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HD28
.M414
no.
Heavy
Traffic
Analysis
Stochastic
Inventory-Routing
Martin
I.
the
of
Dynamic
Problem
Reiman, Rodrigo Rubio, and
Lawrence M. Wein
#3881-96-MSA
February,
1996
^ASs,
r^t'^-l^S
APR
X 1
^996
Heavy
Dynamic Stochastic
Inventory-Routing Problem
Traffic Analysis of the
Martin
AT&T
I.
Reiman
Bell Laboratories
Murray
NJ 07974
Hill,
Rodrigo Ruhio
Operations Research Center, M.I.T.
Cambridge,
MA
02139
Lawrence M. Wein
Sloan School of Management, M.I.T.
Cambridge,
MA
02139
ABSTRACT
We
analyze three queueing control problems that model a dynamic stochastic
tribution system, where a single capacitated vehicle serves a finite
make-to-stock fashion. The objective
is
to
in
of retailers in a
each of these vehicle routing and inventory problems
minimize the long run average inventory (holding and backordering) and transporta-
tion cost.
In all three problems, the controller dynamically specifies
the warehouse should idle or embark with a
travel along a prespecified
how many
(TSP) tour
units to deliver to each retailer.
The
By assuming
whether a vehicle at
problem, the vehicle must
and the controller dynamically decides
In the second problem, the vehicle delivers
and the controller decides which
third problem allows the additional
shipping options.
In the first
full load.
of all retailers,
entire load to one retailer (direct shipping)
next.
number
dis-
dynamic choice between the
an
retailer to visit
TSP and
direct
that the system operates under heavy traffic conditions,
we
approximate these queueing control problems by diffusion control problems, which are explicitly solved in
case.
the fixed route problems, and numerically solved in the dynamic routing
Simulation experiments confirm that the heavy
rate over a broad range of problem parameters.
about the behavior of
this
Our
traffic
approximations are quite accu-
results lead to
complex system.
February 1996
some new observations
Introduction
1
A
prototypical example of the inventory-routing problem (IRP)
large oil
company
as
it
distributes gasoline to
its
is
the challenge faced by a
various gas stations: several warehouses,
which hold inventory of a particular item (gasoline), serve a
set of retailers (stations) in a
make-to-stock fashion; arriving customers (automobiles) consume the product at these
and a
sites,
fleet of finite
capacity vehicles (tanker trucks)
from the warehouse to the various
The management
are many-fold
warehouses and
and operation of such a system
Traditionally, a hierarchical decomposition of the
used to allow for a solvable model at each of the levels
strategic level, the
used to transport the product
retailers.
decisions involved in the design
and complex.
is
managers of
retail
Simchi-Levi 1992).
is
At the
system must determine the location and number of
this
retailers, as well as
(e.g.,
problem
the assignment of retailers to warehouses. At a tactical
they must decide on the number of vehicles to operate, and possibly on the assignment
level,
of vehicles to service districts.
At the operational
send a particular vehicle out or
let it idle,
which of the
how much
retailers should each vehicle visit,
deliver to each of the retailers on
its
level
the decisions include: whether to
of the capacity of the vehicle to use,
and how much of
load should a vehicle
its
route.
At the tactical and operational
levels,
the essence of the
IRP
is
the tradeoff between
inventory costs and transportation costs: in order to reduce inventory levels at the retail
sites
without affecting the service
level,
more frequent replenishment
thereby increasing the transportation costs. In
to a lesser extent, vehicle travel times)
such cases, a stochastic model
this
is
is
many
applications, customer
field of
operations research
is
sometimes
in
criticized
Ackoff 1987).
(e.g.,
terexample to this perception: while heuristics
some spectacularly
In
required to accurately capture the inventory costs, and in
have lagged behind theoretical progress
led to
demand (and
subject to considerable stochastic variation.
paper we focus on the operational aspects of the IRP
The
deliveries are required,
a dynamic stochastic setting.
because
The IRP
real- world applications
is
an important coun-
for this notoriously difficult
problem have
successful industrial applications at both the operational (e.g., the
1
Edelman Prize winning work
of Bell et
1983, Golden et
al.
1988) levels, a concomitant mathematical theory for the
IRP
al.
in
1984) and tactical (Larson
a dynamic and stochastic en-
vironment has not been forthcoming. Federgruen and Zipkin (1984) analyze a single-period
IRP with stochastic demand, and Dror and
Ball (1987) develop a heuristic technique to
reduce the long run average problem to a single period problem. Recent studies that con-
IRP
sider the operational aspects of the stochcistic
include Trudeau and Dror (1992),
who
develop heuristics for the case of an external supplier, where retailer inventories are only
observable at delivery times; Minkoff (1993),
Markov
decision
itinerary
model that dispatches
who
constructs a decomposition heuristic for a
vehicles on a prespecified set of itineraries,
where each
characterized by an inventory allocation to a subset of customers; and
is
Schwarz and Ward (1995), who develop myopic
static
and dynamic strategies
Kumar,
for allocating
the contents of a vehicle to the various retailers on a predetermined tour. Chan, Federgruen
and Simchi-Levi's (1994) probabilistic analysis of random instances of the deterministic IRP
is
useful for addressing tactical
aspects of the
IRP with
and strategic
stochastic
issues,
but has no bearing on the operational
demand.
Our system model has one warehouse and one capacitated
assume that the higher
level decisions
An ample amount
available at the warehouse, and the cost of holding this inventory
demand and
and the objective
may
is
differ
by
is
to
retailer)
vehicle travel times are
and operating the
send the vehicle out with a
less
than one, then an
Two
routing,
when a
We
is
not included in the model.
random, unsatisfied demand
if
when the
full
load or
One
vehicle.
vehicle
let
is
is
backordered,
of the crucial decisions in our
problem
at the warehouse, the controller can either
the vehicle
sit idle;
because we employ a long run
the vehicle does not idle and the "traffic intensity" of the system
infinite
amount
is
of retailer inventory will build up over the long run.
types of IRP's are analyzed: the
dynamic routing.
is
of inventory
minimize costs due to holding and backordering inventory (cost rates
the vehicle idling policy:
average cost criterion,
we effectively
have been made to assign a single warehouse and a
single vehicle to serve all retailers in a particular region.
Retailer
vehicle; hence,
first
assumes fixed routing and the second allows
consider two variants of the fixed routing IRP: in the
vehicle leaves the warehouse
it
IRP with TSP
uses a pre-optimized tour of the
m
retailers,
which we
TSP
refer to as the
idling policy, the controller
decision
is
(travelling salesman
must decide how many units
based on the current inventory
The second
of units in the vehicle.
problem) tour. In addition to the vehicle
levels at all retailers
variant
time the vehicle leaves the warehouse
it
is
the
IRP with
TSP
and
this
and on the remaining number
direct shipping] in this case, each
delivers all of its contents to a single retailer,
the controller dynamically specifies which retailer to
routing, the controller decides, based
to deliver to each retailer,
In the
visit next.
upon the current inventory
and
IRP with dynamic
levels,
whether to use a
tour or direct shipping.
By
TSP
only allowing
we avoid an
tours or direct shipping,
assault on the combi-
embedded routing problem, and model these problems
natorial aspects of the
as queueing
control problems. Since these control problems appear to be analytically intractable, heavy
traffic analysis is
employed
in order to
nontrivial limiting control problem,
of the time to
is
make
further progress.
we assume that the
To obtain an
server (vehicle)
meet average demand, the vehicle capacity
is
conditions are stated
theorems
in
sition in the
more
precisely later in the paper.
is
and
must be busy most
large, the tour
long and nearly deterministic, and the vehicle operating cost
interesting
completion time
large; these
heavy
Guided by the heavy
traffic
traffic limit
Coffman, Puhalskii and Reiman (1995a,b), we uncover a time scale decompo-
heavy
traffic limit;
however, no weak convergence proofs are provided because
they would be very demanding and would distract us from our primary objectives:
gain insights into the nature of the optimal policies for the IRP, and
(ii)
(i)
to
to develop effective
pohcies for the operation- of these systems. Under the traditional heavy traffic normalization,
where time and inventory are compressed by a factor
going to
infinity),
of
n and
y/n, respectively (and
the (embedded or averaged) total retailer inventory process
proximated by a one-dimensional reflected Brownian motion on the interval
the control parameter
idling policy.
The
w
drift
— 00, tf], where
represents an aggregate base stock level that dictates the vehicle
is
TSP
being used, and in the dynamic routing problem the controller
dynamically switches between two Brownian motions (one
its
is
well ap-
and variance of the Brownian motion depend on whether the
or direct shipping policy
ping), each possessing
(
is
n
own
drift,
for
TSP and
variance and cost structure.
If
one for direct ship-
we take the heavy
traffic
normalization and slow
time
fcister
scale.
A
down time by a
factor of ^/n, then a fluid limit
is
obtained on this
deterministic analysis at this time scale allows us to optimize over the
other operational decisions.
Our
results are quite explicit; the derived controls for the fixed routing
given in closed form or
in
that
of the optimal policy for the diffusion control
IRP with dynamic
routing,
we analyze a
cleiss
problem
of triple threshold policies
conjectured to contain the optimal policy, and numerically compute the optimal
is
policy to the diffusion control problem.
the accuracy of the heavy
A
traffic analysis
computational study
and allows us
importance of the various operational decisions
dynamic
allocation,
In §2
and
§3,
The performance
routing
is
TSP
vs. direct
of these
analyzed in
§5.
two routing schemes
is
dynamic
compared
Our computational study
summary
carried out that confirms
the vehicle idling policy, static vs.
(e.g.,
shipping, fixed vs.
is
to obtain insights into the relative
we analyze the IRP with TSP routing and
concluding remarks, including a
routing).
direct shipping, respectively.
in §4
and the IRP with dynamic
§7 contains
described in §6.
is
some
of our key findings.
The IRP with TSP Routing
2
2.1
Problem Formulation
Consider a system where a single vehicle with capacity
product to
m
at the central
geographically dispersed retailers.
warehouse at no
a make-to-stock fashion, and
When
the vehicle
is
cost.
demand
full
is
inlinitc
used to distribute a standard
supply of the product
is
is
kept
retailer inventories in
that cannot be served immediately
load and then visits
retailers are visited could
V
Customers arc served from the
before returning empty. Alternatively, the vehicle
which
An
operating the following policy
(indexed as station 0) with a
in
either
terms of parameters that are solutions to certain equations. Because
we cannot characterize the form
associated with the
IRPs are
is
backordered.
used: the vehicle leaves the warehouse
all
may
the retailers in a predefined sequence
idle at the depot.
be arbitrary, we assume that
it
is
Though the order
the solution to the
TSP
implied TSP, and refer henceforth to this service scheme as the
we assume
generality,
in the
=
i
that retailers are indexed from
through
1
TSP
tour.
Two
sources of variability are considered: customer
l,...,m, customer demand at retailer
process {Di{t)^t
>
0} with rate
Without
m according to their position
demand and
denoted by D{t)
= ^,
= ^,
and A
Di{t),
coefficient of variation c^- (variance of the
demand
the total
A, is
of this paper the index runs over the set of retailers {1,2,...,
Our
otherwise.)
cesses; see §6 of
i
and j
is
rate. (In all
m}, unless
given by
samples of the random variable
iid
coefficient of variation
demand streams and
The sequence
(1984) for details.
cf^
{i,j
run from
of each other.
to m).
[0, i]
summations
explicitly indicated
of travel times
between
which has mean
Tij,
in
compound renewal
results easily generalize to cases with correlated
Reiman
For
travel times.
interdemand time divided by the square of the mean). The cumulative total demand
is
loss of
occurs according to an independent renewal
i
and squared
A,-
policy.
