Evaluating Clustering Methods for Multi-Echelon (r,q) Policy Setting

Transcription

1 Proceedngs of the 2007 Industral Engneerng Research Conference G. Bayraksan W. Ln Y. Son and R. Wysk eds. Evaluatng Clusterng Methods for Mult-Echelon (r) Polcy Settng Vkram L. Desa M.S.; Manuel D. Rossett Ph.D. P.E. Department of Industral Engneerng Unversty of Arkansas 4207 Bell Engneerng Center Fayettevlle AR 7270 Abstract In ths research we develop a segmentaton methodology for reducng the computatonal tme requred for settng the (r ) nventory control polces n a large-scale mult echelon nventory system. The segmentaton methodology uses a clusterng algorthm to compress the nventory data nto clusters of dentcal SKUs at each nventory control pont and uses the reduced dataset of clusters for optmzng the control polces. Whle the segmentaton methodology reduces the computatonal tme requred for settng the nventory control polces n a mult echelon nventory system t also nduces a penalty cost n the form of ncreased nventory nvestment due to loss of dentty for SKUs whle clusterng. We tested the segmentaton methodology on dfferent mult echelon nventory system scenaros to understand the degree to whch the computatonal tme can be reduced and to understand the effect of the resultng penalty cost. Keywords Inventory polcy settng mult-echelon clusterng. Introducton and Background A sngle nventory control pont n a mult echelon nventory system stocks a wde range of stock keepng unts (SKUs) for a vared customer base. The demand rate across the SKUs may vary from as lttle as unt per year to thousands of unts per year. Whereas the lead tmes for procurng these SKUs could vary from few days to months. Settng nventory stockng polces for such a dverse set of SKUs held at dfferent nventory control ponts n order to provde the desred servce level at the lowest possble cost s a key am for many organzatons. The stochastc demand stochastc falures stochastc lead-tmes and repar tmes of the SKUs makes modelng a mult echelon nventory system for settng nventory stockng polces extremely dffcult especally when we are dealng wth a large-scale nventory system have hundreds of thousands of SKUs. Organzatons havng mult echelon supply chans vz; tech companes lke IBM Hewlett-Packard; retalers such as Wal-Mart Target Albertsons; government agences lke the US avy US Ar-Force etc set stockng polces for ther mllons of SKUs held at dfferent echelons usng commercally avalable or n-house bult nventory optmzaton software solutons. Due to the complex nature of the nventory control polcy settng algorthms n ths software the total computatonal tme requred to set polces ncreases as the number of SKUs ncreases. For a typcal mult echelon nventory system consstng of multple depots and multple bases wth thousands of SKUs stocked at each locaton lke the one mantaned by the US avy the total computatonal tme for settng the nventory control polces usng mult echelon nventory optmzaton software s n hours. Consderng the number of busnesses around the globe mantanng mult echelon nventory systems and usng mult echelon nventory optmzaton software solutons there s a serous need to brng down the computatonal tme from hours to mnutes. Ths research examnes the use of clusterng technques n combnaton wth polcy settng algorthms to reduce the computatonal tme for mult echelon nventory optmzaton software solutons. The obectve of mult-echelon nventory optmzaton s often defned n terms of mnmzng the system nventory level subect to achevng a desred customer servce level. Mult- echelon nventory optmzaton models examne the entre system searchng for better solutons for the entre chan not each stage ndependently. Ths coordnaton 352

2 has the advantage of achevng a better global soluton. A full revew of mult-echelon polcy settng algorthms s beyond the scope of ths paper. We refer the nterested reader to Al-Rfa and Rossett (2007) and the references theren for an up to date lterature summary. Work on combnng clusterng and polcy settng s lmted. The prmary paper n ths area s the work presented by Ernst and Cohen (990). Whle developng a model for settng optmal nventory stockng polces for a maor automoble manufacturer that stocked over part-types n an extensve network wth approxmately 50 dstrbuton centers and thousands of dealer locatons Ernst and Cohen (990) developed a segmentaton methodology called ORG (Operatons Related Groups) as they realzed the advantage of groupng the part-types based on operatonally relevant attrbutes and defnng generc group based polces for controllng nventory could be substantal. Rossett and Achlerkar (2004) llustrated the applcaton of clusterng technques