Harmony search algorithms for inventory management problems

Transcription

1 Afrca Joural of Busess Maageme Vol.6 (36), pp , 2 Sepember, 202 Avalable ole a hp:// DOI: /AJBM2.54 ISSN Academc Jourals Revew Harmoy search algorhms for veory maageme problems Aa Allah alezadeh Faculy of Idusral ad Mechacal Egeerg, Qazv Brach, Islamc Azad Uversy, Qazv, Ira. E-mal: aa.alezadeh@gmal.com. Acceped 2 March, 202 I rece years, harmoy search (HS) algorhms have gaed sgfca aeos for her ables o solve dffcul problems egeerg. hs paper roduces he applcaos of HS veory maageme problems. Four specfc veory problems cera ad ucera evromes are cosdered: cosra mul-produc ewsboy problem wh fuzzy demad, ecoomc order quay problem wh advaced payme, b-obecve ewsboy problem wh fuzzy coss ad perodc revew problem wh sochasc perod legh ad dyamc demads. he compuaoal performace of he HS mehod o solvg hese four opmzao problems wll be compared wh oher mea-heursc algorhms such as geec algorhm (GA), smulaed aealg (SA), ad parcle swarm opmzao (PSO) mehods. HS mehod s show o acheve he bes performace. Key words: Iveory corol, ewsboy, ecoomc order quay, perodc revew sysem, harmoy search, sochasc veory model, fuzzy veory model. INRODUCION hs paper revews four praccal veory maageme problems. hese four problems cosder cera ad ucera evromes. hey are: cosra mulproduc ewsboy problem wh fuzzy demad, bobecve ewsboy problem wh fuzzy coss, ecoomc order quay problem wh advaced payme, ad perodc revew problem wh sochasc perod legh ad dyamc demads. I fac hs paper preses a comparso bewee some Mea-heursc algorhms used o solve he four classcal ad he mos famous ad mpora veory maageme problems. Harmoy search algorhm, geec algorhm, parcle swarm opmzao ad smulaed aealg are a group of mea heursc algorhms whch are used o solve he proposed models. hese mea heursc algorhms are commoly ad wdely use o solve he veory maageme problems. he ewsboy problem s a perodc veory maageme problem where uceray demad durg a sgle perod s cosdered. Whle he probably dsrbuo of demad s kow, he acual umber of demad wll o be kow ul afer he decso. he amou sold or delvered wll be he mmum order or demad quaes. I he examples, HS s used o derve he order quay uder maxmum beef ad servce level fucos. he classcal ecoomc order quay (EOQ) formula s he smples model for he cycle sock. he demad, orderg ad holdg coss are cosa over me, he bach quay may o be a eger, he whole bach quay s delvered a he same me ad o shorage are allowed. I hs case, HS s used o deermg he bes order quay of he exeded EOQ model. Le us cosder a veory sysem ha s corolled by a perodc revew polcy sead of coues revew polcy. Perodc revew meas ha he veory poso s speced a he begg of each perod ad ha all repleshmes are rggered a hese revews. A perod may, for example, be a day or a week. If he revew perod s shor, here s o dfferece bewee perodc ad coues revew. I such a case, coues revew resuls ca be used as a approxmao perodc revew sysems ad vce versa. Somemes, he revew perod cao be dsregarded. I hs case HS s used o oba he bes veory level. INVENORY MANAGEMEN he oal vesme veores s eormous ad he corol of capal ed up raw maeral, work--process, ad fshed goods offers a very mpora poeal for mproveme. Scefc mehods for veory maageme

