EOQ Model with Time Induced Demand, Trade Credits and Price Discount on Shortages: A Periodic Review

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1 Global Journal of Pure and Applied Mahemaics. ISSN Volume 3, Number 8 (07, pp Research India Publicaions hp:// EOQ Model wih ime Induced Demand, rade Credis and Price Discoun on Shorages: A Periodic Review *H.S.Shukla, R.P.ripahi and *A. Siddiqui Deparmen of Mahemaics, Graphic Era Universiy, Dehradun (UK India *Deparmen of Mahemaics and Saisics, DDU Gorakhpur Universiy, Gorakhpur (UP India Auhor corresponding Absrac Invenory conrol is he mos imporan applicaion of operaions research. In radiional EOQ models demand rae is considered consan. Bu in acual pracice or any ype of business ransacion i is in dynamic sage. In his sudy a periodic review invenory model wih ime induced demand and is nonincreasing funcion of ime under rade credis is developed. We formulae and analyze he mehod of deermining he opimal order quaniy and oal profi. Invenory manager offers price discoun when here is no sock in hand o cusomer who is ineresed o backorder heir demand. Shorages are allowed and fully backlogged. he model maximizes he oal profi. Numerical examples are provided o illusrae he model. Sensiiviy analysis has also been provided wih he help of several key parameers. he second and hird order approximaions are used o find closed from soluion. Keywords: Invenory model; shorage; price discoun; rade credis; imedependen demand. INRODUCION In he radiion EOQ models i was considered ha buyer mus pay for purchase producs suddenly receiving hem. Bu in pracice a vendor frequenly offers heir reailers a rade credis for seling he amoun owed o hem. Generally, here is no ineres charged, if he ousanding amoun is paid wihin he rade credi. he rade

2 396 H.S.Shukla, R.P.ripahi and A. Siddiqui credi is beneficial for seller as well buyer. During he pas few decades several invenory modelers have sudied heir invenory models wih permissible delay in paymens. Goyal [] explored an EOQ model under rade credis. eng [] provided an appropriae pricing and lo- sizing model for a reailer when he supplier gives a rade credis. Aggarwal & Jaggi [3] generalized Goyal s model for deerioraing producs. Hwang and Shinn [4] esablished he opimal pricing and lo sizing for he reailer under he condiion of rade credis. Khanra e al [5] considered an economic order quaniy model for a deerioraing iem wih ime induced demand under permissible delay in paymens. eng e al. [6] esablished an EOQ under rade credi financing wih increasing demand. Chung [7] explored an alernaive approach o deermine he economic order quaniy under permissible delay in paymens. Relaed research papers can be found in Chung [8], Chung & Liao [9], eng and Chang [0],Huang and Hsu [], Liao e al [], Ouyang e al. [3], eng and Chung [4], Soni [5], ripahi and Kumar [6] and here references. In Classical invenory model demand rae is considered consan. However, in realiy he demand is in dynamic sae. Silver & Meal [7] considered an EOQ model for variable demand. Donaldon [8] was o firs o provide a fully analyical soluion o he problem of invenory replenishmen wih a linearly ime dependen demand. ripahi and Pandey [9] considered an invenory model for deerioraing iems wih Weibull disribuion ime- dependen demand rae under permissible delay in paymens. Min e al. [0] developed a lo- sizing model for deerioraing iems wih a curren sock- dependen demand and delay in paymen. ripahi & omar [] esablished he possible effecs of a emporary price discoun offered by a supplier on a reailer s replenishmen policy for deerioraing iems wih linear ime- dependen demand rae. Research work in his direcion came from Dave & Pael [], Chung & ing [3], Gowsami & Chaudhuri [4], Jalan e al. [5], Lin e al [6] and ohers. Generally, buyers have o wai for some daily life useful producs in case of unavailabiliy. he reason is he ousanding qualiy of he iem or specific characerisics. Ghiami e al [7] invesigaed a wo echelon supply chain model for deerioraing invenory in which he reailer s warehouse has a limied capaciy. Pal & Chandra [8] sudied a periodic review invenory model wih sock- dependen demand under shorages and rade credis. ripahi [9] developed an EOQ model for deerioraing iem wih linearly ime dependen demand rae under inflaion and ime discouning over a finie planning horizon under shorages. Yang [30], Law and Wee [3], Dye [3], Jaggi e al [33], Ouyang & Chang [34], Wee e al [35], Luong& Karim [36] developed heir EOQ models under shorages.

