Multiple Can-order Level for Can-order Policies under Carrier Capacity and Correlated Demands

Transcription

1 J Jpn Ind Manage Assoc 67, , 2016 Orgnal Paper Mulple Can-order Level for Can-order Polces under Carrer Capacy and Correlaed Demands Kesuke NAGAAWA 1 and Takash IROHARA 2 Absrac: In hs paper, we consder a form of mul-em nvenory managemen known as he Jon Replenshmen Problem (JRP). In our model, he arge warehouse sells mulple ems o mee he realer s demand and replenshes hese ems from he suppler. A each replenshmen, he warehouse decdes whch ems should be ordered and how much should be ordered. Carrers wh a fne capacy ranspor ems from he suppler o he warehouse wh a fxed lead-me. The fxed orderng cos of each carrer s charged accordng o he number of carrers used, no he order volume. Because of he sepwse orderng cos, fullload replenshmens reduce he cos, even f an unscheduled em s ordered. In addon, f a relaon exss among he ems demanded, he can-order polcy s more suable. Under he can-order polcy for JRP, some ems are re-ordered when he nvenory volume s below he re-order level and any ems wh an nvenory volume below he can-order level can be ncluded n he order. everal sudes have consdered how o se he parameers of he can-order polcy. Bu, correlaed demands have no receved suffcen aenon n he leraure because s dffcul o fnd he opmal parameers. In hs paper, we propose o fnd he opmal parameers of he can-order polcy, he can-order level and he order-up-o level for each correlaed demand em by applyng a genec algorhm. The man obecves n our model are mnmzng em sorage, sockous and carrer fees. In numercal expermens, we smulaed nvenory movemen and orderng ems whle consderng he capacy of he carrers. The expermenal resuls of our proposed model are compared wh he resuls of oher models. Key words: nvenory, on replenshmen problem, can-order polcy, carrer capacy, correlaed demands 1 INTRODUCTION Effecve nvenory managemen has played an mporan role n he success of supply chan managemen. I s mporan n nvenory managemen o decde when and how many ems should be ordered. Usually, effecve nvenory managemen reduces he amoun of nvenory and shorages hrough he use of proper orderng polces for each em. For mul-em nvenory conrol, he on replenshmen problem (JRP) coordnaes he orderng quany of several ems n a warehouse. For example, le us consder a warehouse wh ems suppled by only one suppler. When he nvenory manager orders many ems from he same suppler, he orderng cos charged depends on he number of orders and he number of dfferen ems ordered. Therefore, he orderng quanes of dfferen ems canno be decded ndependenly. The ypcal assumpons of he 1 Hroshma Unversy 2 opha Unversy Receved: Augus 31, 2015 Acceped: May 27, 2016 JRP are smlar o hose n he convenonal economc orderng quany model. In relaed work, Goyal [1] developed an algorhm o fnd he opmal soluon under he classcal JRP. Goyal and ar [2] and Khoua and Goyal [3] nvesgaed a varey of JRP algorhms. lver [4] developed an effcen heursc algorhm for classcal assumpons, and Kasp and Rosenbla [5] mproved he algorhm. Wang e al. [6] proposed a new dfferenal evoluon algorhm for on replenshmen. Wang e al. [7] proposed an effcen hybrd dfferenal evoluon algorhm for on replenshmen wh delvery. A can-order polcy s one ype of nvenory managemen polcy appled o he JRP. Can-order polces were orgnally suggesed by Balnfy [8], and hen van Es [9] showed ha can-order polces are a leas as good as oher coordnaed replenshmen polces. A can-order polcy s represened by a se of hree parameers, (s, c, ), or smply (s, c, ). When he nvenory level of em drops o or below s, called he re-order level, an order s placed o brng he nvenory level up o, called he order-up-o level. For 114 J Jpn Ind Manage Assoc

