Chapter 12.2. On (R, NQ) Policies in Serial Inventory Systems
1. Introduction
Supply chain management, which is about the material and information flows in multi-stage productiondistribution networks, has recently attracted much attention from academics and practitioners alike.
Driven by fierce global competition and enabled by advanced information technology, many companies
have taken initiatives to revamp their supply chains in order to reduce costs and at the same time increase
responsiveness to changes in the marketplace. A remarkable success story is Wall-Mart, whose
replenishment practices are hailed as the center piece of its competitive strategy.
Multi-echelon inventory models are central to supply chain management. The multi-echelon inventory
theory began when Clark and Scarf (1960) published their seminal paper. The theory is now voluminous
with categories by, e.g., demand characteristics (deterministic, stochastic) and network structures (series,
assembly, distribution). This chapter focuses on a multi-stage, serial inventory system with stochastic
demand.
Consider a production-distribution system where material is processed sequentially before being used
to satisfy uncertain customer demand. The system consist of multiple stages representing the different
stocking points in the production-distribution process. Material flow from one stage to the next requires a
leadtime and incurs a fixed cost (in addition to a variable cost proportional to the flow quantity). Due to
the value added, inventory becomes more expensive to carry as it moves downstream (close to the
customer). Customer demand unsatisfied from on-hand inventory is backlogged, incurring penalty costs.
We assume that the entire supply chain is controlled by a central planner whose goal is to satisfy customer
demand with minimum long-run average system-wide costs. (When the different stages are run by
independent managers, the centralized solution can be used as a benchmark.)
The above model was originally proposed by Clark and Scarf (1962) as a generalization of the now
classic Clark-Scarf (1960) model which does not allow setup costs at any stages except the most upstream
stage. They introduced the important concept of echelon stock: a stage's echelon stock is the inventory
position of the subsystem consisting of the stage itself and all its downstream stages.
The serial model can also be viewed as a generalization of the deterministic models studied by Roundy
(1985, 86), Maxwell and Muckstadt (1985), and Atkins and Sun (1995). They show that the so-called
power-of-two policies are close to the optimal solution. Under the power-of-two structure, the reorder
intervals (or order quantities) at all stages are restricted to be power-of-two multiples of a base time (or
quantity) unit. This facilitates time/quantity co-ordination among the different stages.
For our serial model with random demand and setup cost, Clark and Scarf (1962) have pointed out
correctly that the optimal policy does not have a simple structure. Thus, an optimal policy, even if it exists
and is identified, would not be easy to implement. In other words, the "optimal" policy is no longer
optimal or even attractive once the managerial effort of implementation is taken into account. Therefore
we turn to simple, cost-effective heuristic policies. Specifically, we consider ( R, nQ) policies. An
( R, nQ) policy operates as follows: whenever the inventory position (to be defined) at a stage is at or
n is the minimum integer required to increase the inventory position
to above R . We call R the reorder point and Q the base order quantity. As with power-of-two
below R , order nQ units where
policies, the base order quantities at the different stages satisfy an integer-ratio constraint: the base order
quantity at an upstream stage is an integer multiple of the base order quantity at a downstream stage. If
each stage uses a ( R, nQ) policy to control its echelon stock, then we have an echelon-stock
( R, nQ) policy. Otherwise, if each stage controls its installation stock (i.e., its local inventory position),
then we have an installation-stock ( R, nQ) policy.
Both types of policies are easy to implement. Installation-stock policies only require local inventory
information, while echelon-stock policies require centralized demand information. Note that although the
initial measurement of a stage’s echelon stock requires the inventory information at every downstream
