Contents lists available at ScienceDirect

Int. J. Production Economics

journal homepage: www.elsevier.com/locate/ijpe

Optimality of (s, S) policies in EOQ models with general cost structures

Sandun Pereraa,⁎, Ganesh Janakiramanb, Shun-Chen Niub

a School of Management, University of Michigan–Flint, Flint, MI 48502, USAb Naveen Jindal School of Management, The University of Texas at Dallas, Richardson, TX 75080, USA

A R T I C L E I N F O

Keywords:EOQ models(s, S) optimalityGeneral ordering/procurement cost structures

A B S T R A C T

The classical economic-order-quantity (EOQ) model is at the heart of supply chain optimization and the theoryof inventories. We study an extension of the original EOQ model that permits a minimal set of assumptions onthe ordering/procurement and holding/backorder costs, and establish necessary and sufficient conditions forthe existence of an optimal policy of the s S( , ) type. Our work lends theoretical credibility to the practice of usings S( , ) policies for virtually any cost structure of practical interest. Our proof is constructive and elementary, inthe sense that it is based on first principles that do not rely on advanced mathematical machinery. We also provethat an optimal policy, not necessarily of the s S( , ) type, always exists within our general EOQ framework.

1. Introduction

The economic-order-quantity (EOQ) model (Harris, 1913)1, alongwith the Newsvendor model, is at the heart of the practice of supplychain optimization and the theory of inventories. The EOQ modelassumes that demand arises continuously and at a constant rate.Several versions of this model of trading off inventory-holding costs(and, in some cases, shortage costs) with procurement/logistics costs(e.g., fixed ordering costs, volume discounts, etc.) are presented inbasic texts and courses in Operations Research and OperationsManagement. The typical analysis of the EOQ model involves assuminga policy with the following structure: Place a constant order quantityevery time the net-inventory level decreases to a threshold. The orderquantity and the threshold are then optimized. This approach usuallyleads to square-root formulas or algorithms to determine thesequantities (Porteus, 2002; Winston, 2004; Zipkin, 2000). In this paper,we take a step back and investigate the important issue of when/whysuch a policy structure—commonly referred to as s S( , )—is optimal inthe EOQ model.

We present an elementary proof for the existence of an optimalpolicy of the s S( , ) type in the EOQ model under general ordering/procurement and holding/shortage cost structures. By elementaryproof, we mean a rigorous proof from first principles that do notrequire significant mathematical machinery. This is the main contribu-

tion of our work. To shed some light on this contribution relative to theliterature, we will begin with a discussion both about our work and onthe “EOQ proofs” that exist in the literature.

The assumptions in the standard textbook version of the EOQmodel are as follows. The ordering cost includes a set-up cost plus alinear variable cost. Inventory holding costs are also linear.Backordering is either not allowed or is allowed with a linear shortagecost. This is the model for which convenient square-root formulas areavailable for the optimal re-order point, s, and the optimal re-orderquantity, S s− . Generalizations to accommodate volume discounts inordering costs are also presented frequently; in such cases, the optimalvalues of s and S s− do not take simple square-root forms, butalgorithms are presented for computing them. What is striking is thatthe use of an s S( , ) policy is usually taken as granted; and in fact, formaldiscussions of whether or not there exists an optimal policy of the s S( , )type in the EOQ model appeared in the literature only recently (relativeto the history of this model). For example, an s S( , )-optimality proof fora particular type of EOQ model without backordering is given inLippman (1971). In this paper, the ordering cost structure is that of aset-up cost for placing an order plus a set-up cost for every new truckused, where a truck has a finite capacity. Beyer and Sethi (1998)appear2 to be the first authors that attempted an s S( , )-optimality prooffor the EOQ model under a more general cost structure. Other extantoptimality analyses for the standard EOQ model and a variety of

http://dx.doi.org/10.1016/j.ijpe.2016.09.017Received 4 July 2016; Received in revised form 5 September 2016; Accepted 21 September 2016

⁎ Corresponding author.E-mail addresses: sperera@umich.edu (S. Perera), ganesh@utdallas.edu (G. Janakiraman), scniu@utdallas.edu (S.-C. Niu).

1 We refer the reader to Erlenkotter (1989, 1990) for interesting historical accounts on the EOQ model.2 Beyer and Sethi state: “Every OR textbook contains a calculus derivation of the EOQ formula. However, such a derivation does not establish formally that the EOQ formula provides

a production policy that minimizes the long-run average cost over the class of all admissible policies. Certainly there are many possible ways to prove the formula: But there does notseem to be a publication devoted directly to providing a mathematically rigorous proof of the famous formula. It is also possible that there are a number of papers dealing with complexinventory models, of which the simple lotsize model is a special case.”

International Journal of Production Economics 187 (2017) 216–228

Available online 23 September 20160925-5273/ Published by Elsevier B.V.

MARK

deterministic extensions can be found in Adelman and Klabjan (2005),Brimberg and Hurley (2006), Hassin and Meggido (1991), and Sun(2004).

In contrast, it is interesting to note that multiple proofs of s S( , )optimality in stochastic models exist and are celebrated (Scarf, 1960;Iglehart, 1963a, 1963b; Veinott, 1966; Zheng, 1991, 1994). Theseproofs are either for discrete-time (i.e., periodic review) models or forcontinuous-time models in discrete-space (i.e., demands and orderquantities are integer-valued). The proofs in this literature do notreadily accommodate the continuous-space, continuous-time featuresof the EOQ model.

The proof proposed by Beyer and Sethi (1998) is based on themethod of Quasi-Variational Inequalities (QVI). Their formulationallows a cost structure that is weaker than that in the standard EOQmodel; but the QVI approach makes it necessary for the authors tointroduce several technical assumptions on the cost functions (e.g.,continuity and differentiability). In a subsequent paper, Presman andSethi (2006) also apply the QVI method to analyze a stochastic versionof the EOQ model in which the demand process is the sum of adeterministic process and a compound Poisson process. In this latterpaper, the ordering-cost function is restricted to have the standardstructure of a fixed set-up cost plus a linear variable cost.

Our proof of s S( , )-optimality is entirely different and is built fromfirst principles, which allows us to work with a rather general modelwith minimal cost assumptions. The proof is based on a lower-bounding approach, and it involves two steps.

The first step is to consider a restricted class of policies that orderonly when net-inventory is non-positive and always raise it to a non-negative level upon ordering. That this restriction is without loss ofoptimality requires a proof. To prove this, we develop a coupling thattakes any policy that violates the desired property and constructs a newpolicy that satisfies the property while always having a lower amount ofinventory and a lower amount of backorder than the original. Thus, byconstruction, the latter policy has lower cumulative holding andshortage costs than the former at any time epoch. To complete theproof, the ordering costs accumulated by the two policies also need tobe considered. To this end, we develop an inequality that compares theordering costs for these two policies over two suitably-chosen nestedtime intervals, and combine it with the other cost comparison above toshow that the constructed policy has a long-run average total cost thatis no greater than the original.

In the second step, we show that the total costs for policies in therestricted class of policies above can be naturally decomposed intothose in successive “ordering cycles”, defined as time periods that spantwo consecutive zero-inventory epochs (i.e., time instants when the netinventory is zero). We then exploit this cost decomposition to show thatthe long-run average costs of policies in this restricted class arebounded from below by the infimum of the average costs over any ofsuch cycles. This lower bound then yields a simple characterization thatan s S( , ) policy is optimal under our cost assumptions if and only if anoptimal solution exists for the problem of minimizing the long-runaverage cost within the class of s S( , ) policies (i.e., whenever theinfimum above is attained by an s S( , ) policy).

To cover the scenario where the above infimum is not attained, thatis, to have a full closure for the s S( , )-optimality problem within theEOQ framework, we also provide a simple sufficient condition on theordering-cost function (lower semi-continuity) that ensures the ex-istence of an optimal s S( , ) policy. Furthermore, whenever this ex-istence does not prevail, we show that our approach enables us toconstructively establish that an optimal policy, not of the s S( , ) type,always exists.

