OPTIMIZATION OF A PRODUCTION LOT SIZING PROBLEM WITH QUANTITY DISCOUNT

Transcription

1 NERNAONAL JOURNAL OF OPMZAON N CVL ENGNEERNG n. J. Opm. Cvl Eng., 26; 6(2:87-29 OPMZAON OF A PRODUCON LO SZNG PROBLEM WH QUANY DSCOUN S. hosrav and S.H. Mrmohammad *, Deparmen of ndusral and sysems Engneerng, sfahan Unversy of echnology, sfahan, ran ABSRAC Dynamc lo szng prolem s one of he sgnfcan prolem n ndusral uns and has een consdered y many researchers. Consderng he quany dscoun n purchasng cos s one of he mporan and praccal assumpons n he feld of nvenory conrol models and has een less focused n erms of sochasc verson of dynamc lo szng prolem. n hs paper, sochasc dynamc lo szng prolem wh consderng he quany dscoun s defned and formulaed. Snce he consdered model s mxed neger non-lnear programmng, a pecewse lnear approxmaon s also presened. n order o solve he mxed neger non-lnear programmng, a ranch and ound algorhm are presened. Each node n he ranch and ound algorhm s also MNLP whch s solved ased on dynamc programmng framework. n each sage n hs dynamc programmng algorhm, here s a su-prolem whch can e solved wh lagrangan relaxaon mehod. he numerc resuls found n hs sudy ndcae ha he proposed algorhm solve he prolem faser han he mahemacal soluon usng he commercal sofware GAMS. Moreover, he proposed algorhm for he wo dscoun levels are also compared wh he approxmae soluon n menoned sofware. he resuls ndcae ha our algorhm up o 2 perods no only can reach o he exac soluon, consumes less me n conras o he approxmae model. eywords: dynamc lo szng prolem; oal quany dscoun; ranch and ound algorhm; dynamc programmng; lagrangan relaxaon mehod. Receved: 2 Sepemer 2; Acceped: 8 Novemer 2. NRODUCON One of he major and asc responsles n he ndusral uns s producon plannng and * Correspondng auhor: Deparmen of ndusral and sysems Engneerng, sfahan Unversy of echnology, ran, P.B. BO: E-mal address: h_mrmohammad@cc.u.ac.r

2 88 S. hosrav and S.H. Mrmohammad nvenory conrol. he ssue of nvenoryng maeral and plannng for hgh qualy producon wh favorale volume a suale me and reasonale prce are of he major concerns of he managers. Economc order quany models or lo szng has een developed o acheve hs goal. Economc order quany deermnes how much and when a specal produc should e ordered so ha he sysem coss, whch ofen nclude holdng, orderng and purchasng coss, are mnmzed []. "Dynamc lo szng programmng" refers o hose ssues where plannng horzon s lmed and dscree, or n eer words, s assumed perodcally and he demand s dfferen from one perod o anoher []. Many elemens affec he varey of lo szng models, and y defnon, n he area of producon plannng and nvenory conrol, dfferen specfcaons and assumpons are consdered for he model. Among hese specfcaons ype of demand, capacy consrans, he numer of ems, plannng horzon and he purchase cos can e noed. Dscoun on good purchase has ofen een rased n deermnsc ssues. Callerman and Whyark [2] presened a Mxed-neger Programmng (MP model for orderng prolem wh quany dscoun hrough whch opmal orderng polcy s oaned y nary decson varales. On he ssue of deermnng deermnsc lo szng and consderng dscoun, Chung e al. [3] proved ha here s an opmal polcy ha order quany eween any wo consecuve re-orderng pons, excep for ha las order, s equal o one of he dscoun levels. Usng hs feaure, an algorhm ased on dynamc programmng algorhm was presened ha solve he prolem more effcenly han Callerman and Whyark's algorhm. Mrmohammad e al. [4] presened a ranch and ound algorhm for deermnng he quany of orders, n deermnsc sngle em cases whle consderng dscoun ha s more effcen n solvng large-scale prolems (many perods and hgh dscoun levels compared o prevous mehods. Goossens e al. [] demonsraed ha here s no polynomal algorhm o solve mul em lo szng prolem consderng oal quany dscoun. n oher words, hs prolem known as QD s n NP-hard class. here are wo approaches o conrol unflled demand n sochasc lo szng models. Sandard approach s nroducng he penaly cos for acklogged sales n he ojecve funcon. n some cases, calculang hs parameer, f no mpossle, s oo dffcul ha leads o he use of echncal performance sandards. he second approach s usng servce level consrans. he decson makers deermne he level of sasfacon wh hese sandards. n he leraure of he ssue, varous performance sandards are consdered whch he mos mporan of hem are α, β, γ servce levels [6]. he frs sudes n he feld of random demand and consderng n lo-szng prolems was carred ou y Slver n 978 [7]. Slver offered a heursc hree-sage mehod o deermne lo-sze wh random demand. Booknder and an [8] modeled he sochasc lo szng prolem n a sngle-sage sae wh regard o α servce level consran. n order o conrol he randomness of demand over me, accordng o he condons of nvenory and producon sysems, hree sraeges have een denfed: dynamc uncerany sraegy, sac-dynamc uncerany sraegy, sac uncerany sraegy. hey showed ha he mahemacal srucure of sochasc prolem wh α servce level and sac uncerany sraegy s he equvalen o a deermnsc lo-sze model and deermnsc lo szng prolem solvng mehods can e used o solve he sochasc verson. Vargas [9] presened an opmal algorhm for solvng he sochasc un-capacaed lo szng prolem. hs s

