Cutting stock problem

In operations research, the cutting-stock problem is the where the objective is not to minimise the waste, but to
problem of cutting standard-sized pieces of stock ma- maximise the total value of the produced items, allowing
terial, such as paper rolls or sheet metal, into pieces each order to have a different value.
of specified sizes while minimizing material wasted. It In general, the number of possible patterns grows expo-
is an optimization problem in mathematics that arises
nentially as a function of m, the number of orders. As
from applications in industry. In terms of computational the number of orders increases, it may therefore become
complexity, the problem is an NP-complete problem re- impractical to enumerate the possible cutting patterns.
ducible to the knapsack problem. The problem can be
formulated as an integer linear programming problem. An alternative approach uses delayed column-generation.
This method solves the cutting-stock problem by starting
with just a few patterns. It generates additional patterns
1 Formulation and solution ap- when they are needed. For the one-dimensional case, the
new patterns are introduced by solving an auxiliary op-
proaches timization problem called the knapsack problem, using
dual variable information from the linear program. The
The standard formulation for the cutting-stock problem knapsack problem has well-known methods to solve it,
(but not the only one) starts with a list of m orders, each such as branch and bound and dynamic programming.
requiring qj , j = 1, . . . , m pieces. We then construct The Delayed Column Generation method can be much
a list of all possible combinations of cuts (often called more efficient than the original approach, particularly as
“patterns”), associating with each pattern a positive inte- the size of the problem grows. The column generation
ger variable xi representing how long each pattern is to approach as applied to the cutting stock problem was pi-
be used. The linear integer program is then: oneered by Gilmore and Gomory in a series of papers
published in the 1960s.[1][2] Gilmore and Gomory showed
that this approach is guaranteed to converge to the (frac-
∑
n
tional) optimal solution, without needing to enumerate all
min ci xi
the possible patterns in advance.
i=1
A limitation of the original Gilmore and Gomory method
∑
n
s.t. aij xi ≥ qj , ∀j = 1, . . . , m is that it does not handle integrality, so the solution may
i=1 contain fractions, e.g. a particular pattern should be pro-
xi ≥ 0 duced 3.67 times. Rounding to the nearest integer of-
ten does not work, in the sense that it may lead to a
where aij is the number of times order j appears in pat-
sub-optimal solution and/or under- or over-production of
tern i and ci is the cost (often the waste) of pattern i . The
some of the orders (and possible infeasibility in the pres-
precise nature of the quantity constraints can lead to sub-
ence of two-sided demand constraints). This limitation
tly different mathematical characteristics. The above for-
is overcome in modern algorithms, which can solve to
mulation’s quantity constraints are minimum constraints
optimality (in the sense of finding solutions with mini-
(at least the given amount of each order must be pro-
mum waste) very large instances of the problem (gener-
duced, but possibly more). When ci = 1 the objective
ally larger than encountered in practice[3][4] ).
minimises the number of utilised master items and, if the
constraint for the quantity to be produced is replaced by The cutting-stock problem is often highly degenerate, in
equality, it is called the bin packing problem. The most that multiple solutions with the same waste are possi-
general formulation has two-sided constraints (and in this ble. This degeneracy arises because it is possible to move
case a minimum-waste solution may consume more than items around, creating new patterns, without affecting the
the minimum number of master items): waste. This gives rise to a whole collection of related
problems which are concerned with some other criterion,
such as the following:
∑
n
qj ≤ aij xi ≤ Qj , ∀j = 1, . . . , m • The minimum pattern count problem: to find
i=1
a minimum-pattern-count solution amongst the
This formulation applies not just to one-dimensional minimum-waste solutions. This is a very hard prob-
problems. Many variations are possible, including one lem, even when the waste is known.[5][6] There

1
2 4 CUTTING-STOCK PROBLEM IN PAPER, FILM AND METAL INDUSTRIES

is a conjecture that any equality-constrained one- this case the minimum number of patterns with this level
dimensional instance with n orders has at least one of waste is 10. It can also be computed that 19 different
minimum waste solution with no more than n + 1 such solutions exist, each with 10 patterns and a waste of
patterns. No upper bound to the number of patterns 0.401%, of which one such solution is shown below and
is known either, examples with n + 5 are known. in the picture:

• The minimum stack problem: this is concerned with

the sequencing of the patterns so as not to have
too many partially completed orders at any time.
This was an open problem until 2007, when an effi- 3 Classification
cient algorithm based on dynamic programming was
published.[7]
Cutting-stock problems can be classified in several
ways.[8] One way is the dimensionality of the cutting: the
• The minimum number of knife changes problem
above example illustrates a one-dimensional (1D) prob-
(for the one-dimensional problem): this is con-
lem; other industrial applications of 1D occur when cut-
cerned with sequencing and permuting the patterns
ting pipes, cables, and steel bars. Two-dimensional (2D)
so as to minimise the number of times the slitting
problems are encountered in furniture, clothing and glass
knives have to be moved. This is a special case of
production. Not many three-dimensional (3D) applica-
the generalised travelling salesman problem.
tions involving cutting are known; however the closely
related 3D packing problem has many industrial appli-
cations, such as packing objects into shipping contain-
2 Illustration of one-dimensional ers (see e.g. containerization - the related sphere packing
cutting-stock problem problem has been studied since the 17th century (Kepler
conjecture)).
A paper machine can produce an unlimited number of
master (jumbo) rolls, each 5600 mm wide. The following
13 items must be cut: 4 Cutting-stock problem in paper,
film and metal industries
Industrial applications of cutting-stock problems for high
production volumes arise especially when basic material
2.1 Solution
is produced in large rolls that are further cut into smaller
units (see roll slitting). This is done e.g. in paper and
plastic film industries but also in production of flat met-
als like steel or brass. There are many variants and ad-
ditional constraints arising from special production con-
straints due to machinery and process limits, customer
requirements and quality issues; some examples are:

• Two-stage, where the rolls produced in the first stage

are then processed a second time. For instance, all
office stationery (e.g. A4 size in Europe, Letter size
in US) is produced in such a process. The compli-
cation arises because the machinery in the second
stage is narrower than the primary. Efficient utilisa-
tion of both stages of production is important (from
an energy or material use perspective) and what is
efficient for the primary stage may be inefficient for
the secondary, leading to trade-offs. Metallised film
(used in packaging of snacks), and plastic extrusion
A minimum-waste solution, sequenced to minimise knife changes,
on paper (used in liquid packaging, e.g. juice car-
shown as small white circles tons) are further examples of such a process.

