Comments on two models for operating two-warehouse inventory systems with deteriorating items and inflationary effects

Transcription

1 Cmmen n w mdel f eaing w-waehue inveny yem wih deeiaing iem and inflainay effe ( be ublihed in he junal IJORIS Huahun Xing, Jinxing Xie *, B Niu Deamen f Mahemaial Siene, Tinghua Univeiy, Beijing 00084, China Cendene : Jinxing Xie Deamen f Mahemaial Siene Tinghua Univeiy Beijing 00084, China [email protected] Vie:( Fax:( * Cending auh. Tel.: ; fax: adde: [email protected] (Jinxing Xie.

2 Cmmen n w mdel f eaing w-waehue inveny yem wih deeiaing iem and inflainay effe Aba Thi ae deal wih he w-waehue aial baklgging inveny blem unde inflain f a deeiaing du wih a nan demand ae ve an infinie hizn. In na he adiinal mdel in whih eah elenihmen yle a wih an inan elenihmen and end wih hage, an alenaive mdel ha been ed in een lieaue in whih eah yle a wih hage, and i i ven be le exenive eae han he adiinal mdel in em f he een value f he e uni ime. The een ae in u ha he ieia f minimizing he e uni ime i uneanable when he inflainay effe i aken in nideain, and inead, he ieia f minimizing he een value f he al ve he whle infinie lanning hizn huld be ued. Alng hi line, we hange he bjeive funin f hee w mdel and ve ha he mdel wih hage a he a f he yle i al le exenive eae han he adiinal mdel in em f he een value f he al, bu he imal luin f he mdel minimizing he e uni ime indiae ignifianly highe al. Keywd: Inveny; Tw-Waehue; Paial baklgging; Deeiain; Inflain

3 Induin The inveny blem f deeiaing iem wih deeminii demand ae ha been udied exenively in he lieaue. Nahmia (98, and Gyal and Gii (00 vided exellen eview f eeah wk in hi field befe 980 and 000 eeively. The blem aa gwing inee fm eeahe heeafe (Dye, Ouyang & Hieh, 007a & 007b; Shah, Shah & Wee, 009; Teng, Chang, Dye & Hung, 00. In na he udie ha aume an ganizain wn nly a ingle waehue wih unlimied aaiy, he la deade ee an exlive numbe f udie fuing n w waehue, i.e., an wned waehue (OW wih limied aaiy and a ened waehue (RW whih i aumed be available wih abundan aaiy. An ealy diuin n he effe f w waehue an be aed bak Haely (976,.5-7, and eenly he w-waehue inveny mdel have been nideed by many he eeahe (Chung, He & Lin, 009; Dey, Mndal & Maii 008; Dye, Ouyang & Hieh, 007b; Gayen & Pal, 009; Hieh, Dye & Ouyang, 008; Lee & Hu, 009; Niu & Xie 008; Rng, Mahaaa & Maii 008; Wee, Yu & Law, 005;, 004 & 006; Zhu, 00; Zhu &, 005. In adiinal inveny mdel, i i geneally aumed ha eah elenihmen yle a wih an inan elenihmen and end wih hage. In a een ae, (004 nideed he w-waehue inveny blem f deeiaing iem wih hage unde inflain, and ed an alenaive mdel in whih eah yle begin wih hage and end wihu hage. Unde me aumin, he ved ha he mdel wih hage a he a f he yle i le exenive

4 eae han he adiinal mdel unde he bjeive f minimizing he e uni ime. Me eenly, (006 exended he mleely baklgging mdel inae aial baklgging. Amng all lieaue n inveny managemen, he mdel vided by (004 & 006 ae he nly mdel whee eah yle begin wih hage and end wihu hage. Thee mdel ae unique in lieaue and he aial manage migh have a eial inee in hem if hey ae eally le exenive eae han adiinal mdel. Hweve, aeful examinain f hee mdel eveal ha hee ae me eiu hming in he bjeive funin ued in hee mdel. Fi, he bjeive funin be minimized in hee mdel i he een value f he e uni ime. When he inflainay effe i aken in nideain and he lanning hizn i infinie, he bjeive funin be minimized huld be he een value f he al ve he whle lanning hizn (Dye, Ouyang & Hieh, 007a; Hieh, Dye & Ouyang, 008; Kim & Philia, 986. Send, hee mdel igne he uhaing in he bjeive funin, ju a he mdel wihu he inflainay effe uually d ine he uhaing i a nan. Bu when he inflainay effe exi and he hage ae aially mleely baklgged, he al uhaing i n a nan and hu huld be aken in nideain exliily (Teng, Chang, Dye & Hung, 00; Wee, Yu & Lau, 005. Finally, he een value f uniy due l ale i evaluaed a he ime when he nex elenihmen u in (004 & 006. Hweve, when he inflainay effe i nideed, hi huld be evaluaed ju a he ime when he aual l 4

