E. Teimoury *, I.G. Khondabi & M. Fathi

Transcription

1 Iteratioal Joural of Idustrial Egieerig & Produtio Researh eptember 2 Volume 22 umber 3 pp I: A Itegrated Queuig Model for ite eletio ad Ivetory torage Plaig of a Distributio Ceter with Customer Loss Cosideratio Dowloaded from iiepr.iust.a.ir at 9:4 IRDT o uday May 3th 28 E. Teimoury * I.G. Khodabi & M. Fathi E. Teimoury Assistae professor of Idustrial Eg-Ira Uiversity of iee ad Tehology I.G. Khodabi M.. studet of Idustrial Eg-hahed Uiversity M. Fathi PhD studet of Idustrial Eg-Ira Uiversity of iee ad Tehology KEYWRD Disrete faility loatio Distributio eter Logistis Ivetory poliy Queueig theory Marov proesses ATRACT The distributio eter loatio problem is a ruial questio for logistis deisio maers. The optimizatio of these deisios eeds areful attetio to the fixed faility osts ivetory osts trasportatio osts ad ustomer resposiveess. I this paper we study the loatio seletio of a distributio eter whih satisfies demads with a M/M/ fiite queueig system plus balig ad reegig. The distributio eter uses oe for oe ivetory poliy where eah arrival demad orders a uit of produt to the distributio eter ad the distributio eter refers this demad to its supplier. The matrix geometri method is applied to model the queueig system i order to obtai the steady-state probabilities ad evaluate some performae measures. A ost model is developed to determie the best loatio for the distributio eter ad its optimal storage apaity ad a umerial example is preseted to determie the omputability of the results derived i this study. 2 IUT Publiatio IJIEPR Vol. 22 o. 3 All Rights Reserved.. Itrodutio Total logistis osts ivetory plus trasportatio are importat parts of a produt ost i may outries. Aordig to ostly ad diffiult to reverse ature of faility loatio problems FLPs ad its log time horizo impat there is a ritial maagemet deisio i the desig of effiiet logistis systems whih disuss about the hoie of loatios for distributio eters DCs to ehae operatio effiiey ad logistis performae. Distributio eter is defied as a etity that lis a eterprise with its suppliers ad ustomers. Idetifiatio of a distributio eter loatio should be based o expeses as ivetory osts trasportatio osts ostrutio osts operatig osts ad i some ases the servie level osts. There are may artiles of aalyti study o FLP where loatig distributio eters is oe of the mai * Correspodig author. Ebrahim. Teimoury Teimoury@iust.a.ir Paper first reeived Jue. 2 2 ad i revised form April motivatios for them. I a artile by Aies 985 ie basi loatio models where all of them are to miimize the fixed ivestmet osts ad trasportatio osts are surveyed. May appliatios ad methods for faility loatio problems ad loatio models for distributio systems are surveyed i Dreser 995 ad Klose ad Drexl 25 respetively. Moreover yam 22 ivestigated a FLP model ad some methodologies osiderig logistial ompoets. ome models osidered the demad uertaity effets o DCs optimal loatio. A otable wor i suh topi is the published artile by Cole 995 who osiders ormal distributio for demad ad a required safety sto for a speifi ustomer servie level i order to idetify the DC loatio ad ustomer alloatio. Also simple dyami ad stohasti loatio models are developed for osiderig the dyami ature of FLP ad the stohasti ature of demad by we ad Dasi 998. I a related artile og 26 proposes a model where he miimizes the sum of all the assoiated osts of a

