CHAPTER 4 TWO-COMMODITY CONTINUOUS REVIEW INVENTORY SYSTEM WITH BULK DEMAND FOR ONE COMMODITY

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1 Unvety of Petoa etd Van choo C de Wet 6 CHAPTER 4 TWO-COMMODITY CONTINUOU REVIEW INVENTORY YTEM WITH BULK DEMAND FOR ONE COMMODITY A modfed veon of th chapte ha been accepted n Aa-Pacfc Jounal of Opeatonal Reeach

2 Unvety of Petoa etd Van choo C de Wet 6 4. INTRODUCTION Wth the advent of advanced computng ytem many ndute and fm deal wth mult-commodty ytem. In dealng wth uch ytem model wee ntally popoed wth ndependently etablhed eode pont. In tuaton whee eveal poduct compete fo lmted toage pace o hae the ame tanpot faclty o tem ae poduced on pocued fom the ame eupment upple the above tategy oveloo the potental avng aocated wth ont eplenhment educton n odeng and etup cot and allowng the ue to tae advantage of uantty dcount. In contnuou evew nventoy ytem Ballntfy 964 and lve 974 have condeed a coodnated eodeng polcy whch epeented by the tplet c whee the thee paamete c and ae pecfed fo each tem wth c. In th polcy f the level of -th commodty at any tme below an ode placed fo tem and at the ame tme fo any othe tem wth avalable nventoy at o below t can-ode level c an ode placed o a to bng t level bac to t maxmum capacty. ubeuently many atcle have appeaed wth model nvolvng the above polcy. Anothe atcle of nteet due to Fedeguen Goenevelt and Tm 984 whch deal wth the geneal cae of compound Poon demand and non-zeo lead tme. A evew of nventoy model unde ont eplenhment povded by Goyal and at 989. Kalpaam and Avagnan 993 have ntoduced polcy wth a ngle eode level defned n tem of the total numbe of tem n the toc. Th polcy avod epaate odeng fo each commodty and hence a ngle poceng of ode fo both commodte ha ome advantage n tuaton whee n pocuement made fom the ame upple tem ae poduced on the ame machne o tem have to be uppled by the ame tanpot faclty. 5

3 Unvety of Petoa etd Van choo C de Wet 6 A natual extenon of polcy to two-commodty nventoy ytem to have two eode level and to place ode fo each commodty ndependent of othe. But th polcy wll nceae the total cot a epaate poceng of two ode eued. Anbazhagan and Avagnan have condeed a two commodty nventoy ytem wth ndependent eode level whee a ont ode fo both the commodte placed only when the level of both commodte ae le than o eual to the epectve eode level. The demand pont fom an ndependent Poon poce and the lead-tme dtbuted a negatve exponental. They alo aumed unt demand fo both commodte. In th chapte the above wo extended by aumng unt demand fo one commodty and bul demand fo the othe commodty. The numbe of tem demanded fo the latte commodty aumed to be a andom vaable Y wth pobablty functon p P{Y } A eode made fo both commodte when the nventoy level of thee commodte ae at o below the epectve nventoy level. et of eode level Fgue 4.: pace of Inventoy Level 6

4 Unvety of Petoa etd Van choo C de Wet 6 The ont pobablty dtbuton of the two nventoy level obtaned n both tanent and teady tate cae. Vaou meaue of ytem pefomance and the total expected cot ate n the teady tate ae alo deved. 4. MODEL DECRIPTION Conde a two commodty nventoy ytem wth the maxmum capacty unt fo - th commodty. We aume that demand fo ft commodty fo ngle tem and demand fo econd commodty fo bul tem. The euence of epectve demand pont fo commodte and fo both commodte ae aumed to fom ndependent Poon pocee wth paamete and epectvely. The numbe of tem demanded fo the econd commodty at any demand pont a andom vaable Y wth pobablty functon p P{Y } 3. The eode level fo the -th commodty fxed at and odeng uantty fo -th commodty > tem when both nventoy level ae le than o eual to the epectve eode level. The euement > enue that afte a eplenhment the nventoy level wll be alway above the epectve eode level. Othewe t may not be poble to place eode whch lead to pepetual hotage. That f L t epeent nventoy level of -th commodty at tme t then a eode made when L t and L t. The lead-tme aumed to be dtbuted a negatve exponental wth paamete µ >. The demand that occu dung toc out peod ae lot. The tochatc poce {L t L t t } ha the tate pace E E E whee E {... } and E {... }. 7

