EOQ Model for Deteriorating Items with Linear Time Dependent Demand Rate under Permissible Delay in Payments

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1 International Journal of Operations Resear International Journal of Operations Resear Vol. 9, No., (0) EOQ oel for Deteriorating Items wit Linear ime Depenent Deman Rate uner Permissible Delay in Payments R.P. ripati Department of atematis, Grapi Era University, Deraun, (UK) INDIA Reeive Otober 0; Revise February 0; Aepte February 0 Abstrat is stuy presents an inventory moel for eteriorating items wit linearly time-epenent eman rate uner trae reits. atematial moels ave been erive uner four ifferent situations i.e. Case : e yle time is greater tan or equal to, to get a as isount, Case : e yle time is less tan, Case : e yle time is greater tan or equal to, an Case : e yle time is less tan. Computational proeures are propose to obtain optimal yle time of all four ases. Numerial example an sensitivity analysis sows te appliability of te propose moel. Keywors Inventory, eteriorating item, as-isount, linear time-epenent-eman rate.. INRODUCION Large number of resear papers / artiles as been presente by many autors for ontrolling te inventory of eteriorating items. Deteriorating items su as fasion goos, bloo banks, meiines, volatiles, green vegetable, raioative material, potograpi films, et. In many inventory systems te prout generate ave inefinitely long lives. Generally, almost all items eteriorate over time. Often te rate of eterioration is low an tere is little nee to onsier te eterioration for etermining te eonomi lot size. Hene te effet of eterioration annot be ignore in te eision proess of proution lot size. In past few years, great interest as been sown in eveloping matematial moels in te presene of trae reit. In many ases ustomers are onitione to a sipping elay an may be willing to wait for a sort time in orer to get teir first oie. For fasionable items, te lengt of te waiting time for te next yle time woul etermine weter te baklogging will be aepte or not. us te baklogging rate soul be variable an epenent on te lengt of te waiting time for te next yle time. e main objetive of inventory management eals wit minimization of te inventory arrying ost for wi it is require to etermine te optimal stok an optimal time of replenisment of inventory to meet te future eman. In a realisti prout life yle, eman is inreasing wit time uring te growt pase. In lassial inventory moels te eman rate is assume to be a onstant. In reality eman for pysial goos may be time-epenent, stok epenent an prie epenent. An inventory system of ameliorating items for prie epenent eman rate was onsiere by anal et al. (00). You (005) evelope an inventory moel wit prie an time epenent eman. Hou an Lin (00) onsiere an orering poliy wit a ost minimization proeure for eteriorating items uner trae reit an time isounting. Huang (00) erive an eonomi orer quantity uner onitionally permissible elay in payments. Goyal (95) erive an EOQ moel uner te onition of permissible elay in payments. Cang (00) propose an inventory moel uner a situation tat te supplier provies te puraser a permissible elay in payments if te quantity of te puraser s orer is large. Cung an Liao (00) evelope, uner te onition of permissible elay in payments by te quantity orere, a moel etermining te eonomi orer quantity for exponentially eteriorating items. Cung et al. (005) evelope te problem of etermining te eonomi orer quantity uner te onition of permissible elay in payments by te quantity orere. An EOQ moel for eteriorating items uner trae reits is evelope by Ouyang et al. (005). Gare an Sraer (9) evelope a moel for an exponentially eaying inventory. Gare an Sraer s moel was extene by Covert an Pilip (9) by onsiering onstant eterioration rate to a two-parameter Weibull istribution. Hariga (99) generalize te eman pattern to any onave funtion. eng et al. (999), Yang et al. (00) an eng an Yang (00) furter generalize te eman funtion to inlue any non-negative, ontinuous funtion tat flutuates wit time. Wile etermining te optimal orering poliy, te effet of inflation an time value of money annot be ignore. e resear in tis iretion was one by Buzaott (95), wo evelope an EOQ moel wit inflation subjet to ifferent types of priing poliies. Oter relate artiles an be foun by isra (9) an Roy an Cauuri (99), Liao et al. (000) an Cung an Lin (00). Hou an Lin (009) stuie a as flow oriente EOQ moel wit eteriorating items uner permissible elay in payments, an minimum total osts is obtaine. Corresponing autor s tripati_rp0@reiffmail.om -X Copyrigt 0 ORSW

