An Economic Lot-Sizing Problem with Perishable Inventory and Economies ofScale Costs: Approximation Solutions and Worst Case Analysis

Leon Yang Chu,1 Vernon Ning Hsu,2 Zuo-Jun Max Shen3

1 Department of Industrial & Systems Engineering, University of Florida, Gainesville, Florida 32611

2 School of Management, George Mason University, Fairfax, Virginia 22030

3 Department of Industrial Engineering & Operations Research, University of California, Berkeley, California 94720

Received 27 May 2004; revised 19 December 2004; accepted 29 April 2005DOI 10.1002/nav.20096

Published online 16 June 2005 in Wiley InterScience (www.interscience.wiley.com).

Abstract: The costs of many economic activities such as production, purchasing, distribution, and inventory exhibit economiesof scale under which the average unit cost decreases as the total volume of the activity increases. In this paper, we consider aneconomic lot-sizing problem with general economies of scale cost functions. Our model is applicable to both nonperishable andperishable products. For perishable products, the deterioration rate and inventory carrying cost in each period depend on the ageof the inventory. Realizing that the problem is NP-hard, we analyze the effectiveness of easily implementable policies. We showthat the cost of the best Consecutive-Cover-Ordering (CCO) policy, which can be found in polynomial time, is guaranteed to beno more than (4�2 � 5)/7 � 1.52 times the optimal cost. In addition, if the ordering cost function does not change from periodto period, the cost of the best CCO policy is no more than 1.5 times the optimal cost. © 2005 Wiley Periodicals, Inc. Naval ResearchLogistics 52: 536–548, 2005.

Keywords: perishable inventory; approximation algorithms; Consecutive-Cover-Ordering policies; Economic Lot-Sizing prob-lem

1. INTRODUCTION

The costs of many economic activities such as produc-tion, purchasing, distribution, and inventory can be repre-sented by a so-called economies of scale function, whichsatisfies the following two conditions: (i) The total cost isnondecreasing in the total volume; and (ii) the average unitcost is nonincreasing.

The term economies of scale or increasing return to scaleis widely used in economic and business literature to char-acterize the production cost in an environment where thereare significant setup efforts required for each productionrun; or there are learning effects in the production process.For example, the well-known Cobb-Douglas productionfunction, Y � aKbLc, is widely used to demonstrate therelationship between output volume Y and the amount ofinput (K and L). When b � c � 1, the Cobb-Douglasproduction function exhibits the property of economies of

scale. Readers are referred to Earl [6, pp. 161–162 and pp.252], for more discussions on the production function witheconomies of scale.

High overhead costs of warehousing and transportationtypically lead to economies of scale inventory holding anddistribution costs. A large inventory/transportation volumeenables fixed costs to be spread across more units and thusresults in reduced average unit costs. Finally, volume dis-count schemes offered by sellers often result in economiesof scale in purchasing.

Economies of scale in ordering and inventory is typicallymodelled in the Economic Lot-Sizing (ELS) literature byconcave ordering and inventory functions (see, for example,Aggarwal and Park [1] and Hsu [8]). It is easy to see that aconcave function is a special case of the economies of scalefunction. The former is more restrictive than the latter inthat it requires diminishing marginal rates of increase. Forexample, the modified all-unit discount freight cost struc-ture (Chan et al. [2]) and the less-than-full-truck-load (LTL)and full-truck-load (TL) freight cost (Lee [9] and Li, Hsu,and Xiao [10]) exhibit the economies of scale, but are notCorrespondence to: Z.-J. Max Shen (shen@ieor.berkeley.edu)

© 2005 Wiley Periodicals, Inc.

concave functions. Thus, although ELS problems with con-cave cost functions are efficiently solvable with polynomialtime dynamic programming (DP) solutions, they do notcapture many cost structures in practical applications suchas the price discount schemes offered by suppliers/retailersand transportation service providers, and other more com-plicated cost structures in production and inventory.

Motivated by various quantity discount schemes in pur-chasing and distribution, a number of ELS problems in theliterature propose some special forms of nonconcave econ-omies of scale ordering cost structures. Federgruen and Lee[7] consider two versions of the ELS problems with all-unitdiscount (under which a discount rate is applied to all unitsof a purchase volume) and incremental discount (underwhich different discount rates are applied to incrementalranges of the purchased quantity) ordering cost structures.They develop polynomial-time algorithms to solve theirmodels optimally (see also Xu and Lu [12] for additionaldiscussions on the algorithms).

Lee [9] studies an ELS problem with a restrictive order-ing cost structure that includes a full-truck-load (TL) freightcost. The ordering cost function is assumed to be stationary(i.e., an identical ordering cost function in every period).Lee solves his problem with a polynomial time DP algo-rithm. Li, Hsu, and Xiao [10] generalize Lee’s model byconsidering a more general ELS problem with a time-varying ordering cost function that includes a fixed setupcost, a variable purchase cost, and a freight discount coststructure that includes both less-than-full-truck-load (LTL)and TL charges. They solve their general problem in poly-nomial time through a DP approach.

Chan et al. [2] present an ELS problem with a modifiedall-unit discount freight cost structure. This ordering costfunction typifies transportation costs charged by many LTLcarriers. They demonstrate the NP-hardness of their prob-lem and develop worst case performance bounds for aneasy-to-implement approximation solution. Their approxi-mation solution is the minimum cost solution that satisfiesthe Zero-Inventory-Ordering (ZIO) policy under which anorder is placed only when the inventory level drops to zero.Chan et al. [3] extend Chan et al. [2] to a single-warehousemultiretailer setting, in which the shipment costs from thesupplier to the warehouse are approximated by piecewiselinear concave functions and the shipment costs from thewarehouse to the retailer are represented by the modifiedall-unit discount cost structure. Since the problem is NP-hard in strong sense, their objective is to design simpleinventory policies and transportation strategies to minimizesystem-wide costs by taking advantage of quantity dis-counts in the transportation cost structures. They show thatthe cost of the best ZIO policy in a single-warehouse mul-tiretailer scenario, in which the warehouse serves as a cross-

dock facility, is no more than 4/3 (5.6/4.6 if costs arestationary) times the optimal cost.

Chu and Shen [5] consider an ELS model with perishableinventory and quantity discount cost functions. They as-sume the order cost functions follow modified all-unitsdiscount scheme (modified from the cost functions in Fed-ergruen and Lee [7]), and an item’s deterioration rate and itscarrying cost in each period depend on the age of the item.They are able to solve the corresponding problem in O(T3).

