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:

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 pieces of lengths 19, 25, and 10 meters each. Then further cuts are made as follows: The resulting contribution is

Thus with one woven length, contribution can be increased by $1.05 by merely changing the cutting pattern. With a daily output of 1000 such woven lengths (two defects on average), substantial savings could be realized by improved cutting procedures.

a) Formulate an integer program to maximize the contribution from cutting the woven length described above.

 b) Formulate an integer program to maximize the contribution from cutting a general woven length with no more than four defects. Assume that the defects occur at integral numbers of meters.

 c) How does the formulation in (b) change when the defects are not at integral numbers of meters?