A PRODUCTION INVENTORY MODEL WITH DETERIORATING ITEMS AND SHORTAGES

Transcription

1 Yugoslav Journal of Oeraions Research 4 (004), Number, 9-30 A PRODUCTION INVENTORY MODEL WITH DETERIORATING ITEMS AND SHORTAGES G.P. SAMANTA, Ajana ROY Dearmen of Mahemaics Bengal Engineering College (D. U.), Howrah 703, INDIA Received: Ocober 003 / Acceed: March 004 Absrac: A coninuous roducion conrol invenory model for deerioraing iems wih shorages is develoed. A number of srucural roeries of he invenory sysem are sudied analyically. The formulae for he oimal average sysem cos, sock level, backlog level and roducion cycle ime are derived when he deerioraion rae is very small. Numerical examles are aken o illusrae he rocedure of finding he oimal oal invenory cos, sock level, backlog level and roducion cycle ime. Sensiiviy analysis is carried ou o demonsrae he effecs of changing arameer values on he oimal soluion of he sysem. Keywords: Deerioraing iem, shorage, economic order quaniy model.. INTRODUCTION In recen years, he conrol and mainenance of roducion invenories of deerioraing iems wih shorages have araced much aenion in invenory analysis because mos hysical goods deeriorae over ime. The effec of deerioraion is very imoran in many invenory sysems. Deerioraion is defined as decay or damage such ha he iem can no be used for is original urose. Food iems, drugs, harmaceuicals, radioacive subsances are examles of iems in which sufficien deerioraion can ake lace during he normal sorage eriod of he unis and consequenly his loss mus be aken ino accoun when analyzing he sysem. Research in his direcion began wih he work of Whiin [6] who considered fashion goods deerioraing a he end of a rescribed sorage eriod. Ghare and Schrader [7] develoed an invenory model wih a consan rae of deerioraion. An order level invenory model for iems deerioraing a a consan rae was discussed by Shah and Jaiswal [5]. Aggarwal [] reconsidered his model by recifying he error in he work of Shah and Jaiswal [5] in calculaing he average invenory holding cos. In all hese models, he demand rae and he deerioraion

2 0 G.P. Samana, A. Roy / A Producion Invenory Model wih Deerioraing Iems rae were consans, he relenishmen rae was infinie and no shorage in invenory was allowed. Researchers sared o develo invenory sysems allowing ime variabiliy in one or more han one arameers. Dave and Pael [5] discussed an invenory model for relenishmen. This was followed by anoher model by Dave [4] wih variable insananeous demand, discree ooruniies for relenishmen and shorages. Bahari- Kashani [] discussed a heurisic model wih ime-roorional demand. An Economic Order uaniy (EO) model for deerioraing iems wih shorages and linear end in demand was sudied by Goswami and Chaudhuri [8]. On all hese invenory sysems, he deerioraion rae is a consan. Anoher class of invenory models has been develoed wih ime-deenden deerioraion rae. Cover and Phili [3] used a wo-arameer Weibull disribuion o reresen he disribuion of he ime o deerioraion. This model was furher develoed by Phili [3] by aking a hree-arameer Weibull disribuion for he ime o deerioraion. Mishra [] analyzed an invenory model wih a variable rae of deerioraion, finie rae of relenishmen and no shorage, bu only a secial case of he model was solved under very resricive assumions. Deb and Chaudhuri [6] sudied a model wih a finie rae of roducion and a ime-roorional deerioraion rae, allowing backlogging. Goswami and Chaudhuri [9] assumed ha he demand rae, roducion rae and deerioraion rae were all ime deenden. Deailed informaion regarding invenory modelling for deerioraing iems was given in he review aricles of Nahmias [] and Rafaa [4]. An order-level invenory model for deerioraing iems wihou shorages has been develoed by Jalan and Chaudhuri [0]. In he resen aer we have develoed a coninuous roducion conrol invenory model for deerioraing iems wih shorages. I is assumed ha he demand rae and roducion rae are consans and he disribuion of he ime o deerioraion of an iem follows he exonenial disribuion. The main focus is on he srucural behaviour of he sysem. The convexiy of he cos funcion is esablished o ensure he exisence of a unique oimal soluion. The oimum invenory level is roved o be a decreasing funcion of he deerioraion rae where he deerioraion rae is aken as very small and he cycle ime is aken as consan. The formulae for he oimal average sysem cos, sock level, backlog level and roducion cycle ime are derived when he deerioraion rae is very small. Numerical examles are aken and he sensiiviy analysis is carried ou o demonsrae he effecs of changing arameer values on he oimal soluion of he sysem.. NOTATIONS AND MODELLING ASSUMPTIONS (i) (ii) (iii) (iv) (v) (vi) The following noaions and assumions are used for develoing he model. a is he consan demand rae. (> is he consan roducion rae. C is he holding cos er uni er uni ime. C is he shorage cos er uni er uni ime. C 3 is he cos of a deerioraed uni. (C,C and C 3 are known consans) C is he oal invenory cos or he average sysem cos.