Finally, there are ELS problems in the literature that dealwith cost structures that do not exhibit economies of scale.For example, Chen, Hearn, and Lee [4] propose an ELSproblem with piecewise-linear cost structure. They solve theproblem with a DP approach. Lippman [11] and, morerecently, Li, Hsu, and Xiao [10] have analyzed ELS prob-lems with various piecewise-concave production/transpor-tation cost structures.

In this paper, we study an ELS problem for perishableproducts with economies of scale ordering and inventoryholding cost functions. In addition to fairly general coststructures, we also explicitly consider stock deterioration tomake our model applicable to both nonperishable (when thedeterioration rate is zero) and perishable products. Ourmodel is a generalization of the model in Chan et al. [2].Since the latter has been shown to be NP-hard (transforma-tion from partition problem), it is unlikely that an efficientexact solution exists for our ELS problem. By exploring thespecial structures of the problem, we propose an approxi-mation solution and analyze its effectiveness through thedevelopment of worst case performance bounds and throughcomputational experiments. Although our model is a gen-eralization of the model in Chan et al. [2], the solutionstrategy developed in our paper is not a trivial extension oftheir strategy. For example, we can show that the ZIOapproximation solution used by Chan et al. [2] to solve theirmodel would perform very poorly if it is applied to ourmodel.

The remainder of this paper is organized as follows.Section 2 presents our ELS problem with stock deteriorationand economies of scale ordering and inventory cost func-tions. We then give a number of structural properties of theproblem in this section. Section 3 proposes an approxima-tion solution and analyzes its worst case performance. Sec-tion 4 demonstrates the effectiveness of the proposed ap-proximation solution through a computational study. Sec-tion 5 concludes the paper and offers a few future researchtopics.

2. THE MODEL AND ITS STRUCTURALPROPERTIES

Consider an ELS problem with T periods. For 1 � j �T, define:

537Chu, Hsu, and Shen: Lot-Sizing Problem with Perishable Inventory and Economies of Scale

● Dj: the amount of demand in period j;● Xj: the order quantity in period j, which is called an

ordering period if Xj � 0;● Cj(Xj): the cost of ordering Xj units in period j.

For 1 � i � t � T, define

● Iit: the amount of inventory at the beginning ofperiod t, which was ordered in period i.

● �it: the fraction of Iit that is lost during period t.Thus, only (1 � �it)/Iit units of the item ordered inperiod i remain at the end of period t.

● Hit(Iit): the cost of holding Iit units of inventory,which were ordered in period i, in period t. This costexcludes the value of the lost inventory which iscaptured explicitly by the stock deterioration in ourmodel.

● Zit: the amount of demand in period t that is satisfiedby the order in period i.

A cost function, F(X), defined on [0, ��) is called aneconomies of scale function if:

(1) F(0) � 0.(2) F(X) is nondecreasing on [0, ��).(3) The average cost function defined as F� (X) �

F(X)/X, for X � 0, is a nonincreasing function on(0, ��).

In our problem, the ordering cost functions Ct(•), 1 � t �n, are assumed to be general economies of scale functions.

We note that, for perishable stocks, the longer they havebeen held, the faster they may deteriorate in a period and thehigher the inventory holding cost may be in that period.Thus, for any given period t, we assume that the stockdeterioration rates satisfy

ASSUMPTION 1: �it � �jt, 1 � i � j � t.

For inventory costs, we make the following assumption:

ASSUMPTION 2: Hit( x � �) � Hit( x) � Hjt( y ��) � Hjt( y), where i � j � t and x, y, � � 0.

Assumption 2 states that the marginal cost of holding ad-ditional units of inventory in period t for a newer stock(regardless of its current stock level) is no bigger than thatfor an older stock. This assumption includes the followingspecial case: Hit(Iit) � hitIit, where hit � 0 is a constant,and hit � hjt for i � j � t. This more restricted conditionagain says that the marginal cost of holding additional unitof inventory in period t for a newer stock (ordered in periodj) is no bigger than that for an older stock (ordered in period

i). This condition would hold true in many real-worldapplications.

We assume that there is no inventory available at thebeginning of period 1 and no inventory is required at the endof period T. Ordering and demand fulfillment occur at thebeginning of each period. Backlogging is not allowed.

The following formal statement of our perishable stockELS problem, which we will call problem PELS, is similarto the perishable inventory ELS problem considered by Hsu[8], except that the ordering and inventory cost functions areassumed to be concave in the latter.

����: Minimize �t�1

T

Ct(Xt) � �i�1

t

Hit(Iit)

subject to Xt � Ztt � Itt, 1 � t � T,

1 � �i,t�1Ii,t�1 � Zit � Iit, 1 � i � t � T,

�i�1

t

Zit � Dt, 1 � t � T,

Xt, Iit, Zit � 0, 1 � i � t � T.

We denote � � {X, I, Z} as a feasible solution to problemPELS and �(�) as the corresponding objective functionvalue.

It is easy to see that the model considered by Chan et al.[2] is a special instance of problem PELS with �it � 0 andHit(Iit) � htIit, where ht � 0 is a constant. Since thespecial case has been shown to be NP-hard, it is thereforeunlikely that efficient exact solutions exist for problemPELS. We now establish two structural properties of PELSthat allow us to develop an approximation solution andanalyze its worst case performance.

Define Akti � 1/[ l�k

t�1 (1 � �il)], for i � k � t and Aiii

� 1, and Akti is positive infinity if �il � 1 for some l

satisfying k � l � t. By definition,

Ajti � Ajk

i Akti , for i � j � k � t. (1)

By Assumption 1, we have

Akti � Akt

j , for 1 � i � j � k � t � T. (2)

We also note that, to satisfy one unit demand in period t byan order in period i, i � t, we need to order Ait

i units inperiod i and carry Akt

i units of inventory in every period k,where i � k � t � 1.

538 Naval Research Logistics, Vol. 52 (2005)

THEOREM 1: There is an optimal solution �* � {X*,I*, Z*} to PELS such that, for two order periods i � j, ifZ*jk � 0 for some k, j � k, then Z*it � 0 for all t, k � t �T.