at a sngle large-scale nventory locaton. In what follows we descrbe our basc methodology ncludng a bref overvew of our own polcy settng algorthm. Then we present a set of experments to llustrate the effect of clusterng on the polcy settng procedures. Fnally we descrbe on-gong and future work n ths area. 2. Methodology Wthout loss of generalty we consder a mult-echelon nventory system as one consstng of two echelons. The lowest echelon conssts of bases that experence drect demand from the customers whereas the hghest echelon conssts of depots whch fll the orders placed by the bases n the lowest echelon. We assume that a depot can support multple bases but each base s supported by one and only one depot and that there s no lateral transshpment between bases or depots. The bases are non-dentcal.e. a part-type stocked n dfferent bases say Base A and Base B can have dfferent attrbutes vz. dfferent annual demand lead-tmes etc. A contnuous revew (r ) control polcy s assumed at both the echelons. Our soluton methodology uses a clusterng algorthm for reducng the computatonal tme requred for settng nventory control polces n mult echelon nventory systems. Clusterng s a data-mnng tool used n varous applcatons to dentfy patterns compress the data or segment the data. The clusterng algorthm employed n the soluton methodology groups the SKUs wth smlar attrbutes (Annual demand Unt Cost Lead Tme etc) as a sngle pseudo-tem. Ths reduces the nventory data of hundreds of thousands of SKUs nto few thousands of pseudo-tems for use n mult echelon nventory optmzaton software thus reducng the computatonal tme. Wth ths soluton methodology nstead of settng stockng polces for each SKU at all the locatons ndvdually the polcy makers would have to deal wth a reduced number of pseudo-tems. The segmented nventory data generated by ths soluton methodology would also empower the decson makers wth more control over all the SKUs for strategc analyss as the number of SKUs would be reduced from hundreds of thousands to very few pseudo-tems. Although clusterng of nventory has several advantages from a manageral perspectve there s a trade-off nvolved. A penalty cost s expected for groupng the SKUs nto pseudo-tems due to loss of dentty as the generc groupbased polcy (reorder pont reorder quantty) s appled to the SKU nstead of the SKU s optmal control polcy. Ths penalty cost s reflected n the total nventory nvestment of the system. Also some dscrepancy n the servce levels of the SKUs s expected due to loss of dentty. In ths research we experment to see the effects of clusterng the nventory data on the entre supply chan s performance metrcs. Our am s to understand the amount of reducton n computatonal tme that could be attaned through the use of clusterng algorthm and ts effects on the total nventory cost average system fll rate total system backorders and average system order frequency. The results obtaned from the experments wll help n dentfyng the factors to be used for clusterng the nventory dataset. Ths may allow a reducton n the computatonal tme at the same tme mantan the penalty cost wthn acceptable lmts. Out man contrbuton s our overall methodology whch s a combnaton of well establshed approaches. Due to space lmtatons we wll only brefly descrbe our mult-echelon nventory model ts nteracton wth our segmentaton methodology and some ntal expermental results. 2. Mult-Echelon Inventory Model To set stockng polces for SKUs before clusterng and for pseudo-tems after clusterng durng our expermental procedure we have developed a mult echelon nventory optmzaton model. The model we have formulated s based on Duermeyer and Schwarz s (98) model to evaluate performance measures of a mult- echelon nventory system. They assume a generc (r ) polcy at both retalers and warehouses as s the case n most of the real world 353

3 nventory systems and as opposed to (S- S) base-stock polces pervasve n the lterature of mult-echelon nventory systems whch s a specal model for low demand and hgh cost parts. Our model assumes a contnuous revew (r ) polcy at both the depots and the bases. Our argument s that not all SKUs stocked n a mult echelon nventory system have hgh cost low demand characterstcs to apply base-stock polcy across all the SKUs. A basestock polcy s the specal case of (r ) polcy where reorder quantty = and reorder pont(r) s base stock level(s) -. Hence f a SKU has a Low Demand Hgh Cost characterstc the generc (r ) polcy wll automatcally assgn t a base sock polcy wth =. The generc (r ) mult-tem mult echelon nventory optmzaton model s gven as follows: Set of bases at the lowest echelon { M} k Set of depots at the hghest