CONTINUOUS REVIEW INVENTORY MODELS UNDER TIME VALUE OF MONEY AND CRASHABLE LEAD TIME CONSIDERATION

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1 Yugoslav Journal of Opratons Rsarch (), Numbr, OI: 98/YJOR93H CONTINUOUS REVIEW INVENTORY MOES UNER TIME VAUE OF MONEY AN CRASHABE EA TIME CONSIERATION Kuo-Chn HUNG partmnt of ogstcs Managmnt, Natonal fns Unvrsty, Tawan, ROC Rcvd: Aprl 8 / Accptd: Octobr Abstract: A stock s an asst f t can ract to conomc and sasonal nfluncs n th managmnt of th currnt assts Th fnancal managr must calculat th nput of funds to th stock ntllgntly and th amount of mony cycld through stocks, takng nto account th tm factors n th futur Th purpos of ths papr s to propos an nvntory modl consdrng ssus of crash cost and currnt valu Th snstvty analyss of ach paramtr, n ths rsarch, dffrs from th tradtonal approach W utlz a cours of dducton wth sound mathmatcs to dvlop svral lmmas and on thorm to stmat optmal solutons Ths study frst trs to fnd th optmal ordr quantty at all lngths of lad tm wth componnts crashd at thr mnmum duraton Scond, a smpl mthod to locat th optmal soluton unlk tradtonal snstvty analyss s dvlopd Fnally, som numrcal xampls ar gvn to llustrat all lmmas and th thorm n th soluton algorthm Kywords: Invntory modl, crashabl lad tm, tm valu of mony MSC: 9B5 INTROUCTION From th prspctv of fnancal managmnt, stocks oftn comprs a vry larg proporton of a balanc sht Funds nvstd n stock cannot b usd lswhr bcaus thy ar not lqud assts Thy bcom lqud only whn th stocks ar sold Consdrng captal runnng factors, stocks must b turnd ovr fast, so ntrprss must

2 94 KC Hung / Contnous Rvw Invntory Modls dtrmn approprat nvntory polcs n ordr to rduc dlnss of th stocks, and dad and scrap stocks n ordr to sll and produc ffctvly Studyng nvntory modls and consdrng tm and valu, Moon and Yun [3] mployd th dscountd cash flow approach to fully rcognz th tm valu of mony and constructd a fnt plannng horzon EO modl n whch th plannng horzon s a random varabl Jagg and Aggarwal [8], n ordr to dscuss an optmal rplnshmnt polcy wth an nfnt plannng horzon, rportd that a dtroratng product undr th mpact of a crdt prod dd not allow shortags Bos t al [] and Harga [6] dvlopd two nvntory modls, whch ncorporatd th ffcts of nflaton and tm valu of mony wth a constant rat of dtroraton and tm proportonal dmand Moon and [] nvstgatd th ffct of nflaton and tm-valu of mony n an nvntory modl wth a random product lf cycl W and aw [] mployd th concpts of nflaton and th tm valu of mony n a modl whr dmand s prcdpndnt and shortags allowd Chung and Tsa [3] drvd an nvntory modl for dtroratng tms wth th dmand of lnar trnds and shortags durng th fnt plannng horzon, consdrng th tm valu of mony Sun and uyrann [9] nvstgatd gnral mult-product, mult-stag producton and an nvntory modl usng th nt tm valu of mony wth ts total cost as th obctv functon Balkh [] consdrd a producton lot sz nvntory modl wth dtroratd and mprfct products, takng nto account nflaton and th tm valu of mony Moon t al [9] dvlopd nvntory modls for amloratng and dtroratng tms wth a tm-varant dmand pattrn ovr a fnt plannng horzon, takng nto account th ffcts of nflaton and th tm valu of mony Shah [7] drvd an nvntory modl by assumng a constant rat of dtroraton of unts n an nvntory and th tm valu of mony undr th condtons of prmssbl dlay n paymnts W t al [] dvlopd an optmal rplnshmnt nvntory stratgy to consdr both amloratng and dtroratng ffcts, takng nto account th tm valu of mony and a fnt plannng horzon Both th amloraton and dtroraton rat wr assumd to follow Wbull dstrbuton y t al [5] consdrd an nvntory modl for a dtroratng tm wth tm dpndnt dmand and ntrval-valud lad-tm ovr a fnt tm horzon Th nflaton rat and tm valu of mony ar takn nto account In addton, J [9] constructd a gnral framwork of an nvntory systm for non-nstantanous dtroratng tms wth shortags, th tm valu of mony, and nflaton as t al [4] dvlopd a two-warhous producton-nvntory modl for dtroratng tms consdrd undr nflaton and th tm valu of mony ovr a random tm horzon Hou t al [7] prsntd an nvntory modl for dtroratng tms wth a stock-dpndnt sllng rat undr nflaton and th tm valu of mony ovr a fnt plannng horzon Howvr, Kumar Mat [] also has dvlopd an nvntory modl ncorporatng customrs crdt-prod dpndnt dynamc dmand, nflaton, and th tm valu of mony, whr th lftm of th product s mprcs n natur In th rcnt studs, dcomposng th lad tm nto svral crashng prods s a controllabl approach to lad tm rducton Ouyang t al [5] constructd a varabl lad tm from a mxd nvntory modl wth backordrs and lost sals In ths artcl, w xtnd th nvntory modl of Ouyang t al [5] Whn th dstrbuton of lad tm dmand s normal, w consdr th tm valu of a contnuous rvw nvntory modl wth a mxtur of backordrs and lost sals

