ISSN 394 3386 Augus 07 An EPQ Invenory Model wih Variable Holding Cos and Shorages under he Effec of Learning on Seup Cos for Two Warehouses Monika Vishnoi*, S.R.Singh C.C.S.Universiy, Meeru, U.P., India Corresponding Auhor: * Absrac A wo warehouse producion invenory model for deerioraing iems is explored in an inflaion induced demand and parial backlogging under he effec of learning on se-up cos. Holding cos is reaed as a linear funcion of ime and producion rae is aken as a linear combinaion of demand and on-hand invenory. A finie planning horizon invenory problem having wo separae warehouses, one is an own warehouse (OW) and oher a rened warehouse (RW) is developed wih deerioraion rae of he iems o be differen in differen warehouses. Here i is assumed ha he socks of RW are ranspored o OW in coninuous release paern. Furher, we use numerical example o illusrae he model and sensiiviy analysis on some parameers is made. Keywords: Muli-variae producion rae, Learning effec, Linear holding cos, Inflaion induced demand, Parial backlogging, Varying rae of deerioraion, Two-warehouses, coninuous release paern. Inroducion I has been noed ha he performance of a person, group of persons, or an organizaion, engaged in a repeiive ask improves wih ime. Such a phenomenon is referred, in he lieraure, as he Learning Phenomenon, which implies a reducion in he cos or he ime required for producing each uni. For insance, he familiariy wih operaional asks and heir environmens, and he effecive use of ools and machines are usually increased wih repeiion. Jaber and Bonney (997) sudied he effec of learning on he opimal manufacured quaniy wih he consideraion of inracycle, wihin cycle, backorders. Jaber and Bonney (997) sudied he differences and similariies of hree previously proposed models of learning. The opimal lo sizing under he bounded learning case has been considered by so many researchers. The aricles on his aspec are of Zhou and Lau (998). Jaber and Bonney (998) developed hree models for he infinie and finie planning horizon. They showed ha when he sysem experiences a parial ransmission of learning he opimal policy was o carry fewer invenories in laer los. For more deails abou learning and forgeing, see he ousanding review of Jaber and Bonney (999). Singh and Vishnoi (0) developed a research paper eniled opimal replenishmen policy for deerioraing iems wih ime dependen demand under he learning effec. In many real-life siuaions, he pracical experiences reveal ha some bu no all cusomers will wai for backlogged iems during a shorage period, such as for fashionable commodiies or high-ech producs wih shor produc life cycle. The longer he waiing ime is, he smaller he backlogging rae would be. According o such phenomenon, aking he backlogging rae ino accoun is necessary. However, mos of he invenory models unrealisically assume ha during sockou eiher all demand is backlogged or all is los. In realiy ofen some cusomers are willing o wai unil replenishmen, especially if he wai will be shor, while ohers are more impaien and go elsewhere. The backlogging rae depends on he ime o replenishmen-he longer cusomers mus wai, he greaer he fracion of los sales. Abad (996) developed a pricing and losizing EOQ model for a produc wih a variable rae of deerioraion and parial backlogging. Abad (000) hen exended he opimal pricing and lo-sizing EOQ model o an economic producion quaniy (i.e. EPQ) model. Papachrisos and Skouri (000) discussed an opimal replenishmen policy for deerioraing iems Monika Vishnoi, S.R.Singh
