# Extension of the TOPSIS Method for Decision Making Problems under Complex Fuzzy Data Based on the Central Point Index

Authors: Salameh Barbat; Mahnaz Barkhordariahmadi; Vahidmomenaei Kermani
Journal: Advances in Fuzzy Systems
 (2022)

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## Abstract

This paper presents the CP-TOPSIS Model in group decision-making using complex Fuzzy Data. Complex numbers were employed in this model, and the central point index was used to define both the negative and positive ideals as well as the distance between each option. In this approach, the options are graded using complex data (due to replacing linguistic variables). One of the advantages of this model in decision-making is the capability that creates a complex fuzzy technique for investigating, grading, and selecting the best option related to complex fuzzy data. The results show that this model effectively rates and grades the complex fuzzy data through an alternative period. Quantum mechanics wave functions could not be analyzed, nor could signals or time series or stock exchange transactions predict factors of a multiperiod alternation, nor could predictions be made about any of these variables. As a result, there are numerical results in rating with high precision.


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## Body

## 1. Introduction

In everyday life, different criteria are usually measured implicitly. Such decisions may be made solely based on intuition and judgment. There are several crises that must be dealt with; therefore, it is critical to recognize the situation accurately. The site of a nuclear power facility, for instance, is a highly complex subject involving numerous parameters. Therefore, multicriteria decision-making (MCDM) is related to constructing and solving decision-making and planning problems with various criteria. MCDM was developed as a part of operation research in designing computational and mathematical tools to support the subjective assessment of the criteria performance by decision-makers [1].In the last few years, MCDM tools and their applications have been used in different studies to solve problems associated with the industry [2], science [3], and engineering [4]. MCDM offers the best choice for decision-making out of a finite number of options [5]. Besides, due to the changes and developments over the past decades, MCDM is one of the fastest tools and methods for selecting the best option in the decision-making business. The decision-makers need help deciding between the fast and superior options. Decision-making methods are well accepted in all areas of the decision-making process with the help of computers. MCDM is vital in operations research and management science and is the basis for making and developing various computer-based methods [6]. MCDM issues can be classified into two categories: Multiple Attribute Decision-Making (MADM) and Multiple Objective Decision-Making (MODM). MADM is the most well-known branch of decision-making and a general class of operations research models dealing with decision-making problems under some criteria.MADM approach requires selecting among decision alternatives described by their attributes. MADM problems are assumed to have a predetermined and limited number of decision alternatives. Solving aMADM problem involves sorting and ranking. As an alternative to the decision-making matrix and additional information provided by the decision-maker, MADM methodologies can be used to rank, screen, or select among the options. Moreover, MODM is a powerful tool to help find decisions that will best satisfy several conflicting objectives. Contrary to the MADM, the decision alternatives are not given in the MODM approach. Instead, MODM provides a mathematical framework for designing a set of decision alternatives. A solution’s suitability for implementation is determined by how closely it approaches a given set of goals. The MADM can be classified into two categories of compensatory and noncompensatory methods. Compensatory MADM methods incorporate trade-offs between high and low performance in the analysis. Those methods that do not perform such incorporation are termed noncompensatory. Hwang and Yoon [5] presented several MADM methods. Two methods used in the present work are briefly explained as follows:TOPSIS: the principle behind TOPSIS is simple; the chosen alternative should be as close to the ideal and far from the negative ideal solution as possible.Distance from Target: this method and its results are also straightforward for graphical description. First, target values for each attribute are chosen, not requiring to be exhibited by any available alternative.TheTOPSIS method has some advantages, including its simplicity, comprehensibility, ease of calculation, and ability to measure the relative performance of each alternative using a simple mathematical form. Fuzzy TOPSIS is one of the most effective techniques in MCDM fuzzy, which was introduced by Wang Yoon [5]. The TOPSIS method is based on the premise that the alternative should simultaneously have the shortest distance with the PIS positive ideal solution and the farthest distance with the NIS negative ideal solution. In the TOPSIS method, for each criterion, the relative weighted distances between the various options are listed in relation to PIS and NIS. The relative distances represent efficiency. Therefore, the efficiency of each option against each criterion can be operated in this way. Torfi et al. [7] proposed a hierarchical AHP process to determine the relative weighting of evaluation criteria and a fuzzy TOPSIS to rank alternatives. Liao and Cao [8] presented an integrated fuzzy TOPSIS approach and multichoice goal programming MCGP to select support in supply chain management. The problem of selecting multicriteria group support was simulated by Zouggari and Benyoucef [9] based on the fuzzy TOPSIS approach. Doukas et al. [10] developed a formula for separating weights to rank alternatives. A ranking formula was proposed by Kuo [11] based on the method of Dukas et al. by normalizing the performance measure of each alternative relative to the cost and profit criteria, that is, di+ and di−. In both methods, the weights assigned to the separations have crisp values, and the use of crisp weights limits the range of applications, as linguistic weights are commonly used in fuzzy TOPSIS. In this paper, we used complex language weights to weight the complex data described in sections.This paper is devoted to the discussion of complex fuzzy number ranking and is expected to have wide applications in other areas of complex fuzzy data ranking. The following is how this article is organized: Section2 discusses the basic concept of complex fuzzy numbers and their other applications. In Section 3, a detailed discussion of the concepts of complex fuzzy numbers (including definitions and fuzzy distances) is devoted. In this section, a new method for ranking complex fuzzy numbers with a central point index is introduced, and finally, an example of this ranking method is introduced. In Section 4, the results of the new ranking method are presented. In Section 5, managerial insights are presented, and finally in Section 6, the work’s conclusion is fundamentally summarized, and the next field of research is offered.


## 2. Literature Review

The concept of the type-1 fuzzy set was introduced by LotfiZadeh [12]. In 1975, he introduced the type-2 fuzzy set, in which its membership function was a fuzzy number [13]. A complex fuzzy set was presented by Ramot et al. [14]. In recent years, big data has become a growing research concern. High and low data have often been associated with fluctuations and uncertainty. Uncertainty may be because of factors such as corruption or loss of data components, and a cyclical and repetitive pattern may cause fluctuations in data. For example, there is more temperature in summer and less in winter or more traffic during rush hours. Fuzzy sets have been recently used for ranking the fuzzy numbers with many applications in numerous sectors of production, industry, and management. Some applications of fuzzy sets in industry and management include flow workshop scheduling (FSS), interactive fuzzy solution method, self-adapting artificial fish swarm algorithm (SAAFSA) [15], hybrid fuzzy and data-driven robust optimization for resilience and sustainable health care supply chain [16], viable medical waste chain network design [17], two-level solid planning for the location of renewable energy in conditions of uncertainty [18], sustainable supply chain network design [19], a robust optimization model for sustainable and resilient closed-loop supply chain network design [20], a robust time-cost-quality-energy-environment trade-off with resource-constrained due to uncertainty [21], Viable Supply Chain Network Design by considering Blockchain Technology and Cryptocurrency [22], robust optimization of risk-aware, resilient and sustainable closed-loop supply chain network design [23], an extended robust mathematical model to project the course of COVID-19 epidemic in Iran [24], linear programming of complex integers (CILP) to handle location decisions [25], developing a sustainable operational management system [26], a Covering Tour Approach for Disaster Relief Locating and Routing with Fuzzy Demand [27], a parallel machine sequence-dependent group scheduling problem with the goal of minimizing total weighted [28], hybrid artificial intelligence and robust optimization [55], and Location Improved Harmony Search Algorithm [29]. However, despite their many applications, prioritizing the existing options was not considered in ranking the complex fuzzy numbers. Therefore, in the present paper, we examined the available options for ranking complex fuzzy numbers by the TOPSIS decision method in polar conditions while considering the periodicity of these numbers.The recent complex fuzzy sets are innovative with the increasing number of applications. Among the applications of complex fuzzy sets is the analysis of solar activity based on the number of recorded sunspots [14], signal processing [14], stock trading in the New York Stock Exchange [30], voter turnout forecast [30], multiperiodic factor prediction [31], prediction of voter turnout in elections [32], complex fuzzy logic systems [30, 33–37], image restoration [38], an integrated approach based on artificial intelligence, and novel metaheuristic algorithms to predict demand for dairy products [39].The complex fuzzy set modifies the fuzzy membership function’s initial premise. In certain instances, however, it is argued that a second dimension must be added to the concept of membership function. The fundamental principle of the phase is unaffected by the second characteristic of membership. In both the ordinary and the complex fuzzy sets, the membership function is the same fuzzy function. In the apparent dimension of the membership function, the novelty of complex fuzzy sets is the membership function, the power factor, or the fuzzy period. According to the definition of a complex fuzzy set, each grade of membership is defined by an amplitude term and a phase term. Therefore, the difference between the two complex fuzzy sets can be simply calculated by combining the difference between the amplitude terms and the phase terms. In this regard, Zhang et al. [40] proposed a distance measure to determine the equality of complex fuzzy sets. Alkouri and Salleh [41] introduced several distance measures for complex fuzzy sets. Nonetheless, in the distance measure determined by Zhang et al. [40, 41], the periodicity of the phase term of a complex fuzzy set as a function was disregarded. The cyclic representation of the complex membership in a complex fuzzy set is ignored. We modified the distance measure for complex fuzzy sets.In this research, we defined and presented a new strategy for prioritizing a model of decision-making, in which complex fuzzy numbers play a role. A rating algorithm was constructed based on the central point index and the TOPSIS-oriented technique. We also employ linguistic terminology to analyze possibilities. In this study, we labeled this method, the so-called CP-TOPSIS. For ranking and picking the best options based on complex fuzzy data, a complex fuzzy technique is developed. These options may have many applications, including analyzing wave functions in quantum mechanics and many physical quantities. This paper presents a new rating method through a central point index based on the center of mass in complex fuzzy numbers used in the TOPSIS method. The modified distance function, which is applicable for the complex fuzzy numbers based on the complex membership function, is described in the TOPSIS method and is also shown with a practical instance in the TOPSIS complex method.


