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Linear Programming : Trim and Surplus Waste

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International Cellulose Corporation produces paper products. In one of its processes reels of 100 inch wide paper are cut into smaller width reels of the same length. Each week, orders for different width reels are received. This week the company has orders for 30, 50 25, and 90 reels, respectively, of the 60, 48 36 and 24 inch widths. The objective is to cut the 100 inch reels to meet demand and minimize waste. In this problem waste is defined as the sum of trim loss and surplus. If for example 10 reels were cut using a pattern that produced 4 inches of waste, the loss would be 4x10. If 5 more 24 inch reels were produced than needed, the waste would be 5x24. Determine all possible ways to cut the 100 inch reels to yield reels of sizes 60, 48, 36 and 24 and determine the trim waste for each. Letting xj denote the number of reels cut according to pattern j, formulate and solve an LP to minimize the sum of the trim and surplus waste.