a staggered ordering policy for one-warehouse, multiretailer systems · 2014. 7. 30. · policy”...

A STAGGERED ORDERING POLICY FOR ONE-WAREHOUSE,MULTIRETAILER SYSTEMS

FANGRUO CHENGraduate School of Business, Columbia University, New York, New York 10027, [email protected]

RUNGSON SAMROENGRAJABooz, Allen & Hamilton Inc., 101 Park Avenue, New York, NY 10178, [email protected]

(Received June 1996; revisions received June 1997, May 1998; accepted July 1998)

We consider a one-warehouse, N-identical-retailer model. Random demands occur at the retailers with complete backlogging. Theretailers replenish their inventories from the warehouse, which in turn orders from an outside supplier with unlimited stock. Eachretailer places an order every N periods according to a base-stock policy, and the reorder intervals of the retailers are staggered sothat only one retailer places an order in each period. The warehouse orders according to an (s, S) policy based on its own inventoryposition. We consider two allocation policies, past priority allocation (PPA) and current priority allocation (CPA), which specify howthe retailer orders are filled at the warehouse. For the PPA model, we provide an exact procedure for computing the long-runaverage total cost. Based on the exact procedure, we develop an approximate model that can be used to determine near-optimalcontrol parameters for both the PPA and the CPA model. We conduct a computational study to test the effectiveness of theapproximate model and to compare the performance of the two allocation policies.

1. INTRODUCTION

We consider a distribution system with one ware-house and N retailers. Random demands occur atthe retailers only, and excess demands are completelybacklogged. The retailers replenish their inventories fromthe warehouse, which in turn orders from an outside sup-plier assumed to have unlimited stock. We assume that theretailers are identical, i.e., they have identical cost struc-tures, leadtimes, and demand distributions. The warehousefollows a periodic-review (s, S) policy based on its localinventory position: As soon as its inventory position (itson-hand inventory plus outstanding orders minus back-logged retailer orders) falls to or below s, it places anorder to raise its inventory position up to S. A uniquefeature of our model is that the replenishment decisions atthe retailers are coordinated in the following fashion. Eachretailer is allowed to order only once every N periods, andthe reorder intervals of the retailers are staggered so thatonly one retailer orders in each period. Also, the ware-house is allowed to ship to a retailer only when the retaileris scheduled to order. For example, if there are sevenretailers, then each retailer follows a weekly order sched-ule so that retailer 1 orders on every Sunday, retailer 2orders on every Monday, etc., and the warehouse cannotship to retailer 1 unless it is a Sunday. (The case where theretailers order in groups can be analyzed similarly as longas every group has the same number of retailers.) Eachretailer follows a base-stock policy with the same order-up-to level Y: An order is placed every N periods to raiseits inventory position (its on-hand inventory plus outstand-ing orders minus customer backorders) to the constant

level Y. Retailer orders are filled by the warehouse accord-ing to an allocation policy, which will be described shortly.Therefore, we envision a decentralized distribution systemwhere the only information communicated from the lowerechelon to the upper echelon is through the orders placedby the retailers in the above staggered fashion. The plan-ning horizon is infinite, and the objective is to minimizethe long-run average system-wide cost.

The above one-warehouse, multiretailer model is moti-vated by the Norton Auto Supply case (Hammond 1989).(Through the case writer, we have identified the companyfeatured in the case. Preliminary discussions are under wayto identify joint research opportunities.) The company hasone central distribution center (CDC) and 20 regional dis-tribution centers (RDC). The market is partitioned so thatthe RDCs face similar demand patterns. The RDCs arethen divided into five groups according to their geograph-ical proximity. There are four RDCs in each group. Eachgroup follows a weekly order schedule, and the orderschedules for the different groups are staggered so thatonly one group of RDCs orders on each day from Mondayto Friday. The staggered ordering schedule is reasonablewhen, say, the CDC has limited capacity. For example, thetransportation fleet at the CDC can only deliver to onegroup of RDCs in a single day.

Recently, we visited a distribution center in the NewYork area for a high-end fashion retailer. The managerthere showed us a staggered replenishment policy theyused to coordinate deliveries to the company’s departmentstores on the East Coast. Because each store is replenishedin constant intervals, it is easier for the stores to plan for

Subject classifications: Inventory/production: multi-echelon, stochastic, coordination, heuristics.Area of review: LOGISTICS AND SUPPLY CHAIN OPERATIONS.

281Operations Research, q 2000 INFORMS 0030-364X/00/4802-0281 $05.00Vol. 48, No. 2, March–April 2000, pp. 281–293 1526-5463 electronic ISSN

the incoming shipments so that the merchandise is prop-erly received and promptly displayed on the floor. Anotherbenefit of the staggered policy is that it smooths the work-load at the distribution center and provides the truck driv-ers with a simple, fixed schedule. Given the benefits of thestaggered policy, it is not surprising that it is also used inother industries such as food and tobacco. In a differentcontext, the staggered ordering policy has been suggestedfor its ability to reduce the so-called bullwhip effect; seeLee et al. (1997).

We consider two different types of allocation policiesthat specify how the retailer orders are going to be filled bythe warehouse. One is first-come, first-served. Under thispolicy, when a retailer order exceeds the warehouse on-hand inventory, the unfilled portion of the retailer order isbacklogged at the warehouse. When the warehouse re-ceives a shipment from the outside supplier, the receivedquantity is first used to satisfy the backlogged retailer or-ders on a first-come, first-served basis, and the remainingquantity becomes the on-hand inventory at the warehouse,which is used to fill future retailer orders. Because priorityis always given to the previous backlog, we refer to theabove allocation policy as past priority allocation (PPA).Note that under PPA, it is possible that the inventory usedto fill the backlogs from retailer 1, for example, stays in thewarehouse for several periods before it is shipped to re-tailer 1. This occurs if the retailer is not scheduled to orderwhen the allocation takes place. Therefore, there may besituations in which the warehouse is unable to satisfy aretailer order but at the same time has inventories ear-marked for the other retailers. This motivates the secondtype of allocation policy, called current priority allocation(CPA). Under CPA, when the warehouse receives a ship-ment from the outside supplier, it is immediately added tothe warehouse on-hand inventory. In each period thewarehouse considers only the designated retailer for theperiod and uses its on-hand inventory to fill the current aswell as the backlogged orders from that retailer. The ad-vantage of CPA is that it eliminates the unpleasant sce-nario where inventories sit idle at the warehouse while anincoming retailer order has to be backlogged. However,under CPA, there is a possibility that some retailers arebacklogged successively for several order epochs while atthe same time orders from other retailers are satisfied bythe warehouse without any backlogging. The advantage ofPPA is that it leads to a tractable model, and it seems fairin one way.