ISSN 394 3386 Augus 07 wih ime-varying demand and parial-exponenial-ype backlogging. Oher aricles relaed o his research were wrien by Abad (00), Ouyang e al. (005), Jolai e al. (006) and Yang (007) and Vishnoi and Shon (00) developed wo levels of sorage model for non-insananeous deerioraing iems wih sock dependen demand, ime varying parial backlogging under permissible delay in paymens. Deerioraion is he change, damage, decay, spoilage, evaporaion, obsolescence, pilferage, and loss of uiliy or loss of marginal value of a commodiy ha resuls in decreasing usefulness from he original one. Mos producs such as medicine, blood, fish, alcohol, gasoline, vegeables and radioacive chemicals have finie shelf life, and sar o deeriorae once hey are replenished. Li e al. (00) provided a comprehensive inroducion abou he deerioraing iems invenory managemen research saus, his paper reviews he recen sudies in relevan fields.vishnoi and Singh (05) developed wo sorage invenory model wih rade credis, inflaion in bulk release. I is necessary o consider he effecs of inflaion on he invenory sysem, as many counries experience high annual inflaion rae. Besides his, inflaion also influences demand of cerain producs. The fundamenal resul in he developmen of EOQ model wih inflaion is ha of Buzaco (975) who discussed EOQ model wih inflaion subjec o differen ypes of pricing policies. Bose e al. (995) presened a paper on deerioraing iems wih linear ime dependen rae and shorages under inflaion and ime discouning. Wee and Law (999) addressed he problem wih finie replenishmen rae of deerioraing iems aking accoun of ime value of money. Chang (004) proposed an invenory model for deerioraing iems under inflaion under a siuaion in which he supplier provides he purchaser a permissible delay of paymens if he purchaser orders a large quaniy. Jaggi e al. (006) presened he opimal invenory replenishmen policy of deerioraing iems under inflaionary condiions. The demand rae was assumed o be a funcion of inflaion; shorages were allowed and compleely backlogged. Vishnoi e al. (04) presened progressive ineres scheme under inflaion and ime value of money in wo echelon supply chain model. The variabiliy in he holding cos was inroduced firs in he model developed by Muhlemann and Valis-Spanopoulous (980). In an EOQ model wih a consan demand rae, hey expressed he holding cos as a percenage of he average value of capial invesed in he sock. Van der Veen (967) developed an invenory sysem wih consan demand rae, aking he holding cos as a nonlinear funcion of invenory. Weiss (98) sudied he same model reaing holding cos per uni as a nonlinear funcion of he lengh of ime for which he iem was held in sock. Naddor (966) gave a deailed derivaion of he oal invenory cos for a consan demand-rae lo-size-sysem by considering he holding cos as q m n, q being he amoun of sock held for a ime and m, n being posiive inegers. Naddor's presenaion can be regarded as a generalizaion of boh Van der Veen's (967) and Weiss's (98) work. Laer on, Goh (994) discussed he model of Baker and Urban (988) relaxing he assumpion of a consan holding cos. The lineariy in ime for he holding cos is jusified for he invenory sysem in which no only he cos of holding an iem in sock increases bu also he value of he unsold invenory decreases wih each passing day. In he exising lieraure i is observed ha here is almos a huge vacuum in he invenory models which is based on wo warehouse producion models. Few researchers have considered he same bu hey have aken he consan rae of producion or he producion rae is demand dependen only bu when