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A company produces two products, A and B. The sales volume for A is at least 80%
of the total sales of both A and B. However, the company cannot sell more than 100 unit of A per day.
Both products use one raw material whose maximum daily availability is limited to 240 lb a day.
The usage rates of the raw material are 2 lb per unit of A and 4 lb per unit of B.
The unit prices for A and B are $20 and $50, respectively.
Determine the optimal product mix for the company.
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Looking at the unit prices ($20 per A and $50 per B), you start think that the best strategy
is to make as many of products B as possible.
Looking at the usage rate of the raw material (2 lb for A and 4 lb for B) and comparing it
with the unit prices, you only become stronger in your opinion.
So, the optimal strategy is (a) to produce as many Bs as possible;
(b) to make as few As as needed by the restriction A >= 0.8(A+B),
which gives A = ~ 4B.
(c) then to spend the rest of the raw material (if any) to make As,
still remaining under this restriction A >= 4B.
Let's calculate the number of As and Bs. Write an inequality for the raw material
2A + 4B <= 240 lbs.
Use A = 4B. You will get
2*(4B) + 4B <= 240
8B + 4B <= 240
12B <= 240 ---> B <= 240/12 = 20.
So, make 20 Bs and 4*10 = 80 As. You will spend 2*80 + 4*20 = 160 + 80 = 240 lbs of the raw material.
so, you will spend ALL the raw material.
All other restrictions will be satisfied. The problem is just solved (using mental reasoning).
ANSWER. The optimal strategy is to make 80 As and 20 Bs.