(a) A company's main expense is its workforce, and for it at the end of each month the company has to pay $300,000 worth of salaries. The money comes from a payroll account that is empty at the beginning of the year and receives a variable in ow from the sales revenues net of the other operating costs. On a daily basis, $10,000 ow into the account on average, normally distributed with a variance of 100,000 $^2. If at the end of each month there is not enough money to pay all the employees, the account balance will go negative, and some will be paid late. (xi) What is the distribution and the parameters of the monthly in ow? (xii) What is the probability that at the end of each month all employees are paid on time? (xiii) What is the initial (safety) capital that we need to borrow from other accounts to raise this probability to 99%? (xiv) How would this probability change if the $10,000 mean increases and the 100,000 $^2 variance decreases? (xv) What the probability that at the end of the year the account has zero balance?
(b) For its most sold product, a department store estimates a stable mean demand rate of 200 units per week and a weekly variance of 1,000 units^2, for the whole year. If the store is able to place and receive orders almost immediately and whenever it desires, (xvi) what would be the service level with zero safety inventory? (xvii) What would be the safety inventory for a service level of 80%? If the lead time is now one week, (xviii) what would be the safety inventory for a service level of 80%? If the store places one order every two weeks, (xix) what would be the safety inventory for a service level of 80%? If the lead time is one week and the store places one order every two weeks, (xx) what would the safety inventory be for a service level of 80%?