A Probabilistic Inventory Model for Deteriorating Items with Ramp Type Demand Rate under Inflation

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1 Americn Journl of Operionl Reserch 06, 6(): 6-3 DOI: 0.593/j.jor A Probbilisic Invenory Model for Deerioring Iems wih Rmp ype Demnd Re under Inflion Sushil Kumr *, U. S. Rjpu Deprmen of Mhemics & Asronomy, Universiy of Lucknow, Lucknow, U.P. Indi Absrc In his pper we developed generl invenory model for deerioring iems wih consn deeriorion re nd rmp ype demnd under sock dependen consumpion re. Shorges re llowed nd prilly bcklogged. he bcklogging re of unsisfied demnd is ssumed s decresing funcion of wiing ime. he effec of inflion nd ime vlue of money is inroduced ino he model. he deeriorion of he produc is ssumed s probbilisic o mke he model more relisic one. he prilly bckorder re is ssumed s decline exponenil funcion of wiing ime. he purpose of our sudy is o deermine he opiml replenishmen policy for mximize he ol profi. Numericl exmples re lso given o demonsre he developed model. Keywords Probbilisic deeriorion, Pril-bcklogging, Shorge, Rmp ype demnd re, Inflion nd sock dependen consumpion re. Inroducion In rel life siuions here is relisic phenomenon of he deeriorion of goods in sorge period. Deeriorion is defined s dmge, decy or spoilge of he iems h re sored for fuure use lwys loose pr of heir vlue wih pssge of ime, so deeriorion cnno be voided in ny business orgnizion. over nd Philip [973] developed n invenory model for exponenilly decying iems. Buzco [975] developed n economic order quniy model nd firs ime he ssumed he effec of inflion in his invenory model. Donldson [979] deermine n opiml replenishmen policy for finding he compuionl soluion of he invenory model of deerioring iems. Deb nd hudhuri [987] exended he Donldson [979] model by llowing shorges. In clssicl invenory models he demnd re is eiher consn, linerly incresing, decresing, exponenilly incresing, decresing funcion of ime or sock dependen. Ler i hs been observed h in he super mrke he bove demnd pern do no precisely depic he demnd of cerin iems such s newly lunched producs, fshionble grmens, hrdwre devices, cosmeics, elecronic iems, mobiles ec. increses wih ime nd fer some ime i becomes consn. In such cses he concep of rmp-ype demnd is inroduced. Rmp-ype demnd is demnd which increses up o cerin ime nd fer cerin ime i * orresponding uhor: sushilmh4444@gmil.com (Sushil Kumr) Published online hp://journl.spub.org/jor opyrigh 06 Scienific & Acdemic Publishing. All Righs Reserved becomes consn. Du nd Pl [99] discussed he effec of inflion nd ime vlue of money in his invenory model by considering linerly ime dependen demnd re. Mndl nd Pl [998] were he firs uhors who discussed he rmp-ype demnd re in his order level invenory model for deerioring iems. hng nd Dye [999] developed n EOQ model for deerioring iems in which bcklogging re is reciprocl of liner funcion of wiing ime. Wu, K.S., Ouyng, L.Y. (000) A replenishmen policy for deerioring iems wih rmp-ype demnd re. Wu [00] formuled n EOQ model for weibull deerioring iems wih rmp-ype demnd re nd pril bcklogging. eng e l. [00] deermines n opiml replenishmen policy for deerioring iems wih ime vrying demnd re by llowing shorge. Mnn, S.K. nd houdhuri, K.S. (003) An EOQ model wih rmp-ype demnd re, ime dependen deeriorion re, uni producion cos nd shorges. Jggi e l. [006] deermine n opiml order policy for deerioring iems under inflion nd discouned csh flow pproch over finie plnning horizon. Dye e l. [006] considered n invenory model for deerioring iems wih ime vrying demnd re nd shorge. Deng e l. (007) A noe on invenory models for deerioring iems wih rmp-ype demnd re. Pnd e l. (008) Opiml replenishmen policy for perishble sesonl producs in seson wih rmp-ype ime dependen demnd. Kun-Shn e l. (008) developed reiler s opiml ordering policy for deerioring iems wih rmp-ype demnd under sock-dependen consumpion re. hung nd Wee [008] formuled policy pricing inegred producion invenory model for deerioring iems by considering imperfec producion, inspecion plnning nd sock dependen

