Question

# Sanchez Trucking has been experiencing delays at its warehouse operations. Management hired a consultant to find...

Sanchez Trucking has been experiencing delays at its warehouse operations. Management hired a consultant to find out why service deliveries to local businesses have taken longer than they should. The consultant narrowed down the problem to the number of work crews loading and unloading trucks. Each crew consists of 6 employees who work as a team on a variety of​ tasks; each employee works a full 40 hours a week.​ However, costs are also a concern. The consultant advised management that they could supplement work crews with​ short-term employees, at a higher​ cost, to cover unexpected needs on a weekly basis. Each work crew permanently hired by Sanchez costs ​\$3,100 per week in wages and​ benefits, while a crew of​ short-term employees costs ​\$4,800. per week. Complicating the decision is the fact that the weekly hourly requirements for work crews is uncertain because of the volatility in the number of deliveries to be made. Deliberating with​ management, the consultant arrived at the following data.

 Requirements​ (man-hours) 960 1,200 1,440 1,680 Probability 0.1 0.4 0.4 0.1 Number of crews 4 5 6 7

If the consultant wants to offer a solution that minimizes the expected weekly costs for​ Sanchez, how many work crews should Sanchez have on its permanent​ payroll?

A payroll containing ____ work crews is the lowest cost​ alternative, with expected weekly costs to Sanchez of

First we can calculate the cumulative probabilities of the number of crews required as shown in the table below

 Crews Probability Cumulative 4 0.1 0.1 5 0.4 0.5 6 0.4 0.9 7 0.1 1.00

Cost of underhiring = Cs = 4,800. Cost of Overhiring = Co = 3,100. The optimal service level will be given by Cs/(Cs + Co) = 4800/(4800 + 3100) = 0.607. Since 0.5 < 0.607 < 0.9 we see that the optimal number of crews as per the above table is at 6. Hence the weekly expected costs are 3100*6 = 18,600.