SIMULATION MODEL OF CURRENT STOCK OF DIVISIBLE PRODUCTS IN EXTENDSIM ENVIRONMENT

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1 SIMULATION MOEL OF CURRENT STOCK OF IVISIBLE PROUCTS IN EXTENSIM ENVIRONMENT Eugene Kopytov and Avars Muravjovs Transport and Telecommuncaton Insttute, Lomonosova Stree Rga, LV-9, Latva E-Mal: KEYWORS Current stock, dvsble producton, smulaton, contnuous model, optmzaton, ExtendSm package ABSTRACT In the gven paper we nvestgate the problem of constructng a smulaton model for the optmzaton of current stock of dvsble productons n the warehouse Crteron of optmzaton s mnmum of average expenses for goods holdng, orderng and losses from defct and damage to the goods per tme of season The ExtendSm 8 package has been used as the means of smulaton The numercal example of the problem solvng s presented INTROUCTION One of the central problems of the nventory control theory s to fnd an optmal soluton to the task of orderng productons to be suppled, and man result of the task s the answer to two basc questons: how much to order and when to order Of no less nterest t s the task of determnng the current stock of certan producton (sold by the pece or ndvsble producton and dry or dvsble producton) at any gven moment of a fxed tme perod, wth any random factors taken nto account By "current stock" we denote the quantty of the producton accumulated n the stock, whch s used for dstrbuton n the lght of the crcumstances Qute a lot of dfferent types of models of varyng complexty, purpose and adequacy have been developed n the nventory control theory (Chopra and Mendl, Magableh and Mason 9) We can classfy these models takng n account dfferent ther propertes: determnstc and stochastc, lnear and nonlnear, sngle- and mult-produc dscrete and contnuous models, etc Most of the exstng mathematcal models n ths theory consder ndvsble productons (Stewart 4, Chopra and Mendl, Kopytov et al 7, Kopytov and Muravjov ) In the present research we nvestgate the problem of the nventory control system of dvsble productons In prevous works we have nvestgated the problem of constructng contnuous and unsteady mathematcal models for determne the volumes of current stock of dvsble productons n one or several nterconnected warehouses usng apparatus of mathematcal physcs and contnuum prncple (Kopytov et al ) The smple models are constructed usng the theory of ordnary dfferental equatons; for constructon of more complex models the theory of partal dfferental equatons are appled (Mlsten 995, Kuznetsov 7, Tkhonov and Samarsky 4) It should be noted that the practcal mplementaton of ths approach and fndng a numercal soluton s a rather complcated and tme-consumng task For some proposed models we have found an analytcal soluton n the closed form, and for some of proposed models the dscretzaton s carred out usng stable dfference schemes (Guseynov et al ) In the gven paper we nvestgate the problem of constructng smulaton model for the optmzaton of current stock of dvsble productons Ths approach s certanly easer to mplemen but t has a lower accuracy of the obtaned optmal solutons For the consdered problem solvng the authors have appled ExtendSm package, whch s wdely used for varous systems modellng, but has not been appled for smulaton of nventory control system of dvsble producton Therefore, the authors set the goal to show the effectveness of constructng contnuous smulaton model of current stock of dvsble products n ExtendSm envronment ESCRIPTION OF THE MOEL We consder a stochastc nventory control model for the stock wth homogeneous dvsble producton The schema of the current stock of dvsble producton replenshment and dstrbuton s shown n Fgure Fgure : Flows of Producton n the Stock enote as z (t) the quantty of products n stock n the tme moment t escrbng a contnuous replenshment and dstrbuton of the current stock we consder the Proceedngs 7th European Conference on Modellng and Smulaton ECMS Webjørn Rekdalsbakken, Robn T Bye, Houxang Zhang (Edtors) ISBN: / ISBN: (C)

2 dz ( t) change rate of the current stock volume at a gven dt tme t (Kopytov et al ) Let us consder the functons whch determne the dz ( t) change rate : dt - functon S ( determnes contnuous replenshment of the current stock characterzed by nput flows of producton q ( t), q( t), q3( t) ; - functon S ( determnes contnuous dstrbuton of the current stock characterzed by output flows of producton x t), x ( t), x ( t), x ( ) ) ( 3 4 t The dfference S( S( s a measure of the change of the current stock volume, e d t) S( S( dt The product replenshment conssts of three addtve flows (components), namely: from regular replenshment of the stock, whch s desgnated as q ( ) ; from rregular replenshment by sngle orders q ( ) ; and from random replenshment q 3( t ) (for nstance, a random stock replenshment due to an exceptonally hgh qualty of producton or an exceptonally low prce, or because of an