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

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1 An Economc Lot-Szng Problem wth Pershable Inventory and Economes of Scale Costs: Approxmaton Solutons and Worst Case Analyss Leon Yang Chu, 1 Vernon Nng Hsu, 2 Zuo-Jun Max Shen 3 1 Department of Industral & Systems Engneerng, Unversty of Florda, Ganesvlle, Florda School of Management, George Mason Unversty, Farfax, Vrgna Department of Industral Engneerng & Operatons Research, Unversty of Calforna, Berkeley, Calforna 9420 Receved 2 May 2004; revsed 19 December 2004; accepted 29 Aprl 2005 DOI /nav Publshed onlne 16 June 2005 n Wley InterScence ( Abstract: The costs of many economc actvtes such as producton, purchasng, dstrbuton, and nventory exhbt economes of scale under whch the average unt cost decreases as the total volume of the actvty ncreases. In ths paper, we consder an economc lot-szng problem wth general economes of scale cost functons. Our model s applcable to both nonpershable and pershable products. For pershable products, the deteroraton rate and nventory carryng cost n each perod depend on the age of the nventory. Realzng that the problem s NP-hard, we analyze the effectveness of easly mplementable polces. We show that the cost of the best Consecutve-Cover-Orderng (CCO) polcy, whch can be found n polynomal tme, s guaranteed to be no more than (42 5)/ 1.52 tmes the optmal cost. In addton, f the orderng cost functon does not change from perod to perod, the cost of the best CCO polcy s no more than 1.5 tmes the optmal cost Wley Perodcals, Inc. Naval Research Logstcs 52: , pershable nventory; approxmaton algorthms; Consecutve-Cover-Orderng polces; Economc Lot-Szng prob- Keywords: lem 1. INTRODUCTION The costs of many economc actvtes such as producton, purchasng, dstrbuton, and nventory can be represented by a so-called economes of scale functon, whch satsfes the followng two condtons: () The total cost s nondecreasng n the total volume; and () the average unt cost s nonncreasng. The term economes of scale or ncreasng return to scale s wdely used n economc and busness lterature to characterze the producton cost n an envronment where there are sgnfcant setup efforts requred for each producton run; or there are learnng effects n the producton process. For example, the well-known Cobb-Douglas producton functon, Y ak b L c, s wdely used to demonstrate the relatonshp between output volume Y and the amount of nput (K and L). When b c 1, the Cobb-Douglas producton functon exhbts the property of economes of Correspondence to: Z.-J. Max Shen (shen@eor.berkeley.edu) scale. Readers are referred to Earl [6, pp and pp. 252], for more dscussons on the producton functon wth economes of scale. Hgh overhead costs of warehousng and transportaton typcally lead to economes of scale nventory holdng and dstrbuton costs. A large nventory/transportaton volume enables fxed costs to be spread across more unts and thus results n reduced average unt costs. Fnally, volume dscount schemes offered by sellers often result n economes of scale n purchasng. Economes of scale n orderng and nventory s typcally modelled n the Economc Lot-Szng (ELS) lterature by concave orderng and nventory functons (see, for example, Aggarwal and Park [1] and Hsu [8]). It s easy to see that a concave functon s a specal case of the economes of scale functon. The former s more restrctve than the latter n that t requres dmnshng margnal rates of ncrease. For example, the modfed all-unt dscount freght cost structure (Chan et al. [2]) and the less-than-full-truck-load (LTL) and full-truck-load (TL) freght cost (Lee [9] and L, Hsu, and Xao [10]) exhbt the economes of scale, but are not 2005 Wley Perodcals, Inc.

2 Chu, Hsu, and Shen: Lot-Szng Problem wth Pershable Inventory and Economes of Scale 53 concave functons. Thus, although ELS problems wth concave cost functons are effcently solvable wth polynomal tme dynamc programmng (DP) solutons, they do not capture many cost structures n practcal applcatons such as the prce dscount schemes offered by supplers/retalers and transportaton servce provders, and other more complcated cost structures n producton and nventory. Motvated by varous quantty dscount schemes n purchasng and dstrbuton, a number of ELS problems n the lterature propose some specal forms of nonconcave economes of scale orderng cost structures. Federgruen and Lee [] consder two versons of the ELS problems wth all-unt dscount (under whch a dscount rate s appled to all unts of a purchase volume) and ncremental dscount (under whch dfferent dscount rates are appled to ncremental ranges of the purchased quantty) orderng cost structures. They develop polynomal-tme algorthms to solve ther models optmally (see also Xu and Lu [12] for addtonal dscussons on the algorthms). Lee [9] studes an ELS problem wth a restrctve orderng cost structure that ncludes a full-truck-load (TL) freght cost. The orderng cost functon s assumed to be statonary (.e., an dentcal orderng cost functon n every perod). Lee solves hs problem wth a polynomal tme DP algorthm. L, Hsu, and Xao [10] generalze Lee s model by consderng a more general ELS problem wth a tmevaryng orderng cost functon that ncludes a fxed setup cost, a varable purchase cost, and a freght dscount cost structure that ncludes both less-than-full-truck-load (LTL) and TL charges. They solve ther general problem n polynomal tme through a DP approach. Chan et al. [2] present an ELS problem wth a modfed all-unt dscount freght cost structure. Ths orderng cost functon typfes transportaton costs charged by many LTL carrers. They demonstrate the NP-hardness of ther problem and develop worst case performance bounds for an easy-to-mplement approxmaton soluton. Ther approxmaton soluton s the mnmum cost soluton that satsfes the Zero-Inventory-Orderng (ZIO) polcy under whch an order s placed only when the nventory level drops to zero. Chan et al. [3] extend Chan et al. [2] to a sngle-warehouse multretaler settng, n whch the shpment costs from the suppler to the warehouse are approxmated by pecewse lnear concave functons and the shpment costs from the warehouse to the retaler are represented by the modfed all-unt dscount cost structure. Snce the problem s NPhard n strong sense, ther objectve s to desgn