VRP solution methodologies in the literature

Due to their simplicity and fast computations, VRP heuristics are very popular for obtaining sat- isfactory, though only approximately Pareto-optimal, solutions to small-scale problem instances. Solving large-scale VRP instances with multiple complex objectives is, however, very difficult using only heuristic techniques if the goal is to attain a good approximation of the Pareto front. VRPs of this nature call for the use of alternative, more powerful solution search techniques.

11_{A route failure is said to occur whenever the travel time and/or travel distance associated with a route}

exceeds the assigned vehicle’s distance and/or time autonomy levels. Such a route is then said to be incomplete or unfinished.

2.6. The vehicle routing problem 45

There is consensus in the literature that VRPs are computationally very hard to solve, to the point where instances of the capacitated, single-objective VRP (the most basic VRP formulation) with 10 vehicles and more than 30–40 customers can only be solved approximately with the use of metaheuristics [119]. Of course, this computational complexity typically increases when solving more specialised VRP variants, such as solving VRPs with multiple depots in multiobjective space. Moreover, VRPs with a heterogeneous fleet of vehicles are considered much harder to solve than those with a homogeneous fleet [109].

The multi-depot VRP is a variant of the capacitated VRP in which routes are simultaneously sought for several vehicles originating from multiple depots, serving a fixed set of customers and then returning to their original depots. Compared to simpler VRPs and their variants, for which abundant literature exists, only a relatively small volume of research has been done on multi-depot VRPs. A small number of good optimisation techniques have nevertheless appeared in the literature for this problem. Most of these techniques tend to break down an instance of the problem into a series of single-depot sub-instances and/or only solve it for a single objective and/or for a homogeneous fleet of vehicles [66, 67, 109, 140, 157]. Because vehicles always start and complete their routes at the same depots, it is easy to merge the set of customer vertices with that of the depot vertices in such simplified model formulations.

In [66] and [140], for example, the solution process for a single-objective, multi-depot VRP with a homogeneous vehicle fleet is decomposed into a three-phase heuristic approach. The first phase involves grouping the set of customers to be served by each depot (a clustering sub-problem), while the second phase entails assigning customers served from the same depot to several routes so that the capacity constraint associated with the vehicles is not violated (a routing sub- problem). The third phase consists of determining the visitation sequence of customers located on the same route (a scheduling sub-problem). The chief difference between considering multiple depots simultaneously and solving multiple single-depot sub-problems separately is that the local search operators in metaheuristics tailored to these approaches differ significantly in nature [157]. According to Salhi et al. [123], however, the decomposition of a multi-depot VRP into several single-depot sub-problems by (naively) assigning the customers to the depots closest to them (distance-wise), and solving these sub-problems individually, using a suitable approach, is easy to carry out, but usually leads to poor, significantly sub-optimal solutions. As a result, the mathematical modelling approach proposed in [123] is perhaps best suited to solving instances of the MLE response selection problem in a centralised or intermediate decision making paradigm (where the problem is solved globally). Salhi et al. [123] propose a complete mixed-integer, linear formulation for a generic multi-depot VRP with a heterogeneous vehicle fleet. Moreover, formulation variants are also offered for alternative VRP scenarios. For instance, multi-depot VRP variants are considered in which the number of vehicles of a given type is known (i.e. a heterogeneous fleet with defined classes of vehicles); in which certain types of vehicles cannot be accommodated at certain depots; in which a vehicle is not required to return to the same depot from whence it originated; or in which a maximum route length constraint associated with each class of vehicle is imposed (i.e. a distance-constrained VRP). Salhi et al. [123] then solve the problem (approximately) using a variable neighbourhood search applied to the notion of borderline customers, using six different local search operators. Unfortunately, the model formulation and search technique presented in [123] is applicable to single-objective optimisation only. Consequently, it is unlikely that this variable neighbourhood search heuristic may be as effective for solving similar problems in multiobjective space.

Many studies favour the use of genetic algorithms for solving the above-mentioned multiobjec- tive VRP variants, often suggesting specific or unique genetic operators [66, 67, 88, 140, 157]. Moreover, due to the high complexity level associated with problems of this kind, many stud-

ies favour the use of hybrids over more traditional evolutionary computation approaches in an attempt to improve algorithmic performance. The innovative hybrid genetic search with adap- tive control proposed by Vidal et al. [157], for example, offers an optimisation methodology for solving (among other problems), the multi-depot, periodic VRP with capacitated vehicles and time windows. This hybrid includes a number of advanced features in terms of chromosome fitness evaluation, offspring generation and improvement, and effective population management. Two sub-populations are managed in [157] (one containing feasible individuals and the other infeasible ones), and the mating pool is populated using parents from both sub-populations. Furthermore, the selection operator mechanism takes into account both the fitness of individ- uals and the level of contribution they provide to the diversity of the population. Offspring solutions are enhanced fitness-wise using so-called education and repair local search procedures. In [66], another hybrid genetic algorithm is proposed for solving difficult multi-depot VRPs. Here, three external heuristics are combined with a traditional genetic algorithm to design a hybrid between the Clarke-Wright algorithm [26], the nearest neighbour heuristic [145] and the iterated swap procedure [86]. While the former two heuristics are solely used for generating high-quality initial populations, the iterated swap procedure is applied to offspring solutions which, if found to have better fitness values than one (or both) of their parents, replace that (or those) parent solution(s) in the next generation of candidate solutions.

Although not as popular for solving complex multiobjective VRPs, certain authors have also adopted traditional, sequential simulated annealing techniques as a means to finding solutions to less complex VRPs [80, 87, 156, 164]. In these approaches, the very popular inter-route and intra-route solution transformation techniques [156] are often used to provide the necessary exploration and exploitation features to this metaheuristic for the purpose of generating a good non-dominated front when tuned appropriately. Harnessing the processing power of parallel computing has also been proposed for solving complex multiobjective VRPs [4].

Combining genetic algorithms with simulated annealing is not an uncommon strategy in the literature for both single-objective and multiobjective optimisation VRPs [25, 108, 115, 128]. The hybrid proposed by Chen and Xu [25], for example, first employs the crossover and mutation operations of genetic algorithms to produce offspring solutions, after which the fitness values of the offspring solutions are evaluated, accepting offspring solutions in the next generation of candidate solutions according to the Metropolis acceptance rule of simulated annealing.

Other metaheursitics employed in the literature for solving multiobjective VRPs include tabu search, ant colony optimisation and particle swarm optimisation (see_{§2.5). Szymon and Dominik} [141], for instance, emphasise the value of using a parallel tabu search algorithm for solving a multiobjective distance-constrained VRP. In [5], an ant colony system is put forward for solving a tri-objective VRP in which it is required to minimise the number of vehicles used, the total travel time and the overall delivery time. Finally, in [23], a particle swarm algorithm incorporating certain features from multiobjective optimisation is used to allow particle solutions to conduct a dynamic inter-objective trade-off in order to evolve seemingly poor infeasible routes to very good ones using the benefits of swarm intelligence.