The Bradford Wire Company produces wire screening woven on looms in a process essentially identical to that of weaving cloth. Recently, Bradford has been operating at full capacity and is giving serious consideration to a major capital investment in additional looms. They currently have a total of 43 looms, which are in production all of their available hours. In order to justify the investment in additional looms, they must analyze the utilization of their existing capacity.

 Of Bradford’s 43 looms, 28 are 50 inches wide, and 15 are 80 inches wide. Normally, one or two widths totaling less than the width of the loom are made on a particular loom. With the use of a ‘‘center-tucker,’’ up to three widths can be simultaneously produced on one loom; however, in this instance the effective capacities of the 50-inch and 80-inch looms are reduced to 49 inches and 79 inches, respectively. Under no circumstance is it possible to make more than three widths simultaneously on any one loom.

Figure gives a typical loom-loading configuration at one point in time. Loom #1, which is 50 inches wide, has two 24-inch widths being produced on it, leaving 2 inches of unused or ‘‘wasted’’ capacity. Loom #12 has only a 31-inch width on it, leaving 19 inches of the loom unused. If there were an order for a width of 19 inches or less, then it could be produced at zero marginal machine cost along with the 31-inch width already being produced on this loom. Note that loom #40 has widths of 24, 26, and 28 inches, totaling 78 inches. The ‘‘waste’’ here is considered to be only 1 inch, due to the reduced effective capacity from the use of the center-tucker. Note also that the combination of widths 24, 26, and 30 is not allowed, for similar reasons.

The total of 3833 /4 inches of ‘‘wasted’’ loom capacity represents roughly seven and one-half 50-inch looms; and members of Bradford’s management are sure that there must be a more efficient loom-loading configuration that would save some of this ‘‘wasted’’ capacity. As there are always numerous additional orders to be produced, any additional capacity can immediately be put to good use.

 The two types of looms are run at different speeds and have different efficiencies. The 50-inch looms are operated at 240 pics per second, while the 80-inch looms are operated at 214 pics per second. (There are 16 pics to the inch). Further, the 50-inch looms are more efficient, since their ‘‘up time’’ is about 85% of the total time, as compared to 65% for the 80-inch looms.

 The problem of scheduling the various orders on the looms sequentially over time is difficult. However, as a first cut at analyzing how efficiently the looms are currently operating, the company has decided to examine the loom-loading configuration at one point in time as given in Fig. If these same orders can be rearranged in 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).

Loom number 1 2 3 4 tnorwagEDMASKEKATEREG 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 50" Total 50' Looms Widths 24-24 30 40 301 48 32-14 28 26 30 30 35 31 344 36 32 34-16 29 30 36 47/ 30 20-30 48 28-22 30-14 42 32-18 28/ Waste 2 20 10 192 2 34 22 234 20 20 15 19 153 14 18 20 20 8 0 21/ 3372 Loom number 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 80" Total Grand total 80" Looms Widths 48-32 24-26-28 48-32 48-32 36-36 36-36 60-18 36-36 42-34 24-26-28 24-26-28 24-26-28 48-32 36-36 42-34 Waste 0 1 0 0 8 8 2 8 0 8 4 46 337 383