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AN EFFICIENTALGORITHMFOR COMPUTINGAN OPTIMAL(r, Q)
POLICYIN CONTINUOUSREVIEWSTOCHASTIC
INVENTORYSYSTEMS
AWI FEDERGRUEN
Columbia University, New York, New York
YU-SHENGZHENG
University of Pennsylvania, Philadelphia, Pennsylvania (Received May 1990; revisions received November 1990, April 1991; accepted May 1991)
The reorder point/reorder quantity policies, also referred to as (r, Q) policies, are widely used in industry and extensively studied in the literature. However, for a period of almost 30 years there has been no efficient algorithm for computing optimal control parameters for such policies. In this paper, we present a surprisingly simple and efficient algorithm for the determination of an optimal (r, Q) policy. The computational complexity of the algorithm is linear in Q. For the most prevalent case of linear holding, backlogging and stockout penalty costs (in addition to fixed order costs), the algorithm requires at most (6r + 13Q) elementary operations (additions, comparisons and multiplications), and hence, no more than 13 times the amount of work required to do a single evaluation of the long-run average cost function in the point (r, Q*).
In this paper,we derivea simpleand efficientalgo-
rithm for computing optimal (r, Q) inventory
control policies. These policies are also known as reorderpoint/order quantity policies. We will restrict ourselves to the case when demands arise on a unit-
by-unit basis. Under an (r, Q) policy, the inventory
position (= inventory on-hand + orders outstanding-
backorders) of the item in question is continuously
reviewed,and an order of fixed quantity Q is placed
as soon as the inventory position drops to a reorder
point r.
Such (r, Q) policies are widely used in inventory
systems with uncertain demands and lead times. For single item inventory systems under standardassump- tions, it is well known that an optimal policy exists
within the class of (r, Q) policies. Many multi-item or
multilocation systems are designed such that each
item's (facility's) inventory is governed by an (r, Q)
policy. Other planning models consist of a large num-
ber of single item (r, Q) systems, tied together by
aggregate inventory constraints. These models are decomposed into single item models via Lagrangian relaxation. Highly efficient solution methods are essential here. Similarly, Atkins and lyogun (1988) propose a decomposition method to derive a tight lower bound for stochasticjoint replenishmentmodels
in which optimal (r, Q) policies need to be computed
repeatedly for each of the items involved; (r, Q) poli- cies are also optimal in many (generalized)stochastic clearing systems with point arrivalprocessesthat arise in other settings than those involving physical inven- tories. See Federgruenand Zheng (1988) for details. The use of (r, Q) policies has been propagatedsince the seminal paper of Galliher, Morse and Simmond (1959), and the classical textbook by Hadley and Whitin (1963) appeared 30 years ago. Nevertheless, and as mentioned in Browne and Zipkin (1991), "until recently,there was no reliable,straightforwardmethod for computing an optimal (r, Q) policy, even in the simple case of Poisson demand processes."Instead, a large number of heuristics have been proposed (see Lee and Nahmias 1989). The only existing algorithm, to our knowledge, was presented in Zipkin's (1988) classnotes.This procedureis based on a result in Sahin (1982); see also Sahin (1990). Our algorithm is based on the observation that the long-run averagecost C(r, Q) of an (r, Q) policy is of the form:
r+Q (^) /
Q(r, Q)=^ K+^ E^ G(y) /Q-^ (1)
y=r+ I
Here K> 0 is a given constant and -G(.) is a unimodal function with limlyl=OG(y) = (^) oo. Our approach is
Subject classification: Inventory/production: stochastic policies. Areaof review:STOCHASTICPROCESSESANDTHEIRAPPLICATIONS.