3 G.P. Samana, A. Roy / A Producion Invenory Model wih Deerioraing Iems (vii) () is he invenory level a ime ( 0). (viii) Relenishmen is insananeous and lead ime is zero. (ix) T is he fixed duraion of a roducion cycle. (x) Shorages are allowed and backlogged. (xi) The disribuion of he ime o deerioraion of an iem follows he exonenial disribuion g() where e, for > 0, g () = 0, oherwise. is called he deerioraion rae; a consan fracion ( 0< << ) of he onhand invenory deerioraes er uni ime. I is assumed ha no reair or relacemen of he deerioraed iems akes lace during a given cycle. Here we assume ha he roducion sars a ime = 0 and sos a ime =. During [0, ], he roducion rae is and he demand rae is a ( < ). The sock aains a level a ime =. During [, ], he invenory level gradually decreases mainly o mee demands and arly for deerioraion. The sock falls o he zero level a ime =. Now shorages occur and accumulae o he level a ime = 3. The roducion sars again a a rae a = 3 and he backlog is cleared a ime = T when he sock is again zero. The cycle hen reeas iself afer ime T. This model is reresened by he following diagram: Invenory O 3 Time T

4 G.P. Samana, A. Roy / A Producion Invenory Model wih Deerioraing Iems 3. THE MATHEMATICAL MODEL AND ITS ANALYSIS Le () be he on-hand invenory a ime ( 0 T). Then he differenial equaions governing he insananeous sae of () a any ime are given by d() + () = a, 0 () d d() + () = a, () d d() = a, 3 (3) d d() = a, 3 T (4) d The boundary condiions are (0) = 0, ( ) =, ( ) = 0, ( 3 ) =, (T) = 0 (5) The soluions of equaions () (4) are given by () = ( ( e ), 0 (6) a a ( = + ( ) + ) e, (7) = a ( ), (8) 3 = ( ( ), T (9) From (5) and (6), we have 3 3 a e = ( ) = ( )( ) e = [ ] ( = log[ + { + }] ( ( = + a ( (0 (0b) (neglecing higher owers of, 0< << ).

5 G.P. Samana, A. Roy / A Producion Invenory Model wih Deerioraing Iems 3 Again from (5) and (7), we have a a 0 = ( ) = + ( + ) e ( ) = log ( + ) () a = log[( + ){ + + }] a ( ( (using (0) ) () Using he condiion ( 3 ) = -, we have from (8) a ( 3 ) = 3 = + (3) a 3 = + log[( + ){ + + }] a a ( ( (4) From (9) and (T) = 0, we have ( (T 3 ) = (5) Therefore, oal deerioraion in [0, T] = {( } + { a( )} ( a = [ log{ + + } ] + [ log( + )] a ( a = a log{ } log[{ }( )] + a + ( + a + ( + a = { + } a ( ( a { } a ( a a( a ( (Neglecing higher owers of ) = a( (6) The deerioraion cos over he eriod [0, T] C3 = a( (7)

6 4 G.P. Samana, A. Roy / A Producion Invenory Model wih Deerioraing Iems The shorage cos over he eriod [0, T] T = C { ( )} d 3 T = C [ a( ) d+ {( ( ) } d] (by (8) and (9)) 3 3 C = a( (by using (3) and (5) ) (8) The invenory carrying cos over he cycle [0, T] = C d 0 () ( a a ( ) [ ( ) { ( ) } ] 0 = C e d e d (9) Now, ( ( e ) d 0 ( = ( ) (neglecing higher owers of ) 3 3 = + ( 3( (using (0) and neglecing higher owers of ) (0) a a ( ) { ( ) } a a ( ) + + e d ( ) ( ) { e = + + } a a = log ( + ) + ( + ) { ( + ) } (by ) ) a a = a a log{ ( )} ( 3 ) { ( )} 3 a a + a + + a + a a = a a { ( )} ( 3 3 )( ) 3 a + a a a a + + a a + a = (neglecing higher owers of ) () a Therefore, he invenory carrying cos over he cycle [0, T ] 3 3 = C{ + + } = C { + } ( 3( a a( 3( ()