For the PPA model, we develop an exact procedure forevaluating the long-run average system-wide cost. Thisprocedure is efficient and is based on a cost-accountingscheme that enables us to compute the average costs at theretailers recursively. Our method is different from thewidely used approach based on the characterization of thedelay (or retard) that a retailer order experiences at thewarehouse (see, e.g., Svoronos and Zipkin 1988). It is alsodifferent from the disaggregation method based on allocat-ing the warehouse backorders among the retailers (see,

e.g., Graves 1985 and Chen and Zheng 1997). The exactevaluation procedure indicates that the average cost is arather complicated function of the control parameters, Yand (s, S). To help determine the optimal control param-eters that minimize the average cost (of the PPA model),we develop an approximation. The approximate cost func-tion has a component that has the same structure as thecost function of a two-stage, serial system (i.e., one-warehouse, one-retailer system). This observation simpli-fies optimization significantly. A computational studyshows that the policy parameters based on the approxi-mate cost function are indeed very close to being optimal.

On the other hand, for the CPA model, it is extremelydifficult to evaluate the average cost exactly. Fortunately, asimulation study suggests that the optimal control param-eters based on the PPA model are still very close to beingoptimal for the CPA model. In other words, even if wedecide to use CPA, we can still rely on the PPA model forfinding near-optimal control parameters. A numericalcomparison between the two allocation policies indicatesthat CPA is slightly better than PPA on average, but thereis no dominance.

One-warehouse, multiretailer systems have attractedmuch research attention in the literature. The discrete-time models of one-warehouse, multiretailer systems havebeen studied under different assumptions by, e.g., Eppenand Schrage (1981), Federgruen and Zipkin (1984a, b),Jackson (1988), and Nahmias and Smith (1994). Some as-sume that the warehouse does not hold any inventory, thusan incoming shipment to the warehouse is allocatedamong the retailers immediately. Others allow the ware-house to hold inventory so that the retailers may be re-plenished within a warehouse order cycle. Most assumecomplete backlogging of customer demand at the retaillevel—except for Nahmias and Smith, who consider partiallost sales. However, none of these papers considers stag-gered ordering policies for the retailers. There are alsocontinuous-time models of one-warehouse, multiretailersystems, see, e.g., Duermeyer and Schwarz (1981). In thesemodels it is unlikely that two retailers order at the sametime. Therefore, the retailer orders are “staggered,” but ina completely random and uncoordinated fashion. A recentpaper by Cachon (1996) does consider a model where theretailers follow staggered order schedules. However, in hismodel the shipping schedules for the retailers are not co-ordinated in the same fashion because the warehouse isallowed to ship to a retailer in any period to fill a previ-ously backlogged order from that retailer. Assuming thatboth echelons use (R, nQ) policies, he develops an exactevaluation procedure. Note that while base-stock policiesare special cases of (R, nQ) policies, (s, S) policies are not.(The above list of papers is by no means complete. Werefer the reader to the review articles by Axsater 1993,Federgruen 1993, and Nahmias and Smith 1993 for furtherreferences.)

Graves (1996) develops a multi-echelon inventory modelwhere each facility is allowed to order only at preset times.

282 / CHEN AND SAMROENGRAJA

Therefore, staggered ordering policies fall under hisframework. The key difference between our model and hisis in the allocation policy. He assumes a “virtual allocationpolicy” under which the warehouse inventories are virtu-ally committed to the retail sites as customer demandsoccur at the retail level. Note that under both PPA andCPA, the allocation decisions are postponed until a re-tailer places an order. For multi-echelon inventory sys-tems, it is well known that the postponement of allocationdecisions will in general lead to better decisions. This iscalled statistical economies of scale. The idea is that bydelaying the allocation decisions, one can hope to havemore (thus better) demand information before allocating.We believe that both the PPA and CPA models will pro-vide lower costs than the virtual allocation model. (Notethat Graves makes the virtual allocation assumption pri-marily for tractability.)

The rest of the paper is organized as follows. Section 2introduces the model and notation. Section 3 develops anexact evaluation procedure. Section 4 presents an approx-imate cost function that can be easily minimized. Section 5reports a computational study. Section 6 concludes thepaper with an extension. All the sections except §5 aredevoted to the PPA model. Whenever possible, we remarkon how the CPA model differs from the PPA model andpoint out why the former is intractable.

2. MODEL AND NOTATION

The model consists of one warehouse and N identical re-tailers. The warehouse orders from an outside supplierwith unlimited stock. Each order by the warehouse incursa fixed cost and arrives after a constant leadtime. Thetransportation leadtime from the warehouse to the retail-ers is constant. (Lateral transshipment between the retail-ers is not allowed.) Random customer demands arise atthe retailers only, and they are independent and identicallydistributed across time periods and across retailers. Unsat-isfied customer demands are completely backlogged. Lin-ear holding costs are assessed at each facility, and linearbackorder costs are assessed at each retailer.

The warehouse follows a periodic-review (s, S) policybased on its own inventory position; i.e., when its inventoryposition is at or below s, it places an order to increase itsinventory position to S. Each retailer places an order onceevery N periods according to a base-stock policy withorder-up-to level Y. The retailers’ reorder intervals arestaggered so that only one retailer orders in each period.For example, suppose there are seven retailers and eachretailer follows a weekly order schedule. Under the stag-gered ordering policy, retailer 1 orders every Sunday, andretailer 2 orders every Monday, etc. A shipment can bemade to a retailer only when the retailer is scheduled toorder. The retailer orders are filled at the warehouse on afirst-come, first-served basis, i.e., PPA is in place.

To fully understand the material flow through the ware-house, imagine that there are N 1 1 bins numbered 0, . . . ,

N in the warehouse (see Figure 1). When retailer N placesan order, the warehouse attempts to fill this order as muchas possible by transferring inventory from bin 0 to bin n. Incase bin 0 has insufficient inventory, the warehouse createsan outstanding order for the retailer for the unfilledamount. When a shipment arrives at the warehouse, it isfirst used to fill the outstanding orders for the retailers ona first-come, first-served basis. The inventory used to fillthe outstanding orders for retailer i is placed in bin i, i 51, . . . , N; the remainder goes to bin 0. The amountshipped to retailer n on its order occasion is the inventoryin bin n after the warehouse completes the inventory allo-cation for the period.

REMARK. The material flow is somewhat different underCPA (current priority allocation). Here, all incoming or-ders to the warehouse go directly to bin 0. Suppose retailern places an order in period t. The warehouse tries to fillthis order as well as the cumulative outstanding orders forretailer n by transferring inventory from bin 0 to bin n. Incase bin 0 has insufficient inventory, the amount short be-comes the new cumulative outstanding orders for retailern. Note that all bins except for bins 0 and n have zeroinventories in period t.

Define the following:

K 5 fixed cost for placing an order with the outsidesupplier.

H 5 echelon holding cost per unit per period at thewarehouse.

L 5 transportation leadtime from the outside supplier tothe warehouse, a nonnegative integer representingthe number of periods.

h 5 echelon holding cost per unit per period at theretailers, h Ä 0.

Figure 1. The one-warehouse multiretailer system.

283CHEN AND SAMROENGRAJA /

p 5 backorder penalty cost per unit per period at theretailers.

l 5 transportation leadtime from the warehouse to eachretailer, a nonnegative integer representing thenumber of periods.

D 5 one-period demand at any retailer, a discreterandom variable with mean m.

