ISE 203 OR I Chapter 4 Solving Linear

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

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Big M Method & Two-Phase Method 3

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Fig. 4. 3 Equality constraint 5

• The Big M Method 6

Fig. 4. 4 Sequence of CPF solutions 7

Nonzero coefficient of x 5 in the objective function row. We should make it zero. 8

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Surplus Variable Slack Variable 21

Fig. 4. 5 CP Solutions 22

Fig. 4. 6 Feasible region and the sequence of operations 23

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Two-Phase Method The Big M method can be thought as having two stages: 1. Drive all artificial variables to the value of zero (because of the large penalty, M) 2. While keeping artificial variables at their zero values, find the optimal solution. Another method (Called the Two-Phase Method) does this in two phases, without introducing penalties. 26

Two-Phase Method 27

Two-Phase Method 28

Two-Phase Method for Radiation Therapy Example 29

Two-Phase Method for Radiation Therapy Example 30

Fig. 4. 7

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Post- Optimality Analysis

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QUESTION: 1. What happens when we increase b 2 above 18? Will it differ to have b 2 = 18 or b 2 = 19? 2. What maximum amount would you be willing to pay for an extra unit of resource 2?

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• There is a surplus of resource 1! • Therefore increasing b 1 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. 46

QUESTION: What maximum amount would you be willing to pay for an extra unit of resource 2? 47

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Sensitivity Analysis for ci

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EXCEL SENSITIVITY REPORT

LINDO SENSITIVITY REPORT

LINDO SENSITIVITY REPORT (cont. )