STAR Co. provides paper to smaller companies whose volumes are not large enough to warrant dealing directly with the paper mill. STAR receives 100-feet-wide paper rolls from the mill and cuts the rolls into smaller rolls of widths 12, 15, and 30 feet. The demands for these widths vary from week to week. The following cutting patterns have been established:     Number of: Pattern 12ft. 15ft. 30ft. Trim Loss 1 5 0 1 10 ft. 2 0 0 3 10 ft. 3 8 0 0 4 ft. 4 4 3 0 7 ft. 5 2 1 2 1 ft.   Trim loss is the leftover paper from a pattern (e.g., for pattern 4, 4(12) + 3(15) + 0(30) = 93 feet used resulting in  foot of trim loss). Orders in hand for the coming week are 5,670 12-foot rolls, 1,680 15-foot rolls, and 3,350 30-foot rolls. Any of the three types of rolls produced in excess of the orders in hand will be sold on the open market at the selling price. No inventory is held.   (a) Formulate an integer programming model that will determine how many 100-foot rolls to cut into each of the five patterns in order to minimize trim loss. If your answer is zero enter “0” and if the constant is "1" it must be entered in the box.   Let x j = number of 100-foot-wide rolls using cutting pattern j, j = 1, 2, 3, 4, 5.   Optimal Solution:       Min  x 1 +  x 2 +  x 3 +  x 4 +  x 5       s.t.                    x 1 +  x 2 +  x 3 +  x 4 +  x 5     12-foot rolls    x 1 +  x 2 +  x 3 +  x 4 +  x 5     15-foot rolls    x 1 +  x 2 +  x 3 +  x 4 +  x 5     30-foot rolls           x 1, x 2, x 3, x 4, x 5 are integers and nonnegative         (b) Solve the model formulated in part a. What is the minimal amount of trim loss?   Total Trim Loss:  feet       How many of each pattern should be used and how many of each type of roll will be sold on the open market? If your answer is zero enter “0”.   Pattern Number of Rolls Used for Each Pattern 1   2   3   4   5     Type of Roll Number of Rolls Sold on the Open Market 12-foot rolls   15-foot rolls   30-foot rolls

STAR Co. provides paper to smaller companies whose volumes are not large enough to warrant dealing directly with the paper mill. STAR receives 100-feet-wide paper rolls from the mill and cuts the rolls into smaller rolls of widths 12, 15, and 30 feet. The demands for these widths vary from week to week. The following cutting patterns have been established:

Trim loss is the leftover paper from a pattern (e.g., for pattern 4, 4(12) + 3(15) + 0(30) = 93 feet used resulting in  foot of trim loss). Orders in hand for the coming week are 5,670 12-foot rolls, 1,680 15-foot rolls, and 3,350 30-foot rolls. Any of the three types of rolls produced in excess of the orders in hand will be sold on the open market at the selling price. No inventory is held.