Application of Memory effects In an Inventory Model with Linear Demand and No shortage

Transcription

1 Interntionl Journl of Reserch in Advent echnology Vol.6 No.8 August 8 E-ISSN: -967 Appliction of Memory effects In n Inventory Model with Liner Demnd nd No shortge Rituprn Pkhir Uttm Ghosh b Susmit Srkr c Deprtment of Applied Mthemtics University of Clcutt Kolkt 79 West Bengl Indi Emil:rituprn.pkhir@gmil.com Emil:uttm_mth@yhoo.com b Emil:susmit6@.co.in c Abstrct-In this pper our purpose is to bring out the trditionl thoughts of the clssicl inventory model nd to overcome some difficulty of the trditionl clssicl inventory model. Clssicl inventory model is not ble to include memory or pst experience effects. It is known tht physicl mening of the frctionl order is n index of memory. his pper wnts to develop three type generliztion s frctionl order inventory model using frctionl clculus ccording s (i) only the rte of chnge of inventory level frctionl order (ii) demnd rte s frctionl polynomil of degree nd is the rte of chnge of the inventory level.(iii)demnd rte s frctionl polynomil of degree m my be different from the order.he frctionl models re governed by the Cputo frctionl order derivtive. Keywords: Euler gmm function; Memory dependent derivtive; Frctionl order inventory model.. INRODUCION In 695 Gottfried Leibnitz gve letter to Guillume L Hopitl with question ws it possible if the order of derivtive is some irrtionl frctionl or complex number? Drem commnds life nd this ide gve direction to open new brnch of clculus which is nmed s frctionl clculus. he erliest systemtic studies were ttributed by Leibnitz Riemnn- Liouville for long time. Due to lck of physicl interprettion use of frctionl clculus ws not relm in pplied field. But in recent trends it is going rpid developments in the pplied science becuse physicl intuition of frctionl clculus hs been found which signifies index of memory. In ordinry clculus hs so mny dvntges but it is not ble to include memory of the system but it is pproprite for frctionl clculus. Due to the bove dvntge of frctionl clculus it is rpidly used to the memory ffected system which re ffected by memory or pst experience otherwise it is not logicl. Memory mens it depends not only present stte of the system but lso pst stte of the system. Some field like biologicl system[]physics[]finncil process[45] hs gret importnce of memory effect becuse there re some endogenous nd exogenous vribles of the inventory system re very much depend on the memory or pst experience of the system. Inventory system is one of the most wrthful exmple s memory ffected system. Inventory mens stock of goods or resources. Inventory model is formulted for the business purpose to determine the optimum ordering intervl with minimized totl verge cost nd optimum level of inventories. Hrris ws the first person who ttributed to develop the EOQ model nd Wilson lso gve the ttention to mke up nlyticl result of the EOQ model. here re listed so mny reserchers like Dve nd ptel [6] McDonld[7] Silver nd Mel[8]Donldson[9] Agrwl [] gve their contribution to give more nd more relistic ide bout inventory system depending on demnd rte shortge etc. Here inventory system is considered s memory ffected system. Why? In inventory system optiml ordering intervl nd minimized totl verge cost depend on some endogenous or exogenous vrible of the inventory system. It depends on the environment of the shop or compny i.e. position of the shop or compny politicl or socil sitution. Moreover decrese or increse of profit depends on the deling of the stff of the compny or shopkeeper becuse bd behvior is not suitble to del with customer properly. On the other hnd if customers gin some poor experience 85

