# Setting and solving linear programming question

A furniture manufacturer produces sofas, tables, and chairs. The profits per item are, respectively, $70, $120, and $80. The pieces of furniture require the following labor-hours for their manufacture.

Carpentry Upholstery Finishing

Sofas 3 5 1

Tables 8 0 2

Chairs 6 2 1

The following amounts of labor-hours are available each month: carpentry, at most 768 hours; upholstery, at most 216 hours; and finishing, at most 144 hours. You are to determine how many each of sofas, tables, and chairs should be manufactured to maximize profit. (Please show all work)

a) Identify your three variables

b.) Determine the objective function

c.) Determine the constraints

d.) Construct the initial simplex tableau

e.) Using the simplex method, determine the solution that maximizes profit. Be sure to present your solutions in terms of the amount of each furniture type and any unused type of labor-hours.

#### Solution Preview

a) Identify your three variables

Let x, y, and z be number of sofas, tables and chairs purchased respectively.

b.) Determine the objective function

We need to maximize P such that P=70x + 120y + 80z

c.) Determine the constraints

3x + 8y + 6z ≤ 768

5x + 0y + 2z ≤ 216

1x + 2y + 1z ≤ 144

x, y, z ≥ 0

d.) Construct the initial simplex tableau

Initial simplex tableau is:

...... x .... y ... z u v w P

[ ... 3 .... 8 ... 6 1 0 0 0 | 768 ]

[ ... 5 .... 0 ... 2 0 1 0 0 | 216 ]

[ ... 1 .... 2 ... 1 0 0 1 0 | 144 ]

[ −70 −120 −80 0 0 0 1 | 0 ]

where u, v and ...

#### Solution Summary

The solution gives detailed steps on setting and solving linear programming question using the example of furniture order. A simplex method is used in steps.