PROOF: Suppose in an optimal solution, �� � {X�,I�, Z�}, we have, respectively, Zjk

� � 0 and Zit� � 0 for

some k and t, j � k � t � T. Let � � min{Zit�, Zjk

�/Aktj } �

0. We can construct a new solution, �* � {X*, I*, Z*},as follows: Set

Z*it � Zit� � �, Z*jt � Zjt

� � �,

Z*ik � Zik� � Akt

j �, Z*jk � Zjk� � Akt

j �,

X*i � Xi� � Aik

i Akti � Akt

j �,

I*il � Iil� � Alk

i Akti � Akt

j �, for i � l � k � 1,

I*il � Iil� � Alt

i �, for k � l � t � 1,

I*jl � Ijl� � Alt

j �, for k � l � t � 1.

All other variables of �* are the same as those of ��. It iseasy to verify that solution �* is feasible. Furthermore, if� � Zit

�, we have Z*it � 0; and if � � Zjk�/Akt

j , we haveZ*jk � 0.

We now show that �(�*) � �(��). First, note that theonly change in the ordering cost is in period i. Since Ci(•)is nondecreasing, the total ordering cost corresponding tosolution �* would not be larger than that of solution ��.We also note that there is a net reduction of the inventory,Iil�, in periods l � i, . . . , k � 1. Thus,

��� � ��* � �l�k

t�1

HilIil� � HilIil

� � Alti �

� HjlIjl� � Alt

j � � HjlIjl�.

By (2), we have Alti � Alt

j , l � t � 1. By Assumption 2 andthe fact that Hil� is a nondecreasing function, we have

��� � ��* � �l�k

t�1

HilIil� � HilIil

� � Altj �

� HjlIjl� � Alt

j � � HjlIjl� � 0.

The theorem can be proved by repeatedly applying theearlier modification of the solutions. �

Theorem 1 suggests that there is an optimal solution toproblem PELS where inventory stocks ordered in differentperiods are used to satisfy demands in first-in-first-out(FIFO) fashion. We call such solution a FIFO solution.Before presenting the next property, we have the followingresults for the average cost function:

LEMMA 1: Suppose F(X) and G(X) are economies ofscale functions defined on [0, ��). Then:

(a) F(�X) and F(X) � G(X) are also economies ofscale functions, where � � 0 is any given constant.

(b) F(X � Y) � F(X) � F(Y), for all X, Y � [0,��).

(c) F(X � �) � F(X) � �F� (X), for all X � 0 and� � 0.

PROOF: (a) is trivial. To show (b), note that F(•) is aneconomies of scale function. We have F� (X � Y) � F� (X)and F� (X � Y) � F� (Y). Thus,

FX � Y � X � YF� X � Y � XF� X � YF� Y

� FX � FY.

Finally to show (c), note that F� (X � �) � F� (X). We have

FX � � � FX � X � �F� X � � � XF� X

� �F� X.

�

THEOREM 2: There is an optimal FIFO solution toPELS where each demand, Dj, 1 � j � T, is satisfied byorders from at most three ordering periods.

PROOF: Suppose in an optimal FIFO solution, �� �{X�, I�, Z�}, a demand Dj, 1 � j � T, is satisfied byorders from periods t1 � t2 � . . . � tr � j, where r � 3.We will show that orders from t2 through tr�1 can becombined into one order in a new FIFO solution withreduced total costs. First note that by FIFO policy, ordersfrom t2 through tr�1 are entirely used to satisfy part ofdemand Dj. Suppose dtl

units of Dj are satisfied by the orderfrom period tl, l � 2, . . . , r � 1. For p � t2, . . . , tr�1,the total ordering and inventory costs associated with thefulfillment that uses the order in period p to satisfy dp unitsof period j demand is

Tpdp � CpApjp dp � �

l�p

j�1

HplAljpdp. (3)

539Chu, Hsu, and Shen: Lot-Sizing Problem with Perishable Inventory and Economies of Scale

By Lemma 1, we know that Tp(•), p � t2, . . . , tr�1, areall economies of scale functions.

Let p0 � arg minp�t2,. . .,tr�1T� p(dp). We construct a new

FIFO solution by combining orders in periods t2, . . . , tr�1

into a single order in period p0 to satisfy Mj � ¥l�2r�1 dtl

units of period j demand. Since

Tp0Mj � MjT� p0Mj � MjT� p0dp0 � �l�2

r�1

dtlT� tldtl

� �l�2

r�1

Ttldtl,

the new solution has an objective value no larger than thatof the original solution. �

In the rest of the paper, we consider only optimal solu-tions that satisfy Theorems 1 and 2. To conclude thissection, we show the following property for a restrictedversion of problem PELS where the ordering cost functionis stationary.

THEOREM 3: If Cj(•) � C(•) for all j, 1 � j � T, thereexists an optimal FIFO solution to problem PELS whereeach demand Dj, 1 � j � T, is satisfied by orders from atmost two ordering periods.

PROOF: Suppose in an optimal FIFO solution, �� �{X�, I�, Z�}, to PELS, demand Dj is satisfied by ordersfrom periods i, k and t, i � k � t � j. We obtain a newFIFO solution, �* � {X*, I*, Z*}, to PELS by combiningorders in periods k and t into a single order in period t.Suppose that the order in period k was used to satisfy dk

units of period j demand. We have

��* � ��� � CXt� � Atj

t dk � CXt� � �

l�t

j�1

HtlItl�

� Aljt dk � HtlItl

� � CAkjk dk � �

l�k

j�1

HklAljkdk.

By (b) of Lemma 1,

��* � ��� � CAtjt dk � �

l�t

j�1

HtlAljt dk � CAkj

k dk

� �l�k

j�1

HklAljkdk.

By (2), we have Aljt � Alj

k , for k � t � l � j � 1;furthermore, by (1), Akj

k � Aktk Atj

k � Atjt . Together with the

fact that C(•) is a nondecreasing function, we haveC( Atj

t dk) � C( Akjk dk) � 0. Now, by Assumption 2 and the

fact that Htl(•), t � l � j � 1, are nondecreasing functions,we have

��* � ��� � �l�t

j�1

HtlAljkdk � Htl0

� �l�t

j�1

HklAljkdk � Hkl0 � 0.

The theorem can be proved by repeatedly applying theabove modification of the solution. �

3. THE APPROXIMATION SOLUTIONS ANDWORST CASE ANALYSIS

Chan et al. [2] consider an approximation solution with aZero-Inventory-Ordering (ZIO) policy to solve a specialinstance of problem PELS. In their ZIO solution, there iszero inventory at the beginning of each ordering period. Thefollowing example shows that there are instances of prob-lem PELS where the objective value of the best ZIO solu-tion may have an arbitrarily large error compared to that ofthe optimal solution.

