A branch‐and‐dive heuristic for single vehicle snow removal

Abstract

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...

Review of Reduced Salt, Snow, and Ice Management Practices for Commercial Businesses

“…In contrast, a contract without these details may result in unnecessary use of materials or time spent by private contractors ( 11 ), increasing the potential to use more salt than is necessary to maintain safe conditions. In addition, planning routes efficiently may decrease the time needed to complete service routes for single vehicles ( 30 ).…”

A rolling horizon framework for the time‐dependent multi‐visit dynamic safe street snow plowing problem

“…The problem is reformulated into an asymmetric traveling salesman problem and solved using a heuristic. Hajizadeh and Holmberg [7] extend this study by creating a hybrid heuristic and a MIP relaxation. The authors use branching techniques in both methods, which allows sharing of information between the hybrid heuristic and the MIP relaxation.…”

“…Constraints (5) guarantee that no additional traversals take place during this time; with 𝑑 a − 1 being sufficiently large big-Ms. The service of an arc is determined by Constraints (6), while the levels of snow are calculated using Constraints (7) and (8). Constraints (9) define the safety of the arcs.…”

“…Constraints (9) define the safety of the arcs. The largest value needed for the big-Ms in Constraints (7) for a specific arc a and time t is M = l a0 + ∑ t n=0 i an , as it enables clearing the largest amount of snow that would be possible on arc a at time t. Similarly, for Constraints (9) with arc a and time t, a big-M of M > l a0 + ∑ t n=0 i an − − − H is sufficient. In addition, when only using a single big-M, the smallest sufficient value for both types of constraints would be M = max a∈A S {l a0 + ∑ t∈T i at }, which is the largest amount of snow that can accumulate on a single arc if no plowing is performed.…”

Preface: Special issue on arc routing problems and other related topics

“…Hajizadeh and Holmberg [6] also focus on snow management. In particular, they present a branch‐and‐dive heuristic, based on branch‐and‐bound principles, for the problem of finding the best tour for one snow remover.…”