Reverse Multistar Inequalities and Vehicle Routing ... - IASI-CNR

which are already known from the literature? This is a difficult question to
answer since there are many classes of inequalities for the CVRP. The best
we can say is that from this large class of inequalities, and as far as we know,
only the degree constraints use information on the number of vehicles, and
thus it is highly unlikely that the ERMS inequalities are equivalent to other
inequalities.
6 Computational results
In this section we present computational results to evaluate our contributions
for solving BVRP instances. We show that using the new inequalities
one can solve larger instances. To this end we compare three approaches:
an ILP solver on a compact SCF model given in Section 3, and two branchand-cut
implementations. The first branch-and-cut implementation uses the
standard inequalities for the capacitated VRP and only the RMS constraints
(8). The second branch-and-cut implementation uses the standard inequalities
for the CVRP as in the first implementation, but uses the ERMS (10)
and the rounded ERMS (15) instead of the RMS constraints. The standard
inequalities for the CVRP that we use in both implementations are
the subtour elimination constraints (12) and the rounded MS inequalities
(11). These inequalities are separated through an integrated procedure. We
describe next several steps detailing this procedure. Each step is repeated
while it generates violated inequalities before proceeding to the next step.
Step 1: Find violated subtour elimination constraints (12) as it is standard
in the literature (see, e.g., [14] ).
Step 2: Find violated rounded ERMS inequalities (15). For separating
these constraints we use a heuristic approach. It consists of finding a
most violated weaker version of inequality (15). This weaker inequality
is
x(S ∪ {1} : S ∪ {1}) ≤ |S| + |S|
Q ,
which can also be rewritten in a cut-form as
x(S ∪ {1} : S ′ ) + |S|
Q
≥ x(1 : V \ {1}).
These inequalities can be exactly separated by using a min-cut algorithm
on the following capacitated directed network. Let x ∗ be the
given fractional solution. Let the node set be V and a dummy node
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