Proeedings of he Inernaional onferene on Indusrial Engineering Operaions Managemen Dhaka, Bangladesh, January 9, An Invenory Model wih Linear Dem Rae, Finie Rae of Produion wih Shorages omplee Baklogging. K. Sahoo S. K. Sahoo Insiue of Mahemais Appliaion Bhubaneswar-757, Orissa, India Absra This paper deals in developing invenory model wih linear dem rae allowing shorages in he invenory. These shorages are onsidered o be ompleely baklogged. We have assumed ha he produion rae is finie proporional o he dem rae. The analyial soluion of he model has been done o obain he opimal soluion of he problem. Suiable numerial example has been disussed o unders he problem. Furher we have made sensiiviy analysis of he opimal soluion wih respe o he hanges in he values of he sysem parameers. This model is suiable in ase of seady inrease or derease in he dem in he marke for some produs. Keywords Invenory; Eonomi order quaniy; dem; Shorage.. Inroduion In radiional invenory model he dem rae of he iem was assumed o be onsan whih is no generally o in real life siuaion. The firs modifiaion for varying dem was suggesed by have Silver Meal [] afer whih a lo of work been done by many researhers suh as Silver Meal [] developed an approximae soluion proedure, known as he Silver-Meal heurisi for general ase of a deerminisi, imedependen dem paern, for he firs ime, he lassial no-shorage invenory poliy for he ase of linear, ime-dependen dem, signifian onribuions in his direion has been due o researhers namely Rihi[], Shorages baklogging has also been onsider along wih varying dem in ourse of ime by many researhers suh as Deb haudhuri [] was he firs o inorporae shorages ino he invenory lo sizing wih a linearly problem linearly inreasing ime-varying dem. EOQ models for deerioraing iems wih rended dem have also onsidered by several researhers like Bhari Kashani [], Goswami hodhuri [6] hung Ting [], Hariga [7], Giri hakrabary [5], Jalan hodhuri [8] Lin e al. [9].A group of researhers have also devoed heir aenion o invenory replenishmen problems wih exponenially ime-varying dem paerns. Some of he onribuions in his direion have ome from Agrawal Bhari Kashani [], e. In he presen paper, we assume ha ime- dependene of dem follows a linear dem. Also he produion rae is assumed o be finie proporional o he dem rae. Shorages are allowed are ompleely baklogged. An analyial soluion no he mode is disussed illusraed wih he help of numerial examples. Sensiiviy of he opimal soluion wih respe o hanges in differen parameer values is also examined. Assumpion. The dem rae a any ime is : R. The produion rae is K R where <. The on-h invenory does no deeriorae wih ime.. Lead ime is zero. 5. Shorages are allowed are ompleely baklogged. Noaions arrying os per uni per uni ime. Shorage os per uni per uni ime. Seup os per produion run > a onsan is also.therefore K > R. 99
,, are all assumed o be known fixed during produion yle. he oal average os for a produion yle.. Mahemaial Model The invenory level a differen insans of ime is iniially i.e. a ime, he sok level is zero. The shorage sars a aumulaes up o he level P a. The produion sars a he baklog is leared a. The sok level aains a level S a, when produion is sopped. The invenory level gradually dereases due o dem beomes zero a. The yle hen repeas iself. Our problem is o deermine he opimum values of S, P. Now if Q be he afer ime insananeous invenory level a any ime, he differenial equaions desribing he insananeous saes of Q in he inerval, are R d K R d K R d K R d wih he boundary ondiions Q, Q P, Q, Q S, R K R Q. 5 Now subsiuing in he equaions are solving hem using he boundary ondiions 5, we ge he soluions as follows: Q Q Q Q Q P in 6, we ge Using he ondiion in P Q P in 7, we ge P Similarly using he ondiion Equaing hese wo values of P, we ge Again, using he ondiion in Q S 8 9 we ge respeively 6 7 8 9
S S Equaing hese wo values of S we ge 5 Now we found he differen he oss involved in he sysem. The oal shorage os in he sysem is d Q d Q S 6 6 The oal invenory holding os in he sysem is d Q d Q H 7 Therefore he average os of he sysem is H S 6 8 Subsiuing he values of from from 5, beomes a funion of he variables. Therefore, will be minimum if,, Provided > 9 From 8 9, we ge he equaions