2 Interntionl Journl of Reserch in Advent echnology Vol.6 No.8 August 8 E-ISSN: -967 from ny compny or shop further they will not gree to purchse products from those compnies or shop inspite of its Product s populrity will decrese. Another type of memory is regrded corresponding crrying or holding cost which refers to totl cost of holding or crrying the totl inventory. here is included trnsporttion system. he delings of the trnsporttion driver hve effect on the business. If it provides poor service then further compny or shopkeeper do not wnt to tke this poor service in future. Above ll resons imply tht Inventory system is memory ffected system. Here we hve suggested frctionl order derivtive to tke into ccount memory effect. But why? It is known tht the time rte of chnge of integer orders re determined by the property of differentible functions of time only in infinitely smll neighborhood of the considered point of time. Hence there is ssumed n instnt chnges of the mrginl output when the input level chnges. herefore dynmic memory effect is not present in clssicl clculus nd it is not ble to discuss ll stte of the system i.e (present system depend on the pst).but in frctionl derivtive the rte of chnge is ffected by ll points of the considered intervl so it is memory dependent derivtive [] nd frctionl order is physiclly treted s n index of memory. It cn remove mnesi from the system s frctionl differentition involves integrtion over time from pst up to the present point of interest or present to future point of interest In this pper three type of generliztion with frctionl clculus hs been proposed from the liner type demnd rte no shortge type inventory model. hree frctionl order inventory models hve been considered s (i) only the rte of chnge of inventory level frctionl order (ii) demnd rte s frctionl polynomil of degree where is the rte of chnge of the inventory level.(iii)demnd rte s frctionl polynomil of degree m my be different from the order.here left-cputo frctionl order derivtive hs been refered.o solve the frctionl differentil eqution frctionl lplce trnsform method[4] hs been used nd to evlute crrying cost frctionl order integrtion hs been pplied. Frctionl derivtive nd integrtion ply mjor role to derive the ll frctionl model. But in the clssicl model ordinry integer order derivtive nd integrtion hve been pplied. It is nlogous to the physicl mening of speed. It is lso known to us tht derivtive of integer order is determined by the property of differentible functions of time only in infinitely smll neighborhood of the mesured point of time. Our nlysis estblishes tht Clssicl inventory system is prticulr cse of the frctionl order inventory system nd the business is very much ffected for delings nd environment nd politicl nd socil sitution nd it is lso observed tht the system is not so ffected by the service of the trnsporttion driver. In the system once criticl memory effect hs been found where business policy flls down. Rest prt of the pper is rrnged by the following wys in the section-review of frctionl clculus clssicl inventory model hs been given in section in the section-.4 frctionl order inventory model hs been served Numericl exmple hs been furnished in the section-4 Some conclusions re cited in the section-5.. REVIEW OF FRACIONAL CALCULUS. Euler Gmm Function Euler s gmm function is one of the best tools in frctionl clculus which ws proposed by the Swiss mthemticins Leonhrd Euler (77-78).he gmm function xis continuous extension from the fctoril nottion. he gmm function is denoted nd defined by the formule x x t x e t x is extended for ll rel nd complex numbers nd the gmm function stisfies some bsic x x x x x x properties Numericlly x! cn be evluted for ll positive integer vlues numericlly but evluted for rel vlues. x cn be. Riemnn-Liouville frctionl derivtive(r-l) Left Riemnn-Liouville frctionl derivtive of order is denoted nd defined s follows 854

3 Interntionl Journl of Reserch in Advent echnology Vol.6 No.8 August 8 E-ISSN: -967 ( ) d Dx f x x f ( ) d m dx where x m x m () Right Riemnn-Liouville frctionl derivtive of order is defined s follows x ( ) d Db f x x f ( ) d m dx where x m b m (b) Riemmn-Liouville frctionl derivtive of ny constnt function is not equl to zero which cretes distnce between ordinry clculus nd frctionl clculus. his definition cretes difficulty tht ction of derivtive of constnt term is not zero.. Cputo frctionl order derivtive Left Cputo frctionl derivtive [] for the function f (x) which hs continuous bounded derivtives in b is denoted nd defined s follows C x ( ) x m Dx f x x f ( ) d () m where m m m Right Cputo frctionl derivtive for the function f (x) which hs continuous nd bounded derivtives in b is defined s follows C x b ( ) m m Db f x x f ( ) d m where m m C D x A x o where A constnt..4frctionl Lplce trnsforms Method he Lplce trnsform of the function f (t) is defined s st F s L( f ( t)) e f ( t) () where s> nd s is clled the trnsform prmeter. he Lplce trnsformtion of n th order derivtive is defined s n n n nk k L f t s F s s f (b) f n t k where denotes n th derivtive of the function f with respect to t nd for non integer m it is defined in generlized form[] s n mk m m k L f t s F s s f (c) k Where m is the lrgest integer such tht ( n ) m n..5 Memory dependent derivtive Derivtive of ny function using the kernel cn be written in the following form [] x D f x K x s f s ds (4) For integer order derivtive the kernel is considered s K(x s) (x s) nd it gives the memory less derivtive. o derive the concept of memory effect using definition (4) we consider m x s K x s nd expressed in the m following form x m D f x K x s f s ds (4b) m Where f denotes the common m-th order derivtive which hs specific physicl mening. he integer order derivtive is locl property but the th order frctionl derivtive is not locl property. he totl effects of the commonly used th derivtive on the intervl [ x] describes the vrition of system in which the instntneous chnge rte depends on the pst stte is clled the memory effect. Here the rte of memory kernel decys depending on. Hence the strength of the memory is controlled by. When it becomes week in the sense of memory nd when. the system becomes memory less. he low vlue of indictes long memory of the system.. MODEL FORMULAIONS In this section the clssicl inventory model hs been introduced nlyticlly. All mthemticl models in 855