2 Americn Journl of Operionl Reserch 06, 6(): demnd re. Skouri e l. [009] developed n invenory model for weibull deerioring iems wih rmp-ype demnd re nd shorge. rdens Brron [009] proposed n invenory model wih rework process single sge mnufcuring sysem wih plnned bckorders. Jin, S. nd Kumr, M. (00) formuled n EOQ model for hree prmeer weibull deerioring iems wih rmp-ype demnd nd shorges. Sn [00] developed muli iem EOQ model for boh deerioring nd melioring iems. Srkr e l. [00] described producion policy o find ou n opiml sfey sock, producion lo size nd relibiliy prmeers. Sn [00] formuled lo size invenory model for deerioring iems wih ime vrying deeriorion re nd pril bcklogging. hng,.. (0) developed n invenory model for weibull deerioring iems wih rmp-ype demnd re nd pril bcklogging. Wee e l. [0] deermines n opiml replenishmen cos of life nlysis of deerioring green producs. rdens- Brron [0] considered n invenory model wih shorge nd find ou n pproxime soluion by using bsic lgebric procedure. Srkr nd Moon [0] exended he economic producion quniy model in n imperfec producion sysem. Se e l. [0] formuled wo wrehouse invenory model for ime vrying deerioring iems nd sock dependen demnd re. Ahmd e l. (03) developed n invenory model wih rmp-ype demnd re, pril bcklogging nd generl deeriorion re. rdens e l. [03] deermines n improved soluion procedure of he replenishmen policy for he EMQ model wih rework nd muliple shipmens. Srkr nd Mjumder [03] developed n inegred vendor buyer supply chin invenory model wih he reducion of vendors se up cos. rdens e l. [03] derived wo esy nd improved lgorihms o deermine joinly he replenishmen lo size nd number of shipmens for n EPQ model. Krmkr, B. nd houdhuri, K.D. (04) developed n invenory model for deerioring iems wih rmp-ype demnd re, pril bcklogging nd ime vrying holding cos. Srkr e l. [04] developed n invenory model wih rde credi policy nd vrible deeriorion re for fixed life ime producs. Srkr e l. [04] developed n EMQ model wih price nd ime dependen demnd under inflion. Srkr e l. [05] derived coninuous review mnufcuring invenory model wih se up cos reducion, quliy improvemen nd service level consrin. Kumr e l. (05) developed wo wrehouse prilly bcklogging invenory model for deerioring iems wih rmp-ype demnd re. he bles nd show h he vriion of ol profi when wih respec o he chnge in deeriorion prmeerθ nd inflion prmeer ρ, he bles 3 nd 4 show h he vriion of ol profi when wih respec o he chnge in deeriorion prmeer θ nd inflion prmeer ρ.he figures I nd II re correspond o he developed model in wo cses nd. he figures 3 nd 4 show h he vriion of ol profi when wih respec o he chnge in deeriorion prmeer θ nd inflion prmeer ρ nd he figures 5 nd 6 show h he vriion of ol profi when wih respec o he chnge in deeriorion prmeer θ nd inflion prmeer ρ. In his pper we developed prilly bcklogging invenory model for deerioring iems wih probbilisic deeriorion re nd rmp ype demnd under sock dependen consumpion re. Shorges re llowed nd compleely bcklogged, he bcklogging re of unsisfied demnd is ssumed s funcion of wiing ime. he effec of inflion nd ime vlue of money is inroduced in he model. he rmp ype demnd is demnd which increses up o cerin ime nd fer h i becomes sble or consn. In he cse of rel ese, elecronics iems, fshionble grmens, cosmeics, hrdwre devices, food grins ec. he demnd is unknown wih ceriny while he supply is dependen on consumers need so in his pper we consider n invenory model wih deerminisic demnd. Alhough here re so mny reserch ppers reled o he rmp-ype demnd re of newly lunched producs in he super mrke. his pper dels wih he sme ype problem nd he purpose of our sudy is o provide n pproxime soluion procedure for n opiml replenishmen policy o mximizing he ol profi.. Assumpions nd Noions We consider he following ssumpions nd noions corresponding o he developed model. b [ ( ) H( )]. he rmp ype demnd re is R() = e, where is he iniil demnd re nd b consn governing exponenil demnd re nd H ( ) is he Heviside uni sep funcion of ime defined by, H ( ) = 0,. he selling re is influenced by he demnd re nd on hnd invenory sock by, R () + ki (), I () 0 S () = R ( ), I( ) 0 3. Where, k is posiive consn. 4. θ is he probbilisic deeriorion re. 5. δ is he bcklogging prmeer. 6. O is he ordering cos per order. 7. h is he holding cos per uni 8. s is he shorge cos. 9. p is he purchse cos per uni. 0. c is he los sell cos per uni.. is he selling price per uni. c

3 8 Sushil Kumr e l.: A Probbilisic Invenory Model for Deerioring Iems wih Rmp ype Demnd Re under Inflion. r is he discoun re represen he ime vlue of money. 3. i is he inflion re per uni ime. 4. ρ = r i is he discoun re minus inflion re. 5. is he prmeer of rmp-ype demnd re. 6. Iis () he invenory level ny ime in [0,]. 7. is he ime of zero invenory level. 8. is he lengh of ech ordering cycle. 9. Q is he order quniy per cycle. 0. he replenishmen re is infinie.. Shorges re llowed nd prilly bcklogged.. he led ime is zero. 3. P(,, ) is he ol profi per uni ime for model I. 4. P(,, ) is he ol profi per uni ime for model II. 3. Mhemicl Formulion Suppose n invenory sysem consiss he mximum invenory level ime =0 in he beginning of ech cycle. During he ime inervl [ 0, ] he invenory level decreses due o boh demnd nd deeriorion nd becomes zero =. he shorge srs = nd shorge inervl is he end of curren order cycle. During he shorge inervl [, ] shorges re llowed nd prilly bcklogged. he unsisfied demnd is bcklogged re of δ ( ), where is he wiing ime nd he posiive consn δ is he bcklogging prmeer. he insnneous invenory level ny ime in[0, ] is governed by he following differenil equions wih boundry condiion wih boundry condiion di = S ( ), 0 () d I(0) = I mx di = S ( ) δ ( ), () d I ( ) = 0 he soluion of hese equions is ffeced by he selling re. Now we discuss he following wo cses nd se I When hen in he inervl [0, ] he selling re R () + ki (), I () 0 S () = is defined s R (), I () 0 Figure. For cse I Using S () in equions nd b e + ki ( ), 0 b S() = e, b e, di b = [ e + ki ( )], 0 (3) d

