Dynamic Multi-Level Capacitated and Uncapacitated Location Problems: an approach using primal-dual heuristics

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1 Dynamc Mul-Leel Caacaed and Uncaacaed Locaon Problems: an aroach usng rmal-dual heurscs JOANA DIAS (), M. EUGÉNIA CAPIVO () AND JOÃO CLÍMACO () ()Faculdade de Economa and INESC-Combra Unersdade de Combra A. Das da Sla, Combra Porugal () Unersdade de Lsboa, Faculdade de Cêncas Cenro de Inesgação Oeraconal Camo Grande, Bloco C6, Pso Lsboa Porugal Absrac: In hs aer seeral dynamc mul-leel locaon roblems are formulaed as mxed-neger lnear rograms. Boh uncaacaed and caacaed ersons of he roblem are suded. he models resened are more comlee han he ones known from he leraure: hey are dynamc and consder he ossbly of a facly beng oen, closed and reoen more han once durng he lannng horzon. hey may nclude boh uer and lower lms on he used caacy of each facly and may also consder he suaon where here s no flow conseraon n he nermedae facles. Prmal-dual heurscs were deeloed o sole effcenly he roosed models. Comuaonal resuls are resened and dscussed. Inroducon Mul-leel locaon roblems hae been wdely suded n he leraure. Auhors sudyng and wrng abou hese roblems desgnae hem dfferenly: herarchcal, mul-leel, mul-echelon, mul-sage, ec. Generally, he desgnaon of he roblem ndcaes he maxmum number of leels consdered: k-herarchcal, k-leel, k-echelon locaon roblems refer o roblems wh, a mos, k leels of facles. he models deeloed n hs research reor are desgnaed by mul-leel, because s he auhors undersandng ha hs s he mos general desgnaon (a mul-leel locaon roblem may no necessarly reresen a herarchcal locaon roblem; n he laer case facles hae o be organzed n a herarchcal srucure, whle n he frs case hs s no necessarly he case). hs research was arally suored by research roec POCI/ISFL-/5 and POCI/MA/39/00.

2 here are seeral examles n our daly les ha show he morance of consderng mul-leel facly locaon roblems: he herarchcal healh serce sysem, he herarchcal educaon sysem, he mul-leel srucure of bank and os-offces organzaons, ec. hese facly srucures hae some common feaures and some moran dfferenang characerscs. One can say ha a mul-leel facly locaon model s needed wheneer he facles o be locaed can be groued n ses (leels) hang dfferen characerscs (by offerng dfferen serces o clens, for nsance) and nerac wh each oher so ha s no ossble o locae facles n each se ndeendenly from he ohers. he exsng lnks beween he dfferen knds of facles can be mlc (n he form of a consran ha lms he global budge ha can be sen globally), or exlcly deermnng he flow of clens beween facles (Daskn, 995). Narula (984) aems o classfy he herarchcal locaon-allocaon roblems. He dfferenaes he roblems hrough he facly herarchy (successely ncluse a facly n leel m offers all serces of facles n leels o m- lus serces of leel m; successely excluse a facly n leel m offers serces unque o ); arc and node flow dsclne (flow s negraed f s from any lower leel o any hgher leel or s dscrmnang f he flow s from leel m o leel m; a unah nework s a nework where he ose degree of eery node s less han or equal o one, whereas n a mulah nework he ose degree of a leas one node s greaer han one). o comleely secfy an herarchcal locaon-allocaon roblem s needed o sae he number of leels, he ye of facly herarchy, he arc and flow dsclne. Neerheless he classfcaon scheme resened s no exhause, as he auhor remarks. Daskn (995, ) ges he basc noons of herarchcal facles, and descrbes some models: medan-based, coerage-based herarchcal locaon formulaon and some exensons o hose cases. he models descrbed n he leraure are, almos exclusely, sac roblems. Some consder he uncaacaed ohers he caacaed erson of he mul-leel roblem. Moore and ReVelle (98) descrbe a real case of successely ncluse herarchcal serce locaon roblem. hey consder as obece he mnmzaon of he oulaon ha lacks access o one or more comonens of serce. A oulaon s coered by a gen leel of serce f some member of he facly herarchy elgble o rode ha serce s locaed whn a maxmum esablshed dsance. he lnear relaxaon of he roblem s soled, followed by a branch and bound rocedure wheneer necessary. en e al (983) deelo and sole wo models wh alcaons n he herarchcal healh facly locaon-allocaon roblem. cha and Lee (984) resen a branch and bound algorhm based on a dual ascen rocedure (smlar o he one deeloed by Erlenkoer, 978) for he mul-leel uncaacaed facly locaon roblem. en years laer, a aer by Barros and Labbé (994) roe ha he mul-leel uncaacaed facly locaon roblem s no submodular, and queson he aldy of he branch and bound rocedure descrbed n cha and Lee

3 (984). Ro and cha (984) descrbe a branch and bound algorhm for resoluon of he wo-leel uncaacaed facly locaon wh addonal consrans ha reresen he adunc relaonsh of some warehouses o a ceran lan (f a lan s oen, here s a se of warehouses ha hae o be also oen). Narula and Ogbu (985) use lagrangean relaxaon wh subgraden omsaon o sole a wo-herarchcal uncaacaed successely ncluse locaon roblem. he model consders he locaon of facles n leel and facles n leel. Only a fracon of he demand from leel facles s referred o leel facles. Gao and Robnson (99) nesgae he use of dual-based rocedures smlar o he ones deelo by Erlenkoer (978) n he resoluon of he wo-echelon uncaacaed facly locaon roblem. Aardal e al (996) nesgae srucural roeres of he uncaacaed wo-leel facly locaon roblem. wo dfferen formulaons are suded, ha use dfferen decson arables (he auhors call hese wo formulaons he sngle and he mul commody flow formulaons). Edwards (00) sudes seeral roeres and descrbes aroxmaon algorhms for he mul-leel facly locaon roblem. Bumb (00) and Zhang and Ye (00) descrbe aroxmaon algorhms for he wo leel uncaacaed facly locaon roblem wh a maxmzaon obece. Galão e al (00) reas a 3-leel successely ncluse facly locaon model aled o a case sudy. here s an uer bound on he maxmum number of facles o locae a each leel k, k=,,3. he auhors do no consder he exsence of fxed oenng coss. wo heurscs are deeloed o fnd feasble soluons o he roblem. Eseo e al (003) rea he maxmal coerng wo-leel locaon roblem. A combned Lagrangean-surrogae relaxaon s mbedded no a subgraden omzaon algorhm o fnd lower and uer bounds o he omal soluon. Some references on caacaed mul-leel roblems are also aalable. Ean e al (99) resen a mxed-neger lnear rogrammng model, whch allows dfferen herarchcal relaonshs o be smulaneously resen, caacy consrans o be laced boh on serce yes and serce grous and he consderaon of boh fxed and arable coss. he model s aled o seeral leraure-based roblems and also o a new large-scale roblem ha fully llusraes he model caables. Bloemhof-Ruwaard e al (996) resen a caacaed erson of he wo-leel facly locaon roblem. hey consder wo dfferen model formulaons, and comare he qualy of he lower bounds obaned by her lnear relaxaons wh he lower bounds obaned wh a lagrangean relaxaon. he auhors use LP round-off heurscs and also sequenal caacaed facly locaon heurscs o fnd feasble rmal soluons. raganalerngsak e al (997) deelo and comare sx dfferen lagrangean relaxaons o he wo-echelon, sngle source caacaed facly locaon roblem. he lagrangean relaxaons are used whn a subgraden omsaon algorhm, and feasble soluons are consruced by heurscs based on a general assgnmen roblem formulaon. In a followng aer (raganalerngsak e al, 000), a branch and bound 3

