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Understanding the Generalized Programming (GP) Framework in Multi-Criteria Decision Making

This paper delves into the Generalized Programming (GP) approach, a widely recognized Multi-Criteria Decision Making (MCDM) technique. We explore how different GP variants, including weighted, minmax, and lexicographic methods, influence decision-making outcomes. The theoretical foundations are illustrated through a hypothetical production planning problem. Each GP variant's effect on goal satisfaction and the role of meta-goals in optimizing results are discussed, underscoring the need for a flexible approach in accommodating decision-maker preferences.

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Understanding the Generalized Programming (GP) Framework in Multi-Criteria Decision Making

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  1. Index • Introduction • Analytic Framework • Illustrative Example • Conclusions End

  2. Introduction • GP is the most widely used MCDM approach • Realistic Satisficing Philosophy • Variant used: (Tamiz et al., 1995) • 64%, lexicographic • 21%, weighted • Rest, minmax • The variant chosen critically affects the final solution.

  3. (t1, t2) Feasible set Introduction Weighted Minmax Lexicographic

  4. Analytic Framework • where: • all functions gj are convex, • all functions fi are concave, • all goals derive from attributes “more is better”

  5. Weighted: • Minmax: • Lexicographic. Levels 1,..., l Analytic Framework Classical GP variants:

  6. Type 1. The percentage sum of unwanted deviation variables cannot surpass a certain bound Q1. • Type 2. The maximum percentage deviation cannot surpass a certain bound Q2. Q1 Meta-Satisfying Set Q2 Feasible set Analytic Framework Meta-Goal type 2 Meta-Goal type 1 META-GOALS

  7. Type 3. The percentage of unachieved goals cannot surpass a certain bound Q3 Analytic Framework META-GOALS

  8. Type 1 goal on a set • Type 3 goal on a set • Type 2 goal on a set Analytic Framework

  9. Analytic Framework • General Formulation of the Meta-Goal • Programming model, with: • m1 type 1 meta-goals, • m2 type 2 meta-goals, • m3 type 3 meta-goals

  10. Environmental Impact Gross margin-break-even point g5: x1 + n5 - p5 = 300 Unwanted deviation variables Production capacities g6: x2 + n6 - p6 = 200 Employment Gross margin Illustrative Example Hypothetical Production Planning Problem g1: x1 + 2x2 + n1 - p1 = 300 g2: 100x1 + 300x2 + n2 - p2 = 50000 g3: 100x1 + 300x2 + n3 - p3 = 90000 g4: x1 + x2 + n4 - p4 = 500

  11. Illustrative Example min f ( p1, n2, n3, n4, p5, p6) • Lexicographic Variant • Level 1: Goals 2, 5 and 6 • Level 2: Goals 1, 3 and 4

  12. Illustrative Example SOLUTION: • Decision Variables: • x1 = 300; x2 = 66,66 • Level 1: • n2 = 0; p5 = 0; p6 = 0 • Level 2: • p1 = 133,33; n3 = 40000; n4 = 133,33

  13. Illustrative Example • D.M. says: • With respect to the number of satisfied goals: • - Satisfy goals 2, 5 and 6; • - Maximize the number of satisfied goals among 1, 3 and 4; • Aggregate percentage deviation of not more than a 100% in the second level; • Maximum percentage disagreement of not more than a 75% in the second level

  14. Illustrative Example INFEASIBLE

  15. Illustrative Example (GP)2 Model

  16. Illustrative Example SOLUTION: • Level 1: ( n2 = p5 = p6 = 0 ) • 1 = 0; • Level 2: ( p1 = 400; n3 = n4 = 0 ) • - 2 = 1/3; (1 unsatisfied goal) • - 3 = 0.33; (133% aggregate disagreement) • - 4 = 0.58; (133% maximum disagreement)

  17. Using a mixture of variants instead of a single one • Target values for several achievement functions META-GP MODEL Conclusions • Choosing a single GP variant can be a too mechanistic way of incorporating the DM’s preferences into the model

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