4 Americn Journl of Operionl Reserch 06, 6(): wih boundry condiion wih boundry condiion wih boundry condiion he equion (3) cn lso be wrien s For he soluion of equion (3) he inegring fcor is I(0) = I mx di b = e δ ( ), (4) d I ( ) = 0 di b = e δ ( ), (5) d I ( ) = q di ki () e b, 0 d + = IF.. = e kd nd he soluion of equion (3) is b I *( I. F.) = ( e )* I. F. d he soluion of equion (3) is ( b θ k) ( θ + k)( b+ θ + k) 3 I= [ + ] + Imx{ ( θ + k ) }, 0 (6) For he soluion of equion (4) i cn be wrien s he soluion of equion (4) is For he soluion of equion (5) i cn be wrien s he soluion of equion (5) is b I = e δ ( ) d, [ ], I = δ (7) b I = e δ ( ) d, Using he condiion I ( ) = 0 nd from he equion (6) we obin Using δ ( + b ) I = [ { + } q], (8) ( b+ θ + k) 3 Imx = [ + ( θ + k) ], (9) Imx in he equion (6) we obin he soluion of equion (3) ( b θ k) ( θ k)( b θ k) ( b θ k) I = [ ( θ + k) ( θ + k) ] [ + 3 ( θ + k )( b + θ + k ) 3 ], 0 (6A) When = hen from he equions (7) nd (8)

5 0 Sushil Kumr e l.: A Probbilisic Invenory Model for Deerioring Iems wih Rmp ype Demnd Re under Inflion δ 3 q = [ + + b + b b ], (0) Using he vlue of q in equion (8) we obin he soluion of equion (5) he mximum order quniy Q is Q= Imx q δ 3 I = [ + + b + b + b b ], (8A) 3 3 Q= [ + δ δ + ( b+ θ + k ) δ + δ bδ ( θ + k) bδ + bδ ] () he ordering cos per cycle is he holding cos per cycle is he shorge cos per cycle is he purchse cos per cycle is he los sles cos per cycle is ρ O = o e d 0 ρ O = o[ ], () ρ H = h I() e d 0 h 3 H = [3 + ( b+ θ + k ρ) ], (3) 6 ρ S = s[{ I ( ) + I ( )} e d] δ s S = [ ], (4) 6 ρ P = p Qe d 0 p 3 P = [ + δ δ + ( b + θ + k) δ + δ ρ ], (5) b ρ b ρ LS = c [ e e { δ( )} d + e e { δ( )} d] c 6 he sles revenue per cycle is 3 LS = [6 6 3( b + δ ρ) 3( δ + ρ ) + 6b + 6 δ + b( ρ δ ) + 3bδ + ( bρ bδ + δρ) + 3 b( δ + ρ ) + δρ + 3 δ ( b ρ) ], (6) 3 3 ρ ρ ρ SR = c[ S() e d + S() δ( ) e d + S() δ( ) e d] 0

6 Americn Journl of Operionl Reserch 06, 6(): 6-3 c 6 [6 3( ) 3 3( ) 6 3 SR = + + b + δ ρ + b + δ ρ + bδ b + 3 bδ δρ + (b+ θ + k + ρ bρ + bδ δρ) ], (7) 3 3 he ol profi per uni ime under he effec of inflion nd ime vlue of money is P (,, ) = [ S ( ] R O + H + S + P + LS P c c o c c b h b c (,, ) = [ ( + ) ( ) { + + ( + + δ ρ ) + ( + δ ρ )} + { δ ( c + c ) + ρ ( o + c )} { p + δc + δc ( b + δ ρ )} 3 bc + { bδ( c + c) δs bρ } + { c(b + θ+ k + ρ bρ bδ δρ) h( b+ θ + k ρ) + δ s c( bρ bδ + δρ)} { b( c + δ c) δ 3 δ p} { ρ( c + c) + (s + 3 p)} + { bδ( c c) + δ( s p) 6 h ρ p bρ c} + + ( h + δ p) { p( b + θ + k) + δ( b ρ) c } ], (8) Now our objecive is o deermine he opiml vlue of nd for which he ol profi P(,, ) is mximum. he necessry condiion for P(,, ) o be mximum is h P(,, ) P(,, ) = 0 nd = 0 nd solving hese equions we find he opimum vlues of nd sy for which profi is mximum nd he sufficien condiion is P (,, ) (,, ) (,, ) (,, ) ( P )( P P ) ( ) 0 ( ) 0 nd P(,, ) [( c c ) { c ( b ) h c ( b )} { p c c ( b = δ ρ + + δ ρ + δ + δ + δ ρ} + { c (b+ θ + k + ρ bρ + bδ δρ) h( b+ θ + k ρ) + δ s h ρ p c( bρ + δρ bδ )} + + h ( + δ p) { p( b+ θ + k) + δc ( b ρ )} ], (9) P (,, ) = [ c ( + b + δ ρ) + c ( b + δ ρ) h + { c ( b + θ + k + ρ b ρ + b δ δρ) h ( b+ θ + k ρ) + δ s c ( bρ + δρ bδ )} ], (0) ( P(,, )) = [ ( o + c ) + { δ ( c + c ) + ρ ( o + c )} { p + δc + δc ( b + δ ρ )} δ bc { b( c + δc) δ p} { ρ( c + c) + (s + 3 p)} + { bδ( c c)

