The Cost Function for a Two-warehouse and Three- Echelon Inventory System

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1 Intrnational Journal of Industrial Enginring & Production Rsarch Dcmbr 212, Volum 23, Numbr 4 pp IN: Th Cost Function for a Two-warhous and Thr- Echlon Invntory ystm M. Aminnayri * & M. Hajiaghai-Kshtli Majid Aminnayri, Dpartmnt of Industrial Enginring, Amirkabir Univrsity of Tchnology, Thran, Iran Mostafa Hajiaghai-Kshtli, Dpartmnt of Industrial Enginring, Amirkabir Univrsity of Tchnology, Thran, Iran KEYWORD Thr-chlon invntory systm, On-for-on Rplnishmnts, Poisson Dmands ABTRACT In this papr, th cost function for a thr-chlon invntory systm with two warhouss is drivd. Transportation tims ar constant and rtailrs fac indpndnt Poisson dmand. Rplnishmnts ar on-for-on. Th lad tim of a rtailr is dtrmind not only by th constant transportation tim but also by th random dlay incurrd du to th availability of stock at th warhouss. W considr two warhouss in th scond chlon which may lads to having mor dlays which wr incurrd in th warhouss and facing diffrnt bhaviors of indpndnt Poisson dmands. Bcaus th rplnishmnt policy is bas stock, th obtaind function can also b usd in diffrnt ordring policis to comput th invntory holding and shortag costs. 212 IUT Publication, IJIEPR, Vol. 23, No. 4, All Rights Rsrvd. 1. Introduction upply chains ar gnrally complx and ar charactrizd by numrous activitis sprad ovr multipl functions and organizations (Arshindr & Dshmukh, 28. upply chain has volvd vry rapidly sinc 199s showing an xponntial growth in paprs in diffrnt journals of intrst to acadmics and practitionrs (Burgss, t. al. 26. Rsarchrs wr abl to nhanc som of th prviously dvlopd modls, mthods and optimization tchniqus for modling and improving th prformanc of th supply chain. Multi-chlon systms hav rcivd considrabl attntion during th last dcads. Comptitiv forcs and th high costs of unsatisfid dmands hav rsultd in th us of multi-chlon invntory systms in providing srvic support for products with customrs distributd ovr an xtnsiv gographical rgion. Ths systms usually consist of a numbr of local * Corrsponding author: Majid Aminnayri mjnayri@aut.ac.ir Papr first rcivd July. 5, 212, and in rvisd form Oct. 9, 212. stocking sits that srv as a first lvl of product support for customr dmands. Ths sits will act, in turn, as customrs to a highr lvl stocking sit, or a warhous, for rplnishmnt of thir stock aftr thy ar dpltd by customr dmands. inc most of th rpairabl products ar of high valu with infrqunt failurs, most of th past studis hav mployd a on-for-on, (-1,, invntory ordring policy (Moinzadh t al., Exampls of ths studis includ imon [16], ravs [5], Higa t, al [1], and Hajiaghai-Kshtli and ajadifar [7] Poisson modls with on-for-on ordring policis can b solvd vry fficintly. Karush [13] prsnts th lost-sals modl with bas-stock, and also shows that mutually indpndnt lad tims can b solvd by Erlang s loss formula, and th formula is convx. Convxity has latr bn provd diffrntly by Jagrs and Van Doorn [12]. imon [16] drivs th stadystat distribution for invntory lvls at ach sit. Higa t al. [1] show tady-stat distribution functions thos ar drivd for waiting tim in an ( - 1, invntory systm. Axsätr [2] provids a simpl rcursiv procdur to dtrmin holding and shortag costs. H considrs an invntory systm with on warhous and N rtailrs.