TechnicalNotes / 809
based on the following observations:The unimodality of -G(.) implies: for fixed Q, C(Q) =^ minr C(r, Q) is achieved when the sum in (1) consists of the Q smallest values of this function; and these values are achieved in Q contiguous points and the optimal corresponding reorder level r is trivially identified. Next it is easy to verify that -C(.) is unimodal as shown in Sahin (1982), and Q, the optimal order size, is obtained as the largest value of Q for which C(Q - 1) > GQ with GQ the Qth smallest G(-) value, so that C(Q + 1) = [QC(Q) + GQ+i]/(Q+ 1) >^ C*(Q), and
C(Q)= [(Q-^ 1)C(Q- 1) + GQ]/Q<^ C*(Q-^ 1)
(see Figure 1). These observations may be exploited in an efficient algorithm whose complexity is linear in Q. In the case of linear holding and backloggingcosts the computational complexity of the algorithm is no larger than (6r + 13Q) elementary operations (additions, comparisons and multiplications) when r >^ 0, and hence, no more than 13 times the amount of work required to do a single evaluation of the C(., ) function in the point (r, Q). (Similar complexity counts apply when r^ <^ 0 or when a more general one-step expected cost function is used.) The (r, Q) policies are a special case of (s, S) policies, under which the item's inventory position is ordered
to S whenever it is observed to have fallen to or below
the level.s (s < S). This more general structure arises when demands occur in batches of random size. In a related paper (Zheng and Federgruen 1991) we develop an efficient algorithm for finding optimal (s, S) policies. Since the cost function of an (s, S) policy fails, in general, to be quasiconvex, except under a restrictive assumption on the demand size distribution (see Stidham 1977, and Sahin 1982), that algorithm has to use a different and more complex
search procedure. Its complexity is at least quadratic
in (r* + Q*), even when applied to models in which
(r, Q) policies are optimal, i.e., where the cost function
is given by (1). Another commonly used generalization of the
(r, Q) policy, to accommodate for random demand
sizes, is the (r, nQ) policy: Here as soon as the
inventory position drops to or below r, an (integer)
multipleof a fixed quantity Q is ordered to raise
the inventory position back to the interval of
[r + 1, .. ., r^ +^ Q]. As for the simpler (r, Q) policies,
the steady-statedistribution of the inventory position is uniformly distributed under standard assumptions (see Hadley and Whitin, and Richard 1975). The cost
function of an (r, nQ) policy is therefore of a form similar to (1). An extension of the algorithm in this note may thus be employed (see Zheng and Chen 1990 for details). In Section 1 we introduce the notation and pre- liminaries. The proposed algorithm is derived and discussed in Section 2.
1. NOTATIONAND PRELIMINARIES
Considera single item whose inventory may be replen- ished by placing ordersof unlimited size. Ordersarrive after a given lead time. Stockouts are backlogged. In this section, we briefly review the main inventory models in which (r, Q) policies are optimal and their averagecost is of the form given by (1) because these results are scatteredthroughout or are not available in the open literature. For any t > (^) 0, let D(t) = the total demand in [0, t); IP(t) =^ the inventory position at time t; IL(t) =^ the inventory level at time t. Consider first the simplest of all models for which optimal (r, Q) policies exist, namely the case of Poisson demands and constant lead times. It is well known (see, e.g., Hadley and Whitin, and Zipkin 1986a) that the inventory position process IP(t) and the inventory level process (^) IL(t) have limiting distri- butions. Indeed, with IP(oo) and IL(oo) denoting
G(y)
L (^) u
Figure 1. G(.) function. If G(L), G(L + 1),. , G(U) represent (^) the q smallest values of the G(-) function, then the q +^ 1st smallest value is foundfory=L- 1 or U+ 1.
TechnicalNotes / 811
L, LD(oo) representsthe numberof demands(renew-
als) that occur in the time interval[0, L] underthe
equilibriumrenewalprocess.In some settings,a spe-
cific distributionfor the lead time demandLD(oo) is
directly estimated or surmised,rather than being
derivedfrom an underlyingdemandand lead time
process.
2. THE OPTIMIZATIONALGORITHM
Consideran inventorymodel in which the long-run
averagecost of an (r, Q) policy is of the form of (1)
with -G(.) a unimodalfunction.In this section,we
describeour optimizationprocedure.
As we point out in the Introduction,the unimodal-
ity of -G(.) impliesthat for fixed Q, the Q smallest
G(.) valuescan be achievedin contiguouspointsand
C*(Q) = minr C(r, Q) is achieved if the sum in (1)
consists of these values. These values and, hence,
the optimal reorderlevels r(1),.. ., r(Q) (for given order quantities 1,.. ., Q) are easy to identify by the
following procedure. Let y, be an integer that
minimizes G(y) over all integers. We generate a
sequence (^) YI, Y2, ...,} inductively. Assuming that (^) Iy', ..., (^) I have been generated, let L(Q) = min{y,,. .. , (^) YQ}, R(Q) = max yi,. .. , YQJ.Then let
YQ+'
_ JL(Q) -1 if G(L(Q) -1) <^ G(R(Q) + 1)
IR(Q) + 1 otherwise.
