Solution approaches to the marker layout problem. Kath Dowsland Gower Optimal Algorithms Ltd.

References (see Further details: J.A. Bennell and K.A. Dowsland, ‘A tabu thresholding implementation for the irregular stock cutting problem.’ Int. J. Prod. Res. 37 (1999) J.A. Bennell and K.A. Dowsland, ‘Hybridising tabu search with optimisation techniques for irregular stock cutting’. Management Science 47 (2001) K.A. Dowsland, W.B. Dowsland and J.A. Bennell, ‘Jostling for position: local improvement for irregular cutting patterns.’ J. Opl. Res. Soc. 49 (1998) K.A. Dowsland, S. Vaid and W.B. Dowsland, ‘An algorithm for polygon placement using a bottom-left strategy.’ EJOR 141 (2002) A general survey: K.A Dowsland and W.B. Dowsland, 'Solution Approaches to Irregular Nesting Problems', European Journal of Operational Research 84 (1995) Handling the geometry J.A. Bennell, K.A. Dowsland and W.B. Dowsland,’The irregular cutting stock problem – A new peocedure for deriving the nofit polygon. Computers and OR 28 (2001)

Approaches. 1.Placement policy applied to pieces in a given order. Range from single pass with given ordering to search techniques (e.g. GA) to find ‘optimal’ ordering. 2.Compaction (and/or expansion) of a given layout. Start from (several) random layouts, layouts produced nesting pieces in simpler shapes, or from existing layouts of similar pieces. 3.Neighbourhood searches based on moving one or more pieces at a time. Usually start from random placements with overlapping pieces and use SA, TS etc to find feasible solutions in minimal length.

Each approach has its own advantages BUT each also has drawbacks.

Placement policy applied to pieces in a given order. Only as good as the placement method. A ‘good’ placement method with accurate geometry can be computationally expensive. Sensitive to ordering. A search of the ordering space (e.g. by a GA) combined with good placement policy may be very slow.

Compaction (and/or expansion) of a given layout. Sensitive to starting solution. Requires ‘optimisation’ over non convex space. Result sensitive to choices at optimisation phase e.g. with LP approach need to choose convex sub- region and step size. If starting solution has overlap may not find feasible solution. Optimisation phase often computationally expensive.

Neighbourhood searches. Need to balance feasibility (i.e. removing overlap) with optimality (i.e. minimising length). Many ‘slightly infeasible’ local optima. Random descent approaches e.g. SA tend to perform poorly. Steepest descent approaches need to deal with infinite neighbourhoods.

Observations. Many local optima in neighbourhood search could be made feasible by a chain of moves. Such a chain is often compatible with a compaction / expansion routine. Bottleneck for compaction / expansion is lack of moves that allow pieces to ‘jump over’ each other.

Potential Solution: combine the two approaches. Use a local search approach to reach a (possibly infeasible) local optima and then remove overlap and gaps using a compaction routine. Local search component should be aggressive i.e. able to seek out many good local optima.

Chosen approach. A core algorithm based on tabu thresholding. Avoid infinite neighbourhoods by using a grid of feasible positions. Apply expansion / compaction to sufficiently good local optima throughout the search. Two different compaction approaches were tried; 1. Move pieces back/forward one by one, 2. Use LP routines to move pieces simultaneously.

Problems. Tendency to cycle. Need to split clusters of small pieces. Time wasted failing to remove overlap from similar configurations. Sophisticated compaction slow and sensitive to parameters. Simple modifications were able to overcome all but speed problems.

Experiments. 5 data sets from the literature. 3 widths. evaluation function = length + overlap in x-direction. neighbourhood = move a single piece. no moves = max (5000, 2000 w/o improvement).

Results. Mean % utilization TT Simple comp.TT LP comp. TT onlySeq. Placement (Oliviera)* Seq. placement (Dowsland)SA (Dowsland) TS (Blazewicz)*

What about placement / ordering policies? For predefined single pass orderings and placement policies that can fill ‘behind’ the current packing front ‘big’ first is best. Possibilities: Length, width, or area of enclosing rectangle. Area or perimeter of piece. Aspect ratio of enclosing rectangle or % fit in enclosing rectangle. Better results can be obtained from mixed orderings but searching for a good ordering can be slow.

Jostle: a compromise. Based on the idea that shaking a tube of granular material will tend to flatten the surface. Using initial (random) ordering pack using a leftmost placement policy. Order pieces by rightmost coordinate and pack using a rightmost policy. Order pieces by leftmost coordinate and pack using a leftmost policy. Repeat several times.

The best known layout for this data set was produced by jostle in 1992.

Results. Mean % utilization based on 100 runs of 20 jostles. Jostle is also more consistent than purely random orderings and the distribution of lengths is skewed towards better solutions TT Simple comp.TT LP comp. TT onlySeq. Placement (Oliviera)* Seq. placement (Dowsland)SA (Dowsland) TS (Blazewicz)* Jostle 20 x 1 Jostle 20 x 50

Conclusions. The irregular packing problem is a difficult problem made more complex by the need to handle the geometry efficiently. There are a variety of solution strategies, each with advantages and drawbacks. Solution quality can be improved by intelligent hybridisations and modifications. There is no need to get too sophisticated – simple ideas can give good results!