## 3. Problem Statement

Definition 1.
LetU be a universe of discourse. A complex fuzzy set A on U with a membership function μx for every x∈U specifies that each element is assigned a complex membership degree in the complex set of numbers C. The complex fuzzy set A can be represented as a set of ordered pairs(1)A=x,μAx:x∈U,where the membership function has a complex form μAx=rAx.eiωAx, the amplitude term rAx and the phase term wAx are both real-valued, and rAx∈0,1,i=−1. The function μAx maps U into unit disk of the complex plane, and the function eiωAx is a periodic function with periodicity 2π.(2)ArgAx=ωAx+2kπ,k=0,±1,±2,…,in which 0≤ωAx<2π is the main argument of the function. Consider distance measure for complex fuzzy sets:(3)dA,B=12∑i=1nrAxi−rBxi+12π∑i=1nωAxi−ωBxi,where(4)A=x,μAx:x∈U,μAx=rAx.eiωAx,B=x,μBx:x∈UμB=rBx.eiωBx.
Complex fuzzy sets are a generalization of ordinary fuzzy sets. Any ordinary fuzzy set can be represented as a complex fuzzy set by simply setting the argument component of the membership function for each element,ωx, to zero. As a result, the membership function argument is the distinction between ordinary and complex fuzzy sets. Indeed, without an argument for the membership function, the complex fuzzy set becomes a regular fuzzy set, and the membership function range is [0, 1].Example 1.
LetA=−1,0,1,2 and its complex membership function are(5)A=0.6ei1.2π−1+1.0ei2π0+0.8ei1.6π1+0.5eiπ2.Definition 2.
LetA,B be two complex fuzzy sets on universal set U with membership function μAx=rAxeiωAx and μBx=rBxeiωBx, and then,(i)
μA∪Bx=rA∪BxeiωA∪Bx=maxrAx,rBxeimaxωAx,ωBx(ii)
μA¯x=rA¯xeiωA¯x=1−rAxei2π−ωAx(iii)
μA∪Bx=rA∪BxeiωA∪Bx=minrAx,rBxeiminωAx,ωBx(iv)
μA∘Bx=rA∘BxeiωA∘Bx=rAx.rBxei2πωAx/2π.ωBx/2π,complex fuzzy product(v)
μA+^B=rA+^BxeiωA+^B=rAx+rBx−rAxrBxei2πωAx/2π+ωAx/2π−ωAx/2π.ωAx/2π. ,complexfuzzy sum.
Zhang et al. [40] defined the distance function as follows:Definition 3.
LetU=x1,x2,⋯,xn and A,B be two complex fuzzy numbers:(6)A=x,μAx:x∈U,B=x,μBx:x∈U,where the membership functions are(7)μAx=rAx.eiωAxandμBx=rBx.eiωBx,where rAxrBx,ωAx,ωBx are real-valued and i=−1. Let dz:U×U⟶R be distance the function such that(8)dzA,B=maxsupx∈UrAx−rBx,12πsupx∈UωAx−ωBx.
The following is a counterexample of distance measure 2:Example 2.
LetA=ei0.2π,B=ei2.3π,C=ei1.7π, and then, by using relation 2, we have(9)dA,B=0.525dA,C=0.375.
So,dA,B>dA,C. Based on Figure 1 (the coordinates of points A, B, and C are obtained using the polar form of complex numbers), we have(10)dA,B>dA,C.
As shown in Figure1, this inequality is not correct [42].
Therefore, we can use the following definition to correct the inaccuracy of relations 2 and 3.Figure 1Definition 4.
LetA be a complex fuzzy number with a membership function μAx=rAx.eiωAx, and then, we have(11)ωArx=ωAxi,ifωAxi≤2π,ωAxiMOD2π,ifωAxi>2π,ωAxiMOD2π is the residue modulo 2π. Thus, the modified distance functions 2 and 3 for complex fuzzy sets are as follows:(12)dA,B=12∑i=1nrAxi−rBxi+12π∑i=1nωArxi−ωBrxi,(13)dzA,B=maxsupx∈UrAx−rBx,12πsupx∈UωArx−ωBrx.
Also, the normalized form 5 is as follows:(14)dnA,B=12n∑i=1nrAxi−rBxi+12π∑i=1nωArxi−ωBrxi.
Then, the function of the normalized Euclidean distance is(15)dnEA,B=12n∑i=1nrAxi−rBxi2+14π2ωArxi−ωBrxi21/2,so, we have(16)0≤dnA,B≤1,0≤dzA,B≤1,0≤dnEA,B≤1.


### 3.1. Distances for Complex Fuzzy Sets

LetU=x1,x2,…,xn be a universe of discourse, and the complex fuzzy set A may be represented as the set of ordered pairs:(17)A=x,ÎA1/4x|x∈U,where the membership function ÎA1/4x is of the form rAxÂ·eiωAx,i=−1, and the amplitude term rAx and the phase term ωAx are both real-valued. We know that the complex exponential function eiωAx is a periodic function with periodicity 2π. Therefore, we consider ωAx∈0,2π; otherwise, we take the residue modulo 2π into account.LetA,B be two complex fuzzy sets in U and let p be a parameter satisfying 1≤p≤∞. The Minkowski distances [43]:(18)dpA,B=12∑i=1nrAxi−rBxip+1πpminωAxi−ωBxi,2π−ωAxi−ωBxip1/p.The normalized Minkowski distances [43]:(19)lpA,B=12∑i=1nrAxi−rBxip+1πpminωAxi−ωBxi,2π−ωAxi−ωBxip1/p.Ifp=1, then d is called the Hamming distance of complex fuzzy sets [43]:(20)d1A,B=12∑i=1nrAxi−rBxi+1πminωAxi−ωBxi,2π−ωAxi−ωBxi,and the normalized Hamming distance of complex fuzzy sets is [43](21)l1A,B=12n∑i=1nrAxi−rBxi+1πminωAxi−ωBxi,2π−ωAxi−ωBxi.Ifp=2, then d is termed the Euclidean distance of complex fuzzy sets:(22)d2A,B=12∑i=1nrAxi−rBxi2+1π2minωAxi−ωBxi,2π−ωAxi−ωBxi21/2,and the normalized Euclidean distance of complex fuzzy sets is [43](23)l2A,B=12n∑i=1nrAxi−rBxi2+1π2minωAxi−ωBxi,2π−ωAxi−ωBxi21/2.


