148 Chapter6. Metaheuristics for the Capacitated VRP
The probability associated with each ring is dynamically adjusted as the algorithm
unfolds. At the start, the distance between the current vertex and the closest unit on the
ring plays the dominant role. Later on, the capacity constraint is taken into account, as
rings that cannot accommodate the current vertex without violating this constraint (due to
the tentatively assigned vertices) becomes less likely to be selected. At the end, only feasi-
ble rings have a nonnegligible probability of being selected. In a later study, Ghaziri [27]
extended this model to address the VRP with maximum route time constraints through
a modification of the probability distribution over the rings. Computational results on
the Christofides, Mingozzi, and Toth [11] test set have shown that these models pro-
duce solutions of relatively good quality but are not competitive with alternative meta-
heuristics, in particular tabu search (Gendreau, Hertz, and Laporte [23] and Rochat and
Taillard [56]).
6.8 Conclusions
This survey of metaheuristics for the VRP shows that the best of these methods can find
excellent and sometimes optimal solutions to instances with a few hundred customers,
albeit at a significant cost in computation time. Tabu search now emerges as the most
effective approach. Procedures based on pure genetic algorithms and on neural networks
are clearly outperformed, while those based on simulated or deterministic annealing and on
ant systems are not quite competitive. Considering the performance improvements obtained
with successive implementations of any given approach, it appears, however, that hybrid
ant systems and genetic algorithms may, in the future, be able to match the effectiveness of
existing tabu search heuristics, since these approaches have not been fully exploited. Another
observation is that the data sets currently used as benchmarks are made up of instances that
are too small to allow one to differentiate sharply between the various implementations of
some of the metaheuristics, tabu search in particular. Data sets corresponding to larger
instances are thus required. In the same vein, one may wonder how these metaheuristics
would perform on the much larger instances often encountered in practical applications.
Given their computing requirements, heuristics with such a level of sophistication may
be unable to solve satisfactorily these large instances in any reasonable amount of time,
especially if real-time applications are contemplated. With respect to the classical heuristics
presented in Chapter 5, metaheuristics are rather time consuming, but they also provide much
better solutions. Typically, classical methods yield solution values between 2% and 10%
above the optimum (or the best known solution value), while the corresponding figure for the
best metaheuristic implementation is often less than 0.5%. This is illustrated in Figure 6.7.
It is time to develop simpler methods capable of quickly providing good quality solutions.
This will probably be achieved by speeding up the best available metaheuristics, rather than
investing more effort on classical approaches. The GTS algorithm proposed by Toth and
Vigo is an important step in the right direction. It draws from the vast expertise accumulated
in the field of metaheuristics and exploits some of their best concepts. Yet, by carefully
exploiting the problem structure, it manages to avoid most of the unnecessary computations
carried out in previous tabu search algorithms.

8.5. Heuristic Algorithms 215
thus delaying the union of linehaul and backhaul routes. The backhaul saving is defined as
where S is an estimate of the maximum saving stj and p is a real penalty between 0 and 1.
The Clarke and Wright method, and hence algorithm DB, does not allow for the
control of the number of routes of the final solution. Indeed, the solution found for a given
instance can require more than K routes to serve all the vertices, thus being infeasible. From
a practical point of view, both the routing cost of the solution obtained with algorithm DB
and the probability that this solution is feasible are strongly related to the number of route
mergings executed. It is then evident that, even if delayed, the route orientation arising
in the mixed routes, and the consequent decrease of possible route merging combinations,
reduces the effectiveness of this method in facing the VRPB in terms of both the overall
routing cost and the number of feasible solutions found. It can be noted that algorithm DB
may be easily adapted to consider AVRPB instances (see, e.g., Vigo [35] for a discussion
on the extension of the Clarke and Wright method to the asymmetric CVRP).
Deif and Bodin [9] tested their algorithm on randomly generated problem instances
with 100 to 300 customers and a backhaul percentage between 10% and 50%. Several p
values were tested, and the results obtained show that values of p between 0.05 and 0.20
produced the best solutions.
8.5.2 Algorithms of Goetschalckx and Jacobs-Blecha
Goetschalckx and Jacobs-Blecha [19] proposed an algorithm, called SF herein, for the VRPB
with Euclidean cost matrix. The approach is based on the concept of space-filling curves,
previously used by Bartholdi and Platzman [3] for the solution of the planar TSP. Using
the space-filling curve transformation, linehaul and backhaul customers are, separately,
transformed from points in the plane into points along a line. The two separate sequences
of points are then partitioned to form feasible routes each containing customers of only one
type. Each linehaul route is, in turn, merged with the nearest backhaul route (according to
the space-filling mapping), thus obtaining the final set of vehicle routes. Also in this case,
the method does not guarantee building solutions using exactly K routes. Goetschalckx and
Jacobs-Blecha tested both the DB and SF algorithms on 57 Euclidean instances with 25 to
150 vertices, 20% to 50% of which are backhauls. The results presented by Goetschalckx
and Jacobs-Blecha [19] show that DB solutions are generally better than those obtained by
SF, while SF is faster than DB, mainly for large problems.
More recently, Goetschalckx and Jacobs-Blecha [20] presented a heuristic algorithm,
called LHBH, for the Euclidean version of VRPB. The approach is based on the extension
of the generalized assignment heuristic proposed by Fisher and Jaikumar [15] for CVRP.
Initially, a partial solution made up of K route primitives is obtained as follows: first, K
seed radials are determined by iteratively solving a capacitated location-allocation problem;
then for each such radial a route primitive is obtained by considering the customers located
close to the radial (i.e., within a 10-degree angle) and sequencing the linehaul customers by
increasing distance to the depot and the backhaul customers by decreasing distance. The
customers are then grouped together into K clusters by heuristically solving generalized
assignment problems, whose cost matrices contain the insertion cost of every vertex into each
B or viece versa
otherwise,