Lucky Smokes currently operates a warehouse that serves the Virginia market. Some trucks arrive at the warehouse filled with goods to be stored in the warehouse. Other trucks arrive at the warehouse empty to be loaded with goods. Based on the number of trucks that arrive at the warehouse in a week, the firm is able to accurately estimate the total number of labor hours that are required to finish all of the loading and unloading. The following histogram plots these estimates for each week over the past two years. (There are a total of 1 04 weeks recorded in the graph.) For example, there were three weeks in this period that required 600 total labor hours and only one week that recorded 1,080 hours of required labor.
The mean of the data is 793 and the standard deviation is Ill. Labor is the primary variable cost in the operation of a warehouse. The Virginia warehouse employed 20 workers, who were guaranteed at least 40 hours of pay per week. Thus, in weeks with less than 800 hours of required labor, the workers either went home early on some days or were idle. On weeks with more than 800 hours of required labor, the extra hours were obtained with overtime. Workers were paid time and a half for each hour of overtime. You have been placed in charge of a new warehouse scheduled to serve the North Carolina market. Marketing suggests that the volume for this warehouse should be comparable to the Virginia warehouse. Assume that you must pay each worker for at least 40 hours of work per week and time and a half for each hour of overtime. Assume there is no limit on overtime for a given week. Further, assume you approximate your workload requirement with a normal distribution.
a. If you hire 22 workers, how many weeks a year should you expect to use overtime?
b. If you hire 18 workers, how many weeks a year will your workers be underutilized?
c. If you are interested in minimizing your labor cost, how many workers should you hire (again, assuming your workload forecast is normally distributed)?
d. You are now concerned the normal distribution might not be appropriate. For example, you can't hire 20.5 workers. What is the optimal number of workers to hire if you use the empirical distribution function constructed with the data in the above histogram?