Optimizing Replenishment Intervals for Two-Echelon Distribution … · 2018-10-17 · Optimizing...
Transcript of Optimizing Replenishment Intervals for Two-Echelon Distribution … · 2018-10-17 · Optimizing...
Optimizing Replenishment Intervals for Two-EchelonDistribution Systems with Fixed Order Costs
Kevin H. Shang • Sean X. Zhou
Fuqua School of Business, Duke University, Durham, North Carolina 27708, USA
Systems Engineering & Engineering Management, Chinese University of Hong Kong, China
[email protected] • [email protected]
March 17, 2010
We consider a periodic-review inventory system with one warehouse and N non-identical re-tailers. Each location has a positive lead time and orders according to a base-stock policy infixed time interval. A fixed cost is incurred for each inventory reorder. The warehouse fills theretailers’ orders in the same sequence as the original demand. The objective is to minimize thelong-run average system cost per period. This paper provides an approach to obtain the optimalbase-stock levels and reorder intervals. Specifically, for fixed reorder intervals, we show that theoptimal base-stock levels can be obtained by generalizing a result in the literature. To find theoptimal reorder intervals, we propose a complete enumeration on feasible solutions. The feasibleregion is determined based on two analytical results: (1) We decouple the total system cost intoeach stage and construct a lower bound on the decoupled stage cost; (2) we construct a lower boundon the total system cost by converting the distribution system into a single-location model withthe original problem data. These stage and system lower bounds together generate bounds for theoptimal reorder intervals. Since the computation of the optimal reorder intervals is quite extensive,we further suggest a simple heuristic by solving a set of serial systems. The heuristic appears tobe near-optimal in the tested problems. The primary findings from a numerical study are that theoptimal reorder intervals tend to follow integer-ratio relations and that the optimal reorder intervalof the warehouse is no less than the minimum of the optimal reorder intervals of the retailers.
1. Introduction
Materials are often produced and shipped according to fixed schedules in a supply chain. For exam-
ple, manufacturers often plan production according to material requirement planning (MRP) sys-
tems. The MRP system generates replenishment orders weekly, bi-weekly, or monthly, which results
in periodic inventory replenishment. Most consumer and finished goods are distributed in the same
fashion. Materials are often delivered in fixed days of the week. In these production/distribution
systems, fixed costs, such as inventory review costs, shipping costs, and transportation costs, are
often associated with each inventory reorder. Since these costs account for a significant portion of
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total supply-chain cost, it is important for managers to determine effective replenishment schedules
and inventory stocking levels across the supply chain.
We consider a one-warehouse-multi-retailer system that models the above periodic replenish-
ment practice. There are N non-identical retailers, each facing independent Poisson demand with
complete backlogging. The retailers replenish their inventories from the warehouse, which in turn
orders from an outside supplier with unlimited stock. Each location implements a stationary ech-
elon (s, T ) policy: Stage j reviews its echelon inventory order position (= inventory on order +
inventory on hand + inventory at or in transit to all downstream stages - backorders at the retail-
ers) in every Tj periods and orders up to an echelon base-stock level Sj , j = 0, ..., N . We assume
that the order schedules are synchronized, i.e., a retailer, whenever possible, places an order when
the warehouse receives a shipment (see §2 for an example). We also assume that the warehouse
applies the virtual allocation rule (e.g., Axsater 1993a, Graves 1996), in which the retailers’ or-
ders are filled in the same sequence as the occurrence of the demand at the retailers. The virtual
allocation rule may not be the most efficient way for the warehouse to allocate stocks. However,
this rule is convenient to implement and thus commonly seen in practice. For example, Wal-Mart’s
distribution center can receive real-time demand information from its retail stores. The distribution
center assigns replenishment stocks to the demands as they occurred. The assigned stocks will be
loaded onto a truck and shipped to the retail stores according to some fixed schedule (Chandran,
2003). We assume that a fixed order cost is incurred for each inventory reorder. In addition, a
linear holding cost per period is incurred for each unit of on-hand inventory held at each stage,
and a linear backorder cost per period is charged for each unit of backlogs incurred at the retailer.
The objective is to obtain the echelon (s, T ) policy such that the average total cost per period is
minimized.
We first characterize several key inventory variables and their dynamics. We then provide a
bottom-up recursion to evaluate the average system cost. It is worth mentioning that our model
assumes that the reorder intervals are integer multiples of some base period. In other words, the
reorder interval of the warehouse may be shorter than that of the retailer. With fixed reorder
intervals, the optimal echelon base-stock levels can be found by generalizing the result of Axsater
(1990). We show this generalization in Appendix A. Our main contribution is to provide a method
for obtaining the optimal reorder intervals. This is achieved by two results we establish. First,
we allocate the system cost into each stage by decomposing the retailer’s cost. We then establish
a lower bound to each of the allocated stage cost functions. Specifically, the lower bound for
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the allocated retailer cost is generated from a single-stage (s, T ) model; the lower bound for the
allocated warehouse cost is obtained by regulating the retailers’ reorder intervals. These stage cost
bounds are a function of a stage’s control parameters. Second, we construct a lower bound to
the system cost by converting the distribution system into a single-stage system with the original
problem data. These results together generate bounds for the optimal reorder intervals. The
optimal reorder intervals can therefore be obtained by a numerical search over the feasible region.
The computation for finding the optimal policy is quite extensive, however. In a numerical
study, we observe that the optimal reorder intervals always satisfy integer-ratio relations, i.e., the
ratios of retailer’s reorder interval to warehouse’s and to other retailers’ reorder intervals or the
reciprocals of the ratios are positive integers. In addition, we find that the optimal reorder interval
of the warehouse is always no less than the minimum of the optimal reorder intervals of the retailers.
These observations motivate us to propose a simple heuristic by minimizing the sum of the lower-
bound cost functions subject to integer-ratio constraints on the reorder intervals. Because there
is no order relation on the optimal reorder intervals, solving the above approximate problem is
equivalent to solving a set of serial systems. For each of the serial system, we apply Shang and
Zhou’s (2009) heuristic to generate a solution. The final heuristic solution is the one that leads to the
minimum system cost. A numerical study suggests that the heuristic is near-optimal: The average
(maximum) percentage error between the optimal cost and the heuristic cost is 0.28% (2.07%). We
also identify the conditions under which the heuristic performs less effectively. Finally, unlike the
serial system studied by Shang and Zhou (2009), the ratio of fixed order cost to the holding cost
seems to have less impact on the optimal reorder interval. We provide a detailed discussion in §6.
Our model is a generalization of the joint replenishment problem (JRP) in the literature. The
JRP is described below: Consider a retailer who manages N items. A major fixed cost is incurred
if any of the items is ordered and a minor, item-specific fixed cost is incurred if item j is ordered,
j = 1, ..., N . The retailer has to decide which item(s) should be replenished and how much to order
for each item. Our model reduces to the JRP by assuming that (1) the warehouse lead time is
zero, (2) the warehouse does not hold inventory, (3) the reorder intervals satisfy the integer-ratio
relations, and (4) the reorder interval of the warehouse is equal to the minimum of those of the
retailers. Under these constraints, the warehouse will place an order whenever any of the retailers
places an order. The major setup cost in JRP is equivalent to the fixed cost incurred at the
warehouse. Consequently, our optimization algorithm may be revised to solve the stochastic JRP.
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Literature Review
The literature related to multi-echelon, distribution systems is quite extensive. We only focus
on the most related work, i.e., the stochastic demand models with general reorder intervals. For
the deterministic demand model, we refer the reader to Muckstadt and Roundy (1993) for a review.
For the stochastic demand model with base-stock policies (i.e., the reorder intervals are one for all
stages), we refer the reader to Axsater (1990) for an exact optimization algorithm and Gallego et
al. (2007) for an example of heuristic solutions. For the stochastic demand model with reorder
point, order quantity policies, we refer readers to Axsater (1993a) and Cachon (2001) for local
control systems, and Chen and Zheng (1997) for echelon control systems. See Axsater (2003) for a
complete review.
For the distribution system with fixed reorder intervals, Graves (1996) considered a general
distribution system that may have more than two echelons. The inventory is controlled by a local
(s, T ) policy. His model assumes the virtual allocation rule and does not require any pattern of
the order schedule other than it is fixed and known. Graves found that most of the safety stock
should be held at the retailer sites and that the virtual allocation rule is near optimal in a numerical
study. Axsater (1993b) studied a special case of Graves’s model in that the retailers are identical
and the order schedule is nested, i.e., the reorder interval of the warehouse is an integer multiple
of those of the retailers. He demonstrated that the local (s, T ) policy under the virtual allocation
rule in Graves is essentially the same as the echelon (s, T ) policy. In addition, he showed that
the algorithm in Axsater (1990) can be applied to find the optimal base-stock levels. Lee et al.
(1997) demonstrated that the relationships between ordering schedules and bullwhip effect. They
showed that balanced ordering (i.e., the same number of retailers order each period) will lead to
the minimum level of order variability to the supplier. Cachon (1999) considered a one supplier, N
identical retailer system where retailers’ orders are balanced. He showed that the supplier’s demand
variance can be reduced when the retailer order intervals are lengthened. Chen and Samroengraja
(2004) compared a staggering order policy (a special case of balanced ordering) with an (r,Q)
policy. They found that the staggering policy leads to a higher total cost although it dampens the
bullwhip effect. Cheung and Zhang (2008) compared synchronized ordering with balanced ordering
and obtained a similar conclusion as Chen and Samroengraja, i.e., synchronized ordering leads to
a smaller system cost than balanced ordering.
