selecting review periods for a coordinated multi-item inventory model with staggered deliveries

Selecting Review Periods for a Coordinated Multi-ItemInventory Model with Staggered Deliveries

James Flynn

Department of Operations Management and Business StatisticsCleveland State University

Cleveland, Ohio 44115

Received January 2000; revised August 2000; accepted February 11, 2001

Abstract: Consider anN -item, periodic review, infinite-horizon, undiscounted, inventory modelwith stochastic demands, proportional holding and shortage costs, and full backlogging. For 1 ≤j ≤ N , orders for item j can arrive in every period, and the cost of receiving them is negligible(as in a JIT setting). Every Tj periods, one reviews the current stock level of item j and decideson deliveries for each of the next Tj periods, thus incurring an item-by-item fixed cost kj . Thereis also a joint fixed cost whenever any item is reviewed. The problem is to find review periodsT1, T2, . . . , TN and an ordering policy satisfying the average cost criterion. The current articlebuilds on earlier results for the single-item case. We prove an optimal policy exists, give conditionswhere it has a simple form, and develop a branch and bound algorithm for its computation.We also provide two heuristic policies with O(N) computational requirements. Computationalexperiments indicate that the branch and bound algorithm can handle normal demand problemswith N ≤ 10 and that both heuristics do well for a wide variety of problems with N ranging from2 to 200; moreover, the performance of our heuristics seems insensitive to N . c© 2001 John Wiley& Sons, Inc. Naval Research Logistics 48: 430–449, 2001

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

1. INTRODUCTION

How often should one review the inventory status, place orders, and schedule deliveries? Mostapproaches to this problem assume only one delivery can be scheduled for each review; however, aspointed out in Flynn and Garstka [6, 7] and Flynn [4], unlinking the delivery and review intervalscan lead to substantial cost savings, lower inventory levels, and less frequent reviews. Thosearticles address the problem of finding an optimal review period T for a single-item model whendeliveries are more frequent than reviews. First Flynn and Garstka [6] characterize an optimalordering policy where, for arbitrary T , one observes the current stock level every T periods, andorders deliveries for each of the next T periods. Next they [7] find a best T , assuming one employsthe optimal ordering policy of [6] and incurs a fixed order cost every T periods. Recently, Flynn[4] removed a major limitation of [6, 7]: the assumption that the fixed order cost is incurred everyT periods—even when the stock level is high enough for placing orders to be suboptimal. Thisarticle extends [6, 7] to a coordinated multi-item model where when any item is reviewed there

c© 2001 John Wiley & Sons, Inc.

Flynn: Multi-Item Inventory with Staggered Deliveries 431

is a joint fixed cost plus an item-by item fixed cost for each item reviewed. (We do not extend [4]because its policies are computationally demanding; moreover, results in [4, 5] suggest that thepolicies of [6, 7] perform almost as well as those of [4], given normal or gamma demands withµ/σ ≥ 2.)

Silver et al. [15, Chap. 11] reviews the coordinated multi-item replenishment literature. Much ofthe work on stochastic models focuses on continuous review (S, c, s) or can-order policies. Suchpolicies have been shown to be considerably better than uncoordinated policies, often producing20% savings; however, they are generally suboptimal (see Federgruen, Groenevelt, and Tijms [3],Ingall [9], and Silver [16]). Atkins and Iyogun [1] propose periodic review policies that outperformcan-order policies and are easier to compute. Flynn [4] discusses the order splitting literature andreports on a numerical study that finds substantial cost savings from allowing staggered deliveriesfor single-item exponential and normal demand models.

This article considers an N -item, periodic review, stochastic inventory model with stationaryparameters. Costs are undiscounted, and the planning horizon is infinite. The demands for eachitem are independent, identically distributed, nonnegative random variables. Holding and shortagecosts are proportional, and all shortages are backordered. The decision variables are the reviewperiod Tj and the ordering policy for item j, 1 ≤ j ≤ N . Orders can arrive in any period, andthe cost of receiving them is negligible. Every Tj periods, one observes the current stock levelof item j and decides on deliveries for each of the next Tj periods, thus incurring a proportionalpurchase cost and an item-by-item fixed cost kj . A joint fixed cost K is also incurred wheneverany item is reviewed. The fixed costs do not depend on T1, . . . , TN . The goal is to find reviewperiods and ordering policies that minimize the (long-run expected) average cost per period.

Assuming items are ordered when they are reviewed, K includes the fixed (or header) costof placing a joint replenishment order, and kj includes the line cost of adding line j to theorder [15]. Scheduling frequent deliveries is consistent with the JIT philosophy (Groenevelt [8]),e.g., our model might be of interest to a manufacturer, who places weekly orders, but wants toreduce stock levels by scheduling daily deliveries. Ordering a sequence of deliveries might entailtiming separate order releases or placing orders once with a supplier, who would agree to staggerthe arrivals of the deliveries (Schonberger [14, Chap. 7]). The cost of timing order releases isconsidered part of the cost of receiving an order, which is assumed negligible. Note that, althoughtransactions reporting is popular, periodic review systems are important (Lee and Nahmias [11]).The advantages of periodic over continuous review include lower administrative costs, easiercoordination of the orders of related items, improved workload planning by the supplier andbuyer, and a regular opportunity to adapt the ordering policy to changes in the demand pattern([15]). Furthermore, as Porteus [12] observes, transactions reporting systems that keep inventoryrecords current, but order periodically are equivalent to periodic review systems.

This article is organized as follows. Section 2 defines notation and assumptions. Section 3uses results in [6, 7] to characterize optimal ordering policies when T1, . . . , TN are fixed andprovide formulas that allow one to compute V (T1, . . . , TN ), the optimal average cost per periodgiven T1, . . . , TN . Section 4 proves there exist optimal review periods T ∗∗

1 , . . . , T ∗∗N , i.e., ones

where V (T ∗∗1 , . . . , T ∗∗

N ) = V ∗∗ ≡ infT1,...,TNV (T1, . . . , TN ). Section 4 also gives bounds on

(V (T1, . . . , TN ) − V ∗∗)/V ∗∗ and obtains sufficient conditions for T1, . . . , TN to be optimal.Section 5 introduces approximations based on the myopic ordering policy of [6] and providesadditional bounds.

Section 6 presents an O(N) algorithm for computing a ‘‘good’’ heuristic policy. This algorithmis based on Atkins and Iyogun [1], who solve, for each j, the single-item model with item j afterallocating a portion αjK of the joint fixed cost to the item-by-item fixed cost kj . Their basicidea is to choose αj’s that keep the smallest Tj’s in balance. For normal demand problems, the

432 Naval Research Logistics, Vol. 48 (2001)

computations can be done on a spreadsheet. Section 7 presents a branch and bound algorithmfor computing T ∗∗

1 , T ∗∗2 , . . . , T ∗∗

N . Its memory requirements are modest, but its running time isexponential. Section 7 also introduces a second heuristic policy with O(N) requirements. Section8 contains an illustrative example.

Section 9 describes a numerical study involving normal and gamma demand test problems,which assesses the performance of our optimal branch and bound algorithm and our two heuristicpolicies. Our results suggest that our branch and bound algorithm can handle normal demandproblems with N ≤ 10 and that both heuristics do well for a wide variety of problems with Nranging from 2 to 200; furthermore, the performance of our heuristics seems insensitive to N .This article closes with Section 10, which summarizes our results and states our conclusions.

