methodology utilized in the applications to find the best alternative is shown in Figure1.Figure 1
Flowchart of selecting the best option.


## 5.3.mF-ELECTRE-I Method

In this section, we applymF-ELECTRE-I approach [27] to the problem (performance evaluation of commercial banks, Section 5.2) (Tables 3–5).(1)
Table3 describes the 4F decision matrix.(2)
Tables4 and 5, respectively, describe the 4F concordance and discordance values.(3)
The 4F concordance matrix is calculated by(57)F=−00.840.620.620.341−11110.160−0.160.3400.3801−0.560.220.6600.660.44−00.66010.781−.(4)
The 4F concordance levelf¯=0.498.(5)
The 4F discordance matrix is computed by(58)G=−10.5220.2890.748410−000011−111110.413−10.8303110.70101−10.950010−.(6)
The 4F discordance levelg¯=0.65.(7)
The 4F concordance and discordance matrices are given by(59)H=−001101−111100−000001−101010−010111−,L=−011000−111100−000001−000000−001101−.(8)
The 4F aggregated dominance matrix is constructed as(60)M=−001001−111100−000001−000000−000101−.(9)
Figure2 shows the preference relations between the banks.Table 3
4F weighted decision matrix.
S1S2S3S4B1(0.196, 0.168, 0.196, 0.196)(0.17, 0.238, 0.204, 0.238)(0.11, 0.066, 0.11, 0.11)(0.08, 0.064, 0.064, 0.096)B2(0.252, 0.196, 0.224, 0.196)(0.272, 0.306, 0.272, 0.238)(0.176, 0.154, 0.154, 0.132)(0.112, 0.112, 0.128, 0.112)B3(0.112, 0.224, 0.14, 0.168)(0.17, 0.204, 0.204, 0.136)(0.11, 0.132, 0.11, 0.038)(0.096, 0.096, 0.08, 0.048)B4(0.224, 0.196, 0.168, 0.14)(0.238, 0.17, 0.136, 0.204)(0.11, 0.11, 0.132, 0.11)(0.064, 0.08, 0.096, 0.08)B5(0.196, 0.196, 0.168, 0.196)(0.17, 0.204, 0.136, 0.136)(0.154, 0.11, 0.11, 0.038)(0.08, 0.08, 0.096, 0.08)B6(0.224, 0.224, 0.196, 0.168)(0.238, 0.204, 0.204, 0.17)(0.176, 0.11, 0.11, 0.038)(0.096, 0.096, 0.112, 0.096)Table 4
4F concordance set.
j123456F1j—{}{1, 2, 3}{1, 2}{1, 2}{2}F2j{1, 2, 3, 4}—{1, 2, 3, 4}{1, 2, 3, 4}{1, 2, 3, 4}{1, 2, 3, 4}F3j{4}{}—{4}{2}{}F4j{3, 4}{}{1, 2, 3, 4}—{2, 3}{3}F5j{1, 3, 4}{}{1, 3, 4}{1, 4}—{}F6j{1, 3, 4}{}{1, 2, 3, 4}{1, 2, 4}{1, 2, 3, 4}—Table 5
4F discordance set.
j123456G1j—{1, 2, 3, 4}{4}{3, 4}{1, 3, 4}{1, 3, 4}G2j{}—{}{}{}{}G3j{1, 2, 3}{1, 2, 3, 4}—{1, 2, 3, 4}{1, 3, 4}{1, 2, 3, 4}G4j{1, 2}{1, 2, 3, 4}{4}—{1, 4}{1, 2, 4}G5j{1, 2}{1, 2, 3, 4}{2}{2, 3}—{1, 2, 3, 4}G6j{2}{}{}{3}{}—From Figure2, it is clear that B2 is the best option (Tables 6 and 7).Figure 2
Outranking relation of banks.Figure 3
Comparison of first application inSection 5.1.Figure 4
Comparison of second application inSection 5.2.Table 6
Comparison ofmF Dombi AOs with mF Hamacher AOs [27] in agriculture land selection.
AOSs^1Ss^2Ss^3Ss^4Ss^5Ranking orderProposedmFDWA0.68140.75460.64590.69680.7108Y2>Y5>Y4>Y1>Y3ProposedmFDWG0.58820.52660.59070.56430.5154Y3>Y1>Y4>Y2>Y5mFHWA [27]0.54030.62870.51510.57250.6327Y5>Y2>Y4>Y1>Y3mFHWG [27]0.50840.58810.49080.54450.5909Y5>Y2>Y4>Y1>Y3Table 7
Comparison ofmF Dombi AOs with mF Hamacher AOs [27] in bank selection.
AOSs^1Ss^2Ss^3Ss^4Ss^5Ss^6Ranking orderProposedmFDWA0.61520.80070.57620.61910.60530.6945B2>B6>B4>B1>B5>B3ProposedmFDWG0.49290.26520.50080.48080.48680.4269B3>B1>B5>B4>B6>B2mFHWA [27]0.58510.76810.53420.57270.54900.6355B2>B6>B1>B4>B5>B3mFHWG [27]0.57110.75920.52670.56110.53810.6262B2>B6>B1>B4>B5>B3