5 ale u. Owing hee hming, hee mdel and he euling imal elenihmen yle deived fm hem ae uiiu. In fa, he imal elenihmen hedule bained in Theem f he ae by (006 i nly a lal minimum luin, and he glbal minimum luin de n exi f he mdel (he een value f he e uni ime nvege ze when he elenihmen yle ge infiniy. In de avid mileading he aial manage, i i iman e hee mdel and ndu a fai main f hee mdel wih he adiinal mdel. In hi ae, we efmulae and mae he w mdel in (006 unde he bjeive f minimizing he een value f he al ve he whle lanning hizn. In he nex w ein, he abve-menined hming in (006 will be diued in deail, and he mdified mdel wih he bjeive f minimizing he een value f he al ve he whle lanning hizn will be develed veme hee hming. Then we haaeize he imal luin f he mdified mdel, and daw a nluin ha he mdel wih hage a he a f he yle i al le exenive eae in em f he een value f he al. Finally, me numeial examle ae eened hw ha he imal luin f he mdel minimizing he e uni ime lead ignifianly highe al han he mdified mdel. Analyi f he exiing mdel F imliiy, we efe he mdel ha a wih an inan elenihmen and 5

6 end wih hage a Mdel, and efe he mdel ha a wih a hage and end wihu hage a Mdel. Thughu hi ae, we fllw he nain and aumin ued in (006 exe ha he een value f he e uni ime f Mdel i ( i =, in (006 i dened by i. Beide, we indue CC i, i,, i =, (leae ee belw. i f ( = he demand ae a ime, whih i aumed be deeminii a a nan ae f D uni e uni ime, i.e., f = D (. W = he aaiy f OW (he aaiy f RW i aumed be unlimied. δ ( = e σ, he baklgging ae whee σ 0 i a nan and i he waiing ime. α = he deeiain ae in OW, whee 0 < α <. β = he deeiain ae in RW, whee 0 < β <. = he inflain ae. = he elenihmen e de. = he uhaing e uni. h = he hlding e uni e uni ime in OW. h = he hlding e uni e uni ime in RW. h + α < h + β. b = he baklgging e uni e uni ime, if he hage i baklgged. l = he uni uniy due l ale, if he hage i l. Seifially, l = + g, whee i he elling ie f he du, and g i he f,, l gdwill. Clealy, l >. = he ime a whih he inveny level eahe ze in RW. = he ime a whih he inveny level eahe ze in OW. 6

7 , = he ime a whih he hage level eahe he lwe in in he elenihmen yle. i = he een value f he elevan e uni ime f Mdel i in (006, i =,. CC i = he een value f he f he fi yle f he mdified Mdel i, i=,. i = he een value f he al ve he whle lanning hizn f he mdified Mdel i, i=,. i = he een value f he al ve he whle lanning hizn f he mdified Mdel i, i=,, when he aaiy f OW i abundan ha he RW i n be ued any me. Ne ha he inveny (inluding hlding and deeiain in RW ae aumed be highe han he in OW (i.e., h + β > h + α. Thu, he invenie ae fi ed in OW wih veflw ging RW. Bu when eieving f numin, i i alway fm RW when available befe eieving fm OW. Pleae al ne ha i i me eanable aume 0 and, he han > > > 0 and > > a in (006. The inveny iuain f Mdel and Mdel an be deied gahially in Figue and (aded fm (006, eeively. 7

8 Figue. Gahial eeenain f a w-waehue inveny yem f Mdel. Figue. Gahial eeenain f a w-waehue inveny yem f Mdel. 8

9 In (006, he bjeive funin be minimized ae he een value f he elevan e uni ime duing he fi yle. Seifially, f Mdel and, he bjeive funin ae exeed a fllw: + β + α D = + + β + β α + h β h Δ { [ ( e ( e ] D [ W e ( e ] ( +Δ +Δ b ( + e D [ ( e δ( + l( δ( ] d} ( +Δ +Δ ( whee Δ α = = ln( + αwe / D / α, Δ =, ( h + β βδ Δ = e { + [ ( e ( e ] D β + β h + α D Δ Δ4 + [ W e ( e ] α + b ( + D [ ( e δ( ( ( ] } ( 0 + l δ d +Δ +Δ4 ( whee αδ Δ =, Δ = = ln( + αwe / / α. (4 4 D Fm Enq. ( ( endingly, Eqn. (4, i i bviu ha ( an be exliily exeed a a funin wih nly w deiin vaiable and Δ ( Δ and. Theem. (a F fixed 0, lim = 0. Δ Pf. See Aendix. (b F fixed Δ 0, lim = 0. Theefe, he luin bained fm (006 ae nly lal minimum luin. In fa, neihe n ha a glbal minimum. F examle, 9

