Joint Pricing and Inventory Replenishment Decisions in a Multi-level Supply Chain
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- Leon Ross
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1 Engining Ltt 8:4 EL_8_4_09 Joint Piing and Inntoy Rnint iion in a Muti- Suy Cain YUN HUAN EORE Q HUAN Abtat i a oodinat iing and inntoy nint diion in a uti- uy ain ood of uti ui on anufatu and uti tai W od ti ob a a t- ntd Na ga w a t ui fouat t botto- Na ga t wo ui to ay t idd- Na ga wit t anufatu and bot to a a gou ay fouat t to- Na ga wit t tai Anaytia tod and oution agoit a dod to dtin t quiibiu of t ga A nuia tudy i ondutd to undtand t infun of diffnt aat on t diion and ofit of t uy ain and it ontitunt b Sa intting a finding a bn obtaind Indx iing nint uti- uy ain Na ga I INROUCION In t dntaizd uy ain inonitn and inoodination xitd btwn oa objti and t tota yt objti a t uy ain o it otitin inaingy ([0]) Many a a ondd oganizationa oodination fo anaging uy ain ffiinty ([ 3 5 ]) Intgating iing wit inntoy diion i an iotant at to anufatuing and tai induti Coodinating iing and inntoy diion of uy ain (CPISC) a bn tudid by a at about fifty ya ago Wng and Wong [5] and Wng [4] oo a od of ui-tai ationi and onfi tat oodinatd diion on iing and inntoy bnfit bot t indiidua ain b and t nti yt Rfn [] anayz t ob of oodinating iing and inntoy nint oii in a uy ain oniting of a woa on o o gogaiay did tai y ow tat otiay oodinatd oiy oud b intd ooatiy by an inntoy-onignnt agnt Pafua t a [] nt a t of od of oodination fo iing and od quantity diion in a on anufatu and on tai uy ain y ao diu t adantag and diadantag of aiou oodination oibiiti a w a diud abo ainy fou on oodination of indiidua ntiti o two-tag ann In aity a uy ain uuay onit of uti fi (ui anufatu tai t) Jabb and oya [6] onid oodination of od quantity in a uti ui a ing ndo and Manuit id Stb 6 00 Yun Huang i wit t atnt of Indutia and Manufatuing Syt Engining Uniity o Hong ong Hong ong (on: (0085) ; -ai: uangyun@u) og Q Huang i wit t atnt of Indutia and Manufatuing Syt Engining Uniity o Hong ong Hong ong (-ai: gquang@u) uti buy uy ain i tudy fou on oodination of inntoy oii in t- uy ain wit uti fi at a tag Rnty a oy a bn ud a an atnati to anayz t ating and inntoy oii in uy ain wid Wng [3] tudy a uy ain wit on anufatu and uti idntia tai H ow tat t Stabg ga guaantd ft oodination oniding quantity diount and fani f Yu t a [7] iutanouy onid iing and od inta a diionaiab uing Stabg ga in a uy ain wit on anufatu and uti tai Eaii t a [4] oo a ga od of -buy ationi to otiiz iing and ot izing diion a-toti aoa a oyd to oodinat iing and inntoy oii in t abo a but t auto ti fou on t two-tag uy ain In ti a w intigat t CPISC ob in a uti- uy ain oniting of uti ui a ing anufatu and uti tai anufatu ua diffnt ty of aw atia fo i ui Sing ouing tatgy i adotd btwn t anufatu and t ui n t anufatu u t aw atia to odu diffnt odut fo diffnt indndnt tai wit iitd odution aaity In ti uy ain a t ain b a ationa and dtin ti iing and nint diion to axiiz ti own ofit non-ooatiy W dib t CPISC ob a a t- ntd Na ga wit t to t oa uy