On-Line Algorithms for Truck Fleet Assignment and Scheduling under Real-Time Information

Transcription

1 On-Line Algorithms for Truck Fleet Assignment and Scheduling under Real-Time Information Jian Yang Department of Management Science and Information Systems The University of Texas at Austin Austin, Texas Patrick Jaillet Hani S. Mahmassani Department of Civil Engineering and Department of Management Science and Information Systems The University of Texas at Austin Austin, Texas July Abstract With greater availability of real-time information systems, algorithms are needed to support commercial fleet operators in their decisions to assign vehicles and drivers to loads in a dynamic environment. We present a rolling horizon framework for the dynamic assignment and sequencing of trucks to jobs consisting of picking up and delivering full truckloads when requests for 1 Submitted for presentation at the 78th Annual Meeting of the Transportation Research Board, January 1999, Washington, D.C. and publication in Transportation Research Board. 1

2 service arise on a continuous basis. A mathematical formulation of the problem faced at each stage is presented; its solution allows for the dynamic reassignment of trucks to loads, including diversion to a new load of a truck already en-route to pick up another load, as well as for the dynamic resequencing of the order in which loads are to be served, as new loads arrive and conditions unfold. Loads have associated time windows for pick up and delivery, and the objective function includes explicit penalty cost for not serving a particular load. A solution algorithm is presented and implemented, and computational results are presented, yielding insight into various operational trade-offs in dynamic fleet operations. Because applicability of the solution algorithm is at present limited to relatively small problems, and given the stochastic dynamic nature of these systems, numerical experiments are performed to compare the quality of the solution obtained using this approach to the performance of simpler and less computationally demanding local rules. Keywords: Dynamic fleet management, commercial vehicle operations, dynamic vehicle routing, real-time information, Intelligent Transportation Systems, truckload trucking, freight transportation. 2

3 1 Introduction With greater availability of real-time information systems, algorithms are needed to support commercial fleet operators in their decisions to assign vehicles and drivers to loads in a dynamic environment. The focus of this paper is primarily on truckload trucking operations, though the formulation and algorithms can be readily extended to other types of fleet services. Existing procedures are primarily intended for static formulations with complete a priori knowledge of loads to be served over a certain period, and as such are more applicable when shippers call in their requests sufficiently in advance of the intended pick-up times. However, the environment for truckload trucking and many other services is characterized by less predictability, and a continuous flow of incoming requests, often involving time-sensitive shipments and short time windows. Dispatchers typically deal with these situations using experience and ad hoc decision aids, though the competitive nature of the freight industry calls for more systematic and efficient procedures that can take advantage of real-time information on vehicle positions and status through positioning and two-way communication systems. While the vehicle routing problem has been widely studied in the literature and applied in practice [1][2] [4][5], dynamic network and routing models, including realtime vehicle routing problems, remain in the early stages of development [7]. Powell, Jaillet, and Jaillet [6] presented works on stochastic and dynamic vehicle routing and some related problems. In our paper, we take a continuous time, continuous space, and infinite horizon approach, which is more difficult than the works mentioned there. In previous work, Reagan, Mahmassani, and Jaillet [9][10] [11][12] introduced and investigated various local rules for the dynamic assignment of vehicles to loads under real-time information. These were introduced as part of a general operational process for real-time fleet decisions, which explicitly recognized the expanded set of choice dimensions available to the operator as a result of real-time information. These computationally efficient procedures were evaluated using simulation experiments that illustrated the relatively good performance attainable through these heuristics in a stochastic dynamic environment. However, it is necessary to benchmark these procedures against optimal procedures, which can also form the basis of optimization-based approaches for this important class of problems. 3

4 We present a rolling horizon framework for the dynamic assignment and sequencing of trucks to jobs consisting of picking up and delivering full truckloads when requests for service arise on a continuous basis. A mathematical formulation of the problem faced at each stage is presented; its solution allows for the dynamic reassignment of trucks to loads, including diversion to a new load of a truck already en-route to pick up another load, as well as for the dynamic resequencing of the order in which loads are to be served, as new loads arrive and conditions unfold. Loads have associated time windows for pick up and delivery, and the objective function includes explicit penalty cost for not serving a particular load. A solution algorithm is presented and implemented, and computational results are presented, yielding insight into various operational trade-offs in dynamic fleet operations. Because applicability of the solution algorithm is at present limited to relatively small problems, and given the stochastic dynamic nature of these systems, numerical experiments are performed to compare the quality of the solution obtained using this approach to the performance of simpler and less computationally demanding local rules. The contributions of this paper are to (1) introduce a mathematical formulation of the truckload pick-up and delivery problem with time windows (TPDP), which was previously addressed through tour construction and modification heuristics (with no explicit underlying optimization formulation), and which unifies all previous strategies (diversion, reassignment and resequencing); (2) generalize the previously introduced heuristic rules by including wait time explicitly in the specification; and (3) confirm previous tentative substantive results, and develop new substantive insights into truckload truck fleet operations under real-time information, particularly with regard to the value of advance information and the trade-off between empty distance and wait time in the dynamic assignment decision. 2 Problem Formulation The problem considered in this paper is a special vehicle routing problem, the truckload pickup-and-delivery problem (TPDP) with time windows. Consider a truckload trucking company operating a fleet of trucks to pick up and deliver demands in a certain region. Each truck can only be loaded with one demand, and cannot serve other demands until the current demand is delivered to its destination. Each demand has a 4

5 window of pickup time during which pickup must be made. Since a truck cannot take any other demand once it picks up one, it will normally follow the shortest path from a demand s origin to its destination under any reasonable cost structure. Therefore, a demand s delivery time window is implied in its pickup time window. The trucking company can reject a demand. The cost of rejection is the revenue it otherwise could make from accepting the demand. We assume that the revenue a demand generates is proportional to the shortest distance from its origin to destination. The main operational cost incurred by the trucking company is the total empty distance its trucks have to travel, from one load s delivery point to the next load s pickup location. Most of the on-line strategies considered for this problem involve solving an off-line problem when exogenous information is updated, that is, when the trucking company receives request for service. The natural off-line problem is the problem formed by all the known-yet-unserved demands. The problem is designed as follows. There are K trucks (labeled 1,..., K). Truck k is first available at time θ k at location o k. At the moment when the off-line problem is called by an on-line strategy, θ k and o k are the current time and location of truck k if it is idle or moving empty. They correspond to the time and location at which truck k finishes delivering its current load when it is moving loaded. There are also N known demands (labeled 1,..., N) requiring truck service. Each demand i is a requirement to move a certain load from an origin a i to a destination b i. The load for demand i can only be picked up in the time window [τ, τ + ]. The cost of empty travel between two points a and b is C(a, b). In this formulation, C(a, b) = 1 D(a, b), the Euclidean distance between these two points. Here, 1 is the coefficient for the cost incurred by empty distance. The Euclidean distance is used in this formulation with no loss of applicability; it could alternatively be replaced by distances (or travel times), computed over a highway network. The vehicles speed is scaled to be 1, again with no loss of generality. We denote the required loaded distance to serve demand i by w i (= D(a i, b i )). We assume that within the time window, early pickup is welcomed by customers. This is reflected in a linear penalty on the difference in time from a demand s real pickup time to its early pickup time. We denote the penalty-time coefficient by p. If a truck is assigned a sequence of demands, its task is to serve the demands in the order defined by the sequence, i.e., start from its origin, go to the first demand s origin, move the demand s load to its destination, then move to the second demand s origin, and so on, until the 5

6 last demand in its sequence is served, unless a new sequence assignment is received in the interim. The operational cost of serving a sequence of demands is taken as the total empty distance traveled by the truck and the total delay penalty from all the demands. The trucking company also has the option of rejecting a demand. The lost revenue for rejecting demand i is αw i, where α is a positive constant. The trucking company can earn α Total Loaded Distance Total Empty Distance after serving some demands. Finally, the distance and cost matrices are defined as follows: For any truck k = 1,..., K and any demand i = 1,..., N, let d k 0i = D(o k, a i ) and c k 0i = C(o k, a i ). Also, for any other demand j = 1,..., N, let d ij = D(b i, a j ) and c ij = C(b i, a j ). The problem is formulated as an assignment problem without considering the temporal side of the formulation. The assignment problem, in turn, is one of finding the least-cost set of cycles that involve all the nodes (1,..., K, K + 1,..., K + N), where node k corresponds to truck k and node K + i corresponds to demand i. In the formulation, the binary variable x uv, for u, v = 1,..., K + N, indicates whether arc (u, v) is selected in one of the cycles. Interpreted in the truckload trucking context, the binary variable x k,k+i is to indicate whether truck k first serves demand i; binary variable x K+i,K+j is to indicate whether there is a truck that serves demands i and j consecutively; binary variable x kk = 1 means that truck k serves no demand; and x K+i,K+i = 1 means that demand i is rejected. The timing constraints will disallow a cycle to be formed solely by demand nodes, i.e., nodes (K + 1,..., K + N). So, when the cycles are obtained in the solution, they can be readily translated into the solution that is required. For instance, if a cycle goes as 1, K +1, K +2, 2, K +3, K +4, K +5, 1, then the interpretation is that truck 1 serves demands 1 and 2, truck 2 serves demands 3, 4, and 5. Real variable t i represents the pickup time of demand i. The formulation is presented as below: K (T P DP ) min N N c k i 1 N N 0ix k,k+i + (αw i x K+i,K+i + ( + )c ij x K+i,K+j ) + p t i k=1 i=1 i=1 j=1 j=i+1 i=1 subject to K+N v=1 x uv = 1 u = 1,..., K + N (1) K+N v=1 x vu = 1 u = 1,..., K + N (2) 6

7 x uv = 0, 1 u, v = 1,..., K + N (3) K (d k 0i + θ k )x k,k+i + t i 0 i = 1,..., N (4) k=1 (w i + d ij )x K+i,K+i T x K+i,K+j t i + t j T + w i + d ij i, j = 1,..., N (5) τi t i τ i + i = 1,..., N (6). The objective is to minimize the total cost of processing all the demands, which is a combination of the cost of empty distance traveled, of penalty cost for delay, and of lost revenue due to loads rejected. Due to the first three groups of constraints, the solution will constitute a feasible assignment. The last three groups of constraints are timing constraints. There T is a large number. Constraints (4) ensure that truck k reaches demand i s origin later than d k 0i + θ k if i is the first demand to be served by k. Constraints (5) dictate that a truck must reach demand j s origin w i + d ij after reaching demand i s origin if j is to be served after i. Because T is large enough, when x K+i,K+j = 0, constraints (5) are non-restrictive. And, due to constraints (1) and (2), when x K+i,K+i = 1, we must have x K+i,K+j = 0 at j i, while at j = i, we get the trivial inequality w i + d ii T w i + d ii T. Constraints (6) simply enforce that a load s pickup time is within its time window. As noted, an on-line strategy envision solving the above problem in a rolling horizon framework. Each horizon consists of all the known-yet-served demands. The problem is solved every time a new request for service is received; as a result, new assignments are made. The new solution may often entail resequencing the demands associated with a particular truck, or even reassignment of loads to trucks. Whenever a different demand becomes the first demand in the sequence to be served by a truck, an incidence of diversion occurs. 3 Real-Time Strategies The problem formulation described in the previous section will determine an optimal assignment and sequencing of loads to trucks when the loads to be moved are known a priori for the entire time horizon of interest. However, normally, the trucking company does not know the requests until some time before the requested job is available for pickup. Instead of a whole schedule for the entire time horizon determined prior to 7

8 any actual implementation, the schedule must be periodically updated to accommodate incoming new demands and situations. To do so, we can design rolling-horizon strategies that have strong intuitive appeal. To fully utilize any known information, the strategies should update the vehicle operations decision at the moment that a new request becomes known. At such a moment, the state of the system includes the following variables: information about the new request, each truck s position and state, the sequence and characteristics of the jobs that each truck has yet to serve. A truck s state can be idle, moving empty toward the next job s origin, or moving loaded toward the current job s destination. A truck is idle when it finishes some load and has no assigned load to serve next. The decision to be made at this moment consists of the following: whether to accept or reject the new load, and the new sequences of jobs to be served by each truck, which may involve resequencing all existing loads that remain to be served, including reassignment of loads to different trucks, so that the new request, if accepted, will be satisfied in some future time point. As noted, the solution to the TPDP problem formulated in the previous section allows for diversion, identified by Reagan et al. [9] as an important new operational dimension under real-time information. Diversion refers to a decision by which a truck moving empty toward its pickup point may be diverted to a new job. Following Reagan et al. [9], a truck moving loaded is not allowed to divert. The first, and most general real-time assignment and sequencing strategy considered here consists of solving the problem formulated in the previous section every time a new load request is received (or possibly when changes in supply occur). This strategy is referred to as OPTIMAL, and it makes the currently optimal decision for the new request and the requests not yet served by the trucks. As explained, the explicit problem formulation underlying the OPTIMAL strategy unifies and generalizes as special cases the various local load acceptance, assignment and reassignment rules developed by Reagan et al. [12]. However, this strategy still does not ensure long-run optimality because of the uncertainty in future demands; therefore there is no guarantee that it will always outperform, in hindsight, the strategies developed previously. Furthermore, computational limitations remain to be overcome to attain real-time performability of solution algorithms for the OPTIMAL strategy for large problems. Therefore, we have continued the refinement of those strategies, both as 8

9 strategies in their own right as well as approximate solution approaches for the OP- TIMAL formulation. The principal areas of refinement include the specification of the cost measures that form the assignment criteria, as well as the computational procedures to perform the algorithm steps. Following Reagan, Mahmassani, and Jaillet s framework [11] [12], we consider strategies to select a truck to which the new job is to be assigned. The three assignment criteria considered before are used here though with a different cost measure. The three alternative criteria are: Criterion TT (total cost): assign new load to the truck with the least total cost to serve all the demands already in its queue in addition to the new load Criterion AV (average cost): assign new load to the truck with the least average cost per demand (obtained by dividing the total cost by the number of demands in the queue, including the new load) Criterion IN (incremental cost): select truck with the least additional cost needed for the vehicle to serve the new demand In previous work, the empty distance was the only cost component included in the cost measure [12]. In this paper, we extend this measure to include a penalty for the total delay (waiting) time to pick up the load, similarly to the objective function presented in the previous section. In computing the cost measures for the assignment decision, the new load can either be placed at the end of the truck s current job queue, or the latter can be resequenced to best serve the new load along with the existing ones. While strategies involving resequencing are expected to vastly outperform those that do not involve resequencing in terms of empty travel distance, this is not generally the case with regard to waiting time. Strategies involving resequencing are denoted by SE, while NS denotes the non-resequencing strategies. The third dimension (the cost measure TT, AV, IN and whether resequencing is performed, i.e., SE vs. NS, are the first two) in which strategies that allow resequencing might differ is whether reassignment of previously assigned loads to trucks is allowed; such strategies are denoted as AS, while those with no reassignment are denoted by NA. Strategies may also vary in the extent of reassignment allowed. For example, the OPTIMAL strategy allows reassignment of any load to any truck if it 9

10 improves the overall solution. In previous examples, Reagan et al. [12] allowed reassignment of only the last load in each truck s queue. A similar strategy is followed in the present simulations. The TPDP problem formulation in the OPTIMAL strategy can be solved using a branch-and-cut procedure in the CPLEX solver [3], which is also used in conjunction with several of the SE strategies where optimization is needed, as the problem can be viewed as a restricted version of the TPDP formulation given in the previous section. However, the combinatorial nature of the problem precludes solvability in real time, even for small problems. To get around this limitation, we can limit the demands being resequenced to the first few in the queue of each vehicle. Following the above dimensions, we have a total of 10 different strategies. Their names are self-explanatory: TT/NS, AV/NS, IN/NS, TT/SE/NA, AV/SE/NA, IN/SE/NA, TT/SE/AS, AV/SE/AS, IN/SE/AS, and OPTIMAL. For instance, TT/NS refers to an assignment based on the total cost criterion, with no resequencing, whereas TT/SE/AS refers to an assignment based on total cost with resequencing and reassignment. When implementing the above resequencing and optimization strategies, the problem parameters need to be extracted from the current state of the system. For example, if truck k is idle or moving empty, θ k (in the formulation of the previous section) is the current time; while if truck k is moving loaded, θ k is the time when the truck completes its current load. These strategies are illustrated and tested in simulation experiments described in the next section. 4 The Simulation Experiments A set of simulation experiments are conducted to illustrate the performance of the real-time solution strategies described in the previous section, and to obtain a preliminary assessment of how competitive the various local assignment rules are relative to the more general OPTIMAL strategies. In addition, the experiments are intended to gain insight into fundamental aspects of real-time fleet operation under information, particularly the value of advance information on load availability and the trade-off between empty distance and delay time in serving the demands. The context for the simulation consists of truckload trucking operations to serve demands generated according to a stationary space-time stochastic process in a unit 10

11 (1 1) square region, with both origins and destinations independently and uniformly distributed over the square. Note that the results can be readily scaled up to any realistic dimension; the use of a square is intended for convenience, experimental control, and ease of interpretation. Demands arise according to an exponentially distributed inter-arrival time distribution, with mean T int. Travel distances within the region are Euclidean, and travel takes place at unit speed, again without loss of generality. At the beginning of each simulation, the K trucks available for service are located in a central depot located in the middle of the square (0.5,0.5). A total of 500 demands are generated and served in each simulation run; no load rejection is allowed in this particular set of experiments, which is equivalent to setting the parameter α to a very large number in the objective function of the formulation. Also set to be a very large number is the length of the time window, so that the time-window constraints become nonrestrictive. An important parameter of the simulation is the time between the instant that a load is known to the time when it is first available for pickup; we assume this time to be a constant T adv in a given simulation run. By varying the value of this parameter, the value of advance information can be investigated. In addition to K, T int, and T adv, the fourth parameter characterizing a simulation run is the penalty weight p associated with the delay time in the definition of the cost measure in the assignment strategies (and the objective function of the TPDP formulation in the OPTIMAL strategy). For each parameter set (K, T int, T adv, p), 10 replicates (of 500 randomly generated demands each) are performed. Simulation runs are performed with all 10 real-time strategies described in the previous section, applied in conjunction with each set of parameter values. The parameter values considered in these experiments are the following: K = 4 (fleet size of 4 trucks); T int = 0.5 (time units the time required to traverse one side of the square region); Three values of T adv : T adv = 0.0 (loads become known only when they are ready to be picked up), 2.0 and 5.0 (time units before loads are available for pickup after the request is made); and 11

12 Three values of p: p = 0.01, 1.00 and 10.00, reflecting very low, moderate, and very high importance of wait time relative to empty distance in the cost measure. In the SE and OPTIMAL strategies implemented in these simulation experiments, only up to 10 demands of the first few in the queues of the trucks getting involved are re-optimized at any given time. Additional experiments are performed for a subset of the strategies, which emerged as the most promising from the first series of experiments described above. These experiments are also performed for K = 4, and T int = 0.25 and 0.5 (to consider a case of more rapid demand arrival and hence greater congestion); T adv = 0.0, 2.0, and 5.0; and p = 0.4. The principal measures of performance examined in the analysis include, for each simulation run, the following quantities: The average loaded distance per job, D L The average empty distance per job, D E The average delay time per job, T D The average cost per job, C AV Interval estimates for each measure were obtained from the 10 replicate runs for each set of parameter values; 68%-confidence intervals are reported for each measure. In all cases, the average loaded distance (which does NOT depend on the routing strategy) was found to be equal to 0.522(1.000 ± 0.007), which is the average distance between two random points uniformly distributed over the unit square. The next section discusses the simulation results. 12

13 Table 1: Simulation Results: K = 4, T int = 0.5, T adv = 0.0 Strategy D E T D c p = 0.01: IN/NS 0.292(1.000 ± 0.009) 0.561(1.000 ± 0.025) 0.297(1.000 ± 0.009) IN/SE/NA 0.266(1.000 ± 0.013) 0.696(1.000 ± 0.018) 0.273(1.000 ± 0.013) IN/SE/AS 0.262(1.000 ± 0.010) 0.726(1.000 ± 0.021) 0.269(1.000 ± 0.010) OPTIMAL 0.252(1.000 ± 0.010) 0.719(1.000 ± 0.032) 0.259(1.000 ± 0.009) p = 1.00: IN/NS 0.315(1.000 ± 0.008) 0.397(1.000 ± 0.012) 0.712(1.000 ± 0.010) IN/SE/NA 0.304(1.000 ± 0.010) 0.410(1.000 ± 0.011) 0.713(1.000 ± 0.010) IN/SE/AS 0.356(1.000 ± 0.008) 0.470(1.000 ± 0.009) 0.826(1.000 ± 0.008) OPTIMAL 0.294(1.000 ± 0.010) 0.398(1.000 ± 0.011) 0.692(1.000 ± 0.009) p = 10.00: IN/NS 0.347(1.000 ± 0.006) 0.396(1.000 ± 0.007) 4.309(1.000 ± 0.007) IN/SE/NA 0.335(1.000 ± 0.006) 0.386(1.000 ± 0.009) 4.195(1.000 ± 0.008) IN/SE/AS 0.412(1.000 ± 0.008) 0.517(1.000 ± 0.010) 5.582(1.000 ± 0.009) OPTIMAL 0.326(1.000 ± 0.005) 0.373(1.000 ± 0.009) 4.058(1.000 ± 0.009) 5 Simulation Results Through the simulations, we first find that almost all TT and AV strategies are inferior to their IN counterparts. This is due largely to that reducing the incremental cost directly contributes to reducing the total and average cost. For clearance of presentation, we will not display the results on those strategies. The simulation results are summerized in Tables 1 through 3 for the four remaining strategies. For each strategy, average values of the empty distance D E, delay time T D, and composite cost C AV are reported, along with the corresponding 68%-confidence intervals. These tables correspond to T adv = 0.0, 2.0, and 5.0 respectively. For each value of T adv, the corresponding table reports results for the three values of waiting time penalty p (0.01, 1.00, and respectively). In Tables 4, 5, and 6, we rearrange the results for ease of study. Table 4 shows the average empty distances, Table 5 shows the average waiting times, and Table 6 shows the average composite costs, of all the simulations. In these tables, each triplet entry represents the simulation results under the three different T adv s. From those results, it is notable that for all strategies under any scenario, the average costs decrease dramatically when T adv increases from 0.0 to 2.0. However, 13

14 Table 2: Simulation Results: K = 4, T int = 0.5, T adv = 2.0 Strategy D E T D c p = 0.01: IN/NS 0.292(1.000 ± 0.009) 0.288(1.000 ± 0.039) 0.295(1.000 ± 0.009) IN/SE/NA 0.203(1.000 ± 0.007) 1.370(1.000 ± 0.024) 0.217(1.000 ± 0.007) IN/SE/AS 0.202(1.000 ± 0.016) 1.781(1.000 ± 0.025) 0.220(1.000 ± 0.015) OPTIMAL 0.191(1.000 ± 0.013) 1.178(1.000 ± 0.033) 0.203(1.000 ± 0.013) p = 1.00: IN/NS 0.334(1.000 ± 0.006) 0.060(1.000 ± 0.043) 0.431(1.000 ± 0.010) IN/SE/NA 0.323(1.000 ± 0.007) 0.059(1.000 ± 0.059) 0.383(1.000 ± 0.008) IN/SE/AS 0.344(1.000 ± 0.011) 0.091(1.000 ± 0.052) 0.435(1.000 ± 0.017) OPTIMAL 0.300(1.000 ± 0.007) 0.052(1.000 ± 0.039) 0.352(1.000 ± 0.009) p = 10.00: IN/NS 0.374(1.000 ± 0.006) 0.041(1.000 ± 0.059) 0.784(1.000 ± 0.032) IN/SE/NA 0.374(1.000 ± 0.008) 0.041(1.000 ± 0.058) 0.787(1.000 ± 0.035) IN/SE/AS 0.421(1.000 ± 0.006) 0.086(1.000 ± 0.044) 1.278(1.000 ± 0.030) OPTIMAL 0.348(1.000 ± 0.007) 0.043(1.000 ± 0.065) 0.776(1.000 ± 0.033) Table 3: Simulation Results: K = 4, T int = 0.5, T adv = 5.0 Strategy D E T D c p = 0.01: IN/NS 0.292(1.000 ± 0.009) 0.288(1.000 ± 0.039) 0.295(1.000 ± 0.009) IN/SE/NA 0.175(1.000 ± 0.010) 2.659(1.000 ± 0.068) 0.201(1.000 ± 0.042) IN/SE/AS 0.176(1.000 ± 0.010) 2.462(1.000 ± 0.044) 0.200(1.000 ± 0.011) OPTIMAL 0.165(1.000 ± 0.012) 1.680(1.000 ± 0.034) 0.182(1.000 ± 0.013) p = 1.00: IN/NS 0.334(1.000 ± 0.006) 0.060(1.000 ± 0.043) 0.394(1.000 ± 0.010) IN/SE/NA 0.323(1.000 ± 0.013) 0.059(1.000 ± 0.792) 0.383(1.000 ± 0.349) IN/SE/AS 0.349(1.000 ± 0.009) 0.079(1.000 ± 0.045) 0.428(1.000 ± 0.012) OPTIMAL 0.296(1.000 ± 0.007) 0.068(1.000 ± 0.069) 0.364(1.000 ± 0.033) p = 10.00: IN/NS 0.374(1.000 ± 0.006) 0.041(1.000 ± 0.059) 0.784(1.000 ± 0.032) IN/SE/NA 0.374(1.000 ± 0.013) 0.041(1.000 ± 0.797) 0.783(1.000 ± 0.708) IN/SE/AS 0.409(1.000 ± 0.008) 0.080(1.000 ± 0.042) 1.207(1.000 ± 0.030) OPTIMAL 0.378(1.000 ± 0.009) 0.058(1.000 ± 0.055) 0.953(1.000 ± 0.049) 14

15 Table 4: Empty Distance: K = 4, T int = 0.5 Strategy p = 0.01 p = 1.00 p = IN/NS IN/SE/NA IN/SE/AS OPTIMAL Table 5: Waiting Time: K = 4, T int = 0.5 Strategy p = 0.01 p = 1.00 p = IN/NS IN/SE/NS IN/SE/AS OPTIMAL Table 6: Total Cost: K = 4, T int = 0.5 Strategy p = 0.01 p = 1.00 p = IN/NS IN/SE/NS IN/SE/AS OPTIMAL

16 when T adv is increased from 2.0 to 5.0, no obvious improvement in the average cost is obtained, and in several cases an increase occurs. To appreciate these results, it should be noted that T adv = 2.0 time units exceeds the average time required for a vehicle to serve two loads, and that new loads arrive and trigger new solutions every 0.5 time units on average. In this dynamic environment, advance information beyond two or three loads ahead does not appear to be of much additional value. It can also be noted that, except for p = 0.01, the advance information does not improve, and in fact worsens, the empty distance incurred under the strategies that do not involve reassignment (denoted by NA) of loads to trucks. For these strategies, improvement in waiting time appears to be achieved at the expense of additional empty distance. Hence the level of service offered to customers is improved, but at additional cost to the operator. On the other hand, with reassignment, the empty distance is reduced as well as the waiting time when T adv increases from 0.0 to 2.0. Therefore the fleet operator can offer better service at lower cost. This win-win feature of strategies with reassignment is rather appealing. These results highlight the importance of using appropriate solution strategies in conjunction with additional information. In previous work, strategies with resequencing of loads were shown to generally outperform strategies without resequencing in terms of reducing empty distance [12]. This is again the case in these results, whereby SE/AS strategies are found to be better at reducing empty distance than IN/NS strategies, but not at reducing delay times. This is largely due to the memorylessness of the delay term used in the objective function (i.e., in the cost measure), which does not depend on how long the demand has been delayed. Therefore when the penalty weight p is small, the SE/AS strategies are better overall than IN/NS. The specification of the objective function could be fine-tuned to address the particular needs of a specific operation. The results also confirm that the OPTIMAL strategy outperforms all the others. This is not surprising given that it encompasses many of the other strategies as special cases, and that it makes the best decision for the available information. However, it is also noteworthy that several of the other, simpler strategies, come quite close to it in performance, suggesting that further refinement in specification of the cost measure lead to competitive simple decision rules that can execute rapidly to meet real-time computational performability. 16

17 Table 7: Simulation Results: K = 4, T int = 0.5, p = 0.40 Strategy D E T D c T adv = 0.0: IN/NS 0.303(1.000 ± 0.012) 0.431(1.000 ± 0.016) 0.475(1.000 ± 0.012) IN/SE/NS 0.292(1.000 ± 0.008) 0.443(1.000 ± 0.012) 0.470(1.000 ± 0.009) IN/SE/AS 0.321(1.000 ± 0.011) 0.461(1.000 ± 0.012) 0.506(1.000 ± 0.011) OPTIMAL 0.287(1.000 ± 0.011) 0.423(1.000 ± 0.015) 0.456(1.000 ± 0.012) T adv = 2.0: IN/NS 0.310(1.000 ± 0.009) 0.103(1.000 ± 0.030) 0.351(1.000 ± 0.010) IN/SE/NS 0.293(1.000 ± 0.010) 0.130(1.000 ± 0.032) 0.344(1.000 ± 0.009) IN/SE/AS 0.294(1.000 ± 0.009) 0.133(1.000 ± 0.038) 0.347(1.000 ± 0.012) OPTIMAL 0.265(1.000 ± 0.010) 0.111(1.000 ± 0.040) 0.310(1.000 ± 0.008) T adv = 5.0: IN/NS 0.310(1.000 ± 0.009) 0.103(1.000 ± 0.030) 0.351(1.000 ± 0.010) IN/SE/NS 0.289(1.000 ± 0.010) 0.129(1.000 ± 0.028) 0.340(1.000 ± 0.011) IN/SE/AS 0.298(1.000 ± 0.008) 0.127(1.000 ± 0.031) 0.348(1.000 ± 0.010) OPTIMAL 0.260(1.000 ± 0.007) 0.112(1.000 ± 0.043) 0.305(1.000 ± 0.017) Strategy IN/NS requires little computational effort and generates fairly good results. Strategy OPTIMAL delivers mostly the best results, yet it is most computationally demanding. The IN/SE strategies are in between. As described in the previous section, the additional experiments are conducted for a fleet size K = 4 and penalty index p = 0.4 for those five strategies. The results are shown in Tables 7 and 8, corresponding to lower demand case of T adv = 0.5 and higher demand case of T adv = 0.25 respectively. At the low congestion level, it was noted previously that, when T adv increases from 0.0 to 2.0, considerable decrease in the average waiting times is observed for all strategies. But when T adv further increases to 5.0, no additional gain is attained, reflecting the limited value of additional information beyond a certain threshold for such myopic strategies. Under the high congestion level, the effect of decreasing waiting time by increasing T adv is even dwindled. Additional information is even less valuable when demand opportunities arise at a higher rate. In summary, the IN/NS strategy takes the least time to execute, yet performs competitively with respect to the next two when the congestion level is low. It is not particularly sensitive to advance information: when T adv increases, its performance does not improve much. The IN/SE strategies outperform the IN/NS strategy when 17

18 Table 8: Simulation Results: K = 4, T int = 0.25, p = 0.40 Strategy D E T D c T adv = 0.0: IN/NS 0.327(1.000 ± 0.012) 0.878(1.000 ± 0.016) 0.678(1.000 ± 0.012) IN/SE/NS 0.274(1.000 ± 0.007) 0.685(1.000 ± 0.012) 0.548(1.000 ± 0.009) IN/SE/AS 0.281(1.000 ± 0.011) 0.731(1.000 ± 0.012) 0.574(1.000 ± 0.011) OPTIMAL 0.262(1.000 ± 0.011) 0.642(1.000 ± 0.015) 0.519(1.000 ± 0.012) T adv = 2.0: IN/NS 0.328(1.000 ± 0.010) 0.546(1.000 ± 0.100) 0.546(1.000 ± 0.050) IN/SE/NS 0.253(1.000 ± 0.009) 0.467(1.000 ± 0.048) 0.440(1.000 ± 0.023) IN/SE/AS 0.263(1.000 ± 0.012) 0.433(1.000 ± 0.037) 0.436(1.000 ± 0.018) OPTIMAL 0.256(1.000 ± 0.008) 0.648(1.000 ± 0.061) 0.515(1.000 ± 0.032) T adv = 5.0: IN/NS 0.328(1.000 ± 0.010) 0.546(1.000 ± 0.100) 0.546(1.000 ± 0.050) IN/SE/NS 0.259(1.000 ± 0.007) 0.442(1.000 ± 0.045) 0.436(1.000 ± 0.024) IN/SE/AS 0.263(1.000 ± 0.012) 0.429(1.000 ± 0.037) 0.435(1.000 ± 0.018) OPTIMAL 0.257(1.000 ± 0.011) 0.531(1.000 ± 0.041) 0.469(1.000 ± 0.033) the congestion level is high and T adv is large. The OPTIMAL strategy remains the best under most circumstances, but requires the most time to execute. At high congestion and with abundant advance information, the IN/SE strategies are better than OPTIMAL. Under this circumstance, it is more often for the OPTIMAL strategies that the decision problem it faces involves more than 10 demands so that the decision it makes is even not myopically optimal than for the IN/SE strategies. As noted previously, the simpler and faster strategies offer a comprehensive approach for real-time operation of large fleets serving a large number of demands. 6 Concluding Comments This paper has presented a rolling horizon approach for the real-time assignment of loads to trucks in truckload trucking operation. Several real-time strategies for the assignment decisions are presented, including the solution of a math programming problem formulation that optimizes load assignment and sequencing using all available information on vehicle locations and currently unserved loads every time a new request is received. 18

19 Simulation experiments conducted to compare the performance of the OPTIMAL strategy to that of less computationally demanding local assignment rules provided valuable insights regarding the value of additional information and trade-off between empty distance and waiting time to serve a demand. The results suggested that strategies that allow the reassignment of previous loads to vehicles can profit from advance information to reduce both the waiting time and the empty distance. However, there appears to be a threshold above which additional advance information may not be beneficial and might even be counter-productive. This threshold also depends on overall congestion in the system. Overall, while the OPTIMAL strategy outperforms the others in the cases tested, several of the simpler rules appear to be quite competitive in performance yet require only a fraction of the computational effort, which makes them promising for the real-time operation of large fleets. In future work, additional simulations can be performed to further explore the threshold beyond which increasing T adv does not provide any performance improvement, and the factors that affect this threshold, such as the congestion level. Different shapes of region, different demand behaviors, different cost structures can also be considered in subsequent work. Computationally, special solution techniques can be implemented so that the OPTIMAL and SE strategies may be used to consider a large number of demands at the same time. However, perhaps the most challenging task ahead is to develop more far-sighted strategies, which take into account the stochastic characteristics of the future demands. This can be accomplished through a stochastic programming formulation that builds upon the problem formulation presented in this work to accommodate the mean effect of future effects. Simultaneously, intelligent heuristics must be developed to appreciate the performance of elaborate algorithms to solve more complete model formulations, keeping in mind real-time execution requirements under real-time information. 7 Acknowledgements The paper is based on work supported by the National Science Foundation grant DMI and the State of Texas through the U.S. DOT Region 6 Southwest University 19

20 Transportation Center (SWUTC). The authors have benefited from discussion with Yonjin Kim, Amelia Reagan, and Gang Yu through their various collaborative efforts. The authors are of course responsible for the contents of this paper. References [1] Bodin, L.D., B.L. Golden, A.A. Assad, and M. Ball (1983), Routing and Scheduling of Vehicle and Crews, the State of the Art, Computers and Operations Research, 10, pp [2] Christofides, N. (1985), Vehicle Routing, in Lawler, E.L., J.K. Lenstra, A.H.G. Rinnooy Kan, and D. Shmoyes (eds), The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, John Wiley and Sons, New York. [3] Using the CPLEX Callable Library, CPLEX Optimization Inc., [4] Fisher, M.L. (1995), Vehicle Routing, in Ball, M.O., T.L. Magnanti, C.L. Monma, and G.L. Nemhauser (eds), Handbooks in Operations Research and Management Science, Vol 8, Network Routing, Elsevier, Amsterdam, pp [5] Golden, B.L. and A.A. Assad (1988) (eds), Vehicle Routing: Methods and Studies, Elsevier (North-Holland), Amsterdam. [6] Powell, W.B., P. Jaillet, and A. Odoni (1995), Stochastic and Dynamic Networks and Routing, in Ball, M.O., T.L. Magnanti, C.L. Monma, and G.L. Nemhauser (eds), Handbooks in Operations Research and Management Science, Vol 8, Network Routing, Elsevier, Amsterdam, pp [7] Psaraftis, H.N. (1988), Dynamic Vehicle Routing Problems, in Golden, B.L. and A.A. Assad (eds), Vehicle Routing: Methods and Studies, Elsevier (North- Holland), Amsterdam. [8] Reagan, A.C. (1997), Real-Time Information for Improved Efficiency of Commercial Vehicle Operations, Ph.D. dissertation, University of Texas, Austin. 20

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