Surface cutting problems in two dimensions are considered for nonrectangular items. An exact solution method is discussed. Outlines of several possible heuristic algorithms are also presented. For the heuristic methods a first approximation to the optimal solution is obtained by encompassing each item by a rectangle and then using some available strategy for this standard problem. Different approaches are then suggested for more accurate methods

Murray, J.

English

Mathematics in Industry Information Service Eprints Archive

59
OPTIMAL SURFACE CUTTING
Surface cutting problems in two dimensions are considered for nonrectangular
items. An exact solution method is discussed. Outlines of several possible heuris-
tic algorithms are also presented. For the heuristic methods a first approximation
to the optimal solution is obtained by encompassing each item by a rectangle and
then using some available strategy for this standardproblem. Different approaches
are then suggested for more accurate methods.
1. Introduction
ProActive Technology Pty Ltd is a small computer software company situated in a
southern suburb of Sydney. One of the application programs it has developed involves
determining the cutting strategy of one dimensional sections to satisfy given orders and
so that the associated trim loss is kept to a minimum. The company's request of the
Study Group was to evaluate methods for the corresponding two dimensional problem.
There are many applications of these types of problems to industry - paper trimming,
bin packing, container loading, capital budgeting - and several classifications according
to the specifics of the problem. For a general overview of cutting and packing problems
and the relevant literature the reader should refer to Dyckhoff (1990).
The one dimensional problem for cutting sections can be described as follows. A
company has in stock several standard length pipe sizes. For example it may have 100
lengths of 4 metre pipe, 80 lengths of 6 metre pipe and 20 lengths of 10 metre pipe.
Orders are placed with the company for 15 lengths of 0.3 metre pipe, 27 lengths of 0.7
metre pipe, 32 lengths of 0.8 metre pipe and 16 lengths of 1.2 metre pipe. How should
the company cut the pipes from the stock lengths so as to minimise the wastage? The
wastage is defined to be the end pieces of pipe that remain after several ordered lengths
have been cut from that pipe. If the endpieces are not too small they can be added to
stock, but if they are below a certain size they must be discarded.
The two dimensional problem is similar except instead of dealing with lengths we
allow two dimensional objects. Just about all applications consider rectangular stock
items, and many only have rectangular ordered items as well. For instance, the glass
cutting industry cuts rectangular pieces of glass from rectangular stock sheets. The
cutting process for that application is special since the glass is cut by guillotines. This
type of cut is used in the paper industry as well. Figure 1 illustrates various cuts.
In this discussion paper we will consider rectangular sheets, general ordered items,
and general cuts.
60 ProActive Technology Ply Ltd
-
-
Figure 1: A 2-stage guillotine pattern and a non-guillotine pattern
The variety of applications and the size of the possible problems can be extremely
large, especially when we are dealing with nonrectangular shapes. Let us therefore state
the range of problems we will consider. In line with the applications that ProActive
Technology meets in the metal plate cutting area, we assume the following
1. The problems are small to medium in size; approximately 100 items requiring 10
to 20 sheets.
2. There are no restrictions on the cutting patterns, so we are not limited to guillotine
cuts. In the metal plate industry the items are cut from the plates using oxy-
acetylene torches.
3. The ordered items can be any shape. These may include rectangular shapes as
well as several types of nonrectangular ones.
4. There will only be one size of stock sheet. This is in general not true. However
if we pack the items appropriately with a preference for packing into rectangular
regions on the plate, then, at the end, instead of ordering that complete sheet, the
smallest plate that encompasses the used area is ordered. So after a packing has
been determined, a variety of plate sizes are used making this a less restrictive
assumption.
Even with these assumptions we still face a daunting task. This is based on the
knowledge that even the rectangular problem is NP-complete. NP-complexity refers to
the computational complexity of algorithms. Most efficient algorithms in combinatorial
optimisation are those where the number of computations required to determine the
optimal solution is polynomially related to the size of the problem. Such algorithms are
called polynomial algorithms and they have the property that, as larger problems are
encountered, the time taken to compute a solution grows at a reasonable rate.
