A certain small car-wash business is currently being analyzed to see if costs can be reduced. Customers arrive according to a Poisson process at a mean rate of 15 per hour, and only one car can be washed at a time. At present the time required to wash a car has an exponential distribution, with a mean of 4 minutes. It also has been noticed that if there are already 4 cars waiting (including the one being washed), then any additional arriving customers leave and take their business elsewhere. The lost incremental profit from each such lost customer is $6.
Two proposals have been made. Proposal 1 is to add certain equipment, at a capitalized cost of $6 per hour, which would reduce the expected washing time to 3 minutes. In addition, each arriving customer would be given a guarantee that if she had to wait longer than ½ hour (according to a time slip she receives upon arrival) before her car is ready, then she receives a free car wash (at a marginal cost of $4 for the company). This guarantee would be well posted and advertised, so it is believed that no arriving customers would be lost.
Proposal 2 is to obtain the most advanced equipment available, at an increased cost of $20 per hour, and each car would be sent through two cycles of the process in succession. The time required for a cycle has an exponential distribution, with a mean of 1 minute, so total expected washing time would be 2 minutes. Because of the increased speed and effectiveness, it is believed that essentially no arriving customers would be lost.
The owner also feels that because of the loss of customer goodwill (and consequent lost future business) when customers have to wait, a cost of $0.20 for each minute that a customer has to wait before her car wash begins should be included in the analysis of all alternatives.
Evaluate the expected total cost per hour E(TC) of the status quo, proposal 1, and proposal 2 to determine which one should be chosen.