## 6. Comparison Analysis and Discussion

This section gives a comparison of the developed Dombi AOs withmF Hamacher AOs [27] and an mF-ELECTRE-I model [46] to show their feasibility and practicality.(1)
We compare the results of developedmF Dombi AOs with mF Hamacher AOs [27]. The results computed by applying both operators in first application are explained by Table 6 and Figure 3. In a similar way, the results computed using both operators in second application are explained by Table 7 and Figure 4. Clearly, the results of mF Hamacher weighted average (mFHWA) and mF Hamacher weighted geometric (mFHWG) operators are different from our newly constructed mFDWA and mFDWG operators. The results of mFHWA and mFHWG operators are the same. Therefore, our developed AOs are more generalized and versatile than some existing models to handle mF MCDM problems.(2)
From the second application, it can be observed that the final rankings by applying themFDWA and mFDWG operators are B2>B6>B4>B1>B5>B3 and B3>B1>B5>B4>B6>B2, respectively. However, the final score values are not the same. When mF-ELECTRE-I method is applied, the best option is B2. Clearly, the optimal decision using mF-ELECTRE-I method and mFDWA operator is B2.(3)
When a number ofmFNs are aggregated with the help of mFDWA and mFDWG operators, different computations will increase rapidly. But our developed AOs can explain the assessed data more flexibly for decision making. The developed method ranks every objects in a given problem in comparison with mF-ELECTRE-I approach [46].


