New Heuristic Algorithms for the Windy Rural Postman Problem

[EN] In this paper we deal with the windy rural postman problem. This problem generalizes several important arc routing problems and has interesting real-life applications. Here, we present several heuristics whose study has lead to the design of a scatter search algorithm for the windy rural postman problem. Extensive computational experiments over different sets of instances, with sizes up to 988 nodes and 3952 edges, are also presented. (c) 2004 Elsevier Ltd. All rights reserved.Benavent, E.; Corberán, A.; Piñana, E.; Plana. I.; Sanchís Llopis, JM. (2005). New Heuristic Algorithms for the Windy Rural Postman Problem. Computers & Operations Research. 32(12):3111-3128. doi:10.1016/j.cor.2004.04.007S31113128321

Min-Max K-vehicles Windy Rural Postman Problem

[EN] In this article the Min-Max version of the windy rural postman problem with several vehicles is introduced. For this problem, in which the objective is to minimize the length of the longest tour in order to find a set of balanced tours for the vehicles, we present here an ILP formulation and study its associated polyhedron. Based on its partial description, a branch-and-cut algorithm has been implemented and computational results on a large set of instances are finally presented. (C) 2009 Wiley Periodicals, Inc. NETWORKS, Vol. 54(4),216-226 2009Contract grant sponsor: Ministerio de Education y Ciencia of Spain: Contract gram number: MTM2006-14961-C05-02Benavent López, E.; Corberan, A.; Plana, I.; Sanchís Llopis, JM. (2009). Min-Max K-vehicles Windy Rural Postman Problem. Networks. 54(4):216-226. https://doi.org/10.1002/net.20334S216226544D. Ahr Contributions to multiple postmen problems 2004D. Ahr G. Reinelt “New heuristics and lower bounds for the min-max k -Chinese postman problem” Algorithms-ESA 2002, 10th Annual European Symposium, Rome, Italy, 2002, Lecture Notes in Computer Science 2461 R. Möring R. Raman Springer Berlin 2002 64 74Ahr, D., & Reinelt, G. (2006). A tabu search algorithm for the min–max k-Chinese postman problem. Computers & Operations Research, 33(12), 3403-3422. doi:10.1016/j.cor.2005.02.011D. Applegate R.E. Bixby V. Chvátal W. Cook Finding cuts in the TSP 1995Barahona, F., & Grötschel, M. (1986). On the cycle polytope of a binary matroid. Journal of Combinatorial Theory, Series B, 40(1), 40-62. doi:10.1016/0095-8956(86)90063-8Belenguer, J. M., & Benavent, E. (1998). Computational Optimization and Applications, 10(2), 165-187. doi:10.1023/a:1018316919294Benavent, E., Carrotta, A., Corberán, A., Sanchis, J. M., & Vigo, D. (2007). Lower bounds and heuristics for the Windy Rural Postman Problem. European Journal of Operational Research, 176(2), 855-869. doi:10.1016/j.ejor.2005.09.021N. Christofides V. Campos A. Corberán E. Mota An algorithm for the rural postman problem 1981Christofides, N., Campos, V., Corberán, A., & Mota, E. (1986). An algorithm for the Rural Postman problem on a directed graph. Netflow at Pisa, 155-166. doi:10.1007/bfb0121091Corberán, A., Plana, I., & Sanchis, J. M. (2008). The Windy General Routing Polyhedron: A Global View of Many Known Arc Routing Polyhedra. SIAM Journal on Discrete Mathematics, 22(2), 606-628. doi:10.1137/050640886Corberán, A., Plana, I., & Sanchis, J. M. (2007). A branch & cut algorithm for the windy general routing problem and special cases. Networks, 49(4), 245-257. doi:10.1002/net.20176Eiselt, H. A., Gendreau, M., & Laporte, G. (1995). Arc Routing Problems, Part II: The Rural Postman Problem. Operations Research, 43(3), 399-414. doi:10.1287/opre.43.3.399Frederickson, G. N., Hecht, M. S., & Kim, C. E. (1978). Approximation Algorithms for Some Routing Problems. SIAM Journal on Computing, 7(2), 178-193. doi:10.1137/0207017G. Ghiani D. Laganá G. Laporte R. Musmanno A branch-and-cut algorithm for the undirected capacitated arc routing problem 2007Ghiani, G., & Laporte, G. (2000). A branch-and-cut algorithm for the Undirected Rural Postman Problem. Mathematical Programming, 87(3), 467-481. doi:10.1007/s101070050007Golden, B. L., & Wong, R. T. (1981). Capacitated arc routing problems. Networks, 11(3), 305-315. doi:10.1002/net.3230110308Padberg, M. W., & Rao, M. R. (1982). Odd Minimum Cut-Sets andb-Matchings. Mathematics of Operations Research, 7(1), 67-80. doi:10.1287/moor.7.1.67Pearn, W. L. (1994). Solvable cases of the k-person Chinese postman problem. Operations Research Letters, 16(4), 241-244. doi:10.1016/0167-6377(94)90073-

