Compromise methods: TOPSIS and VIKOR

From my doctoral course

This note comes from the doctoral course “Multi-Criteria Decision Making”. I rewrote the explanations in my own words and recomputed the numerical examples. Main source of the course: Asgharpour, Multi-Criteria Decision Making (University of Tehran).

The shared idea: close to the ideal

Both methods ask: which alternative is closer to an “ideal solution”? The ideal is an imaginary alternative holding the best value on every attribute. They differ in how distance is measured.

TOPSIS (Hwang and Yoon, 1981)

The chosen alternative should have the least distance from the positive ideal and the greatest distance from the negative ideal. Each alternative is a point in $n$-dimensional space.

Steps:

1. Decision matrix; benefit versus cost attributes are specified.
2. Norm normalisation: $r_{ij}=x_{ij}/\sqrt{{\sum }_i{x}_{ij}^2}$.
3. Multiply by weights: $v_{ij}=w_j{r}_{ij}$.
4. Positive ideal $A^{+}$ (the best value of each column) and negative $A^{-}$ (the worst).
5. Euclidean distance of each alternative from the two: $D_i^{+}$ and $D_i^{-}$.
6. Relative closeness: $C_i=D_i^{-}/(D_i^{+}+D_i^{-})$; larger is better. Alternatives are ordered by descending $C_i$.

VIKOR (Opricovic, 1998)

It is built on agreement for conflicting criteria and gives a “compromise” answer: the closest possible solution to the ideal. Three indicators:

• $S_i$ = weighted sum of relative distance from the best (group utility)
• $R_i$ = largest weighted relative distance (individual regret: the worst attribute)
• $Q_i=v\frac{S_i-S^{\ast }}{S^{-}-S^{\ast }}+(1-v)\frac{R_i-R^{\ast }}{R^{-}-R^{\ast }}$

where $v$ is the weight of the “majority of criteria” strategy (usually 0.5). The alternative with the smaller $Q$ is better, provided two conditions: (1) acceptable advantage: $Q(A_2)-Q(A_1)\ge 1/(m-1)$ and (2) stability: $A_1$ must also be best in $S$ or $R$. If a condition fails, a set of alternatives is offered as the “compromise”. Weighting in VIKOR is not part of the method itself; it comes from outside.

Example: the same four laptops

With weights $(0.40,0.35,0.25)$ (see Scoring methods: SAW, WPM and WASPAS):

	TOPSIS: $C_i$	Rank	VIKOR: $Q_i$	Rank
A	0.622	1	0.037	1
B	0.493	3	0.322	2
C	0.515	2	0.889	4
D	0.463	4	0.697	3

Both put “A” first. (In VIKOR, though, the acceptable-advantage condition fails: $Q(B)-Q(A)\approx 0.285<1/(m-1)\approx 0.333$; so strictly, “A” and “B” together form the compromise set.) But the rest of the ranking differs (TOPSIS puts “C” second and VIKOR last) because VIKOR is more sensitive to regret (the worst attribute) and “C” is weak on performance.

Which one, and when?

• TOPSIS: simple and widely used; assumes complete compensation.
• VIKOR: when you want to take agreement, or avoiding regret on a single attribute, into account.

A TOPSIS caveat: “rank reversal” may occur (adding an alternative shifts the ideals).

Related: Outranking methods: ELECTRE and PROMETHEE and Normalizing criteria.

TOPSISVIKORmulti_criteria_decision_making