Question

Question 4 113 points! The Bridgeport city manager and the chief of polis e d the size of the police force necessary for normal daily operations. However, they need assistance in determining the number of additional police officers needed to cover daily absences due to injuries, sickness, Vacations, and personal leave. Record over the past 3 years show that the daily demand for aditional police officers is normally distributed with a mean of 50 offices and a standard deviation of 10 officers. The cost of an additional police officer based on the average pay rate of $150/day. If the daily demand for additional police officers exceeds the number of additional officers available, the EXCESS demand will be covered by over time at the pay rate of 5240/day for each overtime officer, Bridgeport would like to take a single period inventory management approach to help them solve their staffing problem. (a) (8 points) What is the optimal number of addicional police officers that should be included in police force? (b) (2 points) On a typical day, what is the probability that overtime will be necessary? (e) (3 points) Suppose the Chief feels that the probability of having overtime is too high and wants to reduce it to 25%. What is the number of additional police officers that should be included in the police force instead of the number found in part (a)?

Need help parts a-c

Answer 1

Use the newsboy/newsvendor model for single period inventory

a)

The cost of overstaffing (Co) = 150

The cost of under staffing (Cu) = 240

The critical ratio = Cu/(Cu+Co) = 240/(150+240) = 0.6153

This means the probability of facing understaffing or overtime charges are 0.6153. Thus it is better to staff more than less. Based on this, the z value is 0.29

The optimal number is given by mean + z*standard deviation

This means the optimal number is 50 + 10*0.29 = 52.9 or 53 officers

b)

With the current solution, the possibility that we will still face shortage and overtime is 1-0.6153 = 0.3847 or 38.47% probability.

c)

If the probability of facing overtime needs to be reduced to 25% then we need to consider a critical ratio of 0.75. That means z = 0.675

This would means the optimal staffing be 50 + 0.675*10 = 56.75 or 57 officers

This is an increase of 4 officers.