Zigzag inequalities:a new class of facet-inducing inequalities for Arc Routing Problems

[EN] In this paper we introduce a new class of facet-inducing inequalities for the Windy Rural Postman Problem and the Windy General Routing Problem. These inequalities are called Zigzag inequalities because they cut off fractional solutions containing a zigzag associated with variables with 0.5 value. Two different types of inequalities, the Odd Zigzag and the Even Zigzag inequalities, are presented. Finally, their application to other known Arc Routing Problems is discussed.The authors wish to thank the Ministerio de Ciencia y Tecnología of Spain (project TIC2003-05982-C05-01) and the Generalitat Valenciana (Ref: GRUPOS03/189) their support.Corberán, A.; Plana, I.; Sanchís Llopis, JM. (2006). Zigzag inequalities:a new class of facet-inducing inequalities for Arc Routing Problems. Mathematical Programming. 108(1):79-96. https://doi.org/10.1007/s10107-005-0643-yS79961081Benavent, E., Carrotta, A., Corberán, A., Sanchis, J.M., Vigo, D.: Lower Bounds and Heuristics for the Windy Rural Postman Problem. Technical Report TR03-2003. Department of Statistics and OR, University of Valencia (Spain). Submitted to EJOR 2003Benavent, E., Corberán, A., Piñana, E., Plana, I., Sanchis, J.M.: New Heuristics for the Windy Rural Postman Problem. To appear in Comput. Oper. Res. 2005Chopra, S., Rinaldi, G.: The Graphical Asymmetric Traveling Salesman Polyhedron: Symmetric Inequalities. SIAM J. Discrete Math. 9 (4), 602–624 (1996)Christofides, N., Benavent, E., Campos, V., Corberán, A., Mota, E.: An Optimal Method for the Mixed Postman Problem. In: P. Thoft-Christensen (ed.) System Modelling and Optimization. Lecture Notes in Control and Information Sciences, 59. Berlin: Springer-Verlag 1984Christofides, N., Campos, V., Corberán, A., Mota, E.: An Algorithm for the Rural Postman Problem. Report IC.OR. 81.5. Imperial College, London 1981Corberán, A., Mejía, G., Sanchis, J.M.: New Results on the Mixed General Routing Problem. To appear in Oper. Res. 2005Corberán, A., Mota, E., Sanchis, J.M.: A Comparison of Two Different Formulations for Arc Routing Problems on Mixed Graphs. To appear in Comput. Oper. Res. 2005Corberán, A., Plana, I., Sanchis, J.M.: On the Windy General Routing Polyhedron. In preparation 2005Corberán, A., Romero, A., Sanchis, J.M.: The Mixed General Routing Problem Polyhedron. Math. Programming 96, 103–137 (2003)Cornuèjols, G., Fonlupt, J., Naddef, D.: The traveling salesman problem on a graph and some related integer polyhedra. Math. Programming 33, 1–27 (1985)Eiselt, H.A., Gendreau, M., Laporte, G.: Arc-Routing Problems, Part 2: the Rural Postman Problem. Oper. Res. 43, 399–414 (1995)Ford, L.R., Fulkerson, D.R.: Flows in Networks. Princeton University Press, Princeton, NJ 1962Grötschel, M., Win, Z.: On the Windy Postman Polyhedron. Report No. 75, Schwerpunktprogram der Deutschen Forschungsgemeinschaft, Universität Augsburg, Germany 1988Grötschel, M., Win, Z.: A Cutting Plane Algorithm for the Windy Postman Problem. Math. Programming 55, 339–358 (1992)Guan, M.: On the Windy Postman Problem. Discrete Applied Mathematics 9, 41–46 (1984)Letchford, A.: New inequalities for the General Routing Problem. Eur. J. Oper. Res. 96, 317–322 (1997)Minieka, E.: The Chinese Postman Problem for Mixed Networks. Management Sci. 25, 643–648 (1979)Naddef, D., Rinaldi, G.: The Symmetric Traveling Salesman Polytope and its Graphical Relaxation: Composition of Valid Inequalities. Math. Programming 51, 359–400 (1991)Nobert, Y., Picard, J.C.: An Optimal Algorithm for the Mixed Chinese Postman Problem. Networks 27, 95–108 (1996)Ralphs, T.K.: On the Mixed Chinese Postman Problem. Oper. Res. Lett. 14, 123–127 (1993)Win, Z.: Contributions to Routing Problems. PhD Dissertation, University of Augsburg, Germany 198