3 KC Hung / Contnous Rvw Invntory Modls 95 Ths papr s organzd as follows In th nxt scton, w dfn th notaton of th nvntory modl and ts assumptons In scton 3, frst w construct th nvntory modl, takng nto account th tm valu Thn w prov that th total xpctd annual cost s pc-wsly concav down wth rspct to lad tm, and convx n ordr quantts W apply a smpl mthod to dvlop four lmmas and on thorm, and locat th optmal soluton for constructng th procdur of solvng a rplnshmnt polcy n scton 4 Ths approach dffrs from th tradtonal mthods In scton 5, numrcal xampls ar offrd to llustrat our algorthm Scton 6 summarzs th artcl and prsnts som conclusons NOTATION AN ASSUNPTIONS W us th followng notaton and assumptons to dvlop nvntory modls wth crashng componnt lad tm and th tm valu of mony A : Fxd ordrng cost pr ordr : Avrag dmand pr yar h : Invntory holdng cost pr tm pr yar : ad tm that has n mutually ndpndnt componnts Th th componnt has a mnmum duraton a and normal duraton b wth a crashng cost c pr unt tm undr th assumpton c c cn Th componnts of ar crashd on at a tm, startng from th componnt of th last c and so forth Hnc, th rang for n n s from a to b = = : Th lngth of lad tm wth componnts,,, ar crashd to thr n mnmum duratons W dfn = b and n = a and = + b a, for n t t = = t= + =,, n Snc b > a, t follows that >, for =,, n R ( ): Th lad tm crashng cost pr cycl for a gvn [, ] s gvn by R ( ) = c( ) + ct( bt at) t= : Ordr quantty : Th optmal ordr quantty whn lad tm s X : ad tm dmand that follows a normal dstrbuton wth man μ and standard drvaton σ r : Rordr pont Snc r = xpctd dmand durng lad tm + safty stock, σ r = μ+ k Invntory s contnuously rvwd Rplnshmnts ar mad whnvr th nvntory lvl falls to th rordr pont r q : Allowabl stockout probablty durng n n

4 96 KC Hung / Contnous Rvw Invntory Modls k : Safty factor that satsfs PX ( > r) = PZ ( > k) = q, Z rprsntng th standard normal random varabl Br (): Expctd shortag at th nd of th cycl W quot th rsults of Ouyang t al [5], Br () = σ Ψ () k whr Ψ ( k) = ϕ( k) k[ Φ ( k) ] as φ, whr Φ dnots th standard normal probablty dnsty functon and cumulatv dstrbuton β : Th fracton of th dmand durng th stockout prod that wll b backordrd π : Fxd pnalty cost pr unt short π : Margnal proft pr unt : Th ntrst rat pr yar 3 MATHEMATICA FORMUATION Frst, w study th total xpctd annual cost of th nvntory modl wth backordrs and lost sals for varabl lad tm W quot th Equaton () of Ouyang t al [5], for [, n ], who drvd th total xpctd annual cost, EAC(, ), wthout consdrng th tm valu of mony as follows: EAC(, ) = EAC (, ) () as for,, wth =,,, n W rwrt th total xpctd annual cost h EAC (, ) = R ( ) p( ) ( ) + + +Ω () Whr [ β ] Ω ( ) = hσ k + ( ) Ψ ( k), [ β π ] p ( ) = σ π + ( ) Ψ ( k) + A and R ( ) = c ( ) + c ( b a ) t t t t= for, Scondly, w consdr th nvntory modl, takng nto account th tm valu Th xpctd nt nvntory lvl, ust bfor th ordr arrvs, s kσ + ( β) B( r),

