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Use TOPSIS as a transparent distance model, then test whether the ranking survives its design choices
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Full chapter index · Independent practice · Judgement subject guide
The Technique for Order Preference by Similarity to Ideal Solution ranks alternatives by closeness to a constructed positive ideal and distance from a constructed negative ideal. The arithmetic is accessible; the judgement lies in criteria, scales, weights, normalisation and reference points.
This guide builds one fictional TOPSIS result from raw values, shows where rank reversal can enter, and turns a single ranking into an auditable sensitivity record.
Continue from The Gold Standard of Judgement within The Gold Standard of Education: learning that transfers into life and the How X Works learning library.
Full chapter index
See the mechanism · Chapters 1–2
Build the worked model · Chapters 3–5
Test what changes · Chapters 6–8
Check evidence and limits · Chapters 9–10
Practise and continue · Chapters 11–12
Name the decision owner, purpose, feasible alternatives and constraints. TOPSIS compares the options supplied; it does not discover a missing option or decide whether a non-negotiable threshold should be traded away.
Use mutually understandable criteria tied to the decision. Separate outcomes from proxies and avoid counting the same benefit twice under correlated labels.
Declare whether each criterion is a benefit, where more is preferred, or a cost, where less is preferred. If a target range is best, a simple benefit-or-cost treatment may be unsuitable.
Preserve raw data, source date and uncertainty. A precise closeness coefficient cannot rescue weak measurements.
Evidence check: for Define the decision and alternatives, preserve the observation before the interpretation. Record decision owner, alternatives and constraints. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
Weights express importance after scaling and normalisation. A weight of 0.45 does not mean that a criterion alone determines 45% of the real-world decision.
Elicit weights with a documented method and participants. Test whether people understand the implied trade-offs. Normalise weights to sum to one only after checking their meaning.
Decide whether the ideal points will come from the observed alternatives or fixed external aspiration and rejection levels. The classical observed-set approach is relative: adding or removing an option can move the ideals.
Keep vetoes and legal constraints outside compensatory arithmetic unless the method explicitly represents them.
Evidence check: for Elicit weights and reference meaning, preserve the observation before the interpretation. Record criteria, directions, weights and references. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
Criteria use different units, so TOPSIS converts them to comparable dimensionless values. Vector normalisation divides each entry by the square root of the sum of squared entries in its column.
Min-max, sum and other normalisations produce different geometry. Their choice can affect distances and rankings. Use one method because it fits the interpretation, not because it happened to be in a spreadsheet template.
For costs, classical TOPSIS can retain the normalised values and select the minimum as the positive ideal. Avoid ad hoc inversion unless its behaviour is checked, especially near zero.
Archive the normalisation formula and full-precision values. Rounding intermediate values can alter close rankings.
Evidence check: for Choose and document normalisation, preserve the observation before the interpretation. Record normalisation formula and cost handling. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
Suppose options A, B and C are scored on quality, access and annual cost. Quality and access are benefits; cost is a cost. Raw values are A=(8,6,7), B=(7,9,9), C=(9,5,6). Weights are (0.45,0.30,0.25).
Vector-normalised values are approximately A=(0.5744,0.5035,0.5433), B=(0.5026,0.7553,0.6985), C=(0.6462,0.4196,0.4657). Multiplying each column by its weight gives the weighted matrix.
The positive ideal uses maximum weighted quality, maximum weighted access and minimum weighted cost. The negative ideal uses the opposite values.
This is a teaching calculation. The scores do not describe a real institution, and the decision would still need constraints, evidence quality and stakeholder judgement.
| Option | Quality benefit | Access benefit | Annual cost |
|---|---|---|---|
| A | 8 | 6 | 7 |
| B | 7 | 9 | 9 |
| C | 9 | 5 | 6 |
| Weight | 0.45 | 0.30 | 0.25 |
Evidence check: for Work a fictional three-option matrix, preserve the observation before the interpretation. Record every raw and normalised matrix value. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
For the fictional matrix, the positive ideal is about (0.2908,0.2266,0.1164) and the negative ideal is (0.2262,0.1259,0.1746).
Using Euclidean distance, A is 0.0844 from the positive ideal and 0.0564 from the negative. B is 0.0870 and 0.1007. C is 0.1007 and 0.0870.
Relative closeness is distance from the negative ideal divided by the sum of both distances. The coefficients are about A=0.401, B=0.537 and C=0.463, so B ranks first, then C, then A.
Check every column direction and ideal. A cost mistakenly maximised can create a fluent but inverted result.
Evidence check: for Calculate ideals and distances, preserve the observation before the interpretation. Record positive and negative ideals and distances. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
A coefficient near one means close to the constructed positive ideal and far from the constructed negative ideal under this model. It is not a probability of success, an approval percentage or a measure of absolute quality.
The ranking is compensatory: strength on one criterion can offset weakness on another. If that is unacceptable, apply feasibility rules first or choose a non-compensatory method.
Report distances and criterion contributions, not only the final order. Two options can have similar closeness for different reasons.
If coefficients are close, present a near-tie and sensitivity range rather than a theatrical winner.
Evidence check: for Interpret closeness without overclaiming, preserve the observation before the interpretation. Record closeness interpretation and compensation. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
Vary weights across a defensible range and record where the order changes. Use scenarios tied to real stakeholder views, not arbitrary plus-or-minus percentages alone.
Repeat with plausible measurement intervals. An option should not be called robust if ordinary data uncertainty reverses its position.
