Optimal Inventory Control Problem with Inflation-Dependent Demand Rate Under Stochastic Conditions

Transcription

1 sarch Journal Applid Scincs, Enginring and Tchnology 4(4: 6-5, ISSN: Maxwll Scintific Organization, Submittd: August, Accptd: Sptmbr 8, Publishd: Fbruary 5, Optimal Invntory ontrol Problm with Inflation-Dpndnt Dmand at Undr Stochastic onditions A. Mirzazadh Dpartmnt of Industrial Enginring, Islamic Azad Univrsity, Karaj Branch, Karaj, Iran Abstract: Th practical xprincs rval that th Supply hain Managmnt (SM is undr uncrtain and variabl conditions. On of th most important parts of SM is invntory systm managmnt which is inhrntly in non-rministic situation. Th many dpartmnts of organization such as warhous, markting, sal, purchasing, financial, planning, production, maintnanc and tc. ar rlvanc to th invntory problm. Sinc 975 a sris of rlatd paprs appard that considrd th ffcts of inflation on th invntory systm. Thr ar a fw works in th inflationary invntory rsarchs undr stochastic conditions, spcially with multipl stochastic paramtrs. Thrfor, a nw mathmatical modl for th optimal production for an invntory control systm is formulatd undr stochastic nvironmnt. Th dmand rat is a function of inflation and tim valu of mony whr th inflation and tim horizon i.., priod of businss, both ar random in natur. In th ral situation, som but not all customrs will wait for backloggd itms during a shortag priod, such as for fashionabl commoditis or high-tch products. Thus, th modl incorporats partial backlogging. A numrical mthod has bn usd and th numrical xampl has bn providd for valuation and validation of th thortical rsults and som spcial cass of th modl ar discussd. Th rsults show th importanc of taking into account stochastic inflation, tim horizon and dmand. Ky words: Inflation-dpndnt dmand, invntory, optimization, stochastic, supply chain managmnt INTODUTION In th past dcads, th rplnishmnt schduling problms wr typically attackd by dvloping propr mathmatical modls that considr practical factors in ral world situations, such as uncrtain conditions, physical charactristics of invntorid goods, ffcts of inflation and tim valu of mony, partial backlogging of unsatisfid dmand, tc. Invntorid goods can b broadly classifid into four mta-catgoris basd on: Obsolscnc: frs to itms that los thir valu through tim bcaus of rapid changs of tchnology or th introduction of a nw product by a comptitor. For xampl, spar parts for military aircraft ar styl goods, and thy bcom obsolt whn a rplacmnt modl is introducd. Dtrioration: frs to th damag, spoilag, drynss, vaporization, tc. of th products. For xampl, th commonly usd goods lik fruits, vgtabls, mat, foodstuffs, prfums, alcohol, gasolin, radioactiv substancs, photographic films, lctronic componnts, tc. whr rioration is usually obsrvd during thir normal storag priod. Amlioration: frs to itms whos valu or utility or quantity incras with tim. It is a practical xprinc th valu of Prsian carpt incrass by ag. Othr xampls can b win manufacturing industry and fast growing animals lik broilr, shp, pig, tc. in farming yard. Th last on rfrs no obsolscnc, rioration and amlioration. Th shlf-lif of som products can b indfinit and hnc thy would fall undr th no obsolscnc/rioration/amlioration catgory. Th invntory modls by considring th tim valu of mony hav bn causd by conomic changs and inflationary conditions. According