2 52 E. Teimoury I.G. Khodabi & M. Fathi A Itegrated Queuig Model for ite eletio ad Dowloaded from iiepr.iust.a.ir at 9:4 IRDT o uday May 3th 28 supply hai etwor subet to a variety of related ostraits. Ivetory osts should be osidered oitly with other faility trasportatio ad operatio osts i determiig the optimal loatios for DCs. A istae for metioed topi is ozi ad Turquist 998 artile where they desribe a method iludig ivetory osts withi a fixed harge faility loatio model for developig a optimal system desig. Furthermore they preset a ase study ivolvig the distributio of fiished vehiles by a automotive maufaturer i aother artile whih provides a itegrated view with a areful attetio to faility osts trasportatio osts ivetory ad ustomer resposiveess osts ozi ad Turquist 2. Applyig queueig theory i FLPs first was disussed by Larso 974 where he aalyzed problems of vehile loatio-alloatio ad respose distrit desig i emergey respose servies that operate i the server to ustomer mode. Followig Larso 974 study providig probabilisti models that osider queuig theory i FLPs has bee developed by may researhers. Istaes of suh models are desribed by atta 989 Mariaov ad erra 22 yder 26 ad Mariaov 28. The distributio eter loatio problem that is osidered as a basi model i this paper is omposed of a supplier with stohasti repleishmet lead time a distributio eter with stohasti servie time ad a ifiite soure of ustomers who a bal ad reege with stohasti demad arrival times. We deal with the M/M/ queue with fiite apaity of impatiet ustomers. The behavior of impatiet ustomers whih upo arrival may or may ot go i the queue for servie depedig o the umber of ustomers i the system ad those whih o goig to the queue depart the queue without beig served is ivestigated i this study. The queueig systems with balig ad reegig have bee disussed i may artiles. Examples of suh studies a be see i Wag et al. 27 Yue ad u 28 ad Al-eedy et al. 29. This paper aims to study the impat of impatiet ustomers ad their demad uertaities o the optimal loatio of a distributio eter ad the size of its storage apaity whih is eeded to be established. The primary obetives of the study are: Developig the steady-state solutios for the M /M/ queueig system with fiite apaity reegig ad balig. Developig a ost model to idetify the optimal distributio eter loatio i order to miimize the steady-state expeted ost per uit time. btaiig the optimal storage apaity i eah adidate loatio site whih miimizes the steady-state related ivetory expeted osts per uit time. The remaider of the paper is orgaized as follows. I etio 2 we desribe the system defiitio ad more expliitly queueig relatios. etio 3 is dediated to alulatig some performae measures ad derivig the ost aalysis is disussed i etio 4. We have show the ovexity of the expeted total ost futio by a umerial example i etio 5 ad fially etio 6 provides olusios ad diretios for future researh. 2. ystem Defiitio We osider a distributio eter with oe server whih satisfies the arrival demads usig a queueig system with fiite apaity. The arrival demads sigal out from i demad poits ad the time poits of these demads ourrees form a Poisso proess with parameter i i 2... a for eah demad poit. o the total rate of demads referrig to the distributio a eter equals with. ee Fig. for a diagram i depitig the model. Eah ustomer who omes ito the distributio eter has a demad ad satisfyig this demad eeds a o had ivetory ad a proess set up must be doe by distributio eter whih taes some time. Eah ustomer eeds exatly a produt of uit size ad DC uses oe for oe ivetory poliy. Therefore the o had ivetory plus i order ivetory must be hold at a pre-speified level so the demad patter is trasferred exatly to the supplier. The DCs orderig quatity to supplier is of uit size ad it ours whe a demad refers to it. It is otable that the order satisfyig times from the DC to the demad poits are expoetially distributed with mea > whih are idepedet from the distaes to the demad poits. Also repleishmet lead time from the supplier is expoetially distributed with mea >. Fig.. Distributio eter loatio problem with stohasti repleishmet time servie time ad demad The ertai time that a ustomer wait for servie to begi before gettig impatiet is radom variable whih is distributed as a egative expoetial i Iteratioal Joural of Idustrial Egieerig & Produtio Researh eptember 2 Vol. 22 o. 3

