A STUDY OF INFLATION EFFECTS ON AN EOQ MODEL FOR WEIBULL DETERIORATING/AMELIORATING ITEMS WITH RAMP TYPE OF DEMAND AND SHORTAGES

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1 Yugoslav Journal of Operaions Research 3 (3) Numer 3, DOI:.98/YJOR838V A SUDY OF INFLAION EFFECS ON AN EOQ MODEL FOR WEIBULL DEERIORAING/AMELIORAING IEMS WIH RAMP YPE OF DEMAND AND SHORAGES M. VALLIAHAL Chikkaiah Naicker College, Erode,amilnadu, 6384, India, alal_a@yahoo.com R. UHAYAKUMAR Gandhigram Universiy, Gandhigram, amilnadu, 643, India, uhayagri@gmail.com Received: Augus / Acceped: March 3 Asrac: his paper deals wih he effecs of inflaion and ime discouning on an invenory model wih general ramp ype demand rae, ime dependen (Weiull) deerioraion rae and parial acklogging of unsaisfied demand. he model is sudied under he replenishmen policy, saring wih shorages under wo differen ypes of acklogging raes, and heir comparaive sudy is also provided. We hen use he compuer sofware, MALAB o find he opimal replenishmen policies. Duraion of posiive invenory level is aken as he decision variale o minimize he oal cos of he proposed sysem. Numerical examples are hen aken o illusrae he soluion procedure. Finally, sensiiviy of he opimal soluion o changes of he values of differen sysem parameers is also sudied. Keywords: Invenory, ramp ype of demand, inflaion, parial acklogging. МSC: 9B5.. INRODUCION For long, invenory models have deal wih he case where demand is eiher a consan or a monoonic funcion. Consan demands ake place in he fully developed

2 44 М. Valliahal, R. Uhayakumar / A Sudy of Inflaion Effecs on an EOQ Model sage of he iem, and monoonic in he eginning or a he las sage of he cycle of life. Exising invenory models deal wih hese ypes of demand funcions, u in pracice, i is no possile. he demands of fashionale goods increase up o a cerain level and afer ha ecome seady. Such ype of demand funcions is known as ramp ype of demand. (See Skouri e al. []). Researches relaed o his field are: Goyal and Giri [9], Mahaa and Goswami [], Manna and Chaudhuri [3], Panda e al. [8], ec. Now- a- days many counries have een confroned wih flucuaing inflaion raes ha are ofen far from eing negligile. Buzzaco [4] developed he firs EOQ model aking inflaion ino accoun. Researches relaed o his field are: Mishra [4], [5], Biermann and homas [], Yang e al. [], Moon and Yun [6], Brahmha [3] and so on. Some cusomers would like o wai during he shorage period, u ohers would no. In he los sales case, he cusomer s demand for he iem is los and presumaly filled y a compeior, which can e conemplaed as he loss of profi on he sales. Consequenly, he opporuniy cos resuling from los sales should e delieraed in he model. In he previous papers on he opic, shorages were endorsed and assumed o e compleely acklogged or compleely los. Many recen sudies have muaed invenory policies y considering he parial acklogging rae. In some invenory sysems, for many socks like fashionale commodiies, he amoun of demand acklogged ecomes he main facor of deermining if i is o e acceped or no. Mainly, researchers use wo ypes of acklogging such as consan acklogging and ime dependen parial acklogging, i.e. decreasing funcion of a waiing ime up o he nex replenishmen. Aad [] and Dye and Ouyang [7] have considered he parial acklogging rae dependen on he waiing ime up o he nex replenishmen. Recenly, many researchers have sudied he effecs of oh ypes of acklogging on he opimal replenishmen policies of he invenory sysem. Some of he recen works relaed o his concep are: Chang e al [5], Ouyang e al. [7], San Jose e al. [9], Wu e al. [] and Dye e al. [8], ec. Here, we exend and modify Chern e al. [6] in wo direcions: Firsly, y considering he effec of inflaion and ime discouning on he replenishmen policies. Secondly, we also discuss amelioraing iems in he descried model. (See Hwang [], []). Comparaive analysis eween he models of non-exponenial acklogging rae wih linear demand funcion and exponenial acklogging rae wih non-linear demand funcion hrough numerical resuls is also given. he res of his paper is organized as follows. In secion, he noaions and assumpions are given. In secion 3, we presen he mahemaical model. In secion 4, numerical examples are given o illusrae he model. Finally, we give he conclusion of his paper.. NOAIONS AND ASSUMPIONS o develop he Mahemaical model, he following noaions and assumpions are eing made.. Noaions K d h he ordering cos of invenory per order he deerioraion cos per uni per uni ime he holding cos per uni per uni ime

