DEWEY
HD28
.M414
^
ffiAUG 181988
1.CCO
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Make-To-Order vs. Make-To-Stock:
The Role of Inventory
in
Delivery-Time Competition
Lode Li
Sloan School of Management
Massachusetts Institute of Technology
Cambridge,
MA
02139
42000
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
)
Make-To-Order vs. Make-To-Stock:
The Role
of Inventory in Delivery-Time Competition
Lode Li
Sloan School of Management
Massachusetts Institute of Technology
Cambridge, MA 02139
#2000
Make-To-Order
The Role
Make-To-Stock:
vs.
of Inventory in Delivery-Time Competition
Lode
Li*
Sloan School of Maiiaprmont
Massarhusetts Institute of Technology
Cambridf^e,
MA
02139
September, 1987
Abstract
The paper formulates
a set of stochastic
models
to study the i)roduction
policies of a firm that considers explicitly the behavior of customers
We
start with a single firm i)roduction control problem.
explicitly,
The
and inventory
and com])eting
ojjtimal jiolicy
is
firms.
solved
and the optimal mix between make-to-nrder and make-to-stock operations
determined with customers characterized by patience
mula. The model
is
levels in a simple newsl)oy-like for-
then extended to a n-firm market game
orders in the aspect of early delivery.
One could
economic phenomena such as ecouomi(>s
numerical methods which can compute an
which firms compete
for
The
demand
for produce-to-stock,
of scale, uncertainty, or seasonality
can induce production to inventory, and that com])etition of
welfare while decreases the producer's welfare.
in
think of this setting as an oligopoly racing
market. The analysis shows that competition can breed a
just as other
is
this
kind increases the buyer's
pai)er also suggests the analytical
e(iuilil)rium of the stocking
and
game.
*This paper was stinmlafed by the prosenf atioii given liy John nolirrt"^ and the related di«r\i'i>iion' during the Production
and Operations Manapemeiil Summer Camp at Stanford University. 1987. am also indebted to the comments of David Kreps,
Charles Fine and the seminar attendaut<i at MIT and UniversitN- of Iowa.
I
1.
Introduction
Inventories held in the United States exceed one-half trillion dollars in value. Inventory holding
is
a prominent economic
phenomenon
that concerns economists as well as operations mcinagement
academics and practitioners.
The make-to-order versus make-to-inventory question
fmidamental one
a
is
controlling production, but the one that has been subject to relatively
management
either economics or operations
literatures.
One
Milgrom and Roberts (1987) that uses a newsboy-type model
and information
aboTit
that firms should
produce
demand
little
excejjtion
is
in
organizing and
formal modeling in
the recent paper by
to formali7,e the idea that inventories
are substitutes for one another in production. Their results suggest
employ only one
of the
two strategies:
firms either produce for inventory or
to order, but not the both.
Many examples
support
this conclusion,
many
but
Some
others do not.
one alternative to the other; other firms constantly emi)Iny a mix of
example,
in a
in
two alternatives.
Burger King Restaurant, much of the sandwich and drink production
by the counter hostess
However,
th(-
firms often shift from
is
calling in the order to the kitchen during any very slow period of
high-demand time
intervals, a finished
goods inventory of
i>rincii)al
For
triggered
demand.
binger products
is
maintained, and the broiler loading operation responds to the finished burger inventory situation
rather than to the call-in of orders. (See Schmenner (1981)).
In the Burger
King
case, the production activity
is
not a newsboy-like, one-shot decision prob-
lem as in the Milgrom and Roberts paper. Therefore, the dynamics of the environment, for example,
variations in
demand, may provide incentives
been resolved
holding inventory even when the imcertainty has
for
at the design stage. Uncertainty
can
arise
from many sources.
both the quantity demanded and the demand timing may be
from stochastic variability
luicertain.
in the different stages of production.
On
the demaind side,
Uncertainties also arise
Thus a complete resolution
of
micertainties via commujiication of information might not be possible, or could be very expensive.
Li other words, the convexity of survey costs
may
result in a
mixed use
of
two alternatives.
We
conclude that determining optimal use of inventory requires detailed operational analysis as well
as the static design strategy
examined by Milgrom and Roberts.
make-to-order versus make-to-stock cjucstion in
many
Hence, in order to answer the
productive organi/.ations, dynamic models
are necessary.
Why
do firms make to stock instead making pinely to order? Because of the obvious costs of
holding inventory, we
ecouomic justification
may phrase
the question more directly:
why hold
inventories?
The
traditional
for inventories aro decreasing costs or iucreasiiig rettims to scale (the pres-
ence of ordering cost or set-up cost) in procurement and jirodnction, seasonality (the anticipated
variability in requirements or supplies),
and uncertainty (the stochastic
or supplies).
1
variability in requirements
Most invrnfory control
policies air obtaiiiff!
(adjustinout costs), boldini^ costs,
other reasons besides.
necessitate holding inventory
customer are
stockoiit or lost sn\c costs (shortage costs)
aii'i
First,
the j)roductioii lead time
if
may become
and characteristics
number
a waiting
witli
customer.
alsci l)e
market competing on the
which can deliver the product the
firm
measures such
in
advance what the requirements
arrival
luirertainty lu-cause no matter
will be,
However,
into cosf.s of short^gr (see Scarf (19C3)).
among
the least computable costs
above, obviously, no one knows
model
of the
when
if
there
lead time
is
we know
enough
far
deliveries accordingly as long as the
Traditional inventory models lumj) these economic effects of timing uncertainty
capacity allows.
of
what the lead time,
we can scheduh-
as a
as increasing the capacity
(shortening the production lead time). Certainly, these effects occur only
demand
may
a decisive factor for the
Then holding inventoi7 can be used
tactic to facilitate j^romjjt delivery as ojiposed to other
or
long, then the early
of sales that arc completed.
rival firms in a
The
quality of service, particularly, the speed of delivery
completes the sale
siiffirieiifly
may
Second, the characteristics of the
competing firms may
of
firms use of inventory. For example, sujipose there are
earliest
But there are
customers are im])atient. then holding inventories
if
help to reduce the customer waiting time and to increase the
Finally, the existence
is
a reason to liav( inventory
For example,
iniiiortaiif
sct-uj) costs
firms ro)ni)i'te on sjjced of delivery, then jirndnrtion lead times
If
realization of sales itself
hy analyzinp; tradeoffs aiMoup the
this simplification
makes
rusts of shortage
those that concern inventory control. For the reasons mentioned
how
to
do
economic environment
it
in
correctly even in a very rough sense without
some kind
which the productive organization operates. Therefore,
incorporating the behavior of other agents
in the
economy
is
an important task for modeling the
of
economic issues regarding make-to-
strategic use of inventory.
In this paper,
wc work toward a better luiderstanding
order versus make-to-stock, particularly taking into accoiuit
comi)etiug firms.
We
up
first set
policy of a single firm, and
show
The
average production lead time
is
that the o])timal
oi)tim?iI
to the time value of
differences in patience
mix between mak<--to-order and makc-to-stock
In this case, the firm simjily sets an inventory limit
inventory limit increases as
longer, the sale's contribvition
In the case that the firm can backlog
due
money.
We
all
of
its
demand, the
is
tin-
larger,
also study the case in
demand
rate
is
higher, the
and the holding cost
i)ossil)ility of
and show how increasing imjiatience
produce-to-stock regime.
chaiacteristics of customers and
a continuous time, stochastic control modi^l to study the stocking
operations can be determined in our setting.
via a newsboy-like formula.
th(-
holding inventory
is
is
lower.
purely
which customers are characterized by
shifts the firm's oj>timal policy
toward
These comparative statics results are consistent with and illuminate a
variety of observed behavior of productive organizations, for example, the Burger
mentioned above. The nuidel
is
then extended to an n-firm market game
for orders in the aspect of early delivery, as in
in
an oligopoly racing market.
King Restaurant
which firms compete
It
is
noteworthy that
mode
this
of competition
is
very
common
for
microchip producers
Li that industry, customers often place dupHcate orders with a
the
makes a
The
of suppHers
and buy from
sale.
analysis shows that competition can induce inventory holding
when
number
the
of competitors increases (competition
and numerical methods to solve
analytical
likely to
We
intensified).
is
for the ecjuilibrium in a
as other
jiist
Firms are more
reasons like economics of scale, seasonality, or imcertainty.
tory
muuber
one that can deliver. Duplicate orders are then cancelled so that only the winner of the
first
"race"
semiconductor industry*.
in the
economic
hold inven-
also discuss the
duopoly racing market.
