Analytic Hierarchy Process – Wikipedia

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L’ analytic hierarchy process (AHP) It is a technique of support for the multicryngal decisions developed in the seventies by the American Iraqi mathematician from the US Thomas L. Saaty mathematician.

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The methodology allows to compare more alternatives in relation to a plurality of criteria, of a quantitative or qualitative type, and to obtain a global evaluation for each of them. This allows:

  • order alternatives in order of preference;
  • Select the best alternative;
  • assign alternatives to predefined subsets [ not clear ] .

The main strengths are the comparison to pairs of the decision -making alternatives and the separation between the importance of the criterion and the impact on the decision.

The AHP provides for a distinction between the subjective component of the evaluation and the objective data. The decision maker identifies a set of criteria on the basis of which to evaluate the decision -making alternatives and assigns to each criterion a peso percentage, then assigns a score which is the impact of the criterion on the decision. The score of each decision -making alternative is the weighed average of the scores of each criterion on the decision for the weight assigned to any criterion.

The criteria are compared to couples by assigning a score of relative importance compared to the other. The sum of the weights on the whole table must be 100%. The score of each criterion is obtained by adding what it achieves compared to all the others. The scores obtained are usually normalized.

A similar comparison to couples is then operated on among the decision -making alternatives.

The scores are included in an arbitrary scale, for example 0-100, 1-3, 1-10, corresponding to as many quality levels. In general, a “high”, “medium”, “low” scale is adopted; Or, for a fineer evaluation: “high”, “medium-high”, “medium”, “medium-low”, “low”.

The AHP has decision -making alternatives in input and k criteria of decision. It is composed of a table k*k of (weights of) criteria and by k tables n*n decisions. All the tables are square matrices (i.e. having the same number of lines and columns) and mutual (among whose elements the report always exists Aij = 1/Aji). The matrix is ​​a table A, where a “autonomous” component is in all the theory of linear systems: in fact, the judgments are at the discretion of the decision maker. For Aij elements, it is worth that. For i = j, i.e. for those of the main diagonal, aij = 1. Having said the header, the sizing of the tables, we can speak of the positioning, of how they are populated.

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For each criterion, a double entry table is built with decision -making alternatives, generated with external methods of the HP. Then the decision -making alternatives are compared to couples, filling the entire table with a finished number of equal to I and 1/I, with i = 1, .., 9. The scores from 0 (or 1) up to 9 translate a linguistic judgment into numbers relative importance between the two decisions.

In the case of many decision -making alternatives, we start from scratch because the tables with many null values ​​are processed faster by calculators.

This is less subjective than directly indicating a ranking of the most important decisions, compares only some of the possible couples (each element with the previous one), rather than with all.

For a very important decision of the row compared to column J, the score will be 9. Conversely, the score of the decision is the line compared to column I, it will be 1/9.

The relative importance of (comparing) every decision compared to itself is 1 (i = j, the same decision in the line and column considered). This is also achieved with the calculation, having to be i = (1/j) for i = j has the only solution of 1.

So if

p i j {displaystyle p_{ij}}

It is the relative score of the criterion

i {displaystyle i}

in the decision -making alternative

j {DisplayStyle J}

, it is worth that:

Thus the scores are established, the impact of the criteria on decisions. To establish the weights of the criteria, a comparison is made to couples. A double -entered table with the criteria becomes a square matrix, in which numbers are attributed on a scale from 1 to 9 for the relative importance of each criterion. The table is normalized, dividing each score by the sum of the scores of the relative column. The scores in fact vary from 1 to 9, while in middle school the weights are always between 0 and 1.

The final score of each decision is a weighted average (on the weights of the criteria) of the impact of the criterion on the decision. The decisions tables are read by line, adding the scores of the decision (i-Esima) compared to all the others and multiplying it by the weight of the criterion. The score of the decision compared to the criterion is added to those calculated for the following criteria.

A barrier threshold is identified: the decisions that have a lower score are excluded. If the decisions are exclusive and it must be only one chosen, obviously one takes the one with the greatest score.

  • Thomas L. Saaty, Multicriteria decision making – the analytic hierarchy process. Planning, priority setting, resource allocation , RWS Publishing, Pittsburgh, 1988.
  • Thomas L. Saaty, Decision Making for Leaders – The Analytic Hierarchy Process for Decisions in a Complex World , RWS Publishing, Pittsburgh, 1990.

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