selecting t for a periodic review inventory model with staggered deliveries

Selecting T for a Periodic Review Inventory Modelwith Staggered Deliveries

James Flynn

Department of Operations Management and Business Statistics,Cleveland State University, Cleveland, Ohio 44115

Received May 1998; revised December 1999; accepted 21 December 1999

Abstract: Consider a single-item, periodic review, infinite-horizon, undiscounted, inventorymodel with stochastic demands, proportional holding and shortage costs, and full backlogging.Orders can arrive in every period, and the cost of receiving them is negligible (as in a JIT setting).Every T periods, one audits the current stock level and decides on deliveries for the next T periods,thus incurring a fixed audit cost and—when one schedules deliveries—a fixed order cost. Theproblem is to find a review period T and an ordering policy that satisfy the average cost criterion.The current article extends an earlier treatment of this problem, which assumed that the fixed ordercost is automatically incurred once every T periods. We characterize an optimal ordering policywhen T is fixed, prove that an optimal review period T ∗∗ exists, and develop a global searchalgorithm for its computation. We also study the behavior of four approximations to T ∗∗ basedon the assumption that the fixed order cost is incurred during every cycle. Analytic results froma companion article (where µ/σ is large) and extensive computational experiments with normaland gamma demand test problems suggest these approximations and associated heuristic policiesperform well when µ/σ ≥ 2. c© 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 329–352,2000

Keywords: stochastic inventory model; periodic review; optimal review period; order splitting

1. INTRODUCTION

How often should one review the inventory status, place orders, and schedule deliveries? Mostapproaches to this problem assume that only one delivery can be scheduled for each review;however, as pointed out in Flynn and Garstka [4, 5], unlinking the delivery and review intervalscan lead to substantial cost savings, lower inventory levels, and less frequent reviews. Those twoarticles addressed the problem of finding an optimal review period T when deliveries are morefrequent than reviews. The first [4] characterized an optimal ordering policy where, for somearbitrary T , one observes the current stock level every T periods, and orders deliveries for eachof the next T periods. The next [5] found a best T under the assumption that one employed theoptimal ordering policy determined in [4]. The current article removes a major limitation of [4, 5]:the assumption that the fixed order cost is incurred once every T periods—even when the stocklevel is high enough for it to be optimal not to place orders.

Most of the other research on splitting a single order into several shipments focuses on splittingan order over several suppliers (Chiang and Benton [1], Kelle and Silver [10, 11], Lau and

c© 2000 John Wiley & Sons, Inc.

330 Naval Research Logistics, Vol. 47 (2000)

Zhao [12], Ramasesh et al. [15], Sculli and Wu [17]). For the case of a single supplier, Chiangand Chiang [2] report that splitting a single order into two equally sized orders can yield costsavings of up to 20%, while splitting the order into three equally sized orders can yield anadditional savings of 10% (assuming a continuous review (s,Q) ordering policy and the P2service criterion). More recently, Janssen, De Kok, and Van der Duyn Schouten [9] considerorder splitting from the perspective of the supplier under similar assumptions. They indicatethat order splitting can decrease demand variability and provide useful information about futuredeliveries.

This article considers a single-item, periodic review, stochastic inventory model with station-ary parameters. Costs are undiscounted and the planning horizon is infinite. The demands areindependent, identically distributed, nonnegative random variables. Holding and shortage costsare proportional, and all shortages are backordered. Orders can arrive in any period, and the costof receiving them is negligible. Every T periods, one observes the current stock level and decideson deliveries for the next group of T periods, thus incurring a proportional order cost, a fixed costfor auditing the stock level, and—when one schedules deliveries—a fixed order cost. The fixedcosts do not depend on T . The objective is to find a review period T and an ordering policy thatminimize the (long run expected) average cost per period.

Note that stockouts provide free automatic audits in applications where one knows when thesystem is out of stock and the cost of determining the backorder level is negligible. Section 9 of[5] provides one way of incorporating free audits when the review period is fixed at T . See [5]for a discussion of this issue.

Scheduling frequent deliveries is consistent with the JIT philosophy (Groenevelt [6]). Forexample, our model might be of interest to a manufacturer who places weekly orders, but wantsto reduce inventory levels by scheduling daily deliveries. Ordering a sequence of T deliveriesmight entail timing T separate order releases or placing orders for the T periods once with asupplier who would agree to stagger the arrivals of the T deliveries appropriately (see Chapter7 of Schonberger [16]). For the jth delivery, the cumulative order-up-to-level would equal themean j-period demand plus a safety stock. The cost of timing order releases is considered part ofthe cost of receiving an order, which is assumed negligible.

Although transactions reporting is popular, periodic review systems are important (Lee andNahmias [13]). The advantages of periodic over continuous review include lower administrativecosts, easier coordination of the orders of related items, improved workload planning by thesupplier and buyer, and a regular opportunity every T periods to adapt the ordering policy tochanges in the demand pattern (Silver, Pyke, and Peterson [18]). Furthermore, as Porteus [14]observes, transactions reporting systems that keep inventory records current, but order periodicallyare equivalent to periodic review systems. Of course, given transactions reporting, one shouldignore the fixed audit cost.

The distinction between review and delivery intervals is the key feature of our model. This isimportant, because there are different tradeoffs for the review period and delivery decisions. EveryT periods, one audits the stock levels and orders a sequence of deliveries. This takes resources,e.g., labor, computer time, etc. (see [18]), reflected in the fixed audit and order costs. LargeT increases the random variability in the stock levels. The cost associated with this variabilityincreases with the holding and shortage costs, and the demand uncertainty. Thus, it appears thatT should increase as the setup cost increases, and T should decrease as the holding and shortagecosts, and the demand variance σ2 increase. Furthermore, under our assumptions, it is plausiblethat the optimal T should be insensitive to the mean demand µ. (If σ = 0, one should neveraudit and one should order only once.) On the other hand, the delivery frequency should dependstrongly on the mean demand. To see this, consider the deterministic version of our model. Under

Flynn: Periodic Review Inventory Model 331

a policy that schedules deliveries once every T periods, the average holding and shortage cost perperiod is proportional to the mean T -period demand µT .

Other attempts at determining T require that only one delivery can be scheduled for each reviewperiod (see the introduction to [5]). Our approach, which uncouples the review period and deliverydecisions and allows deliveries in every period, can lead to substantial savings, particularly whenthe mean T -period demand is large and the fixed cost of receiving an order is negligible. Section 8reports on computational experiments that allow one to estimate the possible cost savings. Section6 provides an illustrative example.

Reference [4] studies ordering policies for an extension of Veinott’s [20] basic model whereordering occurs once every T periods, but deliveries arrive in every period. The quantity T isfixed, there are no fixed ordering costs, and the holding and shortage costs are convex. Costs maybe discounted or undiscounted. That article formulates a dynamic programming model, wherestage n contains periods (n − 1)T + 1 through nT , and finds that a simple rule describes theoptimal deliveries for each of the T periods of a stage as a function of the beginning inventorylevel and the cumulative T -period order, and shows that a base stock policy determines the optimalcumulative T -period order. It also presents methods for computing optimal policies, develops anexpression for the optimal average cost per period as a function of T , and studies the behaviorof an easily computable myopic base stock policy, whose cost is often close to the optimal cost.Section 3 below extends [3] by including a fixed cost for ordering a sequence of T deliveries andestablishes that an (s, S)-policy determines the optimal cumulative T -period order.

Reference [5] seeks a T that minimizes the average cost per period, under the assumption thatthe system incurs the fixed order and audit costs once every T periods and the ordering policyis optimal for the no fixed cost model discussed in the previous paragraph. This assumption thatthe ordering policy is optimal for a model with no setup cost and that the setup cost is incurredonce each cycle is common when seeking to determine an optimal T (e.g., see [5] for references),since it makes the task more manageable; however, it is suboptimal, since there may be reviewepochs where the stock levels are high enough for it to be optimal to avoid the fixed order costby not scheduling orders.

The rest of this article is organized as follows. Section 2 defines notation and assumptions. Sec-tion 4 proves that an optimal review period T ∗∗ exists, and introduces four quantities from [5] thatapproximate it. One of these, T ∗, is optimal for [5]. The others—TM , T l, and T#—approximateT ∗. (The formula for T# is very simple.) Section 5 presents a global search algorithm for com-puting T ∗∗. Section 6 provides a numerical example. Section 7 presents results for normal andgamma demand problems. Section 8 reports on a numerical study that comparesT ∗∗, T ∗, TM , T l,and T# in 4500 normal demand problems with µ/σ ≥ 2 and 4500 gamma demand problems withµ/σ ≥ 1. Section 9 discusses miscellaneous issues. Finally, Section 10 states our conclusions.

