Essays on Joint Replenishment and Multi-Echelon Inventory Systems

Transcription

1 26:16 LICENTIATE T H E S I S Essays on Jont Replenshment and Mult-Echelon Inventory Systems Andreas Nlsson Luleå Unversty of Technology Department of Busness Admnstraton and Socal Scences Dvson of Industral Logstcs 26:16 ISSN: ISRN: LTU-lc SE

2 Essays on Jont Replenshment and Mult-Echelon Inventory Systems ANDREAS NILSSON

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4 Preface Ths Lcentate thess has been carred out at the dvson of Industral Logstcs at Luleå Unversty of Technology. I would lke to start by expressng my grattude to my supervsor and co-author, Professor Anders Segerstedt, for hs encouragement and gudance durng my tme here. Greatest thanks also to my other co-authors for ther work wth two of the research papers; Rolf Forsberg at the dvson and Erk van der Slus at Unversty of Amsterdam, The Netherlands. Further more, many thanks to Peter Söderholm for helpng out wth the desgn of the thess and to all my present and former colleagues for the many nsprng chats and conversatons. Fnally, great apprecaton to Jan Wallander and Tom Hedelus research foundatons for fnancng ths research project J1-23 Supply Chan Management. I

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6 Abstract Ths Lcentate thess addresses the topcs of Jont Replenshment and Mult- Echelon Inventory Systems. Both are mportant parts n the concepts and current trends n Supply Chan Management and Logstcs. The objectve of Supply Chan Management s to manage the flow of nformaton and goods from the suppler to the fnal customer wth respect to hgh customer value and low costs. A common nventory and coordnaton problem s the so called Jont Replenshment Problem. The problem often occurs when several tems are replenshed at a sngle stockng pont. Ths can happen n many dfferent stuatons, e.g. when several tems are procured from the same suppler or when a product after manufacturng, s packaged n dfferent quanttes. Cost savngs can be obtaned by coordnatng the replenshments, compared to when tems are replenshed ndependently. Coordnaton of several levels n a supply chan, often referred to as mult-echelon nventory control, s another mportant part of Supply Chan Management. A frequently encountered problem n practce s the One-warehouse N-retaler problem, whch can found n dvergent dstrbuton systems where a central warehouse supples several retalers wth goods. The thess contans an ntroductory part and three research papers. The frst two papers deal wth the Jont Replenshment Problem and the second wth the Onewarehouse N-retaler problem. Paper I provdes a novel heurstc method to solve jont replenshment problems usng a spread-sheet technque. The prncple of the recurson procedure s to fnd a balance between the replenshment and nventory holdng costs for the dfferent tems by adjustng the replenshment frequences. Paper II s an extenson of the area and presents a new method that may help reduce peak nventory levels and arrval quanttes n jont replenshment problems. The replenshments are re-scheduled durng the cycle perods and f necessary ndvdual replenshments are delayed sngle tme perods. Paper III deals wth the problem of prortzng retalers when there s a shortage of supply at a warehouse. All customers are often not equally mportant and a warehouse that suffers from stock-outs may therefore want to gve hgher prorty to some retalers when new goods are ready for delvery. The paper presents an approxmaton of the nventory holdng and shortage cost when retalers are prortzed accordng to two groups, hgh and low prorty retalers. Keywords: Jont Replenshment; Mult-Echelon; Inventory Management; Inventory Control; Supply Chan Management; Heurstcs; Warehouse; Retalers III

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8 Table of Contents 1. Introducton Inventory Management The Jont Replenshment Problem Constant demand Indrect groupng strateges Drect groupng strateges Extensons and varants Stochastc demand Contnuous revew systems Perodc revew systems The One-warehouse N-retaler problem Constant demand Stochastc demand Summary and Extensons References...24 Paper I Nlsson A., Segerstedt A. and Slus E. A New Iteratve Heurstc to Solve the Jont Replenshment Problem Usng a Spread-sheet Technque. Accepted for publcaton n Internatonal Journal of Producton Economcs. Paper II Nlsson A. Reducng Peak Inventory Levels and Arrval Quanttes n the Jont Replenshment Problem. Submtted for publcaton. In preprnts and presented at the 14th workng Semnar on Producton Economcs, Innsbruck, Austra, Feb 26. Paper III Forsberg R. and Nlsson A. Prortzng Retalers n a Two-Echelon Inventory System. Submtted for publcaton. Presented at the IFORS Trennal, Hawa, USA, July 25. V

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10 1. Introducton Logstcs and Supply Chan Management concern the assocated actvtes how to plan and control the flow and storage of goods and servces on the way to the fnal customer. As the number of actvtes can be large, there are often contradctory goals at dfferent parts of the logstcs system. Axsäter (2a) states that nventory management plays a crucal and central role to balance these conflctng goals. The purchase manager may want to order large quanttes to obtan volume dscounts and the producton manager prefers large producton batches to avod tme-consumng setups. An dle producton due to shortage of materals s expensve so both managers lke to have a large supply n stock. The marketng manager as well would lke to keep hgh stock levels to quckly meet customer orders. Understandably, hgh nventory levels cost money as well. Not only the captal ted up n stock costs money. Lower nventory levels of raw materal, components and fnshed goods requre smaller buldngs and less manpower for admnstraton and handlng of physcal goods. Less work n progress requres less factory space, makes nternal transportaton more effcent and enables shorter throughput tmes. For nformaton on these areas, see Hopp and Spearman (1996). Accordng to Mentzer et al. (21) there are three dfferent categores of defntons and vewponts on Supply Chan Management (SCM). They can be summarzed nto: A management Phlosophy. A systems approach to vewng the supply chan as a whole, and to managng the total flow of goods nventory from the suppler to the ultmate customer. SCM as a set of actvtes to mplement a management phlosophy. The focus on actvtes to consttute Supply Chan Management. Thus the actvtes to mplement a management phlosophy. SCM as a set of management processes. Ths focus s on processes to consttute SCM nstead of the actvtes. SCM s the process to manage relatonshps, nformaton and materals flow to enhance customer servce and economc value through synchronzed management of flows of physcal goods and assocated nformaton. 1

