NEWTON METHOD FOR DETERMINING THE OPTIMAL REPLENISHMENT POLICY FOR EPQ MODEL WITH PRESENT VALUE

Transcription

1 Yugoslav Joural of Operaios Research 8 (2008, Number, 53-6 DOI: 02298/YUJOR080053W NEWTON METHOD FOR DETERMINING THE OPTIMAL REPLENISHMENT POLICY FOR EPQ MODEL WITH PRESENT VALUE Jeff Kuo-Jug WU, Hsui-Li LEI 2, Sug-Te JUNG 2, Peer CHU 3 Deparme of Logisics Maageme, Naioal Defese Uiversiy, Taiwa, ROC Ady @homailcom 2 Deparme of Commuicaio Egieerig, Orieal Isiue of Techology, Taiwa, ROC {FO06, FO05}@mailoieduw 3 Deparme of Traffic Sciece, Ceral Police Uiversiy, Taiwa, ROC ua2@mailcpueduw Received: March 2004 / Acceped: April 2008 Absrac: This paper is a respose for he paper of Dohi, Kaio ad Osaki, ha was published i RAIRO: Operaios Research, 26, -4 (992 for a EPQ model wih prese value The purpose of his paper is hreefold Firs, he covex ad icreasig properies for he firs derivaive of he objecive fucio are proved Secod, we apply he Newo mehod o fid he opimal cycle ime Third, we provide some umerical examples o demosrae ha he Newo mehod is more efficie ha he bisecio mehod Keywords: Ecoomic producio quaiy, iveory model, prese value, Newo mehod INTRODUCTION The Newo mehod ad he bisecio mehod are he wo mos popular mehods o fid he zero of a equaio Wa ad Chu [5] demosraed ha he Newo mehod is beer ha he bisecio mehod for EOQ iveory models wih prese value The purpose of his paper is exeded discussio for EPQ iveory models For he applicaio of he bisecio mehod, we eed a lower boud ad a upper boud for he

2 54 Jeff K-J Wu, e al / Newo Mehod for Deermiig he Opimal Repleishme opimal soluio O he oher had, whe uilizig Newo mehod, we oly eed oe boud Oly a upper boud is required for miimizaio of moooically icreased covex fucio For decreasig ad covex fucio, we eed a lower boud Therefore, iuiively he Newo mehod should be more easily applied by researchers However, he explaaio of Newo mehod is more complicaed ha he bisecio mehod Hece, may researchers sill preferred o use he bisecio mehod Chug ad Tsai [7] sudied Goswami ad Chaudhuri [2] o reveal ha Newo mehod may diverge wih a improper sarig poi so ha a approximaed soluio of he firs derivaive could o saisfy cosrais Hece, he aalyical srucure of he objecive fucio should be examied case by case i order o validae he legiimacy of he Newo mehod Moreover, how o decide a appropriae sarig poi mus be checked o isure a fas coverge sequece o he opimal soluio Several papers, for examples, Dohi e al [0], Cormier ad Gu [9], Chu e al [] Chug e al [3], ad ohers have applied Newo mehod i order o derive he opimal soluio Dohi e al [0] did o explai how o selec a sarig poi o execue he Newo mehod Chug [2] also cosidered he EPQ iveory model of Dohi e al [0] He disapproved of usig he Newo mehod firs ad he adoped a bisecio algorihm o compue he opimal cycle ime Chug e al [5] revised Chug [2] o prepare a simple bisecio algorihm ad obai he opimal cycle ime wihou uecessary assumpios Chug [2] ad Chug e al [5] cosidered ha he ieraive sequece spaed by Newo mehod may o coverge o a opimal soluio whe he firs derivaive has wo roos There are six oher papers, Dohi e al [], Chug e al [6], Chug ad Li [4], Chug ad Tsai [8], Huag ad Chug [3], ad Moo e al [4] ha have referred o Dohi e al [0], bu oe of hem provided furher discussio of he Newo mehod for he opimal soluio I his paper, we will show ha Newo mehod is applicable o his kid of EPQ iveory model of Dohi e al [0], wih ay sarig poi Uder he cosideraio of he speed of moder compuer, i is uecessary o cosider a good sarig poi o accelerae he covergece of he ieraive procedure We compare umerical examples i Dohi e al [0], Chug [2] ad Chug e al [5], o illusrae ha he Newo mehod is efficie o derive he opimal soluio ad uder a small hreshold, he Newo mehod ouperforms boh bisecio mehods suggesed i Chug [2] ad Chug e al [5] i order o illusrae 2 REVIEW OF PREVIOUS RESULTS To be compaible wih Dohi e al [0], we apply he same oaio ad assumpios for a deermiisic producio iveory model wihou shorage uder he codiio of prese value The oaios used i his paper are lised below D is he demad rae; H is he holdig cos; K is he order cos; r is he ieres rae; S is he producio rae, wih S > D; ad TCr ( is he prese value of oal cos Wih he followig assumpios, he EPQ model wih ifiie plaig horizo is developed A sigle iem is cosidered ad he repleishme lead ime is assumed o be zero There is o shorage The begiig ad edig iveory level are boh zero Sice he ieres rae, r is assumed sable over he ime, so he repleishme cycle, say 0, is cosa over ime 0 = d + s where d is he producio period ad s is he cosumpio period

