MULTI-ITEM FUZZY INVENTORY MODEL FOR DETERIORATING ITEMS WITH FINITE TIME-HORIZON AND TIME-DEPENDENT DEMAND

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1 Yugoslav Journal of Operatons Research 16 (2006), Number 2, MULI-IEM FUZZY INVENORY MODEL FOR DEERIORAING IEMS WI FINIE IME-ORIZON AND IME-DEPENDEN DEMAND S. KAR 1,. K. ROY 2, M. MAII 3 1 Department of Engneerng Scence, alda Insttute of echnology, alda , West engal, Inda 2 Department of Mathematcs, engal Engneerng and Scence Unversty, owrah , West engal, Inda 3 Department of Appled Mathematcs wth Oceanology and Computer Programmng, Vdyasagar Unversty, Paschm Mdnapore , West engal, Inda Receved: March 2003 / Accepted: December 2005 Abstract: hs paper develops a fnte tme-horzon fuzzy mult-deteroratng nventory model wth/wthout shortage, where the demand ncreases lnearly wth tme. ere, the total proft s to be maxmzed under the lmtaton on nvestment. In ths problem, total proft, total nvestment cost and the tme-horzon are fuzzy n nature. he mprecseness n the above objectve and constrant goals have been expressed by fuzzy lnear/nonlnear membershp functons and vagueness n tme-horzon by dfferent types of fuzzy numbers. Results are llustrated wth numercal examples. Keywords: Fuzzy nventory, deteroratng tems, backlogged shortages, dynamc demand, fnte tme-horzon. 1. INRODUCION Most of the classcal determnstc nventory models consder the demand rate to be constant, ndependent of tme t. owever, for certan types of nventory, partcularly consumer goods (vz. food grans, olseeds, etc.), the demand rate ncreases wth tme. In real lfe, the harvest of food grans s perodcal. A large number of landless people of developng countres have a constant demand of food grans throughout the year. Margnal farmers or landless labourers produce food-gran n ther own pece of land or n the land

2 162 S. Kar,.K. Roy, M. Mat / Mult-Item Fuzzy Inventory Model for Deteroratng Items of ther land-lords by share croppng. Due to varous reasons, many of them are bound to sale a part of ther food grans mmedately after producton. hs secton of people cannot produce enough food grans to meet ther need for the entre producton cycle after the partal sale. As a result, the demand of food grans remans partly constant and partly ncreases wth tme for a fxed tme-horzon. Stanfel and Svazlan (1975) dscussed a fnte tme-horzon nventory problem for tme dependent demands. Slver and Meal (1973) developed an approxmate soluton technque of a determnstc nventory model wth tme varyng demand. Donaldson (1977) frst developed an exact soluton procedure for tems wth a lnearly ncreasng demand rate over a fnte plannng horzon. owever, hs soluton procedure was computatonally complcated. Removng the complexty, many other researchers have proposed varous other technques to solve the same nventory problem. In recent years, Dave (1989), Goyal et. al. (1992) and Datta and Pal (1992) developed models wth shortages assumng demand to be tme proportonal. All these models are based on the assumpton that there s no deteroraton effect on nventory. he most mportant assumpton n the extng lterature s that lfe tme of an tem s nfnte whle t s n storage. ut n realty, many physcal goods deterorate due to dryness, spolage, vaporsaton etc. and are damaged due to stayng longer than ther normal storage perod. he deteroraton also depends on preservng facltes and envronmental condtons of warehouse/storage. So, due to deteroraton effect, a certan fracton of the tems are ether damaged or decayed and are not n a perfect condton to satsfy the future demand of customers for good tems. Deteroraton for such tems s contnuous and constant or tme-dependent and/or dependent on the on-hand nventory. A number of research papers have already been publshed on above type of tems by Dave and Patel (1981), Sachan (1989), Goswam and Chowdhury (1991), Kang and Km(1983). It has been recognsed that one s ablty to make precse and sgnfcant statement concernng an nventory model dmnshes wth ncreasng complextes of the marketng stuaton. Generally, n nventory systems only lngustc (vague) statements are used to descrbe the model and t may not be possble to state the objectve functon and the constrants n precse mathematcal form. It may not also be possble to express the objectve n certan terms because the objectve goal s not defnable precsely. Smlarly, length of tme-horzon and average storage cost may be mprecse n nature. ere, the phenomena of such model may be descrbed n a fuzzy way. he theory of fuzzy sets was developed for a doman n whch descrpton of actvtes and observatons are fuzzy n the sense that there are no well defned boundares of the set of actvtes or observatons to whch the descrpton s appled. he theory was ntated by Zadeh (1965) and later appled to dfferent practcal systems by several researchers. Zadeh showed the ntenton to accommodate uncertanty n the nonstochastc sense rather than the presence of random varables. ellman and Zadeh (1970) frst appled fuzzy set theory n decson-makng processes. Zmmerman (1976) used the concept of fuzzy set n decson-makng processes by consderng the objectve and constrants as fuzzy goals. e frst appled fuzzy set theory wth sutable choce of membershp functons and derved a fuzzy lnear programmng problem. Currently, the fuzzy programmng technques are appled to solve lnear as well as non-lnear programmng problems (rappgy, et-al (1988), Carlsson and Korhonen (1986), etc.). La and wang (1992, 1994) descrbed the applcaton of fuzzy sets to several operaton research problems n two well-known books.

