Optimal Replenishment Policy for Ameliorating Item with Shortages under Inflation and Time Value of Money using Genetic Algorithm

Transcription

1 Inernaional Journal of Compuer Applicaions ( ) Volume 7 No., Augus Opimal Replenishmen Policy for Amelioraing Iem wih Shorages under Inflaion and ime Value of Money using Geneic Algorihm S.R. Singh Deparmen of Mahemaics D.N. College, Meeru (U.P.) India arun Kumar Deparmen of Compuer Science Banashali Universiy, Banashali (Rajashan), India C. B. Gupa Deparmen of Mahemaics Birla Insiue of echnology and Science, Pilani (Rajashan), India ABSRAC he implemenaion of supply chain has o reduce he oal cos of sysem, bu generally each componen of a supply chain ries o find he bes policy for is company and consequenly ries o find a local opimum. Knowing ha he sum of local opimum canno consiue he global opimum, i is necessary o consider all coss of sysem simulaneously o find he opimal replenishmen policy for all he componens of a supply chain. Demand rae of he iem is assumed o be a funcion of ime known as ramp ype funcion. Shorages are permied and parially back-ordered. he back-ordering fracion is aken o be decreasing funcion of waiing ime. We consider inflaion and apply discouned cash flow in he problem analysis. oal cos of he sysem is formulaed and opimal replenishmen policy is derived, keeping in view he above facors of he sysem. We use Geneic Algorihm (GA) o solve he models. A numerical example and sensiiviy analysis is shown o illusrae he models. Keywords Replenishmen policy, ramp ype demand, amelioraion iems, Opimizaion, Geneic algorihm. INRODUCION he classical economic producion quaniy (EPQ) model assumes consan demand and infinie lifeime of iems in invenory. Subsequen research effors have led o he removal of hese wo resricions in consideraion of ime varying demand funcions and finie lifeimes for invenoried iems. his is in line wih common experience in day-o-day producion and invenory managemen. he ime-varying demand funcions considered in mos EPQ models are unidirecional, i.e. hey are eiher coninuously nondecreasing or coninuously non-increasing funcion of ime. he main objecive of an invenory model is o calculae how much o replenish and when o replenish. Afer reviewing many invenory models of he pas, i has been observed ha many researchers included differen facors o basic invenory model in order o sudy differen environmens/scenarios. One of he basic requiremens o develop any replenishmen policy for invenory sysem is is demand. he concep of boh deerminisic and probabilisic ypes of demand was discussed in much earlier phase of developmen of his sudy. In case of deerminisic demand, consan demand was in vogue in earlier days. Since here are differen invenory sysems in real life, herefore researchers have had o choose demand paerns accordingly. One of he mos common demand paerns ha is suiable for almos all ypes of invenory sysems is ime varying demand paern and a lo of work has been done in his direcion ill now. In order o represen he sales in differen phases of he produc life cycle in marke, ime dependen demand is suiable, e.g. he demand for invenory iems increases over he ime in he growh phase and decreases in decline phase. Silver and Meal (969), Daa and Pal(99), Chakrabary e al. (997) are few examples who considered linearly ime dependen demand. Goyal(987), Hwang (995), Hwang(997), Skouri and Papachrisos (3), Moon e al. (5), Lin and Lin (6) considered demand as coninuous funcion of ime. Benkherhouf and Balki (997), Zhou, Lang and Yang(4) considered demand as coninuously increasingly funcion of ime. Daa and Pal(988) used power paern in ime dependen demand. As far as concerned wo ypes of ime dependen demand can be frequenly found in papers: ) linearly posiive/negaive rend in demand and ) exponenially increasing/decreasing demand rae. his concep resuls in coninuously increase/decrease in demand over ime which is no realisic. Demand may increase during cerain ime periods and becomes consan afer ha. his ype of demand can be represened by ramp ype funcion. Hill (995), Mondal and Pal(998), Wu and Quyang (), Panda e al. (8), Skouri e al. (9) considered ramp ype ime demand paern. Besides demand, here are oher imporan facors which affec an invenory model. Shorages are one of hem. Any invenory model which is developed wih no shorage can be redeveloped aking shorage ino consideraion. In above cied papers, he concep of shorages is excluded in some while included in ohers. When shorages arise, quesion of backordering arise simulaneously. Again wo ypes of backordering can be experienced in invenory models, namely full backordering and parial backordering. Models considered by Goyal (987), Daa and Pal (99), Hariga (995,997), Chakrabary e al. (997), Wu and Quyang (), Moon e al. (5) are some of he examples in which shorages are fully backordered. When cusomers have o face shorages, heir response is differen according he ype of commodiy and marke environmen. For a compeiive marke parial backordering plays a pracical role. Wu (), Skouri and Papachrisos (3), Yang (4), Lin and Lin (6), Skouri e al. (9) are few of researchers who considered shorages as parially backordered. he replenishmen problem has been radiionally reaed from a muli-echelon and muli-produc perspecive (Jen- Ming and sung-hui 5). A muli-echelon replenishmen problem focuses on channel coordinaion issues for invenory replenishmen, beween upsream and downsream componens of a supply chain, wih he objecive of minimizing oal sysem coss (Sıla e al. 5). Moreover, muli-produc replenishmen problems aim o coordinae he replenishmen of various iems in he same family or same caegory in order o reduce he frequency of major seups and 5

