A TOPSIS method based on intuitionistic fuzzy values: a case study of North African airports

Article history: Received: October 1, 2016 Received in revised format: November 16, 2016 Accepted: April 14, 2017 Available online: April 16, 2017 In this paper we develop a fuzzy TOPSIS method based on intuitionistic fuzzy values to solve multiple criteria decision making problems in which the performance rating of alternatives values and the weights of criteria are given by linguistic terms. Arithmetic operations between intuitionistic fuzzy values are used for normalizing imprecise ratings and weights of criteria. In order to demonstrate the effectiveness of the suggested method, we propose a case study aiming to evaluate and compare the service quality of five major airports in North Africa. The suggested method helps manager of airports know the needs of passengers and the priority of enhancing service items. © 2017 Growing Science Ltd. All rights reserved.


Introduction
TOPSIS (technique for order preference by similarity to ideal solution), initially developed by Hwang and Yoon (1981), is an important classical multiple criteria decision making method (MCDM).It consists to calculate a closeness coefficient that can be used for ranking the set of alternatives with respect to the set of criteria.In the classical TOPSIS method, the performance rating and the weights of criteria are given by crisp values.However, with the complexity of the environment and under many conditions, crisp values cannot model adequately some real world situations because human judgment and preference are often ambiguous and cannot be estimated exactly (Kuo et al., 2007).Fuzzy set theory (Zadeh, 1965) emerged as an alternative way to treat information from human judgment and preference, it has been successfully applied to handle the imprecision in the TOPSIS method.The obtained technique, called fuzzy TOPSIS, has been investigated in many works (Chen, 2000;Chen et al., 2006;Chen & Hwang, 1992;Chen & Tzeng, 2004;Jahanshahloo et al., 2006;Liang, 1999;Wang & Elhag, 2006;Wang & Lee, 2007;Wang et al., 2007;Yeh et al., 2000;Yeh & Deng, 2004).A review on fuzzy TOP-SIS methods can be found in Kuo et al. (2007).Atanassov (1986) generalized the idea of fuzzy set to intuitionistic fuzzy set (IFS) which takes into account the hesitation of decision makers.IFS treats vague information by considering membership function and non-membership function and this can minimize the imprecision degree in complex systems.Intuitionistic fuzzy TOPSIS has been applied in some studies (Boran et al., 2009;Xu, 2007;Tan & Chen, 2010;Wei et al., 2013;Aloini et al., 2014;Zhang & Yu, 2012;Joshi & Kumar, 2014).
In this paper, we develop a fuzzy TOPSIS method based on intuitionistic fuzzy values (IFV-TOPSIS) to solve MCDM problems in which the performance rating values and the weights of criteria are given by linguistic terms.Arithmetic operations between intuitionistic fuzzy values are used for normalizing imprecise ratings and weights of criteria.In order to demonstrate the effectiveness of the suggested method, we propose a case study aiming to evaluate and compare the service quality of major five airports in North Africa.
The rest of the paper is organized as follows.Section 2 introduces the TOPSIS method.Section 3 illustrates intuitionistic fuzzy set (IFS).Section 4 describes developed TOPSIS method to solve MCDM problems when ratings and weights of criteria are considered as intuitionistic fuzzy values.Section 5 proposes a case study aiming to evaluate the service quality of five airports in North Africa.The paper is concluded in Section 6.

TOPSIS method
TOPSIS method consists to calculate a closeness coefficient for each alternative based on distances between the target alternative and the positive and the negative-ideal solutions.The best alternative has the shortest distance from the positive-ideal solution (PIS) and the farthest from the negative-ideal solution (NIS). .The TOPSIS method can be summarized on the following steps: i.First, normalize the decision matrix using the following transformation for each Then, multiply the columns of the normalized decision matrix by the associated weights.The weighted and normalized decision matrix is obtained as: where j w represents the weight of the jth criterion.
ii. Determine a positive-ideal and a negative -ideal solution: the positive-ideal and the negative-ideal alternatives are determined, respectively, as follows: , ,...., max ; , min ; , ,...., min ; , max ; where b  is the set of benefit criteria and c  is the set of cost criteria.
iii.Calculate the distance of each alternative from positive-ideal and negative-ideal alternatives: the Euclidean distances for each alternative are, respectively, given by: iv. Calculate a closeness coefficient for each alternative as , 1,...., ; 0 1 v. Rank the alternatives according to the closeness coefficients.

