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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.

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