2 Inernaional Journal of Compuer Applicaions ( ) Volume 7 No., Augus he relaed coss. his can be obained by choosing an appropriae common replenishmen frequency and lo-sizes wihin he family of iems (Bahloul e al. 8). Several previous works have sudied he problem of muli-echelon, muli-produc Supply Chain. Chen e al. (Cheng-Liang e al. 4) have sudied a muli-iem invenory and ranspor problem wih join seup coss, referred o a join replenishmen problem.. ASSUMPIONS AND NOAIONS he following assumpions and noaions are used in formulaing he models:. Assumpions. he invenory sysem involves single iem.. Demand rae is depending on ime given by D, f < = f, where f() is a posiive coninuous funcion of [,]. he funcion defined above is known as ramp ype funcion. 3. Shorages are permied and backordered a a rae B(), which is a non-increasing funcion of wih B(), where is he waiing ime up o nex replenishmen. 4. Replenishmen rae is infinie. 5. Deerioraion and amelioraion occur a consan rae 6. he deerioraion and amelioraion occur when he iem is effecively in sock. 7. Inflaion and money-value is considered.. Noaions: D() : Demand rae : Parameer for ramp ype demand funcion θ() : Deerioraion rae which is defined in erms of wo parameer Weibull disribuion as θ()= αβ β- α : Scale parameer of deerioraion rae β : Shape parameer of deerioraion rae γ() : Amelioraion rae which is defined in erms of wo parameer Weibull disribuion as γ() = gh h- g : Scale parameer of amelioraion rae h : Shape parameer of amelioraion rae r : Consan represening he difference beween discoun rae and inflaion rae c p : Purchase cos per uni c o : Ordering cos per order c h : Holding cos per uni per uni ime c a : Amelioraion cos per uni c d : Deerioraion cos per uni c s : Shorage cos per uni per uni ime c l : Los sale cos per uni W : Maximum invenory level a iniial poin I i () : Invenory level a any ime for i h phase, where i=,,3,4 : ime up o which invenory amelioraes. : ime a which invenory level becomes zero : ime for one replenishmen cycle B(τ) : Fracion of demand backordered, which is decreasing funcion of waiing ime, given by /+δ(τ), <δ< 3. MODEL FORMULAION: Replenishmen is made a he beginning of he cycle i.e., a =, which brings he invenory level maximum equals o W. Amelioraion/deerioraion occurs as soon as iem is received ino invenory. Due o amelioraion invenory accumulaes ill =. Due o demand and deerioraion invenory level sars declining and reaches o zero a =. Shorage occurs during he inerval [,], which is parially backordered. Since he demand funcion is divided ino wo inervals, herefore hree cases can be considered in order o represen he syse Case: when > Invenory Level W ime Fig. : Variaion of invenory level wih ime for Case. 6