stage, its update only requires the demand information at the point of sales, which is readily available for
most companies with advanced communication networks such as EDI (Electronic Data Interchange).
This chapter summarizes the key results for both types of ( R, nQ) policies in serial systems. Since
installation-stock ( R, nQ) policies are special case of echelon-stock ( R, nQ) policies (as we will see
later), most of the results are presented for echelon-stock policies only.
The following notation will be used throughout this chapter:
E[ X ]  expectation of random variable X
V [ X ]  variance of X
(x)   max{x, 0}
(x)   max{-x, 0}
2. (R, NQ) Policies in Single-Location Systems
Consider the following single-location inventory system. Customer demand for a single item arises
periodically. The demands in different periods are independent and identically distributed random
variables. These random variables are nonnegative and integer-valued. If demand exceeds the on-hand
inventory, the excess is backlogged. Linear holding and backorder costs are assessed with h and p being
the holding and backorder cost per unit per period respectively. Inventory is replenished from an outside
source; each order arrives after a constant leadtime of L periods. Replenishment is controlled by a
( R, nQ) policy: at the beginning of each period, if the inventory position y (on-hand inventory +
outstanding orders - backorders) is at or below R , an order of size nQ is placed where n is the
minimum integer such that y  nQ  R ; otherwise, no order is placed. A fixed setup cost K is
incurred for each Q units of order. The decision variables are the reorder point R and the base order
quantity Q . The planning horizon is infinite, and the objective is to minimize the long-run average total
cost in the system.
For clarity, we assume that for each period, the ordering decision is made at the beginning of period,
customer demand arrives during the period, and the holding and backorder costs are assessed at the end of
the period.
Let IP (t ) be the inventory position at the beginning of period
t after order placement (if any) and
before demand occurrence. Thus R  1  IP (t )  R  Q . Let IL (t ) be the inventory level (on-hand
inventory minus backorders) at the end of period t . Write
D[t1 , t 2 ] for the total demand in periods
t1 , t1  1,    , t 2 . The following inventory balance equation is well known
IL(t  L)  IP (t )  D[t , t  L]
Notice that IP (t ) and D[t , t  L] are independent. Therefore, given IP (t )  y , the expected
t  L can be expressed as
G( y)  E[h( y  D[t , t  L])   p( D[t , t  L]  y)  ]
Clearly, G () is convex. If the Markov chain {IP (t )} is irreducible then its steady state distribution is
uniform over R  1,    , R  Q (see Hadley and Whitin 1961). This condition is satisfied when, for
holding and backorder costs in period
def
example, the one-period demand equals 1 with a positive probability. This is a mild condition satisfied by
most demand distributions. We make this assumption in this chapter, for the sake of brevity. Therefore,
the long-run average holding and backorder cost is
setup cost is
K / Q
where
1/ Q y r 1 G( y) . Since the long-run average
r Q
 is the mean of the one-period demand, the long-run average total cost
associated with the ( R, nQ) policy is
K   y  R 1 G ( y )
R Q
def
C ( R, Q ) 
Q
(12.1)
This cost function has the same form as the cost function of ( R, Q) policies, see Federgruen and Zheng
(1992) and Zheng (1992). The following results follow directly from this observation. [An ( R, Q)
policy works in continuous-review systems where demand is for one unit at a time. Here the size of each
order is exactly Q units.]
2.1 Optimization
Since C ( R, Q ) is jointly convex in the decision variables, it can be easily minimized. Here is one way to
do it. Fix Q and minimize C ( R, Q ) over R . Since G () is convex, the Q smallest values of
G () are achieved at Q contiguous points, which can be easily determined for each Q . The optimal
R is the one so that the sum in (11.1) is over those Q points. The optimal Q is obtained by
def
minimizing C (Q)
 min R C ( R, Q) , a unimodal function.
2.2 Sensitivity Analysis
The cost performance of ( R, nQ) policies is insensitive to the choice of Q . Under the continuous
approximation (i.e., both the customer demand and the inventory are represented by continuous variables),
one can rewrite the cost function as
C ( R, Q ) 
K  
R Q
R
Q
G( y )dy
.
Following Zheng (1992), we have
C (Q* ) 1
1
 (  ) for all   0 .
*