2. ASSUMPTIONS AND NOTATION

Recall that costs are undiscounted, the planning horizon is infinite, and the objective is tominimize the average cost per period. The decision variables are the review period T and theordering policy. Orders can arrive in any period, and cost of receiving them is negligible. Forn = 1, 2, . . ., one observes the current stock level at the beginning of period (n − 1)T + 1 andplaces orders for periods (n− 1)T + 1 through nT , thus incurring a proportional purchase costc ≥ 0, a fixed cost KA ≥ 0 for auditing the stock level and—when the total order quantityis positive—a fixed cost KO ≥ 0 for ordering a sequence of T deliveries. The fixed costs donot depend on T . Audits provide perfect information about current stock levels. Holding andshortage costs are assessed on end-of-period inventories and are proportional with unit shortage


cost p > 0 and unit holding cost h > 0. All shortages are backordered. Demands are independent,nonnegative, and stationary with a finite positive mean µ and variance σ2. This article employsthe following notation:

dj = generic cumulative j-period demand, where j ≥ 1, (1)

Fj(z) = P [dj ≤ z], for z ∈ R, (2)

Φ(·) = the standard normal cumulative distribution function, (3)

φ(·) = the standard normal density function, (4)

K = KA +KO, where K > 0, (5)

α = KA/K, where 0 ≤ α ≤ 1, (6)

τ = p/(p+ h) and θ = K/[σ(p+ h)φ(Φ−1(τ))]. (7)

Note that KA = αK and KO = (1− α)K. In particular, α = 0 in the transactions reportingcase. The results of [5] apply directly to the case where α = 1. The function Fj is the j-foldconvolution of F1 with itself. Distributions for which the Fj’s are readily obtainable include thePoisson, the negative binomial, the exponential, the gamma, and the normal.

3. THE MODEL WITH TTT FIXED

This section assumes T is fixed and ignores the fixed audit cost KA. Following [4], consideran infinite stage dynamic programming model where stage m (m = 1, 2, · · ·) contains periods(m−1)T+1 throughmT . The statex in stagem is the initial inventory level in period (m−1)T+1,while the decision determines the orders for periods (m−1)T+1 throughmT . The state is knownwith certainty. The decision first specifies y ∈ R, where y equals x plus the cumulative ordersplaced for theT periods in stagem, and then specifies z = (z1, z2, . . . , zT ) ∈ RT , where zi equalsx plus the cumulative orders for periods (m − 1)T + 1 through (m − 1)T + i for 1 ≤ i ≤ T .Decision y and z are feasible at state x if x ≤ z1 ≤ · · · ≤ zT = y. The fixed order cost KO isincurred in stages where y − x > 0; dT is the generic single-stage demand.

If y and z are selected in state x, then the end-of-stage inventory level equals y − dT by ourbackordering assumption (negative inventories indicate backorders), and the single-stage costequals

T∑j=1

Gj(zj) +KOδ(y − x), (8)

where δ(z) = 0 for z = 0 and δ(z) = 1 for z > 0, and

Gj(zj) ≡ pE{dj − zj}+ + hE{dj − zj}− (9)

denotes the expected holding and shortage cost in the jth period of the stage. It is easy to see thatthe assumption below holds (e.g., see Section 2 of [5]).


ASSUMPTION 1: For 1 ≤ j ≤ T,Gj is a positive and convex function on R andlim|y|→∞ Gj(y) =∞; moreover, there exists (z∗1 , z

∗2 , . . . , z

∗T ) ∈ RT such that

Gj(z∗j ) = min{Gj(z) : z ∈ R}, j = 1, . . . , T and 0 ≤ z∗1 ≤ z∗2 ≤ · · · ≤ z∗T <∞. (10)

Note that if the single-period demands are continuous, then

Fj(z∗j ) = τ, j = 1, . . . , T. (11)

One can ‘‘optimize out’’ the decision variable z. Theorem 1 of [4] establishes that the formula

z(x, y) = (zj(x, y)) = (max{x,min{z∗1 , y}}, . . . ,max{x,min{z∗T−1, y}}, y), (12)

optimally selects z as a function of x and y. That is, given that the state is x and the total orderfor the stage is y − x ≥ 0, the following cumulative order rule optimally distributes the ordersfor the T periods of the stage: In period 1, order z1(x, y) − x. In period 2 ≤ j ≤ T , orderzj(x, y) − zj−1(x, y). Note that y need not be optimal. Assume without loss of generality thatz = z(x, y). There is now one decision variable y.

An ordering policy is a sequenceπ = (π1, π2, · · ·) of (Borel measurable) mappingsπm:R→ Rsuch that πm(x) ≥ x for x ∈ R, and m ≥ 1: πm(x) is the decision that π selects for state xin stage m. If, for each m ≥ 1, there exist real numbers sm ≤ Sm such that πm(x) = Sm forx < sm and πm(x) = x for x ≥ sm, then π is called an (s, S) policy and sm and Sm are calledthe critical numbers for stage m.

Following [4], this article assumes the following terminal condition of Veinott [20] whendealing with an n stage model. Any stock left over at the end of stage n can be salvaged at theoriginal purchase cost c. Similarly, any backordered shortages are made up by purchases at thesame cost. (This assumption has little effect when n is large. Section 9.1 of [4] discusses the effectof assuming other terminal conditions.)

For n = 1, 2, . . . , x1 ∈ R and any policy π, let Vn,T (x1, π) denote the expected total n-stagecost under the policy π when the initial state is x1. Form = 1, 2, . . ., let xm and ym, respectively,denote the values of x and y for stage m under π.

LEMMA 1:

Vn,T (x1, π) = c(nTµ− x1) + C1T (x1)

+ E

{n−1∑m=1

[C0T (ym) +KOδ(ym − xm)] + C2T (yn) +KOδ(yn − xn)

}, (13)

where

C1T (y) =T−1∑j=1

[Gj(max{y, z∗j })−Gj(z∗j )], (14)

C2T (y) =T−1∑j=1

Gj(min{y, z∗j }) +GT (y), (15)


C0T (y) = C2T (y) + E{C1T (y − dT )}. (16)

PROOF: Add E{∑nm=1KOδ(ym − xm)} to the expression for the expected total n-stage

cost in Lemma 3 of [4] to obtain (13). Conditions (22), (23), and (18) of [4] give us (14), (15),and (16), respectively.

The next lemma gives information aboutC1T , C2T , andC0T . References [4, 5] provide formulasthat expedite the computation of these objects.

LEMMA 2:

(a) C1T , C2T , and C0T are nonnegative, real valued, continuous, and convex.(b) There exists a y∗T ∈ R such that

C0T (y∗T ) = min{C0T (y): y ∈ [z∗1 , z∗T ]} and y∗T ∈ [z∗1 , z

∗T ]. (17)

(c)

C2T (z∗T ) = min{C2T (y): y ∈ R} =T∑j=1

Gj(z∗j ). (18)

(d)

C0T (y∗T ) ≥T∑j=1

Gj(z∗j ). (19)

(e)

lim|y|→∞

C0T (y) =∞. (20)

(f) There exist sT and ST that satisfy

sT ≤ y∗T ≤ ST and C0T (ST ) = C0T (sT ) = KO + C0T (y∗T ). (21)

PROOF: Parts (a)–(d) follow from Lemma 2 of [4]. Part (e) follows from lim|y|→∞Gj(y) =∞, (15), (16) and the nonnegativity of C1T . Finally, (f) follows from (b), (e), K0 ≥ 0, and thecontinuity of C0T .

For x1 ∈ R and any policy π, let CT (x1, π) = lim infn→∞ Vn,T (x1, π)/n − cTµ. Thatis, cTµ + CT (x1, π) denotes the average cost per stage under π at state x1. [Since our orderingdecisions do not affect cTµ, we do not include cTµ inCT (x1, π).] Defineπ∗ to be average-optimalif CT (x1, π

∗) = infπ CT (x1, π) for x1 ∈ R. By the next lemma, a policy is average-optimal if itis average-optimal for a stationary version of our model where the single-stage cost of selectingy in state x always equals C0T (y) +KOδ(y − x).


LEMMA 3: Let x1 ∈ R and the policy π be arbitrary. For m = 1, 2, . . ., let xm and ym,respectively, denote the values of x and y for stage m under π. Then

lim infn→∞Vn,T (x1, π)/n = cTµ+ lim inf

n→∞E

{n∑

m=1

[C0T (ym) +KOδ(ym − xm)]

}/n. (22)

PROOF: Condition (13) and the nonnegativity of C2T and KO imply that

lim infn→∞Vn,T (x1, π)/n ≥ cTµ+ lim inf

n→∞E

{n−1∑m=1

[C0T (ym) +KOδ(ym − xm)]

}/n,

which implies that the left-side of (22) ≥ the right-side of (22). By (16) and the nonnegativity ofC1T , we have C0T ≥ C2T , which together with (13) implies the left-side of (22)≤ the right-sideof (22). This proves (22).

Consider a stationary version of our model where the single-stage cost of selecting y in state xequals C0T (y) + KOδ(y − x) for each stage n. It is well known that the unimodality of −C0T(which follows from the convexity ofC0T ) and lim|y|→∞ C0T (y) =∞ ensure the existence of anaverage-optimal (s, S) policy π∗T with the same critical numbers (s∗T , S

∗T ) for each stage, where

sT ≤ s∗T ≤ y∗T ≤ S∗T ≤ ST ; (23)

furthermore, the optimal average cost per stage is independent of the initial state (e.g., see Veinott[21, p. 1081], Heyman and Sobel [8], and Remark 1 below). Using this result, Lemma 3, and theremarks in the sentence preceding Lemma 3, one can prove the following theorem.

THEOREM 1:

(a) There exists an average-optimal (s, S) policy π∗T where the critical numbers(s∗T , S

∗T ) are the same for each stage and satisfy (23).

(b) π∗T is average-optimal for a stationary version of our model where the single-stage cost of selecting y in state x equals C0T (y) +KOδ(y − x).

(c) The optimal average cost per stage, cTµ + CT (x1, π∗T ), is independent of the

initial state x1.

