Waiting Time Distribution of Demand Requiring Multiple Items under a Base Stock Policy

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1 Joural of Servce Scece ad Maagemet Publshed Ole October 23 ( Watg Tme Dstrbuto of Demad Requrg Multple Items uder a Base Stoc Polcy Naoa Morta Koch Naade Departmet of Idustral Egeerg ad Systems Maagemet Nagoya Isttute of Techology Nagoya Japa. Emal: aade@tech.ac.jp Receved July 3 th 23; revsed September 2 d 23; accepted September 28 th 23 Copyrght 23 Naoa Morta Koch Naade. Ths s a ope access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese whch permts urestrcted use dstrbuto ad reproducto ay medum provded the orgal wor s properly cted. ABSTRACT I demadg for repar tems or stadard products customers sometmes requre multple commo tems. I ths study the watg tme of demad whch requres multple tems uder base stoc polcy s aalyzed theoretcally ad umercally. It s assumed that the producto tme has a epoetal dstrbuto ad the demad s satsfed uder a FCFS rule. Each demad wats utl all of ts requremets are satsfed. A arrvg process of demad s assumed to follow a Posso process ad the umbers of requred tems are depedet ad geerally dstrbuted. A method for dervg watg tme dstrbuto of demad s show by usg methods for computg M X /M/ watg tme dstrbuto ad for dervg the dstrbuto o the order correspodg to the last tem the curret customer requres. By usg the aalytcal results propertes of watg tme dstrbuto of demad ad optmal umbers of base stocs are dscussed through umercal epermets. Keywords: Base Stoc Polcy; M X /M/ Queue Watg Tme Dstrbuto. Itroducto It s mportat for producto ad vetory systems to atta the mmal vetory ad shortage costs. I Zp [] may vetory ad producto polces are dscussed. As a vetory polcy producto ad vetory systems wth stochastc demad a base stoc polcy s mportat ad well-ow. der ths polcy the total umber of wor--process ad fshed tems the system retas costat. That s whe the demad arrves at the same amout of products the demad requres are ordered at the same tme ad f there are fshed tems eough to satsfy the demad the t s satsfed ad otherwse t wats for the completo of processg tems. The producto ad vetory system wth demad requrg multple tems uder base stoc polcy has the followg two types. Frst s the model whch the demad s decomposed to multple uts each of whch correspods to each tem requred ad each ut s satsfed mmedately whe ths tem s produced. Ths s the case that multple demads arrve at the system at the same tme. It s foud whe a retaler orders tems o multple customers ad each customer s satsfed whe the tem o hs order s produced. The secod case s that each customer requres multple tems ad orders them to the system. I ths case the customer s satsfed whe all tems she/he orders are produced. For eample f some products fal ad repar parts are requred the multple parts are eeded for repar. I the frst case the watg tme for each ut has bee aalyzed. The arrval process s assumed to be a compoud Posso process whch the terarrval tme of demad forms Posso process ad the umbers of arrvals at the same tme have the same probablty fucto ad they are mutually depedet. I Feeey ad Sherbrooe [2] the watg tme dstrbuto of the demad s aalyzed ad recetly Zhao [3] the etwor wth multple producto ad vetory systems s aalyzed. I the secod model Hga et al. [4] derve the watg tme dstrbuto of demad whe a probablty mass fucto of tems the demad requres s a geometrc dstrbuto whch s ot practcal may stuatos. I the other model Ko et al. [5] aalyze a appromate lead tme dstrbuto a assemble-to-order producto system. I ths study the watg tme dstrbuto of demad the secod model wth a geeral dstrbuto o the umbers of tems the demad requres s aalytcally Copyrght 23 ScRes.

