Determination of a Desirable Inventory Policy in a three Echelon Multilayer Supply Chain with Normal Demand

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1 Internatonal Journal of Industral Engneerng & Producton Research March 1, Volume 3, Number 1 pp ISSN: Determnaton of a Desrable Inventory Polcy n a three Echelon Multlayer Supply Chan wth Normal Demand M. Amr *, M. Sef barghy, L. Olfat & S. H. Razav Haagha Maghsoud. Amr, Assocate Professor, Industral Management Department, Faculty of Management and Accountng, Allameh Tabatabae Unversty, Tehran, Iran Mehd Sef barghy, Assstant Professor, Techncal and Engneerng Department, Alzahra Unversty Laaya Olfat, Assocate Professor, Industral Management Department, Faculty of Management and Accountng, Allameh Tabatabae Unversty, Tehran, Iran Seyed Hossen Razav Haagha PhD Canddate, Industral Management Department, Faculty of Management and Accountng, Allameh Tabatabae Unversty, Tehran, Iran KEYWORDS Supply Chan; Mult Echelon Inventory System; Normal Dstrbuton; N-Fold Convoluton ABSTRACT Inventory control s one of the most mportant ssues n supply chan management. In ths paper, a three-echelon producton, dstrbuton, nventory system composed of one producer, a set of wholesalers and retalers s consdered. Costumers' demands can be approxmated by a normal dstrbuton and the nventory polcy s a knd of contnuous revew (R, Q). In ths paper, a model based on standard cost structure of nventory systems s developed and a heurstc algorthm s desgned to optmze the developed model. The applcaton of model s examned n a seres of desgned experments that are compared wth smulaton results. These comparsons verfy the valdty of the model. Regardng to real complextes n three-echelon systems analyss, the proposed method can have a wde applcaton n practcal problems wth the same consderatons and assumptons. In addton, ths method can be used to approxmate those systems that follow a Posson demand. 1 IUST Publcaton, IJIEPR, Vol. 3, No. 1, All Rghts Reserved. 1. Introducton Supply chan has become the maor and domnant paradgm of busness and competton. More than 4 years ago, Forrester n 1958 ntroduced the elements of a theory that today s called supply chan management (SCM) [1]. The concept of supply chan s whether that many experts beleved that competton s transferred from companes to chans. The extenson of SCM s * Correspondng author: Maghsoud. Amr Emal: amr@atu.ac.r Paper frst receved Dec. 11, 11, and n revsed form Feb. 5, 1. whether that n the past decade large nternatonal corporatons such Csco, Dell Computer, llette, Kodak, LEO, Motorola, Sony, 3M, Xerox and Wal- Mart have mplemented SCM. Moreover, nternatonal consultancy frms such as IDM busness Consultng Servces, A.T. Kearney, Cap emn, etc. have adopted SCM as an mportant busness area. In addton, a large number of unverstes and busness schools have ncluded SCM courses n ther currcula []. Many scholars and experts gave dfferent defntons for SCM that depend on ther vewpont and atttude. Walker [3] defned supply chan as the global network s used to delver goods and servces from raw

2 66 M. Amr, M. Sef barghy, L. Olfat & S. H. Razav Haagha Determnaton of a Desrable Inventory materal to fnal consumers by a desgned flow of nformaton, fscal goods and funds. There are many challenges that are latent n supply chan concept. The decsons made n SCM are manly about the flows between chans stages. Therefore, many scholars express the challenges and problems that SCM have tred to answer them. The scholars pont to the tme dmensons and effects of these challenges over performance of supply chans. From the vewpont of these theores, nventory control s one of the man challenges of SCM [4, 5, 6, and 7]. These theores have emphaszed that the nventory control and ts polces are the most mportant problems of supply chans. The man complexty of supply chans nvetory control s to acheve coordnaton between chans stages. Supply chans