AN INVENTORY MODEL WITH TIME DEPENDENT DETERIORATION RATE AND EXPONENTIAL DEMAND RATE UNDER TRADE CREDITS

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1 International Journal of athematis an Computer Appliations Researh (IJCAR ISSN Vol., Issue Sep 5-6 JPRC Pvt. Lt., AN INVENORY ODEL WIH IE DEPENDEN DEERIORAION RAE AND EXPONENIAL DEAND RAE UNDER RADE CREDIS H.S.SHUKLA, R.P.RIPAHI & SUSHIL KUAR YADAV, Department of athematis, Du Gorakhpur University, Gorakhpur (Up Inia epartment of athematis, Graphi Era University, Dehraun (Uk Inia ABSRAC his paper eals with the optimal poliy for the ostumers to obtain its minimum ost when supplier offers both permissible elay as well as ash isount. In this paper eterioration is onsiere as time epenent an eman rate is an exponential funtion of time. Four ifferent ases have been isusse. runate aylor s series expansion is use to obtain lose form optimal solution. athematia software is use for fining optimal solution. Finally, numerial examples an sensitivity analysis are given to valiate the purpose moel. KEYWORDS: Inventory, Cash Disount, rae Creits, ime Depenent Deterioration, Exponential Deman Rate INRODUCION In lassial inventory moels the eman rate is assume to be either onstant or time-epenent. In a buyer-seller situation, an inventory moel onsiers a ase in whih epletion of inventory is ause by onstant eman rate. any researh papers have been publishe onerning the ontrol of eteriorating an non-eteriorating inventory items. Deteriorating items inlue suh items as volatile liquis, meiines; bloo, fashion goos, raioative material an photographi films an non-eteriorating items inlue suh items as rie fruits, wheat an rie. It is a ommon assumption in too many inventory systems that prouts have inefinitely long lives. Generally, however, almost all items eteriorate over time but often the rate of eterioration is low an there is little nee to onsier the eterioration when etermining eonomi lot size. In past few eaes great interest has been shown in eveloping mathematial moels in presene of trae reits. In toay s business translations, it is ommon to fin that the buyers are allowe some reit perio before they settle the aount with the whole seller. his provies an avantage to the ustomers, ue to the fat that they o not have to pay the whole seller immeiately after reeiving the prout, but elay there payment until the en of allowe perio. In this paper, the eterioration rate is onsiere as time epenent an eman rate is exponential time epenent. At present infletion has beome a permanent feature of eonomy. Lot of researhers has shown infletion effet on inventory.

2 5 An Inventory oel with ime Depenent DeterioratioRate an RExponential Deman Rate Uner rae Creits Goyal [] erive an EOQ moel uner the onitions of permissible elay in payment. Aggrawal an Jaggi [] extene Goyal s moel for eteriorating items. Jammal et al [] then generalize the moel to allow shortages. Hwang & Shim [] ae eman rate is prie epenent an evelope the optimal prie an lot-sizing for a retailer uner the onition of a permissible elay in payment. Chung [5] evelope an alternative approah to fin the EOQ uner the onitions of permissible elay in payment being garnte.eoq moel with trae reit finaning for non-ereasing eman is presente by eng et al.[6].chung et al. [7] evelope an EOQ moel for eteriorating items uner supplier reits linke to orering quantity. In this regar,most of researhers like Covert & Philip [8],Dave & Patel [9],Sahan [],Datta & Pal [],Goswmi & Chauhari [],Raafat [],Hariga [],Goyal & Giri [5].Chung & Dye [6],Skouri & Papahristos [7],Ghare & Shraer [8],Skouri et al.[9],onsiere either a onstant or exponential eterioration rate. In all the mentione above moels the inflations, time value of money an stok is isregare. Gupta & Vrat [] first evelope a moel for onsumption environment to minimize the ost with assumption that stok epenent onsumption rate is a funtion of the initial stok level. Pamanabhan & Vrat [] efine stok-epenent onsumption rate as a funtion of inventory level at any instant of time an evelope moels for non-sales environment. Inventory moels for permissible items with stok-epenent selling rate are evelope by Pamanabhan & Vrat []. It is assume that selling rate is a funtion of urrent inventory level; Sarker et al. [] evelope an orer level lot size inventory moel with inventory level epenent eman an eterioration. In paper [] Sarker assume that there is the nature of ereasing eman, the replenishment rate is regare to be finite an the planning horizon is infinite. Balkhi & Benkheronf [] evelope an inventory moel for eteriorating items with stok-epenent an time varying eman rate over a finite planning horizon. It has happene mostly beause of the belief that the inflation an the time value of money woul not influene the inventory poliy to any signifiant egree. But most of the ountries have suffere from large sale inflation an sharp eline in the purhasing power of money last several years. hus the effets of inflation an time value of money an t be ignore. Buzaott [5] presente an EOQ moel with inflation subjet to ifferent types of priing poliies. ishra [6] evelope an EOQ moel inorporating inflationary effets. Ray & Chauhari [7], Chen [8],Sarker et al.[9],chung& Lui [] an Wee& Law [] all have evelope the effets of inflation, time value of money an eterioration on inventory moel. An EOQ moel for eteriorating items uner trae reit is evelope by Ouyang et al. [] ripathi & Kumar [] presente reit finaning in eonomi orering poliies of time- epenent eteriorating items. In paper [] three ifferent ases has been onsiere an obtaine minimum present value of all future ash flow. ripathi [] evelope an EOQ moel with time epenent eman rate an time-epenent holing ost funtion an minimum total inventory ost is obtaine. ripathi et al. [5] evelope a ash flow oriente EOQ moel of eteriorating items with time-epenent eman rate uner permissible elay in payments.