2 Watg Tme Dstrbuto of Demad Requrg Multple Items uder a Base Stoc Polcy 267 derved. The epresso of the M X /G/ watg tme dstrbuto wth servce tme dstrbuto some class whch s dscussed Chadhry ad Gupta [6] ad the probablty dstrbuto o the order ut correspodg to a tem each ut of the demad requres whch s obtaed Zhao [3] are used our aalyss. The aalytcal method for dervg the watg tme dstrbuto of demad s developed. Through umercal epermets propertes of the watg tme dstrbutos are dscussed. It s used for fdg the optmal umber of base stocs whch s a mmal umber of base stocs satsfyg the codto that the fracto of demad who wats for tems for perod less tha or equal to a predetermed tme must be greater tha a pre-specfed value. The orgazato s as follows. I Secto 2 the model aalyzed ths paper s descrbed. I Secto 3 the watg tme dstrbuto for each demad s aalyzed theoretcally. I Secto 4 the umercal aalyss s gve ad Secto 5 cocludes the paper. 2. Model Descrpto 2.. Model A producto ad vetory system s cosdered wth a sgle product ad demads for requrg multple tems. The umber of tems each demad requres s stochastc ad has dstrbuto p = P(X = ) = 2 where s ts mamal sze. The epectato s deoted by E[X]. The szes of tems whch successve demads requre are mutually depedet ad demad arrvals form a Posso process wth rate λ. As a result the arrval process o tem requremets forms a compoud Posso process. The system follows a base stoc polcy. Fgure llustrates the system. The umber of base stoc s deoted by s. Whe the demad arrves ad requres multple tems the system orders the same umber of tems for producto. At the same tme f there are tems eough for the demadto requre the t s fulflled mmedately ad t receves tems ad leaves the system. If t s ot eough or the other demad wats for tems the t wats for all tems whch t wll receve to be processed adplaced at the vetory. Note that the demad s fulflled uder a frst come frst served rule. I the followg t s assumed that each demad cossts of multple uts each of whch requres oe fshed tem.ts are umbered 2 whe demad taes order uts ad ut meas the frst ut of the demad ut 2 meas the secod ut of the demad ad so o. The ut s called the last ut. Whe a order s receved a producto process produces tems. If there are multple order uts each ut wats for producto a queue. The processg tme s mutually depedet ad detcally epoetally dstrbuted wth rate μ. Therefore the uts the producto Fgure. Producto ad vetory system uder a base stoc polcy (s = 2). process form a M X /M/ process wth arrval rate λ the batch sze dstrbuto p ad epoetal servce rate μ. Here t s assumed that E X whch assures that the umber of orders watg for process s fte almost surely. The objectve of ths paper s to aalytcally derve the watg tme dstrbuto of demad a steady state that s the tme terval from the demad arrval epoch to the fshg epoch of process for all uts satsfyg ths demad Watg Tme of Demad The relatos amog watg tme of demad terarrval tme of demad ad processg tme of tems are dscussed. They are llustrated Fgure 2. Specfc demad s fulflled ad departs the system whe all uts o the demad are satsfed. The watg tme of the demad s defed as the sojour tme of specfc demad from ts arrval to ts departure whe all uts are fulflled. We mae atteto to the last ut of the demad (whch s pated blac Fgure 2). Ths order ut s satsfed whe the processg of product correspodg to the s-th prevous ut from ths ut s completed because the system s uder base stoc polcy wth base stoc s. The followg otatos are used: W: the watg tme of specfc demad steady state : the terarrval tme from the ( )st demad to the th demad umbered bacward from the specfc demad ( = 2 ). X : the batch sze of uts o the th demad bacward before the specfc demad Sm : the processg tme o the mth ut of the th demad bacward from the specfc demad. (m = 2 = 2 ). W q : the tme terval from the arrval tme of order whch cludes the s-th prevous ut from the last ut of the specfc demad to the start of processg o the frst ut of ths order. Note that W q depeds oly o the arrval process before ths order arrval ad thus W q ad S m are depedet of { = 2 } ad from the assumpto of compoud Posso process { = 2 } ad Copyrght 23 ScRes.

