Particle Swarm Procedure for the Capacitated Open Pit Mining Problem

Transcription

1 Parcle Swarm Procedure for he Capacaed Open P Mnng Problem Jacques A Ferland, Jorge Amaya 2, Melody Suzy Dumo Déparemen d nformaque e de recherche opéraonnelle, Unversé de Monréal, Monréal, Québec, Canada ferland@roumonrealca, dumoys@roumonrealca 2 Cenro de Modelameno Maemáco, Unversdad de Chle, Sanago, Chle amaya@dmuchlecl Absrac In he Capacaed Open P Mnng Problem, we consder he sequenal exracon of blocs n order o maxmze he oal dscouned prof under an exracon capacy durng each perod of he horzon We propose a formulaon closely relaed o he Resource-Consraned Proec Schedulng Problem (RCPSP) where he genoype represenaon of he soluon s based on a prory value encodng We use a GRASP procedure o generae an nal populaon (swarm) evolvng accordng o a Parcle Swarm Procedure o search he feasble doman of he represenaons Numercal resuls are nroduced o analyse he mpac of he dfferen parameers of he procedures Keywords: Open p mnng, GRASP, Parcle Swarm, evoluonary process, RCPSP, prory encodng Inroducon Consder he problem where a mnng ndusry s analyzng he prof of exracng he ore conaned n some se In fac, he analyss ncludes wo sages where wo dfferen problems have o be solved he frs problem s o deermne he maxmal open p correspondng o he maxmal gan ha he mnng ndusry can ge from he exracon a he se hs problem can be formulaed as denfyng he maxmal closure of an assocaed orened graph Pcard proposes a very effcen procedure o solve hs problem n [] he second problem, denoed bloc exracon problem, s o deermne he exracon order leadng o he maxmum prof accounng for he dscoun facor and parcular consrans relaed o he exracng operaon In general, he mnng se s paroned no blocs characerzed by several numbers One of hese s he ne value of he bloc esmaed from prospec on daa hs ne value s equal o he prof assocaed wh he bloc correspondng o he dfference beween he ore conen value and he cos of exracng he bloc Hence hs value can be negave f he ore conen value s smaller han he cos of exracng Furhermore, he physcal naure of he problem may requre exracng blocs havng negave ne values n order o have access o valuable blocs In her poneerng wor, Lerchs and Grossman [2] deals wh hs problem wh an approach generang a sequence of nesed ps hen several oher approach have been proposed : heurscs [3,4], Lagrangan relaxaon [3], paramerc mehods [5, 6, 7, 8], dynamc programmng [9], mxed neger programmng [3, 5, 0], and meaheurscs or arfcal nellgence mehods [, 2] In hs paper, we rely on an analogy wh he Resource-Consraned Proec Schedulng Problem (RCPSP) [3] o propose a parcle swarm procedure [4] o solve a specfc varan of he problem referred o as he Capacaed Open P Mnng Problem Recall ha a parcle swarm procedure s an evoluonary populaon based process he swarm (populaon) s evolvng hrough he feasble doman searchng for an opmal soluon of he opmzaon problem In he concludng remars, we ndcae how o exend o oher varans of hs problem he paper s organzed as follows In Secon 2, we formulae he wo problems and we summarze Pcard s procedure o solve he maxmal open p problem Our soluon approach relyng on an analogy wh he RCPSP o deal wh he schedulng bloc problem s summarzed n Secon 3, and Secon 4 ncludes our parcle swarm procedure In Secon 5, we nroduce numercal resuls o analyse he mpac of he dfferen parameers of he procedures he numercal resuls allow verfyng ha he qualy of he soluons generaed wh he parcle swarm procedure s beer han hose obaned wh greedy genoype prory vecors 2 Problem formulaons In hs secon we frs nroduce a mahemacal model for he maxmal open p problem and a mehod o solve hen he schedulng problem specfyng he exracon order of he blocs s