4 mehod based on he mos effcen lagrangean relaxaon s resened. Prkul and Jayaraman (998) deelo a mul-commody, caacaed, wo-leel facly locaon model. hey consder he locaon of lans and warehouses. he roblem s soled heurscally, consrucng a rmal feasble soluon from he soluon of a lagrangan relaxaon (whn a subgraden mehod omsaon). Chardare (999) reas boh uncaacaed and caacaed wo-leel locaon roblems n he elecommuncaons nework-lannng conex. Lagrangean relaxaon and smulaed annealng are boh used o fnd feasble soluons. Klose (995, 998, 000) consders a wo-leel sngle sourcng caacaed facly locaon roblem (clens are sered from deos ha, n urn, are sered by lans). he locaon decson arables consder only he roblem of locang deos ( s consdered ha lans are already locaed a fxed ses). he roblem s soled usng lagrangean relaxaon followed by a heursc rocedure. he auhor res o sole he same roblem usng a LP-based heursc (Klose, 999): nroducng seeral ald nequales moaed by he subroblems conaned n he mul-leel roblem, he LP relaxaon s srenghen and soled. A heursc rocedure bulds an admssble soluon from he omal lnear soluon calculaed. Seeral comuaonal ess where erformed o assess he erformance of boh resoluon aroaches. Jayaraman e al (003) consder a herarchcal facly srucure where clens hae wo dfferen knds of demand: demand for basc and secalzed serces. he model resened nends o maxmze he oal demand coerage, allocang leels of serces o oen facles and consderng caacy consrans. A lagrangean relaxaon followed by a rmal heursc s deeloed, and he ably of he aroach s demonsraed hrough he resuls of exense comuaonal exermens. Dynamc mul-leel locaon roblems are descrbed n a few number of more recen aers. Melachrnouds and Mn (000) resen a real case of dynamc relocang an exsng facly ha belongs o a wo-leel facles srucure. he ranson beween he exsen and he new locaon has o be done gradually. he auhors consder boh caacy and budge consrans and more han one obece funcon. No dedcaed rocedure s deeloed: he mul-obece mxed-neger lnear roblem s soled usng a general soler. Hnoosa e al (000) model he dynamc woechelon mulcommody caacaed lan locaon roblem. he model consders he ossbly of oenng a facly a he begnnng of any me erod (remanng oen unl he end of he lannng horzon), and closng an already exsng facly a he end of any me erod (remanng closed unl he end of he lannng horzon). he calculaon of admssble soluons s done usng heurscs based on a lagrangean relaxaon. Canel e al (00) deelo a model for he same knd of roblem, bu consder he ossbly of a facly beng oen n more han one me erod, no sequenally. he auhors consder reoenng and closng coss, bu resen a non-lnear obece funcon. he aer descrbes an algorhm o sole he roblem. 4

5 here are oher neresng asecs of mul-leel facly locaon roblems ha hae also been reaed by some auhors. Hodgson (98) sudes a dfferen roblem ha can be consdered a knd of mul-leel locaon roblem: he locaon of ublc facles nermedae o he ourney o work. I s consdered ha clens hae o rael daly from home o work. he obece s o locae ublc facles such ha he exra me needed o rael o hose facles s mnmzed. Madsen (983) sudes he roblem of combned locaon-roung roblems n a sysem comosed of clens, deos and, a mos, one facory. Serra e al (99) sudy he roblem of locang facles wh a herarchcal srucure when here s comeon n he regon of neres. he model deeloed allows for boh he locaon of new facles and he relocaon of exsng facles. Marano and Serra (00) sudy a herarchcal locaon-allocaon model where he congeson roblem s reaed exlcly: here are suaons where clens hae o wa on queue for some me before beng sered. he auhors descrbe a b-leel heursc for consrucon of feasble soluons, and resen he resuls of comuaonal ess erformed. In hs research reor seeral dynamc mul-leel locaon roblems are addressed. In secon he uncaacaed erson of he roblem s descrbed, n secon 3 he caacaed erson wh maxmum caacy resrcons and, n secon 4, he caacaed erson wh maxmum and mnmum caacy resrcons bu wh no flow conseraon a he nermedae facles. In secons o 4, rmal-dual heurscs are descrbed ha can calculae admssble soluons for he corresondng roblem, based on he work of Erlenkoer (978), Van Roy and Erlenkoer (98) and Gugnard and Selberg (979). All roblems reaed n hs research reor hae he followng characerscs ha dsngush hem from he roblems usually referred o n he leraure:. he roblems are dynamc and consder ha a facly can be oen, closed and reoen more han once durng he lannng horzon. I s ossble o consder exlcly dfferen fxed oenng and reoenng coss (ha are, mos of he mes, clearly dfferen). I s ossble o consder he exsence of already oen facles. I s ossble o esablsh mnmum funconng me nerals for a gen facly (n some suaons, a facly ha s oen n me erod should be oen for a mnmum number of me erods before beng closed).. he models can be used for boh successely ncluse and excluse herarchcal roblems. he model s also ald for all he arcs and nodes flow dsclnes defned n Narula (984). 3. he model can deal wh ahs of facles wh a dfferen number of arcs. In a k-leel facly locaon roblem, s ossble o consder ahs wh a number of arcs from o k. hs s an neresng feaure for boh successely ncluse and excluse roblems. Imagne, for nsance, a healh facly serce sysem. Paens are adsed o go o her neares healh cenre, from here hey may be sen o a regonal hosal, and from here o a cenral hosal (wheneer s 5

6 needed). Somemes aens refer o go drecly o regonal or cenral hosals. In he models here resened, all hese suaons can be consdered. 4. I s ossble o consder more han one ah consued by exacly he same facles, bu wh dfferen characerscs. hs s an moran feaure, esecally f dealng wh more han one obece funcon. Imagne a roblem where here are wo obeces: mnmzng oal cos and oal rael me. Consder a ah consued by locaons. One can consder he locaons conneced by a hghway and by a naonal road. he frs oon wll be more exense bu faser, he second oon wll be less exense bu slower. Boh ahs can be smulaneously consdered n he model formulaon. In secon 5 some conclusons and fuure work drecons are addressed. Mul-Leel Uncaacaed Locaon Problem Fgure llusraes a ossble -leel nework confguraon. Clens are assgned o ahs, and no o sngle facles Fgure : Examle of a -leel facly nework. he symbol reresens clens, and reresens facles. Consder he followng defnons: J = {,...,,, n} se of ndexes corresondng o he clens locaons; I = {,...,,, m} se of ndexes corresondng o facles ossble locaons; P = {,, q } se of all ossble ahs; P( ) = { P: belongs o ah }; = number of me erods consdered n he lannng horzon ( ); K = maxmum number of leels n he mul-leel facly srucure; K wll also be he maxmum ah lengh. 6

