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MCDM Methods

Step-by-Step Algorithms

Spherical Fuzzy VIKOR Method

Spherical Fuzzy VIKOR Method

Video Demonstration

Step-by-Step Algorithm

1

Form the decision matrix

Decision maker \(k\) evaluates alternative \(A_i\) with respect to criterion \(C_j\) using a spherical fuzzy number:

\[ \widetilde{x}_{ij}^{(k)}=\left( \mu_{ij}^{(k)}, \nu_{ij}^{(k)}, \pi_{ij}^{(k)} \right) \]

\(\mu\), \(\nu\), and \(\pi\) represent membership, non-membership, and hesitancy degrees, respectively.

\[ \widetilde{X}_{S}^{(k)}=\begin{bmatrix} \widetilde{x}_{11}^{(k)} & \widetilde{x}_{12}^{(k)} & \cdots & \widetilde{x}_{1n}^{(k)} \\[6pt] \widetilde{x}_{21}^{(k)} & \widetilde{x}_{22}^{(k)} & \cdots & \widetilde{x}_{2n}^{(k)} \\[6pt] \vdots & \vdots & \ddots & \vdots \\[6pt] \widetilde{x}_{m1}^{(k)} & \widetilde{x}_{m2}^{(k)} & \cdots & \widetilde{x}_{mn}^{(k)} \end{bmatrix} \]

Weights of criteria :

\[ \widetilde{w}_{j}^{(k)}=\left( \mu_{j}^{(k)}, \nu_{j}^{(k)}, \pi_{j}^{(k)} \right) \]

Each decision-maker \(DM_k\) may have a different importance weight \(w_k\). The weights represent the relative importance of the decision-makers in the group decision-making process.

\[ w_k \geq 0 \]
\[ \sum_{k=1}^{K} w_k=1 \]
2

Aggregated spherical fuzzy decision matrix

Formula

Aggregate the judgments of each decision maker (DM) and the criteria weights using Spherical Weighted Arithmetic Mean (SWAM) or Spherical Weighted Geometric Mean (SWGM) operators.


\[ \widetilde{x}_{ij}=\operatorname{SWAM} \left( \widetilde{x}_{ij}^{(1)}, \widetilde{x}_{ij}^{(2)}, \ldots, \widetilde{x}_{ij}^{(K)} \right) \]

\[ \operatorname{SWAM} \left( \widetilde{X}_{S1}, \widetilde{X}_{S2}, \ldots, \widetilde{X}_{SK} \right)=\left\{ \begin{array}{c} \left[ 1- \displaystyle\prod_{i=1}^{n} \left(1-\mu_{\widetilde{X}_{Si}}^{\,2}\right)^{w_i} \right]^{\frac{1}{2}} \ ,\ \\[12pt] \displaystyle \prod_{i=1}^{n} \left(v_{\widetilde{X}_{Si}}\right)^{w_i} \ ,\ \\[12pt] \left[ \displaystyle\prod_{i=1}^{n} \left(1-\mu_{\widetilde{X}_{Si}}^{\,2}\right)^{w_i} - \displaystyle\prod_{i=1}^{n} \left( 1-\mu_{\widetilde{X}_{Si}}^{\,2} -\pi_{\widetilde{X}_{Si}}^{\,2} \right)^{w_i} \right]^{\frac{1}{2}} \end{array} \right\} \]

\[ \widetilde{x}_{ij}=\operatorname{SWGM} \left( \widetilde{x}_{ij}^{(1)}, \widetilde{x}_{ij}^{(2)}, \ldots, \widetilde{x}_{ij}^{(K)} \right) \]

\[ \operatorname{SWGM} \left( \widetilde{X}_{S1}, \widetilde{X}_{S2}, \ldots, \widetilde{X}_{SK} \right)=\left\{ \begin{array}{c} \displaystyle \prod_{i=1}^{n} \left(\mu_{\widetilde{X}_{Si}}\right)^{w_i} \ ,\ \\[12pt] \left[ 1- \displaystyle\prod_{i=1}^{n} \left(1-\nu_{\widetilde{X}_{Si}}^{\,2}\right)^{w_i} \right]^{\frac{1}{2}} \ ,\ \\[12pt] \left[ \displaystyle\prod_{i=1}^{n} \left(1-\nu_{\widetilde{X}_{Si}}^{\,2}\right)^{w_i} - \displaystyle\prod_{i=1}^{n} \left( 1-\nu_{\widetilde{X}_{Si}}^{\,2} -\pi_{\widetilde{X}_{Si}}^{\,2} \right)^{w_i} \right]^{\frac{1}{2}} \end{array} \right\} \]

