Session 8
Agenda
Solve a dynamic (adaptive) optimization problem as a static one
Simple policy in dynamic optimization
Applications:
–Optimal ordering policy for fashion products
–Test marketing a new product
Decision Rule
There is one major difference between stochastic dynamic programs
and stochastic static optimization.
For static solution: all future decisions are determined now, and are
fixed.
In a stochastic dynamic solution, the actual decision path will depend
on the way the random aspects play out.
“Solving‘” a stochastic dynamic program involves giving a decision
rule for every possible state, not just along an optimal path.
The Laptop-Bag Pricing Problem
A static solution for this multi-stage problem:
Charge $103/bag for the first 6 weeks
Charge $80/bag for the last 3 weeks
Stick to this solution until all bags are sold out
The Laptop-Bag Pricing Problem
An (adaptive) dynamic solution for the problem
Charge $103/bag for the first 6 weeks
For week 7,
if the remaining inventory is in [370, 530], charge $103/bag
if the remaining inventory is in [530, 550], charge $80/bag
For week 8,
if the remaining inventory is in [265, 325], charge $103/bag
if the remaining inventory is in [330, 410], charge $80/bag
If the remaining inventory is in [415, 425], charge $75/bag
For week 9 (more complicated)
The American Option Pricing Problem
A
14
15
16 Down Moves
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
B
C
D
E
F
G
H
2
$ 62.989
$ 50.000
$ 39.689
-
3
$ 70.699
$ 56.120
$ 44.547
$ 35.361
-
4
$ 79.353
$ 62.989
$ 50.000
$ 39.689
$ 31.505
-
5
$ 89.066
$ 70.699
$ 56.120
$ 44.547
$ 35.361
$ 28.069
-
$
$
$
$
$
$
$
-
0
1
2
3
5.523 $ 3.330 $ 1.707 $ 0.669
$ 7.875 $ 5.058 $ 2.804
$ 10.908 $ 7.465
$ 14.639
-
4
$ 0.152
$ 1.213
$ 4.489
$ 10.655
$ 18.495
-
5
$
$ 0.311
$ 2.163
$ 6.960
$ 14.639
$ 21.931
-
$
$
$
$
$
$
$
-
Stock Prices
0
1
0 $ 50.000 $ 56.120
1
$ 44.547
2
3
4
5
6
7
8
9
10
-
I
6
99.967
79.353
62.989
50.000
39.689
31.505
25.008
J
7
8
9
$
$
$
$
$
$
$
$
$
$
$
$
$ 1.302 $
$
$
$ 6.378 $ 2.664 $
$
$ 14.639 $ 10.311 $ 5.453 $
$ 21.931 $ 18.495 $ 14.639 $
$ 27.719 $ 24.992 $ 21.931 $
$ 30.149 $ 27.719 $
$ 32.314 $
$
10
10.311
18.495
24.992
30.149
34.242
$
$
$
$
$
$
$
$
$
$
-
9
141.352
112.203
89.066
70.699
56.120
44.547
35.361
28.069
22.281
17.686
M
10
158.654
125.937
99.967
79.353
62.989
50.000
39.689
31.505
25.008
19.851
15.758
$
$
$
$
$
$
$
$
$
-
8
125.937
99.967
79.353
62.989
50.000
39.689
31.505
25.008
19.851
L
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
-
7
112.203
89.066
70.699
56.120
44.547
35.361
28.069
22.281
K
Put Value
0
1
2
3
4
5
6
7
8
9
10
$
-
6
0.636
3.771
10.362
18.495
24.992
Static solution: exercise the option on period 6, no matter what
Adaptive dynamic solution:
•Do not exercise the option in the first three periods
•Period 4: exercise if the price drops to $35.36 or lower
•Period 5: exercise if the price drops to $31.50 or lower
Dynamic Decision via Solving a Static Problem
Adaptive dynamic decision:
• Do not exercise the option in the first three periods
• Period 6: exercise if the price drops to $36.07 or lower
• Period 7: exercise if the price drops to $39.14 or lower
• Period 8: exercise if the price drops to $42.47 or lower
• Period 9: exercise if the price drops to $46.08 or lower
• …
The dynamic decision rule is defined by cutoff points (prices).
