1. Part 3 Linear Programming 3.3 Theoretical Analysis

2. Matrix Form of the Linear Programming Problem

3. Feasible Solution in Matrix Form

4. Tableau in Matrix Form (without the objective column)

5. Criteria for Determining A Minimum Feasible Solution

8. Theorem (Improvement of Basic Feasible Solution) Given a non-degenerate basic feasible solution with corresponding objective function f0, suppose for some j there holds cj-fj<0. Then there is a feasible solution with objective value f<f0. If the column aj can be substituted for some vector in the original basis to yield a new basic feasible solution, this new solution will have f<f0. If aj cannot be substituted to yield a basic feasible solution, then the solution set K is unbounded and the objective function can be made arbitrarily small (negative) toward minus infinity.

9. Optimality Condition for a Minimum!

10. Symmetric Form of Duality (1)

11. Symmetric Form of Duality (2)

12. Alternative Form of Duality

13. Example Batch Reactor A Batch Reactor B Batch Reactor C Raw materials R1, R2, R3, R4 Products P1, P2, P3, P4 P1P2P3P4 capacity time A1.51.02.41.02000 B1.05.01.03.58000 C1.53.03.51.05000 profit /batch $5.24$7.30$8.34$4.18 time/batch

14. Example: Primal Problem

15. Example: Dual Problem

16. Property 1 For any feasible solution to the primal problem and any feasible solution to the dual problem, the value of the primal objective function being maximized is always equal to or less than the value of the dual objective function being minimized.

17. Proof

18. Property 2

19. Proof

20. Duality Theorem If either the primal or dual problem has a finite optimal solution, so does the other, and the corresponding values of objective functions are equal. If either problem has an unbounded objective, the other problem has no feasible solution.

21. Alternative Form of Duality

22. Additional Insights Shadow Prices!

23. Matrix Form of the Linear Programming Problem

24. Feasible Solution in Matrix Form

25. Tableau in Matrix Form (without the objective column!)

26. Relations associated with the Optimal Feasible Solution of the Primal (Minimization) Problem This is the optimality condition of the primal minimization problem! Property 2 is satisfied!

27. Example PRIMAL DUAL

29. Tableau in Matrix Form of Primal Problem

31. Example: The Primal Diet Problem

32. Primal Formulation

33. Alternative Form of Duality

34. The Dual Diet Problem

35. Dual Formulation

36. Shadow Prices How does the minimum cost in the primal problem change if we change the right hand side b (lower limits of nutrient j)? If the changes are small, then the corner which was optimal remains optimal, i.e. –The choice of basic variables does not change. –At the end of simplex method, the corresponding m columns of A make up the basis matrix B.