There are papers aiming to find the optimal reorder interval in a context of distribution systems.
Cetinkaya and Lee (2000) considered a warehouse that replenishes its stock according to an (s, S)
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policy and ships stocks to retailers in every fixed interval. They provided a method to find the
optimal control parameters. Gurbuz et al. (2007) considered a new replenishment policy in which
retailers are replenished simultaneously in every fixed interval. They compared this new policy with
three existing policies in the literature. The reorder interval decision in these models is a single
variable. This is different from ours, where we determine a set of reorder intervals for the entire
system.
A closely related paper is Marklund (2008), in which a one-warehouse-multi-retailer distribution
system is considered under continuous review scheme. The warehouse implements a local (r,Q)
policy and allocates the stock to the retailers according to the virtual allocation rule. The retailers
implement a (s, T ) policy. There is a fixed order cost for each inventory reorder at the retailer
but no fixed order cost at the warehouse. Marklund provided a method to evaluate the average
system cost and proposed a heuristic to generate the policy parameters. Our model is different
from Marklund’s in that our model assumes (1) echelon control, (2) that the warehouse uses (s, T )
policy, and (3) that there is a fixed order cost incurred at the warehouse. In terms of the result,
we develop an approach to obtain the exact optimal solution, while Marklund only proposed an
effective heuristic.
One important issue for managing distribution systems is how the warehouse allocates stocks
to the retailers. This subject has attracted much research attention in the literature. Eppen and
Schrage (1981) assumed that the warehouse does not hold stock and the retailers are identical.
Under the assumption that the stocks can be freely transmitted between the retailers (the so-called
“balanced assumption”), they showed how to obtain the optimal base stock levels. Federgruen
and Zipkin (1984) considered a similar model in a discrete time setting. By approximating the
dynamic program, they proposed several approaches to generate near-optimal order and allocation
policies. Jackson (1988) extended Eppen and Schrage’s model by assuming that the warehouse can
hold stocks. Thus, the warehouse can allocate stock during the warehouse order cycle. Schwarz
(1989) examined the value of warehouse risk pooling over the warehouse lead time. Chen and
Samroengraja (2000) considered a similar system as Cachon (1999) and compared two allocation
policy: Past priority allocation (PPA) and current priority allocation (CPA). They found that the
CPA, which utilizes the current retailer inventory information, performs slightly better than the
PPA on average, but there is no significant dominance.
Finally, as stated, our model reduces to JRP under some conditions. A policy suggested to
manage JRP in the early literature is called “can-order” policy or an (s, c, S) policy that aims to
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coordinate orders. This policy were studied by, e.g., Silver (1981) and Federgruen et al. (1984).
They found that the policy is considerably more effective than uncoordinated policies. Atkins
and Iyogun (1988) studied a periodic-review JRP in which each item is replenished according to
an (s, T ) policy. They showed the periodic review (s, T ) policy outperforms the can-order policy
especially when the major fixed cost is large. The JRP resulted from our model is the same as that
of Atkins and Iyogun. Thus, our optimization method can be used for obtaining the exact optimal
solution in Atkins and Iyogun.
The rest of the paper is organized as follows. §2 introduces the model and the notation.
§3 characterizes the inventory variables and evaluates the total cost per period. §4 provides an
approach for finding the optimal reorder intervals. §5 proposes a simple heuristic. §6 conducts a
numerical study to examine the optimal solution and the effectiveness of the heuristic. §7 concludes.
Appendix A shows how to optimize the base-stock levels when the reorder intervals are fixed.
Appendix B provides an evaluation scheme for a serial system with non-nested (s, T ) policies.
Appendix C presents the proofs.
2. Model and Assumptions
We consider a periodic-review, two-echelon distribution system in which a single warehouse supplies
N non-identical retailers. Time is divided into equal length of one and periods are indexed 0, 1, 2, ....
Let [t, t + τ) and [t, t + τ ] denote the time interval over periods t, t + 1, ..., t + τ − 1 and periods
t, t + 1, ..., t + τ , respectively. We use j as the stage index, where 0 represents the warehouse and
j represents the retailer j, j = 1, ..., N . Retailer j faces a Poisson demand with stationary rate λj .
The retailers’ demands are independent of each other. Let Dj [t, t+ τ) and Dj [t, t+ τ ] denote the
cumulative demand over time in [t, t+ τ) and [t, t+ τ ]. There is a constant lead times Lj for stage
j, j = 0, ..., N . Let L[0,j] = L0 + Lj . Each stage implements a stationary echelon (s, T ) policy,
where Sj is the echelon base-stock level and Tj is the reorder interval for stage j. The policy is
operated as follows: At the beginning of every Tj periods, stage j orders up to Sj if the echelon
inventory order position is less than Sj . For the retailer, the echelon inventory order position is
equal to its local inventory order position, i.e., inventory on order plus inventory on hand minus
backorders. For the warehouse, the echelon inventory order position is the inventory on order plus
inventory on hand plus inventory in transit to and held at the retailers minus backorders at the
retailers. We term these Tj-th periods as order periods and the moment of placing an order as order
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epoch. Let hj be the echelon holding cost rate for stage j, j = 0, ..., N , and the local holding cost
rate Hj = h0 + hj for j = 1, ..., N . Unmet demand is fully backlogged at all stages. Let bj be the
backorder cost rate for stage j, j = 1, ..., N . Finally, there is a fixed cost Kj associated with each
inventory reorder. The objective is to obtain the policy such that the total system cost per period
is minimized.
We assume that the reorder intervals are an integer multiple of some base period. Thus, T0
may be smaller than Tj , j = 1, ..., N . Without loss of generality, let the base period be one.
The ordering activities between the warehouse and the retailers are coordinated in a synchronized
manner: Whenever possible, retailer j places an order when the warehouse receives a shipment. For
example, consider a one-warehouse-two-retailer system with L0 = 3, T0 = 2, T1 = 1, and T2 = 3.
Suppose that the warehouse places an order at the beginning of period t, t+ 2, t+ 4, ..... Consider
the order epoch t. This order placed at t will arrive at the beginning of period t + L0 = t + 3.
This is the moment that both retailers place an order. Thus, the order periods for the retailer 1 is
t+3, t+4, t+5, ..., and for the retailer 2 is t+3, t+6, t+9, .... We term such a order coordination
as synchronized ordering rule. Let
T = lcm{T0, T1, ..., TN},
where lcm is an operator that generates the least common multiplier of Tj ’s. If we define t and
t+L0 as the starting time of a cycle, clearly, the next cycle will start in T periods later under the
synchronized ordering rule. (That is, the next time when the warehouse receives a shipment and
all retailers place an order occurs in T periods later.)
Allowing the possibility of T0 less than Tj implies a non-nested ordering policy. (A ordering
policy is nested if each retailer orders every time when the warehouse receives a shipment. The
nested ordering policy implies T0 = njTj , j = 1, ..., N, nj some positive integer. See Maxwell
and Muckstadt 1985). It is interesting to note that for the serial system under the nested policy
assumption, Chao and Zhou (2008) showed that the synchronized ordering rule is the best among
all possible scheduling rules for the stationary echelon (s, T ) policy; Shang et al. (2009) showed
the same result for the stationary local (s, T ) policy. An intuitive explanation for these results is
that a shipment arrived at an upstream stage can be immediately transferred to its downstream
stage under the synchronized ordering rule. This helps to meet the demand more effectively, which,
in turn, results in a smaller total system cost. One critique for synchronized ordering under the
distribution system is that it may lead to a significant bullwhip effect when the system is controlled
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by local policies (Lee et al. 1997). However, Cheung and Zhang (2008) pointed out that high
bullwhip effect may not necessarily lead to a high system cost. Moreover, the bullwhip effect has
less impact in our model because the demand is learned by the warehouse when it occurs (i.e.,
echelon control).
Under the echelon (s, T ) policy, the retailer fills the incoming demand as if it were a single-stage
system. That is, when a unit of demand arrives, the retailer fills the demand and the retailer’s
net inventory level (= inventory on hand minus backorders) is reduced by one unit. The retailer
does not place an order until its next order epoch. On the other hand, the warehouse immediately
learns this arrived demand and assigns one unit to this demand. (In other words, the inventory
order position for the retailer is always equal to its base-stock level.) Similarly, the warehouse does
not order until its next order epoch. In short, the downstream ordering is for triggering a shipment
sent from the warehouse, not for transmitting the demand information. Thus, one may view the
retailers’ order epoches as “shipping times” planned by the warehouse.
Because of the centralized information available, we assume that the warehouse commits its
inventory, if available, to fill the incoming orders according to the sequence of the demands occurring
at the retailer site. If the warehouse does not have an uncommitted unit to fill an arrived demand,
the warehouse creates a backorder and adds this to the current list of outstanding orders. When
inventory becomes available, the outstanding orders are filled in the sequence in which they are
created. This is the so-called virtual allocation rule (Graves 1996).
We end this section by listing the sequence of events in a period: (1) A stage receives a shipment;
(2) the stage places an order if the period is its order period; (3) Poisson demand arrives during
the period; (4) costs are evaluated at the end of the period. The objective is to obtain the policy
parameters such that the average total cost per period is minimized.
3. Evaluation
Under the echelon (s, T ) policy, the system forms a regenerative process with a cycle of T periods.