3 Inernaional Journal of Compuer Applicaions ( ) Volume 7 No., Augus Invenory sysem can be represened by he following differenial equaions: di h + = + d I = W wih () β ( αβ gh ) I ( a b) di h + = + d I ( ) = I ( ) wih β ( αβ gh ) I ( a ) di3 β + αβ I3 = ( a+ ) d I ( ) = wih 3 ( a+ ) di 4 d = + δ ( ) I wih 4( ) = he soluion of he above equaions can be given as: β+ h+ β h + g g β+ h+ g α α α I = e a + b + + W β+ h+ β+ h+, () () (3) (4) (5) h h β h β+ + β+ + α + g α g α g I = e ( a+ ) + + b + + W β + h+ ( β+ )( β + ) ( h+ )( h+ ) (6) β+ β+ β α α α 3 = ( + ) + + I e a b β+ β+, ( a+ ) + δ( ) δ + δ( ) I4 = log, (7) (8) I ( ) = I ( ) Since 3 e a+ b + + e a+ b + β + β+ β+ h+ h h β + β + β β g g β + + α α α α + α α g + b + + W = ( β + )( β + ) ( h+ )( h+ ) (9) Now, he coss conribuing he oal cos of he sysem in case are given as follows: ) Presen worh ordering and purchase cos : PC= c + c W β+ h+ o ) Presen worh holding cos : p r r( + ) r( + ) HC= c h I e d+ I e d+ I3 e d 3) Presen worh amelioraion cos : () () 7

4 Inernaional Journal of Compuer Applicaions ( ) Volume 7 No., Augus h r h r( + ) AC= c a gh I e d+ gh I e d () 4) Presen worh deerioraion cos : β r β r( + ) β r( + ) DC= c d αβ I e d+ αβ I e d+ αβ I3 e d (3) 5) Presen worh shorage cos : r( + ) SC= cs ( I4 ) e d (4) 6) Presen worh los sales cos : + δ ( ) ( ) r( + ) LC= cl a+ b e d Our problem is o minimize (,,, ) C W = PC+ HC+ AC+ DC+ SC+ LC < < < subjec o and consrain given by equ.(9).,,, W Case : when < Invenory Level (5) (6) W ime Fig. : Variaion of invenory level wih ime for Case Invenory sysem can be represened by he following differenial equaions: di h + = + d I W wih () β ( αβ gh ) I ( a b) = β I ( a b) di + αβ = + d I ( ) = I ( ) wih di3 β + αβ I3 = ( a+ ) d I ( ) = wih 3 (7) (8) (9) 8

5 Inernaional Journal of Compuer Applicaions ( ) Volume 7 No., Augus ( a+ ) di 4 d = + δ ( ) I wih 4( ) = he soluion of he above equaions can be given as: β+ h+ β h + g g β+ h+ g α α α I = e a + b + + W β+ h+ β+ h+, () h β β+ β+ β+ + h α α bα b g α g I = e ( a+ b) + a e a + β + ( β+ )( β+ ) β+ h+ β+ h+ β β α + + α α g b b + b + W a b a ( β ) ( h ) β+ ( β + )( β + ) β+ β+ β α α α 3 = ( + ) + + I e a b β+ β+, ( a+ ) + δ( ) δ + δ( ) I4 = log, () () (3) (4) I ( ) = I ( ) Since 3 β+ β+ h+ β+ h+ α h g α g α g ( a + ) + + e a + + b + W β+ β+ h+ β+ h+ β+ β+ α bα b + ( a+ b) + a+ = β + β+ β+ Now, he coss conribuing he oal cos of he sysem in case are given as follows: ) Presen worh ordering and purchase cos : PC= c + c W o ) Presen worh holding cos : p r r( ) + r( + ) HC= c h I e d+ I e d+ I3 e d 3) Presen worh amelioraion cos : h r a 4) Presen worh deerioraion cos : (5) (6) (7) AC= c gh I e d (8) β r r( ) r DC cd I e d β + β + = αβ + αβ I e d+ αβ I3 e d (9) 5) Presen worh shorage cos : r( + ) SC= cs ( I4 ) e d (3) 6) Presen worh los sales cos : + δ ( ) ( ) r( + ) LC= cl a+ b e d Our problem is o minimize (3) 9