New Facets and an Enhanced Branch-and-Cut for the Min-Max K-Windy Rural Postman Problem

[EN] The min-max windy rural postman problem is a multiple vehicle version of the windy rural postman problem, WRPP, which consists of minimizing the length of the longest route to find a set of balanced routes for the vehicles. In a previous paper, an ILP formulation and a partial polyhedral study were presented, and a preliminary branch-and-cut algorithm that produced some promising computational results was implemented. In this article, we present further results for this problem. We describe several new facet-inducing inequalities obtained from the WRPP, as well as some inequalities that have to be satisfied by any optimal solution. We present an enhanced branch-and-cut algorithm that takes advantage of both these new inequalities and high quality min-max K-WRPP feasible solutions obtained by a metaheuristic. Computational results on a large set of instances are also reported. © 2011 Wiley Periodicals, Inc.Contract grant sponsor: Ministerio de Ciencia e Innovacion of Spain; Contract grant numbers: MTM2006-14961-C05-02, MTM2009-14039-C06-02Benavent López, E.; Corberán, A.; Plana, I.; Sanchís Llopis, JM. (2011). New Facets and an Enhanced Branch-and-Cut for the Min-Max K-Windy Rural Postman Problem. Networks. 58(4):255-272. https://doi.org/10.1002/net.20469S255272584D. Ahr Contributions to multiple postmen problems 2004Ahr, D., & Reinelt, G. (2002). New Heuristics and Lower Bounds for the Min-Max k-Chinese Postman Problem. Lecture Notes in Computer Science, 64-74. doi:10.1007/3-540-45749-6_10Ahr, D., & Reinelt, G. (2006). A tabu search algorithm for the min–max k-Chinese postman problem. Computers & Operations Research, 33(12), 3403-3422. doi:10.1016/j.cor.2005.02.011D. Applegate R. E. Bixby V. Chvátal W. Cook Finding cuts in the TSP 1995Benavent, E., Carrotta, A., Corberán, A., Sanchis, J. M., & Vigo, D. (2007). Lower bounds and heuristics for the Windy Rural Postman Problem. European Journal of Operational Research, 176(2), 855-869. doi:10.1016/j.ejor.2005.09.021Benavent, E., Corberán, A., Plana, I., & Sanchis, J. M. (2009). Min-Max K -vehicles windy rural postman problem. Networks, 54(4), 216-226. doi:10.1002/net.20334Benavent, E., Corberán, Á., & Sanchis, J. M. (2009). A metaheuristic for the min–max windy rural postman problem with K vehicles. Computational Management Science, 7(3), 269-287. doi:10.1007/s10287-009-0119-2Corberáan, A., Letchford, A. N., & Sanchis, J. M. (2001). A cutting plane algorithm for the General Routing Problem. Mathematical Programming, 90(2), 291-316. doi:10.1007/pl00011426Corberán, A., Plana, I., & Sanchis, J. M. (2007). A branch & cut algorithm for the windy general routing problem and special cases. Networks, 49(4), 245-257. doi:10.1002/net.20176Corberán, A., Plana, I., & Sanchis, J. M. (2008). The Windy General Routing Polyhedron: A Global View of Many Known Arc Routing Polyhedra. SIAM Journal on Discrete Mathematics, 22(2), 606-628. doi:10.1137/050640886Frederickson, G. N., Hecht, M. S., & Kim, C. E. (1978). Approximation Algorithms for Some Routing Problems. SIAM Journal on Computing, 7(2), 178-193. doi:10.1137/0207017Pearn, W. L. (1994). Solvable cases of the k-person Chinese postman problem. Operations Research Letters, 16(4), 241-244. doi:10.1016/0167-6377(94)90073-6I. Plana The windy general routing problem 200

