A fuzzy AHP-VIKOR approach for evaluation of educational use simulation software packages

Abstract

In this study, an intelligent, integrated approach is presented to help educators select the best simulation software package. Selecting the best simulation software package for educational use is a complex multiple criteria decision making (MCDM) problem with several potentially conflicting quantitative and qualitative criteria. In this paper, two fuzzy MCDM methods; fuzzy Analytic Hierarchy Process (F-AHP) and fuzzy VIsekriterijumska optimizacija i KOmpromisno Resenje (F-VIKOR) are integrated to evaluate educational use simulation software package alternatives. In the proposed fuzzy AHP-VIKOR approach, F-AHP is used to determine the fuzzy criteria weights and F-VIKOR is applied to rank simulation software package alternatives with respect to these criteria. A case study is given where several educational use simulation software packages in Turkey are evaluated and ranked.

Keywords

Simulation software package selection fuzzy multiple-criteria decision making AHP VIKOR

1 Introduction

Today, simulation has a wide range of applications from manufacturing operations to service operations. Extensive calculations needed to simulate a real life problem makes the usage of simulation software inevitable. Due to advances in technology, nowadays simulation software packages are widely available to practitioners, educators and students. Evaluating and selecting the most appropriate simulation software package for educational use is a difficult process educators face due to many criteria that need to be taken into consideration and due to availability of many software packages with varying properties. Since this process is often costly and time consuming, a structured approach is required [29]. Therefore, the main goal of this study is to present a structured approach to help educators select the best educational use simulation software package. In fact, this is a complicated MCDM problem with the existence of many conflicting quantitative and qualitative criteria and alternatives. Thus, an efficient MCDM method is needed to evaluate simulation software package alternatives for educational use and select the best one.

In this research, as the MCDM method, F-AHP and F-VIKOR are integrated to evaluate and rank educational use simulation software packages. In fact, F-AHP and F-VIKOR are integrated to have both methods’ advantages. At the proposed fuzzy AHP-VIKOR, first, F-AHP is used to determine the weights of the evaluation criteria, since with F-AHP, through pairwise comparisons of quantitative and qualitative evaluation criteria, reliable importance weights can be determined. Then, F-VIKOR is used to rank the competing alternatives, using the fuzzy criteria weights obtained with F-AHP, since this method is widely used to find compromise solution(s) in large size applications, especially with many alternatives and when there are conflicting criteria. With F-VIKOR, an accepted compromise solution is obtained with a maximum group utility of the majority and a minimum of individual regret of the opponent [41 , 47–50]. In general, F-VIKOR is easy to use for applications with conflicting criteria, however, it does not provide specific guidelines for determining the weights of criteria. Therefore, a systematic method such as F-AHP is needed to determine the weights. However, when there are many alternatives and criteria, F-AHP, by itself, without the integration, can be burdensome due to large number of pairwise comparisons. With the proposed fuzzy AHP-VIKOR, simulation software packages are ranked without too many repetitive pairwise comparisons and complicated calculations. In the next sections, related literature is given and steps of the fuzzy AHP-VIKOR are presented in detail, along with a case study inspired from the work of Ayag [10].

2 Literature review

The AHP method was introduced by Thomas L. Saaty in 1980 [56]. In AHP, alternatives are evaluated with respect to various quantitative and qualitative criteria in a hierarchical and multi-level structure, and then ranked based on a total weighted score. AHP has been implemented to various MCDM problems such as selection of: Electric power plants in Jordan [2], grid-connected photovoltaic power plants sites in Spain [6], a solar– wind hybrid power station site in China [68], solar-thermal power plant investment projects [5] and main renewable energy resources in Malaysia [1]. To capture the uncertainty and vagueness on judgments of decision-makers (DMs), its fuzzy extension, F-AHP, is used in many real life applications such as evaluation of: Machine tools in a manufacturing system [12, 25], concepts in a NPD environment [11], the benefits of information-sharing decision problems in a supply chain [51], notebook computers for buyers [59], thermal power plant locations [21], disassembly line balancing solutions [8] and power substation locations [35].

The VIKOR method was developed to solve MCDM problems with conflicting or non-commensurable criteria. In VIKOR, compromise-ranking is performed, comparing the alternatives according to the closeness to the ideal solution [48 , 61]. Different fuzzy extensions of VIKOR was proposed throughout the literature. Triangular fuzzy numbers were used by Opricovic and Tzeng [49] and Opricovic [46] and triangular intuitionistic fuzzy numbers were utilized by Wan, Wang and Dong [63]. An extended group decision making F-VIKOR, based on 2-tuple linguistic model, was proposed by Ju and Wang [34]. An attitudinal-based interval 2-tuple linguistic VIKOR (ITL-VIKOR) was developed by Liu et al. [42]. A F-VIKOR under type-2 fuzzy environment was proposed by Qin, Liu, and Pedrycz [52] and Yazici and Kahraman [66]. An intuitionistic hesitant fuzzy VIKOR method with entropy weights was developed by Narayanamoorthy and Geetha [45].

F-VIKOR was used in several MCDM evaluation and selection problems. Some of these applications are: Material selection in an engineering application [33], water resources planning [46], evaluation of the vulnerability of the water supply to climate change and variability in South Korea [38], hospital service evaluation in Taiwan [20], site selection in waste management [42], ranking schools’ academic performance [44], CNC machine tool selection problem [64] and evaluation of green supplier development programs [9]. An extensive literature review about VIKOR and F-VIKOR applications can be found in the studies of Yazdani and Graeml [65] and Gul et al. [27].

AHP and VIKOR and their fuzzy extensions were integrated to solve several MCDM problems in the literature. Kaya and Kahraman [37] utilized fuzzy AHP-VIKOR to determine the best renewable energy alternative for Istanbul and to select the best energy production site in Istanbul. Ilangkumaran et al. [30] used F-AHP to determine the criteria weights and F-VIKOR to select the best machine tool alternative for the manufacturing sector. Sakthivel et al. [57] studied the selection of the best biodiesel blend for IC engines with integrated F-AHP and VIKOR and with integrated F-AHP and Technique for order performance by similarity to ideal solution (TOPSIS). Dincer and Hacioglu [24] implemented AHP and F-VIKOR to analyze the performance levels of Turkish banks registered in Borsa Istanbul. Vinodh and Jayakrishna [62] utilized F-AHP to determine the criteria weights and then F-VIKOR to select the best tyre recycling process. Ghadikolaei, Esbouei and Antucheviciene [26] used F-AHP to determine the weights of evaluation criteria and F-VIKOR, Fuzzy Additive Ratio Assessment (ARAS-F) and Fuzzy Complex Proportional Assessment (Fuzzy COPRAS) to rank the financial performance of several Iranian companies. Rezaie et al. [53] integrated F-AHP and VIKOR to evaluate performance of Iranian cement firms. Anojkumar, Ilangkumaran and Sasirekha [4] implemented F-AHP to determine the weights of evaluation criteria and then VIKOR, TOPSIS, ELECTRE and PRO-METHEE to select pipe material in sugar industry. Aydin and Kahraman [13] studied the problem of bus selection for public transportation with integrated F-AHP and F-VIKOR. Bhosale and Kant [16] used fuzzy AHP-VIKOR to select the best knowledge flow practicing organization. Büyüközkan and Görener [18] evaluated possible partners in product development process using an integrated AHP-VIKOR method. Zhu et al. [71] integrated AHP and VIKOR based on rough number and evaluated design concepts. Arunachalam, Idapalapati and Subbiah [7] utilized AHP and F-VIKOR to evaluate and select compliant polishing tool.

In the literature, software evaluation and selection has been studied in several MCDM applications. Davis and Williams [23] used AHP for evaluation and selection of simulation software in manufacturing at a medium-size UK engineering company. Cochran and Chen [22] determined fuzzy weighted scores to select object-oriented simulation software for analysis of production systems. Büyüközkan and Ruan [19] used F-VIKOR to evaluate the performance of enterprise resource planning (ERP) software products. Gupta, Singh and Verma [28] implemented AHP to compare manufacturing simulation software. Jadhav and Sonar [32] compared a hybrid knowledge based system (HKBS) approach with AHP and weighted scoring method (WSM) for the problem of evaluation of software packages. Kara and Cheikhrouhou [36] first determined criteria weights with F-AHP and then ranked collaborative software with TOPSIS, presenting a case study in Switzerland. Zaidan et al. [70] evaluated open-source electronic medical record (OS-EMR), especially open-source electronic health record (EHR)/EMR software packages with AHP integrated with Weighted product model (WPM), WSM, Simple additive weighting (SAW), Hierarchical adaptive weighting (HAW), and TOPSIS. Rouhani and Ravasan [54] applied fuzzy TOPSIS to select suitable business process management software in a port and maritime organization. An extensive review about evaluating and selecting software packages is given by Jadhav and Sonar [31], and a comprehensive review of evaluation methods for simulation software packages is given by Alomair, Ahmad and Alghamdi [3].