ain ui fouat t botto- Na ga and a a wo ay t idd- Na ga wit t anufatu Lat t ui and t anufatu bing a gou fouat t to- Na ga wit t tai t- ntd Na ga tt an quiibiu oution u tat any ain b annot io i ofit by ating uniatay witout dgading t foan of ot ay W oo bot anaytia and outationa tod to o ti ntd Na ga i a i oganizd a foow nxt tion gi t CPISC ob dition and notation to b ud Stion 3 do t t- ntd Na ga od fo t CPISC ob Stion 4 oo t anaytia and outationa tod ud to o t CPISC ob in Stion 3 In tion 5 a nuia tudy and oonding nitiity anayi fo o td aat a bn ntd Finay ti a onud in Stion 6 wit o uggtion fo fut wo (Adan onin ubiation: 3 Nob 00)
2 Engining Ltt 8:4 EL_8_4_09 II PROBLEM SAEMEN AN NOAIONS A Pob dition and aution In t t- uy ain w onid t tai faing t uto dand of diffnt odut wi an b odud by t anufatu wit diffnt aw atia uad fo t ui non-ooati ui a an quiibiu and a a wo ngotiat wit t anufatu on ti iing and inntoy diion to axiiz ti own ofit Aft t ui and t anufatu a an agnt t anufatu wi ua t aw atia to odu diffnt odut fo t tai Ngotiation wi ao b ondutd btwn t anufatu and t tai on ti iing and inntoy diion Wn an agnt i aid btwn t t tai wi ua t odut and tn ditibut to ti uto W tn gi t foowing aution of ti a: () Ea tai ony on ty of odut tai at a aud to b indndnt of a ot annua dand funtion fo a tai i t daing and onx funtion wit t to i own tai i () Soo ouing tatgy i adotd btwn ui and anufatu at i to ay a ui oid on ty of aw atia to t anufatu and t anufatu ua on ty of aw atia fo ony on ui (3) intg utii ani [9] fo nint i adotd at i a ui y ti i an intg utii of t y ti of t anufatu and t anufatu nint ti i t intg utii of a t tai (4) inntoy of t aw atia fo t anufatu ony ou wn odution i t u (5) Sotag a not ittd n t annua odution aaity i gat tan o qua to t tota annua at dand ([4]) B Notation A t inut aat and aiab ud in ou od wi b tatd a foow Au t foowing ant aat fo t tai: L: ota nub of tai : Indx of tai A : A ontant in t dand funtion of tai wi nt i at a : Coffiint of t odut dand atiity fo tai : Rtai i agd to t uto by tai : Rtai annua dand R : Rtai annua fixd ot fo t faiiti and oganization to ay ti odut : iion to t of tai x i i diion to : Objti (ayoff) funtion of tai tai diion aiab a: : Rtai ofit agin : intg diio ud to dtin t nint y of tai anufatu ant aat a: : Indx of anufatu : Hoding ot unit of odut inntoy : Hoding ot unit of aw atia uad fo ui S : Stu ot odution O : Oding oing ot od of aw atia P : Annua odution aaity odut wi i a nown ontant R : Manufatu annua fixd ot fo t faiiti and oganization fo t odution of ti odut : Woa i agd by t anufatu to t tai : Podution ot unit odut fo t anufatu : iion to t of t anufatu x i i diion to : Objti (ayoff) funtion of t anufatu anufatu diion aiab a: : Manufatu ofit agin fo odut : Manufatu tu ti inta ant aat fo t ui a: V: ota nub of ui : Indx of ui = V : Hoding ot unit of aw atia inntoy fo ui : Raw atia ot aid by ui R : Sui annua fixd ot fo t faiiti and oganization to ay t aw atia O : Od oing ot fo ui od : Uag of ui aw atia to odu a unit odut : Raw atia i agd by ui to t anufatu : iion to t of ui x i i diion to : Objti (ayoff) funtion of ui ui diion aiab a: : Sui ofit agin : intg utii ud to dtin t nint y of ui (Adan onin ubiation: 3 Nob 00)
