Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

A distributed routing concept for vehicle routing problems

A distributed routing concept for vehicle routing problems Logist. Res. (2009) 1:45–52 DOI 10.1007/s12159-008-0003-4 OR IGINAL PAPER Henning Rekersbrink Æ Thomas Makuschewitz Æ Bernd Scholz-Reiter Received: 19 March 2008 / Accepted: 6 October 2008 / Published online: 4 November 2008 Springer-Verlag 2008 Abstract Traditional solution concepts for the vehicle 1 Introduction routing problem (VRP) are pushed to their limits, when applied on dynamically changing vehicle routing scenar- One opportunity to handle growing dynamics and com- ios—which are more close to reality than the static plexity of logistics systems is to shift from central planning formulation. By contrast, the introduced distributed routing to decentral, autonomous control strategies. The concept of concept is designed to match packages and vehicles and to autonomous control is the research area of the German continuously make route decisions especially within a Collaborative Research Centre (CRC) 637 ‘Autonomous dynamic environment. In this autonomous control concept, Cooperating Logistic Processes—A Paradigm Shift and its each of these objects makes its own decisions. The Limitations’. This CRC develops a new concept for developed algorithm was entitled Distributed Logistics dynamic transport networks, which is designed to match Routing Protocol (DLRP). But in spite of the restricted goods and vehicles and to continuously make route deci- suitability of the traditional VRP concepts for dynamic sions within a dynamic transport environment. Here, each environments, they are still the benchmark for any VRP- object makes its own decisions. It is called Distributed similar task. Therefore, we first present a description of the Logistics Routing Protocol (DLRP). developed DLRP. Then an adapted vehicle routing problem In order to evaluate this new concept, we compare it to is defined, which both sides, static and dynamic concepts, the traditional solutions for the vehicle routing problem can cope with. Finally, both concepts are compared using a (VRP), shown in this article. tabu search algorithm as a well working instance of tradi- To describe the different approach of autonomous con- tional VRP-concepts. For a quantitative comparison, four trol to transport problems, the developed DLRP is solutions are given for the same adapted problem: the described at first. In contrast to traditional algorithms for optimal solution as a lower bound, the DLRP solution, a the VRP problem, which do static optimisation, this tabu search solution and a random-like solution as an upper approach tries to control an ongoing dynamic transport bound. process. Therefore the problem definitions of both sides are dif- Keywords Vehicle routing problem (VRP)  ferent in principal. One basic point is that the VRP is a Autonomous control  Distributed logistics routing static problem, because all customers are known at the protocol (DLRP)  Tabu search  Optimisation  beginning. In contrast, the problem for the DLRP is a Routing algorithm  Transport logistic dynamic network formulation: the customers appear con- tinuously and are not known from the beginning. In order to make the results for both sides comparable, an adapted VRP scenario is described which can be handled by the H. Rekersbrink (&)  T. Makuschewitz  B. Scholz-Reiter DLRP and traditional VRP algorithms. BIBA an der Universitat Bremen, Several versions of this adapted VRP scenario were Hochschulring 20, 28359 Bremen, Germany solved by the DLRP on the one hand and by a tabu search e-mail: rek@biba.uni-bremen.de algorithm on the other hand. The tabu search algorithm was URL: www.biba.uni-bremen.de 123 46 Logist. Res. (2009) 1:45–52 taken as a typical traditional VRP algorithm which can inspired by internet routing protocols, which are able to find manage big solution spaces (see [1, p. 275]). routes through a permanently changing and unknown net. In To rate the results and to evaluate the new concept in a addition, these concepts are able to deal with very large nets more objective way, four solutions are given for the same without a central perspective. The basic concept for one adapted problem. In addition to the DLRP and the tabu data package wanting to get to its destination is the search solution, lower and upper bounds for the overall RouteRequest/RouteReply mechanism. This package sends vehicle distance were calculated. The optimal vehicle ways a RouteRequest to all its neighbour vertices, which for are given as best and some kind of random vehicle ways as themselves sent it ahead to their neighbours. If one vertex worst case values. Therefore the new concept is evaluated notices that it is the destination for this RouteRequest, it relatively to a traditional algorithm and secondary to an sends back a RouteReply to the asking package (for a more absolute scale. detailed description, see, e.g. [7]). One part of the DLRP is based on this concept. On the basis of Fig. 1 the fundamental procedure of the developed 2 Distributed Logistics Routing Protocol protocol can be illustrated: When a package makes a route decision, it first disannounces its old route (see Fig. 1: Real life scenarios of transport processes require a kind of RouteDisAnnouncement) and announces its actual planned continuous control of logistic objects. Objects like pack- routes to the vertices involved (see Fig. 1: RouteAn- ages and vehicles appear and disappear continuously—the nouncement). An individual vertex thus has information scenario is a dynamic one. After a close consideration on about when how many packages with what destinations static routing problems like TSP, TRP, VRP, or PDP, to will be at its position. Additional information such as name a few, the need of research on dynamic problems had restrictions concerning the transport of the packages (e.g. been stressed in literature recently (see [2–5]or[6] for cooling freight) is stored likewise. examples). In the majority of cases, dynamic means that If a vehicle needs a route, it sends a RouteRequest to the not all customer orders are known in advance, in contrast to net—the RouteRequest/RouteReply mechanisms are the the traditional static scenario. same as described above. After receiving several Rou- The second crucial property of real life scenarios is their teReplies, which are route suggestions with appropriate size. Our global viewpoint and vision is the control of additional information, a vehicle decides on a route—for nearly all transports for example in Germany—to show our example the route with the maximum expected utilisation. long range perspective. Under these circumstances, it is not This Route is then announced to the involved vertices (see possible to calculate any optimum—it is not even possible Fig. 1: RouteAnnouncement). This leads to a continuous cooperative structure. The objects in a transport net do not to receive all relevant information for one point of time. All approaches mentioned above follow a central strategy, plan their route at the same time. Packages emerge con- which has strong restrictions to the scenario size (the tinuously or reach their destination, vehicles replan their number of orders in the mentioned scenarios vary between routes and so on. At each time there is enough information 100 and 1,000). for any route decision. Against this background, an autonomous control concept The whole DLRP concept offers outstanding advantages for transport nets was developed. The concept was initially for real life applications such as: self-adaptation, manual Package Vehicl Vehicle e Vertex Package send send one one send send one one R RouteRequest outeRequest Ro RouteRequest uteRequest pass information about vehicle and package r receive several eceive several rec receive several eive several routes R RouteReplys outeReplys R RouteReplys outeReplys Ro Route Decision ute Decision R Route Decision oute Decision [R [RouteDisAnnouncement] outeDisAnnouncement] [R [RouteDisAnnouncements] outeDisAnnouncements] collect and update send send send send several