2 alezadeh 9865 ca gve a sgfca compeve advaage. Advaces formao echology have drascally chaged he possbles o apply effce veory maageme echques. Moder veory maageme s based o advaced ad complex decso models whch may requre cosderable compuaoal effors (Axsaer, 2006). Iveory maageme s a commo problem o all orgazao ay secor of he ecoomy. he problems of veory do o cofe hemselves o prof-makg suos bu also ecouered by socal ad oprof suos. Iveores are commo farms, maufacurg plas, wholesaler warehouses, realer shops, hospals, zoos, uverses, ec (erse, 994). he obecve of veory maageme s o balace coflcg goals. Oe goal s o keep sock level dow o make cash avalable for oher purposes. he purchasg maager may wsh o ge volume dscous for larger baches order. he produco maager may prefer a large raw maerals veory o avod produco soppage due o mssg maerals. Hgher sock of fshed goods also provdes hgher servce level. Maagg veory volves effcely ad effecvely coordag boh he formao ad maerals flow he supply cha (Bradmare ad Zoer, 2007). Iveory maageme models help o acheve such goals. Due o uceray, safey socks are requred. wo ypes of uceray drecly mpac veory polcy. Demad Uceray s due o a ucera rae of sale or demad durg lead me. Performace cycle uceray volves veory repleshme me varao (Bowersox e al., 2007). Uceraes demad ad cycle legh ogeher wh lead-mes produco ad rasporao evably creae a eed for safey socks. Mos orgazaos ca reduce her veores usg veory maageme models. here are some veory maageme problems a supply cha. We show hree basc kds: Newsboy problem, ecoomc order quay model, ad sochasc perodc revew problem wh creasg ad decreasg demad. Newsboy problem I hs problem, he maeral maager orders a he begg of he perod based o hs esmao of he demad. If hs order s more ha he acual demad, holdg cos exss, oherwse los sales or backorder resuls. he obecve of hs model s o maxmze he expeced prof hrough order. I he real world, may producs have a lmed sellg perod, so he ewsboy problem s ofe used o ad decso-makg. I real world, may producs (fasho, sporg, ad servce dusres) have a lmed sellg perod, so he ewsboy problem s ofe used o ad decso-makg. he classcal ewsboy problem are wdely exeded o dffere obecves ad uly fucos, dffere dscou polces, mul-produc mul- cosra, mul-perod models ad fuzzy cosderaos, Avar ad Kusy, 990; Chug, 990; Akso, 979; Abdel-Malek ad Moaar, 2005; Masuyama, 2006; Alfares ad Elmorra, 2005; Lushu e al., 2002; J ad Shao, 2006; Shao ad J, 2006; alezadeh e al., 2009, 2008). Example : Mul-produc ewsboy wh fuzzy demad o defe a problem for mul-produc ewsboy wh fuzzy demad, we assume here are producs ad he ewsboy makes order a he begg of he perod. he demads are assumed o be fuzzy ad he ewsboy orders are based esmao. he orders ca be ake a he begg of each perod ad o opporuy s allowed o replesh he sock ha perod. he order quay should be a mulpler of a defed bach sze. he demad rae for he compay s assumed o be a ragular fuzzy umber. here are resrcos o warehouses capacy, budge ad servce level. Shorage los sale s allowed. he coss of shorage ad holdg ad he remaed ems are assumed o be deermsc. he holdg cos s a fraco of purchasg cos ad he goal s o deerme he opmal order quay of each produc. he model of hs problem s formulaed by alezadeh e al. (2009) as follows: : Z = P % ξ h ( Q % ξ ) X + PQ ˆ π ( % ξ Q ) ( X ) WλQ = = = = S. : Q % ξ - MX, =,2, L, Q < % ξ + M ( X ), =, 2, L, = f. m F ( Cλ ) Q W = = ( Cλ ) Q W = = α [ ( ξ + 2 ξ2 + ξ3 )] Q UL, =, 2, L, 4 Q = K m, =, 2, L, 0 < Q q λ, =, 2,..., q, λ, < Q qλ,,, =,2,...,, = 2,3, L, qλ < Q, =,2,..., X = 0,, =,2, L, λ =, =,2,..., = λ = 0,;, =,2,...,, =,2, L, Q, m 0, eger, =, 2, L, ; Where, M s a bg posve umber ()

