Effect of Probabilistic Backorder on an Inventory System with Selling Price Demand Under Volume Flexible Strategy

Transcription

1 Inernaional Transacions in Mahemaical Sciences and compuers July-December 0, Volume 5, No., pp ISSN-(Prining) , (Online) AACS. ( All righ reserved. Effec of Probabilisic Backorder on an Invenory Sysem wih Selling Price Demand Under Volume Flexible Sraegy Surbhi Singhal and S.R. Singh Deparmen of Mahemaics, D.N. College, Meeru ABSTARCT In his paper, an invenory sysem for decaying iems wih selling price demand in flexible environmen has been discussed. Holding cos is linear increasing funcion of ime. Shorages are allowed wih parial backorder and backorder rae is aken as random variable wih bea disribuion of firs kind. Finally, numerical example is presened o demonsrae he developed model. Sensiiviy analysis of he opimal soluion wih respec o major parameers is carried ou. Keywords: Volume flexible manufacuring sysem, variable holding cos, probabilisic backorder. Inroducion The Economic Producion Lo Size model assumes ha and he producion rae is o be predeermined and inflexible. This is no suiable for realisic business siuaion. I is quie appropriae o consider he flexible producion rae in producion model. The producion rae should be flexible o adjus iself wih he variabiliy in he marke demand. I may hen become necessary o purchase more machines or o hire more workers and he producion rae should be reaed as a decision variable. The reamen of producion rae as a decision variable is especially appropriae for auomaed echnologies ha are volume flexible. In he real-world applicaion he demand is affeced by he selling price, since as low is he selling price move hen demand increases, when he selling prices is high hen i decline demand. The dependence of he sale of any iem on is selling price is no a new concep, bu a common sense conclusion. The occurrences of shorages in invenory are naural phenomenon in realiy. In real business environmen, some cusomers are able o wai for he nex order o saisfy heir demands during

2 Effec of Probabilisic Backorder on an Invenory Sysem wih Selling Price Demand Under Volume Flexible 98 he sock ou period, while ohers do no wish or canno wai and hey have o fill heir demands from oher sources. Shorages are unavoidable due o uncerainies. Though, in pracice, he backorder rae may also be influenced by subsiue, brand loyaly, cusomer s preferences ec., which makes is exac value difficul o be measured. Therefore, he backorder rae should be random variable wih probabiliy disribuion. The Economic Producion Lo Scheduling (EPLS) model deals wih he availabiliy of invenories a a consan rae where he producion rae may be changed. The concep of flexibiliy in he machine producion rae was firs inroduced by Schweizer and Seidmann (99) and hey discussed he processing rae opimizaion for a flexible manufacuring sysem (FMS). FMS can be described simply as he manufacuring environmen where he producion is made according o he demand and he producion rae can be slowed down or increased according o he decision maker s desire. No only ha here should be a beer co-ordinaion among he differen secions of he enire producion uni. Silver (990) sudied he effecs of slowing down he producion rae for he producion of a family of iems assuming a common producion cycle for all iems and Gallego (99) exended i o differen producion cycle for differen iems. Invenory models which consider a mixure of backorders and los sales for non-perishable produc were suggesed and solved by Park (98). They assumed only a fixed fracion of demand during he sock ou ime is backordered. Abad (000) developed an opimal lo size for a perishable good under condiions of finie producion and parial backordering and los sale. Daa & Paul (00) discussed an invenory sysem wih sock-dependen, price-sensiive demand rae. Singh, S.R. (008) have coordinaed perishable invenory model wih quadraic demand and parial backlogging. Singh, S.R. e al. (009) developed an invenory model for perishable iems wih power demand and parial backlogging. Zhou (009) considered a noe on an EOQ model wih sock and price sensiive demand. Singh, S.R. & Kavia Devi (00) discussed an invenory sysem for deerioraing iems wih shorages. Sarkar (0) developed an EOQ model for finie replenishmen rae where demand and deerioraion rae are boh ime-dependen. A volume flexible producion sysem for decaying iems wih selling price demand and ime dependen holding cos has been developed. Shorages are allowed wih probabilisic backlogging rae. Cos minimizaion echnique is used in his model. Numerical examples ogeher wih sensiiviy analysis wih respec o differen parameers are also provided.. Assumpions and Noaions To develop he mahemaical model, he following assumpions are being made:. The demand for he iem is a downward slopping funcion of he price.. Shorages are allowed wih parial backlogging.. The fracion of he backlogging rae is a random variable and has a probabiliy densiy funcion g( ), where Holding cos is linear increasing funcion of ime.