Next, we briefly summarize the cost assumptions in our model, andindicate, in particular, the breadth of ordering cost structures that canbe accommodated in our formulation. We only assume that theordering-cost function is uniformly bounded from below by a strictlypositive constant for all strictly positive order quantities, and that the

inventory-holding and backordering cost functions are monotone.These assumptions are rather minimal, potentially the weakest reason-able assumptions within the EOQ framework. Hence, any cost structureof practical interest is subsumed in our formulation. Some well-knownexamples are all-unit discounts, incremental discounts, and truckloaddiscounts. These are discussed, for example, in Altintas et al. (2008),Benton and Park (1996), Chen (2009), Federgruen and Lee (1990), Liet al. (2004, 2012), and Zipkin (2000). Other variants of these coststructures, such as modified all-unit discounts (cf. Chan et al., 2002)and generalized truckload discount (cf. Li et al., 2004), are alsocovered. The widely-studied multiple set-up costs structures (Alpet al., 2014; Lippman, 1969, 1971) and the interesting quantity-dependent fixed costs scheme discussed in Caliskan-Demirag et al.(2012) are also allowed under our ordering-cost assumptions.Moreover, when a supplier imposes stringent constraints on the sizeof the orders, e.g., batch-ordering3, minimum-order-quantity (MOQ)4

and a combination of both batch-ordering and MOQ (cf. Zhu et al.,2015) restrictions, the cost of ordering the forbidden amounts could be(theoretically) considered to be infinite; our results also apply to theseand other scenarios where the ordering costs could be infinite in someregions. Finally, we emphasize that some of the interesting ordering-cost functions noted above are not increasing5, e.g., all-unit discounts6,batch-ordering, and MOQ. To the best of our knowledge, an attempt tostudy s S( , )-optimality for not-necessarily-increasing ordering-costfunctions does not exist in the literature.

The remainder of the paper is organized as follows. We present themodel, assumptions, and the main results formally in Section 2.Section 3 is devoted to proving the main results. Finally, in Section4, we present some concluding remarks.

2. Model formulation and the main results

In this section, we present our cost assumptions and state our mainresults. We first introduce the following notation:

λ:=the constant demand rate, λ > 0,c q( ):=non-negative cost for ordering q units, q≥0,g x( ):=non-negative holding/shortage cost rate when net-inventory

level is x, −∞ < x <∞, andI0:=net-inventory level at time 0.

In the standard EOQ model with backorder, it is assumed that c(·) iscomposed of a fixed set-up cost and a linear variable cost (i.e.,c q K νq1( ) = +q{ >0} , where K > 0 and ν ≥ 0 are constants), and g x( ) isof the form h x b xmax(0, ) + max(0, − ) for some positive constants hand b. In contrast, we only need the following minimal assumptions onc(·) and g(·).

Assumption 1. The function c(·) satisfies c(0) = 0 and K c q0 < ≤ ( )for all q > 0, where c(q) is not necessarily finite-valued.

3 This means that there is a constraint on the order size that stipulates that ordersmust be integer multiples of a constant B; see Chen (2000) and Li et al. (2004). Such aconstraint can be interpreted as saying that the ordering cost, c(·), is finite at all integermultiples of B, but is infinite elsewhere (see Fig. A.1 in Appendix A).

4 This type of order constraint is widely used. We refer the reader to Zhao andKatehakis (2006), where the authors note that: “Minimum Order Quantity (MOQ) iswidely used in many industries (e.g., apparel, pharmaceutical, and consumer packagedproducts). In these industries, customers (e.g., distributors or retailers) typically havetwo choices, either not to order from their suppliers or to order at least a minimumquantity, namely the MOQ”. Such a cost scheme can be reinterpreted as saying that thecost of ordering q units, c(q), is infinite when q is strictly positive but is less than MOQ,and is finite elsewhere (see Fig. A.2 in Appendix A). The reader is also referred to Hellionet al. (2012), Kiesmüller et al. (2011), and Zhou et al. (2007) for further discussionregarding the importance of the MOQ constraint.

5 Throughout this paper, by “increasing” and “decreasing”, we mean “non-decreasing”and “non-increasing”, respectively.

6 The ordering-cost function for the all-unit discount scheme is illustrated in Fig. A.3of Appendix A.

S. Perera et al. International Journal of Production Economics 187 (2017) 216–228

217

Assumption 2. The function g(·) is increasing on [0, ∞) anddecreasing on (−∞, 0], with g xlim ( ) = ∞x→±∞ .

Now, for a b0 ≤ ≤ , letC a b[ , ]π denote the total cost (i.e., the sum ofordering, holding and backordering costs) incurred over the timeinterval [a,b]7 for a given policy π . Here, a policy π is a specificationof an infinite sequence of ordering time instants and order sizes. Wewill measure the performance of a policy π by its long-run average cost,which is defined as

f C TT

= lim sup [0, ] .π

T

π

→∞ (1)

Our objective is to minimize f π over an admissible policy class, whichwe specify next.

Let tnπ and qn

π respectively be the n-th ordering epoch and the size ofthe n-th order under a given policy π . Observe that if tlim < ∞n n

π→∞ ,

then f = ∞π 8. Hence, we will restrict our attention to policies thatsatisfy the condition that tlim = ∞n n

π→∞ ; and formally define an

admissible policy as follows:

Definition 1. An admissible policy π is a sequence t q n{( , ): ≥ 1}nπ

nπ ,

where t ≥ 0π1 , t t<n

πnπ+1, tlim = ∞n n

π→∞ , and q > 0n

π for all n.

Let Π denote the set of all admissible policies. Our goal is to identifyconditions under which an optimal policy exists in Π; that is, we seek apolicy π* ∈ Π that satisfies f f= infπ

ππ*

∈Π . In addition, of particularinterest is the question of whether or not the search for an optimalpolicy in Π can be limited to policies of the s S( , ) type, where an s S( , )policy is defined to be one that raises the net-inventory level to S everytime it decreases to s.

Next, observe that for a given s S( , ) policy with s=x and S=y, wherex y< , the long-run average cost α x y( , ) is given by

∫α x y

c y x g z dzλ

y x λ( , ) =

( − ) + ( )

( − )/.

x

y

(2)

Let α α x y*≔inf ( , )x y x y{( , ): < } ; and note that, in general, α* may not befinite. To have a clean setup, we shall assume that there exists at leastone policy π in Π with f < ∞π . Clearly, this assumption can be madewithout loss of generality; and in particular, it implies that there mustexist x y<0 0 such that c y x( − )0 0 , the right limit of g(·) at x0, and the leftlimit of g(·) at y0 are all finite. It then follows from Assumption 2 thatα x y( , ) < ∞0 0 , which ensures the finiteness of α*. In the same spirit, wewill further assume that if I0 is positive, then the left limit of g(·) at I0 isfinite.

We are now ready to state the main results of this paper. Unlessexplicitly indicated otherwise, we assume henceforth that Assumptions1 and 2 hold. The first result establishes a lower bound for f π for allπ ∈ Π.

Theorem 1. The long-run average cost of any policy π ∈ Π isbounded from below by α*.

It follows immediately from Theorem 1 that if there exists a policyin Π whose long-run average cost exactly equals α*, then that policy isoptimal. In particular, if a minimizer x y( *, *) for the function α(·, ·)exists in the set x y x y{( , ): < }, then the s S( , ) policy with s x= * andS y= * is optimal in Π. Thus, the following characterization holds.

Corollary 1. An optimal policy of the s S( , ) type exists in Π if and only

if the function α(·, ·) has a minimizer.

Our second result provides a simple sufficient condition for theexistence of a minimizer for α(·, ·).

Theorem 2. If c(·) is lower semi-continuous9, then, α(·, ·) isguaranteed to have a minimizer; hence, there exists an s S( , ) policythat is optimal in Π.

Since a minimizer for α(·, ·) may not exist, the remaining question iswhether or not there exists a policy in Π (not necessarily of the s S( , )type) with long-run average cost α*. Our final result answers thisquestion in the affirmative.

Theorem 3. An optimal policy always exists in Π.

3. Proofs of the theorems

3.1. Proof of Theorem 1

The first step in our proof is to restrict consideration from the classof all policies Π to a subset of policies that are more amenable toanalysis. In light of Assumption 2, an immediate intuition is that anyattempts to maintain the net-inventory level “as close to zero aspossible” should result in a reduction in total cost. When backorderingis not allowed, this intuition can be formalized as: Order only when theinventory level decreases to zero. Indeed, in that case, it is a well-known result (Beyer and Sethi, 1998; Brimberg and Hurley, 2006;Lippman, 1971; Porteus, 2002; Zipkin, 2000) that restricting attentionto this smaller policy class, which we denote by Π p

0 , can be madewithout loss of generality. When backordering is allowed, it is a naturalconjecture that the corresponding class of policies to consider is Π0,defined as policies that order only when net-inventory is non-positiveand always raise the inventory to a non-negative level upon ordering.(Clearly, Π ⊂ Πp

0 0.) The following lemma rigorously confirms thisconjecture10.

Lemma 1. It is sufficient to restrict attention to policies in Π0; that is,

f finf = inf .π

π

π

π

∈Π ∈Π0

Proof. Clearly, it is sufficient to show that for a given π ∈ Π⧹Π0, there

exists a corresponding policy π ∈ Π∼0 that satisfies f f≤π π∼

.For n ≥ 1, denote by xn

π and ynπ , respectively, the net-inventory levels

just before and after the n-th order under a policy π in Π. Note that bydefinition, q y x= −n

πnπ

nπ . Observe that for any n ≥ 1, the vector x y( , )n

πnπ

associated with the n-th order must belong to one of the following threecategories: (i) x y0 < <n

πnπ , (ii) x ≤ 0n

π and y ≥ 0nπ with x y<n

πnπ , or (iii)

x y< < 0nπ

nπ . Clearly, every order for a π ∈ Π0 is by definition in

Category (ii). In contrast, any π ∈ Π⧹Π0 must have at least one orderin either Category (i) or Category (iii).

