1. Introduction

1.7. Decision Matrix

A matrix format is the simple way to express an MCDM problem. A decision matrix $\mathbf{A}$ is an $(m\times n)$ matrix in which element $a_{ij}$ represents the performance of alternative $A_i$ when evaluated in terms of decision criterion $C_i$ (for $i=$$1,2,3,\dots ,m$, and $j=1,2,3,\dots ,n).$ It is also assumed that the decision maker determined the relative performance weights of the decision criteria (denoted as $w_j$, for $\left.j=1,2,3,\dots ,n\right)$.

Table 1. A Typical Decision Table.

Table 1. A Typical Decision Table.

There are no clear guidelines for which MCDM method should be used to solve a specific MCDM problem. This is a controversial topic that has been researched in the literature for many decades. It is true that the solution may differ depending on the MCDM method used, especially when the alternatives are very similar [2].

As a result, in this paper, we aim to evaluate the properties of various Multi Criteria Decision Making (MCDM) methods and compare their outcomes in terms of assessing the performance of consumer goods companies using financial ratios and financial experts’ perspectives.

2. Literature Review

3. Data and Methodology

3.2. Methodology

In this section, we will go through multi-criteria decision-making methods in depth. These MCDM methods are techniques that help stakeholders choose the appropriate selection tool in order to make the best decision. The various processes for applying selected approaches and their comparative evaluation would be discussed.

Similarly, Buckley’s model, which uses triangular fuzzy numbers to express decision makers’ evaluations of alternatives with regard to each criterion, will be described in this chapter.

3.2.1. Comparative Review of MCDM Methods

Despite the huge number of MCDM methods available, no single method is regarded as best suited for all types of decision-making problems [15,16]. This leads to the paradox that choosing an acceptable approach for a particular problem leads to an MCDM problem [17]. The fact that different approaches might produce different outcomes when applied to the same problem is a key criticism of MCDM [18]. Identifying and selecting an effective MCDM technique is obviously not a straightforward undertaking, and some thought must be devoted to method selection. A variety of practical applications of comparison studies of different MCDM approaches are presented in the literature [19,20].

Additionally, a number of scholars [15,16] have created criteria to aid in the selection of an acceptable MCDM technique. However, it has been noted that numerous strategies may be theoretically relevant for a given decision-making circumstance; there is not always a compelling reason to choose one strategy over another. One of the most essential criteria in adopting an MCDM technique appears to be its compliance with the problem’s purpose [20].

The problem proposed in this study is to examine the financial performance of some consumer goods companies listed on the NSE. To accomplish this, a ranking of options must be identified. As a result, the goal of this problem is to rate alternatives. Consequently, an MCDM approach capable of providing a complete ranking of options (showing the position of each alternative) is necessary. Furthermore, the approach must be capable of handling both benefit and cost criteria, as well as quantitative and qualitative criteria. Similarly, the MCDM strategy must be simple to use and understand so that any interested parties can readily adopt the proposed method. This research compares the performance of different applicable MCDM methodologies, including the TOPSIS, VIKOR and ELECTRE.

These strategies are used on the data from the practical case studies provided later in the final chapter of this thesis. The goal of each technique is to determine the relative importance of each alternative under consideration, as well as to establish the priority order of the alternatives in relation to one another. The selected methods for comparative analysis differ in their fundamental concepts, the type of data normalization process used, and how the criteria values and criteria weights are combined into the evaluation procedure.

Because criteria have multiple units of measurement in general, MCDM approaches use a sort of normalization to remove the units of criteria values (e.g., ratio, points, percentage, price) so that all criteria are non-dimensional [18]. Normalization procedures vary, but in many circumstances, this stage is critical to the method’s consistent and proper use. The normalisation procedures of the WSM, WPM, and revised AHP methods are essentially comparable, with the exception of TOPSIS.

It is to be noted that the advantages and disadvantages of these methods are crucial in selecting the efficient one amongst them. For instance, the advantage of the AHP method is that it does not necessitate the use of an additional instrument to determine the weight of the criteria, while its disadvantage is that as the number of criteria and possibilities rises, the process becomes more sophisticated. On the other hand, the merit of the TOPSIS method is that it is relatively straightforward, and the solution procedure remains constant regardless of the number of selection criteria and alternatives, while its demerit is that the correlation among criteria is not taken into account while calculating Euclidean distance. Furthermore, vector normalisation may be required when handling multi-dimensional problems.

Similarly, the VIKOR method which is a modernized version of TOPSIS becomes difficult in the face of a competing scenario. In addition, the positive effect of the ELECTRE method is that it can yield a solution even when data is missing, but because of the complicated evaluation techniques involved, the methodology is computationally difficult in the absence of software [21].

3.2.2. Analytic Hierarchy Process (AHP) Method

The analytic hierarchy process (AHP) is a prominent analytical technique for difficult decision-making situations. AHP was invented by Saaty ([22,23]), which breaks down a decision-making problem into a system of hierarchies of objectives, attributes (or criteria), and alternatives.

An AHP hierarchy can include as many levels as necessary to accurately represent a decision circumstance. AHP is a valuable methodology due to a variety of functional properties. These include the ability to deal with decision circumstances including subjective judgments, multiple decision-makers, and the ability to offer measurements of preference consistency [17].

AHP, which was designed to mirror how people actually think, remains the most highly acclaimed and extensively used decision-making approach. AHP can deal with both tangible (i.e., objective) and non-tangible (i.e., subjective) features efficiently, especially when the subjective judgments of diverse individuals play a key role in the decision process.

To make a decision in an organised way in order to generate priorities we need to decompose the decision into the following steps.

• Define the problem and the type of information needed.
• Structure the decision hierarchy from the top with the choice’s purpose, then the objectives from a broad perspective, via the intermediate levels (criteria on which following elements rely) to the bottom level (which usually is a set of the alternatives).
• Create a collection of pairwise comparison matrices. Each element in an upper level is used to compare the components in the level directly below it.
• Use the comparison priorities to weigh the priorities in the level directly below. Repeat for each element. Then, for each element on the level below, add its weighted values to determine its overall or global priority. Continue weighing and adding until the final priorities of the alternatives at the lowest level are obtained.

To perform comparisons, we need a numerical scale that specifies how many times more important or dominating one element is over another one in relation to the criterion or feature being compared. The scale is shown in Table 2.

A fair assumption is that if activity i is compared to activity j and has one of the preceding non-zero numbers given to it, then j has the reciprocal value when compared to i.

Similarly, if the activities are very close, assigning the optimal value may be challenging; but, when compared to other contrasting activities, the size of the small numbers would not be too visible, and they can still show the relative importance of the activities.

Moving forward, we are going to discuss about Fuzzy Analytic Hierarchy Process. But before we do that, we would first need to give an insight on Fuzzy Sets.

3.2.3. Fuzzy Sets

Zadeh [25] firstly proposed a mathematical theory known as fuzzy set in order to overcome the vagueness and imprecise condition of human cognitive processes [26]. Apart from classical set theory based on binary logic, fuzzy set describe actual objects similar to human language [27]. A fuzzy set which is extension of crisp one allows partial belonging of element by membership function denoted by ${\mu }_M\left(x\right)$. Membership values of objects in a fuzzy set range from 0 (nonmembership) to 1 (complete membership) and values between these boundaries are called intermediate membership degrees and show degree to which an element belongs to a set [28].

There are several definitions involving fuzzy sets which are extensions definitions for ordinary set. Now, suppose X is a space of points, and let x be an element of X, then:

Definition 1. 

A fuzzy set is empty if and only if its membership function is identically zero on X [25].

Definition 2. 

Two fuzzy sets M and N are equal, written as M = N, if and only if ${\mu }_M\left(x\right)={\mu }_N\left(x\right)$ for all $x\in X$.

