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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