3 E. Teimoury I.G. Khodabi & M. Fathi A Itegrated Queuig Model for ite eletio ad 53 Dowloaded from iiepr.iust.a.ir at 9:4 IRDT o uday May 3th 28 distributio with parameter. It is importat to say there is o physial queue i the system ad the arrival demads go to a order queue whih does ot eed a physial lie. The trasportatio proess is performed usig oe vehile. The DC a serve ust oe ustomer at a time ad the servie proess assumed to be idepedet of ustomer arrivals. The problem is to fid the best loatio for distributio eter ad its optimal storage apaity aordig to defied system parameters. Fig. 2. tate trasitio rates diagram 2-. The Marov Chai The system a be desribed by a quasi birth-ad-death Marov proess with states. Cosider the otiuous Marov hai { } where is the umber of ustomers i the system iludig the oe beig served ad is the umber of produts whih are available i the distributio eter storage. The state spae with trasitio rates is depited i Fig teady state results We desribed the state of the system by the pairs { }. ow let deote the probability that a ustomer eters the queue whe there are ustomers i the system. is defied as follows hat 28: e ie the waitig time before gettig impatiet is a radom variable whih follows a expoetial distributio with mea ad ustomers deisios are idepedet of eah other the average reege rate is. I order to develop the steady-state probabilities π we get help from the matrix geometri method whih was first itrodued by euts 98. The geerator matrix of the uder study Marov hai is give as: C Q A A C C A Where A ad C are blo square matrixes of order whih are displayed i Appedix A. Aordig to the fiite apaity of the queue there is o eed to he the stability oditio for the system uder osideratio. It is otable that A ad C givig the rate at whih the umber of the ustomer orders i the system irease by oe stay at Iteratioal Joural of Idustrial Egieerig & Produtio Researh eptember 2 Vol. 22 o. 3

4 54 E. Teimoury I.G. Khodabi & M. Fathi A Itegrated Queuig Model for ite eletio ad Dowloaded from iiepr.iust.a.ir at 9:4 IRDT o uday May 3th 28 the same level or derease by oe. is the matrix rate at whih the ustomer orders i the system moves from zero to oe. We defie the steady-state probability vetor π [ π π... π ] for the Marov hai { }. Eah π a be alulated usig π Q ad π where π π π π π π π π π π π π π π π π π π π π π π π π [ π π... π ] is a row vetor. π deotes the steady-state probability assoiated with the oditio that there are ustomers ad produts i the system. Referrig to the state trasitio diagram for the fiite M / M / queueig system with balig ad reegig whih is show i Fig. 2 the followig balae equatios are derived: π π π π π π π π I order to solve π Q it is ot possible to defie a ostat rate matrix R suh that π π R πr as disussed i euts 98 beause of the asymmetri struture of Qs sub-matrixes. o due to the fiite umber of sub-matrixes i geerator matrix Q balae equatios are solved diretly by MATLA 7. the laguage of tehial omputig i order to alulate the steady-state probabilities. 3. ystem Performae Measures I this setio we derive a umber of performae measures of the system uder osideratio i the steady-state. 3.. Mea Ivetory Level Let EI represet the average ivetory of produts i the steady state. The we have E I π Iteratioal Joural of Idustrial Egieerig & Produtio Researh eptember 2 Vol. 22 o. 3