3 S. Kar,.K. Roy, M. Mat / Mult-Item Fuzzy Inventory Model for Deteroratng Items 163 owever, as far as we know, fuzzy set theory has been used n few nventory models. Sommer (1981) appled fuzzy dynamc programmng to an nventory and producton-schedulng problem. Kacprzyk and Stanewsk (1982) consdered a fuzzy nventory problem n whch, nstead of mnmzng the total average cost, they reduced t to a mult-stage fuzzy-decson-makng problem and solved by a branch and bound algorthm. Park (1987) examned the EOQ formula wth fuzzy nventory costs represented by rapezodal fuzzy number (rfn). Recently, Lam and Wong (1996) solved the fuzzy model of jont economc lot sze problem wth multple prce breaks. Roy and Mat (1995) solved the classcal EOQ model n a fuzzy envronment wth fuzzy goal, fuzzy nventory costs and fuzzy storage area by FNLP method usng dfferent types of membershp functons for nventory parameters. hey (1997) examned the fuzzy EOQ model wth demand dependent unt prce and mprecse storage area by both fuzzy geometrc and non-lnear programmng methods. hey (1997, 1998) also dscussed sngle and mult-perod fuzzy nventory models usng fuzzy numbers. It may be noted that none has consdered the tme-horzon as fuzzy number and attacked the fuzzy optmzaton problem drectly usng fuzzy non-lnear programmng technques. ll now, no lterature s avalable for the mult-tem nventory models for fnte tme-horzon n fuzzy envronment wth or wthout shortages. In ths paper, we have developed a mult-tem nventory model ncorporatng the constant rate of deteroraton effect assumng the demand to be a lnearly ncreasng functon wth tme and shortages to be allowed for the prescrbed fnte tme-horzon. In the extng lterature of nventory models, the tme-horzon s assumed to be fxed. ut n realty, tme-horzon s normally lmted but mprecse, uncertan and flexble. hs may be better represented by some fuzzy numbers. he problem s reduced to a fuzzy optmzaton problem assocatng fuzzness wth the tme-horzon, objectve and constrant goal. he fuzzy mult-tem nventory problem s solved for dfferent fuzzy numbers and fuzzy membershp functons. he model s llustrated wth a numercal example and the results for the fuzzy and crsp model are compared. 2. MODEL AND ASSUMPIONS We use the followng notatons n proposed model: n = numbers of tems, = total nvestment for replenshment. For -th. tem ( = 1, 2, 3,..,n) D (t) = demand rate (functon of tme), C 1 = nventory holdng cost per unt tem per unt tme, p = cost prce per unt tem, s = sellng prce per unt tem, C 2 = shortage cost per unt tem per unt tme, = constant rate of deteroraton, = prescrbed tme-horzon, m = total number of replenshment to be made durng the prescrbed tmehorzon,.e. m must be a postve nteger (decson varable), = length of each cycle.e., = /m q j = nventory level at tme t for j-th cycle, (j = 1, 2,, m )

