SIMULATION OF INVENTORY CONTROL SYSTEM FOR SUPPLY CHAIN PRODUCER WHOLESALER CLIENT IN EXTENDSIM ENVIRONMENT

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1 SIMULATION OF INVENTOY CONTOL SYSTEM FO SUPPLY CHAIN PODUCE WHOLESALE CLIENT IN EXTENDSIM ENVIONMENT Eugene Kopytov and Avars Muravjovs Transport and Telecommuncaton Insttute, Lomonosov Street, ga, LV-09, Latva E-Mal: KEYWODS Inventory control, supply chan, wholesaler, omer, smulaton, ExtendSm package ABSTACT There s consdered a two-level sngle-product nventory control system whch controls correspondngly the wholesale s warehouse and the omers warehouses. For delverng the product there has been organzed a supply chan producer wholesaler omer. It s assumed that the omers and the wholesaler shape ther stocks havng n mnd the mnmzaton of the costs of the product orderng, holdng and losses from defct per tme unt. The omers demands for the product and the tme of delverng the cargo from the producer to the wholesaler and from the wholesaler to each of the omers are random values wth known laws of dstrbuton. The wholesaler orders the goods at the fxed, equally dstant, moments of tme. The order of goods s performed by the omer at a random moment of tme, when the remans of the goods n hs warehouse have reduced up to a fxed level called the reorder pont. In the gven paper there s suggested a smulaton model of the above nventory control. The ExtendSm 8 package has been used as the means of smulaton. The numercal examples of the problem solvng are presented. INTODUCTION In the gven paper we consder a stochastc sngleproduct nventory control model for the chan producer wholesaler omer (Chopra and Mendl 00; Magableh and Mason 009). In practce there are many cases when a wholesaler s nvolved n orderng process and a stuaton producer wholesaler omer takes place. We have to take nto account that the sum of costs for goods orderng, holdng and losses from defct per tme unt should be mnmal. In proposed crtera costs are sum of correspondng costs (losses) for all subjects takng part n the orderng process, n our case for omers and a wholesaler. The wholesaler and omers can use dfferent orderng strateges. Two models of orderng process used by omers were consdered n the authors work (Kopytov et al. 007). The frst model s a model wth fxed reorder pont and fxed order quantty, second model s the model wth fxed perod of tme between the moments of placng the neghborng orders. The wholesaler s orderng algorthms were presented n the paper (Kopytov et al. 005). The smplest algorthm consders the followng stuaton: every omer s order s sent by the wholesaler to the producer at once and receved goods are sent back to the omers at once too. In the second varant the wholesaler constructs a common order for group of omers takng n account or an nventory level for each omer, or a tme of recevng the omers orders and a quantty of them. In the gven paper we have proposed the thrd varant when the wholesaler has hs own warehouse wth the defnte quantty of goods. In practce suggested models are realzed usng analytcal and smulaton approaches. In the gven paper the consdered task s solved usng a smulaton method. DESCIPTION OF THE MODEL Let us consder a sngle-product nventory system for the supply chan producer wholesaler omer wth two stages n orderng process (see Fg.).The frst stage s executed by n omers. In the moment of tme, when the omer s stock level falls to a certan level, a new order s sent to the wholesaler. The second stage s executed by the wholesaler. A new order s sent to the producer n the fxed moments of tme. We assume that the wholesaler has hs own warehouse. Fgure : Chan of Product Orderng The prncpal am of the consdered problem s to defne the exact orderng strategy for n omers and the wholesaler to acheve the mnmum expenses n nventory control system per tme unt (Kopytov et al. 007). Takng nto account stochastc character of the nventory control problem, the crteron of optmzaton Proceedngs 5th European Conference on Modellng and Smulaton ECMS Tadeusz Burczynsk, Joanna Kolodzej Aleksander Byrsk, Marco Carvalho (Edtors) ISBN: / ISBN: (CD)

2 s mnmum average expenses E per tme unt, whch are calculated by followng formula: = n + wh = E E E, () where E s -th omer s average expenses for goods holdng, orderng and losses from defct per tme wh unt, =,,..., n; E s wholesaler s average expenses for goods holdng, orderng and losses from defct per tme unt. Let us consder n detal the stages n the presented chan of product orderng. Frst stage of orderng process The demand for goods D of -th omer s Posson process wth ntensty λ. Tme L of goods delvery from the wholesaler to -th omer has a normal dstrbuton wth parameters μ andσ. The polcy of order formng for -th omer s as follows: A new order s placed at the moment of tme, when the stock level falls to certan level. The order quantty