manufacurers already have sufficien sock o fulfill he marke demand so in ha siuaion producion rae is demand dependen or consan is no realisic. In he presen sudy considering his realisic approach, we have aken producion rae as he linear combinaion of on-hand invenory and demand rae a any ime. In our sudy, we have srived o sudy a wo warehouse producion invenory model wih parially backordered shorages, in which unis are, deeriorae wih differen ime dependen raes in boh he warehouses and he demand rae is increasing exponenially due o inflaion, over a finie planning horizon under he learning effec on se-up cos. In his paper, a more realisic scenario was assumed where par of he shorage was backordered and he res was los. In his paper we have aken variable holding cos for boh warehouses due o differen preservaion condiions. I is assumed ha he socks of RW are ranspored o OW in coninuous release paern so ha ransporaion cos is negleced in his paper. 3 Monika Vishnoi, S.R.Singh
ISSN 394 3386 Augus 07 The whole combinaion of he seup is very unique and more pracical. Finally, numerical example is presened o demonsrae he developed model and he soluion procedure. Sensiiviy analysis of he opimal soluion wih respec o major parameers is carried ou. The final oucome shows ha he model is no only economically feasible, bu sable also.. Assumpions and Noaions Assumpions: The following assumpions have been used hroughou he sudy. (i) The demand rae is exponenially increasing and is represened by λ () = λ 0 e δ, where 0 δ is a consan inflaion rae and λ 0 is he iniial demand rae. (ii) Lead ime is zero and no replenishmen or repair of deerioraed iems is made during a given cycle. (iii) Producion rae is he linear combinaion of on-hand invenory and demand rae a any ime s.. P () = I () + η λ (), η > and P () > λ () (iv) () Shorages are allowed and he backlogging rae is T, when invenory is in shorages, he backlogging parameer is a posiive consan s.. 0 < <. (v) A single iem is considered over he prescribed period T unis of ime, which is subjec o variable deerioraion rae. (vi) Learning effec is o be considered on seup cos. CN ( ) The replenishmen cos, $/order, is parly consan and parly decreasing in each cycle due o learning effec of employees and is of he form = ( ( / N, φ > 0. N is he cumulaive number of replenishmens of he invenory. C (vii) The owned warehouse (OW) has a fixed capaciy of W unis, he rened warehouse (RW) has unlimied capaciy. (viii) The goods of OW are consumed only afer consuming he goods kep in RW. (ix) The uni invenory coss (including holding cos) per uni ime in RW are higher han hose in OW. (x) Holding coss C OW () and C RW () per iems per ime uni is linearly ime dependen and are assumed in OW and RW respecively as C OW () = C OW + and C RW () = C RW +, where 0 < < and 0 < <. (xi) Deerioraion rae of he iems is considered o be differen in differen warehouses. In OW ime dependen deerioraion rae is θ () = θ while in RW Weibull disribuion deerioraion rae αβ β- where α, β > 0, > 0. (xii) Finie planning horizon is considered. Noaions: The following noaions have been used hroughou he sudy. H: Toal planning horizon W: Fixed capaciy level of OW C 0: Fix amoun of he replenishmen coss per $ per order C : Amoun of he se up cos on which learning effec is applied. C RW: Carrying cos per invenory uni held in RW per uni ime C OW: Carrying cos per invenory uni held in OW per uni ime C : Deerioraion cos per uni ime C 3 : Shorage cos for backlogged iems C 4 : The uni cos of los sales C 0 + 4 Monika Vishnoi, S.R.Singh
