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An easily implementable hierarchical heuristicfor a two-echelon spare parts distribution system
WALLACE J. HOPP1, RACHEL Q. ZHANG2 and MARK L. SPEARMAN3
1Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL 60208, USAE-mail: [email protected] of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI 48109, USAE-mail: [email protected] of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USAE-mail: [email protected]
Received September 1997 and accepted October 1998
This paper addresses a two-echelon spare parts stocking and distribution system consisting of a central Distribution Center (DC)and regional facilities. Because the primary purpose for holding inventory is to provide timely repairs of customer's equipment, weset as our objective to minimize total inventory investment subject to constraints on the delay due to parts outages. We decomposethe resulting problem by level and by facility. By simplifying the expressions for the delay constraints and applying previouslydeveloped heuristics for the single-level problem [1], we are able to derive closed-form expressions for the inventory controlparameters. We then develop a search algorithm (on DC ®ll rate) to approximate the parameters (Lagrange multipliers) in theclosed-form expressions. Numerical comparisons against an analytic lower bound and, for small problems, exact solutions showthe approximation to be quite accurate. We also found that it outperforms methods currently in use by the ®rm that motivated thiswork. Finally, because it yields closed-form expressions for inventory control parameters and the parameters are only updatedperiodically, the policy is ``easily implementable'' once suitable Lagrange multipliers have been computed.
1. Motivation and background
This research was motivated by a manufacturer of mailprocessing equipment, which o�ers service contracts onits equipment and stocks spare parts (there are almost30 000 distinct part numbers) to support this mainte-nance function in four types of locations:
(1) At the main Distribution Center (DC) located inthe Chicago area.
(2) At approximately 60 facilities across the country.(3) At customer sites whose service contracts guaran-
tee on-site inventory.(4) With customer representatives (who generally work
out of a facility, but sometimes also carry inventoryin their service vans).
For purposes of analysis, we regard customer rep-resentative inventories as part of facility inventory.Furthermore, because relatively few customers haveon-site inventory, and only for a few part numbers, wedo not explicitly treat these inventories either. There-fore, the basic problem is to manage the two-echeloninventory system involving the DC and facilities. Atthe time this research project began, total inventory
was divided roughly equally between the DC and thefacilities.Regardless of where they are held, the reason for spare
parts inventories is to support customer service contractsthat guarantee outages due to failures will not exceed acertain number of hours per month. Thus it is essentialthat the likelihood of a part being out of stock whenrequired be kept low. However, because inventory is ex-pensive and can become obsolete as equipment modelschange, the ®rm does not want to hold excessive amountsof stock. Therefore, understanding the inventory versuscustomer service tradeo� and striking a reasonable bal-ance were the ®rm's primary concern and the mainmotivation for this research.In support of this objective, the ®rm maintains several
levels of data for each part number, including
(1) Demand rate (units per year).(2) Lead time (days) required to replenish inventory.(3) Price (dollars) of the item.(4) Shipping times (hours) from the distribution center
to the facilities and from the facilities to the sites.
The policies in use by the ®rm at the time we began thisresearch were generated in an ad hoc fashion and had no
0740-817X Ó 1999 ``IIE''
IIE Transactions (1999) 31, 977±988
explicit link to cost, replenishment lead time, shippingtime, and customer service. Although probably ine�ec-tive, this approach had the bene®t of being very simple toimplement in their mainframe-based information system.Without substantial system and cultural changes, a large-scale optimization procedure was simply not feasible onour client's system. We suspect this is the case in manyother real-world systems as well. Therefore, we took it asa constraint that anything we proposed be easily imple-mentable, in the sense that
(1) The policies do not depend on di�cult-to-estimateparameters (e.g., ordering cost, backorder cost).
(2) The policies lead to closed-form expressions forstocking parameters that can be computed using asimple program (e.g., spreadsheet).
(3) Small changes in the list of part numbers beingmanaged do not require re-running an algorithm toadjust the control parameters. Of course, as we willsee, the parts must be considered together for thepurposes of computing constraints (i.e., Lagrangemultipliers in the closed-form expressions). How-ever, as long as this computation can be done outsidethe system and updated infrequently, we considerit to be within the spirit of easily implementable.
The environment in which this tradeo� is faced is afairly standard continuous review system. The charac-teristics of this system make it amenable to variants of thewell-known �Q; r� model (where an order of ®xed quan-tity Q is placed as soon as the inventory position drops toa ®xed reorder point r). The challenge is to make ap-propriate use of available data (i.e., demand rate, leadtime, cost, and shipping times) to identify e�ective andeasily implementable policies for the multi-level probleminvolving the DC and facilities.