Al-
lowing duplicate orders increases the buyer's welfare and decreases the prodiicers welfare. Hence,
may
duplicate ordering and induced delivery-time competition
or
may
not be socially desirable.
Formulation of the Model
2.
We
consider a firm that produces and
fashion. Orders that occur
others
must wait
when
for processing.
sells
Demand
a homogenen\is good.
arrives in a
random
the products are available in inventory are fulfilled immediately;
The
The
firm processes orders on a first-come-first-served basis.
cumulative input (demand) and output (production) are represented by two increasing non-negative
integer-valued stochastic processes
A =
>
{A{t).t
D =
0} and
denote the amount of demand and the amount of jiroductiou
order waiting and inventory level at time
t
Z(t)
where x
is
inventory
the
i
if
amovmt
A,
B
amount
the
<
0.
of inventory at
t
if
is
Z(t)
the
<
0.
in
>
O},
where A{t) and B{t)
the time intei-val
[0,<].
Then
the
can be represented by
= x+ A{t)-
of order waiting at time zero
Similarly, Z{t)
{D(t),t
amount
>
j
if
(2.1)
and
of order waiting at
We assume
are independent Poisson processes with
D{t),
t
—x
is
the
if it
is
positive
amoimt
of initial
and —Z{t)
is
that
random
intensities {nii,t
>
>
0} and {/?/,t
O}.
(2.2)
That
is
is,
{A(t)
-
/pQ,rf.t,t
>
0} and {D{t)
the actual production rate at time
the firm cam achieve
{{ftt,i
^
is
fi.
A
is
ft, (1)^,1
>
controllal)le
is
0} are martingales,
by the
firm.
hi the model,
The maximum
then defined as a stochastic process
continuous and have right-hand limits,
(2.4)
ft is
adapted with respect to Z,
(2.5)
0<
ft,
<
fiioT
all
(
>
0.
indebted to Professor R. Learhman of University of Califoruia at Berkolpy for ronversations on this topic.
3
/?,
rate that
O}) that satisfies the following:
ft is left
am
which
J^
feasible operating policy
(2.3)
*I
t
-
ft
On
if
thr other
wc assume that
Qi
—
X,t
"> 0, i.e., /I i?
rrflcrt? tlic artion of
rnstonuTs. For
cxaiiii)lc,
a Poisson iirorcs? witli ron!>laiit iiitrusity A, then
demands
cbf\rartori?tir<! of prorc?!'
h'-\iid,
rt
that arrive will wait indrfijiitcjy uo matter what the firiiiV ()i)eratinp i>oliry
many
not realistic for
cases, as
we allow customers change
The process a must
information.
is
left
(2 7)
Q
is
adapted with respect to Z,
<
a,
<
A for
all
t
>
assume that customers behave according
and that the maxiiinim deniaml
is
to the historical informa-
characterizerl by a constant arrival rate A.
mathematical justification of the fomnilatiou. see Dremaiid (1981) and
Note that the monotone increasing assumption
this
i.e.,
historical
0.
Essentially, these conditions
renege,
is
continuous and have right-hand limits,
a
tion
on the
their behavior based
this
satisfy the conditions:
(2.C)
(2.8)
Obviously
is.
of jiroress
a customer can not cancel the order once
assumption
will
relieved in Sertioii
lie
it
Li (198C).
there
imi)lies
places the order with
where renegiu};
5.
A
is
For further
is
However,
firm.
tjie
endogeiiized
no customer
an equilibrium
in
model.
To complete our formulation, we
order
iniit
fulfilled is
;>
dollars.
A
Fjierify the cost
The contribution
strueture as follows
i)hysical holding cost of h dollars j)er unit time
of jiroduct held in inventory.
Assume
continuously, on the fluids which are required for production operations.
Given an
over an infinite time hori7,ou.
profit
initial
incurred for each
is
that the firm earns interest at rate
r
>
0,
compounded
Production
order waiting or inventory of
x,
of each
is
planned
the expected firm
is
t(j)
=
E,{r e-''[l(o.^){Z(t))p,lD(t)
+ l,_«,o)(2(<));"M(0 - hZ{trdt]},
(2.9)
Jo
where E^ denotes the exjjectation conditional on Z(0) =
S.
and Z{t)~
to stock
=
max(0,
when Z <
0.
— Z(()).
Note that the firm
when Z <
realization of processes A. D.
and Z
making
i)roblem of the firm
is
is
0,
)
'^
'"^i'
to order
and neither of them
constructed
in
Figiue
to choose a control process
/?
such that assumption (2.1)-(2.2), and feasibility constraints
S.
ls(
indicator function for set
when Z >
and
Thus, a comjiletion of the jirocessing generates a sale when
order arrival generates a sale
The
is
x.
generati^s any sah"
is
making
Z >
when Z =
0.
0.
an
A
1.
to
maximize the expected
profit (2.9)
(2.3)-(2.!i) are satisfied.
The Single-Firm Problem
We
first
consider the case that that customers have no alternative and definitely need the
processing service from the firm,
i.e.,
ni
=
A, for
business seems quite attractive since backlogged
However, the firm
completion of
still
sales.
t
>
0.
demand
hi the situation like that,
is
not lost and there
is
make-to-order
no waiting
cost.
could hold inventory as shown later, trathng off the holding cost for a quicker
,
The form
operates at
of tbe optimal policy
capacity
its full
imtil the inventory level b
Z
process
luitil
=
/?,
and we denote the value fimction by
Under
\init.
is,
the firm
and resumes operation
a l^arrier jioliry with
parameter
6,
Ml(-t,oo)(^(«-)),
where the
(x)
t;
—h(h>0)
hits a lower barrier
depleted by one
is
That
rather simjile, namely, a Earner policy.
is
be used only when
sujjcrscript h will
it
is
necessary.
Proposition 3.1. The value function under a
harrier policy with
parameter
h
>
(h
0)
is
of the
form
jGp^ + '^p,
^\Epl + Fpl +
"^^^
where p
=
and
Pi, Pi
ifx>0;
+ (A^ +
A;,
^
<
Pi
<
1,
p2
P.
=
P2
=
^
_
>
I,
>
for r
+
More
0.
fi
+
M+r
+
^/{X
+
r'
+
r
VT^^—
X +
+
fi
(3.2)
r
explicitly.
A + M+r- \/{X + n +
-—
A
(3.1)
0;
two roots of the quadratic equation
P2 are the
X
with
<
ifx
x)^,
r)^
- iXn
(3.3)
,
+ n + ry -iXfi
-^
(3.4)
,
d(p, p,)^lh
-
a[P2)x{{n
a{p,)X{[ti
-
A)(l
-x){i-p]-
-p)-
r)p-,\r
T)p-\p + ?) -
+
;)
d{p, p,)tih
re{b)
G=E+
—
X
u
h
,
-{p+
F-h
T
fi'iPi
-
r
)h
P2
^
-)
-
A((A
-
-p)^
,t)(l
r)[a{p,)p-C
-
-h\
a[p2)pl'')[v
+
^)
(3.7)
rc(ft)
+
A
-
h
/i
(;'+-),
r
Proof.
r
a(^)
=
A(l -/).•)
t[h)
=
a{pi)d(p,p2)p^''
By
the
+
rf(/>,/>,)
r,
Markov property
-
=
a{p2)d{p,pi)p:^
of process
Z under
of difference equations that the value functions
v(x)
=
t;(x
X
+ n+
r
+
1)
A(l-/.) + /x(l-^-')
+
v{x
A
-
+ /i+r
t-
=
l,2,
a barrier policy,
1)
(3.8)
(3.9)
satisfy,
+^+
we can
write
down
the system
i.e.,
+
A
5
r,
.
should
t;
+
/),
r
for
x
>
1,
(3.10)
v(0)=-
^
v(\)+
A-f/i+r
v(t)
+
1)
+
X
=
v(-f>)
X
v(x
+
v{-l).
+
(3.11)
r
u
-
u(j
Y—v(-h
A 4 r
+
+
1)
A
+
before a comi'letion of prodnrtioii,
luiit,
i
T
lengtli of
time
firm
tlic
f\ilfils
Z(T) = j +
e.,
inventory by one unit by time T,
inventory for
+
<
1
<
z
-1, (3.12)
— +
/<
rciiiaiiiiup ('(juation?
first
1
<
7
jtiiiii)
< — 1.
time T. pet p
at
time of Z.
fh)]Iars,
lias
If
ran he
Under a
demand comes
a
and the inventory
no pain but increases the
In either case, the firm holds |z| unit of
I.
Thus,
+
v{t) =i<:,l.-'^i,,,,)(Z(r))i,.(T
+
the
is
Otherwise, the firm
1.
