Question: A railroad company paints its own railroad cars as needed. The company is about to make a significant overhaul of the painting operations and needs to decide between two alternative paint shop configurations.

Alternative 1: Two "wall-to-wall" manually operated paint shops, where the painting is done by hand (one car at a time in each shop). The annual joint operating cost for each shop is estimated at $150,000. In each paint shop, the average painting time is estimated to be 6 h per car. The painting time closely follows an exponential distribution.

Alternative 2: An automated paint shop at an annual operating cost of $400,000. In this case, the average paint time for a car is 3 h and exponentially distributed.

Regardless of which paint shop alternative is chosen, the railroad cars in need of painting arrive to the paint shop according to a Poisson process with a mean of 1 car every 5 h (= the interarrival time is 5 h). The cost for an idle railroad car is $50 per hour. A car is considered idle as soon as it is not in traffic; consequently, all the time spent in the paint shop is considered idle time. For efficiency reasons, the paint shop operation is running 24 h, 365 days a year, for a total of 8760 h/year.

a. What is the utilization of the paint shops in alternative 1 and 2, respectively? What are the probabilities, for alternative 1 and 2, respectively, that no railroad cars are in the paint shop system?

b. Provided the company wants to minimize the total expected cost of the system, including operating costs and the opportunity cost of having idle railroad cars, which alternative should the railroad company choose?