Question 1203984
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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).


<U>ANSWER</U>.  The optimal strategy is to make 80 As and 20 Bs.
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