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

3. Big M Method & Two-Phase Method

5. Fig. 4.3 Equality constraint

6. The Big M Method

7. Fig. 4.4 Sequence of CPF solutions

8. Nonzero coefficient of x5 in the objective function row.
We should make it zero.

21. Surplus Variable
Slack Variable

22. Fig. 4.5 CP Solutions

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

26. Two-Phase Method The Big M method can be thought as having two stages:
Drive all artificial variables to the value of zero (because of the large penalty, M)
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.

27. Two-Phase Method

28. Two-Phase Method

29. Two-Phase Method for Radiation Therapy Example

30. Two-Phase Method for Radiation Therapy Example

34. Fig. 4.7

37. Post- Optimality Analysis

44. 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?

46. 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.

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

52. Sensitivity Analysis for ci

55. EXCEL SENSITIVITY REPORT

56. LINDO SENSITIVITY REPORT

57. LINDO SENSITIVITY REPORT (cont.)