1. Introduction

2. Materials and Methods

2.2. The Fuzzy TOPSIS Method

“Technique for Order Preference by Similarity to Ideal Solution (TOPSIS)” is a “comprehensive distance-based evaluation method” developed by Hwang and Yoon (1981) and can be expressed as a method for ranking by ideal solution [43,53]. TOPSIS focuses on locating the solution point nearest to the positive ideal solution while being the furthest away from the negative perfect solution [54]. The ideal solution or the most suitable alternative is the one that maximizes the benefit criterion and minimizes the cost criterion.

Fuzzy TOPSIS is a decision-making method used to select, rank, and evaluate alternatives in quantitative and qualitative multi-criteria decision-making problems [55,56,57,58,59,60,61,62]. Fuzzy decision methods are necessary to incorporate human thoughts and evaluations in the solution process and make more realistic evaluations in ambiguous situations and events. In applying the fuzzy TOPSIS method, decision-makers verbally express their judgments about the decision criteria and alternatives. The similarity coefficient for the alternatives is calculated by converting the decision makers’ assessments about the criteria and alternatives into fuzzy numbers. The alternatives are listed using the calculated similarity coefficients, and the solution is presented. The fuzzy TOPSIS method developed by Chen (2000) is a method that can be applied in individual or group decision-making. The linguistic evaluations and fuzzy number equivalents proposed by Chen and used in evaluating alternatives are shown below [63].

Table 2. Linguistic variables for the ratings of alternatives [63].

Table 2. Linguistic variables for the ratings of alternatives [63].

Linguistic Variable	Triangular Fuzzy Number
Very Poor (VP)	(0, 0, 1)
Poor (P)	(0, 1, 3)
Medium Poor (MP)	(1, 3, 5)
Fair (F)	(3, 5, 7)
Medium Good (MG)	(5, 7, 9)
Good (G)	(7, 9, 10)
Very Good (VG)	(9, 10, 10)

In a group of K decision makers, the aggregated fuzzy importance weight of the decision criteria affecting the problem and the aggregated fuzzy ratings of the alternatives according to each criterion can be calculated with the following equations.

$$\widetilde{W_{ij}}=\frac{1}{k}\left[{\tilde{w}}_{ij}^1+{\tilde{w}}_{ij}^2+\cdots +{\tilde{w}}_{ij}^K\right]$$

(2.8)

$$\widetilde{X_{ij}}=\frac{1}{k}\left[{\tilde{x}}_{ij}^1+{\tilde{x}}_{ij}^2+\cdots +{\tilde{x}}_{ij}^K\right]$$

(2.9)

The decision matrix ($\tilde{D})$ and criterion weights ($\tilde{W})$ of a multi-criteria decision problem can be represented as follows:

$$C_1\hspace{1em}\hspace{1em}\hspace{1em}{C}_2\hspace{1em}\hspace{1em}\cdots \hspace{1em}{C}_3$$

$$\tilde{D}=\begin{array}{c}{A}_1\\ A_2\\ \begin{array}{c}⋮\\ \cdots \\ A_m\end{array}\end{array}\left[\begin{array}{c}\begin{array}{cccc}{\tilde{x}}_{11}& {\tilde{x}}_{12}& ⋮& {\tilde{x}}_{1n}\end{array}\\ \begin{array}{cccc}{\tilde{x}}_{21}& {\tilde{x}}_{22}& ⋮& {\tilde{x}}_{2n}\end{array}\\ \begin{array}{cccc}⋮& ⋮& ⋮& ⋮\\ ⋮& ⋮& ⋮& ⋮\\ {\tilde{x}}_{m1}& {\tilde{x}}_{m2}& \cdots & {\tilde{x}}_{mn}\end{array}\end{array}\right],\tilde{W}=\left[{\tilde{w}}_1,\hspace{1em}{\tilde{w}}_2,\cdots \cdots \cdots ,{\tilde{w}}_n\right]$$

(2.10)

where ${\tilde{x}}_{ij},$ $\left(\forall i,j\right)$ ve ${\tilde{w}}_j,$ $j=(1,2,3,\cdots ,n)$ are linguistic variables and $A_1,A_2,A_3,\cdots ,A_m$, are alternatives; K decision makers and their number; $C_1,C_2,C_3,\cdots ,C_n$ are the decision criteria; ${\tilde{x}}_{ij}$ is the criterion value of the alternative $A_i$ with respect to a decision criterion$C_j$; and ${\tilde{w}}_j$ is the importance weight of the criterion $C_j$. $\tilde{D}$ is expressed as a fuzzy decision matrix and $\tilde{W}$ as a matrix of fuzzy weights. The elements and weights of the matrix are represented as fuzzy numbers ${\tilde{x}}_{ij}=(a_{ij},b_{ij},c_{ij})$ and ${\tilde{w}}_j=(w_{j1},w_{j2},w_{j3})$.

