Single-Echelon Supply Chain Two Stage. Distribution Inventory Optimization Model. for the Confectionery Industry

Transcription

1 Alied Maematical Sciences, Vol. 5, 0, no. 50, Single-Echelon Suly Chain Two Stage Distribution Inventory Otimization Model for e Confectionery Industry K. Balaji eddy, S. arayanan and P. Pandian School of Mechanical and Building Sciences, VIT University, Vellore-4, Tamilnadu, India School of Advanced Sciences, VIT University, Vellore-4, Tamilnadu, India andian6@rediffmail.com Abstract In is aer, a suly chain two stage distribution inventory otimization model wi interactive-lateral transshiment among retailers in a articular eriod is considered. Interactive-lateral transshiment among retailers means at once e stocks are allocated to retailers from ware houses, we allow lateral stocks transfer among retailers if e furer otimization is ossible which will enable us to furer minimize overall cost in e suly chain. The roosed model is formulated as a linear rogramming model which is validated wi hel of a confectionary industry data. This model is to rovide an otimal inventory level for e warehouses and retailers and also, minimizing e total cost of e entire suly chain for a finite lanning horizon. Maematics Subject Classification: 90B05, 90B06, 90B50, 90C05, 6P0 Keywords: Suly Chain Management, Lateral Transshiment, Linear Programming, Inventory Otimization. Introduction ow-a-days, e Suly Chain Management (SCM) is attracting much research attention, because of its ability to rea more benefits by efficiently managing it. Suly Chain Management (SCM) is a set of synchronized decisions

2 49 K. Balaji eddy, S. arayanan and P. Pandian and activities utilized to efficiently integrate suliers, manufacturers, warehouses, transorters, retailers, and customers so at e right roduct or service is distributed at e right quantities, to e right locations, and at e right time, in order to minimize system-wide costs while satisfying customer service level requirements. The objective of SCM is to achieve sustainable cometitive advantage. To stay cometitive, organizations should imrove customer service, reduction of costs across e suly chain and efficient use of resources available in e suly chain One-warehouse and multi-retailer systems have attracted much research attention in e literature because of eir alication in all business models. The discrete time models of one-warehouse, multi-retailer systems have been studied under different assumtions [ 4, 5, 7]. Archibald et al. [] develoed a model which was motivated by e inventory roblem faced by a UK car art retailer at grous its deots into airs. Avijit Banerjee et al. [] studied e effects, in terms of some selected criteria, of two lateral transshiment aroaches in a two-echelon suly chain network, wi a single suly source at e higher echelon and multile retail locations at e lower, rough a series of simulation exeriments under different oerating conditions. oorul Haq and Kannan [9] considered a two-echelon distribution inventory suly chain model for a bread industry using genetic algorim. Young Hae Lee et al. [] roosed a new lateral transshiment olicy, called Service Level Adjustment (SLA), which differs from revious olicies by integrating emergency lateral transshiment wi reventive lateral transshiment to efficiently resond to customer demands. Additionally, e roosed olicy considers e service level to decide e quantity for lateral transshiment. Fredrick Olsson [6] develoed a model in which e transshiment rule is to always transshi when ere is a shortage at one location and stock on hand at e oer. They assumed at e locations aly (, Q) olicies for normal relenishments and showed at e otimal olicies are not necessarily symmetric even ough e locations are identical. Emmanouil et al. [] roosed a model to determine e timing and size of e relenishment services togeer wi e vehicle routes, so at e total transortation and inventory holding cost of e system is minimized. Sheng-Yuan Shen and Masakazu Honda [0] focused on develoing an integrated relenishment and routing lan at takes into account lateral transfers of bo vehicles and inventory for a ree-echelon suly chain system including a single lant, multile distribution centers and multile retailers by mixed integer rogramming techniques. Lorenzo Tiacci and Stefano Saetta [8] considered a reventive transshiment case, where e inventory of different locations at e same echelon is balanced rough lateral transshiments, before a shortage haens. A heuristic for deciding on transshiment olicy trying to minimize overall exected costs was resented.