Z 5 order by any retailer, p(j) 5 Pr(Z 5 j), j 5 0,1, . . . .

In addition, for any t1 ¶ t2, let D[t1, t2] be the totalcustomer demand at a retailer in periods t1, t1 1 1, . . . , t2and Zn(t1, t2] be the total orders by retailer n in periodst1 1 1, . . . , t2. Note that the notation (t1, t2] is reallyequivalent to [t1 1 1, t2], and the former is adopted herefor brevity. (Of course, Zn(t, t] [ 0.) Let Z(t1, t2] 5 ¥n51

N

Zn(t1, t2].For clarity, we assume that the following events occur

sequentially at the beginning of each period t:(i) The designated retailer, say n, places an order.(ii) The warehouse reviews its inventory position and

places an order, if necessary.(iii) The warehouse order due this period arrives and is

used to fill the outstanding orders for the retailers (if any)under the PPA rule. Inventories are then transferred tothe retailer bins, and the remainder goes to bin 0. Anoutstanding order (or any portion thereof) for a retailer,once filled, is no longer considered outstanding, even if itis not yet shipped.

(iv) The warehouse fills the current order from retailern as much as possible by transferring inventory from bin 0to bin n. If bin 0 has insufficient inventory, the warehousecreates an outstanding order for retailer n for the unfilledportion.

(v) The contents in bin n are shipped to retailer n.(vi) The shipment due to a retailer this period is re-

ceived. (Only one retailer can receive a shipment in eachperiod.) Outstanding customer backorders (if any) aresatisfied.

Demand accrues throughout the rest of the period. At theend of the period, holding and backorder costs are assessed.

We next introduce additional notation to describe theinventory state of the system. For each period t, let t2 bethe time epoch after all the events at the beginning of theperiod have occurred but before demand arises, and let t1

represent the end of the period. For some inventory vari-ables, this distinction is unnecessary since they remain un-changed from t2 to t1. These variables are defined ateither t2 or t1. Define the following:

IP0(t) 5 warehouse inventory position at t2 or t1.

5 inventory on hand at the warehouse (i.e., inbin 0) plus orders in transit to the warehouseminus outstanding orders for the retailers.

I0(t) 5 inventory on hand at the warehouse (i.e., inbin 0) at t2 or t1.

B0(t) 5 total outstanding orders for the retailers at thewarehouse at t2 or t1.

IL0(t) 5 warehouse inventory level at t2 or t1.

5 I0(t) 2 B0(t).B0

n(t) 5 total outstanding orders for retailer n at thewarehouse at t2 or t1.

ICn(t) 5 inventory committed to retailer n at thewarehouse at t2 or t1 (i.e., inventory in bin n).

ITn(t) 5 inventory in transit to retailer n at t2 or t1.

IPn(t) 5 inventory position at retailer n at t2.

5 inventory on hand at retailer n plus inventoryin transit to retailer n minus customerbackorders at retailer n.

NIPn(t) 5 nominal inventory position at retailer n at t2.

5 outstanding orders for retailer n at thewarehouse plus inventory committed to retailern at the warehouse (i.e., in bin n) plus theinventory position at retailer n.

In(t) 5 inventory on hand at retailer n at t1.

Bn(t) 5 customer backorders at retailer n at t1.

ILn(t) 5 inventory level at retailer n at t1.

5 In(t) 2 Bn(t).

Note that neither IP0(t) nor IL0(t) include the invento-ries in the retailer bins. In other words, for the purpose ofdetermining the warehouse inventory position/level, onecan imagine that the retailer bins are placed “outside” thewarehouse.

With the above definitions, we can describe the (Y, s, S)policy more precisely. At each period t, the designatedretailer, say retailer n, orders to increase its nominal inven-tory position up to Y. Thus NIPn(t) 5 Y. Then, the ware-house reviews its inventory position. If it is at or below s,the warehouse orders to increase its inventory position upto S. Thus, s 1 1 ¶ IP0(t) ¶ S for all t.

For the rest of the paper we assume S Ä 0. We makethis assumption in order to facilitate certain regenerativearguments to be used later. Moreover, the assumption isintuitively appealing for the following reason. Suppose S ,0. In this case, each unit ordered by a retailer is includedin a subsequent warehouse order. The time between thesetwo orders is referred to as the order delay for this unit.The warehouse order takes L periods to arrive from theoutside supplier. When the warehouse order arrives, theunit is allocated to the bin associated with the retailer.This unit will not be shipped until the retailer’s next orderoccasion. This represents another delay, called the shippingdelay. When the unit is shipped, it takes l periods to arriveat the retailer. The sum of the order delay, the shippingdelay, and the two transportation leadtimes is the totalleadtime for the given unit. Clearly, the transportationleadtimes are not affected by the value of S. We believethat the shipping delay is a result of the staggered struc-ture, which is obviously unaffected by the value of S. But ifwe increase S to zero (with S 2 s fixed), then the orderdelay is reduced. Therefore, this change of S shortens thetotal leadtime for the unit and thus lowers holding andbackorder costs at the retailers. On the other hand, theabove change in S clearly does not affect the inventory in


bin 0 (which remains zero), and it does not seem to affectthe average inventory levels in the other bins (this is con-firmed in simulation examples). Moreover, the warehouseaverage fixed costs remain unchanged since S 2 s is fixed.In other words, increasing S to zero does not change thecosts at the warehouse. The above arguments suggest thatthe system as a whole benefits from changing S to zero andit thus makes sense to restrict ourselves to control param-eters with S Ä 0. However, we allow the reorder point s tobe negative. (In fact, many numerical examples have anoptimal reorder point that is negative.)

3. COST EVALUATION

In this section we consider the PPA model and show howto evaluate its long-run average cost exactly. We begin byintroducing a cost-accounting scheme to organize the costsinto a convenient form. Then, it becomes apparent thatthe most challenging step is to evaluate the average costsat the retailers, which, as it turns out, can be computedrecursively and exactly.

At the end of each period, we assess holding and back-order costs based on the inventory status of the system.First, determine the system on-hand inventory and chargeH for each unit. The system on-hand inventory consists ofthe on-hand and committed inventories at the warehouse(i.e., the contents in bins 0, . . . , N), inventories in transitto the retailers, and inventories on hand at the retailers. (Itis easy to see that the long-run average holding cost asso-ciated with the inventories in transit to the retailers isconstant and is independent of the control parameters.The inclusion of this cost component, while not essentialfor determining the optimal control parameters, simplifiespresentation.) Second, charge h for each unit of on-handinventory at the retailers. Finally, charge p for each unit ofcustomer backorder at the retailers. Therefore, the totalholding and backorder cost assessed in period t is

HF I 0 ~t! 1 On51

N

~IC n ~t! 1 IT n ~t! 1 I n ~t!!G1 h O

n51

N

I n ~t! 1 p On51

N

B n ~t! .