6 Inernaional Journal of Compuer Applicaions ( ) Volume 7 No., Augus (,,, ) C W = PC+ HC+ AC+ DC+ SC+ LC < < < subjec o and consrain given by equ.(5).,,, W Case 3: when < < (3) Invenory Level S ime Fig. 3 : Variaion of invenory level wih ime for Case3 Invenory sysem can be represened by he following differenial equaions: di h + = + d I W wih () β ( αβ gh ) I ( a b) = β I ( a b) di + αβ = + d I ( ) = wih ( a+ b) di 3 d = + δ ( ) I wih 3( ) = di ( 4 ) ( a+ ) = d + δ ( ) I ( ) = I ( ) wih 4 3 he soluion of he above equaions can be given as: β+ h+ β+ h+ β h α + g α g α g I = e a + b + + W β+ h+ β+ h+, (33) (34) (35) (36) (37)

7 Inernaional Journal of Compuer Applicaions ( ) Volume 7 No., Augus α α bα b α bα b I e a b a a b a β + ( β+ )( β+ ) β + ( β+ )( β+ ) β+ β+ β+ β+ β = ( + ) + + ( + ) + + ( ) + δ b I3 = a+ b + log + ( ) δ δ + δ δ, a+ + δ + δ b I4 = log a b log ( ) δ δ δ δ δ + + δ, I ( ) = I ( ) Since (38) (39) (4) β+ h+ β+ h+ β+ β+ h g α g α g α bα e a + b + + W + ( a+ b) β+ h+ β+ h+ β+ β+ β+ β+ β+ b α bα b + a+ ( a+ b) + a = β+ β+ β + Now, he coss conribuing he oal cos of he sysem in case are given as follows: ) Presen worh ordering and purchase cos : PC= c + c W o ) Presen worh holding cos : p r r( + ) HC= ch I e d+ I e d 3) Presen worh amelioraion cos : h r a 4) Presen worh deerioraion cos : (4) (4) (43) AC= c gh I e d (44) r β αβ β r( + ) DC= cd αβ I e d+ I e d 5) Presen worh shorage cos : ( ) ( ) ( ) ( ) SC c r r = s I e + d I e d ) Presen worh los sales cos : r( + ) r( + ) LC= cl ( a+ b) e d+ ( a+ ) e d + δ( ) + δ( ) Our problem is o minimize (,,, ) C W = PC+ HC+ AC+ DC+ SC+ LC subjec o and consrain given by equ.(4).,,, W < < < 4. GENEIC ALGORIHM Geneic Algorihm was developed by Holland and his colleagues in he 96s and 97s. Geneic Algorihms are inspired by he evoluionis heory explaining he origin of species. In naure, weak and unfi species wihin heir (47) (45) (46) (48) environmen are faced wih exincion by naural selecion. he srong ones have greaer opporuniy o pass heir genes o fuure generaions via reproducion. In he long run, species carrying he correc combinaion in heir genes become dominan in heir populaion. Someimes, during he slow