Z(T) = x —
i.e.,
T
for
/i
it
The
rx<iiiii)|(\
Suppose Z(0) = x and
expoiionfia) with parameter A
decreases by one
+
(3.13)
in a similar fashion.
level
6
r
obtained
is
-
for
,
-^—-.
ouly tbf drrivafioii of cciuatioii (3.12) a? an
T
+ xh
X]i
Wr show
barrier pohcy,
—
+
1)
Xp-hh
X
-
fx
1)
+
£;,le-''"i,,_,,(z(T))lr(T
+ £;,[/
£;,lr-'^i,,+,,(z(r))i,.
-
i)
r-''dt]zh.
Jo
Equation (3.12) follows by noticing that
^,[e-''"l,,+,,(Z(T))]
=
XEaI
c~"dt].
Jo
EAr-'''U.-i)(Z(T))]^^EAl
c-^'dt].
Jo
/^
EA
1
e-"dt]
-
Jo
where the
/(
)
7——1
=
1
X
r
two equations come from
first
f"'^
= EA
Lemma
3.5.
Li
in
-i ^t
-i-
r
(198G) taking /(•)
=
1(^4.1)
() and
=
l(,_i)(
It
ran be verified that the general solution of the above differenre etjuation must have the form
)
respectively.
of that defined in (3.1) (sec
C?, i?,
F
Levy and Lessman (19G1)). The remaining
usinp the boundary conditions (3.11), (3,13) and the
G+
//
-/'
=
.C;
+
F+
r
which follows from the
From
X
+
r
fact that v{0) satisfies
the proof of the above proposition,
A
-;»
both forms
we can
—
]5rol)lem
will
use throvighout the i)aper.
optimal barrier policy.
to
u
li
-)-.
(
r
determine
<'(iu;\tion
(3.14)
r
in (3.1).
see that the comptitation of the value functions
vinder a barrier policy requires solving a system of difference equations.
technique we
is
The following
i)roi)osition
This
is
the standard
shows there
is
a miique
Proposition S.2. There
with the iiivrntary Hmit
exists an opfinia/ hHrricr poliry
b*
that
.
is
uniquely determined by the condition
k{h'
+ l)<
'-^, and
+
p
4—,
>
k(ir)
h/r
p
+
(3.15)
h/r
where
k{b)
And
k(b)
is
decresising in
Compute
Proof.
-
a{piMP2)((X
=
m)(1
+ r)(p-' "^'""
-r)
f-')
'^
'\
/:,;,:
r{(l(Pl)cl(p,P2)(l - Pl)P2 - (l(P2)fi(P^Pl){l - P2)P\]
>
^
(3.16)
.
,
b
with k(oo)
=
the difference
v'ix)
-
v'-'{x)
=
-
(G(h)
-
G{h
\))p\
=
C[k{h)
-
where
„
_
^i^[P).
'
-
-
P2^)HPi)'^iP^P2){^
-
Pi)Pi
<i(p2)<l(P^Pi)(l
-
P2)p2
](/'
+
h/r)
e(b)e(b-l)
So
b
as a function of
>
b* since
b,
the vakic function
can be verified that k
it
is
is
strictly increasiuR for
ft
<
ft*
and
strictly decreasing for
strictly decreasinp; to 7,ero.
I
Note that the
policies
ojitinial
barrier policy deterininrd above
and that we have actually solved the
newsboy problem which
among
all
adaj^tcd
intensity control i)roblem formulated in the preceding
section (see Li (1984) for a proof). Siuj)risingly. the solution
of a
also oi)timal
is
fixes a fractile of the
demand
is
very simple and
is
similar to that
distribiition equal to a critical ratio (a
function of the overage cost and the underage cost). Here, the firm simply fixes a inventory limit
to balance a decreasing fmiction k
we can
and a
To
ratio.
get the
economic intuition of formula
(3.15),
rewrite (3.15) as follows either through a policy impiovement logic as in Chapter 3.3. of Li
(1984) or through direct verification. Let
T{y]
e{x,y)
Av{x)
Then, (3.15)
is
^(-(6
= mi{t>0:Z(i) =
=
=
u}.
EAe-'''^%
v{x)
-
v(x
0,
+
I).
equivalent to
+
l),oo)At^(oo)
-
At;(-(ft
+
1))
<
1))
>
0,
and 0(-b. oo)Av(oo)
- Av(-b) >
0,
or simply,
Av{-(b +
since At;(oo)
=
0.
Therefore, the firm builds
and At>(-ft) <
up inventoiy
0,
just to the point
equals zero.
Direct observation gives the following comparative statics result.
7
where marginal
profit
Corollary S.l.
In this
Tin' optiniHl invrnlory limit h'
paprr
want to study
v/c
tlir
ratiouair
iiirrrm^m or h dcrrrascs.
}>
and makc-to-ordcr. TLerc-
iiiakc-to-stork
l)rliin(l
undrr
forr w(- arc niorr intcrcstrd in the conditions
to-ord**r
incrmsrs as
wliirli tlir
Imiuidary
drawn bptwcou make-
is
and makc-to-stork.
Proposition
3.3.
Tlir firm Imlds inventory if
ami only
if
h
9(Pi.X,ti.r) > -,
(3.17)
P
where
= X(l- ^,^'^~/!
9{p.X.,,.r)
A(l
Proof.
Usinp the same technique
-
+
r)
in flie jiroof of Projjosition 3
a."!
(3.18)
).
r
1, we can compute the value
function? fur the policies of zero inventory and only (^nc unit of inventory. That
,,,
A(l
-
+
/),)
'
/i((/i
/t
= ~p
J
V Ix]
—
^\
v°N = -rr
+
for i
>
+
0,
(3.19)
pI
r)p
r)(l -/),)
+
w>^+ r)(l
w, — Tin..
^i) + (A +
/i
A(A
r'
is,
+
h)
(A
+
/i
+
(3.20)
i
r)r'
Then,
-
"'^'^
''^'^
=
A(A
So, v^(x)
>
v'^{x)
if
and only
proved
property of w
('ondifiou (3.17J
in
(3.17) holds.
if
Proposition 3
inii)lies
We
capacity,
fi
is
Lemma
3.1.
would
follows from the simi)le
Finiction p{X,n,r)
is
^=
n
=
demand
or
hold inventory or solely
of a sale
rate. A.
is
dX
p(l
fip
^
- p)
— Xp
to
(//),
In particular,
(/)).
hipher, the processing
fiie iii>l(hni^ rost. h. is
smaller.
lemmas stated below.
X
fi,
and drrreasing
+ ^i+r-^/(\—+
^l
in A
and
r,
dp
dfi
- p
— Xp
\
^ip
^
where
+ r)^-4XJl
(3.21)
.
explicitly,
dp
make
rate (A), the processing capacity
demand
if tlir
is lyif^grr.
increasing in
pi
p
like to
and the contribution
firm tends to hold inventory
f{X,ti,r)
More
(/i),
lower, the sale's contribution, p,
The proof
-).
1.
order depends on the parameters of the model: the
Corollary 3.2. The
-
^
r)r
conclude the i)roof hy observing the monotonicity
that whether the firm
the interest rate (r), the holding cost
(9{Pi.y^^r]
+
-.
dp
dX
p
HP
^
—
Xp
Lemma
3.2.
Function g{p.
A,
/i,
Xfir
dp
(A(l -/.)
J/l
Proof.
A(l
fi.,r),
iu the
+
-
~ ~A(1 -
3.S. Function g{p(X,
Usiug the result
calculate
increasing in
dg
dg _
Lemma
r) is
X,
p)
r)2'
+
^,r)
r'
is
A
nnd
r,
nn'J drcrcHsing in
-
^3
_ ._
Mf-il
d\
'
(A(l -/))
_
~
57
dg^
p)
p.
A/i(l
-
(A(l -/))
fi.
More
explicitly,
p)
+
r)2'
p)
+
r)2'
increasing in X and decreasing in
/i.
above two lemmas and the rhain rule of differentiation, we can
,
Corollary 3.3.
Wr
4.
optimal invnitory limit
TIip
omit the proof
lirrr since
wr Inokcd
the previous section,
uiifiilfillod
who has waited
or docs not place the order in the
formulation
at
that can backlog
all
therefore wait for ])rocessiuR for a long time.
