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MCDM Methods

Step-by-Step Algorithms

Fuzzy TOPSIS Method

Fuzzy Technique for Order Preference by Similarity to Ideal Solution

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Step-by-Step Algorithm

Calculated cell
Source cells in formula
1

Form the 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.

Alternative$C_1$ (Benefit)$C_2$ (Cost)$C_3$ (Benefit)
$A_1$(6, 7, 8)(4, 5, 6)(7, 8, 9)
$A_2$(3, 4, 5)(7, 8, 9)(5, 6, 7)
$A_3$(8, 9, 9)(2, 3, 4)(6, 7, 8)
2

Combined group decision matrix

Formula

The aggregated group rating is obtained by taking the minimum lower value, the arithmetic mean of the modal values, and the maximum upper value provided by all decision-makers.

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

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.

Alternative$C_1$$C_2$$C_3$
$A_1$(6, 7, 8)(4, 5, 6)(7, 8, 9)
$A_2$(3, 4, 5)(7, 8, 9)(5, 6, 7)
$A_3$(8, 9, 9)(2, 3, 4)(6, 7, 8)
3

Normalize the fuzzy decision matrix

Formula

Normalize the fuzzy decision matrix so that benefit and cost criteria become comparable.

For benefit criteria :

\[ \widetilde{r}_{ij}=\left( \frac{a_{ij}}{c_j^{*}}, \frac{b_{ij}}{c_j^{*}}, \frac{c_{ij}}{c_j^{*}} \right) \]
\[ c_j^{*}=\max_i c_{ij} \]

For cost criteria :

\[ \widetilde{r}_{ij}=\left( \frac{\min a_{ij}}{c_{ij}}, \frac{\min a_{ij}}{b_{ij}}, \frac{\min a_{ij}}{a_{ij}} \right) \]
\[ \min a_{ij}=\min_i a_{ij} \]
Alternative$C_1$$C_2$$C_3$
$A_1$(0.67, 0.78, 0.89)(0.33, 0.40, 0.50)(0.78, 0.89, 1.00)
$A_2$(0.33, 0.44, 0.56)(0.22, 0.25, 0.29)(0.56, 0.67, 0.78)
$A_3$(0.89, 1.00, 1.00)(0.50, 0.67, 1.00)(0.67, 0.78, 0.89)
4

Construct the weighted normalized fuzzy matrix

Formula

Multiply each normalized fuzzy rating by the fuzzy weight of its corresponding criterion.

\[ \widetilde{v}_{ij}=\widetilde{r}_{ij} \otimes \widetilde{w}_{j} \]

For positive triangular fuzzy numbers :

\[ (a_1,b_1,c_1) \otimes (a_2,b_2,c_2)=(a_1a_2,b_1b_2,c_1c_2) \]

The result is the weighted normalized fuzzy decision matrix \(\widetilde{V}\).

Alternative$C_1$ ($w_1=(0.5,0.6,0.7)$)$C_2$ ($w_2=(0.3,0.4,0.5)$)$C_3$ ($w_3=(0.5,0.6,0.7)$)
$A_1$(0.34, 0.47, 0.62)(0.10, 0.16, 0.25)(0.39, 0.53, 0.70)
$A_2$(0.17, 0.26, 0.39)(0.07, 0.10, 0.15)(0.28, 0.40, 0.55)
$A_3$(0.45, 0.60, 0.70)(0.15, 0.27, 0.50)(0.34, 0.47, 0.62)
5

Fuzzy positive ideal solution and fuzzy negative ideal solution

Formula

Determine the fuzzy positive ideal solution (FPIS) and fuzzy negative ideal solution (FNIS) from the weighted normalized fuzzy decision matrix.

\[ \widetilde{V}=[\widetilde{v}_{ij}]_{m\times n} \]

For each criterion \(C_j\), determine the maximum and minimum fuzzy values among all alternatives.

\[ \widetilde{v}_{j}^{+}=\max_{i} \left\{ \widetilde{v}_{ij} \right\} \]
\[ \widetilde{v}_{j}^{-}=\min_{i} \left\{ \widetilde{v}_{ij} \right\} \]

