MAT 119
FINITE MATHEMATICS NOTES
PART 1 – LINEAR ALGEBRA CHAPTER 4
LINEAR PROGRAMMING:SIMPLEX METHOD
4.1 The Simplex Tableau: Pivoting
Standard form of a Maximum Linear Programming Problem 1. All variables are nonnegative.
2. All other constraints are written as a linear expression that is less than or equal to a positive constant.
Initial simplex tableau – from augmented matrix
objective row –
BV (basic variables) –
RHS (right hand side) -
For a maximum problem is standard form:
1. The constraints are changed from inequalities to equations by the introduction of additional variables – one for each constraint and all nonnegative – slack variables 2. These equations and the objective function are placed in the initial simplex
tableau
Pivoting – to pivot a matrix about a given element, called the pivot element, is to apply certain row operations so that the pivot element is replaced by a 1 and all other entries in the same column, the pivot column, becomes 0’s.
Steps for pivoting
1. In the pivot row (where the pivot element appears), divide each entry by the pivot element ( 0). Pivot element becomes 1.
2. Obtain 0’s elsewhere in the pivot column by performing row operations involving the pivot row.
Row operations used in pivoting
1. Replace the pivot row by a positive multiple of that same row.
2. Replace a row by the sum of that row and a multiple of the pivot row.
DOUGLAS A. WILLIAMS, ARIZONA STATE UNIVERSITY, DEPARTMENT OF MATHEMATICS 2
Analyzing a tableau
To obtain the current values of the objective function and the basic variables in a tableau, follow
1. From the tableau write the equations corresponding to each row.
2. Solve the bottom equation for P and the remaining equations for the basic variables.
3. Set each nonbasic variable equal to 0 to obtain the current values of P and the
basic variables.
4.2 The Simplex Method: Solving Maximum Problems in Standard Form
Steps
1. Begin with the initial simplex tableau of a maximum problem in standard form.
2.
3.
)
4.
5.
6. The pivot element is the entry in the pivot row and pivot column. (can never be in the objective row. Pivot and repeat the process from step 2 until a stop is
obtained.
DOUGLAS A. WILLIAMS, ARIZONA STATE UNIVERSITY, DEPARTMENT OF MATHEMATICS 4
If it is nonnegative, STOP.
A solution has been found.
If it is negative, the pivot column is found. (if tie, choose any one Identify the smallest
entry in the objective row
If all in the pivot column are zero or negative, STOP. The problem is unbounded and has no solution.
For each possible entry above the objective row in the pivot column, compute the value of the entry in the far right hand column (RHS) divided by the corresponding positive entry of the pivot column.
The pivot row is the one having the smallest
nonnegative quotient. (if tie
choose any one)
The choice of pivot column forces us to pivot the variable that apparently improves the value of the objective function most effectively.
The choice of the pivot row prevents us from making this variable too large to be
feasible.
4.3 Solving Minimum Problems in Standard Form; The Duality Principle
Standard Form of a Minimum Linear Programming Problem Conditions
1. All the variables are nonnegative.
2. All other constraints are written as linear expressions that are greater than or equal to a constant.
3. The objective function must be expressed as a linear expression with nonnegative coefficients.
Von Neumann Duality Principle
Suppose a minimum linear programming problem is standard form has a solution. The minimum value of the objective function of the minimum problem is standard form equals the maximum value of the objective function of the dual problem, a maximum problem in standard form.
Steps for obtaining the Dual Problem
1. Write the minimum problem in standard form.
2. Construct a matrix that represents the constraints and the objective function.
3. Interchange the rows and columns to form the matrix of the dual problem.
4. Translate this matrix into a maximum problem in standard form.
DOUGLAS A. WILLIAMS, ARIZONA STATE UNIVERSITY, DEPARTMENT OF MATHEMATICS 6
Solving a Minimum Problem in Standard Form using the Dual Problem 1. Write the dual (maximum) problem.
2. Solve this maximum problem by the simplex method.
3. The minimum value of the objective function (C) will appear in the lower right
corner of the final tableau; it is equal to the maximum value of the dual objective
function (P). The values of the variables that give ride to the minimum value are
located in the objective row in the slack variable columns.
4.4 The Simplex Method with Mixed Constraints
Steps
1. Write each constraint, except the nonnegative constraints, as an inequality with the variables on the left side of a sign.
2. Introduce nonnegative slack variables on the left side of each inequality to form an equality.
3. Set up the initial simplex tableau.
Alternative Pivoting Strategies
4. Whenever negative entries occur in the right hand column RHS of the constraints equations, the pivot element is selected as follows:
Pivot Row:
Identify the negative entries on the RHS and their corresponding basic variables (BV).
Ignore the objective row. If any of the basic variables is an x-variable, choose the one with the smallest subscript. Otherwise choose the slack variable with the smallest subscript. The basic variable BV chosen identifies the pivot row. Because the objective row is ignored, it can never be the pivot row.
Pivot Column
Go from left to right along the pivot row until a negative entry is found. Ignore the RHS.
This entry identifies the pivot column and is the pivot element. If there are no negative entries in the pivot row except the one in the RHS column, then the problem has no solution.
5. Pivot
1. If, in the new tableau, negative entries appear on the RHS of the constraint equations, repeat step 4.
2. If, in the new tableau, only nonnegative entries appear on the RHS of the
constraint equations, then the tableau represents a maximum problem in standard form and the regular method outlined earlier should be used.
DOUGLAS A. WILLIAMS, ARIZONA STATE UNIVERSITY, DEPARTMENT OF MATHEMATICS 8