several information about vehicle R RouteAnnouncement outeAnnouncement R RouteAnnouncements outeAnnouncements and package routes Fig. 1 Scheme of the distributed logistics routing protocol, DLRP (from [8]) 123 Logist. Res. (2009) 1:45–52 47 intervention, estimation of future net conditions, implicit objects have their own objective function, e.g. shortest uncertain knowledge, arbitrary decision processes and way for packages and the best utilisation for vehicles arbitrary kind and quantity of information (see [8]). For the It is not reasonable for a DLRP implementation to deal solution of the described adapted vehicle routing problems with full nets. In full nets, the RouteRequest/RouteReply- (see next section), the concept was simplified to exclusively mechanism leads to a factorial growing number RouteRe- optimise the overall vehicle distance. But for all that, the quest objects—a combinatorial explosion. Because it is algorithm is still very complex and has many points, directly easy for most traditional VRP-algorithms to build them for affecting performance; e.g. which packages are loaded into a not-full-nets, we decided to restrict the net to feasible edges vehicle at the vertex: all with the same direction, all with for the adapted problem. their destination vertex on the vehicle route or other pack- We decided to take a real world network. In Fig. 2 you ages. Hence, a detailed description is not possible here. For a can see the chosen topology, which is the basic autobahn- more detailed description refer to [8–10]. net of Germany. The topology contains 18 vertices, the biggest cities in Germany, and 35 undirected edges. The scenario edge lengths match the real ones. 3 Adapted vehicle routing problem All traditional VRP optimisation algorithms have a static nature. They can only handle dynamic environments, if they Unfortunately, it is hardly possible to compare the different are embedded in a replanning algorithm. On the other side, approaches for dynamic scenarios. On the one hand, each the DLRP can cope with these static cases, even though it was approach deals with different scenarios [1]: regards the created especially for the control of dynamic environments. case of vehicle capacity of 1, which leads to some kind of For a good performance, the DLRP has to be adopted for this TRP [2] has no net but coordinates, [3] does not allow a special static case. Hence the adapted VRP was created as a replanning of a truck with an order, [4] does not allow a static problem, all orders are known from the beginning. transhipment of packages and [5] takes stochastic infor- In the DLRP, the routes of the vehicles are never-end- mation into account (which is an improvement here). Due ing. Because of this character, it is almost impossible to to this large diversity of dynamic problem formulations, we find the point where a vehicle has finished its work within a decided to compare our new approach to the most basic bounded scenario. On the other hand, it is not too difficult VRP instances in this field. An additional advantage of this for the traditional VRP-algorithms, to transfer the objective approach is that we were able to compare our approach to function from closed to non-closed vehicle routes. So the traditional algorithms and to an absolute scale with the adapted VRP treats the distance of non-closed vehicle optimal solution—which normally cannot be calculated for routes as objective to minimise. the dynamic scenarios. The fourth point is not a conflict, but needs to be The approach to the transportation problem taken for the mentioned. The original [12] as well as the most discussed DLRP is basically different to the approach for the tradi- VRP-formulations (e.g. [13]) take the sum of driven tional VRP. The developed protocol is not an optimisation vehicle distances as objective to minimise—other objective algorithm for a static scenario, but an autonomous control functions are also discussed, see e.g. [1, p. 276]. But for the algorithm, designed for a continuous changing process. In traditional objective, we can easily show that the overall order to compare both concepts, it is necessary to execute vehicle distance ( d ), package distances (p ) and vehicle j i them on one scenario which both sides can cope with. utilisation (u ) are connected: To draw such an adapted vehicle routing problem for both sides, let us first have a look at major differences between the traditional vehicle routing problem formula- tion and the scenarios for the DLRP: • the VRP instances have only coordinates so every possible path is allowed—the DLRP scenario has few edges and no fully connected net • the DLRP is designed for a dynamic scenario, orders may appear at every vertex and at every time—the VRP orders are known from the beginning • the VRP enforces closed routes—because of its dynamic control nature, the DLRP has ongoing, unclosed routes • the objective function for the most VRP instances is the sum of all vehicle distances—within the DLRP, all Fig. 2 Topology for the adapted VRP (from [11]) 123 48 Logist. Res. (2009) 1:45–52 X 4 Computational results i vehicle capacity all packages i ¼  d ð1Þ P j P package size j all vehicles j all vehicles j 4.1 Random like solution This equation requires uniform vehicles and uniform In order to give something like an upper bound for rea- packages concerning capacity and size. So the DLRP will sonable solutions to the scenarios, a random-like heuristic indirectly minimise the requested objective function even was implemented. though it primarily minimises the package distances and In the first step, all packages are assigned randomly to maximises the vehicle utilisation. the vehicles. A uniform distribution is used for this step The new adapted VRP scenarios were built like distri- with the restriction that every vehicle has to carry at least bution scenarios in this first application. All vehicles start one package. The second step is to find an optimal way for at a central vertex, the city of Kassel, and all orders start one vehicle and its load of packages. To save computing form there. In order to keep the optimum value comput- time, only loop-free routes are considered. This restriction able, the size of the scenarios is not too large. The number makes the route non optimal in some cases, but it is of vehicles can be 3, 6 or 9, while the number of packages assumed that the optimal solution is not too far (see above, can vary between 17, 34, 51 or 68. In addition we created a the difference between the shortest way with and without large scenario with 68 packages and 12 vehicles, which loops is 10 km or 0.4%). was the largest optimal solvable scenario. The amount of This algorithm was calculated 10,000 times for each packages was matched to the topology. All 17 vertices, subset. The resulting mean values are shown in the next except Kassel, were supposed to be costumers. The pack- table (Table 2). age destinations for the larger scenario sets are uniformly The described algorithm has some analogies with the distributed, whereas each vertex has one package at least. real world transport market: the different forwarder com- Therefore the 34, the 51 and the 68-scenarios have 10 panies receive their orders randomly and each company subsets with different package destinations. The vehicle tries to optimise its vehicle routes on its own. An overall capacity was chosen in that way, that every vehicle is optimum cannot be expected from a procedure like this. needed, if no vehicle comes back to Kassel. All different Note that the average utilization which was reached by the scenarios are shown in Table 1. DLRP in the largest scenario 68-12, about 70%, would be a For an overview, three indicator values can be given: very good value for real world forwarder companies. The • the shortest way from Kassel via all 17 vertices is vision of the DLRP is to implement this protocol inde- 2,235 km long pendently from different companies. In this vision, an • the shortest way without any loop is 2,245 km long overall optimisation can happen without taking any deci- • the sum of the direct shortest ways