2 any oher ems,, whose nvenory levels are less han her respecve can-order levels, c, orders wll also be placed so ha nvenory levels of ems are rased o. Federgruen e al. [10] modeled a can-order polcy as a sem-markov decson problem wh compound Posson demands and posve lead-mes. They showed ha a can-order polcy s consderably beer han uncoordnaed polces and ofen provde as much as a 20% savngs. Mos research on can-order polces focuses on how o derve he opmal (s, c, ) values for each em based on an algorhm (Akns and Iyogun [11], Panumsncha [12], Dellaer and Poel [13], Melchors [14], and Johansen and Melchors [15]). A good summary of oher applcaons of can-order polces can be found n Kayş e al. [16]. Ths lne of research assumes ha he probables beween any wo ems are already known or can be derved manually (see, for example, Lu and Yuan [17]). Tsa e al. [18] proposed an assocaon cluserng algorhm o evaluae he correlaon of em demands. They ulzed he suppor concep o measure he demand smlary among ems. Correlaed demand s he endency of demand ha ncludes wo or more ems a once by hgh probables. Iems whch nclude exensve correlaed demand are called correlaed ems. For correlaed ems, he can-order polcy s suable because consders orderng mulple ems a once and correlaed em ordered ndrecly. However, mos of he above JRP research dd no consder he consrans of he orderng quany. Thus, more research of he mul-em nvenory conrol problem has been conduced. The full ruckload problem consders he delvery of producs by rucks havng a fne capacy. When fxed ransporaon coss are charged regardless of he volume ordered, effecve coordnaon of he orders and full ruckload shpmens can be a benef. Kesmüller [19] proposed a dynamc order-up-o polcy resulng n full ruckloads. Kesmüller [20] provded approxmaon formulas for compung he parameers of he replenshmen polcy, whch akes no accoun he capacy resrcons of he oal order volume so ha gven arge servce levels can be me. Nagasawa e al. [21] proposed he addng and subbng orderng quany rule solved by a genec algorhm (GA) for reducng he number of carrers used, he oal nvenory and he los-sales quany. Recenly, Amaya e al. [22] proposed a new heursc algorhm for solvng he JRP under deermnsc demand and resource consrans. Mos of hese sudes consder he capacy of he orderng quany, no he correlaon of em demand. If he capacy of an orderng quany has consrans on he number of carrers used and he varey of ems ordered, addonal orderng of a small varey of hgh-volume ems correlaed o he ordered rgger em, could be more benefcal han orderng of a low volume or many varees of ems unrelaed o he rgger em. In hs case, applyng he can-order polcy consderng he correlaed ems and capacy consrans could be benefcal because he concep s he same as he can-order polcy. However, o consder em correlaon of he can-order polcy, we should nclude mulple levels of can-order levels wh all combnaons of every em because he convenonal can-order polcy uses a sngle can-order level for every sngle em ndvdually. For hs exenson, we nroduce he can-order level and order-up-o level, whch are relaed o he order rgger em. We opmze he combnaon of can-order levels and orderup-o levels of each em. In our research, he re-order, can-order and order-up-o levels of many ems mus be se. o, we opmze each level of each em usng a GA. We compare he number of carrers used, he oal nvenory, and he oal shorage. We defne hs problem as a mul-obecve opmzaon problem. The remander of hs paper s organzed as follows. The mahemacal formulaon for he deermnsc demand model s presened n econ 2. In econ 3, we propose a GA for obanng he re-order level, he can-order level and he order-up-o level for he sochasc demand problem for each em. The compuaonal resuls are descrbed n econ 4. Fnally, conclusons are presened n econ 5. 2 PROBLEM FORMULATION We consder a perodc revew nvenory polcy. Under perodc revew he nvenory polcy nvenory manager revews nvenory, and orders ems o suppler perodcally ((e.g.) day, week, monh or perod), whle connuous revewng polces and orderng ems whenever need. everal correlaed and ndependen demand ems are n he nvenory: he demand of boh em and, he demand of em whou em, and he demand of only em whou em. The probables of hese demands are p, p and p respecvely, where p +p + p 1. Thus, for each day, here s demand of em and, here s demand of em whou em, here s demand of em whou em, or demand does no occur. Vol.67 No.2E (2016) 115

