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Chapter 19

Q. 19.11

A specialty chemicals facility manufactures two products A and B in barrels. Products A and B utilize the same raw material; A uses 120  kg / bbl, while B requires 100  kg / bbl. There is an upper limit on the raw material supply of 9000  kg / day. Another constraint is warehouse storage space \left(40 m ^{2}\right. total; both A and B require \left.0.5  m ^{2} / bbl \right) . In addition, production time is limited to 7  h per day. A and B can be produced at 20  bbl / h and 10  bbl / h, respectively. If the profit per bbl is \$ 10 for A and \$ 14 for B, find the production levels that maximize profit.


Verified Solution

Let x_{A} be bbl/day of A produced x_{B} be bbl/day of B produced

Objective is to maximize profit

\max P=10 x_{A}+14 x_{B}          (1)

Subject to

Raw material constraint: \quad 120 x_{A}+100 x_{B} \leq 9,000            (2)

Warehouse space constraint: 0.5 x_{A}+0.5 x_{B} \leq 40           (3)

Production time constraint: (1 / 20) x_{A}+(1 / 10) x_{B} \leq 7                (4)

X_{A} X_{B}
Initial values 1 1
Final values 20 60
max P = 1040
120X_{A} + 100X_{B} 8400
0.5X_{A} + 0.5X_{B} 40
(1/20)X_{A} + (1/10)X_{B} 7

Table S19.11. Excel solution

Thus the optimum point is x_{A}=20 and x_{B}=60

The maximum profit =\$ 1040 / day