1. Vehicle Routing Problem with Backhauls Jhon Jairo Santa Chvez Estudiante Doctorado en Ingenieras1

2. Seminario de Investigacin I I. II.III.The VRPB has a long history VRP with services on nodes or arcs of a network : A. TSP B. VRP An Exact Method for the Vehicle Routing Problem with Backhauls 2 3. I. The VRPB has a long history3 4. Abstract VRPBis generalization of delivery and collection service (Linehaul and Backhaul). In the literature many forms have been proposed to model VRPB. Following is a discussion of these models. 5. Introduction Capacity Vehicle Routing Problem (CVRP) has to deliver the goods from one warehouse to a set of customers using a homogeneous fleet. Routes should be built from start and end of the tank to minimize the total travel distance and considering the capacity limits of the vehicles. The complexity increases when not only must serve but also collects customer products and bring them to the tank. 6. Introduction This problem can solve two independent CVRPs. One for delivery (linehaul) and one for the collection (backhaul), but the solutions are not of good quality. VRPB use the same vehicle for the collection and delivery of customers on the same route. VRPB model is a generalization of the CVRP. 7. Applications VRPB Food is delivered to supermarkets and grocery stores in central distribution center and food are collected from the production centers and taken to the distribution center. Management of returnable bottles, where the filled bottles are brought to customers and the empty bottles are returned to the brewery, to be recycled (Environmental Issues). In order to obtain the maximum benefit from the vehicle fleet and reducing empty transport, is attractive to address conceptually different transport tasks in the same fleet. 8. The VRPB is subject to the following limitations: A.B. C.D. E. F.Linehaul customers must be addressed before customers Backhaul (Difficulties in unloading and delivery schedules and harvesting early in the afternoon). A route must not only meet customers Backhaul. The sum of deliveries to customers Linehaul and the sum of the collections Backhaul customers do not exceed the vehicle capacity. The number of vehicles is given or calculated at the beginning of the problem. Customers are served from a single tank. The fleet is homogeneous. 9. Overview of the Articles of the VRP with Backhauls Literature The following describes an overview of the articles in the literature devoted to Ruteamiento Vehicle Problem with Backhauls - VRPB. The single-vehicle problems have been studied eg Gendreau et al. [1], Ghaziri and Osman [2] and Sral and binder [3].[1] M. Gendreau, G. Laporte, D. Vigo, Heuristics for the traveling salesman problem with pickup and delivery, Computers & Operations Research 26, 699714 (1999). [2] H. Ghaziri, I.H. Osman, A neural network algorithm for the traveling salesman problem with backhauls, Computers & Industrial Engineering 44, 267281 (2003). [3 H. Sral, J.H. Bookbinder, The single-vehicle routing problem with unrestricted backhauls, Networks 41, 127136 (2003). 10. Formulating VRPB most common variants: 1.2. 3.4. 5. 6. 7.The Vehicle Routing Problem with Backhauls (VRPB). The Mixed Vehicle Routing Problem with Backhauls (MVRPB). The Multiple Depot Mixed Vehicle Routing Problem with Backhauls (MDMVRPB). The Vehicle Routing Problem with Backhauls and Time Windows (VRPBTW). The Mixed Vehicle Routing Problem with Backhauls and Time Windows (MVRPBTW). The Vehicle Routing Problem with Simultaneous Deliveries and Pickups (VRPSDP). Other backhauling problems 11. 1. VRPB with Multiple Fleet VRPB a study was presented by Toth and Vigo [4]. The first accurate methods for VRPB are proposed by Mingozzi et al. [5] and Toth and Vigo [6]. Regarding metaheurisiticos heuristics and have been developed by Anily [7], Hull et al. [8], Crispim and Brandao [9], and Jacobs-Blecha Goetschalckx [10], [11] and Toth and Vigo [12]. 12. 