smple nventory polces and transportaton strateges to mnmze system-wde costs by takng advantage of quantty dscounts n the transportaton cost structures. They show that the cost of the best ZIO polcy n a sngle-warehouse multretaler scenaro, n whch the warehouse serves as a crossdock faclty, s no more than 4/3 (5.6/4.6 f costs are statonary) tmes the optmal cost. Chu and Shen [5] consder an ELS model wth pershable nventory and quantty dscount cost functons. They assume the order cost functons follow modfed all-unts dscount scheme (modfed from the cost functons n Federgruen and Lee []), and an tem s deteroraton rate and ts carryng cost n each perod depend on the age of the tem. They are able to solve the correspondng problem n O(T 3 ). Fnally, there are ELS problems n the lterature that deal wth cost structures that do not exhbt economes of scale. For example, Chen, Hearn, and Lee [4] propose an ELS problem wth pecewse-lnear cost structure. They solve the problem wth a DP approach. Lppman [11] and, more recently, L, Hsu, and Xao [10] have analyzed ELS problems wth varous pecewse-concave producton/transportaton cost structures. In ths paper, we study an ELS problem for pershable products wth economes of scale orderng and nventory holdng cost functons. In addton to farly general cost structures, we also explctly consder stock deteroraton to make our model applcable to both nonpershable (when the deteroraton rate s zero) and pershable products. Our model s a generalzaton of the model n Chan et al. [2]. Snce the latter has been shown to be NP-hard (transformaton from partton problem), t s unlkely that an effcent exact soluton exsts for our ELS problem. By explorng the specal structures of the problem, we propose an approxmaton soluton and analyze ts effectveness through the development of worst case performance bounds and through computatonal experments. Although our model s a generalzaton of the model n Chan et al. [2], the soluton strategy developed n our paper s not a trval extenson of ther strategy. For example, we can show that the ZIO approxmaton soluton used by Chan et al. [2] to solve ther model would perform very poorly f t s appled to our model. The remander of ths paper s organzed as follows. Secton 2 presents our ELS problem wth stock deteroraton and economes of scale orderng and nventory cost functons. We then gve a number of structural propertes of the problem n ths secton. Secton 3 proposes an approxmaton soluton and analyzes ts worst case performance. Secton 4 demonstrates the effectveness of the proposed approxmaton soluton through a computatonal study. Secton 5 concludes the paper and offers a few future research topcs. 2. THE MODEL AND ITS STRUCTURAL PROPERTIES Consder an ELS problem wth T perods. For 1 j T, defne:

3 538 Naval Research Logstcs, Vol. 52 (2005) D j : the amount of demand n perod j; X j : the order quantty n perod j, whch s called an orderng perod f X j 0; C j (X j ): the cost of orderng X j unts n perod j. For 1 t T, defne I t : the amount of nventory at the begnnng of perod t, whch was ordered n perod. t : the fracton of I t that s lost durng perod t. Thus, only (1 t )/I t unts of the tem ordered n perod reman at the end of perod t. H t (I t ): the cost of holdng I t unts of nventory, whch were ordered n perod, n perod t. Ths cost excludes the value of the lost nventory whch s captured explctly by the stock deteroraton n our model. Z t : the amount of demand n perod t that s satsfed by the order n perod. A cost functon, F(X), defned on [0, ) s called an economes of scale functon f: (1) F(0) 0. (2) F(X) s nondecreasng on [0, ). (3) The average cost functon defned as F (X) F(X)/X, for X 0, s a nonncreasng functon on (0, ). In our problem, the orderng cost functons C t ( ), 1 t n, are assumed to be general economes of scale functons. We note that, for pershable stocks, the longer they have been held, the faster they may deterorate n a perod and the hgher the nventory holdng cost may be n that perod. Thus, for any gven perod t, we assume that the stock deteroraton rates satsfy ASSUMPTION 1: t jt,1 j t. For nventory costs, we make the followng assumpton: ASSUMPTION 2: H t ( x ) H t ( x) H jt ( y ) H jt ( y), where j t and x, y, 0. Assumpton 2 states that the margnal cost of holdng addtonal unts of nventory n perod t for a newer stock (regardless of ts current stock level) s no bgger than that for an older stock. Ths assumpton ncludes the followng specal case: H t (I t ) h t I t, where h t 0 s a constant, and h t h jt for j t. Ths more restrcted condton agan says that the margnal cost of holdng addtonal unt of nventory n perod t for a newer stock (ordered n perod j) s no bgger than that for an older stock (ordered n perod ). Ths condton would hold true n many real-world applcatons. We assume that there s no nventory avalable at the begnnng of perod 1 and no nventory s requred at the end of perod T. Orderng and demand fulfllment occur at the begnnng of each perod. Backloggng s not allowed. The followng formal statement of our pershable stock ELS problem, whch we wll call problem PELS, s smlar to the pershable nventory ELS problem consdered by Hsu [8], except that the orderng and nventory cost functons are assumed to be concave n the latter. T : Mnmze t1 t C t (X t ) H t (I t ) 1 subject to X t Z tt I tt, 1 t T, 1,t1 I,t1 Z t I t, 1 t T, t Z t D t, 1 t T, 1 X t, I t, Z t 0, 1 t T. We denote {X, I, Z} as a feasble soluton to problem PELS and () as the correspondng objectve functon value. It s easy to see that the model consdered by Chan et al. [2] s a specal nstance of problem PELS wth t 0 and H t (I t ) h t I t, where h t 0 s a constant. Snce the specal case has been shown to be NP-hard, t s therefore unlkely that effcent exact solutons exst for problem PELS. We now establsh two structural propertes of PELS that allow us to develop an approxmaton soluton and analyze ts worst case performance. Defne A kt 1/[ t1 lk (1 l )], for k t and A 1, and A kt s postve nfnty f l 1 for some l satsfyng k l t. By defnton, A jt A jk A kt, for j k t. (1) By Assumpton 1, we have A kt A j kt, for 1 j k t T. (2) We also note that, to satsfy one unt demand n perod t by an order n perod, t, we need to order A t unts n perod and carry A kt unts of nventory n every perod k, where k t 1.