EXAMPLE 1: Consider an instance of PELS with threeperiods where D1 � D3 � and D2 � 1 � 2 ( is asmall amount). An inventory stock deteriorates completelyafter one period. Thus, we cannot use the order in period 1to satisfy the demand in period 3. The holding cost is zero.The ordering cost in period 1 has a fixed charge of 1 for anysize of order. The ordering cost in period 2 is 1/ per unit.The ordering cost in period 3 is prohibitively high, say 1/3

per unit. The optimal policy is to order 1 � units in period1 to satisfy demands in periods 1 and 2, and order units inperiod 2 to satisfy the period 3 demand. The total cost of thisoptimal solution is (1 � 1) � 2. Consider the optimal ZIOpolicy, which orders units in period 1 for the demand inperiod 1 and orders 1 � units in period 2 for demands inperiods 2 and 3. The total cost of the best ZIO solution is1 � (1 � )/ � 1/. Note that as 3 0, ZZIO/Z* 3 �.In other words, the ZIO policy could be as poor as we couldwant.

We now propose a new easily implementable approxima-tion solution to solve problem PELS. A feasible solution toPELS is called a Consecutive-Cover-Ordering (CCO) solu-tion if each order is used to satisfy (cover) all demands froma number of consecutive indexed periods. Denote ZCCO as

540 Naval Research Logistics, Vol. 52 (2005)

the minimum objective value of the best CCO solutions toproblem PELS. Hsu [8] shows that ZCCO is the optimalobjective value of PELS if all of the ordering and inventorycosts, Ct(•) and Hit(•), are concave functions, which arespecial cases of the economies of scale function. He alsodevelops a dynamic programming (DP) algorithm to obtainthe best CCO solution corresponding to ZCCO. To imple-ment the dynamic programming algorithm, first we need toknow the costs to satisfy demands from period j to k byperiod i for every triplet (i, j, k) (i � j � k). Thecomputational complexity of these costs is bounded byO(T4). Then, we can recursively calculate V(i, r), theoptimal CCO policy cost to satisfy demand through period1 to r with the largest indexed production period i. Thecomputational complexity of the DP solution to the generalproblem PELS is bounded by O(T4). The DP algorithmdoes not depend on the cost structure, and readers arereferred to Hsu [8] for more details of the algorithm.

To analyze the effectiveness of the CCO solutions, wefirst present a transformation procedure that will transforman optimal solution to problem PELS into a CCO solution.Let �� � {X�, I�, Z�} be an optimal solution to PELS.We define, for i � j,

Si,j � C� iXi�Aij

i � �l�i

j�1

H� ilIil�Alj

i . (4)

Si, j is the average cost to satisfy one unit demand in periodj by the order in period i.

Given an optimal FIFO solution, ��, to problem PELSthat satisfies the condition in Theorem 2 (i.e., the demand ineach period is satisfied from at most three orders), denoteR(��) as the set of period index for the period for whichthe demand is satisfied by orders from two or three differentperiods in ��. For any period, j, whose demand is satisfiedby two order periods, i and t, in the optimal solution, ��,we could introduce a pseudo-period k between periods i andt, with zero demand and a prohibitively high ordering cost,no holding cost, and no deterioration. Thus, we may assumewithout loss of generality that for any j � R(��), there arethree ordering periods i( j), k( j), and t( j) that satisfy �jDj,jDj, and (1 � �j � j) Dj units of period j demand,respectively, where 0 � �j � 1, 0 � j � 1, and i( j) �k( j) � t( j) � j. It is easy to verify that

��� � �j�R��

�jSij,j � jSkj,j � 1 � �j � jStj,jDj.

(5)

Define

�1,j � 1 � �jSij,j � jSkj,jDj,

�2,j � 1 � jSkj,jDj,

and

�3,j � �j � jStj,j � jSkj,jDj.

Let �i, j (i � 1, 2, 3) be the same as �� except that Dj,demand in period j, is satisfied entirely by ordering inperiods i( j), k( j), and t( j). Then, �i, j (i � 1, 2, 3) are theupper bounds of the incremental costs, �(�i, j) � �(��)(i � 1, 2, 3). We now present the following transformationprocedure to construct a near-to-optimal CCO policy from��:

Transformation Procedure:

● Step 1: Let p � 0 and set �p � ��.● Step 2: For solution �p, set j � jp, the smallest

index in R(�p). Suppose orders from periods i( j),k( j), and t( j), i( j) � k( j) � t( j) � j, are used tosatisfy �jDj, jDj, and (1 � �j � j) Dj units ofperiod j demand, respectively.

● Step 3: Construct a new solution �� by satisfying Dj,demand in period j, by the period that the corre-sponding �i, j (i � 1, 2, 3) is minimum (break tiesarbitrarily).

● Step 4: Set R(��) � R(�p)�{ j}. If R(��) � A, set�̂ � �� and stop; otherwise, increase p by 1, set�p � ��, R(�p) � R(��), and return to Step 2.

It is easy to see that in each iteration, p � 0, of the aboveTransformation Procedure, we construct a new FIFO solu-tion, �p � {Xp, Ip, Zp}, in which the three orders used tosatisfy the demand in a certain period, jp, are combined intoa single order from one of the three ordering periods i( jp),k( jp), and t( jp). Clearly, the overall procedure consists ofr � �R(��)� iterations. Furthermore, upon termination ofthe procedure, the solution �̂ is a CCO solution to problemPELS. Define for each p, 1 � p � r,

p � � 1, if iteration p uses Combine 1 or 2,0, if iteration p uses Combine 3,

and

Lp � �Ctjp(Xtjpp ) � Ctjp(Xtjp

� ) � �l�tjp

jp�1

(Htjp,l(Itjp,lp )

� Htjp,l(Itjp,l� ))� .