Specifically, consider a warehouse order epoch t at which the warehouse orders up to S0. This order
will arrive at the beginning of period t+ L0. We assume that this is the moment that all retailers
will simultaneously place an order up to Sj . Thus, the regenerative cycle that we consider in the
subsequent analysis is [t, t+ T − 1] for the warehouse and [t+L0, t+L0 + T − 1] for retailer j. We
call t and t + L0 regenerative epoch for the warehouse and the retailers, respectively. To evaluate
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the total cost per period, we only need to characterize the distribution of the inventory variables in
the above regenerative cycles. The long-run average total cost per period is equal to the expected
total cost incurred in the cycle divided by the cycle length T .
We first define inventory variables below: For stage j, j = 0, ..., N ,
IOPj(n) = echelon inventory order position after ordering at stage j at the
beginning of period n,
IL−j (n) = echelon inventory level at stage j at the beginning of period n,
ILj(n) = echelon inventory level at stage j at the end of period n.
We describe the inventory dynamics in the regenerative cycle. Let r be the period index in the
considered regenerative cycle, i.e.,
r = 0, 1, 2, ..., T − 1.
Also, define ⌊a⌋ as the roundoff operator, which returns the greatest integer less than or equal to
a, a real number. Define
rj(r) =
⌊r
Tj
⌋Tj , j = 0, 1, 2, ..., N.
Thus, for r = 0, 1, . . . , T − 1,
rj(r) = 0, Tj , 2Tj , ...,
⌊T − 1
Tj
⌋Tj .
Since t and t + L0 are a regenerative epoch for the warehouse and the retailers, respectively, the
warehouse’s order periods are t+ r0(r), and the retailer j’s order periods are t+L0 + rj(r). As we
shall see, the order decision for the warehouse at the beginning of period t+ r0(r) will directly or
indirectly determine the retailer j’s inventory levels ILj(t+ L[0,j] + r).
Let D0[s, t) =∑N
j=1Dj [s, t) and D0[s, t] =∑N
j=1Dj [s, t]. The inventory dynamics for the
warehouse are
IOP0(t+ r0(r)) = S0, (1)
IOP0(t+ r) = IOP0(t+ r0(r))−D0[t+ r0(r), t+ r) (2)
= S0 −D0[t+ r0(r), t+ r),
IL−0 (t+ L0 + r) = IOP0(t+ r)−D0[t+ r, t+ L0 + r), (3)
IL0(t+ L0 + r) = IOP0(t+ r)−D0[t+ r, t+ L0 + r]. (4)
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Equation (1) means that the warehouse’s echelon inventory order position after ordering is equal
to S0. Equation (2) shows the warehouse’s echelon inventory order position at any given time t+ r
in the regenerative cycle. Equations (3) and (4) show the echelon inventory level at the beginning
and end of period t+ L0 + r, respectively.
Now, we consider retailer j. Since retailer j may not be able to obtain what it ordered from
the warehouse, we define IPj , B0 and B0j to reflect this fact:
IPj(n) = inventory in-transit position at the beginning of period n at retailer j,
B0(n) = total number of warehouse backorders at the beginning of period n,
B0j(n) = number of warehouse backorders that belong to retailer j.
Clearly, B0(n) =∑N
j=1B0j(n).
The corresponding regenerative cycle for stage j includes periods t+ L0 + r, r = 0, 1, .., T − 1.
Suppose that IPj(t+ L0 + r) is known, it will determine ILj(t+ L[0,j] + r) as follows:
ILj(t+ L[0,j] + r) = IPj(t+ L0 + r)−Dj [t+ L0 + r, t+ L[0,j] + r]. (5)
After all ILj in each period of the regenerative cycle is obtained, the total cost per period is
C(S,T) =
N∑j=0
Kj
Tj+
1
T
T−1∑r=0
E[h0IL0(t+ L0 + r) +
N∑j=1
hjILj(t+ L[0,j] + r)
+(bj +Hj)[ILj(t+ L[0,j] + r)]−]. (6)
The first term in C(S,T) represents the average total fixed cost per period1. The rest represents
the average inventory holding and backorder cost per period.
Our remaining task is to characterize IPj(t+ L0 + r). First, for retailer j, notice that IPj(t+
L0 + r) depends on IPj(t+ L0 + rj(r)), i.e.,
IPj(t+ L0 + r) = IPj(t+ L0 + rj(r))−Dj [t+ L0 + rj(r), t+ L0 + r). (7)
Since retailer j may not be able to receive the full quantity it ordered in period t + L0 + rj(r),
IPj(t+ L0 + rj(r)) is not necessary equal to Sj . More specifically,
IPj(t+ L0 + rj(r)) = IOPj(t+ L0 + rj(r))−B0j(t+ L0 + rj(r))
= Sj −B0j(t+ L0 + rj(r)). (8)
1Technically, the fixed cost term should be∑N
j=0 Kj Pr(D[Tj) > 0)/Tj . Following Section 5 of Shang and Zhou(2009), it is easy to show that all of our results can be carried over to the exact cost expression. We ignore Pr(D[Tj) >0) in the cost expression here because this probability should be reasonably close to one.
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To characterize B0j(t+L0+rj(r)), we first have to characterize B0(t+L0+rj(r)). Define IL′0(n) and
IOP′0(n) the local inventory level and inventory order position for the warehouse at the beginning
of period n, respectively. By definition,
B0(t+ L0 + rj(r)) = [IL′0(t+ L0 + rj(r))]
−
= [IOP′0(t+ rj(r))−D0[t+ rj(r), t+ L0 + rj(r))]
−
=[IOP0(t+ rj(r))−D0[t+ rj(r), t+ L0 + rj(r))−
N∑i=1
IOPi(t+ rj(r))]−
=[IOP0(t+ rj(r))−D0[t+ rj(r), t+ L0 + rj(r))−
N∑i=1
Si
]−, (9)
where [x]− = min{0,−x}. The last term in (9) is due to the fact that IOPi(t+ rj(r)) = Si under
the virtual allocation rule.
Furthermore, from (2), we can re-write (9) as follows:
B0(t+ L0 + rj(r))
=
[S0 −D0[t+ r0(rj(r)), t+ rj(r))−D0[t+ rj(r), t+ L0 + rj(r))−
N∑j=1
Sj
]−
=
[s0 −D0[t+ r0(rj(r)), t+ rj(r))−D0[t+ rj(r), t+ L0 + rj(r))
]−
=
[s0 −D0[t+ r0(rj(r)), t+ L0 + rj(r))
]−(10)
where s0 = S0−∑N
j=1 Sj , the local base-stock level for the warehouse. Note that r0(rj(r)) ≤ rj(r),
which implies that the time interval in D0 in (10) must be at least L0 periods.
Under the virtual allocation rule, we can apply binomial disaggregation on B0 to obtain the
distribution of B0j because demand arrival at each retailer follows an independent Poisson process.
That is, for any period n, we have
P(B0j(n) = k|B0(n) = m) =
(m
k
)(λj
λ0
)k(λ0 − λj
λ0
)m−k
, k = 0, 1, ...,m, (11)
where λ0 =∑N
j=1 λj . Since this conditional probability is independent of time, we shall omit the
period index n in the further analysis. Consequently, the distribution of B0j(t+ L0 + rj(r)) is
P (B0j (t+ L0 + rj(r)) = k) =
∞∑m=k
(P(B0
(t+ L0 + rj(r)
)= m
)P (B0j = k|B0 = m)
). (12)
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From Equations (10)-(12), we can obtain the distribution for B0j at the retailer j’s order period.
After B0j is characterized, we can use (7) and (8) to further obtain the distribution of IPj . Finally,
we apply (5) to obtain the distribution of ILj for each period in the considered regenerative cycle.
The average total cost per period is shown in (6).
A Bottom-up Evaluation Scheme
We provide a bottom-up recursion to simplify the evaluation of C(S,T). This procedure is to
first evaluate the retailer j’s cost by assuming that the retailer has ample supply. Then, we evaluate
the echelon cost of the warehouse, which is equivalent to the total system cost. More specifically,
let Mx(y) be an operator that returns the remainder of y divided by x, where x is a positive integer
and y is a nonnegative integer.
Let Dj [τ) and Dj [τ ] denote the total demand in τ and τ +1 periods, respectively, at retailer j.
For j = 1, ..., N , define
Gj(y, Tj , r) = E[hj(y −Dj [Lj +MTj (r)]) + (bj +Hj)(y −Dj [Lj +MTj (r)])−], (13)
and
G0(S,T, r) = E
h0(S0 −D0[L0 +MT0(r)]) +
N∑j=1
Gj (Sj −B0j (L0 + rj(r)) , Tj , r)
, (14)
where B0j(L0 + rj(r)) is the steady-state distribution for B0j(t+ L0 + rj(r)), which can be found
by removing the time index t in (10)-(12). Let
G(S,T)def=
1
T
T−1∑r=0
G0(S,T, r).
Proposition 1 For given echelon base-stock policies with parameters (S,T), the average total cost
per period is
C(S,T) =
N∑j=0
Kj
Tj+G(S,T),
where∑N
j=0(Kj/Tj) is the average total fixed order cost per period and G(S,T) is the average
inventory holding and backorder cost per period.
Proposition 1 can be proven by comparing the total cost obtained by using the above recursion
with the total cost obtained in (6). We omit the details for brevity.
12
4. Optimization
This section discusses how to obtain the optimal (s, T ) policies. When the reorder intervals are
fixed, it can be shown that Axsater’s (1990) algorithm can be revised to obtain the optimal echelon
base-stock levels. See Appendix A for the revised algorithm. As a result, our focus is to optimize
the reorder intervals.