3 Convenonal can-order polcy Invenory 1 c 1 s 1 Tme Invenory 2 c 2 s Tme Invenory 3 c 3 s Tme Invenory c 21 c 31 s 1 Tme Invenory c 12 c 32 s 2 Invenory c 23 c 13 s 3 Proposed can-order polcy Tme Tme Fg. 1 Convenonal and proposed can-order polcy. Generally, he convenonal can-order polcy s represened by a se of hree parameers, (s, c ). The nvenory level of an em s l. When hs nvenory level l drops o or below s, an order s placed and orderng quany, x, brngs he nvenory level up o, where x = l regardless of he demand correlaon. We need o adop an orderng polcy ha consders he correlaon of ems, f he demands of each em are correlaed. Therefore, n hs research, we change he canorder level and he order-up-o level dependng on he orderng rgger em. The dfference n convenonal and proposed can-order polces s shown n Fg. 1. In hs model, for each em,, he orderng polcy s ( s, c, ). Whenever he nvenory level of em drops o s re-order level s, rggers a replenshmen order ha rases em s nvenory levels o s order-up-o level. A he same me, f he nvenory level of anoher em k ( k ) s a or below s can-order level when em s a s reorder level, hen he can-order level s c k, and em k s ncluded n he addonal order o replensh o he order-upo level for em, k. In he assumpon n hs paper, f some em,, s rggered by mulple ems, and k, (l < c and l < c k ), we prorze and order a he order-up-o level from he can-order level, (f l < c < c k, hen x = k - l ). Frs, we model he problem as he well-known nvenory model of he sngle warehouse mul-realer problem, as repored by Chan e al. [23]. Yang e al. [24] appled a genec algorhm for a smlar problem. In our problem, he nvenory manager knows he exernal fuure deermnsc demand of ems over a fne plannng horzon. The goal of hs problem s o fnd he opmal replenshmen mng and quany. The mahemacal model of he mul-em nvenory problem, where he replenshmen of ems s carred ou by carrers havng a fne capacy, s a smlar problem o ha of Nagasawa e al. [25]. The assumpons of our research and our formulaon of he problem are as follows. We consder a mul-em nvenory managemen problem n whch a warehouse sells mulple ems wh no quany dscouns. The replenshmen of ems s accomplshed by carrers havng a fne capacy. Because carrer fee penales are charged based on he number of carrers used, smaller carrer 116 J Jpn Ind Manage Assoc

4 replenshmens can be a benef. Iems can be kep n sorage. Non-pershables or ems ha do no deerorae can be kep n sorage, alhough hey are subec o a holdng penaly. When demands canno be sasfed, ems on hand are shpped and he shorage demand nduces a los-sales penaly. In hs problem, a carrer fee penaly, an nvenory penaly, and a shorage penaly are he obecve funcons. Our can-order polcy s formulaed as follows: mnmze f y (1) mnmze mnmze subec o l x l f f T o d, I T I T o l I, 1 T I 1 l l l l 1 c v x by 1 1 q q c I x d d l l c s k o 1 1 d d k r q l l d r, k o o d d o o 1 M( r, q I c Mr s M(1 r o o Mk x x M(1 k q ) s M(1 r M(1 r c ) c ) ) T, I, T, I, T, I, T, I, T,, I,, I,, I, T,, I, T,, I, T,, I, T, I, T,, I, T, 1 d o x M(1 q ), I, T, 1 d o x M(1 q ), I, T, I q {0,1}, y, l, x, o, s, c, Z Decson varables y : number of carrers used on he s c o : he : he reorder level of em : he order up o ) I, T, I, T,, I, T, I, T day can order level of em when em s ordered : he los sales h level of em when em s ordered quany of em on he h l: nvenory level of em on he day x : he orderng quany of em on he 1, f l drop below s r : 0, oherwse on he h day h h day, day (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) h 1, f l drop below c on he day, k : 0, oherwse 1, l drop below c, h a leas one em s ordered a day( max( r ) 1) I q : and addonal order wll be placed o, 0, oherwse Parameers h d :demand