3 268 Watg Tme Dstrbuto of Demad Requrg Multple Items uder a Base Stoc Polcy Fgure 2. Relatos amog watg tme of demad terarrval tme of demad ad processg tme of tems. {X = 2 } are also mutually depedet. 3. Aalyss of Watg Tme of Demad The N()-th demad bacward before the specfc demad cludes the order o the tem whch the -th ut of the specfc demad receves ad M() deote the mared umber of ths ut the N()-th demad. From the total probablty law we have = = = PX PW t M m N X PW t P X PW t X = = = ( )= ( )= = = m P M m N X where X s the sze of uts of the specfc demad. Sce W s depedet of X whe M() ad N() are gve t follows that = PWt M = mn =. PWt M = mn = X = As show Fgure 2 the watg tme of the specfc demad W s equal to W q the watg tme of the correspodg order whch cludes the ut whch processes the product of specfc demad plus the totalservce tme o requremets frot of ths ut mus the terarrval tme betwee the demad ad ths order. That s PWt M = mn = m = PWq S t = = I the followg ths dstrbuto s aalyzed theoretcally. 3.. M X /M/ Watg Tme Dstrbuto From the result o M X /M/ queue watg tme dstrbuto aalyzed Chadhry ad Gupta [6] the model of ths paper t s derved that () A t PW q t= e for t> = where α α deotethe solutos ecept zero whch satsfy the ( + )-st order equato ad by lettg P f A f f j j j j. = P (2) If ths equato has z pars of cojugate comple solutos ad -2z real solutos the for some real costats a b c J K L t s epressed that for t 2z at q = z PW t = Je = bt bt Ke cosct Le s c t 3.2. Aalyss of Watg Tme Dstrbuto m Whe = the S ( m ) follows the Er- lag dstrbuto. Thus for t m t m m PW S t PW t e d. m! q q = (3) I the followg the case s cosdered. The value of S m may be postve or egatve. ) Whe t ca be obtaed that after some calculatos where m P S = = m m t = P t e m! m a =! e m m r m a = m r r m t dt!!! for 2 m 2. Ths leads to the followg desty fucto. r Copyrght 23 ScRes.

4 Watg Tme Dstrbuto of Demad Requrg Multple Items uder a Base Stoc Polcy 269 m d P S d = = = = m m a a e e!! = = 2 m m a a e!! e = = m m 2 a m a a m a =!! m m e e m = = m e j!!!!! m e m l l m m l m j m l! = e e!! =!! m l where f = the frst term o the rght had s zero. 2) Whe < t follows that for 2 m 2. m m m t t P S P t e dt = = m! Thus m m 2 m m m m a a a = = = =! d P S e e e d!! m m v e j! v! l m! j!!! v l! j l m v e j! j l v l m! j!!! m v e j! v! v l l m! j!!! v l! j l v m!!!! m! j j!!!!! v l m!! m v e j e j! v! where v m. From these equatos t fol- These ca be computed by the followg recursve lows that for equatos. m P Wq S t Is t I s tcicos t = = c = PW q tdpc m Icos t ci s ticos t c = PW q tdpc m t wth PW q tdpcm c Is t sct cos ct c c where C c = m m S. Icos t cosct sct c c Note that by (3) ths equato cludes the followg types of tegrals: ad the same way ˆ c t I Is t t e s t e s ctd c I cos t e cosctd I s tci cos t c ad ˆ t Icos t t e ˆ I c s t e sctd t ˆ I cos tci I s cos t cos t e c t d t c v l v l v Copyrght 23 ScRes.

