Case $10\mathrm{.}1$ Minnetonka Warehouse

Wayne Schuller managed a warehouse in Minnetonka, Minnesota. His major concern was the number of workers toassign to his single unloading dock. After he began contracting with motor carriers for deliveries, he found that they were assessing him stiff penalties if their trucks had to wait to be unloaded. Wayne started adding larger crews at the unloading dock, but often they seemed idle because there were no trucks to unload. Wayne recalled from college that queueing theory might be applicable to such a problem.

The theory of queueing is an analysis of the probabilities associated with waiting in line, assuming that orders, customers, and so on arrive in some pattern $($often a random pattern$)$ to stand in line. A common situation is that on the average a facility may have excess capacity, but often it is more than full, with a backlog of work to be done. Often, this backlog has costs associated with it$,$ including penalties to be paid or customers who walk away rather than wait. If a firm expands its capacity to reduce waiting times, then its costs go up and must be paid even when the facility is idle. Queueing theory is used to find the best level of capacity, the one that minimizes the costs of providing a service and the costs of those waiting to use the service.

$$

After some further research specific to his firm, Wayne determined the following facts:

$$

$·$ Trucks arrive randomly at the average rate of four per hour, with a deviation of plus or minus one.

$·$ A team of two warehouse workers can unload trucks at the rate of five per hour, or one every $12$ minutes.

$$

$·$ A team of three warehouse workers can unload trucks at the rate of eight per hour, or one every $7\mathrm{.}5$ minutes.

$·$ A team of four warehouse workers can unload trucks at the rate of $10$ per hour, or one every $6$ minutes.

$·$ A team of five warehouse workers can unload trucks at the rate of $11$ per hour, or one every $4\mathrm{.}45$ minutes.

$·$ The unloading times given in the preceding items $(1–5)$ are average figures.

$$

$·$ Each warehouse worker receives $$14$ per hour, must be paid for an entire shift, and$—$ because of union workrules$—$cannot be assigned to other tasks within the warehouse.

$·$ Because of its contract with the carriers, the Minnetonka warehouse must pay the motor carriers that own idle trucks at the rate of $$60$ per hour while the trucks stand idle, waiting to be unloaded.

Note: Make sure you show the formulas used and show your calculations.

Questions

$$

$1.$ For each of the four work team sizes, calculate the expected number of trucks waiting in the queue to be unloaded.

$2.$ For each of the four work team sizes, calculate the expected time in the queue$—$that is$,$ the expected time a truck has to wait in line to be unloaded.

$3.$ For each of the four work team sizes, what is the probability that a truck cannot be unloaded immediately?

$$

$4.$ Which of the four work team sizes results in the lowest cost to Wayne?

$5.$ Wayne is also considering rental of a forklift to use in truck unloading. A team of only two would be needed, butthe hourly cost would be $$38$ per hour $($$$28$ for the workers and

$$10$ for the forklift$).$ The two workers could unload a truck in $5$ minutes. Should Wayne rent the forklift?

$6.$ Disregard your answer to question $5.$ Labor negotiations are coming up$,$ and Wayne thinks he can get the union to give way on the work rule that prohibits warehouse workers on the unloading dock from being given other assignments when they are not unloading trucks. How much would Wayne save in unloading dock costs if he could reassign warehouse workers to other tasks when they are not unloading trucks, assuming he has picked a good team of workers and each worker works $8$ hours a day?