WSM - Weighted Sum Model (Simple Additive Weighting)

Overview

WSM (SAW): score = Σ w_j × r_ij. Higher score means the alternative is closer to ideal across all weighted criteria. Simplest and most transparent MCDM method - appropriate when criteria are commensurate or pre-normalised.

Output

weighted sum score, higher is better

Data

Crisp, complete numeric matrix

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Alternative selection, Supplier evaluation, site selection, Personnel selection

How it works

1. 1

   Direction-aware linear normalisation: for max criteria map to [0,1] by (x−min)/(max−min); for min criteria invert by (max−x)/(max−min). If all criteria are already in the same unit and all max, this step may be skipped (as in Triantaphyllou 2000 Example 2-1 which uses raw values directly).

   $$r_{ij}=\{\begin{array}{ll}{x}_{ij}& \text{if all }j\in J^{+}\text{ and same unit (Triantaphyllou 2000 Eq.(2-1) shortcut)}\\ \frac{x_{ij}-{\min }_i{x}_{ij}}{{\max }_i{x}_{ij}-{\min }_i{x}_{ij}}& j\in J^{+}\text{ (engineering normalisation, mixed units)}\\ \frac{{\max }_i{x}_{ij}-x_{ij}}{{\max }_i{x}_{ij}-{\min }_i{x}_{ij}}& j\in J^{-}\text{ (engineering normalisation, mixed units)}\end{array}$$

   Fishburn 1967 Operations Research 15(3):537-542; Triantaphyllou 2000 Ch.2 §2.2.1 Eq.(2-1)

2. 2

   Compute WSM score for each alternative: weighted sum of normalised performance values. Higher score = better. Rank alternatives by descending score.

   $${\text{score}}_i=\sum _{j=1}^n{w}_j\cdot r_{ij},i=1,\dots ,m$$

   Fishburn 1967 Operations Research 15(3):537-542 Eq.(1); Triantaphyllou 2000 Ch.2 §2.2.1 Eq.(2-1)

Look elsewhere when

Assumptions to verify

• Criteria are commensurate or have been pre-normalised (additive utility assumption).
• Compensation is acceptable: high score on one criterion can fully offset low on another.
• Criterion weights accurately reflect relative importance.
• Decision matrix is complete (no missing values).

Edge cases and pitfalls

• •If all criteria are already in the same unit and all max, this step may be skipped (as in Triantaphyllou 2000 Example 2-1 which uses raw values directly).

• Dimensional inconsistency: WSM assumes criteria are commensurate (same unit). Mixing different units violates additive utility - normalise first (Triantaphyllou 2000 p.7: 'equivalent to adding apples and oranges').

• Full compensation: a very high score on one criterion can fully compensate for a very low score on another. Use outranking methods (ELECTRE, PROMETHEE) if non-compensatory preferences are intended.

Works with

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