Intuitionistic fuzzy EDAS method: an application to solid waste disposal site selection

. Evaluation based on Distance from Average Solution (EDAS) is a new multicriteria decision making (MCDM) method, which is based on the distances of alternatives from the average scores of attributes. Classical EDAS has been already extended by using ordinary fuzzy sets in case of vague and incomplete data. In this paper, we propose an interval-valued intuitionistic fuzzy EDAS method, which is based on the data belonging to membership, nonmembership, and hesitance degrees. A sensitivity analysis is also given to show how robust decisions are obtained through the proposed intuitionistic fuzzy EDAS. The proposed intuitionistic fuzzy EDAS method is applied to the evaluation of solid waste disposal site selection alternatives. The comparative and sensitivity analyses are also included.


Introduction
EDAS method uses average solution for appraising the alternatives. It was developed by Keshavarz Ghorabaee et al. (2015). In this method, two measures which is called PDA (positive distance from average) and NDA (negative distance from average) are considered for the appraisal. This method is very useful when we have some conflicting criteria. In the compromise MCDM methods such as VIKOR and TOPSIS, the best alternative is obtained by calculating the distance from positive and negative ideal solutions. The best alternative has least distance from positive ideal solution (PIS) and largest distance from negative ideal solution (NIS) in these MCDM methods. In EDAS method, the best alternative is selected with respect to the distances from average solution: The positive distance from average (PDA) and the negative distance from average (NDA). An alternative having higher values of PDA and lower values of NDA is better (Keshavarz Ghorabaee et al. 2015). uncertain information environment. IFS can consider the membership and nonmembership degrees of the elements of a set at the same time, which their sum is not necessarily equal to one. Multicriteria decision making with interval-valued intuitionistic fuzzy sets has received a great deal of attention from researchers recently Oztaysi et al. 2015). Later, intuitionistic fuzzy sets have been considered as a special subset of neutrosophic sets by Smarandache (1998) and have been extended by some researchers (Liu, Shi 2015;Liu, Wang 2014).
It is evident from the previous studies in the literature, there has been no work extending EDAS to IVIF EDAS model to solve a multicriteria problem under fuzzy environment. The proposed method can be used to compare electronic waste or solid waste disposal technologies, disposal site alternatives, and hazardous material removal technologies for environmental protection.
To the best of our knowledge, this is the first paper extending EDAS method to IVIF EDAS. The performance of the proposed IVIF EDAS is illustrated through a solid waste disposal site selection problem. Solid waste is the unwanted or useless solid materials generated from combined residential, industrial and commercial activities in a given area. Management of solid waste reduces or eliminates adverse impacts on the environment and human health and supports economic development and improved quality of life. A number of processes are involved in effectively managing waste for a municipality. These include monitoring, collection, transport, processing, recycling and disposal.
The rest of this paper is organized as follows. Section 1 includes a literature review on EDAS method and gives the steps of classical EDAS. Section 2 presents the preliminaries of ordinary fuzzy sets and intuitionistic fuzzy sets together with the steps of ordinary fuzzy EDAS and intuitionistic fuzzy EDAS. Section 3 presents the applications of classical (crisp), ordinary fuzzy, and intuitionistic fuzzy EDAS for the solution of a solid waste disposal site selection problem. The last Section concludes the paper.

Edas method
Curiosity of researchers to inventing new MCDM methods is getting competitive. Trends of MCDM methods development since initial days showed enormous interest among researchers in this area to build robust structures in order to handle complex decisions. Their precise effort was concentrated on realizing requirements of decision system, then implementing variables and parameters, and interconnecting those variables in a logical manner to establish specific tool. Many MCDM methods have been created and then extended. Analytical hierarchy process (Deng et al. 2014), ANP (Van Horenbeek, Pintelon 2014), TOPSIS (Büyüközkan, Çifçi 2012;Seçme et al. 2009), VIKOR (Mardani et al. 2016), ARAS (Turskis et al. 2013), COPRAS (Turanoglu Bekar et al. 2016), SAW (Hashemkhani Zolfani et al. 2012, ELECTRE (Figueira et al. 2013), PROMETHEE (Corrente et al. 2013), DEMA-TEL (Yeh, Huang 2014), MOORA (Stanujkic et al. 2014;Yazdani 2015), WASPAS (Chakraborty, Zavadskas 2014), SWARA (Kouchaksaraei et al. 2015), and so on are examples of MCDM methods which are applied in different situations. Sort of methods have been invented, developed and expanded conjoining with fuzzy theories or in any other scale of integration as hybrid MCDM models. The method Evaluation Based on Distance from Average Solution (EDAS) has been developed by Keshavarz Ghorabaee et al. (2015). TOPSIS and VIKOR are among top and most popular MCDM methods (Behzadian et al. 2012;Yazdani, Payam 2015). The logical function of these methods considers the optimal solution based on maximum distance from negative solution simultaneously minimum distance from best or positive ideal solution. However, the best alternative in the EDAS method is corresponded to the distance from average solution (AV). In this work we first present the crisp and ordinary fuzzy EDAS methods and then extend to its intuitionistic fuzzy version.
In EDAS method, first two measures are delivered as the positive distance from average (PDA), and the negative distance from average (NDA). These measures can show the difference between each solution (alternative) and the average solution. Therefore higher values of PDA and lower values of NDA will indicate optimal solution. In fact, the higher values of PDA and/or lower values of NDA represent that the solution (alternative) is better than average solution. Classical algorithm of EDAS can be followed using the following steps: step 1. Choose the most relevant attributes which describe decision alternatives for specific decision problem.  (2) step 4. The positive distance from average (PDA) and the negative distance from average (NDA) matrixes need to be calculated in this step according to lower and upper values of matrix as shown: and max(0,( )) (4) In this way ij PDA and ij NDA represent the positive and negative distance of i th alternative from average solution in terms of j th criterion for the lower level of decision matrix, respectively. step 5. Obtain weighted summation of the positive distance and negative distances from average matric: and 1 Max ( ) where 01 i AS ≤≤ .
step 8. Rank the alternatives according to the decreasing values of appraisal score () i AS . The alternative with the highest i AS is the best choice among the candidate alternatives.