2 M. Aminnayri & M. Hajiaghai-Kshtli Th Cost Function for a Two-warhous and Thr- 286 Lad tims ar constant and th rtailrs fac indpndnt Poisson dmand. H limits this invntory systm to on-for-on Rplnishmnt policis. Axsätr [3] first xprssd costs as a wightd man of costs for on-for-on ordring polics. voronos and Zipkin [17] study bas-stock policis for similar multi-chlon invntory systms. Thy show that lad tim variancs play an important rol in systm prformanc. Hausman [9] considrs low dmand, high cost itms controlld on an ( - 1, basis, with all warhous stock outs mt on an mrgncy ordring basis. Hill [] considrs continuous-rviw lost-sals invntory modls with no fixd ordr cost and a Poisson dmand procss, holding cost pr unit pr unit tim and a lost sals cost pr unit. Hajiaghai-Kshtli and ajadifar [7] considrd a thr-chlon invntory systm with two warhouss and N rtailrs. Transportation tims ar constant and rtailrs fac indpndnt Poisson dmand. Rplnishmnts ar on-for-on. Th lad tim of th rtailr is dtrmind not only by th constant transportation tim but also by th random dlay incurrd du to th availability of stock at th warhouss. Thy xtnd Axsatr [2] work and add a warhous as third chlon and considr having on mor dlay in shipmnt which may incurrd in th nw warhous. Thy obtaind th cost function for this invntory systm and tstd it by svral xampls. In this papr, w ar to driv th xpctd total holding and shortag costs for a unit dmand in thrchlon invntory systm with a on-for-on ordring policy for a thr-chlon invntory systm, using th ida of Hajiaghai-Kshtli and ajadifar [7]. W hav two warhouss in scond chlon. Thrfor, this may lads to having mor dlays which wr incurrd in th warhouss and facing diffrnt bhaviors of indpndnt Poisson dmand. Th rtailrs fac indpndnt Poisson dmands. Unfilld dmand is backordrd and th shortag cost is a linar function of tim until dlivry, or quivalntly, a tim avrag of th nt invntory whn it is ngativ. Transportation tims ar constant and ach chlon follows a bas stock, or ( - 1,, or onfor-on rplnishmnt policis. Ordrs that cannot b dlivrd instantanously from th warhous ar ultimatly dlivrd on a first com, first srv basis. W obtaind th cost function for th prsntd invntory systm. Also, bcaus th rplnishmnt policy is bas stock, th obtaind function can also b usd in diffrnt invntory policy to comput th invntory holding and shortag costs. For instanc, Hajiaghai-Kshtli t al. (21, usd th cost function which obtaind from bas stock policy, to comput (R,Q ordring policy in thrchlon invntory systm. o this mans that th obtaind function in this papr is vry usful for futur rsarchs and rsarchrs who want to dtrmin th cost of othr ordring policis in multi-chlon invntory systms or who want to prsnt nw ordring policy. For futur invstigations in this ara on can s how Haji t al. (28 and Moinzadh (22 prsntd nw ordring policis and drivd cost function. 2. Driving th Cost Function In this sction, w considr a thr-chlon invntory systm with two warhouss (supplirs in scond chlon and two rtailrs blong to ach warhous as shown in Fig. 1. Warhous L L 22 L Warhous Warhous 22 L L 12 Echlon: (3 (2 (1 Rtailr Rtailr 12 Fig. 1.Thr-chlon Invntory ystm with Two Warhouss Axsätr [2] found th xpctd total holding and shortag costs for a unit dmand, that is, c (, 1 for th on-for-on ordring policy in two-chlon invntory systm. Hajiaghai-Kshtli and ajadifar [7] found th xpctd total holding and shortag costs for a unit dmand in thr-chlon invntory systm with two warhouss (supplirs and on rtailr. In thir prsntd invntory systm, with two warhouss (supplirs and on rtailr, as shown in Fig. 2, transportation tims from an outsid sourc to th warhous 3, btwn warhouss, and also from th warhous 2 to th rtailrs ar constant. Thy assum that th rtailr facs Poisson dmand. Unfilld dmand is backordrd and th shortag cost is a linar function of tim until dlivry, or quivalntly, a tim avrag of th nt invntory whn it is ngativ. Each chlon follows a bas stock, or (-1,, or on-foron rplnishmnt policis. This mans ssntially that thy assum that ordring costs ar low and can b disrgardd. Whn a dmand occurs at a rtailr, a nw unit is immdiatly ordrd from th warhous 2 to warhous 3 and also warhous 3 immdiatly ordrs a nw unit at th sam tim, that is, ach chlon facs Intrnational Journal of Industrial Enginring & Production Rsarch, Dcmbr 212, Vol. 23, No. 4

