Multi-Criteria Decision Making
Multi-Criteria Decision-Making (MCDM) · Choosing with several criteria at once
How do you pick the most suitable alternative while weighing several criteria at once? In real decisions it is hard to settle on the best option using a single criterion. The cheapest laptop is not also the best performer, and the lowest-priced supplier is not also the best on quality and delivery. The approach used when several criteria must be considered at the same time is Multi-Criteria Decision-Making (MCDM). It compares several alternatives on multiple criteria and supports the choice by combining the importance of each criterion with how each alternative performs.
What is multi-criteria decision making?
If a company is choosing office laptops, it might consider price, performance, battery life and weight. Just as some products are cheap but heavy and others are light but expensive, each product has different strengths and weaknesses on each criterion.
What matters in multi-criteria decision making is that not all criteria move in the same direction. You need to distinguish criteria where lower is better, such as price or weight, from criteria where higher is better, such as performance.
The basic building blocks of MCDM
| Element | Meaning | Example |
|---|---|---|
| Alternatives | The candidates you can choose | Laptops A, B, C |
| Criteria | The items used to compare alternatives | Price, performance, battery, weight |
| Weights | The relative importance of each criterion | Price 30%, performance 35%, etc. |
Assigning numbers to weights arbitrarily does not make them an objective right answer. They can be set from expert judgment, survey responses, real data or the purpose of the analysis, and you should clearly explain how the weights were set.
What is the weighted sum model?
The most intuitive method is the Weighted Sum Model (WSM). It multiplies each criterion's score by its weight and adds them all up to get an overall score for each alternative (Ch. 01).
Si: overall score of alternative i · wj: weight of criterion j · rij: score of alternative i on criterion j · m: number of criteria
Criteria with larger weights have a bigger influence on the overall score. However, values in different units, such as price (dollars), battery (hours) and weight (kg), must first be converted to a comparable scale. A typical method is min–max normalization, which sets the minimum to 0 and the maximum to 100; for criteria where lower is better, the direction is flipped, as in (max − value) / (max − min).
A worked example: comparing three laptops
All criterion scores have been converted to a 100-point scale, and a higher score is assumed to be better on that criterion. The weights are price 30%, performance 35%, battery 20% and portability 15%.
| Alternative | Price | Performance | Battery | Portability |
|---|---|---|---|---|
| Laptop A | 90 | 80 | 70 | 85 |
| Laptop B | 75 | 95 | 90 | 70 |
| Laptop C | 85 | 75 | 85 | 95 |
SB = 0.30(75) + 0.35(95) + 0.20(90) + 0.15(70) = 22.5 + 33.25 + 18 + 10.5 = 84.25
SC = 0.30(85) + 0.35(75) + 0.20(85) + 0.15(95) = 25.5 + 26.25 + 17 + 14.25 = 83.00
With these weights and scores, laptop B ranks first with 84.25 points, C second with 83.00, and A third with 81.75. B has the lowest price score, but its performance and battery scores are high, and it pulled ahead by earning 33.25 points on performance, the criterion with the largest weight (35%). This result does not mean B is the best product for everyone; if the weights and scores change, the ranking can change too.
Figure 3 shows the result of changing only the performance weight. With a performance weight below about 31%, C ranks first; above it, B. The lesson's assumption (35%) is not far from this boundary, so the conclusion is fairly sensitive to the performance weight.
If it is hard to settle on a single set of weights, you can draw weights at random many times and look at how often each alternative ranks first. As on the left of Figure 4, with no information about the weights, C ranks first 54% of the time and B 36%, so C wins more often. If the weights are only jittered around the lesson's weights (30 : 35 : 20 : 15), B keeps first place 62% of the time, but C still ranks first in 30% of cases.
How do you set the weights?
Even with the same alternatives and scores, the final ranking can change when the weights change, so setting the weights is an important step in multi-criteria decision making.
| Method | Basic idea |
|---|---|
| Direct weighting | The decision maker or experts rate the importance of each criterion directly |
| AHP | Compare criteria pairwise to calculate relative importance and weights (Ch. 05) |
| Entropy weights | Calculate weights from the distribution and information content of the data |
| CRITIC | Calculate weights from each criterion's variability and the correlations between criteria |
AHP reflects the judge's preferences, while entropy and CRITIC use characteristics of the data. In Figure 5, the entropy method gives 36% to portability, where the three laptops' scores differ a lot, and 13% to price, where they differ little. This is because a criterion that separates the alternatives well is seen as carrying more information. As a result, the winner changes to C. Even an objective method contains the assumption that "a criterion with larger score differences is more important," and whether that assumption matches the decision maker's actual preferences is a separate question. If a survey shows that customers value performance more than price, that preference can be built into the weights; if you only have performance data on several suppliers, you can consider an objective weighting method.
Strengths and limitations of the weighted sum model
The weighted sum model is easy to calculate and its results are easy to explain. If you disclose the alternatives' scores and the weights, anyone can see why a particular alternative scored high. It does have some limitations, however.
Values in different units need an appropriate score conversion, and the ranking can change depending on the conversion method.
A low score on one criterion is offset by high scores on other criteria.
The result depends on whose judgment the weights reflect.
Even small changes in the weights or scores can change the ranking.
Hypothetical laptop D in Figure 6 scores only 30 points on battery, but all its other criteria are 100, so it ranks first with a weighted sum of 86.00. If battery life must be above a certain level, that condition should be applied first to exclude alternatives that fail it, and then the rest compared. Then B is first again.
How MCDM relates to other analysis methods
| Method | Role | Covered in |
|---|---|---|
| AHP | Calculates the relative importance of the criteria | Ch. 05 |
| TOPSIS | Ranks alternatives by their distance from an ideal alternative | Ch. 06 |
| PROMETHEE · VIKOR | Ranks alternatives considering preference relations between them or conflicting criteria | — |
| Optimization models | Finds the best choice while also accounting for constraints such as resources or budget | Later lessons |
You can also get the weights with AHP and then rank the alternatives with TOPSIS. Not every problem needs several methods combined, however; what matters is choosing the method that fits the purpose of the analysis and the data.
A weighted sum was computed for three laptops using price, performance, battery and portability scores converted to a 100-point scale with weights 30 : 35 : 20 : 15; the ranking was B (84.25), C (83.00), A (81.75). When the performance weight was below about 31%, C ranked first, and in 20,000 random variations of the weights around the lesson's weights, B ranked first 62% of the time.
Align the direction and scale of the criteria, state the weights, and then shake those weights.
- MCDM: An approach that compares alternatives by considering several criteria at the same time.
- Weighted sum model: Multiply each criterion's direction- and scale-aligned score by its weight and add them up.
- Weights: Set by direct assignment, AHP, entropy, CRITIC and other methods; the ranking can differ by method.
- Sensitivity: Change the weights or jitter them randomly to check how robust the winner is.
- Must-have conditions: To prevent the compensation effect, apply conditions that must be met separately, before the weighted sum.
Next is 05 Analytic Hierarchy Process (AHP). We will look at how to calculate criterion weights systematically through pairwise comparisons.