10 uing he daa f Examle f (006 and fixing ( Δ =0.490 (he lal minimum in, we an l he bjeive funin and in em f Δ and eeively (ee Figue and 4. I eay ee ha Δ ( =0.054 i nly a lal minimum, and ( nvegene ze when Δ ( ge infiniy. Fm abve bevain, he mdel in (006 and he nluin dawn by hem ae uiiu. The iial in i ha, when he inflainay effe i aken in nideain and he lanning hizn i infinie, he bjeive f minimizing he een value f he e uni ime i uneanable. In he nex ein, we will efmulae he mdel in em f he een value f he al ve he whle lanning hizn Δ ( Figue. ( a a funin f Δ ( when Δ ( i mall 0

11 Δ ( Figue 4. ( a a funin f Δ ( when Δ ( i lage Fmulain f he mdified mdel In hi ein, we e mdified mdel f he ame blem a in (006. Le u fily nide he een value f he elevan f he fi yle (ne-yle f Mdel. We make w mdifiain f he ne-yle in (006 a fllw. Fi, he een value f he uniy due l ale i exeed a e l [ δ ( ] f( d (A.8 in Aendix A in (006, whih imlie ha he i evaluaed a ime (i.e., he end f he yle. Thi i uneanable beaue f ha he huld be evaluaed ju a he ime he aual l ale u. Theefe, hi huld be exeed a e [ δ ( ] f ( d. Send, when he inflainay effe i nideed and he hage ae aially l

12 mleely baklgged, he al uhaing i n a nan and huld be aken in nideain exliily (Teng, Chang, Dye & Hung, 00; Wee, Yu & Lau, 005. Beide, when he uhae i added he elevan, he f he deeiaed iem ((A.9 in Aendix A in (006 de n need exliily add in any me. Nie ha he iem numed and deeiaed fm ime 0 ae uhaed a he beginning f he yle (ime 0, and he iem ued fulfill he bakdeed demand duing ime and ae uhaed a he end f he yle (. Theefe, he een value f he uhaing in he fi yle an be exeed β a [ W + D( e / β + e D δ( d]. By he abve analye and wih a imila mehd a in Aendix A in (006, we an me u een value f he al f he fi yle f Mdel : D β ( +Δ +Δ CC = { + [ W + ( e ] + e D δ ( d β D + e e D+ W e e β + β α + h β h Δ [ ( ( ] [ ( ] ( +Δ +Δ b ( ( + e D [ ( e δ( + e l ( δ( ] d}. ( 5 Sine he lanning hizn f he inveny blem i infinie and he inflainay effe i nideed, we huld ue he een value f he al f he infinie lanning hizn a u bjeive funin be minimized. The imila aah ha been ued inveny mdel unde inflain (Dye, Ouyang & Hieh, 007a; Hieh, Dye & Ouyang, 008; Kim & Philia, 986. Making ue f he ne-yle in Eqn. (5, he een value f he al ve he infinie lanning hizn f Mdel i

13 CC e n ( +Δ +Δ =. (6 n= 0 Niing n( +Δ+Δ e =, (7 ( +Δ+Δ n= 0 e and ubiuing Eqn. (5 in Eqn. (6, we bain D β ( +Δ +Δ = { [ ( ] ( ( + W + e + e D δ d +Δ +Δ ( e β D + + β + β α + h β h Δ [ ( e ( e ] D [ W e ( e ] ( +Δ +Δ b ( ( + e D [ ( e δ( + e l ( δ( ] d}, (8 whee Δ α α = = ln( + We / D /, Δ = α. (9 Similaly, he een value f he al elevan ve he infinie lanning hizn f Mdel an be exeed a whee e D βδ ( 4 δ +Δ +Δ 0 = { + [ W + ( e ] + D ( d ( e β h βδ Δ h D Δ Δ4 + [ ( e ( e ] D+ [ W e ( e ] β + β α + + D e + e d b ( ( [ ( δ( ( ( ] }, 0 l δ (0 αδ Δ =, Δ = = ln( + αwe / / α. ( 4 D Sluin he inveny mdel Fm Eqn. (9, we knw Δ i a funin f, i.e., Δ = Δ ( = α ln( + αwe / D / α (in he e f he ae, we dene Δ ( by Δ if n

