Sport Obermeyer
 What to order?
 What are the issues?

A Sample Problem
 Commit 10,000 units before show
 Commit 10,000 units after show
 Minimum of 600 units

A First Approach
 Ignore differences in
 Profit margins
 Salvage values

 Ignore minimum lot sizes
 Consider only first order cycle

Sample Problem
Style
Mean Forecast Std Deviation in Demand
Gail
1,017
388
Isis
1,042
646
Entice
1,358
496
Assault
2,525
680
Teri
1,100
762
Electra
2,150
807
Stephanie
1,113
1,048
Seduced
4,017
1,113
Anita
3,296
2,094
Daphne
2,383
1,394

Normal Distribution
0.45

0.4

0.35

0.3

0.25

0.2

0.15

Std Dev.s

0.1

0.05

0
-5

-4

-3

-2

-1

0

1

2

3

4

5

Idea 1
 Make all products equally likely to sell out
 Choose a single std dev. To set production quotas for all products

What should the Std. Dev. Be?
Style
Gail
Isis
Entice
Assault
Teri
Electra
Stephanie
Seduced
Anita
Daphne

Mean
Forecast
1,017
1,042
1,358
2,525
1,100
2,150
1,113
4,017
3,296
2,383

Std Deviation in
Demand
388
646
496
680
762
807
1,048
1,113
2,094
1,394
Total Production

Order
Amount
1,017
1,042
1,358
2,525
1,100
2,150
1,113
4,017
3,296
2,383
20,001

Std. Devs
0
0
0
0
0
0
0
0
0
0
0
Probability of Sell out

Probability of Sell Out
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
Number of Standard Deviations
50%

Normal Distribution
0.45

Set order Qty to this many std. devs
0.4

Probability we discount last item =

Probability we stock out =
Probability
demand exceeds over qty =
0.86

0.35

0.3

0.25

Probability demand is smaller than order quantity =

0.2

0.15

Std Dev.s

0.1

0.05

0.14
0
-5

-4

-3

-2

-1

0

1

2

3

4

5

What’s Wrong with This?
 What else should we be looking at?
 Still just worried about
 Order up to 10,000
 One order cycle
 No minimum order qty.

A Second Idea
 Look at 1 Product
 How to trade off risks of overstock
(discounting) vs risks of understock (lost sales)?  If we order Q
 The last item faces what risk of being discounted?
 Probability Demand < Q = F(Q)

 The last item faces what risk of selling out
 Probability Demand > Q = 1 - F(Q)

We want to be indifferent
 We want two to be equal
 Expected loss from Overstock =
CO*F(Q)
 Expected loss from Lost Sale = CL*(1F(Q))
 A little Algebra:
 F(Q) = CL/(CO+CL)

Example
 Oversimplification
 Lost Sale: CL = Selling Price - Cost
 Discount: CO = Cost - Salvage Value
 Electra:
 Selling Price $173
 Cost $ 50
 Salvage
$ 0

 Lost Sale: CL = $123
 Discount: CO = 50
 Want Probability of Discount = F(Q) = 123/173 = 0.71
 Find Q with this cumulative probability: ~2,599

Balancing Risks
Style
Gail
Isis
Entice
Assault
Teri
Electra
Stephanie
Seduced
Anita
Daphne

Mean
Forecast
1,017
1,042
1,358
2,525
1,100
2,150
1,113
4,017
3,296
2,383

Probability of
Style
Sell Out
Gail
0.86
Isis
0.86
Entice
0.86
Assault
0.86
Teri
0.86
Electra
0.86
Stephanie
0.86
Seduced
0.86
Anita
0.86
Daphne
0.86

Std Deviation in
Demand
388
646
496
680
762
807
1,048
1,113
2,094
1,394
Total Production

Expect Cost of
Lost Sale
$
51.33
$
41.92
$
25.67
$
34.22
$
62.45
$
105.23
$
71.01
$
19.68
$
36.79
$
83.84

Order
Amount
Std. Devs
606
(1.06)
357
(1.06)
832
(1.06)
1,804
(1.06)
292
(1.06)
1,294
(1.06)
2
(1.06)
2,837
(1.06)
1,075
(1.06)
905
(1.06)
10,003
-1.0605
Probability of Sell out
Probability
Last Item is Discounted
0.14
0.14
0.14
0.14
0.14
0.14
0.14
0.14
0.14
0.14

Expect Cost of
Discount
$
7.22
$
7.22
$
7.22
$
7.22
$
7.22
$
7.22
$
7.22
$
7.22
$
7.22
$
7.22

Salvage
Price
Cost
Value
$
110 $
50 $
$
99 $
50 $
$
80 $
50 $
$
90 $
50 $
$
123 $
50 $
$
173 $
50 $
$
133 $
50 $
$
73 $
50 $
$
93 $
50 $
$
148 $
50 $