Definition 3. 

The complement of a fuzzy set M is denoted by $M^{\prime }$ and is defined by

$${\mu }_{M^{\prime }}\left(x\right),=1-{\mu }_M\left(x\right)$$

(1)

Definition 4. 

We say that a set M is contained in N (or, equivalently, M is a subset of N, or M is smaller than or equal to N) if and only if ${\mu }_M\left(x\right)\le {\mu }_N\left(x\right)$. That is,

$$M\subset N\iff {\mu }_M\left(x\right)\le {\mu }_N\left(x\right)$$

(2)

Definition 5. 

We say that the union of two fuzzy sets M and N with respective membership functions ${\mu }_M\left(x\right)$ and ${\mu }_N\left(x\right)$ is a fuzzy set C, which is written as $L=M\cup N$ whose membership function is related to those of M and N by

$${\mu }_L\left(x\right)=Max\left[{\mu }_M\left(x\right),{\mu }_N\left(x\right)\right],x\in X$$

(3)

Definition 6. 

We say that the intersection of two fuzzy sets M and N with respective membership functions ${\mu }_M\left(x\right)$ and ${\mu }_N\left(x\right)$ is a fuzzy set L, which is written as $L=$$M\cap N$, whose membership function is related to those of M and N by

$${\mu }_L\left(x\right)=Min\left[{\mu }_M\left(x\right),{\mu }_N\left(x\right)\right],x\in X$$

(4)

3.2.4. Fuzzy Numbers

Definition 7 

([29]). A fuzzy number $\left(FN\right)$ is a fuzzy set in $\mathbb{R}$, namely a mapping $\mu :\mathbb{R}\to [0,1]$, with the following properties:

• μ is convex, i.e. $\mu \left(x\right)\ge min\left\{\mu \right(s),\mu (r\left)\right\}$, for $s\le x\le r$;
• μ is normal, i.e. $(\exists )$$x_0\in \mathbb{R}:\mu \left(x_0\right)=1$;
• μ is upper semi-continuous, i.e.

$$(\forall )x\in \mathbb{R},(\forall )\alpha \in (0,1]:\mu \left(x\right)<\alpha ,(\exists )\delta >0\mathit{such}\mathit{that}|s-x|<\delta \Rightarrow \mu \left(s\right)<\alpha$$

(5)

Furthermore, triangular and trapezoidal fuzzy numbers are mostly used in application fields. Triangular fuzzy numbers are employed in this study due to their ease of computation and effectiveness as a representation.

3.2.5. Triangular Fuzzy Number

A triangular fuzzy number $\tilde{M}=(l,m,u)$ is a feature expressed by means of the membership function ${\mu }_{\tilde{M}}:\mathcal{R}\to [0,1]$ such that ${\mu }_{\tilde{M}}\left(x\right)$ expresses the degree (determined by an expert or a group of experts) to which x possesses this feature and

$${\mu }_{\tilde{M}}\left(x\right)=\left\{\begin{array}{cc}0\hfill & x\le l\hfill \\ {\displaystyle \frac{x-l}{m-l}}\hfill & l<x\le m\hfill \\ {\displaystyle \frac{u-x}{u-m}}\hfill & m\le x<u\hfill \\ 0\hfill & x\ge u\hfill \end{array}\right.$$

(6)

Membership of triangular fuzzy number is defined by three real numbers expressed as $(\mathrm{l},\mathrm{m},\mathrm{u})$ indicating smallest possible value, the most promising value and the largest possible value respectively [30].

Figure 1. Triangular fuzzy number diagram.

Figure 1. Triangular fuzzy number diagram.

3.2.6. Basic Operations on Triangular Fuzzy Numbers

Let $M=\left(l_1,m_1,u_1\right)$ and $N=\left(l_2,m_2,u_2\right)$ be two positive triangular fuzzy numbers then:

• Addition: $M+N=\left(l_1,m_1,u_1\right)+\left(l_2,m_2,u_2\right)=\left(l_1+l_2,m_1+m_2,u_1+u_2\right)$
• Subtraction: $M-N=\left(l_1,m_1,u_1\right)-\left(l_2,m_2,u_2\right)=\left(l_1-u_2,m_1-m_2,u_1+l_2\right)$
• Multiplication: $M·N=\left(l_1,m_1,u_1\right)·\left(l_2,m_2,u_2\right)=\left(l_1·l_2,m_1·m_2,u_1·u_2\right)$
• Division: ${\displaystyle \frac{M}{N}}={\displaystyle \frac{\left(l_1,m_1,u_1\right)}{\left(l_2,m_2,u_2\right)}}=\left({\displaystyle \frac{l_1}{u_2}},{\displaystyle \frac{m_1}{m_2}},{\displaystyle \frac{u_1}{l_2}}\right)$
• Inverse: $M^{-1}={\left(l_1,m_1,u_1\right)}^{-1}\approx \left(1/u_1,1/m_1,1/l_1\right)$
• Scalar Multiplication:
  $\forall \lambda >0,\lambda \in R,\lambda M=\left(\lambda l_1,\lambda m_1,\lambda u_1\right)$,
  $\forall \lambda <0,\lambda \in R,\lambda M=\left(\lambda u_1,\lambda m_1,\lambda l_1\right).$

3.2.7. One-Sided Fuzzy Numbers

One-sided fuzzy numbers (left-sided or right-sided) are triangular fuzzy numbers such that

a. For a left-sided fuzzy number, $l=-\infty$

b. For a right-sided fuzzy number, $u=\infty$.

It’s essential to specify which values of the membership function accepts positive values, i.e., the support for fuzzy numbers.

3.2.8. the Support of the Triangular Fuzzy Number

The support of the triangular fuzzy number $\tilde{M}=(l,m,u)$ is the open interval $(l,u)$, the support of the left-sided fuzzy number is the half-line $(\infty ,u)$ and that of the right-sided fuzzy number is the half-line $(l,\infty )$.

For example, in the banking sector [31] the experts of each bank can define their understanding of such terms as “very low", “low", “average", “high" and “very high" financial security, or “very low", “low", “average", “high" and “very high" yearly income of the potential borrower. As financial stability is difficult to quantify, experts would be requested to describe the first five elements using a predetermined scale (e.g., from 0 to 5). Also because the borrower’s annual income is quantifiable, the scale for the last five elements would correspond to the income values directly. Figure 2 presents example definitions of the terms, generated on the basis of expert opinions.

Five examples of fuzzy numbers are shown in Figure 2: three triangular, one left-sided, and one right-sided. They represent a number of different aspects of financial security or income (measured on the horizontal axis using units which must be predefined, such as hundreds of thousands of dollars for the income or an arbitrary scale for financial security). Each income value may have one or more of the six characteristics in whole, in part, or not at all. For example, income of $1.5$ is neither extremely high nor very low; it is medium to the degree of $0.25$, low to the degree of $0.75$, and very low to the degree of $0.75$ (according to the experts).

3.2.9. Fuzzy AHP

Since AHP can not reflect human thinking style in inaccurate and subjective environment due to unbalanced scale of judgments, inability to adequately handle inherent uncertainty and imprecise pair-wise comparisons. In order to effectively handle subjective perceptions and impreciseness, fuzzy numbers are integrated with AHP, allowing the appropriate expression of linguistic evaluation [33]. In examining real-world decision-making problems, fuzzy numbers are also utilized to deal with uncertainties affecting subjective choices.

For that reason, fuzzy analytic hierarchy process (FAHP) extension of traditional AHP was developed to solve hierarchical fuzzy problems in interval judgment matrix [34].

The fuzzy set theory allows respondents to explain semantic judgments subjectively [27]. As a result, in terms of triangular fuzzy numbers, Saaty’s 9-point scale is turned into the fuzzy ratio scale.