2 formulaed he soluon approach for hs problem s gven n he nex secon 2 Maxmal open p o formulae he maxmal open p model used o deermne he maxmal gan expeced from he exracon, we denoe by N he se of blocs n he mnng se and by b he ne value of exracng bloc N A bnary varable x s assocaed wh each bloc : f bloc s exraced x 0 oherwse he maxmal open p corresponds o he soluon x maxmzng he obecve funcon b x N he only physcal consrans consdered are relaed o he maxmal p slope leadng o denfy a se of predecessors B ncludng he blocs of he precedng layer above ha have o be removed before bloc can be exraced he problem can be formulaed usng an orened graph (open p graph) G(V, A) where V { N : node corresponds o bloc } and he se of arcs A {(, : B, N} used o ndcae he precedence order of he bloc exracon Hence he mahemacal model assocaed wh he maxmal open p problem can be summarzed as follows (MOP) Max Subec o b x N x x 0 x 0 ou (, N () (2) Now recall ha he closure of an orened graph [] s a subse of nodes such ha all successors of any of s node also belong o he se I s easy o see ha he problem (MOP) s equvalen o deermne he maxmal closure of he open p graph G(V, A) In [], Pcard shows ha he problem of fndng he maxmal closure of G(V, A) s equvalen o deermne a mnmum cu ( S, S ) of he assocaed Pcard s graph G ( V, A) specfed as follows he se on nodes V V U { s, } where s and correspond o a super source and and a super sn, respecvely + he se of arcs A AU A U A, where + A {( s, ) : V, > 0}, A {(, ) : V, 0} b Furhermore, he capacy d of arc specfed as follows: d + b b f (, f s and ( s, f and (, ) b (, s + Pcard [] shows ha a maxmal closure of G s he se (S {s}) I follows ha he Ford and Fulerson labellng procedure [5] can be appled o deermne he maxmal flow n he Pcard s graph assocaed wh G, and hen he maxmal open p s equal o N (S {s}) where ( S, S ) s he mnmal cu assocaed wh he maxmal flow 22 Schedulng bloc exracon he maxmal open p ndcaes he se of blocs o exrac n order o maxmze he oal prof, bu does no gve any ndcaon of he order for exracng he blocs under operaonal consrans, nor does accoun for he dscoun facor durng he exracon horzon hs problem s more complex and hence more dffcul o solve In hs paper, we consder he capacaed open p mnng problem where only one addonal operaonal consran relaed o he maxmal quany ha can be exraced durng each perod of he horzon In he concludng remars we ndcae how o exend he soluon procedure o nclude oher operaonal consrans Denoe by p he wegh of bloc N, by C he maxmal wegh ha can be exraced durng perod,, and he dscoun rae per perod A +α bnary varable each perod : x x s assocaed wh each bloc for f bloc s exraced durng perod 0 oherwse Referrng o he noaon nroduce n Secon 2, he schedulng bloc exracon problem can be formulaed as follows: (SBE) Subec o Max x l N x l x x 0 p x C b ( + ) x N α 0 ou N (,,, L,, L, N,, L, (3) ( 4) (5) (6) (7) he obecve funcon (3) accouns for he dscoun facor n evaluang he ne values of he blocs when hey are exraced Consrans (4) guaranee ha each bloc s exraced a mos once durng he horzon he exracon precedence s enforced by consrans (5), and he consrans (6) are relaed o he exracon capacy durng each perod of he horzon I can be shown ha he se of blocs exraced n an opmal soluon of (SBE) s a subse of he maxmal

3 open p Hence, n he model (SBE), we can replace N by N (S {s}) deermned n Secon 2 3 Soluon approach o fnd a schedule for he bloc exracon problem, we consder he smlary beween (SBE) and he well nown Resource Consran Proec Schedulng Proble m (RCPS) [3] he open p mne exracon corresponds o he proec, and he exracon of each bloc, o an acvy of he proec he precedence relaonshp beween he acves (blocs) s derved from he open p graph G (V, A): for all N P { bloc predecessors} { N :(, } ) he capacy consrans correspond o consrans (6) n problem (SBE) Hence, for any perod, le E { N :bloc s exraced n perod }, and he exracon capacy consrans can be wren as E of he exracon perod p C, L, he reward assocaed wh acvy (bloc) depends b : ( + α) he genoype represenaon PR [ pr, ] L, pr N ha we use s smlar o he prory value encodng nroduced by Harman n [3] o deal wh he RCPSP he h elemen pr [0,] corresponds o he prory of schedulng bloc o be exraced Hence he prory of exracng bloc ncreases wh he value of pr, and hese values are such ha pr N Sarng from he genoype vecor PR, a