Aircraft selection using fuzzy ANP and the generalized choquet integral method: The Turkish airlines case

Abstract

Airway transportation is the fastest type of transportation. Advances in aviation technologies, planes’ increased speed, and the large number of companies in the airway transportation sector cause rising competition and lower prices. In such an important sector, each type of investment has to be feasible and logical, e.g. airport location, aircraft purchase, services, crews and their assignment, etc. Whether purchasing, renting or chartering aircraft, airline companies’ biggest cost is the maintenance of their fleet. For this reason, in this paper we study the purchase of aircraft in a Turkish airline company using fuzzy numbers. Multi criteria-decision making techniques, fuzzy ANP and Choquet integral methods are used for evaluation and the results of both algorithms are compared by using fuzzy AHP. The main contribution of this paper is to determine the interdependency among main criteria and subcriteria, the nonlinear relationship among them and the environmental uncertainties. To the authors’ knowledge, this will be the first study which compares the results of fuzzy ANP and Choquet integral methods using fuzzy AHP.

Keywords

Aircraft fuzzy AHP fuzzy ANP choquet integral multi-criteria decision making (MCDM)

1 Introduction and literature review

In recent years, a decrease in ticket prices in the aviation sector has meant that passengers prefer airline transport to the alternatives. Prices have been competitive compared with the ticket prices of other transportation types.

Selecting or prioritizing alternatives from a set of available alternatives with respect to multiple criteria is often referred to multi-criteria decision-making (MCDM). MCDM is a well-known branch of a general class of operation research models which deal with decision problems in the presence of a number of decision criteria. This class is further divided into multi-objective decision-making (MODM) and multi-attribute decision-making (MADM). There are several methods in each of the above categories. Priority-based, outranking, distance-based and mixed methods are also applied to various problems. Each method has its own characteristics and such methods can also be classified as deterministic, stochastic and fuzzy methods [1].

In this paper we apply two MCDM methods to select the best aircraft alternative, namely fuzzy ANP and Choquet integral. This is the first paper in the literature to apply these two techniques in the aviation sector. The motivation for the research is the need of multicriteria decision making method, which can handle the inner and/or outer dependencies and the interactions between the elements of an aviation network. For this reason, we aim at comparing the performances of ANP and Choquet integral under fuzziness. We study the purchase of aircraft using these techniques in a Turkish airline company.

The analytic network process (ANP) is one of the common methods with which to solve MCDM problems. The decision problem is structured hierarchically at different levels in the methodology [2]. The local priorities in ANP are established by means of pairwise comparisons and judgments [3]. The analytical network process is a generalization of Saaty’s analytical hierarchy process, which is one of the most widely employed decision support tools [4]. The priorities in the ANP are assessed indirectly by means of pairwise comparison judgments [5].

ANP is a useful tool for prediction and for representing a variety of competitors, their assumed interactions and their relative strengths to wield influence in making a decision [6].

The Choquet integral is a fuzzy integral with a numerical structure which is used to evaluate selection criteria by dividing them into parts. Successful establishment of a Choquet integral depends on results that the fuzzy criteria impose, which in turn establish the importance of each criterion or their combination [7].

In this paper, the selection of the best aircraft in relation to the criteria determined from the alternative aircrafts for an airline company in Turkey was handled as an MCDM problem and we used fuzzy ANP and generalized Choquet integral methodology to solve the problem. Then to evaluate the results of these methodologies, fuzzy AHP was used with the same data of the Choquet integral and the fuzzy ANP. We aimed to find a useful way to handle fuzzy MCDM problems in a more flexible and more intelligent manner.

The rest of this paper is organized as follows: the problem definition is described in Section 2. Fuzzy ANP methodology and Choquet integral methodology are presented in Section 3 and Section 4, respectively. In Section 5, we show a real application of fuzzy ANP methodology and Choquet integral methodology in aircraft selection. Computational results and evaluation of the results using fuzzy AHP are given in this section. Finally, comparison of the results and future research directions are discussed in Section 6, which concludes the paper.