5 E. Teimoury I.G. Khodabi & M. Fathi A Itegrated Queuig Model for ite eletio ad Mea aorder Level Let E deote the mea umber of baorders i the steady-state. The we have E π where Idex of the distributio eter adidate sites Maximum storage apaity whih a be established i adidate site Dowloaded from iiepr.iust.a.ir at 9:4 IRDT o uday May 3th Mea Customer rder Fulfillmet Delay Let EW represet the average ustomer order fulfillmet delay i the steady-state. I order to alulatig EW we have to obtai the mea ustomer orders i the system EL that is ahievable as follows E L π The via Little s law we have EW E L π 3.4. Mea Rate of Customer Loss Let EA ERE ad EL deote the average balig rate the average reegig rate ad the average rate of ustomer loss. Usig the oept of Aer ad Gafaria 963 these average rates are obtaied as follows E A π E RE π E L E A E RE 4. ptimal DC Loatio ad torage Capaity The expeted total ost i the steady-state for the osidered logisti model is defied to be: a EL TC F rii rs i E hei EW EL b w l F r i r s b h w l Fixed ost of opeig a distributio eter i adidate site Costat oeffiiet for trasformig distae to ost Eulidia distae betwee demad poit i ad th adidate site Eulidia distae betwee supplier ad th adidate site The ost of establishig produt storage apaity per uit produt per uit time i adidate site The fixed baorder ost per order per uit time The holdig ost per uit produt per uit time i adidate site The ost of order fulfillmet delay per uit produt per uit time The loss ost of oe ustomer reege or bal per uit time It is otable that the ostat E L deotes the peretage of satisfied orders for eah demad poit. Replaig the values of mea performae measures we get the followig expeted total ost futio. TC F h i a r i i π w π π π l r s π b π π Aordig to reursive omputatio of the π s it is quite diffiult to show the ovexity of the expeted total ost futio. However we preset a umerial example to prove the omputability of the results derived i this study. Iteratioal Joural of Idustrial Egieerig & Produtio Researh eptember 2 Vol. 22 o. 3

6 56 E. Teimoury I.G. Khodabi & M. Fathi A Itegrated Queuig Model for ite eletio ad Dowloaded from iiepr.iust.a.ir at 9:4 IRDT o uday May 3th umerial Example For a distributio eter with a queueig system we set the system apaity 8. We assume the followig parameter values: 35 produts per moth 33 uit repleishmets per moth 2 dollars per ilometer 35 dollars per order per moth w Cadidate site b 7 dollars per uit produt per moth ad Demad poit Tab.. Demad poits iformatio Demad poit oordiates Kilometers dollars per ustomer per moth. The waitig l time before gettig impatiet is a radom egative expoetial distributed variable with parameter.3. The repleishmet supplier is loated i oordiates 4 whih is expressed i ilometers. ther eessary iformatio about demad poits ad adidate sites for loatig the distributio eter are provided i Table ad Table 2. Demad arrival rate rder per moth Cadidate site oordiates Kilometers Fixed ost of opeig a distributio eter dollars Tab. 2. Cadidate sites iformatio Dollars per uit Dollars per uit h produt per moth produt per moth Maximum possible storage apaity The values of expeted total osts are give i Table 3. The optimal storage apaity is show i bold for eah distributio eter adidate site. The umerial values show that expeted total ost futio is ovex i. Established storage apaity Tab. 3. umerial example results The optimum value whih represets the miimum possible ost for the distributio eter is obtaied i adidate site with 3 produts storage apaity. Expeted total ost for eah adidate site Dollars Usig the preseted model we a selet the ear optimal deisios if the optimal oe a ot be performed. More expliitly if we a ot ope the DC i adidate site we ow the ext optimal solutio whih is establishig the DC i adidate site 3 with 3 produts storage apaity. 6. Colusio A distributio eter loatio problem with uertai demad ad impatiet ustomers is studied. Produt repleishmet time i DC demad satisfyig time ad the time betwee demad arrivals follow a Iteratioal Joural of Idustrial Egieerig & Produtio Researh eptember 2 Vol. 22 o. 3