Brenner, Shacham and Cutlip [17] studied the application of modelling and simulation software packages in environmental engineering education. They mentioned that learning by simulation is very effective since it encourages students to ask “what if” questions, to test the effects of various parameters on processes, to try different scenarios and understand the consequences and to learn by “discovery”. Komulainen et al. [40] discussed the benefits of two commercial dynamic simulation software, D-SPICE and K-Spice for three different chemical engineering courses and confirmed that these software provide realistic training and can successfully be used in undergraduate and graduate teaching. Belton [15] highlighted that it is convenient to include simulation software since the first year at higher education. Ruis-Ramoz et al. [55] discussed the use of simulation package ASPEN Plus for undergraduate and graduate chemical engineering and environmental engineering courses and mentioned that it is useful to scale-up studies.

The closest research to the presented one are by Azadeh, Shirkouhi and Rezaie [14] and Ayağ [10]. Azadeh, Shirkouhi and Rezaie [14] used F-AHP and Ayağ [10] applied fuzzy Analytic Network Process (F-ANP) for simulation software package selection problem. With F-AHP and F-ANP, pairwise comparisons can be burdensome for the DM(s) if there are many alternatives and criteria. With the proposed fuzzy AHP-VIKOR integration, educational use simulation software packages are ranked without too many repetitive pairwise comparisons and assessments and without complicated calculations. At present, there does not appear to be a research paper in the literature that focuses on simulation software package selection (for educational use) utilizing F-AHP integrated with F-VIKOR. In the next section, steps of the fuzzy AHP-VIKOR are presented in detail.

3 Proposed fuzzy AHP-VIKOR approach

3.1 Definitions related to fuzzy numbers

Fuzzy set theory is a mathematical theory of classes with unsharp boundaries [39, 43]. Any crisp theory can be fuzzified by generalizing the concept of a set within that theory to the concept of a fuzzy set [69]. In this paper, due to its simplicity, triangular fuzzy numbers (TFNs) are used in F-AHP and F-VIKOR. A fuzzy number is a special fuzzy set F ={ (x, μ_F (x)), x ∈ R }, where x is a real number, R : -∞ < x < + ∞ and μ_F (x) is a continuous mapping from R to [0, 1]. A TFN denoted as $\tilde{M} = (l, m, u)$ , where l ≤ m ≤ u, has the following triangular type membership function;

$μ_{F} (x) = {\begin{matrix} 0 \\ x - l / - m - l \\ u - x / - u - m \\ 0 \end{matrix} \begin{matrix} x < l \\ l \leq x \leq m \\ m \leq x \leq u \\ x > u \end{matrix}$ (1)

Basic operations between two positive TFNs $\tilde{A} = (l_{1}, m_{1}, u_{1})$ , $\tilde{B} = (l_{2}, m_{2}, u_{2})$ , l₁ ≤ m₁ ≤ u₁, l₂ ≤ m₂ ≤ u₂ and a crisp number C are explained as [4, 64]: $\tilde{A} + \tilde{B} = (l_{1} + l_{2}, m_{1} + m_{2}, u_{1} + u_{2})$ (2) $\tilde{A} + C = (l_{1} + C, m_{1} + C, u_{1} + C)$ (3) $\tilde{A} - \tilde{B} = (l_{1} - u_{2}, m_{1} - m_{2}, u_{1} - l_{2})$ (4) $\tilde{A} - C = (l_{1} - C, m_{1} - C, u_{1} - C)$ (5) $\tilde{A} * \tilde{B} = (l_{1} * l_{2}, m_{1} * m_{2}, u_{1} * u_{2})$ (6) $\tilde{A} * C = (l_{1} * C, m_{1} * C, u_{1} * C) for C ⩾ 0$ (7) $\frac{\tilde{A}}{\tilde{B}} = (\frac{l_{1}}{u_{2}}, \frac{m_{1}}{m_{2}}, \frac{u_{1}}{l_{2}})$ (8)

$\begin{matrix} \frac{\tilde{A}}{\tilde{B}} = (\min (\frac{l_{1}}{l_{2}}, \frac{l_{1}}{u_{2}}, \frac{u_{1}}{l_{2}}, \frac{u_{1}}{u_{2}}), \frac{m_{1}}{m_{2}},) \\ \max (\frac{l_{1}}{l_{2}}, \frac{l_{1}}{u_{2}}, \frac{u_{1}}{l_{2}}, \frac{u_{1}}{u_{2}}) \end{matrix}$ (9)

if $\tilde{A}$ and $\tilde{B}$ are two TFNs (not necessarily positive TFNs). $\frac{\tilde{A}}{C} = (\frac{l_{1}}{C}, \frac{m_{1}}{C}, \frac{u_{1}}{C}) for C > 0$ (10) ${\tilde{A}}^{- 1} = (\frac{1}{u_{1}}, \frac{1}{m_{1}}, \frac{1}{l_{1}})$ (11) $\begin{matrix} MAX (\tilde{A} + \tilde{B}) = \\ (max (l_{1}, l_{2}), max (m_{1}, m_{2}), max (u_{1}, u_{2})) \end{matrix}$ (12) $\begin{matrix} MIN (\tilde{A} + \tilde{B}) = \\ (min (l_{1}, l_{2}), min (m_{1}, m_{2}), min (u_{1}, u_{2})) \end{matrix}$ (13)

The graded mean integration approach [67] is used as the defuzzification method to convert TFNs into crisp numbers. Here, $Crisp (\tilde{A}) = (4 m_{1} + l_{1} + u_{1}) / 6$ (14)

3.2 Finding the importance weights of criteria with F-AHP

After constructing the hierarchical structure of the problem, the decision makers (DMs) make pairwise comparisons of the criteria and estimate their relative importance in relation to the element at the immediate proceeding level. During the process of evaluation of criteria, the pairwise comparisons are made by using the linguistic terms and scale presented in Table 1.

Table 1
Linguistic terms and TFNs for the evaluation of criteria in F-AHP

Linguistic terms Triangular fuzzy number (TFN)

Absolutely strong (AS) (2, 5/2, 3)

Very strong (VS) (3/2, 2, 5/2)

Fairly strong (FS) (1, 3/2, 2)

Slightly strong (SS) (1, 1, 3/2)

Equal (E) (1, 1, 1)

Slightly weak (SW) (2/3, 1, 1)

Fairly weak (FW) (1/2, 2/3, 1)

Very weak (VW) (2/5, 1/2, 2/3)

Absolutely weak (AW) (1/3, 2/5, 1/2)

Linguistic terms	Triangular fuzzy number (TFN)
Absolutely strong (AS)	(2, 5/2, 3)
Very strong (VS)	(3/2, 2, 5/2)
Fairly strong (FS)	(1, 3/2, 2)
Slightly strong (SS)	(1, 1, 3/2)
Equal (E)	(1, 1, 1)
Slightly weak (SW)	(2/3, 1, 1)
Fairly weak (FW)	(1/2, 2/3, 1)
Very weak (VW)	(2/5, 1/2, 2/3)
Absolutely weak (AW)	(1/3, 2/5, 1/2)

3.2.1 Computational steps of F-AHP [4 , 58]

Step 1. Form a decision group of K people. Identify n criteria; and select the suitable linguistic terms for the pairwise comparison of criteria. Calculate the aggregated ${\tilde{x}}_{ij} = \frac{1}{K} ({\tilde{x}}_{ij}^{1} (+) {\tilde{x}}_{ij}^{2} (+) \dots (+) {\tilde{x}}_{ij}^{K})$ where ${\tilde{x}}_{ij}^{K} = (a_{ij}^{K}, b_{ij}^{K}, c_{ij}^{K}) \forall i, j, k$ is the TFN corresponding to the evaluation of the K^th DM.

Step 2. $\tilde{X} = [\begin{matrix} (1, 1, 1) & \tilde{x_{12}} & . . & . . & \tilde{x_{1 n}} \\ \tilde{x_{21}} & (1, 1, 1) & . . & . . & \tilde{x_{2 n}} \\ . . & . . & . . & . . & . . \\ . . & . . & . . & . . & . . \\ \tilde{x_{n 1}} & \tilde{x_{n 2}} & . . & . . & (1, 1, 1) \end{matrix}]$ with elements ${\tilde{x}}_{ij} = (a_{ij}, b_{ij}, c_{ij})$ is normalized and $\tilde{S}$ is obtained. $\tilde{S} = [\begin{matrix} \tilde{s_{11}} & \tilde{s_{12}} & . . & . . & \tilde{s_{1 n}} \\ \tilde{s_{21}} & \tilde{s_{22}} & . . & . . & \tilde{s_{2 n}} \\ . . & . . & . . & . . & . . \\ . . & . . & . . & . . & . . \\ \tilde{s_{n 1}} & \tilde{s_{n 2}} & . . & . . & \tilde{s_{nn}} \end{matrix}]$ , where ${\tilde{s}}_{ij} = (\frac{a_{ij}}{\sum_{i} c_{ij}}, \frac{b_{ij}}{\sum_{i} b_{ij}}, \frac{c_{ij}}{\sum_{i} a_{ij}})$ . Fuzzy priority weight vector ${\tilde{w}}_{criteria} = ({\tilde{w}}_{1}, {\tilde{w}}_{2}, \dots, {\tilde{w}}_{n})$ is calculated by averaging the entries on each row of $\tilde{S}$ .