3 EOQ Model wih ime Induced Demand, rade Credis and Price Discoun NOAIONS AND ASSUMPION he following noaions are adoped: A p C C β0 β Ie Ip s : ordering cos/ order : purchase cos/ uni/uni ime : back order cos/ uni/ uni ime : cos of a los sale : marginal profi/ uni : price discoun on uni backorder offered : ineres earned/ uni ime : ineres payable/ uni ime beyond he rade credi (Ip > Ie : selling price/ uni α0 : upper bound on backorder raio, 0 α0 α : fracion of he demand during sock-ou ime which is acceped o be backlogged Im Ib Q( λ : ime aken from sock in hand 0 : lengh of a replenishmen cycle : maximum sock heigh in a replenishmen cycle : maximum shorage (backorder : invenory level a ime : deerioraion rae he assumpions are as follows: (i Lead ime is negligible (ii Shorages are allowed and fracion α of unme demands in he sock ou is backlogged. (iii Demand rae D( a ime is (iv Single iem is considered. a b, for, 0, a 0, a > b > 0 D (. a, for, (v Back order fracion α is proporional o he price discoun β offered by vendor. hus 0, where 0 β β0. 0

4 3964 H.S.Shukla, R.P.ripahi and A. Siddiqui 3. MAHEMAICAL MODEL AND OPIMAL SOLUION Le us consider ha he saring of he firs reorder ime, he sock level is zero before ordering, he order quaniy during he period (0, is Im. he planning horizon is divided ino reorder ime inervals, each of lengh. Orders are placed a ime poins,, 3, , he order quaniy being jus sufficien o bring he sock heigh o a cerain level Im. Decrease of invenory level Q( occurs due o boh demand and deerioraion in inerval (0,. he shorages occurs during (, in which a fracion α is backlogged. he change of invenory level wih respec o ime is considered as: dq( Q( ( a b., 0 ( d and dq( d a, ( under he condiion Q( = 0 (3 he soluion of ( and ( wih he help of (3 are b ( ( Q a e b ( e and Q( a ( (5 respecively. b I Q a e b e hus m (0 And Ib Q( a ( (7 he sales revenue SR during [0, ] is b s a b d a d s a a ( ( (8 0 he holding cos HC during [0,] h b e e h Q( d a b (9 0 oal number of backorders BO during [, ] is (4 (6

5 EOQ Model wih ime Induced Demand, rade Credis and Price Discoun C Q( d (0 a C ( oal number of los sales LS during [, ] is C a( ( ( wo cases may arise regarding he permissible delay in paymens (m and m > Case I (m Since credi period is shorer han ime for sock in hand, hus vendor can use he sales revenue o earn ineres a rae Ie in [0, ]. he ineres earned IE by he buyer is si e b e e sie Q( d a b ( 0 and he ineres payable IP by he vendor beyond he fixed credi period is ( m e m ( m pi r b e pir Q( d a ( m b (3 m herefore, he oal profi P(, per uni cycle ime is P (, SR A HC BO LS IP IE (4 Case II (m > Since credi period is longer han, seller pays no ineres, bu earns ineres IE wih rae Ie. hus m b e e a m ( ( e (5 IE sie Q d a m d si a b 0 Hence, he oal profi P(, per uni cycle ime is P (, SR A HC BO LS IE (6 Wih he help of above discussion we ge he following properies: Propery : he opimal cycle ime is an increasing funcion of ime for posiive invenory: Proof: I is obvious from (A5, (A3 and (A33.

6 3966 H.S.Shukla, R.P.ripahi and A. Siddiqui Propery : he opimal cycle ime is convex funcion of. Proof: I is obvious from (A6, (A3 and (A34. Propery 3: If α =, he oal profi is greaer han ha of 0 < α <. Proof: In case of α =, LS = 0. he oal profi for boh cases become P (, SR A HC BO IP IE (7 P (, SR A HC BO IE (8 From (4 and (7, we ge LS P (, P (, 0 P (, P (, (9 From (6 and (8, we ge (, (, LS P P 0 P (, P (, (0 4. NUMERICAL EXAMPLES Case I: Consider he parameer values s = 50, a = 5, b =.5, α = 0.7, θ = 0.05, C = 50, C = 0, h = 40, Ie = 0.03, Ir = 0.05, A = 00, p = 50, m = 0. in appropriae unie. We ge, * = year, * = year, Q = unis, P* = $ Case II: Consider he parameer values s = 50, a = 5, b =.5, α = 0.7, θ = 0.05, C = 50, C = 0, h = 400, Ie = 0.03, Ir = 0.05, A = 00, p = 50, m = 0. in appropriae unie. We ge * = year, * = year, Q = unis and P* = $ he following figures (Case & and show ha he oal profi is concave wih respec o cycle ime: Fig (Case can be drawn considering he following parameer values s = 50, a = 5, b =.5, α = 0.7, λ = 0.05, C = 50, C = 0, h = 50, Ie = 0.03, Ir = 0.05, A = 00, p = 50, m = 0., = 0., in appropriae unis.