541Chu, Hsu, and Shen: Lot-Sizing Problem with Perishable Inventory and Economies of Scale

Note that if p � 1, we have Xt( jp)p � Xt( jp)

� , and Itp( jp),l �

It�

( jp),l for all l, t( jp) � l � j � 1. Thus,

pLp � p�Ctjp(Xtjpp ) � Ctjp(Xtjp

� ) � �l�tjp

jp�1

(Htjp,l(Itjp,lp )

� Htjp,l(Itjp,l� ))� � 0. (6)

We have the following result:

THEOREM 4: For any p, 1 � p � r,

��p � ��� � �l�1

p

min��1,jl, �2,jl

, �3,jl� � pLp. (7)

PROOF: See the Appendix. �

LEMMA 2: For arbitrary �, , and ci, i � 1, 2, 3, where� � 0, � 0, � � � 1, and ci � 0, we have

min�1 � �c1 � c2, 1 � c2, � � c3 � c2�

�4�2 � 2

7�c1 � c2 � 1 � � � c3 (8)

and

min�1 � �c1, �c2� �1

2�c1 � 1 � �c2. (9)

PROOF: See the Appendix. �

Let Z* be the optimal objective value to problem PELS.We are now ready to present the worst case performance forthe approximation solution ZCCO.

THEOREM 5: For every instance of problem PELS,ZCCO � [(4�2 � 5)/7]Z*, and this bound is tight.

PROOF: Since ZCCO is the minimum value among all ofthe CCO solutions and �̂, the solution obtained by theTransformation Procedure is a CCO solution and we haveZCCO � �(�̂). By Theorem 4 and (6), we have

ZCCO � Z* � �j�R��

min��1,j, �2,j, �3,j�.

By the definition of �s and Lemma 2, we have

�j�R��

min��1,j, �2,j, �3,j� �4�2 � 2

7 �j�R��

�jSij,j

� jSkj,j � 1 � �j � jStj,jDj.

Finally by (5), we have

4�2 � 2

7 �j�R��

�jSij,j � jSkj,j � 1 � �j � jStj,jDj

�4�2 � 2

7Z*.

Thus, ZCCO � [(4�2 � 5)/7]Z*.To show that this bound is tight, we only need to dem-

onstrate that there exist instances of problem PELS forwhich the ratio ZCCO/Z* is arbitrarily close to (4�2 � 5)/7.Consider an instance of PELS with four periods whereD1 � D2 � D4 � and D3 � 1 � 3 ( is a smallamount). The orders from periods 1 and 2 deteriorate com-pletely after period 3. That is, we cannot use orders inperiod 1 and 2 to satisfy the demand in period 4. There is noholding cost and the ordering costs in periods 1, 3, and 4have a fixed charge of (3�2 � 2)/14 for any order with asize no larger than �2 � 1; and a rate of (5�2 � 8)/14 perunit for orders with sizes larger than �2 � 1. The orderingcost in period 2 has a fixed charge of (�3�2 � 5)/7 for anyorder with a size no larger than 3 � 2�2; and a rate of (3 ��2/7 per unit for orders with sizes greater than 3 � 2�2.

In the optimal solution to this instance, �2 � 1 units areordered in periods 1 and 3, while 3 � 2�2 units are orderedin period 2. The total cost is Z* � 1. However, the bestCCO solution is to order 2 units in period 1 and 1 � 2units in period 3, with a total cost of ZCCO � [(4�2 � 5)/7]� [(5�2 � 8)/7]. Hence,

ZCCO

Z*�

4�2 � 5

7�

5�2 � 8

7 3

4�2 � 5

7as 3 0.

�

As mentioned in Theorem 3, if the ordering cost isstationary over the T-period planning horizon, there existsan optimal FIFO solution ��, where each demand Dj issatisfied by orders from at most two ordering periods. De-note R̂(��) as the set of period index for period j whosedemand, Dj, is satisfied by two orders from periods i( j) andt( j) with sizes of �jDj and (1 � �j) Dj, respectively, wherei( j) � t( j) � j and 0 � �j � 1. Similar to (5), it is easyto see that

��� � �j�R̂��

�jSij,j � 1 � �jStj,jDj. (10)

542 Naval Research Logistics, Vol. 52 (2005)

THEOREM 6: If there exists an optimal FIFO solution�� to PELS in which each demand Dj is satisfied by ordersfrom at most two ordering periods, we have ZCCO � (3/2) Z* and this bound is tight.

PROOF: Similar to the proof of Theorem 5, we can show

ZCCO � Z* � �j� R̂��

min�1 � �jSij,j, �Stj,j�Dj.

By (9) and (10),

ZCCO � Z* �1

2 �j�R̂��

�jSij,j � 1 � �jStj,jDj �1

2Z*.

It remains to be shown that there are instances of PELS inwhich the ratio ZCCO/Z* is arbitrarily close to 3/2. Consideran instance of PELS with three periods where D1 � D3 � and D2 � 2 � 2 ( is a small amount). The inventorydeteriorates after one period; i.e., we cannot use the order inperiod 1 to satisfy the demand in period 3. There is noholding cost and all ordering cost have a fixed charge of 1for any order with a size no greater than 1; and a rate of 1per unit for orders with sizes larger than 1.

In the optimal solution to this instance, 1 unit is orderedin period 1 and 1 in period 2 with a total cost of Z* � 2.However, the best CCO solution is to order units in period1 and 2 � units in period 2, with a total cost of ZCCO �3 � . Hence,

ZCCO

Z*�

3 �

23

3

2as 3 0.

�

4. COMPUTATIONAL RESULTS

In this section, we demonstrate the computational perfor-mance of the proposed CCO approximation solution. Werandomly generate a large number of test problems with

wide ranges of problem characteristics (see discussions be-low). For each test problem, we obtain the optimal solutionusing CPLEX by IP formulation and compare it with thebest CCO solution.

We consider eight different classes of instances repre-senting different combinations of planning horizon, costfunctions and deterioration rate, as described in Table 1. Foreach class, we generate 100 random instances. The numberof periods is either 10 or 12. The demand in each period isgenerated from a normal distribution with a mean of 10 anda standard deviation of 2.5. The ordering cost functionsconsidered in each case are stationary; i.e., they do notchange over time. Two different ordering cost structures arestudied: The first structure, as illustrated in Figure 1, hasbreak points at 0, 20, 40, 60, 80, 100, and �, and the fixedcosts for the different intervals are 15, 0, 30, 0, 40, and 0,correspondingly. The second structure has break points at 0,10, 20, 40, 60, 90, and �, and the fixed costs for the differentintervals are 15, 0, 30, 0, 45, and 0, correspondingly. Theholding costs are linear at 0.1 or 0.2 per unit for all theperiods.

We use two different types of deterioration schemes. Forany two periods, i and j, define �ij � ( j � i), where

Table 1. Problem classes.

