MANUFACTURER-SUPPLIER COOPERATIVE INVENTORY MODEL FOR DETERIORATING ITEM WITH TRAPEZOIDAL TYPE DEMAND

Transcription

1 Yugoslav Journal of Operaions Research 6 (6) Number, 3- DOI:.98/YJOR4S MANUFACURER-SUPPLIER COOPERAIVE INVENORY MODEL FOR DEERIORAING IEM WIH RAPEZOIDAL YPE DEMAND Narayan SINGH Deparmen of Mahemaics D.N. (P.G) College, Meeru (U.P), India narayansingh98@gmail.com Bindu VAISH Deparmen of Mahemaics D.N. (P.G) College, Meeru (U.P), India bindu58@rediffmail.com S. R. SINGH Deparmen of Mahemaics D.N. (P.G) College, Meeru (U.P), India shivrajpundir@gmail.com Received: January 4 / Acceped: June 4 Absrac: his paper invesigaes a Supply Chain Sysem for deerioraing iems in which a supplier supplies a manufacurer wih raw maerial, and he manufacurer produces he finished goods. Demand rae is assumed o be ime-sensiive in naure (rapezoidal ype), which allows hree-phase variaion in demand, and producion rae is demand dependen. Our adopion of rapezoidal ype demand reflecs a real marke demand for newly launched produc. We show ha he oal cos funcion is convex. Wih he convexiy, a simple soluion algorihm is presened o deermine he opimal order quaniy and opimal cycle ime of he oal cos funcion. Numerical examples are given and he resuls are discussed. Keywords: Supply chain, rapezoidal ype demand, Deerioraion. MSC: 9B5.

2 4 N.Singh, B.Vaish, S.R.Singh / Manufacurer-Supplier. INRODUCION here has been a growing ineres in supply chain managemen in recen years. he supply chain, which is also referred o as he logisic nework, consiss of supplier disribuion cenres and reailer oules, as well as raw maerial, work in process invenory and finished goods ha flow beween he faciliies. Quie a lo of researchers and enerprises have shown a growing ineres for efficien supply chain managemen. his is due o he rising cos of manufacuring, ransporaion, he globalizaion of marke economies and he cusomer demand for diverse producs of shor life cycles, which are all facors ha increase compeiion among companies. For reducing he coss, a common sraegy is made and hrough is coordinaion, he number of deliveries is derived o achieve a minimum overall inegraed cos. Clark and Scarf (96) were he firs auhors o consider he muli echelon supply chain in invenory research wih he assumpion of consan demand rae. In he growh and/or end sage life cycle, demand rae may well be approximaed by a linear funcion. Resh e al. (976) and Donaldson (977) were he firs who sudied a model wih linearly ime varying demand. In he mos of he papers, wo ypes of ime varying demand rae have been considered: (i) Linear posiive/ negaive rend in demand rae (ii) Exponenially increasing/decreasing demand rae (canno increase coninuously over ime). Hill (995) proposed an invenory model wih increasing demand followed by a consan demand. However, i is observed ha he demand rae of a new brand of consumer goods increases a he beginning of he season up o a cerain momen (say, μ), and hen remains o be consan for he res of he ime. For example, he demand rae is increasing during he growh sage and hen he marke grows ino a sable sage ha he demand becomes consan unil he end of he invenory cycle. he erm ramp ype is used o represen such a demand paern. herefore, a ramp ype demand rae funcion has wo differen ime segmens. In is firs segmen, he demand is an increasing funcion of ime. Bu he demand remains consan in is second ime segmen. I is obvious ha any ramp ype demand funcion has a leas one break poin μ beween wo ime segmens a which i is no differeniable. his non-differeniable break poin μ makes he analysis of he problem more complicaed. his has urged researchers o sudy invenory models wih ramp ype demand paerns. Mandal and Pal (998) exended he invenory model wih ramp ype demand for deerioraion iems and allowing shorage. Wu and Ouyang () exended he invenory model o include wo differen replenishmen policies: (a) models saring wih no shorage and (b) models saring wih shorage. Deng, Lin and Chu (7) poined ou some quesionable resuls of Mandal and Pal (998) and Wu and Ouyang () and hen resolved he similar problem by offering a rigorous and efficien mehod o derive he opimal soluion. Wu () furher invesigaed he invenory model wih ramp ype demand rae such ha he deerioraion is followed by Weibull disribuion. Giri, Jalan and Chaudhari (3) exended he ramp ype demand invenory model wih a more generalized Weibull deerioraion disribuion. Various ypes of order level invenory models for deerioraing iems wih ime dependen demand were discussed by several auhors as Manna and Chaudhari (6), Panda, Senapai and Basu (8). Afer ha, several auhors have discussed he ime dependen demand in EOQ/EPQ invenory models as well as in muli-echelon supply chain models for invenory like Goyal and Gunasekaran (995), hey observed an inegraed producion