2 R.P.ripati: EOQ oel for Deteriorating Items wit Linear ime Depenent Deman Rate uner Permissible Delay in Payments IJOR Vol. 9, No., (0) Reently, Aggrawal et al. (009) evelope a moel on integrate inventory system wit te effet of inflation an reit perio. In tis paper te eman rate is assume to be a funtion of inflation. ripati an isra (00) evelope EOQ moel on reit finaning in eonomi orering poliies of non-eteriorating items wit time-epenent eman rate in te presene of trae reit using isounte as flow (DCF) approa. Jaggi et al. (00) evelope a moel retailer s optimal replenisment eisions wit trae reit linke eman uner permissible elay in payments. Jaggi et al. (00) evelope a moel on retailer s optimal orering poliy uner two stage trae reit finaning. is paper evelops an inventory moel uner two levels of trae reit poliy by assuming te eman is a funtion of reit perio offere by te retailer to te ustomers using isounte as flow (DCF) approa. Hwang an Sinn (99) evelope retailer s priing an lot sizing poliy for exponentially eteriorating prouts uner te onition of permissible elay in payments. In paper Hwang an Sinn analyze ow a retailer an etermine te optimal retail prie an lot size simultaneously wen te supplier permits elay in payments for an orer of a prout wose eman rate is represente by a onstant prie elastiity funtion. Inventory moel wit time-epenent eman rate uner inflation wen supplier reit linke to orer quantity is evelope by ripati (0). In tis paper ripati establise an inventory moel for non- eteriorating items an time- epenent eman rate uner inflation wen supplier offers a permissible elay to te puraser, if te orer quantity is greater tan or equal to a preetermine quantity. e aim of te paper is to evelop an EOQ moel for eteriorating items wit linear time-epenent eman rate uner permissible elay in payment. In tis stuy sortages are not allowe. atematial moels are erive uner four ifferent irumstanes i.e. Case. is less tan or equal to yle time; Case : Cyle time is less tan ; Case : Cyle time is greater tan or equal to an Case : Cyle time is less tan. e expressions for an inventory systems total relevant osts an erive for tese four ases. Finally, we provie numerial example an sensitivity analysis for illustration of te propose moel. e rest of te paper is organize as follows: In setion notation an assumptions are given. In setion we evelop matematial formulation for te solution of total relevant ost. aylor s series expansion is use to fin lose form solution of te optimal values of yle time, Orer quantity an total relevant osts wit regar to four ifferent ases followe by numerial examples in setion. We provie sensitivity analysis in setion 5 followe by onlusion an future resear in te last setion.. ASSUPIONS AND NOAIONS e following assumptions are being mae trougout te paper: () e eman for te item is linearly time-epenent. () Replenisment is instantaneous. () Sortages are not allowe. () ime orizon is infinite. (5) If te aount is not settle uring te time, generate sales revenue is eposite in an interest bearing aount. At te en of reit perio, te aount is settle as well as te buyer pays off all units sol an starts paying for te interest arges on te item in stok. In tis ase, supplier provies a as isount if te full payment is pai witin time, oterwise, te full payment is pai witin time. e aount is settle wen te payment is pai ( > ). In aition, te following notations are use trougout te manusript: : te unit oling ost per year exluing interest arges p : te selling prie per unit : te unit purasing ost, wit < p I : te interest arge per ollar in stoks per year by te supplier I : te interest earne per ollar per year s : te orering ost per orer r : te as isount rate θ : te onstant eterioration rate, were 0 θ : te perio of as isount : te perio of permissible elay in settling aount wit > : te replenisment time interval D : te eman rate per year i.e. D = D(t) = a + bt, a > 0, 0 < b < I(t) : te level of inventory at time t, 0 θ Q : te orer quantity Z () : te total relevant ost per year for ase Z () : te total relevant ost per year for ase Z () : te total relevant ost per year for ase Z () : te total relevant ost per year for ase,,, : te optimal yle times for ase, ase, ase an ase respetively

