This paper deals with planning of a tour for a vehicle to clear a certain set of streets in a city of snow. Our previous results on the problem contain a heuristic based on reformulation to an asymmetric traveling salesman problem (ATSP) which yields feasible solutions and upper bounds, and a relaxation of a MIP model for obtaining lower bounds. The goal now is to try to improve the solutions and bounds. In this paper we describe a branch-and-dive heuristic which is based on branch-and-bound principles. We discuss how branching can be done so that the fixations can be utilized in both the relaxation and the ATSP model, and how the search for better solutions can be done. The heuristic has been implemented and applied to real life city networks. The method is shown to outperform two other heuristics for the ATSP with precedence constraints. KEYWORDS arc routing, asymmetric traveling salesman, branch-and-bound, heuristic, precedences, turning penalties 1 INTRODUCTION Snow removal is an important problem in Nordic countries. In Figure 1 the maximal snow depth (December to April) in Stockholm (Observatorielunden) is given for each year from 1904 to 2018, and the large variation is evident. (The data are taken from Stockholm Stad [26]. The horizontal line is the average, and the slightly sloping horizontal line is the linear trend.) The total costs for snow removal in the city of Stockholm for each year from 2008 to 2013 (as reported by Stockholm Stad) were 130, 146, 265, 238, 224, and 192 million SEK (10 SEK ≈ 1 Euro). Note that this is only for one city, not the whole country, and that Stockholm is not situated in the northern parts of Sweden, where there is much more snow. The costs of course depend on the maximal snow depth, but also on how and when the snow comes. Large amounts of snow in a short time give high costs. It seems hard to predict the amount of snow next year, as a year with much snow is often followed by a year with much less, and vice versa. Using last year's plan is often not possible, so one must be able to quickly plan new tours. Our work is aimed at Swedish circumstances, as specified in communication with the contractors that do the work, and with the municipalities that pay for it. In this paper we deal with a few aspects (described later) that are different from previous work from other countries. We consider removing snow after a snowfall, which means that each street only needs to be cleared once. Furthermore, we consider snow removal in urban areas, not in rural areas. In general, the streets in a city need to be cleared of snow by a limited number of vehicles within a certain time. Tours for the vehicles should be planned in order to minimize the time or/and cost. Snow should be cleared from all the streets including bicycle paths, pedestrian paths, bus stops, intersections, and so forth. Roghayeh Hajizadeh and Kaj Holmberg contributed equally to this study. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and repr...

Snow removal is, in Sweden, an infrequently occurring challenge. Doing the snow removal more efficiently could give much benefits for society. Since the amounts of snow vary a lot from day to day, and from year to year, fixed plans are not the best. Optimization of the snow removal tours could save much money. In this paper, we study the multi-vehicle urban snow removal problem from a mixed integer programming perspective. It is a very hard problem, and obtaining the exact optimum seems to be out of reach. Therefore, we study relaxations of the problem. Our goal is simply to find the best bounds for the optimal objective function value that is possible in limited time. We present some promising possibilities, verified by extensive computational tests.

Snow removal problem in cities is a challenging task in Nordic countries. The problem is finding optimal tours for a certain number of vehicles with some circumstances in order to clear a number of streets in a city. We have formulated the urban snow removal problem as a time-indexed mixed integer linear programming model which is huge and complicated. In our previous work, we studied the model and its different relaxations which show that the problem is not solvable in practice. Since the problem has many sets of constraints with complicated structures, relaxing them with Lagrangian relaxation might be beneficial. In this paper, we discuss different possibilities of relaxing sets of constraints and develop a Lagrangian heuristic which consists of a suitable Lagrangian relaxation of the problem, a subgradient optimization method for solving the Lagrangian dual, and procedures for obtaining feasible solutions. The heuristic has been implemented and applied to artificial and real life city networks. The results show that the bounds have been improved.

Removing snow in a city is an unavoidable task in Nordic countries like Sweden. A number of streets in an area need to be cleared of snow by a limited number of vehicles and the tours for the vehicles must be planned in order to minimize the time and/or cost. Since the amount of snow can vary significantly from one year to another, the plans/tours of one year cannot be used for the next year. Hence, new tours need to be planned each time. Snow removal can be done in rural or urban areas and in addition during snowfall or after a snowfall. In this thesis, we study urban snow removal after a snowfall.There are different relevant specifics of the urban snow removal problem. For instance, there are different types of streets which need different numbers of sweeps in order to remove the snow. In addition, some tasks must be done before other tasks can be started. This leads to precedence constraints. Furthermore, each vehicle needs a certain time to switch from a task to another task. The problem can be formulated as a huge time-indexed mixed integer programming which often is not directly solvable in practice.The contributions of this thesis include the study of different relaxations and heuristics to find feasible solutions and improve the bounds on the optimal objective function values which are discussed in five papers. Paper I deals with single vehicle snow removal. A branch-and-dive heuristic based on branch-and-bound principles is given in order to improve the solutions and bounds. In Paper II, feasible solutions for the snow removal problem with a limited number of identical vehicles are obtained. First, the work is broken down into smaller parts, one for each vehicle. Based on the obtained allocation, a feasible tour for each single vehicle snow removal is obtained. Finally, combined solution approaches and coordination of the vehicles to find a feasible solution for the original problem are discussed.In order to improve the computational efficiency, one can take advantage of the tree structure, since modern real life city networks often contain parts that are trees. In Paper III, tree parts are studied and a tree elimination procedure is given for the snow removal problem, to be used before searching for optimal tours. Two variations encountered in practice for normal streets are compared in Paper IV. The first variant is doing a middle sweep before the two side sweeps and the second one is doing only side sweeps. Paper V studies the problem from modeling perspective. The problem is formulated as a mixed integer programming model and different relaxations of it are investigated. Finally, Lagrangian relaxation of the problem is studied in Paper VI. Different possibilities for Lagrangian relaxations are investigated and subgradient optimization is used to solve the Lagrangian dual.iii iv When I was 30 years old, I packed those 30 years in a 30 kg luggage and moved to Sweden to start a new journey with an ambiguous future. Although I had lots of ups and downs during this journey and my life in Sweden, I met so ...