REMARK 1: The problem of proving the existence of optimal (s, S) policies has been widelystudied. Zheng [22] surveys the literature and provides his own proof, which is simpler thanprevious attempts, but is still quite involved (and assumes discrete demand). Note that Theorem1 continues to hold when (9) is replaced by Assumption 1. Also, it is easy to extend Theorem 1to the discounted cost and the finite stage cases (see [21]). (The critical numbers can vary fromstage to stage in the finite stage case.)

4. THE OPTIMAL REVIEW PERIOD T ∗∗

This section obtains a lower bound on the optimal average cost per period, proves the existenceof an optimal review period T ∗∗, and introduces four quantities from [5]—T ∗, TM , T l, and T#—that can be used to approximate T ∗∗. For T = 1, 2, . . . , let C∗(T ) denote the optimal average


cost per period when the review period is fixed at T , i.e.,

C∗(T ) = [CT (x1, π∗T ) +KA]/T, for T = 1, 2, . . . , (24)

where CT (x1, π∗T ) equals the optimal average cost per stage of Theorem 1 (which is independent

of x1). Theorem 2 establishes that C∗(T ) is bounded below by

CL(T ) = [C0T (y∗T ) +KOP{dT > ST − sT }+KA]/T, for T = 1, 2, . . . . (25)

(This result is used in Section 9.2 of [5], but its proof is not provided there.) The function CL

plays an important role in our algorithm for computing T ∗∗.

THEOREM 2:

C∗(T ) ≥ CL(T ), for T = 1, 2, . . . . (26)

PROOF: Let π∗ be an average-optimal (s, S) policy with critical numbers (s∗T , S∗T ) satisfying

(23). To prove (26), one need only show that

[lim infn→∞Vn,T (x1, π

∗)/n+KA]/T ≥ CL(T ). (27)

By (17) and (22),

lim infn→∞Vn,T (x1, π

∗)/n+KA ≥ C0T (y∗T ) +KO lim infn→∞

n∑m=1

E{δ(ym − xm)}/n+KA.

(28)

This, (25), and (27) allow us to prove (26) by showing that

lim infn→∞

n∑m=1

E{δ(ym − xm)}n

≥ P{dT > ST − sT }. (29)

Under π∗, we have δ(yn − xn) = 1 if xn < s∗T and δ(yn − xn) = 0 if xn ≥ s∗T . Hence,

E{δ(yn − xn)} = P{xn ≤ s∗T } ≥ P{xn−1 ≤ S∗T , xn ≤ s∗T }= P{xn−1 ≤ S∗T }P{xn ≤ s∗T | xn−1 ≤ S∗T }. (30)

By the strong law of large numbers, 1/n times the cumulative n-stage demand approachesE{dT } > 0 as n → ∞ with probability 1. Hence, the cumulative n-stage demand exceedsx1 − s∗T for large n with probability 1. This implies limn→∞ P{xn−1 ≤ S∗T } = 1. AlsoP{xn ≤ s∗T | xn−1 ≤ S∗T } ≥ P{dT ≥ S∗T − s∗T }. These two results, (30), and (23) implylim infn→∞E{δ(yn − xn)} ≥ P{dT ≥ S∗T − s∗T } ≥ P{dT > ST − sT }, which proves (29),since lim infn→∞

∑nm=1E{δ(ym − xm)}/n ≥ lim infn→∞E{δ(yn − xn)}.

Next, define the optimal review period T ∗∗ as the smallest integer minimizing C∗.


THEOREM 3: T ∗∗ exists and is finite. Hence, the policy ‘‘never review’’ is never optimal.

PROOF: Lemma 2(d) implies that CL(T ) is bounded below by∑Tj=1Gj(z

∗j )/T , which

by Lemma 3.1(c) of [5] approaches ∞ as T → ∞. Applying Theorem 2, we conclude thatlimT→∞ C∗(T ) =∞, which implies that T ∗∗ exists and is finite.

Turn to the task of approximating T ∗∗. For the case where the review period is fixed at T ,[4] proves that when KO = 0, the following policy is average-optimal: When in state x, choosey = max{x, y∗T }. That is, follow a base stock policy with the same base stock level y∗T for allstages. As an alternative to an optimal policy, Section 4 of [4] introduces the myopic base stockpolicy, i.e., one where the base stock level always equals z∗T . Ignoring fixed costs, the averagecost per stage under the optimal and myopic base stock policies, respectively, equal C0T (y∗T )and C0T (z∗T ). Reference [5] assumes the fixed cost K is incurred once each stage (which is trueunder a base stock policy with continuous demands since the probability of zero demand equals0). Following [5], let

W ∗(T ) = [C0T (y∗T ) +K]/T, for T = 1, 2, . . . , (31)

WM (T ) = [C0T (z∗T ) +K]/T, for T = 1, 2, . . . , (32)

W l(T ) =

T∑j=1

Gj(z∗j ) +K

/T, for T = 1, 2, . . . , (33)

T# = floor((3θ)2/3) + 1, (34)

where θ satisfies (7) and floor(x) denotes the greatest integer less than or equal to the real numberx.The quantityW ∗(T ) represents the average cost per period under the optimal base stock policy

of [5] when the review period equals T and the fixed cost K is incurred once every T periods.Similarly, WM (T ) represents the average cost per period under the myopic base stock policy.Define T ∗, TM , and T l as the smallest integers minimizing W ∗,WM and W l, respectively.Section 6 of [5] derives T# as an approximation to T l (with maximum error 1) when demandsare normal.

Under the assumptions of [5], T ∗ coupled with an optimal base stock policy is optimal. Undera myopic base stock policy, the best review period is TM . Sections 4 and 8 of [5] cover thecomputation of T ∗, TM , and T l. Given normal demands, T l can be computed by spreadsheet (seeTheorem 6.1 of [5] and Section 6 below). The numerical study of Section 8 finds that T ∗, TM , T l,and T# perform well as approximations to T ∗∗, for normal or gamma demands with µ/σ ≥ 2.Furthermore, [3] establishes that given normal demands, the percentage relative errors of usingT ∗, TM , or T l instead of T ∗∗ approaches 0 as µ/σ →∞.

5. COMPUTING T ∗∗

This section presents a search algorithm for computing T ∗∗. Our description employs pseudo-Pascal terminology and uses the notation T for the current review period, T for the incumbentreview period, and C for C∗(T ). When appropriate, the algorithm calls the procedure below toupdate T and C.


Procedure update:begin if C∗(T ) < C then C := C∗(T ) and T := T ; end;

Before describing our algorithm, we show how to construct a finite interval [TL, TU ] thatcontains T ∗∗. Define T0 as the smallest integer minimizing CL0 , where

CL0 (T ) =

T∑j=1

Gj(z∗j ) +KA

/T, for T = 1, 2, . . . . (35)

(T0 exists by Lemma 3.1 of [5].) The next lemma characterizes CL0 .

LEMMA 4:

(a) CL0 is strictly decreasing on [1, T0] and nondecreasing on [T0,∞).(b) limT→∞ CL0 (T ) =∞.(c) CL(T ) ≥ CL0 (T ), for T = 1, 2, . . . . (36)

PROOF: Parts (a) and (b), respectively, follow from Theorem 4.1(a) and Lemma 3.1(c) of[5]. Part (c) follows from (19), KO ≥ 0, and the definitions of CL and CL0 .

Recall that T ∗ denotes the optimal review period for the model of [5] (see Section 4). Let TU

denote the smallest T ≥ T0 such thatCL0 (T +1) > C∗(T ∗). Lemma 4(b) ensures that TU exists.Next, if CL0 (1) ≤ C∗(T ∗), let TL = 1; otherwise, let TL denote the largest T ∈ [2, T0] suchthat CL0 (T − 1) > C∗(T ∗). Since T0 ≥ 2 when CL0 (1) > C∗(T ∗), TL exists. By Lemma 4(a),CL0 (T ) > C∗(T ∗) when T /∈ [TL, TU ]. This, C∗(T ∗) ≥ C∗(T ∗∗), Theorem 2, and Lemma 4(c)imply that T ∗∗ ∈ [TL, TU ].

A GLOBAL SEARCH ALGORITHM FOR COMPUTING T ∗∗:

STEP 1. (Initialize). Compute TL, TU , T ∗ and C∗(T ∗); T := T ∗; C := C∗(T ∗);STEP 2. (Search T ∈ [TL, T ∗)). T := T ∗ − 1;

while T ≥ TL dobegin

while T ≥ TL and CL(T ) ≥ C do T := T − 1;if T ≥ TL then begin compute C∗(T ); call update; end;T := T − 1;

end;STEP 3 (Search T ∈ (T ∗, TU0 ]). T := T ∗ + 1;

while T ≤ TU dobegin

while T ≤ TU and CL(T ) ≥ C do T := T + 1;if T ≤ TU then begin compute C∗(T ); call update; end;T := T + 1;

end.

This algorithm stops with T = T ∗∗ and C = C∗(T ∗∗). Step 1 computesTL, TU , T ∗, andC∗(T ∗),and then uses T ∗ and C∗(T ∗) to initialize T and C. Step 2 searches {TL, TL + 1, . . . , T ∗ − 1}for a better T . This step exploits our lower bound CL(T ) on C∗(T ) by skipping the calculation


of C∗(T ) whenever CL(T ) ≥ C. Finally, Step 3 uses the approach of Step 2 to search {T ∗ +1, T ∗ + 2, . . . , TU}.