to inflation rat, it is important to invstigat how th tim valu of mony influncs various invntory policis. Sinc 975 a sris of rlatd paprs appard that considrd th ffcts of inflation on th invntory systm. Thr ar a fw problms in th inflationary invntory systms on obsolscnc and amlioration itms which hav bn addrssd by th rsarchrs, bcaus, w will not us obsolscd itms in th futur and th amlioration products ar limitd in th ral world. For xampl, Moon and Giri (5 considrd amliorating/riorating itms with a tim-varying dmand pattrn. Anothr rsarch for amliorating itms has bn don by Sana (. Thr ar som rsarchs on invntory systm for no obsolscing, riorating and amliorating products. Buzacott (975 dalt with an conomic ordr quantity modl with inflation subjct to diffrnt typs of pricing 6

2 s. J. Appl. Sci. Eng. Tchnol., 4(4: 6-5, policis. Misra (979 dvlopd a discountd cost modl and includd intrnal (company and xtrnal (gnral conomy inflation rats for various costs associatd with an invntory systm. Sarkr and Pan (994 survyd th ffcts of inflation and th tim valu of mony on ordr quantity with finit rplnishmnt rat. Som fforts wr xtndd th prvious works to considr mor complx and ralistic assumption, such as Uthayakumar and Gtha (9, Maity (8, Vrat and Padmanabhan (99, Datta and Pal (99, Hariga (995, Hariga and Bn-Daya (996 and hung (. Thr ar svral studis of riorating invntory modls undr inflationary conditions. hung and Tsai ( prsntd an invntory modl for riorating itms with th dmand of linar trnd considring th tim-valu of mony. W and Law ( drivd a riorating invntory modl undr inflationary conditions whn th dmand rat is a linar dcrasing function of th slling pric. hn and Lin ( discussd an invntory modl for riorating itms with a normally distributd shlf lif, continuous tim-varying dmand, and shortags undr an inflationary and tim discounting nvironmnt. hang (4 stablishd a riorating EOQ modl whn th supplir offrs a prmissibl dlay to th purchasr if th ordr quantity is gratr than or qual to a prrmind quantity. Yang (6 discussd th two-warhous invntory problm for riorating itms with a constant dmand rat and shortags. Maity ( proposd an invntory modl with stock-dpndnt dmand rat and two storag facilitis undr inflation and tim valu of mony. Lo t al. (7 dvlopd an intgratd production-invntory modl with assumptions of varying rat of rioration, partial backordring, inflation, imprfct production procsss and multipl dlivris. A Two storag invntory problm with dynamic dmand and intrval valud lad-tim ovr a finit tim horizon undr inflation and tim-valu of mony considrd by Dy t al. (8. Othr fforts on inflationary invntory systms for riorating itms hav bn mad by Hsih and Dy (, Su t al. (996, hn (998, W and Law (999, Sarkr t al. (, Yang t al. (,, Liao and chn (, Balkhi (4a, b, Hou and Lin (4, Hou (6, Jaggi t al. (6, hrn t al. (8 and Sarkar and Moon (. In abov cass, it has bn implicitly assumd that th rat of inflation is known with crtainty. Yt, inflation ntrs th invntory pictur only bcaus it may hav an impact on th futur invntory costs, and th futur rat of inflation is inhrntly uncrtain and unstabl. Horowitz ( discussd an EOQ modl with a normal distribution for th inflation rat and Mirzazadh and Sarfaraz (997 prsntd multipl-itms invntory systm with a budgt constraint and th uniform distribution function for th xtrnal inflation rat for no obsolscnc, rioration and