Inventory Planning with Batch Ordering in Multi-echelon Multi-product Supply Chain by Queuing Approach
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1 Inventory Planning wit Batc Ordering in Multi-ecelon Multi-product Supply ain y ueuing Approac Eraim Teimoury, Ali Mazlomi, Raele Nadafioun, Iman G. Kondai, and Medi Fati Astract In tis paper, we apply queuing models for performance evaluation analysis in multi-product multiecelon manufacturing supply cain network wit atc ordering. Te analysis is clued wit an inventory optimization model, wic can e used for designing inventory policies for eac product. We consider a tree-ecelon supply cain: retailers, wareouses and manufacturing plants supply types of products to various retailers. Production system is MTS and we use queue operating under inventory control rule to analyze te performance of any manufacturing plant. Proposed model determines te optimal inventory level at te wareouse of eac product tat minimizes total expected cost. Moreover, we extend te proposed model in order to analyze te logistics process. Index Terms ueuing system, Supply cain, production/inventory systems, order atcing, inventory control. I. INTRODUTION supply cain is an integrated manufacturing process Awere in raw materials are converted to final products ten delivered to customers. A supply cain consists of all parties involved in fulfilling customer s demands. Te supply cain includes not only te manufacturer and suppliers, ut also transporters, wareouses, retailers, and even customers temselves. A supply cain is consisted of two asic integrated processes: te production planning and inventory control process and te distriution and logistics process. Te supply cain is dynamic and involves te constant flow of information, product and funds etween different stages. Te principal purpose of any supply cain is to fulfill customer s demands and generate profit for itself. In reality, a manufacturer may receive material from several suppliers and ten supply several distriutors. Tus, most supply cains are actually networks. Te oective of every supply cain sould e to maximize te overall generated value. Te value a supply cain generates is te difference etween wat te final product is wort to te Manuscript received Novemer, 00; revised January 06, 0. E.Teimouri, R.Nadafioun, M.Fati, Department of Industrial Engineering, Iran University of Science and Tecnology, Teran, Iran (corresponding autor, M. Fati, mfati@iust.ac.ir) A. Mazlomi, Department of Mecanical and Manufacturing Engineering, Universiti Putra Malaysia 3400 UPM, Serdang MAAYSIA- SAPO OMPANY, Teran, Iran. I.G. Kondai, Department of Engineering, Saed University, Teran, Iran. customer and te costs te supply cain incurs in filling te customer s demands. Also, a manufacturing supply cain can e viewed as a network of suppliers, manufacturing sites, distriution centers, and customer locations, troug wic components and products flow. Trougout tese networks, tere are different sources of uncertainties, including supply (availaility and quality), process (macine reakdown, operator variation), and demand (arrival time and volume). Also, tese variations will transmit from upstream stages to downstream stages and will lead longer cycle time and lower fill-rates. One of te callenges in supply cain management is to control te capital in inventories. Te oective of inventory control is terefore to alance conflicting goals like keeping stock levels down to ave cas availale for oter purposes and aving ig stock levels for te continuity of te production and for providing a ig service level to customers. A good inventory management system as always een important in te workings of an effective manufacturing supply cain. ueuing systems are te natural models wen dealing wit prolems were te main caracteristics are congestion and ams. In tis paper, we use GI /G / queue as tool for performance measures of te manufacturing supply cain and also, we use queue to analyze logistics processes. We review te articles of inventory management in logistics cains including single-product multi-stage systems and multi-product systems (Tale). Tale. Related articles in inventory management in logistics cains Reference Article s code (prolem article(s) definition/constraints/outputs/oective functions) [] and [] [3] ueueing, Single Production, Decomposition Metod ueueing, Assemle, Single Production, Stocastic, M / M /, ongest Pat Analysis Inventory ueueing, Single Production, Decomposition Metod [4] [5] Multi Production, M / G /,ead Time [6] ueueing, Multi Production, Decomposition Metod [7] Inventory ueueing, Multi Production, ontinuous, M X / G / [8] ueueing, Single Production, Batc, GI X / G / [9] [0] Production Inventory, Multi Production, Stocastic, Batc Production Inventory, Stocastic, Batc, Decomposition Metod, Monte arlo simulation