of em on he day v : volume of em b :capacy of carrer M : bg M I : se of ems T :se of days The frs obecve funcon, Eq. (1), sums he number of carrers used n he plannng perod. The second obecve funcon, Eq. (2), sums he number of shorages, whch s he amoun of los sales, whn he plannng perod. The hrd obecve funcon, Eq. (3), sums he oal nvenory whn he plannng perod. Equaon (4) s he nvenory balancng equaon for all ems. Equaon (5) s a consran on orderng ems whn each plannng perod. By Eqs. (6) and (7), f he nvenory level drops o or below he re-order level, an order wll be placed. Furhermore, by Eqs. (8) and (9), when he nvenory level drops o or below he re-order level, he order quany s se as he dfference beween he order-up-o level and he nvenory level. Equaon (10) denoes ha each can-order level s no more han he orderup-o level. Equaon (11) denoes ha each re-order level s no more han he can-order level. Equaons (12) and (13) denfy wheher he nvenory level of an em has dropped o or below s can-order level. By Eqs. (14-17), when he nvenory level of a leas one em has dropped o or below s re-order level, an order s placed ha ncludes ems for whch he nvenory level has dropped o or below he canorder level. The can-order level s se o he maxmum canorder level of he rgger em. By Eqs. (18) and (19), he ems for whch he nvenory level has dropped o or below he can-order level, whch s relaed o he order rgger em, s ordered n he quany equal o he dfference beween he order-up-o level, whch s relaed o he order rgger em, and he nvenory level. Equaon (20) denoes ha ems for whch he nvenory level has dropped o or below he can-order or he re-order level can be ordered. Equaon (21) denoes ha he shorage amoun of each em s no more han he demand of he correspondng perod. Because of he mul-obecve funcons and dfferen uns, he opmzaon problem nvolves hree obecve Vol.67 No.2E (2016) 117

5 funcons. The mahemacal opmzaon problem model s known as NP-hard. Therefore, we propose a real-value genec algorhm for decdng he re-order level, he canorder level and he order-up-o level of each em. 3 IMPLEMENTATION OF GENETIC ALGORITHM In hs sudy, because he mahemacal formulaon n he prevous secon s a mul-obecve opmzaon model, we canno solve he problem usng a mahemacal opmzaon solver. Alhough varous combnaons of Pareo opmal soluons are possble, he opmal soluon of he mahemacal formulaon, whch s known as he NPhard problem, akes a long me o calculae, even f we use a sngle obecve funcon. Thus, for applyng a can-order polcy o an em se, we use a heursc mehod o oban a beer soluon nsead of decdng he orderng quany of each em separaely. The obecve funcons are he number of carrers used, he amoun of he shorage, or he amoun of he nvenory. Genec operaon ar Inalzaon Calculae fness funcon elecon Crossover Muaon Termnaon? End Yes Fg. 2 Oulne of a genec algorhm. The GA s an adapve heursc search algorhm based on he evoluonary concep of naural selecon. The processes represen an nellgen exploaon of a random search whn a defned search space for he purpose of resolvng a problem. GAs are ypcally used o solve combnaoral problems ha canno be handled by exhausve, mul-obecve, or exac mehods due o her prohbve complexy. The basc procedure of he GA approach s o code he decson varables of he problem as a fne-lengh array No (referred o as a chromosome) and calculae he obecve value (fness) of each chromosome. Based on he fness, he probably of survval for each chromosome s calculaed. The survvng chromosomes hen reproduce and form he chromosomes of he nex generaon hrough crossover and muaon processes. Fgure 2 shows he oulne of a GA. 3.1 Chromosome represenaon A chromosome s a se of genes relaed o he soluon of he problem. A crcal ssue when applyng GAs o an opmzaon problem s he selecon of a suable encodng scheme for ransformng feasble soluons no genec represenaons. In he presen paper, we decde he re-order level, s, he order-up-o level, whch s relaed o he order rgger em,, and he can-order