An Inventory Model for Weibull Time-Dependence. Demand Rate with Completely Backlogged. Shortages

Transcription

1 Inernaional Mahemaial Forum, 5, 00, no. 5, An Invenory Model for Weibull Time-Dependene Demand Rae wih Compleely Baklogged Shorages C. K. Tripahy and U. Mishra Deparmen of Saisis, Sambalpur Universiy Jyoi Vihar, Sambalpur-76809, Deparmen of Mahemais, P.K.A.College of Engg. Chakarkend Bargarh-76808, India Absra This paper deals wih in developing an invenory model wih imedependen Weibull demand rae where shorages are allowed. Here we have onsidered shorages are ompleely baklogged. The produion rae is assumed o be finie and proporional o he demand rae. The analyial soluion of he model has been done o obain he opimal soluion of he problem. Numerial examples are given o illusrae he model developed. Sensiiviy analysis of he deision variables has been done o examine he effe of hanges in he values of he parameers and affe of hese hanges on he opimal poliy of invenory. Mahemais Subje Classifiaion: 90B05 Keywords: Invenory; Eonomi order quaniy; demand; Shorage; Weibull disribuion. Inroduion Demand is a major faor in invenory managemen. In invenory models, four ypes of demand are basially assumed i.e. onsan demand, ime-dependen

2 676 C. K. Tripahy and U. Mishra demand, probabilisi demand and sok-dependen demand. Invenory models dealing wih sok-dependen demand have more relevane in presen siuaion. Therefore, many auhors have sudied suh models. Iniially he demand rae of he iem was assumed o be onsan. However, in real marke siuaions, demand varies wih ime. Silver and Meal [8], were he firs o sugges a simple modifiaion of he EOQ formula for he ase of varying demand. Laer, Silver and Meal [7] developed an approximae soluion proedure, known as he Silver- Meal Heurisi for general ase of a deerminisi, ime dependen demand paern. However, Donaldson [] disussed, for he firs ime, he lassial noshorage invenory poliy for he ase of a linear, ime-dependen demand. Following Donaldson [], signifian onribuions in his direion ame from researhers like Silver [9], Rihie [6] and Sahan [8], e. The quesion of invenory shorages and baklogging was no onsidered by he above researhers. Deb and Chaudhuri [] were he firs o inorporae shorages ino he invenory lo sizing problem wih a linearly inreasing ime-varying demand. Subsequen onribuions in his ype of modeling ame from Goyal [], Hariga [6] and many ohers. EOQ models for deerioraing iems wih rended demand were onsidered by several researhers like Bahari-Kashani [0], Goswami and Chaudhuri [] Chung and Ting [], Hariga [5], Jalan e al. [], Giri and Chaudhuri [], Jalan and Chaudhuri [] and Lin e al. [5]. Some of he reen works on ime-varying demand for deerioraing iems are due o Skouri and Papahrisos [], Teng e al. [], Khanra and Chaudhuri [9], Sana e al. [0], e. Furher a group of researhers have devoed heir aenion o invenory replenishmen problems wih exponenially ime-varying demand paerns and works in his direion are due o Aggarwal and Bahari- Kashani [], Hollier and Mak [7], e. In he presen paper, we have developed an invenory model assuming ha ime-dependene of demand follows a hree-parameer Weibull disribuion. Here he produion rae is assumed o be finie and proporional o he demand rae. In his model shorages are allowed and are ompleely baklogged. An analyial soluion of he mode is disussed and illusraed wih he help of numerial examples. We have also done he sensiiviy analysis of he opimal soluion wih respe o hanges in various parameri values.. Assumpion and Noaion The following assumpions & noaions are used o develop he proposed model. Assumpion The demand rae a any ime is: R () = ( ), where (> 0), (> 0) and (> 0) are sale parameer, shape parameer and loaion parameer, respeively.

