Yugoslav Journal of Operaions Research 2 (22), Number, 6-7 DEERMINISIC INVENORY MODEL FOR IEMS WIH IME VARYING DEMAND, WEIBULL DISRIBUION DEERIORAION AND SHORAGES KUN-SHAN WU Deparmen of Bussines Adminisraion amkang Universiy, amsui, aipei, aiwan kunshan@email.ku.edu.w Absrac: In his paper, an EOQ invenory model is depleed no only by ime varying demand bu also by Weibull disribuion deerioraion, in which he invenory is permied o sar wih shorages and end wihou shorages. A heory is developed o obain he opimal soluion of he problem; i is hen illusraed wih he aid of several numerical examples. Moreover, we also assume ha he holding cos is a coninuous, non-negaive and non-decreasing funcion of ime in order o exend he EOQ model. Finally, sensiiviy of he opimal soluion o changes in he values of differen sysem parameers is also sudied. Keywords: Invenory, ime-varying demand, Weibull disribuion, shorage.. INRODUCION Deerioraion is defined as decay, change or spoilage ha prevens he iems from being used for is original purpose. Foods, pharmaceuicals, chemicals, blood, drugs are some examples of such producs. In many invenory sysems, deerioraion of goods in he form of a direc spoilage or gradual physical decay in he course of ime is a realisic phenomenon and hence i should be considered in invenory modeling. Deerioraing invenory has been widely sudied in recen years. Ghare and Schrader [9] were wo of he earlies researchers o consider coninuously decaying invenory for a consan demand. Shah and Jaiswal [2] presened an order-level invenory model for deerioraing iems wih a consan rae of deerioraion. Aggarwal [] developed an order-level invenory model by correcing and modifying he error in Shah and Jaiswal's [2] analysis in calculaing he average invenory holding cos. Cover and Philip [4] used a variable deerioraion rae of wo-parameer Weibull
62 K.-S. Wu / Deerminisic Invenory Model for Iems wih ime Varying Demand disribuion o formulae he model wih assumpions of a consan demand rae and no shorages. However, all he above models are limied o he consan demand. ime-varying demand paerns are commonly used o reflec sales in differen phases of a produc life cycle in he marke. For example, he demand for invenory iems increases over ime in he growh phase and decreases in he decline phase. Donaldson [8] iniially developed an invenory model wih a linear rend in demand. Afer ha, many researchers' works (see, for example, Silver [22], Goel and Aggarwal [2], Richie [2], Deb and Chaundhuri [7], Dave and Pal [5-6], Chung and ing [3], Kishan and Mishra [5], Giri e al. [], Hwang [4], Pal and Mandal [9], Mandal and pal [6], and Wu e al. [25-26]) have been devoed o incorporaing a ime-varying demand rae ino heir models for deerioraing iems wih or wihou shorages under a variey of circumsances. Recenly, Wu e al. [26] invesigaed an EOQ model for invenory of an iem ha deerioraes a a Weibull disribuion rae, where he demand rae is a coninuous funcion of ime. In heir model, he invenory model sars wih an insan replenishmen and ends wih shorages. In he presen paper, he model of Wu e al. [26] is reconsidered. We have revised his model o consider ha i sars wih zero invenories and ends wihou shorages. Comparing he opimal soluions for he same numerical examples, we find ha boh he order quaniy and he sysem cos decrease considerably as a resul of is saring wih shorages and ending wihou shorages. 2. ASSUMPIONS AND NOAIONS he proposed invenory model is developed under he following assumpions and noaions.. Replenishmen size is consan and replenishmen rae is infinie. 2. Lead ime is zero. 3. is he fixed lengh of each producion cycle. 4. he iniial and final invenories are boh zero. 5. he invenory model sars wih zero invenories and ends wihou shorages. 6. he demand rae D () a any insan is posiive in (, ] and coninuous in [, ] 7. he invenory holding cos c per uni per uni ime, he shorage cos c 2 per uni per uni ime, and he uni-deerioraed cos c 3 are known and consan during he period. 8. he deerioraion rae funcion, θ (), represens he on-hand invenory deerioraion per uni ime, and here is no replacemen or repair of deerioraed unis during he period. Moreover, in he presen model, he funcion θ() =, >, >, >, θ() < (also see Cover and Philip [4]).