### 3.2. Centroid-Index Ranking Method of Complex Fuzzy Numbers

The ranking of fuzzy numbers has been a concern in fuzzy Multiple Attribute Decision-Making (MADM) since its inception. More than 20 fuzzy ranking indices have been proposed since 1976. Various techniques are applied to compare the fuzzy numbers. The fuzzy number ranking, including complex fuzzy numbers to choose the best option from different options, is completely based on ranking or comparing the data. Various fuzzy ranking methods have been proposed ranging from one to several fuzzy number properties. In ranking fuzzy numbers with a center point index, the geometric center of a fuzzy number is found. Each geometric center has two componentsx˜,y˜, where x˜ is on the horizontal axis, and y˜ is on the vertical coordinate axis.Ranking fuzzy numbers are considered either by valuesx˜ alone or with the values x˜ and y˜. Several methods have been proposed in this regard, including Yager’s method to rank fuzzy numbers based onx˜. Yager [44] proposed a centroid-index ranking method to calculate the value x˜ for a fuzzy number A as follows:(24)x˜=∫01wxfAxdx∫01fAxdx,where wx is a weighting function measuring the importance of the value x, and fA denotes the membership function of the fuzzy number A and fA:X⟶0,1. Murakami et al. (1983) proposed a centroid-index ranking method for calculating the center of gravity COG point x⋇,y⋇ for each fuzzy number. The larger the value of x⋇, the better the ranking of fuzzy numbers. We proposed a centroid-index ranking method to calculate the centroid point of complex fuzzy numbers. Since any complex fuzzy number contains a polar representation, in polar coordinates R and θ, respectively, the directional distance is used from the pole and the angle between this half-line and the polar axis. Therefore, we consider ranking based on both components. In polar coordinates, the relationship between Cartesian (x, y) and polar R,θ coordinates is as follows:(25)x=Rcosθ,y=Rsinθ.Consider a segment with the radius ofR and polar angle of θ, given that the center of mass is located 2/3R from the pole, as depicted below. The area of a polar sector with a radius R and the polar angle θ equals A=1/2R2θ. Thus, the element area is equal to dA=1/2R2dθ. In the triangle, the centroid is located 2/3 of the distance from the vertex. Hence, the centroid of this segment is at 2/3 of the distance from the pole. Considering the coordinates of the center of mass xc,yc, we can calculate the centroid of this element (Figure 2):(26)xc=23Rcosθ,yc=23Rsinθ.Figure 2As a result, the center of mass of the whole sector with radiusR and angle θ is(27)x˜=∫0θxdA∫0θdA=∫0θ2/3Rcosθ1/2R2dθ∫0θ1/2R2dθ,=1/3R3∫0θcosθdθ1/2R2∫0θdθ=1/3R3sinθ1/2R2θ=2/3Rsinθθ.Also,(28)y˜=∫0θydA∫0θdA,=∫0θ2/3Rsinθ1/2R2dθ∫0θ1/2R2dθ,=1/3R3∫0θsinθdθ1/2R2∫0θdθ,=1/3R3−cosθ0θ1/2R2θ=23R1−cosθθ.So,(29)x˜=23Rsinθθ,y˜=23R1−cosθθ.To rank complex fuzzy numbers, we can use the distance function between the center pointsx˜,y˜ and the origin as shown in Table 1:(30)DA=x˜2+y˜2.Table 1Complex numberCentroid pointx˜i,y˜iDistanceDA=x˜2+y˜2A1−0.0623,0.19190.2018A20,00A3−0.1009,0.07330.1247A40,0.21220.2122LetA1,A2,⋯,An be complex numbers then for every Ai,Aj, and then, we have(i)
IfDAi<DAj,1≤i≤n,1≤j≤n then Ai<Aj(ii)
IfDAi=DAj,1≤i≤n,1≤j≤n then Ai=Aj(iii)
IfDAi>DAj,1≤i≤n,1≤j≤n then Ai>Aj.Example 3.
LetA1=0.6ei1.2π/−1,A2=1.0ei2π/0,A3=0.8ei1.6π/1,A4=0.5eiπ/2, and then


### 3.3. Linguistic Variable of Complex Fuzzy Sets

An original element in human knowledge representation is the linguistic variables, which is important when using indicators to measure variables and get numbers. Besides, when we asked human experts to describe these variables, they gave us words such as good, slightly, more, or less to describe the variables.Definition 5.
LetA be a complex fuzzy set in U, and then, very A, very-very A, indeed A, a little A, slightly A, more or less A, and extremely A are defined as a complex fuzzy set in U with membership functions [41]:(i)
veryA=x,μveryAx,μveryAx=rAx2ei2ωAx(ii)
very−veryA=x,μvery−veryAx,μvery−veryAx=rAx4ei4ωAx(iii)
indeedA=x,μindeedAx where(31)μindeedAx=2μAx2eiωAx,0≤μAx≤0.5,1−21−μAx2eiωAx,0.5≤μAx≤1,(iv)
alittleA=x,μalittleAx,μalittleAx=rAx1.3ei1.3ωAx(v)
slightlyA=x,μslightlyAx,μslightlyAx=rAx1.7ei1.7ωAx(vi)
extremelyA=x,μextremlyAx,μextremlyAx=rAx3ei3ωAx(vii)
moreorlessA=x,μmoreorlessAx,μmoreorlessAx=rAx1/2ei1/2ωAx
By representing the relative performance using the expressed complex fuzzy numbers, the fuzzy quantity set will be consideredWorst,VL,L,M,H,VH,Best. The complex fuzzy rankings for the language variables are presented in Table 2.Table 2
The complex fuzzy rankings for linguistic variables.
Fuzzy numbersOption evaluationWeightsμvery−veryVery−VeryV−VWorstWμextremlyextremlyextverylowVLμveryVeryVlowLμindeedindeedindmediumMμslightlyslightlyShighHμalittlealittlealitveryhighVHμmoreorlessmoreorlessmolBestB


