A new hybrid simulation-based assignment approach for evaluating airlines with multiple service quality criteria

Journal of Air Transport Management 63 (2017) 45e60 Contents lists available at ScienceDirect Journal of Air Transport Management journal homepage: www.elsevier.com/locate/jairtraman A new hybrid simulation-based assignment approach for evaluating airlines with multiple service quality criteria Mehdi Keshavarz Ghorabaee a, *, Maghsoud Amiri a, Edmundas Kazimieras Zavadskas b, Zenonas Turskis b, Jurgita Antucheviciene b a b Department of Industrial Management, Faculty of Management and Accounting, Allameh Tabataba'i University, Tehran, Iran Department of Construction Technology and Management, Vilnius Gediminas Technical University, Sauletekio al. 11, LTe10223 Vilnius, Lithuania a r t i c l e i n f o a b s t r a c t Article history: Received 7 March 2017 Received in revised form 10 April 2017 Accepted 24 May 2017 Evaluation of airlines based on service quality criteria can help to improve the processes of airlines, and also can give guidance to travel agencies to provide better choices for passengers and tourists. In this study, a hybrid simulation-based assignment approach is proposed to deal with multi-criteria decisionmaking problems with a group of decision-makers. A probability distribution is used to model decisionmakers’ opinions and constructing a stochastic decision matrix. Then some efficient multi-criteria decision-making methods are utilized for evaluating alternatives in a simulation process. The proposed approach is applied to a problem of evaluation of five airlines with respect to opinions of 58 experts on 28 criteria. The results show the efficiency of the proposed to handle decision-making problems with a large number of experts. Moreover, the evaluation results are more reliable than the other decisionmaking approaches because of simulating decision-makers’ opinions, using multiple methods and evaluating based on aggregative results. © 2017 Elsevier Ltd. All rights reserved. Keywords: Airline evaluation Service quality TOPSIS COPRAS WASPAS EDAS MCDM 1. Introduction Service quality can be defined as consumer's overall feeling of the relative superiority or inferiority of an organization and its services which is a result of comparing between customers' expectations and actual services performed (Rust and Oliver, 1993). Service quality is an important factor for airlines and many researchers have applied service quality related theories and methods in the airline industry. Providing high quality services which satisfy passengers and tourists is a core competitive advantage for an airline to reach profitability and sustainable development (Chen, 2008). Most of the studies in airline service quality evaluation presumed the quality of services as a multidimensional factor and measured it by a well-known instrument called SERVQUAL (Saha and Theingi, 2009). SERVQUAL is a multidimensional measuring instrument which is designed to capture consumer expectations and perceptions of a service in terms of five dimensions including reliability, assurance, tangibles, empathy and responsiveness that are believed to represent the quality of services * Corresponding author. E-mail address: m.keshavarz_gh@yahoo.com (M. Keshavarz Ghorabaee). http://dx.doi.org/10.1016/j.jairtraman.2017.05.008 0969-6997/© 2017 Elsevier Ltd. All rights reserved. (Parasuraman et al., 1988). Because of the multi-dimensional nature of SERVQUAL, this instrument can be integrated with multi-criteria decision-making (MCDM) approaches for evaluation of quality of service (Mardani et al., 2015c). In this field, Awasthi et al. (2011) developed hybrid approach based on fuzzy TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) method and SERVQUAL model for evaluation of transportation service quality. They used the dimensions of the SERVQUAL model as criteria for evaluation and ranking some alternatives. Kuo (2011) proposed a novel intervalvalued fuzzy MCDM approach for evaluation of Chinese crossstrait airlines based on service quality criteria. The approach was based on combining VIKOR (in Serbian: VlseKriterijumska Optimizacija I Kompromisno Resenje) and grey relational analysis (GRA) methods, and the SERVQUAL model was used to development of evaluation criteria. In this study, we also use the SERVQUAL model and propose an MCDM approach based on its dimensions. Chou et al. (2011) presented a fuzzy weighted SERVQUAL model and applied it to the evaluation of airline service quality. The dimensions of their SERVQUAL model are used in this study as the evaluation criteria. It is important to make the evaluation of airlines with multiple 46 M. Keshavarz Ghorabaee et al. / Journal of Air Transport Management 63 (2017) 45e60 service quality criteria based on opinions of people who have experience of traveling by the considered airline. Different opinions can result in different evaluations, and it may be more complicated when the number of people increases. The group decision-making approaches are efficient in such situations. In the group decisionmaking approaches, all the people, who are involved in the evaluation process, are considered as a group of decision-makers. What a majority of individuals prefer should be reflected as a solution in the group decision-making (Kacprzyk, 1986). In this study, a new hybrid simulation-based assignment approach is developed to deal with multi-criteria decision-making problems with a group of decision-makers. In the proposed approach, the PERT distribution is used to model opinions of the group of decision-makers. This distribution is a special case of the beta distribution and has three parameters (minimum, most likely and maximum) in its standard format. This is a very flexible distribution for modelling expert opinions and can be viewed as a smooth version of the uniform distribution or triangular distribution. After defining a stochastic MCDM problem by this distribution, the Monte Carlo simulation process is started with a predefined number of iterations. In the simulation process, a random MCDM problem generated in each iteration, and the alternatives are evaluated using some MCDM methods. Although the TOPSIS, COPRAS (COmplex PRoportional ASsessment), WASPAS (Weighted Aggregated Sum Product Assessment) and EDAS (Evaluation based on Distance from Average Solution) methods are used in this study, the proposed approach is not limited to these methods, and we