Evolutionary algorithms, simulated annealing and tabu search

15 Pages·2001·0.38 MB·English

EngineeringApplicationsofArtificialIntelligence14(2001)167–181
Evolutionary algorithms, simulated annealing and tabu search:
a comparative study
Habib Youssef*, Sadiq M. Sait, Hakim Adiche
DepartmentofComputerEngineering,KingFahdUniversityofPetroleumandMinerals,KFUPMBox#673,Dhahran-31261,SaudiArabia
Received1July1999;receivedinrevisedform1June2000;accepted1July2000
Abstract
Evolutionary algorithms, simulated annealing (SA), and tabu search (TS) are general iterative algorithms for combinatorial
optimization. The term evolutionary algorithm is used to refer to any probabilistic algorithm whose design is inspired by
evolutionarymechanismsfoundinbiologicalspecies.Mostwidelyknownalgorithmsofthiscategoryaregeneticalgorithms(GA).
GA,SA,andTShavebeenfoundtobeveryeffectiveandrobustinsolvingnumerousproblemsfromawiderangeofapplication
domains. Furthermore, they are even suitable for ill-posed problems where some of the parameters are not known before hand.
Thesepropertiesarelackinginalltraditionaloptimizationtechniques.InthispaperweperformacomparativestudyamongGA,
SA, and TS. These algorithms have many similarities, but they also possess distinctive features, mainly in their strategies for
searchingthesolutionstatespace.Thethreeheuristicsareappliedonthesameoptimizationproblemandcomparedwithrespectto
(1)qualityofthebestsolutionidentifiedbyeachheuristic,(2)progressofthesearchfrominitialsolution(s)untilstoppingcriteria
aremet,(3)theprogressofthecostofthebestsolutionasafunctionoftime(iterationcount),and(4)thenumberofsolutionsfound
atsuccessiveintervalsofthecostfunction.Thebenchmarkproblemusedisthefloorplanningofverylargescaleintegrated(VLSI)
circuits.Thisisahardmulti-criteriaoptimizationproblem.Fuzzylogicisusedtocombineallobjectivecriteriaintoasinglefuzzy
evaluation(cost)function, which isthenused torate competing solutions. # 2001 Elsevier ScienceLtd. All rightsreserved.
Keywords: Geneticalgorithms;Simulatedannealing;Tabusearch;Fuzzylogic;Floorplanning;Combinatorialoptimization;VLSI
1. Introduction 2. They are blind, in that they do not know when they
reached an optimal solution. Therefore they must be
This paper is concerned with one class of combina- told when to stop.
torial optimization algorithms: general iterative al- 3. They have ‘‘hill climbing’’ property, i.e., they
gorithms, which constitute a class of approximation occasionally accept uphill (bad) moves.
algorithms. The growing interest in this class of 4. Theyaregeneral,i.e.,theycaneasilybeengineeredto
algorithms is attributed to their generality, ease of implement any combinatorial optimization problem;
implementation, and mainly, their robustness in solving all that is required is to have a suitable solution
a wide variety of problems from a range of application representation, a cost function, and a mechanism to
domains. We shall limit ourselves to three popular traverse the search space.
iterative algorithms,(1)geneticalgorithm,(2)simulated 5. Under certain conditions, they asymptotically con-
annealing, and, (3) tabu search. All three optimization verge to an optimal solution.
heuristics have several similarities, namely (Sait and
The question of how to compare two constructive
Youssef, 1999a):
algorithms has long been settled. Two measures are
1. They are approximation (heuristic) algorithms, i.e., typically used: (1) the time complexity of the algorithm,
they do not guarantee finding an optimal solution. and, (2) the quality of solution, i.e., how far would the
solution be from optimal in the worst or best case. In
most cases, the time complexity of a constructive
algorithm is usually easy to derive; however, determin-
*Corresponding author. Tel.: [phone]; fax: [phone]-
ing the quality of solution relative to the optimum is
3059.
E-mailaddress:[email protected](H.Youssef). usually very difficult, and in some cases not possible.
0952-1976/01/$-seefrontmatter # 2001 ElsevierScienceLtd.Allrightsreserved.
PII:S0952-1976(00)00065-8
168 H.Youssefetal./EngineeringApplicationsofArtificialIntelligence14(2001)167–181
For iterative approximation algorithms, the time * wi;hi: width and height of bi, which are
complexity is mostly a function of the number of constants for rigid blocks and variable for
iterations allowed for the algorithm as well as the flexible blocks.
complexity of the evaluation function. The various * ri: desirable aspect ratio about each variable-
operators of the algorithm as well as the problem to be shape block where 1=r4h=w4r; r ¼1 if
i i i i i
solved can also have determining effect on the time block i is rigid.
complexity of the algorithm. However, typically, all * ai: area of bi (i.e., ai ¼wi(cid:5)hi), ai is constant.
implementations of approximation iterative algorithms 2. A set of nets N ¼fn ;n ;...;n;...;n g describ-
1 2 i k
have polynomial time complexity. Thus the time ing the connectivity information.
complexity isnot of concernwhen comparing heuristics 3. Desirable floorplan aspect ratio r such that
in this category. Of major importance however is the 1=r4H=W4r; where H and W are the height
quality ofsolution(s) found by a particular heuristic for and width of the floorplan, respectively.