feasble phenoype soluon x of (SBE) s generaed usng he followng seral decodng scheme where he blocs are scheduled sequenally o be exraced o nae he frs exracon perod, we frs remove he bloc among hose havng no predecessor (e, n he op layer) havng he larges prory Durng any perod, a any sage of he decodng scheme, he nex bloc o be removed s one of hose wh he hghes prory among hose havng all her predecessors already exraced such ha he capacy C s no exceeded by s exracon If no such bloc exss, hen a new exracon perod ( + ) s naed We denoe by v(pr) he value of he feasble soluon x of (SBE) assocaed wh he genoype vecor PR I s worh nocng ha hs decodng scheme s closely relaed o he Seral SGS procedure used o generae phenoype schedule from genoype represenaon for he RCPSP In order o assocae a prory o a bloc, we need o consder no only s ne value b, bu also s mpac on he exracon of oher blocs n fuure perods One such measure proposed by olwns and Underwood n [9] s he bloc looahead value b hs value s deermned by referrng o he spannng cone SC of bloc n he opmal open p N specfed as follows: SC { N : here exss a pah from o n G} Hence, he value b b SC Referrng o he looahead values, several dfferen genoype prory vecors can be generaed usng he followng GRASP procedure [6] o deermne sequenally he bloc prores A he h (, L, N ) eraon, a bloc s seleced as follows: deermne he subse of he β% bes (n erms of her b values) blocs no seleced ye selec randomly one bloc n hs subse hen, pr ( N + ) 2( N + ) L+ N N ( N + ) Noe ha he percenage β% s a parameer of he procedure Several feasble soluons of (SBE) can be generaed by decodng dfferen genoype vecors generaed wh he GRASP procedure Furhermore, oher process can be used o generae oher genoype vecors In he nex secon, we nroduce an evoluonary process evolvng n he se of genoype vecors n order o converge o an mproved feasble phenoype soluon of (SBE) 4 Parcle Swarm Procedure In hs secon, we nroduce a parcle swarm procedure [7] evolvng hrough he se of genoype vecors n order o search he feasble doman of he problem (SBE) Consder a populaon (swarm) P of M genoype vecors (parcles): P { PR, L, PR M } he M nal genoype vecors are generaed usng he GRASP procedure descrbed n he precedng secon o nalze he frs eraon of he parcle swarm procedure, he genoype vecor PR corresponds boh o he curren vecor and he bes acheved vecor PR of he ndvdual (e, a he sar of he frs eraon, PR PR ) Durng he eraons of he procedure, he ndvduals (genoype vecors) of he populaon are evolvng, and we denoe by PRb he bes overall genoype vecor acheved so far Hence, a he end of each eraon, PR,, L, M, and PRb are upd aed as follows:

4 PR : ArgMax PRb : ArgMax M { v( PR ), v( PR )} (8) { v( PR )} (9) Now we descrbe a ypcal eraon of he parcle swarm procedure Each curren vecor PR evolves ndvdually o a new curren genoype vecor accordng o a probablsc process accounng for s curren value PR, for s bes acheved value PR, and for he bes overall acheved vecor PRb More specfcally, wh each componen of each vecor, we assocae a velocy [7] facor vc evolvng a each eraon of he procedure Is value s nalzed a 0 (e, vc 0) when he procedure sars A each eraon, evolves as follows: vc : wvc and we defne + c r ( pr pr 2 2 ) + c r ( prb pr ), ppr : vc + pr (0) () he probablsc naure of he procedure follows from he fac ha r [ ] 0, and r [ ] 2 0, are dfferen unform random number seleced a each eraon he mpac of he erms s scaled by so called acceleraon coeffcens c and c 2 he nera wegh w was nroduced n [8] o mprove he convergence rae Usng he vecor PPR ppr, L, ppr N, we deermne he new curren genoype PR Frs, we have o verfy f all componens of PPR are non negave Hence, deermne ppr N and f ppr < 0, hen replace ppr { ppr } Mn, : ppr ppr hus he new curren genoype vecor specfed as follows: pr ppr : N ppr N where pr 0,, L, N, and pr,as requred PR s 5 Numercal Resuls Numercal resuls are now nroduced o analyse he mpac of he parameer values on he effcency of he soluon approach he dfferen procedures are mplemened n Java 5, and he ess are