The next step is to calculate the normalized fuzzy decision matrix. This matrix is represented as follows:

$$\tilde{R}={\left[{\tilde{r}}_{ij}\right]}_{mxn}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}i=\mathrm{1,2},3,\cdots ,m,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}j=\mathrm{1,2},3,\cdots ,n.$$

(2.11)

Each element of the normalized fuzzy decision matrix is calculated by the following equations, with B representing the benefit and C the cost criteria:

$${\tilde{r}}_{ij}=\left(\frac{a_{ij}}{c_j^{*}},\frac{b_{ij}}{c_j^{*}},\frac{c_{ij}}{c_j^{*}}\right),\hspace{1em}\hspace{1em}{c}_j^{+}=max{c}_{ij},{\hspace{1em}\forall }_j\in B$$

(2.12)

$${\tilde{r}}_{ij}=\left(\frac{a_j^{-}}{c_{ij}},\frac{a_j^{-}}{c_{ij}},\frac{a_j^{-}}{c_{ij}}\right),\hspace{1em}\hspace{1em}{a}_j^{-}=min{a}_{ij},{\hspace{1em}\forall }_j\in C$$

(2.13)

It is ensured that the fuzzy number values in a normalized matrix are in the range of [0,1]. After calculating the normalized decision matrix, the weighted normalized decision matrix is calculated by considering the importance weight of the criteria. The weighted normalized decision matrix is shown as

$$\tilde{V}={\left[{\tilde{v}}_{ij}\right]}_{mxn}\hspace{1em}\hspace{1em}\hspace{1em}i=\mathrm{1,2},3,\cdots ,m,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}j=\mathrm{1,2},3,\cdots ,n.$$

(2.14)

where ${\tilde{v}}_{ij}={\tilde{r}}_{ij}.{\tilde{w}}_j$.

The weighted normalized fuzzy decision matrix is obtained by multiplying the normalized fuzzy decision matrix with the fuzzy weights matrix. In this case, the calculated $\tilde{V}$ matrix is shown as:

$$\tilde{V}=\left[\begin{array}{ccc}\begin{array}{c}{\tilde{w}}_1{\tilde{r}}_{11}\\ {\tilde{w}}_1{\tilde{r}}_{21}\\ ⋮\end{array}& \begin{array}{c}\begin{array}{cc}{\tilde{w}}_2{\tilde{r}}_{12}& ⋮\end{array}\\ \begin{array}{cc}{\tilde{w}}_2{\tilde{r}}_{22}& ⋮\end{array}\\ \begin{array}{cc}⋮& \hspace{1em}⋮\end{array}\end{array}& \begin{array}{c}{\tilde{w}}_n{\tilde{r}}_{1n}\\ {\tilde{w}}_n{\tilde{r}}_{2n}\\ ⋮\end{array}\\ ⋮& \begin{array}{cc}\hspace{1em}⋮& \hspace{1em}⋮\end{array}& ⋮\\ {\tilde{w}}_1{\tilde{r}}_{m1}& \begin{array}{cc}{\tilde{w}}_2{\tilde{r}}_{m2}& ⋮\end{array}& {\tilde{w}}_n{\tilde{r}}_{mn}\end{array}\right]$$

(2.15)

After calculating the weighted normalized fuzzy decision matrix $\tilde{V}$, the fuzzy positive ideal solution $A^{*}$ and the fuzzy negative ideal solution $A^{-}$ can define as;

$$A^{*}=\left\{v_1^{*},v_2^{*},\cdots \cdots ,v_n^{*}\right\}$$

(2.16)

$$A^{-}=\left\{v_1^{-},v_2^{-},\cdots \cdots ,v_n^{-}\right\}$$

(2.17)

where ${\tilde{v}}_j^{\mathrm{*}}=\left(\mathrm{1,1},1\right) and {\tilde{v}}_j^{-}=\left(\mathrm{0,0},0\right)j=1,2,3,\cdots ,n.$