3 М. Valliahal, R. Uhayakumar / A Sudy of Inflaion Effecs on an EOQ Model 443 s he shorage cos per uni per uni ime he parameer of he ramp ype demand funcion (ime poin) π he opporuniy cos due o los sales per uni r he ne inflaion rae I he invenory level a ime, [, ] f he demand rae a ime, [, ] θ he deerioraion rae a ime, [, ] he lengh of he replenishmen cycle he ime a which he shorage sars, C he oal cos per uni ime... Assumpions he replenishmen rae is infinie and lead ime is zero. he uni cos and he invenory carrying cos are known and consan. he selling price per uni and he ordering cos per order are known and consan Shorages are allowed. he fracion of shorages acklogged is a non-increasing funcion B(x), where x is he waiing ime up o he nex replenishmen, and B(x) wih B =.ogeher wih his, we assume ha B(x) + B (x) >, where B (x) is he derivaive of B(x).When B(x) =(or ), i corresponds o a complee acklogging (or complee los sales) model. here is no repair or replacemen of he deerioraed iems during he producion cycle. 3. MODEL FORMULAION Due o he posiion of he ime poin, he model can e classified ino wo cases as follows: Case : and Case : We shall discuss firs Case, and hen Case. 3. Case : A ypical ehavior of he invenory in a cycle is depiced in Figure. he insananeous saes of he invenory level I a ime ( ) can e descried y he following differenial equaion: di d = f B ( )], wih he oundary condiion I ( ) =. Solving he differenial equaion, he invenory level is I = f( u) B( u) du,

4 444 М. Valliahal, R. Uhayakumar / A Sudy of Inflaion Effecs on an EOQ Model he insananeous saes of he invenory level I a ime ( ) can e descried y he following differenial equaion: di d ( ) ( ) = f θ I,, I = I (3) + Figure. Graphical represenaion of invenory sysem wih he oundary condiion I( ) I( ) invenory level is =. Solving he differenial equaion (3), he + a ax ax I = e e f ( x) dx + f ( ) e dx,, I = I ( + ) ( ) (4) he insananeous saes of he invenory level I a ime ( ) can e descried y he following differenial equaion: di d ( ) = f( ) θ I( ),, I = (5) wih he oundary condiion I ( ) =. Solving he differenial equaion (5), he invenory level is a ax I = e f( ) e dx, (6) he presen worh of he holding cos for carrying invenory over he period [, ] is given y

5 М. Valliahal, R. Uhayakumar / A Sudy of Inflaion Effecs on an EOQ Model 445 HC = h I e d hi e d hi e d = + a ax ax a ax = he e f( x) dx+ f( ) e dx e d+ h e f( ) e dx e d he presen worh of he shorage cos over he period [, ] is given y (7) SC = s I e d = s f( u) B( u) due d he presen worh of he opporuniy cos due o los sales during he period [, ] is given y (8) OC = π f B e d ( ) ( ) = π f B e d (9) he presen worh of he deerioraion cos over he period [, ] is given y e f e dx e f x dx e f e d f e d a r DC = d ( ) ax + ax ( ) ( ) ( ) herefore, he presen worh of he oal cos during he period [, ] is given y C( ) = he presen worh of he holding cos + shorage cos +opporuniy cos due o los sales + deerioraion cos

6 446 М. Valliahal, R. Uhayakumar / A Sudy of Inflaion Effecs on an EOQ Model a ax ax a ax = h e e f ( x) dx + f ( ) e dx e d + h e f ( ) e dxe d ( ) ( ) π ( ) a ax ax r r + s f u B u du e d + f B e d + d e f( ) e dx+ e f( x) dx e r f( ) e d f( ) e d he soluions for he opimal values of (say *) can e found y solving he following equaions simulaneously: C ( ) C ( ) C ( ) = and = Provided hey saisfy he condiions: C = ( ) ( *) > (3) a ax ax r h ( e ) f( x) e dx + f( ) e dx e + s f( u) B( u) du e + π (( B( )) f( )) e + df( ) e r r r a ( ) ( ) r r + d e r e + e ( a) a e f ( ) e ax dx + f ( x) e ax dx ( ) ( ) db db u + π f e d + s f ( u) du e d = d d (4)

