A new variant of a vehicle routing problem: Lower and upper bounds

This paper deals with a combinatorial optimization problem that arises in the design of a semi-automatic system for cutting leather skins called the laser multi-scanner problem (LMSP). The problem occurs in the first part of the cutting process: the parts to be cut are placed on the cutting board by using a new technique, the so-called ‘lasernest’; the outlines of the pieces are projected onto the board by several laser rays. The optimization goal is to assign the pieces to the laser rays such that the maximum length of a laser route is minimal. This can be modelled as a new variant of a vehicle routing problem. Due to severe time restrictions we are only interested in fast heuristic solutions. We construct feasible solutions by opening heuristics, namely, insertion algorithms and greedy variants. In a second step we improve the solutions using local search based on arc exchanges. We evaluate the quality of solutions by lower bounds which are derived from combinatorial and LP-relaxations. Thereby, we examine directed Q-forests on a multi-digraph which have matroid structure. Finally, we present some computational results on real-world data.