Two warehouse inventory policy with price dependent demand and deterioration under partial backlogging

Transcription

1 Decisin Science Letters 6 (27) 22 Cntents lists available at GrwingScience Decisin Science Letters hmepage: Tw warehuse inventry plicy with price dependent demand and deteriratin under partial backlgging Mhit Rastgi a*, S.R. Singh b, Prashant ushwah a and Shilpy Tayal b a Centre fr Mathematical Sciences, Banasthali University, Banasthali, Rajasthan, India b Department f Mathematics, CCS University, Meerut, India C H R O N I C L E A B S T R A C T Article histry: Received February 25, 26 Received in revised frmat: March 28, 26 Accepted August 22, 26 Available nline August eywrds: Inventry Deteriratin Warehuse Shrtages Partial backlgging In tday s era f higher cmpetitin in the business, there are many cnditins such as ffered cncessin in bulk purchasing, seasnality, higher rdering cst, etc., which frce a retailer t purchase mre quantities than needed r exceed the strage capacity. S in this situatin the retailer has t purchase an extra warehuse named as rented warehuse t stck the extra quantity. In this paper an inventry mdel fr deterirating prducts with selling price dependent rate is develped. The ccurring shrtages are assumed t be partially backlgged and cycle time is als variable. The purpse f the develpment f this mdel is t cmpute the amunt and time f rder which can ptimize the ttal average cst f the system. A slutin prcedure and numerical example are presented t illustrate the implementatin f the prpsed study. Sensitivity analysis cncerning with distinct system parameters is als presented t demnstrate the mdel. 27 Grwing Science Ltd. All rights reserved.. Intrductin Generally it is seen that the enterprisers are frced t buy mre than their strage capacities due t ffered cncessin in bulk purchasing, t avid the rdering cst, etc. In these situatins the business enterprises have t purchase a rented warehuse t stck the extra rdered quantity. Nrmally per unit hlding and deteriratin csts in rented warehuse are greater than the cst in wned warehuse wing t the additinal cst f safeguarding and hlding material s the items shuld be stred first in O.W. and nly surplus stck shuld be stred in R.W. Hence t reduce the ttal inventry cst it is necessary t finish the stck f rented warehuse first and then t cnsume the stck in wned warehuse. Als the increased capacity f wned warehuse decreases the ttal system cst. S while develping the inventry mdels the study f a tw warehuse system cannt be verlked. * Crrespnding authr. address: mhitrastgi84@gmail.cm (M. Rastgi) 27 Grwing Science Ltd. All rights reserved. di:.5267/j.dsl

