Irregular stock cutting with guillotine cutsHan Wei, Julia Bennell

NanJing ,China

Summary• Problem definition

• Current practice

• Direct packing approach

– Generate candidate combinations– Evaluate candidate combinations– Evolve layouts in a self adaptive forest

• Example solutions/demo

• Future work

Problem description• Problem arises in the glass cutting industry, in this

instance, specifically for conservatories (glass houses)

Problem description• All pieces are convex

• Single size stock sheet (multiple sizes and off-cuts will be considered in the future)

• Pieces are broken out using guillotine cuts (orthogonal and non-orthogonal)

• No limit to the number of stages/cuts

• Pieces can be continuously rotated

• Demand met exactly

Current practice• Nest pairs of all non-rectangular pieces into a

rectangle

• Select best pairs according to following ratio

• Use standard rectangle bin packing with guillotine constraints

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Pi

Research aimCurrent software encloses two pieces into rectangles

then packs the rectangles to be guillotine cut-able.

Our aim

1. Investigate nesting more than two pieces into rectangles

2. Removing the requirement to nest into rectangles while meeting guillotine constraint.

Direct packing approach overviewIn words …..

Evolve many layouts by recursively combining pieces, or configurations of subsets of pieces, together.

Layouts evolve via a forest structure, where the size is controlled by an acceptance threshold

An accepted combination of pieces becomes a single piece defined by the convex hull of the combination

Pieces must be combined in such a way to result in a feasible layout

Direct packing approach overview1. Generate candidate combinations of two (sets of)

pieces

1. Non overlapping2. Guillotine cut-able

2. Evaluate candidate configurations

3. Evolve solutions in self adaptive forest search

Generating candidate combinationsFor each i in P1, and j in P2, and sliding distance d,

Attach

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Evaluating candidate configurations

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Match algorithm1. Initialise best_i, best_j, best_d, max_U

2. For each i in P1, and j in P2, Attach(P1,P2,i,j)

If

do d = 0, max{0,length(ei)-length(ej)} Slide(d)If Uwf(P1,P2)> max_U

set best_i, best_j, best_d, max_U

3. Attach(P1,P2,best_i,best_j) , Slide(best_d)

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Feasibility1. Given all pieces are

convex and the described attach procedure, the no-overlap and guillotine constraints are clearly met for two original pieces.

2. Once the best match is found the combination is defined by its convex hull, hence combining sets of pieces, defined by their convex hull is the same as 1.

Guillotine cut

More assumptions and definitions

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The maximum population of D given θ and maxg:

At level m, the candidate set of configurations are:

The mth generation of the population is described by:

Example results

Example results

Software demo

Future workExplore the following

• How scalable is the approach

• Sensitivity to parameters; w, θd and θ

• Incorporate full constraints

Additional constraints• Minimum angle of 30°for cuts intersecting an edge

• No more than two cuts can intersect

• Both are permitted if a 20mil gap is added

Thank youQuestions