Replenishment Policies for Multi-Product Stochastic Inventory

Replenishment Policies for Multi-Product StochasticInventory Systems with Correlated Demand and

Joint-Replenishment Costs

Haolin FengLingnan College, Sun Yat-sen University, 510275 Guangzhou, China, [email protected]

Qi WuWeatherhead School of Management, Case Western Reserve University, Cleveland, Ohio 44106, USA, [email protected]

Kumar MuthuramanMcCombs School of Business, University of Texas, Austin, Texas 78712, USA, [email protected]

Vinayak DeshpandeKenan-Flagler Business School, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599, USA

[email protected]

T his study analyzes optimal replenishment policies that minimize expected discounted cost of multi-product stochas-tic inventory systems. The distinguishing feature of the multi-product inventory system that we analyze is the exis-

tence of correlated demand and joint-replenishment costs across multiple products. Our objective is to understand thestructure of the optimal policy and use this structure to construct a heuristic method that can solve problems set in real-world sizes/dimensions. Using an MDP formulation we first compute the optimal policy. The optimal policy can only becomputed for problems with a small number of product types due to the curse of dimensionality. Hence, using the insightgained from the optimal policy, we propose a class of policies that captures the impact of demand correlation on thestructure of the optimal policy. We call this class (s, c, d, S)-policies, and also develop an algorithm to compute good poli-cies in this class, for large multi-product problems. Finally using an exhaustive set of computational examples we showthat policies in this class very closely approximate the optimal policy and can outperform policies analyzed in prior litera-ture which assume independent demand. We have also included examples that illustrate performance under the averagecost objective.

Key words: multi-item inventory management; joint replenishment; stochastic inventory control; correlated demand; fixedordering costHistory: Received: September 2012; Accepted: May 2014 by Eric Johnson, after 2 revisions.

1. Introduction

Stochastic inventory control refers to the problem ofstocking a warehouse/inventory system that is sub-ject to random demand. Most practical inventory sys-tems consist of more than one product andconsiderable savings may be achieved by coordinat-ing the replenishment of inventory across multipleproducts. In addition, in many cases, demand for themultiple products in an inventory system may becorrelated. If products are complementary, then apositive correlation may be observed between thedemands of these complementary products. If prod-ucts are substitutable, then negative correlation maybe observed. This demand correlation can be

exploited for coordinating the inventory replenish-ment across multiple products.The problem of managing inventories with corre-

lated demand is important in practice. Examples ofinventory systems with correlated demand arenumerous. For example, demand for electronics prod-ucts such as ipod, iphone, etc., and their accessoriessuch as case covers, software apps, are positively cor-related. While demand between various competingportable music players would be negative. It has alsobecome extremely popular for retailers to offer infor-mation on what other customers may buy, in order tobetter inform customers of complementary and sub-stitutable products. For example, on Amazon.com,under product descriptions, there is a historical

Please Cite this article in press as: Feng, H., et al. Replenishment Policies for Multi-Product Stochastic Inventory Systems with Corre-lated Demand and Joint-Replenishment Costs. Production and Operations Management (2014), doi 10.1111/poms.12290

Vol. 0, No. 0, xxxx–xxxx 2014, pp. 1–18 DOI 10.1111/poms.12290ISSN 1059-1478|EISSN 1937-5956|14|00|0001 © 2014 Production and Operations Management Society

summary of “customer who bought this item alsobought...” products. It also provides a list of similarproducts. These shopping advices further reinforcethe correlations among products demand. Xu (1999)also provides other examples of correlated demandsin practice such as Make-to-Order assembly-dis-assembly systems, Make-to-stock assemble-to-ordersystems, and synchronized communication networks.With this correlated demand, an interesting questionis how firms should adapt their inventory replenish-ment policies accordingly.While there exists a large stream of literature on sin-

gle-product inventory systems, literature on optimalreplenishment policies for multi-product stochasticinventory systems is scarce, particularly whendemand is correlated. Most standard inventory mod-els either assume that demand of each product isindependent of other products, or do not explicitlytake into account the correlation between demands bytreating each product independently (product-basedapproach described in Song 1998). The distinguishingfeature of the multi-product inventory system that weanalyze is the existence of correlated demand and joint-replenishment costs across multiple products.The decision maker is posed with the problem of

striking a balance between ordering costs, storage costs,and stock-out costs. Ordering costs usually contain afixed and a proportional component that makes veryfrequent ordering unacceptable. For a system with mul-tiple products, the fixed ordering cost can again be oftwo types: A fixed component which is product depen-dent, and another fixed component which is indepen-dent of the assortment of products in the order (and ishence called the fixed joint replenishment cost). Forexample, the fixed joint replenishment cost can includethe transportation cost of sending out an empty truck.Additional costs incurred from adding another productinto the truck, such as extra loading, additional gas,insurance etc., would be accounted for by the propor-tional component. Another interpretation is that bigretailers typically have their own regional warehousesthat take incoming products from different suppliers.Then the products are sorted and sent to local retailstores. In this case, the transportation costs from suppli-ers to warehouses correspond to individual productordering costs, while the transportation costs fromwarehouses to local stores are shared among all theproducts ordered and hence are counted as joint order-ing costs. A similar idea about interpreting the jointreplenishment problem as a multi-location problem,but in a different context, is proposed in Nielsen andLarsen (2005).The storage cost, often called the holding cost, is

incurred throughout time and is proportional to theinventory level in the warehouse. We assume thatunmet demand is backordered and at every point in

time incurs a penalty cost that is proportional to thequantity backordered.Typically, an inventory manager considers the

tradeoff between the inventory holding cost andinventory ordering cost. With multiple items, besidesthis tradeoff, we also need to consider the joint order-ing to exploit the advantage of the fixed joint replen-ishment costs. However, to achieve cost savings in theordering cost by joint replenishment, sometimes theorder may include certain products that need not bereplenished immediately when considered individu-ally, and hence may incur an extra holding cost.Therefore, in addition to the tradeoffs considered insingle-item case, a good inventory replenishmentdecision of multiple products needs to balance thefixed joint replenishment costs and holding costs.

1.1. Contributions and OutlineIn This study our objective is to understand the struc-ture of the optimal policy and use this structure toconstruct a heuristic method that can solve problemsset in real-world sizes/dimensions. Since we need tocompute the optimal policy, at least in lower dimen-sions, we first formulate the inventory system withmultiple products, joint-replenishment costs andcorrelated demand as a Markov Decision Process(MDP) and solve it with policy iteration. Realizingthat existing heuristic approximations in the literaturefail to capture the structure of optimal policies whendemand is correlated, we propose the (s, c, d, S) pol-icy. We evaluate the performance of the (s, c, d, S)structure by computing the closest (s, c, d, S) policyto the optimal policy, and compare it to the optimalpolicy and other heuristic policies available in litera-ture. The purpose of this comparison is to show thatthe (s, c, d, S) structure captures all the primary fea-tures of the optimal policy. By way of examples, wealso provide insight into why correlation dictates achange in structure of the policy. Realizing that the(s, c, d, S) structure is extremely close to the optimalpolicy for such inventory systems, we present analgorithm that can construct heuristic (s, c, d, S) poli-cies for very large inventory systems. We compare theresults of this algorithm using a benchmark 12-prod-uct example provided in Atkins and Iyogun (1988).To the best of our knowledge, there exists no work inliterature that focuses on constructing and testing pol-icies for inventory systems that have several producttypes with correlated demand and joint replenish-ment costs.

To summarize, the contributions of the study are asfollows:

• A model formulation for multi-product inven-tory systems with joint replenishment costsand correlated demand processes.

Feng, Wu, Muthuraman, and Deshpande: Multi-Product Stochastic Inventory Systems2 Production and Operations Management 0(0), pp. 1–18, © 2014 Production and Operations Management Society


• An understanding of how demand correlationaffects the structure of optimal policies formulti-product inventory systems.

• A class of policies, called the (s, c, d, S), to cap-ture the structural impact of demand correla-tion.

• A computational algorithm that constructsheuristic (s, c, d, S) policies for systems with alarge number of products (dimensions).

• Computational examples that show the perfor-mance of (s, c, d, S) policy as compared to theoptimal policy and other polices proposed inprior literature.

• Discussions that provide insight into thedependence of the policy on various problemparameters.

• Comparison against our (s, c, d, S) policies foraverage cost objectives and a large 12-productbenchmark example.

Of the contributions listed above, we believe that,the key contributions are the insights provided in theconstruction of the (s, c, d, S) policies and the algo-rithm that computes good policies in this class. Thecomputational examples seek to highlight the perfor-mance of the (s, c, d, S) policies in both problemsinvolving few items as well as problems involvinglarge number of items, with both discounted andaverage cost objectives.In the rest of this section, we provide an overview

of relevant literature. In section 2, the problem formu-lation is presented along with illustration of how posi-tive lead times can be handled within theformulation. We study the structure of the optimalpolicy and propose the (s, c, d, S) policy in section 3.Numerical experiments that quantify the performanceof (s, c, d, S) policies with respect to other heuristicpolicies and the optimal policy is presented in section4. Section 5 provides a numerical algorithm to com-pute a heuristic (s, c, d, S) and evaluates its perfor-mance on a 12-product benchmark example providedin Atkins and Iyogun (1988).