4 Chu, Hsu, and Shen: Lot-Szng Problem wth Pershable Inventory and Economes of Scale 539 THEOREM 1: There s an optmal soluton * {X*, I*, Z*} to PELS such that, for two order perods j, f Z* jk 0 for some k, j k, then Z* t 0 for all t, k t T. PROOF: Suppose n an optmal soluton, {X, I, Z }, we have, respectvely, Z jk 0 and Z t 0 for some k and t, j k t T. Let mn{z t, Z jk /A j kt } 0. We can construct a new soluton, * {X*, I*, Z*}, as follows: Set Z* t Z t, Z* jt Z jt, Z* k Z k A j kt, I* l I l A lk X* X A k A kt A kt Z* jk Z jk A j kt, A j kt, A j kt, for l k 1, I* l I l A lt, for k l t 1, I* jl I jl A j lt, for k l t 1. All other varables of * are the same as those of.its easy to verfy that soluton * s feasble. Furthermore, f Z t, we have Z* t 0; and f Z jk /A j kt, we have Z* jk 0. We now show that (*) ( ). Frst, note that the only change n the orderng cost s n perod. Snce C ( ) s nondecreasng, the total orderng cost correspondng to soluton * would not be larger than that of soluton. We also note that there s a net reducton of the nventory, I l, n perods l,..., k 1. Thus, t1 * H l I l H l I l A lt lk H jl I jl A j lt H jl I jl. By (2), we have A lt A j lt, l t 1. By Assumpton 2 and the fact that H l s a nondecreasng functon, we have t1 * H l I l H l I l A j lt lk H jl I jl A j lt H jl I jl 0. The theorem can be proved by repeatedly applyng the earler modfcaton of the solutons. Theorem 1 suggests that there s an optmal soluton to problem PELS where nventory stocks ordered n dfferent perods are used to satsfy demands n frst-n-frst-out (FIFO) fashon. We call such soluton a FIFO soluton. Before presentng the next property, we have the followng results for the average cost functon: LEMMA 1: Suppose F(X) and G(X) are economes of scale functons defned on [0, ). Then: (a) F(X) and F(X) G(X) are also economes of scale functons, where 0 s any gven 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) s trval. To show (b), note that F( ) san economes of scale functon. 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. Fnally to show (c), note that F (X ) F (X). We have FX FX X F X XF X F X. THEOREM 2: There s an optmal FIFO soluton to PELS where each demand, D j,1j T, s satsfed by orders from at most three orderng perods. PROOF: Suppose n an optmal FIFO soluton, {X, I, Z }, a demand D j,1 j T, s satsfed by orders from perods t 1 t 2... t r j, where r 3. We wll show that orders from t 2 through t r1 can be combned nto one order n a new FIFO soluton wth reduced total costs. Frst note that by FIFO polcy, orders from t 2 through t r1 are entrely used to satsfy part of demand D j. Suppose d tl unts of D j are satsfed by the order from perod t l, l 2,...,r 1. For p t 2,...,t r1, the total orderng and nventory costs assocated wth the fulfllment that uses the order n perod p to satsfy d p unts of perod j demand s T p d p C p A pj p d p lp H pl A p lj d p. (3)

5 540 Naval Research Logstcs, Vol. 52 (2005) By Lemma 1, we know that T p ( ), p t 2,..., t r1, are all economes of scale functons. Let p 0 arg mn pt2,...,t r1 T p (d p ). We construct a new FIFO soluton by combnng orders n perods t 2,...,t r1 nto a sngle order n perod p 0 to satsfy M j r1 l2 d tl unts of perod j demand. Snce r1 T p0 M j M j T p0 M j M j T p0 d p0 d tl T tl d tl l2 r1 T tl d tl, l2 the new soluton has an objectve value no larger than that of the orgnal soluton. In the rest of the paper, we consder only optmal solutons that satsfy Theorems 1 and 2. To conclude ths secton, we show the followng property for a restrcted verson of problem PELS where the orderng cost functon s statonary. THEOREM 3: If C j ( ) C( ) for all j, 1 j T, there exsts an optmal FIFO soluton to problem PELS where each demand D j,1 j T, s satsfed by orders from at most two orderng perods. PROOF: Suppose n an optmal FIFO soluton, {X, I, Z }, to PELS, demand D j s satsfed by orders from perods, k and t, k t j. We obtan a new FIFO soluton, * {X*, I*, Z*}, to PELS by combnng orders n perods k and t nto a sngle order n perod t. Suppose that the order n perod k was used to satsfy d k unts of perod j demand. We have * CX t A t tj d k CX t H tl I tl lt By (b) of Lemma 1, A t lj d k H tl I tl CA kj * CA tj t d k lt k d k lk H kl A k lj d k. H tl A t lj d k CA k kj d k H kl A k lj d k. lk t By (2), we have A lj A k lj, for k t l j 1; furthermore, by (1), A k kj A k kt A k tj A t tj. Together wth the fact that C( ) s a nondecreasng functon, we have C( A t tj d k ) C( A k kj d k ) 0. Now, by Assumpton 2 and the fact that H tl ( ), t l j 1, are nondecreasng functons, we have * H tl A k lj d k H tl 0 lt H kl A k lj d k H kl 0 0. lt The theorem can be proved by repeatedly applyng the above modfcaton of the soluton. 