14 ambiguiy eul fm i, hu an be exliily exeed a a funin wih nly w deiin vaiable and Δ. Similaly, an be exliily exeed a a funin f Δ and. F nveniene, dene D β h β A ( = 0 + W+ ( e ( e ( e D β + β β + D α + h Δ + W e ( e, ( b y y B D Δ ( Δ = δ ( y + ( e δ ( y + le ( δ ( y dy 0. ( I eay hek ( +Δ +Δ = (, Δ = A( ( ( + e B Δ +Δ +Δ e, (4 e = ( Δ, = A( ( B( Δ +Δ 4+ Δ +. (5 e F Mdel, fm Eqn. (9, we have dδ αwe d D We α αδ = = e α + α. (6 The fi de ndiin f in Eqn. (4 be minimized ae (, Δ = + Δ + ( +Δ +Δ ( e ( { ( ( } ( e +Δ +Δ e α Δ A( B( e +Δ +Δ A ( = 0, ( 7 (, Δ Δ = + Δ + Δ ( +Δ +Δ ( e ( +Δ +Δ { ( A( B( ( e B( } = 0. (8 By Eqn. ( and (, we have 4

15 β h β h ( + α Δ A ( = De + ( e e + e ( e D, β + α + (9 Δ Δ ( Δ = ( Δ + ( ( Δ + [ ( Δ ] b B D δ e δ le δ. (0 Dene K ( ( + Δ = e α e A (, ( Δ K ( Δ = e B ( Δ. ( Then we have he fllwing equain by Eqn. (7 and (8: K ( = K ( Δ. ( Lemma. (a Sue ha ( α β h ( β + and h + β > h + α. K ( i a ily ineaing funin f. (b Sue b > l. K ( Δ i ineaing n 0, Δ ] and deeaing n [ [ Δ, +, whee Δ ( + σ ( b = ln. ( b l Pf. See Aendix. In Pa (a f Lemma, he ndiin > ( α β h ( β + i n a igu ndiin and an be aified in ealiy f many ae, e.g., f he ae f α β, whih mean he deeiain ae in RW i n le han ha in OW. In Pa (b, he ndiin > an be al eaily aified ine he inflainay ae i uually b l vey lw. Afe baining he eul in Lemma, we an me he fllwing lemma, whih haaeize he luin Eqn. (7 and (8. Lemma. Unde he aumin > ( α β h ( β +, h + β > h + α and >, hee ae a m w iive luin f Eqn. (7 and (8: b l 5

16 ( ( ( ( (, Δ and (, Δ, whee Δ (0, Δ ] and Δ ( Δ [, +. ( Pf. See Aendix. By Lemma, we an haaeize he minimum in f in he fllwing heem. Theem. Aume > ( α β h ( β +, h + β > h + α, and b > l. * * ( ( Then (, Δ i minimized a (, Δ = (, Δ, whee Δ ( (0, Δ ], n * he bunday f he dmain: Δ 0 * = 0. Pf. See Aendix 4. = F Mdel, he fi de ndiin f in Eqn. (5 be minimized ae Δ ( Δ, e { ( Δ +Δ 4+ ( e ( Δ +Δ 4+ ( e A( } ( ( ( ( Δ +Δ 4+ αδ4 = e e A Δ + B + Δ = 0, (4 ( Δ, ( Δ +Δ 4+ ( e ( 4 { ( ( } Δ +Δ + A( B( e B( e = Δ + + = 0. (5 Fllwing he imila aahe analyzing Mdel, we have he fllwing eul f Mdel (he f ae mied ave ae ine hey ae imila he ued f Mdel. Lemma. Unde he aumin > ( α β h ( β +, h + > h β + α and >, hee ae a m w iive luin f Eqn. (4 and (5: b l 6