7 Sushil Kumr e l.: A Probbilisic Invenory Model for Deerioring Iems wih Rmp ype Demnd Re under Inflion + δ( s p) bρ c} + ρp + ( h + δ p) { p( b + θ + k) + δ( b ρ) c} ] [ ( c + c) ( o + c) + { c( + b + δ ρ) h + c ( b + δ ρ)} + { δ( c + c ) + ρ( o )} { ( )} + c p + δc + δc b + δ ρ 3 bc + { bδ( c + c) δs bρ } + { c(b + θ+ k + ρ bρ bδ δρ) h( b+ θ + k ρ) + δ s c( bρ bδ + δρ)} { b( c + δ c) δ 3 δ p} { ρ( c + c) + (s + 3 p)} + { bδ( c c) + δ( s p) 6 h ρ p bρ c} + + ( h + δ p) { p( b + θ + k) + δ( b ρ) c } ] () ( P(,, )) δ = [{ δ ( c + c ) + ρ ( o + c )} { ρ ( c + c ) + ( s + 3 p )} + { bδ ( c c ) + δ( s p) bρ c} + ρp] [ ( o + c) + { δ( c + c) + ρ( o + c)} { p + δc + δc ( b + δ ρ )} bc { b( c + δc ) δ p } δ{ ρ( c+ c) + (s + 3 p )} + b { δ( c c) + δ( s p ) bρ c} + 3ρp + ( h + δ p ) { p ( b + θ + k) + δ( b ρ) c } ] + 3 [ ( c + c ) ( o + c ) + { c ( + b + δ ρ ) h + ( b )} c + δ ρ + { δ ( c + c ) + ρ ( o + c )} { p + δc + δc ( b + δ ρ )} bc 3 + { bδ ( c + c) δ s bρ } + { c(b+ θ + k + ρ bρ + bδ δρ) h( b+ θ + k ρ) + δ s c( bρ bδ + δρ)} { b( c + δ c) δ p} δ 3 { ρ ( c + c) + (s + 3 p)} + { b δ ( c c) + δ ( s p) b ρ c} 6 h ρ p + + ( h + δ p) { p( b + θ + k) + δ( b ρ) c} ] () P(,, ) = [ { p + δc + δc( b+ δ ρ)} + ρp + h ( + δ p) { p( b+ θ + k) + δc ( b ρ)}] [( c + c ) + { c ( + b+ δ ρ) h + c ( b+ δ ρ)} { p + δ c + δ c( b+ δ ρ)} + { c(b+ θ + k + ρ bρ + bδ δρ)

8 Americn Journl of Operionl Reserch 06, 6(): h ρ p h( b+ θ + k ρ) + δ s c ( bρ + δρ bδ )} + + h ( + δ p ) { p ( b+ θ + k) + δc( b ρ)} ] (3) se II When hen in he inervl [0, ] he selling re is defined s e b + ki ( ), 0 b S() = e + ki (), b e, R () + ki (), I () 0 S () = R (), I () 0 Figure. For cse II hen he insnneous invenory level ny ime in [0, ] Wih boundry condiion Wih boundry condiion Wih boundry condiion re governed by he following differenil equions di b + θ I = [ e + ki ( )], 0 (4) d I(0) he soluion of he equion () is = I mx di b + θ I = [ e + ki ( )], (5) d I ( ) = 0 di b = e δ ( ), (6) d I ( ) = 0 ( b θ k) ( θ + k)( b+ θ + k) 3 I= [ + ] + Imx{ ( θ + k)}, 0 (7) he soluion of he equion () is 3( θ + )( ) 3( ) ( ) [( ) k + b θ + ( )( ) k + b I = + b k b ] [( b ) θ ( θ + k)( + b ) ( θ + k) ( + b ) 3 ], (8)