expected sudden defct of partcular products, etc), whch can be descrbed mathematcally as a random quantty that desgnatng the total volume of producton that has been delvered nto a partcular warehouse from random and/or non-random sources by the tme t t The product dstrbuton conssts of four addtve flows (components) namely: regular dstrbuton whch s denoted as x ( ) ; rregular dstrbuton x ( ) ; possble t losses x 3( t ) of dvsble productons whch take place durng holdng and dstrbuton processes (for example, for petroleum productons t s evaporaton, for gran man reasons of losses are gnawng anmals and nundaton); and random (rare event) dstrbuton (smlar to random replenshmen there can be crcumstances due to whch random dstrbuton takes place) that can be mathematcally presented as a random flow x 4 ( t ) desgnatng the total volume of productons that was taken away from the warehouse by the tme t due to random crcumstances We assume that man parameters of nput and output producton flows are constant (unchanged) durng fxed tme span T [ t s, te ], where t s and te are day of start and day of the end of the perod T, respectvely Usually for petroleum and agrcultural dvsble productons (whea rce, meal, etc) tme perod T s the season perod occupyng 3 months or 9 days Let us consder the ntroduced components n detal The product replenshment components The component q ( t ) can be nterpreted as guaranteed replenshment of the current stock of dvsble producton, that takes place regularly n fxed moments of tme t, t, t,, t k accordng to a contract durng the tme perod T wth the constant volume of products Q const The quantty Q s one of control parameters of the optmzaton model The component q ( ) obvously depends on random demand for products durng tme perod and also R, whch desgnates the mnmal on a certan quantty volume of stock n a partcular warehouse necessary for admnsterng unregulated stock replenshment on condton that such replenshment s guaranteed In other words, n the moment of tme, when the stock level falls tll certan level R, a new order s placed The quantty R s called as reorder pont We assume that demand has a normal dstrbuton wth a mean and a standard devaton In consdered task the reorder pont s calculated by followng formula: R ( t) [ ( L) X k( L)] S, () where L s lead tme (tme between placng an order and recevng t); (L) s average demand for products durng lead tme L ( n consdered task lead tme L s constant); k(l) s number of cases of regulated (accordng to contracts) dstrbuton X of products durng lead tme L, (number k(l) depends on the moment of tme t, when the order for delvery s placng); S s a safety coeffcent whch determnes certan reserve stock of products, S We suppose that n case of producton defct the last cannot be covered by expected order In consdered optmzaton model safety coeffcent S s the second control parameter The flow q ( ) determnes the volume of producton 3 t Q3 that s delvered nto the warehouse by the tme t due to random (rare event) crcumstances from random and/or non-random sources In consdered task, we assume that the probablty p3 of occurrence of ths event durng tme unt s known, and t s a qute rare event; for example, for one day we assume that p 3 So, the vector Q { Q ( T), Q ( T), Q3 ( T)} determnes total volume of products replenshment delvered durng

3 tme perod T, where Q ( T), Q ( T), Q3( T) are regular, rregular and random (rare event) replenshments durng perod T The products dstrbuton components The component x ( t ) can be nterpreted as "strong" (guaranteed) constant dstrbuton of the current stock of dvsble productons, e the volume of the current stock s regularly taken away from the warehouse n fxed moments of tme t, t, t,, t k accordng to a contract durng the tme perod T wth the constant volume of product X The component x ( ) depends on random demand for products durng tme unt and regular dstrbuton, whch determnes the stock volume of dvsble productons allowng for ts unregulated dstrbuton, The component x 3( t ) descrbes possble losses of the dvsble productons n current stock n the processes of storage and dstrbuton For nstance, f we have the ol productons stock, losses wll result from the evaporaton and/or from the leakage through the reservors; f we have the agrcultural productons stock (whea rce, meal, etc), there wll be unavodable losses caused by pests, flood, strong wnds, etc Apparently, the value of these losses s a random one The flow x 4( t ) determnes quantty X 4 desgnates the total volume of productons (unexpected dstrbuton wth a large proft) that has been removed from the warehouse by the tme t due to random (rare event) crcumstances In consdered task we assume that the probablty p4 of occurrence of ths event durng tme unt s known, and t s a qute rare event; we assume that for one day p 4 In the consdered problem we suppose that the followng economc parameters are known: For -th component