3 OPMZAON OF A PRODUCON LO SZNG PROBLEM WH 89 called as sochasc verson of Wagner Whn prolem. Sox n [] deal wh opmal solvng of sochasc dynamc lo szng prolem y consderng non-saonary purchase cos. empelmeer [] has revewed he mahemacal models of sochasc lo szng prolems and developed a model wh sac-dynamc sraegy consderng fll rae β n whose soluon nvenory on hand s used nsead of ne nvenory. Vargas and Mers [2] developed he heursc PDLA algorhm o solve he sochasc sngle em un-capacaed lo szng prolem n sngle-sage sae y consderng penaly cos for unflled demand n a rollng plannng horzon. hs algorhm s an exenson of opmal algorhm of shores pah prolem n sac uncerany sraegy. Quany dscoun has newly een suded n sochasc dynamc lo szng models and few arcles have een pulshed n hs regard. Hajj e al. suded he quany dscoun n sngle-perod model (he newsoy prolem n 27 wh he random nal nvenory. For hs prolem, he opmum quany of order s deermned o maxmze prof and he prolem s rewren wh random demand and nvenory varales n normal mode. n ang and Lee [4], sngle em dynamc lo szng prolem wh random demand y consderng he oal quany dscoun n suppler selecon feld has een nvesgaed, a heursc mehod ased on Dynamc Programmng (HDP has also een developed o solve he prolem. he remander of he paper has een organzed as follows. n Secon 2, he prolem s defned and esde wo mahemacal non-lnear models of he prolem, a pecewse lnear model of he prolem s presened. Secon 3 presens he soluon approach of he prolem whch s ased on decomposng he prolem n four levels. n each level, a proper approach s appled o handle he su-prolem of he level. n Secon 4 some es prolems randomly generaed are solved y wo approaches o evaluae her effcency relavely. Concludng remarks and resuls are appeared n Secon. 2. PROBLEM DEFFNON AND FORMULAON n asc models of sochasc dynamc lo szng prolem, s assumed ha he un prce of he ordered ems wll no change y he quany of each order. n hs paper, a model s nvesgaed n whch un em prce depends on he quany of each order. hs means ha realers and produc supplers of commody offer ha f he order quany x reaches a ceran value q, hey are wllng o sell oal value of x o a prce lower han c o he uyer. A hs pon, he newly announced prce ncludes he oal value of order x. hs prce srucure s called All-un dscoun. hs dscoun cos srucure can e defned as nonsaonary for he case ha he purchase prce s non-saonary. n oher words, dscouns polcy s dfferen a any perod compared wh he ohers (oh n erm of prce and dscoun levels. n hs sudy, s assumed ha he numer of dscoun levels s he same for all perods, u whou hs assumpon he presened model wll sll e vald. n he nended prolem, he demand s consdered for one em, and he me horzon s fne. Orderng cos n each perod s consdered only n case of orderng and resources are unlmed. Demand s assumed o e random and connuous. Shorage s allowed n form of ackloggng and he amoun of shorage s conrolled as shorage penaly cos n he ojecve funcon. Demand s random and s densy funcon s known and n any perod s ndependen of oher

4 9 S. hosrav and S.H. Mrmohammad perods. Shorage, holdng and orderng coss can vary from one perod o anoher. he goal s o mnmze he expeced cos of holdng cos, shorage cos, orderng cos and purchase cos n oal plannng horzon. A he egnnng of he plannng horzon, me and amoun of orderng s deermned for he enre plannng horzon. n hs sudy, he ssue Sochasc Sngle em Dscouned Lo Szng Prolem s expressed as wh he arevaon SSDLSP. 2.. Mxed neger nonlnear modes n hs secon we formulaed he prolem n wo dfferen ways whch lead o wo dfferen mxed neger nonlnear models. he followng noaon s used n he mahemacal formulaon of he prolem: Numer of perods n plannng horzen Numer or dscoun level and for perod,,2,...,, A h M D x Orderng cos Holdng cos Backorder penaly cos A suffcenly large nulmer Demand Orderng quany Cumulave order quany hrough perods o ( Y Cumulave demand hrough perod o ( j Y D j x j j f Y ( y p.d.f of Y F Y ( y c.d.f of Y L ( oal expeced holdng and penaly coss ncurred a he end of perod s A nary varale whch s f an orderng occuredn perod, oherwse q k he mnmum accepale quany o deserve for dscoun level k n perod c k Un Purchasng cos n perod and n dscoun level k u A nary varale whch s f an order performed n perod n dscoun level k, k oherwse he prolem can e formulaed as follows. Mn E{ c} A s L ( c ( (