3. THE APPROXIMATION SOLUTIONS AND WORST CASE ANALYSIS Chan et al. [2] consder an approxmaton soluton wth a Zero-Inventory-Orderng (ZIO) polcy to solve a specal nstance of problem PELS. In ther ZIO soluton, there s zero nventory at the begnnng of each orderng perod. The followng example shows that there are nstances of problem PELS where the objectve value of the best ZIO soluton may have an arbtrarly large error compared to that of the optmal soluton. EXAMPLE 1: Consder an nstance of PELS wth three perods where D 1 D 3 and D ( s a small amount). An nventory stock deterorates completely after one perod. Thus, we cannot use the order n perod 1 to satsfy the demand n perod 3. The holdng cost s zero. The orderng cost n perod 1 has a fxed charge of 1 for any sze of order. The orderng cost n perod 2 s 1/ per unt. The orderng cost n perod 3 s prohbtvely hgh, say 1/ 3 per unt. The optmal polcy s to order 1 unts n perod 1 to satsfy demands n perods 1 and 2, and order unts n perod 2 to satsfy the perod 3 demand. The total cost of ths optmal soluton s (1 1) 2. Consder the optmal ZIO polcy, whch orders unts n perod 1 for the demand n perod 1 and orders 1 unts n perod 2 for demands n perods 2 and 3. The total cost of the best ZIO soluton s 1 (1 )/ 1/. Note that as 3 0, Z ZIO /Z* 3. In other words, the ZIO polcy could be as poor as we could want. We now propose a new easly mplementable approxmaton soluton to solve problem PELS. A feasble soluton to PELS s called a Consecutve-Cover-Orderng (CCO) soluton f each order s used to satsfy (cover) all demands from a number of consecutve ndexed perods. Denote Z CCO as

6 Chu, Hsu, and Shen: Lot-Szng Problem wth Pershable Inventory and Economes of Scale 541 the mnmum objectve value of the best CCO solutons to problem PELS. Hsu [8] shows that Z CCO s the optmal objectve value of PELS f all of the orderng and nventory costs, C t ( ) and H t ( ), are concave functons, whch are specal cases of the economes of scale functon. He also develops a dynamc programmng (DP) algorthm to obtan the best CCO soluton correspondng to Z CCO. To mplement the dynamc programmng algorthm, frst we need to know the costs to satsfy demands from perod j to k by perod for every trplet (, j, k) ( j k). The computatonal complexty of these costs s bounded by O(T 4 ). Then, we can recursvely calculate V(, r), the optmal CCO polcy cost to satsfy demand through perod 1 to r wth the largest ndexed producton perod. The computatonal complexty of the DP soluton to the general problem PELS s bounded by O(T 4 ). The DP algorthm does not depend on the cost structure, and readers are referred to Hsu [8] for more detals of the algorthm. To analyze the effectveness of the CCO solutons, we frst present a transformaton procedure that wll transform an optmal soluton to problem PELS nto a CCO soluton. Let {X, I, Z } be an optmal soluton to PELS. We defne, for j, S,j C X A j H l I l A lj. (4) l S, j s the average cost to satsfy one unt demand n perod j by the order n perod. Gven an optmal FIFO soluton,, to problem PELS that satsfes the condton n Theorem 2 (.e., the demand n each perod s satsfed from at most three orders), denote R( ) as the set of perod ndex for the perod for whch the demand s satsfed by orders from two or three dfferent perods n. For any perod, j, whose demand s satsfed by two order perods, and t, n the optmal soluton,, we could ntroduce a pseudo-perod k between perods and t, wth zero demand and a prohbtvely hgh orderng cost, no holdng cost, and no deteroraton. Thus, we may assume wthout loss of generalty that for any j R( ), there are three orderng perods ( j), k( j), and t( j) that satsfy j D j, j D j, and (1 j j ) D j unts of perod j demand, respectvely, where 0 j 1, 0 j 1, and ( j) k( j) t( j) j. It s easy to verfy that Defne j S j,j j S kj,j 1 j j S tj,j D j. jr (5) and 1,j 1 j S j,j j S kj,j D j, 2,j 1 j S kj,j D j, 3,j j j S tj,j j S kj,j D j. Let, j ( 1, 2, 3) be the same as except that D j, demand n perod j, s satsfed entrely by orderng n perods ( j), k( j), and t( j). Then,, j ( 1, 2, 3) are the upper bounds of the ncremental costs, (, j ) ( ) ( 1, 2, 3). We now present the followng transformaton procedure to construct a near-to-optmal CCO polcy from : Transformaton Procedure: Step 1: Let p 0 and set p. Step 2: For soluton p, set j j p, the smallest ndex n R( p ). Suppose orders from perods ( j), k( j), and t( j), ( j) k( j) t( j) j, are used to satsfy j D j, j D j, and (1 j j ) D j unts of perod j demand, respectvely. Step 3: Construct a new soluton by satsfyng D j, demand n perod j, by the perod that the correspondng, j ( 1, 2, 3) s mnmum (break tes arbtrarly). Step 4: Set R() R( p ){ j}. If R() A, set ˆ and stop; otherwse, ncrease p by 1, set p, R( p ) R(), and return to Step 2. It s easy to see that n each teraton, p 0, of the above Transformaton Procedure, we construct a new FIFO soluton, p {X p, I p, Z p }, n whch the three orders used to satsfy the demand n a certan perod, j p, are combned nto a sngle order from one of the three orderng perods ( j p ), k( j p ), and t( j p ). Clearly, the overall procedure conssts of r R( ) teratons. Furthermore, upon termnaton of the procedure, the soluton ˆ s a CCO soluton to problem PELS. Defne for each p, 1 p r, and L p p C p tj p(x tjp 1, f teraton p uses Combne 1 or 2, 0, f teraton p uses Combne 3, ) C tjp(x tjp j p1 ) ltj p p (H tjp,l(i tjp,l) H tjp,l(i tjp,l)).

7 542 Naval Research Logstcs, Vol. 52 (2005) p Note that f p 1, we have X t( jp ) X p t( jp ), and I t( jp ),l I t ( jp ),l for all l, t( j p ) l j 1. Thus, p p L p C p tj p(x tjp ) C tjp(x tjp We have the followng result: j p1 ) ltj p THEOREM 4: For any p, 1 p r, p (H tjp,l(i tjp,l) H tjp,l(i tjp,l)) 0. (6) p p mn 1,jl, 2,jl, 3,jl p L p. () l1 PROOF: See the Appendx. LEMMA 2: For arbtrary,, and c, 1, 2, 3, where 0, 0, 1, and c 0, we have mn1 c 1 c 2, 1 c 2, c 3 c c 1 c 2 1 c 3 (8) and mn1 c 1, c c 1 1 c 2. (9) PROOF: See the Appendx. Let Z* be the optmal objectve value to problem PELS. We are now ready to present the worst case performance for the approxmaton soluton Z CCO. THEOREM 5: For every nstance of problem PELS, Z CCO [(42 5)/]Z*, and ths bound s tght. PROOF: Snce Z CCO s the mnmum value among all of the CCO solutons and ˆ, the soluton obtaned by the Transformaton Procedure s a CCO soluton and we have Z CCO (ˆ ). By Theorem 4 and (6), we have Z CCO Z* jr mn 1,j, 2,j, 3,j. By the defnton of s and Lemma 2, we have 42 2 mn 1,j, 2,j, 3,j j S j,j jr jr Fnally by (5), we have j S kj,j 1 j j S tj,j D j j S j,j j S kj,j 1 j j S tj,j D j jr 42 2 Z*. Thus, Z CCO [(42 5)/]Z*. To show that ths bound s tght, we only need to demonstrate that there exst nstances of problem PELS for whch the rato Z CCO /Z* s arbtrarly close to (42 5)/. Consder an nstance of PELS wth four perods where D 1 D 2 D 4 and D ( s a small amount). The orders from perods 1 and 2 deterorate completely after perod 3. That s, we cannot use orders n perod 1 and 2 to satsfy the demand n perod 4. There s no holdng cost and the orderng costs n perods 1, 3, and 4 have a fxed charge of (32 2)/14 for any order wth a sze no larger than 2 1; and a rate of (52 8)/14 per unt for orders wth szes larger than 2 1. The orderng cost n perod 2 has a fxed charge of (32 5)/ for any order wth a sze no larger than 3 22; and a rate of (3 2/ per unt for orders wth szes greater than In the optmal soluton to ths nstance, 2 1 unts are ordered n perods 1 and 3, whle 3 22 unts are ordered n perod 2. The total cost s Z* 1. However, the best CCO soluton s to order 2 unts n perod 1 and 1 2 unts n perod 3, wth a total cost of Z CCO [(42 5)/] [(52 8)/]. Hence, Z CCO Z* as 3 0. As mentoned n Theorem 3, f the orderng cost s statonary over the T-perod plannng horzon, there exsts an optmal FIFO soluton, where each demand D j s satsfed by orders from at most two orderng perods. Denote Rˆ ( ) as the set of perod ndex for perod j whose demand, D j, s satsfed by two orders from perods ( j) and t( j) wth szes of j D j and (1 j ) D j, respectvely, where ( j) t( j) j and 0 j 1. Smlar to (5), t s easy to see that j S j,j 1 j S tj,j D j. (10) j ˆR