3 Invenory model for Weibull ime-dependene demand rae 677 The produion rae is K( = λr( where ( λ > ) a onsan is also. Therefore K ( > R(. The on-hand invenory does no deeriorae wih ime. Lead ime is zero. Shorages are allowed & are ompleely baklogged. Noaions Carrying os per uni per uni ime. Shorage os per uni per uni ime. Seup os per produion run.,, are all assumed o be known & fixed during produion yle. C he oal average os for a produion yle.. Formulaion & Soluion The invenory level is iniially (i.e. a ime = 0 ) zero. The shorage sars a = 0 whih aumulaes up o he level P a ime =. The produion sars a = and he baklog is leared a =. The sok level aains a level S a =. The produion is sopped a his poin of ime i.e. a =. Therefore he invenory level gradually dereases due o demand and beomes zero a = afer whih he yle is ompleed and i repeas iself. We are ineresed in deermining he opimum values of S, P and C. Q () be he insananeous invenory level a any ime ( 0 ). The differenial equaions desribing he insananeous saes of Q () in he inerval 0, ) are ( dq( d dq( d dq( d = R( 0 () = K( R( () = K( R( () dq( = R( () d

4 678 C. K. Tripahy and U. Mishra wih he boundary ondiions Q ( 0) = 0, Q( ) = P, Q ( ) = 0, Q ( ) = S, Q ( ) = 0. (5) Subsiuing R () = ( ) and K( = λr( in he equaions ()-() and solving hem using he boundary ondiions (5), we ge he following soluions: {( ) } Q () = ( ) 0 (6) {( ) } Q () = ( λ ) ( ) (7) {( ) } Q () = ( λ ) ( ) (8) {( ) } Q () = ( ) (9) Using he ondiion in Q( = P in equaion (6), we ge {( ) } P= ( ) (0) Similarly using he ondiion Q( ) = P in equaion (7), we ge {( ) } P= ( λ ) ( ) () Equaing hese wo values of P, we ge λ = ( ) + ( ) + λ λ () Again, using he ondiion Q ( ) = S in equaion (8) and equaion (9) we ge {( ) } S = ( λ ) ( ) () and {( ) ( ) } S = ()

5 Invenory model for Weibull ime-dependene demand rae 679 Equaing hese wo values of S we ge + λ ( ) ( ) ( ) = + λ (5) Now we shall ry o find differen oss involved in he sysem. The oal shorage os in he sysem is SC = Q( d + Q( d ( ) ( ) ( λ ) ( ) ( ) = ( ) ( ) ( + + (6) The oal invenory holding os in he sysem is HC = Q( d + Q( d ( ) ( ) ( ) ( ) = ( λ ) ( )( + ( )( + + (7) Therefore he average os of he sysem is C = ( SC + HC + ) ( ) ( ) λ ( ) ( ) ( ) = ( ) ( ) ( ( ) ( ) ( ) ( ) + ( λ ) ( )( + ( )( (8) Puing he values of and in equaion (8), C beomes a funion of variables and. Therefore he average os of he sysem is

6 680 C. K. Tripahy and U. Mishra + λ ( ) ( ) + + λ λ ( ) λ C = ( ) ( ) + ( ) ( λ) + + λ λ λ ( ) ( ) + + ( ) λ λ λ ( λ ) ( ) + ( ) + ( ) + + λ λ + ( ) ( ) ( ) + + λ ( ) ( ) + ( ) + ( λ ) ( ) + λ + ( ) ( ) ( ) + λ ( ) + ( ) + ( ) ( ) λ + ( ) ( ) ( ) + ( ) ( ) ( ) ( ) + λ + λ ( ) + + λ + λ + (9) + Here C will be minimum if C = 0 C, = 0 (0) Provided C C C > 0 () From equaion (9) and equaion (0), we ge he equaions

7 Invenory model for Weibull ime-dependene demand rae 68 λ λ λ λ λ λ λ ( ) ( ) ( ) λ ( ) + ( ) ( ) λ λ λ λ λ λ λ ( ) + ( ) ( ) ( ) ( ) + ( ) λ λ λ λ + ( ) + ( ) ( ) ( ) + ( ) ( ) + ( ) λ λ λ λ λ ( ) ( ) ( ) + λ ( ) + ( λ ) ( ) ( λ ) ( ) λ λ ( ) ( ) ( ) ( ) + λ ( ) ( λ ) ( ) + ( ) λ λ + ( ) ( ) ( ) + λ λ λ ( ) ( ) ( ) ( ) ( ) ( ) + λ + λ λ = λ and ( ) ( λ ) 0 () λ λ λ ( ) ( ) ( ) ( ) ( ) ( ) + λ + λ ( λ ) ( ) ( ) ( ) + ( ) ( ) ( ) λ λ ( ) ( ) ( ) + λ

