Estimation of Retail Demand Under Partially-Observed Out-

of-Stocks

Agenda

• Motivation• Big Picture• Contribution• Model & Methodology• Empirical Results• Managerial Implications• Extensions• Conclusions

Motivation: iPhone

How would the analyst with Apple store data know whetherwhen you went to buy the product the store was OOS?

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– Environment: Out-of-stocks– Other applications: Shopping paths

• What fraction of consumers were exposed to an out-of-stock (OOS)?

• How many choose not to buy? (money left on the table)

• How many choose to buy another product?

• Can we reduce lost sales via improved inventory methods?

• What is the impact of these policies on the retailer’s profits?

• Can OOS’s lead to misleading demand estimates? (assortment planning, inventory decisions)

Managerial Issues:

…Motivation

• Dealing with OOS’s:

– Operations Management: • Tools for assortment and inventory management (e.g., Mahajan and

van Ryzin 2001) given a choice model.

– Economics:• Conlon and Mortimer (2007): ECM algorithm, E-step becomes

harder to derive/implement as the number of simultaneous out-of-stocks increases.

– Marketing:• Most applications of demand estimation in the marketing literature

ignore out-of-stocks (OOS) or treat it as an outcome to be modeled exogenously.

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 OOSs.

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

– However, it is important to determine when the product became out-of-stock.

– Why?

Mktg Variables OOS dummy variable

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

8 A 5 4 no no

9 A 4 4 no no

10 O 3 4 no no

11 A 3 4 no no

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

Available information:

• N= total number of customers=20.

• SA= number of customers buying A = 10.

• SB= number of customers buying B =3.

• IA= inventory at the beginning and the end of the period for brand A: 100.

• IB= inventory at the beginning and the end of the period for brand B: 52.

Example:

Example:

Available information:

• N= total number of customers=20.

• SA= number of customers buying A = 10.

• SB= number of customers buying B =3.

• IA= inventory at the beginning and the end of the period for brand A: 100.

• IB= inventory at the beginning and the end of the period for brand B: 52.

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

8 A 5 4 no no

9 A 4 4 no no

10 O 3 4 no no

11 A 3 4 no no

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

Demand Model:

• Multinomial Logit Model with heterogeneous customers.

1

( )1

itm jtm jtm

itm ktm ktm

xijtm

itm Jx

iktmk

a eP y j

a e

consumer

product

period

choice

availability indicator

marketing variables

market

demand shock

Demand Model:

• Multinomial Logit Model with heterogeneous customers.

• Heterogeneity:

~ MVN( , ), 'itm m m mZ

demographics

1

( )1

itm jtm jtm

itm ktm ktm

xijtm

itm Jx

iktmk

a eP y j

a e

consumer

product

period

choice

availability indicator

marketing variables

market

demand shock

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

1

1

1

01

ijtm

N

ijtm jtmi

i

ijtm jtm hjtmh

ijtm I

w S

I I w

a

choice indicator

Choices

Inventory

Product Availability

initial inventory

sales

inventory faced by customer i

product availability indicator

Constraints

• 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

8 A 5 4 no no

9 A 4 4 no no

10 O 3 4 no no

11 A 3 4 no no

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

Available information:

• N= total number of customers=20.

• NA= number of customers buying A = 10.

• NB= number of customers buying B =3.

• IA= inventory at the beginning and the end of the period for brand A: 100.

• IB= inventory at the beginning and the end of the period for brand B: 52.

Out-of-Stocks (OOS)

Available information:

• N= total number of customers=20.

• NA= number of customers buying A = 10.

• NB= number of customers buying B =3.

• IA= inventory at the beginning and the end of the period for brand A: 100.

• IB= inventory at the beginning and the end of the period for brand B: 52.

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

8 A 6 4 no no

9 A 5 4 no no

10 O 4 4 no no

11 A 4 4 no no

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)

15

*7

15 15

*7 7

( *)( | *)

( *) ( )

i

i i

iy ii

iy i iy ii i

p ap swap

p a p a

Estimation

Gibbs Sampling:• The choices of the consumers in a given pair

are swapped according to the following full-conditional probability:

choices in new sequence product availability

based on new sequence

Estimation:

Initial Values: Sequence of Choices,

Availability and Demand Parameters

IndividualChoices & Availability

Individual Parameters

Hyper Parameters

Gibbs Sampler:

MCMC Simulation

DemandShocks

Simulation Study:

• Choice Set: J=10 products + no-purchase.• Markets: M=12 markets• Utility function:

– Covariates: • X1-X3: dummy variables (2 brands, purchase option)

• X4: continuous variable~N(2,1)