3 R.P.ripati: EOQ oel for Deteriorating Items wit Linear ime Depenent Deman Rate uner Permissible Delay in Payments IJOR Vol. 9, No., (0) Q ( ), Q ( ), Q ( ), Q ( ): te optimal orer quantities for ase,, an respetively. Z ( ), Z ( ), Z ( ), Z ( ): te optimal total relevant osts per year for ase,, an respetively. e total relevant ost onsists of () ost of plaing orer, () ost of eteriorate units, () ost of arrying inventory exluing interest arges, () as-isount earne if te payment is mae at (5) ost of interest arges for unsol items after te permissible elay or an () interest earne for sales revenue uring te permissible elay perio [0, ] or [0, ].. AHEAICAL FORULAION e inventory level I(t) at any time t generally ereases mainly to meet eman an partially ue to eterioration. e variation of inventory wit respet to time t an be esribe by te following ifferential equation: I() t +θ I(t) = D (= a + bt), 0 t () t Wit te bounary onition I() = 0. e solution of equation () is given by a a I(t) = ( t b ) ( t e e ) t, 0 t () e orer quantity Q is given by a a Q = b e e () b e total eman uring one yle is (a + ). us te number of eteriorating items uring a replenisment yle is a b be b e a () e total relevant ost per year onsists of te following elements: () Cost of plaing orers = s (5) Q a b be () Cost of purasing units = e () a b e b e () Cost of arrying inventory = I() t t () 0 Regaring as isount, interest arges an earne, te four possible ases base on te ustomer s two oies (i.e. pays at or ) an te lengt of. In ase, te payment is pai at to get a as isount an. For ase, te ustomer pays in full at but <. In te same manner, if te payments are pai at time to get te permissible an, ten it is ase. As to ase, te ustomer pays in full at but <. All four ases sown in figure. Inventory Inventory Inventory Inventory Q Q Q Q 0 ime 0 ime 0 ime 0 Case : Case : < Case : Case : < ime Figure. Grapial representation of four ifferent situations

4 R.P.ripati: EOQ oel for Deteriorating Items wit Linear ime Depenent Deman Rate uner Permissible Delay in Payments IJOR Vol. 9, No., (0) Case : e isount saving per year by ustomer is rq r a b be e () e ustomer pays off all units orere at time to get te as isount. us te items in te stoks ave to be finane at interest rate I after time, te interest payable per year in tis ase is ( ) ( ri ) ( ri ) a b e ( ) I() t t b e + (9) During [0, ], te ustomer sells prouts an eposits te revenue into an aount tat earns I per ollar per year. us interest earne per year is pi ( ) pi a a bt tt b 0 (0) e total relevant ost per year Z () is given by Z () = ost of plaing orer + ost of purasing + ost of arrying inventory isount saving per year + interest payable per year interest earne per year. = s + ( r) a b be e a b e b e + ( ( ri ) ) a b e ( ) + b e + pi a b () Case : < b b In tis ase te ustomer sells (a + ) units in total at time, an as ( r) (a + ) to pay te supplier in full at te time. us tere is no interest payable wile te as isount is te same as tat in ase. e interest earne per year is b pi a () e total relevant ost per year Z () is Z () = s + ( r) a b be e a b e b e + b pi a () Case : In tis ase, te payment is pai at time, tere is no as isount. e interest payable per year is ( ) I a b e ( ) b e + () e interest earne per year is pi (a b ) e total relevant ost per year Z () is Z () = s + a b be e a b e b e + ( ) I a b e ( ) b e + pi (a b ) Case : < In tis ase, tere is no interest arge. e interest earne per year is (5) ()