2 Problem definition

As we explained above, aircraft transportation is crucial in today’s world. An aircraft selection problem was chosen for this study and fuzzy ANP and Choquet integral approaches were used. We asked three aviation sector experts about the problem of selecting an aircraft type. Three main criteria, 10 sub-criteria and three alternatives were determined and weighted accordingly.

In the numerical example, the biggest Turkish airline company, Turkish Airlines, needs to purchase a middle-of-the-range standard body, and single corridor aircraft. For this problem, decision criteria and alternatives were defined by experts, as seen in Fig. 1. In this paper the main criteria are cost, time, physical attributes, etc. The arrows in Fig. 1 represent the interactions among the subcriteria.

Cost criteria include sub-criteria on financial decisions: “Purchasing Cost”, “Operation and Spare Cost”, “Maintenance Cost”, and “Salvage Cost”.

Time criteria include the following sub-criteria: “Delivery Time” and the “Useful Life” of the aircraft.

Physical attributes etc. include these sub-criteria: “Dimensions”, “Security”, “Reliability”, and “Suitability for Service Quality”.

As seen in Fig. 1 the alternatives for aircraft selection are A, B, and C.

3 Fuzzy ANP methodology

The analytic network process (ANP) is a generalization of the analytic hierarchy process (AHP) which can take the inner and outer dependencies among multiple criteria into consideration. ANP is used to determine the priorities of the elements in the network and the alternatives of the goal. ANP allows modeling complex and dynamic environments which are influenced by changing external factors [8]. ANP is an excellent methodology which can deal with several issues by considering dependencies between nodes and clusters of criteria [9].

Buckley’s fuzzy AHP algorithm [10 –12] based fuzzy ANP is used for selecting the best aircraft in this paper. Buckley extended Saaty’s AHP to the case where the evaluators are allowed to employ fuzzy ratios in place of exact ratios to handle the difficulty for people to assign exact ratios when comparing two criteria and derive the fuzzy weights of criteria by geometric mean method [10]. Fuzzy ANP allows measuring qualitative factors by using fuzzy numbers instead of crisp values in order to make decisions easier and obtain more realistic results [13].

In the literature, the fuzzy ANP method has been used to solve problems like research and development project selection [14], performance evaluation [15], quality function deployment implementation [16], enterprise resource planning (ERP), and software selection [17], etc.

Öztayşi et al. [9] compared the CRM performances of e-commerce firms using a multiple criteria decision making (MCDM) approach - ANP. A sensitivity analysis also provided in order to monitor the robustness of the proposed ANP framework to changes in the weights of evaluation criteria. Their results showed that the ranking among the alternatives are sensitive to changes in the parameters.

Tuzkaya et al. [6] proposed an integrated fuzzy multi-criteria decision making methodology for selecting material handling equipment. The proposed approach utilizes fuzzy sets, ANP and the preference ranking organization method for enrichment evaluations (PROMETHEE).

Tuzkaya and Onut [18] proposed a model for selecting the most convenient transportation mode by considering the effects of criteria on the alternative modes and relations among the criteria clusters and subcriteria using fuzzy ANP.

Buyukozkan et al. [19] used fuzzy ANP to prioritize design requirements by taking into account the degree of the interdependence between customer needs and design requirements and their dependence.

Ebrahimnejad et al. [20] studied a construction project problem with multiple criteria in a fuzzy environment and proposed a new two-phase group decision-making approach. This approach integrated a modified analytic network process (ANP) and an improved compromise ranking method, VIKOR.

Zhou et al. [21] proposed a flexibility measurement model of enterprise resources planning (ERP) based on a fuzzy analytic network process (FANP). Hung et al. [22] applied the fuzzy analytic network process model to evaluate the strategic impact of new integrated circuit (IC) manufacturing technologies within Taiwan’s packaging industry.