3 N.Singh, B.Vaish, S.R.Singh / Manufacurer-Supplier 5 invenory markeing model o deermine economic producion quaniy and economic order quaniy for raw maerials in a muli-echelon producion sysem. Zhou e al. (8) addressed a wo echelon supply chain coordinaion model wih one manufacurer and one reailer where he demand for he produc a he reailer is dependen on he on-hand invenory. Skouri e al. (9) developed an invenory model wih general ramp ype demand rae, Weibull deerioraion rae and parial backlogging of unsaisfied demand. hey discussed wo cases in heir model. Firs is saring wih no shorage, and he second is saring wih shorage. Singh and Singh () discussed supply chain model wih sochasic lead ime under imprecise parially backlogging and fuzzy ramp-ype demand for expiring iems. He e al. () developed a wo echelon supply chain invenory model of deerioraing iems for manufacurers selling goods o muliple markes wih differen selling seasons. Singh e al. () discussed a ime sensiive demand, Pareo disribuion for deerioraion and backlogging under rade credi policy. Skouri e al. () analysed a supply chain models for deerioraing producs wih ramp ype demand rae under permissible delay in paymens. Recenly, aleizadeh e al. () invesigaed an invenory model for a muli-produc, muli-chance consrain, muli-buyer and single-vendor, considering uniform disribuion demand and lo size dependen lead ime parial backlogging. Singh e al () discussed shorage in an economic producion lo-size model wih rework and flexibiliy. Sarkar () exended an EOQ model for ime-varying demand, and deerioraing iems wih discouns on purchasing coss under he environmen of delay-in-paymens. Goyal e al. (3) discussed a producion policy for amelioraion/deerioraing iems wih ramp ype demand. In he following, we exend Hill s ramp ype demand rae o rapezoidal ype demand rae. his ype of demand paern is generally seen in he case of any fad or seasonal goods coming o marke. he demand rae for such iems increases wih ime up o a cerain momen hen ulimaely sabilizes and becomes consan, finally i approximaely decreases o a consan or zero, and hen he nex replenishmen cycle begins. his ype of demand rae is quie realisic and a useful invenory replenishmen policy. In his paper, we propose a wo echelon invenory model wih rapezoidal ype producion rae and demand rae for boh supplier and manufacurer, and a consan rae for deerioraion is also considered. Is sudy requires exploring he feasible ordering relaions beween he imes parameers appeared, which leads o muliple models. For each model, he opimal replenishmen policy is deermined. he necessary and he sufficien condiions of he exisence and uniqueness of he opimal soluions are also provided. An easy-o-use algorihm is proposed o find he opimal replenishmen/producion policy and he opimal order quaniy. Numerical examples are presened o demonsrae he developed model and he soluion procedure. he paper is organized as follows: he noaion and assumpions used are given in Secion. In Secion 3 and 4, models are formulaed for he supplier of raw maerial and he manufacurer, and Secions 5 is devoed o he formulaion of he inegraed supply chain models and heir derivaion of he opimal producion policy. Numerical examples highlighing he resuls obained are given in Secion 6. he paper closes wih concluding remarks in secion 7.

4 6 N.Singh, B.Vaish, S.R.Singh / Manufacurer-Supplier. ASSUMPIONS AND NOAIONS he following assumpions and noaions are considered o develop he model... Assumpions. A Single supplier, a Single manufacurer and a single iem are considered.. Replenishmen rae is infinie, hus, replenishmen is insananeous. 3. Deerioraion rae is consan, and deerioraed iems are no repaired or replaced during a given cycle. 4. he planning horizon is finie. 5. Shorages are no allowed. 6. Lead ime is assumed o be negligible. 7. Producion rae P() is demand dependen and demand rae d() is a rapezoidal ype funcion of ime given by.. Noaions C C C C C C w m w m 3w 3m f() μ d = d μ δ where f() is a posiive coninuous g() δ increasing funcion of [, μ] and g() is a coninuous decreasing funcion of ime [δ, ] and P = k d where k >, δ > μ. he consan cycle ime deerioraion rae for raw maerial deerioraion rae for manufacure s finished goods supplier s ordering cos per order cycle manufacurer s ordering and se-up cos per order cycle raw maerial per uni holding cos per uni ime manufacurer s finished goods per uni holding cos per uni ime he cos incurred from he deerioraion of one uni for raw maerial he cos incurred from he deerioraion of one uni for manufacurer Iw raw maerials invenory level a any ime, Im manufacurer s finished goods invenory level a anyime Cw Cm C oal cos per uni ime for raw maerial oal cos per uni ime for manufacurer oal sysem cos per uni ime