5 R.P.ripati: EOQ oel for Deteriorating Items wit Linear ime Depenent Deman Rate uner Permissible Delay in Payments IJOR Vol. 9, No., (0) 5 b pi a () erefore, te total relevant ost per year Z () is Z () = s + a b be e a b e b e + b pi a () It is iffiult to obtain te optimal solution in expliit form for equations (), (), () an (). erefore, te moel will be solve approximately by using a trunate aylor s series for te exponential terms i.e. e, e et. (9) wi is vali approximation for smaller values of θ, θ, et. Wit te above approximation, te total relevant ost per year in all four ases are given by s b ( a b) ( ri ) ( ) Z () ( r) a ( ) (0) pi - a b Z () s + b b ( r) a ( ) ( a b) pi a () Z () s + b ( a b) I ( ) a ( ) pi a b () Z () s + b b a ( ) ( a b) pi a () Note tat te purpose of tis approximation is to obtain te unique lose form solution for te optimal solution. By taking first an seon orer erivatives of Z i (), i =,,,, wit respet to, we obtain Z ( ) s ( ) bi r = + a b( ) ( a b) I ( a b) Z ( ) Z ( ) Z ( ) Z ( ) Z Z = = = pi a b + () s ( r) pi (5) s I a b( ) ( a b) a b () s ( ) ( ) a b a b a b () pi (a b ) s ( r) ai b ( r)( I >0 () + a b( ) ( a b) a b + + s = + ( ) s bpi = b ( r) > 0 (9) ( ) s = + pi (a b ) s I a b I > 0 (0) Z ( ) s bpi = b ( ) > 0 ()

6 R.P.ripati: EOQ oel for Deteriorating Items wit Linear ime Depenent Deman Rate uner Permissible Delay in Payments IJOR Vol. 9, No., (0) Z( ) i Sine > 0, i =,,,, te optimal (minimum) values of = i, i =,,, are obtaine on solving 0, i =,,, from equations (), (5), () an () respetively, we obtain b{ ( I )( r)} { a ( r)( b a b I ai )} {s ai ( r) Z ( ) i = pi (a b )} 0 () b{ ( r) pi } { a ( r)( b a ) pi ( a b )} s = 0 () b( I ) { a ( b a) I ( a b )} {s ai pi (a b )} 0 () b{ pi } { a ( b a ) pi ( a b )} s = 0 (5) Speial ase (a) If s ai ( r) = pi (a b ). From equation (), we obtain ' = = ( r )( b I ai b a ) a b { ( I)( r)} ' ' erefore Z () = Z ( ) is bi ' ' Z ( ) = ( r) a( I) { ( r)( b I a b a) a} b { ( I )( r)} Speial ase (b) If a = ( r) ( b I a b a). From Eq. (), we obtain {s ai ( r) pi (a b )} '' = = b { ( I)( r)} '' '' erefore Z () = Z ( ) is / '' '' b / / Z ( ) = ( r)( I ) / s a( r) I pi (a b ) () bi + ( r) a( I) Speial ase. If pi ( b a) a ( r)( b a). From equation (), we obtain / ' s = = b{ ( r) pi } ' ' erefore Z () = Z ( ) is / ' ' ( s) / Z ( ) = ( r) pi a ( r) pi () Speial ase (a) If pi ( a b ) s ai. From equation (), we obtain ' bi a a b ai = () b( I ) ' ' erefore Z () = Z ( ) is ' ' bi bi a a b ai Z ( ) = a( I) b ( I ) Speial ase (b) If b I = a + (b + aθ) + a I. From equation (), we obtain () () () (9) (0) ()