7 c = cos of fully assgnng clen o ah n erod ; FA = fxed cos of oenng a facly a he begnnng of erod, and closng a he end of erod (he facly wll be n oeraon from he begnnng of o he end of ); FR = fxed cos of reoenng a facly a he begnnng of erod, and closng a he end of erod (he facly wll be n oeraon from he begnnng of o he end of ); and le us defne he arables: a r f facly s oen a he begnnng of erod and says oen unl he end of erod = 0 oherwse f facly s reoen a he begnnng of erod and says oen unl he end of erod =, > 0 oherwse f clen s assgned o ah durng erod x =. 0 oherwse he defnon of arables x was moaed by he work of cha and Lee (984). he man dfference has o do wh he fac ha each ah can hae any number of arcs from o K, whle n he referred o aer all ahs are of exacly he same lengh. Each ah can be consued by one or more facles. Each facly can belong o one or more ahs. Consder he wo-leel examle deced n fgure. All ahs reresened are ald. A ah wll be reresened by an ordered se (,,, k ), wh k K. If clen s assgned o ah, hs means ha clen s sered by facly, hen goes o facly, and so on, unl reachng facly k. Defnon : A ah s sad o be oen durng erod f and only f all facles belongng o are oen durng erod. Defnon : A ah s sad o be arally oen f and only f s no oen and a leas one facly belongng o s oen durng erod. Defnon 3: A ah s sad o be closed durng erod f and only f all facles belongng o are closed durng erod. I s no necessary o consder all ossble alues for (,),. If, for nsance, a facly has o be oerang for, a leas, s me erods erods afer beng oen, hen only ars (,) wh s should be consdered. 7

8 . Prmal Problem he dynamc, uncaacaed, mul-leel locaon roblem can be formulaed as DUMLP: DUMLP Mn c x = FA a > = FR r ( ) subec o: x =,, ( ) ( a r ) x 0 = = = = a = =,,,, P( ) ( 3 ) a r 0,, > ( 4 ) =, ( 5 ) ( a r ) = = a,x r { 0, },,, ( 6 ) { 0, },,,,, >, he obece funcon mnmzes he oal fxed and assgnmen coss. Consrans () guaranee ha, n eery me erod, each clen s fully assgned o exacly one ah; consrans (3) assure ha, n eery me erod, a clen can only be assgned o oen ahs; consrans (4) guaranee ha a facly can only be reoen a he begnnng of erod f has already been oen earler and s no n oeraon a he begnnng of erod ; consrans (5) mose ha a facly can only be oen once durng he lannng horzon; consrans (6) assure ha, n eery me erod, only one facly can be oen n each locaon. Consrans (5) and (6) need o be consdered exlcly only when here are negae fxed coss. Consrans (3) could be relaced by he aggregaed consrans: ( 7 ) ( a r ) x 0 = = P( ),, ( 3 ). Dual Problem and Comlemenary Condons Mullyng consrans (5) and (6) by -, and assocang dual arables wh consrans (), dual arables w wh consrans (3), dual arables u wh consrans (4), dual arables ρ 8

9 wh consrans (5 ) and dual arables π wh consrans (6 ), he dual roblem of DUMLP can be formulaed as D-DUMLP: D-DUMLP Max ρ π ( 8 ) subec o: w c,,, ( 9 ) w u = P ( ) = w = P( ) w ρ π u, u,, π 0 = π = FR FA,,, =, L, (0),, >, =, L, () ρ,,,, P( ) Consderng w = max{ 0, c } condensed formulaon s obaned: CD-DUMLP Max subec o: η, wh η = and η 0,,, P(), an equalen ρ η = P ( ) η = P ( ) u SA SR π { 0, c } FA u ρ max π,,, =, L, () = { 0, c } FR u max π,, >, =, L,, ρ, π 0,,, Le us defne: = FA = FR u = u { SA SR } ρ = π = π η = = P ( ) η = P ( ) = { 0, c } (3) max,,, =, L, (4) { 0, c } max,, >, =, L, ( 5 ) S = mn,,,, =, L, ( 6 ) Consderng he rmal roblem DUMLP and s dual CD-DUMLP, he followng comlemenary condons hold f n resence of omal rmal and dual soluons o he resece roblems (when here s no dualy ga): 9

10 ( a ) r x w = 0, = = = = = =,, P(), ( 7 ) a r u = 0,, > ( 8 ) = a ρ = 0, ( a ) r π = 0, = =, ( 9 ) ( 0 ) SA = 0,,, =,..., ( ) a SR = 0,,, =,..., ( ) r.3 Prmal-Dual Heursc he rmal-dual heursc ha has been deeloed bulds admssble rmal soluons based on admssble dual soluons o roblem CD-DUMLP, ryng o force he sasfacon of he comlemenary condons. If he heursc calculaes a ar of admssble and comlemenary rmal and dual soluons, hen he rmal omal soluon has been found. When hs s no ossble, he bes dual soluon found wll esablsh a ald lower lm o he omal alue of he rmal obece funcon. he heursc funconng scheme s as follows:. Inalsaon of dual arables;. Dual Ascen Procedure for dual arables 3. Prmal Procedure; ; 4. Dual Adusmen Procedure for dual arables ρ. If he dual soluon s changed go o ; 5. Reea he Prmal-Dual Adusmen Procedure for arables unl here s no mroemen n he dual obece funcon alue; 6. Dual Adusmen Procedure for dual arables ρ. If he dual soluon s changed go o ; 7. Dual Ascen Procedure for dual arables 8. Dual Descen Procedure for dual arables u. If he dual soluon s changed go o ; u. If he dual soluon s changed go o ; 9. Dual Adusmen Procedure for arables π. If he dual soluon s changed go o. hs heursc funconng scheme s equal o he funconng scheme resened n Das e al (004b). In fac, he only rocedures ha are dfferen from he ones descrbed n Das e al (004b) are he Dual Ascen Procedure and he Prmal-Dual Adusmen Procedure for arables Prmal Procedure. For hs reason, only hese hree rocedures are gong o be descrbed here., and he 0

11 .3. Dual Ascen Procedure for arables hs rocedure res o ncrease all dual arables, J, J J. When hs rocedure s execued n se of he rmal-dual heursc, hen J s equal o J. Oherwse, he se J s defned before hs rocedure s execued. hs rocedure s a sraghforward adaaon of he one descrbed n Van Roy and Erlenkoer (98). he only dfference s n he udang se of slacks SA and SR wh such ha more han one ah P( ), slacks c SA and and P( ). I s neresng o noe ha f c for SR, wh, wll be decremened more han once (hs s a comleely dfferen behaor, when comared wh he sngle-leel case (Erkenkoer, 978; Van Roy and Erlenkoer, 98). Dual Ascen Procedure for Varables. Consder any admssble nal soluon { } k each (, ), defne k(,) = mn{ k : c }. If q. J. (, ) (, ) ; ; δ 0 3. If (, ), hen go o 7. such ha c ( ) S k c (, ),,, 0, = hen (, ) k(, ) k.,,. For S 4. mn : (). If = 0, go o 7. η P c k ( ) 5. If, k (, ) > c hen c ; δ ; k(, ) k(, ). 6. For each facly, SR SR η and SA SA η,.. 7. If # J,(, ) (, ) ; q q q q. Go o If δ = go o. Else SOP. () P c () P c he calculaon of he η arameers can be done n seeral dfferen ways. One ery smle way s o consder η = n, where n s equal o he number of serces belongng o ah. Anoher calculaon rocedure s moaed by he analyss of se 4 of he dual ascen rocedure. I s sraghforward o conclude ha η should ake smaller alues for serces such ha he quoen beween ' mn{ S } S =,, and he number of ahs ncludng such ha c s smaller. hs rocedure s more me consumng han he frs, bu, n general, calculaes beer dual soluons.

