Fuzzy similarity based TOPSIS

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. This was done by computing the distances to both ideal solutions and then forming so called closeness coefficient from these distance. Originally it was developed in 1980's. After that several researches have made improved modifications to it e.g. Similarity-based-TOPSIS. Fuzzy extension to Technique for Order Performance by Similarity to Ideal Solution (TOPSIS) was proposed by Chen in 2000. Later fuzzy similarity based approach to Fuzzy TOPSIS and extensions to forming ideal solutions and was proposed. New ways of forming closeness coefficients has also been examined.

Preliminaries

• A set of K decision-makers is called [math]\displaystyle{ E=\left\{D_1,D_2, \cdots,D_K \right\} }[/math];
• A set of m possible alternatives is called [math]\displaystyle{ A=\left\{A_1, A_2, \cdots,A_m \right \} }[/math];
• A set on n criteria, [math]\displaystyle{ C=\left\{C_1, C_2, \cdots,C_n \right \} }[/math], with which alternatives performance are measured;
• A set of performance ratings of [math]\displaystyle{ A_i (i=1,2,\cdots,m) }[/math] w.r.t. criteria [math]\displaystyle{ C_j(j=1,2, \cdots ,n) }[/math]

called [math]\displaystyle{ X=\left \{ x_{ij}, i=1,2,\cdots,m, j=1,2,\cdots,n \right\} }[/math]

Method

Assume that a decision group has [math]\displaystyle{ k }[/math] decision-makers, and the fuzzy rating of each decision-maker [math]\displaystyle{ D_k(k=1,2,\cdots,K) }[/math] are represented using positive trapezoidal fuzzy number [math]\displaystyle{ \hat{X}_k (k=1,2,\cdots,K) }[/math] with membership function [math]\displaystyle{ \mu_{\hat{X}_k}(x) }[/math]. Let the fuzzy ratings of all decision-makers be trapezoidal fuzzy numbers [math]\displaystyle{ \hat{R}_k=(a_k,b_k,c_k,d_k) }[/math], [math]\displaystyle{ k=1,2,\cdots,K }[/math]. Then the aggregated fuzzy rating can be defined as [math]\displaystyle{ \hat{R}=(a,b,c,d) }[/math] where

[math]\displaystyle{ a=min_{k} \{a_k\} \quad b=\frac{1}{K}\sum_{k=1}^{K} b_k \quad c=\frac{1}{K}\sum_{k=1}^{K} c_k \quad d=max_{k}\{d_k\} }[/math]

Let the fuzzy rating and importance weight of the [math]\displaystyle{ k }[/math]th decision-maker be [math]\displaystyle{ \hat{x}_{ijk}=(a_{ijk},b_{ijk},c_{ijk},d_{ijk}) }[/math] and [math]\displaystyle{ \hat{w}=(w_{jk1},w_{jk2},w_{jk3},w_{jk4}) }[/math]; [math]\displaystyle{ i=1,2,\cdots,m }[/math], [math]\displaystyle{ j=1,2, \cdots ,n }[/math], respectively. Hence, the aggregated fuzzy ratings [math]\displaystyle{ (\hat{x}_{ij}) }[/math] of alternatives w.r.t. each criterion can be calculated as [math]\displaystyle{ \hat{x}_{ij}=(a_{ij},b_{ij},c_{ij},d_{ij}) }[/math]

where [math]\displaystyle{ a_{ij}=min_{k} \{a_{ijk}\} \quad b_{ij}=\frac{1}{K}\sum_{k=1}^{K} b_{ijk} \quad c_{ij}=\frac{1}{K}\sum_{k=1}^{K} c_{ijk} \quad d_{ij}=max_{k}\{d_{ijk}\} }[/math]

in similar manner the aggregated fuzzy weight [math]\displaystyle{ (\hat{w}_j) }[/math] of each criterion can be calculated as