8 Chu, Hsu, and Shen: Lot-Szng Problem wth Pershable Inventory and Economes of Scale 543 Problem classes Number of perods Table 1. Problem classes. Type of order cost structure Holdng cost h Class 1 10 Type Type 1 Class 2 10 Type Type 2 Class 3 10 Type Type 1 Class 4 10 Type Type 2 Class 5 12 Type Type 1 Class 6 12 Type Type 2 Class 12 Type Type 1 Class 8 12 Type Type 2 Type of deteroraton rate THEOREM 6: If there exsts an optmal FIFO soluton to PELS n whch each demand D j s satsfed by orders from at most two orderng perods, we have Z CCO (3/ 2)Z* and ths bound s tght. PROOF: Smlar to the proof of Theorem 5, we can show Z CCO Z* By (9) and (10), Z CCO Z* 1 2 j ˆR j ˆR mn1 j S j,j, S tj,j D j. j S j,j 1 j S tj,j D j 1 2 Z*. It remans to be shown that there are nstances of PELS n whch the rato Z CCO /Z* s arbtrarly close to 3/2. Consder an nstance of PELS wth three perods where D 1 D 3 and D ( s a small amount). The nventory deterorates after one perod;.e., we cannot use the order n perod 1 to satsfy the demand n perod 3. There s no holdng cost and all orderng cost have a fxed charge of 1 for any order wth a sze no greater than 1; and a rate of 1 per unt for orders wth szes larger than 1. In the optmal soluton to ths nstance, 1 unt s ordered n perod 1 and 1 n perod 2 wth a total cost of Z* 2. However, the best CCO soluton s to order unts n perod 1 and 2 unts n perod 2, wth a total cost of Z CCO 3. Hence, wde ranges of problem characterstcs (see dscussons below). For each test problem, we obtan the optmal soluton usng CPLEX by IP formulaton and compare t wth the best CCO soluton. We consder eght dfferent classes of nstances representng dfferent combnatons of plannng horzon, cost functons and deteroraton rate, as descrbed n Table 1. For each class, we generate 100 random nstances. The number of perods s ether 10 or 12. The demand n each perod s generated from a normal dstrbuton wth a mean of 10 and a standard devaton of 2.5. The orderng cost functons consdered n each case are statonary;.e., they do not change over tme. Two dfferent orderng cost structures are studed: The frst structure, as llustrated n Fgure 1, has break ponts at 0, 20, 40, 60, 80, 100, and, and the fxed costs for the dfferent ntervals are 15, 0, 30, 0, 40, and 0, correspondngly. The second structure has break ponts at 0, 10, 20, 40, 60, 90, and, and the fxed costs for the dfferent ntervals are 15, 0, 30, 0, 45, and 0, correspondngly. The holdng costs are lnear at 0.1 or 0.2 per unt for all the perods. We use two dfferent types of deteroraton schemes. For any two perods, and j, defne j ( j ), where Z CCO Z* as COMPUTATIONAL RESULTS In ths secton, we demonstrate the computatonal performance of the proposed CCO approxmaton soluton. We randomly generate a large number of test problems wth Fgure 1. The economc of scale order cost structure.

9 544 Naval Research Logstcs, Vol. 52 (2005) Table 3. Comparson of computatonal tmes for dfferent number of perods. Problem classes Average computatonal tme for 10 perods Average computatonal tme for 15 perods Rato of tme Class Class Class Class Fgure , 0.04, 0.08, 0.16, 0.24, 0.36, 0.54, 0.81, 0.9, 1, 1 for the frst scheme, and The deteroraton structure , 0.03, 0.05, 0.08, 0.13, 0.21, 0.34, 0.55, 0.89, 1, 1 j for the second scheme. Note that A j l (1 l ). Fgure 2 shows the relatonshp between the age of nventory t and log 2 A 1t. For the nstances wth only 10 perods, we choose the frst 10 values from the above lsts. Table 2 gves, for each of the problem classes, the average and maxmum ratos of the cost of the best CCO polcy to the cost of the optmal soluton. These computatonal results ndcate that our approxmaton approach s very effectve. The maxmal dervaton from the optmal soluton s only 4% and the average dervaton s less than 1% for all classes. Furthermore, the runnng tme of the approxmaton soluton s less than 0.1 seconds n all cases, whle the runnng tme of the optmal algorthm ncreases drastcally as the length of the plannng horzon and the number of break ponts n the cost functons Table 2. Computatonal results of