# Recitation 12

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 $4000s_1 + 5000s_2 + 3000s_3 + 3500s_4 + 50b_1+9y_1+350b_2+6y_2 \le 20500$ $x_1 \le 700s_1$ $x_2 \le 700s_2$ $x_3 \le 700s_3$ $x_4 \le 700s_4$ $y_1 \le 100b_1$ $y_2 \le Mb_2$ $y_1 + y_2 = x_1 + x_2 + x_3 + x_4$ $b_1 + b_2 \le 1$ $0.5x_1+0.5x_2+0.4x_3+0.4x_4 \le 800$