In
or expects to wait for a
jilace.
first
and tnnis
which customers have limited jiatiencc
in
of a firm
stockiii;^ i)nlirirs
tlir
The customer orders may
demand.
reality, a custoiiuT
of orders waiting exceeds certain ])arameter
issues the order
and waits
each customer
Siiiiixisc that
the order waiting hits an ujijier barrier
Here, the parameter
is.
c
0).
c.
we
=
demand
if
r,
will
in
the numlier
the customer
be turned
off
Al,_^,,,(Z(f_)).
more
backlogging
they
])atient
are.
a sjiecial case by setting
is
prove that the ojitimal policy of the firm with imiiatienf customers
h.
than
who comes
can be viewed as a measurement of the patience of customers. The larger
solver! witji jierfert
certain jiarametcr
is le.ss
look at a
Mathematically, we assume,
the longer customers are willing to wait, and the
the i)roblem
and leaves
lias
the backing
If
cancel the order,
We now
to other alternatives.
In other words, the arrival of
for j^rocessiug.
a,
>
c, (r
may want
Inn^: tinir
with an order can see the numtx-r of l)ackorders the firm cuiTeufly
c
<irrrrascs.
The Single-Firm Problem with Impatient Customers
111
if
//
only of ."^traiKliffoiwaici cojiipufations.
cou-'^ist.'!
if
inrrrn^r;^ as X j/iTfasr.; or
ft*
r
=
a matter of fact,
CO. Again,
we can
a barrier policy with
still
is
As
the o])timaI inventory limit.
Proposition 4.1. Thr value
fuiirtion iiiidrr a barr/er foliry with j>nrniiirtrr
ft
(ft
>
O)
if^
of the
form,
where
pi
fCp\ + Hrl+'^p,
,/A-M^
17 7 _^ r 7 ^ X
F(>2 + jp + i-;^ +
Epi +
^
"(^)
\
[
hikI p2 are defined in (3.3)
and
and
(3.4).
(i
.
iTt>0;
-f
^
ifj<0:
>/,
(4.1)
f.
t)~.
H, E.
F
iukI
are
uij/fjiie
solution of the
following linear equations,
G
-^
H+
X
II
Gp[ + Hp'^ =
+
a + H +-p=
A
r
X
+
+
fi
+
(Gpi + Hp2
+
the values of G,
X
X
+ -;.+
-^{Ep-'^' +
H, E, F here
+
(4.3)
r
r
F^j- '
—
+
to avoid
7^
in
-It
T
•
-.
(4.5)
r
very similar to that of Projiosition 3.1. Also we
complex
proceeding section, we have
(4.4)
r
r
X
is
h
11
^-1)-).
(
F/>,-*+')
r
omit the proof of the proposition which
list
/'
r
->t
-^7..
+ -p)
r
r
X
do not
r
Hp[-' + ^/O +
,
(E/^r'
Ep-' + Fp-' =
We
/x
^
(4.2)
r
,
+
,
-
•
r
u
-^^
h
II
r
-^(Gp\-' +
/i
—
—
r
r
+
X
-p = E + F + -p +
exi)ressions.
Parallel to the results
from
Proposition 4.2. There
an optimal
cxistf^
with iiivrnfory limit
/larr/rr pnliry
h'
,
that
if^
uniquely
determined by the condition
-^,
+
+ l,c)<
ki{h'
-7^,
+ h/r
and k,{b\c) >
n/r
p
(4.G)
p
where
j^
„
^
.
a'(Pi)a'(P2){p7'
-
^2"')((A
-
-
^)(1
P2)
+
-
r)a4p,)p\
-
((A
m)(1
-
Pi)
+
r)a.(/>2)^^)
'
r(a.(^i)rf(/'2,^i)(l-/'2)/'^^
+
«.(/'2)'/(/'i./'2)(l-/'i)/'2^')
(4.7)
where
a.(r,)^Hl-p,) +
and d{pi,pj)
And
defined in (3.8).
is
ki{b,c)
is
= p(l-p-') +
a'{pi)
r,
decreasing in
h
(4.8)
r,
with ^1(00,
=
r)
0.
Proof. Similar to the proof of Proposition 3.2.
I
Corollary 4.1. The
opfiina/ inventory huiit h' determined
of the patience measure
(4.C)
is
an nonincreasing function
c.
Denote the denominator
Proof.
}iy
of the fractional expression of ki{h.c) in (4.7)
by Ci(b,c).
We
have
ki{b,c)- ki(h.c-l)
^
aAPi)(i*(P2)p\P2{p'[^
-
—
1)
ei{b, c)ci(b. c
(((A
•
<
That
is,
-
-
m)(1
+
Pi)
P2^)
r)d(p2, Pi)(l
-
P2)p\
+
((A
-
/i)(l
-
P2)
+
r)d(p,,p2)(l
-
^i)^)
0.
^1(6,)
is
decreasing.
decreasing implies that b'{c)
is
Condition
(4.G) together
decreasing in
with the fact that ki{-,c) and ki(b,) are
c.
I
Proposition 4.3. The
firm hohis inventory
if
and only
if
g(4>(c),X,ti.r)> -,
where function g
is
(4.9)
defined in (3.18), and
^^^^^P2a.{pM-Pi-AP2)p'2
a*(Pi)p[
Proof.
That
Compute
is,
for
-
(4 10)
<^'{P2)P2
the value functions for the policies of zero inventoi-y and only cue unit of inventory.
any x
>
0,
0,
.
V (x)
M
= —p
r
1,
,
V (x)
M
= -p
((lApl)p[p2
-
(i'(p2)p2P^i
)W
,.,,.
(411)
,
a'(p2)n,(pi)p\- n'(pi)n4p2)P2
(a.(pi)p[pl-(i.{p2)P2p'i)t^{{t^
-—
+
r)p
+
h)
,
V(c)
r
11
(4.12)
wlicrc
V'(c)
=
(X{X
+
r)(l
-
+
(A
(x)
=
P2)
-f y/
4 r)r)n.(p^)p\
-
(A(A
+
r)(l -/>,)
+
(A
+
+ r)r)a.(p2)pl
/i
Thru.
(x)-v
V
—
(?('A('-),
A,/i,r)
).
V'(c)
So, v'li)
> u°(i)
if
and only
p
(4.9) bold?.
if
I
Lemma
4.1.
Fimrtinn
<f>{c) is
decreasing
and
in c,
lini
<f>{c)
=
pi
Proof. Simply notire tb^t
,,
X
^(C)
-
J.I
^(C
-
Corollary 4.2. Function
inventory
Pronf.
if
Tbe
customers are
first
i\
1)
-02')'
12
a4Pi)<iAr2)p[^'p2^'ip:'
1-2
-"vri
/"vri//i
w'l
=
,
{(^>(Pi)p'i
-
g{<i>(r). X, fi.r)
is
,
,
,
less patient, that
,
.
,w
,
,
n.{p2)P2){'iApi)p\~^
drcrcasing in
is.
c is
r.
^
I
r-l
-
,
,
.-1,
,,
<
0-
n.(p2)p'2'
and hence,
tlie
firm
tends to hold
smaller.
assertion follows from Lcinina 3.2. and 4
1.,
and siToiid
is
a rorollary of Proposition
43
I
With
inipaticnt ctistoniers, in addition to the
two
Section 3 (earlier realization of
demands and inrnrrenre
reduction of
facinp; less patient
lost sales.
Hence,
lost in the lonp run, the firm
What happens
it
to a
holding inventory mentioned in
of lioldinp rost), there
is
a third effect: a
customers, a situation of having more demands
tends to liold more inventory.
demand
goes to other alternatives.
rffrrts of
lost to
One can
without exj)anding the srojje of a
siiifrle
the firm? hi reality,
it
d(ic,«
not arfuaily (lisa])pear; rather,
not precisely capt\ire the cost of lost sales to the firm
firm setting to equilibrium models to study the behavior
of customers as well as rival firms,
5.
Competitive Stocking
In this section,
we extend the
single-firm
impact of competition on inventory
policies.
model to a multi-firm market setting and study the
The
12
comi)etition
we
introdiice here
is
only on the
timely delivery rather than price, product quality, or other
show that competition may breed a demand
Assume
of exactly one firm are fulfilled immediately
complete a
will
by that
there
is
no penalty
the product
If
is
required and
is
it
to
It is
We
fi.
arrives in a Poisson
is
have non-empty inventory,
not immediately available in
b\it
com])lrtes the sale only
rest of the firms arc cancelled.
noteworthy that
this
is
a
common
Production yields of chips arc very uiiprrdictabie.
the semi-conductor industry.
reprocessing
Orders with
first.
for order cancellation.
is
the ])roducts arc available in the inventory
any of the firms, then the buyer places the same order with each firm,
with the firm which finishes the order
goal
processing rate
demand
firm. If several firms
with equal probability.
sale
maximum
hidustry
order racing,
-
The demands that occur while
fashion with intensity A.
