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Multi-criteria decision making from first principles: the table, what a method does to it, the data types, the words, the families, and how to defend the result you get.
Reading paths
Student path
What is a credible decision route?
Start with the table, then the five stages and the concepts.
02Academic path
How do I defend the result?
Go to How to defend the result, then the report section of the guide.
03Method path
Which assumptions shaped the route?
Read the families and the goal table, then open the library.
The decision problem in one table
Every MCDM problem is a table. Rows are the alternatives you choose between; columns are the criteria you judge them on; each cell is how an alternative does on a criterion. Two more things belong to the table: a direction per criterion (is more better, or less) and a weight per criterion (how much it counts).
| Alternative | Price (USD) | Quality | Delivery (days) |
|---|---|---|---|
| Supplier A | 120 | 8 | 7 |
| Supplier B | 95 | 6 | 9 |
| Supplier C | 110 | 9 | 12 |
| Direction | minimize | maximize | minimize |
| Weight | 0.5 | 0.3 | 0.2 |
Three suppliers on three criteria. Price is a cost criterion, so lower is better; the other two are benefit criteria.
In DecisionMind: Data check shows the roles and directions; Criterion weights sets the weights.
How a method turns the table into a ranking
Methods differ in their formulas, but almost all of them pass the same five stages. Knowing the stages tells you what a method's page is talking about and where a result can go wrong.
- 1
Normalize
Criteria come in different units: money, days, a score out of ten. Normalization puts them on one scale so they can be combined. Min-max, vector, linear and sum normalization are the common forms; the method's manifest states which one it uses.
In DecisionMind: The report's Method part and the normalisation appendix.
- 2
Weight
Each normalized value is multiplied by the weight of its criterion. Subjective weighting asks you (equal, typed in, pairwise); objective weighting reads the spread of the data.
In DecisionMind: The Criterion weights step and the Criteria weights panel.
- 3
Aggregate
The weighted values of one alternative are combined into a score, or alternatives are compared in pairs. This is where the families part ways: a distance to an ideal, a weighted sum, an outranking relation.
In DecisionMind: The formula steps on the method's page.
- 4
Rank
Scores are ordered, classes are assigned, or the outranking graph is read. The shape of the answer follows the goal: a ranking, a class per alternative, a weight vector, an efficiency score.
In DecisionMind: The result card and the Ranking results panel.
- 5
Check robustness
A ranking is only as good as its inputs. Sensitivity nudges each weight and watches the order; weight scenarios draw many weight sets at once; running a second method on the same data shows whether the answer depends on the method.
In DecisionMind: The Sensitivity analysis and Weight scenarios panels, and Method Lab.
Data types
A cell does not have to be one number. When a judgement is uncertain, several cell formats carry that uncertainty into the method instead of hiding it. The system detects the type from the cells; Templates gives a blank file per type.
| Type | A cell looks like | Use it when |
|---|---|---|
| Crisp | 7.5 | You have measured or agreed numbers. |
| Interval and grey | [4, 7] | You know a range, not a point. |
| Fuzzy (triangular) | (3, 5, 7) | Linguistic judgements such as low, medium, high, mapped to three numbers. |
| Intuitionistic, Pythagorean, Fermatean | (0.6, 0.3) (0.7, 0.4) (0.8, 0.5) | A judgement has a degree of membership and a degree of non-membership, with hesitation between them. |
| Neutrosophic | (0.6, 0.2, 0.3) | Truth, indeterminacy and falsity are judged separately. |
| Hesitant | {0.4, 0.6, 0.7} | Several possible values are given for one cell. |
Key concepts
- Alternative
- One of the options being compared: a supplier, a site, a design, a candidate. A row of the decision matrix.
- Criterion
- A property the alternatives are judged on. A column of the decision matrix, with a direction and a weight.
- Direction
- Whether more of a criterion is better (benefit, maximize) or less is better (cost, minimize). A wrong direction reverses the ranking.
- Decision matrix
- The table of alternatives against criteria. Every method starts from it.
- Normalization
- Scaling criteria to a common unit so they can be combined. Types: min-max, vector, linear, sum.
- Weighting
- Assigning relative importance to criteria. Subjective weighting asks the decision maker; objective weighting reads the data.
- Aggregation
- Combining several values, or several rankings, into one. Inside a method it builds the score; across methods it builds a consensus ranking.
- Compensation
- Whether a strong criterion can make up for a weak one. Scoring methods compensate fully; outranking methods limit it with thresholds.
- Outranking
- Pairwise comparison of alternatives without full compensability: A outranks B when it is at least as good on enough criteria and not much worse on any.
- Ideal solution
- The best value on every criterion, and its opposite. Distance-based methods rank alternatives by how close they come to the ideal and how far from the anti-ideal.
- Rank reversal
- The order of two alternatives changing when a third one is added or removed. Some methods are prone to it; the robustness tests look for it.
- Sensitivity
- How much the ranking moves when a weight or a value is nudged. A stable ranking is easier to defend.
- Consistency
- Whether pairwise judgements agree with each other. Pairwise weighting reports a consistency ratio and asks you to revise when it is too high.
- Efficiency
- How much output a unit produces for its inputs, compared with the best units in the set. Efficiency methods need input and output columns instead of weights.
- Group decision
- Several experts judge the same table; their judgements are combined before or after the method runs, and their agreement is measured.
Method families
Families group methods by how they aggregate. Knowing the family tells you what kind of answer to expect and which assumptions come with it. Each name below opens the method's page.
Distance to an ideal
Score each alternative by its closeness to the best possible profile and its distance from the worst.
Weighted scores
Sum or multiply the weighted normalized values into one score per alternative; fully compensatory.
Outranking
Compare alternatives in pairs with preference thresholds; the answer can be a partial order.
Weighting
Produce the criteria weights themselves, from judgements or from the data, to feed a ranking method.
Efficiency
Measure each unit's output for its inputs against the frontier formed by the best units.
Which family for which goal
The goal sentence in the wizard does this selection for you; the table shows the logic behind it.
| You want to | Look at |
|---|---|
| Rank the alternatives | Distance to an ideal, weighted scores, outranking. |
| Pick the best one and defend it | A ranking method plus a second one to compare, and the sensitivity panel. |
| Sort the alternatives into classes | Sorting. |
| Find the criteria weights only | Weighting, subjective or objective. |
| Measure the units' efficiency | Efficiency, with input and output columns. |
| Combine the experts' opinions | Consensus methods in Method Lab. |
How to defend the result
A reviewer asks five things of an MCDM result. The report answers each of them, and this is where.
- 1
Name the method and its source
The References part cites the source paper and gives a BibTeX entry; the method's page states the assumptions.
- 2
State the normalization and the weights
The Method part gives the formula steps; the Application part gives the criteria table with directions and weights.
- 3
Show the ranking is stable
The Findings part reports the sensitivity analysis; the Weight scenarios panel adds the random-weight view.
- 4
Show it does not depend on the method
Add methods to compare in the wizard, or run them in Method Lab; the agreement coefficient and the comparison table go into the report.
- 5
Make it reproducible
The imprint carries the run id, the manifest fingerprint and the reproduction statement, so anyone with the same data and method can produce the same numbers.
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