Optimal surface cutting 61
The worst problems as far as computational complexity are those that require expo-
nential algorithms, that is the number of computations required to solve the problem for
a worst case scenario is an exponential of the problem size. Since exponentials grow
exceedingly quickly, a small increase in problem size may make the problem computa-
tionally intractable.
NP-complete problems fall in between these two classes except that there are no
known polynomial algorithms that will solve the problems. These may exist but at
present we must rely on exponential algorithms or use heuristic algorithms where the
optimal solution might not be found, but perhaps a good solution will be determined in
a reasonable amount of time.
With this knowledge, we see that a search for an optimal solution will in general fail.
The hope is that the assumptions we have described above will make the calculation of
a good solution computation ally reasonable.
2. Solution methods
One method for these problems encountered in the literature generates patterns and
then determines the best collection of patterns to use. This is based on the succesful
approach for one dimensional cutting problems proposed by Gilmore & Gomory (1961).
A pattern contains a specific number of ordered items that will be cut from a sheet in a
prescribed manner. For example a pattern might contain 3 pieces of item 1, 2 pieces of
item 3, and 8 pieces of item 6. The way these are cut from the sheet is also given. The
cost of the pattern equals the amount of wastage.
Let us describe this formally. Define
Cj = the cost ofpatternj
Xj = the number of times pattern j is used to fill an order
aij = the number of times item i is used in pattern j
d, = numbers of item i ordered
The optimisation problem describing the cutting stock problem is then
minimise 2:CjXj
j
2:aijXj ~ dj
Xj ~ 0
Xj integer
subject to
62 PttiActive Technology Ply Ltd
One of the first difficulties encountered with this method is determining the patterns.
Some simple patterns may be generated but a good pattern may be difficult to find. If
the method is to work well it requires a large number of patterns to choose from. Most
of these patterns however will be discarded from the final solution.
For industries that require many pieces of each item and where the number of items
is large, this is a reasonable approach. For the problem types we investigate however,
the great deal of initial work in generating the patterns cannot be justified. Hence we
did not consider this to be a fruitful technique.
3. An exact method
One approach that was discussed was to convert the placement of items on sheets to
an optimisation problem. The model can be defined precisely but to give the basic idea
define the following: for each item A (I) let y(l) represent a measure of the placement
of the item on the plate. This may include several components such as height, unused
space generated, etc. Then the problem can be stated as
minimise
subject to Vi,j (1)
We can specify the objects by their corner points, approximating any rounded edges by
piecewise linear ones. The constraint (1) would be implemented via the separation of
the corner points of each object. Except for the approximation of items with rounded
edges by ones with piecewise linear edges, this method: in combination with a gen-
eral optimisation algorithm: would be exact. However it would entail large computing
needs.
Some of the properties of the problem that would contribute to the lengthy comput-
ing time are
Rotation: each item must be considered in all orientations.
Constraint feasibility: no two items can overlap so there is considerable constraint
checking involved.
Local optima: because of the constraints there will be many local optima which will
repeatedly stop the search for the true minimum.
Nevertheless this approach is discussed for rectangular items by Beasley (1985).
For small problems it may very well be the best approach and even for large problems
may contribute by its application to solve small subproblems.
Optimal surface cutting 63
4. Heuristic methods
Rather than trying to solve the problem exactly, we can approach our task with the
more modest aim of trying to find an acceptable solution in a reasonable amount of time.
The methods discussed below are heuristic and have a similar structure to each other but
with different implementations. The steps for these methods follow the template given
below
Step 0 specify the items.
Step 1 presort the sequence of items in some way such as
a) into groups with area approximately the same as the area of the sheets, or
b) by area, or diameter, or ...
Step 2 place the items onto the sheets using some method.
