An Ant Colony Algorithm for Time-Dependent Vehicle Routing Problem with Time Windows

Conference paper


In this paper, we address the Vehicle Routing Problem with Time Windows, both time-independent and -dependent cases. In the timeindependent case, our objective is to minimize the total distance. To solve this problem, we propose an Ant Colony Optimization algorithm. Then we implement the algorithm to solve the time-dependent case where the objective is to minimize the total tour time. The time dependency is embedded in this model by using a deterministic travel speed function which is a step function of the time of the day. An experimental evaluation of the proposed approach is performed on the well-known benchmark problems Optimizing a distribution network has been and remains an important challenge both in the literature and in real-life applications and the routing of a eet of vehicles is the most widely addressed problem in a distribution network. The Vehicle Routing Problem (VRP) determines a set of vehicle routes originating and terminating at a single depot such that all customers are visited exactly once and the total demand of the customers assigned to each route does not violate the capacity of the vehicle. The objective is to minimize the total distance traveled. An implicit primary objective is to use the least number of vehicles. The Vehicle Routing Problem with Time Windows (VRPTW) is a variant of VRP in which lower and upper limits are imposed to the delivery time of each customer. The arrival at a customer outside the specified delivery time is either penalized (soft time windows) or strictly forbidden (hard time windows). The interested reader is referred to [1] for more details on VRPTW. In the Stochastic Vehicle Routing Problem, the customer demands and/or the travel times between the customers may vary. Although stochastic travel times and demand distributions have been frequently used in the literature, time-varying travel speeds and time-dependent VRPTW (TDVRPTW) have seldom been addressed. In the literature, time dependency is taken into consideration in two ways: stochastic travel times and deterministic travel times. First introduced by [2], stochastic travel times are mainly examined by [4] and [3]. [5] proposeda deterministic travel time based model in which the important nonpassing property is introduced. [6] and [7] also use deterministic travel times in a setting where the day is divided into time intervals.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Cordeau J, Desaulniers G, Desrosiers J, Solomon MM, Soumis F (2001) VRP with Time Windows. In: Toth P and Vigo D (Eds) The vehicle routing problem. SIAM Monographs on Discrete Mathematics and Applications. SIAM Publishing, Philadelphia, PA, pp. 157–193Google Scholar
  2. 2.
    Laporte G, Louveaux F, Mercure H (1992) The vehicle routing problem with stochastic travel times. Transportation Science 26(3):161–170CrossRefGoogle Scholar
  3. 3.
    Potvin J, Xu Y, Benyahia I (2006) Vehicle routing and scheduling with dynamic travel times. Computers & Operations Research 33:1129–1137CrossRefGoogle Scholar
  4. 4.
    Kenyon AS, Morton DP (2003) Stochastic vehicle routing with random travel times. Transportation Science 37(1):69–82CrossRefGoogle Scholar
  5. 5.
    Ahn BH, Shin JY (1991) Vehicle-routeing with time windows and time-varying congestion. Journal of Operations Research Society 42(5):393–400Google Scholar
  6. 6.
    Ichoua S, Gendreau M, Potvin JY (2003) Vehicle dispatching with time-dependent travel times. European Journal of Operations Research 144:379–396CrossRefGoogle Scholar
  7. 7.
    Donati AV, Montemanni R, Casagrande N, Rizzoli AE, Gambardella LM (2008) Time-dependent vehicle routing problem with a multi ant colony system. European Journal of Operational Research 185:1174–1191CrossRefGoogle Scholar
  8. 8.
    Dorigo B, Stützle T (2004) Ant colony optimization. MIT Press, Cambridge MassachusettsGoogle Scholar
  9. 9.
    Solomon MM (1987) Algorithms for the vehicle routing and scheduling problems with time window constraints. Operations Research 35(2):254–265CrossRefGoogle Scholar
  10. 10.
    Bullnheimer B, Hartl RF, Strauss C (1999) An Improved Ant System Algorithm for the Vehicle Routing Problem. Annals of Operations Research 89:319–328CrossRefGoogle Scholar
  11. 11.
    Alvarenga GB, Mateus GR, Tomi G (2007) A genetic and set partitioning two-phase approach for the vehicle routing problem with time windows. Computers & Operations Research 34:1561–1584CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Orhanli, TuzlaSabanci UniversityIstanbulTurkey

Personalised recommendations