Five visible transformations
Equations and symbol key
wⱼ = bⱼ/Σb · sⱼ = √Σᵢxᵢⱼ²
rᵢⱼ = xᵢⱼ/sⱼ · vᵢⱼ = wⱼrᵢⱼ
Dᵢ⁺ = √Σⱼ(vᵢⱼ − vⱼ⁺)² · Dᵢ⁻ = √Σⱼ(vᵢⱼ − vⱼ⁻)²
Cᵢ = Dᵢ⁻/(Dᵢ⁺ + Dᵢ⁻)
| Symbol | Meaning |
|---|---|
| m, n | Number of alternatives and criteria, each 2–6 |
| xᵢⱼ | Alternative i performance on criterion j, in the stated column unit |
| bⱼ, wⱼ | Raw and sum-normalized nonnegative criterion weights |
| sⱼ | Euclidean norm of criterion j’s input column |
| rᵢⱼ, vᵢⱼ | Vector-normalized and weighted normalized performance |
| vⱼ⁺, vⱼ⁻ | Ideal and nadir coordinates, selected by benefit/cost direction |
| Dᵢ⁺, Dᵢ⁻ | Euclidean distances from alternative i to ideal and nadir |
| Cᵢ | Relative closeness, dimensionless; larger ranks ahead when defined |
Record an m × n decision matrix X, benefit (maximize) or cost (minimize) directions and raw weights bⱼ. Normalize the weights: wⱼ = bⱼ/Σb. A weight of zero removes a criterion’s contribution but remains visible in the data.
For each criterion calculate sⱼ = √Σᵢxᵢⱼ² and rᵢⱼ = xᵢⱼ/sⱼ. The resulting column vector has Euclidean norm 1 unless explicitly inactive. This vector normalization is not min–max scaling, reciprocal cost conversion or sum normalization.
Calculate vᵢⱼ = wⱼrᵢⱼ. For a benefit, the ideal is the largest vᵢⱼ and the nadir the smallest; for a cost, reverse those choices. Cost data are not inverted. The reference points are assembled from this alternative set and may not describe a feasible alternative.
Calculate Dᵢ⁺ = √Σⱼ(vᵢⱼ − vⱼ⁺)² and Dᵢ⁻ = √Σⱼ(vᵢⱼ − vⱼ⁻)². Then Cᵢ = Dᵢ⁻/(Dᵢ⁺ + Dᵢ⁻), from 0 to 1, when the denominator is positive. Higher closeness ranks ahead under this model.
If ideal and nadir coincide, every effective alternative is identical: both distances are zero and closeness is 0/0. This tool displays an indeterminate tie, with no invented 0, 0.5 or 1 score and no numerical rank.
Worked article: compare fictional ship-maintenance programs
Imagine three fictional programs for the same ship-maintenance scope. The invented criteria are avoided failure demands in 100 comparable demands (higher preferred), downtime in hours (lower preferred), and labor in person-hours (lower preferred). No observed reliability or cost data are claimed.
Programs A, B and C have rows (8,16,40), (6,8,32) and (9,24,56). Raw weights (4,2,1) normalize to (4/7,2/7,1/7), matching the deliberately consistent AHP teaching priorities. This is an illustrative hybrid, not a universal recommended weighting.
The norms are √181, √896 and √5760. Divide each input by its column norm, then multiply by the normalized weight. For example, vA,1 = (8/√181) × (4/7). The result tables below carry every value through to ideal/nadir, distances and ranking.
Program A has the highest closeness in this particular example. Its lead is conditional on the values, directions and weights. Change a weight or datum and review the sensitivity rows; do not interpret closeness as a probability of success.
