Dynamic Revenue Management Through Cross-Selling in E-Commerce Retailing

Sergei Savin

Graduate School of Business, Columbia University

(Joint work with Sergei Netessine, The Wharton School Wen-Qiang Xiao, Columbia GSB)

Cross-Selling in Retailing

“Brick-and-Mortar” Retailing

– Product A and Product B may be sold separately as well as part of package P

» Shampoo and conditioner

» Christmas gift preparations (wrapping paper, tape, holiday cards, etc.)

» Starter kits for kid’s bicycles (bicycle, helmet, knee pads)

– Package contents and package price are static

Online retailing: packages can be composed and priced dynamically

$20.37

$51.87

– Example: buying books on Amazon.com

$31.50

$174.95

$174.95$143.45

Cross-Selling Configuration Number of Books Involved

7

51

6

14

6

4

4

8

- book on the top100 list, - book outside of top 100 list

Cross-Selling of Books on Amazon.com: Top 100 on 09/17/03

– Survey be E-tailing group: cross-selling is used by

» 62% of top internet retailers

» 100% of top internet retailers specializing in computers, books/music, consumer electronics, pet supplies

– Travel web sites: dynamic cross-selling of “air+hotel”, “air+car”, “air+hotel+car” packages

Cross-Selling in e-Retailing: Personalized Sales Recommendations

• “Personalization” approach to selecting packaging complements

• Designed to maximize probability of buying a package

Product Cross-Selling in the Literature

Economics and Marketing (review by V. Rao (1993))

» Static “demand-side” analysis: under what conditions cross-selling is profitable (Adams and Yellen (1976))

» No “supply-side” considerations

Operations

» Optimal pricing of product bundles in static environment (Hanson and Martin (1990))

» Stocking decisions for individual products and product bundles (Ernst and Kouvelis (1999))

» Multi-Product Revenue Management (Gallego and Van Ryzin (1997), Cooper and Zhang (2003))

Dynamic Cross-Selling Model

A company sells m items

We consider a cross-selling problem between two successive inventory replenishments

N decision epochs

At the beginning of each decision epoch, the inventory levels of all products are observed and packaging and pricing decisions are made

Class i customer is offered item i the (fixed) price pi and a an “item i – item j” (for some j) package at the price pi ≤ pij ≤ pi + pj.

During each decision epoch at most 1 customer arrives, asking for item i with probability i

Class i customer buys only item i with probability Fij(pij) and buys “item i – item j” package with probability 1-Fij(pij)

Objective: at each decision epoch select packages and their prices to maximize expected revenues over remaining horizon

Dynamic Packaging and Pricing Model

1 11

11

1

For 1, 1,..., , 1,..., ,

max max

1

"Boundary condition": 0 for any .

i ij i j

i

m

ijn i ij ij i n i ij ij n i jj i p p p p

i

m

i ni

N

I i m n N

V F p p V F p p V

V

V

I I e I e e

I

I I

What happens if we run out of one or more products?

• Emergency Replenishment Model (ERM): missing item i is procured at an additional cost bi

• Lost Sales Model (LSM): missing item is “out”

How to Price Static Packages?

i=1

i=3

i=2

Packaging Complement of Porteus’s book = Zipkin’s book:

C(2)=3

Packaging Primary of Porteus’s book = {Zipkin’s book, Heyman-Sobel’s book}:

P(2)={1,3}

Decomposition under Emergency Replenishment Model

1

1 1( )

1 1( )

The optimal expected revenue is a separable function: ,

where

1 1

max 1 , 1, 1,..., ,

and

ji

mi

n n n ii

i i in i i i n i i j n i

j P i

i ijij ji ji n i ji ji j n i i

pj P i

in

V V G I

G I p G I G I

F p G I F p p p G I I n N

G

I I

1 1( )

1 1( )

1

0 0 1 0

max 0 0 , 1,..., ,

0.

ji

i ii i n i i j n

j P i

i ijij ji ji n ji ji j n i

pj P i

iN i

p G b G

F p G F p p p G b n N

G I

Proposition

ERM Decomposition

i=1

i=3

i=2

12

32

2 22 2 2 1 2

2 2121 12 12 1 2 12 12 1 1 2

2 2323 32 32 1 2 32 32 3 1 2

21 2 3 1 2

1

max 1

max 1

1 .