4 Interntionl Journl of Reserch in Advent echnology Vol.6 No.8 August 8 E-ISSN: -967 this pper re developed on the bsis of the following nottions nd ssumptions.. Nottions ( i) R : Demnd rte ( ii) Q: otl order quntity ( iii) U : Per unit cost ( iv) C : Inventory holding cost per unit ( v) C : Ordering cost ( vi) I( t) : Stock level or or setup cost inventory level ( vii) : Ordering intervl. ( viii) HOC : Inventory holding cost per cycle for the clssicl inventory model. ( ix) : Optiml ( x) OC : otl verge ordering intervl cost during the totl time intervl ( xi) OC: Minimized ( xii) OC : totl verge cost during Minimized totl verge the totl time intervl cost during the totl time for clssicl intervl for inventory model. frctionl order model s defined in the section-.4.. ( xiii ) : Optiml ( xiv) HOC : ordering intervl for frctionl order inventory model s defined in the section-.4.. ( xv ) : Optiml m ordering intervl for frctionl order inventory model s defined in section-.4.. ( xvii ) HOC m Inventory holding cost per cycle for frctionl order inventory model s defined in section-.4.. ( xix) : Optiml ordering intervl for frctionl order : Inventory holding cost per cycle for frctionl order inventory model s defined in the section.4.. ( xvi) OC m Minimized totl verge cost during the totl time intervl s defined in section-.4.. ( xviii) B : Bet function. ( xx) HOC : Inventory holding cost per cycle for frctionl order inventory : inventory model s defined in the section-.4.. ( xxi) OC Minimized totl verge cost during the totl time intervl s defined in section-.4.. : model s defined in the section-.4.. ( xxii) : function. gmm ble-: Used different symbols nd items for developing the models..assumptions (i) Led time is zero. (ii)ime horizon is infinite. (iii)here is no shortge. (iv)here is no deteriortion.. Clssicl inventory model During the period the inventory level depletes due to the demnd with demnd rte bt b where shortge is not llowed. Hence the ordinry differentil eqution governing the inventory level t ny time t during the period [] is given by d I t bt for t (5) with boundry conditions re I( ) nd I( ) Q. Inventory level is obtined by solving (5) with the boundry condition I( ) in the following form I b t t t (6) Using initil condition I() quntity cn be obtined s t b Q the optiml order I (7) Corresponding totl inventory holding cost over the time intervl [] is b HOC C t t b C (8) 856

5 Interntionl Journl of Reserch in Advent echnology Vol.6 No.8 August 8 E-ISSN: -967 Hence the totl cost per unit time is s OC = ((Purchsing cost ( PC )) +(holding cost( HOC ))+(set up cost ( C ))) OC b b (9) U C C herefore the totl verge cost per unit time per cycle is OCv b b U C C () hus the objective of the clssicl EOQ model cn be represented in the form s UQ HOC( ) C MinOC () Subject to We now wnt to find its optiml ordering intervl using the necessry condition (i) d( OC) d d OC provided (ii). d he necessry condition (i) gives bu C C bc () he optiml ordering intervl hs been found from the non-liner polynomil eqution. hus OC is the minimum vlue of OC ( ) which is obtined t this optiml ordering intervl..4frctionl Generliztion of the Clssicl EOQ Models Now we re going to generlize the bove clssicl inventory model considering frctionl order rte of chnge of the inventory level with different type demnd rte (i) qudrtic type polynomil (ii) frctionl order polynomil whose order is sme s frction order rte of chnge of inventory level (iii) frctionl order polynomil whose order my not be sme s frctionl order rte of chnge of inventory level. In order to study the study of memory effect on the inventory model we consider the frctionl generliztion of the clssicl inventory model or memory less inventory model with liner type demnd rte bt nd no shortge. All other ssumptions re sme s for the clssicl inventory model. Due to observe the influence of memory effects first the differentil eqution (5) cn be written using the kernel function s follows []. t di t k t t' bt' ' () in which k( t t ') is the kernel function. For Mrkov process it is equl to the delt function ( tt') nd it will generte the eqution (5). Indeed ny rbitrry function cn be succeed by sum of delt functions thereby leding to given type of time correltions. An pproprite choice in order to include in longterm memory effects cn be power-lw function which displys slow decy such tht the stte of the system t quite erly times lso contributes to the evolution of the system[]. his type of kernel gurntees the existence of scling fetures s it is often intrinsic in most nturl phenomen. hus to generte the frctionl order model we consider k t t' t t' where nd ( ) denotes the gmm function. Using the definition of frctionl derivtive [4] we cn re-write the Eqution () to the form of frctionl differentil equtions with the Cputo-type derivtive in the following form di t D bt t (4) Now pplying frctionl Cputo derivtive of order ( ) on both sides of the Eq. (4) nd using the fct the Cputo frctionl derivtive nd frctionl integrl re inverse opertors the following frctionl differentil equtions cn be obtined for the model C D t bt Model-I formultion with memory kernel 857