5 N.Singh, B.Vaish, S.R.Singh / Manufacurer-Supplier 7 3. SUPPLIER S RAW MAERIAL INVENORY SYSEM he supplier s raw maerials invenory level I w, from Fig. (a), Fig. (b) and Fig. (c) a any ime can be represened by he following differenial equaion: w P I w () wih boundary condiion I w = here are hree possible relaions beween parameers μ, δ and : (I) μ δ (II) μ δ and (III) μ δ Invenory Level Qw O Invenory Level Qw δ μ δ O μ O ime ime Fig. (a) Fig. (b) Fig. (c) Invenory Level Qw μ δ ime Case I when μ δ In his case, eq. () reduces o he following hree: w kf I w () w kd I w (3) w3 kg I w3 (4) I I I I I wih boundary condiions, and w w w w3 he soluion of eqs. () - (4) are x x x Iw ke e f xdx d e dx e g xdx x x Iw ke d e dx e g xdx w3 x w3 (5) (6) I ke e g x dx (7) he oal amoun of deerioraed iems during [, ] is

6 8 N.Singh, B.Vaish, S.R.Singh / Manufacurer-Supplier w D I P() w (8) D k e f kd e k e g w he oal invenory carried during he inerval [, ] is (9) H I I I I w w w w w3 hen, he average oal cos of raw maerial per uni ime under he condiion μ δ can be given by Cw cw cwh w cw3dw () Case II when μ δ are: In his case, eq. () reduces o he following wo: w w w kf I w kd I wih boundary condiions I I and w w x x Iw ke e f x dx d e dx x w I, heir soluions w () () (3) I kd e e dx (4) he oal amoun of deerioraed iems during [, ] is () (5) D I P k e f kd e w w he oal invenory carried during he inerval [, ] is (6) H I I I w w w w hen, he average oal cos of raw maerial per uni ime under he condiion μ δ can be given by

7 N.Singh, B.Vaish, S.R.Singh / Manufacurer-Supplier 9 Cw cw cwh w cw3dw (7) Case III when μ δ In his case, eq. () becomes: w kf I w (8) I, he soluion is wih boundary condiion w x Iw ke e f x dx (9) he oal amoun of deerioraed iems during [, ] is w3 w () () D I P k e f he oal invenory carried during he inerval [, ] is () H I I w3 w w hen, he average oal cos of raw maerial per uni ime under he condiion μ δ can be given by Cw3 cw cwh w3 cw3dw3 () 4. MANUFACURER S FINISHED GOODS INVENORY SYSEM he manufacure s invenory sysem is shown in Fig. 3(a), Fig.3 (b) and Fig. 3 (c). During ime period here is an invenory build up, and hence deerioraion becomes effecive. A he = he producion sops and he invenory level increases o is maximum MI m. here is no producion during ime period( ) and invenory level decreases due o demand and deerioraion and becomes zero a =. he manufacurer s finished goods invenory level I m, from Fig. 3 (a), Fig. 3 (b) and Fig. 3 (c) a any ime can be represened by he following differenial equaion; m P d Im wih boundary condiion I m = and m d Im (3) (4)

8 Invenory level Invenory level Invenory level N.Singh, B.Vaish, S.R.Singh / Manufacurer-Supplier wih boundary condiion I m = here are hree possible relaions beween parameers μ, δ, and : () μ δ () μ δ and (3) μ δ Case when μ δ In his case, eq. (3) and (4) reduce o he following four: m m m3 m4 k f I (5) m k d I (6) m k g I m3 wih boundary condiions Im4 g Im4 (7) (8) Im, Im Im, Im Im3 and MI m MI m MI m O μ δ O μ δ O μ δ Fig. 3 (a) Fig. 3 (b) Fig. 3 (c) he soluion of eqs. (5) - (8) are m x I k e e f x dx (9) Im k e e f x dx d e dx x x Im3 k e e f x dx d e dx e g x dx (3) x x x (3)

