Method and assumptions
This implementation is only the weighted MOORA ratio system. It does not implement the reference-point approach, full multiplicative form or MULTIMOORA synthesis. Each column is divided by its Euclidean norm. Benefit contributions are added and cost contributions subtracted. Higher y is preferred; a negative result is not an error.
This tool accepts nonnegative ratio-scale criterion values. An all-zero column is rejected because its norm is zero. Multiplying a column by a positive unit-conversion factor leaves the result unchanged; adding a constant may change it. Adding an alternative changes column norms and can cause rank reversal.
rᵢⱼ = xᵢⱼ / √(Σᵢ xᵢⱼ²)
yᵢ = Σbenefit wⱼrᵢⱼ − Σcost wⱼrᵢⱼ
i indexes alternatives, j criteria, m is the alternative count, x the raw value, a an entered weight and w its sum-to-one normalization. Higher scores are preferred. A 10⁻¹² score tolerance identifies tied ranks; ties are not arbitrarily converted into a winner.
Worked example
Invented maintenance-program comparison, not measured vessel data: cost in thousand EUR; quality and support are illustrative scores. Smaller cost and larger other scores are preferred. They are assumed cardinal for this exercise.
| Cost | Quality | Support | |
|---|---|---|---|
| A | 100 | 80 | 6 |
| B | 120 | 90 | 8 |
| C | 90 | 70 | 5 |
A: −0.4 × 100/√32500 + 0.35 × 80/√19400 + 0.25 × 6/√125 = 0.11331228
| Rank | Alternative | Score |
|---|---|---|
| 1 | B | 0.13878616 |
| 2 | A | 0.11331228 |
| 3 | C | 0.088011077 |
Bar length shows absolute magnitude; the number retains its sign. Scores from different methods are not on a common scale.
| Criterion | Weight |
|---|---|
| Cost | 0.4 |
| Quality | 0.35 |
| Support | 0.25 |
| Cost | Quality | Support | |
|---|---|---|---|
| norm | 180.27756 | 139.28388 | 11.18034 |
| Cost | Quality | Support | |
|---|---|---|---|
| A | 0.5547002 | 0.57436653 | 0.53665631 |
| B | 0.66564024 | 0.64616234 | 0.71554175 |
| C | 0.49923018 | 0.50257071 | 0.4472136 |
| Cost | Quality | Support | |
|---|---|---|---|
| A | -0.22188008 | 0.20102828 | 0.13416408 |
| B | -0.26625609 | 0.22615682 | 0.17888544 |
| C | -0.19969207 | 0.17589975 | 0.1118034 |
One-way weight sensitivity
The first weight is varied while the ratios of the remaining weights are preserved. These are sampled scenarios, not a probability, confidence interval or proof of robustness. If all remaining weights are zero, the scan is undefined.
| First criterion weight | A | B | C | Leading alternatives |
|---|---|---|---|---|
| 0 | 0.55865394 | 0.67507043 | 0.47950525 | B |
| 0.1 | 0.44731852 | 0.54099936 | 0.3816317 | B |
| 0.25 | 0.2803154 | 0.33989276 | 0.23482139 | B |
| 0.4 | 0.11331228 | 0.13878616 | 0.088011077 | B |
| 0.5 | 0.0019768711 | 0.0047150974 | -0.0098624652 | B |
| 0.75 | -0.27636166 | -0.33046257 | -0.25454632 | C |
| 0.9 | -0.44336478 | -0.53156917 | -0.40135663 | C |
| 1 | -0.5547002 | -0.66564024 | -0.49923018 | C |
Limits and interpretation
The result depends on the preference model and supplied alternative set; it does not prove a uniquely correct decision. Justify weights and criterion directions, avoid double counting and examine sensitivity. Measurement uncertainty is not modeled here. Do not sum scores across methods into a new truth score. Qualified people must evaluate safety and regulatory constraints before ranking.