Our
of the market.
for produce-to-stock.
there are n firms in the market and each has a
consider a particular form of competition
then each
apiirct,"?
Assume
occurrence in
A
great deal of
often difficult to meet delivery schedules of the buyers
who
primarily the assemblers of electronic equipment. As a result, assembly plants often issue the
are
same
orders to several suppliers and later cancel the orders with the late supi>liers.
We
first
look for the necessary and sufficient condition under which firms will have inventory
in the equilibrium, or equivalently, the condition
under which there
an ecinilibrium in which no
is
firm holds inventory.
Proposition 5.1. For any
firm
t's
best response
is
t,
t
=
1,
.
.
.
,
n,
suppose Rniis
j
j.
an barrier policy with inventory limit
^
b'
do not held inventory.
i.
,
thnt
Then
uniquely determined by
is
the condition
hlr
p
+
-
hlr
-.
hlr
]i
+ h/r
where
- /^)(^ - ^") + '•)(^r' - p7^)
r[a(pi)d(r]„,p2)(l - pi)p^2 - (^{Pi^in,,^ Pi){^ - P7)p\\'
A + n/i + r - \/{X -f nil + rf - AnX^i
_
—
= p(X,nfi,r) =
-
hb)
rin
And
k{h)
is
satisfy the
Vi(x)
t
The
with ^(co)
=
=
v[x
+
(5.3)
0.
plays a barrier policy with parameter
n/i-fr
Therefore, the solution
appears.
b
system of equations (3.10)
A
(5 2)
,
decreasing in
Proof. Suppose firm
t, Uj's,
"^^i)^(^2)((-^
is
rest of the
+
v(x
is
same form
>
0).
Then
the payoff fimction to
(3.13) except eciuation (3.10) altered to
A-t-n/i
of the
proof
-f 1)
-
h (h
+
-
1)
+
A-t-n/t
r
as in (3.1)
-
(3.9)
but p
is
+
r,
for I
1,
(5.4)
r
replaced by
just a repetition of the proof of Proposition 3.2.
13
>
be
rj,,
wherever
it
Proposition
5.2.
Then- rau/mf /» an
rqiu/i/inii/ji
wiijV/j
)/j
no iiivrntory
j.«
lirl<l if
Ami only
if
h
?(';„.
> -,
/'.»)
-^
(5.5)
r
wlirrr thr fitnction g
Proof.
we can show
ai)pears,
only
samr
the
U.<;ing
is
dcfinri] in (3.18).
argiiiiKut
th.vt
fiiiii
t
a.''
m
Proposition
by
apjaiu witli p iTi)Iarod
3.3.
zrm
has inrcnfive to drviafr finm
inv(iitf)ry
wherever
fj„
it
if
and
would
like
arraiigemcut
(5.5) holds. Tin- asprrtinn of the proj)osition follows dirrrtly
if
I
We
compare
to
the
rail
Proposition
Sharing Market. By Monopoly Market, wr
3.3.,
the
monopoly holds inventory
(?(/.,,
In a
Demand
demand
the
get the order
A.
/i,
>
r)
Earh time
in the following fashion.
is
1/n, whirli
is
in
in the
tind that
market aloue. By
if
-.
(5.C)
somehnw
is
jirohibited
and n firms share
a donianfl orrurs, the prol)ability for each firm to
inflependent of the arrival i)rorrss A.
demand stream each
Market
a M'uiojioly
one firm
iiiran
and only
if
Sharing Market, jilacing duplicate orders
process implies that the
is
market with that
firms" stocking i>olicy in a raring
Demand
in a
We
gaming situation fornnilafcd above an Olipojioly Rucini^ Mnrkct.
firm fares
is
The d(Tomj)osition
of a Poisson
an Poisson Process with rate A/n whirh
independent of the others. Thus, earh firm's decision jiroblem
is
reduced to a single-firm decision
problem with a Poisson demand of rate A/n. The demand sharing firms hold inventory
if
and only
if
g(r{\/n,^l,r].^/u.^l,r) > -.
The Demand Sharing Market represents the
situation in which du])lirate order
a customer arrives places an order to a firm on a i)urely
consideration, and hence, there
is
(5.7)
random
no delivery-time comi)etition
at
is
basis without
not allowed,
any strategic
all.
Proposition 5.3.
> g(p{X/n.(i.r).X/n,ii.r).
g(pi,X,fi,r)
g{p(X/n,^l.r),X/n,^l.r) > g(p(X/{n
for
n
>
2,
Tijaf
and g(p{X/n,
is,
demand
Proof.
A
r),
the firm as a
multi-firm market
in a
/i,
and the
A/n,
/i,
—
r)
monopo/y
likeliness of
is
»
a-s
n
—
»
more Ukely
+
1).
,/.
r).
A/(m +
(5.8)
l).//.r).
(5.9)
oo.
to hold inventory than in a
demand
sharing
holding inventory decreases as the numlier of firms increases
sharing market.
direct corollary of
Lemma
3.3.
I
14
Proposition 5.4.
>
p(r?„,A,/i,r)
(/(»/„+,. A,
for
n
>
2,
Tijaf
and g(nnis,
monopoly market and the
A
>
more
are
finjis
—
-^.M^ r)
a.s
likely
n
—
>
(5.10)
>
(5-11)
5(f7„,-^. /'•').
oo.
have inventory
to
likeJiness of
M,r)
(7(/'i,A,/i.r),
in
an
racing market than in a
oi/fi-o/ui/y
holding inventory increases as the competition
is
intensified
(the numlier of firms increases).
Proof. Note that by
Lemma
'Jf.+
i
=
p{>^,{^^
two assertions
So, the first
3.1.,
+
IIMi*-)
>
n,,
=
/'(A,n/i,r)
in the proposition follows directly
>
p{X.fi,r)
from
3.2.
= d.
The
limiting result
is
easy
to verify.
I
To explain the word
Siii)pose the
the above propositions,
"likely"" iisod in
parameters of the model.
A,
/z,
r,p,
and
we provide
the following examples.
are so rhnsen that
/i,
g(p{X/m,^l,r),X/m,^i.r)
=
h
-.
V
Then, by conditions
(5.C),
(5.5),
demand sharing market
inventory in any other
in
and the aljove propositions, no hiventory
(5.7)
which the number of firms
demand
is
greater than or equal to
is
held in any
m, while there
is
sharing market, in the monopoly market, or in any oligopoly racing
market. Suppose that the parameters are such that
9{pi,X,n,r)
=
h
-.
7'
Then, no inventory held
in
any
of the
demand
sharing markets or monr^jioly market, but there
is
inventory in any of the oligojtoly racing markets. Suijjiosc that the parameters are such that
^
g{t),„,X,ti,r)
Then, no inventory held
in
racing market in which the
number
of firms
is
is
less
than or
we
We
call
CEin
that occurs in the third example
draw here
is
etiual to
m. but firms hold inventory
in
bigger.
is
wouldn't have any inventory in a monopoly market or a
sion
h
-.
r
any demand sharing market, the monopoly market, or any oligopoly
any other ohgopoly racing market which
The inventory
=
solely
induced
demand
l)y
comi)etition since a firm
sharing market. The basic conclu-
that inventory holding can he used as part of a firm's comjietitive strategy.
the inventory of this kind, the Competitive Stock, analogous to the
common
terms safety
stock Jind cycle stock.
To further
illustrate
competitive stocking, we comiiare the stocking
competition when the discoimt rate
is
small.
15
i)olicies
with and without
Proposition 5.5. Fit
> A
ri/i
>
n
hikI
1.
liin(7(/)(A/n,/i,r),A/ri,/i,r)
=
0.
(5.12)
r|0
For
>
n(i
X
and
>
ri
2,
hmg(n„.X/n,^i.r) = A(l - -).
Proof By
L'liospital's nile,
=/x-A,
liiii
>
A.
lini
rjo 1
if
n/(
>
^
=
-
fi
—
A/n,
aiifl
liiii
p(X/n,ti.r)
r\()
g(p(X/n.
fi.
A/n.
r),
/t.
r)
=
r\0
liin
A
=
no
lim A
no
The assumption
1
-
-I-
l-^(A/r.,/i,r)
^— =
+ j3;^y
A
\^
i
\
that the joint capacity of firms
A(l
no inventory
that in that case,
firm marlcet
when
hi the
little
the interest rate
we
demand
time from order to
its
sharing cases,
fj\ieueing
is
completion.
i.e.,
and
in
>
a
—
1/n) >
A,
A, is
jirojjosition
a (h'liiand sharing multi-
tlie cas(^ in
an oligoiioly racing
h/]>.
do have incentive
sharing market.
