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MCDM Methods

Step-by-Step Algorithms

Fuzzy VIKOR Method

Fuzzy VIKOR Method

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Step-by-Step Algorithm

1

Form the fuzzy decision Matrix

Each decision-maker evaluates the performance of every alternative with respect to each criterion using linguistic terms. Each linguistic assessment is represented by a triangular fuzzy number (TFN).

\[ \widetilde{X}^{(k)}=[\widetilde{x}_{ij}^{(k)}]_{m\times n}, \qquad k=1,2,\ldots,K \]

The fuzzy rating provided by decision-maker \(D_k\) for alternative \(A_i\) with respect to criterion \(C_j\) is :

\[ \widetilde{x}_{ij}^{(k)}=(a_{ij}^{(k)},b_{ij}^{(k)},c_{ij}^{(k)}) \]

\(A_i\)=alternative, \(C_j\)=criterion, \(D_k\)=decision-maker, and \(K\)=number of decision-makers.

2

Combined group decision matrix

Formula
\[ \widetilde{x}_{ij}=\left( \min_{k} a_{ij}^{(k)}, \frac{1}{K}\sum_{k=1}^{K} b_{ij}^{(k)}, \max_{k} c_{ij}^{(k)} \right) \]

The resulting triangular fuzzy number is \(\widetilde{x}_{ij}=(a_{ij},b_{ij},c_{ij})\), where \(a_{ij}\) is the minimum lower value, \(b_{ij}\) is the average modal value, and \(c_{ij}\) is the maximum upper value.

3

Fuzzy best value and fuzzy worst value

Formula

Determine the fuzzy best value and fuzzy worst value for each criterion from the combined group decision matrix. Benefit criteria are maximized, while cost criteria are minimized.

For benefit criteria :

\[ \widetilde{f}_j^{*}=\max_i \left\{ \widetilde{x}_{ij} \right\}=\left( \max_i a_{ij}, \max_i b_{ij}, \max_i c_{ij} \right) \]
\[ \widetilde{f}_j^{-}=\min_i \left\{ \widetilde{x}_{ij} \right\}=\left( \min_i a_{ij}, \min_i b_{ij}, \min_i c_{ij} \right) \]

For cost criteria :

\[ \widetilde{f}_j^{*}=\min_i \left\{ \widetilde{x}_{ij} \right\}=\left( \min_i a_{ij}, \min_i b_{ij}, \min_i c_{ij} \right) \]
\[ \widetilde{f}_j^{-}=\max_i \left\{ \widetilde{x}_{ij} \right\}=\left( \max_i a_{ij}, \max_i b_{ij}, \max_i c_{ij} \right) \]

\(\widetilde{f}_j^{*}\) is the fuzzy best value and \(\widetilde{f}_j^{-}\) is the fuzzy worst value for criterion \(C_j\).

4

Calculate \(\widetilde{S}_i\) and \(\widetilde{R}_i\) values

Formula
Calculate \(\widetilde{S}_i\) and \(\widetilde{R}_i\) Values :
\[ \widetilde{S}_i=\sum_{j=1}^{n} \widetilde{w}_j \left( \frac{ \widetilde{f}_j^{*}-\widetilde{x}_{ij}}{ \widetilde{f}_j^{*}-\widetilde{f}_j^{-}} \right) \]
\[ \widetilde{R}_i=\max_j \left[ \widetilde{w}_j \left( \frac{ \widetilde{f}_j^{*}-\widetilde{x}_{ij}}{ \widetilde{f}_j^{*}-\widetilde{f}_j^{-}} \right) \right] \]

The three components are calculated separately :

\[ S_i^a=\sum_{j=1}^{n} \left( w_j^a \frac{ f_j^{*a}-a_{ij}}{ f_j^{*a}-f_j^{-a}} \right) \]
\[ S_i^b=\sum_{j=1}^{n} \left( w_j^b \frac{ f_j^{*b}-b_{ij}}{ f_j^{*b}-f_j^{-b}} \right) \]
\[ S_i^c=\sum_{j=1}^{n} \left( w_j^c \frac{ f_j^{*c}-c_{ij}}{ f_j^{*c}-f_j^{-c}} \right) \]
\[ \widetilde{S}_i=(S_i^a,S_i^b,S_i^c) \]
\[ R_i^a=\max_j \left[ w_j^a \frac{ f_j^{*a}-a_{ij}}{ f_j^{*a}-f_j^{-a}} \right] \]
\[ R_i^b=\max_j \left[ w_j^b \frac{ f_j^{*b}-b_{ij}}{ f_j^{*b}-f_j^{-b}} \right] \]
\[ R_i^c=\max_j \left[ w_j^c \frac{ f_j^{*c}-c_{ij}}{ f_j^{*c}-f_j^{-c}} \right] \]
\[ \widetilde{R}_i=(R_i^a,R_i^b,R_i^c). \]
\(\widetilde{w}_j=(w_j^a,w_j^b,w_j^c)\) is the fuzzy criterion weights.
5

Determine \(\widetilde{S}^+\), \(\widetilde{S}^-\), \(\widetilde{R}^+\) and \(\widetilde{R}^-\)

Formula

Determine the best and worst values among all alternatives for the \(\widetilde{S}_i\) and \(\widetilde{R}_i\) measures.

\[ \widetilde{S}^{+}=\min_i \widetilde{S}_i=\left( \min_i S_i^a, \min_i S_i^b, \min_i S_i^c \right) \]
\[ \widetilde{S}^{-}=\max_i \widetilde{S}_i=\left( \max_i S_i^a, \max_i S_i^b, \max_i S_i^c \right) \]
\[ \widetilde{R}^{+}=\min_i \widetilde{R}_i=\left( \min_i R_i^a, \min_i R_i^b, \min_i R_i^c \right) \]
\[ \widetilde{R}^{-}=\max_i \widetilde{R}_i=\left( \max_i R_i^a, \max_i R_i^b, \max_i R_i^c \right) \]

\(\widetilde{S}^+\) and \(\widetilde{R}^+\) represent the best fuzzy group utility and minimum fuzzy regret, respectively. \(\widetilde{S}^-\) and \(\widetilde{R}^-\) represent the corresponding worst fuzzy values.