aim to increase the accuracy of the evaluation by using multiple methods and reach to more reliable results. The normalized ranking scores obtained from these evaluations at the end of iterations are aggregated and used as the parameters of a linear assignment model. Solving the assignment model, we can determine the final rank of alternatives. The proposed approach is applied to a case study of evaluation and prioritization of airlines with multiple service quality criteria defined in the SERVQUAL model of Chou et al. (2011). The rest of this paper is organized as follows. In Section 2, we briefly review the literature on the airline service quality and multicriteria decision-making approaches. In Section 3, the methodological components of the study and the proposed approach are presented in detail. Section 4 describes the application of the proposed approach in evaluation of airlines with multiple service quality criteria. Section 5 presents discussion, and finally conclusions and future directions are presented in Section 6. 2. Literature review In this section, we present a brief review of some studies on airline service quality and multi-criteria decision-making methods. 2.1. Airline service quality There have been many studies in the field of airline service quality and the researchers have worked on different aspect of this field and used different methodologies in their studies. In the following we summarize some of the important studies in this field. Tsaur et al. (2002) applied the fuzzy set theory for evaluation of the service quality of airline. They used the analytic hierarchy process (AHP) method for determination of criteria weights. Then the TOPSIS method is utilized for ranking the alternatives. The dimensions of the SERVQUAL model were used to define the evaluation criteria, and the tangibles and empathy were found as the most and the least important criteria of their study, respectively. Park et al. (2004) studied on understanding of air passengers' behavioral intentions by testing a conceptual model. Their model considers some variables including service perception, service expectation, airline image, passenger satisfaction, service value and behavioral intentions simultaneously. They applied path analysis via maximum likelihood estimator to data collected from Korean passengers, and found that passenger satisfaction, service value and airline image have a direct effect on air passengers’ behavioral intentions. Chen and Chang (2005) examined the gaps between the service expectations of passengers and two other variables of a Taiwanese airline: the real service received and the perceptions of the expectations by frontline managers and employees. Then for determining areas for improvement, they applied the importanceperformance analysis to construct service attribute evaluation maps. Results showed that the passengers were more concerned about the responsiveness and assurance dimensions from airline frontline staff. The tangibles dimension was identified as an important dimension for evaluation of in-flight service quality. Pakdil and Aydın (2007) studied on expectations and perceptions in airline services. Based on data collected at a Turkish airline, they measured airline service quality using a weighted SERVQUAL model and factor analysis. The results of their research showed that the responsiveness was the most important dimension and the availability was the least important dimension of service quality. The educational level of passengers was an important variable in their study affecting the expectations and perceptions of them. An and Noh (2009) investigated the impact of the in-flight service quality on airline customer satisfaction and loyalty. Data from two classes of passengers including prestige (business) and economy were analyzed in their study. The results showed that different factors are important in the in-flight service quality according to the passengers' class. The findings implied that different delivery strategies should be chosen by airline companies' in-flight service based on the passengers’ class. Liou et al. (2011) applied a modified VIKOR method to improve service quality of domestic airlines in Taiwan. Their model helps decision-makers to identify the gaps between alternatives and aspired levels in practice. To establish a comprehensive service quality evaluation framework and reduce the gaps for achieving the aspired-level, a large sample was used by them. They also provided some managerial implications to improve the level of service quality of different airlines. Baker (2013) studied on the service quality and customer satisfaction of the top 14 U.S. airlines between 2007 and 2011. His study had two objectives: comparison of customer satisfaction and service quality based on service quality dimensions of the airlines and examination of the relationships between the dimensions of service quality and passengers’ satisfaction. Implications related to operating costs, market share, infrastructure and customer service confirmed that the service quality of low cost airlines was higher than that of traditional legacy airlines. Muturi et al. (2013) examined the impact of airline service quality on passenger satisfaction and loyalty in Uganda. Their study used random sampling technique and with 303 respondents. The results of their study showed that the quality of pre-flight, in-flight and post-flight services had a statistically significant effect on passenger satisfaction, and also passenger satisfaction had a significant effect on passenger loyalty. They suggested that airlines should consider different strategies based on characteristics of the customers such as occupation, age, gender and education level, to improve their service quality. Choi et al. (2015) applied a service quality-adjusted data envelopment analysis (SQ-adjusted DEA) to study operational efficiency of US airlines. They found that, in the long-term, a focus on service quality can help to increase customer satisfaction and improve service productivity and overall organizational M. Keshavarz Ghorabaee et al. / Journal of Air Transport Management 63 (2017) 45e60 performance even though there were short-term tradeoffs between service quality and productivity. They also showed that the proposed SQ-adjusted DEA was better than the standard DEA to explore service productivity. Suki (2014) examined the impact of airline service quality dimensions such as terminal