for each of the 17 sion possibilities from the single forwarders [8]. packages is 4,965 km 4.2 Optimal solution For a more detailed description of the scenarios con- cerning the topology distances and the distribution of the On the other side, a lower bound for the overall vehicle package destinations, feel free to contact the authors or distance should be given. To calculate optimal solutions for refer to [14]. Tables and scenario data can be found on the given instances, the specified problem was formulated ‘‘http://dlrp.biba.uni-bremen.de’’. as a mixed-integer program (MIP, see below). The objec- tive of the program is based on the formulation of Dantzig and Ramser [12] and minimizes the distance driven by all Table 1 Chosen scenarios and corresponding vehicle capacities vehicles. In order to cope with the characteristics described number of vehicles 369 12 8 pck/veh 3 pck/veh 2 pck/veh Table 2 Vehicle distances calculated by the random-like algorithm 1 subset Number of packages Number of vehicles 16 pck/veh 6 pck/veh 4 pck/veh - 369 12 10 subsets 25 pck/veh 10 pck/veh 6 pck/veh - 17 3,927 5,118 5,474 10 subsets 34 5,051 7,594 9,072 51 5,583 8,940 11,225 33 pck/veh 13 pck/veh 8 pck/veh 6 pck/veh 68 5,943 9,897 12,756 14,876 10 subsets number of packages Logist. Res. (2009) 1:45–52 49 Table 3 Optimal vehicle distances capacitated vehicle routing problem (CVRP) and the split order vehicle routing problem (SDVRP). Within the Number of packages Number of vehicles considered network only one depot exists and the verti- ces are connected by undirected edges. Thus the associated distance between two directly connected ver- 17 2,245 2,565 3,095 tices is the same in either way. The demand of each 34 2,245 2,868 3,310 vertex is a priori known. In this context the demand of a 51 2,245 2,608 3,362 certain vertex can be split and met by either one or 68 2,245 2,683 3,400 4,097 multiple deliveries. According to the CVRP the capacity of the vehicles is limited. For our analysis we assume the same capacity r  for every vehicle v. All vehicles in chapter 3, we adapt the formulation of the VRP. The start their route at the depot and have to be used. In our proposed formulation was implemented in GAMS and modeling approach the route of a vehicle is described by could be solved with CPLEX 11. Because of the quite a number of consecutive segments. In this way a seg- small size of the instances, the MIP could be solved within ment represents either a movement of a vehicle from acceptable time—the scenario 68-12 seems to be the vertex i to i or the end of the route. Note that the total largest one which can be reasonably solved optimal. number of permitted segments of a route is a critical Table 3 shows the optimal values for the overall vehicle constraint of the problem. For our analysis we have distances. Note that in this solution, routes with loops are chosen the number of segments as high as the number of allowed and each vehicle has to be used. vertices within the network. Every vehicle is allowed to visit each vertex several times during its route. This 4.2.1 MIP-formulation characteristic of the model permits a vehicle on the one hand to serve a remote vertex from a given vertex and to 4.2.1.1 Nomenclature return afterwards to the vertex before it continues its Sets route through the network and on the other hand to pick up packages several times from the depot. Furthermore I vertices of the network D D each vertex can be used to store packages. This means a I depots of the considered network (I , I); in our case vertex can receive more packages than requested and only one depot exists S S that these packages can be picked up and delivered to I vertices that are directly connected to vertex i (I 7 I) i i other vertices. At the end of their route the vehicles S segments of vehicle routes remain at the vertex of their final delivery and do not V vehicles have to return to the depot. Parameters 4.2.1.3 Mathematical model Problem constraints: The c distance between vertex i and i i;i first segment of the route of each vehicle starts at the depot. d demand of packages at vertex i p provided packages at vertex i; in our case packages 0 i x ¼ 1 i 2 I ; v 2 V; s ¼ 0 ð2Þ v;i;i ;s 0 S are only provided at the depot i [ I with p = d i 2I i i[I i r  transportation capacity of vehicle v The route of each vehicle can be terminated only once at r required transportation capacity of one package any vertex of the network. M a very large number XX z ¼ 1 ðÞ v 2 V ð3Þ v;i;s Variables s2S i2I u amount of transported packages from vertex i to i v;i;i ;s In every segment the route of a vehicle can be either by vehicle v in s continued or terminated. x binary variable denoting that vehicle v drives in s v;i;i ;s XX X x þ z  1ðÞ s 2 S; v 2 V ð4Þ from vertex i to i v;i;i ;s v;i;s i2I 0 i2I i 2I z binary variable denoting that vertex i is the end of i v,i,s the route of vehicle v in s The route of a certain vehicle is described by a set of consecutive segments. This means a vehicle has either to leave its current vertex in the successive segment or to 4.2.1.2 Model assumptions The applied formulation of terminate its route. The introduction of segments ensures the vehicle routing problem has characteristics of the that a vehicle can visit a certain vertex several times and no 123 50 Logist. Res. (2009) 1:45–52 independent sub cycles occur. Note that the depot can be Table 4 Vehicle distances calculated by the tabu search algorithm visited more than once as well. Number of packages Number of vehicles X X 0 0 0 x  x  z ¼ 0 v;i;h;s v;h;i ;s v;h;s 0 S i2I: i 2I h2I 17 2,410 2,610 3,095 0 0 ðÞ h 2 I; s; s 2 S : s ¼ s þ 1; v 2 V ð5Þ 34 3,335 3,689 3,775 51 3,988 4,637 5,026 The load of each vehicle has to be less than or equal to 68 4,409 5,782 6,580 7,058 the maximum transportation capacity. 0 S u r  r  i 2 I; i 2 I ; s 2 S; v 2 V ð6Þ v;i;i ;s v 4.4 DLRP solution Within the network the packages are transported by the The DLRP solution was calculated with the DLRP roughly vehicles between directly connected vertices. In this context every vertex can receive, store and ship described above. To increase the performance, the protocol packages. Furthermore each vertex has a deterministic was adapted to this special static situation. Additionally the decision functions for the different objects were simplified demand of packages. In our case only the depot i [ I has a stock of packages p in the beginning. and harmonized: packages only choose their routes by the route length, vehicles only by the estimated utilisation. This s s XX X XXX p þ u ¼ d þ u means that the packages do not adjust to the vehicle routes, i v;h;i;r i v;i;i ;r v2V r¼0 v2V r¼0 0 h2I: i 2I in contrast to the dynamic version. Only vehicles choose i2I their route dependent on the package route situation. ðÞ i 2 I; s 2 S ð7Þ The results are shown in Table 5. Routes with loops are allowed here and each vehicle does not need to carry a Packages can only be shipped between two directly package, but do so. connected vertices i and i if a vehicle v serves this segment s on its route. Equation 8 simplifies the problem in a way that avoids a formulation as a mixed integer non-linear 5 Conclusions program. 0 S 0 0 u  x Mi 2 I; i 2 I ; s 2 S; v 2 V ð8Þ v;i;i ;s v;i;i ;s i For a better overview, all results are shown as line charts in the following Figs. 3, 4, 5 and 6. Compared to the optimum Objective function: The objective of the formulation is to and the random-like solution, the tabu search heuristic minimize the distance driven by all vehicles. leads to near optimal results with small scenarios and gets P PP P 0 0 Min: x c v;i;i ;s i;i worse with larger scenarios. This is a normal behaviour for ð9Þ v2V s2S i2I 0 i 2I tabu search algorithms and is due to the exponential growing solution space. 