Let π be an arbitrary policy in Π⧹Π0; and let π∼ be the policyconstructed from π according to the following prescription:

t txλ

q q x

t tyλ

q q y

t t q q

= + and = , if > 0;

= + and = , if < 0; and

= and = , otherwise.

nπ

nπ n

π

nπ

nπ

nπ

nπ

nπ n

π

nπ

nπ

nπ

nπ

nπ

nπ

nπ

∼ ∼

∼ ∼

∼ ∼

For t ≥ 0, denote by I t( )π the net-inventory level at time t under apolicy π in Π (with I I(0) =π

0 for all π ∈ Π). Clearly, the inventory

7 Similar notation for other types of intervals, namely a b[ , ), a b( , ], and (a,b), will beadopted throughout this paper.

8 Suppose that t Mlim = < ∞n nπ

→∞ for a policy π. Then, we have that t M≤nπ for all

n ≥ 1, and this implies that for every N, the number of orders placed in the time intervalM[0, ] exceeds N. For any T M> , let N T= 2. Then, we have C T KT[0, ] ≥π 2. Therefore,

C T T KT[0, ]/ ≥π for all T M> . Letting T → ∞ then yields f π is infinite for all π withtlim < ∞n n

π→∞ .

9 A function h: → [ − ∞, ∞]n is called lower semi-continuous at x0 if for every ε > 0there exists a neighborhood B ⊂ n of x0 such that h h εx x( ) ≥ ( ) −0 for all Bx ∈ . Thefunction h is said to be lower semi-continuous on E ⊂ n if it is lower semi-continuous atevery point in E.

10 When backordering is allowed, the necessary argument to extend the result by theabove-cited authors is new, and not immediate.

S. Perera et al. International Journal of Production Economics 187 (2017) 216–228

218

trajectory I t t{ ( ), ≥ 0}π is completely determined by I0 and thesequence t q n{( , ), ≥ 1}n

πnπ . The inventory trajectories under both π

and π∼ are illustrated in Fig. 1. It is easily seen from this figure that ifx > 0n

π for any n, then x = 0nπ∼ (see, e.g., epochs t π

2 and t π2∼); and similarly

that if y < 0nπ for any n, then y = 0n

π∼ (see, e.g., epochs t π8 and t π

8∼).

Therefore, π ∈ Π∼0 by construction.

We now compare the costs under π and π∼. Consider a given timeinterval T[0, ]. Observe that if an order in π is relocated to arrive at onein π∼, then this corresponding order in π∼ could be either a postpone-ment or an early placement of the original. Consequently, the totalnumber of orders under π∼ in T[0, ] can be larger than that of π over thesame interval. To facilitate cost comparison, we will therefore considerthe total cost under π for a possibly larger interval T δ[0, + ], where δ isan appropriately-chosen non-negative increment to T that depends onboth π and T.

Our intent is to pick δ so as to ensure that every order placed by π∼

within T[0, ] originates from an order placed by π within T δ[0, + ]. Forthis purpose, let τ T( )∼π be the epoch at which the last order under π∼ in

T[0, ] is placed; let τ T( )π be the epoch at which the “corresponding”order under π (the one with the same order index) is placed; anddenote by y T( )π the net-inventory level under π immediately after theplacement of this corresponding order. Then, we will let δ y T λ= ( ( )) /π − ,where y T y T( ( )) ≔ − min( ( ), 0)π π− (the negative part of y T( )π ), and claimthat the following inequality holds:

C T C T y T λ T[0, ] ≤ [0, + ( ( )) / ] for all ≥ 0.π π π −∼(3)

Before proving (3), we will first show that

τ T T y T λ( ) ≤ + ( ( )) /π π − (4)

always holds. Obviously, (4) is valid if τ T T( ) ≤π . Suppose on the otherhand that T τ T< ( )π . Then, since τ T T( ) ≤∼π by assumption, we neces-sarily have y T( ) < 0π (e.g., when T is located in the interval t t[ , )π π

8 8∼

inFig. 1), implying that I T( )π must not be positive. Hence, we haveλ τ T T y T y T( ( ) − ) ≤ − ( ) = ( ( ))π π π −; and this is equivalent to (4).

We now prove (3). To facilitate the argument, it will be convenientto split the total cost into two parts and then compare them separately.For a b0 ≤ ≤ and any given π , let R a b[ , ]π and H a b[ , ]π be thecumulative ordering cost and, respectively, the cumulative sum of bothholding and shortage costs over the time interval a b[ , ]. By definition,we have C a b R a b H a b[ , ] = [ , ] + [ , ]π π π . Recall that τ T( )∼π is the lastordering epoch under π∼ in T[0, ]. It follows that

R T R τ T R τ T R T y T λ[0, ] = [0, ( )] = [0, ( )] ≤ [0, + ( ( )) / ],∼π π π π π π π −∼ ∼(5)

where the inequality is due to (4). Clearly, we also haveH T H T H T y T λ[0, ] ≤ [0, ] ≤ [0, + ( ( )) / ]π π π π −∼

. Summing this inequality

and (5) yields (3).We are finally ready to show that f f≤π π∼

holds under ourconstruction. If f = ∞π , we obviously have nothing to prove; we willtherefore assume that f π is finite.

We begin by noting that (3) can be rewritten as:

C TT

C T y T λT y T λ

T y T λT

[0, ] ≤ [0, + ( ( )) / ]+ ( ( )) /

+ ( ( )) / .π π π

π

π−

−

−∼

(6)

Upon taking lim sup, (6) yields

ff λ

y TT

≤ 1 + 1 lim sup ( ( )) .π

πT

π

→∞

−∼

(7)

Clearly, if y T( )π is uniformly bounded from below for all T, then (7)implies11 that f f≤π π∼

. (Note in particular that if backordering is notallowed, then this conclusion is immediate.) Therefore, to complete theproof, it is sufficient to show that for any given increasing sequenceT{ }n n=1

∞ satisfying Tlim = ∞n n→∞ , y Tlim ( ( )) = ∞nπ

n→∞− , and y T( ( ))π

n− is

strictly increasing in n with y T( ( )) > 0π1

− (the last requirement iswithout loss of generality since we are concerned with lim sup), we have

y TT

lim sup( ( ))

= 0.n

πn

n→∞

−

(8)

In words, this means that whenever there exists a sequencey T n{ ( ), ≥ 1}π

n that strictly descends to −∞ as n → ∞, the averagegrowth of the magnitude y T( ( ))π

n− is negligible relative to that of Tn.

Since g(·) is assumed to satisfy g xlim ( ) = ∞x→−∞ and f π is finite, it isintuitive that (8) should hold. We next formalize this intuition andcomplete the proof by establishing (8).

First, observe that whenever y T( )π is negative (e.g., when T t= π8 in

Fig. 1), we have τ T τ T y T λ( ) = ( ) − ( ( )) /∼π π π − . This scenario is illustratedin Fig. 2 (T′ in the figure indicates another possible location for T).From this figure, we see that the inventory trajectory under π in theinterval τ T τ T[ ( ), ( ))∼π π (with the right endpoint open) is no higher thanthe diagonal border of the shaded region spanning the same interval.That is, I t λ t τ T( ) ≤ − ( − ( ))∼π π holds for any t in τ T τ T[ ( ), ( ))∼π π . It theneasily follows that the shortage cost under π during τ T τ T[ ( ), ( ))∼π π is noless than g y T y T λ( ( )/2)(( ( )) /2)(1/ )π π − (see corresponding interval mar-kers in Fig. 2). Therefore,

Fig. 1. Construction of a policy π ∈ Π∼0 from a given π ∈ Π⧹Π0.

11 In a previous version of this paper, the requirement of having a uniform lowerbound was introduced. The authors are grateful to two anonymous reviewers of thatearlier version for demonstrating that this requirement is not necessary. Their contribu-tions led to a significantly shortened proof of Theorem 1.