ISSN 394 3386 Augus 07 I i : Invenory level in OW a ime wih I i : Invenory level in RW a ime wih I i3: I i4 : Invenory level in RW a ime wih Invenory level in OW a ime wih [0, ] [0, ] [0, ] 3 [0, ] I i5 : Invenory level in OW a ime wih [0, 3] 4 I i6 : I i7 : Invenory level in OW a ime wih Invenory level in OW a ime wih [0, ] 5 [0, T], : 3, 4 : 5 : T : The producion periods for OW and RW The non producion periods The Shorage period Toal cycle ime 3. Formulaion and Soluion of he Model The behavior of invenory level in a producion sysem wih inflaion induced demand for deerioraing iems is depiced in Fig.. In Fig., he invenory level during a producion cycle in which boh OW and RW are used. Iniially, he invenory level is zero. The producion sars a ime = 0 and iems accumulae from 0 up o W unis in OW in unis of ime. Afer ime any producion quaniy exceeding W will be sored in RW. Afer his producion sopped and he invenory level in RW begins o decrease a and will reach 0 unis a 3 because of demand and deerioraion. The invenory level in OW comes o decrease a and hen falls below W a + 3 due o deerioraion. The remaining socks in OW will be fully exhaused a 4 owing o demand and deerioraion, he invenory becomes zero. A his ime shorage sars developing and a ime 5 i reaches o maximum shorage level, a his ime fresh producion sars o clear he backlog by he ime T. 5 Monika Vishnoi, S.R.Singh
ISSN 394 3386 Augus 07 The differenial equaions saing he invenory levels wihin he cycle are given as follows: dii () ( ) Ii ( ) P( ) ( ), d 0 () dii() Ii( ) P( ) ( ), d 0 () dii3() Ii3( ) ( ), d 0 3 (3) dii4() ( ) Ii4( ) ( ), d 0 4 (4) dii5() ( ) Ii5( ) 0, d 0 + 3 (5) di () () i6, d ( T ) 0 5 (6) di () i7 P( ) ( ), 0 T (7) d Wih he boundary condiions Ii (0) 0, Ii (0) 0, Ii3( 3) 0, Ii4( 4) 0, I (0), i5 W and I ( T) 0 respecively, he above equaions can be solved successively as follows: i7 0 I i6 Ii ( ) e e 0 (8), (0) 0, Ii( ) 0 0 (9), 3 3 Ii3( ) 0 e e 3 3 e e, 0 3 (0) 4 4 4 4 Ii4() 0 e e 3 4 e e 4e e e e, 0 4 () I ( ), i5 We 0 + 3 () Ii6( ) 0 e T T e, 0 T Ii7( ) e e, 0 5 (3) 0 T (4) 6 Monika Vishnoi, S.R.Singh
ISSN 394 3386 Augus 07 4. Presen worh Ordering Cos Ordering cos per cycle is given b C( N) C 0 C N (5) 5. Presen worh Holding Cos in RW The oal invenory level in RW can be derived as HRW CRW Ii d Ii d 3 0 ( ) 0 3( ) 3 3 3 CRW 0 3 3 3 3 4 4 3 4 0 3 4 4 3 4 3 CRW0 3 3 3 3 3 3 4 0 3 3 6 3 4 (6) 6. Presen worh Holding Cos in OW The oal invenory level in OW can be derived as HOW COW Ii d Ii d Ii d 3 4 0 ( ) 0 5( ) 0 4( ) 3 4 COW0 3 8 3 4 5 0 COWW 3 3 4 0 6 W 4 8 3 3 COW0 3 3 3 4 4 4 4 3 3 3 4 5 4 4 5 4 0 6 8 40 (7) 7 Monika Vishnoi, S.R.Singh
ISSN 394 3386 Augus 07 7. Presen worh Deerioraed Iems The oal quaniy of deerioraed iems during he period (0, T) is given by 3 4-3 - ( ) DC C I 0 0 5( ) 0 4( ) 0 ( ) i d Ii d Ii d Ii d I 0 i3( ) d 4 5 4 0 3 3 C W 3 4 0 8 0 6 8 8 3 4 5 4 4 4 0 3 3 3 0 (8) 8. Presen worh Shorage Cos Toal quaniy of shorage unis (I S ) during he period (0,T) is given by 5 T SC C 3 - I 0 i6( ) d - I 0 i7( ) d 3 5 5 T T C30 T 3 (9) 9. Presen worh Los Sales Cos The oal amoun of los sales (I L ) during he period (0, T) can be obained as 3 5 5 5 LS C4-0 0e d C40 T5 T (0) ( T - ) 3 0. Presen worh Toal Cos The oal average cos (TC) of he invenory sysem is given by TC(, 4) C H RW HOW DC SC LS () T. Soluion Procedure To minimize oal average cos per uni ime (TC), he opimal values of and 4 can be obained by solving he following equaions simulaneously TC 0 () and TC 0 (3) 4 8 Monika Vishnoi, S.R.Singh