4.3 Tabu search solution In contrast, the DLRP solutions are not very good with small scenarios, but get better with larger scenarios. The As a representative of established solution techniques (see large scenarios 68-6, 68-9 and 68-12 show that the DLRP [1, 15]) for vehicle routing problems, a tabu search algo- gets better than the tabu search concept with a growing rithm was applied to the scenarios. network size. This algorithm is similar to the random-like solution Because the DLRP is originally a control method, its technique. One solution set for the tabu search is one main advantages point to dynamic and close to reality assignment set. This set assigns all packages to the vehi- scenarios: self-adaptation to changing situations, possible cles. After the assignment, an optimal route or each vehicle is calculated (again with loop-free routes only like the random-like algorithm). Table 5 Vehicle distances calculated by the DLRP For the neighbourhood generation, the k -interchange Number of packages Number of vehicles generation mechanism by Osmand [16] was implemented. The k was set to 1 and insured that no vehicle is empty. The maximum age of elements within the tabulist was set to 6 17 4,530 2,875 4,400 and the search was aborted after 18 moves. Note that tabu 34 5,184 5,928 7,022 search is very close to the optimal solution in the small 51 4,918 4,807 6,310 scenarios and moves away with greater scenarios (Table 4). 68 4,558 5,586 6,331 6,550 123 Logist. Res. (2009) 1:45–52 51 17 packages 68 packages 12,000 12,000 tabu search ttabu search ab u search optimum op optimum t imum 10,000 10,000 random-like ra random-like ndo m-like DLRP DL DLRP RP 8,000 8,000 6,000 6,000 4,000 4,000 2,000 2,000 369 12 369 12 no of vehicles no of vehicles Fig. 3 Results for 17 packages Fig. 6 Results for 68 packages 34 packages and dynamic scenarios with 2,500 packages emphasise this 12,000 assumption. These results are shown in Fig. 7. tabu search optimum 10,000 random-like 5.1 New evaluation chart DLRP 8,000 For larger scenarios, the classification and evaluation of algorithms for the VRP or the adapted VRP gets more and 6,000 more difficult. For these scenarios, it is not possible to calculate optimal solutions, so a lower bound is missing. 4,000 Additionally, it is not possible to compare two different scenarios, because the optimal way lengths can be very 2,000 369 12 different. On the basis of Eq. (1), we suggest an alternative no of vehicles comparison approach. The vehicle utilisation and the Fig. 4 Results for 34 packages package distances are, with some restrictions, directly connected to the overall vehicle distance. The vehicles utilisation has a natural best value and upper bound: it 51 packages 12,000 cannot be greater than one. The package distances have a ttabu search ab u search natural best value and lower bound, which is easy to cal- op optimum t imum 10,000 culate: their individual shortest way to their destination. ra random-like ndom-like DL DLRP RP Therefore we can define a relative package distance: driven 8,000 package distance by shortest possible distance. The lower bound of this relative package distance is one. 6,000 Now we can illustrate these two values in one plane, exemplarily shown in Fig. 7. The upper right corner rep- 4,000 resents one theoretical extreme: each package has one vehicle of capacity 1 and drives its shortest way. The 2,000 horizontal line at utilisation = 0.5 is another extreme: 369 12 when one vehicle with capacity for all packages brings out no of vehicles all packages, the utilisation goes to 0.5 and the relative Fig. 5 Results for 51 packages package distance increases infinitely. In this suggested evaluation chart, on can see that an manual interventions at runtime, implementation of optimal solution must be somewhere in the upper right uncertain knowledge and complex and context driven corner: a relative package distance of nearly 1 and a high decision functions [8]. The described results show that the utilisation of 80–90% (see optimum of scenario 68-12). DLRP has a high potential. Apart from the described In Fig. 7, some of the larger scenario results are shown. advantages the DLRP can be coequal to classical VRP With this kind of chart, it is possible to compare the large concepts in real world scenarios. DLRP results from large dynamic scenario results as well. Even though there are no overall vehicle distance [km] overall vehicle distance [km] overall vehicle distance [km] overall vehicle distance [km] 52 Logist. Res. (2009) 1:45–52 1.00 optimum 51-9 optimum 68-9 optimum 68-12 0.75 DLRP 51-9 DLRP 68-9 DLRP 68-12 tabu search 51-9 0.50 tabu search 68-9 tabu search 68-12 random-like 51-9 0.25 random-like 68-9 random-like 68-12 big scenarios DLRP 0.00 3.00 2.50 2.00 1.50 1.00 relative package distance [-] Fig. 7 Alternative evaluation chart for the comparison of VRP solutions 6. Bent RW, Van Hentenryck P (2004) Scenario-based planning for optimal or tabu search values for these scenarios, one can partially dynamic vehicle routing with stochastic customers. Oper say that the DLRP works well with these large dynamic Res 52(6):977–987 environments. Additionally, one can see that it is possible 7. Perkins CE (2001) Ad hoc networking. Addison-Wesley, Boston to change the DLRP parameters in a way that either the 8. Scholz-Reiter B, Rekersbrink H, Freitag M (2006) Internet routing protocols as an autonomous control approach for trans- vehicle utilisation or the relative package distance is port networks. In: Proceedings of the 5th CIRP international preferred. seminar on intelligent computation in manufacturing engineering, Another practical advantage of this evaluation chart is pp 341–345 that the suggested values can be treated as a state function. 9. Scholz-Reiter B, Rekersbrink H, Freitag M (2006) Kooperierende Routingprotokolle zur Selbststeuerung von Transportprozessen. It is possible to measure the vehicle utilisation and the Industrie Management 22/3, pp 7–10 relative package distance continuously. Consequently it is 10. Wenning B-L, Rekersbrink H, Timm-Giel A, Go ¨ rg C, Scholz- possible to control these values. Reiter B (2007) Autonomous control by means of distributed routing. In: Understanding Autonomous Cooperation & Control Acknowledgments This research is funded by the German in Logistics—The Impact on Management, Information and Research Foundation as part of the Collaborative Research Centre Communication and Material Flow. Springer, Berlin, pp 325–336 637 ‘Autonomous Cooperating Logistic Processes—A Paradigm Shift 11. Wenning B-L, Rekersbrink H, Becker M, Timm-Giel A, Go ¨ rg C, and its Limitations’. Scholz-Reiter B (2007) Dynamic transport reference scenarios. In: Understanding autonomous cooperation & control in logis- tics—the impact on management, information and communication and material flow. Springer, Berlin, pp 337–350 References 12. Dantzig GB, Ramser JH (1959) The truck dispatching problem. Manage Sci 6(1):80–91 1. Vahrenkamp R, Mattfeld D (2007) Logistiknetzwerke. Gabler, 13. Solomon MM (1987) Algorithms for the vehicle routing and Wiesbaden scheduling problems with time window constraints. Oper Res 2. Fleischmann B, Gnutzmann S, Sandvoß E (2004) Dynamic 35(2):254–265 vehicle routing based on online traffic information. Transp Sci 14. Scholz-Reiter B, Rekersbrink H, Wenning, B-L, Makuschewitz T 38(4):420–433 (2008) A survey of autonomous control algorithms by means of 3. Jaillet P, Wagner MR (2006) Online routing problems: value of adapted vehicle routing problems. In: Proceedings of the 9th advanced information as improved competitive ratios. Transp Sci Biennial ASME conference on engineering systems design and 40(2):200–210 analysis ESDA 08 (on CD), Haifa, Israel 4. Hiller B, Krumke SO, Rambau J (2006) Reoptimization gaps 15. Laporte G (1992) The vehicle routing problem: an overview of versus model errors in online-dispatching of service units for exact and approximative algorithms. Eur J Oper Res 59:345–358 adac. Discrete Appl Math 154:1897–1907 16. Osman IH (1993) Metastrategy simulated annealing and tabu 5. Savelsbergh M, Sol M (1998) Drive: dynamic routing of inde- search for the vehicle routing problem. Ann Oper Res 41:421– pendent vehicles. Oper Res 46:474–490 vehicle utilisation [-] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Logistics Research Springer Journals

A distributed routing concept for vehicle routing problems

Loading next page...
