inventory competition for newsvendors under the profit (and revenue) satisficing objective

SBE Supply Chain Seminar Series Nov. 30, 2007 Slide #1

Inventory Competition for Newsvendors under the Profit (and Revenue) Satisficing Objective

Xuan Zhao

Joint work with Victor ShiSchool of Business and Economics

Wilfrid Laurier University


Agenda

• Introduction and Motivation

• Results

• Conclusions


Why Newsvendors under Competition?

• Newsvendor model is the basic building block of stochastic inventory theory

• Understanding newsvendors under competition is a foundation to understand more complicated competitive system such as supply chain competition


Literature on Inventory Competition

• Research in this stream:

Parlar (1987),

Lippman and McCardle (1997),

Netessine and Rudi (2003),

Zhao and Atkins (2007)


What’s new?

• We consider risk-averse newsvendors

• Several ways to model risk aversion:

- mean variance analysis (Markowitz 1959)

- through measures of downside risk

- Semi-variance

- critical probability


What’s new?

In this research, we consider newsvendors striving to achieving certain profit and revenue targets: i.e., maximizing the probability to achieve both targets simultaneously (satisficing objective)


Literature on Target-based Decision Making: Satisficing objective

• First invented by Nobel Prize winner Simmon (1959): individuals and firms settle for “good enough target” performance measures

• Lanzillotti (1958) interviews 20 large companies: the most typical goal of managers was a target return on investment

• Shipley (1981) studies the objective of 728 British manufacturing firms: 2/3 of the firms use profit target or target rate of return on capital as important measures


More Literature

• Brown and Tang (2006) survey 250 MBA students and 6 big retail store (e.g., Sears and J.C. Penny): meeting targets on both profit and sales

• The classical utility-based decision making is equivalent to target-based decision making (Bordley and Kirkwood 2004)


Real Examples

• eBay (2004): profit of 33 cents a share vs. the target 34 cents a share stock price down 12%

• Yahoo! (2005): revenue 875 million vs. the target 881 million stock price down 10%


What are the other characteristics in our model?

• Newsvendors selling substitutable products, complementary products, or both simultaneously

• Newsvendors maximizing profit probability, revenue probability and both simultaneously

.


What are the other characteristics in our model?

• Newsvendors having general demand functions

- model I e.g., Netessine, S., N. Rudi. 2003

- model II e.g., Wang, Y., Y. Gerchak. 2001

Q = 1 2, , nq q q iQ = 1 1 1, ,i i nq q q q

( )ei iD Q

.

( )eiD Q

qi is the inventory level of newsvendor i


Agenda

• Introduction and Motivation

• Results

• Conclusions


A single newsvendor’s problem (with both profit and revenue targets)

• NV maximizes her profit and revenue probability, which is defined as

( )peP q = ( ) & ( )p p e eP q t q t

e (q) = ( )p q cq

( )p q = ( )r c q - ( )r v ( )q D

Profit target

Revenue target


oq

( )pP q

1p

Dt

Fr c

pt

r c

a.

( )peP q

p

e

t

t

r c

r

>

( )p q=

Theorem 1:


b.

p

e

t

t

r c

r

oq

( )1

( )

e p

D

c v t vtF

c r v

e pt t

c

1e

D

tF

r

et

r

( )peP q

Theorem 1:


oq

( )pP q

1p

Dt

Fr c

pt

r c

•

•

( )peP q

p

e

t

t

r c

r

>

p

e

t

t r c

r

oq

( )1

( )

e p

D

c v t vtF

c r v

e pt t

c

1e

D

tF

r

et

r

( )peP q( )p q=

Theorem 1:

Profit target

P&R targets


Inventory competition: demand model I


oq

( )pP q

1p

Dt

Fr c

pt

r c

•

•

( )peP q

p

e

t

t

r c

r

>

p

e

t

t r c

r

oq

( )1

( )

e p

D

c v t vtF

c r v

e pt t

c

1e

D

tF

r

et

r

( )peP q( )p q=


( )1

( )ei

e pi i i i i

Di i i

c v t v tF

c r v

1- ei

pi

Di i

tF

r c

( )ei iD Q

Profit target

P&R targets

Theorem 2:


1. For each NV adopting the only profit target, increasing her profit target will decrease the probability that substitutable NVs achieve targets, but increase the probability that complementary NVs achieve targets

2. For each NV adopting the P &R targets, increasing her profit target will increase the probability that substitutable NVs achieve targets, but decrease the probability that complementary NVs achieve targets

Theorem 3:



Inventory Competition: Demand Model II



• Additive model:

• Multiplicative model:

( )eiD Q

( )eiD Q = ( )i ig Q

( )eiD Q = ( )i ig Q

( )ig Q is increasing and concave function of qi given Q-i.

Petruzzi and Dada (1999)


a. For the additive models, there exists a pure-strategy Nash equilibrium.

Where solves

*piq = max ,

pi

ii i

tq

r c

*pe

iq = max ,e pi i

ii

t tq

c

( , )i i i

i

g q Q

q

=

i i

i i

c v

r v

iq

Theorem 4:



b. The sufficient condition for a unique pure-strategy Nash equilibrium to exist is

2 2

2

( ) ( )i i

i j ij

g Q g Q

q q q

=

Theorem 4:



For multiplicative model, there exists a pure-strategy Nash equilibrium.

The equilibrium can be characterized in a similar way as the additive case.

Theorem 5



A special case of multiplicative model


• A special case of multiplicative model: symmetrical NVs under the proportional allocation model (Wang and Gerchak 2001, Cachon 2003).

where

and D the total industry demand.

( )iD Q

=

/iDq q

q

=

1

nii

q



Suppose n symmetrical NVs using only profit target compete under the proportional allocation model. Then for any NV, her optimal stocking level and maximal profit probability depends on the number of NVs,

Theorem 6

*piq =

pt

r c

*piP = 1-

p

Dnt

Fr c

*piq =

( 1) pn t

c v

*piP = 1-

2 p

Dn t

Fr v

cr

vrn p

Demand Model II: A Special Case


Suppose n symmetrical NVs using both P & R targets compete under the proportional model. Then for any NV , her optimal stocking level and maximal probability depends on the number of NVs.

Theorem 7

*peiq =

e pt t

c

*peiP =

( )1

( )

e p

D

c v t vtF n

c r v

*peiq =

( 1) pn t

c v

*peiP = 1-

2 p

Dn t

Fr v

c

v

t

t

c

vn

p

epe )1(



• Observation:

when each NV uses P & R targets, the market can incorporate more NVs before it becomes highly competitive


pen pn


Conclusions

• Risk-averse newsvendors under competition• A single newsvendor with both profit and

revenue targets• With demand model I (no assumptions on

substitutable/complementary product), each NV optimally stocks as if she is independent from other NVs. However, her probabilities to achieve the targets depend on the stocking levels (as well as targets) of other NVs.


• With demand model II, we characterize the equilibrium stocking levels and explore conditions for a unique equilibrium

• We study symmetrical NVs under the proportional allocation model as a specific example

• We characterize the critical threshold number of newsvendors where there is a change in the competitive situation of the market


• This strongly indicates the results and managerial insights based on the profit maximizing objective may not be generalized to other objectives, including the risk averse agents with satisficing objectives (target based decision-making).

inventory competition for newsvendors under the profit (and revenue) satisficing objective

Documents