Oij
pro-
facilities
and squared
These travel times are independent of the
Keeping with the convention
in the literature,
we assume
that pickup and delivery of units occur instantaneously; in practice, load/unload times tend
to
be dwarfed by the travel times. (Although non-zero load/unload times can be incorpo-
rated in a straightforward manner, the analysis becomes
cloud the basic issues). Hence, the
the
TSP
tour are given by 0t
spectively,
=
E.To'
where the subscript "T"
coefficient of variation of the tour
denote the counting process for
continuously active in
is
^.,.+i
is
+
^mo and
mnemonic
for
idle; (ii)
The busy/idle
while the vehicle
control
be interrupted, and so the sequence
ini {t
I
TSP. For
later use,
=
we
s^/^j, and
t
re-
define the squared
let
{^^(i),^
>
assuming the vehicle
two operating control decisions remain:
is
0}
is
STiBxit))
>
k}.
The
r^.,
is
busy,
how much
(i)
whether
of the load to leave
expressed in terms of the cumulative process Bj{t),
which represents the amount of time the vehicle
=
would
4 = ZfJ^' Ol^+Aj+i + ^moC^o,
tour completions up to time
fixed, only
the vehicle should be busy or
Tfc
inclusion
variance of the total time required to complete
completion time as Cj
TSP
its
[0,t].
Because the route
at each retailer.
mean and
more tedious and
k
=
is
1,2,
..
busy
.
in [0,<].
We
do not allow tours to
of tour completion epochs
delivery allocations are
is
given by
modeled by the m-dimensional
amount
control process L,{t), which represents the cumulative
time
nominal delivery
We
let
developments,
In anticipation of future
t.
=
Vi
size for retailer
A,V/A
for all
V among
allocation process
defined by
and a dynamic allocation process
V^,
=
L,{t)
-
V,ST{BT{t)) for
>
t
amount dehvered during the current
deliveries to retailer
up
i
we need only
to time
cycle for retailer
ef (0)
=
{^^)
>
^i
ej{t)
>
ef(rjt_i)
=
eriTk)
"-"
where the superscripts
The number
XI,
Qi(0-
at time
If
t
=
(1)
size over past tours,
Because the tour completion
(.J{t)
to determine the total
we assume
J2i ^Ji^) represents
the total amount
that the vehicle leaves the warehouse
(^~)
i,
only
for
(2)
if
retailer
<G
is
visited at time
is
andall
(r;t_i,rfc)
is
(3)
/,
i,
(4)
and
(5)
0,
(6)
and "+" denote the times just before and after an epoch.
if
this quantity
denoted by Qi{t), and the total inventory
we assume
that Q,{0)
=
(J
(wliich
is
without
is
negative) at
at the retailers
is
Q{t)
loss of generality, since a long
=
run
being used) then the current inventory Q,{t) equals the cumulative
minus the cumulative demand, which by
Q,it)
i
of units in inventory (or backordered
average cost criterion
deliveries
for all
= V
iT{T^)
i
load
load and returns empty, the dynamic load allocation process must satisfy
^i
retailer
The
Notice that deviations from the nominal allocation
t.
delivered during the current cycle. Because
full
i.
specify the value of
cancel out across the retailers and the process er(i)
with a
tf{t).
0,
which represents the cumulative deviations from the nominal delivery
history can be observed,
up to
us express this control in terms of a
the retailers according to their relative demands.
eJit)
plus the
i
so that the nominal delivery size corresponds to allocating
i,
the vehicle capacity
is
denoted by
i,
let
delivered to retailer
= V,STiBT{t))-D,{t) +
eJ{t)
(1)
for
given by
is
/
=
l,...,m,
t
>
0.
(7)
Define the cumulative vehicle idle time process I{t) by
I{t)
=t-
so that the control policy Bj{t),tJ{t)
BT^
Our
must
>
t
0,
satisfy
are nonanticipating with respect to Q,
e,
in
(9)
Bt
is
nondecreasing and continuous with -Sj(O)
/
is
nondecreasing with 7(0) ==0.
The
=
(10)
0,
(11)
cost,
travel cost rate per unit time, which includes vehicle depreciation, fuel
is r.
Note that these costs can be combined because we are ignoring the
load/unload times (only the driver, but not the vehicle,
ing).
(8)
objective function includes transportation costs and inventory holding and back-
ordering costs.
and driver
Brit) for
is
busy while loading and unload-
Inventory costs are assumed to be piecewise-linear, with the holding cost rate (per unit
inventory per unit time) at retailer
i
denoted by
and the backorder cost rate by
h^
Because travel costs are incurred whenever the vehicle
6,-.
busy, the travel cost rate r can be
is
equivalently treated as a reward for exerting idleness. Hence the problem reduces to finding
a control poHcy fB7'(i), ef (i)j to minimize
limsup
—E
T— CO T
subject to (2)
-
(11),
The dynamic
[
where the
Y.ih^{Q^{t)]''
"-|-"
"-"
h,{Q,[t)]-)dt
for the
-
rI{T)
(12)
denote the positive and negative parts.
stochastic IRP, as formulated in (2)
Even under Markovian assumptions
is
and
-f
underlying
-
(12), does not
random
seem
to be tractable.
processes, the control space
enormous and the state space has m-\-2 dimensions: the inventory/backorder
retailer
and the location and
the problem,
2.2
We
we analyze
Heavy
it
Traffic
begin our heavy
traffic
total contents of the vehicle.
when
To gain
level at
each
further understanding of
the system operates in the heavy traffic regime.
Normalizations
development by centering the service completion and demand
processes; define the centered processes «Sr(i)
=
Sxit)
—
Oj^t and T)[t)
=
D[t)
—
Xt.
It is
convenient to define the process
"
'^^^^
we
refer to this quantity as the
(^ " ^J
V^5T(5r(/))
-
netput process, although
it
'
to the netput processes constructed in the
networks
retailers
(e.g.,
Peterson 1991).
Summing
and substituting the relevant
"^
heavy
V(t)-
(13)
does not correspond precisely
traffic analysis of
conventional queueing
the inventory evolution equations (7) over
definitions yields
Q{t}=x{t)-^m + eT{t).
The heavy
(14)
may be found
approximation for the IRP
traffic
as the
of a sequence of systems indexed by the heavy traffic parameter n.
will
are non-traditional,
we index
off
This indexing
(as
will
quantities with
is
n
(in
Even though no weak
an appropriate place) to make the scalings
be confined to this subsection;
the index, with the understanding that
associated value of n.
Hmit (as n -^ oo)
be undertaken here, because some of the scalings that we introduce
convergence proofs
clear.
all
we
analysis are independent of n.
according to standard heavy
of as a large integer (e.g., 100) but
recommendations that emerge from our heavy
The parameter n
traffic
paper we leave
are considering a single system that has an
The parameter n can be thought
typically the case) the policy
for the rest of the
is
traffic
used to normalize the various processes
Bj
undergoes
9^,
y^
(15)
conventions (notice that only the process
a "fluid" scaling):
W}-Ht)
=
9i:^
W^-\t)
for alii,
'
^/^
\/n
processes ly'"' and
V'*"'
Y,w!'^\t) =
y
5n<) =
'
'
(16)
yjn
yjn
mi) = V^,
The
™
=
and
4"'(')=^£^.
(17)
represent the normalized inventory and idleness, respectively; to
reduce the amount of notation, the normalized versions of the remaining processes contain
8
a "hat".
Employing these
and
scalings in (13)
(14) yields expressions for the uormalized
netput process
'T
and the normalized inventory process
y(n)
=
H/(")(0
To obtain a
parameters
- -^Y(-\t) +
x^-\t)
problem
nontrivial control
in
F^"'
(19)
we normalize the system
traffic,
The demand and inventory
in a particular fashion.
and the other parameters are normalized
scaled,
heavy
e?)(0.
cost parameters are not
as follows:
= ^^,
(20)
n{n)
op =
%.
(21)
/T/(n)
(n)
\
^\^J^B
/'r
^*"'
(n)
>
(22)
0'
T
4(n)
=
^f^c\[n),
(23)
r'"'
=
—
(24)
n
We
assume that
all
and positive hmits
the quantities on the
as
specify, in a unified
n
^ oo.
manner
traffic
conditions and they
via the heavy traffic parameter n, the relative magnitudes of
These conditions are more extensive than those enforced
queueing systems, and therefore warrant some discussion. Since the natural
definition of the traffic intensity
traffic condition,
Now we
side of definitions (20)-(24) converge to finite
Equations (20)-(24) are the iieavy
the various system parameters.
in traditional
left
is
pj
=
X9j-/V, condition (22)
which requires that pj be close
to,
but
less
is
the "traditional" heavy
than, unity.
turn to conditions (20)-(21). Because the state space
is
compressed by a factor
of y/n in the heavy traffic normalization, the vehicle capacity in terms of scaled inventory
units
is
V*"'/\/"'-
Hence,
would reduce to a variant
if
V*"'
was 0(1)
it
would vanish
of the multiclass make-to-stock
Although such a model would be tractable, a
9
limit that
in the limit,
queue analyzed
and our system
in
Wein
(1992).
employs infinitesimal vehicle
sizes
to capture the essence of the behavior of the original system.
fails
condition (20), so that
in
Therefore,
we enforce
0[yjn) and the bulkiness of the retailer deliveries
V'*"' is
the limit. However, since the
demand
rate A^"'
lengths according to (21) to ensure that the ratio
is
unsealed,
we need
K^"V^r converges
retained
is
to also scale the tour
to a finite
and positive
limit.
Turning to
note that since the vehicle capacity
(2.3),
is
0(\/n),
if
the Cj(n)
is
not
scaled then a standard calculation shows that the variance term for the normalized netput
process x^"'
is
0{n) and hence approaches
c^{n)6j{n), by (21
)
and (23) we obtain
enforcing condition (23),
in contrast, this
way
we assume
Sj'{n)
is
to
on a two-dimensional
y/n.
Finally,
relative
travel times
=
=
Thus, by
9j'C^{n).
is
0{y/n);
were simply multiplied by y/n. One
travel time of the tour
is
the
and
we need
is
map
sum
with a fine grid, in such a
likely
of y/n
iid finite
way
that the tour
is
problematic
be independent and the necessary data would
not pursued here; see Rubio (1995) for further details.
to normalize the cost parameters to account for distortions in the
magnitudes of the transportation and inventory costs that
traffic scaling.
n times
^/nsj[n), where Sj-{n)
grid points. However, this modeling artifice
(because adjacent travel times would not
collect)
if
assume that the
passes through approximately
be tedious to
Since 5j(n)
traffic limit.
This construction could arise by superimposing the warehouse and
variance travel times.
retailer locations
=
heavy
that the variance of the tour completion time
quantity would be 0{n}
to achieve (23)
infinity in the
The appropriate
scaling
is
result
from the heavy
to allow the travel cost rate r^"' to be approximately
larger than the inventory cost rates, as in condition (24); see
Rubio
for a detailed
explanation.
In
summary, the heavy
traffic
conditions assume that the vehicle must be busy the
great majority of the time to meet average
demand, the
vehicle capacity
must be
large, the
tour completion time must be large and nearly deterministic, and the travel cost rate must
be very large relative to the inventory cost
rates.
The computational study
that our results are rather insensitive to these conditions.
10
in
§6 reveals
System Behavior
2.3
Heavy
in
Traffic
This subsection considers the limiting behavior of equations (18)-(19). Following Harrison
(1988),
we
replace Bjit), which
for this substitution
is
is
the fluid scaled busy time process, by pjt; the justification
that any pohcy that does not utilize the vehicle for a fraction pj of
the time over a sufficiently long time interval will generate extremely large inventory costs.
In addition,
we consider the normalized netput
Without some embedding
because
exist
it
X{t)
With
is
at tour
completion epochs.
would not
by 0(1) on a time of length 0(l/>/n).
The
thus defined as
= ^{-0--^)
+ y^TipTTk-i) -
^^-^
'DiT,_,) for
t
€ [n-uTk)
.
standard tools of weak convergence (the functional central hmit
this definition the
theorem
embedded
or averaging the limit of the normalized netput process
varies (after normalization)
we consider
process
process
for renewal processes, the
random time change theorem and the continuous mapping
theorem; see Billingsley 1968) can be used to show that the normahzed netput process
embedded
drift
at tour completion epochs
pt and variance
Now we
=
cry
A(c^
is
well
time units under the heavy
of size 0(1)
whenever a delivery occurs and
traffic
m— dimensional normalized
However,
if
we
start with the
then a fluid scaling
\/n.
