Throughput Loss and Profitability

A new technology of hand-making glass products became available, but it requires that special components are added to the glass and that the hot glass products are cooled in special tanks immediately after hand-shaping by a skilled worker.

One skilled worker shapes the glass products manually. Since a large variety of glass products is made, the time to make a unit is assumed to have exponential distribution with an average of 1 minute per unit. The worker works 8 hours a day, 20 days a month.

As soon as the shaping of a unit is finished, it has to be put in one of the cooling tanks. The cooling time is random with a mean of 0.5 minute and a standard deviation of 0.2 minute. After cooling of a unit is completed, an assistant moves the unit to a storage area. If no tank is available when a unit needs to be cooled, the unit is cooled without a tank (no waiting is allowed), but it is then considered a lower quality and sold for only 20% of the normal price. The normal price is $5 per unit on average. Process is capacity constrained (demand and input are unlimited).

A manager has a problem of deciding the number of cooling tanks to use. The tanks are relatively cheap to buy, but the chemicals used for cooling are expensive and have to be replaced regularly. As a result, each cooling tank costs $2,000/month.

1. Consider four cases: installing 1, 2, 3, and 4 cooling tanks. What percentage of glass products are going to be of lower quality in each case? Circle the four results. (Hint: Tanks are servers, cooling is the service, and the arrival process is determined by the output of the worker.)
2. What is the most profitable number of cooling tanks? Express profits in $ per month and circle each profit. Also, circle the most profitable number of tanks.