5 KC Hung / Contnous Rvw Invntory Modls 97 and th xpctd nt nvntory at th bgnnng of th cycl s + kσ + ( β) B( r) Thrfor, th xpctd avrag nvntory lvl s + kσ + ( β) B( r) t for t, Hnc, th nvntory carryng cost for th frst cycl quals t= t kσ ( β) B( r) t + + h dt h kσ ( β) B( r) h = (3) W adopt th dscountd cash flow approach followng Moon and Yun [4] At th bgnnng of ach cycl wll b cash outflows for th ordrng cost, stockout cost and lad tm crashng cost Thrfor, th total rlvant cost for th frst cycl s h A+ ( π + π ( β)) σ Ψ ( k) + R( ) + kσ ( β) B( r) + h + + Rfrrng to Slvr and Ptrson [8], w gt that th tm valu of mony of th xpctd total rlvant cost ovr an nfnt tm horzon, C (, ), s gvn by A ( π π ( β)) σψ( k) R( ) h h + [ r μ ( β) B( r) ]( ) ( ) W can rwrt C(, ) as follows: f( ) h C (, ) = + g ( ) + for < < and <, whr f( ) = p( ) + R( ), p( ) = σ π + ( β) π Ψ ( k) + A, for [ ] (4),, =,, n, R( ) = R( ), Ω( ) h R ( ) = c ( ) + c ( b а ), g ( ) = t t t t =

6 98 KC Hung / Contnous Rvw Invntory Modls and [ β ] Ω ( ) = hσ k + ( ) Ψ ( k) Thrd, w us R ( ) to dnot th crashng cost W hav that R ( ) = R( ) whr R ( ) = c ( ) + c ( b a ) for,, t t t t= wth =,,, n Snc R ( ) s a lnar dcrasng functon on, = b a, w gt and R ( ) = c ( ) + c ( b a ) = c ( b a ) = R ( ) t t t t t t + t= t=, t follows that R ( ) s a pc-ws lnar dcrasng and contnuous functon on [, n ] At th ponts { : =,,, n }, R ( ) has dffrnt slops c and c + of th tangnt ln from th rght and lft, rspctvly Hnc, R ( ) s not dffrntabl at thos ponts, so w must dvd th doman of from [, n ] nto subntrvals,, wth =,,, n Accordng to Rachamadugu [6], n ordr to compar our rsults wth th prvous modl of Ouyang t al [5], w us A(, ) = C (, ), an altrnat but quvalnt masur A(, ) rprsnts th quvalnt unform cash flow stram that gnrats th sam C (, ) From h h h lm + =, w hav h lm A(, ) = + R ( ) + p ( ) +Ω ( ) That s quaton () for th total xpctd annual cost of Ouyang t al [5] Hnc, w xtnd thr modl Now, w bgn to fnd th mnmum valu of th total xpctd annual cost C (, ) for < < and < Takng th frst and scond partal drvatvs of C (, ) wth rspct to gvs

7 KC Hung / Contnous Rvw Invntory Modls 99 and C (, ) σ Ψ( k) = ( π π ( β)) c + + hσ C (, ) [ k+ ( β ) Ψ( k) ] σπ ( + π( β)) ψ( k) = hσ 4 3 [ k + ( β ) ψ ( k )] (5) (6) C (, ) From <, C (, ) s concav n, Hnc, w can n n rduc th mnmum problm from C (, ): a b, < to th = = boundary of ach pc-ws dfnd doman as { C (, ) : =, for =,,, n, < < } Fxng =, wth =,,, n, takng th frst and scond partal drvatv of C (, ) wth rspct to, gvs = + C (, ) f( ) h and C (, ) + h + + ( ) = f Rachamadugu [] drvd that x x >, for x > Hnc, w know that + x C s convx n th scond trm of th scond partal drvatv s postv, so (, ) (, ) wth th mnmum pont at such that = f( ) (7) h