Note that T ∗ is not the only possible initial value for T ; however, the numerical study of Section8 found that T ∗ is often close to T ∗∗. In particular, T ∗∗ = T ∗ for all the test problems of Section 8with µ/σ ≥ 3. Note also that our algorithm searches [TL, TU ] for a global minimum of C∗. Onecan simplify computations by searching for a local minimum of C∗, starting from T ∗; however,this approach failed to produce a global minimum for 8% of the exponential demand problemsof Section 8 and for the example below.

EXAMPLE 1: (A local minimum ofC∗ is not a global minimum.) Let demand be exponentialwithµ = 10, and letKA = 0,KO = 400, p = 5, andh = 5. Our algorithms findT ∗ = 10, T ∗∗ =5, C∗(T ∗) = 129.151, and C∗(T ∗∗) = 126.828. A local search starting from T = T ∗ would notfind T ∗∗, since C∗(10) = 129.151, C∗(9) = 129.017, and C∗(8) = 129.037.

REMARK 1: Our global search algorithm does not require that the Gjs satisfy (3.2) (i.e.,proportional holding and shortage costs). It is sufficient that Gj(z∗j ) is nondecreasing in j,limi→∞Gj(z∗j ) =∞, and Assumption 1 holds for all T (see Remark 1).

5. 1. Computing Optimal (s, S) Policies

To get C∗(T ), one must compute the optimal (s, S) policy of Theorem 1. In Sections 6 and 8,optimal (s, S) policies are computed using Zheng and Federgruen [23, 24], which assumes discretedemands and entails calculations involving dT ∈ [0, ST −sT ], sT ∈ [sT , y

∗T ], and ST ∈ [y∗T , ST ].

Since the demands in Sections 6 and 8 are continuous, one must discretize dT , sT , and ST . Ourapproach defines the gridsize ∆ via an integer M ≥ 3 and the formula

∆ ≡ (ST − sT )/(M − 2). (37)

The discretized dT , sT , and ST are restricted to the sets {0,∆, . . . ,M∆}, {y∗T − j∆: j =0, . . . , a}, and {y∗T + j∆: j = 0, . . . , b}, respectively, where a = ceiling((y∗T − sT )/∆), b =ceiling((ST − y∗T )/∆), and ceiling(x) denotes the least integer greater than or equal to thereal number x. Both accuracy and computation time increase with M . (Given our discretizationscheme, [23, 24] require O(M) evaluations of the C0T -function and O(M2) elementary oper-ations: see Theorem 1 of [23]). After trying M values ranging from 250 to 8000, we settled onM = 1000 for the computational experiments of Section 8 andM = 8000 for our examples. Thischoice reflects the primary goal of Section 8, which is the accurate assessment of the performanceof our approximations to T ∗∗, not the most efficient code for computing T ∗∗.

6. A NUMERICAL EXAMPLE

This section provides a numerical example illustrating our results. Let the single-period de-mands be normal with µ = 30 and σ = 10, and let KA = 0,KO = 80, p = 6, h = 4. ThenK = 80, α = 0.0, θ = 2.070702, and τ = 0.6 [see (6)–(7)]. (The calculations of this section usedouble precision.)

Consider, first, a simple approximation to an optimal policy: the myopic base stock policywith review period T l. This can be computed using a spreadsheet when demands are normal (theExcel function NORMSINV computes Φ−1). We begin with T l. Define the increasing functionJ(T ) = (T + 1)3/2 −∑T+1

j=1√j, for T ≥ 0. Assuming normal demands, Theorem 6.1 of [5]

implies the following. First, T = T l if and only if J(T − 1) < θ ≤ J(T ). Second, T l ∈


{T# − 1, T#, T# + 1}. By (33), T# = floor((3θ)2/3) + 1 = 4. Hence, one need only searchthe set {3, 4, 5} for T l. Now, J(3) ≈ 1.85 < θ ≤ 2.80 ≈ J(4) gives us T l = 4. Next, by(5.2) of [5], z∗j ≡ σ

√jΦ−1(τ) + µj = 2.533

√j + 30j, for 1 ≤ j ≤ 4, which determines the

myopic base stock policy (see Remark 2 below). One can approximate WM (T l), the mean cost

of this policy, by W l(T l) ≡ σ(p + h)φ(Φ−1(τ)){∑T l

j=1√j + θ}/T l = 79.3641 (see Section

5 of [5]). Using the formulas in [5] (which require numerical integration), one can also obtainWM (T l) = 79.7684. Note that the myopic base stock policy with review period T l has relativeerror 100[WM (T l)/C∗(T ∗∗)− 1] = 0.027%. (The quantity C∗(T ∗∗) is calculated below). Notealso that for this example TM also equals 4.

REMARK 2: To obtain the orders for periods 1 through T under the myopic policy, substitutey = max{x, z∗T } into (12) and follow the cumulative order rule. Note that if the initial state is lessthan or equal to z∗T , then all subsequent states are also less than or equal to z∗T and the cumulativeorder rule takes the following form: In period 1, order max{x, z∗1}. In period 2 ≤ j ≤ T , ordermax{x, z∗j } −max{x, z∗j−1}.

Second, consider a more complicated approximation: the optimal base stock policy of [5]with review period T ∗. Using the approach outlined at the end of Section 7 below, we find thatT ∗ = 4 and the optimal base stock level, y∗T∗ = 124.17. Moreover, this policy has mean costW ∗(T ∗) = 79.7469 and relative error 100[W ∗(T ∗)/C∗(T ∗∗) − 1] = 0.000%. (This becomespositive if one uses more decimal places). Its computation time on a 300 Mhz Pentium II underthe MS-Dos mode of Windows 95 was 0.03 s.

Third, we turn to an optimal (s, S) policy with review period T ∗∗. Using the algorithm ofSection 5, we obtain T ∗∗ = 4, s∗T∗ = 95.27, S∗T∗ = 124.17, andC∗(T ∗∗) = 79.7469. Moreover,for this example—and for most of the normal demand test problems of Section 8—s∗T∗ is closeto sT∗ , S

∗T∗ is close to y∗T∗ , and P{dT∗ ≤ S∗T∗ − s∗T∗} is close to 0, which helps explain why

base stock policies do so well for such test problems. This pattern is most pronounced for largeµ/σ and starts to break down when µ/σ drops below 3. (Corollary 6.1 of [3] proves that whendemands are normal and both θ > 0 and τ are fixed, P{dT∗ ≤ S∗T∗ − s∗T∗} → 0 as µ/σ →∞.)Incidentally, the computation time for an optimal (s, S) policy with review period T ∗∗ is highlysensitive to M (see the end of Section 5). It equaled 3.5 s for M = 500, 7.5 s for M = 1000, and251.2 s for M = 8000. All three M -values yielded the same C∗(T ∗∗) to nine digit accuracy.

Finally, we determine the cost advantage of allowing staggered deliveries. For the standardmodel without staggered deliveries, the optimal ordering policy is an (s, S) policy; furthermore,KA = 0 implies that the optimal review period equals 1, the optimal average cost per pe-riod equals C∗(1), and the percentage cost penalty for not allowing staggered deliveries equals100[C∗(1)/C∗(T ∗∗) − 1]. Hence, for our current example, s∗1 = 11.57, S∗1 = 36.59, C∗(1) =111.9095, and 100[C∗(1)/C∗(T ∗∗)− 1] = 40.33%.

7. NORMAL AND GAMMA DEMANDS

This section summarizes results needed for the numerical study of Section 8, which dealswith normal and gamma demands. Because of its wide applicability and analytic tractability, thenormal is the most commonly studied continuous demand distribution; however, its use as anapproximation to demand requires that µ/σ ≥ 2. The gamma is also popular and applies to caseswhere µ/σ is large or small. Both distributions can fit a wide variety of situations.

The numerical study of Section 8 examines the effect of varying α, µ/σ, τ , and θ for normaland exponential demand test problems. This approach to selecting parameters needs motivation.For the case of normal demands, Section 5 of [5] shows that T ∗ and TM are functions of µ/σ, τ ,


and θ, and T l is a function of θ. Using similar arguments, one can show that T ∗∗ is a functionof α ≡ KA/K, µ/σ, τ , and θ if demands are normal. For the case of gamma demands, Theorem4 below and µ/σ =

√r imply that T ∗, TM , and T l are functions of µ/σ, τ , and θ; and T ∗∗ is a

function of α, µ/σ, τ , and θ.Next, suppose that the single period demands are gamma with scale parameter λ and shape

parameter r. Note that µ = r/λ, σ2 = r/λ2, and µ/σ =√r (see [19]). (Also if r is a positive

integer, then the gamma is equivalent to the well-known Erlang distribution, which reduces to theexponential if r = 1.) Temporarily change notation by adjoining λ, r to fj , Fj , z∗j , and y∗T . Thenthe cumulative j-period demands are gamma with density function,

fj,λ,r(x) = λe−λx(λx)jr−1/Γ(jr), for x ≥ 0 and j ≥ 1. (38)

LEMMA 5: Let the single period demands have a gamma distribution with scale parameter λand shape parameter r. Then, for j ≥ 1, T ≥ 1, x ∈ R, and z ∈ R,

Fj,λ,r(x) = Fj,1,r(λx), (39)

z∗j,λ,r =1λz∗j,1,r, (40)

Gj(z) =1λ

(p+ h)Gj(λz), (41)

C0T (z) =1λ

(p+ h)C0T (λz), (42)

y∗T,λ,r =1λy∗T,1,r, (43)

C0T (z) +KOδ(z − x) ≡ 1λ

(p+ h)[C0T (λz) + K0δ(λz − λx)], (44)

where Gj depends on j, τ , and r; C0T , and y∗T,1,r depend on T, τ , and r; and K0 depends onα, θ, τ , and r.