s constant. We suppose that. Note that order reorder pont and quantty are control parameters of the frst stage model. Dynamcs of the nventory level of product for -th omer durng one cycle * T = T + L (tme nterval between two neghborng order delveres for -th omer) s shown n Fg.. There s a possble stuaton of defct, when demand D( L) durng lead tme L exceeds the reorder pont. We suppose that n case of defct the last cannot be covered by expected order. Z 0 ϕ (t) * T Fgure : Dynamcs of -th Customer s Inventory Level Durng One Cycle Denote as Z the quantty of goods n stock n the tme moment mmedately after order recevng. Ths random varable Z s determned as a functon of demand D( L) durng lead tme L : L t Formula () allows expressng dfferent economcal ndexes of consdered process. We suppose that the wholesaler has hs own warehouse wth a defnte quantty of goods q. If omer s order quantty s less or equal than quantty of products n the stock ( q) the wholesaler performs ths order n full volume at once. Otherwse when the order quantty exceeds the stock quantty ( > q) the omer wll receve only a part of goods, and there s the stuaton of defct of q unts of products n the wholesaler s warehouse. Orderng cost C ( ) 0 has two components: constant c, whch ncludes cost of the order formng and constant part of expenses of order transportaton, and varable component c ( ), whch depends on the order quantty,.e. C 0 ( ) = c + c ( ). We assume that for all omers the holdng cost s proportonal to quantty of goods n stock and holdng tme wth coeffcent of proportonalty C H ; losses from defct are proportonal to quantty of defct wth coeffcent of proportonalty C SH. For -th omer the average cost n nventory system durng one cycle E ( T) s calculated by the followng formula: E ( T) = C( ) + E ( T) + E ( T), =,,..., n, 0 H SH where T s average cycle tme; E H ( T ) s average holdng cost durng one cycle; E SH ( T ) s average shortage cost durng one cycle, and cost E per tme unt for -th omer can be found as dvded by average cycle tme T (oss 99): E ( T) E =, =,,..., n. () T Note that E H ( T ) and E SH ( T ) depend on control parameters and. Analytcal formulas for these economcal characterstcs are presented n the paper (Kopytov and Greenglaz 004). For problem solvng we have to mnmze crtera () by and. Second stage of orderng process We assume that producer supples ts producton to wholesaler accordng to a fxed schedule. In ths case we consder orderng process wth constant perod of tme T between the moments of placng neghbourng wholesaler s orders; and order quantty q s determned as dfference between fxed stock level S and quantty of goods n the moment of orderng r (see Fg.),.e. q = S r. + D( L), f D( L) < ; Z =, f D( L). ()

3 ) r = S and Dt () = 0, where Dt () s the demand for goods durng the tme t; 0 t T ; ) r < S and Dt () = 0, where L t T. Takng nto account that n case of defct t can t be covered by expected order, we can obtan the expresson for goods quantty at the moment of tme mmedately after order recevng: r+ q DL ( ), f DL ( ) < r; Z = q, f D( L) r, where DL ( ) s the demand durng lead tme L. Fgure : Dynamcs of Wholesalers Inventory Level Durng One Cycle Let lead tme L from the producer to the wholesaler have a normal dstrbuton wth a mean μ L and a standard devatonσ L. We suppose that lead tme essentally less as tme of the cycle: μl + σ L << T. We suppose that durng tme T the wholesaler has receved orders from n omers, these orders can be descrbed by the vector{,,..., n}. There s a possble stuaton of the defct, when the demand n DT ( ) = durng tme T exceeds the quantty of = goods n stock Z n the tme moment mmedately after order recevng. Analogously to frst stage we suppose that n case of defct the last cannot be covered by expected order. We denote as S the goods quantty whch s needed deally for one perod and t equals to the sum S = D( T) + S0, (4) where DT ( ) s average demand durng cycle tme; S 0 s some safety stock (emergency stock). We suppose that deally S gves n future the mnmum of expendture n nventory control system per unt of tme. So, for the second stage n suggested model tme perod T and stock level S are control parameters. We suppose that n the moment of tme, when a new order has to be placed, may be stuaton, when the stock level s so bg that a new orderng doesn t occur. However for generalty of model we ll keep the concepton of lead tme and quantty of goods at the tme moment mmedately after order recevng n such case too. It corresponds to real stuaton when the wholesaler uses the transport means, whch depart at the fxed moments of tme not dependng on exstence of the order and whch have the random lead tme; for example transportaton by tralers whch depart the st and 5th day of each month. In real stuaton n the moment of tme t the stock level ϕ(t) s equal to S only n two cases: Usng (4) we have: S D( L), f