There are 308 possible patterns for this small instance. • Winder constraints where the slitting process has
The optimal answer requires 73 master rolls and has physical or logical constraints: a very common con-
0.401% waste; it can be shown computationally that in straint is that only a certain number of slitting knives
 3

are available, so that feasible patterns should not 7 History

contain more than a maximum number of rolls. Be-
cause winder machinery is not standardised, very The cutting stock problem was first formulated by
many other constraints are encountered. Kantorovich in 1939.[10] In 1951 before computers be-
came widely available, L. V. Kantorovich and V. A.
• An example of a customer requirement is when a
Zalgaller suggested[11] solving the problem of the eco-
particular order cannot be satisfied from either of
nomical use of material at the cutting stage with the help
the two edge positions: this is because the edges of
of linear programming. The proposed technique was
the sheet tend to have greater variations in thickness
later called the Column Generation method.
and some applications can be very sensitive to these.

• An example of a quality issue is when the master roll

contains defects that have to be cut around. Expen- 8 Software
sive materials with demanding quality characteris-
tics such as photographic paper or Tyvek have to be
carefully optimised so that the wasted area is min- 9 References
imised.
[1] Gilmore P. C., R. E. Gomory (1961). A linear program-
• Multi-machine problems arise when orders can be ming approach to the cutting-stock problem. Operations
produced on more than one machine and these ma- Research 9: 849-859
chines have different widths. Generally availability
[2] Gilmore P. C., R. E. Gomory (1963). A linear program-
of more than one master roll width improves the ming approach to the cutting-stock problem - Part II. Op-
waste considerably; in practice however additional erations Research 11: 863-888
order splitting constraints may have to be taken into
account. [3] Goulimis C (1990). Optimal solutions for the cutting stock
problem. European Journal of Operational Research 44:
• There is also a semi-continuous problem, where the 197-208
produced rolls do not have to be of the same diam-
[4] de Carvalho V (1998). Exact solution of cutting stock prob-
eter, but can vary within a range. This typically oc-
lems using column generation and branch-and-bound. In-
curs with sheet orders. This is sometimes known as ternational Transactions in Operational Research 5: 35–
a 1½ dimensional problem. This variant also occurs 44
in the production of corrugated fiberboard, where
it is called, somewhat confusingly, the corrugator [5] S. Umetani, M. Yagiura, and T. Ibaraki (2003). One di-
scheduling problem. mensional cutting stock problem to minimize the number of
different patterns. European Journal of Operational Re-
• In the metals industry one key difference is that typ- search 146, 388–402
ically the master rolls are produced earlier and are
[6] A. Diegel, E. Montocchio, E. Walters, S. van Schalkwyk
generally different from each other (both in terms of and S. Naidoo (1996). Setup minimizing conditions in the
width and length). Therefore there are similarities trim loss problem. European Journal of Operational Re-
with the multi-machine problem mentioned above. search 95:631-640
The presence of length variations creates a 2D prob-
lem, because waste can occur both width-wise and [7] Maria Garcia de la Banda, P. J. Stuckey. Dynamic Pro-
length-wise. gramming to Minimize the Maximum Number of Open
Stacks. INFORMS Journal on Computing, Vol. 19, No.
4, Fall 2007, 607-617.

5 Cutting-stock problem in the [8] Wäscher, G.; Haußner, H.; Schumann, H. An Improved
Typology of Cutting and Packing Problems. European
glass industry Journal of Operational Research Volume 183, Issue 3,
1109-1130
The guillotine problem is a problem of cutting sheets of
glass into rectangles of specified sizes, using only cuts that [9] Raffensperger, J. F. (2010). “The generalized assort-
ment and best cutting stock length problems”. Inter-
continue all the way across each sheet.
national Transactions in Operational Research 17: 35.
doi:10.1111/j.1475-3995.2009.00724.x.

6 The assortment problem [10] L. V. Kantorovich Mathematical methods of organizing

and planning production. Leningrad State University.
1939
The related problem of determining, for the one-
dimensional case, the best master size that will meet given [11] Kantorovich L. V. and Zalgaller V. A. . (1951). Calcula-
demand is known as the assortment problem.[9] tion of Rational Cutting of Stock. Lenizdat, Leningrad.
4 11 EXTERNAL LINKS

10 Further reading
• Chvátal, V. (1983). Linear Programming. W.H.
Freeman. ISBN 978-0-7167-1587-0.
• Hatem Ben Amor, J.M. Valério de Carvalho, Cut-
ting Stock Problems in Column Generation, edited
by Guy Desaulniers, Jacques Desrosiers, and Mar-
ius M. Solomon, Springer, 2005, XVI, ISBN 0-387-
25485-4

11 External links
• European Special Interest Group on Cutting &
Packing
• A rudimentary brute-force algorithm for cutting
stock
• Bin Packing and Cutting Stock Solver Algorithm
 5

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