5 OPMZAON OF A PRODUCON LO SZNG PROBLEM WH 9 s.,,..., (2 M s,,..., (3 u k k u k k ck,,..., (4 c,,..., ( ( q, k M( uk,,...,, k,..., (6 q, k M( uk,,...,, k,..., (7 u k, s {,},,...,, k,..., (8 c,,,..., (9 hs model s an exenson of Sox [] model. Because of exsence of quany dscoun, n hs model he un purchase prce n perod ( c s consdered as a decson varale and he expresson of he equaon ( s consdered as a non-lnear expresson. hs equaon represens he expeced holdng, shorage, orderng, and purchase coss n he oal plannng horzon. Consran (2 saes ha he order amoun n perod mus a leas e equal o cumulave order value n he prevous perod. Consran (3 s presened o se correc amoun o orderng varale ( s. n hs equaon, he value of orderng can only e greaer han zero when orderng varale ges value one and s cos s consdered n he ojecve funcon. Consran (4 shows ha he order quany n each perod elongs only o one of he levels. Consran ( specfes purchase cos n each perod accordng o he level se for he order. Consrans (6 and (7 fulfl he quany dscoun lms n orderng n each perod. n hese consrans, f an order s gven a dscoun level k and n perod, he order quany s lmed eween q, k and q, k.oherwse, he consran for perod and level dscoun k s relaxed usng he large numer M. Generally, q and q s assumed n he models and dscouns. n he presened model, L ( s as he oal funcon of expeced shorage and holdng coss n accordance wh [] s defned as follows: k h ( E[ Y ] ( h ( y f ( y dy f L ( ( E[ Y ] f ( f he nal nvenory s negave and unl perod he amoun of nvenory has no ecome posve hen s negave. n hs case, perod does no ncur any holdng cos whle he shorage cos s equal o he shorage cos of he amoun acklogged ll perod ( E[ Y ]. Snce he numer of zero-one varales has an mporan effec on compuaonal me, we ry o ncrease he effcency of he model va reducng he numer of zero-one varales n he second model. Furhermore, he purchase cos n ojecve funcon has rewren n lnear form.

6 92 S. hosrav and S.H. Mrmohammad Mn E{ c} [ A ( uk L ( ( lk ck ] ( k s. lk k k u k k,..., (2,..., (3 qk uk lk,..., k,..., (4 l k ( qk uk,..., k,..., ( uk {,},..., k,..., (6,..., (7,..., k,..., (8 lk n hs model, orderng cos s deermned ased on he varale deermnng level of dscoun ( u k, and varales s,,..., are omed from he model. n hs model varale l k s he amoun ordered n dscoun level k a perod. n consran (2, he order sze of perod s calculaed va sum of l k on all dscoun levels. Consran (3 forces orderng o e occurred a mos from one of dscoun levels. Consrans (4 and ( are se o deermne he allowed lms of valung o l k. he frs model s more represenave han he second one ecause of smplcy u, n our expermenal compuaons, we adoped he second model due o s me effcency. 2.2 Pecewse lnear approxmaon model n he ojecve funcon of he prevously presened models, he erm L ( s nonlnear and makes he whole models nonlnear. o reach a soluon wh a conrollale error, we esmae L ( y lnear approxmaon and presen a lnear u approxmae model. L can e wren as follows: ( L ( h E[ ] E[ ] (9 n equaon (9, s posve nvenory a perod and acklog n perod,.e. Y and Y s negave nvenory or. herefore, we have E [ ] ( y f ( y and E[ ] ( y f ( y. E [ ] s named he frs order loss Y funcon and E [ ] as s he complemenary funcon n he lraure and hey can e wren on as follows. Y E[ Y ] G ( (2

7 OPMZAON OF A PRODUCON LO SZNG PROBLEM WH 93 (2 ( ] [ Y G E n equaons (2 and (2, ( Y G s consdered as nonlnear funcon for each perod, and for he parcular case of normal dsruon s defned as follows ased on reference []: (22 Y G ( n equaon (22, Y s he random varale of cumulave demand unl perod wh normal dsruon wh mean and sandard devaon and (z s he cumulave dsruon funcon of he sandard normal dsruon. Consder B pons on he - axs n perod named e, B...,,, ha s he value of n h pon n perod. Approxmaon funcon wh B lnear peces presenng oal expeced holdng and shorage coss, s as follows (see [6]: (23 B B w w h L ( n equaon (23 he slope of each pece s defned as follows: (24 Y Y e e e G e e G e,,, ( ( (2 Y Y e e e G e G,, ( ( Fgure. he pecewse approxmaon of average on-hand nvenory and are he average nvenory and average shorage a he prmary pon e,

8 94 S. hosrav and S.H. Mrmohammad respecvely. w s he decson varale defned as he cumulave order amoun hrough he h nerval such ha e e. Fg. represen he lnear approxmaon of E [ ] n w, whch and. n hs fgure E [ ] has een approxmaed va lnear peces (B=. Snce he expeced funcons are convex, he slopes and are ncreasng on,,...,b. herefore, due o he mnmzng he ojecve funcon of he model, w ecome posve only when he prevous varale, w, reaches s maxmum value,.e. w, e, e 2,. herefore, we can se w n he mahemacal model whou addng any axllary consrans. So pecewse lnear approxmaon mahemacal model of SSDLSP s as follows: B B B Mn E{ c} [ A ( uk h w w ( lk ck ] (26 k k s.. l k k B w B u k k w,,..., (27,..., (28 qk uk lk,..., k,..., (29 l k ( qk uk,..., k,..., (3 w e e,,...,,..., B (3 uk {,},..., k,..., (32 w,...,,..., B (33 l,..., k,..., (34 k 3. SOLUON APPROACH f Sox s model [] s consdered as a ase model wh specfc soluon, he proposed model SSDLSP s more complex han he asc model n wo ways. hey are deermnng he opmal dscoun levels n each perod (deermnng he opmal amoun of varales u k and deermnng he opmal order quany consderng upper and lower lms of permed orderng. Our sraegy o solve he prolem s ased on decomposon echnques. n hs paper, he prolem SSDLSP s solved y ranch and ound mehod. n each node of hs algorhm a su-prolem called P, s solved y dynamc programmng approach. A each sage of hs algorhm a su-prolem called P 2 s rased whch s solved y a ranch and ound mehod. n each node of he second ranch and ound algorhm, su-prolem P 3 s solved y Lagrange relaxaon mehod. n he remander of hs secon, he soluon

