Weighting criteria

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).

Where do weights come from?

The weight of an attribute says “how important is this attribute to the decision”. It has two sources:

• Subjective: it comes from experts or the decision maker.
• Objective: it is extracted from the data of the decision matrix itself.

Subjective methods

1. Relative frequency: we ask experts to “vote” for some attributes; the weight of an attribute is its votes divided by all votes. (30 experts, 3 votes each from 6 attributes: 90 votes; an attribute with 19 votes gets a weight of 0.21.)
2. Arithmetic mean on a Likert scale: each expert rates the importance of an attribute from 1 to 5 (very low to very high); we average and normalise so the weights sum to 1.
3. Ranking: the expert orders the attributes from most to least important and the rank is converted into a weight (for example with the reverse rank-sum formula).
4. Pairwise comparison (AHP): we compare every two attributes; details in AHP: the analytic hierarchy process.
5. Best-Worst Method (BWM): the decision maker names the best and the worst attribute and compares only those with the rest. For $n$ attributes it needs $2n-3$ comparisons (instead of $n(n-1)/2$ in AHP) and has better consistency. (Rezaei, 2015)
6. Others: SWARA (Kersuliene, 2010), CRITIC (Diakoulaki, 1995) and LINMAP.

Objective methods

Shannon entropy: the idea is that an attribute that separates the alternatives more (more dispersion) carries more information and gets more weight. An attribute on which all alternatives are nearly alike adds nothing to the decision.

Steps:

1. $p_{ij}=x_{ij}/{\sum }_i{x}_{ij}$
2. entropy of each attribute: $E_j=-k{\sum }_i{p}_{ij}\ln p_{ij}$, with $k=1/\ln m$
3. degree of divergence: $d_j=1-E_j$
4. weight: $w_j=d_j/\sum d_j$

Example: four jobs and five attributes (income, social standing, hardship, distance, job security):

	Income	Standing	Hardship	Distance	Security
Job 1	15	7	4	10	7
Job 2	12	5	5	3	9
Job 3	20	9	3	30	5
Job 4	30	3	1	1	3

The entropies come to about $(0.956,0.948,0.913,0.625,0.948)$ and the weights to about (0.072, 0.086, 0.143, 0.615, 0.086). “Distance” gets a very high weight because it has the most dispersion, not because it matters more to me.

Standard deviation: simpler: normalise the matrix with the norm method, take the standard deviation of each column and make the weight proportional to it.

A warning

An objective weight equates importance with dispersion. In the example above “distance” won, but job security might be the most important to me. I think the best approach is a combination: the subjective weight (what matters to me) and the objective weight (what separates the alternatives), and then looking at the gap between them.

Related: AHP: the analytic hierarchy process and Normalizing criteria.

weightingshannon_entropymulti_criteria_decision_making