8 3 KC Hung / Contnous Rvw Invntory Modls t φ ( ) = for < W know that φ ( ) s a strctly ncrasng functon from φ () = to lm φ( ) = Thrfor, gvn an, thr xsts a unqu pont satsfyng = f( ) h W hav shown that C (, ) s concav down n, In addton, for =, wth =,,, n, C(, ) s concav up n So th mnmum problm s to consdr th ponts (, ) for =,,, nw construct an algorthm as follows () Fnd th local mnmum ponts (, ) = n along th boundars of ach subntrval for,,, () For ach pont (, ), valuat th total xpctd annual cost C (, ) for =,,, n () Solv th mnmum of { C (, ):,,, n} = 4 MONOTONIC PROPERTY AN PROPOSITIONS W dtrmn a crtron to rduc th computaton of fndng th local mnmum for th nvntory modl In addton, w construct a nw functon as th dffrnc of th total xpctd annual cost functon valuatd at two adacnt local mnmum ponts Thn w vrfy f t s an ncrasng functon of th fracton of backordrs Thrfor, w can rduc th calculaton for locatng th optmal soluton Our purpos n ths scton s to dvlop a procdur that lmnats th nd to comput th xact valus of { : =,, n} and { C (, ): =,, n} W stablsh a crtron to compar and mplctly Morovr, w chang th valu of β to nvstgat th snstv analyss of backordrd rato pr cycl Our nw mthod sgnfcantly rducs th amount of computaton Frst, w offr such a crtron that w can mplctly compar wth All th proofs for th mmas and th thorm ar n th Appndx mma : Gvn a backordrd fracton rato β, thn < c ( + ) < σ π + ( β) π Ψ ( k) [ ] Scondly, w stat th monoton proprty btwn C (, ) and

9 KC Hung / Contnous Rvw Invntory Modls 3 mma : For a gvn β, f <, thn C (, ) < C (, ) From th Tabl of Ouyang t al [5], f >, w know that thr s no rgulaton btwn C (, ) and C (, ) Howvr, f w trat ( β ) and ( ) β as functons of β, thn w can stll masur th dffrnc btwn C ( ( β ), ) and C ( ( β ), ) mma 3: For a gvn ntrval β [ β, β], f ( β) ( β), thn for β [ β, β], C ( ( β ), ) C ( ( β ), ) s an ncrasng functon of β Hr, w show th monoton proprty of ( β) ( β) wth rspct to β mma 4: Gvn a fxd β, f ( β) ( β), thn ( β) ( β) for th ntrval β [ β, ] Fnally, w drv a crtron to compar C ( ( β ), ) wth C ( ( β ), ) Thorm : If C ( ( β), ) > C ( ( β), ) and ( β) ( β) for a fxd β, thn C ( ( β), ) > C ( ( β), ) for th ntrval β [ β, ] 5 NUMERICA EXAMPES Th followng numrcal xampls xplan how th abov mmas and th thorm smplfy th soluton procdur Usng th numrcal xampl from Ouyang t al [5], w hav th followng data: = 6 unts/yar, k = 845, A = $ /pr ordr, h = $ /pr tm pr yar, π = $5 /pr unt short, π = $5 /pr unt, σ = 7 unts/pr wk, q = (n ths stuaton, from th normal dstrbuton, w fnd k = 845 and ψ ( k) = ), and th lad tm has thr componnts wth data shown n Tabl W assum that th ntrst rat = Followng th soluton algorthm, w obtan Tabl Whn β = n Tabl, w slghtly chang th dcmal xprsson of, so apparntly t mpls () < () Tabl : ad tm data ad tm 3 componnt, R( ) Normal duraton, b (days) 6 Mnmum duraton, a (days) b a (wks) Unt crashng cost, c ($/wk)

10 3 KC Hung / Contnous Rvw Invntory Modls Tabl : Summary of solutons ( n wks) β = β = 5 β = 8 β = C (, ) C (, ) C (, ) C (, ) Consdrng th cass for β =, 5, 8 and, w us Tabl 3 to valuat c ( ) σ π + ( β) π Ψ ( k) + along wth [ ] Tabl 3: ata for comparson c( + ) β σ[ π + ( β) π ] Ψ ( k) Whn β =, w fnd σ[ π β π ] c ( + ) < + ( ) Ψ ( k) for all < for all,, 3 =,, 3 By mma, w gt () () = For all =,, 3, mma mpls C ( (), ) < C ( (), ), so th optmal soluton s ( 3(), 3) = (6,3) Whn β = 5,8 and, w fnd c ( ) < σ π + ( β) π Ψ ( k) for all =, Thus, by mma, w hav + [ ] C ( ( β), ) < C ( ( β), ) whn β = 5,8 and wth =, Thrfor, w nd to calculat only mn C ( ( β ), ) nstad of mn C ( ( β ), ) n ordr to gt th =, 3 =,,,3 optmal soluton for β = 5,8 and Furthrmor, w fnd C ( 3(5), 3) = 3496 > = C ( (5), ) and 3(5) = 76 > 73 = (5) Usng Thorm, w can conclud that mn EAC( ( β ), ) = EAC( ( β ), ) for =, 3 β = 5,8 and Consquntly, mmas,, 3 and 4, and Thorm, can smplfy th soluton procdur Wth our crtron, t s vry asy to compar th local mnmum