1. VRPB with Multiple Fleet [4] P. Toth, D. Vigo, VRP with backhauls, In P. Toth and D. Vigo (eds.): The Vehicle Routing Problem, SIAM Monographs on Discrete Mathematics and Applications 9, SIAM, Philadelphia, 195-221, (2002). [5] A. Mingozzi, S. Giorgi, R. Baldacci, An exact method for the vehicle routing problem with backhauls, Transportation Science 33, 315329 (1999). [6] P. Toth, D. Vigo, An exact algorithm for the vehicle routing problem with backhauls, Transportation Science 31 372-285 (1997). [7] S. Anily, The vehicle-routing problem with delivery and back-haul options, Naval Research Logistics 43 415434 (1996). [8] D.O. Casco, B.L. Golden, E.A. Wasil, Vehicle routing with backhauls: models, algorithms and case studies, in Vehicle Routing: Methods and Studies (Edited by B. Golden and A. Assad), North-Holland, Amsterdam 127147 (1988). [9] J. Crispim, J. Brandao, Reactive tabu search and variable neighbourhood descent applied to the vehicle routing problem with backhauls, MIC2001 4th Metaheuristic International Conference, Porto, Portugal July 1620 (2001). [10] M. Goetschalckx, C. Jacobs-Blecha, The vehicle routing problem with backhauls, European Journal of Operational Research 42 3951 (1989). [11] C. Jacobs-Blecha, M. Goetschalckx, The vehicle routing problem with backhauls: properties and solution algorithms, Technical Report, 1992-1998, Georgia Tech Research Corporation. [12] P. Toth, D. Vigo, A heuristic algorithm for the symmetric and asymmetric vehicle routing problems with backhauls, European Journal of Operational Research 113 528543 (1999). 13. 2. The Mixed Vehicle Routing Problem with Backhauls (MVRPB)the constraints (A), (B) and (D). Relaxing Linehaul customers can mix and Backhaul clients freely within a route. B. You can have only customers backhaul routes. D. You can use as many vehicles as you like. A. It must respect the capacity limit of the vehicles (Their calculation is more difficult). Some MVRPB have limitation in time of care and Backhaul Linehaul customers, and the length of time of the route. 14. 2. The Mixed Vehicle Routing Problem with Backhauls (MVRPB) He name Vehicle Routing problem with pickups and deliveries (VRPPD) is sometimes used instead of MVRPB. Heuristics for this problem are presented by Halse [13], Nagy and Salhi [14], [15] and Wade and Salhi [16], [17]. 15. 2. The Mixed Vehicle Routing Problem with Backhauls (MVRPB) [13] K. Halse, Modeling and solving complex vehicle routing problems, PhD thesis, Institute of Mathematical Statistics and Operations Research (IMSOR), Technical University of Denmark (1992). [14] G. Nagy, S. Salhi, Heuristic algorithms for single and multiple depot vehicle routing problems with pickups and deliveries, Working Paper no. 42, Canterbury Business School, 2003. [15] S. Salhi, G. Nagy, A cluster insertion heuristic for single and multiple depot vehicle routing problems with backhauling, Journal of the Operational Research Society (1999) 50, 10341042. [16] A. Wade, S. Salhi, An ant system algorithm for the mixed vehicle routing problem with backhauls, in M.G.C. Resende and J.P. de Sousa (eds.): Metaheuristics: Computer DecisionMaking, Chapter 33, 699-719, Kluwer (2003). [17] A. Wade, S. Salhi, An ant system algorithm for the vehicle routing problem with backhauls, MIC2001 - 4th Metaheursistic International Conference. 16. 3. The Multiple Depot Mixed Vehicle Routing Problem with Backhauls (MDMVRPB) It is a generalization of the MVRPB. It relaxes the limitation MDVRPB (E) giving the possibility that a customer can be served by more than one deposit. In each tank there is a limited number of vehicles. Each vehicle must start and end their turn in the same tank. Heuristics for the problem are proposed by Nagy and Salhi [18], [19]. They indicate that the problem Multi Depot Vehicle Routing Problem with collection and delivery. 17. 3. The Multiple Depot Mixed Vehicle Routing Problem with Backhauls (MDMVRPB) [18] G. Nagy, S. Salhi, Heuristic algorithms for single and multiple depot vehicle routing problems with pickups and deliveries, Working Paper no. 42, Canterbury Business School, 2003. [19] S. Salhi, G. Nagy, A cluster insertion heuristic for single and multiple depot vehicle routing problems with backhauling, Journal of the Operational Research Society (1999) 50, 1034-1042. 