7 E. Teimoury I.G. Khodabi & M. Fathi A Itegrated Queuig Model for ite eletio ad 57 Dowloaded from iiepr.iust.a.ir at 9:4 IRDT o uday May 3th 28 expoetial distributio. The servie proedure is modeled as a fiite queueig system with ustomer loss reege ad bal. The matrix geometri method is used to model the queueig system. This paper geeralizes a method to obtai a optimal loatio for a distributio eter amog some adidate sites ad its optimal storage apaity applyig a itegrated total ost futio. Usig this method deisio maer eables to selet ear optimal solutios simultaeously if the optimal oe a ot be performed. Aalyzig the problem disussed i this artile assumig the queueig system has o fiite apaity ad ustomers a oey betwee more tha oe distributio eters would be a good topi for future researh. Aother iterestig extesio ould be made by relaxig the assumptios of expoetially distributed repleishmet time servie time ad the time betwee demad arrivals. Referees [] Aies C.H. Faility Loatio Models for Distributio Plaig. Europea Joural of peratioal Researh 22: [2] Al-eedy R.. El-herbiy A.A. El-hehawy.A. Ammar.I. Trasiet olutio of the M/M/ Queue with alig ad Reegig. Computers & Mathematis with Appliatios 57: [3] Aer C.J. Gafaria A.V. ome Queuig Problems with alig ad Reegig. peratios Researh : [4] atta R. A Queueig-Loatio Model with Expeted ervie Time Depedet Queueig Disiplies. Europea Joural of peratioal Researh 39: [5] hat U.. A Itrodutio to Queueig Theory: Modelig ad Aalysis i Appliatios. irhauser: osto 28. [6] Cole M.I.L. ervie Cosideratios ad the Desig of trategi Distributio ystems. Ph.D. thesis hool of Idustrial ad ystems Egieerig Georgia Istitute of Tehology Atlata GA 995. [7] Dreser Z. Faility Loatio: A urvey of Appliatios ad Methods. priger: ew Yor 995. [8] Klose A. Drexl A. Faility Loatio Models for Distributio ystem Desig. Europea Joural of peratioal Researh 62: [9] Larso R.C. A Hyperube Queueig Model for Faility Loatio ad Redistritig i Urba Faility ervie Computers ad peratios Researh : [] Mariaov V. Rios M. Iaza M.J. Faility Loatio for Maret Capture whe Users Ra Failities by horter Travel ad Waitig Times. Europea Joural of peratioal Researh 9: [] Mariaov V. erra D. Loatio-Alloatio of igle ad Multiple-erver ervie Ceters with Costraied Queues or Waitig Times. Aals of peratios Researh: [2] euts M.F. Matrix-geometri olutios i tohasti Models: A Algorithmi Approah. Johs Hopis Uiversity Press: altimore 98. [3] ozi L.K. Turquist M.A. Itegratig Ivetory Impats ito a Fixed-Charge Model for Loatig Distributio Ceters. Trasportatio Researh Part E Logistis ad Trasp Rev. 34 3: [4] ozi L.K. Turquist M.A. Ivetory Trasportatio ervie Quality ad the Loatio of Distributio Ceters. Europea Joural of peratioal Researh 29: [5] we.h. Dasi M.. trategi Faility Loatio: a Review. Europea Joural of peratioal Researh : [6] og.h. Multi-Period Itegrated Ivetory ad Distributio Plaig with Dyami distributio Ceter Assigmet. IEM :-. [7] yder L.V. Faility Loatio Uder Uertaity: A Review. IIE Trasatios Istitute of Idustrial Egieers 38: [8] yam.. A Model ad Methodologies for the Loatio Problem with Logistial Compoets. Computers ad peratios Researh 29: [9] Wag K.H. Ke J.. Ke J.C. Profit Aalysis of the M/M/R Mahie Repair Problem with alig Reegig ad stadby swithig failures. Computers ad peratios Researh 34: [2] Yue D.Q. u Y.P. Waitig Time of M/M// Queuig ystem with alig reegig ad multiple syhroous vaatios of partial servers. ystems Egieerig - Theory & Pratie 28: Iteratioal Joural of Idustrial Egieerig & Produtio Researh eptember 2 Vol. 22 o. 3

8 58 E. Teimoury I.G. Khodabi & M. Fathi A Itegrated Queuig Model for ite eletio ad Appedix A A. 2 A.2 A.3 A.4 A.5 A I A.6 C Iteratioal Joural of Idustrial Egieerig & Produtio Researh eptember 2 Vol. 22 o. 3 Dowloaded from iiepr.iust.a.ir at 9:4 IRDT o uday May 3th 28