17 ( ( ( ( ( ( ( ( ( Δ, and ( Δ,. Fuheme, ( Δ, = (, Δ and ( ( ( ( ( Δ, = (, Δ. Theem. Aume > ( α β h ( β +, h + > h β + α and b > l. Then * * ( ( ( Δ, i minimized a ( Δ, = ( Δ,, whee (0, Δ ], n he ( * bunday f he dmain: Δ 0 * = 0. = Cmain beween he mdel In hi ein, we diu whih mdel i le exenive eae. F nveniene and wihu l f genealiy, we an aume ha he vaiable, Δ in Mdel ae he ame a he imal luin * Δ, * in Mdel. In he wd, we mae he value f (, Δ and * * (, Δ. * * * Fi, if Δ 0, hen hee i n hage in he whle lanning hizn. In hi = ae, i i lea ha * * (, Δ = (, Δ. * * * * ( ( Send, we nide he ae ha (, Δ = (, Δ. By he aumin menined abve, and making ue f Enq. (9, (7 and (8, we have (, Δ A( + e B( Δ B ( Δ B( Δ, (6 A B B ( ( ( T ( ( ( ( ( Δ Δ = e = e ( ( ( ( ( (, Δ ( + ( Δ ( Δ T + Δ. By Pa (b f Lemma, we have e y B ( y = + Δ ( ( ( whee ( i ( iive and ineaing n y 0, Δ ] [0, ], whih imlie ha [ Δ ( ( Δ ( Δ y y 0 0 B( Δ = B( y dy = e e B( y dy ( ( Δ ( Δ Δ ( y ( e 0 < e B( Δ e dy = B( Δ. (7 Fm (6 and (7, we have 7

18 ( Δ ( e ( (, Δ B ( Δ B( Δ ( B ( Δ = > =. (, B ( B ( ( ( ( ( ( ( Δ Δ e e ( ( ( ( Δ Δ Δ Theefe, he fllwing heem i ue. * * ( ( Theem 4. If (, Δ = (, Δ, hen (, Δ > ( Δ,. * * * * Theem 4 indiae ha when he aaiy f OW i limied and RW huld be ued, Mdel i le exenive eae han Mdel in em f he een value f he al ve he whle lanning hizn, unle hee ae n hage in bh mdel. Finally, nide he ae f * = 0, whih mean he aaiy f OW i abundan ha hee i n need ue RW (i.e., ne-waehue ae. In hi ae, he w-waehue inveny blem edue he ne-waehue (i.e., nly OW i available inveny blem. Reenly, Dye, Ouyang and Hieh (007b udied he ne-waehue (i.e., nly OW i available ae f he adiinal inveny mdel (like Mdel in u ae unde he bjeive f maximizing he diuned fi. They d n nide he inflainay effe (i.e., he inflain ae = 0, bu he iing deiin ae inluded. When he ie i n a deiin vaiable bu a fixed nan, he imila aah ued in u ae an al be exended he ne-waehue ae wih inflainay nideain and wih hage a he a f he elenihmen yle (like Mdel in u ae. Unde he w-waehue ae, we alway aume ha he aaiy W f he OW i fully uilized. Hweve, unde he ne-waehue ae, even he aaiy f OW migh n be fully uilized in de minimize he een value f he al, hu we huld emve he aaiy nain W. Alng hi line, he bjeive funin i an be deived f 8

19 Mdel i (i=, eeively f he ne-waehue inveny blem (ee Aendix 5 f deail. Uing he imila aahe a w-waehue ae, ne an bain he imila eul a Theem 4: When he aaiy f OW i abundan ha hee i n need ue RW, he ne-waehue Mdel i le exenive eae han ne-waehue Mdel in em f he een value f he al ve he whle lanning hizn, unle hee ae n hage in bh mdel. Numeial examle The main beween he w mdel i illuaed by he fllwing numeial examle. Examle (, 006. Le D= W = δ = e = = , 00, (, 00, h 0., h = 0.6, b =, l = 5, = 0, α = 0.05, β = 0.0, and =0.06 in aiae uni. The muainal eul f he w mdel ae hwn in Table. A main Figue and 4, he funin and ae led in Figue 5 and 6, a a funin f Δ f fixed ( Δ =0.57 (he glbal imize. ( Table. Cmuainal eul f Examle Mdel i Objeive funin Oimal luin Cyle lengh Tal i i= Thi ae =0.57 Δ =0.47 Δ = (006 =0.490 Δ =0.45 Δ = i= Thi ae Δ =0.57 Δ 4 =0.47 (006 Δ = Δ 4 =0.45 = =

20 7.8 x Δ ( Figue 5. ( a a funin f Δ ( when Δ ( i lage 0 x Δ ( Figue 6. and a a funin f Δ ( when Δ ( i lage 0

