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Operations Research Models and Methods
Models Section
Penny Fab

The Penny Fab problem is taken from the book Factory Physics (Wallace J. Hopp and Mark L. Spearman, Irwin/McGraw-Hill, 1996). The following is a partial quotation from that book, modified slightly to suit the purposes of this example.

Penny Fab represents a simple production line that makes giant one-cent pieces used extensively in Fourth of July parades. The line consists of four machines in sequence. The first machine is a punch press that cuts penny blanks, the second stamps Lincoln’s face on one side and the Lincoln Memorial on the back, the third puts a rim on the penny, and the fourth cleans away any burrs. The times required to process a penny on each machine are random variables. After each penny is processed, it is moved immediately to the next machine. The line runs 24 hours per day, with breaks and lunches covered by spare operators. For our purposes, the market for giant pennies can be assumed to be unlimited, so that all product made is sold; thus, more throughput is unambiguously better for this system.

The system is pictured in Figure 14. Coins are shown as circles and machines as rectangles. Blanks enter station 1 and when cutting is complete move to station 2. The process continues through all four stations until finally the completed pennies leave station 4. Each station consists of a single machine. The processing rates in the stations may differ. Each machine operator processes the partially completed coins whenever a coin is available for processing. Because of variability in the processing times or arrivals of blanks into the system, queues may form in front of each machine. At any time a coin may be in a queue or at a machine. Material that has entered the system, but is not yet finished is called WIP (Work In Process).

Figure 14. The Penny Fab System


The system uses a production control policy called CONWIP (Constant Work In Process). The system begins empty and penny blanks are allowed to enter the system one at a time. A reasonable plan allows a blank to enter station 1 whenever station 1 is empty. When some predetermined number of pennies are in the system, say K, the rule for allowing blanks to enter changes. Now, a new blank is allowed to enter only when a finished penny leaves. In this way the WIP is maintained at a constant level, justifying the name of this policy, CONWIP.

To completely describe this system one must specify the probability distributions for the machine times at each station and the number of coins allowed in WIP, K. The most important output measure is the average processing rate of finished coins, called the throughput rate. As one might expect the processing rate is minimized when K = 1 and increases as K increases. Since this is a closed queueing system (the others in this section were open queueing systems), there can be at most 4 coins in actual production. When processing times are constant the K equal to 4 provides the maximum throughput and there is no point in providing more. Because of random processing times, however, throughput will be increased by increasing the WIP the to a value greater than 4. Variability inevitably leads to queues and idle times at the machines.

Another important measure is the production cycle time. This is the time required by a single penny to pass through the system. With constant processing times, the cycle time is constant. With random variability in machine times, the cycle time becomes a random variable. It is interesting to observe the mean and standard deviation of the cycle time. It is also interesting to observe statistics on the queues at the stations. Of course, for a CONWIP system, the total number or parts in the queues and receiving processing must remain constant after the start-up period, but it is instructive to see the distribution of the coins among the various machines.
Figure 15 shows an Extend model that can be used to experiment with the parameters of the CONWIP system. The model reports the throughput rate and average WIP in the system. For the case shown, the material available is equal to 10 units. Each station has a processing time governed by the exponential distribution with a mean time of 2. The average WIP is less than 10 because it takes an initial startup time to fill the system. The ten green circles in Figure 15 indicate that 10 parts were in the system when the simulation was stopped.

Figure 15. Penny Fab Model


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Operations Research Models and Methods
by Paul A. Jensen
Copyright 2004 - All rights reserved

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