Transcript O - Duke University`s Fuqua School of Business

Structural Estimation of the
Effect of Out-of-Stocks
Andrés Musalem
Marcelo Olivares
Eric T. Bradlow
Christian Terwiesch
Daniel Corsten
Duke U. (Fuqua)
Columbia U. (CBS)
U. of Pennsylvania (Wharton)
U. of Pennsylvania (Wharton)
IE Business School
Agenda
•
•
•
•
•
•
•
Motivation & Managerial issues
Contribution
Model & Methodology
Empirical Results
Managerial Implications
Conclusions
Big picture
Motivation
Managerial Issues:
• What fraction of consumers were exposed to an out-ofstock (OOS)?
• How many choose not to buy? (money left on the table)
• How many choose to buy another product?
• Can we reduce lost sales?
• What is the impact of these policies on the retailer’s profits?
• Can OOS’s lead to misleading demand estimates?
(assortment planning, inventory decisions)
…Motivation
• Dealing with OOS’s:
– Operations Management:
• Tools for assortment and inventory management (e.g.,
Mahajan and van Ryzin 2001) given a choice model.
– Marketing:
• Most applications of demand estimation in the marketing
literature ignore out-of-stocks (OOS)
• But…
…Motivation
• Marketing:
– Assume:
• 0 sales => no availability
• Positive sales => availability (e.g., ACV weighted distribution)
– Anupindi, Dada and Gupta (1998):
• Vending Machines Application / EM
• Jointly model sales and availability
• One-Stage Substitution assumption.
– Kalyanam et al. (2007):
• COM-Poisson, reduced-form model of substitution, categorical variables.
– Bruno and Vilcassim (2008) extension of BLP:
• ACV as a proxy for product availability
• P(OOS Brand A) independent of OOS for Brand B.
• Zero sales issues (slow-moving items).
– Conlon and Mortimer (2007):
• EM method becomes more difficult to implement as the # of products
simultaneously OOS increases.
Contribution: What’s new?
1. Joint model of sales and availability consistent with utility
maximization (structural demand model)
2. No restrictive assumptions about availability (e.g., OOS
independence)
3. No restrictive assumptions about substitution (e.g., one-stage
substitution)
4. Multiple stores / relatively large number of SKUs
5. Heterogeneity: Observed (different stores) / Unobserved (within
stores)
6. Products characteristics: categorical and continuous
7. Simple expressions to estimate lost sales / evaluate policies to
mitigate the consequences of OOS’s.
Modeling the impact of OOS:
• A simple way to capture the effect of an OOS
(reduced-form):
– If an OOS is observed in period t:
f(Salesjt)=Xjt’+ OOSjt+jt
Mktg Variables
OOS dummy variable
– However, it is important to determine when the
product became out-of-stock.
– Why?
consumer
choice
beg inv A
beg inv B
oos A
oos B
1
A
10
5
no
no
2
A
9
5
no
no
3
A
8
5
no
no
4
B
7
5
no
no
5
A
7
4
no
no
6
O
6
4
no
no
7
A
6
4
no
no
SA= number of customers
buying A = 10.
8
A
5
4
no
no
9
A
4
4
no
no
SB= number of customers
buying B =3.
10
O
3
4
no
no
11
A
3
4
no
no
IA= inventory at the beginning
and the end of the period for
brand A: 100.
12
A
2
4
no
no
13
A
1
4
no
no
14
O
0
4
yes
no
15
B
0
4
yes
no
16
O
0
3
yes
no
17
O
0
3
yes
no
18
B
0
3
yes
no
19
O
0
2
yes
no
N=20
O
0
2
yes
no
Example:
Available information:
•
•
•
•
•
N= total number of
customers=20.
IB= inventory at the beginning
and the end of the period for
brand B: 52.
consumer
choice
beg inv A
beg inv B
oos A
oos B
1
A
10
5
no
no
2
A
9
5
no
no
3
A
8
5
no
no
4
B
7
5
no
no
5
A
7
4
no
no
6
O
6
4
no
no
7
A
6
4
no
no
SA= number of customers
buying A = 10.
8
A
5
4
no
no
9
A
4
4
no
no
SB= number of customers
buying B =3.
10
O
3
4
no
no
11
A
3
4
no
no
IA= inventory at the beginning
and the end of the period for
brand A: 100.