tangibles, empathy and airline tangibles on satisfaction levels of customers in Malaysia. The other investigation of the research was related to the relationship between the levels of satisfaction and the general perceptions about service quality. The relationship between customer satisfaction and airline service quality was shown by using structural equation modeling (SEM) approach. Moreover, the results revealed that empathy highly affects customer satisfaction. Liou et al. (2016) applied the multivariate statistical analysis and multi-criteria decision-making methods to the improvement of service quality. The rough set theory with a flow graph approach was used by them to identify customer attitudes towards service quality and a large sample of airline customers was used to define a set of rules. They demonstrated that the proposed approach can assist in identifying the needs of customers and determining their characteristics and can help managers to develop airline strategies for improving the quality of services and satisfying customers’ needs. Chen (2016) proposed an approach to select airline service quality improvement criteria for the Taiwanese airline industry. A combined MCDM model based on decision-making trial and evaluation laboratory (DEMATEL) and analytic network process (ANP) is utilized for the selection process. The proposed approach provides a direction for airlines to measure and improve their service quality for developing their competitive advantages. 2.2. Multi-criteria decision-making approaches Many studies have been conducted in the field of MCDM methods during the past years. These methods have been applied to many problems in science and engineering. In the following, because of using the TOPSIS, COPRAS, WASPAS and EDAS methods in the proposed approach, we briefly review some studies on these methods. However, there are some other MCDM methods like ARAS (Additive Ratio ASsessment), VIKOR and MULTIMOORA (abbreviation of ‘multi-objective optimization by ratio analysis plus the full multiplicative form’) which have been widely used. Interested readers are referred to the review articles (Mardani et al., 2015a, 2015b, 2016a, 2016b, 2016c). The TOPSIS method is one of the popular MCDM methods which was proposed by Hwang and Yoon (1981). This method has been extended in many types of uncertain environments. A fuzzy extension of TOPSIS method was introduced by Chen (2000) and has been applied to many problems till now. Some other extensions of this method in fuzzy environment were proposed by Jahanshahloo et al. (2006) and Wang and Elhag (2006). Moreover, this method has been extended in the other types of fuzzy sets such as intuitionistic fuzzy sets (Boran et al., 2009), interval-valued fuzzy sets (Chen and Tsao, 2008), interval-valued intuitionistic fuzzy sets (Ye, 2010), interval type-2 fuzzy sets (Chen and Lee, 2010) and hesitant fuzzy sets (Beg and Rashid, 2013; Xu and Zhang, 2013). This method and its extensions have been used in many studies to deal with multi-criteria decision-making problems. Interested readers are referred to the review performed by Zavadskas et al. (2016). The COPRAS method which was proposed by Zavadskas et al. (1994) has been applied to many MCDM problems. Das et al. (2012) proposed a framework by integrating fuzzy AHP and COPRAS methods to measure relative performance of Indian technical institutions. Mulliner et al. (2013) applied the COPRAS method to a multi-criteria decision-making problem of sustainable housing 47 affordability in three residential areas. Tavana et al. (2013) developed a hybrid MCDM approach for social media platform selection using the fuzzy ANP method and extended COPRAS method with grey numbers. Nguyen et al. (2015) used a fuzzy linguistic preference based AHP and integrated it with the fuzzy COPRAS for machine tool evaluation. Mulliner et al. (2016) conducted a comparative analysis of some MCDM methods for the process of sustainable housing affordability assessment. Rathi and Balamohan (2016) developed a mathematical model based on the COPRAS method for fuzzy multi-criteria group decision making with subjective evaluation. Mousavi-Nasab and Sotoudeh-Anvari (2017) proposed a hybrid MCDM approach based on the data envelopment analysis, TOPSIS and COPRAS methods and applied it to a material selection problem. A review on applications and extensions of the COPRAS method was performed by Stefano et al. (2015). The WASPAS method is a relatively new method which was proposed and optimized by Zavadskas et al. (2012). This method has been used by many researchers in the past years. Madi
c et al. (2015) applied the WASPAS method to the selection of cutting inserts for aluminum alloys machining. Ghorshi Nezhad et al. (2015) proposed a hybrid MCDM approach based on the step-wise weight assessment ratio analysis (SWARA) and WASPAS method for planning high tech industries. Chakraborty et al. (2015) used the WASPAS method to solve some MCDM problems in manufacturing environment. Hashemkhani Zolfani et al. (2015) applied a hybrid SWARA-WASPAS approach for evaluation of strategies in multiple Nash equilibriums. Turskis et al. (2015) developed a hybrid model based on the fuzzy AHP and WASPAS methods for construction site selection. Bozorg-Haddad et al. (2016) used WASPAS and evolutionary algorithms for benchmarking in optimal reservoir optimization problems. The WASPAS method has also been applied to other areas such as asset redevelopment (Pavlovskis et al., 2016), green supplier selection (Keshavarz Ghorabaee et al., 2016a; Yazdani et al., 2016), maintenance performance analysis (Ighravwe and Oke, 2016) and personnel selection (Karabasevi
c et al., 2016). The EDAS method is a new and efficient method which was proposed by Keshavarz Ghorabaee et al. (2015) for inventory classification problem. Keshavarz Ghorabaee et al. (2015) demonstrated the efficiency of this method for solving MCDM problems. A fuzzy extension of this method was proposed and applied to the supplier selection problem (Keshavarz Ghorabaee et al., 2016b). Furthermore, the EDAS method has been extended using grey interval numbers (Stanujkic et al., 2017), interval type-2 fuzzy sets (Keshavarz Ghorabaee et al., 2017) and intuitionistic fuzzy sets (Kahraman et al., 2017). Peng and Liu (2017) developed some algorithms for soft decision-making with neutrosophic sets based on the EDAS method, new similarity measure and level soft set. Stevi