amlioration itms. Maity t al. (6 dvlopd a numrical approach to a multiobjctiv optimal invntory control problm for riorating multi-itms undr fuzzy inflation and discounting. Mirzazadh (7 compard th avrag annual cost and th discountd cost mthods in th invntory systm's modling with considring stochastic inflation. Th rsults show that thr is a ngligibl diffrnc btwn two procdurs for wid rang valus of th paramtrs. Furthrmor, Mirzazadh (8 in anothr work, proposd an invntory modl undr timvarying inflationary conditions for riorating itms. In th abov mntiond rsarch, on of ths assumptions has bn considrd for th dmand rat: onstant and wll known Tim-varying Stock dpndnt Pric-dpndnt Furthrmor, in som practical situations, th dmand rat is dpndnt to th changs in th invntory systm costs. Thrfor, in this papr, dmand is a function of th inflation rat. In th xisting litratur, inflationary invntory modls ar usually dvlopd undr th assumption of constant and wll known tim horizon. Howvr, thr ar many ral lif situations whr ths assumptions ar not valid,.g., for a sasonal product, though tim horizon is normally assumd as finit and crisp in natur, but, in vry yar it fluctuats dpnding upon th nvironmntal ffcts and it is bttr to stimat this horizon as a stochastic paramtr, which has bn considrd in this papr. In many ral situations, during a shortag priod, th longr th waiting tim is, th smallr th backlogging rat would b. For instanc, for fashionabl commoditis and high-tch products with th short product lif cycl, th willingnss for a customr to wait for backlogging is diminishing with th lngth of th waiting tim. Thrfor, th partial backlogging has bn considrd in this papr. Additionally, th rplnishmnt rat is finit and riorating itms ar survyd with considring rioration cost. ASSUMPTIONS AND NOTATIONS For th dvlopd modl, following assumptions and notations ar usd: H is takn to b th stochastic tim horizon and f(h is th pdf of H. 7

3 s. J. Appl. Sci. Eng. Tchnol., 4(4: 6-5, Shortags ar allowd. Unsatisfid dmand is backloggd, and th fraction of shortags backordrd is a diffrntiabl and dcrasing function of tim t, dnotd by *(t, whr t is th waiting tim up to th nxt rplnishmnt, # *(t# with *( = and *(4 =. Not that if *(t = (or for all t, thn shortags ar compltly backloggd (or lost. W hr assum that *(t = -"t whr ". At tim t =, c 4 and c 5, rspctivly, ar dnotd as th backlogging cost pr unit pr unit tim, if th shortag is backloggd and th unit opportunity cost du to lost sal, if th shortag is lost. All of th systm costs will b incras ovr tim horizon via stochastic inflation rat which is dnotd by i with th pdf of f(i. 4 r is th discount rat and is th discount rat nt of inflation: = r-i 5 Th dmand rat has bn shown with (i and is assumd hr a linar function of th inflation rat: i ( = a+ bi a>, b< ( 6 Dtrioration of units occurs only whn th itm is ffctivly in stock and thr is no rpair or rplnishmnt of th rioratd itms during th invntory cycl. Th constant rioration rat pr unit tim is dnotd by J(# J <. Th rioration cost pr unit of th rioratd itm is c 6 at tim zro 7 Th constant annual production (plnishmnt rat, P, is finit. Th plnishmnt rat is highr than th sum of consumption and rioration rats. 8 At tim t=, c is th ordring cost pr ordr, c is pr unit cost of th itm and c is th invntory holding cost pr unit pr unit tim 9 Lad tim is ngligibl. Also, th initial and final invntory lvl