3 H.S.Shukla,R.P.ripathi & sushil Kumar Yaav 5 In this paper eman rate is taken as exponential time epenent an time-epenent eterioration rate. Disount rate is onsiere an shortages are not allowe in this paper. he main objetive of present work is to minimize the total relevant ost of all four ifferent ases. runate aylor s series expansion is use to etermine lose form solution of optimality. he rest of the paper organize as follows: In the next setion.notations an assumptions is given followe by mathematial formulation in setion. In setion, numerial examples are given followe by sensitivity analysis is in setion 5. Finally onluing remarks an future researh iretion is given in the last setion 6. NOAIONS AND ASSUPIONS he following notations are use throughout the paper p: selling prie per unit : the unit purhasing ost with p > I : the interest harge per ollar in stok per year by the supplier I : the interest earne per ollar per year s : the orering ost per orer Q: the orer quantity r: the ash isount rate < r < h: the unit holing ost per year exluing the interest harges : the perio of ash isount : the perio of permissible elay in settling aount with > : the replenishment time interval I(t: the level of inventory at time t, t,,, an : the optimal replenishment time for ase I, II, III an IV respetively Z(: the total relevant ost per year Z (, Z (, Z (an Z (: the total relevant ost per year for ase I, II, III an IV respetively Z * (, Z * (, Z * ( an Z * ( : Optimal total relevant ost per year for ase I, II, III an IV respetively Q*(,Q*(,Q*( an Q*( : the optimal orer quantity for ase I, II, III an IV respetively

4 5 An Inventory oel with ime Depenent DeterioratioRate an RExponential Deman Rate Uner rae Creits In aition following assumption is being mae: (i the eman rate is exponential time epenent i.e. D D(t= t e α where is the initial eman an >, < α < (ii the eterioration rate is time epenent i.e. θ θ(t = θt. < θ < (iii lea time is zero (iv time horizon is infinite (v shortages are not allowe (vi when the aount is not settle the generate sells revenue is eposite in an interest bearing aount. At the en of or, the aount is settle as well as the buyer pays of all units sol an starts paying for the interest harges on the items in the stok. AHEAICAL FORULAIONS he variation of inventory with respet to time an be esribe by the following ifferential equation ( I ( t α t + θ ti ( t = e, t t With bounary onitions I ( = Q, I(=. he solution of ( is given by ( θ t α ( θ + α I ( t = ( t + ( t + ( t e, t 6 he orer quantity is α ( θ α + Q = ( otal eman uring one yle is α ( θ α he total relevant ost per year onsists of the following elements: (a ost of plaing orer = s (

5 H.S.Shukla,R.P.ripathi & sushil Kumar Yaav 5 (b ost of purhasing = Q α ( θ α + = (5 ( ost of arrying inventory = h α α I t t h 6 ( = ( + + (approx. (6 Case I., sine the payment is mae at time the ustomer save rq per yle ue to prie isount. From ( we know that the isount per year is given by rq α ( θ + α r = (7 Aoring to the assumption ( the ustomer pays off all units orere at time to obtain the ash isount. Consequently, the items in stok have to be finane (at interest rate after time an hene the interest payable per year is ( r I ( r I ( ( + + θ α ( θ + α ( + ( + + α I ( t t = (8 Finally uring [, ] perio,the ostumer sells the prout an eposits the revenue into an aount that earns interest I per ollar per year. herefore, the interest earne per year is pi α t pi α e tt = e ( α + α α (9 otal relevant ost is given by Z ( = ost of plaing orer + ost of purhasing + ost of arrying inventory + interest year interest earne per year payable per s α ( θ + α α α = + r h ( ( r I ( ( + + θ α ( θ + α ( + ( + + α pi α e ( α + α α ( Case II. < αt αt, In this ase the ustomers sells e. units in total at time an has (-r e. to pay the supplier in full at time onsequently there is no interest payable, while the ash isount is the same as that in ase (. However, the interest earne per year is