11 There exst many defntons of Supply Chan Management. Chrstopher (1998) provdes the followng defnton wth an overarchng management perspectve: The management of upstream and downstream relatonshps wth supplers and customers to delver superor customer value at less cost to the supply chan as a whole. Steven (1989) for nstance, presents a more specfc defnton whch s geared towards nventory management. The objectve of managng the supply chan s to synchronze the requrements of the customer wth the flow of materals from supplers n order to effect a balance between what are often seen as conflctng goals of hgh customer value, low nventory management, and low unt cost. Persson (1997) reveals the stated objectves of Supply Chan Management and concludes that t can be seen as a set of fundamental belefs; coordnaton and ntegraton along the materal flow, wn-wn relatons and end customer focus. However, n comparson wth forerunners Persson fnds that the deas and belefs of Supply Chan Management also can be found n pror works on the area. She also states that there s much emprcal evdence on how the work wth Supply Chan Management affects companes and the outcome shows an overwhelmng success when usng t. Chrstopher (1998) dscusses the supply chan and compettve performance. He means that the Supply Chan can be vewed as a network of organzatons that are nvolved wth upstream and downstream lnkages n dfferent processes and actvtes that produce value to the products or servces. Tradtonally organzatons have vewed themselves as ndependent enttes. Ths has changed over the years and today many supply chans are more ntegrated. Ths can be done n dfferent stages; the frst step s to have a functonal ntegraton where the companes have recognzed a lmt degree of ntegraton between adjacent functons. The natural next step s to reach nternal ntegraton wthn a company and fnally external ntegraton when all supplers and customers are lnked together wth nternal operatons. Supply Chan Management and logstcs are changng and there are several trends towards the areas are progressng. Waters (23) dscusses 12 dfferent trends n logstcs today. Improvng communcatons. Better communcaton wthn the supply chan wth new technologes such as Electronc Data Interchange (EDI). 2

12 Improvng customer servce. Makng logstcs cost as low as possble and ncreasng hgh servce levels at low costs. Globalzaton. Frms operatons are becomng more and more nternatonal when trade and competton are rsng. Reduced number of supplers. The number of supplers s reduced and long term relatonshps are created. Concentraton of ownershp. When the companes experence economcs of scale the competton s concentrated to fewer players on the market. Outsourcng. More and more logstcs operatons are outsourced to specalzed companes. Postponement. The movement of almost-fnshed goods nto the dstrbuton system and delays of fnal modfcatons. Cross-dockng. When ncomng goods are shpped agan wthout beng stored at the warehouse. Drect delvery. Drect shppng from manufacturer to fnal customer. Other stock reducton methods. Newer methods such as Just-n-Tme and Vendor Managed Inventores. Increasng envronmental concerns. The movement towards greener practces among logstcs operatons. More collaboraton along the supply chan. Amng towards the same objectves and no nternal competton wthn the supply chan. The papers n ths thess are a consequence of two of the logstcs trends above; mprovng customer servce and reduced number of supplers. When the number of supplers are becomng fewer and fewer each suppler may have to offer more types of raw materal or components than they used to. Procurng several products from a suppler requres well functonng coordnaton regardng transportaton and nventory management. The frst two papers n the thess deal wth a famous nventory and coordnaton problem, The Jont Replenshment Problem. Lookng at Supply Chan Management out of an Inventory Management perspectve t s possble to obtan hgh servce levels and reduce costs by consderng more than one stockng ponts n a Supply Chan System. Ths s called mult-echelon nventory control. Muckstadt and Thomas (198) show that mplementng mult-level methods may result n cost savngs up to 5 %. The thrd paper deals wth a common mult-echelon nventory control problem, the One-warehouse N-retaler problem. Both problems, the Jont Replenshment Problem and the One-Warehouse N- retaler are related under constant demand. Graves (1979) dscusses the smlartes regardng cost functons and solutons procedures for the Jont 3

13 Replenshment Problem, The Economc Lot Schedulng Problem (ELSP) and the One-warehouse N-retaler problem. The research objectve of ths thess s to develop new models n nventory management wth ether constant or stochastc demand and evaluate these based on performance. The outlne of the thess s the followng: Chapter 2 presents an overvew of the nventory management, ncludng types of nventores, nventory control systems and dfferent mult-echelon nventory systems. Chapter 3 and 4 gve a revew of earler research on the two areas. The constant and stochastc demand jont replenshment problem, ncludng varants and extenson are revewed n Chapter 3. Papers on the one-warehouse N-retaler problem wth constant and stochastc demand are revewed n chapter 4. Paper I, II, III are brefly summarzed n the revew where they contrbute to earler research on the area. Fnally, chapter 5 presents a summary and extensons. 4

