Multi-Criteria Decision Aid

This is a tool that helps a decision maker to choose a solution when he is facing conflicting criteria and cannot decide.

For example, you want to buy a new car:

  • One is expensive, speed is good;
  • another is cheap but slow and with little comfort.
  • These criteria (cost vs speed) are conflicting.

We need to find a compromise that answers the expectation of a decision maker


  • Step 1: Define the set of alternatives $A = \{a_1, ..., a_n\}$
  • Step 2: Define the set of criteria $G = \{g_1, ..., g_k\}$
  • Step 3: Define the Preferences (the expectations of a decision maker)
  • Step 4: Apply methods to find the best alternative


$A$ - set of alternatives (actions, options, items, decisions, etc)

$A$ can be

  • finite or infinite
  • countable or uncountable
  • stable (always the same) or evolving


A criterion $g_i$ is a mapping from the set of alternatives $A$ to some totally ordered set $E_i$:

  • $g_i: A \mapsto E_i$
  • $g_i \in G$ form a set of criteria

With $E_i$ we can rank all elements of $A$ from best to worst


  • $E = \mathbb{R}$
  • $E = \{\text{VB}, \text{B}, \text{M}, \text{G}, \text{VG}\}$

A set can be:

  • ordinal (operations $<, =, >$)
    • $E = \{\text{VB}, \text{B}, \text{M}, \text{G}, \text{VG}\}$
  • interval (operations $<, =, >, +, -$)
    • temperature
  • ratio (operations $<, =, >, +, -, \cdot, / $)
    • $E = \mathbb{R}$

Restrictions on $G$:

Dominance Principle

Some alternatives can be eliminated by Dominance principle

  • If for two alternatives $a$ and $b$ for all criteria they are equally good
  • but there exists one criteria at which $a$ is better than $b$
  • then $b$ is dominated by $a$ and will never be chosen

Consider this example

  • we're choosing a car
  • there are 4 criteria: price, power, consumption, comfort
  • there are 6 alternatives
Price Power Consumption Comfort
Avg A. 18 75 8 3
Sport 18.5 110 9 2
Avg B. 17.5 85 7 3
Lux 1 24 90 8.5 5
Exonomic 12.5 50 7.5 1
Lux 2 22.5 85 9 4

By Dominance principle:

  • We see that Avg B is always better than Avg. A
  • then nobody will ever choose Avg A: A is dominated by B
  • but no other alternative can be eliminated this way

How to chose which one is the best?

  • Need subjective preferences


To be able to find the best solution we need to know subjective preferences

Given two alternatives $a$ and $b$ a decision maker can say if

  • $a \ P \ b$ or $b \ P \ a$: $a$ is preferred to $b$ - the preference relation
  • $a \ I \ b$ - the indifference relation (not transitive! see Luce's Coffee Cups)
  • $a \ J \ b$ - the incomparability relation, when you cannot compare things


How to represent a Decision Maker's preferences in some model?

With Modeling Preferences:

Important condition when modeling preferences:


There some important families of criteria:


Outranking methods perform pair-wise comparisons (like in the Condorcet's Rule)

Most famous methods:

Problems of outranking methods:

Multi-Objective Optimization

Once we found the Pareto-optimal set of solutions in a problem, we need to find the best solution, and MCDA can help with it