Dinkelbach s Algorithm as an Efficient Method for Solving a Class of MINLP Models for Large-Scale Cyclic Scheduling Problems

Transcription

1 Dnkelbach s Algorhm as an Effcen Mehod for Solvng a Class of MINLP Models for Large-Scale Cyclc Schedulng Problems Fengq You, 1 Pedro M. Casro, 2 Ignaco E. Grossmann 1 1 Deparmen of Chemcal Engneerng, Carnege Mellon Unversy, Psburgh, PA Deparameno de Modelação e Smulação de Processos, INETI, Lsboa, Porugal Absrac January, 2009 In hs paper we consder he soluon mehods for mxed-neger lnear fraconal programmng (MILFP models, whch arse n cyclc process schedulng problems. We frs dscuss convexy properes of MILFP problems, and hen nvesgae he capably of solvng MILFP problems wh MINLP mehods. Dnkelbach s algorhm s nroduced as an effcen mehod for solvng large scale MILFP problems for whch s opmaly and convergence properes are esablshed. Exensve compuaonal examples are presened o compare Dnkelbach s algorhm wh varous MINLP mehods. To llusrae he applcaons of hs algorhm, we consder ndusral cyclc schedulng problems for a reacon-separaon nework and a ssue paper mll wh byproduc recyclng. These problems are formulaed as MILFP models based on a connuous me Resource-Task Nework (RTN. The resuls show ha orders of magnude reducon n CPU mes can be acheved when usng hs algorhm compared o solvng he problems wh commercal MINLP solvers. Keywords: Cyclc Schedulng; Mxed-Ineger Lnear Fraconal Programmng; Mxed-neger Nonlnear Programmng; Dnkelbach s Algorhm; Resource-Task Nework; To whom all correspondence should be addressed. E-mal: grossmann@cmu.edu -1-

2 1. Inroducon Cyclc schedulng s approprae wherever produc demands are sable over exended perods of me. In such cases, cyclc schedulng s usually mplemened o fnd he bes resource assgnmens as well as asks sequencng, and hen repea such producon paern over an exended perod of me (Sahnds & Grossmann, 1991; Pno & Grossmann, 1994; Casro and co-workers, 2003, 2005, 2009; Poche & Warche, Cyclc schedulng problems can be encounered n refnery operaon (Pno e al., 2000, chemcal manufacurng (Sahnds & Grossmann, 1991, pulp mlls (Casro e al., 2003 and paper producon (Casro e al., If connuous-me formulaons are used, he objecve funcon ypcally nvolves he maxmzaon of a ceran economc ndcaor (e.g. revenue, prof dvded by he cycle me, whch s a model varable. If here are lnear consrans and he objecve funcon s gven by he rao of lnear erms dvded by he cycle me, he resulng mahemacal problem s a specal ype of mxed-neger nonlnear program (MINLP ha s known as a mxed-neger lnear fraconal program (MILFP. An MILFP ncludes boh connuous and dscree varables, has an objecve funcon n general form as he rao of wo lnear funcons and all he consrans are lnear. Mahemacally, an MILFP can be formulaed as followng problem (P: (P max a b ax bx s.. c0 j + cjx = 0, j x 0, I1 and { 0,1} x, I2 and I 1 I 2 = I where s assumed ha b0 + bx > 0 for all feasble soluons and where all nequales are convered no equales hrough he use of slack varables. Due o he nonlneares n he objecve funcon and he combnaoral naure because of he presence of 0-1 varables, he -2-

3 MILFP problems can become compuaonally nracable, especally for large scale nsances wh more han hundreds of bnary varables. Snce (P has boh connuous and dscree varables, and ncludes a nonlnear objecve funcon wh nonconvexes, s also a mxed-neger nonlnear programmng (MINLP problem. Thus, opmzaon mehods for MINLP problems can also be ulzed for MILFP problems. Global opmzaon mehods for MINLP problems nclude he branch and reduce mehod (Ryoo & Sahnds, 1996; Tawarmalan & Sahnds, 2004, he α-bb mehod (Adjman, e al., 2000, he spaal branch and bound search mehod for blnear and lnear fraconal erms (Quesada & Grossmann, 1995; Smh & Paneldes, 1999 and he ouer-approxmaon mehod by Kesavan e al. (2000. All hese mehods rely on a branch and bound soluon procedure. The dfference among hese mehods les on he defnon of he convex envelopes for compung he lower bound, and on how o perform he branchng on he dscree and connuous varables. In addon o global opmzaon mehods, MINLP mehods ha rely on convexy assumpons can also be ulzed. Alhough global opmaly canno be guaraneed, hese mehods can usually oban opmal or near opmal soluons for MILFP problems much faser han global opmzers. These MINLP mehods nclude he branch & bound mehod (Leyffer, 2001, generalzed Benders decomposon (Geoffron, 1972, ouer-approxmaon (Duran & Grossmann, 1986; Vswanahan & Grossmann, 1990, LP/NLP based branch and bound (Quesada & Grossmann, 1992, and exended cung plane Mehod (Weserlund and Peersson, A number of compuer codes are avalable ha mplemen hese mehods. The program DICOPT s an MINLP solver ha s based on he Ouer-Approxmaon algorhm, and s avalable n he modelng sysem GAMS (Brooke, The code α-ecp mplemens he exended cung plane mehod by Weserlund and Peersson (1995, Codes ha mplemen he branch-and-bound mehod nclude he code MINLP_BB (Leyffer,

