Time-Dependent Multiple Depot Vehicle Routing Problem on Megapolis Network under Wardrop's Traffic Flow Assignment

______________________________________________________PROCEEDING OF THE 22ND CONFERENCE OF FRUCT ASSOCIATION Time-Dependent Multiple Depot Vehicle Routing Problem on Megapolis Network under Wardrop’s Traffic Flow Assignment Alexander V. Mugayskikh, Victor V. Zakharov Tero Tuovinen Saint-Petersburg State University Saint-Petersburg, Russia , University of Jyväskylä Jyväskylä, Finland tero.tuovinen@jyu.fi Abstract—In this work multiple depot vehicle routing problem is considered in case of variable travel times between nodes on a metropolis network. This variant of the classic multiple depot vehicle routing problem is motivated by the fact that in urban contexts variable traffic conditions play an essential role and can not be ignored in order to perform a realistic optimization. Time-travel matrices corresponding to each period of planning horizon were formed by solving the traffic assignment problem in conjunction with shortest path problem. Routing problem instances include from 20 to 100 customers randomly chosen from a road network of Saint-Petersburg. The results demonstrate that taking into account traffic flow information can reduce route time by 8-37% depending on number of customers in the problem instance. I. I NTRODUCTION The present article is devoted to possible ways of reducing costs of freight forwarding companies by taking into account road networks traffic load while planning delivery route. Basing upon research results of Russian leading specialists one can draw a conclusion that constantly increasing level of traffic congestion is becoming as essential reason of economic losses. Shorter transportation time would make transportation more efficient and increase the probability of improving the service level. Routing problems on a megapolis network can be solved by constructing routes in terms of mathematical modelling. Generated routing plans will provide the minimum travel time and shortest-time path for vehicles travelling between given depot and customers at different times of the day. The main obstacle in applying such methods is that most of the models assume that the travel speeds are constant, and ignore the fact that travel speeds can change throughout the day. But in practice, solutions become not optimal, and non-feasible in some cases. It causes late arrivals at customers and additional hiring costs for the truck drivers. Proposed in this article approach of planning delivery routes can reduce costs as it considers traffic information and avoids well-predictable traffic congestion in off-line vehicle routing. We focus on delays caused by traffic congestion during peak hours since they constitute a large part (from 70 up to 87%) of all traffic congestion delays [1]. We consider both traffic assignment and vehicle routing problems. By modelling traffic flow assignment on road net- works, travel time matrices for 7 periods of a day are formed. Information on travel times is be used while constructing routes in the time-dependent variant of multiple depot vehicle routing problem. Road network of the Saint Petersburg is considered as an example to test the impact of our approach and show that the time-dependent model provides significant improvements over the model with fixed travel times. The rest of the paper is organised as follows. In the next section, there is a literature review on time dependent models in vehicle routing. Sections 3 is devoted to description of general model of the time-dependent multiple depot vehicle routing problem (TD-MDVRP). We briefly describe Wardrop’s principles of equilibrium assignment of traffic flows on road network and consider the formulation of the traffic assignment problem (TAP) in section 4. Section 5 reports experimental results on the road network of Saint-Petersburg. Firstly, the traffic assignment problem is solved by Frank-Wolf algorithm and travel time matrices are obtained by Dijkstra’s algorithm. Secondly, we consider randomly generated TDMDVRP instances and demonstrate the effectiveness of timedependent approach in vehicle routing in comparison with static formulation. Section 6 concludes research and proposes future avenues of our study. II. L ITERATURE REVIEW Time Dependent Vehicle Routing problem (TDVRP) is the variant of the classic Vehicle Routing Problem (VRP) motivated by the fact that in a congested urban environment variable traffic conditions play an essential role and should not be ignored in order to perform a realistic optimization. Vehicle routing problem consists in finding a set of routes for identical vehicles based at the depot, such that each of the customers is visited exactly once minimizing the total routing cost. Since the introduction of VRP in work [2] developing real life applications of the routing problems have led to the emergence of a wide range of VRP flavors. This paper is focused on problems in which speeds are not constant and the travel time between two points is not a function of only the distance travelled. Time dependent vehicle routing problems have received considerably little attention among researchers. Although these problems represent an urban congested environment more accurately than do their ISSN 2305-7254 ______________________________________________________PROCEEDING OF THE 22ND CONFERENCE OF FRUCT ASSOCIATION nontemporal counterparts. The time dependent vehicle routing problem formulation was first introduced in study [3]. Randomly generated small-sized instances were solved by nearest neighbour heuristics for the time-dependent vehicle routing problem without time windows. Travel times were represented by step functions of two or three time periods and defined by uniform distribution for each period. In work [4] authors developed a restricted dynamic programming approach based on heuristic algorithm for solving the time-dependent traveling salesman problem. The algorithm was extended to handle with a special case of travel time step functions for which the principle of optimality holds, i.e. partial path of minimum arrival time necessarily leads to a minimum tour. Authors in [5] formulated node-based time dependent vehicle routing problem. In this formulation, constant speed r is assigned to each location for each time period. Thus, rijt is an average travel speed for a move from i to j starting at period t. But in fact such definitions as time-dependent travel time and time-dependent travel speed are equivalent since it is always possible to deduce these values from each other. The time-dependent travel speed model has been validated in a vehicle scheduling package used to schedule bank couriers in a number of large metropolitan cities in the United States, but no details are provided in the article. The major weakness of the above models is that they do not satisfy the FIFO property firstly mentioned in [6]. This is intuitive and desirable property that guarantees if a vehicle leaves a node i for a node j at a given time t, any identical vehicle with th (...truncated)