2 Te remainder of tis te paper is organized as follows. In section II, we descried te principal caracteristics of te model. In section III, we perform te proposed model and provide computational results. Finally, we give some concluding remarks in section IV. II. PROBEM DESRIPTION AND FORMUATION We consider a tree ecelon supply cain network including n retailers, wareouses and manufacturing plants as sown in Fig.. Tis network offers types of product to te customers arrived into retailers' node. ustomers' demands enter to te retailers and te wole demand accumulation for eac product is forwarded to wareouses of tat product. We apply production autorization (PA) system to produce eac production. Te PA system is a generalized pull-ased production control system. We assume tat products in inventories are stored in atces for eac product, and tere is a PA card attaced to eac atc. In tis paper, we consider te case wen te numer of PA cards is te same as te numer of atces. Te PA system operates in te following way wenever units are depleted from a atc in te inventory; te corresponding PA card is transmitted to te manufacturing plant. And also serve as new production orders tat trigger te manufacturing plant to egin its production process. In general, te manufacturing plant uses a FFS discipline to produce tese orders. Once te manufacturing plant produces units, te finised units and te PA card are sent to te wareouse. In te event wen a customer places an order and tere is no production inventory availale, we assume tat tis customer wait until te product ecomes availale. In our model, we assume tat te set-up time is incurred wen te processor egins its production for eac atcing order. Fig. Tree stage supply cain network A. Assumptions Assumptions of te developed model are as follow: ustomers' demand includes all types of products. We assume te orders of retailer i are as an independent renewal process wit a constant rate i 0 and coefficient of variation i. Te proaility vector qi ( qi, qi,..., qi ) defines customers' demand from eac kinds of product at retailer i ( q, 0 q ; i,,..., n ). Te i orders of wareouse are as an independent renewal process wit a constant rate q and coefficient of variation i n i i n iqii / i i. In our prolem it is assumed tat eac wareouse old one type product in atc size wic maximum numer of atces is K. Terefore, maximum capacity of wareouses for eac product is given y Z K. Production policy is MTS strategy for wareouses. In te following, we use GI /G / queue operating under K, K ) inventory ( control rule to analyze te performance of te singleproduct type. We assumed tat unit production times at manufacturing plants for product are i.i.d. generally distriuted random variale, denoted y B, wit E( B ) and coefficient of variation.tus, mean production time for atc product is and coefficient of variation. In our model, eac manufacturing plant produces one type product, in oter words, eac wareouse ave single sourcing constraint from manufacturing plants. Arriving orders from different retailers deplete te on-and inventory at wareouses, if any. Oterwise, (in a stock-out situation) te arriving orders ave to wait to e fulfilled; te waiting process consists of manufacturing te desired units to e produced at te manufacturing plant of product and eing sipped to te wareouse of product. We refer tis situation as ack order. In our prolem, we assume tat ack orders can e infinite. Wareouse of product places orders to manufacturing plant of product wen one ucket ( units) is depleted from te inventory at te wareouse. Tus, arrival process of product is caracterized y,. (,,..., ) We assume tat all manufacturing plants ave infinite waiting line capacity. Te system incurs a olding cost per unit of inventory of product per unit time, a ackordering cost per unit of product ackordered per unit time and order set up cost for product ( $ per set up) s. Te goal of modeling suc supply network is to minimize supply cain total cost in order to find optimal values of K,. osts contain inventory olding cost ( ), ack ordering cost ( ) and order set up cost s ). (
