Production Policies of Perishable Product and Raw Materials

Transcription

1 Poduction Policies of Peishable Poduct and Raw Mateials Yu-Hsuan Lin, Jei-Zheng Wu, and Jinshyang Roan Depatment of Business Administation, Soochow Univesity 56, Section, Kuei-yang Steet, aipei 0048, aiwan, R.O.C. Abstact - Peishability is an impotant featue of inventoy management. his study developed a two-echelon poduction model to exploe the optimal poduction policy consideing both peishable finish poduct and peishable aw mateials. A numeical seach method was developed to find the optimal poduction cycle time, poduction un size, and aw mateial supply ate. Numeical studies and sensitivity esults showed that when the peish ate of finish poduct inceases, the manufactue tends to shoten poduction cycle time and decease poduction un size to avoid the peishable of poducts. On the othe hand, when the peish ate of aw mateial inceases, the manufactues tend to incease cycle poduction time and poduction un size since manufactues need moe time to pepae aw mateials fo poduction. Because of the complicate natue of this type of two-echelon model, appoximation methods, such as 3d aylo seies and 4th aylo seies, wee commonly used fo simplification. A model compaison showed that when the demand ate is closed to poduction ate and the poduct peishable ate inceases, the appoximation eo is significant by using 4th aylo seies. Keywods: Supply chain management, echelon inventoy, peishability, constant supply, disposal cost.. INRODUCION Many inventoy models have been developed in ecent yeas. Most existing models wee based on economic ode quantity (EOQ) models. Goyal (977) modeled EOQ by consideing total vaiable cost minimization with constant demand ate fo a single poduct system. Some studies futhe incopoated with inflation on ode quantity decisions (Meha et al. 99) vaying demand (Bill and Chaouch 995). An extension of the EOQ model is the economic poduction quantity (EPQ), also known as economic manufactuing quantity (EMQ), model which assumes odes be eceived continuously duing the poduction pocess. he common featue of EOQ and EPQ models is the tadeoff between holding cost and fixed odeing/poducing cost. Most inventoy models assumed that stock items can be stoed indefinitely to meet futue demands. hat is, conventional inventoy models assume that stock items can be stoed without expiation dates. Little eseach has been done egading peishable poducts. Paticulaly, among few studies discussing with peishability, finished poducts ae mainly focused on. Howeve, peishable poducts o aw mateial could be found anywhee in ou life. Fo example, gape and aisin is the kind of easily peishable aw mateial and difficultly peishable poduct. Flou and cake is the kind of difficult peishable aw mateial and easily peishable poduct. heefoe, this study aims to develop a two-echelon poduction model to exploe the optimal poduction policy consideing peishable of poduct and aw mateials. In addition, a numeical seach method is developed to help decision-makes to find the optimal poduction cycle time, poduction un size, and aw mateial supply ate. Futhemoe, we analyze diffeent models with o without consideing finished poduct peishability and aw mateial peishability to illustate and validate the poposed model. 2. LIERAURE REVIEW Main classification featues of convention inventoy model included single/multiple items, static/vaying demand, single/multiple peiods, puchase/poduction models, with/without backodeing, single/multiple buyes, constant/changing deteioation ates (Raafat, 99), fixed/andom lifetime types (Nahmias, 982). Paticulaly, the fixed lifetime peishable items ae those could be stoed in specified fixed time and afte the time they must be discaded. On the othe hand, andom life time peishable items ae those will be discaded an uncetain expiation time. Exponential decay is the common assumption of peishability. Chu and Chen (2002) investigated inventoy eplenishment policies fo deteioating items with fixed patial backodeing in a declining maket. eng et al. (2002) develop an inventoy model fo deteioating items with time-vaying demand in which unsatisfied demand is patially backodeed. hey impose an additional condition Coesponding Autho