nventory control problem can be analyzed n the context of mult echelon nventory systems whch s a generalzed form of classc nventory model that ts orgnal form s proposed by Hars n 1913 [8]. Accordng to classcal nventory control, two man questons that these models try to answer are when and how much to order an tem such that the warehouse costs s mnmzed and satsfes the demand synchronously. The researchs n the feld of mult-echelon nventory systems formaly are orgnated n 195s and 196s. Clarck and Scarf [9] have done one of the most promnent ntal works n ths feld, whch n many researches are known as the founders of ths feld. One of the other promnent works n ths feld s Sherbrocke's [1] METR method whch s used greatly n further works. Contnuous revew models of multechelon nventory systems n 198s concentrated more on reparable tems n a Depot-Base system than on consumable tems. Lee and Monzade [11, 1] examned the problem of determnaton of optmal order quantty and nventory levels n such systems. Axsater [13, 14, and 15] consdered a two echelon system and proposed dfferent solutons to such a system wth dfferent assumptons on ts demand pattern or nventory polcy. Cachon [16] examned a two-echelon system wth random and dscrete demand, batch orderng, perodc revew and constant transferrng tme. rubbström and Wang [17] examned a multechelon producton nventory system and consdered the approxmaton of net present value as the obectve functon. Ben-Daya and Harga [18] also consdered an ntegrated producton nventory system wth random demand and a lnear relaton between lead tme and batch quantty. Sefbarghy and Akbar oker [19] examned a two echelon nventory system, conssts of one central warehouse and N retalers, wth posson demand and lost sales. Zhao et al. [] proposed fxed partton and power of two approach for a threeechelon system conssts of one manufacturer, one central warehouse and several retalers. ao and Wang [1] consdered a three echelon system and tred to mnmze ts total cost. Leeune and Margot [] also consdered an nteger lnear programmng model for a three-echelon nventory producton dstrbuton system. Haaghae and Saadfar [3] also studed a three echelon system conssts of two warehouse and N retalers. Also, Angelus [4], Nranan and Carallo [5], and Haaghae-Keshtel et al. [6] are nstances of the latests studes n ths feld. Some of authors used normal dstrbuton as an approxmaton of demand n mult echelon systems. In ths paper, a multechelon multlayer producton nventory dstrbuton system s consdered n whch customer demands n retalers can be approxmated by a normal dstrbuton. To analyse such a system, cost functon of each level of the system, ncludng manufacturer, wholesalers and retalers are estmated by usng approxmaton of each stages demand and nventory level dstrbuton. The rest of the paper s organzed as follow. The secton conssts of the problem defnton and ntroducton of modelng parameters and varables. The problem modelng and optmzaton are expressed n secton 3. In secton 4, statstcal desgn that s used to verfy the model s explaned and the results of smulaton studes, whch are desgned and done based on these experments, are shown and compared wth the proposed model s results. The paper s ended wth concluson and suggestons for future researchs n secton 5.. Problem Defnton Fgure 1 llustrates the problem under the study n ths paper. As can be seen n Fg.1, the system conssts of a central manufacturer that serves N wholesalers. These wholesalers themselves serve several reatalers. There are no nterrelatonshps between the retalers of one wholesaler wth other wholesalers or ther retalers. It s assumed that the customers demand, that retalers faced wth, can be approxmated based on a normal dstrbuton. Also, the nventory polcy of retalers and wholesalers follows a contnuos revew (R, Q) polcy. Unsatsfed demands are assumed backordered wth a certan cost for each unsatsfed unt n retalers and wholesalers, and these backordered orders wll be satsfed based on a frst come, frst serve (FCFS) approach. Also, there s no backorderng cost n manufacturer. It s assumed that the batch szes of retalers and wholesalers are known and are determned based on a determnstc model, as many smlar papers such as Axsater [13], and Sefbarghy and Akbar okar [19] done before to smplfy the problem. Another assumpton s that the retalers and wholesalers leadtme s constant. Internatonal Journal of Industral Engneerng & Producton Research, March 1, Vol. 3, No. 1