The Aggregated matrix :

\[ \widetilde{X}_{S}=\begin{bmatrix} \widetilde{x}_{11} & \widetilde{x}_{12} & \cdots & \widetilde{x}_{1n} \\[6pt] \widetilde{x}_{21} & \widetilde{x}_{22} & \cdots & \widetilde{x}_{2n} \\[6pt] \vdots & \vdots & \ddots & \vdots \\[6pt] \widetilde{x}_{m1} & \widetilde{x}_{m2} & \cdots & \widetilde{x}_{mn} \end{bmatrix} \] \[ \widetilde{x}_{ij}=\left( \mu_{\widetilde{x}_{ij}}, \nu_{\widetilde{x}_{ij}}, \pi_{\widetilde{x}_{ij}} \right) \]
\[ \widetilde{W}=\begin{bmatrix} \widetilde{w}_{1} & \widetilde{w}_{2} & \cdots & \widetilde{w}_{n} \end{bmatrix} \] \[ \widetilde{w}_{j}=\left( \mu_{\widetilde{w}_{j}}, \nu_{\widetilde{w}_{j}}, \pi_{\widetilde{w}_{j}} \right) \]
3

Positive ideal solution PIS and negative ideal solution NIS

Formula

Two ideal solutions are proposed.

The first is the absolute ideal solution which is independent of the decision variables :

\[ X^{+}=(1,0,0) \]
\[ X^{-}=(0,0,1) \]

The second is the relative ideal solution that depends on the ratings of the alternatives for the assessment criteria :

\[ X_{j}^{+}=\left( \mu_{P}, \nu_{P}, \pi_{P} \right) \]
\[ \mu_{P}=\max_{i}\mu_{i}, \qquad \nu_{P}=\min_{i}\nu_{i}, \qquad \pi_{P}=\min_{i}\pi_{i} \]
\[ X_{j}^{-}=\begin{cases} \left( \mu_{N}, \nu_{N}, \pi_{N} \right) & \text{if} \hspace{10px} \mu_{N}^{2} + \nu_{N}^{2} + \pi_{N}^{2} \leq 1, \\[12pt] \left( \mu_{N}, \nu_{N}, \sqrt{ 1-\mu_{N}^{2}-\nu_{N}^{2}} \right), & \text{otherwise}. \end{cases} \]
\[ \mu_{N}=\min_{i}\mu_{i}, \qquad \nu_{N}=\max_{i}\nu_{i}, \qquad \pi_{N}=\max_{i}\pi_{i} \]
4

Distances between each alternative and PIS

Formula

Calculate the distance between each alternative \(\widetilde{X}_{ij}\) and the spherical fuzzy positive ideal solution \(\widetilde{X}_{j}^{+}\).

\[ d\left(\widetilde{X}_{ij},\widetilde{X}_{j}^{+}\right)=\sqrt{ \frac{1}{2} \left[ \left( \mu_{\widetilde{X}_{ij}} - \mu_{\widetilde{X}_{j}^{+}} \right)^2 + \left( \nu_{\widetilde{X}_{ij}} - \nu_{\widetilde{X}_{j}^{+}} \right)^2 + \left( \pi_{\widetilde{X}_{ij}} - \pi_{\widetilde{X}_{j}^{+}} \right)^2 \right]} \]
5

Distances between PIS and NIS

Formula

Calculate the distance between the spherical fuzzy positive ideal solution \(\widetilde{X}_{j}^{+}\) and the spherical fuzzy negative ideal solution \(\widetilde{X}_{j}^{-}\).

\[ d\left(\widetilde{X}_{j}^{+},\widetilde{X}_{j}^{-}\right)=\sqrt{ \frac{1}{2} \left[ \left( \mu_{\widetilde{X}^{+}} - \mu_{\widetilde{X}^{-}} \right)^2 + \left( \nu_{\widetilde{X}^{+}} - \nu_{\widetilde{X}^{-}} \right)^2 + \left( \pi_{\widetilde{X}^{+}} - \pi_{\widetilde{X}^{-}} \right)^2 \right]} \]
6

Weighted distances \(\widetilde{R}_{ij}\)

Formula

The weighted distance \(\widetilde{R}_{ij}\) is calculated.

