Non-compensatory methods in multi-attribute decision making

From my doctoral course

This note comes from the doctoral course “Multi-Criteria Decision Making”. I rewrote the explanations in my own words and recomputed the numerical examples. Main source of the course: Asgharpour, Multi-Criteria Decision Making (University of Tehran).

What does “not compensated” mean?

In non-compensatory methods, weakness on one attribute is not compensated by strength on another. They usually do not need normalisation, since each attribute is measured separately from the rest.

1. Dominance

Alternative $A$ dominates $B$ if it is worse on no attribute and better on at least one. This method gives no ranking; it is a first filter: dominated alternatives are removed and the remainder are “efficient”. If several efficient alternatives remain, the decision maker chooses one as the “preferred solution”.

2. Maximin: for the pessimist

First the attributes are quantified and normalised linearly (so that negative attributes also become positive). Then for each alternative we find its worst value and pick the alternative whose worst is better than all the others’. Suited to a cautious, risk-averse decision maker.

3. Maximax: for the optimist

For each alternative we find its best value and pick the one with the best of the bests. Hurwicz combines the two with an optimism coefficient $\alpha$: $\alpha \cdot \max +(1-\alpha )\cdot \min$.

The flaw of these two methods: they see only part of the matrix’s information (the minimum or maximum of each row).

4. Conjunctive (general satisficing)

Each attribute has a threshold and an alternative must pass it on all attributes, like a driving licence. Suited to screening and “acceptable/unacceptable” classification.

5. Disjunctive (specific satisficing)

The alternative must meet the threshold on at least one attribute, like picking a football player who is enough if outstanding in one particular position. And if we have $n$ independent tests and want the probability of passing, $q=(1-P_a{)}^n$ is the probability of failing all of them.

6. Lexicographic

We rank the attributes. Alternatives are compared on the most important attribute; if one wins we are done, and if tied we go to the next attribute. It is like ordering words in a dictionary (hence “lexicographic”). Semi-lexicographic: small differences below a threshold are ignored (indifference) so the next attribute gets its turn too.

7. Elimination

Threshold plus order: first the most important attribute, and alternatives that miss its threshold are eliminated; then the second attribute, and so on until one alternative remains.

8. Permutation

With $m$ alternatives there are $m!$ possible orders. For each order we compute its “concordance” with the attributes (using their weights): entry $(i,j)$ of the pairwise matrix is the sum of the weights of attributes on which alternative $i$ is better than or equal to $j$; the concordance of an order = (sum of entries above the diagonal) − (sum of entries below it). The order with the highest concordance wins. This gives a full ranking but is practical only for few alternatives (10 alternatives mean 3.6 million permutations).

Where are they useful?

To me their strongest use is the first screen: with 50 alternatives, dominance and conjunctive satisficing bring them down to 5, and then I rank with a compensatory method. And where there is a “red line” (safety, law), non-compensatory is the only right way.

Related: Scoring methods: SAW, WPM and WASPAS and Introduction: what is multi-attribute decision making?.

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