### 3.4. CP-TOPSIS Method

We proposed a systematic method for ranking complex fuzzy numbers based on TOPSIS in this section after briefly discussing the underlying ideal of TOPSIS. This is termed the developed CP-TOPSIS method. Geometrically speaking, the rank of the option selected as the best from a set of multiple alternatives should have the lowest distance from the ideal solution and the greatest distance from the negative ideal solution, according to the TOPSIS algorithm’s fundamental principle. The TOPSIS method evaluates the following decision matrix, which refers tom alternatives based on n criteria (Table 3).(i)
Let there beK decision-makers D1,D2,⋯,DK responsible for evaluating m alternatives Ai,i=1,2,…,m under n qualitative criteria Cj,j=1,2,…,n.(a)
A1,A2,⋯,Am , m possible alternatives for decision-makers.(b)
C1,C2,⋯,Cn , n criteria for measuring the effectiveness of options.(c)
Letzijk,i=1,2,⋯m,j=1,2,…,n1,k=1,2,⋯,K , be the rating assigned to alternative Ai under criterion Cj by decision-maker Dk.To show the importance of each criterion, we used the weight vectorsWk=w1k,w2k,⋯,wnk assigned by the decision-maker Dk to criterion Cj such that(32)w1k+w2k+⋯+wnk=1.Now, we calculate the real and imaginary parts of these components.(ii)
LetZijk=xijk,yijk,i=1,⋯,m,j=1,⋯,n,k=1,⋯,K be a rank assigned to Ai under Cj by decision-maker Dk. Linguistic variables and their corresponding complex fuzzy numbers are given in Definition 5. The overall rating of Zij=xij,yij is as follows:(33)Zij=1K∘Zij1+^Zij2+^⋯+^ZijK,where(34)xij=∑k=1Kxijkk,yij=∑k=1Kyijkk.(iii)
Letwjkujk,vjk,j=1…,k,k=1,…,K be the important weights assigned by the decision-maker Dk to criterion Cj. The importance weights are quantified by linguistic values represented by complex fuzzy numbers, as stated in Definition 5. The averaged weight of each criterion wj=uj,vj can be evaluated as(35)wj=1K∘wj1+^wj2+^⋯+^wjk,whereuj=∑k=1Kujk/k,vj=∑k=1Kvjk/k,j=1,⋯n.Moreover, we convertedwjk to its polar state.(iv)
We normalized the decision matrix by the following equations. LetZij=xij,yij be the decision matrix components ordered in pairs. We obtained the maximum and minimum of each column for each component and normalized them using the following equations. The components x˜ij and y˜ij are obtained as follows:(36)x˜ij=xij−minixijmaxixij−minixij,x˜ij=xij−minixijmaxixij−minixij,y˜ij=yij−miniyijmaxiyij−miniyij.Thus, the normalized components areZ˜ij=x˜ij,y˜ij. Then, the complex form of the normalized ranking matrix was calculated. The weighted normalized decision matrix was denoted as follows (by Definition 2):(37)Q˜ij=Z˜ij∘wj,i=1,2,…,m,j=1,2,…,n.whereQij s are weighted normalized values.(v)
Here, in the centroid method proposed by 4, the complex fuzzy weighted ratings of alternatives were defuzzified because formulas of the defuzzification procedure could improve the execution of decision-making. LetQij be the complex fuzzified value of Q˜ij with the real and imaginary components pij and qij by the centroid using equations (29) and (30):(38)Qij=pij,qij,pij=23Rsinθθ,qij=23R1−cosθθ.Now, we display a complex fuzzy form of a matrix to calculate the values ofQj+,Qj−.It should be noted that, in each step, we used the following equations to convert the polar coordinates of complex numbers to a regular pair of that numbers:When(39)z=reiθthenx=rcosθ,y=rsinθ.But, whenz=x,y, then(40)θ=argz=arctanyx+πifx>0,y≥0,arctanyx+πifx<0,y≥0,arctanyx−πifx<0,y<0,π2ifx=0,y>0,−π2ifx=0,y<0,∞ifx=0,y=0,r=z=x2+y2.Nowz=reiθ.(vi)
The positive ideal solution (PIS),A+, and negative ideal solution (NIS), A−, can be obtained as follows. To obtain these answers, we used the polar area of the factor Qij. For this purpose, suppose Qij=rijeiwij, and then, the polar area is obtained from the following formula:(41)AQij=∫0ωij12r2dθ.Ifωij is negative, we change the boundary of integral. The positive ideal solution is as follows:(42)A+=Q1+,Q2+,⋯,Qn+=maxiAQij|j∈I,miniAQij|j∈J,The negative ideal solution is as follows:(43)A−=Q1−,Q2−,⋯,Qn−=miniAQij|j∈I,maxiAQij|j∈J,(vii)
The distance of each alternative from (PIS) is as follows:(44)di+=∑j=1nDQij−Qj+2,i=1,2,…,m.The distance of each alternative from (NIS) is as follows:(45)di−=∑j=1nDQij−Qj−2,i=1,2,…,m.(viii)
The ranking order of all alternatives from the highest-ranking index to the lowest is determined by calculating the closeness coefficients of alternatives by the following equation. However, as the optionAi gets closer to A+ and the further away from A−, the R˜i value tends to be 1.(46)Ri=di−di++di−,i=1,2,…,m.Table 3C1C2CnA1z11kz12k⋯z12kA2z21kz22k⋯z2nk⋮⋯⋯⋯⋱Amzm1kzm2k⋯zmnkNow, all the steps of this new ranking method are shown in Figure3.Figure 3
The flow chart with steps of the proposed algorithm.Example 4.
LetU=1,2,3 and M, N be two complex fuzzy sets:(47)M=0.6ei0.2π1+0.4ei0.5π2+0.5ei1.7π3,N=0.45ei0.5π4+0.66ei1.3π5+0.37ei0.3π6.
The set of decision makers isD=D1,D2. The following is a table of alternative ranking by Tables 4–16 decision-maker D1:
The following is a table of alternative ranking by decision-makerD2:
The next table is the weighting of criteria by decision makers:
The following table shows the complex fuzzy values assigned to the alternatives forD1:
The following table shows the complex fuzzy values assigned to the alternatives forD2:
Now, we get the real and imaginary parts of these elements:
And, we show also the values ofD2:
Now, we get the weighted values of the decision makers:
The ordered form of weighted values is as follows:
According to 21, the complex form of weighted average is equal to the following:
Now, by using 20, we get the ranking matrix as follows:
Now, normalize the ranking matrix using relations 22 and 23:
To normalize aggregated matrix, we require the polar form of values because it is difficult to normalize complex forms directly. Thus, we use the polar form and then normalize values. In order to use 24, we obtain the complex form of the normalized ranking matrix:
Now, we return to the complex form because we should build the weighed matrix since we can easily multiply complex numbers. Note that multiplication of the complex numbers was defined in previous equations. We use the complex form of the weights and construct a weighted matrix using complex multiplication. The weighted matrix in Table17 is in complex form by using 24, and we obtain the normalized weighted matrix:
We determine positive and negative ideals. For this purpose, we should perform a defuzzification process because it helps determine the maximum and minimum of each column. Due to the difficulty of determining the maximum and minimum values of complex numbers, we use two formulas. First, data are written as ordered pairs, as presented in Table18. This is the first defuzzification step, which is not the main step. We do the phase defuzzification process using centroid method 25:
In the second defuzzification step, we try to calculate the polar area of factors. Therefore, we should transform Table18 into the complex form as presented in Table 19. Now, we display complex fuzzy form of matrix in order to calculate the values of Qj+,Qj−:
Table19 is used to determine the polar area of the factors after the matrices had been converted to polar form. Accordingly, Table 20 is obtained, which provides a criterion to select positive and negative ideals in complex forms. The following areas are obtained in micro:
Now,PIS is as follows:(48)A+=16.11∗10−6,0.29514,0.001280,so, we have(49)Q1+=0.01849e−i0.03π,Q2+=0.2317ei0.35π,Q3+=0.0655ei0.19π.
NIS is as follows:(50)A−=0.0,0.0,0.0,so, we have(51)Q1−=0.01093ei0.0π,Q2−=0.0ei0.0π,Q3−=0.02566ei0.0π.
The distance between each alternative and positive and negative ideals should be calculated. Since the distance formula is based on the complex form of numbers, and ideals are complex as shown in Table21, we use the table of complex data and calculate the differences, as given in Tables 22 and 23. The values of di+ are as follows:
The values ofdi− are as follows:
We get the closeness coefficients of each alternative according to relation 31:
Ultimately, we use the closeness coefficient formula and rank the alternatives, as presented in Table24. So, the rating alternatives are as follows:Table 4Deceision makerD1C1C2C3A1valitextA2alitextmolA3sindvA4molvsA5v−vextalitA6molalitsTable 5Deceision  makerD2C1C2C3A1v−vindextA2vsmolA3indalitextA4molv−valitA5svv−vA6indmolvTable 6D1D2C1LWC2VLBC3HMTable 7C1C2C3A10.36ei0.4π0.51ei0.26π0.21ei0.6πA20.30ei0.65π0.06ei1.5π0.63ei0.25πA30.3ei2.89π0.5ei1.7π0.25ei3.4πA40.67ei0.25π0.2eiπ0.25ei0.85πA50.18ei5.2π0.28ei3.9π0.58ei1.69πA60.6eiπ0.27ei2.6π0.18ei0.51πTable 8C1C2C3A10.12ei0.8π0.68ei0.2π0.21ei0.6πA20.16eiπ0.21ei0.85π0.63ei0.25πA30.5ei1.7π0.4ei2.21π0.12ei5.1πA40.67ei0.25π0.04ei2π0.35ei0.65πA50.49ei2.21π0.43ei2.6π0.18ei5.2πA60.27ei2π0.6eiπ0.13ei0.6πTable 9
Values ofD1.
C1C2C3A10.1112,0.34230.3491,0.3717−0.0648,0.1997A2−0.1361,0.26730.0,−0.060.4454,0.4454A3−0.2822,−0.10160.2938,−0.4045−0.0772,−0.2377A40.4737,0.4737−0.2,0.0−0.2022,0.1469A5−0.1456,−0.10580.2662,−0.08650.3260,−0.4797A6−0.6,0.0−0.0834,0.2567−0.0056,−0.1799Table 10
Values ofD2.
C1C2C3A1−0.0970,0.0705−0.3214,0.5065−0.0648,0.1997A2−0.16,0.0−0.1871,0.09530.4454,0.4454A30.2938,−0.40450.3160,0.2451−0.1141,−0.0370A40.4737,0.47370.04,0.0−0.1588,0.3118A50.3871,0.3003−0.1328,0.4089−0.1456,−0.1058A60.27,0.0−0.6,0.0−0.0401,0.1236Table 11D1D2C1L=0.0257i3.39πW=0.0006ei6.78πC2VL=0.0041ei5.08πB=0.4007ei0.84πC3H=0.0446ei2.88πM=0.0551ei1.69πTable 12D1D2C10.00870,−0.0241−0.00046,0.00038C20.0039,−0.0010−0.3511,0.1930C3−0.0414,0.01640.0316,−0.0450Table 13Complex form of weighted averageC10.0262e−i1.32πC20.4031ei3.78πC30.0972ei2.13πTable 14
Ranking matrix.
Aggregated ranking matrixA10.43ei1.04π0.84ei0.43π0.37e1.02πA20.41ei1.3π0.25ei1.71π0.86ei1.06πA30.65ei2.33π0.7ei2.03π0.34e−i0.17πA40.89ei0.46π0.23ei2π0.51ei1.22πA50.58ei1.66π0.58ei1.43π0.65ei0.7πA60.7ei2π0.70ei2.3π0.28ei0.95πTable 15
The aggregated matrix ofD1 and D2 experts’ opinions.
A10.0,0.32660.3761,1.00000.4180,0.3547A20.1648,0.12690.3397,0.26590.0,0.1926A30.6722,0.76741.0000,0.45591.0,0.1786A40.4776,1.00000.4330,0.40840.3972,0.0A50.6266,0.00.0,0.00.4068,1.0A61.000,0.36530.6533,0.81710.4996,0.4334Table 16complex form of normalized ranking matrixA10.3266ei0.5π1.0683ei0.38π0.5482ei0.22πA20.2079ei0.2π0.4313ei0.21π0.1926ei0.5πA31.0201ei0.27π1.099ei0.13π1.0158ei0.05πA41.1081ei0.35π0.5952ei0.24π0.3972ei0.0πA50.6266ei0.0π0.0ei0.0π1.0795ei0.37πA61.0646ei0.11π1.0461ei0.28π0.6613ei0.22πTable 17Q˜ij=z˜ij∘wjnormalized weighted matrixA10.00855e−i0.33π0.4306ei0.71π0.0531ei0.23πA20.0054e−i0.13π0.1738ei0.39π0.0186ei0.53πA30.0267e−i0.17π0.4430ei0.24π0.0985ei0.05πA40.0290e−i0.23π0.2399ei0.45π0.0385ei0.0πA50.0164ei0.0π0.0ei0.0π0.1047ei0.39πA60.0278e−i0.07π0.4216ei0.52π0.0641ei0.23πTable 18Qijdeffuzification matrixA10.00473,−0.002680.10296,0.207570.03239,0.01224A20.00352,−0.000720.08897,0.062530.00741,0.00814A30.01696,−0.004640.26813,0.106160.06539,0.00514A40.01769,−0.006680.11173,0.095430.02566,0.0A50.01093,0.00.0,0.00.05360,0.03767A60.01838,−0.002020.17171,0.182850.03911,0.01477Table 19Qijpolar matrixA10.00543e−i0.16π0.2317ei0.35π0.03462ei0.11πA20.00359e−i0.06π0.10874ei0.19π0.011ei0.26πA30.01758e−i0.08π0.2883ei0.11π0.0655ei0.02πA40.0189−i0.011π0.14693ei0.22π0.02566ei0.0πA50.01093ei0.0π0.0ei0.0π0.0655ei0.19πA60.01849e−i0.03π0.2508ei0.25π0.0418ei0.11πTable 20AQijPolarAreaA17.410×10−60.0295140.000207A21.214×10−60.0035290.0000494A338.83×10−60.0143690.000134A461.72×10−60.007460.0A50.0×10−60.00.00128A616.11×10−60.0247010.000301Table 21d1+d2+d3+d4+d5+d6+0.0364160.1575610.1938400.1194960.2343980.078236Table 22d1−d2−d3−d4−d5−d6−0.2322210.1108170.2912490.1472440.0469000.251748Table 23R1R2R3R4R5R60.8644410.4129130.6004030.5520130.1667270.762909Table 24A1A2A3A4A5A6Ri0.8644410.4129130.6004030.5520130.1667270.762909Ranking153462