3 Engining Ltt 8:4 EL_8_4_09 III MOEL FORMULAION A t- nt Na ga W od t CPISC ob a a t- ntd Na ga wit V+ L+ ay i V ui on anufatu and L tai Ea ui onto diion to t (= V) to axiiz i ayoff funtion A diion to x inud ofit agin and nint diion to Hi diion to fo diffnt anufatu onto t diion to t axiiz i ayoff funtion x onit of ofit agin odut (= L) and tu ti inta Ea tai onto t diion to t wo diion to x i ood of ofit agin and nint diion to axiiz i ayoff funtion In ou ga fawo fity t V ui fouat t botto- Na ga in t anufatu and t tai diion to a ui diion to x ai wit t ang of t ot ui diion to to axiiz i ayoff funtion Wn non of t woud i to at ti diion t botto- Na quiibiu i obtaind Sondy t ui in quiibiu bing a ay fouat t idd- Na ga wit t anufatu In ti ga gin t tai diion to t ui adjut ti diion to x x x wit t ang of t V anufatu diion wi t anufatu ai i diion to x a t ui diion anging unti non of t oud io i ayoff funtion by uniatay ating i diion u t idd- Na quiibiu ai Laty t to- Na ga i ayd btwn t ui and t anufatu in quiibiu and a t tai Ea tai diion to x ai wit t ang of t ui and t anufatu diion and t ot tai diion ui and t anufatu ao adjut ti quiibiu diion to x x wit t ang of t tai diion o ontinu unti t ui t anufatu and t tai annot ina ti ayoff by anging ti diion at i t to- Na quiibiu a B tai od W fit onid t objti (ayoff) funtion fo t tai tai objti i to axiiz i nt ofit by otiizing i ofit agin and nint diion A indiatd in t fout oint of t aution in tion t intg utii ani i oyd btwn t anufatu and t tai Sin t tu ti inta fo t anufatu i aud to b t nint y fo tai i / oud b a oiti intg u t annua oding ot i / ( Fig (a)) and t oding o ot i O / u tai fa t oding ot t oding ot and an annua fixd ot fo t tai objti funtion i gin by t foowing quation: O ax R () Subjt to { 3 } () (3) A (4) 0 (5) 0 P (6) Containt () ow t dand funtion Containt (3) gi t au of t diio ud to dtin t tai nint y ti Containt (4) indiat t ationi btwn t i (t tai i and t woa i) and tai ofit agin Containt (5) nu tat t au of i nonngati Containt (6) gi t bound of t annua dand wi annot xd t annua odution aaity of t odut (Adan onin ubiation: 3 Nob 00)
4 Engining Ltt 8:4 EL_8_4_09 C anufatu od anufatu objti i to dtin i diion to x ood of t ofit agin fo a t odut and t tu ti inta fo odution to axiiz i nt ofit anufatu fa annua oding ot tu and oding ot and an annua fixd ot annua oding ot fo t anufatu i ood of two at: t ot of oding aw atia ud to ont to odut t ot of oding odut uing t odution otion t aag inntoy of aw atia ud fo odut i / odution ti in a ya i / P uing t non-odution otion of t y t aw atia inntoy do to zo and t oding ot i zo aoding to ou aution Hn t annua oding ot fo aw atia i / P annua inntoy fo odut i gin by P (a uggtd by [8]) baio of t inntoy fo t anufatu i iutatd a Fig (b) tu ot S and oding ot O ou at t bginning of a odution u w an aiy di t anufatu objti (ayoff) funtion : ax L P (7) S O R P Subjt to fo a = L (8) 0 fo a = L (9) 0 Containt (8) gi t ationi btwn t i (t woa i and