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.
Chapter 19
Q. 19.11
Step-by-Step
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