7 R.P.ripati: EOQ oel for Deteriorating Items wit Linear ime Depenent Deman Rate uner Permissible Delay in Payments IJOR Vol. 9, No., (0) = '' = s ai pi (a b ) ( b I ) " " erefore Z () = Z ( ) is / Z " " ( ) = b / / I ) s ai pi (a b ) / () Speial ase = / + a( I bi If pi ( b a ) a b a. From equation (5), we obtain ' s b( pi ) erefore Z () = Z ( ) is ( s) ' ' / / ' ' / Z ( ) = ' pi a( pi ) () o ensure >, we substitute () into inequality >, we obtain (a + b ) + ( r) ( b + ai + b + aθ) < 0 () ' Sine o < r < inequality () oes not exist. erefore speial ase (a) oes not exist for s ai ( r) = pi (a b ) Again, to ensure s ai ( r) '' '' >, we substitute () into inequality >, we obtain > pi ( a b ) b b ( I )( r ) (9) In equality (9) is vali, if a = ( r) (b I a b aθ ) ' ' o ensure >, we substitute (0) into inequality <, we obtain s < b ( r) pi (50) Speial ase (a) oes not exists for s ai pi ( a b ) (as speial ase (a)) '' o ensure >, we substitute () into inequality >, we obtain s ai > pi (a b ) b b ( ) I '' (5) e inequality (5) is vali if b I = a + (b + aθ) + ai ' o ensure <, we substitute () into inequality <, we obtain s < b pi ' (5) e inequality (5) is vali if pi (b a) = a + b + aθ. () (5) (). NUERICAL EXAPLES Given a = 500 units/year, b = 0.5 unit, = $ 5/unit/year, I = $0.09/year, I = 0.0/year, = $ 5 per unit, p = $ 0 per unit, r = 0.0, θ = 0.0, = 5 ays = years an = 0 ays = 0.09 years. Case : > For s = 5, = Case : < For s =, = Case : For s =, = Case : < For s = 5, = = years, Q = = 0.0 years, Q = = 0.0 years, Q = = 0.09 years, Q = Q =.9, Q = 9.50, Q =., Q =.99, Z = $ 0.0 Z = $ 5. Z = $ 9. Z = $ 0.5

8 R.P.ripati: EOQ oel for Deteriorating Items wit Linear ime Depenent Deman Rate uner Permissible Delay in Payments IJOR Vol. 9, No., (0) 5. SENSIIVIY ANALYSIS We ave performe sensitivity analysis by anging s an an keeping te remaining parameters at teir original values. e orresponing variations in te yle time, eonomi orer quantity an total relevant ost per year are exibite in able (able.a, able.b) for ase I, able (able.a, able.b) for ase, able (able.a, able.b) for ase, an able (able.a, able.b) for ase respetively. s able. Case able.a Sensitivity analysis on s ( = 5) Replenisment yle time Eonomi orer quantity (in years) Q ( ) units otal relevant ost Z ( ) in ollars able.b Sensitivity analysis on (s = 5) Replenisment yle time Eonomi orer quantity (in years) Q ( ) units otal relevant ost Z ( ) in ollars s able. Case able.a Sensitivity analysis on s ( = 5) Replenisment yle time Eonomi orer quantity (in years) Q ( ) units otal relevant ost Z ( ) in ollars able.b Sensitivity analysis on (s = ) Replenisment yle time Eonomi orer quantity (in years) Q ( ) units otal relevant ost Z ( ) in ollars

9 R.P.ripati: EOQ oel for Deteriorating Items wit Linear ime Depenent Deman Rate uner Permissible Delay in Payments IJOR Vol. 9, No., (0) 9 s Replenisment yle time (in years) able. able.a Sensitivity analysis on s ( = 5) Eonomi orer quantity Q ( ) units otal relevant ost Z ( ) in ollars able.b Sensitivity analysis on (s = 5) Replenisment yle time Eonomi orer quantity (in years) Q ( ) units otal relevant ost Z ( ) in ollars s able. able.a Sensitivity analysis on s ( = 5) Replenisment yle time Eonomi orer quantity (in years) Q ( ) units otal relevant ost Z ( ) in ollars able.b Sensitivity analysis on (s = 5) Replenisment yle time Eonomi orer quantity (in years) Q ( ) units otal relevant ost Z ( ) in ollars From te above tables te following results ave been obtaine: (a) e omputational results are sown in able.a, iniates tat iger value of orering ost s implies iger values of replenisment yle time, orer quantity Q ( ) an total relevant ost Z ( ). (b) e omputational results are sown in able.b, iniates tat iger value of unit oling ost implies lower values of replenisment yle time, orer quantity Q ( ) an total relevant ost Z ( ).

