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Part 3. Linear Programming

Part 3. Linear Programming. 3.2 Algorithm. General Formulation. Convex function. Convex region. Example. Profit. Amount of product p. Amount of crude c. Graphical Solution. Degenerate Problems. Non-unique solutions. Unbounded minimum. Degenerate Problems – No feasible region.

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Part 3. Linear Programming

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  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 ProcedureStep 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 (particular solution). • A basic feasible solution is a basic solution in which the values of the basic 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

  17. 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.

  18. Implication of Fundamental Theorem

  19. Extreme Point

  20. Theorem (Equivalence of extreme points and basic solutions)

  21. 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.

  22. Step 2 x1 and x2 are selected as non-basic variables

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

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

  25. Step 4: Transformation of the Equations

  26. =0

  27. Repeat step 4 by Gauss-Jordan elimination

  28. 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

  29. Basic variables

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

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

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

  33. 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.

  34. New Basic Variables

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

  36. Step 3 – Step 5

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