12 Calculaon of η arameers. ;. If c hen go o 3. Else go o If n =, hen η, P(). Go o If P( ) hen go o 6, else go o 7., f c 6. e, n e, η 0, oherwse P() 7.. If > m, hen go o 8. Else go o 5. η 8. D η. Udae η,. D 9.. If > q hen so. Else go o. S mn. n Examle : Consder a roblem wh four clens, wo me erods and hree serces organzed n wo leels as deced n fgure. 3 Fgure : Clen; Serce. he admssble ahs are: =(,3), =(,3) and 3 =(3). ables and show he assgnmen coss for each me erod and able 3 shows he fxed (re) oenng coss. Varables are nalzed as showed n able 4. he dual ascen rocedure for arables begns by consderng = and =. For ah equal o, he rocedure calculaes 0 8 η = = and η = =.. Varable can be ncreased o 8. 8 For arable, he rocedure calculaes η = 0 56 and η = If. 3. was ncreased by 7 ( ) k, ( c ), hen slack SA would become equal o ha s less han zero, so s calculaed as beng equal o. No more dual arables are changed because slacks become as deced n able 5.

13 able able FA FA FA FR able FA FA able 4 able 5 3. For equal o and equal o, η = η = 0 5. If he alue (, ) k c = 4 was consdered, hen slack SA would be changed o 0.5 4, ha s less han zero. For hs reason s calculaed as beng equal o. Afer he dual ascen rocedure, he fnal resul s deced n ables 6 and FA FA FA FR able 6 able 7.3. Prmal-Dual Adusmen Procedure for arables he Prmal-Dual adusmen rocedure for arables deecs olaons of he comlemenary condons (7), and decreases he alues of some arables, ncreasng slacks and allowng oher arables o ncrease. hs rocedure res o reduce he number of comlemenary condons olaons and, smulaneously, mroe he alue of he dual obece funcon. 3

14 Comlemenary condon (7) wll be olaed f here exss a leas wo oen or arally oen ahs and such ha c > and c >. Noce ha sn necessary ha ahs are oen. hey only need o be arally oen o olae he comlemenary condons. Dmnshng arable wll ncrease all slacks SA and SR, wh, such ha > c, wh P(). If here are wo oher dual arables blocked exclusely due o slacks ha are ncreased, hen s ossble o mroe he dual obece funcon alue. I s neresng o noce ha he olaon may occur due o a sngle facly ha belongs o more han one ah (n he sngle case here had o be a leas wo dfferen facles noled). In he sngle-leel case, he dual ascen algorhm res frsly o ncrease arables blocked exclusely by slacks corresondng o a sngle facly (Das e al, 004a). he dual adusmen rocedure n he mul-leel case res frsly o ncrease all arables blocked exclusely due o slacks ha are gong o be ncreased (een f corresondng o more han one facly). Consder he followng defnons: I * = { (,, ): S = 0} I * = { : (,, ) I * and } I = { : facly s oen durng erod } I * = * { : (, ) wh and P( ) (,, ) I and c } { : I and P() such ha c } I = > { : such ha I and c } * {( ) * ( ) *, : I < < } {( ) I and, γ, I, I, γ or γ, : I I } { c : c } P = > J = c = max > Se I * corresonds o (,,), such ha SA and/or each clen, all oen or arally oen ahs such ha SR are equal o zero. Se P s greaer han he assgnmen cos ndcaes, for c. A olaon of he comlemenary condons (7) s deeced by he exsence of, a leas, one ar (,) such ha he number of elemens of P s greaer han one. he se I ndcaes, for each clen, all oerang facles durng erod ha belong o any ah such ha s greaer han he assgnmen cos c. hs means ha all slacks SA and SR,, wh I wll be ncreased wh he decrease n. he se * I corresonds o he se of all facles such ha here exss a leas one slack SA or SR,, blockng arable. Se J reresens all dual 4

15 arables ha can be ncreased wh he decrease of arable belongs o se ascen rocedure s called. Consder arables. I s ossble ha arable self J. Neerheless, hs arable won be consdered n se J a he frs me he dual organzed as a sequence of ars (,). Prmal-Dual Adusmen Procedure for Dual Varables. Inalze (, ) (, ) ; q ; δ 0.. If # hen go o 9. P 3. If J \{(, ) } 4. For each (, ) c. 5. J = J \{(, ) } =, hen go o 9.,, SA SA ( c ), SR SR ( c ), {( ) } () P > c. Execue he dual ascen rocedure for arable J = J,. Execue he dual ascen rocedure for arable J = J. Execue he dual ascen rocedure for arable. P() ; > c 6. If has been changed, go o. 7. Execue he rmal rocedure. 8. If here hae been mroemens n he dual or rmal obece funcon alue, hen δ 0. Else δ δ. 9. If he rmal obece funcon alue s equal o he dual obece funcon alue, or δ=δ max, or q # J hen so; else q q ; (, ) (, ) q, go o..3.3 Prmal Procedure he rmal rocedure consrucs rmal feasble soluons based on dual feasble soluons, ryng o force comlemenary condons o be sasfed. Consder he ses I *, I and I * defned n he reous secon, and also: I A = { (,, ) :.. a = }, I R = { (,, ) : r = }, P = {: s oen durng erod }. Ses I * and I are no necessarly equal, because he rmal rocedure wll always ry o oen he mnmum number of serces, guaraneeng ha all clens wll be assgned o one oen ah n eery me erod. Furhermore, s ofen necessary o nser n se I serces ha don belong o I *. Ses I A and I R are bul durng he rmal rocedure and deermne whch serces wll be (re) oened, when and for how long. Defnon 4: A ah s consdered essenal durng erod f here s a leas one clen ha has o be assgned o ah durng erod. hs haens f and only f J : c < c, ' P, '. ' 5

