Water Resources Development and Management

Optimization

(Linear Programming)

CVEN 5393

Mar 4, 2011

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

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

SIMPLEX METHOD (MATRIX FORM)

MATRIX ALGEBRA BASICSREVISED SIMPLEX

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,

Matrix Operations

To multiply a matrix by a number (denote this number by k )

Let and be two matrices having the same number of rows and the same number of columns.

To add two matrices A and B

For example,

Subtraction of two matrices

Matrix multiplication

Matrix division is not defined

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.

Special kinds of matrices

Identity matrix

where I is assigned the appropriate number of rows and columns in each case for the multiplication operation to be defined.

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).

Null matrix

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

Inverse of matrix

Vectors

(We use boldface lowercase letters to represent vectors)

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

Example : Calculate AB, given

Matrix Form of Linear Programming

Original Form of the Model Augmented Form of the Model

Solving for a Basic Feasible SolutionFor initialization,

For any iteration,

Solving for a Basic Feasible Solution

For initialization,

BxB + NxN = bBxB = b - NxN xB = B-1b – B-1NxN

xB = B-1b

Z = cB xB + cNxN = cB xB = cB B-1b

xB = B-1bZ = cB B-1b

xB = x = I-1b = bZ = cB I-1b = cB b

For any iteration,

Example

xB = B-1bZ = cB B-1b

xB = x = I-1b = bZ = cB I-1b = cB b

Matrix Form of the Set of Equations in the Simplex TableauFor the original set of equations, the matrix form is

xB = B-1bZ = cB B-1b

For any iteration,

Example

For Iteration 2

Summary of the Revised Simplex Method

1. Initialization (Iteration 0)

Optimality test:

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

Thus, x4 is chosen to be the entering variable.

So the number of the pivot row r =2

The new set of basic variables is

To obtain the new B-1,

So the new B-1 is

Step 3: Determine the new BF solution

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.

Step 2: Determine the leaving basic variable

The ratio 4/1 > 6/3 indicate that x5 is the leaving basic variable

Therefore, the new B-1 is

The new set of basic variables is

Step 3: Determine the new BF solution

Optimality test:

The nonbasic variables are x4 and x5.

Relationship between the initial and final simplex tableaux

For iteration 1:

S* = B-1 = y* =

For iteration 2:

S* = B-1 = y* =

DUAL SIMPLEX

HUGHES-MCMAKEE-NOTES\CHAPTER-05.PDF

Duality Theory and Sensitivity Analysis

Duality Theory

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

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.

Summary of Primal-Dual Relationships

Summary of Primal-Dual Relationships

F

G

H

C

D

B

A E

C

DB

AE

G

F

H

Primal Problem

Dual Problem

Suboptimal

Superoptimal

Optimal

Superptimal

Suboptimal

Optimal

Neither feasible nor superoptimal

Neither feasible nor superoptimal

Adapting to Other Primal Forms

subject to subject to

(1)

(2)

: dual variables corresponding to (1)

: dual variables corresponding to (2)

subject to

: unconstrained in sign

subject to subject to

: dual variables corresponding to (2)

subject to

Dual Simplex Method

Suboptimal

Superoptimal

Optimal

Superptimal

Suboptimal

Optimal

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.

Summary of Dual Simplex Method

subject to

Sensitivity Analysis

Case 1 : Changes in bi

Incremental analysis

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

The allowable range of bi to stay feasible

The solution remains feasible only if

A

B

C

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.

The allowable range of the coefficient ci of a nonbasic variable

Case 2b : Introduction of a new variable

Case 3 : Changes in the coefficients of a basic variable

The allowable range of the coefficient ci of a basic variable

Case 4 : Introduction of a new constraint

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 is a parameter and j are given input constants representing the relative rates at which the coefficients are to be changed.

Example:To illustrate the solution procedure, suppose 1=2 and 2 = -1 for the original Wyndor Glass Co. problem, so that

Beginning with the final simplex tableau for = 0, we see that its Eq. (0)

If ≠ 0, we have

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:

Summary of the Parametric Linear Programming Procedure for Systematic Changes the cj

Parameters

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 i are given input constants. Thus the problem becomes

subject to

The goal is to identify the optimal solution as a function of

Example:

subject to

Suppose that 1=2 and 2 = -1 so that the functional constraints become

This problem with = 0 has already been solved in the table, so we begin with the final tableau given there.

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

Summary of the Parametric Linear Programming Procedure for Systematic Changes the bi

Parameters