We suggest a complete enumeration to find the optimal reorder intervals. To facilitate the
search, we construct bounds for the optimal reorder interval T ∗j , j = 0, ..., N . These solution
bounds can be obtained by constructing a lower-bound function for each retailer’s cost and for the
entire system cost. More specifically, in §4.1, we allocate the total system cost into each stage by
decomposing the retailer’s cost. In §4.2, we construct a lower bound to the allocated warehouse’s
cost and retailer’s cost. These lower bounds are a function of a stage’s control parameters. In
§4.3, we show that the total cost for the distribution system is bounded below by the total cost
obtained from a single-stage system with the original problem data. Finally, in §4.4, we show how
to construct upper and lower bounds for the optimal reorder intervals by using these stage and
system cost bound functions.
4.1 Decomposition of the System Cost
We provide an approach to allocate G(S,T) into each stage. This approach is to decompose the
retailer’s cost function in each period r in the considered regenerative cycle. More specifically, for
j = 1, 2, ..., N and r = 1, 2, ..., T − 1, define gj(y, Tj , r) = Gj(y, Tj , r). Let
Sj(Tj , r) = argminygj(y, Tj , r),
and
gjj(y, Tj , r) =
{gj(Sj(Tj , r), Tj , r), if y ≤ Sj(Tj , r),gj(y, Tj , r), otherwise.
Let
gj0(y, Tj , r) = gj(y, Tj , r)− gjj(y, Tj , r).
Notice that, for fixed Tj and r, gjj(y, Tj , r) is a constant before Sj(Tj , r) and convex and increasing
in y after Sj(Tj , r). The minimum value is gj(Sj(Tj , r), Tj , r). On the other hand, gj0(y, Tj , r) is
convex and decreasing in y with minimum value zero after Sj(Tj , r).
13
We provide an explanation for this decomposition. The gj(y, Tj , r) function is the expected
inventory holding and backorder cost for retailer j in period t + L[0,j] + rj(r) + MTj (r), referred
to as period r below, when retailer j’s IPj(t + L0 + rj(r)) = y. Sj(Tj , r) represents the IPj level
that leads to retailer j’s minimum cost in period r. Notice that Sj(Tj , r) is different in each period
r. From retailer j’s perspective, it would be the best if the IPj(t + L0 + rj(r)) level can achieve
Sj(Tj , r) in period r. Unfortunately, according to the considered (s, T ) policy, achieving different
Sj(Tj , r) in period r is impossible because retailer j is only allowed to place an order in period
t + L0 + rj(r). For this reason, we define gjj(y, Tj , r) as minimum period cost for retailer j, and
g0j(y, Tj , r) as policy-induced period penalty cost from retailer j or perid penalty cost for short.
There are two reasons that attribute to this penalty cost function. The first reason is that the
warehouse may be short of stock, making IPj less than a desired level; the second reason is due to
the nature of the (s, T ) policy – it is just impossible to have a different IPj for each period r.
Note that gj0(y, Tj , r) is different from the so-called induced-penalty cost function discussed
in the multi-echelon literature, e.g., Chen and Zheng (1998), Shang and Zhou (2009), in that the
penalty cost function charged to the warehouse here is a function of Sj(Tj , r), not the retailer j’s
target base-stock level Sj at its ordering epoch.
Since gj(y, Tj , r) = gj0(y, Tj , r) + gjj(y, Tj , r) for any y with fixed Tj and r, the total inventory
and backorder cost can be rewritten as
G(S,T) =1
T
T−1∑r=0
E
[h0(S0 −D0[L0 +MT0(r)]) +
N∑j=1
gj0(Sj −B0j(L0 + rj(r)), Tj , r)
+N∑j=1
gjj(Sj −B0j(L0 + rj(r)), Tj , r)
].
We allocate the penalty cost gj0 to the warehouse in each period r, and define the allocated
warehouse cost as
g0(S,T) =1
T
T−1∑r=0
E
[h0(S0 −D0[L0 +MT0(r)]) +
N∑j=1
gj0(Sj −B0j(L0 + rj(r)), Tj , r)
]. (15)
Furthermore, we define the allocated retailer cost as
gj(S,T) =1
T
T−1∑r=0
E
[gjj(Sj −B0j(L0 + rj(r)), Tj , r)
]. (16)
Then, we have G(S,T) = g0(S,T)+∑N
j=1 gj(S,T). This completes the decomposition of the total
system holding and backorder cost.
14
4.2 A Lower Bound on the Stage Cost
We proceed to construct a lower bound function for gj(S,T) and g0(S,T). As we shall see, the
lower-bound cost functions are a function of each stage’s own control parameters and therefore they
are independent. The independence feature is important for us to construct the solution bounds.
We first construct a lower bound for the retailer j’s cost. Recall (16), we have
gj(S,T) ≥ 1
T
T−1∑r=0
gjj(Sj(Tj , r), Tj , r) =1
Tj
Tj−1∑r=0
gjj(Sj(Tj , r), Tj , r)
=1
Tj
Tj−1∑r=0
gj(Sj(Tj , r), Tj , r)def= g
j(Tj).
The inequality holds because gjj(Sj(Tj , r), Tj , r) is the minimum cost for the retailer in period
t+ L0 + rj(r) +MTj (r). Proposition 2 summarizes this result.
Proposition 2 gj(S,T) ≥ gj(Tj).
We next construct a lower bound to g0(S,T) in (15). The first term in (15) is a function of
(S0, T0). Our goal is to construct a lower bound that is a function of (S0, T0) to the second term
and then as a result, we can also reduce 1/T∑T−1
r=0 to 1/T0∑T0−1
r=0 and the whole function will
depend only on (S0, T0).
First, for any retailer j, notice that Sj−B0j(t+L0+rj(r)) = IPj(t+L0+rj(r)). By definition,
IPj(t+ L0 + rj(r)) ≤ IL−0 (t+ L0 + rj(r))
= IOP0(t+ rj(r))−D0[t+ rj(r), t+ L0 + rj(r)) [from (3)]
= S0 −D0[t+ r0(rj(r)), t+ L0 + rj(r)). [from (2)]
Since gj0(y, Tj , r) is convex and decreasing in y, we have
Proposition 3 For fixed Tj and r,
E [gj0(Sj −B0j(L0 + rj(r)), Tj , r)] ≥ E [gj0(S0 −D0[L0 +MT0(rj(r))), Tj , r)] .
The right-hand side of the inequality in Proposition 3 is a lower bound to the period penalty cost
incurred from retailer j. This lower bound still depends on the retailer j’s reorder interval Tj . The
next result develops a lower bound to the above right-hand side term by removing this dependency.
15
Proposition 4
E
[gj0(S0 −D0[L0 +MT0(rj(r))), Tj , r)
]≥ E
[gj0(S0 −D0[L0 +MT0(r0(r))), T0, r)
].
Proposition 4 states that by regulating the retailer j’s reorder interval Tj to T0, the resulting period
penalty cost function is smallest among all other possibilities. Since MT0(r0(r)) = 0, we have
E
[gj0(S0 −D0[L0 +MT0(r0(r))), T0, r)
]= E
[gj0(S0 −D0[L0), T0, r)
]. (17)
It is interesting to compare this result with the one established in Shang and Zhou (2009), who
studied a serial system with nested reorder interval policies (i.e., the reorder interval at an upstream
stage is an integer multiple of that of its immediate downstream stage) under the synchronized
ordering rule. They constructed a lower bound to the induced penalty cost function by regulating
the reorder intervals of the downstream stages. Although the idea of constructing bounds is similar,
our result here is different from Shang and Zhou’s. First, we construct a lower bound to each period
penalty cost by regulating the retailer’s reorder interval, not to the induced penalty cost. Second,
the reorder interval of the warehouse may be smaller than the reorder interval of the retailer in our
model.
We substitute the right-hand-side term in (17) for the exact period penalty cost function in
g0(S,T), and define the resulting function as g0(S0, T0), i.e.,
g0(S0, T0) =
1
T0
T0−1∑r=0
E
[h0(S0 −D0[L0 +MT0(r)]) +
N∑j=1
gj0(S0 −D0[L0), T0, r)
].
The g0(S0, T0) function is the lower bound to the warehouse cost g0(S,T).
Proposition 5 For fixed T0, g0(S0, T0) is convex in S0.
Set S0(T0) = argminS0g0(S0, T0), and g
0(T0) = g
0(S0(T0), T0). With Propositions 2 - 5, we have
G(S,T) =N∑j=0
gj(S,T) ≥N∑j=0
gj(Tj).
Now, define
cj(Tj)def=
Kj
Tj+ g
j(Tj).
We have
C(S,T) ≥N∑j=0
cj(Tj). (18)
16
The following unimodality property is useful to construct the solutions bounds for T ∗j , j ≥ 1 in
§4.4.
Proposition 6 For j = 1, ..., N , cj(Tj) is unimodal in Tj.
Unfortunately, c0(T0) is not unimodal in T0. In order to construct bounds for T ∗0 , we provide
a different approach based on establishing a lower bound to the system cost demonstrated in the
next section.
4.3 A Lower Bound to the System Cost
This section provides a lower bound to the average system cost. As we shall see at the end, this
lower bound is a function of warehouse’s control parameters (S0, T0).
Define
h = min{h1, ..., hN}, H = h0 + h, b = min{b1, ..., bN},
IPr(τ) = the sum of inventory in-transit positions of the retailers in period τ =N∑j=1
IPj(τ).