At
this faster
ual inventory levels
is
time
scale, the
move
at delivery epochs.
value,
and equals zero
is
with
at the
end of the
0{\/n) by (21), a tour takes only 0(l/\/n)
heavy
traffic
is
compressed by the factor
hmit. Consequently, neither tj{t)
inventory process converge to a limit in the usual sense.
traffic
normalization and expand time by a factor of y/n,
obtained, where both time and space are compressed by the factor
jumps
This
heavy
is
normahzation (where time
n); hence, tours occur instantaneously in the
nor the
X
Vc^-).
In addition, because the tour length
cycle.
approximated by a Brownian motion
turn our attention to the process ey. This process equals zero at tour com-
and has jumps
pletion epochs
+
x
Brownian motion
X
remains constant, and the individ-
in a deterministic fashion, decreasing at a finite rate
The
process tj traverses through
many
between the
tours before
X
changes
at each tour completion epoch.
similar to the state of affairs in the heavy traffic results of CofFman, Puhalskii
11
and
Reiriiaii.
In their
exhaustive polHng system, the total queue length process behaves as a
one-dimensional diffusion under the slow time scale associated with the heavy
and the individual queues move as a
fluid limit.
(HTAP)
traffic scaling,
under the faster time scale associated with the
fluid
This time scale decomposition gives
rise to
a heavy
traffic
averaging principle
that implies the following: for purposes of calculating performance measures for the
individual queues, one can analyze the deterministic fluid cycle for each fixed value of the
diffusion process.
and
in
in
the polling system
the IRP. First, the fluid trajectories are different. In the polling problem, the fluid
paths look
down
There are four key differences between the IITAP
like
those for the economic production quantity
at a finite rate. In the IRP, the paths look like those
(EOQ) model:
they go
down
at finite rate but
go up
(EPQ) model: they go up and
from the economic order quantity
in
jumps
at delivery epochs.
The
second key difference relates to the issue of "control". In the polling system the exhaustive
discipline guarantees that
whenever the server switches from a queue, that queue
is
empty.
This exerts a type of control that keeps the multidimensional process well behaved. There
is
no such natural mechanism
in the
IRP; we must introduce a dynamic allocation scheme
to keep the multidimensional process well behaved.
emerged
scale decomposition in the polling system
traffic
normalization, whereas in the
This gives
rise to
IRP
it
the fourth key difference
(This
is
done below.) Third, the time
as a consequence of the standard
heavy
from the scaling assumptions (20)-(23).
follows
— the proof of the HTAP. In the polling context
a difficult proof involving a threshold queue was needed.
For the
IRP the proof
follows
from assumptions (20)-(23) and the properties of the dynamic control (we do not, however,
provide the proof).
2.4
The Limiting Control Problem
As described above, the
scales.
On
analysis of our limiting control problem decomposes onto two time
the slow time scale associated with the diffusion limit,
effects of the controlled allocation process er
is
and choose the vehicle
generated by the normalized cumulative idleness
12
)
.
we can average out the
idling policy.
which we as.sume
is
This policy
nondecreasing and
right continuous.
=
Let Z{t)
—
X{t)
g-Y{t); this
is
the process that would be obtained
one were to observe the total inventory only at tour completion epochs.
process as the total
At the
embedded inventory
faster time scale
process, to differentiate
where the
embedded inventory
total
the individual inventories behave as a fluid,
it
we must
from
is
We
if
refer to this
W.
fixed at Z{t)
find the optimal allocation
=
x and
pohcy that
minimizes inventory costs per unit time. The limit cycle associated with an allocation policy
can be viewed as a closed
to the
m— dimensional
problem of optimally placing a deterministic cycle
inventory cost per unit time that
We
routing:
Z{t)
=
X,
R™.
in
placement
find the optimal cycle
(i)
and
its
Z{t)
=
x.
IRP with TSP
level
choose the nondecreasing
(ii)
to minimize
limsup
T—oo
—E
subject
to
cycle placement problem
control problem for
for the
when
embedded inventory
for a given total
corresponding inventory cost rate g{x); and
Y
Let g{x) represent the
achieved by optimally locating a cycle
is
can now state the limiting stochastic control problem
right continuous process
The
path, and the optimal allocation policy reduces
/ 9iZit))dt-rY(T)
(25)
Jo
J
Z{t)
=
X{t)
-
—V y'(0-
(26)
a nonhnear program and problem (25)- (26)
is
Brownian motion; these two problems are solved
a singular
is
in the next
two sub-
sections.
2.5
Optimal Cycle Placement and Dynamic Allocation
To optimally place the
Wein (1995)
limit cycle,
for the stochastic
and denote the individual
we
follow the approach used in Markowitz,
economic
lot
fluid inventory levels
used to denote quantities introduced for the
in
many ways and we
scheduling problem (ELSP). Let us
choose to specify
it
by the vector
fix
Z{t)
=
x,
=
Qi{y/nt)l s/n (a "bar" will be
The
cycle placement can be defined
by Wi{t)
fluid limit).
Reiman and
(xi, X2,
.
,
.
.
x^), where x^ represents
the lowest point during the cycle of Wi[t).
The
choose (xj,
choice of optimal
.
,
.
.
(.Tj,
.
.
,
.
Xm)
is
a constrained optimization problem:
Xm) to minimize the inventory cost rate subject
13
we want
to
to consistency with the total
WAt)
-^^^,
A
Figure
1:
Fluid inventory evolution at retailer
embedded inventory
level.
be described below.
We
To
inventory cost rate will
come from the averaging
TSP
of the inter-site
and
i
mean
we need
to introduce
path between any two
travel times
=
Of-^^
some new
sites i,j
G
0, 1,
is
model by
Ofj^^
time over a cycle so that the vehicle leaves the warehouse at
to its corresponding cycle placement value x,
=
1,
.
.
,
.
.
.
,m by
compressed by a factor of ^yn
define the corresponding travel times for the fluid
z
.
m. Summing these inventory
i
by (see Figure
9j]^^; in
levels over all retailers,
=
Oj^^^ /y/n.
=
0,
If
we measure
then VV,(0)
=
VV',(0)
x,
+
is
ventory cost rate.
solution.
to retailer
size
is
The
Long term
i
V,
=
.
,Xm) satisfying
.
bcisic intuition of
^(dof^
Vi
=
Vi/y/n.
for
we obtain
(27)
(27),
we want
to determine the associated in-
the time scale decomposition appears to provide the
stability requires that, in the long run, the average
per cycle be
given by
.
related
i
t
(xi,
terms
YliZi ^k.k+i
^x. = x-X:A.^T/^
Given a vector
total
in the fluid limit,
=
1)
and the
Denote the mean
notation.
these quantities are defined by Of^^^
6,j,
Since time
0.
principle to
deal with the consistency issue.
first
level x,
travel time aiong' the
>
during a nominal allocation cycle.
establish the relationship between the cycle placement variables i,
embedded inventory
for j
The
i
A,K/A; under the diffusion and
Viewing the
fluid
14
amount deUvered
fluid scalings, this delivery
inventory of retailer
i
in isolation
with a
we
delivery of Vi on each tour cycle,
delivery, decreasing at a constant rate until
Figure
The inventory
1).
considering the normalized inventory process
Although
[xi, Xi-\-Vi].
this
Simply delivering
VF, to
retailer
be uniformly distributed on the interval
Vi to retailer
i
on every
is
simple allocation scheme, the
drift of
VJOf
the inventory "arrival" rate
rate,
visit will result in
an
infinite long
this,
finite.
run average cost
note that under this
Wi{t) does not depend on Wi{t) and equals zero, since
equals the
demand
rate A^
when pj =
with
I]
<
/^j
generated with the effective arrival rate being prVilOj. With a zero
is
grow
central limit theorem arguments indicate that the inventory or backlog will
fact, similar
turns out that
it
needed to keep the long run average cost
because this allocation leads to a null recurrent process. To see
similar result
can then be calculated by
i
approach provides the correct inventory cost
a dynamic (state-dependent) delivery size
immediately after
V^
reached just prior to the next delivery (see
is
a-,
component associated with
cost
+
see a fluid starting out at Xi
arguments can be used to explain some numerical
results in
1
a
drift,
as \/t. In
Federgruen and
Katalan (1994) and Wein, where a state-independent policy performs poorly
a stochastic
in
setting.
A
at the
simple dynamic allocation policy escapes this
warehouse as
follows.
Given a
We determine delivery sizes
difficulty.
fluid inventory level [w\^
.
.
.
,to„)
when the
at the warehouse, the fluid limit of the inventory immediately before delivery
If
possible,
we would
like to deliver
(/,
=
Xi
+
Vi
—
Wi
-f
A,^^"^^ to retailer
the fluid inventory level immediately after dehvery to x,
then this delivery allocation
have
d^
Yl
>
V. This
is
is
feasible. If d^
<
for
some
+
i,
li d,
V,.
i,
>
in
—
is
—TSP
XiOQ^
.
order to bring
for
then, since Yli di
Wi
is
vehicle
=
?'
=
1,
.
.
.
,m
V, we must
a transient state for the fluid limit; within a finite
number
of
{i:d,>0}
cycles
we
will
have
d,
>
for all
i.
This transient interval has no
average inventory cost, and an averaging principle
will
when
between
Xi
and
Xi
+
The average
Z{t)
=
x, W{{t)
on the long run
hold under this dynamic allocation
scheme. In summary, the essence of the averaging principle here
allocation scheme,
effect
can be treated as
is
if it
that,
is
under
this
dynamic
uniformly distributed
Vi.
inventory cost per unit time
divided by the corresponding cycle length.
The
15
is
equal to the cost incurred over a cycle
cost at retailer
i
may be obtained by
simple
geometric arguments
for
any cycle placement
When
i^.
the cycle placement
is
sufficiently
high (low) so that the inventory remains positive (negative) for the duration of the cycle, the
cost
simply the holding (backordering) rate multiplied by the absolute value of
is
which
is
When
the average inventory level over a cycle.
i,
+
K72,
the inventory changes sign during the
cycle the total holding (backordering) cost over a cycle equals the area of one of the triangles
above (below) the time axis multiplied by
when there
length.
is
a sign change we
In the
heavy
sum
traffic limit,
amount demanded per
which
cycle,
/t,
To obtain the time average inventory
(6,).
cost
the areas of these two triangles and divide by the cycle
amount
the
of fluid delivered per cycle, V, equals the
X6t] hence,
is
we
set the cycle length in the fluid
model
equal to V/X, rather than Oj. In summary, we have the following expression for retailer
+ ^)
hi{xi
9i{xi)
=
!^x]
+
2V,
<
Notice that
problem
is
S',(x,) is
a convex function of
to minimize
Let us
make
conventions he
=
h
H,
iixi>0
h,x,
+ f^ \i-V,<x,<0
+ ^)
6.(.r.
With equation
x,.
hi
and
bp
=
b
=
form solution to the cycle placement problem
b,
min,
is
in
h, for all
where
x.
Not
i,
=
£
—
rn
1
1.
The solution
Region
1:
surprisingly, g{x)
x<ar =
j: X^'' -
Oi
+
i
Vi
for
hi
16
I
^p,
is
L ^K,
+
0,
= -
A
closed-
The
solution
cost as a function of the
quadratic with linear edges in the
is
placement problem
i
X,
allowed.
for further details.
and g{x), the inventory
to the cycle
\s
variables; readers are referred to
inventory level x.
Proposition
p
and define the labeling
found by using constraint (27) to turn the
Markowitz, Reiman and Wcin
yields the vector of optimal placements x'
embedded inventory
>
6;,
problem into one of unconstrained optimization over
total
(28) in hand, the cycle placement
^,(x,) subject to (27).
min,
an analogous optimization
(28)
.