6 Interntionl Journl of Reserch in Advent echnology Vol.6 No.8 August 8 E-ISSN: -967 d I t C Dt I t bt (5) t with boundry conditions I( ) nd I( ) Q. he strength of memory is controlled by. When memory of the system becomes wek. Model-II In the second frctionl model both rte of chnge of the inventory level nd the demnd rte hs been generlized. Frctionl index in both the cses remins sme. We consider frctionl order inventory model where the rte of chnge of the inventory level is the frctionl type nd the demnd rte is the frctionl type with the sme index. he model cn be posed s d I t C Dt I t bt (6) t With boundry conditions re I( ) nd I( ) Q. Model-III In the third frctionl model the rte of chnge of the inventory level nd the demnd rte re both generlized but their frctionl indexes re considered different. he inventory problem will then be d I t C m Dt I t bt (7) where m t with boundry conditions I( ) nd I( ) Q. In ll the bove three generlized cses of frctionl order differentil equtions boundry conditions re used sme s of the clssicl differentil eqution. his is the right pltform for the use of left-cputo frctionl derivtive. Hence we hve been used it nd get memory effect of the system..4. Anlytic solution of model-i Here we consider the frctionl order inventory model-i which cn be solved by using Lplce trnsform method with the initil condition given in the problem. In opertor form the frctionl differentil eqution in (5) cn be represented s C Dt I t bt (8) d D where t Using Lplce trnsform nd the corresponding inversion formul on the eqution (8) we get the inventory level for this frctionl order inventory model t time t which cn be written s t bt I t Q (9) Using the boundry condition I on the eqution (9) we get the totl order quntity s b Q () nd corresponding the inventory level t time t being It he b t t th cost is denoted s order totl inventory holding HOC nd defined s HOC C D I t C t b t C B t Cb B () () (here is considered nother memory prmeter corresponding crrying cost. It is known to us tht crrying cost is referred to the totl cost for crrying or holding of the totl inventory. Hence it is trnsporttion relted cost. In the trnsporttion system trnsporttion driver my be good or bd. he effect of bd service lwys hs bd impct on the business due to the bove reson memory effect will be found on the business) 858

7 Interntionl Journl of Reserch in Advent echnology Vol.6 No.8 August 8 E-ISSN: -967 herefore the totl verge cost per unit time per cycle is OC UQ HOC C U bu C B () Cb B C Here four cses hve been studied for the chrcteriztion of this frctionl order inventory model (i)<α.<β. (ii)β. nd <α. (iii)α. nd <β. (iv) α.β.. U bu Where A B C B C Cb B D E C (A)Priml Geometric progrmming method he bove inventory model(5) is solved by priml geometric progrmming method [6]. (i)cse-: <α. <β.. In this cse the totl verge cost becomes v OC U bu C B Cb B C (4) he dul form of the bovepriml(5) model is s Mxd w w w w w4 w5 A B C D E ( ) (6) w w w w4 w5 Orthogonl condition is s w w w w w 4 5 Normlized condition is s w w w w w (8) 4 5 Priml dul reltions re (7) o evlute the minimum vlue of the totl verge v OC we propose the corresponding cost non-liner progrmming problem in the following form nd solve priml geometric progrmming method re discussed bellow A B C MinOC D E v (5) Subject to 4 A w d w B wd w C w d w D w d w E w5d w From the bove priml dul reltions we get 859