the CCO polcy. Problem classes Maxmum rato Z CCO /Z* Average rato Class Class Class Class Class Class Class Class ncrease. Table 3 shows the computatonal tme n mnute for obtanng optmal solutons for four test classes as we ncrease the number of perod from 10 to 15. We thus conclude that the CCO polcy s very effectve n solvng our problem snce t allows us to fnd the almost optmal soluton n a neglgble amount of tme whle the effort to determne the optmal soluton s unduly hgh. 5. CONCLUSION Ths paper studes an mportant varant of the ELS problem wth pershable nventory and general economes of scale orderng and nventory functons. Snce the problem s NP-hard, we focus on dentfyng an easy to mplement replenshment strategy wth a guaranteed level of soluton qualty. In partcular, we show that the cost of the best CCO polcy, whch can be found effcently n polynomal tme, s guaranteed to be no more than (42 5)/ tmes the overall optmal soluton and ths bound s tght. In addton, f the orderng cost functon s statonary, the cost of the best CCO polcy s no hgher than 1.5 tmes the optmal cost. Our computatonal experments further confrm the effectveness of the proposed approxmaton soluton. In the future research, we could extend our model to allow backorders. It would also be nterestng to study the sngle-warehouse multretaler problem or other fnte-perod plannng problems wth general economes of scale cost functons. APPENDIX PROOF OF THEOREM 4: We wll prove the theorem by nducton on the teraton ndex p. For smplcty of exposton, we wll drop the ndex j p from jp, jp, ( j p ), k( j p ), and t( j p ) f there s no confuson about p. In the frst teraton ( p 1), we obtan a soluton, 1 {X 1, I 1, Z 1 }, by executng one of the three combne operatons n Step 3 of the procedure. Note that snce soluton s a FIFO soluton, the order n perod k s entrely used to satsfy D j unts of perod j demand;.e., we have X k D j A k kj and I kl D j A k lj, k l j 1. Suppose Combne 1 s executed n Step 3. In ths case, we have Z 1 j D j, Z 1 kj Z 1 tj 0,

10 Chu, Hsu, and Shen: Lot-Szng Problem wth Pershable Inventory and Economes of Scale 545 X 1 X 1 D j A j, I 1 l I l 1 D j A lj, l j 1, X 1 k X k D j A k kj 0, lt H tl I 1 tl H tl I tl 1 D j A k kj C k X k lk 1 D j A k lj H kl I kl L 1 2,j 1 L 1. (13) I 1 kl I kl D j A k lj 0, k l j 1, X 1 t X t 1 D j A t tj, I 1 tl I tl 1 D j A t lj, t l j 1. Thus, we have 1 C X 1 C X H l I l l 1 H l I l C k X k 1 C k X k lk lt l H kl I 1 kl H kl I kl C t X 1 t C t X t H tl I 1 tl H tl I tl C X 1 D j A j C X H l I l 1 D j A lj H l I l S kj D j L 1. (11) Snce C ( ) and H l ( ) are the economes of scale functons, by (c) of Lemma 1, we have 1 1 D j A j C X l 1 D j A lj H l I l S kj D j L 1 1,j 1 L 1. (12) Suppose now Combne 2 s executed n Step 3. We have z 1 kj D j, z 1 j z 1 tj 0, X 1 X D j A j, I 1 l I l D j A lj, l j 1, X 1 k X k 1 D j A k kj, I 1 kl I kl 1 D j A k kj, k l j 1, X 1 t X t 1 D j A tj, I 1 tl I tl 1 D j A lj, t l j 1. Furthermore, we have X 1 X and I 1 tl I tl for all l, t l j 1. Thus, 1 C X 1 C X H l I l l 1 H l I l Fnally, suppose Combne 3 s executed n Step 3. We have z 1 tj D j, z 1 j z 1 kj 0, X 1 X D j A j, I 1 l I l D j A lj, l j 1, X 1 k X k D j A k kj 0, I 1 kl I kl D j A k kj 0, k l j 1, X 1 t X t D j A tj, I 1 tl I tl D j A lj, t l j 1. Note that, n ths case, 1 0. Smlar to earler dscussons, we have 1 D j A t tj C t X t lt D j A t lj H tl I tl S kj D j 3,j 1 L 1. (14) Combnng (12) (14), we see that () holds for p 1. Assume that () holds for p 1,..., m 1. We now consder the mth teraton of the Transformaton Procedure. By the nductve assumpton, we have m m m1 m1 m We wll now show that m1 m1 l1 m l1 m mn 1,jl, 2,jl, 3,jl m1 L m1. mn 1,jl, 2,jl, 3,jl m L m. (15) Frst note that s a FIFO soluton. Therefore, we have t( j m1 ) ( j m ) k( j m ). Thus, n the mth teraton, X m1 k X k, X m1 t X m1 t, I kl I kl, k l j 1, and I m1 tl I tl, t l j 1. Suppose X m1 X and I m1 l I l for all l, l j 1 n the mth teraton. By the same arguments as (12) (14), we can show that m m1 mn 1,jm, 2,jm, 3,jm m L m. As m1 L m1 0, we have m m m1 m1 m1 mn 1,jm, 2,jm, 3,jm m L m l1 mn 1,jl, 2,jl, 3,jl m1 L m1 C k X k 1 C k X k lk H kl I 1 kl H kl I kl C t X 1 t C t X t m l1 mn 1,jl, 2,jl, 3,jl m L m.