9 4 Sushil Kumr e l.: A Probbilisic Invenory Model for Deerioring Iems wih Rmp ype Demnd Re under Inflion b b I = δ [ + + b b ], (9) From he coninuiy of I, () puing = in he equion (4) Using Imx ( b+ θ + k) 3( θ + k) ( θ + k) I = [ ( b+ θ + k) + ( + b) mx ( θ + k) 3( θ + k)( b+ θ + k) 3( θ + k) ( θ + k) ( θ + k) ( θ + k)( + b) + + b b( θ + k) ( θ + k)( b + θ + k) ], (30) in he equion (4), he soluion of he equion () is (4θ + 4 k b) ( b+ θ + k) 3( θ + k) I = [ ( + b) ( θ + k) ( θ + k)( b+ θ + k) 3( θ + k)( b+ θ + k) ( b + θ + k) + + 3( θ + k) 3 (3b+ θ + k + )( θ + k) (3 b+ θ + k) + ( θ + k) + + b ( θ + k)( b + θ + k) ], 0 (A) he mximum moun of demnd bcklogged per cycle is obined by puing = in he equion (6) δ q = [ b b 3 b ], + (3) he mximum order quniy is Q= Imx q ( b+ θ + k) 3( θ + k) ( θ + k) Q= [ ( b+ θ + k) + ( + b) he ordering cos per cycle is he holding cos per cycle is he shorge cos per cycle is ( θ + k) 3( θ + k) 3( θ + k)( b+ θ + k) ( θ + k) ( θ + k) ( + b)( θ + k) δ + + b b( θ + k) ( θ + k)( b+ θ + k) ] [ + b b 3 b ], (3) 0 ρ ρ H = h [ I() e d + I() e d] ρ O = o e d 0 ρ O = o[ ], (33) h 3 3 ρ 3 3 H = [ b ( θ + k ) + 3b 3( θ + k ) ( + )], (34) 6 6

10 Americn Journl of Operionl Reserch 06, 6(): ρ S = s[ I() e d] 3 3 S = δ s[ + ], (35) 3 3 he purchse cos per cycle is ρ P = p Qe d 0 p P = [ ( b+ θ + k ) 9( θ + k ) + ( θ + k ) 3( b+ θ + k) ( + b) 3( θ + k ) 3 δ( ) ρ (3 + 3 )], (36) he los sles cos per cycle is 3 3 b ρ LS = c e e { δ ( )} d c [6 6 3( ) 3( ) LS = δ + ρ δ ρ + δ + b b δρ + δρ he sles revenue per cycle is + δρ 3( b δ ρ ) 3( b δ ρ ) + 6 bδ ], (37) 0 3 ρ ρ S = c [ S() e d + S() δ ( ) e d] R b ρ SR = c[ + ( b + k + kθ + k ) + (3b + 6k + 6θ 8kθ 5kb + bρ + ρk ρ 6 8 k ) + (7ρ kθ kb k ) + (k + kb kθ k b ) ( kb kθ k 4 ρk ρ ) δ c + ] + [ ρ ρ + 3b b + 3 ρ ], (38) he ol profi per uni ime is P(, ) = [ ( c + c) ( c0 + c) { bc 7 p( b + θ + k) h} + { δ( c + c) ρ( c + c) h} + { δ( c + c) + ρc + p + ρc0} + { c( b+ k + kθ b 4kθ + k ) + bc} { δ ( c + c) + p} { p + bc} + { c( + k + θ 3 5kb bρ ρk ρ 4k 3 7 ) ) h ρ kθ kb k ( ρ b)} + c { ( δρc ( θ + kh ) ρh ρs δρc 3 5δρc 5δ p } + { } + { c( bk k kθ k b ) + bδ( c + c ) b( h + ρ c )} + { c ( kb kθ k

11 6 Sushil Kumr e l.: A Probbilisic Invenory Model for Deerioring Iems wih Rmp ype Demnd Re under Inflion 4 ρk ρ) + h( θ + k )} + { bδ( c + c) + p( b+ θ + k + δ + ρ) bρ c} + { δρ( c c) + p(3θ + 3 k δ ) δ s} { b( p + δ c) + p} + { p( θ + k + ρ + 3 δ) + δs} ], (39) Now our objecive is o deermine he opiml vlue of for which he ol profi P(,, ) is mximum. he necessry condiion for P(,, ) o be mximum is h P(,, ) P(,, ) = 0 nd = 0 nd solving hese equions we find he opimum vlues of nd sy for which profi is mximum nd he sufficien condiion is P (,, ) (,, ) (,, ) (,, ) ( P )( P P ) ( ) 0 ( ) 0 nd P = c + c + δ c + c ρ c + c h c b k k k bc c c θ + + δ + [( ) { ( ) ( ) } { ( ) } { ( ) c (7ρ kθ kb k ) h ( θ + k) ρh + p} + { δρc δ s δρc} + { c( kb + k kθ k b ) + bδ( c + c) b( h + ρ c)} + { c( kb kθ k 4 ρk ρ) h( θ k )} { δρ( c c) p(3θ 3 k δ ) δ s} {( bp + δ c ) + p} + { p( θ + k+ 3 δ + ρ) + δs} ], (40) P ρh = [ δ ( c + c) ρ( c + c) h + { c(7ρ kθ kb k ) δρc + ( θ + k) h δ( s ρc )} + { c ( bk + k kθ k b ) + bδ( c + c ) b( h + ρ c )} + { δρ( c c ) + p (3θ + 3 k δ ) δ s } ], (4) P(, ) = [ ( c + c ) + { δ ( c + c ) + ρc + p + ρc } { δ ( c + c ) + p } 0 0 { p + bc} + { bδ ( c + c) + p( b + θ + k + δ + ρ) bρ c} + { δρ( c c) + p (3θ + 3 k δ) δs} bp { ( + δ c) + p} + { p ( θ + k+ ρ + 3 δ) + δs} ] [ ( c + c) ( c0 + c) { bc 7 p( b + θ + k) h} + { δ( c + c) ρ( c + c) h} + { δ( c + c) + ρc + p + ρc0} + { c( b+ k b 4kθ + kθ + k ) + bc} { δ ( c + c) + p} { p + bc} + { c( + k + θ 3 5kb bρ ρk ρ 4k 3 7 ) ) h ρ kθ kb k ( ρ b)} + c { (