10 R.P.ripati: EOQ oel for Deteriorating Items wit Linear ime Depenent Deman Rate uner Permissible Delay in Payments IJOR Vol. 9, No., (0) 0 () () (e) (f) (g) e omputational results are sown in able.a, iniates tat iger value of orering ost s implies iger values of replenisment yle time, orer quantity Q ( ) an total relevant ost Z ( ). e omputational results are sown in able.b, iniates tat iger value of unit oling ost implies lower values of replenisment yle time, orer quantity Q ( ) an total relevant ostz ( ). e omputational results are sown in able.a, iniates tat iger value of orering ost s implies iger values of replenisment yle time, orer quantity Q ( ) an total relevant ost Z ( ). e omputational results are sown in able.b, iniates tat iger value of unit oling ost implies lower values of replenisment yle time, orer quantity Q ( ) an total relevant ostz ( ). e omputational results are sown in able.a, iniates tat iger value of orering ost s implies iger values of replenisment yle time, orer quantity Q ( ) an total relevant ost Z ( ). () e omputational results are sown in able.b, iniates tat iger value of unit oling ost implies lower values of replenisment yle time, orer quantity Q ( ) an total relevant ostz ( ). e main ifferene between Hwang an Seong (99) paper an tis paper is as follows: () In Hwang an Seong (99) paper eman rate is a funtion of retail prie wile in tis paper eman rate is a funtion of time. () In Hwang an Seong (99) paper two ifferent ases ave been onsiere i.e. ase. Creit perio t is less tan or equal to yle time an ase. Creit perio t is greater tan yle time, wile in tis paper four ifferent ases ave been onsiere i.e. ase : <,, ase :, <, were is te replenisment time interval, is te perio of as isount an is te perio of permissible elay in settling te aount wit >. () In Hwang an Seong (99) paper as isount is not onsiere wile in tis paper as isount is onsiere. () In Hwang an Seong (99) paper maximum annual net profit ave been obtaine wile in tis paper minimum total relevant ost per is obtaine. (5) In Hwang an Seong (99) paper te annual net profit is a onave funtion of yle time, wile in tis paper total relevant ost is a onvex funtion of yle time. Also te main ifferene between ripati (0) paper an tis paper is as follows: () In R.P.ripati (0) paper, lengt of planning orizon H = n ave been onsiere, were n is te number of replenisment an is an interval of time between replenisments, wile is tis paper only yle time is onsiere. () In R.P.ripati (0) paper four, ifferent ases ave been onsiere i.e.ase: 0 <<, ase. < < m, ase. m an ase. m, were is te time interval tat Q units are eplete to zero, Q is te minimum orer quantity at wi te elay in payments is permitte, an m is te permissible elay in settling aount, wile in tis paper four ifferent ases is onsiere in ifferent ways: i.e. ase :, ase : <, ase : an ase : <, were is replenisment time interval, is te perio of as isount, is te perio of permissible elay in settling aount wit >. () In R.P.ripati (0) paper speial ases ave been not onsiere, wile in tis paper four speial ases are onsiere. () In R.P.ripati (0) paper EOQ moel ave been evelope for non eteriorating items, wile in tis paper EOQ moel is evelope for eteriorating items. (5) In R.P.ripati (0) paper eman rate is time epenent, wile in tis paper te eman rate is linearly time epenent (wi is more useful in real life). CONCLUSION AND FUURE RESEARCH We evelope EOQ moel for eteriorating items an time epenent eman rate to fin te optimal orering poliy wen te supplier provies a as isount an (or) trae reit. We use aylor s series approximation to obtain te expliit lose-form solution of te optimal replenisment yle time. We also araterize te effet of te value of parameters on te optimal replenisment yle time. Numerial example an sensitivity analysis is given to illustrate te moel. Numerial tenique meto is applie to obtain optimal yle time. e propose moel an be extene in several ways. For instane, we may exten te eman rate to a quarati time-epenent eman rate. We oul also onsier te eman rate as a funtion of quantity, selling prie, prout quality an oters. Finally we oul generalize te moel to allow for sortages, quantity isount an time-epenent eterioration rate, et.. ACKNOWLEDGEENS e autor eeply appreiates anaging Eitor ing-jong Yao for is enouragement an patiene. I also tank two anonymous referees for improving te quality of te paper.

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