Question

A Company sells a particular model of component, with most of the sales being made in the fall season. Company makes a one-time purchase of the components prior to each fall season at a cost of $40 for each component and sells for $60 each of them. Any component unsold at the end of the fall season are marked down to $29 and sold in a special winter sale. Sales of the past 10 season are given below:

1. A Company sells a particular model of component, with most of the sales being made in the fall season. Company makes a one-time purchase of the components prior to each fall season at a cost of $40 for each component and sells for $60 each of them. Any component unsold at the end of the fall season are marked down to $29 and sold in a special winter sale. Sales of the past 10 season are given below. 1 23 4 56 7 8 9 10 Sales 50 30 60 10 40 30 30 40 30 20 a. Assume that demand follows a normal distribution. Determine the optimal number of components for company to buy prior to each fall season based on service level calculations. (15 points) b. Based on the observed 10 values of the prior demand, construct an empirical probability distribution and determine the optimal number of components based on this empirical distribution considering service level. (10 points) c. According to your results for parts (a) and (b), would you say that the approximation of normal distribution provides an adequate solution? Why or why not? (5 points)

Answer 1

ANSWER:

a)

Assuming demand follows a normal distribution, we can calculate the mean (m) and standard deviation (s) of the above data set and use that to arrive at an optimal order value based on the target service levels.

Year	1	2	3	4	5	6	7	8	9	10
Sales	30	50	30	60	10	40	30	30	20	40
Mean	34	Std Dev	14

The mean of the above set is 34 with a standard deviation of approximately 14. For service level calculation we need to find the cost of understocking = Unit Profit = Cs = 60 - 40 = 20. Cost of Over stocking is the purchase price net of salvage value = Co = 40 - 29 = 11. The service level for optimal profit is then given by Cs/(Cs+Co) = 20/31 = 64%. The z value corresponding to this service level is approximately z= 0.37. Hence the order quantity should be m +s*z = 39

b)

Since the number of units are finite over the last 10 years we can approximate it to the distribution of 10 to 60 units with the frequencies from the given table and apply the same principle of service levels as calculated above. The frequency table has been reconstructed below

Demand	Frequency	Probability	Cumulative Probability
10	1	0.10	0.10
20	1	0.10	0.20
30	4	0.40	0.60
40	2	0.20	0.80
50	1	0.10	0.90
60	1	0.10	1.00
Total	10

Here the optimal service level at 64% corresponds to the demand at 40 (0.6 < 0.64 < 0.8). Hence the order quantity calculated by this method is about 40 which we see is close to the value calculated by approximating a normal distribution.

c)

The approximation of a normal distribution provides a very close value to the second method so is probably a good approximation. Also we observe the trend in the distribution is also centrally oriented with 30 and 40 being the central values.

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