Step 3. $\tilde{X}$ is defuzzified by using Equation (14) and w_cr = (w₁, w₂, …, w_n) (approximate crisp criteria weights) is calculated by averaging the entries on each row of normalized X. So the normalized principal eigen vector is w_cr. The largest eigen value, called the principal eigen value (λ_max) is determined with Equation (15). ${Xw}_{cr}^{T} = λ_{max} w_{cr}^{T}$ (15)

The measure of inconsistency of pairwise comparisons is called the consistency index (CI) and it is calculated as: $CI = \frac{λ_{max} - n}{n - 1}$ (16)

The consistency ratio (CR) is used to estimate the consistency of pairwise comparisons, and the CR is calculated by dividing CI by Random Consistency Index (RI); $CR = \frac{CI}{RI}$ (17)

RI is the average index for randomly generated weights [56]. If the CR is less than 0.10, the comparisons are acceptable, otherwise they are not [56].

3.3 Ranking of alternatives with F-VIKOR

In the previous section, fuzzy priority weight vector ${\tilde{w}}_{criteria} = ({\tilde{w}}_{1}, {\tilde{w}}_{2}, \dots, {\tilde{w}}_{n})$ was obtained with F-AHP. After the determination of ${\tilde{w}}_{criteria}$ with F-AHP, in order to rank the alternatives, F-VIKOR is used. During the process of evaluation of alternatives with F-VIKOR, the linguistic terms and scale presented in Table 2 is used.

Table 2
Linguistic terms and TFNs for the ratings of alternatives in F-VIKOR

Linguistic terms Triangular fuzzy number (TFN)

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)

Linguistic terms	Triangular fuzzy number (TFN)
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)

3.3.1 Computational steps of F-VIKOR [37 , 64]

Step 1. Identify the m alternatives; and select the suitable linguistic terms for the evaluations of alternatives with respect to each criterion. Calculate the aggregated ${\tilde{r}}_{ij} = \frac{1}{K} ({\tilde{r}}_{ij}^{1} (+) {\tilde{r}}_{ij}^{2} (+) \dots (+) {\tilde{r}}_{ij}^{K})$ where ${\tilde{r}}_{ij}^{K} = (a_{ij}^{K}, b_{ij}^{K}, c_{ij}^{k})$ is the TFN corresponding to the evaluation of the Kth DM. After the aggregation, the fuzzy MCDM problem with m alternatives that are evaluated in terms of n criteria can be expressed in a fuzzy matrix format as: $\tilde{D} = [\begin{matrix} \tilde{r_{11}} & \tilde{r_{12}} & . . & . . & \tilde{r_{1 n}} \\ \tilde{r_{21}} & \tilde{r_{22}} & . . & . . & \tilde{r_{2 n}} \\ . . & . . & . . & . . & . . \\ . . & . . & . . & . . & . . \\ \tilde{r_{n 1}} & \tilde{r_{n 2}} & . . & . . & \tilde{r_{mn}} \end{matrix}],$ where ${\tilde{r}}_{ij} = (a_{ij}, b_{ij}, c_{ij}) \forall i, j$ are positive TFNs.

Step 2. Determine the fuzzy best value (FBV, ${\tilde{f}}_{j}^{*}$ ) and the fuzzy worst value (FWV, ${\tilde{f}}_{j}^{-}$ ) for each criterion. ${\tilde{f}}_{j}^{*} = max_{i} {\tilde{r}}_{ij} \forall j, {\tilde{f}}_{j}^{-} = min_{i} {\tilde{r}}_{ij} \forall j$ (18)

Step 3. Calculate the separation measures of each alternative from the FBV ( ${\tilde{S}}_{i}$ ) and FWV ( ${\tilde{R}}_{i}$ ). ${\tilde{S}}_{i} = \sum_{j = 1}^{n} {\tilde{w}}_{j} ({\tilde{f}}_{j}^{*} - {\tilde{r}}_{ij}) / - ({\tilde{f}}_{j}^{*} - {\tilde{f}}_{j}^{-}) \forall i$ (19) ${\tilde{R}}_{i} \underset{j}{= max} [w˜j (f˜j* - r˜ij) / - (f˜j* - f˜j-)] \forall i$ (20)

Step 4. Calculate ${\tilde{S}}^{*}, {\tilde{S}}^{-}, {\tilde{R}}^{*}, {\tilde{R}}^{-}$ values as: ${\tilde{S}}^{*} = min_{i} {\tilde{S}}_{i}, {\tilde{S}}^{-} = max_{i} {\tilde{S}}_{i}$ (21) ${\tilde{R}}^{*} = min_{i} {\tilde{R}}_{i}, {\tilde{R}}^{-} = max_{i} {\tilde{R}}_{i}$ (22)

Step 5. Calculate ${\tilde{Q}}_{i}$ values for each alternative. $\begin{matrix} {\tilde{Q}}_{i} = ν ({\tilde{S}}_{i} - {\tilde{S}}^{*}) / ({\tilde{S}}_{-} - {\tilde{S}}^{*}) + \\ (1 - ν) ({\tilde{R}}_{i} - {\tilde{R}}^{*}) / ({\tilde{R}}^{-} - {\tilde{R}}^{*}) \forall i \end{matrix}$

Here, ν is the weight of the strategy of the maximum group utility (majority of criteria) and 1 - ν is the weight of the individual regret. ν is usually assumed to be 0.5 (by consensus) [4, 37].

Step 6. Defuzzify the ${\tilde{Q}}_{i}$ values with Equation (14) and rank the alternatives based on crisp Q_i values so that the alternative with the minimum Q_i value is the best.

4 Case study

In this study, 15 criteria similar to Ayağ’s research [10] are determined for the evaluation of educational use simulation software packages. These are listed in Table 3.

Table 3
Evaluation criteria for simulation software packages

1 pedigree (PE) 9 simulation testability (TE)

2 coding aspects (CA) 10 experimentation facilities (EX)

3 software compatibility (SC) 11 statistical facilities (SF)

4 financial and technical features (FT) 12 Input / Output capabilities (IO)

5 general modeling features (GF) 13 analysis capabilities (AC)

6 modeling assistance (MA) 14 documentation and tutorial support (DT)

7 visual simulation aspects (VA) 15 Maintenance and Update (MU)

8 simulation efficiency (EF)

The alternatives that are going to be evaluated with respect to these criteria are: Arena (A1), Simio (A2), ProModel (A3), AnyLogic (A4), and Simul8 (A5). DMs are three academicians who teach undergraduate /graduate level simulation courses at universities in Turkey. In order to determine the fuzzy criteria weights, F-AHP is used. In F-AHP, first DMs do pairwise comparison of criteria using the linguistic terms presented in Table 1. Comparisons of 3 DMs are presented in Table 4. After the aggregation of the corresponding TFNs of the DMs evaluations, the fuzzy evaluation matrix for the criteria weights ( $\tilde{X}$ ) is obtained as in Table 5. Afterwards, fuzzy priority weight vector ${\tilde{w}}_{criteria} = ({\tilde{w}}_{1}, {\tilde{w}}_{2}, \dots, {\tilde{w}}_{n})$ is calculated by averaging the entries on each row of normalized $\tilde{X}$ ( $\tilde{S}$ ). The ${\tilde{w}}_{criteria}$ is presented in Table 6. In order to check the consistency ratio (CR) of the evaluation matrix $\tilde{X}$ , Equation (14) is utilized for defuzzification. Then from Equations (15 and 16), CI is calculated as CI = (15.682 - 15)/14 = 0.0487 and from Equation (17), CR is determined as CR = 0.0487/1.59 = 0.0306. Since CR is less than 0.1, the comparison results is considered to be consistent.