3 Otimization model 49 In is aer, a suly chain distribution two stage inventory otimization model is roosed wi interactive-lateral transshiment among retailers. The overall objective of e model is to minimize e total cost of e suly chain for a finite lanning horizon. Interactive-lateral transshiment among retailers means at once e stocks are allocated to retailers from ware houses, lateral stocks transfer among retailers may be allowed if e furer otimization is ossible. This will enable us to furer minimize overall cost in e suly chain. The key cost comonents which considered here are, inventory carrying cost at warehouse and retailers, transortation costs from ware house to retailers and among retailers, and e back order cost at retailers in a articular eriod. The roosed work is formulated as a linear rogramming model in two stages and is validated by considering e case study of a confectionary industry.. Model Descrition The roosed model is a single-echelon two stage distribution inventory model wi w warehouses and r retailers wi limited caacities. It is assumed at a single roduct is being distributed from e warehouse to retailers in e system. The demand for e roduct will be forecasted before beginning of every eriod and will be used as e reference for ware houses to transfer stocks from em to retailers in a articular eriod. All retailers and customers in e downstream are identical; i.e., ey have identical cost structures and demand distributions. Any demand at is not met is considered as e back order. All warehouses assume e instantaneous suly from an outside source. All Warehouses and retailers will be managed by a single organisation only. andom demands occur at retailers, which were en transferred to warehouses for relenishing inventories. Excess demands at retailers are comletely backlogged. Furer more, e lead times of retailers from warehouses are different and deending on eir geograhical ositions. This model is suitable to any kind of industry at follows e two-stage suly chain model wi finite number of ware houses and retailers. The roblem is divided into two stage otimization model, and in e first stage we consider only ware houses and retailers and interaction between em. Based on e forecasted demand values, warehouses will try to satisfy e need of all retailers by transferring stocks to em. The caacities of all warehouses and retailers will also be taken into consideration in otimising e model. Once e otimal solution is found en ese values will be used in e second stage to furer otimize e model. In e second stage, we consider e interaction among retailers. We use otimal values from first stage to find excess inventories and back ordered units at each retailer. We calculate stage inventories and stage back orders for each retailer after realizing e actual demand, and a second stage constrained otimization model is develoed to furer otimize e total cost by considering

4 494 K. Balaji eddy, S. arayanan and P. Pandian e lateral transhiment among retailers. For modelling simlicity, we did not consider e lead times between retailers at e end of every eriod, and we assume it as negligible. That is, we assume at is lateral transhiment of stocks among retailers will be taken lace before starting of e next eriod. The urose of is model is to rovide an otimal inventory level for e warehouses and retailers and also minimising e total cost of e suly chain at each eriod. In designing e roosed model, e ree subsystems need to be analysed. (i) The inventory level at each warehouse for all eriods. (ii) The inventory level at each retailers for all eriods and (iii) oute between entities where e roduct is transferred from warehouse to retailers and among retailers wi back orders for all eriods.. Two Stage Constrained Otimization Model The roosed model deals wi otimizing e inventory levels of e suly chain entities wi consideration of single-echelon warehouse to retailers and among retailers. The diagrammatic reresentation of e roosed model for e eriod is shown in Figure. Let, TC, TC and TC be e minimum total cost in a articular eriod, e total cost in e first stage of eriod and e total cost in e second stage of eriod resectively. Then, TC = MinimizeTC + MinimizeTC Figure Two Stage Constrained Otimization Model First Stage roblem : Let ICWH be total inventory carrying cost of all warehouses in eriod and TPW be total transortation cost from all warehouses to all retailers in eriod

5 Otimization model 495 Let IWH j be inventory at UICCW j be unit inventory carrying cost at j warehouse to transferred from unit transortation cost from j ware house at e end of e eriod, j warehouse, UTW jk be units k retailer in eriod and UTCW jk be j ware house to k retailer. ow, e first stage objective function is given below: TC = w j= IW j UICCW j + w r j = k = UTW In e first stage, we have e following constraints: jk UTCW jk. The sum of e units transferred from a ware house to all retailers should be less an or equal to e ware house inventory for a articular eriod r k = UTW jk IWH j where UTW jk is units transferred from eriod and IWH j is inventory of j ware house to k retailer in j ware house in eriod. Inventory at e warehouse should be less an or equal to e ware house caacity in eriod. IWH j WHCAP j IWH j is inventory of where Caacity of e j ware house in eriod and WHCAP j is j ware house in eriod.. The sum of e units transferred from all ware houses to a articular retailer should be greater an or equal to e forecasted demand of at articular retailer in eriod. w j= UTW jk FD FD k is e forecasted demand of e where is units transferred from k j ware house to k retailer in eriod and UTW jk k retailer in eriod 4. The total retailers forecasted demand for a eriod should be less an or equal to all ware houses inventory in at articular eriod. w j= r IWH j FD k = k