[math]\displaystyle{ \hat{w}_j=(w_{j1},w_{j2},w_{j3},w_{j4}) }[/math]

where [math]\displaystyle{ w_{j1}=min_{k} \{w_{jk1}\} \quad w_{j2}=\frac{1}{K}\sum_{k=1}^{K} w_{jk2} \quad w_{j3}=\frac{1}{K}\sum_{k=1}^{K} w_{jk3} \quad w_{j4}=max_{k}\{w_{jk4}\} }[/math]

Now multi-criteria decision making problem can be expressed in matrix form as

[math]\displaystyle{ \hat{X}= \begin{pmatrix} \hat{x}_{11} & \hat{x}_{12} & \ldots & \hat{x}_{1n} \\ \hat{x}_{21} & \hat{x}_{22} & \ldots & \hat{x}_{2n} \\ \vdots & \vdots & {} & \vdots \\ \hat{x}_{m1} & \hat{x}_{m2} & \ldots & \hat{x}_{mn} \end{pmatrix} }[/math]

Next linear scale transformation is used to transform the various criteria scales into comparable scales. The set of criteria can be divided into benefit criteria (the larger the rating, the greater the preference) and cost criteria (the smaller the rating, the greater the preference). Therefore, the normalized fuzzy decision matrix can be represented as [math]\displaystyle{ \hat{R}=[\hat{r}_{ij}]_{m \times n} }[/math] where B and C are the sets of benefit criteria and cost criteria, respectively, and

[math]\displaystyle{ \hat{r}_{ij}=\left(\frac{a_{ij}}{d_j^\oplus},\frac{b_{ij}}{d_j^\oplus},\frac{c_{ij}}{d_j^\oplus},\frac{d_{ij}}{d_j^\oplus} \right),j \in B }[/math] for benefit criteria and [math]\displaystyle{ \hat{r}_{ij}=\left(\frac{a_j^\ominus}{d_{ij}},\frac{a_j^\ominus}{c_{ij}},\frac{a_j^\ominus}{b_{ij}},\frac{a_j^\ominus}{a_{ij}} \right),j \in C }[/math] for cost criteria. where [math]\displaystyle{ d_j^\oplus=max_{i} (d_{ij}) }[/math], [math]\displaystyle{ j\in B }[/math] and [math]\displaystyle{ a_j^\ominus=min_{i}( a_{ij}) }[/math], [math]\displaystyle{ j\in C }[/math].

Considering the importance of each criterion, the weighted normalized fuzzy decision matrix can now be expressed as [math]\displaystyle{ \hat{V}=(\hat{v}_{ij})_{m \times n} }[/math]

where [math]\displaystyle{ \hat{v}_{ij}=\hat{r}_{ij}(\cdot)\hat{w}_j }[/math].

Next the fuzzy positive-ideal solution (FPIS,[math]\displaystyle{ v^\oplus }[/math]) and fuzzy negative-ideal solution (FNIS,[math]\displaystyle{ v^\ominus }[/math]) needs to be defined. There exists several ways how these can be defined.

By considering a finite set of given criteria [math]\displaystyle{ C=\{C_{j}|j=1, 2, ..., n\} }[/math], the ways to select the FPIS([math]\displaystyle{ v^{+} }[/math]) and the FNIS([math]\displaystyle{ v^{-} }[/math]) come from the weighted normalized decision matrix [math]\displaystyle{ V=(v_{ij})_{m \times n} }[/math], where the obtained weighted normalized values [math]\displaystyle{ v_{ij} }[/math] are fuzzy numbers expressed as:

[math]\displaystyle{ v_{ij}=(v_{ij1}, v_{ij2}, v_{ij3}, v_{ij4}) }[/math]

The fuzzy positive-ideal solution [math]\displaystyle{ v^{+} }[/math] and the fuzzy negative-ideal solution [math]\displaystyle{ v^{-} }[/math], respectively are:

[math]\displaystyle{ v^{+}=[v_{1}^{+}, v_{2}^{+}, ..., v_{n}^{+}] }[/math] [math]\displaystyle{ v^{-}=[v_{1}^{-}, v_{2}^{-}, ..., v_{n}^{-}] }[/math]

