A Branch – and – Cut Algorithm for the Capacitated Open Vehicle Routing Problem

                                                                            

A Branch – and – Cut Algorithm for the Capacitated Open Vehicle Routing Problem

Adam N. Letchford, Jens Lysgaard and Richard W. Eglese

          In open vehicle routing problems, the vehicles are not required to return to the depot after completing service. In this paper, the first exact optimization algorithm for the open version of the well-known capacitated vehicle routing problem (CVRP). The algorithm is based on branch – and – cut. Even though the open CVRP initially looks like a minor variation of the standard CVRP, the integer programming formulation and cutting planes need to be modified in subtle ways.

             Computatuonal results are given for several standard test instances, which enables us for the first time to assess the quality of existing heuristic methods, and to compare the relative difficulty of open and closed versions of the same problem.

Formulation and Valid Inequalities

Definition 1 The partially asymmetric CVRP (PACVRP) is the generalization of the CVRP in which the cost of travel c_0i is permitted to be different from c_i0.

Proposition 1 The COVRP and the PACVRP are equivalent.

Symmetric inequalities

Proposition 2 Let   be valid for the CVRP. Then ) ≤ β is valid for the COVRP.

Asymmetric inequalities

Theorem 1 Let  H  V_c (the handle) and T₁ , . . . , T_t    V (the teeth) be such that :

·    Every tooth property intersects with handle, i.e., T_i   H   and  T_i \ H  are non – empty for all i ;

·    If any pair of teeth intersect, then either all vertices in the intersection lie in the handle or all lie outside, i.e., for 1  , either T_i  T_j    H  or  T_i  T_j  H  = .

The Branch – and – Cut Algorithm

A.  Separation of symmetric inequalities

B.  Separation of balancing inequalities

C.  Saparation of mixed strengthened comb inequalities

D.  Separation strategy

Computational Experiments

             Our algorithm has been coded in the C programming language using the Microsoft Visual C++ v. 6.0 compiler. For solving LPs you have used the CPLEX callable library v. 9.0. All ecperiments have been done on a PC with a 1.6 GHz Intel Pentium M processor and 512 MB of RAM running Microsoft Windows XP .