the same batch.
(4) Min/max  heating  duration:  heating  duration  interval  that
depends  on the homogenization  code of the batch.
With regard  to preferences,  a penalty  must  be paid  if they
are not satisfied.  Only one type of preferences  is considered
for the furnaces,  namely:
(1)  Codes  and  products:  Preferences  may be expressed  for a
particular  furnace  type, depending  on the homogenizing
code and product  type.
2.2. Hot  rolling mill
Hot  rolling considerably  reduces  the thickness  of  a metal
piece  by pushing  it many  times  through  two extremely  hard
cylinders  or rolls  that  spin  at high  speed.  As its name  points
out, this operation  must  be done at high  temperature  to ease
thickness  reduction.  Ingots are unloaded  one by one from
the  furnace  and  are  directly  rolled  to avoid  excessive  cooling.
Although  a plant might  have many  rolling  mills,  a single  mill
is considered  in this study.
2.2.1. Alloy groups  and roll life.  Alloys are  classified  into
hardness  groups  from softer  to harder.  Here, we only have
two  groups (soft  and hard) but  the  same  ideas can  be
applied  for more groups. Each alloy group  has  its own roll
wear coefficient. This constant coefficient  is  the  average
amount of wear  that a particular  ingot of this alloy group
causes when  it is rolled. Rolls must be changed when they
are  too  worn;  the  time  between  two  consecutive  roll
changes  is called a roll life.
2.2.2.  Hard  and  soft  constraints.  To  meet  quality
standards during rolling,  some  rules must  be  followed.
Here  is the list of hard  constraints  that must be taken  into
account by the hot rolling mill:
(1)  Wear  interval:  Each  ingot of a particular  hardness  group
has a wear  interval  within  which  it can be rolled.  These
intervals  are not disjoint  in general.
The  list  of  preferences  for  the  hot  rolling mill  is  the
following:
(1)  Hardness  transition:  As much as possible, soft  ingots
must be rolled before hard ones in  the  same roll life
because  soft alloys require  less damaged  rolls.
(2)  Width:  Ingots should be  processed  in  non-decreasing
order  of width  to avoid damage  to the sheet  edges  (due
to roll deformation).
(3)  Final  thickness  or gauge: Ingots  should be  rolled in
non-decreasing order  of  gauge.  Furthermore,  large
transitions  between  consecutive  ingots  should  be avoided.
This  is  because  abrupt  transitions require manual
adjustments  on the mill.
In practice,  it  is almost  impossible  to satisfy  all preferences
because  they  are  often  contradictory. Furthermore,  the
sequence  on  the  rolling  mill  highly  depends  on  the homogenization  furnaces,  since  ingots must  be heated  before
they are  rolled. 启发式算法热轧机铝英文文献和中文翻译(3):http://www.751com.cn/fanyi/lunwen_16688.html