level, whch s relaed o he order rgger em, c. Table 1 Example chromosome of he genec represenaon scheme. Iem : 1 1 n n Index of gene 1 m n 2 m Gene Order-up-o level: Iem : 1 2 n (n-1) Index of gene 1 m n(n-1)m Gene Can-order level : c Iem : 1 n Index of gene 1 m nm Gene Reorder level : s 7 5 As shown n Table 1, f we defne he number of ems as n and he number of members of he se of bnary srng genes for each em as m, he can-order level of all ems can be represened by a 2n 2 m-lengh chromosome. Each level of each em s deermned from he correspondng m- lengh srng of he bnary genes. The value of each gene s 0 or 1, and each gene s referred o as a b. For example, f m = 8 and he gene bnary srngs of em are , he decmal number n he Gray code number s J Jpn Ind Manage Assoc

6 In some GAs, encodng and decodng are carred ou usng bnary srngs. Tradonal bnary codng for funcon opmzaon s consdered weak due o he large change n he real parameer values when a sngle b n he bnary srng of he parameer s changed. To avod hs weakness, we apply he Gray code [26]. The Gray code has he propery ha an ncrease by one sep n he value of a desgn varable corresponds o he change of a sngle b n he bnary srng of he desgn varable. Andre e al. [27] compared bnary codng and Gray codng for GAs usng a wo-pon crossover. They concluded ha Gray codng helped o mprove he speed of bnary codng. Therefore, we decode he 2n 2 m-lengh gene secon of each em o a re-order level, can-order level, and order-up-o level as Gray code. 3.2 Inal populaon Durng he nalzaon process, a predefned number of chromosomes s randomly generaed o represen he reorder levels, can-order, levels and order-up-o levels for all ems. In hs nalzaon phase, we also creae a specal chromosome n whch he can-order levels of each em are se o zero, and he chromosomes represen usng only he order-up-o level and he re-order level, bu no he canorder level. 3.3 Fness funcon A fness funcon s a parcular ype of obecve funcon ha prescrbes he opmaly of a chromosome. Opmal chromosomes, or a leas chromosomes wh nearopmal values, are allowed o breed and mx her daases by followng one of he avalable echnques n order o produce a new generaon wh mproved characerscs. In hs sudy, he fness funcon s defned n erms of he orgnal obecve funcons. The fness funcon of he chromosome, f (chromosome), s represened as follows: 3 mn f e chromosome fe f chromosome 1, (22) max mn e1 fe fe where f e(chromosome), e = 1, 2, 3, denoes he obecve values defned by funcons, Eqs. (1-3) n econ 2, respecvely. These values are he resuls of he smulaon when he chromosome s appled wh he orderng polcy. Here, f e max s he maxmum value n he curren generaon of obecve funcon f e calculaed by he smulaon resuls. mlarly, f e mn s he mnmum value. We calculae he fness values a each generaon and use hese values o mprove he pool of poenal soluons n he selecon sep. 3.4 Genec operaons Once nalzaon and fness calculaons have been performed, he genec operaons, whch nclude selecon, crossover, and muaon, are carred ou. Along wh new fness calculaons, hese operaons are repeaed unl he ermnaon condons are sasfed. A he end of each round of genec operaons, a new generaon of chromosomes s obaned from he processes descrbed below for he nex eraon, and s assumed hs process evenually yelds an opmal ndvdual elecon The selecon operaor selecs chromosomes n he populaon for producng he nex generaon. Well-fng chromosomes are lkely o be seleced for he nex generaon. In hs sudy, we use he roulee wheel selecon mehod, n whch he probably of beng seleced s drecly proporonal o he fness of he chromosome Crossover The crossover operaor roughly mmcs bologcal recombnaon beween wo sngle-chromosome organsms. In he presen sudy, we use a wo-pon crossover mehod. The crossover operaon s llusraed n Fg. 3. Paren 1 Paren 2 Chld 1 Chld 2 Crossover pon Crossover pon Iem 1 Iem 2 Iem 3 Iem Fg. 3 Basc procedure of wo-pon crossover. Frs, we selec wo paren chromosomes, ndcaed n he fgure by red dashed lnes ( ) and blue double lnes ( ). In he wo-pon crossover, he sub-sequences of genes are randomly seleced and exchanged relave o he parens o creae wo chldren. (In he fgure, he source paren of genes n a chld s ndcaed by he same color/lne desgnaons as hose of he parens.) The crossover pons where he chromosomes are broken are randomly seleced. However, wh some probably, no crossover occurs and he parens are coped Vol.67 No.2E (2016) 119