Assume
to
i.'^suc
iiiultiplc
/i,
r)
than that
in a
demand
<
h/]i.
and n
demand
>
2,
where
sharing market, that
w = nu —
W W
1
1
and
X
— A/n
/x
orders instead of sjjlitting
that customers are l)etter off by spending less
Also assume no inveutoiy held
condition, p('Jn. A,
w" tifi
not
in
hoth oligojioly racing and
Then,
holds
formula implies that the average time a customer waits
strictly less
for
as long as A(l
ol)serve that customers
their orders in a
demand
tliis is
The
>
presence of competition, the firms alway.s have incentive to hold inventory no matter
discounting there
Finally,
sufficient small, hut
in the loiifr run.
market or
will l)e lield in a iiiono])(i|y
i?
).
greater than the deinaud rate, n/i
is
necessary to avoid having the oider waiting l>Iow up to infinity
how
A,
r}„
5
1
rlO
lim j(r7„, A/n, /i,r)
market
= nu —
-
1
Therefore,
A
lini
say.<:
'
that
iini)li('s
Till.';
(5.14)
l-/,(A,/i,r)
r|o
if /i
(5.13)
in
ai)i)lication of the
simple
an oligo])oly racing market
is
is
=
n n/i
—
1
<
0.
A
are average waiting times in an oligopoly racing market
sharing market respectively.
From the
duplicate orders increases the buyer's welfare.
16
social welfare point of view, allowing
However, the increase
To
sharing cases and
demand
and
v"
let
v'
——
= -p- —
-
v'(x)
both oligopoly racing and demand
ojjtimal in
market and
in a o!igoi>oly racing
is,
= -p-
^"{x)
is
is
be the expected profits of a firm
sharing market. That
where the function p
achieved hy scarifying the jirodiicer's welfare.
is
we again assume no inventory holding
see this,
in a
in the buyer's welfare
^7—r ^(^' "Z^-'-))'-
r^^^^
+
r
A(l
p{X,n^i.r))
r
A(l
p(X/n,fi,r))
defined in (3.21).
r—+—r /'(A/n,
r))',
/i,
And
v'(x)-v''(x)
-
p(X/n.fi,rY)
+ \T^^V^^^^^^'^r{Kn^l,r) A(l - p(X/n,n)) + r
p{X/n,^i,r)))
X{1
-
+
rV
p(X,nfi,r))
XpiXIn,
II,
((p{X,nn,ry
r
>
since
p{X,nn,r)
~
+
X
nfi
+
r
p{X/n,fi,r)
—
\/(X
+
nfi
+
—
r)^
+
A
4riA/i
+ nr —
n/t
\/(X
2A
(n
>/(A
for
n
>
2.
+
—
nfi-h
n/j
+
nr)"^
—
4nA/i
2A
+
l)r(/)(A, n/i, r)
nry -
+
p{X/n,
fi,r))
>
4nA/i
+
0,
x/(XT"n^rT7)2^^^'4nA//
In the case that there are inventories in the eqnilil)rium of an oligopoly racing
(remember that there
is
no inventoiy held
in a
demand
when
sharing market
market
the interest rate
is
small), the holding costs are the further losses to the prod\icer"s welfare.
In
summary, allowing duplicate ordering
ducer's welfare.
From
increases the buyer's welfare and decreases the pro-
a policy-making point of view, a
more
careful study of the situation (the
preferences of buyers and producers, the parameters of the market, etc.)
is
needed
in order to
decide whether duplicate orders shoiild be allowed or not. Sometimes, a cancellation fee
may be
appropriate as a measure to deter duplicate orders or as a welfare transfer mechanism.
In the remainder of the section,
racing game.
We
we
shall discuss briefly
restrict the strategies of the players to
best response to a barrier policy
is
a barrier policy
instantaneously but not the other firms' inventory
with
6,-,
t
=
The
1,2, in a two-player game.
equations. Note that the states are the
of
both firms
if
there
is
and whoever generates an output
'''('')
=
A,
^
+
o\
2/i-l-r
first
"^li^
If
level.
to obtain a solution to a
be the barrier
Supjjose firm
1,
Note that the
own
inventory level
employs a barrier policy
t
denoted by
of orders waiting
if
Z >
duopoly
jiolicies.
firms obsei-ve their
payoffs to firm
number
no order waiting.
if
how
Vj, satisfy the following
0,
or the inventory levels
there are orders waiting, then firms race for the order,
gets the first order in line. Thus.
+
1)
"
+ ttt4-*''('
A-f-2/x + r
17
^)
+ TTT-r-^''
X + 2fi-^ r
(^-^^^
—
for 2
>
1
operating
is
—2
where
in the
z
is
the uuniher of order waifinp.
make-to-?tock regime, aud Ui(i,j/)
aiul the rival? niventory level
t'i(0)=
"'(^'V)
for
-62 <
1/
<
=
^.
+ 2/i ^
+
,
A
—y
is
<
for i
i'i(l)+-
^
t'i(n.t/+l)
+
there
If
the payoff to finu
is
0,
no order waitinp, theu firms are
i?
<
j/
or 7
<
0,
<
(/
level
Th\is,
0.
(ri(-l,0) + .;j(0.-l)),
("i(-i.v) +
TTV-—
A +
+
r
own inventory
if its
1
(G.IC)
-
«'i(n,i/
(517)
i)).
r
2/i
0,
t,,(n.
=-
-62)
rjfn. -/,2
+
A
+
+
1)
t;,(-l, -/.2).
T
/i4-r
\
(5.18)
-^ r
n
->r
A
=T
»'»(^'0)
A
+
+
2/i
fi(^+
1.0)
r
u
+
^,2/1 +
^
+
A,
for
-61 < J <
+ ih
Af)
(t^i(^
-
1.")
+
.'il^.
-1)) +
^
A.^^
+ 2/i +
r
(5.19)
,
r
0,
fi{-''i.O)=-
+
A
/i
+ 1,0)+
t;,(-6i
+
+
t-i(-/M,-l)+
/;
+
r
t;,(x,}/
+
I))
A
r
~
A
—,
+
//
+
(5.20)
r
A/2
•^il^;.!/)
=-;
+
A
+
-61 < z <
0,
-62 <
y
+
+
(fil^
l,y)
+
r
—^^(".(x M
A
for
2//
<
+
2//
+
1, J/)
+
.•i(:r,
- D) +
V
r
+ xh
A;»/2
TV^—
+
+
A
2/i
(5.21)
-.
r
0,
A/2
t^ii-'-i.v)
=T
A
+
+
/i
("i(-^ +
l.jy)
+
+
t.,(-ft,,y
1))
r
(5.22)
A
for
-^2 <
1/
<
+
/z
+
i'i(-''i. V
-
1)
+
+
/i
+
"1(3-. -''2
+
A
r
r
0-
A/2
v,(a:.
-^2)
=T
A
+
+
/i
(fi(^
+
1-
-''2)
+
u
A
for -fti
< X <
+
/i
1))
r
+
Af)/2
+
A
r
+
yi
(5.23)
j/i
+
r
0,
A/2
t^i(-fci,-/'2)
A7)/2
= -^(i'i(-6i +
A + r
l,-i2) + "i(-'M.''2-f 1))+
In order to get the value functions for each player,
difference equation on an rectangle with specified
solution for (5.15)
is
we have
uj(2)
= An +
-p,
r
18
-
f>,/i
,
,
(5-24)
r
to solve a second order,
boundary conditions
simple,
-
-^A +
(see Fig\ire 2.).
two variable
The
general
where
'
and
>4 is
X
=
+
2^l
+
r
-
y/(X
+
r
The
a constant to determine.
and ^
+
r)2
-
8A/i
n
'
general solution to (5.21)
Jr
where
2/i
is
r
r
r
satisfy the following relations,
A
and B's are parameters
Theoretically, one
+
M(^"'
+ r').
to determine.
may
boiuidary conditions (5.1G)
of this kind can
+ 2M + r=-(r +
pick
-
up proper base
(5.20)
and
(5.22)
{ti,T2,.
.
.}
and determine the constants by the
(5.24). Nnmerically,
-
system of difference equations
be solved by two-color nniltigrid method (Causs-Seidel) very
efficiently (see
(1987)). After the payoffs to each player are obtained, searching for an equilibrium
Lemke and Howson
game
6.
here.
More
(1964)).
