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Jacquenot et al.[6] introduced 2D multi-objective placement
method for complex
geometry components !heir proposed relaxed placement technique
is based on the
hybridation of a genetic algorithm and a separation algorithm"
and it allo#s to solve
placement problem #ith several types of placement constraints
!hey claimed that #ith
appropriate parameters of genetic algorithms" high quality
solutions can be obtained
$onsecutive #or% by them involves a study on the influence of
initial population and
parameters in the genetic algorithm
&opper and !urton [2] introduced t#o hybrid genetic
algorithms for solving 2D rectangle
pac%ing problem !he first algorithm uses the heuristic technique
called 'ottom-(eft
)'(* routine" #here the components are moved to the bottom and
as far as possible to the
left side of the bin !he major disadvantage of the '(-routine is
the creation of empty
areas in the layout" #hen larger items bloc% the movement of
successive one +n order to
overcome this dra#bac%" the '( algorithm has been modified as
'ottom-(eft-,ill )'(,*
placement algorithm
!his algorithm allo#s placing each item at the lo#est available
position of the object
ince" the generation of the layout #ith the '(, algorithm is
based on the allocation of
the lo#est sufficiently large region in the partial layout
rather than on a series of bottom-
left moves" it is capable of filling existing .gaps. +n order to
achieve high quality layouts
in an industrial placement problem" the improved '(, heuristic
is recommended over a
sufficient number of iterations
'e%rar and /acem [01] have considered t#o-dimensional strip
pac%ing problem under
the guillotine constraint !he pac%ing problem described is a set
of rectangular items on
one strip of #idth and infinite height !he items pac%ed #ithout
overlapping must be
extracted by a series of cuts that go from one edge to the
opposite edge guillotine
constraint !he main contribution consists in the elaboration of
ne# tight lo#er and upper
bounds +n order to solve this problem" a dichotomic algorithm is
proposed that useslo#er bound" an upper bound" and a feasibility
test algorithm !he upper bounds are based
on ne# rules for solving the problem under the above constraint
!he lo#er bounds are
based on a linear formulation using a set of various valid
inequalities #ith a connection to
scheduling on parallel machines uch bounds #ere very useful to
build an efficient
dichotomic method" #hich are compared to an existing
branch-and-bound method
$omputational results obtained sho# that the dichotomic
algorithm" using the ne#
bounds gives good results compared to existing methods
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Do#sland and Do#sland [03] revie#ed the researches carried out
on modelling and
solution of layout problems in t#o and three dimensions and the
research carried out on
exact and heuristic solution approaches !hey have reported
several #or%s on pac%ing
problems such as t#o-dimensional rectangular pac%ing" pallet
loading" strip bin pac%ing"
t#o dimensional bin pac%ing" three dimensional pac%ing and
non-rectangular pac%ing
4oreover" they have recommended that there is still plenty of
scope for the researcher
into pac%ing problems in spite of the extent of existing
methodology
$hen and &uanga [06] developed a t#o-level search algorithm
to solve t#o-dimensional
rectangle-pac%ing problem +n this algorithm" the rectangles are
placed ina container one
by one and each rectangle is pac%ed at a position by a
corner-occupying action !his
action touches t#o items #ithout overlapping the other already
pac%ed rectangles 5t thefirst level of the algorithm" a simple
algorithm called 5" selects and pac%s one rectangle
according to the highest degree first rule at every iteration of
pac%ing 5t the second level"
5is itself used to evaluate the benefit of a $andidate
$orner-7ccupying 5ction ($$75*
more globally $omputational results obtained in this paper sho#s
that the resulted
pac%ing algorithm called 50produces high-density solutions
#ithin short running times
Jain and 8ea [0] present a technique for applying genetic
algorithms )85* on t#o-
dimensional pac%ing problems !his approach is applicable to not
only convex shaped
objects" but can also accommodate any type of concave and
complex shaped objects
including objects #ith holes +n this approach" a ne# concept of
a t#o-dimensional
genetic chromosome is introduced !he total layout space is
divided into a finite number
of cells for mapping it into this 2D genetic algorithm
chromosome !he mutation and
crossover operators have been modified and are applied in
conjunction #ith connectivity
analysis for the objects to reduce the creation of faulty
generations 5 ne# feature has
been added to the 8enetic 5lgorithms in the form of a ne#
operator called compactioneveral examples of 85-based layout are
presented
/ier%os9 and (uc9a% [0:] presented a hybrid evolutionary
algorithm for the t#o-
dimensional non-guillotine pac%ing problem !he problem consists
of pac%ing many
rectangular pieces into a single rectangular sheet in order to
maximi9e the total area of the
pieces pac%ed 4oreover" there is a constraint on the maximum
number of times that a
piece may be used in a pac%ing pattern !hree mutation operators
and t#o types of quality
functions are used in the algorithm !he best solution obtained
by the evolutionary
algorithm is used as the initial solution in a tree search
improvement procedure !his
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approach is tested on a set of benchmar% problems ta%en from the
literature and compared
#ith the results published by other authors