One of the fundamental problems in fuzzy optimization and fuzzy decision-making is ranking fuzzy numbers in an imprecise and ambiguous environment. Fuzzy values are rated using different fuzzy set criteria, such as the center of attraction, an area under the membership degree function, and some intersecting points [1]. As a result, the FAHP methodology is well adapted to handling subjective evaluation decision-making problems, and it is currently one of the most extensively used MCDM methods in the domains of business, management, manufacturing, industry, and government.

The first FAHP method was proposed by Van Laarhoven [35] using triangular fuzzy numbers in the pairwise comparison matrix.

Fuzzy pairwise comparison matrices were created in the FAHP approaches by leveraging linguistic evaluations of the decision makers’ judgments. Using triangular fuzzy numbers, the logarithmic regression method is utilized to create fuzzy weights or fuzzy performance scores through AHP operations [35]. Table 3 displays linguistic variables for pairwise comparisons of each criterion.

Buckley [37] developed a model to state decision maker’s evaluation on alternatives with respect to each criterion by using triangular fuzzy numbers.

3.2.10. Buckley’s Column Geometric Mean Method

The steps for Buckley’s Column Geometric Mean method are illustrated as follows:

• Establishing the hierarchical structure and comparing criteria or alternatives via fuzzy scale for constructing pairwise comparison matrix.

  $${\overline{A}}^k=\left[\begin{array}{cccc}{\overline{a}}_{11}^k& {\overline{a}}_{12}^k& ...& {\overline{a}}_{1n}^k\\ {\overline{a}}_{21}^k& {\overline{a}}_{22}^k& ...& {\overline{a}}_{2n}^k\\ ...& ...& ...& ...\\ {\overline{a}}_{m1}^k& {\overline{a}}_{m2}^k& ...& {\overline{a}}_{mn}^k\end{array}\right]$$

  (7)
• Taking the average of preferences of all decision-makers and obtain a new pairwise comparison matrix.

  $${\overline{a}}_{ij}={\displaystyle \frac{{\sum }_{k=1}^K{a}_{ij}^k}{K}}$$

  (8)

  $$\overline{A}=\left[\begin{array}{cccc}{\overline{a}}_{11}& {\overline{a}}_{12}& ...& {\overline{a}}_{1n}\\ {\overline{a}}_{21}& {\overline{a}}_{22}& ...& {\overline{a}}_{2n}\\ ...& ...& ...& ...\\ {\overline{a}}_{m1}& {\overline{a}}_{m2}& ...& {\overline{a}}_{mn}\end{array}\right]$$

  (9)
• Calculating the Geometric Mean of each criterion.

  $$\overline{g_i}={\left[\prod _{j=1}^n{\overline{a}}_{ij}\right]}^{1/n},i=1,2,...,m$$

  (10)
• Obtaining the fuzzy weights of each criterion.

  $${\overline{w}}_i={\overline{z}}_i\otimes {({\overline{z}}_1\oplus {\overline{z}}_2\oplus ...\oplus {\overline{z}}_n)}^{-1}=(l_i,m_i,u_i)$$

  (11)
• Transformation of fuzzy weights composed of fuzzy triangular numbers into crisp ones using center of area defuzzification techniques.

  $$S_i={\displaystyle \frac{l_i+m_i+u_i}{3}}$$

  (12)
• Normalizing the obtained crisp weights.

  $$N_i={\displaystyle \frac{S_i}{{\sum }_{i=1}^m{S}_i}}$$

  (13)

3.2.11. TOPSIS Method

TOPSIS (for the Technique for Order Preference by Similarity to Ideal Solution) is a well known, classic ranking method, which was developed by Hwang [38]. It was actually created to analyze MCDM problems. Basically, this technique is to choose an alternative having the shortest Euclidean distance from positive ideal solution (PIS) which maximizes benefits and minimizes cost, and the farthest distance from negative ideal solution (NIS) which maximizes cost and minimizes benefit.

The TOPSIS methodology posits that each criterion has a tendency to increase or decrease utility in a monotonic manner. As a result, defining ideal and negative-ideal solutions is simple. To determine how close the alternatives are to the ideal solution, the Euclidean distance method was introduced. As a result, the method focuses on calculating both the best, or ideal scenario, and the worst, or negatively ideal case. TOPSIS chooses the alternative whose value is closest to the ideal solution and farthest from the negatively ideal solution [39]. The following are the significant highlights of TOPSIS:

• It’s a really clear and understandable method, with each step of the computation being very logical and understandable.
• Simple and uncomplicated computations are involved.
• In addition to the development of the most optimal example, this technique includes the generation of the ideal and negative ideal cases (practically feasible solution).

The major steps involved are:

• Forming decision matrix $A={\left(a_{ij}\right)}_{mxn}$ for m alternatives and n criteria. Where $(i=1,2,m)$ and $(j=1,...,n)$.

  $$A=\left[\begin{array}{cccccc}{a}_{11}& a_{12}& \cdots & a_{1j}& \cdots & a_{1n}\\ a_{21}& a_{22}& \cdots & a_{2j}& \cdots & a_{2n}\\ ⋮& ⋮& \cdots & ⋮& \cdots & ⋮\\ a_{i1}& a_{i2}& \cdots & a_{ij}& \cdots & a_{in}\\ ⋮& ⋮& \cdots & ⋮& \cdots & ⋮\\ a_{m1}& a_{m2}& \cdots & a_{mj}& \cdots & a_{mn}\end{array}\right]$$

  (14)
• Normalizing the decision matrix.

  $$r_{ij}={\displaystyle \frac{a_{ij}}{\sqrt{{\sum }_{i=1}^m{a}_{ij}^2}}},i=1,2,\cdots ,nj=1,2,\cdots ,m.$$

  (15)
• Weighting normalized decision matrix.

  $$v_{ij}=r_{ij}\times w_{ij},i=1,2,\cdots ,nj=1,2,\cdots ,m.$$

  (16)
• Finding positive and negative ideal solutions.

  $$PIS=A^{*}=\left\{v_1^{*},v_2^{*},\cdots ,v_m^{*}\right\}=\left\{\left(\underset{i}{max}{v}_{ij}\mid j\in {\Omega }_b\right),\left(\underset{i}{min}{v}_{ij}\mid j\in {\Omega }_c\right)\right\}$$

  (17)

  $$NIS=A^{-}=\left\{v_1^{-},v_2^{-},\cdots ,v_m^{-}\right\}=\left\{\left(\underset{i}{min}{v}_{ij}\mid j\in {\Omega }_b\right),\left(\underset{i}{max}{v}_{ij}\mid j\in {\Omega }_c\right)\right\}$$

  (18)
• Calculating the Euclidean distance of alternatives from positive and negative ideal solutions.

  $$d_i^{*}=\sqrt{{\sum }_{j=1}^m{(v_{ij}-v_j^{*})}^2},i=1,2,\cdots ,n.$$

  (19)

  $$d_i^{-}=\sqrt{{\sum }_{j=1}^m{(v_{ij}-v_j^{-})}^2},i=1,2,\cdots ,n.$$

  (20)
• Calculating the relative closeness $\left(R{C}_i\right)$ of each alternative to ideal solutions.

  $$R{C}_i={\displaystyle \frac{d_i^{-}}{d_i^{-}+d_i^{*}}},i=1,2,\cdots ,n{RC}_i\in [0,1]$$

  (21)
• Finally, ranking alternatives according to their $R{C}_i$ values in descending order from 1 to 0 and choosing the highest one.