2. Literature review
Two-echelon inventory systems have been widely studiedin the literature (see Nahmias and Smith [2] and Fed-ergruen [3] for surveys). One of the simplest approaches iscalled the METRIC (Multi-Echelon Technique for Re-coverable Item Control) developed by Sherbrooke [4].This model assumes �S ÿ 1; S� continuous review policiesat facilities (i.e., order up to S whenever inventory fallsbelow S) and that all stocks are conserved, so there is nosystem reordering once initial stocks are established. Theobjective of the model is to determine the policies at fa-cilities and the DC that minimize total expected base levelbackorders subject to an investment constraint. The re-sulting expression for the stationary distribution ofbackorders is approximated by the Poisson and used toshow that system performance depends on mean transittimes. For this system, Graves [5] determined exactexpressions for both the average and variance of back-
orders at facilities, and approximated the distribution bya negative binomial. Sherbrooke [6] considered the com-bined case of multi-indenture, multi-echelon systems.Simon [7] obtained exact expressions for the steady-statedistribution of backorders allowing condemnations. Ax-saÈ ter [8] presented an approach using an inventory costfunction that re¯ects costs incurred on an average unit.These cost expressions can be used for e�cient determi-nation of an optimal one-for-one replenishment policy bya recursive procedure. Svoronos and Zipkin [9] consid-ered a more general system than the METRIC-typemodel, with stochastic transportation times that aregenerated exogenously and showed that transit-timevariance can cause large backorder levels.For high cost and low demand items, the �S ÿ 1; S�
policy is often appropriate. In general, however, thisone-for-one replenishment may not be optimal. Forrepairable items, Lee and Moinzadeh [10] derived a verye�ective two-parameter approximation to the distributionof backorders when the facilities employ a batch orderingpolicy. Moinzadeh and Lee [11] used regression tech-niques and one-pass search procedures to determine theoptimal batch size and stocking levels, respectively, inthese systems. Lee and Moinzadeh [12] proposed an ap-proximation scheme for the distribution of backorders atfacilities for di�erent repair-time distributions in systemswhere items can either be repaired or condemned on thebasis of extensive simulation results. For consumableitems, Muckstadt [13], derived the steady-state distribu-tion for backorders at each facility for two systems: twoidentical facilities and a ``large'' number of identical fa-cilities (where ``large'' means a number su�cient to en-sure that the depot demand process is approximatelyPoisson). Deuermeyer and Schwarz [14] presented andtested an analytical model that generalizes the exact singleechelon �Q; r� model to estimate the expected ®ll rate andbackorder level for the system where external demand isPoisson at facilities, all the facilities are identical, and�Q; r� policies are used. Simulation showed the model'sestimates to closely approximate the empirically observed®ll rate of the system. Svoronos and Zipkin [15] proposeda mixture of two translated Poisson distributions to ap-proximate the steady-state distribution of lead time de-mand at both DC and facilities in cases with identicalfacilities. They compared their model numerically againstthose of Deuermeyer and Schwarz [14] and Lee andMoinzadeh [10,12] and all three models against estimatescalculated through simulation. They found their model toperform substantially better than the other models.All of the above models focus only on single product,
®xed order quantities at facilities, and identical facilities.Some heuristic models have been developed to determineinventory policies in multi-echelon systems in real in-dustrial settings, including Rosenbaum [16] for EastmanKodak Company, Kostic and Pendic [17] for a multilevelmaintenance system, and Cohen et al. [18] for IBM. The
978 Hopp et al.
®rst two models treat items independently and hence donot capture interactions between items (i.e., in terms oftheir e�ect on service). The third model does considerinteractions between items by decomposing the modelinto three stages. The decomposition starts with thelowest echelon where demand occurs and passes up to thenext level. At each level, it is assumed that there is alwaysan ample supply at the replenishment sources (i.e., theservice requirement at all sites is high). Hausman andErkip [19] explored an improved single-echelon model forlow-demand, high cost items controlled on a base stockbasis and applied a search procedure for service at eachlocation to improve the performance of the model. Erkipet al. [20] consider a model to allow demands to be cor-related both across facilities and also correlated in time.Hopp et al. [1] proposed procedures for computing an
easily implementable policy in a single level system byformulating the problem as a constrained optimizationproblem to minimize inventory investment subject toconstraints on order frequency and service level. By sim-plifying representations of the inventory and service levelexpressions, they were able to derive closed-form expres-sions for the inventory control parameters. Numericalcomparisons showed that even the simplest heuristic workswell under the condition of high service level requirement.Amore sophisticated, andmore complex, heuristic is morerobustly accurate. Tests on industrial data indicate thatthis method also outperforms techniques currently in useby the ®rm. In this paper, we extend the work by Hoppet al. [1] to a two-echelon inventory system.
3. Problem formulation
Our previous paper modeled a single-echelon system asan optimization problem to minimize inventory invest-ment subject to a service constraint [1]. This formulationis not directly applicable to the two-echelon system be-cause, among other di�erences, customers are notpromised a service level; they are promised a limit on theaverage number of hours per year of downtime due torepairs. Therefore, to more closely match the servicecontracts the ®rm o�ers its customers, we formulate theproblem at the facility level as an optimization problemto minimize inventory investment subject to ensuring thatthe average total delay due to failures is below a speci®edlevel. Verbally, this formulation can be expressed as:
Minimize Annual inventory investment �1�Subject to:
Average order frequency per year per item at the DC� F
�2�Average total delay at facilitym per year�Tm; m� 1; . . . ;M
�3�
where F is the target order frequency at the DC, Tm is thetotal delay per year allowed at facility m, and M is thenumber of facilities. Tm must be established consistentlywith the service contracts between facility m and itscustomers. Since these contracts address downtime dueto repairs, not just part shortages, the total number ofhours of delay, Tm, must account for this. Also, since inany given year the delay will be distributed unevenlyamong customers, it makes sense to set this target belowthat allowed by the service contracts. While this isn'tan ideal representation of the service contracts, it ismuch closer than a conventional ®ll rate constraint, andcan therefore be expected to generate more reasonablesolutions.
3.1. Assumptions and notation
To develop a practical model, we make use of the fol-lowing assumptions:
· Demand at each facility is Poisson and the pro-curement lead times at the DC are constant. Becausedemand seen by the facilities is the superposition ofindependent demand processes from many customersites and we had no information whatever concern-ing variance of demand, we chose to model thedemand processes as Poisson. Estimates of procure-ment lead times were available in our client's database, but they were far from reliable. For modelingpurposes, we assumed that these lead times were®xed, but we reviewed and revised the data to adjustwhat we and our client agreed were overly optimisticlead times.
· Because we are considering repair parts that are ul-timately needed by the customer, we assume thatdemands that cannot be ®lled from stock are back-ordered. While an occasional demand is satis®edfrom outside the system (e.g., from a hardwarestore), most customer demands are backordered (i.e.,a temporary repair is eventually done properly andrequires the part).
· Each part replacement represents a separate repairincident. That is, if several di�erent parts cause de-lays, we assume the total delay is given by the sum ofthe individual delays. In reality, it may be the casethat some of the delays are overlapping (e.g., twoout-of-stock parts are required for the same repair).However, modeling multi-part repair incidents iscomplex and requires much more detailed informa-tion about the correlation between parts usage thanis currently collected by the ®rm. Moreover, sincetreating part demands as separate repair incidents isconservative (i.e., it overestimates total delay andhence will push the model toward better-than-expected customer service), we felt this to be a rea-sonable assumption for practical purposes.