18. 4. The Vehicle Routing Problem with Backhauls and Time Windows (VRPBTW) 1. 2. 3.VRPB is assigned to a time slot to each client, making the travel time associated with each pair of locations, and making the service time associated with the clients. Visits to a customer must start within the time window. If the vehicle arrives too early at a customer who has to wait until the beginning of the time window. If the vehicle arrives too late the path is invalid. Limitations (B) and (D) of the relax VRPB (only customers backhaul routes and anticipated number of vehicles). VRPBTW objective function: Minimize the total distance traveled. Minimizing the number of vehicles. Minimize the total distance traveled with option 2. 19. 4. The Vehicle Routing Problem with Backhauls and Time Windows (VRPBTW)An exact algorithm for the generation based VRPBTW columns proposed by Gelinas et al. [20], and heuristics were proposed by Duhamel et al. [21] Hasama et al. [22], Reimann et al. [23] Thangiah et al. [24] and Zhong et al [25]. 20. 4. The Vehicle Routing Problem with Backhauls and Time Windows (VRPBTW)Tomado: http://www.iiia.csic.es/udt/en/comment/reply/230 21. 4. The Vehicle Routing Problem with Backhauls and Time Windows (VRPBTW) [21] S. Gelinas, M. Desrochers, J. Desrosiers, M.M. Solomon., A new branching strategy for time constrained routing problems with application to backhauling, Annals of Operations Research 61 91109 (1995).[21] C. Duhamel, J.-Y. Potvin, J.-M. Rousseau, A tabu search heuristic for the vehicle routing problem with backhaulsand time windows, Transportation Science 31 4959 (1997).[22] T. Hasama, H. Kokubugata, H. Kawashima, A heuristic approach based on the string model to solve vehicle routing problem with backhauls, Proceedings of the 5th World Congress on Intelligent Transport Systems (ITS), Seoul, 1998.[23] M. Reimann, Doerner K., Hartl R.F., Insertion based ants for vehicle routing problems with backhauls and time windows, LNCS 2463 135148 (2002).[24] S.R. Thangiah, J.-Y. Potvin, Sun T., Heuristic approaches to vehicle routing with backhauls and time windows, Computers & Operations Research 23 10431057 (1996).[25] Y. Zhong, M.H. Cole, A vehicle routing problem with backhauls and time windows: a guided local search solution, Transportation Research Part E, Article in press (2004). 22. 5. The Mixed Vehicle Routing Problem with Backhauls and Time Windows (MVRPBTW) He relaxes constraint (A), being able to mix and Backhaul Linehaul customers clients freely within a route. The objective considered in the literature is: 1. Reducing the number of vehicles. 2. Reduce the distance as the second priority. Two heuristics have been proposed in Kontoravdis and Bard [26] and Zhong et al [25]. [26] G. Kontoravdis, J.F. Bard, A GRASP for the vehicle routing problem with time windows, ORSA Journal on Computing 7 1023 (1995). 23. 6. The Vehicle Routing Problem with Simultaneous Deliveries and Pickups (VRPSDP) Customers are required to deliver and collect their products simultaneously. The pickup and delivery must be made at the same time so that each customer is visited only once by a vehicle. The download is done, obviously, before loading to these customers. The collection and delivery operation simultaneously decreasing costs and customer inconvenience associated with handling vehicles. May result in longer routes. 24. 6. The Vehicle Routing Problem with Simultaneous Deliveries and Pickups (VRPSDP) This problem was first introduced by Min [27]. Halse [28] presented exact methods and heuristics for this problem and Dethloff [29], [30] presented other heuristics. Nagy and Salhi [31] uses heuristics to solve the problem MVRPB. This is discussed in more detail by Dethloff [30]. Two variants of the problem have recently been proposed. Nagy and Salhi [31] by introducing a multitank version of the problem, while Angelelli and Mansini [32] proposed column generation. 25. 