Step 3 evaluate the placement according to some objective.
Step 4 perturb the sequence and return to Step 2.
This strategy can be implemented in two ways. In both of these however, we try to
fit each item by either a rectangle or a union of rectangles. The preference for rectan-
gular items stems from three reasons. First there are many heuristic methods that are
available in the literature for rectangular regions while those for nonrectangular regions
are few. Second, it is quite easy to determine good placements of rectangular items
next to each other; simply place an edge of one item next to the edge of the other item.
On the other hand this is not as straightforward for nonrectangular shapes, and since
a great deal of the method relies on close packing of items this causes some difficulty.
Third, rectangular regions have only two orientations - on their side or standing up -
while nonrectangular items can have an infinite number of orientations. Fitting an item
according to a particular orientation may not be as good if it were oriented in a different
way. When all different orientations for all the items are considered the computation
becomes large.
Our guide as to "good" placement of items onto a sheet is the minimisation of waste.
However not all unused area is waste. We started with the assumption that the stock
sheets are all one size; in the end however, we order the smallest size sheet available
that covers the items placed on the standard one. Hence we do not want to count as
wasted space any part of the sheet that can be omitted when the sheet is purchased.
Therefore when we are considering placing items on a sheet, we try to put them into
a rectangular subregion of the plate. To achieve this, we put them as far to the left as
64
----------, ,,,,,,
_______ - __ J,,,,----,..-~,,r----'
I '
I '
I 'I ,
I I
I I
PraActivc Technology Pty Ltd
,,,,,,
L _
,
I
I,
------~, ,
, I, ,
, I, ,, ,, ,, ,, ,
r----
I,,,,
IL _
Figure 2: Good and bad placements on a sheet
,/
.......v
..,..-V
7'
.......7
Figure 3: Converting a triangle to a union of squares
possible and as far to the bottom as possible. Combining this approach with some other
appropriate strategies helps achieve our aim.
Implementation 1
The first approach we consider is where we immediately try to specify the items in a
rectangularised fashion as closely as possible. For this we choose a grid on which to lay
each nonrectangular item. Those squares of the grid containing any part of the item are
then used for the description of the item. The new description then consists of a union of
squares. For example if the item were a right angle triangle with the two sides forming
the right angle running parallel to the lines of the grid, then the hypotenuse would be
replaced by a sequence of vertical and horizontal segments - it would now look like a
set of stairs.
Now that the items have been rectangularised, we must determine their placement
on the sheets. If we are using method (a) in Step 1, then the items within a group can
be placed on the sheets in any order. If we are using method (b), then we place them on
the sheets according to whichever is next in line, if at all possible.
Optimal surface cutting 65
The placement strategy can be any of the heuristics found in the literature. For
example we could use a bottom left heuristic where each item is placed to the bottom of
the sheet and as far left as possible. Or we could use a shelf heuristic where the sheet
is divided up into shelves of various height and the items are placed on the lowest shelf
where they will fit and as far to the left as possible.
Since we are here allowing the items not to be rectangles but more general shapes
then we should decide how to place the items next to each other. For the simplest case
when we have two items only on the sheet with the first placed at the bottom left corner
and the second on the bottom of the sheet as well, we can put them together so as to
minimise the intervening space, or maximise it, or according to some other strategy.
Trying to minimise the space seems an obvious approach because we then have as much
space remaining in one connected section as we can obtain. However the other approach
of maximising the intervening space between two items that are placed next to each other
also has its advantages. At the end of the entire placement we will probably have some
small items that could fit into these large gaps. Making these gaps as usable as possible
makes the task easier.
-u
Figure 4: Maximising and minimising spaces
Implementation 2
Another approach that can be tried is to only use rectangles. As an initial approxi-
mation to the optimal solution, this is not a bad start. Let us consider the process a metal
plate cutting company follows to bid for a contract for supplying the components for say
a bridge. First they are given rough plans ofthe bridge which an engineer studies to de-
termine what components are needed. These are not necessarily included in the plans.