3 M. Amr, M. Sef barghy, L. Olfat & S. H. Razav Haagha Determnaton of a Desrable Inventory Modelng Process In ths secton, the modelng process that s used to model the above problem s explaned Retalers Model Takng nto account one of the retalers n fg.1, lke th retaler of wholesaler. the method used here s taken from axsater [7]. In ths system, each retaler can be analysed as an ndependent depot that faces wth customers demand. The followng notatons are defned: x max x, )1( x max x, () Fg. 1. Three echelon multlayer nventory system The followng notatons wll be used: m : Number of wholesalers; n : Number of retalers that ordered to wholesaler 1,,,m ; N : Demand dstrbuton of th retaler s of, wholesaler that follows a normal dstrbuton wth mean and varance ; R : Reorder pont of th retaler of wholesaler ; Q : Common batch sze of retalers of wholesaler ; R : Reorder pont of wholesaler ; Q : Batch sze of wholesaler ; : Holdng cost per unt at th retaler of wholesaler ; : Penalty cost per backordered unt at th retaler of wholesaler ; : Lead tme of orders at th retaler of wholesaler ; : Holdng cost per unt at wholesaler ; : Penalty cost per backordered unt at wholesaler ; : Lead tme of orders at wholesaler ; : Holdng cost per unt at manufacturer; Q : Producton batch sze of Manufacturer; P : Processng rate of manufacturer; By attenton to nventory level (IL), holdng cots wll be IL and backorderng cost wll be IL Snce x x as follows: IL IL. x, the sum of costs can be expressed IL IL IL IL Here, E IL s the expected value of backorders. If shows the mean demand n per tme unt, then the watng tme for each ordere s equal to E IL. Ths formula shows an applcaton of lttle formula n queng theory. Based on thrd relaton n Eq. (3), the expected costs of ths retaler are: EIL π EIL C r R Q μ π F x Where, retaler dx (4) s the average demand n lead-tme of, and x accordng to Eq. (5). F s dstrbuton functon (3) The problem here s to determne retalers and wholesalers reorder ponts, whle ther batch sze s constant, such that ther holdng and backorderng costs are to be mnmzed and to determne producton batch sze of manufacturer such that ts costs are to be mnmzed. F F x PIL x x 1 Q RQ R 1 Q RQ R u x μ du σ u x μ 1 υ du σ (5) Internatonal Journal of Industral Engneerng & Producton Research, March 1, Vol. 3, No. 1

4 68 M. Amr, M. Sef barghy, L. Olfat & S. H. Razav Haagha Determnaton of a Desrable Inventory In whch, (x) s called normal dstrbuton s loss functon as follows and ts values are gven n Axsater [7]: x υ x υdυ x x1 υx x )6( By nsertng related relatons and defntons, the followng relaton s acheved: r R Q μ C R Q σ π Q R u μ σ du )7( That s lead-tme demand s varance. Now, the H(x) functon s defned as follows that ts values are gven n Axsater [7]: H 1 x vdv x 1 1-υx x Snce, H x x xx )8(, H(x) s decreasng and convex. By nsertng Eq. (8), the cost functon (7) can be rewrtten as follow: C r R R H H Q R Q Q (9) Snce H(x) s a convex functon, the above functon s convex related to R. Then: dc dr R Q Q R )1( Because the dc dr s ncreasng related to R, C s a convex functon of R that ts optmal value can be derved by lettng dc dr that can be achved based on functon s value and by a smple search algorthm. For each retaler, the fll rate can be measured as follows: FR 1 F 1 σ R μ R Q σ Q σ μ )11( Based on Eq. (1) and (11), the fll rate n optmal soluton s: FR (1) 3-. Wholesalers Model The second stage