Natural and extended formulations for the Time-Dependent Traveling Salesman Problem

Consider a graph G=(V,A), where V ={1,2,..,n} and A={(i,j): i,j=1,..,n, i ≠ j}. The TimeDependent Travelling Salesman problem (TDTSP) is to find a minimum cost Hamiltonian circuit, starting and ending on node 1, where arc costs depend on its position in the tour. Thus, to each arc (i,j) in A and each possible position h of the arc in the tour we associate a cost h ij c . Clearly, an arc (1,j) leaving the depot can be only in position 1 and an arc (i,1) entering the depot can be only in the last position. Every other arc (i,j), i,j ≠ 1, can be located in positions h=2,...,n-1. Several formulations for the TDTSP described in the literature can be obtained by using the binary variables h ij z for all ( ) , i j A ∈ and 1,.., h n = , indicating whether or not arc ( , ) i j A ∈ is in the th h position of the circuit. A formulation that uses only the h ij z variables is called a natural formulation. We strart by reviewing the well known formulation by Picard and Queyranne (1978), whose main feature is that it uses, as a subproblem, an exact description of the n-circuit problem. An n-circuit is a circuit with n arcs which may repeat nodes and even arcs. Then we introduce new models for the problem. These models are built on two features: i) use a stronger subproblem a n-circuit subproblem with the additional constraint that a given node is not repeated in the circuit and ii) combine these subproblems for all nodes. The new formulation will use extra variables (besides the h ij z variables) and thus, it will