9 N.Singh, B.Vaish, S.R.Singh / Manufacurer-Supplier x Im4 e e g x dx (3) he cumulaive invenory carried in he inerval. is (33) H I I I I I m m m m m3 m4 he oal amoun of deerioraed iems during. is m D P() d( ) Dm k f d g g (34) hen, he average oal cos of finished produc per uni ime under he condiion μ δ can be given by Cm cm cmh m cm3dm (35) Case when μ δ In his case, eq. (3) and (4) reduce o he following four: m m m3 m4 k f I (36) m k d I m d Im3 wih boundary condiions I m4 (37) (38) g Im4 (39) he soluions of eqs. (36) - (39) are m x Im, Im Im, Im3 Im4 and I k e e f x dx (4) Im k e e f x dx d e dx x x (4)

10 N.Singh, B.Vaish, S.R.Singh / Manufacurer-Supplier x x Im3 e d e dx e g x dx m4 x (4) I e e g x dx (43) he cumulaive invenory carried in he inerval. is (44) H I I I I I m m m m m3 m4 he oal amoun of deerioraed iems during. is m D P() d( ) Dm k f d d g (45) hen, he average oal cos of finished produc per uni ime under he condiion μ δ can be given by Cm cm cmh m cm3dm (46) Case 3 when μ δ In his case, eq. (3) and (4) reduce o he following four: m m m3 m4 k f I m f () Im m3 (47) (48) d I (49) wih boundary condiions I m4 m4 g I (5) he soluions of eqs. (47) - (5) are x Im, Im Im3, Im3 Im4 and Im k e e f x dx (5)

11 N.Singh, B.Vaish, S.R.Singh / Manufacurer-Supplier 3 Im e e f x dx d e dx e f x dx x x x x x Im3 e d e dx e g x dx m4 x (5) (53) I e e g x dx (54) he cumulaive invenory carried in he inerval, is (55) H I I I I I m3 m m m m3 m4 he oal amoun of deerioraed iems during, is m3 D P() d( ) () () (56) D k f f d g m3 hen, he average oal cos of finished produc per uni ime under he condiion μ δ can be given by Cm3 cm cmh m3 cm3dm 3 (57) 5. INEGRAED SUPPLY CHAIN MODEL I is obvious ha he resuls of secion 3 & 4 are non-inegraed. he deerminaion of he oal cos funcion requires furher examinaion of he ordering relaions beween δ,,. his work aims o minimize he annual inegraed cos C defined as C = C w + C m We rea C as defined in k, >. Fixing k, he following resuls are obained C C C where (58) w m C C C where (59) w m C C C where (6) 3 w3 m3 By seing f = a + b and g = a b 5..he opimal producion policy for he Case ( μ δ ) he resuls in he previous sub-secion lead o he following oal sysem cos funcion:

12 4 N.Singh, B.Vaish, S.R.Singh / Manufacurer-Supplier C ( ) = C w ( ) + C m ( ) he firs order condiion for a minimum of C ( ) is: dc h where d c w c m k c3w e a b b k c3m h cmdk c c m k a b a b b e ka c m 3m e cm k a b a b b I is easily verified ha < e c k a b a b b ka c (since m 3m cm c m cmd k k a b a b b e ), > c c k c e a b b k c ka c w (Because m 3w 3m 3m cmd k e cm k a b a b b cm c a b b e m ) (6) k a b and! >. So he derivaive dc ( ) d vanishes a. wih δ. which is roo of =. For his., we have d C cw c. m k c 3w a. b e b k c3m d c. a b b e m herefore, we have

13 N.Singh, B.Vaish, S.R.Singh / Manufacurer-Supplier 5 Propery. he deerioraing supply chain invenory model under he condiion δ, C ( ) obains is minimum value a =., where. = if δ <.. On he oher hand, C ( ) obains is minimum a. = δ if. < δ. herefore, he opimal order quaniy of raw maerial for supplier, Q w = I w (). * Qw k e f, and he opimal producion quaniy of finished good. * Qm P() 5..he opimal producion policy for he Case ( μ δ ) he resuls in he previous sub-secion lead o he following oal sysem cos funcion: C ( ) = C w ( ) + C m ( ) he firs order condiion for a minimum of C ( ) is dc h where d c h kd c e k c a b d d c w w m m 3 a b b a b b cm e e kd cm I is easily verified ha < (since m c a b d kd c 3m (6) a b b a b b c d e e m c d c e c a b d d c w > (because 3w m 3m a b b a b b! cm d e e ) and >. So he derivaive dc ( ) d vanishes a. wih μ. δ which is roo of =. For his., we have d C d kd c c e c d k 3 w 3w m herefore, we have