• For a particular period, if the price is lower than the cutoff, then
exercise.
As long as the threshold prices are not affected by changes in the
random variables (stock prices), we can determine the threshold prices
now by solving a static optimization problem.
Optimal Ordering Policy for Fashion Products
Assume that Zarah has to place an order now to its supplier for women’s
leather biker jackets for the current selling season.
It can place a second order after week 3.
What should be the optimal order quantity (now and after week 3)?
Zarah has estimated that the demand for the first three weeks will be
1000 units, but forecasts tend to be wrong.
Additional information to consider:
•Full price $180/unit
•Ordering cost now: $80/unit
•Ordering cost in week 3: $90/units
•Salvage value after the selling season: $60/unit
•Fixed cost of 2nd order: $3000
Single Order Case
What if the supplier allows Zarah to order only once?
In this case, we need to model the demand of the entire selling season.
For the new product:
•We have the forecast for the first the first three weeks: 1000 units
•We need to find out the relation between
•The forecast of the first three weeks
•The actual demand of the entire season
Forecast vs. Actual Sales
A
B
1
Forecast
2 Product for Week 1-3
3
1
2063
4
2
1498
5
3
1848
6
4
1299
7
5
445
8
6
853
9
7
467
10
8
1345
11
9
1584
12
10
1483
C
D
Week 1
510
424
632
326
106
210
146
443
527
382
E
F
Actual Sales
Week 2
Week 3
688
720
349
430
528
571
281
301
160
129
297
258
153
123
433
403
421
431
320
462
G
Rest of the Season
6509
4453
5459
2984
1327
2310
1409
4536
4660
4570
12000
10000
y = 3.9676x - 189.7
R² = 0.9132
8000
Actual Sales
6000
Whole Season
4000
2000
0
0
500
1000
1500
Forecast Weeks 1-3
2000
2500
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.9556
0.9132
0.9123
719.1
100
ANOVA
df
Regression
Residual
Total
Intercept
Forecast
SS
1 532,789,821.1
98 50,671,947.4
99 583,461,768.5
Coefficients Standard Error
-189.7001
184.6856
3.9676
0.1236
MS
532,789,821.1
517,060.7
F
Significance F
1030.4
0.0000
t Stat
P-value
-1.0272 0.3069
32.1002 0.0000
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.9818
0.9639
0.9636
362.6
100
ANOVA
df
Regression
Residual
Total
Intercept
Actual
SS
1 344,289,858.6
98 12,884,459.9
99 357,174,318.5
Coefficients Standard Error
-17.8641
88.0907
3.4801
0.0680
MS
344,289,858.6
131,474.1
F
Significance F
2618.7
0.0000
t Stat
P-value
-0.2028 0.8397
51.1731 0.0000
9000
y = 3.4801x - 17.864
R² = 0.9639
8000
7000
6000
Actual Sales 5000
Whole Season 4000
3000
2000
1000
0
0
500
1000
1500
Actual Sales Weeks 1-3
2000
2500
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.9818
0.9639
0.9636
362.6
100
ANOVA
df
Regression
Residual
Total
Intercept
Actual
SS
1 344,289,858.6
98 12,884,459.9
99 357,174,318.5
Coefficients Standard Error
88.0907
-17.8641
0.0680
3.4801
MS
344,289,858.6
131,474.1
Significance F
F
0.0000
2618.7
P-value
t Stat
-0.2028 0.8397
51.1731 0.0000
Single Order Case
For the new product, the forecast for first three weeks is 1000 units.
Actual total sales = 3.9676*Forecast first three weeks – 189.7
= 3.9676* 1000 -189.7
= 3778
Should we order 3778 units if we want to maximize the expected profit?