is zro. Additional notations will b introducd latr. THE MODEL FOMULATION For th dvlopd modl, at tim t =, production will b startd. Initial and final invntory lvls ar both zro. Th ral tim horizon (H has bn dividd into n qual and full cycls of lngth T. Each invntory cycl xcpt th last cycl can b dividd into four parts (A ralization of th invntory lvl in th systm is givn in Fig.. Th production starts at tim zro and Thraftr, as tim passs, th invntory lvl gradually incrasing du to production, dmand and rioration rats (s assumption 7. This fact continus till th production stops at tim 8. Thn th invntory lvl gradually dcrasing mainly du to consumption and partly du to rioration and rachs zro at tim kt and shortags occur and ar accumulatd until tim 8. During th tim intrval [kt,t], w do not hav any rioration and thrfor, shortags lvl linarly chang. At tim 8 th production starts again and shortags lvl linarly dcrass until th momnt of T. Th partially backordrd quantity is supplid to customrs during th tim intrval [8,T]. At tim T, th scond cycl starts and this bhavior continu till th nd of th (n--th cycl. In th last cycl shortags ar not allowd and th invntory cycl can b dividd into two parts. Th production stops at tim (n-t+8 and thn th invntory lvl dcrass to lad zro at th nd of th tim horizon. Lt I i (t i dnot th invntory lvl at any tim t i in th ith part of th first to (n--th cycls (i=,,,4. Th diffrntial quations dscribing th invntory lvl at any tim in th cycl ar givn as: di( t + I( t = P ( i, t λ di ( t + I ( t = ( i, t kt λ di ( t =δ( λ kt t ( i, t λ kt di 4 ( t4 = P (, i t4 T λ 4 ( ( (4 (5 In th last cycl shortags ar not allowd and th invntory lvl is govrnd by th following diffrntial quations (I i (t i dnot th invntory lvl at any tim t i in th (i-4th part of th last cycl that i = 5, 6: di5( t5 + I5( t5 = P ( i, t5 λ 5 di6( t6 + I6( t6 = ( i, t6 Tλ 6 (6 (7 Th solution of th abov diffrntial quations along with th boundary conditions I (=, I (kt-8 =, I ( =, I 4 (T-8 =, I 5 ( = and I 6 (T-8 =, ar: I I P ( i t ( t = (, t λ i ( kt t ( t ( ( = λ, t kt λ (8 (9 8

4 s. J. Appl. Sci. Eng. Tchnol., 4(4: 6-5, Fig. : Graphical rprsntation of th invntory systm I ( λ kt α i ( αt ( t = (, t λ kt α ( [ P (( i k T] λ = P (5 I ( t = ( P ( i( t T+ λ, t T λ ( Finally, solving I 5 (8 = I 6 ( for 8 w hav I P ( i t5 ( t = (, t λ ( λ = Ln P i T (( P (6 I i ( T t6 ( t ( ( = λ, t Tλ ( Th valus of 8, 8 and 8 5 can b calculatd with rspct to k and T, using th abov quations. Solving I (8 = I ( for 8 w hav: λ = Ln P i kt (( P 8 can b calculatd by solving I (8 -kt = I 4 ( (4 Lt E as th Expctd Prsnt Valu (EPV of rplnishmnt costs, EP as th (EPV of purchasing costs, EH as th EPV of carrying costs, ES as th EPV of shortags costs (backordring and lost sal and ED as th EPV of rioration costs, rspctivly. Th aild analysis is givn as follows: Th xpctd prsnt valu of ordring cost (E: Assum as th ordring cost: n = c + jt j= ( + λ (7 9