6 55 An Inventory oel with ime Depenent DeterioratioRate an RExponential Deman Rate Uner rae Creits pi α t α t pi α α e tt + ( e t = e ( ( ( e α + + α α o ( As a result, the total relevant ost per year Z ( is given by s α ( θ + α α α pi ( ( α α Z = + r h + + e ( + + ( ( e 6 6 α α α ( Case III. sine the payment is mae at time there is no isount in this ase i.e. r =. he interest payable per year is given by I I ( ( + + θ α ( θ + α ( + ( + + α I ( t t = ( he interest earne per year is p I α t p I α e. t t = e ( α + α α ( he total relevant ost is given by s α ( θ + α α α Z ( = + r h ( I ( ( + + θ α ( θ + α ( + ( + + α pi e ( α α α (5 α + Case IV. < In this ase there is no interest harge. he earne per year is pi α t α t pi α α e. t t + ( e t = e ( + + ( ( e α α α (6 he total relevant ost Z ( is given by s α ( θ + α α α pi Z (= ( α α + r h + + e ( + + ( ( e 6 α α α (7

7 H.S.Shukla,R.P.ripathi & sushil Kumar Yaav 56 HEOREICAL RESULS From equation (, (, (5 & (7, it is iffiult to obtain optimal solutions expliitly. hus, the moel is solve approximately using a trunate aylor s series expansion for the exponential terms, i.e. α α e α, e α, et + + Whih is a vali approximation for the smaller values of α, α et. With the above approximation, the total relevant osts Zi ( ; i =,,, is given by s α ( θ + α α α ( r I ( Z ( + r h ( θ α ( θ + α ( + ( + + α pi (8 s α ( θ + α α α Z r h p I 6 6 (9 ( ( + + s α ( θ + α α α I ( Z ( + r h ( θ α ( θ + α ( + ( + + α pi ( ( s α ( θ + α α α Z r h p I 6 6 ( ( + + Differentiating (8, (9,(, & ( with respet to two times, we get Z ( s α ( θ + α α α = + r + + h ( ( r I α α θα α ( θ + α θ ( θ + α pi ( + + ( ( Z ( s α ( θ + α α α = + r + + h + + ( ( Z ( s α ( θ + α α α = + r + + h( α α θα α ( θ + α θ ( θ + α I ( ( + + ( Z ( s α ( θ + α α α = + r h (5 ( pi

8 57 An Inventory oel with ime Depenent DeterioratioRate an RExponential Deman Rate Uner rae Creits Z ( s r ( θ + α α = + + h ( + α + ( r I. α α ( θ + α θ ( θ + α p I > (6 Z ( s r ( θ + α α h( α = > (7 Z ( s r( θ + α α α α ( θ + α θ ( θ + α = + + h( + α + I pi > (8 Z ( s r ( θ + α α = + + h ( + α > (9 he optimal (minimum value of = i* ; i =,,,. Z obtaine by solving i ; i,,, α + θ + α + α + { α θ + α + θ θ + α } { } 8 h r( h I ( r ( ( = =, we have + 8 rα + 8 h + ( r I (8 8α + θα 6S 6 ( r I ( + α + 6pI = ( { } hα + r ( θ + α + hα + ( rα + h 6 S = ( α + θ + α + α + { α θ + α + θ θ + α } { } 8 h r( h I ( ( + 8 rα + 8 h + I (8 8α + θα 6S 6 I ( + α + 6 pi = ( { } hα + r ( θ + α + hα + ( rα + h 6S = ( NUERICAL EXAPLES Example (ase I. Let α =.; h = $ /unit/year; I =.9 / year ; I.6 / year = ; = $ /unit; p=$5/unit; θ=.; r=.; =.95 year; = 5; s=$5/orer. Substituting these values in equation ( an solving; we get optimal = =.987years; an optimal eonomi orere quantity Q*( = 9.9 units; an total relevant ost Z*( = $ , whih verifie ase I i.e. Example (ase II. Let α =.; h = $ /unit/year; = $ /unit; θ=.; r=.;.95year; = ; s=$5/orer. Substituting these values in equation ( an solving; we get optimal = =.585 years,