4 avalable n AMPL, and he program SBB whch s also avalable n GAMS. Recenly, he open source MINLP solver Bonmn (Bonam, e al. 2008, whch s par of he COIN-OR projec, mplemens an exenson of he branch-and-cu ouer-approxmaon algorhm ha was proposed by Quesada and Grossmann (1992, as well as he branch and bound and ouer-approxmaon mehod. The res par of hs paper s organzed as follows. In Secon 2, we dscuss he specal properes of solvng MILFP problems wh convex MINLP mehods. The Dnkelbach s algorhm s nroduced and dscussed n Secon 3. In Secon 4 and Secon 5, we presen he numercal examples and cases sudes. Secon 6 concludes hs paper. 2. MINLP Mehods for MILFP Due o he specal srucure of MILFP problems, we can oban global opmal soluons by fndng he local opma wh ceran MINLP mehods. Before addressng he soluon mehods, le us frs consder problem (RP, whch s he nonlnear relaxaon of (P wh all he bnary varables { 0,1} x, 2 I relaxed as connuous varables 0 x 1, I2. (RP max a b ax bx s.. c0 j + cjx = 0, j b 0 + bx > 0 x 0, I1 and 0 x 1, I2 and I 1 I 2 = I I s easy o see ha (RP s a lnear fraconal program (LFP, n whch he objecve funcon s he rao of wo lnear funcons, all he consrans are lnear and all he varables are connuous. The LFP problems are well suded and some of s mporan properes wll asss he soluon of MILFP wh MINLP mehods. -4-

5 Lemma 1: Le Qx = ( a0 + ax ( b0 bx +, and le F be a convex se such ha b over F. Then, Qx s boh pseudoconvex and pseudoconcave over F. 0 + bx 0 Proof: See Chaper 11.4, Lnear Fraconal Programmng n Bazaraa e al. (2004. Lemma 2: Qx s srcly quasconvex and srcly quasconcave over F. Proof: Based on Lemma 1 ha Qx s pseudoconvex and pseudoconcave, we can conclude ha Qx s also srcly quasconvex and srcly quasconcave over F (Bazaraa e al., Lemma 3: Every local opmum of Qx over F s also s global soluon. Proof: Lemma 2 and he properes of srcly quasconvex and srcly quasconcave funcons (Bazaraa e al., 2004 lead o Lemma 3. Based on Lemma 3, we can conclude ha he local maxmum obaned by solvng he LFP problem (RP wh any nonlnear programmng (NLP solver s also he global maxmum of (RP. Furhermore, we can oban he global opmal soluon of he MILFP problem (P wh an MINLP mehod ha can handle pseudoconvex/pseudoconcave objecve funcon (eg. branch-and-bound, α-ecp mehods. Lemma 4: A global opmal soluon of he MILFP problem (P can be obaned by solvng wh MINLP mehods ha can handle pseudoconvex/pseudoconcave objecve funcon (eg. branch-and-bound, α-ecp. Proof: Based on Lemma 3, he NLP relaxaon of he MILFP problem (P or an NLP subproblem wh a subse of fxed 0-1 varables can be solved o global opmaly by usng a local NLP solver. Snce he branch-and-bound mehod reles on solvng a sequence of relaxed -5-

6 NLP subproblems, global opmaly can be guaraneed wh ha mehod. The α-ecp mehod by defnon can handle pseudoconvex/pseudoconcave funcons, and herefore can also guaranee he global opmaly. Lemma 4 allows us o oban global opmal soluons of MILFP problems wh convex MINLP solvers, such as SBB and α-ecp, whch are poenally faser han usng a global opmzer. Noe ha he Generalzed Benders Decomposon (Geoffron, 1972 algorhm and Ouer-Approxmaon (Duran & Grossmann, 1986; Vswanahan & Grossmann, 1990 mehod do no guaranee global opmaly, snce n boh cases he lower/upper bounds predced by he maser problem rely on he assumpon ha he funcons are convex. Besdes he MINLP mehods, an alernave o solve he MILFP problems s o use Dnkelbach s algorhm by successvely solvng a number of mxed-neger lnear programmng (MILP problems. 3. Dnkelbach's Algorhm for MILFP By explong he relaonshp beween nonlnear fraconal programmng and nonlnear paramerc programmng, Dnkelbach (1967 developed an algorhm o solve convex nonlnear fraconal programmng problems (whou dscree varables by successve solvng a sequence of smplfed NLP problems. As an exenson, Dnkelbach s algorhm has recenly been mplemened o solve he MILFP problems (Bradley and Arnzen, 1999; Poche & Warche, 2008, bu he opmaly and convergence properes of usng Dnkelbach s algorhm o solve MILFP problems have no been fully addressed (Warche, In hs secon, we frs nroduce he algorhmc scheme for solvng MILFP problems wh Dnkelbach s algorhm, and hen show ha smlar opmaly and convergence properes (such as superlnear convergence rae of usng Dnkelbach s algorhm for convex nonlnear fraconal programmng problems (Dnkelbach, 1967; Schable, 1976 also exs n usng hs -6-