Clearly, for any given Q, Iyi,.. ., yQ are contig-
uous, and theirG(.) valuesconstitutethe Q smallest
ones due to the unimodalityof -G(.). This leads
directlyto the followinglemma.
Lemma 1. For any given integer Q >^ 1, r*(Q) = L(Q)- 1.
Proof. The proof is by (1).
Clearly, L(Q) = L(Q - 1) or L(Q) = L(Q - 1) -
- This implies the following corollary.
Corollary 1. r(Q) -^1 <^ r(Q + 1) < r*(Q) for all
integers Q > 1.
This corollary may be derived (with considerably more effort) from the results in Sahin (1982) (see Zipkin 1988). We conclude that
C*(Q)= K^ +^ G(yi)] Q,
so that C(Q + 1) = [QC(Q) + G(yQ+,)]/(Q + 1). (6)
Note that C(Q + 1) < C(Q) if and only if
G(yQ+,)< C*(Q). This suggeststhe following exceed-
ingly simple characterizationof Q*,the optimal order
size.
Lemma 2. Q* is the smallestintegerq withtheprop-
ertyC*(q) < G(yq+,).
Proof. It follows from (6) that C*(Q)is decreasingfor
Q <^ Q. Moreover, for Q > Q,
C(Q) -^ C(Q*)
Q/
= (^) -i=Q+lG(yj) - (Q -^ Q)C*(Q)] Q
> (Q -^ Q)[G(yQ+,)-^ C(Q)]/Q 2 0.
Lemmas 1 and 2 clearly suggest an efficient algo- rithm for finding an optimal reorder level and order
size (r, Q). Below we give a detailed algorithm for
the case where AG(y) = G(y + 1) - (^) G(y) is easy to compute (this is, for example, the case in the Poisson demand model, see (5)). For notational convenience only, we restrict ourselves to the most common case, where Y, > 0.
Algorithm OPT
Step0. Calculate G(0) and AG(O);L 0;
while AG(L) <^0 do
begin L := L +^ 1, evaluate zAG(L),G(L + 1) G(L) + AG(L) end;
S =^ K + G(L), Q:= 1, C* :=S, r:= L - 1,
R:= L + 1;
Step1.^ Repeat
begin if G(r) < G(R) then if C* < G(r) then stop.
else begin S:= S + G(r), r:= r- 1,
if r < 0, evaluate AG(r) and G(r) G(r + 1) -^ AG(r), end; else if C* < G(R) then stop. else begin S:= S + G(R), evaluate AG(R),
G(R+ 1):=G(R)+AG(R),R:=R+ 1
end;
Q:=Q+ 1,C*:=S/Q
end.
812 / FEDERGRUEN AND ZHENG
Complexity of the Algorithm
Assume that the probabilitydensity function of
LD(oo),i.e., the numbersIpj:j = 0, 1,.. .4,aregiven
as input to the problem.We first analyzethe com-
plexityof the algorithmfor the (mostprevalent)case
wherethe cost structureis linear,i.e., whereit consists
of linearholding,backloggingand stockoutpenalty
costs (in additionto the fixed ordercosts). We first
analyzethe case wherer* > 0. The entirealgorithm
consistsof:
i. evaluationof the r* + Q* first values of the
difference function AG(y) = (h + p)Py - (^) X1rpy-
p: each such evaluationrequiresfive elementary
operations(three additionsand two multiplica-
tions;in the absenceof stockoutpenaltycostsonly
threeoperationsarerequired);
ii. r* + Q* additionsto compute the IG(1),
G(r+ Q)}values;
iii. at most, 7Q* elementary operations (3Q*
comparisons;3Q* additionsand Q* divisions)to
executethe remainingworkin Step 1.