 
/lp/springer-journals/a-distributed-routing-concept-for-vehicle-routing-problems-glZ7NevpjX
Publisher
Springer Journals
Copyright
Copyright © 2008 by Springer-Verlag
Subject
Engineering; Engineering Economics, Organization, Logistics, Marketing; Logistics; Industrial and Production Engineering; Simulation and Modeling; Operation Research/Decision Theory
ISSN
1865-035X
eISSN
1865-0368
DOI
10.1007/s12159-008-0003-4
Publisher site
See Article on Publisher Site

Abstract

Logist. Res. (2009) 1:45–52 DOI 10.1007/s12159-008-0003-4 OR IGINAL PAPER Henning Rekersbrink Æ Thomas Makuschewitz Æ Bernd Scholz-Reiter Received: 19 March 2008 / Accepted: 6 October 2008 / Published online: 4 November 2008 Springer-Verlag 2008 Abstract Traditional solution concepts for the vehicle 1 Introduction routing problem (VRP) are pushed to their limits, when applied on dynamically changing vehicle routing scenar- One opportunity to handle growing dynamics and com- ios—which are more close to reality than the static plexity of logistics systems is to shift from central planning formulation. By contrast, the introduced distributed routing to decentral, autonomous control strategies. The concept of concept is designed to match packages and vehicles and to autonomous control is the research area of the German continuously make route decisions especially within a Collaborative Research Centre (CRC) 637 ‘Autonomous dynamic environment. In this autonomous control concept, Cooperating Logistic Processes—A Paradigm Shift and its each of these objects makes its own decisions. The Limitations’. This CRC develops a new concept for developed algorithm was entitled Distributed Logistics dynamic transport networks, which is designed to match Routing Protocol (DLRP). But in spite of the restricted goods and vehicles and to continuously make route deci- suitability of the traditional VRP concepts for dynamic sions within a dynamic transport environment. Here, each environments, they are still the benchmark for any VRP- object makes its own decisions. It is called Distributed similar task. Therefore, we first present a description of the Logistics Routing Protocol (DLRP). developed DLRP. Then an adapted vehicle routing problem In order to evaluate this new concept, we compare it to is defined, which both sides, static and dynamic concepts, the traditional solutions for the vehicle routing problem can cope with. Finally, both concepts are compared using a (VRP), shown in this article. tabu search algorithm as a well working instance of tradi- To describe the different approach of autonomous con- tional VRP-concepts. For a quantitative comparison, four trol to transport problems, the developed DLRP is solutions are given for the same adapted problem: the described at first. In contrast to traditional algorithms for optimal solution as a lower bound, the DLRP solution, a the VRP problem, which do static optimisation, this tabu search solution and a random-like solution as an upper approach tries to control an ongoing dynamic transport bound. process. Therefore the problem definitions of both sides are dif- Keywords Vehicle routing problem (VRP)  ferent in principal. One basic point is that the VRP is a Autonomous control  Distributed logistics routing static problem, because all customers are known at the protocol (DLRP)  Tabu search  Optimisation  beginning. In contrast, the problem for the DLRP is a Routing algorithm  Transport logistic dynamic network formulation: the customers appear con- tinuously and are not known from the beginning. In order to make the results for both sides comparable, an adapted VRP scenario is described which can be handled by the H. Rekersbrink (&)  T. Makuschewitz  B. Scholz-Reiter DLRP and traditional VRP algorithms. BIBA an der Universitat Bremen, Several versions of this adapted VRP scenario were Hochschulring 20, 28359 Bremen, Germany solved by the DLRP on the one hand and by a tabu search e-mail: rek@biba.uni-bremen.de algorithm on the other hand. The tabu search algorithm was URL: www.biba.uni-bremen.de 123 46 Logist. Res. (2009) 1:45–52 taken as a typical traditional VRP algorithm which can inspired by internet routing protocols, which are able to find manage big solution spaces (see [1, p. 275]). routes through a permanently changing and unknown net. In To rate the results and to evaluate the new concept in a addition, these concepts are able to deal with very large nets more objective way, four solutions are given for the same without a central perspective. The basic concept for one adapted problem. In addition to the DLRP and the tabu data package wanting to get to its destination is the search solution, lower and upper bounds for the overall RouteRequest/RouteReply mechanism. This package sends vehicle distance were calculated. The optimal vehicle ways a RouteRequest to all its neighbour vertices, which for are given as best and some kind of random vehicle ways as themselves sent it ahead to their neighbours. If one vertex worst case values. Therefore the new concept is evaluated notices that it is the destination for this RouteRequest, it relatively to a traditional algorithm and secondary to an sends back a RouteReply to the asking package (for a more absolute scale. detailed description, see, e.g. [7]). One part of the DLRP is based on this concept. On the basis of Fig. 1 the fundamental procedure of the developed 2 Distributed Logistics Routing Protocol protocol can be illustrated: When a package makes a route decision, it first disannounces its old route (see Fig. 1: Real life scenarios of transport processes require a kind of RouteDisAnnouncement) and announces its actual planned continuous control of logistic objects. Objects like pack- routes to the vertices involved (see Fig. 1: RouteAn- ages and vehicles appear and disappear continuously—the nouncement). An individual vertex thus has information scenario is a dynamic one. After a close consideration on about when how many packages with what destinations static routing problems like TSP, TRP, VRP, or PDP, to will be at its position. Additional information such as name a few, the need of research on dynamic problems had restrictions concerning the transport of the packages (e.g. been stressed in literature recently (see [2–5]or[6] for cooling freight) is stored likewise. examples). In the majority of cases, dynamic means that If a vehicle needs a route, it sends a RouteRequest to the not all customer orders are known in advance, in contrast to net—the RouteRequest/RouteReply mechanisms are the the traditional static scenario. same as described above. After receiving several Rou- The second crucial property of real life scenarios is their teReplies, which are route suggestions with appropriate size. Our global viewpoint and vision is the control of additional information, a vehicle decides on a route—for nearly all transports for example in Germany—to show our example the route with the maximum expected utilisation. long range perspective. Under these circumstances, it is not This Route is then announced to the involved vertices (see possible to calculate any optimum—it is not even possible Fig. 1: RouteAnnouncement). This leads to a continuous cooperative structure. The objects in a transport net do not to receive all relevant information for one point of time. All approaches mentioned above follow a central strategy, plan their route at the same time. Packages emerge con- which has strong restrictions to the scenario size (the tinuously or reach their destination, vehicles replan their number of orders in the mentioned scenarios vary between routes and so on. At each time there is enough information 100 and 1,000). for any route decision. Against this background, an autonomous control concept The whole DLRP concept offers outstanding advantages for transport nets was developed. The concept was initially for real life applications such as: self-adaptation, manual Package Vehicl Vehicle e Vertex Package send send one one send send one one R RouteRequest outeRequest Ro RouteRequest uteRequest pass information about vehicle and package r receive several eceive several rec receive several eive several routes R