An operations manager is trying to determine a production plan for the next week.

There are three products $($say$,$ P$,$ Q$,$ and R$)$ to produce using four machines $($say$,$ A and B$,$ C$,$ and

D$).$ Each of the four machines performs a unique process. There is one machine of each type, and

each machine is available for $2400$ minutes per week. The unit processing times for each

machine is given below:

Machine Product P Product Q Product R Availability

$($minutes$)$

A $2010102400$

B $1228162400$

C $156162400$

D $101502400$

Total processing

time $5759429600$

The unit revenues and maximum sales for the week are indicated in the following table: Item Product P Product Q Product R

Revenue per unit $$90$ $$100$ $$70$

Material cost per unit $$45$ $$40$ $$20$

Profit per unit $$45$ $$60$ $$50$

Maximum sales $1004060$

Storage from one week to the next is not permitted. The operating expenses associated with the

plant are $$6000$ per week, regardless of how many components and products are made. The

$$6000$ includes all expenses except for material costs. The operations management wants to find

the best product mix. Help him$/$her formulate the problem as a mathematical program.