2. ASSUMPTIONS AND NOTATION

Costs are undiscounted, the planning horizon is infinite, and the objective is to minimize theaverage cost per period. The decision variables are the review period Tj and the ordering policyfor item j, 1 ≤ j ≤ N . Orders can arrive in any period, and cost of receiving them is negligible.For j = 1, 2, . . . , N and m = 1, 2, . . ., one reviews the current stock level of item j at thebeginning of period (m − 1)Tj + 1 and places orders for periods (m − 1)Tj + 1 through mTj ,thus incurring a proportional purchase cost and an item-by-item fixed cost kj > 0. A joint fixedcost K > 0 is also incurred whenever any item is reviewed. The fixed costs, which arise fromauditing the stock levels and ordering sequences of deliveries, are independent of T1, . . . , TN .Audits yield perfect information about the current stock levels. Holding and shortage costs areassessed on end-of-period inventories and are proportional with unit shortage cost pj > 0 andunit holding cost hj > 0 for item j. All shortages are backordered. The demands for item j indifferent periods are independent, nonnegative, and stationary with a finite positive mean µj andvariance σ2

j . (The demands for different items in the same period may be dependent.) This paperemploys the following notation:

djn = generic cumulative n-period demand for item j, where 1 ≤ j ≤ N and n ≥ 1, (1)

F jn(z) = P [dj

n ≤ z], for z ∈ R, (2)

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

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

τj = pj/(pj + hj), (5)

βj = σj(pj + hj)φ(Φ−1(τj)), (6)

θj(z) = (zK + kj)/βj , for z ∈ R, (7)

θ =

(K +

N∑i=1

ki

)/ N∑i=1

βi. (8)

Our decisions do not affect the purchase cost, so we ignore it. The function F jn is the n-fold

convolution of F j1 with itself. Distributions for which the F j

n’s are readily obtainable include thePoisson, the negative binomial, the exponential, the gamma, and the normal.

3. THE MODEL WITH T 1, T 2, . . ., T N FIXED

Our approach to finding an optimal policy involves two parts. The first part obtains an optimalordering policy given fixedT1, . . . , TN , while the second part finds best values ofT1, . . . , TN given


the ordering policy of the first part. The first part is covered here. Assuming fixed T1, . . . , TN , theperiods where the item-by-item and the joint fixed costs occur are fixed, so the optimal orderingpolicy for item j depends only on Tj , pj , hj , and demand parameters for item j. In particular, theoptimal ordering policies for the various items are independent of one another when T1, . . . , TN

are fixed. This section assumes that j and Tj are fixed and uses results in [6, 7] to characterizean optimal ordering policy for item j. All fixed costs are ignored. To simplify the exposition, wedrop the subscript j from Tj .

Consider an infinite stage dynamic programming model where stage m (m = 1, 2, · · ·) containsperiods (m−1)T +1 through mT . The state x in stage m is the initial inventory level of item j inperiod (m−1)T +1, while the decision determines the orders for item j in periods (m−1)T +1through mT . The state is always known with certainty. The decision first specifies y ∈ R, wherey equals x plus the cumulative orders for item j in the T periods of stage m, and then specifiesz = (z1, . . . , zT ) ∈ R

T , where zi equals x plus the cumulative orders for item j in periods(m − 1)T + 1 through (m − 1)T + i for 1 ≤ i ≤ T . Decision y and z is feasible at state x ifx ≤ z1 ≤ · · · ≤ zT = y. Of course, dj

T is the generic single-stage demand for item j.If y and z are selected in state x, then the end-of-stage inventory level equals y − dj

T by ourbackordering assumption (negative inventories indicate backorders), and the single-stage costequals

T∑i=1

Gji (zi), (9)

where

Gji (zi) ≡ pjEdj

i − zi+ + hjEdji − zi− (10)

denotes the expected holding and shortage cost in the ith period of the stage. For 1 ≤ i ≤ T, Gji

is positive and convex. Let

zji = minz : F j

i (z) ≥ τj, i = 1, . . . , T. (11)

If demands are continuous, then F ji (zj

i ) = τj , i = 1, . . . , T . It is well known that

Gji (z

ji ) = minGj

i (z) : z ∈ R, i = 1, . . . , T. (12)

Also, using the nonnegativity of demand, one can prove

0 ≤ zj1 ≤ · · · ≤ zj

T < ∞. (13)

One can simplify the decision process by ‘‘optimizing out’’ z. Theorem 1 of [6] establishesthat

z(x, y) = (zi(x, y)) = (maxx, minzj1, y, . . . ,maxx, minzj

T−1, y, y) (14)

optimally selects z as a function of x and y. That is, if the state is x and the total order for thestage is y − x ≥ 0, then the following cumulative order rule optimally distributes the orders


for the T periods of the stage: In period 1, order z1(x, y) − x. In period 2 ≤ i ≤ T , orderzi(x, y) − zi−1(x, y). Note that y need not be optimal. Henceforth, assume without loss ofgenerality that z = z(x, y). There is now one decision variable y. By [6, Lemma 2], the single-stage cost of selecting decision y when the state is x equals

Cj1T (x) + Cj

2T (y), (15)

where the functions Cj1T (·) and Cj

2T (·) are positive and convex, and

Cj1T (x) ≡

T−1∑i=1

[Gji (maxx, zj

i ) − Gj(zji )], (16)

Cj2T (y) ≡

T−1∑i=1

Gji (miny, zj

i ) + GjT (y). (17)

Decomposition (15) plays a key role in the analysis of [6].An ordering policy for item j is a sequence πj = (πj

1, πj2, · · ·) of (Borel measurable) mappings

πjm : R → R such that πj

m(x) ≥ x for x ∈ R and m ≥ 1 : πjm(x) is the decision that πj selects

for state x in stage m. Theorem 3 of [6] proves there exists a yjT such that the following policy is

optimal for item j: When in state x, choose y = maxx, yjT . That is, follow a base stock policy

with the same base stock level yjT for all stages. By results in Sections 2 and 3 of [6], yj

T is anysolution to

Cj0T (yj

T ) = minCj0T (y) : y ∈ [zj

1, zjT ] and yj

T ∈ [zj1, z

jT ], (18)

where Cj0T (·) is positive and convex and

Cj0T (y) = Cj

2T (y) + ECj1T (y − dj

T ). (19)

[Section 2 of [7] provides formulas that expedite the computation of yjT and Cj

0T (·).]Ignoring fixed costs and purchase costs, Cj

0T (y) represents the average cost per stage for itemj under a base stock policy where the base stock level always equals y. To obtain the average costper period, one must divide the average cost per stage by T . Specifically, Cj

0T (yjT )/T represents

the average cost per period under an optimal base stock policy.As an alternative to an optimal policy, Section 4 of [6] introduces the myopic base stock policy,

i.e., one where the base stock level always equals zjT . This policy would be optimal if at the end

of each stage all stocks of item j were salvaged at the original purchase price, and all backorderedshortages were made up by purchases at the same price. The myopic policy is easy to computesince it depends only on the zj

T ’s. In particular, to obtain the orders for periods 1–T under themyopic policy, substitute y = maxx, zj

T into (14) and follow the cumulative order rule. Notethat if the initial state is less than or equal to zj

T , then all subsequent states are also less than orequal to zj

T and the cumulative order rule takes the following form: In period 1, order maxx, zj1.

In period 2 ≤ i ≤ T , order maxx, zji − maxx, zj

i−1.Again, ignoring fixed costs and purchase costs, the average cost per period for item j under

a myopic base stock policy equals Cj0T (zj

T )/T , which differs from the average cost per period


under an optimal base stock policy by (Cj0T (zj

T )−Cj0T (yj

T ))/T . The numerical study of [6] findsthat Cj

0T (zjT ) closely approximates Cj

0T (yjT ), given normal demands with µj/σj ≥ 3. (In the

worse normal demand case found, the difference equals 1%.) Another useful approximation is∑Ti=1 Gj

i (zji ). Theorem 7.2 of [7] shows that for normal demands,

∑Ti=1 Gj

i (zji ), C

j0T (zj

T ), andCj

0T (yjT ) are asymptotically equal for large µj/σj . Note that (2.15) of [7] and the positivity of

the Gji ’s ensure that

Cj0T (zj

T ) ≥ Cj0T (yj

T ) ≥T∑

i=1

Gji (z

ji ) > 0. (20)

4. OPTIMAL REVIEW PERIODS T ∗∗1 , T ∗∗

2 , . . ., T ∗∗N

Section 3 characterized an optimal ordering policy under the assumption that T1, . . . , TN arefixed at arbitrary positive integer values. Let V (T1, . . . , TN ) denote the average cost per periodunder such a policy. Define a collection of review periods T ∗∗

1 , . . . , T ∗∗N to be optimal if

V (T ∗∗1 , . . . , T ∗∗

N ) = V ∗∗ ≡ infT1,...,TN

V (T1, . . . , TN ). (21)

Theorem 1 gives sufficient conditions for T1, . . . , TN to be optimal and provides useful bounds.Corollary 1 proves that optimal T1, . . . , TN always exist. An algorithm for computing optimalT1, . . . , TN appears in Section 8.