2 2 Tw warehuses system was first addressed by Hartely (976) under the assumptin that R.W. causes a higher inventry hlding cst than wned warehuse. As a result, prducts in R.W. are shifted t O.W. until stck level in R.W. becmes zer and after that prducts in O.W. are cnsumed. Gswami and Chaudhuri (992) extended this mdel fr shrtages and time dependent functin f demand rate. They als applied a transprtatin cst t shift the stck frm R.W. t O.W. But this mdel was develped fr nn-deterirating prducts nly. Fr deterirating prducts, a tw warehuses inventry system was intrduced by Pakkala and Achary (992). The authrs cnsidered backlgging shrtages and finite prductin fr such mdel. Yang (24) cnsidered the rate f inflatin in tw warehuse inventry mdels and presented a mdel fr deterirating items with allwable shrtages. The demand rate is cnsidered as fixed in this mdel. Dye et al. (28) studied prblem f tw strage inventry assuming dynamic demand ver finite time hrizn. Fr deterirating items a tw-warehuse inventry mdel with shrtages under inflatin was ffered by Singh et al. (29). Jaggi and Verma (2) presented a tw-warehuse system by cnsidering inflatinary envirnment and linear trend in demand. Shrtages were permissible and backlgged cmpletely in this mdel. A tw-warehuse inventry system cnsidering time varying deteriratin was derived by Sett et al. (22). The derived mdel cnsidered in quadratic frm as demand increases and by cnsidering finite replenishment rate. Singh et al. (23) intrduced an inventry mdel f imperfect quality items with inflatin under tw limited strage capacity. Agrawal et al. (23) suggested a tw warehuse system with ramp-type demand fr deterirating prducts. T develp this mdel zer lead-time is cnsidered and shrtages are backlgged partially at an unvarying rate. Bhunia et al. (24) explred a single item, tw warehuse deterirating inventry mdel with distinct preserving facilities by cnsidering partially backlgged shrtages ver infinite planning hrizn. Fr the frmulatin f the mdel the rate f demand is cnsidered as fixed and well-knwn and lead time is als assumed as cnstant. On tw-warehuse inventry mdel, Jaggi et al. (25) studied the effect f deteriratin with imperfect quality. Authrs stated retailer was required t hire RW t decrease the lsses caused by deteriratin with imprved preserving facilities, because f nt having gd facilities in OW. In this study, rate f screening is cnsidered t be mre than the demand rate, which means n shrtages are permitted. Palanivel et al. (26) frmulated a tw warehuse system with nn instantaneusly deterirating items by cnsidering demand rate as stck-dependent. Shrtages are permissible and backlgged partially in their mdel. But the cncentratin t price sensitive demand is nt given in this mdel. During the last few decades, many inventry practitiners bradly have studied numerus facets f inventry mdeling by assuming demand rate as cnstant. Hwever in realism, demand f an item has been fr all time in a dynamic state. This catches the attentin f researchers t feel regarding the variability f demand rate. In the tday s cmpetitive market, the selling price f an item is ne f the vital factrs in chsing the prduct. The selling price factr accunts fr the fact that an increase in the selling price f the cmmdity discurages a repeat demand. Varius demand patterns have been used in the inventry mdeling such as cnstant, time dependent, stck dependent and selling price dependent. Nrmally it is seen that the selling price f the prducts is mst affecting factr f demand. Fr illustratin, firms may vigrusly regulate their prices t enhance demand and increase incmes. Therefre, the prduct s demand has t depend n the selling price, which makes the study mre realistic. In this area, Wee (997) presented a replenishment plicy fr items with a price-dependent demand and a varying rate f deteriratin. Mndal et al. (23) suggested an inventry system f amelirating items in which demand rate is price dependent. Maiti et al. (29) presented an inventry mdel with price-dependent demand fr an item in stchastic envirnment. Singh et al. (2) intrduced a sft cmputing based inventry mdel with deteriratin and price dependent demand. Jaggi et al. (24) presented credit financing fr deterirating items in a tw-warehuse envirnment with price-sensitive demand. In this mdel, shrtages were backlgged cmpletely. Tayal et al. (25) intrduced an inventry mdel fr deterirating items with seasnal prducts and an ptin f an alternative market.

3 M. Rastgi et al. / Decisin Science Letters 6 (27) 3 In this mdel the demand fr the prducts is taken as a functin f price and seasn. Sharma et al. (25) suggested a deterirating inventry mdel by intrducing price sensitive demand and shrtages. Alfares and Ghaithan (26) explred an inventry and pricing mdel by cnsidering price-dependent demand. In this mdel, hlding cst is cnsidered as dependent upn time. In many f the develped mdels the attentin is nt given t the shrtages during stck ut and if the researchers cnsidered shrtages they assumed it cmpletely backlgged r cmpletely lst. Bth f these cnditins d nt satisfy the cnditin f backlgging cmpletely. Since sme custmers cme back t cmplete their demand ccurring during stck ut and sme ther impatient custmers make their purchases frm any ther places. Dave (989) presented a lt-size inventry mdel with allwable shrtages and linear trend in demand. Ouyang et al. (999) develped lead time and rdering cst reductins plicies in cntinuus review inventry systems with partial backrdering f ccurring shrtages. An inventry mdel fr deterirating items with expnential declining demand and partial backlgging was develped by Ouyang et al. (25). Singh and Singh (27) suggested a mdel with partial backrdering and ramp type demand. In this mdel deteriratin fllws Weibull distributin (tw parameters). Dye et al. (27) studied pricing and rdering plicy with cnstant rate f partial backlgging fr deterirating prducts. Chern et al. (28) studied a partial backlgging lt-size mdel fr deterirating cmmdities with fluctuating demand. Skuri et al. (29) suggested a mdel by cnsidering ramp type demand, partial backlgging and Weibull deteriratin. Taleizadeh and Pentic (23) intrduced an ecnmic rder quantity mdel with a knwn price increase and partial backrdering. Tayal et al. (24) presented a tw echeln supply chain mdel with effective investment in preservatin technlgy fr deterirating items. In this mdel, the shrtages are allwed and the happening shrtages are partially backlgged. Shastri et al. (25) explred an inventry mdel by cnsidering trade credit effect and ramp type demand fr deterirating items. Shrtages are permitted and unsatisfied demand is backlgged partially, als rate f deteriratin is cnsidered as linear increasing functin f time. San-Jsé et al. (25) analyzed an ecnmic rder quantity inventry mdel with partial backrdering. During the perid f stck ut, shrtages are permissible and simply a prtin f demand is cnsidered as backrdered partially. Recently, hanna et al. (26) presented an inventry mdel cnsidering permissible delay in payments with allwable shrtages fr imperfect quality deterirating items and ccurring shrtages are assumed as fully backlgged in this mdel. Frm abve literature it is bserved that less interest has been paid by the researchers in develping tw-warehuse inventry mdel with price-sensitive demand. S, in this present mdel we cmbine all mentined factrs with the selling price dependent rate f demand. This is an EOQ mdel fr deterirating prducts with tw warehuse and allwable shrtages and ccurring shrtages are cnsidered as backlgged partially. The numerical example and sensitivity analysis are presented t illustrate this study. 2. Assumptins and Ntatins 2. Assumptins In rder t develp the present mdel, sme specific assumptins used as fllws,. The replenishment rate is assumed t be infinite. 2. The wned warehuse has a limited capacity f W units. 3. The rented warehuse has unlimited capacity. 4. The lead time is assumed t be zer. 5. The demand rate is a functin f selling price. 6. The items cnsidered in this mdel are deterirating in nature. 7. The items are stred in R.W. nly after filling wned warehuse. 8. The items kept in R.W. will be cnsumed first.

