Specialty faces the decision of how many Weather Teddy units to order for coming

Question

Specialty faces the decision of how many Weather Teddy units to order for coming holiday season. Members of the management team suggested order quantities of 15,000, 18,000, 24,000, 28,000 units. The wide range of order quantities suggested indicates considerable disagreement concerning the market potential. The product management team ask you for analysis of the stock-out probabilities for various order quantities, an estimate of the product potential, and to help make an order quantity recommendation. Specialty expects to sell Weather Teddy for $24 based on cost of $16 per unit. If inventory remains after the holiday season, Specialty will sell all surplus inventory for $5 per unit. After reviewing the sales history of similar products, Specialtys sales forecaster predicted and expected demand of 20,000 units with a .95 probability that demand would be between 10,000 units and 30,000 units. Prepare a managerial Report that addresses the following issues and recommends an order quantity for the Weather Teddy product.

Explanation / Answer

Let X be the demand for the toy. Then X follows normal distribution with mean = 20000 and standard deviation . Then
P(10000 < X < 30000) = 0.95
P((10000-20000)/ < (X-20000)/ < (30000-20000)/) = 0.95

From tables of areas under the standard normal curve (30000-20000)/ = 1.96
= (30000-20000)/1.96 =10000/1.96 = 5102

The demand distribution can be approximated by a normal distribution with mean µ = 20000 and standard deviation = 5102.

Probability of stock out with an order of K units is P(X > K) = P(Z > (K-20000)/5102), where Z is distributed as standard normal

Order (K)

(K-20000)/5102

P(X > K)

15000

-0.98001

0.836458876

18000

-0.392

0.652472052

24000

0.784006

0.216518215

28000

1.568013

0.058439102

3. The projected profit for the different order quantities and scenarios are given in the following table.

Order

     Scenario 10000

Scenario 20000

Scenario 30000

15000

8*10000-11*5000 =25000

8*15000=120000

8*15000 = 120000

18000

8*10000-11*8000

= -8000

8*18000 = 144000

8*18000 = 144000

24000

8*10000-11*14000

= -74000

8*20000-11*4000

=116000

8*24000 = 192000

28000

8*10000-11*18000

= -118000

8*2000-11*8000

=72000

8*28000 = 224000

4. The order quantity to meet 70% demand is found by solving

P(X < K) =0.70

P(Z < (K-20000)/5102 ) = 0.70

(K-20000)/5102 = 0.5244

K = 20000 + 5102 * 0.5244 = 20000 + 2675 = 22675.

Order (K)

(K-20000)/5102

P(X > K)

15000

-0.98001

0.836458876

18000

-0.392

0.652472052

24000

0.784006

0.216518215

28000

1.568013

0.058439102

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