RouteReplys outeReplys R RouteReplys outeReplys Ro Route Decision ute Decision R Route Decision oute Decision [R [RouteDisAnnouncement] outeDisAnnouncement] [R [RouteDisAnnouncements] outeDisAnnouncements] collect and update send send send send several several information about vehicle R RouteAnnouncement outeAnnouncement R RouteAnnouncements outeAnnouncements and package routes Fig. 1 Scheme of the distributed logistics routing protocol, DLRP (from [8]) 123 Logist. Res. (2009) 1:45–52 47 intervention, estimation of future net conditions, implicit objects have their own objective function, e.g. shortest uncertain knowledge, arbitrary decision processes and way for packages and the best utilisation for vehicles arbitrary kind and quantity of information (see [8]). For the It is not reasonable for a DLRP implementation to deal solution of the described adapted vehicle routing problems with full nets. In full nets, the RouteRequest/RouteReply- (see next section), the concept was simplified to exclusively mechanism leads to a factorial growing number RouteRe- optimise the overall vehicle distance. But for all that, the quest objects—a combinatorial explosion. Because it is algorithm is still very complex and has many points, directly easy for most traditional VRP-algorithms to build them for affecting performance; e.g. which packages are loaded into a not-full-nets, we decided to restrict the net to feasible edges vehicle at the vertex: all with the same direction, all with for the adapted problem. their destination vertex on the vehicle route or other pack- We decided to take a real world network. In Fig. 2 you ages. Hence, a detailed description is not possible here. For a can see the chosen topology, which is the basic autobahn- more detailed description refer to [8–10]. net of Germany. The topology contains 18 vertices, the biggest cities in Germany, and 35 undirected edges. The scenario edge lengths match the real ones. 3 Adapted vehicle routing problem All traditional VRP optimisation algorithms have a static nature. They can only handle dynamic environments, if they Unfortunately, it is hardly possible to compare the different are embedded in a replanning algorithm. On the other side, approaches for dynamic scenarios. On the one hand, each the DLRP can cope with these static cases, even though it was approach deals with different scenarios [1]: regards the created especially for the control of dynamic environments. case of vehicle capacity of 1, which leads to some kind of For a good performance, the DLRP has to be adopted for this TRP [2] has no net but coordinates, [3] does not allow a special static case. Hence the adapted VRP was created as a replanning of a truck with an order, [4] does not allow a static problem, all orders are known from the beginning. transhipment of packages and [5] takes stochastic infor- In the DLRP, the routes of the vehicles are never-end- mation into account (which is an improvement here). Due ing. Because of this character, it is almost impossible to to this large diversity of dynamic problem formulations, we find the point where a vehicle has finished its work within a decided to compare our new approach to the most basic bounded scenario. On the other hand, it is not too difficult VRP instances in this field. An additional advantage of this for the traditional VRP-algorithms, to transfer the objective approach is that we were able to compare our approach to function from closed to non-closed vehicle routes. So the traditional algorithms and to an absolute scale with the adapted VRP treats the distance of non-closed vehicle optimal solution—which normally cannot be calculated for routes as objective to minimise. the dynamic scenarios. The fourth point is not a conflict, but needs to be The approach to the transportation problem taken for the mentioned. The original [12] as well as the most discussed DLRP is basically different to the approach for the tradi- VRP-formulations (e.g. [13]) take the sum of driven tional VRP. The developed protocol is not an optimisation vehicle distances as objective to minimise—other objective algorithm for a static scenario, but an autonomous control functions are also discussed, see e.g. [1, p. 276]. But for the algorithm, designed for a continuous changing process. In traditional objective, we can easily show that the overall order to compare both concepts, it is necessary to execute vehicle distance ( d ), package distances (p ) and vehicle j i them on one scenario which both sides can cope with. utilisation (u ) are connected: To draw such an adapted vehicle routing problem for both sides, let us first have a look at major differences between the traditional vehicle routing problem formula- tion and the scenarios for the DLRP: • the VRP instances have only coordinates so every possible path is allowed—the DLRP scenario has few edges and no fully connected net • the DLRP is designed for a dynamic scenario, orders may appear at every vertex and at every time—the VRP orders are known from the beginning • the VRP enforces closed routes—because of its dynamic control nature, the DLRP has ongoing, unclosed routes • the objective function for the most VRP instances is the sum of all vehicle distances—within the DLRP, all Fig. 2 Topology for the adapted VRP (from [11]) 123 48 Logist. Res. (2009) 1:45–52 X 4 Computational results i vehicle capacity all packages i ¼  d ð1Þ P j P package size j all vehicles j all vehicles j 4.1 Random like solution This equation requires uniform vehicles and uniform In order to give something like an upper bound for rea- packages concerning capacity and size. So the DLRP will sonable solutions to the scenarios, a random-like heuristic indirectly minimise the requested objective function even was implemented. though it primarily minimises the package distances and In the first step, all packages are assigned randomly to maximises the vehicle utilisation. the vehicles. A uniform distribution is used for this step The new adapted VRP scenarios were built like distri- with the restriction that every vehicle has to carry at least bution scenarios in this first application. All vehicles start one package. The second step is to find an optimal way for at a central vertex, the city of Kassel, and all orders start one vehicle and its load of packages. To save computing form there. In order to keep the optimum value comput- time, only loop-free routes are considered. This restriction able, the size of the scenarios is not too large. The number makes the route non optimal in some cases, but it is of vehicles can be 3, 6 or 9, while the number of packages assumed that the optimal solution is not too far (see above, can vary between 17, 34, 51 or 68. In addition we created a the difference between the shortest way with and without large scenario with 68 packages and 12 vehicles, which loops is 10 km or 0.4%). was the largest optimal solvable scenario. The amount of This algorithm was calculated 10,000 times for each packages was matched to the topology. All 17 vertices, subset. The resulting mean values are shown in the next except Kassel, were supposed to be costumers. The pack- table (Table 2). age destinations for the larger scenario sets are uniformly The described algorithm has some analogies with the distributed, whereas each vertex has one package at least. real world transport market: the different forwarder com- Therefore the 34, the 51 and the 68-scenarios have 10 panies receive their orders randomly and each company subsets with different package destinations. The vehicle tries to optimise its vehicle routes on its own. An overall capacity was chosen in that way, that every vehicle is optimum cannot be expected from a procedure like this. needed, if no vehicle comes back to Kassel. All different Note that the average utilization which was reached by the scenarios are shown in Table 1. DLRP in the largest scenario 68-12, about 70%, would be a For an overview, three indicator values can be given: very good value for real world forwarder companies. The • the shortest way from Kassel via all 17 vertices is vision of the DLRP is to implement this protocol inde- 2,235 km long pendently