Following Atkins and Iyogun [1, 2], we get a lower bound on V (T1, . . . , TN ) by examiningthe situation where the joint fixed cost K is allocated to the various items so that item j receivesαjK, where

αj ≥ 0, for 1 ≤ j ≤ N, andN∑

j=1

αj = 1. (22)

Consider any period n. Let xj = 1 if item j is reviewed in period n and xj = 0 otherwise (i.e.,xj = 1 if and only if n − 1 is an integer multiple of Tj). Then, as argued in [2], the joint fixedcost for period n equals

Kδ

N∑

j=1

xj

≥

N∑j=1

(αjK)xj , (23)

where δ(x) = 0 if x = 0 and δ(x) = 1 if x > 1. The assumption below ensures that (23) holdsas equality.

ASSUMPTION 1: Each Tj is an integer multiple of a base period Tb. Also αj = 0 whenTj > Tb.

LEMMA 1: Let (22) and Assumption 4.1 hold. Then (23) holds as equality for each period n.

PROOF: We consider two cases.


Case 1: n − 1 is an integer multiple of Tb. Here Assumption 1 implies that n − 1 is an integermultiple of Tj for all 1 ≤ j ≤ N such that αj > 0, which implies that xj = 1 for all such j. Thisand (22) give us δ(

∑Ni=1 xi) = 1 and

∑Ni=1 αixi = 1, which imply equality in (23).

Case 2: n − 1 is not an integer multiple of Tb. Then n − 1 is not an integer multiple of Tj forany 1 ≤ j ≤ N , which implies that xj = 0 for 1 ≤ j ≤ N and thus both sides of (23) equal0.

Sections 6 and 7 propose heuristic policies that satisfy Assumption 1. Under this assumption,each Tj is an integer multiple of min1≤i≤N Ti. Policies where this occurs are called strictly cyclic(e.g., see van Eijs [17]). Note that the restriction to such policies is not necessarily onerous: Fora deterministic model, results in Jackson, Maxwell, and Muckstadt [10] and Roundy [13] implythat one can achieve an average cost that is within 6% of the optimal average cost under a policywhere each Tj is a power-of-2 multiple of min1≤i≤N Ti. Incidentally, the algorithm of Section 7does not limit consideration to strictly cyclic policies.

For 1 ≤ j ≤ N , let Vj(αj , T ) denote the optimal average cost per period for a single-itemmodel with item j when the review period equals T and the fixed cost equals kj + αjK, i.e.,

Vj(αj , T ) = (Cj0T (yj

T ) + kj + αjK)/T (24)

[see (16)–(17)]. Also let T ∗j (αj) denote the smallest integer T minimizing Vj(αj , T ) (which

exists by Theorem 3.1 of [7]), and define

V ∗j (αj) = Vj(αj , T ∗

j (αj)) ≡ minT

Vj(αj , T ). (25)

[Note that T ∗j (αj) is an optimal review period for a single-item model with item j when one

follows an optimal base stock policy and the fixed cost equals kj + αjK.] The assumptionsof Section 2 ensure that the V ∗

j (αj)’s are positive. The theorem below provides a simple ex-pression for V (T1, . . . , TN ) when Assumption 1 holds, gives sufficient conditions for T ∗

1 (α1),T ∗

2 (α2), . . . , T ∗N (αN ) to be optimal, and obtains some useful bounds.

THEOREM 1: Let T1, . . . , TN be arbitrary and let α1, . . . , αN satisfy (22).

(a) V (T1, . . . , TN ) ≥ ∑Nj=1 Vj(αj , Tj), with equality holding under Assumption

1.(b) (Atkins and Iyogun) V ∗∗ ≥∑N

j=1 V ∗j (αj).

(c) Let Assumption 1 hold whenTj = T ∗j (αj), for1 ≤ j ≤ N . ThenT ∗

1 (α1), . . . , T ∗N (αN )

is an optimal collection of review periods, and V ∗∗ =∑N

j=1 V ∗j (αj).

(d) Let Assumption 1 hold. Then

V (T1, . . . , TN ) − V ∗∗

V ∗∗ ≤∑N

i=1(Vi(αi, Ti) − V ∗i (αi))∑N

i=1 V ∗i (αi)

. (26)

PROOF: Except for the joint fixed cost components, V (T1, . . . , TN ) and∑N

j=1 Vj(αj , Tj)are the same. Inequality (23) implies that the joint fixed cost component of V (T1, . . . , TN ) isgreater than or equal to the joint fixed cost component of

∑Nj=1 Vj(αj , Tj), giving us the in-

equality in (a). If the Tj’s satisfy Assumption 1, then (23) holds as equality by Lemma 1,


implying V (T1, . . . , TN ) =∑N

j=1 Vj(αj , Tj) and finishing (a). Part (b) follows easily from

(a). The proof of (c) is more involved. First, (b) and (25) imply V ∗∗ ≥ ∑Nj=1 V ∗

j (αj) =∑Nj=1 Vj(αj , T

∗j (αj)). Second, (a) and the T ∗

j (αj)’s satisfying Assumption 1 imply∑N

j=1 Vj(αj ,T ∗

j (αj)) = V (T ∗1 (α1), . . . , T ∗

N (αN )). Third, (21) implies V (T ∗1 (α1), . . . , T ∗

N (αN )) ≥ V ∗∗. Us-

ing these three results, one easily sees that V ∗∗ =∑N

j=1 V ∗j (αj) = V (T ∗

1 (α1), . . . , T ∗N (αN )),

which establishes (c). Finally, (a) and Assumption 1 imply V (T1, . . . , TN ) =∑N

i=1 Vi(αi, Ti);and (b) implies V ∗∗ ≥∑N

i=1 V ∗i (αi). These results and the positivity of the V ∗

i (αi)’s imply (26),proving (d).

COROLLARY 1: There exists an optimal collection of review periods T ∗∗1 , . . . , T ∗∗

N .

PROOF: Let T1, . . . , TN be arbitrary positive integers and let V = V (T1, . . . , TN ). ByLemma 3.1(c) of [5], Vj(1, Tj) → ∞ as Tj → ∞, for 1 ≤ j ≤ N . Thus, for 1 ≤ j ≤ N ,there exist Mj < ∞ such that Vj(1, Tj) > V when Tj ≥ Mj . Next, consider any T1, . . . , TN

where Tj ≥ Mj for some j. By Theorem 1(a), V (T1, . . . , TN ) ≥ Vj(1, Tj) ≥ V , so T1, . . . , TN

dominates T1, . . . , TN . Hence, when minimizing V (T1, . . . , TN ), one can restrict each Tj to thefinite set of positive integers less than Mj . Since a function restricted to a finite set always achievesits minimum, we conclude that an optimal T ∗∗

1 , . . . , T ∗∗N exists.

Note that given minor modifications, our arguments prove that Theorem 1 and its corollarieshold for any discrete time model where the V ∗

j (αj)’s are positive and limT→∞ Vj(1, T ) = ∞,for 1 ≤ j ≤ L.

5. SOME APPROXIMATIONS

Approximations to T ∗1 (α1), . . . , T ∗

N (αN ) are important. For the single-item model, [7] definesan optimal review period T ∗ and introduces three review periods—TM , T I , and T#—that ap-proximate it. The review period TM minimizes the average cost per period when one employsthe myopic ordering policy of Section 3. The other two review periods approximate TM and areeasier to compute than TM and T ∗. Computational experiments in [4, 7] for single-item modelswith normal and gamma demands indicate that T ∗, TM , T I , and T# tend to be close to oneanother when µ/σ ≥ 2. The quantity T ∗

j (αj) defined in Section 4 equals the value of T ∗ for thesingle-item model with item j when αjK is added to kj . Analogues of T I and T# are definedhere. (To simplify matters, we do not develop an analogue of TM .) Theorem 2 below providesbounds on performance under the myopic policy.