Subtracting and adding H ¥n51N B0

n(t), which is equal toHB0(t) by definition, to the above expression, we have

H@I 0 ~t! 2 B 0 ~t!#

1 H On51

N

@B 0n~t! 1 IC n ~t! 1 IT n ~t! 1 I n ~t!#

1 h On51

N

I n ~t! 1 p On51

N

B n ~t! ,

which, after subtracting and adding H ¥n51N Bn(t), becomes

HIL 0 ~t! 1 H On51

N

NIP n ~t 1!

1 On51

N

@hI n ~t! 1 ~ p 1 H! B n ~t!# , (1)

where NIPn(t1) is the nominal inventory position at re-

tailer n at the end of period t.Take any retailer n, and consider the long-run average

value of NIPn(t1). Suppose retailer n places an order in

period t. Thus NIPn(t) 5 Y. Since NIPn(t1) 5 NIPn(t) 2

D[t, t], we have NIPn(t1) 5 Y 2 D[t, t]. Because the

next order by retailer n occurs in period t 1 N, we haveNIPn((t 1 i)

1) 5 Y 2 D[t, t 1 i] for i 5 0, . . . , N 2 1.Therefore the long-run average value of NIPn(t

1) is

1N

Oi50

N21

E~Y 2 D@t , t 1 i#! 5 Y 2~N 1 1!

2m.

Because the above long-run average value is the same forevery retailer, the long-run average value of H ¥n51

N

NIPn(t1) is

HNSY 2 ~N 1 1!2 mD . (2)For the rest of the section, we concentrate on the long-runaverage values of the remaining terms in (1).

We begin by introducing a cost-accounting scheme forcharging holding and backorder costs. First note that

IL 0 ~t 1 L! 5 IP 0 ~t! 2 Z~t, t 1 L#. (3)

(We use Z(t, t 1 L] because IP0(t) is assessed after theretailer order in period t.) Because the orders from theretailers are independent and identically distributed, IP0(t)is independent of Z(t, t 1 L]. Given IP0(t), the expectedvalue of IL0(t 1 L) is determined. It is then natural tocharge HIL0(t 1 L) to period t.

The above time-shifting idea can also be used to chargethe holding and backorder costs at the retailers. Considerany retailer n. Suppose it orders at period t with IPn(t) 5y. (Thus y ¶ Y.) Because ILn(t 1 l ) 5 IPn(t) 2 D[t, t 1 l]and IPn(t) is independent of D[t, t 1 l] (the retailer facesiid demands), we have

IL n ~t 1 l ! 5 y 2 D@t, t 1 l#.

Moreover, because the next order by the retailer occurs inperiod t 1 N, the retailer does not receive any shipment inperiods t 1 l 1 1, . . . , t 1 l 1 N 2 1. Thus

IL n ~t 1 l 1 i! 5 IL n ~t 1 l ! 2 D@t 1 l 1 1, t 1 l 1 i#5 y 2 D@t, t 1 l 1 i#

for i 5 1, . . . , N 2 1. In other words, the distributions ofILn(t 1 l 1 i) for i 5 0, . . . , N 2 1 are all determined byy. As a result, the expected holding and backorder costs atretailer n over periods t 1 1, . . . , t 1 l 1 N 2 1 can beexpressed as the following function of y:

g~ y!def5 Oi50

N21

E@h~ y 2 D@t, t 1 l 1 i#! 1

1 ~ p 1 H!~ y 2 D@t, t 1 l 1 i#! 2],

because In 5 (ILn)1 and Bn 5 (ILn)

2 by definition. Wehave thus grouped the holding and backorder costs in-curred at retailer n over an N-period interval into a singleterm.


Now we are ready to introduce a cost-accountingscheme for charging the holding and backorder costs at theretailers: For each period t, first determine which retaileris scheduled to order in period t 1 L, and let it be retailern; and then charge g(IPn(t 1 L)) to period t. This cost-accounting scheme represents a particular way of groupingthe holding and backorder costs at the retailers as well as ashift in time. While the time-shifting idea is commonlyused in most stochastic inventory models, the grouping ofretailer costs is new, the validity of which is explainedbelow.

One way to determine the holding and backorder costsat the retailers is to count the costs across all the retailersin the first period, then in the second period, and so on.Summing the results across all the periods gives the totalholding and backorder costs at the retailers. However, ourapproach is different. Take any period t. First, determinethe retailer who is scheduled to receive a shipment fromthe warehouse in period t. (There is, of course, exactly onesuch retailer.) Let it be retailer 1. Now count the holdingand backorder costs at retailer 1 in periods t, . . . , t 1 N 21. Call the total the “cost” in period t. Do the same forevery period. The total holding and backorder costs at theretailers are determined by summing the “costs” across allthe periods. The two approaches lead to the same long-runaverage holding and backorder costs at the retailers. Fig-ure 2 illustrates the above two approaches for groupingcosts. In this example there are three retailers. Ovals rep-resent holding and backorder costs. The shaded ovals andthe arrows represent how costs are grouped. For an alter-native explanation of the equivalence of the two groupingtechniques, see the Appendix.

There is also a shift in time embedded in our cost-accounting scheme. Recall that in period t, we identify theretailer that is ordering in period t 1 L, say retailer n, andcharge g(IPn(t 1 L)) to period t. By the definition of g[,g(IPn(t 1 L)) represents the (conditional) expected hold-ing and backorder costs at retailer n in the interval [t 1L 1 l, t 1 L 1 l 1 N 2 1]. Figure 3 illustrates the timeepochs when cost assessments take place and the intervalscovered by each assessment. The rationale for charging

g(IPn(t 1 L)) to period t is that a statistical link can beestablished, as we shall see below, between IPn(t 1 L) andIP0(t), which is determined by the warehouse’s orderingdecision in period t.

In sum, to period t we charge a warehouse holding cost,HIL0(t 1 L), and a retailer holding-backorder cost,g(IPn(t 1 L)), where n is the retailer who orders in periodt 1 L.

We proceed to derive the long-run average system-widecost. Define a cycle to be the time interval between twoconsecutive orders by the warehouse. At the beginning ofeach cycle, the warehouse inventory position is S. As inde-pendent and identically distributed orders from the retail-ers arrive at the warehouse, the warehouse inventoryposition decreases. Because the retailer orders unsatisfiedfrom the warehouse on-hand inventory are fully back-logged, each retailer order reduces the warehouse inven-tory position by the amount ordered. As soon as thewarehouse inventory position falls to or below s, an orderis placed with the outside supplier and a new cycle begins.Therefore, the warehouse inventory position evolves in ex-actly the same manner as in the standard single-location (s,S) model with iid demands. This enables us to use thefollowing standard results. (We refer the reader to Veinottand Wagner 1965 and Zheng and Federgruen 1991 fordetails.) Let m(0) 5 1/(1 2 p(0)). (Recall that p( j) is theprobability mass function of the size of a retailer order,which equals the total demand in N periods at the retailer.Since the demands are iid, p( j) can be obtained by simpleconvolutions of the demand distribution.) Define recur-sively

m~ j! 5 Oi50

j

p~i!m~ j 2 i! and M~ j! 5 Oi50

j21

m~i!,

j 5 1, 2, · · ·

with M(0) 5 0. Note that m( j) and M( j) are the renewaldensity and the renewal function associated with a certainrenewal process, and they have the following intuitive ex-planation: m( j) is the average amount of time the ware-house inventory position equals S 2 j, and M( j) theaverage amount of time it takes the warehouse inventoryposition to fall to or below S 2 j since the beginning of awarehouse replenishment cycle. Recall that we charge

Figure 2. Illustration of the grouping techniques. Figure 3. Illustration of the cost-accounting scheme:time-shift.