3.2.12. Fuzzy TOPSIS

Chen [40] introduced the Fuzzy TOPSIS approach, which is an extension of the TOPSIS method using triangular fuzzy numbers in order to address multi-criteria decision-making problems under uncertainty. The decision-makers employ linguistic variables, $Dr$$(r=1,\dots ,k)$, to evaluate the weights of the criteria and the ratings of the alternatives. As a result, $\tilde{W}{r}^j$ denotes the weight of the jth criterion, $C_j$$(j=1,\dots ,m)$, as determined by the rth decision-maker. Similarly, ${\tilde{x}}_{ij}^r$ denotes the evaluation of the ith alternative, $A_i$$(i=1,\dots ,n)$, in terms of criterion j, as determined by the rth decision-maker. That being said, the Fuzzy TOPSIS method comprises the following steps [41]:

• Aggregate the weights of criteria and ratings of alternatives given by k decision makers, as expressed in Equations (22) and (23) respectively:

  $${\tilde{w}}_j={\displaystyle \frac{1}{k}}\left[{\tilde{w}}_j^1+{\tilde{w}}_j^2+\cdots +{\tilde{w}}_j^k\right]$$

  (22)

  $${\tilde{x}}_{ij}={\displaystyle \frac{1}{k}}\left[{\tilde{x}}_{ij}^1+{\tilde{x}}_{ij}^r+\cdots +{\tilde{x}}_j^k\right]$$

  (23)
• Arrange the fuzzy decision matrix of the alternatives $\tilde{D}$ and the criteria $\tilde{W}$ as follows:
  $$\begin{array}{cccc}{C}_1& C_2& C_j& C_m\end{array}$$

  $$\tilde{D}=\begin{array}{c}{A}_1\\ A_i\\ A_n\end{array}\left[\begin{array}{cccc}{\tilde{x}}_{11}& {\tilde{x}}_{12}& {\tilde{x}}_{1j}& {\tilde{x}}_{1m}\\ ⋮& ⋮& ⋮& ⋮\\ {\tilde{x}}_{n1}& {\tilde{x}}_{n2}& {\tilde{x}}_{nj}& {\tilde{x}}_{nm}\end{array}\right]$$

  $$\tilde{W}=\left[{\tilde{w}}_1+{\tilde{w}}_2+\cdots +{\tilde{w}}_m\right]$$

  (24)
• Normalize the fuzzy decision matrix of the alternatives $\tilde{D}$ using linear scale transformation. The normalized fuzzy decision matrix $\tilde{R}$ is given by:

  $$\tilde{R}={\left[{\tilde{r}}_{ij}\right]}_{m\times n}$$

  (25)

  $${\tilde{r}}_{ij}=\left({\displaystyle \frac{l_{ij}}{u_j^{+}}},{\displaystyle \frac{m_{ij}}{u_j^{+}}},{\displaystyle \frac{u_{ij}}{u_j^{+}}}\right)\mathrm{and}{u}_j^{+}=\underset{i}{max}{u}_{ij}(\mathrm{benefit}\mathrm{criteria})$$

  (26)

  $${\tilde{r}}_{ij}=\left({\displaystyle \frac{l_j^{-}}{u_{ij}}},{\displaystyle \frac{l_j^{-}}{m_{ij}}},{\displaystyle \frac{l_j^{-}}{l_{ij}}}\right)\mathrm{and}{l}_j^{-}=\underset{i}{max}{l}_{ij}(\cos \mathrm{t}\mathrm{criteria})$$

  (27)
• Compute the weighted normalized decision matrix, $\tilde{V}$, by multiplying the weights of the evaluation criteria, ${\tilde{w}}_j$, by the elements ${\tilde{r}}_{ij}$ of the normalized fuzzy decision matrix

  $$\tilde{V}={\left[{\tilde{v}}_{ij}\right]}_{m\times n}\mathrm{where}{\tilde{v}}_{ij}\mathrm{is}\mathrm{given}\mathrm{by}{\tilde{v}}_{ij}={\tilde{x}}_{ij}\times {\tilde{w}}_j$$

  (28)
• Compute the weighted normalized decision matrix, $\tilde{V}$, by multiplying the weights of the evaluation criteria, ${\tilde{w}}_j$, by the elements ${\tilde{r}}_{ij}$ of the normalized fuzzy decision matrix

  $$\tilde{V}={\left[{\tilde{v}}_{ij}\right]}_{m\times n}\mathrm{where}{\tilde{v}}_{ij}\mathrm{is}\mathrm{given}\mathrm{by}{\tilde{v}}_{ij}={\tilde{x}}_{ij}\times {\tilde{w}}_j$$

  (29)
• Define the Fuzzy Positive Ideal Solution (FPIS, $A^{+}$) and the Fuzzy Negative Ideal Solution (FNIS, $A^{-}$) as follows:

  $$A^{+}=\left\{{\tilde{v}}_1^{+},{\tilde{v}}_j^{+},\dots ,{\tilde{v}}_m^{+}\right\}\mathrm{and}{A}^{-}=\left\{{\tilde{v}}_1^{-},{\tilde{v}}_i^{-},\dots ,{\tilde{v}}_m^{-}\right\}$$

  (30)

  where ${\tilde{v}}_j^{+}=(1,1,1)$ and ${\tilde{v}}_j^{-}=(0,0,0)$.
• Compute the distances $d_j^{+}$ and $d_i^{-}$ of each alternative from respective ${\tilde{v}}_j^{+}$ and ${\tilde{v}}_j^{-}$:

  $$d_i^{+}=\sum _{j=1}^n{d}_v\left({\tilde{v}}_{ij},{\tilde{v}}_j^{+}\right)\mathrm{and}{d}_i^{-}=\sum _{j=1}^n{d}_v\left({\tilde{v}}_{ij},{\tilde{v}}_j^{-}\right)$$

  (31)

  where $d(..)$, according to the vertex method, represents the distance between two fuzzy numbers. For triangular fuzzy numbers, this is stated as:

  $$d(\tilde{x},\tilde{z})=\sqrt{{\displaystyle \frac{1}{3}}\left[{\left(l_x-l_z\right)}^2+{\left(m_x-m_z\right)}^2+{\left(u_x-u_z\right)}^2\right]}$$

  (32)
• Compute the closeness coefficient, $C{C}_i$

  $$C{C}_i={\displaystyle \frac{d_i^{-}}{d_i^{+}+d_i^{-}}}$$

  (33)
• Define the ranking of the alternatives according to the closeness coefficient, $C{C}_i$, in decreasing order. The optimal alternative is the one that is closest to the FPIS but farthest away from the FNIS.

3.2.13. VIKOR Method

The VIKOR method developed by Opricovic [42] is, as TOPSIS method, also based in the idea of the distances to “ideal solutions”. This method is oriented for selecting and ranking alternatives in case of conflicting criteria. The compromised solution is the closest to the ideal one.

The VIKOR Method can be used for solving MCDM problems if the following conditions are satisfied:

• Compromised solution should be accepted in order to overcome conflict.
• Decision maker is willing to accept the closest solution to ideal one.
• A linear relationship between benefit and each criteria function for decision maker.
• Alternatives should be evaluated in terms of each criteria.
• Preferences of decision makers are expressed by weights.
• Decision makers are responsible for approving the final solution.

