MCDMAKER-logo

MCDM Methods

Step-by-Step Algorithms

EXPROM Method

Extended PROMETHEE

1991 Diakoulaki and Koumoutsos 8 Steps

EXPROM (Extended PROMETHEE) is an outranking-based Multi-Criteria Decision-Making (MCDM) method introduced by Diakoulaki and Koumoutsos in 1991. It was developed to resolve information loss issues found in traditional outranking approaches like PROMETHEE, offering enhanced accuracy when ranking competing alternatives.

Video Demonstration

Step-by-Step Algorithm

Calculated cell
Source cells in formula
1

Form the decision matrix

Create the decision matrix with $m$ alternatives and $n$ criteria :

\[ X=\begin{bmatrix} x_{11} & x_{12} & \cdots & x_{1n}\\ x_{21} & x_{22} & \cdots & x_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ x_{m1} & x_{m2} & \cdots & x_{mn} \end{bmatrix} \]

\(x_{ij}\) presents the performance value of ith alternative on jth criterion, m and n are the numbers of alternatives and criteria respectively.

Alternative$C_1$$C_2$$C_3$
$A_1$102030
$A_2$152535
$A_3$251535
Criterion type: C₁ Benefit C₂ Cost C₃ Benefit
Weights: $w_1=0.7279$ · $w_2=0.2009$ · $w_3=0.0712$ (Entropy method)
2

Ideal and anti-ideal solutions

Formula

The ideal solution \(IS_j\) and anti-ideal solution \(AIS_j\) are determined for each criterion according to whether the criterion is a benefit or non-benefit criterion.

\[ IS_j=\begin{cases}\displaystyle\max_i x_{ij}, & \text{benefit criterion}\\[6pt]\displaystyle\min_i x_{ij}, & \text{non-benefit criterion}\end{cases} \]
\[ AIS_j=\begin{cases}\displaystyle\min_i x_{ij}, & \text{benefit criterion}\\[6pt]\displaystyle\max_i x_{ij}, & \text{non-benefit criterion}\end{cases} \]

where \(i=1,\ldots,m\) represents the alternatives and \(j=1,\ldots,n\) represents the criteria.

Solution$C_1$ (B) $C_2$ (C) $C_3$ (B)
Ideal ($IS_j$)251535
Anti-ideal ($AIS_j$)102530
3

Weak preference index

Formula

For every pair of alternatives \(A_i\) and \(A_k\), calculate the difference for each criterion.

$$ d_j(A_i,A_{k})=x_{ij}-x_{kj} \qquad i,k \in \{1,\ldots,m\} $$

The value of the weak preference function is calculated based on the function :

\[ P_j(A_i,A_{k})=f_j\left[d_j(A_i,A_{k})\right], \qquad 0\leq P_j(A_i,A_{k})\leq1 \]

The six commonly used preference functions are :

1. Usual criterion (Type I). No parameter
$$ P(d)=\begin{cases} 0, & d\leq0\\ 1, & d>0 \end{cases} $$
Quasi-criterion (Type II). Parameter : l
$$ P(d)=\begin{cases} 0, & d\leq l\\ 1, & d>l \end{cases} $$
3. V-shape criterion (Type III). Parameter : m
$$ P(d)=\begin{cases} 0, & d\leq0\\[4pt] \dfrac{d}{m}, & 0 < d \leq m\\[8pt] 1, & d>m \end{cases} $$
4. Level criterion (Type IV). Parameters : q , p
$$ P(d)=\begin{cases} 0, & d\leq q\\[4pt] \frac12, & q < d \leq p\\[4pt] 1, & d>p \end{cases} $$
5. Linear criterion (Type V). Parameters : s , r
$$ P(d)=\begin{cases} 0, & d\leq s\\[6pt] \dfrac{d-s}{r-s}, & s < d \leq r\\[10pt] 1, & d>r \end{cases} $$
6. Gaussian criterion (Type VI). Parameter : u
$$ P(d)=\begin{cases} 0, & d\leq0\\[6pt] 1-e^{-\frac{d^2}{2u^2}}, & d>0 \end{cases} $$
1
Usual criterion (Type I)
Type I — No threshold
P (d)1.00-1012
P (d )=0 if d ≤ 0 ; P (d )=1 if d > 0
2
Quasi-criterion (Type II)
Type II — Indifference threshold
P (d)1.00l
P (d )=0 if d ≤ l ; P (d )=1 if d >l
3
V-shape criterion (Type III)
Type III — Preference threshold
P (d)1.000m
P (d )=d /m if 0 <d ≤ m ; 1 if d > m
4
Level criterion (Type IV)
Type IV — Indifference & preference thresholds
P (d)1.00.500qp
0 if d ≤ q ; ½ if q < d ≤p ; 1 if d >p
5
Linear criterion (Type V)
Type V — Linear ramp after indifference
P (d)1.000sr
0 if d ≤s ; (d −s )/(r −s ) if s < d ≤ r ; 1 if d > r
6
Gaussian criterion (Type VI)
Type VI — Smooth S-curve
P(d)1.0001 − e ^ (−d²/2u²)
P (d )=1 − exp (−d ² / 2u ²) for d > 0
Calculate the Weak Preference Index :
$$ WP(A_i,A_k)=\sum_{j=1}^{n} w_jP_j(A_i,A_k) $$
where \(w_j\) is the weight of criterion \(C_j\).
$WP(A_i,A_k)$$A_1$$A_2$$A_3$
$A_1$—0.20090.0000
$A_2$0.7991—0.0000
$A_3$1.00000.9288—
4