7 EOQ Model wih ime Induced Demand, rade Credis and Price Discoun Fig : (Case : Graph beween cycle ime and oal profi P Fig (Case consider he following parameer values s = 00, a = 0, b =.5, α = 0.7, λ = 0.05, C = 50, C = 0, h = 50, Ie = 0.03, Ir = 0.05, A = 00, p = 50, m = 0., = 0.05 in appropriae unis. Fig.. (Case Graph beween cycle ime and oal profi P

8 3968 H.S.Shukla, R.P.ripahi and A. Siddiqui 5. SENSIIVIY ANALYSIS In real life he and business managemen he fuure planning is uncerain. Seller wans o more profi for he producs kep in hand. he sensiiviy analysis is beneficial for vendor and buyer boh. We consider he variaion of,, Q and P wih he variaion of s, h, A, C, C,m and p. he numerical dae is aken from numerical examples and for case I and II respecively keeping remaining parameers same. able : Variaion of,, Q and P wih several parameers Case I Case II s Q P s Q P h Q P a Q P A Q P C Q P m Q P C Q P

9 EOQ Model wih ime Induced Demand, rade Credis and Price Discoun C Q P h Q P C Q P A Q P p Q P m Q P From he above discussion he inferences can be made: Increase in, P and decrease in, Q wih he increase in s. Decrease in, P, and Q wih increase in h and p respecively. Increase in, and decrease in P,, Q wih increase in C and C respecively. Increase in,, Q and decrease in P wih increase in A. Decrease in, and increase in Q, P, wih increase in a. 6. CONCLUSION I is difficul for invenory manager o decide when and how many iems are kep in he shop. I depends on he siuaion and public demand. In mos of he cases he demand in dynamic sae. In his sudy, we have considered he demand in ime dependen and deerioraion is consan. Shorages are allowed. wo differen cases have been considered. Mahemaical model is provided for finding opimal soluion. based on he opimal soluion wo properies have been obained based on he opimal

10 3970 H.S.Shukla, R.P.ripahi and A. Siddiqui soluion. We have proved ha he oal profi in concave wih cycle ime. From managerial poin of view he oal profi have decrease wih increase of uni holding cos, back order cos, los sell cos and purchase cos, bu increase wih he increase of iniial demand, selling price and ordering cos. he possible exension of he model for including weibull disribuion deerioraion. We may also add he adverisemens charges and freigh charges. APPENDIX ( SR / s b ( SR / s a a b a a,, ( SR / sb s b ( SR / a a 3 ( SR / s, a a b ( HC / h( a b e. HC h b e e a 3 b ( / ( HC / h a b e (, ( HC / h a b e b e (. ( BO / a C ( LS / Ca, ( LS / C a, ( BO / a C ( LS / 0. ( IE / sie a b e,, ( BO / a C( and ( LS / Ca ( BO / a C and ( IE / sie a b e b e ( IE/ si e e b e a b..

11 EOQ Model wih ime Induced Demand, rade Credis and Price Discoun ( IE/ si e e b e a 3 b ( IE / sie a b e. ( m ( IP / pi ( r b e m ( m a e b e ( IP / pi r ( m ( m ( m ae be e e ( m ( IP / pir b ( m ( m a e b e ( m ( IP/ pi ( r b e m m a e m b ( IE / si e b e a e b e a m ( IE / si e ae be e a ( IE / si e b e a e b e a m ( IP / s b a a a b a C( Ca ( m pi ( r b e m m a e m b (A ( IP / 0, gives h si e b e e.,