8 68 C. K. Tripahy and U. Mishra ( ) + ( ) ( λ ) ( ) ( ) ( ) + λ ( ) ( ) ( ) ( ) λ λ λ λ + λ ( ) ( ) + + λ λ ( ) λ ( ) ( ) + ( ) ( λ) + + λ λ λ ( ) ( ) + + ( ) λ λ ( λ ) ( ) + ( ) + ( ) + + λ λ + ( ) ( ) ( ) + + λ ( ) ( ) + ( ) + ( λ ) ( ) + λ + + λ ( ) ( ) ( ) + λ ( ) + ( ) + ( ) ( ) λ + ( ) ( ) ( ) + ( ) ( ) ( ) ( ) + λ + λ ( ) + + = 0 λ + λ + (). Numerial examples Le = 00, λ =. 05, = 0, = 0, 0 =, = 0. in appropriae unis. From () and (), we obain he opimum values of and. Puing he opimum values of and in equaion () and (5) we obain he opimum values of and respeively. Example-: Suppose = 0, he opimum values of i ( i =,,, ) are = , = , =. 7059, =. 79. Subsiue hese opimum values of,, and in equaion (8), we ge he opimum average

9 Invenory model for Weibull ime-dependene demand rae 68 os C = The opimum values of P and S have been obained from equaion (0) and equaion () respeively and he values are P = 0. 5 and S = Example-: Suppose =, he opimum values of i ( i =,,, ) are = , =. 0686, =. 99, =. 96. Subsiue hese opimum values of,, and in equaion (8), we ge he opimum average os C = 76.. The opimum values of P and S obained from equaion (0) and equaion () respeively and he values are P = and S = Sensiiviy Analysis To sudy he effes of hanges in he sysem of parameers,,,,,, and λ on he opimal os derived by he proposed mehod, sensiiviy analysis has been performed by hanging (inreasing or dereasing ) he parameers by 0 % and 50 % and hanging one parameer a a ime and keeping he remaining parameers a heir original values. Inrease in he value of he parameers hen yle lengh,,, and C is inreased. The resuls are shown in Table - given below.

10 68 C. K. Tripahy and U. Mishra Table - Parameer % Change Change in Change in Change in Change in Change in C λ

11 Invenory model for Weibull ime-dependene demand rae 685 On he basis of he resuls of Table -, he following observaions an be made: (i) Derease in he values of eiher of he parameers, will resul in derease of,,, and C. (ii) Derease in he value of he parameers will resul in inrease in,, and derease in and C. (iii) Derease in he value of he parameers will resul in derease in,,, and inrease in C. (iv) Derease in he value of he parameers λ will resul in derease in, and inrease in,, C. (v) Derease in he value of he parameer will resul in derease in,, and inrease in, C. 6. Conlusion Time-dependene of demand is usually non-linear in naure and he degree of non-lineariy deermines he inensiy of is inrease or derease. Power law funional form in ime is more appropriae for demand rae of a produ. Considering hese fas we have assumed ha he ime-dependene of demand R ( follows a hree parameers Weibull disribuion in his paper. Here he produion rae has been assumed o be finie and proporional o he demand where shorages are allowed, whih is ompleely baklogged. Analyial soluions of he model are illusraed wih he help of suiable numerial examples. Sensiiviy analysis has also been made in he presen paper. Referenes [] A. Goswami and K.S. Chaudhuri, An EOQ model for deerioraing iems wih shorages and a linear rend in demand, Journal of Operaional Researh Soiey,, pp. 05 0, 99. [] A.K. Jalan and K.S. Chaudhuri, Sruural properies of an invenory sysem wih deerioraion and rended demand, Inernaional Journal of Sysems Siene, 0, pp. 67 6, 999. [] A.K. Jalan, R.R. Giri and K.S. Chaudhuri, EOQ model for iems wih Weibull disribuion deerioraion, shorages and rended demand, Inernaional Journal of Sysems Siene, 7, pp , 996.