D( L) < r; Z = S r, f D( L) r. Fnally average cost for tme unt for the wholesaler s expressed by the followng formula E wh wh wh H + E C ( q) SH + 0 E =. (5) T Unlke stage, n the consdered stage expendtures wh wh EH and E SH depend on control parameters S and T. SIMULATION MODEL IN EXTENDSIM 8 ENVIONMENT For the consdered problem solvng, the authors have used a smulaton model realzed n smulaton package ExtendSm 8, whch s the most powerful and flexble smulaton tool for analyzng, desgnng, and operatng complex systems n the market. It enables the researcher to test the hypotheses wthout havng to carry them out. ExtendSm has repeatedly proven ts beng capable of modelng large complex systems (Krahl 007). Assume that n consdered system we have three omers. The created smulaton model for the supply chan producer wholesaler omers s shown n Fg. 4, 5, 7 and 8. The man screen of the smulaton model s presented n Fg.4. Each zone of the model has a numerc label. In zone an executve block, that controls all dscreet events n Extend models, s placed. Zones and contan blocks whch are responsble for modelng result representaton: to a plotter block s placed n zone, and n zone expenses calculaton and data export to Excel spreadsheet are executed. Zone 4 contans a block whch s ntended for scheduled transact generaton; lead tme and transport actvty for goods transportaton to the man store are smulated n blocks placed n zone 5. In the man storehouse zone there are placed: a block for holdng actvty smulaton, a block for order quantty calculaton, and an ntalzaton block that performs queue ntalzaton tasks before the model starts. In ths stuaton all stocks are ntalzed before startng to

4 represent a typcal stuaton of goods quantty n the warehouse. Fgure 4: Man Screen of the Smulaton Model After goods delvery to the man warehouse they are transferred to omers warehouses accordng to ther orders. The herarchcal blocks shown n Fg.5 realze the reorder pont strategy n goods orderng. demand for goods, shortage, delvery and holdng costs, order quantty and reorder pont. Specfyng these parameters we can receve approprate results, such as quantty of sold goods, amount of defct, costs that nclude orderng, holdng and shortage costs. These result parameters are used for cost calculaton. Order quantty and reorder pont are control parameters and have to be changed durng the smulaton procedure. Usng output connectors for goods quantty n stocks and plotter block, ExtendSm bulds graphcal representaton of the dynamcs of nventory level n all stocks shown n Fg.6. Fgure 6: The Example of Smulaton of the Inventory Control Process n All Stocks Fgure 5: eorder Pont Store Ths herarchcal block s made n the way, whch allows usng t n any necessary Extend model that needs such functonalty. In the created model there are three dentcal reorder pont blocks, for three omers stocks modelng respectvely. For ths reason, all control parameters and results are realzed as nput and output connectors. The nternal parameters for ths type of block are: stochastc lead tme of goods delvery and Fg.7 llustrates nternal structure of eorder pont herarchcal block. Frst block n zone 6 s called Gate, whch allows or dsallows transact entrance to ths part of the model. Behavor of ths block s controlled by Equaton block that collects nformaton about stock level, reorder pont and placed order status. Based on the calculaton of these parameters, Equaton block sends Boolean value to Gate (0 close and open). If transact s allowed for entrance, than t s passed to actvty transport block (zone 7), after approprate delay to the end store (zone 8). Blocks of zone 9 are used for expenses calculaton. Zone 0 s used for nternal communcaton between herarchcal block together Fgure 7: eorder Pont Herarchcal Block (nternal)

5 wth ExtendSm database. Structure of the fnal herarchcal block s shown n Fg.8. Fgure 8: Herarchcal Block Customer Store In zone transactons are arrvng to the warehouse, where they assgned the holdng cost value. Zone s desgned for the defct modelng wth dummy suppler and approprate attrbute assgnng. Zone represents a market place where goods are sent to omers. For end users facltaton a specalzed user nterface was desgned. Usng ths nterface user can change control parameters of the model and get fnal results of smulaton. There are several tools for user nterface development n ExtendSm. One of them s Notebook wndow that can be called from any place of model and other s clonng tool that allows clone core control elements from ExtendSm block and place t nto Notebook. The example of Notebook s wndow wth ntal data and results s shown n Fg.9. Fgure 9: Example of Notebook s Wndow EXAMPLE There s consdered a two-level nventory control system, whch ncludes correspondngly a wholesale warehouse and the warehouses of omers. For delverng the product there has been organzed a supply