Question

The Bridgeport City Police need assistance in determining the number of additional police officers needed to cover daily absences due to injuries, sickness, vacations, and personal leave.Records indicate that the daily demand for additional police officers is normally distributed with a mean of 50 officers and a standard deviation of 10 officers.The cost of an additional police officer is based on the average pay rate of $150 per day.If the daily demand for police officers exceeds the number of additional officers available, the excess demand will be covered by overtime at the pay rate of $240 per day for each overtime officer.

a.If the number of additional police officers available is greater than demand, the additional officers will have to be paid anyway.If the number of officers is less than demand, overtime will have to used and paid for.What are Cu and Co in this instance?

b.On a typical day, what is the probability that overtime will be necessary?

c.On a typical day, what is the probability that there will be excess additional police officers?

d.What is the optimal daily number of additional officers that should be hired?

Answer 1

Here its given that the cost of an additional police officer is based on the average pay rate of $150 per day and excess demand will be covered by overtime at pay rate of $240 per day for each overtime officer.

a. Cost of overestimating demand = C_o= 150$ and Cost of understimating demand= C_u= 240-150= 90

b.

probability of not needing overtime = C_u/ C_{o +}C_{o =} 90/ 90+150 = 90/240 = 0.375

probability of overtime = 1-0.375 = 0.625

c. probability that there will be excess additional police officers

0.375

d. The optimal daily number of additional officers that should be hired

probability of overtime = 0.625, corresponding z value = 0.31864

Q= mean - (z* standard deviation) = 50 - (0.31864*10) = 50- 3.18= 46.81 = 47

so 47 additional officers should be hired