Strict preference index

Formula

The Strict Preference Function :

\[ P'_j(A_i,A_{k})=\max\left\{0,\,\frac{d_j(A_i,A_{k})-L_j}{dm_j-L_j}\right\} \]

\(L_j\) is the limit of preference equal to zero, \(\qquad dm_j=IS_j-AIS_j, \qquad j=1,\ldots,n\)


The Strict Preference Index :

\[ SP(A_i,A_{k})=\frac{\displaystyle\sum_{j=1}^{n}P'_j(A_i,A_{k})\,w_j}{\displaystyle\sum_{j=1}^{n}w_j}, \qquad i,k\in\{1,\ldots,m\} \]

Using $L_j=0$ for all criteria. For cost criteria, the sign of $d_j$ is reversed: $P'_j=\max\{0,\,-d_j/dm_j\}$. Thus $dm_1=25-10=15$, $dm_2=15-25=-10$, $dm_3=35-30=5$.

Strict Preference Functions $P'_j(A_i,A_k)$

Pair$C_1$$C_2$$C_3$
$A_1$ vs $A_2$000
$A_1$ vs $A_3$00.50
$A_2$ vs $A_1$0.33330.51
$A_2$ vs $A_3$010
$A_3$ vs $A_1$101
$A_3$ vs $A_2$0.666700

Strict Preference Index $SP(A_i,A_k)$

$SP(A_i,A_k)$$A_1$$A_2$$A_3$
$A_1$—0.00000.1005
$A_2$0.4143—0.2009
$A_3$0.79910.4853—
5

Total preference index

Formula

Combine the weak and strict preference index to obtain the overall preference of \(A_i\) over \(A_k\).

$$ TP(A_i,A_k)=\min\left[1,\,WP(A_i,A_k)+SP(A_i,A_k)\right] $$

\(TP(A_i,A_k)\) expresses the overall strength with which \(A_i\) is preferred to \(A_k\).

$TP(A_i,A_k)$$A_1$$A_2$$A_3$
$A_1$—0.20090.1005
$A_2$1.0000—0.2009
$A_3$1.00001.0000—
6

Entering and leaving flows

Formula

The leaving flow measures how strongly alternative \(A_i\) prefers the other alternatives.

$$ \phi^{+}(A_i)=\frac{1}{m-1}\sum_{\substack{k=1\\k\neq i}}^{m}TP(A_i,A_k) $$

A larger leaving flow is better.


The entering flow measures how strongly the other alternatives prefer \(A_i\).

$$ \phi^{-}(A_i)=\frac{1}{m-1}\sum_{\substack{k=1\\k\neq i}}^{m}TP(A_k,A_i) $$

A smaller entering flow is better.

AlternativeLeaving Flow $\phi^{+}$Entering Flow $\phi^{-}$
$A_1$0.15071.0000
$A_2$0.60050.6005
$A_3$1.00000.1507
7

Ranking EXPROM I

Formula

The priority of alternatives to each other is determined based on the leaving and entering flows.


\(\,P\,\) Preference : \(A_i\,P\,A_k\) means \(A_i\) is better than \(A_k\).

\[ \begin{aligned} P \quad if \quad \begin{cases}\phi^{+}(A_i)>\phi^{+}(A_k)\quad and \quad \phi^{-}(A_i)<\phi^{-}(A_k)\\\phi^{+}(A_i)=\phi^{+}(A_k)\quad and \quad \phi^{-}(A_i)<\phi^{-}(A_k)\\\phi^{+}(A_i)>\phi^{+}(A_k)\quad and \quad \phi^{-}(A_i)=\phi^{-}(A_k)\end{cases}\end{aligned} \]

\(\,I\,\) Indifference : \(A_i\,I\,A_k\) means means neither alternative is preferred over the other.

\[ I \quad if \quad \begin{cases}\phi^{+}(A_i)=\phi^{+}(A_k)\quad and \quad \phi^{-}(A_i)=\phi^{-}(A_k)\end{cases} \]

\(\,R\,\) Incomparability : \(A_i\,R\,A_k\) means no clear preference can be established.

\[ \begin{aligned} R \quad if \quad \begin{cases}\phi^{+}(A_i)>\phi^{+}(A_k)\quad and \quad \phi^{-}(A_i)>\phi^{-}(A_k)\\\phi^{+}(A_i)<\phi^{+}(A_k)\quad and \quad \phi^{-}(A_i)<\phi^{-}(A_k)\end{cases}\end{aligned} \]
PairCondition CheckRelation
$A_2$ vs $A_1$$P$$A_2 \succ A_1$
$A_3$ vs $A_1$$P$$A_3 \succ A_1$
$A_3$ vs $A_2$$P$$A_3 \succ A_2$
Reverse pairs——
EXPROM I Final ranking: $A_3 \succ A_2 \succ A_1$
8

Ranking EXPROM II

Formula

Calculate the net flow

The net flow combines the leaving and entering flows.

$$ \phi(A_i)=\phi^{+}(A_i)-\phi^{-}(A_i) $$
A larger \(\phi(A_i)\) indicates a better alternative.

Ranking

Sort the alternatives according to their net flow values.

$$ \phi(A_i)>\phi(A_k)\quad\Longrightarrow\quad A_i >A_k $$
AlternativeNet Flow $\phi(A_i)$
$A_1$−0.8493
$A_2$0.0000
$A_3$0.8493
Net flows: $\phi(A_3)=0.8493 >\phi(A_2)=0.0000 >\phi(A_1)=-0.8493$
EXPROM II Final ranking: $A_3 \succ A_2 \succ A_1$

EXPROM Algorithm Flowchart