echelon { L} Set of part-types stocked n the system { } C Unt cost of part-type l Demand Rate of part-type at base l Demand Rate of part-type at depot k F k Target average order frequency at depot k F Target average order frequency at base B Target backorder level at base B Target backorder level at depot k k ( r k ) k ( r ) ( r k ) k ( r ) I Avg. nventory of part-type at depot k I Avg. nventory of part-type at base B Avg. backorder level of part-type at depot k B Avg backorder level of part-type at base L M Mn C I ( r ) + C I ( r ) s.t. å = å B = å B = å å k = = l å = l k = = F F å å k " k =... L " =... M ( r ) B " k L k =.. ( r ) B " M =.. r r 0 ³ k > r Reorder pont for part-type at depot k Reorder quantty for part-type at depot k r Reorder pont for part-type at base Reorder quantty for part-type at base In order to be more consstent wth real world mult-echelon nventory systems we allow non-dentcal bases at the lowest echelon meanng each base at the lowest echelon stocks vared number of part-types and common part-types have dfferent characterstcs and attrbutes at dfferent base locatons. The customer-demand for part-type at base s stochastc and assumed to be Posson wth mean l. As the nventory poston reaches r an order of sze s placed at the respectve depot k. The base lead tmes are assumed to be the sum of two components: fxed transportaton tme and average backorder watng tme. Any order that s not met at the base s backordered and s fulflled on frst come frst serve (FCFS) bass. We approxmate the average backorder watng tme usng the METRIC (Sherbrooke 986) approxmaton as s commonly done n the lterature. At the depot f the on-hand nventory s greater than or equal to the demanded order quantty the order s fulflled or else the entre order s backordered as no order-splttng s allowed. As the nventory poston at the depot reaches r an order of sze s placed at the depot s suppler whch s receved after a constant transportaton tme.e. depot lead tme as we assume that the suppler s wth nfnte resource and replenshes the depot mmedately. Constrants are ncluded on the average order frequency and the total backorder level for depots and bases respectvely. The (r ) polcy settng model s an ntegral part of our expermental procedure and serves as a tool for comparatve analyss to test the affects of clusterng the SKUs on the performance measures of the system. We do not am at developng an optmzaton model that would set global optmal stockng polces for SKUs n a mult echelon nventory system. For our purposes an nventory control polcy settng model generatng near-optmal soluton s suffcent. From ths research s pont of vew the (r ) polcy settng model s solely used for comparatve analyss of the performance measures before and after clusterng. The followng secton explans our methodology to set stockng polces at each locaton n the echelon. k 354

4 2.2 Segmentaton Methodology The step-by-step procedure for settng stockng polces n the entre network usng clusterng concept for reducng the computatonal tme s as follows: ) For each base { 2 M} n a network compute the ntal values for order quanttes for each parttype Î { K} based on the average order frequency constrant F at that base = l F " = to and " = to M 2) Compute the expected lead-tme demand (n terms of orders) and varance of the lead tme demand for each part-type at depot k of the network. See Desa (2006) for the detaled formulas. 3) Use SAS s FastClus procedure (K Mean clusterng algorthm) to compress the depot s part-type dataset nto the desred number of cluster (pseudo tems) n terms of percentage of actual dataset. The followng attrbutes of the part-types stocked at the depot are used for clusterng: unt cost demand rate depot lead tme base locaton ndex. 4) Set the attrbutes for pseudo tems (clusters) based on the Group Polcy Decdng Crtera (Mn Avg Max). If the Group polcy decdng crtera at a partcular desgn pont of the expermental setup s Average then the attrbutes of the cluster (pseudo tem) wll have the average attrbute value of all the part types n the cluster. 5) Set the stockng polces (r k k ) for all the pseudo tems (clusters) at the depot K Î {2 L} of the network usng Mult-Product Backorder Model (Hopp Spearman and Zhang 995) so as to mnmze the total nventory cost subect to the depot servce constrants. 6) Apply the generc group based nventory control polces of the pseudo tems (clusters) to all the part-types nsde t. Expand the clustered dataset back to the orgnal dataset. 