<-V;
ifx,
the innocuous assumption that
=
i:
hi
^v
g{x)
= -bx +
ai,
nTSP
Region
,
aT<x<h = Y.
2:
l^{b + h,Y
Iv-.tV
>^^~^l''^
E r^^'
+
'
i
X:
=
«2
=
7,\1^
2
V
+
h^
,
i
h.
0,
/?,
%r
03
bih,Vj
04
bi
Region
x
3:
x]
>
-\-
hi
/Sj,
= -r^nr^
Oj +
Z^''
^^
^'
ft,
g[x)
a^
=
/ix
+
a,5,
= -hTx.'eir
+
Oi
Z^
^ -y.hM
2
^
^
In the region
where the
cycle holds (backorders)
so,
is
much
most of the inventory
'
-E
2 ^
^!'
6,
~
+
f^'
site
(6),
depends upon two
and
its
factors:
nominal delivery
particular retailer represents).
greater (smaller) than zero, the optimal
at retailer i (p),
the difference between
size (or equivalently, the
In the region
where the
cycle placement at each retailer varies linearly with the
It is
worth noting that,
for the
where
its
cheapest to do
is
it
The exact
demand
proportion of
is
when
hi
that the
close to zero, the
embedded inventory
(i.e.,
level for
holding (backordering) cost
total inventory
symmetric cost case
17
^-
/i,
while the cycle for the rest of the retailers remains close to zero.
each
h
total inventory
'
=
in the system.
h and
b,
—
b for
all
=
2
1, ...
,
m), the more natural solution of
{x-ZkhO^r)
also optimal.
is
Optimal Base Stock Level
2.6
Now
that g{x)
known, we can proceed with the solution to the one-dimensional Brownian
is
The
control problem.
Proposition
some
following proposition
The optimal solution
2.
is
proved
Appendix A
in
to (25)-(26) is Y'^t)
=
of Rubio.
suPq^^^^{X{s)
—
z^}'^ for
base stock level Zj.
Hence, the optimal solution
rier 2j,
(
&T<x<^T
if
is
the local time of the Brownian motion at the bar-
and the optimally controlled process
— oo,2j]
Z
a reflected Brownian motion
is
The remainder
(see §2.2 of Harrison 1985 for a definition).
(RBM) on
of this subsection
is
devoted to the derivation of Zj, which can be found by using two well known facts regarding
an
RBM
df^j/V
on (— oo,z].
>
for fij
First, for
which
0,
Y
independent of the base stock
is
portation cost does not affect the selection of
limsup-r^^
^E
[Jq
The steady
if
X
>
2,
where
z,
Hence, the trans-
level z.
and the problem
=
simplifies to
minimizing
g{Z{t))dt] subject to (26).
state density for
{/j
we have \\mt^oot~^Ex[Y{t)]
defined in Proposition 2
—
2p.'r/crj
>
0.
Z
is
given by pz{x)
=
j/je"^'^"^'
\{
x
<
z
and pz{x)
=
Therefore, the optimal base stock level can be found by
minimizing
= n{-bx^a,)vTe^''^'=-'Ux+j\a2X^ +
Ft{z)
r [hx + a5)C'Te^''^'~'^dx
+
a^x^a^)uTe^-'^''"^dx
z
>
/Sj,
{a2X^
+
a:iX
for
and
(29)
Jpr
Ft{z)
= n{-bx +
ai)C'Te^^^'-'Ux
+r
J—oo
for
Or <
z
<
fir-
The constants
in (29)
is
linear
a4)he^''^'~'^dx
and (30) have the same definitions as
that while the optimal base stock level Zj always satisfies zj
fact that g{x)
+
(30)
Jaf
and has a negative slope
18
for
x
<
qj),
> aj
it
(this
is
in §2.5.
easily seen
Note
by the
need not be larger than 0t-
The
following proposition
and then taking the
Proposition
first
The value that minimizes Ft{z)
3.
\b
vj
is
derived by using integration by parts on (29) and (30),
two derivatives of Fxi^) with respect to
h
otherwise, z^
is
+
\
(
yri^T
^3-^f
+
is
a unique
(31) that satisfies zf
2.7
4>/5t;
if
+ as-'^ =
^
2a2ZT
is
Ft{zx)
In this subsection
the fact that Ft{z)
optimum base
>
/?x or
The Proposed
=
We
is
hzj
+
0.
(32)
05 if Zj
is,
a solution Zy to (32) that
we map the
satisfies
IRP with TSP
idle,
routing.
The
zf
is
is
size of the
proposed policy
/?y,
is
is
but not both.
control concerns two decisions:
best delayed until the vehicle arrives at the
how much
site.
full load,
Let
to
is
correspond to
and consider the epoch
i.
The mapping from heavy
At time t~ the
,
.
.
,Qm{l-7))i
traffic solution to
straightforward: the proposed pohcy attempts to track the heavy traffic
solution (in particular, the optimal cycle placement) as closely as possible.
to be addressed
of the load to
given by the inventory levels at the retailers, iQi{t~),.
remaining load, L{t7).
retailers
first.
the point in time just before the vehicle arrives at retailer
state of the system
and the
<
and how to assign the load among the
the epoch at which the vehicle leaves the warehouse with a
which
=
solution of the approximating heavy traffic control problem
address the load allocations
leave at each retailer
to,
^j, and Fr{zj)
either there exists a solution Zj to
Since the system evolves dynamically in time, the decision of
>
>
convex and continuously difFerentiable)
stock level; that
whether the vehicle should be busy or
t~
(31)
Policy
into a policy for the original
during a tour.
+ aT
I
^4 otherwise.
One can show (from
that there
—
the solution to
Furthermore, the predicted optimal cost
+
is
Q^r)
h J ye^TWr-oiT)
2a2^_0rizT-&T)
^2{^t)^
—
z.
that the heavy traffic solution
19
is
The key
issue
expressed in terms of normalized space
and time and
terms of the total embedded inventory process, whereas the proposed policy
in
must be expressed
terms oi {Qi{t~),
in
,Qm{t~),L{t'~)).
...
Recall that equation (27) relates the scaled cycle placement vector x, and the normalized total
each retailer
=
'^'"^
embedded inventory
<
=
q
If
time <~, where
i.
taken when
Because
level.
y/nx and the unsealed cycle placement vector
Summing
over
by the heavy
under a deterministic evolution
this relation
i,
=
<o
+
+
qj
we
all retailers,
<0)
then
over the
arrives at retailer
it
Therefore, the retailer inventories relate to the cycle
^oi'^^-
=
averaging principle,
traffic
for the retailer inventories
the vehicle leaves the warehouse at time
placement parameters by Qj{t~)
j
corresponding embedded inventory
l-''^'
\/nxi. In keeping with the behavior predicted
course of a cycle.
at
is
to reverse the scalings in the solution to the heavy traffic control problem, let us
we develop
i
Since the load allocation decision
x.
establish a relationship between the current total system
first
= ^jQji^T)
define the unsealed
9i
we
reached,
is
inventory Q(ti')
we need
=
embedded inventory Z{t)
^j&Jf^
for j
>
i
and Qj{t~)
=
+
qj
V^
—
^j^Ji^^ for
get
Qit-)
= V.+E<l:^
(33)
J
where
T]i
=
12j<i{Vj
~
^j^Jf^)
+
J2j>i
Making the substitutions qi/y/n =
x,,
'^
^j^If^
q/y/n
=
^ri
epoch locator constant
x and Ojf^ I \/n
=
for retailer
i.
Ojf^ into (27) yields the
unsealed version of constraint (27),
E9. =
9-L^.<—
(34)
t
t
Using equations (33) and (34), we can express the total inventory at time
vehicle
was
^o (i-c,
when the
at the warehouse) as a translation of the inventory vector at time t~:
Q{to)
=
q
= EQAt7) -
V.
+
J2^Mr
f«^
^
=
0,...,m.
(35)
This equation maps the current inventory levels Qj{t~) into the one-dimensional quantity q
that
is
required to interpret the heavy traffic results. In particular, for a given value of q
can find q'
=
we
y/nx', the corresponding optimal (unsealed) cycle placeincnl parameters by
reversing the normalizations
in
Proposition
1.
As
20
before, this
is
done by
letting q,/\/n
=
a:,-,
x, Vif \pn
=
terms of the state
q:
=
ql y/n
and Off^/ \/n
Vi,
Region
1:
Region
Region
g
=
< ar =
Off^ to obtain the following expressioiTs
E -^>C " E rVT:^'
^^^^
«.-
=
-^Kfor.^,,
(37)
€
=
9-E^.C + E^V;;
(38)
2:
3:
oj <
9
<
q,
/^r
0,-
,;
=
+
< fr =
E ^.C " E f^'^'
/ij-
,-EvS- + E^^'..
(44)
n.
can now use these results to determine the delivery
size at retailer
deterministic inventory evolution for the optimal cycle placement, Qi{tf)
level at retailer
i
just after the delivery
is
made)
Q,{tt)
If
we
deliver
Qi{t~)
+
di;
(fj
(39)
(42)
which are independent of the scaling parameter
We
for q* in
units to retailer
i
=
q*
(i.e.
i.
Under the
the inventory
satisfies
+
(45)
K-.
then the actual inventory after the delivery
equating this quantity to (45) yields the desired delivery
is
made
is
size,
d,=q^ + V,-Q,{t-).
(46)
Because we cannot allocate more than the available load and do not want to make negative
deliveries, the
proposed delivery
0?*
=
max[c?„0]
size
+
is
given by
min[0,L(<,") -(/,] for
21
z
=
1,2,
.
.
.
,m -
1.
(47)
Finally, to
guarantee that the vehicle returns to the warehouse empty, we set
=
d'm
We
could
at retailer
under
its
ill
(48)
principle allow negative deliveries, as long as there
and the
i
L{t-J-
total
total capacity.
amount
of load the vehicle carries as
it
is
inventory available
leaves this retailer
is
kept
However, because the vehicle returns empty, the items accrued from
negative deliveries would most likely be shifted to the last few retailers of the tour, which will
not necessarily bring the state of the system closer to the optimal cycle; hence, we disallow
negative deliveries.
To
dynamic delivery allocations are derived by the following
recapitulate, the proposed
procedure:
(i)
observe the current inventory levels {Q\{t~)
unsealed embedded inventory q via (35),
placement parameters
q',
and
(iii)
.
.
.
and compute the
^Q,ri{iT))
use (36)-(44) to derive the optimal unsealed cycle
(ii)
observe the current remaining load L{t~) and compute
the proposed delivery size via (46)-(48).
We now
the vehicle
turn our attention to the busy/idle policy, which has decision epochs
at the warehouse.
is
total inventory level IZjQjit)
otherwise,
it
is
base stock level solely
=
is
in
^
hi
+ aT
1
Ii
j
\e''T(PT-o:T)
qj
=
the
\/n^f)
-
if
qT>0T;
(49)
1
the solution to
Furthermore, the predicted optimal cost
—
level
if
Proposition 3 yields the optimal unsealed
^e-.r(,r-r) + 2a2qT +
^V('7f)
new tour
terms of the original problem parameters:
in
J>+
otherwise, q^
a
in time, the vehicle starts
below the unsealed aggregate base stock
Reversing the normalizations
idles.
qr
At these points
when
<^2(?f)^
+ ^^^r +
'^'1
otherwise.
counterparts of the ones defined
is
03
-
given by Friqj-)
The constants
in
earlier:
Ut
=
—=
2(1
-ptW
xerici
22
+ vc^y
=
0.
h(jj
(50)
+
05
if
q^
>
(3t,
and by
these expressions are the unsealed
«.
=
4E^-i:A,<'H^iE^
Z^
We
oz
t
I
2
^
2
'
'
^
6,
+
and
/li
have completely characterized a dynamic control policy that depends exclusively
on the original system parameters. The policy
two controllable aspects of the
specifies the
the aggregate base stock level defined in (49)-(50) determines the vehicle idhng
system:
pohcy, and the delivery sizes d* defined
in (47)- (48) characterize
the allocation of units to
retailers.