6. The Vehicle Routing Problem with Simultaneous Deliveries and Pickups (VRPSDP) [27] H. Min, The multiple vehicle routing problem with simultaneous delivery and pickup, Transportation Research Part A 23 377386 (1989).[28] K. Halse, Modeling and solving complex vehicle routing problems, PhD thesis, Institute of Mathematical Statistics and Operations Research (IMSOR), Technical University of Denmark (1992).[29] J. Dethloff, Relation between vehicle routing problems: an insertion heuristic for the vehicle routing problem with simultaneous delivery and pickup applied to the vehicle routing problem with backhauls, Journal of the Operational Research Society 53 115118 (2002).[30] J. Dethloff, Vehicle routing and reverse logistics: the vehicle routing problem with simultaneous delivery and pick-up, OR Spektrum 23 79-96 (2001).[31] S. Salhi, G. Nagy, A cluster insertion heuristic for single and multiple depot vehicle routing problems with backhauling, Journal of the Operational Research Society (1999) 50, 1034-1042.[32] E.Angelelli, R. Mansini, The vehicle routing problem with time windows and simultaneous pick-up and delivery, in Quantitative Approaches to Distribution Logistics and Supply Chain Management, (edited by A. Klose, M. G. Speranza, L. N. Van Wassenhove), Springer-Verlag, 249267 (2002) 26. 7. Other backhauling problems 7.1 Wade and Salhi [33] introduces a problem that generalizes VRPB and MVRPB. Not mix linehaul and backhaul customers on a route freely. A vehicle can only begin to serve backhaul customers after a certain percentage of linehaul load has been traversed. 1. Percentage = 0%, we MVRPB. 2. Percentage = 100%, we VRPB. 3. Percentage between 0% and 100%, are mixed together and MVRPB VRPB. [33] A.C. Wade, S. Salhi, An investigation into a new class of vehicle routing problem with backhauls, Omega 30 497487 (2002). 27. 7.1 Mezclas entre VRPB y MVRPBPorcentaje 100%Porcentaje 0% 28. 7.2 VRPB with Lasso Halskau et al. [34] proposes a VRPB with so-called loop routes. 1. The problem most customers require both collection and delivery. 2. In the first customers are given only to free up space in the vehicle (Lazo). 3. In the following clients and gather hands them simultaneously (Loop Radio - Honda). 4. At the end of the route will be collected at the earliest. 5. Therefore loop method is called. [34] . Halskau; I. Gribkovskaia; K.N.B. Myklebost. Models for pick-up and deliveries from depots with lasso solutions. Proceedings of the 13th Annual Conference on Logistics Research - NOFOMA 2001, Collaboration in logistics : Connecting Islands using Information Technology. Reykjavik, Iceland, 2001-06-14 - 2001-06-15. Chalmers University of Technology, Gteborg, Sweden. 279293 (2001). 29. Solution of the vehicle routing problem for the potato distribution in Colombia It addresses the problem of designing optimal routes that meet the demand of potato in major cities of Colombia minimizing fleet travels current (no load). Strategies were used, including: Model of Transportation, the nearest neighbor heuristic and ant colony technique, a technique MDVRPBS multiple vehicle routing Depot Supply technical Bakhauls. We analyzed two objective functions: minimizing the distance A (Transporter interest) and another minimizing user costs charge. [35] E. Toro, J. Santa, M. Granada, Solution of the vehicle routing problem for the potatoe distribution in Colombia, Scientia et Technica Ao XVIII, Vol. 18, No 1, Abril de 2013. Pag 128-139.http://revistas.utp.edu.co/index.php/revistaciencia/article/view/8373/5305 http://repositorio.utp.edu.co/dspace/handle/11059/3192 30. THE VEHICLE ROUTING PROBLEM WITH BACKHAULS: A MULTI-OBJECTIVE EVOLUTIONARY APPROACH Abel Garcia-Najera (2012)It is considered in the VRPB as a multi-objective optimization problem, as follows: 1. The number of vehicles. 2. Transport costs. 