In the F-ANP, to evaluate the decision-makers’ preferences, pairwise comparisons are structured using triangular fuzzy numbers (a^l, a^m, a^u) since the most used fuzzy numbers in the literature are triangular because of their computational simplicity. Decision makers’ preferences can be also represented using trapezoidal fuzzy numbers (a^l, a^m, aⁿ, a^u). A trapezoidal fuzzy number becomes a triangular fuzzy number when a^m= aⁿ. The m x n fuzzy matrix can be given as in Equation 1. The element a_mn represents the comparison of the component m (row element) with component n (column element). If $\tilde{A}$ is a pairwise comparison matrix Equation 1, it is assumed that the reciprocal, and the reciprocal value, i.e. 1/a_mn, is assigned to the element a_mn [6, 18]: $\tilde{A} = (\begin{matrix} (1, 1, 1) & a_{12}^{l}, a_{12}^{m}, a_{12}^{u} & \dots & a_{1 n}^{l}, a_{1 n}^{m}, a_{1 n}^{u} \\ (\frac{1}{a_{11}^{u}}, \frac{1}{a_{11}^{m}}, \frac{1}{a_{11}^{l}}) & (1, 1, 1) & \dots & a_{2 n}^{l}, a_{2 n}^{m}, a_{2 n}^{u} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ (\frac{1}{a_{1 n}^{u}}, \frac{1}{a_{1 n}^{m}}, \frac{1}{a_{1 n}^{l}}) & (\frac{1}{a_{2 n}^{u}}, \frac{1}{a_{2 n}^{m}}, \frac{1}{a_{2 n}^{l}}) & \dots & (1, 1, 1) \end{matrix})$ (1)

The steps of fuzzy ANP can be listed as follows [23]:

Step 1: Determine alternatives, criteria and subcriteria to be used in the model

Step 2: Create a network including alternatives, criteria, subcriteria, inner and outer dependencies among the model.

Step 3: Construct pairwise matrices of the components by the experts with fuzzy numbers.

Step 4: Construct the fuzzy comparison matrix by using triangular fuzzy numbers:

Zadeh [24] introduced the fuzzy set theory to deal with the uncertainty due to imprecision and vagueness. A major contribution of fuzzy set theory is its capability of representing vague data. A triangular fuzzy number that defined as (l,m,u), where l≤m≤u, denote the smallest possible value, the most promising value and the largest possible value.

Step 5: Calculate fuzzy eigen value to find whether the constructed matrix is consistent or not:

To verify the consistency of the comparison matrix, Saaty proposed a consistency index (C.I.) and consistency ratio (C.R.). The consistency index of a matrix is given by $C . I . = (λ max - n) / (n - 1)$ (2) $C . R = C . I . / R . I$ (3) where, R.I is Random Consistency Index. The consistency index should be less than or equal to 0.10.

Step 6: Forming initial supermatrix of the network of ANP is composed by listing all nodes horizontally and vertically.

Step 7: Obtaining weighted supermatrix by multiplying the unweighted supermatrix with the corresponding cluster priorities.

Step 8: Calculating limited supermatrix by limiting the weighted supermatrix by raising it to sufficiently large power so that it converges into a stable supermatrix (i.e, all columns being identical).

4 Choquet integral methodology

Choquet integral is a sort of general averaging operator that can represent the notions of importance of a criterion and interactions among criteria. A set of values of importance is composed of the values of a fuzzy measure. The success of a Choquet integral depends on an appropriate representation of fuzzy measures, which captures the importance of individual criterion or their combination [25].

Chen and Tzeng [26] tried to construct a traffic assignment model using a fuzzy travel cost function based on the possibility concept instead of precise calculation of traffic volumes. The techniques of fuzzy measure and fuzzy integral are applied to calculate the subjectively perceived travel costs during traffic assignment. An example with 22 nodes and 36 links were used to illustrate their study.