Heuristic for a two-echelon spare parts distribution system 979
· The DC makes use of a continuous review �Q; r�policy, while the facilities use base stock policies (i.e.,Q � 1) for procurement of units from the DC. The�Q; r� model is an accurate representation of the waythe DC was operating at the time this study wasinitiated. The base stock model is a good approxi-mation of facility behavior because the DC makesshipments to the facilities on a frequent (i.e., almostdaily) basis, so it is reasonable to assume one-at-a-time replenishment.
· The facilities are resupplied only from the DC. Thatis, lateral supply among the facilities is not allowed.In practice, some lateral shipments do occur, al-though on an informal basis. We ignore these forplanning purposes in order to be conservative. Usingsuch practices in the ®eld may permit the system toslightly outperform model predictions. Again, thistype of conservativeness is probably a good thing,since there are undoubtedly factors that degrade thereal system (e.g., lost inventory, data entry errors,etc.) that are not addressed by the model.
We summarize below the notation needed to make ourformulation precise:
General notation
N = number of items;ci = cost of item i ($).
Decision variables
Qi0 = order quantity for item i at the DC;
ri0 = reorder point for item i at the DC;
Sim = base stock level for item i at facility m.
Facility notation
M = number of facilities;Tm = maximum allowable total delay at facility
m (yrs/yr);dm = shipping time for parts from the DC to
facility m (yrs);ki
m = expected annual demand for item i at fa-cility m (parts/yr);
lim = expected replenishment lead time for item
i at facility m (yrs);hi
m = expected lead time demand for item i atfacility m (parts);
Bim�Si
m�= expected number of backorders for item iat facility m at any point in time (parts);
him�Si
m�= Sim ÿ hi
m � Bim�Si
m� = expected on handinventory of item i at facility m at anypoint in time (parts).
DC notation
ki0 =
PMm�1 ki
m=expected annual demand foritem i at the DC (parts/yr);
li0 = replenishment lead time for item i at the
DC (yrs);
hi0 = ki
0li0 = expected demand for item i at the
DC during lead time li0 (parts);
Ai0�ri
0;Qi0�= probability of stockout for item i at the
DC;Bi0�ri
0;Qi0�= expected number of backorders for item i
at the DC at any point in time (parts);si0�ri
0;Qi0�= 1ÿ Ai
0�ri0;Q
i0� = service level of item i at
the DC;hi0�ri
0;Qi0�= ri
0 ÿ hi0 �Qi
0=2� 1=2� Bi0�ri
0;Qi0� � expec-
ted on hand inventory of item i at the DCat any point in time (parts).
The quantities Ai0�ri
0;Qi0� and Bi
0�ri0;Q
i0� can be computed
using the well-known results of Hadley and Whitin [21].See Appendix A for details.
3.2. Model description
With this notation we can derive a mathematical for-mulation. Note that Bi
m���, the expected number ofbackorders for item i at facility m at any point in time, isalso the expected unit years of shortage incurred per yearfor item i at facility m, i.e., the expected total time delayseen by the customers for item i at facility m.We can now formulate the combined DC and facility
problem to minimize total inventory investment in thesystem subject to a constraint on order frequency at theDC and constraints on the delays seen by customers atthe facilities as:
MinimizeXN
i�1ci hi
0 ri0;Q
i0
ÿ ��XMm�1
him Si
m
ÿ �( ); �4�
Subject to:
1
N
XN
i�1
ki0
Qi0
� F ; �5�
XN
i�1Bi
m�Sim� � Tm; m � 1; . . . ;M ; �6�
ri0 � ri
0; Qi0 � 1; i � 1; . . . ;N ; �7�
Sim � Si
m; i � 1; . . . ;N ; m � 1; . . . ;M ; �8�ri0;Q
i0; S
im : integer; i � 1; . . . ;N ; m � 1; . . . ;M : �9�
Note that we have added constraints on the decisionvariables to prevent unreasonable outcomes (e.g., someparts having very low ®ll rates). Of course, this is notnecessary, but we found it useful in adjusting the solu-tions to meet the needs of the ®rm that motivated thisresearch project. In that case we chose ri
0 � ÿ1 andSi
m � 0.The average backorders at the facilities, Bi
m�Sim�, can be
computed using the method suggested by AxsaÈ ter [8].However, this recursive procedure is too complex to beeasily implementable, so we seek approximations. Thesimplest way to approximate Bi
m�Sim� is to replace the
980 Hopp et al.
replenishment lead times at facilities by their expectationsas suggested by many researchers [6,14]. While this is verysimple, it is easy to show that it consistently causesbackorders to be understated, and in some cases thediscrepancy can be large. So, instead we estimate thevariance of the lead times and approximate the lead timedemand distribution at the facilities with a negative bi-nomial distribution (a two-parameter approximation)suggested by Graves [5] and Svoronos and Zipkin [15].See Appendix A for details.
4. DC-facility heuristic
Even with an approximation for Bim�Si
m�, (4)±(9) repre-sents a large-scale, nonlinear, discrete optimizationproblem. Hence, there is no practical exact solutionmethod and our goal is to ®nd a good approximation. Inthis section, we ®rst decompose the problem by level andfacility. We then develop closed-form expressions for thecontrol parameters at the facilities and DC, and a hier-archical heuristic to ®nd the parameters in closed-forms.Note that this type of procedure is widely used and mayrequire an extensive search routine to ®nd a good solu-tion. However, as we mentioned earlier, as long as thesearch is done outside the system and only periodically,we consider it within the spirit of easily implementable.
4.1. Facility heuristic
To develop a heuristic, we ®rst determine policies for thefacilities assuming we know the policy at the DC (i.e., weknow Qi
0 and ri0 for all i). Then problem (4)±(9) becomes:
MinimizeXN
i�1ci
XMm�1
him Si
m
ÿ �( ); �10�
Subject to:
XN
i�1Bi
m Sim
ÿ � � Tm; m � 1; . . . ;M ; �11�
Sim � Si
m; i � 1; . . . ;N ; �12�Si
m : integer; i � 1; . . . ;N : �13�Solving problem (10)±(13) is equivalent to solving the
following M subproblems:
MinimizeXN
i�1cihi
m Sim
ÿ �; �14�
Subject to: XN
i�1Bi
m Sim
ÿ � � Tm; �15�
Sim � Si
m; i � 1; . . . ;N ; �16�
Sim : integer; i � 1; . . . ;N : �17�
That is, it is equivalent to solving M single facilityproblems.We can derive simple formulas for Si
m by using thefollowing approximations:
(1) Inventory for item i is given by him�Si
m� � Sim ÿ hi
m.(2) Demand for item i during lead time is approxi-
mated by the normal distribution to match the twomoments of the demand distribution.