7) Compute the average backorder watng tme BWT for all the part-types stocked n depot k of the network under consderaton. See formulas n Desa (2006) 8) The (r k k ) polces set for pseudo tems (clusters) at depot k are n terms of orders and not n terms of unt tem. In order to set polces n terms of unt tem we multply both reorder pont (r k ) and reorder quantty ( k ) wth the smallest of the order quantty Mn( r receved at the depot from all the bases for part-type ) 9) Compute the effectve base lead tme L for an order placed by base for part-type by ncludng the expected backorder watng tme BWT for part-type at depot K 0) Use SAS s FastClus procedure (K Mean clusterng algorthm) to compress the base s SKU dataset nto the desred number of cluster n terms of percentage of actual dataset. ) Set the attrbutes for Pseudo tems (clusters) based on the Group Polcy Decdng Crtera (Mn Avg Max). If the Group polcy decdng crtera at a partcular desgn pont of the expermental setup s Average then the attrbutes of the cluster (pseudo tem) wll have the average attrbute value of all the SKUs n the cluster. 2) Wth the updated lead-tme L set the stockng polces (r ) for all the part-types stocked at base n the network usng Mult-Product Backorder Model (Hopp Spearman and Zhang 995) so as to mnmze the total nventory cost subect to the base servce constrants. 3) Apply the generc group based nventory control polces of the pseudo tems (clusters) to all the SKUs nsde t. Expand the clustered dataset back to the orgnal dataset. 4) Repeat step -3 for all the bases n the network under consderaton Wth optmzed base order quanttes( ) n Step Iterate -4 by updatng effectve base lead-tme L at each teraton untl the effectve lead tme L for 75% of SKUs n the network converges wthn an absolute percentage dfference of 0.0 from ts value of the prevous teraton. Ths methodology allows the user to defne the number of clusters he/she would lke the dataset to be reduced to for settng the nventory control polces. The computatonal tme requred for settng the nventory control polces by usng clustered dataset through ths segmentaton methodology would be consderably less than the computatonal tme requred usng the (r ) polcy settng model. 3. Illustratve Experments In ths secton we brefly dscuss the results of applyng the methodology overvewed n Secton 2 to a set of llustratve cases. We expermented wth varous mult echelon scenaros varyng the number of SKUs stocked at each locaton and the number of bases supported by a sngle depot whle observng the response varables Reducton n Tme (% dfference) and Penalty Cost (% dfference). In order to dentfy the levels of the clusterng factors that would reduce the computatonal tme for settng the nventory control polces whle mantanng the penalty cost wthn acceptable lmts we set up a full factoral 355

5 expermental desgn wth two factors vz; Group Polcy Decdng Crtera and Clusters (%). The factor Clusters (%) ndcates the percentage to whch the nventory dataset at each nventory control pont s reduced to after usng the segmentaton methodology. Factor Group Polcy Decdng Crtera determnes how the attrbutes of the pseudo tems (Clusters) formed are assgned the value after the dataset s clustered. Based on the Group Polcy Decdng Crtera set at each level of the experment the attrbute of a pseudo-tem can take the mnmum maxmum or average attrbute value of all the SKUs wthn the cluster. Fgure shows the average penalty cost observed at varous levels of factor Clusters (%) wth the Group Polcy Decdng Crtera factor set at Mean. We observed that clusters (%) set at 60% results lowest penalty cost of 7.50%. Penalty Cost (% Dfference) for Group Polcy Decdng Crtera "MEA" Penalty Cost (% Dfference) 20.00% 5.00% 0.00% 5.00% 0.00% Mean =5.72 % Mean =.04 % Mean =7.50 % 40% 40% 40% 50% 50% 50% 60% 60% Clusters (%) Fgure : Penalty Cost (% Dfference) for Group Polcy Decdng Crtera "MEA" Our experments revealed that as the Factor Cluster (%) decreases the total computatonal tme requred for polcy settng also decreases whle mantanng the Order Frequency constrant and Total Backorder constrant at the bases and the depot. On average a reducton of 55.92% n computatonal tme can be acheved when the nventory dataset at each locaton s reduced to 40% of ts orgnal sze usng the segmentaton methodology. Table llustrates ten experments ran on dfferent mult echelon scenaros wth factor Clusters (%) set at 40%. Table : Reducton n Tme (% Dfference) wth Clusters (%) = 40% Scenaro Cluster (%) Group Polcy Decdng Crtera Reducton n Tme (% Dfference) 40% Maxmum 5.29% 40% Mean 5.69% 2 40% Maxmum 64.37% 2 40% Mean 64.30% 3 40% Maxmum 48.26% 3 40% Mean 45.76% 4 40% Maxmum 62.60% 4 40% Mean 64.2% 5 40% Maxmum 47.30% 5 40% Mean 47.46% 4. Conclusons and Future Research Wth the results observed through expermentaton we can conclude that the segmentaton methodology s a potent tool for reducng the computatonal tme requred for settng the (r ) nventory control polces n large and dverse mult echelon nventory systems. We also observed that the segmentaton methodology does not negatvely affect the supply chan performance metrcs. Wth the clusterng algorthm and the factors explored n ths research we showed that the computatonal tme can be reduced by about 55.92% by usng the segmentaton methodology. But 356