Provided, hey saisfy he following condiions TC TC 0, 0 4 and Inernaional Journal of Engineering Technology Science and Research ISSN 394 3386 Augus 07 TC TC TC 4 4 Equaion () is our objecive funcion which needs o be minimized. For his, we use he classical opimizaion echniques. The equaions () and (3) obained hereafer are highly non-linear in he coninuous variable, 4 and he discree variables, 3, 5 and T. However, if we give paricular values o he discree variable, 3, and 5, our objecive funcion becomes he funcion of hree variables and 4 and T. We have used he mahemaical sofware MATHEMATICA 5. o arrive a he soluion of he sysem in consideraion. We can obain he opimal values of differen values of he ime wih he help of sofware. Wih he use of hese opimal values equaion () provides minimum oal average cos per uni ime of he sysem in consideraion. The oal planning horizon H has been divided ino N equal cycles where he lengh of each cycle is exacly T = H/N. Here numerical illusraion is o be given for s cycle. 0. Numerical Illusraions and Analysis To illusrae he resuls, le us apply he proposed mehod o efficienly solve he following numerical example. For convenience, he values of he parameers are seleced randomly. Example : α = 0.05, β =, λ 0 = 000, η =, θ = 0.05, δ = 0.05, γ = 0.05, C = $00, C RW = $5, C OW = $5, C = $0, C 3 = $, C 4 =$ 5, H= 40 Monhs, W = 600 unis, = 5, 3 = 7, 5 = 8, N = Table : Opimal Soluion of he Proposed Model 4 Toal Cos (TC) 5.636504 4.3670 4305 The graph shows he variaion of he sysem cos wih and 4. From he figure 6. i is very clear ha he oal cos funcion is convex wih respec o he wo variables. Hence he soluion obained is no only opimum bu also unique. 48000 46000 TC 44000 4000 4 6 8 3 4 5 4 6 Fig.. Graphical represenaion of a convexiy of he oal cos funcion (TC) 9 Monika Vishnoi, S.R.Singh
ISSN 394 3386 Augus 07 3. Sensiiviy Analysis We now sudy he effecs of changes in he values of he sysem parameers α, β, γ, η, θ on he oal sysem cos in consideraion. The sensiiviy analysis is performed by changing each of he parameers by 50%, 5%, 5% and 50%, aking one parameer a a ime and keeping he remaining parameers unchanged. The analysis is based on he adjused resuls obained from Example. From Tables 6, we observe some ineresing facs, which are quie obvious when considered in he ligh of realiy. Table : Effecs of Scale Parameer (α) of Deerioraion Rae in RW: α T 4 Toal Cos % Variaion in Toal Cos -50% 5.6365 4.367 343-8.066-5% 5.6365 4.367 3343-4.063 5% 5.6365 4.367 53030 4. 50% 5.6365 4.367 6387 8.3007 Table 3: Effecs of Shape Parameer (β) of Deerioraion Rae in RW: β T 4 Toal Cos % Variaion in Toal Cos -50% 5.6365 4.367 4758.7048-5% 5.6365 4.367 45445 0.9999 5% 5.6365 4.367 4068-0.9864 50% 5.6365 4.367 38444 -.8809 Table 4: Effecs of Backlogging Parameer (γ) γ 4 Toal Cos % Variaion in Toal Cos -50% 5.6365 4.367 506.98-5% 5.6365 4.367 46638.4909 5% 5.6365 4.367 3939 -.4909 50% 5.6365 4.367 35768 -.98 Table 5: Effecs of ime dependen parameer in holding cos for RW (ξ) ξ 4 Toal Cos % Variaion in Toal Cos -50% 5.6365 4.367 4330.9-3.567-5% 5.6365 4.367 0934-5.39 5% 5.6365 4.367 376688 5.334 50% 5.6365 4.367 5036 3.397 30 Monika Vishnoi, S.R.Singh