3 B. Notations Te notations used in tis paper are as follow: Numer of units in one ucket of product ; K Z i A I N B,,..., Total numer of uckets at wareouse of product Maximum inventory at wareouse of product ; K Demand arrival rate at retailer i Numer of orders arrived at manufacturing plant of product Production rate of manufacturing plant of product units/unit time Inventory level at wareouse of product Numer of orders at manufacturing plant of product eing processed Numer of ack orders at wareouse of product R Numer of orders arrived at wareouse of product, ut after te last atc was released for processing s l i Inventory olding cost for product ($ per unit per unit time) Back order cost for product ($ per unit per unit time) Order set up cost for product ( $ per set up) Intensity of te manufacturing plant of product ead time of logistics for retailer i to receive items from wareouses of product, l l... l l Expected numer of orders in te queue M / M / in steady state Service rate of logistics process Intensity of te logistics u W i i a, Expected waiting time at wareouse of product due to ack ordering alone Mean lead time (including ack ordering delay) for an order of items from retailer i to e filled from wareouses of product, n... n Expected demand for product during replenisment lead time for item at retailer i ( ) i. Prolem formulation In tis paper, we would like to minimize te expected total cost at te wareouses, i.e. Minimize Total ost = Expected i inventory olding cost + Expected ack ordering cost + Expected ordering set up cost Matematically, we can express, Min s s. t. K, T(, K ) Z ( E I EB () ( )) Te goal of modeling suc a supply cain is minimizing te total cost were results can e used to otain te optimal values of, Z. For computing inventory and ack orders, we need to develop stocastic equations wic capture te properties of te system as in []. Oserve tat, A () R A,,,..., B maxn R K,0,,,..., (3) K N R,0,, I max,..., (4) Te corresponding steady state proaility distriution for R, N, B, I are as follows: R is uniformly distriuted from 0 to. Tus, P { R n}, n 0,,..., aracterizing te proaility distriution of queue size in a GI/G/ is difficult in general. Terefore, we use a development descried in [] to approximate te proaility distriution of atces in te system using a geometric distriution of te following form: n 0 PN n PN ( n) n (6) ( ) n,,... Were ( Nˆ ) / Nˆ, Nˆ w w ( B) w0 and a Wˆ, 0 GI / G /,,, 0 ( (5). ) ( ) ( ) a, ( ) To otain te steady state utilization of te production system, wic we denote y, as follows In continue, we can otain steady state proaility distriutions I, B as follow: (7)
4 Z n PB n PN ( ) ; n,,... (8) Z n PI n PN ( ) n,,..., K (9) I E as, And also, we can calculate E and E I E Z B i Z i ( ) E I PN i (0) ( ) K [( ) ( ) K ( ) [(K ) ]] () B P ( ) i E B i Z i 0 () N i K K (3) [ ] [ ] K ( ) D. Performance measure of wareouses Te stock-out proaility at wareouse of product is te fraction of time tat te on-and inventory at wareouse of product is zero and is otained as follows: P I 0 P Z N R k ( ) k (4) And also, te fill rate at wareouse of product is te fraction of time tat te on-and inventory at wareouse of product is greater tan zero: P I 0 P Z N R P I 0 k ( ) k (5) Also te lead time of product at its manufacturing plant is given y ( ) Ws ( ) w0 ( ) (6) ( ) Were ( ) is atc forming time of product and product atc. is mean production time for E. Te squared coefficient of variation of te interdeparture times is produced from te wareouses We use te approximation of squared coefficient of variation (SV) of te inter-departure times for te atces from te wareouse of product wit atc setups in te GI /G / queue, given in [] as sown in (7): a, d, ( B) ( ) (7) ( ) And also, from [], we use te following approximation of te squared coefficient of variation of te inter-departures of individuals from te wareouse of product : (8) Were fixed atces size of product is. Wen a product departs from te wareouse of product, tere is a proaility q i, tat te product will e routed to retailer i. Terefore, te mean inter-arrival time and squared coefficient of variation for arrivals to retailer i is given y i iqi (9) a i i d,, q ( I) q (0) F. ogistics process i In continue, we extend te model y adding logistics processes. we assume tat tere is some logistics time to supply products from wareouses to retailers.we model te logistics process of product y using c M / M / queue in continuous time, were c is veicle capacity wic is deterministic and logistics time is exponential. We assume tat te logistics process is depending on te demands of customers for its arrival process. Fig. tree stage supply cain network wit logistics u and co-located retailers M / M c / For te performance analysis of queue, we use te results of [3] and [4]. We otain mean lead time of product from wareouse of product at retailer i, law as following on: EB W i l i W y using ittle s ()