21 Examle (, 006. Le D= W = δ = e = = , 00, (, 00, h.0, h =.5, b =, l = 5, = 0, α = 0.05, β = 0.0, and =0.06 in aiae uni. The muainal eul f he w mdel ae hwn in Table. Table. Cmuainal eul f Examle Mdel i Objeive funin Oimal luin Cyle lengh Tal i i= Thi ae =0.04 Δ =0.47 Δ = (006 =0.86 Δ =0.46 Δ = i= Thi ae Δ =0.04 Δ 4 =0.47 (006 Δ =0.86 Δ 4 =0.46 = = Examinain f Table and eveal ha he imal yle lengh minimizing he een value f he al ve he whle lanning hizn ignifianly diffe fm ha minimizing he e uni ime in (006, and he luin deived fm (006 eul in ignifianly highe een value f he al. Theefe, he mdel unde inflain huld be baed n minimizing he een value f he al he han baed n minimizing he e uni ime. Examle : Le W=000 and he he aamee emain he ame a in Examle in aiae uni. The muainal eul f he w mdel ae hwn in Table. We an ee ha he imal =0 ( =0 and u mdel eun ne-waehue mdel. Examle 4: Le W=000 and he he aamee emain he ame a in Examle in aiae uni. The muainal eul f he w mdel ae hwn in Table

22 4. We an ee ha he imal =0 ( =0 and u mdel eun ne-waehue mdel. Table. Cmuainal eul f Examle f he bjeive funin in hi ae Mdel = 0 Δ =.557 Δ = 0.50 = Mdel Δ = 0 Δ 4 =.557 = 0.50 = Mdel (ne-waehue Mdel (ne-waehue = Δ = 0.55 Δ 4 = = 0.55 = = Table 4. Cmuainal eul f Examle 4 f he bjeive funin in hi ae Mdel = 0 Δ =.557 Δ = = Mdel Δ = 0 Δ 4 =.557 = = 899. Mdel (ne-waehue Mdel (ne-waehue = 0.4 Δ = Δ 4 = 0.4 = = = The eul in Table and 4 hw ha Mdel i alway le exenive eae han Mdel in em f he een value f he al, n mae i i neeay ue RW n. Cnluin In hi ae, we nide he w-waehue aial baklgging inveny blem unde inflain f a ingle deeiaing du wih a nan demand ae ve an infinie hizn. We in u hee eiu hming in he mdel f a een ae by (006, and hange he bjeive fm minimizing he e uni ime minimizing he een value f he al ve he infinie hizn.

23 Unde he new bjeive, main f he w mdel nfim ha if he inflain ae i geae han ze he mdel wih hage a he a f he yle i ill le exenive eae han he adiinal mdel wihu hage a he a f he yle. Hweve, he luin deived fm (006 eul in ignifianly highe een value f he al f he infinie hizn. Theefe, he mdel unde inflain huld be baed n minimizing he een value f he al ve he whle lanning hizn he han baed n minimizing he e uni ime. Obviuly, imila diuin a hi ae al aly he iuain wih mlee baklgging (, 004, whih i nly a eial ae f aial baklgging. Fuue eeahe an be uued nide he iuain whee he demand and/ he inflain ae vay ve ime. Aendix. Pf f Theem Dene D A % + β + α = + e e D+ W e e β + β α + ( h β [ ( ( ] h Δ [ ( ] b y B % Δ ( Δ = D [ ( e δ( y + l ( δ( y] dy 0. I an be eaily veified ha, = A % e B% +Δ +Δ ( ( +Δ +Δ + ( Δ. Sine % ( Δ = [ ( δ( Δ + ( δ( Δ ], b Δ B D e l

24 % Dl, if σ >, σ +, if σ <, b lim B ( Δ = D ( l +, if =, Δ + hen % Δ = B % y dy =+. Thu, by LHôial ule, we have Δ lim B( lim ( Δ + Δ + 0 B% Δ ( lim e B% Δ ( Δ = lim Δ + Δ + Δ e b Δ Δ = lim D[ ( e δ( Δ + e l ( δ( Δ ] = 0. Δ + Theefe, f fixed 0, lim Δ + = 0. Thi mlee he f f Pa (a. The f f Pa (b i imila and hu i i mied ave ae. Aendix. Pf f Lemma. ( Pa (a. Diffeeniaing ( α + Δ K ( = e e A, we have ( α+ Δ ( β+ α + ( β+ dδ K( = e ( ( β + + h e + ( α + + h( e + h. β + d Sine dδ d = e αδ > and h + β > h + α, hen α + K ( > e ( ( β + + e ( α + + ( e + β + ( α+ Δ ( β+ ( β+ h h h ( α+ Δ ( β+ α + > e ( e ( β + + h h. β + Thu, we have K ( 0 > by he ndiin ha ( α β h ( β +. Δ Pa (b. Diffeeniaing K ( Δ = e B( Δ, we have Δ Δ b b K( Δ = e δ( Δ σe l + ( σ +. Sine b > l and l >, i i eay veify ha K ( Δ > 0 if Δ 0, and [ Δ K ( Δ < 0 if Δ ( Δ, +. 4