The Steps of VIKOR Method are as follows:

• Identification of best $\left(f_a^{*}\right)$ and worst $\left(f_a^{-}\right)$ values for each criteria. If evaluation criteria $(b=1,2,\cdots ,n)$ is based on benefit;

  $$f_b^{*}=\underset{a}{max}{x}_{ab}{f}_b^{-}=\underset{a}{min}{x}_{ab}$$

  (34)
  If evaluation criteria $(b=1,2,\cdots ,n)$ is based on cost;

  $$f_b^{*}=\underset{a}{min}{x}_{ab}{f}_b^{-}=\underset{a}{max}{x}_{ab}$$

  (35)
• In order to make comparisons we obtained a normalization matrix. During the normalization process, the Decision matrix X composed of k criteria and l alternatives was transformed into normalization matrix S with same dimensions. The decision matrix X with elements $\left(x_{kl}\right)$ before normalization is given as:

  $$X=\left[\begin{array}{cccccc}{x}_{11}& x_{12}& \cdots & x_{1b}& \cdots & x_{1l}\\ x_{21}& x_{22}& \cdots & x_{2b}& \cdots & x_{2l}\\ ⋮& ⋮& \cdots & ⋮& \cdots & ⋮\\ x_{a1}& x_{a2}& \cdots & x_{ab}& \cdots & x_{al}\\ ⋮& ⋮& \cdots & ⋮& \cdots & ⋮\\ x_{k1}& x_{k2}& \cdots & x_{kb}& \cdots & x_{kl}\end{array}\right]$$

  (36)
  The normalization matrix S with elements $\left(s_{kl}\right)$ obtained after normalizing is given as:

  $$S=\left[\begin{array}{cccccc}{s}_{11}& s_{12}& \cdots & s_{1b}& \cdots & s_{1l}\\ s_{21}& s_{22}& \cdots & s_{2b}& \cdots & s_{2l}\\ ⋮& ⋮& \cdots & ⋮& \cdots & ⋮\\ s_{a1}& s_{a2}& \cdots & s_{ab}& \cdots & s_{al}\\ ⋮& ⋮& \cdots & ⋮& \cdots & ⋮\\ s_{k1}& s_{k2}& \cdots & s_{kb}& \cdots & s_{kl}\end{array}\right]s_{ab}={\displaystyle \frac{f_b^{*}-x_{a}b}{f_b^{*}-f_b^{-}}}$$

  (37)
• We find normalized decision matrix. By multiplying criteria weights $\left(w_b\right)$ and normalized decision matrix elements $\left(s_{ab}\right)$.

  $$T=\left[\begin{array}{cccccc}{t}_{11}& t_{12}& \cdots & t_{1b}& \cdots & t_{1l}\\ t_{21}& t_{22}& \cdots & t_{2b}& \cdots & t_{2l}\\ ⋮& ⋮& \cdots & ⋮& \cdots & ⋮\\ t_{a1}& t_{a2}& \cdots & t_{ab}& \cdots & t_{al}\\ ⋮& ⋮& \cdots & ⋮& \cdots & ⋮\\ t_{k1}& t_{k2}& \cdots & t_{kb}& \cdots & t_{kl}\end{array}\right]t_{ab}=s_{ab}\times w_{ab}$$

  (38)
• We calculate $S_a$ (mean group score) and $R_a$ (worst group score) values for each alternatives.

  $$S_a=\sum _{b=1}^l{w}_b{\displaystyle \frac{f_b^{*}-x_{ab}}{f_b^{*}-f_b^{-}}}{R}_a=\underset{b}{max}\left[w_b{\displaystyle \frac{f_b^{*}-x_{ab}}{f_b^{*}-f_b^{-}}}\right]$$

  (39)
• We calculate $Q_a$ values for each alternatives. The values of $Q_a$ is obtain through the values of $S^{*}$, $S^{-}$, $R^{*}$, $R^{-}$. We also use additional y parameter which states the weight of alternative providing maximum group benefit and $(1-y)$ parameter for minimum individual regret. Generally, we assume $y=0.5$.
  $S^{*}={min}_a{S}_a{S}^{-}={max}_a{S}_a{R}^{*}={min}_a{R}_a{R}^{-}={max}_a{R}_a$

  $$Q_a=y{\displaystyle \frac{S_a-S^{*}}{S^{-}-S^{*}}}+(1-y){\displaystyle \frac{R_a-R^{*}}{R^{-}-R^{*}}}$$

  (40)
• We rank the values of $S_a$, $R_a$, and $Q_a$ from lower to higher. We also control alternative having minimum $Q_a$ values by two conditions to know if the ranking is accurate. The conditions are known as acceptable advantage and acceptable stability.
  Acceptable advantage condition: According to $Q_a$ values, if first $Q\left(C_1\right)$ and second alternative $Q\left(C_2\right)$ satisfied significant difference, we calculate threshold value $DQ$ which depend on alternative number. If the number of alternative is lower than 4, then the value of $DQ$ equals to $0.25$ [43]

  $$Q\left(C_2\right)-Q\left(C_1\right)\ge DQDQ={\displaystyle \frac{1}{k-1}}$$

  (41)
  Acceptable stability condition: According to $Q_a$ values, first alternative $Q\left(C_1\right)$ should get at least one best score for values of S and R. If these two conditions are not satisfied, then we formed a compromised solution set in two ways:

  (a)

  If the second condition is not satisfied, then the first and second alternatives are accepted as the compromised solution.

  (b)

  If the first condition is not satisfied, then $C_1,C_2,\cdots ,C_k$ alternatives are contained in compromised solution set according to $Q\left(C_2\right)-Q\left(C_1\right)\ge DQ$ [44]

3.2.14. ELECTRE Method

The ELECTRE (for Elimination and Choice Translating Reality; English translation from the French original) method was devised in 1966 [45], which is an outranking method and is based on partial aggregation. The idea of this method is to rank alternatives based on concord and discord indexes that are calculated with extracted data from a decision table. As Mateo [39] mentioned this method has 4 main steps. In the first phase, weight must be assigned to each criterion based on a normalization theory such that the sum of all weights equals 1, and a threshold function must be developed. The concordance and discordance indexes for a pair of alternatives must be determined in the second phase. The outranking degree for each pair of alternatives must then be determined using the concordance and discordance indexes.

Finally, a partial ranking will be determined by taking into account all pairs of alternatives. According to Hui-Fen Li [39], “the weakness of normal ranking of ELECTRE is that it requires the introduction of an additional threshold, and the ranking of the alternative is dependent on the size of this threshold for which there is no correct value". ELECTRE, on the other hand, has the advantage of being able to manage both quantitative and qualitative data when outranking alternatives.

In order to solve decision problems more than two criteria of ELECTRE methods can be preferred to other ones if at least one of these conditions is satisfied: a) Performances of criteria are expressed in different units and decision maker does not want to use complex and difficult common scale. b) The problem does not tolerate a compensation effect. c) If there is requirement to use indifference and preference thresholds such that sum of small differences is decisive apart from insignificant small differences. d) If alternatives are weak interval or any order scale in which it is difficult to compare differences [45].

ELECTRE also allows decision makers to avoid compensation between criteria and any normalization process that can distort original data, also uncertain conditions are being considered. On the other hand these methods require difficult technical parameters which are not easily understandable [45].

