A hybrid generational genetic algorithm for the periodic vehicle routing problem with time windows

Abstract

We propose a new population-based hybrid meta-heuristic for the periodic vehicle routing problem with time windows. This meta-heuristic is a generational genetic algorithm that uses two neighborhood-based meta-heuristics to optimize offspring. Local search methods have previously been proposed to enhance the fitness of offspring generated by crossover operators. In the proposed method, neighborhood-based meta-heuristics are used for their capacity to escape local optima, and deliver optimized and diversified solutions to the population of the next generation. Furthermore, the search performed by the neighborhood-based meta-heuristics repairs most of the constraint violations that naturally occur after the application of the crossover operators. The genetic algorithm we propose introduces two new crossover operators addressing the periodic vehicle routing problem with time windows. The two crossover operators are seeking the diversification of the exploration in the solution space from solution recombination, while simultaneously aiming not to destroy information about routes in the population as computing routes is NP-hard. Extensive numerical experiments and comparisons with all methods proposed in the literature show that the proposed methodology is highly competitive, providing new best solutions for a number of large instances.

Appendices

Appendix 1: The UTS and RVNS meta-heuristics

This Appendix briefly presents the two meta-heuristics that have been implemented for the offspring education procedure.

1.1 Unified tabu search

UTS (Cordeau et al. 2004) uses its own penalty coefficients, $\alpha _1,\,\alpha _2$, and $\alpha _3$, for violations of vehicle capacity, route duration and customer service time window constraints, respectively. The cost function of a solution $s$ then becomes $f(s)$ = $c(s)$ + $\alpha _1 q(s)$ + $\alpha _2 d(s)$ + $\alpha _3 w(s)$.

An attribute set $B(s) =$ {$(i,k,l):$ customer $i$ is visited by vehicle $k$ on day $l$} is associated to each solution $s$. Let $b_{rl}$ be a binary constant equal to 1 if and only if day $l$ belongs to pattern $r$, for each pattern $r \in R$ and every day $l \in \mathcal{T}$ of solution $s$. The neighborhood $N(s)$ of a solution $s$ is then defined by two transformations to (1) relocate a customer within a day (routing modification), and (2) replace the pattern of a customer (pattern modification):

• (1) Remove customer $i$ from route $k$ on day $l$ and insert it into another route $k'$.

• (2) Replace pattern $r$ currently assigned to customer $i$ with another pattern $r' \in R_i$; Then, for $l=1, \ldots ,t$ do

  • If $b_{rl}$ = 1 and $b_{r'l}$ = 0, remove customer $i$ from its route of day $l$;

  • If $b_{rl}$ = 0 and $b_{r'l}$ = 1, insert customer $i$ into the route of day $l$ minimizing the increase in fitness $f(s)$.

UTS starts from a given offspring $s$ and chooses, at each iteration, the best non-tabu solution in $N(s)$. After each iteration, the values of the parameters $\alpha _1,\,\alpha _2$, and $\alpha _3$ are modified by a factor $1+ \delta$; multiplied by the factor if the solution is feasible with respect to the respective constraint, divided, otherwise. We set $\delta = 0.5$, the best value reported by the authors (Cordeau et al. 2004).

To diversify the search, any solution $\overline{s} \in N(s)$ such that $f(\overline{s}) \ge f(s)$ is penalized by a factor $p(\overline{s})$ proportional to the addition frequency of its attributes, $p(\overline{s}) = \lambda c(\overline{s})\sqrt{nm}\sum _{(i,k,l) \in B(\overline{s})} {\rho _{ikl}}$, where $\rho _{ikl}$ is the number of times attribute $(i,k,l)$ has been added to the solution during the search process and $\lambda = 0.015$.

The tabu length was adjusted to a smaller number of iterations performed and set to $\theta$ = $1.5log_{10}(n)$. The post-optimization heuristic is not implemented in the education procedure. The UTS procedure returns the best feasible solution $s_1$ if it exists, the best infeasible solution $s_2$, otherwise. The procedure is summarized in Algorithm 3.

1.2 Random Variable Neighborhood Search (RVNS)

RVNS (Pirkwieser and Raidl 2008) uses three different neighborhood structures. For each of these structures, the authors defined six moves, hence resulting in a total of 18 neighborhoods: (1) randomly changing patterns of customers, (2) moving a random segment of customers of a route to another route on the same day, and (3) exchanging two random segments of customers between two routes on the same day. In the latter two cases, the segments are reversed with a small probability, $p_{rev} = 0.1$. The neighborhoods are summarized in Table 10.

Table 10 Neighborhood structures of RVNS

RVNS starts with a random neighborhood ordering and generates a new ordering each time a full VNS iteration is completed. For intensification, RVNS applies 2-opt in a best-improvement fashion. Additionally, each new incumbent solution is subject to a 2-opt*. RVNS accepts solutions which degrade the objective value under the Metropolis criterion, like in simulated annealing. Thus, an inferior solution $s'$ is accepted with probability $e^{-(f(s') - f(s))/T}$, depending on the cost difference to the current solution $s$ relative to the temperature $T$. A linear cooling scheme is applied, $T$ being decreased every $\tau$ iterations by an amount of $(T*\tau )/\tau _{max}$, where $\tau _{max}$ denoted the maximal VNS iterations. The value of $\tau$ was adjusted to the smaller number of iterations performed and set to $\tau =10$, the initial temperature value $T_0 = 10$, and $\tau _{max}$ = [100, 800]. The detailed description of the implementation is given in Algorithms 4 and 5.

Appendix 2: Detailed results

Table 11 summarizes the HGGA results for the Pirkwieser and Raidl (2009a) instances with and without travel cost truncation. The algorithm was run 10 times per instance. Best results, average results, standard deviations, and computation time are reported. Vidal et al. (2013) are the only one to provide the best solutions for their algorithm (VCGP13), and for the truncation case only.

Table 11 HGGA results on Pirkwieser and Raidl (2009a) instances with and without travel cost truncation

We therefore also include these best results of the truncation case in the column VCGP13 for comparison purposes. The “GAP Best-to-Best” column displays the gaps of the best solutions obtained by HGGA with respect to those of VCGP13.