Problem classes Number of periodsType of order cost

structure Holding cost hType of

deterioration rate

Class 1 10 Type 1 0.1 Type 1Class 2 10 Type 1 0.1 Type 2Class 3 10 Type 2 0.2 Type 1Class 4 10 Type 2 0.2 Type 2Class 5 12 Type 1 0.1 Type 1Class 6 12 Type 1 0.1 Type 2Class 7 12 Type 2 0.2 Type 1Class 8 12 Type 2 0.2 Type 2

Figure 1. The economic of scale order cost structure.

543Chu, Hsu, and Shen: Lot-Sizing Problem with Perishable Inventory and Economies of Scale

1 . . . 11

� �0.02, 0.04, 0.08, 0.16, 0.24, 0.36, 0.54, 0.81, 0.9, 1, 1�

for the first scheme, and

1 . . . 11

� �0.02, 0.03, 0.05, 0.08, 0.13, 0.21, 0.34, 0.55, 0.89, 1, 1�

for the second scheme. Note that Aij � �l�ij (1 � l).

Figure 2 shows the relationship between the age of inven-tory t and �log2A1t. For the instances with only 10 periods,we choose the first 10 values from the above lists. Table2 gives, for each of the problem classes, the average andmaximum ratios of the cost of the best CCO policy to thecost of the optimal solution.

These computational results indicate that our approxima-tion approach is very effective. The maximal derivationfrom the optimal solution is only 4% and the averagederivation is less than 1% for all classes. Furthermore, therunning time of the approximation solution is less than 0.1seconds in all cases, while the running time of the optimalalgorithm increases drastically as the length of the planninghorizon and the number of break points in the cost functions

increase. Table 3 shows the computational time in minutefor obtaining optimal solutions for four test classes as weincrease the number of period from 10 to 15. We thusconclude that the CCO policy is very effective in solvingour problem since it allows us to find the almost optimalsolution in a negligible amount of time while the effort todetermine the optimal solution is unduly high.

5. CONCLUSION

This paper studies an important variant of the ELS prob-lem with perishable inventory and general economies ofscale ordering and inventory functions. Since the problem isNP-hard, we focus on identifying an easy to implementreplenishment strategy with a guaranteed level of solutionquality. In particular, we show that the cost of the best CCOpolicy, which can be found efficiently in polynomial time, isguaranteed to be no more than (4�2 � 5)/7 times theoverall optimal solution and this bound is tight. In addition,if the ordering cost function is stationary, the cost of the bestCCO policy is no higher than 1.5 times the optimal cost.Our computational experiments further confirm the effec-tiveness of the proposed approximation solution.

In the future research, we could extend our model toallow backorders. It would also be interesting to study thesingle-warehouse multiretailer problem or other finite-pe-riod planning problems with general economies of scalecost functions.

APPENDIX

PROOF OF THEOREM 4: We will prove the theorem by induction onthe iteration index p. For simplicity of exposition, we will drop the indexjp from �jp

, jp, i( jp), k( jp), and t( jp) if there is no confusion about p.

In the first iteration ( p � 1), we obtain a solution, �1 � {X1, I1, Z1},by executing one of the three combine operations in Step 3 of the proce-dure. Note that since solution �� is a FIFO solution, the order in period kis entirely used to satisfy �Dj units of period j demand; i.e., we have Xk

�

� DjAkjk and Ikl

� � DjAljk , k � l � j � 1.

Suppose Combine 1 is executed in Step 3. In this case, we have

Zij1 � Dj, Zkj

1 � Ztj1 � 0,

Figure 2. The deterioration structure.

Table 2. Computational results of the CCO policy.

Problem classesMaximum ratio

ZCCO/Z* Average ratio

Class 1 1.04007 1.0054Class 2 1.03338 1.0037Class 3 1.01721 1.0008Class 4 1.01029 1.0008Class 5 1.03018 1.0052Class 6 1.03068 1.0043Class 7 1.01238 1.0005Class 8 1.00545 1.0004

Table 3. Comparison of computational times for different num-ber of periods.

Problem classes

Averagecomputational

time for 10periods

Averagecomputational

time for 15periods Ratio of time

Class 1 0.9736 7.2512 7.4478Class 2 0.8708 6.1480 7.0602Class 3 0.9728 7.2552 7.4581Class 4 1.0600 9.3692 8.8389

544 Naval Research Logistics, Vol. 52 (2005)

Xi1 � Xi

� � 1 � �DjAiji ,

Iil1 � Iil

� � 1 � �DjAlji , i � l � j � 1,

Xk1 � Xk

� � DjAkjk � 0,

Ikl1 � Ikl

� � DjAljk � 0, k � l � j � 1,

Xt1 � Xt

� � 1 � � � DjAtjt ,

Itl1 � Itl

� � 1 � � � DjAljt , t � l � j � 1.

Thus, we have

��1 � ��� � CiXi1 � CiXi

� � �l�i

j�1

HilIil1 � HilIil

�

� CkXk1 � CkXk

� � �l�k

j�1

HklIkl1 � HklIkl

� � CtXt1 � CtXt

�

� �l�t

j�1

HtlItl1 � HtlItl

� � CiXi� � 1 � �DjAij

i � CiXi�

� �l�i

j�1

HilIil� � 1 � �DjAlj

i � HilIil� � SkjDj � L1. (11)

Since Ci(•) and Hil(•) are the economies of scale functions, by (c) ofLemma 1, we have

��1 � ��� � 1 � �DjAiji C� iXi

� � �l�i

j�1

1 � �DjAlji H� ilIil

�

� SkjDj � L1 � �1,j � 1L1. (12)

Suppose now Combine 2 is executed in Step 3. We have

zkj1 � Dj, zij

1 � ztj1 � 0,

Xi1 � Xi

� � �DjAiji ,

Iil1 � Iil

� � �DjAlji , i � l � j � 1,

Xk1 � Xk

� � 1 � DjAkjk ,

Ikl1 � Ikl

� � 1 � DjAkjk , k � l � j � 1,

Xt1 � Xt

� � 1 � � � DjAtji ,

Itl1 � Itl

� � 1 � � � DjAlji , t � l � j � 1.

