Specialty Chemical Incorporated (SCI) makes three chemicals used in the biotech industry. For the purposes of this example, we call them A, B, and C. The firm has observed that for every additional dollar it charges per barrel for chemical A, demand drops about 0.3%. Extrapolating this effect back to a zero price, SCI estimates that if it distributed chemical A for free, demand would be 1000 barrels/week. Thus, demand per week for chemical A is given by the formula
D = 1000(1 - .003)P = 1000(.997)P.
Demand for products B and C behaves similarly, according to the respective formulas
D = 1500(1 - .002)P = 1500(.998)P and D = 2000(1 - .0045)P = 2000(.9955)P.
Four production processes called P1, P2, P3, and P4 are required to make the the chemicals as follows:
| Process Hours per Barrel | ||||
| Chemical | P1 | P2 | P3 | P4 |
| A | 1 | 0.5 | 0 | 0.5 |
| B | 0.25 | 2 | 1 | 0.5 |
| C | 1 | 1 | 1.5 | 0.5 |
| Hours Available: | 336 | 336 | 168 | 168 |
| Cost per Hour: | $ 185 | $ 215 | $ 89 | $ 45 |
Making one barrel of chemical A requires one hour of process P1, 0.5 hours of process P2, and 0.5 hours of process P4 (process P3 is not required for chemical A). SCI operates around the clock (168 hours/week), and has two P1 machines, two P2 machines, one P3 machine, and one P4 machine. Thus, the available hours for the four processes are 336, 336, 168, and 168 per week, respectively. The costs per hour to operate the various machines are as shown in the table, and are the only variable costs in SCI's production process.
Assume that SCI does not want to route production into long-term inventory, and therefore wants to produce exactly the amount of chemical demanded each month. Estimate the price should it charge for each chemical, and how much should it produce in order to maximize its weekly profit.