11 546 Naval Research Logstcs, Vol. 52 (2005) We conclude that (15) holds f X m1 X and I l m1 I l for all l, l j 1 n teraton m. The only possblty for X m1 X or I l m1 I l for some l s that t( j m1 ) ( j m ). In ths case, L m1 C tjm1x m1 tjm1 C tjm1x tjm1 H tjm1li tjm1l C jmx jm jm11 ltjm1 m1 C jmx jm jm11 H jmli jml C X m1 C X l H tjm1li m1 tjm1l jm11 ljm H jmli m1 jml H l I l m1 H l I l m m1 C X m C X m1 H l I l l m H l I m1 l S kj D j L m C X m C X H l I l l m H l I l S kj D j m L m C (X m1 ) C (X ) (H l (I l l m1 ) H l (I l )) C X m C X l H l I l m H l I l S kj D j m L m L m1. C X m1 C X l H l I l m1 H l I l. The last equaton comes from the fact that the nventory levels are unchanged after perod j m1 n both solutons, m1 and. We now consder the followng dfferent cases n the mth teraton. CASE 1: Suppose Combne 1 s used n mth teraton. We wll show that m m1 mn 1,jm, 2,jm, 3,jm m L m m1 L m1. (16) Frst, smlar to (11) (usng the fact that X m1 k X k, X m1 t X m1 t, I kl I kl, k l j 1, and I m1 tl I tl, t l j 1), we have m m1 C X m C X m1 l H l I l m H l I l m1 S kj D j L m. We further consder the followng two subcases n whch I m1 0 and I m1 1. CASE 1.1: If m1 0, Combne 3 s used n the (m 1)th teraton. Snce t( j m1 ) ( j m ), we have X m1 X and I l m1 I l. Furthermore, by (c) of Lemma 1 and the fact that C ( ) s an economes of scale functon, C X m C X m1 C X m1 1 D j A j C t X m1 t 1 D j A j C X m1 1 D j A j C X. Smlarly, for each l, l j 1, Snce m1 0 and H l I m l H l I m1 l 1 D j A lj H l I l. 1 D j A j C X l we have 1 D j A lj H l I l 1,jm S kj D j, m m1 1,jm L m 1,jm m L m m1 L m1. (1) CASE 1.2: If m1 1, Combne 1 or 2 s used n the (m 1)th teraton. Snce t( j m1 ) ( j m ), we have X m1 X and I l m1 I l. Thus, By (c) of Lemma 1 and the fact that C ( ) s an economes of scale functon, C X m C X X m X C X X m X m1 C X Smlarly, for each l, l j 1, Note that H l I m l H l I l 1 D j A lj H l I l. 1 D j A j C X l We have 1 D j A j C X. 1 D j A lj H l I l 1,jm S kj D j. m m1 1,jm m L m m1 L m1. (18) Equatons (1) and (18) complete the proof of (16). We can now conclude that f Combne 1 s used n the mth teraton, m1 m m m1 l1 m1 m1 L m1 l1 mn 1,jl, 2,jl, 3,jl mn 1,jl, 2,jl, 3,jl 1,jm m L m. (19) CASE 2: Suppose Combne 2 or 3 s used n the mth teraton. Smlar to the proofs of (13) and (14) and together wth the fact X m1 k X m1 k, X t X t, I m1 kl I kl, k l j 1, and I m1 tl I tl, t l j 1, we can easly show that Thus, m m1 mn 2,jm, 3,jm m L m. m1 m mn 1,jl, 2,jl, 3,jl mn 2,jm, 3,jm m L m l1 m1 m1 L m1 l1 By (19) and (20) n Cases 1 and 2, mn 1,jl, 2,jl, 3,jl mn 2,jm, 3,jm m L m. (20)

12 Chu, Hsu, and Shen: Lot-Szng Problem wth Pershable Inventory and Economes of Scale 54 m l1 Ths completes the proof of the theorem. m mn 1,jl, 2,jl, 3,jl m L m. (21) PROOF OF LEMMA 2: Assume that (8) s not true and, thus, c 1 c 2 c 1 c 2 1 c 3, c 2 c 1 c 2 1 c 3, 42 2 c 3 c 2 c 1 c 2 1 c 3. Denotng Q [(42 2)/](c 1 (1 )c 3 ) and P Q [(42 5)/]c 2, we have 1 c 1 P, (22) c 2 P, (23) c 3 P. (24) By (23), c 2 Q [(42 5)/]c 2 or (1 [(42 5)/])c 2 Q. Snce both c 2 and Q are postve, (1 [(42 5)/]) s also postve. Thus, c 2 Q/{1 [(42 5)/]} and hence 42 5 P Q c 2 Q (25) Multplyng both sdes of (22) and (24) by ( ) and (1 )(1 ), respectvely, and addng them together, we obtan 1 c 1 1 c P. By (25), we have 1 or 1 Q Q, Snce [(1 )/2] 2 (1 )( ) and ( ) (1 )(1 ) 1 2(1 ) (1 ) [(1 ) 2 /2] [(1 )(1 )/2], we obtan: or , , 2 whch mples that Ths s a contradcton. We have just shown (8). To prove (9), assume that t does not hold. We have 1 c 1 c 1 1 c 2, 2 c 2 c 1 1 c 2. 2 We multply the rght-hand sdes and left-hand sdes of both nequaltes to obtan whch mples that Ths s agan a contradcton. 4c 1 1 c 2 c 1 1 c 2 2, c 1 1 c ACKNOWLEDGMENT Ths work was sponsored n part by the NSF CAREER Award DMI We thank the Assocate Edtor and two referees for constructve comments that led to ths mproved verson. REFERENCES [1] A. Aggarwal and J.K. Park, Improved algorthms for economc lot sze problems, Oper Res 41 (1993), [2] L.M.A. Chan, A. Murel, Z.-J. Shen, and D. Smch-Lev, On the effectveness of zero-nventory-orderng polces for the economc lot szng model wth pecewse lnear cost structures, Oper Res 50 (2002), [3] L.M.A. Chan, A. Murel, Z.-J. Shen, D. Smch-Lev, and C.P. Teo, Effectveness of zero-nventory-orderng polces for a one-warehouse mult-retaler problem wth pecewse lnear cost structures, Management Sc 48 (2002), [4] H.D. Chen, D.W. Hearn, and C.Y. Lee, A dynamc programmng algorthm for dynamc lot sze models wth pecewse lnear costs, J Global Optm 4 (1994), [5] L.Y. Chu and Z.-J. Shen, Pershable stock lot-szng problem wth modfed all-unt dscount cost structures, Global supply

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