Furthermore, we have Xi1 � Xi

� and Itl1 � Itl

� for all l, t � l � j � 1.Thus,

��1 � ��� � CiXi1 � CiXi

� � �l�i

j�1

HilIil1 � HilIil

�

� CkXk1 � CkXk

� � �l�k

j�1

HklIkl1 � HklIkl

� � CtXt1 � CtXt

�

� �l�t

j�1

HtlItl1 � HtlItl

� � 1 � DjAkjk C� kXk

�

� �l�k

j�1

1 � DjAljkH� klIkl

� � L1 � �2,j � 1L1. (13)

Finally, suppose Combine 3 is executed in Step 3. We have

ztj1 � Dj, zij

1 � zkj1 � 0,

Xi1 � Xi

� � �DjAiji ,

Iil1 � Iil

� � �DjAlji , i � l � j � 1,

Xk1 � Xk

� � DjAkjk � 0,

Ikl1 � Ikl

� � DjAkjk � 0, k � l � j � 1,

Xt1 � Xt

� � � � DjAtji ,

Itl1 � Itl

� � � � DjAlji , t � l � j � 1.

Note that, in this case, 1 � 0. Similar to earlier discussions, we have

��1 � ��� � � � DjAtjt C� tXt

� � �l�t

j�1

� � DjAljt H� tlItl

�

� SkjDj � �3,j � 1L1. (14)

Combining (12)–(14), we see that (7) holds for p � 1. Assume that (7)holds for p � 1, . . . , m � 1. We now consider the mth iteration of theTransformation Procedure. By the inductive assumption, we have

��m � ��� � ��m � ��m�1 � ��m�1 � ��� � ��m

� ��m�1 � �l�1

m�1

min��1,jl, �2,jl, �3,jl� � m�1Lm�1.

We will now show that

��m � ��� � �l�1

m

min��1,jl, �2,jl, �3,jl� � mLm. (15)

First note that �� is a FIFO solution. Therefore, we have t( jm�1) �

i( jm) � k( jm). Thus, in the mth iteration, Xkm�1 � Xk

�, Xtm�1 � Xt

�, Iklm�1

� Ikl�, k � l � j � 1, and Itl

m�1 � Itl�, t � l � j � 1.

Suppose Xim�1 � Xi

� and Iilm�1 � Iil

� for all l, i � l � j � 1 in themth iteration. By the same arguments as (12)–(14), we can show that

��m � ��m�1 � min��1,jm, �2,jm, �3,jm� � mLm.

As m�1Lm�1 � 0, we have

��m � ��� � ��m � ��m�1 � ��m�1 � ���

� min��1,jm, �2,jm, �3,jm� � mLm � �l�1

m�1

min��1,jl, �2,jl, �3,jl� � m�1Lm�1

� �l�1

m

min��1,jl, �2,jl, �3,jl� � mLm.

545Chu, Hsu, and Shen: Lot-Sizing Problem with Perishable Inventory and Economies of Scale

We conclude that (15) holds if Xim�1 � Xi

� and Iilm�1 � Iil

� for all l, i �

l � j � 1 in iteration m.The only possibility for Xi

m�1 � Xi� or Iil

m�1 � Iil� for some l is that

t( jm�1) � i( jm). In this case,

Lm�1 � Ctjm�1Xtjm�1m�1 � Ctjm�1Xtjm�1

� � �l�tjm�1

jm�1�1

Htjm�1lItjm�1lm�1

� Htjm�1lItjm�1l� � CijmXijm

m�1 � CijmXijm� � �

l�ijm

jm�1�1

HijmlIijmlm�1

� HijmlIijml� � CiXi

m�1 � CiXi� � �

l�i

jm�1�1

HilIilm�1 � HilIil

�

� CiXim�1 � CiXi

� � �l�i

j�1

HilIilm�1 � HilIil

�.

The last equation comes from the fact that the inventory levels are un-changed after period jm�1 in both solutions, �m�1 and ��.

We now consider the following different cases in the mth iteration.

CASE 1: Suppose Combine 1 is used in mth iteration. We will show that

��m � ��m�1 � min��1,jm, �2,jm, �3,jm� � mLm � m�1Lm�1.

(16)

First, similar to (11) (using the fact that Xkm�1 � Xk

�, Xtm�1 � Xt

�, Iklm�1

� Ikl�, k � l � j � 1, and Itl

m�1 � Itl�, t � l � j � 1), we have

��m � ��m�1 � CiXim � CiXi

m�1 � �l�i

j�1

HilIilm � HilIil

m�1

� SkjDj � Lm.

We further consider the following two subcases in which Im�1 � 0 andIm�1 � 1.

CASE 1.1: If m�1 � 0, Combine 3 is used in the (m � 1)th iteration.Since t( jm�1) � i( jm), we have Xi

m�1 � Xi� and Iil

m�1 � Iil�. Further-

more, by (c) of Lemma 1 and the fact that Ci(•) is an economies of scalefunction,

CiXim � CiXi

m�1 � CiXim�1 � 1 � �DjAij

i � CtXtm�1

� 1 � �DjAiji C� iXi

m�1 � 1 � �DjAiji C� iXi

�.

Similarly, for each l, i � l � j � 1,

HilIilm � HilIil

m�1 � 1 � �DjAlji H� ilIil

�.

Since m�1 � 0 and

1 � �DjAiji C� iXi

� � �l�i

j�1

1 � �DjAlji H� ilIil

� � �1,jm � SkjDj,

we have

��m � ��m�1 � �1,jm � Lm � �1,jm � mLm � m�1Lm�1.

(17)

CASE 1.2: If m�1 � 1, Combine 1 or 2 is used in the (m � 1)thiteration. Since t( jm�1) � i( jm), we have Xi

m�1 � Xi� and Iil

m�1 � Iil�.

Thus,

��m � ��m�1 � CiXim � CiXi

m�1 � �l�i

j�1

HilIilm � HilIil

m�1

� SkjDj � Lm � CiXim � CiXi

� � �l�i

j�1

HilIilm � HilIil

� � SkjDj

� mLm � �Ci(Xim�1) � Ci(Xi

�) � �l�i

j�1

(Hil(Iilm�1) � Hil(Iil

�))� � CiXim

� CiXi� � �

l�i

j�1

HilIilm � HilIil

� � SkjDj � mLm � Lm�1.

By (c) of Lemma 1 and the fact that Ci(•) is an economies of scalefunction,

CiXim � CiXi

� � Xim � Xi

�C� iXi� � Xi

m � Xim�1C� iXi

�

� 1 � �DjAiji C� iXi

�.

Similarly, for each l, i � l � j � 1,

HilIilm � HilIil

� � 1 � �DjAlji H� ilIil

�.

Note that

1 � �DjAiji C� iXi

� � �l�i

j�1

1 � �DjAlji H� ilIil

� � �1,jm � SkjDj.