Recall the average inventory holding and backorder cost G(S,T) in (6).
G(S,T) =1
T
T−1∑r=0
E
[h0IL0(t+ L0 + r) +
N∑j=1
(hjILj(t+ L[0,j] + r) + (bj +Hj)[ILj(t+ L[0,j] + r)]−
)]
≥ 1
T
T−1∑r=0
E
[h0IL0(t+ L0 + r) +
N∑j=1
(hILj(t+ L[0,j] + r) + (b+H)[ILj(t+ L[0,j] + r)]−
)]
≥ 1
T
T−1∑r=0
E
[h0IL0(t+ L0 + r) + h
(IPr(t+ L0 + r)−
N∑j=1
Dj [Lj ]
)
+(b+H)
(IPr(t+ L0 + r)−
N∑j=1
Dj [Lj ]
)−]def= GS(S,T). (19)
The cost function GS(S,T) in (19) can be viewed as the inventory holding and backorder cost
generated from a two-stage serial inventory system, in which the upstream stage has the parameters
as those in the warehouse and the downstream stage has holding cost rate h, backorder cost rate
b, and inventory in-transit position IPr(t+ L0 + r).
In Appendix B, we provide a simple recursion to evaluate GS(S,T). Here, we give an example
to see the difference between G(S,T) and GS(S,T). Consider a two-retailer system that has the
following parameters: h0 = 0.5, h1 = h2 = 1, b1 = b2 = 9, L0 = L1 = L2 = 1, λ1 = λ2 = 3, and
17
(S0, T0) = (45, 6), (S1, T1) = (15, 3), and (S2, T2) = (10, 2). In this example, G(S,T) = 29.74, and
GS(S,T) = 24.14.
Shang and Zhou (2009) developed a lower bound to the average inventory holding and backorder
cost for a serial system. The lower bound cost is obtained from a single-stage (s, T ) system. We
show that their result can be extended to the above two-stage cost function.
We first define this single-stage system. Consider a single-stage system with holding cost rate
h0, backorder cost rate b, and the demand during lead time D0[L0 +MT0(r)) +∑N
j=1Dj [Lj ]. The
cost function for this single-stage system for a given period r, r = 0, 1..., T0 − 1, is
F (y, r) = E
[h0
(y −D0[L0 +MT0(r)) +
N∑j=1
Dj [Lj ])+ (b+ h0)
(y −D0[L0 +MT0(r)) +
N∑j=1
Dj [Lj ])−]
.
Let
Gℓ(S0, T0) =1
T0
T0−1∑r=0
F (S0, r).
Following Rao (2003) or Shang and Zhou (2009), by assuming continuous approximation on the
demand and control variables, one can show Gℓ(S0, T0) is jointly convex in (S0, T0).
Let Sℓ0(T0) = argminS0 G
ℓ(S0, T0), and Gℓ(T0)def= Gℓ(Sℓ
0(T0), T0). Also, let π = h0∑N
j=1 λjLj .
We have
Proposition 7 GS(S,T) ≥ Gℓ(S0, T0) + π ≥ Gℓ(T0) + π.
Notice that Gℓ(T0) is a convex function of T0 and is independent of Tj , j = 1, ..., N .
4.4 Bounds on the Optimal Reorder Intervals
We shall use the system cost bound Gℓ(T0) + π to find the bounds for T ∗0 , and use the retailer cost
bound cj(Tj) to find the bounds for T ∗j , j = 1...., N .
We start with the construction of bounds for T ∗0 . Let C
h be the cost from any heuristic policy
(see §5 for a heuristic policy). Then,
Ch ≥ C(S∗,T∗) ≥ K0
T ∗0
+Gℓ(T ∗0 ) + π.
Since Gℓ(T0) is convex in T0 and Gℓ goes to infinity when T0 → ∞, a lower bound T 0 and an upper
T 0 can be obtained as follows:
T 0 = max
{T0|
K0
T0+Gℓ(T0) ≤ Ch − π
},
T 0 = min
{T0|
K0
T0+Gℓ(T0) ≤ Ch − π
}.
18
We next construct the bounds for T ∗j , j = 1, ..., N . First, we can obtain the minimum value of
c0(T0) by searching T0 ∈ [T 0, T 0]. Denote the resulting minimum value c0. Then,
Ch ≥ C(S∗,T∗) ≥ c0(T∗0 ) +
N∑j=1
cj(T∗j ) ≥ c0 +
N∑j=1
cj(T∗j ).
From Proposition 6, we can find the minimal value of cj(Tj), referred to as cj , j = 1, ..., N . Thus,
an upper bound T j and a lower bound T j to T ∗j can be obtained from the following inequalities:
T j = max
Tj | cj(Tj) ≤ Ch − c0 −∑i=j
ci
, T j = min
Tj | cj(Tj) ≤ Ch − c0 −∑i=j
ci
.
To find the optimal reorder intervals, we have to evaluate the policies such that Tj ∈ [T j , T j ],
and their corresponding optimal base-stock levels. Apparently, the effectiveness of the optimization
algorithm depends on Ch. In the next section, we provide a heuristic that seems to work well.
5. Heuristic
In our numerical experience, we observe that the optimal reorder intervals always follow integer-
ratio relations. That is,
T ∗j
T ∗0
∈ R, j = 1, ..., N, (20)
where R is the set of all positive integers and their reciprocals. Based on this observation, we propose
a heuristic that minimizes the sum of the lower bound cost functions subject to the integer-ratio
constraints. More specifically, we solve the following problem:
(P ) minT
C(T) =N∑j=0
(Kj
Tj+ g
j(Tj)
)
s.t.Tj
T0∈ R, j = 1, ..., N.
Since there are no order relations between Tj ’s, we need to consider all possible combinations.
Taking a two-retailer system as an example, we have to consider the following six possible combi-
nations:
(1) T0 = nT1, T1 = mT2, (2) T0 = nT2, T2 = mT1, (3) T1 = nT0, T0 = mT2,
(4) T1 = nT2, T2 = mT0, (5) T2 = nT1, T1 = mT0, (6) T2 = nT0, T0 = mT1,
19
where n and m are positive integers. In general, when there are N retailers, we have to consider
(N + 1)! possible combinations.
For each of the possible constraints, we solve a problem with a sum of unimodal functions2
subject to integer-ratio constraints. Continuing the two-retailer example, suppose that we consider
the first constraint above, the problem is:
(P1) minT
C(T)
s.t. T0 = nT1, T1 = mT2.
This problem has exactly the same structure as the approximate problem suggested by Shang and
Zhou (2009) for generating an effective heuristic solution in serial inventory systems. We therefore
apply their two-step procedure to generate an effective solution for (P1). We refer the reader to
Shang and Zhou (2009) for a detailed explanation. After solving the problem with each possible
constraint, we will have (N+1)! feasible solutions. Our final heuristic solution is the one that leads
to the smallest system cost.
6. Numerical Study
This section aims to examine the behaviors of the optimal reorder intervals and the effectiveness
of the proposed heuristic.
6.1 Properties of the Optimal Policy
We use the following four two-retailer cases to illustrate the behaviors of the optimal reorder
intervals. These four instances have the following common parameters: h0 = h1 = h2 = 1,
L0 = L1 = L2 = 1, λ1 = λ2 = 1, and b1 = b2 = 18. The fixed cost parameters and the optimal
reorder intervals are shown in Table 1 below.
Case K0 K1 K2 (T ∗0 , T
∗1 , T
∗2 ) Optimal Cost
1 16 4 1 (4, 2, 2) 25.23
2 1 4 16 (3, 3, 3) 25.31
3 0.5 0.5 16 (1, 1, 3) 23.81
4 0.5 1 16 (2, 2, 4) 24.11
Table 1: Optimal solutions for two-retailer systems
2Although g0(T0) is not unimodal, the cost function tends to be unimodal in our numerical experience.
20
Notice that the observations obtained from these four instances generally hold true for the
instances we tested in this section.
We first provide four interesting observations for the optimal reorder intervals.
(1) The optimal reorder intervals always satisfy the integer-ratio relations.
(2) We cannot find any instance such that T ∗0 < min{T ∗
1 , ..., T∗N}. For example, in Case 2, one
would expect T ∗1 to be larger than T ∗
0 since K1/h1 is four times larger than K0/h0. However,
the optimal reorder intervals are T ∗0 = T ∗
1 = 3. This seems to suggest that the reorder interval
of warehouse should be at least equal to the minimum of those of the retailers at optimality.
To further verify this statement, we reduce both K0 and K1 to 0.5 in Case 3 and the optimal
reorder intervals are T ∗0 = T ∗
1 = 1. Finally, we increase K1 from 0.5 to 1 in Case 4, T ∗1
increases from 1 to 2. Interestingly, T ∗0 in Case 4 also increases from 1 to 2. Thus, the
findings from these experiments are consistent with the statement.
(3) The cost ratios Kj/hj seem to have less impact on the optimal reorder intervals. For example,
in Case 1, one would expect that T ∗0 > T ∗
1 > T ∗2 since K0/h0 > K1/h1 > K2/h2. The optimal
reorder intervals are T ∗1 = T ∗
2 = 2, although K1/h1 is four times larger than K2/h2. This
observation suggests that the optimal reorder intervals of the retailers tend to be the same
when their corresponding cost ratios are not significantly different.
It is worth noting that this conclusion is different from that observed in Shang and Zhou
(2009) for the serial system. In their numerical study, Shang and Zhou found that the cost
ratio Kj/hj has direct impact on the optimal reorder interval T ∗j . More specifically, let
j = N (j = 1) represent the most upstream (downstream) stage in an N -stage serial system.