PROOF: Using fj,λ,r(x) ≡ λfj,1,r(λx) and some integration, one can show that (39) holds,which implies (40). By (9),Gj(0) = pjµ. This, the continuity of demand and [4, (61)] imply that

Gj(z) ={pjµ+

∫ z0 (p+ h)(−τ + Fj,λ,r(w)) dw, for z ≥ 0,

p(jµ− z). for z < 0. (45)

Also (39) implies that∫ z

0 Fj,λ,r(x) dx = 1λ [∫ λz

0 Fj,1,r(w) dw, for z ≥ 0. Using this result, (45),τ = p/(p+ h), and r = λµ, one can show that (41) holds with

Gj(λz) ={τjr − τλz +

∫ λz0 Fj,1,r(w) dw, for z ≥ 0,

τ(jr − λz), for z < 0.(46)

Next, using (40)–(41) and (14)–(15), one can verify that C1T (z) ≡ 1λ (p + h)C1T (λz) and

C2T (z) ≡ 1λ (p + h)C2T (λz), where C1T and C2T depend on T, r, and τ . Using these results,


(16), and fT,λ,r(x) ≡ λfT,1,r(λx), one can establish (42), which implies (43). Finally,σ2 = r/λ2

and (7) imply that

K =1λ

(p+ h)φ(Φ−1(τ))√rθ. (47)

This and K0 = (1− α)K imply that K0 = (1− α)φ(Φ−1(τ))√rθ satisfies (44).

REMARK 3: Note that Gj , C0T , and K0 equal Gj , C0T , and K0, respectively, when λ =p+ h = 1.

THEOREM 4: Let the single period demands have a gamma distribution with scale parameterλ and shape parameter r. Then, for T ≥ 1,

W ∗(T ) =1λ

(p+ h)W ∗(T ), (48)

WM (T ) =1λ

(p+ h)WM (T ), (49)

W l(T ) =1λ

(p+ h)W l(T ), (50)

C∗(T ) =1λ

(p+ h)C∗(T ), (51)

where W ∗(T ), WM (T ), and W l(T ) are functions of T, r, τ , and θ; and C∗(T ) is a function ofT, r, τ, α, and θ. Furthermore, T ∗, TM , and T l are functions of r, τ , and θ; and T ∗∗ is a functionof r, τ, α, and θ.

PROOF: Now (47), (42), and (43) imply that W ∗(T ) ≡ [C0T (y∗T ) + φ(Φ−1(τ))√rθ]/T

satisfies (48). Also Lemma 5 ensures that C0T (y∗T ) depends on T, r, and τ . Hence, W ∗(T )depends on T, r, τ , and θ; and T ∗ depends on r, τ , and θ. Similar arguments establish the claimsabout WM (T ), W l(T ), TM , and T l. Next, according to Theorem 1, CT (x1, π

∗T ) is independent

of x1 and equals the optimal average cost per stage for a stationary model where the single-stagecost of selecting y in state x equals C0T (y) + KOδ(y − x). Using this, (44), (24), KA = αK,and (47), one can show that (51) holds for some CT that is a function of T, r, τ, α, and θ, whichimplies that T ∗∗ is a function of r, τ, α, and θ.

7. 1. Computations for the Normal and Gamma

Our calculations for the normal obtain Fj(z), fj(z), z∗j , Gj(z), andGj(z∗j ) from (5.1)–(5.5) of[5] and then use (2.8)–(2.14) of [5] to getC0T (z) andC ′0T (z). This entails numerical integration.To obtain y∗T , we locate a zero of C ′0T in [z∗1 , z

∗T ]. Also, we use the short procedure of Section

4 of [5] to get T ∗. Our test problems for the gamma, use λ = 1 (without sacrificing generality)and integer r. For such parameters, Fj(z) = 1 − e−z∑jr−1

i=0 zi/i! and Gj(z) = h(z − jr) +(p+h)e−z

∑jr−1i=0 zi(jr− i)/i!, for z ≥ 0 (see (65)–(67) of [5]). One can use these expressions,

z∗j = F−1j (τ), and (2.8)–(2.14) of [5] to findC0T (z) andC ′0T (z). We avoid numerical integration

by using the approach of Section 5 of [4] to compute H ′jT and HjT of (2.11) and (2.13) of [5](see Remark 4 below). Our method of obtaining y∗T and T ∗ is the same as for the normal.


REMARK 4: There are typographical errors in Section 5 of [4], which are corrected as follows.First, add c(1 − β)z to the right-side of (62) and replace −c(1 − β) by +c(1 − β) in (66) and(67). Second, multiply the right-side of (68) by i! and move (T + i)! from the numerator tothe denominator of (70). These errors did not affect other parts of [4, 5]. Our computationalexperiments used the right formulas.

8. COMPUTATIONAL EXPERIENCE

Some issues are difficult to resolve analytically. Recall that Section 4 defined an optimal reviewperiod T ∗∗ and introduced four quantities from [5]—T ∗, TM , T l, and T#—which can be used toapproximate T ∗∗. This section describes a numerical study involving normal and gamma demandtest problems that is designed to answer the following questions.

Q1. How close are T ∗, TM , T l, and T# to T ∗∗?Q2. How doesC∗(T ∗∗) (the optimal average cost) compare withC∗(T ) [the average

cost of an optimal (s, S) policy with review period T ] when T = T ∗, TM , T l,or T#?

Q3. How does C∗(T ∗∗) compare with W ∗(T ) (the average cost of an optimal basestock policy with review period T ) when T = T ∗, TM , T l, or T#?

Q4. What about C∗(T ∗∗) versus WM (T ) (the average cost of a myopic base stockpolicy with review period T ) when T = TM , T l, or T#?

Q5. How doesC∗(T ∗∗) compare with the optimal average cost for a standard modelwithout staggered deliveries? That is, what is the cost advantage of our approachover the conventional one?

Q6. What are the effects of the model parameters?Q7. What are the computational requirements?

Note that [3] provides partial answers to Q1–Q4 for the normal demand case by establish-ing that the following hold for large µ/σ when θ > 0 and τ are fixed. First, T ∗∗, T ∗, andTM ∈ {T l, T l + 1}. (By [4], |T l − T#| ≤ 1 and both T l and T# depend only on θ.) Sec-ond, C∗(T )/C∗(T ∗∗),W ∗(T )/C∗(T ∗∗), and WM (T )/C∗(T ∗∗) are asymptotically equal to 1when T = T ∗, TM , and T l. And, third,C∗(T#)/C∗(T ∗∗),W ∗(T#)/C∗(T ∗∗), andWM (T#)/C∗(T ∗∗) are asymptotically equal toW l(T#)/W l(T l). The numerical study below suggests thatthe patterns associated with large µ/σ can occur with µ/σ as small as 3.

These results suggest that when demands are normal and µ/σ is large, near optimum perfor-mance results from using an optimal or a myopic base stock policy and setting T = T ∗, TM , orT l. The effect on performance of setting T = T# depends on W l(T#)/W l(T l) ≥ 1, which isa function only of θ. This ratio approaches 1 as θ → ∞, but can be as high as 1.03 for small θ(see Section 6 of [5]). Section 6 illustrates that little effort is required to compute the myopic basestock policy, the review period T l, or the asymptotic value of the optimal average cost W l(T l).Note that θ depends on σ and neither θ nor τ depend on µ. The simplest way to make µ/σ largewhile keeping θ and τ fixed is to fix σ and make µ large.

Concerning Q6, Section 1 states that it is plausible for the optimal review period to increasewith K, decrease with σ, p, and h, and be insensitive to µ. The behavior of T ∗∗, T ∗, TM , and T l

tend to fit this pattern when demands are normal and µ/σ ≥ 2. Our justification for this is thecloseness of T ∗∗, T ∗, TM , and T l to T# in the experiments below and remarks in Section 1 of [5]about the properties of θ, which imply that T# tends to increase linearly with [K/σ]2/3, decreasewith p and h, and be independent of µ (see Section 8 of [5]). Note that for gamma demands,σ =√r/λ =

√µ/λ, so T# is not independent of µ.


Table 1a. Mean and maximum absolute deviations for gamma.

µ/σ = 1 µ/σ =√

2 µ/σ =√

3 µ/σ = 2 µ/σ = 3

Mean Max Mean Max Mean Max Mean Max Mean Max

|T ∗∗ − T ∗| 1.237 19 0.158 7 0.048 2 0.003 1 0.000 0|T ∗∗ − TM | 1.369 18 0.307 6 0.208 2 0.162 1 0.108 1|T ∗∗ − T l| 1.498 19 0.361 7 0.207 3 0.134 1 0.097 1|T ∗∗ − T#| 1.692 20 0.494 7 0.330 3 0.257 1 0.204 1

We computed the following quantities for each of our normal and gamma demand test problems:T ∗∗, T ∗, TM , T l, and T#;C∗(T )/C∗(T ∗∗), for T = T ∗, TM , T l, T#, and 1;W ∗(T )/C∗(T ∗∗),for T = T ∗, TM , T l, and T#; andWM (T )/C∗(T ∗∗), for T = TM , T l, and T#. The algorithmsand formulas, which are discussed in Sections 5 and 7 above, were coded in Borland Pascal 7.0and run on a 300 Mhz Pentium II under the MS-DOS mode of Windows 95. All calculations weredone in double precision.