8 Inernaional Journal of Compuer Applicaions ( ) Volume 7 No., Augus process of evoluion, random changes may occur in genes. If hese changes provide addiional advanages in he challenge for survival, new species evolve from he old ones. Unsuccessful changes are eliminaed by naural selecion. GA erminology, a soluion vecor x X is called an individual or a chromosome. Chromosomes are made of discree unis called genes. Each gene conrols one or more feaures of he chromosome. In he original implemenaion of GA by Holland, genes are assumed o be binary digis. In laer implemenaions, more varied gene ypes have been inroduced. Normally, a chromosome corresponds o a unique soluion x in he soluion space. his requires a mapping mechanism beween he soluion space and he chromosomes. his mapping is called an encoding. In fac, GA works on he encoding of a problem, no on he problem iself. GA operaes wih a collecion of chromosomes, called a populaion. he populaion is normally randomly iniialized. As he search evolves, he populaion includes fier and fier soluions, and evenually i converges, meaning ha i is dominaed by a single soluion. Holland also presened a proof of convergence (he schema heorem) o he global opimum where chromosomes are binary vecors. GA use wo operaors o generae new soluions from exising ones: crossover and muaion. he crossover operaor is he mos imporan operaor of GA. In crossover, generally wo chromosomes, called parens, are combined ogeher o form new chromosomes, called offspring. he parens are seleced among exising chromosomes in he populaion wih preference owards finess so ha offspring is expeced o inheri good genes which make he parens fier. By ieraively applying he crossover operaor, genes of good chromosomes are expeced o appear more frequenly in he populaion, evenually leading o convergence o an overall good soluion. he muaion operaor inroduces random changes ino characerisics of chromosomes. Muaion is generally applied a he gene level. In ypical GA implemenaions, he muaion rae (probabiliy of changing he properies of a gene) is very small and depends on he lengh of he chromosome. herefore, he new chromosome produced by muaion will no be very differen from he original one. Muaion plays a criical role in GA. As discussed earlier, crossover leads he populaion o converge by making he chromosomes in he populaion alike. Muaion reinroduces geneic diversiy back ino he populaion and assiss he search escape from local opima. Reproducion involves selecion of chromosomes for he nex generaion. In he mos general case, he finess of an individual deermines he probabiliy of is survival for he nex generaion. A more complee discussion of GAs including exensions o he general algorihm and relaed opics can be found in books by Davis (99),, Holland (975), Michalewicz (994) and Goldberg (989). 5. NUMERICAL EXAMPLE In his secion, we solved a numerical example of he proposed model using he above-described GA. for running GA, we se he populaion size equal o, elie coun equal o, crossover fracion equal o.8 wih Gaussian muaion funcion and he sopping crieria include he maximal ieraions of each cycle is Case when > We consider he values of he parameers in appropriae unis such ha a=, b=, α =.3, β=., g=.4, h=.5, δ =.6, r=.6, c o = 5, c p =, c h =.5, c a =.5, c d = 5, c s =, c l =, =.Opimal value of =.847, =.758, =.9396, W=.474. he minimized oal cos C = is calculaed from equ. (6). 5.. Sensiiviy Analysis able.sensiiviy analsysis based on example when case wih differen parameers. A+5% 6.666e % % % B+5% 4.38e % % % Α+5% 3.959e e % % % β+5% % % %

9 Inernaional Journal of Compuer Applicaions ( ) Volume 7 No., Augus g+5% 3.79e % % % h+5% % % % δ+5% % %.8638e %.399e-8.399e r+5% % % % e c o +5% % % %.55578e e c p +5% % % % c h +5% % % % c s +5% % % 6.779e %.399e-8.399e c d +5% % % e % c l +5% % % % e c a +5% % % % Case when < We consider he values of he parameers in appropriae unis such ha a=, b=, α=., β=3., g=.4, h=.5, δ=.6, r=.6, c o =5, c p =, c h =3, c s =, c d =5, c l =, c a =3, =. Opimal value of,i.e., *=., =.5766, =.8977, W= he minimized oal cos C =3.885 is calculaed from equ.(3). 3