Intuitionistic fuzzy sets
In this section, we review the basic concepts relates to intuitionistic fuzzy sets.

Definition 3.1 Atanassov (1986). An intuitionistic fuzzy set
is given by: where, x are respectively the degree of membership and non-membership function of x to A .
If we use a membership function A t and a non-membership function A f to denote the lower bounds on A  , then, the degree of membership of x in the intuitionistic fuzzy set A is bounded to a subinterval

   
,1  is considered as intuitionistic fuzzy value and Eq. ( 1) can be replaced with

 
,1 be two IFV.The basic operations between as intuitionistic fuzzy values can be summarized as follow:

The proposed interval-valued fuzzy TOPSIS
Suppose a decision process composed by k decision-makers which are responsible for selecting n alternatives   1 2 , ,..., n a a a under m criteria   1 2 , ,..., .

m C C C
Criteria can be classified into benefit criteria (B) and cost criteria ( ).
C Further assume that the performance ratings and the weights of the criteria are evaluated in linguistic terms represented by intuitionistic fuzzy values.The suggested IFV-TOPSIS method is described as follow:

 
,1 , be the performance rating assigned to alternative i A by decision maker k D for criterion j C .The aggregated performance rating,

 
,1 of alternative i A under criterion j C can be evaluated as: The aggregated, i.e.

 
,1 , can be normalized as follow:

 
,1 be the importance weight given by decision maker k D to criterion j C .The aggregated importance weight,

 
,1 of criterion j C can be calculated as: The aggregated weights can be normalized as follows, where ' j w denotes the normalized j w

Construct the normalized decision matrix
The weighted normalized fuzzy decision matrix is given by ij The multiply operator can be applied as:

Determinate ideal and ideal negative solutions
Positive-Ideal and negative-ideal solutions can be defined as: According to definition (2.2), the distance between each alternative and A  (and A  ) can be obtained as: where i d  denotes the distance between each alternative and A  and i d  denotes the distance between each alternative and A  .

Calculate the closeness coefficient
The closeness coefficient of each alternative can be obtained as: , 1,..., According to the closeness coefficient, we can determine the ranking order of all alternatives and select the best one from them.

A case study of North Africa airports
The quality of service is a basic performance indicator for the operation of an airport.High service quality may have a significant impact in promoting future tourism and business activities.Thus, the evaluation of the quality of services has become an important issue for airport management.As we know, tourism is an economic factor of the North African countries.Thus, governments of these countries have recently become interested in evaluating the service quality of their airports in order to confront the intense competition that characterizes this sector.
In this paper, the proposed IFV-TOPSIS method is applied to evaluate the service quality of five major airports in north Africa; Houari Boumedienne Airport (A1), Cairo International Airport (A2), Carthage International Airport (A3) Mohammed V International Airport (A4) , and Tripoli International Airport (A5), .7 benefic criteria (B) and 3 cost criteria (C) have been used in the study; Safety record (B), security (B), Seating comfort (B) , Courtesy of employees (B), Neat appearance of employees (B), Availability of non-stop flight (B), Promptness and accuracy of baggage delivery (B), pollution (C), ticket price (C) and redtardness (C).
To conduct the study, 200 questionnaires are sent out to licensed tour guides in 15 general travel agencies in Tunisia.The reason of the choice of these respondents was that we wished respondents had the experience of traveling with all airlines to be evaluated.The licensed tour guides were the most natural choices due to their frequent travels.Each decision maker has presented his assessment based on linguistic variable for rating performance and importance of each criterion by a linguistic variable.For evaluated with respect to the m criteria.All the values/ratings are assigned to alternatives with respect to decision matrix denoted by  