1. Water Resources Development and Management Optimization (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 Simplex Method for other forms
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 Basics Revised 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. To multiply a matrix by a number (denote this number by k )
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, 19

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
xB = B-1b
Z = cB B-1b
BxB + NxN = b
BxB = b - NxN
xB = B-1b – B-1NxN
xB = B-1b

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

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. Example
For Iteration 2

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

25. Step 1: Determine the entering basic variable
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. The nonbasic variables are x1 and x4.
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

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 C D B A E G
Dual Problem F A E F D B C G H

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

46. Adapting to Other Primal Forms

49. subject to 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

53. Dual Simplex Method Suboptimal Superptimal Optimal Optimal
Superoptimal
Suboptimal

54. Simplex Method : Keep the solution in any iteration suboptimal ( not satisfying the condition for optimality, but the condition for feasibility).
Dual Simplex Method : Keep the solution in any iteration superoptimal ( not satisfying the condition for feasibility, but the condition for optimality).
If a solution satisfies the condition for optimality, the coefficients in row (0) of the simplex tableau must nonnegative.
If a solution does not satisfy the condition for feasibility, one or more of the values of b in the right-side of simplex tableau must be negative.

55. Summary of Dual Simplex Method
subject to

56. Sensitivity Analysis

60. Case 1 : Changes in bi

62. Incremental analysis

63. The dual simplex method now can be applied to find the new optimal solution.

64. The allowable range of bi to stay feasibleC
The solution remains feasible only if

65. Case 2a : Changes in the coefficients of a nonbasic variable
Consider a particular variable xj (fixed j) that is a nonbaic variable in the optimal solution shown by the final simplex tableau.
is the vector of column j in the matrix A .
We have
for the revised model.
We can observe that the changes lead to a single revised constraint for the dual problem.

66. The allowable range of the coefficient ci of a nonbasic variable

67. Case 2b : Introduction of a new variable

68. Case 3 : Changes in the coefficients of a basic variable

70. The allowable range of the coefficient ci of a basic variable

71. Case 4 : Introduction of a new constraint

72. Parametric Linear Programming
Systematic Changes in the cj Parameters
The objective function of the ordinary linear programming model is
For the case where the parameters are being changed, the objective function of the ordinary linear programming model is replaced by
where q is a parameter and aj are given input constants representing the relative rates at which the coefficients are to be changed.

73. Example:
To illustrate the solution procedure, suppose a1=2 and a2 = -1 for the original Wyndor Glass Co. problem, so that
Beginning with the final simplex tableau for q = 0, we see that its Eq. (0)
If q ≠ 0, we have

74. Because both x1 and x2 are basic variables, they both need to be eliminated algebraically from Eq. (0)
The optimality test says that the current BF solution will remain optimal as long as these coefficients of the nonbasic variables remain nonnegative:

75. Summary of the Parametric Linear Programming Procedure for Systematic Changes the cj Parameters

76. Systematic Changes in the bi Parameters
For the case where the bi parameters change systematically, the one modification made in the original linear programming model is that is replaced by, for i = 1, 2, …, m, where the ai are given input constants. Thus the problem becomes
subject to
The goal is to identify the optimal solution as a function of q

77. Example:
subject to
Suppose that a1=2 and a2 = -1 so that the functional constraints become
This problem with q = 0 has already been solved in the table, so we begin with the final tableau given there.

78. Using the sensitivity procedure, we find that the entries in the right side column of the tableau change to the values given below
Therefore, the two basic variables in this tableau
Remain nonnegative for

79. Summary of the Parametric Linear Programming Procedure for Systematic Changes the bi Parameters