Given the engineer's assessment, an estimate is made of the amount of metal plate and
sections needed. This is not accurate because a detailed analysis would require a signifi-
cant investment in time by the engineer which would be wasted if the company's tender
for the job failed. So it is an educated guess based on experience and with a margin for
66 ProActive Technology Ply Ltd
Figure 5: Enclosing a triangle in a rectangle
error. Using rectangles for the items at this initial stage is probably as accurate or more
so than what is done in practice.
If the company wins the tender they use detailed plans to get an accurate specifica-
tion of the shape and size of each component. They then estimate as precisely as they
can the amounts of plates and sections they will need. This also builds in an estimate of
the waste that will be generated by their cutters. The profit on the contract will be the
difference between the amount they submitted in the tender and the cost of the metal
with labour that is actually used.
At this last stage. if they can reduce their estimate of the plate and section required
then they will increase their profit - so it is important to have a good surface cutting
scheme available. For this stage. we therefore want to improve on the initial estimate as
much as possible. In terms of the scheme proposed here. we try to do this by considering
groups of items enclosed in single rectangles. rather than by simply encompassing each
nonrectangular item in a rectangle.
For example. suppose two of the components are a triangle and a circle. The initial
process places each in a rectangle and solves that problem via some heuristic. The
second process may combine these two items into one rectangle. and along with other
combinations. uses an heuristic to obtain a solution. Since the space used for this one
rectangle will be less than that used by one rectangle for each item. there should be some
improvement.
Taking several attempts at this combination process and resolving will in general
find a better solution than the original one.
So the strategy is as follows:
Step 1 place rectangles around each item.
Step 2 use a plate cutting method for rectangles to find a solution. If an appropriate
Optimal surface culling 67
o
Figure 6: Enclosing a triangle and a circle in a rectangle
convergence condition is satisfied then Stop. Otherwise go to Step 3.
Step 3 group items together into single rectangles. Go to Step 2.
Steps 1 and 2 give a suitable answer for the initial tendering stage. Canying on with
several passes through Step 3 will in general improve the solution and give the best
cutting strategy.
Orientation of objects is a smaller concern here than in previous strategies because
the only time it occurs is when we place individual items or groups of items in rectangles.
This is a smaller scale problem than for the other approaches.
5. Discussion
This is an easily stated problem but - as so often occurs in combinatorial optimi-
sation problems - it hides a difficult computational problem. We have listed several
strategies that could be implemented; until these methods have been programmed and
tested on representative examples we can make few assertions about which will perform
best. In fact it may very well be that several of these strategies should be combined to
take advantage of each of their strengths and cover more possibilities.
The Study Group has therefore provided suggestions that ProActive Technology
might follow, even though most of the work remains to be done.
Acknowledgements
Many people contributed to the discussion and made suggestions of strategies for
this problem. Our guides in this were Paul Saunders and Tim Young from ProActive
Technology and also Wal Stinson from the metal plate cutting industry. The other sub-
stantial contributors were Roderick van Beelen, Natashia Boland, Andreas Emst, John
68 ProActive Technology Ply Ltd
Hewitt, Mohan Krishnamoorthy, Nian Li, John Lions, Fawang Liu, Rhys Newman,
Duncan Roper, Tony Watts, Mark Westcott, Kevin Wilkins, Matt Williams and Simone
Zmood.
References
J .E. Beasley, "An exact two-dimensional non-guillotine cutting tree search procedure",
Op. Res. 33 (1985),49-64.
H. Dyckhoff, "A typology of cutting and packing problems", Eur. J. Op. Res. 44
(1990), 145-159.
P.C. Gilmore & R.E. Gomory, "A linear programming approach to the cutting-stock
problem", Op. Res. 9 (1961), 849-859.


Optimal surface cutting

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