12
A
2
4
no
no
13
O
1
4
no
no
14
O
1
4
no
no
15
B
1
4
no
no
16
O
1
3
no
no
17
O
1
3
no
no
18
B
1
3
no
no
19
O
1
2
no
no
N=20
A
1
2
no
no
Example:
Available information:
•
•
•
•
•
N= total number of
customers=20.
IB= inventory at the beginning
and the end of the period for
brand B: 52.
Demand Model:
• Multinomial Logit Model with heterogeneous
marketing demand
customers.
availability
variables
shock
indicator
choice
product
P( yitm  j ) 
consumer
market
period
aijtm e
J
itm x jtm  jtm
1   aiktm e
k 1
itm xktm ktm
Demand Model:
• Multinomial Logit Model with heterogeneous
marketing demand
customers.
availability
variables
shock
indicator
choice
product
P( yitm  j ) 
consumer
market
period
aijtm e
J
itm x jtm  jtm
1   aiktm e
itm xktm ktm
k 1
• Heterogeneity:
itm ~ MVN( m , ),
demographics
m   ' Zm
Estimation:
•
If availability and individual choices were observed (aijtm)
=> standard methods
•
Solution: data augmentation conditional on aggregate data
(following Chen & Yang 2007; Musalem, Bradlow & Raju 2007, 2008)
Key elements:
1. Use aggregate data to formulate constraints on the
unobserved individual behavior.
2. Define a mechanism to sample availability & choices from
their posterior distribution.
Simulating Sequence of Choices
• Constraints:
choice
indicator
N
Choices
w
i 1
inventory faced
by customer i
Inventory
ijtm
sales
 S jtm
initial
inventory
i 1
I ijtm  I jtm   whjtm
h 1
product availability
indicator
Product
Availability
aijtm  1I 0
ijtm
Constraints
consumer
choice
beg inv A
beg inv B
1-aiA
1-aiB
1
A
10
5
no
no
2
A
9
5
no
no
3
A
8
5
no
no
4
B
7
5
no
no
5
A
7
4
no
no
6
O
6
4
no
no
7
A
6
4
no
no
SA= number of customers
buying A = 10.
8
A
5
4
no
no
9
A
4
4
no
no
SB= number of customers
buying B =3.
10
O
3
4
no
no
11
A
3
4
no
no
IA= inventory at the beginning
and the end of the period for
brand A: 100.
12
A
2
4
no
no
13
A
1
4
no
no
14
O
0
4
yes
no
15
B
0
4
yes
no
16
O
0
3
yes
no
17
O
0
3
yes
no
18
B
0
3
yes
no
19
O
0
2
yes
no
N=20
O
0
2
yes
no
Out-of-Stocks (OOS)
Available information:
•
•
•
•
•
N= total number of
customers=20.
IB= inventory at the beginning
and the end of the period for
brand B: 52.
consumer
choice
beg inv A
beg inv B
1-aiA
1-aiB
1
A
10
5
no
no
2
A
9
5
no
no
3
A
8
5
no
no
4
B
7
5
no
no
5
A
7
4
no
no
6
O
6
4
no
no
7
B
6
4
no
no
SA= number of customers
buying A = 10.
8
A
6
4
no
no
9
A
5
4
no
no
SB= number of customers
buying B =3.
10
O
4
4
no
no
11
A
4
4
no
no
IA= inventory at the beginning
and the end of the period for
brand A: 100.
12
A
3
4
no
no
13
A
2
4
no
no
14
O
1
4
no
no
15
A
1
4
no
no
16
O
0
3
yes
no
17
O
0
3
yes
no
18
B
0
3
yes
no
19
O
0
2
yes
no
N=20
O
0
2
yes
no
Out-of-Stocks (OOS)
Available information:
•
•
•
•
•
N= total number of
customers=20.
IB= inventory at the beginning
and the end of the period for
brand B: 52.
Estimation
Gibbs Sampling:
• The choices of the consumers in a given pair
are swapped according to the following fullconditional probability:
choices in new
sequence
15
p( swap | *) 
p
i 7
15
p
i 7
iyi *
iyi *
product availability
based on new sequence
(ai *)
15
(ai *)   piyi (ai )
i 7
Estimation:
Gibbs Sampler:
Initial Values:
Sequence of Choices,
Availability and
Demand Parameters
Individual
Choices &
Availability
Hyper
Parameters
MCMC
Simulation
Individual
Parameters
Demand
Shocks
Numerical Example:
• Choice Set: J=10 products + no-purchase.