11 KC Hung / Contnous Rvw Invntory Modls 33 ponts and Usng th monoton proprty btwn C (, ) and, and th dffrnc of th total xpctd annual cost functon valuatd at two adacnt local mnmum ponts as an ncrasng functon of th fracton of backordrs, our computaton rsults bcom much smplr 6 CONCUSION Usually, thr ar thr knds of stocks n a company: raw matrals, work-nprocss, and fnshd goods Ths stocks all nd funds to b managd Th currnt assts ar th most dffcult to b cashd Good nvntory managmnt s oftn th mark of a wll-run frm Ths artcl consdrs th tm valu of mony of a contnuous rvw nvntory modl wth a mxtur of backordrs and lost sals, whr lad tm dmand has a normal dstrbuton W fnd th optmal ordr quantty and optmal lad tm of th total xpctd annual costs at all lngths of lad tms wth componnts crashd to thr mnmum duraton, and construct a procss for an optmal soluton W dvlop a prncpl to compar th optmal ordr quantts (β) at ponts ( (β), ) for all =,,, n Our approach, whn solvng most stuatons lk ths, dffrs from th tradtonal snstvty analyss W dduc th optmal valus va complt procdurs that ar mathmatcally sound Acknowldgmnts Ths papr s supportd by th Natonal Scnc Councl of th Rpublc of Chna, Tawan (NSC 99-4-H-66-5) REFERENCES [] Balkh, ZT, An optmal soluton of a gnral lot sz nvntory modl wth dtroratd and mprfct products, takng nto account nflaton and tm valu of mony, Intrnatonal Journal of Systms Scnc, 35 (4) [] Bos, S, Goswam, A, and Chaudhur, KS, An EO modl for dtroratng tms wth lnar tm-dpndnt dmand and shortags undr nflaton and tm dscountng, Journal of th Opratonal Rsarch Socty, 46 (995) [3] Chung, KJ, and Tsa, SF, Invntory systms for dtroratng tms wth shortags and a lnar trnd n dmand-takng account of tm valu, Computr & Opratonal Rsarch, 8 () [4] as,, Kar, MB, Roy, A, and Kar, S, Two-warhous producton modl for dtroratng nvntory tms wth stock-dpndnt dmand undr nflaton ovr a random plannng horzon, Cntral Europan Journal of Opratons Rsarch, Onln Octobr, pp - 3 [5] y, JK, Mondal, SK, and Mat, M, Two storag nvntory problm wth dynamc dmand and ntrval valud lad-tm ovr fnt tm horzon undr nflaton and tm-valu of mony, Europan Journal of Opratonal Rsarch, 85 () (8) 7-94 [6] Harga, M, Effcts of nflaton and tm valu of mony on and tm-dpndnt dmand rat and shortags, Europan Journal of Opratonal Rsarch, 8 (995) 5-5