The IRP with Direct Shipping
3
The only
difference between the direct shipping (DS) case
scheme used when the vehicle
is
operating.
We retain
all
TSP
and the
case
is
the routing
notation from §2, occasionally using
the subscript "D" (for direct shipping) in place of the subscript "T" (for TSP). Moreover,
since the procedure
is
very similar in both cases,
we omit
nearly
case, describing only the distinctive aspects of the analysis.
between
DS and TSP
is
that the
consequences, one of which
is
DS
all
of the details for the
The most
DS
significant difference
case does not have a cyclic structure. This has several
that the results are not as theoretically solid as in the
TSP
case.
3.1
In the
Heavy
DS
Traffic Analysis
case, the vehicle always leaves the
and returns empty, so that every time a
V
units.
nominal
As
before,
policy.
it is
retail site
is
full
load, visits a single retailer
visited its inventory level increases
by
convenient to express the dynamic allocation as deviations from a
The nominal
policy
the nominal policy an amount Vi
denote the number of
warehouse with a
DS
is
deliveries
we consider
is
not achievable:
delivered to retailer
made by a
23
i
in
we assume that under
every delivery.
We
continuously active vehicle during
let
Soit)
[0,i].
We
then
be defined by the analog of equation
let cf^{t)
Then
=
c/j(0
is
i.i.d.,
structure
or
amount
DS
TSP
and each tour
results in the delivery of
as a
Markov
same reason that
would be
The DS
case would have a cychc
Neither of these can be used
chain).
fixed delivery sizes could not
infinite over the long run.
A dynamic
be used
A,/A for retailer
i.
The
traffic intensity for
use the same heavy
traffic
demand
the
is
thus pp
normalizations as in §2.2.
<
< m, and
2
the
does not
must be
= 2^^ KOoi/V.
The heavy
conditions
traffic
are given by (20)-(24), with (21) and (23) understood to hold, respectively, for
1
i
at all retailers, this fraction
DS system
in
in
policy, described
used. For this policy the fraction of total shipments that go to retailer
vary over times of order n. To satisfy average
We
units.
V
TSP
the sequence of retailers visited followed a cyclic pattern (such as a polling table),
if
case: inventory costs
is
Z)'s.
once 7"s are replaced by D's. The problem
case lacks the natural cyclic structure of the TSP. Tour times in the
case, for exactly the
in §3.2,
obtained by replacing 7"s by
delivered during the current trip. These processes
(7) still holds as well,
had a regenerative structure (such
the
1
thus nearly identical to (2)-(12).
The DS
are
the total
is
Equation
satisfy (2)-(6).
formulation
ef (0
IC,
(i
% and
Cq,-,
(22) replaced by
^(^-')>»-
'- =
As mentioned above, the DS case does not have the
''"
TSP
cyclic structure of the
case.
Thus we
cannot use functional central limit theorems based on renewal processes to prove convergence
to a
Brownian motion.
5£j, is
given by
were chosen
in
sj)
an
=
A~^
i.i.d.
times that a retailer
In the
is
DS
case the parameter corresponding to
^ his expression would clearly
J^i ^i^oi^oi-
manner.
It
can be shown to hold
for
Sj- in
arise
if
the
TSP
case,
the next retailer
any policy where the fraction of
visited does not vary over times of order
n using the
Random Time
Change Theorem.
The
lack of a cyclic structure
netput process. Indeed,
we
will
if
makes
we embed
at
impossible to consider an
embedded normalized
epochs during which the vehicle
not obtain a meaningful process.
process to obtain a meaningful limit.
it
is
at the warehouse,
VVc must thus average the normalized netput
By arguments
24
similar to those in §2.3, this averaged
normalized netput process
is
well
approximated by a Brownian motion
X
with
drift /i^
and
variance
The averaged
total inventory process
m
Now we
face the
slow
down time by
=
is
defined by
-
X{t)
problem of optimally placing the
facilitate this optimization.
We
=
60, \\
3,
=
A2
2
(52)
.
Again we
limit cycles for the deterministic evolution of the
In contrast to the
V =
Y{t)
a factor of ^/n and turn to the fluid model.
retailer inventories in R"*.
that
^\
TSP
case,
we introduce an approximation
motivate this approximation by use of an example. Suppose
and each
was exactly
retailer
six
time units away from the
warehouse. Then a continually busy vehicle using the polling table (12121) could
average, retailer
1
to
visit,
on
every 20 time units and retailer 2 every 30 time units, thereby satisfying
average demand. Although optimally placing a limit cycle for a small polling table such as
this
one
is
manageable, the optimization problem gets unwieldy very quickly as the
the table grows.
make a
Our approximation assumes the
delivery to retailer
example, we assume that
i
size of
existence of an idealized policy that would
every Vj \i time units in the fluid model; in the context of our
retailer 1 (2) receives a
dehvery exactly every 20 (30) time units,
even though the (12121) polling table cannot achieve such perfect regularity
in
the fluid
model.
Hence, our approach
scale,
and then track
is
to optimally place an idealized fluid cycle at the fast time
this cycle as closely as possible
with our proposed policy.
deliveries are perfectly regular in the idealized cycle, the use of this
Because
approximation causes
us to underestimate the inventory cost incurred over a cycle; however, simulation results
in §6.1
show that the heavy
traffic analysis
incorporating this approximation appears to be
very accurate, at least for the five-retailer cases considered there. Moreover, the use of an
idealized cycle allows us to avoid the task of determining the actual behavior of the individual
inventory levels on the fluid time scale. This task
of a cyclic structure
it
appears to be extremely
25
is
more than
difficult.
just tedious:
due to the lack
Now we
placement by the vector
(xi,X2.,.
.
.
are similar to the
we
paths pictured
deliver a full load of
V /X,
which equals
The next
T,
TSP
and the averaged
in
retailer
Z{t)
=
i
sum
Under the
i.
Figure
units
is
idealized policy, the fluid inventories
the only differences are the delivery size (in
1;
on each
a retailer) and the visit frequency,
visit to
to establish the relationship
=
total inventory level Z{t)
between individual and total inventory
equals the
between the cycle placement variables
x.
The
levels takes the
constraint related to consistency
form that the averaged total inventory
of the average individual inventory levels.
over an idealized cycle
X, the cycle
define the cycle
order to maintain a balanced flow.
in
step
V
still
,Xm), where x, represents the lowest point during the
cycle of the fluid inventory level at retailer
this case
We
turn to the optimal placement of the idealized cycle.
is
x,
+ V /2.
The average
fluid inventory at
Hence, when the averaged total inventory
placement parameter must satisfy
E-.=--'fBecause the
gi{xi)
and
TSP
is
(53),
it
is
V
under the
is
replaced by V.
fluid delivery size equals
given as in (28), except that
V,
clear that the optimal cycle
case, except that
we
The computation
replace
K
by
V
(53)
placement
and
J2i
DS
is
case, the inventory cost function
Comparing constraints
precisely the solution given in the
^i^of^ by mV/2.
of the optimal vehicle idling policy
is
identical to that in §2, except
that the parameter of the exponential stationary density of the
Hence, Proposition 3 characterizes the optimal base stock
replacing
i>t,
and with the substitutions described above
the constants
3.2
a,
and the thresholds a and
RBM
level for the
in the
is
i>d
DS
=
'Ifio/f^o-
case, with i>D
corresponding definitions of
$.
The Proposed Policy
The mapping from heavy
the
(27)
TSP
case.
We
traffic solution to
proposed policy uses the same philosophy as
begin by establishing a relationship between the total inventory
in
in
the
system at the current decision epoch and the cycle placement parameters. Although there
are several possible ways to do this,
we keep
track of the vector process (rj(0,.
26
.
.
,rm(0)i
which
time of the most recent
specifies the
current time by
retailer
The
i.
parameter
qi
=
then
<,
—
t
ri{t)
each retailer. Hence,
visit to
inventory at retailer
at time
i
relates to the unsealed cycle
t
—
ri{t))
<
=
V- Ut - r,{t)).
+
q,
(54)
expect that Qt{t)
in
>
=
thereby contradicting the definition of
level,
+
Qi
Aj^oi,
we modify equation
this modification leads to considerable
Summing
the simulation study.
u{t)
J2i T^SuX
[V
—
it
which would make the cycle placement of the
0,
the current inventory
ri{t)), XiOoi];
placement
\/n Xi via
Since stochastic effects can lead to unusually long intervisit periods,
Xi{t
we denote the
represents the elapsed time since the vehicle last visited
Qi{t)
V—
if
\i{t
—
over
all retailers
Combining
r,(i)), Aj^Oi]
(54) to Qi{t)
improvements
we obtain
YliQi
than
Because one would
qi
in
possible to have
retailer higher
g,.
=
is
+ max[V —
A,(i
—
system performance
=
Q{i)
~~
u{t),
where
equation with the unsealed version
this
of (53) yields
1
= T.Qdt)-u{t) + '^.
which relates the unsealed averaged inventory
total inventory in the
normalizations,
vector q*
=
DS
r,
q (this quantity represents the unsealed average
Reversing the heavy
case) to the current inventory level.
we obtain the
y/nx* given
(55)
following formulas for the unsealed optimal cycle placement
q:
.
Region
1:
< Qp =
g
jrK-^
-V
}^
= -
+ h,
V
6, +
b
+
b
,
+
,
0,
i
q,
hi
,
+
mV
"1^'
2
h,
.
tor
I
^p,
tit
2
Region
^d <
2:
9
b,
,^p
<
/?D
=
+
h,'
+ ^T'
^E mT
k+
2
h,
i
hi
1
Region
3:
mV
2a7V (
.
«7
traffic
=
+
bt
f^
W E
(3d
<
2
\
ysr^h,-hk\
h
bk + ^A;
^
fc
1
q,
27
\-^
OA:
-t-
,
which are independent of the scaling factor
We
and
u{t),
n.
use this cycle placement as an "ideal" m-dimensional inventory state for a given q
and choose the next
retailer so as to bring the current inventory vector as close as
possible to this ideal state. Let
be the time epoch at which the vehicle
^o
is
ready to depart
from the warehouse, and consider the inventory evolution over the next delivery
a deterministic inventory evolution, the vehicle
next) at time
i,
=
^o
+
reach retailer
will
q:
+V
Q'^itf)
(if it
is
made
optimal cycle
to retailer
i
is
given by
and
= max
+
[q*
V- X^ito +
-
0o.
r,{to)), q]
where the maximization makes the adjustment
+
A,(^o,
+
for long intervisit
for ;
^o,)]
under the deterministic evolution the actual inventory vector
to retailer
is
given by Qx[it)
=
Qi[i-o)
+ V—
and Qj{tt)
X^Oq,
^
i,
times as discussed above.
In contrast,
i
Under
chooses to go there
In the deterministic tour corresponding to the
^oi-
placement, the retailer inventory levels right after a delivery
Qntf) =
i
trip.
=
Qj{to)
—
after a delivery
X-jOq,
for j
^
i.
Therefore, the resulting Euclidean distance between the ideal and actual inventory vectors
after a delivery to retailer
the vehicle to retailer
k,
Finally, as in the
i
is
A{i)
=
where k
TSP
= y Hj
argmin,
~
Q'ji^t)]
is
The proposed
control sends
A(?').
case, the busy/idle control
The only added complexity
traffic results.
[Qjiif)
that for the
is
DS
a direct unsealing of the heavy
case the vehicle visits the ware-
house after every delivery and so has many possible idling decision epochs. By equation
our proposed policy
idles
a vehicle at the warehouse whenever
Q{t)
where
tions,
Wq =
y/n
we have
z'^ is
(55),
-
u{t)
+
—
>
the unsealed idling threshold.
w'o,
Making the
suitable scaling substitu-
that
\ ( ^d{0d - cud)
b-V h) \e''D(PD-aD) - 1
f'
=
Wn
'D
In
I'D
28
+ Qo
if
w])
>
i3d;
(56)
otherwise,
w^
is
the solution to
1
-e
-.n(wn-an)
-i'c(tu£)-a£i)
vd
+
.
1
__
r.