• Markets: M=12 markets
• Utility function:
Product
– Covariates:
1
2
• X1-X3: dummy variables (2 brands, purchase option)
3
• X4: continuous variable~N(2,1)
4
5
– Preferences in each market ~ N( ,):
6
m
• m  Z m ,
Z1m =1; Z2 m ~ U ( 1.5,1.5)
7
8
9
10
• =diag( 0, 0, 0.8, 2)
– jtm~N(0,0.5)
x1 x2 x3
1
1
1
0
0
0
0
0
0
0
0
0
0
1
1
1
0
0
0
0
1
1
1
1
1
1
1
1
1
1
x4
0.04
-0.20
-0.02
0.16
-0.60
0.61
0.57
-0.50
-0.48
-0.12
…Numerical Example
•
Two models:
1. Ignoring OOS (Benchmark): all products are
available all the time
2. Full model: jointly modeling demand and
availability
First Case: OOS=29%
mean of pref. coefficients
interaction with z2
heterogeneity var()
Second Case: OOS=1.3%
mean of pref. coefficients
interaction with z2
heterogeneity var()
Simulation Study: 50 replications
Summary statistics for the posterior mean for each model across 50 replications.
mean of pref. coefficients
interaction with z2
heterogeneity
var()
Estimating Lost Sales:
• Let A*: Set of all products
• Let Ai: Set of missing products
• Probability of a given consumer having chosen
one of the missing alternatives had it been
available:
Estimating Lost Sales:
• Lost Sales:
MCMC draws
Data Set:
• M=6 stores from a major retailer in Spain
• J=24 SKUs (shampoo)
• T=15 days
• Sales and price data for each SKU in each day
and periodic inventory data
• Demographics (income)
Summary Statistics
Empirical Results:
Empirical Results:
Estimating Lost Purchases:
Store 1
Store 2
Store 3
Store 4
Store 5
Store 6
% Lost Sales
% Lost Sales vs. OOS incidence
30%
9.5%
Number of OOS products
Dynamic Pricing: Sales Improvement
• Lost sales reduction after a temporary price
promotion:
– It’s not equal to the anticipated change in sales!
– Instead, it’s equal to the fraction of consumers who
meet the following 3 requirements:
• Did not buy any products
• Would have purchased a product had all alternatives been
available
• Would purchase one of the available alternatives if a discount
is offered.
Lost Sales Reduction
• Market 5, Day 3 (p=-20%):
– 10 Missing products: 4 (Timotei), 9 (Other), 10-13
(Pantene), 14 (Other), 18-19 (H&S), 23 (Cabello
Sano)
Market 5, Day 3 (10 products missing)
16.0%
15%
14.0%
12.0%
10.0%
Herbal Essence (17)
8.0%
All other products
6.0%
4.0%
2.0%
3.5%
2.6%
0.6%
0.0%
Lost Sales Reduction
Profit Change
Lost Sales Reduction
• Market 2, Day 15 (p=-20%):
– Only 1 missing product: SKU 15 (Pantene)
Market 2, Day 15 (1 product missing)
10.00%
5.00%
4.50%
3.20%
0.00%
-5.00%
Lost Sales Reduction
Profit Change
-10.00%
-8%
Pantene (13)
-15.00%
Herbal Essence (17)
-20.00%
-25.00%
-30.00%
-35.00%
-40.00%
-33%
Conclusions:
• Bayesian methods / data augmentation enable us to jointly model
choices and product availability w/o restrictive assumptions on:
– Joint probability of out-of-stocks / substitution
• Key: use available information to formulate constraints on
unobserved individual data:
– Constraints and Data Augmentation
• As a byproduct, we obtain simple expressions to:
– Estimate the magnitude of lost sales
– Assess effectiveness of policies aimed at mitigating the costs of OOS’s
• Several extensions are possible
Big Picture:
• Many situations in which we don’t observe
individual behavior, but we may have some
aggregate or limited information.
• Key: use aggregate data to formulate constraints
on the unobserved individual behavior.
–
–
–
–
Dependent variables: Choices
Independent variables: Coupon promotions
Shopping Environment: Out-of-stocks
Other applications: Shopping paths