3. Meet the demand of customers. It solves these problems with multiple objectives with an evolutionary algorithm previously proposed adapted and evaluate their performance with the proper tools. [36] Garcia-Najera, A. (2012), The Vehicle Routing Problem with Backhauls: a Multi-objective Evolutionary Approach. In: 12th European Conference on Evolutionary Computation in Combinatorial Optimisation, LNCS 7245, pp. 255-266. Springer 31. Mitigation of greenhouse gas emissions in vehicle routing problems with backhauling - Praderas, Oportus and Parada (2013) We study the reduction of the emission of greenhouse gases Backhauls and VRP with time windows. Considering the energy required for each route and estimating the load and the distance between the customers. Resolved an issue with up to 100 clients distributed randomly. The variables: distance, transportation costs, energy requirements, fuel consumption and emission of greenhouse gases. [37] Pradenas, L., Oportus B., Parada V., Mitigation of greenhouse gas emissions in vehicle routing problems with backhauling. Expert Systems with Applications 40 (2013) 29852991. 32. A unified heuristic for a large class of vehicle routing problems with backhauls - Ropke and Pisinger (2006) Present a study of different variants of VRPB and develops a unified model that is capable of handling most of the problem variants literature. The unified model can be seen as a set of delivery and collection problems with time windows, which can be solved by an improved version of the great sweep heuristic search proposed by Ropke and Pisinger [Great Local search heuristics for gathering adaptation and delivery problem with time windows, Technical Report, DIKU, University of Copenhagen, 2004].[38] S. Ropke and D. Pisinger. A unified heuristic for a large class of vehicle routing problems with backhauls. European Journal of Operational Research, 171: 750775, 2006. 41 33. Articles in the Database = 143 Articles VRPB = 39 (27,27%) CHARACTERIST HEURISTIC METAHEURISTIC EXACT ANT COLONY GENETICOS ANT COLONY HIBRIDOS MONO-OBJETIVO MULTI-OBJETIVOQUANTITY 19 18 2 5 5 5 3 35 4PERCENTAGE 28,48% 29,80% 8,61% 27,78% 27,78% 27,78% 16,67% 89,74% 10,26%European Journal of Operational Research and Institute of Electrical and Electronics Engineers (IEEE) 42 34. 20132012201120102009200820072006200520042003200220012000199919981997199619951994VRP with Backhauls - Aos987654 Artculos 39321043 35. VRP with Backhauls - Paises 987654 Artculos321044 36. II. Vehicle Routing Problems with customers on nodes or arcs of a network45 37. A graph, which is only a few nodes with few connections called edges, in the case of vehicle routing problem are the customers and routes or clients located in the arches46 38. Introduction Pickup and Delivery is a frequent activity in companies and communities. According to the nature of the service will have two types of models: 1. Client Representation by the nodes of a network (Node Routing Problem- NRP). 2. Representation of clients associated with all arcs of a network (Arc Routing Problem ARP). 47 39. Exact and Heuristic Method for NRP and ARP Mostof the variants of the NRP and the ARP are NP-hard. When the problem size grows the exact method does not solve it, or it is too costly. Heuristic methods will give an optimal approach to the problem. 48 40. 1. Node Routing Problem 1.1. Traveling Salesman Problem The traveling salesman problem was probably mentioned for the first time in a mathematical circle by A.W. Tucker in 1931. Problem: A vendor, leaving home (or a deposit), you must visit a set of customers and then return home, the travel distance should be as small as possible. 49 41. 1.1.1. Constructive Heuristics 1.1.1.1. Nearest Neighbour algorithm1.1.1.2. Cheapest Insertion algorithm 1.1.1.3. Christofides algorithm50 42. 1.1.1.1. Nearest Neighbour Algorithm 1.2. 3.Choose v as the closest customer to depot and consider a partial stroke = T (depot, v, depot). Determine the client w as the v nearest customer is still T. Add the client w at the end of the tour of T. If all clients have been visited, then STOP, otherwise go back to 2. 51 43. Nearest Neighbour algorithm52 44. Example 1: Calculate the minimum cost path to the nearest neighbor algorithm53 45. Solution: Minimum Path With Nearest Neighbor 0-3-2-4-1-0 = 6+5+6+13+9 = 3954 46. 