14 2. Inventory Management It has been known for a long tme that economc growth fluctuates n a cycle, called the busness cycle. Perods of expanson are followed by perods of recesson and vce versa. For a number of reasons fluctuatons are very much unwanted for ndustry, the economy and the socety as a whole. Inventores play a crucal role n the busness cycle. Durng perods of expanson ndustry nvests n nventores and durng perods of expanson ndustry dsnvests n nventores. For nstance, Aley (1995) clams that the advances n nventory management and producton plannng and schedulng have contrbuted to less volatlty n the busness cycles n later years. Increasng effcency n nventory management has made t possble to ncrease producton of goods and servces wthout the need of growng nventory nvestments. In the Swedsh economy, the nvestments n nventores have been kept on a relatve constant level durng the last decades whle the growth n the Gross Domestc Product (GDP) has been a few percent per year. See fgure 1. Between 198 and 23 the real GDP growth was 59.6% whle the nventores decreased by 1.1 %. The progress made on Inventory Management n has enabled us to produce more wthout tyng up more captal n nventores. An mportant contrbuton to ncreased productvty, better economc growth and a hgher standard of lvng Growth n GDP vs Inventores GDP Inventores Fgure 1. Real growth n Gross Domestc Product (GDP) and nventores = Index 1. Source: Statstcs Sweden (SCB)

15 Many authors have ponted out the purpose and mportance of nventores n ndustry. For nstance, Aschner (199) gves these fve reasons: Demand/supply fluctuatons. Safety stocks or buffer stocks are held to absorb varatons n demand, and suppler performance uncertanty. Antcpaton. Inventory allocatons made to meet seasonal demand, sales promoton and to meet customer requrements durng perods n whch the producton faclty s noperable. Transportaton. There s a trade-off between the speed of the transportaton and the cost of nventory n a ppelne system where products are n transt. Hedgng. An ssue of procurement economes and the cost of holdng nventores versus the mpact of prce ncrease, specal procurement offers or speculaton. Lot sze. Replenshment volumes that exceed mmedate demand rates n order that economes may be obtaned from longer producton runs or lower transportaton costs. Inventores can be categorzed n dfferent ways. For nstance, Lambert & Stock (1993) categorze sx types of nventores as follows: Cycle Stock. Inventory that s the result from a replenshment process and s requred n order to meet demand under condtons of uncertanty. In-transt nventores. Items that are on a route from one locaton to another. Safety or Buffer stock. Stock that s held n excess of cycle stock because of uncertanty n demand or lead tme. Speculatve stock. Inventory held for other reasons than satsfyng current demand. Seasonal stock. Another type of speculatve stock that nvolves accumulaton of stocks before a hgh season begns. Dead stock. The set of tems for whch no demand has been regstered for some specfc perod of tme. Throughout the year s research on nventory control have progressed. Stll however, some of many old models and strateges are stll successfully used today. The classc Economc Order Quantty (EOQ) s used to calculate lot szes and can be used as an nventory control system when demand s constant and known. Harrs (1913) s frst to suggest the approach but the model s also known after the publcaton of Wlson (1934). The EOQ s a smple optmal calculaton of the replenshment quantty based on two costs for an ndvdual tem n an nventory system; one replenshment and one nventory holdng cost. 6

16 When demand s not constant but can be determned over a set of dscrete tme perods t s tme-varyng. Ths means that demand s determnstc but can vary over tme. As wth the EOQ there are stll one ndvdual tem wth a replenshment and nventory holdng cost. Wagner and Whtn (1958) present an optmal soluton for ths problem. The method has been crtczed snce t s rather complex to understand and use, and more mportant because t s backwards lookng from a clear defned endng pont. Several heurstc methods have been suggested over the years. The most famous and best performng n terms of low cost solutons s the Slver-Meal heurstc from the publcaton of Slver and Meal (1973). The method s also known as least perod cost heurstc and s a forward lookng technque whch enables the method to work n combnaton wth Materal Requrements Plannng (MRP) systems. See Baker (1989) for a revew on the area and the test performance of dfferent methods where he states that the Slver-Meal outperforms other heurstcs. Many authors and publcatons have dealt wth stochastc demand models. Slver et al. (1998) deal wth the four common contnuous revew and perodc revew models. In contnuous revew models, the nventory poston s updated contnuously meanng that whenever the poston changes, t s nstantly vsble n the system. In perodc revew models, there s a fxed revew nterval when the nventory poston s updated whch makes the poston vsble only at the revew opportunty. One of the oldest contnuous revew models s the (r, Q) system When the nventory poston reaches the reorder pont r, a fxed order quantty Q s ordered. The order quantty s preferably calculated usng the EOQ formula. Another common system s the (s, S) system whch uses order-up-to S polces. Ths means that when ever the nventory poston reaches or passes the reorder pont s, an order of varable sze s placed to fll up the nventory to ts S-level. Ths system has an advantage over the (r, Q) snce t s possble to better take account of the sze of the customer order that reaches or passes the reorder pont. A specal case of the (s, S) system s the (s-1, S) system where a new order s mmedately placed as soon as the nventory poston drops below the order-upto S-level. The (s-1, S) system s used for products wth relatvely low demand, hgh value and low replenshment costs. As for perodc revew there are two common models. The frst and most smple model s the (R, S) system where there s an nspecton nterval of the nventory poston every R unts of tme. At every nspecton a varable order s placed to fll up the nventory to ts S-level. The other model s the (R, s, S) system whch s a combnaton of (s, S) and (R, S) systems. The nventory poston s checked 7