3 Jeff K-J Wu, e al / Newo Mehod for Deermiig he Opimal Repleishme 55 Le us recall he Eq (4 of Dohi e al [0], he prese value of oal coss over he eire ime horizo is expressed as TC r ( H rd K + S exp D exp r 2 r S = exp ( r ( ( Hece, i Eqs (5 ad (6 of Dohi e al [0], hey obaied ha where r ( exp( = ( exp( r dtc r d ( 2 ( ( ( HD r S D rd HS rd = exp exp rk exp (3 r S S r S I Theorem of heir paper, hey derived ha, wih he codiio S > D, ( is a sricly icreasig ad coiuous fucio Accordig o ( 0 = rk < 0 ad ( =, hey obaied ha here is a uique soluio, say, for ( = 0 (2 ad 3 OUR ANALYTICAL APPROACH I he followig, we begi o develop some properies for ( o derive ha ( S D r S D S D rd ( = HD exp + HD exp S S S S ( 2 2 S D r S D D rd ( = rhd exp + rh( S D exp S S S S Recall he properies of expoeial fucio, we kow ha ( 0 > Moreover, ( 0 is a icreasig fucio, for 0 fucio, for > 0 We summarize our fidigs i he ex Theorem (4 (5 > so ( > he ( is a covex Theorem Uder he codiio S > D, for > 0, ( is a icreasig ad a covex fucio To fid he opimal soluio for he objecio fucio, TC (, ha is o solve r

4 56 Jeff K-J Wu, e al / Newo Mehod for Deermiig he Opimal Repleishme dtcr d ( = 0 Accordig o Eq (2, sice expoeial fucio is always posiive, we have o cosider he soluio of he followig equaio ( = 0 (6 The radiioal approach is o discuss how o selec a good sarig poi as close o he opimal cycle ime,, as possible O he oher had, if we really udersad he properies of ( beig coiuous ad covex, he we will demosrae ha selecig ay sarig poi will coverge o he desired opimal cycle ime, We will apply he Newo mehod o fid he soluio of ( = 0 If we arbirarily selec a posiive umber, say, he here cases may happe: Case (a ( = 0, Case (b ( > 0, ad Case (c ( < 0 For Case (a, is he opimal soluio so we already fid i as = For Case (b, we will show ha here exiss a decreasig sequece wih = j+ j ( j ( j for j =,2, ad < j+ < j If we ca show ha < 2 < he i yields ha ( 2 > 0 ad < 3 < 2 By iducio, i will imply ha < < for j =,2, We assume ha y ( is he age lie of ( wih he age poi (, ( j+ y ( ( ( ( We will show ha y ( ( ( y( = ( ( ( ( = ( α( ( ( ( β( α ( j so ha = + (8 ad if, he y ( ( < We compue =, (9 where α is derived from Mea Value Theorem wih α is bewee ad, ad β is agai derived from Mea Value Theorem wih β is bewee α ad We cosider he relaios amog, ad α o divide he problem io followig hree cases: Case ( >, Case (2 =, ad Case (3 < For Case (, >, he < α < o imply ha α > 0 ad > 0, (7