from different companies. In this vision, an • the shortest way without any loop is 2,245 km long overall optimisation can happen without taking any deci- • the sum of the direct shortest ways for each of the 17 sion possibilities from the single forwarders [8]. packages is 4,965 km 4.2 Optimal solution For a more detailed description of the scenarios con- cerning the topology distances and the distribution of the On the other side, a lower bound for the overall vehicle package destinations, feel free to contact the authors or distance should be given. To calculate optimal solutions for refer to [14]. Tables and scenario data can be found on the given instances, the specified problem was formulated ‘‘http://dlrp.biba.uni-bremen.de’’. as a mixed-integer program (MIP, see below). The objec- tive of the program is based on the formulation of Dantzig and Ramser [12] and minimizes the distance driven by all Table 1 Chosen scenarios and corresponding vehicle capacities vehicles. In order to cope with the characteristics described number of vehicles 369 12 8 pck/veh 3 pck/veh 2 pck/veh Table 2 Vehicle distances calculated by the random-like algorithm 1 subset Number of packages Number of vehicles 16 pck/veh 6 pck/veh 4 pck/veh - 369 12 10 subsets 25 pck/veh 10 pck/veh 6 pck/veh - 17 3,927 5,118 5,474 10 subsets 34 5,051 7,594 9,072 51 5,583 8,940 11,225 33 pck/veh 13 pck/veh 8 pck/veh 6 pck/veh 68 5,943 9,897 12,756 14,876 10 subsets number of packages Logist. Res. (2009) 1:45–52 49 Table 3 Optimal vehicle distances capacitated vehicle routing problem (CVRP) and the split order vehicle routing problem (SDVRP). Within the Number of packages Number of vehicles considered network only one depot exists and the verti- ces are connected by undirected edges. Thus the associated distance between two directly connected ver- 17 2,245 2,565 3,095 tices is the same in either way. The demand of each 34 2,245 2,868 3,310 vertex is a priori known. In this context the demand of a 51 2,245 2,608 3,362 certain vertex can be split and met by either one or 68 2,245 2,683 3,400 4,097 multiple deliveries. According to the CVRP the capacity of the vehicles is limited. For our analysis we assume the same capacity r  for every vehicle v. All vehicles in chapter 3, we adapt the formulation of the VRP. The start their route at the depot and have to be used. In our proposed formulation was implemented in GAMS and modeling approach the route of a vehicle is described by could be solved with CPLEX 11. Because of the quite a number of consecutive segments. In this way a seg- small size of the instances, the MIP could be solved within ment represents either a movement of a vehicle from acceptable time—the scenario 68-12 seems to be the vertex i to i or the end of the route. Note that the total largest one which can be reasonably solved optimal. number of permitted segments of a route is a critical Table 3 shows the optimal values for the overall vehicle constraint of the problem. For our analysis we have distances. Note that in this solution, routes with loops are chosen the number of segments as high as the number of allowed and each vehicle has to be used. vertices within the network. Every vehicle is allowed to visit each vertex several times during its route. This 4.2.1 MIP-formulation characteristic of the model permits a vehicle on the one hand to serve a remote vertex from a given vertex and to 4.2.1.1 Nomenclature return afterwards to the vertex before it continues its Sets route through the network and on the other hand to pick up packages several times from the depot. Furthermore I vertices of the network D D each vertex can be used to store packages. This means a I depots of the considered network (I , I); in our case vertex can receive more packages than requested and only one depot exists S S that these packages can be picked up and delivered to I vertices that are directly connected to vertex i (I 7 I) i i other vertices. At the end of their route the vehicles S segments of vehicle routes remain at the vertex of their final delivery and do not V vehicles have to return to the depot. Parameters 4.2.1.3 Mathematical model Problem constraints: The c distance between vertex i and i i;i first segment of the route of each vehicle starts at the depot. d demand of packages at vertex i p provided packages at vertex i; in our case packages 0 i x ¼ 1 i 2 I ; v 2 V; s ¼ 0 ð2Þ v;i;i ;s 0 S are only provided at the depot i [ I with p = d i 2I i i[I i r  transportation capacity of vehicle v The route of each vehicle can be terminated only once at r required transportation capacity of one package any vertex of the network. M a very large number XX z ¼ 1 ðÞ v 2 V ð3Þ v;i;s Variables s2S i2I u amount of transported packages from vertex i to i v;i;i ;s In every segment the route of a vehicle can be either by vehicle v in s continued or terminated. x binary variable denoting that vehicle v drives in s v;i;i ;s XX X x þ z  1ðÞ s 2 S; v 2 V ð4Þ from vertex i to i v;i;i ;s v;i;s i2I 0 i2I i 2I z binary variable denoting that vertex i is the end of i v,i,s the route of vehicle v in s The route of a certain vehicle is described by a set of consecutive segments. This means a vehicle has either to leave its current vertex in the successive segment or to 4.2.1.2 Model assumptions The applied formulation of terminate its route. The introduction of segments ensures the vehicle routing problem has characteristics of the that a vehicle can visit a certain vertex several times and no 123 50 Logist. Res. (2009) 1:45–52 independent sub cycles occur. Note that the depot can be Table 4 Vehicle distances calculated by the tabu search algorithm visited more than once as well. Number of packages Number of vehicles X X 0 0 0 x  x  z ¼ 0 v;i;h;s v;h;i ;s v;h;s 0 S i2I: i 2I h2I 17 2,410 2,610 3,095 0 0 ðÞ h 2 I; s; s 2 S : s ¼ s þ 1; v 2 V ð5Þ 34 3,335 3,689 3,775 51 3,988 4,637 5,026 The load of each vehicle has to be less than or equal to 68 4,409 5,782 6,580 7,058 the maximum transportation capacity. 0 S u r  r  i 2 I; i 2 I ; s 2 S; v 2 V ð6Þ v;i;i ;s v 4.4 DLRP solution Within the network the packages are transported by the The DLRP solution was calculated with the DLRP roughly vehicles between directly connected vertices. In this context every vertex can receive, store and ship described above. To increase the performance, the protocol packages. Furthermore each vertex has a deterministic was adapted to this special static situation. Additionally the decision functions for the different objects were simplified demand of packages. In our case only the depot i [ I has a stock of packages p in the beginning. and harmonized: packages only choose their routes by the route length, vehicles only by the estimated utilisation. This s s XX X XXX p þ u ¼ d þ u means that the packages do not adjust to the vehicle routes, i v;h;i;r i v;i;i ;r v2V r¼0 v2V r¼0 0 h2I: i 2I in contrast to the dynamic version. Only vehicles choose i2I their route dependent on the package route situation. ðÞ i 2 I; s 2 S ð7Þ The results are shown in Table 5. Routes with loops are allowed here and each vehicle does not need to carry a Packages can only be shipped between two directly package, but do so. connected vertices i and i if a vehicle v serves this segment s on its route. Equation 8 simplifies the problem in a way that avoids a formulation as a mixed integer non-linear 5 Conclusions program. 0 S 0 0 u  x Mi 2 I; i 2 I ; s 2 S; v 2 V ð8Þ v;i;i ;s v;i;i ;s i For a better overview, all results are shown as line charts in the following Figs. 3, 4, 5 and 6. Compared to the optimum Objective function: The objective of the formulation is to and the random-like solution, the tabu search heuristic minimize the distance driven by all vehicles. leads to near optimal results with small scenarios and gets P PP P 0 0 Min: x c v;i;i ;s i;i worse with larger scenarios. This is a normal behaviour for ð9Þ v2V s2S i2I 0 i 2I tabu search algorithms and is due to the exponential growing solution space. 