For 1 ≤ j ≤ N, αj ≥ 0, and T ≥ 1, let V Mj (αj , T ) equal the right side of (24) when Cj

0T (zjT )

replaces Cj0T (yj

T ), and let V Ij (αj , T ) equal the right side of (24) when

∑Ti=1 Gj

i (zji ) replaces

Cj0T (yj

T ). Note that V Mj (αj , T ) equals the average cost per period for a single-item model with

item j when the review period equals T , the fixed cost equals kj + αjK, and the ordering policyis the myopic policy of Section 3. By (20),

V Mj (αj , T ) ≥ Vj(αj , T ) ≥ V I

j (αj , T ) > 0. (27)

Let T Ij (αj) denote the smallest integer minimizing V I

j (αj , ·) and let

T#j (αj) = floor((3θj(αj))2/3) + 1, (28)


where θj(·) satisfies (7) and floor(x) denotes the greatest integer less than or equal to x. Also, let

V Ij (αj) = V I

j (αj , TIj (αj)). (29)

[Note that T Ij (αj) and T#

j (αj) are approximations to an optimal review period for a single-item model with item j when one follows a myopic base stock policy and the fixed cost equalskj +αjK.] Ceteris paribus, T ∗

j (αj), T Ij (αj), and T#

j (αj) are nondecreasing in αj (see Theorem3.2 of [7]). By Section 5 of [7], given normal demands, each T I

j (αj) is nondecreasing in θj(αj)and satisfies

T Ij (αj) equals the smallest T ∈ T#

j (αj) − 1, T#j (αj) + 1

minimizing

T∑

i=1

√i + θj(αj)

/T. (30)

Recall that Section 3 characterizes a myopic ordering policy when T1, . . . , TN are fixed atarbitrary values. Let V M (T1, . . . , TN ) denote the average cost per period under such a policy.Theorem 2(b) below obtains two bounds on (V M (T1, . . . , TN ) − V ∗∗)/V ∗∗, which are helpfulwhen evaluating the performance of the myopic policy. Section 9 illustrates the usefulness ofthese bounds.

THEOREM 2: Let T1, . . . , TN be arbitrary and let α1, α2, . . . , αN satisfy (22).

(a) V M (T1, . . . , TN ) ≥∑Nj=1 V M

j (αj , Tj), with equality holding under Assump-tion 1.

(b) Let Assumption 1 hold. Then

V M (T1, . . . , TN ) − V ∗∗

V ∗∗ ≤∑N

i=1(VMi (αi, Ti) − V ∗

i (αi))∑Ni=1 V ∗

i (αi)

≤∑N

i=1(VMi (αi, Ti) − V I

i (αi))∑Ni=1 V I

i (αi). (31)

PROOF: Part (a) follows from the arguments for Theorem 1(a). Next, (27) and (29) implyV I

j (αj) is a positive valued lower bound on V ∗j (αj), for 1 ≤ j ≤ N . Using this, one can

easily establish the second inequality in (31). The first inequality in (31) follows from (a) and thearguments for Theorem 1(d).

6. THE HEURISTIC POLICY: HM

Atkins and Iyogun [1, 2] propose a heuristic for determining the review periods for items1–N , which entails solving the N single-item models after increasing the fixed cost of item jby αjK, 1 ≤ j ≤ N , for appropriate αj’s that satisfy (22). To obtain the αj’s, they consider adeterministic version of the problem and proceed as follows: First, solve the single-item modelsafter setting each αj equal to 0 and calculate the runout times. Second, increase αj for any itemj with the smallest runout time. This will increase the runout time for item j and could lead to


ties for the shortest runout time. When more than one j has the shortest runout time, increase theαj’s for each, keeping the shortest runout times in balance. Continue until

∑Nj αj = 1. They call

the minimum runout time, Tb, the base period and B = j : αj > 0 the base set. For j ∈ B,they set the review period equal to Tb. For j /∈ B, they set it equal to an integer multiple of Tb

closest to its runout time. Here we adapt the Atkins-Iyogun procedure to our model. Section 6.1indicates how to obtain the αj’s. Section 6.2 provides an O(N) algorithm for the review periods.The heuristic policy, HM, combines the review periods of Section 6.2 with the myopic base stockordering policy of Section 3.

6. 1. Finding α∗1, . . ., α∗

N and B∗

The T#j (αj)’s of Section 5 play the same role here that deterministic runout times play in [1,

2]. That is, we adjust the αj’s so that the smallest T#j (αj)’s are in balance. By (28), this reduces

to choosing αj’s so that the smallest θj(αj)’s are equal and the αj’s satisfy (22). Theorem 3below indicates that one can make all the θj(αj)’s equal if and only if θ ≥ max1≤i≤L θi(0). Theproblem of adjusting the αj’s so that smallest θj(αj)’s are equal to some number ρ and the αj’ssatisfy (22) reduces to the following:

PROBLEM 1: Find ρ ≥ min1≤i≤N θi(0) and αj , for 1 ≤ j ≤ N , such that

αj = 0 if θj(0) ≥ ρ, (32)

θj(αj) = ρ if θj(0) < ρ, (33)

N∑i=1

αi = 1. (34)

As shown here, Problem 1 has a unique solution, whose computation is straightforward. For1 ≤ j ≤ N and ρ ≥ min1≤i≤N θi(0), define αj(ρ) as the unique αj ∈ [0,∞) that satisfies (32)and (33). By examining the cases where θj(0) ≥ ρ and θj(0) < ρ and applying (6) and (7), onecan verify that

αj(ρ) = (ρ − θj(0))+βj/K. (35)

Next, define

α(ρ) =N∑

j=1

αj(ρ). (36)

For 1 ≤ j ≤ N, αj(ρ) is continuous, αj(ρ) = 0 for ρ ∈ [min1≤i≤N θi(0), θj(0)], andαj(ρ) is strictly increasing for ρ ∈ (θj(0),∞). These observations, (35), and (36) imply thatα(min1≤i≤N θi(0)) = 0 and α(ρ) is continuous and strictly increasing for ρ ∈ [min1≤i≤N θi(0),∞]. Also, limρ→∞ α(ρ) = ∞. Hence, α(ρ) = 1 has a unique solution ρ∗, which can be found byusual search methods. Clearly ρ∗ and α∗

j ≡ αj(ρ∗), 1 ≤ j ≤ N , solve Problem 1. Incidentally,all calculations can be done on an Excel spreadsheet. [The function NORMSINV computes Φ−1

and the solver can solve α(ρ) = 1 for ρ∗.] The theorem and corollary below characterize ρ∗ andgive conditions where the θj(α∗

j )’s are equal.


THEOREM 3:

(a) θj(α∗j ) = maxρ∗, θj(0), for 1 ≤ j ≤ N .

(b) ρ∗ ≤ θ.(c) ρ∗ = θ if and only if θ ≥ max1≤i≤N θi(0).

PROOF: Clearly, θj(0) < ρ∗, α∗j = αj(ρ∗), and (33) imply θj(α∗

j ) = ρ∗, while θj(0) ≥ ρ∗

and (32) give us αj(ρ∗) = 0, which implies θj(α∗j ) = θj(0). This proves (a). Next, by (7) and

(8), θ∑N

i=1 βi −∑Ni=1 θi(0)βi = K. This, (35) and (36) imply that

α(θ) =N∑

i=1

(θ − θi(0))+βi

/K ≥

N∑i=1

(θ − θi(0))βi

/K =

(θ

N∑i=1

βi −N∑

i=1

θi(0)βi

)/K = 1.

(37)

Hence α(θ) ≥ 1. Since ρ∗ is the unique solution to α(ρ) = 1, this gives us ρ∗ ≤ θ, proving (b).Next, (37) implies that α(θ) = 1 if and only if θ ≥ max1≤j≤N θj(0), which implies (c).

The next corollary follows directly from Theorem 3 and (28).

COROLLARY 2: For 1 ≤ j ≤ N, T#j (α∗

j ) ≤ floor((3θ)2/3)+1, with equality holding whenρ∗ = θ.

REMARK 1: Using Theorem 3(c), (7), and (8), one can show that the θj(α∗j )’s are equal when

K is large or when the ki’s are close to one another and the βi’s are close to one another (which istrue of identical products). This is important. First, normal demand models with equal θj(α∗

j )’soften have equal T ∗

j (α∗j )’s and always have equal T I

j (α∗j )’s. Second, the heuristic policy HM

tends to be close to optimal for equal T Ij (α∗

j )’s, while the heuristic policy HO of Section 7.1below is optimal for equal T ∗

j (α∗j )’s.

Define

B∗ = j : α∗j > 0. (38)

Note that (32) and (33) imply θj(α∗j ) = ρ∗ for j ∈ B∗.

