On a Discrete-In-Time Order Level Inventory Model for Items with Random Deterioration

Transcription

1 Journal of Agriculure and Life Sciences Vol., No. ; June 4 On a Discree-In-Time Order Level Invenory Model for Iems wih Random Deerioraion Dr Biswaranjan Mandal Associae Professor of Mahemaics Acharya Jagadish Chandra Bose College Kolkaa Absrac The presen paper deals wih an order level invenory model wih coninuous in unis and discree-in-ime for iems having a ime dependen random deerioraion rae. Shorages are assumed which are fully backlogged. Lasly a numerical example is given o illusrae he model wih is sensiiviy sudy. Keywords: Discree-in-ime, random deerioraion, shorages. Inroducion The effec of deerioraion plays an imporan role in many invenory sysems. I is defined as decay, damage or spoilage such ha he iems can no be used for is original purpose. Food iems, phoo films, drugs, chemicals ec are few examples of iems in which sufficien deerioraion may occur during he normal sorage period of he unis and consequenly his loss mus be aken ino accoun when analyzing he sysems. Effecs of ime dependen or consan deerioraion on invenory models have been invesigaed by Ghare and Schrader[], Goel and Aggarwal[], Cover and Philip[3], Daa and Pal[4], Mandal and Pal[5], Mandal[6] ec. Bu deerioraion of iems(drugs, foods) depends upon he flucuaion of humidiy, emperaure ec and so i is more reasonable and realisic if we assume he deerioraion funcion o depend upon a parameer α in addiion o ime which ranges over space in which probabiliy densiy funcion p(α) is defined. Such ype of deerioraion funcion has been developed firs by Pal and Mandal[7]. Furher developmen has been made by Mandal[ 8 ] Recenly here has been considerable ineres in developing mahemaical models for describing opimal policies for iems in invenory whose uiliy does no remain he same wih he passage of ime. Even more ime has been considered as a coninuous variable which is no exacly he case in pracice. In real life problem, ime is always reaed as a discree variable, in erms of complee uni of days, weeks, monhs ec. Dave[9] has developed firs ime an invenory model for deerioraing iems assuming as he ime variable is discree. Furher developmen in his regard has been made by Daa and Pal[]. In he presen paper, an order level invenory model for random deerioraing funcion has been developed in which ime variable is assumed o be discree one. The deerminisic model wih insananeous delivery is considered. This work is generalised by allowing shorages and lasly an example is given o illusrae he model along wih is sensiiviy analysis. The Mahemaical Model The mahemaical model is developed under he following assumpions: (i) The demand rae R unis per uni ime is known and consan (ii) T is he fixed duraion of each producion cycle. (iii) Lead ime is zero. (iv) Replenishmen size is consan and is rae is infinie. (v) Shorages are allowed and fully backlogged. (vi) The uni cos C, he invenory holding cos C per uni per ime uni and he shorage cos C per uni per ime uni are known and consan during he period. 5

2 Cener for Promoing Ideas, USA (vii) There is no repair or replacemen of he deerioraed invenory during a given cycle. (viii) The fixed lo size q raises he invenory a he beginning of each scheduling period o he order level S. of he on-hand invenory deerioraes per uni ime. (ix) A variable fracion (, ) In he presen problem, (, ) is aken as (, ) ( ), ( < ( ) <<)...() I is some funcions of he random variable which ranges over a space and in which a probabiliy densiy funcion p( ) is defined such ha p( ) d. A he ime of a cycle a bach of q unis eners he invenory sysem, from which ( q-s) unis are delivered owards backorders leaving S unis as he iniial invenory level. As ime is going on, he invenory level gradually decreases mainly due o demands bu parially due o deerioraion of unis in invenory up o and including ime -. A, he invenory level reaches zero. Furher demands for he remaining period (, T) are backlogged. If I() denoes he number of unis a he beginning of he ime uni, he difference equaions governing he invenory sysem during he cycle of ime T are given by I ( ) (, ) I( ) R,,,, () I () - R,, +...T (3) Where (, ) ( ), ( < ( ) <<) The soluions of he above difference equaions () and (3) are found o be he following ( neglecing he erms conaining square and higher power of as < <<) ( ) ( ) ( ) I( ) ( K ( R ) R )[ ] - ( ) R ( )( ) - ( ) ( ),,,... - (4) 6 And I() K R,, +...T (5) Where K and K are consans of inegraions. The boundary condiions are I() S and I ( ) Therefore using I ( ), equaions (4) and (5) become K ( ) ( ) ( ) [ R{ ( )} + ( )( ) + ( ) (6) ( ) ( ) 6 And K R Again from equaion (4), I() S gives S K Therefore from equaion (6) and neglecing second and higher powers of ( ) ( ) 3 S( ) R[ ] 6 6 we ge Now subsiuing he values of K and K from he relaions (6) and (7) in he equaions (4) and (5) respecively, we ge he following (7) (8) ( ) 6