4.3 Tabu search solution In contrast, the DLRP solutions are not very good with small scenarios, but get better with larger scenarios. The As a representative of established solution techniques (see large scenarios 68-6, 68-9 and 68-12 show that the DLRP [1, 15]) for vehicle routing problems, a tabu search algo- gets better than the tabu search concept with a growing rithm was applied to the scenarios. network size. This algorithm is similar to the random-like solution Because the DLRP is originally a control method, its technique. One solution set for the tabu search is one main advantages point to dynamic and close to reality assignment set. This set assigns all packages to the vehi- scenarios: self-adaptation to changing situations, possible cles. After the assignment, an optimal route or each vehicle is calculated (again with loop-free routes only like the random-like algorithm). Table 5 Vehicle distances calculated by the DLRP For the neighbourhood generation, the k -interchange Number of packages Number of vehicles generation mechanism by Osmand [16] was implemented. The k was set to 1 and insured that no vehicle is empty. The maximum age of elements within the tabulist was set to 6 17 4,530 2,875 4,400 and the search was aborted after 18 moves. Note that tabu 34 5,184 5,928 7,022 search is very close to the optimal solution in the small 51 4,918 4,807 6,310 scenarios and moves away with greater scenarios (Table 4). 68 4,558 5,586 6,331 6,550 123 Logist. Res. (2009) 1:45–52 51 17 packages 68 packages 12,000 12,000 tabu search ttabu search ab u search optimum op optimum t imum 10,000 10,000 random-like ra random-like ndo m-like DLRP DL DLRP RP 8,000 8,000 6,000 6,000 4,000 4,000 2,000 2,000 369 12 369 12 no of vehicles no of vehicles Fig. 3 Results for 17 packages Fig. 6 Results for 68 packages 34 packages and dynamic scenarios with 2,500 packages emphasise this 12,000 assumption. These results are shown in Fig. 7. tabu search optimum 10,000 random-like 5.1 New evaluation chart DLRP 8,000 For larger scenarios, the classification and evaluation of algorithms for the VRP or the adapted VRP gets more and 6,000 more difficult. For these scenarios, it is not possible to calculate optimal solutions, so a lower bound is missing. 4,000 Additionally, it is not possible to compare two different scenarios, because the optimal way lengths can be very 2,000 369 12 different. On the basis of Eq. (1), we suggest an alternative no of vehicles comparison approach. The vehicle utilisation and the Fig. 4 Results for 34 packages package distances are, with some restrictions, directly connected to the overall vehicle distance. The vehicles utilisation has a natural best value and upper bound: it 51 packages 12,000 cannot be greater than one. The package distances have a ttabu search ab u search natural best value and lower bound, which is easy to cal- op optimum t imum 10,000 culate: their individual shortest way to their destination. ra random-like ndom-like DL DLRP RP Therefore we can define a relative package distance: driven 8,000 package distance by shortest possible distance. The lower bound of this relative package distance is one. 6,000 Now we can illustrate these two values in one plane, exemplarily shown in Fig. 7. The upper right corner rep- 4,000 resents one theoretical extreme: each package has one vehicle of capacity 1 and drives its shortest way. The 2,000 horizontal line at utilisation = 0.5 is another extreme: 369 12 when one vehicle with capacity for all packages brings out no of vehicles all packages, the utilisation goes to 0.5 and the relative Fig. 5 Results for 51 packages package distance increases infinitely. In this suggested evaluation chart, on can see that an manual interventions at runtime, implementation of optimal solution must be somewhere in the upper right uncertain knowledge and complex and context driven corner: a relative package distance of nearly 1 and a high decision functions [8]. The described results show that the utilisation of 80–90% (see optimum of scenario 68-12). DLRP has a high potential. Apart from the described In Fig. 7, some of the larger scenario results are shown. advantages the DLRP can be coequal to classical VRP With this kind of chart, it is possible to compare the large concepts in real world scenarios. DLRP results from large dynamic scenario results as well. Even though there are no overall vehicle distance [km] overall vehicle distance [km] overall vehicle distance [km] overall vehicle distance [km] 52 Logist. Res. (2009) 1:45–52 1.00 optimum 51-9 optimum 68-9 optimum 68-12 0.75 DLRP 51-9 DLRP 68-9 DLRP 68-12 tabu search 51-9 0.50 tabu search 68-9 tabu search 68-12 random-like 51-9 0.25 random-like 68-9 random-like 68-12 big scenarios DLRP 0.00 3.00 2.50 2.00 1.50 1.00 relative package distance [-] Fig. 7 Alternative evaluation chart for the comparison of VRP solutions 6. Bent RW, Van Hentenryck P (2004) Scenario-based planning for optimal or tabu search values for these scenarios, one can partially dynamic vehicle routing with stochastic customers. Oper say that the DLRP works well with these large dynamic Res 52(6):977–987 environments. Additionally, one can see that it is possible 7. Perkins CE (2001) Ad hoc networking. Addison-Wesley, Boston to change the DLRP parameters in a way that either the 8. Scholz-Reiter B, Rekersbrink H, Freitag M (2006) Internet routing protocols as an autonomous control approach for trans- vehicle utilisation or the relative package distance is port networks. In: Proceedings of the 5th CIRP international preferred. seminar on intelligent computation in manufacturing engineering, Another practical advantage of this evaluation chart is pp 341–345 that the suggested values can be treated as a state function. 9. Scholz-Reiter B, Rekersbrink H, Freitag M (2006) Kooperierende Routingprotokolle zur Selbststeuerung von Transportprozessen. It is possible to measure the vehicle utilisation and the Industrie Management 22/3, pp 7–10 relative package distance continuously. Consequently it is 10. Wenning B-L, Rekersbrink H, Timm-Giel A, Go ¨ rg C, Scholz- possible to control these values. Reiter B (2007) Autonomous control by means of distributed routing. In: Understanding Autonomous Cooperation & Control Acknowledgments This research is funded by the German in Logistics—The Impact on Management, Information and Research Foundation as part of the Collaborative Research Centre Communication and Material Flow. Springer, Berlin, pp 325–336 637 ‘Autonomous Cooperating Logistic Processes—A Paradigm Shift 11. Wenning B-L, Rekersbrink H, Becker M, Timm-Giel A, Go ¨ rg C, and its Limitations’. Scholz-Reiter B (2007) Dynamic transport reference scenarios. In: Understanding autonomous cooperation & control in logis- tics—the impact on management, information and communication and material flow. Springer, Berlin, pp 337–350 References 12. Dantzig GB, Ramser JH (1959) The truck dispatching problem. Manage Sci 6(1):80–91 1. Vahrenkamp R, Mattfeld D (2007) Logistiknetzwerke. Gabler, 13. Solomon MM (1987) Algorithms for the vehicle routing and Wiesbaden scheduling problems with time window constraints. Oper Res 2. Fleischmann B, Gnutzmann S, Sandvoß E (2004) Dynamic 35(2):254–265 vehicle routing based on online traffic information. Transp Sci 14. Scholz-Reiter B, Rekersbrink H, Wenning, B-L, Makuschewitz T 38(4):420–433 (2008) A survey of autonomous control algorithms by means of 3. Jaillet P, Wagner MR (2006) Online routing problems: value of adapted vehicle routing problems. In: Proceedings of the 9th advanced information as improved competitive ratios. Transp Sci Biennial ASME conference on engineering systems design and 40(2):200–210 analysis ESDA 08 (on CD), Haifa, Israel 4. Hiller B, Krumke SO, Rambau J (2006) Reoptimization gaps 15. Laporte G (1992) The vehicle routing problem: an overview of versus model errors in online-dispatching of service units for exact and approximative algorithms. Eur J Oper Res 59:345–358 adac. Discrete Appl Math 154:1897–1907 16. Osman IH (1993) Metastrategy simulated annealing and tabu 5. Savelsbergh M, Sol M (1998) Drive: dynamic routing of inde- search for the vehicle routing problem. Ann Oper Res 41:421– pendent vehicles. Oper Res 46:474–490 vehicle utilisation [-]

Journal

Logistics ResearchSpringer Journals

Published: Nov 4, 2008

References