7 G.P. Samana, A. Roy / A Producion Invenory Model wih Deerioraing Iems 5 Hence he oal invenory cos of he sysem (using (7), (8) & () ) = C (, ) 3 C C C3 = { + } + + T a( 3( at( at( (3) From (4) and (5), we have at( ( a ) = a( (4) Therefore, using (3) and (4), he oal invenory cos of he sysem C C at ( = C ( ) = { + } + { T a( 3( at( 3 ( a ) C3 } + a ( a ) at ( a ) (5) Theorem : The average sysem cos funcion C( ) is sricly convex when 0< <<. Proof: Using (5), we have dc( ) C = { + } d T a( ( C ( a ) C3 { } at ( a( at ( + + ( ) ( ) { } [{ } dc C C a = d T a( ( at( a( ( a ) C ] 3 + > 0 a( at( (6) (as 0< << and > (7) Therefore C( ) is sricly convex when 0< <<. As C( ) is sricly convex in, here exiss an unique oimal sock level dc ha minimizes C( ). This oimal is he soluion of he equaion 0. d = We, herefore, find from (6) ha is he unique roo of he following equaion in : C C ( a ) C T a( ( at( a( at( 3 { + } { + } + = 0 (8) where is given by (4).

8 6 G.P. Samana, A. Roy / A Producion Invenory Model wih Deerioraing Iems Afer some calculaions, neglecing higher owers of, we have a( C T { CCT( + C ( C + C )} [ ] 3 = C ( + C) ( C+ C) which is a decreasing funcion of, where 0< <<. From (4), he oimal backlog level is given by (for fixed T ) : at( C a( CT { CCT ( + C3 ( C + C)} = + [ C ( + C) C ( + C) ( C + C) (30) ( CT + ] C ( + C) Therefore is an increasing or decreasing funcion of if CCT( + C ( C + C ) ( C T + > or < 0 resecively. ( ) 3 ( C + C) C+ C If is fixed and T varies, hen also vary and is given by (4). In his case he average sysem cos is a funcion of T alone and given by 3 C C at ( CT ( ) = { + } + { T a( 3( at( (3) ( a ) C3 } + a( at( Theorem : The average sysem cos funcion C(T), given by (3), is sricly convex when 0< <<. Proof: Here and 3 dc( T ) C = { + } dt T a( 3( C a( T ( { + } at ( a( C a( T ( C3 + { + } T a( at ( (9) (3) 3 dct ( ) C = { + } 3 dt T a( 3( + C ( { 3 at ( a( } + C 3 > 0 3 at ( (as 0< << and > (33)

9 G.P. Samana, A. Roy / A Producion Invenory Model wih Deerioraing Iems 7 Hence C(T) is sricly convex when 0< <<. Since C(T) is sricly convex in T, here exiss an unique oimal cycle ime T ha minimizes C(T). This oimal cycle ime T dc is he soluion of he equaion 0. dt = Therefore, he oimal cycle ime T is he unique roo of he following equaion in T (using (3) ) : 3 C C a( T { + } { T a( 3( at ( (34) ( C a( T ( C3 + } + { + } = 0 a( T a( at ( Afer some calculaions, neglecing higher owers of, we have T = [( C + C ) + { Ca + 3 C ( a ) + 3 C a( }] / 3 a( C 3 (35) Therefore, we conclude ha T is an increasing or decreasing funcion of if Ca + 3 C( a ) + 3 Ca( > or < 0 resecively NUMERICAL EXAMPLES Here we have calculaed oimal sock level, oimal backlog level,and he minimum average sysem cos C for given values of roducion cycle lengh T and oher arameers and T, and C for given values of and oher arameers by considering wo examles. Examle : Le = , C = 4, C =0, C 3 = 40, = 0, a = 8, and T = 80 in aroriae unis.based on hese inu daa, he comuer ouus are as follows : = , = and C = Examle : Here we have aken = , C = 4, C = 0, C 3 = 40, = 0, a = 8 and = 60 in aroriae unis. The comuer ouus are as follows : = 5.509, T = and C = SENSITIVITY ANALYSIS I. Here we have sudied he effecs of changes in he values of he arameers, C, C, C 3,, a and T on he oimal oal invenory cos, sock level and backlog level derived by he roosed mehod.the sensiiviy analysis is erformed by changing he value of each of he arameers by 50%, 5%, 5%, and 50%, aking one arameer a a ime and keeing he remaining six arameers unchanged. Examle is used. On he basis of he resuls shown in able, he following observaions can be made.