7 М. Valliahal, R. Uhayakumar / A Sudy of Inflaion Effecs on an EOQ Model 447 C ( ) = + + a ( ) ax ax r h e f( x) e dx f( ) e dx e db( u) r s( r) f( u) B( u) du f( u) du f( ) e d ( ) ( r) ( a)( ) + a r ax ( ) + + ( r)( a) r r r ( ) (( )) ( ) ( ) d B( ) d B( u) ( ) ax e e f( ) e dx f( x) e dx d + r f( ) e + f '( ) e + e r + a f( ) + π f e d + s f u du e d > d d Summarizing he aove resuls, we now esalish he following soluion procedure o oain he opimal soluion of our prolem. 3.. Soluion procedure for Case Sep : ake =. Sep : Solving (4) find. Sep 3: Check equaion (3); if yes, go o sep ; oherwise, go ack o sep. Sep 4: Using equaion o find C Value. 3. Case : A ypical ehavior of he invenory in a cycle is depiced in Figure. (5) Figure Graphical represenaion of invenory sysem

8 448 М. Valliahal, R. Uhayakumar / A Sudy of Inflaion Effecs on an EOQ Model he insananeous saes of he invenory level I a ime ( ) can e descried y he following differenial equaion: di d = f ( ) θ I, (6) wih he oundary condiion I( ) =. Solving he differenial equaion (6), he invenory level is as ax (7) a I = e f( ) e dx, he insananeous saes of he invenory level I a ime ( ) can e descried y he following differenial equaion: di d = f B( )], (8) wih he oundary condiion I ( ) =. Solving he differenial equaion (8), he invenory level is as I = fub ( ) ( udu ), (9) he insananeous saes of he invenory level I a ime ( ) can e descried y he following differenial equaion: di d = f ( ) B ( )], wih he oundary condiion I( ) I( ) invenory level is as =. Solving he differenial equaion, he + I( ) = f( u) B( u) du+ f( ) B( u) du, he presen worh of he holding cos for carrying invenory over he period [, ] is given y HC = h I e d a ax = h e f( ) e dxe d he presen worh of he shorage cos over he period [, ] is given y

9 М. Valliahal, R. Uhayakumar / A Sudy of Inflaion Effecs on an EOQ Model 449 SC = s I e d = s f ( u) B( u) du e d + s f ( u) B( u ) du f ( ) B( u ) du e d he presen worh of he opporuniy cos due o los sales during he period [, ] is given y OC = π f B e d + f B e d (3) ( ) ( ) ( ) ( ) (4) he presen worh of he deerioraion cos over he period [, ] is given y d e e f e dx e f d a r ax DC = ( ) ( ) (5) herefore, he presen worh of he oal cos during he period [,] is given y C( ) = he presen worh of he holding cos + shorage cos +opporuniy cos due o los sales + deerioraion cos = h e f e dx e d + s f u B u du e d a ax ( ) ( ) ( ) ( ) ( ) ( ) ( ) + s f u B u du f B u du e d + π f ( ) B( ) e d+ f ( ) B( ) e d a r ax + d e e f( ) e dx e f ( ) d he soluions for he opimal values of (say *) can e found y solving he following equaions simulaneously: C ( ) = C Provided hey saisfy he condiions: ( ) ( * ) > (6) (7) (8)

10 45 М. Valliahal, R. Uhayakumar / A Sudy of Inflaion Effecs on an EOQ Model C (, ) = h e f e e dx s f u du e d r ( a ax db u) ( ) ( ) + ( ) d + π + d db( u) db( u) e f( u) du f( ) du d d + r e f( u) B( u) du f( ) B( u) du db( ) db( ) f e d + f ( ) e d df ( ) e d + d a ( ) ( ) r r d e r e e ( a) a e f( ) e ax dx + + = C (, ) = ( )( ) ( ) r r a ax h e f( ) + re e + a f( ) e dx d B( u) + s f( u) du e d d d B( u) d B( u) + e f( u) du f( ) du d d d r db( u) db ( u) + e f( u) du f( ) du d d r re f ( u) B( u) du f ( ) B( u) du r (9)