5 s. J. Appl. Sci. Eng. Tchnol., 4(4: 6-5, By rplacing Eq. (5 in (7 and taking th xpctd valu w hav: [ ( ( ] Tbi + arbira k + p E = ce + p Tn T (8 Th xpctd prsnt valu of purchasing cost (EP: Lt EP and EP as th EPV of th purchas cost in th first to (n--th cycls and in th last cycl, rspctivly. Th first purchas cost that is ordrd at tim zro quals to: c P8. Thn, th nxt purchas will occur at tim 8 and thrfor, th first cycl purchas cost is: λ [ λ + ( λ ] c P T (9 Th purchas cost for j-th cycl, (j =,,, n- is similar to th abov quation with considring th discount factor, thrfor, th EPV of th purchas cost in th first (n--th cycls is: EP Ln P i (( ( P ( i( k T + T P P = c PE kt ( P( i( k T P T( n T ( Th production quantity in th last cycl will occur at tim (n-t and quals to 8 P. Thrfor, th EPV of th purchas cost in th last cycl will b: EP c PE Ln P i T (( ( n = P T ( Th total xpctd purchas cost ovr th tim horizon would b: EP = EP + EP ( Expctd prsnt valu of holding cost (EH: onsidr EH as th EPV of th holding cost during th first to (n-- th cycls. Th EPV of th holding cost during th last cycl can b dfind with EH. In th first priod, th holding costs for j-th cycl is λ kt λ λ [ ] ( j T H = c I ( t t + I ( t t, j =,,..., n j ( Aftr som complx calculations and taking th xpctd valu w hav: { [ ] } λ λ ( p ( i ( + + ( + EH= ce λ KT KT λ( + i ( ( ( + For th last cycl, holding cost will b: [ ] T( n T (4

6 s. J. Appl. Sci. Eng. Tchnol., 4(4: 6-5, λ T λ Hn = c I t + I t 5( 5 5 6( 6 6 t5 ( n T t6 (( n T + λ (5 Aftr som complx calculations and taking th xpctd valu w hav: EH =c E λ ( P ( i[ ( + ( λ ] + [ λ ] + ( ( T ( λ ( Tλ i ([ ( ] ( + + n T (6 So, th total EPV of th holding costs ovr th tim horizon is: EH = EH + EH (7 Th xpctd prsnt valu of shortags cost (ES: ES shows th EPV of th shortags cost, including backordr and lost sals, during th first to (n--th cycls. Shortags ar not allowd in th last cycl. Thrfor, or ES = n j= ( λ [ σ( + ( σ( ][ ( ] ( λ [ σ( ( σ( ][ ( ] λ kt c t c t I t E 4 5 Tλ + c t + T I t t kt kt t4 T λ ( λ [ σ( + ( σ( ][ ( ] ( λ [ σ( ( σ( ][ ( ] λ kt c t c t I t 4 5 ES = E T λ + c t + c t I t t kt kt t4 T λ ( j T T( n T (8 (9 Th xpctd prsnt valu of riorating cost (ED: Dnot DI th quantity of invntory itms which hav bn rioratd pr cycl in th first to th (n--th cycls: λ kt λ [ ( ( ] DI = I t + I t λ ( kt λ ( P a bi( λ ( a + bi( + ( kt λ = ( Now, assum ED as th EPV of th rioration cost during th first to th (n--th cycls. Also, ED is dfind th EPV of th rioration cost during th last cycl. ED aftr taking th xpctd valu will b: n λ ( j T ( ktλ ( j + λ T [( ( λ ( ( ( λ ] ED c 6 = E Pabi a+ bi + kt j= c 6 E [ P a bi a bi kt kt T λ ( λ λ = ( ( λ ( + ( + ( λ ] T( n T (

7 s. J. Appl. Sci. Eng. Tchnol., 4(4: 6-5, For th last cycl, rioration cost will b: λ kt λ ( n T ( n+ λ T { [ ( 6( 6 6 ]} ED = c E I t + I t λ ( Tλ λ ( n T {[( ( λ ( ( ( λ ] } c6 = E P a bi + a + bi + T ( Thrfor, th total EPV of th rioration cost ovr th tim horizon is: ED = ED +ED ( onsidring th abov mntiond analysis, th EPV of th total systm costs ovr th tim horizon for a givn valu of H, is as follow: ET(n, k = E+EP+EH+ES+ED (4 Not that th tim horizon H has a p.d.f. f(h. So, th prsnt valu of xpctd total cost from n complt cycls, ETV (n,k, is givn by: ( n+ T nt n= ETV ( n, k = n = ET ( n, k f ( h dh (5 Thrfor, T ( ( ( ( [ bi + a rb i ra k + pk] M ( H EVT n, k = ce + T p E KT (( p i Ln P cp ( (( P i K T ( P (( i K T P + T P λ ( ( + ( + P i λ T + E MnT ( + c ( + T λ KT i + kt λ + ([ ( + + ( ( + λ kt t ( ( ( ( λ kt KT c 4 σ t + c5 σ t [ I( t ] + T λ t ( ( ( ( Tλ λ c 4 t + c t [ I( t ] σ σ c + 6 P ( P a bi λ ( ( ( ( + + λ KT λ λ T a bi KT λ