fuzzy extension of Edas
Fuzzy decision making models alongside development of MCDM methods are growing rapidly. The crucial importance of fuzzy decision models is evidence among research groups. Different MCDM methods with different instruments transformed to a fuzzy model. Fuzzy 2.1. ordinary fuzzy Edas (Keshavarz Ghorabaee et al.

2016)
Let us assume the fuzzy decision matrix as this: . So, to solve fuzzy EDAS model following steps must be pursued: step 1. Determine the fuzzy average decision matrix regarding to all of the criteria: step 2. The optimal solution should have maximum distance from negative feasible solutions while as the same time minimum distance by best and ideal solutions, so, identify the fuzzy positive distance from average and the fuzzy negative distance from average: and and and (17) step 3. Compute the fuzzy weighted summation of the positive distance and negative distances from average matric: and step 4. Determine the fuzzy normalized values of and step 5. Calculate the fuzzy appraisal score for all alternatives. Then this score must be deffuzified and finally highest appraisal score is the best choice among the candidate alternatives: (22) step 6. Rank the alternatives according to the decreasing values of defuzzified appraisal scores. In other words, the alternative with the highest appraisal score is the best choice among the candidate alternatives.

The proposed IVIf Edas
In the fuzzy set theory, the membership of an element to a fuzzy set is a single value between zero and one. However, the degree of non-membership of an element " ( ) " in a fuzzy set may not be equal to "1 " since there may be some hesitation degree. Therefore, a generalization of fuzzy sets was proposed by Atanassov (1986) as intuitionistic fuzzy sets (IFS) which incorporate the degree of hesitation, which is defined as "1 . Let X ≠∅ be a given set. An intuitionistic fuzzy set in X is an object A given by for every xX ∈ . Hesitancy is equal to " The definition of interval-valued intuitionistic fuzzy sets (IVIFS) is given as follows. Let 0,1 D ⊆   be the set of all closed subintervals of the interval and X be a universe of discourse. An interval-valued intuitionistic fuzzy set in A  over X is an object having the form: where 0,1 , vx v x , respectively. Interval-valued intiutionistic fuzzy set A  is then denoted by where For each element x, we can compute the unknown degree (hesitancy degree) of an interval-valued intuitionistic fuzzy interval of xX ∈ in A  defined as follows: . Figure 2 illustrates an interval-valued intuitionistic fuzzy set. Some arithmetic operations with interval-valued intuitionistic fuzzy sets and 0 λ≥ are given in the following: Let ,, , ,, 11 Eq. (28) Eq. (29) Multiplication by a fixed value, λ ≥ 0, is given by Eq. (32).
The necessary definitions used in the proposed methodology are stated as follows: In the following, the steps of the proposed intervalvalued intuitionistic fuzzy EDAS method are given: step 1. Choose the most relevant attributes which describe decision alternatives for specific decision problem.
step 4. The positive distance from average (PDA) and the negative distance from average (NDA) matrixes need to be calculated in this step according to lower and upper values of matrix as below: In this way ~i j PDA and ~i j NDA represent the positive and negative distance of i th alternative from average solution in terms of j th criterion for the lower level of decision matrix, respectively.
We propose the following ranking method for determining the maximum term in Eqs (39)     is the best choice among the candidate alternatives.

application
Various environmental protection problems have been solved by using multi criteria decision making techniques (Turskis et al. 2012;Zavadskas et al. 2009;Khan, Samadder 2015). Solid waste disposal site selection is also an important environmental problem that we have to use many tangible, intangible, and conflicting criteria in its solution.
In a multicriteria solid waste disposal site selection problem, there are three alternatives and three criteria which are water pollution (W), distance to residential areas (D), and slope (S). The evaluation scales for crisp case, ordinary fuzzy case, and intuitionistic fuzzy case are given in Table 1.