c et al. (2016) proposed a hybrid MCDM approach based on the AHP and EDAS methods for logistics evaluations. Turskis and Juodagalviene_ (2016) used the EDAS method to propose a hybrid MCDM approach for assessing a stairs shape for dwelling houses. 3. Methodology The proposed approach has a framework with different steps which need to be elucidated. In this section, preliminaries of the proposed approach are presented first, and then we describe the steps and framework of it. 3.1. Preliminaries As previously mentioned, the PERT distribution and multicriteria decision-making methods are the main components of the proposed approach. In the following, we define the PERT 48 M. Keshavarz Ghorabaee et al. / Journal of Air Transport Management 63 (2017) 45e60 distribution and present the steps of using the considered MCDM methods. 3.1.1. PERT distribution A PERT distribution is a modification to the beta distribution. The beta distribution has two parameters (a1 and a2 ) and its domain is between zero and one. The probability density function (PDF) of the beta distribution is as follows (Gullco and Anderson, 2009): Betaðx; a1 ; a2 Þ ¼ Gða1 þ a2 Þ a1 x Gða1 ÞGða2 Þ 1 ð1 xÞa2 1 (1) where, GðzÞ ¼ Z∞ xz 1 e x dx (2) 0 However, the PERT distribution is defined by its minimum, most likely and maximum values which can be any real numbers (Murray, 2004; Vally et al., 2014). Suppose that a, b and c denote the minimum, most likely and maximum values of a PERT distribution, respectively. Then the equation of the PDF of a PERT distribution is related to the beta distribution as follows: PERTðx; a; b; cÞ ¼ Betaðx; a1 ; a2 Þ  ðc aÞ þ a (3) where, a1 ¼ ðb aÞ  ð2b a cÞ mÞ  ðc aÞ (4) a1  ðc mÞ m a (5) a þ ðg  bÞ þ c gþ2 (6) a2 ¼ m¼ ðm In the PERT distribution g is the weighting factor of the mean (m) and can affect the shape of distribution. The value of this factor is equal to four (g¼4) in the standard PERT distribution (Vose, 2008). It should be noted that the standard version of PERT distribution is used in this research. In comparison with the triangular distribution, the PERT distribution gives a more natural shape. Moreover, the standard deviation of the PERT distribution is less sensitive to the estimate of the extreme (minimum and maximum) values. Therefore, it is not influenced as much by these values, especially if the distribution is skewed (Murray, 2004). This distribution is very useful to model expert opinion (Murray, 2004; Vally et al., 2014; Vose, 2008). As an example, Fig. 1 shows the PDF of a standard PERT distribution with parameters a ¼ 4, b ¼ 7 and c ¼ 13. 3.1.2. MCDM methods The proposed approach is based on using multiple MCDM methods to increase the accuracy and reliability of the evaluation results. Here, four efficient MCDM methods including TOPSIS, COPRAS, WASPAS and EDAS are used in this research for evaluation of alternatives. However, the proposed approach is not limited to these methods. Any of these four methods can be replaced with other MCDM method, and also we can use another method together with these four methods, provided that the method has similar nature. However, in this study, we try to consider some old Fig. 1. An example of the PDF of a PERT distribution. and new MCDM methods which have been applied in this field. Accordingly, the TOPSIS and COPRAS methods are chosen as two old MCDM methods which have been widely used by many researchers in the past years, and the WASPAS and EDAS methods are selected as two new methods which have been recently given scholarly attention. We have briefly reviewed some studies which used these four methods in decision-making processes. In this subsection, the steps of using these methods are presented. Suppose that we have a multi-criteria decision-making problem with n alternatives and m criteria, and the decision-matrix is defined as follows: x11 6 x21 6 6 « X¼6 6 xi1 6 4 « xn1 2 x12 x22 « xi2 « xn2 / / 1 / 1 / x1j x2j « xij « xnj / / 1 / 1 / 3 x1m x2m 7 7 « 7 7 xim 7 7 « 5 xnm (7) In this decision-making problem, xij shows the performance value (rating) of i th alternative with respect to j th criterion (i¼1,2, …,n and j ¼ 1,2, …,m). Also, wj is used to define the weight of j th criPm terion, and j¼1 wj ¼ 1. According to the mentioned definition, many MCDM methods can be used for evaluating and ranking of alternatives. In the following, the steps of TOPSIS, COPRAS, WASPAS and EDAS methods are summarized. 3.1.2.1. TOPSIS method. TOPSIS method was presented by Hwang and Yoon (1981). The process of evaluation of alternatives in this method is based on the distance of them from the ideal and antiideal (nadir) solution. The procedure of the TOPSIS method is presented in the following steps. Step 1. Determine the normalized values of the decision-matrix, as follows: xij xij ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 i¼1 xij (8) Step 2. Use the following equation to calculate the weighted normalized values: M. Keshavarz Ghorabaee et al. / Journal of Air Transport Management 63 (2017) 45e60 xw ij ¼ wj  xij Step 3. Determine the relative significance of each alternative (<i ) according the values of Sþ and Si , and calculate the utility i degree (N i ) of each alternative as follows: (9) Step 3. Obtain the ideal and anti-ideal solutions using the calculated weighted normalized values shown as follows:      n o    w* w w j2BC ; min j2NC ¼ max I * ¼ xw* ; …; x x x 1 m ij  ij  i i (10)      o  n   w w w I ¼ xw ¼ min ; …; x x x j2BC ; max j2NC 1 m ij  ij  i i <i ¼ Sþ i þ N i¼ (12) vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX 2 u m  w xij xw Di ¼ t j (13) j¼1 CCi ¼ Di D*i þ Di 3.1.2.2. COPRAS method. The COPRAS method is an MCDM method in which direct and proportional dependence of significance and priority of investigated alternatives on a system of criteria are considered (Zavadskas et al., 1994). This method has four steps as follows: wj xij xij ¼ Pn i¼1 xij (15) <i max <k xij ¼ 8 x ij > > > > max > < i xij if j2BC (20) > min xij > > i > > : xij if j2NC where BC and NC are the sets of beneficial and non-beneficial criteria, respectively. ð1Þ ð2Þ Step 2. Determine the measures of WSM (Vi ) and WPM (Vi ) for each alternative as follows: ð1Þ Vi ¼ m X wj xij (21) wj (22) j¼1 ð2Þ ¼ m Y xij j¼1 Step 3. Compute the combined measure of the WASPAS method for each alternative as follows: ð1Þ Vi ¼ wVi Step 2. Calculate the sum of weighted normalized values of each alternative with respect to beneficial (Sþ ) and non-beneficial i (Si ) criteria using