## 3.1. Distances for Complex Fuzzy Sets

LetU=x1,x2,…,xn be a universe of discourse, and the complex fuzzy set A may be represented as the set of ordered pairs:(17)A=x,ÎA1/4x|x∈U,where the membership function ÎA1/4x is of the form rAxÂ·eiωAx,i=−1, and the amplitude term rAx and the phase term ωAx are both real-valued. We know that the complex exponential function eiωAx is a periodic function with periodicity 2π. Therefore, we consider ωAx∈0,2π; otherwise, we take the residue modulo 2π into account.LetA,B be two complex fuzzy sets in U and let p be a parameter satisfying 1≤p≤∞. The Minkowski distances [43]:(18)dpA,B=12∑i=1nrAxi−rBxip+1πpminωAxi−ωBxi,2π−ωAxi−ωBxip1/p.The normalized Minkowski distances [43]:(19)lpA,B=12∑i=1nrAxi−rBxip+1πpminωAxi−ωBxi,2π−ωAxi−ωBxip1/p.Ifp=1, then d is called the Hamming distance of complex fuzzy sets [43]:(20)d1A,B=12∑i=1nrAxi−rBxi+1πminωAxi−ωBxi,2π−ωAxi−ωBxi,and the normalized Hamming distance of complex fuzzy sets is [43](21)l1A,B=12n∑i=1nrAxi−rBxi+1πminωAxi−ωBxi,2π−ωAxi−ωBxi.Ifp=2, then d is termed the Euclidean distance of complex fuzzy sets:(22)d2A,B=12∑i=1nrAxi−rBxi2+1π2minωAxi−ωBxi,2π−ωAxi−ωBxi21/2,and the normalized Euclidean distance of complex fuzzy sets is [43](23)l2A,B=12n∑i=1nrAxi−rBxi2+1π2minωAxi−ωBxi,2π−ωAxi−ωBxi21/2.