application of crisp Edas
The crisp decision matrix is given as in Table 2. All criteria are equally weighted.  (3) and (4), crisp PDA and NDA values are obtained as in Table 3.   Table 4, ranking of the alternatives are obtained as in Table 5.

application of ordinary fuzzy Edas
In this case, the decision matrix in Table 6 is filled in by using the linguistic scale given in Table 1.  The fuzzy positive and negative distances from average are found as follows: For SWDS-1, The ranking of alternatives is as follows: 1 > 2 = 3. The best alternative is the same as when the crisp EDAS was used. However, the other alternatives are equivalent when ordinary fuzzy EDAS is used. This is caused because of the vagueness in the linguistic evaluations.

application of Interval-Valued Intuitionistic fuzzy Edas
In this case, the decision matrix is filled in by using the linguistic scale given in Table 1 as in Table 7. The average values with respect to all criteria are determined as in Table 8: PDA and NDA values of Alternatives 1, 2, and 3 are calculated as in Table 9. According to the obtained results, the ranking of the alternatives is 1 > 2 > 3. The best alternative is again SWDS-1. However, the second best alternative is SWDS-2 with intuitionistic fuzzy EDAS where as it is 3 with crisp EDAS. Alternatives SWDS-2 and SWDS-3 are equivalent with ordinary fuzzy EDAS. These differences come from the structures of the data, which are exact in crisp case, trapezoidal in ordinary fuzzy case, and interval-valued in intuitionistic fuzzy case. A decision maker should select the proper approach with respect to the data he/she has.

sensitivity analysis
The weights of the alternatives have been significantly changed as it can be seen in Table 10. However, the ranking has not changed. SWDS-1 is the best alternative in all cases. Alternative 2 is the second best alternative in all cases.   Table 10 indicates that the ranking result obtained by IVIF EDAS is very robust. From slight changes to extreme changes in criteria weights, SWDS-1 is selected without any doubt. Smaller weights of the criterion "water pollution (W)" cause the scores of alternatives 2 and 3 get closer. . SWDS-2 becomes the best alternative in this case.
conclusions EDAS method needs fewer computations with respect to most of the other multiattribute decision-making methods while it can produce the same ranking of alternatives. The evaluation of alternatives in this method is based on distances of each alternative from the average solution with respect to each criterion. Ordinary fuzzy extension of crisp EDAS has been developed based on trapezoidal fuzzy numbers. We have proposed the interval-valued intuitionistic fuzzy EDAS (IVIF EDAS) method in this paper. The linguistic evaluations have been represented by interval-valued intuitionistic fuzzy sets and their arithmetic operations have been applied. Since the vagueness included by trapezoidal fuzzy sets and interval-valued intuitionistic fuzzy sets are basically different, the obtained rankings may change from ordinary fuzzy EDAS (OF EDAS) to intuitionistic fuzzy EDAS. The calculations of IVIF EDAS are more tedious with respect to ordinary fuzzy EDAS. Different ranking equations must be used in OF EDAS and IVIF EDAS.
For further research, the other extensions of fuzzy sets such as type-2 fuzzy sets, hesitant fuzzy sets, or neutrosophic sets may be used to develop the other versions of fuzzy EDAS. references cengiz KaHraMan is a full Professor at Industrial Engineering Department of Istanbul Technical University (ITU). His research areas are engineering economics, quality management, statistical decision making, multiple criteria decision making, and fuzzy decision making. He published about 200 journal papers, about 160 conference papers and 70 book chapters. He is in the editorial boards of 20 international journals and edited 10 international books from Springer. He guest-edited many special issues of international journals and organized international conferences. He was the vice dean of ITU Management Faculty between the years 2004-2007 and the head of ITU Industrial Engineering Department between the years 2010-2013.
Mehdi KEsHaVarz GHoraBaEE received the BS degree in electrical engineering from the University of Guilan, Rasht, Iran in 2010 and the MS degree in production management from the Allame Tabataba'i University, Tehran, Iran in 2013. He is currently working toward the PhD degree in management (Operations Research) at Allame Tabataba'i University. He has published some papers in leading international journals. His research interests include multi-criteria decision making (MCDM), multi-objective evolutionary algorithms, genetic algorithm, fuzzy MCDM, inventory control, supply chain management, scheduling and reliability engineering. Basar ozTaysI is a full time Associate Professor at Industrial Engineering Department of Istanbul Technical University (ITU). His research interests include multiple criteria decision making, data mining and intelligent systems. He has published 25 articles, 32 conference papers and 10 book chapters. He has edited a book on Supply Chain Management which is published by Springer. Başar Öztayşi teaches courses on data management, information systems management and business intelligence and decision support systems.