*
.
X
and Fj:){w^)
=
=
^^
,
'^
aj{w*p)
.
+ a^Wp +
ug otherwise.
The constants
m
—
—
k + k
h,
2Var (y:
,^
where (ciDi/^D^Oy) are defined
4
in
in these
equations are
+
\
and
2
h,
y,-^
)
'
kh.
2^b, +
h,'
the unsealed cycle placement formulas above.
Comparison of TSP and DS Routing
In this section
we compare the
relative
performance of the two fixed routing schemes (TSP
and DS). The predicted cost functions Ft, Fq derived
tory
.
w},>f3o,
2
mV\
h,
b,
,
if
= '1-Pd)XV
vd
/
(57)
given by
is
v^(hi-h)^
,^y y ^,
+ -^h,--Y:^-^-^
hwh-h^
,
o.
hi
i
Furthermore, the predicted optimal cost
FD{w},)
"^'^
T.V+ ^Er^ =
-^-T:-^
ud
k +
component
of the system cost.
and by C(3?) the
total cost for the
in §2
and §3 represent only the inven-
Denote a generic fixed routing scheme by
system under
this
3J
G {TSP, DS},
scheme. The total system cost
is
ob-
tained by adding the transportation cost (or equivalently, subtract the idleness reward) to
the inventory cost; that
cost
is
is,
we
set C(9?)
= F^{w^) — r{l — p^n).
independent of the stochastic nature of the system.
A
crucial observation
is
that po
<
pr] this fact (see
consequence of the triangle inequality. Although
important implications.
TSP
Notice that the transportation
policy.
This
is
First, the
DS
Rubio
for a proof)
trivial to prove, this inequality
is
a simple
has several
policy achieves lower transportation costs than the
quite interesting since, at
first
glance, one might expect the converse
to hold. However, minimization of the steady state transportation cost in the
29
IRP context
is
amount
equivalent to maximizing the
full
of items delivered per unit time traveled; hence,
load direct shipping provides the highest transportation efficiency of any fixed routing
scheme. More importantly,
until
pt
=
1
and po <
I;
TSP
be stable while the
for
that
any given problem instance, the demand rate can be increased
is,
there exist
some demand
We
policy would not.
<
1
it is
DS
form of stable dynamic allocation
is
The remainder
in
will
used
dominate
TSP
DS
case.
in
the
I
is
a.
necessary
not sufficient. In particular,
is
the absence of adequate dynamic
mind that while the
r
and
b.
/Sj
—
>
as long as
1
performance of the
some
DS and
However, readers should keep
qualitative statements below are true, these results are not exact,
because our calculation of F£){wq)
is
the true heavy traffic cost under the
TSP
routing as
of this section investigates the relative
schemes as a function of the cost parameters
preferred to the
in
it
<
possible to accumulate inventory at one retail site while backorders grow
without bound at another. Hence
TSP
but
keep the total inventory stable but,
will
load allocation,
3J
DS poHcy would
where the
should note that p^
condition for stability of any fixed routing scheme
having p^
levels
policy
if
approximate and represents a
DS
policy.
the transportation cost
may be found
pp < pj implies that DS
inequality
is
underestimate of
is
DS
high enough. In particular, the
> {Ft{wj) — Fd{w}j))/{pt —
policy achieves a lower overall cost for any r
value of this threshold cost
The
slight
po)- While the
numerically for any particular problem instance,
a more precise characterization requires a better understanding of the relationship between
the inventory costs in both systems. Unfortunately,
performance comparisons
levels,
and
it is
hard to make simple inventory cost
for the different routing schemes, primarily
and hence the predicted inventory
costs, are not in closed
because the base stock
form (see equations (50)
(57)).
To study the
relative inventory cost
performance of the
TSP and DS
consider the case where the inventory costs at the retailers are symmetric
hi
=
b for all i)
and
incretising in b/h,
increased above
These
b
becomes
large.
Because the value of
one expects that there
them
critical values
exist
some
ii^j
and
xr'p in
critical values br-ibo
(while leaving h fixed) the optimal base stock
is
schemes,
(i.e.,
(49)
/),
=
and
such that
let
h
us
and
(56)
is
6
is
if
given in closed form.
indeed exist and, for the case of symmetric costs, have the following
30
closed form expressions:
lyxVe
e^TV
>
For b
will
In
^
6
_
and
bo
=
'i/Dmye"^'"^
^I'om V
h
1
— Fd{wq)
ma.x{bx,bD}, the inventory cost difference Ft(wj)
n
As
i/j'V
GO the value of (58)
is
^i/omV
o^T^
—
+
+
1
In
utV
—
2
+
UDmV
l^D
dominated by the term
be the same as the sign of uq
_
—
[uq-^
can be expressed as
[m-\)V^
(58)
i^£,^)ln(l
+
whose sign
b/h)^
vx- Define the critical value
l/rp
e""^^
K
=
exp[.T]
cost
and only
e^.
if
distribution of the
(Tq
> aj
times
both
is
Then
ud
—
fiD
for b
>
>
(.^OmV
-l\
vprnV
^D-'T
'2{yT
J
max{67',6£),6c}, the
where v^
0,
is
=
TSP
DS dominates
more frequent
in
counterintuitive:
deliveries to each retailer,
terms of inventory
cost.
However,
it
5
5.1
in
achieves the lower inventory
Because pn < pT, the condition
we
high backorder case. Moreover,
for large
if
distance travelled must be 0{n~^'^)]
actually have cr^
since the
TSP
policy
=
Uj.
makes smaller and
might be expected to outperform the
backorder penalties, the
efficiency over the long run via its smaller drift,
much time
3J.
in the
mean
then the difference in
somewhat
-h,
i^d)
policy to be preferred. For the case of deterministic travel
aj, and so
finite
is
DS pohcy
-
— \)V
the exponential parameter for the steady state
see equation (62). Hence, in the heavy traffic limit,
This result
i/£)UT{m
associated with routing scheme
required for the
and ht are
/
\
J
i't
RBM
follows that a\)
it
l\ "T-'D
vjV
where
if
—
and causes the
TSP
DS scheme
policy sacrifices
total inventory to
spend too
the expensive backorder regions.
The IRP with Dynamic Routing
Formulation of the Limiting Control Problem
Consider now a situation where, once the vehicle
on either a
full
load
TSP
is
loaded at the warehouse,
it
can embark
tour or a direct shipment to some retailer. All other aspects of the
31
problem
Ccises.
Because the formulation of the dynamic problem
problems described
fixed routing
Rubio
U
earlier,
=
Qi{0)
Qi{t)
is
as in the fixed routing
a natural extension of the two
we only sketch the argument and
much
a detailed treatment; furthermore,
for
same
sources of uncertainty, cost structure) remain the
(e.g.
refer readers to
of the earlier notation will be reused.
then the system state equations are given by
=
V,ST{BT{t))
and the cumulative
idle
+
V,SD{BD{t))
lime process
is
=
I{t)
D,{i)
t
—
+
ef (0
Brit)
—
+
ej{t)
Boit)
for
for
>
t
>
t
sents the cumulative deviation from the nominal allocation over past
plus the
amount
The
first
delivered over the current cycle/trip at retailer
step in the development of a heavy traffic approximation
denote the cumulative fraction of busy time that the
netput for the dynamic routing IRP system
= (^^^ ~
'^^^^^
+
2T^\Af
^^^
-
DS
+
cf (f) repre-
cycles/DS trips
i.
the influence of the routing control on the total netput. Let 6{t)
^^^^
TSP
Notice that,
0.
according to the previous definitions of the delivery allocation controls, cj{t)
(59)
0,
=
is
to characterize
+
Bu{i)l{Bj[t)
service has been used.
Boit))
Then the
is
aJ
i
Notice that this equation reduces to (13) when
+ V [Sr{BT{t)) +
TSP
routing
is
Su[Bo{i))]
always used.
-
V{t).
Summing
(60)
the
equations in (59) over the retailers and substituting the relevant definitions into (60), we
obtain the following expression for the total inventory in terms of the netput and controls:
Q{t)
where
t[t)
=
er(i)
The heavy
problems.
DS
+
=
X{i)
- (^(1 -
+ 2^^I.go/ ^^))
<^(0)
+
(^^^
'^^)'
(.o{t)-
traffic
conditions are a union of the conditions for the two fixed routing
Conditions (22) and (51) require the
policies to
^(^^
approach one
because the condition
/^d
>
in
traffic intensity
the limit; however,
now we do not
ensures that a stable control
is
under both the TSP and
require
i)ossible.
{.ij
to be positive,
Hence,
in
terms of
the problem data, the retailers are required to be located fairly close together relative to
their distance to the warehouse.
^7-2^0.
More
precisely, the average travel times
^^/1\
32
(.^^^.^^1,
,^i^...^,„.
must
satisfy
(62)
One consequence
appearing
are equal
only
in (61)
and the
of the control
of equation (62)
by 0(n~^/^); hence,
differ
V/Ox and AV'/(?^j
that the quantities
is
in the
term
coefficient in front of the idleness
pohcy employed. Nonetheless, we
will
heavy
traffic
in (61) is
Aj^oj)
hmit, these two terms
a constant and independent
maintain equation (61) as
is,
and view
the time-dependent coefficient as a refinement of the heavy traffic Hmit.
Let
3?(i)
G {TSP,DS} denote the routing mode that
control problem.
We want
used at time
is
to consider intervals of time during
in the limiting
t
which one of the two controls
is
used exclusively, and approximate the resulting normalized netput process by a diffusion with
control-dependent
and variance.
drift
definitions of the netput process that are
TSP
netput process in the
of
embedding
in the
TSP
netput process from the
DS
case was
TSP
approximated
we need
in
to reconcile the different
the two cases:
made
for convenience.
DS
an averaged netput process x[t). This translation
=
x{t)
-f
77
constant will be used to translate the normalized total inventory process for the
well,
and can be calculated
Xj
developed
in §2.5.
inventory level we are seeking. Then, since the average of the
sum
of the averages,
with (27) yields x
=
x
- E.
we have x = ^i{xi + VJ2) =
A,^^^"^
+
V
.
The same
TSP
case as
between total embedded
in that context using the relationship
inventory x and cycle placement variables
equal to the
The
case.
Thus we translate the embedded
which needs to be determined: \(<)
?7,
in the
embedded
the
case forced us to use averaging there, while the choice
case, x{t)-> to
consists of adding a constant,
this
and the averaged netput process
case,
lack of a cyclic structure in the
do
In order to
Let x denote the average
sum
of the inventory levels
Ylt
+ V fi- Combining
-^i
is
this
V/2, so that
=
^-E^^^or"
(63)
i
The
role of this translation constant in the
implementation of the proposed policy
is
described
in the next subsection.
By
the arguments used in the
averaged total netput process x
control-dependent
^(^(^^)
drift
= [,r
is
TSP and DS
well
cases,
we can deduce that the normalized
approximated by a diffusion process X{t,^{t)) with
and variance given by
If
m
= TSP
^"^
33
"^^(^^^
=
14
if
m
= TSP
'
respectively.
In other words, the routing control switches the
netput of the system from
one Brownian motion to another. Furthermore, these two Brownian motions have the same
parameters as the diffusion limits of the corresponding fixed routing
(Tq
and
o-j.
By equation
ccises.
only differ by 0{n~^^'^); once again, we retain this refinement of the heavy
(62),
traffic
limit.
Because the normalized idleness process Y{t)
we
zero,
e{t)
=
traffic
are free to choose ^{t) for
e{nt)/y/n,
let 6{t)
=
times
all
t,
will
only be exerted on a set of measure
not just the nonidling times.
6{nt) be the fraction of busy time devoted to
we
let
the heavy
in
system, and define the averaged total inventory level for the system as
then we obtain the same time scale decomposition as before: under the
fixed
DS
If
and the individual
The exact
fluid levels {Wi{t),
evolution of {Wi{t),
.
,
.
.
cycle placement and the routing
dynamic routing IRP
into:
Wm{t))
scheme
.
