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    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.
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