12 Americn Journl of Operionl Reserch 06, 6(): δρc ( θ + kh ) ρh ρs δρc 3 5δρc 5δ p } + { } + { c( bk k kθ k b ) + bδ( c + c ) b( h + ρ c )} + { c ( kb kθ k 4 ρk ρ) + h( θ + k )} + { bδ( c + c) + p( b+ θ + k + δ + ρ) bρ c} + { δρ( c c) + p(3θ + 3 k δ ) δ s} { b( p + δ c) + p} + { p( θ + k + ρ + 3 δ) + δs} ] (4) P(, ) = [{ δ ( c + c ) + ρc + p + ρ c0 } { p + bc } + { bδ ( c + c ) + p ( b + θ + k + δ + ρ) bρ c} + { p( θ + k + ρ + 3 δ) + δs} ] [ ( c 0 + c) + { δ( c + c ) + ρc + p + ρc } { δ ( c + c ) + p } { p + bc } 0 + b { δ ( c+ c) + p ( b+ θ + k+ δ + ρ) bρ c} + { δρ( c c) + p (3θ + 3 k δ) δs } { b( p + δ c ) + p } + { p ( θ + k + ρ + 3 δ) + δs } ] + 3 [ ( c + c ) ( c0 + c ) { bc 7 p ( b + θ + k ) h } + { δ( c + c) ρ( c + c) h} + { δ( c + c) + ρc + p + ρc0} + { c( b+ k b 4kθ + kθ + k ) + bc} { δ ( c + c) + p} { p + bc} + { c( + k + θ 3 P 5kb bρ ρk ρ 4k 3 7 ) ) h ρ kθ kb k ( ρ b)} + c { ( δρc ( θ + kh ) ρh ρs δρc 3 5δρc 5δ p } + { } + { c( bk k kθ k b ) + bδ( c + c ) b( h + ρ c )} + { c ( kb kθ k 4 ρk ρ) + h( θ + k )} + { bδ( c + c) + p( b+ θ + k + δ + ρ) bρ c} + { δρ( c c) + p(3θ + 3 k δ ) δ s} { b( p + δ c) + p} + { p( θ + k + ρ + 3 δ) + δs} ] (43) = [ { δ ( c + c) + p} + { δρ( c c) + p(3θ + 3 k δ ) δ s} { b( p + δ c) + p} + { p ( θ + k + 3 δ + ρ) + δs } ] [( c + c ) + { δ( c + c ) ρ( c + c ) h } c (7 ρ kθ kb k ) θ δ δρ + { c ( b + k + k + k ) + bc } { ( c + c ) + p } + { c

13 8 Sushil Kumr e l.: A Probbilisic Invenory Model for Deerioring Iems wih Rmp ype Demnd Re under Inflion h( θ + k) ρh δ s δρc} + { c( kb + k kθ k b ) + bδ ( c + c) b( h + ρ c)} + { c( kb kθ k 4 ρk ρ) + h( θ + k )} + { δρ( c c) + p(3θ + 3 k δ) δs} { b( p + δ c) + p} + { p( θ + k + 3 δ + ρ) + δs} ] (44) 4. Numericl Prmeers Le us consider he following prmeers in he pproprie unis nd wo differen vlues of he rmp-ype demnd prmeer = 0, b = 0.5, o = $5 / order, h = $5 / uni / uni ime, s = $8 / uni, p = $0 / uni, c = $4 / uni, c = $6 / uni, θ = 0.05, δ =, k = 3, ρ = 0.3, = 0.4 Numericl exmple - when hen solving he P P equions = 0 nd = 0 we find he opimum vlue of nd for differen vlues of θ nd ρ. Since he sign of ol profi comes ou o be negive nd P he vlue of 0 so he ol profi is minimum. As we increse he inflion prmeer ρ hen he ol profi increses. Figure 3. Wih respec o θ ble. Vriion in ol profi wih respec o he chnge in deeriorion prmeer θ θ P Since he sign of ol profi comes ou o be negive nd P he vlue of 0 so he ol profi is minimum. As we increse he deeriorion prmeer θ hen he ol profi decreses. ble. Vriion in ol profi wih respec o he chnge in inflion prmeer ρ ρ P Figure 4. Wih respec o ρ Numericl exmple - when hen solving he P P equions = 0 nd = 0 we find he opimum vlue of nd for differen vlues of θ nd ρ. ble 3. Vriion in ol profi wih respec o he chnge in deeriorion prmeer θ θ P As we increse he deeriorion prmeer θ hen he vlue of he ol profi increses.

