APLOCO - Automatic Pairwise Linear Order Combination

Overview

S_i can be any real number. Higher S means stronger net pairwise dominance over other alternatives. S_i > 0 means the alternative dominates more than it is dominated; S_i < 0 means the reverse. APLOCO is similar in spirit to PROMETHEE but uses a simpler linear (max-0) preference function without threshold parameters.

Output

utility, higher is better

Data

Crisp, complete numeric matrix

Weights

Needs a weight source

Size

2+ alternatives, 3-12 criteria works best

Used for

Alternative selection, Supplier evaluation

How it works

1. 1

   Min-max normalisation to [0,1].

   $$r_{ij}=\frac{x_{ij}-x_j^{\min }}{x_j^{\max }-x_j^{\min }}\text{ (benefit)};r_{ij}=\frac{x_j^{\max }-x_{ij}}{x_j^{\max }-x_j^{\min }}\text{ (cost)}$$

   Konstantinos et al. 2020, p.3726

2. 2

   Compute pairwise dominance score D(A_i, A_k) for each pair: weighted sum of positive normalised differences.

   $$D(A_i,A_k)=\sum _{j=1}^n{w}_j\max (0,r_{ij}-r_{kj})$$

   Konstantinos et al. 2020, p.3727 Eq.(1)

3. 3

   Compute APLOCO score S_i = net dominance (sum of D(A_i,A_k) minus sum of D(A_k,A_i)). Rank in descending order.

   $$S_i=\sum _{k\ne i}D(A_i,A_k)-\sum _{k\ne i}D(A_k,A_i)$$

   Konstantinos et al. 2020, p.3728 Eq.(2)

Look elsewhere when

Assumptions to verify

• Criteria preferences are independent (no synergistic interactions)
• Compensation is acceptable: high score on one criterion can offset low on another
• Decision matrix is complete (no missing values)

Edge cases and pitfalls

• Constant criterion column: normalisation denominator zero - always check for non-constant columns.

Works with

How to cite