12 397 H.S.Shukla, R.P.ripahi and A. Siddiqui b a C( sa a a b C a h si e b e e ( m pi ( r b e m m a e m b 0 Differeniaing (A w.r.. wo imes, we ge ( a b s( a b a ( ( m h si e e d pir e a C a C Ca( = 0 d (A 3 sb ( h si ae b b( e - pi r (A 4 e ae b b( e ( m ( m From (A 3 and (A 4, we ge d d = 0. d d a C a C a C ( m e r d * ( a b ( h si e pi e s( a b a a C Ca( d ac > 0 (A 5 and d ( m ( m r * ( h sie ( a b e b e pi ( a b e b e d ac Again ( sb ac ac d d > 0 (A 6 ac ( h si ( a b e ( IP / s( a b a e ac ( Ca( pi r ( ( m a b e (A 6 (A ( IP / 0, gives

13 EOQ Model wih ime Induced Demand, rade Credis and Price Discoun ( m r s( a b a ( h si ( a b e ac ( C a( pi ( a b e = 0 (A e Differeniaing (A w.r.. wo imes we ge d sb ( h si b be ( a b e ac ac e d pi b be ( a b e 0 r ( m ( m (A ( m ( m e r e r d ( a b ( h si e pi e b ( h si e pi e ac 0 (A d From (A and (A, we ge ( m ( m r d * ( h sie ae b b( e pi ae b b( e ( sb ac d ac > 0 (A 3 and d d Again * ( ( m a b b ( h sie e piee > 0 (A 3 a C ( P / s b ( h si e e b e a a a b d ac ac si a m ( P / 0 d e, gives b ( h si e b e e a C s a a a b siea ( m Ca( 0 (A 3 Differeniaing (A 3 wo imes wih respec o, we ge ( h si d s a b a a b e ac ac e d C a( si a( m 0 (A 3 e

14 3974 H.S.Shukla, R.P.ripahi and A. Siddiqui ( h si d d ( e 0 d d e sb ae b b e a C a C a C si a (A 3 From Equaions (A 3 and (A 3, we ge d * ( h sie( a b ( e s( a b a a C Ca( siea ( m d ac and d d (A 34 > 0 (A 33 * ( ( h sie ae b b e sb ac siea ac d ac > 0 d REFERENCES [] Goyal, S.K. (985. Economic Order Quaniy under condiions of permissible delay in paymens. Journal of Operaional Research Sociey, 36, [] eng, J.. Chang, C.. and Goyal, S.K. (005. Opimal pricing and ordering policy under permissible delay in paymens. Inernaional Journal of Producion Economics, 97, -9. [3] Aggarwal, S.P. and Jaggi, C.K. (995. Ordering policies of deerioraing iems under permissible delay in paymens. Journal of Operaional Research Sociey, 46, [4] Hwang, H. and Shinn, S.W. (997. Reailer s pricing and lo sizing policy for exponenially deerioraing producs under he condiion of permissible delay in paymens. Compuers and Operaions Research, 4, [5] Khanra, S. Ghosh, S.K. and Chaudhuri, K.S. (0. An EOQ model for a deerioraing iem wih ime dependen quadraic demand under permissible delay in paymen. Applied Mahemaics and Compuaion, 8, -9. [6] eng, J.. Min, J. and Pan, Q. (0. Economic order quaniy model wih rade credi financing for non- decreasing demands. Omega, 40, [7] Chung,K.J. (998. A heorem on he deerminaion of economic order quaniy under condiions of permissible delay in paymens. Compuers and Operaions Research, 5, [8] Chung,K.J. (008. Commen on he EPQ model under reailer parial rade credi policy in he supply chain. Inernaional Journal of Producion Economics, 4,

