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Question

A paper mill produces rolls of paper used in making cash registers.Each roll of paper is 100m in length and can be used in widths of 3,4,6 and 10cm.The company's production process results in rolls that are 24cm in width.Thus, the company must cut its 24cm roll to the desired widths.It has six basic cutting alternatives as follows:
Cutting
alternatives
Width of Rolls(cm)
3
Width of Rolls(cm)
4
Width of Rolls(cm)
6
Width of Rolls(cm)
10
Waste(cm)
143---
2-32--
311111
4--212
5-41-2
6321-1
The minimum demand for the four rolls is as follows:
Roll width(cm)Demand
22,000
43,600
61,600
10 500
The paper mill wishes to minimize the waste resulting from trimming to size.Formulate the L.P model.

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Solution

Formulation of L.P model:
Key decision is to determine how the paper rolls be cut to the required widths so that trim losses(wastage) are minimum.
Let xj(j=1,2,...6) represent the number of times each cutting alternative is to be used.These alternatives result/do not result in certain trim loss.
Objective is to minimize the trim losses.
i.e., minimize Z=x3+2x4+2x5+x6.
Constraints are on the market demand for each type of roll width:
For roll width of 3cm, 4x1+x3+3x62,000
For roll width of 4cm, 3x1+3x2+x3+4x5+2x63,600
and For roll width of 10cm, x3+x4500
Since the variables represent the number of times each alternative is to be used,they cannot have negative values.
x1,x2,x3,x4,x5,x6,each0

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