under consderaton s related to wholesalers. Suppose the wholesaler whch serves n retalers. The man pont here s to determne the wholesaler s demand dstrbuton. If IL(t+L) s the wholesaler s nventory level at tme t+l, IP(t) wll be ts nventory poston at tme t, and D(t,t+L) s random demand at tme nterval (t, t+l), then the followng relaton s hold: IL t L IPt Dt,t L (13) In fact, by lettng an arbtrary tme t, nventory level at tme t+l wll be nventory poston at tme t mnus demand n (t, t+l) nterval. Also, accordng to Axsater [13, 15, 7], nventory poston of wholesaler at tme t, IP(t), unformly s dstrbuted on R 1,R,,R Q. Now, let the th retaler that orders to wholesaler. ths retaler s demand follows a normal dstrbuton N R,. Now, and ts nventory polcy s, Q the probablty that a certan retaler fll an order n a perod of F that can be tme unt s equal to calculated from Eq. (5) by lettng and. Now, the random varable O s defned as follows: O Q f retaler fll anorderna tment ervalwth lenghtτ otherwse R (14) The probablty that a certan retaler fll an order n F. In fact: ths tme nterval s equal to P Q 1 FR P O (15) Now, the random varable R X s defned as follow: 1 O Q X (16) O Internatonal Journal of Industral Engneerng & Producton Research, March 1, Vol. 3, No. 1

5 M. Amr, M. Sef barghy, L. Olfat & S. H. Razav Haagha Determnaton of a Desrable Inventory 69 Ths random varable has a bernoll probablty functon wth success probablty P. So, the number of wholesaler s order s equal to D n X 1, whch s the sum of n bernoll random varables wth dfferent success probabltes. Probablty dstrbuton of D s n fact the n-fold convoluton of a set of bernollan varables. The probablty that n retalers ordered s: n 1 p x n 1 x p, 1 x n (17) 1 Now, the probablty D n P s the summaton of n n dfferent combnatons that n whch n retalers ordered and each combnaton's probablty s calculated based on Eq. (17). Note that varable D can take ts values between 1,,,n and t s notable that D can take these values n dfferent confguratons. So, the wholesaler s demand, Wd, take ts values from Q, Q,,n Q and ts expected value s: E n Wd Q 1 p n (18) Then, the dstrbuton of nventory level wll be obtaned. Frst, t s necessary to note that nventory poston takes ts values from the set J R 1,R,,R Q. Based on Eq. of (13), the probablty functon IL obtaned as: P IL PIP k PD L 1 Q R Q kmax R P Wd 1, k R Q kmax R 1, P wll be k (19) by attendng to frst relaton n EQ. (3), wholesaler s average cost s equal to: C w, R EIL EIL PIL PIL J Note that n Eq. (), IL J () E s the dfference between mean nventory poston and average demand n leadtme. Also, J J s the set of postve values and J J s the set of negatve values of nventory poston. In ths case, the lower bound of wholesaler s reorder pont s n Q and the optmal reorder pont can be found by a smple search procedure and lettng dfferent values of R n Q n Eq. () that n each step, reorder pont s ncreased by Q unt and fnally the reorder pont wth lowest cost s chosen. For per wholesaler, the fll rate can be calculated as follows: J IL PIL FR P (1) It s noted that dstrbuton functon (17) can be approxmated by a bnomal dstrbuton, when all retalers have a same demand pattern. In addton, ths dstrbuton can be approxmated based on a normal dstrbuton wth mean (Eq. ()) and varance (Eq. (3)), when the number of retalers s ncreased. n Q p () n 1 1 p 1 p (3) 3-3. Manufacturer Model The last stage n model development s manufacturer model. Accordng to classcal economc producton quantty (EPQ), the producton costs of manufacturer wll be [7]: C P Q 1 d ~ p d ~ A (4) p Where, d ~ s the expected value of manufacturers demand and, accorng to Eq. (), obtaned m as d ~. By replacng ths value and takng 1 dervatve of Eq. (4) and let t equal zero, the optmal producton batch quantty wll be obtaned: A E d ~ 1 E d ~ p (5) Q 4. Model Verfcaton In order to determne the power of approxmatons that are used n ths paper, a set of problems are desgned and solved wth proposed approach. Snce there are not any prevous numercal problems as a reference to compare presented approach wth, a problem set s developed to verfy the presented approach. These experments are desgned based on Internatonal Journal of Industral Engneerng & Producton Research, March 1, Vol. 3, No. 1