7 algorhm for MILFP problems. Consder N( x a0 ax = +, Dx = b0 + bx, Qx Nx / Dx = and he feasble regon of (P as S: { D( x > 0 and c 0 j + cjx = 0, j and x 0, I1 and x = 0, 1, I2 and I I = I }, he Dnkelbach s algorhm for MILFP problem (P: { Qx x S} max 1 2 s as follows: Sep 1: choose arbrary x1 S and se q 2 ( 1 N x = (or se q 2 = 0, le k = 2. D x 1 Sep 2: solve he MILP problem F( qk max { N( x qkd( x x S} =, and denoe he opmal soluon as x k. Sep 3: If F( qk < δ (opmaly olerance, sop and oupu x k as he opmal soluon; If F( qk δ, le q k + 1 ( k N x = and go o Sep 2 o replace k wh 1 D x k + and q k wh q k + 1. k Proposon 1 (Opmaly: F( q { N x q D x x S} N x = = max { / }. D x q N x D x x S Proof: See appendx. = max = 0 f and only f Proposon 2 (Convergence: Dnkelbach s algorhm converges superlnearly for MILFP problem (P. Proof: See appendx. -7-

8 Based on Proposon 1 and Proposon 2, we can solve a sequence of MILP subproblems o oban he global opmal soluon of an MILFP problem. A major advanage of usng hs algorhm s ha no NLP solver s requred, and he memory usage durng he compuaonal process ends o be raher small compared wh usng MINLP mehods, especally for large-scale MILFP problems. In he nex wo secons, we presen compuaonal resuls for solvng MILFP problems wh Dnkelbach s algorhm and varous MINLP mehods. 4. Compuaonal Resuls In order o llusrae he applcaon of he proposed soluon sraeges, we presen compuaonal expermens for 20 large scale nsances. The compuaonal expermens are carred ou on an Inel 3.2 GHz machne wh 512 MB memory. The proposed soluon procedure s coded n GAMS (Brooke, The MILP problems n Dnkelbach s soluon procedure are solved usng CPLEX 10, he MINLP solvers nclude SBB (specal branch-and-bound algorhm, DICOPT (ouer-approxmaon algorhm, and α-ecp (α - exended cung-plane mehod, and he global opmzer used n he compuaonal expermens s BARON The relave opmaly olerance s se o be We solve he MILFP problems (CE as follows: (CE max a + a1 x + a2 y 0 b + b1 x + b2 y 0 j J j j j J j j, k K s.. c + c 0 1 x + c 2 y = 0 k k j J jk j b + b 0 1 x + b 2 y j j x 0, I j J y, j J and { 0,1} whch ncludes I connuous varables, J bnary varables, and K + 1 consrans/equaons. The values of I, J and K range from 100 o 2,000. Gven he large sze of problems, he npu daa are generaed randomly. a 0 s generaed unformly on U[-10, 10], b 0 s generaed unformly on U[-10, 10], a 1 s generaed unformly on U[0, 10], a 2 j s generaed unformly on U[0, 10], b 1 s generaed unformly on U[0, 10], b 2 j s j -8-

9 generaed unformly on U[0, 10], c 0k s generaed unformly on U[-15, 20], c 1 k s generaed unformly on U[-10, 10], c 2 jk s generaed unformly on U[-20, 15]. For each nsance, we solve Problem (CE wh Dnkelbach s algorhm usng CPLEX (he nal value of q s se o 0, and MINLP solvers SBB, DICOPT, α-ecp, as well as he global opmzer BARON. The compuaonal resuls are gven n Table 1. The opmal soluons obaned wh all he mehods are he same and equal o he global maxmum, alhough DICOPT does no guaranee global opmaly. The CPU mes vary from one approach o he oher. Specfcally, solvng MILFP problems wh Dnkelbach s algorhm requres up o 7 eraons and s usually faser han usng he global opmzer BARON and he MINLP solver α-ecp. Dnkelbach s algorhm usually requres slghly more compuaonal mes han SBB and DICOPT for medum nsances, bu usually less CPU me han hese MINLP solvers for large scale nsances. More mporanly, for very large scale nsances wh up o housands of consrans and varables (nsances 18-20, DICOPT and SBB boh exceed he memory lm whle Dnkelbach s algorhm can sll reurn global opmal soluons. Ths s due o he smaller memory requremen when solvng MILP raher han MINLP problems. As for he comparson beween he MINLP solvers and he global opmzer, SBB and DICOPT requre smlar compuaonal resources and are boh much faser han α-ecp and BARON, especally for large scale nsances (nsances The compuaonal sudes sugges ha SBB and DICOPT mgh be he mos effcen solvers for medum sze MILFP problems, and Dnkelbach s algorhm may have hgher compuaonal effcency for very large scale nsances. In all runs, DICOPT was able o fnd he global opmal soluons, despe he fac ha he ouer-approxmaon mehod canno guaranee global opmaly for pseudoconvex/pseudoconcave funcons. The resuls also show ha Dnkelbach s algorhm wh CPLEX and he convex MINLP solvers (SBB, DICOPT and α-ecp are far more effcen han global opmzer BARON for solvng MILFP problems. -9-