Shang and Zhou found that (i) if Kj/hj ≥ Kj−1/hj−1, T∗j ≥ T ∗
j−1 for j = 2, ..., N ; (ii) if
Kj/hj ≤ Kj−1/hj−1, T∗j tends to be the same between stages. Apparently, the impact of the
cost ratios on the optimal reorder intervals is less obvious for the distribution system.
(4) Based on the observation (3), it is natural to ask how “significant” the difference between
K1/h1 and K2/h2 should be in order to see that T ∗1 is different from T ∗
2 . In Case 4, we find
that when K2/h2 is 16 times larger than K1/h1, we see a difference. In Case 3, we reduce
both K0 and K1 to 0.5, and now K2/h2 is 32 times larger than K0/h0. In this case, we see
T ∗0 = T ∗
1 = 1, which is smaller than T ∗2 = 3. To our surprise, although K2/h2 is 32 times
larger than K0/h0, T∗2 is only three times larger than T ∗
0 .
21
6.2 Effectiveness of the Heuristic
This section examines the effectiveness of the heuristic. We focus on the two-retailer system. We
fix the retailer 1’s parameters and vary the parameters for the warehouse and retailer 2. More
specifically, the parameters for retailer 1 are K1 = h1 = L1 = λ1 = 1 and b1 = 3. For retailer 2, we
set h2 = λ2 = 1. The rest of the parameters for the warehouse and the retailer 2 are chosen from
the following sets:
h0 ∈ {1, 2}, K0,K2 ∈ {0.25, 16}, L0, L2 ∈ {1, 3}, b2 ∈ {3, 18}.
The total number of instances is 64. The percentage error is defined as below:
ξ% =C(T)− C(T∗)
C(T∗)× 100%,
where T = (T0, T1, ..., TN ) is the vector of the heuristic reorder intervals.
In this test, the average percentage error is 0.28% with maximum error of 2.07%. The heuristic
generates the optimal solution in 33 instances. The heuristic appears to be effective. We find that
the heuristic performs less effectively when (K0,K1,K2) = (16, 1, 16). For the 16 instances with
these fixed order costs, the average percentage error is 0.82%.
7. Concluding Remarks
This paper studies a one-warehouse, multi-retailer system in which base-stock, reorder interval
policies are implemented. The feature of our model is that we explicitly consider fixed order costs.
Specifically, we assume that a fixed order cost is incurred for each inventory reorder. We show how
to evaluate the system cost. We provide a method to obtain the optimal policy parameters. This
result is established by constructing stage independent lower bounds on the cost for each stage and
a lower bound to the system cost. We also suggest a heuristic that generates effective solution in a
numerical study. We observe that the optimal reorder intervals often follow integer-ratio relations.
Finally, unlike the serial inventory model studied by Shang and Zhou (2009), we find that the cost
ratio of fixed order cost to holding cost has less impact on the optimal reorder interval.
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24
Appendix A: Optimization of Base-Stock Levels with Fixed T
This appendix shows how to optimize the optimal echelon base-stock levels when the reorder in-
tervals T are fixed. It is worth mentioning that Axsater (1993) provides an approach (which is
a generalization of Axsater (1990) on base-stock systems) for finding the optimal local base-stock
levels with fixed reorder intervals. Axsater’s model is a special case of ours because he assumes
identical retailers and that the reorder interval of the warehouse is an integer multiple of that of
the retailer. Below we show that his approach can be generalized to our model.
We present the analysis from the local policy perspective, because B0 is a function of s0, and
Sj is the same as the local base-stock levels sj .
Define the local inventory holding and backorder cost for the retailer j as follows:
fj(y, r) = E[Hj(y −Dj [Lj +MTj (r)]) + (bj +Hj)(y −Dj [Lj +MTj (r)])−], j = 1, ..., N,
and
f0(y, r) = h0E
y −D0[L0 +MT0(r)] +N∑j=1
Dj [Lj +MTj (r)] +N∑j=1
λj
λ0(y −D0[L0 +MT0(rj(r))))
−
.
Proposition 8
C(S,T) =
N∑j=0
Kj
Tj+
1
T
T−1∑r=0
(f0(s0, r) +
N∑j=1
E[fj(sj −B0j(L0 + rj(r)), r)])+ h0
N∑j=1
(λjLj).
Note that f0(s0, r) is the warehouse inventory holding cost; fj(sj−B0j(L0+rj(r)), r) is the retailer
j’s inventory holding and backorder cost; the last term is the average pipeline cost per period,
which is constant.
For convenience, let us define
f0(s0) =1
T
T−1∑r=0
f0(s0, r), fj(s0, sj) =1
T
T−1∑r=0
E[fj(sj −B0j(L0 + rj(r)), r)].
Here, fj(·, ·) is a function of s0 because B0j is a function of s0.
Proposition 9 For fixed T and s0, fj(s0, sj) is convex in sj.
With Proposition 9, we can find the best local base-stock level sj(s0) for each retailer j. That
is,
sj(s0) = argminsj
fj(s0, sj).
25
Substituting sj(s0) for sj in C(S,T), the objective function becomes a function of s0, i.e., C(s0) =
f0(s0) +∑N
j=1 fj(s0, sj(s0)). Unfortunately, C(s0) is not convex in s0, so we have to construct
bounds for the optimal s0, denoted as s∗0, and conduct a search over the feasible interval.
Following Axsater’s (1990) approach, we next provide bounds for s∗0. Let sℓj = sj(∞) and
suj = sj(0). Define
su0 = argmins0
f0(s0) +
N∑j=1
fj(s0, sℓj)
and
sℓ0 = argmins0
f0(s0) +N∑j=1
fj(s0, suj )
.
Then,
Proposition 10 (1) sℓj ≤ s∗j ≤ suj ; (2) sℓ0 ≤ s∗0 ≤ su0 .
With Proposition 10, we can search over all possible s0 between sℓ0 and su0 . After s∗0 is found,
the optimal base-stock level for retailer j is s∗j = sj(s∗0).
Appendix B: A Recursion to Evaluate GS(S,T)
We provide a simple recursion to evaluate GS(S,T). Let SR =∑N
j=1 Sj . Then,
GS(S,T) =1
T
T−1∑r=0
F0(S,T, r),
where
F0(S,T, r) = E
[h0(S0 −D0[L0 +MT0(r)])
+F1
min
SR, S0 −D0[L0 +MT0(r)) +
N∑j=1
Dj [MTj (r))
,T, r
],F1(y,T, r) = E
hy −
N∑j=1
Dj [Lj +MTj (r)]
+ (b+H)
y −N∑j=1
Dj [Lj +MTj (r)]
− .
We omit the proof for brevity. A complete proof is available from the authors.
Notice that the recursion is similar to the one presented in Shang and Zhou (2009) to evaluate
the inventory holding and backorder cost for a serial system with nested echelon (s, T ) policies. In
fact, the above recursion can be used to evaluate a two-stage serial system with nonnested echelon
(s, T ) policies by setting N = 1 (one retailer) in the recursion.
26
Appendix C: Proofs
Proposition 3
Note that, for any order period t of stage j,
Sj −B0j(L0 + rj(r)) = IPj(t+ L0 + rj(r)),
S0 −D0[L0 +MT0(rj(r))) = IL−0 (t+ L0 + rj(r)),
and IL−0 (t + L0 + rj(r)) ≥ IPj(t + L0 + rj(r)). For simplicity, we suppress (t + L0 + rj(r)) from
IPj and IL−0 since we also consider the same time period. Thus,
E
N∑j=1
gj0(IPj , Tj , r)
=
N∑j=1
Sj∑n=−∞
gj0(n, Tj , r)P(IPj = n)
≥
N∑j=1
S0∑n0=−∞
[n0∑
n=−∞gj0(n0, Tj , r)
[P(IPj = n|IL−
0 = n0)P(IL−0 = n0)
]]
=
N∑j=1
S0∑n0=−∞
gj0(n0, Tj , r)P(IL−0 = n0)
[n0∑
n=−∞P(IPj = n|IL0 = n0)
]
= E
N∑j=1
gj0(IL−0 , Tj , r)
,
where the inequality follows from that gj0 is nonincreasing.
Proposition 4
By definition
E[gj0(y −D0[L0 +MT0(r0(r))), T0, r)] = E [gj (min{y −D0[L0), Sj(r, T0)}, T0, r)− gj(Sj(r, T0), T0, r)]
and
E
[gj0
(y −D0
[L0 +MT0
(⌊r
Tj
⌋Tj
)), Tj , r
)]= E
[gj
(min
{y −D0
[L0 +MT0
(⌊r
Tj
⌋Tj
)), Sj(r, Tj)
}, Tj , r
)− gj(Sj(r, Tj), Tj , r)
].
So to prove the result, we need to show, for r = 0, 1, ..., T − 1,
E
[gj
(min
{y −D0
[L0 +MT0
(⌊r
Tj
⌋Tj
)), Sj(r, Tj)
}, Tj , r
)− gj(Sj(r, Tj), Tj , r)
]≥ E [gj (min{y −D0[L0), Sj(r, T0)}, T0, r)− gj(Sj(r, T0), T0, r)] . (21)
27
It is clear that we just need to verify r = 0, 1, ..., lcm(T0, Tj)− 1. Let lcm(T0, Tj) = nT0 = mTj .
Consider the following two cases differentiated by n ≥ m or n < m.
