Goal Programming: Minimise deviations from multiple aspiration levels

Overview

Multi-objective optimisation: weighted/lexicographic goal deviation minimisation. Output typically weighted_deviation (lower value = preferred).

Strengths

• •Method-specific: Multi-objective optimisation: weighted/lexicographic goal deviation minimisation

Limitations

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

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)

When not to use

• •Criteria strongly correlated → consider DEMATEL/ANP for interdependence
• •Non-compensatory preferences → consider outranking (ELECTRE/PROMETHEE)

Edge cases

• •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.

Common pitfalls

• •Hatalı: 'GOAL-PROGRAMMING bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Criteria preferences are independent (no synergistic interactions)
• •Hatalı: 'GOAL-PROGRAMMING bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Compensation is acceptable: high score on one criterion can offset low on another
• •Hatalı: 'GOAL-PROGRAMMING bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix is complete (no missing values)
• •Hatalı: GOAL-PROGRAMMING'yi 'Criteria strongly correlated → consider DEMATEL/ANP for interdependence' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: GOAL-PROGRAMMING'yi 'Non-compensatory preferences → consider outranking (ELECTRE/PROMETHEE)' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Validate inputs: decision matrix non-empty, all values finite, weights provided (Σ w_k = 1, w_k > 0), criterion directions specified. Formül: w_k > 0,\ \sum_{k=1}^{n} w_k = 1,\ x_{ij} \in \mathbb{R}\ \forall i,j Anchor: Charnes & Cooper 1955 Management Science 1(2):138-151
2. 2.Adım 2 (F2): 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. Formül: g_k = \begin{cases} \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{cases} Anchor: Charnes & Cooper 1955 Management Science 1(2):138-151; Romero 1991 Handbook of Critical Issues in Goal Programming, pp.1-20
3. 3.Adım 3 (F3): 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. Formül: d_{ik}^{+} = \max(0,\ f_k(x_i) - g_k),\quad d_{ik}^{-} = \max(0,\ g_k - f_k(x_i)) Anchor: Charnes & Cooper 1955 Management Science 1(2):138-151, Eq. (model constraint); Romero 1991 p.4 Eq.(1.2)
4. 4.Adım 4 (F4): 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. Formül: \text{score}_i = \sum_{k=1}^{n} w_k \left(d_{ik}^{+} + d_{ik}^{-}\right) Anchor: Charnes & Cooper 1955 Management Science 1(2):138-151; Martel & Aouni 1998 J.Global Optim. 12:127-138 §2 Eq.(2)
5. 5.Adım 5 (F5): 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. Formül: \text{rank}_i = 1 + |\{j : \text{score}_j < \text{score}_i\}| Anchor: Standard WGP ranking convention

Commonly paired with

• •AHP + GOAL-PROGRAMMING (high)
• •BWM + GOAL-PROGRAMMING (high)
• •ENTROPY + GOAL-PROGRAMMING (high)
• •CRITIC + GOAL-PROGRAMMING (high)
• •SWARA + GOAL-PROGRAMMING (high)

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