Research Article Order Level Inventory Models for Deteriorating Seasonable/Fashionable Products with Time Dependent Demand and Shortages

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1 Hindawi Publishing Corporaion Mahemaical Problems in Engineering Volume 29, Aricle ID , 24 pages doi:.55/29/ Research Aricle Order Level Invenory Models for Deerioraing Seasonable/Fashionable Producs wih Time Dependen Demand and Shorages K. Skouri and I. Konsanaras, 2 Deparmen of Mahemaics, Universiy of Ioannina, 45 Ioannina, Greece 2 Hellenic Army Academy, Vari 6673, Aica, Greece Correspondence should be addressed o I. Konsanaras, ikons@cc.uoi.gr Received 8 Sepember 28; Revised 3 July 29; Acceped 2 July 29 Recommended by Wei-Chiang Hong An order level invenory model for seasonable/fashionable producs subjec o a period of increasing demand followed by a period of level demand and hen by a period of decreasing demand rae hree branches ramp ype demand rae is considered. The unsaisfied demand is parially backlogged wih a ime dependen backlogging rae. In addiion, he produc deerioraes wih a ime dependen, namely, Weibull, deerioraion rae. The model is sudied under he following differen replenishmen policies: a saring wih no shorages and b saring wih shorages. The opimal replenishmen policy for he model is derived for boh he above menioned policies. Copyrigh q 29 K. Skouri and I. Konsanaras. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied.. Inroducion I is observed ha he life cycle of many seasonal producs, over he enire ime horizon, can be porrayed as a period of growh, followed by a period of relaively level demand and finishing wih a period of decline. So researchers commonly use a ime-varying demand paern o reflec sales in differen phases of produc life cycle. Resh e al. and Donaldson 2 are he firs researchers who considered an invenory model wih a linear rend in demand. Thereafer, numerous research works have been carried ou incorporaing imevarying demand paerns ino invenory models. The ime dependen demand paerns, mainly, used in lieraure are, i linearly ime dependen and, ii exponenially ime dependen Dave and Pael 3, Goyal 4,Hariga5, Hariga and Benkherouf 6,Yangeal. 7. The ime dependen demand paerns repored above are unidirecional, ha is, increase coninuously or decrease coninuously. Hill 8 proposed a ime dependen demand paern by considering i as he combinaion of wo differen ypes of demand in wo successive

2 2 Mahemaical Problems in Engineering ime periods over he enire ime horizon and ermed i as ramp-ype ime dependen demand paern. Then, invenory models wih ramp ype demand rae also are sudied by Mandal and Pal 9, Wueal. and Wu and Ouyang, Wu2, Giri e al. 3, and Manna and Chaudhuri 4. In hese papers, he deerminaion of he opimal replenishmen policy requires he deerminaion of he ime poin, when he invenory level falls o zero. So he following wo cases should be examined: his ime poin occurs before he poin, where he demand is sabilized, and 2 his ime poin occurs afer he poin, where he demand is sabilized. Almos all of he researchers examine only he firs case. Deng e al. 5 reconsidered he invenory model of Mandal and Pal 9 and Wu and Ouyang and sudied i exploring hese wo cases. Skouri e al. 6 exend he work of Deng e al. 5 by inroducing a general ramp ype demand rae and considering Weibull disribued deerioraion rae. The assumpion ha he goods in invenory always preserve heir physical characerisics is no rue in general. There are some iems, which are subjec o risks of breakage, deerioraion, evaporaion, obsolescence, and so forh. Food iems, pharmaceuicals, phoographic film, chemicals, and radioacivesubsances are few iems in which appreciable deerioraion can ake place during he normal sorage of he unis. A model wih exponenially decaying invenory was iniially proposed by Ghare and Schrader 7. Cover and Phillip 8 and Tadikamalla 9 developed an economic order quaniy model wih Weibull and Gamma disribued deerioraion raes, respecively. Thereafer, a grea deal of research effors have been devoed o invenory models of deerioraing iems, he deails can be found in he review aricles by Raafa 2, and Goyal and Giri 2. In mos of he above-menioned papers, he demand during sockou period is oally backlogged. In pracice, here are cusomers who are willing o wai and receive heir orders a he end of sockou period, while ohers are no. In he las few years, considerable aenion has been paid o invenory models wih parial backlogging. The backlogging rae can be modelled aking ino accoun he cusomers behavior. The firs paper in which cusomers impaience funcions are proposed seems o be ha by Abad 22. Chang and Dye 23 developed a finie horizon invenory model using Abad s reciprocal backlogging rae. Skouri and Papachrisos 24 sudied a muliperiod invenory model using he negaive exponenial backlogging rae proposed by Abad 22. Teng e al. 25 exended Chang and Dye s 23 and Skouri and Papachrisos 24 models, assuming as backlogging rae any decreasing funcion of he waiing ime up o he nex replenishmen. Research on models wih parial backlogging coninues wih Wang 26 and San Jose e al. 27 and 28. Manna and Chaudhuri 4 noed ha ramp ype demand paern is generally followed by new brand of consumer goods coming o he marke. Bu for fashionable producs as well as for seasonal producs, he seady demand will never be coninued indefiniely. Raher i would be followed by decremen wih respec o ime afer a period of ime and becomes asympoic in naure. Thus he demand may be illusraed by hree successive ime periods ha classified ime dependen ramp-ype funcion, in which in he firs phase he demand increases wih ime and afer ha i becomes seady, and owards he end in he final phase i decreases and becomes asympoic. Chen e al. 29 proposed a search procedure based on Nelder-Mead algorihm o find a soluion for he case of invenory sysems wih shorage allowance and nonlinear demand paern. Also, Chen e al. 3 proposed a ne presen value approach for he previous invenory sysem wihou shorages. For boh models, he demand rae is a revised version of he Bea disribuion funcion and so is a differeniable wih respec o ime.