Table 4

Pairwise comparison of evaluation criteria

PE	CA	SC	FT	GF	MA	FA	EF	TE	EX	SF	IO	AC	DT	MU
	PE	E	E	SS	SW	SS	E	FS	VW	FW	FS	VW	FW	FW	FW	FW
	E	E	FW	FW	SW	E	SS	SW	AW	SS	SW	AW	AW	AW	AW
	E	E	SW	FW	SW	E	SS	SW	FW	SS	SW	FW	FW	FW	FW
CA	E	E	SS	SW	SS	SS	VW	SW	FW	FW	FS	E	FW	FW	FW
	E	E	FW	FW	SW	FW	SW	FW	VW	VW	SS	E	VW	VW	VW
	E	E	SS	FW	SW	SW	SW	FW	FW	FW	SS	SS	FW	FW	FW
SC	SW	SW	E	SS	E	E	E	FW	VW	SS	FS	VW	VW	VW	SS
	FS	FS	E	FW	SS	E	SS	E	SW	FW	SS	SW	SW	SW	FW
	SS	SW	E	SW	E	E	E	SW	SW	SW	SS	SW	SW	SW	SW
FT	SS	SS	SW	E	FS	FS	FS	VW	FS	SW	SW	E	SW	SW	SW
	FS	FS	FS	E	SS	E	SS	FW	SS	SW	SW	E	SW	SW	SW
	FS	FS	SS	E	SS	SS	SS	SW	SS	E	E	E	E	E	E
GF	SW	SW	E	FW	E	E	E	VW	VW	VW	VW	FS	VW	VW	VW
	SS	SS	SW	SW	E	SW	E	SW	FW	FW	FW	SS	FW	FW	FW
	SS	SS	E	SW	E	E	E	SW	SW	SW	SW	SS	SW	SW	SW
MA	E	SW	E	FW	E	E	SW	SS	VW	FS	VW	VW	VW	VW	E
	E	FS	E	E	SS	E	SS	FW	SW	SS	SW	SW	SW	SW	E
	E	SS	E	SW	E	E	E	SW	SW	SS	SW	SW	SW	SW	E
FA	FW	VS	E	FW	E	SS	E	FW	SS	AW	E	AW	SS	AW	AW
	SW	SS	SW	SW	E	SW	E	SW	FW	FW	E	FW	FW	FW	FW
	SW	SS	E	SW	E	E	E	SW	SW	SW	SS	SW	SW	SW	SW
EF	VS	SS	FS	VS	VS	SW	FS	E	SW	SW	SW	FS	SW	E	SW
	SS	FS	E	FS	SS	FS	SS	E	SW	SW	SW	SS	SW	E	SW
	SS	FS	SS	SS	SS	SS	SS	E	E	E	E	SS	E	E	E
TE	FS	FS	VS	FW	VS	VS	SW	SS	E	E	SW	E	AW	AW	VW
	AS	VS	SS	SW	FS	SS	FS	SS	E	E	SW	E	FW	FW	FW
	FS	FS	SS	SW	SS	SS	SS	E	E	E	E	SS	SW	SW	SW
EX	FW	FS	SW	SS	VS	FW	AS	SS	E	E	AW	AW	VW	SW	VW
	SW	VS	FS	SS	FS	SW	FS	SS	E	E	FW	FW	SW	SW	SW
	SW	FS	SS	E	SS	SW	SS	E	E	E	SW	SW	SW	E	SW
SF	VS	FW	FW	SS	VS	VS	E	SS	SS	AS	E	SW	SW	SW	E
	SS	SW	SW	SS	FS	SS	E	SS	SS	FS	E	SW	SW	SW	E
	SS	SW	SW	E	SS	SS	SW	E	E	SS	E	E	E	E	E
IO	FS	E	VS	E	FW	VS	AS	FW	E	AS	SS	E	VW	SS	VW
	AS	E	SS	E	SW	SS	FS	SW	E	FS	SS	E	SW	FW	SW
	FS	SW	SS	E	SW	SS	SS	SW	SW	SS	E	E	SW	SW	SW
AC	FS	FS	VS	SS	VS	VS	SW	SS	AS	VS	SS	VS	E	VW	AW
	AS	VS	SS	SS	FS	SS	FS	SS	FS	SS	SS	SS	E	SW	FW
	FS	FS	SS	E	SS	SS	SS	E	SS	SS	E	SS	E	SW	SW
DT	FS	FS	VS	SS	VS	VS	AS	E	AS	SS	SS	SW	VS	E	AW
	AS	VS	SS	SS	FS	SS	FS	E	FS	SS	SS	FS	SS	E	FW
	FS	FS	SS	E	SS	SS	SS	E	SS	E	E	SS	SS	E	SW
MU	FS	FS	SW	SS	VS	E	AS	SS	VS	VS	E	VS	AS	AS	E
	AS	VS	FS	SS	FS	E	FS	SS	FS	SS	E	SS	FS	FS	E
	FS	FS	SS	E	SS	E	SS	E	SS	SS	E	SS	SS	SS	E

Table 5

The fuzzy evaluation matrix for the criteria weights ( $\tilde{X}$ )

PE	CA	SC	FT	GF	MA	FA	EF
PE	(1.000, 1.000, 1.000)	(1.000, 1.000, 1.000)	(0.723, 0.890, 1.167)	(0.557, 0.780, 1.000)	(0.780, 1.000, 1.167)	(1.000, 1.000, 1.000)	(1.000, 1.167, 1.667)	(0.580, 0.833, 0.890)
CA	(1.000, 1.000, 1.000)	(1.000, 1.000, 1.000)	(0.833, 0.890, 1.333)	(0.557, 0.780, 1.000)	(0.780, 1.000, 1.167)	(0.723, 0.890, 1.167)	(0.580, 0.833, 0.890)	(0.557, 0.780, 1.000)
SC	(0.890, 1.167, 1.500)	(0.780, 1.167, 1.333)	(1.000, 1.000, 1.000)	(0.723, 0.890, 1.167)	(1.000, 1.000, 1.167)	(1.000, 1.000, 1.000)	(1.000, 1.000, 1.167)	(0.723, 0.890, 1.000)
FT	(1.000, 1.333, 1.833)	(1.000, 1.333, 1.833)	(0.890, 1.167, 1.500)	(1.000, 1.000, 1.000)	(1.000, 1.167, 1.667)	(1.000, 1.167, 1.500)	(1.000, 1.167, 1.667)	(0.523, 0.723, 0.890)
GF	(0.890, 1.000, 1.333)	(0.890, 1.000, 1.333)	(0.890, 1.000, 1.000)	(0.613, 0.890, 1.000)	(1.000, 1.000, 1.000)	(0.890, 1.000, 1.000)	(1.000, 1.000, 1.000)	(0.580, 0.833, 0.890)
MA	(1.000, 1.000, 1.000)	(0.890, 1.167, 1.500)	(1.000, 1.000, 1.000)	(0.723, 0.890, 1.000)	(1.000, 1.000, 1.167)	(1.000, 1.000, 1.000)	(0.890, 1.000, 1.167)	(0.723, 0.890, 1.167)
FA	(0.613, 0.890, 1.000)	(1.167, 1.333, 1.833)	(0.890, 1.000, 1.000)	(0.613, 0.890, 1.000)	(1.000, 1.000, 1.000)	(0.890, 1.000, 1.167)	(1.000, 1.000, 1.000)	(0.613, 0.890, 1.000)
EF	(1.167, 1.333, 1.833)	(1.000, 1.333, 1.833)	(1.000, 1.167, 1.500)	(1.167, 1.500, 2.000)	(1.167, 1.333, 1.833)	(0.890, 1.167, 1.500)	(1.000, 1.167, 1.667)	(1.000, 1.000, 1.000)
TE	(1.333, 1.833, 2.333)	(1.167, 1.667, 2.167)	(1.167, 1.333, 1.833)	(0.613, 0.890, 1.000)	(1.167, 1.500, 2.000)	(1.167, 1.333, 1.833)	(0.890, 1.167, 1.500)	(1.000, 1.000, 1.333)
EX	(0.613, 0.890, 1.000)	(1.167, 1.667, 2.167)	(0.890, 1.167, 1.500)	(1.000, 1.000, 1.333)	(1.167, 1.500, 2.000)	(0.613, 0.890, 1.000)	(1.333, 1.667, 2.167)	(1.000, 1.000, 1.333)
SF	(1.167, 1.333, 1.833)	(0.613, 0.890, 1.000)	(0.613, 0.890, 1.000)	(1.000, 1.000, 1.333)	(1.167, 1.500, 2.000)	(1.167, 1.333, 1.833)	(0.890, 1.000, 1.000)	(1.000, 1.000, 1.333)
IO	(1.333, 1.833, 2.333)	(0.890, 1.000, 1.000)	(1.167, 1.333, 1.833)	(1.000, 1.000, 1.000)	(0.613, 0.890, 1.000)	(1.167, 1.333, 1.833)	(1.333, 1.667, 2.167)	(0.613, 0.890, 1.000)
AC	(1.333, 1.833, 2.333)	(1.167, 1.667, 2.167)	(1.167, 1.333, 1.833)	(1.000, 1.000, 1.333)	(1.167, 1.500, 2.000)	(1.167, 1.333, 1.833)	(0.890, 1.167, 1.500)	(1.000, 1.000, 1.333)
DT	(1.333, 1.833, 2.333)	(1.167, 1.667, 2.167)	(1.167, 1.333, 1.833)	(1.000, 1.000, 1.333)	(1.167, 1.500, 2.000)	(1.167, 1.333, 1.833)	(1.333, 1.667, 2.167)	(1.000, 1.000, 1.000)
MU	(1.333, 1.833, 2.333)	(1.167, 1.667, 2.167)	(0.890, 1.167, 1.500)	(1.000, 1.000, 1.333)	(1.167, 1.500, 2.000)	(1.000, 1.000, 1.167)	(1.333, 1.667, 2.167)	(1.000, 1.000, 1.333)
TE	EX	SF	IO	AC	DT	MU
PE	(0.443, 0.580, 0.833)	(1.000, 1.167, 1.667)	(0.580, 0.833, 0.890)	(0.443, 0.580, 0.833)	(0.443, 0.580, 0.833)	(0.443, 0.580, 0.833)	(0.443, 0.580, 0.833)
CA	(0.467, 0.613, 0.890)	(0.467, 0.613, 0.890)	(1.000, 1.167, 1.667)	(1.000, 1.000, 1.167)	(0.467, 0.613, 0.890)	(0.467, 0.613, 0.890)	(0.467, 0.613, 0.890)
SC	(0.580, 0.833, 0.890)	(0.723, 0.890, 1.167)	(1.000, 1.167, 1.667)	(0.580, 0.833, 0.890)	(0.580, 0.833, 0.890)	(0.580, 0.833, 0.890)	(0.723, 0.890, 1.167)
FT	(1.000, 1.167, 1.667)	(0.780, 1.000, 1.000)	(0.780, 1.000, 1.000)	(1.000, 1.000, 1.000)	(0.780, 1.000, 1.000)	(0.780, 1.000, 1.000)	(0.780, 1.000, 1.000)
GF	(0.523, 0.723, 0.890)	(0.523, 0.723, 0.890)	(0.523, 0.723, 0.890)	(1.000, 1.167, 1.667)	(0.523, 0.723, 0.890)	(0.523, 0.723, 0.890)	(0.523, 0.723, 0.890)
MA	(0.580, 0.833, 0.890)	(1.000, 1.167, 1.667)	(0.580, 0.833, 0.890)	(0.580, 0.833, 0.890)	(0.580, 0.833, 0.890)	(0.580, 0.833, 0.890)	(1.000, 1.000, 1.000)
FA	(0.723, 0.890, 1.167)	(0.500, 0.690, 0.833)	(1.000, 1.000, 1.167)	(0.500, 0.690, 0.833)	(0.723, 0.890, 1.167)	(0.500, 0.690, 0.833)	(0.500, 0.690, 0.833)
EF	(0.780, 1.000, 1.000)	(0.780, 1.000, 1.000)	(0.780, 1.000, 1.000)	(1.000, 1.167, 1.667)	(0.780, 1.000, 1.000)	(1.000, 1.000, 1.000)	(0.780, 1.000, 1.000)
TE	(1.000, 1.000, 1.000)	(1.000, 1.000, 1.000)	(0.780, 1.000, 1.000)	(1.000, 1.000, 1.167)	(0.500, 0.690, 0.833)	(0.500, 0.690, 0.833)	(0.523, 0.723, 0.890)
EX	(1.000, 1.000, 1.000)	(1.000, 1.000, 1.000)	(0.500, 0.690, 0.833)	(0.500, 0.690, 0.833)	(0.580, 0.833, 0.890)	(0.780, 1.000, 1.000)	(0.580, 0.833, 0.890)
SF	(1.000, 1.000, 1.333)	(1.333, 1.667, 2.167)	(1.000, 1.000, 1.000)	(0.780, 1.000, 1.000)	(0.780, 1.000, 1.000)	(0.780, 1.000, 1.000)	(1.000, 1.000, 1.000)
IO	(0.890, 1.000, 1.000)	(1.333, 1.667, 2.167)	(1.000, 1.000, 1.333)	(1.000, 1.000, 1.000)	(0.580, 0.833, 0.890)	(0.723, 0.890, 1.167)	(0.580, 0.833, 0.890)
AC	(1.333, 1.667, 2.167)	(1.167, 1.333, 1.833)	(1.000, 1.000, 1.333)	(1.167, 1.333, 1.833)	(1.000, 1.000, 1.000)	(0.580, 0.833, 0.890)	(0.500, 0.690, 0.833)
DT	(1.333, 1.667, 2.167)	(1.000, 1.000, 1.333)	(1.000, 1.000, 1.333)	(0.890, 1.167, 1.500)	(1.167, 1.333, 1.833)	(1.000, 1.000, 1.000)	(0.500, 0.690, 0.833)
MU	(1.167, 1.500, 2.000)	(1.167, 1.333, 1.667)	(1.000, 1.000, 1.000)	(1.167, 1.333, 1.833)	(1.333, 1.667, 2.167)	(1.333, 1.667, 2.000)	(0.890, 1.000, 1.000)