## 3.2. Centroid-Index Ranking Method of Complex Fuzzy Numbers

The ranking of fuzzy numbers has been a concern in fuzzy Multiple Attribute Decision-Making (MADM) since its inception. More than 20 fuzzy ranking indices have been proposed since 1976. Various techniques are applied to compare the fuzzy numbers. The fuzzy number ranking, including complex fuzzy numbers to choose the best option from different options, is completely based on ranking or comparing the data. Various fuzzy ranking methods have been proposed ranging from one to several fuzzy number properties. In ranking fuzzy numbers with a center point index, the geometric center of a fuzzy number is found. Each geometric center has two componentsx˜,y˜, where x˜ is on the horizontal axis, and y˜ is on the vertical coordinate axis.Ranking fuzzy numbers are considered either by valuesx˜ alone or with the values x˜ and y˜. Several methods have been proposed in this regard, including Yager’s method to rank fuzzy numbers based onx˜. Yager [44] proposed a centroid-index ranking method to calculate the value x˜ for a fuzzy number A as follows:(24)x˜=∫01wxfAxdx∫01fAxdx,where wx is a weighting function measuring the importance of the value x, and fA denotes the membership function of the fuzzy number A and fA:X⟶0,1. Murakami et al. (1983) proposed a centroid-index ranking method for calculating the center of gravity COG point x⋇,y⋇ for each fuzzy number. The larger the value of x⋇, the better the ranking of fuzzy numbers. We proposed a centroid-index ranking method to calculate the centroid point of complex fuzzy numbers. Since any complex fuzzy number contains a polar representation, in polar coordinates R and θ, respectively, the directional distance is used from the pole and the angle between this half-line and the polar axis. Therefore, we consider ranking based on both components. In polar coordinates, the relationship between Cartesian (x, y) and polar R,θ coordinates is as follows:(25)x=Rcosθ,y=Rsinθ.Consider a segment with the radius ofR and polar angle of θ, given that the center of mass is located 2/3R from the pole, as depicted below. The area of a polar sector with a radius R and the polar angle θ equals A=1/2R2θ. Thus, the element area is equal to dA=1/2R2dθ. In the triangle, the centroid is located 2/3 of the distance from the vertex. Hence, the centroid of this segment is at 2/3 of the distance from the pole. Considering the coordinates of the center of mass xc,yc, we can calculate the centroid of this element (Figure 2):(26)xc=23Rcosθ,yc=23Rsinθ.Figure 2As a result, the center of mass of the whole sector with radiusR and angle θ is(27)x˜=∫0θxdA∫0θdA=∫0θ2/3Rcosθ1/2R2dθ∫0θ1/2R2dθ,=1/3R3∫0θcosθdθ1/2R2∫0θdθ=1/3R3sinθ1/2R2θ=2/3Rsinθθ.Also,(28)y˜=∫0θydA∫0θdA,=∫0θ2/3Rsinθ1/2R2dθ∫0θ1/2R2dθ,=1/3R3∫0θsinθdθ1/2R2∫0θdθ,=1/3R3−cosθ0θ1/2R2θ=23R1−cosθθ.So,(29)x˜=23Rsinθθ,y˜=23R1−cosθθ.To rank complex fuzzy numbers, we can use the distance function between the center pointsx˜,y˜ and the origin as shown in Table 1:(30)DA=x˜2+y˜2.Table 1Complex numberCentroid pointx˜i,y˜iDistanceDA=x˜2+y˜2A1−0.0623,0.19190.2018A20,00A3−0.1009,0.07330.1247A40,0.21220.2122LetA1,A2,⋯,An be complex numbers then for every Ai,Aj, and then, we have(i)
IfDAi<DAj,1≤i≤n,1≤j≤n then Ai<Aj(ii)
IfDAi=DAj,1≤i≤n,1≤j≤n then Ai=Aj(iii)
IfDAi>DAj,1≤i≤n,1≤j≤n then Ai>Aj.Example 3.
LetA1=0.6ei1.2π/−1,A2=1.0ei2π/0,A3=0.8ei1.6π/1,A4=0.5eiπ/2, and then


## 3.3. Linguistic Variable of Complex Fuzzy Sets

An original element in human knowledge representation is the linguistic variables, which is important when using indicators to measure variables and get numbers. Besides, when we asked human experts to describe these variables, they gave us words such as good, slightly, more, or less to describe the variables.Definition 5.
LetA be a complex fuzzy set in U, and then, very A, very-very A, indeed A, a little A, slightly A, more or less A, and extremely A are defined as a complex fuzzy set in U with membership functions [41]:(i)
veryA=x,μveryAx,μveryAx=rAx2ei2ωAx(ii)
very−veryA=x,μvery−veryAx,μvery−veryAx=rAx4ei4ωAx(iii)
indeedA=x,μindeedAx where(31)μindeedAx=2μAx2eiωAx,0≤μAx≤0.5,1−21−μAx2eiωAx,0.5≤μAx≤1,(iv)
alittleA=x,μalittleAx,μalittleAx=rAx1.3ei1.3ωAx(v)
slightlyA=x,μslightlyAx,μslightlyAx=rAx1.7ei1.7ωAx(vi)
extremelyA=x,μextremlyAx,μextremlyAx=rAx3ei3ωAx(vii)
moreorlessA=x,μmoreorlessAx,μmoreorlessAx=rAx1/2ei1/2ωAx
By representing the relative performance using the expressed complex fuzzy numbers, the fuzzy quantity set will be consideredWorst,VL,L,M,H,VH,Best. The complex fuzzy rankings for the language variables are presented in Table 2.Table 2
The complex fuzzy rankings for linguistic variables.
Fuzzy numbersOption evaluationWeightsμvery−veryVery−VeryV−VWorstWμextremlyextremlyextverylowVLμveryVeryVlowLμindeedindeedindmediumMμslightlyslightlyShighHμalittlealittlealitveryhighVHμmoreorlessmoreorlessmolBestB