(3) Backorder level for item i is given by
Bim Si
m
ÿ � � Z1Si
m�1=2
xÿ Sim ÿ 1=2
ÿ �dU
xÿ him
rim
� �;
where rim is the standard deviation of the demand dis-
tribution. (We have added the 1=2 term because demandis treated as continuous and it gives better approxima-tion).Using these approximations and di�erentiating the
Lagrangian yield:
Sim � hi
m � 0:5� rimUÿ1 1ÿ �ci=gm�� � if ci � gm;
Sim else,
��18�
where gm, m � 1; . . . ;M , is the Lagrange multiplier. Notethat the inverse cdf of the normal is included in mostspreadsheets or can easily be calculated using very accu-rate polynomial approximations [22].To approximate gm, we adjust gm, 1 � m � M , and
compute Sim using (18) to achieve the highest delay that
satis®es the maximum time delay constraint. In order toensure a feasible solution to the original problem, we usethe exact expression for delay at the facilities (not thesimpli®ed version) when searching for appropriateLagrange multipliers.
4.2. DC heuristics
To ®nd a policy for the DC, we form a single levelproblem that minimizes the total inventory investment atthe DC subject to constraints on average order frequencyand service level:
MinimizeXN
i�1cihi
0 ri0;Q
i0
ÿ �; �19�
Subject to:
1
N
XN
i�1
ki0
Qi0
� F ; �20�
1
K
XN
i�1ki0si
0 ri0;Q
i0
ÿ � � S; �21�
Heuristic for a two-echelon spare parts distribution system 981
ri0 � ri
0;Qi0 � 1; i � 1; . . . ;N ; �22�
ri0;Q
i0 : integer; i � 1; . . . ;N : �23�
where S is a speci®ed service level and K �PNi�1 ki
0. ThisDC problem can be easily solved using any of the heu-ristics presented in our previous paper [1] for any given S,0 < S < 1. Basic assumptions and a summary of the re-sulting closed-form expressions for the various modelsare given in Appendix B. The following simple one-dimensional search routine is used to ®nd the parametersin the closed-form expressions for Qi
0 and ri0. Compute Qi
0
using (A1) and the Lagrange multiplier m to achieve thehighest order frequency that satis®es the order frequencyconstraint (20). Compute ri
0 (using (A2) or (A3)) and theLagrange multiplier l to achieve the lowest service levelthat satis®es the service constraint (21). Note that theresulting solution is feasible to problem (19)±(23).
4.3. DC-facility heuristic
We can now state a heuristic for the combined DC-Fa-cility problem as follows:
AlgorithmStep 1. Compute order quantities Qi
0, i � 1; . . . ;N , and mfor a given F using (A1) in Appendix B:
Qi0 � max
���������2mki
0
ciN
s; 1
8<:9=;; i � 1; 2; . . . ;N ;
where m can be computed through the simpleone-dimensional search routine described inSection 4.2.
Step 2. For any given S,(a) Solve for ri
0 and l using (A2) or (A3) inAppendix B and the heuristic procedure de-scribed in Section 4.2. Compute hi
0�ri0;Q
i0�,
i � 1; . . . ;N .(b) Solve the M facility problems (14)±(17) to
®nd Sim and gm using (18) and the search
procedure described in Section 4.1. Computehi
m�Sim�, m � 1; . . . ;M ;
(c) Compute total inventory investment:
total inv �XN
i�1ci hi
0�ri0;Q
i0� �
XMm�1
him�Si
m�( )
:
Because, as pointed out by Moinzadeh and Lee [11], totalinventory investment is in general not a convex functionof service level S, we rely on direct search, rather than amore elegant procedure, to ®nd the value of S that min-imizes total inventory investment. Therefore, we repeatSteps 2(a)±(c) and ®nd the parameters l and gm,m � 1; 2; . . . ;M , in the closed-form expressions (l in (A2)
or (A3), and gm in (18)) that achieve the lowest overallinventory investment.
5. A lower bounding procedure for ®xed Qi0
To evaluate the performance of the above heuristic, it isuseful to be able to compute a lower bound on the optimalcost. Recall that in the DC model, we actually determineorder quantities and reorder points separately (i.e., weobtain order quantities ®rst and then compute reorderpoints given the order quantities). Our numerical tests [1],as well as the theoretical results of Zheng [23], suggest thatthe heuristic order quantities do not introduce inordinateerror into the solution. Therefore, to facilitate a tractablelower bound, we will take the order quantities at the DC,Qi
0, i � 1; . . . ;N , as given and establish a lower bound ontotal cost under this restriction. The lower boundingprocedure is presented in Appendix C.
6. Performance evaluation
To test the performance of our combined DC-facilityheuristic, we consider examples with ten items, two fa-cilities (Case 1±Case 4) and two items, ®ve facilities (Case5±Case 7) in Tables 1, 2 and 3. These examples cover arange of possible scenarios at the facilities, where Cases 1and 5 have identical facilities, Cases 2 and 6 have varyingshipping times, Case 3 has unbalanced demand betweenfacilities with the unbalance proportional across items,Case 4 has unbalanced demand between facilities with the
Table 1. DC data sets for cases 1±7
Cases 1±4 Cases 5±7
i 1 2 3 4 5 6 7 8 9 10 1 2
ci 20 18 16 14 12 10 8 6 4 2 10 5ki0 4 8 12 16 20 24 28 32 36 40 20 40
li0 100 90 80 70 60 50 40 30 20 10 100 50
Qi0 2 2 3 4 4 5 6 8 10 15 5 10
Table 2. Facility data sets for cases 1±4
m k1m k2m k3m k4m k5m k6m k7m k8m k9m k10m dm �hours�Case 1 1 2 4 6 8 10 12 14 16 18 20 5