3 Mahemaical Problems in Engineering 3 The purpose of he presen paper is o sudy an order level invenory model when he demand is described by a hree successive ime periods ha classified ime dependen ramp-ype funcion. Any such funcion has poins, a leas one, where differeniaion is no possible, and his inroduces exra complexiy in he analysis of he relevan models. The unsaisfied demand is parially backlogged wih ime dependen backlogging rae, and unis in invenory are subjec o deerioraion wih Weibull-disribued deerioraion rae. The res of he paper is organized as follows. In he nex secion he assumpions and noaions for he developmen of he model are provided. The model saring wih no shorages is sudied in Secion 3, and he corresponding one saring wih shorages is sudied in Secion 4. For each model he opimal policy is obained. Numerical examples highlighing he resuls obained are given in Secion 5. The paper closes wih concluding remarks in Secion Noaion and Assumpions The following noaions and assumpions are used in developing he model: Noaions T The consan scheduling period cycle The ime when he invenory level reaches zero S The maximum invenory level a each scheduling period cycle c The invenory holding cos per uni per uni ime c 2 The shorage cos per uni per uni ime c 3 The cos incurred from he deerioraion of one uni c 4 The per uni opporuniy cos due o he los sales The ime poin ha increasing demand becomes seady The ime poin, afer, unil he demand is seady and hen decreases I The invenory level a ime, T. Assumpions The ordering quaniy brings he invenory level up o he order level S. Replenishmen rae is infinie. 2 Shorages are backlogged a a rae βx, which is a nonincreasing funcion of xβ x wih βx, β, and x is he waiing ime up o he nex replenishmen. Moreover i is assumed ha βx saisfies he relaion βx Tβ x, where β x is he derivae of βx. The cases wih βx or correspond o complee backlogging or complee los sales models. 3 The ime o deerioraion of he iem is disribued as Weibull a, b; hais,he deerioraion rae is θ ab b a>,b >, >. There is no replacemen or repair of deerioraed unis during he period T. For b, θ becomes consan, which corresponds o exponenially decaying case.

4 4 Mahemaical Problems in Engineering 4 The demand rae D is a ime dependen ramp-ype funcion and is of he following form: f, <<, D f ( ) g ( ),, g, <, 2. where f is a posiive, coninuous, and increasing funcion of, and g is a posiive, coninuous and decreasing funcion of. 3. The Mahemaical Formulaion of he Model Saring wih No Shorages The replenishmen a he beginning of he cycle brings he invenory level up o S. Dueo demand and deerioraion, he invenory level gradually deplees during he period, and falls o zero a. Thereafer shorages occur during he period,t, which are parially backlogged. Consequenly, he invenory level, I, during he ime inerval T, saisfies he following differenial equaions: di θi D, d,i, 3. di DβT, d T, I. 3.2 The soluions of hese differenial equaions are affeced from he relaion beween,, and hrough he demand rae funcion. Since he demand has hree componens in hree successive ime periods, he following cases: i <<<T, ii < <<T,and iii << <Tmus be considered o deermine he oal cos and hen he opimal replenishmen policy. Case <<<T. In his case, 3. becomes di d Equaion 3.2 leads o he following hree: ab b I f,,i. 3.3 di fβt, d, I, 3.4 di f ( ) βt, d, I ( ) ( ) I, 3.5 di gβt, d T. 3.6