3 287 M. Aminnayri & M. Hajiaghai-Kshtli Th Cost Function for a Two-warhous and Thr- th sam dmand intnsity. If dmands occur whil th warhouss ar mpty, shipmnts ar dlayd. Whn units ar again availabl at th warhouss, dlivrd according to a first com, first srvd policy. In such situation th individual unit is, in fact, alrady virtually assignd to a dmand whn it occurs, that is, bfor it arrivs at th warhous. Thy found C( 3, 2, 1 as th cost function which rlats to 3, 2, and 1 that indicat th warhous 3, warhous 2, and th rtailr invntory positions rspctivly in thir systm. For th on-for-on ordring policy, an arbitrary customr consums ( th, ( th and th 1, ordr placd by th warhous 3, warhous 2, and th rtailr, rspctivly, just bfor his arrival to th rtailr. If th ordrd unit arrivs prior to its (assignd dmand, it is kpt in stock and incurs carrying cost; if it arrivs aftr its assignd dmand, this customr dmand is backloggd and shortag costs ar incurrd until th ordr arrivs. This is an immdiat consqunc of th ordring policy and of our assumption that dlayd dmands and ordrs ar filld on a first com, first srvd basis. In this papr, basd on th on-for-on ordring policy as dscribd abov, w want to obtain th xact valu of C(,, 22,, 12, th xpctd total holding and shortag costs pr tim unit for th invntory systm as shown in Fig. 1. Warhous I L 3 Warhous II L 2 Rtailr Echlon: (3 (2 (1 Fig. 2.Thr-chlon rial Invntory ystm W fix th two rtailrs ( and 12, th two warhouss ( and 22, and th warhous, to chlon on, two and thr rspctivly as shown in Fig. 2. In ordr to driv th cost function, th following notations ar usd for this invntory systm: Invntory position at warhous Invntory position at warhous 22 Invntory position at warhous 22 Invntory position at rtailr 12 Invntory position at rtailr 12 L Transportation tim from th Warhous to th rtailr L 12 Transportation tim from th Warhous 22 to th rtailr 12 L Transportation tim from th Warhous to th Warhous L 22 Transportation tim from th Warhous to th Warhous 22 L 1 L Transportation tim from th outsid sourc to th Warhous (Lad tim of th Warhous T Random dlay incurrd du to th shortag of stock at th Warhous T Random dlay incurrd du to th shortag of stock at th Warhous T 22 Random dlay incurrd du to th shortag of stock at th Warhous 22 λ Dmand intnsity at rtailr λ 12 Dmand intnsity at rtailr 12 λ Dmand intnsity at th warhous h Holding cost pr unit pr unit tim at rtailr h 12 Holding cost pr unit pr unit tim at rtailr 12 h Holding cost pr unit pr unit tim at warhous h 22 Holding cost pr unit pr unit tim at warhous 22 h Holding cost pr unit pr unit tim at warhous β hortag cost pr unit pr unit tim at th rtailr β 12 hortag cost pr unit pr unit tim at th rtailr 12 W charactrizs our on-for-on rplnishmnt policy by th (,, 22,, 12 of ordr-up-to invntory positions which,, 22,, 12 ar th invntory position at warhous (chlon 3, th invntory position at warhous and 22 (chlon 2, and th invntory position at rtailr and 12 (chlon 1, rspctivly. o w considr a on-for-on rplnishmnt rul with ( 3,, 22,, 12 as th vctor of ordr-up-to lvls. Whn a dmand occurs at rtailr with a dmand dnsity, λ, a nw unit is immdiatly ordrd from th warhous to warhous and also warhous immdiatly ordrs a nw unit at th sam tim. This action also occurs for rtailr 12, warhous 22 and warhous with dmand dnsity λ rspctivly. For th on-for-on ordring policy as dscribd abov, any unit ordrd by th rtailr is usd to fill th th dmand following this ordr, hraftr, rfrrd to as its dmand. It mans that, an arbitrary customr consums th ordr placd by th rtailr just bfor his arrival to th rtailr and w can also say th that th customr consums ( th ordr placd by th warhous (warhous just bfor his arrival to th rtailr. If th ordrd unit arrivs prior to its (assignd dmand, it is kpt in stock and incurs carrying cost; if it arrivs aftr its assignd dmand, this customr dmand is backloggd and shortag costs ar incurrd until th ordr arrivs. This is an immdiat consqunc of th ordring policy and of Intrnational Journal of Industrial Enginring & Production Rsarch, Dcmbr 212, Vol. 23, No. 4