the following equations: (19) Step 1. Calculate normalized performance values by linear normalization, as follows: Vi Step 1. Compute the weighted normalize values the decisionmatrix elements as follows: 1 i¼1 Si 3.1.2.3. WASPAS method. Zavadskas et al. (2012) proposed the WASPAS method as an MCDM method by integration of the weighted sum model (WSM) and weighted product model (WPM). The following steps are used in this method for decision-making process. (14) Step 6. Rank the alternatives in decreasing order of calculated closeness coefficient values. (18) Pn Step 4. Determine the rank of alternatives according to the values of the relative significance (<i ) or utility degree (N i ). The greater the value of <i (or N i ), the higher the priority. j¼1 Step 5. Calculate the closeness coefficient (CCi ) of each alternative, as follows: i¼1 Si Si It should be noted that if we have no non-beneficial criteria in the MCDM problem, the second term of Eq. (18) is omitted, i.e. <i ¼ Sþ . i where BC and NC are the sets of beneficial and non-beneficial criteria, respectively. vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX 2 u m  w D*i ¼ t xij xw* j Pn k (11) Step 4. Calculate the Euclidean distance of alternatives from the ideal (D*i ) and anti-ideal (Di ) solutions: 49 þ ð1 ð2Þ wÞVi (23) (16) In the above equation, w is the trade-off parameter of the WASPAS method and can be varied between zero and one. Using w ¼ 1 leads to weighted sum model, and when w ¼ 0 WASPAS method is transformed to weighted product model. (17) Step 4. Determine rank of the alternatives according to decreasing values of Vi : where BC and NC are the sets of beneficial and non-beneficial criteria, respectively. 3.1.2.4. EDAS method. The EDAS method is a new and efficient MCDM method which proposed by Keshavarz Ghorabaee et al. Sþ i ¼ X xij j2BC Si ¼ X xij j2NC 50 M. Keshavarz Ghorabaee et al. / Journal of Air Transport Management 63 (2017) 45e60 (2015). Evaluation process in this method is based on positive and negative distances from the average solution. The steps of using this method are presented as follows: Step 1. Determine the average solution elements (T j ) with respect to each criterion shown as follows: T j ¼ Pn i¼1 xij (24) n Step 2. Calculate the positive distance (P dij ) and negative distance (N dij ) of each elements of the decision-matrix from the calculated elements of the average solution using the following equations: P dij ¼ 8 max 0; xij > > > > < T j T j > > max 0; T j > > : T j xij 8 max 0; T j > > > > < T j xij N dij ¼ > > max 0; xij > > : T j T if j2BC (25) if j2NC if j2NC where BC and NC are the sets of beneficial and non-beneficial criteria, respectively. Step 3. Compute the weighted summation of the calculated positive and negative distances for each alternative as follows: S Pi ¼ m X wj P dij (27) m X wj N dij (28) j¼1 N Pi ¼ j¼1 Step 4. Calculate the normalized values of S P i and N P i as follows: ðnÞ S Pi ¼ S Pi max S P k (29) k ðnÞ N Pi ¼1 N Pi max N P k (30) k Step 5. Calculated the appraisal score of each alternative using the following equation: A si ¼ 1 ðnÞ ðnÞ S Pi þN Pi 2 (31) Step 6. Rank the alternatives according to decreasing values of A si . 3.1.3. Monte Carlo simulation Monte Carlo simulation is a 1. Determine the domain of the variables of the model. 2. Generate random values over the domain of the variables from a probability distribution. 3. Run the computational steps of the model iteratively and obtain the values of the output variables. 4. Aggregate the results of running in different iterations. 3.2. Proposed approach if j2BC (26) j incorporates using computer-generated random numbers and theory of probability into the solving process of problems. Usually the Monte Carlo simulation is an alternative to analytical mathematics which uses repeated sampling to determine the properties of some phenomenon or behavior (Chang, 2010). In this method, the computational model (or any other type of model) should be run in a large number of iterations with random sampling. Random values are generated for each input variable in each iteration, and running the model results random outcomes on each output variable (Thomopoulos, 2012). Although there have been many studies on the Monte Carlo simulation and its application in different fields of science and engineering, most of them tend to use a common pattern with the following steps: method that commonly In this section, we present a simulation-based approach based on the PERT distribution, the described MCDM methods (TOPSIS, COPRAS, WASPAS and EDAS) and a mathematical assignment model for evaluation of airlines with multiple service quality criteria based on the opinion of multiple experts (decision-makers). Suppose that we have p decision-makers. We describe the steps of the proposed approach as follows: Step 1. Design the evaluation problem by determination of alternatives and evaluation criteria.Choosing suitable criteria for evaluation process is very important in this step and every decision-maker should be in agreement on the chosen criteria because in the following step the alternatives are evaluated with respect to these criteria by each decision-maker. Step 2. Get the score of the evaluation criteria from each decision-maker.The 9-point Likert scale is used in this step to elicit the opinion of the decision-makers. Let us denote by wsjk the given score of j th criterion by k th (k¼1,2, …, p) decisionmaker. Then the following equation is used to normalize the scores and transform them to some values between zero and one. wsjk wjk ¼ Pm s j¼1 wjk (32) where wjk is the weight of j th criterion with respect to k th P decision-maker, and m j¼1 wjk ¼ 1. Step 3. Obtain the performance values of alternatives with respect to each criterion and each decision-maker.The 9-point Likert scale is also used in this step for rating the performance of the alternatives. Here and subsequently, xijk stands for the performance value of i th alternative on j th criterion given by k th decision-maker. Step 4. Define the stochastic multi-criteria decision-making problem according to the weights of criteria and performance values of alternatives elicited from decision-makers and the PERT distribution.Let waj , wbj and wcj denote the minimum, most likely and maximum values of weights of each criterion in a PERT distribution, respectively. Also, xaij , xbij and xcij show the minimum, most likely and maximum values of the performance