### 6.1. Effectiveness Test

To examine the validity of the provided algorithm, we verify it with test criteria developed by Wang and Triantaphyllou [21] as follows (Tables 8 and 9):(i)
Test criterion I: if we change the membership grades of nonoptimal object with worse membership values without effecting criteria, then the optimal object should not change.(ii)
Test criterion II: MCDM approach should satisfy transitive property.(iii)
Test criterion III: when a designated problem is resolved into different small issues and the similar MCDM technique has been utilized, then the rank order of the objects should be similar to the original ranking order. We have checked these test criteria on developed MCDM approach undermF Dombi AOs as below.(1)
Effectiveness test by criterion I: using this test, if we change only the membership grades of alternativeY3 with Y3′=0.3,0.6,0.2,0.4,0.3,0.2,0.5,0.7,0.3,0.5,0.1,0.6 in Table 1 (that is, 3F decision matrix), then the new 3F decision matrix is shown in Table 8. By using mFDWA operator, the score values of the alternative are Ss^1=0.6814,Ss^2=0.7546,Ss^3′=0.6459,Ss^4=0.6968,Ss^5=0.7108. Clearly, Ss^2>Ss^5>Ss^4>Ss^1>Ss^3′; consequently, the ranking of the objects is Y2>Y5>Y4>Y1>Y3′. Thus, Y2 is the best alternative. According to above information, the presented AOs have been employed, and the decision alternative is Y2 which is similar to the original optimal object. Similarly, if we change the membership grades of alternative Y4 with Y4′=0.4,0.5,0.2,0.7,0.5,0.5,0.8,0.7,0.1,0.8,0.2,0.5 in Table 1 (that is, 3F decision matrix), then the new 3F decision matrix is shown in Table 9. By applying the mFDWA operator, the ranking order of the alternatives is Y2>Y5>Y1>Y3>Y4′. Thus, the optimal alternative is Y2 which is same as that of the original ranking. Therefore, the proposed algorithm is feasible under test criterion I.(2)
Effectiveness test by criteria II and III: based upon these test criteria, if we dissolved the designated problem (Application 1) into the subissuesY1,Y2,Y3,Y2,Y3,Y4,Y3,Y4,Y5, and Y4,Y5,Y1 and employed the procedure steps of Algorithm 1, then we obtain the ranking of these smaller issues as Y2>Y1>Y3, Y2>Y4>Y3, Y5>Y4>Y3, and Y5>Y4>Y1, respectively. Hence, by uniting above criteria II and III, we obtain the overall ranking order of the alternatives as Y2>Y5>Y4>Y1>Y3, which is exactly same as the original ranking order. Therefore, the developed algorithm is feasible under test criteria II and III.Algorithm 1: Computing maximum score value.
(1)
Input:S^, an mF decision matrix having k objects and n attributes.Θ=Θ1,Θ2,…,Θn, the vector having weights.(2)
Apply themFDWA operators to aggregate the data in mF decision matrix S^ and calculate the preference values s^i, where “i” varies from 1 to k for the mFNsζi.s^i=mFDWAΘζ^i1,ζ^i2,…,ζ^in=⊕j=1nΘjζ^ij=1−1/1+∑j=1nΘjp1∘ζij/1−p1∘ζijk1/k,…,1−1/1+∑j=1nΘjpm∘ζij/1−pm∘ζijk1/k.When we usemFDWG operators,s^i=mFDWGΘζ^i1,ζ^i2,…,ζ^in=⊗j=1nζ^ijΘj,=1−1/1+∑j=1nΘj1−p1∘ζij/p1∘ζijk1/k,…,1−1/∑j=1nΘj1−pm∘ζij/pm∘ζijk.(3)
Compute the score valuesSs^i,i=1,2,…,k.(4)
Rank the objectsui,i=1,2,…,k with respect to their scores Ss^i. When the scores of two objects are equal, we apply the accuracy function to find the order of alternatives.Output: the object containing maximum score value in last step will be the decision.Table 8
3F decision matrix.
S1S2S3S4Y1(0.7, 0.6, 0.3)(0.5, 0.7, 0.2)(0.6, 0.3, 0.7)(0.9, 0.5, 0.4)Y2(0.9, 0.6, 0.5)(0.8, 0.5, 0.3)(0.4, 0.5, 0.8)(0.5, 0.8, 0.5)Y3′(0.3, 0.6, 0.2)(0.4, 0.3, 0.2)(0.5, 0.7, 0.3)(0.5, 0.1, 0.6)Y4(0.5, 0.6, 0.3)(0.8, 0.6, 0.5)(0.7, 0.8, 0.2)(0.9, 0.3, 0.7)Y5(0.9, 0.7, 0.6)(0.8, 0.4, 0.3)(0.6, 0.5, 0.7)(0.4, 0.6, 0.5)Table 9
3F decision matrix.
S1S2S3S4Y1(0.7, 0.6, 0.3)(0.5, 0.7, 0.2)(0.6, 0.3, 0.7)(0.9, 0.5, 0.4)Y2(0.9, 0.6, 0.5)(0.8, 0.5, 0.3)(0.4, 0.5, 0.8)(0.5, 0.8, 0.5)Y3(0.3, 0.6, 0.2)(0.4, 0.3, 0.2)(0.5, 0.7, 0.3)(0.5, 0.1, 0.6)Y4′(0.4, 0.5, 0.2)(0.7, 0.5, 0.5)(0.8, 0.7, 0.1)(0.8, 0.2, 0.5)Y5(0.9, 0.7, 0.6)(0.8, 0.4, 0.3)(0.6, 0.5, 0.7)(0.4, 0.6, 0.5)