5 Jeff K-J Wu, e al / Newo Mehod for Deermiig he Opimal Repleishme 57 hece accordig o Eq (9, i yields ha ( y( > 0 =, sice y ( = (, i shows ha ( y( For Case (2, = 0 For Case (3, <, he < α < o imply ha α < 0 ad < 0, herefore based o Eq (9 agai, i derives ha ( y( 0 > We combie our resuls i he ex lemma Lemma If f( is a icreasig ad a covex fucio ad y( is he age lie of f( wih age poi, f (, he f ( > y( for ( Accordig o Eq (7, 2 is he iersecio of he age lie, y (, wih age poi, (, (, wih he x-axis so y( 2 = 0 From Lemma, i shows ha f ( 2 > y( 2 = 0 so < 2 O he oher had, usig Eq (7 agai, ad ( > 0, uder he codiio of Case (b ( 0 < < We >, such ha 2 < Cosequely, i derives ha 2 summarize our fidigs i he ex lemma Lemma 2 If f ( is a icreasig ad a covex fucio, ad we selec a poi, say, wih f ( > he he Newo mehod will geerae a decreasig sequece, ( wih 0 + < < for =,2, Now we cosider Case (c, uder he cosrai ( < 0, ad we apply he Newo mehod of Eq (7, he we will prove ha ( 2 > 0 Owig o Eq (7, we recall ha ( ( 2 = = + (0 ( ( From ( > 0 Uder he <, hece, Eq (0 implies ha 2 > From Lemma, for 2, becomes a decreasig sequece, < + < for = 2,3, Now, we cosider he decreasig sequece, ( for = 2,3,, o prove ha is ideed coverge o From he compleeess axiom of real umber, he bouded below decreasig sequece wih < + < for = 2,3,, he i mus coverges o = Φ Accordig o Eq (7, i yields ha icreases he ( > 0 ad he ( codiio of ( 0 ( > y( = Therefore, ( is mos lower boud, say Φ, as lim

6 58 Jeff K-J Wu, e al / Newo Mehod for Deermiig he Opimal Repleishme lim + = lim ( ( We kow ha ( ad ( are boh coiuous fucios so ha hey ca ierchage wih he limi o imply ha Φ Φ ( Φ ( ( Φ = (2 Based o Eq (2, i obais ha ( Φ = 0 O he oher had, from he sricly icreasig propery of ( Φ = ad, we kow ha ( has a uique roo a Hece, lim = We summarize our fidigs i he ex heorem Theorem 2 If we selec a poi, say, as he sarig poi for he Newo mehod of a icreasig ad a covex fucio, f (,wih uique zero a he he decreasig sequece, ( 2 will coverge o Based o he above discussio, sice ( is a icreasig ad a covex fucio wih ( 0 < 0 ad ( =, so ( has a uique roo, say I Theorem 2, we prove ha ( 2 will coverge o ha is idepede of he sarig poi However, if we selec he sarig poi ha saisfies ( > 0 ha will accelerae he covergece Therefore, we develop a algorihm o fid he opimal cycle ime We will use a hreshold, say ε, o corol he ieraed compuaio for he Newo mehod wih ( > 0 as he Sep : Give ε > 0 as he hreshold, ad pick a poi, say sarig poi for he Newo mehod ( j Sep 2: To assume ha j = j for j = 2,3, ( j { } Sep 3: To defie ha mi : r ( j r ( j = j TC TC < ε Sep 4: To accep ha he opimal cycle ime is TCr Sice he compuer compuaio abiliy is improved so ha he compuaio sep is o loger a impora issue for researchers We will demosrae his fac by he CPU ime for our umerical examples ad opimal value is (