Table 6

Fuzzy criteria weights ${\tilde{w}}_{criteria} = ({\tilde{w}}_{1}, {\tilde{w}}_{2}, \dots, {\tilde{w}}_{n})$ determined with F-AHP

Criteria	Fuzzy weights	Criteria	Fuzzy weights
PE	(0.035, 0.052, 0.079)	TE	(0.047, 0.069, 0.102)
CA	(0.036, 0.052, 0.081)	EX	(0.044, 0.066, 0.095)
SC	(0.041, 0.061, 0.086)	SF	(0.050, 0.070, 0.100)
FT	(0.046, 0.068, 0.098)	IO	(0.049, 0.072, 0.102)
GF	(0.037, 0.056, 0.079)	AC	(0.054, 0.078, 0.120)
MA	(0.042, 0.061, 0.082)	DT	(0.056, 0.080, 0.123)
VA	(0.039, 0.057, 0.081)	MU	(0.060, 0.086, 0.130)
EF	(0.050, 0.073, 0.104)

${\tilde{w}}_{criteria} = ({\tilde{w}}_{1}, {\tilde{w}}_{2}, \dots, {\tilde{w}}_{n})$ determined with F-AHP is used in F-VIKOR to rank simulation software package alternatives. In F-VIKOR, first DMs evaluate alternatives with respect to evaluation criteria using the linguistic terms presented in Table 2. These evaluations are presented in Table 7. After the aggregation of the corresponding TFNs of the DMs’ evaluations, the fuzzy evaluation matrix $\tilde{D}$ is presented in Table 8. Also, in Table 8, the fuzzy best value (FBV, ${\tilde{f}}_{j}^{*}$ ) and the fuzzy worst value (FWV, ${\tilde{f}}_{j}^{-}$ ) for each criterion are presented. The separation measures of each alternative ${\tilde{S}}_{i}$ and ${\tilde{R}}_{i}$ are presented in Table 9, along with ${\tilde{S}}^{*}, {\tilde{S}}^{-}, {\tilde{R}}^{*}, {\tilde{R}}^{-}$ values. Based on these, ${\tilde{Q}}_{i}$ value for each alternative is calculated and ${\tilde{Q}}_{i}$ values are defuzzified with Equation (14) (Q_i) to rank the alternatives. The results of the fuzzy AHP-VIKOR are presented in Table 10. As seen from these results, evaluated educational use simulation software packages in Turkey are ranked from best to worst as Simio (A2), Arena (A1), Simul8 (A5), AnyLogic (A4) and ProModel (A3).

Table 7

3 DMs’ evaluation scores of the simulation software package alternatives with respect to each criterion

PE	CA	SC	FT	GF	MA	FA	EF	TE	EX	SF	IO	AC	DT	MU
A1	G	MG	G	F	G	G	F	MG	G	MG	MG	G	MG	G	F
	G	MG	G	F	G	G	MG	G	MG	F	MG	F	MG	MG	MG
	G	MG	G	F	MG	MG	MG	MG	MG	MG	F	MG	G	G	MG
A2	MG	F	MG	F	G	F	MG	F	G	F	G	F	G	G	MG
	MG	F	MG	MG	MG	MG	MG	MG	MG	MG	G	F	G	G	MG
	MG	MG	F	MG	MG	F	F	MG	MG	F	G	F	VG	G	MG
A3	VG	P	VG	MG	VG	P	MG	P	MP	VG	P	P	VG	F	F
	G	P	VG	MG	G	P	MG	MP	MP	G	P	P	G	F	MG
	G	P	VG	MG	G	VP	MG	P	P	G	P	MP	G	F	F
A4	MP	MP	MG	MP	MP	F	MG	F	MG	MP	MP	P	MP	MG	G
	F	P	MG	MP	F	F	G	F	G	F	P	VP	F	MG	G
	MP	MP	MG	F	MP	F	MG	F	MG	MP	MP	VP	MP	MG	G
A5	G	P	F	G	G	F	MG	P	MG	G	F	VP	F	G	MG
	MG	P	F	G	MG	MG	MG	MP	MG	MG	MG	VP	MG	MG	MG
	G	P	F	G	G	MG	MG	P	MG	G	MG	VP	MG	MG	F

Table 8

Fuzzy evaluation matrix ( $\tilde{D}$ ) for the alternatives and fuzzy best values (FBV) and fuzzy worst values (FWV)