1.2. Related Literature on Single and Multi-Product Inventory SystemsFor a single-product system the optimal policy hasbeen shown to be an (s, S) policy, in very generalsettings. With an (s, S) policy, whenever the inven-tory position is at or below s, an order is placedto raise the inventory position up to S. The opti-mality of (s, S) was proved by Iglehart (1963) forperiodic review systems. Veinott and Wagner(1965) proposed tight bounds for the optimal sand S as well as a method to compute the opti-mal (s, S) policy. This method essentially appliedfull enumeration of the two dimensional grid on

the (s, d) plane (d � S � s). Zheng and Federgruen(1991) derived a new algorithm based on a num-ber of new properties of the infinite horizon costfunction as well as tighter bounds for optimal sand S. Based on these, they first fixed either s orS, and then moved the other iteratively toimprove the value function until convergence.They considered a periodic review model with along-run average cost criteria. They also describeda way to apply it to continuous review and dis-counted cost models.However, in multi-product systems with joint

replenishment costs, optimal policies do not havesimple structure in general. Ignall (1969) gave asimple two-product example to show that the opti-mal policy fails to have a simple form. A class ofsimple coordinated control rules, called an (s, c, S)policy, was introduced by Balintfy (1964). Thispolicy has three parameters si, ci, and Si for eachproduct i with si ≤ ci < Si. An order is triggeredwhen any product has its inventory level at orbelow its reorder point si. Any product j with theinventory level at or below its can-order point cjwill be included in the order when an order istriggered. The inventory of each product l includedin the order is replenished up to its order-up-topoint Sl.There exists some literature analyzing how to com-

pute a good (s, c, S) policy. For the special case ofunit-Poisson demand, Silver (1974) provided an itera-tive method to compute a suboptimal (s, c, S) rule. Ina closely related paper, Federgruen et al. (1984) stud-ied the problem in a more general setting, and pro-posed a heuristic algorithm. They used a decouplingprinciple, as in Silver (1974), to decouple the problemto multiple single-product problems. For each of theresulting single-product problems, they used the con-cept of a special replenishment opportunity to obtain(s, c, S) policy structures. Normal replenishmentsoccur when the inventory level is at or below s. Spe-cial ordering replenishment opportunities arrive as aPoisson process and they provide an opportunity toreplenish the inventory with a lower setup cost. Whenthe special ordering opportunity occurs, the productwill be replenished if the inventory level is at or belowc. S is the order-up-to level for both normal and spe-cial opportunity orders. Their algorithm is efficientdue to the decomposition of the N dimensional prob-lem into N one dimensional problems. Although themethod is not exact due to the assumption of constantarrival rate for the special ordering opportunities, themethod more than compensates for the approxima-tion with efficiency. When correlations in demand arepresent it is only obvious that approximationsbrought about by such decoupling would becomeworse.

Feng, Wu, Muthuraman, and Deshpande: Multi-Product Stochastic Inventory SystemsProduction and Operations Management 0(0), pp. 1–18, © 2014 Production and Operations Management Society 3


Atkins and Iyogun (1988) proposed periodicreplenishment policies where all the products areordered upto a base-stock level. Two types of policieswere proposed: the periodic policy where all productsare ordered up to the base-stock level every period,and a modified policy, where products belonging to abase set are ordered every period, while other prod-ucts have an order period which is an integer multipleof the base period. Atkins and Iyogun (1988) showedthat periodic review policies can outperform can-order policies. Viswanathan (1997) suggested twonew classes of policies, namely the P(s, S) and Q(s, S)policies. Nielsen and Larsen (2005) studied the Q(s, S)policy and showed that it performed the best amongthe considered policies. Larsen (2009) extended the Q(s, S) policy to the joint replenishment problem, fol-lowed by several numerical results for two-productinventory systems. Ozkaya et al. (2006) providedanalysis of the (Q, S, T) policy. Other relevant papersinclude Zheng (1994), Schultz and Johansen (1999),Lee and Chew (2005).For the case of correlated demand processes, there

exists some literature discussing the practicality ofcorrelated demand, as well as how to model correla-tions in the stochastic inventory problem. Eppen(1979) analyzed single period stochastic inventorysystems with normal-dependent demand in the con-text of risk pooling, while Corbett and Rajaram (2006)generalized these results to non-normal-dependentdemand processes. There exists various ways of cap-turing correlated demand across multiple products ina continuous review setting. Xu (1999) captured corre-lated arrivals of multi-class demand in a assemble-to-order system, where each class is composed of one ormore job types (products). The arrival process foreach demand class is modeled as a Poisson process,while the number of units of each job type is an inde-pendent draw from a distribution. This leads to amarginal arrival process for each product to be a com-pound Poisson process. The demands for each prod-uct are, however, now correlated due to joint arrivals(a demand class having more than one product). Song(1998) provided a generalized version of Xu (1999) tocapture correlated demand. In her model, customerdemand arrivals are modeled as a Poisson processwith potentially multiple products in each demandrequest. However, a general distribution for batchsizes of each product is assumed (no independencerequirement for batch size of each product needed ina customer demand request). This allows the repre-sentation of very generalized correlated processesthat can capture correlated Poisson demand with unitarrivals, assembly systems and distribution systems.We use a demand model, similar to Song (1998), forcapturing correlated demand between multipleproducts.

2. Model Formulation

Consider a continuously monitored inventory sys-tem with N different products. LetΩ = {1, 2, . . ., N} and the set of all subsets of Ω tobe S. The null set / denotes arrival of windowshopping customers, who do not buy any of theproducts. The state of the inventory system is rep-resented by X = {X1, X2, . . ., XN}, where Xi denotesthe inventory level of product i.

2.1. Correlation and Joint demandsIn multiple product inventory systems, there aredifferent types of demand arrivals. For any subsetof products K 2 S, a demand arrival is called atype-K demand if it requires aKi units of producti 2 K and zero units of products in Ω/K. Type-Kdemand has Poisson arrivals with rate kK. Thenumber of units of different products in a type-Kdemand has a joint density distribution fK, that is,given that a demand is type-K, the probability thatit is aK ¼ ðaKi Þi2K is fK(aK).The demand model presented above is similar to

the one considered in Song (1998). It can also bethought of as a generalized version of the model con-sidered in Xu (1999). The model considered in Xu(1999) restricts fK(aK) to functions of the formPi2KgiðaKi Þ, that is, the quantity aKi of different prod-ucts in an arriving demand is independent of demandclass K and the quantity of other products demandedin that customer order. This formulation assures thatthe marginal demand for any specific product consti-tutes a compound Poisson process. Note that such anarrival process is rightly termed as correlated-arrivalswhile we term our arrival process more generally ascorrelated-demand. We do not impose any restrictionon fK. However special and important cases pointedout in Song (1998) are worth mentioning.

• Unit demand. fK(aK) = 1 if aKi ¼ 1 for all i andfK(aK) = 0 otherwise. This restricts demandarrivals to contain a maximum of one order ofany product. An example would be textbookdemand faced by a university bookstore.

• Assembly systems. fK(aK) = gK(b) if aKi ¼ bKi bfor all i and fK(aK) = 0 otherwise. Here bdenotes a scalar random variable and gK is itsdensity distribution. An arrival, in this case, isa request for b quantities of an assembledproduct K, each of which requires bKi numberof component i.

• Independent demand. fKðaKÞ ¼ Pi2KgiðaKi Þ.Here, gi(�) is the arrival density distribution forproduct i. This model has been considered tobe a reasonable approximation for demandarriving at distribution systems.



The overall arrival rate is � ¼ PK2S �

K and theprobability that an arriving demand is of type K isqK = kK/k. Aggregate demand arrival for product i is�i ¼

PK2S �

K1fi2Kg and the probability that an arriv-ing demand has a request for product i is qi = ki/k.Here 1{} denotes the indicator function. Also for nota-tional simplicity, as in Song (1998), when K containsonly a single product we abbreviate type-K by type iand similarly we say an order is of type ij if K = {i, j}.Note that there is an important distinction

between joint arrival probabilities and correlateddemands though they are related. As is standard inthese systems, the fundamental modeling parameterwe use is the joint arrival probability. Neither theprobability distribution of joint arrival fK(aK) northe probability of a type-K arrival qK, by definitioncan take non-positive values. However, this doesnot imply that correlation amongst possibly substi-tutable products cannot be modeled to be negative.For example, consider the two-product case withunit demand. If the demands for products 1 and 2always occur separately and the joint probability is0, then arrival of one product implies non arrivalof the other. This implies a negative correlationwhile qK is non-negative for all K.With N products, the marginal probability for ai

units demand for product i is piðaiÞ ¼ Pai¼aK

ifKðaKÞ.

