Predictive route choice control of destination coded vehicles with mixed integer linear programming optimization

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1 Delf Universiy of Technology Delf Cener for Sysems and Conrol Technical repor Predicive roue choice conrol of desinaion coded vehicles wih mixed ineger linear programming opimizaion A.N. Tarău, B. De Schuer, and J. Hellendoorn If you wan o cie his repor, please use he following reference insead: A.N. Tarău, B. De Schuer, and J. Hellendoorn, Predicive roue choice conrol of desinaion coded vehicles wih mixed ineger linear programming opimizaion, Proceedings of he 12h IFAC Symposium on Transporaion Sysems, Redondo Beach, California, pp , Sep Delf Cener for Sysems and Conrol Delf Universiy of Technology Mekelweg 2, 2628 CD Delf The Neherlands phone: (secreary) fax: URL:hp:// This repor can also be downloaded viahp://pub.deschuer.info/abs/09_030.hml

2 Predicive roue choice conrol of desinaion coded vehicles wih mixed ineger linear programming opimizaion A.N. Tarău B. De Schuer, J. Hellendoorn Delf Cener for Sysems and Conrol Delf Universiy of Technology, Mekelweg 2, 2628 CD Delf, The Neherlands Marine and Transpor Technology Deparmen Delf Universiy of Technology, The Neherlands, Absrac: Sae-of-he-ar baggage handling sysems ranspor luggage in an auomaed way using desinaion coded vehicles (DCVs). These vehicles ranspor he bags a high speeds on a mini railway nework. In his paper we consider he problem of conrolling he roue of each DCV in he sysem. This is a nonlinear, nonconvex, mixed ineger opimizaion problem. Nonlinear model predicive conrol (MPC) for mixed ineger problems is usually very expensive in erms of compuaional effor. Therefore, in his paper we presen an alernaive approach for reducing he complexiy of he compuaions by simplifying and approximaing he nonlinear opimizaion problem by a mixed ineger linear programming (MILP) problem. The advanage is ha for MILP opimizaion problems solvers are available o allow us o efficienly compue he global opimal soluion. The soluion of he MILP problem can hen be used as a good iniial saring poin for he original nonlinear opimizaion problem. To assess he performance of he proposed formulaion of he MPC opimizaion problem, we consider a benchmark case sudy, he resuls being compared for several scenarios. Keywords: Baggage handling sysems, roue choice conrol, model predicive conrol. 1. INTRODUCTION Modern baggage handling sysems in airpors ranspor luggage a high speeds using desinaion coded vehicles (DCVs). These vehicles ranspor he bags a high speed on a mini railway nework. Low-level conrollers ensure he coordinaion and synchronizaion when loading a bag ono a DCV, in order o avoid damaging he bags or blocking he sysem, and when unloading i o he corresponding end poin. Low-level conrollers also compue he velociy of he DCVs such ha collisions are avoided. Currenly, he DCVs are roued hrough he sysem using rouing schemes based on preferred roues. These rouing schemes can be adaped o respond on he occurrence of predefined evens. However, as argued by de Neufville (1994), he paerns of loads on he sysem are highly variable, depending on e.g. he season, ime of he day, ype of aircraf a each gae, number of passengers for each fligh. Therefore, in he research we conduc we do no consider predefined preferred roues. Insead we develop advanced conrol mehods o deermine he opimal rouing in case of dynamic demand. For applicaions such as auomaed guided vehicles roue planning or raffic roue guidance, he roue assignmen problem has been addressed by e.g. Gang e al. (1996); Kaufman e al. (1998). Bu, in our case we do no deal wih a shores-pah or shores-ime problem, since we need he bags a heir corresponding end poin wihin a given ime window. Fay (2005) solved he rouing problem of DCVs ransporing bags using an analogy of how daa are ransmied via inerne, bu wihou presening any experimenal resuls. Also, Hallenborg and Demazeau (2006) presen a muli-agen approach for he conrol sofware of a DCV-based baggage handling sysem. However, his muli-agen sysem is faced wih maor challenges due o he exensive communicaion required. The goal of our work is o develop and compare efficien conrol approaches for roue choice conrol of each DCV on he rack nework. Theoreically, he maximum performance of such a DCV-based baggage handling sysems would be obained if one compues he opimal roues using opimal conrol (Lewis, 1986). However, as shown by Tarău e al. (2008), his conrol mehod becomes inracable in pracice due o he heavy compuaion burden. Therefore, in order o make a rade-off beween compuaional effor and opimaliy, in (Tarău e al., 2009), we have also implemened cenralized and decenralized model predicive conrol (MPC), and also a decenralized heurisic approach. As he resuls confirmed, cenralized MPC requires high compuaion ime o deermine a soluion. The use of decenralized conrol lowers he compuaion ime, bu a he cos of subopimaliy. In his paper we invesigae wheher he compuaional effor required for compuing he roue of each DCV by using MPC can be lowered even more by using mixed ineger linear programming (MILP). The large compuaion ime obained in previous work comes from solving he nonlinear, nonconvex, mixed ineger opimizaion problems. Noe ha such problems may also have muliple local minima and are NP hard, and herefore, difficul o solve. So, in his paper we rewrie he roue choice problem as an MILP problem for which efficien solvers are available. The soluion of his MILP can hen be

3 used as an iniial saring poin for he original nonlinear opimizaion problem. The paper is organized as follows. Secion 2 briefly inroduces he conceps of MPC ha will be laer on used in solving he roue choice problem. In Secion 3, we briefly recapiulae an even-driven roue choice model ha we have developed (Tarău e al., 2008). Aferwards, in Secion 4 we approximae he model by using MILP equivalences. Boh he nonlinear and MILP model are hen used o deermine he roue of DCVs using MPC. The analysis of he simulaion resuls and he comparison of he proposed formulaions are elaboraed in Secion 6. Finally, Secion 7 draws he conclusions for he paper. 2. BACKGROUND Since laer on we will use model predicive conrol (MPC) for deermining he roues of he DCVs in he nework, in his secion we briefly inroduce he basic MPC conceps. MPC is an on-line model-based predicive conrol design mehod (Macieowski, 2002) ha uses a receding horizon principle. As illusraed in Fig. 1, in he basic MPC approach, given a horizon N, a sep k 0, where k is ineger valued, corresponding o he ime insan k = k wih he sampling ime, he fuure conrol sequence u(k),u(k+1),...,u(k+n 1) is compued by solving a discree-ime opimizaion problem over he period[ k, k +N ] so ha a performance index defined over he considered period [ k, k + N ] is opimized subec o he operaional consrains. Afer compuing he opimal conrol sequence, only he firs conrol sample is implemened, and subsequenly he horizon is shifed. Nex, he new sae of he sysem is measured or esimaed, and a new opimizaion problem a ime k+1 is solved using his new informaion. In his way, a feedback mechanism is inroduced. pas u fuure Fig. 1. Predicion horizon in MPC. u(k+ 1) u(k+ N 1) u(k) k k+ 1 k+ N horizon 3. MODELS 3.1 Sysem descripion and original model In his secion we briefly recapiulae he even-driven roue choice model of a baggage handling sysem ha we have developed in (Tarău e al., 2008). The DCV-based baggage handling sysem operaes as follows: given a demand of bags and he nework of racks, he roue of each DCV (from a given loading saion o he corresponding unloading saion) has o be compued subec o operaional and safey consrains such ha he performance of he sysem is opimized. The model of he baggage handling sysem we have developed in (Tarău e al., 2008) consiss of a coninuous par describing he movemen of he individual vehicles ransporing he bags hrough he nework, and of he following discree evens: loading a new bag ono a DCV, unloading a bag ha arrives a is end poin, updaing he posiion of he swiches ino and ou of a uncion, and updaing he speed of a DCV. The sae of he sysem consiss of he posiions of he DCVs in he nework and he posiions of each swich of he nework. According o he discree-even model of (Tarău e al., 2008), as long as here are bags o be handled, given he curren sae, he sysem evolves as follows: we shif he curren ime o he nex even ime, ake he appropriae acion, and updae he sae of he sysem. The operaional consrains derived from he mechanical and design limiaions of he sysem are he following: he speed of each DCV is bounded beween 0 and v max, while a swich a a uncion has o wai a leas ime unis beween wo consecuive swiches in order o avoid he quick and repeaed back and forh movemens of he swich which may lead o mechanical damage. 3.2 Simplified roue choice model Nework We represen he mini railway nework ha DCVs use o ranspor he luggage as a direced graph. Then he nodes via which he DCVs ener he nework are called loading saions, he nodes via which he DCVs unload he ranspored bags are called unloading saions, while all oher nodes in he nework are called uncions. The secion of rack beween wo nodes is called rack segmen (or link). For each rack segmen a free-flow ravel ime is assigned. This free-flow ravel ime represens he ime period ha a DCV requires o ravel hrough a rack segmen in case of no congesion, using, hence, maximum speed. In order o simplify he explanaion of our approach we assume ha he free-flow ravel ime of a link is always a muliple of. We assume wihou loss of generaliy ha in our nework each uncion has maximum wo incoming and maximum wo ougoing links indexed by l {0,1} as illusraed in Fig. 2. This assumpion of a nework corresponds o curren pracice in sae-of-he-ar baggage handling sysems. Exra model assumpions In order o ransform he roue choice problem ino an MILP problem, we firs simplify i by assuming he following: We only deermine he posiion of he swiches ou of uncions. We do no conrol he posiion of he swiches ino uncions. For hese swiches we assume ha lowlevel conrollers are insalled o oggle he posiion such ha a DCV can ener he uncion as soon as possible. This assumpion lowers he compuaional complexiy. Noe however ha an exension o also conrolling he swich ino he uncion is sraighforward. Fig. 2. Incoming and ougoing links a a uncion. Boh swiches are posiioned on.

4 D i () Fig. 3. Demand profile. The DCVs run wih maximum speed along he rack segmen and, if necessary, hey wai before crossing he uncion in a verical queue. The dynamic demand D i of loading saion L i, i {1,...,L}, where L is he number of loading saions, is approximaed wih a piecewise consan demand as illusraed in Fig. 3. The piecewise consan demand D i has level changes occurring only a ineger muliples of. This is necessary in order o easily combine he ime when a bag reaches a queue a a uncion wih he ime when he demand changes. So, in he ime inerval [ k, k+1 ), wih k = k, he demand is D i (k). Simplified model In order o illusrae he derivaion of he roue choice model le us now consider he mos complex cell a nework can conain, which is depiced in Fig. 4 where uncion S r has wo neighboring uncions S s and S p conneced via is incoming links. Nex we presen how he evoluion of he queue lengh a uncion S r is deermined. The conrol ime sep for each uncion in he nework is. So, a each sep k 0 he conrol acions u s (k) and u p (k) are compued for uncions S s and S p. A conrol acion a sep k corresponds o he posiion of he swich on he ougoing or 1 of a uncion during he period [ k, k+1 ). So, a sep k each of he conrol signals u s (k) and u p (k) is eiher 0 or 1. Le q r (k) denoe he lengh of he queue a uncion S r a ime sep k. Recall ha each link in he nework has been assigned a given free-flow ravel ime. Le us denoe he link beween wo nodes a and b as a b. Then, as illusraed in Fig. 4, he free-flow of he link S s S r is r and he free-flow of he link S p S r is T pr. Hence, he conrol signals u s (k) and u p (k) influence q r afer r and respecively T pr ime seps. The evoluion of queue q r, he lengh of which is always greaer han or equal o 0, is given by: q r (k+ 1)=max ( 0, f r (k) ) (1) where f r (k) is defined as: f r (k)=q r (k)+ ( I r (k) O max ) Ts S s r ld0 Fig. 4. Nework elemens. S r ld1 T pr S p wih I r (k) denoing he number of vehicles ha ener uncion S r or he verical queue a S r during he period [ k, k+1 ) and O max he maximum ouflow 1 of a uncion. The variable I r (k) is defined as follows: I r (k)=u s (k r )O s (k r )+ ( 1 up (k T pr ) ) O p (k T pr ) (2) where O s (k) and O p (k) are he ouflow of uncion S s and respecively S p during [ k, k+1 ). If k < 0, hen O (k) is equal o 0 by definiion. The erm u s (k)o s (k) represens he inflow 2 of he link S s S r a sep k due o he conrol acion u s (k). So, if u s (k) = 0 he inflow ( of he link S s S r a sep k is 0. Similarly, he erm 1 up (k) ) O p (k) represens inflow of he link S p S r a sep k. Noe ha, in (2), hese erms appear wih a delay of r and respecively T pr ime seps due o he free-flow of links S s S r and respecively S p S r. For k 0 he ouflow O (k) wih {s, p}, is defined as: ( O (k)=min O max, q ) (k) + I (k) 4. MIXED INTEGER LINEAR PROGRAMMING In his secion we ransform he model presened above using mixed ineger linear programming (MILP) heory. 4.1 Background To remove he nonlineariies of (1)-(3) we will use he following equivalences, see (Bemporad and Morari, 1999), where f is a funcion defined on a bounded se X wih upper and lower bounds M and m for he funcion values, δ is a binary valued scalar variable, y is a real valued scalar variable, and ε is a small olerance (ypically he machine precision): P1: [ f(x) 0] [δ = 1] is rue if and only if { f(x) M(1 δ) f(x) ε+(m ε)δ, P2: y=δ f(x) is equivalen o y Mδ y mδ y f(x) m(1 δ) y f(x) M(1 δ). The olerance ε is needed o ransform a consrain of he form y > 0 ino y 0, since in MILP problems only nonsric inequaliies are allowed. 4.2 MILP model In his secion we use he MILP properies presened above in order o obain an MILP model for he simplified roue choice model given by equaions (1)-(3). 1 The ouflow of a uncion is defined as he number of vehicles ha cross ha uncion per ime uni. 2 The inflow of a link equals he number of vehicles ha enered ha link per ime uni. (3)

5 We sar by ransforming (3) using Propery P1. So, we inroduce he binary variable δ ou (k) wih {s, p} which equals 1 if and only if O max q (k) + I (k). Then we rewrie (3) as follows: O (k)=δ ou (k)o max + ( 1 δ ou (k) )( q (k) + I (k) ) (4) where he condiion δ ou = 1 if and only if O max q (k) I (k) 0 is equivalen o (conform Propery P1): + I (k) O max δ ou (k) q (k) O max q (k) I (k) ε+(o max q max I max ε)δ ou (k) wih q max he maximum possible lengh of he queue and I max he maximum possible value for I wih {s, p}. Bu (4) is no ye linear, so, we use Propery P2 and inroduce he real-valued scalar variable (k) such ha: or equivalenly: (k)=δ ou (k)q (k) (k) q max δ ou (k) (k) 0 (k) q (k) (k) q (k) q max (1 δ ou (k)). and he real-valued scalar variable y inflow (k) such ha: y inflow (k)=δ ou (k)i (k) or is equivalen se of inequaliies of Propery P2 for f(x) = I (k), M = I max, and m=0. Hence, one obains: O (k)=δ ou which is linear. (k)o max + 1 q (k)+i (k) 1 (k) y inflow Now, in order o ransform (2), we inroduce he exra variables y us (k) = u s (k)o s (k) and y up (k) = u p (k)o p (k) and he corresponding se of linear inequaliies of Propery P2 for f(x) = O s (k) and respecively f(x) = O p (k), wih M = O max, and m=0, and we obain he linear equaion: (k) I r (k)=y us (k r )+O p (k T pr ) y up (k T pr ) (5) Finally, we wan o ransform (1). So, we inroduce he binary variable δ r (k) which equals 1 if and only if f r (k) 0 and we rewrie (1) as: q r (k+ 1)= ( 1 δ r (k) ) f r (k)) (6) ogeher wih he se of linear inequaliies of Propery P1 for M = q max + O max and m= O max. However (6) is no ye linear. Therefore, we inroduce an addiional variable y r (k) = δ r (k) f r (k) and he se of linear inequaliies of Propery P2 for f(x) = f r (k), M = q max + O max, and m= O max, and we obain: which is linear. q r (k+ 1)= f r (k) y r (k) (7) If we now collec all he variables for he model (i.e. q r (k), f r (k), I r (k), y r (k), y us (k), y up (k), s (k), y inflow (k), p (k), s q k (k+ 1) (a) Fig. 5. Two siuaions for queue evoluion. q k (k+ 1) y inflow p (k), u s (k), u p (k), δ r (k), δs ou (k), δp ou (k)) in one vecor v(k), we can express q r (k+ 1) as an affine funcion of v(k): q r (k+ 1)=av(k)+b wih a vecor properly defined a and a scalar b, where v(k) saisfies a sysem of linear equaions Cv(k) = e and linear inequaliies Fv(k) g, sysem which corresponds o he linear equaions and consrains inroduced above by he MILP ransformaions. 5. MODEL PREDICTIVE ROUTE CHOICE CONTROL In his secion we define he MPC opimizaion problem for boh he nonlinear and he MILP case. Recall ha we wan o assess he performance of MPC when using he original nonlinear model and when using he approximaed MILP model. Therefore, he performance index should be linear or piecewise affine. In his paper we consider minimizing he oal ime spen in he queue for a nework wih S uncions. This performance index has been considered since he ime spen in he queue τ queue a a uncion S, wih {1,2,...,S}, can be approximaed o a linear one, see e.g. (van den Berg e al., 2008) for road raffic. However, noe ha he piecewise affine performance index used in (Tarău e al., 2009) can also be used afer linearizing i using he MILP equivalences presened above. When he lengh of a queue decreases during he sampling period, we deal wih he wo siuaions skeched in Fig. 5 where a sep k+1 eiher he queue lengh q (k+1) > 0 or where q becomes 0 before sep k+1. Noe ha in his second case if he queue vanishes before sep k+ 1, hen q says equal o 0 a leas unil (k+1) since is he sampling period, and boh he demand and he conrol acion are piecewise affine. Then, as in De Schuer (2002), we approximae he gray area under he curve of Fig. 5(b) wih he dashed area A (k)= 1 2 (q (k)+q (k+ 1)), a formula which holds also for Fig. 5(a). Then, he oal ime ha he DCVs raveling oward uncion S spend in he queue a uncion S from he beginning of he simulaion (sep 0) unil he las prediced ime insan (sep k+ N 1) is given by: Le J k,n = S =1 w τ queue,k,n τ queue k+n 1,k,N = i=0 (b) A (i) (8) O max denoe he performance index a sep k, for a predicion horizon N, where w is a weighing parameer ha represens he penalizaion of DCVs waiing a uncion S. Then he nonlinear MPC opimizaion problem is defined as: min J k,n(u(k)) u(k)

6 subec o he sysem dynamics operaional consrains where u(k) = [u(k)u(k+1)... u(k+n 1)] T wih u(k) = [u 1 (k)u 2 (k)... u S (k)] T, while he ime spen in he queue is deermined via simulaion. In order o solve his mixed ineger nonlinear opimizaion problem above one could use e.g. geneic algorihms, simulaed annealing, or abu search see e.g. Reeves and Rowe (2002), Dowsland (1993), and Glover and Laguna (1997). Similarly, he linear MILP MPC opimizaion problem is defined as: min J k,n(v(k)) v(k) subec o MILP model operaional consrains where v(k) = [v(k)v(k+1)... v(k+n 1)] T, while he ime spen in he queue is compued using (8). To solve he MILP opimizaion problem one could use solvers such as CPLEX, Xpress-MP, GLPK, see e.g. (Aamürk and Savelsbergh, 2005). We expec ha compuing he roue for each DCV in he nework when solving he nonlinear opimizaion problem will give beer performance han when solving he approximaed MILP, bu a he cos of much higher compuaional effors. So, one could use MILP o compue a good iniial poin for he nonlinear opimizaion problem and his would reduce he compuaion ime. 6. CASE STUDY We consider as benchmark case sudy he nework depiced in Fig. 6. This nework consiss of four loading saions and one unloading saion conneced via single direcion rack segmens, where he free-flow ravel ime is provided for each link. 3 L 1 S 1 L 2 L S U 1 S 4 3 Fig. 6. Case sudy for a DCV-based baggage handling sysem. Then he evoluion of queue q, for = 1,2,3,4 is given by: q (k+ 1)=max ( 0, f (k) ) where f (k) is defined as follows: f 1 (k)=q 1 (k)+(d 1 (k 3)+ ( 1 u 2 (k 6) ) O 2 (k 6) O max ) f 2 (k)=q 2 (k)+(d 2 (k 2)+D 3 (k 2) O max ) 6 S 3 L 4 4 demand T load demand T load demand a) b) c) Fig. 7. Demand profile. T load f 3 (k)=q 3 (k)+(d 4 (k 4)+u 2 (k 7)O 2 (k 7) O max ) f 4 (k)=q 4 (k)+(o 1 (k 5)+O 3 (k 6) O max ) To compare he resuls we have considered 18 scenarios where 460 bags have o be handled for differen iniial saes of he sysem, queues on differen links, and differen weighing parameers. For hese scenarios we consider ha he bags arrive a loading saions according o he hree differen classes of demand profiles skeched in Fig. 7, where T load is he oal loading ime. Le us now compare he resuls obained when using he proposed predicive conrol mehod wih differen formulaions of he opimizaion problem. To solve he original mixed ineger nonlinear MPC opimizaion problem we have chosen a geneic algorihm wih muliple runs since simulaions show ha his opimizaion echnique gives good performance wih he shores compuaion ime. For solving he MILP opimizaion we have used he CPLEX solver. As predicion horizon we have considered N = for all MPC opimizaion problems. Noe ha we have chosen his horizon since simulaions show ha his value gives accepable compuaional effor and performance index for all problem formulaions. Based on simulaions we now compare, for he given scenarios, he resuls obained for he proposed formulaions of he opimizaion problem. The resuls of he simulaions are repored in Fig. 8 where he oal performance of he sysem is defined as J = S =1 i Λ w τ queue i,, wih Λ he se of bags ha wai a uncion S during he simulaion, and τ queue i, he real ime ha bag i spends in he queue a uncion while being ranspored o is corresponding end poin. These resuls confirm ha compuing he roue choice using he original nonlinear formulaion for he MPC opimizaion problem gives ypically beer performance han using he MILP formulaion, bu a he cos of higher compuaional effor. However, i can be noed ha for some scenarios, he use of MILP formulaion resuls in beer performance. This happens due o he fac ha he predicion horizon is no sufficienly large. Bu, increasing he predicion horizon will resul in increasing he compuaional effor even more. Recall ha we have used a geneic algorihm for solving he original nonlinear MPC opimizaion problem. Bu, geneic algorihms do no allow a given iniial guess, herefore, o furher reduce he compuaional effor, a each MPC sep, we have solved he MILP opimizaion problem and we have used his soluion as a feasible iniial guess for compuing a soluion of he original nonlinear MPC problem wih simulaed annealing. As illusraed in Fig. 8, he resuls confirm ha his las mehod offers a good rade-off beween performance and compuaional effor.

7 J (s) CPU ime (s) 6 5 GA SA + ini. guess MILP only MILP scenario index (a) performance GA SA + ini. guess MILP only MILP scenario index (b) compuaion ime Fig. 8. Comparison of he proposed approaches. 7. CONCLUSIONS In his paper we have considered he problem of efficienly compuing roues for desinaion coded vehicle (DCV) ha ranspor bags in an airpor on a mini railway nework. This is a nonlinear, nonconvex, mixed ineger opimizaion problem, and very expensive o solve in erms of compuaional effor. Therefore, we have used an alernaive approach for reducing he complexiy of he compuaions by simplifying and approximaing he nonlinear opimizaion problem by a mixed ineger linear programming (MILP) problem. The advanage is ha for MILP opimizaion problems he global opimal soluion can be efficienly compued wih available solvers. These wo formulaions of he opimizaion problem have been used o compue he roue of DCVs using model predicive conrol (MPC) for a benchmark case sudy. Simulaion resuls confirm ha compuing he roue choice using he original nonlinear formulaion for he MPC opimizaion problem gives usually beer performance han using he MILP formulaion, bu a he cos of significanly higher compuaional effors. To reduce he compuaion ime while obaining good resuls, one can solve he original MPC opimizaion problem, bu using a each sep he local soluion of he corresponding MILP formulaion as iniial guess. In fuure work we will apply his mehod o more complex case sudies where we will also consider conrolling he swich ino uncions. ACKNOWLEDGEMENTS Research suppored by he STW-VIDI proec Muli-Agen Conrol of Large-Scale Hybrid Sysems, he BSIK proec Nex Generaion Infrasrucures, he Transpor Research Cenre Delf, he Delf Research Cener Nex Generaion Infrasrucures, and he European 7h framework STREP proec Hierarchical and Disribued Model Predicive Conrol. REFERENCES Aamürk, A. and Savelsbergh, M. (2005). Inegerprogramming sofware sysems. Annals of Operaions Research, 140(1), Bemporad, A. and Morari, M. (1999). Conrol of sysems inegraing logic, dynamics, and consrains. Auomaica, 35, de Neufville, R. (1994). The baggage sysem a Denver: Prospecs and lessons. Journal of Air Transpor Managemen, 1(4), De Schuer, B. (2002). Opimizing acyclic raffic signal swiching sequences hrough an exended linear complemenariy problem formulaion. European Journal of Operaional Research, 139(2), Dowsland, K. (1993). Simulaed annealing. In C. Reeves (ed.), Modern heurisic echniques for combinaorial problems, chaper 2, John Wiley & Sons, Inc., New York, USA. Fay, A. (2005). Decenralized conrol sraegies for ransporaion sysems. In Proceedings of he Inernaional Conference on Conrol and Auomaion, Budapes, Hungary. Gang, H., Shang, J., and Vargas, L. (1996). A neural nework model for he free-ranging AGV roue-planning problem. Journal of Inelligen Manufacuring, 7(3), Glover, F. and Laguna, F. (1997). Tabu Search. Kluwer Academic Publishers, Norwell, Massachuses, USA. Hallenborg, K. and Demazeau, Y. (2006). Dynamical conrol in large-scale maerial handling sysems hrough agen echnology. In Proceedings of he Inernaional Conference on Inelligen Agen Technology, Hong Kong, China. Kaufman, D., Nonis, J., and Smih, R. (1998). A mixed ineger linear programming model for dynamic roue guidance. Transporaion Research Par B: Mehodological, 32(6), Lewis, F. (1986). Opimal Conrol. John Wiley & Sons, Inc., New York, USA. Macieowski, J. (2002). Predicive Conrol wih Consrains. Prenice Hall, Harlow, UK. Reeves, C. and Rowe, J. (2002). Geneic Algorihms - Principles and Perspecives: A Guide o GA Theory. Kluwer Academic Publishers, Norwell, Massachuses, USA. Tarău, A., De Schuer, B., and Hellendoorn, J. (2008). Travel ime conrol of desinaion coded vehicles in baggage handling sysems. In In he IEEE Inernaional Conference on Conrol Applicaions, San Anonio, Texas, USA. Tarău, A., De Schuer, B., and Hellendoorn, J. (2009). Roue choice conrol of auomaed baggage handling sysems. In Proceedings of he 88h Annual Meeing of he Transporaion Research Board. Washingon DC, USA. van den Berg, M., De Schuer, B., Hellendoorn, J., and Hegyi, A. (2008). Influencing roue choice in raffic neworks: A model predicive conrol approach based on mixed-ineger linear programming. In Proceedings of he 17h IEEE Inernaional Conference on Conrol Applicaions, San Anonio, Texas.