6. 2. The Algorithm

Before proceeding, use the approach of Section 6.1 to find ρ∗. Then use (35) to computeα∗

j ≡ αj(ρ∗), 1 ≤ j ≤ N , and use (38) to compute the set B∗. (Note that α∗j represents the

proportion of the joint fixed cost that our algorithm assigns to item j.) Our algorithm, whichconsists of Steps 1 and 2 below, yields review periods T1, T2, . . . , TN that satisfy Assumption 1.Its running time and storage requirements are O(N).

STEP 1: For 1 ≤ j ≤ N , compute T Ij (α∗

j ) and V Ij (α∗

j ) ≡ V Ij (α∗

j , TIj (α∗

j )). Set T ′ :=mini∈B∗ T I

i (α∗j ) and T ′′ := maxi∈B∗ T I

i (α∗i ). Find a Tb that minimizes

∑i∈B∗ V I

i (α∗i , Tb)

subject to T ′ ≤ Tb ≤ T ′′.

STEP 2: Do the following for 1 ≤ j ≤ N : If j ∈ B∗, then set Tj := Tb and computeV M

j (α∗j , Tj); else if j /∈ B∗, then set Tj equal to a positive integer multiple of Tb closest to


T Ij (α∗

j ) and compute V Mj (α∗

j , Tj). [Given a tie for the multiple of Tb closest to T Ij (α∗

j ), choosea Tj with the smallest V M

j (α∗j , Tj) value.]

Under this algorithm, Tj = Tb for j ∈ B∗. If the T Ij (α∗

j )s are equal for j ∈ B∗, then Tb equalstheir common value. Otherwise, Tb minimizes

∑i∈B∗ V I

i (α∗i , Tb) over Tb ∈ [T ′, T ′′]. The aver-

age cost per period under the heuristic HM equals V M (T1, . . . , TN ). Theorem 2(a) ensures thatV M (T1, . . . , TN ) =

∑Nj=1 V M

j (α∗j , Tj), and Theorem 2(b) gives bounds on (V M (T1, . . . , TN )−

V ∗∗)/V ∗∗. Although HM is suboptimal, it often achieves near optimal performance when Tj =T I

j (α∗j ), for 1 ≤ j ≤ N , and demands are normal or gamma, with µj/σjs ≥ 2. (The numerical

study of [4, 7] suggests that the bounds of Theorem 2(b) are close to 0 under these conditions.)Note that Tj = T I

j (α∗j ), for 1 ≤ j ≤ N , if T I

j (α∗j ) = Tb for j ∈ B, and T I

j (α∗j ) is a integer

multiple of Tb for j /∈ B. This is true when the T Ij (α∗

i )’s are equal.When demands are normal, one can use a spreadsheet to solve (30) for T I

j (α∗j ), to compute

V Ij (α∗

j ), and to calculate the myopic policy (see Section 8); moreover, the T Ij (α∗

j )’s are the samewhen j ∈ B, so Tb equals their common value. Formulas that expedite calculations for normaland exponential demand problems appear in [4, 7]. For normal demands, (5.2)–(5.5) of [7] giveus F j

i (z) ≡ Φ((z − iµj)/√

iσj),

zji ≡ σj

√iΦ−1(τj) + iµj , (39)

Gji (z

ji ) ≡ σj(pj + hj)ϕ(Φ−1(τj))

√i. (40)

7. AN OPTIMAL ALGORITHM FOR T ∗∗1 , T ∗∗

2 , . . ., T ∗∗N

This section provides a branch and bound algorithm for T ∗∗1 , . . . , T ∗∗

N . Section 7.1 obtainsa initial incumbent, T1, . . . , TN . If this satisfies the optimality condition of Theorem 1(c), westop. Else, we proceed to Section 7.2, which constructs a finite set containing T ∗∗

1 , . . . , T ∗∗N , and

then to Section 7.3, which describes the branch and bound algorithm. Throughout this section,α∗

1, . . . , α∗N and B∗ satisfy Section 6.1 (see the first paragraph of Section 6.2). Note that Section

7.1 naturally leads to an alternative heuristic, HO, which combines T1, . . . , TN with an optimalbase stock ordering policy.

7. 1. The Initial Incumbent T 1, T 2, . . ., T N

Our procedure for finding an initial incumbent solution, which consists of Steps 1 and 2 below,yields T1, . . . , TN satisfying Assumption 1. Its running time and storage requirements are O(N).(One could use the heuristic of Section 6.2 for an initial incumbent; however, the quantitiescomputed here are needed later.)

STEP 1: For 1 ≤ j ≤ N , compute T ∗j (α∗

j ) and V ∗j (α∗

j ) ≡ Vj(α∗j , T

∗j (α∗

j )). Set T ′ :=mini∈B∗ T ∗

i (α∗i ) and T ′′ := maxi∈B∗ T ∗

i (α∗i ). Find a Tb that minimizes

∑i∈B∗ Vi(α∗

i , Tb)subject to T ′ ≤ Tb ≤ T ′′.

STEP 2: Do the following for 1 ≤ j ≤ N : If j ∈ B∗, then set Tj := Tb and computeVj(α∗

j , Tj); else if j /∈ B, then set Tj equal to a positive integer multiple of Tb closest to T ∗j (α∗

j )and compute Vj(α∗

j , Tj). [Given a tie for the multiple of Tb closest to T ∗j (α∗

j ), choose a Tj withthe smallest Vj(α∗

j , Tj) value.]


By Theorem 1(c), this produces an optimal collection T1 = T ∗1 (α∗

1), . . . , TN = T ∗N (α∗

N ),when T ∗

j (α∗j ) = Tb for j ∈ B∗, and T ∗

j (α∗j ) is an integer multiple of Tb for j /∈ B∗. This is true if

the T ∗i (α∗

i )’s are equal [in which case, Step 2 need not compute any Vj(α∗j , Tj)’s]. Theorem 1(a)

ensures V (T1, . . . , TN ) =∑N

j=1 Vj(α∗j , Tj). If T1, . . . , TN is suboptimal, Theorem 1(d) gives a

bound on (V (T1, . . . , TN ) − V ∗∗)/V ∗∗.Formulas for calculating the quantities of interest appear in [4, 7]. In particular, one can use

(2.8)–(2.14) of [7] to get Cj0T (z) and (d/dz)Cj

0T (z). This entails numerical integration. One canget yj

T by locating a zero of (d/dz)Cj0T (z) in [zj

1, zjT ]. To find T ∗

j (αj), one can follow Section 4of [7].

7. 2. Constructing a Finite Set That Contains T ∗∗1 , T ∗∗

2 , . . ., T ∗∗N

Let T1, . . . , TN satisfy Section 7.1. This section constructs a finite interval [TLj , TU

j ] that con-tains T ∗∗

j , for 1 ≤ j ≤ N . (Our notation ignores the dependence of these objects on α∗1, . . . , α

∗N .)

We require the next lemma, which follows easily from Theorem 1(a) and Lemma 3.1(c) of [7].

LEMMA 2: Suppose that 1 ≤ j ≤ N is fixed. Then V Ij (α∗

j , T ) is a strictly decreasing in T on[1, T I

j (α∗j )] and nondecreasing in T on [T I

j (α∗j ),∞). Furthermore, limT→∞ V I

j (α∗j , T ) = ∞.

Define V = V (T1, . . . , TN ). Consider any T1, T2, . . . , TN and 1 ≤ j ≤ N . By Theorem 1,

V (T1, T2, . . . , TN ) ≥∑i /=j

V ∗i (α∗

i ) + Vj(α∗j , Tj). (41)

This implies that V (T1, T2, . . . , TN ) > V and thus T1, . . . , TN strictly dominates T1, . . . , TN if

Vj(α∗j , Tj) > V −

∑i /=j

V ∗i (α∗

i ). (42)

We obtain TLj and TU

j so that (42) holds for Tj /∈ [TLj , TU

j ]. First, find T 0j , the smallest Tj ≥ 1

such that

V Ij (α∗

j , Tj) > V −∑i /=j

V ∗i (α∗

i ) (43)

fails. Second, find TLj , the smallest Tj ≥ T 0

j such that (42) fails. These two steps are justi-fied because (27) implies that (43) fails when (42) fails and Theorem 1 implies that (42) failswhen Tj = T ∗

j (α∗j ). Third, find T 1

j , the smallest Tj ≥ T Ij such that (43) holds. [T 1

j exists sincelimT→∞ V I

j (α∗j , T ) = ∞.] Fourth, find TU

j , the largest Tj < T 1j such that (42) fails. By con-

struction, (42) holds for T 0j ≤ Tj < TL

j and TUj < Tj < T 1

j . Lemma 2 and (27) ensure (42) holdswhen Tj ≥ T 1

j and Tj < T 0j . Finding TL

j and TUj entails getting T 0

j and T 1j and then evaluating

Vj(α∗j , Tj) for T 0

j ≤ Tj ≤ TLj and TU

j ≤ Tj < T 1j . Our branch and bound algorithm searches

(T1, . . . , TN ) : TLj ≤ Tj ≤ TU

j for 1 ≤ j ≤ N for T ∗∗1 , . . . , T ∗∗

N .