Follow every calculation step
| Criterion | Unit | Preference | Raw weight | wⱼ | sⱼ |
|---|---|---|---|---|---|
| Avoided failure demands | count / 100 | Benefit: higher is preferred | 4 | 0.5714285714 | 13.45362405 |
| Downtime | h | Cost: lower is preferred | 2 | 0.2857142857 | 29.93325909 |
| Labor | person·h | Cost: lower is preferred | 1 | 0.1428571429 | 75.89466384 |
| Avoided failure demands | Downtime | Labor | |
|---|---|---|---|
| Program A | 8 | 16 | 40 |
| Program B | 6 | 8 | 32 |
| Program C | 9 | 24 | 56 |
| Avoided failure demands | Downtime | Labor | |
|---|---|---|---|
| Program A | 0.594635317 | 0.5345224838 | 0.5270462767 |
| Program B | 0.4459764877 | 0.2672612419 | 0.4216370214 |
| Program C | 0.6689647316 | 0.8017837257 | 0.7378647874 |
| Avoided failure demands | Downtime | Labor | |
|---|---|---|---|
| Program A | 0.3397916097 | 0.1527207097 | 0.07529232524 |
| Program B | 0.2548437073 | 0.07636035483 | 0.06023386019 |
| Program C | 0.3822655609 | 0.2290810645 | 0.1054092553 |
| Criterion | vⱼ⁺ | vⱼ⁻ |
|---|---|---|
| Avoided failure demands | 0.3822655609 | 0.2548437073 |
| Downtime | 0.07636035483 | 0.2290810645 |
| Labor | 0.06023386019 | 0.1054092553 |
| Alternative | D⁺ | D⁻ | Closeness | Rank |
|---|---|---|---|---|
| Program A | 0.08866621505 | 0.1181273863 | 0.5712332757 | 1 |
| Program B | 0.1274218536 | 0.1592621471 | 0.5555320376 | 2 |
| Program C | 0.1592621471 | 0.1274218536 | 0.4444679624 | 3 |
Tables display up to 10 significant digits; CSV preserves the computed binary64 values. Ties are grouped within 10⁻¹² of each descending group’s highest score, with competition ranks (1, 1, 3). Display rounding never determines rank.
Weight sensitivity: one raw weight −20% or +20%
Each positive raw weight is multiplied by 0.8 or 1.2 in turn, then the entire vector is normalized again. Zero weights remain zero. Inputs, directions and alternative set stay fixed. These are hypothetical scenarios, not confidence bounds or a proof of robustness.
| Changed criterion / factor | New normalized weights | Closeness in input alternative order | Ranks in input alternative order |
|---|---|---|---|
| Avoided failure demands × 0.8 | 0.5161290323 | 0.3225806452 | 0.1612903226 | 0.5565062666 | 0.6097334329 | 0.3902665671 | 2 | 1 | 3 |
| Avoided failure demands × 1.2 | 0.6153846154 | 0.2564102564 | 0.1282051282 | 0.5845052698 | 0.5101802797 | 0.4898197203 | 1 | 2 | 3 |
| Downtime × 0.8 | 0.6060606061 | 0.2424242424 | 0.1515151515 | 0.5892052607 | 0.5055090076 | 0.4944909924 | 1 | 2 | 3 |
| Downtime × 1.2 | 0.5405405405 | 0.3243243243 | 0.1351351351 | 0.5572633115 | 0.5969863384 | 0.4030136616 | 2 | 1 | 3 |
| Labor × 0.8 | 0.5882352941 | 0.2941176471 | 0.1176470588 | 0.5696110307 | 0.5519002983 | 0.4480997017 | 1 | 2 | 3 |
| Labor × 1.2 | 0.5555555556 | 0.2777777778 | 0.1666666667 | 0.5731411457 | 0.5598228973 | 0.4401771027 | 1 | 2 | 3 |
Before applying a ranking
- TOPSIS is compensatory: strength on one criterion may offset weakness on another. Check mandatory safety, feasibility and legal requirements first. Ranking cannot certify equipment or authorize maintenance or operation.
- Vector normalization is invariant to multiplying a whole criterion by a positive unit-conversion factor, within numerical bounds. It is generally not invariant to adding a constant, such as changing an arbitrary scale origin. Ordinal labels such as “good = 3” do not automatically become ratio-scale evidence.
- Adding or removing alternatives changes norms and reference points and can change the order of existing alternatives. Correlated criteria can double-count effects. Data uncertainty, expert disagreement and normalization variants require additional analysis.
- This tool does not train an ANN, estimate probabilities, infer weights from data or implement fuzzy TOPSIS. The ±20% weight scenarios are limited one-at-a-time comparisons; unchanged ranks are not a global robustness guarantee.
Examine how AHP derives the example weights
Primary methods and further reading
- Hwang & Yoon (1981), Multiple Attribute Decision Making: Methods and Applications
- Kacprzak (2020), An extended TOPSIS method based on ordered fuzzy numbers for group decision making, §3 classical TOPSIS equations
Sources checked 8 October 2026. All ship-maintenance values are invented teaching inputs, not measured data. Model version 1.0.0.