HIL0(t 1 L) to period t. Thus, given IP0(t) 5 y, the ex-pected warehouse holding cost in period t is HE( y 2 Z(t,t 1 L]) 5 H( y 2 LNm) (see (3)). The expected warehouseholding cost in a cycle can thus be expressed as

Oy50

S2s21

m~ y!@H~S 2 y! 2 HLNm#

5 Oy50

S2s21

m~ y! H~S 2 y! 2 M~S 2 s!~HLNm! . (4)

Let k(Y, s, S) be the expected holding and backorder costsat the retailers in a cycle. Note that a setup cost K isincurred for each cycle, and the expected cycle length isM(S 2 s). From (2) and (4), we have the following expres-sion for the long-run average system-wide cost:

c~Y, s, S!def5

K 1 Oy50

S2s21

m~ y! H~S 2 y! 1 k~Y, s, S!

M~S 2 s!

1 HNY 2 c 0 , (5)

where

c 0 5 HNm@~N 1 1!/ 2 1 L# .

Now it remains only to evaluate k(Y, s, S).Take any period t, and let n be the retailer who orders in

period t 1 L. Recall that the holding and backorder costsat the retailers charged to period t are g(IPn(t 1 L)). Nowsuppose IP0(t) is given. To determine IPn(t 1 L), noticethat IP0(t) may include some outstanding orders in transitfrom the outside supplier to the warehouse. But all (andonly) these orders will have arrived by period t 1 L. Be-sides, the retailer orders are satisfied at the warehouse ona first-come, first-served basis with full backlogging. There-fore, as far as IPn(t 1 L) is concerned, the exact deliverytimes of those outstanding orders are not important. Forconvenience, we adopt the following convention for therest of this section:

Delivery Convention: To determine IPn(t 1 L), one canassume that all the orders placed by the warehouse beforeor in period t have been delivered by period t.

REMARK. Under CPA, the above Delivery Convention is nolonger valid because the distribution of IPn(t 1 L) de-pends not only on IP0(t) but also on the exact pattern ofdeliveries from the outside supplier to the warehouse inperiods t 1 1, . . . , t 1 L. This is the main reason why,under CPA, the exact evaluation of the average costs isextremely complicated.

Suppose IP0(t) 5 y. Let us try to determine the value ofIPn(t 1 L), where n is the retailer who orders in periodt 1 L. (In other words, we establish the statistical linkagebetween IP0(t) and IPn(t 1 L), which was mentioned whenwe introduced the accounting scheme.) First, suppose y Ä0. In this case, the warehouse has nonnegative on-handinventory at the beginning of period t, which implies thatall the previous retailer orders have been covered by pe-

riod t. Since there are no deliveries to the warehouse inperiods t 1 1, . . . , t 1 L (as per the Delivery Convention),the value of IPn(t 1 L) is uniquely determined by y andthe retailer orders in periods t 1 1, . . . , t 1 L. Becausethe distributions of these retailer orders are known, thedistribution of IPn(t 1 L) is determined by y. Now supposey , 0. Because the warehouse has a negative inventorylevel at the beginning of period t, some of the previousretailer orders have not been covered by period t. More-over, all the subsequent retailer orders placed in periodst 1 1, . . . , t 1 L will be completely backlogged. In orderto determine the value of IPn(t 1 L), we need to know theexact portion of the warehouse backlog in period t (i.e.,2y) that belongs to retailer n, which depends on the his-tory of retailer orders before period t. Therefore, the valueof IP0(t), if negative, is not sufficient for determining thedistribution of IPn(t 1 L). The above two cases suggestthat we need to have two different approaches for comput-ing the expected holding and backorder costs at the retail-ers, depending on the sign of the warehouse inventoryposition. (Recall that S was assumed to be nonnegative.Note that at the beginning of a warehouse cycle, the ware-house inventory position is S and all the backlogged re-tailer orders incurred previously have been cleared (as perthe Delivery Convention) because S is nonnegative. Thisinitial state and the order process from the retailers, bothof which are the same for every warehouse cycle, uniquelycharacterize the evolution of the warehouse inventory po-sition and the backlogged retailer orders for the rest of thecycle. This regenerative argument facilitates the determi-nation of the expected holding and backorder costs at theretailers. On the other hand, if S , 0, then the aboveregenerative argument fails because the initial state of thebacklogged retailer orders changes from cycle to cycle.This complicates the analysis.)

We divide the warehouse cycle into two parts: The non-negative (resp., negative) part consists of those periodswhere the warehouse inventory position is nonnegative(resp., negative). (Note that if s Ä 21, then the warehouseinventory position will never become negative.) Let k1(Y,s, S) (resp., k2(Y, s, S)) be the expected holding and back-order costs at the retailers in the nonnegative (resp., neg-ative) part of a cycle. Thus k(Y, s, S) 5 k1(Y, s, S) 1 k2(Y,s, S). We derive these functions separately.

Take any period t in the nonnegative part of a cycle. LetIP0(t) 5 y Ä 0. Let n be the retailer who orders in periodt 1 L. Define

G~ y! 5 E@ g~IP n ~t 1 L!! uIP 0 ~t! 5 y#.

We provide below a recursive procedure for computing theabove G[ function.

We begin by introducing a sequence of functions. Forany x Ä 0, define

U(x, i) 5 expected value of g(IPn(t 1 L)) given thewarehouse on-hand inventory at the beginning of period(t 1 i) is x, i 5 0, . . . , L.Thus G( y) 5 U( y, 0). Also define, for any x , 0,


U(x, i) 5 expected value of g(IPn(t 1 L)) given that theretailer order in period t 1 i is the first to exceed thewarehouse on-hand inventory, and the excess is 2 x, i 5 1,. . . , L.

It is rather easy to determine U(x, i) for any x , 0.Because the warehouse on-hand inventory is depleted inperiod t 1 i (for the first time), all the retailer ordersplaced in periods t 1 i 1 1, . . . , t 1 L are backordered. Inparticular, all the orders by retailer n in this interval, thetotal of which is Zn(t 1 i, t 1 L], are not satisfied byperiod t 1 L. Let the shortfall at retailer n be the totalunsatisfied orders from the retailer in period t 1 L. Ifretailer n orders in period t 1 i, then the shortfall atretailer n is 2x 1 Zn(t 1 i, t 1 L]; otherwise, the shortfallis just Zn(t 1 i, t 1 L]. Combining these two cases, wehave

U~ x, i! 5 HEg~Y 1 x 2 Z n ~t 1 i , t 1 L#!if retailer n orders in period t 1 i,Eg~Y 2 Z n ~t 1 i , t 1 L#!otherwise.

for any x , 0 and i 5 1, . . . , L. (To determine whether ornot retailer n orders in period t 1 i, simply note that eachretailer orders once every N periods and that retailer norders in period t 1 L.)