4 164 S. Kar,.K. Roy, M. Mat / Mult-Item Fuzzy Inventory Model for Deteroratng Items Q j = lot sze for j-th cycle, (j = 1, 2,, m ) R j = backlogged quantty for j-th cycle, (j = 1, 2,, m ) F j = replenshment cost for j-th cycle, (j = 1, 2,, m ) k = fracton of scheduled perod of whch no shortage occur,.e. k be a real number n [0, 1] (decson varable), (m, k) = otal proft of the system. (Where m and k are the vectors of n decson varables m ( = 1, 2,,n) and k ( = 1, 2,,n) respectvely.) he basc assumptons about the model are: () Replenshment rate s nstantaneous, () are allowed and fully backlogged, () Lead tme s zero, (v) he demand rate D (t) at any nstant t s a lnear functon of t such that D (t) = a b t, a, b 0, 0 t, (v) he replenshment cost F j for j-th cycle (j = 1, 2,., m ) s lnearly dependent on tme and s of the followng form F j = A r (j 1), j = 1, 2,., m where A > 0 and r s the addtonal replenshment cost per unt of tems, (v) We assume that the perod for whch there s no shortage n each nterval [(j 1), j ] s a fracton of the schedulng perod and s equal to k (0 < k <1). occur at tmes (k j 1), (j = 1, 2,, m -1) where (j 1) < (k j 1) < j, (j = 1, 2,.m -1). Last replenshment occurs at tme (m -1) and shortages are not allowed n the last perod [(m -1), ]. Our problem s to derve the optmal reorder and shortage ponts and hence to determne the optmal values of m and k ( = 1, 2,., n) whch maxmze the total proft over the tme-horzon [0, ].

5 S. Kar,.K. Roy, M. Mat / Mult-Item Fuzzy Inventory Model for Deteroratng Items MAEMAICAL FORMULAION OF E PROLEM Let q j (t) be the amount n nventory at tme t durng the jth cycle [(j-1) t j, j = 1, 2, 3,., m ]. hen the dfferental equatons descrbng the system durng the jth cycle are dq j () t q j () t D () t = 0, (j-1) t (k j-1), (1) dt dq j () t D () t = 0, (k j 1) t j, j = 1, 2,., (m -1), (2) dt and the dfferental equaton governng the stock status for the last replenshment cycle [(m -1) t ] s dq m () t q m () t D () t 0 dt =, (m -1) t, (3) wth the boundary condtons q j (t) = 0 at t = (k j-1), j = 1, 2,..., (m -1) and q m () t = 0 at t =. he solutons of (1), (2), and (3) are { ( k j1) t b } 1 {( 1) } () e b k j t q t a ( k j 1) e j = t (4) ( j1) t ( k j1), b { } { } q j () t = a ( k j1) t 2 ( k j1) t, (k j 1) t j, (5) j = 1, 2,., (m 1), { t} b e 1 b { t} () q t a e m = t If Q j be the nventory level n tme (j-1), then k b e 1 b k Q a ( k j 1) e j = ( j1) and (m 1) t. (6) j = 1, 2,, (m 1), (7)

6 166 S. Kar,.K. Roy, M. Mat / Mult-Item Fuzzy Inventory Model for Deteroratng Items b e 1 b Q a e m = ( m 1). (8) Let R j be the total amount of backlogged quanttes over the perod [(k j 1), j ], j = 1, 2,.., (m 1). hen j Rj = q j () t dt= ( K 1) a b j ( K 1), ( 1) 2 3 k j j = 1,2,,(m 1). (9) oldng cost n j-th [j = 1, 2,., (m 1)] cycle s V j = C 1 G j ( k j-1) Gj = q j () t dt (10) ( j1) b a k k 2 1 b e e 1 k 2( j 1) k = k ( k j 1) 2 2 oldng cost n the last cycle s j = 1, 2,, (m 1). (11) where V m = C 1 q m () t dt = ( m 1) C 1 G m, (12) G m = q m () t dt ( m 1) b a e 1 b 1 e (2m 1) = 2 2. (13) otal number of deteroratng tems over the perod [(j-1), j ], j = 1, 2,.., (m 1), s S Dj = G j, j = 1, 2,.., (m 1), (14) and total number of deteroratng tems n the last cycle s