7 (unchanged) no he new populaon. The probably of a leas one crossover occurrng for a paren chromosome par, p, s usually se beween 40% and 90% Muaon The muaon operaor s used o manan genec dversy from one generaon of a populaon of chromosomes o he nex. Muaon s an algorhm for avodng local opma. The muaon operaor s appled o each chld soluon resulng from he crossover operaon and s usually defned as a change n he values of genes n a chromosome. In hs sudy, as shown n Fg. 4, he muaon operaor changes one randomly seleced gene, ndcaed by he red dashed lne ( ). Muaon can occur on any chromosome wh some small probably. We se he probably as 0.01%. Fg. 4 Basc procedure of muaon. 4 EXPERIMENT AND REULT In hs sudy, we use an (s, c, ) polcy and opmze he re-order level, s, he can-order level, whch s relaed o he order rgger em, c, and he order-up-o level, whch s relaed o he order rgger em,, of each em by he GA. We apply he GA for random demand daa. We decode each gene and calculae he fness values from he smulaon resuls. To show he effecveness of consderng he can-order level, c, and he order-up-o level,, we compared fve dfferen can-order polces, shown n Table 2. Table 2 Comparson mehod and relaon of each level. Mehod Re-order Can-order Order-up-o level level level Fx-Nc Fxed c * Fx-Cc (npu parameer) c * Nc-N c * Cc-N Proposed Fxed (npu parameer) s * c * * * * opmzed by GA The frs wo polces are he fxed re-order level and he order-up-o level of each em and he opmzed can-order polcy. The las hree polces are opmzed for each level. From he comparson of he frs wo polces, we can check he effecveness of nroducng he proposed canorder level. By he comparng of he las hree polces, we can check he effecveness of nroducng he proposed can-order level and order-up-o level. The orderng polces are descrbed as follows. Fx-Nc : In hs polcy, he re-order level and he orderup-o level of each em are npu parameers and he can-order level of each em s opmzed by he GA. Ths polcy does no change he can-order level by em combnaon. (c = c, ) Fx-Cc : Dfferen from Fx-Nc, he can-order level, whch s relaed o he order rgger em, of each em s mplemened and opmzed by he GA. Ths polcy changes he can-order level by em combnaon. Nc-N : Ths polcy opmzes he re-order level, he canorder level and he order-up-o level by he GA, bu does no change he can-order level and he order-upo level by em combnaon. (c = c = c and = =, ) Cc-N : Dfferen from Fx-Cc, he can-order level, whch s relaed o he order rgger em, of each em s mplemened and opmzed by he GA. Bu, he order-up-o level does no change by em combnaon. ( = =, ) Proposed (= Cc-C ): Ths polcy opmzes he re-order level, can-order level, and order-up-o level by he GA. 4.1 Daa se In he numercal and smulaon expermens, we use he random demand daa. For each em, average of em demand,, s generaed from unform dsrbuon, U[100, 3000]. The demand amoun of each em s generaed by Normal dsrbuon, N(, /5). Correlaed em ses are generaed by randomly dvdng ems o hree groups, and he demand of a sngle em group occurs every day. As a resul, hese 25 correlaed ems are ordered o one specfc suppler, and we apply he mehod proposed o hese 25 ems only. We smulae (s, c, ) polces from he abovemenoned shppng daa for 10,000 days and se he parameers as descrbed below. The lead-me (.e., he me from when he ems were ordered o when he ems were delvered) s se o wo days. The capacy of he carrer s changed o 10,000, 40,000, and 100, J Jpn Ind Manage Assoc