17 every R unts of tme but an order s only placed at the revew opportunty f the nventory poston s at the tme above the s-level. More complex nventory systems are so called mult-echelon nventory systems. An echelon n ths case s refereed to a sngle level n a supply chan. In multechelon nventory systems several stockng ponts s evaluated together comparng to sngle echelon control where each level s evaluated ndependently. Informaton on dfferent mult-echelon nventory systems can be found n Slver et al. (1998), Axsäter (2a), and Zpkn (2). There are dfferent systems of mult-echelon nventores. The most smple system s a seres system where two or more nventores are coupled. Fgure 2 llustrates such a system. Ths system can be encountered n dfferent stuatons. For nstance where the frst nventory holds the stock of a subassembly and the second nventory holds the fnal parts. The second nventory can n any stuaton be consdered as a customer of nventory one. Fgure 2. A seres system wth two nventores. In a dvergent dstrbuton system each stockng pont has at least one predecessor. An llustraton s gven n fgure 3. A typcal stuaton n practce s when a central warehouse supples goods to several retalers. Fgure 3. A general two-echelon dstrbuton system. A convergent assembly system s just the opposte of a general dstrbuton system. Each stockng pont has at least one mmedate successor. Ths s llustrated n fgure 4. 8

18 Fgure 4. An assembly system. General systems n a supply chan can of course be of more complex nature and be a combnaton of dfferent systems descrbed above. Fgure 5 llustrates a general system whch s a complex structure of all three basc systems. Fgure 5. A general system. Henrch and Schneewess (1986) present prevous research and a survey of methods that have been proposed to solve the mult-stage lot-szng problem. For treatng ths problem t s mportant to be aware of and dstngush between the basc product structures; seral structure, assembly structure, dvergent structure, and general structure. Suggested soluton methods are often dedcated to a specal structure. In ths thess the One-warehouse N-retaler problem s studed, whch belongs to dvergent systems. The problem s descrbed n chapter 4. 9

19 3. The Jont Replenshment Problem The Jont Replenshment Problem (JRP) has been a famous research topc for many years snce t s a common real-world problem that occurs n dfferent stuatons. For nstance when a group of tems are purchased from the same suppler or when a product after manufacturng s packaged n dfferent quanttes. See Goyal (1973a, 1974b) for defntons and descrptons of the problem. The problem occurs where there are two dentfed replenshment costs nvolved. One major replenshment cost when a famly of tems s replenshed. Even though the defntons major and mnor replenshment costs are wdely used, the major cost does not necessarly have to be bgger than mnor costs. The word major can be seen as an overarchng cost durng the replenshment. In each replenshment stuaton, the major replenshment cost does not depend on the number of tems that are replenshed. The mnor replenshment costs are ndvdual for each tem and wll only occur when a partcular tem s replenshed. For each tem there s of course an ndvdual demand rate and nventory holdng cost as well. There s an ndvdual nventory holdng cost for each tem n the famly as well. Slver et al. (1998) dscuss the cost savngs that can be acheved by coordnatng the replenshment of several tems. 3.1 Constant demand There are two dfferent types of strateges to deal wth the JRP and constant demand. Indrect groupng strateges and drect groupng strateges. When usng the former, jont replenshment opportuntes are scheduled at constant ntervals of tme and the quantty ordered of each tem s suffcent to last for an nteger multple of the basc tme perod. Ths s the most common approach to the problem. The strategy wth assumptons, notaton and formulas and the many solutons procedures s descrbed n In drect groupng strateges, dfferent tems that are economcally reordered together are grouped. Each group has ts own base perod tme. More on drect groupng strateges s found n Indrect groupng strateges When usng ndrect groupng strateges, all replenshments occur at constant ntervals of a base tme perod. The replenshment quantty of each tem must last an nteger multple of the basc tme perod. The major replenshment cost 1

20 occurs durng every base tme perod and the mnor replenshment costs occur when a partcular tem s ordered. When usng so called strct cycle strateges at least one tem n the famly must be replenshed durng each base tme perod. In the more relaxed cycle strateges a replenshment opportunty does not have to be used, meanng that n some problems there wll be empty replenshments. There are several assumptons for the orgnal determnstc JRP: The demand rate for each tem s constant and known. Holdng costs for all tems are at a constant rate per unt and unt tme Lead tmes are known and ndependent of the sze of the replenshment For the orgnal problem lead tmes are neglgble for all tems No shortages are allowed The entre order quantty s replenshed at the same tme There are no quantty dscounts on the replenshments The tme horzon s nfnte There s no ntal stock n nventory Notaton for the JRP: d A a h N T m TRC Demand rate n unts per unt tme for tem Major replenshment cost for the famly of tems Mnor replenshment cost for tem Inventory holdng cost per unt and tme unt for tem Number of tems n the famly Base tme perod, wthn zero, one or several tems can be replenshed The nteger number of T ntervals that the replenshment quantty of tem wll last Total cost functon The total cost functon of the JRP dependng on the tme perod and m-values: 1 TRC ( A T N 1 a ) T ( m N 1 d h m ) 2 (1) The tme perod can be found by dervng Eq. (1). N a T ( m, m,,, m ) 2( A ) 1 2 N m N m d h (2) 11