12 686 C. K. Tripahy and U. Mishra [] B.C. Giri and K.S. Chaudhuri, Heurisi models for deerioraing iems wih shorages and ime-varying demand and oss, Inernaional Journal of Sysems Siene, 8, pp. 5 59, 997. [5] C. Lin, B. Tan and W.C. Lee, An EOQ model for deerioraing iems wih ime-varying demand and shorages, Inernaional Journal of Sysems Siene,, pp. 9 00, 000. [6] E. Rihie, An EOQ for linear inreasing demand: a simple opimal soluion, Journal of he Operaional Researh Soiey, 5, pp , 98. [7] E.A. Silver and H.C. Meal, A heurisi for seleing lo size quaniies for he ase of a deerminisi ime-varying demand rae and disree opporuniies for replenishmen, Produion and Invenory Managemen,, pp. 6 7, 97. [8] E.A. Silver and H.C. Meal, A simple modifiaion of he EOQ for he ase of a varying demand rae, Produion and Invenory Managemen, 0, pp. 5 65, 969. [9] E.A. Silver, A simple invenory replenishmen deision rule of a linear rend in demand, Journal of he Operaional Researh Soiey, 0, pp. 7 75, 979. [0] H. Bahari-Kashani, Replenishmen shedule for deerioraing iems wih ime-proporional demand, Journal of he Operaional Researh Soiey, 0, pp. 75 8, 989. [] J.T. Teng, H.J. Cheng, C.Y. Dye and C.H. Hung,. An opimal replenishmen poliy for deerioraing iems wih ime-varying demand and parial baklogging, Operaions Researh Leers, 0, pp. 87 9, 00. [] K. Skouri and S. Papahrisos, A oninuous review invenory model, wih deerioraing iems, ime varying demand, linear replenishmen os, parially ime varying baklogging, Applied Mahemaial Modelling, 6, pp ,00. [] K.J. Chung and P.S. Ting, A heurisi for replenishmen of deerioraing iems wih a linear rend in demand, Journal of he Operaional Researh Soiey,, pp. 5, 99. [] M. Deb and K.S. Chaudhuri, A noe on he heurisi for replenishmen of rended invenories onsidering shorages, Journal of he Operaional Researh Soiey, 8, pp. 59 6, 987. [5] M. Hariga, An EOQ model for deerioraing iems wih shorages and ime-varying demand, Journal of he Operaional researh Soiey, 6, pp. 98 0, 996. [6] M. Hariga, The invenory lo-sizing problem wih oninuous ime varying demand and shorages, Journal of he Operaional Researh Soiey, 5, pp , 99. [7] R.H. Hollier and K.L. Mak, Invenory replenishmen poliies for a deerioraing iems in a delining marke, Inernaional Journal of Produion Researh,, pp. 8 86, 98.

13 Invenory model for Weibull ime-dependene demand rae 687 [8] R.S. Sahan, On (T, Si) poliy invenory model for deerioraing iems wih ime proporional demand, Journal of he Operaional Researh Soiey, 5, pp. 0 09, 98. [9] S. Khanra and K.S. Chaudhuri, A noe on an order-level invenory model for a deerioraing iem wih ime-dependen quadrai demand, Compuers and Operaions Researh, 0, pp , 00. [0] S. Sana, S.K. Goyal and S.K. Chaudhuri, A produion invenory model for a deerioraing iem wih rended demand and shorages, European Journal of Operaional Researh, 57, pp. 57 7, 00. [] S.K. Goyal, A heurisi for replenishmen of rended invenories onsidering shorages, Journal of he Operaional Researh Soiey, 9, pp , 988. [] V. Aggarwal and H. Bahari-Kashani, Synronized produion poliies for deerioraing iems in a delining marke, AIIE Transaion,, pp , 99. [] W.A. Donaldson, Invenory replenishmen poliy for a linear rend in demand: an analyial soluion, Operaional Researh Quarerly, 8, pp , 977. Reeived: April, 00