8 For comparng he effecveness of nroducng he canorder level and he order-up-o level, whch are relaed o he order rgger em, and he re-order level, we use predeermned re-order levels and pre-deermned order-up-o levels n he expermenal resuls. In hese cases, he predeermned re-order level, s, of each em s se o he mean wo-day demand plus 95% safey sock for wo days. The pre-deermned order-up-o level,, of each em s se o he mean 28-day demand. The GA parameers are se as follows. The probably of crossover occurrng s 80%. The probably for muaon s 0.01%. The maxmum number of generaons s The number of chromosomes n each generaon s 1,000. Roulee selecon s used for each generaon. In hs sudy, a hgher fness value represens a beer soluon. Each of he hree obecve values s evaluaed n he smulaon. 4.1 Expermenal resuls The smulaon resuls for he model proposed are shown n Fgs. 5 and 6 and Table 3. The fgures show he Pareo soluons afer he 3,000 h GA eraon for nvenory vs. shorage and nvenory vs. orderng, respecvely. In Fgs. 5 and 6, he resuls of Fx-Nc, Fx-Cc, Nc-N, Cc-N and Proposed are denoed by blue crcles, purple rangles, yellow recangles, green plus sgns, and red mes sgns, respecvely. 16,000,000 Fx-Nc 14,000,000 Fx-Cc 12,000,000 Nc-N 10,000,000 Cc-N 8,000,000 Proposed 6,000,000 4,000,000 2,000, ,000 1,000,000 Invenory Fg. 5 Pareo soluons of nvenory vs. shorage. horage Carrer 1,400,000 1,200,000 1,000, , , , ,000 0 Fx-Nc Fx-Cc Nc-N Cc-N 0 500,000 1,000,000 Invenory Proposed Fg. 6 Pareo soluons of nvenory vs. carrers used. In Fgs. 5 and 6, he Pareo soluons of Fx-Nc and Fx-Cc appear a he rgh-boom corner. Because we fx he re-order level and he order-up-o level of every em, he resuls vary slghly. Because some Pareo soluons of Fx-Cc domnae he Pareo soluon of Fx-Nc, nroducng he can-order level has an effec. mlarly, because some Pareo soluons of Proposed domnae he Pareo soluons of Fx-Nc, Fx-Cc, Nc-N, Cc-N, and he soluons are spread n he drecon of he orgn. Thus smulaneously nroducng he can-order level and he order-up-o level s effecve. As shown n Fg. 6, s dffcul for Proposed o domnae oher mehods. Bu, we can propose some choces for he nvenory manager o reduce nvenory nsead of ncreasng he number of carrers. If he nvenory manager wans o reduce he number of carrers, he nvenory manager can choose he level of can-order polcy by ncreasng some exen of nvenory or shorage. Table 3 shows he capacy of he carrers used, he varaon of he can-order polcy, he average fness of Pareo soluons a he 3,000 h generaon, and he maxmum and mnmum values of each obecve funcon. Here, he fness values are calculaed by Eq. (22), bu normalzed by he maxmum value, f e max, and he mnmum value, f e mn, of each obecve value n all mehods. In he column of Avg. fness, a hgher value s preferable. Compareng of he Avg. fness values, he Proposed can-order polcy obans he mos preferable resul because he values of he Proposed rows have hgher values for each carrer capacy. In addon o Proposed, he mehods Nc-N and Cc-N, whch nclude he can-order level and he order-up-o level, whch are relaed o he order rgger em, have hgh fness values. Capacy of carrer 10,000 40, ,000 Table 3 Resuls of he numercal expermens. Mehod Avg. Fness Toal carrer f 1 Toal shorage f 2 Toal nvenory f 3 max. mn. max. mn. max. mn. Fx-Nc , ,633 93,948 19, , ,409 Fx-Cc , ,605 89,883 27, , ,630 Nc-N ,304, ,755 1,743, ,214 91,895 Cc-N ,305, ,407 14,759, ,813 18,363 Proposed ,302, ,073 14,321, ,352 22,101 Fx-Nc ,755 37,258 93,982 19, , ,555 Fx-Cc ,761 54,028 86,327 32, , ,568 Nc-N ,296 38,428 13,001, ,347 11,705 Cc-N , ,597 5,142, ,998 69,154 Proposed ,017 49,598 14,142, ,346 18,920 Fx-Nc ,819 14,615 93,135 17, , ,498 Fx-Cc ,874 20,307 80,300 28, , ,248 Nc-N ,937 29,788 2,104, ,530 75,303 Cc-N ,004 92,060 7,812, ,800 55,911 Proposed ,544 20,322 14,403, ,229 25,905 Vol.67 No.2E (2016) 121