Given normal and gamma demands, our numerical study does not sacrifice generality by exam-ining the effect of varying onlyα, µ/σ, τ , and θ (see Section 7). Furthermore, all our test problemsuse α = 0, since it provides a worst case for most of our performance measures. Intuitively, theeffectiveness of base stock policies increases as α moves from 0 to 1, since KO moves from Kto 0 while KA moves from 0 to K. The same holds for T ∗ as an approximation to T ∗∗ (givencontinuous demands, T ∗∗ = T ∗ for α = 1). Formally, α = 0 maximizes W ∗(T )/C∗(T ∗∗) andWM (T )/C∗(T ∗∗) for all T . This is because of the lemma below and because both W ∗(T ) andWM (T ) are independent of α.

LEMMA 6: C∗(T ∗∗) is minimized at α = 0, ceteris paribus.

PROOF: Let T be fixed. One can write the expected one-stage cost under any policy (s, S) inany state x as Cs,S(x) = Cs,S(x) + (1− α)δ(s− x)K + αK, where Cs,S(x) does not dependon α or K. Since 0 ≤ δ(s − x) ≤ 1, Cs,S(x) is nondecreasing in α (ceteris paribus), implyingthat C∗(T ) is minimized at α = 0. Since T is arbitrary, this implies that α = 0 minimizesC∗(T ∗∗) ≡ minT C∗(T ).

Following [5], 10 demand distributions were tested: the gamma with λ = 1 and µ/σ ≡ √r =1,√

2,√

3, 2, and 3; and the normal withµ/σ = 2, 3, 4, 5, 10. (Whenµ/σ ≥ 4, the normal approx-imates the gamma.) For each distribution, problems were generated with all combinations of thefollowing parameters: c = 0;α = 0; τ = 0.01, 0.05, 0.10, 0.20, 0.30, 0.40, 0.45, 0.50, 0.55, 0.60,0.70, 0.80, 0.90, 0.95, 0.99; θ = (j−1)/10, for 1 ≤ j ≤ 10, θ = j, for 1 ≤ j ≤ 20, θ = 20+5j,

Table 1b. Mean and maximum absolute deviations for normal.

µ/σ = 2 µ/σ = 3 µ/σ = 4 µ/σ = 5 µ/σ = 10


|T ∗∗ − T ∗| 0.033 1 0.000 0 0.000 0 0.000 0 0.000 0|T ∗∗ − TM | 0.178 1 0.104 1 0.073 1 0.054 1 0.016 1|T ∗∗ − T l| 0.149 1 0.098 1 0.093 1 0.087 1 0.060 1|T ∗∗ − T#| 0.232 1 0.181 1 0.177 1 0.170 1 0.146 1


Table 2a. Optimal (s, S) policies for gamma: Mean and maximum of percent of C∗(T ) above C∗(T ∗∗).

µ/σ = 1 µ/σ =√

2 µ/σ =√

3 µ/σ = 2 µ/σ = 3


T = T ∗ 1.706 17.28 0.326 7.18 0.056 3.87 0.002 1.08 0.000 0.00T = TM 1.685 17.28 0.323 7.18 0.057 3.87 0.003 1.08 0.000 0.02T = T l 1.861 19.00 0.350 8.59 0.066 3.87 0.006 1.07 0.002 0.23T = T# 2.206 19.00 0.532 9.85 0.169 4.95 0.073 2.77 0.030 1.03

for 1 ≤ j ≤ 16, θ = 100 + 10j, for 1 ≤ j ≤ 10, and θ = 200 + 25j, for 1 ≤ j ≤ 4. This sampleof 9000 problems covers a wide variety of situations.

Tables 1a and 1b address Q1 by reporting the mean and maximum values of |T ∗∗ − T | forT = T ∗, TM , T l, and T#. Notice that the values tend to decrease rapidly as µ/σ increases. Inparticular, T ∗∗ = T ∗ in all of the problems with µ/σ ≥ 3. For the problems with µ/σ < 3, thevalues tend to decrease as θ increases. For instance, T ∗∗ = T ∗ for θ ≥ 25 when µ/σ = 1, forθ ≥ 12 when µ/σ =

√2, and for θ ≥ 5 when µ/σ =

√3 or 2. Overall T ∗∗ = T ∗ in almost

two-thirds of the problems with µ/σ = 1—the worst case in the tables. This validates our choiceof T ∗ as the initial value in our algorithm. Incidentally, we were unable to find any cases whereT ∗∗ > T ∗. This supports the conjecture Section 9 of [5] that T ∗ is an upper bound on T ∗∗. It alsosuggests reducing computations by eliminating Step 3 from our algorithm.

Q2 deals with the performance of the optimal ordering policy when combined with the reviewperiod T = T ∗, TM , T l, and T#. Tables 2a and 2b respond to Q2 by reporting on the meanand maximum values of the percentage relative error, 100[C∗(T )/C∗(T ∗∗) − 1]. As before,the means and maximums decrease rapidly as µ/σ increases. The four review periods performsurprisingly well for normal demands with µ/σ ≥ 3: All the relative errors equal 0 for T = T ∗;the overall means equal 0.000% for T = TM , 0.001% for T = T l, and 0.025% for T = T#; andthe overall maximums equal 0.03% for T = TM , 0.26% for T = T l, and 1.32% for T = T#.For gamma demands with µ/σ = 3, the values are basically the same as for normal demands withµ/σ = 3. At the other extreme, performance is good but not outstanding for gamma demandswith µ/σ = 1: The means lie between 1.7% and 2.2% and the maximums between 17.3% and19.0%. Again performance tends to improve as θ increases. For instance, the maximum relativeerrors are under 0.5% for θ ≥ 11 when µ/σ = 1, and for θ ≥ 3 when µ/σ =

√2,√

3 or 2.Q3 and Q4 cover the optimal and myopic base stock policies of [4]. Tables 3a, 3b, 4a, and 4b

report on the mean and maximum values of the percentage relative errors, 100[W ∗(T )/C∗(T ∗∗)−1] and 100[WM (T )/C∗(T ∗∗) − 1]. Those tables follow the same pattern as Tables 2a and 2b,but the numbers are higher. The comments below apply to normal demands with µ/σ ≥ 3.

Table 2b. Optimal (s, S) policies for normal: Mean and maximum of percent of C∗(T ) above C∗(T ∗∗).

µ/σ = 2 µ/σ = 3 µ/σ = 4 µ/σ = 5 µ/σ = 10


T = T ∗ 0.019 1.91 0.000 0.00 0.000 0.00 0.000 0.00 0.000 0.00T = TM 0.020 1.91 0.000 0.03 0.000 0.03 0.000 0.01 0.000 0.00T = T l 0.023 1.91 0.002 0.26 0.001 0.17 0.001 0.07 0.000 0.03T = T# 0.121 4.05 0.036 1.32 0.024 0.80 0.021 0.57 0.019 0.51


Table 3a. Optimal base stock policies for gamma: Mean and maximum of percent ofW ∗(T ) aboveC∗(T ∗∗).

µ/σ = 1 µ/σ =√

2 µ/σ =√

3 µ/σ = 2 µ/σ = 3


T = T ∗ 2.927 19.39 0.687 8.11 0.185 3.94 0.050 2.10 0.000 0.03T = TM 2.931 19.39 0.688 8.10 0.186 3.94 0.051 2.10 0.001 0.07T = T l 2.943 19.92 0.695 8.66 0.190 3.94 0.054 2.10 0.002 0.23T = T# 3.033 19.92 0.754 9.99 0.237 4.99 0.095 2.77 0.030 1.03

First, the approximations to T ∗∗ continue to perform well. Second, the optimal base stock policywith review period T ∗ has overall mean and maximum relative errors of 0.002% and 0.52%,respectively. Table 6 reports a maximum computation time of 0.7 s, which makes this policy aviable alternative to an optimal policy. Third, the myopic policy with review period T l has overallmean and maximum relative errors of 0.091% and 0.70% respectively. This policy is especiallyattractive, since as Section 6 illustrates, it can be computed by spreadsheet.

Q5 addresses the fundamental question: What are the cost advantages of allowing staggereddeliveries? To simplify the analysis, assume α = 0. This ensures that C∗(1) equals the optimalaverage cost per period for the ordinary model without staggered deliveries. Our numerical re-sults, summarized in Tables 5a and 5b, indicate that the percentage cost penalty for not allowingstaggered deliveries, 100[C∗(1)/C∗(T ∗∗) − 1], is sensitive to and increases with µ/σ and θ.Tables 5a and 5b find substantial cost penalties for moderate and large values of θ. One expectsthe savings to increase with θ, since θ increases with K. That the penalties increase with µ/σ isalso intuitive when one considers that a model with large µ/σ resembles a deterministic model.(In a deterministic model with staggered deliveries, it is optimal to order only once.) Results in[3] imply that limµ/σ→∞ C∗(1)/C∗(T ∗∗) = W l(1)/W l(T l) when demands are normal. Thelatter was used to compute the values in Table 5b for µ/σ =∞.