C (Q ) 2
Therefore, for example, if we use Q 
2Q * instead of the optimal Q* then the relative cost
increase is no more than 6%. Deviations from the optimal base order quantity may be necessary due to
physical constraints (e.g., a full truckload) or coordination among different locations. Notice that the
above equality becomes an equality for the EOQ model (see, e.g., Hadley and Whitin 1963), suggesting
that ( R, nQ)
model is even more robust than the EOQ model with respect to the lot size. This result is
very useful for multi-echelon, multi-location systems where quantity coordination is beneficial.
Remarks:
1.
It was assumed above that a fixed setup cost is incurred for each Q units ordered. Therefore,
for example, the setup cost for an order of 2 Q units 2 K . An alternative assumption is that a
fixed cost is incurred for each order, independent of its size, as in Zheng and Chen (1992). The
resulting cost function is no longer jointly convex in the decision variables. Consequently, a
more complex algorithm is needed to determine the optimal control parameters. The sensitive
analysis, although more complex, still suggests that the cost performance is insensitive to the
choice of Q . Moreover, under the new setup cost, it is known that (s, S) policies are optimal.
2.
The single-location ( R, nQ) model originated in Morse (1959). Hadley and Whitin (1961)
showed that the steady state distribution of IP (t ) is uniform over R  1,    , R  Q under
some minor condition.
3. (R, NQ) Policies in Serial Systems
This section defines ( R, nQ) policies in serial systems. Unlike the single-location systems ( R, nQ)
policies in serial systems have two variations with different informational requirements. Basic
assumptions and preliminary notation are also introduced.
Consider a serial inventory system with
N stages. Stage 1 orders from stage 2, 2 from 3, …, and
stage N orders from an outside supplier with unlimited stock. For convenience, the outside supplier is
also referred to as stage N  1 . Each stage represents a stocking point in a production-distribution system.
The customer demands in different periods are independent and identically distributed random variables,
which are nonnegative and integer-valued. When stage 1 runs out of stock, the excess demand is
backlogged.
The replenishment policy is of the following type. Each stage controls a stage-specific inventory
position according to a stage-specific ( R, nQ) policy: when the inventory position falls to or below a
reorder point R , the stage orders a minimum integer multiple of Q (base quantity) from its upstream
stage to increase the inventory position to above R . In case the upstream stage does not have sufficient
on-hand inventory to satisfy this order, a partial shipment is sent with the remainder backlogged at the
upstream stage. The production/transportation leadtimes from one stage to the next are constant.
Let
Qi be the base quantity at stage i, i  1,    , N . We assume
Qi 1  ni Qi , i  1,    , N  1
where
(12.2)
ni is a positive integer, This integer-ratio constraint simplifies analysis significantly. It also
simplifies material handling (e.g., packaging and bulk breaking) by restricting the shipments to each stage
to multiples of a fixed quantity, which may represent a truckload or the size of a standard container.
Moreover, the cost increase due to the constraint is likely to be insignificant because inventory costs tend
to be insensitive to the choice of order quantities (see section 2).
We consider two variations of the above ( R, nQ) policy. One is based on echelon stock: each stage
replenishes its echelon stock with an echelon reorder point. The echelon stock at a stage is the inventory
Ri be the echelon
reorder point at stage i  1,    , N . Therefore, under an echelon-stock ( R, nQ) policy, stage i orders a
position of the subsystem consisting of the stage and all the downstream stages. Let
multiple of
Qi from stage i 1 every time its echelon stock falls to or below Ri . The other variation
is based on installation stock: each stage controls its installation stock with an installation reorder point.
The installation stock at a stage is just the inventory position of the stage itself (outstanding orders plus
on-hand inventory minus backorders). Let be the installation reorder point at stage i  1,    , N .
Qi from stage
i 1 every time its installation stock falls to or below ri . Note that echelon-stock ( R, nQ) policies
require centralized demand information, while installation-stock ( R, nQ) policies only require local
Therefore, under an installation-stock ( R, nQ) policy, stage i orders a multiple of
'demand' information, i.e., orders from the immediate downstream stage.
A fixed cost is incurred for each base quantity ordered at every stage. Linear holding costs are incurred
at every stage, and linear backorder costs are incurred at stage 1 only. The decision variables are the
reorder points and the base quantities. The objective is to minimize the long-run average total cost in the
system.
The system has the following parameters:

= mean of the demand in one period
Li = leadtime from stage i 1 to stage i , a given nonnegative integer
K i = fixed cost for each Qi ordered at stage i
H i = installation holding cost per unit per period at stage i
hi = echelon holding cost per unit per period at stage i  H i  H i 1  0 with H N 1  0
p = backorder cost per unit per period (at stage 1)
For any t i
 t 2 , let D[ti , t 2 ] be the total demand in periods t1 ,    , t2 , D[t1 , t2 ) the total demand
t1 ,    , t2 1, D(t1 , t2 ) the total demand periods t1  1,    , t2 1 .
in periods
For clarity, we assume that replenishment activities - ordering, shipping, and receiving - in a period
occur at the beginning of the period. Demand occurs during the period. Holding and backorder costs are
assessed at the end of the period. The following notation will be used for the rest of the chapter. Let t
be an epoch after the replenishment activities but before demand in period t , and let t


be an epoch
after demand in period t . Define
I i (t ) = echelon inventory at stage i at t 
= on-hand inventory at stage i plus inventories on hand at, and in transit to stages 1,    , i -1
B (t ) = backorder level at stage 1 at t 
ILi (t ) = echelon inventory level at stage i at t   I i (t )  B (t )
ILi (t ) = echelon inventory level at stage i at t 
= on-hand inventory at stage i plus inventories in transit to or on hand at stages 1,    , i -1
minus backorders at stage i at t

IPi (t ) = echelon inventory position at stage i at t 
=

ILi (t ) plus orders in transit to stage i
ESi (t ) = echelon stock at stage i at t 
=
IPi (t ) plus outstanding orders of stage i that are backlogged at stage i 1
IS i (t ) = installation stock at stage i at t 
= outstanding orders at stage i (in transit to stage i or backlogged at stage i  1 ) plus onhand inventory at stage i minus backlogged orders from stage
= ESi (t )  ESi 1 (t ) ,
i  2 ; and IS1 (t )  ES1 (t )
i 1
Since the outside supplier has unlimited supply,
ES N (t )  IPN (t ) for all t . Note also that the echelon
stock at a stage decreases as customer demands arrive at stage 1. To continuously monitor its echelonstock level, a stage must have access to the demand information at stage 1 on a real-time basis. On the
other hand, the installation stock at a stage is local information. When the time index
t is suppressed,
the notation represents the corresponding steady state variables.
We assume that the initial on-hand inventory at stage i is an integer multiple of
Qi 1 , i =2,…, N .
i 1 is an integer multiple of Qi 1 and thus
there is no incentive for stage i to keep a fraction of Qi 1 on hand. This assumption and the integerratio constraint (see (12.2)) imply that the installation stock at stage i is always integer multiple of Qi 1 .
This observation is true whether an echelon-stock ( R, nQ) policy or an installation-stock ( R, nQ)
policy is in place. Consequently, without any loss of generality, we restrict ri to be an integer multiple
of Qi 1 , i ≥2. Of course, r1 can be any integer. No such restrictions are placed on Ri , i  1,    , N .
This is reasonable because each order placed by stage
N
It can be shown that any installation-stock ( R, nQ) policy, ( ri , Qi ) i 1 , is equivalent to an echelonN
stock ( R, nQ) policy, ( ri , Qi ) i 1 , as long as its echelon reorder points satisfy
(12.3)
R1  r1 , and Ri  Ri 1  Qi 1  ri , i  2,    , N .
In other words, installation-stock ( R, nQ) policies are special cases of echelon-stock ( R, nQ)
policies. See Axsater and Rosling (1993).
3.1. Related Literature
Section 2 describes the literature on single location ( R, nQ) models. To our knowledge, Chen and
Zheng (1994a) are the first to adapt the single-location ( R, nQ) policy to a multi-echelon system.
There is, however, a large body of research on ( R, Q) policies in multi-echelon systems. (An
( R, nQ) policy reduces to an ( R, Q) policy when demand is for a single unit at a time and a
continuous-review system is in place.) De Bodt and Graves (1985) study echelon-stock ( R, Q) policies
in serial systems. They develop approximate cost expressions under the nestedness assumption: whenever
a stage receives a shipment, a batch must be immediately sent to its downstream stage. Badinelli (1992)
considers installation-stock ( R, Q) policies in serial systems. He proves exact long-run average
holding and backorder cost under the assumption that the installation stock at each stage is nonnegative.
Installation-stock ( R, Q) policies have also been studied in the context of one-warehouse multi-retailer
systems, see, e.g., Deuermeyer and Schwarz (1981), Moinzadeh and Lee (1986), Lee and Moinzadeh
(1987a,b), Svornos and Zipkin (1988), and Axsater (1991,93a). The major reason why installation stock
policies have received much attention is perhaps their modest informational requirement: installation
stock is local inventory information, whereas echelon stock requires a certain degree of information
centralization. But this advantage is quickly disappearing as more and more companies are equipped with
advanced information technologies such as EDI.
Compared with serial systems, assembly systems with stochastic demand have attracted relatively less
attention in the literature. Schmidt and Nahmias (1985) characterize an optimal policy for a system where
two components are assembled into one and item. Rosling (1989) shows that a general assembly system
(with any number of items) is equivalent to a serial system and thus can be solved by using the ClarkScarf (1960) result. Both papers assume zero setup costs. Chen (1996b) shows that ( R, nQ) policies
are optimal for assembly systems where inventory transfers from one stage to another must be in integer
multiples of a fixed base quantity (due to, e.g., fixed setup costs). He demonstrates that such an assembly
system is still equivalent to a serial with batch ordering, extending Rosling’s result to assembly systems
with batch transfers.
4. Performance evaluation
This section shows how to determine the long-run average system-wide cost of an ( R, nQ) policy in
the serial system. Since installation-stock ( R, nQ) policies are special cases of echelon-stock
( R, nQ) policies, it suffices to consider the latter with control parameter ( Ri , Qi ) iN1 .
First, recall that the on-hand inventory at stage
i 1 is always a nonnegative integer multiple of Qi ,
i = 1,…,N - 1. By definition, ILi 1 (t )  IPi (t ) is the on-hand inventory at stage i 1 . Therefore,
ILi 1 (t )  IPi (t )  mQi , m  0 integer.
On the other hand, the echelon-stock policy implies that
(12.4)
IPi (t ) is above Ri whenever stage i 1