2 2 4 6 8 10 12 14 16 18 20 5Case 2 1 2 4 6 8 10 12 14 16 18 20 5
2 2 4 6 8 10 12 14 16 18 20 24Case 3 1 3 6 9 12 15 18 21 24 27 30 5
2 1 2 3 4 5 6 7 8 9 10 5Case 4 1 3 6 9 12 15 6 7 8 9 10 5
2 1 2 3 4 5 18 21 24 27 30 5
982 Hopp et al.
unbalance disproportionate across items, and Case 7 hasvarying demand and shipping times.We set F � 4 for all cases and compute the order
quantities, Qi0. The resulting values of Qi
0 are reported inTable 1 with f � 3:8733 in Cases 1±4 and f � 4 in Cases5±7, where f denotes the actual order frequency achievedat the DC. Using these order quantities, we compute thereorder points for the DC (using Hybrid formula) andfacilities for di�erent values of the delay constraints. Inall tests, we assume ri
0 � ÿ1 and Sim � 0. Since the total
delay allowed at each facility depends on the expecteddemand, it makes sense to set the target delay for eachfacility based on the demand rate (e.g., if we consider thetarget delay to be 0:5 hours per demand at facilities, weactually set the total target delay T1 � 1� 165 � 165hours or 165=24=365 years of delay per year at Facility 1in Case 3 since the total expected demand at Facility 1 is165 per year, while T2 � 1� 55 � 55 hours at Facility 2since the demand rate is 55 per year). To simplify com-parisons, we report the results based on average delay(total delay is simply the average delay times the demandrate and is thus equivalent). Since we are dealing withsmall delays, we report delays in hours instead of years.We let avg-T denote the average target delay at facilities.For avg-T � 0:5; 1; 2; 3 hours (we convert hours into
years in our computation), the test results are shown inTable 4, where tm denotes the actual delay (hours) perdemand at facility m, S denotes the average service levelrequired at the DC, and inv represents the total inventoryinvestment. Since the target delays are not achieved ex-actly due to integrality, we compute lower bounds basedon both target delay (LB (unadjusted)) and the actualdelay (LB (adjusted)). We feel that LB (adjusted) is themore appropriate measure of the accuracy of a heuristic,since even an exact algorithm could show large gaps with
LB (unadjusted) due to integrality of the decision vari-ables.The main observation we can make about these test
results is that errors relative to the adjusted lower boundsare less than 5%, while errors relative to the unadjustedlower bound are less than 17%. Hence, to the extent thatthe heuristic for computing Qi
0 and the approximation offacility level service are accurate, our heuristic for com-puting inventory policies in these two echelon systems isquite accurate.Finally, to test the overall accuracy of the heuristic (i.e.,
including errors due to order quantities), we consider foursmall examples (Case 8±Case 11) with two items and twofacilities shown in Table 5. Supposing F � 4 and avg-T� 1 hour, we compare the solutions yielded by the heu-ristic against the optimal solutions determined via com-plete enumeration. The results are shown in Table 6. Intwo of the four cases, our heuristic found the optimalorder quantities for the DC. The largest error is 4:6% andin one of the cases the heuristic found the optimal solu-tion. These results give further support to the accuracy ofour method.Finally, to provide a more realistic test of our heuristic,
we used data from our client and compared its perfor-mance to that of the policy currently in use. Since we hadcomplete data for the DC but not all of the facilities, wefocused on two facilities with a total of 1263 part num-bers and shipping time dm � 24 hours (i.e., overnightdelivery) for m � 1; 2.For this example, we computed the inventory invest-
ment based on current policies in use at the DC and thefacilities. The current policies resulted in total delay atFacilities 1 and 2 of 7980 and 7763 hours per year. Wetook these as our target delay and used current orderfrequency at the DC as the target order frequency in theheuristic. The results shown in Table 7 are the total in-ventory investment (in dollars), the average service leveland order frequency at the DC, and the total delay ateach facility. Our heuristic results in a 36.63% reductionin inventory investment, which is worth at least $66 216dollars for the two facilities. Note that our policy has alsohigher service at the DC than the current policy andachieves lower delay at the facilities. Multiplying thesesavings by 30 (i.e., to approximate the savings over a 60facility system) results in a total savings of approximately$1986 000. Since this does not completely consider the
Table 4. Solutions to cases 1±7 using hybrid heuristic
Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7avg-T Unadj Adj Unadj Adj Unadj Adj Unadj Adj Unadj Adj Unadj Adj Unadj Adj
0.5 opt., 0.71 1.73, 2.84 1.53, 3.49 2.58, 3.90 opt., 0.23 opt., 4.77 0.06, 9.811 0.00, 0.99 opt., 2.81 2.56, 4.28 1.96, 3.37 opt., 0.09 opt., 4.41 1.30, 7.972 opt., 0.82 1.48, 3.57 2.87, 5.74 4.21, 5.23 opt., 3.45 opt., 9.97 0.29, 16.473 opt., 6.20 0.83, 3.77 4.51, 5.43 3.79, 6.23 2.68, 8.86 opt., 4.07 opt., 5.57
Table 3. Facility data sets for cases 5±7
Case 5 Case 6 Case 7
m k1m k2m dm k1m k2m dm k1m k2m dm
1 4 8 5 4 8 5 10 2 52 4 8 5 4 8 10 7 2 103 4 8 5 4 8 15 1 8 154 4 8 5 4 8 20 1 12 205 4 8 5 4 8 24 1 16 24
Heuristic for a two-echelon spare parts distribution system 983
joint coverage e�ects of holding inventory at the DC, wewould expect the actual savings from applying the heu-ristic to the 60 facility system directly to be greater thanthis.Further work needs to be done to implement this
method in the actual system. The memory requirementsfor the input data for the 60 facility system are consid-erable. In practice, therefore, it may make sense to simply®x the service level at the DC (e.g., at 98:86% on the basisof test runs like that in Table 7) and solve the facilitiesseparately. This would make the facility computationseasily manageable on PC's.