Note that the results in Tables 1–5 indicate that both the cost advantage of allowing staggereddeliveries and the performance of our approximations to T ∗∗ improve µ/σ increases beyond 1.Tests on a limited number of gamma problems with µ/σ = 0.5 suggest that this pattern extendsto smaller µ/σ.

Q7 addresses computational requirements. For the case of normal demands with µ/σ = 3,Table 6 reports on the computation times for the optimal base stock policy of [5] with T = T ∗,and for optimal (s, S) policies with T = T ∗ and T = T ∗∗. On the average, optimal (s, S) policieswithT = T ∗∗ took over 500 times as long to compute as optimal base stock policies withT = T ∗;however, the time to compute optimal (s, S) policies could be reduced significantly by using acoarser gridsize (see the last part of Section 5). Our choice of gridsize emphasized accuracy over

Table 3b. Optimal base stock policies for normal: Mean and maximum of percent ofW ∗(T ) aboveC∗(T ∗∗).

µ/σ = 2 µ/σ = 3 µ/σ = 4 µ/σ = 5 µ/σ = 10


T = T ∗ 0.129 2.85 0.009 0.52 0.000 0.04 0.000 0.00 0.000 0.00T = TM 0.130 2.85 0.010 0.52 0.001 0.04 0.000 0.01 0.000 0.00T = T l 0.132 2.85 0.011 0.52 0.001 0.17 0.001 0.07 0.000 0.03T = T# 0.190 4.22 0.040 1.32 0.024 0.80 0.021 0.57 0.019 0.51


Table 4a. Myopic base stock policies for gamma: Mean and maximum of percent of WM (T ) aboveC∗(T ∗∗).

µ/σ = 1 µ/σ =√

2 µ/σ =√

3 µ/σ = 2 µ/σ = 3


T = TM 3.818 20.54 1.267 8.56 0.622 4.05 0.401 2.10 0.172 0.68T = T l 3.856 21.45 1.287 9.40 0.635 4.05 0.410 2.10 0.176 0.69T = T# 3.963 21.45 1.353 10.52 0.687 5.10 0.455 2.83 0.206 1.08

computation time. Note that the times for the normal tend to decrease with µ/σ and increaserapidly with θ. (The relationship to θ is expected, since both T ∗ and T ∗∗ increase with θ and thework required to evaluate C0T increases with T .) A similar pattern holds for gamma demandsexcept that the computation times are longer and tend to increase rapidly with µ/σ. (The averagetimes for a gamma with µ/σ = 1 are almost twice as long as for a normal with µ/σ = 3.)

REMARK 5: We restricted the search for T ∗∗ to the set {1, 2, . . . , 120}. This seems reason-able, since the largest T ∗∗ obtained was 95. Note also that Section 8 of [5] used an approximationto C0T (z∗T ) when calculating TM for normal demands. This paper uses the exact value, whichleads to minor inconsistencies in the numerical results reported in the two papers. Another minorinconsistency, arises from the computational experiments of [5] overestimating T# by 1 whenθ = 75 and θ = 275 because of rounding errors.

REMARK 6: For the normal and exponential test problems, we used the global search algo-rithm of Section 5 to locate T ∗∗. For the gamma problems with µ/σ ≥ √2, we used local searchstarting from T ∗ when θ ≥ 35. (Our computations found T ∗∗ = T ∗ for θ ≥ 25 when µ/σ = 1;furthermore, local search always yielded T ∗∗ = T ∗ when θ ≥ 12 and µ/σ ≥ √2.)

9. MISCELLANEOUS ISSUES

9. 1. Forecast Errors

By independence, σdT , the standard deviation of dT , equals√Tσ. In practice, one uses a

forecast dT to estimate dT , and forecast errors tend to be positively correlated. Hax and Candea[7, p. 177] indicate that σdT can often be approximated by T bσd1

, where b lies between 0.5 and1.0. We believe that when b > 0.5, our approach tends to overestimateT ∗∗, since it underestimatesσdT . If one assumes that dj is normal with mean jµ and standard deviation jbσ for some b ≥ 0.5and all j ≥ 1, then using the arguments for Lemma 5.1 and Theorem 5.1(a) of [5], one can

Table 4b. Myopic base stock policies for normal: Mean and maximum of percent of WM (T ) aboveC∗(T ∗∗).

µ/σ = 2 µ/σ = 3 µ/σ = 4 µ/σ = 5 µ/σ = 10


T = TM 0.517 2.85 0.193 0.69 0.098 0.46 0.057 0.33 0.007 0.09T = T l 0.526 2.85 0.197 0.70 0.101 0.46 0.058 0.33 0.007 0.09T = T# 0.588 4.23 0.227 1.39 0.124 0.81 0.079 0.57 0.026 0.51


Table 5a. Nonstaggered versus staggered models: Mean of percent of C∗(1) above C∗(T ∗∗) for gammademands as a function of µ/σ and θ (taken over all values of τ ).

θ µ/σ = 1 µ/σ =√

2 µ/σ =√

3 µ/σ = 2 µ/σ = 3

0–0.4 0 0 0 0 00.5 0 0 0 0.5 2.80.6 0 0 0.5 2.5 6.00.7 0 0 1.6 4.2 8.90.8 0 0.3 2.6 5.6 11.60.9 0 0.7 3.5 6.7 13.91 0 1.2 4.2 7.5 16.12 1.1 5.1 10.5 16.1 34.33–5 4.3 11.5 20.2 27.5 50.86–10 9.1 21.6 33.0 41.8 70.5

11–20 15.8 34.2 47.5 57.8 91.425–45 29.4 52.8 68.6 80.8 120.550–95 45.3 72.4 90.5 104.6 150.2

100–190 62.8 93.7 114.3 130.2 181.9200–300 78.0 112.6 135.4 153.1 210.0

0–300 30.6 49.6 63.2 73.8 108.1

show that

W l(T ) = σ(p+ h)φ(Φ−1(τ))

T∑j=1

jb + θ

/T, for T ≥ 1. (52)

Table 5b. Nonstaggered versus staggered models: Mean of percent of C∗(1) above C∗(T ∗∗) for normaldemands as a function of µ/σ and θ (taken over all values of τ ).

θ µ/σ = 2 µ/σ = 3 µ/σ = 4 µ/σ = 5 µ/σ = 10 µ/σ =∞0–0.4 0 0 0 0 0 00.5 0 2.3 2.9 2.9 2.9 2.90.6 1.2 5.3 6.1 6.2 6.2 6.20.7 3.1 8.1 9.1 9.2 9.2 9.20.8 4.7 10.5 11.9 12.0 12.0 12.00.9 6.2 12.8 14.4 14.6 14.7 14.71 7.4 14.9 16.8 17.1 17.2 17.22 15.8 33.8 43.1 46.1 47.3 47.33–5 26.9 50.5 68.9 82.0 100.3 101.86–10 41.3 70.1 94.0 113.5 170.3 195.9

11–20 57.3 91.0 119.2 143.4 226.5 342.225–45 80.5 120.1 153.7 183.0 290.8 644.550–95 104.5 149.9 188.4 222.3 351.5 1101.4

100–190 130.3 181.7 225.4 263.8 412.4 1802.8200–300 153.1 209.8 257.8 300.1 465.2 2640.6

0–300 73.6 107.8 136.2 160.5 249.1 838.1


Table 6. Mean and maximum computation time in seconds for normal demands with µ/σ = 3 as a functionof θ.

Mean MaxOrdering Reviewpolicy period 0 ≤ θ ≤ 20 25 ≤ θ ≤ 95 100 ≤ θ ≤ 300 0 ≤ θ ≤ 20 25 ≤ θ ≤ 95 100 ≤ θ ≤ 300

OptimalBaseStock T = T ∗ 0.09 0.26 0.41 0.2 0.4 0.7

Optimal(s, S) T = T ∗ 5.15 15.68 23.96 11.8 22.8 37.1

Optimal(s, S) T = T ∗∗ 16.24 137.32 350.44 52.7 281.5 602.7

Next, using (52) and the approach of Section 6 of [5], one can derive the following approximationto T l:

T#,b = floor

((1 + b

bθ

)1/(1+b))

+ 1. (53)

(This coincides with T# when b = 0.5.) The lemma below implies that T#,0.5 ≥ T#,b whenb > 0.5. We conjecture that our approach also overestimates T ∗, TM , T l, and T ∗∗ for such b.

LEMMA 7: b1 > b2 > 0 implies T#,b1 ≤ T#,b2 .

PROOF: By way of contradiction, assume b1 > b2 and T#,b1 > T#,b2 . By T#,b1 >T#,b2 and (53), T#,b1 ≥ 2, which implies that (1+b1

b1θ) ≥ 1. The latter and b1 > b2 im-

ply that ( 1+b1b1

θ)1/(1+b1) ≤ ( 1+b1b1

θ)1/(1+b2). Now b1 > b2 > 0 implies that ( 1+b1b1

θ)1/(1+b2) ≤( 1+b2b2

θ)1/(1+b2). Hence, ( 1+b1b1

θ)1/(1+b1) ≤ ( 1+b2b2

θ)1/(1+b2). This contradicts T#,b1

> T#,b2 .