8 Interntionl Journl of Reserch in Advent echnology Vol.6 No.8 August 8 E-ISSN: -967 Aw Bw B w Cw Aw Cw 4 B w Dw Aw Ew 4 B w Dw 5 Along with Aw B w () (9) he Non-liner equtions (789) for w w w w w cn be solved to obtin 4 5 w w w w w.optiml ordering intervl nd 4 5 minimized totl verge cost cn be solved from () nd(5) nlyticlly. (ii)cse-:. nd.. herefore totl verge cost in this cse is s follows U bu C B v OC () Cb B C In this cse the generlized inventory model (5) will be in the following form v A B C D E Min OC () U bu Where A B C B C () Cb B D E C In the similr mnner s in cse (i) of model-i priml geometric progrmming lgorithm cn provide the minimized totl verge cost nd optiml ordering intervl OC. (iii)cse-: α. nd <β.. otl verge cost per unit time per cycle is OC v U bu C B Cb B C In this cse the generlized inventory model (5) becomes s v Min OC A B C D E Subject to U bu A B C B C where Cb B D E C In the similr mnner s in cse (i) of model-i priml geometric progrmming lgorithm cn provide the minimized totl verge cost nd optiml ordering intervl OC. respectively (iv) Cse-4:... In this cse the generlized inventory model (5) will be (4) 86

9 Interntionl Journl of Reserch in Advent echnology Vol.6 No.8 August 8 E-ISSN: -967 v MinOC A B C D E Subject to (5) I t t t b (9) U bu Where A B C B C Cb B D E C he expression (5) coincides to the expression () for... In the similr wy s in cse (i) of model-i priml geometric progrmming lgorithm cn provide the minimized totl verge cost OC nd optiml ordering intervl..4. Anlytic solution of model-ii Here we consider frctionl order inventory model is described by the eqution (6) where the rte of chnge of the inventory level It () is of frctionl order nd the demnd is frctionl polynomil of order.his frctionl order differentil eqution hs been solved using Lplce trnsform method. In opertor form the eqution (6) becomes s D I t bt d D where t (6) Using Lplce trnsform nd the corresponding inversion formul on the eqution (6) we get the inventory level for this frctionl order inventory model t time t which cn be written s t b t I t Q (7) Using the boundry condition I on the eqution (7) the totl order quntity is obtined s b Q (8) he inventory level becomes s he th holding cost is denoted HOC order totl frctionl inventory C D I t C HOC nd defined s t t b t C B C b B herefore the totl verge cost is denoted nd defined s OC UQ HOC C U Cb B bu C B C Now we shll consider four cses to study the behvior of this frctionl order model 86 (4) (4)

10 Interntionl Journl of Reserch in Advent echnology Vol.6 No.8 August 8 E-ISSN: -967 ( i).nd. ( ii). nd. ( iii). nd. ( iv). nd.. (i)cse-:... o find the minimum vlue of the totl verge cost v OC the corresponding non-liner progrmming problem cn be used in the following form(model-cse-(i)) nd then solve it nlyticlly. A B C MinOC D E v (4) U bu Where A B C B C Cb B D E C (ii) Cse-:.. In this cse totl verge cost is presented s follows OC UQ HOC C U bu C B (4) Cb B C herefore the frctionl order inventory model (4) in this cse will be Min A B C OC D v (44) Subject to bu U Where A B C B Cb B C D C (iii) Cse-:... In this cse totl verge cost becomes s follows OC UQ HOC C U bu C B (45) Cb B C herefore the eqution (4) reduces s v A B C D E Min OC (46) Subject to U bu Where A B C B C Cb B D E C In the similr wy s in cse (i) of model-i minimum vlue of the totl verge cost OC nd optiml ordering intervl nd nlyticlly. (iv) Cse-4:... herefore the totl verge cost is s follows 86

11 Interntionl Journl of Reserch in Advent echnology Vol.6 No.8 August 8 E-ISSN: -967 OC UQ HOC C U bu C B Cb B C herefore the eqution (4) reduces s Min A B C D v OC (47) Subject to U Where A bu B C B Cb B C D C he expression (47) coincides to the expression () for... In the similr mnner s in cse (i) of model-i priml geometric progrmming lgorithm cn provide the minimum vlue of the totl verge cost OC nd optiml ordering intervl..4.anlytic solution of model III Here we consider the frctionl order inventory model which is described by the eqution (7).he frctionl order differentil eqution (7) cn be solved by using Lplce trnsform method with the initil condition re given in the problem. In opertor form the eqution (7) becomes m D I t bt d (48) D t Using Lplce trnsform nd the corresponding inversion formul on the eqution (48) the inventory level t time t cn be obtined s m t b m t It Q m (49) whereα my be different from m <α.. where is the memory prmeter After using the boundry condition I the eqution (49) the totl order quntity for this type frctionl order inventory model cn be obtined s b m Q m m (5) th Corresponding memory dependent order totl inventory holding cost is HOC m C D I t C t ( t) bm m m t m C B m Cb m B m m herefore the totl verge cost is s OC m UQ HOC m C m U bum m C B (5) m Cb m B m m C in (5) 86