10 8 G.P. Samana, A. Roy / A Producion Invenory Model wih Deerioraing Iems Table : Sensiiviy analysis Parameer % change % change in % change in % change in C C C C a T I is seen from able ha he soluion is insensiive o changes in he arameers and C 3, while i is considerably sensiive o changes in he arameers C, C,, a and T.

11 G.P. Samana, A. Roy / A Producion Invenory Model wih Deerioraing Iems 9 II. We now sudy he effecs of changes in he values of he arameers, C, C, C 3,, a, and on he oimal oal invenory cos, cycle ime and backlog level by using examle. Table : Sensiiviy analysis Parameer % change %change in T % change in % change in C C C C a I is observed from able ha he soluion is insensiive o changes in he arameer, slighly sensiive o changes in he arameer C 3 while i is considerably sensiive o changes in he arameers C, C,, a and.

12 30 G.P. Samana, A. Roy / A Producion Invenory Model wih Deerioraing Iems Therefore he above sensiiviy analysis indicaes ha sufficien care should be aken o esimae he arameers C, C,, a and T(or ) in marke sudies. 6. CONCLUDING REMARKS In he resen aer, we have deal wih a coninuous roducion conrol invenory model for deerioraing iems wih shorages. I is assumed ha he demand and roducion raes are consan and he disribuion of he ime o deerioraion of an iem follows he exonenial disribuion. This model is alicable for food iems, drugs, harmaceuicals ec. Here we have sudied he srucural roeries of his invenory sysem. The sensiiviy analysis shows ha sufficien care should be aken o esimae he arameers C, C,, a and T (or ) in marke sudies. Acknowledgemen: The auhors would like o hank he referee for helful commens. REFERENCES [] Aggarwal, S.P., A noe on an order-level invenory model for a sysem wih consan rae of deerioraion, Osearch, 5 (978) [] Bahari-Kashani, H., "Relenishmen schedule for deerioraing iems wih ime-roorional demand", Journal of he Oeraional Research Sociey, 40 (989) [3] Cover, R.P., and Phili, G.C., "An EO model for iems wih Weibull disribuion deerioraion", AIIE Transacion, 5 (973) [4] Dave, U., "An order-level invenory model for deerioraing iems wih variable insananeous demand and discree ooruniies for relenishmen", Osearch, 3 (986) [5] Dave, U., and Pael, L.K., "(T,S i ) olicy invenory model for deerioraing iems wih ime roorional demand", Journal of he Oeraional Research Sociey, 3 (98) [6] Deb, M., and Chaudhuri, K.S., "An EO Model for iems wih finie rae of roducion and variable rae of deerioraion", Osearch, 3 (986) [7] Ghare, P.M., and Schrader, G.P., "A model for exonenially decaying invenories", Journal of Indusrial Engineering, 4 (963) [8] Goswami, A., and Chaudhuri, K.S., "An EO model for deerioraing iems wih shorages and a linear rend in demand", Journal of he Oeraional Research Sociey, 4 (99) [9] Goswami, A., and Chaudhuri, K.S., "Variaions of order-level invenory models for deerioraing iems", Inernaional Journal of Producion Economics, 7 (99) -7. [0] Jalan, A.K., and Chaudhuri, K.S., "Srucural roeries of an invenory sysem wih deerioraion and rended demand", Inernaional J. of Sysems Science, 30 (999) [] Mishra, R.B., "Oimum roducion lo-size model for a sysem wih deerioraing invenory", Inernaional Journal of Producion Research, 3 (975) [] Nahmias, S., "Perishable invenory heory: A review", Oeraions Research, 30 (98) [3] Phili,G.C., "A generalized EO model for iems wih Weibull disribuion deerioraion", AIIE Transacion, 6 (974) [4] Rafaa, F., "Survey of lieraure on coninuously deerioraing invenory model", Journal of he Oeraional Research Sociey, 4 (99) [5] Shah, Y.K., and Jaiswal, M.C., "An order-level invenory model for a sysem wih consan rae of deerioraion", Osearch, 4 (977) [6] Whiin, T.M., Theory of Invenory Managemen, Princeon Universiy Press, Princeon, NJ, 957.