9 OPMZAON OF A PRODUCON LO SZNG PROBLEM WH 9 approach s presened a four levels. n he frs level, su-prolem P s presened and ranch and ound algorhm s descred. n he second level, su-prolem P 2 s defned and soluon of he su-prolem P s presened. n he hrd level, he soluon o su-prolem P 2 wh he defnon of P 3 s dscussed and a he las level, he soluon o su-prolem P 3 s descred. 3.. Frs level n hs secon, we frs defne he su-prolem P and hen he soluon of he prolem wh ranch and ound mehod s provded. 3.. Defnon of su-prolem P n hs su-prolems, s assumed ha he dscounng s permed only for a se of perods, call D, and for oher perods, call R, purchase prce s fxed o cheapes case (hghes dscouns level wh no lm n orderng. hus, he perods are consdered n wo ses R and D. he collecon of hese wo ses ncludes he complee plannng horzon. n oher words, he consrans (4 o (8 are appled only for he perods of D. n each node of he ranch and ound algorhm he dscoun level for each perod of D s specfed,.e. he varale u k n perod, D, s only for a specfc dscoun level, say m, and for oher dscoun levels, k m, s zero. Consequenly, consrans (6 and (7 change no q q. m, m For smplcy n model P, he allowed upper and lower lms for orderng n perod are shown y u and l, respecvely. he un purchase cos c s defned as follows: c c c m D R (3 he mahemacal model of he su-prolem P s as follows: Mn E c P s L c A (36 s. - u D (37 - l D (38 -,..., (39 - Ms,..., (4 s,,..., (4,..., (42 n hs model, he ojecve funcon s defned as n [] and can e rewren as follows:

10 96 S. hosrav and S.H. Mrmohammad Mn E c P A s L c c (43 Propery : f we se D as an empy se, he opmal soluon of he prolem P s a lower ound for he prolem SSDLSP. Empness of se D means ha he prolem has no lms for orderng wh regard o he dscoun polces. hus, y reducng he numer of consrans, he soluon space wll ge greaer. On he oher hand, he es purchase prce s consdered for all perods. So he oaned soluon wll e he es possle soluon for he expeced oal cos. Propery 2: he opmal soluon of su-prolem P s an upper ound for he prolem SSDLSP. he dscouned cos srucure and relaed consrans are ncorporaed only for he perods of D and he purchase cos of oher perods s se o he lowes case. Hence, s ovous ha n hs crcumsance he oaned soluon s a lower ound for he more resrced general case of SSDLSP. he man enef of applyng a B&B algorhm for solvng SSDLSP s ha enumeraes all possle crcumsances of varale u k,,...,, k,...,, and solvng he relaed suprolem. herefore, he opmal soluon o he prolem SSDLSP s oaned y hs approach. n a complee enumeraon, here s quanes for dscoun level n each perod,,...,. Hence, here are possle quanes for uk varale. he presened B&B approach enumerae hese cases mplcly. 3.. he frs level ranch and ound algorhm n hs secon, he seps of he ranch and ound algorhm whch reaks down SSDLSP o P are presened. o oan an nal soluon for he prolem and use as an lower ound from he egnnng of he B&B algorhm a he nal level, s assumed ha here are no orderng consrans and he un purchase prce n all perods s he lowes possle value (maxmum dscoun. he soluon space wh hs assumpon s much larger han he orgnal prolem and s ojecve funcon value s a lower ound for he prolem. he prolem wh he menoned assumpon s n fac, he prolem nroduced y Sox []. n Branchng sep of he algorhm, a perod s seleced and s nsered o se D. he sraegy of selecng he perod depends on he value of orders n he nal soluon whch s n he roo node. n oher words, he perods wh posve orderng value n he nal soluon are n prory of ranchng. More precsely, a he roo node, a ls of prorzng wh he menoned creron s deermned for ranchng and ranchng happens accordng o hs ls for dfferen perods. For each node, as a paren node, nodes, as chldren node, are generaed y addng he relaed consrans for each dscoun level k, k,...,. More precsely, a perod, say perod, s seleced from he menoned ls and for each dscoun level k, k,...,, nodes n perod s generang such ha n each node a su prolem P wh he upper and lower lm consrans, correspondng o he dscoun level k, s added for he orderng value on perod. Branchng connues from he acve node ha has he es