has positive on-hand inventory. This has the following implications. Suppose ILi 1 (t )  Ri . From (12.4),
IPi (t )  Ri since m  0. Thus the on-hand stock at stage i 1 cannot be positive, i.e., m = 0


or ILi 1 (t )  IPi (t ) . Now suppose ILi 1 (t )  Ri .
In this case, if
IPi (t )  Ri then m > 0 (see (12.4))
i 1 has positive on-hand inventory which in turn implies that IPi (t )  Ri , a
contradiction. Thus IPi (t ) must belong to the set {Ri  1,    , Ri  Qi } . Note that there is a unique
implying that stage

point in the set that satisfies (11.4). In short, ILi 1 (t ) uniquely determines IPi (t ) . Formally,
IPi (t )  Oi [ ILI 1 (t )] , for i =1,… ,N-1
where
(12.5)
if x  Ri
x
Oi [ x]  
 x  nQi otherwise
n being the largest integer so that x  nQi  Ri .
with
Notice that
ILi (t  Li )  IPi (t )  D[t , t  Li ]
(12.6)
ILi (t  Li )  IPi (t )  D[t , t  Li )
(12.7)
and

t  Li .
since ILi (t  Li ) is assessed before the demand in period
We now have a recursive procedure for determining the steady-state distributions of the key inventory
variables in the serial system. First, consider stage N . Since the outside supplier is perfectly reliable, the
N inventory position behaves as if the whole serial system were a single location following the
( RN , nQN ) policy. Thus, IPN , the steady-state echelon N inventory position, is uniformly
echelon
distributed over
{RN  1,    , RN  QN } (see section 2). From (12.7), we have the distribution of ILN .
Now proceed to stage N 1 . Use (12.5) to obtain the distribution of IPN 1 , and then (12.7) to obtain the

distribution of IL N 1 . Continuing in this fashion, we obtain the steady-state distributions of the echelon
inventory positions at all stages. The steady-state distributions of the echelon inventory levels are
obtained from (12.6).
We close this section with an expression for the long-run average cost of the echelon-stock ( R, nQ)
policy. First, note that the system-wide holding and backorder cost incurred in period
t can be expressed
as
N
 h I (t )  pB(t ) .
i i
i 1
Since
I i (t )  ILi (t )  B(t ) by definition, the above expression can be rewritten as
N
 h IL (t )  ( p  H ) B(t )
i
i 1
Let
i
i
li (t )   j i L j for i ,, N with l N 1  0 , i.e., li is the total leadtime from the outside
N
supplier to stage i . In period t , charge
N
 h IL (t  l )  ( p  H ) B(t  l )
i 1
i
i
i
i
1
(12.8)
Cleary, this cost-accounting system merely shifts cost in time and thus does not affect the long-run
average cost.
t in steady state. For i  1,, N , define
Take any period
IPi  IPi (t  li 1 )
ILi  ILi (t  li )
ILi  ILi (t  li )
B  B(t  l1 )  ( IL1 ) 
Di  Di [t  li 1 , t  li ]
Di  Di [t  li 1 , t  li )