25 Aendix. Pf f Lemma. By a (b f Lemma, we have ha f any 0, hee ae a m w value ( f Δ aifying Eqn. (. Dene he w value by Δ ( [0, Δ] and ( ( Δ ( [ Δ, + if hey exi. By ubiuing Δ in Eqn. (8 by Δ (, we knw ha Eqn. (8 i equivalen ( ( ( +Δ +Δ ( ( ( ( ( ( ( A + B Δ + e B ( Δ ( = 0. (A The deivaive f he lef hand ide f Eqn. (A wih ee i B ( Δ + e = = ( ( ( ( +Δ+Δ ( ( dδ ( d e α Δ + A ( ( dδ ( + d + ( ( +Δ+Δ ( ( e B ( Δ ( Δ ( ( ( ( ( +Δ ( ( ( ( ( Δ ( +Δ Δ d e B e e ( ( ( +Δ ( ( ( +Δ Δ dk ( e e > 0. d d ( ( B ( Δ ( ( ( dδ ( d ( ( ( Thu, hee i a m ne luin (, Δ ( Δ (0, Δ ] f Eqn. (A. Similaly, ( ( ( hee i a m ne luin (, Δ ( Δ Δ [, + f ( ( ( +Δ +Δ ( ( ( ( ( ( ( A + B Δ + e B ( Δ ( = 0. (A Theefe, hee ae a m w iive luin f Eqn. (7 and (8. Aendix 4. Pf f Theem. Ne ha he minimum in f ae eihe he iial in n he bunday f he dmain. In hi f, we fi ague ha (a (, Δ i a lal ( ( minimum in (if i exi and ( ( Δ Δ and (, Δ i n a lal minimum ( ( 5

26 in (if i exi and ( ( Δ Δ, and hen ague ha (b will n be minimized when + Δ +. F imliiy, we dene T = + Δ + Δ. By Eqn. (4, we have e e e e e e α ( e = + + T αδ T αδ T αδ αδ T T T ( e ( e ( e T αδ e e ( A + BΔ A T + A T ( ( ( (, ( e e e e e e = + + Δ Δ T αδ T αδ T T ( e ( e T αδ T e e e B( Δ + A(, T ( e T ( e T T T T Δ ( e ( e T T e e B Δ T + B Δ T ( A ( B( e e = + + Δ By Eqn. (7 and (8, we have ( A ( B( ( (. ( e e ( ( (, Δ T αδ ( A( T αδ e e d e e = > 0, (A T e d ( = Δ ( ( (, Δ = 0, (A4 Δ ( B( Δ e d e = > 0, (A5 Δ Δ ( +Δ T ( ( e (, Δ d ( Δ =Δ whee he inequaliie in (A and (A5 ae imlied by Lemma. Sine (A, (A4 ( ( and (A5 indiae ha he Heian maix f a (, Δ i iive definie, ( ( (, Δ i a lal minimum. Similaly, we have Δ ( B( Δ e d e = < 0, (A6 Δ Δ ( +Δ T ( ( e (, Δ d ( Δ =Δ 6

27 ( ( whih indiae ha (, Δ i n a lal minimum. Thi end he f f Pa (a. I an be eaily veified ha lim (, Δ =+, whih indiae ha + will n be minimized when +. I i al eay knw ha ( +Δ e l (, + = lim (, Δ = A( +, Δ + (,0 = A(. ( e +Δ Le be he uni elling ie f eah du. Then we an exe he een value f al fi ve he whle lanning hizn a + D TP(, Δ = De d (, Δ = (, Δ. 0 T ve ha will n be minimized when Δ +, we nly need ve ha TP will n be maximized when Δ +. Sine l >, hen D TP(, + = (, + ( +Δ ( +Δ e e = D A( ( l D ( +Δ e > D A(. Beide, we have Sue ( ( +Δ * * +Δ ( ( +Δ e TP (,0 = D A(. ( +Δ e e D A( i minimized a ( ( * ( 0 *. If e D A, hen TP (, 0 + < f all 0. Thu, TP will lealy n maximized when Δ +, a lng a he whle yem i fiable a me Δ <+ (i i eanable aume hi ine i i meaningle udy a 7

