Example: Esquire Products will produce four new lines in the next month. The respective per-unit profits on the lines are $200, $220, $185, and $190. They are basically testing the market and do not wish to produce more than 700 of any one line. The respective fixed start-up costs for the products are $4,000, $5,000, $3,000, and $3,500. Also included in the start-up costs is the purchase of a part, one of which is needed for every item produced. The supplier of the part charges according to the following schedule:
- $50 ordering charge
- $9 each for the first 100
- $6 each for all additional
Esquire has budgeted $20,500 for the start-up costs. Lines 1 and 2 require a half hour of production time per item while lines 3 and 4 require 0.4 hour per item. There will be 800 hours of production time available during the month. How many of each line should they produce to maximize their profit?
Solution: Let x_1, x_2, x_3, x_4 be the number of units produced from the new lines and s_1, s_2, s_3, s_4 be the binary variables indicating whether the lines are open or not. For the piecewise linear pricing, let y_1 be the number of parts if we were to order no more than 100 parts and let y_2 be the number of parts if we were to order more than 100. Let b_1, b_2 indicate if 0 < y_1 \le 100 and if 100 < y_2 respectively. Using these variables, we can formulate the problem as a linear program as follows.
| Maximize | 200x_1 + 220x_2 + 185x_3 + 190x_4 |
| Subject to | 4000s_1 + 5000s_2 + 3000s_3 + 3500s_4 + 50b_1+9y_1+350b_2+6y_2 \le 20500 |
| x_1 \le 700s_1 | |
| x_2 \le 700s_2 | |
| x_3 \le 700s_3 | |
| x_4 \le 700s_4 | |
| y_1 \le 100b_1 | |
| y_2 \le Mb_2 | |
| y_1 + y_2 = x_1 + x_2 + x_3 + x_4 | |
| b_1 + b_2 \le 1 | |
| 0.5x_1+0.5x_2+0.4x_3+0.4x_4 \le 800 |