9 H.S.Shukla,R.P.ripathi & sushil Kumar Yaav 58 an optimal eonomi orere quantity Q*( = units; an total relevant ost Z*( = $ 8.776, whih verifie ase II i.e. < Example (ase III. Let α =.; h = $ /unit/year; I =.9 / year ; I.6 / year = ; = $ /unit; p=$5/unit; θ=.; r=.; =.89 year, =; s=$5/orer. Substituting these values in equation ( an solving; we get optimal = =.68 years; an optimal eonomi orere quantity Q*( =.59 units; an total relevant ost Z*( = $9.9695, whih verifie ase III i.e. Example (ase Iv. Let α =.; h = $ /unit/year; = $ /unit; θ=.; r=.; =.89 year; = 5; S=$/orer. Substituting these values in equation ( an solving; we get optimal = =.5697years; an optimal eonomi orere quantity Q*( = 7.56 units; an total relevant ost Z*( = $.75, whih verifie ase IV i.e. < SENSIIVIY ANALYSIS We have performe sensitivity analysis by hanging s, I, I & r an keeping the remaining parameters at their original values. he orresponing variations is the replenishment yle time, eonomi orer quantity an total relevant ost are exhibite in able (.a,.b,., for ase ( able (able.a,.b for ase (, able (able.a,.b,., for ase ( an able (able.a,.b, for ase ( respetively Case I able, able.a Variation of s keeping all the parameters same as in Ex.. s Replenishment yle time (in Eonomi orer quantity otal relevant ost years Q*( Z*( able.b Variation of I keeping all the parameters same as in Ex.. I Replenishment yle ime (in years Eonomi orer quantity Q*( otal relevant ost Z*(

10 59 An Inventory oel with ime Depenent DeterioratioRate an RExponential Deman Rate Uner rae Creits able. Variation of r keeping all the parameters same as in Ex.. r Replenishment yle time ( in Eonomi orer quantity otal relevant ost years Q*( Z*( Case II able. able.a Variation of s keeping all the parameters same as in Ex.. s Replenishment yle time ( in Eonomi orer quantity otal relevant ost years Q*( Z*( able.b Variation of r keeping all the parameters same as in Ex.. r Replenishment yle time ( in Eonomi orer quantity otal relevant ost years Q*( Z*( Case III able. able.a Variation of s keeping all the parameters same as in Ex.. S Replenishment yle time ( in Eonomi orer quantity otal relevant ost years Q*( Z*( able.b Variation of I keeping all the parameters same as in Ex.. I Replenishment yle time (in Eonomi orer quantity otal relevant ost years Q*( Z*(

11 H.S.Shukla,R.P.ripathi & sushil Kumar Yaav 6 able. Variation of I keeping all the parameters same as in Ex.. I Replenishment yle time ( in Eonomi orer quantity otal relevant ost years Q*( Z*( Case IV able.able.a Variation of s keeping all the parameters same as in Ex.. S Replenishment yle time ( in Eonomi orer quantity otal relevant ost years Q*( Z*( able.b Variation of r keeping all the parameters same as in Ex.. r Replenishment yle time ( in Eonomi orer quantity otal relevant ost years Q*( Z*( he all above results an be summe up as follows: (A. (i.case I from able (a.an inrease of s results inrease of replenishment yle time, eonomi orer quantity Q*( an otal relevant oat Z*(, keeping all other parameters same. (ii. From able (b. An inrease if I results erease of replenishment yle time, eonomi orer quantity Q*( an slight inrease of otal relevant ost Z*(, keeping all other parameters same. (iii From able (.An inrease of r results, slight inrease of replenishment yle time, eonomi orer quantity Q*( an inrease of otal relevant ost Z*(, keeping all other parameters same. (B. (iv. From able (a.an inrease of s results inrease of replenishment yle time, eonomi orer quantity Q*( an otal relevant ost Z*(, keeping all other parameters same. (v.from able (b.an inrease of r results, slight erease of replenishment yle time, eonomi orer quantity Q*( an inrease of the total relevant ost Z*(, keeping all other parameters same. (C. (vi From able (a.an inrease of s results inrease of replenishment yle time, eonomi orer quantity Q*( an otal relevant ost Z*(, keeping all other parameters same.