We next present a recursive formula for computing U(x,i) for any x Ä 0 and i 5 0, . . . , L 2 1. Let Z be theretailer order in period t 1 i 1 1. Thus the warehouseinventory level at the beginning of period (t 1 i 1 1) isx 2 Z. If x 2 Z Ä 0 then by definition, the conditionalexpected value of g(IPn(t 1 L)) is U(x 2 Z, i 1 1);otherwise, the order in period t 1 i 1 1 is the first toexceed the warehouse on-hand inventory with excess Z 2 xand by definition, the conditional expected value of g(I-Pn(t 1 L)) is again U(x 2 Z, i 1 1). Therefore,

U~ x, i! 5 EU~ x 2 Z , i 1 1! , x > 0, i 5 0, . . . , (6)L 2 1.

On the other hand, it follows by definition that

U~ x, L! 5 H g~Y!,g~Y 1 x! , x > 0,otherwise.The recursive procedure (6), together with the aboveboundary condition, can be used to compute G( y), i.e.,U( y, 0), for any y Ä 0.

Now we are ready to give an expression for k1(Y, s, S). Ifs Ä 21 then by the standard renewal argument, we have

k 1 ~Y, s, S! 5 Oy50

S2s21

m~ y!G~S 2 y! .

On the other hand, if s , 21, then the nonnegative partends when the warehouse inventory position is at 21 orbelow. In this case,

k 1 ~Y, s, S! 5 Oy50

S

m~ y!G~S 2 y! .

Combining the above two cases, we have

k 1 ~Y , s, S! 5 Oy50

S2max$21, s%21

m~ y!G~S 2 y!. (7)

We proceed to derive k2(Y, s, S), the expected holdingand backorder costs at the retailers charged to the nega-tive part of a cycle. (Clearly, this is necessary only if s ,21.) Take any period t in the negative part of a cycle.Thus s , IP0(t) , 0. Define the shortfall at a retailer to bethe total unfilled orders by that retailer. (Thus the DeliveryConvention implies that the total shortfall across all theretailers in period t is 2IP0(t) , 2s.) Suppose that retailern orders in period t 1 L. For any xn Ä 0 and x2n Ä 0 withxn 1 x2n , 2s, defineV(xn, x2n) 5 expected holding and backorder costs at re-

tailer n charged to periods t, t 1 N, etc., untilthe end of the cycle, given that the shortfallat retailer n in period t is xn and the totalshortfall at the other retailers in period t isx2n.

Also define

V~ x n , x 2n ! 5 0, x n 1 x 2n > 2 s. (8)

As we shall see shortly, once the V( z , z ) function isdetermined, it is rather easy to compute k2(Y, s, S).

We now present a recursive procedure for computingV(xn, x2n) for any xn Ä 0 and x2n Ä 0 with xn 1 x2n ,2s. First, consider the expected holding and backordercosts at retailer n charged to period t. Note that the re-tailer orders placed in periods t 1 1, . . . , t 1 L are stillbacklogged at the warehouse in period t 1 L. In particu-lar, the orders by retailer n in periods t 1 1, . . . , t 1 L,the total of which is Zn(t, t 1 L], are still backlogged inperiod t 1 L. Thus IPn(t 1 L) 5 Y 2 xn 2 Zn(t, t 1 L].As a result, the expected holding and backorder costs (atretailer n) charged to period t are

Eg~Y 2 x n 2 Z n ~t , t 1 L#! .

Now consider the expected holding and backorder costs atretailer n charged to periods t 1 N, t 1 2N, etc., until theend of the cycle. Recall that Zn(t, t 1 N] is the total ordersplaced by retailer n from period t 1 1 to t 1 N. Let Z2n(t,t 1 N] be the total orders placed by the other retailers inthe same time interval. Let x9n 5 xn 1 Zn(t, t 1 N] andx92n 5 x2n 1 Z2n(t, t 1 N]. If x9n 1 x92n Ä 2s, then anorder has already been placed by the warehouse at orbefore period t 1 N. In this case, the expected holding andbackorder costs charged to periods t 1 N, etc., until theend of the cycle are zero, which is equal to V(x9n, x92n) (see(8)). On the other hand, if x9n 1 x92n , 2s, then by defini-tion, the expected holding and backorder costs at retailer ncharged to periods t 1 N, etc., until the end of the cycleare V(x9n, x92n). Therefore,

V~ x n , x 2n ! 5 Eg~Y 2 x n 2 Z n ~t, t 1 L#!

1 EV~ x n 1 Z n ~t, t 1 N#, x 2n

1 Z 2n ~t, t 1 N#). (9)


The above formula, together with the boundary conditionin (8), can be used to compute the V( z , z ) functionefficiently.

The above V( z , z ) function can be used to computek2(Y, s, S). Suppose retailer 1 places an order in period 1that decreases the warehouse inventory position from anonnegative value to 2X, where X Ä 1. (The labeling hereis arbitrary. The X is the excess of a renewal process andits distribution can be easily obtained, see, e.g., Ross1983.) Recall that the holding and backorder costs chargedto period 1 are associated with a particular retailer. Let itbe retailer n1, i.e., retailer n1 is scheduled to order inperiod 1 1 L. Now consider the expected holding andbackorder costs at retailer n1 charged to the negative partof the cycle. To do this, we first need to assess the shortfallat retailer n1 in period 1, xn1, as well as the total shortfall atthe other retailers in period 1, x2n1. Note that if n1 5 1then xn1 5 X and x2n1 5 0; otherwise, xn1 5 0 and x2n1 5X. For the former case, the expected holding and back-order costs at retailer n1 charged to the negative part ofthe cycle are EV(X, 0); and for the latter, EV(0, X). Todetermine k2(Y, s, S), we need to carry out the aboveanalysis for every retailer (we just did for retailer n1). Notethat retailer 1 orders in period 1, retailer 2 orders in pe-riod 2, etc., and retailer N orders in period N. For m 52, . . . , N, let Zm be the retailer order quantity in period mand let nm be the retailer whose costs are charged to pe-riod m (i.e., retailer nm is scheduled to order in periodm 1 L). For m 5 1, . . . , N, define

W m 5 HX,Z nm ,0,

n m 5 1,2 < n m < m,otherwise.

Note that Wm is the shortfall at retailer nm in period m.(Figure 4 illustrates the above definitions of nm and Wmfor the case with N 5 3 and L 5 2.) Because the totalshortfall across all the retailers in period m is X 1 ¥i52

m Zi,the total shortfall across all the retailers except retailer nmin period m is X 1 ¥i52

m Zi 2 Wm. Consequently, theexpected holding and backorder costs at retailer nmcharged to the negative part of the cycle are

EV(W m , X 1 Oi52

m

Z i 2 W m )

Summing over m 5 1, . . . , N, we have

k 2 ~Y , s, S! 5 Om51

N

EV(W m , X 1 Oi52

m

Z i 2 W m ). (10)

With (5), (7), and (10), the development of the exact eval-uation procedure is now complete.

