Chapter 3
Continuous Probability Distributions
Chapter 3 Continuous Probability Distributions Case Problem: Problem: Specialty Toys
1.
Information provided by the forecaster
. 025
. 95
10,000
20,000
At x = 30,000,
z =
x − µ
=
30,000 30,000 − 20,000 20,000
σ
∴σ =
30,000 30,000 − 20,000 20,000 1.96
Normal distribution 2.
= 1.96
σ
µ =
=
5102
20,000
σ = 5102
@ 15,000
z =
15,000 15,000 − 20,000 20,000 5102
= −0.98
P(stockout) = 0.3365 + 0.5000 = 0.8365 @ 18,000
z =
18,000 18,000 − 20,000 20,000 5102
= −0.39
P(stockout) = 0.1517 + 0.5000 = 0.6517 @ 24,000
CP - 18
. 025
30,000
Chapter 3
Continuous Probability Distributions
z =
24,000 − 20,000 5102
=
0.78
P(stockout) = 0.5000 - 0.2823 = 0.2177
@ 28,000
z =
28,000 − 20,000 5102
= 1.57
P(stockout) = 0.5000 - 0.4418 = 0.0582 3.
Profit projections for the order quantities under the 3 scenarios are computed below: Order Quantity: 15,000 Sales Unit Sales
Total Cost
at $24
at $5
Profit
10,000 20,000 30,000
240,000 240,000 240,000
240,000 360,000 360,000
25,000 0 0
25,000 120,000 120,000
Order Quantity: 18,000 Sales Unit Sales
Total Cost
at $24
at $5
Profit
10,000 20,000 30,000
288,000 288,000 288,000
240,000 432,000 432,000
40,000 0 0
-8,000 144,000 144,000
Order Quantity: 24,000 Sales Unit Sales 10,000 20,000 30,000
Total Cost 384,000 384,000 384,000
at $24 240,000 480,000 576,000
at $5 70,000 20,000 0
Profit -74,000 116,000 192,000
Order Quantity: 28,000 Sales Unit Sales
Total Cost
at $24
at $5
Profit
10,000
448,000
240,000
90,000
-118,000
CP - 19
Chapter 3
Continuous Probability Distributions
20,000 30,000
4.
448,000 448,000
480,000 672,000
40,000 0
72,000 224,000
We need to find an order quantity that cuts off an area of .70 in the lower tail of the normal curve for demand.
30 % 70 %
20,000 Q
z = 0.52
z =
Q − 20,000
0.52
=
5102
Q = 20,000 + 0.52(5102) = 22,653
The projected profits under the 3 scenarios are computed below. Order Quantity: 22,653 Sales Unit Sales
Total Cost
at $24
at $5
Profit
10,000 20,000 30,000
362,488 362,488 362,488
240,000 480,000 543,672
63,265 13,265 0
-59,183 130,817 181,224
CP - 20
Chapter 3
5.
Continuous Probability Distributions
A variety of recommendations are possible. The students should justify their recommendation by showing the projected profit obtained under the 3 scenarios used in parts 3 and 4. An order quantity in the 18,000 to 20,000 range strikes a good compromise between the risk of a loss and generating good profits. While the students don't have the benefit of the following, a single-period inventory model (sometimes called the news vendor model) shows how to find an optimal solution. We outline that solution below. A single-period inventory model recommends an order quantity that maximizes expected profit based on the following formula: P(Demand ≤ Q* ) =
cu cu
+ co
* where P(Demand ≤ Q ) is the probability that demand is less than or equal to the recommended * order quantity, Q . cu is the cost of underestimating demand (having lost sales because of a stockout)
and co is the cost per unit of overestimating demand (having unsold inventory). Specialty will sell Weather Teddy for $24 per unit. The cost is $16 per unit. So, cu = $24 - $16 = $8. If inventory remains after the holiday season, Specialty will sell all surplus inventory for $5 a unit. So, co = $16 $5 = $11. P(Demand ≤ Q* ) =
8
=
8 + 11
0.4211
0.4211
0.5789
Q* z =
z =
Q
*
Q
*
− 20,000
5102 =
-0.20
= −0.20
20,000 − 0.20(5102) = 18,980
The profit projections for this order quantity are computed below: Order Quantity: 18,980 Sales Unit Sales
Total Cost
at $24
at $5
CP - 21
Profit
Chapter 3
Continuous Probability Distributions
10,000 20,000 30,000
303,680 303,680 303,680
240,000 455,520 455,520
44,900 0 0
CP - 22
-18,780 151,840 151,840