10 No. Con. Var. I Dsc. Var. J Table 1 Compuaonal resuls for MILFP Problems wh Dnkelbach s algorhm and MINLP mehods Eqs. K +1 Dnkelbach s Algorhm (CPLEX SBB DICOPT α-ecp BARON Ier. Obj. CPU (s Obj. CPU (s Obj. CPU (s Obj. CPU (s Obj. CPU (s ~ , , , ~ , , , ~ , , , ~ , ,000 1,000 1, , , ,000 1,000 1, , , ,000 2,000 1, , , ,000 1,000 2, >memo --- > memo --- 3, , ,000 2,000 1, >memo --- > memo --- 3, , ,000 2,000 2, >memo --- > memo --- 3, ,600 : Subopmal soluons (or lower and upper bounds for BARON obaned afer 1 hour (3,600 seconds : No soluon was reurned afer 1 hour (3,600 seconds >memo: ermnaes because memory upper lm exceed -10-

11 5. Cyclc Schedulng Problems Cyclc schedulng problems can be formulaed as mxed-neger lnear fraconal programs (Bradley and Arnzen, 1999; Poche & Warche, In hs secon, we employ he Dnkelbach s algorhm o solve large scale MILFP models for cyclc producon schedulng from wo case sudes when compared o he MINLP commercal solvers Case sudy 1: Cyclc Schedulng for a Reacon-Separaon Nework The frs case sudy deals wh a varan of he mos suded schedulng problem n he process sysems engneerng leraure (Kondl e al., I nvolves a mulpurpose pharmaceucal bach plan producng wo producs from hree dfferen raw-maerals hrough varous reacon and separaon operaons. Fgure 1 shows he assocaed RTN (Paneldes, 1994, annoaed wh nformaon on ask duraon and mnmum and maxmum ask sze. Noce ha he duraon of he reacon asks s dependen on he ype of equpmen beng used. The produc values are equal o $12,000/kg and $10,000/kg, respecvely, as n Schllng and Paneldes (1999. The problem was frs nvesgaed by Schllng & Paneldes (1999 and more recenly addressed by Casro e al. (2003 and Wu & Ieraperou (2004. Dfferen connuous-me mulpurpose formulaons can be used o solve he problem. The frs wo conrbuons reled on a sngle me grd o keep rack of evens akng place, whle he laer s a un-specfc formulaon. In hs paper, we wll be usng he one proposed by Casro and co-workers (2003, 2005, whch has been shown o be more effcen han he one by Schllng & Paneldes (1999, prmarly due o he need of fewer even pons o fnd he opmal soluon. Bear n mnd, however, ha he formulaon of Wu & Ieraperou (2004 shows beer performance. The small dfferences observed n he values of he repored soluons beween he papers resul from slghly dfferen daa. -11-

12 Produc1 0.4 HoA 0.4 Reacon2_1 Dur.= Sze Sze=20-80 Reacon2_2 Dur.= Sze Sze= InAB Separaon Dur.= E-2Sze Sze=40-80 Sll 0.9 Produc2 FeedB 0.5 InBC Reac1 ImpureE 1.0 Reac2 FeedC Reacon1_1 Dur.= Sze Sze=20-80 Reacon3_1 Dur.= Sze Sze= Reacon1_2 Dur.= Sze Sze= Reacon3_2 1.0 Dur.= E-2Sze Sze=30-80 Fgure 1 Resource-Task Nework represenaon of case sudy 1. Duraon n h, sze n kg. The schedulng formulaon uses he RTN process represenaon, whch reles on generc enes lke resources (se R and asks (se I o descrbe he process/produc recpe. Through he use of he wrap-around operaor Ω( (Shah e al., 1993,,1 T Ω( = + T, < 1 he cyclc schedulng formulaon can be wren n a very compac form, eqs 1-7. In addon, an anchorng consran should be used for reducng soluon degeneracy arsng from cyclc schedulng permuaons. max v Δ r r H (1 FP r R Rr, = Rr, Ω( 1 + [ ( μr, N,, + ν r, ξ,, + ( μr, N,, + ν r, ξ, ( Δ r RM r R Δ r FP r R T < Δ+ Δ+ T = 1 r R, T T Δ < Δ + T ', ] + (2 R = EQ r, T 1+ ( μ r, N,, + μr, N,, r R (3 T T T < Δ+ Δ+ T < Δ+ -12-

13 N mn max μ V ξ N μ V I, T, T, < Δ + (4,, r, r,,,, r, r T EQ EQ r R r R T T α N + βξ I, T, T, < Δ + (5 ' T T H (,, ',, ' (,,,, α N + β ξ I, T, T, Δ + T (6 mn max max H H H T = 0;0 T H ; N {0,1}; ξ, R, Δ 0 (7 ; 1,,,, r, r The bnary varables N denfy he sar of ask a even pon and endng a even,, pon, whle he connuous varables ξ gve he amoun processed, whch mus le,, whn he equpmen un s (r R EQ mnmum ( V and maxmum capaces ( V mn r max r, see eq 4. In hs problem, he amoun handled by he ask affecs s duraon, wh parameers α (h and β (h/kg, gvng he fxed and bach sze dependen processng mes, respecvely. The excess amoun of resource r a even pon s gven by R r,. Raw-maerals (r R RM and fnal producs (r R FP have ne consumpon/producon over he cycle, wh varables Δ r holdng he exac amouns. The laer have a ceran marke value v r ($/kg, and he objecve s o maxmze he revenue whle mnmzng he cycle me, H (h,.e. maxmzng he hourly revenue ($/h, see eq 1. Varable T holds he me of even pon relave o he sar of he cycle. Fnally, srucural parameers μ r,, μ conan he consumpon/producon of resource r a he sar/end (over bar of he ask ha s ndependen on he amoun processed. These are lnked o he dashed arrows n Fgure 1. Parameers ν r, and ν r, are used for he bach sze dependen values and are lnked o he sold arrows. r, Cycle N Cycle N+1 Slo 1 Slo 2 Slo 3 Slo T Slo T T+1=1 2 Fgure 2 Cycle Tme H Even pons and me grds n connuous-me formulaon -13-

