ROV Method
Range of Value
ROV (Range of Value) is a simple MCDM method used to rank alternatives by calculating their best and worst utility values across benefit and cost criteria. The final score is obtained from the range between these two utility values.
Video Demonstration
Step-by-Step Algorithm
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 | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 10 | 20 | 30 |
| $A_2$ | 15 | 25 | 35 |
| $A_3$ | 25 | 15 | 35 |
Normalize the decision matrix
For benefit criteria , the normalized value is calculated as :
For cost criteria :
Where \(x_{ij}\) is the original performance value of alternative \(i\) with respect to criterion \(j\).
C₁ and C₃ are benefit criteria. C₂ is a cost criterion.
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.0000 | 0.5000 | 0.0000 |
| $A_2$ | 0.3333 | 0.0000 | 1.0000 |
| $A_3$ | 1.0000 | 1.0000 | 1.0000 |
Calculate the weighted normalized decision matrix
Multiply each normalized value by the corresponding criterion weight :
Where \(w_j\) is the weight of criterion \(C_j\).
The resulting matrix is the weighted normalized decision matrix :
Using weights: $w_1=0.7279$, $w_2=0.2009$, $w_3=0.0712$
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.0000 | 0.1005 | 0.0000 |
| $A_2$ | 0.2426 | 0.0000 | 0.0712 |
| $A_3$ | 0.7279 | 0.2009 | 0.0712 |
Utility functions and ranking
Calculate the utility of each alternative using its weighted normalized values for the benefit criteria.
\(J^{+}\) represents the set of benefit criteria.
Calculate the utility of each alternative using its weighted normalized values for the cost criteria.
\(J^{-}\) represents the set of cost criteria.
Calculate the final range of value score by combining the positive and negative utility components.
\(U_i\) is the overall ROV utility value of alternative \(A_i\).
Rank the alternatives according to their final ROV utility values in descending order.
Benefit criteria: \(J^{+}=\{C_1, C_3\}\) — sum their weighted values for \(u_i^{+}\). Cost criteria: \(J^{-}=\{C_2\}\) — sum their weighted values for \(u_i^{-}\).
| Alternative | $u_i^{+}$ (Benefit) | $u_i^{-}$ (Cost) | $U_i$ (Score) |
|---|---|---|---|
| $A_1$ | 0.0000 | 0.1005 | 0.0503 |
| $A_2$ | 0.3138 | 0.0000 | 0.1569 |
| $A_3$ | 0.7991 | 0.2009 | 0.5000 |
Final ranking: $A_3 \succ A_2 \succ A_1$