Case 1: n < m and so T0 > Tj . Thus we can write
T0 =⌊mn
⌋Tj +
(mn
−⌊mn
⌋)Tj ,
where Tj must be divisable by n as m and n are relatively prime. Note that (mn −⌊mn
⌋) could take
value 1/n, . . . , (n−1)/n. For ease of presentation, we assume it is 1/n or T0 =⌊mn
⌋Tj+
1nTj . Other
cases can be similarly proved using the following procedure.
To prove the inequality (21) for all r = 0, 1, ..., lcm(T0, Tj)− 1, we consider different regions of
r.
Subcase 1. For r = 0, .., Tj − 1,⌊
rTj
⌋Tj = 0 and from the definition of gj(y, T, r) and Sj(r, T ),
E [gj (min {y −D0[L0), Sj(r, T0))} , T0, r)− gj(Sj(r, T0), T0, r)]
= E
[gj
(min
{y −D0
[L0 +MT0
(⌊r
Tj
⌋Tj
)), Sj(r, Tj)
}, Tj , r
)− gj(Sj(r, Tj), Tj , r)
],
because Sj(r, Tj) = Sj(r, T0) and gj(y, Tj , r) = gj(y, T0, r) in this case.
Subcase 2. We next consider r = Tj , . . . , T0 − 1. Note that, for r < T0, Sj(r, T0) is increasing
in r because
gj(y, T0, r) = E[hj(y −Dj [Lj + r]) + (bj +Hj)(Dj [Lj + r]− y)+]
is submodular in y and r as ∂gj(y, T0, r)/∂y = hj − (bj +Hj)Pr(Dj [Lj + r] > y) is decreasing in r.
Moreover, in this case,
MT0
(⌊r
Tj
⌋Tj
)=
⌊r
Tj
⌋Tj ,
from the definition of gj(y, T0, r) and MT0(r) = MTj (r) +⌊
rTj
⌋Tj ,
E
[gj
(min
{y −D0
[L0 +
⌊r
Tj
⌋Tj
), Sj(r, Tj)
}, Tj , r
)]≥ E
[gj
(min
{y −D0[L0)−Dj
[⌊r
Tj
⌋Tj
), Sj(r, Tj)
}, Tj , r
)]= E
[gj
(min
{y −D0[L0), Sj(r, Tj) +Dj
[⌊r
Tj
⌋Tj
)}, T0, r
)]≥ E[gj (min{y −D0[L0), Sj(r, T0)}, T0, r)], (22)
where the first inequality follows from that gj(min{y, Sj(r, Tj)}, Tj , r) is decreasing in y and the
second inequality from the convexity of gj(y, T0, r) and the optimality of Sj(r, T0) for gj(y, T0, r).
28
Meanwhile, as gj(Sj(r, T0), T0, r) ≥ gj(Sj(r, Tj), Tj , r) for Tj ≤ r < T0 beacause MT0(r) > MTj (r),
the following inequality is valid,
E
[gj
(min
{y −D0
[L0 +
⌊r
Tj
⌋Tj
), Sj(r, Tj)
}, Tj , r
)− gj(Sj(r, Tj), Tj , r)
]≥ E [gj (min{y −D0[L0), Sj(r, T0)}, T0, r)− gj(Sj(r, T0), T0, r)] ,
and so is inequality (21).
Subcase 3. Now we examine r = ℓT0, ℓT0 + 1, ..., (ℓ+ 1)T0 − 1 for 1 ≤ ℓ ≤ n− 1.
(a) First consider r = ℓT0 + Tj , ℓT0 + Tj + 1, ..., (ℓ + 1)T0 − 1. Note that for r = ℓT0 + (k −
1)Tj , . . . , ℓT0 + kTj − 1 for 1 < k ≤⌊mn
⌋Tj ,
MT0(r) = (k − 1)Tj , (k − 1)Tj + 1, . . . , kTj − 1, (23)
MTj (r) (24)
=
{ℓnTj ,
ℓnTj + 1, . . . , Tj − 1, r = ℓT0 + (k − 1)Tj , . . . , T0 + (k − 1)Tj +
n−ℓn Tj − 1,
0, 1, . . . , ℓnTj − 1, r = ℓT0 + (k − 1)Tj +
n−ℓn Tj , . . . , ℓT0 + kTj − 1,
and
MT0
(⌊r
Tj
⌋Tj
)(25)
=
{(k − 1)Tj − ℓ
nTj , , . . . (k − 1)Tj − ℓnTj , r = ℓT0 + (k − 1)Tj , . . . , ℓT0 + (k − 1)Tj +
n−ℓn Tj − 1,
kTj − ℓnTj , . . . , kTj − ℓ
nTj , r = ℓT0 + (k − 1)Tj +n−ℓn Tj , . . . , ℓT0 + kTj − 1.
With these, we can show that MT0(r) = MTj (r)+MT0
(⌊rTj
⌋Tj
). Following the arguments as those
in deriving (22),
E
[gj
(min
{y −D0
[L0 +MT0
(⌊r
Tj
⌋Tj
)), Sj(r, Tj)
}, Tj , r
)]≥ E[gj (min{y −D0[L0), Sj(r, T0)}, T0, r)].
Morover, as in this range, MT0(r) ≥ MTj (r) and so gj(Sj(r, T0), T0, r) ≥ gj(Sj(r, Tj), Tj , r). Thus,
(21) is true.
For r = ℓT0 +⌊mn
⌋Tj , ℓT0 +
⌊mn
⌋Tj + 1, ..., (ℓ+ 1)T0 − 1,
MT0(r) =⌊mn
⌋Tj ,⌊mn
⌋Tj + 1,
⌊mn
⌋Tj +
1
nTj − 1,
MTj (r) =ℓ
nTj ,
ℓ
nTj + 1, . . . ,
ℓ+ 1
nTj − 1,
MT0
(⌊r
Tj
⌋Tj
)=
⌊mn
⌋Tj −
ℓ
nTj .
29
As MT0(r) = MTj (r) + MT0
(⌊rTj
⌋Tj
), the inequality (21) is also valid following the preceding
argument.
(b) For r = ℓT0, ℓT0+1, ..., ℓT0+Tj − 1. Note that in this case, MT0(r) and MTj (r) are given in
(23) and (24) respectively with k = 1. Therefore, it can be seen that when r = ℓT0, ℓT0+1, . . . , ℓT0+
n−ℓn Tj−1, MT0(r) ≤ MTj (r) and so Sj(r, Tj) ≥ Sj(r, T0) but gj(Sj(r, T0), T0, r) ≤ gj(Sj(r, Tj), Tj , r).
However, we can show that
E
[gj
(min
{y −D0
[L0 +MT0
(⌊r
Tj
⌋Tj
)), Sj(r, Tj)
}, Tj , r
)− gj(Sj(r, Tj), Tj , r)
]≥ E [gj (min {y −D0[L0), Sj(r, Tj)} , Tj , r)]− gj(Sj(r, Tj), Tj , r)
≥ E [gj (min {y −D0[L0), Sj(r, T0)} , Tj , r)]− gj(Sj(r, T0), Tj , r)
≥ E [gj (min{y −D0[L0), Sj(r, T0))}, T0, r)]− gj(Sj(r, T0), T0, r),
where the first inequality follows from that gj0(y, Tj , r) is decreasing in y; the second inequality
follows from the convexity of gj(y, Tj , r) and the optimality of Sj(r, Tj) for gj(y, Tj , r); the third
inequality follows from that E[hj(y −Dj [Lj + r]) + (bj +Hj)(Dj [Lj + r] − y)+] is submodular in
(y, r) and MT0(r) ≤ MTj (r).
For r = ℓT0 +n−ℓn Tj , . . . , ℓT0 + Tj − 1, from (23) and (24), MT0(r) ≥ MTj (r). Moreover, as
MT0
(⌊rTj
⌋Tj
)= Tj − ℓ/nTj , MT0(r) = MTj (r) +MT0(
⌊rTj
⌋Tj) and so the inequality (21) follows
the same argument as we prove the case r = ℓT0 + Tj , . . . , (ℓ+ 1)T0 − 1.
Case 2: n ≥ m and so Tj ≥ T0 and
Tj =⌊ nm
⌋T0 +
( n
m−⌊ nm
⌋)T0.
Similar to the analysis of Case 1, we discuss different ranges of r. And although nm −
⌊nm
⌋can take
values 1m , . . . , m−1
m , we again only consider the case where it is 1/m, or Tj =⌊nm
⌋T0 +
1mT0.
Subcase 1. For r = 0, 1, ..., T0 − 1, the inequality (21) still holds as equality.
Subcase 2. For r = T0, T0 + 1..., Tj − 1, then MT0(r) < MTj (r) and so
E
[gj
(min
{y −D0
[L0 +MT0
(⌊r
Tj
⌋Tj
)), Sj(r, Tj)
}, Tj , r
)− gj(Sj(r, Tj), Tj , r)
]= E[gj (min{y −D0[L0), Sj(r, Tj)}, Tj , r)− gj(Sj(r, Tj), Tj , r)]
≥ E [gj (min{y −D0[L0), Sj(r, T0)}, Tj , r)− gj (Sj(r, T0), Tj , r)]
≥ E [gj (min{y −D0[L0), Sj(r, T0))}, T0, r)− gj(Sj(r, T0), T0, r)] ,
30
where the first inequality follows from the convexity of gj(y, Tj , r) and the optimality of Sj(r, Tj)
while the second inequality follows from the submodularity of gj(y, T, r) on (y, r) when r < T .