2 on this function to guaantee the existence of an optimal solution. Dye et al. (2006) futhe developed a seach method to find the optimal eplenishment schedule. Goyal and Gii (2003) studied the inventoy model consideing that demand, poduction, and deteioation ate vaied ove time when backlog was allowed. hey used bivaiate seach methods to solve the numeical equations. Feguson and Koenigsbeg (2007) consideed two-peiod inventoy model to find the optimal poduction and picing decision. Rong et al. (2008) and Dye et al. (2007) studied two - waehouse inventoy models fo deteioating items with consideation of backlog. Peisability and deteioation have simila meanings to expess that poducts ae gadually out of function. Paticulaly, peishability is usually used fo food o phamaceuticals to emphasize the impotance of feshness. In this pape, we use peishability to expess the phenomenon when poduct becomes out of egulaly function unless specially mentioned. he taditional EPQ model used the bounday condition between poduction time and est time stages to deive the time function. When poduct may peish, the elation between two stages can hadly be deived by a simple elation due to complex exponential tems on time vaiables and cost functions. Luo (998) used the bounday condition of poduction and est stages to veify the time vaiable. Additionally, the exponential paametes and time vaiable can be expessed by using aylo seies that ignoed tems highe than 3d ode by assuming the multiplication between deteioation ate and time vey small (Misa, 975). Yang and Wee (2005) extended Misa (975) to veify the time vaiable and the exponential paametes in multiple-buye model. On the othe hand, Huang and Yao (2006) used the aylo seies expession tems highe than 4th ode to veify the time vaiable and exponential paametes in multiple-buyes poblem. When peishability happened on aw mateial, the quantity of aw mateial puchased and aw mateial stock level will change since peishable aw mateial will diectly incease the poduction cost and the holding cost. Pak (983) developed a poduction inventoy model fo decayed aw mateial fo a single poduct system. Roan (200) detemined pocess mean and poduction policies with constant supply of exponentially peishable aw mateial fo a containe-filling pocess. He found that the optimal pocess mean and supply ate of aw mateial ae less sensitive to the peishability. Nevetheless, it is lack of compehensive discussions on poduction systems with poduct and aw mateial peishability compaed with those without peishability. 3. MODELS AND SEARCH MEHOD - poduction time and decision vaiable - total poduction time and decision vaiable - the poduction ate pe unit time D - the demand of poducts ate pe unit time S - the setup cost pe poduction cycle α - the added value of aw mateials to poduct pe unit inventoy β - the supply ate of aw mateials pe unit time and decision vaiable ε - the peishable ate of aw mateials pe unit time λ - the peishable ate of inventoy pe unit time c - the cost of aw mateials pe unit aw mateial C - the decay cost of poduct pe unit inventoy C 2 - the decay cost of aw mateials pe unit aw mateial a - the aw mateials equied fo poducing a poduct h - the holding cost each unit of the aw mateial fo unit time h - the holding cost of aw mateials pe unit time, h = h c H - the holding cost pe unit inventoy, H = h αca = hαa PSC - aveage setup cost of poduct PHC - aveage holding cost of poduct PDC - aveage decay cost of poduct RPC - aveage poduction cost of aw mateial RHC - aveage holding cost of aw mateial RDC - aveage disposal cost of aw mateial C n - aveage total cost of Poblem(n), n =, 2,, 7 Assumptions ) otal cycle time, poduction time, est time and aw mateial supply ate ae decision vaiables. All these vaiables can be inteelated and thus some of them become afte efomulation. 2) Poduction and demand ates ae constant and given. 3) Finished poducts fom poduction ae immediately available. 4) Poducts o aw mateials stat to peish once they ae stocked as inventoy. 5) Peishable poduct and aw mateial can be esold o need to be disposed of. 6) Peishable ate of poducts and aw mateial ae positive and small constants. 7) Raw mateial supply ate is constant and needs to be detemined. his study developed a famewok to examine the elationship among two-echelon poduction system with o without peishable poduct and aw mateials (Figue ). As When λ 0, optimal solutions of Poblem will appoach those of Poblem 2 which epesents the system with peishable aw mateials. On the othe hand, when ε 0, optimal solutions of Poblem will appoach those of

3 Poblem 3 which epesents the system with peishable aw mateials. In addition, when λ 0 and ε 0, all solutions of Poblem to 3 will appoach those of Poblem 4, the system without consideing peishability. Followings ae models of Poblem to 4. Paticulaly, the optimal solution of Poblem 4 is given by Eq. (2). We will illustate the elation among these poblems in the chapte of numeical study. Poblem min () C PSC PHC PDC RPC RHC RDC whee () Figue : he elation of two-echelon poduction models.