7. Conclusions
We have developed an approach for computing inventorycontrol policies for a two-echelon spare parts distributionsystem. Our formulation directly addresses the concern
for customer service by stating constraints at the facilitiesin terms of expected delay due to parts outages. By sim-plifying these constraints and inventory expressions, weare able to apply previously developed closed-form ex-pressions for computing policies at the DC and developclosed-form expressions for computing reorder points atfacilities. Once a service level has been speci®ed for thedistribution center, our formulation decomposes by leveland by facility, yielding simple sub-problems. We usedthis decomposition to develop a search algorithm to ®nda system-wide policy that meets customer service con-straints with minimal inventory investment.The heuristic resulting from solution of our approxi-
mate optimization problem is easily implementable in thesense that it is stated in terms of closed-form expressionsfor the reorder points and order quantities. Because theLagrange multipliers in these expressions are relativelyinsensitive to small changes in the list of part numbers inlarge systems (see discussions in Hopp et al. [1]), policiesfor newly introduced parts can be computed withoutre-running the algorithm (i.e., by simply using theclosed-form reorder point and order quantity expres-sions). More extensive changes might require re-solvingthe individual subproblems (i.e., provided DC servicelevel computed by the initial search is still valid).
Table 6. Optimal and hybrid heuristic solutions for cases 8±11
inv f t1 t2 i Qi0 ri
0 Si1 Si
2
Optimal solutionsCase 8 217.4032 3.25 0.8778 0.8778 1 5 4 2 2
2 4 5 1 1Case 9 217.1997 3.25 0.8714 0.9381 1 5 4 2 2
2 4 5 1 1Case 10 217.3896 3.6667 0.7621 0.9673 1 5 4 2 1
2 3 5 1 2Case 11 226.1608 3.6667 0.2454 0.7731 1 5 5 2 1
2 3 5 1 2Heuristic solutionsCase 8 217.4088 3.6667 0.8560 0.8560 1 5 5 2 2
(0.00%) 2 3 5 1 1Case 9 217.2069 3.6667 0.9228 0.9332 1 5 6 1 2
(0.00%) 2 3 5 1 1Case 10 217.3882 3.6667 0.5269 0.7075 1 5 5 2 1
(4.6%) 2 3 4 1 3Case 11 226.1608 3.6667 0.2454 0.7731 1 5 5 2 1
(opt.) 2 3 5 1 2
Table 5. DC and facility data sets for cases 8±11
DC Facility 1 Facility 2
Partnumber
ci li
(days)ki1 d1
(hours)ki2 d2
(hours)
Case 8 1 10 50 10 10 10 102 20 100 5 10 5 10
Case 9 1 10 50 10 5 10 242 20 100 5 5 5 24
Case 10 1 10 50 15 10 5 102 20 100 2 10 8 10
Case 11 1 10 50 15 24 5 482 20 100 2 24 8 48
Table 7. Comparison of existing policy with the heuristic
Inventoryinvestment
T1 T2 F S
Currentpolicy
180 753 7930.2779 7762.6237 2.4136 0.9376
Heuristic 114 537 7716.2183 7690.6309 2.4118 0.9886
984 Hopp et al.
Finally, truly major changes may require re-solving theentire problem to update the Lagrange multipliers. Sincethis need be done only infrequently (e.g., annually) andcan be done o�ine, this approach meets our goal toprovide an easily implementable solution to this chal-lenging problem.
Acknowledgment
This work was supported in part by the National ScienceFoundation under grants SES-9119621, DDM-9322830and DMI-9501740.
References
[1] Hopp, W.J., Spearman, M.L. and Zhang, R.Q. (1997) Easilyimplementable �Q; r� inventory control policies. Operations Re-search, 45, 327±340.
[2] Nahmias, S. and Smith, S. (1992) Mathematical models of retailerinventory systems: a review, Perspectives in Operations Manage-ment: Essays in Honor of Elwood S. Bu�a, Sarin, R.K. (ed.),Kluwer, Boston, MA, pp. 249±278.
[3] Federgruen, A. (1993) Centralized planning models for multi-echelon inventory systems under uncertainty, in Logistics ofProduction and Inventory, Graves, S.C., Rinnooy Kan, A.H.G.and Zipkin, P.H. (eds.), North-Holland, New York, pp. 133±173.
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[6] Sherbrooke, C. (1986) Vari-metric: improved approximations formulti-indenture, multi-echelon availability models. OperationsResearch, 34, 311±319.
[7] Simon, R. (1971) Stationary properties of a two-echelon in-ventory model for low demand items. Operations Research, 19,761±777.
[8] AxsaÈ ter, S. (1990) Simple solution procedures for a class of two-echelon inventory problems. Operations Research, 38, 64±69.
[9] Svoronos, A. and Zipkin, P. (1991) Evaluation of one-for-onereplenishment policies for multiechelon inventory systems. Man-agement Science, 37, 68±83.
[10] Lee, H. and Moinzadeh, K. (1987) Two-parameter approxima-tions for multi-echelon repairable inventory models with batchordering policy. IIE Transactions, 19, 140±149.
[11] Moinzadeh, K. and Lee, H. (1986) Batch size and stochastic levelsin multi-echelon repairable systems. Management Science, 32,1567±1581.
[12] Lee, H. and Moinzadeh, K. (1987) Operating characteristics of atwo-echelon inventory system for repairable and consumableitems under batch ordering and shipment policy. Naval ResearchLogistics, 34, 365±380.
[13] Muckstadt, J.A. (1977) Analysis of a two-echelon inventory sys-tem in which all locations follow continuous review �s; S� policies.Technical report No. 337, School of Operations Research andIndustrial Engineering, Cornell University.
[14] Deuermeyer, B. and Schwarz, L.B. (1981) A model for the anal-ysis of system service level in warehouse/retailer distribution sys-tems: the identical retailer case, in Multi-Level Production/Inventory Control Systems, Schwarz, L.B. (ed.), North-Holland,Amsterdam, pp. 163±193.
[15] Svoronos, A. and Zipkin, P. (1988) Estimating the performance ofmulti-level inventory systems. Operations Research, 36, 57±72.
[16] Rosenbaum, B.A. (1981) Inventory placement in a two-echeloninventory system: an application, in Multi-Level Production/Inventory Control Systems, Schwarz, L.B. (ed.), North-Holland,Amsterdam, pp. 195±207.