12 Interntionl Journl of Reserch in Advent echnology Vol.6 No.8 August 8 E-ISSN: -967 Now we shll consider eightcses to study this type frctionl order model s follows ( i) m... ( ii) m. nd.. ( iii). nd m.. (iv). nd m.. ( v) m. nd. ( vi) m. nd. ( vii). m. (viii m.. (i)cse-:. m... otl verge cost is s OC m UQ HOC m C m U bu m C B m (5) m Cb m B m C m o find the minimum vlue of the totl verge cost v OC we proposed priml geometric m progrmming method s model-i cse-(i). m m v A B C D E Min OC m (54) Subject to U bu A B C B m m Where m C C b m B m D E C (ii)cse-: m.... otl verge cost becomes s OC UQ HOC C U bu C B (55) Cb B C hen system (54) reduces to v A B C D E Min OC (56) Subject to bu U Where A B C B C Cb B D E C In the similr mnner s in cse (i) of model-i OC nd cn be found nlyticlly. iiicse : m..nd.. In this cse totl verge cost is s 864

13 Interntionl Journl of Reserch in Advent echnology Vol.6 No.8 August 8 E-ISSN: -967 OC m UQ HOC m C m U bu m C B m (57) m Cb m B m C m For this prmetric vlues the system (54) reduces to m m v A B C D E Min OC m (58) Subject to U bu m Where A B m C B C Cb m B m D m E C In the similr wy s in cse (i) of model-i it helps to give the results of minimized totl verge cost nd optiml ordering intervl OC m nd m respectively nlyticlly. (iv) Cse-4:. m... In this cse totl verge cost is s OC m UQ HOC C m m U bum m C Bm m Cb B C hen the system (54) will reduce to m m v A B C D E Min OC m (59) Subject to m U bu m Where A B C B C (6) Cb B m D E C Using priml geometric progrmming lgorithm we cn find minimized totl verge cost OCm nd the optiml ordering intervl cse- of model-i. m s describe in (v)cse-5: m. nd.. In this cse totl verge cost becomes s OC UQ HOC C U bu C B (6) Cb B C herefore generlized inventory model (54) reduces to v A B C D Min OC (6) Subject to 865

14 Interntionl Journl of Reserch in Advent echnology Vol.6 No.8 August 8 E-ISSN: -967 U Where A bu C B B Cb B C E C In the similr wy s in the cse (i) of model-i nlyticlly priml geometric progrmming lgorithm cn give the minimized totl verge cost OC nd optiml ordering intervl. (vi) Cse-6: m... In this cse totl verge cost is s OC UQ HOC C U bu C B 6 Cb B C herefore the generlized inventory model (54) reduces to Min A B C D E OC v (64) Subject to U bu Where A B C B C Cb B D E C In the similr mnner s in cse (i) of model-i priml geometric progrmming lgorithm gives the nlyticl results of the minimized totl verge cost OC nd optiml ordering intervl. (vii)cse-7:. nd m.. In this cse totl verge cost becomes s OC m UQ HOC m C m U bum C B m 65 m Cb m Bm C m In this cse the generlized inventory model (54) becomes s m m A B C D E v Min OC m (66) Subject to In the similr mnner s in cse (i) of model-i priml geometric progrmming lgorithm cn give the minimized totl verge cost OC nd optiml m. ordering intervl (viii)cse-8: m.. In this cse totl verge cost is s OC m UQ HOC C U bu C B Cb B C In this cse the generlized inventory model (54) becomes s Min A B C D v OC (68) Subject to 866 (67)