5 27 Watg Tme Dstrbuto of Demad Requrg Multple Items uder a Base Stoc Polcy ˆ c t Is ( t) sctccos ct e c c ˆ t Icos t cosctcs ct e. c c 3.3. Probablty o Posto of the Correspodg Order ad t The probablty o the posto of the correspodg order ad ut satsfyg the last ut of the specfc demad ca be obtaed by Zhao [3] whch s gve by P M = m N = X = f s ad m s P X sm f s ad m 2. s s l d z l z PX s l m l dz l! f s 2 s ad m m s where z s a probablty geeratg fucto of the -fold covoluto of p(). From the above equatos derved Sectos 3.2 to 3.3 the watg tme dstrbuto of the demad the steady state ca be obtaed by Equato (). 4. Eamples I ths secto watg tmes ad optmal base stocs are llustrated through eamples. It s set as = ad the mamal sze s = 3 through eamples. Eample P =.3 P 2 =.4 P 3 =.3 E[X] = 2 λ =.4 ρ =.8; Eample 2 P =. P 2 =.8 P 3 =. E[X] = 2 λ =.4 ρ =.8; Eample 3 P =.3 P 2 =.4 P 3 =.3 E[X] = 2 λ =.45 ρ =.9; Eample 4 P =. P 2 =.3 P 3 =.6 E[X] = 2.5 λ =.32 ρ =.8. The watg tme dstrbutos of demad are derved by the aalytcal method obtaed Secto 3 ad Wolfram Mathematca 8.. For eample eample wth base stoc s = the watg tme dstrbuto of demad s derved as.23993t PW t=.9469e.238t.6883e cos.28344t.238t e s.28344t where there are other terms such as e t te t ad t 2 e t o the rght had sde but they are removed because ther coeffcets are eglgble small as ther absolute values are less tha. I eample the solutos of (2) ecept zero are ad where s a purely magary umber. I eample 2 all solutos are real umbers ad.498. Tables ad 2 show the probabltes that the demad has o watg tme ad those that the watg tme of demad s o more tha uder base stoc polces wth base stocs to 5 eamples to 4 respectvely. Frst the result Eample s compared wth that Eample 2 whch has the same epected umber of tems requred by each demad ad the same arrval rate but dfferet varaces o the umbers of requred tems. As show Table 2 f the varace s smaller the the probablty that watg tme of demad s o more tha s smaller. Ths effect s usually see vetory ad queueg processes that more varace of compoets the process leads to the greater watg tme. As Table shows however whe s = the probablty that demad has o watg tme Eample s greater tha that Eample 2. The reaso s as follows. Whe s = f the ut sze of the demad s 2 or 3 the the last ut of demad s fulflled by producg the tem ordered by the former ut of the same order ad so the demad requrg for multple tems wats for completo of processg almost surely. If the ut sze of the demad s the the demad s satsfed by producg the tem ordered by some ut of the former demad so there s a possblty that the demad wth oe ut has o watg tme. Eample has the greater probablty of the demad wth oe ut tha Eample 2 ad so the probablty that the demad has o watg tme s greater Eample. For Eample 3 whch has the greater arrval rate the above probabltes are much smaller tha those Eample. Compared wth Eample Eample 4 whch has the same testy ad more epected tems requred leads to more watg tme ad the above probabltes become small. Ths s because the terarrval tmes o the orders have more varaces ad the process o the umber of fshed tems vetory alsoluctuates more. Thus Eample 4 has more chaces that whe the demad arrves there s o tem vetory ad t wats for fshed tems. Table 3 shows the probabltes that demad wth each sze of uts has watg tme o more tha for s = 2 3 Eample. Whe the sze s greater the probablty s smaller because the demad wth more uts has to wat for completo of the tem ordered by recet demad or the latter ut compared wth the demad wth oe ut. Note that the probabltes for pars (s ) are the same whe the value of s- s the same. It s because whe s- s the same the last ut of the specfed de- Copyrght 23 ScRes.