Therefore, the expected demand arrival for product iis EðDiÞ ¼ P1

ai¼0 aipiðaiÞ and the variance is

VarðDiÞ ¼ P1ai¼0 a

2i piðaiÞ � E2ðDiÞ. Similarly, let us

denote the marginal probability for bj units demandfor product j by pj(bj) and the expectations and vari-ance of demand by E(Dj) and Var(Dj). For example, inthe two-product, single unit demand case,

p/ ¼ f/ ¼ q/

p1ð1Þ ¼ ff1gð1Þ þ ff1;2gð1; 1Þ ¼ qf1g þ qf1;2g

p2ð1Þ ¼ ff2gð1Þ þ ff1;2gð1; 1Þ ¼ qf2g þ qf1;2g

p1;2ð1; 1Þ ¼ ff1;2gð1; 1Þ ¼ qf1;2g:

2.2. Cost FunctionPositive inventory levels incur a holding cost of hi perunit time for each unit of product i. Unfulfilleddemands are back ordered and incur a penalty cost ofpi per unit time for each unit of back ordered producti. Thus the instantaneous cost, G(X), incurred atinventory level X is

GðXÞ ¼ �X

i:Xi\0

piXi þX

i:Xi [ 0

hiXi: ð1Þ

Ordering costs incur a setup cost apart from price.The per unit price of product i is mi. The setup cost

has a fixed component j0 and product specificcomponents ji. Hence, a type K order of aK will costj0 þ P

i2Kðji þ aKi miÞ.An ordering or inventory management policy is

denoted by U ¼ fðnn; gnÞ1n¼1g, a sequence of ordersmade at time gn 2 R for nni units of product i for alli 2 Ω. With XU representing the inventory levelunder a policy U andD(0, t] representing the demandup to time t,

XUðtÞ ¼ Xð0Þ �Dð0; t� þX

n:gn � t

nn; ð2Þ

where X(0) 2 RN is the initial state of inventorylevel. Equation (2) assumes zero lead time for placedorders. For the sake of simplicity we will formulatethe model with zero lead times. Positive lead timescan be easily incorporated by interpreting X as theinventory position and considering a modified hold-ing cost function presented later.Defining the set of non anticipatory strategies by U,

our objective is to choose the U that minimizes theinfinite horizon discounted cost. Denoting the dis-count factor by a > 0, and the evolution of inventorylevel under U buy XU, the value function is definedby,

VðXÞ ¼ infU2U

E

(Z 1

0

e�atGðXUðtÞÞdt

þ sum1n¼1e

�agn ½j0 þX

i:nni [ 0

ðji þ minni Þ�

):

ð3Þ

This is clearly a Markovian system. Kushner (1971)shows that for a Markovian system, there exists anoptimal pure Markovian Control which is at least asgood as any non-anticipative control. Therefore, ourinventory problem has an optimal control that onlydepends on the inventory level for different products.Hence the value function V(X) is dependent only onthe state. This also implies that the current time canalways be taken as 0 and optimal policy is completelydetermined if the optimal decision at the current timeis known for all possible states X. Therefore, the opti-mal policy can simply be represented by a functionΦ(X) that gives the optimal order quantity when thecurrent state is X.Given current time is 0, let the random variable s

denote the time of next demand arrival, that is,s = {t ≥ 0: D(t) > 0}. The value of placing an instanta-neous order for ξ>0 (that is, ∑iξi > 0 and ξi ≥ 0 for alli) and then following an optimal policy is,

MnVðXÞ ¼ VðXþ nÞ þ j0 þX

i:ni [ 0

ðji þ miniÞ" #

ð4Þ



and the value of not placing any order, until thenext demand arrival and then following an optimalpolicy is given by

LVðXÞ ¼ E

" Z s

0

e�atGðXðtÞÞdtþ e�asXY

PðXðsÞ

¼ YjXð0Þ ¼ XÞVðYÞ#: ð5Þ

Let MVðXÞ ¼ minn2A MnVðXÞ, where the set Acomprises of all ξ 2 RN s.t. ∑iξi > 0 and ξi ≥ 0 forall i. Then from the definition of V(�), VðXÞ�MVðXÞand VðXÞ�LVðXÞ. At optimality,

VðXÞ ¼ min½LVðXÞ;MVðXÞ�: ð6ÞThe above Bellman equation can be interpreted asa Markov Decision Process (MDP) with impulsecontrols and can be solved using policy iteration(Puterman 2005).To accommodate positive lead times we will fol-

low the convention in literature (Muthuraman et al.2014) to interpret X(t) as the inventory position atany given time t, which is the total on hand inven-tory plus the inventory on order minus backorders.Denote si the lead time for product i, we couldthen re-defined a new ‘holding function’~GðXÞ ¼ P

i~GiðXiÞ in which

~GiðXiÞ ¼ e�asiEDið0;si� Gi Xi�Dið0; sþ si� þX

gi2ð0;s�ni

0@

1A

24

35:ð7Þ

Re-defining the operator LV in Equation (5) byreplacing G(�) with ~Gið�Þ, the Bellman Equation (6)becomes a unified MDP formulation for the prob-lem, whose solution procedure is exactly the samefor both with and without lead time.1

3. Inventory Policy Structures

This section first illustrates the optimal policy for thetwo-product case. We illustrate that the structure ofthe optimal policy deviates significantly from the(s, c, S) policy and argue why a structural modifica-tion of the (s, c, S) policy will perform better for caseswith correlated demands. Using the gained intuitionwe then propose a better structural approximation forthe optimal policy in this section.Numerical evaluations of the performance of the

(s, c, S) policy against that of the optimal policy is col-lected in the next section, along with comparisonsagainst the proposed policy. While this section usesonly the two-product case for the sake of illustration,the next section does include examples with multi-

product cases, lead-time cases and compound Poissonarrival cases.

3.1. Optimal PolicyUsing the formulation and method from the previoussection, we first explore a two-product unit demandcase with zero lead times. For the two-product casewe will take as holding and penalty costs, hi = 1 andpi = 100 for both products. Costs of ordering are takento be mi = 0.50 and ji = 5 for both products, with ajoint setup cost of j0 = 6. The discount rate in theobjective is set to a = 10%.In Figure 1, we show the optimal inventory order-

ing policy with joint demand arrival of probabilities0.0, 0.2, 0.4, and 1.0. We consider the symmetric casewith overall arrival rate k = 20. In each plot, thepoints that trigger the replenishment and the inven-tory level after the replenishment are plotted with thesame markers. For example, in plot 1 of Figure 1, a“star” at point (�1, 1) indicates that an order will beplaced when the inventory level is (�1, 1), and theinventory level is raised up to (12, 12), which is also a“star” in the same plot. Another example is the point(�1, 7) which will be replenished to (11, 7), both mar-ket with a “square.” Due to the symmetry in parame-ters, all these plots are symmetric across the diagonal.For no joint arrival case, observe the first plot

(q{1,2} = 0) in Figure 1 shows a policy that is close tothe (s, c, S) policy but is not exactly an (s, c, S) policy.This is understandable since the (s, c, S) policy is notoptimal even for the independent arrival case but is,however, close and most importantly simple and easyto implement. This closeness to the (s, c, S) structurevanishes when q{1,2} is non-zero. Consider the secondplot in Figure 1 illustrating the optimal policy for thecase with joint arrival probability 0.2. In this case, theorder from points still have an (s,c) type of partition,that is, the optimal policy order product 1 only ifX1 < s1 and X2 > c2 while it orders both products ifX1 < s1 and X2 ≤ c2. However, when only product 1 isordered the optimal policy does not elevate the prod-uct 1 inventory level to S1. The optimal policyattempts to account for the fact that future arrivalswill be correlated. The following example, similar tothe one presented in the introduction, with perfectlycorrelated demand clarifies the intuition on the struc-ture of the optimal policy.For the sake of illustration, consider a perfectly cor-

related demand case where q{1,2} = 1. This capturesperfect correlation since product 1 arrivals are alwaysaccompanied by product 2 arrivals (and vice versa).We use this example is to illustrate the mechanism ofthe optimal policies shown in Figure 1. Here, the opti-mal policy moves the state space to the diagonal thefirst time it places an order (after which the systemwill always remain on the diagonal because demand



is perfectly correlated). To see why this is optimal,first consider the (s, c, S) policy for this case. Say theinventory levels fall to (�1, 12). The (s, c, S) policydictates an order of 17 units of product 1 to moveinventory levels to (16, 12). Due to perfect correlation,13 demand arrivals will bring the state to (3, �1) atwhich instance the (s, c, S) policy elevates the inven-tory level to (16, 16). Now consider what the optimalpolicy does in contrast. It moves the state initiallyfrom (�1, 12), first to (12, 12) by ordering only 13units. The same 13 demand would eventually bringthe state to (�1, �1) at which point the optimal policyelevates the state to (16, 16). Notice that in movingfrom a beginning state of (�1, 12) eventually to(16, 16), both the (s, c, S) policy and the optimal pol-icy require the same amount of ordering cost and time(13 demand arrivals), but the optimal policy incursmuch less holding cost than the (s, c, S) policy, sinceit does not hold the 4-extra (17 vs. 13) units of product1 during this time.One of the most interesting aspects of all the four

plots in Figure 1 is that the optimal order up-to levelswhen Xi > ci are simply the discretized version of alinear order up-to level. The linear interpolation of

these order up-to levels are also plotted in each of thesubplots of Figure 1 as straight lines. An exhaustiveset of numerical experiments align with this observa-tion and the policy structure that we propose in thenext section primarily relies on this observation. Ofcourse while we use this observation as a startingpoint, we will be evaluating the performance of the tobe proposed policy later in the paper using valuefunction comparisons. We also do not claim that thediscretized linear order up-to level is optimal. How-ever, it will make our to be proposed policy an extre-mely good approximation that is very simple andease to implement.