21 If Eq (2) s substtuted back nto Eq. (1) the total cost functon gven a set of m- values s the followng: TRC N a 2( A ) ( m N ( m, m,,, m ) 1 2 N m d h ) (3) The dffculty of the JRP s to fnd the best set of m-values. An early author to propose a heurstc method s Brown (1967). He uses an teratve procedure to fnd the set of m-values, where the startng soluton s when all tems are replenshed each base tme perod. Shu (1971) gves a method for determnng the jont replenshment of two tems. The fast movng tem s supposed to be replenshed every opportunty and he then determnes a relatve optmum orderng frequency for the slow movng tem. Goyal (1973a) and (1974a) ntroduce teratve procedures to fnd the set of m-values by usng an upper and lower bound for T. The bounds of optmal T gve the upper and lower bounds for each m-value. All possble combnatons are then evaluated and the combnaton havng the lowest cost s chosen. Goyal (1974b) presents an optmal soluton, thus the lowest possble cost soluton, whch s an enumeratve procedure wth a strct cycle polcy. The upper and lower bound for T s determned and all ntervals of T when m s unchanged are evaluated wthn these bounds. Determnng the lower bound of T s a dffcult task and the method does not guarantee the optmal soluton n all stuatons. See for nstance Andres and Emmons (1976). Goyal (1988a) modfes Goyal (1974b) become optmal by determnng a new lower bound. Slver (1976) ntroduces a dfferent smple approach where he modfes the economc orderng quantty (EOQ) to handle the coordnaton of several tems. The algorthm ranks the tem n ascendng order of a / h d. The tem wth the lowest rankng s replenshed every T. The m-values of the other tems are calculated as rounded nteger multple of the frst tem. Goyal and Belton (1979) modfy the way how to calculate the multples. Kasp and Rosenblatt (1983) suggest a combnaton of Slver (1976) and Goyal (1974a). The frst value of T s calculated usng Slver s method and then Goyal s teratve method s used to fnd the best set of m-values. Goyal (1988b) gves an teratve procedure wthout the use of an upper and lower bound for T and the m-value. Kasp and Rosenblatt (1991) present a heurstc method called the RANDmethod. In the method an upper and lower s determned for T and the nterval s dvded nto a number of equally spaced values. For each potental value of T the procedure the method n Kasp & Rosenblatt (1983) s appled. For larger numbers of the equally spaced values, the procedure s bascally a complete 12

22 enumeraton of the search procedure. Goyal and Deshmukh (1993) margnally mprove the method by provdng a better estmate for the lower bound for T. Van Ejs (1993) presents a smlar enumeratve optmal algorthm as Goyal (1988a). He defnes a new lower bound for T whch enables optmal solutons for any cycle polcy. Vswanathan (1996) ntroduces a new algorthm that reduces the computatonal effort requred by Goyal s and Van Ejs s optmal algorthms by mprovng the bounds of T n an teratve fashon. Wldeman et al. (1997) present a new approach where they use the Lpschtz optmsaton technque to solve the problem. The method fnds solutons wth an arbtrary small devaton from the optmal value. The procedure s farly complex but requres less computatonal effort than other procedures wth full enumeraton. Kasp and Rosenblatt (1985, 1991) test the performance of dfferent heurstc methods. The frst paper concludes that Kasp and Rosenblatt (1983) produce most optmal solutons, followed by Goyal (1974a), Brown (1967), Goyal and Belton (1979) and fnally Slver (1976). The second paper shows that the RAND procedure n Kasp and Rosenblatt (1991) performs best followed by Kasp and Rosenblatt (1983), Goyal (1988), Goyal (1974a), Goyal and Belton (1979) and Slver (1976). Vswanathan (22) tests dfferent optmal methods and shows that Vswanathan (1996) on average requres least processng tme. In general all optmal solutons and the most well performng heurstcs are rather complex to use. In most heurstcs the underlyng prncple s the same. Gven a tme nterval for the jont replenshments, optmal order frequences are determned. Then a new tme nterval s determned. These steps are repeated untl the soluton converges. Paper I presents a completely dfferent approach n dealng wth the Jont Replenshment Problem (JRP). The method s relatvely smple and solves the problem n an teratve procedure usng on a spread-sheet technque. The prncple of the recurson procedure s to fnd a balance between the replenshment and nventory holdng costs for the dfferent tems by adjustng the replenshment frequences. The heurstc performs well n comparson wth many other heurstcs, especally concernng number of optmal solutons reached Drect groupng strateges In drect groupng strateges, dfferent tems that are economcally reordered together are grouped. Ths means that tems are collected to dfferent groups whch are treated ndependently. Each group has ts own base perod tme and all tems wthn the group are replenshed together. The challengng ssue of drect groupng strateges s to dvde the number of tems nto a certan number of dfferent groups, snce there can easly be a large amount of combnatons to 13

23 consder. Chakravarty (1981) proves a theorem that he called consecutveness property. Ths means that the tems are rearranged accordng to decreasng order of a / hd. Usng the same method of rankng the tems, several authors ncludng Page and Paul (1976), Chakravarty (1981, 1985) and Bastan (1986) present dfferent algorthms wth drect groupng strateges. Van Ejs et al. (1992) compare drect and ndrect groupng strateges on dfferent problems. They conclude that the ndrect groupng methods produce lower cost solutons than drect groupng n cases where the major replenshment cost s large relatve to the mnor replenshment costs. It should be noted however, that the heurstcs tested for ndrect groupng strateges were older ones. Olsen (25) uses an evolutonary algorthm to solve the problem and whch slghtly outperform the RAND method n Kasp and Rosenblatt (1991) Extensons and varants Goyal (1973b, 1974c) present methods when there are a known frequency of replenshment orders. Goyal (1975a) solves the JRP when there s a fnte horzon. Goyal (1976) presents a model where the tems have dependent replenshment costs. Goyal (1977) deals wth the problem when the tems are pershable. Goyal (198) suggests a method when the tems have more than one brand name. Khouja et al. (25) develops a method when the tems are under a contnuous unt cost change. Jackson et al. (1985) ntroduce a so called Power-of-Two (PoT polcy) for the Jont Replenshment Problem. Ther PoT polcy s the usage of a power-of-two multples nstead of nteger values. They show that the cost penalty for such an approach s wthn 6 % of the orgnal problem. Roundy (1986) and Federgruen and Zheng (1992) present better methods that guarantee solutons wthn 2 % of the optmal value. Lee and Yao (23) derve a global optmum search algorthm wth the use of the PoT polcy. Harga (1994a) presents two new heurstc methods wth a relaxaton of the problem, where the reorder ntervals do not need to be an nteger multple of the base tme perod. Harga (1994b) shows that the second of the two heurstcs n Harga (1994a) outperforms the modfed RAND-procedure n Goyal and Deshmukh (1993). Ths s however not a far comparson snce the RAND procedure solves the JRP wthout any relaxaton. Goyal (1975b) adds a constrant to the problem of how much that can be ted up n stock. Hoque (25) presents an optmal soluton method wth constrants 14