7. 3. Branch and Bound

This section presents a branch and bound algorithm for computing T ∗∗1 , . . . , T ∗∗

N , starting fromT1, . . . , TN of Section 7.1. The algorithm’s memory storage requirement is O(N) and its running


time is exponential in N . The numerical study of Section 9 suggests that it can handle problemswith N ≤ 10.

Consider a tree Ω whose root is Φ and whose other nodes are of the form (T1, T2, . . . , Tm) where1 ≤ m ≤ N and TL

q ≤ Tq ≤ TUq for 1 ≤ q ≤ m. The root has branches TL

1 , TL1 + 1, . . . , TU

1 ,with branch T leading to node (T ). Node (T1, . . . , Tm) is terminal if m = N ; otherwise, ithas branches TL

m+1, TLm+1 + 1, . . . , TU

m+1, with branch T leading to node (T1, . . . , Tm, T ). Callnode (T1, . . . , Tm, T ′′) a right-sibling of (T1, . . . , Tm, T ′) if T ′′ > T ′. We use the notation(T1, . . . , TN ) and V = V (T1, . . . , TN ) for the current incumbent solution. By Theorem 1, thecost associated with any node (T1, T2, . . . , Tm) and each of its descendants is bounded below by

L(T1, . . . , Tm) =m∑

i=1

Vi(α∗i , Ti) +

N∑i=m+1

V ∗i (α∗

i ). (44)

Hence one can prune (T1, . . . , Tm) when L(T1, . . . , Tm) ≥ V . Sometimes one can do more. For1 ≤ m ≤ N , let TH

m denote the smallest integer in [T ∗m, TU

m ] such that Vm(α∗m, j) ≥ Vm(α∗

m, i)for TH

m ≤ i < j ≤ TUm . (This suppresses the dependency of TH

m on α∗m.) When Vm(α∗

m, T ) isunimodal in T, TH

m = T ∗m. One can prove

L(T1, . . . , Tm−1, j) ≥ L(T1, . . . , Tm−1, i), for THm ≤ i < j ≤ TU

m . (45)

Thus one can also prune each right-sibling of (T1, . . . , Tm) when L(T1, . . . , Tm) ≥ V andTm ≥ TH

m . Next, for any terminal node, (T1, . . . , TN ),

V (T1, . . . , TN ) =N∑

i=1

Vi(α∗i , Ti) − K

N∑i

α∗i

/Ti + Kr(T1, . . . , TN ), (46)

where r(T1, . . . , TN ) is the long-run proportion of periods where at least one item is reviewed,given review periods T1, . . . , TN . If each Tj is a multiple of min1≤i≤N Ti, then r(T1, . . . , TN ) =1/ min1≤i≤N Ti. Otherwise, one can find r(T1, . . . , TN ) by counting the number of review peri-ods in the cycle determined by T1, . . . , TN . This entails computing the cycle length, i.e., the leastcommon multiple of T1, . . . , TN. On reaching a terminal node that is better than the currentincumbent node, one makes that terminal node the new incumbent.

Our tree search procedure outlined below uses a depth-first search of Ω; e.g., when N =2, TL

1 = TL2 = 1, and TU

1 = TU2 = 2, it attempts to traverse Ω in the following order: (1), (1, 1),

(1, 2), (2), (2, 1), (2, 2). This reduces memory requirements and allows the use a stack to simplifycomputations (see below).

STEP 0 (Preprocessing): Compute and save the following quantities: TLi , TU

i , and THi , for

1 ≤ i ≤ N,∑N

i=m+1 V ∗i (α∗

i ), for 1 ≤ i ≤ N − 1, and Vi(α∗i , Ti), for 1 ≤ i ≤ N and

TLi ≤ Ti ≤ TU

i .

STEP 1 (Initialize): m := 1;T1 := TL1 ; (T1, . . . , TN ) := (T1, . . . , TN ).

STEP 2 (Main branch):

(a) Bound test: If L(T1, T2, . . . , Tm) ≥ V , then begin if Tm ≥ THm , then go to

Step 4, else go to Step 3, end;


(b) Move down the tree: If m < N , then begin m := m + 1;Tm := TLm; go to

Step 2(a), end;(c) Terminal node: If V (T1, . . . , TN ) < V , then set V := V (T1, . . . , TN ) and

Tj := Tj for 1 ≤ j ≤ N .

STEP 3 (Move to a right-sibling): If Tm = TUm , then go to Step 4, else begin Tm := Tm + 1;

go to Step 2, end.

STEP 4 (Move up the tree): m := m − 1; if m /= 0, then go to Step 3, else set T ∗j := Tj , for

1 ≤ j ≤ N and V ∗∗ := V and Stop.

Step 0 calculates and stores objects that never change. Both the memory storage (i.e., 5N −1+∑Nj=1 TU

j − TLj ) and computation time required by this step appear to be roughly proportional

to N . (This is confirmed in our numerical study.) Step 1 initializes the current node to (TL1 ). Step

2 examines all current nodes: 2(a) invokes the bound test and when appropriate transfers controlto 3 or 4; if the current node is nonterminal, 2(b) moves down the tree and repeats 2(a); if thecurrent node is terminal, control goes to 2(c), which updates the incumbent node and continuesto 3. Step 3 either branches to 4 or selects a right-sibling and branches to 2. Finally, 4 moves upthe tree and either stops or branches to 3.

Note that we use a stack to expedite the calculations for L(T1, . . . , Tm) and V (T1, . . . , TN ).Let

Γ(T1, . . . , Tm) =

(m∑

i=1

Vi(α∗i , Ti),

m∑i

α∗i

/Ti

), for (T1, . . . , Tm) ∈ Ω. (47)

The Γ-values are stored on a stack, where the top of this stack contains Γ(T1, . . . , Tm), the nextposition of the stack contains the Γ-value of the parent of (T1, . . . , Tm), and so forth.

8. A NUMERICAL EXAMPLE

This section provides a numerical example illustrating our optimal and heuristic algorithms. Letthe single-period demands be normal with N = 3, K = 160, σ1 = 10, µ1 = 30, p1 = 3, h1 =2, k1 = 60, σ2 = 10, µ2 = 50, p2 = 1, h2 = 1, k2 = 80, σ3 = 10, µ3 = 40, p3 = 3, h3 = 5, andk3 = 40.

Begin with α∗1, α∗

2, α∗3, and B∗ of Section 6.1. Applying (5)–(8), τ1 = 3/5, β1 = 50φ

(Φ−1(3/5)) = 19.3171, θ1(0) = 3.1061, τ2 = 1/2, β2 = 20φ(Φ−1(1/2)) = 7.9788, θ2(0) =10.0266, τ3 = 3/8, β3 = 80φ(Φ−1(3/8)) = 30.3356, θ3(0) = 1.3186, and θ = 340/57.6315 =5.8996. One can solve α(ρ) = 1 for ρ∗ = 5.2363 [see (36)]. By (35), α∗

1 = α1(ρ∗) =0.2572, α∗

2 = α2(ρ∗) = 0.0000, and α∗3 = α3(ρ∗) = 0.7428. Hence B∗ = 1, 3. All these

calculations can be done on a spreadsheet (see Section 6.1).Turn to the heuristic HM of Section 6. Step 1 yields T#

1 (α∗1) = T#

3 (α∗3) = T I

1 (α∗1) =

T I3 (α∗

2) = 7 and T#2 (α∗

2) = T I2 (α∗

2) = 10 using (28) and (30). Then one gets V I1 (α∗

1) =51.6428, V I

2 (α∗2) = 25.9271, and V I

3 (α∗3) = 81.0999, using (40). Of course, Tb = T ′ = T ′′ =

7. Step 2 gets T1 = T2 = T3 = 7, V M1 (α∗

1, T1) = 52.0758, V M2 (α∗

2, T2) = 26.8129, andV M

3 (α∗3, T3) = 81.2630. By Theorem 2, V M (T1, T2, T3) =

∑3i=1 V M

i (α∗i , Ti) = 160.1517

and (V M (T1, T2, T3)−V ∗∗)/V ∗∗ ≤∑3i=1(V

Mi (α∗

i , Ti)−V Ii (α∗

i ))/∑3

i=1 V Ii (α∗

i ) = 0.934%.Note that given T1 = T2 = T3 = 7, a myopic ordering policy for items 1, 2, and 3 is determined


by the zji ’s of (39) (see Section 3). The only object described here that cannot be computed

on a spreadsheet is V M (T1, T2, T3) =∑3

i=1 V Mi (α∗

i , Ti); however, it can be approximated by∑3i=1 V I

i (α∗i , Ti) = 159.5335.