7 S. Kar,.K. Roy, M. Mat / Mult-Item Fuzzy Inventory Model for Deteroratng Items 167 S D m = G m. (15) herefore, the total proft of the system for the entre tme horzon s Crsp model: m -1 (m, k ) = ( s p )( Q j R j ) V j F j s S D C 2 R j j= 1 j ( s p ) Q m V m F m s S D. m ence, our object s to maxmze the total proft subject to the lmtaton on nvestment cost,.e. Max (m, k) = n ( m, k ) (16) = 1 n = {( s p ) Q F ( C 1 s ) G ( s p C 2 ) R } = 1 subject to n p ( Q ) = 1 Q > 0, = 1, 2,,n, where, m Q = Qj = j= 1 k m k 1 b b e m ( m 1) a e ( 2k m 2) ( m 2 ) 2 m m 1 b b e m a e ( m 1) m = 1, 2,,n.

8 168 S. Kar,.K. Roy, M. Mat / Mult-Item Fuzzy Inventory Model for Deteroratng Items F = m F j j= 1 r ( m 1) m A = 2m = 1, 2,,n. m G = G j j= 1 k k b m k b m m 2m e 1 e 1 = ( m 1) a ( 2k m 2) ( 2 ) m k k m b a m m 2 e 1 b 1 e (2m 1) 2 m = 1, 2,..., n. R = m 1 R j = m ( -1) ( -1) 2 K m a b ( K -1) j= = 1, 2,,n. Fuzzy Objectve and Constrant goal: In most of the programmng model, the decson maker s not able to artculate a precse aspraton level to an objectve or constrant. owever, t s possble for hm to state the desrablty of achevng an aspraton level n an mprecse nterval around t. An objectve wth nexact target value (aspraton level) s termed as a fuzzy goal. Smlarly, a constrant wth mprecse aspraton level s also treated as a fuzzy goal. Fuzzy Decson: A fuzzy decson s defned as the fuzzy set of alternatves resultng from the ntersecton of the objectve goal and the constrants. More formally, gven a fuzzy goal G and a fuzzy constrant C n the space alternatves X, a decson D s defned as the fuzzy set, G C. he membershp functon of the fuzzy decson μ D s gven by mn( μ, G μ C ).

9 S. Kar,.K. Roy, M. Mat / Mult-Item Fuzzy Inventory Model for Deteroratng Items 169 Mathematcal Formulaton of the fuzzy model: When the above proft goal, average storage cost and total tme horzon becomes fuzzy then the sad crsp model (16) s transformed to Max (m, k) = n ( m, k ) = 1 n = {( s p ) Q F ( C 1 s ) G ( s p C 2 ) R } (17) = 1 subject to n pq = 1 where, k Q m k e 1 b b = ( 1) m ( 2 2 ) ( 2 ) m a e k m m 2 m m e 1 b b m a e ( m 1) m k k G m b m e 1 k b e 1 = ( m 1) a ( 2k m 2 ) m 2m ( 2) k k m m b a m m 2 e 1 b 1 e (2m 1) m 2 m F = m F j j= 1 r ( m 1) = m A 2m

10 170 S. Kar,.K. Roy, M. Mat / Mult-Item Fuzzy Inventory Model for Deteroratng Items 2 m 1 R 1 = R j = 2 2 ( -1) ( -1) m K m a b ( K -1) j= 1 2 m m 2 3. (where wavy bar ( ) represents the fuzzy charactersaton). In ths fuzzy model, the fuzzy objectve goal and fuzzy nvestment cost constrant are represented by ther membershp functons, whch may be lnear or nonlnear. ere, μ and μ are assumed to be non-decreasng and non-ncreasng contnuous lnear/non-lnear membershp functon for objectve proft goal and storage cost constrant as follows: 1 for x> x μ ( x) = 1 for P < x < P 0 for x< P 1 for x> 2 x μ ( x) = 1 for P < x < P 0 for x< P