Should we order more than this? Or less?
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
A
Parameters
Regular Price
Unit Order Cost Now
Unit Order Cost Later
Fixed Cost of 2nd Order
Salvage Value
Week 1-3 Forecast
B
$ 180
$
80
$
90
$ 3,000
$
60
1000
C
D
E
Week 1-3
Week 4-End
Decisions
First Order Quantity
Cutoff Point
On Hand at end of week 3
N/A
5000
N/A
N/A
Revenues
Full-price Unit Sales Weeks 1-3
Full-price Unit Sales Rest of season
Leftover Unit sales
Full-price revenue
Leftover revenue
Total revenue
N/A
N/A
N/A
Costs
First order cost
Fixed cost of second order
Cost of second order items
Total cost
Total Profit
N/A
H
I
N/A
N/A
J
Regression 1 (Weeks 1-3)
Intercept
Slope
Std Error
-39.13
0.89
137.35
Regression 2 (Week 4-End)
Intercept
Slope
Std Error
-17.86
3.48
362.59
Regression 3 (Whole Season)
Intercept
Slope
Std Error
Whole Season
Implied Second Stage Decision
Actual 2nd Order Quantity
F
G
Forecasts CB Green Cells
N/A
N/A
-189.70
3.97
719.07
A
1
2
3
4
5
6
7
Parameters
Regular Price
Unit Order Cost Now
Unit Order Cost Later
Fixed Cost of 2nd Order
Salvage Value
Week 1-3 Forecast
B
$ 180
$
80
$
90
$ 3,000
$
60
1000
2
3
4
5
6
7
8
9
10
11
12
13
14
15
I
Regression 1 (Weeks 1-3)
Intercept
Slope
Std Error
-39.13
0.89
137.35
Regression 2 (Week 4-End)
Intercept
Slope
Std Error
-17.86
3.48
362.59
Regression 3 (Whole Season)
Intercept
Slope
Std Error
J
-189.70
3.97
719.07
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
A
Parameters
Regular Price
Unit Order Cost Now
Unit Order Cost Later
Fixed Cost of 2nd Order
Salvage Value
Week 1-3 Forecast
Decisions
First Order Quantity
Cutoff Point
On Hand at end of week 3
B
$
$
$
$
$
C
180
80
90
3,000
60
1000
D
E
Week 1-3
F
G
Forecasts CB Green Cells
N/A
N/A
Week 4-End
N/A
4500
N/A
N/A
Whole Season
Implied Second Stage Decision
Actual 2nd Order Quantity
Revenues
Full-price Unit Sales Weeks 1-3
Full-price Unit Sales Rest of season
Leftover Unit sales
Full-price revenue
Leftover revenue
Total revenue
N/A
3777.945
N/A
N/A
N/A
722.0551154
$ 680,030.08
$ 43,323.31
$
723,353
Costs
First order cost
Fixed cost of second order
=MAX(B10-G13,0)
Cost of second order items
=B2*MIN(B10,G13)
Total cost
=B6*B20
=SUM(B21:B22)
Total Profit
$ 360,000
N/A
N/A
$ 360,000
$ 363,353
3777.944885
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
A
Parameters
Regular Price
Unit Order Cost Now
Unit Order Cost Later
Fixed Cost of 2nd Order
Salvage Value
Week 1-3 Forecast
B
$
$
$
$
$
C
180
80
90
3,000
60
1000
D
E
Week 1-3
Week 4-End
Decisions
First Order Quantity
Cutoff Point
On Hand at end of week 3
F
G
Forecasts CB Green Cells
N/A
N/A
N/A
N/A
4500
N/A
N/A
Whole Season
Implied Second Stage Decision
Actual 2nd Order Quantity
N/A
Revenues
Full-price Unit Sales Weeks 1-3
Full-price Unit Sales Rest of season
Leftover Unit sales
Full-price revenue
Leftover revenue
Total revenue
N/A
N/A
722.0551154
$ 680,030.08
$ 43,323.31
$
723,353
Costs
First order cost
Fixed cost of second order
Cost of second order items
Total cost
Total Profit
3777.945
$ 360,000
N/A
N/A
$ 360,000
$ 363,353
3777.944885
=B3*B10
=SUM(E18:E20)
H
I
J
Regression 1 (Weeks 1-3)
Intercept
Slope
Std Error
-39.13
0.89
137.35
Regression 2 (Week 4-End)
Intercept
Slope
Std Error
-17.86
3.48
362.59
Regression 3 (Whole Season)
Intercept
Slope
Std Error
-189.70
3.97
719.07
OptQuest comes up with ~3850 as the optimal order quantity.