K.-S. Wu / Deerminisic Invenory Model for Iems wih ime Varying Demand 63 3. MAHEMAICAL MODEL AND SOLUION he objecive of he invenory problem here is o deermine he opimal order quaniy in order o keep he oal relevan cos as low as possible. he behavior of he invenory sysem a any ime during a given cycle is depiced in Fig.. he invenory sysem sars wih zero invenories a = and shorages are allowed o accumulae up o. Procuremen is done a ime. he quaniy received a is used parly o make up for he shorages ha accumulaed in he pervious cycle from ime o. he res of he procuremen accouns for he demand and deerioraion in [, ]. he invenory level gradually falls o zero a. Figure : An illusraion of invenory cycle he invenory level of he sysem a ime over he period [, ] can be described by he following differenial equaions: d I () = D (), d () and di() + θ () I () = D (), d (2) where θ() =, >, >, > (3)
64 K.-S. Wu / Deerminisic Invenory Model for Iems wih ime Varying Demand By virue of equaion (3) and (2), we ge di() + I() = D(), d (4) he soluions of differenial equaions () and (4) wih he boundary condiions I ( )= and I ( ) = are and I () = Dudu ( ), (5) u I () = e Due ( ) du, (6) is Using equaion (6), he oal number of iems ha deerioraed during [, ] = ( ) () = D I Dd e De () d Dd () (7) he invenory ha accumulaes over he period [, ] is u I = e D( u) e du d (8) Moreover, from equaion (5), he amoun of shorage during he inerval [, ) is given by B = D( u) dud = ( u) D( u) du (9) Using equaions (7)-(9), we can ge he average oal cos per uni ime (including holding cos, shorage cos and deerioraion cos) as C ( ) = [ ci + cb 2 + cd 3 ] c u c2 = e D( u) e du d + ( u) D( u) du c 3 + u e D( u) e du D( u) du () he firs and second order differenials of ( ) C wih respec o are respecively as follows:
K.-S. Wu / Deerminisic Invenory Model for Iems wih ime Varying Demand 65 and ( ) dc = () ( + ) c2 Dd c c3 e De () d d 2 ( ) dc 2 ( ) = + + c ( ) 3 c e D e d 2 d D ( ) ( + c + c2 + c3 ) () (2) 2 dc ( ) Because 2 > for, herefore, he opimal value of (we denoe i by ) d which minimizes he average oal cos per uni ime can be obained by solving he dc( ) equaion: =. ha is, saisfies he following equaion: d Now, we le () ( + ) c D d c c e D() e d = (3) 2 3 ( ) = () ( + ) f c Dd c c e De () d 2 3 because f ( )< and f( )>, hen by using he Inermediae Value heorem here exiss a unique soluion (, ) saisfying (3). Equaion (3) in general canno be solved in an explici form; hence we solve he opimal value by using Maple V, a program developed by he Waerloo Maple Sofware Indusry, which can perform he symbolic as well as he numerical analysis. Subsiuing = in equaion (6), we find ha he opimal ordering quaniy Q (which is denoed by Q ) is given by ( ) Q = I ( ) + Dd () = e De () d+ Dd () (4) Moreover, from equaion (), we have ha he minimum value of he average oal cos per uni ime is C C( ). = 4. EQQ INVENORY WIH IME VARYING OF HOLDING COS In he Secion 3 he holding cos is assumed o be consan. In pracice, he holding cos may no always be consan because he price index may increase wih
66 K.-S. Wu / Deerminisic Invenory Model for Iems wih ime Varying Demand ime. In order o generalize he EQQ invenory model, various funcions describing he holding cos were inroduced by several researchers, such as Naddor [8], Van der Veen [23], Muhlemann and Valis Spanopoulos [7], Weiss [24], Goh [3], Giri e al. [], Giri and Chaudhuri [], Beyer and Sehi [2], Wu e al. [26], and among ohers. herefore, in his secion we assume ha he holding cos h () per uni per uni ime is a coninuous, nonnegaive and non-decreasing funcion of ime. hen, he average oal cos per uni is replaced by u c2 C ( ) = he ( ) Due ( ) du d+ ( ududu ) ( ) c + 3 u e D( u) e du D( u) du Hence, he necessary condiion ha he average oal cos C ( ) be minimum dc( ) is replaced by =, which gives d (5) () ( ( ) + ) c D d h c e D() e d = (6) 2 3 Similarly, here exiss a single soluion [, ] ha can be solved from equaion (6). Moreover, he sufficien condiion for he minimum average oal cos is 2 ( ) dc 2 ( ) ( ) ( ) = + + + c ( ) 3 h h e D e d 2 d (7) D ( ) ( ( ) + h + c2 + c3 ) > ( for ) would be saisfied. In addiion, from equaion (5), we have ha he minimum value of he average oal cos per uni ime is C = C( ). Finally, he opimal order quaniy is he same as equaion (4). 