S. Perera et al. International Journal of Production Economics 187 (2017) 216–228

219

g y T y T λ C τ T y T λ τ T

C τ T

C T y T λ

( − ( ( )) /2)( ( )) /(2 ) ≤ [ ( ) − ( ( )) / , ( )]

≤ [0, ( )]

≤ [0, + ( ( )) / ],

π π π π π π

π π

π π

− − −

− (9)

where the last inequality is again due to (4).Now, for our given π , let T{ }n n=1

∞ be a sequence satisfying thestipulations in conjunction with (8). (An example can be constructedto show that such sequences could exist.) Then, after setting T T= n in(9) and dividing the resulting inequality by T y T λ+ ( ( )) /n

πn

− , we see, inlight of the assumed finiteness of f π , that for any ϵ > 0,

g y T y T λT y T λ

f( − ( ( )) /2)( ( )) /(2 )

+ ( ( )) /≤ + ϵ

πn

πn

nπ

n

π− −

− (10)

holds for all sufficiently large n; and furthermore, g y T( − ( ( )) /2)πn

− isnecessarily finite. Since y T( ( ))π

n− is positive, a rearrangement of (10)

now yields

⎡⎣⎢

⎤⎦⎥λ

g y Tf

Ty T

1 ( − ( ( )) /2)2( + ϵ)

− 1 ≤( ( ))

.π

nπ

nπ

n

−

−(11)

Finally, since g xlim ( ) = ∞x→−∞ , we can further assume without lossof generality that the left-hand side of (11) is positive. It follows that

⎡⎣⎢

⎤⎦⎥

y TT

λg y T

f0 <

( ( ))≤

( − ( ( )) /2)2( + ϵ)

− 1 .π

n

n

πn

π

− − −1

(12)

Letting n → ∞ in (12) then completes the proof. □

Given the reduction established in Lemma 1, the next step is toanalyze the average costs of policies in Π0. Our approach is to first

decompose the total cost over a given interval under a policy into thoseover successive ordering cycles, defined as intervals that span twoconsecutive zero-inventory epochs; and then bound the total cost inevery ordering cycle from below by α* times the length of the samecycle.

We begin with an analysis of the total costs in successive orderingcycles. Consider an arbitrary policy π in Π0. Let u t x λ≔max[0, + / ]π π π

1 1 1and v t y λ≔ + /π π π

1 1 1 ; and for i ≥ 2, let u t x λ≔ + /iπ

iπ

iπ and v t y λ≔ + /i

πiπ

iπ . We

will formally refer to the closed time interval u v[ , ]iπ

iπ , i ≥ 1, as the i-th

ordering cycle. Note that, since π ∈ Π0, we have x ≤ 0iπ , y ≥ 0i

π , andx y<i

πiπ for all i ≥ 1; therefore, these intervals are all well defined (and

have positive durations). It is also easily seen that v u=iπ

iπ+1 holds for all

i ≥ 1. These notations are illustrated in Fig. 3.Next, let

∫κ c y x g z dzλ

i≔ ( − ) + ( ) , for ≥ 1.iπ

iπ

iπ

x

y

iπ

iπ

(13)

For i ≥ 2, we will assign (or define) κiπ as the total cost associated with

the i-th order, or alternatively with the i-th ordering cycle12. For i=1,

observe that t x λ+ /π π1 1 could be strictly negative so that u v[ , ]π π

1 1 may notbe a “full” cycle; therefore, we should and will assignδ c y x H u v≔ ( − ) + [ , ]π π π π π π

1 1 1 1 1 ( κ≤ π1 ) as the total cost associated with the

first ordering cycle.Given the above cost assignments, we will next establish a lower

bound for the total cost over a time interval. Surprisingly, for a giventime epoch T, instead of working with the default choice of interval

T[0, ], the key idea turns out to be the consideration of a suitably-enlarged time interval. Details are now developed.

Consider the last order in T[0, ] under π . For t ≥ 0, denote by N t( )π

the total number of orders in t[0, ]; formally,

N t n t t( )≔max{ ≥ 1: ≤ }.πnπ

Whenever N T( ) ≥ 1π , denote by vN Tπ

( ) the right endpoint of the orderingcycle associated with the N T( )π -th order; that is, let v v≔N T

πN Tπ

( ) ( )π .Clearly, we have v T I T λ= + ( )/N T

π π( ) .

We will next examine the total cost under π over the intervalT I T λ[0, + ( ( )) / ]π + , where I T I T( ( )) ≔max( ( ), 0)π π+ (the positive part of

I T( )π ). The motivation behind this enlargement of T[0, ] is thatwhenever vN T

π( ) overshoots T, which occurs if and only if I T( ) > 0π ,

Fig. 2. Lower bound for shortage cost over τ T τ T[ ( ), ( ))∼π π .

Fig. 3. Illustration of ordering cycles.

12 Note that the ordering cycles are not disjoint; hence, the language here may appearinappropriate. However, the intersection between any two consecutive cycles is a singlepoint. Therefore, the total holding and shortage costs over any time interval are notaffected by these intersections. If we adopted disjoint ordering cycles, then the indexingof the successive ordering costs would be complicated by the fact that orders couldappear at either end of a cycle (see Fig. 3). Our cost assignment scheme here is intendedto overcome this notational complication, by allowing intersections in the definition ofordering cycles.

S. Perera et al. International Journal of Production Economics 187 (2017) 216–228

220

the entire N T( )π -th ordering cycle would be included in the costcalculation, and this facilitates the development of a “clean” lowerbound.

It is easily seen that13

∑C T I T λ H u δ κ H v T T[0, + ( ( )) / ] = [0, ) + + + [min( , ), ],π π π π π

i

Nπ T

iπ π

N Tπ+

1 1=2

( )

( )(14)

where we have assumed without loss of generality that T is sufficientlylarge so that N T( ) ≥ 2π . Observe further that, for i ≥ 2, the average costincurred in the i-th ordering cycle is given by κ v u/( − )i

πiπ

iπ . Since

v u y x λ− = ( − )/iπ

iπ

iπ

iπ , it follows from (2) and (13) that

κ α x y v u α v u= ( , )( − ) ≥ *( − )iπ

iπ

iπ

iπ

iπ

iπ

iπ . Noting that the first two and

the last terms on the right-hand side of (14) are non-negative, we thenhave

⎛⎝⎜

⎞⎠⎟

∑C T I T λ κ α v u

α T I Tλ

u

[0, + ( ( )) / ] ≥ ≥ *( − )

= * + ( ) − .

π π

i

N T

iπ

N Tπ π

ππ

+

=2

( )

( ) 2

2

π

(15)

Since π ∈ Π0, it is easily seen that if I ≤ 00 (i.e., u = 0π1 ), then u q λ≤ /π π

2 1 ;and that if I > 00 (i.e., u > 0π

1 ), then

u u q λ I q λ= + / =( + )/ .π π π π2 1 1 0 1

Therefore,

uI q

λ≤

( ) +.π

π

20

+1

(16)

Combining (15) and (16) then yields the following lemma.

Lemma 2. For any policy π ∈ Π0,

⎛⎝⎜⎜

⎞⎠⎟⎟C T I T λ α T I T

λI q

λ[0, + ( ( )) / ] ≥ * + ( ) −

( ) +π ππ π

+ 0+

1

(17)

holds for all sufficiently large T.

Recall that our aim is to show that f α≥ *π . In light of thisobjective, we see that the significance of Lemma 2 is that it uncoversthe fact that the successful development of a tight lower bound for theaverage cost under π hinges on the asymptotic behavior of I T T( )/π asT → ∞. We now take up this task.

Dividing both sides of (17) by T I T λ+ ( ( )) /π + and using the fact thatI T I T I T( ) = ( ( )) − ( ( ))π π π+ − lead to

αC T I T λ

T I T λI T λ

T I T λI q λ

T I T λ

I T λT

I q λT

1*

[0, + ( ( )) / ]+ ( ( )) /

≥ 1 − ( ( )) /+ ( ( )) /

−(( ) + )/

+ ( ( )) /

≥ 1 − ( ( )) / −(( ) + )/

.

π π

π

π

π

π

π

π π

+

+

−

+0

+1

+

−0

+1

Upon taking lim sup, this yields

fα λ

I TT* ≥ 1 − 1 lim sup ( ( )) .

π

T

π

→∞

−

(18)

Finally, we claim that whenever f π is finite, we have

I TT

lim sup ( ( )) = 0.T

π

→∞

−

(19)

The proof of this claim is similar to that for (8). Suppose that there exista positive ϵ and an increasing sequence T{ }n n=1

∞ with Tlim = ∞n n→∞ thatsatisfy I T T( ( )) / > ϵπ

n n− for all n. Consider the time interval

T I T λ T[ − ( ( )) / , ]π − (which would be a single point when I T( ) ≥ 0π ),and observe that

C T H T I T λ T

g I T I T λ

[0, ] ≥ [ − ( ( )) / , ]

≥ ( − ( ( )) /2)(( ( )) /2)(1/ ).