The algorithmthus requiresno more than 6r* +
13Qelementaryoperations(4r + 11Q*operations
in the absenceof stockoutpenaltycosts).In caser* -
0, the total numberof operationsis reducedto 13Q*
because only Q* values of the G(.) function are
computed.It is useful to comparethis complexity
bound with a lowerbound for the amount of work
that is requiredto do a singleevaluationof the cost
function C(., ) at the optimal point (r, Q*). Even
when the cost structureis linear this evaluation
requires the computation of Pr+Qand the latter
requiresr* + Q* additions.We concludeas follows.
Theorem1. AlgorithmOPT determinesthe optimal
(r, Q)policy. Underlinearcoststructuresand when
r > 0, the amountof work(measuredas the number
of elementaryoperationsrequired)is at worst13 times
the amountof workrequiredto evaluatePr+Q(andat
worst11timesin theabsenceofstockoutpenaltycosts).
We concludethis paperwith a few commentson
the case wherethe cost structureconsistsof general
nonlinearholding,backloggingor stockoutpenalty
cost functions(the nonlinearcase).In the linearcase,
the functionG(.) is most easy to evaluatevia AG(.)
(see (5)). In the nonlinearcase,it is moreefficientto
evaluatethe G(.) valuesdirectly.The algorithmneeds
to be modifiedaccordingly.We mayreplaceStep0 by
a bisectionsearchto locatethe minimumpointYi. To
gain efficiency,{G(r), ..., G(r + Q*)) shouldbe
evaluatedin Step 1 when needed.The resultingalgo-
rithmrequires(see Federgruenand Zheng(1988):at
most rlog2Ul + Q* + 1 evaluationsof the G(.)
functionwith U an prioriupperboundfor yi; and at
most 8Q* elementary operations to execute the
remainingwork.Note that a singleevaluationof the
averagecost function C(., ) requiresat least Q
evaluationsof the functionG.).
Note Added in Proof
Shortlyprior to receivingthe galley proofs of this
paper,we becameawareof G. Rubal'sky's"Calcula-
tionsof OptimumParametersin an InventoryControl
Problem"(Eng.Cybern.10, 182-187;translatedfrom
Russian),which describesa similarmethod for the
determinationof optimalr- and q-values.
REFERENCES
ATKINS,D., AND P. IYOGUN1988. Priodic VersusCan- OrderPoliciesforCoordinatedMulti-itemInventory Systems.Mgmt. Sci. 34, 791-795. BROWNE,S., AND P. (^) ZIPKIN. 1991. Inventory Models With Continuous,StochasticDemands.Anns.Appl. Prob. 1 (3), 419-435. FEDERGRUEN,A., AND Y. S. (^) ZHENG. 1988. A Simpleand EfficientAlgorithmfor ComputingOptimal (r, Q) Policiesin Continuous-ReviewStochasticInventory Systems. Working Paper (Unabridged Version). Decision Sciences Department, The Wharton School, Universityof Pennsylvania,Philadelphia. GALLIHER, H., P. MORSE AND M. SIMMOND. 1959. Dynamics of Two Classes of Continuous-Review InventorySystems.Opns.Res. 7, 362-384. HADLEY, G., AND T. M. WHITIN. 1963. Analysis of InventorySystems.Prentice-Hall,EnglewoodCliffs, N.J. LEE, H., AND S. NAHMIAS. 1989. Single Product, Single-Location,Models, Chap. 2. In Handbookin Operations Research and Management Science, Vol. 4: Logistics of Production and Inventory, S. Graves, A. Rinnooy Kan and P. Zipkin (eds). North Holland,Amsterdam. RICHARD, (^) F. 1975. Comments of the Distribution of InventoryPosition in a Continuous Review (s, S) InventorySystem.Opns.Res. 23, 366-371. SAHIN, I. 1982. On the ObjectiveFunction Behaviorin (s, S) InventoryModels. (^) Opns.Res. 30, 709-725. SAHIN, I. 1990. Regenerative Inventory Systemf, Springer-Verlag,New York. STIDHAM, S. 1977. Cost Models for StochasticClearing Systems.Opns.Res. 25, 100-127. ZHENG, Y. S. 1989. Propertiesof StochasticInventory Systems.Mgmt. Sci. (to appear). ZHENG, Y. S., AND F. CHEN. 1990. InventoryPolicies With QuantizedOrdering.WorkingPaper,Decision Sciences Department. The Wharton School, Universityof Pennsylvania,Philadelphia.
