Splitting a Giant Tour using Integer Linear Programming

An integration of Lagrangian split and VNS: The case of the capacitated vehicle routing problem

“…Ls is a subgradient method that solves Split(T ) based on a Lagrangian relaxation LB(λ) of this model where λ is a Lagrangian multiplier (see Section 3.2). In a previous contribution (see [12]), LB(λ) was proved to be solvable in polynomial time. Initially, the Lagrangian multiplier is set to a value λ 0 .…”

“…Indeed, an optimal solution to LB(λ) can be found simply by selecting the m smallest ∆ i , assigning 1 to the corresponding y i and 0 to the remaining ones. This can be performed in O(nm) (see [12]).…”

“…Note that LB(λ) has the integral property, which means that the optimal value of LB(λ) is not altered by dropping the integrality conditions on its variables (see Proposition 2.2 in [12]). Therefore, the maximum value attainable by z LB (λ) equals the optimal value of the LP relaxation of Split(T ) when it is feasible (see Theorem 2 in Geoffrion [18]).…”

“…Recently, the partitioning of a giant tour for the multiple traveling salesman problem (mTSP) was formulated as an integer linear program (ILP) (see [12]). It was proven that the obtained formulation is solvable in polynomial time.…”