16 Pahs consdered as essenal are he frs o be consdered oen. o oen a ah a me erod all facles belongng o ha ah hae o be oen. hs s acheed by nserng all hose facles n se I. When oenng a ah, he rmal rocedure wll ofen olae comlemenary resrcons () and (), by ncludng n se I facles I *. Wheneer ah s oen, all oher ahs such ha f hen wll also be oen. Pahs no consdered essenal wll only be exlcly oen durng me erod f here are clens ha canno be assgned o already oen ahs. In hs case, he rocedure wll oen he ah ha corresonds o he smalles assgnmen cos. Ses I A and I R are buld based on ses I,. hese ses are bul usng exacly he same rocedures descrbed n Das e al (004a). Prmal Procedure. I A = I R =. I =,. Buld ses I * and I *. Num = 0;. For = unl, nclude n se P all ahs such ha c and < c, '. : ' Udae ses P,, ncludng n P all oen ahs P. 3. For each clen such ha c, P, nclude n se P ah such ha c c ' = mn c. Num Num. ' < 4. Include n se I all facles belongng o ah P,. 5. If Num = hen I * I and I, P,, go o. Else go o Buld ses I A and I R. Udae I and P. 7. For = unl, assgn each clen o ah P such ha c = mn { c } ' P. Calculae Z as beng he rmal obece funcon alue. 8. G = I A I R. 9. Choose arbrarly a arable a or r belongng o G and change s alue from one o zero. If he soluon remans admssble, recalculae he assgnmens of clens o oen ahs. Calculae Z as he rmal obece funcon alue of he new soluon. If Z < Z, hen remoe arable a (or r ) from se I A (or I R ) and se Z = Z. 0. Remoe arable a (or r ) from se G. If G =, go o. Else go o 9.. es comlemenary condons (9)-(). Ses 8-0 ry o mroe he rmal soluon calculaed by decreasng he number of facly locaon arables ha are consdered equal o one. hs has roed o decrease sgnfcanly he alue of he rmal obece funcon. hese ses could also be relaced by a dro heursc: from all arables equal o one, choose he one whch, when s alue s changed o zero, leads o he greaes mroe n he rmal obece funcon alue. Reea he rocess unl here s no mroemen n he obece funcon alue. Such a dro heursc was esed, bu he resuls obaned showed ha s much more me consumng and he alue of he rmal obece funcon 6

17 obaned s he same as wh he execuon of ses 8-0. hs s usfed by he fac ha he rmal rocedure consders more arables equal o one han he ones srcly needed (because he oenng of a ah consss n oenng a se of facles, ha mos of he mes oen mlcly oher ahs). hese arables wll always be consdered equal o zero, een f chosen arbrarly. Se res o change he rmal soluon n order o guaranee he sasfacon of comlemenary condons (8)-(0) ha are beng olaed. hs es s equal o he one already deeloed by he auhors o he sngle-leel case (Das e al, 004b). Consderng he dual soluon resened n ables 6 and 7, he rmal rocedure would oen ah, consderng facles and 3 oen from erod one o wo ( a = a ). hs rmal 3 = soluon has an obece funcon alue equal o 73, hus s he omal soluon. 3 Includng maxmum caacy consrans In almos all real suaons, a facly has an uer lm on he demand can sere. So, he ncluson of maxmum caacy resrcons n he roblem DUMLP s a naural exenson. Consder: Q = maxmum caacy of facly durng a me erod, x = fracon of cusomer s demand sered by ah, durng me erod. d = oal demand of cusomer durng me erod. he mul-leel, dynamc, caacaed locaon roblem (DCMLP) can be formulaed smly by ncludng n DUMLP consran (3) below and changng (7) o (7 ): ( a r ) Q = = { 0, } { 0, } d x 0,, ( 3 ) P() a,,, r,, >, ( 7 ) x 0,,, he addonal se of resrcons (3) esablsh an uer lm on he oal flow ha reaches an oen facly n each me erod. o deelo a rmal-dual heursc, s necessary o obsere he changes hese addonal resrcons brng o he dual condensed roblem formulaon. Assocang dual arables λ o resrcons (3), he condensed dual roblem formulaon becomes: 7

18 CD-DCMLP Max ρ π subec o: max 0, c d η = P( ) ηmax, c d = P( ) u λ u = FA ρ π Q = = λ,,, =, L, ( 4 ) 0 λ FR u π Q λ, = =, ρ, π, λ 0,,,,, =, L, ( 5 ) A decrease or ncrease n dual arable λ wll nfluence all slacks S ', wh, such ha P() P( ), I (n he calculaon of S ', he sum oer all ahs P( ) wll consder all alues of λ such ha belongs o ). he rmal-dual heursc s funconng scheme resened for he uncaacaed case remans ald for he caacaed case, wh wo addonal ses: 0. Dual Ascen Procedure for dual arables λ. If he dual soluon s changed, go o.. Dual Descen Procedure for dual arables λ. If he dual soluon s changed, go o. he dual and descen rocedures for arables λ wll be descrbed n he nex wo secons. he deelomen of hese rocedures followed he work of Gugnard and Selberg (979), Saldanha da Gama (00) and Das e al (004b). he rmal rocedure wll hae o be changed ( s necessary o es he sasfacon of he addonal caacy resrcons). he dual ascen and Prmal-Dual adusmen rocedures for arables he assgnmen coss equal o c d reman ald beng only necessary o consder λ. All he remanng rocedures are no changed. 3. Dual Ascen Procedure for Varables λ If arable λ s ncreased, hen he lef hand sde of consrans (4) and (5) wll dmnsh. he maxmum change ha should be consdered s such ha all hose alues became less han or 8

19 equal o zero. Consderng ha arable λ s ncreased by δ, hen δ should be less han or equal o () such ha: () c d λ = max J P () d ( 6 ) Consder he followng defnons: (,', δ ) = (, ) : J P( ) P( ' ) c d JP d c d JP(,', δ ) = (, ) : J P( ) P( ' ) d λ λ δ, ( 7 ) > δ. ( 8 ) Prooson : If dual arable λ s ncreased by δ ]0, ()], hen slacks I such ha P() P( ), wll be changed by: SA' and SR ',, Ω(δ,, ) = η ' max 0, c d λ η' d δ E ( 9) (,) JP(,', δ ) (,) JP (,', δ ) where: Q, f = ' E = δ. 0, oherwse hs rooson follows smlar resuls ha can be found n Gugnard and Selberg (979). hs rooson moaes he dual ascen rocedure ha s now descrbed. Dual Ascen Procedure for Varables δ 0; δ. 4. δ max max 0, c d λ. J () d P max 0, c d λ < δ' d λ 5. Calculae ses JP(,,δ) and JP (,,δ) as n (7) and (8), and Ω(,,δ) as n (9), I, wh P() P( ). 6. If Ω(δ,, ) < 0, I, wh P() P( ), hen go o. Else go o If I, wh P() P( ), such ha SA ' Ω(δ,, ) < 0 or SR ' Ω(δ,, ) < 0,, hen go o 8. Else go o 9. 9

20 8. If δ = 0, hen go o. Else δ δ and go o λ λ δ; SA ' SA ' Ω(δ,, ) and SR' SR ' Ω(δ,, ),, I such ha P() P( ). 0. Execue he dual ascen rocedure for arables, wh J = J... If > M hen go o. Else go o 3... If > hen so. Else go o. Examle : Consder a roblem wh sx clens, hree serces organzed as n fgure, and wo me erods. he clens hae a known demand ha s equal o 56,, 76, 58, 9 and 57 n me erod and equal o 59,, 78, 6, 8 and 58 n me erod, for =,,6. Serces hae maxmum caaces equal o 49, 6 and 45. Imagne ha he frs me he dual ascen rocedure for arables λ s execued arables are equal o zero and slacks hae alues resened n able 9. ake on he alues resened n able 8, all oher dual arables SA SA able 8 SA SR able 9 Consderng arable λ : δ ; P()={}. he alue of δ s calculaed as: δ Ω max max max { 0, }, max{ 0474, 034}, max{ 0, }, 9 { 0, }, max{ 0, 53 53}, max{ 0, } = ( 8. 69,, ) = ( ) η ( ) η ( ) ( ) η η Consderng η = 0. 3, η = 0. 76, η 3 = and η 6 = 0. 87, hen ( 8. 69,, ) = Ω. Varable λ canno be ncreased by δ because he rocedure deecs ha slack SA would become less han zero. he alue of δ s changed o 8.69 and he rocedure s reeaed, obanng δ equal o 0. 0

