Weighted sum model

In decision theory, the weighted sum model (WSM), ^{[1] [2]} also called weighted linear combination (WLC) ^[3] or simple additive weighting (SAW), ^[4] is the best known and simplest multi-criteria decision analysis (MCDA) / multi-criteria decision making method for evaluating a number of alternatives in terms of a number of decision criteria.

Description

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 w_j denotes the relative weight of importance of the criterion C_j and a_ij is the performance value of alternative A_i when it is evaluated in terms of criterion C_j. Then, the total (i.e., when all the criteria are considered simultaneously) importance of alternative A_i, denoted as A_i^WSM-score, is defined as follows:

${\displaystyle A_{i}^{\text{WSM-score}}=\sum _{j=1}^{n}w_{j}a_{ij},{\text{ for }}i=1,2,3,\dots ,m.}$

For the maximization case, the best alternative is the one that yields the maximum total performance value. ^{[2] [ clarification needed ]}

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."

Example

For a simple numerical example suppose that a decision problem of this type is defined on three alternative choices 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:

	Criteria				WSM Score
	C1	C2	C3	C4
Weighting	0.20	0.15	0.40	0.25	–
Choice A1	25	20	15	30	21.50
Choice A2	10	30	20	30	22.00
Choice A3	30	10	30	10	22.00

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 (i.e., A₁) 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:

${\displaystyle A_{1}^{\text{WSM-score}}=25\times 0.20+20\times 0.15+15\times 0.40+30\times 0.25=21.50.}$

Similarly, one gets:

${\displaystyle A_{2}^{\text{WSM-score}}=22.00,{\text{ and }}A_{3}^{\text{WSM-score}}=22.00.}$

Thus, the best choice (in the maximization case) is either alternative A₂ or A₃ (because they both have the maximum WSM score which is equal to 22.00). These numerical results imply the following ranking of these three alternatives: A₂ = A₃ > A₁ (where the symbol ">" stands for "greater than").

Choosing the weights

The choice of values for the weights is rarely easy. The simple default of equal weighting is sometimes used when all criteria are measured in the same units. Scoring methods may be used for rankings (universities, countries, consumer products etc.), and the weights will determine the order in which these entities are placed. There is often much argument about the appropriateness of the chosen weights, and whether they are biased or display favouritism.
One approach for overcoming this issue is to automatically generate the weights from the data. ^[5] This has the advantage of avoiding personal input and so is more objective. The so-called Automatic Democratic Method for weight generation has two key steps:

(1) For each alternative, identify the weights which will maximize its score, subject to the condition that these weights do not lead to any of the alternatives exceeding a score of 100%.

(2) Fit an equation to these optimal scores using regression so that the regression equation predicts these scores as closely as possible using the criteria data as explanatory variables. The regression coefficients then provide the final weights.

See also

• Decision-making software
• Weighted product model

References

1. Fishburn, P.C. (1967). "Additive Utilities with Incomplete Product Set: Applications to Priorities and Assignments". Journal of the Operations Research Society of America. doi: 10.1287/opre.15.3.537 .
2. Triantaphyllou, E. (2000). Multi-Criteria Decision Making: A Comparative Study. Dordrecht, The Netherlands: Kluwer Academic Publishers (now Springer). p. 320. ISBN   0-7923-6607-7.
3. Malczewski, Jacek; Rinner, Claus (2015). Multicriteria Decision Analysis in Geographic Information Science. New York (USA), Heidelberg (Germany), Dordrecht (The Netherlands), London (UK): Springer. doi:10.1007/978-3-540-74757-4. ISBN   978-3-540-86875-0. S2CID   126734355 . Retrieved May 31, 2020.
4. Churchman, Charles W.; Ackoff, Russell L.; Smith, Nicolas M. (1954). "An approximate measure of value". Journal of the Operations Research Society of America. 2 (2): 172–187. doi:10.1287/opre.2.2.172 . Retrieved May 31, 2020.
5. Tofallis, Chris (2022). "Objective Weights for Scoring: The Automatic Democratic Method". Multiple Criteria Decision Making. 17: 69–84. doi:10.22367/mcdm.2022.17.04. SSRN   4465402 – via SSRN.