5 Mahemaical Problems in Engineering 5 The soluions of 3.3, 3.4, 3.5,and3.6, are, respecively, I I e ab fxeaxb dx,, 3.7 I fxβt xdx,, 3.8 I f ( ) βt xdx gxβt xdx fxβt xdx f ( ) fxβt xdx,, 3.9 βt xdx, T. 3. The oal amoun of deerioraed iems during, is D fe ab d fd. 3. The cumulaive invenory carried in he inerval, is found from 3.7 and is I Id e ab fxe axb dx d. 3.2 Due o 3.8, 3.9,and3., he ime-weighed backorders during he inerval,t are I 2 Id Id Id Id ( ) ( ) fβt d f gxβt xdx d βt xdx d fxβt xdx d fxβt xdx d f ( ) βt xdx d. 3.3 The amoun of los sales during,t is L ( βt ) fd f ( ) ( ) T ( ) βt d βt gd. 3.4

6 6 Mahemaical Problems in Engineering The oal cos in he ime inerval,t is he sum of holding, shorage, deerioraion, and opporuniy coss and is given by TC c I c 2 I 2 c 3 D c 4 L { } { c e ab fxe axb dx d c 3 fe ab d fd { ( ) ( ) c 2 fβt d f βt xdx d fxβt xdx d gxβt xdx d fxβt xdx d f ( ) } βt xdx d { ( ) ( ) ( ) T } ( ) c 4 βt fd f βt d βt g d. } 3.5 Case 2 < <<T. In his case, 3. reduces o he following wo: di ab b I f, d, I ( ) I ( ), di ab b I f ( ), d, I. 3.6 Equaion 3.2 leads o he following wo: di f ( ) βt, d, I, di gβt, d T, I ( ) I ( ). 3.7 Their soluions are, respecively, I e ab fxe axb dx f ( ) e axb dx,, I e ab f ( ) e axb dx,, I f ( ) I βt xdx,, gxβt xdx f ( ) βt xdx, T. 3.8

7 Mahemaical Problems in Engineering 7 The oal amoun of deerioraed iems during, is D I Dd fe ab d f ( ) e ab d fd f ( )( ). 3.9 The oal invenory carried during he inerval, is I Id Id e ab Id fxe axb dx f ( ) e axb dx d f ( ) e ab e axb dx d. 3.2 The ime-weighed backorders during he inerval,t are I 2 Id Id f ( ) βt xdx d Id gxβt xdx d f ( ) βt xdx d. 3.2 The los sales in he inerval,t are L f ( ) βt d βt gd The invenory cos for his case is TC 2 c I c 2 I 2 c 3 D c 4 L { c e ab fxe axb dx f ( ) e axb dx d f ( ) } e ab e axb dx d { c 2 f ( ) βt xdx d f ( ) } βt xdx d gxβt xdx d { c 3 fe ab d f ( ) e ab d { c 4 f ( ) βt d fd f ( )( )} βt gd }. 3.23

8 8 Mahemaical Problems in Engineering Case 3 << <T. In his case, 3. reduces o he following hree: di d ab b I f,,i ( ) I ( ), di d ab b I f ( ),, I ( ) I ( ), 3.24 di d ab b I g,,i. Equaion 3.2 leads o he following: di d gβt, T, I Their soluions are, respecively, I e ab fxe axb dx f ( ) e axb dx I e ab f ( ) e axb dx gxe axb dx gxe axb dx,, 3.26,, 3.27 I e ab gxe axb dx,, 3.28 I gxβt xdx, T The oal amoun of deerioraed iems during, is D I Dd fxe axb dx f ( ) e axb dx gxe axb dx 3.3 fd f ( )( ) gd.

9 Mahemaical Problems in Engineering 9 The oal invenory carried during he inerval,,using3.26, 3.27, and3.28 is I Id Id Id e ab Id fxe axb dx f ( ) e axb dx e ab f ( ) e axb dx gxe axb dx gxe axb dx d d e ab gxe axb dx d. 3.3 The ime-weighed backorders during he inerval,t are I 2 Id gxβt xdx The los sales in he inerval,t are L βt gd The invenory cos for his case is TC 3 c I c 2 I 2 c 3 D c 4 L { c e ab fxe axb dx f ( ) e axb dx gxe axb dx d e ab f ( ) } e axb dx gxe axb dx d e ab gxe axb dx d c 2 { gxβt dx } c 4 { βt gd } { c 3 fxe axb dx f ( ) e axb dx fd f ( )( ) gd gxe axb dx Finally he oal cos funcion of he sysem over,t akes he following form: TC, if, TC TC 2, if < <, TC 3, if. }