4 4 9. The shrtages are allwed and partially backlgged.. Hlding cst per unit in R.W. is greater in cmparisn f hlding cst per unit in O.W. 2.2 Ntatins The fllwing are the ntatins used thrughut this mdel. Ir(t) I(t) a b t W v s Q Q2 T p θ hr h d c2 c3 Inventry level at time t in R.W. Inventry level at time t in O.W. rate f deteriratin initial demand rate psitive demand parameter the time at which inventry level in R.W. becmes zer stck capacity f O.W. time at which inventry level becmes zer in O.W. selling price per unit initial stck level backrdered quantity during stck ut cycle time purchasing cst per unit rate f backlgging hlding cst per unit in R.W. hlding cst per unit in O.W. per unit deteriratin cst per unit shrtage cst per unit lst sale cst ttal average cst 3. Mathematical Mdelling In the beginning Q units are received in stck, ut f which Q2 units are utilized t satisfy backlgged demand and Q units are the initial stck level. Since capacity f O.W. is nly W units and Q>W, s remaining (Q-W) units are stred in a rented warehuse. Nw since hlding cst in R.W. is greater cmpared with hlding cst in O.W., the items in R.W. will be cnsumed first. In this duratin inventry level in O.W. is reduced because f deteriratin nly. At t=t inventry level in R.W. becmes zer after satisfying the demand and deteriratin. During [t, v] stck is available nly in O.W. At t=v inventry level in O.W. als becmes zer and after that shrtage ccurs and is shwn in Fig.. Inventry (Q W) W t v T Q2 Time Fig.. Inventry time graph f tw warehuse inventry mdel

5 M. Rastgi et al. / Decisin Science Letters 6 (27) 5 Differential equatins shwing the behavir f inventry with time are as fllws: di r() t () Ir() t ( a bs) t t dt with bundary cnditin Ir ( t) di () t I() t t t (2) dt with bundary cnditin I() W di() t I() t ( a bs) dt t t v (3) with bundary cnditin I() v Slutin f the abve mentined differential equatins are given by: ( a bs) ( t t I () ( ) r t e ) t t (4) I () t We t t t (5) ( a bs) ( v t) I() t ( e ) t t v (6) At initial stage an rder f Q+Q2 units is made ut f which the Q2 units are used t meet the backrdered quantity and the remaining Q units are stred as the initial stck level fr next cycle. Since the wned warehuse has a limited capacity f W units, s if the stck levelq W, then the remaining quantity ( Q W) will be stred in rented warehuse. Frm Eq. (5) we knw that: I () ( Q W) r ( a bs) ( t ) Q W e (7) Frm Eq. (5) and Eq. (6): ( a bs) ( v t W e e ) (8) Since << S by slving Eq. (8) we get: W t v ( a bs) Different Assciated Csts Purchasing Cst Since Q is the initial stck level and Q2 is the backrdered quantity. S the purchasing cst will be: PC.. ( Q Q ) p 2 (9) where Q is given in Eq. (7) and Q2 can be calculated as fllw:

6 6 T Q2 ( a bs) dt v Q2 ( a bs) ( T v) Then purchasing cst will be ( a bs) t P. C. {( W ( e )) ( a bs) ( T v)} p () Hlding Cst The stck is stred in wned warehuse and rented warehuse. t t ( a bs) e HC. h I ( t) dt h ( t) and R r r r () t v ( ) ( ) v Wh t t a bs e (2) t HC. h{ I() tdt I() tdt} ( e ) h ( ( t v)) Deteriratin Cst The stck f items is stred in wned warehuse and in rented warehuse s the deteriratin will ccur in bth the places. Here D[,t ] dentes the demand during [,t ]. D. C d( I () D[, t ]) R r a bs t DC. R d{ ( e ) ( a bs) t} (3) and D. C d( W D[ t, v]) D. C d{ W ( a bs)( v t )} (4) Shrtage Cst The shrtage cst during the shrtage perid [v, T] is T S. C. c ( a bs) dt c ( a bs)( T v) (5) Lst Sale Cst 2 2 v The lst sale cst due t the partial backlgging during stck ut is given by: T LSC... c ( )( a bs) dt c ( )( a bs)( T v) 3 3 v (6) Nw the fr the presented mdel during a given cycle is: T. AC. ( v, T) [ PC.. H. CR H. C DC. R DC. SC.. LSC...] (7) T

7 M. Rastgi et al. / Decisin Science Letters 6 (27) 7 Here is a functin f tw variables v and T. S t cmpute the minimum value f we have t cmpute ptimal value f v and T. Fr ptimal value f v and T TAC... TAC... and v T 2 2 TAC... TAC..., and v T With the help f these equatins, ptimal values f T, v and can be fund. The belw mentined Fig.2 presents that mdel is cnvex. 4. Numerical Example A numerical example is demnstrated with the help f fllwing input parameters. a 5 units, b.5, s 3 rs / unit,.8, c rs / unit, c 7 rs / unit, p 5 rs / unit 2 3.5, W units, d 6 rs / unit, h.6 rs / unit, h.5 rs / unit Crrespnding t these input values, ptimal values f v & T are days and days, respectively and with the minimum value f is rs. r Fig. 2.Cnvexity f the functin 5. Sensitivity Analysis With regard t varius assciated parameters a sensitivity analysis is given which is shwn belw numerically as well as graphically. Table Effect n ptimal slutin fr distinct values f demand parameter (a) % change in a a T v -2% % % % % % % % %

8 8 a b Fig. 3.Variatin in ttal average cst w.r.t. demand parameter (a) Fig. 4.Variatin in ttal average cst w.r.t. demand parameter (b) Table 2 Effect n ptimal slutin fr distinct values f demand parameter (b) % change in b b T v -2% % % % % % % % % Table 3 Effect n ptimal slutin fr distinct values f deteriratin parameter () % change in T v -2% % % % % % % % % s Fig. 5.Variatin in ttal average cst w.r.t. deteriratin parameter () Fig. 6.Variatin in ttal average cst w.r.t. selling price (s)

9 M. Rastgi et al. / Decisin Science Letters 6 (27) 9 Table 4 Effect n ptimal slutin fr distinct values f selling price (s) % change in s s T v -2% % % % % % % % % Table 5 Effect n ptimal slutin fr distinct values f backlgging parameter (θ) % change in θ θ T v -2% % % % % % % % % θ W Fig. 7.Variatin in ttal average cst w.r.t. backlgging parameter (θ) Fig. 8.Variatin in ttal average cst w.r.t. warehuse capacity (W) Table 6 Effect n ptimal slutin fr distinct values f warehuse capacity (W) % change in W W T v -2% % % % % % % % %