ISSN 394 3386 Augus 07 Table 6: Effecs of ime dependen parameer in holding cos for OW (ζ) ζ 4 Toal Cos % Variaion in Toal Cos -50% 5.6365 4.367 4330.9-3.6588-5% 5.6365 4.367 0934-7.8760 5% 5.6365 4.367 376688 7.09954 50% 5.6365 4.367 5036 3.776 Table 7: Effecs of Sock Parameer in Producion (η) η 4 Toal Cos % Variaion in Toal Cos -50% 5.6365 4.367 4330.9-89.9879-5% 5.6365 4.367 0934-55.0060 5% 5.6365 4.367 376688 55.0060 50% 5.6365 4.367 5036 0.0 Table 8: Effecs of Deerioraion Parameer in OW (θ) θ 4 Toal Cos % Variaion in Toal Cos -50% 8.87.36 6984.0388-5% 6.5868 3.336 5764 6.086. 5% 4.9886 4.650 5756-7.00 50% 4.508 4.8 05848-5.94 4. Observaions The main observaions drawn from he sensiiviy analysis are as follow: The values of percenage variaion in oal coss are he mos sensiive o he sock parameer η in producion. When η increases by 50%, he value of percenage variaion in oal cos increases by over 0%. This shows he drasic variaion wih respec o sock parameer in producion which proves he validiy of he proposed analyical model and need o be more aenion of he company s producion managemen. When ξ decreases by 50%, he value of percenage variaion in oal cos decreases by over 3%. When ζ increases by 50%, he value of percenage variaion in oal cos increases by over 3%. The values of percenage variaion in oal coss are quie sensiive o he parameers α, θ. When α increases by 50%, he value of percenage variaion in oal cos increases by over 8%. When θ increases by 50%, he value of percenage variaion in oal cos decreases by over 5%. I shows ha proper concenraion is o be given o he correc esimaion of he parameers α, θ from he marke analysis. 3 Monika Vishnoi, S.R.Singh
ISSN 394 3386 Augus 07 The values of percenage variaion in oal coss are no so sensiive o he parameers β, γ. When β decreases by 50%, he value of percenage variaion in oal cos increases by over %. When γ decreases by 50%, he value of percenage variaion in oal cos increases by over %. The sysem shows a very good sabiliy wih respec o backlogging parameer and shape parameer of he deerioraion rae of RW. Hence, in order o decrease he oal cos per uni ime, he reailer should increase he value of backlogging parameer γ. When γ equals o zero, he model reduces o he case of complee backlogging, and has he minimum cos per uni ime. All hese facors ulimaely give way o he fac ha he proposed model is very suied o presen day marke condiions. The mos sriking observaion from he above ables o 5 is ha he opimum value of he decision variables and 4 are no affeced by any parameer excep θ (able 6), he deerioraion parameer in OW. As he deerioraion parameer increases, obviously, he opimum ime and 4 will be changed. Graphical represenaion of he sensiiviy resuls wih respec o differen sysem parameers have been ploed in Fig 3 o 7. Fig.3 Variaion in oal Cos w.r. Scale Parameer in RW (α) Fig. 4 Variaion in oal Cos w.r. Shape Parameer in RW (β) Fig.5 Variaion in oal Cos w.r. Backlogging Fig.6 Variaion in oal Cos w.r. Demand Parameer parameer in (γ) in Producion (η) o 3 Monika Vishnoi, S.R.Singh
ISSN 394 3386 Augus 07 Fig. 7 Variaion in oal Cos w.r. Deerioraion Parameer in OW (θ) 5. Concluding Remarks The proposed model incorporaes some realisic and pracical feaures ha are likely o be associaed wih he invenory of cerain ypes of goods, such as: demand rae is Inflaion-dependen, he invenory deerioraes a a variable rae over ime and producion rae is flexible and considers wo warehouses o reflec realisic business siuaions. Mos producs experience a period of rapid demand increase during he inroducion phase of produc life cycle, level off in demand afer reaching heir mauriy period, and will ener a period of sales decline due o new compeing producs or changes in consumer preference. The wowarehouse invenory conrol is an inriguing ye pracicable issue of decision science when inflaion induced demand is involved. Furhermore, producion learning has he greaes influence on he opimal oal cos. The effecs of producion learning on he