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

1 Introduction
Several variations of routing problems arise in real-world applications. The
basic problem is the one of designing a set of minimum-cost routes originating
and finishing at a given location. This problem has a depot (where
several capacitated vehicles are located) and a set of customers, each one
requiring a known demand of a commodity which is available at the depot.
The objective is to design routes with minimum length (or cost) to serve the
customers. Applications impose several constraints on this basic problem.
Among others we may consider:
1. capacity constraints stating that each vehicle cannot carry more than
a certain number of goods;
2. distance constraints stating that the length of the route of each route
cannot exceed a given length;
3. time windows stating that each client cannot be visited after a certain
time;
4. precedence constraints requiring that a given customer can be visited
only after another has been visited.
Here we study a variant which considers a lower limit on the capacity of each
vehicle to simply state that a vehicle can be used only if it visits, at least, a
certain number of clients. Similar constraints have already been considered
in, for example, Groër, Golden and Wasil [7] and Jozefowiez, Semet and
Talbi [11] where the routes of a solution must be balanced.
Single Commodity Flow (SCF) formulations provide a basic framework
for modelling most of these problems (see, for instance, Gavish and Graves
[5], Letchford and Salazar-González [13] and Toth and Vigo [15]). The
SCF model is an example of an extended model, where there are two set
of mathematical variables. One set is composed by the decision variables
describing the design of the routes. The other set is composed by flow
variables to capture the requirements on the customer demands and on the
vehicle capacities. Given a SCF model, we can use a tool of projection to
create a natural model based only on the route-design variables. Although
the linear programming (LP) relaxation bounds from both models coincide
and the natural models require cutting-plane generation approaches, they
are very successful for solving some routing problems (see, e.g., Naddef and
Rinaldi [14]).
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