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

The other set of constraints imposes the connectivity and the capacity
constraints, and involves the use of the continuous variables:
fa −
fa = di for all i ∈ V (3)
a∈δ − (i)
a∈δ + (i)
q a xa ≤ fa ≤ q axa for all a ∈ A. (4)
As one specific example, a model for the unit-demand CVRP with vehicle
capacity Q ≥ 2 can be defined by setting
1 if i ∈ V \ {1}
di :=
1 − |V | if i = 1
and
q a = q a = 0 for all a ∈ δ − (1)
q a = 1 and q a = Q for all a ∈ δ + (1)
q a = 1 and q a = Q − 1 for all a ∈ δ + (1) ∪ δ − (1).
Different variations of vehicle routing problems can be formulated by
changing some of the parameters given in the previous generic SCF formulation
or setting new ones (see, e.g., Toth and Vigo [15]). The SCF model
is an example of compact model since it involves a number of variables and
constraints that is bounded by a polynomial function defined on the size
(number of nodes and number of arcs) of the input instance. We can create
a natural model involving only the xa variables and with a LP relaxation
bound equal to the LP relaxation bound of the SCF model, by using the
tool of projection. The procedure to project out the continuous variables
from the LP relaxation of the SCF model is based on the following result:
Theorem 2.1 (Hoffman 1960 [10]) Given xa values, there is a solution
of the linear system (3)–(4) on the fa variables if and only if
a∈δ − (S)
q axa ≥
a∈δ + (S)
q xa +
a
di for all S ⊂ V. (5)
We can give some intuition on how to generate these inequalities from
the SCF model. Suppose we add the flow conservation constraints (3) for
all nodes i in a set S and cancel equal terms, leading to:
fa =
fa +
di.
a∈δ − (S)
i∈S
a∈δ + (S)
5
i∈S