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Two-Phase Method for Radiation Therapy

This chapter explores the Two-Phase Method for solving linear programming problems, specifically in the context of radiation therapy. It discusses the Big M Method, equality constraints, surplus and slack variables, feasible regions, and post-optimality analysis.

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Two-Phase Method for Radiation Therapy

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  1. ISE 203 OR I Chapter 4 Solving Linear Programming Problems: Continued Asst. Prof. Dr. Nergiz Kasımbeyli

  2. Big M Method&Two-Phase Method

  3. Fig. 4.3 Equality constraint

  4. TheBig M Method

  5. Fig. 4.4 Sequence of CPF solutions

  6. Nonzerocoefficient of x5 in theobjectivefunctionrow. Weshouldmake it zero.

  7. SurplusVariable SlackVariable

  8. Fig. 4.5 CP Solutions

  9. Fig. 4.6 Feasible region and the sequence of operations

  10. Two-Phase Method • TheBig M method can be thought as havingtwostages: • Driveallartificialvariablestothevalue of zero (because of thelargepenalty, M) • Whilekeepingartificialvariables at theirzerovalues, findthe optimal solution. • Anothermethod (CalledtheTwo-PhaseMethod) doesthis in twophases, withoutintroducingpenalties.

  11. Two-Phase Method

  12. Two-Phase Method

  13. Two-Phase Method for Radiation Therapy Example

  14. Two-Phase Method for Radiation Therapy Example

  15. Fig. 4.7

  16. Post- Optimality Analysis

  17. QUESTION: • What happens when we increase b2 above 18? Will it differ to have b2 = 18 or b2 = 19? • What maximum amount would you be willing to pay for an extra unit of resource 2?

  18. There is a surplus of resource 1! • Therefore increasing b1 beyond 4 does not effect the optimal Z value. • The constraints on resources 2 and 3 are binding at the optimal solution. • Since the limited supply of these resources bind Z from being increased further, they have positive shadow prices. In such a case, the economists say Resources 2 and 3 are scarce resources, and Resource 1 is a free resource.

  19. QUESTION: What maximum amount would you be willing to pay for an extra unit of resource 2?

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