5 i l W () i qi i i l i q i ( qi ) (3) (4) We can compute expected demand of product at retailer i during replenisment lead time as (5) i a,, i i Tale4. Total cost variation y increasing if III. NUMERIA EXAMPE In tis section, we analyze te model y a numerical example. We consider a supply cain network wic produces tree products. Te supply cain includes two retailers wic teir exponential arrival demands are caracterized y 0. 6, 0. 8 and, a. Te proaility vectors q define q (0.,0.3,0.5), (0.4,0.3,0.3) customers demands for tree products at two retailers. Information of tree manufacturing plants to produce te products and costs of tree wareouses is sowed in Tale and information of logistics processes is sowed in Tale 3: Product type Tale. Information of manufacturing plants S Tale3. Information of logistics processes Product type l c We solved te prolem wit Matla7 software. We otain optimum value K y varying values of atc sizing for tree products. In te condition tat, we increase, and optimum maximum inventory level and total cost of tree products are increasing. Te results imply tat if ack order costs are greater tan olding costs, system tends to old more inventories (Tale4). In condition tat 0, we increase, and optimum numer of atces ( K ) are otained. (System does not * tend to old more inventories) and optimum maximum inventory level and total cost of tree products are increasing (Tale5). By comparing Tale5 and Tale6, we sow tat optimum maximum inventory level and total cost of tree products in Tale5 are greater tan Tale6 (To decrease total costs). Tale5. Total cost variation y increasing if 0
6 In te conditions tat 0 we increase S, optimum maximum inventory levels of tree produces are otained and only total cost of tree products are increasing (Tale6). Tale6. Total cost variation y increasing 0 S and if paper, we surveyed te effect of order atcing in multiproduct multi-ecelon supply cains. In future researces, we can consider a central wareouse tat in te stock-out condition in eac wareouse, customers demands are satisfied (adding transmittal cost). Also, pricing prolem can e added to te presented model. REFERENES [] Y.J. ee, P.H. Zipkin, Tandem queues wit planned inventories. Oper Res, 99, 40: [] Y.J. ee, P.H. Zipkin, Processing networks wit inventories: Sequential refinement systems, Oper Res 43(6), 995, [3] A. Azaron, H. Katagiri, K. Kato, M. Sakaw Modelling complex assemlies as a queueing network for lead time control, Eur J Oper Res 74, 006, [4].M. iu, X.M. iu, D.D. Yao, Analysis and optimization of a multistage inventory-queue system, Manage Sci 50(3), 004, [5] U.W. Tonemann, J.R. Bradley, Te Effect of Product Variety on Supply ain-performance, European Journal of Operational Researc, 00. [6] G.R. Bitran, D. Tirupati, Multi-product queueing networks wit deterministic routing : decomposition approac and te notion of interference, Manage Sci34(), 988, [7] M. Ettl, G.E. Feigin, G.Y. in, D.D. Yao, A supply network model wit ase-stock control and service requirements, Oper Res48, 000, 6 3. [8] M. Dong, Inventory planning of supply cains y linking production autorization strategy to queueing models, Prod Plan ontrol 4(6), 003, [9] Y.D. u, J.S. Song, D.D. Yao, Backorder minimization in multiproduct assemle-to-order systems, IIE Trans 37, 005, [0] Y. Zao, D. Simci-evi, Performance analysis and evaluation of assemle-to-order systems wit stocastic sequential lead times, Oper Res 54(4), 006, [] J.A. Buzacott, J.G. Santikumar, Stocastic models of manufacturing systems. Prentice Hall, Englewood liffs, NJ, 993. [] G.. urry, R.M. Feldman, Manufacturing Systems Modeling and Analysis. Springer-Verlag Berlin Heidelerg, 009. [3] B.R.K. Kasyap,. iu, J.G.. Templeton, On te GI X /G/ system, J Appl Proa 7, 990, [4] P. Purdue, D. inton, An infinite-server queue suect to an extraneous pase process and related models. J Appl Proa 8, 98, IV. ONUSION In tis paper, we presented a model for te analysis of a tree-layer supply cain wic produces more tan one product. We used GI / GI / queue operating under ( K, K ) inventory control rule to analyze te performance of wareouses. We otained performance of measures suc as stock-out proaility, fill-rate and lead time of wareouses in proposed model. In te model, we c used M / M queue for analyze logistics process. In tis
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