8 s. J. Appl. Sci. Eng. Tchnol., 4(4: 6-5, cp Ln P i T (( P λ λ [ + ] ( P ( i ( ( + ( + λ c + E T ( λ ( Tλ i ( [ ( ] ( + c6 ( T + ( P a bi( λ + ( a + bi( + ( T λ [ λ λ λ ] T MnT ( (6 which M H (- is th momnt gnrating function of H. Th solution procdur: Th problm is to rmin th optimal valus of n, th numbr of rplnishmnts to b mad during priod H, and k, th proportion of tim in any givn invntory cycl which ordrs can b filld from th xisting stock <k#. Sinc ETV(n, k is a function of a discrt variabl n and a continuous variabl k ( < k <, thrfor, for any givn n, th ncssary condition for th minimum of ETV(n,k is: detv ( n, k = dk (7 For a givn valu of n, driv k* from Eq. (7. ETV(n, k* drivs by substituting (n, k* into Eq. (6. Thn, n incras by th incrmnt of on continually and ETV(n, k* calculat again. Th abov stags rpat until th minimum ETV(n, k* b found. Th (n*, k* and ETV(n*, k* valus constitut th optimal solution and satisfy th following conditions whr Δ ETV ( n *, k * < < Δ ETV ( n *, k* Δ ETV ( n *, k* = ETV ( n * +, k* ETV( n *, k * (8 (9 To nsur convxity of th objctiv function, th drivd valus of (n*,k* must satisfy th following sufficint condition: d ETV ( n, k dk (4 Numrical xampl: Optimal rplnishmnt and shortags policy to minimiz th xpctd prsnt worth total systm cost may b obtaind by using th mthodology proposd in th prcding sctions. Th following numrical xampl is illustratd th modl. Lt Th constant annual production rat P = 5 units/yar Th company intrst rat r = $./$/yar Tabl : Th optimal solution of ETV(n,k,k for th numrical xampl N k ETV (n, k n k ETV (n, k *.668 * * Th rioration rat of th on-hand invntory pr unit tim J =.5/unit/yar Th backlogging rat *(t = -.5t Th dmand paramtric valus a= units/yar and b = - Th ordring, production, holding, backordring, lost sals and rioration costs at th bginning of th tim horizon ar: c = $/ordr; c = $8/unit; c = $/unit/yar c4 = $/unit/yar; c5 = $/unit and c6 = $/unit Th inflation rat is stochastic with Uniform distribution:i -U($.8/$/yars, $.5/$/yars. Also, th tim horizon has Normal distribution with man of yars: H - N(,.5 onsidring th abov mntiond paramtrs valus and using th numrical mthods, th problm is solvd and th rsults ar illustratd in Tabl. It can b sn that th minimum xpctd cost is 764.8$ for n* = 7 and k* =.668 (Th shortags occur aftr lapsing 6.68% of th cycl tim. Spcial cass: An attmpt has bn mad in this sction to study thr important spcial cass of th modl. as of no-shortags: If shortags ar not allowd, k= can b substitutd in xprssion (6 and th xpctd prsnt worth of th total cost, ETV(n, can b obtaind. Th minimum solution of ETV(n for discrt variabl of n must satisfy th following quation: ETV(n##ETV(n+ (4

9 s. J. Appl. Sci. Eng. Tchnol., 4(4: 6-5, whr ETV(n = ETV(n-ETV(n-. Using th rcnt inquality and considring th abov mntiond numrical data, th following solution was obtaind: n* =, ETV(n= 76.47$. It shows that th n and ETV incras in th without shortags cas. as of constant dmand rat: Assum th dmand rat in indpndnt of th inflation rat ovr th tim horizon. Th total prsnt valu of costs, ETV (n, k, can b obtaind with placing b = Eq. (6. Th optimal solution in this cas, with considring th prvious numrical xampl, is as follows: n* = 9, k* =.67, ETV (n, k = W can s that k is insnsitiv to dpndnc btwn dmand rat and inflation. as of constant and wll known tim horizon: In th cas of constant tim horizon, w can us Eq. (5 in th solution procdur for optimizing total systm cost with considring a rmind valu for th lngth of tim horizon. Lt, H = yars. Th optimal solution in this