Applying Constraint Logic Programming to the Resolution of Nesting Problems

Nesting problems are combinatorial optimisation problems where one or more pieces of material or space must be divided into smaller irregular pieces, minimising the waste. In OR, the state-of-the-art in nesting problems is to solve them by means of heuristic algorithms; the use of mixed integer programming models can handle only rather small instances. The satisfaction of geometric constraints is at the core of nesting problems. We propose here a CLP formulation of the problem that makes use of the geometric constraints generated by the use of nofit polygons. This approach reveals itself as very flexible with respect to the use of problem features for extra constraints and control of the search. Ongoing work involves comparing solutions obtained with CLP with both integer programming and heuristic approaches. 1 The nesting problem The nesting problem belongs to the more generic class of combinatorial optimization problems, the cutting and packing problems. In these problems, one or more pieces of material or space must be divided into smaller pieces. The objective is usually to minimize the waste, i.e. the portion of the big pieces that is not used to produce small pieces. Nesting problems are a subclass of the cutting and packing problems where only two dimensions are relevant, the length and the width, the other dimension being common to all the pieces. They are characterised by the irregular, non-rectangular shape of the small pieces. The variant of the problem that we are going to deal with considers only one big piece, the plate, with fixed width and infinite length. The objective is the minimisation of the length of the plate that is used to place a given set of pieces. Both pieces with convex and non-convex shapes will be considered. A more detailed description of nesting problems and cutting and packing problems can be found in [5]. Several approaches [4, 3, 8, 7] have been proposed for the resolution of nesting problems. However none of them is able to generate global optimal solutions. Solution techniques range from simple heuristics to local optimization techniques, including, more recently, meta-heuristics. Even for small instances of the problem the mathematical programming model of the problem is not solvable in a reasonable amount of time.