Question

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

$____(Enter your responses as integers.)

Answer #1

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 = C_{s} = 4,800. Cost of Overhiring
= C_{o} = 3,100. The optimal service level will be given by
C_{s}/(C_{s} + C_{o}) = 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.

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