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Hybrid adaptive predictive control for a dynamic pickup and delivery problem including traffic congestion

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This paper presents a hybrid adaptive predictive control approach to incorporate future information regarding unknown demand and expected traffic conditions, in the context of a dynamic pickup and delivery problem with fixed fleet size. As the
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  Vol. 43, No. 1, February 2009, pp. 27–42 issn 0041-1655  eissn 1526-5447  09  4301  0027 informs  ®  doi 10.1287/trsc.1080.0251©2009 INFORMS Hybrid Adaptive Predictive Control for a DynamicPickup and Delivery Problem Cristián E. Cortés Civil Engineering Department, Universidad de Chile, Avenue Blanco Encalada 2002,Santiago, Chile, ccortes@ing.uchile.cl Doris Sáez, Alfredo Núñez, Diego Muñoz-Carpintero Electrical Engineering Department, Universidad de Chile, Avenue Tupper 2007, Santiago, Chile{dsaez@ing.uchile.cl, alfnunez@ing.uchile.cl, dimunoz@ing.uchile.cl} T his paper presents a hybrid adaptive predictive control approach that includes future information in real-time routing decisions in the context of a dynamic pickup and delivery problem (DPDP). We recognize inthis research that when the problem is dynamic, an additional stochastic effect has to be considered withinthe analytical expression of the objective function for vehicle scheduling and routing, which is the extra costassociated with potential rerouting arising from unknown requests in the future. The major contributions of thispaper are: first, the development of a formal adaptive predictive control framework to model the DPDP, andsecond, the development and coding of an ad hoc particle swarm optimization (PSO) algorithm to efficientlysolve it. Predictive state-space formulations are written on the relevant variables (vehicle load and departuretime at stops) for the DPDP. Next, an objective function is stated to solve the real-time system when predictingone and two steps ahead in time. A problem-specific PSO algorithm is proposed and coded according to thedynamic formulation. Then, the PSO method is used to validate this approach through a simulated numericalexample. Key words : pickup-and-delivery system; dynamic vehicle routing problem; hybrid predictive control; particleswarm optimization  History : Received: March 2005; revisions received: April 2006, July 2007, October 2007; accepted: October 2008. 1. Introduction One of the most studied problems in the literatureon logistics is the well-known pickup and deliv-ery problem (with or without time windows), whichinvolves the satisfaction of a set of transportationrequests by a vehicle fleet initially located at sev-eral depots (Desrosiers, Soumis, and Dumas 1986;Savelsbergh and Sol 1995). A transportation requestconsists of picking up a certain number of customersat a predetermined pickup location during a depar-ture time interval and taking them to a predetermineddelivery location within an arrival time interval. Load-ing and unloading times are incurred at each vehiclestop. The problem can be generalized to the dynamiccase, in which a subset of the requests is not knownin advance and dispatch decisions have to be madein real time. The dynamic pickup and delivery prob-lem (DPDP) has become of great interest in the lastdecade, mainly due to the fast growth in communi-cation and information technologies, as well as thecurrent interest in real-time dispatching and routing.The problem can be characterized as a real-time routedtransit and has been treated mostly heuristically bymany authors under different policy schemes in thepast (as representative references see Psaraftis 1988;Madsen, Raven, and Rygaard 1995; Bertsimas andVanRyzin1991,1993a,b;MalandrakiandDaskin1992;Dial 1995; Gendreau et al. 1999).In this scenario, if the objective were to transportpassengers, inefficient routing decisions could greatlyaffect the performance of the system as perceived bythe users, resulting in a poor level of service, lowdemand, and insufficient productivity. One of themajor issues for improving efficiency is the correct def-inition of a decision objective function for dispatch-ing, including total travel and waiting times for usersas well as a performance measure for vehicles. How-ever, when the problem is dynamic, an additionalstochastic effect has to be considered when comput-ing an analytical expression for any decision objectivefunction (whether it affects the user or the operator).In other words, we recognize that current dispatchactions taken in real time can be affected by poten-tial rerouting decisions decided in the future, affectingmost customers already in the system, those waiting,as well as those traveling.The importance of this issue has been underesti-mated in the dynamic vehicle-routing literature. Oneassumption behind most of the proposed schedul-ing-routing rules is that travel and waiting timesexperienced by customers are considered fixed in theobjective function expressions, independent of future 27  Cortés et al.:  Hybrid Adaptive Predictive Control for a Dynamic Pickup and Delivery Problem 28  Transportation Science 43(1), pp. 27–42, ©2009 INFORMS reroutings. In other words, as stated by Spivey andPowell (2004), the complexity of real-time routingschemes have generally restricted research to myopicmodels (for example, see Wilson and Weissberg 1976;Wilson and Colvin 1977; Psaraftis 1980, 1988; Madsen, Raven, and Rygaard 1995; Gendreau et al. 1999;Swihart and Papastavrou 1999).However, some recent studies in the field of vehiclerouting and dispatching have tried to exploit infor-mation about future events to improve decision mak-ing (Ichoua, Gendreau, and Potvin 2006; Spivey andPowell 2004). Solution approaches found in this lineof research are diverse, with formulations based upondynamic network models (Powell 1988), dynamicand stochastic programming schemes (Godfrey andPowell 2002, Topaloglu and Powell 2005), etc. Cortésand Jayakrishnan (2004) propose a scheme for making better dynamic decisions by estimating the effectivecost of a real-time request insertion based upon futureinformation. The authors realized that the problemconceptually fits within a stochastic predictive controlframework, although they did not enter into the con-trol formulation details.In this paper, we formalize the approach suggested by Cortés and Jayakrishnan (2004) by developing aconsistent framework based upon predictive controltheory for optimizing the performance of a DPDPthat is mainly oriented to passenger movements. Theformulation turned out to be highly nonlinear, with acombination of integer/discrete and continuous vari-ables to properly describe the future behavior of therouting process. Hence, an efficient ad hoc algorithmfrom the computational intelligence literature (parti-cle swarm optimization, PSO) is developed to solvethe proposed formulation and to test the benefits of incorporating demand patterns’ prediction in currentrouting decisions under different scenarios.Unlike others’ nonmyopic dynamic vehicle-routingapproaches, this formulation is based on state-spacevariables. The system state is defined in terms of departure time and vehicle loads (stochastic state-space variables), the system inputs (control actions)are routing decisions, the system outputs are effec-tive departure time to stops, and the demand requestsare modeled as disturbances. We use a discrete modelwith variable step size equal to the time between suc-cessive calls (events). In order to include future andunknown demand in the current decision, we solvean objective function incorporating the predictiveeffect via probabilities computed from historical dataregarding typical demand patterns.In summary, we highlight two major contributionsof this paper: the development of a hybrid predictivecontrol framework to model the DPDP, and the devel-opment of an ad hoc PSO algorithm to efficiently solvethe proposed formulation for real-size problems. Thisline of research represents an innovative attempt todevelop control-based algorithms for modeling andsolving dynamic transportation problems in a realis-tic context. Specifically, in this application we havedeveloped a new version of the PSO algorithm (src-inally conceived for solving continuous problems) inorder to add integer variables into the solution andsolve the DPDP efficiently. It is important to mentionthat the proposed algorithm was conceived from thehybrid predictive control scheme (HPC) to deal withthe DPDP developed here, and depends exclusivelyon the structure of the HPC formulation, as shownin §3.4.The structure of the paper is as follows. In the nextsection, the relevant background on dynamic vehi-cle routing is presented. In §3, the dynamic pickupand delivery problem is described in context, andis formulated under an adaptive-predictive controlscheme. Thus, the specific state-space formulation forthe problem is developed, the associated dispatchobjective function is shown, and the solution algo-rithms are developed to solve the proposed hybridpredictive control scheme. In §4, a numerical exampleis presented to show the benefits of applying predic-tive control at least two steps ahead in time. Finally,in §5, analysis, comments, and further research linesare presented. 2. The Dynamic Vehicle-RoutingProblem: Approaches andSolution Methods In this section, the objective is to provide a reviewon the most relevant dynamic vehicle-routing prob-lem (DVRP) variants, intensely studied by differentauthors with different applications over the past 15to 20 years. DVRPs are characterized by routes thatare constructed as unknown requests enter the systemin real time. Thus, DVRPs are formulated by assum-ing that inputs may change or have to be updatedduring the execution of the solution algorithm. Larsen(2000) develops a nice characterization of the dif-ferent dynamic problems, starting again from theTSP (traveling salesman problem), which yields thedynamic TSP (DTSP) introduced by Psaraftis (1988).This work motivates the development of the dynamictraveling repairman problem (DTRP), introduced byBertsimas and Van Ryzin (1991) and next extended byBertsimas and Van Ryzin (1993a, b). Lately, Swihartand Papastavrou (1999), and Thomas and White (2004)formulate and solve two variants of the DTRP.The dynamic pickup and delivery problem (DPDP)that is designed to solve the dynamic dial-a-rideProblem (DDRP) has been intensely studied in thelast 20 years (Psaraftis 1980, 1988; Gendreau et al. 1999; Savelsbergh and Sol 1995). The final output of   Cortés et al.:  Hybrid Adaptive Predictive Control for a Dynamic Pickup and Delivery Problem Transportation Science 43(1), pp. 27–42, ©2009 INFORMS  29such a problem is a set of routes for all vehicles,which dynamically change over time. With regard toreal applications, Madsen, Raven, and Rygaard (1995)adapt the insertion heuristics by Jaw et al. (1986) andsolve a real-life problem for moving elderly and hand-icapped people in Copenhagen, whereas Dial (1995)proposes a modern approach to the many-to-few dial-a-ride transit operation ADART (autonomous dial-a-ride transit), currently implemented in Corpus Christi,TX, USA.With regard to solution methods to handle differentDVRPs, Gendreau et al. (1999) modify the tabu searchheuristics to solve the DVRP with soft time windowsmotivated from courier service applications, whichis implemented in a parallel platform. Tabu searchmethods are derived in more sophisticated versions,such as granular tabu search (Toth and Vigo 2003) andadaptive memory-based tabu search (Tarantilis 2005).Tighe, Smith, and Lyons (2004) propose a priority- based solver that considers subproblems of real-timevehicle routing in order to obtain an optimal solutionin less time by using fuzzy decisions.Evolutionary computation techniques have also been proposed to handle such problems. Specifically,genetic algorithms (GA) are applied for various VRP,considering different chromosome representation andgenetic operators according to the particular problem(Skrlec, Filipec, and Krajcar 1997 for the single vehiclecapacity VRP; Haghani and Jung 2005 for the multive-hicle DVRP with time-dependent travel time and softtime windows). Zhu et al. (2006) propose an adaptedpartial swarm optimization (PSO) algorithm to solvea static VRP with time windows. Jih and Yung-Jen (1999) and Osman, Abo-Sinna,and Mousa (2005) present a successful comparison of the GA against dynamic programming (DP) in termsof computation time. The former solve the DVRP withtime windows and capacity constraints, while thelatter solve a multiobjective VRP. Additionally, antcolony methods, as new metaheuristics inspired bythe behavior of real ant colonies, have been appliedto DVRP (Montemanni et al. 2005, Dréo et al. 2006). In dynamic as well as stochastic problems, twoapproaches (myopic and nonmyopic) are found in theliterature; these differ based on how the future infor-mation is considered in the generation of real-timedecisions. The myopic research line does not explicitlyconsidertheexpectedfutureinformationofthesystemto improve the current solution (as shown the afore-mentioned papers), whereas the nonmyopic optionconsiders a mechanism to update information regard-ing the future to make better decisions at present. Suchfuture data may be imprecise or unknown, and there-fore developing consistent information update toolsare essential for getting good predictions and making better real-time dispatch decisions.Powell and his team have worked for many years ina nonmyopic line of research that incorporates explicitstochastic and dynamic algorithms with the currentinformation and probabilities of future events to pro-duce more efficient solutions than those obtainedthrough myopic deterministic strategies. They solvethe problem of dynamically assigning drivers to loadsthat arise randomly over time, a scenario motivatedfrom long-haul truckload trucking applications.Powell (1988) first considers the potential advan-tages of relocating vehicles in anticipation of futuredemands. He writes a two-stage stochastic programincluding a recourse function representing the futurecost. Powell, Jaillet, and Odoni (1995) studies a mixedassignment and fleet management problem, modeledas a dynamic-stochastic network, which they solvewith a network simplex algorithm on a rolling horizon basis. Spivey and Powell (2004) propose a very generalclass of dynamic assignment models, and propose anadaptive, nonmyopic algorithm that iteratively solvessequences of assignment problems. Topaloglu andPowell (2005) propose a distributed solution approachto a certain class of dynamic resource allocation prob-lems. Topolaglu and Powell (2007) show how to coor-dinate the decisions on pricing and fleet managementof a freight carrier. The objective is to find the set of prices that maximize the total expected profit over thetime horizon, considering random loads (whose distri- butions depend on the prices) and the cost associatedwith repositioning the empty vehicles. The authorspresent a tractable method to obtain sample path- based directional derivatives of the objective func-tion with respect to the prices to search for a goodset of prices. Numerical experiments show that theirapproach yields high-quality solutions.In his thesis, Larsen (2000) investigates the use of future information by relocating empty vehicles inanticipation of future demands. Ichoua, Gendreau,and Potvin (2006) develop a strategy based on prob-abilistic knowledge about future request arrivals to better manage a fleet of vehicles for real-time vehicledispatching. This problem is solved using a paralleltabu search technique.Figliozzi, Mahmassani, and Jaillet (2007) introducethe VRP in a competitive environment (VRPCE) asan extension of the traveling salesman problem withprofits (TSPP) to a dynamic competitive auction envi-ronment. The authors develop a dynamic model tocompute optimal price expressions for the VRPCEconsidering both, the expected change due to alteringthe current fleet assignment scheme and the oppor-tunity costs on future profits created by servicing anew contract. Analytically, they propose an approx-imate solution approach, using a finite look-aheadhorizon based on backward induction, which is com-pared against a static approach with no look ahead.  Cortés et al.:  Hybrid Adaptive Predictive Control for a Dynamic Pickup and Delivery Problem 30  Transportation Science 43(1), pp. 27–42, ©2009 INFORMS A simulation-based approach to evaluate service costsis proposed, which not only outperforms a static pric-ing, but it also price discriminates by market arrivalrate, time windows, and shipment features.The analysis of these nonmyopic models that incor-porate future information is crucial for our purposes, because this paper formalizes the use of future infor-mation in dynamic vehicle-routing problems througha hybrid predictive control scheme. In the next sec-tion, this scheme is presented in detail. 3. Hybrid Predictive ControlApproach to Solve the DynamicPickup and Delivery Problem(DPDP) In the context of control theory, the notion of hybridsystems arises when the problem conditions are char-acterized by both continuous and discrete/integervariables. In the last decade, hybrid systems have been studied more intensely by researchers from sev-eral study areas, such as computer science and auto-matic control. A systematic methodology for a generalcontrol design of hybrid systems has been devel-oped by Bemporad and Morari (1999) and Bemporad,Borrelli, and Morari (2002). Specifically, a hybrid sys-tem can be expressed as a nonlinear state-space modelgiven by xk + 1  = fxkukyk = gxk (1)where  xk  are the continuous and/or discrete (inte-ger) state-space variables,  uk  are the continuousand/or discrete input or manipulated variables,  yk define the continuous and/or discrete system out-puts, and  fg  are nonlinear functions. In general, ahybrid predictive control design minimizes the fol-lowing generic objective function:min uk J   ukuk + N   − 1   xk + 1   xk + N  yk + 1   yk + N    (2)where  J   is an objective function;  k  is the currenttime;  N   the prediction horizon;   xk + t  and   yk + t are, respectively, the expected state-space vectorand the expected system output at instant  k + t ;and  ukuk + N   − 1   represents the controlsequence, which corresponds to the vector of opti-mization variables. Once expression (2) is optimized,only the first element of the control vector  uk  isused to update the system conditions, based upon thereceding-horizon methodology.Next, we characterize the dynamic pickup anddelivery problem (DPDP) as a hybrid system to showthe advantages of this approach when predictingfuture conditions under unknown dynamic demand. 3.1. Problem Statement In this paper, we formulate a generic DPDP as ahybrid predictive control problem, following the the-ory explained above, recognizing that the dynamicrouting process behind the real-time dispatch deci-sions includes discrete/integer and continuous state-space variables, as well as discrete input variables.Conceptually, the hybrid predictive control frame-work used to model the DPDP incorporates stochastic-ity into the routing dispatch rules by considering theimpact of future reassignments on the performanceof already-scheduled customers. The stochastic pre-diction allows the dispatcher to incorporate a morerealistic measure of effective travel (waiting) timeexperienced by the users into the decision objectivefunction expression (see §3.3 for details). The focushere is on passenger routing; however, the schemecould be generalized to freight, too.Let us assume an influence area  A , with a servicenetwork of length  D  in distance units. Suppose wehave a set of vehicles  V   of size  F  . The fleet of vehi-cles is currently in operation traveling within the areaaccording to predefined routing rules. The demandfor service is unknown and comes up in real-time(assume a rate    of calls per time unit). Quick routingand scheduling decisions are needed to handle suchdemand with the available vehicles. At any time  k ,we assume that each vehicle  j   ∈ V   has been assigneda control action that includes pickups and deliveries,and can be represented by a function  u j  k  =  S  j  k  = s 1 j  k ··· s i j  k ··· s w j  k j  k T  , in which the  i th element of the sequence represents a specific  i th stop along vehi-cle  j  ’s route, and  w j  k  is the number of stops.The complete control action or manipulated vari-able  uk  =  Sk , as the dispatching decision, can berepresented by the set of sequences assigned to everyvehicle at instant  k . Analytically, uk = Sk =  S  1 kS  j  kS  F  k    (3)Vehicles will travel according to the predefinedsequence vector  Sk − 1   while no new calls arereceived. When a new service request (call) comes in,the controller or central dispatcher calculates the con-trol sequence in the next step  Sk  for the fleet of vehicles, including the stops requested by the newcustomer. Then, each sequence  S  j  k  remains fixedduring the whole time interval  kk  +  1  , unless avehicle reaches a predefined pickup or delivery stopduring such an interval, in which case its sequencewill decrease in size to show that the scheduled taskhas been accomplished. Thus, in this scheme it is nec-essary to formulate the problem in terms of a vari-able time step (triggered by events), which representsthe time interval between two consecutive requests,that is to say, the predictive controller makes a routingdecision when a new call enters the system.  Cortés et al.:  Hybrid Adaptive Predictive Control for a Dynamic Pickup and Delivery Problem Transportation Science 43(1), pp. 27–42, ©2009 INFORMS  31The state of the system at instant  k  is associatedwith the previous sequences  Sk − 1   (the new call isnot considered). In the DPDP problem, the state-spacevariables include the clock time of departure  T  ij   k and the vehicle load  L ij  k , after vehicle  j   leaves stop  i , both computed at instant  k . At this point, let us de-fine, for each vehicle  j   ∈  V  , the load and departure-time vectors as follows: L j  k =  L 0 j  k L 1 j  k  ···  L w j  k − 1 j   k  T w j  k − 1  + 1  × 1  (4) T  j  k =  T  0 j   k T  1 j   k  ···  T  w j  k − 1 j   k  T w j  k − 1  + 1  × 1  (5)Thus, the set of state-space variables for the entiresystem at instant  k  can be written as  xk  = LkTk , where  Lk  and  Tk  represent the setof load and departure-time vectors, respectively;that is,  Lk  =  L 1 kL j  kL F  k  and  Tk  = T  1 kT  j  kT  F  k  The output set  yk  is rep-resented by the vector of observed departure times of vehicles at stops,  Tk .In summary, under this hybrid predictive controlapproach, Equation (1) can be written for the DPDP by recognizing the dependence of the routing pro-cess on the following associated variables:  xk  = LkTk yk = Tk uk = Sk . The hybrid pre-dictive control scheme proposed in this paper can berepresented by a generic flow chart shown in Figure 1.In the figure, the predictive controller is represented by the dispatcher and the routing is solved by min-imizing an objective function that considers the usercost based upon total travel and waiting time spent by the users, and a component as a proxy of theoperational cost, as explained in §3.3. The routingprocess is defined by the online dispatching deci-sion ( Sk − 1  ) under uncertain demand (  ), whichresults in observed departure times ( yk ). An adap-tive mechanism is also added in the figure due to the Objective functionPredictive controller(dispatcher)AdaptivemechanismRouting process  x  ( k  ) = {  L ( k  ),  T  ( k  )}  y ( k  ) S  ( k  –1) µ Figure 1 Overall Block Diagram of a Hybrid Predictive Approach forDPDP variant parameters of the system and dimension of thedeparture-time and vehicle load models  LkTk .Next, this model is analytically described, highlight-ing the treatment of both the departure-time and loadcomponents. 3.2. Predictive Dynamic Model This research considers a predictive dynamic model based on state-space representation for both the vehi-cle load and the departure time at stops (as a functionof segment travel times). Both the clock time of depar-ture T  ij   k andthevehicleload L ij  k arestochasticvari-ables, because they depend on the evolution of thesystem affected by uncertain demand. Therefore, andin order to work with deterministic values, reason-able estimations of the load and departure-time vec-torshavetobeobtained.Thepredictionofwhenanewrequest will occur is given by the expected value of thestate-space vector for vehicle  j  ,   x j  k + 1  . Analytically,  x j  k + 1   =  EL j  k + 1 /kET  j  k + 1 /k  =  ˆ L j  k + 1   T  j  k + 1   =  f  L L j  kS  j  kf  T  T  j  kS  j  k   ∀ j   = 1 F   (6)where the functions  f  L  and  f  T   are the state-space mod-els to be defined in Equations (8) and (9).The dynamic system for a specific vehicle  j   can begraphicallyrepresentedbyitssequence S  j  k computedat certain instant  k , and the associated expected valuesof the state-space variables in the next instant  k  +  1, isshowninFigure2.The components of   S  j  k  are S  j  k =  r  1 j   k  1 − r  1 j   k   1 j   k  label 1 j   kr  i j   k  1 − r  i j   k   i j   k  label i j  kr  w j  k j   k  1 − r  w j  k j   k   w j  k j   k  label w j  k j   k   (7)  L 0  j ( k  + 1), T  0  j ( k  + 1)  L 1  j ( k  + 1), T  1  j ( k  + 1) i v  j ˆ ˆ  L 2  j ( k  + 1),  T  2  j ( k  + 1)ˆ ˆ  L i j ( k  + 1),  T  i j ( k  + 1)ˆˆ  L i j +1 ( k  + 1), T  i j +1 ( k  + 1)ˆˆ  L  jw  j ( k  ) ( k  + 1), T   jw  j ( k  ) ( k  + 1)ˆˆ i + 1 S   j ( k  ) Figure 2 Typical Vehicle Route at Time  k   and State-Space VariablesEstimated at  k   + 1
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