Therefore, the fuzzy Ppositive ideal solution (FPIS) and fuzzy negative ideal solution (FNIS) are :

\[ A^{+}=\left\{ \widetilde{v}_{1}^{+}, \widetilde{v}_{2}^{+}, \ldots, \widetilde{v}_{n}^{+} \right\} \]
\[ A^{-}=\left\{ \widetilde{v}_{1}^{-}, \widetilde{v}_{2}^{-}, \ldots, \widetilde{v}_{n}^{-} \right\} \]
CriterionFPIS ($A^+$)FNIS ($A^-$)
$C_1$(0.45, 0.60, 0.70)(0.17, 0.26, 0.39)
$C_2$(0.15, 0.27, 0.50)(0.07, 0.10, 0.15)
$C_3$(0.39, 0.53, 0.70)(0.28, 0.40, 0.55)
6

Distance from each alternative to the fuzzy positive ideal solution

Formula

Calculate the distance between each alternative and the fuzzy positive ideal solution (FPIS). The distance is calculated for every criterion and then summed over all criteria.

\[ d_i^{+}=\sum_{j=1}^{n} d\left( \widetilde{v}_{ij}, \widetilde{v}_{j}^{+} \right) \]

The distance between two triangular fuzzy numbers \(\widetilde{v}_{ij}=(a_{ij},b_{ij},c_{ij})\) and \(\widetilde{v}_{j}^{+}=(a_j^{+},b_j^{+},c_j^{+})\) is calculated using the vertex distance:

\[ d\left( \widetilde{v}_{ij}, \widetilde{v}_{j}^{+} \right)=\sqrt{ \frac{1}{3} \left[ (a_{ij}-a_j^{+})^2 + (b_{ij}-b_j^{+})^2 + (c_{ij}-c_j^{+})^2 \right]} \]

A smaller \(d_i^{+}\) means that alternative \(A_i\) is closer to the fuzzy positive ideal solution.

AlternativeDist to FPIS ($D^+$)
$A_1$0.27
$A_2$0.67
$A_3$0.06
7

Distance from each alternative to the fuzzy negative ideal solution

Formula

Calculate the distance between each alternative and the fuzzy negative ideal solution (FNIS). The distance is calculated for every criterion and then summed over all criteria.

\[ d_i^{-}=\sum_{j=1}^{n} d\left( \widetilde{v}_{ij}, \widetilde{v}_{j}^{-} \right) \]

The distance between the weighted normalized fuzzy value \(\widetilde{v}_{ij}=(a_{ij},b_{ij},c_{ij})\) and the fuzzy negative ideal value \(\widetilde{v}_{j}^{-}=(a_j^{-},b_j^{-},c_j^{-})\) is:

\[ d\left( \widetilde{v}_{ij}, \widetilde{v}_{j}^{-} \right)=\sqrt{ \frac{1}{3} \left[ (a_{ij}-a_j^{-})^2 + (b_{ij}-b_j^{-})^2 + (c_{ij}-c_j^{-})^2 \right]} \]

A larger \(d_i^{-}\) means that alternative \(A_i\) is farther from the fuzzy negative ideal solution and is therefore preferable.

AlternativeDist to FNIS ($D^-$)
$A_1$0.40
$A_2$0.00
$A_3$0.61
8

Closeness coefficient and ranking

Formula

Calculate the closeness coefficient \(CC_i\) for each alternative using its distance from the fuzzy positive ideal solution \(D_i^{+}\) and its distance from the fuzzy negative ideal solution \(D_i^{-}\).

\[ D_i^{+}=\sum_{j=1}^{n} d\left( \widetilde{v}_{ij}, \widetilde{v}_{j}^{+} \right) \]
\[ D_i^{-}=\sum_{j=1}^{n} d\left( \widetilde{v}_{ij}, \widetilde{v}_{j}^{-} \right) \]

The closeness coefficient of alternative \(A_i\) is then calculated as :

\[ CC_i=\frac{D_i^{-}} {D_i^{+}+D_i^{-}} \]

Finally, rank the alternatives according to their closeness coefficients in descending order.

\[ CC_{(1)} \geq CC_{(2)} \geq \cdots \geq CC_{(m)} \]
AlternativeCloseness Coeff ($CC_i$)
$A_1$0.60
$A_2$0.00
$A_3$0.91
Closeness coefficients: $C_3^*=0.91 >C_1^*=0.60 >C_2^*=0.00$
Final ranking: $A_3 \succ A_2 \succ A_1$

FTOPSIS Algorithm Flowchart