,
.
is
Wm{t)) move deterministically
at
a
finite rate.
determined by the averaged inventory
in use at
given Z{t)
(i)
.
fluid scaling, Z{t) is
=
time
t.
We may
x and routing
mode
level,
the
therefore
decompose the
^{t) G
{TSP,DS}, use
the results in §2 and §3 to determine the optimal cycle placement and the corresponding
inventory cost function
Y
is
^3e(j)(a;); (ii)
choose the nonanticipating control (y'(f),3?(i)) (where
nondecreasing and right continuous) to minimize
limsup
T— CO
—E
i
rg^(,){Z{t,^{t)))dt-rY{T)
(65)
•/O
subject to (64).
5.2
The
Optimization of a Triple Threshold Policy
diffusion control
problem (64)-(65) appears to be
coefficient in front of the control Y(t) in (64)
the control-dependent cost function .g»(x)
section to niimrrically
compute the
is
difficult to tackle for
depends upon the routing control ^{t), and
very complex.
An
solution to (64)-(65). Here,
the following triple threshold policy that
is
two reasons: the
algorithm
we
is
used
the next
specialize our analysis to
characterized by the parameters ^i
31
in
<
^2
^
^3^
the
vehicle
is
busy whenever the
total averaged inventory Z{t)
while busy, the vehicle uses the
DS mode whenever
Z{t)
<
Zi
TSP
Our
diffusion control
the ELSP, the
is
in
goal in this subsection
ELSP
of the triple threshold
policy corresponds to using large lot sizes
to using small lot sizes: large lot sizes
vehicle here) in a
more
Z{t)
>
Zs;
to find the
is
with setup times
Markowitz,
in
that the most general form of the optimal policy to the
problem (64)-(65)
DS
when
idles
{zi,Z2.,Zs).
Motivated by our analysis of the stochastic
Reiman and Wein, we conjecture
and
23,
routing scheme whenever Z{t) € [^1,22)? and uses the
or Z{t) G [22,23).
optimal values for the parameters
<
form described above. In terms of
and the
TSP scheme corresponds
and the DS policy both use the server (which
efficient fashion.
The most
general form of the
Markowitz, Reiman and Wein corresponds to the
ELSP
is
the
solution derived
Figure 5 of
triple threshold policy (see
that paper), and in some cases (see Figure 6 of that paper) the solution could be described
with fewer thresholds.
We conjecture that
a double threshold policy with 22
the
we
TSP
poficy {zi
=
—00,
22
=
=
-^3,
the
IRP
solution
either a triple threshold policy,
is
or a single threshold policy, which could
23) or the
DS
policy (21
=
22
=
23 or 21
=
=
22
be either
—00). In §7
describe the rationale behind our conjecture.
Our
analysis of the triple threshold policy requires knowledge of the stationary distri-
bution of the controlled reflected diffusion process and the long run expected average idleness
rate.
The
stationary distribution 7r(x) must satisfy the following system of differential equa-
tions (see Karlin
and Taylor 1981):
^^^(a:)-/x(x)—
a'^{x)
d
-j-T^[x)
z
-
7r(.T)
=
x<23,
ii{x)w{x)
=
x
=
ax
(66)
(67)
23,
where the dilFusion parameters are given by
K-) = l'^
fiD
[
if-e[2„22)
it
X < zi OT x e
^^^
[22,23)
a^x) =
\i
[
35
afj
if-^[-i'-^)
\t
x
<
zi
ov
X e [22,23)
The only continuous
density that satisfies (66) and (67)
7r(a;)
=
i
.
is
k^he^''^^-''''
if
x G (21,22]
k-^C'DC^'^^^-''^
if
a;
G
(68)
,
(22,23]
w here
Ux
k,=
^
^
i>x(l
+
e^T{'2-zi)gUD(z3-:2)
-
e'^r{z2-2i)) 4- j)£,(e'^r(z2-zi)
_
and
1)
j>^e''7-(22-Zl)gi>D(23-22)
A;.-,=
i>T{l
this
is
+
e''^(^2-2i)e^D(23-22)
—
-E
lim
(-.00
t
and taking the
first
_
1)'
-
<
—
>
00 we have that
Y{t)
lim -E[Z{t,^{t))].
(69)
term on the RIIS of (69) corresponds to the asymptotic growth rate of the netput
process under the proposed policy.
time that the vehicle uses the
DS
If
we
let S
=
=
=
r
K[x)dx
J — CO
(1
-
6)fiT
+
SuD.
is
(70)
t
the definition of the triple threshold policy,
6
//m(_oo6(i) denote the long run fraction of
routing policy, then the long run growth rate
lim -E[X{t,^{t))]
i—'OO
left
limit as
!^^-^<+^(^^TO
t
lim -E[X{t,^{t))]
The
i>p(e''r(22-2i)
turn to the expected idleness rate. Taking expectations on both sides of (64),
rearranging terms, dividing through by
By
_(-
the stationary density for the total inventory process.
Now we
The
e''T(22-2i))
+
/''
K{x)dx
it
follows that
=
k^
+
^-,(1
-
e''^^''^-''')).
J Z2
side of (69) can be expressed as
)'-<''^S£)^'^^
36
(71)
Finally, according to the steady state distribution for Z{t) in (68), z
=
limt_oo E[Z{t)] exists
Hence, the second term on the right side of (69) vanishes in the
and
is
this
term and substituting (70) and (71) into
finite.
y
=
We may now
Canceling
(69) gives
V^[i-
;un-^Elnt)] =
limit.
,i_„2j;_,,,„,^,,,^
write an expression for the steady state cost of the
)
(72)
IRP under
threshold dynamic routing policy as a function of the control parameters.
the triple
The problem
is
hence reduced to finding {zl,Z2,zl) that minimize
/•'2
/'I
F{zi,Z2,Z3)=
gD{x)Tr{x)dx
J — (X>
+
/"ZS
gT{x)-K{x)dx
+
J Zl
Unfortunately, F(zi, 22,23)
is
gD{x)Tr{x)dx
I
-
ry.
(73)
J Z1
rather complicated and a closed form solution for the
optimal control parameters does not seem possible. Furthermore, an explicit expression for
F(2i, 22,23)
not available in general because
is
between [aT^f^r) and
(cidiI^d),
its
exact form depends on the relationship
which are the parameters that define the three characteristic
regions for the cycle placement and inventory cost solutions.
parameters, one
5.3
The
=
get either 13d
The Proposed
first
step in
dynamic IRP
Wi
may
y/n
Zi
is
> ^t
or f^o
<
By
considering different problem
Pt-
Policy
mapping the
solution of the diffusion control problem into a policy for the
to obtain unsealed threshold levels.
If
we
and make the parameter scaling substitutions
define the unsealed thresholds as
as in the fixed routing cases, then
the scaling factor n cancels out of the expression for
F{wi,W2,uJ3)
and what remains
is
= \/nF(^=., --=,—=),
\/n Wn Jn
(74)
a function only of the original system parameters. In §6.2, the thresholds
minimizing (74) are determined numerically, and we
refer to these
cost-minimizing thresholds
as {wl,wl,wX).
We now
describe
mode, the DS mode or
how
these thresholds dictate whether the vehicle employs the
idles;
this decision
is
37
made
at the
epochs when the vehicle
TSP
is
at
the warehouse. Because of the translation of the embedded total inventory process into the
averaged total inventory process
shipping option.
If
DS was most
inventory q in (55) and compare
the TSP/DS/idle policy.
and
in (63),
it
TSP
If
we must keep
recently used then
track of the most recently employed
we
calculate the unsealed averaged
to the unsealed threshold levels {wl,W2,w'^) to determine
was most recently used then we define q by combining (35)
(63):
9
= E^.(^r)-'/. + EvJ''' +
(75)
'7,
w here
rj
is
= ^f,= ^-~'£X,el'''
(76)
.
the unsealed translation constant. In this case the TSP/DS/idling decision
by comparing q
in (75) to
the unsealed threshold levels {w'[,W2.,w^).
allocation decisions (which retailer to visit next in the
to each retailer in the
we now use the value
TSP
DS
case and
TSP
determined
Finally, the detailed
how many
case) are determined exactly as in Sections 2
of q in (75) in the
is
and
units to deliver
3,
except that
case.
Computational Results
6
This section contains a series of computational experiments aimed at assessing the accuracy
of the
heavy
traffic analysis
and determining what aspects of the control policy are most
important for good system performance. The computational study
the fixed routing
6.1
have
IRP
simulation experiments performed
five retailers
r equal to zero
We
and Poisson demand processes.
and concentrate on the inventory
in this
cost.
fixed so that Aj
=
A/5, A2
=
A/10, A3
=
A/10,A4
38
=
subsection consider systems that
also set the transportation cost rate
The
obtain different utilization rates; however, the fraction of
is
described in two parts:
IRP and the dynamic routing IRP.
Fixed Routing
The Monte Carlo
is
total arrival rate A
demand
A/5 and A5
is
varied to
represented by retailer
=
2A/5.
The
i
travel time
random
109t
The mean
variables T,j are iid second order Erlang.
= V
travel times are adjusted so that
always holds; this allows us to consider several vehicle sizes while maintaining the
traffic intensity at
We
=
pT
O.IA.
perform four simulation experiments aimed at various aspects of system perfor-
mance.
Experiment
The
1:
obtained under the
TSP
first
set of simulation runs quantifies the cost
policy by recalculating the cycle placement at each retailer, as
opposed to determining these values only once per cycle
We
warehouse).
times ^01
=
^so
let bi
=
=
b
=
^r/lO, ^12
5, hi
=
=
h
=
^34
^23
with a pentagon structure, where the
and the warehouse
is
located
=
l
=
for
^45
midway between
of three vehicle sizes (100,10,5)
i
=
=
1, ...
(i.e.,
,5
first
more with
is
at the
and consider the mean
stations
1
and
travel
5.
which are generated by the different combinations
and three
traffic intensities (0.5,0.7,0.9);
For
all
notice that
cases with px
three replications of 36,000 time units (starting with an
discarding the
vehicle
placed at the vertices of a pentagon,
of these scenarios grossly violate the heavy traffic conditions.
three
when the
^t/5- These travel times are consistent
five retailers are
We consider nine different scenarios,
we simulated
improvement
=
<
0.7,
empty system and
2000 time units) with cycle placement recalculation at the
calculation only at the warehouse; for the px
some
retailers
and
0.9 instances, the length of
each replication was increased to 240,000 time units (discarding the
first
20,000 time units).
This simulation design was used throughout our study and allowed us to keep the standard
deviation of the cost estirhate under
Table
1
summarizes the
1%
of
its
mean.
results of the experiment.
when
the increase in the average inventory cost
The
entries in the table represent
delivery sizes are calculated only at the
warehouse, and not adjusted over the course of the tour. As predicted by heavy
the advantage obtained by recalculation at the retailers becomes quite small
intensity
is
high (about
1% when pr =
with small vehicle sizes at lower
0.9).
when the
traffic
However, the recalculation advantage increases
traffic intensities.
Kumar, Schwarz and Ward, who
traffic theory,
These observations complement those
in
focus on this issue (calculating delivery allocations once
per cycle or at each retailer) using a
much
different model. All subsequent
39
TSP
simulations
,'Pd=0.63
/ yPD=0.45
1/
rpD=0-8I
10%
Figure
2: Sensitivity of
inventory cost to base stock level.
44
A
=
9.0
to (64)-(65). In three of the 36 cases the
tion algoritlim
poHcy generated by the Markov chain approxima-
of the triple threshold form; in the remaining 33 cases, the solution
is
is
a
degenerate form of the triple threshold policy that can be described by one or two thresholds.
Table 5 specifies the proposed triple threshold policies
for the three cases,
scheme
is
used
terms of the unsealed inventory)
(in
DS
along with the proportion of time that
is
used.
