tioned as well as many more methods.
(Silva et al., 2010), consider waste, but lack the
third dimension in their problem formulation. They
also mention the value of the surplus material on a
conceptual level, but do not show it in their problem
formulation. (Burke et al., 2011), present an iterative
packing methodology based on squeaky wheel opti-
mization. (Furini and Malaguti, 2013), make similar
formulations as previously found in the literature, but
focus on the run-time of the solution.
The recent results from (Andrade et al., 2013),
lay the foundation for our problem formulation.
(Andrade et al., 2013), investigate two-stage two-
dimensional guillotine cutting stock problems with
usable leftover. We will extend their formulation for
two-stage three-dimensional guillotine cutting stock
problems with usable leftover. The term “stage” will
be explained when we describe the constrains that are
imposed by our production machinery.
There are three properties that distinguish our
problem formulation from others. First, we allow the
items to be rotated.
Second, we consider the case in which leftover
material is to be reused in subsequent production cy-
cles. I.e., after one production cycle, leftover material
is added to the set of slabs.
Third, we solve the problem in three dimensions.
On the one hand, this is needed to calculate the
weight, which is required in our objective function.
On the other hand, this opens up for the possibility to
cut an item from a slab that is thicker than necessary,
if the value of the leftover material permits or even
dictates this decision.
The remainder of the present paper is organized
as follows. In Section 2, we present the problem we
want to solve and the model we use. In Section 3, we
propose an optimization program to solve the prob-
lem. In Section 4, we discuss some examples and
their solutions. Finally, we summarize our results in
Section 5.
2 PROBLEM FORMULATION
AND MODEL DESCRIPTION
The overall goal is to fulfill customer orders of blocks
of steel. We call the ordered blocks of steel items.
These items are cut from larger slabs of steel. As the
size of an item is defined by the customer, the size is,
in general, different and arbitrary from item to item.
The problem we solve in this work is the decision
which item is cut from which slab and how items are
geometrically placed on each slab. Due to the produc-
tion process, or technical and economic restrictions,
(a) (b) (c)
Figure 1: Guillotine cutting in (b) and (c).
these problems can have an abundance of constraints.
It is not important that the material we are working
with is steel. However, the machines that are used
for cutting the slabs create certain constraints in our
problem formulation.
In general, when items are cut from slabs, the orig-
inal slab is cut up into items and surplus material. The
surplus material can either be useful in the future or it
is so small that it is thrown away. If it is kept we call
it leftover material and it will be placed in the set of
slabs for future usage. If it is thrown away we call it
scrap material.
One of our goals is to use surplus material more
frequently than new slabs. Otherwise, the slabs in
stock could increase over time. Our solution for this
problem is to attach a value per kilogram to all the
slabs. The less a slab weights in total, the smaller its
value per kilogram.
The machines cutting the slabs can only do full
straight cuts, parallel to the edges of the slab and not
stop halfway through the material. This is called guil-
lotine cutting.
Figure 1a shows how the blue item can not be cut
from the gray slab if it is placed in the bottom left cor-
ner. The item has either to be cut as seen in Fig. 1b or
Fig. 1c. Leftover material will have different shapes,
not only depending on the exact geometrical place-
ment of the item on the slab, but also depending on the
exact cuts being made. Currently, in order to calculate
the dimensions of the surplus material, the problem
formulation assumes a strict cutting order, which will
be explained shortly.
Our model follows (Lodi and Monaci, 2003), in
using the notion of shelves, as does (Andrade et al.,
2013). Shelves are a way to connect the restriction on
guillotine cutting and geometric placement of items.
Figure 2 shows the general layout of items and shelves
on a slab.
In this figure, item 1 and item 2 are on the same
shelf, while item 3 is on a separate shelf. To make this
clear, a shelf is just a notion to group and place items
on a slab. Further, shelf ν is opened by item ν. The
height of all items, that are placed in this shelf after
item ν must be less than or equal to the height of item
ν. I.e., the height of shelf ν is equivalent to the height
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