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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.

##### Solution Summary

Linear programming is used minimize waste. All constraints and tableaux are shown. The solution is detailed and well presented.

Solution provided by:
###### Education
• BSc , Wuhan Univ. China
• MA, Shandong Univ.
###### Recent Feedback
• "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
• "excellent work"
• "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
• "Thank you"
• "Thank you very much for your valuable time and assistance!"

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