11 S. Kar,.K. Roy, M. Mat / Mult-Item Fuzzy Inventory Model for Deteroratng Items for x< x μ ( x) = 1 for < x < P P 0 for x> P

12 172 S. Kar,.K. Roy, M. Mat / Mult-Item Fuzzy Inventory Model for Deteroratng Items 1 for x< 2 x μ ( x) = 1 for < x < P P 0 for x> P Agan, as the tme-horzon be near about ( = 1, 2,., n), these fuzzy coeffcents may be represented by dfferent type of fuzzy numbers (e.g. FN, rfn, N and PrFN) and μ ( = 1, 2,., n) are represented the membershp functons of these coeffcents. Let μ D be the membershp functon of the fuzzy set decson of the model. Snce, μ ( = 1, 2,., n), μ are the membershp functons of fuzzy coeffcents, constrant goal and μ s the membershp functon of fuzzy objectve goal. he decson space n fuzzy envronment s the ntersecton of fuzzy sets correspondng to the fuzzy proft goal and fuzzy constrant goals. ence our problem s Max μ D (x) (18) subject to μ μ D (x), μ μ D (x), where m, k are the decson vectors as n (18), μ D (x) = Mn [ μ, μ, μ, μ,..., μ ], = 1, 2,., n. 1 2 Defnng α for μ D (x) and usng the expressons for the membershp functons μ, μ, μ (Consderng as FN), the followng equvalent crsp problem can be defned as a mxed nteger non-lnear programmng problem: n

13 S. Kar,.K. Roy, M. Mat / Mult-Item Fuzzy Inventory Model for Deteroratng Items 173 Max α (19) subject to 1 (m, k) > ( α ) μ 1 (m, k) > μ ( α ) -1 μ ( α ) μ μ -1 ( α ) = 1, 2,.., n. L U where m, k are the decson vectors as n (18), 1 μ ( α ) 1/ 0 = - (1 α) n P 1 μ ( α ) 1/ 0 = - (1 α) n P -1 μ ( α ) = L 1 α ( 2 1 ) -1 μ ( α ) = U 3 - α ( 2 1 ) n 0 1 or 2, α [0, 1], for FN (= [ 1, 2, 3 ]). Smlarly, for rfn we can express the equvalent crsp problem. he problems n (16) and (19) are solved usng a mxed nteger-programmng algorthm n FORRAN NUMERICAL EXAMPLE o llustrate the above crsp model (16) and the correspondng fuzzy model (17) we assume the followng nput data shown n table-1 and present the results for crsp and fuzzy models n table 2, and table 3, 4, 5, 6 respectvely. Dfferent fuzzy models are due to dfferent fuzzy membershp functons and fuzzy numbers for total proft, total nvestment cost and tme horzon respectvely. able 1: Input data for crsp and fuzzy numbers Items S I ($) p ($) a b A ($) C 1 ($) C 2 ($) r w = $ 6720 able 2: Results for crsp model Cases ($) m 1 m 2 m 3 k 1 k 2 k 3 ($) Wth Wthout

14 174 S. Kar,.K. Roy, M. Mat / Mult-Item Fuzzy Inventory Model for Deteroratng Items able 3: Optmal result for fuzzy model-1 When the membershp functon of total proft and total nvestment cost are lnear and tme-horzon ( = 1, 2, 3.) are trangular fuzzy number,.e., = ($650, $800), = ($6000, $7500), 1 = (10, 12, 15), 2 = (12, 14, 16), 3 = (8, 11, 14). Cases α ($) m 1 m 2 m 3 k 1 k 2 k ($) Wth Wthout able 4: Optmal result for fuzzy model-2 When the membershp functon of total proft, total nvestment cost are lnear and tme-horzon ( = 1, 2, 3.) are trapezodal fuzzy number,.e, where = ($650, $800), = ($6000, $7500), 1 = (8, 11, 13, 15), 2 = (12, 14, 15, 17), 3 = (8, 11, 12, 14). Cases α ($) m 1 m 2 m 3 k 1 k 2 k ($) Wth Wthout able 5: Optmal result for fuzzy model-3 When the membershp functon of total proft, total nvestment cost are parabolc and tme-horzon ( = 1, 2, 3.) are trangular fuzzy number,.e, where = ($650, $800), = ($6000, $7500), 1 = (8, 11, 13, 15), 2 = (12, 14, 15, 17), 3 = (8, 11, 12, 14). Cases α ($) m 1 m 2 m 3 k 1 k 2 k ($) Wth Wthout able 6: Optmal result for fuzzy model-4 When the membershp functon of total proft s parabolc, nvestment cost s lnear and total tme-horzon 1, 2 are trangular fuzzy number but 3 s trapezodal fuzzy number,.e, where = ($650, $800), = ($6000, $7500), 1 = (10, 12, 15), 2 = (12, 14, 16) and 3 = (8, 11, 12, 14). Cases α ($) m 1 m 2 m 3 k 1 k 2 k ($) Wth Wthout