\[ \widetilde{R}_{ij}=\widetilde{w}_j \left( \frac{ d\left(\widetilde{X}_{j}^{+},\widetilde{X}_{ij}\right)}{ d\left(\widetilde{X}_{j}^{+},\widetilde{X}_{j}^{-}\right)} \right)=\widetilde{w}_j d_{ij} \]
\[=\left\{ \begin{array}{c} \left[ 1- \left( 1-\mu_{\widetilde{w}_j}^{\,2} \right)^{\ d_{ij}} \right]^{\frac{1}{2}} \\[12pt] \nu_{\widetilde{w}_j}^{\ d_{ij}} \\[12pt] \left[ \left( 1-\mu_{\widetilde{w}_j}^{\,2} \right)^{\ d_{ij}} - \left( 1-\mu_{\widetilde{w}_j}^{\,2} - \pi_{\widetilde{w}_j}^{\,2} \right)^{\ d_{ij}} \right]^{\frac{1}{2}} \end{array} \right\} \]
7

Weighted distances \(\widetilde{S}_{i}\)

Formula

The total weighted distance is calculated for each alternative.

\[ \widetilde{S}_i=\sum_{j=1}^{n} \widetilde{R}_{ij} \]
\[ \widetilde{R}_{it} \oplus \widetilde{R}_{i(t+1)}=\left\{ \begin{array}{c} \left[ \mu_{\widetilde{R}_{it}}^{\,2} + \mu_{\widetilde{R}_{i(t+1)}}^{\,2} - \mu_{\widetilde{R}_{it}}^{\,2} \mu_{\widetilde{R}_{i(t+1)}}^{\,2} \right]^{\frac{1}{2}} \\[12pt] \nu_{\widetilde{R}_{it}} \nu_{\widetilde{R}_{i(t+1)}} \\[12pt] \left[ \left( 1-\mu_{\widetilde{R}_{i(t+1)}}^{\,2} \right) \pi_{\widetilde{R}_{it}}^{\,2} + \left( 1-\mu_{\widetilde{R}_{it}}^{\,2} \right) \pi_{\widetilde{R}_{i(t+1)}}^{\,2} - \pi_{\widetilde{R}_{it}}^{\,2} \pi_{\widetilde{R}_{i(t+1)}}^{\,2} \right]^{\frac{1}{2}} \end{array} \right\} \]
8

\(\widetilde{R}_{ij}\) score

Formula

Use the score function to form the score matrix.

\[ \operatorname{Score} \left( \widetilde{R}_{ij} \right)=\left( \mu_{\widetilde{R}_{ij}} - \pi_{\widetilde{R}_{ij}} \right)^2 - \left( \nu_{\widetilde{R}_{ij}} - \pi_{\widetilde{R}_{ij}} \right)^2 \]
9

\( R_i\)

Formula

The regret measure $R_i$ is calculated.

\[ R_i=\max_j \operatorname{Score} \left( \widetilde{R}_{ij} \right) \]
10

\( S_i\)

Formula

The utility measure $S_i$ is calculated.

\[ S_i=\operatorname{Score} \left( \widetilde{S}_i \right)=\left( \mu_{\widetilde{S}_{i}} - \pi_{\widetilde{S}_{i}} \right)^2 - \left( \nu_{\widetilde{S}_{i}} - \pi_{\widetilde{S}_{i}} \right)^2 \]
11

\( S^+\), \( S^-\), \( R^+\) and \( R^-\)

Formula

Determine the best and worst values among all alternatives for the \(S_i\) and \(R_i\) measures.

\[ S^{+}=\min_i S_i, \qquad S^{-}=\max_i S_i, \qquad R^{+}=\min_i R_i, \qquad R^{-}=\max_i R_i \]
12

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

Formula

Combine the group utility measure \(S_i\) and the individual regret measure \(R_i\) into the VIKOR index.

\[ Q_i=v \frac{S_i-S^+} {S^- - S^+} + (1-v) \frac{R_i-R^+} {R^- - R^+} \]

The parameter \(v\) represents the weight of the group utility , usually set to \(v=0.5\).

A smaller \(Q_i\) represents a better compromise solution.


Rank the alternatives according to \(S_i\), \(R_i\), and \(Q_i\), with the VIKOR index \(Q_i\) providing the main compromise ranking.

\[ Q_i < Q_k \quad\Longrightarrow\quad A_i >A_k \]

The alternative with the lowest \(Q_i\) is the preferred compromise alternative, subject to VIKOR's acceptable-advantage and stability conditions.

13

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.

SFVIKOR Algorithm Flowchart