Next, consider the optimal algorithm of Section 7, which starts by computing the initial incum-bent using the methods of Section 7.1. Following the approach of [7, Section 4], Step 1 obtainsT ∗

1 (α∗1) = T ∗

3 (α∗3) = 7, T ∗

2 (α∗2) = 10, V ∗

1 (α∗1) = 52.0303, V ∗

2 (α∗2) = 25.9704, V ∗

3 (α∗3) =

81.2556, and Tb = T ′ = T ′′ = 7. Step 2 gets T1 = T2 = T3 = 7, V1(α∗1, T1) = 52.0303, V2(α∗

2,T2) = 26.8123, and V3(α∗

3, T3) = 81.2556. Next, Section 7.2 produces TL1 = 6, TL

2 = 7, TL3 =

6, TU1 = 8, TU

2 = 14, and TU3 = 8. Finally, the branch and bound algorithm of Section 7.3 finds

T ∗∗1 = 6, T ∗∗

2 = 12, T ∗∗3 = 6, and V ∗∗ = V (T ∗∗

1 , T ∗∗2 , T ∗∗

3 ) = 159.5937. This implies that thetrue error of the heuristic HM is (V M (T1, T2, T3) − V ∗∗)/V ∗∗ = 0.350%.

The heuristic HO of Section 7 performs slightly better than HM, but is harder to compute.Applying Theorem 1, V (T1, T2, T3) =

∑3i=1 Vi(α∗

i , Ti) = 160.0982 and (V (T1, T2, T3) −V ∗∗)/V ∗∗ ≤ 0.529%. [The true error of HO is (V (T1, T2, T3) − V ∗∗)/V ∗∗ = 0.316%.] GivenT1 = T2 = T3 = 7, an optimal ordering policy for items 1, 2, and 3 is determined by the zj

i ’s of(39) and by y1

7 = 214.02, y27 = 349.49, y3

7 = 270.65 (see Section 3).

9. COMPUTATIONAL EXPERIENCE

This section describes a numerical study, involving normal and gamma test problems, whichassesses the performance of our optimal branch and bound algorithm and our two heuristic policies,HM of Section 6 and HO of Section 7. Our results suggest that our branch and bound algorithmcan handle normal demand problems with N ≤ 10 and that both heuristics do well for a widevariety of parameters; furthermore, the performance of our heuristics appears to be insensitive toN . Note that our algorithms and formulas, which are discussed in Sections 6 and 7, were codedin Borland Pascal 7.00 and run on a 300 MHz Pentium II under the MS-DOS mode of Windows98. All calculations used double precision.

In their numerical study, Atkins and Iyogun [2] simplify the parameter selection process byassuming identical products (i.e., constant µi’s, σi’s, pi’s, hi’s, and ki’s) and restricting K tolarge values. Here, their approach would be biased in favor of our heuristics, which tend to dovery well for such problems (see Remark 1). To describe our test bed, define Λ(N), for N ≥ 2,as the set of N -item problems where each item has a normal demand or each item has a gammademand, µi ∈ 30, 50, σi = 10, pi ∈ 1, 3, 10, hi ∈ 1, 2, 5, and ki ∈ 5, 10, 20, 40, 60, 80,for 1 ≤ i ≤ N , and K ∈ 5, 10, 20, 40, 80, 120, 160, 200. This set contains 16 · 108N problemsthat cover a wide variety of situations.

To evaluate the performance of the optimal branch and bound algorithm, we solved 200 ran-domly generated normal problems from Λ(N) for N = 2, 3, . . . , 12. We stopped the algorithmwhen the node count reached the upper limit of 2,000,000 without locating an optimal solution.This affected 2.5% of the cases with N = 12, but none of the cases with N < 12. Table 1 reportson the relationship between N and the mean and maximum node count, the mean and maxi-mum computation time, and the percentage of cases where the branch and bound part is skipped.[Branch and bound is skipped and the node count equals 0 if the initial incumbent satisfies theoptimality condition of Theorem 1(c).] Our results indicate that the optimal algorithm can easilysolve problems with N ≤ 8 and can handle problems with N ≤ 10, but experiences difficultywith larger N .


Table 1. Optimal solutions: the % of cases where branch and bound is skipped, and the mean and maximumcomputation time and node count versus N .

Time (s) Node CountN % of Cases Skipped Mean Maximum Mean Maximum

2 76.5 0.44 2.3 1.6 253 54.0 0.84 3.4 7.5 1104 38.0 1.24 3.9 25.5 2585 25.0 1.90 8.0 149.8 2,7456 21.5 2.46 25.2 334.7 7,0587 19.5 3.77 107.8 1,141.0 61,6858 14.5 5.15 115.5 2,236.5 35,4189 22.5 32.71 3,316.5 10,352.2 477,015

10 16.5 28.17 1,636.1 17,787.4 746,75711 22.0 220.90 26,263.7 64,427.7 1,446,12812 16.5 483.91 12,236.5 165,263.6 2,000,000

Turn to our heuristic policies. To measure performance of HO and HM, we computed thefollowing:

ERRO =∑N

i=1(Vi(α∗i , Ti) − V ∗

i (α∗i ))∑N

i=1 V ∗i (α∗

i )and ERRM =

∑Ni=1(V

Mi (α∗

i , Ti) − V ∗i (α∗

i ))∑Ni=1 V ∗

i (α∗i )

.

(48)

By Theorems 1 and 2, 100·ERRO and 100·ERRM, respectively, are upper bounds on the percent-age amounts by which the average costs of HO and HM exceed the optimal averagecost.

The first phase of our testing examines normal problems with N = 2. We compute ERRO andERRM for all 93,312 normal demand problems in Λ(2). For these test problems, performanceis good and there almost no difference between ERRO and ERRM. Table 2 reports that therespective mean and maximum values of ERRO equal 0.12% and 2.51%, while the respectivemean and maximum values of ERRM equal 0.15% and 2.52%. Moreover, ERRO and ERRM arerelatively insensitive to the µi’s, pi’s, and hi’s, but highly sensitive to the ki’s and to K. On theaverage, they tend to decrease with K and increase with the ki’s. Problems with small K coupledwith both large and small ki’s tend to have larger ERRO and ERRM.

Table 2. Mean and maximum values of 100-ERRO and 100-ERRM for normal and gamma test problemswith N = 2.

Normal GammaERRO ERRM ERRO ERRM

Parameter set Mean Max Mean Max Mean Max Mean Max

Λ(2) with large µj/σj 0.13 2.59 0.13 2.59 — — — —Original Λ(2) 0.12 2.51 0.15 2.52 0.13 2.34 0.15 2.36Λ(2) with µj/σj ≡ 2 0.12 2.30 0.36 2.52 0.12 2.21 0.30 2.37Λ(2) with µj/σj ≡ 1 — — — — 0.11 2.36 0.88 2.73Λ(2) with µj/σj ≡ 0.5 — — — — 0.11 1.25 1.76 3.12


For normal problems in Λ(2) with K ≥ 80, the respective mean and maximum values of ERROequal 0.03% and 1.32%, while the respective mean and maximum values of ERRM equal 0.08%and 1.51%. On the other hand, both ERRO and ERRM have means equal to 0.24% and achievetheir maximums for K = 5. Note that performance does not deteriorate as K becomes small. Weverified this by solving many additional 2-item, normal problems with K close to 0. Note alsothat our heuristics do well for some problems with K near 0. For instance, problems with K = 0are more likely to have base periods equal to 1. For such problems, HO is always optimal andHM tends to produce a solution that is close to optimal.