6 Watg Tme Dstrbuto of Demad Requrg Multple Items uder a Base Stoc Polcy 27 Table. Probabltes that demad has o watg tme. s Eample Eample 2 Eample 3 Eample Table 2. Probabltes that demad has watg tme o more tha. s Eample Eample 2 Eample 3 Eample Table 3. Probabltes that demad whch has /2/3 uts has watg tme o more tha. s ut 2 uts 3 uts mad wth uts requres the same tem uder the base stoc polcy wth s base stocs. For eample demad wth 2 uts requres the tem ordered by the last ut of ts prevous demad uder s = 2 whch s the oe that demad wth 3 uts requres uder s = 3. Table 4 shows the probabltes that the watg tmes are o more tha t for t = cases s = to 5 of Eample. As s creases the probabltes also crease but the dffereces are the smaller whe t s greater. Table 5 shows that the optmal umber of base stocs s also derved Eample. I ths eample the optmal umber of base stocs meas the mmal umber of base stocs wth whch the probablty that watg tme of demad s o more tha s greater tha.8. Table 5 shows the optmal base stocs s * for varous arrval rates. Ths shows that the optmal base stoc s small whe ρ s small but as arrval rates crease from.7 to.9 the umbers of optmal base stoc crease rapdly. Ths shows that the optmal umber of bases stocs hghly depeds o the arrval rate whe ρ s hgh. It s oted that the computato tme for dervg the equato o watg tme dstrbuto by usg the method Secto 3 ad Mathematca 8 hghly depeds o the umber of base stoc. For Eample the compu- Table 4. probabltes that demad has watg tme o more tha t(t = 5 5 2) Eample. s P(W ) P(W 5) P(W ) P(W 5) P(W 2) Table 5. Optmal umbers of base stocs Eample s * 4 7 tato tme for dervg the watg tme for s = s about secods but the computato tme for dervg the watg tme for s = 7 s about 2 hours whe t s computed o a PC wth Core 7-262M (2.7 GHz) ad 8 GB RAM. Ths s because whe s = 7 there are more tha terms to be computed for dervg the dstrbuto. 5. Coclusos I ths study the watg tme dstrbuto of demad whch requres radom umbers of tems s aalyzed theoretcally uder the base stoc polcy. sg the aalyss umercal eamples gve the propertes of the dstrbutos ad optmal umbers of base stocs a smple problem s dscussed. Ths study does ot cosder the vetory ad baclog costs o demad because the aalyss of the epected amouts o vetory of tems ad baclogs of demad seems dffcult. For eample the order ut assged to a specfed tem s radom ad the ut sze of the order cludg ths ut s also radom. Thus the holdg tme dstrbuto wll be much complcated ad the aalyss of such optmzato problems s left future. The aalyss of the process uder the other types of producto/vetory polces s also left for future research. REFERENCES [] P. H. Zp Foudatos of Ivetory Maagemet McGraw-Hll Compaes New Yor 2. [2] G. J. Feeey ad C. C. Sherbrooe The (s-s) Ivetory Polcy uder Compoud Posso Demad Maagemet Scece Vol. 2 No pp [3] Y. Zhao Evaluato ad Optmzato of Istallato Base-Stoc Polces Supply Chas wth Compoud Posso Demad Operatos Research Vol. 56 No pp Copyrght 23 ScRes.

7 272 Watg Tme Dstrbuto of Demad Requrg Multple Items uder a Base Stoc Polcy [4] I. Hga A. M. Feyerherm ad A. L. Machado Watg Tme a (s-s) Ivetory System Operatos Research Vol. 23 No pp [5] S.-S. Ko J. Youg Cho D.-W. Seo Appromatos of Lead-Tme Dstrbuto a Assemble-to-Order System uder a Base-Stoc Polcy Computers ad Operatos Research Vol. 38 No. pp [6] M. L. Chaudhry ad. C. Gupta Eact Compoud Aalyss of Watg-Tme Dstrbutos of Sgle-Server Bul-Arrval Queues: MX/G/ Europea Joural of Operatos Research Vol. 63 No pp Copyrght 23 ScRes.