10 Inernaional Journal of Compuer Applicaions ( ) Volume 7 No., Augus 5.. Sensiiviy Analysis able.sensiiviy analsysis based on example when case wih differen parameers. a+5% % % % b+5% % % % e α+5% % % % β+5% % e % % g+5% % % % h+5% e % e % % e δ+5% % % % e r+5% % % % c o +5% % % % c p +5% % e % e % c h +5% e % % e %

11 Inernaional Journal of Compuer Applicaions ( ) Volume 7 No., Augus c s +5% % % % c d +5% % % % c l +5% % % % c a +5% % % % Case 3 when < < We consider he values of he parameers in appropriae unis such ha a=, b=.5, α=.4, β=., g=.4, h=.5, δ=.6, r=.6, c o =5, c p =, c h =3, c s =, c d =5, c l =, c a =3, =. Opimal value of,i.e., *=.635, =, =.49, W=.954. he minimized oal cos C 3 =.53 is calculaed from equ.(48) Sensiiviy Analysis: able 3.Sensiiviy analsysis based on example when case 3wih differen parameers. W C PCI (%) a+5% e % % % w C PCI(%) b+5% % % % α+5% % % % e β+5% % % % g+5%.5887e %.437e % e % 4.569e h+5% % % %

12 Inernaional Journal of Compuer Applicaions ( ) Volume 7 No., Augus δ+5% % % % r+5% % % % c o +5% % % % c p +5% % % % c h +5% 6.959e % % % c s +5% %.9854e % % c d +5% % % % c l +5% % % % c a +5% % % % e CONCLUSION his work is an aemp for analyzing an order level invenory model for amelioraing iems. he analysis helps us formulae a generic model for furher work. Hence, all he effors have been carefully direced owards he possible fuurisic enhancemens of he model. he demand rae has been generically seleced o represen any funcion of ime ill he sabilizaion insan arrives (general ramp ype demand rae), and he backlogging rae has been generically chosen o represen any non-increasing funcion of he waiing ime, up o he nex replenishmen. he invenory model has been analyzed for he scenarios of replenishmen policy, saring wih no shorages. he opimal replenishmen policy for he model is derived for he above menioned invenory sysem. A numerical example is also presened o illusrae his model. 7. REFERENCES [] Skouri, K. and Papachrisos, S. (3), Opimal sopping and resaring producion imes for an EOQ model wih deerioraing iems and ime-dependen parial backlogging, Inernaional Journal of Producion Economics, Vol.8-8, pp [] Panda, S., Senapai, S. and Basu, M. (8). Opimal replenishmen policy for perishable seasonal producs in a season wih ramp-ype ime dependen demand. Compuers & Indusrial Engineering 54, [3] Benkherouf, L. and Balkhi, Zaid. (997). On an Invenory Model for Deerioraing Iems and ime- 6