Subcase 3. For r = ℓTj + (k − 1)T0, . . . , ℓTj + kT0 − 1, for 1 ≤ ℓ ≤ m− 1 and 1 ≤ k ≤⌊nm
⌋,
MT0(r) =
{ℓmT0,
ℓmT0 + 1, . . . , T0 − 1, r = ℓTj + (k − 1)T0, . . . , ℓTj + (k − 1)T0 +
m−ℓm T0 − 1
0, 1, ..., ℓmT0 − 1, r = ℓTj + (k − 1)T0 +
m−ℓm T0, . . . , ℓTj + kT0 − 1,
(26)
MTj (r) = (k − 1)T0, (k − 1)T0 + 1, . . . , kT0 − 1, (27)
MT0
(⌊r
Tj
⌋Tj
)=
ℓ
mT0. (28)
In this case, note that for k > 1, as MT0(r) ≤ MTj (r), we can use the same arguments as in the
subcase 2 of this case to verify (21).
For k = 1, if r = ℓTj , . . . , ℓTj + m−ℓm T0 − 1, MT0(r) ≥ MTj (r) and MT0(r) = MTj (r) +
MT0
(⌊rTj
⌋Tj
). So gj(Sj(r, T0), T0, r) ≥ gj(Sj(r, Tj), Tj , r) and
E
[gj
(min
{y −D0
[L0 +MT0
(⌊r
Tj
⌋Tj
)), Sj(r, Tj)
}, Tj , r
)− gj(Sj(r, Tj), Tj , r)
]
≥ E
[gj
(min
{y −D0[L0), Sj(r, Tj) +Dj
[MT0
(⌊r
Tj
⌋Tj
)]}, T0, r
)− gj (Sj(r, Tj), Tj , r))
]≥ E [gj (min{y −D0[L0), Sj(r, T0))}, T0, r)− gj(Sj(r, T0), T0, r)] ,
where the last inequality follows also from the optimality of Sj(r, T0).
Otherwise, if r = ℓTj +m−ℓm T0, . . . , ℓTj + T0 − 1, MT0(r) < MTj (r) and the inequality (21) can
be proved similarly as subcase 2.
Finally for r = ℓTj +⌊nm
⌋T0, ℓTj +
⌊nm
⌋T0 + 1, . . . , ℓTj +
⌊nm
⌋T0 +
1mT0 − 1, MT0
(⌊rTj
⌋Tj
)is
the same as (28) while
MT0(r) =ℓ
mT0,
ℓ
mT0 + 1, . . . ,
ℓ+ 1
mT0 − 1
MTj (r) =⌊ nm
⌋T0,⌊ nm
⌋T0 + 1, . . . ,
⌊ nm
⌋T0 +
1
mT0 − 1,
which shows that MT0(r) < MTj (r) and so again the inequality (21) can be proved similarly as
subcase 2.
Therefore, the proof is complete.
Proposition 5
It is clear that, for each Tj and r, gj0(y, Tj , r) is convex in y from its definition. Thus, from its
definition, the convexity of g0(S0, T0) in S0 follows.
31
Proposition 6
To show cj(Tj) is unimodal, we need to show cj(Tj + 1) − cj(Tj) is positive after some Tj . It can
be shown that gj(Sj(Tj), Tj) is increasing in Tj . Thus,
cj(Tj + 1)− cj(Tj) =1
Tj(Tj + 1)
(Tjgj(Sj(Tj), Tj)−
Tj−1∑r=0
gj(Sj(r), r)−Kj
)(29)
Let
∆(Tj)def= Tjgj(Sj(Tj), Tj)−
Tj−1∑r=0
gj(Sj(r), r).
Notice that
∆(Tj + 1) = (Tj + 1)gj(Sj(Tj + 1), Tj + 1)−Tj∑r=0
gj(Sj(r), r)
≥ gj(Sj(Tj + 1), Tj + 1) + Tjgj(Sj(Tj), Tj)−Tj∑r=0
gj(Sj(r), r)
= [gj(Sj(Tj + 1), Tj + 1)− gj(Sj(Tj), Tj)] + ∆(Tj) ≥ ∆(Tj).
So the bracketed terms in (29) are increasing in Tj , which implies the unimodality of cj(Tj).
Proposition 7
For simplicity, let D′j(r) = Dj [Lj + MTj (r)) and D′
j [r] = Dj [Lj + MTj (r)] for j = 0, ..., N . From
Appendix B,
GS(S,T) =1
T
T−1∑r=0
E
[h0(S0 −D′
0[r]) + F1
min
SR, S0 −D′0(r) +
N∑j=1
Dj [MTj (r))
, r
]
=1
T
T−1∑r=0
E
[h0(S0 −D′
0[r]) + h
min
SR, S0 −D′0(r) +
N∑j=1
Dj [MTj (r))
−N∑j=1
D′j [r]
+(b+H)
min
SR, S0 −D′0(r) +
N∑j=1
Dj [MTj (r))
−N∑j=1
D′j [r]
− ]
≥ 1
T
T−1∑r=0
E
[h0(S0 −D′
0[r])
+(b+ h0)
min
SR, S0 −D′0(r) +
N∑j=1
Dj [MTj (r))
−N∑j=1
D′j [r]
− ]
≥ 1
T
T−1∑r=0
E
[h0(S0 −D′
0[r]) + (b+ h0)
S0 −D′0(r) +
N∑j=1
Dj [MTj (r))−N∑j=1
D′j [r]
− ]
32
=1
T0
T0−1∑r=0
E
[h0(S0 −D′
0(r)−N∑j=1
Dj [Lj ])
+(b+ h0)
S0 −D′0(r)−
N∑j=1
Dj [Lj ]
− ]+ h0
N∑j=1
λj(Lj + 1)− h0λ0
= Gℓ(S0, T0) + h0
N∑j=1
λjLj ,
where the first inequality follows by dropping a positive term h()+ and the second inequality from
that ()− is decreasing.
Proposition 8
Note that
Gj(y, Tj , r) = E[hj(y −Dj [Lj +MTj (r)]) + (bj +Hj)(y −Dj [Lj +MTj (r)])−],
and
G0(S,T, r)
= E
h0(S0 −D0[L0 +MT0(r)]) +N∑j=1
Gj (Sj −B0j (L0 + rj(r)) , r)
= E[h0(S0 −D0[L0 +MT0(r)])] +
N∑j=1
E[hj(sj −B0j (L0 + rj(r))−Dj [Lj +MTj (r)])
+(bj +Hj)(sj −B0j (L0 + rj(r))−Dj [Lj +MTj (r)])−]
= E[h0(S0 −D0[L0 +MT0(r)])]−N∑j=1
E[h0(sj −B0j (L0 + rj(r))−Dj [Lj +MTj (r)])
+
N∑j=1
E[Hj(sj −B0j (L0 + rj(r))−Dj [Lj +MTj (r)])
+(bj +Hj)(sj −B0j (L0 + rj(r))−Dj [Lj +MTj (r)])−]
= E[h0(s0 −D0[L0 +MT0(r)])] +N∑j=1
E[h0(B0j (L0 + rj(r)))]−N∑j=1
E[h0(−Dj [Lj +MTj (r)])
+
N∑j=1
E[fj(sj −B0j(L0 + rj(r)), r)]
= h0E
(s0 −D0[L0 +MT0(r)] +
N∑j=1
(Dj [MTj (r)])) +
N∑j=1
λj
λ0(s0 −D0[L0 + rj(r)− r0(rj(r))))
−
33
+
N∑j=1
h0(λj(Lj)) +
N∑j=1
E[fj(sj −B0j(L0 + rj(r)), r)].
As h0 = H0, so from Proposition 1, the result follows.
Proposition 9
It is straightforward to prove the convexity from the definition of fj(s0, sj).
Proposition 10
We need to first show that fj(s0, sj) is supermodular in s0 and sj , or
fj(s0, sj + 1)− fj(s0, sj) ≤ fj(s0 + 1, sj + 1)− fj(s0 + 1, sj). (30)
For notational simplicity, we denote Dj [Lj +MTj (r)] by Dj . In addition, to emphasize the depen-
dency of B0j on s0 and for brevity, let B0j(s0) denote B0j(L0 + rj(r)). Note that
fj(s0, sj + 1)− fj(s0, sj)
= Hj +1
T
T−1∑r=0
(bj +Hj)E[(sj + 1−B0j(s0)−Dj)− − (sj −B0j(s0)−Dj)
−]
≤ Hj +1
T
T−1∑r=0
(bj +Hj)E[(sj + 1−B0j(s0 + 1)−Dj)− − (sj −B0j(s0 + 1)−Dj)
−]
= fj(s0 + 1, sj + 1)− fj(s0 + 1, sj)
because ()− is convex and B0j(s0+1) is smaller than B0j(s0). So the inequality (30) is valid. With
this result and the convexity of fj(s0, sj) is sj , part (1) follows.
To show part (2), from (30), we have, for any sj ≥ sℓj ,
0 ≥ f0(su0) +
N∑j=1
fj(sℓj , s
u0)− [f0(s
u0 + 1) +
N∑j=1
fj(sℓj , s
u0 + 1)]
≥ f0(su0) +
N∑j=1
fj(sj , su0)− [f0(s
u0 + 1) +
N∑j=1
fj(sj , su0 + 1)]
where the first inequality follows from the definition of su0 . As the optimal s∗j will be greater than
sℓj , su0 is an upper bound for s∗0. Similarly, we can verify the lower bound sℓ0.
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