number of producion runs and oal cos are more influenial han ha of seup learning. Mos of he researchers have ill now ignored he effecs of deerioraion in boh he warehouses or have considered a consan rae of deerioraion bu we have aken a linear ime dependen deerioraion rae in OW and Weibull disribuion ype deerioraion rae in RW. In he model, he holding cos is regarded as linear funcion of he lengh of ime he iem is held in sock. This ype of assumpion is quie appropriae when he value of he unsold iems decreases wih ime. Reailers in supermarke face his problem while selling producs like green vegeables, fruis and breads whose qualiy drops wih each passing day. As a resul, increasing holding coss are incurred o arrange beer sorage faciliies o preven spoilage and o mainain freshness of he iems in sock. The funcional form of linear ime-dependen holding cos is quie realisic from ha poin of view. This assumpion is jusified for he producs such as elecronic componens, radioacive subsances, volaile liquids ec. which are no only cosly bu also require more sophisicaed arrangemens for heir securiy and safey. In his sudy, we have aken ha producion rae as he linear combinaion of on-hand invenory and demand rae a any ime because when manufacurer s have sufficien sock o fulfill he marke demand hen from business poin of view he will sop he producion a he same ime so ha said producion rae is more realisic oher han ha. Shorages in invenory are allowed and he backlogging rae is aken as variable and inversely proporional o he waiing ime up o he arrival of nex lo. Cos minimizaion echnique is used o ge he expressions for oal cos and oher parameers. A numerical assessmen of he heoreical model has been done o illusrae he heory. The soluion obained has also been checked for sensiiviy wih he resul ha he model is found o be quie suiable and sable. The variaions in he sysem saisics wih a variaion in sysem parameers has also been illusraed graphically. All hese facs ogeher make his sudy very unique and maer-of-fac. 33 Monika Vishnoi, S.R.Singh
ISSN 394 3386 Augus 07 The proposed model can be exended in numerous ways. For example, we may exend he inflaion dependen demand o inflaion and sock dependen demand rae. Also, we could exend he model o incorporae some more feaures, such as quaniy discoun, inflaion and permissible delay in paymen. References. Abad, P.L. (996). Opimal pricing and lo sizing under condiions of perishabiliy and parial backordering, Managemen Science; 4: 093-04.. Abad, P.L. (000). Opimal lo size for a perishable good under condiions of finie producion and parial backordering and los sale. Compuers & Indusrial Engineering; 38(4): 457-465. 3. Abad, P.L. (00). Opimal price and order size for a reseller under parial backordering, Compuers and Operaions Research; 8: 53-65. 4. Baker, R.C., Urban, T.L. (988). A deerminisic invenory sysem wih an invenory level dependen demand rae, Journal of he Operaional Research Sociey; 39: 83-83. 5. Bose, S., Goswami, A., Chaudhuri, K.S. (995). An EOQ model for deerioraing iems wih linear imedependen demand rae and shorages under inflaion and ime discouning, Journal of he Operaional research Sociey; 46:77-78. 6. Buzaco, J.A. (975). Economic order quaniy wih inflaion, Operaions Research Qr; (3): 553-558. 7. Chang, C.T. (004). An EOQ model wih deerioraing iems under inflaion when supplier credis linked o order quaniies, Inernaional Journal of Producion Economics; 88: 307-36. 8. Goh, M. (994). EOQ models wih general demand and holding cos funcions, European Journal of Operaional Research; 73: 50-54. 