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Integer Programming 9.8 such a way as to free up one or two looms, then
it would be clear that a closer look at the utilization of existing
equipment would be warranted before additional equipment is purchased.
a) Since saving an 80-inch loom is not equivalent to saving a 50-inch
loom, what is an appropriate objective function to minimize? b)
Formulate an integer program to minimize the objective function
suggested in part (a). 23. In the export division of Lowell Textile
Mills, cloth is woven in lengths that are multiples of the piece-length
required by the customer. The major demand is for 18-meter
piece-lengths, and the cloth is generally woven in 54-meter lengths.
Cloth cannot be sold in lengths greater than the stipulated
piece-length. Lengths falling short are classified into four groups. For
18-meter piece-lengths, the categories and the contribution per meter
are as follows: Length Contribution/Category Technical term (Meters)
Meter A First sort 18 1.00 B Seconds 11–17 0.90 C Short lengths 6–10
0.75 D Rags 1–5 0.60 J Joined parts? 18 0.90 ? Joined parts consist of
lengths obtained by joining two pieces such that the shorter piece is at
least 6 meters long. The current cutting practice is as follows. Each
woven length is inspected and defects are flagged prominently. The cloth
is cut at these defects and, since the cloth is folded in exact meter
lengths, the lengths of each cut piece is known. The cut pieces are then
cut again, if necessary, to obtain as many pieces of first sort as
possible. The short lengths are joined wherever possible to make joined
parts. Since the process is continuous, it is impossible to pool all the
woven lengths and then decide on a cutting pattern. Each woven length
has to be classified for cutting before it enters the cutting room. As
an example of the current practice, consider a woven length of 54 meters
with two defects, one at 19 meters and one at 44 meters. The woven
length is first cut at the defects, giving three...
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