[ 6. Numerial Examples Le.5,,, in appropriae unis. From, we obain he opimum values of i i,,,. Taking one parameers used in model are analyzed in he following able. suppose, he opimum values of i i,,, are 9.,.995,.5598,.69. Subsiue hese opimum values of,, in equaion 8 using Mahemaia 5., we ge he opimum average os * 98.. The opimum values of P S obained from respeively S* 9. P* 9.568. Table hanging %hange Parameer sysem hange in hange hange in hange in hange in.96.96.5987.66797 98.85.898.898.9688.675 98.6.7889.7889.8999.687 99.8.68.68.6695.66 99. 5.5.5.59.5979 9.9 -.65.65.55956.7875 97.79 -.6579.6579.587979.75 97.579 -.77.77.668.7896 97.67 -...6679.86877 97.98-5.7.7.7779.89687 96.66.86.585.857.6 9..677..5858.59977 889.7.59.995.96759.566 877.57.7959.6975.67.585 867.77
5..78.69.9867 859.7 -..8.68.7765 95.88 -.6.595.6795.85 955.786 -.8595.58955.779.895778 977.6 -.7776.7689.975.6 997.967-5.769.9.556.55 997.967.8.8965.559.666 9989.956.88.997.665 66.6.8996.7.889.687.7.856.68.7.595 8.5 5.888.66.69.578 95.7 -.8.77.55889.765 85.69 -.8997.7.5878.769567 775.79 -.66.7.67.856 78.89 -.656.658.676.8976 65.59-5.857.975.7798.989 59.7.76.95.5566.755 97.87.8.956.576575.7599 957.755..966.59669.78 997.77.998.8858.668.8969.8 5.77.858.659.8 8. -.7.88857.55.65955 9.55 -.99.787.9578.669 88.87 -.86.678.76.59 868.7 -.759.5556.69.55679 85.9-5.6599.9.575.5977 8..65.797.557.767.6.98.887.57797.796 9.9.56.565.5978.76 8..676.7.66859.7869 75.6 5.89.9.6586.866 65.97 -.98.785.5695.66556 8. -.7658.5668.8866.67 7.5 -.655.7.6.67786 69.585 -.97.957.66.5765 58.99-5.9889.9686.98.5595 6.76. Sensiiviy Analysis To sudy he effes of hanges in he sysem parameers,,,,, on he opimal os derived by he proposed mehod, sensiiviy analysis is performed by hanging inreasing or dereasing he parameers by -5 % 5 % aking one parameer a a ime, keeping he remaining parameers a heir original values. On he basis of he resuls of able, he following observaion an be made. * * i Derease in he value of eiher of he parameers, hen,,, is inreased is dereased. ii Derease in he values of eiher of he parameers, hen * *,,, is dereased. iii Derease in he value of he parameer hen,, *, * is inreased. 5. onlusion In his paper we assumed ha ime dependen linear dem shorages are ompleely baklogged. Here he produion rae assumed o be finie proporional o he dem shorage wih ompleely baklogged. In real marke siuaions, dem is unlikely o vary wih a rae whih is so high as exponenial. Time-dependene of dem is usually nonlinear in naure. The advanage of he linear funional form of he dem ake are of seady inreasing or seady dereasing onsan dem for differen ranges of values of is parameer.
Referenes. Aggarwal, V., Bahari-Kashani, H., 99 Synhronized produion poliies for deerioraing in a delining marke, AIIE Transaion, vol., pp. 85-97.. Bahari-Kashani, H., 989, Replenishmen shedule for deerioraing iems wih ime-proporional dem, Journal of he Operaional Researh Soiey, vol., pp. 75-8.. hung, K.J., Ting, P.S., 99 A heurisi for replenishmen of deerioraing iems wih a linear rend in dem, Journal of he Operaional Researh Soiey, vol., pp. 5-.. Deb, M., haudhuri, K.S.,987, A noe on he heurisi for replenishmen of rended invenories onsidering shorages, Journal of he Operaional Researh Soiey, vol. 8, pp. 59-6. 5. Giri, B.., hakrabari, T., haudhuri, K.S., 997, Heurisi models for deerioraing iems wih shorages ime-varying dem oss, Inernaional Journal of Siene, vol. 8, pp. 5-59. 6. Goswami, A., haudhuri, K. S., 99, An EOQ model for deerioraing iems wih sorages a linear rend in dem, Journal of Operaional Researh Soiey, vol., pp. 5-. 7. Hariga, M.A., Bankherouf, L., 99 Opimal heurisi invenory replenishmen models for deerioraing iems wih exponenial ime-varying dem, European Journal of Operaions Researh, vol. 79, pp. -7. 8. Jalan, A.K., haudhuri, K.S., 999 Sruural properies of an invenory sysem wih deerioraion rended dem, Inernaional Journal of Sysems Siene, vol., pp. 67-6. 9. Lin,., Tan, B., Lee, W..,, An EOQ model for deerioraing iems wih ime-varying dem shorages, Inernaional Journal of Sysems Siene, vol., pp. 9-.. Rihie, E., 98, An EOQ for Linear inreasing dem: a simple opimal soluion, Journal of he operaional Researh Soiey, vol. 5, pp. 99-59.. Silver, E.A., Meal, H.., 97 A heurisi for seleing lo size quaniies for he ase of a deerminisi ime-varying dem rae disree opporuniies for replenishmen, Produion Invenory Managemen, vol., pp. 6-7.. Silver, E.A., Meal, H.., 969 A simple modifiaion of EOQ for he ase of a varying dem rae, Produion Invenory Managemen, pp. 5-65.