1.1.1.2. Cheapest Insertion Algorithm 1.2.3.Choose two clients v and w and consider a partial tour T visiting only these clients, that is, T=(depot, v, w, depot). For each client z who is not yet in T, is the calculation of inserting z distance between two consecutive customers in T. Z choose a client that is not yet in T and minimizing the distance traveled. If all customers are visited, then STOP, otherwise go back to 2. 55 47. Cheapest Insertion Algorithm56 48. Example 2: Cheapest Insertion algorithm57 49. It calculates distances by inserting each of the nodes, we choose minimum 1958 50. It calculates distances from the reservoir, incorporating each of the nodes to the path 0-3-0 as well: Calculate the distances traveled by inserting each of the nodes, Node 2 is the most economical insert: 6 +5 +8 = 19.59 51. Inserting routes: node 1: 0-3-2-1-0 = 6+5+8+9 = 28 node 4: 0-3-4-2-0 = 6+8+6+8 = 2860 52. Routes inserting node 1 (Original: 0-3-2-1-0): Finally insert node 4: 0-3-4-2-1-0 = 6+8+6+8+9 = 37 0-3-2-4-1-0 = 6+5+6+13+9 = 3961 53. 1.1.1.3. Christofides algorithm, 1976 1.2. 3.Determine a maximal tree of minimum cost A connecting the clients. Determine the odd degree nodes in the tree A. They connect at minimal cost. These unions induce tours, then remove the arches that pass through the same client twice. 62 54. Example 3: Spanning Tree maximum at minimum cost (Kruskal)63 55. Add the edges of a perfect matching of minimum weight on the vertices of odd degree spanning tree: Route 0-3-2-4-1-0 = 6+5+6+13+9 = 3964 56. 1.1.2. Exact Method 1.1.2.1. Exact Branch and Bound Method The Branch and Bound technique is to start building a tree with all possible solutions but when a branch is not the best, it is left to build the tree by this branch, to save computational resources, so you can reach the optimal solution without having to explore each and every one of the possible solutions. In the case of VRP must have an initial feasible solution with a total distance associated and thus perform cutting tree branches that exceed this distance. 65 57. Example 4: Calculate the minimum cost path with Branch and Bound method66 58. Graph with all possible solutions: Our starts each of the possible paths, taking into account that when you exceed the initial feasible basic solution or a solution of better quality earlier probe that route.67 59. B & B Save the tour of all the branches of the tree having an Initial Basic Feasible Solution: S.B.F.I. = 40In red all routes that pass the initial solution 40 to be probed 68 60. Possible Routes: 0-1-2-4-3-0 o 0-3-4-2-1-0 : Distance = 37 - Optimal Solution 0-1-4-2-3-0 o 0-3-2-4-1-0 : Distance = 39 - Better Solutions69 61. 1.2 Vehicle tours with capacity constraints Mtodo Clarke y Wright, 1964 To make delivery of merchandise must take into account the capacity limits of the vehicles. Delivery Planning below requires the solution of two sub-problems : 1. Determine the set of customers that each vehicle must serve; 2. Determine the order in which each vehicle serves its customers. 70 62. 1.2 Vehicle tours with capacity constraints Mtodo Clarke y Wright, 1964Vehicles have the same load capacity. Vehicle starts and finishes its tour at the depot. The deliveries to a client are made with the help of a single vehicle. The goal is to minimize the total distance. 71 63. DEFINITION.- Given two clients v and w, the saving s(v, w) is defined as the gain in length obtained by delivering v and w in the same tour (depot, v, w, depot) instead of using two tours (depot, v, depot) and (depot, w, depot).s(v,w) = D(v, depot) + D(depot, w)-D(v,w) 72 64. 1.2.1 Clarke and Wright Algorithm 1.2.3.Determine the savings s (v, w) for all pairs of customers, and organize these values in descending order. Choose the pair of nodes savings S (v, w) is not yet serve, and not to exceed the capacity of the vehicle. Create a tour T = (depot, v, w, depot). Repeat the previous step until all nodes serve. 