14 Table 2 T Δ H mn H max Dsc. Var. Resuls for Case Sudy 1 wh Dnkelbach s algorhm and MINLP mehods Con. Var Eqs. Obj. (k$/h H(h CPU (s Solver (3 er. Dnkelbach SBB DICOPT α-ecp BARON (4 er. Dnkelbach SBB DICOPT α-ecp BARON (3 er. Dnkelbach SBB DICOPT α-ecp BARON (3 er. Dnkelbach ~ ,200 SBB DICOPT α-ecp ~ ,200 BARON (3 er. Dnkelbach ~ ,200 SBB DICOPT α-ecp ~ ,200 BARON (3 er. Dnkelbach ~ ,200 SBB , DICOPT α-ecp ~ ,200 BARON (3 er. Dnkelbach ~ ,200 SBB ,200 DICOPT ,200 α-ecp ~ ,200 BARON , (3 er. Dnkelbach ~ ,200 SBB ,200 DICOPT ,200 α-ecp ~ ,200 BARON : Subopmal soluons (or lower and upper bounds for BARON and SBB obaned afer 2 hours (7,200 seconds -14-

15 The drawback of a me grd formulaon s ha he soluon reurned depends on he number of even pons, T (as shown n Fgure 2, and also on he number of nervals ha a ask can span, whch s specfed hrough parameer Δ. Only for a suffcenly large number of boh parameers can we fnd he global opmal opmum. The soluon also depends on he cycle me range H [H mn ; H max ] as llusraed by Wu & Ieraperou (2004. In hs paper, however, he man goal s o compare he compuaonal performance of varous solvers/algorhms for MILFP problems, so we smply esed hem for dfferen values of hese four parameers. We solve hs problem usng he same machne and sofware envronmen as nroduced n Secon 3 wh Dnkelbach s algorhm (solver CPLEX, MINLP solvers SBB, DICOPT and α-ecp, as well as global opmzer BARON. Table 2 lss he resuls obaned. As we can see, he problem szes ncrease as he number of even pons T and he number of nervals ha a ask can span Δ ncrease. For he frs hree nsances wh T =4, he opmal soluons obaned by usng Dnkelbach s algorhm are he same as he global opmal soluons obaned by usng BARON, and Dnkelbach s algorhm requres much less compuaonal me han BARON. Alhough he opmal soluons from SBB and α-ecp are also he same, hese wo solvers requre longer CPU mes han Dnkelbach s algorhm. DICOPT can oban soluons faser han Dnkelbach s algorhm, bu for he second and he hrd nsances n Table 2, he soluons are no he global opmum. For hose wo nsances wh T =5, he compuaonal resuls n Table 2 show ha boh SBB and BARON faled o converge n 2 hours, alhough subopmal soluons closed o global maxmum are reurned. We can also see ha α-ecp reurned he same global opmal soluons as hose obaned by Dnkelbach s algorhm, alhough α-ecp requres more CPU mes. DICOPT s he fases solver, bu agan reurned subopmal soluons for hese wo nsances. I s neresng o noe ha for he second, hrd and fourh nsances, f he lower bound of he cycle me s reduced -15-

16 from 40 o 20, hen DICOPT does converge o he global opmum, because he lower bound predced by he MILP maser problem s a vald bound for hese new nsances. For large scale nsances wh T =6 and T =7, mos of he MINLP solvers faled o converge n 2 hours (DICOPT and α-ecp can boh solve he sxh nsance, bu Dnkelbach s algorhm can solve all he hree nsances whn 1 hour. Overall, Dnkelbach s algorhm s clearly he bes approach for solvng large scale MILFP problems for hs cyclc process schedulng problem. I requres much less compuaonal mes han mos of he MINLP solvers and guaranees he global opmaly of he soluons. Alhough DICOPT s a fas solver for hs ype of problems, may reurn subopmal soluons for some nsances Case sudy 2: Cyclc Schedulng for a Tssue Paper Mll wh Byproduc Recyclng The second case sudy s aken from Casro e al. (2009 and nvolves a connuous mulproduc ssue-paper plan. The cyclc schedulng problem, occurrng a a Scandnavan ssue paper mll consss of maxmzng he prof of he plan and fndng he opmal cycle me whle meeng mnmum demands for all producs. There are bascally wo producon sages, each feaurng wo connuous lnes n parallel, he sock preparaon lnes n he frs sage, and he ssue-paper machnes n he second sage, see Fgure 3. In he frs sage, he dfferen raw-maerals (ONP=old newspaper, MOW=mxed offce wase, VF=vrgn fber ogeher wh some recycled fber (broke are prepared for he ssue machnes. Fve nermedae producs resul from sock preparaon, wh wo of hem havng dedcaed sockyard ples of lmed capacy. These are hen processed n he ssue machnes o ge he fve quales, from leas o mos valuable produc: P50, P60, P75, P80 and P85, where he number represens he brghness n %ISO. -16-