15 Interntionl Journl of Reserch in Advent echnology Vol.6 No.8 August 8 E-ISSN: -967 bu U A B C B Cb B C D C he expression (68) coincides to the expression () for m.. In the similr wy s in cse (i) of model-i priml geometric progrmming lgorithm cn provide the minimum vlue of the totl verge cost OC nd optiml ordering intervl. 4. NUMERICAL ILLUSRAIONS (i)o illustrte numericlly the developed clssicl nd frctionl order inventory model we consider empiricl vlues of the vrious prmeters in proper units s b 6 U 5 C 7 C.he optiml ordering intervl minimized totl verge cost of the clssicl inventory model is found.units nd 6.9 units respectively. (ii) Here we provide numericl illustrtion for the frctionl order inventory models considering sme prmeters s used in clssicl inventory model. OC (growing memory effect) ble-(): Optiml ordering intervl nd minimized totl verge costoc for. nd vries from.to. s defined in section.4.(frctionl model-i) cse-. It is cler from the tble-() tht there is criticl vlue of the memory prmeter (here it is. 6 )for which the minimized totl verge cost becomes mximum nd then grdully decreses bellow nd bove. For the bove cse low vlues of signifies lrge memory of the inventory problem. here is nother criticl vlue of memory prmeter (here it is.464 )for which minimum vlue of the totl verge cost becomes equl to the memory less minimized totl verge cost but optiml ordering intervl is different. For lrge memory effect(which is. 464 )the system needs more time to rech the minimum vlue of the totl verge cost compred to memory less system i.e. for the lrge memory effect the system will tke longer time to sell the optimum lot size compred to the memory less inventory system. Hence rte of selling decreses in lrge memory ffected system. herefore to rech the sme profit like memory less system the shopkeeper should be chnged his business policy such s ttitude of public deling environment of shop or compny product qulity etc. An observtion hs lso been observed tht there is some sitution for which the system is very much ttcked by the bd memory nd then the compny or shopkeeper hs recovered his business policy. he bove described fcts hppening in rel life inventory system but we re not ble to include in terms of clssicl model. Initilly the business strted with reputtion with mximum profit minimizing the totl verge cost. As time goes on the compny strts to lose it s reputtion due to the vrious unwnted cuses. Accordingly the compny strts to downfll of its business when downfll becomes mximum t.6.attining highest vlue t tht point the compny chnges its business policy nd tkes cre to recover its reputtion. OC

16 Interntionl Journl of Reserch in Advent echnology Vol.6 No.8 August 8 E-ISSN: (growing memory effect) ble-(b): Optiml ordering intervl nd minimized totl verge costoc for. nd vries from.to. s defined in section.4.(frctionl model-i) cse-. It is cler from the tble-(b) tht there is criticl memory vlue of the memory prmeter (here it is.5 ) for which minimized totl verge cost OC becomes mximum nd then grdully decreses below nd bove. here is lso cler tht for the memory vlue for which minimized totl verge cost becomes equl to the memory less minimized totl verge cost but optiml ordering intervl keeps smll difference between them. Hence for the lrge memory effect corresponding the system does not tke significntly more time to rech the minimum vlue of the totl verge cost compred to the memory less system. Prcticlly is the memory prmeter corresponding inventory holding cost or crrying cost. Here memory or pst experience is considered s bd or good ttitude of the shopkeeper to the trnsporttion driver. But in generl the trnsporttion driver does not rect corresponding bd ttitude of the shopkeeper. On the other hnd trnsporttion driver my be bd s service mn i.e. he my not serious for his duty. Due to the bove reson the system is ffected by the poor service of the trnsporttion but this is not effective too which is lso proved from the tble-(b). In tble-(b) OC does not show similr behvior s in tble-() without the pt (.nd. nd in this cse the ordering intervl is less i.e for. nd frctionlly vrying lesser time is required to rech the minimum vlue of the totl verge cost. OC (growing memory effect) ble-(c): Optiml ordering intervl nd minimized totl verge costoc for.4 nd vries from.to. s defined in section.4.(frctionl model-i) cse-. ( symbolizes for incresing grdully) It is cler from tble-(c) tht for the criticl memory vlue (here it is.7. 4 ) where minimized totl verge cost becomes mximum nd then grdully decreses below nd bove. When exponent of holding cost is frctionl. 4 nd nother memory prmeter vries minimized totl verge costoc is sme to the minimized totl verge cost OC.. 4 but in the optiml ordering intervl hs difference. For the lrge memory effect of the memory prmeter ( ) system tkes more time to rech the minimum vlue of the totl verge cost compred to the low memory effect. In tble-(c) the minimized totl verge cost OC shows similr behvior s in tble-() but the optiml ordering intervl does not behve similrly s in tble-(). he following tble-() nd tble-(b) hs been constructed for the model-ii where rte of chnge of inventory level is of frctionl order nd the 868