5 Unvety of Petoa etd Van choo C de Wet 6 Notaton: : zeo matx ' I N :... N I : an dentty matx x f < x > f x > x Fom the aumpton made on demand and on eplenhment pocee t follow that {L t L t t } a Maov poce. To detemne the nfntemal geneato A a E we ue the followng agument: The demand fo the ft commodty tae the tate of the poce fom to and the ntenty of tanton a gven by.... A bul demand of tem fo econd commodty tae the tate fom to < > and the epectve ntenty of tanton ae gven by p and. A ont demand fo ngle tem of ft commodty and fo tem of econd commodty tae the ytem fom the tate to < > u p u and the epectve ntenty of tanton ae gven by p and u p. Fom the tate a eplenhment tae the ont nventoy level to and the ntenty of tanton fo th gven by µ. Fo othe tanton fom to when zeo. To obtan the ntenty of paage a of tate we note that the ente n any ow of th matx add to zeo. Hence the dagonal enty eual to the negatve of the um of the othe ente n that ow. Moe explctly a a 8

6 Hence we have Othewe p p p p a µ µ µ µ µ 9 Unvety of Petoa etd Van choo C de Wet 6

7 Unvety of Petoa etd Van choo C de Wet 6 whee p p. Denotng m m m... m m fo m... the nfntemal geneato A can be convenently expeed a a bloc pattoned matx: whee

8 Unvety of Petoa etd Van choo C de Wet 6 wth d µ and wth d - µ.

9 Unvety of Petoa etd Van choo C de Wet TRANIENT ANALYI Defne t P [L t X t L X ] E. Let t denote a matx whoe th element t and Φt denote a bloc pattoned matx wth the ub matx t at th poton. The Kolmogoov dffeental euaton can be wtten a Φ t Φ t A the oluton of whch gven by Φ t e At whee e At epeent I At! A t! Altenatvely f we ue the notaton A* α to denote the Laplace tanfom of the functon o matx A t then we have Φ * α α I A

10 Unvety of Petoa etd Van choo C de Wet 6 The matx α I A ha the followng bloc pattoned fom Whee D α I D α I A α I A Note that the ow and column have been numbeed n deceang ode f magntude. It may be obeved that α I A an almot lowe tangula matx n bloc pattoned fom. That f we denote the th ub matx of P α I A by P then we have P... >. To compute P α I A - we poceed a decbed below: Conde a lowe tangula matx 3

11 Unvety of Petoa etd Van choo C de Wet 6 wth U... and an almot lowe tangula matx uch that PU R. We fnd the ub matce U and R by computng the poduct PU and euatng t to R. The th ub matx of PU denoted by [PU] gven by By euatng the ub matce of PU to the coepondng element of R we get and The euaton PU R mple PU R U P R P UR. 4

12 Unvety of Petoa etd Van choo C de Wet 6 It can be vefed that the nvee of R gven by nce the expeon fo R nvolve the tem R ext. t demontated that the latte Fom PU R we get detpu detr detpdetu det R detbdetb detb. nce U a lowe tangula matx and B a uppe tangula matx the detemnant value ae not eual to zeo. Hence det extence of the nvee of R R denoted by P of P α I A - and t gven by not eual to zeo. Th pove the. Fom P UR we can compute the th ub matx 5

13 4.4 teady tate Analy It can be een fom the tuctue of A that the homogeneou Maov poce {L t L t t } on the tate pace E educble. Hence the lmtng dtbuton Φ 4. wth whee denote the teady tate pobablty fo the tate of the nventoy level poce ext and gven by m m m m Φ E and A. 4. The ft euaton of the above yeld the followng et of euaton:. C A C A B A B A B D B mplfcaton yeld the followng: whee can be obtaned by olvng 6 Unvety of Petoa etd Van choo C de Wet 6

14 and C A that and { } AB CB A B DB AB A B DB CB AB A B DB AB A B DB A B DB The magnal pobablty dtbuton {... } of the ft commodty gven by and the magnal pobablty dtbuton {... } of the econd commodty gven by The expected nventoy level n the teady tate fo the -th commodty gven by L E. ] [ Unvety of Petoa etd Van choo C de Wet 6

