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 启发式算法热轧机铝英文文献和中文翻译(2):http://www.751com.cn/fanyi/lunwen_16688.html