3 9866 Afr. J. Bus. Maage. able. Bes resuls of obecve fuco by dffere hybrd Algorhms. Hybrd algorhms Producs' order quay mum prof ($) Smulaed aealg ad fuzzy smulao Geec algorhm ad fuzzy smulao Parcle swarm opmzao ad fuzzy smulao Harmoy search ad fuzzy smulao Sce he model s eger aure, reachg a aalycal soluo (f ay) o he problem s dffcul (Ge ad Cheg, 997). So we eed o use mea heursc algorhms. he proposed model for sx producs s solved usg mea-heursc approach, four hybrd ellge algorhms of harmoy search-fuzzy smulao (HS-FS) alezadeh ad Nak (2009), parcle swarm opmzao-fuzzy smulao (PSO-FS) alezadeh e al. (2009), smulaed aealg- fuzzy smulao (SA-FS) alezadeh e al. (2009), ad geec algorhm-fuzzy smulao (GA-FS) alezadeh e al. (2009). A comparso of he resuls able shows he PSO- FS hybrd algorhm performs beer ha he GA-FS ad SA-FS algorhms erms of he obecve fuco values, ad he proposed HS-FS mehod of hs research performs he bes. I he erm of CPU me he expeced values of SA-FS, GA-FS, PSO-FS ad HS-FS are respecvely 67, 59, 59 ad 52 secods showg HS-FS performs beer ha oher do. Example 2: B-obecve mul-produc ewsboy wh fuzzy cos Cosder a compay ha orders producs oly oce ad oly a he sar of a perod. he cusomer demad for each produc follows a Posso dsrbuo. here s o eforced cosra o he suppler o supply a order. he ere capacy of he warehouse s assged o he producs. he shorage ad holdg coss are fuzzy ad deployed a he ed of he perod ad crease quadrac fasho. he rasporao cos o carry he producs has wo compoes: fxed cos for each shpme ad varable cos for each u of he producs. Dscou for purchasg ems s allowed ad follows cremeal dscou rule. Sce he rasporao ad he order-processg mes are relavely small as compared o he cycle legh, he lead-me s cosdered equal o zero, whch s a commo pracce he ewsboy problems. Smlarly o example, he order quay of each produc should oly be eger mulples of packes ad shorage los sale s cosdered. he goal s o deerme he order quay of each produc such ha he cosras are sasfed ad boh he expeced prof ad servce rae are maxmzed. he model of hs problem s formulaed by alezadeh e al. (2009) as follows: Q + : ZP = P. X. fx ( x ).. ( ) + P Q f X x CW = X = 0 = X = Q = = : ZSR s.: Q λ X ˆ ˆ 2 e λ ( h ( Q X ) + h2 ( Q X ) ) X! = X = 0 λ X m 2 e λ ( ˆ ( ) ˆ π X Q + π2 ( X Q ) ) K Q kay k X! = X = Q + = k = Q ( Q X ) f X ( x ) λ = X = 0 = = f B F Q = V B, =, 2,..., q λ2 W q λ, =,2,..., M ( q q ) λ W ( q q ),, λ,, = 2,..., ad, =, 2,..., 0 W M λ, =, 2,...,,M s a bg umber 0 < f ˆ B fy = = ( k ) fy ˆ ˆ < f B kfy k ; k = 2,3, L, m m Y k =, Y k = 0, ; k =, 2,..., m k = λ λ2 L λ, =, 2, L, λ = 0,, =, 2, L ad, =, 2, L, B 0 ad eger, =, 2, L, Sce he model 2 s eger aure, reachg a aalycal soluo (f ay) o he problem s dffcul (Ge ad Cheg, 997). So we eed o use mea heursc algorhms. o solve he model wh e producs uder (2)

