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code is not exceeded.
The plant considered in this study has one pusher and two
soaking pits. These two types of furnaces are described below.
2.1.1. Soaking pit. The capacity of a soaking pit varies
between 15 and 20 ingots depending on their dimensions.
Ingots are loaded vertically in the pit and a minimal spacing
between them is required to ensure uniform heating. As the
capacity depends on ingot dimensions, a bin-packing
problem should be solved to fill a furnace with minimum
residual capacity. In practice, pre-engineered tables that
indicate capacity versus ingot dimensions are used.
Although some manual adjustments are needed when a
batch is made of ingots of different sizes, this approxima-
tion is in general satisfactory.
When loading is completed, the furnace is started and all
ingots are heated at the same time. After the minimal heating
time is reached, it is possible to leave ingots in the furnace, as
long as the maximum heating time is not exceeded. Beyond
that limit, ingots are considered as scrap. It is thus primordial
to achieve a good synchronization between the furnaces and
the hot mill. It should also be noted that all ingots in a batch
must be rolled before the furnace can be used again.
2.1.2. Pusher. As opposed to soaking pits, pushers are
pided into independent compartments and can be fed with
ingots on a more continuous basis. Basically, ingots are
loaded in the first compartment and are pushed from one
compartment to the next at time intervals that depend on the
homogenization code. When the current compartment is
empty, it is possible to load it immediately, if the code is
compatible with the previous one, or to leave it empty and
fill the next one, otherwise. It is thus beneficial to minimize
the number of transitions between non-compatible codes.
Pushers can also work in batch mode, like soaking pits.
That is, all compartments are filled with compatible ingots
and are heated at the same time. This is how the pusher is
modelled in this study. The capacity of the pusher is 25
ingots (five compartments of five ingots each), independently
of the ingot dimensions.
2.1.3. Hard and soft constraints. For each furnace type,
there are different preferences and constraints, which are
used to assess production quality. This information is kept in
tables that are used daily. Here is the list of hard constraints
that must be taken into account during homogenization:
(1) Min/max dimensions: allowable dimension ranges for an
ingot.
(2) Authorized codes: incompatibilities between homogeni-
zation codes and furnace types.
(3) Code compatibility: incompatibilities between codes in