6 7 M. Amr, M. Sef barghy, L. Olfat & S. H. Razav Haagha Determnaton of a Desrable Inventory two factoral desgns n the context of desgn of experments (DOE). Thrteen factors are consdered as affectve on the problem. Among these factors, the holdng cost of wholesalers and retalers s consdered equal to one and the lead-tme of them s consdered as one unt of tme. Table1 llustrates the hgh and low levels of remanng 9 factors. When the quantty of factors s hgh, the numbers of experments that must be desgned are ncreased dramatcally. In ths experment, by 9 factors, the numbers of desgned experments based on factoral desgns are 51 experments. Therefore, a fractonal factoral 9 4 desgn of s used. Such a desgn s consttuted by IV 5 development of a full factoral desgn by A, B, C, D and E factors as the basc desgn and then the four columns of F=BCDE, = ACDE, H=ABDE, and J=ABCE are attached to desgn. The generators of ths desgn are I=BCDEF=ACDE=ABDEH=ABCEJ [8]. In the next step, 3 problems are desgned and solved wth proposed approach. Every problem s soluton ncludes the values of reorder ponts for each retalers and supplers wth the producton batch sze of manufacturer (because of the lmtaton n space, these results are not presented here). On the other hand, problems wth above confguratons are smulated usng ARENA and the resultng costs of smulated systems are compared wth costs of proposed method. Table llustrates the costs of proposed method and resulted costs of problems smulaton. Accordng to table, all of the resulted costs based on proposed method are n the ntrerval of average smulaton cost ± half wdth that can be nterpreted as an evdence of proposed method s credt. Also, f a 95% confdence nterval of pared observaton s calculated for the mean of dfference between proposed method and smulaton costs, ths nterval wll be μd that ncludes zero and shows that the assumpton of dfference between costs s nonsgnfcant: Tab. 1. Postve and negatve levels of experments factors Code Factor Postve level (+) Negatve level (-) A Number of manufacturer s producton lnes 6 1 B Manufacturer s holdng cost 1 1 C Number of work statons n each lne 15 5 D Producton capacty of each staton 5 1 E Number of wholesalers 13 F Wholesalers backorderng cost 1 5 6, 15 un,5 Number of retalers for each wholesaler un * H Demand rate for each retaler un,5 un.5, J Retalers backorderng cost *un: unform dstrbuton Run Tab.. Comparson of the results of proposed method and smulaton Costs Costs Approxmaton Smulaton Half wdth Run Approxmaton Smulaton Half wdth Internatonal Journal of Industral Engneerng & Producton Research, March 1, Vol. 3, No. 1

7 M. Amr, M. Sef barghy, L. Olfat & S. H. Razav Haagha Determnaton of a Desrable Inventory 71 References [1] Mentzer, J.T., Supply Chan Management. Sage Publcaton, Calforna, 1. [] Jespersen, B.D., Skoett-larsen, T., Supply Chan Management: n Theory and Practce. Copenhagen Busness School Press, Copenhagen, 5. [3] Walker, W.T., Supply Chan Archtecture: a Blueprnt for Networkng the Flow of Materal, Informaton and Cash. CRC Press, Florda, 5. [4] Smch Lev, D., Kamnsky, P., Smch Lev, E., Managng the Supply Chan: the Defntve ude for the Busness Professonal. Mc-raw Hll, New York, 4. [5] Chandra, C., Kamran, A.K., Mass Customzaton: a Supply Chan Approach. Kluwer Academc Publshers, New York, 4. [6] Chopra, S., Mendel, P., Supply Chan Management. Prentce Hall, New York, 7. [7] Wsner, J.D., Tan, K.C., Leong,.K., Prncples of Supply Chan Management: A Balanced Approach. South-Western Cengage Learnng, Oho, 8. [8] Roach, B., Orgn of the Economc Order Quantty Formula: Transcrpton or