PE	CA	SC	FT	GF	MA	FA	EF
A1	(7.000, 9.000, 10.00)	(5.000, 7.000, 9.000)	(7.000, 9.000, 10.00)	(3.000, 5.000, 7.000)	(6.333, 8.333, 9.667)	(6.333, 8.333, 9.667)	(4.333, 6.333, 8.333)	(5.667, 7.667, 9.333)
A2	(5.000, 7.000, 9.000)	(3.667, 5.667, 7.667)	(4.333, 6.333, 8.333)	(4.333, 6.333, 8.333)	(5.667, 7.667, 9.333)	(3.667, 5.667, 7.667)	(4.333, 6.333, 8.333)	(4.333, 6.333, 8.333)
A3	(7.667, 9.333, 10.00)	(0.000, 1.000, 3.000)	(9.000, 10.00, 10.00)	(5.000, 7.000, 9.000)	(7.667, 9.333, 10.00)	(0.000, 0.667, 2.333)	(5.000, 7.000, 9.000)	(0.333, 1.667, 3.667)
A4	(1.667, 3.667, 5.667)	(0.667, 2.333, 4.333)	(5.000, 7.000, 9.000)	(1.667, 3.667, 5.667)	(1.667, 3.667, 5.667)	(3.000, 5.000, 7.000)	(5.667, 7.667, 9.333)	(3.000, 5.000, 7.000)
A5	(6.333, 8.333, 9.667)	(0.000, 1.000, 3.000)	(3.000, 5.000, 7.000)	(7.000, 9.000, 10.00)	(6.333, 8.333, 9.667)	(4.333, 6.333, 8.333)	(5.000, 7.000, 9.000)	(0.333, 1.667, 3.667)
FBV	(7.667, 9.333, 10.00)	(5.000, 7.000, 9.000)	(9.000, 10.00, 10.00)	(7.000, 9.000, 10.00)	(7.667, 9.333, 10.00)	(6.333, 8.333, 9.667)	(5.667, 7.667, 9.333)	(5.667, 7.667, 9.333)
FWV	(1.667, 3.667, 5.667)	(0.000, 1.000, 3.000)	(3.000, 5.000, 7.000)	(1.667, 3.667, 5.667)	(1.667, 3.667, 5.667)	(0.000, 0.667, 2.333)	(4.333, 6.333, 8.333)	(0.333, 1.667, 3.667)
TE	EX	SF	IO	AC	DT	MU
A1	(5.667, 7.667, 9.333)	(4.333, 6.333, 8.333)	(4.333, 6.333, 8.333)	(5.000, 7.000, 8.667)	(5.667, 7.667, 9.333)	(6.333, 8.333, 9.667)	(4.333, 6.333, 8.333)
A2	(5.667, 7.667, 9.333)	(3.667, 5.667, 7.667)	(7.000, 9.000, 10.00)	(3.000, 5.000, 7.000)	(7.667, 9.333, 10.00)	(7.000, 9.000, 10.00)	(5.000, 7.000, 9.000)
A3	(0.667, 2.333, 4.333)	(7.667, 9.333, 10.00)	(0.000, 1.000, 3.000)	(0.333, 1.667, 3.667)	(7.667, 9.333, 10.00)	(3.000, 5.000, 7.000)	(3.667, 5.667, 7.667)
A4	(5.667, 7.667, 9.333)	(1.667, 3.667, 5.667)	(0.667, 2.333, 4.333)	(0.000, 0.333, 1.667)	(1.667, 3.667, 5.667)	(5.000, 7.000, 9.000)	(7.000, 9.000, 10.00)
A5	(5.000, 7.000, 9.000)	(6.333, 8.333, 9.667)	(4.333, 6.333, 8.333)	(0.000, 0.000, 1.000)	(4.333, 6.333, 8.333)	(5.667, 7.667, 9.333)	(4.333, 6.333, 8.333)
FBV	(5.667, 7.667, 9.333)	(7.667, 9.333, 10.00)	(7.000, 9.000, 10.00)	(5.000, 7.000, 8.667)	(7.667, 9.333, 10.00)	(7.000, 9.000, 10.00)	(7.000, 9.000, 10.00)
FWV	(0.667, 2.333, 4.333)	(1.667, 3.667, 5.667)	(0.000, 1.000, 3.000)	(0.000, 0.000, 1.000)	(1.667, 3.667, 5.667)	(3.000, 5.000, 7.000)	(3.667, 5.667, 7.667)

Table 9

${\tilde{S}}_{i}$ , ${\tilde{R}}_{i}$ , ${\tilde{S}}^{*}, {\tilde{S}}^{-}, {\tilde{R}}^{*}, {\tilde{R}}^{-}$ values.

${\tilde{S}}_{i}$	${\tilde{R}}_{i}$	${\tilde{Q}}_{i}$
A1	(– 1.147, 0.296, 2.776)	(0.000, 0.069, 0.512)	(– 1.266, 0.201, 1.142)
A2	(– 1.022, 0.337, 3.091)	(0.004, 0.057, 0.415)	(– 1.191, 0.063, 1.117)
A3	(– 0.846, 0.599, 3.356)	(0.020, 0.086, 0.663)	(– 1.549, 0.964, 1.069)
A4	(– 0.423, 0.623, 4.703)	(0.019, 0.078, 0.610)	(– 1.692, 0.858, 1.005)
A5	(– 0.928, 0.501, 3.400)	(0.023, 0.073, 0.469)	(– 1.309, 0.585, 1.079)
${\tilde{S}}^{*}$	(– 1.147, 0.296, 2.776)	${\tilde{R}}^{*}$	(0.000, 0.057, 0.415)
${\tilde{S}}^{-}$	(– 0.423, 0.623, 4.703)	${\tilde{R}}^{-}$	(0.023, 0.086, 0.663)

Table 10

Fuzzy AHP-VIKOR results for simulation software package selection

	${\tilde{Q}}_{i}$	Q _i	Rank
A1	(– 1.266, 0.201, 1.142)	0.113	2
A2	(– 1.191, 0.063, 1.117)	0.029	1
A3	(– 1.549, 0.964, 1.069)	0.563	5
A4	(– 1.692, 0.858, 1.005)	0.457	4
A5	(– 1.309, 0.585, 1.079)	0.352	3

5 Conclusion

In this paper, an integrated fuzzy AHP-VIKOR method is proposed to evaluate simulation software package alternatives for educational use. In this integration, F-AHP is used to determine the fuzzy weights of criteria, and F-VIKOR is used to rank alternatives using these fuzzy weights.

At present, there does not appear to be a study in the literature that integrates F-AHP and F-VIKOR for the evaluation of educational use simulation software packages. Adoption of fuzzy numbers in AHP and VIKOR captures the uncertainty and vagueness on judgments of the DMs and with the proposed integration, DMs can take the advantages of both F-AHP and F-VIKOR. With fuzzy AHP-VIKOR, in the presence of conflicting criteria, simulation software packages are evaluated and ranked at a reasonable time and effort, without too many repetitive pairwise comparisons and without complicated calculations.

As a result of the evaluation, Simio is selected as the best simulation software package for educational use among popular software packages in Turkey.

For future research, outer-dependence, inner-dependence, and feedback relationships between criteria can be investigated with F-ANP and F-ANP can be integrated with F-VIKOR for evaluation and ranking problems.

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1	pedigree (PE)	9	simulation testability (TE)
2	coding aspects (CA)	10	experimentation facilities (EX)
3	software compatibility (SC)	11	statistical facilities (SF)
4	financial and technical features (FT)	12	Input / Output capabilities (IO)
5	general modeling features (GF)	13	analysis capabilities (AC)
6	modeling assistance (MA)	14	documentation and tutorial support (DT)
7	visual simulation aspects (VA)	15	Maintenance and Update (MU)
8	simulation efficiency (EF)