15 S. Kar,.K. Roy, M. Mat / Mult-Item Fuzzy Inventory Model for Deteroratng Items CONCLUSION In ths paper, we have solved a tme-horzon nventory problem for deteroratng tems havng a lnear tme-dependent demand under storage cost constrant n fuzzy envronment. he model permts nventory shortage n each cycle (except the last cycle), whch s completely backlogged wthn the cycle tself. Optmal results of both the crsp and fuzzy models for two cases (wth shortages and wthout shortages) for dfferent fuzzy numbers are presented n tables 2-6. ere t s observed that the results of the fuzzy model are better than the respectve crsp ones. We also mentoned herewth that n most of the fxed tme-horzon problems, optmum number of replenshment s evaluated by tral and error method,.e., substtutng m = 1, 2, 3,.. and then choosng that value of m for whch cost s mnmum or proft s maxmum. In ths paper, number of replenshment has been taken as a decson varable (nteger) and the optmum value has been evaluated usng the mxed nteger-programmng algorthm. Stll, there s a lot of scope to make the nventory problems much more realstc by consderng some parameters of the objectve/constrants are probablstc, and other are mprecse. hese fuzzy stochastc problems are to convert nto an equvalent determnstc/crsp problem usng probablty dstrbuton and fuzzy membershp functons and then solved by dfferent programmng methods. REFERENCE [1] ellman, R.E., and Zadeh, L.A., Decson-makng n a fuzzy envronment, Management Scence, 17 (1970) [2] Carlsson, C., and Korhonen, P., A parametrc approach to fuzzy lnear programmng, Fuzzy sets and Systems, 20 (1986) [3] Donaldson, W.A., Inventory replenshment polcy for a lnear trend n demand- an analytcal soluton, Operaton Research Quarterly, 28 (1977) [4] Dave, U., A determnstc lot-sze nventory model wth shortages and a lnear trend n demand, Novel Research Logstst, 36 (1989) [5] Datta,.K., and Pal, A.K., A note on a replenshment polcy for an nventory model wth lnear trend n demand and shortages, Journal of Operatonal Research Socety, 43 (1992) [6] Dave, U., and Patel, L.K., (, S ) polcy nventory model for deteroratng tems wth tme proportonal demand, Journal of Operatonal Research Socety, 32 (1981) [7] Dubos, D., and Prade,., Operaton on fuzzy numbers, Internatonal Journal of Systems Scence, 9(6) (1978) [8] Goal, S.K., et. al., he fnte horzon trended nventory replenshment problem wth shortages, Journal of Operatonal Research Socety, 43 (1992) [9] Goswam, A., and Chowdhury, K.S., A EOQ model for deteroratng tems wth shortages and lnear trend n demand, Journal of Operatonal Research Socety, 42 (1991) [10] Kang, S., and Km, A., A study on the prce and producton level of the deteroratng nventory system, Internatonal Journal of Producton Research, 21 (1983) [11] Kacprzyk, J., and Stanewsk, P., Long term nventory polcy makng through fuzzy decson makng models, Fuzzy Sets and Systems, 8 (1982) [12] La, Y.J., and wang, C.L., Fuzzy Mathematcal Programmng, Methods and Applcatons, Sprnger-Verlag, edelberg, 1992.

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