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## 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 Constraints $120X_{A} + 100X_{B}$ 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