10 Mahemaical Problems in Engineering I is easy o check ha his funcion is coninuous a and. The problem now is he minimizaion of his, hree branches, funcion TC. This requires, separaely, sudying each of hese branches and hen combining he resuls o sae he algorihm giving he opimal policy. 3.. The Opimal Replenishmen Policy In his subsecion we presen he resuls, which ensure he exisence of a unique opimal value for,say, which minimizes he oal cos funcion. Alhough he opimaliy procedure requires he consrained opimizaion of he funcions TC,TC 2,andTC 3, we will, firsly, search for heir unconsrained minimum. The firs- and second-order derivaives of TC, TC 2, and TC 3 are, respecively, dtc d f h, d 2 TC df h 2 f dh, d d d dtc 2 f ( ) h, d d 2 TC 2 f ( )dh, 2 d d dtc 3 g h, d d 2 TC 3 d 2 dg h g dh, d d 3.36 where h c e a b ) e ab d c 3 (e a ( b c 2 T βt c 4 βt ) Equaion 3.37 is he same as 6 of he paper of Skouri e al. 6. So, following he mehodology proposed by Skouri e al. 6, he algorihm, which gives he opimal replenishmen policy, is as follows. Sep. Compue from h. Sep 2. If, hen he opimal order quaniy is given by Q fe ab d βt fd f ( ) βt d βt gd 3.38 and he oal cos is given by TC.

11 Mahemaical Problems in Engineering If < <,hen he opimal order quaniy is given by Q fe ab d f ( ) e ab d f ( ) βt d gβt d 3.39 and he oal cos is given by TC 2. If < <T,hen he opimal order quaniy is given by Q fe ab d f ( ) e ab d ge ab d gβt d 3.4 and he oal cos is given by TC 3. Remark 3.. The previous analysis shows ha is independen from he demand rae D. This very ineresing resul agrees wih he classical resul, in many order level invenory sysems, ha he poin is independen from he demand rae Naddor 3, page The Special Case βx and a If we are considering he case ha here is no deerioraion of he produc a and unsaisfied demand is complee backlogged βx, hen he oal cos funcion of he model saring wih no shorages over,t akes he following form: c fxdx d c 2 fxdx d c 2 f ( ) ( ) d c2 c 2 gxdx d d c 2 f ( )( )( T ) c 2 T fxdx d, fxdx d <<<T, c TC fxdx d c T c 2 c 2 gxdxd f ( ) dx d c f ( ) dx d, c fxdx d c f ( ) dxd c f ( ) dx d c 2 f ( ) dx d < <<T, gxdxd c f ( ) dx d c gxdx d c gxdx d c 2 gxdx d, << <T. 3.4

12 2 Mahemaical Problems in Engineering Following he previous procedure for he opimal replenishmen policy, he opimal value of,say, is given by he very simple and known, in classical order level invenory sysem Naddor 3, equaion: c 2T c c The Mahemaical Formulaion of he Model Saring wih Shorages In his secion he invenory model saring wih shorages is sudied. The cycle now sars wih shorages, which occur during he period,, and are parially backlogged. A ime a replenishmen brings he invenory level up o S. Demand and deerioraion of he iems deplee he invenory level during he period,t unil his falls o zero a T. Again he hree cases <<<T, < <<T,and << <Tmus be examined. Case 4 <<<T. The invenory level, I, T saisfies he following differenial equaions: di fβ, d,i, di ab b I f, d, I ( ) ( ) I, di ab b I f ( ), d, I ( ) ( ) I, di ab b I g, d T, IT. 4. The soluions of 4., are, respecively, I fxβ xdx,, 4.2 I e αb e αxb fxdx f ( ) e αxb dx e αxb gxdx,, 4.3 I e αb f ( ) e αxb dx e αxb gxdx,, 4.4 I e ab e axb gxdx, T. 4.5 The oal amoun of deerioraed unis during,t is D e ab fxe axb dx f ( ) e αxb dx fxdx ( ) f ( ) gxdx. e αxb gxdx 4.6

13 Mahemaical Problems in Engineering 3 The oal invenory carried during he inerval,t is found using 4.3, 4.4,and4.5 and is I e ab e αxb fxdx f ( ) e αxb dx e αxb gxdx d e αb f ( ) e αxb dx e αxb gxdx d e axb gxdx d. e ab 4.7 Due o 4.2 he ime-weighed backorders during he ime inerval, are I 2 fxβ xdx d. 4.8 The amoun of los sales during, is L β fd. 4.9 The invenory cos during he ime inerval,t is he sum of holding, shorage, deerioraion, and opporuniy coss and is given by TC c I c 2 I 2 c 3 D c 4 L ( c e ab e αxb fxdx f ( ) ) e αxb dx e αxb gxdx d ( c e αb f ( ) e αxb dx e αxb gxdx d e ab ) e axb gxdx d c 2 fxβ xdx d c 4 β fd c 3 e ab fxe axb dx f ( ) e αxb dx e αxb gxdx fxdx ( ) f ( ) gxdx. 4.