π π π

π π

−

− − (20)

After setting T T= n in (20), dividing both sides of the resultinginequality by Tn, and then taking lim sup, we obtain

fλ

g I T∞ >ϵ

≥ 12

lim sup ( − ( ( )) /2).π

n

πn

→∞

−

(21)

Since the right-hand side of (21) is unbounded when n → ∞, we havearrived at a contradiction. This establishes (19) (because ϵ is arbitrary),which together with (18) imply f α≥ *π ; and hence the proof ofTheorem 1 is complete.

3.2. Proof of Theorem 2

Consider any x y( , ) with x y< . First, observe from (2) andAssumption 2 that when x > 0 or y < 0, we have α x y α y x( , ) ≥ (0, − )or α x y α x y( , ) ≥ ( − , 0), respectively. Hence, we can limit our search ofx y( *, *) to x y x y x y≔{( , ): < and ≤ 0 ≤ }0 . We will next argue that wecan further restrict the search to a closed and bounded subset of 0.The idea is to show that α x y( , ) is uniformly bounded from below wheny x− is sufficiently large.

For a given q > 0, let x y( , ) be any vector in 0 such that y x q− = .Then, from (2), we have

∫ ∫ ∫α x y

c q g z dzλ

q λ

g z dz

q

g z dz

q( , ) =

( ) + ( )

/≥

( )+

( ).

x

y

x

y0

0

(22)

Observe that with q fixed, we have either y x≥ − or x y− > . We willnext examine these two scenarios.

Suppose first that y x≥ − . Clearly, (22) implies that

∫α x y

g z dz

q( , ) ≥

( ).

y

0

(23)

Since g z( ) → ∞ as z → ∞, we have that for a given M > 0, there exists apositive z0 such that g z M( ) > for all z z> 0. It follows that, for all y z> 0,

∫ ∫g z dz M dz M y z( ) > = ( − ).z

y

z

y

00 0

This implies that

∫ ∫g z dz g z dz M y z( ) ≥ ( ) > ( − ).y

z

y

00

0

Dividing this inequality by q and using (23) then yield

⎛⎝⎜

⎞⎠⎟α x y M y

qzq

( , ) > − .0

Observe that the condition y x≥ − implies that y y x q2 ≥ − = , ory q/ ≥ 1/2; therefore,

⎛⎝⎜

⎞⎠⎟α x y M

zq

( , ) > 12

− .0

(24)

Next, for any ε > 0, there exists a q > 00 such that z q ε/ <0 wheneverq q> 0. Hence, from (24), we have α x y M ε( , ) > (1/2 − ) whenever q q> 0and y z> 0. Note that, when y x≥ − , the requirement q q z> max( , 2 )0 0guarantees that q q> 0 and y z> 0. Therefore, α x y M ε( , ) > (1/2 − ) holdsif q q z> max( , 2 )0 0 and y x≥ − .

The analysis for the other case with x y− > is parallel. Again, from(22), we have

∫α x y

g z dz

q( , ) ≥

( ).x

0

An argument similar to that for the previous case then shows that thereexist positive constants 1 and z1 such that α x y M ε( , ) > (1/2 − ) holds ifq q z> max( , 2 )1 1 and x y− > . This now allows us to conclude that

13 If u = 0π1 , the interval u[0, )π

1 is vacuous; otherwise, this interval does not containany order.

S. Perera et al. International Journal of Production Economics 187 (2017) 216–228

221

α x y M ε( , ) > (1/2 − ) holds if q y x q q z q z= − > ≔ max ( , 2 , , 2 )m 0 0 1 1 , re-gardless of the relative sizes of x− and y.

Next, recall that there exists an x y( , )0 0 such that α x y( , ) < ∞0 0 (seethe paragraph following (2)). Without loss of generality, we can furtherassume that x y( , )0 0 is in 0. Then, since M and ε are arbitrary, they canbe chosen to satisfy M ε α x y(1/2 − ) ≥ ( , )0 0 . Therefore, for anyx y( , ) ∈ 0 such that y x q− > m, we have that α x y α x y( , ) > ( , )0 0 . Sincex y( , ) ∈0 0 0, we have established that

α x y α x yinf ( , ) = inf ( , ),x y x y< ( , )∈ (25)

where x y y x q x y≔{( , ): 0 < − ≤ and ≤ 0 ≤ }m .Recall that α x y( , ) is originally defined for x y< . This original

domain can be extended to include (0, 0) with the convention thatα(0, 0) = ∞. That is, we now consider α x y( , ) on the closed and bounded(i.e., compact) set = ∪ (0, 0). Since c(·) is lower semi-continuous,and y x− is continuous on , it can be easily shown, using thedefinition of lower semi-continuity, that c y x( − ) is lower semi-con-tinuous on . Next, since c y x( − ) and y x1/( − ) are both lower semi-continuous, c y x y x( − )/( − ) is also lower semi-continuous on .Observe further that ∫ g z dz y x( ) /( − )

x

yis continuous. Therefore, α x y( , )

is lower semi-continuous on . Note that from Assumption 1, we havec q Klim ( ) ≥q→0+ , implying that α x y αlim ( , ) = ∞ = (0, 0)x y( , )→(0 ,0 )− + . This

means that α x y( , ) is continuous at (0, 0). Therefore, α x y( , ) is lowersemi-continuous on , and hence it attains its minimum on .However, since x y( , ) ∈0 0 and α x y α( , ) < ∞ = (0, 0)0 0 , the point(0, 0) cannot be a minimizer of α x y( , ) over . Hence, α x y( , ) attainsits minimum on . Thus, from (25), we conclude that there exists anx y( *, *) ∈ such that α α x y α x y*≔ ( *, *) = inf ( , )x y< .

3.3. Proof of Theorem 3

In light of (25), we will assume that α* is not attained by anyx y( , ) ∈ , as otherwise this theorem is a consequence of Theorem 1.Under this assumption, we will construct a policy in Π whose long-runaverage cost exactly equals α*. Specifically, our strategy is to show thatthere exists a policy π* whose cumulative cost function satisfies, for allsufficiently large T, an inequality of the form

C TT

α β T[0, ] ≤ * + ( ),π*

(26)

where β T( ) is a (non-negative) function that converges to 0 as T → ∞.

Clearly, (26) implies that f α≤ *π*holds. Since f α≥ *π*

also holds

(Theorem 1), we will then have the desired result that f α= *π*.

Before proceeding, we will make the additional assumption that theordering-cost function is sub-additive, i.e.,

c q q c q c q( + ) ≤ ( ) + ( )1 2 1 2

for all positive q1 and q2. This assumption is without loss of generalityas any cost function that is not sub-additive can be modified to one thatis, by splitting an order of size q q+1 2 into two orders of sizes q1 and q2each and placing these orders consecutively with little time delay toachieve a lower ordering cost14.

Our starting point is the vector x y( , )0 0 in with

α α x y≔ ( , ) < ∞,0 0 0

defined in Section 3.2. We will begin by letting π* follow the x y( , )0 0policy for k0 ordering cycles, where k0 is a positive integer. To initializethis x y( , )0 0 policy, we will assume I = 00 and claim that this assumptioncan be made without loss of generality. To see the validity of this claim,observe that if I = 00 does not hold, then there are three possible

scenarios: (i) I > 00 , (ii) I x0 > ≥0 0, and (iii) x I0 ≥ >0 0. For scenario(i), we can always do nothing until inventory hits level 0 (recall that theleft limit of g(·) at I0 was assumed to be finite whenever I0 ispositive). For scenario (ii), we will simply follow the x y( , )0 0 policyand let it run until the net-inventory level hits level 0 for the firsttime. For scenario (iii), we will immediately place an order of sizen y x( − )0 0 , where n is the smallest positive integer such thatn y x I x( − ) + >0 0 0 0. Note that the cost of placing this order is finitedue to the sub-additivity of c(·) and the finiteness of c y x( − )0 0 ;moreover, since this order takes the net-inventory level up to aposition higher than x0 (but no greater than y0), we can thenduplicate the action prescribed for scenario (ii). It follows that, forall three scenarios, we will be able to reach a zero-inventory epochin finite time. Since the total cost involved is guaranteed to befinite, this establishes the claim.

Next, let ε α α≔ − *0 0 , which is positive. Now, with

s S x y( , ) ≔ ( , ),0 0 0 0

the definition of infimum implies that there exists a sequences S n{( , ), ≥ 0}n n in such that α α s S α ε≔ ( , ) ≤ * +n n n n, where ε ε≔ /2n

n0 ,

for all n ≥ 0; moreover, this sequence can be chosen so that the αn's aredecreasing in n. Observe that, since α α ε≤ * +n 0 and S s q− ≤n n m for alln ≥ 0, we have

∫c S s c S s g z dz λ

α ε S s λ

κ

( − ) ≤ ( − ) + ( ) /

≤ ( * + )( − )/

≤ ,

n n n ns

S

n n0

n

n

(27)

where κ α q λ≔ /m0 , a finite constant. Note in addition that since s S( , )n nis in for every n ≥ 0, we always have s S≤n n+1 .

For n ≥ 0, define w S s λ≔( − )/n n n . Since I = 00 , the exact length of theperiod in which policy π* follows the s S( , )0 0 policy is ζ k w≔0 0 0. We shallrefer to the time interval ζ[0, ]0 as the 0-th policy segment, or just the 0-th segment, of π*; and from epoch ζ0 onward, we will let π* consist of asequence of contiguous segments, where, for n ≥ 1, the n-th segment isa time interval of length ζ k w≔n n n, where kn is a positive integer, inwhich the s S( , )n n policy is employed. Let ζΓ ≔ ∑n j

nj=0 , for n ≥ 0; then,

note that Γn is the epoch at which the n-th segment terminates and atwhich the n( + 1)-th segment begins.

To “connect” the s S( , ) policies in successive policy segments, noticehowever that there exists a complication, namely that for any n ≥ 1, it ispossible to have S s= 0 =n n−1 . If such a scenario should occur, we willlet π* place a single order of size S s−n n−1 at Γn−1, as opposed to the“inadmissible” action of placing two concurrent orders of respectivesizes S s s− = −n n n−1 −1 −1 and S s S− =n n n. If, on the other hand,S s= 0 =n n−1 does not apply, then, at most one order could exist atΓn−1 and it is easily seen that the s S( , )n n−1 −1 policy would “automati-cally” switch into the s S( , )n n policy at that epoch without any interven-tion; this is because Γn−1 is a zero-inventory epoch if and only ifS s= 0 =n n−1 does not hold.

To complete our definition of policy π*, what remains is thespecification of the kn's. Recall that with our choice of εn's above, theconvergence of the αn's to α* is at least at an exponential rate. The basicintuition behind this last part of our construction, then, is that thelengths of the successive policy segments in π*, which are determinedby the kn's, can be made to increase at a sufficient pace so that the long-run average cost of π* actually achieves α*.

We will now proceed by setting k = 10 ; this means that, startingfrom time 0, the s S( , )0 0 policy is followed for a time interval of lengthζ w=0 0. Hence, the total cost contribution (see (13)) associated withthe ordering cycle spanning [0, Γ ]0 equals α w0 0, or α Γ0 0. Next, at epochΓ0, observe that if S s= 0 =0 1 holds, then the “combined” order of sizeS s−1 0 there creates a larger (or merged) ordering cycle of durationw w+0 1. Moreover, since c(·) is sub-additive, the total cost contributionassociated with this combined order is no greater than α w α w+0 0 1 1,which, interestingly, precisely equals the total of the cost contributions

14 For discrete-time inventory models with a constraint on total production capacity,ordering q1 and q2 units concurrently could be infeasible; and this would result in anexample of an ordering-cost function that is not sub-additive. However, in ourcontinuous-time EOQ setting, these two orders could be placed “back-to-back”; that is,within very little time of each other.

S. Perera et al. International Journal of Production Economics 187 (2017) 216–228

222

associated with the two “scheduled” orders of sizes S s−0 0 and S s−1 1 atΓ0 if we had not been “forced” into combining these orders. Recall thatour goal is to develop an upper bound on the cumulative total cost overtime. It follows that for bounding purposes, we can and will con-veniently “pretend” that the ordering cost for the single order at Γ0 isequal to c S s c S s( − ) + ( − )0 0 1 1 if S s= 0 =0 1. With this device, we thenhave that the total cost associated with the order(s) during the “w w+0 1interval” that covers Γ0 always equals α w α w+0 0 1 1, regardless ofwhether or not S s= 0 =0 1; in addition, we will assign c S s( − )0 0 as acost increment that belongs to the interval [0, Γ ]0 and c S s( − )1 1 as onethat belongs to [Γ , Γ]0 1 . This assignment scheme will also be adoptedwithout further comment for all future segments.

Next, let k1 be sufficiently large so that ζ ζΓ − Γ = ≥ = Γ1 0 1 0 0.Clearly, if there is no combined order at Γ1, then the increment tothe total cost in the interval [Γ , Γ]0 1 would be exactly α ζ1 1; otherwise,there exists an additional cost c S s c S( − ) = ( )2 2 2 at Γ1 that is from thenext segment [Γ, Γ ]1 2 . Recall from (27) that c S s κ( − ) ≤n n for all n ≥ 0;hence, we have c S κ( ) ≤2 . It follows that

∑C α ζ κ[0, Γ] ≤ +π

jj j

*1

=0

1

(28)

always holds. Since we also have α α α ε≤ = * +1 0 0, α α ε≤ * +1 1, andζ ζ≥1 0, it is easily seen that (28) further implies that15

Cα ε ε

κ[0, Γ]Γ

≤ * + 12

( + ) +Γ

.π*

1

10 1

1 (29)

Note that, since ε ε= /21 0 , the term ε ε(1/2)( + )0 1 can be rewritten asε n( + 2)/2n

0+1 with n=1.

Our intent beyond k1 should now be clear. For any n ≥ 2, we will letkn be sufficiently large so that the length of the n-th segment (i.e., ζn) isno shorter than the total length of all previous segments (i.e., Γn−1).That is, we will require that the kn's be chosen so that Γ − Γ ≥ Γn n n−1 −1holds for every n ≥ 1; and this completes our construction of π*. Itshould also be clear that this construction is designed to ensure that,for all n ≥ 1, we have

Cα ε n κ[0, Γ ]

Γ≤ * + + 2

2+

Γ.

πn

nn

n

*

0 +1 (30)

This inequality is an immediate extension of (29), obtained bydecomposing C [0, Γ ]π

n*

into the sum of C [0, Γ ]πn

*−1 and C [Γ , Γ ]π

n n*

−1and then carrying out a simple induction on n.

Consider now an arbitrary T >Γ0. Let θ T( ) be the index of the lastcompleted segment under π* by time T; and correspondingly, defineχ T j j w T( )≔max{ ≥ 0: Γ + ≤ }θ T θ T( ) ( )+1 . Also, denote χ T wΓ + ( )θ T θ T( ) ( )+1

by Γθ T( ). Then, by definition, we have T wΓ ≤ Γ ≤ < Γ +θ T θ T θ T θ T( ) ( ) ( ) ( )+1.Since αθ T( )+1 is no greater than αθ T( ) (which, in turn, is no greater thanany αn for n θ T0 ≤ ≤ ( )), it immediately follows that16

C C[0, Γ ]

Γ≤

[0, Γ ]Γ

πθ T

θ T

πθ T

θ T

*( )

( )

*( )

( ) (31)

holds. Next, note that the cost associated with the intervalw[Γ , Γ + ]θ T θ T θ T( ) ( ) ( )+1 equals α wθ T θ T( )+1 ( )+1, and that, in light of (27), this

cost is bounded from above by κ . Hence, we have

C T C κ[0, ] ≤ [0,Γ ] + .π πθ T

* *( )

Dividing this inequality by Γθ T( ) and then invoking (31) and (30) nowyield (26) with

β T ε θ T κ( )≔ ( ) + 2

2+

2Γ

.θ Tθ T

0 ( )+1( )

Finally, observe that θ T( ) → ∞ as T → ∞; hence, we indeed haveβ T( ) → 0 as T → ∞. This completes the proof of Theorem 3.

4. Concluding remarks

Our definition of the policy class Π in Section 2 only allowsdeterministic policies. However, in the interest of generality, it is alsoworthwhile to consider randomized and history-dependent policies.Denote this larger class of policies by Π͠. We now show that wheneverα* is attained by an x y( *, *), then the corresponding x y( *, *) policy is, infact, also optimal over Π͠. Somewhat surprisingly, such an extension isan easy consequence of our proof of Theorem 1.

For π ∈ Π͠, define

f C TT

E= lim sup ( [0, ]) ,π

T

π

→∞

where E denotes expectation17; and let this be our performancemeasure. Observe that

⎛⎝⎜

⎞⎠⎟

f C TT

C TT

C TT

E E

E

= lim sup ( [0, ]) ≥ lim inf ( [0, ])

≥ lim inf [0, ] ,

π

T

π

T

π

T

π→∞ →∞

→∞ (32)

where the second inequality is due to Fatou's Lemma. Next, note thatour proof of Theorem 1 directly applies to every realized inventorytrajectory under π ; and moreover, the argument remains valid whenlim sup in (1) is replaced by lim inf . That is, C T T αlim inf [0, ]/ ≥ *T

π→∞

holds for every sample path. This implies that the right-hand side of(32) is also bounded below by α*. Since f α= *π*

, we see that π* isoptimal over Π͠.

Finally, we note that under stronger assumptions on the coststructure, simple and direct proofs for our Theorem 1 exist. Sincesuch proofs have been desperately absent in standard textbooks, webelieve they are of pedagogical interest. We will articulate twoexamples, but, in the interest of brevity, delegate the details to theAppendix. The first example is for the special case when c(·) isincreasing and backordering is not allowed (i.e., g x( ) = ∞ for allx < 0). We have uncovered an extremely short proof. The argument,which complements the extant literature (see, e.g., Beyer and Sethi(1998) and Lippman (1971)), is given in Appendix B. When comparedwith our full proof of Theorem 1, it can be seen from Appendix B thatthe shortness of the proof is primarily due to the no-backorderingconstraint. In the absence of this requirement, it turns out that a directproof of Theorem 1 that does not rely on the reduction in Lemma 1 alsoexists if the cost structure is restricted to that of the standard EOQmodel; and this is our second example. The argument for this extensionof the original EOQ model in Harris (1913) is provided in Appendix C.

15 This follows from the fact that a b≥ ≥ 0 and y x≥ > 0 together imply thatax by x y a b( + )/( + ) ≤ (1/2)( + ).16 Note that the right-hand-side ratio is no less than αθ T( ). This inequality then follows

from the fact that for any pair of positive x and y, a ax by x y≥ ( + )/( + ) if and only ifa b≥ ≥ 0.

17 Conceptually, this expectation is on the probability space of all inventory trajec-tories generated by π .

S. Perera et al. International Journal of Production Economics 187 (2017) 216–228

223

Appendix A. Well-known examples of ordering-cost functions that are not increasing.

Fig. A.1. Ordering-cost function with batch-ordering constraint.

Fig. A.2. Ordering-cost function with MOQ constraint.

Fig. A.3. Ordering-cost function with all-unit discounts.

S. Perera et al. International Journal of Production Economics 187 (2017) 216–228

224

Appendix B. Proof of Theorem 1 with an increasing c(·) and no backordering

Recall that Π p0 denotes the class of policies that place an order only when the inventory level decreases to zero. Clearly, Lemma 1 remains valid

after replacing Π0 by Π p0 . The proof for this special case is well known and much shorter, as we do not have the complications that arise from

allowing backorders.Let π be a policy in Π⧹Π p

0 . Then, π must place at least an order at some epoch t with a positive I t( )π ; clearly, this is equivalent to the existence ofx > 0i

π for some i ≥ 1. Let π∼ be the policy constructed from π by setting:

t txλ

q q x

t t q q

= + and = , if > 0; and

= and = , otherwise.

nπ

nπ n

π

nπ

nπ

nπ

nπ

nπ

nπ

nπ

∼ ∼

∼ ∼

An illustration of the inventory trajectories under both π and π∼ is shown in Fig. B.1. It is easily seen from this figure (see the shaded region) thatthe cumulative sum of holding and shortage costs under π∼ over time is dominated by that under π . Moreover, it is clear that for a given T ≥ 0, everyorder placed within T[0, ] under π∼ corresponds to an order placed under π within the same interval (and furthermore π may have additional ordersin T[0, ], e.g., the order at epoch t π

6 in Fig. B.1). It follows that the total ordering cost under π∼ is no greater than that under π within T[0, ]. Takentogether, these two observations imply that C T T C T T[0, ]/ ≤ [0, ]/π π∼

for all T ≥ 0. By letting T → ∞, we then have f f≤π π∼.

Now, consider a policy π ∈ Π p0 . Let T be a time epoch that is sufficiently large so that N T( ) ≥ 2π ; then, the total cost under π in T[0, ] can be

written as:

∑C T C t C t t C t T[0, ] = [0, ) + [ , ) + [ , ] .π π π

i

N Tπ

iπ

iπ π

N Tπ

1=1

( )−1

+1 ( )

π

(B.1)

This cost decomposition is illustrated in Fig. B.2.Now, it is immediate that

C t t α y t t α t t[ , ) = (0, )( − ) ≥ *( − ).πiπ

iπ

iπ

iπ

iπ

iπ

iπ

+1 +1 +1 (B.2)

Moreover, since c(·) and g(·) are increasing functions, we see from the shaded area in Fig. B.2 that if T t− N Tπ

( ) is positive, then, with yN Tπ

( ) defined

similar to tN Tπ

( ), we have C t T α y I T T t[ , ]≥ (0, − ( ))( − )πN Tπ

N Tπ π

N Tπ

( ) ( ) ( ) and hence

C t T α T t[ , ] ≥ * ( − ).πN Tπ

N Tπ

( ) ( ) (B.3)

Notice in addition that this lower bound continues to hold even if T t− = 0N Tπ

( ) . Since C t[0, ) ≥ 0π π1 and t I λ= /π

1 0 , we then conclude from (B.1)–(B.3)that

Fig. B.1. Construction of a policy π∼ from a given π ∈ Π⧹Π p0 .

Fig. B.2. Cost decomposition over T[0, ] for π ∈ Π p0 .

S. Perera et al. International Journal of Production Economics 187 (2017) 216–228

225

C T α T I λ[0, ] ≥ * ( − / ).π0 (B.4)

Dividing both sides of (B.4) by T and then letting T → ∞ now yield f α≥ *π , completing the proof for this special case of Theorem 1.Finally, we note that our proof in this section is similar in spirit to the approach in Lippman (1971). Lippman studies a single-product, single-

location model without backordering. The holding cost is linear, and the ordering-cost structure allows multiple set-up costs. Under these specificcost assumptions, Lippman proves s S( , )-optimality using a lower-bounding argument. Since the multiple set-up cost function is increasing and leftcontinuous, (B.4) and our Theorem 2 also imply that an optimal s S( , ) policy exists in Lippman's model.

Appendix C. A simple proof of the standard EOQ formula with backorder

We first recall that the cost assumptions in the standard EOQ model with backorder are: c q K νq g x h x b x1( ) = + and ( ) = max(0, ) + max(0,− )q{ >0}for some constants K > 0, ν ≥ 0, h > 0, and b > 0. Now, for a given policy π , let Q T[0, ]π and η T[0, ]π be the cumulative order quantities and,respectively, the total number of orders placed under π in the time interval T[0, ]. Then,

f ν Q T Kη T H TT

ν Q T Kη T H TT

ν Q TT

Kη T H TT

= lim sup [0, ] + [0, ] + [0, ]

≥ lim inf [0, ] + [0, ] + [0, ]

≥ lim inf [0, ] + lim inf [0, ] + [0, ] .

π

T

π π π

T

π π π

T

π

T

π π

→∞

→∞

→∞ →∞ (C.1)

Next, we claim that we can restrict attention to policies that satisfy

Q TT

λlim inf [0, ] ≥ .T

π

→∞ (C.2)

The intuition behind this claim is obvious: If the long-run rate of procurement is less than the demand rate, then the amount of backorder willdiverge to infinity. This is formalized in the following lemma.

Lemma 3. The long-run average cost f π for any policy π that violates inequality (C.2) is infinite.

Proof. Suppose Q T T λlim inf [0, ]/ <Tπ

→∞ for a given policy π . Then, by the definition of lim inf , there exist an ε λ∈ (0, ), an n > 00 , and an increasingsequence T{ }n n=1

∞ with Tlim = ∞n n→∞ such that Q T T λ ε[0, ]/ < −πn n for all n n≥ 0. Observe that I T I Q T λT( ) = + [0, ] −π

nπ

n n0 ; and this, together withthe previous inequality, implies that I T I εT( ) < −π

n n0 for all n n≥ 0. (Hence, I T( )πn diverges to −∞ as n → ∞.) Suppose further that n ( n≥ 0) is

sufficiently large so that I εT− n0 is strictly negative and T T εT I λ ε λ T I λ≔ − ( − )/ = (1− / ) + /nl

n n n0 0 is non-negative. The time epochs Tnl and Tn are

illustrated in Fig. C.1. From this figure, we also see that εT I λ( − )/n 0 is the amount of time necessary for the net-inventory level to drop down toI εT− n0 , assuming that it starts from level 0 at time Tn

l and no future orders are placed. Now, consider the time interval T T( , ]nl

n . Clearly, thecumulative shortage cost in T[0, ]n is no less than that within the subinterval T T( , ]n

ln . Moreover, the latter cost is, in turn, greater than

b εT I εT I λ(1/2)( − )(( − )/ )n n0 0 , which is the total shortage cost in T T( , ]nl

n if the inventory level under π were zero at epoch Tnl and no future orders were

placed. (The diagonal border of the shaded area in Fig. C.1 always lies entirely above the inventory trajectory under π in T T( , ]nl

n .) It follows thatC T T b ε I T εT I λ[0, ]/ > (1/2)( − / )(( − )/ )π

n n n n0 0 . Finally, letting n → ∞ in this inequality yields that f = ∞π . □

From Lemma 3, we have that the first limit in (C.1) is no less than νλ. We now develop a lower bound for the second limit in (C.1). Observe thatthis limit can be interpreted as the objective function in a standard EOQ model with ν = 0. Denote by α x y( , )0 the function (2) for this special case. Itis easily shown that α x y( , )0 attains its minimum at s λKh b b h*= − 2 /[ ( + )]0 and S λKb h b h*= 2 /[ ( + )] .0 We will next show that, for any π ∈ Π (notnecessarily limited to Π0),

Fig. C.1. Lower bound for backordering cost over T[0, ]n .