4 M. Aminnayri & M. Hajiaghai-Kshtli Th Cost Function for a Two-warhous and Thr- 288 our assumption that dlayd dmands and ordrs ar filld on a first com, first srvd basis. W confin ourslvs to th cas whr invntory position in all warhouss and rtailrs is qual or gratr than zro. To find th total cost, following th Axsätr [3], Hajiaghai-Kshtli and sajadifar [7] s ida, lt g (. (=,12,,22, dnot th dnsity function of Erlang (λ xy, distribution of th tim lapsd btwn th placmnt of an ordr and th occurrnc of its assignd dmand unit: g xy 1 λ t (t ( 1! λt xy=,12 =,12,,22, Th corrsponding cumulativ distribution function: (t is: ( t (1 k xy xy ( t t (2 k k! An ordr placd by th rtailr, arrivs aftr L +T tim units, and an ordr placd by warhous, arrivs aftr L +T tim units, whr T (=,,22 is th random dlay ncountrd at chlon 2 and 3 in cas a warhous in th chlon 2 or 3 is out of stock. As w mntiond arlir, w want to show driving th cost function clarly in this sction. o, lt trac on path;,, and. Lt ( t dnots th xpctd rtailr carrying and shortag costs incurrd to fill a unit of dmand at rtailr whn invntory position at rtailr is. W valuat this quantity by conditioning on T =t. Not that th conditional xpctd cost is indpndnt of and, and is givn by: Th conditional distribution of T, on condition that T =t, obtaind from: P T T t 1 k 1 k t k! t k ( L t. (5 Also th conditional dnsity function f(t for T L +t is givn by: f T t ( T t t t 1! g 1 ( t L t t t (6 Th xprssion (6 shows th probability of tim of rciving th dmand; that is aftr rciving ( -1 th dmand, th dmand occurs at th tim of L +t -t. On th othr viw, w can say th tim distanc btwn rciving th dmand and rciving th ordr from warhous (L +t is t and w call it th dlay tim that occurrd in warhous. As w mntiond arlir th warhouss and 22 facs a Poisson dmand procss with rat λ and λ 12 rspctivly, lik th rtailrs and 12. But th warhous, facs a Poisson dmand procss with rat λ, which is qual to λ plus λ 12. Thrfor w us th xprssion (5 in third chlon as follows: P(T 1 k k ( L L 1 (7 k! k Th dnsity function f(t for t L,bcaus w assum that invntory positrons at all facilitis in this systm ar qual or gratr that zro, is givn by: t 1 ( Lt f ( t g t ( 1! (8 h ( t Lt ( s L Lt t g t s g ( s ds, ( s ds ; t t ; (4 ( (3 Lt, dnots th xpctd rtailr ( carrying and shortag costs incurrd to fill a unit of dmand at rtailr whn,, and ar th invntory position at warhous, warhous and th rtailr, rspctivly. Considring both stats that w hav dlay tim or hav not in both warhouss, w obtain th cost that incurrd to fill a unit of dmand at rtailr, as follows: L ( g, ( 1 t L t g L g t t t ( t dt ( t dt ( 1 ( 1 t ( ( dt. (9 Intrnational Journal of Industrial Enginring & Production Rsarch, Dcmbr 212, Vol. 23, No. 4

5 289 M. Aminnayri & M. Hajiaghai-Kshtli Th Cost Function for a Two-warhous and Thr- Th long-run avrag shortag and rtailr carrying costs is clarly givn by (,. Th conditional xpctd warhous holding s cost,, on condition that T =t, is ( t indpndnt of and givn by: t h ( s L t g ( s ds, ; (1 ( Lt Thrfor w find th avrag warhous holding cost pr unit for warhous whn th invntory position at warhous is as follows: L ( g t ( t dt (1 (. ( paramtrs in thir papr. Also w us ths valus, bcaus most of prvious works lik Haji t al. [6], and Moinzadh [14], and som othr works usd th sam valus for th sam paramtrs. W tstd th cost function by giving som valus to variabls. Th function has fiv variabls, and thrfor w hav six dimnsions. In ordr to show th function s bhavior w fix fiv variabls and s th function in rmaind two dimnsions. W giv valu 2 and 3 to, 12,, and 22 variabls and s th function s bhavior with th chang of as w can s in Fig. 3. As dpictd in Fig. 3, ths rsults show th cost function s bhavior and also nsur us to hav a minimum cost in a spcific invntory position in chlons for th invntory systm. Also th avrag warhous holding costs pr unit (, which dpnds only on th invntory position is: ( h g ( s( s L L3 ds (12 and (. W conclud that th long-run systm-wid cost for th path in this invntory systm as shown in fig. 2, by adding th costs which occurrd in ach chlon and is givn by: C(, (, ( ( (, (13 o, by considring th cost of anothr path (12,22,, which can b obtaind similarly, th total cost for th whol invntory systm shown in fig. 2 is as