Notice that DN , DN 1 ,, D2 , D1 are independent. From (12.6) and (12.7),
ILi  IPi  Di , i  1,, N
(12.9)
and
ILi  IPi  Di , i  1,, N
Note that
(12.10)
IPi is independent of Di and Di for i  1,, N . Moreover, the top-down recursive
procedure is used for determining the distributions of IPi , which are then easily used to determine the
distributions of
ILi via (12.9). The long-run average value of (12.8) is thus equal to
N
E[ hi ILi  ( p  H 1 ) B]
(12.11)
i 1
Let
R  ( R1 ,, RN ) and Q  (Q1 ,, QN ) . Combining the long-run average fixed costs with
(12.11), we have the following long-run average system-wide cost
def
C (R, Q) 

N
K i
i 1
Qi

N
i 1
K i
Q
i 1
N
 E[ hi ILi  ( p  H1 ) B]
i
N
  hi E[ ILi ]  EG1 ( IP1 )
i 2
where
G1 ( y )  E[h1 IL1  ( p  H1 ) B | IP1  y ]
 E[h1 ( y  D1 )  ( p  H 1 )( y  D1 )  ]
(12.12)
5. Optimization
This section has three subsections. The first and second subsections each present an algorithm for finding
the optimal echelon and installation reorder points respectively. Both algorithms assume that the base
order quantities are given. The third subsection describes several approaches to determining the optimal
base order quantities for both echelon-stock and installation-stock policies.
5.1 Echelon Reorder Point
N
Consider the echelon-stock ( R, nQ) policy, ( Ri , Qi ) i 1 . Let its base order quantities be fixed. Thus
the long-run average fixed cost is independent of the decision variables Ri . Let C ( R) be the long-run
average system-wide holding and backorder cost, i.e., it is expression (12.12) without the fixed costs. We
present an efficient procedure to identify the optimal echelon reorder points R
*
 ( R1 ,, RN* ) .
*
The following random variables are useful. Define
Pr(U i  u ) 
1
, u  1,, Qi , i  1,, N
Qi
and
Pr( Z i  z ) 
1
, z  0,, ni  1, i  1,, N  1
ni
Qi 1  ni Qi where ni is a positive integer, see (12.2).) We assume that these uniform
random variables are independent, and they are all independent of the demand Process. Let VN  RN .
(Recall that
Define recursively
Vi  min {Ri ,Vi 1  Z i Qi  Di1}, i  1,, N  1
(12.13)
Since the U 's are independent of the demand process as well as the Z 's, the V 's are independent of
the U 's. From the above recursive definition, it is also clear that
Vi is independent of Di for
i  1,, N and that Vi 1 is independent of Z i for i  1,, N  1 .
d
Write
X 1  X 2 if the two random variables X 1 and X 2 have the same distribution. From Chen
(1995),
d
IPi Vi  U i i  1,, N
(12.14)
IPi is precisely the steady-state inventory position of the standard single-location ( R, nQ)
model with R  Vi and Q  Qi . A key feature here is that the reorder point of the single-location
Therefore,
model is random, and it is jointly determined by the control parameters (echelon reorder points and base
quantities) at stages i, i  1,, N and the leadtime demands at stages i  1,, N . We will refer to
Vi
as the effective reorder point at stage i . Consequently, the
N -stage model can be decomposed into N
single-stage ( R, nQ) models with reorder point Vi and base quantity Qi , i  1,, N . The linkage
between these single-stage models is captured by (12.13).
*
We now present an algorithm for identifying R . Note that
G1 () is convex and has a finite
minimum point. Define
G1 ( y)  EG1 ( y  U1 )
G1 () is convex. Let Y1 be its minimum point. Note that
Thus
N
 h E[ IL ]  EG ( IP )
C (R ) 
j
j 2
(4.14)


i
1
N
 h E[ IL ]  EG (V )
j
j 2
(4.13)
j
j
1
(12.15)
1
N
 h E[ IL ]  EG (min {R ,V
j
j 2
j
1
1
2
 Z1Q1  D2 })
From section 4.4, the distribution of IL j , j  2 , is independent of R1 . As a result, the first term in
(12.15),