21 Ω ( 0,, ) = ( ) 0. 3 ( ) ( ) = hs change s sll no admssble, so δ s changed o 0 and δ wll become equal o. Ω (,, ) = ( ) ( ) = All slacks P() P() = S wll connue greaer han or equal o zero. Slacks S are no changed because ( 3,, ) = ( ) ( ) Ω = hs means ha slacks S ,, are gong o be ncreased. he udaed slacks alues, are shown n able 0. he ncrease n arable λ allows he mroemen n he dual obece funcon alue, because arable s ncreased o able shows he slacks alues afer he execuon of he dual ascen rocedure for arables. SA SA SA SR SA SA SA able 0 able 3. Dual Descen Procedure for Varables λ SR If dual arable λ s decreased all alues c d λ, wh P(), wll be ncreased. Consder a decrease n λ such ha all hose alues ha are less han zero reman ha way. hs means ha λ can be decreased by δ such ha: 0 < δ mn c d λ J, P () d c d < λ 0 ( 30 ) changed. Once agan all slacks SA' and SR ',, I such ha P() P( ), wll be Prooson : If arable λ s decreased by a alue δ n he neral defned by (30), hen SA' and SR ',, I such ha P() P( ), wll be changed by:

22 (,',) δ = δ E d η', ( 3 ) J P () P( ' ) λ 0 Φ c d where Q, f = ' E =. 0, oherwse hs rooson follows smlar resuls ha can be found n Gugnard and Selberg, (979). Prooson moaes he followng dual descen rocedure. Dual Descen Procedure for Varables λ δ mn λ, mn c d λ. J, P () d c d λ < 0 4. Calculae Φ(δ,,) as n (3), I, wh P() P( ). 5. If Φ(δ,,) < 0, I, wh P() P( ), hen go o 0. Else go o If I, wh P() P( ), such ha SA ' Φ(δ,,) < 0 or SR ' Φ(δ,,) < 0,, hen go o 7. Else go o 8. S ' 7. δ mn Φ( ). If δ = 0 hen go o 0. Else go o 8. δ,', δ 8. λ S ' Φ ( δ,', ) λ - δ; < 0 SA SA Φ(δ,,) and ' ' SR' SR ' P( ). 9. Execue dual ascen rocedure for arables, wh J = J. 0.. If > M hen go o. Else go o 3... If > hen so. Else go o. Φ (δ,,),, I such ha P() Examle 3: Consderng examle and arable λ equal o. hen, for equal o he rocedure calculaes: = d. c d 0 η and Φ (,, ) = = 0. 7 Consderng equal o 3 ( s no needed o consder equal o because P() P() = ) he rocedure calculaes Φ (, 3, ) = hs means ha slacks S ',, wll ncrease for

23 equal o and decrease for equal o. Slack equal o zero n se 7 of he rocedure, so he dual arable s no changed. 3.3 Prmal Procedure SA 3 s equal o zero, whch means ha δ wll be he rmal rocedure ha bulds admssble soluons o DCMLP s ery smlar o he one descrbe for DUMLP. Afer buldng ses I A and I R, s necessary o es he sasfacon of he maxmum caacy resrcons. he assgnmen of cusomers o ahs s acheed by solng ranshmen roblems. If he ranshmen roblem s no feasble for some me erod hen wll be necessary o oen more ahs (noce ha s no ossble o esablsh a necessary and suffcen condon of he form oal caacy of oen facles greaer han or equal o clen s oal demand o es he admssbly of a gen soluon, as haens n he sngle leel case). o choose wha ah o oen, he rocedure calculaes he mnmum cos of oenng a ah by consderng he coss of oenng all serces belongng o ha are no oeraonal durng me erod. Consder: F = mnmum cos of oenng facly I durng erod ; H = mnmum cos of oenng ah P durng erod ;, f P e =,, ; 0, oherwse n = e. P () Prmal Procedure. Execue se -6 of he DUMLP s rmal rocedure Sole a ranshmen roblem consderng as sources he se J of clens (wh sules d ), as desnaons he se of facles I belongng o leel K (wh demands Q ), and as ranshmen ons he se of facles I belongng o leels o K- (wh demand and suly equal o Q ). If he roblem s no feasble go o 4. Else go o If P, hen so. he roblem s nfeasble. Else go o Calculae H = F and I Q Ca = mn, P. n 6. Choose ah such ha H ' Ca ' = H mn Ca P. 7. Oen ah, ncludng n se P and n se I,. Go o 3. 3

24 8.. If > hen go o 9. Else go o Calculae Z as he curren rmal obece funcon alue. 0. Execue ses 8-0 of DUMLP s rmal rocedure. he mnmum cos of oenng a facly durng erod (F ) s calculaed as descrbed n Das e al (004b). Calculaon of Ca res o llusrae he maxmum caacy of ah. he maxmum caacy of ah s deermned by he facly wh mnmum caacy. If facly belongs o more han one ah, s caacy s consdered o be equally dded by all oen ahs. 4 Pahs Whou Flow Conseraon Consder ha facles belongng o leels,,k- hae a arameer θ such ha f a flow of d uns reaches hen hs facly wll only ass o he nex facly a flow d θ (see fgure 3). hs s a generalsaon of DCMLP, where θ s equal o one for all facles. d Fgure 3 d θ here are seeral examles where one can fnd hs knd of behaor n nermedae facles. In sold wase reamen sysems, ncneraors recee a ceran amoun of wase o burn, and only a small ar of (n he form of ashes) has o be laced n landflls. In a healh facly srucure, only a small grou of clens ha are sered n local healh cenres hae o be conduced o local hosals. he consderaon of arameers θ assocaed wh nermedae facles comlcaes exremely he roblem, secally he deermnaon of omal assgnmens of clens o ahs (when he locaon arables are already fxed o one or zero). In fac, hs assgnmen can no longer be calculaed omally hrough he resoluon of ransshmen roblems. In he nex aragrahs, he rmal and condensed-dual formulaons are resened, and he rmal-dual heursc deeloed s descrbed. he comlemenary condons are no saed because hey are ery smlar o (7)-(). he only dfference beween hs roblem and DCMLP les n he maxmum caacy resrcons. Consder he followng defnon: Defnon 5: Le and be wo facles and a ah such ha and. hen f facly aears before facly n he ordered se. < ' f and only 4