We have

��m � ��m�1 � �1,jm � mLm � m�1Lm�1. (18)

Equations (17) and (18) complete the proof of (16). We can nowconclude that if Combine 1 is used in the mth iteration,

��m � ��� � ��m � ��m�1 � �l�1

m�1

min��1,jl, �2,jl, �3,jl�

� m�1Lm�1 � �l�1

m�1

min��1,jl, �2,jl, �3,jl� � �1,jm � mLm. (19)

CASE 2: Suppose Combine 2 or 3 is used in the mth iteration. Similarto the proofs of (13) and (14) and together with the fact Xk

m�1 � Xk�, Xt

m�1

� Xt�, Ikl

m�1 � Ikl�, k � l � j � 1, and Itl

m�1 � Itl�, t � l � j � 1, we

can easily show that

��m � ��m�1 � min��2,jm, �3,jm� � mLm.

Thus,

��m � ��� � �l�1

m�1

min��1,jl, �2,jl, �3,jl� � min��2,jm, �3,jm� � mLm

� m�1Lm�1 � �l�1

m�1

min��1,jl, �2,jl, �3,jl� � min��2,jm, �3,jm� � mLm. (20)

By (19) and (20) in Cases 1 and 2,

546 Naval Research Logistics, Vol. 52 (2005)

��m � ��� � �l�1

m

min��1,jl, �2,jl, �3,jl� � mLm. (21)

This completes the proof of the theorem. �

PROOF OF LEMMA 2: Assume that (8) is not true and, thus,

1 � �c1 � c2 4�2 � 2

7�c1 � c2 � 1 � � � c3,

1 � c2 4�2 � 2

7�c1 � c2 � 1 � � � c3,

� � c3 � c2 4�2 � 2

7�c1 � c2 � 1 � � � c3.

Denoting Q � [(4�2 � 2)/7](�c1 � (1 � � � )c3) and P � Q �[(4�2 � 5)/7]c2, we have

1 � �c1 P, (22)

c2 P, (23)

� � c3 P. (24)

By (23), c2 � Q � [(4�2 � 5)/7]c2 or (1 � [(4�2 � 5)/7])c2 �Q. Since both c2 and Q are positive, (1 � [(4�2 � 5)/7]) is alsopositive. Thus, c2 � Q/{1 � [(4�2 � 5)/7]} and hence

P � Q �4�2 � 5

7c2

Q

1 �4�2 � 5

7

. (25)

Multiplying both sides of (22) and (24) by �(� � ) and (1 � � �)(1 � �), respectively, and adding them together, we obtain

1 � �� � �c1 � 1 � � � c3

�� � � 1 � � � 1 � �P.

By (25), we have

1 � �� � Q

4�2 � 2

7

�� � � 1 � � � 1 � �Q

1 �4�2 � 5

7

,

or

1 � �� �

�� � � 1 � � � 1 � �

4�2 � 2

7

1 �4�2 � 5

7

.

Since [(1 � )/2]2 � (1 � �)(� � ) and �(� � ) � (1 � � � )(1 ��) � 1 � � 2�(1 � � �) � (1 � ) � [(1 � )2/2] � [(1 � )(1 �)/2], we obtain:

�1 �

2 � 2

1 � 1 �

2

4�2 � 2

7

1 �4�2 � 5

7

,

or

�1 �

2 ��1 �4�2 � 5

7� �

4�2 � 2

71 � 0,

which implies that

�4�2 � 5

14 � 3 � 2�22 0.

This is a contradiction. We have just shown (8).To prove (9), assume that it does not hold. We have

1 � �c1 �c1 � 1 � �c2

2,

�c2 �c1 � 1 � �c2

2.

We multiply the right-hand sides and left-hand sides of both inequalities toobtain

4�c11 � �c2 �c1 � 1 � �c22,

which implies that

�c1 � 1 � �c22 � 0.

This is again a contradiction. �

ACKNOWLEDGMENT

This work was sponsored in part by the NSF CAREERAward DMI-0348209. We thank the Associate Editor andtwo referees for constructive comments that led to thisimproved version.

REFERENCES

[1] A. Aggarwal and J.K. Park, Improved algorithms for eco-nomic lot size problems, Oper Res 41 (1993), 549–571.

[2] L.M.A. Chan, A. Muriel, Z.-J. Shen, and D. Simchi-Levi, Onthe effectiveness of zero-inventory-ordering policies for theeconomic lot sizing model with piecewise linear cost struc-tures, Oper Res 50 (2002), 1058–1067.

[3] L.M.A. Chan, A. Muriel, Z.-J. Shen, D. Simchi-Levi, andC.P. Teo, Effectiveness of zero-inventory-ordering policiesfor a one-warehouse multi-retailer problem with piecewiselinear cost structures, Management Sci 48 (2002), 1446–1460.

[4] H.D. Chen, D.W. Hearn, and C.Y. Lee, A dynamic program-ming algorithm for dynamic lot size models with piecewiselinear costs, J Global Optim 4 (1994), 397–413.

[5] L.Y. Chu and Z.-J. Shen, Perishable stock lot-sizing problemwith modified all-unit discount cost structures, Global supply

547Chu, Hsu, and Shen: Lot-Sizing Problem with Perishable Inventory and Economies of Scale

chain management, Jian Chen (Editor), Tsinghua UniversityPress, Beijing, 2002, pp. 13–18.

[6] P.E. Earl, Microeconomics for business and marketing: Lec-tures, cases and worked essays, Edward Elgar, Cheltenham,UK, 1995.

[7] A. Federgruen and C.-Y. Lee, The dynamic lot size modelwith quantity discount, Naval Res Logist 37 (1990), 707–713.

[8] V.N. Hsu, Dynamic economic lot size model with perishableinventory, Management Sci 46 (2000), 1159–1169.

[9] C.-Y. Lee, A solution to the multiple set-up problem withdynamic demand, IIE Trans 21 (1989), 266–270.

[10] C.L. Li, V.N. Hsu, and W.Q. Xiao, Dynamic lot sizing withbatch ordering and truckload discounts, Oper Res 52 (2004),639–654.

[11] S.A. Lippman, Optimal inventory policy with multiple set-upcosts, Management Sci 16 (1969), 118–138.

[12] J.F. Xu and L.L. Lu, The dynamic lot size model withquantity discount: Counterexamples and correction, NavalRes Logist 45 (1998), 419–422.

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