10 2 6. Observatins. Table shws the effect f demand parameter (a) n v, T and n Frm this, we have nticed that an increase in demand parameter (a) shws a reverse effect f decrement in v and T and the same effect f increment in 2. Table 2 presents the effect f demand parameter b n v, T and n It is nticed that with an increment in demand parameter b, values f v and T increase while value f decreases. 3. Table 3 shws the deviatin f deteriratin parameter () at distinct pints and it is nticed that as deteriratin parameter (θ) increases, the values f v and T decrease while increases. 4. Table 4 lists the variatin in selling price s and frm this, we have nticed that an increment in selling price shws maintains a reverse effect f decrement in 5. Table 5 presents the deviatin f backlgging parameter (θ) at distinct pints and it is nticed that as backlgging parameter (θ) increases, the values f v, T and decrease. 6. In Table 6 the variatin in stck capacity (W) f wned warehuse is discussed. Here it is shwn that with an increase in capacity f the wned warehuse the f the system gradually decreases. 7. Cnclusin The main mtive f the develpment f inventry mdel is t find ut that the quantity and the time f rder, which can ptimize the ttal average cst f the system. Due t ffered cncessin in bulk purchasing and different cnditins the vendr purchases the quantity greater than the warehuse capacity. S in this cnditin vendr has t stck the extra quantity in any rented warehuse. eeping in mind that the inventry cst such as hlding cst in R.W. is higher than that f O.W. this paper has presented a significant utcme that additinal cst f safeguarding and hlding material etc. culd be reduced using the prpsed methd. But in rder t reduce the inventry cst, it will be mre realistic and prfitable fr rganizatins t stre items in O.W. befre R.W., but utilize the stcks in R.W. befre O.W. Frm this analysis it is cncluded that an increase capacity f O.W. decreases ttal system cst. Shrtages are permitted in this mdel and the ccurring shrtages are partially backlgged. The cnvexity f ttal average cst functin has als been presented. Fr future scpe, the mdel can be develped with trade credit perid, stck dependent demand, time value f mney and many mre. Als the inventry hlding cst, the unit purchase cst, and thers cst can be cnsidered as dependent upn time. A numerical example and sensitivity analysis with respect t varius system parameters is als presented. References Agrawal, S., Banerjee, S., & Papachrists, S. (23). Inventry mdel with deterirating items, ramptype demand and partially backlgged shrtages fr a tw warehuse system. Applied Mathematical Mdelling, 37(2-2), Alfares, H.., & Ghaithan, A. M. (26). Inventry and pricing mdel with price-dependent demand, time-varying hlding cst, and quantity discunts. Cmputers & Industrial Engineering, 94, Bhunia, A.., Jaggi, C.., Sharma, A., & Sharma, R. (24). A tw-warehuse inventry mdel fr deterirating items under permissible delay in payment with partial backlgging. Applied Mathematics and Cmputatin, 232, Dave, U. (989). A deterministic lt-size inventry mdel with shrtage and a linear trend in demand. Naval Research Lgistics, 36(4),

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12 22 Singh, S. R., umar, T., & Gupta, C. B. (2). A sft cmputing based inventry mdel with deteriratin and price dependent demand. Internatinal Jurnal f Cmputer Applicatins, 36(4), -7. Singh, S. R., & Singh, T. J. (27). An EOQ inventry mdel with Weibull distributin deteriratin, ramp type demand and partial backlgging. Indian Jurnal f Mathematics and Mathematical Sciences, 3(2), Taleizadeh, A. A., & Pentic, D. W. (23). An ecnmic rder quantity mdel with a knwn price increase and partial back rdering. Eurpean Jurnal f Operatinal Research, 228(3), Tayal, S., Singh, S. R., & Sharma, R. (25). An inventry mdel fr deterirating items with seasnal prducts and an ptin f an alternative market. Uncertain Supply Chain Management, 3(), Tayal, S., Singh, S. R., Sharma, R., & Chauhan, A. (24). Tw echeln supply chain mdel fr deterirating items with effective investment in preservatin technlgy. Internatinal Jurnal f Mathematics in Operatinal Research, 6(), Wee, H. M. (997). A replenishment plicy fr items with a price-dependent demand and a varying rate f deteriratin. Prductin Planning & Cntrl, 8(5), Yang, H. L. (24). Tw-warehuse inventry mdels fr deterirating items with shrtages under inflatin. Eurpean Jurnal f Operatinal Research, 57(2), by the authrs; licensee Grwing Science, Canada. This is an pen access article distributed under the terms and cnditins f the Creative Cmmns Attributin (CC-BY) license (