4 alezadeh 9867 able 2. Bes resuls of obecve fucos by dffere hybrd algorhms. Hybrd algorhms Smulaed aealg, fuzzy smulao ad goal programmg Geec algorhm, fuzzy smulao ad goal programmg Geec algorhm, fuzzy smulao ad pareo selecg Harmoy search ad fuzzy smulao ad goal programmg Producs' order quay mum prof ($) mum servce level mea-heursc approach, four hybrd ellge algorhms of harmoy search, fuzzy smulao ad goal programmg (HS-FS-GP) alezadeh ad Nak (2009), smulaed aealg, fuzzy smulao ad goal programmg (SA-FS-GP) alezadeh e al. (2009), alezadeh ad Nak (2009), geec algorhm, fuzzy smulao ad goal programmg (GA-FS+GP) alezadeh e al. (2009), ad geec algorhm, fuzzy smulao ad Pareo selecg (GA-FS+PS) alezadeh e al. (2009), are used. A comparso of he resuls able 2 shows he proposed HS-FS+GP mehod performs he bes. Resuls show ha HS has he bes opmal soluos. he prof ad servce level are $3250 ad respecvely. I should be also oed ha hybrd mehod of GA-FS-pareo selecg performs beer ha he hybrd mehod of GA-FS-GP ad SA-FS-GP. I he erm of CPU me he expeced values of SA-FS-GP, GA-FS-GP, GA- FS-PS ad HS-FS+GP are respecvely 22, 08, 08 ad 04 secods showg HS-FS+GP performs beer ha oher do. Ecoomc order quay model he classcal ecoomc order quay (EOQ) formula s he smples model for cycle sock. he square-rooformula for he ecoomc order quay (EOQ) has bee ced he veory leraure sce 95. hs formula s based o he assumpo of a cosa demad. he dscree case of he dyamc verso of EOQ was frs dscussed by Wager ad Wh (958). Regardg he couous-me dyamc EOQ models, Slver ad Meal (969) were he frs o sugges a smple modfcao of he classcal square-roo-formula he case of mevaryg demad. Hece, he classcal ecoomc order quay model s wdely exeded Dffere ways by Harga (994), Chug ad g (994), Km (997), L e al. (2000), Grubbsrom ad Erdem (999), alezadeh e al. (2008) ec. Example 3: Jo cosra EOQ wh advaced payme Ecoomc order quay (EOQ) model wh advaced payme s usually developed o purchase hgh-prce raw maerals. A o polcy of repleshmes ad prepaymes s employed o supply he maerals. he rae of demad ad lead me are ake o be cosa ad s assumed ha shorage does o occur. he cycle s dvded o hree pars; he frs par s he me bewee he prevous repleshme-me o he ex order-me ( 0 0 ), he secod par s he perod bewee o a payme-me ( lc ), ad he hrd par s he perod lc bewee o he ex repleshme-me. A he sar of he secod par ( 0 ), α% of he purchasg cos s pad. he ( α)% remag purchasg cos s pad a he sar of he hrd par ( lc ) (Fgure ). he cos of he model s purchasg wh cremeal dscou for each order, clearace cos, fxed-order cos, rasporao cos, holdg ad capal coss. Holdg cos s for o had veory ad capal cos s for capal ha s pad for he ex order. he cosras of he problem are space, budge ad upper lm for he umber of orders per year. Also lead-me s cosdered less ha a cycle me. he model of hs problem s formulaed by alezadeh e al. (2008) as follows:

5 9868 Afr. J. Bus. Maage. M : Z = C + C + C + C + C P C A H K P P P P P c A h D 2 3 kay k C Q C D lc 0 lc 0 2 h D h D L k = = = = = = = = ( ) + ( ) K P P P P P h D c 2 3 = A + kay k + CQ + + C ( lc 0) [ ( lc 0) ] 2 D + h D + h D L k = = = = = = = P S. : CQ B = = P f Q = N F P 0 < ˆ f m fy = P fy ˆ ˆ 2 < f m 2 fy 2 = M P ( K ) fy ˆ ˆ k < f m KfY k = D = m Q = Q + Q Q q λ2 Q q λ ( q q ) λ Q ( q q ) λ2 M 0 Q M λ Y + Y 2 + L + Y k = λ λ2 L λ λ = 0, ; Y k = 0, k =,2, L, K,, =,2, L, P,, =, 2, L, m λ λ2 λ [ ] L (3) Sce he model 2 s eger aure, reachg a aalycal soluo (f ay) o he problem s dffcul (Ge ad Cheg, 997). So we eed o use mea heursc algorhms. o solve he model uder mea-heursc approach, four hybrd ellge algorhms of harmoy search alezadeh e al. (2008), smulaed aealg, alezadeh e al. (2008), alezadeh e al. (2009a), geec