73 65. Example 5: You have four demand nodes(24, 25, 31, 28). Vehicle capacity is 60 unitsTotal Distance : d01+d10+d02+d20+d03+d30+d04+d40 = 9+9+8+8+13+13+6+6=72 74 66. Join nodes 2 and 4: Capacity 25 +28 = 53, Feasible. Join nodes 1 and 3: Capacity 24 +31 = 55, Feasible.75 67. Solutions: Route 1: 0-2-4-0 y Route 2: 0-1-3-076 68. 1.2.2 Two stage method of Clarke and Wright algorithm The two-stage methods is an improvement to the method of Clarke and Wright and outperform constructive algorithm. 1. First Method: Phase 1: customers are divided into groups based on each total demand of each group does not exceed the vehicle capacity. Phase 2: It solves a TSP problem in each group. 2. Second Method: Phase 1: It solves a giant TSP problem (visiting all clients) without taking into account the capacity of the vehicle. Phase 2: The course is divided into smaller paths that meet all the capacity constraints of the vehicle. 77 69. Graph of the Two-Phase method78 70. 2. Arc Routing Problem2.1 Chinese postman problem 2.2 Rural Postman Problem 79 71. 2.1 The Chinese postman problem The Chinese postman problem was first introduced by the mathematician Meigu Guan in 1962. 1. Determining a minimum total length circuit. 2. It must traverse each arc of the graph at least once. 3. It must reach the starting node. 4. You can have a directed graph with no arcs, priorities, so that the NP-hard problem. 80 72. 2.1.1 Algorithm for the non oriented Chinese postman problem, where each node has an even number of neighbours 1. 2.3.Determine a cycle C in the graph. Choose a node v belonging to C and perform a second cycle nodes without repeating the previous step. Repeat step 2 until it covers all the arcs.81 73. Example 6:82 74. Ejemplo 7: Time 113 minutes (1 hour and 53 minutes)83 75. 2.1.2 Algorithm for the non oriented Chinese postman problem (odd nodes) 1. 2.3.Find all the odd nodes in the network. Consider all the routes joining pairs of odd nodes. Choose the routes with the shortest total distances. Add in these edges again. This will give a network with only even nodes. 84 76. Example 8:85 77. Example 9: There are four nodes with odd degree86 78. Better Pairing: CG+AE= 16 min87 79. Time 113 minutes (1 hour and 53 minutes)88 80. 2.2 The Rural Postman Problem It is the Chinese postman problem which not only requires going through a series of arches, on which you must pass at least once. The problem of finding a minimum cost is known as the rural postman problem. Orloff was introduced in 1974.89 81. 2.2.1 Algorithm for the non oriented rural postman problem where the clients form a connected sub-networkThere is a graph composed exclusively of arcs required. W determine the set of nodes that have an odd number of neighbors. 2. Consider all the routes joining pairs of odd nodes. Choose the routes with the shortest total distances. 3. Add in these edges again. This will give a network with only even nodes. 4. Determine a tour by solving the Chinese postman problem in the augmented graph. 1.90 82. a. Required Arcsb. Odd Degree NodesExample 10: Steps a. and b. 91 83. c. Better Pairing: hi+cj= 2+1=3d. Eulerian tourExample 10: Steps c. and d. 92 84. 2.2.2 Fredricksons Algorithm for the non oriented rural postman problem 1. 2. 3.4.It has a partial Graph composed exclusively of arcs required. Build a Complete Graph arcs joining with minimum length. Determining a tree of minimum cost maximum expansion. Solving the rural postman problem with the above algorithm. 93 85. a. Required Arcsb. Subgraph: Route is the minimum of the arches not requiredExample 11: Steps a. and b. 94 86. c. Determine a maximal tree of minimum costd. Nodes connected odd degreeExample: Steps c. and d. 95 87. Conclusion: Regarding algorithms must be PNR: The nearest neighbor algorithm is very simple but the solution is low quality. The algorithm of the cheapest insertion is more complex but give a better quality solution. The Christofides algorithm uses an additional algorithm for finding the maximum expansion tree at the lowest cost and delivers a better quality of response that the nearest neighbor algorithm. The exact method of branch and bound gives the optimal solution is computationally expensive and although, this can be improved with a good quality initial solution, which allows probing tree branches efficiently. 