Transformaton?. Management Decson, Vol. 43, Issue 9, 5, pp [9] Clark, A.J., Scarf, H., Optmal Polces for a Mult- Echelon Inventory Problem. Management Scence, Vol. 6, No. 4, 196, pp [1] Sherbrooke, C.C., Metrc: A Mult-Echelon Technque for Recoverable Item Control. Operatons Research, Vol. 16, No. 1, 1968, pp [11] Lee, H.L., Monzadeh, K., Operatng Characterstcs of a Two-Echelon Inventory System for Reparable and Consumable Items under Batch Orderng and Shpment Polcy. Naval Research Logst, Vol. 34, Issue 3, 1987, pp [1] Lee, H.L., Monzadeh, K., Two-Parameter Approxmatons For Mult-Echelon Reparable Inventory Models Wth Batch Orderng Polcy. IIE Transactons, Vol. 19, Issue, 1987, pp [13] Axsater, S., Evaluaton of Installaton Stock Based (R,Q)-Polces for Two-Level Inventory Systems wth Posson Demand. Operatons Research, Vol. 46, No. 3, 1998, pp [14] Axsater, S., Exact Analyss of Contnuous Revew (R, Q) Polces n Two-Echelon Inventory Systems wth Compound Posson Demand. Operatons Research, Vol. 48, Issue 5,, pp [15] Axsater, S., Approxmate Optmzaton of a Two-Level Dstrbuton Inventory System. Internatonal Journal of Producton Economcs, Vol. 81-8, Issue 1, 3, pp [16] Cachon,.P., Exact Evaluaton of Batch-Orderng Inventory Polces n Two-Echelon Supply Chans wth Perodc Revew. Operatons Research, Vol. 49, No. 1, 1, pp [17] rubbström, R.W., Zhpng, W., A Stochastc Model of Mult-Level/Mult-Stage Capacty-Constraned Producton Inventory Systems. Internatonal Journal of Producton Economcs, Vol. 81-8, Issue 1, 3, pp [18] Ben-Daya, M., Harga, M., Integrated Sngle Vendor Sngle Buyer Model wth Stochastc Demand and Varable Lead Tme. Internatonal Journal of Producton Economcs, Vol. 9, Issue 1, 4, pp [19] Sefbarghy, M., Akbar Jokar, M.R., Cost Evaluaton of a Two-Echelon Inventory System wth Lost Sales and Approxmately Posson Demand. Internatonal Journal of Producton Economcs, Vol. 1, Issue, 6, pp [] Zhao, Q.H., Chen, S., Zangb, C.X., Model and algorthm for nventory/routng decson n a three-echelon logstcs system. European Journal of Operatonal Research, Vol. 191, Issue 3, 8, pp [1] ao, J., Wang, D., Smulaton-Based Optmzaton Method for Three-Echelon Network Inventory System of a Supply Chan. Proceedng of Control and Decson Conference, Chnese, Yanta, 8, pp [] Leeune, M.A., Margot, F., Integer Programmng Soluton Approach for Inventory-Producton Dstrbuton Problems wth Drect Shpments. Internatonal Transactons n Operatonal Research, Vol. 15, Issue 3, 8, pp [3] Haaghae-Keshtel, M., Saadfar, S.M., Dervng the Cost Functon for a Class of Three-Echelon Inventory System wth N-Retalers and One-for-One Orderng Polcy. The Internatonal Journal of Advanced Manufacturng Technology, Vol. 5, No. 1-4, 1, pp [4] Angelus, A., A Mult Echelon Inventory Problem wth Secondary Market Sales. Management Scence, Vol. 57, No. 1, 11, pp [5] Nranan, S., Carallo. F.W., Supply Performance n Mult-Echelon Inventory Systems wth Intermedate Product Demand: A Perspectve on Allocaton. Decson Scence, Vol. 4, Issue 3, 11, [6] Haaghae-Keshtel, M., Saadfar, S.M., Ha, R., Determnaton of the Economcal Polcy of a Three- Echelon Inventory System wth (R, Q) Orderng and Informaton Sharng. The Internatonal Journal of Advanced Manufacturng Technology, Vol. 5, No. 5-8, Internatonal Journal of Industral Engneerng & Producton Research, March 1, Vol. 3, No. 1

8 7 M. Amr, M. Sef barghy, L. Olfat & S. H. Razav Haagha Determnaton of a Desrable Inventory [7] Axsäter, Sven. Inventory Control. Kluwer Academc Publshers, Boston, 6. [8] Montgomery, D.C., Desgn and Analyss of Experments. John Wley & Sons, New York, 8. [9] umus. A.T., uner, A.F., Mult-Echelon Inventory Management n Supply Chans wth Uncertan Demand and lead Tmes: Lterature Revew from an Operatonal Research Perspectve. Proceedngs of Insttuton of Mechancal Engneers Part B: Journal of Engneerng Manufacture, Vol. 1, No. 1, 7, pp Internatonal Journal of Industral Engneerng & Producton Research, March 1, Vol. 3, No. 1