5. NUMERICAL EXAMPLES o illusrae he proceeding heory, he following examples are considered. Example. Linear rend in demand Le he values of he parameers of he invenory model be c = $ per uni 3 per year, c 2 = $ 5 per uni per year, c 3 = $ 5 per uni, = 2, = 5,. = year, and linear demand rae D () = a+ b, a= 2, b = 2. Under he given parameer values and according o equaion (3), we obain ha he opimal value. 49555 year. aking =
K.-S. Wu / Deerminisic Invenory Model for Iems wih ime Varying Demand 67 =. 49555 ino equaion (4), we can ge ha he opimal order quaniy Q is 25.2369 unis. Moreover, from equaion () we have ha he minimum average oal cos per uni ime is C = $ 9. 7465. Example 2. Consan demand he parameer's values in he example are idenical o example excep for he consan demand rae D ()= 5. By using a similar procedure, we obain ha he opimal values =. 48662 year, Q = 6. 22576 unis and he minimum average oal cos per uni ime C is $26.87344. Example 3. Exponenial rend in demand he parameer's values in he example are idenical o example excep for 98. he exponenial demand rae D () = 5e. By using a similar procedure, we obain ha he opimal values =. 39773 year, Q = 39. 7469 unis and he minimum average oal cos per uni ime C is $ 59. 6552. Example 4. Linear rend in holding cos he parameer's values in he example are idenical o example 3 excep for he holding cos rae h ()= 3+ 2 per uni per year. By using a similar procedure, we obain ha he opimal values =. 499 year, Q = 38. 64943 unis and he minimum average oal cos per uni ime C is $ 64. 6384. Example 5. he parameer's values in he example are idenical o example excep for = 25., = 25. and =. By using a similar procedure, he compued resuls are shown in able. able shows ha each of, Q and C increases wih an increase in he value of. Nex, comparison of our resuls wih hose of Wu [26] for four examples is shown in able 2 and 3. hey show ha C all decrease in our model. ha is, i is esablished ha his model, where he invenory sars wih shorages and ends wihou shorages, is economically beer han he model of Wu e al. [26] (where he invenory sars wihou shorages and ends wih shorages). able : Opimal resuls of he various values of =. 25.33668 23.4848 87.8862 = 25..4965 24.2743 5.2536 = 5..49555 25.2369 9.7465 =.58636 25.999 5.99 Q C
68 K.-S. Wu / Deerminisic Invenory Model for Iems wih ime Varying Demand able 2: Opimal resuls of he proposed model Q Example 25.2369 9.7465 Example 2 6.22576 26.87344 Example 3 39.7469 59.6552 Example 4 38.64943 64.6384 able 3: Opimal resuls of Wu's model Q Example 27.66787 6.2575 Example 2 66.36872 278.46337 Example 3 45.57834 89.6897 Example 4 45.38 89.5684 C C 6. SENSIIVIY ANALYSIS We are now o sudy he effecs of changes in he sysem parameers c, c2, c 3,, a, b and on he opimal value ( ), opimal order quaniy ( Q ) and opimal average oal cos per uni ime ( C ) in he EOQ model of Example. he sensiiviy analysis is performed by changing each of he parameers by 5%, 25%, + 25 % and +5 %, aking one parameer a a ime and keeping he remaining parameers unchanged. he resuls are shown in able 4.. 2. 3. and On he basis of he resuls of able 4, he following observaions can be made. C increase while parameer c. However,, C increase while Q decreases wih an increase in he value of he model C are lowly sensiive o changes in c. decreases wih an increase in he value of he model parameer c 2. he obained resuls show ha and C are moderaely sensiive whereas Q is lowly sensiive o changes in he value of c 2. and C increase while Q decreases wih an increase in he value of he model parameer c 3. Moreover, and C are moderaely sensiive whereas sensiive o changes in he value of c 3. 4. Each of, Q is lowly C increases wih an increase in he value of he parameer. he obained resuls show ha and C are moderaely sensiive whereas Q is lowly sensiive o changes in he value of.