M. Keshavarz Ghorabaee et al. / Journal of Air Transport Management 63 (2017) 45e60 values of alternative in a PERT distribution, respectively. Then the following equations are used to define the parameters of the PERT distributions of each element for simulation process of the decision-making problem. waj ¼ min wjk k (33) (34) wcj ¼ max wjk (35) i¼1 n P yiz ¼ 1 cz yiz ¼ 1 ci (41) where yiz ¼ xaij ¼ min xijk k xbij ¼ p 1X xijk p (36) k (37) (38) Step 5. Start the Monte Carlo simulation process, set iteration counter to one (r¼1) and define the number of iterations (N).In this step, four variables are defined to measure the ranking score of each alternative with respect to use of each MCDM method. We will denote by Sizl the score of i th alternative in z th rank (z¼1,2, …,n) in l th MCDM method (l¼1,2, …, L). In this study, we use four MCDM methods (L¼4), and Siz1 , Siz2 , Siz3 and Siz4 show the ranking score variables related to the TOPSIS, COPRAS, WASPAS and EDAS, respectively. These variables are set to zero in this step (Sizl ¼0). Step 6. Generate random values for the weights of criteria and performance values of alternatives according the PERT distriðrÞ bution (Eqs. (1) to (6)) and Eqs. (33)e(38).Let us denote by wj ðrÞ and xij the randomly generated values of the weights of criteria and performance values of alternatives in r th iteration, respectively. Step 7. Solve the randomly generated MCDM problem using all of the considered methods. In this study we use Eqs. (8)e(31) described in previous sections. a) If i th alternative is placed in z th rank in l th MCDM method, increase the value of Sizl by one (Sizl ¼Sizl þ1). b) Increase the iteration counter by one (r¼rþ1). Step 8. If the iteration counter is less than or equal to N (r  N), go to Step 6, otherwise continue. Step 9. Normalize the obtained ranking scores of different MCDM methods as follows: SN izl ¼ Sizl N (39) Step 10. Calculate the aggregated ranking scores of the alternatives as follows: SAG iz ¼ L 1X SN izl L  1 0 if ith alternative is assigned to zth rank otherwise To clear the procedure, the flowchart of the proposed approach is also depicted in Fig. 2. k¼1 xcij ¼ max xijk SAG iz yiz subject to, z¼1 k¼1 n X n X i¼1 z¼1 n P p 1X wbj ¼ wjk p k Max f ¼ 51 (40) l¼1 Step 11. Solve the following linear assignment model and find the optimal rank of alternatives: 4. Application of the proposed approach In this section, the proposed approach is applied to an example of evaluation of airline with multiple service quality criteria. For this aim, five airlines (alternatives) are considered for evaluation. To avoid advertising for these airlines, names of them are not called in this research, and we refer to them by A1 to A5 . According to the research of Chou et al. (2011) a questionnaire was designed with 28 sub-criteria (in five main criteria). The evaluation criteria and sub-criteria are presented in Table 1. We contacted 32 travel agencies and got the email address of 186 tour leaders from them. These tour leaders worked for the travel agencies for more than a year. We sent an email to these tour leaders and requested them to cooperate with us in evaluation process if they have had some experience with the considered airlines. In the questionnaire, which sent by email, the participants were asked to score the importance of the evaluation sub-criteria and also the performance of each airline on each sub-criterion by using a 9-point Likert scale. During a month, we received some replies from 58 persons of the invited tour leaders. The received data are provided as supplementary material. The 58 respondent of the questionnaire are considered as decision-makers of the problem to use the proposed approach for evaluation of the airlines. According to the steps 2 to 4 of the proposed approach, we determine the parameters of the PERT distributions for the normalized criteria weights and performance values of the alternatives. Tables 2 and 3 represent the parameters for criteria weights and performance values, respectively. It should be noted that the minimum, most likely and maximum values in these tables are calculated using Eqs. (33)e(38) and the data provided as supplementary material. Using the values of Tables 2 and 3 we can go to the step 5 of the proposed approach and start the simulation process with any number of iterations (N). In this study, we run the proposed approach with different values of N to show the effect of this parameter on the final evaluation results. For this aim, the proposed approach is run with N ¼ 5, 10, 20, 50, 100, 500, 1000 and 50000. By running the proposed approach the aggregated ranking scores are calculated. These values are shown in Table 4 for the defined numbers of iterations. The effect of increasing the number of iterations on the normalized ranking scores of different MCDM methods and aggregated ranking scores for different ranks (z¼1,2, …,5) depicted in Figs. 3e7. According to these figures, the values of the aggregated ranking scores in different ranks (i.e. z ¼ 1,2, …,5) are not stable at lower 52 M. Keshavarz Ghorabaee et al. / Journal of Air Transport Management 63 (2017) 45e60 Fig. 2. The flowchart of the proposed approach. Table 1 The service quality criteria for airline evaluation. Criteria C1 Tangibles C2 Responsiveness C3 Reliability and assurance C4 Empathy C5 Flight pattern Sub-criteria C11 C12 C13 C14 C15 C16 C17 C21 C22 C23 C24 C25 C26 C27 C28 C31 C32 C33 C41 C42 C43 C44 C45 C46 C51 C52 C53 C54 Comfort and cleanness of seat Quality of food and beverage In-flight newspapers, magazines and books In-flight washroom facility In-flight entertainment facilities and programs Availability of waiting lounges Size of airplane Courtesy of crew Handling of delay Efficient check-in/baggage handling services Crew's speed handling request Quality of the reservation services Crew's approach against unexpected situations Crew's willingness to help Appearance of crew Safety On-time departure and arrival Consistent ground/in-flight services Crew's behavior to delayed passenger Individual attention to passenger Understanding of passenger's specific needs Extent travel services Convenient ticketing process Customer complaint handling Flight problems Convenient flight schedules Frequency of flight Non-stop flight M. Keshavarz Ghorabaee et al. / Journal of Air Transport Management 63 (2017) 45e60 53 Table 2 The PERT distribution parameters for criteria weights. Criteria C1 C11 C12 C13 C14 C15 