of product replenshment (,, 3 ) the orderng cost of product C ( Q ) s a known functon of the products quantty Q, delvered durng tme perod T, and conssts of two addtve components, namely: constant c whch ncludes cost of the order formng and constant part of expenses of products transportaton, and varable component c ( Q ), whch O depends on the order quantty Q, C O ( Q ) c c ( Q ),,, 3 e We suppose that n the consdered nventory control system for,, 3 coeffcents c and c are () () (3) () () (3) dfferent: c ; c ; c () c () (), c c where c () s determned for one unt of delvered producton () () (3) Therefore we can wrte: C () C () C () The total orderng cost for tme perod T s determned by the followng formula: () () (3) E T) C ( Q ( T)) C ( Q ( T)) C ( Q ( T)) O ( O O O 3 The holdng cost of the product s proportonal to ts quantty n the stock and the holdng tme wth the coeffcent of proportonalty C The losses from the defct of the product are proportonal to the quantty of ts defct wth the coeffcents of proportonalty C whch are dfferent for each type of product dstrbuton At the same tme losses from the defct of the product for regular dstrbuton are the larges but for random (unplanned, rare event) dstrbuton these losses (e lost proft) are the lowes e CSH CSH CSH 3 Losses from damage and loss of product are proportonal to the cost of product unt C CS The total cost E(T ) n nventory system durng the season perod T s calculated by the followng formula: H SH j Fgure : Stock Smulaton

4 E( T), () O H where E O (T ) s orderng cost; (T ) s holdng cost; E SH (T) s shortage cost; E CS (T ) s losses from damage or loses of products durng tme perod T Prncpal am of the consdered task s to defne the optmal values of regular order quantty Q and safety coeffcent S for rregular replenshmen whch are control parameters of the model Crtera of optmzaton s mnmum of average total cost E(T ) durng tme perod T, whch can be calculated by formula () for average costs and loses E O (T ), E H (T ), E SH (T ) and E CS (T ) SH E H SIMULATION MOEL IN EXTENSIM 8 ENVIRONMENT For solvng the problems consdered above we have used smulaton method realzed n the ExtendSm 8 envronment (Strckland ) The package ExtendSm can be used to model contnuous, dscrete even dscrete rate, and agent based systems ExtendSm s desgn facltates every phase of the smulaton projec from creatng, valdatng, and verfyng the model, to the constructon of a user nterface that allows others to analyze the system (Kopytov and Muravjov ) Smulaton tool developers can use ExtendSm s bult-n compled language ModL to create reusable smulaton components All of ths s done wthn a sngle selfcontaned software program, whch does not requre external nterfaces, complers, or code generators For ths task mplementaton we have chosen contnuous smulaton model The created model conssts of four man parts: Stock, emand, Orderng costs and Total costs calculaton that are represented on Fgure -5 The purposes of blocks shown n Fgure -5 are gven n captons Let us consder the man sectons of the smulaton model CS Secton Stock (see Fgure ) In area # there are placed blocks that are responsble for scheduled delvery smulaton Area # s used for generaton emergency delvery orders (rregular replenshment) based on current stock level and tme between scheduled orders Next area #3 generates random delveres cheap that occurs one out of hundred cases ( p 3 )The stock s realzed n area #4 Secton emand (see Fgure3) s created for product dstrbuton smulaton and conssts of the blocks responsble for demand generaton There are four demand sources: random demand s realzed n area #5, scheduled demand (regular dstrbuton) n area #6, random demand wth dfferent dstrbuton n area #7, and holdng n area #8 Fgure 4: Costs Calculatons Next two sectons Costs and Orderng Costs, shown n Fgure 4 and Fgure 5 accordngly, nclude costs calculatons blocks, namely: holdng, orderng and losses costs for all delvery sources descrbed above The total holdng cost E H (T ) s calculated n the blocks of areas #9 and # Current stock s calculated n blocks of area # Fgure 3: emand Generaton Fgure 5: Orderngs Costs Calculatons

5 Fgure 6: Example of Smulaton Process Blocks n area # are used for order costs calculatons from each delvery sources The total cost E(T ) n nventory system s calculated n blocks n area #3 An example of the nventory control process smulaton (one realzaton) s shown n Fgure 6 The plot shows the current stock of certan producton durng perod of season T Usng created smulaton model we can fnd the optmal soluton for nventory control of stock of dvsble producton One of examples s consdered n the next secton EXAMPLE AN OPTIMIZATION Let s consder a stochastc nventory control model for the stock wth homogeneous dvsble producton shown n Fgure Table and Table descrbe man parameters of the products replenshment and dstrbuton Table : Intal ata of Product Replenshment Source