First proposal for choosing the FPIS ([math]\displaystyle{ v^{+} }[/math]) and the FNIS ([math]\displaystyle{ v^{-} }[/math]) have been given as for [math]\displaystyle{ v^{+} }[/math], the maximum, and [math]\displaystyle{ v^{-} }[/math], the minimum of the weighted normalized values:

[math]\displaystyle{ v_{j}^{+}=\max_{i}v_{ij4} }[/math]

[math]\displaystyle{ v_{j}^{-}=\min_{i}v_{ij1} }[/math]

In addition, two other ways of selecting the FPIS ([math]\displaystyle{ v^{+} }[/math]) and the FNIS ([math]\displaystyle{ v^{-} }[/math]) have been give as follow:

• [math]\displaystyle{ v^{+} }[/math] as a vector of ones, and [math]\displaystyle{ v^{-} }[/math] as a vector of zeros:

[math]\displaystyle{ v_{j}^{+}=(1, 1,\cdots 1) }[/math]

[math]\displaystyle{ v_{j}^{-}=(0, 0, \cdots, 0) }[/math]

• Every element of [math]\displaystyle{ v^{+} }[/math] is the maximum for all [math]\displaystyle{ i }[/math] weighted normalized value , and every element of [math]\displaystyle{ v^{-} }[/math] is the minimum for all [math]\displaystyle{ i }[/math] weighted normalized value:

[math]\displaystyle{ v_{j}^{+}=(\max_{i}v_{ij1}, \max_{i}v_{ij2}, \max_{i}v_{ij3}, \max_{i}v_{ij4}) }[/math]

[math]\displaystyle{ v_{j}^{-}=(\min_{i}v_{ij1}, \min_{i}v_{ij2}, \min_{i}v_{ij3}, \min_{i}v_{ij4}) }[/math]

After ideal vectors are computed, next step is to calculate similarities of the [math]\displaystyle{ i^{th} }[/math] alternative and both ideal solutions.

[math]\displaystyle{ s_{i}^{+}=s(v_{ij}, v_{j}^{+})=(1-\frac{\sum_{t=1}^{4}|v_{ijt}-v_{jt}^{+}|}{4}) \times \frac{min(P(v_{ij}), P(v_{j}^{+}))+min(\omega_{v_{ij}}, \omega_{v_{j}^{+}})}{max(P(v_{ij}), P(v_{j}^{+}))+max(\omega_{v_{ij}}, \omega_{v_{j}^{+}})} }[/math]

[math]\displaystyle{ s_{i}^{-}=s_{3}(v_{ij}, v_{j}^{-})=(1-\frac{\sum_{t=1}^{4}|v_{ijt}-v_{jt}^{-}|}{4}) \times \frac{min(P(v_{ij}), P(v_{j}^{-}))+min(\omega_{v_{ij}}, \omega_{v_{j}^{-}})}{max(P(v_{ij}), P(v_{j}^{-}))+max(\omega_{v_{ij}}, \omega_{v_{j}^{-}})} }[/math]

and average similarities i.e. by [math]\displaystyle{ s_{i}^{-}=\frac{\sum_{j=1}^n s_3(v_{ij},v_j^-)}{n} }[/math]. Notice that here only one example on how to calculate similarities is given see reference 4 for more about this.

The closeness coefficient can be applied to similarity measures in straightforward way. The closeness coefficients of the [math]\displaystyle{ i^{th} }[/math] alternative with respect to the positive ideal solution by using the similarity measures matrix ([math]\displaystyle{ CCS_{i}^{+} }[/math]) are defined as:

[math]\displaystyle{ CCS_{i}^{+}=\frac{s_{i}^{+}}{s_{i}^{+} + s_{i}^{-}}, i=1, 2, ..., m }[/math]

Fuzzy similarity based TOPSIS has been implemented in matlab