[17] Kostic, S. and Pendic, Z. (1990) Optimization of spare parts in amultilevel maintenance system. Engineering Costs and ProductionEconomics, 20, 93±99.
[18] Cohen, M., Kamesam, P.V., Kleindorfer, P., Lee, H. and Teker-ian, A. (1990) Optimizer: IBM's multi-echelon inventory systemfor managing service logistics. Interfaces, 20, 65±82.
[19] Hausman, W.H. and Erkip, N.K. (1994) Multi-echelon vs. single-echelon inventory control policies for low-demand items. Man-agement Science, 40, 597±602.
[20] Erkip, N., Hausman, W. and Nahmias, S. (1990) Optimal cen-tralized ordering policies in multi-echelon inventory systems withcorrelated demands. Management Science, 30, 381±392.
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Appendices
Appendix A: Performance evaluation
De®ne the following random variables:
P ��; v�= the complementary cdf of the Poisson dis-tribution with parameter v;
W im = lead time for item i at facility m;
Dim = lead time demand for item i at facility m;
Bi0 = number of backorders for item i at the DC at
any point in time.
Then the statistical properties of the above randomvariables can be obtained by the following expressions:
At the DC:
Bi0�ri
0;Qi0� �
1
Qi0
�bi0�ri
0� ÿ bi0�ri
0 � Qi0��;
bi0�v� �
X1u�v�1
�uÿ vÿ 1�P�u : hi0�;
� hi0
2
2P�vÿ 1 : hi
0� ÿ hi0vP�v; hi
0�
� v�v� 1�2
P�v� 1; hi0�;
Ai0�ri
0;Qi0� �
1
Qi0
�ai0�ri
0� ÿ ai0�ri
0 � Qi0��;
Heuristic for a two-echelon spare parts distribution system 985
ai0�v� �
X1u�v�1
P�u; hi0� � hi
0P�v; hi0� ÿ vP�v� 1; hi
0�:
At the facilities:
E�W im� � li
m � dm � Bi0�ri
0;Qi0�
ki0
;
V �W im� �
E�Bi0�Bi
0 ÿ 1�� ÿ �Bi0�ri
0;Qi0��2
ki0
2
� 0 if ri0 � 0;
�ri0 � 1�=ki
0Qi0 else:
(E�Di
m� � kimE�W i
m� � him;
V �Dim� � ki
mE�W im� � ki
m2V �W i
m�;where
E�Bi0�Bi
0ÿ1��� 1
Qi0
�ci0�ri
0�ÿci0�ri
0;Qi0��;
ci0�v��
X1u�v�1
f�uÿ�v�1��2ÿ�uÿ�v�1��g P�u;hi0�;
�hi0
3
3P�vÿ2;hi
0�ÿhi0
2vP�vÿ1;hi
0��hi
0v�v�1�P�v;hi0�
ÿ1
3v�v�1��v�2�P�v�1;hi
0�:With these we can compute the expected number ofbackorders and inventory levels at the facilities as fol-lows:
Bim�Si
m� �X1
u�Sim�1
P im�u�;
� nimqi
m
pim
P im�Si
mjnim � 1� ÿ Si
mP im�Si
m � 1jnim�;
him�Si
m� � Sim ÿ hi
m � Bim�Si
m�;where
nim �
�E�Dim��2
V �Dim� ÿ E�Di
m�;
pim �
E�Dim�
V �Dim�;
qim � 1ÿ pi
m
pim�jjni
m� �ni
m � jÿ 1
j
� �pi
mni
m qim
j;
P im�xjni
m� �X1j�x
pim�jjni
m�:
Appendix B: Summary of heuristicsfor single echelon problem (19)±(23)
Type I formulas
The simplest heuristic from Hopp et al. [1] uses the fol-lowing approximations to derive closed form expressionsfor Qi
0 and ri0:
� hi0�ri
0;Qi0� � ri
0 ÿ hi0 � Qi
0=2.� Demand during lead time is normally distributedwith mean and variance hi
0.
� si0�ri
0;Qi0� � U ri
0 ÿ hi0
ÿ �. �����hi0
q� �(i.e., service is given
by the Type I (base stock) formula).
Di�erentiating the Lagrangian corresponding to (19)±(23) with respect to ri
0 and Qi0 and solving yields:
Qi0 � max
���������2mki
0
ciN
s; 1
8<:9=;; �A1�
ri0 � hi
0
��������������������������������������������ÿ2hi
0 ln����������2phi
0
qci
ki0
Kl
� �s; if
����������2phi
0
qci
ki0
Kl � 1
ri0; otherwise
8><>:�A2�
where m and l are Lagrange multipliers for the orderfrequency constraint (20) and service constraint (21), re-spectively.
Hybrid formulas
A slightly more complex approximation makes use of theType I formula (A1) to compute Qi
0, but uses the fol-lowing approximations to compute ri
0:
� hi0�ri
0;Qi0� � ri
0 ÿ hi0 � Qi
0=2.� Demand during lead time is normally distributedwith mean and variance hi
0.
� si0�ri
0;Qi0��1ÿ
hR1ri0
h1ÿU
��tÿhi
0
�. �����hi0
q �idti.
Qi0
� �(i.e., service is given by the Type II formula).
Di�erentiating the Lagrangean corresponding to (19)±(23) with respect to ri
0 and solving yields:
ri0 � hi
0 � Uÿ1�1ÿ KQ i0ci
lki0
������hi0
q; if KQi
0ci � lki0,
ri0; otherwise,
(�A3�
where l is the Lagrange multiplier for the service con-straint (21).
