1. Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc4
Linear Programming: An Algebraic Approach
The Simplex Method with
Standard Maximization Problems
Standard Minimization Problems
Nonstandard Problems
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The Simplex Method
The simplex method is an iterative process. Starting at some initial feasible solution (a corner point – usually the origin), each iteration moves to another corner point with an improved (or at least not worse) value of the objective function. Iteration stops when an optimal solution (if it exists) is found.
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A Standard (maximization) Linear Programming Problem:
The objective function is to be maximized.
All the variables involved in the problem are nonnegative.
Each constraint may be written so that the expression with the variables is less than or equal to a nonnegative constant.
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Ex. A standard maximization problem:
Maximize P = 4x + 5y
Subject to
First introduce nonnegative slack variables to make equations out of the inequalities:
Next, rewrite the objective function Set = 0 with a 1 on P:
–4x – 5y + P = 0
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Form the system:
Basic variables
Write as a tableau:
Nonbasic variables
Variables in non-unit columns are given a value of zero, so initially x = y = 0.
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Choose a pivot:
Ratios:
Select column: select most negative entry in the last row (to left of vertical line).
Select row: select smallest ratio: constant/entry (using only entries from selected column)
Next using the pivot, create a unit column
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Ratios:
Repeat steps
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All entries in the last row are nonnegative therefore an optimal solution has been reached:
Assign 0 to variables w/out unit columns (u, v).
Notice x = 1, y = 5, and P = 25 (the max).
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The Simplex Method:
Set up the initial simplex tableau.
If all entries in the last row are nonnegative then an optimal solution has been reached, go to step 4.
Perform the pivot operation: convert pivot to a 1, then use row operations to make a unit column. Return to step 2.
Determine the optimal solution(s). The value of the variable heading each unit column is given by the corresponding value in the column of constants. Variables heading the non-unit columns have value zero.
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Multiple Solutions
There are infinitely many solutions if and only if the last row to the right of the vertical line of the final simplex tableau has a zero in a non-unit column.
No Solution
A linear programming problem will have no solution if the simplex method breaks down (ex. if at some stage there are no nonnegative ratios for computation).
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Minimization with Constraints
Ex. Minimize C = –4 x – 5y
Subject to
Notice if we let P = –C = 4x + 5y we have a standard maximization problem.
If we solve this associated problem we find P = 25, therefore the minimum is C = –25.
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A Standard (minimization) Linear Programming Problem:
The objective function is to be minimized.
All the variables involved in the problem are nonnegative.
Each constraint may be written so that the expression with the variables is greater than or equal to a nonnegative constant.
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The Dual Problem
Maximization problems can be associated with minimization problems (and vice versa). The original problem is called the Primal and the associated problem is called the Dual.
Theorem of Duality
A primal problem has a solution if and only if the dual has a solution.
Both objective functions attain the same optimal value.
The optimal solution of the primal appears under the slack variables in the last row of the final simplex tableau associated with the dual.
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Ex.
Minimize C = 10x + 11y
Subject to
Write down the primal information:
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Interchange the columns and rows and use variables u, v, w.
This can be represented by the problem:
Maximize P = 300u + 300v + 250w
Subject to
This is a standard maximization problem
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Create the initial simplex tableau adding slack variables x and y.
pivot
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This is the final tableau with x = 15, y = 5, and P = C = 205.
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Ex. A nonstandard maximization problem:
Maximize P = 8x + 3y
Subject to
First change the inequalities to less than or equal to.
Now proceed with the simplex method
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Introduce slack variables to make equations out of the inequalities and set the objective function = 0:
The initial tableau and notice v = –2 (not feasible):
We need to pivot to a feasible solution
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Ratios 8 2
Locate any negative number in the constant column ( –2). Now go to the first negative to the left of that constant (–1). This determines the pivot column. The pivot row is found by examining the positive ratios. So –1 is our pivot.
Create unit column
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New pivot since it is the only positive ratio
Note: now we have a feasible solution proceed with simplex
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This is the final tableau: x = 2, y = 4, P = 28
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The Simplex Method for Nonstandard Problems
If necessary, rewrite as a maximization problem.
If necessary, rewrite inequalities as less or equal to.
Introduce slack variables and write simplex tableau.
If no negative constants (upper column) use simplex method, otherwise go to step 5.
Pick a negative entry in a row with a negative constant (this is the pivot column). Compute positive ratios to determine pivot row. Then pivot and return to step 4.
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