chan producer wholesaler omer wth two stages n the orderng process. It s assumed that all the omers are fnancally ndependent and organze the whole polcy of orderng and holdng the product by themselves; the wholesaler also acts only wth the account of the mnmzaton of hs costs, losses from the product defct ncluded. It s requred to fnd such values of the reorder ponts,, wth the omers and such value of the desred product stock S wth the wholesaler at whch the sum of the costs of the goods orderng and holdng and the losses from the defct per a unt of tme unt would be mnmal. The omers demands for goods D,( =,,) are Posson processes wth ntensty λ, and tme L of goods delvery from wholesaler to -th omer has normal dstrbuton wth parameters μ andσ (see Table ). Orderng costs C 0 (ncludng expenses of order transportaton) for each omer are presented n Table too. Customer, Lead tme, L, days μ =; σ =.5 μ =; σ = μ =4; σ =.7 Table : Intal Data Demand, λ unts/day Order quantty,, unts Intal stock, Z, unts Orderng cost, C 0, EU The producer supples ts producton to wholesaler accordng fxed schedule, and tme perod T between the moments of placng neghbourng orders s constant and equals 0 days. The polcy of order formng for -th omer s follows. A new order s placed n the moment of tme, when the stock level falls to a certan level.the tme L of goods delvery from producer to wholesaler has a normal dstrbuton wth a mean μ L = and a standard devatonσ L =. Orderng cost C 0 (ncludng expenses of order transportaton) for wholesaler equals to 900 EU. For omers and for wholesaler holdng cost C H equals to 0,005 EU per unt per day, losses from defct C SH equal to 0 EU per unt. Intal stock n wholesaler s warehouse s equal 4000 unts. Fxed stock level n wholesaler s warehouse S s the control parameter of the model. To solve the gven task, we ll be usng the smulaton model descrbed above and presented n Fg. 4, 5, 7 and 8. The perod of smulaton s one year and the number of replcatons s 00. There can be two strateges of optmzaton. The goal of the frst strategy s summary mnmzaton of all expenses for all model partcpants. The second strategy s optmzaton of ndvdual omer and wholesaler actvty. In ths paper we ll look more closely at the frst strategy. After havng performed the modelng of the ntal varant of the system presented n Table, we ll get the followng values of the control parameters: the reorder ponts wth the omers, correspondngly, are

6 =00, =70 and =00 unts; the level of the desred product stock wth the wholesaler S =50 unts. And the value of the average cost per year n the nventory system (see formula ()) equals 6 EU. The gven varant has been taken as the basc one. Let us perform the optmzaton of the basc varant of the nventory control system. Note that due to lmted volume of the gven paper, we ll use only one control parameter from each par: wth the omer t s the reorder pont (the second parameter order quantty s fxed and determned n Table ), wth the wholesaler t s the stock level (the nterval between orders, as t has been notced, equals to 0 days). Wth the account of the above assumptons about the economc ndependence of the partcular omers and the wholesaler, t s suggested to use the algorthm of the step-by-step optmzaton. At each step of the proposed algorthm there s determned the value of the control parameter for the selected structural enterprse (frst omers, then wholesaler), whch, n the consdered range of ts change, gves the mnmal value of the average cost per year n the nventory system. Due to the llustratve character of the gven artcle, the step of change of the control parameters wth the omer s taken as 0 unts, and wth the wholesaler 50 unts. Let us consder the soluton of the task n more detal. Step Usng the data of the basc varant (see Table ), let us perform the smulaton of the stock system by changng the value of the reorder pont, wth Customer, n the range from 0 to 0 unts wth the step of 0, gettng for each of the ponts 00 realzatons. The results of the smulaton are shown n Fg.0. Note that for the gven steps of the control parameter changng the best result s acheved for reorder pont =90 unts, where for 00 realzatons average cost E for one year perod (see () equals 88 EU. Step Fgure 0: Average Total Expenses per Year n Inventory Control System (Step ) Usng the data receved at step, let us perform the smulaton of the stock system, changng the value of the reorder pont, wth Customer, n the range from 0 to 0 unts. The results of the smulaton are shown n Fg.. For the gven steps of the control parameter changng the best result s acheved for reorder pont =00 unts, where average cost E for one year perod equals 8 EU. Step Fgure : Average Total Expenses per Year n Inventory Control System (Step ) Usng the data receved at step, let us perform the smulaton of the stock system, changng the value of the reorder pont, wth