25 he mul-leel, dynamc caacaed locaon roblem whou flow conseraon on he nermedae facles (DCMLP) can be formulaed as DCMLP wh consrans (3) below nsead of consrans (3). Q ( a r ) dx θ' 0,, ( 3 ) = = P() ' < Consderng he same dual arables as n CD-DCMLP, he condensed-dual roblem becomes: CD-DCMLP Max subec o: ρ π max 0, c d θ' λ FA u ρ π Q λ ' < = = =, η = P( ),, =, L, ( 33 ) max 0, c d θ' λ FR u π Q λ, ' < = = η = P( ) u, ρ, π λ 0,,,,,, =, L, ( 34 ) he rmal-dual heursc funconng scheme s he same resened n secon 3, wh an addonal se:. Calculae he omal assgnmen of clens o ahs consderng ses I and P ha corresond o he bes rmal soluon calculaed hus far. Some rocedures had o be slghly changed. Consder ' ' < d = d θ, P(),. he dual ascen and rmal-dual adusmen rocedures for arables do no need o be changed, beng only necessary o consder assgnmen coss c = c d ' ' λ,, P(),. he changes n dual ascen and descen rocedures for arables λ are a drec consequence of d defnon. In he dual ascen rocedure he defnons (6) o (8) should be relaced by (35) o (37): 5

26 () c d' λ = max, ( 35 ) J P () d' ' (,', δ ) = (, ) : J P( ) P( ' ) c d JP d' c d ' ' JP(,', δ ) = (, ) : J P( ) P( ' ) d' λ λ δ, ( 36 ) > δ, ( 37 ) In hs case Ω(δ,, ) s gen by: Ω(δ,, ) = η ' max 0, c d' λ η' d' δ E ( 38) (,) JP(,', δ ) (,) JP (,', δ ) where: Q, f = ' E = δ. 0, oherwse 4. δ Se 4 of he dual ascen rocedure for arables λ should be relaced by: 0 max d max max 0, c d' λ J () d P ' ', c d ' λ < δ' In he dual descen rocedure for arables λ, he followng changes should be consdered: δ = δ E d ' η ', ( 39 ) J P () P( ' ) c d ' λ 0 (,',) Φ where: 0 < δ mn c d () ' λ. ( 40 ) J, P d ' c d λ < 0 ' 6

27 Se 3 should be changed o: 3. δ mn λ, mn () J, P ' c d ' λ < 0 d' c d ' λ. he rmal rocedure s ery smlar o he one already resened n secon 3. Se 5 s changed o: 5. Calculae H = F and I Ca Q = mn, P. n θ' ' < hs change res o accoun no only for he number of ahs o whch facly belongs, bu also for he relae oson of facly n hose ahs. he assgnmen roblem canno be soled as a ranshmen roblem (se 3 of he rmal rocedure). Afer calculang ses I and P, he followng lnear rogram should be soled o fnd he omal assgnmens of clens o ahs, for each me erod : AP() Mn ( 4 ) subec o: P c x x =, ( 4 ) x d θ ' Q, I ( 43 ) P() P ' < x 0,,, Solng hs roblem omally (usng a general soler) n each execuon of he rmal rocedure (and for all me erods) s ery me consumng. So, a heursc was deeloed o fnd feasble soluons o AP(). hs heursc rocedure s used n se 3 of he rmal rocedure (nsead of he resoluon of he ransshmen roblem). Problem AP(),, s only soled omally a se of he rmal-dual heursc (usng a general soler), consderng he locaon arables ha corresond o he bes rmal soluon calculaed by he heursc unl ha momen. 7

28 Heursc Procedure for he resoluon of roblem AP() Q. D d,. Q θ ' ' <,, P(), Ca mn{ Q }, P. x 0,,.. If Ca = 0, P and such ha D > 0, hen so: he roblem s mossble. c mn c and 3. For each clen J, wh D >0, calculae { } c { c : c c } mn P Ca > 0 4. Choose clen and ah such ha c c = max{ c ' c ' } ' J 5. δ mn{ D, }, x x δ Q Ca Q, δ 6., P,, such ha, Q 7. If D = 0,, hen so. Else, go o... P Ca > 0 Q,, Ca δ, D D δ. Q δ θ ' ' < θ ' ' < and mn{ Ca, Q } Ca. Calculaon of Q exresses he maxmum flow ha can be assgned o ah due o he maxmum caacy of facly. he mnmum of hese caaces corresonds o he maxmum caacy of ah desgnaed by Ca. Obseraon : Consder wo ses (S and S ) of locaon arables a and r. Consder wo admssble soluons o DCML consruced by consderng equal o one all locaon arables belongng o S and S, resecely, beng all he oher locaon arables equal o zero. Le us consder ha Z * <Z *, where Z * * and Z reresen he omal obece funcon alue corresondng o soluon S and S, resecely. If Z and Z reresen he obece funcon alue obaned by consderng soluons S and S, resecely, and solng he assgnmen roblems AP() heurscally, hen Z Z * and Z Z *, bu s no ossble o guaranee ha Z < Z. hs has an moran consequence: s ossble ha he rmal-dual heursc fnds soluons beer han he fnal soluon resened, bu ha he heursc s no able o denfy as beng beer. hs shor come canno be soled, unless he omal soluons o AP() are calculaed for eery me erod, wheneer he rmal rocedure s execued. As has already been sad, ha would be rohbe n erms of execuon mes. Examle 4: Consder a roblem wh four clens and fe oenal serces such ha serce and are n leel, serces 3 and 4 are n leel and serce 5 s n leel 3. here are fe admssble ahs: 8

29 =(,3,5), =(,4,5), 3 =(,4,5), 4 =(,5), 5 =(4,5). A erod he clens demands are as follows: d =5; d =0; d 3 =8; d 4 =. Serces o 5 hae maxmum caaces equal o 0, 50, 80, 70 and 00, resecely, and θ alues equal o 0.8, 0.5, 0.5, 0.. Se : Q = 0 ; Q = = ; Q = = ; { Q,Q, } 0 Ca ; = mn 3 Q5 = Q = 0 ; Q 4 = =. ; Q 00 5 = = ; { Q,Q, } 0 Ca ; = mn 4 Q5 = Q 3 = 50 ; Q = = ; Q = =.. ; { Q,Q, } 50 Ca ; 3 = mn 3 43 Q53 = Q = 50 ; = = 0. 5 Ca ; 4 = mn Q4, Q54 = Q ; { } Q 45 = 70; 55 = = Q ; = mn{ Q, Q } 70 Ca = Se 3: hs nformaon s summarzed n able Ca able c c c able 3 Se 4: Choose clen equal o 3 and ah equal o. Se 5: δ mn{8, 0}=8. D 3 0; Values c Q and Ca are changed accordng wh able 4. 9

30 = 0-8= = =. 0 = = = = = 97. = Ca able 4 Se 4: Choose clen equal o and ah equal o. Se 5: δ mn{0, }=; D 8; Values Se 3: Q and Ca are changed accordng wh able 5 (he bold alues are he ones ha are changed) Ca able 5 c c c c able 6 Se 4: Choose clen equal o and ah equal o 3. Ses 5 and 6 : δ mn{5, 50}=5 D 0; Values able 7 (he bold alues are he ones ha are changed). Se 4: Choose clen equal o and ah equal o 5. Ses 5 and 6: δ mn{8, 46.5}=8. D 0; Values able 8. Q and Ca are changed accordng wh Q and Ca are changed accordng wh 30