The Choquet integral has been used for the solution of multiple criteria decision-making problems in the literature. Chiou et al. [27] proposed the non-additive fuzzy integral (Choquet integral) to cope with evaluation of fuzzy MCDM problems while there is dependence among considered criteria. The sustainable development strategy for aquatic product processors in Taiwan was investigated.

Meyer and Roubens [28] presented a multiple criteria decision support approach in order to build rankings and suggest the best choice from a set of alternatives using the Choquet integral.

Tsai and Lu [29] generalized the standard Choquet integral whose measurable evidence and fuzzy measures are real numbers. They showed that their proposed generalization can deal with fuzzy number types of measurable evidence and fuzzy measures.

Kong et al. [30] presented a fuzzy Choquet integral approach to evaluate the capability of supplier and to deal with the supplier selection problem. Firstly, the basic of fuzzy measures and Choquet integral were introduced and then linguistic terms expressed in trapezoidal fuzzy numbers were used to assess the ratings for the suppliers’ quality and the importance of criteria. Lastly an example was given to demonstrate their model.

Saad et al. [31] introduced the Choquet integral for dealing with MCDM and used in optimization flexible job-shop scheduling problems.

Demirel et al. [25] proposed a multi-criteria decision-making method using a fuzzy integral for the evaluation of alternative warehouse locations. They first determined the main and sub-criteria and the hierarchy for the warehouse location selection problem, then made a multi-criteria evaluation of the warehouse location alternatives to illustrate how the generalized Choquet integral can be used todo this.

Göztepe [32] aimed to develop a decision model based on Analytic Network Process (ANP)-Choquet integral integration that select an appropriate operating system for critical computer systems by taking subjective judgments of decision makers into consideration. An ANP is used for determining the weights of the criteria by decision makers and then Choquet integral is applied in ranking of the operating systems.

Yayla et al. [7] presented a case study on the selection of the optimal subcontractor for a Turkish textile firm using generalized Choquet integral methodology.

Grabisch et al. [33] analyzed the discrete Choquet integrals as aggregation tools to integrate a function with respect to a fuzzy measure. They studied the properties of aggregation function methods (triangular norms and conorms, copulas, means and averages, and those based on nonadditive integrals) together with construction methods and practical identification methods. Without going deeper into details, they defined nonadditive integrals (e.g., the Choquet integral) with respect to capacities (nonadditive monotone measures), and generalized the notion of expected value.

Wang and Klir [34] presented a general overview on fuzzy integration theory. They included Sugeno integral, Pan-integrals, Choquet integrals, and upper and lower integrals in their book.

Pap [35] introduced a unified approach to the non-additive set functions with applications. Their approach includes triangular set functions, fuzzy measures, decomposable measures, possibility measures, distorted probabilities, autocontinuous set functions, etc.

Relationship between trapezoidal fuzzy numbers and degrees of linguistic importance on a nine-linguistic-term scale can be seen fromTable 2.

The methodology is composed of eight steps [25, 29]:

Step 1. Given criterion i, respondents’ linguistic preferences for the degree of importance, perceived performance levels of alternative locations, and tolerance zone are surveyed.

Step 2. In view of the compatibility between perceived performance levels and the tolerance zone, trapezoidal fuzzy numbers are used to quantify all linguistic terms. Given respondent t and criteria i, linguistic terms for the degree of importance is parameterized by

${\tilde{A}}_{i}^{t} = (a_{i 1}^{t}, a_{i 2}^{t}, a_{i 3}^{t}, a_{i 4}^{t})$ , perceived performance levels by ${\tilde{p}}_{i}^{t} = (p_{i 1}^{t}, p_{i 2}^{t}, p_{i 3}^{t}, p_{i 4}^{t})$ , and the tolerance zone by ${\tilde{e}}_{i}^{t} = (e_{i 1 L}^{t}, e_{i 2 L}^{t}, e_{i 3 U}^{t}, e_{i 4 U}^{t})$ .