Customer, n the range from 5 to 40 unts. The results of the smulaton are shown n Fg.. For the gven steps of the control parameter changng the best result s acheved for reorder pont =00 unts, where average cost for one year perod E equals 65 EU. Fgure : Average Total Expenses per Year n Inventory Control System (Step ) Step 4 Usng the data receved at step, let us change the level of the desred stock S wth the wholesaler n the range from 900 to 450 unts and perform the smulaton for dfferent S values. The results of the smulaton are shown n Fg.. Note that for the gven steps of the control parameter S changng the best result s acheved for reorder pont S=000 unts, where for 00 realzatons average cost for one year E perod equals 57 EU (see () and (5)). The results of the consdered steps are presented n Table. Note, that the optmal values of parameters receved after each step are underlned. We cannot but notce that due to the change of values for the selected control parameters, we have managed to receve the best varant from the consdered ones whch provdes the reducton of cost E n the nventory control system, as compared wth the source varant, by 094 EU or by 4,84%. It s clear that the gven value cannot be seen as the mnmal one snce we have changed only

7 one from the each pare of control parameters and used qute a large step of the parameters changng. Fgure : Average Total Expenses per Year n Inventory Control System (Step 4) Table : The results of optmzaton process Parameters eorder pont, unts eorder pont, unts eorder pont, unts Desred stock level S, unts Total expanses E, EU Base Values of control parameters Optmzaton steps n the model Step Step Step Step CONCLUSIONS The gven paper has shown the possblty of usng the ExtendSm 8 package for the smulaton of a two-level nventory control system for the homogenous product stocks wth the wholesaler s and omers warehouses, characterzed by a random demand for the product and random tme of product delvery. The man advantages of the consdered smulaton method of nventory control problems solvng are as follows: the clarty of the presentaton of results; frstly, t touches the case of analyss of expenses dependence on one control parameter wth fxng others; the possblty of fndng optmum soluton of an nventory problem n the case when realzaton of analytcal model s rather dffcult; the descrptve user nterface, and ablty to control any necessary parameter. The developed model can be used for examnng the dynamcs of the stocks level at the warehouses of the omers and the wholesaler and for searchng an optmal decson for the company havng ts own wholesale warehouse and a network of ts own enterprses-omers. The future plan s nvestgaton of dfferent varants of wholesaler s orderng polcy. Further gudelnes of the current research are the followng: to nvestgate mult-product nventory control problem wth lmtaton on the exstng resources (warehouse volume and monetary means). EFEENCES Chopra, S. and P. Mendl. 00. Supply Chan Management. Prentce Hall, London. Magableh G. M., Mason S J An ntegrated supply chan model wth dynamc flow and replenshment requrements. Journal of Smulaton, Vol., Kopytov, E.; Greenglaz, L.; Muravjov, A. and E. Puznkevch Modelng of Two Strateges n Inventory Control System wth andom Lead Tme and Demand. Computer Modelng & New Technologes, Vol. (), ga: Transport and Telecommuncaton Insttute, -0. Kopytov E., Greenglaz L., Tssen F Inventory Control Model for the Chan Producer Wholesaler Customer. Proceedngs of the Internatonal Symposum on STOCHASTICS MODELS n ELIABILITY, SAFETY, SECUITY and LOGISTICS, (Beer Sheva, Israel, Feb. 5-7), oss, Sh. M. 99. Appled Probablty Models wth Optmzaton Applcatons. Dover Publcatons, INC., New York. Krahl D ExtendSm 7. Proceedngs of the 9th conference on Wnter smulaton: 40 years! (Dec. 09-). S.G. Henderson, B. Bller, M.-H. Hseh, J. Shortle, J.D. Tew, and.. Barton (Eds.). Washngton D.C. 6-. AUTHO BIOGAPHIES EUGENE A. KOPYTOV was born n Lgnca, Poland and went to the ga Cvl Avaton Engneerng Insttute, where he studed Computer Mantenance and obtaned hs engneer dploma n 97. Canddate of Techncal scence degree (984), Kev Cvl Avaton Engneerng Insttute. Dr.sc.ng. (99) and Dr.habl.sc.ng. (997), ga Avaton Unversty. Professor (999). Present poston: Chancellor of Transport and Telecommuncaton Insttute, professor of Computer Scence Department. Member of Internatonal Telecommuncaton Academy. Felds of research: statstcal recognton and classfcaton, modelng and smulaton, modern database technologes. Publcaton: 50 scentfc papers and teachng books, certfcate of nventons. AIVAS MUAVJOVS was born n ga, Latva and went to Transport and Telecommuncaton Insttute where he studed Computer Scences and obtaned Master of Natural Scences n Computer Scence n 009. Present studyng PhD student n Telematcs and Logstcs. Present poston: Deputy Head of IT Department.