14 Americn Journl of Operionl Reserch 06, 6(): ble 4. Vriion in ol profi wih respec o he chnge in inflion prmeer ρ ρ P As we increse he inflion prmeer ρ hen he vlue of he ol profi increses. he ol profi is mximum. hus when hen he deeriorion prmeer θ nd he inflion prmeer ρ become more sensiive in comprison of deeriorion prmeer θ nd inflion prmeer ρ in cse of becuse in he cse I he shorge inervl is greer hn he shorge inervl in cse II. Furher his model cn be generlized by considering he fuzzy ype demnd re. REFERENES [] over, R.P. nd Philip, G.. (973) An EOQ model for Weibull deerioring iems. AIIE rnscions, 5, hp://dx.doi.org/0.080/ [] Buzco, J.A. (975) Economic order quniy model for deerioring iems under inflion. Operion Reserch Qurerly, 6, hp://dx.doi.org/0.057/jors Figure 5. Wih respec o θ [3] Mcdonld, J.J. (979) Invenory replenishmen policy for compuionl soluions. Journl of Operion Reserch Sociey, 30, hp://dx.doi.org/0.307/ [4] Deb, M. nd hudhuri, K.S. (987) A noe on he Heurisic invenory model for he replenishmen of rended invenories considering shorges. Journl of Operion Reserch Sociey, 38, [5] D,.K. nd Pl, A.K. (99) Effec of inflion nd ime vlue of money on n invenory model wih liner ime dependen demnd re nd shorges. Europen Journl of Operion Reserch, 5, hp://dx.doi.org/0.06/ (9) Figure 6. Wih respec o ρ 5. Sensiiviy Anlysis hus from he bles,, 3 nd 4 we see h he prmeers θ nd ρ re more sensiive in cse of in comprison of becuse in he cse I shorge inervl is greer hn he shorge inervl in cse II nd so he ol profi is minimum in cse I nd he ol profi is mximum in cse II. 6. onclusions In his pper we developed probbilisic invenory model for deerioring iems wih rmp-ype demnd re under he effec of inflion. Shorges re llowed nd prilly bcklogged. We sudied he bove developed model in wo cses nd. In cse I from he bles nd we see h s we increse he prmeers θ nd ρ hen he ol profi is minimum. In cse II from he bles 3 nd 4 we see h s we increse he prmeers θ nd ρ hen [6] Mndl, B. nd Pl, A.K. (998) An order level invenory sysem wih rmp-ype demnd re for deerioring iems. Journl of Inerdisciplinry Mhemics,, hp://dx.doi.org/0.080/ [7] hng, H.J. nd Dye,.Y. (999) An EOQ model for deerioring iems wih ime vrying demnd nd pril bcklogging. Journl of Operion Reserch Sociey, 50, hp://dx.doi.org/0.307/ [8] Wu, K.S. nd Ouyng, L.Y. (000) A replenishmen policy for deerioring iems wih rmp-ype demnd re. Proceeding of he Nionl Science ouncil, Republic of hin, 4(4), [9] Wu, K.S. (00) An EOQ model for Weibull deerioring iems wih rmp-ype demnd re nd pril bcklogging. Producion Plnning nd onrol,, hp://dx.doi.org/0.080/ [0] eng, J.., hng, H.J. Dye,.Y. nd Hung,.H. (00) An opiml replenishmen policy for deerioring iems wih rmp-ype demnd nd pril bcklogging. Operion Reserch Leers, 30, hp://dx.doi.org/ 0.06/S (0) [] Mnn, S.K. nd houdhuri, K.S. (003) An EOQ model wih rmp-ype demnd re, ime dependen deeriorion re, uni producion cos nd shorges. Inernionl Journl of Sysems Science, 34(3),

