Matrix Algebra Basics for Linear Programming: Optimization Techniques in Water Resources Management

1. Water Resources Development and ManagementOptimization(Linear Programming) CVEN 5393 Mar 7, 2011

2. Acknowledgements Dr. Yicheng Wang (Visiting Researcher, CADSWES during Fall 2009 – early Spring 2010) for slides from his Optimization course during Fall 2009 Introduction to Operations Research by Hillier and Lieberman, McGraw Hill

3. Today’s Lecture • Simplex Method • Recap of algebraic form • Simplex Method in Tabular form • Simplex Method for other forms • Equality Constraints • Minimization Problems (Big M and Twophase methods) • Sensitivity / Shadow Prices • Simplex Method in Matrix form • Basics of Matrix Algebra • Revised Simplex • Dual Simplex • R-resources / demonstration

4. Simplex Method (MAtrix Form)Matrix Algebra BasicsRevised Simplex

5. Matrices and Matrix Operations A matrix is a rectangular array of numbers. For example is a 3x2 matrix (matrices are denoted by Boldface Capital Letters) In more general terms,

6. Matrix Operations Let and be two matrices having the same number of rows and the same number of columns. To multiply a matrix by a number (denote this number by k ) To add two matrices A and B For example,

7. Subtraction of two matrices Matrix multiplication

8. Matrix division is not defined

9. Matrix operations satisfy the following laws. The relative sizes of these matrices are such that the indicated operations are defined. Transpose operations This operation involves nothing more than interchanging the rows and columns of the matrix.

10. Special kinds of matrices Identity matrix The identity matrix I is a square matrix whose elements are 0s except for 1s along the main diagonal. ( A square matrix is one in which the number of rows is equal to the number of columns). where I is assigned the appropriate number of rows and columns in each case for the multiplication operation to be defined.

11. Null matrix The null matrix 0 is a matrix of any size whose elements are all 0s .

12. Inverse of matrix

13. Vectors (We use boldface lowercase letters to represent vectors)

14. Partitioning of matrices Up to this point, matrices have been rectangular arrays of elements, each of which is a number. However, the notation and results are also valid if each element is itself a matrix. For example, the matrix

16. Example : Calculate AB, given

17. Matrix Form of Linear Programming Original Form of the Model Augmented Form of the Model

19. Solving for a Basic Feasible Solution For initialization, For any iteration,

20. Solving for a Basic Feasible Solution For initialization, xB = x = I-1b = b Z = cB I-1b = cB b For any iteration, Z = cB xB + cNxN = cB xB = cB B-1b BxB + NxN = b BxB = b - NxN xB = B-1b – B-1NxN xB = B-1b xB = B-1b Z = cB B-1b

21. Example xB = x = I-1b = b Z = cB I-1b = cB b xB = B-1b Z = cB B-1b

22. Matrix Form of the Set of Equations in the Simplex Tableau For the original set of equations, the matrix form is xB = B-1b Z = cB B-1b For any iteration,

23. For Iteration 2 Example

24. Summary of the Revised Simplex Method 1. Initialization (Iteration 0) Optimality test:

25. 2. Iteration 1 Step 1: Determine the entering basic variable So x2 is chosen to be the entering variable. Step 2: Determine the leaving basic variable So the number of the pivot row r =2 Thus, x4 is chosen to be the entering variable.

26. Step 3: Determine the new BF solution The new set of basic variables is To obtain the new B-1, So the new B-1 is

27. Optimality test: The nonbasic variables are x1 and x4. 3. Iteration 2 Step 1: Determine the entering basic variable x1 is chosen to be the entering variable.

28. Step 2: Determine the leaving basic variable The ratio 4/1 > 6/3 indicate that x5 is the leaving basic variable Step 3: Determine the new BF solution The new set of basic variables is Therefore, the new B-1 is Optimality test: The nonbasic variables are x4 and x5.

29. Relationship between the initial and final simplex tableaux

30. For iteration 1: y* = S* = B-1 =

31. For iteration 2: y* = S* = B-1 =

32. Dual Simplex

33. Hughes-McMakee-notes\chapter-05.pdf

34. Duality Theory and Sensitivity Analysis Duality Theory

36. is the surplus variable for the functional constraints in the dual problem.

37. If a solution for the primal problem and its corresponding solution for the dual problem are both feasible, the value of the objective function is optimal. If a solution for the primal problem is feasible and the value of the objective function is not optimal (for this example, not maximum), the corresponding dual solution is infeasible.

38. Summary of Primal-Dual Relationships

39. Summary of Primal-Dual Relationships

42. Primal Problem H G Dual Problem F B C D A E F D A E B C G H

44. Neither feasible nor superoptimal Neither feasible nor superoptimal Suboptimal Superptimal Optimal Optimal Superoptimal Suboptimal

46. Adapting to Other Primal Forms

49. subject to subject to (1) (2) : dual variables corresponding to (1) : dual variables corresponding to (2) subject to : unconstrained in sign

50. subject to subject to : dual variables corresponding to (2) subject to