Optimization of Snow Removal in Cities

Abstract: 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 co-ordination 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.