Expected profit ~$356,000.
Define CO and CU to be the “costs” of over-ordering
and under-ordering, respectively.
In this case:
C = $80.00
P = $180.00
V = $60.00
Define CO and CU to be the “costs” of over-ordering
and under-ordering, respectively.
In this case:
CO
 V  C 
 60  80 
 $20.00
CU
 P C
 180  80
 $100.00
It can be shown that the optimal order quantity is
the value in the demand distribution that
corresponds to the “critical probability”:
Critical probability
CU

CU  CO
100

120
 0.1667
From the standard normal table, the z-value corresponding to a
0.1667 probability is -0.97.
z = -0.97
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Q
   0.1667
 3,777.94  0.1667719.07
 3,658
z = -0.97
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
What if a Second Order is Allowed?
Decisions
Expected Demand Weeks 1-3
Simulated Demand Weeks 1-3
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
A
Parameters
Regular Price
Unit Order Cost Now
Unit Order Cost Later
Fixed Cost of 2nd Order
Salvage Value
Week 1-3 Forecast
B
$
$
$
$
$
C
180
80
90
3,000
60
1000
D
E
Week 1-3
Week 4-End
Decisions
First Order Quantity
Cutoff Point
On Hand at end of week 3
3114.25
Revenues
Full-price Unit Sales Weeks 1-3
Full-price Unit Sales Rest of season
Leftover Unit sales
Full-price revenue
Leftover revenue
Total revenue
3000
5000
2000
4100
Whole Season
Implied Second Stage Decision
Actual 2nd Order Quantity
F
G
Forecasts CB Green Cells
847.04
900
N/A
I
Costs
First order cost
Fixed cost of second order
Cost of second order items
Total cost
Total Profit
$ 400,000
0
$
$ 400,000
Expected Demand 4-End
$ 368,000
J
Regression 1 (Weeks 1-3)
Intercept
Slope
Std Error
-39.13
0.89
137.35
Regression 2 (Week 4-End)
Intercept
Slope
Std Error
-17.86
3.48
362.59
Regression 3 (Whole Season)
Intercept
Slope
Std Error
Simulated Demand 4-End
0
900
3000
1100
$ 702,000
$ 66,000
$ 768,000
N/A
H
-189.70
3.97
719.07
F
G
1 Forecasts CB Green Cells
=J3+J4*B7
2
847.04
900
3
4
5
6
7
=J8+J9*G2
8
3114.25
3000
9
10
11
12
13 N/A
N/A
14
15
H
I
J
Regression 1 (Weeks 1-3)
Intercept
Slope
Std Error
-39.13
0.89
137.35
Regression 2 (Week 4-End)
Intercept
Slope
Std Error
-17.86
3.48
362.59
Regression 3 (Whole Season)
Intercept
Slope
Std Error
-189.70
3.97
719.07
F
G
1 Forecasts CB Green Cells
2
847.04
900
3
4
5
6
7
8
3114.25
3000
9
10
11
12
13 N/A
N/A
14
15
H
I
J
Regression 1 (Weeks 1-3)
Intercept
Slope
Std Error
-39.13
0.89
137.35
Regression 2 (Week 4-End)
Intercept
Slope
Std Error
-17.86
3.48
362.59
Regression 3 (Whole Season)
Intercept
Slope
Std Error
-189.70
3.97
719.07
A
14 Implied Second Stage Decision
15 Actual 2nd Order Quantity
16
17 Revenues
18 Full-price Unit Sales Weeks 1-3
19 Full-price Unit Sales Rest of season
20 Leftover Unit sales
21 Full-price revenue
22 Leftover revenue
23 Total revenue
B
C
0
D
=IF(B12<=B11, MAX(J3*G2+J4-B12,0),0)
Costs
900
3000
1100
$ 702,000
$ 66,000