31 Se 4: Choose clen equal o 4 and ah equal o 3 Ses 5 and 6: δ mn{, 35}=; D 4 0; Se 7: Values D are equal o zero, for all, whch means ha an admssble soluon has been found (he soluon found s llusraed n fgure 4): P I Ca Ca able 7 able Fgure 4 Problem DCMLP can be generalzed consderng a new se of consrans mosng mnmum caacy resrcons: Q' ( a r ) d x θ' 0 = = P (),, ( 44 ) ' ' < whereq' reresens he mnmum flow ha has o reach an oen facly, durng he me erods hs facly s oerang. Assocang dual arables β wh consrans (44), he condensed dual formulaon s slghly changed: CD-DCMLP Max subec o: ρ π 3

32 max 0, c d θ' λ β ' <,,, =, L, (45) FA u ρ π Q λ β Q' η = P( ) = = η max 0, c d θ' λ β = P( ) ' <,,, =, L, (46 ) FR u π Q λ β = = Q' = u =, ρ, π, λ, β 0,,, = As can be seen, he behaor of dual arables β and λ s symmerc, so he dual ascen rocedure descrbed for dual arables λ consues he dual descen rocedure for β, and he dual descen rocedure for dual arables λ consues he dual ascen rocedure for arables β. he rmal rocedure has o be changed, because s necessary o sasfy a new se of consrans. Neerheless, he new rmal rocedure consruced s, n essence, smlar o he one reously descrbed. he only dfference consss n he heursc rocedure o sole he assgnmen roblem (roblem AP() has also a new se of mnmum caacy consrans), and also n some addonal ses ha ry o close or nerchange oen and closed facles, when an admssble soluon s no found by oenng new facles (usng he rocedure already descrbed). he omal assgnmen of clens o facles, consderng he locaon arables fxed o one or zero, can be found by solng he followng lnear roblem: AP() Mn subec o: P Q' c x x =, ( 4 ) x d P P ' < () θ Q, I ( 47 ) ' x 0,,, 3

33 Heursc o fnd an admssble soluon o AP(). Calculae an admssble soluon o AP(). If he calculaed soluon s admssble o AP() so. Else go o.. 0,, ; Camn Q' and Camax Q, I ; D d,. x 3. If Ca 0, I, hen go o If D = 0, and I such ha Camn >0, hen so: he heursc canno fnd an admssble soluon. Else go o 5. Camn 5. Choose facly, wh Camn > 0, such ha Camn = max '. Camax ' I Camax' 6. If P P() such ha Camn > 0, hen go o 7. Else go o 8. c c ' ' 7. Choose ah P P() and clen such ha = mn. Go o 9. θ ' ' P() P θ ' ' < Camn > 0, ' ' < ' ':D ' > 0 c c ' ' 8. Choose ah P P() and clen such ha = mn. θ ' ' P() P θ ' ' < ' :D ' > 0 ' < ' 9. δ Camax ' Camn mn D,mn, ' θ'' θ ; x δ x ; D D δ; ' '' < ' ' < Camn Camn δ θ and Camax Camax δ θ,. Go o 3. ' ' < ' ' < 0. If such ha D > 0, hen sole heurscally roblem AP() consderng Q equal o Camax, I and d equal o D,. Else so. If an admssble soluon s no found n se of hs heursc, hen res o assgn flow o ahs n order o sasfy he mnmum caacy consrans of oen facles. Se 5 of hs heursc chooses facly such ha he relaon beween s maxmum and mnmum caaces s more dffcul n erms of sasfyng boh resrcons. he heursc wll ry frs o assgn flow o ahs P() such ha no facly belongng o ah has s mnmum caacy resrcon sasfed (se 6). When hs s no ossble, he heursc consders all oen ahs P() (se 7). In ses 6 or 7 he heursc chooses a clen and an oen ah ha corresond o he mnmum cos er un ha reaches facly. he ncrease n he flow ha s assgned o ah s calculaed as beng he mnmum beween he remanng demand of clen and he necessary flow o sasfy he mnmum caacy resrcon of facly, guaraneeng ha he maxmum caacy resrcons for all oher facles belongng o ah reman admssble. When he heursc reaches se 0, all mnmum 33

34 caacy resrcons for oen facles hae been sasfed. If here are sll clens hang a remanng demand greaer han zero, hen wll be necessary o assgn hese demands o ahs usng he already descrbed heursc for roblem AP(). he descrbed heursc canno guaranee he consrucon of an admssble soluon o AP() wheneer he roblem s feasble. I s always ossble o check f he roblem AP() s nfeasble by usng a general soler (whch wll be more me consumng bu wll fnd ou, for sure, f he roblem s or s no nfeasble, when he heursc s unable o fnd an admssble soluon). Examle 5 Consder he roblem descrbed n examle 4, wh facles hang he followng mnmum caaces: 5, 0, 0, 60, 5. he mnmum caacy resrcon s no sasfed for facly 4, so he soluon found n se of he heursc s no admssble. Se : Camn 5; Camn 0; Camn 3 0; Camn 4 60; Camn 5 0. Camax 0; Camax 50; Camax 3 80; Camax 4 70; Camax D 5; D 0; D 3 8; D 4. Se 5: Choose facly 4 because = max,,, Se 7: Choose ah, 3 or 5 and a clen accordng o he calculaons resened n able 9. he heursc wll choose ah and clen able Se 9: δ mn 8, 0,,, = 8 ; x 3 8 ; D 3 0; Camn 5 8= 3; Camn =45.60; Camn =3.56. Camax ; Camax ; Camax Se 5: Choose facly 4 because = max,, Se 7: he heursc wll choose ah 5 and clen 4. 34

35 Se 9: δ mn, ,, = ; x ; D 4 0; Camn ; Camn 5.36; Camax ; Camax Se 5: Choose facly 4 because = max,, Se 7: he heursc wll choose arbrarly beween clen and ah 3 or clen and ah 5 (he heursc wll neer choose clen and ah because ah has facles wh he mnmum caacy resrcon sasfed). Consder ha he heursc chooses clen and ah Se 9: δ mn 5, 50,,, = 5 ; x 3 5 ; D 0; Camn 5; Camn 4 6.0; Camn 5 0.6; Camax 35; Camax 4 6.0; Camax Se 5: Choose facly 4 because = max, Se 7: he heursc chooses ah 5, because s he only ah ha s consued by facles whose mnmum caacy resrcon s no ye sasfed, and chooses clen Se 9: δ mn 0, 6.,, 6. 0 = 6. 0 ; x ; D 3.9; 0. Camn 4 0; Camn 5 ; Camax 4 0; Camax Se 0: he curren soluon s already admssble because Camn 0,. As D > 0, hen wll be necessary o use he heursc o sole roblem AP(). If he mnmum caacy resrcons were consdered n roblem DCMLP (wh θ equal o one, for all facles) hen he assgnmen roblem AP() would be a ransshmen roblem, wh each nermedae facly corresondng o wo ransshmen ons: one hang demand and suly equal o s mnmum caacy and he oher hang demand and suly equal o s maxmum caacy mnus s mnmum caacy. Consder: f = smalles cos ncurred by closng I durng erod. 35