6

Calculate the \(Q_i\) value

Formula

Calculate the VIKOR index \( \widetilde{Q}_i\), which combines the group utility measure \( \widetilde{S}_i\) and the individual regret measure \( \widetilde{R}_i\).

\[ \widetilde{Q}_i=v \frac{ \widetilde{S}_i- \widetilde{S}^{+}}{ \widetilde{S}^{-}- \widetilde{S}^{+}} + (1-v) \frac{ \widetilde{R}_i- \widetilde{R}^{+}}{ \widetilde{R}^{-}- \widetilde{R}^{+}} \]
\[ Q_i^a=v \left( \frac{ S_i^a-S^{+a}}{ S^{-a}-S^{+a}} \right) + (1-v) \left( \frac{ R_i^a-R^{+a}}{ R^{-a}-R^{+a}} \right) \]
\[ Q_i^b=v \left( \frac{ S_i^b-S^{+b}}{ S^{-b}-S^{+b}} \right) + (1-v) \left( \frac{ R_i^b-R^{+b}}{ R^{-b}-R^{+b}} \right) \]
\[ Q_i^c=v \left( \frac{ S_i^c-S^{+c}}{ S^{-c}-S^{+c}} \right) + (1-v) \left( \frac{ R_i^c-R^{+c}}{ R^{-c}-R^{+c}} \right) \]

Therefore, the fuzzy VIKOR index for alternative \(A_i\) is :

\[ \widetilde{Q}_i=(Q_i^a,Q_i^b,Q_i^c) \]

Equivalently :

\[ \begin{aligned} \widetilde{Q}_i=\Bigg( &v \frac{S_i^a-S^{+a}} {S^{-a}-S^{+a}} +(1-v) \frac{R_i^a-R^{+a}} {R^{-a}-R^{+a}}, \\[6pt] &v \frac{S_i^b-S^{+b}} {S^{-b}-S^{+b}} +(1-v) \frac{R_i^b-R^{+b}} {R^{-b}-R^{+b}}, \\[6pt] &v \frac{S_i^c-S^{+c}} {S^{-c}-S^{+c}} +(1-v) \frac{R_i^c-R^{+c}} {R^{-c}-R^{+c}} \Bigg) \end{aligned} \]

The commonly used value of the VIKOR strategy weight is :

\[ v=0.5 \]
7

Determine \(S_i\), \(R_i\), \(Q_i\) values and ranking

Formula

After defuzzification, obtain one crisp \(S_i\), \(R_i\), and \(Q_i\) value for every alternative.

\[ S_i=\frac{ S_i^a+2S_i^b+S_i^c}{4} \]
\[ R_i=\frac{ R_i^a+2R_i^b+R_i^c}{4} \]
\[ Q_i=\frac{ Q_i^a+2Q_i^b+Q_i^c}{4} \]

Create three rankings by sorting the alternatives in ascending order:

\[ S_{(1)} \leq S_{(2)} \leq \cdots \leq S_{(m)} \]
\[ R_{(1)} \leq R_{(2)} \leq \cdots \leq R_{(m)} \]
\[ Q_{(1)} \leq Q_{(2)} \leq \cdots \leq Q_{(m)} \]

The alternative with the smallest \(Q_i\) is the best-ranked alternative according to the VIKOR index. Smaller \(S_i\) and \(R_i\) are also preferable.

8

Acceptable advantage and acceptable stability

Formula

VIKOR does not select the best compromise solution solely from the numerical value of \(Q_i\). The leading alternative must satisfy the acceptable-advantage and acceptable-stability conditions.


Acceptable advantage :

The best alternative must have a sufficiently large advantage over the second-best alternative.

\[ Q(A^{(2)})-Q(A^{(1)}) >DQ \]

The threshold \(DQ\) is calculated as :

\[ DQ=\frac{1}{m-1} \]

\(m\)=number of alternatives.

If the acceptable-advantage condition is not satisfied, a set of alternatives \(A^{(1)},...,A^{(k)}\) are proposed as the best choices while \(Q(A^{(k)})-Q(A^{(1)})\leq DQ\)


Acceptable Stability :

The alternative ranked first according to \(Q_i\) must also have a strong position according to at least one of the two measures : group utility \(S_i\) or individual regret \(R_i\).

Acceptable Stability is satisfied when : \(Q\)-rank 1 and \(S\)-rank 1 and/or \(Q\)-rank 1 and \(R\)-rank 1. \(A^{(1)}\) is the best choice.

If \(A^{(1)}\) is not ranked first by either \(S_i\) or \(R_i\), the acceptable stability condition is not satisfied. \(A^{(1)}\) and \(A^{(2)}\) are proposed as the best choices.

FVIKOR Algorithm Flowchart