Part 3 Linear Programming

1. Part 3 Linear Programming 3.3 Theoretical Analysis

2. Matrix Form of the Linear Programming Problem

3. LP Solution in Matrix Form

4. Tableau in Matrix Form

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 jthere 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 If for some basic feasible solution cj-fj or rj is larger than or equal to zero for all j, then the solution is optimal.

10. Symmetric Form of Duality (1)

11. Symmetric Form of Duality (2) • MAX in primal; MIN in dual. • <= in constraints of primal; >= in constraints of dual. • Number of constraints in primal = Number of variable in dual • Number of variables in primal = Number of constraints in dual • Coefficients of x in objective function = RHS of constraints in dual • RHS of the constraints in primal = Coefficients of y in dual • f(xopt)=g(yopt)

12. Symmetric Form of Duality (3)

13. Example Batch Reactor B Batch Reactor C Batch Reactor A Products P1, P2, P3, P4 Raw materials R1, R2, R3, R4 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. Additional Insights

22. Symmetric Form of Duality (3)

23. LP Solution in Matrix Form

24. Relations associated with the Optimal Feasible Solution of the Primal problem

25. Example PRIMAL DUAL

27. Tableau in Matrix Form

29. Example: The Primal Diet Problem How can we determine the most economical diet that satisfies the basic minimum nutritional requirements for good health? We assume that there are available at the marketn different foods that the ith food sells at a price ci per unit. In addition, there are m basic nutritional ingredients and, to achieve a balanced diet, each individual must receive at least bj unit of the jth nutrient per day. Finally, we assume that each unit of food icontains aji units of the jth nutrient.

30. Primal Formulation

31. The Dual Diet Problem Imagine a pharmaceutical company that produces in pill form each of the nutrients considered important by the dietician. The pharmaceutical company tries to convince the dietician to buy pills, and thereby supplies the nutrients directly rather than through purchase of various food. The problem faced by the drug company is that of determining positive unit pricesy1, y2, …, ym for the nutrients so as to maximize the revenue while at the same time being competitive with real food. To be competitive with the real food, the cost a unit of food made synthetically from pure nutrients bought from the druggist must be no greater than ci, the market price of the food, i.e. y1 a1i + y2 a2i + … + ym ami <= ci.

32. Dual Formulation

33. Shadow Prices How does the minimum cost change if we change the right hand side b? If the changes are small, then the corner which was optimal remains optimal. 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.