## 6.1. Effectiveness Test

To examine the validity of the provided algorithm, we verify it with test criteria developed by Wang and Triantaphyllou [21] as follows (Tables 8 and 9):(i)
Test criterion I: if we change the membership grades of nonoptimal object with worse membership values without effecting criteria, then the optimal object should not change.(ii)
Test criterion II: MCDM approach should satisfy transitive property.(iii)
Test criterion III: when a designated problem is resolved into different small issues and the similar MCDM technique has been utilized, then the rank order of the objects should be similar to the original ranking order. We have checked these test criteria on developed MCDM approach undermF Dombi AOs as below.(1)
Effectiveness test by criterion I: using this test, if we change only the membership grades of alternativeY3 with Y3′=0.3,0.6,0.2,0.4,0.3,0.2,0.5,0.7,0.3,0.5,0.1,0.6 in Table 1 (that is, 3F decision matrix), then the new 3F decision matrix is shown in Table 8. By using mFDWA operator, the score values of the alternative are Ss^1=0.6814,Ss^2=0.7546,Ss^3′=0.6459,Ss^4=0.6968,Ss^5=0.7108. Clearly, Ss^2>Ss^5>Ss^4>Ss^1>Ss^3′; consequently, the ranking of the objects is Y2>Y5>Y4>Y1>Y3′. Thus, Y2 is the best alternative. According to above information, the presented AOs have been employed, and the decision alternative is Y2 which is similar to the original optimal object. Similarly, if we change the membership grades of alternative Y4 with Y4′=0.4,0.5,0.2,0.7,0.5,0.5,0.8,0.7,0.1,0.8,0.2,0.5 in Table 1 (that is, 3F decision matrix), then the new 3F decision matrix is shown in Table 9. By applying the mFDWA operator, the ranking order of the alternatives is Y2>Y5>Y1>Y3>Y4′. Thus, the optimal alternative is Y2 which is same as that of the original ranking. Therefore, the proposed algorithm is feasible under test criterion I.(2)
Effectiveness test by criteria II and III: based upon these test criteria, if we dissolved the designated problem (Application 1) into the subissuesY1,Y2,Y3,Y2,Y3,Y4,Y3,Y4,Y5, and Y4,Y5,Y1 and employed the procedure steps of Algorithm 1, then we obtain the ranking of these smaller issues as Y2>Y1>Y3, Y2>Y4>Y3, Y5>Y4>Y3, and Y5>Y4>Y1, respectively. Hence, by uniting above criteria II and III, we obtain the overall ranking order of the alternatives as Y2>Y5>Y4>Y1>Y3, which is exactly same as the original ranking order. Therefore, the developed algorithm is feasible under test criteria II and III.Algorithm 1: Computing maximum score value.
(1)
Input:S^, an mF decision matrix having k objects and n attributes.Θ=Θ1,Θ2,…,Θn, the vector having weights.(2)
Apply themFDWA operators to aggregate the data in mF decision matrix S^ and calculate the preference values s^i, where “i” varies from 1 to k for the mFNsζi.s^i=mFDWAΘζ^i1,ζ^i2,…,ζ^in=⊕j=1nΘjζ^ij=1−1/1+∑j=1nΘjp1∘ζij/1−p1∘ζijk1/k,…,1−1/1+∑j=1nΘjpm∘ζij/1−pm∘ζijk1/k.When we usemFDWG operators,s^i=mFDWGΘζ^i1,ζ^i2,…,ζ^in=⊗j=1nζ^ijΘj,=1−1/1+∑j=1nΘj1−p1∘ζij/p1∘ζijk1/k,…,1−1/∑j=1nΘj1−pm∘ζij/pm∘ζijk.(3)
Compute the score valuesSs^i,i=1,2,…,k.(4)
Rank the objectsui,i=1,2,…,k with respect to their scores Ss^i. When the scores of two objects are equal, we apply the accuracy function to find the order of alternatives.Output: the object containing maximum score value in last step will be the decision.Table 8
3F decision matrix.
S1S2S3S4Y1(0.7, 0.6, 0.3)(0.5, 0.7, 0.2)(0.6, 0.3, 0.7)(0.9, 0.5, 0.4)Y2(0.9, 0.6, 0.5)(0.8, 0.5, 0.3)(0.4, 0.5, 0.8)(0.5, 0.8, 0.5)Y3′(0.3, 0.6, 0.2)(0.4, 0.3, 0.2)(0.5, 0.7, 0.3)(0.5, 0.1, 0.6)Y4(0.5, 0.6, 0.3)(0.8, 0.6, 0.5)(0.7, 0.8, 0.2)(0.9, 0.3, 0.7)Y5(0.9, 0.7, 0.6)(0.8, 0.4, 0.3)(0.6, 0.5, 0.7)(0.4, 0.6, 0.5)Table 9
3F decision matrix.
S1S2S3S4Y1(0.7, 0.6, 0.3)(0.5, 0.7, 0.2)(0.6, 0.3, 0.7)(0.9, 0.5, 0.4)Y2(0.9, 0.6, 0.5)(0.8, 0.5, 0.3)(0.4, 0.5, 0.8)(0.5, 0.8, 0.5)Y3(0.3, 0.6, 0.2)(0.4, 0.3, 0.2)(0.5, 0.7, 0.3)(0.5, 0.1, 0.6)Y4′(0.4, 0.5, 0.2)(0.7, 0.5, 0.5)(0.8, 0.7, 0.1)(0.8, 0.2, 0.5)Y5(0.9, 0.7, 0.6)(0.8, 0.4, 0.3)(0.6, 0.5, 0.7)(0.4, 0.6, 0.5)