11 OPMZAON OF A PRODUCON LO SZNG PROBLEM WH 97 lower ound. f wo nodes wh equal lower ound found, ranchng s done n he node ha has more deph n he search ree. Afer oanng lower ound of each node y solvng P, he values of orderng are calculaed wh her relaed prces accordance wh he dscoun cos srucure o sasfy he dscoun level consrans of ha perod. n hs way, an upper ound s oaned for he prolem. n hs algorhm, nodes are fahomed wh he followng rules:. f he upper and lower ounds are equal n a node, means ha he resulng soluon s feasle and ranchng wll no eer he answer 2. f he lower ound of a node s greaer han he es so far upper ound. should e noed ha f he node s a he lowes level of he ree, he frs rule s rue aou and he node s fahomed. A he end of he algorhm, all nodes are fahomed and he node wh he lowes ojecve funcon whose s upper and lower lms are equal s he opmal soluon Second level o e an opmal algorhm, he B&B algorhm mus solve he su prolem P n each node opmally. For hs purpose, a dynamc programmng algorhm s appled ha n each sage a su-prolem, named P 2, s rased Defnon of su prolem P 2 (s, Consder a su prolem P rased n a node of B&B algorhm. Solvng hs su prolem y our dynamc programmng approach lead o e rased a su prolem P 2 ( s, n each sage of he DP algorhm. n P 2 ( s, he goal s o fnd he mnmum expeced cos of holdng, orderng, purchasng and shorage from perod s o he end of perod, assumng ha he orderng occurs only n he perod s and f here are any dscoun consrans, n he perod o m wh followng condons:. Perod s and + are wo perods of su prolem P whch do no have any orderng consran. (.e. s, R 2. M {,... m } ( m s s he se of all he perods eween s and ha have consrans n orderng. ( s,..., m m, M D Mahemacal model of su prolem P ( s, s as follows: Mn P s. s m 2 As A s g, ( (44 m u s,..., m (4 l,..., m (46 s {,},..., m (47,..., m (48 n equaon (44, funcon g (, j expresses he expeced oal holdng, shorage and

12 98 S. hosrav and S.H. Mrmohammad purchase coss from perod o he end of perod j when cumulave order from perod o he end of perod j s equal o he fxed amoun. So g ( s as follows., j g, j ( c c j ( c j j j L ( L ( j (49 n addon, oal orderng cos s equal o he sum of orderng cos n perod s and f any, he orderng coss n and m. Equaon (4 s a comnaon of he wo consrans (37, (4 and saes ha n he mddle perods, f here s an order, s amoun should no exceed he upper lm of relaed dscoun level value of ha perod. s noale ha for he perods ha lower lm of dscoun level for hem ( l s greaer han zero, he relaed varale s wll e one. hus, ha par of orderng cos whch s varale s only consdered for a suse of D z M whose lower permssle lm of he dscoun level for hem s zero snce for perods whose orderng amoun are locaed on he frs dscoun level we have l. Base on wha s saed aove, we can rewre he ojecve funcon of he model as equaon (. Mn P s A s A Dz, M Dz A s m g (, ( Dynamc programmng algorhm n hs secon, a forward dynamc programmng s presened whch solves he su prolem P and n each s sage a su prolem P 2( s, mus e solved. As menoned earler, a suprolem P ( s, deermnes he opmal orderng polcy from perod s o he end of perod named y 2 op s. n fac, op s s a vecor of opmal orders amoun from perod s o perod,, whch mnmzes he expeced oal cos durng menoned perods P ( s,. Le he mnmum expeced cos, op * * *.e. s ( s, s,..., under he menoned assumpon for su prolem op op menoned aove, s denoed y P s ( s and suppose w, w,..., W, e he perods of se R where we have s. n hs case, he ojecve funcon of he su prolem can e w 2 rewren as he sum of several ojecve funcons of he su-prolem P ( s, : 2 E{ c p } W P w op, (, w w w w ( (, w, w w w op n equaon ( P shows he opmal value of ojecve funcon of suprolem P 2 from perod w o perod w, when he opmal order quany s equal o

13 OPMZAON OF A PRODUCON LO SZNG PROBLEM WH 99 op, w w. n hs equaon. hs equaon ndcaes ha su prolem P s separale w o su prolems P 2( w, w. he proposed dynamc programmng algorhm has he followng elemens: Sage: calculaon of he mnmum cos o perod ( R. Sae: Perod s ( s R as he las orderng perod efore he perod. Recursve relaonshp: f s he mnmum cos assocaed wh he orderng polcy s form perod s o when he las order s occurred among he perods of R n perod s and nex order n hs se occurs a perod, we have f s op Ps( s mn { fk, s : k, s( s ks, kr op op s ( s} (2 f op op op hen f. hs suaon means ha n he opmal polcy { k, s k, s s } s here s no polcy y whch s possle o reach s, and hen go o, ecause hs pah volaes Consran (39. he pseudo code for he proposed algorhm s as follows For =,, and R For s =, and s R op op Solve P2 ( s, and compue s and P s ( s f s =, MN Else op op f { k k s, k R : k, s ( s s ( s}, MN Else op op MN mn { fk, s : k, s( s s ( s} f s ks, kr op s P ( MN s Whle f f mn { f } s j j k, kr op for j = s,, ( j ( s 3.3 hrd level k j s f he opmal values of nary orderng varales are deermned n su prolem P 2( s,, hen he prolem wll ecome a non-lnear non-neger programmng ha convex opmzaon mehods can e used for opmal solvng of he model Defnon of su prolem P 3 As prevously menoned, he collecon D z s a seres of perods eween s and where he lower lm of order s zero. So n hese perods orderng may e ssued or no. A su-