4. APPROXIMATION AND HEURISTIC SOLUTION

The exact evaluation procedure developed in the previoussection indicates that c(Y, s, S) is a complex function ofthe three control parameters Y, s, and S. To help deter-mine the optimal values of these control parameters, wedevelop an approximate cost function in this section whichhas a simple structure and thus is easy to minimize. Theoptimal solution based on the approximate cost functioncan be used as a heuristic solution to the original problem,or it can be used as a starting point in the search for theoptimal solution to the original problem using the exactcost function. The goodness of the heuristic solution willbe tested in a computational study reported in the nextsection.

Recall from the previous section that the most difficultstep in evaluating the long-run average cost of any given(Y, s, S) policy is to compute the expected holding andbackorder costs at the retailers. We simplify this step by anapproximation. Take any period t, and let n be the retailerwho orders in period t 1 L. Let IP0(t) 5 y. From (3),

IL 0 ~t 1 L! 5 y 2 Z~t, t 1 L#.

To determine IPn(t 1 L), consider two cases. If IL0(t 1L) Ä 0, then retailer n’s order in period t 1 L is com-pletely satisfied and thus IPn(t 1 L) 5 Y. Otherwise, ifIL0(t 1 L) , 0, then the warehouse has a list of outstand-ing orders for the retailers in period t 1 L. If we assumethat all these outstanding orders belong to retailer n, thenIPn(t 1 L) 5 Y 1 IL0(t 1 L). (Because some of theoutstanding orders may belong to the other retailers, thisapproximation overestimates the shortfall at retailer n.)Combining the above two cases, we have

IP n ~t 1 L! < min$Y , Y 1 IL 0 ~t 1 L!%5 min$Y , Y 1 y 2 Z~t, t 1 L#% ,

which leads to the following approximate holding andbackorder costs (at retailer n) charged to period t:

Eg~min$Y , Y 1 y 2 Z~t, t 1 L#%! .

As a result, we can approximate the expected holding andbackorder costs at the retailers in a warehouse cycle by

k~Y , s, S!

< Oy50

S2s21

m~ y! Eg~min$Y , Y 1 S 2 y 2 Z~t, t 1 L#%!. (11)

Let s9 5 s 1 Y and S9 5 S 1 Y. Define

Figure 4. Illustration of the definition of nm and Wmfor the case with N 5 3 and L 5 2.


G̃~ y, Y! 5 Eg~min$Y, y 2 Z~t, t 1 L#%! .

Thus (11) can be written as

k~Y, s, S! < Oy50

S92s921

m~ y!G̃~S9 2 y, Y! .

Using the above expression in (5), we have

c~Y, s, S! < c̃~Y, s9 , S9! def5 c̃ 0 ~Y, s9 , S9!

1 H~N 2 1!Y 2 c 0 , (12)

where

c̃ 0 ~Y, s9, S9!

5

K 1 Oy50

S92s921

m~ y!@H~S9 2 y! 1 G̃~S9 2 y, Y!#

M~S9 2 s9!.

Let (Ỹ, s̃9, S̃9) be the minimum point of c̃(Y, s9, S9), whichis easy to find, as we show next.

We first note that c̃0(Y, s9, S9) has exactly the same formas the cost function of a two-stage, serial system (or aone-warehouse, one-retailer system). In this system, theupper stage uses the (s9, S9) policy based on the system (orechelon) inventory position and the lower stage uses abase-stock policy with order-up-to level Y. This is preciselythe Clark and Scarf (1960) model. (It is easy to check thatif N 5 1, the approximate cost function is exact!) Let Y0 bethe minimum point of g[. Because for any fixed y, G̃( y,Y) is minimized at Y 5 Y0, the optimal Y that minimizesc̃0(Y, s9, S9) is Y0, which is independent of the values of s9and S9! Also note that G̃( y, Y0) is convex in y. Therefore,

c̃ 0 ~Y 0 , s9, S9!

5

K 1 Oy50

S92s921

m~ y!@H~S9 2 y! 1 G̃~S9 2 y, Y 0 !#

M~S9 2 s9!

has the same form as the cost function of the standardsingle-location (s, S) model. As a result, we can use theefficient algorithm in Zheng and Federgruen (1991) tominimize c̃0(Y0, s9, S9) over (s9, S9). In short, the minimumpoint of c̃0(Y, s9, S9) is very easy to find. (This is basicallythe sequential algorithm given in Chen and Zheng 1994for multistage serial systems.)

To minimize the approximate cost function c̃(Y, s9, S9),we notice that it is the sum of two components: c̃0(Y, s9,S9) and H(N 2 1)Y (the constant term c0 can be ignoredin optimization). Because H(N 2 1)Y is increasing in Y(assuming N . 1), we know from (12) and the abovediscussion on c̃0(Y, s9, S9) that Ỹ ¶ Y0. Now take any Y ¶Y0. It is easy to verify that G̃( y, Y) is still convex in y.Therefore, c̃0(Y, s9, S9) as a function of s9 and S9 still hasthe same form as the cost function of the standard single-location (s, S) model. As a result, the optimal (s9, S9) forthe given Y is easy to find. It is then clear that a one-dimensional search over Y ¶ Y0 will lead to the optimal

solution (Ỹ, s̃9, S̃9). This solution is then used to generatethe following heuristic solution to the original problem:

Y a 5 Ỹ , s a 5 s̃9 2 Ỹ , and S a 5 S̃9 2 Ỹ .

REMARK. As mentioned earlier, the approximation overes-timates the shortfall at retailer n in period t 1 L or equiv-alently, underestimates IPn(t 1 L). Because IPn(t 1 L)never exceeds Y and g( y) is decreasing in y for y ¶ Y0, weknow that the approximation overestimates the expectedholding and backorder costs at the retailers for any Y ¶Y0, or c̃(Y, s9, S9) Ä c(Y, s, S) for any Y ¶ Y0.

5. COMPUTATIONAL STUDY

We conducted a computational study to answer the follow-ing questions.

Y Is the heuristic solution developed in §4 close to opti-mal for both the PPA system and the CPA system?

Y Does the optimal policy based on PPA still perform wellfor the CPA system? Is it better than the heuristic solu-tion for the CPA system?

Y Is one allocation policy better than the other?

For the computational study, we assumed that D (thedemand at a retailer in one period) can be described by anegative binomial distribution with parameters (n, r), i.e.,

Pr~D 5 x! 5 ~x 1 n 2 1n 2 1 !r n~1 2 r! x, x 5 0, 1, 2, . . . .

(See Nahmias and Smith 1994 for more discussions on theappropriateness of the negative binomial distribution.) Weconstructed 324 numerical examples by taking all possiblecombinations of N 5 2, 3, 5, 7; K 5 10, 100, 1000; H 5 1;L 5 1, 3, 5; h 5 1; p 5 10, 20, 30; l 5 3; and (n, r) 5 (8,0.5), (2, 0.5), and (1, 0.75). The coefficients of variation ofthe three demand distributions are 0.5, 1, and 2,respectively.

