EFFECT OF PERMISSIBLE DELAY ON TWO-WAREHOUSE INVENTORY MODEL FOR DETERIORATING ITEMS WITH SHORTAGES

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1 Volume, ssue 3, Mach 03 SSN EFFEC OF PERMSSBLE DELAY ON WO-WAREHOUSE NVENORY MODEL FOR DEERORANG EMS WH SHORAGES D. Ajay Singh Yadav, Ms. Anupam Swami Assisan Pofesso, Depamen of Mahemaics, SRM Univesiy NCR Campus, Ghaziabad, U.P Assisan Pofesso, Depamen of Mahemaics, Gov. Degee College, Sambhal, U.P ABSRAC n his pape we developed an invenoy sysem wih he effec of pemissible delay in paymens and sock dependen demand. he occuences of shoages ae naual phenomenon allowed in invenoy. heefoe, shoages ae occuing wih paial backlogging. Backlogging ae is aken as waiing ime fo he nex eplenishmen. Holding cos is vaiable and i is linea inceasing funcion of ime. Numeical example is pesened o illusae he model and he sensiiviy analysis of he opimal wih espec o paamees of he sysem is also caied ou.. NRODUCON n oday's business ansacions, i is fequenly obseved ha a cusome is allowed some gace peiod befoe seling he accoun wih he supplie o he poduce. he cusome does no have o pay any inees duing his fixed peiod bu if he paymen ges beyond he supplie will chage he peiod inees. his aangemen comes ou o be vey advanageous o he cusome as he may delay he paymen ill he end of he pemissible delay peiod. Duing he peiod he may sell he goods, accumulae evenues on he sales and ean inees on ha evenue. hus, i makes economic sense fo he cusome o delay he paymen of he eplenishmen accoun up o he las day of he selemen peiod allowed by he supplie o he poduce. his concep is known as pemissible delay in paymens. Goyal (985) was he fis o develop he economic ode quaniy unde condiions of pemissible delay in paymens. Auho has assumed ha he uni selling pice and he puchase pice ae equal. he uni selling pice should be geae han he uni puchasing pice. Aggawal and Jaggi (995) developed odeing policies of deeioaing iems unde pemissible delay in paymens. he demand and deeioaion wee consumed as consan. Jamal e al. (997) developed a model o deemine an opimal odeing policy fo deeioaing iems unde pemissible delay of paymen and allowable shoage. Diffeen faces of he pemissible delays in paymen ae discussed, and his genealized model exhibis a se of soluions ha educes o an exising model. Kun-Jen Chung (998) discussed he economic quaniy unde condiions of pemissible delay in paymens. Jamal e al. (000) pesened opimal paymen ime fo a eaile unde pemied delay of paymen by he wholesale. he wholesale allowed a pemissible cedi peiod o pay he dues wihou paying any inees fo he eaile. n he sudy, a eaile model was consideed wih a consan ae of deeioaion. Dye (00) developed a deeioaing invenoy model wih sock-dependen demand and paial backlogging. he condiions of pemissible delay in paymens wee also aken ino consideaion. Chung and Liao (004) deals he poblem of deemining he economic ode quaniy fo exponenially deeioaing iems unde he condiions of pemissible delay in paymens. n addiion, he objecive funcion is modeled as a oal vaiable cosminimizaion poblem. eng e al. (005) developed vaious EOQ models fo a eaile when he supplie offes a pemissible delay in paymens. n his pape, hey complemen he shocoming of he pevious models by consideing he diffeence beween he selling pice and he puchase cos. Soni e al. (006) fomulae opimal odeing policies fo he eaile when he supplie offes pogessive cedi peiods o sele he accoun. he objecive funcion o be opimized is consideed as pesen value of all fuue cash-ou-flows. Singh, S.R. and Singh,.J. (008) developed he peishable invenoy model wih quadaic demand, paial backlogging and pemissible delay in paymens. Soni, H. e al. (008) developed a mahemaical model o fomulae opimal odeing policies fo eaile when demand is paially consan and paially dependen on he sock, and he supplie offes pogessive cedi peiods o sele he accoun. his chape poposed a wo soage invenoy model fo deeioaing iems wih invenoy level dependen demand. Shoages ae allowed and paially backlogged. Backlogging ae is aken as waiing ime fo he nex eplenishmen. he effec of pemissible delay in paymens is also aken in his sudy. Holding cos is vaiable and i is linea Volume, ssue 3, Mach 03 Page 65

