# Linear Programming : Simplex Method, pivoting and maximizing

1. Consider the following linear programming problem:

Maximize 10x + 7y subject to:

X + 3y (less than or equal to symbol) 10

2x -y (less than or equal to symbol) 8

x (greater than or equal to symbol) 0, y (greater than or equal to symbol) 0

The initial simplex tableau is:

(for choices, please see attachment)

2. The particular solution corresponding to the simplex tableau (see attachment) is:

a. x = 2, y = 6, u = 10, v = 0, M = 0

b. x = 2, y = 6, u = 0, v = 0, M = 10

c. x = 6, y = 0, u = 2, v = 0, M =10

d. x = 6, y = 0, u = 6, v = 0, M = 10

e. none of the above

3. The result of pivoting the simplex tableau (see attachment) about -2 (1st row, 1st column) is:

(for choices, please see attachment)

4. Consider the simplex tableau (see attachment). The tableau is the final one in a problem to maximize x + 2y + 3z.

The maximum value of x + 2y + 3z is:

a. -19

b. -12

c. 0

d. 19

e. none of the above

5. In the following simplex tableau, the next pivot element is (see attachment for tableau):

a. 4 in the second row, first column

b. -3 in the third row, first column

c. 8 in the second row, fourth column

d. 1 in the first row, second column

e. none of the above

6. Consider the following linear programming problem:

Maximize 3x -y subject to:

6x -5y (less than or equal to symbol) 24

x + 9y (greater than or equal to symbol) 63

x (greater than or equal to symbol) 0

y (greater than or equal to symbol) 0

Which of the following is the initial simplex tableau?

(for choices, please see attachment)

https://brainmass.com/math/linear-programming/linear-programming-simplex-method-pivoting-maximizing-35365

#### Solution Summary

Linear programming problems are solved. Pivoting and maximizing values are investigated.

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.

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