6 496 K. Balaji eddy, S. arayanan and P. Pandian where IWH j is inventory at FD k is demand of j ware house at e end of e eriod and k retailer in eriod. 5. The total number units transferred from all ware houses to a articular retailer should be less an or equal to at retailer caacity. w j= UTW jk CAP where CAP k is e caacity of e retailer for making stock and units transferred from j ware house to k retailer in eriod. k UTW jk is Therefore, e maematical model for e first stage roblem is to minimize e objective function TC subjected to above said constraints. Second Stage roblem : Let, ICC be total inventory carrying cost of all retailers in eriod TPC be total transortation cost from all retailers to all retailers in eriod and BOC be total back order cost of all retailers in eriod where eriod, ow, k ICC P = c k = r IC IC = sic k UT km k m= k UICC sic is e stage inventory at UT km is units transferred from unit Inventory carrying cost at where ow, eriod, BOC k = r k = BO k r k, k =,,... r is inventory at k retailer to k retailer UCBO BO = sbo k U km m= sbo is e stage back order at U km is units received by k cost of back order at k retailer. k k retailer from k retailer at e end of e k retailer at e end of e stage, k =,,... r is back order at m retailer and UICC k is k retailer at e end of k retailer at e end of e stage, m retailer and UCBO k is unit

7 Otimization model 497 ow, TPC = r r k = m= UT km UTC km k, m =,,... where UT km is units transferred from k retailer to m retailer in eriod wi UT km = 0, if k = m and UTC km is Unit Transortation cost from k retailer to m retailer wi UTC = UTC. km Therefore, e second stage objective function is given below: TC = ICC + TPC + BOC. mk ( ) In e second stage, we have e following constraints. P. The inventory at a articular retailer at e end of stage one, is equal to e total number units received from all warehouses by at retailer minus e actual demand of at articular retailer in eriod. sic k w = UTW j= jk AD ( If is value is negative en, sicu = 0 ). The back order of a retailer at e end of e stage is equal to e actual demand of at articular retailer minus e total number units received from all warehouses by at retailer in e eriod. sbo k = AD k w j= k UTW ( If is value is negative en, sbo = 0 ). The sum of e units transferred from a retailer to all retailers should be less an or equal to e excess inventory of a retailer at e end of e stage where, r m= UT km sicu k UT is units transferred from km and sic k = Inventory of e jk k k k where, UT = 0, if k = m km k retailer to r m retailer in eriod k retailer at e end of e stage 4. The sum of e units received by a retailer from all retailers should be less an or unequal to e back order of at retailer for a articular eriod r m= U km sbo k k =,,... r

8 498 K. Balaji eddy, S. arayanan and P. Pandian where, U km is units received by and sbo k is back order of k retailer from m retailer in eriod k retailer at e end of e stage 5. The sum of all transortation costs from all retailers to a articular retailer should be less an or equal to e transortation cost from any ware house to at retailer lus e back ordering cost of at retailer. r m= TP km TPW jk + BOC where, TP km = Ukm UTC km is e transortation cost to k retailer from m retailer in eriod, TPW jk = sbo k UTCW jk is e transortation cost from j ware house to k retailer in eriod, BOC k = sbo k BOCk is Back Order cost of k retailer in eriod Therefore, e maematical model for e second stage roblem is to minimize e objective function TC subjected to above said constraints. At e end of e eriod, we have TC = MinTC + MinTC k 4. Industry Case study for validation of model The comany chosen for e alication of suly chain distribution inventory model in is work is a famous confectionery comany located in e souern art of India. The comany has lanned to build a maematical model to minimize e total cost of e suly chain. The total cost can be minimized by otimizing e inventory levels at warehouse and retailers and otimizing e transortation cost in e suly chain. This can be done by transorting e roduct rough e otimum route and by maintaining an otimum inventory at ware houses and retailers. The following data is collected for validating e above roosed model. umber of eriods =, umber of Ware houses w = and umber of retailers r =. The inut data for ware houses and retailers are given in Tables to 5 resectively. Table Inut Data for Warehouse (DPG) Warehouse Inventory carrying cost er units in uees ICWH 5 Warehouse Caacity in units WHS 5,00,000 P

9 Otimization model 499 Table etailer Demand etailers Demand in Units Forecasted Actual Forecasted Actual Forecasted Actual Period P Period P Period P Table Inut Data for retailers etailers Inventory Carrying cost in ruees er unit Table 4 Transortation Cost from Warehouse to etailers er unit in s etailers Transortation Cost in uees from Warehouse TPWC to 9 5 Table 5 Transortation Cost from etailers to etailers er unit in s etailers and back Order Cost at each retailer er Kg = s esults and Discussions After solving e model for two stages, using Premium Solver Software, e otimal solutions and results for e case study are obtained. The roblem is solved for eriods, by using e industry data by considering a single warehouse and ree retailers. These results are tabulated, comared wi e real time existing solutions of e industry and are given in tables 6 to resectively. Table 6 Otimal Ware House Stock Period umber of units at W Proosed Meodology Existing(Industry)