12 6 An Inventory oel with ime Depenent DeterioratioRate an RExponential Deman Rate Uner rae Creits (vii. From able (b.an inrease of I results, slight erease of replenishment yle time, eonomi orer quantity Q*( an inrease of otal relevant ost Z*(, keeping all other parameters same. (viii From able (.An inrease of I results, erease of replenishment yle time, eonomi orer quantity an otal relevant ost Z*(, keeping all other parameters same. (D. (ix From able (a.an inrease of s results, inrease of eonomi orer quantity Q*( an otal relevant ost Z*(, keeping all other parameters same. (x.from able (b.an inrease of r results, slight erease of replenishment yle time, eonomi orer quantity Q*( an inrease of otal relevant ost Z*(, CONCLUSIONS AND FUURE RESEARCH DIRECION his moel inorporates some realisti features with some kins of inventory. First time epenent eterioration over time is a natural feature for goos. Seonly, ourrene of ash flow in inventory is a marketing strategies an natural phenomena in real situation. hirly, time epenent eman at that time. It is important to onsier the effets of inflation an the time value of money in formulating inventory replenishment poliy. he moel is very useful in the retail business an inustries where the eman is influene by politial fators an natural alamities. We have given a mathematial formulation of the problem presente an optimal proeure for fining optimal replenishment poliy. Four ifferent ases have been isusse. We have also verifie that the effet of inflation in formulating replenishment poliy. Finally, the sensitivity analysis of the solution to hange in the values of ifferent parameters has been isusse. It is seen that hanges in the orer ost (s, the ash isount rate (r, the interest harge (I, an the interest earne (I lea to signifiant effet on the orer quantity as well as otal relevant ost. his paper an be extene for several ways. For instane we may exten the paper for stok epenent eman rate as well as prie epenent eman rate. We may also exten this paper for allowing shortages. Finally we oul generalize the moel for non- eteriorating items. APPENDIX he solution of ( is θ t / ( α t + θ t / I ( t e = C e t = ( C ( t θ + α = + α + t t ( α t + θ t / ( θ + α [ e ( + α t + t, A pprox.] Where C is a onstant of integration whih is obtaine by using the onition I ( =. herefore solution beomes

13 H.S.Shukla,R.P.ripathi & sushil Kumar Yaav 6 θ t ( + α θ α I ( t = ( t + ( t + ( t e 6 REFERENCES [] Goyal, S.K. (985. Eonomi orer quantity uner onition of permissible elay in payment. Journal of Operations Researh Soiety, 6, 5-8 [] Aggarwal, S.P. & Jaggi, C.K. (995.Orering poliies of eteriorating items uner permissible elay in payments. Journal of Operations Researh Soiety, 6, [] Jamal, A..,Sarkar, B.R. & Wan S. (997. An orering polies for eteriorating items with allowable shortage an permissible elay in payment. Journal of Operations Researh Soiety, 8, [] Hwang H.Shinn.S.W. (997. Retailers priing an lot sizing poliy for exponentially eteriorating prouts uner the onition of permissible elay in payment.computers & Operations Researh,, [5] Chung ; K.J.(998. A theorem on the etermination of eonomi orer quantity uner onition of permissible elay in payments. Computers & Operations Researh, 5, 9-5 [6] eng, J., im, J. & Pan, Q. (. Eonomi orer quantity moel with trae reit finaning for non ereasing eman. Omega,, 8-5 [7] Chang, C..Ouyanng, L.Y. & eng, J.. (. An EOQ moel for eteriorating items uner supplier reits linke to orering quantity. Applie athematial oeling, 7, [8] Covert, R.P., Philip,G.C. (97. An EOQ moel for items with waybill istributions eterioration, AIIE rans. 5, -6. [9] Dave, U, Patel, L.K. (, Si. (98. Poliy inventory moel for eteriorating items with time proportional eman, Journal of Operations Researh Soiety,, 7- [] Sahan, R.S.On (, Si (98. Poliy inventory moel for eteriorating items with time proportional eman. Journal of Operations Researh Soiety, 5, -9. [] Datta, J.K. & Pal, A.K. (988. Orer level inventory systems with power eman pattern for items with variable rate of eterioration. Inian Journal of Pure & Applie athematis, 9, -5. [] Goswami, A & Chauhuri, K.S. (99. An EOQ moel for eteriorating items with shortage an a linear tren in eman. Journal of Operations Researh Soiety,, 5- [] Raafal, F. (999 Survey of literature on ontinuously eteriorating inventory moel. Journal of Operations Researh Soiety,, 7-7.

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