We do
technical readers
is
Kuo
routine (see
not intend to discuss in dejjth the equilibria of the stocking
may
find the Apj)endix interesting.
Conclusion
In this paper,
we model formally
the role of inventory in the competition of timing.
long been recognized that imcertainty justifies safety stock, but not
much has been done
to exjiand
the scope of inventory models to include the interaction between a productive organization
other agents in the economy.
in the decision
competition.
With lead time uncertainty, we
iflentify three
on make-to-ordcr and make-to-stock: discoimting, customer
Though
the Poisson assumptions
qualitative results obtained should prevail.
may
We
not
also
fit
and
important factors
characteristics
and
real-life situations,
the
show that allowing duplicate orders increase
the buyer's surplus while decreases the producer's surplus.
to study the welfare
many
exactly in
has
It
Thus,
otir
and policy implications mider the competition
analysis provides a basis
of this kind.
The
possible
extension of the work would be to include price competition as well. In that case, we conjecture
that firms would use pricing to mitigate the inventory effect induced by delivery-time competition.
Our model
also provides a
Eind suppliers, for
framework
to further study the other strategic interaction
example, the dynamic selection of suppliers, bribing
19
for
among buyers
an early service,
etc.
Appendix
Theorem
Alternate Stocking and a Folk
WV
yliall
only one firm
is
allowed to
make
and the
to stf)ck.
uaiiicly, ultcrnfitc
t^niiir,
opi>oifiuiity of storking
To make the game symmetric, wc assign
inventory the
first
time
that player
ecjual proliahility for
Denote by v](x) the payoff to player
starter of stocking.
Assnmc
a nioclificd vcrsiou of a duviioly raring
at
In the sotting of the altrniHtr stocking g^iitr. wlirurvor tbrrc
tbi? .occtiou.
alternately.
look
Game
the Stocking
in
t
Z =
and
(x) that
t',
j)layer
if
t
=
uo order waiting,
given to the firms
i?
each player to be the
j and player
t
can hold
ran hold inventory the second time
adopts a barrier policy with jiarameter
Proposition A.l. The value functions under
Z{t)
if
»
i?
stocking iu
t
/',,
=
Z =
0.
1,2.
barrier policies with j)nrnnieters
l>,
>
(l>,
0)
t
=
1,2,
arc,
''•^^^
"
''^''''~
for
i
=
1,2,
anJ j
^
t.
+
EAh,.h,)p:
I
whr^rr-
^;.
(^ +
+
\E2(h,.b,)r',+F2{f>,.h,)pl
=
r;
+
F,{hi,b^)pl
^(A,
and
2/i, r), p^
<
,fx
x)'j.
^^^^
0;
^^'^
i{x<(V.
P2 arr th<^ twf rant^ nf the qtiadratic equation
(3.2).
Ei(bi.b,)
= -r^[((A
+
F,(b,.b,)
=
+
E2{b,.b,)
F2(6.,ft;)
+
//
+
r)h,{b,)
-
+
(Ar,
iiP2')h2{b,))ti-
(A.3)
a{p2)P2'''(hAf^,)ci-h2(b,)c2)].
-—L-
+M+
-[((A
9{t>„bj)
r)h,{b,)
-
(Ar,
+ HP:')h2{bj))nr
(AA)
a{p,)p:'-(h,(b,)c,-h2(bj)r2)].
= "^f'^?/
gV'i^bj)
[/'2(^)o
-
h,(b,)r2
-
+
(A
;.
+
r){p;'
- ^7^/-).
= -^%^[/i2(M<-. -
h,(b,]r2
-
(A
+
//
+
r]{p;'
-
=
+
^2
/>7')/''-I.
{AC)
»
gibt^bj)
Gi(6.,6;)
(A5)
r
^2
(A.l)
(X
G2{b,.bj)
=
E:-\- F,
+
^^(;>+
_
(A
+
/i
+
fi-i-r)g(b,.bj)
-)
r
r
r)(^7'
+
-
P2')h2{b,)^i''-
+
/.i(ft.)(/.2(fty)r2
-
/»i(ft;>i)
(A.S)
(A
+
(P+
r
+
-).
r
20
/i
+
r)g(b,.bj)
—
,
h,{b)
=
(X
^2(6)
=
(Arj
g(bi,b,)
=
/),(6,>i(ft>)
ci
=
-[A(A
+ n + r)(a(/.,K* +
-
tiP2')a(pi)r-[''
-
(A.9)
a(p2)P2'').
+
(Af,
(A.IO)
tip:')n{p2)p^''
(A.U)
/i2(ft.)/i2(6y).
+ nn +
+
r)p
+n+
(X
-
r)(X
(^12)
n)h/r].
r
=
C2
+
-[A(A
-
m(2
r7))r
+
((A
+
/i)(A
-
,1]
-
[A.U)
tir)h/r].
r
Proof.
For notatioual simplicity, we ouly calculate the value fiuiction? for firm
proof of Propositiou
we can write dowu the system
3.1.,
functions v\ should satisfy,
•'i(^)
,^o\
+
+
=
A
A
=
.}(-!)
1
vlix)
=
2/i
+
+
/i
-
^i(^
r
+
+
+
—^^'AO)
A + /i+r
A
+
1
A
.^(-1)
^2(2.)
+
2
<
=
\
A
+ //+r
2/i
of difference equations that the value
+
''i^^
—^vK.
u
A
+
1)
+
TTT^^''
A + 2/i + r
^°' ^
^
1'
(^-^^^
r
/i
+
-
r
+
Y-^t.}(-2)
A +
+ r
1
+
/t
-
-^p^.
X
+
-\-
Ai»
1)
+
r
(A16)
r
fi
+ xh
T^-^.
A + /i+r
(A17)
-2,
A
r
=
vl{-b2)
It
<
A
/i
,
1)
,JS
^
+
+
+
1)
A
r
-v}(x +
T—
+ /x+r
fci
Similar to the
i.e.,
A
for
1.
vliO)
+
+
1)
t;J(j
/
r
+
(A19)
v}(-2).
+
A
= j^/A-f>2 +
+
/i
+
vl(x
-
1).
for
-
^2
+
1
<
z
<
-2,
(A.20)
r
(^-21)
l)-
can be verified that the general solution of the al)ove difference etiuatious nuist have the form of
that defined in (A.l) and (A. 2). Parameters G,-,^,-,F, are determined
(A.15), (A.18), (A. 21)
and the
liy
the boiuidary conditions
relations,
Gi + -p
=
E2
+
(A.22)
F2,
r
u
X
r
r
Gj + -p = ^1 + ^1 + -r +
X
—
(i
h
,
.
r
,
(>l-23)
r
imposed by equations (A. 16) and (A. 19).
I
The
following two propositions show that there
the alternate stocking game.
21
is
a symmetric stationary
Nash equilibrium
in
Proposition A. 2.
(livni that firm 2 ,v/o/)fs a Imrrirr
j>'<liry
response of Bnii
a barrier [lohry witii iiivontory
''1(^2) siirli tlint
1 is
+
k(bi
<
0.
P2){ci
-
1,^2)
liiiiit
with invrutory hinit
>
aiKi k(liy,l,2)
0,
^2.
the best
(>4.24)
where
-
k(f>i'h) =r<i(PiU(P7){Pi
-
Proof I'siug the valnr
-
fihp\'p','((h,(h,)
7(''2)<'2)
h:(by
- D) -
7(ft2)(/«2(''i)
A
fuiirtinne nbtaincfl in Propositinii
1
.
-
/»2(ti
-
I)),
(^.25)
we have
,2
41.63/
'l— l.'ji
V
\
\
j.^i.'ij/
^(
l.t'\—it'1
\
M
/
where the positive constant
M(Mt_2) +
^^
Note that
for 6
>
/«2(62))/»l(''2)(^'r'
-
=
h2(h)
where functions a and d are defined
-
a{ri)d(rj,P2)p;^
a? in (3.8),
and
a{p2]d(fi.ri)P2'
>
0,
{A.27)
also the defcrenre in (A.27) increases to infinity
Therefore,
as h increases.
<
-,({,]
<
and 7(M
1,
i
—
A
the only term in
p\'P2'(f^ii''i)
^2'')/'r'V2~''
2,
hi{b)
Then
-
^(''i,''2)
- hAbi -
1))
which depends on
- libiWMf'i) -
/.2(''i
^^^.
+
/J
+
as
/-
|
(A.28)
00.
r
/'j.
