1. Part 3. Linear Programming
3.2 Algorithm

2. General Formulation
Convex function
Convex
region

3. Example

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 form

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 Procedure Step 2: find a basic 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.
A basic feasible solution is a basic solution in which the values of the basi variables are nonnegative.
Every corner point of the feasible region corresponds to a basic feasible solution of the constraint equations. Thus, the optimum solution is a basic feasible solution.

16. Full Rank Assumption

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

19. Implication of Fundamental Theorem

20. Extreme Point

21. Theorem (Equivalence of extreme points and basic 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 and x2 are selected as
non-basic variables

24. Step 3: select new basic and non-basic variables
new basic variable

25. Which one of x3, x4, x5 should be selected as the new non-basic variables?

26. Step 4: Transformation of the Equations

27. =0

28. Repeat step 4 by Gauss-Jordan elimination

29. N N B B B
Step 3: Pivot Row
Select the smallest positive ratio
bi/ai1
Step 3: Pivot Column
Select the largest positive element in the objective function.
Pivot element

30. Basic variables

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

33. It is necessary to obtain a first feasible solution!
Infeasible!

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

35. Phase-I Procedure Step 0 and Step 1 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.

36. New Basic Variables

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

38. Step 3 – Step 5