Compare vector and a justified alternative normalisation. If the ranking depends on scaling choice, that dependence is part of the result.
Inspect correlated criteria and outliers. An extreme value changes a vector norm and may compress other differences.
Evidence check: for Test weights, data and normalisation, preserve the observation before the interpretation. Record weight, data and normalisation sensitivity. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
Rank reversal occurs when the order of existing alternatives changes after another alternative is added or removed. In classical TOPSIS, the observed alternatives help define normalisation and ideal points, so the geometry can change.
Adding a clearly poor or duplicate-like option is a diagnostic test. If A and B reverse even though their raw data and weights did not change, identify which column norm or ideal moved.
Rank reversal is not automatically a software bug. It follows from a context-dependent model. The decision is whether that context dependence matches the intended question.
Fixed external reference levels or alternative TOPSIS variants can reduce some reversals, but they add assumptions and do not guarantee universal stability.
Evidence check: for Understand rank reversal, preserve the observation before the interpretation. Record the exact mechanism of rank reversal. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
CHAPTER 9 OF 12 · Check evidence and limits
9. Audit irrelevant and dominated alternatives
Check dominance before ranking. If one option is no better on any criterion and worse on at least one, ask why it is included and whether its presence changes normalisation.
Test deletion of each option, addition of a dominated option and duplication of a near-identical option. Record all reversals among the original options.
Do not quietly remove an inconvenient alternative after seeing the result. Use a predeclared eligibility rule and preserve an audit trail.
If the option set will change frequently, observed-set ideals may make longitudinal comparison difficult. Stable reference levels may better fit monitoring.
| Test | What changes | What stays fixed | Signal |
|---|---|---|---|
| Weight scenario | Declared weights | Raw data | Preference sensitivity |
| Normalisation test | Scaling formula | Options and weights | Geometry sensitivity |
| Deletion test | One option removed | Old options raw data | Rank reversal |
| Dominated addition | Inferior option added | Old options raw data | Context dependence |
Evidence check: for Audit irrelevant and dominated alternatives, preserve the observation before the interpretation. Record dominance, deletion and added-option tests. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
CHAPTER 10 OF 12 · Check evidence and limits
10. Compare neighbouring methods and limits
VIKOR separates group utility from maximum regret and adds compromise-acceptance rules. ELECTRE can represent concordance and veto. PROMETHEE uses preference functions and outranking flows. No method is automatically superior; each encodes a different decision logic.
TOPSIS is useful when distance from ideal and anti-ideal is meaningful, criteria can be placed on defensible scales, and compensation is acceptable.
It is weaker when uncertainty dominates, thresholds are non-negotiable, criteria are highly dependent or stakeholders cannot agree on trade-offs.
Use the model to expose reasoning. Do not let a ranking replace responsibility for the decision.
Explain the result back in the original units. A reader should be able to see that option B’s first place comes from stronger access offsetting its weaker quality and higher cost under the declared weights. If that trade-off sounds unacceptable when spoken plainly, the numerical model has revealed a design problem rather than solved it.
Separate technical robustness from social agreement. A ranking may remain stable through every numerical test while affected people reject the criteria or the decision-maker lacks authority to impose the trade-off. Conversely, disagreement does not make the calculation wrong; it shows that computation answers only the question encoded.
For repeated decisions, compare TOPSIS outputs only when criteria, definitions, normalisation and reference construction remain comparable. A coefficient of 0.60 this year is not automatically better than 0.55 last year if the alternative set or column norms changed. Preserve the entire matrix and method version with each result.
Where consequences are serious, pair the ranking with qualitative review, conflict screening and a written decision record. The record should name any departure from the numerical order and the lawful or practical reason for it. Transparency includes explaining when judgement properly overrides a model.
Evidence check: for Compare neighbouring methods and limits, preserve the observation before the interpretation. Record method choice and responsibility limits. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
Case A is the fictional matrix. B ranks first under vector normalisation and the declared weights.
Case B doubles the weight on cost and reduces quality proportionally. Recalculate from raw data; do not adjust the final coefficients. C may move upward because it has the lowest cost.
Case C adds option D with extreme access and very poor quality and cost. Column norms and ideals change. If B and C reverse, report rank reversal and its mechanism.
Case D fixes aspiration and rejection levels before new options arrive. The old options keep the same reference frame, but values outside the declared range need an explicit capping or distance rule.
Evidence check: for Run a changed-context transfer test, preserve the observation before the interpretation. Record the weight, extreme-option and fixed-reference contexts. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
Archive the decision, alternatives, eligibility rules, criteria, raw data, uncertainty, benefit-cost directions, weights, normalisation, weighted matrix, ideal points, distances, closeness, rank, dominance, deletion tests, added-option tests, sensitivity ranges and exact recommendation.
Common questions: Is closeness a probability? No. Do weights have meaning before scaling is fixed? Not fully. Can a dominated option affect the ranking? Yes, under some formulations. Does rank reversal prove TOPSIS is useless? No; it reveals context dependence. Should constraints be traded away? No. Is one normalisation universally correct? No.
Continue with VIKOR compromise solutions, analytic hierarchy and rank reversal, and swing weighting and reference levels. Read the primary research on rank reversal in TOPSIS, normalisation and distance choices, and the comparison of five normalisation methods.
Evidence check: for Keep the evidence route and continue, preserve the observation before the interpretation. Record the narrowest defensible TOPSIS recommendation. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.