Weighted sum model


In decision theory, the weighted sum model, or also called weighted linear combination or simple additive weighting, is the best known and simplest multi-criteria decision analysis / multi-criteria decision making method for evaluating a number of alternatives in terms of a number of decision criteria.

Description

It is very important to state here that it is applicable only when all the data are expressed in exactly the same unit. If this is not the case, then the final result is equivalent to "adding apples and oranges."
In general, suppose that a given MCDA problem is defined on m alternatives and n decision criteria. Furthermore, let us assume that all the criteria are benefit criteria, that is, the higher the values are, the better it is. Next suppose that wj denotes the relative weight of importance of the criterion Cj and aij is the performance value of alternative Ai when it is evaluated in terms of criterion Cj. Then, the total importance of alternative Ai, denoted as AiWSM-score, is defined as follows:
For the maximization case, the best alternative is the one that yields the maximum total performance value.

Example

For a simple numerical example suppose that a decision problem of this type is defined on three alternatives A1, A2, A3 each described in terms of four criteria C1, C2, C3 and C4. Furthermore, let the numerical data for this problem be as in the following decision matrix:
C1C2C3C4
Alts.0.200.150.400.25
A125201530
A210302030
A330103010

For instance, the relative weight of the first criterion is equal to 0.20, the relative weight for the second criterion is 0.15 and so on. Similarly, the value of the first alternative in terms of the first criterion is equal to 25, the value of the same alternative in terms of the second criterion is equal to 20 and so on.
When the previous formula is applied on these numerical data the WSM scores for the three alternatives are:
Similarly, one gets:
Thus, the best alternative is alternative A2. Furthermore, these numerical results imply the following ranking of these three alternatives: A2 = A3 > A1.