9. 2. Shortage Costs

One problem with our model is the difficulty of estimating the shortage cost p. The shortage costincludes added information processing costs, loss of goodwill and future sales because of customerdissatisfaction, lost revenue when unsatisfied demand is lost, and the cost of delayed revenue,priority shipments, and other expediting actions when unsatisfied demand is backordered (see[17]). For a producer, a shortage could cause a line shut down. Factors like customer dissatisfactionare nebulous. One approach to this problem is to ignore p and impose a service level constraintsuch as the P1 (cycle service level), the P2 (fill rate), or the P3 (ready rate) constraint (see [17,pp. 244–246]). Silver, Pyke, and Peterson [18, p. 318] recommends against this approach whendealing with class A items.

Incidentally the ready rate (i.e., the fraction of periods where no stockouts occur) is never belowτ ≡ p/(p+h) under the myopic base stock policy of [5] when demands are continuous. To justifythis, notice that under the myopic policy, the stock level at the beginning of the jth period of stagem is always greater than or equal to z∗j , for m ≥ 1 and 1 ≤ j ≤ T . Therefore, the probability of

no stockout during period j is greater than or equal to P{dj < z∗j }, which equals τ by (11) andthe continuity of demand.


10. SUMMARY AND CONCLUSIONS

This article assumes orders can arrive in every period and the cost of receiving them is negligible(as in a JIT setting). EveryT periods, one audits the current stock level and decides on deliveries foreach of the next T periods, thus incurring a fixed audit cost and—when one schedules deliveries—a fixed order cost. The problem is to find a review period T and an ordering policy that satisfythe average cost criterion.

For the case of fixed T , Section 3 formulates a dynamic programming model, which groupsblocks of T periods into stages, and presents a simple rule describing the optimal deliveries forthe T periods of a stage as a function of the starting inventory level and the cumulative T -periodorder, and shows that a stationary (s, S) policy determines the optimal cumulative T -periodorder. Section 4 proves that an optimal review period T ∗∗ exists and introduces four quantities—T ∗, TM , T l, and T#—that approximate it. The quantity T ∗ is optimal for the problem addressedin [5]. The others approximate T ∗. Note that T# ≡ floor((3K/[σ(p+h)φ(Φ−1(τ))])2/3) + 1 iseasy to compute. Section 5 develops a global search algorithm for T ∗∗.

In [5], which assumes the fixed order cost is incurred every T periods, a base stock policydetermines the optimal cumulativeT -period order; moreover, there is an easily computable myopicbase stock policy, whose cost is often close to the optimal cost. These objects form the basis oftwo heuristic policies: the myopic base stock policy of [5] with review period T l, and the optimalbase stock policy of [5] with review period T ∗. Section 6 describes a numerical example withnormal demands, which depicts the computation of these policies. By mimicking that example,one can use a spreadsheet to calculate the myopic base stock policy with review period T l and anapproximation to its cost for any normal demand problem.

Sections 7 and 8 deal with a numerical study that compares T ∗∗, T ∗, TM , T l, and T# in 4500normal demand problems with µ/σ ≥ 2 and 4500 gamma demand problems with µ/σ ≥ 1. Theresults of this study and analytic results from [3] (which proves that the cost differences approach0 as µ/σ → ∞) suggest that our approximations to T ∗∗ tend to do better for larger values ofµ/σ, do well for µ/σ ≥ 2, and do very well for µ/σ ≥ 3. Furthermore, Section 8 reports thatthe difference between the optimal cost and the cost of the optimal base stock policy with reviewperiod T ∗ has mean and worst case values, respectively, of 2.93% and 19.39% for µ/σ = 1,0.090% and 2.85% for µ/σ = 2, and 0.002% and 0.52% for µ/σ ≥ 3, while the differencebetween the optimal cost and the cost of the myopic base stock policy with review period T l hasmean and worst case values, respectively, of 3.86% and 21.45% for µ/σ = 1, 0.468% and 2.85%for µ/σ = 2, and 0.108% and 0.70% for µ/σ ≥ 3.

For our normal test problems withµ/σ = 3, the mean computation times on a 300 Mhz PentiumII equals 0.21 s for the optimal base stock policy of [5] with T = T ∗ and 130.06 s for an optimal(s, S) policy with T = T ∗∗. Furthermore, the differences in the times are greater for gamma testproblems. This suggests that one would only want to use the exact procedure when the errors ofthe heuristics are high, e.g., when µ/σ < 2 or µ/σ < 3. Among the two heuristics, the optimalbase stock policy with T = T ∗ is more accurate but it requires some programming effort.

For the standard model without staggered deliveries, the optimal policy is an (s, S) policy.By comparing the cost of the latter with the cost of this article’s optimal (s, S) policy withreview period T ∗∗, one can determine the cost advantage of allowing staggered deliveries. Forthe example of Section 6, the cost advantage was 40%. The numbers in Tables 5a and 5b suggestthat the cost advantage of allowing staggered deliveries is sensitive to and increases with µ/σ and


θ ≡ K/[σ(p + h)φ(Φ−1(τ))]. Incidentally, the cost of both of our heuristic policies was lowerthan the cost of an optimal (s, S) policy for the standard model in all test problems with µ/σ ≥ 3and θ ≥ 0.5 or with µ/σ = 2 and θ ≥ 0.7. This is interesting because our heuristic policies areless demanding computationally.

The model developed here might be of interest to a JIT manufacturer, who places weeklyorders with a vendor, but wants to reduce inventory levels by scheduling daily deliveries from thevendor. Its advantage over the standard model is that it uncouples the review period and deliverydecisions and allows multiple deliveries for each review instead of just a single delivery. Asindicated above, the cost savings resulting from our model can be substantial. Furthermore, weprovide two heuristic policies which perform well for normal and gamma problems with µ/σ ≥ 3and are also easy to compute and implement. (One of these can be computed on a spreadsheetwhen demands are normal.)

REFERENCES

[1] C. Chiang and W. Benton, Sole sourcing versus dual sourcing under stochastic demands and leadtimes, Nav Res Logistics 41 (1994), 609–624.

[2] C. Chiang and W. Chiang, Reducing inventory costs by order splitting in the sole sourcing environment,J Oper Res Soc 47 (1996), 446–456.

[3] J. Flynn, The asymptotic behavior of the optimal review period as µ/σ → ∞ in an inventory modelwith staggered deliveries, and normal demands, working paper, 1998.

[4] J. Flynn and S. Garstka, A dynamic inventory model with periodic auditing, Oper Res 38 (1990),1089–1103.

[5] J. Flynn and S. Garstka, The optimal review period in a dynamic inventory model, Oper Res 45 (1997),736–750.

[6] H. Groenevelt, ‘‘The just-in-time system,’’ Handbooks in operations research and management Science.Volume IV: Logistics of production and inventory, S. Graves, A. Rinnooy Kahn, and P. Zipkin (Editors),North-Holland, Amsterdam, 1993, pp. 629–670.

[7] A. Hax and D. Candea, Production and inventory management, Prentice-Hall, Englewood Cliffs, NJ,1984.

[8] D. Heyman and M. Sobel, Stochastic models in operations research, McGraw-Hill, New York, 1984,Vol. II.

[9] F. Janssen, A. De Kok, and F. Van der Duyn Schouten, Order splitting from a suppliers perspective,working paper, 1998.

[10] P. Kelle and E. Silver, Decreasing expected shortages through order splitting, Eng Costs Prod Econ 19(1990), 351–357.

[11] P. Kelle and E. Silver, Safety stock reduction by order splitting, Nav Res Logistics 37 (1990), 725–743.[12] H. Lau and L. Zhao, Optimal ordering policies with two suppliers when lead times and demand are all

stochastic, Eur J Oper Res 68 (1993), 120–133.[13] H. Lee and S. Nahmias, ‘‘Single-product, single-location models,’’ Handbooks in operations research

and management science. Volume IV: Logistics of production and inventory, S. Graves, A. RinnooyKan, and P. Zipkin (Editors), North-Holland, Amsterdam, 1993, pp. 3–55.

[14] E. Porteus, Numerical comparisons of inventory policies for periodic review systems, Oper Res 33(1985), 134–152.

[15] R. Ramasesh, J. Ord, J. Hayya, and A. Pan, Sole versus dual sourcing in stochastic lead-time (s,Q)inventory models, Manage Sci 37 (1991), 428–443.

[16] R.J. Schonberger, Japanese manufacturing techniques, The Free Press, New York, 1982.[17] D. Sculli and S. Wu, Stock control with two suppliers and normal lead times, J Oper Res Soc 32 (1981),

1003–1009.[18] E. Silver, D. Pyke, and R. Peterson, Inventory management and production planning and scheduling,

Ed. 3, Wiley, New York, 1998.[19] H. Tijms, Stochastic models: An algorithmic approach, Wiley, New York, 1995.


[20] A.F. Veinott, Jr., The optimal inventory policy for batch ordering, Oper Res 13 (1965), 424–432.[21] A.F. Veinott, Jr., On the optimality of (s, S) inventory policies: New conditions and a new proof, SIAM

J Appl Math 14 (1966), 1067–1083.[22] Y.-S. Zheng, A simple proof for the optimality of (s, S) policies in infinite-horizon inventory systems,

J Appl Probab 28 (1996), 802–810.[23] Y.-S. Zheng and A. Federgruen, Finding optimal (s, S) policies is about as simple as evaluating a

single policy, Oper Res 39 (1991), 654–665.[24] Y.-S. Zheng and A. Federgruen, Corrections to ‘‘Finding optimal (s, S) policies is about as simple as

evaluating a single policy,’’ Oper Res 40 (1992), 192.

selecting t for a periodic review inventory model with staggered deliveries

Documents