14 2 S. hosrav and S.H. Mrmohammad prolem P 3 s he same as su-prolem P 2 where s decded eforehand for orderng n he perods relang o se D z. n hs case; perods of D z are dvded no hree caegores. he frs se s D A where he cos of orderng for s memer perods s deermned accordance wh he erms defned n he su-prolem P 2 and consdered n he ojecve funcon of he prolem (wheher here s an order n hose perods or no, n oher words s. he second se s D B where he cos of orderng for s perods s consdered as zero, and he order s consdered free n hose perods. n oher words s ( DB. he hrd se s DC ha s assumed no o perform any orders n hose perods, n oher words s ( D. Communy of hese hree ses composes hen for ease n model P 3 he cos Dz se. A s defned as follows: c A A D D A B (3 Wh hs dvson for perods of D z he su prolem P 2 changes no a prolem where here are no zero-one varales of orderng. Moreover, orderng occurs only n he perod s and n he perods of he seres N {,... n }. hs collecon s a suse of M where orderng varale for all s perods n he model P 2 s equal o one ( s. n oher words, he perod of DC se has een removed from he collecon of o m ( N M DC. Mahemacal programmng model of he prolem P3 s as follows: Mn PR s. s n A g (, (4 u,..., n ( l,...,n (6,..., n (7 n hs model, perods are assumed as follows: ( s,..., n n,,..., n N Propery 3: n P 3, f all perods of Dz are placed n se D B, he opmal soluon for su prolem P 3 would e a lower ound for su prolem P 2( s,. Propery 4: From any opmal soluon of he su prolem P 3, s possle o reach an upper ound for he su prolem P 2 y modfyng he order coss wh regard o he soluon oaned. Propery : By complee enumeraon of dfferen values of varale s ( D and solvng he relaed su prolems, he opmal soluon of P 2 wll e deermned he hrd level ranch and ound algorhm n hs secon, n an algorhm smlar o he menoned ranch and ound algorhm, a ree z

15 OPMZAON OF A PRODUCON LO SZNG PROBLEM WH 2 search ased on he ranch and ound approach s presened y whch he s 's are deermned are deermned n su prolem P 2 and opmal value of orderng are oaned. he nal soluon for he roo node of he ranch and ound algorhm s consdered as follows. n P 2, varale s has appeared as a complcang varale. f we assume ha n each perod, one can freely order and here s no charge for orderng, hen he lower ound of he prolem s specfed. herefore, a he roo node, orderng cos wll e consdered for none of he perods of D Z and all he perods of D Z are placed n D B. A he ranchng sep of he algorhm, paren node s ranched for a perod from among he perods of D Z. hs ranchng generaes wo chld nodes for a paren node where n one he orderng cos s ncurred and orderng s occurred n he menoned perod (he menoned perod s placed n D A and he oher one, he orderng cos s se o zero and no orderng s occurred n he menoned perod. (he menoned perod s placed n D C. he sraegy of node selecon for ranchng s as follows. n hs algorhm, among he acve nodes, he node wh he es lower ound s seleced for ranchng. f wo nodes have equal lower ounds, ranchng s done on he node wh larger deph n he search ree. n hs algorhm, nodes are fahomed wh he followng fahomng rules: f he upper and lower ounds are equal n a node, means ha he resulng soluon s feasle and ranchng would no lead o eer soluon. f he lower ound of a node s greaer han he upper ound oaned so far. f for all perods of D Z he decson of orderng s deermned. means he node s nacve and no furher ranchng s needed. he opmal soluon of he su prolem wll e oaned a he end of he algorhm. Among he fahomed nodes, he node wh he lowes ojecve funcon whose upper and lower lms are equal s he opmal soluon. 3.4 Fourh level he Lagrange relaxaon mehod s used n cases wherey relaxng a numer of consrans of he prolem leads o a more smple prolem. n su prolem P 3, f he permssle lower and upper lms of he order n equaon ( and (6 are relaxed, hen he soluon of he relaxed prolem can easly e oaned y dervave of he ojecve funcon. n hs secon, he Lagrange relaxaon mehod (LR s presened o solvng he su prolem P 3 n hree man seps Sep : Solvng he relaxed verson of P 3 (RPP he Lagrange relaxed verson of he su prolem P 3 can e saed as follows: Mn RPP s n A g, n u u l l s..,, n (8 (9

16 22 S. hosrav and S.H. Mrmohammad Lagrange mulplers assocaed wh he upper and lower lms of he orderng consrans are u and l, respecvely. We assume ha u l l n un. herefore, for each perod lke, here are hree varale funcon of RPP can e rewren as follows., l and u. he ojecve where f f Mn RPP n s A s defned as follows. f s.. l l u u,, n c u. l c u l L j j (6 (6 (62 hus, he relaxed verson of he su prolem P3 s an unconsraned non-lnear mahemacal programmng where s ojecve funcon s concave. Hence, s opmal soluon can easly e oaned y dervave. he paral dervave of he ojecve funcon relave o varale s as follows. RPP s h ( h ( F ( c u l c u l j j j j y j (63 o fnd he roos of hs funcon a comnaon of secon mehod and false poson eraon mehods are used Sep 2: Updang he Lagrange mulplers he man sep n Lagrange relaxaon s updang Lagrange mulplers for whch n he relaed leraure, varous mehods have een developed. n hese mehods, he goal s o maxmze he dual prolem of he orgnal relaxed model va changng some coeffcens of Lagrange mulpler whch susequenly leads o mnmze orgnal prolem he orgnal varales. One of he man and general mehods of updang Lagrange mulplers s usng su-graden mehod. For he su-prolem of P 3, Lagrange mulplers are updaed as follows: ( u max, ( l max, ( u ( l ( ( G G ( u ( l (64 (6

