Sensitivity Analysis

Sensitivity analysis answers a question asked after a ranking method such as TOPSIS, VIKOR or SAW has already produced a result: "how much does this ranking depend on the weights I gave it?" The method does not produce a new ranking on its own; it takes an existing ranking method, increases and decreases the criterion weights one by one by small amounts, reruns the same calculation repeatedly, and looks at each alternative's average rank across these repetitions. Its output is a robustness score for every alternative: does the alternative keep staying near the top even as weights are perturbed, or does a small weight change shift its position. Saltelli, Tarantola, Campolongo and Ratto's 2004 book is the standard reference systematically treating how input uncertainty carries through to output in engineering and decision models; DecisionMind adapts this general approach to multi-criteria decision rankings.

The idea behind sensitivity analysis is simple: stating "this is the best alternative" in a decision report is not enough, because that result usually rests on the weights the decision-maker gave, and weights are rarely uncontested. Sensitivity analysis does not ask "is the result correct"; it asks "how fragile is the result." If a ranking does not change at all when weights are perturbed by ten per cent, it can be defended with confidence; if the gap between first and second place changes places under a small weight change, this should be presented not as "a definitive result" but as "a preference sensitive to the weights."

This carries a consequence: sensitivity analysis is not itself a preference method but a layer of audit. It is added on top of any ranking method and neither confirms nor refutes that method's result; it only shows how reliably that result can be presented.

The method proceeds through three steps.

First, selecting the base calculation. Sensitivity analysis first fixes which ranking method (by default, TOPSIS in DecisionMind) and which weights will be used. This base calculation is the reference point for "what would the result have been had the weights not changed."

Second, perturbing the weights in turn. The method takes each criterion's weight one by one, first decreasing it by a set percentage (by default ten per cent) and then increasing it by the same amount; each time, the other weights are rescaled so that the total remains one. This creates 2n weight scenarios for an n-criterion problem, and the base ranking method is rerun from scratch in every scenario.

Third, summarising robustness. For every alternative, the rank numbers across these 2n scenarios are summed and averaged. The further ahead (the smaller) an alternative's average rank, the more robustly it stays near the top against weight changes. The method ranks the alternatives by this average robustness, and it can also be traced criterion by criterion which weight change caused the ranking to shift.

DecisionMind holds the perturbation rate fixed at ±10 per cent by default; a user can change this rate if desired. The formulas behind each step and the intermediate tables are given on the DecisionMind method page; this card carries no formulas.

The robustness score sensitivity analysis gives shows how high an alternative stays on average across the weight changes; it does not prove that the base ranking is "correct." A high robustness score for an alternative means "even if the weights are not known exactly, this alternative is likely to stay near the top"; a low one means "this alternative's rank depends on the values given to the weights." The score should not be confused with the base method's own score (TOPSIS, SAW…); the two measure different things.

Thus instead of writing:

"Sensitivity analysis confirmed that alternative A is the best"

the report should read:

"In the base ranking, A was first; when the weights were perturbed by ±10 per cent, A stayed first throughout, which shows the ranking is not sensitive to the choice of weights"

Sensitivity analysis is built on top of a base ranking method that works with crisp data; it does not define a data type of its own, it inherits the data type of the method it sits on. DecisionMind currently holds no separate extension of this auxiliary method.

You need a decision table, criterion directions, an initial weight set, and information on which base ranking method will be used. These initial weights may be equally distributed, or may come from expert opinion or an objective method; sensitivity analysis does not care where they come from. The perturbation rate is an optional parameter; raising it puts robustness through a harsher test. The computational load rises in proportion to the number of criteria, because the base method is rerun twice for every criterion; as the number of criteria grows, the total number of runs grows too, though each run is light.

Before presenting a ranking result to a decision-maker or senior management, sensitivity analysis is suitable, particularly where the weights are contested or the gap between first and second place is small. It strengthens the report's credibility anywhere the result might be questioned, such as supplier selection, investment prioritisation or policy evaluation.

The case where it should not be used is a quick preliminary assessment where the result is already treated as provisional; the repeated computational load is unnecessary at such a stage. Also, if the base method itself is not reliable, for instance because data is missing or criteria are misdirected, sensitivity analysis does not correct this underlying fault; it only shows weight uncertainty.

A ranking result is going into a decision, and the weights are contested → sensitivity analysis

A quick preliminary assessment, the result is already provisional → sensitivity analysis can be deferred

The uncertainty lies not in the weights but in the measurement data → a different approach: methods that carry data uncertainty (fuzzy, grey, probabilistic extensions)

The question of how the weights themselves should be found → AHP, BWM, SWARA (subjective) · Entropy, CRITIC (objective)

Sensitivity analysis's most important advantage is that it adds the answer to "how fragile is this result" to a decision report. It can be used together with any ranking method; it is method-independent. It is simple to apply: an existing calculation is repeated with small changes, no new theory or parameter is needed. It lets results be conveyed to the decision-maker more honestly, as "the ranking is this, and it is this robust," rather than simply "the ranking is this."

Its limitations come from how it is applied. First, the choice of perturbation rate (five per cent, ten, twenty) affects the result; a small rate can make robustness look better than it is, a large one can make it look worse (Triantaphyllou and Sánchez, 1997). Second, it tests only the weights; it does not catch error in the measurement data or misdefinition of the criteria. Third, it perturbs each criterion one at a time; it does not on its own cover realistic scenarios in which more than one weight changes simultaneously, for instance a case where all the weights are redistributed at once. Fourth, interpreting the result also inherits the base method's own limitations; if the base method is compensatory, sensitivity analysis operates through that same compensatory logic.

The most common mistake is confusing sensitivity analysis's result with the base method's own result; the alternative with the highest robustness score need not be the alternative that comes first under the base method, and the two must be reported separately. A second mistake is working with a single perturbation rate (say, only 10 per cent) and treating that rate as fixed and correct; it should be retried with different rates. A third mistake is looking only at weight uncertainty and disregarding data uncertainty; if measurement error is large, the "robustness" sensitivity analysis finds can be misleading. A fourth mistake is presenting the result as "the method has been validated"; sensitivity analysis is a robustness test, not a validation.

The governing principle is this:

Sensitivity analysis's robustness score shows how resistant a ranking is to weight changes; it passes no judgement on the base method's correctness.