algorhm alezadeh e al. (2008), alezadeh e al. (2009a), ad parcle swarm opmzao alezadeh e al. (2009a), are used. able 3 shows he proposed HS-FS+GP mehod performs he bes. Resuls show ha HS reached he opmal soluos wh bes obecve fuco value (Prof=24,820,000). I should be also oed ha parcle swarm opmzao performs beer ha he geec algorhm ad smulaed aealg approaches based o obecve fuco values. I he erm of CPU me he expeced values of SA, GA, PSO ad HS are respecvely 6, 7, 7 ad 5 secods showg HS performs beer ha oher do. Perodc revew model I mul-perodc veory corol models, he couous revew ad he perodc revew are he maor vasly used polces. However, he uderlyg assumpos of he proposed models resrc her correc ulzao realworld evromes. I couous revew polcy, he user has he freedom o ac a ayme ad replesh orders based upo he avalable veory level. Whle he perodc revew polcy, he user s allowed o replesh he orders oly specfc ad predeermed mes. he

6 alezadeh 9869 I Q D ROP 0 lc Q 0 lc Lead me Fgure. Iveory corol pcure. able 3. Bes resuls of obecve fuco by dffere algorhms. Hybrd algorhms Mmum cos ($) Smulaed aealg ,456,000 Geec algorhm ,956,000 Parcle swarm opmzao ,450,000 Harmoy search ,820,000 mul-perodc veory corol problems have bee vesgaed deph dffere researches by Chag Chag (2003), Bylka (2005), Chag (2006), Feg ad Rao (2007), Eya ad Kropp (2007) ad alezadeh e al. (2008, 2009a, 2009). coss assocaed wh he veory corol sysem are holdg, back-order, los-sales ad purchasg coss. Furhermore, he servce level of each produc, warehouse space ad budge are lmed ad he decso varables are eger dgs. he goal s o defy he veory levels each cycle such ha he expeced prof s maxmzed. alezadeh e al. (2009) roduced a veory maageme model wh sochasc perod legh wh varyg demad fucos. Fgures 2 ad 3 show he veory dagram decreasg demad case. I he secod case, he me perod bewee repleshmes s greaer ha he amou of me requred for he veory level o reach zero. he creasg ad decreasg demad fucos are gve Equaos 4 ad 5 by alezadeh e al. (2009), respecvely. Example 4: Sochasc perod legh wh dyamc demad α D ( ) = D ; α, D > 0 (4) Cosder a perodc veory corol model wh sochasc repleshme me. Le he me-perods bewee wo repleshmes be decal ad depede radom varables; case of shorage, a fraco s cosdered back-order ad a fraco as los-sale. he demads of he producs are dyamc ad varyg. he ( ) D = D ; D > he models of hs problem for creasg ad decreasg demads are formulaed by alezadeh e al. (2008) as follows: (5)

7 9870 Afr. J. Bus. Maage. D α α : Z( R) = ( P W ) D d d + R + β D d d = M 0 M D D M S. : + D D D α + D α + h ( R ) d. d ( ). 0 α + d = M α 0 + M D M α π ( ). ( ) ˆ β D d f ( ). d π β D d d = D = D D D M = = f R W Q F B ( ) α + α + R D R d SL : =, 2, L, M d 0, Ieger : =, 2, L, (6) Ad D : Z ( R ) = ( P W ) D d d + R + β D d d = M 0 M D D M S. : = D D h ( R ) d. d + d. = M 0 M D 0 M π. ˆ β D d d π ( β ) D d. d = D D M = D D M = f R W Q F B ( ) α + α + R D R d SL : =, 2, L, M 0, Ieger :, 2,, d = L (7) Sce he model 2 s eger aure, reachg a aalycal soluo (f ay) o he problem s dffcul (Ge ad Cheg, 997). So we eed o use Mea heursc algorhms. o solve he models uder mea-heursc approach, four hybrd ellge algorhms of harmoy search (alezadeh e al., 2009), smulaed aealg (alezadeh, 2008, 2009), geec algorhm alezadeh (2008, 2009), ad parcle swarm opmzao (alezadeh e al., 2009) are used. A comparso of he resuls ables 4 ad 5 for creasg ad decreasg demad show ha HS mehod performs he bes. From ables 4 ad 5, for creasg ad decreasg demad fucos, erm of obecve fuco's values HS performs beer ha oher algorhms do. Smlarly, oher examples, parcle swarm opmzao mehod has a beer soluo ha geec algorhm ad smulaed