3 99 Surbhi Singhal and S.R. Singh 5. The uni producion cos is a funcion of producion rae. 6. The rae of producion is considered o be decision variable. NOTATIONS. D: The annual demand as a decreasing funcion of price; D( P).. P: Selling price per uni. h+γ: Holding cos per uni ime 4. C S: The shorages cos per uni backordered per uni ime. 5. C LS: The cos of los sales per uni. P, where, 0 6. : The fracion of he demand during he sock-ou period ha will be backordered and a 7. random variable, 0. ( ) 0 : The p.d.f. of. M g d 8. g( ) The probabiliy densiy funcion of and follows Bea disribuion of firs kind 9. TC(, T)= Toal cos. EAC(, T) E[ TC(, T)] 0. ( ),, 0,0 g( ) B(, ) 0, oherwise = Expeced oal cos. C P : Uni producion cos of an iem P energy and labor cos and H is ool or die cos.. Model Formulaion and G C N HK and where N is maerial cos, G is K In his sudy, producion invenory model for decaying iems wih selling price demand has been sudied. Iniially, he sock is zero. A =0 producion sars and a = he sock reaches a level S afer meeing demands. Producion is sopped a ime =. The invenory level decreases mainly o mee up demand and parly for deerioraion during he ime inerval (, ). A, invenory level is zero. A his ime shorage sars developing and sock reaches a maximum shorage level a ime. In his ime producions sars again and clear he backlog a he ime T. Here our aim is o find he opimum values of,,, T ha minimize he oal average cos over he ime horizon (0, T). The invenory sysem I() a ime (0 T) saisfies he following differenial equaions: I ' ( ) I( ) K P 0 ()

4 Effec of Probabilisic Backorder on an Invenory Sysem wih Selling Price Demand Under Volume Flexible 00 I ' ( ) I( ) P ' I () e P I ' () K P T Wih he boundary condiions I(0)=0, I( )=S, I( )=0, I(T)=0 Soluions of hese equaions are: ( K P ) I( ) [ e ] P I( ) ( S ) e ( ) P I e e ( ) ( ) P I( ) ( K P )( T) T From equaion (5) and (6), one can ge S ln( ) K P From equaion (7) and (8), one can ge 0 () () (4) (5) (6) (7) (8) (9) P ( K P ) T K K Nex, he relevan invenory cos per cycle consiss of he following elemens: Holding cos occurs during [0, ] Invenory is carried over [0, ] and [, ]. Hence he cos of carrying invenory becomes: HC h I d h I d [ ( ) ( ) ( ) ( ) ] 0 ( ) K P e e e P [ h( ) ( )] h[( S ) ( ) ( ) ( ) e P ( ) P e e P ( ) ( ) ] [( S )( ) ] Deerioraion cos occurs during [0, ] Deerioraion is carried over [0, ] and [, ]. Hence he cos of deerioraion becomes: () (0)

5 0 Surbhi Singhal and S.R. Singh d DC C I d I d [ ( ) ( ) ] 0 e P Cd K P S e P ( [( )( ) ( )( ) ) ( )] Shorage cos occurs during [, ] and [, T] Shorage of he invenory occurs in [, ] and [, T]. So he shorage cos is S SC C I d I d [ ( ( )) ( ( )) ] T P e e T CS[ ( e ( ) ) ( K P )( T)] Los sale cos occurs during [, ] Los sale of he invenory occurs in [T, T ]. So he los sale cos is LS LSC C e P d [ ( ) ] e e CLSP [ ] Producion cos occurs during [0, ] and [,T] Invenory is produced in [0, ] and [,T]. As he cos of producing invenory, we obain G PC N HK Kd Kd K T ( )[ ] 0 ( NK G HK )( T ) Toal cos of he sysem is Therefore, he oal cos is he sum of he equaions (), (), (), (4) and (5). TC(,T)=HC+DC+SC+LSC+PC () () (4) (5) (6) Since he backorder rae is a random variable wih p.d.f. g( ), he expeced backorder rae is M g( ) d 0 EAC(, T) E[ TC(, T)]. Therefore, he expeced oal cos is: (7) To minimize he expeced oal cos, he opimal values of and T can be obained by solving he following equaions simulaneously EAC(, T) 0 (8)

6 Effec of Probabilisic Backorder on an Invenory Sysem wih Selling Price Demand Under Volume Flexible 0 and EAC(, T) 0 T provided, hey saisfy he following condiions and (9) EAC(, T ) EAC(, T ) 0, 0 T EAC(, T ) EAC(, T ) EAC(, T ) 0 T T (0) Equaions (8) and (9) are he objecive funcion which needs o be minimized. For his, we use he opimizaion echniques. These equaions are non-linear and hence are solved wih he help of mahemaical sofware MATHEMATICA 5.. We can obain he opimal values. Wih he use of hese opimal values equaion (7) provides minimum expeced oal cos of he sysem in consideraion. Numerical illusraions: To illusrae he preceding heory, le us consider he proposed mehod o efficienly solve he following numerical example wih he following daa: =5000 = P=0 G=500 H=0.0 N=50 K=500 S=50 h= C S=4 C LS=6 =0.5 =0.5 C 0=500 M =0.75 The compuaional resul shows he following opimal values: = =.4058 = T=8.95 EAC(, T ) =76.00 Sensiiviy analysis Table for Sochasic backorder EAC(, T) µ M T Using he numerical example given in he preceding secion, he sensiiviy analysis of various parameers has been done. The resuls of sensiiviy analysis are summarized in he able. Table 8. Effec of changing various parameers % change T EAC(, T)