14 6 N.Singh, B.Vaish, S.R.Singh / Manufacurer-Supplier Propery. he deerioraing supply chain invenory model under he condiion μ δ, C ( ) obains is minimum value a =., where. = if μ <. < δ. On he oher hand, C ( ) obains is minimum a. = μ if. < μ and C ( ) obains is minimum a. = δ if δ <.. herefore, he opimal order quaniy of raw maerial for supplier, Q w = I w (). * Qw k e f, and he opimal producion quaniy of finished goods. * Qm P() 5.3.he opimal producion policy for he Case ( μ δ ) he resuls in he previous sub-secion lead o he following oal sysem cos funcion: C 3 ( ) = C w3 ( ) + C m3 ( ) he firs order condiion for a minimum of C 3 ( ) is: dc3 h3 where d h 3 c c w k c e e a b c m k k e a b c m k b a b b d e kc3wb c m a b b a b b d e e w 3 3m (63) I can be easily verified ha 3 < c 3 a b kc3 wb ka c3m cm w (since c a b b d e a b b e a b b d e 3 >, Because m ),

15 N.Singh, B.Vaish, S.R.Singh / Manufacurer-Supplier 7 cw c3w a b k c3m e e a b cmk k e cm k b kc3wb c m a b b d a b b a b b d e e e and 3! ( ) >. So, he derivaive dc 3 ( ) d vanishes a.3 wih.3 μ which is roo of 3 =. For his.3, we have d C3 b w d kc a b e e b kc3w a b e e k b b cm k a e c3mka ) herefore, we have Propery3. he deerioraing supply chain invenory model under he condiion μ, C 3 ( ) obains is minimum value a =.3, where 3.3 = if.3 < μ. On he oher hand, C 3 ( ) obains is minimum a.3 = μ if.3 μ. herefore, he opimal order quaniy of raw maerial for supplier, Q w = I w ().3 * Qw k e f, and he opimal producion quaniy of finished goods.3 * Qm P() hen a procedure is proposed o deermine he opimal replenishmen/producion policy: Sep. Find he global minimum of C( ), say.,.,.3 as follow Sep. Compue from (6), (6) and (63) if δ < <, μ < < δ and < < μ respecively, hen se =., =. and =.3 and compue C., C. andc 3 (.3 ) else go o sep 3. Sep3. Find he min C., C., C 3 (.3 ) value for and accordingly selec he opimal 6. NUMERICAL EXAMPLES In his secion, we provide some numerical examples o illusrae he heoreical resuls obained in he previous secions.

16 8 N.Singh, B.Vaish, S.R.Singh / Manufacurer-Supplier Example (δ < < ). he values of inpu parameers are given as follow; c w = $, c m = $5, c w = $ per uni, c m = $5 per uni, c 3w = $6 per uni, c 3m = $8 per uni, θ =., θ =.4, μ = weeks, δ = 4 weeks, = weeks, k =, f = a + b, g = a b were a = uni, b = 5 uni, a = 3 uni, b = 5 uni,. Based on he soluion procedure as he above, and wih he help of sofware (Mahemaica-7.) from (6), we have δ = <! and h = 59.4 >, hen i yields ha he opimal replenishmen or producion ime =. = 5.8 weeks and he oal minimum sysem cos C. = $ Example (μ < < δ). he values of inpu parameers are given as follow; c w = $, c m = $5, c w = $4 per uni, c m = $ per uni, c 3w = 5 per uni, c 3m = $3 per uni, θ =., θ =.4, μ = weeks, δ = 6 weeks, = weeks, k =, f = a + b, g = a b were a = uni, b = 5 uni, a = 4 uni, b = 5 uni,. Based on he soluion procedure as he above, and wih he help of sofware (Mahemaica-7.) from (6), we have μ = <! and h δ = >, hen i yields ha he opimal replenishmen or producion ime =. = 4.98 weeks and he oal minimum sysem cos C. = $ Example 3 ( < < μ). he values of inpu parameers are given as follow; c w = $, c m = $5, c w = $3 per uni, c m = $ per uni, c 3w = $5 per uni, c 3m = $6 per uni, θ =., θ =.4, μ = 4 weeks, δ = 6 weeks, = weeks, k =, f = a + b, g = a b were a = uni, b = 5 uni, a = 5 uni, b = 5 uni,. Based on he soluion procedure as he above, and wih he help of sofware (Mahemaica-7.) from (63), we have 3 = 38.8 <! and h 3 μ = >, and hen i yields ha he opimal replenishmen or producion ime =.3 =.56 weeks and he oal minimum sysem cos C 3.3 = $ CONCLUDING REMARK In his paper, a Supplier-Manufacurer Cooperaive Invenory Model for deerioraing iems wih rapezoidal ype Demand and Producion rae is sudied., where ordering relaions beween he imes parameers appeared leads o hree differen siuaions. For each siuaion, he opimal producion policy has been derived, and convexiy has also been provided. An easy o use algorihm is proposed o find he opimal producion policy and opimal producion ime, and numerical examples are sudied o illusrae he model. Our model can be used for he deerminaion of he oal sysem cos and opimal producion ime when all he paries work ogeher. his paper may be exended by using a wo-parameer Weibull disribuion deerioraion rae and under he condiion of permissible delay.