8 alezadeh 987 Fgure 2. Preseg he veory cycle whe M D. < Fgure 3. Preseg he veory cycle whe D. able 4. Bes resuls of obecve fuco by dffere algorhms for creasg demad. Hybrd algorhms Producs' maxmum veory level mum prof ($) Smulaed aealg ,239 Geec algorhm ,57 Parcle swarm opmzao ,833 Harmoy search ,90

9 9872 Afr. J. Bus. Maage. able 5. Bes resuls of obecve fuco by dffere algorhms for decreasg demad. Hybrd algorhms Producs' maxmum veory level mum prof ($) Smulaed aealg ,320 Geec algorhm ,270 Parcle swarm opmzao ,644 Harmoy search ,950 aealg approaches do. I he erm of CPU me he expeced values of SA, GA, PSO ad HS are respecvely 8, 8, 7 ad 7 secods showg HS performs beer ha oher do. SUMMARY AND CONCLUSION hs paper has vesgaed varous HS applcaos veory maageme. Specfc examples clude cosra mul-produc ewsboy problem wh fuzzy demad, o ecoomc order quay problem wh advaced payme, cosra b-obecve ewsboy problem wh fuzzy coss, ad perodc revew problem wh sochasc perod legh ad dyamc demads. he goals of he frs ad hrd examples were o defy he order quay of each produc ucera ad cera evrome respecvely. he goals of secod example were o deerme he order quay of each produc so as o maxmze he oal expeced prof ad servce level. I he fourh example, he veory maager deermed he opmal veory level of each produc ad maxmzed he oal expeced prof. o solve he models, four kds of mea-heursc algorhms or mea-heursc hybrd algorhms: harmoy search, parcle swarm opmzao, geec algorhms, smulaed aealg, fuzzy smulao, goal programmg ad Pareo selecg were used. Compuaoal resuls showed ha HS, hybrd of HS ad fuzzy smulao, ad hybrd mehod of HS, fuzzy smulao ad goal programmg performed he bes, based o obecve fuco values respec o oher kd of he algorhms. Oher applcaos of HS veory ad supply cha maagemes ca be exeded o cosder prcg problems. REFERENCES Abdel-Malek LL, Moaar R (2005). A aalyss of he mul-produc ewsboy problem wh a budge cosra. I. J. Prod. Eco., 97: Alfares HK, Elmorra HH (2005). he dsrbuo free ewsboy problem: Exeso o he shorage pealy case. I. J. Prod. Eco., 93-94: Avar M, Kusy M (990). Rsk veory models: revew ad mplemeao. Eg. Coss Prod. Eco., 9: Akso AA (979). Iceves, uceray, ad rsk he ewsboy problem. Decs. Sc., 0: Axsaer S (2006). Iveory corol. Secod ed., Sprger, New York, USA. Bowersox DJ, Closs DJ, Cooper MB, (2007). Supply cha logscs maageme. Secod Ed., Mc Graw. Hll, New York, USA. Bradmare P, Zoer G (2007). Iroduco o dsrbuo Logscs. Frs ed., Joh wly & Sos, New Jersey, USA. Bylka S (2005). urpke Polces for Perodc Revew Iveory Model wh Emergecy Orders. I. J. Prod. Eco., 93-94: Chag C (2003). Opmal Repleshme for a Perodc Revew Iveory Sysem wh wo Supply Modes. Eur. J. Oper. Res., 49: Chag C (2006). Opmal Orderg Polces for Perodc-Revew Sysems wh Repleshme Cycles. Eur. J. Oper. Res., 70: Chug KH (990). Rsk veory models: he case of he ewsboy problem opmaly codos. J. Oper. Res. Soc., 4: Chug KJ, g PS (994). O repleshme schedule for deerorag ems wh me proporoal demad. Prod. Pla. Corol, 5: Eya A, Kropp D (2007). Effecve ad Smple EOQ-lke