1. Nesting
by Shachaf Ben Jakov

2. Overview Nesting definition Nesting categories (related problems)
Complexity analysis
Solution types (placement methods)
Constraints
7/21/2019

3. Nesting
Used to describe a wide variety of two-dimensional cutting and packing problems.
They all involve a non-overlapping placement of a set of irregular two dimensional shapes within some region of two dimensional space
The objective may vary.
7/21/2019

4. Categories Decision problem: Knapsack problem: Bin packing problem:
Decide whether a set of shapes fit within a given region.
Knapsack problem:
Given a set of shapes and a region, find a placement of a subset of shapes that maximizes the utilization (area covered) of the region.
Bin packing problem:
Given a set of shapes and a set of regions, minimize the number of regions needed to place all shapes.
Examples:
Knapsack problem – animal hides in leather industry
Strip packing problem – cloth-strip in textile industry
7/21/2019

5. Categories cont. Strip packing problem: Repeated pattern problem:
Given a set of shapes and a width W, minimize the length of a rectangular region with width W such that all shapes are contained in the region.
Repeated pattern problem:
A variant of the strip packing problem, where the packing layout is reused repeatedly in 1 or 2 directions.
Examples:
Knapsack problem – animal hides in leather industry
Strip packing problem – cloth-strip in textile industry
7/21/2019

6. Complexity 1D bin-packing: RECT-PACK:
Given a set of integers 𝑋, a bin capacity 𝐵 and an integer 𝐾, determine whether we can partition 𝑋 into 𝐾 disjoint sets such that the sum of integers in each set is at most 𝐵.
RECT-PACK:
Determine whether a given set of integer rectangles (not necessarily identical) can be placed (without rotation) inside a rectangular region without overlapping.
1D bin-packing reduction to RECT-PACK:
Convert each integer 𝑥∈𝑋 into a rectangle of height 1 and width 𝑥.
Decide whether the rectangles can be placed in a rectangle region of width 𝐵 and height 𝐾.
Therefore, RECT-PACK is NP-complete.
7/21/2019

7. Complexity cont. 3-Partition problem:
Given a multiset of 3𝑛 integer numbers, decide whether they can be split into triples of equal sum.
This problem is NP-complete.
3-Partition reduction to 1D bin-packing:
Denote the sum of numbers as 𝑛𝐵.
Decide whether the 3𝑛 numbers can be partitioned into 𝑛 bins of size 𝐵.
Therefore, bin-packing is NP-complete.
7/21/2019

8. Some definitions Stencil – a piece/shape/polygon to be packed.
Material – the packing region.
Placement – the positioning of stencils on the material.
Legal placement – stencils don’t overlap, and are placed within the material’s limits.
7/21/2019

9. Some definitions cont. NFP – No-Fit-Polygon 𝑄 𝑃
A polygon which describes the legal/illegal placements of one polygon in relation to another.
Correlates with the Minkowski sum 𝑄⊕ −𝑃 . 𝑄 𝑃
Reference point
7/21/2019

10. Solution Types Legal placement methods Relaxed placement methods
Never violate the overlay constraint.
Relaxed placement methods
Allow overlay as part of the solution process.
A legal placement is achieved when overlay is 0.
7/21/2019

11. Legal Placement Methods
Usually sequential:
Determine a sequence of stencils (randomly or sorting according to some measure).
Place the stencils with some first/best fit algorithm (typically on the contour of the placement done so far).
Evaluate the quality of the placement, and if necessary, restart with a different sequence (randomly or using some heuristic).
Non-sequential are very similar to the other placement method.
“Fast and dumb”.
Another possibility is to pick some initial stencil (based on measurements), and then iteratively add stencils, picking the next best one each time.
7/21/2019