3 Journal of Agriculure and Life Sciences Vol., No. ; June 4 ( ) ( ) 3 ( ) ( ) I(, ) R[ 6 6 ( ) ( ) ( ) 3 ],,,... - (9) 6 3 And I(, ) R( ),, +...T () (neglecing O( ( )) as < ( ) <<) The average number of unis in invenory per uni ime during a cycle is H( α), I (, ) T R 3 ( ) 6 ( ) ( ) ( ) 3 4 [ ](neglecing T 6 6 The average shorage per uni ime during a cycle is G( α), T I (, ) T R ( T )( T ) ( T ) The average number of unis ha deerioraes per uni ime is o( ( )) as < ( ) <<) () () D( α), T [ S( ) R ] R ( ) ( 3 ) 6T Therefore he oal average cos of he sysem per uni ime during a cycle is given by (3) K( α) C, H( α) + C, G( α) + C D(, α), Hence he mean average oal cos of he sysem per uni ime during he cycle is K( ) < K( α)> C, < H( α)> + C, <G( α)> + C <D(, α)> (4), where < K( α)> K(,, ) p( ) d Now < H( α)> he mean average number of unis carrying in invenory per uni ime, R 3 4 [(3 A) (6 A) A A ] ( T ) <G( α)> he mean average shorage per uni ime G(, ) R ( T )( T ) ( T ) AR 3 <D( α)> he mean average amoun of invenory ha deerioraes per uni ime (, ) 6T where A ( ) p( ) d (5) Therefore from (4), subsiuing he above values we obain 7

4 Cener for Promoing Ideas, USA K( ) [(3 A) (6 A) A A ] + ( T )( T ) ( T ) ( T ) A 3 + ( ) 6T (6) Since is a non-negaive ineger, he condiions for o have an absolue minimum a are K( ) K( ) K ( (7) ) and K ( ), for all,,...t (8) vide Sasieni e al[] K( ) K( ) K( ) Now 3 [(3 A) (6 A)( ) A(3 3 ) A(4 6 4 )] - ( T ) T ( ) + ( T ) And A ( ) T K( ) ( K( )) K K ( ) ( ) (9) RC RC CAR [(6 A) 36A 6A A ] ( )] ( T ) T T () Since <A<, we observe ha K ( ) for all,,...t. Hence K ( would ) have an absolue value a * if he condiion (7) is saisfied. Now using (7) and (9), he condiion for opimaliy of becomes * M( ) C T C M ( ) Where + () C 3 AC M ( ) (3 A 3 A A ) C ( T )( ) 3 T a K ( ) * () Therefore he mean ordering quaniy is A A 3 S( ) S( ) R[ ] (3) 6 6 Special Case: If he deerioraion of he iems is swiched off he ( ) Then he value of A becomes zero. In his case he mean average cos equaion (6) reduces o he following K( ) ( ) + ( T )( T ) ( T ) ( T ) (4) The value of for which K( ) given by (4)would be minimum saisfying he following inequaliy C T C T (5) 8