14 4 Mahemaical Problems in Engineering Case 5 < <<T. The invenory level, I, T saisfies he following differenial equaions: di fβ, d, I, di f ( ) β, d,i ( ) I ( ), di ab b I f ( ), d, I ( ) ( ) I, di ab b I g, d T, IT. 4. The soluions of 4., are, respecively, I I I e αb fxβ xdx,, fxβ xdx f ( ) β xdx,, f ( ) e αxb dx e αxb gxdx,, 4.2 I e ab e axb gxdx, T. The oal cos of his case is obained wih a similar way of he previous cases and is, ( TC 2 c e αb f ( ) ) e αxb dx e αxb gxdx d ( ) e ab e axb gxdx d ( c 3 e ab f ( ) ) e axb dx e axb gxdx f ( )( ) T gxdx c 2 fxβ xdx d fxβ xdx f ( ) β xdx d c 4 β fd f ( ) β d. 4.3

15 Mahemaical Problems in Engineering 5 Case 6 << <T. The invenory level, I, T for his case saisfies he following differenial equaions: di d di d di d di d fβ,, I, 4.4 f ( ) β,, I ( ) I ( ), 4.5 gβ,, I ( ) I ( ), 4.6 ab b I g, T, IT. 4.7 The soluions of 4.4, 4.5, 4.6, and4.7, are, respecively, I I I fxβ xdx,, fxβ xdx f ( ) β xdx,, gxβ xdx fxβ xdx f ( ) β xdx,, 4.8 I e ab e axb gxdx, T. The oal cos of his case is obained wih a similar way of he previous cases and is, TC 3 c e αb e axb gxdx d c 3 e ab e axb gxdx gxdx c 2 fxβ xdx d fxβ xdx f ( ) β xdx d ( c 2 gxβ xdx fxβ xdx f ( ) ) β xdx d c 4 β fd f ( ) β d β gd. 4.9

16 6 Mahemaical Problems in Engineering Finally he oal cos funcion of he sysem over,t akes he following form: TC, if, TC TC 2, if < <, TC 3, if. 4.2 I is easy o check ha his funcion is coninuous a and. The problem now is he minimizaion of his, hree branches, funcion TC. This requires, separaely, sudying each of hese branches and hen combining he resuls o sae he algorihm giving he opimal policy. 4.. The Opimal Replenishmen Policy In his subsecion we derive he opimal replenishmen policy, ha is, we calculae he value, say, which minimizes he oal cos funcion. Taking he firs-order derivaive of TC, say K, and equaing i o zero gives: ( ) K c c 3 αb b e αb e αxb fxdx f ( ) e αxb dx c2 β c 2 β c 4 β fd. e αxb gxdx 4.2 If is a roo of 4.2, for his roo he second-order condiion for minimum is c c 3 αb ( ) b c3 b ( ) αb ( ) b e α b e αxb fxdx f ( ) e αxb dx e αxb gxdx c c 3 αb ( ) b f ( ) c2 c 4 β f ( ) c2 β ( ) c 2 ( ) β ( ) c 4 β ( ) fd >. So, if 4.22 holds and, hen he value of order level, S, is S I ( ) e α b e αxb fxdx f ( ) e αxb dx e αxb gxdx, 4.23

17 Mahemaical Problems in Engineering 7 he ordering quaniy is Q fxβ ( x) dx S, 4.24 and he oal cos is TC. Equaing he firs-order derivaive of TC 2,sayK 2, o zero gives ( ) K 2 c c 3 αb b e αb ( f ( ) e αxb dx ) e αxb gxdx c2 β c 2 β c 4 β fd 4.25 f ( ) c2 β c 2 β c 4 β d. If is a roo of 4.25, for his roo he second-order condiion for minimum is c c 3 αb b αb b e α b ( f ( ) c 3 αbb b 2 e α b ( f ( ) f ( )( c 2 c 4 β ) e αxb dx e αxb dx ) e αxb gxdx ) e αxb gxdx c c 3 αb ( ) b f ( ) 2c2 β ( ) c 2 ( ) β ( ) c 4 β ( ) fd f ( ) 2c2 β ( ) c 2 ( ) β ( ) c 4 β ( ) d > So, if 4.26 holds and < <, hen he value of S is S I ( ) e α b f ( ) e αxb dx e αxb gxdx he ordering quaniy is Q fxβ ( x) dx f ( ) β ( x) dx S, 4.28 and he oal cos is TC 2.

