An EOQ Model with Verhulst s Demand and Time-Dependent Deterioration Rate

Transcription

1 An EOQ Model wih Verhuls s Demand and Time-Dependen Deerioraion Rae Yuan-Chih Huang Kuo-Hsien Wang Deparmen of Business Managemen, Takming niversiy of Science and Technology, Taiwan Yu-Je Lee, Deparmen of Markeing Managemen, Takming niversiy of Science and Technology, Taiwan ABSTRACT In his sudy, we boldly propose a so-called Verhuls s populaion increase rae as a demand rae for perishable producs o explore invenory managemen wih profi opimaliy concern, hrough which i helps o simulaneously analyze a variey of ime poins of EOQ problem in he respec of a manufacurer s maximum profi per uni ime by deermining is opimal producion run ime and opimal cycle ime under assumpions ha he manufacurer s producion rae is in linear proporion o produc demand rae, deerioraion rae is a ime-dependen funcion and uni producion cos is in he form of reciprocal demand. Finally, hree specific ime poins of opimizaion models for new produc, produc in he middle of growh sage and maure produc are respecively demonsraed. Keywords: EOQ; Verhuls s demand; Deerioraion; Invenory INTRODCTION In he area of invenory managemen, demand of produc plays a pivoal role when formulaing economical order quaniy model since i dramaically dominaes invenory characerisics. As saed by Lau and Lau (3), a small change in demand paern may resul in a large change in opimal invenory decisions; hus hey suggesed ha decision makers should invesigae all possible facors ha may have an impac on demand before decisions are made. A number of demand forms have previously been appearing in he exan lieraure, from where ime is usually considered as a major facor ha has influence on produc demand, ha is, demand may vary hroughou is enire produc lifespan, and producs for holiday seasons are he examples. Barbosa and Friedman (978) presened a demand funcion of ime in a polynomial form of D()= wih γ> for increasing demand, γ< for decreasing demand and γ= for consan demand; also Sachan (984) exended an invenory model for deerioraing iems wih ime-proporional demand. Mandal and Pal (988) developed an order level invenory sysem in which demand rae is assumed o be a ramp ype funcion of ime for new producs coming o he marke, which is described as D, D () () D, where represens a mauriy ime of producs. A common phenomenon o promoe sales in a selling season is a discoun price sraegy ha enices more poenial buyers, so obviously selling price is anoher facor ha affecs demand behavior. The

2 following wo price-dependen demands are widely adoped. The firs is D( p) p, where p is selling price; α is a primary demand and is a price-sensiiviy. Wang e al. () based heir research on his demand rae o ackle an economical order quaniy model for deerioraing iems wih resalable reurns; and laer on, modified heir previous model o non-insananeous resalable reurns wih he same demand rae. The second is ha Ho e al. (8) addressed opimal pricing, shipmen and paymen policy by assuming demand is in he form of D( p) p, where > is a price-elasiciy coefficien. Yao e al.(8) considered he impac of price-sensiiviy facors on reurn policy in coordinaing a supply chain under he assumpion of sochasic demand form of D( p, ) p, where is a random variable wih normal disribuion. Recenly, Zhang e al.() discussed opimal fences and join price and invenory decisions in wo disinc price markes allowing demand leakage from a high-priced marke o a low-priced one, in which deerminisic demand funcions are given by D ( p, p) a b p ( p p) and D ( p, p) a b p ( p p), where > represens demand leakage rae and = means zero demand leakage. Empirically, he amoun of on-hand invenory also has been idenified o have influences on demand behavior, showing ha demand may increase wih he presence of producs on shelves, which equivalenly means ha he more producs on shelves, he more demand ou here; consequenly i makes on-hand sock level a facor ha affecs produc demand, excep for ime and selling price. Baker and rban (988) sudied a deerminisic, single period model wih an invenory-dependen demand rae in a form of D( Q) Q, where Q is sock level. Bhunia and Maii (998) esablished a deerminisic model for deerioraing iems in a condiion where replenishmen rae R() is dependen on on-hand sock level Q() and demand rae D(), such ha R( ) Q( ) D( ), where,, and D( ) a b, a, b. You and Hsieh (7) developed an EOQ model aking sock and price ino accoun wih assumpion ha demand is a linear form of D( p, ) p Q( ). Meanwhile, many researchers have also reaed produc demand as a mulivariae funcion of ime, price and invenory level; hus rban and Baker (997) examined an opimal ordering and pricing policies wih demand as D( p,, Q) p Q,,,,. Raher han he aforemenioned demand forms, we, inspired by he Verhuls s populaion increase rae, will develop a produc demand model, during which ime is he only facor o be considered; no exogenous facors ha may affec demand would be aken ino consideraion. Conribuions of he demand model are ha i helps porray an inegraed demand behavior wih a single mahemaical expression, and moreover, i will make our proposed EOQ models valid for any specific ime poin during he enire produc lifespan, oher han growh and mauriy sage. limaely, he objecive of his sudy is o explore he manufacurer s opimal profi per uni ime by deermining producion run ime and cycle ime under he proposed demand model. The remainder of his sudy is organized as follows. A brief descripion abou Verhuls s populaion increase rae is given, and he proposed demand model is accordingly developed in Secion. In Secion 3, assumpions and noaion are clarified, followed by formulaion of objecive funcion associaed wih heoreical analyses in response o hree differen ime poins of profi opimizaion problem. Finally, a conclusion and direcions for furher research are provided in Secion 4 o close his work.