17 Interntionl Journl of Reserch in Advent echnology Vol.6 No.8 August 8 E-ISSN: -967 highest degree of the demnd polynomil is lso frctionl order. For. nd vries from.to. the model-ii nd model-i re identicl. he sme numericl results hve been displyed in the tble- (b). OC (gro wing memory effect) ble-(): Optiml ordering intervl nd minimized totl verge cost OC for. nd vries from.to. s defined in section.4.(frctionl model-ii) cse-. ( uses for incresing grdully) It is obvious from tble-() tht the minimized totl OC where. is verge cost mximum t the memory prmeter. 7 nd grdully decreses below nd bove. Since for.4 the vlue of the minimized totl verge cost OC is mximum t. 7 in model-ii but it ttins mximum t.6 in model-i. his fct hppens for considertion of frctionl polynomils in demnd rte in the model-ii. For the demurrge of inventory for long of the optiml ordering intervl there is less significnce of the optiml ordering intervl.in this cse the.. model will relistic on or fter.. OC..x x x x (growing memory effect) ble-(b): Optiml ordering intervl nd minimized totl verge cost OC for.4 nd vries from.to. s defined in section.4.(frctionl model-ii) cse-. ( uses for incresing grdully) From tble-(b) it is cler tht for lrge memory effect (here it is..4) the system tkes more time to rech the minimum vlue of the totl verge cost compred to the low memory effect( here it is..4).hence the business sty long time to rech the minimum vlue of the totl verge cost. In this cse the model will relistic on or fter.6 nd.4.hence in this cse short memory hs been observed compred to the other cse. he following tble-4() hs been constructed for model-iii where rte of chnge of inventory level is of frctionl order nd the highest degree of the demnd polynomil is lso frctionl m. For m. nd vries from.to. in frctionl model-iii (described in section.4. cse-v) coincides with model-i (cse-).he obtined numericl results re sme s given in tble-(b) OC..x x x x (growing memory effect)

18 Interntionl Journl of Reserch in Advent echnology Vol.6 No.8 August 8 E-ISSN: ble-4(): Optiml ordering intervl nd minimized totl verge cost OC m for m.4 nd vries from.to. s defined in section.4.(frctionl model-iii) cse-.( uses for incresing grdully) From the tble-4() it is cler tht for lrge memory effect (here it is. m.4) the system tkes more time to rech the minimum vlue of the totl verge cost compred to the low memory effect(here it is. m.4).hence for lrge memory effect the business sty long time to rech the minimum vlue of the totl verge cost. Here lso short memory hs worked s in the tble- () nd tble-(b). Now if we consider m. nd vries from.to. in the frctionl model-iii (described in section.4. cse-vi) coincides with model-i (cse-).he obtined numericl results re sme s given in tble- (b). 5. CONCLUSIONS he purpose of the rticle is to describe different type generliztion of n inventory model to tke into ccount memory effect vi frctionl clculus. An observtion hs been found tht for ll situtions of ll frctionl order inventory model there is criticl memory effect for which minimized totl verge cost is mximum nd then grdully decreses bellow nd bove. his criticl memory effect indictes tht in tht sitution profit of the system is low for poor memory effect or poor experiences. he memory prmeter plys mjor role to tke into ccount memory of the system compred to the nother memory prmeter. Due to consider frctionl type demnd rte short term memory hs been observed which is pproprite for newly strted business. For long memory ffected business our first frctionl model is suitble. More work with vision needs to be crried out for future spects. ACKNOWLEDGEMENS he uthors would like to thnk the reviewers nd the editor for the vluble comments nd suggestions. he work is lso supported by Prof R.N.Mukerjee Retired professor of Burdwn University. he uthors lso would like to thnk the Deprtment of Science nd echnology Government of Indi New Delhi for the finncil ssistnce under AORC Inspire fellowship Scheme towrds this reserch work. REFERENCES []M.SeedinM.Khlighi Azimi- freshin.g.r.jfrim.ausloos. Memory effects on epidemic evolution: he susceptibleinfected-recovered epidemic model. Physicl Review E [].Ds U.Ghosh S.Srkr nd S.Ds. ime independent frctionl Schrodinger eqution for generlized Mie-type potentil in higher dimension frmed with Jumrie type frctionl derivtive. Journl of Mthemticl Physics59 8. [] R.Pkhir. U.Ghosh S.Srkr. Study of Memory Effects in n Inventory Model Using Frctionl Clculus.Applied Mthemticl Sciences Vol. pges [4]V.E.rsov. V.V.rsov. Long nd short memory in economics frctionl-order difference nd differentition. IRA-Interntionl Journl of Mngement nd Socil Sciences. Vol. 5. No.. P [5]V.V.rsov V.E.rsov. Logistic mp with memory from economic model. Chos Solitons nd Frctls Pges [6]U.Dve nd L.K.Ptel. ( Si) policy inventory model for deteriorting items with time proportionl demnd.journl of the Opertionl Reserch Society. pges [7] J.J.McDonld. Inventory replenishment policies-computtionl solutions. Journl of the opertion Reserch Qurterly; pges

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