6 wth ths reducton n the computatonal tme we also observed a penalty cost of about 5.72% for usng the segmentaton methodology. The penalty cost (% Dfference) resultng from usng the segmentaton methodology s not drastcally hgh but we recommend future work to further brng down the resultng penalty cost. For future research we recommend the exploraton of addtonal clusterng algorthms so as to use the most sutable algorthm and mprove the qualty of groups obtaned usng the segmentaton methodology. For expermentaton we explored Maxmum Mean and Mnmum as group polcy decdng crtera for the clusters formed usng the segmentaton methodology. Where Mnmum faled to mantan the targeted performance constrants for most of the scenaros we observed the smallest penalty cost for usng the segmentaton methodology was obtaned usng Mean as the group polcy decdng crtera. We recommend expermentng wth Mode and Medan as group polcy decdng crtera for the clusters formed usng the segmentaton methodology to see ts effects on the penalty cost. The attrbutes selected to cluster the SKUs at the base; unt cost demand rate base lead tme and depot lead tme are based on the screenng experments performed by Achlerkar (2004) whch ndcated ther sgnfcance n clusterng at a sngle locaton. We recommend examnng other attrbutes such as a base locaton ndex (BLI) order frequency constrant ndex (OFI) and back order constrant ndex (BCI). The BLI ndcates how many locatons a part type has n common the BCI s the mnmum of the total back order constrant across the locatons for a part type and the OFI s the mnmum order frequency constrant across the locatons for a part type. We also recommend nvestgatng other possble attrbutes for clusterng SKUs at bases and depots to mprove the qualty of clusters formed and further reduce the penalty cost. The qualty of the output gven by the mult echelon nventory optmzaton software solutons that set optmzed (r ) or (s S) nventory control polces n a mult-echelon nventory system depends on the ntal values used for reorder pont (r) reorder quantty () or Order-Up to-level (S). Instead of assumng ntal values for hundreds of thousands of SKUs n the mult echelon nventory system we recommend usng the soluton provded by the segmentaton methodology for settng the ntal values. We observed that the segmentaton methodology mantans the system performance constrants whle clusterng the nventory data for reducng the computatonal tme. Hence the ntal values set usng the segmentaton methodology wll be wthn the constrants and provde a good startng pont to further optmze the nventory control polcy parameters. Acknowledgements Ths materal s based upon work supported by the atonal Scence Foundaton under Grant o Any opnons fndngs and conclusons or recommendatons expressed n ths materal are those of the author(s) and do not necessarly reflect the vews of the atonal Scence Foundaton. References. Achlerkar A. (2004) Evaluaton of Segmentaton Technques for Spare Parts Inventory Management unpublshed Master s Thess Department of Industral Engneerng Unversty of Arkansas Fayettevlle Arkansas. 2. Al-Rfa M. H. and Rossett M. D. (2007) An Effcent Heurstc Optmzaton Algorthm for a Two- Echelon (R ) Inventory System to appear n the Internatonal Journal of Producton Economcs 3. Deuermeyer B. and L. B. Schwarz (98) A Model for the Analyss of System Servce Level n Warehouse/ Retaler Dstrbuton Systems: the Identcal Retaler Case n: L. B. Schwarz (Ed.) Studes n Management Scences Vol. 6 Mult-Level Producton/ Inventory Control Systems orth-holland Amsterdam pp Ernst R. and Cohen M.A. (990) Operatons Related Groups (ORGs): A Clusterng Procedure for Producton / Inventory Systems Journal of Operatons Management 9(4) Hopp W. J.; Zhang R..; and Spearman M. L An easly mplementable herarchcal heurstc for a two-echelon spare parts dstrbuton system IIE Transactons Rossett M. and Achlerkar A. (2004) A Constraned Clusterng Algorthm for Spare Parts Segmentaton The Proceedngs of the 2004 Industral Engneerng Research Conference R. Kng B. orman (eds.) Houston TX. 7. Sherbrooke C.C. (986) VARI-METRIC: Improved Approxmatons for Mult-Indenture Mult-Echelon Avalablty Models Operatons Research 34 (2)