7 Jeff K-J Wu, e al / Newo Mehod for Deermiig he Opimal Repleishme 59 4 NUMERICAL EXAMPLES We cosider he followig example, wih he daa of parameers, r = 03, S = 9, K = 365, H = 605 ad D = 3 ha is relaed o examples i Dohi e al [0], Chug [2] ad Chug e al [5] For example, we ake ε = 000, wih he sarig poi, =, he afer wo ieraio, i shows ha 4 = = ad TCr ( 4 = Table : The Newo mehod for ε = 000 k= k=2 k=3 k=4 k ( k TC r ( k For example 2, we ake ε = , wih he sarig poi, =, he afer hree ieraio, i shows ha 5 = = ad TCr ( 5 = Table 2: The Newo mehod for ε = k= k=2 k=3 k=4 k=5 k ( k TC r ( k From he above hree examples, i reveals ha he Newo mehod is a efficie mehod o fid he opimal soluio Moreover, he value of ε will slighly ifluece he ieraio umber ad compuer CPU secods Here, we cosider he impac of differe sarig poi o lis he resuls i he ex able Table 3: The effec of arbirary sarig poi Iem Iiial value Sep Time = = = = = Based o Table 3, we may say ha he Newo mehod is a effecive mehod for fidig he opimal soluio ha is idepede of he sarig poi Nex, we compared he Newo mehod wih he bisecio mehod Firs, i Chug [2], uder he resricio S 2D, he foud a upper boud, KS HD S 2 U = ( D,

8 60 Jeff K-J Wu, e al / Newo Mehod for Deermiig he Opimal Repleishme ad ook zero as he lower boud for O he oher had, i Chug e al [5], hey S 2K ried o improve he bisecio mehod o derive a upper boud, TU = ad S D HD 2 a lower boud, T L = 2 2HD S D r+ r + + K D S To compare i wih bisecio mehod, we ake he upper boud of Chug [2] as our sarig poi for he Newo mehod o lis he ieraio umber i he ex able Table 4: Compariso of ieraio umbers Mehod ε =0 3 ε =0 6 Newo mehod 2 3 Bisecio [2] 8 20 Bisecio [5] 7 8 From Table 4, i demosraes ha whe he accuracy is emphasized he he Newo mehod is more efficie ha he bisecio mehod 5 CONCLUSIONS I his paper, whe he objecive fucio is icreasig ad covex, we prove ha he Newo mehod is efficie o fid he roo of he objecive fucio Based o umerical examples ha are relaed o Dohi e al [0], Chug [2] ad Chug e al [5], we show ha Newo mehod wih arbirary sarig poi will geerae a very fas coverge sequece o he opimal soluio REFERENCES [] Chu, P, Chug, KJ, ad La, SP, The crierio for he opimal soluio of a iveory sysem wih a egaive expoeial crashig cos, Egieerig Opimizaio, 32 ( [2] Chug, KJ, Opimal orderig ime ierval akig accou of ime value, Producio Plaig ad Corol, 7 ( [3] Chug, KJ, Chu, P, ad La, SP, A oe o EOQ models for deerioraig iems uder sock depede sellig rae, Europea Joural Operaioal Research, 24 ( [4] Chug, KJ, ad Li, CN, Opimal iveory repleishme models for deerioraig iems akig accou of ime discouig, Compuers & Operaios Research, 28 ( ( [5] Chug, KJ, Li, SD, Chu, P, ad La, SP, The producio ad iveory sysems akig accou of ime value, Producio Plaig ad Corol, 9 ( [6] Chug, KJ, Liu, J, ad Tsai, SF, Iveory sysems for deerioraig iems

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