Сравнительный анализ системных компонентов педагогического процесса в системе высшего образования Великобритании и Республики Беларусь

We describe a weighted version of the k-Chinese or k-rural postman problem that occurs in the context of snow removal. The problem concerns the questions of which vehicle shall do each task and how the vehicles shall travel between tasks. We also consider different numbers of vehicles, in view of a fixed cost for each vehicle. We describe and discuss heuristic solution approaches, based on usable substructures, such as Chinese/rural postman problems, meta-heuristics, k-means clustering and local search improvements by moving cycles. The methods have been implemented and tested on real life examples

Modeling and Solving Arc Routing Problems in Street Sweeping and Snow Plowing

In arc routing problems, the goal is to determine an optimal path, or set of paths, that traverse a required subset of arcs on a graph with respect to a set of constraints and objective function. The Chinese Postman Problem (CPP) forms the basis for many arc routing problems. Let graph G =(V,A), where V is a set of vertices and A = {(i,j) | i,j in V} is a set of arcs that each connect exactly two vertices, each with its own cost of traversal cij. The objective of the CPP is to construct a least cost path that traverses each arc in A at least once. There are many practical applications for variants of the CPP, including winter street maintenance, and street sweeping that incorporate: [Rural Instances] Rural Postman Problems (RPP) stipulate that only a subset $A_R \subset A$ require traversal, allowing for non-servicing traversal on the rest of the graph. In the context of street sweeping, a street sweeper isn't responsible for sweeping all the streets. [Windy Graphs] In the CPP, the cost of traversal of an arc is the same, regardless of the direction of traversal. In the Windy Postman Problem (WPP), the cost of traversal is asymmetric. That is, it is possible for cij not equal cji. In the context of snow plowing, it is harder to plow uphill than downhill. [Multi-Vehicle] Instead of a single vehicle with a single tour, multiple tours are found for multiple vehicles. This is often accompanied with an objective function that seeks to minimize the cost of the largest cost route. This is motivated by practical applications, which seek to balance the cost of each route. In the case where route cost is measured in time, route balancing minimizes, for example, paid overtime. [Turn Penalties] UPS reported that it saved three million gallons of gasoline annually by avoiding unnecessary left-hand turns, which take longer to perform than going straight or turning right. Instances with turn penalties incorporate costs of turning, in addition to costs of traversal. The Windy Postman Problem (WPP) incorporates windy graphs and the Rural Postman Problem (RPP) incorporates rural instances. The RPP can be extended to include turn penalties (RPPTP). The Windy Rural Postman Problem (WRPP) incorporates instances that are both windy and rural. The WRPP can be extended to the MM k-WRPP which adds k plows. In this dissertation, we extend these variants to new problems with new problem attributes that are practically motivated. Our new attributes are listed below. [Multi-Period] The CPP solves for a single route, which can be interpreted to be traversed in a single day. It is possible that the set of required arcs is too long to service in a single day and therefore must be split among multiple days. In this case, we need to decide which day to assign service to each arc, before routing can take place. [Downhill Instances] In street snow plowing, it is faster to deadhead (traverse without servicing) a street rather than plowing it. In this case, there are different costs for deadheading and plowing a street. Moreover, it takes longer to plow uphill, resulting in four costs: plowing uphill, plowing downhill, deadheading uphill, and deadheading downhill. [Precedence] When considering downhill instances, the snow may be so deep that it is impossible for a snowplow to deadhead a street before the street is plowed. In this dissertation we present a variety of heuristics to solve these problems, all adaptations of the concept of cycle permutation based on Euclidean cycle decomposition. To our knowledge, the use of moving or permuting sub-cycles as a way to change and improve a Eulerian cycle is novel and we show that it is very robust at improving solutions