15 30 Sushil Kumr e l.: A Probbilisic Invenory Model for Deerioring Iems wih Rmp ype Demnd Re under Inflion [] Giri, B.., Jln, A.K. nd hudhuri, K.S. (003) Economic order quniy model for Weibull deerioring iems wih rmp-ype demnd nd shorge. Inernionl Journl of Sysem Science, 34, hp://dx.doi.org/0.080/ [3] Jggi,.K., Aggrwl, K.K. nd Goel, S.K. (006) Opiml order policy for deerioring iems under inflion induced Demnd. Inernionl Journl of Producion Economic, 03, hp://dx.doi.org/0.06/j.ijpe [4] Dye,.Y., hng, H.J. nd eng, J.. (006) A deerioring invenory model wih ime vrying demnd nd shorge dependen pril bcklogging. Europen Journl of Operion Reserch, 7, hp://dx.doi.org/0.06 /j.ejor [5] Deng, P.S., Lin, RH-J., hu, P. (007) A noe on invenory models for deerioring iems wih rmp-ype demnd re. Europen Journl of Operion Reserch, 78, -0. [6] Pnd, S., Senpi, S. nd Bsu, M. (008) Opiml replenishmen policy for perishble sesonl producs in seson wih rmp-ype ime dependen demnd. ompuers nd Indusril Engineering, 54, [7] Kun-Shn, W., Ling-Yuh, O. nd hih-e, Y. (008) A reiler s opiml ordering policy for deerioring iems wih rmp-ype demnd under sock-dependen consumpion re. Informion nd Mngemen Science, 9(), [8] hung,.j. nd Wee, H.M. (008) An inegred producion invenory model for deerioring iems wih pricing policy considering imperfec producion, inspecion plnning nd wrrny period nd sock level dependen demnd. Inernionl Journl of Sysem Science, 39, hp://dx.doi.org/0.080/ [9] Skouri, K., Konsnrs, I., Ppchrisos, S. nd Gns, I. (009) Invenory model for Weibull deerioring iems wih rmp-ype demnd re nd pril bcklogging. Europen Journl of Operion Reserch, 9, hp://dx.doi.org/0.06/j.ejor [0] rdens-brron, L.E. (009) Economic producion quniy model wih rework process single-sge mnufcuring sysem wih plnned bckorders. ompuer nd Indusril Engineering, 57, hp://dx.doi.org/0.06/j.cie [] Jin, S. nd Kumr, M. (00) proposed n EOQ model for hree prmeer weibull deerioring iems wih rmp-ype demnd re nd shorges. Yugoslv Journl of Operion Reserch, 0(), [] Sn, S.S. (00) Demnd influenced by enerprises iniiive muli iem EOQ model for deerioring melioring iems. Mhemicl nd ompuer Modeling, 5, hp://dx.doi.org/ 0.06/j.mcm [3] Srkr, B., Sn, S. nd hudhuri, K. (00) Opiml relibiliy, producion lo size nd sfey sock. An economic mnufcuring quniy model. Inernionl Journl of Mngemen Science nd Engineering Mngemen, 5, 9-0. [4] Sn, S.S. (00) Opiml selling price nd lo size invenory model wih ime vrying deeriorion nd pril bcklogging. Applied Mhemics nd ompuion, 7, [5] hng,.. (0) developed n invenory model for weibull deerioring iems wih rmp-ype demnd re nd pril bcklogging. msui Oxford Journl of Informion nd Mhemicl Sciences, 7(), [6] Wee, H.M., Lee, M.., Yu, J..P. nd Wng,.E. (0) An opiml replenishmen policy for deerioring green produc, lifecycle cosing nlysis. Inernionl Journl of Producion Economics, 33, hp://dx.doi.org/0.0 6/j.ijpe [7] rdens-brron, L.E. (0) An economic producion quniy (EPQ) wih shorge derived lgebriclly. Inernionl Journl of Producion Economics, 70, hp://dx.doi.org/0.06/s (00) [8] Srkr, B. nd Moon, I. (0) An equivlen producion quniy (EPQ) model wih inflion in n imperfec producion sysem. Applied Mhemics nd ompuion, 7, hp://dx.doi.org/0.06/j.mc [9] Se, B.K., Srkr, B. nd Goswmi, A. (0) A wo wrehouse invenory model wih incresing demnd nd ime vrying deeriorion. Scieni Irnic. E, 9, hp://dx.doi.org/0.06/j.scien [30] Ahmd, M.A., Al-khn,.A., Benkherouf, L. (03) Invenory models wih rmp-ype demnd re, pril bcklogging nd generl deeriorion re. Applied Mhemics nd ompuion, 9, [3] rdens-brron, L.E., Srkr, B. nd revino Grz, G. (03) An improved soluion o he replenishmen policy for he EMQ model wih rework nd muliple shipmens. Applied Mhemicl Modeling, 37, hp://dx.doi.org/0. 06/j.pm [3] Srkr, B. nd Mjumder, A. (03) Inegred vendor buyer supply chin model wih vendors seup cos reducion. Applied Mhemics nd ompuion, 4, hp://dx.doi.org/0.06/j.mc [33] rdens-brron, L.E., Srkr, B. nd revindo-grz, G. (03) Esy nd improved lgorihms o join deerminion of he replenishmen lo size nd number of shipmens for n EPQ model wih rework. Mhemicl nd ompuionl Algorihms, 8, [34] Krmkr, B. nd houdhury, K.B. (04) proposed n invenory model for deerioring iems wih rmp-ype demnd re, pril bcklogging nd ime vrying holding cos. Yugoslv Journl of Operion Reserch, 4(), [35] Srkr, B., Sren, S. nd rdens-brron, L.E. (04) An invenory model wih rde credi policy nd vrible deeriorion for fixed lifeime producs. Annls of Operion Reserch, hp://dx.doi.org/0.007/s [36] Srkr, B., Mndl, P. nd Srkr, S. (04) An EMQ model wih price nd ime dependen demnd under he effec of relibiliy nd inflion. Applied Mhemics nd ompuion, 3, hp://dx.doi.org/0.06/j.mc [37] Srkr, B., hudhuri, K. nd Moon, I. (05) developed disribuion free coninuous review invenory model wih service level consrin for he reducion of mnufcuring se up cos. Journl of Mnufcuring Sysem, 34, 74-8.

16 Americn Journl of Operionl Reserch 06, 6(): hp://dx.doi.org/0.06/j.msy [38] Kumr, V., Shrm, A. nd Gup,.B. (05) formuled wo wrehouse pril bcklogging invenory model for deerioring iems wih rmp-ype demnd. Innovive Sysems Design nd Engineering, 6(),