- D)
1))
(>1.29)
=
n(Pl)'i{n'l{h)^p2lV'2))(i
increases to infinity as
-
increases.
ftj
-
Pl)P2
This implies that
ib(''i./'2)
(A. 29) has a negative sipn in k(hi,h2). Hence, ^1(^2) does the
Proposition A.S. Thrre
inventory limit
b'
where
is
h' is
-
'^{P2)<i{ni(l>2)-Pil(l'7))(i
/'2)/'t'-
—00
I
as
//i
|
00 since expression
jol).
a stationary symmetric Nasli eqiiihl>ri\ini in wliirli each firm sets
its
determined by
k(b'
+
l,b')
<
0,
an(/it(^*,ft*)
22
>
0.
{A.30)
rropf. Suppose 62
best response
is
=
meaning firm
CO.
no upper
2 sets
Ijairior for lioldinp inventory.
Then
firm I's
determined by
+
k(hi
<
0,
p2)(ci
-
1,00)
and k(hi,oo) >
0,
where,
-
A:(6i,oo) =ra{pi)a(p2)(pi
-
=
where 7(00)
(Ary
+
-
tihp\'p\^({h,(b,)
+
fip2^)/(X
+
/i
-
/i,(6,
1))
fti(oo)
is
-
-
7(cx))(/i2(^,)
/i2(6i
-
1)),
and k(lii,oo) decreases to negative
r) as in (A. 28),
That implies
as 61 increases to infinity.
7(00)02)
bounded from above, and hence, there
such that (A. 30) holds. The symmetry of the game
infinity
exists a h*
imjjlies the assertion in the proposition.
I
However, there are many other possible equilibria
game described
by many game
in the
paper
theorists,
is
may
for this
game,
hi particular, the stocking
a strategic rivalry in a long term relationship which, as recognized
differ
from that of a one-shot game. As A\imann and Shapley (197C),
Rubinstein (1977), Fiidenberg and Maskin (198G) and others have shown, any individually rational
outcome,
i.e.,
in infinitely
known
an outcome that Pareto dominates the minimax point, can
repeated games with sufficiently
little
play of a one-shot game, and as a matter of fact,
that the
phenomenon addressed by
game we studied
the stocking
a stochastic
is
it
Theorem
the Folk
Nash equilibrium
discounting. This assertion constitutes the well-
Though
"Folk Theorem" for repeated games.
arise as a
is still
game
is
of timing,
not a repeated
we
shall
tnie.
For simplicity and illustration, we only i)rove a weak version of the Folk Theorem. That
the case that X
can
arise as a
<
fi,
v/c
Nash equilibrium
sets
Lemma
no upper
game with
in the alternate stocking
firm that employs a barrier policy with parameter
it
is,
for
show that any outcome that Pareto dominates the zero inventory outcome,
Denote by v" the payoff when no firm holds inventory,
and
show
limit for the inventory,
i.e.,
h
t;°^
i.e.,
when
=
t'
'
v"
=
it
is
sufficient little discoimtiug.
v^' (D),
and v°° the payoff to one
the rival's turn to hold inventory
'°^(0).
A.l.
A
r
t;°°
=
(A
1
-n) +
-f /x
+
(X
X
X
r
where 7(00)
=
(Af? -f fip2
^)/(A
r)
-1
r)(Pi
—
uh
r
r
+n+
-
„-l\_/^
\..7h
/>^')7(oo)/^ 7
^l
+
+
rlo
rt;
2
-
ci)
,
r).
= — and
,
hi{h)(')(co)c2
r)(h^{h)-'1(oo)h2(l>)]
For
/x
>
A
lim
+
lim
rt;°°
r|o
A,
= —
2A'
-:
(X
23
,
-h.
+ n)(2fi-X)
{A.33)
J
Proof. Equation (A. 31) follow? from (3 19) by roplariup;
=
v°^
he
limit in (A. 33) caii
Lemma
A. 2. F^r
>
/i
A,
ol>taiiirfl
/>!
—
by uotiring the fart?
85 r
By L'Hospitals
In the Folk
make
r
2/i
Theorem, the individually rational outroinc
We
arisi-s a?
any feasible payoff pairs
game
the stocking
//
>
A.
h,
f^"*'
•''"<'^'
fi'i,i'2i
is
There exists
is i)ayoff
^i(^)
is
G
r*
(0,OO)
>"".« =
b'
=
so,
00.
What we would
t
deviates from
show
like to
is
1,
,
'^
'
is
sufficient
Each firm
loss of generality, otir discussion
and
A
=
firm
is
it
V
/
2/i
+
+
r
VA(1 -r;) +
A
+M+
r
V
A
+
+
r
/i
1
concerns firm
if it
2/i
,
'^
'
vUO) +
+
r
and
/»,
for
i)lay
ft,
payoff
i
=
1,2.
as long as the rival
it
by inlaying
small.
r is sufficient
Sui)i)os('
if it
i;,,
it
is
deviates to
firm
b\
Is turn.
respectively
Tlinii,
-
A(l
2,
only.
1
stay with
turn to hold inventory.
A
in the
the other firm will j^unish
b'-,
and
(0,r*)
is r.
which n-snlts
t
G
exists a Na,sii equilibriuin of
that no firm wants to deviate wIkui
Denote by v\(0) and v\{0) the payoffs to firm
when Z =
say, to
h,,
thnr
wiirn thr f/jsruimf rate
i>,-
the strategy emi)loyed by firm
and once firm
Without
v\(0)
?uj)ported by
inventory
tlmt. far nil r
siirii
1.2.
strategies that su])port the etjnilibrium arc as follnwp.
doing
rfiuilil)riuin
will see that the strategy of sotting infinite
in whicli firm
Proof. Suppose that
is
an
the outcome that dominates the zero inventory oiitroiiir be an e(juilibriuiii outcome.
Proposition A. 4. Suppose
The
A
nilo.
certain penalizing strategy.
to
-
/J
0.
»
Proof.
the folIf)wiiip Icmnia.
in
,1
an (I
1
Pi
r
—
T
^2 ~* mZ-^i
1,
I-
aii<l
r).
lini(<72(ft.ft')4 -/•).
b'\oo
Till
witli
/),
+
r?)
r
-
A(l
r?)
+
+
2//
\
fi
^
-/)
r'
r
"
0,(0)
'
=
(A
+
/i(l
VA(1
-
r/)
I
/i
'
'^
+ M+
^
\
-
+
,,2"')
r)a(p,)p:'"-''
V
+
2/x
+
+
-
r?)
+
Ha[p,)pf^^' - a[p,)p-/^^')[v^ (A
+
,i(l
Using the above expression and
-
^-')
Lemma
+ r)a{p,)pf^^' A. 2.,
+
(A
/i(l
we have
for
2/i
+
(^,.
(A
/i
+
>
-
/-fM
r
-y
r'j
+
^^))
;
A(l
r
-
,i(l
-
+
/>:')
+ r)n(p,)p;'"^'
[p-'
+
-
J
p-')n-''-
r)a[p2)p-''^^'
A,
limr(.;|(0)-p|(0))
''"
=
2
limr(t;J(0)
r|0
24
-
(^.34)
u°°)
By assumption and Lemma
lim r(u?(0)
for
auy
h\.
-
A.I.,
u°°)
>
lim r(v^
-
,;°^)
=
Therefore, there exists a r* such that for
25
r
A
-;,
G
+
2A'
(".r*), w}(0)
h
>
>
0,
t)}(0), for
any
b[
^
ftj.
References
1.
Auinami, R. and
L. Sliai>lfy,
"Long Term
Conii)ctitioii:
A Game
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Point rrocrssrs an-/ Qunirs, Sjjrinjrci-Vrrlap;,
Nrw
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3
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westem
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Dyukin. E B and A. A YtisLkevicli. Controlled M.irkov Processes, Springer- Verlag, 1979.
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M
Brownian Motion and Buffered Flow,
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Prabhu, N.U., Queues and Inventories:
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A
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John
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Applied ProltaUilistic Models with OjttiniizHtion
Ai>j>lirations.
Holden-Day, San
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IG. Rubinstein,
A
,"E(}uilibriniii
in
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thr-
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17. Scarf, H. E.,
"A Survey
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Srarf,
Cilford,
and Shelly
(ed.),
Theory', Multistage Inventory
Stanford University Press, 19G3,
pp. 185-225.
18.
Schmenner, R. W., Production/Operations Management
Research Associates, Lie, 1981.
2G
.
Concepts and Situations, Science
X
B
r
n
^k—
—o
o X X X
Z =
o o
A
figure
t>)<
- B
1
—b
p
9
O
X
-
y
-^i
^x
figure 2
/
V*
'-^
K
's^
\J
Date Due
MAY
2
MAR.
133QJ
'
=>
l"'"
3
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