2 Volume, ssue 3, Mach 03 SSN inceasing funcion of ime. Numeical example is pesened o illusae he model and he sensiiviy analysis of he opimal wih espec o paamees of he sysem is also caied ou, which is followed by concluding emaks.. ASSUMPONS AND NOAONS he mahemaical model is based on he following assumpions:. Lead-ime is zeo.. he iniial invenoy is zeo. 3. he demand ae D () is deeminisic and is a known funcion of insananeous sock level; he funcion D () is given by: Whee > 0 and 0 < <. 4. Replenishmen ae is infinie and eplenishmens ae insananeous. 5. he owned waehouse (OW) has a fixed limied capaciy of W unis. 6. he ened waehouse (RW) has unlimied capaciy. 7. he iems of OW ae saed o consume when RW is empy. 8. he invenoy coss (including holding cos and deeioaion cos) in RW ae highe han hose in OW. 9. Shoages ae pemied and he backlogging ae is defined o be /[+δ(-)] when he invenoy is negaive. he backlogging paamee δ is posiive consan. n addiion, he following noaions ae used houghou his pape: L epesens an invenoy sysem wih an OW only. L epesens an invenoy sysem wih boh OW and RW. c 0 he eplenishmen cos pe ode. c d deeioaion cos pe uni. c h he invenoy holding cos pe uni pe uni ime in OW. he invenoy holding cos pe uni pe uni ime in RW. c h ( ), 0 D,, Noe ha implies assumpion 6, c h + c d > c h + c d. c s shoage cos pe uni ime. he deeioaion ae in OW, whee 0 < <. he deeioaion ae in RW, whee 0 < <. S he highes sock level a RW and OW. B he maximum shoage level. P puchase cos pe uni. M pemissible delay peiod in seled he accouns. c inees chages pe upee pe yea. e inees ha can be eaned on he sales evenue of unis sold duing he pemissible delay peiod ( e < c ). W soage capaciy of OW, fixed consan and W < S. 0 () he invenoy level in OW a any ime. () he invenoy level in RW a any ime. 3. MAHEMACAL FORMULAON Hee, we discuss he deeminisic invenoy model fo deeioaing iems wih wo-waehouse whee shoages occu a he end of he cycle. Fo a L sysem (see fig. (a)), a ime =0, a lo size of S unis enes he L sysem in which W unis ae kep in OW and S-W unis in RW. he goods of OW ae consumed only when RW is empy. Duing he ime ineval [0, ], he invenoy S-W in RW deceases due o demand and deeioaion and i vanishes a =. n OW, he invenoy W deceases duing [0, ] due o deeioaion only, bu duing [, ] he invenoy is depleed due o boh demand and deeioaion. A ime =. he invenoy in OW eaches zeo and heeafe he shoages occu duing he ime ineval [, ]. he shoage quaniy is supplied o cusomes a he beginning of he nex cycle. he objecive of he invenoy sysem is o deemine he imings of, and in ode o keep he oal elevan cos pe uni of ime as low as possible. As o a L sysem (see fig. (b)), he fim eceives W unis in OW a =0. he invenoy W depleed due o boh demand and deeioaion, and eaches zeo a =, and heeafe he shoages occus duing [, ]. Noe ha he L sysem hee is, in fac, equivalen o he L sysem wih =0. Volume, ssue 3, Mach 03 Page 66

3 Volume, ssue 3, Mach 03 SSN Fo a L sysem, he invenoy level a RW duing he ime ineval [0, ] is depleed by he combined effec of demand and deeioaion, he invenoy level a ime [0, ], (), is govening by he following diffeenial equaion: d, 0 d (7.) wih he bounday condiion he ( )=0. Solving he diffeenial equaion (), we have e, 0 (7.) Duing he ime ineval [0, ], as he demand is mee fom RW, he sock a OW deceases due o deeioaion only. hus, he invenoy level a ime [0, ], 0 () is govened by he following diffeenial equaion: d 0, 0 (7.3) d 0 wih he iniial condiion 0 (0)=W. Again, duing he ime ineval [, ], he invenoy level a OW is depleed by he combined effec of demand and deeioaion, he invenoy level a ime [, ], 0 (), is govened by he following diffeenial equaion: d 0 0, (7.4) d wih he bounday condiion 0 ( )=0. Solving he diffeenial equaion (7.3) and (7.4), we have W e, 0 (7.5) 0 0 e, (7.6) Due o coninuiy of 0 () a =, if follows eq. (7.5) and (7.6), we have 0 W e e (7.7) Fuhemoe, duing he peiod [, ], he behavio of he invenoy sysem can be descibed by d 0, d (7.8) wih iniial condiion 0 ( )=0, we have 0 n n, (7.9) Fom he equaions (7.), (7.5), (7.6) and (7.0), he oal pe cycle consiss of he elemens:. Odeing cos pe cycle = c 0. Holding cos pe cycle in RW ( F h ) d F h ( ) ( ) 3. Holding cos pe cycle in OW 0 ( ) e ( ) ( ) ( H ) 0 d ( H ) 0 d 0 W H e ( e ) e e Volume, ssue 3, Mach 03 Page 67