n n

n np

n np

n

G I p G I

F p G I F p p p G I

F p G I F p p p G I

G I

*

a) is a non-decreasing concave function of ,1 .

b) Optimal bundle price , is non-increasing in ,

and is independent of , .

in i i

ij

j k

G I I n N

p n n

I I k j

I

Proposition

* *12 2 12 2

* *12 2 12 2

( 1, ) ( , ), 1,..., 1.

( , 1) ( , ), 1,..., .

p n I p n I n N

p n I p n I n N

i=1 i=2

i=3

Optimal Packaging is State-Dependent…

11 I

1

2

3 1

2

3

1

2

3 1

2

3

1

2

3

1

2

3

21 I 31 I

41 I 51 I 61 I

10,1,4,2 32 NnII

… and Dynamic

1n

1

2

3

2n 3n

4n 5n 6n

10,2,2,1 321 NIII

1

2

3 1

2

3

1

2

3 1

2

3 1

2

3

“Infinite Supply” Heuristic

*1 1, arg max

i ij i j

ijij ij ij i n i n i jp p p p

p n F p p p V V

I I e I e e

• Optimal Package Pricing

Heuristic Packaging/Pricing Approaches

arg maxi ij i j

Mijij ij ij i

p p p pp F p p p

Mijp

• “Infinite Supply” Pricing

is a lower bound on the optimal price, optimal when Ij is high

• “Infinite Supply” Packaging

( ) arg maxM M Mij ij ij i

j iC i F p p p

“Two-Stage” Heuristic

*1 1, arg max

i ij i j

ijij ij ij i n i n i jp p p p

p n F p p p V V

I I e I e e

1 1 1j

N n kN n kT Tn i n i j j j j

k I

N nV V b

k

I e I e e

• In period n, assume that there is no product packaging in periods n+1 to N

Heuristic Packaging/Pricing Approaches

'j

T N nij k N n kijTij i j j jT

k Iij ij

F p N np p b

kF p

• “Two-Stage” Pricing

• “Two-Stage” Packaging

2

'arg maxT

ij ijT

Tj ij ij

F pC i

F p

Testing “Infinite Supply” and “Two Stage” Heuristics

• Three products

• Prices of individual products are fixed and equal:

1 2 3 1p p p

expij ij ij ij iF p p p

• Exponential reservation price functions for packages:

• Various combinations of demand intensities i , i=1,2,3

1 , 0.8, 0.3,0,0.3,0.8.i iI N • Initial product inventories:

• Emergency replenishment prices: , 0.2,0.5,0.8.i ib p

• About 7000 problem instances

• “Infinite Supply” heuristic:

Average performance gap (as compared to optimal) is about 10%

• “Two Stage” heuristic:

Average performance gap is 0.12%

Worst performance gap is 0.69%

Impact of Initial Inventory

Inventory

parameter -0.8 -0.5 0 0.3 0.8

Average gap, “Infinite Supply”,% 18.47 11.50 10.18 7.50 5.48

Average gap, “Two Stage”,% 0.03 0.14 0.20 0.15 0.08

Impact of Emergency Replenishment Price

ER price

parameter 0.2 0.5 0.8

Average gap, “Infinite Supply”,% 2.63 9.38 19.87

Average gap, “Two Stage”,% 0.05 0.13 0.18

Summary

• Dynamic Cross-Selling Model

• Two types of “boundary” conditions: Emergency Replenishment and Lost Sales

• Emergency Replenishment:

Static Packaging: Decomposable Value Function

Dynamic Packaging: “Two Stage” Heuristic

• Lost Sales:

Few structural properties

Best Cross-Selling Approaches

Inventory « Exp. Demand

Inventory ≈ Exp. Demand

Inventory » Exp. Demand

ER Model Dynamic Cross-SellingInfinite Supply Heuristic

LS ModelNo Cross-Selling

Dynamic Cross-Selling

Infinite Supply Heuristic