17 Raw maeral ONP Raw maeral MOW De-nkng lne 1 Raw maeral VF Sludge & rejec Tssue Machne 1 Ash P50 P60 P75 P80 P85 De-nkng lne 2 Tssue Machne 2 P50 P60 P75 P80 P85 Broke Sorage Inermedae Sorage 1 Inermedae Sorage 2 Fgure 3. Flowshee of he ssue mll. In order o be more envronmenally frendly and lower he operang coss, a proper broke recyclng sraegy should be mplemened. Broke s manly generaed when processng fbers n he ssue machnes and n he downsream converng lnes. Despe he fac ha he laer are no shown n he flowshee, her fber losses are accouned for n he ssue machnes. Three scenaros were analyzed n Casro e al. (2009: ( no recyclng; ( ncorporae all broke n he lowes qualy producs (P50 and P60; ( raher han mxng all broke, recycle accordng o qualy. More specfcally, broke from P85 wll parly replace VF, broke from P80, MOW, whle broke from he remanng producs wll be mxed wh ONP. The RTN correspondng o hs opon, he mos cos effcen, s shown n Fgure 4. Smlar o case sudy 1, we wll be dfferenang he resources. Equpmen uns (R EQ nclude he sock preparaon lnes (SP1-2, ssue machnes (TM1-2, dewaerng lne ha prepares he nermedaes for sorage (DW, and he wo dedcaed sorage lnes (L67, L80. The subse of raw-maerals (R RM conans ONP, MOW and VF; he nermedaes comprse ONP50, ONP60, ONP67, MOW80, VF85, S67 and S80; and he fnal producs (R FP ake n P50-P85. The elemens of he subse of broke resources (R BK vary accordng o he recyclng polcy. As for he asks, all are of he connuous ype and are characerzed by maxmum -17-

18 processng rae ρ max. A pecular problem feaure s ha we canno se beforehand he proporon of broke o be ncorporaed n a ceran nermedae, snce he amoun produced wll be deermned as par of he opmzaon procedure and we wan o use as much as possble. In oher words, here are flexble proporons of npu maerals for he asks execued n he sock preparaon lnes. Such varable recpe asks (I VR wll be subjec o addonal ses of consrans. VF SP2_VF85 VF 85 TM1_P85 TM2_P P BR 85 MOW SP1_MOW80 SP1 SP2_ONP67 L80 DW MOW 80 ONP 67 GFS_80 GTS_80 RaeMax=48 /day GTS_67 RaeMax=48 /day S TM1_P80 TM2_P80 TM1_P75 TM2_P TM1 P80 P BR 80 ONP SP2 SP2_ONP60 ONP 60 L67 GFS_67 S67 TM2_P60 TM1_P TM2 P BR SP2_ONP50 ONP 50 TM1_P50 TM2_P P Fgure 4. Resource-Task Nework represenaon of bes broke recyclng polcy. In conras o her bach counerpars, can be assumed whou loss of generaly ha connuous asks las a sngle me nerval. As a consequence, only one me ndex s needed n he ask-relaed model varables. Bnary varables N, assgn he sar of ask o even pon, whle he oal amoun produced s gven by ξ,. For varable recpe asks, he par due o resource r s accouned for by ξ r,,. In order o denfy changeovers n he ssue -18-

19 machnes, varables X r,,, are employed. The remanng varables were already presened n case 1. The model consrans are gven nex. The objecve funcon s he maxmzaon of he annual prof, gven n relave moneary uns (r.m.u./year, eq 8. The frs erm ncludes he maerals value, whch s negave for raw-maerals and broke resources and posve for fnal producs. The second erm ncludes he oal operang cos and he hrd, he oal changeover cos. Parameer wd gves he number of workng days per year. Noce ha he numeraor s lnear whle he denomnaor, he cycle me H, s a varable, makng hs model an MILFP. The oher consrans are smlar o case 1, wh dfferences arsng from handlng connuous raher han bach asks. In parcular, noe ha srucural parameers λ r, and λ r, have replaced ν r, and ν r,, n he excess resource balances (compare eq 9 o eq 2. Furhermore, he upper bound on he amoun processed s now deermned by he produc of he maxmum processng rae and he upper bound on he cycle me (eq 12. In addon, he mng consrans, eq 17, are wren for every equpmen resource raher han every ask, where he duraon of he ask s deermned by dvdng he amoun by he maxmum processng rae. Specfc consrans for hs example are gven n eqs The exen of a ceran varable recpe ask mus be equal o he sum of he paral exens, where parameer ff (fber facor can be vewed as a yeld. Eq 14 lms broke ncorporaon o 20% and eq 15 acvaes he changeover varables. Fnally, eq 16 ensures ha he produc demands are me. max v Δ coξ ccg X / r r,, ' r,, ', wd H RM BK FP I T TM r R R R r R ' I T (8 R = R r RM r R ( μ N r FP BK r R R = 1 + μ N r, r, Ω( 1 r,, r,, Ω( 1 r,, Ω( 1 r, r,, Ω( 1 ( Δ Δ + r R, T + λ ξ + λ ξ + (9 R EQ r, T 1 + μ r, N, T r R (10 = -19-

