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

khshang@duke.edu • zhoux@se.cuhk.edu.hk

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

1

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

5

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.

References

[1] Atkins, D., P. Iyogun. 1988. Period versus ‘can-order’ policies for coordinated multi-item in-

ventory systems. Management Science 34 791-796.

22

[2] Axsater, S. 1990. Simple solution procedure for a class of two-echelon inventory problems.

Operations Research 38 64-69.

[3] Axsater, S. 1993a. Exact and approximate evaluation of batch-ordering policies for two-level

inventory systems. Operations Research 41 777-785.

[4] Axsater, S. 1993b. Optimization of order-up-to-S policies in two-echelon inventory systems

with periodic review. Naval Research Logistics 40 245-253.

[5] Axsater, S. 2003. Supply chain operations: serial and distribution inventory systems, Chapter

10 in Graves, S. C. and T. de Kok (Eds.), Handbooks in Operations Research and Management

Science, Vol. 11: Supply Chain Management: Design, Coordination and Operation, Elsevier,

525-559.

[6] Cachon, G. 1999. Managing supply chain demand variability with scheduled ordering policies.

Management Science 45 843-856.

[7] Cachon, G. 2001. Exact evaluation of batch-ordering policies in two-echelon supply chains with

periodic review. Operations Research. 49(1). 79-98.

[8] Cetinkaya, S., C.-Y. Lee. 2000. Stock replenishment and shipment scheduling for vendor-

managed inventory systems. Management Science 46 217-232.

[9] Chandran, M. 2003. Wal-Mart’s supply chain management practice. ICFAI Center for Man-

agement Research, Nagarjuna Hills, Hyderabad, India.

[10] Chao, X., S. Zhou. 2009. Optimal policy for multi-echelon inventory system with batch ordering

and fixed replenishment intervals. Operations Research 57 377-390.

[11] Chen, F., R. Samroengraja. 2000. A staggered ordering policy for one-warehouse multi-retailer

systems. Operations Research 48 281-293.

[12] Chen, F., R. Samroengraja. 2004. Order volatility and supply chain costs. Operations Research

52 707-722.

[13] Chen, F., Y.-S. Zheng. 1997. One-warehouse multi-retailer systems with centralized stock

information. Operations Research 45, 275-287.

[14] Cheung, K.-L., S.-H. Zhang. 2008. Balanced and synchronized ordering in supply chains. IIE

Transactions 40 1-11.

[15] Eppen, G., L. Schrage. 1981. Centralized ordering policies in a multi-warehouse system with

lead times and random demand. TIMS Studies in the Management Sciences 16 51-67.

[16] Federgruen, A., H. Groenevelt, H. Tijms. 1984. Coordinated replenishments in a multi-item

inventory system with compound Poisson demands. Management Science 30 344-357.

[17] Federgruen, A., P. Zipkin. 1984. Allocation policies and cost approximations for multilocation

inventory systems. Naval Research Logistics Quarterly 31 97-129.

23

[18] Gallego, G., O. Ozer, P. Zipkin. 2007. Bounds, heuristics and approximations for distribution

systems. Operations Research 55, 503-517.

[19] Graves, S. 1996. A multiechelon inventory model with fixed replenishment intervals. Manage-

ment Science 42 1-18.

[20] Gurbuz, M., K., Moinzadeh, Y.-P. Zhou. 2007. Coordinated replenishment strategies in inven-

tory/distribution systems. Management Science 53 293-307.

[21] Jackson, P. L. 1988. Stock allocation in a two-echelon distribution system or ‘what to do until

your ship comes in’. Management Science 34 880-895.

[22] Lee, H., V. Padmanabhan, S. Whang. 1997. Information distortion in a supply chain: the

bullwhip effect. Management Science 43 546558.

[23] Marklund, J. 2008. Inventory control in divergent supply chains with time based dispatching

and shipment consolidation. Working paper. Lund University, Lund, Sweden.

[24] Maxwell, W., J. Muckstadt. 1985. Establishing consistent and realistic reorder intervals in

production-distribution systems. Operations Research 33 1316-1341.

[25] Muckstadt, J., R. Roundy. 1993. Analysis of multistage production systems. S. C. Graves, A.

H. G. Rinnooy Kan, P. H. Zipkin (Eds.), Handbook in Operations Research and Management

Science, Vol. 4: logistics of Production and Inventory. North Holland, Amsterdam 59-131.

[26] Rao, U. S. 2003. Properties of the period review (R, T ) inventory control policy for stationary,

stochastic demand. Manufacturing & Service Operations Management 5 37-53.

[27] Schwarz, L. 1989. A model for assessing the value of warehouse risk-pooling: Risk-pooling over

outside-supplier leadtimes. Management Science 35 828-842.

[28] Shang, K., S. Zhou. 2009. Optimal and heuristic (r, nQ, T ) policies in serial inventory systems

with fixed costs. Forthcoming in Operations Research.

[29] Shang, K., S. Zhou, G.-J. van Houtum. 2009. Improving supply chain performance: real-

time demand information and flexible deliveries. Forthcoming in Manufacturing and Service

Operations Management.

[30] Silver, E. 1981. Establishing reorder points in the (S, c, s) coordinated control system under

compound Poisson demand. International Journal of Production Research 9 743-750.

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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