t aw atia i) and t anufatu ofit agin Containt (9) and nu tat t au of and a nonngati ui od Ea ui ob i to dtin an otia diion to x (inuding nint diion and ofit agin ) to axiiz i nt ofit Aoding to t fout oint of t aution in tion t intg utii ani i adotd btwn t ui and t anufatu So t nint y ti fo ui i aw atia inntoy do y ya by tating fo ( ) a Fig () ow fo t oding ot i ( ) ( ) ( ) ( ) ( ) ( ) wi i qua to ui fa oding ot oding ot and an annua fixd ot u t ui objti (ayoff) funtion i: ( ) O ax R () Subjt to 3 () (3) 0 (4) Containt () gi t au of ui utii ud to dtin i nint y ti Containt (3) indiat t ationi btwn t aw atia i and t ui ofit agin Containt (4) nu t non-ngatin of IV SOLUION ALORIHM In ti a w ainy bad on anaytia toy ud by [7] to out Na quiibiu In od to dtin t t- ntd Na quiibiu w fit u anayti tod to auat t bt ation funtion of a ay and oy agoit odu to buid t Na quiibiu A Ration funtion ) tai ation W x t tai dand funtion by t oonding ofit agin Subtituting (3) (8) (3) w an wit (5) a: A (5) Now uo tat t diion aiab fo ui and anufatu a fixd n t tai ob of finding t otia nint y bo: O in U (6) au of tat iniiz U i by t at tat atifi: (7) O bt ation an b xd a [3]: / (8) O H w dfin a a t agt intg no ag tan a W tn onid t otia au of Fo ontaint (6) and (7) w an obtain ow bound and t u bound of : (Adan onin ubiation: 3 Nob 00)
5 Engining Ltt 8:4 EL_8_4_09 A P ax 0 (9) A Subtitutd (6) into () w an tat i a quadati funtion of Bau t ond diati of wit t to i ngati w a: 0 () u i a ona funtion of St t fit diati of wit t to qua to zo n an b obtaind a: C 4 () w C A If obtaind fo () i in t inta of it i obiouy t otia ation of t tai Otwi w a to ubtitut t bound (9) and into () t bound tat oid ig ofit i t bt ation ) anufatu ation Au tat t diion aiab fo t ui and t tai a fixd anufatu ob of finding t otia tu inta in ti a bo: S O in U P P (3) Sin t ond diation of (3) U S O 0 t otia fo t iniu 3 of U an b did fo: U S O 0 P P (4) o S O P P (5) Obiouy t otia obtaind fo (5) atifi ontaint () nt ofit i t quadati funtion about Fo ontaint (7) and w an obtain ow and t u bound of : A P ax 0 (6) A (7) ond diation about i P (8) If 0 (9) t otia an b obtaind fo t fit od ondition of : 0 Subtitut (6) into (30) w a: W P P w If A W (30) W (3) obtaind fo (3) i in t inta of it i t otia ation of t anufatu Otwi a it axia au wn i at it u bound o ow bound bound tat oid ig ofit i t otia ation If 0 oid axia au of 3) ui ation w ao a to find t bound tat at i t bt ation Laty w onid t ation funtion fo t ui Suo tat t diion aiab fo tai and anufatu a fixd ui ob of finding t otia nint y bo: (Adan onin ubiation: 3 Nob 00)
6 Engining Ltt 8:4 EL_8_4_09 ( ) O in U (3) otia tat iniiz U an b xd a foow: 8 O / (33) W tn onid ui otia ation fo ond od ondition fo 0 (34) u t nay ondition to axiiz t ui nt ofit i: ( ) 0 (35) Subtitut (6) into (35) w an obtain: E ( ) 4 u u u V u (36) w E A Fo ontaint (7) and (5) w an obtain t u bound and ow bound of : A P ax 0 u u u u (37) A u u u u (38) If obtaind fo (36) i in t inta of it i t bt ation of ui Otwi w a to ubtitut t bound (37) and (38) into () t bound tat oid ig