12 34 KC Hung / Contnous Rvw Invntory Modls [7] Hou, K, Huang, YF, and n, C, An nvntory modl for dtroratng tms wth stock-dpndnt sllng rat and partal backloggng undr nflaton, Afrcan Journal of Busnss Managmnt, 5 () () [8] Jagg, CK, and Aggarwal, SP, Crdt fnancng n conomc ordrng polcs of dtroratng tm, Intrnatonal Journal of Producton Economcs, 34 (994) 5-55 [9] J,, Rsarch on nvntory modl for non-nstantanous dtroratng tms takng nto account tm valu of mony, Intrnatonal Confrnc on Intrnt Tchnology and Applcatons, ITAP - Procdngs, [] Kumar Mat, M, A fuzzy gntc algorthm wth varyng populaton sz to solv an nvntory modl wth crdt-lnkd promotonal dmand n an mprcs plannng horzon, Europan Journal of Opratonal Rsarch, 3 () () 96-6 [] Moon, I, Gr, BC, and Ko, B, Economc ordr quantty modls for amloratng dtroratng tms undr nflaton and tm dscountng, Europan Journal of Opratonal Rsarch, 6 (3) (5) [] Moon, I, and, S, Th ffcts of nflaton and tm-valu of mony on an conomc ordr quantty modl wth a random product lf cycl, Europan Journal of Opratonal Rsarch, 5 () [3] Moon, I, and Yun, W, A not on valuatng nvstmnts nvntory systms: A nt prsnt valu framwork, Th Engnrng Economst, 39 () (993) [4] Moon, I, and Yun, W, An conomc ordr quantty modl wth a random plannng horzon, Th Engnrng Economst, 39 () (993) [5] Ouyang, Y, Yh, NC, and Wu, KS, Mxtur nvntory modls wth backordrs and lost sals for varabl lad tm, Journal of th Opratonal Rsarch Socty, 47 (996) [6] Rachamadugu, R, Error bounds for EO, Naval Rsarch ogstcs, 35 (988) [7] Shah, NH, Invntory modl for dtroratng tms and tm valu of mony for a fnt tm horzon undr th prmssbl dlay n paymnts, Intrnatonal Journal of Systms Scnc, 37 () (6) 9-5 [8] Slvr, EA, and Ptrson, R, cson Systms for Invntory Managmnt and Producton Plannng, John Wly, Nw York, 985 [9] Sun, N, and uyrann, M, Producton and nvntory modl usng nt prsnt valu, Opratons Rsarch, 5 () [] W, HM, and aw, ST, Rplnshmnt and prcng polcy for dtroratng tms takng nto account th tm-valu of mony, Intrnatonal Journal of Producton Economcs, 7 () 3- [] W, HM, o, ST, Yu, J, and Chn, HC, An nvntory modl for amloratng and dtroratng tms takng account of tm valu of mony and fnt plannng horzon, Intrnatonal Journal of Systms Scnc, 39 (8) (8) 8-87 APPENIX Proof of mma : From = f( ) and φ ( ) =, whch s a h strctly ncrasng functon, w gt th followng crtron for comparng and : < f( ) < f( ) p ( ) + R ( ) < p ( ) + R ( ) [ β π ] c ( + ) < σ π + ( ) Ψ ( k)

13 KC Hung / Contnous Rvw Invntory Modls 35 Proof of mma : From Equatons (4) and (7), w hav σ [ β ] C (, ) = h h k ( ) ( k) h + + Ψ (8) Usng Equaton (8) and = b a >, w gt mma Proof of mma 3: From quaton (8), w obtan h C ( ( β), ) C ( ( β), ) = p ( ) p ( ) + Snc σ[ β ψ ] ( β) ( β) (9) p ( ) p ( ) = h k+ ( ) ( k) ( ) and >, w gt that p ( ) p ( ) s an ncrasng functon of β ( β ) From quaton (7), ( β ) = f( ) and h f ( ) p( ) R( ) p ( ) = σ π + ( β) π Ψ ( k) + A and = +, wth [ ] R ( ) = c( b a) follows: k k k k = ; thn w tak th drvatv of Equaton (7) wth rspct to β as ( ) ( ) ( ) β d β σ π ψ k = dβ h Hnc, w gt d ( β ) σπ ψ ( k) = dβ ( β ) h () x x W assum Η ( x) = x, whch shows that Η ( x) = < x ( ) From ( β) ( β), w hav ( β) ( β) ( β) ( β) Combnng Equaton () wth > shows ()

14 36 KC Hung / Contnous Rvw Invntory Modls ( β) ( β) > ( β) ( β) () ( β ) ( β ) Now, w comput th drvatv of wth rspct to β as follows: d ( β) ( β) ( β) d( β) ( β) d ( β) dβ = dβ dβ Usng Equatons () and (), w know d dβ ( β) ( β) ( β) ( β) σ π ψ ( k ) = h ( β) > ( β) ( β ) ( β ) Thrfor, s an ncrasng functon of β From quaton (9), C ( ( β), ) C ( ( β), ) s th sum of two ncrasng functons of β, so w fnsh th proof of mma 3 Proof of mma 4: Gvn a fxd β, f ( β) ( β), from th dual statmnt of mma, c( + ) σ[ π + ( β) π] Ψ ( k) Thrfor, c( + ) σ[ π + ( β) π] Ψ ( k), for β [ β,] Smlarly, from th dual statmnt of mma, ( β) ( β), for β [ β, ] Proof of Thorm : Usng mma 4 nducs that ( β) ( β), for β [ β, ] Hnc, from mma 3, w complt th proof of Thorm