As expected, the
TSP
an interval containing zero.
in
The Analytical Double Threshold
Because the numerical results
Policy.
in
33 of the
36 cases can be described by one or two thresholds, and because the Markov chain procedure
only offers an approximate solution to problem (64)-(65)
independent of the heavy
traffic
parameter
[^i, 23)
and
idles
when
equal to Z3 in the results
Z{t)
>
we turn
n),
The parameters
23.
in §5.2; in particular,
<
Z{t)
Zj
the solution
is
not
an analytical derivation of the
to
DS when
double threshold policy, whore the vehicle uses
^(0 ^
(in particular,
Zi,
and
the function
F
uses
TSP
routing
by setting
Zj are derived
in (74)
when
22
must be minimized.
For the symmetric cost, wedge topology cases, this function can take eight different possible
functional forms, and a steepest descent
this function; see
Appendix B
riie results for
policy
is
optimal
of 6'
is
degenerate
in 6 cases.
in
of
Rubio
method was used
for
complete
the 36 cases are presented
22 of the 36 cases, with the
In the remaining 14 cases
minimum
of
details.
Table
in
to find the global
TSP
6.
The
derived double threshold
and DS
policy optimal in 16 cases
where two thresholds are required, the value
often very close to zero, suggesting that the
TSP
policy
is
close to optimal.
\\ liilc
results are not reported here, similar findings continued to hold for different values of the
backordering-to-holding cost ratio b/h, as well as for wider or narrower wedges
(i.e.,
for other
values of 601 /0 12)-
For the 33 cases where the Markov chain procedure generated a single or double
threshold solution, these solutions matched the analytical solutions
hence, the
Markov chain
solutions are not included hrrr.
Markov chain procedure generated a
the
Markov chain
triple threshold policy,
in
Table 6 very
closely;
For the three cases where the
we used simulation
to
compare
policy in Table 5 to the analytically derived policy in Table 6. Table 7 shows
that the double threshold solution out pci forms the policy generated
46
i)y
the
Markov chain
—
V=
\ =
r
to deliver full loads to either a single retailer (direct shipping or
(TSP)
tour,
we avoid the combinatorial complexities
DS) or along a prespecified
inherent in the problem and maintain a
sharp focus on the crucial tradeoff between inventory costs and transportation costs that
at the heart of the IK P.
problem
offers
Our modeling
dynamic stochastic IRP
of the
lies
as a queueing control
a new perspective on the problem: rather than view the IRP as a variant of
the vehicle routing problem,
we
see
it
as a variant of a production/inventory control
problem
(where the capacitated vehicle plays the role of the production system); a^ such, this paper
is
a natural descendant of Wein and Markowitz, Rciman and Wein, which consider more
conventional production/inventory control problems.
By assuming that
the system
we approximate the queueing
TSP
is
operating in the (suitably defined) heavy
traffic
control problem by a diffusion control problem.
regime,
When
only
tours are allowed, this modeling approach, together with the application of a heavy
time scale decomposition, allows us to
traffic
fully characterize the solution to the diffusion
By assuming
control problem, thereby generating an operating policy for the original system.
the existence of a fixed sequence of retailer visits than can achieve constant inter- deli very
times to each retailer
in
the fluid limit, we perform a similar analysis for the
control policy in both cases
is
which
which
specifies
how many
retailer to visit next in the
sit idle
or set out with a
full
load,
is
diffusion control
analyzed.
The
problem
is
and a dynamic allocation
units to leave off at each retailer under a
DS
TSP
scheme, and
scheme.
We also consider the case where dynamic route selection
The
case.
characterized by a vehicle idling policy, which dictates whether
a vehicle at the warehouse should
policy,
DS
(either
TSP or DS)
is
allowed.
solved numerically and a class of triple threshold policies
Finally, a series of simulation studies
is
performed to complement the heavy
traffic analysis.
Our key
•
findings can be
The inventory component
summarized
as follows:
of the total long run average cost
characteristics of the system, while the transportation
scheme
is
determined solely from
first
moment
50
depends on the stochastic
component
information.
for
a fixed routing
•
The
vehicle idhng
idles at the
poHcy
is
characterized by an aggregate base stock level: the vehicle
warehouse whenever the
total retailer inventory exceeds a certain threshold
The value of the optimal base stock
level.
level
is
independent of the transportation cost
Although the existing IRP literature does not typically address the vehicle
rate.
issue,
our simulation results show that system performance
value of the base stock level, deteriorating rapidly
is
idling
quite sensitive to the
when the base stock
level differs
from
the optimal value by more than the vehicle capacity. Moreover, simulation results also
confirm that the system cost under our derived base stock levels are typically within
several percent of the cost achieved by the best (found by exhaustive search using
simulation) base stock level, unless the heavy
equals 0.5 and vehicle capacity
(e.g., traffic intensity
•
The
traffic
allocation of load
among
the retailers
is
conditions are grossly violated
is less
than or equal to 10
units).
dictated by the desire to concentrate most
of the total inventory (backorders) at the site with the smallest holding (backorder)
cost rate.
•
Dynamic
(i.e.,
state-dependent or closed-loop) delivery allocations greatly outperform
their static (state-independent or open-loop) counterparts in a stochastic environment.
In fact, central limit theorem arguments indicate that static delivery allocations cause
the absolute value of the inventory levels to grow as the square root of time, thereby
leading to unbounded costs over the long run.
•
The
TSP
relative
tour, as
advantage of recalculating the load allocation at each
opposed to setting
utilization increases,
•
The
once at the beginning of each tour, decreases as
in the
heavy
traffic limit.
policy that achieves the lowest transportation cost
largest
amount per
fore, direct
fact
and vanishes
it
is
shipping
is
the one that delivers the
unit time travelled (subject to meeting average
is
retailer within a
demand). There-
the most transportation-efficient routing scheme; although this
a trivial consequence of the triangle inequality,
it is
perhaps our most important
observation. This fact helps highlight the basic cost tradeoff in the IRP:
51
DS
leads to
TSP
smaller transportation costs, hut
may
deliveries,
routing, by
demand
for
is,
where the
rates to a level
TSP
routing scheme has a traffic
DS scheme
Moreover, our analysis highlights the danger
in
minimizes current inventory costs
(for
has an intensity
less
in-
than one.
employing a myopic policy that always
example, have each outgoing vehicle satisfy
backorders as possible, regardless of location); such a policy, which
in spirit
has
any given problem instance, one can
tensity greater than or equal to one and the
many
DS
lead to smaller inventory costs. This result also implies that
a larger stability region than TSP; that
increase the
making smaller and more frequent
is
«is
similar
(and consequence) to a production lot-sizing policy that frequently breaks a
setup in order to satisfy backordered demand, can easily become unstable.
•
Heavy
traffic analysis
backorder costs,
•
DS
shows that
will
costs
policies.
cost-symmetric systems with sufficiently high
be preferred to
Simulation results show that there
DS and TSP
for
is
Although the
TSP
routing.
often a large difference in performance between the
traffic intensity,
backorder costs and transportation
play a significant role, the topology probably plays the largest role in the
all
relative attractiveness of each policy. For systems with relatively high loads,
that
TSP
could only be a desirable alternative
when the tour
is
it
appears
wedge-shaped, as
is
often the case in practice.
•
The heavy
traffic analysis
TSP
or fixed
schemes.
accurately predicts the relative cost of using the fixed
DS
Hence, our procedure can be used as an aid in higher level
decisions, as discussed at the end of this paper.
• If
one can dynamically choose between the
the most general form of the solution
Wi
it
<
is
W2
in
<
Wj-.
if
the interval
where W3
is
is
DS and TSP
then
TSP
the idling threshold, then
is
is
is
than wi then
preferred.
the absolute value of the total retailer inventory
52
less
preferred and
DS
we conjecture
that
a triple threshold policy characterized by
the total retailer inventory
[101,102)
options,
is
if it is
Our
in
DS
is
preferred,
the interval
rationale
is
large then the routing
if
[102,^^3)1
as follows:
if
scheme may
have
little effect
DS may
on the rate at which inventory costs are incurred.
be preferable because
DS
-efficiency of
incurs smaller transportation costs. In addition, the
it
has a tendency to increase the total inventory, level relative to
the diffusion control problem the
and
DS
so
zero, as
in
is
it
will help to
TSP
option has a larger drift than the
decrease future backorders. However,
is
much
when the
TSP
(in
option),
less
than
total inventory
the interval [iui,W2), which should contain zero in the nondegenerate case, the
TSP
lead to less backorders and smaller inventory costs,
making
the more attractive alternative. Finally, because the effective penalty for using the
inefficient
TSP
when the
policy decreases
most cases the optimal solution
pohcy, where W2
ELSP
We
=
This state of
W3.
correspond to
DS
is
large,
we
believe that in
be no more complex than a double threshold
affairs
is
somewhat analogous
to the stochastic
(TSP).
computed the numerical
to the
will
total inventory
with setup times analyzed in Markowitz, Reiman and Wein, where large (small)
lot sizes
•
DS
be even more attractive when the total inventory
will
frequent deliveries of
it
In tlicse cases,
dynamic IRP
for a
solution to the diffusion control problem corresponding
number
of instances,
and the
results
were consistent with
our conjectures: the most general optimal policy was of the triple threshold form, and
in
most cases a degenerate form of the pohcy was optimal: either the
{wi
=
i«2
= —00
or
lOi
double threshold policy
we
=
W2
(102
=
=
did not find a numerically
W3), the fixed
TSP
case
{ivi
=
fixed
=
—00, W2
DS
case
w^) or the
^3)- Moreover, in our limited simulation experiments,
computed
triple threshold policy that
outperformed
the analytically derived double threshold policy (although we did not search beyond
the computed triple threshold values).
simulation,
of the
we
By performing many exhaustive
also found that the best double threshold policy differed
two fixed routing
this range, the best
policies in only a
TSP and DS
searches using
from the better
narrow range of system parameter space;
policies achieve fairly similar
in
performance. Hence,
coupling this observation with a previous one suggests that finding the best fixed
route policy
is
very important while allowing for dynamic routing provides a
53
much
less substantial benefit; this
is
particularly true in light of the increased complexity of
implementing a dynamic routing scheme.
In
summary, the important operational
level,
levers for the
IRP iiulude the aggregate base stock
the dynamic allocation policy and the choice of fixed routing scheme, but not the dy-
namic routing
policy.
Moreover, these key decisions are interrelated and a unified stochastic
control model, such as the one considered here,
required for achieving reliable system
is
performance.
Two
topics for future research naturally
dynamic routing scheme
so as to allow
K
come
TSP
Although
tour through xetailers
in
2
1,
and
The
different types of routes
to consider cyclic routes that use a combination of
of a
to mind.
3,
DS and TSP
first
(where
(e.g.,
is
to extend the
K
>
2)
and/or
a cycle could consist
followed by a direct shipment to retailer 2).
theory these extensions could be incorporated and system improvements could
be achieved, the analysis would be tedious and
is
it
doubtful that any additional insights
would be found.
Perhaps the most
fruitful area for future research
would be to develop the necessary
steps for a hierarchical approach to the general (multi-vehicle, multi-warehouse) IRP; such
an approach would be similar
Levi, but
the vehicle routing analysis performed by Simchi-
in spirit to
would also incorporate the inventory cost component. Our
any system given a particular assignment
policies provide estimates for the operating cost for
of retailers to vehicles
and vehicles
results for fixed route
to warehouses.
Motivated by our observation that the
best fixed route policy performs nearly as well as the best dynamic policy over a broad
range of parameters, the
optimization algorithm
first level
(e.g.
up
in
the hierarchy could implrmont
a k-opt algorithm as used
in
some interexchange
the detenninistic vehicle routing
literature) to find the best such route. Higher levels in the hierarchy could then be used to
select the best possible
assignment of vehicles and
to have in the system.
At an even higher
number and
level,
location of warehouses.
54
retailers,
and the
total
number
of vehicles
these results could be used to decide on the
Acknowledgment
This research was supported by a grant from the Leaders for Manufacturing Program at
MIT, NSF grant DDM-9057297 and EPSRC grant GR/J71786. The
Statistical
Laboratory at the University of Cambridge
work was carried
out.
55
last
author thanks the
for its hospitality
while some of this
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