Appendix C: a lower bounding procedurefor ®xed Qi
0
Write the Lagrangian dual of the problem (4)±(9) forgiven Qi
0 as:
986 Hopp et al.
Maximize L�l1;l2; . . . ; lM�; �A4�Subject to:
lm � 0; m � 1; 2; . . . ;M ; �A5�where L�l1;l2; . . . ; lM �
� infri0;Si
m
XN
i�1ci hi
0�ri0� �
XMm�1
him�Si
m�" #(
�XMm�1
lm
XN
i�1Bi
m�Sim� ÿ Tm
" #:
1
N
XN
i�1
ki0
Qi0
� F ; ri0 � ri
0; Sim � Si
mg;
�XN
i�1Li�l1;l2; . . . ; lM� ÿ
XMm�1
lmTm; �A6�
and Li�l1;l2; . . . ; lM� �
infri0;Si
m
cihi0�ri
0;Qi0� �
XMm�1
cihim�Si
m� � lmBim�Si
m�� �
:
(1
N
XN
i�1
ki0
Qi0
� F ; ri0 � ri
0; Sim � Si
m
): �A7�
Rewrite
Li�l1;l2; . . . ; lM �
� infri0;Si
m
(cihi
0�ri0;Q
i0� �
XMm�1
Gim�Si
mjri0� : ri
0 � ri0; r
im � ri
m
);
�A8�where
Gim�Si
mjri0� � cihi
m�Sim� � lmBi
m�Sim�;
� ci�Sim ÿ hi
m� � �ci � lm�Bim�Si
m�; �A9�is a convex function of Si
m���.Obviously, for any l1; l2; . . . ;lM , L�l1; l2; . . . ; lM� is a
lower bound on the objective of the original problemsubject to the restriction on the Qi
0 values. SolvingL�l1;l2; . . . ; lM � for a given set of l1; l2; . . . ; lM is equiv-alent to solving N subproblems of Li�l1; l2; . . . ;lM �. Tosolve Li�l1;l2; . . . ; lM �, we use the following procedure:for any ®xed ri
0, we ®nd Sim, for m � 1; . . . ;M , which is easy
due to the convexity of Gim�Si
mjri0�. Then we can ®nd the
optimal ri0 values via a search routine to minimize the term
inside the f g in (A8) for any ®xed l1; . . . ; lM . Finally, weuse Powell's method [24] to adjust lm, m � 1; . . . ;M inorder to ®nd as tight a lower bound as possible.To make a search for ri
0 possible, we need to bound theoptimal values of ri
0. From the formulation we know thatri0 � ri
0. To ®nd an upper bound on the optimal value ofri0, let
~Gim�Si
m� � ci�Sim ÿ ~hi
m� � �ci � lm�Bim�Si
m�; �A10�
where ~him � ki
mdm � ~him can be interpreted as the lead time
demand of part i at facility m when the DC has amplesupply (i.e., li
m � dm). Therefore, ~Gim�Si
m� is independentof ri
0. Also let ~Sim be the optimal solution to ~Gi
m�Sim�. It is
easy to show that Sim, 1 � m � M , decreases as ri
0 in-creases. The search for ri
0 is done when Sim converges to ~Si
mor ri
0 reaches a preset high number.Thus, computing Li�l1; . . . ; lM� can be accomplished
via a simple one dimensional search on ri0 over �ri
0;~ri0�.
Since ri0 is restricted to integers, complete enumeration is
a practical method for this.
Biographies
Wallace J. Hopp is the Breed University Professor of ManufacturingManagement in the Department of Industrial Engineering and Direc-tor of the Master of Management in Manufacturing Program, a jointprogram of the Kellogg Graduate School of Management and theMcCormick School of Engineering, at Northwestern University. Hisresearch focuses primarily on the design and control of manufacturingsystems. He has won a number of research and teaching awards, in-cluding the 1985 Nicholson Prize (for best student paper in OperationsResearch), the 1989 McCormick Teaching Award (for best engineeringprofessor), the 1990 Scaife Award (with Mark Spearman, for the paperwith the ``greatest potential for assisting an advance of manufacturingpractice), the Pentair-Nugent professorship in Business Leadership (forleadership in manufacturing management) in 1993, the Kellogg TopFive Professors Award in 1998 (for outstanding management teach-ing), and the 1998 IIE Joint Publishers Book-of-the-Year Award (forthe book Factory Physics: Foundations of Manufacturing Management).He is Department Editor for Manufacturing, Distribution and ServiceOperations for the journalManagement Science, an Associate Editor ofIIE Transactions, and a member of IIE, INFORMS, POMS, and SME.He is an active industry consultant, whose clients have included An-ixter, Bell & Howell, Black & Decker, Eli Lilly, Ford, General Motors,John Deere, IBM, Motorola, Owens Corning, S & C Electric, SPX,Whirlpool, Zenith, and others.
Rachel Q. Zhang is an Assistant Professor of Industrial and Opera-tions Engineering at the University of Michigan. She holds an B.S. inApplied Mathematics, an M.S. and Ph.D. in Industrial Engineeringfrom Northwestern University. Her research interests center on pro-duction and inventory control, and interdisciplinary problems relatedto manufacturing and distribution systems. She has received anCAREER award from the National Science Foundation in 1995 andan honorable mention at the Nichoson student paper completion in1994.
Mark L. Spearman is an Associate Professor of Industrial and SystemsEngineering at Georgia Tech. He also serves as Chair of the Coalitionfor Manufacturing Control Systems in the Logistics Institute atGeorgia Tech. Prior to earning a Ph.D. degree in Industrial Engi-neering at Texas A & M University, he worked over 5 years as anindustrial engineer in the petro-chemical and electronics industry. Hehas been an active consultant working with more than 20 companies inthe areas of cycle time reduction, throughput enhancement, and in-ventory management. He has had over $1 million in funded researchfrom both industrial and government sponsors. Before coming toGeorgia Tech, he was the Director of the Master of EngineeringManagement Program at Northwestern University with appointments
Heuristic for a two-echelon spare parts distribution system 987
in both the McCormick School of Engineering and Applied Scienceand the Kellogg School of Management. He has numerous publica-tions on production and inventory control in respected journalsand has recently completed a book on the subject with ProfessorWallace Hopp titled, Factory Physics: Foundations of ManufacturingManagement. This book received the IIE Joint Publishers Book of theYear Award in 1998. He has served as Secretary and Member of
the Board of the Production and Operations Management Societyand as President of the Manufacturing and Service OperationsManagement Section of the Institute for Operations Research and theManagement Sciences.
Contributed by the Manufacturing Systems Control Department.
988 Hopp et al.