$ 768,000
=MIN(G2,B10)
First order cost
=MIN(B12+B15,G8)
Fixed cost of second order
=MAX(B12+B15-B19,0)
Cost of second order items
=B2*SUM(B18:B19)
Total cost
=B6*B20
=SUM(B21:B22)
Total Profit
D
17
18
19
20
21
22
23
Costs
First order cost
Fixed cost of second order
Cost of second order items
Total cost
Total Profit
E
$ 400,000
0
$
$ 400,000
$ 368,000
F
=B3*B10
=IF(B15>0,B5,0)
=B15*B4
=SUM(E18:E20)
=B23-E21
G
OptQuest Best Solution:
Cutoff Point ~3,300
First Order Quantity ~4,460
Mean of Total Profit ~$355,742
Test Marketing a New Product
Q and H is a large consumer goods corporation.
They are thinking of marketing a new barbecue sauce nationwide.
The cost of nationally rolling out the product is estimated at $3.2 million.
The consensus view is that annual sales will equal 510,000 units.
Each unit sells for $4.50 and costs $3.20 to produce.
We believe the product will sell for ten years.
Assume a discount rate of 10%.
Annual sales = 510,000*($4.5-3.2) = $663,000
Total NPV of sales for 10 years =-PV(10%, 10, $663000)=$4.074 million
Should they market the new product nationwide?
But
The consensus view of the annual unit sales is not the actual unit sales.
Product
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Consenus
Forecast
559,502
964,473
984,187
443,883
704,814
220,327
192,038
453,347
321,447
225,307
120,672
362,419
935,354
285,593
772,230
242,632
379,353
952,275
629,192
Actual Sales
267,987
1,138,569
780,247
372,377
663,592
290,172
140,381
380,997
203,257
45,359
66,285
196,949
956,951
391,144
698,553
178,235
295,806
1,236,329
635,897
Test Marketing a New Product
If the actual annual units sales is only 300,000 units.
Annual sales = 300,000*($4.5-3.2) = $390,000
Total NPV of sales for 10 years =-PV(10%, 10, $390000)=$2.396 million
Which is about 0.8 million less than the cost of national rollout.
High risk if the consensus view is not accurate.
Test marketing to reduce uncertainty on (customer reaction and) actual
units sales. Cost of test marketing $90,000.
What is the optimal strategy?
Forecast After Test Marketing
Product
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Consenus Forecast
Before Test
559,502
964,473
984,187
443,883
704,814
220,327
192,038
453,347
321,447
225,307
120,672
362,419
935,354
285,593
772,230
242,632
379,353
952,275
629,192
Forecast
After Test Actual Sales
271,369
267,987
1,047,338
1,138,569
856,606
780,247
393,328
372,377
659,044
663,592
279,390
290,172
149,119
140,381
389,536
380,997
210,441
203,257
57,193
45,359
56,415
66,285
186,467
196,949
943,745
956,951
355,475
391,144
668,937
698,553
162,547
178,235
260,884
295,806
1,072,461
1,236,329
588,195
635,897
Consensus
Forecast
Test Market?
Yes
Update
Week 1-3 Forecast
No
Decide 1st Order Quantity
Decide Order Quantity
Observe 1-3
Demand
Decide 2nd Order Quantity
Observe
4-End Demand
Observe Demand
Consensus
Forecast
Regression?
Regression?