There are many versions of ELECTRE Method such as ELECTRE (II, III, IV etc.) which can be used according to the types of decision problem. The steps of ELECTRE method can be illustrated as follows [46]:

• Forming decision matrix $A={\left(a_{ij}\right)}_{mxn}$ for m alternatives and n criteria. Where $(i=1,2,m)$ and $(j=1,...,n)$.

  $$A=\left[\begin{array}{cccccc}{a}_{11}& a_{12}& \cdots & a_{1j}& \cdots & a_{1n}\\ a_{21}& a_{22}& \cdots & a_{2j}& \cdots & a_{2n}\\ ⋮& ⋮& \cdots & ⋮& \cdots & ⋮\\ a_{i1}& a_{i2}& \cdots & a_{ij}& \cdots & a_{in}\\ ⋮& ⋮& \cdots & ⋮& \cdots & ⋮\\ a_{m1}& a_{m2}& \cdots & a_{mj}& \cdots & a_{mn}\end{array}\right]$$

  (42)
• Forming normalized decision matrix $X={\left(x_{ij}\right)}_{m\times n}$ for m alternatives and n criteria.

  $$X=\left[\begin{array}{cccccc}{x}_{11}& x_{12}& \cdots & x_{1j}& \cdots & x_{1n}\\ x_{21}& x_{22}& \cdots & x_{2j}& \cdots & x_{2n}\\ ⋮& ⋮& \cdots & ⋮& \cdots & ⋮\\ x_{i1}& x_{i2}& \cdots & x_{ij}& \cdots & x_{in}\\ ⋮& ⋮& \cdots & ⋮& \cdots & ⋮\\ x_{m1}& x_{m2}& \cdots & x_{mj}& \cdots & x_{mn}\end{array}\right]$$

  (43)
  Where:

  $$x_{ij}={\displaystyle \frac{a_{ij}}{\sqrt{{\sum }_{i=1}^m{a}_{ij}^2}}},\mathrm{for}\mathrm{maximization}\mathrm{objective}$$

  (44)

  $$x_{ij}={\displaystyle \frac{\frac{1}{a_{ij}}}{\sqrt{{\sum }_{i=1}^m{\left(\frac{1}{a_{ij}}\right)}^2}}},\mathrm{for}\mathrm{minimization}\mathrm{objective}$$

  (45)
• Calculating weighted normalized decision matrix $V={\left(v_{ij}\right)}_{m\times n}$. By multiplying weight of criterion $w_j$ and elements of normalized decision matrix $x_{ij}$

  $$v_{ij}=w_j\times x_{i}j$$

  (46)
• Determining concordance and discordance sets for each pairs of the alternatives $A_p$ and $A_q$ If alternative $A_p$ is preferred to alternative $A_q$ for all criteria concordance set $C(p,q)$, then the collection of criteria where $A_p$ is better than or equal to $A_q$ is formed as:

  $$C(p,q)=\left\{j\mid v_{pj}\ge v_{qj}\right\}$$

  (47)
  Similarly, if alternative $A_p$ is worse than $A_q$ for all criteria discordance set $D(p,q)$, then the collection of criteria where $A_p$ is worse than $A_q$ is formed as:

  $$D(p,q)=\left\{j\mid v_{pj}<v_{qj}\right\}$$

  (48)
  According to the above formulations $V_{pj}$ is defined as the weighted normalized rating of alternative $A_p$ with regard to jth criterion and $V_{qj}$ is defined as weighted normalizing rating of alternative $A_q$ with respect to jth criterion.
• Calculating the concordance $C_{pq}$ and discordance $D_{pq}$ indexes as follows:

  $$C_{pq}=\sum _{j^{*}}{w}_{j*},\mathrm{where}{j}^{*}\mathrm{is}\mathrm{criteria}\mathrm{in}\mathrm{C}(\mathrm{p},\mathrm{q})$$

  (49)
  And

  $$D_{pq}={\displaystyle \frac{{\sum }_{j^{+}}|v_{p{j}^{+}}-v_{q{j}^{+}}|}{{\sum }_j|v_{pj}-v_{qj}|}}\mathrm{where}{j}^{+}\mathrm{is}\mathrm{criteria}\mathrm{in}\mathrm{D}(\mathrm{p},\mathrm{q})$$

  (50)
• Relationship between alternatives are outranked by computing the averages of $C_{pq}$ and $D_{pq}$ which are represented by $\overline{C}$ and $\overline{D}$ respectively. According to this method, the alternative of $A_p$ outrank the alternative of $A_q$ if and only if $C_{pq}\ge \overline{C}$ and $D_{pq}\le \overline{D}$ conditions are satisfied.
• Calculating Net concordance $C_p$ and discordance $D_p$ indexes. The ultimate ranking is obtained according to ordering $C_p$ values from higher to lower and $D_p$ values from lower to higher.

  $$C_p=\sum _{k=1}^m{C}_{pk}-\sum _{k=1}^m{C}_{kp}k\ne p$$

  (51)

  $$D_p=\sum _{k=1}^m{D}_{pk}-\sum _{k=1}^m{D}_{kp}k\ne p$$

  (52)

4. Research and Result

4.1. Research

In this section, we are going to consider a practical case for our study. That is we are going to select three MCDM methods which are ELECTRE, TOPSIS and VIKOR, and use them to evaluate the financial performance of 10 selected consumer goods companies listed on the Nigerian Stock Exchange.

We will start by weighting the criteria that were acquired from the survey using Buckley’s column geometric mean approach through the following steps:

Step 1: We first create a pair-wise comparison matrix using Saaty’s scale from AHP Method.

Table 4. Pairwise comparison matrix

Table 4. Pairwise comparison matrix

Criteria	Current Ratio	Acid Test Ratio	Cash Ratio	Leverage Ratio	Asset Turnover Ratio	Net Profit /Total Asset	Net Profit/Capital	Net Profit/Net Sales
Current Ratio	1	1/7	1/8	8	1/6	1/7	1/7	1/6
Acid Test Ratio	7	1	1/8	9	1/5	1/7	6	1/6
Cash Ratio	8	8	1	9	4	1/9	7	6
Leverage Ratio	1/8	1/9	1/9	1	1/6	1/9	2	1/5
Asset Turnover Ratio	6	5	1/4	6	1	1/6	5	3
Net Profit/Total Assets	7	7	9	9	6	1	8	5
Net Profit/Capital	7	1/6	1/7	1/2	1/5	1/8	1	1/7
Net Profit/Net Sales	6	6	1/6	5	1/3	1/5	7	1

Step 2: The next is fuzzification; that is we turn our pair-wise comparison matrix to triangular fuzzy form. In other words, we want to replace the crisp numeric values to fuzzy numbers:

Table 5. Fuzzified pairwise comparison matrix

Table 5. Fuzzified pairwise comparison matrix

Criteria	Current Ratio	Acid Test Ratio	Cash Ratio	Leverage Ratio	Asset Turnover Ratio	Net Profit /Total Asset	Net Profit/Capital	Net Profit/Net Sales
Current Ratio	(1,1,1)	(1/8,1/7,1/6)	(1/9,1/8,1/7)	(7,8,9)	(1/7,1/6,1/5)	(1/8,1/7,1/6)	(1/8,1/7,1/6)	(1/7,1/6,1/5)
Acid Test Ratio	(6,7,8)	(1,1,1)	(1/9,1/8,1/7)	(9,9,9)	(1/6,1/5,1/4)	(1/8,1/7,1/6)	(5,6,7)	(1/7,1/6,1/5)
Cash Ratio	(7,8,9)	(7,8,9)	(1,1,1)	(9,9,9)	(3,4,5)	(1/9,1/9,1/9)	(6,7,8)	(5,6,7)
Leverage Ratio	(1/9,1/8,1/7)	(1/9,1/9,1/9)	(1/9,1/9,1/9)	(1,1,1)	(1/7,1/6,1/5)	(1/9,1/9,1/9)	(1,2,3)	(1/6,1/5,1/4)
Asset Turnover Ratio	(5,6,7)	(4,5,6)	(1/5,1/4,1/3)	(5,6,7)	(1,1,1)	(1/7,1/6,1/5)	(4,5,6)	(2,3,4)
Net Profit/Total Assets	(6,7,8)	(6,7,8)	(9,9,9)	(9,9,9)	(5,6,7)	(1,1,1)	(7,8,9)	(4,5,6)
Net Profit/Capital	(6,7,8)	(1/7,1/6,1/5)	(1/8,1/7,1/6)	(1/3,1/2,1/1)	(1/6,1/5,1/4)	(1/9,1/8,1/7)	(1,1,1)	(1/8,1/7,1/6)
Net Profit/Net Sales	(5,6,7)	(5,6,7)	(1/7,1/6,1/5)	(4,5,6)	(1/4,1/3,1/2)	(1/6,1/5,1/4)	(6,7,8)	(1,1,1)

Step 3: Now, we can calculate the fuzzy geometric mean value $\overline{g_i}$.