PE	CA	SC	FT	GF	MA	FA	EF	TE	EX	SF	IO	AC	DT	MU
	PE	E	E	SS	SW	SS	E	FS	VW	FW	FS	VW	FW	FW	FW	FW
	E	E	FW	FW	SW	E	SS	SW	AW	SS	SW	AW	AW	AW	AW
	E	E	SW	FW	SW	E	SS	SW	FW	SS	SW	FW	FW	FW	FW
CA	E	E	SS	SW	SS	SS	VW	SW	FW	FW	FS	E	FW	FW	FW
	E	E	FW	FW	SW	FW	SW	FW	VW	VW	SS	E	VW	VW	VW
	E	E	SS	FW	SW	SW	SW	FW	FW	FW	SS	SS	FW	FW	FW
SC	SW	SW	E	SS	E	E	E	FW	VW	SS	FS	VW	VW	VW	SS
	FS	FS	E	FW	SS	E	SS	E	SW	FW	SS	SW	SW	SW	FW
	SS	SW	E	SW	E	E	E	SW	SW	SW	SS	SW	SW	SW	SW
FT	SS	SS	SW	E	FS	FS	FS	VW	FS	SW	SW	E	SW	SW	SW
	FS	FS	FS	E	SS	E	SS	FW	SS	SW	SW	E	SW	SW	SW
	FS	FS	SS	E	SS	SS	SS	SW	SS	E	E	E	E	E	E
GF	SW	SW	E	FW	E	E	E	VW	VW	VW	VW	FS	VW	VW	VW
	SS	SS	SW	SW	E	SW	E	SW	FW	FW	FW	SS	FW	FW	FW
	SS	SS	E	SW	E	E	E	SW	SW	SW	SW	SS	SW	SW	SW
MA	E	SW	E	FW	E	E	SW	SS	VW	FS	VW	VW	VW	VW	E
	E	FS	E	E	SS	E	SS	FW	SW	SS	SW	SW	SW	SW	E
	E	SS	E	SW	E	E	E	SW	SW	SS	SW	SW	SW	SW	E
FA	FW	VS	E	FW	E	SS	E	FW	SS	AW	E	AW	SS	AW	AW
	SW	SS	SW	SW	E	SW	E	SW	FW	FW	E	FW	FW	FW	FW
	SW	SS	E	SW	E	E	E	SW	SW	SW	SS	SW	SW	SW	SW
EF	VS	SS	FS	VS	VS	SW	FS	E	SW	SW	SW	FS	SW	E	SW
	SS	FS	E	FS	SS	FS	SS	E	SW	SW	SW	SS	SW	E	SW
	SS	FS	SS	SS	SS	SS	SS	E	E	E	E	SS	E	E	E
TE	FS	FS	VS	FW	VS	VS	SW	SS	E	E	SW	E	AW	AW	VW
	AS	VS	SS	SW	FS	SS	FS	SS	E	E	SW	E	FW	FW	FW
	FS	FS	SS	SW	SS	SS	SS	E	E	E	E	SS	SW	SW	SW
EX	FW	FS	SW	SS	VS	FW	AS	SS	E	E	AW	AW	VW	SW	VW
	SW	VS	FS	SS	FS	SW	FS	SS	E	E	FW	FW	SW	SW	SW
	SW	FS	SS	E	SS	SW	SS	E	E	E	SW	SW	SW	E	SW
SF	VS	FW	FW	SS	VS	VS	E	SS	SS	AS	E	SW	SW	SW	E
	SS	SW	SW	SS	FS	SS	E	SS	SS	FS	E	SW	SW	SW	E
	SS	SW	SW	E	SS	SS	SW	E	E	SS	E	E	E	E	E
IO	FS	E	VS	E	FW	VS	AS	FW	E	AS	SS	E	VW	SS	VW
	AS	E	SS	E	SW	SS	FS	SW	E	FS	SS	E	SW	FW	SW
	FS	SW	SS	E	SW	SS	SS	SW	SW	SS	E	E	SW	SW	SW
AC	FS	FS	VS	SS	VS	VS	SW	SS	AS	VS	SS	VS	E	VW	AW
	AS	VS	SS	SS	FS	SS	FS	SS	FS	SS	SS	SS	E	SW	FW
	FS	FS	SS	E	SS	SS	SS	E	SS	SS	E	SS	E	SW	SW
DT	FS	FS	VS	SS	VS	VS	AS	E	AS	SS	SS	SW	VS	E	AW
	AS	VS	SS	SS	FS	SS	FS	E	FS	SS	SS	FS	SS	E	FW
	FS	FS	SS	E	SS	SS	SS	E	SS	E	E	SS	SS	E	SW
MU	FS	FS	SW	SS	VS	E	AS	SS	VS	VS	E	VS	AS	AS	E
	AS	VS	FS	SS	FS	E	FS	SS	FS	SS	E	SS	FS	FS	E
	FS	FS	SS	E	SS	E	SS	E	SS	SS	E	SS	SS	SS	E

PE	CA	SC	FT	GF	MA	FA	EF	TE	EX	SF	IO	AC	DT	MU
A1	G	MG	G	F	G	G	F	MG	G	MG	MG	G	MG	G	F
	G	MG	G	F	G	G	MG	G	MG	F	MG	F	MG	MG	MG
	G	MG	G	F	MG	MG	MG	MG	MG	MG	F	MG	G	G	MG
A2	MG	F	MG	F	G	F	MG	F	G	F	G	F	G	G	MG
	MG	F	MG	MG	MG	MG	MG	MG	MG	MG	G	F	G	G	MG
	MG	MG	F	MG	MG	F	F	MG	MG	F	G	F	VG	G	MG
A3	VG	P	VG	MG	VG	P	MG	P	MP	VG	P	P	VG	F	F
	G	P	VG	MG	G	P	MG	MP	MP	G	P	P	G	F	MG
	G	P	VG	MG	G	VP	MG	P	P	G	P	MP	G	F	F
A4	MP	MP	MG	MP	MP	F	MG	F	MG	MP	MP	P	MP	MG	G
	F	P	MG	MP	F	F	G	F	G	F	P	VP	F	MG	G
	MP	MP	MG	F	MP	F	MG	F	MG	MP	MP	VP	MP	MG	G
A5	G	P	F	G	G	F	MG	P	MG	G	F	VP	F	G	MG
	MG	P	F	G	MG	MG	MG	MP	MG	MG	MG	VP	MG	MG	MG
	G	P	F	G	G	MG	MG	P	MG	G	MG	VP	MG	MG	F

PE	CA	SC	FT	GF	MA	FA	EF	TE	EX	SF	IO	AC	DT	MU
	PE	E	E	SS	SW	SS	E	FS	VW	FW	FS	VW	FW	FW	FW	FW
	E	E	FW	FW	SW	E	SS	SW	AW	SS	SW	AW	AW	AW	AW
	E	E	SW	FW	SW	E	SS	SW	FW	SS	SW	FW	FW	FW	FW
CA	E	E	SS	SW	SS	SS	VW	SW	FW	FW	FS	E	FW	FW	FW
	E	E	FW	FW	SW	FW	SW	FW	VW	VW	SS	E	VW	VW	VW
	E	E	SS	FW	SW	SW	SW	FW	FW	FW	SS	SS	FW	FW	FW
SC	SW	SW	E	SS	E	E	E	FW	VW	SS	FS	VW	VW	VW	SS
	FS	FS	E	FW	SS	E	SS	E	SW	FW	SS	SW	SW	SW	FW
	SS	SW	E	SW	E	E	E	SW	SW	SW	SS	SW	SW	SW	SW
FT	SS	SS	SW	E	FS	FS	FS	VW	FS	SW	SW	E	SW	SW	SW
	FS	FS	FS	E	SS	E	SS	FW	SS	SW	SW	E	SW	SW	SW
	FS	FS	SS	E	SS	SS	SS	SW	SS	E	E	E	E	E	E
GF	SW	SW	E	FW	E	E	E	VW	VW	VW	VW	FS	VW	VW	VW
	SS	SS	SW	SW	E	SW	E	SW	FW	FW	FW	SS	FW	FW	FW
	SS	SS	E	SW	E	E	E	SW	SW	SW	SW	SS	SW	SW	SW
MA	E	SW	E	FW	E	E	SW	SS	VW	FS	VW	VW	VW	VW	E
	E	FS	E	E	SS	E	SS	FW	SW	SS	SW	SW	SW	SW	E
	E	SS	E	SW	E	E	E	SW	SW	SS	SW	SW	SW	SW	E
FA	FW	VS	E	FW	E	SS	E	FW	SS	AW	E	AW	SS	AW	AW
	SW	SS	SW	SW	E	SW	E	SW	FW	FW	E	FW	FW	FW	FW
	SW	SS	E	SW	E	E	E	SW	SW	SW	SS	SW	SW	SW	SW
EF	VS	SS	FS	VS	VS	SW	FS	E	SW	SW	SW	FS	SW	E	SW
	SS	FS	E	FS	SS	FS	SS	E	SW	SW	SW	SS	SW	E	SW
	SS	FS	SS	SS	SS	SS	SS	E	E	E	E	SS	E	E	E
TE	FS	FS	VS	FW	VS	VS	SW	SS	E	E	SW	E	AW	AW	VW
	AS	VS	SS	SW	FS	SS	FS	SS	E	E	SW	E	FW	FW	FW
	FS	FS	SS	SW	SS	SS	SS	E	E	E	E	SS	SW	SW	SW
EX	FW	FS	SW	SS	VS	FW	AS	SS	E	E	AW	AW	VW	SW	VW
	SW	VS	FS	SS	FS	SW	FS	SS	E	E	FW	FW	SW	SW	SW
	SW	FS	SS	E	SS	SW	SS	E	E	E	SW	SW	SW	E	SW
SF	VS	FW	FW	SS	VS	VS	E	SS	SS	AS	E	SW	SW	SW	E
	SS	SW	SW	SS	FS	SS	E	SS	SS	FS	E	SW	SW	SW	E
	SS	SW	SW	E	SS	SS	SW	E	E	SS	E	E	E	E	E
IO	FS	E	VS	E	FW	VS	AS	FW	E	AS	SS	E	VW	SS	VW
	AS	E	SS	E	SW	SS	FS	SW	E	FS	SS	E	SW	FW	SW
	FS	SW	SS	E	SW	SS	SS	SW	SW	SS	E	E	SW	SW	SW
AC	FS	FS	VS	SS	VS	VS	SW	SS	AS	VS	SS	VS	E	VW	AW
	AS	VS	SS	SS	FS	SS	FS	SS	FS	SS	SS	SS	E	SW	FW
	FS	FS	SS	E	SS	SS	SS	E	SS	SS	E	SS	E	SW	SW
DT	FS	FS	VS	SS	VS	VS	AS	E	AS	SS	SS	SW	VS	E	AW
	AS	VS	SS	SS	FS	SS	FS	E	FS	SS	SS	FS	SS	E	FW
	FS	FS	SS	E	SS	SS	SS	E	SS	E	E	SS	SS	E	SW
MU	FS	FS	SW	SS	VS	E	AS	SS	VS	VS	E	VS	AS	AS	E
	AS	VS	FS	SS	FS	E	FS	SS	FS	SS	E	SS	FS	FS	E
	FS	FS	SS	E	SS	E	SS	E	SS	SS	E	SS	SS	SS	E