3.2. The (s, c, d, S) PolicyNow that we understand the structure of the optimalpolicy, we can compare and quantify the performanceof the (s, c, S) polices. We collect these numericalevaluations in the next section along with othernumerical studies. From the results presented in sec-tion 1 we will realize that the (s, c, S) policies signifi-cantly under perform optimal polices. This is notsurprising and is also understandable since (s, c, S)policies were not originally designed to handle

−2 0 2 4 6 8 10 12−2

0

2

4

6

8

10

12

X1

X2

2−D Poisson: Joint Arrival Prob.=0

−2 0 2 4 6 8 10 12 14−2

0

2

4

6

8

10

12

14

X1

X2

2−D Poisson: Joint Arrival Prob.=0.2

−2 0 2 4 6 8 10 12 14−2

0

2

4

6

8

10

12

14

X1

X2

2−D Poisson: Joint Arrival Prob.=0.4

−2 0 2 4 6 8 10 12 14 16−2

0

2

4

6

8

10

12

14

16

X1

X2

2−D Poisson: Joint Arrival Prob.=1

Figure 1 Optimal Policy for a Two-Product Case



inventory systems with correlated demand arrivals.We will hence propose a better structural approxima-tion to the optimal policy by starting with the simplic-ity of the (s, c, S) policy and altering it to account forcorrelation in arrivals. In the spirit of notational simi-larity to (s, c, S) policies, we term our proposed policyas an (s, c, d, S) policy. Let us begin with the formaldefinition of the (s, c, S) policy. The formal definitionof an (s, c, S) policy for the N product case is as fol-lows:

DEFINITION 1 (AN S, C, S POLICY). A replenishmentorder is triggered whenever the inventory level Xi

of any product i falls to or below its threshold levelsi. Let O and O be sets such that O = {j: Xj ≤ cj}and O ¼ fj : Xj [ cjg. If ∃ i such that Xi ≤ si, thenthe size of the replenishment order quantity ξ isgiven by:

nj ¼Sj � Xj if j 2 O

0 if j 2 O

�ð8Þ

The above definition states that an order is trig-gered when the inventory level of a product i falls tosi. When an order is triggered all products j with theirinventory levels not greater than cj are also includedin the order. The set of products included in the orderis called O. Finally all products j in O are elevated totheir order up-to level Sj. We next define an (s, c, d, S)policy as follows:

DEFINITION 2 (AN S, C, D, S POLICY). A replenishmentorder is triggered whenever the inventory level Xi ofany product i falls to or below its threshold level si.Let O and O be sets such that O={j: Xj ≤ cj} andO ¼ fj : Xj [ cjg. If ∃i such that Xi ≤ si, then the sizeof the replenishment order quantity is then given by:

nj ¼ Sj�Xj� 1fO 6¼Xg maxk2OðSj�djÞðSk�XkÞ

ðSk�ckÞh in o

if j 2O

0 if j 2 O

(

ð9Þ

In the above definition the orders are triggered andthe products included in an order are just the same asthe (s, c, S) case. However, instead of elevating theinventory levels of all products j included in the orderto Sj, the (s, c, d, S) policy sometimes elevates them toa level that is lower than Sj in order to avoid unneces-sary storage costs. Due to joint order arrivals, it is bet-ter to keep inventory levels of products closer to oneanother rather than always elevate them always to Sjand this is where the cost savings occur. For eachproduct j in O the third term of Equation (9) gives theamount of the order up-to lowering. The third terms

calculates a linear interpolation between 0 and Sj � djand hence dj can be thought of the lowest order up-tolevel for product j, if it was included in the order. Aswas discussed previously, the linearity gives us a verysimple policy that is easy to implement. Our perfor-mance evaluations will later show that this simplicity,fortunately for us, turns out to be a very good approx-imation to the optimal policy as well.Due to the indivisibility of the products, the ordering

quantity will be rounded to the closest integer. Figure 2illustrates the structure of an (s, c, S) and the (s, c, d, S)policy for a case with two products having correlateddemand. In both cases, when the inventory level strikesor crosses the line segment between (si, sj) and (ci, sj),an order is placed for both products to replenish theinventory level up to (Si, Sj). However, under the(s, c, d, S) policy, when the inventory strikes or crossesthe line segment between (ci, sj) and (Si, sj), an order isplaced only for product j to elevate inventory level tothe line segment that connects (ci, dj) and (Si, Sj),where, i 6¼ j. It is easily seen that the (s, c, S) policy is aspecial case of the (s, c, d, S) policy with d = S.The visualization of the polices for the three-prod-

uct case, illustrated in Figure 3, provides a betterinsight into the structure of the (s, c, d, S) policy inhigher dimensions. Under both the (s, c, S) and the(s, c, d, S) policies, orders are only triggered whenone of the product inventory levels strikes or fallsbelow its order trigger level si. Let us take a three-product case and say the inventory level of product 3strikes s3 and hence triggering an order. Now whenan order is triggered we have three structurally dis-tinct possibilities (a) Both product 1 and product 2 areon or below their respective can order levels c1 and c2,(b) Product 1 is on or below c1 but product 2 is abovec2, (c) Both product 1 and product 2 have inventorylevels above their can order levels. Again the types ofproducts included in an order does not differ betweenthe two policies: (a) all three products are ordered; (b)only products 1 and 3 are ordered; (c) only product 3is ordered. What changes between the policies is thesize of the order and hence the inventory level afterthe order. Now in the case of (s, c, S) policies, as illus-trate in Figure 4, if a product is ordered, then theorder size is always Si�Xi. However, in the case of(s, c, d, S) policies the order size is given by Equation(9) which can be better understood by looking at itcase by case. In case (a) all three products are ordered,that is, O = Ω and hence from Equation (2), this doesnot differ from (s, c, S) order. In case (b), O = {1,3}and hence the order sizes for product 1 and 3 areðSi � diÞðS2 �;X2Þ

ðS2 � c2Þ less than Si � Xi, where i is 1 and 3. TheðSi � diÞðS2 �X2Þ

ðS2 � c2Þ is the inventory quantity that is unneces-

sary at the current time due to the presence of correla-tion. A quick insight into this unnecessary inventory



is the following. The presence of correlation tells usthat the lower the X2 value, the sooner the next orderwill occur and hence larger the unnecessary inventoryof product i. This unnecessary inventory incurs aholding cost that can be completely avoided using an(s, c, d, S) policy. In case (c) when only product 3 isordered, the (s, c, S) policy dictates an order quantity

of S3�s3. However, since there is positive correlation

between 1 and 3, we again have ðS3 � d3ÞðS1 �X1ÞðS1 � c1Þ as the

unnecessary inventory size. Similarly due to product3’s correlation with product 2, the unnecessary inven-

tory is ðS3 � d3ÞðS2 �X2ÞðS2 � c2Þ . The overall unnecessary inven-

tory is the maximum of these two quantities.

Figure 2 Illustration of a Two-Product (s, c, S) and (s, c, d, S) PolicyThe dotted arrows represent the inventory jumps caused by replenishments triggered when product 2 hits s2.

(a) (b) (c)

Figure 3 (s, c, d, S) Policy for the Three-Product Case

(a) (b) (c)

Figure 4 (s, c, S) Policy for the Three-Product Case



4. Quantifying the Performance ofPolicies

The goal of this section is to quantify the performanceof (s, c, d, S) policies. The exhaustive results pre-sented in this section establish that the (s, c, d, S)structure is extremely close to the optimal policy interms of its performance. With the objective ofevaluating the performance of the best (s, c, d, S) pol-icy, for all comparisons in this section, we use the(s, c, d, S) policy that is closest in value function tothe optimal policy. This comparison, while providingus insight into the value of the (s, c, d, S) structure,does necessitate the computation of the optimal pol-icy in the first place. Hence for multi-product systemswith a large number of product types, it is not feasibleto compute the (s, c, d, S) policies as in this section.Section 5 will focus on constructing, evaluating, andcomparing a proposed heuristic (s, c, d, S) policy forsuch large systems.This section first compares the optimal policy to the

(s, c, S) policy for several two- and three-productcases, including those with lead times and compoundPoisson demand arrivals. The rest of this section per-forms the evaluation of the (s, c, d, S) policies in threeways. Section 4.2 provides an exhaustive comparisonof both the (s, c, S) and the (s, c, d, S) policy againstthe optimal policy. Section 4.3 presents comparisonsagainst the P(s, S) and Q(s, S) policies proposed byViswanathan (1997). Finally, section 4.4 uses asimulation model to evaluate the performance of the(s, c, d, S) policy against the (s, c, S) policy when oneuses a average cost objective instead of the discountedcost objective. Due to the extreme computational runtimes in computing optimal policies for the three-product case, a significant fraction of the examples inthis section use the two-product case (Figure 4). How-ever, many examples of three product inventory sys-tems are also included. Section 5 considers a larger12-product system.

4.1. Evaluating the (s, c, S) PolicyTo quantify the cost savings yielded by the optimalpolicy over the (s, c, S) policy, a comparison is shownin Table 1 for various values of q{1,2}. The (s, c, S) pol-icy here is computed using the method by Federgruenet al. (1984) and is shown in the second column. Our

objective in these comparisons is not to show that theoptimal is better than the (s, c, S) since this is obvious.Here, and later when we compare a new class of poli-cies, we are primarily interested in understandinghow these policies structurally account for correlationin demand and how much this buys us in cost sav-ings. In order to do this we identify states Xm at whichthe largest of the value function differences occur(shown in column 3) since these indicate the statewhere the impact of the structural change in the pol-icy is largest. Column 4 titled Vopt(Xm) shows the opti-mal value function at this state and column 5, titledDscS(Xm), shows the difference between the valuefunction associated with the (s, c, S) policy and theoptimal value function at Xm. As is evident, while(s, c, S) can be considered as performing well for theindependent demand case, as the demand correlationincreases it tends to diminish in performance. How-ever, note that for the no joint arrival case, while theoptimal policy is itself an (s, c, S) policy with the val-ues (�1, 1, 6), the (s, c, S) policy computed by themethod in Federgruen et al. (1984) results in(�1, 1, 7). At the extreme case of perfectly correlatedarrivals, the performance of an (s, c, S) is significantlypoor in comparison to the optimal policy.Next we consider a three-product case with a posi-

tive lead-time Li = 0.2. Here, we take parameters simi-lar to those used in Atkins and Iyogun (1988). Thediscount factor a = 0.10, ordering costs are given byj0 = 150, ji = 20, mi = 0, penalty cost pi = 1200, andholding cost hi = 6. The marginal arrival rate for eachproduct is ki = 40. Table 2 again compares the differ-ence between the (s, c, S) policy and the optimal pol-icy for various values of q{1,2,3}. We set q{i}=(1�q{1,2,3})/3. As evident from Tables 1 and 2, the dif-ference between the (s, c, S) and the optimal areamplified as the dimension increases.Moving up in complexity, let us consider a case

with three-products and compound Poisson arrivals.We will take the same set of parameters used above,however, with ki = 20 and Li = 0. For modeling thecompound Poisson arrivals, for individual productdemand we will take 0.5, 0.3, 0.2 as the probabilitiesof the individual demand request being for products1, 2, and 3, respectively. For multi-product demandrequests, that is, joint arrivals we will use 0.5, 0.3, and0.2 as the probability that the demand is a request for

Table 1 Comparison of (s, c, S) Policies to Optimal Policies. Two-Product Case with No Lead Time

q{1,2} (s, c, S) Xm Vopt(Xm) DscS(Xm)

0.00 (�1, 1, 7) (3, �1) 352 1.370.20 (�1, 1, 7) (2, �1) 389 2.990.60 (�1, 2, 9) (4, �1) 443 11.201.00 (�1, 2, 10) (6, �1) 470 75.70

Table 2 Comparison of (s, c, S) Policies to Optimal Policies. Three-Product Case with Positive Lead Time

q{1,2,3} (s, c, S) Xm Vopt(Xm) DscS(Xm)

0.00 (10, 36, 53) (37, 10, 36) 6997 2480.60 (10, 36, 53) (36, 10, 37) 6618 3080.80 (10, 36, 53) (37, 36, 10) 6534 3191.00 (10, 36, 53) (38, 37, 11) 6451 373



1 unit, 2 units, and 3 units of all three products,respectively. This parameter set allows the marginaldemand distribution for each product to be the samefor all joint arrival probabilities as well as for zero-joint-arrival case. The computational results areshown in Table 3).

4.2. The (s, c, d, S) Policy and Its Dependence ofProblem ParametersThis section provides comparisons of the optimalpolicy, the (s, c, S) and the (s, c, d, S) policies fordifferent parameters with the objective to demon-

strate the robustness of the (s, c, d, S) approximation.Visualizations that are aimed at providing the readeran insight into how the policies depend on problemparameters are also provided. We restrict ourselves tothe two-product case here (Table 12).Table 4 and Figure 5 compare the approximations

yielded by (s, c, d, S) policies to the approximationsyielded by (s, c, S) policies. Notice that, by definitionof d, we allow d to take non-integer values. However,when using Equation (9) to compute the orderingquantity at a specific point, the ordering quantity is

Table 3 Comparison of (s, c, S) Policies to Optimal Policies. Three-Product Case with Compound Poisson Demand Arrivals

q{1,2,3} (s, c, S) Xm Vopt(Xm) DscS(Xm)

0.00 (�1, 21, 36) (�1, 21, 22) 6158 1540.60 (�1, 21, 36) (�1, 21, 22) 5578 2590.80 (�1, 21, 36) (�1, 21, 22) 5413 2921.00 (�1, 21, 36) (�1, 21, 22) 5098 346

Table 4 Poisson Demand with Base Parameter Set

q{1,2} (s, c, S) (s, c, d, S) Xm Vopt DscS DscdS

0.00 (�1, 1, 6) (�1, 1, 4.33, 6) (2, �1) 441.00 1.44 0.040.20 (�1, 1, 6) (�1, 2, 4.57, 6) (2, �1) 481.00 3.08 0.000.40 (�1, 2, 7) (�1, 2, 3.71, 6) (3, �1) 514.00 5.77 0.000.60 (�1, 2, 7) (�1, 2, 3.17, 7) (3, �1) 545.00 5.66 0.000.80 (�1, 2, 8) (�1, 2, 2.00, 7) (3, �1) 569.00 17.00 0.001.00 (�1, 2, 8) (�1, 2, 2.00, 7) (4, �1) 574.00 111.00 0.00

−2 0 2 4 6 8−2

0

2

4

6

8

X1

X2

q{1,2}=0

−2 0 2 4 6 8−2

0

2

4

6

8

X1

X2

q{1,2}=0.2

−2 0 2 4 6 8−2

0

2

4

6

8

X1

X2

q{1,2}=0.4

−2 0 2 4 6 8−2

0

2

4

6

8

X1

X2

q{1,2}=0.6

−2 0 2 4 6 8−2

0

2

4

6

8

X1

X2

q{1,2}=0.8

−2 0 2 4 6 8−2

0

2

4

6

8

X1

X2

q{1,2}=1

Figure 5 Base Parameter Set



rounded to the closest integer. For Table 4, in eachcase, we first find the point at which the maximumdifference between the optimal policy and the (s, c, S)policy is observed. This point is denoted by Xm andthe value of the optimal value function at this pointdenoted by Vopt. DscS and DscdS denote the differencebetween the optimal value function and the valueobtained by the (s, c, S) and (s, c, d, S) policies at Xm,respectively. In each of the figures we plot the(s, c, d, S) policy. Both the regression ‘order-upto’ lev-els and optimal points are plotted. The (s, c, d, S) pol-icy is prescribed to order-upto the closest discretepoint to the regression line.We first consider a symmetric two-product Poisson

demand problem with a demand arrival rate k = 10,holding and penalty cost for both products given byhi = 3 and pi = 80, ordering costs for both productsgiven by mi = 0.5, j0 = 10, ji = 5 and a discountingfactor of a = 10%. Let q{1,2} ranges from 0 to 1.Table 4 compares the performance of the (s, c, S)

and the (s, c, d, S) policy for the above parameter set.The table shows that the performance of the (s, c, S)policy compared to the optimal policy deteriorates athigh values of correlation. In contrast, the table showsthat the (s, c, d, S) policy is a close approximation tothe optimal policy, even at high values of correlation.Thus, the (s, c, d, S) approximation offers a significantimprovement over (s, c, S) policies.Figure 5 plots the structure of the optimal policy

and the (s, c, d, S) policy for the above parameter set.The figure shows that, even for the independentdemand case (q{1,2} = 0), the optimal policy does nothave the structure of an (s, c, S) policy. The lines inthis figure, which are linear approximations of thejump to inventory levels, illustrates the prescribed(s, c, d, S) policy. It is evident from this figure that the(s, c, d, S) policy closely tracks the optimal policy instructure. The perfectly correlated case (q{1,2} = 1)clearly illustrates the difference in the structure of theoptimal policy and the (s, c, S) policy. If demand forthe two products are perfectly correlated, then thesample paths will always move along the diagonal.The optimal policy recognizes this and places anorder so as to reach the diagonal, instead of placingan order upto S as prescribed by the (s, c, S) policy.Tables 5–7, show the impact of m on the (s, c, S) and

(s, c, d, S) policies. As can be seen, the (s, c, S) policyperforms quite well for the independent demand arri-val case. However, as the correlation in demandbetween the two products increases, the performanceof the (s, c, S) policy deteriorates. However, (s, c, d, S)policy continues to perform well for different valuesof m and demand correlations.Tables 8–10 measure the impact of changes in the

individual setup cost, holding cost, and the demandrate, respectively, for q{1,2} = 0.6. As seen, an increase

in the setup cost or the demand rate, or a decrease inthe holding cost rate leads to a deterioration in theperformance of the (s, c, S) policy. This is because a

Table 5 Impact of m with q{1,2} = 0

m (s, c, S) (s, c, d, S) Xm Vopt DscS DscdS

0.10 (�1, 1, 6) (�1, 1, 4.33, 6) (2, �1) 399.00 1.29 0.030.50 (�1, 1, 6) (�1, 1, 4.33, 6) (2, �1) 441.00 1.44 0.041.00 (�1, 1, 6) (�1, 1, 5.00, 5) (2, �1) 493.00 1.89 0.0010.00 (�1, 1, 5) (�1, 1, 3.57, 5) (2, �1) 1430.00 0.40 0.0020.00 (�1, 1, 4) (�1, 1, 4.00, 4) (2, �1) 2478.00 0.00 0.0040.00 (�1, 0, 3) (�1, 0, 3.00, 3) (2, �1) 4554.00 0.00 0.00

Table 6 Impact of m with q{1,2} = 0.6


0.10 (�1, 2, 8) (�1, 2, 3.17, 7) (3, �1) 479.00 14.66 0.000.50 (�1, 2, 7) (�1, 2, 3.17, 7) (3, �1) 545.00 5.66 0.001.00 (�1, 2, 7) (�1, 2, 3.17, 7) (3, �1) 628.00 5.71 0.0010.00 (�1, 1, 6) (�1, 2, 2.86, 6) (2, �1) 2120.00 12.20 0.0020.00 (�1, 1, 6) (�1, 1, 2.43, 5) (2, �1) 3770.00 19.30 0.0040.00 (�1, 1, 5) (�1, 1, 2.20, 4) (2, �1) 7040.00 18.80 0.00

Table 7 Impact of m with q{1,2} = 1


0.10 (�1, 2, 9) (�1, 3, 3.00, 7) (5, �1) 493.00 108.32 0.00

0.50 (�1, 2, 8) (�1, 2, 2.00, 7) (4, �1) 574.00 111.00 0.00

1.00 (�1, 2, 8) (�1, 2, 2.00, 7) (4, �1) 676.00 111.00 0.00

10.00 (�1, 2, 7) (�1, 2, 2.00, 6) (3, �1) 2520.00 126.00 0.00

20.00 (�1, 1, 6) (�1, 2, 2.00, 5) (3, �1) 4560.00 143.00 0.00

40.00 (�1, 1, 5) (�1, 1, 1.00, 4) (2, �1) 8650.00 167.00 0.00

Table 8 Impact of ji, i = 1, 2, with q{1,2} = 0.6

ji (s, c, S) (s, c, d, S) Xm Vopt DscS DscdS

2.00 (�1, 3, 6) (�1, 2, 3.20, 5) (4, �1) 464.00 2.58 0.008.00 (�1, 1, 8) (�1, 2, 3.31, 8) (4, �1) 610.00 14.00 0.00

Table 9 Impact of h with q{1,2} = 0.6

h (s, c, S) (s, c, d, S) Xm Vopt DscS DscdS

1.00 (�1, 4, 13) (�1, 5, 6.38, 12) (6, �1) 358.00 9.15 0.006.00 (�1, 1, 5) (�1, 1, 2.20, 4) (2, �1) 721.00 8.69 0.00

Table 10 Impact of k with q{1,2} = 0.6

k (s, c, S) (s, c, d, S) Xm Vopt DscdS DscdS

2 (�1, 0, 3) (�1, 0, 2.00, 2) (1, �1) 221.00 4.47 0.005 (�1, 1, 5) (�1, 1, 2.20, 4) (2, �1) 368.00 6.09 0.0010 (�1, 2, 7) (�1, 2, 3.17, 7) (3, �1) 545.00 5.66 0.0025 (�1, 4, 12) (�1, 4, 5.38, 11) (5, �1) 933.00 11.50 0.00



change in these parameters shifts the optimal policytoward the diagonal, while the (s, c, S) policy isinflexible to accommodate such a shift. The (s, c, d, S)policy again closely approximates the optimal policyfor various h, ji (j1 = j2) and k parameters.Figure 6 shows the impact of the change in demand

rate on the structure of the optimal. It shows thatwhen demand rate is small, the optimal policy has an(s, c, S) structure. As the demand rate increases theoptimal policy moves toward the diagonal and as aresult (s, c, S) departs further away from the optimal.Table 11 measures the impact of the asymmetry in

demand between the two products on the perfor-mance of the two policies. Here, r is defined as theratio of the probability of type-{2} demand arrival

over type-{1} demand arrival, that is, r ¼ qf2g

qf1g, andq{1,2} is set to 0.6. The asymmetry is also illustrated inFigure 7.We next relax the Poisson demand assumption to

consider compound Poisson demand. Again taking ademand arrival rate of k = 10, holding and penaltycost for both products given by h = 3 and pi = 80,ordering costs for both products given by mi = 0.5,j0 = 10, ji = 5 and a discounting factor of a = 0.1. Themass distribution of demand for each product is asfollows: f{1}(1) = f{2}(1) = 0.5, f{1}(2) = f{2}(2) = 0.3and f{1}(3) = f{2}(3) = 0.2. For a joint arrival ofdemand, the joint distribution is: f{1,2}(1,1) = 0.5,f{1,2}(2,2) = 0.3 and f{1,2}(3,3) = 0.2. The table shows asimilar trend for compound Poisson demand, that is,

0 5 10−2

0

2

4

6

8

10

12

X1

X2

λ=2

0 5 10−2

0

2

4

6

8

10

12

X1

X2

λ=5

0 5 10−2

0

2

4

6

8

10

12

X1

X2

λ=10

0 5 10−2

0

2

4

6

8

10

12

X1

X2

λ=25

Figure 6 Impact of k with q{1,2} = 0.6

Table 11 Different r with q{1,2}

r (s1,c1,S1) (s2,c2,S2) (s1,c1,d1,S1) (s2,c2,d2,S2) Xm Vopt DscS DscdS

1.00 (�1, 2, 7) (�1, 2, 7) (�1, 2, 3.17, 7) (�1, 2, 3.17, 7) (3, �1) 545.00 5.66 0.002.50 (�1, 2, 7) (�1, 2, 8) (�1, 2, 2.50, 6) (�1, 2, 4.43, 7) (3, �1) 544.00 10.70 0.00



the performance of the (s, c, S) policy is significantlyworse at high values of correlation as compared tolow values of correlation. The (s, c, d, S) policy con-tinues to be a good approximation to the optimal pol-icy for the compound Poisson demand case also.

4.3. Comparison to Other Policies Proposed inLiteratureIn this sub-section we compare the proposed(s, c, d, S) policy with other policies analyzed in priorliterature. We consider the P(s, S) and Q(s, S) policiesproposed by Viswanathan (1997). The Q(S, s) policyhas been shown to perform very well by Nielsen andLarsen (2005), when demand for the multiple prod-ucts is independent. We demonstrate that the intro-duction of correlation in demand can causesignificant deterioration in performance of the abovepolicies, and the (s, c, d, S) policy, which incorporatesdemand correlation, can improve policy performance.We use the same two and three product problem setswith the following problem parameters, the dis-counted factor is a = 0.10, j0 = 150, ji = 20, the pen-alty cost rate pi = 1200, the holding cost rate hi = 6,the proportional replenishment cost is zero, and Leadtime is Li = 0.2. The marginal arrival rate for eachproduct is ki = 40.The Q(s, S) policy for our numerical study is

obtained from the algorithm described in Nielsen andLarsen (2005). Since the marginal demand arrival rate

for each product is unchanged over different jointarrival probabilities, the value of Q, si, and Si remainunchanged. The P(s, S) policy is obtained accordingto the algorithm described in Viswanathan (1997). Forthe two-product case, the optimal Q(s, S) policy forour parameter set is Q = 68, si = 45, and Si = 49, whilethe optimal P(s, S) policy is T = 0.803, si = 43 andSi = 48. For the three products case, the best Q(s, S)policy is Q = 85, si = 40, and Si = 44, while the P(s, S)policy is T = 0.686, si = 40, and Si = 45.Note that P(s, S) policy described by Viswanathan

(1997) and the Q(s, S) policy analyzed by Nielsen andLarsen (2005) uses a average cost criteria, while weconsider the discounted cost criteria in this section.We use simulation to obtain the discounted cost forthe P(s, S) and Q(s, S) policies for the above policyparameters. We used a discrete-event simulation (c.f.Law and Kelton (2000)) based methodology. Thenumerical comparison of the (s, c, d, S) policy and theP(s, S) and Q(s, S) policies is provided in the next twotables. In Tables 13 and 14, we look at the point atwhich the value function of the corresponding (s, c, S)differs most from the optimal, and report that point asXm. The value function of the optimal policy is evalu-ated at Xm, and we compare all policies at Xm.For our parameter set, the above tables show that

P(s, S) and Q(s, S) policies, which ignore demand cor-relation, can be far off from the optimal. The differ-ence from the optimal policy from these policies

−2 0 2 4 6 8−2

0

2

4

6

8

X1

X2

r=1

−2 0 2 4 6 8−2

0

2

4

6

8

X1

X2

r=2.5

Figure 7 Impact of r with q{1,2} = 0.6

Table 12 Compound Poisson Demand

q{1,2} (s, c, S) (s, c, d, S) Xm Vopt DscS DscdS

0.00 (�1, 2, 7) (�1, 2, 7.00, 7) N/A0.20 (�1, 2, 8) (�1, 2, 6.47, 8) (3, �1) 663.00 0.95 0.000.40 (�1, 2, 9) (�1, 3, 5.83, 8) (4, �1) 713.00 5.77 0.000.60 (�1, 2, 10) (�1, 3, 4.50, 8) (3, �1) 757.00 17.30 0.000.80 (�1, 3, 10) (�1, 3, 3.00, 9) (4, �1) 786.00 13.60 0.001.00 (�1, 3, 11) (�1, 3, 3.00, 9) (5, �1) 793.00 40.20 0.00



which ignore demand correlation can be as high as20%. Whereas, in all instances, the (s, c, d, S) policythat incorporates demand correlation comes veryclose to the optimal policy.

4.4. Average Cost ComparisonThe objective of this section is to demonstrate that theperformance improvements yielded by the (s, c, d, S)structure on problems with correlation and jointreplenishment costs are not in anyway an artifact ofthe discounted cost objective and the improvementsextend to the average cost objective as well. The(s, c, d, S) structure primarily takes advantage of jointorder opportunities to save on ordering cost. Thisargument in general holds for the average cost objec-tive as well. Consider the two-product poissondemand problem example. Assume that the demandfor those two products are symmetric and the overallarrival rate k of the demand is 3, the storage costh = 1, and shortage cost p = 80. Ordering costs forboth products are set to be: variable ordering costmi = 0, joint fixed ordering cost j0 = 10, individualfixed cost ji = 8. In Table 15, we list the (s, c, S) and(s, c, d, S) policies and their respective average costsobtained by simulation. We see that the (s, c, d, S)does significantly better with increasing correlation,with cost reductions as much as 40%.Let us take the q = 1 case and examine it to provide

insight into why the (s, c, d, S)’s structural advantage

does not get wiped out in the long run, that is, in theaverage cost. Consider the initial state (4, �1). Underthe (s, c, S) policy the inventory levels cycle through:(4, �1) ? (4, 8) ? (�1, 3) ? (8, 3) ? (4, �1) (SeeFigure 8 for an visualization of this). During eachcycle, since the average total inventory level of thetwo products is 8 and unit holding cost is 1, the aver-age holding cost is then 8. There are two orders placedduring each cycle, for 9 units of product 1 and 9 unitsof product 2. Therefore, the ordering cost per cycle is2 � (10 + 8) = 36. Since the rate of the arrival for bothproducts is 3, the average cycle time is 9/3 = 3. There-fore, the average total cost during a cycle under the(s, c, S) policy is 36/3 + 8 = 20. While under the(s, c, d, S) policy the inventory is initially brought to(4, 4), then the inventory level cycles as follows(4, 4) ? (�1, �1) ? (8, 8) ? (�1, �1). We can showthe average inventory cycle time for this case is also 3

Table 13 2D Poisson Demand with Li = 0.2

q{1,2} (s, c, S)F (s, c, d, S) Xm Vopt DscS DscdS DQsS DPsS

0.00 (10, 38, 55) (10, 36, 39.46, 49) (39, 10) 5190 85 1 234 7460.60 (10, 38, 55) (10, 35, 37.10, 48) (39, 10) 5026 112 3 216 8980.80 (10, 38, 55) (10, 35, 36.44, 48) (39, 10) 4977 119 4 251 9381.00 (10, 38, 55) (11, 36, 36.50, 48) (40, 11) 4918 145 0 331 987

Table 14 3D Poisson Demand with Li = 0.2

q{1,2,3} (s, c, S)F (s, c, d, S) Xm Vopt DscS DscdS DQsS DPsS

0.00 (10, 36, 53) (10, 33, 36, 44) (37, 10, 36) 6997 248 8 341 7260.60 (10, 36, 53) (10, 33, 34, 43) (36, 10, 37) 6618 308 5 391 11040.80 (10, 36, 53) (10, 33, 34, 43) (37, 36, 10) 6534 319 9 464 11911.00 (10, 36, 53) (11, 33, 34, 43) (38, 37, 11) 6451 373 0 480 1283

Table 15 Two-Product Case: Comparison of Average Costs under(s, c, S) and (s, c, d, S) Policy

q1,2 (s, c, S) (s, c, d, S) VscdS VscS

0.00 (�1, 1, 6) (�1, 1, 6.00, 6) 20.21 20.210.20 (�1, 1, 7) (�1, 1, 5.36, 7) 19.45 19.300.40 (�1, 2, 7) (�1, 2, 4.17, 7) 18.86 18.870.60 (�1, 2, 7) (�1, 2, 3.50, 7) 18.21 25.110.80 (�1, 2, 8) (�1, 2, 2.55, 8) 17.64 23.621.00 (�1, 2, 8) (�1, 2, 2.00, 8) 16.66 20.00

X2

X1(-1,-1) S=8

S=8

X0=(4,-1)

(4,8)

(-1,3)(8,3)

83 mm

Figure 8 Inventory Evolution under the (s, c, S) for the PerfectlyCorrelated Case



and average holding cost is still 8. However, theordering cost within the cycle is (8 � 2 + 10) = 26, andtherefore the average total inventory cost is 8 + 26/3 = 16.66. The difference between the two policies isexactly the fixed joint cost per cycle.The q = 1 is an extreme case, and it is unlikely that

the inventory state would get stuck in one cycle or theother for non-perfect correlations. However, withincreasing correlations such opportunities to save onordering costs occur more often and their effects staylonger. This is evident from Table 15 as well. For theq = 1 case the long-term average cost could dependon the initial point in which simulation is begun,unlike for the other cases. The row in Table 15 corre-sponding to the perfect correlation case is obtainedusing the initial state (4, �1).

5. Algorithm to Construct Heuristic(s, c, d, S) Policies

The previous sections have demonstrated the per-formance of the (s, c, d, S) structure and its close-ness to the optimal policy. However, the (s, c, d, S)policies used for such comparisons was simply theclosest one to the optimal policy. The primaryobjective in such comparisons was to understandthe ability of the structure to capture the featuresof the optimal policy. This, however, will not be afeasible computational methodology to obtainusable policies based on the (s, c, d, S) structure,for inventory systems with a large number of prod-uct types (Figure 8).In this section, we present a heuristic method to

compute good (s, c, d, S) policies that achieve the costsaving by coordinating the ordering of the productsthat have joint arrival probabilities. The algorithm, inthe spirit of the one proposed in Federgruen et al.(1984), is to compute good (s, c, d, S) policies for largeinventory systems. We evaluate the (s, c, d, S) policiesprovided by our algorithm both in terms of closenessto our earlier computed (s, c, d, S) policies that wereclosest to the optimal policy and in terms of its perfor-mance against the (s, c, S) policy in a very largebenchmark inventory system considered in Atkinsand Iyogun (1988).The central idea to the dimension decoupling algo-

rithms (Federgruen et al. 1984, Silver 1974), for theefficient computation of good (s, c, S) policies, is toconsider each product separately but allow for specialordering opportunities that allow the inventory to getreplenished at lower fixed costs. These special order-ing opportunities are then iteratively computed as afunction of the frequency with which the other prod-ucts placed their orders. However, (s, c, S) policiesalways elevate the inventories to the level S for all

products ordered. In our case, however, we have toconsider correlations amongst the products to choos-ing the right level to bring inventories to. Based onthe heuristic (s, c, S) policy computed by Federgruenet al. (1984), we need to find an appropriate choice ofdi for product i.To choose di, let us first define a matrix D:

Di;j ¼ ci þ ð1� pi;jÞðSi � ciÞ; for i 6¼ j; ð10Þ

where pi,j is the probability of the joint demand forproduct i and product j. We then choose di = median{Di,j}j 6¼i.For numerical evaluation, let us first consider two-

product example in section 4.2, and reconstruct Table4 in Table 16 which compares both the average costsand discounted costs under the policy computed bythe algorithm and the one closest to the optimal policy(called “best” in Table 16). We see that, for both aver-age cost and discounted cost, the algorithm gives aclose approximation. The maximum difference in theinventory cost shown in Table 16 is <1%.Let us next consider the 12 product example from

Atkins and Iyogun (1988). The demand arrival rate ki,individual ordering cost ki and lead time Li for eachproduct are listed in Table 17. The storage cost for allthe products is 6 and the shortage cost is 1200. Thejoint ordering cost k0 is 150. Our objective is to evalu-ate under the presence of correlation and to avoiddefining an 12 9 12 covariance matrix, we takePðxijxkÞPðxjjxkÞ ¼

PðxijxmÞPðxjjxmÞ, for all products, where PðxijxjÞ denotes

the conditional probability of ith product’s arrivalgiven jth product arrives. We also take that there areno window shoppers, that is, P(x1 = 0, x2 = 0,⋯, x12 = 0) = 0. Table 18 shows the (s, c, d, S) policycomputed by our algorithm. We test for the differentjoint probability of demand arrivals and show theresults in Table 19.We simulate the inventory system with the com-

puted (s, c, d, S) policy and the (s, c, S) policy fromFedergruen et al. (1984). We consider different valuesof joint arrival probabilities. For example, we set theprobabilities of joint demand arrival for 9-th item and

Table 16 Two-Product Case: Comparison of the Computed (s, c, d, S)to the One Closest to the Optimal

Discounted obj. Average cost obj.

q{1,2} Best Algorithm Best Algorithm

0 440.6 442.1 42.5 42.60.2 480.9 482.6 46.4 46.70.4 515.1 517.2 50.0 50.00.6 545.0 547.2 52.8 52.90.8 568.2 572.4 55.2 55.61 577.1 578.2 56.2 56.5



10-th item to be 0.5 respectively, that is, P(jointdemand is for i-th item) = 0.5, for i = 9, 10. Using thealgorithm developed earlier in this section, theresulting heuristic d’s for the 12 items are(35.31, 49.31, 35.27, 30.27, 35.01, 61.50, 49.00, 49.00,65.00, 50.00, 51.00, 51.00).We simulate the discounted and the average inven-

tory cost for this system under both the (s, c, S) andthe (s, c, d, S) policies. Under the (s, c, S) policy, thediscounted cost is 2.59 9 104, and the average cost is2567.55, while using the (s, c, d, S) , the discountedcost is 2.48 9 104 and the average cost is 2424.13. Theimprovement of heuristic (s, c, d, S), compared toheuristic (s, c, S), is around 4.5%. The result of exam-ples for different joint arrival probabilities Q’s areshown in Table 19. Since the costs are simulationresults, the last column in Table 19, titled as r(VscS � VscdS), shows the standard error of differencesin the inventory costs under the two heuristic policies.We find that the improvement of the heuristics(s, c, d, S) is always significant with respect to thecommonly used heuristic (s, c, S). The magnitude ofimprovement ranges up to around 6% among ourexamples. In many retail industry segments, profitsaverage about 3% of sales (Fisher and Raman 1996).

Thus, a 6% cost reduction, although it might appearsmall, has the potential to double profits in manyretail industry segments.In section 4, we sought insights into how the

(s, c, d, S) policy changes with respect to changes inparameter values, using the two-product case. Weused the 12-product case and compute both Federgru-en et al. (1984)’s (s, c, S) heuristic and our (s, c, d, S)heuristic. Numerical comparisons show that theadvantage of using the (s, c, d, S) policy as opposedto the (s, c, S) policy, is robust in multiple-item inven-tory systems.2

6. Concluding Remarks

In this article, we have analyzed the optimal replen-ishment policies for multi-product stochasticinventory systems. The distinguishing feature of themulti-product inventory system we studied is theexistence of joint-replenishment costs and correlateddemand across multiple products. While prior litera-ture on multi-product stochastic inventory systemshas assumed independent demand, our focus was oncomputing and finding the structure of the optimalpolicies for multi-product inventory systems in which

Table 17 12-Product Example, Parameters

pdt 1 pdt 2 pdt 3 pdt 4 pdt 5 pdt 6 pdt 7 pdt 8 pdt 9 pdt 10 pdt 11 pdt 12

ki 40 35 40 40 40 20 20 20 28 20 20 20ki 10 10 20 20 40 20 40 40 60 60 80 80Li 0.2 0.5 0.2 0.1 0.2 1.5 1 1 1 1 1 1

Table 18 Heuristic (s, c, S) for 12-pdt Example

pdt 1 pdt 2 pdt 3 pdt 4 pdt 5 pdt 6 pdt 7 pdt 8 pdt 9 pdt 10 pdt 11 pdt 12

s 9 21 9 4 9 35 23 24 32 23 23 23c 34 44 31 26 29 51 37 37 48 36 35 35S 46 55 48 43 52 63 54 54 72 57 59 59

Table 19 12-pdt Example

Q (Qi: probability of i pdts arrive together) Values VscS VscdS r(VscS�VscdS)

(0, 0, 0, 0, 0, 0, 0.5, 0.5, 0, 0) Discounted 25926.20 24771.89 1690.25Average 2567.55 2424.13 13.10

(0, 0, 0, 0, 0, 0.5, 0, 0, 0, 0, 0, 0.5) Discounted 25364.24 24280.91 1389.71Average 2522.45 2402.61 15.48

(0, 0, 0.5, 0, 0, 0, 0, 0, 0, 0, 0, 0.5) Discounted 25735.57 24807.82 1917.37Average 2548.794 2444.04 15.64

(0, 0, 0.3, 0, 0, 0.3, 0, 0, 0.3, 0, 0, 0.1) Discounted 25305.98 24718.91 1171.12Average 2515.93 2423.22 14.27

(0, 0.2, 0, 0.2, 0, 0.2, 0, 0.2, 0, 0.2, 0, 0) Discounted 25342.56 24735.14 917.74Average 2519.08 2444.17 14.50

(0, 0, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1) Discounted 25254.72 24313.39 1153.20Average 2509.78 2404.86 15.69

(0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0, 0) Discounted 25536.30 24763.43 1177.95Average 2523.12 2453.56 15.43



demand arrivals are correlated across products. Real-izing that existing heuristic approximations in litera-ture fail to capture the structure of optimal policieswhen demand is correlated, we propose a heuristicpolicy called the (s, c, d, S) policy. Apart from closelyapproximating the optimal policy and performingbetter than other heuristics under correlated demand,the (s, c, d, S) policy also provides an insight intohow correlation can be leveraged for cost savings inthe inventory system.This study has important practical implications for

management of a multi-product inventory systemunder uncertain demand. It highlights the importanceof incorporating demand correlation across multipleproducts in managing inventory. The (s, c, d, S) pol-icy we proposed is intuitive and effective in incorpo-rating correlation into ordering decisions. We showthat demand correlation affects optimal policies intwo ways. First, managers need to group and consoli-date replenishment orders of different products tosave on the ordering cost. In particular, a replenish-ment order may include certain products that do notneed an immediate replenishment when consideredindividually. Second, inventory levels after replenish-ment should be set such that, when correlateddemand occurs, the inventories for different productsare likely to diminish and trigger replenishmentorders at around the same time. This way, replenish-ment orders across multiple products will be coordi-nated and, as a result, reduce inventory holding andordering costs.In terms of exciting future research direction, the

issue of constructing policies that can deal with autocorrelated demand would be very interesting. Extend-ing this work for more general holding cost functionsand possibly more general ordering costs are alsointeresting possibilities. Empirical analysis thatexplores the impact on product varieties and theanalysis of existing inventory management methodswould also provide very interesting insight.

Acknowledgments

The authors thank the Department Editor Eric Johnson, theanonymous senior editor, and two anonymous referees fortheir thoughtful and constructive feedback on the study.Haolin Feng would like to acknowledge the support fromthe National Natural Science Foundation of China (GrantNo. 71401187) and the Fundamental Research Funds for theCentral Universities.

Notes

1The detailed derivation and the proof for the lead timecase are available upon request.2More extensive numerical results are omitted due tolength limit. They are available upon request.

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Replenishment Policies for Multi-Product Stochastic Inventory