7 0 Surbhi Singhal and S.R. Singh Observaions: Effec of parameer on θ +0% % % % % % Effec of parameer on K +0% % % % % % Effec of parameer on C h +0% % % % % % Effec of parameer on C S +0% % % % % % Effec of parameer on C LS +0% % % % % % The main conclusions drawn from he sensiiviy analysis are as follow:. We observed ha, if he parameer of deerioraion is increases hen he opimal ime, are decreases and and T are increases. Expeced oal cos of he sysem is increases.. Decreases all he opimal ime,, and T wih he increases of he producion rae. Bu he expeced oal cos is increases.. Wih he incremen of he parameer of holding cos, expeced oal cos is increases. The opimal ime is decreases, and T are increases and here is no effec on ime. 4. If he parameer of shorages cos is increases hen he expeced oal cos is increase. The opimal ime and T are decreases, is increases and here is no effec on ime.

8 Effec of Probabilisic Backorder on an Invenory Sysem wih Selling Price Demand Under Volume Flexible Increases he expeced oal cos wih he increases of he rae of los sale. Bu i does no give any effec on and he opimal ime is increases and and T are decreases. Conclusion: This paper presens an invenory sysem for decaying iems wih selling price dependen demand in which holding cos is a linear increasing funcion of ime. The holding or sorage cos is he cos associaed wih mainaining an invenory unil i is used or sold. Holding or sorage cos includes he cos of mainaining sorage faciliies, he cos of insuring he invenory, axes aribued o sorage, coss associaed wih obsolescence, and coss associaed wih he capial ha is commied o he invenory. The holding or sorage cos is usually relaed o he maximum quaniy, average quaniy, or excess of supply in relaion o demand during a paricular ime period. In order o make effecive his sraegies, i can be ignored he uncerainies ha happen in he real world and hese uncerainies are usually associaed wih he los sale. Therefore, shorages are permied wih random backlogging rae. We used a numerical example o illusrae he model and sensiiviy analysis on some parameers is made. These facs make his sudy ineresing and unique. References Abad, P.L. (000), Opimal lo size for a perishable good under condiions of finie producion and parial backordering and los sale, Compuers & Indusrial Engineering, 8, Daa, T.K., Paul, K. (00), An invenory sysem wih sock-dependen, price- sensiive demand rae, Producion Planning and Conrol,,, -0 Gallego, G. (99), Reduced producion raes in he economic lo scheduling problem. Inernaional Journal of Producion Research (5), Mo, J., Mi, F., Zhou, F. and Pan, H. (009), A noe on an EOQ model wih sock and price sensiive demand, Mahemaical and Compuer Modelling, 49, 9-0, Park, K.S. (98), Invenory models wih parial backorders, Inernaional Journal of Sysem Science,, -7 Sarkar, B., (0), An EOQ model wih delay in paymens and ime varying deerioraion rae, Mahemaical and Compuer Modelling, 55(-4), Schweizer, P.J. and Seidmann, A. (99), Opimizing processing raes for flexible manufacuring sysems, Managemen Sciences, 7, Silver, E.A. (990) Deliberaely solving down oupu in a family producion conex. Inernaional Journal of Producion Research 8(), 7-7. Singh, S.R. and Singh, T.J. (008), Perishable invenory model wih quadraic demand, parial backlogging and permissible delay in paymens, Inernaional Review of Pure and Applied Mahemaics,, 5-66 Singh, S.R., Singh, T.J. & Rajul Du (009), An EOQ Model for Perishable Iems Wih Power Demand And Parial Backlogging, In. J. Operaional and Quaniaive Managemen,Vol.5 No.,March,pp Singh, S.R. & Kavia Devi (00) An Invenory Model for Deerioraing Iems wih Shorages and Inflaion, Inernaional Review of Pure and Applied Mahemaics, vol. 6, no., D Yadav, S Pundir, R Kumari (0)A fuzzy muli-iem producion model wih reliabiliy and flexibiliy under limied sorage capaciy wih deerioraion via geomeric programming Inernaional Journal of Mahemaics in Operaional Research (), 78-98