Ranking Alternatives with TOPSIS
Technique for Order Preference by Similarity to Ideal Solution
Can you find the option that sits closest to the ideal and, at the same time, farthest from the worst? When you evaluate several alternatives, it’s rare for one of them to be best on every criterion. Some products are cheap but perform poorly; others perform well but cost a lot. TOPSIS is a multi-criteria decision-making method that ranks alternatives by considering both how close each one is to an ideal reference point and how far it is from the least favorable one.
The basic idea of TOPSIS
A reference point made of the most desirable performance on each criterion
A reference point made of the least desirable performance on each criterion
The ideal and negative-ideal solutions may not be alternatives that actually exist. If one product is the cheapest and another performs best, the ideal solution is a hypothetical reference point that combines the strengths of both.
In what order is TOPSIS calculated?
A worked example: choosing a supplier
A company is choosing one of three suppliers, and the criteria are price, quality, and on-time delivery rate. Assume every score is out of 10 and higher is better. The price score is not the actual price; it’s a rating converted so that cheaper suppliers get higher scores.
| Supplier | Price score | Quality score | Delivery score |
|---|---|---|---|
| A | 9 | 7 | 8 |
| B | 7 | 9 | 9 |
| C | 8 | 8 | 6 |
| Criterion | Weight |
|---|---|
| Price | 0.40 |
| Quality | 0.35 |
| Delivery | 0.25 |
| Total | 1.00 |
Normalizing and applying weights
TOPSIS usually uses vector normalization. Each score is divided by the square root of the sum of squares of that criterion’s values, and then multiplied by the criterion’s weight.
Example: denominator for price = √(9² + 7² + 8²) = √194 ≈ 13.928 → A’s normalized value 9 / 13.928 = 0.6462
| Supplier | r price | r quality | r delivery | v price | v quality | v delivery |
|---|---|---|---|---|---|---|
| A | 0.6462 | 0.5026 | 0.5946 | 0.2585 | 0.1759 | 0.1487 |
| B | 0.5026 | 0.6462 | 0.6690 | 0.2010 | 0.2262 | 0.1672 |
| C | 0.5744 | 0.5744 | 0.4460 | 0.2297 | 0.2010 | 0.1115 |
This lets you adjust for values measured in different units while still reflecting how important each criterion is. Results can differ depending on the normalization method (vector, min–max, and so on), though, so you should state which method you used.
Calculating the ideal and negative-ideal solutions
Because every score has been converted so that higher is better, the ideal solution is made of the largest weighted normalized value on each criterion, and the negative-ideal solution of the smallest.
The ideal solution’s price comes from A, and its quality and delivery come from B. In a real analysis, if you use raw data where lower is better, such as price, you have to flip the direction so that the minimum becomes the ideal for that criterion. Set a criterion’s direction wrong and the results get read backwards.
Calculating the distances to the two reference points
Di⁺: distance between alternative i and the ideal solution · Di⁻: distance between alternative i and the negative-ideal solution
A good alternative tends to be a short distance from the ideal solution and a long distance from the negative-ideal solution.
Relative closeness and the final ranking
| Supplier | D⁺ | D⁻ | Closeness C | Rank |
|---|---|---|---|---|
| A | 0.0536 | 0.0684 | 0.561 | 2 |
| B | 0.0574 | 0.0751 | 0.566 | 1 |
| C | 0.0676 | 0.0382 | 0.361 | 3 |
B (0.566) edges out A (0.561) for first place, and C (0.361), with the lowest delivery score, sits close to the negative-ideal solution. For reference, a weighted sum with the same scores and weights gives B 8.20, A 8.05, C 7.50, the same ranking.
| Relative closeness | Interpretation |
|---|---|
| Near 1 | Relatively close to the ideal solution |
| Around 0.5 | About equally far from both reference points |
| Near 0 | Relatively close to the negative-ideal solution |
Relative closeness is not an actual probability of success or a rate of hitting a target. It’s a relative score among the alternatives that depends on the criteria, the weights, and the normalization method.
How is TOPSIS different from AHP?
| Aspect | AHP (Ch. 05) | TOPSIS |
|---|---|---|
| Main purpose | Derive the relative importance of criteria and priorities | Evaluate alternatives by their distance from the ideal solution |
| Typical input | Pairwise comparison judgments | Each alternative’s performance on the criteria, plus weights |
| Main output | Criterion weights, alternative priorities | Relative closeness and a ranking of alternatives |
| Notable feature | Lets you check the consistency of judgments | Combines performance on several criteria into a ranking |
You can combine them, deriving criterion weights with AHP and then evaluating alternatives with TOPSIS, but if the weights are already set or were derived some other way, there’s no need to go through AHP.
Cautions when using TOPSIS
First, check how the weights affect the result. In Figure 4, once the price weight goes above about 41%, first place switches from B to A. The example’s assumption (40%) sits just below that boundary, so the gap between A and B is very small. Lower the price weight to 20% and B’s closeness is 0.777 versus A’s 0.445, a clear lead for B.
Second, check each criterion’s direction and the normalization method. Mishandle cost criteria (lower is better) and benefit criteria (higher is better) and the conclusion gets distorted.
Third, a rank doesn’t mean absolute superiority. TOPSIS produces a relative ranking within the chosen set of alternatives and criteria, so the result can change when the set being compared changes. In Figure 5, all we did was add supplier D (5, 5, 5), the worst on every criterion, yet the normalization denominators and the negative-ideal solution shift and A (0.719) moves ahead of B (0.705). This is called rank reversal, and it’s best to finalize the list of candidates before you run the analysis.
Fourth, any must-have requirements need to be handled separately. If a supplier must meet a specific quality standard, apply that condition first and then run TOPSIS.
Three suppliers were evaluated on price, quality, and delivery (weights 0.40 : 0.35 : 0.25) using TOPSIS with vector normalization. Relative closeness ranked B 0.566, A 0.561, and C 0.361. However, the gap between A and B was small, and A became first once the price weight exceeded about 41%, so the rationale for the price weight was reviewed again before the final selection.
Rank alternatives by how close they are to the best and how far from the worst.
- Reference points: TOPSIS evaluates alternatives against an ideal and a negative-ideal solution built from the best and worst on each criterion.
- Calculation: It goes vector normalization → apply weights → find A⁺ · A⁻ → distances D⁺ · D⁻ → closeness C = D⁻/(D⁺ + D⁻).
- Interpretation: Higher closeness means a higher rank, but it’s a relative score, not a probability or achievement rate.
- Sensitivity: The ranking can change with the weights, the normalization method, the criteria directions, and which alternatives are being compared.
- Relationship to AHP: AHP can be used to derive weights, and TOPSIS to rank alternatives.
Up next: 07 Sensitivity Analysis for Decisions. We’ll look at how to check how much a decision changes when assumptions such as weights, probabilities, or scores change.