New results on the Windy Postman Problem

[EN] In this paper, we study the Windy Postman Problem (WPP). This is a well-known Arc Routing Problem that contains the Mixed Chinese Postman Problem (MCPP) as a special case. We extend to arbitrary dimension some new inequalities that complete the description of the polyhedron associated with the Windy Postman Problem over graphs with up to four vertices and ten edges. We introduce two new families of facet-inducing inequalities and prove that these inequalities, along with the already known odd zigzag inequalities, are Chvátal-Gomory inequalities of rank at most 2. Moreover, a branch-and-cut algorithm that incorporates two new separation algorithms for all the previously mentioned inequalities and a new heuristic procedure to obtain upper bounds are presented. Finally, the performance of a branch-and-cut algorithm over several sets of large WPP and MCPP instances, with up to 3,000 nodes and 9,000 edges (and arcs in the MCPP case), shows that, to our knowledge, this is the best algorithm to date for the exact resolution of the WPP and the MCPP. © 2010 Springer and Mathematical Optimization Society.The authors want to thank the three referees for their careful reading of the manuscript and for their many comments and suggestions that have contributed to improve the paper content and readability. In particular, several remarks regarding the discussion of C-G and mod-k inequalities were pointed out by one of the referees. A. Corberan, I. Plana and J.M. Sanchis wish to thank the Ministerio de Educacion y Ciencia of Spain (projects MTM2006-14961-C05-02 and MTM2009-14039-C06-02) for its support.Corberán, A.; Oswald, M.; Plana, I.; Reinelt, G.; Sanchís Llopis, JM. (2012). New results on the Windy Postman Problem. Mathematical Programming. 132(1-2):309-332. https://doi.org/10.1007/s10107-010-0399-xS3093321321-2Benavent E., Carrotta A., Corberán A., Sanchis J.M., Vigo D.: Lower bounds and heuristics for the windy rural postman problem. Eur. J. Oper. Res. 176, 855–869 (2007)Brucker P. The Chinese postman problem for mixed graphs. In Proceedings of international workshop. Lecture Notes in Computer Science 100, 354–366 (1981)Caprara A., Fischetti M.: $${\{0,\frac{1}{2}\}}$$ -Chvátal-Gomory cuts. Math. Program. 74, 221–235 (1996)Caprara A., Fischetti M., Letchford A.N.: On the separation of maximally violated mod-k cuts. Math. Program. 87, 37–56 (2000)Christof, T., Loebel, A.: PORTA—a polyhedron representation algorithm www.informatik.uni-heidelberg.de/groups/comopt/software/PORTA/ (1998)Christofides, N., Benavent, E., Campos, V., Corberán, A., Mota, E.: An optimal method for the mixed postman problem. In Thoft-Christensen, P. (ed.) System Modelling and Optimization. Lecture Notes in Control and Information Sciences 59, Springer (1984)Corberán A., Plana I., Sanchis J.M.: Zigzag inequalities: a new class of facet-inducing inequalities for arc routing problems. Math. Program. 