Chapter 6
Continuous Probability Distributions
Case Problem: Specialty Toys
1. Information provided by the forecaster
At x = 30,000,
30,000 20,000 1.96
x
z
−−
= = =
2. @ 15,000
15,000 20,000 0.98
5102
z−
= = −
P(stockout) = 0.3365 + 0.5000 = 0.8365
@ 18,000
18,000 20,000 0.39
5102
z−
= = −
20,0
00
10,0
00
30,0
00
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
240,000
240,000
25,000
25,000
20,000
240,000
360,000
0
120,000
30,000
240,000
360,000
0
120,000
Order Quantity: 18,000
Sales
Unit Sales
Total Cost
at $24
at $5
Profit
10,000
288,000
240,000
40,000
-8,000
20,000
288,000
432,000
0
144,000
30,000
288,000
432,000
0
144,000
Order Quantity: 24,000
Sales
Unit Sales
Total Cost
at $24
at $5
Profit
10,000
384,000
240,000
70,000
-74,000
20,000
384,000
480,000
20,000
116,000
30,000
384,000
576,000
0
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
20,000
448,000
480,000
40,000
72,000
30,000
448,000
672,000
0
224,000
4. We need to find an order quantity that cuts off an area of .70 in the lower tail of the normal curve for
demand.
20,000 0.52
5102
Q
z−
==
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
362,488
240,000
63,265
-59,183
20,000
362,488
480,000
13,265
130,817
30,000
362,488
543,672
0
181,224
5. 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.
uo
+
where
*
P(Demand )Q
is the probability that demand is less than or equal to the recommended
order quantity,
*
Q
.
u
c
is the cost of underestimating demand (having lost sales because of a stockout)
and
o
c
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,
u
c
= $24 – $16 = $8. If inventory
c
o
20,000
30%
Qz = 0.52
70%
*8
8 11
+
*20,000 0.20
5102
Q
z−
= = −
*20,000 0.20(5102) 18,980Q= − =
The profit projections for this order quantity are computed below:
Order Quantity: 18,980
Sales
Unit Sales
Total Cost
at $24
at $5
Profit
10,000
303,680
240,000
44,900
-18,780
20,000
303,680
455,520
0
151,840
30,000
303,680
455,520
0
151,840
0.5789
Q*
z = -0.20
0.4211