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
Alternatives
$A$ - set of alternatives (actions, options, items, decisions, etc)
$A$ can be
- finite or infinite
- countable or uncountable
- stable (always the same) or evolving
Criteria
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
Examples:
- $E = \mathbb{R}$
- $E = \{\text{VB}, \text{B}, \text{M}, \text{G}, \text{VG}\}$
A set can be:
- ordinal (operations $<, =, >$)
- $E = \{\text{VB}, \text{B}, \text{M}, \text{G}, \text{VG}\}$
- interval (operations $<, =, >, +, -$)
- ratio (operations $<, =, >, +, -, \cdot, / $)
Restrictions on $G$:
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
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:
Methods
There some important families of criteria:
Outranking
Outranking methods perform pair-wise comparisons (like in the Condorcet's Rule)
Most famous methods:
Problems of outranking methods:
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
Links
Sources