18 8 Mahemaical Problems in Engineering Equaing he firs-order derivaive of TC 3,sayK 3, o zero gives ( ) K 3 c c 3 αb b e αb e αxb gxdx c2 β c 2 β c 4 β fd f ( ) c2 β c 2 β c 4 β d 4.29 c2 β c 2 β c 4 β gd. If is a roo of 4.29, for his roo he second-order condiion for minimum is c c 3 αb b αb b e α b e αxb gxdx c 3 αbb b 2 e α b e αxb gxdx c c 3 αb ( ) b g ( ) g ( )( c2 c 4 β ) 2c2 β ( ) c 2 ( ) β ( ) c 4 β ( ) fd f ( ) 2c2 β ( ) ( c 2 ) β ( ) c 4 β ( ) d 4.3 2c2 β ( ) c 2 ( ) β ( ) c 4 β ( ) gd >. So, if 4.3 holds and >,hen he value of S is S I ( ) T e a b e axb gxdx, 4.3 he ordering quaniy is Q fxβ ( x) dx f ( ) β ( x) dx gxβ ( x) dx S, 4.32 and he oal cos is TC 3. Remark 4.. Due o 4.2, 4.25, and4.29 he funcion TC is differeniable a he poin and.

19 Mahemaical Problems in Engineering 9 In he previous analysis here is no guaranee ha exiss and corresponds o he minimum. Is uniqueness is also anoher issue. The proposiion, which follows, provides sufficien condiions for exisence, uniqueness, and validiy of. Le us se ( ) Δ K ( c c 3 αb b ) e αb f ( ) e αxb dx e αxb gxdx Δ 2 K 2 ( ) c2 β ( ) c 2 ( ) β ( ) c 4 β ( ) fd, ( ( c c 3 αb b ) T ) e α b e αxb gxdx 4.33 c2 β ( ) c 2 ( ) β ( ) c 4 β ( ) fd f ( ) c2 β ( ) ( ) c 2 β ( ) c 4 β ( ) d and h c 2 βc 2 β c 4 β. The following proposiion can be easily proved observing ha K < and K 3 T >. Proposiion 4.2. If b<,h >,,T and Δ >, hen4.2 has one roo, say, which is he unique opimal value of he problem. 2 Δ <, and Δ 2 > hen 4.25 has one roo, say, which is he unique opimal value of he problem. 3 Δ <, and Δ 2 < hen 4.29 has one roo, say, which is he unique opimal value of he problem. We noe ha he above proposiion ensures he exisence, uniqueness, and validiy of. If he condiions of his proposiion do no hold, hen he following procedure can be used o calculae he opimal replenishmen policy. Sep. Find he global minimizing poin,,fortc. This will be one of he following poins: a roo of 4.2, an inerior poin of, which saisfies 4.22 b, c. Then calculae TC.

20 2 Mahemaical Problems in Engineering 2 Find he global minimizing,,fortc 2. This will be one of he following poins: a a roo of 4.25, an inerior poin of, T which saisfies 4.26, b, c. Then calculae TC 2. 3 Find he global minimizing,,fortc 3. This will be one of he following poins: a a roo of 4.29, an inerior poin of, T which saisfies 4.3, b, c T. Then calculae TC 3. Sep 2. Find he min {TC,TC 2,TC 3 } and accordingly selec he opimum. Remark 4.3. The analysis shows ha, in his model, D. is dependen from he demand rae 5. Numerical Examples The examples, which follow, illusrae he resuls obained. Example 5.. The inpu parameers are c $3 per uni per year, c 2 $5 per uni per year, c 3 $5 per uni, c 4 $2 per uni,.2 year,.9, a., b 2, T year, f 3e 4.5, g 3e 4.5.8, and βx e.2x. Model Saring wih No Shorages Using 3.37 he opimal value of is <.86 <, and consequenly he opimal ordering quaniy is Q 4.98 from 3.39 and he minimum cos is TC 6.63 from Model Saring wih Shorages Using 4.25 he opimal value of is <.63 <, and consequenly he opimal ordering quaniy is Q 4.98 from 4.28, and he minimum cos is TC from 4.3. Example 5.2. This example is idenical o Example 5., excep ha.7, f 2e 3.5,and g 2e Model Saring wih No Shorages Again.86 bu now <, he opimal ordering quaniy is Q from 3.4, and he minimum cos is TC from 3.34.