Table 6. Fuzzy geometric mean value $\overline{g_i}$

Table 6. Fuzzy geometric mean value $\overline{g_i}$

Criteria	Current Ratio	Acid Test Ratio	Cash Ratio	Leverage Ratio	Asset Turnover Ratio	Net Profit /Total Asset	Net Profit/Capital	Net Profit/Net Sales	$\overline{{\mathit{g}}_{\mathit{i}}}$
Current Ratio	(1,1,1)	(1/8,1/7,1/6)	(1/9,1/8,1/7)	(7,8,9)	(1/7,1/6,1/5)	(1/8,1/7,1/6)	(1/8,1/7,1/6)	(1/7,1/6,1/5)	(0.273,0.308,0.353)
Acid Test Ratio	(6,7,8)	(1,1,1)	(1/9,1/8,1/7)	(9,9,9)	(1/6,1/5,1/4)	(1/8,1/7,1/6)	(5,6,7)	(1/7,1/6,1/5)	(0.740,0.830,0.938)
Cash Ratio	(7,8,9)	(7,8,9)	(1,1,1)	(9,9,9)	(3,4,5)	(1/9,1/9,1/9)	(6,7,8)	(5,6,7)	(2.854,3.191,3.503)
Leverage Ratio	(1/9,1/8,1/7)	(1/9,1/9,1/9)	(1/9,1/9,1/9)	(1,1,1)	(1/7,1/6,1/5)	(1/9,1/9,1/9)	(1,2,3)	(1/6,1/5,1/4)	(0.209,0.241,0.271)
Asset Turnover Ratio	(5,6,7)	(4,5,6)	(1/5,1/4,1/3)	(5,6,7)	(1,1,1)	(1/7,1/6,1/5)	(4,5,6)	(2,3,4)	(1.479,1.805,2.158)
Net Profit/Total Assets	(6,7,8)	(6,7,8)	(9,9,9)	(9,9,9)	(5,6,7)	(1,1,1)	(7,8,9)	(4,5,6)	(5.144,5.698,6.117)
Net Profit/Capital	(6,7,8)	(1/7,1/6,1/5)	(1/8,1/7,1/6)	(1/3,1/2,1/1)	(1/6,1/5,1/4)	(1/9,1/8,1/7)	(1,1,1)	(1/8,1/7,1/6)	(0.309,0.363,0.447)
Net Profit/Net Sales	(5,6,7)	(5,6,7)	(1/7,1/6,1/5)	(4,5,6)	(1/4,1/3,1/2)	(1/6,1/5,1/4)	(6,7,8)	(1,1,1)	(1.173,1.391,1.664)

Step 4: Now, we calculate fuzzy weights $\overline{w_i}$.

Table 7. Fuzzy weights $\overline{w_i}$

Table 7. Fuzzy weights $\overline{w_i}$

Criteria	Fuzzy geometric mean value $\overline{{\mathit{g}}_{\mathit{i}}}$	Fuzzy weights $\overline{{\mathit{w}}_{\mathit{i}}}$
Current Ratio	(0.273,0.308,0.353)	(0.018,0.022,0.029)
Acid Test Ratio	(0.740,0.830,0.938)	(0.048,0.060,0.077)
Cash Ratio	(2.854,3.191,3.503)	(0.186,0.230,0.287)
Leverage Ratio	(0.209,0.241,0.271)	(0.014,0.017,0.022)
Asset Turnover Ratio	(1.479,1.805,2.158)	(0.096,0.130,0.177)
Net Profit/Total Assets	(5.144,5.698,6.117)	(0.334,0.410,0.502)
Net Profit/Capital	(0.309,0.363,0.447)	(0.020,0.026,0.037)
Net Profit/Net Sales	(1.173,1.391,1.664)	(0.076,0.100,0.136)
SUM	(12.180,13.827,15.451)
$SU{M}^{-1}$	(0.065,0.072,0.082)

Step 5 and 6: Transforming fuzzy weights into crisp ones and normalizing the obtained results:

Table 8. Averaged weight criterion $\left(S_i\right)$ and Normalized weight criterion $\left(N_i\right)$

Table 8. Averaged weight criterion $\left(S_i\right)$ and Normalized weight criterion $\left(N_i\right)$

Criteria	Fuzzy weights $\overline{w_i}$	$S_i$	$N_i$
Current Ratio	(0.018,0.022,0.029)	0.023	0.023
Acid Test Ratio	(0.048,0.060,0.077)	0.062	0.061
Cash Ratio	(0.186,0.230,0.287)	0.234	0.230
Leverage Ratio	(0.014,0.017,0.022)	0.018	0.017
Asset Turnover Ratio	(0.096,0.130,0.177)	0.134	0.132
Net Profit/Total Assets	(0.334,0.410,0.502)	0.415	0.408
Net Profit/Capital	(0.020,0.026,0.037)	0.028	0.027
Net Profit/Net Sales	(0.076,0.100,0.136)	0.104	0.102
TOTAL		1.018	1

According to the results of BCGM approach weights of ratios are given in Table 9.

Table 9. Weights of financial ratios

Table 9. Weights of financial ratios

Financial Ratios	Weights	Rank
Currency Ratio	0.023	7
Acid Test Ratio	0.061	5
Cash Ratio	0.230	2
Leverage Ratio	0.017	8
Asset Turnover Ratio	0.132	3
Net Profit/Total Assets	0.408	1
Net Profit/Capital	0.027	6
Net Profit/Net Sales	0.102	4

According to the importance level of financial ratios Net Profit/Total Assets ratio was found as the most important criteria having the value of $0.408$. On the other hand, leverage ratio was obtained as the least important one having the value of $0.017$.