17 OPMZAON OF A PRODUCON LO SZNG PROBLEM WH 23 ( ( n ( UBes RPP ( 2 ( G G l u 2 (66 n hese equaons G ( ( ( u ( and G ( ( ( l l (. Usually, n he leraure we found ha 2 and for eer convergence s reduced durng eraons of he algorhm. UBes s he es upper ound oaned from he prmary prolem P3 ha are oaned n a heursc way y makng he soluon feasle from he relaxed prolem y ( changng order amoun o mee he feasle lms values of consrans. RPPs s he value of he ojecve funcon of he relaxed prolem n v h sep Sep 3: Soppng crera A hs sep, he predeermned convergence crera are check o ensure ha he oaned soluon s suffcenly close o he opmal soluon o sop algorhm. Followng crera are he man condons saed n he leraure. f he dfference of he es lower ound ( RPP es wh he es upper ound ( UB es n he algorhm s less han an error, he algorhm s ermnaed and he soluon s repored as -opmal soluon. f he numer of occurrences s greaer han a specfed lm, he algorhm wll sop. f he vecor of Lagrange mulplers are suffcenly close n he las eraons ( ( ( (, he algorhm wll sop. Overall Lagrange algorhm descred n hs secon s summarzed n form of followng pseudo code. Sep : nalzaon. Se, ( nalze dual varales Se ( down Sep : Soluon of he relaxed prmal prolem. ( Solve he relaxed prmal prolem and oan opmal value of x and s assocaed ( ojecve funcon Updae he lower ound for he ojecve funcon of he prmal prolem, ( ( ( ( f down se down Sep 2: Mulpler updang. Updae mulplers usng su-graden mehod. f possle, updae also he ojecve funcon upper ound. Sep 3: Convergence checkng. ( * f he soppng creron s me, he -opmal soluon s x x and sop. Oherwse se and go o Sep.

18 24 S. hosrav and S.H. Mrmohammad 4. COMPUAONAL EPERENCE Numercal expermens were carred ou wh he programmng he proposed algorhm n C # n Mcrosof Vsual Sudo 2. he resuls of solvng he es prolems are analyzed for evaluang he performance of he algorhm. Also he resuls are compared wh he resuls oaned wh he resuls oaned from solvng he mxed-neger nonlnear model of SSDLSP and mxed-neger lnear approxmaon model whch have een run n n commercal opmzaon sofware GAMS. hs comparson s performed ased on compuaonal me and accuracy of soluon y solvng a se of random prolems. Among solver n GAMS only solvers BONMN, NRO, ALPHAECP and LNDOGLOBAL are avalale for MNLP models n whch he frs-order normal loss funcon s defnale. he only one of hem whch s ale o announce he opmal soluon on a small scale s LNDOGLOBAL. Oher solvers, even f hey have reached he gloal opmum soluon, repor as a local opmal soluon. So he ass of comparson of soluon mehods s o solve he prolem n GAMS wh he solver LNDOGLOBAL. hs solver can also e used n LNGO sofware. 4. Expermenal desgn n generang es prolems, each of he npu daa s a conrollng facor. Among hese facors, he mpac of wo facors and are of more mporance han oher facors n solvng he prolem. Oher npus have een adjused expermenally and hey have fxed hrough all es prolems. Orderng, holdng, and shorage coss and nal nvenory have een se n accordance wh reference []. he expeced value of demand, E ( d, for each perod s deermned randomly. he sandard devaon of demand for each perod s assumed as d.2* E( d lke wha has een done y []. herefore, he expeced value ( and sandard devaon of cumulave demand ll perod, s equal o 2 ( d j j E( and d j j, respecvely. he frs level un prce of he un purchasng prce n dscouned cos srucure n each perod ( c s deermned randomly as shown n ale. Also, ale shows he adjused values of oher npu parameers whch have een randomly generaed. Parameers adjused value ale : Adjused values of he npu parameers c [,8] E ( d [2,2] 98 2 h. A 48 n seng he dscoun polcy parameers, wo facors are have grea mporance generang es prolems. he frs one s k, he proporon of dscoun level o he average demand y whch he mnmum orderng amoun n k h level n he h perod, q k, s

19 OPMZAON OF A PRODUCON LO SZNG PROBLEM WH 2 deermned.e. qk k.. he second mporan parameer s γ, he neres rae of dscoun, whch s an ndcaor o measure he purchase cos savng ased on whch he purchase cos s oaned wh he equaon ck ( k. c. hese wo facors, smlar o he Mrmohammad e al. [4], have een se n deermnsc he lo szng prolem wh quany dscoun. her value have een lsed n ale 2 for fve dscoun levels. k ale 2: Adjused value of parameers for dscounng polcy k o consder he cases for more han wo random parameers and c, random generaon of daa for each prolem wh & s repeaed fve mes and he me o solve prolems s oaned from he average of fve prolems Analyss of es resuls he average me o solve he prolems are recorded from he mplemenaon of he program on he machne wh specs nel (R core (M 7-26 CPU@3.4 GHz and hey have lsed n ale 3. n hs ale, he mean of LG s he solver LNDOGLOBAL and B&B refers o he proposed Branch and ound algorhm. R shows he average compuaonal me for every fve es prolems solved wh varous values of and n seconds. As shown n ale 3, LG solver s unale o solve prolems wh more han 9 perods n he specfed me lm. N shows he numer of nsances of prolems (from fve nsances whch are solved opmally n a me less han 72 seconds. Num R ale 3: Run me comparson of B&B and LG B&B LG R R R N R N 4 R N 4 3 Fg. 2 shows he compuaonal me of B&B n comparson wh solver LG wh wo dscoun levels. As s shown n hs fgure, he compuaonal me of he B&B algorhm s drascally less han he wha s oaned from solver LG. hs solver s no ale o