Step 3. Average ${\tilde{A}}_{i}^{t}$ , ${\tilde{p}}_{i}^{t}$ and ${\tilde{e}}_{i}^{t}$ into ${\tilde{A}}_{i}$ and ${\tilde{e}}_{i}$ respectively using Equation 4.

$\begin{matrix} {\tilde{A}}_{i} = \frac{\sum_{t = 1}^{k} {\tilde{A}}_{i}^{t}}{k} = \\ (\frac{\sum_{t = 1}^{k} a_{i 1}^{t}}{k}, \frac{\sum_{t = 1}^{k} a_{i 2}^{t}}{k}, \frac{\sum_{t = 1}^{k} a_{i 3}^{t}}{k}, \frac{\sum_{t = 1}^{k} a_{i 4}^{t}}{k}) \end{matrix}$ (4)

Step 4. Normalize the location value of each criterion using Equation 5. ${\tilde{f}}_{i} = \underset{α \in [0, 1]}{∥ {\bar{f}}_{i}^{α}} = \underset{α \in [0, 1]}{∥ [f_{i α}^{-}, f_{i α}^{+}]},$ (5) where f_i ∈ F (S) is a fuzzy-valued function. $\tilde{F} (S)$ is the set of all fuzzy-valued functions $f, f_{i}^{α} = [f_{i α}^{-}, f_{i α}^{+}] = \frac{{\bar{p}}_{i}^{α} - {\bar{e}}_{i}^{α} + [1, 1]}{2}, {\bar{p}}_{i}^{α}$ and ${\bar{e}}_{i}^{α}$ are α-level cuts of ${\tilde{p}}_{i}$ and ${\tilde{e}}_{i}$ for all α = [0, 1].

Step 5. Find the location value of dimension j using Equation 6. $(C) \int \tilde{fd} \tilde{g} = \underset{α = [0, 1]}{∥ ⌊ (C) \int f_{α}^{-} {dg}_{α}^{-}, (C) \int f_{α}^{+} {dg}_{α}^{+} ⌋}$ (6) where ${\bar{g}}_{i} : P (S) \to I (R^{+})$ , ${\bar{g}}_{i} = [g_{i}^{-}, g_{i}^{+}]$ , ${\bar{g}}_{i}^{α} = [g_{i, α}^{-}, g_{i, α}^{+}]$ , ${\bar{f}}_{i} : S \to I (R^{+})$ , and $f_{i} = [f_{i}^{-}, f_{i}^{+}]$ for i = 1, 2, 3, …, n_j.

To be able to calculate this location value, a λ value and the fuzzy measures g (A_(i)) , i = 1, 2, 3, …, n are needed. These are obtained from the following Equations 7–9. $g (A_{(n)}) = g ({s_{(n)}}) = g_{n},$ (7) $\begin{matrix} g (A_{(i)}) = g_{i} + g (A_{(i + 1)}) + λ g_{i} g (A_{(i + 1)}), \\ where 1 \leq i < n \end{matrix}$ (8) $\begin{matrix} 1 = g (S) = {\begin{matrix} 1 / λ {\prod_{i = 1}^{n} [1 + λ g (A_{i})] - 1} & if λ \neq 0 \\ \sum_{i = 1}^{n} g (A_{i}) & if λ = 0, \end{matrix} \end{matrix}$ (9) where, A_i∩ A_j = Ø for all i, j = 1, 2, 3, …, n and i ≠ j, and λ ∈ (-1, ∞].

Let μ be a fuzzy measure on (I, P (I)) and an application $f : I \to R^{+} .$ The Choquet integral of f with respect to μ is defined by: $(C) \int_{I} fd μ = \sum_{i = 1}^{n} (f (σ (i)) - f (σ (i - 1))) μ (A_{(i)})$ (10)

where σ is a permutation of the indices in order to have

f (σ (1)) ≤ … ≤ f (σ (n)) , A_(i) = {σ (i) , …, σ (n)} and xf (σ (0)) =0, by convention.

