Linear Programming: Simplex Method and Extreme Point Theorem

1. Part 3. Linear Programming 3.2 Algorithm

2. General Formulation Convex function Convex region

3. Example: Production Planning for a Refinery

4. Profit Amount of product p Amount of crude c

5. Graphical Solution

6. Degenerate Problems Non-unique solutions Unbounded minimum

7. Degenerate Problems – No feasible region

8. Simplex Method – The standard formulation

9. Simplex Method - Handling inequalities

10. Simplex Method - Handling unrestricted variables

11. Simplex Method- Calculation procedure

12. Calculation Procedure- Step 0

13. Calculation Procedure - Step 1

14. Calculation ProcedureStep 2:find a basic feasible solution corresponding to a corner of the feasible region.

15. Remarks • The solution obtained from a cannonical system by setting the non-basic variables to zero is called a basic solution (i.e., a particular solution). • A basic feasible solutionis a basic solution in which the values of the basic variables are nonnegative. • Every corner pointof the feasible region corresponds to a basic feasible solution of the constraint equations. Thus, the optimum solution is a basic feasible solution.

17. Full Rank Assumption

18. Fundamental Theorem of Linear Programming Given a linear program in standard form where A is an m by n matrix of rank m. • If there is a feasible solution, then there must be a basic feasible solution; • If there is an optimal solution, then there must be an optimal basic feasible solution.

19. Implication of Fundamental Theorem

20. Extreme Point

21. Theorem (Equivalence of extreme points and basic feasible solutions)

22. Corollary If there is a finite optimal solution to a linear programming problem, there is a finite optimal solution which is an extreme point of the constraint set.

23. Step 2 x1=0 and x2=0 are selected as initial non-basic (free) variables

24. Step 3: Select new basic and new non-basic (free) variables new basic variable

25. Which one of the basic variables (x3, x4, x5) should be selected as the new non-basic or free variables? (Note: x2=0)

26. Step 4: Transformation of the Linear Equations

27. =0

28. Carry out step 3 and step 4 in matrix form by Gauss-Jordan elimination

29. N N B OBJ B B Step 3.2: Pivot Row Select the smallest positive ratio bi/ai1 Identity Matrix Step 3.1: Pivot Column Select the largest positive element in the objective function. Pivot element

30. OBJ Basic variables

31. Step 5: Repeat Iteration An increase in x4 or x5 does not reduce f Corner C!

32. Simplex Method- Calculation procedure

33. Corner D!

34. It is still necessary to guarantee obtaining the first basic feasible solution in any LP problem! Infeasible!

35. Phase I – Phase II Algorithm • Phase I: generate an initial basic feasible solution; • Phase II: generate the optimal basic feasible solution.

36. Phase-I Procedure • Step 0 and Step1 are the same as before. • Step 2: Augment the set of equations by one artificial variable for each equation to get a new standard form.

37. New Basic Variables Artificial Variables Artificial Variables

38. New Objective Function If the minimum of this objective function is reached, then all the artificial variables should be reduced to 0.

39. Step 3 – Step 5