Determining daily production rate and maximum profit

Question

Corporation A produces two types of products, product X and product Y. Three machines are required to produce each type, namely, MC1, MC2, and MC3. Each product X requires 5 hours in MC1, 3 hours in MC2 and 2 hours in MC3. On the other hand, product Y requires 4 hours in MC1, 2 hours in MC2 and 3 hours in MC3. Each day there are 24 hours available for each machines. The variable cost in producing product X is \$150 per unit and \$120 per unit for product Y. The selling price of product X is \$450 per unit and \$400 per unit for product Y. The company fixed cost is $30 000. Please help me determin The daily production rate and maximum profit.

Answer 1

In general for a problem like this you want to use the Simplex method. But this problem is small enough that you can solve it by just checking a few cases.

Product X takes $5$ hours on MC1 so you can make at most $4$ of X each day. Given that you want to make as many products as possible, the optimal choice will be one of the following (X,Y) tuples:(4,1),(3,2),(2,3),(1,4),(0,6). X gives you a profit of $300$ and Y a profit of $280$ so the profits for the above choices are $1480$, $1460$, $1440$, $1420$, and $1680$, respectively. So the optimal choice is to produce 6 of Y and 0 of X, for a daily profit of $1680$.