ofit i t otia ation B Agoit W dnot L and a t t of diion to of tai anufatu and ui tiy V L and a t tatgy ofi t of t ui t tai t ui and t anufatu and a t ain b W nt t foowing agoit fo oing t t- ntd Na ga od: St 0 Initiaiz x x x x in tatgy t St not a t tatgy ofi of a t ain x b in x xt fo tai Fo a tai fixd x to x find out t otia ation otiiz t tai ayoff funtion I St not b in in it tatgy t x a t tatgy ofi of a t ain x xt fo t anufatu Fixd x x L to I a t tatgy ofi of a t ain find out t otia ation otiiz t ofit funtion in it tatgy t St 3 not x b in x xt fo ui Fo a ui fixd x to x find out t otia ation otiiz t ofit funtion x in it tatgy t I If x 0 t botto Na Equiibiu x obtaind o t 4 Otwi x x at t 3 x x x If x x 0 t idd St 4 Na Equiibiu x obtaind o t 5 Otwi x x go t St 5 x x x If x x 0 t abo Na Equiibiu x obtaind Outut t otia ut and to Otwi x x go t V NUMERICAL EAMPLE AN SENSIIVE ANALYSIS In ti tion w nt a i nuia xa to dontat t aiabiity of t ood oution odu to ou ga od W onid a uy ain oniting of t ui on ing anufatu and two tai anufatu ou t ind of aw atia fo t t ui n t anufatu u t to odu two diffnt odut and ditibut t to two tai atd inut aat fo t ba xa a bad on t uggtion fo ot a ([7 6]) Fo xa t oding ot unit fina odut at any tai oud b ig tan t anufatu anufatu tu ot oud b u ag tan any oding ot aat fo t bad xa a gin a: O 3 O 3 O S O 50 P P A A O 40 O 30 And t fixd ot fo a t ay a 000 By aying t abo oution odu in tion 5 t otia ut fo t ui t anufatu and t tai a own in ab In od to nu tat ou onuion a not bad uy on t on nuia au of t ba xa w ao (Adan onin ubiation: 3 Nob 00)
7 Engining Ltt 8:4 EL_8_4_09 ondut o niti anayi on o aat inuding t at atd aat t odution atd aat and t aw atia atd aat oug t t- ntd Na ga od and t nuia xa o aningfu anagia iiation an b dawn: () ina of at aat wi du t tai ofit but bnfit t ot tai Wn ina t ang of t tai dand i o niti to t ang of i tai i oad wit t ba xa tai ofit an b dud by owing down i tai i But i at dand annot b inad wi a t anufatu fo ig ofit fo ot odut to fi u t o ddud by ti odut / tai at It i good nw to t ot odut / tai bau t anufatu wi ow down i woa i to tiuat ti at dand () Wn t anufatu tu ot S ina t anufatu ofit da o ignifianty tan t tai wi o ui ofit ina ina of S a t anufatu odu o odut wit ig ofit agin (odut ) and du t odution of ow ofitab odut (odut ) uag of aw atia ina a t ang of t anufatu odution tatgy At t a ti o ui bu u ti i tu binging ig ofit to t (3) iat of t ina of ui aw atia ot on i own ofit ay not a ignifiant a tat on t ot ui ina of a t ui ai i aw atia i and ut in an ina ot in fina odut a w a t da in at dand Hn t ot ui wi du ti i to t at and otiiz ti indiidua ofit Sui a t u ow ofit agin tan ot ui o wi not du i ofit agin Hn t ui ofit da at (4) Wn t at aat t anufatu tu ot S o t ui aw atia ot ina t anufatu tu ti inta wi b ngtnd A ig o ut in t tota at dand da a w a a ow inntoy onution at ina of S a t anufatu ot odution i u Hn t anufatu a to ondut i odution fqunty VI CONCLUSION In ti a w a onidd oodination of iing and nint y in a uti- uy ain ood of uti ui on ing anufatu and uti tai Sniti anayi a bn ondutd on at aat odution aat and aw atia aat ut of t nuia xa ao ow tat: (a) wn on tai at bo o niti to ti i i ofit wi b dad wi t ot tai ofit wi ina; (b) t ina of t anufatu odution tu ot wi bing o to if and t tai but ay ina t ofit of o ui; () t ina of aw atia ot au o to a t uy ain b Suiingy t ofit of ti aw atia ui ay not da a ignifiant a t ot ui ; (d) t tu ti inta fo t anufatu wi b ngtnd a t ina of t tai i nitiity t anufatu tu ot o t ui aw atia ot How ti a a t foowing iitation wi an b xtndd in t fut a Atoug ti a onid uti odut and uti tai t otition aong t i not od Und ti otition t dand of on odut / tai i not ony t funtion of i own i but ao t ot odut / tai i Sondy t ui a aud to b td and ing ouing tatgy i adotd In fat it ui tion o uti ouing i an initab at of uy ain anagnt Ao w au tat t odution at i gat tan o qua to t dand at to aoid otag ot Witout ti aution t xta ot oud b inooatd into t futu wo REFERENCES [] Ada 995 Catgoy anagnt A ating ont fo anging ti in: J Hibunn d Mating Enyodia: Iu and tnd aing t futu [] Boyai ago Coodinating iing and inntoy nint oii fo on woa and on o o gogaiay did tai Intnationa Jouna of Podution Eonoi 77() [3] Cuan A A Ladd 000 SAP/R3 buin buint: undtanding nti uy ain anagnt U Sadd Ri: Pnti Ha [4] Eaii M Mi-Baado Ayanzad P ongu A ga toy aoa in -buy uy ain Euoan Jouna of Oation Ra 9() [5] Huang Yun Q Huang a-toti oodination of ating and inntoy oii in a uti- uy ain Ltu Not in Engining and Cout Sin: Poding of Wod Cong on Engining 00 WCE Jun - Juy 00 London U [6] Jab M Y S oya Coodinating a t- uy ain wit uti ui a ndo and uti buy Intnationa Jouna of Podution Eonoi (6) [7] Liu Baoding Stabg-Na Equiibiu fo uti ogaing wit uti foow uing gnti agoit Cout Mat Aiation 36(7) [8] Lu L A on-ndo uti-buy intgatd inntoy od Euoan Jouna of Oationa Ra 8() [9] Moutaz ouja Otiizing inntoy diion in a uti-tag uti-uto uy ain anotation Ra Pat E [0] Pot M 985 Cotiti adantag: ating and utaining uio foan Nw Yo: F P985 [] Pafua Joga Madjid aana Ja Raaot A oni t of Mod of inta and int-oganizationa oodination fo ating and inntoy diion in a uy ain Intnationa Jouna of Intgatd Suy Managnt 006 [] Sando Cobitt R Boyin 00 Enti intgation Nw Yo: Jon Wiy and Son [3] Viwanatan S Q Wang 003 iount iing diion in ditibution ann wit i-niti dand Euoan Jouna of Oationa Ra [4] Wng Cann oodination and quantity diount Managnt Sin 4(9) [5] Wng R Wong na od fo t ui a-unit quantity diount oiy Naa Ra Logiti 40(7) [6] Woo Y Y Su-Lu Hu Souan Wu An intgatd inntoy od fo a ing ndo and uti buy wit oding ot dution Intnationa Jouna of Podution Eonoi (Adan onin ubiation: 3 Nob 00)
8 Engining Ltt 8:4 EL_8_4_09 [7] Yu Yugang Liang Liang og Q Huang Lad-foow ga in ndo-anagd inntoy yt wit iitd odution aaity oniding woa and tai i Intnationa Jouna of Logiti: Ra and Aiation 9(4) ab Rut fo ui anufatu and tai und diffnt aat (a) Podut dand and ofit fo ui anufatu and tai (b) Piing and nint diion fo ui anufatu and tai (0 4 ) 3 (0 4 ) (0 5 ) 3 (0 5 ) (0 5 ) (0 5 ) 3 3 (0 5 ) 3 (0 5 ) Ba xa S (Continud) (Adan onin ubiation: 3 Nob 00)
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