N
j 2
h j E[ IL j ] , is independent of R1 . Therefore, the optimal R1 minimizes
EG1 (min {R1 ,V2  Z1Q1  D2 })
Since
G1 (min{ R1 ,W })  G1 (min{ Y1 ,W }) for any value of W , R1*  Y1 , which is independent of
the reorder points at the upstream stages! Now set R1
 Y1 .
To continue the optimization process, we define a sequence of functions recursively. Suppose Gi ()
is defined, i  1,, N . Let Yi be a finite minimum point of Gi () . Define for i  1,, N  1 ,
Gi 1 ( y )  hi 1 E ( y  U i 1  Di 1 )  EG1 (min {Yi , y  Z i Qi  Di1})
d
IL2  IP2  D2 U 2  V2  D2 and
It is easy to see that Gi () are all convex functions. Note that
that
V2 is independent of U 2 , Z1 , D2 and D2 . Thus, from (12.15) and (12.16)
C (R ) 
N
 h E[ IL ]  EG (V )
j 3
(4.13)

j
j
2
2
N
 h E[ IL ]  EG (min {R ,V
j 3
j
j
2
2
(12.16)
3
 Z 2Q2  D3 })
Similarly,
R2*  Y2 . Continuing in this fashion, we have Ri*  Yi for i  1,, N . Note that we obtain
the optimal echelon reorder points by sequentially minimizing
N convex functions. The minimum
holding and backorder cost is G N  (YN ) .
5.2 Installation Reorder Points
N
Consider the installation-stock ( R, nQ) policy, ( ri , Qi ) i 1 . Let the base order quantities be fixed. The
goal is to determine the optimal installation reorder points, denoted by ri , i  1,, N .
*
As mentioned in section 3, installation-stock ( R, nQ) policies are special cases of echelon-stock
( R, nQ) policies. From (12.3) and the fact that ri is an integer multiple of Qi 1 , for i  2 , one can
formulate the problem of finding ri
*
as
P min C (R)
s.t. Ri 1  Ri  mi Qi , mi integer
i  1,, N  1
Let R  ( R1 ,, RN ) be an optimal solution to P . Thus
0
0
0
r1*  R10 and ri*  Ri0  Ri01  Qi 1
for i  2 . Problem P is, however, difficult to solve. One can develop bounds and search for the optimal
solution, see Chen (1995). This approach becomes computationally infeasible for large problem instances
(e.g., large N ). Below, we describe a heuristic algorithm that finds near-optimal installation reorder
points,
The heuristic algorithm is based on the following empirical observation. In many numerical examples,
*
we found that R , the optimal echelon reorder points, contains useful information about
installation reorder points. Obviously, if
r * , the optimal
R * is a feasible solution to P , then the problem is solved:
r1*  R1* and ri*  Ri*  Ri*1  Qi 1 for i  2 . Interestingly, even when R * is infeasible, one can
still obtain ri , i  2 , by rounding Ri  Ri 1  Qi 1 to an integer multiple of Qi 1 . Formally, take
*
any i  2 . Let ri
*

*

(resp., ri ) be the maximum (resp., minimum) integer multiple of
(resp., larger) than or equal to Ri  Ri 1  Qi 1 . Then, ri
*
*
*
is either ri

Qi 1 that is less

or ri .
Based on this empirical observation, we restrict the installation reorder points at stages 2,, N to
the following set
def
(r2 ,, rN )  R1  {r2 , r2 }   {rN , rN }




Note that if Ri  Ri 1  Qi 1 is an integer multiple of Qi 1 , then ri  ri ; otherwise, ri  Qi 1  ri .
*
*


Thus, the set {ri , ri } contains at most two points, which implies that there are at most
2 N 1
combinations of installation reorder points in R1 .
Here is the algorithm. First, determine ri


and ri , i  2,, N , by rounding Ri  Ri 1  Qi 1
*
*
down and up to an integer multiple of Qi 1 . Then, for each point in R1 , find the optimal corresponding r1 .
This is easy to do since for any fixed installation reorder points at stages 2,, N , the cost function is
convex in
r1 (see Chen 1995). The heuristic solution is the best one obtained in this way. The
computational complexity of this algorithm is about
2 N 1 times the effort of evaluating a single policy.
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