Updated
Week 1-3 Forecast
Observe 1-3
Demand
Regression?
Observe
4-End Demand
Observe Demand
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
A
Test Marketing
Mean
StDev
Input (Deterministic)
Price/unit
Variable Cost/unit
Test Market Cost
Product Life (year)
Discount Rate
Development Cost
Consensus forecast
Profit without TM
Decision with TM Option
Cutoff Point
Objective with TM Option
Profit with TM option
B
C
D
E
F
G
Actual/Consensus Actual/TF TF/Consensus
0.851
1.027
0.821
0.307
0.094
0.273
$
$
$
$
$
4.50
3.20
90,000.00
10
0.1
3,200,000
510,000
785,778
474,557
453,201
=CB.Normal(B3,B4)*B13
=CB.Normal(D3,D4)*B13
=CB.Normal(C3,C4)*E8
3,076,747
$
Random Parameters (units)
Actual Sales without TM
Forecast after TM
Actual Sales with TM
330,140
Implied Second Stage Decision
National Rollout?
1
=IF(E8>=B18,1,0)
H
A
1 Test Marketing
2
3 Mean
4 StDev
5
6 Input (Deterministic)
7 Price/unit
8 Variable Cost/unit
9 Test Market Cost
10 Product Life (year)
11 Discount Rate
12 Development Cost
13 Consensus forecast
14
15 Profit without TM
16
17 Decision with TM Option
18 Cutoff Point
19 Objective with TM Option
20 Profit with TM option
B
C
D
E
Actual/Consensus Actual/TF TF/Consensus
0.851
1.027
0.821
0.307
0.094
0.273
$
$
$
$
$
4.50
3.20
90,000.00
10
0.1
3,200,000
510,000
3,076,747
$
330,140
Random Parameters (units)
Actual Sales without TM
Forecast after TM
Actual Sales with TM
785,778
474,557
453,201
=-(B7-B8)*E7*PV(B11,B10,1)-B12
Implied Second Stage Decision
National Rollout?
1
=-B9+IF(E18=1,-(B7-B8)*E9*PV(B11,B10,1)-B12,0)
F
A
1 Test Marketing
2
3 Mean
4 StDev
5
6 Input (Deterministic)
7 Price/unit
8 Variable Cost/unit
9 Test Market Cost
10 Product Life (year)
11 Discount Rate
12 Development Cost
13 Consensus forecast
14
15 Profit without TM
16
17 Decision with TM Option
18 Cutoff Point
19 Objective with TM Option
20 Profit with TM option
B
C
D
E
Actual/Consensus Actual/TF TF/Consensus
0.851
1.027
0.821
0.307
0.094
0.273
$
$
$
$
$
4.50
3.20
90,000.00
10
0.1
3,200,000
510,000
119,386
467,449
498,422
(2,246,352)
600,000
$
Random Parameters (units)
Actual Sales without TM
Forecast after TM
Actual Sales with TM
Implied Second Stage Decision
National Rollout?
(90,000)
OptQuest: Optimal decision is ~354,000 units.
Estimated average profit ~ $532,375
0
A
1 Test Marketing
2
3 Mean
4 StDev
5
6 Input (Deterministic)
7 Price/unit
8 Variable Cost/unit
9 Test Market Cost
10 Product Life (year)
11 Discount Rate
12 Development Cost
13 Consensus forecast
14
15 Profit without TM
16
17 Decision with TM Option
18 Cutoff Point
19 Objective with TM Option
20 Profit with TM option
B
C
D
E
Actual/Consensus Actual/TF TF/Consensus
0.851
1.027
0.821
0.307
0.094
0.273
$
$
$
$
$
4.50
3.20
90,000.00
10
0.1
3,200,000
510,000
507,175
483,046
533,402
851,282
354,000
$
Random Parameters (units)
Actual Sales without TM
Forecast after TM
Actual Sales with TM
970,779
Implied Second Stage Decision
National Rollout?
1