The second phase of our testing shifts to normal problems in Λ(N) with N > 2. To see howN affects performance, we compute ERRO and ERRM for a random sample of 2000 normalproblems from Λ(N) for N = 3, . . . , 20, 30, 40, 50, 100, and 200. Performance is good for allof these N values: The largest respective mean and maximum values of ERRO equal 0.28% atN = 9 and 2.44% at N = 4, while the largest respective mean and maximum values of ERRMequal 0.29% at N = 9 and 2.44% at N = 4. Furthermore, both the mean and maximum valuesof ERRO and ERRM decrease with N for N ≥ 20. (When N = 200, the respective mean andmaximum values of ERRO equal 0.02% and 0.51%, while the respective mean and maximumvalues of ERRM equal 0.03% and 0.53%.)

The third phase covers gamma problems in Λ(N) with N ≥ 2. We compute ERRO and ERRMfor a random sample of 2,000 gamma problems from Λ(N) for N = 2, . . . , 20, 30, 40, 50. Thepattern is the same as for the normal and the numbers are similar (e.g., see Table 2).

The fourth phase examines large µj/σj’s. First, we compute ERRO and ERRM for a randomsample of 2000 normal problems from Λ(2) after fixing µj ≡ 100 (and thus µj/σj ≡ 10). Then werepeat this after setting µj ≡ 1000. (One need not look at the gamma, since it closely approximatesthe normal for these µj/σj’s.) These two sets of calculations produce almost identical results.The individual values of ERRO and ERRM are very close and the mean values resemble thosefor the first phase (see Table 2).

The fifth phase examines smaller µj/σj’s. First, we compute ERRO and ERRM for a randomsample of 2000 normal problems and 2000 gamma problems from Λ(2) after fixing µj ≡ 20(and thus µj/σj ≡ 2). Second, we make the same calculations for a random sample of 2000gamma problems from Λ(2) after fixing µi = 10. For these test problems, fixing µj ≡ 20 has anegligible effect on ERRO and a slight effect on ERRM, while fixing µj ≡ 10 has a negligibleeffect on ERRO, but increases the mean and maximum values of ERRM (see Table 2). To furtherinvestigate the effect of decreasing the µj/σj’s, we compute ERRO and ERRM for a randomsample of 200 gamma problems from Λ(2) after fixing µj ≡ 5 (yielding µj/σj ≡ 0.5). Likebefore, the effect on ERRO is slight, while the respective mean and maximum values of ERRMincrease to 1.76% and 3.12%. Thus, it appears that the performance of ERRO remains stable andthe performance of ERRM becomes worse as the µj/σj’s gets smaller.

The computation time and the storage requirements of both HO and HM are roughly propor-tional to N . To estimate the time when N = 2, we took random samples of 2000 normal and 2000gamma test problems from Λ(2). For the normal problems, the respective mean times for HO andHM equaled 0.303 s and 0.009 s. For the gamma problems, they equaled 2.100 s and 0.087 s.

Our results indicate that HM performs almost as well as HO for problems with µj/σj ≥2, for 1 ≤ j ≤ N . Since HO is more demanding computationally than HM, this suggeststhat one should use HM for such problems. One caveat when using HM is that it naturallyprovides a bound which is much less accurate than the bound ERRM used here. [If one knowsV ∗

1 (α∗1), V

∗2 (α∗

2), . . . , V∗N (α∗

N ), which are required to compute ERRM, then one might just as


well use algorithm HO.] Specifically, HM provides the bound

ERRM0 =N∑

i=1

(V Mi (α∗

i , Ti) − V Ii (α∗

i ))/ N∑

i=1

V Ii (α∗

i ), (49)

which is looser than ERRM [see (31)]. For the normal test problems in the original Λ(2), therespective mean and maximum values of ERRM0 equal 0.49% and 3.24% (compare with Table2). Things get worse for smaller µj/σj’s. For our 2000 normal test problems with µj/σj ≡ 2,the respective mean and maximum values of ERRM0 equal 1.78% and 4.64%. We use ERRMsince it is more accurate. Usually, when one uses HM, one will measure performance by ERRM0,which yields more pessimistic results.

10. SUMMARY AND CONCLUSIONS

This article considers an N -item, periodic review inventory model with stochastic demands,where for 1 ≤ j ≤ N , orders for item j can arrive in every period and the cost of receiving themis negligible (as in a JIT setting). Every Tj periods, one reviews the current stock level of itemj and decides on deliveries for each of the next Tj periods, thus incurring an item-by-item fixedcost. There is also a joint fixed cost whenever any item is reviewed. The problem is to find reviewperiods, T1, T2, . . . , TN , and an ordering policy for each item so as to minimize the average costover an infinite-horizon.

Our approach to finding an optimal policy involves two parts. The first obtains an optimalordering policy given fixed T1, . . . , TN , while the second finds best values of T1, . . . , TN giventhe ordering policy of the first part. Section 3 covers the first part, exploiting results for the single-item case. For fixed Tj , we formulate a dynamic programming model—which groups blocks ofTj periods into stages; present a simple rule describing the optimal deliveries for the Tj periodsof a stage as a function of the starting inventory level and the cumulative Tj-period order; showthat a base stock policy determines the optimal cumulative Tj-period order; and introduce amyopic base stock ordering policy as an approximation. We also provide formulas for computingV (T1, . . . , TN ), the optimal average cost per period given T1, . . . , TN .

Sections 4–7 cover the second part. Section 4 proves there exist optimal review periodsT ∗∗

1 , . . . , T ∗∗N , i.e., ones that minimize V (T1, . . . , TN ). Section 4 also gives bounds for assessing

the performance of arbitrary T1, . . . , TN and obtains sufficient conditions for T1, . . . , TN to beoptimal. Section 5 introduces approximations based on the myopic base stock ordering policyand provides additional bounds. All of these bounds depend on a weighting scheme that allocatesthe joint fixed cost among the N items. Section 6 presents an O(N) algorithm for computing a‘‘good’’ heuristic policy, HM. This policy utilizes the myopic base stock ordering policy and anadaptation of a weighting scheme of Atkins and Iyogun [1]. For normal demand problems, thecomputations can be done on a spreadsheet. Section 7 presents a branch and bound algorithmfor computing T ∗∗

1 , T ∗∗2 , . . . , T ∗∗

N . Its memory requirements are modest, but its running time isexponential. Section 7 also introduces a second heuristic policy, HO, with O(N) requirements.Both HO and the bounds for our branch and bound algorithm use the same weighting schemefor allocating the joint fixed cost that HM does. Unlike HM, however, HO uses the optimal basestock ordering policy; hence, HO performs better than HM but demands more computationally.Section 8 contains an example illustrating our results.

Section 9 describes a numerical study involving normal and gamma demand test problems,which assesses the performance of our optimal branch and bound algorithm and our two heuristic


policies. Our results suggest that the branch and bound algorithm can easily solve normal demandproblems with N ≤ 8 and can handle problems with N ≤ 10, but experiences difficulty withlarger N . Both heuristics do well for a wide variety of problems with N ranging from 2 to200; moreover, performance seems insensitive to N . For our normal problems with N = 2, therespective mean and maximum errors under HO equal 0.12% and 2.59%, while the respectivemean and maximum errors under HM equal 0.21% and 2.59%. Since HO is more demandingcomputationally than HM, this suggests that one should use HM for normal problems.

The model developed here might be of interest to a JIT manufacturer, who places weekly orderswith a vendor, but wants to reduce inventory levels by scheduling daily deliveries from the vendor.Its advantage over the standard model is that it uncouples the review period and delivery decisionsand allows multiple deliveries for each review instead of just a single delivery. As indicated in [4,Section 8], the cost savings resulting from our model can be substantial. Furthermore, we providetwo heuristic policies which perform well for a wide variety of normal and gamma problems andare also easy to compute and implement.

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selecting review periods for a coordinated multi-item inventory model with staggered deliveries

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