A Total
Integer Model Example
The owner
of a machine shop is planning to expand by purchasing some new
machines –
presses and lathes. The owner has
estimated that each press purchased will increase profit by $100 per day and
each lathe will increase profit by $150 daily.
The number of machines the owner can purchase is limited by the cost of
the machines and the available floor space in the shop. The machine purchase prices and space requirements
are as follows.
Machine 
Required
Floor Space (ft^{2}) 
Purchase Price

Press Lathe 
15 30 
$8,000
4,000 
The owner
has a budget of $40,000 for purchasing machines and 200 square feet of
available floor space. The owner wants
to know how many of each type of machine to purchase in order to maximize the
daily increase in profit.
The linear
programming model for an integer programming problem is formulated in exactly
the same way as the linear programming examples we have already done. The only difference is that in this problem,
the decision variables are restricted to integer values because the owner
cannot purchase a fraction, or portion, of a machine. The linear programming model follows.
maximize
Z = $100x_{1} + 150x_{2}
subject
to
8,000x_{1}
+ 4,000x_{2} < $40,000
15x_{1} + 30x_{2} < 200 ft^{2}
^{ }x_{1}, x_{2} > 0 and
integer
where
x_{1}
= number of presses
x_{2} = number of lathes