3 demand The Verhuls s Demand In 84, Belgium mahemaician Verhuls presened a model for populaion increase rae, where he believed ha populaion increase rae should be direcly proporional o he amoun of populaion a ha ime, and canno exceed a cerain maximum capaciy deermined by he surrounding environmen. Hence, he provided he following differenial equaion o govern populaion increase paern. dp() p( )( p( )) () d where is ime; p() is amoun of populaion a ime and represens a maximum capaciy for populaion. As we all undersand, manufacurers design and manufacure producs for people o purchase and consume, which could be inerpreed as he more populaion, he more produc demand ; hus i is reasonable and feasible o adop he Verhuls s model as a produc demand o explore relaed invenory managemen problems. More deails abou Verhuls s model will be summarized as follows. Firs, variaion of he demand wih respec o ime is assumed o be direcly proporional o he amoun of demand iself a ha ime. Meanwhile, no producs ge infinie demands, which no only implies ha here exiss an upper bound for demand bu also he variaion of demand will gradually urn ino a moderae growh and hen evenually vanish in he long run. Therefore, he demand iself and he difference beween he upper bound and he demand are wo main facors ha conribue o he demand behavior. Nex, we le D () be he demand funcion of ime ; D is a iniial demand ; is a posiive consan ; is a consan represening he upper bound of produc. Based on above saemen, he variaion of demand wih respec o ime is given by dd() D( )( D( )), d (3) wih iniial condiion D() D. Soluion of (3) is given below and Fig. is an example of is behavior. D( ), ke where k D (4) ime Figure : Behavior of he Verhuls s demand

4 Now we need o define a reference ime poin ha will enable us o invesigae perinen economical order quaniy model whenever any specific ime poin of produc is, where [, ) wih = represening he ime poin ha new producs come o he marke, and for he ime poin of maure producs. Firs, le be a mauriy degree of produc defined by he raio of D( ) over D upper bound as follows, and obviously ha [,). D( ) ke (5) Solving (5), he reference ime poin can be expressed in erms of as k ln (6) D Tha way we have = accompanied wih for maure producs. accompanied wih he = for brand new producs, and Assumpions and Formulaion. Noaion and assumpions Excep he noaion menioned in previous secion, he following are clarified. A = seup cos per seup s = uni selling price c = uni holding cos per uni ime = producion run ime (decision variable) = invenory cycle ime (decision variable) Ii( ; ) = sock level wih he reference ime poin a ime, i=, Also, he following assumpions are made hroughou his sudy. () Planning horizon is infinie. () Lead ime is zero. (3) The iniial and final invenory levels are boh zero. (4) Demand rae is assumed o be he Eq. (4). (5) Producion rae is assumed o be D ( ), is a consan. (6) Deerioraion rae is assumed o be,. (7) The uni producion cos is assumed o be ( D ( )),,. (8) There is no repair or replacemen of deerioraing iems, hence no salvage value exiss.. Mahemaical formulaion In his invenory sysem, he manufacurer s producion sars producing producs a ime = wih zero sock level and sops a ime =. Due o facors of demand and deerioraion, he manufacurer s invenory level gradually diminishes over ime inerval [, ] and finally drops o zero a ime =. According o he assumpions, over ime span [, ], deerioraion rae is I ( ; ) ; demand rae and producion rae are respecive D ( ) and D ( ), which makes variaion of he invenory level wih respec o ime for he reference ime poin μ be governed by