## 3.4. CP-TOPSIS Method

We proposed a systematic method for ranking complex fuzzy numbers based on TOPSIS in this section after briefly discussing the underlying ideal of TOPSIS. This is termed the developed CP-TOPSIS method. Geometrically speaking, the rank of the option selected as the best from a set of multiple alternatives should have the lowest distance from the ideal solution and the greatest distance from the negative ideal solution, according to the TOPSIS algorithm’s fundamental principle. The TOPSIS method evaluates the following decision matrix, which refers tom alternatives based on n criteria (Table 3).(i)
Let there beK decision-makers D1,D2,⋯,DK responsible for evaluating m alternatives Ai,i=1,2,…,m under n qualitative criteria Cj,j=1,2,…,n.(a)
A1,A2,⋯,Am , m possible alternatives for decision-makers.(b)
C1,C2,⋯,Cn , n criteria for measuring the effectiveness of options.(c)
Letzijk,i=1,2,⋯m,j=1,2,…,n1,k=1,2,⋯,K , be the rating assigned to alternative Ai under criterion Cj by decision-maker Dk.To show the importance of each criterion, we used the weight vectorsWk=w1k,w2k,⋯,wnk assigned by the decision-maker Dk to criterion Cj such that(32)w1k+w2k+⋯+wnk=1.Now, we calculate the real and imaginary parts of these components.(ii)
LetZijk=xijk,yijk,i=1,⋯,m,j=1,⋯,n,k=1,⋯,K be a rank assigned to Ai under Cj by decision-maker Dk. Linguistic variables and their corresponding complex fuzzy numbers are given in Definition 5. The overall rating of Zij=xij,yij is as follows:(33)Zij=1K∘Zij1+^Zij2+^⋯+^ZijK,where(34)xij=∑k=1Kxijkk,yij=∑k=1Kyijkk.(iii)
Letwjkujk,vjk,j=1…,k,k=1,…,K be the important weights assigned by the decision-maker Dk to criterion Cj. The importance weights are quantified by linguistic values represented by complex fuzzy numbers, as stated in Definition 5. The averaged weight of each criterion wj=uj,vj can be evaluated as(35)wj=1K∘wj1+^wj2+^⋯+^wjk,whereuj=∑k=1Kujk/k,vj=∑k=1Kvjk/k,j=1,⋯n.Moreover, we convertedwjk to its polar state.(iv)
We normalized the decision matrix by the following equations. LetZij=xij,yij be the decision matrix components ordered in pairs. We obtained the maximum and minimum of each column for each component and normalized them using the following equations. The components x˜ij and y˜ij are obtained as follows:(36)x˜ij=xij−minixijmaxixij−minixij,x˜ij=xij−minixijmaxixij−minixij,y˜ij=yij−miniyijmaxiyij−miniyij.Thus, the normalized components areZ˜ij=x˜ij,y˜ij. Then, the complex form of the normalized ranking matrix was calculated. The weighted normalized decision matrix was denoted as follows (by Definition 2):(37)Q˜ij=Z˜ij∘wj,i=1,2,…,m,j=1,2,…,n.whereQij s are weighted normalized values.(v)
Here, in the centroid method proposed by 4, the complex fuzzy weighted ratings of alternatives were defuzzified because formulas of the defuzzification procedure could improve the execution of decision-making. LetQij be the complex fuzzified value of Q˜ij with the real and imaginary components pij and qij by the centroid using equations (29) and (30):(38)Qij=pij,qij,pij=23Rsinθθ,qij=23R1−cosθθ.Now, we display a complex fuzzy form of a matrix to calculate the values ofQj+,Qj−.It should be noted that, in each step, we used the following equations to convert the polar coordinates of complex numbers to a regular pair of that numbers:When(39)z=reiθthenx=rcosθ,y=rsinθ.But, whenz=x,y, then(40)θ=argz=arctanyx+πifx>0,y≥0,arctanyx+πifx<0,y≥0,arctanyx−πifx<0,y<0,π2ifx=0,y>0,−π2ifx=0,y<0,∞ifx=0,y=0,r=z=x2+y2.Nowz=reiθ.(vi)
The positive ideal solution (PIS),A+, and negative ideal solution (NIS), A−, can be obtained as follows. To obtain these answers, we used the polar area of the factor Qij. For this purpose, suppose Qij=rijeiwij, and then, the polar area is obtained from the following formula:(41)AQij=∫0ωij12r2dθ.Ifωij is negative, we change the boundary of integral. The positive ideal solution is as follows:(42)A+=Q1+,Q2+,⋯,Qn+=maxiAQij|j∈I,miniAQij|j∈J,The negative ideal solution is as follows:(43)A−=Q1−,Q2−,⋯,Qn−=miniAQij|j∈I,maxiAQij|j∈J,(vii)
The distance of each alternative from (PIS) is as follows:(44)di+=∑j=1nDQij−Qj+2,i=1,2,…,m.The distance of each alternative from (NIS) is as follows:(45)di−=∑j=1nDQij−Qj−2,i=1,2,…,m.(viii)
The ranking order of all alternatives from the highest-ranking index to the lowest is determined by calculating the closeness coefficients of alternatives by the following equation. However, as the optionAi gets closer to A+ and the further away from A−, the R˜i value tends to be 1.(46)Ri=di−di++di−,i=1,2,…,m.Table 3C1C2CnA1z11kz12k⋯z12kA2z21kz22k⋯z2nk⋮⋯⋯⋯⋱Amzm1kzm2k⋯zmnkNow, all the steps of this new ranking method are shown in Figure3.Figure 3
The flow chart with steps of the proposed algorithm.Example 4.
LetU=1,2,3 and M, N be two complex fuzzy sets:(47)M=0.6ei0.2π1+0.4ei0.5π2+0.5ei1.7π3,N=0.45ei0.5π4+0.66ei1.3π5+0.37ei0.3π6.
The set of decision makers isD=D1,D2. The following is a table of alternative ranking by Tables 4–16 decision-maker D1:
The following is a table of alternative ranking by decision-makerD2:
The next table is the weighting of criteria by decision makers:
The following table shows the complex fuzzy values assigned to the alternatives forD1:
The following table shows the complex fuzzy values assigned to the alternatives forD2:
Now, we get the real and imaginary parts of these elements:
And, we show also the values ofD2:
Now, we get the weighted values of the decision makers:
The ordered form of weighted values is as follows:
According to 21, the complex form of weighted average is equal to the following:
Now, by using 20, we get the ranking matrix as follows:
Now, normalize the ranking matrix using relations 22 and 23:
To normalize aggregated matrix, we require the polar form of values because it is difficult to normalize complex forms directly. Thus, we use the polar form and then normalize values. In order to use 24, we obtain the complex form of the normalized ranking matrix:
Now, we return to the complex form because we should build the weighed matrix since we can easily multiply complex numbers. Note that multiplication of the complex numbers was defined in previous equations. We use the complex form of the weights and construct a weighted matrix using complex multiplication. The weighted matrix in Table17 is in complex form by using 24, and we obtain the normalized weighted matrix:
We determine positive and negative ideals. For this purpose, we should perform a defuzzification process because it helps determine the maximum and minimum of each column. Due to the difficulty of determining the maximum and minimum values of complex numbers, we use two formulas. First, data are written as ordered pairs, as presented in Table18. This is the first defuzzification step, which is not the main step. We do the phase defuzzification process using centroid method 25:
In the second defuzzification step, we try to calculate the polar area of factors. Therefore, we should transform Table18 into the complex form as presented in Table 19. Now, we display complex fuzzy form of matrix in order to calculate the values of Qj+,Qj−:
Table19 is used to determine the polar area of the factors after the matrices had been converted to polar form. Accordingly, Table 20 is obtained, which provides a criterion to select positive and negative ideals in complex forms. The following areas are obtained in micro:
Now,PIS is as follows:(48)A+=16.11∗10−6,0.29514,0.001280,so, we have(49)Q1+=0.01849e−i0.03π,Q2+=0.2317ei0.35π,Q3+=0.0655ei0.19π.
NIS is as follows:(50)A−=0.0,0.0,0.0,so, we have(51)Q1−=0.01093ei0.0π,Q2−=0.0ei0.0π,Q3−=0.02566ei0.0π.
The distance between each alternative and positive and negative ideals should be calculated. Since the distance formula is based on the complex form of numbers, and ideals are complex as shown in Table21, we use the table of complex data and calculate the differences, as given in Tables 22 and 23. The values of di+ are as follows:
The values ofdi− are as follows:
We get the closeness coefficients of each alternative according to relation 31:
Ultimately, we use the closeness coefficient formula and rank the alternatives, as presented in Table24. So, the rating alternatives are as follows:Table 4Deceision makerD1C1C2C3A1valitextA2alitextmolA3sindvA4molvsA5v−vextalitA6molalitsTable 5Deceision  makerD2C1C2C3A1v−vindextA2vsmolA3indalitextA4molv−valitA5svv−vA6indmolvTable 6D1D2C1LWC2VLBC3HMTable 7C1C2C3A10.36ei0.4π0.51ei0.26π0.21ei0.6πA20.30ei0.65π0.06ei1.5π0.63ei0.25πA30.3ei2.89π0.5ei1.7π0.25ei3.4πA40.67ei0.25π0.2eiπ0.25ei0.85πA50.18ei5.2π0.28ei3.9π0.58ei1.69πA60.6eiπ0.27ei2.6π0.18ei0.51πTable 8C1C2C3A10.12ei0.8π0.68ei0.2π0.21ei0.6πA20.16eiπ0.21ei0.85π0.63ei0.25πA30.5ei1.7π0.4ei2.21π0.12ei5.1πA40.67ei0.25π0.04ei2π0.35ei0.65πA50.49ei2.21π0.43ei2.6π0.18ei5.2πA60.27ei2π0.6eiπ0.13ei0.6πTable 9
Values ofD1.
C1C2C3A10.1112,0.34230.3491,0.3717−0.0648,0.1997A2−0.1361,0.26730.0,−0.060.4454,0.4454A3−0.2822,−0.10160.2938,−0.4045−0.0772,−0.2377A40.4737,0.4737−0.2,0.0−0.2022,0.1469A5−0.1456,−0.10580.2662,−0.08650.3260,−0.4797A6−0.6,0.0−0.0834,0.2567−0.0056,−0.1799Table 10
Values ofD2.
C1C2C3A1−0.0970,0.0705−0.3214,0.5065−0.0648,0.1997A2−0.16,0.0−0.1871,0.09530.4454,0.4454A30.2938,−0.40450.3160,0.2451−0.1141,−0.0370A40.4737,0.47370.04,0.0−0.1588,0.3118A50.3871,0.3003−0.1328,0.4089−0.1456,−0.1058A60.27,0.0−0.6,0.0−0.0401,0.1236Table 11D1D2C1L=0.0257i3.39πW=0.0006ei6.78πC2VL=0.0041ei5.08πB=0.4007ei0.84πC3H=0.0446ei2.88πM=0.0551ei1.69πTable 12D1D2C10.00870,−0.0241−0.00046,0.00038C20.0039,−0.0010−0.3511,0.1930C3−0.0414,0.01640.0316,−0.0450Table 13Complex form of weighted averageC10.0262e−i1.32πC20.4031ei3.78πC30.0972ei2.13πTable 14
Ranking matrix.
Aggregated ranking matrixA10.43ei1.04π0.84ei0.43π0.37e1.02πA20.41ei1.3π0.25ei1.71π0.86ei1.06πA30.65ei2.33π0.7ei2.03π0.34e−i0.17πA40.89ei0.46π0.23ei2π0.51ei1.22πA50.58ei1.66π0.58ei1.43π0.65ei0.7πA60.7ei2π0.70ei2.3π0.28ei0.95πTable 15
The aggregated matrix ofD1 and D2 experts’ opinions.
A10.0,0.32660.3761,1.00000.4180,0.3547A20.1648,0.12690.3397,0.26590.0,0.1926A30.6722,0.76741.0000,0.45591.0,0.1786A40.4776,1.00000.4330,0.40840.3972,0.0A50.6266,0.00.0,0.00.4068,1.0A61.000,0.36530.6533,0.81710.4996,0.4334Table 16complex form of normalized ranking matrixA10.3266ei0.5π1.0683ei0.38π0.5482ei0.22πA20.2079ei0.2π0.4313ei0.21π0.1926ei0.5πA31.0201ei0.27π1.099ei0.13π1.0158ei0.05πA41.1081ei0.35π0.5952ei0.24π0.3972ei0.0πA50.6266ei0.0π0.0ei0.0π1.0795ei0.37πA61.0646ei0.11π1.0461ei0.28π0.6613ei0.22πTable 17Q˜ij=z˜ij∘wjnormalized weighted matrixA10.00855e−i0.33π0.4306ei0.71π0.0531ei0.23πA20.0054e−i0.13π0.1738ei0.39π0.0186ei0.53πA30.0267e−i0.17π0.4430ei0.24π0.0985ei0.05πA40.0290e−i0.23π0.2399ei0.45π0.0385ei0.0πA50.0164ei0.0π0.0ei0.0π0.1047ei0.39πA60.0278e−i0.07π0.4216ei0.52π0.0641ei0.23πTable 18Qijdeffuzification matrixA10.00473,−0.002680.10296,0.207570.03239,0.01224A20.00352,−0.000720.08897,0.062530.00741,0.00814A30.01696,−0.004640.26813,0.106160.06539,0.00514A40.01769,−0.006680.11173,0.095430.02566,0.0A50.01093,0.00.0,0.00.05360,0.03767A60.01838,−0.002020.17171,0.182850.03911,0.01477Table 19Qijpolar matrixA10.00543e−i0.16π0.2317ei0.35π0.03462ei0.11πA20.00359e−i0.06π0.10874ei0.19π0.011ei0.26πA30.01758e−i0.08π0.2883ei0.11π0.0655ei0.02πA40.0189−i0.011π0.14693ei0.22π0.02566ei0.0πA50.01093ei0.0π0.0ei0.0π0.0655ei0.19πA60.01849e−i0.03π0.2508ei0.25π0.0418ei0.11πTable 20AQijPolarAreaA17.410×10−60.0295140.000207A21.214×10−60.0035290.0000494A338.83×10−60.0143690.000134A461.72×10−60.007460.0A50.0×10−60.00.00128A616.11×10−60.0247010.000301Table 21d1+d2+d3+d4+d5+d6+0.0364160.1575610.1938400.1194960.2343980.078236Table 22d1−d2−d3−d4−d5−d6−0.2322210.1108170.2912490.1472440.0469000.251748Table 23R1R2R3R4R5R60.8644410.4129130.6004030.5520130.1667270.762909Table 24A1A2A3A4A5A6Ri0.8644410.4129130.6004030.5520130.1667270.762909Ranking153462