cas, with considring th prvious numrical xampl, is as follows: n* =, k* =.655, ETV (n, k = DISUSSION In rality, th valu or utility of goods dcrass ovr tim for riorating itms, which in turn suggsts smallr cycl lngth, whras prsnc of inflation in cost and its impact on dmand suggsts largr cycl lngth. In this articl, invntory modl has bn dvlopd considring both th opposit charactristics (rioration and inflation of th itms, with shortags ovr a stochastic tim horizon. Shortags ar partially backloggd. Furthrmor, in som practical situations, th dmand rat is dpndnt to th changs in th invntory systm costs. Thrfor, in this papr, dmand is a function of th inflation rat. It can b sn in th litratur rviw that th inflation rat, usually, has bn assumd constant ovr th tim horizon. Howvr, many conomic factors may also affct th futur changs of costs; such as changs in th world inflation rat, rat of invstmnt, dmand lvl, labor costs, cost of raw matrials, rats of xchang, rat of unmploymnt, productivity lvl, tax, liquidity, tc. Thrfor, th constant inflation rat assumption is not valid in th ral world situation. From th inflation point of viw, th dvlopd modl will b usful to th stochastic inflationary conditions as it givs a bttr and mor gnral invntory control systm. Th numrical xampl has bn givn to illustrat th thortical rsults and th spcial cass inclusiv as of no-shortags, as of onstant Dmand at and as of constant and wll known tim horizon hav bn discussd. Ths spcial cass ar compard with th main modl through th numrical xampl. Th study has bn conductd undr th Discountd ash Flow (DF approach. EFEENES Balkhi, Z.T., 4a. On th optimality of invntory modls with riorating itms for dmand and onhand invntory dpndnt production rat. IMA J. Manag. Math., 5: Balkhi, Z.T., 4b. An optimal solution of a gnral lot siz invntory modl with rioratd and imprfct products, taking into account inflation and tim valu of mony. Intr. J. Syst. Sci., 5: Buzacott, J.A., 975. Economic ordr quantitis with inflation. Opr. s. Quart., 6: hang,.t., 4. An EOQ modl with riorating itms undr inflation whn supplir crdits linkd to ordr quantity. Intr. J. Prod. Eco., 88: 7-6. hn, J.M., 998. An invntory modl for riorating itms with tim-proportional dmand and shortags undr inflation and tim discounting. Intr. J. Prod. Eco., 55: -. hn J.M.,.S. Lin,. An optimal rplnishmnt modl for invntory itms with normally distributd rioration. Prod. Plann. ontrol : hrn, M.S., H.L. Yang and J.T. Tng, 8. Partial backlogginginvntorylot-siz modlsfor riorating itms with fluctuating dmand undr inflation. Euro. J. Opr. s., 9: 7-4. hung, K.J.,. An algorithm for an invntory modl with invntory-lvl-dpndnt dmand rat. omp. Opr. s., : -7. hung, K.J. and S.F. Tsai,. Invntory systms for riorating itms with shortag and a linar trnd in dmand-taking account of tim valu. omp. Opr. s., 8: Datta, T.K. and A.K. Pal, 99. Effcts of inflation and tim-valu of mony on an invntory modl with linar tim-dpndnt dmand rat and shortags. Euro. J. Opr. s., 5: 6-. Dy, J.K., S.K. Mondal and M. Maiti, 8. Two storag invntory problm with dynamic dmand and intrval valud lad-tim ovr finit tim horizon undr inflation and tim-valu of mony. Euro. J. Opr. s., 85: Hariga, M.A., 995. Effcts of inflation and tim-valu of mony on an invntory modl with tim-dpndnt dmand rat and shortags. Euro. J. Opr. s., 8:

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