4 S PSC (2) PHC h a() D (3) ca C (4) RPC c (5) PDC () D RHC RDC h c() a (6) C () a 2 (7) De D () ln() (8) () Poblem 2 a[() ] e De D min () C PSC PHC whee RPC RHC RDC (9) (0) H PHC2 [()() D] D () 2 () D (2) () Poblem3 a() e e 3 D / min () C PSC PHC PDC whee RPC RHC 2 (3) (4) h RHC2 [()() a ] (5) 2 D De () ln() (6) a () (7) Poblem4 min () C PSC PHC RPC RHC (8) whee 4 () D/ (9) () Da (20) * 4 Da (2) D H DH acd h 2cDh ch Since it is had to deive a closed-fom solution of Poblem, a Golden-section seach method is developed to find the optimal and total cost: Step. Find the optimal solution, 4 *, based on Eq. (2). Step 2. Exhaustively seach possible > 4 * fo the optimal solution of Poblem 2, 2 *, by adding. 2 * is the uppe bounday fo finding *. Step 3. Exhaustively seach possible < 4 * fo the optimal solution of Poblem 3, 3 *, by adding. 3 * is the lowe bounday fo finding *. Step 4. Apply the Golden-section seach within [ 3 *, 2 *] to find *.. Fo given, calculate β. 2. Fo all the possible pais of (, β), computes total cost C (, β) 3. If the total cost found in 2 less than the lowest total cost found peviously, eplace it. 4. If temination condition meets, the and β coesponding to lowest total cost ae optimal solution. Othewise, go to. 4. APPROXIMAION MODELS 2 3 ()() () e f (22) () e () g (23) 2 () 2D 2D () D (23)

5 Poblem 5 ( min ()() C5 C ) () ln(() /) Df / (24) () a[ (() Df /) ] () / 2() / 6 / 2 3 Poblem 6 ( min ()() C6 C ) (25) () ln(() /) Dg / (26) () a[ (() Dg /) ] () / 2 / Poblem 7 ( min ()() C7 C ) () 2D 2D () D Conventional appoximate models utilized aylo seies expansions of exponential function ignoing tems highe than 4 th ode and 3 d ode ae as shown in Eq. (22) and (23) when 0 < λ <, espectively. he appoximation model utilizing Eq. (22) can be deived as Poblem 5 wheeas one with (23) can be expessed as Poblem 6. On the othe hand, Misa-appoximate fom of as in Eq. (28) was developed by Misa (975). Accodingly, an appoximate model is constucted as in Poblem 7 which contains the same β() with Poblem 6. Appoximation eos will be examined in this study. (27) (28) in the appoximate eos analysis, existing appoximation methods ae effective only when λ and ε ae vey small and demand ate is not closed to poduction ate. hat is, a seach method to find optimal solution in geneal case is necessay. able : Relations among Poblem to Poblem 4 Situation C * * Poblem λ = 0.0, ε = , λ = 0.0, ε 0 20, λ 0, ε = , λ 0, ε 0 20, Poblem 2 λ = 0, ε = , λ = 0, ε 0 20, Poblem 3 λ = 0.0, ε = 0 20, Poblem 4 λ 0, ε = 0 20, λ = 0, ε = 0 20, NUMERICAL SUDY his study adopted the paamete settings as those in Roan (200) to conduct numeical study. Paticulaly, (, D, α, S, λ, ε, C, C 2, h, c, a) = ( 7500, 5000,.2, 50, 0.0, 0.0,.5, 0, 0.03,, 4). Optimal solutions of Poblem includes, * = 0.67, q = 3,354 and β = 20,045. able shows elations among Poblem to Poblem 4 validating that models with peishability will appoach models without peishability when peishable ates ae small. * * * e n [()()]/(), C n C5,6,7 C n (29) We futhe examined appoximate eos when peishable ates ae applied. Numeical esults show that the appoximate eos of Poblem 5 to Poblem 6 ae negligible. hat is, the values of Eq. (29) ae close to zeo when n = 5, 6. On the othe hand, the eo of Poblem 7 inceases as λ inceases (Figue 2). Nevetheless, as shown Figue 2: Appoximate eo tend Poblem 7 6. CONCLUSION In this study, we develop a two-echelon poduction model fo peished poduct and peished aw mateial with constant aw mateial supply. We use the bounday condition to simplify the cost function. Fou situations (with peishable poduct and peishable aw mateial, only with peishable poduct, only with peishable aw mateial, with impeishable poduct and impeishable aw mateial) and the elations among them ae showed in this study. Golden-section seach was applied to find the optimal solution of ou poposed model. We use the elation among the thee exteme models to find the bounday fo this