9 As shown n Table 3, he varaon of each obecve funcon of Nc-N, Cc-N, and Proposed becomes large. Because Fx-Nc and Fx-Cc fxed he s and or o he npu orderng level, he varaons of each obecve funcon of he wo polces are lmed. The proposed can-order polcy succeeded n ncreasng he average fness values and average fness values afer runnng he GAs for 3,000 generaons. In Table 3, here s some hgh obecve value. nce he problem s a mul-obecve opmzaon problem, he fness funcon appled s Eq. (22), and here s no penaly for surplus obecve values. Thus, he resuls represens he soluon space. In addon, he mehods proposed fnds a beer soluon n he soluon space because he average fness value s hgh. Based on hese resuls, under he condons consdered here, we demonsraed ha we could reduce he amoun of shorages by applyng he proposed can-order polcy usng a GA. By nroducng he can-order level proposed, he number of carrers wll ncrease and consequenly he numbers of ems n nvenory wll be reduced. 5 CONCLUION In hs paper, a mul-em nvenory problem wh a canorder polcy for correlaed demand and ems wh a fxed carrer capacy was consdered. The obecve of hs problem was o deermne he can-order level of each em n order o mnmze he number of ems n nvenory, he number of shorages, and he number of carrers used. We proposed o se a mulple can-order level for every em consderng he relaon of em demand as correlaed demand, whle prevous sudes consdered o se a sngle can-order level for every em. A mahemacal model was formulaed, and we proposed a GA o decde he proposed can-order polcy ha consss of a mulple can-order level, re-order level, and order-up-o level for each em. In he numercal expermen, we smulaed can-order polces wh he correlaed ems. We showed ha some chromosomes of he can-order polcy proposed ouperform he sngle can-order level orderng polcy. Addonally, nroducng he can-order level, whch consders he demand correlaons of dfferen ems, provdes more effcen nvenory managemen due o he order-up-o level. Furher sudes are requred o check he proper parameer sengs of he crossover rae and he muaon rae of he GA proposed. In addon, he relaon beween em shppng characerscs and he correlaed can-order polcy appled should be nvesgaed more horoughly. REFERENCE [1] Goyal,. K.: Deermnaon of Opmum Packagng Frequency of Iems Jonly Replenshed, Manage. c., Vol. 21, pp (1974) [2] Goyal,. K. and ar, A. T.: Jon Replenshmen Invenory Conrol: Deermnsc and ochasc Models, Eur. J. Oper. Res., Vol. 38, pp (1989) [3] Khoua, M. and Goyal,.: A Revew of he Jon Replenshmen Problem Leraure: , Eur. J. Oper. Res., Vol. 186, pp (2008) [4] lver, E.: A mple Mehod of Deermnng Order Quanes n Jon Replenshmens under Deermnsc Demand, Manage. c., Vol. 22, pp (1976) [5] Kasp, M. and Rosenbla, M. J.: On he economc Orderng Quany for Jonly Replenshmen Iems, In. J. Prod. Res., 29, (1991) [6] Wang, L., He, J., Wu, D. and Zeng, Y.: A Novel Dfferenal Evoluon Algorhm for Jon Replenshmen Problem under Inerdependence and Is Applcaon, In. J. Prod. Econ., Vol. 135, pp (2012) [7] Wang, L., Dun, C., B, W. and Zeng, Y.: An Effecve and Effcen Dfferenal Evoluon Algorhm for he Inegraed ochasc Jon Replenshmen and Delvery Model, Knowl.-Based ys., Vol. 36, pp (2012) [8] Balnfy, J. L.: On a Basc Class of Mul-Iem Invenory Problems, Manage. c., Vol. 10, pp (1964) [9] van Es, M. J.: On he Deermnaon of he Conrol Parameers of he Opmal Can-Order Polcy, ZOR- Mah. Models Oper. Re., Vol. 39, pp (1994) [10] Federgruen, A., Groenevel, H. and Tms, H. C.: Coordnaed Replenshmens n a Mul-Iem Invenory ysem wh Compound Posson Demands, Manage. c., Vol. 30, pp (1984) [11] Akns, D. R. and Iyogun, P. O.: Perodc Versus Can-Order Polces for Coordnaed Mul-Iem Invenory ysems, Manage. c., Vol. 34, pp (1988) [12] Panumsncha, P.: A Comparson of Three Jon Orderng Invenory Polces, Decs. c., Vol. 23, pp (1992) [13] Dellaer, N. and Poel, E.: Global Invenory Conrol n an Academc Hospal, In. J. Prod. Econ., Vol. 46, pp (1996) [14] Melchors, P.: Calculang Can-Order Polces for he Jon Replenshmen Problem by he Compensaon Approach, Eur. J. Oper. Re., Vol. 141, pp (2002) 122 J Jpn Ind Manage Assoc

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