Poly Products sells packaging tape to industrial customers. All tape is sold in 100-foot rolls that are cut in various widths from a master roll, which is 15 inches wide.
The product line consists of the following widths: 3", 5", and 7" . These can be cut in different combinations from a 15-inch master roll. For example, one combination might consist of three cuts of 5 each. Another combination might consist of one 3"cut, one 5" cut, and one 7" cut. Both of these combinations use the entire 15-inch roll without any waste, but other configurations are also possible. For example, another combination might consist of two 5 cuts and a 3 cut. This combination creates two inches of waste for every roll cut this way.
Each week, Poly Products collects demands from its customers and distributors and must figure out how to configure the cuts in its master rolls. To do so, the production manager lists all possible combinations of cuts and tries to fit them together so that waste is minimized while demand is met. (In particular, demand must be met exactly, because Poly Products does not keep inventories of its tape.)
This week’s demands are for 50 3" rolls, 40 5" rolls, and 30 7 "rolls.
a. List the combinations that can be cut from a 15-inch master roll so that there is less than three inches of waste (i.e., the smallest quantity that can be sold) left
on the roll.
b. Find a set of combinations that meets demand exactly and generates the minimum amount of waste. Stated another way, the requirement is to meet or exceed
demand for each size, but excess cuts must be counted as waste.