VIKOR-SMAA (stochastic acceptability)

Four things change; VIKOR's four steps (best/worst value, S and R, Q, comparison) do not change in themselves, they are simply repeated thousands of times.

Cells. The decision matrix's cells are single numbers, as in crisp VIKOR; this extension does not change the cell format. What changes is the weights. Crisp VIKOR asks for the weight as a single number. Here no fixed number is given for the weight at all; the method scans the entire simplex of weight combinations that sum to 1, on the assumption that every criterion's weight can equally likely take any value.

Scale equalisation and S/R/Q. DecisionMind draws a large number of samples from this weight space, ten thousand by default. Each sample is, on its own, a valid weight combination. Crisp VIKOR's four steps, exactly as on the base card, are run separately and in full for each sample's weights. The steps are the same: best/worst value, linear normalisation, S, R and Q with v = 0.5 (changeable). Instead of a single S, R, Q triple, as many S, R, Q triples as there are samples emerge.

Aggregation. In each sample the alternatives are ranked by their own Q. DecisionMind counts these thousands of rankings: it tabulates, for every alternative, in what percentage of samples it came first, in what percentage second, in what percentage third. This is called the rank acceptability distribution.

Result and defuzzification. The rank acceptability distribution is reduced to a single comparison figure. The holistic acceptability index is a weighted sum that gives the highest contribution to the probability of first place and the lowest to last place. Ranking is done by this index. Crisp VIKOR's two compromise conditions, acceptable advantage and acceptable stability, are not calculated here at all. Their place is taken by this distribution itself, spread over the entire sample. Instead of looking at a single Q gap, the user sees a direct answer to the question "what is the probability that this alternative is first."

DecisionMind exposes the number of samples (N, default 10,000) and the compromise coefficient (v, default 0.5) to the user for this extension. The random seed used for sampling is not a parameter open to the user; the engine internally uses a fixed seed. For this reason, two runs with the same N and the same v give exactly the same result. Reproducibility comes not from a setting the user chooses but from the engine's fixed internal seed.

The output is not a ranking but a probability distribution for each alternative: "this alternative is first with such a probability, second with such a probability," and so on. The holistic acceptability index is a single number summarising this distribution, and a larger value is better. This is the exact opposite of crisp VIKOR's rule that a small Q is good; it is a separate scale operating in the opposite direction. If two alternatives' first-place probabilities are close to one another, this is an honest result of VIKOR-SMAA, not a calculation error. If uncertainty about the weights is genuinely large, there is no clear winner between two alternatives.

Thus instead of writing:

"According to VIKOR-SMAA the best alternative is A2"

the report should read:

"A2 comes first in 70.4 per cent of the samples and also has the highest value in the holistic acceptability index. A1 and A3's first-place probabilities are 13.7 per cent and 16.0 per cent respectively"

This extension is suitable when decision-makers cannot agree about the criterion weights, when giving a weight is being avoided, or when nothing is known beyond the range the weight might fall in. If the criterion values themselves are given precisely and only the weights are uncertain, this extension runs directly. If the criterion values themselves are also uncertain, the type of that uncertainty (approximation, a range, or a probability) should first be identified and the relevant data-type card consulted. If the weights are already clear and the decision-makers have agreed on them, crisp VIKOR is sufficient; running VIKOR-SMAA would then be an unnecessary sweep of uncertainty. Base VIKOR's exit condition applies here too: if no compromise at all is acceptable on one criterion, outranking methods should be used instead.

Keeping the number of samples (N) small. If the weight space is not swept thoroughly enough, the acceptability figures come out noisy and can produce different numbers under a different seed even with the same N. This is shown in Case 1 below.

Choosing the compromise coefficient v as 0 or 1. The same error as in crisp VIKOR applies here too. At these extremes the method looks only at total benefit or only at regret, and stops being a compromise.

Looking for crisp VIKOR's two conditions here and failing to find them. This extension never calculates the acceptable-advantage and acceptable-stability conditions; their place is taken by the acceptability distribution itself. Treating the engine as having "left this out" is wrong; the concept is simply different here.

Confusing the holistic acceptability index with crisp VIKOR's Q. In this index a large value is good; in crisp VIKOR's Q a small value is good. Reading the two figures in the same direction reverses the ranking.

The governing principle is this:

VIKOR-SMAA's output is a probability distribution, not a single ranking. Writing down only the top-ranked alternative, without stating the sample count and the compromise coefficient in the report, is a misuse of this extension.