Consolidated Corkscrews (CC) is a multinational manufacturer of precision carbon-steel cor...

Consolidated Corkscrews (CC) is a multinational manufacturer of precision carbon-steel corkscrews for heavy-duty, high-speed use. Each corkscrew is made on a metal lathe, and in order to meet rising consumer demand for their product, CC is planning a new plant with six lathes. They are not sure, however, how this new plant should be constructed, or how the maintenance department should be equipped. Each lathe has its own operator, who is also in charge of repairing the lathe when it breaks down. Reliability data on lathe operation indicate that the “up” time of a lathe is exponentially distributed with mean 75 minutes. When a lathe goes down, its operator immediately calls the tool crib to request a tool kit for repairs. The plant has a fixed number, m, of tool kits, so there may or may not be a kit in the crib when an operator calls for one. If a tool kit is not available, the operator requesting one is placed in a FIFO queue and must wait his or her turn for a kit; when one later becomes available, it is then placed on a conveyor belt and arrives t_i minutes later to lathe i, where t_i might depend on the lathe number, i, requesting the kit. If a kit is available, it is immediately placed on a conveyor belt and arrives at the broken lathe t_i minutes later; in this case the operator’s queue delay is counted as 0. When an operator of a broken lathe receives a tool kit, he or she begins repair, which takes an amount of time distributed as a 3-Erlang random variable with mean 15 minutes. When the repair is complete, the lathe is brought back up and the tool kit is sent back to the tool crib, where it arrives t_i minutes later, if it is sent back from lathe i. Initially, assume that all lathes are up and have just been “freshly repaired,” and that all m tool kits are in the crib. CC wants to know about the projected operation of the plant over a continuous 24-hour day by looking at:

• The proportion of time that each of the six lathes is down

• The time-average number of lathes that are down

• The time-average number of tool kits sitting idle in the crib

• The average delay in queue of operators requesting a tool kit

FIGURE 2.69 The linear design.

FIGURE 2.70 The circular design.

There are two major questions to be addressed:

(a) How should the plant be laid out? Two layouts are under consideration:

(i) In the linear design (see Fig. 2.69), the lathes are placed in a straight line with the tool crib at the left end, and a single conveyor belt for the tool kits can reach all lathes. In this case, t_i = 2i minutes, for i = 1, 2, …, 6.

(ii) In the circular design, the lathes are placed around the tool crib (see Fig. 2.70), and each lathe has its own conveyor belt to the crib; here, t_i = 3 for all lathe numbers i. This is a more expensive design, but results in shorter travel times for the kits.

(b) How many tool kits should there be? As tool kits are quite expensive, CC does not want to purchase more than necessary.

Carry out the necessary simulations and advise CC on questions (a) and (b). In all cases, use stream 1 for the lathe-up times, and stream 2 for repair times.