10 500 K. Balaji eddy, S. arayanan and P. Pandian Table 7 Otimal number of units transferred from ware house to retailer Period etailer W, Proosed Meodology W, Existing(Industry) Table Otimal number of units transferred from retailer to retailer Period Proosed Meodology Existing(Industry) Customer Table 9 Otimal Inventory in units at etailers Period Proosed Meodology Inventory in Units Existing(Industry) Inventory in Units

11 Otimization model 50 Table 0 Period umber of Back Ordered units at etailers Proosed Meodology Back Order in Units Existing(Industry) Back Order in Units Table Otimal Total Cost Comonent Proosed Meodology in s Existing(Industry) in s ICWH 5,50,000 7,75,000 5,00,000 0,00,000 0,00,000 0,00,000 TPWC 46,60,000 5,05,000 45,0,000 5,0,000 58,9,000 49,90,000 ICCU 5,785 0,7,7,0,74,84,4,680 TPCC,77,65,50,60,48, BOC 0 5,98 0 5,65,5,875 0 TC 64,0,050 7,55,758 6,69,656 74,98,475 8,55,709 7,4,680 The otimal cost comarison grah is given in Figure. The major advantage of e roosed model over existing model is e maintenance of e otimum inventory levels at e ware house and retailers, and minimizing e back order costs at retailers level. Because of is, ere is a significant decrease in total cost using e roosed model over existing model, as shown in e Figure. Figure Comarison of Proosed Meodology and Existing (Industry)

12 50 K. Balaji eddy, S. arayanan and P. Pandian 5. Conclusion This aer has resented a suly chain two stage distribution inventory otimization model for a distribution network wi multile ware houses sulying multile retailers, who in turn serve a large number of customers. This model has taken e distribution and inventory carrying costs into account in e suly chain network at each eriod. Wi e validation of case study using e confectionery industry data, it is clear at e results obtained are encouraging and reduced over all system costs. By making retailers to interact and taking a decision on lateral transshiment, e inventory level of different locations at e same echelon is balanced. However, is work has few limitations. Future research can address some of e following issues. The model assumes lead time between retailers is negligible at e end of e eriod, for modeling simlicity. So, better models can be develoed to address is issue. The model also assumes at ere is a single roduct in e study area. Models can be develoed for multi-roduct scenarios. This model is limited to single echelon and extensions of two and more echelons can be addressed wi different aroaches. eferences [] Archibald T.W., Sassen S.A.E. and Thomas L.C., An otimal olicy for a twodeot inventory roblem wi stock transfer, Oerations esearch, 4(997), 7 8. [] Avijit Banerjee, Jonaan Burton and Snehamay Banerjee, A simulation study of lateral transshiments in single sulier, multile buyers suly chain networks, International Journal of Production Economics, 8-8(00), 0-4.

13 Otimization model 50 [] Emmanouil E. Zachariadis, Christos D. Tarantilis and Chris T. Kiranoudis, An integrated local search meod for inventory and routing decisions, Exert Systems wi Alications, 6(009), [4] Een G and Schrage L., Centralized ordering olicies in a multi-warehouse system wi lead-times and random demand, Multi-Level Production/inventory Systems: Theory and Practice. or-holland, ew York, 98, [5] Federgruen A. and Zikin, P., Allocation olicies and cost aroximations for multi-location inventory systems, aval esearch Logistics Quarterly, (984), [6] Fredrick Olsson, Otimal olicies for inventory systems wi lateral transshiments, International Journal of Production Economics, 8(008), [7] Jackson P.L., Stock allocation in a two echelon distribution system or "what to do until your shi comes in", Management Science, 4(988), [8] Lorenzo Tiacci and Stefano Saetta, A Heuristic for balancing e inventory level of different locations rough lateral transshiments, International Journal of Production Economies, Article in Press (00). [9] oorul Haq and Kannan G., Two-echelon distribution-inventory suly chain model for e bread industry using genetic algorim, International Journal of Logistics Systems and Management, (006), [0] Sheng-Yuan Shen and Masakazu Honda, Incororating lateral transfers of vehicles and inventory into an integrated relenishment and routing lan for a ree echelon suly chain, Comuters and Industrial Engineering, 56(009), [] Young Hae Lee, Jung Woo Jung and Young Sang Jeon, An effective lateral transshiment olicy to imrove service level in e suly chain. International Journal of Production Economics, 06(007), 5-6.

14 504 K. Balaji eddy, S. arayanan and P. Pandian eceived: February, 0