9. Jaber, M.Y., Bonney, M. (997). A comparaive sudy of learning curves wih forgeing. Applied Mahemaical Modeling, 53 53. 0. Jaber, M.Y., Bonney, M. (997). The effec of learning and forgeing on he economic manufacured quaniy (EMQ) wih consideraion of inracycle backorders. Inernaional Journal of Producion Economics; 53:.. Jaber, M.Y., Bonney, M. (998). The effec of learning and forgeing on he opimal lo size quaniy of inermien producion runs, Producion Planning and Conrol; 9: 0 7.. Jaber, M.Y., Bonney, M. (999). The economic manufacured/order quaniy (EMQ/EOQ) and he learning curve: pas, presen and fuure, Inernaional Journal of Producion Economics; 59: 93 0. 3. Jaggi, C.K., Aggarwal, K.K., Goel, S.K. (006). Opimal order policy for deerioraing iems wih inflaion induced demand, Inernaional Journal of Producion Economics; 03: 707-74. 4. Jolai, F., Tavakkoli-Moghaddam, R., Rabbani, M., Sadoughian, M.R. (006). An economic producion lo size model wih deerioraing iems, sock-dependen demand, inflaion, and parial backlogging, Applied Mahemaics and Compuaion; 8: 380-389. 5. Li, R., Lan, H., Mawhinney, J. R. (00). A Review on deerioraing invenory sudy, J. Service Science & Managemen; 3:7-9. 6. Muhlemann, A.P. Valis Spanopoulos, N.P. (980). A variable holding cos rae EOQ model, European Journal of Operaional Research; 4: 3-35. 7. Naddor, E. (966). Invenory Sysems, Wiley, New York. 8. Ouyang, L.Y., Wu, K.S., Cheng, M.C. (005). An invenory model for deerioraing iems wih exponenial declining demand and parial backlogging, Yugoslav Journal of Operaions Research; 5(): 77-88. 9. Papachrisos, S., Skouri, K. (000). An opimal replenishmen policy for deerioraing iems wih ime-varying demand and parial-exponenial ype backlogging, Operaions Research Leers; 7(4): 75-84.. 0. Singh, S., Vishnoi, M. (0) Opimal Replenishmen Policy for Deerioraing Iems wih Time Dependen Demand under he Learning Effec Inernaional Transacions in Mahemaical Sciences and Compuer, 4:, pp. 7-86. ISSN 0974-5068.. Van der Veen, B. (967). Inroducion o he Theory of Operaional Research, Philips Technical Library, Springer-Verlag, New York.. Vishnoi, M., Shon, S.K. (00) Two Levels of Sorage Model for Non-Insananeous Deerioraing Iems wih Sock Dependen Demand, Time Varying Parial Backlogging under Permissible Delay in Paymens Inernaional Journal of Operaions Research and Opimizaion, :., pp.33-47. ISSN 0975-3737. 3. Vishnoi, M., Singh, S.R., Vishnoi, G. (04). Progressive ineres scheme under inflaion and ime value of money in wo echelon supply chain model, Proceedings of ICRTET 04, 3rd Inernaional Conference on Recen Trends in Engineering & Technology (Elsevier), 360-367, ISBN: 978-93-507--5. 34 Monika Vishnoi, S.R.Singh
ISSN 394 3386 Augus 07 4. Vishnoi, M., Singh, S.R. (05) Two Sorage Invenory Model wih Trade Credis, Inflaion in Bulk Release Rule Inernaional Journal of Invenory Conrol & Managemen, ISSN 0975-379, Vol. 5, No. -, 05, pp. 44-430 5. Wee, H.M., Law, S.T. (999). Economic producion lo size for deerioraing iems aking accoun of ime value of money, Compuers and Operaional Research; 6: 545-558. 6. Weiss, H.J. (98). Economic order quaniy models wih nonlinear holding cos, European Joumal of Operaional Research; 9: 56-60. 7. Yang, G.K. (007). Noe on sensiiviy analysis of invenory model wih parial backorders, European Journal of Operaional Research; 77: 865-87. 8. Zhou,Y.W., Lau, H.S. (998). Opimal producion lo sizing model considering he bounded learning case and shorages backordered, Journal of he Operaional Research Sociey; 49: 06. 35 Monika Vishnoi, S.R.Singh