Goal Programming - Minimise deviations from multiple aspiration levels

Overview

Goal Programming - Minimise deviations from multiple aspiration levels

Output

weighted deviation, lower 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

   Validate inputs: decision matrix non-empty, all values finite, weights provided (Σ w_k = 1, w_k > 0), criterion directions specified.

   $$w_k>0,\sum _{k=1}^n{w}_k=1,x_{ij}\in \mathbb{R}\forall i,j$$

   Charnes & Cooper 1955 Management Science 1(2):138-151

2. 2

   Set aspiration (goal) level g_k for each criterion k: for max criteria use the best (maximum) observed value; for min criteria use the best (minimum) observed value. These are the ideal points each alternative is measured against.

   $$g_k=\{\begin{array}{ll}{\max }_i{f}_k(x_i)& \text{if }k\text{ is max}\\ {\min }_i{f}_k(x_i)& \text{if }k\text{ is min}\end{array}$$

   Charnes & Cooper 1955 Management Science 1(2):138-151; Romero 1991 Handbook of Critical Issues in Goal Programming, pp.1-20

3. 3

   Compute positive deviation d_ik^+ (overachievement: f_k(x_i) exceeds g_k) and negative deviation d_ik^- (underachievement: f_k(x_i) falls short of g_k) for each alternative i and criterion k. Both deviations are non-negative; exactly one of {d_ik^+, d_ik^-} is nonzero unless f_k(x_i) = g_k.

   $$d_{ik}^{+}=\max (0,f_k(x_i)-g_k),d_{ik}^{-}=\max (0,g_k-f_k(x_i))$$

   Charnes & Cooper 1955 Management Science 1(2):138-151, Eq. (model constraint); Romero 1991 p.4 Eq.(1.2)

4. 4

   Compute total weighted deviation score for each alternative i by summing weighted absolute deviations across all criteria. Lower score indicates the alternative is closer to the aspiration levels across all criteria.

   $${\text{score}}_i=\sum _{k=1}^n{w}_k\left(d_{ik}^{+}+d_{ik}^{-}\right)$$

   Charnes & Cooper 1955 Management Science 1(2):138-151; Martel & Aouni 1998 J.Global Optim. 12:127-138 §2 Eq.(2)

5. 5

   Rank alternatives by ascending score_i (lower total weighted deviation = better). Assign rank 1 to the alternative with the minimum score. Ties broken by alternative_id lexicographic order.

   $${\text{rank}}_i=1+|\{j:{\text{score}}_j<{\text{score}}_i\}|$$

   Standard WGP ranking convention

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

• •empty, all values finite, weights provided (Σ w_k = 1, w_k > 0), criterion directions specified.
• •zero unless f_k(x_i) = g_k.
• •Ties broken by alternative_id lexicographic order.

• Applying GOAL-PROGRAMMING without verifying this assumption.

  Requirement: Criteria preferences are independent (no synergistic interactions)

• Applying GOAL-PROGRAMMING without verifying this assumption.

  Requirement: Compensation is acceptable: high score on one criterion can offset low on another

• Applying GOAL-PROGRAMMING without verifying this assumption.

  Requirement: Decision matrix is complete (no missing values)

• Using GOAL-PROGRAMMING when: Criteria strongly correlated → consider DEMATEL/ANP for interdependence.

  An alternative method is recommended in this situation.

• Using GOAL-PROGRAMMING when: Non-compensatory preferences → consider outranking (ELECTRE/PROMETHEE).

  An alternative method is recommended in this situation.

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