24 regardng storage, transport and budget. Porras and Dekker (25) provde an optmal soluton method for the Jont Replenshment problem wth mnmum order quanttes. Involvng constrants that changes the m-values for a problem may lead to a hgh cost penalty. Paper II presents a smple method that often can help reduce peak nventory level and arrval quanttes. It s based on three steps, where the replenshments are re-schedule durng the cycle perods and f necessary delayed one or more tme perods where there are a large number of replenshments at the same tme. The method ncurs ether zero or a very small cost penalty from the optmal soluton of the orgnal Jont Replenshment Problem. 3.2 Stochastc demand Stochastc jont replenshment systems can be dvded nto contnuous and perodc revew models. As most papers proposng perodc revew systems have also compared the methods to contnuous system, all remarks on performance are n chapter For both systems there are a few assumptons nvolved: Lead tmes are assumed to be determnstc or neglgble. The entre order quantty s replenshed at the same tme. Holdng costs for all tems are at a constant rate per unt and unt tme. There are no quantty dscounts on the replenshments. The horzon s nfnte Contnuous revew systems The most common strategy n contnuous revew system s the (S, c, s) system, also known as can-order polces. As a dfference from sngle product re-order systems, there are three computed levels n the (S, c, s) system. All products n the famly have three ndvdual levels n the re-order system. When any product passes ts s-level, t trggers a general replenshment order whch wll nclude at least that partcular tem. All other tems that are below ts ndvdual c-level wll then also be ncluded n ths replenshment. The remanng tems that are above ts c-level wll not be ncluded. The order quantty s varable and when an tem s replenshed t wll be ordered-up-to ts ndvdual S-level. 15

25 Fndng the control varables for all three levels s a dffcult task and many research papers n ths area have therefore dealt wth dfferent ways to calculate the dfferent levels. The frst author to propose the (S, c, s) system s Balntfy (1964) who calls t the random jont order polcy. He assumes that the dstrbuton nvolved s negatve exponental. Slver (1965) consders a smple problem when there are only two tems havng dentcal cost and demand, the latter accordng to a Posson process. Ignall (1969) also deals wth a two product problem where demand s determned by two ndependent Posson processes. Slver (1973) analyzes three dfferent procedures to obtan the same total cost functon of the problem when havng Posson demand and zero lead tmes. Slver (1974) deals nstead wth constant lead tmes. He shows that by usng the (S, c, s) system t s possble to obtan substantal cost savngs n comparson wth ndvdual orderng polces. Thompstone and Slver (1975) fnd the cost expresson wth compound Posson demand and zero lead tmes. The followng authors suggest dfferent methods to fnd control varables when havng compound Posson demand and non-zero lead tmes; Shaack and Slver (1972), Slver (1981), Federgruen et al. (1984), Schultz and Johansen (1999) and Melchors (22). Much of the research has focused on the (S, c, s) system, however a few authors have dealt wth system wth fxed order quanttes. Smons (1972) uses an teratve procedure to calculate optmal order quanttes and control varables when the dstrbuton functons for the demand are known. Hartfel and Curry (1974) suggest an alternatve soluton procedure to Smon s paper. Chern (1974) present an nventory model based of an ndependent re-orderng polcy wth Posson demand Perodc revew systems Svazlan (197) proposes models on so called mxed orderng polces, where replenshment opportuntes nvolves zero, one or several tems. Naddor (1975) present a heurstc model for a perodc revew mult-tem nventory system. Atkns and Iogyun (1988) propose two replenshment polces. 1) A perodc polcy where all tems are ordered up to the base stock level n every replenshment opportunty. 2) A modfed perodc polcy where tems belongng to a base set are ordered every replenshment opportunty and other tems are ordered once n a certan number of replenshment opportuntes. A lower bound s also suggested by allocatng the major replenshment to the dfferent tems. 16

26 The modfed perodc polcy outperformed the (S, c, s) system n a number of test problems. Pantumsncha (1992) suggests the use of a what he a so called QS-polcy, whch was orgnally studed by Renberg and Planche (1967). The prncple of the QSpolcy s that when the aggregated consumpton snce the prevous order reaches a certan amount, all tems are ordered up to ther base stock level. It performs well n comparson wth the (S, c, s) system when there s a small number of tems wth smlar demand and havng a hgh major replenshment cost. Vswanathan (1997) presents what he defnes as a P(s, S) polcy. At every revew a (s, S) polcy s appled to each tem, meanng that any tem havng an nventory poston below t s-level s ncluded n the replenshment. In the paper he shows that the P(s, S) polcy outperforms earler approaches on the majorty of the test problems. Johansen and Melchors (23) present a method to calculate can-order polces for a perodc revew nventory system. Ther method performs well n comparson to other perodc revew approaches. When the demand s rregular t works especally well. Nelsen and Larsson (25) mprove the P(s, S) polcy by workng out an analytcal soluton procedure. The method s shown to be superor the (S, c, s) system on almost all of the test problems evaluated n the paper. 17

