Question

Specialty faces the decision of how many Weather Teddy units to order for the coming holiday season. Members of the management team recommended order quantities of 15,000, 18,000, 24,000, and 28,000. Considerable disagreement concerning the market potential is evidenced by the different order quantities suggested. The product management team has asked you for an analysis of the stock-out probabilities for various order quantities, an estimate of the profit potential, and help in making an order quantity recommendation. Specialty expects to sell Weather Teddy for $24, and the cost is $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, Specialty’s senior sales forecaster predicted an expected demand of 20,000 units with a 0.95 probability that demand would be between 10,000 units and 30,000 units.

2. Compute the probability of a stock-out for the order quantities suggested by members of the management team.

Answer 1

The expected demand or the mean of the normal distribution µ = 20000.
Margin of error as by prediction = (30000-10000)/2 = 10000
At 0.95 probability, margin of error = 1.96*σ
where σ = standard deviation.
So, 1.96*σ = 10000
or σ =10000/1.96 = 5102

Normal probability distribution of mean µ and standard deviation σ is given as
f(x) =

Just plot this function, after putting values for µ and σ.
f(x) = (exp(-((x-20000)^2) /52060808))/12788.82

probability of a stock-out for the order quantities suggested by members of the management team

At 15000

z=(15000-20000)/5102=-0.98

P(Stockout)=1-0.1635=0.8365

At 18000

z=(18000-20000)/5102=-0.39

P(Stockout)=1-0.3483=0.6517

At 24000

z=(24000-20000)/5102=0.78

P(Stockout)=1-0.7823=0.2177

At 28000

z=(28000-20000)/5102=1.57

P(Stockout)=1-0.9418=0.0582