4 Volume, ssue 3, Mach 03 SSN Shoage cos pe cycle s d s n he amoun of deeioaed iems in boh RW and OW ae And D W 5. Deeioaion cos pe cycle P D D P e W 6. Oppouniy cos due o los sale pe cycle 0 0 O C d n D e Case : when M n his siuaion, since he lengh of peiod wih posiive sock is lage hen he pemissible delay peiod, he buye can use he sale evenue o ean inees a an annual ae e in (0, ). he inees ean E is E P e d d 0 P (7.0) e 3 3 e Howeve beyond he pemissible delay peiod, he unsold sock is supposed o be financial wih an annual ae and inees payable is given by P M P P 0 d e M (7.) M heefoe oal aveage cos pe uni ime is O C H C R W H O O W S C O C D C P E C, F { h c e ( ) ( ) ( ) ( ) ( ) 0 W e e H e ( e ) s n P e W 3 P M P e 3 e e M (7.) Fo minimizing he oal elevan cos pe uni ime, he appoximae opimal values of and (denoed by * and *) can be obained by solving he following equaions: C C 0 and 0 (7.3) which also saisfies he condiions: C C * * 0 and 0, * *, and C C C 0 *, * Nex by using he opimal values * and *, he appoximae opimal values of (denoed by *) and he appoximae minimum oal cos pe uni ime can be obained fom (3) especively. Case-: when M> Volume, ssue 3, Mach 03 Page 68

5 Volume, ssue 3, Mach 03 SSN Since M> he buye pays an inees bu eans inees a an annual ae e duing he peiod (0, M), inees eans in his case, denoed by E, is given by E P e ( ) d d M d d 0 0 P e 3 3 e M 3 e hen he oal aveage cos pe uni ime is (7.4) C, O C H C R W H C O W S C O C D C E F ( ) h W c 0 e ( ) ( ) H e ( e ) ( ) ( ) e e s n P e W P e 3 3 e 3 M e (7.5) Fo minimizing he oal elevan cos pe uni ime, he appoximae opimal values of and (denoed by * and *) can be obained by solving he following equaions: C C 0 a n d 0 (7.6) which also saisfies he condiions: C C * * 0 a nd 0, * *, and C C C 0 * *, Nex by using he opimal values * and *, he appoximae opimal values of (denoed by *) and he appoximae minimum oal cos pe uni ime can be obained fom (7.5) especively. 4. NUMERCAL EXAMPLES o illusae he esuls, we apply he poposed mehod o solve he following numeical example: Le α = 350, β = 0, c o = 60, c h = 8, c h = 0, W = 00, γ = 0.05, θ = 0.06, c s = 3, = 0.5, e = 0., P = 68, M = 0.3, c d = 0.5. he opimal values of,,, C and C have been compued. Compued esuls ae displayed in able 7.. Volume, ssue 3, Mach 03 Page 69

6 Volume, ssue 3, Mach 03 SSN able : M M > = =.0908 = C = = 0.00 = =.0546 C = Paamees Pecenage change in paamees able : Sensiiviy analysis: M M > C Pecenage C Pecenage change in change in oal oal cos cos C c o D W c h OBSERVAONS. Fom able 7. and able 7., i is obseved ha C is always less hen C wih espec o he change in evey paamee. his is due o in he second case M >. So, we have no paid any inees and we ean some inees.. As he puchasing cos (P) inceases, he oal cos is deceases in boh cases. 3. As he odeing cos inceases (c 0 ), he oal invenoy cos is inceases in boh cases. 4. As he demand ae inceases (D), he oal invenoy cos is decease in boh cases. 5. As he capaciy of he own waehouse inceases, he oal invenoy cos is also inceases in boh cases. 6. As he holding cos of own waehouse inceases, he oal invenoy cos is also inceases in boh cases. 7. he oal invenoy cos is vey sensiive wih espec o W and vey less effeced by he vaiaion of c CONCLUSON n his sudy an invenoy sysem is developed fo decaying iems wih wo-waehouses and sock dependen demand. Shoages ae pemiing in his model and paially backlogged. And backlogging ae is ime dependen and i is waiing ime fo he nex eplenishmen. he condiions of pemissible delay in paymens and ime dependen holding cos ae also aken ino accoun. Holding coss and deeioaion coss ae diffeen in OW and RW due o diffeen pesevaion envionmens. he invenoy coss (including holding cos and deeioaion cos) in RW ae assumed o be highe han hose in OW. o educe he invenoy coss, i will be economical fo fims o soe he goods in OW o he maximum level and afe ha he emaining goods soe in RW, bu clea he socks in RW befoe OW. So ha en of ened waehouse is minimum. Fom he viewpoin of he coss, decisions ules o find he opimal ode cycle ime conains wo cases: (i) M (ii) M >. Volume, ssue 3, Mach 03 Page 70

Volume, ssue 3, Mach 03 SSN 39-4847 Finally, a numeical example in able is sudied o illusae he heoeical esuls.