13 Inernaional Journal of Compuer Applicaions ( ) Volume 7 No., Augus Varying Demand. Mahemaical Mehods of Operaions Research,45,-33. [4] Chakrabary,., Giri, B.C. and Chauduri, K.S. (997). An EOQ model for iems wih Weibull disribuion deerioraion, shorages and rended demand: An Exension of Philip s model. Compuers Ops. Res.,5, 7/8, [5] Sıla, C., M. Faih and L. Chung-Yee (5). "A comparison of ou bound dispach policies for inegraed invenory and ransporaion decisions." European Journal of Operaional Research. [6] Jen-Ming, C. and C. sung-hui (5). "he muli-iem replenishmen problem in a wo-echelon supply chain: he effec of cenralizaion versus decenralizaion." Compuers & Operaions Research 3. [7] Bahloul, K., A. Baboli and J.-P. Campagne (8). Opimizaion mehods for invenory and ransporaion problem in Supply Chain: lieraure review. Inernaional Conference on Informaion Sysems, Logisics and Supply Chain (ILS). Madison, WI, U.S.A. [8] Daa,.K., and Pal, A.K. (99). Effecs of inflaion and ime-value of money on an invenory model wih linear ime-dependen demand rae and shorages. European Journal of Operaional Research,5, [9] Daa,.P., and Pal, A.K. (988). Order level invenory sysem wih power demand paerns for iems wih variable rae of deerioraion. Indian J of pure App. Mah.,9(); [] Ghare, P. M., & Schrader, G. F. (963). A model for an exponenially decaying invenory. Journal of Indusrial Engineering, 4, [] Goyal S.K. (987). Economic ordering policies for deerioraing iems over an infinie ime horizon. European Journal of Operaional Research, 8, [] Hariga, M. (995). Effecs of inflaion and ime-value of money on an invenory model wih ime-dependen demand rae and shorages. European Journal of Operaional Research,8,5-5. [3] Hariga, M. (997). Opimal invenory policies for perishable iems wih ime-dependen demand. In. J. Producion Economics,5,35-4. [4] Hill, R.M. (995). Invenory model for increasing demand followed by level demand. Journal of he Operaional Research Sociey 46, [5] Silver E. A. and Meal, H. C., 969. A simple modificaion of he EOQ for he case of a varying demand rae. Producion of Invenory Managemen. (4), [6] Hwang, H. S. (997). A sudy on an invenory model for iems wih Weibull amelioraing. Compuers and Indusrial Engineering, 33, [7] Hwang, H. S. (999). Invenory models for boh deerioraing and amelioraing iems. Compuers and Indusrial Engineering, 37, [8] Lin, Y. and Lin, C. (6). Purchasing model for deerioraing iems wih ime-varying demand under inflaion and ime discouning.,in J Adv Manuf. echnology 7: [9] Mondal, B., & Pal, A. K. (998). Order level invenory sysem wih ramp ype demand for deerioraing iems. Journal of Inerdisciplinary Mahemaics,, [] Moon, I., Giri, B.C., Ko, B.(5). Economic order quaniy models for amelioraing/deerioraing iems under inflaion and ime discouning. European Journal of Operaional Research 6, [] Chih-Yao Lo, (8). Advance of Dynamic Producion- Invenory Sraegy for Muliple Policies using Geneic Algorihm, Informaion echnology Journal, Vol. 7, pp [] Skouri, K., Konsanaras, I., Papachrisos, S., Ganas, I.(9). Invenory models wih ramp ype demand rae, parial backlogging and Weibull deerioraion rae. European Journal of Operaional Research, 9, [3] Wu, K.S. & Ouyang, L.Y. (). A Replenishmen Policy for Deerioraing Iems wih Ramp ype Demand Rae. Proc. Nal. Sci. Counc. ROC(A),4(4), [4] Wee, H. M.(999). Deerioraing invenory model wih quaniy discoun, pricing and parial backordering. Inernaional Journal of Producion Economics, 59, [5] Wu, K., S.(). An EOQ invenory model for iems wih Weibull disribuion deerioraion, ramp ype demand rae and parial backlogging. Producion Planning and Conrol, (8), [6] Zhou, Y.W., Lau, H.S. and Yang S.L. (4). A finie horizon lo-sizing problem wih ime-varying deerminisic demand and waiing-ime-dependen parial backlogging. In. J. Producion Economics,9,9 9. [7] Huang, Guobo, (995). Modelling China's demand for inernaional reserves. Applied Financial Economics, 5, pp [8] Yang, P.C. (4). Pricing sraegy for deerioraing iems using quaniy discoun when demand is price sensiive. European Journal of Operaional Research, 57(), [9] L. Davis, (99). he Handbook of Geneic Algorihms, Van Nosrand Reinhold, New York. [3] D.E. Goldberg,(989). Geneic Algorihms in Search, Opimizaion, and Machine Learning, Addison-Wesley, Reading, MA. [3] J.H. Holl and,(975). Adapaion in Naural and Arificial Sysems, he Universiy of Michigan Press, Ann Arbor, IL. [3] Z. Michalewicz,994, Geneic Algorihms + Daa Srucures = Evoluion Programs. AI Series, Springer, New York. 7