17 N.Singh, B.Vaish, S.R.Singh / Manufacurer-Supplier 9 Acknowledgemen: he auhors would like o hank he anonymous referees and ediors for heir deailed and consrucive commens and he firs auhor is very much hankful o he council of scienific and indusrial research (CSIR) New Delhi, India for he financial assisance in he form of Junior Research Fellowship (JRF). REFERENCES Cleark, A.J., and Scarf, H., Opimal policy for a muli-echelon invenory problem, Managemen Science, 6 (96) Donaldson, W.A., Invenory replenishmen policy for a linear rend in demand: an analyic soluion, Operaional Research quarerly, 8 (977) Deng, P.S., Lin, R. H. J., and Chu, P., A noe on he invenory models for deerioraing iems wih ramp ype demand rae, European Journal of Operaional Research, 78 () (7). Goyal, S.K., and Gunasekaran, A., An inegraed producion invenory markeing model for deerioraing iems, Compuers and Indusrial Engineering, 8 (995) Giri, B.C., Jalan, A.K. and Chandhuri, K.S., Economic order quaniy model wih weibull deerioraion disribuion, shorage and ramp ype demand, Inernaional Journal of Sysem Science, 34 (4) (3) Goyal, S.K., Singh, S.R. and Dem, H., Producion policy for amelioraing/deerioraing iems wih ramp ype demand, Inernaional Journal of Procuremen Managemen, 6 (4) (3) Hill, R.M., Invenory model for increasing demand followed level demand, Journal of he Operaional Research Sociey, 46 (995) He, Y., Wang, S.Y., and Lai, K.K., An opimal producion-invenory model for deerioraing iems wih muliple-marke demand, European Journal of Operaional Research, 3 (), Mandal, B., and Pal, A.K., Order level invenory sysem wih ramp ype demand rae for deerioraing iems, Journal of Inerdisciplinary Mahemaics, (998) Manna, S.K., and Chandhuri, K.S., An EOQ model wih ramp ype demand, ime dependen deerioraion rae, uni producion cos and shorages, European Journal of Operaional Research, 7 (6) Panda, S., Senapai, S., and Basu, M., Opimal replenishmen policy for perishable seasonal producs in a season wih ramp ype ime dependen demand, Compuer Indusrial Engineering, 54 (8)3-34. Resh, M., Friedman, M., and Barbosa, L.C., On a general soluion of he deerminisic lo size problem wih ime proporional demand, Operaions Research, 4 (976) Skouri, K., Konsanaras, I., Papachrisos, S., and Ganas, I., Invenory models wih ramp ype demand rae parial backlogging and weibull deerioraion rae, European Journal of Operaional Research, 9 (9) Singh, S.R., and Singh, C., Supply chain model wih sochasic lead ime under imprecise parially backlogging and fuzzy ramp-ype demand for expiring iems, Inernaional Journal of Operaional Research, 8 (4) (), 5-5. Singh, N., Vaish, B., and Singh, S.R., An EOQ model wih Pareo disribuion for deerioraion, rapezoidal ype demand and backlogging under rade credi policy, he IUP Journal of Compuaional Mahemaics, 3 (4) ()3-53. Skouri, K., Konsanaras, I., Papachrisos, S., and eng, J.., Supply chain models for deerioraing producs wih ramp ype demand rae under permissible delay in paymens, Exper Sysems wih Applicaions, 38 ()

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