Then it is necessary to calculate the distances of the alternatives from $A^{*}$ and $A^{-}$ In this calculation, d represents the distances between fuzzy numbers and the calculation is done with the following equations:

$${\mathit{d}}_{\mathit{i}}^{\mathbf{*}}=\sum _{\mathit{j}=1}^{\mathit{n}}\mathit{d}\left({\tilde{\mathit{v}}}_{\mathit{i}\mathit{j}},{\tilde{\mathit{v}}}_{\mathit{j}}^{\mathbf{*}}\right),\mathit{i}=1,2,3,\cdots \cdots ,\mathit{m}$$

(2.18)

$${\mathit{d}}_{\mathit{i}}^{-}=\sum _{\mathit{j}=1}^{\mathit{n}}\mathit{d}\left({\tilde{\mathit{v}}}_{\mathit{i}\mathit{j}},{\tilde{\mathit{v}}}_{\mathit{j}}^{-}\right),\mathit{i}=1,2,3,\cdots \cdots ,\mathit{m}$$

(2.19)

Finally, the closeness of the alternatives to the ideal solution is calculated. For this, the Vertex method, which is used to calculate the distances to one of the fuzzy numbers, is used. The distance between two triangular fuzzy numbers, such as $\tilde{A}=\left(a_1,a_2,a_3\right)$ and $\tilde{B}=\left(b_1,b_2,b_3\right)$ is calculated according to the Vertex method with the following equation:

$${\mathit{d}}_{\mathit{v}}\left(\tilde{\mathit{a}},\tilde{\mathit{b}}\right)=\sqrt{\frac{1}{3}}\left[{\left({\mathit{a}}_1-{\mathit{b}}_1\right)}^2+{\left({\mathit{a}}_2-{\mathit{b}}_2\right)}^2+{\left({\mathit{a}}_3-{\mathit{b}}_3\right)}^2\right]$$

(2.20)

To evaluate the alternatives and make a choice, the closeness coefficients are calculated for each alternative with the help of the following equation [64].

$${\mathit{C}\mathit{C}}_{\mathit{i}}=\frac{{\mathit{d}}_{\mathit{i}}^{-}}{{\mathit{d}}_{\mathit{i}}^{\mathbf{*}}+{\mathit{d}}_{\mathit{i}}^{-}},$$

(2.21)

Alternatives are decided to rank according to ${CC}_i$ (the closeness coefficient) values. Suppose the closeness coefficient is equal to 1. In that case, the value of the alternative in question equals the fuzzy positive ideal solution. If the closeness coefficient is equal to 0, the value of the alternative is equal to the fuzzy negative ideal solution.

3. Case Study Analyses and Results

3.2. Background: Low-Cost (Carriers) Airlines in Türkiye

The second stage of the study is to determine the criteria to be used to evaluate alternatives. For this purpose, a detailed literature review was conducted, and five main criteria and 18 sub-criteria given in Table 3 were obtained with expert support.

These criteria categories are coded for each primary and sub-criterion as follows; Online Services (OS) (three sub-criteria; OS1 to OS3), Ticketing (T) (four sub-criteria; T1 to T4), Counter Services (CS) (five sub-criteria; CS1 to CS5), Service Personnel (SP) (two sub-criteria; SP1 and SP2) and Flight Schedule and Routes (FSR) (four sub-criteria; FSR1 to FSR4).

Six experts were consulted in the evaluation of the study. In order to categorize the criteria and compare them pairwise with the fuzzy AHP approach, the views of six experts were employed, and the evaluation of the airlines with the fuzzy TOPSIS method conducted. Three of them are professionals and passengers who travel frequently with airlines, and the other two are experts with many years of airline experience in various ground service roles. One is an airline employee with many years of flying experience. The other expert is an industrial/process engineer with experience in process management and specializes in airline and travel issues. The average experience of the experts is 10+ years. The five-point linguistic expression scale given in Table 1 is used for the pairwise comparisons. According to the results of the observations, the data obtained were summarized, and the average estimates of the six experts were used in this article.

Figure 2. The hierarchical structure of the study.

Figure 2. The hierarchical structure of the study.

4. Discussion and Conclusions

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