Solve the linear programming problem

Question

A manufacturer of golf clubs makes a profit of $60 per set on a model A set and $40 per set on a model B set. Daily production of the model A clubs is between 30 and 60 sets, inclusive, and that of the model B clubs is between 20 and 30 sets, inclusive. The total daily production is not to exceed 60 sets. How many sets of each model should be manufactured per day to maximize the profit?

Answer 1

Let: P= Total profit, S= Total sets

A= number of sets of Model A

B= number of sets of Model B

Goal: Maximize: P(A,B) = 60A + 40B

Constraints:

A + B <= 60

30 <= A <= 60

20 <= B <= 30

While plotting A vs. B on a graph is acceptable, I would generate vertices (A,B) directly from the constraints.

Using the theory that the optimal solution in goal programming lies along an edge between 2 vertices (and typically at the vertices), we can deduce:

A=30 --> B=30 (30,30)

note: Any A>30 does not yield an acceptable value of B

B=20 --> A=40 (40,20)

B=30 --> A=30 (same as above)

So the solution lies along the edge (40,20), (30,30)

Plug these values into the profit function (which is to be maximized) P= 60A + 40B

(40,20) --> P = 60(40) + 40(20) = 3200

(30,30) --> P = 60(30) + 40(30) = 3000

Therefore profit is optimized when A = 40 sets and B = 20 sets