3. (50 pts.) A company selling hot rolled steel serves 10 customer locations dis

Question

3. (50 pts.) A company selling hot rolled steel serves 10 customer locations dispersed around the country. The monthly demand at each customer location is normally dis- tributed with mean 5 million metric tons and standard deviation 1 million metric tons The demands at dfferent customer locations are independent of each other. The steel inventory at each customer location is replenished only once each month, just before the month begins. The company's goal is to cover all customer demand in every month at least 95% of the time (a) Assuming that any shortage of steel at one of the customer locations can be made up by transporting excess steel from another of the customer locations, compute the minimum amount of steel that should be stocked in total across all of the locations at the beginning of each month so that the company will be able to accomplish this goal. (Hint: NORMS INV (0.5) function in Excel or scipy. norm. ppf(O.5) function of the scipy package in Python returns z such that P{N(0, 1)- 0.5 NORMSDIST (1. 5) function in Excel or scipy. norm . cdf ( 1 . 5) function of the scipy package in Python returns the probability P{N(0, 1)

Explanation / Answer

a) Mean = 5* 10 =50

Standard deviation = 10^(1/2) * 1

z score for 95% confidence =1.96

hence

Z = (x- mean)/ standard deviation

1.96= (x- 50)/3.16

Hence X = 56.2

b) Mean = 5

Standard deviation = 1

z score for 95% confidence =1.96

hence

Z = (x- mean)/ standard deviation

1.96= (x- 5)/1

X= 6.96

Hence total inventory = 10 * 6.96 = 69.6

c) The inventory is less in 1st part since the combined inventory can be used in any of the plants but in 2nd case each and every location should be self sufficient. So we have to keep enough inventory at every location for a 95% confidence level. Hence the inventory required in 2nd case would be more.

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