20 26 S. hosrav and S.H. Mrmohammad announce he opmal soluon n 9 and perods n a me less han 72 sec.. hs s whle he longes compuaonal me of he proposed algorhm s for he case n whch =2 and =2 wh me seconds. hs ndcaes he hgh performance of proposed algorhms. R B&B LG Fgure 2. Run me comparson of he solver B&B and LG wh wo dscoun level Snce n Fg. 2 changng he values relavely s no angle, he oom par of hs graph s magnfed n Fg. 3. R B&B LG Fgure 3. Paral magnfed of Fgure wo levels dscoun prolem has more praccal aspec han oher varan of hs prolem. hen he ehavor of he proposed algorhm for wo levels of dscouns up o 3 perods s compared. One of he man ssues n he analyss of he ehavor of nonlnear algorhms s comparng hem wh her lnear approxmaon verson. n hs respec, he approxmaon model of he prolem was encoded n GAMS sofware y consderng approxmaon pons n each shorage funcon n he ojecve funcon. he compuaonal resuls are shown n ale 4. n hs ale, R s he average run me for all fve nsances. APP sands for lnear approxmaon model ha runs on GAMS sofware. E means he relave error of soluon of approxmaon model o B & B algorhm. Num shows he prolem numer.

21 OPMZAON OF A PRODUCON LO SZNG PROBLEM WH 27 ale 4: Run me comparson of B&B and APP Num B&B R APP R E As shown n Fg. 4, he B&B algorhm ges he opmal soluon faser han he approxmae model (APP. From hs pon of nersecon of he curves n Fg. 4, he prory of user should e specfed, f he accuracy of he soluon has hgher prory, he proposed B&B algorhm should e appled, and f he soluon me s mporan, gnorng he relave error, he approxmaon model s eer. R B&B APP Fgure 4. Run me comparson of B&B and APP. RESULS AND CONCLUSONS Addng he assumpon of possly of dscoun n maeral purchasng o he Sox s model [] and deffnng he sochasc sngle em lo szng prolem under quany dscoun n purchasng (SSDLSP make he prolem much more complex from wo pons of vew. Addng several nary varales o he model s he frs aspec and he second one s he exra consrans added o he ase model. hus, alhough he ase model has een solved

22 28 S. hosrav and S.H. Mrmohammad wh a dynamc programmng algorhm, he developed model of he algorhm has a more complex soluon approach. he proposed soluon approach s presened a four levels ased on a ranch and ound algorhm hyrdzed wh a dynamc programmng algorhm o oan he opmal soluon of he prolem. hs approach can e used for any arrary dsruon of demand, and s faser han LNDOGLOBAL solver n GAMS. Furhermore, o have a more precse evaluaon of he presened algorhm n large scale prolems, we presened he lnear approxmae model of he model and we compared he presened algorhm wh. he presened algorhm solves he prolem wh 3 perods (=3 opmally n a reasonale me, u, slower han approxmae model. he approxmae model performs more effcen han B&B algorhm u wh a of error o he opmal soluon. n our expermens, he maxmum error of he approxmae model s.44 percen whch seems olerale regardng he speed of hs approach. n hs paper we conaned he shorage of produc y chargng he shorage cos o he ojecve funcon of he prolem. Snce evaluang he shorage cos parameers may e hard n pracce, handlng he shorage of producs va defnng he proper cusomer servce level may me more praccal and s lef as a fuure developmen of he curren work. REFERENCES. arm B, Faem Ghom S, Wlson J. he capacaed lo szng prolem: a revew of models and algorhms, Omega 23; 3: Callerman, Whyark D. Purchase quany dscouns n an MRP envronmen, Proceedng of 8h A nnual Mdwes Conference, Chung, CS, Chang D, Lu CY. An opmal algorhm for he quany dscoun prolem, J Oper Manag 987; 7: Mrmohammad SH, Shadrokh S, anfar F. An effcen opmal algorhm for he quany dscoun prolem n maeral requremen plannng, Compu Oper Res 29; 36: Goossens DR, Maas A, Speksma FC, Van de lunder J. Exac algorhms for procuremen prolems under a oal quany dscoun srucure, European J Oper Res 27; 78: empelmeer H. Sochasc lo szng prolems, Handook of Sochasc Models and Analyss of Manufacurng Sysem Operaons 23, Sprnger, pp Slver E. nvenory conrol under a proalsc me-varyng, demand paern, AE rans 978, : Booknder JH, an JY. Sraeges for he proalsc lo-szng prolem wh servce-level consrans, Manag Sc 988; 34: Vargas V. An opmal soluon for he sochasc verson of he Wagner Whn dynamc losze model, European J Oper Res 29; 98: Sox CR. Dynamc lo szng wh random demand and non-saonary coss, Oper Res Leers 997; 2: empelmeer H. On he sochasc uncapacaed dynamc sngle-em loszng prolem wh servce level consrans, European J Oper Res 27; 8: Vargas V, Meers R. A maser producon schedulng procedure for sochasc demand and rollng plannng horzons, n J Produc Economcs 2; 32:

23 OPMZAON OF A PRODUCON LO SZNG PROBLEM WH Haj M, Haj R, Dara H. Prce dscoun and sochasc nal nvenory n he newsoy prolem, J ndus Sys Eng 27; : ang HY, Lee AH. A sochasc lo-szng model wh mul-suppler and quany dscouns, n J Produc Res 23; : Ross R, lc OA, arm SA. Pecewse lnear approxmaons for he sac dynamc uncerany sraegy n sochasc lo-szng, Omega 2; : 26-4.