96 88. Conclusion: Operational research seeks to optimize the resources used in transport problems ensuring services to customers. We described the main heuristics used to solve vehicle routing problems, where customers are found in both nodes of the problem (NRP) or arcs (ARP problem) of a network. The heuristics described must be adapted to the situations that arise in real life, so only are a basis for more specialized studies. 97 89. III. An Exact Method for the Vehicle Routing Problem with Backhauls 1999 ARISTIDE MINGOZZI and SIMONE GIORGI Department of Mathematics, University of Bologna, Piazza di Porta S. Donato, 5, 40127 Bologna, ItalyROBERTO BALDACCI The Management School, Imperial College, 53 Princes Gate Exhibition Road, London SW7 2PG, United Kingdom 98 90. An Exact Method for the Vehicle Routing Problem with Backhauls There is a fleet of vehicles in a central repository. Should optimally serve two subsets of patients: Linehaul and backhaul. Each route starts and ends at the depot. Backhaul customers must be visited after Linehaul customers. We present a binary integer programming formulation. The proposed algorithms including a LPrelaxation show efficacy in solving problems up to 100 clients.99 91. Fig. 1. Example of a VRP solution100 92. 1. NOTATION101 93. NOTATION:102 94. Anelementary path P in GL starting at vertex 0 (resp. in GB ending at vertex 0) is called a feasible path if its load satisfies the inequalitieswhere QLmin (resp. QBmin ) represents the minimum load of linehaul customers (resp. backhaul customers) of any feasible path in GL (resp. GB). 103 95. The values QLmin and QBmin are computed as follows:We will use t(P) to indicate both the terminal vertex of a feasible path P in GL and the starting vertex of a feasible path P in GB. 104 96. NOTATION: Linehaul feasible routes are created by adding arcs from the tank and discharge to the nodes. Backhaul feasible routes are created by adding arcs from nodes collects and ending in the warehouse. The feasibility is verified with the capacity of the vehicles. They have arcs link Linehaul and backhaul routes. ML = Routes or Linehaul Vehicles. MB = Routes or Backhaul Vehicles. ML MB is assumed, but it is easily extended to ML MB. Routes M of any viable solution viable paths consist105 97. 2. MATHEMATICAL FORMULATION106 98. Exact Model107 99. Equations Components Description A: Sum of costs Linehaul routes in the objective function. B: Sum of the costs of backhaul routes in the objective function. C: Sum of costs Linehaul-Backhaul links in the objective function. D: Routes that pass through the node i Linehaul routes. E: Routes that pass through the node i backhaul routes. F: Routes ending at node i Linehaul routes. G: links that connect to the node i Linehaul routes. H: Routes that start at node i belonging to backhaul 108 100. Example: You have five customers, three of which are linehaul L1, L2, and L3 and 2 backhaul B1 and B2, it has a deposit. Perform a distribution and collection considering that you have a homogenous fleet with Q = 15 units. ML=MB=2, then M=2. Cost MatrixProvision109 101. Nearest Neighbour Heuristic for VRP with Backhaul (VRPB)110 102. CALCULATION OF MINIMUM CAPABILITIES LINEHAUL AND BACKHAUL (Equation 2)111 103. Linehaul Route: R1, R2 and R3Linehaul Route: R4, R5 and R6Set of Linehaul and Backhaul Routes 112 104. Linehaul Route: R7, R8 and R9Backhaul Route: R10 and R11Set of Linehaul and Backhaul Routes 113 105. Arcs Joining Linehaul and Backhaul Routes114 106. MATHEMATICAL FORMULATION115 107. Matlab Code116 108. Mathematical Model: Solution117 109. Optimal Solution118 110. Conclusion ARISTIDE MINGOZZI, SIMONE GIORGI and ROBERTO BALDACCI : IN THIS PAPER, we have described an exact algorithm for the basic Vehicle Routing Problem with Backhauls (VRPB) based on a new (01) integer programming formulation. Problem IP cannot be solved directly, even for problems of moderate size because the number of variables may be too large. 119