Introduction: what is 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).

When one criterion is not enough

Choosing a car, a supplier or a job cannot be settled with one number: we want a low price and high quality, and the two usually conflict. Multi-criteria decision making (MCDM) is exactly this problem: choosing, ranking or classifying alternatives on the basis of several criteria.

Two families: MADM and MODM

• MADM (Multi-Attribute Decision Making): the number of alternatives is finite and known, and we choose among them. Here we have “attributes” and work with a decision matrix.
• MODM (Multi-Objective Decision Making): the decision variables are continuous and the alternatives are countless; the problem takes the form of optimisation with several objective functions (goal programming, ε-constraint and so on).

This section of Systophia is about MADM. (We meet MODM in multi-objective network design: cost, sustainability and resilience.)

The decision matrix

With $m$ alternatives and $n$ attributes, all the information gathers in a matrix $X=[x_{ij}]$: the entry $x_{ij}$ is the value of alternative $i$ on attribute $j$. We usually also have a weight vector $w=(w_1,\dots ,w_n)$ stating the importance of the attributes.

	Price	Performance	Battery
Alternative A	40	8	7
Alternative B	55	9	9
Alternative C	35	6	6

Types of attribute

• Quantitative (price) or qualitative (beauty). A qualitative attribute must be quantified with a scale (a Likert scale, say, or fuzzy numbers).
• Positive (benefit) or negative (cost): the more, the better or the worse. Price and waste are negative; performance and quality are positive.

The general steps, in almost every method

1. Form the decision matrix and define the attributes.
2. Normalise so that units become comparable.
3. Weight the attributes.
4. Apply an aggregation method: non-compensatory, scoring, compromise, outranking or pairwise comparison.
5. Sensitivity analysis: if the weights change slightly, does the ranking change?

Compensatory or non-compensatory?

This is the most important classification of the methods: can weakness on one attribute be compensated by strength on another? In a compensatory method, yes (a low price can cover for low quality); in a non-compensatory method, no (someone without a licence does not become a driver, however good their speed).

My note

To me the most important question in all these methods comes before the mathematics: who decided the attributes, and why these ones? The method does not resolve the conflict; it only makes it transparent. (For the limits of these methods, see MCDM does not understand love.)

Related: Normalizing criteria and Weighting criteria.

multi_criteria_decision_makingMADM