S. Perera et al. International Journal of Production Economics 187 (2017) 216–228

226

Kη T H TT

α α s Slim inf [0, ] + [0, ] ≥ *≔ ( *, *).T

π π

→∞0 0 0 0

(C.3)

The argument relies on a lower bound that is similar in spirit to (B.4).For a given policy π , denote byC π

0 the cumulative cost function under the assumption that ν = 0. Then, (B.1) again holds18 withC π0 replacing Cπ;

and this is illustrated in Fig. C.2.It is easily seen from Fig. C.2 and (2) that, for i N T1 ≤ ≤ ( ) − 1π ,

∫C t t K g z dzλ

α x y t t α t t[ , ) = + ( ) = ( , )( − ) ≥ *( − ).πiπ

iπ

x

y

iπ

iπ

iπ

iπ

iπ

iπ

0 +1 0 +1 +1 0 +1iπ

iπ

+1

Now, consider the particular scenario in which both t π1 and T t− N T

π( ) are positive. Then, similar reasoning also shows that

C t T α I T y T t α T t[ , ] = ( ( ), )( − ) ≥ *( − );πN Tπ π

N Tπ

N Tπ

N Tπ

0 ( ) 0 ( ) ( ) 0 ( ) and that C t α x I t K α t K[0, ) = ( , )( −0) − ≥ * − .π π π π π0 1 0 1 0 1 0 1 Consequently, from the modified (B.1),

we have

Kη T H T C T α T γ[0, ] + [0, ] = [0, ] ≥ * + ,π π π0 0 (C.4)

where γ K= − . Since t π1 and/or T t− N T

π( ) may not be positive, there are three other scenarios. It is easily seen that for all scenarios, the constant γ

equals K− , 0, or K. Dividing both sides of (C.4) by T and letting T → ∞ now yield (C.3).From (C.1)–(C.3), we now have f νλ α≥ + *π

0 . It is easily verified that f νλ α= + *π *0

0 , where π*0 denotes the s S( *, *)0 0 policy. Therefore, π*0 isoptimal over Π.

References

Adelman, D., Klabjan, D., 2005. Duality and existence of optimal policies in generalizedjoint replenishment. Math. Oper. Res. 30 (1), 28–50.

Alp, O., Huh, W.T., Tan, T., 2014. Inventory control with multiple setup costs. Manuf.Serv. Oper. Manag. 16 (1), 89–103.

Altintas, N., Erhun, F., Tayur, S., 2008. Quantity discounts under demand uncertainty.Manag. Sci. 54 (4), 777–792.

Benton, W.C., Park, S., 1996. A classification of literature on determining the lot sizeunder quantity discounts. Eur. J. Oper. Res. 92 (2), 219–238.

Beyer, D., Sethi, S.P., 1998. A proof of the EOQ formula using quasi-variationalinequalities. Int. J. Syst. Sci. 29 (11), 1295–1299.

Brimberg, J., Hurley, W.J., 2006. A note on the assumption of constant order size in thebasic EOQ model. Prod. Oper. Manag. 15 (1), 171–172.

Caliskan-Demirag, O., Chen, Y., Yang, Y., 2012. Ordering policies for periodic-reviewinventory systems with quantity-dependent fixed costs. Oper. Res. 60 (4), 785–796.

Chan, L.M.A., Muriel, A., Shen, Z., Simchi-Levi, D., 2002. On the effectiveness of zero-inventory-ordering policies for the economic lot-sizing model with a class ofpiecewise linear cost structures. Oper. Res. 50 (6), 1058–1067.

Chen, F., 2000. Optimal policies for multi-echelon inventory problems with batchordering. Oper. Res. 48 (3), 376–389.

Chen, X., 2009. Inventory centralization games with price-dependent demand andquantity discount. Oper. Res. 57 (6), 1394–1406.

Erlenkotter, D., 1989. An early classic misplaced: Ford W. Harris's economic orderquantity model of 1915. Manag. Sci. 35 (7), 898–900.

Erlenkotter, D., 1990. Ford Whitman Harris and the economic order quantity model.Oper. Res. 38 (6), 937–946.

Federgruen, A., Lee, C., 1990. The dynamic lot size model with quantity discount. Nav.Res. Logist. 37 (5), 707–713.

Harris, F.W., 1913. How many parts to make at once. Fact. Mag. Manag. 10 (2),135–136, 152.

Hassin, R., Meggido, N., 1991. Exact computation of optimal inventory policies over anunbounded horizon. Math. Oper. Res. 16 (3), 534–546.

Hellion, B., Mangione, F., Penz, B., 2012. A polynomial time algorithm to solve thesingle-item capacitated lot sizing problem withminimum order quantities andconcave costs. Eur. J. Oper. Res. 222 (1), 10–16.

Iglehart, D., 1963a. Optimality of (s, S) policies in the infinite-horizon dynamic inventoryproblem. Management Sci 9 (2), 259–267.

Iglehart, D., 1963b. Dynamic programming and stationary analysis of inventoryproblems (Chapter 1). In: Scarf, H., Gilford, D., Shelly, M. (Eds.), MultistageInventory Models and Techniques. Stanford University Press, Stanford, CA.

Kiesmüller, G.P., de Kok, A.G., Dabia, S., 2011. single item inventory control underperiodic review and a minimum order quantity. Int. J. Prod. Econ. 133 (1), 280–285.

Li, C., Hsu, V.N., Xiao, W., 2004. Dynamic lot sizing with batch ordering and truckloaddiscounts. Oper. Res. 52 (4), 639–654.

Li, C., Ou, J., Hsu, V.N., 2012. Dynamic lot sizing with all-units discount and resales.Nav. Res. Logist. 59 (3–4), 230–243.

Lippman, S.A., 1969. Optimal inventory policy with multiple set-up costs. Manag. Sci. 16(1), 118–138.

Lippman, S.A., 1971. Economic order quantities and multiple set-up costs. Manag. Sci.18 (1), 39–47.

Porteus, E.L., 2002. Foundations of Stochastic Inventory Theory. Stanford UniversityPress, Stanford, CA.

Presman, E., Sethi, S.P., 2006. Inventory models with continuous and Poisson demandsand discounted and average costs. Prod. Oper. Manag. 15 (2), 279–293.

Fig. C.2. Cost decomposition over T[0, ].

18 If there is an order at time zero, i.e., if t = 0π1 , then C t[0, ) = 0π π

0 1 .

S. Perera et al. International Journal of Production Economics 187 (2017) 216–228

227

Scarf, H., 1960. The optimality of (S, s) policies in dynamic inventory problems. In:Arrow, K.J., Karlin, S., Suppes, P. (Eds), Proceedings of the First StanfordSymposium on Mathematical Methods in the Social Sciences. Stanford UniversityPress, Stanford, California, pp. 196–202.

Sun, D., 2004. Existence and properties of optimal production and inventory policies.Math. Oper. Res. 29 (4), 923–934.

Veinott, A.F., 1966. On the optimality of (s, S) inventory policies: new conditions and anew proof. SIAM J. Appl. Math. 14 (5), 1067–1083.

Winston, W.L., 2004. Operations Research: Applications and Algorithms 4th ed..Thomson Brooks/Cole, Belmont, CA.

Zhao, Y., Katehakis, M.N., 2006. On the structure of optimal ordering policies forstochastic inventory systems with minimum order quantity. Probab. Eng. Inf. Sci. 20

(2), 257–270.Zheng, Y.-S., 1991. A simple proof of the optimality of (s, S) policies in infinite-horizon

inventory systems. J. Appl. Probab. 28 (4), 802–810.Zheng, Y.-S., 1994. Optimal control policy for stochastic inventory systems with

Markovian discount opportunities. Oper. Res. 42 (4), 721–738.Zhou, B., Zhao, Y., Katehakis, M.N., 2007. Effective control policies for stochastic

inventory systems with a minimum order quantity and linear costs. In. J. Prod. Econ.106 (2), 523–531.

Zhu, H., Liu, X., Chen, Y., 2015. Effective inventory control policies with a minimumorder quantity and batch ordering. Int. J. Prod. Econ. 168 (1), 21–30.

Zipkin, P., 2000. Foundations of Inventory Management. McGraw-Hill, Boston, MA.

S. Perera et al. International Journal of Production Economics 187 (2017) 216–228

228