Under rather general assumptions over the set of alternatives X, and over the weak orders ${\underline{≻}}_{i},$ there exists a unique fuzzy measure μ over I such that [25]: $\forall x, y \in X, x \underline{≻} y \Leftrightarrow u (x) \geq u (y),$ (11) where $u (x) = \sum_{i = 1}^{n} [u_{(i)} (x_{(i)}) - u_{(i - 1)} (x_{(i - 1)})] μ (A_{(i)}),$ (12) which is simply the aggregation of the monodimensional utility functions using the Choquet integral with respect to μ.

Step 6. Aggregate all dimensional performance levels of the aircraft alternatives into overall performance levels, using a hierarchical process applying the two-stage aggregation process of the generalized Choquet integral Equation 13. The overall performance levels yields a fuzzy number, $\tilde{V} .$ $\begin{matrix} {maincriterion}_{(1)} = (C) \int fdg \\ ⋮ \\ {maincriterion}_{(m)} = (C) \int fdg \end{matrix} 〈 V = (C) \int maincriterion dg$ (13)

Step 7. Assume that the membership of $\tilde{V}$ is μ_v (x); defuzzy the fuzzy number $\tilde{V}$ into a crisp value v using Equation 14 and make a comparison of the overall performance levels of alternative aircrafts. $F (\tilde{A}) = \frac{a_{1} + a_{2} + a_{3} + a_{4}}{4} .$ (14)

Step 8. Compare weak and advantageous criteria among the aircraft alternatives using Equation 4 [25].

5 A real case study

5.1 Computational results of generalized Choquet integral methodology

To solve the problem using Choquet integral, we make the comparisons with experts using trapezoidal fuzzy numbers as shown in Table 1 and the average values can be seen on Table 2.

By using Equation 5, the performance criteria of the alternatives were normalized and the results were given in Tables 3 and 4. For α= 0, results evaluated by the generalized Choquet integral given in Table 3, were calculated using Equation 6.

All of the fuzzy measures and λ values obtained for α= 0 are listed in Table 5 and for α= 1 in Table 6.

By using the collecting process of the generalized Choquet integral total aircraft value belonging to the three alternatives, Equation 14 was simplified as shown in Table 7.

Using the total evaluation results of the application in Table 7, the evaluated list can be seen in Table 8.

When simplified values of the main criteria are taken into consideration, aircraft “C” takes first place according to the “Cost” criteria; aircraft “B” achieved the highest value for “Physical Attributes and Others” criteria, and aircraft “A” only for the “Time” criteria.

According to the results in Table 8 the ranking is obtained as B > A > C.

5.2 Computational results of Fuzzy ANP methodology

To solve the problem using fuzzy ANP, we used fuzzy numbers as shown in Table 9 and compared our results with those of experts. Evaluations of the alternatives by three experts with respect to the criteria were the same as the values of the Choquet integral that can be seen in Table 10. The fuzzy weight matrix of the criteria according to the goal, fuzzy weight matrix of the subcriteria and fuzzy weight matrix of the alternatives with respect to each criterion are given in Tables 11–13, respectively.

Also initial supermatrix, weighted supermatrix and the limited supermatrix can be seen from Tables 14–16. The evaluation and the methodology described above produced the results shown in Table 17.

According to the results in Table 17 the ranking is obtained as B > C > A.

5.3 Comparison of the results using Fuzzy AHP methodology

To evaluate the results of generalized Choquet integral and fuzzy ANP, we used one of the most common MCDM methodologies, fuzzy AHP, using the same data of the Choquet integral and the fuzzy ANP.

