Similarity-based-TOPSIS

Similarity based TOPSIS is a multi-criteria decision-making method. The name TOPSIS is shortening from the Technique for Order Performance by Similarity to Ideal Solution. It is based on the idea of forming two ideal solutions (best possible case and worst possible case) and comparing the current alternative to these two. Unlike the name suggests originally this was done by computing the distances to both ideal solutions and then forming so called closeness coefficient from these distance. Version where similarity is used to compare both ideal solutions and the forming closeness coefficient was published in 2016. Later generalized version from it was developed where aggregation of similarities was done using Bonferroni mean. There exists also fuzzy similarity based TOPSIS which is developed for linguistic evaluations.

To apply similarity based TOPSIS we require a specification of the decision matrix for a set of alternatives over a set of criteria. Given a set of alternatives [math]\displaystyle{ A=\{a_i | i=1,2,\cdots,m\} }[/math], a set of criteria [math]\displaystyle{ C=\{c_j | j=1,2, \cdots,n\} }[/math] and a set of weights [math]\displaystyle{ W=\{w_j | j=1,2,\cdots,n\} }[/math], [math]\displaystyle{ w_j\gt 0 }[/math], [math]\displaystyle{ \sum_{j=1}^n w_j=1 }[/math], where [math]\displaystyle{ w_j }[/math] denotes the weight of the criteria [math]\displaystyle{ c_j }[/math], let [math]\displaystyle{ X=\{x_{ij | i=1,2,\cdots,m, j=1,2,\cdots,n}\} }[/math] denote the decision matrix where [math]\displaystyle{ x_{ij} }[/math] is the performance measure of the alternative [math]\displaystyle{ a_i }[/math] with respect to the criteria [math]\displaystyle{ c_j }[/math].

Method is based on following steps.

• Compute normalized decision matrix [math]\displaystyle{ r_{ij} }[/math] where normalization is done to unit interval. [math]\displaystyle{ X^n \rightarrow [0,1]^n }[/math]

[math]\displaystyle{ 
r_{ij}=\frac{x_{ij}+min_i(x_{ij})}{max_i(x_{ij})-min_i(x_{ij})}, 
 }[/math] 

[math]\displaystyle{ i=1,2,\cdots,m, j=1,2,\cdots,n }[/math]

• Compute the weighted normalized decision matrix [math]\displaystyle{ V=[v_{ij}] }[/math]:

[math]\displaystyle{ v_{ij}=r_{ij}(\cdot)w_j }[/math], [math]\displaystyle{ j=1,\cdots,m }[/math], [math]\displaystyle{ i=1,\cdots,n }[/math] where m is for criteria and n for alternatives.

• Determine positive and negative ideal solutions

[math]\displaystyle{ V^+=\{v_1^+, \cdots, v_m^+\}=\{(\max_i v_{ij}| j \in B),(\min_i v_{ij}| j \in C)\} }[/math]

[math]\displaystyle{ V^-=\{v_1^-, \cdots, v_m^-\}=\{(\min_i v_{ij}| j \in B),(\max_i v_{ij}| j \in C)\} }[/math]

where B is for benefit criteria, and C is for cost criteria.

• Compute similarity of each alternative w.r.t positive ideal solution and negative ideal solution:

[math]\displaystyle{ s_i^+=\frac{1}{n} \sum_{j=1}^m \sqrt[p]{1-|(v_{ij})^p-(v_j^+)^{p}|}, i=1, \cdots, n }[/math]

[math]\displaystyle{ s_i^-=\frac{1}{n} \sum_{j=1}^m \sqrt[p]{1-|(v_{ij})^p-(v_j^-)^{p}|}, i=1, \cdots, n }[/math]

• Compute relative closeness to the ideal solutions:

[math]\displaystyle{ CC_i=\frac{s_i^+}{s_i^++s_i^-}, i=1, \cdots, n }[/math]

Method differs from original TOPSIS in three ways. First normalization is done to unit interval unlike in the original version. Second in computing how similar alternatives and ideal vectors are similarity measure is used instead of distance. Third change is that relative closeness computation in step 5 is adjusted to similarity measure instead distance measure.

Method has been implemented in MATLAB.