15 EOQ Model wih ime Induced Demand, rade Credis and Price Discoun [9] Chung, K.J. and Liao, J.J. (004. Lo- sizing decisions under rade credi depending on he order quaniy. Compuers and Operaions Research, 3, [0] Goyal, S.K. eng, J.. and Chung, C.. (007.Opimal ordering policies when he supplier provides a progressive ineres payable- scheme. European Journal of Operaional Research, 79, [] Huang, Y.F. and Hsu, K.H. (008. An EOQ model under reailer parial rade credi policy in rade credi policy in supply chain. Inernaional Journal of Producion Economics,, [] Liao, H.C. sai, C.H. and Su, C.. (000. An invenory model wih deerioraing iems under inflaion when delay in paymen is permissible. Inernaional Journal of Producion Economics, 63, [3] Ouyang, L.Y. Chang, C.. and eng, J.. (005. An EOQ model for decaying iems under rade credis. Journal of Operaional Research Sociey, 5, [4] eng, J.. and Chang, C.. (009. Opimal manufacurer s replenishmen policies in he EPQ model under wo levels of rade credi policy. European Journal of Operaional Research, 95, [5] Soni, H.N. (03.Opimal replenishmen policies for non- insananeous deerioraing iems wih price and sock- sensiive demand under permissible delay in paymens. Inernaional Journal of Producion Economics, 46, [6] ripahi, R.P. and Kumar, A.K. (004. EOQ model wih cash flow oriened and quaniy dependen under rade credis. Inernaional Journal of Engineering, 7(7, 09-. [7] Silver, E.A. and Meal, H.C. (969. A simple modificaion of EOQ for he case of a varying demand rae. Producion and Invenory Managemen, 0(4, [8] Silver,E.A. (979. A single invenory replenishmen decision rule for a linear rend in demand. Journal of Operaional Research Sociey, 30, [9] ripahi, R.P. and Pandey, H.S. (03. An EOQ model for deerioraing iem wih Weibull ime- dependen demand rae under rade credis. Inernaional Journal of Informaion and Managemen Sciences, 4, [0] Min, J. Zhou, Y.W. and Zhao, J. (00. An invenory model for deerioraing iems under sock- dependen demand and wo level rade credi. Applied Mahemaical Modelling, 34, [] ripahi, R.P. and omar, S.S. (05. Opimal order policy for deerioraing iems wih ime- dependen demand in response o emporary price discoun linked o order quaniy. Inernaional Journal of Mahemaical Analysis, 9(3,

16 3976 H.S.Shukla, R.P.ripahi and A. Siddiqui [] Dave, U. and Pael, L.K. (98. (, Si policy invenory model for deerioraing iems wih ime- proporion demand. Journal of Operaional Research Sociey 3, [3] Chung, K.J. and ing, P.S. (993. A heurisic for replenishmen of deerioraing iems wih a linear rend in demand. Journal of Operaional Research Sociey, 44 (, [4] Goswami, A. and Chaudhuri, K.S. (99. An EOQ model for deerioraing iems wih a linear rend in demand. Journal of Operaional Research Sociey, 4(, [5] Jalan, A.K. Giri, K.S. and Chaudhuri, K.S. (996. EOQ model for iem wih Weibull disribuion deerioraion, shorages and rended demand. Inernaional Journal of Sysem Science, 7(9, [6] Lin,C. an,b. and Lee, W.C.(000. An EOQ model for deerioraing iems wih ime varying demand and shorages. Inernaional Journal of Sysem Science, 3(3, [7] Ghiami, Y. Williams,. and Wu, Y. (03 A wo- echelon invenory model for a deerioraing iem wih sock- dependen demand, parial backlogging and capaciy consrains. European Journal of Operaional Research, 3, [8] Pal, M. and Chandra, S. (04 A periodic review invenory model wih sock dependen demand, permissible delay in paymens and price discoun on backorders. Yugoslav Journal of Operaions Research, 4(, [9] ripahi, R.P. (06 Economic order quaniy for deerioraing iems wih nondecreasing demand and shorages under inflaion and ime discouning. Inernaional Journal of Engineering, 8(9, [30] Yang,H.L. (005 A comparision among various parial backlogging invenory lo-size models for deerioraing iems on he basis of maximum profi. Inernaional Journal of Producion Economics, 96, 9-8. [3] Law, S.. and Wee, H.M. (000. An inegraed producion invenory model for amelioraing and deerioraing iems aking accoun of ime discouning. Mahemaics and Compuer Modelling, 43, [3] Dye, C.Y. (007. Join pricing and ordering policy for a deerioraing invenory wih parial backlogging. he Inernaional Journal of Managemen Science, 35, [33] Jaggi, C.K. Goel, S.K. and Mial, M. (03. Credi financing in economic ordering policies for defecive iem wih allowable shorages. Applied Mahemaics

17 EOQ Model wih ime Induced Demand, rade Credis and Price Discoun and Compuaion, 9, [34] Ouyang,L.Y. and Chang, C.. (03. Opimal producion lo wih imperfec producion process under permissible delay in paymens and complee backlogging. Inernaional Journal of Producion Economics, 44, [35] Wee, H.M. Huang, Y.D. Wang, W.. and Chang, Y.L. (04. An EPQ model wih parial backorders considering wo backordering coss. Applied Mahemaics and Compuaion, 3, [36] Luong, H.. & Karim, R.(07. An inegraed producion invenory model of deerioraing iems subjec o random machine breakdown wih a sochasic repair ime. Inernaional Journal of Indusrial Engineering Compuaions, 8, 7-36.

18 3978 H.S.Shukla, R.P.ripahi and A. Siddiqui