21 Mahemaical Problems in Engineering 2 Table : Sensiiviy analysis for model saring wihou shorages. The iniial cos parameers: c 3, c 2 5, c 3 5, c 4 2 3e 4.5, <<.2, 2e 3.5, <<.2, D f.2 g.9,.2.9, D f.2g.7,.2.7, 3e ,.9 <. 2e ,.7 <. Q TC Q TC c c c c Model Saring wih Shorages Using 4.25 he opimal value of is <.48 <, and consequenly he opimal ordering quaniy is Q from 4.28 and he minimum cos is TC 33.8 from 4.3. In Tables and 2 some sensiiviy analysis for he models saring wihou and wih shorages, respecively, is performed, for he above examples, by changing he cos parameer values 75%, 5%, 5% and, % aking one a a ime and keeping he remaining unchanged. From hese wo examples and he sensiiviy analysis is eviden he following. For he model saring wihou shorages, alhough he ime when shorages occur is idenical for he wo examples, he ordering quaniies and coss are significan differen, obviously because of he demand rae. 2 For he wo models he changes in he oal opimal coss indicae ha he models are highly sensiive o he error on he esimaion of he parameer value c, moderaely sensiive o he error on c 2, while low sensiiviy is o he error on he esimaion of he parameers c 3 and c 4. In presened examples, he coss relaed o sorage invenory c and c 3 are less han coss relaed o unsaisfied demand c 2 and c 4 maybe his explains he low sensiiviy o he error on he esimaion of he parameer c 4. While he small deerioraion rae maybe implies low sensiiviy wih respec o c 3.

22 22 Mahemaical Problems in Engineering Table 2: Sensiiviy analysis for model saring wih shorages. The iniial cos parameers: c 3, c 2 5, c 3 5, c 4 2 3e 4.5, <<.2, 2e 3.5, <<.2, D f.2 g.9,.2.9, D f.2 g.7,.2.7, 3e ,.9 <. 2e ,.7 <. Q TC Q TC c c c c The errors in cos parameers are aenuaed when ranslaed ino changes in he opimal ordering quaniy. 6. Concluding Remarks In his paper, an order level invenory model for deerioraing iems has been sudied. The basic assumpion of he model is based on ime dependen hree branches ramp ype demand rae. The demand of seasonable and fashionable producs can be described well wih his funcion, as he naure of demand of hese producs is increasing a he beginning of he season, seady in he mid of he season, and decreasing a he end of he season. To he bes of our knowledge his demand paern sudied for he firs ime, a leas, using so general funcions for he nonseady periods. In addiion a ime dependen backlogging and deerioraion rae are assumed. The invenory model is sudied under wo differen replenishmen policies: a saring wih no shorages and b saring wih shorages. An algorihm o obain he opimal policy is proposed. Moreover from his model follows as special cases he following ones. If f D,,, gx f, and βx / δx, his model reduces o ha of Wu 2. 2 If in addiion o, βx case of complee backlogging, hen i furher reduces o ha of Wu e al..

