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Weigh methods (additional info)

Ranking methods for weight assessment

Expected value method
Rank sum method

Pairwise comparison

1. Ranking methods for weight assessment

The set of weights of a problem with three criteria (c1, c2, c3) that are non-negative and add up to one is represented as triangle ABC in Figure 1. If criterion c1 is more important than criterion c2 and criterion c2 is more important than criterion c3, then the set of feasible weights S equals the shaded triangle ADE in Figure 1. This shows that the information contained in the ordering is substantial. From the set of weights that are non-negative and add up to 1, only 1/6 proves to remain feasible.

Fig. 1: Feasible weights in a problem with three criteria (after Janssen, 1993). Shaded area W1 ≥ W2 ≥ W3.

1a. Expected value method

The expected value in one method for using the information on the set S of feasible weights to produce quantitative weights.

The expected value method assumes that each set of weights that fits the rank order of criteria has equal probability. The weight vector is calculated as the expected value of the feasible set. In the shaded triangle ADE (set S) of Figure 1, the expected value is found as the centroid of this triangle (Figure 2). This method results in a unique weight vector. In combination with, for example, weighted summation it also results in a complete ranking of the alternatives.

Fig. 2: Centroid of a triangle.

The expected value method calculates the weight, w_k, for criterion k according to Formula 1 where n is the number of criteria. The weights fit the rank order of criteria defined by set S, meaning that w1 ≥ w2 ≥ ... ≥ w_n ≥ 0.

(Formula 1)

Table 1 shows the weight vectors for various numbers of criteria according to Formula 1.

Table 1: Example of expected value of criterion weights.

Number of criteria	Expected value of criterion weights
N	E(w1)	E(w2)	E(w3)	E(w4)	E(w5)	E(w6)
2	0.75	0.25
3	0.61	0.28	0.11
4	0.52	0.27	0.15	0.06
5	0.46	0.26	0.16	0.09	0.04
6	0.41	0.26	0.16	0.10	0.06	0.03

1b. Rank sum method

Another method to generate numerical weights from a rank order of criteria is the rank sum method. This method calculates the weight, w_k, for criterion k according to Formula 2 where n is the number of criteria. Again, the weights fit the rank order of criteria defined by set S, meaning that w1 ≥ w2 ≥ ... ≥ w_n ≥ 0.

(Formula 2)

Table 2 shows the weight vectors for various numbers of criteria according to Formula 2.

Table 2: Example of criterion weights using rank sum method.

Number of criteria	Criterion weights using rank sum method.
N	w1	w2	w3	w4	w5	w6
2	0.66	0.33
3	0.50	0.33	0.17
4	0.40	0.30	0.20	0.10
5	0.33	0.27	0.20	0.13	0.07
6	0.29	0.24	0.19	0.14	0.10	0.05

The rank sum method, combined with a multicriteria method, always leads to complete ranking. The rank order is not always in agreement with all the possible quantitative weights, the weights of set S, and is therefore not entirely certain.

2. Pairwise comparison

Justification of priorities would be much easier if we use a less precise way of expressing judgments, such as words instead of numbers. Suppose we were to use words instead of numbers. Words are often easier to justify than numbers. For example, if you say that, with respect to corporate image, alternative A is 3 times more preferable than alternative B, can you justify why it is exactly 3? Why not 2.9, or 3.1? But if you said, instead, that A is 'moderately' more preferable than B, this can be justified with a variety of arguments, including, perhaps, some hard data.

The Analytical Hierarchy Process (AHP) is based on criteria that are measured on a ratio scale. In AHP the decision maker has to make a comparison for every pair of criteria: first qualitative and then quantitative on a scale from 1 to 9 to make the method operational. This scale is presented here for a binary relation:

Equally preferred	1
		Very weak preference	2
Weak preference	3
		Rather strong preference	4
Strong preference	5
		More strong preference	6
Demonstrable preference	7
		More demonstrable preference	8
Absolute preference	9

The method then creates a matrix containing the pairwise comparison judgments for the criteria, from which a priority vector is derived of relative weights for these elements (the principal eigenvector of the matrix). Moreover, due to the fact that more information than is necessary is retrieved from the decision maker, the method can deliver an inconsistency measure¹. This measure can be used to verify in what measure the judgments supplied are consistent. The decision maker is totally consistent if for all sets of three criteria the triangular relationship holds². AHP is especially designed to assess weights within a hierarchical structure of the criteria. However, due to the fast-growing number of pairwise comparisons it is not sensible to use the method for a large set of criteria.

Saaty T.L. and L.G. Vargas, 1984, Comparison of Eigenvalue and logarithmic least squares and least squares methods in estimating ratios. Mathematical modelling, Vol. 5, pp. 309-324.
The triangular relationship for a set of three criteria c₁, c₂ and c₃ holds if the equation r₁₃ = r₁₂ � r₂₃ holds, where r₁₂ is the quantitative judgment given by the decision maker.

References:

Janssen, R, and Van Herwijnen, M. (1994) Multiobjective decision support for environmental management + DEFINITE DEcisions on an FINITE set of alternatives : demonstration disks and instruction. Kluwer Academic Publishers, Dordrecht (Netherlands). 232 p., 16 p. + two 3,5" disks. ISBN 0-7923-1908-7
Saaty, T. (1980) The Analytical Hierarchy Process. New York, McGraw Hill.