1. Operations Research Assistant Professor Dr. Sana’a Wafa Al-Sayegh 2 nd Semester 2008-2009 ITGD4207 University of Palestine

2. ITGD4207 Operations Research Chapter 4 Linear Programming Duality and Sensitivity analysis

3. Linear Programming Duality and Sensitivity analysis Dual Problem of an LPP Definition of the dual problem Examples (1,2 and 3) Sensitivity Analysis Examples (1 and 2)

4. Given a LPP (called the primal problem), we shall associate another LPP called the dual problem of the original (primal) problem. We shall see that the Optimal values of the primal and dual are the same provided both have finite feasible solutions. This topic is further used to develop another method of solving LPPs and is also used in the sensitivity (or post-optimal) analysis. Dual Problem of an LPP

5. Definition of the dual problem Given the primal problem (in standard form) Maximize subject to

6. the dual problem is the LPP Minimize subject to

7. If the primal problem (in standard form) is Minimize subject to

8. Then the dual problem is the LPP Maximize subject to

9. 1. In the dual, there are as many (decision) variables as there are constraints in the primal. We usually say y i is the dual variable associated with the ith constraint of the primal. 2. There are as many constraints in the dual as there are variables in the primal. We thus note the following:

10. 3. If the primal is maximization then the dual is minimization and all constraints are  If the primal is minimization then the dual is maximization and all constraints are  4.In the primal, all variables are  0 while in the dual all the variables are unrestricted in sign.

11. 5. The objective function coefficients c j of the primal are the RHS constants of the dual constraints. 6. The RHS constants b i of the primal constraints are the objective function coefficients of the dual. 7. The coefficient matrix of the constraints of the dual is the transpose of the coefficient matrix of the constraints of the primal.

12. Example1 Write the dual of the LPP subject to Maximize

13. Thus the primal in the standard form is: subject to Maximize

14. Hence the dual is: subject to Minimize

15. Example2 Write the dual of the LPP subject to Minimize

16. Thus the primal in the standard form is: subject to Minimize

17. Hence the dual is: Maximize subject to

18. Example3 Write the dual of the LPP subject to Maximize

19. subject to Thus the primal in the standard form is:

20. Minimize subject to Hence the dual is:

21. Theorem: The dual of the dual is the primal (original problem). Proof. Consider the primal problem (in standard form) Maximize subject to

22. Sensitivity Analysis Sensitivity analysis is the study of how changes –in the coefficients of a linear program affect the optimal solution or –in the value of right hand sides of the problem affect the optimal solution

23. Sensitivity Analysis cont’d Using sensitivity analysis we can answer questions such as: 1.How will a change in a coefficient of the objective function affect the optimal solution? –We can define a range of optimality for each objective function coefficient by changing the objective function coefficients, one at a time

24. Sensitivity Analysis cont’d 2.How will a change in the right-hand-side value for a constraint affect the optimal solution? The feasible region may change when RHSs (one at a time) are changed and perhaps cause a change in the optimal solution to the problem

25. Sensitivity Analysis cont’d 3.How much value is added/reduced to the objective function if I have a larger/smaller quantity of a scarce resource? Sensitivity analysis is important to the manager who must operate in a dynamic environment with imprecise estimates of the coefficients

26. Example 1 LP Formulation Max 5x 1 + 7x 2 s.t. x 1 < 6 2x 1 + 3x 2 < 19 x 1 + x 2 < 8 x 1, x 2 > 0

27. Example 1 Graphical Solution 87654321 1 2 3 4 5 6 7 8 9 10 2 x 1 + 3 x 2 < 19 x 2 x 2 x1x1x1x1 x 1 + x 2 < 8 Max 5 x 1 + 7x 2 x 1 < 6 Optimal: x 1 = 5, x 2 = 3, z = 46

28. Objective Function Coefficients Let us consider how changes in the objective function coefficients might affect the optimal solution. The range of optimality for each coefficient provides the range of values over which the current solution will remain optimal. Managers should focus on those objective coefficients that have a narrow range of optimality and coefficients near the endpoints of the range.

29. Example 1 Changing Slope of Objective Function 87654321 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 x1x1x1x1 FeasibleRegion 1122 33 44 55 x 2 x 2

30. Range of Optimality Graphically, the limits of a range of optimality are found by changing the slope of the objective function line within the limits of the slopes of the binding constraint lines. The slope of an objective function line, Max c 1 x 1 + c 2 x 2, is -c 1 /c 2, and the slope of a constraint, a 1 x 1 + a 2 x 2 = b, is -a 1 /a 2.

31. Right-Hand Sides Let us consider how a change in the right-hand side for a constraint might affect the feasible region and perhaps cause a change in the optimal solution. The improvement in the value of the optimal solution per unit increase in the right-hand side is called the dual price. The range of feasibility is the range over which the dual price is applicable. As the RHS increases, other constraints will become binding and limit the change in the value of the objective function.

32. Example 2 Consider the following linear program: Min 6x 1 + 9x 2 ($ cost) s.t. x 1 + 2x 2 < 8 10x 1 + 7.5x 2 > 30 x 2 > 2 x 1, x 2 > 0

33. Example 2 Optimal Solution According to the output: x 1 = 1.5 x 2 = 2.0 Objective function value = 27.00

34. Example 2 Range of Optimality Question Suppose the unit cost of x 1 is decreased to $4. Is the current solution still optimal? What is the value of the objective function when this unit cost is decreased to $4?

35. Example 2 Range of Optimality Answer The output states that the solution remains optimal as long as the objective function coefficient of x 1 is between 0 and 12. Because 4 is within this range, the optimal solution will not change. However, the optimal total cost will be affected: 6x 1 + 9x 2 = 4(1.5) + 9(2.0) = $24.00.

36. Example 2 Range of Optimality Question How much can the unit cost of x 2 be decreased without concern for the optimal solution changing?

37. Example 2 Range of Optimality Answer The output states that the solution remains optimal as long as the objective function coefficient of x 2 does not fall below 4.5.

38. Example 2 Range of Feasibility Answer A dual price represents the improvement in the objective function value per unit increase in the right- hand side. A negative dual price indicates a negative improvement in the objective, which in this problem means an increase in total cost because we're minimizing. Since the right-hand side remains within the range of feasibility, there is no change in the optimal solution. However, the objective function value increases by $4.50.