## 7. Conclusions and Future Directions

Aggregation operators are mathematical functions and essential tools of unifying several inputs into single valuable output. Due to the existence of multipolar data and multiple attributes in many real-world problems, classical MCDM methods are not useful to tackle complicated decision-making situations. To overcome the difficulties of existing models, we have combinedmFNs with Dombi AOs.In this article, we have discussed MCDM issues based onmF information. Motivated by Dombi operations, we have proposed certain mF Dombi AOs, namely, mFDWA, mFDOWA, mFDHA, mFDWG, mFDOWG, and mFDHG operators. We have investigated different features of these operators. We have employed these AOs to enlarge the applicability scope of MCDM. We have given real-life applications for the selection of best agricultural land and for the selection of best bank regarding performance. At the end, we have provided a comparison of developed AOs with mF-ELECTRE-I method [46] and mF Hamacher AOs [27] and have authenticated the proposed strategy by effectiveness tests to check its validity. In the comparison, we have seen that the optimal alternative is the same by applying mF-ELECTRE-I method [46], mF Hamacher AOs [27], and proposed mFDWA operator. However, it is different in case of mFDWG operator. In the future, we plan to extend our work to (i) mF Dombi prioritized AOs, (ii) mF soft Dombi AOs, (iii) q-rung orthopair fuzzy Dombi AOs, and (iv) q-rung orthopair fuzzy soft Dombi AOs.



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*Source: 4739567-2020-08-01.xml*