20 max IT Rr, Rr r R, T (11 max max ξ H N I, T (12, ρ, VR ffξ, = λ, rξr,, I, T (13 RM BK r R R VR 0.2 ff ξ λ I, T (14,, rξr,, BK r R TM X r,, ', N', + N, ( 1 1 r R,, ' I T (15 Ω, d T H wd Δ d H r R mn max FP r r r (16 + EQ H T max μ r / r R T = T, ξ, ( ρ, T H mn H H max max ; T1 = 0; 0 T H ; N, {0,1};0 X r,, ', 1; ξ,, ξr,,, Rr,, Δr 0 (18 We consder four nsances, oal number of even pons T =4, T =5, T =6 and T =7 (Δ=1 for all he nsances. We solve hs problem usng he same machne and sofware envronmen as nroduced n Secon 3 wh Dnkelbach s algorhm (solver CPLEX. Gven he problem szes of hese nsances and he fac from he prevous case sudy ha global opmzer BARON usually requres long CPU mes for large scale nsances, we only compare he performance of Dnkelbach s algorhm wh hose of he MINLP solvers SBB, DICOPT and α-ecp. The resuls are lsed n Table 3. From Table 3, we can see ha he same opmal soluons are obaned by usng Dnkelbach s algorhm and MINLP solvers for he nsances ha MINLP solvers can reurn opmal soluon whn 20 hours. Smlar o wha we observed n he prevous case sudy, Dnkelbach s algorhm requres much less compuaonal mes han he MINLP solvers, especally SBB. For he large scale nsances wh T =6 and T =7, only Dnkelbach s algorhm and DICOPT can reurn opmal soluons, whle SBB and α-ecp faled o converge whn 20 hours. For hese wo nsances, Dnkelbach s algorhm sll requres sgnfcanly less CPU me han DICOPT, alhough DICOPT does reurn global opmal soluons for hese nsances. -20-

21 Noce ha he cycle me s convergng o s upper bound, ndcang ha he objecve funcon would connue o ncrease. However, longer cycle mes would also mean longer sorage perods for he nermedaes and hgher degradaon resulng from nsance from drec sunlgh, whch s no desrable. Snce he obaned opmal schedules are he same as n Casro e al. (2009, he readers can refer o hs paper for he dealed soluons. The compuaonal resuls of hs case sudy sugges he advanage of usng Dnkelbach s algorhm o solve MILFP models of cyclc schedulng problems. Table 3 Resuls for Case Sudy 2 wh Dnkelbach s algorhm and MINLP mehods T H mn H max Dsc. Con. Obj. Eqs. Var. Var. (r.m.u./yr H(h CPU (s Solver (4 er. Dnkelbach , SBB DICOPT α-ecp (3 er. Dnkelbach ,000 SBB , DICOPT , α-ecp , (3 er. Dnkelbach ,000 SBB , DICOPT ,000 α-ecp , (3 er. Dnkelbach ,000 SBB , DICOPT ,000 α-ecp : Subopmal soluons obaned afer 20 hours (72,000 seconds 6. Conclusons In hs paper, we have dscussed he convexy properes of MILFP problems and her NLP relaxaon. The capably and opmaly of solvng MILFP problems wh varous MINLP mehods has been addressed. We also show ha Dnkelbach s algorhm, whch s used o solve connuous fraconal programmng problems, can be an effcen mehod for -21-

22 solvng large scale MILFP problems. We have addressed he opmaly and convergence ssues of hs algorhm, and proved ha also has superlnear convergence rae. Exensve compuaonal sudes were performed o demonsrae he effcency of hs algorhm. Case sudes of cyclc schedulng problems for a reacon-separaon nework and for a ssue paper mll wh byproduc recyclng were also consdered o llusrae he praccal capably of hs algorhm. The resuls show ha Dnkelbach s algorhm requred sgnfcanly less compuaonal effor o solve MILFP es problems compared o commercal MINLP solvers, such as DICOPT (Ouer-Approxmaon, SBB (specal branch-and-bound, α-ecp (exended cung plane mehod, and he global opmzer BARON (branch-and-reduce. Acknowledgmen The auhors acknowledge he fnancal suppor from he Naonal Scence Foundaon under Gran No. OCI and from he Luso-Amercan Foundaon. Appendx = +, Dx = b0 + bx, F ( q max { N( x qd( x x S} Le N( x a0 ax =, Qx = Nx / Dx, and he feasble regon be S: { D( x > 0 and c 0 + c x = 0, j and x 0, I1 and 0 x 1, I2 and I1 I2 = I } j j Before provng Proposon 1, frs consder he followng lemmas. Lemma 4: The funcon F ( q max { N( x qd( x x S} Proof: Le = s convex. x be he opmal soluon ha maxmze F( q' ( 1 q'' 0 1. Thus, we have: + wh q' q'' and -22-