5 Journal of Agriculure and Life Sciences Vol., No. ; June 4 Numerical Illusraion To illusrae he presen model we consider he following values of he various parameers : C Rs. per uni per monh, C Rs 9. per uni per monh C Rs 8. per uni, R unis per monh, The funcion ( ) is aken in he form ( ) a + b, < ( ) <<, a, b > where we ake a., b. The probabiliy densiy funcion be defined as follows p( ) ( ),, elsewhere T monhs Using equaion (5), we find A.3 For his sysem, C T - C 7 For differen discree values of, he corresponding values of M( ) and K( ) are given in he Table- M( Table- ) K( ) Analyzing he above able- we find he opimum value of is 3 monhs and he minimum mean average cos is K( ) Rs Using he equaion (3), he opimum value of mean ordering quaniy is S( ) 784 unis. Sensiiviy Analysis and Discussion We now sudy he effecs of changes in he invenory sysem parameers such as,, C, a, b and R on he opimal C Cmean average cos K K( ) and opimal mean ordering quaniy S S( ) in he presen EOQ model. The sensiiviy analysis is performed by changing each of he parameers by -5%, -%, +5% and +% aking one parameer a a ime and keeping remaining parameers unchanged. 9

6 Cener for Promoing Ideas, USA Table-: Effecs of Changes in he Parameers on he Model Change of Changing parameers change(%) C S K C C a b R Commens on he sensiiviy analysis Analyzing he resuls given in he able-, he following observaions may be made (i) Increases/decreases wih increase/decrease in he value of he sysem parameer C. On he oher hand remains unchanged. The resuls obained show ha S is moderaely sensiive while K is insensiive o he changes in he value of C. (ii) K increases/decreases S wih increase/decrease S in he value of he K sysem parameer C. I can be noiced ha changes in he value of is sensiive where as K is almos insensiive o he changes in he value of C. (iii). For increase/decrease in he value of he parameer C, he adjoining S sensiiviy able shows ha decreases/increases and K increases/ decreases. However i can be seen ha S and K are very sensiive o changes in he value of C. (iv). The naure of changes of Sand K owards he changes in he value of a are similar as in he changes of he parameer C.

7 Journal of Agriculure and Life Sciences Vol., No. ; June 4 (v). S is insensiive and K is moderaely sensiive o changes in he value of he parameer b. (vi) K and S increase/decrease wih he increase/decrease in he value of R. The effecs on K and S due o changes in he value of R are very much appreciable. Concluding Remarks In his sudy, we have proposed an invenory model for random deerioraing iems wih discree ime variable. The mehod of solving he problem is analyical as well as compuaional. A numerical example and sensiiviy of he soluion have been performed in his model. We have also discussed a special case of he invenory model having no deerioraion of iems. The proposed discree naure of ime variable is more reasonable and realisic in pracice. The sensiiviy analysis concludes ha he reflecion of he uni cos and demand rae on he model are very significan. Reference Ghare PM, Schrader G (963). A model for exponenially decaying invenories, J. Ind. Engg. 4: Goel VP, Aggarwal SP (98), Order level invenory sysem wih power demand paern for deerioraing iems, proceeding wih All India Seminar on Operaional Research and Decision-making, Universiy of Delhi. Cover R, Philip G (973). An EOQ model for iems wih Weibull disribuion deerioraion. AIIE Trans. 5: Dua TK, Pal AK (988). Order Level invenory sysem wih power demand paern for iems wih variable rae of deerioraion. Ind. J. Pure and Appl. Mah. 9 (): Mandal B. and Pal A.K.(), Order level invenory sysem for perishable iems wih power demand paern, In. J. Mgm. & Sys., 6(3), Mandal B.(4). Opimal invenory managemen for iems wih weibull disribuion deerioraion under alernaing demand raes, In. J. Eng. and Tech Res (4), Pal A.K. and Mandal B.(998), Order level invenory sysem wih power demand paern for iems wih random deerioraion, In J. Mgm and sys., 4(3), 7-4 Mandal B.(3), An invenory model for random deerioraing iems wih a linear rended in demand and parial backlogging, Res. J. Bus Mgm and Accn., (3), 48-5 Dave U.(979), on a discree-in-ime order level invenory model for deerioraing iems, JORS, 3, Daa T.K. and Pal A.K.(99), Order level invenory model for iems wih ime-dependen deerioraion rae, J. Na. Acad. Mahs, 8, Sasieni M., Yaspan A. And Friedman L., Operaions Research Mehods and Problems, Appendix I John Wiley, new York, 959.