The steps of fuzzy AHP can be listed as follows [10, 12]:

Step 1: Evaluate the relative importance of the criteria using pairwise comparisons. Assign linguistic terms to the pairwise comparisons by asking which criterion is more important than the other. $\tilde{A} = [\begin{matrix} 1 & {\tilde{a}}_{12} & \dots & {\tilde{a}}_{1 n} \\ {\tilde{a}}_{21} & 1 & \dots & {\tilde{a}}_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\tilde{a}}_{n 1} & {\tilde{a}}_{n 2} & \dots & 1 \end{matrix}]$ (15) $= [\begin{matrix} 1 & {\tilde{a}}_{12} & \dots & {\tilde{a}}_{1 n} \\ 1 / {\tilde{a}}_{12} & 1 & \dots & {\tilde{a}}_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 1 / {\tilde{a}}_{1 n} & 1 / {\tilde{a}}_{2 n} & \dots & 1 \end{matrix}]$ (16)

Step 2: Define the fuzzy geometric mean and fuzzy weight of each criterion. ${\tilde{r}}_{i} = ({\tilde{a}}_{i 1} \otimes {\tilde{a}}_{i 2} \otimes \dots \otimes {\tilde{a}}_{in})^{1 / n}, {\tilde{w}}_{i}$ (17) $= {\tilde{r}}_{i} \otimes ({\tilde{r}}_{1} \oplus \dots \oplus {\tilde{r}}_{n})^{- 1}$ (18) where ${\tilde{a}}_{in}$ is the fuzzy comparison value of criterion i to criterion n, thus, ${\tilde{r}}_{i}$ is the geometric mean of fuzzy comparison value of criterion i to each criterion, ${\tilde{w}}_{i}$ is the fuzzy weight of the ith criterion.

Step 3: Defuzzify and normalize the fuzzy weights.

Fuzzy numbers, evaluations of the alternatives by three experts with respect to the criteria, the fuzzy weights matrix of the criteria according to the goal, fuzzy weights matrix of the subcriteria and fuzzy weights matrix of the alternatives with respect to each criterion can be seen in Tables 9–13, respectively. The methodology (F-AHP) produced the results shown in Table 18.

According to the results in Table 18 the ranking is obtained as B > C> A.

The impact of interactivity among main criteria and subcriteria and inner and outer dependencies are the reason of the variations in the rankings.

Given these results, it is fair to say that selecting Alternative B is the most reasonable and the most feasible outcome, followed by the others (Table 19).

6 Conclusion

In this paper, multi criteria-decision making techniques, fuzzy ANP and Choquet integral methods are used for the evaluation of aircrafts. Then, the obtained results of these techniques are compared using fuzzy AHP. As a result of evaluation process, these two MCDM methods, fuzzy ANP and Choquet integral, have determined the most suitable result as B. But the ranking of the other alternatives are not same. It is C > A in fuzzy ANP and in fuzzy AHP whereas it is A > C in Choquet integral.

The reason of this difference can be thought of as at ANP and AHP calculation step, there is an assumption that criteria are non-interactive. Even so, main criteria, subcriteria, and alternatives are not entirely non-interactive among themselves. A Choquet integral methodology considers interactivity among main criteria and subcriteria. When interactions among criteria exist, Choquet integral is proved to be an adequate aggregation operator by taking into account the interactions [37]. The main advantage of the proposed model is to indicate the impact of this interactivity. The main contribution of this paper is to determine the interdependency, the nonlinear relationship and the environmental uncertainties.

The general limitation of the proposed model is the costly and exhausting information requested from experts (approx. 120 pairwise comparisons per one expert). Other limitations of the model are the preferences of the expert including uncertainty and conflicts and there is often needed more than one expert to make decisions.

As regards future research, the problem could be solved by other MCDM techniques to explain interdependency among main criteria and subcriteria, and more solutions compared. Also, other MCDM techniques with fuzzy numbers could be used in aircraft selection processes and intelligent software to calculate solutions automatically developed.

Footnotes

Acknowledgments

We appreciate the help and support that we have received from Turkish Airlines and its managerial team. We extend our deepest gratitude to Dr. Ahmet Bolat, Chief Investment and Technology Officer at Turkish Airlines, and Dr. Cengiz Kahraman, Professor at Istanbul Technical University, for their unwavering support and dedication to this study.

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