12. Legal Placement Methods - Simulation
7/21/2019

13. Legal Placement Methods - Examples
Art (1966) – First sequential algorithm.
Blazewicz (1993) – Non-sequential algorithm, uses a meta heuristic function.
Heistermann and Lengaur (1995) – Approximates stencils. Handles irregular material, quality zones.
Oliveira et al. (2000) – TOPOS – A combination of 126 algorithms. Sorts/Best-fits stencils based on length, area and convexity.
Art (1966) – disregard small stencils, use stencils’ convex counterpart, use placement’s envelope (NFP), do bottom-left packing, use meta-stencils (not implemented).
Adamowicz and Albano (1976) – rotate and cluster stencils (using NFP) into rectangles, then pack these.
Albano and Sappupo (1980) – Like Art’s, but handles non-convex stencils and supports a set of rotations.
Blazewicz (1993) – improves an initial placement using translations, rotations and swaps.
Dowsland et al. (1998) – ‘Jostling for position’, uses NFP and splits stencils into x-convex subparts.
Oliveira et al. (2000) – the basic algorithm is leftmost placement without hole filling. Sequence is either sorted by stencil’s length, area and convexity, or is determined iteratively by best fit. Takes a LOT of time.
Gomes and Oliveira (2002) – improved TOPOS, change sequence order using 2-exchanges, focus on finding good order using neighborhood and first/best/random local search strategies. Takes EVEN MORE time.
7/21/2019

14. Relaxed Placement Methods
Steps:
Construct an initial placement (can be random, may use legal placement methods).
Introduce a set of moves (translations, rotations, flips) to define the neighborhood of solutions.
Search the neighborhood of the current placement for a minimum overlap, by iteratively improving the current placement (thus decreasing the total overlap).
Searching the neighborhood involves using a meta heuristic method.
“Slow but smart”.
7/21/2019

15. Relaxed Placement Methods - Simulation
Start with an Initial placement.
Then iteratively search the neighborhood for an improved placement until a solution is found.
7/21/2019

16. Relaxed Placement Methods - Examples
Lutfiyya et al. (1992) – Fine tunes the Simulated Annealing techniques.
Heckmann and Lengauer (1995) – SA in 4 stages. Approximates polygons.
Jakobs (1996) – Genetic algorithm. Packs bounding rectangles.
Egeblad et al. (2001) – Guided Local Search.
Lutfiyya et al. (1992) – Cost function maximizes edgewise adjacency between polygons, overlap is determined using a raster model.
Jain et al. (1992) – SA, Addresses the repeated pattern problem, Changing the repeat distance is part of the neighborhood.
Oliveira et al. (1993) – Raster and polygonal variants of SA, neighborhood only allows translation.
Heckmann and Lengauer (1995) – 4 stages: Rough placement, Eliminate overlaps, Fine placement with approximates, Fine placement with originals.
Theodoracatos and Grimsley (1995) – SA. Packing circles and polygons (separately).
Bennell and Dowsland (1999) – Tabu Thresholding (Tabu Search variant – worsening is allowed, never retreating). Uses horizontal intersection depth.
Egeblad et al. (2001) – Fast neighborhood search. Leaves local minima/maxima.
7/21/2019

17. Considering Constraints
Fixed positions
Material shape
Correspondence between stencils (e.g. vertical/horizontal line).
Pattern symmetries
Limited rotation angles
Quality areas and requirements
Margins between polygons
Folding
Fixed positions – don’t allow translations for the given stencil.
Material shape – make a stencil corresponding to a hold and fix its position.
Correspondence between stencils – force the stencils to translate together when searching the neighborhood.
Pattern symmetries – find a minimum along a legal set of translations instead of a true minimum.
Limited rotation angles – e.g. in fabric. Limit rotation.
Quality areas – divide stencils into a set, corresponding to each quality areas. Move and rotate all set items as a single unit.
Margins between polygons – Required by cutting technologies. Enlarge the polygons, round the corners and remove self intersections.
Folding – symmetric stencils can be placed at the folding line.
7/21/2019

18. Questions?
7/21/2019