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ŠUMARSKI LIST 11-12/2017 str. 48     <-- 48 -->        PDF

S-TOPSIS – S-TOPSIS
The TOPSIS (technique for order performance by similarity to idea solution) was first developed by Hwang & Yoon (1981). According to this technique, the best alternative would be the one that is nearest to the positive-ideal solution and farthest from the negative ideal solution (Ertugrul et al 2007). The positive-ideal solution is a solution that maximizes the benefit criteria and minimizes the cost criteria, whereas the negative ideal solution maximizes the cost criteria and minimizes the benefit criteria (Wang et al 2006). In short, the positive-ideal solution is composed of all best values attainable from the criteria, whereas the negative ideal solution consists of all worst values attainable from the criteria (Wang, 2007). There have been lots of studies in the literature using TOPSIS for the solution of MCDM problems. (Chen, 2000; Chu, 2002; Chu and Lin, 2002;Lai et al., 1994;Olson 2004; Wang et al., 2005; Yang et al 2007; Dağdeviren et al 2009 and Yıldırım et al 2016). The TOPSIS method consists of the following steps (Shyur et al 2006): Variables at the equation sequence of TOPSIS calculation and these variables are defined below:
D = decision matrix
A1, ……, An = value corresponding to jth alternative
F1, ……, Fn = value corresponding to ith criteria (factor)
R(=[rij]) = normalized decision matrix
Vij = weighted normalized matrix
Wi = weight of any criteria (factor)
A+ = positive ideal solution
A– = negative ideal solution
Dj+ = separation measures to positive-ideal solution
Dj– = separation measures to negative-ideal solution
CCj+ = relative closeness to the ideal solution
Step 1: Establish a decision matrix for the ranking. The structure of the matrix can be expressed as follows:
                D =         (1)
where Aj denotes the alternatives j, j = 1, 2,…, J; Fi represents the ith attribute or criterion, i = 1, 2,…, n, related to the ith alternative; and fij is a crisp value indicating the performance rating of each alternative Ai with respect to each criterion Fj.
Step 2: Calculate the normalized decision matrix R (= [rij]). The normalized value rij is calculated as
             ,               j = 1, 2,…, J; I = 1, 2,…, n     (2)
Step 3: Calculate the weighted normalized decision matrix by multiplying the normalized decision matrix by its associated weights. The weighted normalized value vij is calculated as
Vij= wi x rij,              j = 1, 2,, J; I = 1, 2,…, n         (3)
where wi represents the weight of the ith attribute or criterion.
Step 4: Determine the positive-ideal and negative-ideal solutions.
A+ =       (4)
A– =       (5)
where I´ is associated with the positive criteria, and I´´ is associated with the negative criteria.
Step 5: Calculate the separation measures, using the n-dimensional Euclidean distance. The separation of each alternative from the positive-ideal solution  is given as
                       ,      j = 1, 2, …, J       (6)
Similarly, the separation of each alternative from the negative-ideal solution  is as follows:
                       ,      j = 1, 2, …, J       (7)
Step 6: Calculate the relative closeness to the ideal solution and rank the performance order. The relative closeness of the alternative Aj can be expressed as
                                 ,      j = 1, 2,…, J              (8)
Since  and , then clearly , The larger the index value, the better the performance of the alternatives.
As can be seen above, S-TOPSIS is an efficient method in the model of Multicriteria Decision Support Systems. The factor and sub-factor weights were calculated using S-TOPSIS.