C16 C17 C21 C22 C23 C24 C25 C26 C27 C28 C31 C32 C33 C41 C42 C43 C44 C45 C46 C51 C52 C53 C54 C2 C3 C4 C5 Sum waj wbj wcj 0.0201 0.0072 0.0061 0.0146 0.0065 0.0069 0.0067 0.0140 0.0207 0.0064 0.0208 0.0211 0.0140 0.0069 0.0135 0.0211 0.0214 0.0135 0.0066 0.0140 0.0064 0.0063 0.0203 0.0196 0.0216 0.0216 0.0205 0.0138 0.0465 0.0226 0.0166 0.0367 0.0223 0.0222 0.0228 0.0356 0.0479 0.0243 0.0467 0.0486 0.0358 0.0230 0.0347 0.0535 0.0555 0.0359 0.0234 0.0364 0.0181 0.0159 0.0464 0.0466 0.0473 0.0457 0.0514 0.0375 0.0650 0.0538 0.0526 0.0556 0.0493 0.0515 0.0493 0.0588 0.0629 0.0504 0.0629 0.0643 0.0588 0.0496 0.0630 0.0732 0.0709 0.0559 0.0504 0.0584 0.0461 0.0490 0.0643 0.0602 0.0638 0.0652 0.0709 0.0571 e 1 e Table 3 The PERT distribution parameters for performance values of the alternatives. xaij C11 C12 C13 C14 C15 C16 C17 C21 C22 C23 C24 C25 C26 C27 C28 C31 C32 C33 C41 C42 C43 C44 C45 C46 C51 C52 C53 C54 xcij xbij A1 A2 A3 A4 A5 A1 A2 A3 A4 A5 A1 A2 A3 A4 A5 4 3 2 1 4 3 2 2 3 1 1 1 1 1 3 3 3 3 3 1 1 2 2 3 3 3 1 4 2 1 1 1 2 2 1 3 1 1 1 2 3 3 3 1 2 1 1 1 2 1 1 1 3 1 1 1 3 3 3 3 3 2 3 3 3 2 2 2 1 3 3 2 2 1 1 4 1 3 1 2 2 3 2 2 3 1 1 1 3 1 3 1 1 1 1 2 1 2 1 1 1 1 2 1 1 1 3 1 3 1 1 1 1 1 3 3 1 1 1 1 2 3 3 4 2 1 1 3 3 2 1 2 1 1 1 2 1 1 3 1 6.8 7.8 4.8 3.5 6.8 6.9 5.1 5.2 7.7 3.2 2.0 1.8 3.5 2.2 6.8 7.6 6.7 6.7 7.8 3.2 1.9 5.0 5.0 6.4 6.7 7.4 2.2 6.8 5.0 3.1 3.2 2.2 4.7 5.1 3.0 7.1 3.0 2.4 3.1 5.0 8.0 6.6 6.6 3.2 3.4 2.5 3.5 2.7 5.1 2.3 3.2 3.6 6.7 2.3 3.2 3.2 7.6 6.8 6.8 6.7 6.8 5.2 7.7 6.4 6.6 5.0 5.0 3.4 2.2 7.9 7.9 4.9 5.0 3.3 3.3 6.6 3.5 6.9 3.3 5.2 5.0 6.4 5.1 5.0 7.5 2.6 3.4 3.1 7.4 2.2 7.6 3.3 3.6 3.4 3.5 4.7 3.6 5.0 2.4 2.4 2.1 3.4 4.9 2.2 3.6 3.2 6.6 3.6 7.7 2.8 2.3 3.5 3.2 3.1 7.4 7.8 2.4 3.3 2.2 1.9 5.1 6.2 6.9 6.9 4.9 3.8 3.5 6.7 8.2 5.1 2.1 4.9 2.3 3.5 2.4 4.9 3.1 3.1 6.4 2.2 9 9 8 7 9 9 8 8 9 7 7 7 7 7 9 9 9 9 9 7 7 8 8 9 9 9 7 8 8 6 7 7 8 8 7 9 7 7 7 8 9 9 9 7 7 7 7 7 8 7 7 7 9 7 7 7 9 9 9 9 9 8 9 9 9 8 8 7 7 9 9 8 8 7 7 9 7 9 7 8 8 9 7 8 9 7 6 7 9 7 9 7 7 7 7 8 7 8 7 7 7 7 8 7 7 7 9 7 9 7 7 7 7 7 9 9 7 7 7 7 8 9 9 9 8 7 7 9 9 8 7 8 7 7 7 8 7 7 9 7 54 M. Keshavarz Ghorabaee et al. / Journal of Air Transport Management 63 (2017) 45e60 Table 4 The aggregated ranking scores in different numbers of iterations. N z SAG 1z SAG 2z SAG 3z SAG 4z SAG 5z N z SAG 1z SAG 2z SAG 3z SAG 4z SAG 5z 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 0.600 0.350 0.050 0.000 0.000 0.450 0.450 0.100 0.000 0.000 0.488 0.463 0.050 0.000 0.000 0.515 0.445 0.040 0.000 0.000 0.000 0.000 0.150 0.650 0.200 0.000 0.000 0.000 0.525 0.475 0.000 0.000 0.000 0.700 0.300 0.000 0.000 0.015 0.570 0.415 0.250 0.650 0.100 0.000 0.000 0.250 0.450 0.300 0.000 0.000 0.488 0.425 0.088 0.000 0.000 0.460 0.505 0.035 0.000 0.000 0.000 0.000 0.000 0.200 0.800 0.000 0.000 0.000 0.475 0.525 0.000 0.000 0.038 0.300 0.663 0.000 0.000 0.010 0.405 0.585 0.150 0.000 0.700 0.150 0.000 0.300 0.100 0.600 0.000 0.000 0.025 0.113 0.825 0.000 0.038 0.025 0.050 0.900 0.025 0.000 100 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 0.420 0.440 0.140 0.000 0.000 0.380 0.498 0.120 0.003 0.000 0.329 0.572 0.097 0.002 0.000 0.340 0.550 0.108 0.002 0.000 0.000 0.000 0.030 0.528 0.443 0.000 0.003 0.036 0.574 0.388 0.000 0.002 0.041 0.567 0.390 0.000 0.001 0.039 0.571 0.389 0.523 0.433 0.045 0.000 0.000 0.574 0.359 0.067 0.001 0.000 0.637 0.321 0.040 0.002 0.000 0.620 0.330 0.049 0.001 0.000 0.000 0.000 0.070 0.385 0.545 0.000 0.000 0.018 0.372 0.611 0.000 0.000 0.018 0.377 0.604 0.000 0.000 0.021 0.375 0.604 0.058 0.128 0.715 0.088 0.013 0.047 0.141 0.760 0.051 0.002 0.034 0.105 0.803 0.052 0.006 0.040 0.119 0.782 0.052 0.007 10 20 50 500 1000 50000 Fig. 3. Normalized and aggregated ranking scores in z ¼ 1 and different values of N. M. Keshavarz Ghorabaee et al. / Journal of Air Transport Management 63 (2017) 45e60 55 Fig. 4. Normalized and aggregated ranking scores in z ¼ 2 and different values of N. values of N, and the stability of these values increase when we run the proposed approach with higher number of iterations. This fact can also be seen in the normalized ranking scores of TOPSIS, COPRAS, WASPAS and EDAS. The instability of the ranking scores can lead to instability of final ranking of alternatives. However, in these figures we can see little variation in the ranking scores after N ¼ 1000. The linear assignment model which has been described in Step 11 of the proposed approach is used here to determine the final rank of the alternatives with different numbers of iterations. As an example, we formulate the assignment model for N ¼ 50000 as follows: subject to, 5 P i¼1 5 P z¼1 yiz ¼ 1 cz yiz ¼ 1 ci yiz 2f0; 1g The ranking results are presented in Table 5. The effect of instability of ranking scores in lower number of iterations on the final ranking of the alternatives can be seen in Table 5. However, it can be said that the stable optimal rank of the Max f ¼ 0:340y11 þ 0:550y12 þ 0:108y13 þ 0:002y14 þ 0:001y22 þ 0:039y23 þ 0:571y24 þ 0:389y25 þ 0:620y31 þ 0:330y32 þ 0:049y33 þ 0:001y34 þ 0:021y43 þ 0:375y44 þ 0:604y45 þ 0:040y51 þ 0:119y52 þ 0:782y53 þ 0:052y54 þ 0:007y54 56 M. Keshavarz Ghorabaee et al. / Journal of Air Transport Management 63 (2017) 45e60 alternatives is A3 _A1 _A5 _A2 _A4 . Therefore, we can say that A3 is an airline that has the higher level of service quality than the other alternative airlines according to our evaluation based on the selected criteria of service quality. It should be noted that the simulation part of the proposed approach has been coded and run in MATLAB R2014a, and the linear assignment has been solved using LINGO Extended 16 (x64 Educational). All computations have been performed on a PC with 2.4 GHz CPU (Intel® Core™ i5-520M), 4 GB of RAM and Windows 7 (64 bit) operating system. The computational time of running the proposed approach with different number of iterations is presented in Fig. 8. 