108, 79–96 (2006)Corberán A., Plana I., Sanchis J.M.: A branch & cut algorithm for the windy general routing problem and special cases. Networks 49, 245–257 (2007)Corberán, A., Plana I., Sanchis, J.M.: Arc routing problems: data instances. www.uv.es/corberan/instancias.htm (2007)Corberán A., Plana I., Sanchis J.M.: The windy general routing polyhedron: a global view of many known arc routing polyhedra. SIAM J. Discrete Math. 22, 606–628 (2008)Grötschel, M., Win, Z.: On the windy postman polyhedron. Report No. 75, Schwerpunktprogram der Deutschen Forschungsgemeinschaft, Universität Augsburg, Germany (1988)Grötschel M., Win Z.: A cutting plane algorithm for the Windy Postman Problem. Math. Program. 55, 339–358 (1992)Guan M.: On the Windy Postman Problem. Discrete Appl. Math. 9, 41–46 (1984)Minieka E.: The Chinese postman problem for mixed networks. Manage. Sci. 25, 643–648 (1979)Naddef D., Rinaldi G.: The symmetric traveling salesman polytope and its graphical relaxation: composition of valid inequalities. Math. Program. 51, 359–400 (1991)Oswald M., Reinelt G., Seitz H.: Applying mod-k cuts for solving linear ordering problems. TOP 17, 158–170 (2009)Papadimitriou C.H.: On the complexity of edge traversing. J. Assoc. Comput. Mach. 23, 544–554 (1976)Ralphs T.K.: On the mixed Chinese postman problem. Oper. Res. Lett. 14, 123–127 (1993)Wenger, K.: Generic Cut Generation Methods for Routing Problems. PhD Dissertation, University of Heidelberg, Germany (2004)Win, Z.: Contributions to Routing Problems. PhD Dissertation, University of Augsburg, Germany (1987)Win Z.: On the Windy Postman Problem on eulerian graphs. Math. Program. 44, 97–112 (1989)Zaragoza Martínez F.J.: Series-parallel graphs are windy postman perfect. Discrete Math. 308, 1366–1374 (2008

Efficient routing of snow removal vehicles

This research addresses the problem of finding a minimum cost set of routes for vehicles in a road network subject to some constraints. Extensions, such as multiple service requirements, and mixed networks have been considered. Variations of this problem exist in many practical applications such as snow removal, refuse collection, mail delivery, etc. An exact algorithm was developed using integer programming to solve small size problems. Since the problem is NP-hard, a heuristic algorithm needs to be developed. An algorithm was developed based on the Greedy Randomized Adaptive Search Procedure (GRASP) heuristic, in which each replication consists of applying a construction heuristic to find feasible and good quality solutions, followed by a local search heuristic. A simulated annealing heuristic was developed to improve the solutions obtained from the construction heuristic. The best overall solution was selected from the results of several replications. The heuristic was tested on four sets of problem instances (total of 115 instances) obtained from the literature. The simulated annealing heuristic was able to achieve average improvements of up to 26.36% over the construction results on these problem instances. The results obtained with the developed heuristic were compared to the results obtained with recent heuristics developed by other authors. The developed heuristic improved the best-known solution found by other authors on 18 of the 115 instances and matched the results on 89 of those instances. It worked specially better with larger problems. The average deviations to known lower bounds for all four datasets were found to range between 0.21 and 2.61%