PE	CA	SC	FT	GF	MA	FA	EF	TE	EX	SF	IO	AC	DT	MU
A1	G	MG	G	F	G	G	F	MG	G	MG	MG	G	MG	G	F
	G	MG	G	F	G	G	MG	G	MG	F	MG	F	MG	MG	MG
	G	MG	G	F	MG	MG	MG	MG	MG	MG	F	MG	G	G	MG
A2	MG	F	MG	F	G	F	MG	F	G	F	G	F	G	G	MG
	MG	F	MG	MG	MG	MG	MG	MG	MG	MG	G	F	G	G	MG
	MG	MG	F	MG	MG	F	F	MG	MG	F	G	F	VG	G	MG
A3	VG	P	VG	MG	VG	P	MG	P	MP	VG	P	P	VG	F	F
	G	P	VG	MG	G	P	MG	MP	MP	G	P	P	G	F	MG
	G	P	VG	MG	G	VP	MG	P	P	G	P	MP	G	F	F
A4	MP	MP	MG	MP	MP	F	MG	F	MG	MP	MP	P	MP	MG	G
	F	P	MG	MP	F	F	G	F	G	F	P	VP	F	MG	G
	MP	MP	MG	F	MP	F	MG	F	MG	MP	MP	VP	MP	MG	G
A5	G	P	F	G	G	F	MG	P	MG	G	F	VP	F	G	MG
	MG	P	F	G	MG	MG	MG	MP	MG	MG	MG	VP	MG	MG	MG
	G	P	F	G	G	MG	MG	P	MG	G	MG	VP	MG	MG	F

A fuzzy AHP-VIKOR approach for evaluation of educational use simulation software packages

Abstract

Keywords

1 Introduction

2 Literature review

3 Proposed fuzzy AHP-VIKOR approach

3.1 Definitions related to fuzzy numbers

Table 2 Linguistic terms and TFNs for the ratings of alternatives in F-VIKOR Linguistic terms Triangular fuzzy number (TFN) 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)

References

Table 2
Linguistic terms and TFNs for the ratings of alternatives in F-VIKOR

Linguistic terms Triangular fuzzy number (TFN)

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)

PE	CA	SC	FT	GF	MA	FA	EF	TE	EX	SF	IO	AC	DT	MU
	PE	E	E	SS	SW	SS	E	FS	VW	FW	FS	VW	FW	FW	FW	FW
	E	E	FW	FW	SW	E	SS	SW	AW	SS	SW	AW	AW	AW	AW
	E	E	SW	FW	SW	E	SS	SW	FW	SS	SW	FW	FW	FW	FW
CA	E	E	SS	SW	SS	SS	VW	SW	FW	FW	FS	E	FW	FW	FW
	E	E	FW	FW	SW	FW	SW	FW	VW	VW	SS	E	VW	VW	VW
	E	E	SS	FW	SW	SW	SW	FW	FW	FW	SS	SS	FW	FW	FW
SC	SW	SW	E	SS	E	E	E	FW	VW	SS	FS	VW	VW	VW	SS
	FS	FS	E	FW	SS	E	SS	E	SW	FW	SS	SW	SW	SW	FW
	SS	SW	E	SW	E	E	E	SW	SW	SW	SS	SW	SW	SW	SW
FT	SS	SS	SW	E	FS	FS	FS	VW	FS	SW	SW	E	SW	SW	SW
	FS	FS	FS	E	SS	E	SS	FW	SS	SW	SW	E	SW	SW	SW
	FS	FS	SS	E	SS	SS	SS	SW	SS	E	E	E	E	E	E
GF	SW	SW	E	FW	E	E	E	VW	VW	VW	VW	FS	VW	VW	VW
	SS	SS	SW	SW	E	SW	E	SW	FW	FW	FW	SS	FW	FW	FW
	SS	SS	E	SW	E	E	E	SW	SW	SW	SW	SS	SW	SW	SW
MA	E	SW	E	FW	E	E	SW	SS	VW	FS	VW	VW	VW	VW	E
	E	FS	E	E	SS	E	SS	FW	SW	SS	SW	SW	SW	SW	E
	E	SS	E	SW	E	E	E	SW	SW	SS	SW	SW	SW	SW	E
FA	FW	VS	E	FW	E	SS	E	FW	SS	AW	E	AW	SS	AW	AW
	SW	SS	SW	SW	E	SW	E	SW	FW	FW	E	FW	FW	FW	FW
	SW	SS	E	SW	E	E	E	SW	SW	SW	SS	SW	SW	SW	SW
EF	VS	SS	FS	VS	VS	SW	FS	E	SW	SW	SW	FS	SW	E	SW
	SS	FS	E	FS	SS	FS	SS	E	SW	SW	SW	SS	SW	E	SW
	SS	FS	SS	SS	SS	SS	SS	E	E	E	E	SS	E	E	E
TE	FS	FS	VS	FW	VS	VS	SW	SS	E	E	SW	E	AW	AW	VW
	AS	VS	SS	SW	FS	SS	FS	SS	E	E	SW	E	FW	FW	FW
	FS	FS	SS	SW	SS	SS	SS	E	E	E	E	SS	SW	SW	SW
EX	FW	FS	SW	SS	VS	FW	AS	SS	E	E	AW	AW	VW	SW	VW
	SW	VS	FS	SS	FS	SW	FS	SS	E	E	FW	FW	SW	SW	SW
	SW	FS	SS	E	SS	SW	SS	E	E	E	SW	SW	SW	E	SW
SF	VS	FW	FW	SS	VS	VS	E	SS	SS	AS	E	SW	SW	SW	E
	SS	SW	SW	SS	FS	SS	E	SS	SS	FS	E	SW	SW	SW	E
	SS	SW	SW	E	SS	SS	SW	E	E	SS	E	E	E	E	E
IO	FS	E	VS	E	FW	VS	AS	FW	E	AS	SS	E	VW	SS	VW
	AS	E	SS	E	SW	SS	FS	SW	E	FS	SS	E	SW	FW	SW
	FS	SW	SS	E	SW	SS	SS	SW	SW	SS	E	E	SW	SW	SW
AC	FS	FS	VS	SS	VS	VS	SW	SS	AS	VS	SS	VS	E	VW	AW
	AS	VS	SS	SS	FS	SS	FS	SS	FS	SS	SS	SS	E	SW	FW
	FS	FS	SS	E	SS	SS	SS	E	SS	SS	E	SS	E	SW	SW
DT	FS	FS	VS	SS	VS	VS	AS	E	AS	SS	SS	SW	VS	E	AW
	AS	VS	SS	SS	FS	SS	FS	E	FS	SS	SS	FS	SS	E	FW
	FS	FS	SS	E	SS	SS	SS	E	SS	E	E	SS	SS	E	SW
MU	FS	FS	SW	SS	VS	E	AS	SS	VS	VS	E	VS	AS	AS	E
	AS	VS	FS	SS	FS	E	FS	SS	FS	SS	E	SS	FS	FS	E
	FS	FS	SS	E	SS	E	SS	E	SS	SS	E	SS	SS	SS	E

PE	CA	SC	FT	GF	MA	FA	EF	TE	EX	SF	IO	AC	DT	MU
A1	G	MG	G	F	G	G	F	MG	G	MG	MG	G	MG	G	F
	G	MG	G	F	G	G	MG	G	MG	F	MG	F	MG	MG	MG
	G	MG	G	F	MG	MG	MG	MG	MG	MG	F	MG	G	G	MG
A2	MG	F	MG	F	G	F	MG	F	G	F	G	F	G	G	MG
	MG	F	MG	MG	MG	MG	MG	MG	MG	MG	G	F	G	G	MG
	MG	MG	F	MG	MG	F	F	MG	MG	F	G	F	VG	G	MG
A3	VG	P	VG	MG	VG	P	MG	P	MP	VG	P	P	VG	F	F
	G	P	VG	MG	G	P	MG	MP	MP	G	P	P	G	F	MG
	G	P	VG	MG	G	VP	MG	P	P	G	P	MP	G	F	F
A4	MP	MP	MG	MP	MP	F	MG	F	MG	MP	MP	P	MP	MG	G
	F	P	MG	MP	F	F	G	F	G	F	P	VP	F	MG	G
	MP	MP	MG	F	MP	F	MG	F	MG	MP	MP	VP	MP	MG	G
A5	G	P	F	G	G	F	MG	P	MG	G	F	VP	F	G	MG
	MG	P	F	G	MG	MG	MG	MP	MG	MG	MG	VP	MG	MG	MG
	G	P	F	G	G	MG	MG	P	MG	G	MG	VP	MG	MG	F