5. Discussion The multi-criteria decision-making approaches have been used in many fields of science and engineering. However, there are still some issues to deal with MCDM problems with a group of decisionmakers and validation of the evaluation results. In the case of group decision-making, when we are confronted with a few members in the group of decision-making, handling the problem is not difficult. For example, after quantifying their opinions, we can combine the opinions of them by using a simple average. On the other hand, if the number of members increases, the evaluation process will be more complex. In such situations, the opinions of the decisionmakers, which are experts in many practical problems, may follow a distribution with some parameters which should be involved in the decision-making process. However, fitting different distributions and finding the best fit for the elicited data is a timeconsuming problem because if there are m criteria and n alternatives we should find m distributions for importance (weights) of criteria and m  n distributions for performance values (elements of the decision-matrix). To deal with this problem, in this study, the PERT distribution, which is a flexible distribution, has been utilized for modelling decision-makers’ opinions. This distribution uses the minimum, most likely and maximum parameters that can be very helpful to estimate opinions of decision-makers efficiently. It should be noted that the proposed approach can also be used in decision-making problems with small group of decision-makers. To solve the MCDM problems, the values of criteria weights and performance values of alternatives should be specified first. The Fig. 5. Normalized and aggregated ranking scores in z ¼ 3 and different values of N. M. Keshavarz Ghorabaee et al. / Journal of Air Transport Management 63 (2017) 45e60 57 Fig. 6. Normalized and aggregated ranking scores in z ¼ 4 and different values of N. Monte Carlo simulation has been used in the proposed approach to generate random values for criteria weights and performance values of alternatives with respect to defined PERT distribution. Then the generated problems have been solved by four efficient MCDM methods. One of the main issues in using MCDM approaches is the validation of ranking results. In most of the studies in this field, the ranking results are compared with the results of some existing MCDM approaches. As can be seen in the previous section, not only can the proposed approach yield a comparative analysis of the results of different MCDM methods, but also it gives a final ranking result based on an assignment model that involves the evaluation process of all the considered MCDM methods. Therefore, it can be said that the proposed approach helps us to reach a more accurate decision. According to Fig. 8, we can say that the computational time of running the proposed approach has a relatively linear relation with the number of iteration, and the computational time in N ¼ 1000 is about 20 s. Moreover, we have little variation in the ranking scores after N ¼ 1000. Therefore, setting the number of iterations to 1000 can be appropriate to obtain reliable results for this problem. Computational complexity of the proposed approach is depend on number of decision-makers, number of evaluation criteria, number of alternatives, number of MCDM methods used for evaluation and also level of complexity of each MCDM method. 6. Conclusion In this study, we have proposed a new hybrid simulation-based assignment approach to handle decision-making problems with multiple criteria and a group of decision-makers. We have applied the proposed approach to a problem of evaluation airlines with multiple service quality criteria. The PERT distribution has been utilized to define a stochastic MCDM problem according to the opinions of decision-makers. Then a Monte Carlo simulation with four MCDM approaches including TOPSIS, COPRAS, WASPAS and EDAS has been designed to determine normalized and aggregated ranking scores of alternatives. We have use a linear assignment model to find the final rank of alternatives. The simulation process has been done in different numbers of iterations. The results shows 58 M. Keshavarz Ghorabaee et al. / Journal of Air Transport Management 63 (2017) 45e60 Fig. 7. Normalized and aggregated ranking scores in z ¼ 5 and different values of N. Table 5 The ranking results in different numbers of iterations. Alternatives A1 A2 A3 A4 A5 N 5 10 20 50 100 500 1000 50000 1 4 2 5 3 1 4 2 5 3 2 4 1 5 3 1 4 2 5 3 2 4 1 5 3 2 4 1 5 3 2 4 1 5 3 2 4 1 5 3 that the normalized and aggregated ranking scores of alternatives will be more stable when we increase the number of iterations. This stability can be seen in the final ranking of the alternatives after solving the linear assignment model. Comparing the normalized ranking scores of different MCDM approaches with the aggregated values shows the stability and efficiency of the proposed approach. Although in the example of this study we have confronted with a situation that the normalized ranking scores of different MCDM methods and the aggregated ranking scores give the same ranking Fig. 8. The computational time of running the proposed approach. M. Keshavarz Ghorabaee et al. / Journal of Air Transport Management 63 (2017) 45e60 results in the stable range, it can be different in other problems. Therefore, using the aggregated ranking scores can provide more reliable results than using individual MCDM methods. Overall, the proposed approach has two main advantages over the other decision-making approach. The first advantage of the proposed approach is its ability to involve opinions of a large number of decision-makers, and using multiple methods to increase the accuracy of evaluation process is the second advantage of it. However, because the proposed approach uses a Monte Carlo simulation, the computational time of running the process can increase when we are faced with a problem with a large number of alternatives and/or criteria. As previously mentioned, although the TOPSIS, COPRAS, WASPAS and EDAS have been used in the algorithm of the hybrid proposed approach, the proposed approach is not limited to these methods. Accordingly, these methods can be replaced with any other efficient MCDM methods in future studies. Also, future research can incorporate another MCDM method like VIKOR, ARAS and MULTIMOORA into the algorithm to increase the comparability and accuracy of the results. Furthermore, the proposed approach can be applied to many other MCDM problems such as supplier selection, project selection, location selection and market segment evaluation. Appendix A. Supplementary data Supplementary data related to this article can be found at http:// dx.doi.org/10.1016/j.jairtraman.2017.05.008. References An, M., Noh, Y., 2009. Airline customer satisfaction and loyalty: impact of in-flight service quality. Serv. Bus. 3 (3), 293e307. Awasthi, A., Chauhan, S.S., Omrani, H., Panahi, A., 2011. 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