5 di( ; ) I( ; ) ( ) D( ), (7) d wih iniial condiion I (; ). Likewise, during [, ], he invenory level is affeced by deerioraion rae and demand rae, so ha he variaion of invenory level wih respec o ime is governed by di( ; ) I( ; ) D( ), (8) d wih boundary condiion I( ; ). Soluions of (7) and (8) are given by w ( ) e ( ; ), ( w) (9) I e dw ke w e ( ; ), ( w) () I e dw ke I ( ; ) I ( ; ) Wih coninuous condiion a ime so ha w w e e ke ke, and we have dw dw ( w) () ( w) Obviously, Eq.() implicaes ha can be ermed as a funcion of. So aking he derivaive of () wih respec o, we obain ( ) d ( ke ) e ( ) d ke ( ) The above resul will be uilized when solving he firs-order necessary condiion in a purpose of he manufacurer s profi opimizaion. Now he manufacurer s oal profi will be consruced, which consiss of he following four elemens: () seup cos : A () sales revenue : SR = sd( ) d ( ) s e k = ln e (3) producion cos : PC = = k ( D( )) D( ) d (3) d (4) ( ) [ ke ] (4) invenory holding cos : HC = ci ( ; ) d ci ( ; ) d ()

6 = w w ( ) e e ( ) ( w) (5) ce dwd c e dwd w ke ke Thus he manufacurer s oal profi is compued by TP(, ; ) = SR A PC HC = ( ) s e k ln e k A [ ke ] ( ) w w ( ) e e ( ) ( w) (6) ce dwd c e dwd w ke ke And is corresponding oal profi per uni ime over [, ] is TP(, ; ) TPR(, ; ) = Because is a funcion of, TP(, ; ) and TPR(, ; ) can be reduced o a single variable, respecively denoed by TP( ; ) and TPR( ; ) for convenience. Therefore, o maximizetpr( ; ), we ake he firs-order derivaive from (7) wih respec o and se he resul o be zero, hen we have dtpr( ; ) dtp( ; ) d ( ( ; ) TPR ) (8) d d d d (7) Subsiuing he resul of () ino above equaion, he opimaliy necessary condiion (8) hen becomes ( ) dtp( ( ) ; ) TP( ; ) ( ke ) e (9) ( ) d ke Since complexiy of Eq.(9) prevens us from mahemaically acquiring is closed-form soluion, he sofware MATHEMATICA will be borrowed o obain numerical opimal value by simulaneously solving (9),(6) and (). Once is deermined, corresponding value is hen deermined by (), and sequenially generaes opimal oal profi per uni ime by (7). 3. Examples for special ime poins Noe ha he Eqs. (7)~(9) are valid for any specific ime poin hroughou he produc s lifespan ; here we illusrae hree objecive funcions in response o differen ime poin. Time poin : For brand new producs The mauriy degree is = TP( ;) = s e k ln k D associaed wih =. Subsiuing = ino (6), hen A [ ke ] w w ( ) e e w ce dwd ce dwd w ke ke d

7 TPR( ;) TP( ;) () Time poin : For produc in he middle of growh sage The mauriy degree is assumed as = associaed wih ln k and from (6), hen s e TP( ; ln k) = ln A d [ e ] TPR( ; ln k) w w ( ) e e w ce dwd ce w e e dwd TP( ; ln k) () Time poin 3: For producs a mauriy sage The mauriy degree is associaed wih, so from (6) we have TP( ; ) s A TPR( ; ) w w ( ) ce e dwd ce e dwd TP( ; ) () CONCLSIONS To he bes of our knowledge, we are he firs in he lieraure o presen a novel proposal ha incorporaes he so-called Verhuls s demand model ino invenory managemen sysem exploraion, in which manufacurer s opimal producion run ime and cycle ime were invesigaed in a hope of maximizing is profi per uni ime under he proposed demand model. Compared o he ramp-ype demand funcion which is in need of separae discussions in formulaing relevan invenory models when encounering differen ime poins of producion run ime and of produc s mauriy age as well(see Chen and Ouyang, 6), our Verhuls s demand model precisely describes an inegraed demand paern from he birh of a new produc o is mauriy sage wih a single mahemaical expression, which of course enables us o simulaneously analyze a variey of differen ime poins of relaed economical order quaniy problems during he enire produc lifespan. As a resul, i helps decision makers predeermine any specific ime poin of opimal managemen policies before is scheduled ime poin is due. nforunaely, compuer sofware such as he MATHEMATICA fails o numerically solve our problems due o complexiy of he proposed models. Sill, his sudy is a beginning ha hopefully enables o enice oher researchers resonance and hen join us o explore he relaed problems. For fuure sudy, we can exend he Verhuls s demand model by adding oher demand-affecing facors such as selling price and insananeous invenory level, and even modify he demand model o characerize a single-period environmen ha feaures a declining demand behavior for he rear duraion of produc lifespan.

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