## 4. Results

In this paper, by defining a new method, we presented a new method to prioritize a model of decision-making, in which data are of complex fuzzy numbers, and finally, a rating algorithm was designed based on the central point index and TOPSIS-oriented method. This paper aims to establish a complex fuzzy technique for rating and choosing the best options for the complex fuzzy data that have many uses including analyzing wave functions in quantum mechanics and a lot of physical quantities.


## 5. Managerial Insights

As the uncertainty conditions play a vital role in the decision-making process by managers, the complex fuzzy sets have many utilizations in the industry, management, and financial sector. Given the importance of this topic, this paper will grade complex fuzzy options through CP-TOPSIS developmental method. Recently, the complex fuzzy set contains a range of values, in which its membership function may be obtained. In contrast to the traditional fuzzy membership function, this range is not limited [0, 1] because it will be expanded into the unit circle on the complex page. Therefore, the complex fuzzy set presents a mathematical framework for describing a membership on the set in terms of a complex number. The possible applications, which provide a new concept, include the complex fuzzy display, solar activity (by measuring the number of sunspots), signal process usage, problems in forecasting time series and comparing two national economies, identifying and measuring signals, analysis, wave functions within quantum mechanics, and many physical quantities, through the complex fuzzy relation. Today, there is no device such as home appliances created based on its technical structure without using the complex fuzzy sets. We often use and enjoy the complex fuzzy logic in our routine life. Also, it can be referred to as the motor systems (elevator), automotive industry (design of ABS braking system), designing controllers of home appliances (washing machine and refrigerator), and cameras, and artificial intelligence is one of the most interesting applications in the computer games like special cinematic sights.


## 6. Conclusions and Outlook

Grading the complex fuzzy numbers has a prominent role in management, engineering, and industry. The complex fuzzy numbers are shown by a complex form membership function. Due to the presence of real numbers that can be arranged as linearity, the complex fuzzy numbers may have overlapped each other. Therefore, it is difficult to grade all of them. Currently, a novel CP-TOPSIS grading method is proposed in this study that is related to the complex fuzzy numbers. The proposed fuzzy set is a new approach that can complex the membership functions. It is vital to have a profound understanding of the aspects of complex fuzzy sets in order to make effective use of the possibilities those sets offer. As a result, we allocated several sections for the properties and features of the complex membership function, such as complex subscription, multiplication, and collection. One of the advantages of the new method, in some cases, is the difficulty of identifying exact values of properties, and we considered their values as the complex fuzzy data. Accordingly, we proposed grading data based on the central point index. In this regard, we explained the distance functions for the complex fuzzy numbers based on fuzzy period and domain. Finally, we introduced a developmental method, CT-TOPSIS for rating complex fuzzy data. In this research, CT-TOPSIS is created as complex fuzzy sets, and a method for determining the most desirable choice is proposed. While the fuzzy data is complex, the best and the most effective option is presented between all possible options. The fuzzy decision matrix is calculated using the results` concept. In this method, despite the distance considerations, an option is considered regarding the solution of a fuzzy positive ideal and its distance, far from the solution related to a fuzzy negative ideal. It means that the lower the distance of an option evaluated through the solution of a fuzzy positive ideal is, and the more the distance from the solution of a fuzzy negative ideal is, the better the ranking is. This sort of ranking, which is introduced in this study for the first time, corresponds to the complex fuzzy numbers. We hope that a new way is paved for future works upon the complex fuzzy data investigation due to many applications in the industry, management, and financial sectors.



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*Source: 1477098-2022-10-27.xml*