So, we have no paid any inees and we ean some inees. So, we conclude ha he effec of pemissible delay canno be ignoed.

7 Volume, ssue 3, Mach 03 SSN Finally, a numeical example in able is sudied o illusae he heoeical esuls. Fom he above able and, i is obseved ha he oal invenoy cos C is always less hen C wih espec o he change in evey paamee. his is due o in he second case M >. So, we have no paid any inees and we ean some inees. So, we conclude ha he effec of pemissible delay canno be ignoed. hus, his model incopoaes some ealisic feaues ha ae likely o be associaed wih some kinds of invenoy. he model is vey useful in hei eail business. can be used fo eleconic componens, fashionable clohes, domesic goods and ohe poducs which ae moe likely wih he chaaceisics above. n fuue eseach on his poblem, i would be of inees o add effec of moe ealisic demand ae in he model (e. g. ime-vaying and sock-dependen demand paens). On he ohe hand, he possible exension of his wok may elax he assumpion of consan deeioaion ae. REFERENCES [] Aggawal, S.P. and Jaggi, C.K. (995): Odeing policies of deeioaing iems unde pemissible delay in paymens, Jounal of Opeaional Reseach Sociey (J.O.R.S.), 46, [] Chung, K.J. (998): A heoem on he deeminaion of economic ode quaniy unde condiions of pemissible delay in paymens, Compues & Opeaions Reseach, 5,, [3] Chung, K.J. and Liao, J.J. (004): Lo sizing decision unde ade cedi depending on he odeing quaniy, C.O.R., 3, [4] Dye, C.Y. (00): A deeioaing invenoy model wih sock dependen demand and paial backlogging unde condiions of pemissible delay in paymens, Opseach, 39(3&4), [5] Goyal, S.K. (985): Economic ode quaniy unde condiions of pemissible delay in paymens, J.O.R.S., 36, [6] Jamal, A.M.M., Sake, B.R. and Wang, S. (997): An odeing policy fo deeioaing iems wih allowable shoage and pemissible delay in paymen, J.O.R.S., 48, [7] Jamal, A.M.M, Sake, B.R. and Wang, S. (000): Opimal paymen ime fo a eaile unde pemied delay of paymen by he wholesale,.j.p.e., 66, [8] Soni, H. e al. (006): An EOQ Model Fo Pogessive Paymen Scheme Unde DCF Appoach, Asia-Pacific Jounal of Opeaional Reseach, 3, 4, [9] Soni, H. and Shah, N.H. (008): Opimal odeing policy fo sock-dependen demand unde pogessive paymen scheme, E.J.O.R., 84 (), [0] Singh, S.R. and Singh,.J. (008): Peishable invenoy model wih quadaic demand, paial backlogging and pemissible delay in paymens, nenaional Review of Pue and Applied Mahemaics,, [] eng, J.., Chang, C.. and Goyal, S.K. (005): Opimal picing and odeing policy unde pemissible delay in paymens,.j.p.e., 97, -9. D. Ajay Singh Yadav has done M.Sc. in Mahemaics and Ph.D. in invenoy Modelling, he has ove 6 yeas expeience in eaching Mahemaics in deffeen Engineeing Colleges. Pesenly he is Assisan Pofesso in SRM Univesiy NCR Campus Ghaziabad Ms. Anupam Swami has done M.Sc,M.Phil. in Mahemaics and pusing Ph.D in invenoy Modelling, she has ove 5 yeas expeience in eaching Mahemaics in deffeen Degee Colleges. Pesenly she is Assisan Pofesso in Depamen of Mahemaics, Gov. Degee College, Sambhal, U.P Volume, ssue 3, Mach 03 Page 7

An Inventory Model for Two Warehouses with Constant Deterioration and Quadratic Demand Rate under Inflation and Permissible Delay in Payments

An Inventory Model for Two Warehouses with Constant Deterioration and Quadratic Demand Rate under Inflation and Permissible Delay in Payments Ameican Jounal of Engineeing Reseach (AJER) 16 Ameican Jounal of Engineeing Reseach (AJER) e-issn: 3-847 p-issn : 3-936 Volume-5, Issue-6, pp-6-73 www.aje.og Reseach Pape Open Access An Inventoy Model