23 ( ' + ( 1 q'' ( ( ' ( 1 '' ( ( ' ( ( 1 ( '' ( max { ( ' ( } ( 1 max { ( '' ( } ( ' ( 1 F ( q'' F q = N x q + q D x = N x q D x + N x q D x N x q D x x S + N x q D x x S = F q + Lemma 5: For q' q'' srcly monoonc decreasng. Proof: Le have: <, we have F( q' > F( q'',.e. F( q max { N( x qd( x x S} x S be he opmal soluon ha maxmze ( '' ( '' { N( x q D( x x S} N( x q D( x ( ' ( max { ' } ( ' F q = max '' = '' N x q D x = N x q D x x S = F q = s F q. Assumng Dx > 0, we Lemma 6: F( q = 0 has a unque soluon. Proof: I s easy o see ha for > 0 Dx, lm F( q = + and lm F( q q q + based on Lemma 5, we can conclude ha F( q = 0 has a unque soluon. =. Furhermore, Lemma 7: For any x ' S, le q ' = N x D x ( '. Then, we have ( ' F q ' 0. Proof: I s easy o see ha { } F q' = max N x q' D x x S N x' q' D x ' = 0. Proof of Proposon 1: a Le { } x S be an opmal soluon of max N x q D x x S, so we have -23-

24 ( ( N x q D x =0 and Snce > 0 N x q D x N x q D x =0, x S. Dx, we have = ( / ( / q N x D x N x D x, x S. Thus, { N x qd x x S} q s he maxmum of max and x s he opmal soluon of max { N( x / D( x x S}. b Le { N x D x x S} x be an opmal soluon of max / and q be he opmal objecve funcon value, so we have = ( / ( / Snce > 0 q N x D x N x D x, x S. Dx, we have N x q D x N x q D x =0, x S. Ths mples ha { } max N x q D x x S. x s he opmal soluon of Before provng Proposon 2, consder he followng lemmas. Lemma 8: Le ( ' D( x'' D x. x ' and x '' be he opmal soluon of ( ' F q and ( '' F q. If q' < q'', Proof: Snce x ' and '' x be he opmal soluon of F( q ' and F( q '', we have: ( ' ' ( ' ( '' ' ( '' N x q D x N x q D x ( '' '' ( '' ( ' '' ( ' N x q D x N x q D x Add he above wo nequales and rearrange boh sdes of he new one, we have: As '' > ' ( q'' q' D( x' ( q'' q' D( x '' q q, herefore we have D( x' D( x''. Lemma 9: Le x ' and x '' be he opmal soluon of ( ' F q and F( q ''. Then, -24-

25 ( '' Q( x' Q x ( '' ( '' ( '' ( ' F q F q D x D x. Proof: As x '' s he opmal soluon of F( q '', we have: ( '' '' ( '' ( ' '' ( ' N x q D x N x q D x Dvdng boh sdes by D( x ' > 0, we have N x'' q'' D x'' N x' q''. D x' D x' D x' Thus, we have ( '' Q( x' ( '' F( q'' ( '' D( x' Q x N x'' N x' N x'' N x'' D x'' D x'' 1 1 = + q'' = ( F( q'' D x'' D x' D x'' D x' D x' D x'' D( x' D( x'' F q = D x Lemma 10: Le x ' and x '' be he opmal soluon of ( ' F q and F ( q ''. If q' q q '', Then, Q( x' Q( x'', where q sasfes F q = 0. Proof: The proof readly follows from Lemmas 7, 8 and 9. Lemma 11: Le x ' and x '' be he opmal soluon of ( ' F q and F( q ''. Then, 1 1 Q( x'' Q( x' F( q'' + ( q' q'' D( x''. D( x' D( x'' Proof: Smlar o Lemma 9, consderng he opmaly condon for x ', We have N( x' q' D( x' N( x'' q' D( x''. Dvdng boh sdes by D( x ', we have N x ' N x ' '' D x q q' ''. D x' D x' D x' Thus we have, -25-

26 ( '' Q( x' Q x N x'' N x'' D x'' D x'' 1 1 q' = ( N( x'' + q' D( x'' D x'' D x' D x' D x'' D( x' D( x'' 1 1 = F( q'' + ( q' q'' D( x'' D( x' D( x'' Lemma 12: Le x ' and x be he opmal soluon of ( ' F q and F q, where q sasfes F( q = 0. Then we have: q Q x ( q q Proof: Snce prove Lemma 12. D x ( ' ' 1 D x q sasfes F( q = 0, we have Q( x ( ' = q. Based on Lemma 11, s easy o Proof of Proposon 2: Dnkelbach s algorhm updaes q wh he prevous value of Q( x,.e. q Q( x 1 + =, where x s he opmal soluon of F ( q. From Lemma 12, he algorhm converges wh a rae of D x 1 D x. From Lemma 5, we have D x 1 D x s nonncreasng. Therefore, he sequence { q } obaned by Dnkelbach s algorhm converges superlnearly o q for each q < q. References 1 Adjman, C. S.; Androulaks, I. P.; Floudas, C. A. (2000. Global opmzaon of mxed-neger nonlnear problems. AIChE Journal, 16(9, Bazaraa, M. S.; Sheral, H. D.; Shey, C. M. (2004. Nonlnear Programmng: Theory and -26-

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