Amount Schedule Cost / un conventonal unts (CE) Regular 3 Bmonthly Irregular Accordng to stock level 3 Random Random, p= 7 Table : Intal ata of Product strbuton Source Amount Schedule Irregular Regular 5 Random (rare event) emand, normal dstrbuton aly =7; 3 Monday, Wednesday, Fryday Random, p= Holdng loses 5% of daly stock aly For optmzaton process we consder that amount of regular replenshment Q can be changed from to 4 and safety level S from to 5 The perod of smulaton s 3 months (one season perod) and the number of realzaton s The optmzaton model was done by ExtendSm optmzaton tool that gves us flexble soluton for optmal result searchng The Fgure 7 represents optmzaton process n ExtendSm envronment For the gven steps of the control parameter Q and S changng, the best result s acheved at pont Q = 435 unts and S =34, where for replcatons the average total cost of the one season perod equals Fgure 7: Optmzaton Process

6 54695 CE It gave us total costs reducton from CE (for ntal values of control parameters Q =3 and S =3) to CE CONCLUSIONS In the gven paper the smulaton model for the optmzaton current stock of dvsble productons s created For the problem solvng authors have used a smulaton contnuous model realzed n the package ExtendSm 8, whch s the most powerful and flexble smulaton tool for analysng, desgnng, and operatng complex systems n the market The results of smulaton ndcate the good feasblty of the applcaton of ExtendSm 8 n the tasks of nventory control of dvsble productons Comparng wth analytcal approach used n authors prevous works the consdered smulaton model of nventory control of dvsble productons provdes the researcher wth: the clearness of results presentaton; the possblty of fndng optmum soluton of an nventory problem n the case when realzaton of analytcal model s rather dffcult The man problem of the proposed smulaton approach s ntal values of control parameters and ther changng range determnaton n the searchng the best soluton Further gudelne of the current research s to consder the nventory control models for determne the volumes of current stock of dvsble productons n several nterconnected warehouses REFERENCES Chopra, S and P Mendl Supply Chan Management Prentce Hall, London Guseynov, ShE; Kopytov, EA; Puznkevch, E On contnuous models of current stock of dvsble productons ynamcal Systems, fferental Equatons and Applcatons" Vol I,, Publshed by the Amercan Insttute of Mathematcal Scences (AIMS), 6-63 Kopytov, E; Greenglaz, L; Muravjov, A and E Puznkevch 7 Modelng of Two Strateges n Inventory Control System wth Random Lead Tme and emand Computer Modelng & New Technologes, Vol (), Rga: Transport and Telecommuncaton Insttute, -3 Kopytov, E, Guseynov, Sh, Puznkevch, E, Greenglaz, L Contnuous Models of Current Stock of vsble Productons Computer Modellng and New Technologes, Vol 4, No 4, p 9-3 Kopytov, E and A Muravjov Smulaton of nventory control system for supply chan producer wholesaler clent n ExtendSm envronment Proceedngs of the 5 th European conference on modelng smulaton (ECMS-) (Krakow, June 3-4) Poland, Krahl, 7 ExtendSm 7 Proceedngs of the 39th conference on Wnter smulaton: 4 years! (ec 9-) SG Henderson, B Bller, M-H Hseh, J Shortle, J Tew, and RR Barton (Eds) Washngton C 6-3 Kuznetsov, F, 7 Stochastc fferental equatons: theory and practce of numercal solutons St Petersburg: Polytechnc Unversty (In Russan) Magableh, G M, Mason, S J 9 An ntegrated supply chan model wth dynamc flow and replenshment requrements Journal of Smulaton, Vol 3, Mlsten, GN, 995 Numercal ntegraton of stochastc dfferental equatons New York-London: Kluwer Academc Publshers Stewar R 4 Smulaton The practce of model development and use Wley Strckland, J screte Event Smulaton usng ExtendSm 8 Lulu Tkhonov, AN and AA Samarsky 4 Equatons of Mathematcal Physcs Moscow: Lomonosov MSU Press (In Russan) ACKNOWLEGEMENTS The artcle s wrtten wth the fnancal assstance of European Socal Fund Project Nr 9/59/P//9/IPIA/VIAA/6 (The Support n Realsaton of the octoral Programme Telematcs and Logstcs of the Transport and Telecommuncaton Insttute) AUTHOR BIOGRAPHIES EUGENE A KOPYTOV was born n Lgnca, Poland and went to the Rga 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 rscng (99) and rhablscng (997), Rga Avaton Unversty Professor (999) Present poston: Head of Software Engneerng epartment of Transport and Telecommuncaton Insttute, professor of Computer Scence Member of Internatonal Telecommuncaton Academy Felds of research: statstcal recognton and classfcaton, modelng and smulaton, modern database technologes Publcaton: 8 scentfc papers and teachng books, certfcate of nventons AIVARS MURAVJOVS was graduated at Transport and Telecommuncaton Insttute where he studed Computer Scences and obtaned Master of Natural Scences n Computer Scence n 9 Present studyng Ph student n Telematcs and Logstcs Felds of research: nventory control systems, smulaton