Soluos for Sochasc Demad Perodc Revew Sysems. Eur. J. Oper. Res., 80: Feg K, Rao US (2007). Echelo-Sock (R-) Corol wo Sage Seral Sochasc Iveory Sysem. Oper. Res. Le., 35: Ge M, Cheg R (997). Geec Algorhm ad Egeerg Desg. Joh Wley & Sos, New York, U.S.A. Grubbsrom RW, Erdem A (999). he EOQ wh backloggg derved whou dervaves. I. J. Prod. Eco., 59: Harga M (994). he veory lo-szg problem wh couous mevaryg demad ad shorages. J. Oper. Res. Soc., 45: J X, Shao ZH (2006). Model ad algorhm for b-level ewsboy problem wh fuzzy demads ad dscous. Appl. Mah. Compu., p. 72: L C, a B, Lee WC (2000). A EOQ model for deerorag ems wh me-varyg demad ad shorages. I. J. Sys. Sc., 3: Lushu L, Kabad SN, Nar KPK (2002). Fuzzy models for sgle-perod veory problem. Fuzzy ses Sys., 32: Masuyama K (2006). he mul perod veory problem. Eur. J. Oper. Res., 7: Shao ZH, J X (2006). Fuzzy mul-produc cosra ewsboy problem. Appl. Mah. Compu., 80: 7-5. Slver EA, Meal HC (969). A smple modfcao of he EOQ for he case of a varyg demad rae. Prod. Iveory Maage., 0: alezadeh AA, Aryaezhad MB, Nak SA (2008). Opmzg Mul- Produc Mul-Cosra Iveory Corol Sysems wh Sochasc Repleshme. J. Appl. Sc., 8: alezadeh AA, Moghadas H, Nak SA, Efekhar A (2008). A EOQ- Jo Repleshme Polcy o Supply Expesve Impored Raw Maerals wh Payme Advace. J. Appl. Sc., 8: alezadeh AA, Nak AA, Shaf N, Ghavamzadeh-Mebod R, (2009) A Parcle Swarm Opmzao Approach for Cosra Jo Sgle Buyer-Sgle Vedor Iveory Problem wh Chageable Lead-me ad (r,q) Polcy Supply Cha. I. J. Adv. Mauf. Sys., 5(9-2): alezadeh AA, Nak S, Aryaezhad MB (2009). A Hybrd Mehod of Pareo, OPSIS ad Geec Algorhm o Opmze Mul-Produc Mul-Cosra Iveory Corol Sysems wh Radom Fuzzy Repleshmes. Mah. Compu. Model., 49: alezadeh AA, Nak S, Hosse SV (2008). Mul producs, Mul

10 alezadeh 9873 Cosras Newsboy Problem wh Bach Order ad Dscou. Asa J. Appl. Sc., : alezadeh AA, Nak S, Hosse V (2009). Opmzg Mul Produc Mul Cosras B-obecve Newsboy Problem wh dscou by Hybrd Mehod of Goal Programmg ad Geec Algorhm. Eg. Opmz., 4: alezadeh AA, Nak SA (2009). A Hybrd Mehod o Opmze B- Obecve Sgle Perod Newsboy Problem wh Fuzzy Cos ad Icremeal Dscou. J. Id. Eg., Qazv Islamc Azad Uversy, 3(3): -4. alezadeh AA, Nak SA, Aryaezhad MB (2009). Mul-Produc Mul- Cosra Iveory Corol Sysems wh Sochasc Repleshme ad Dscou uder Fuzzy Purchasg Prce ad Holdg Coss. Am. J. Appl. Sc., 6(): -2. alezadeh AA, Nak SA, Aryaezhad MB, (2009a) Mul-produc mul-cosra veory corol sysems wh sochasc perod legh ad oal dscou uder fuzzy purchasg prce ad holdg coss. I. J. Sys. Sc., 4(0): alezadeh AA, Nak SA, Hosse SV (2009). Opmzg mulproduc mul-cosra b-obecve ewsboy problem wh dscou by hybrd mehod of goal programmg ad geec algorhm. Eg. Opmz., 4: alezadeh AA, Nak SA, Seyed Javad MH. (2009). a harmoy search algorhm o opmze chace cosra sochasc veory sysems wh dyamc demad. J. Mauf. Sys., I Press. alezadeh AA, Shavad H, Amr M (2009). Jo Replesh-up-o mul producs EOQ wh fuzzy rough demad. 6 h eraoal dusral egeerg, ehra, Ira. alezadeh AA, Shavad H, Asla S (2009). Developg a Mulproducs Newsboy Problem wh a Hybrd Mehod of Fuzzy Smulao ad Smulaed Aealg, Proceedg of 39 h Ieraoal Coferece o Compuers ad Idusral Egeerg, Pars, Frace, 6-8 July erse RJ (994). Prcples of veory ad maerals maageme. Fourh ed., Prece Hall, New Jersey, USA. Wager HM, Wh M (958). Dyamcal verso of he ecoomc lo sze model, Maag. Sc., 5: