A Road Timetable TM to aid vehicle routing and scheduling

Transcription

1 Computers & Operations Research 33 (2006) A Road Timetable TM to aid vehicle routing and scheduling Richard Eglese a,, Will Maden a, Alan Slater b a Department of Management Science, Lancaster University Management School, Lancaster LA1 4YX, UK b Added Value Logistics Consulting Ltd., Grosvenor House, 45 The Downs, Altrincham, Cheshire, WA14 2QG, UK Available online 3 June 2005 Abstract Both within and between urban conurbations, the time of a journey and the corresponding shortest path in a road network from an origin to a destination may depend on the time of the day, the day of the week and the season of the year. Significant journey time differences occur mainly due to recurring instances and variations in levels of traffic congestion throughout the year. This paper examines the issues involved in constructing a database of road times for a road network that uses time-dependent data on the travel times for individual roads in the network to provide the expected times and distances between locations for journeys starting at different times. The benefits of time-dependent vehicle routing and scheduling systems are demonstrated by using real-world data for the road network in the north west of England Elsevier Ltd. All rights reserved. Keywords: Road networks; Vehicle routing; Vehicle scheduling; Time-dependent networks 1. Introduction There has been considerable research to provide techniques for solving vehicle routing and scheduling problems (see, for example, [1]). Exact optimal solution methods are available for small-scale problems and for some problems with particular structures. However, heuristic methods are often required to solve problems of a realistic size, which include all the constraints and features that are important in practice. The survey accompanying the article by Hall [2] describes the capabilities of commercially available vehicle routing software. Most of the research published is based on models where the time between nodes on a road network is regarded as fixed. In practice, this is generally not the case and the time taken Corresponding author /$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi: /j.cor

2 R. Eglese et al. / Computers & Operations Research 33 (2006) for any journey may vary significantly by the time of the day, the day of the week and the season of the year in which the journey takes place. This is an issue that may have important consequences for the use of the results from fixed-time models by vehicle fleet managers. The majority of current vehicle routing and scheduling software (both commercial and academic) offers solutions to the problem in either minimum mileage or minimum time for each set of routes calculated. As a result of increases in traffic congestion and the emergence of time as the real cost driver, commercial vehicle users have begun to use minimum travel time as their chosen optimum rather than minimum mileage. In these instances the travel time calculation may be based upon a single set travel time for various classifications of road. At best such set travel times on selected road types or road lengths may be reduced by a defined percentage to lower road speeds at particular times of the day when traffic congestion is known to occur. Thus a particular motorway section may, for example, may warrant a normal speed of 50 mph for commercial vehicles except between and h when the speed may be reduced by a factor of 65% to 32.5 mph. Most commercial vehicle operators recognise (from such evidence as tachograph analysis) that individual road speeds differ considerably from hour to hour, by day of the week and season of the year. A study undertaken for one large UK commercial vehicle operator by means of both tachograph analysis and data from an individual vehicle tracking and tracing system showed that on one section of motorway in the North of England the same commercial vehicle speeds varied in one week from 5 mph (at on the Monday) to 55 mph (at on the Wednesday). However, when the recorded speeds were compared over a 10 week period the variation in speed recorded for the same time of day and day of the week was less than 5%. This implies that vehicle speeds may be forecast accurately for any single road length at particular times of the day for any day of the week. It may also be concluded that the greater the variation in vehicle speeds on a particular type of road throughout the day, or between the days of the week, the greater the inaccuracy of vehicle scheduling systems which identify travel time by using set travel times for each type of road. In an on-going survey in the UK of over 50 users of vehicle routing and scheduling systems, with up to 20 issues to choose from, 100% of directors/managers, 100% of vehicle schedulers and 66% of drivers questioned declared one of the top five issues to be significant inaccuracy with their systems due to the credibility of the forecast times quoted for individual vehicle trips. Furthermore, amongst all the respondents, the most popular reason for inaccurate timings was traffic congestion (85%) as a result of peak traffic volumes or road works (factors which are generally predictable). A minority of respondents also commented that the real problem with their routing and scheduling system was that they had to adjust manually after the routes were produced for significant and re-occurring timing variations in local road speeds over the same distance. Typical comments from respondents included: Locally road speeds change on the same stretch of road both at different times of the day and between different days of the week. Transport scheduler for an owner operator Our (routing and scheduling) system gives the management the same forecast time for a route on the day shift as on the night shift, but we know (based upon job and finish) that the night shift driver will complete the route in an average of 62% of the time taken by the day shift driver if the night driver does not encounter any traffic accidents or road closures Transport Manager for a haulier working on retail deliveries

3 3510 R. Eglese et al. / Computers & Operations Research 33 (2006) our (routing and scheduling) system cannot be relied upon to provide accurate results so significant manual adjustments need to be undertaken before we finalise our routes for the next day Logistics Director with a fleet of vehicles delivering white goods nationally Despite all respondents having knowledge of potential timing inaccuracies in their vehicle routing and scheduling solutions only 56% of the respondents declared they manually reviewed and adjusted the vehicle routes provided before settling upon the routes. Those who did not review the routes produced by their routing and scheduling system (45% of the respondents) accepted the implications of regular occurrences of drivers overtime charges (38%), additional overnight allowances (22%) and on multi-drop trips failure to achieve all the deliveries on a shift (14%). These results imply that everyone (managers, schedulers and drivers) understands that current routing and scheduling systems will offer inaccurate timings, but there is little that may be done with current systems except manual review and adjustment of the routes produced. In the past, one of the barriers to incorporate time-dependent travel times in models (apart from increasing the model complexity) was the lack of available data. However, such data is now becoming available through various traffic monitoring systems. These may be provided by static installations for monitoring the speeds of vehicles at different points in the road network. Alternative systems provide the data by monitoring the position and speed of vehicles that are fitted with tracking devices. In the UK, the ITIS Floating Vehicle Data (FVD ) provides a national road network monitoring system. The system can be used to update journey times based on current road conditions, but also provides a record of past conditions so that travel times can be related to the time of the day, the day of the week and the season of the year. This is used to produce information on time-dependent journey times for a road network and is known as a Road Timetable TM. This paper discusses the issues involved in creating a Road Timetable TM that will provide the minimum time and shortest-time path between pairs of nodes on a network at different times of the day, days of the week and seasons of the year using historical time-dependent travel times for the roads in the network. Once time-dependent travel times are used, then not only the minimum time, but also the route of the shortest-time path may vary depending on the time of the journey. This phenomenon will be familiar to drivers who may choose to drive on a fast uncongested road, even though the journey may be longer in distance than a more direct route through a congested urban area. A Road Timetable TM needs to hold a large amount of data. For each origin-destination pair of interest, the minimum time and the shortest-time path is required for many different starting times. When the origin-destination pairs are far apart on a large regional or national road network, there is a significant computational challenge in designing an algorithm to find all the minimum times and shortest-time paths efficiently. This paper describes an effective way of dealing with this issue by dividing journeys into stages, distinguishing between travel on minor roads close to the origin or destination points and travel on major trunk roads or motorways. The approach to vehicle routing and scheduling in this paper uses time-dependent travel times that can be forecast based on observations of travel times in the past at a similar time of day, day of week and season of year to that of the journey being planned. Thus the times used should allow for regular congestion on the road network and delays that occur due to long-term restrictions on the road network such as road works. The current models are not dynamic in so far as they do not include the possibility for modifying routes and schedules according to current traffic conditions that may be abnormal due to accidents or other incidents. This would be a natural extension in further work.

4 R. Eglese et al. / Computers & Operations Research 33 (2006) The rest of the paper is organised as follows. In the next section, there is a discussion of previous work using time-dependent travel times. Section 3 examines issues concerning how the road network is represented and the form of the time-dependent travel time function. Section 4 concerns how to calculate the time required for a Road Timetable TM in an efficient way. Section 5 gives an example illustrating the differences that may arise from using time-dependent travel times in a vehicle routing and scheduling model. The final section summarises some conclusions and directions for further research and applications. 2. Related research on time-dependent travel time models for vehicle routing and scheduling Fleischmann et al. [3] describe modern traffic information systems and how time-dependent travel time information can be derived from them. They then present a framework for the implementation of time-varying travel times in various vehicle routing algorithms. Their testing is carried out in an urban setting, rather than the regional example used in this paper. Computational tests based on travel time data obtained from a traffic information system in Berlin indicate that the use of constant average travel times leads to a significant underestimation of total travel time of about 10%. In addition increased travel times and time window violations occur for up to 10% of the orders. In a subsequent paper, Fleischmann et al. [4] describe the use of dynamic travel time information for a dynamic routing system that dispatches a fleet of vehicles according to customer orders arriving at random during the planning period. The paper presents a planning framework that includes the description of a method to effectively assign orders to vehicles under changing conditions. The proposed system requires the calculation of shortest-time paths dynamically using the latest information from a traffic management centre. The models and methods are tested using data from a local area express delivery service. Ichoua et al. [5] examine a model for vehicle routing and scheduling based on time-dependent travel times. Their model derives the travel times from time-dependent travel speeds and they show that this has the advantage of maintaining the first-in-first-out (FIFO) property that will be discussed in Section 3.2. The proposed model is not tested using real traffic information, but is tested on generated data for use in both static and dynamic environments. The results illustrate substantial benefits that the proposed model can deliver compared to a model based on fixed travel times. Their paper also includes a useful literature review of time-dependent vehicle routing, though they recognise that at the time of writing that the literature related to time-dependent vehicle routing problems is very scarce. Kim et al. [6] develop decision-making procedures for determining driver attendance time, departure times and routing policies under time-varying traffic flows based on a Markov decision process formulation. Their approach is tested based on real traffic information from an urban road network in Southeast Michegan, considering 10 origin and destination pairs for deliveries. The results quantify the savings in costs and vehicle usage from using their approach compared to a fixed-time model. The results also differentiate between the benefits from using historical traffic information and real-time traffic information. In their example the average cost savings from their time-dependent model is about 7% of which the majority of the saving is due to using historical traffic information. In addition they determined that the effect of using real-time traffic information has a proportionately greater effect during rush hours. A further paper [7] develops algorithms to efficiently reduce the amount of data to improve the computation times required.

5 3512 R. Eglese et al. / Computers & Operations Research 33 (2006) Fig. 1. A roundabout junction and its network representation. Fig. 2. A T-junction and its network representation. 3. Data structures for the Road Timetable TM 3.1. Representation of the road network An essential feature of any model for journey planning is an accurate in-depth representation of the road network. This should include information to identify such features as the direction of travel, one-way streets, restricted turns and all forms of limited access to specific roads. The natural network representation is for an arc to represent a section of road between junctions carrying traffic in one direction and the nodes of the network to represent junctions where traffic may move from one road section to another. In order to support calculations for shortest-time and shortest-path journeys, information on the road traffic network must be stored in such a way that the network correctly represents valid transfers from one road to another at junctions. When two roads cross and it is possible for traffic entering the junction from any direction to leave on any of the roads meeting at that junction (including the road from which it entered) as happens when a roundabout is constructed at the junction, then it will be appropriate to represent the junction by a single node in the network as shown in Fig. 1. However, if for example we wish to ensure that U-turns are not allowed at a junction, then this can be achieved by representing the junction by more than one node in the network and using dummy arcs (which have zero distance and require zero time to traverse) to represent the turns that are allowed. Additional turn restrictions can be modelled by removing appropriate dummy arcs and may allow a reduction in the number of nodes needed to represent the junction. An example of a T-junction where U-turns are not allowed is shown in Fig. 2. (The layout of the diagram follows the convention used in the UK of driving on the left.) At first sight, it may appear unnecessary to construct the network to prohibit U-turns when we are interested in finding shortest-time paths. However, when there are two successive junctions on a road, if

6 R. Eglese et al. / Computers & Operations Research 33 (2006) there are turn restrictions into a destination road at the first, it may appear that the best route avoiding the turn restriction is to go to the second junction, perform a U-turn, return to the first junction and then turn into the destination road. It is therefore important to examine the road network that is being modelled carefully to ensure that junctions are represented at an appropriate level of detail. In [3], the situation modelled included no-turn restrictions that depended on the time of day. This feature was modelled by keeping a list of permitted successor arcs at each node according to the time of day. Shortest paths were then calculated using a modified Dijkstra algorithm [8] working on arcs instead of nodes. This approach obviates the need for additional nodes and dummy arcs to represent junctions. However, our approach can be used to model no-turn restrictions that depend on time of day by assigning artificially high travel times for any dummy arc representing a turn that is not allowed during a particular time window. In addition, particularly when modelling urban networks, if the data are available, our approach can be used to model the delay for traffic when executing certain turns at junctions Time-dependent travel time data For each arc in the network, data are needed on the time required to traverse the arc at a given time of day, day of week and season of year. In practice, the time will be different for different individual vehicles so a summary measure will need to be calculated if the information is based on historical data. If the information is to be used for a vehicle routing system for heavy goods vehicles then the summary measure could be a truncated mean, where the mean value is calculated after vehicles travelling faster than the maximum speed of the heavy goods vehicles have been discarded. The time will be continuously varying, but most traffic information systems will calculate summary measures within time bins of several minutes, so that the time to traverse an arc is a step function of the time of day. Usually, information is also recorded for the length of each arc, so distances can be calculated and the time-dependent information may be stored as times or speeds. In the ITIS Floating Vehicle Data (FVD ) covering the UK, the information for each arc in the network is summarised according to 15-min time bins throughout each day. The data is also matched to one of the 7 days of the week or to an eighth day if the data is for a public holiday. However, the times corresponding to the 96 time bins are not all significantly different. For example, late at night and in the early morning, when trunk roads are uncongested, there are no significant differences in the times required to travel along the roads. By examining the average speeds recorded in consecutive time bins, it is possible to reduce the number of time bins needed to give a reasonable approximation to the travel time function. For our model, it was decided to reduce the number of time bins to 15 to cover a 24 h period, the narrowest time bin being 30 min in width during peak traffic times and the widest being 8 h covering the night-time period. The time bins used in our model are illustrated in Fig. 3. When time-dependent travel times are used, consideration must be given to whether the travel time function possesses the FIFO consistency property. This is discussed in [3,5]. In[3] the lack of the FIFO property is referred to as passing. The FIFO property holds if a vehicle travelling from node i to node j in the network starting at any time T from node i, will arrive at node j at a later time than any identical vehicle starting from node i earlier than time T. This property is what one would expect to happen in road networks where vehicles are travelling at the same speed as each other. It implies that vehicles reach the end of a link in the same order as they enter the link and that shortest-time paths do not include any waiting at nodes. However, if a step function is used to represent time-dependent travel times by time of

7 3514 R. Eglese et al. / Computers & Operations Research 33 (2006) Traffic Density 00:00 02:00 04:00 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 Fig. 3. Time bins. day, then if the travel time decreases in consecutive time bins (i.e. the traffic is speeding up over the time period covered by the two time bins) the FIFO property will fail. Ichoua et al. [5] propose avoiding this problem by modelling travel speed as a step function of time of day and then performing a simple calculation to obtain the time implied to travel from one node to another along an arc of the network. The implied travel time function is shown to satisfy the FIFO property. This approach is similar to the one adopted by Sung et al. [9] in what they term their Flow Speed Model. Fleischmann et al. [3] adopt a different approach where the travel time function is modified from a step function by smoothing the original function and representing travel time as a linear function of time of day around the time bin boundaries. They demonstrate that provided the slopes of the linear functions are chosen with care, the resulting travel time function will retain the FIFO property. Ahn and Shin [10] also discuss the FIFO or non-passing property and show how some basic vehicle routing heuristics can be modified efficiently for time-dependent travel times with this property. Our approach is essentially equivalent to that adopted by Ichoua et al. [5]. However, the lengths of the individual arcs in our network and their transit times are relatively short compared to the width of the time bins used, so it is only relatively rare that the time of arrival at the end of an arc needs to be amended from a simple look-up function. Therefore, the calculation has been coded so that it is only necessary to enter the loop to perform the extra calculations needed when it is necessary to revise the arrival time to maintain the FIFO property. This ensures that the algorithm is fast while maintaining the FIFO property. The procedure can be described as follows: suppose we wish to find the arrival time at node j, a j, after travelling from node i to node j along a single arc ij when starting at node i at time s i. Let the time to go from i to j be saved as T k in the database when t k t<t k+1, where t k denotes the boundary of time bin k, k = 1,...,nand n is the number of time bins. Let the distance from i to j be d ij. The arrival time at node j, a j, is calculated as described by the following pseudo code: If s i + T k t k+1 then a j := s i + T k else set t := s i + T k,d := d ij, time := s i,v k := d ij /T k while (t >t k+1 ) d := d v k (t k+1 time) time := t k+1 v k+1 := d ij /T k+1 t := time + (d/v k+1 ) k := k + 1 a j := t.

8 R. Eglese et al. / Computers & Operations Research 33 (2006) For example, suppose a single arc ij is of length 2 km and that there are two time bins: one from 08:00 to 09:00 when the time to traverse the arc is stored as T 1 = 4 min and one from 09:00 to 10:00 when the time to traverse the arc is stored as T 2 = 2 min. If the starting time s i 08:56, then the arrival time at the end of the arc a j will be s i + 4. However, if 08:56 <s i 09:00, the code following the else statement is used. If s i = 08:58, then d := 2km,t:= 08:58 and v 1 := 0.5km/ min. Updating the variables gives d := 2 0.5(09:00 08:58) = 1 km, time := 09:00, v 2 := 2/2 = 1km/ min, t := 09:00 + 1/1 = 09:01, k:= 2 and a j := 09:01. In our case, using data from the ITIS Floating Vehicle Database for the UK, in the network used for the test in the final section, the mean arc length was 2.3 km. The mean time for traffic to traverse a single arc was 2.73 min at the least congested time and 3.15 min at the most congested time. So using time bins of width of at least 30 min meant that the code following the else statement was only entered rarely. 4. Carrying out the network calculations Providing the FIFO property holds, calculating the minimum time and shortest-time path between nodes on a network can be done efficiently using Dijkstra s algorithm [10] using time-dependent travel times for the arcs involved. This is discussed in Horn [11] where several adaptations of Dijkstra s algorithm are tested. In cases where the origins and destinations of journeys are known in advance, the network can be constructed so that the origins and destinations all correspond to nodes on the network. If the relevant network is not too big, this enables calculations for all origin-destination pairs needed for the Road Timetable TM to be completed on a standard PC in a reasonable computing time. The shortest-time between nodes in a time-dependent travel time network depends on the starting time for the journey and may vary even when the starting time remains within one of the time bins for which the travel time for any individual arc is constant. This is because the arrival times at intermediate nodes on the path to the destination node may be within different time bins. Therefore, if we wish to be able to look up a shortest travel time in the Road Timetable TM for any starting time, then there are potentially very many different times and paths to be stored together with the starting times that are on the boundaries. We have therefore opted to approximate the times by storing the shortest-times and paths at regular intervals throughout each day. The tests reported at the end of this paper used min intervals to cover each day. Experiments suggested that this produced an acceptable estimate of the travel times required for any starting time which would not be more than 7.5 min from one of the stored starting time values. The resulting Road Timetable TM was not too big to store within the memory of programme used for the vehicle routing and scheduling. Using this approach, it is of course easy to use more or less than the 15-min time intervals we adopted; reducing the interval will provide more accurate approximations to the travel times required at the expense of greater calculation time and a larger Road Timetable TM to store. However, when the minimum times and shortest paths are required quickly from a large national or regional network for any origin-destination pairs that have been identified, then the calculation may need to be speeded up. We decided to adopt a staged approach based on dividing the road network into a national and local road network using the road classifications. Various methods have been proposed to decrease the calculation time of a shortest-path algorithm and a survey of heuristics for this purpose is provided in [12]. That survey concludes that the hierarchical search strategy has the most potential for

9 3516 R. Eglese et al. / Computers & Operations Research 33 (2006) OP LDN NDN NDN LDN DP Origin Destination Legend OP - Origin Point LDN - Local Decision Node NDN - National Decision Node DP - Destination Point Fig. 4. Stages of calculation. computational time saving by several orders of magnitude, corresponding to the style of approach that we have adopted. The stages in the calculation are represented in Fig. 4. In the figure, the origin point is labelled OP and the destination point is labelled DP. The nodes where the path enters and leaves the road network are referred to as local decision nodes (LDNs). National decision nodes (NDNs) are nodes linking the most important trunk roads in the road network (corresponding to motorways and major A roads in the UK). The minimum times and shortest-time paths between the NDNs may be precalculated and stored as a Road Timetable TM for starting times at the required interval throughout the day for each day required. Dijkstra s algorithm may be used to find the minimum time and shortest-time path from the origin LDN to the nearest p NDNs. Depending on the direction of the journey and the characteristics of the road network, it will not always be quickest to travel from the origin LDN to the nearest NDN, so it is proposed that p>1; a value of p = 3 may be adequate for many road networks in practice. Similarly, Dijkstra s algorithm can be used to find the minimum time and shortest-time path from the nearest p NDNs to the destination LDN. It then remains to find the overall minimum time and shortest-time path from the p 2 possible routes that have been evaluated. The origin and destination may not correspond exactly to LDNs on the road network (e.g. where the origin or destination is a private address on a suburban estate) and may not even lie on roads included in the road network model. It is therefore necessary to decide at which LDN the path from the origin point meets the road network and to calculate the time required. The decision may not simply be the closest LDN, but may also take into account the direction of travel. The time required for a vehicle to travel from any origin point to a node included in the road network will be small compared to the time to the destination point. Therefore, this part of the calculation could be based simply on a crow-fly or straight-line distance divided by an appropriate factor to represent speed on minor roads and the fact that the actual road distance will be longer than the straight-line distance. An analogous calculation will be required at the end of the journey if the destination point does not correspond to an LDN. 5. Testing the Road Timetable TM A test was designed to demonstrate the effect that using a Road Timetable TM might have on a distribution operation using real traffic data. A scenario was modelled where a logistics service provider was assumed to have a distribution centre near Warrington in the north west of England and to be making deliveries from this centre to a set of 18 customers located in the surrounding area as shown in the map in Fig. 5.

10 R. Eglese et al. / Computers & Operations Research 33 (2006) Fig. 5. Locations of distribution centre and customers. A RoadTimetable TM for the distribution centre and customers was constructed in the way outlined in this paper for one day of the week (Friday). The network covered the principal roads in the area and consisted of 3326 arcs and 1666 nodes. For any journey between the distribution centre and a customer or between customers the minimum time and the shortest-time path could be retrieved from the Road Timetable TM array for a journey starting at 15-min time intervals throughout the day. The Road Timetable TM was created in 20.7 s on a PC using a Pentium IV processor operating at 3 GHz. A vehicle routing and scheduling algorithm was written which was capable of using the Road Timetable TM information in creating a distribution schedule. The algorithm is based on a tabu search heuristic and is described in another paper [13]. Customers were assigned different order sizes ranging between 1 and 3 pallets at random and each vehicle could hold up to 21 pallets. Each customer was also assigned a time window for delivery. In this demonstration each customer was assigned delivery time window of 1.5 h in duration. The time for each driver to complete each route was constrained to 6 h 20 min. An initial solution was found for the distribution for the day using times from the Road Timetable TM that corresponded to the least congested time of day, i.e. when the times to travel along arcs were at their shortest. The objective used was to minimise the total driving time required to distribute the goods to

11 3518 R. Eglese et al. / Computers & Operations Research 33 (2006) the customers where each trip was feasible with respect to the capacity of the vehicle and the service time windows of the customers. The solution required four vehicle routes with a total travelling time of 1037 min and a total distance travelled of 1158 km. However, when the routes in the distribution plan were re-evaluated using times from the Road Timetable TM, taking into account the time of day for the journeys, it was found that the actual travelling time increased to 1113 min (an increase of 7.3%). This resulted in one of the driver s requiring 23 min longer than the maximum time to complete the route (which in practice might lead to additional overtime payments) and that for two of the customers their service time windows were missed. The Road Timetable TM was used to construct a distribution plan taking into account the time of day for the journeys that minimised the total actual time travelled and respected the time window constraints. The final solution again used four vehicles; the total travelling time was 1176 min and the total distance travelled was 1250 km. This plan is feasible in that all routes are within the maximum route time constraint and all the customer service time windows are satisfied. In some cases, the routes travelled between customers are different from those used at the least congested time of day, because at other times of day it is preferable to use routes that are longer in distance but quicker. This solution is one that should be feasible in practice. The results quoted here are just one demonstration of the effect of taking account of time-dependent travel times in a distribution schedule. Whether using routes derived from static times will lead to missed time windows or overtime for drivers will depend to some extent on the particular instance examined. More extensive tests, together with a description of the vehicle routing and scheduling algorithm will be the subject of another paper [13]. However, this exercise demonstrates that using real road traffic data, the use of a Road Timetable TM and planning vehicle routes and schedules taking time-dependent journey times into account can make a significant difference to a distribution plan. A plan designed this way can help vehicles to avoid missing delivery time windows and reduce the likelihood of driver overtime. 6. Conclusions The paper has shown how information on time-dependent travel times in a road network can be used to construct a Road Timetable TM that will provide the minimum travel time and shortest-time path for vehicles travelling between given origins and destinations at different times of the day, different days of the week and different seasons of the year. It has also provided a demonstration to show how such data taken for a network of roads in the north west of England can be used to produce vehicle routes and schedules that can avoid missing service time windows and incurring driver overtime. Future work will include how to incorporate travel time information dynamically to make use of the latest information concerning road travel conditions. This paper has demonstrated the use of Road Timetable TM information for vehicle routing and scheduling, but other applications may include traffic information distributed by call-centres, on-board vehicle navigation systems, planning and calculating road toll charges, planning road access limitations (type of vehicle, timing and congestion charging) and in the planning of new roads. References [1] Toth P, Vigo D, editors. The vehicle routing problem. Philadelphia: Society for Industrial and Applied Mathematics; 2002.

12 R. Eglese et al. / Computers & Operations Research 33 (2006) [2] Hall R. On the road to recovery. ORMS Today 2004;31(3):40 9. [3] Fleischmann B, Gietz M, Gnutzmann S. Time-varying travel times in vehicle routing. Transportation Science 2004;38: [4] Fleischmann B, Gnutzmann S, Sandvoß E. Dynamic vehicle routing based on on-line traffic information. Working paper, University of Augsburg, [5] Ichoua S, Gendreau M, Potvin J-Y. Vehicle dispatching with time-dependent travel times. European Journal of Operational Research 2003;144: [6] Kim S, Lewis ME, White CC III. Optimal vehicle routing with real-time traffic information. Working paper, College of Engineering, University of Michigan, [7] Kim S, Lewis ME, White CC III. State space reduction for non-stationary stochastic shortest path problems with real-time traffic information. Working paper, College of Engineering, University of Michigan, [8] Dijkstra EW. A note on two problems in connexion with graphs. Numerische Mathematik 1959;1: [9] Sung K, Bell MGH, Seong M, Park S. Shortest paths in a network with time-dependent flow speeds. European Journal of Operational Research 2000;121:32 9. [10] Ahn B-H, Shin J-Y. Vehicle-routeing with time windows and time varying congestion. Journal of the Operational Research Society 1991;42: [11] Horn MET. Efficient modeling of travel in networks with time-varying link speeds. Networks 2000;36: [12] Fu L, Sun D, Rilett LR, Henderson J. Heuristic shortest path algorithms for ITS: state of the art. Computers & Operations Research, 2004, to appear. [13] Maden W, Eglese RW. A vehicle routing and scheduling method using a road timetable. Working paper, Lancaster University Management School, 2004.