11 М. Valliahal, R. Uhayakumar / A Sudy of Inflaion Effecs on an EOQ Model 45 r db( u) db( u) + e f( u) du f( ) du+ f( ) d d d B( ) d B( ) + π f e d + f ( ) e d rdf ( ) e d d a ( ) d e ( r) e r e r + + ( a) a e f( ) e ax dx > Summarizing he aove resuls, we now esalish he following soluion procedure o oain he opimal soluion of our prolem. 3.3 Soluion procedure Sep: ake =. Sep : Solving (9) find. Sep 3: Check equaion (8); if yes, go o sep ; oherwise go ack o sep. Sep 4: Using equaion (6) o find C Value. Find he opimum order quaniy using he relaion, r (3) Q a ax ax f ( x) B( x) dx+ e e f ( x) dx+ f( ) e dx = a ax B ( x) f( xdx ) + f( ) B ( xdx ) + e f( ) e dx > 4. NUMERICAL EXAMPLES (3) he purpose of his secion is o illusrae he resuls of our models and demonsrae he performance of he soluion procedures presened in secions 3. and 3.4. Example presened elow is for a non-linear demand funcion wih exponenial parial acklogging. Example presened elow is for a linear demand funcion wih nonexponenial parial acklogging. 4. Example o illusrae he proposed model, le us consider he following example: Le s = 5, d = 5, f = e,h = 3, π =, a =., =, =, =., r =.4, B( ( ). ) = e in appropriae unis. Numerical values for deerioraing iems are shown in ale, and ha of amelioraing iems are shown in ale.

12 45 М. Valliahal, R. Uhayakumar / A Sudy of Inflaion Effecs on an EOQ Model ale Opimal replenishmen schedule for deerioraing iems * Q* C* ale Opimal replenishmen schedule for amelioraing iems * Q* C* Example o illusrae he proposed model, le us consider he following example: Le s = 5, d = 5, f = 3 + 5, h = 3, π =, a =., =, =, =., r =.4, B( ) =. + in appropriae unis. Numerical values for deerioraing iems are shown in ale 3, and ha of amelioraing iems are shown in ale 4. ale 3. Opimal replenishmen schedule for deerioraing iems * Q* C* ale 4. Opimal replenishmen schedule for amelioraing iems * Q* C* Comparaive Sudy From he numerical values displayed in ale and ale 3, we conclude ha he resuls of non-exponenial acklogging rae wih linear demand funcion are eer han ha of exponenial acklogging rae wih non-linear demand funcion. Opimum order quaniy and he oal cos of he firs one are lesser han of he second one. Duraion of shorage period of he previous model is longer han he laer one. So, exponenial acklogging model shorens he shorage ime and minimizes he oal profi. he aove resuls are he same as for ha of amelioraing iems. 4.4 Parameric Sudy In order o sudy how various parameers affec he opimal soluion of he proposed invenory model, sensiiviy analysis is performed, keeping all he oher parameers fixed and varying a single parameer a a ime, for he same se of values. Sensiiviy analyses for various parameers involved in he model menioned aove are shown in ale 5.

13 М. Valliahal, R. Uhayakumar / A Sudy of Inflaion Effecs on an EOQ Model 453 ale 5. Effec of various parameers on he oal cos for deerioraing iems Parameer value C r h π s B he purpose of his secion is o give some pracical applicaions of our model. 4.5 Managerial Implicaions Based on our numerical resuls, we oain he following managerial phenomena: Increasing inflaion rae is no suiale for he reailer. ry o reduce he holding cos. Maximize he acklogging rae and minimize he los sales case. Reduce he shorage cos. he aove applicaions are also applicale o suppliers. Some special cases of our model are lised elow: 4.6 Some special cases If we ake r =,he descried model is reduced o ha of an EOQ model for Weiull deerioraing/amelioraing iems wih parial acklogging and ramp ype of demand. If we akeθ = θ,he descried model is reduced o ha of an EOQ model for Weiull amelioraing iems wih parial acklogging and ramp ype of demand. 5. CONCLUSIONS his paper discussed an effec of inflaion and ime discouning on an EOQ model for Weiull deerioraing/amelioraing iems wih parial acklogging and ramp ype of demand. Non-exponenial acklogging rae wih linear demand funcion gives eer resuls han exponenial acklogging rae wih non-linear demand funcion.