follows: 2 1i 2i C(,,22,,12 1i( 1 i (,2i 2i ( ( (14 i1 3. Numrical Exampls In this sction, w prsnt an xampl to rval th convxity tndncy of our formulation. W want to show that our cost function has a minimum in a spcific invntory position in chlons for an invntory systm. W do this for th systm with two warhouss in scond chlon and a rtailr blongs to ach warhous. For this purpos, assuming fiv variabls, 12,, 22,, w fix th valu of th paramtrs L.., h.., β.. and λ.. to 1, 1, 1, and 2, rspctivly, as Hajiaghai-Kshtli and ajadifar [7] valud th sam Fig. 3. Th bhavior of Cost Function with th chang of and fixing othr variabls to 2 and 3 4. Conclusion and Futur Works In this papr a thr-chlon invntory systm with two warhouss is considrd. Transportation tims ar constant and rtailrs fac indpndnt Poisson dmand. Th modl considrd hr is diffrnt from prviously addrssd in considring th numbr of facilitis. In ordr to find th cost function in th prsntd invntory systm, conditional distribution for dlays in shipmnt which may incurrd in warhouss is usd and bhavior of diffrnt Poisson dmands in rtailrs is considrd. Thr ar potntially unlimitd opportunitis for rsarch in multi-chlon invntory systms. For futur rsarchs, it is possibl to invstigat and dvlop th proposd modl with mor assumption. Othr ralistic aspcts of th problm lik ordring policis, rplnishmnt policis and dmand s bhavior ar proposd. Also, bcaus th rplnishmnt policy is bas stock, th obtaind function can also b usd in diffrnt invntory policy to comput th invntory holding and shortag costs. Rfrncs: [1] Arshindr, K.A., Dshmukh,.., upply Chain Coordination: Prspctivs, Empirical tudis and Intrnational Journal of Industrial Enginring & Production Rsarch, Dcmbr 212, Vol. 23, No. 4

6 M. Aminnayri & M. Hajiaghai-Kshtli Th Cost Function for a Two-warhous and Thr- 29 Rsarch Dirctions, Intrnational Journal Production Economics, 5, 28, pp [2] Axsätr,., impl olution Procdurs for a Class of Two-Echlon Invntory Problms, Oprations Rsarch, 38, 199, pp [3] Axsätr,., Exact and approximat valuation of batchordring policis for two-lvl invntory systms, Oprations Rsarch, 41, 1993, pp [16] imon, R. M., tationary Proprtis of a Two-Echlon Invntory Modl for Low Dmand Itms, Oprations Rsarch, 19, 1971, pp [17] voronos, A., Zipkin, P., Evaluation of On-for-On Rplnishmnt Policis for Multi Echlon Invntory ystms, Managmnt cinc, 37, 1991, pp [4] Burgss, K., ingh, P.J., Koroglu, R., upply Chain Managmnt: A tructurd Rviw and Implications for Futur Rsarch, Intrnational Journal of Oprations and Production Managmnt, 26, 26, pp [5] ravs,.c., A Multi-Echlon Invntory Modl for a Rpairabl Itm with On-for-On Rplnishmnt, Managmnt cinc,, 1985, pp [6] Haji, R., ajadifar,. M., Haji, A., Driving th Exact Cost Function for a Two-Lvl Invntory ystm with Information haring, Journal of Industrial and ystms Enginring, 2, 28, pp [7] Hajiaghai-Kshtli, M., ajadifar,. M., Driving th Cost Function for a Class of Thr-Echlon Invntory ystm with N-Rtailrs and On-for-On Ordring Policy, Intrnational Journal of Advancd Manufacturing Tchnology, 5, 21, pp [8] Hajiaghai-Kshtli, M., ajadifar,. M., Haji, R., Dtrmination of th Economical Policy of a Thr- Echlon Invntory ystm with (R, Q Ordring Policy and Information haring, Intrnational Journal of Advancd Manufacturing Tchnology, 21. [9] Hausman. W. H., Erkip. N. K., Multi-Echlon vs. ingl-echlon Invntory Control Policis for Low- Dmand Itms, Managmnt cinc, 4, 1994, pp [1] Higa, I., Fyrhrm, A.M., Arltt, L., Machado, A.L., Waiting Tim in an (-1, Invntory ystm, Oprations Rsarch, 23, 1975, pp [] Hill, R.M., Continuous-Rviw, Lost-als Invntory Modls with Poisson Dmand, a Fixd Lad Tim and no Fixd Ordr Cost, Europan Journal of Oprational Rsarch, 176, 27, pp [12] Jagrs, A.A., Van Doorn, E.A., On th Continud Erlang Loss Function, Oprations Rsarch Lttrs, 5, 1986, pp [13] Karush, W., A Quuing Modl for an Invntory Problm, Opration Rsarch, 5, 1957, pp [14] Moinzadh, K., A Multi-Echlon Invntory ystm with Information Exchang, Managmnt cinc, 48:3, 22, pp [15] Moinzadh, K. and L, Hau L., Batch iz and tocking Lvls in Multi-Echlon Rpairabl ystms, Managmnt cinc, 32, 1986, pp Intrnational Journal of Industrial Enginring & Production Rsarch, Dcmbr 212, Vol. 23, No. 4