23 Mahemaical Problems in Engineering 23 3 If b i.e., consan deerioraion rae, f D, gx f,, and βx, hen he models reduce o hose of Wu and Ouyang, MandalandPal 9, and Deng e al If gx f,, his model gives he one model proposed by Skouri e al. 6. Acknowledgmen The auhors would like o hank he referees for heir valuable commens and suggesions ha improved he paper. References M. Resh, M. Friedman, and L. C. Barbosa, On a general soluion of he deerminisic lo size problem wih ime-proporional demand, Operaions Research, vol. 24, no. 4, pp , W. A. Donaldson, Invenory replenishmen policy for a linear rend in demand: an analyic soluion, Operaional Research Quarerly, vol. 28, pp , U. Dave and L. K. Pael, T, S i policy invenory model for deerioraing iems wih ime proporional demand, Journal of he Operaional Research Sociey, vol. 32, no. 2, pp , S. K. Goyal, On improving replenishmen policies for linear rend in demand, Engineering Coss and Producion Economics, vol., no., pp , M. Hariga, An EOQ model for deerioraing iems wih shorages and ime varying demand, Journal of he Operaional Research Sociey, vol. 46, pp , M. A. Hariga and L. Benkherouf, Opimal and heurisic invenory replenishmen models for deerioraing iems wih exponenial ime-varying demand, European Journal of Operaional Research, vol. 79, no., pp , H.-L. Yang, J.-T. Teng, and M.-S. Chern, Deerminisic invenory lo-size models under inflaion wih shorages and deerioraion for flucuaing demand, Naval Research Logisics, vol. 48, no. 2, pp , 2. 8 R. M. Hill, Invenory models for increasing demand followed by level demand, Journal of he Operaional Research Sociey, vol. 46, no., pp , B. Mandal and A. K. Pal, Order level invenory sysem wih ramp ype demand rae for deerioraing iems, Journal of Inerdisciplinary Mahemaics, vol., no., pp , 998. J.-W. Wu, C. Lin, B. Tan, and W.-C. Lee, An EOQ invenory model wih ramp ype demand rae for iems wih Weibull deerioraion, Inernaional Journal of Informaion and Managemen Sciences, vol., no. 3, pp. 4 5, 999. K.-S. Wu and L.-Y. Ouyang, A replenishmen policy for deerioraing iems wih ramp ype demand rae, Proceedings of he Naional Science Council, Republic of China A, vol. 24, no. 4, pp , 2. 2 K.-S. Wu, An EOQ invenory model for iems wih Weibull disribuion deerioraion, ramp ype demand rae and parial backlogging, Producion Planning and Conrol, vol. 2, no. 8, pp , 2. 3 B. C. Giri, A. K. Jalan, and K. S. Chaudhuri, Economic order quaniy model wih Weibull deerioraion disribuion, shorage and ramp-ype demand, Inernaional Journal of Sysems Science, vol. 34, no. 4, pp , S. K. Manna and K. S. Chaudhuri, An EOQ model wih ramp ype demand rae, ime dependen deerioraion rae, uni producion cos and shorages, European Journal of Operaional Research, vol. 7, no. 2, pp , P. S. Deng, R. H.-J. Lin, and P. Chu, A noe on he invenory models for deerioraing iems wih ramp ype demand rae, European Journal of Operaional Research, vol. 78, no., pp. 2 2, K. Skouri, I. Konsanaras, S. Papachrisos, and I. Ganas, Invenory models wih ramp ype demand rae, parial backlogging and Weibull deerioraion rae, European Journal of Operaional Research, vol. 92, no., pp , P. M. Ghare and G. F. Schrader, A model for exponenially decaying invenories, Journal of Indusrial Engineering, vol. 4, pp , 963.

24 24 Mahemaical Problems in Engineering 8 R. P. Cover and G. C. Philip, An EOQ model for iems wih Weibull disribuion deerioraion, AIIE Transacion, vol. 5, no. 4, pp , P. R. Tadikamalla, An EOQ invenory model for iems wih gamma disribuion, AIIE Transacion, vol., no., pp. 3, F. Raafa, Survey of lieraure on coninuously deerioraing invenory models, Journal of he Operaional Research Sociey, vol. 42, no., pp , S. K. Goyal and B. C. Giri, Recen rends in modeling of deerioraing invenory, European Journal of Operaional Research, vol. 34, no., pp. 6, P. L. Abad, Opimal pricing and lo-sizing under condiions of perishabiliy and parial backordering, Managemen Science, vol. 42, no. 8, pp. 93 4, H.-J. Chang and C.-Y. Dye, An EOQ model for deerioraing iems wih ime varying demand and parial backlogging, Journal of he Operaional Research Sociey, vol. 5, no., pp , K. Skouri and S. Papachrisos, A coninuous review invenory model, wih deerioraing iems, ime-varying demand, linear replenishmen cos, parially ime-varying backlogging, Applied Mahemaical Modelling, vol. 26, no. 5, pp , J.-T. Teng, H.-J. Chang, C.-Y. Dye, and C.-H. Hung, An opimal replenishmen policy for deerioraing iems wih ime-varying demand and parial backlogging, Operaions Research Leers, vol. 3, no. 6, pp , S.-P. Wang, An invenory replenishmen policy for deerioraing iems wih shorages and parial backlogging, Compuers & Operaions Research, vol. 29, no. 4, pp , L. A. San José, J. Sicilia, and J. García-Laguna, An invenory sysem wih parial backlogging modeled according o a linear funcion, Asia-Pacific Journal of Operaional Research, vol. 22, no. 2, pp , L. A. San José, J. Sicilia, and J. García-Laguna, Analysis of an invenory sysem wih exponenial parial backordering, Inernaional Journal of Producion Economics, vol., no., pp , C.-K. Chen, T.-W. Hung, and T.-C. Weng, Opimal replenishmen policies wih allowable shorages for a produc life cycle, Compuers & Mahemaics wih Applicaions, vol. 53, no., pp , C.-K. Chen, T.-W. Hung, and T.-C. Weng, A ne presen value approach in developing opimal replenishmen policies for a produc life cycle, Applied Mahemaics and Compuaion, vol. 84, no. 2, pp , E. Naddor, Invenory Sysems, John Wiley & Sons, New York, NY, USA, 966.