27 4. The One-warehouse N-retaler problem In the One-warehouse N-retaler problem there are N number of retalers that are served from a central warehouse. All facltes hold stock. Demand takes place at the retalers who order the goods from the warehouse when the nventory poston reaches the reorder pont. The warehouse s suppled from an outsde suppler (e.g. factory). See fgure 6 for a descrpton of the One-warehouse N- retaler problem. As wth the Jont Replenshment Problem dfferent models demand can be categorzed usng ether constant or stochastc demand. The ssue wth One-warehouse N-retaler s to fnd models that can fnd low cost solutons for the entre system. L Warehouse L 1 = Lead tme retaler 1 L 2 L N 1 2 N Retalers d 1 = Demand retaler 1 d 2 d N Fgure 6. The One-warehouse N-retaler problem. 4.2 Constant demand The One-warehouse N-retaler problem wth constant demand s related to the Jont Replenshment problem. Graves (1979) dscusses the smlartes regardng 18

28 cost functons and solutons procedures for the Jont Replenshment Problem, The Economc Lot Schedulng Problem (ELSP) and the One-warehouse N- retaler problem. The assumptons for the Jont Replenshment Problem apply to the Onewarehouse N-retaler problem n the same manner, see secton 3.1. A author to suggest complete soluton methods to the One-warehouse N-retaler problem s Schwarz (1973). He shows that a well performng polcy may be very complex snce the order quanttes must vary wth tme even though demand s constant. He ntroduces two restrcted basc polces, an optmal sngle cycle polcy and a so called myopc polcy. The sngle cycle polcy s nested and statonary, whch means that the ordered quanttes are the same and the retalers are restrcted to order whenever the warehouse places an order. The myopc polcy dvdes up the system nto N number of one-warehouse one-retaler systems. Further more; he also suggests a heurstc method for the case when there s one-warehouse and N number of dentcal retalers. Graves and Schwarz (1977) conduct a smlar analyss for dvergent systems, where each stage orders the goods from a predecessor. They fnd optmal sngle cycle polces by usng a branch-and-bound algorthm and examne the myopc polces regardng closeness to optmalty. Muckstadt and Snger (1978) contrbute wth correctons to the above paper. Maxwell and Muckstadt (1985) present a heurstc for a nested and statonary system that produces solutons wthn 6 % of the optmal sngle cycle polcy. The retalers order a Power-of-Two (PoT) multple of a base tme perod, whch s the tme nterval between each replenshment to the warehouse. Roundy (1985) argue that the optmal sngle cycle polcy can result n a hgh cost penalty from the general optmal value. He proposed a dfferent approach wthout the use of a nested and a statonary assumpton. As n Maxwell and Muckstadt s paper the retalers order a Power-of-Two multple of a base tme perod. If the tme perod s fxed the method guarantees a soluton wthn 6 % of optmal value and f the tme perod s varable t guarantees a soluton wthn 2 %. Lu and Posner (1994) present a method gvng the same results but requre less computatonal effort. Abdul-Jalbar et al. (25) derve a soluton method to fnd near optmal solutons usng nteger multples nstead of Power-of-Two multples. Muckstadt and Roundy (1987) present an algorthm that coordnates the tems and levels n the problem and guarantees solutons wthn 6 % of the optmal nested polcy. 19

29 4.3 Stochastc demand In stochastc demand stuatons generally (r, Q) or (s-1, S) polces are used, see chapter 2 for a descrpton of dfferent re-order systems. The assumptons are smlar to the ones for the stochastc Jont replenshment problem: The entre order quantty s replenshed at the same tme. Holdng costs for all tems are at a constant rate per unt and unt tme. Lead tmes may be consdered to be ether determnstc or stochastc. There are no quantty dscounts on the replenshments. The horzon s nfnte. There are no lost sales due to shortage of supply. The frst contnuous models were wth a so called (s-1, S) polcy. As soon as a demand takes place at a retaler a new tem s mmedately placed at the warehouse. Then nstantly as the warehouse receves the request from the retaler, t orders a new unt from the outsde suppler. All orders that can not be delvered from stock are backlogged at both the warehouse and the retalers. Whenever the warehouse suffers from stock-outs and have orders backlogged, the retalers are served on a frst-come-frst-served bass when the warehouse receves new goods agan. Most of the studes are based on so called nstallaton stock polces,whch means that every nstallaton s calculated ndvdually. A few later research papers consder echelon stock polces, whch means that two or more nstallatons are calculated together downwards n the supply chan. All research papers are under contnuous revew f not mentoned otherwse. Sherbrooke (1968) s the frst author to propose a soluton method, called the METRIC approxmaton, whch gves the nventory holdng and shortage costs at the retalers. The model has been shown to underestmate the shortage costs at the retalers when there s a shortage of supply at the warehouse. Smon (1971) fnds the exact steady-state dstrbutons at the retalers and Shanker (1981) shows the model wth compound Poson demand. Graves (1985) presents the exact calculatons of the costs at the retalers. That requres a hgh computatonal effort to do, so he provdes a two-parameter approxmaton model as well. On the contrary to the METRIC approach, Grave s model overestmates the shortage costs at the retalers. However, he shows that the over all performance s much better. Svoronos and Zpkn (1991) consder a more general model wth stochastc transportaton lead tmes. 2