15 Unvety of Petoa etd Van choo C de Wet REORDER AND HORTAGE In th ecton the eode and hotage ae tuded. Th eue the tudy of tme pont at whch a tanton occu n the nventoy level poce. Let T < T < T < be the ntance of tanton of the poce. Let L n L LT n L T n n.... Fom the well nown theoy of Maov pocee { L n L n...} a Maov chan and wth the tanton pobablty matx tpm n n whee P p l El E p l a l / θ l l Hee θ a whch a negatve value. Moeove fo all n we alo have P [L T n L T n l T n T n > t L T n L T n ] p le θt Reode A eode fo both commodte made when the ont nventoy level at any tme t dop to ethe o < o <. We aocate wth a eode a countng poce Nt. Defne 8

16 Unvety of Petoa etd Van choo C de Wet 6 _ whee P [ ] epeent P[ L L ]. The fact that the eode at tme t ethe due to the ft tanton o due to a ubeuent one gve the followng euaton: whee β t gven by In the above expeon we have ued the fact that when... and then the next demand fo commodty wll tgge a eode. When... and... then o moe than demand fo commodty alone wll tgge a eode. A demand fo both commodte wll tgge a eode f the numbe of demanded tem fo the econd commodty... when... and. A the Maov poce {L t L t t } educble and ecuent due to fnte tate pace 9

17 Unvety of Petoa etd Van choo C de Wet 6 ext and wll be eual to the teady tate mean eode ate. Moeove we have fom Cnla975 ~ β π m 4.4 π β t dt / E E whee m the mean ooun tme n the nventoy level and gven by /θ and π the tatonay dtbuton of the Maov chan { L n L n...}. n nce fo a Maov poce. l π m / π m 4.5 l E l we have fom 4.5

18 Unvety of Petoa etd Van choo C de Wet hotage A hotage fo a commodty occu when a demand occu dung a tocout peod. We aocate wth a hotage a countng poce Mt. Defne 4.6 whch atfe the euaton β t ~ θu t β t p l θe β l t u du. l E We have ued the fact that the hotage at tme t due to the ft demand o a ubeuent one. Hence and gven by 4.7 Devaton mla to the one ued to deve β efe ubecton Reode yeld

19 Unvety of Petoa etd Van choo C de Wet Expected Cot The long un expected cot ate C gven by C h E[L ] h E[L ] Kβ bβ whee h and h ae holdng cot fo ft and econd commodty epectvely K the fxed cot pe ode and b the hotage cot. Then we have

20 Unvety of Petoa etd Van choo C de Wet NUMERICAL ILLUTRATION The lmtng pobablty dtbuton of nventoy level computed fo pecfc value of paamete. Fo the ft example we have aumed µ.5 p.5 p. p 3.5 p 4.5 p 5. p 6.5 p 7. h. h.3 b.7 K 5. Commodty I Commodty I Commodty II Commodty II Table 4.: Lmtng pobablty dtbuton of the nventoy level Example Th example gve the followng eult: Expected eode ate Expected hotage ate.479. Expected nventoy level fo the commodty I Expected nventoy level fo the commodty II.458. Total Expected Cot ate

21 Unvety of Petoa etd Van choo C de Wet 6 A a econd example the followng value have been condeed and the calculated ont pobablty dtbuton of the nventoy level gven n Table 4.: µ p.3 p. p 3.5 p 4.5 p 5. p 6.5 h.3 h.3b.9 K 75. Commodty I Commodty II Commodty I Commodty II Table 4.: Jont pobablty dtbuton of the nventoy level Example Th example gve the followng eult: Expected eode ate Expected hotage ate Expected nventoy level fo the commodty I Expected nventoy level fo the commodty II Total Expected Cot ate

22 Unvety of Petoa etd Van choo C de Wet CONCLUIION Th atcle analye a two-commodty nventoy ytem unde contnuou evew. The maxmum toage capacty fo the -th tem. The demand pont fo each commodty ae aumed to fom an ndependent Poon poce. We alo aume that unt demand fo one tem and bul demand fo the othe. The eode level fxed a fo the -th commodty and the odeng polcy to place ode fo tem fo the -th commodty when both the nventoy level ae le than o eual to the epectve eode level. The lead-tme aumed to be exponental. The ont pobablty dtbuton fo both commodte obtaned n both tanent and teady tate cae. Vaou meaue of ytem pefomance and the total expected cot ate n the teady tate ae deved. The eult ae llutated wth numecal example. 5