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Linear Programming : Simplex Method

Please see the attached file for the fully formatted problems.

1. Consider the following maximum problem in standard form:

Maximize Z = 8X1 + 2X2 + 3X3

subject to the constraints

X1 + 3X2 + 2X3 < 10
4X1 + 2X2 + 3X3 < 8

X1 > 0, X2 > 0, X3 > 0

(a) Rewrite the two constraints as equations by adding slack variables S1 and S2.
(b) Set up the initial simplex tableau for this problem.

In problems 2 -4, each tableaux represents a step in the solution of a maximization problem in standard form. Determine if the tableaux:
(i) is the final tableaux
(ii) requires additional pivoting
(iii) indicates no solution to the problem
If the answer is (i), write down the solution; if the answer is (ii), identify the pivot element.

2.

3.

4.

5. A brewery manufactures three types of beer - lite, regular, and dark. Each vat of lite beer requires 6 bags of barley, 1 bag of sugar and 1 bag of hops. Each vat of regular beer requires 4 bags of barley, 3 bag of sugar and 1 bag of hops. Each vat of dark beer requires 2 bags of barley, 2 bag of sugar and 4 bag of hops. Each day the brewery has 800 bags of barley, 600 bag of sugar and 300 bag of hops. The brewery realizes a profit of \$10 per vat of lite beer, \$20 per vat of regular beer, and \$30 per vat of dark beer. For this linear programming problem:

(a) What are the decision variables?
(b) What is the objective function?
(c) What are the constraints?

E.C. Solve the linear program in problem #5.

Solution Preview

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Show your work.

1. Consider the following maximum problem in standard form:

Maximize Z = 8X1 + 2X2 + 3X3

subject to the constraints

X1 + 3X2 + 2X3 < 10
4X1 + 2X2 + 3X3 < 8

X1 > 0, X2 > 0, X3 > 0

(a) Rewrite the two constraints as equations by adding slack variables S1 and S2.

X1 + 3X2 + 2X3 +S1 = 10
4X1 + 2X2 + 3X3 +S2 = 8

(b) Set up the initial simplex tableau for this problem.

|-----------------------------------------|-----
| -8 -2 -3 0 0 1 | 0

In problems 2 -4, each tableaux represents a step in the solution of a maximization problem in standard form. Determine if the tableaux:
(i) is the final tableaux
(ii) requires additional pivoting
(iii) indicates no solution to the problem
If the answer ...

Solution Summary

LP problems are solved using the simplex method.

\$2.19