14 454 М. Valliahal, R. Uhayakumar / A Sudy of Inflaion Effecs on an EOQ Model Opimum order quaniy and he oal cos of he firs one are lesser han ha of he second one. From our numerical values, we concluded ha inflaion definiely plays a major role on he replenishmen policies and he opimum invenory cos. Here, oal cos is more sensiive o inflaion han opimum order quaniy. he proposed model can e exended in several ways. We could exend he deerminisic demand funcion o sochasic demand paerns, as a funcion of selling price, ec. REFERENCES [] Aad, P.L., Opimal pricing and lo sizing under condiions of perishailiy and parial ackordering, Managemen Science, 4(8) (996) [] Bierman, H., and homas, J., Invenory decisions under inflaionary condiion, Decision Science, 8 (977) [3] Brahmha, A.C., Economic order quaniy under variale rae of inflaion and mark-up prices, Produciviy, 3(98) 7 3. [4] Buzaco, J.A., Economic order quaniies wih inflaion, Operaion Research Quarerly, 6 (3 (975) [5] Chang, H.J., eng, J.., Oyuang, L.Y., and Dye, C.Y., Reailer s opimal pricing and lo sizing policies for deerioraing iems wih parial acklogging, European Journal of Operaional Research, 68 (6) [6] Chern, M.S., Yang, H.L., eng. J.., and Papachrisos, S., Parial acklogging invenory losize models for deerioraing iems wih flucuaing demand under inflaion, European Journal of Operaional Research, 9 (8) 7 4. [7] Dye, C.Y., and Ouyang, L.Y., An EOQ model for perishale iems under sock-dependen selling rae and ime- dependen parial acklogging, European Journal of Operaional Research, 63 (3 (5) [8] Dye, C.Y., Hsieh,.P., and Ouyang, L.Y., Deermining opimal selling price and lo size wih a varying rae of deerioraion and exponenial parial acklogging, European Journal of Operaional Research, 8 (7) [9] Goyal, S.K., and Giri, B.C., he producion- invenory prolem of a produc wih ime varying demand, producion and deerioraion raes, European Journal of Operaional Research, 47(3) (3) [] Hwang, H.S., A sudy on an invenory model for iems wih Weiull amelioraing, Compuers and Indusrial Engineering, 33 (3-4) (997) [] Hwang, H.S., Invenory models for oh deerioraing and amelioraing iems, Compuers and Indusrial Engineering, 37 (-) (999) [] Mahaa, G.C., and Goswami, A., A fuzzy replenishmen policy for deerioraing iems wih ramp ype demand rae under inflaion, Inernaional Journal of Operaional Research, 5(3)(9) [3] Manna, S.K., and Chaudhuri, K.S., An EOQ model wih ramp ype demand rae, ime dependen deerioraion rae, uni producion cos and shorages, European Journal of Operaional Research,7 (6)

15 М. Valliahal, R. Uhayakumar / A Sudy of Inflaion Effecs on an EOQ Model 455 [4] Misra, R.B., A noe on opimal invenory managemen under inflaion, Naval Research Logisics Quarerly, 6(979) [5] Misra, R.B., A sudy of inflaion effecs on invenory sysem, Logisics Specrum, 9(3) (997) [6] Moon, I., and Yun, W., A noe on evaluaing invenory sysems: a ne presen value frame work, he Engineering Economis, 39 (993) [7] Ouyang, L.Y., eng. J.., and Chen, L. H., Opimal ordering policy for deerioraing iems wih parial acklogging under permissile delay in paymens, Journal of Gloal Opimizaion, 34 (6) [8] Panda, S., Senapai, S., and Basu, M., Opimal replenishmen policy for perishale seasonal producs in a season wih ramp-ype ime dependen demand, Compuers and Indusrial Engineering, 54 (8) [9] SanJose, L.A., Sicilia, J., and Garcia-Laguna, J., Analysis of an invenory sysem wih exponenial parial ackordering, Inernaional Journal of Producion Economics, (6) [] Skouri, K., Konsanaras, I., Papachrisos, S., and Ganas, I., Invenory models wih ramp ype demand rae, parial acklogging and Weiull deerioraion rae, European Journal of Operaional Research, 9 (9) [] Yang, H. L., eng, J.., and Chern, M. S., Deerminisic invenory lo-size models under inflaion wih shorages and deerioraion for flucuaing demand, Naval Research Logisics, 48(3) [] Wu, K.S., Ouyang, L.Y., and Yang, C.., An opimal replenishmen policy for noninsananeous deerioraing iems wih sock-dependen demand and parial acklogging, Inernaional Journal of Producion Economics, (6)