Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Linear Programming: An Algebraic Approach 4 The Simplex Method with Standard Maximization Problems Standard Minimization Problems Nonstandard Problems (Optional)

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. 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. The iteration stops when an optimal solution (if it exists) is found. 4.1 The Simplex Method: Standard Maximization Problem

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. A Standard (maximization) Linear Programming Problem: 1.The objective function is to be maximized. 2.All the variables involved in the problem are nonnegative. 3.Each constraint may be written so that the expression with the variables is less than or equal to a nonnegative constant. Standard Maximization Problem

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Maximize P = 4x + 5y Subject to A standard maximization problem: First introduce nonnegative slack variables to make equations out of the inequalities: Next, rewrite the objective function so that all the variables are on the left and the coefficient of P is 1. –4x – 5y + P = 0... Example

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Form the system: Write as a tableau: Nonbasic variables Basic variables Variables in non-unit columns are given a value of zero, so initially x = y = 0....

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Choose a pivot: 1.Select column: select most negative entry in the last row (to left of vertical line). 2.Select row: select smallest ratio: constant/positive entry (using only entries from selected column) Ratios: Next using the pivot, create a unit column...

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.... Ratios : Repeat steps

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc....

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. All entries in the last row are nonnegative therefore an optimal solution has been reached: Assign 0 to the nonbasic variables u and v. (Notice that columns u and v are not in unit form.) We have x = 5, y = 1, and P = 25 (the max).

The Simplex Method for Maximization Problems 1.Convert the constraints to equalities by adding slack variables 2.Rewrite the objective function 3.Construct the tableau 4.Check for completion a.If all entries in the last row are non-negative then an optimal solution is found 5.Pivot a.Select the column with the largest negative entry. b.Select the row with the smallest ratio of constant to entry c.Make the selected column a unit column using row operations 6.Go to step 4

Using the TI-83 Calculator The PIVOT program Enter the tableau into matrix A Run the PIVOT program –Asks to pivot or quit –Select pivot –Asks for row and column –Enter pivot row and column –Continue until an optimal solution is found

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. 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).

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. 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 = – The Simplex Method: Standard Minimization Problems

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. A Standard (minimization) Linear Programming Problem: 1.The objective function is to be minimized. 2.All the variables involved in the problem are nonnegative. 3.Each constraint may be written so that the expression with the variables is greater than or equal to a nonnegative constant. Standard Minimization Problems

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. 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. The Dual Problem 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.

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Example Minimize C = 10x + 11y Subject to Write down the primal information:...

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. 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...

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Create the initial simplex tableau adding slack variables x and y. pivot...

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc....

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.... This is the final tableau with x = 15, y = 5, and P = C = 205.