For each example, we computed the following:

(Ya, sa, Sa) 5 heuristic solution that minimizes theapproximate cost function developed inSection 4.

(Yp, sp, Sp) 5 optimal solution that minimizes the exactcost function developed in Section 3 for thePPA system; it was obtained by a searchusing (Ya, sa, Sa) as a starting point.

(Yc, sc, Sc) 5 optimal solution that minimizes the long-run average cost of the CPA system; thelong-run average cost was determined viasimulation, and the optimal solution wasobtained through a search with (Yp, sp, Sp)as a starting point.

Cp(Y, s, S) 5 long-run average cost for the PPA systemfor a given (Y, s, S) policy (same as c(Y, s,S), the subscript is added for easycomparison).

Cc(Y, s, S) 5 simulated cost for the CPA system for agiven (Y, s, S) policy.


Note that the search for (Yp, sp, Sp) and (Yc, sc, Sc) wasperformed over all the three control parameters Y, s, andS. The range of each parameter was set sufficiently wide toensure that the optimal policy was found. To evaluate apolicy in the CPA system, we simulated 50,000 periods.The 95% confidence half-intervals were all less than 1% ofthe average values.

For each example we made the following fourcomparisons:

(1) Cp(Ya, sa, Sa)/Cp(Yp, sp, Sp) 2 1,(2) Cc(Ya, sa, Sa)/Cc(Yc, sc, Sc) 2 1,(3) Cc(Yp, sp, Sp)/Cc(Yc, sc, Sc) 2 1,(4) Cp(Yp, sp, Sp)/Cc(Yc, sc, Sc) 2 1,

and their histograms are in Figures 5 through 8, respec-tively. The numerical results suggest that (i) the heuristicsolution based on the approximate cost function is effec-tive for both the PPA and the CPA system; (ii) (Yp, sp, Sp)is close to optimal for the CPA system, and it is better thanthe heuristic solution; and (iii) CPA is slightly better thanPPA on average, but there is no dominance.

We also found that Yp ¶ Y0 and Yc ¶ Y0 for all theexamples, where Y0 is the minimum point of g[. We can

interpret Y0 as the ideal base-stock level for the retailers.By having a less-than-ideal base-stock level at the retailers,more inventory can be centralized at the warehouse. Thenumerical examples suggest that this is in fact optimal!This can be attributed to the risk-pooling function of thewarehouse.

Note that the different solutions have very differentcomputational requirements. The heuristic solution is ex-tremely easy to obtain with an average computing time ofabout 1 second across all examples. (All the computationswere carried out on a Pentium 200 personal computer.)The heuristic solution is used as a starting point in thesearch for the optimal solutions for both the PPA and theCPA systems. The optimal solution for the PPA system isstill easy to find if the search range is relatively small; theaverage time is about 7.2 seconds across all examples. (Thesearch range increases as the setup cost and/or the averagedemand increases.) The computational effort for findingthe optimal solution for the CPA system is significantlyhigher than that for the other two solutions due to simula-tion; the average time is about 12 minutes across allexamples.

We also looked at the changes in computing times dueto changes in problem parameters. We found that for fixed

Figure 5. Performance of heuristic solution in PPA sys-tem.

Figure 6. Performance of heuristic solution in CPA sys-tem.

Figure 7. Performance of PPA-optimal solution inCPA system.

Figure 8. Comparison between PPA and CPA.


demand distributions, the computing times are most sensi-tive to N, the number of retailers. For the heuristic (resp.,PPA) solution method, doubling N results in a 15% (resp.,80%) increase in computing times on average. The re-maining parameters have negligible effects on the comput-ing times. The spread of the demand distribution seems tohave a significant impact on computing times: the largerthe spread, the larger the search range for both the heuris-tic solution method and the PPA solution method.

Finally, we used certain techniques to enhance compu-tational efficiency. First, the use of the negative binomialdemand distribution simplifies convolutions. Second, forthe PPA solution method, some intermediate results canbe stored and used repeatedly. More specifically, the fol-lowing functions remain unchanged as we search for theoptimal control parameters (Y, s, S): the renewal densitym[, the renewal function M[, the retailer holding-backorder cost function g[, and the U( z , z ) function.

6. CONCLUDING REMARKS

As we mentioned in the introduction, sometimes it may bemore reasonable for the retailers to order in groups. Ifevery group has the same number of retailers (like in theNorton case), then our exact evaluation procedure (for thePPA system) still holds with minor modifications. Let N bethe number of groups, and M the number of retailers ineach group. (So there are a total of NM retailers in thesystem.) Everything previously associated with a retailer isnow associated with a group, e.g., IPn(t) is now the inven-tory position of group n, which is the sum of the individualinventory positions at the retailers in the group. The onlyexception is D, which still represents the demand at oneretailer in a single period. Now suppose group n is sched-uled to order in period t, and let IPn(t) 5 y. Define

ĝ~ y! 5 min Oi51

M

g~ y i !

s.t. Oi51

M

y i 5 y.

Under the Allocation Assumption (see, e.g., Eppen andSchrage 1981), ĝ( y) represents the expected holding andbackorder costs in group n from period t 1 l to t 1 l 1N 2 1. By simply replacing g[ with ĝ[ in §3, we have theexact evaluation procedure for the above group-orderingcase. The approximation developed in §4 can be modifiedsimilarly.

We conclude this paper by suggesting several topics forfuture research. A critical assumption of this paper is thatthe retailers are identical. This assumption, while commonin the literature, is restrictive. How should our results bemodified when the retailers are significantly different? Oneissue raised in the Norton case is the issue of emergencyshipments from the warehouse to the retailers. The com-pany is considering using Federal Express to expeditesome shipments. What is the impact of this? Also in the

case are some different ways to measure customer serviceat the retail level. How to incorporate these into themodel?

APPENDIX

An alternative explanation of the equivalence of the twotechniques for grouping the retailer holding and backordercosts is as follows:

1. At any period t, there is exactly one retailer that wasreplenished by the warehouse (i.e., receiving a ship-ment) i periods ago; i 5 0, . . . , N 2 1.

2. The probability distribution of the inventory level at aretailer i periods after it is replenished by the ware-house is identical for all retailers (since the retailersare identical). Hence the expected holding and back-order costs at a retailer in the ith period after replen-ishment (by the warehouse) are identical for allretailers.

3. Therefore, instead of charging the expected holdingand backorder costs at all the retailers in period t, wecharge the expected holding and backorder costs atone retailer in periods t, t 1 1, . . . , t 1 N 2 1.

In terms of Figure 2, the above arguments show that insteady state, the expected holding and backorder costs atretailer 1 in periods t, t 1 1, and t 1 2 are equal to theexpected holding and backorder costs at all the retailers inperiod t. However, the above explanation does require thatthe retailers be identical.

ACKNOWLEDGMENT

The authors are grateful to Paul Zipkin for bringing theNorton Auto Supply case to their attention, and to JanHammond for discussions on the case. They would alsolike to thank the associate editor and the two anonymousreferees for their comments and suggestions, which haveled to significant improvements in the exposition of thepaper.

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