K.-S. Wu / Deerminisic Invenory Model for Iems wih ime Varying Demand 69 5. Each of 6. Moreover,,, C increases wih an increase in he value of he parameer. C are very highly sensiive o changes in. C increase while decreases wih an increase in he value of he parameer a. he obained resuls show ha Q and C are highly sensiive whereas Q is mos insensiive o changes in he value of a. 7. Each of,, Moreover, C increases wih an increase in he value of he parameer b. C are lowly sensiive o changes in b. able 4: Effec of changes in he parameers of he invenory model %change in parameers %change Q c +5 5 c 2 +5 5 c 3 +5 5 +5 5 +5 5 a +5 5 b +5 5 +5. +2.62 2.75 5.63 4.26 7.95 +.5.4 +.85 +5.84 6.98 5.67 +5.92 +8.64.44 23.39 +48.8 +23.9 23.74 47.5.58.35 +.56 +.64 +.84 +.42.43.44 2.5.7 +.9 +2.5 +6.89 3.6 4. 8.45 4.2 2.23 +3.4 +7.67 +.3 +.87.55 4.8 +62.62 +3.36 28.38 54.65 +47.73 +23.87 23.87 47.73 +2.27 +.3.3 2.27 C +2.94 +.53.65 3.43 +8.3 +9.9 2.43 28.97 +22.34 +.75 3.2 28.3 +.24 +6.68 9.95 24.82 +2.48 +47.3 38.66 68.85 +47.55 +23.78 23.78 47.57 +2.44 +.22.22 2.45
7 K.-S. Wu / Deerminisic Invenory Model for Iems wih ime Varying Demand 7. CONCLUSIONS In his paper we consider ha he invenory model is depleed no only by ime-varying demand bu also by Weibull disribuion deerioraion, in which he invenory model sars wih shorages and ends wihou shorages. herefore, he proposed model can be used in invenory conrol of cerain deerioraing iems such as food iems, elecronic componens, and fashionable commodiies, and ohers. Moreover, he advanage of he proposed invenory model is illusraed wih four examples. On he oher hand, as is shown by able 4, he opimal order quaniy ( Q ) and he minimum average oal cos per uni ime ( C ) are highly sensiive o changes in he value of. Acknowledgmens: he auhor would like o hank anonymous referees for helpful commens and suggesions. REFERENCES [] Aggarwal, S.P., "A noe on an order-level model for a sysem wih consan rae of deerioraion", Opsearch, 5 (978) 84-87. [2] Beyer, D., and Sehi, S., "A proof of he EOQ formula using quasi-variaion inequaliies", Inernaional Journal of Sysems Science, 29 (998) 295-299. [3] Chung, K.J., and ing, P.S., "A heurisic for replenishmen of deerioraing iems wih a linear rend in demand", Journal of he Operaional Research Sociey, 44 (993) 235-24. [4] Cover, R.P., and Philip, G.C., "An EOQ model for iems wih Weibull disribuion deerioraion", AIIE ransacion, 5 (973) 323-326. [5] Dave, U., and Pael, L.K., "Order level invenory sysem wih power demand paern for iems wih variable rae of deerioraion", Indian Journal of Pure and Applied Mahemaics, 9 (988) 43-53. [6] Dave, U., and Pael, L.K., "A noe on a replenishmen policy for an invenory model wih linear rend in demand and shorage", Journal of he Operaional Research Sociey, 43 (992) 993-. [7] Deb, M., and Chaudhuri, K.S., "An EOQ model for iems wih finie rae of producion and variable rae of deerioraion", Journal of he Operaional Research Sociey, 43 (992) 75-8. [8] Donaldson, W.A., "Invenory replenishmen policy for a linear rend in demand: an analyic soluion", Operaional Research Quarerly, 28 (977) 663-67. [9] Ghare, P.M., and Schrader, G.P., "A model for exponenially decaying invenory", Journal of Indusrial Engineering, 4 (963) 238-243. [] Giri, B.C., Goswami, A., and Chaudhuri, K.S., "An EOQ model for deerioraing iems wih ime varying demand and coss", Journal of he Operaional Research Sociey, 47 (996) 398-45. [] Giri, B.C., and Chaudhuri, K.S., "Deerminisic models of perishable invenory wih sockdependen demand rae and nonlinear holding cos", European Journal of Operaional Research, 5 (998) 467-474. [2] Goel, V.P., and Aggarwal, S.P., "Order level invenory sysem wih power demand paern for deerioraing iems", Proceedings all India Seminar on Operaional Research and Decision Making, Universiy of Delhi, Delhi, 98.
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