30 Axsäter (199) suggests a completely dfferent approach to exactly calculate the nventory and shortage costs at the retaler by usng a recursve procedure. The method requres much less computatonal effort than the method by Graves. Forsberg (1995) extends the former model and presents an exact recursve method wth compound Posson demand, whch means that the ordered sze throughout the system s a stochastc nteger number. As for (r, Q) systems, Deurmeyer and Schwarz (1981) present a decomposton technque where they assume all retaler to be dentcal and have the same order quantty. They approxmate the demand at the warehouse to be a Posson process wth demand quantty Q. Monzadeh and Lee (1986) and Monzadeh (1987a, 1987b) suggest better approxmaton models that are closely related to Graves (1985). They drop the assumpton that the retalers are dentcal but use the same order quantty. The demand process at the warehouse s approxmated to be a Posson process whch works well for systems wth a large number of retalers. Svoronos and Zpkn (1988) present a related approxmaton where they assumed demand to be a shfted Posson process at the warehouse. Axsäter (1993a) derves the exact and approxmate cost functons for the nventory levels n a system wth dentcal retalers and compound Posson demand. Axsäter (1993b) presents a smlar procedure as Axsäter (199) but wth a batch orderng polcy under perodc revew. Axsäter (1995) provdes an approxmaton procedure for non dentcal retalers under compound Posson demand. Axsäter (1998) s a generalzed model of Axsäter (1993) whch consders two non-dentcal retalers. Forsberg (1997a) s an extenson to handle N number of retalers. Forsberg (1997b) s a further extenson and generalzed model that uses Erlang dstrbuted customer nter-arrval tmes at the warehouse. Axsäter (2b) fnally presents a model where he provdes the complete probablty dstrbutons for the retaler nventory levels and permts the usage of compound Posson demand. Chen and Zheng (1997) consder echelon stock polces and present a model that gves the costs for a system wth non-dentcal retalers. The model s exact wth Posson demand and an approxmaton wth compound Posson demand. Axsäter (1997) suggests an alternatve approach where he evaluates the exact costs for the compound Posson case as well. Dada (1992) uses prorty nventory poolng to expedte servce when the warehouse s out of stock. Axsäter et al. (24) present a model that sets dfferent servce levels for the retalers and the drect customer demand. 21

31 Paper III presents an approxmaton procedure to prortze retalers n a twoechelon nventory system when there s a shortage of supply at the warehouse. Whenever there s a shortage at the warehouse the retalers are prortzed accordng to two groups, hgh and low prorty retalers. The model uses Posson demand and a one-for-one replenshment polcy from the warehouse. The approxmaton procedure wll n any stuaton gve a more accurate descrpton of the performance of the system n comparson wth the exact frst-come-frst served calculatons where retalers are not prortzed. 22

32 5. Summary and Extensons Ths lcentate thess contans two research papers on the Jont Replenshment Problem and one research paper on the One-warehouse N-retaler problem. The frst two papers consder constant demand and the last paper consders stochastc demand. In ths thess, earler research on the area has been presented and t s clear that both problems have extensvely been studed throughout the years. The reason for ths s the practcal relevance of both problems. When mplemented, cost savngs can be reached n comparson wth more basc models. The three papers are related under determnstc demand as ponted out by Graves (1979). Together however they contrbute to the general scentfc dscusson by provdng new deas to Inventory Management and Supply Chan Management. As dscussed n chapter one, there are several trends n logstcs and Supply Chan Management. The research papers are mportant for two of these trends; reduced number of supplers and mprovng customer servce. The frst two research papers wll help the coordnaton of procurng several tems from the same suppler. An mportant ssue snce companes rely on fewer and fewer supplers, see Waters (23). All three research papers are mportant for obtanng hgher servce levels at lower costs. Paper I presents a novel method to solve the Jont Replenshment Problem usng a spread-sheet technque. The heurstc s well performng and also an argued advantage s the relatve smplcty of the model n conjuncton wth a spreadsheet program. Future extenson may be to mprove performance by dervng a dfferent value for step 1 nstead of usng a fxed value for any problem. Paper II ntroduces a new dea that may help reduce peak nventory levels and arrval quanttes n the Jont Replenshment Problem. Ths can be done wthout usng a constrant to a very low cost. Future work may be to come up wth a model that s less computatonal heavy and the development a stochastc model. Interestng would also be to develop a model that uses Power-of-Two multples and compare the results wth the model presented n the paper. Paper III extends the work on the One-warehouse N-retaler problem. The paper presents an approxmaton method to prortze retalers accordng to hgh- and low prorty retalers. A sgnfcant subject snce n practcal stuatons all customers or markets are often not equally mportant. Future work could be dedcated to mprove performance and extend the model to handle more numbers of prorty groups. 23

33 6. References Abdul-Jalbar B., Guterrez J. and Scla J. 25. Integer-rato polces for dstrbuton/nventory systems. Internatonal Journal of Producton Economcs 93-94, Aley J. (1995). Inventores won t kll growth. Fortune (June 26), Andres F.M., and Emmons H On the optmal packagng frequnecy of jontly replenshed tems, Management scence 22, Atkns D. and Iyogun P.O Perodc versus can-order polces for coordnated mult-tem nventory systems. Management Scence 34, Aschner A Managng and Controllng Logstcs Inventores. In The Gower Handbook of Logstcs and Dstrbuton management, (ed.) J. Gattorna, Aldershot: Gower, Brookfeld. Axsäter S Smple soluton procedures for a class of two-echelon nventory problems. Operatons Research 38, Axsäter S. 1993a. Exact and approxmate evaluaton of batch-orderng polces for two-level nventory systems. Operatons Research 41, Axsäter S. 1993b. Optmzaton of order-up-to-s polces n two-echelon nventory systems wth perodc revew. Naval Research Logstcs Quarterly 4, Axsäter S Approxmate evaluaton of batch-orderng polces for a onewarehouse, N non-dentcal retaler system under compound Posson demand. Naval Research Logstcs 42, Axsäter S Smple evaluaton of echelon stock (R, Q)-polces for twolevel nventory systems. IIE Transactons 29, Axsäter S Evaluaton of nstallaton stock based (R, Q)-polces for twolevel nventory systems wth Posson demand. Operatons Research 46, Axsäter S. 2a. Inventory Control. Kluwer, Boston. 24