execued on an AMD Ahlon(m) 64 Processor 3200, 22 GHz havng 200 Go Ram We are usng 20 dfferen problems randomly generaed over a wo dmensons grd havng 20 layers and beng 60 blocs wde For each problem, 0 clusers ncludng he blocs havng ore nsde are randomly generaed (noe ha hese clusers can overlap) he value b of he blocs belongng o he clusers s seleced randomly n he se {6, 8, 2, 6} he res of he blocs ousde he clusers have no ore nsde, and hey have a negave value b equal o -4 he opmal p of each problem s deermned wh he approach descrbed n Secon 2 he number of blocs N n he opmal p of each problem s ndcaed n able able : Opmal p szes Problem N Problem N he wegh p of each bl oc s equal o, and he maxmal weg h C 3 for ea ch perod We use he frs 0 problems havng smaller maxmal open p o analyse he m pac of he dfferen parameers Each problem s solved usng 2 dfferen ses of parameer values as summarzed n able 2 Noe ha er denoes he number of eraons of he parcle swarm procedure For each se of parameer values l, each problem ρ s solved 5 mes o deermne va lρ : he average of he bes values v(prb) acheved vb lρ : he bes values v(prb) acheved % lρ : he average % of mprovemen va l ρ - v( PRb) a frs er % lρ 00 v(prb) a frs er lρ : he las eraon where an mprovemen of PRb occurs

5 able 2: Parameer values Se β M Ier w c c hen for each se of parameer l, we compue he average values va l, vb l, % l, and l over he 0 problems hese values are summarzed n able 3, and hey are used as crera o evaluae he mpac of he parameer values able 3: Average values of crera Se l % l va l vb l Impac of he β% n he GRASP procedure Comparng rows, 2, and 3 of able 3, we observe ha he values of va l and vb l decrease whle he value of % l ncreases as he value of β ncreases he same observaons apply for rows 4, 5, and 6 Referrng o he defnon of he GRASP procedure n Secon 3, s easy o see ha n general, he prory of blocs wh larger looahead values ge larger when β decreases Hence, we can expec he ndvduals n he nal populaon o have beer prof values nducng ha we can reach beer soluons, bu ha he percenage of mprovemen % l s smaller Impac of he sze M of he populaon he value va l n row s larger han n row 4 he same s rue f we compare rows 2 and 5, and rows 3 and 6 hs ndcaes ha he values of he soluons generaed are beer when he sze of he populaon s larger hs maes sense snce we generae a larger number of dfferen soluons Impac of he values of he parcle swarm parameers Comparng he resuls n rows 2, 7, 8,and 9, and hose n rows 3, 0,, and 2, here s no clear mpac of modfyng he values of he parameers w, c and c 2 Noe ha he values w 07, c 4, and c 2 4 were seleced accordngly o he auhors n [9] who shown ha seng he values of he parameers close o w and c c gves accepable resuls Noe ha he number of eraons l used by he parcle swarm procedure o reach he bes soluon s smaller han he number er of eraons compleed One could as he followng legmae queson How he soluons obaned wh an evoluonary process le parcle swarm compare wh he one obaned by decodng a greedy genoype prory vecor generaed wh a varan of he GRASP procedure where, a each eraon, he bloc wh he larges value b among hose lef s seleced? o complee hs analyss we use he las 0 problems (numbered from o 20) havng larger maxmal open p o oban he resuls n able 4 Usng he parameer values n se, each problem ρ s solved 5 mes o deermne va ρ : he average of he bes values v(prb) acheved for parameer values n se vb ρ : he bes value v(prb) acheved for parameer values n se vw ρ : he wors value v(prb) acheved for parameer values n se v greedy : he value obaned by decodng a greedy genoype prory vecor hs choce of he parameer values s usfed by he resuls n able 3 where he bes value for va l s obaned wh he parameer values n se he resuls n able 4 ndcae ha even he wors values vw ρ s beer han v greedy for all problems Furhermore, he percenage of mprovemen of va ρ over v greedy ranges from 232% o 5224% We can conclude ha usng parcle swarm nduces a gan n he soluon qualy able 4: Improved soluons wh parcle swarm Problem va ρ vb ρ vw ρ v greedy Referrng o able 4, we can also observe ha for each problem, he nerval [vw ρ, vb ρ ] ncludng he values v(prb) s que small Indeed, he smalles rao