6 seaching method. We find that if only the aw mateial peishes, the total cycle time inceases if the aw mateial peishable ate inceases. Because the poduce needs moe time to accumulate the aw mateials fo poduction. In addition, if only the poduct peishes, the total cycle time deceases when the poduct peishable ate deceases. Because the poduce educes the cycle time to avoid the poduct peished. he findings povide the boundaies of ou seaching of optimal cycle time in ou model. We compae conventional appoximation methods including 4th aylo seies and 3d aylo seies. he mathematical appoximation makes the cost function easie to ead and analyze. Howeve, the numeical example has showed the existence of appoximate eos. heefoe, we should be awae using appoximation in a complex system may lead to unexpected esults. hee ae still many possible eseach extensions, which include ) he backodeing of poduct can be consideed; 2) Othe aw mateials models, such as multiple-odes aw mateials model, can be applied; 3) Raw mateial ode quantity discount can be consideed 4) he influence of advetisement to demand can be studied. ACKNOWLEDGMEN his eseach is suppoted by National Science Council (NSC H ). REFERENCES Bill, P. H. and Chaouch, B. A. (995) An EOQ model with andom vaiations in demand, Management Science, 4(5), Chu, P. and Chen, P. S. (2002) A note on inventoy eplenishment policies fo deteioating items in an exponentially declining maket, Computes and Opeations Reseach, 29(3), Dye, C.-Y., Chang, H.-J., and eng, J.-. (2006) A deteioating inventoy model with time-vaying demand and shotage-dependent patial backlogging, Euopean Jounal of Opeational Reseach, 72, Dye, C.-Y., Ouyang, L.-Y., and Hsieh,.-P. (2007) Deteministic inventoy model fo deteioating items with capacity constaint and time-popotional backlogging ate, Euopean Jounal of Opeational Reseach, 78, Feguson, M. E. and Koenigsbeg, O. (2007) How should a fim manage deteioating inventoy, Poduction and Opeations Management, 6(3), Goyal, S. K. (977) An integated inventoy model fo a single poduct system, Opeational Reseach Quately, 28(3), Goyal, S. K. and Gii, B. C. (2003) he poduction - inventoy poblem of a poduct with time vaying demand, poduction and deteioation ates, Euopean Jounal of Opeational Reseach, 47(3), Huang, J.-Y. and Yao, M.-J. (2006) A new algoithm fo optimally detemining lot-sizing policies fo a deteioating item in integated poduction-inventoy system, Computes & Mathematics with Applications, 5(), Luo, W. (998) An integated inventoy system fo peishable goods with backodeing, Computes and Industial Engineeing, 34(3), Misa, R. B. (975) Optimum poduction lot size fo a system with deteioating inventoy, Intenational Jounal of Poduction Reseach, 3(5), Meha, S., Agawal, P., and Rajagopalan, M. (99) Some comments on the validity of EOQ fomula unde inflationay conditions, Decision Sciences, 22(), Nahmias, S. (982) Peishable inventoy theoy: A eview, Opeations Reseach, 30(4), Pak, K. S. (983) Integated poduction inventoy model fo decaying aw mateials, Intenational Jounal of Systems Science, 4(7), Raafat, K. (99) Suvey of liteatue on continuously deteioating inventoy models, he Jounal of the Opeational Reseach Society, 42(), Roan, J. (200) Pocess mean and poduction policies detemination unde constant supply of peishable aw mateial, Fu Jen Management Review, 8(), Rong, M., Mahapata, N. K. and Maiti, M. (2008) A two waehouse inventoy model fo a deteioating item with patially/fully backlogged shotage and fuzzy lead time, Euopean Jounal of Opeational Reseach, 89, eng, J.-., Chang, H.-J., Dye, C.-Y., and Hung, C.-H. (2002) An optimal eplenishment policy fo deteioating items with time-vaying demand and patial backlogging, Opeations Reseach Lettes, 30(6), Yang, P.-C. and Wee, H.-M. (2005) A single-vendo and multiple-buyes poduction-inventoy policy fo a deteioating item, Euopean Jounal of Opeational Reseach, 43(3),