28 * * +Δ nnfi yem. If ( * * ( ( * ( 0 e D A >, hen i i bviu ha TP (, + < TP (,0, whih indiae max TP(, + < max TP(,0. Theefe, 0 0 TP will n be maximized when Δ +. Thi end he f. Aendix 5. One-waehue inveny mdel When OW i abundan ha hee i n need ue RW, Mdel edue he ne-waehue inveny mdel. We emve he aaiy nain f OW, and ewie he een value f he al f he infinie lanning hizn f Mdel in Eqn. (8 a D α ( { ( +Δ h α = ( [ ( ( + e + e D δ d + e +Δ ( e α α + α + + ( ( ] +Δ [ b ( ( ( ( e D e D e δ e l ( δ ( ] d }, whee Δ =. Similaly, he een value f he al f he infinie lanning hizn f Mdel in Eqn. (0 an be ewien a e D αδ 4 h αδ4 ( 4 δ +Δ 0 = { + ( e + D ( d + [ ( e ( e α α + α + + ( Δ 4 ] [ b ( ( ( ( ( ( e D D e δ ] }, 0 e l δ d whee Δ =. 4 Refeene Chung K. J., He C. C., & Lin S. D. (009. A w-waehue inveny mdel wih imefe qualiy duin ee. Cmue and Induial Engineeing, 8

29 56(, Dey, J. K., Mndal, S. K., & Maii, M. (008. Tw age inveny blem wih dynami demand and ineval valued lead-ime ve finie ime hizn unde inflain and ime-value f mney. Euean Junal f Oeainal Reeah, 85(, Dye, C. Y., Ouyang, L. Y., & Hieh, T. P. (007a. Inveny and iing aegie f deeiaing iem wih hage: a diuned ah flw aah. Cmue and Induial Engineeing, 5, Dye, C. Y., Ouyang, L. Y., & Hieh, T. P. (007b. Deeminii inveny mdel f deeiaing iem wih aaiy nain and ime-inal baklgging ae. Euean Junal f Oeainal Reeah, 78, Gayen, M., & Pal, A. K. (009. A w wae hue inveny mdel f deeiaing iem wih k deenden demand ae and hlding. Oeainal Reeah: An Inenainal Junal. 9(, Gyal, S. K., & Gii, B.C. (00. Reen end in mdeling f deeiaing inveny. Euean Junal f Oeainal Reeah, 4, -6. Haley, R.V. (976. Oeain Reeah A Manageial Emhai. Califnia: Gd Yea Publihing Cmany. Hieh, T. P., Dye, C. Y., & Ouyang, L. Y. (008. Deemining imal l ize f a w-waehue yem wih deeiain and hage uing ne een value. Euean Junal f Oeainal Reeah, 9(, 8-9. Kim, Y. H., & Philia, G. C. (986. Evaluaing invemen in inveny liy: a 9

30 ne een value famewk. The Engineeing Enmi,, 9-6. Lee, C. C., & Hu, S.-L. (009. A w-waehue duin mdel f deeiaing inveny iem wih ime-deenden demand. Euean Junal f Oeainal Reeah, 94(, Nahmia, S. (98. Peihable inveny hey: a eview. Oeain Reeah, 0, Niu, B., & Xie, J. (008. A ne n Tw-waehue inveny mdel wih deeiain unde FIFO diah liy. Euean Junal f Oeainal Reeah, 90(, Rng, M., Mahaaa, N. K., & Maii, M. (008. A w waehue inveny mdel f a deeiaing iem wih aially/fully baklgged hage and fuzzy lead ime. Euean Junal f Oeainal Reeah, 89(, Shah, N. H., Shah, B. J., & Wee, H.M. (009. A l ize inveny mdel f he Weibull diibued deeiain ae wih diuned elling ie and k-deenden demand. Inenainal Junal f Daa Analyi Tehnique and Saegie, (4, Teng, J. T., Chang, H. J., Dye, C. Y., & Hung, C. H. (00. An imal elenihmen liy f deeiaing iem wih ime-vaying demand and aial baklgging. Oeain Reeah Lee, 0, Wee, H. M., Yu, J. C. P., & Law, S. T. (005. Tw-waehue inveny mdel wih aial baklgging and Weibull diibuin unde inflain. Junal f he Chinee Iniue f Induial Enginee, (6,

31 , H. L. (004. Tw-waehue inveny mdel f deeiaing iem wih hage unde inflain. Euean Junal f Oeainal Reeah, 57, , H. L. (006. Tw-waehue aial baklgging inveny mdel f deeiaing iem unde inflain. Inenainal Junal f Pduin Enmi, 0, Zhu, Y. W. (00. A muli-waehue inveny mdel f iem wih ime-vaying demand and hage. Cmue and Oeain Reeah, 0, 5-4. Zhu, Y. W., &, S. L. (005. A w-waehue inveny mdel f iem wih k-level-deenden demand ae. Inenainal Junal f Pduin Enmi, 95(, 5-8.