6 ρ vw vb ρ ρ s equal o 774 for problem ρ 20 hs ndcaes ha he Parcle Swarm procedure s sable 6 Concluson In hs paper we consder he Capacaed Open P Mnng Problem where we have o deermne he sequenal exracon of blocs maxmzng he oal dscouned prof under an exracon capacy durng each perod of he horzon We propose a formulaon closely relaed o he Resource-Consraned Proec Schedulng Problem (RCPSP) where he genoype represenaon of he soluon s based on a prory value encodng We use a GRASP procedure o generae an nal populaon (swarm) evolvng accordng o a Parcle Swarm Procedure o search he feasble doman of he represenaons he numercal resuls ndcae ha he beer are he ndvduals n he nal populaon, he beer s he soluon generaed bu he smaller s he benef of usng an evoluonary process le parcle swarm Also, he qualy of he soluon ncreases wh he sze of he populaon he numercal resuls n our expermenaon do no seem o be very sensve o he parameer values of he parcle swarm procedure Fnally, he numercal resuls allow verfyng ha he qualy of he soluons generaed wh he parcle swarm procedure s beer han hose obaned wh greedy genoype prory vecors Addonal expermenaons should be compleed on problems of larger sze closer o real world applcaons Furhermore, s easy o see ha he decodng procedure of he genoype prory vecors can be adaped o accoun for addonal consrans found n oher varans of he schedulng bloc exracon problem References [] Pcard, JC, Maxmal closure of a graph and applcaons o combnaoral problems, Managemen Scence, 22, pp (976) [2] Lerchs, H, Grossman, IF, Opmum desgn for open p mnes, CIM Bullen, 58, pp (965) [3] Cacea, L, Kelsey, P, Gannn, LM, Open p mne producon schedulng, Compuer Applcaons n he Mneral Indusres Inernaonal Symposum (3 rd Regonal APCOM), Ausral Ins Mn Meall Publcaon Seres, 5, pp (998) [4] Gershon, M, Heursc approaches for mne plannng and producon schedulng, Inernaonal Journal of Mnng and Geologcal Engneerng, 5, pp-3 (987) [5] Dagdelen, K, Johnson, B, Opmum open p mne producon schedulng by lagrangan parameerzaon, Proceedngs of he 9 h APCOM Symposum of he Socey of Mnng Engneers (AIME), pp (986) [6] Franços-Bongarçon, DM, Gubal, D, Paramezaon of opmal desgns of an open p begnnng a new phse of research, ransacons SME, AIME, 274, pp (984) [7] Maheron, G, Le paramérage des conours opmaux, echncal repor no 403, Cenre de Géosasques, Fonanebleau, France (975) [8] Whle, J, Four-X user manual, Whle Programmng Py Ld,Melbourne, Ausrala (998) [9] olwns, B, Underwood, R, A schedulng algorhm for open p mnes, IMA Journal of Mahemacs Appled n Busness & Indusry, 7, pp (996) [0]Gershon, M, Mne schedulng opmzaon wh mxed neger programmng, Mnng Engneerng, 35, pp (983) []Denby, B, Schofeld, D, he use of genec algorhms n underground mne schedulng, Proceedngs 25 h APCOM Symposum of he Socey of Mnng Engneers (AIME), pp (995) [2]Denby, B, Schofeld, D, Bradford, S, Neural newor applcaons n mnng engneerng, Deparmen of Mneral Resources Engneerng Mgzne, Unversy of Noongham, pp 3-23 (99) [3]Harmann, S, A compeve genec algorhm for he resource-consraned proec schedulng, Naval Research Logscs, 456, pp (998) [4]Kennedy, J, Eberhar, RC, Parcle swarm opmzaon, Proceedngs of he IEEE Inernaonal Conference on Neural Newors, IV, pp (995) [5]Ford, LR, Fulerson, DR, Flows n Nenors, Prnceon Unversy Press, New Jersey (962) [6]Feo,, Resende, MGC, Greedy Randomzed Adapve Search Procedure, Journal of Global Opmzaon, 2, pp -27 (995) [7]Paque, U, Engelbrech, AP, A new parcle swarm opmser for lnearly consraned opmzaon, Proceedngs of he 2003 Congress on Evoluonary Compuaon, pp (2003) [8]Sh, Y, Eberhar, RC, A modfed parcle swarm opmzer, Proceedngs of he IEEE Congress on Evoluonary Compuaon, Pscaaway, New Jersey, pp (998) [9]Eberhar, RC, Sh, Y, Comparng nera weghs and consrucon facors n parcle swarm opmzaon Proceedngs of he Congress on Evoluonary Compuaon, pp (2000)