5. Conclusion

6. Recommendation

References

1. Chen, S.J.; Hwang, C.L. Fuzzy multiple attribute decision making methods. Fuzzy multiple attribute decision making 1992, pp. 289–486.
2. Ghaleb, A.M.; Kaid, H.; Alsamhan, A.; Mian, S.H.; Hidri, L. Assessment and comparison of various MCDM approaches in the selection of manufacturing process. Advances in Materials Science and Engineering 2020, 2020. [Google Scholar] [CrossRef]
3. Alias, M.A.; Hashim, S.Z.M.; Samsudin, S. Multi criteria decision making and its applications: a literature review. Jurnal Teknologi Maklumat 2008, 20, 129–152. [Google Scholar]
4. Singh, A.; Malik, S.K. Major MCDM Techniques and their application-A Review. IOSR Journal of Engineering 2014, 4, 15–25. [Google Scholar]
5. Rao, R.V. Introduction to multiple attribute decision-making (MADM) methods. Decision Making in the Manufacturing Environment: Using Graph Theory and Fuzzy Multiple Attribute Decision Making Methods, 2007; 27–41. [Google Scholar]
6. Aruldoss, M.; Lakshmi, T.M.; Venkatesan, V.P. A survey on multi criteria decision making methods and its applications. American Journal of Information Systems 2013, 1, 31–43. [Google Scholar]
7. Dooley, A.; Sheath, G.W.; Smeaton, D. Multiple criteria decision making: method selection and application to three contrasting agricultural case studies. Technical report, 2005.
8. Altman, E.I. Financial ratios, discriminant analysis and the prediction of corporate bankruptcy. The journal of finance 1968, 23, 589–609. [Google Scholar] [CrossRef]
9. İÇ, Y.T.; Tekin, M.; PAMUKOĞLU, F.Z.; YILDIRIM, S.E. KURUMSAL FİRMALAR İÇİN BİR FİNANSAL PERFORMANS KARŞILAŞTIRMA MODELİNİN GELİŞTİRİLMESİ. Journal of the Faculty of Engineering & Architecture of Gazi University 2015, 30. [Google Scholar]
10. Feng, C.M.; Wang, R.T. Performance evaluation for airlines including the consideration of financial ratios. Journal of Air Transport Management 2000, 6, 133–142. [Google Scholar] [CrossRef]
11. Yurdakul, M.; Ic, Y. An illustrative study aimed to measure and rank performance of Turkish automotive companies using TOPSIS. Journal of the Faculty of Engineering and Architecture of Gazi University 2003, 18, 1–18. [Google Scholar]
12. Mahmoodzadeh, S.; Shahrabi, J.; Pariazar, M.; Zaeri, M. Project selection by using fuzzy AHP and TOPSIS technique. World Academy of Science, Engineering and Technology 2007, 30, 333–338. [Google Scholar]
13. Wu, H.Y.; Tzeng, G.H.; Chen, Y.H. A fuzzy MCDM approach for evaluating banking performance based on Balanced Scorecard. Expert systems with applications 2009, 36, 10135–10147. [Google Scholar] [CrossRef]
14. Bülbül, S.; Köse, A. EVALUATION OF THE FINANCIAL PERFORMANCE OF TURKISH FOODS COMPANIES WITH MULTI-PURPOSE DECISION MAKING METHODS. Atatürk University Journal of Economics and Administrative Sciences 2011, 25, 71–97. [Google Scholar]
15. Guitouni, A.; Martel, J.M. Tentative guidelines to help choosing an appropriate MCDA method. European Journal of Operational Research 1998, 109, 501–521. [Google Scholar] [CrossRef]
16. Roy, B.; Słowiński, R. Questions guiding the choice of a multicriteria decision aiding method. EURO Journal on Decision Processes 2013, 1, 69–97. [Google Scholar] [CrossRef]
17. Triantaphyllou, E. Multi-criteria decision making methods: A comparative study; Springer, 2000.
18. Zavadskas, E.K.; Turskis, Z. Multiple criteria decision making (MCDM) methods in economics: an overview. Technological and economic development of economy 2011, 17, 397–427. [Google Scholar] [CrossRef]
19. Zanakis, S.H.; Solomon, A.; Wishart, N.; Dublish, S. Multi-attribute decision making: A simulation comparison of select methods. European journal of operational research 1998, 107, 507–529. [Google Scholar] [CrossRef]
20. Mulliner, E.; Malys, N.; Maliene, V. Comparative analysis of MCDM methods for the assessment of sustainable housing affordability. Omega 2016, 59, 146–156. [Google Scholar] [CrossRef]
21. Emovon, I.; Oghenenyerovwho, O.S. Application of MCDM method in material selection for optimal design: A review. Results in Materials 2020, 7, 100115. [Google Scholar] [CrossRef]
22. Saaty, T. The Analytical Hierarchy Process McGraw-Hill. New York 1980. [Google Scholar]
23. Saaty, T.L. Fundamentals of decision making and priority theory with the analytic hierarchy process; Vol. 6, RWS publications, 2000.
24. Saaty, T.L. Decision making with the analytic hierarchy process. International journal of services sciences 2008, 1, 83–98. [Google Scholar] [CrossRef]
25. Zadeh, L.A. Information and control. Fuzzy sets 1965, 8, 338–353. [Google Scholar]
26. Jie, L.H.; Meng, M.C.; Cheong, C.W. Web based fuzzy multicriteria decision making tool. International Journal of the computer, the Internet and management 2006, 14, 1–14. [Google Scholar]
27. Huang, H.C.; Ho, C. Applying the Fuzzy Analytic Hierarchy Process to Consumer Decision-Making Regarding Home Stays. International Journal of Advancements in Computing Technology 2013, 5, 981–990. [Google Scholar] [CrossRef]
28. İrfan Ertuğrul. ; Karakaşoğlu, N. Performance evaluation of Turkish cement firms with fuzzy analytic hierarchy process and TOPSIS methods. Expert Systems with Applications 2009, 36, 702–715. [Google Scholar] [CrossRef]
29. Dzitac, I. The fuzzification of classical structures: A general view. International Journal of Computers Communications & Control 2015, 10, 12–28. [Google Scholar]
30. Deng, H. Multicriteria analysis with fuzzy pairwise comparison. International journal of approximate reasoning 1999, 21, 215–231. [Google Scholar] [CrossRef]
31. Korol, T. Fuzzy logic in financial management 2012.
32. Kuchta, D. Multicriteria Fuzzy Evaluation of Project Success in R&D Projects. Multiple Criteria Decision Making 2019, 14, 44–59. [Google Scholar]
33. Calabrese, A.; Costa, R.; Levialdi, N.; Menichini, T. A fuzzy analytic hierarchy process method to support materiality assessment in sustainability reporting. Journal of Cleaner Production 2016, 121, 248–264. [Google Scholar] [CrossRef]
34. Kahraman, C.; Cebeci, U.; Ulukan, Z. Multi-criteria supplier selection using fuzzy AHP. Logistics information management 2003. [Google Scholar] [CrossRef]
35. Van Laarhoven, P.J.; Pedrycz, W. A fuzzy extension of Saaty’s priority theory. Fuzzy sets and Systems 1983, 11, 229–241. [Google Scholar] [CrossRef]
36. Kannan, D.; Khodaverdi, R.; Olfat, L.; Jafarian, A.; Diabat, A. Integrated fuzzy multi criteria decision making method and multi-objective programming approach for supplier selection and order allocation in a green supply chain. Journal of Cleaner production 2013, 47, 355–367. [Google Scholar] [CrossRef]
37. Buckley, J.J. Fuzzy hierarchical analysis. Fuzzy sets and systems 1985, 17, 233–247. [Google Scholar] [CrossRef]
38. Hwang, C.L.; Yoon, K. Methods for multiple attribute decision making. In Multiple attribute decision making; Springer, 1981; pp. 58–191.
39. Sabaei, D.; Erkoyuncu, J.; Roy, R. A review of multi-criteria decision making methods for enhanced maintenance delivery. Procedia CIRP 2015, 37, 30–35. [Google Scholar] [CrossRef]
40. Chen, C.T. Extensions of the TOPSIS for group decision-making under fuzzy environment. Fuzzy sets and systems 2000, 114, 1–9. [Google Scholar] [CrossRef]
41. Junior, F.R.L.; Osiro, L.; Carpinetti, L.C.R. A comparison between Fuzzy AHP and Fuzzy TOPSIS methods to supplier selection. Applied soft computing 2014, 21, 194–209. [Google Scholar] [CrossRef]
42. Opricovic, S. Multicriteria optimization of civil engineering systems. Faculty of Civil Engineering, Belgrade 1998, 2, 5–21. [Google Scholar]
43. Chen, L.Y.; Wang, T.C. Optimizing partners’ choice in IS/IT outsourcing projects: The strategic decision of fuzzy VIKOR. International Journal of Production Economics 2009, 120, 233–242. [Google Scholar] [CrossRef]
44. Opricovic, S.; Tzeng, G.H. Compromise solution by MCDM methods: A comparative analysis of VIKOR and TOPSIS. European journal of operational research 2004, 156, 445–455. [Google Scholar] [CrossRef]
45. Ishizaka, A.; Nemery, P. Multi-criteria decision analysis: methods and software; John Wiley & Sons, 2013.
46. Yoon, K.P.; Hwang, C.L. Multiple attribute decision making: an introduction; Sage publications, 1995.
47. Vakilipour, S.; Sadeghi-Niaraki, A.; Ghodousi, M.; Choi, S.M. Comparison between Multi-Criteria Decision-Making Methods and Evaluating the Quality of Life at Different Spatial Levels. Sustainability 2021, 13, 4067. [Google Scholar] [CrossRef]