– Preferences in each market ~ N( ,):•

=diag( 0, 0, 0.5, 2)

jtm~N(0,0.5)

m1 2, Z =1; Z ~ ( 1.5,1.5)m m m mZ U

Product x1 x2 x3 x4

1 1 0 1 0.042 1 0 1 -0.203 1 0 1 -0.024 0 1 1 0.165 0 1 1 -0.606 0 1 1 0.617 0 0 1 0.578 0 0 1 -0.509 0 0 1 -0.48

10 0 0 1 -0.12

…Simulation Study

• Two models:

1. Ignoring OOS: all products are available all the time

2. Full model: jointly modeling demand and availability

First Case: OOS=35%

mean of pref. coefficients interaction with z2 heterogeneity var()

Second Case: OOS=1.4%

mean of pref. coefficients interaction with z2 heterogeneity var()

Results:

true true

estim

ated

estim

ated

• Fraction of consumers experiencing an OOS for product 1:

est(%) = 1-0.5*(S1tm+Ntm)/Ntm est(%) : simulated from posterior

R = 0.70

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

Real 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:

Estimating Lost Purchases:

Market 1 Market 2

Market 3 Market 4

Market 5 Market 6

Number of OOS products

% L

ost

Sal

es% Lost Sales vs. OOS

incidence

Dynamic Pricing: Sales Improvement

Missing products in Day 5 & Market 5: 4 (Timotei), 9 (Other), 10-13 (Pantene), 14 (Other), 18-19 (H&S), 23 (Cabello Sano)

Dynamic Pricing: Profit Improvement

Item implied by Lost Revenue ≠ Lost Profit ≠ Most Frequent OOS

Extensions / Next Steps

• Behavioral issues (e.g., complexity, variety)• Backorder effects• Purchase quantity model• Price Endogeneity• Sampling: k components instead of 2• Infer OOS without (periodic) inventory data• In-Store Shopping Behavior

Behavioral Issues:

Choice Complexity:

• Current model: Ui0tm=i0tm

• Instead: Ui0tm= f( ; ) + i0tm1

J

ijtmj

a

Proxy for Complexity

outside good

Backorder Effects:

Backorder:

• Current model: Ui0tm=i0tm

• Instead: Ui0tm= g( ; )+i0tm

1 11

1{ 0} (1 )J

it m ijt mj

y a

Previous OOS

outside good

Previous no purchase

Quantity Decisions

• Sampling Choices and Quantities:

Period

Consumer 1 2 3

1 B AAA 0

2 ? BB AAA

3 AA B A

4 ? A BBB

N=5 AA BBB A

Total A 5 4 5

Total B 3 6 3

choices

• For simplicity: no variety seeking.

• What are the feasible values of the choices and quantities of consumers 2 and 4 in period 1?

(BB, A) & (A, BB)

• Update product inventory for customers 2, 3 and 4.

~ N(0, )tm

tm

Price Endogeneity:

• Very limited price variation for each SKU within market.

• Price endogeneity could arise from price differences across markets.– Bayesian instrumental variables approach (e.g.,

Yang, Chen and Allenby 2003; Conley et al. 2008):

pjtm= zjtm + jtm

Sampling Choices in groups of k components.

• Example: k=3

Period

Consumer 1 2 3

1 B A A

2 ? B A

3 ? B A

4 ? A B

N=5 A B A

Total A 3 2 4

choices• What values of y21, y31 and y41 are consistent with the sales data?

(A,A,B) (A,B,A) (B,A,A)

• Assign (A,A,B) with the following probability:

Prob((A,A,B)|*)=

Note: number of terms in the denominator may increase at k! rate (e.g., ABC, ACA, BAC, BCA, CAB, CBA).

Sampling Choices in groups of k components.

• Example: k=3

Period

Consumer 1 2 3

1 B A A

2 ? B A

3 ? B A

4 ? A B

N=5 A B A

Total A 3 2 4

choices

2A1 3A1 4B1

2A1 3A1 4B1 2A1 3B1 4A1 2B1 3A1 4A1

p p p

p p p p p p p p p

( , , )

( , , ) ( , , ) ( , , )

A A B

A A B A B A B A A

• What values of y21, y31 and y41 are consistent with the sales data?

(A,A,B) (A,B,A) (B,A,A)

• Assign (A,A,B) with the following probability:

Prob((A,A,B)|*)=

In-Store Shopping Behavior

• Using RFID technology it is possible to track the location of shopping carts in a grocery store every 5 seconds (disaggregate data).

• Alternatively: record the number of shopping carts that pass through a measuring point (aggregate data).

– Infer the trajectory of shopping carts using only these aggregate measurements.

A B

CD

qA =10 qB =5

qC =7qD =10

qAB =5

qBC =4qBD =1qAC =3

qAD =2

qCD =7

In-Store Shopping Behavior

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 OOSs

• Several extensions are possible…

Thank You