AnalysisLets consider a normal probaility distribution N( , 2 )
for the sales forecast prediction.Let X be the continuous random
variable that denotes demand for the toy. Then X follows normal
distribution with mean = 20000 (as forecaster predicted an expected
demand of 20000 units) and standard deviation . ThenProbability
that X lies between 10000 and 30000 = P(10000 < X < 30000) =
0.90Let Z be the standard normal variable : Z = (X - )/
P((10000-20000)/ < (X-20000)/ < (30000-20000)/) = 0.90=>
P((10000-20000)/ < Z < (30000-20000)/) = 0.90=> p(-10000/
< Z < 10000/) = 0.90 As normal distribution is symmetric ,
p(-10000/ < Z quantities orderedCase 1: Ordered quantities =
15000Probability of stock out with an order of 15000 units = P(X
> 15000) = P(Z > (15000-20000)/6079) = P(Z > -0.822) = 1-
0.2066 = 0.7933Case 2: Ordered quantities = 18000Probability of
stock out with an order of 18000 units = P(X > 18000) = P(Z >
(18000-20000)/ 6079) = P(Z > -0.329) = 1- 0.3483 = 0.3632
Question 2 (cont.)Case 3: Ordered quantities = 24000Probability
of stock out with an order of 24000 units = P(X > 24000) = P(Z
> (24000-20000)/ 6079) = P(Z > 0.658) = 1- 0.7454 =
0.2546Case 4: Ordered quantities = 28000Probability of stock out
with an order of 18000 units = P(X > 28000) = P(Z >
(28000-20000)/ 6079) = P(Z > 1.316) = 1- 0.9075 =
0.09425Question 3The projected profit for the different order
quantities and scenarios is calculate as :(Selling price - Cost
price) of Sold units * no of sold units + (Selling price - Cost
price) of UnSold/surplus units * no. of surplus units= (24-16) * no
of sold units + (5-16) * no. of surplus units = 8 * no of sold
units - 11 * no. of surplus units =>Equation 1Profit for all the
cases is calculated in the following table :
Question 3 (cont.)30000 units cannot be sold if the ordering
quantities are 15000, 18000, 24000 and 28000. Also, 20000 units
cannot be sold when the ordering quantities are 15000 and 18000. So
in these two cases we can consider that the full ordered quantity
is exhausted (i.e. sold).8Question 4Let K be the number of items
needs to ordered.As K needs to meet 70% of demand,Therefore,P(X
< K) =0.70P(Z < (K-20000)/ 6079) = 0.70From the table, P(Z
< 0.525) = 0.70Hence, (K-20000)/ 6079= 0.525=> K = 20000 +
6079* 0.525 = 20000 + 3191 = 23191.
Question 4(cont.)Equation for the number of items when there is
30% chance of stock-out is given byP(X > K) =0.30=> P(X