the problem class at hand. That is, if for example, a 4. Timing information.
giveniterativeapproximationalgorithmA1performson
average better than a second algorithm A2 over a Output: A legal floorplan is a floorplan satisfying the
numberofprobleminstances,wethenconcludethatA1 following constraints and objectives:
implementation is better than that of A2. Another
1. Each block b is assigned to a location (x;y);
legitimate comparison would be with respect to the i i i
2. no two blocks overlap;
quality of the solution subspace examined by each
3. for each flexible block b; a ¼w (cid:5)h;
algorithm. That is, if the number of solutions obtained i i i i
4. meet aspect ratio constraints, for each flexible
with algorithm A1 with a cost lower than a particular
block b and for the overall floorplan; and
value C is higher than that of algorithm A2, we then i
5. minimize floorplan area, wire length and circuit
conclude that A1 exhibits superior performance than
delay.
A2. Yet another comparison would be with respect to
the progress of the best solution versus the number of Fig. 3(a) is a floorplan with 7 blocks.
iterations. In this paper we report a number of
experiments performed to compare the performance of 3. Solution encoding and cost evaluation
three well-known approximation iterative algorithms
with respect to criteria such as those described above. 3.1. Solution encoding
The test problem used is floorplanning, a hard problem
encounteredduringthephysicaldesignofVLSIcircuits. To have a simple and efficient encoding scheme the
Itisencounteredinseveralotherareasofengineeringas floorplanner constrains the structure to be slicing, but
well. In VLSI design, a floorplanning tool helps decide allows the circuit modules to have flexible shapes and
several important questions such as dimensions, orien- free orientations. The layout is represented as a
tationsandlocationofcells(orblocks),overallrequired repetitive division into basic rectangles by horizontal
areaofthecircuit,andspeedperformance requirements and vertical cut-lines. Such a layout is called a slicing
(Sait and Youssef, 1995). structure.Aslicingstructurecanbemodeledbyabinary
The paper is organized as follows. Section 2 tree with n leaves and n(cid:6)1 nodes, where each node
formulates floorplanning as a combinatorial optimiza- represents a vertical cut-line or horizontal cut-line, and
tion problem. In Section 3 we describe how a floor- each leaf a basic rectangle. This binary tree is called
planning solution is encoded and evaluated. The same slicing tree, and the corresponding floorplan is a slicing
solutionencodingandevaluationfunctionareusedwith floorplan. Letters H and V refer to horizontal and
all three approximation algorithms. Section 4 briefly vertical cut operators, respectively. A postorder traver-
discusses the main features and operators of each of sal of a slicing tree produces a Polish expression with
GA, SA, and TS algorithms. In Section 5, experimental operators H and V, and with operands the basic
results are presented and the performance of the three rectangles 1;2;...;n (Wong and Liu, 1986). Fig. 3 gives
heuristics are compared. We conclude in Section 6. a rectangular dissection, two possible slicing trees, and
thecorrespondingPolishexpressions.Sincethereisonly
2. Floorplanning onewayofperformingapostordertraversalofabinary
tree, there is one-to-one correspondence between floor-
A possible formulation of the floorplanning problem plan trees and their corresponding Polish expressions.
of VLSI circuits is as follows (Sait and Youssef, 1995): The operators H and V carry the following meanings:
Given: ijH means rectangle j on-top-of rectangle i; ijV means
rectangle i to-the-left-of rectangle j. In this work each
1. A set of n rectangular blocks B¼fb ;b ;...; solution is encoded as a Polish expression as in Wong
1 2
b;...;b g; for each b 2B we have and Liu (1986).
i n i
H.Youssefetal./EngineeringApplicationsofArtificialIntelligence14(2001)167–181 169
3.2. Cost evaluation A¼fðx;m ðxÞÞj all x2Xg, where X is a space of
A
points and m ðxÞ is a membership function of x2X
A
Unlike constructive algorithms for physical design, being an element of A. In general the membership
which produce a solution only at the end of the design functionm ð:ÞisamappingfromX totheinterval[0,1].
A
process, iterative algorithms operate with design solu- Ifm ðxÞ¼1,or0,forallx2X,thefuzzysetAbecomes
A
tionsdefinedateachiteration.Anevaluationfunctionis an ordinary set (Zadeh, 1965).
used to compare results of consecutive iterations and to
select a solution based on the maximal (minimal) value Example. As an example, let h refer to height of an
of the objective function. athlete,and‘‘tall’’consideredasaparticularvalueofthe
The cost function is a measure of how good is a fuzzy variable ‘‘height’’. Then for each h in the range
particularfloorplan.Forfloorplanning,threecriteriaare from 0 to h , m ðhÞ2½0;1(cid:13) indicates the extent to
max tall
ofprimeimportance:area,wirelength,andtiming.Area which h is considered to be tall; m ðhÞ is called the
tall
is estimated by the size of the smallest bounding membership function.
rectangle of the floorplan. Wire length is set equal to
thelengthofallthenets(orinterconnectsoftheblocks) Example. As another example, consider the possible
of the circuit. The length of an individual net or heights of sportsmen in feet to be H ¼
interconnect is estimated using the Steiner tree approx- f3:5;4:0;4:5;5:0;5:5;6:0;6:5g. Heights around 4.5 feet
imation of Youssef et al. (1995). The delay along the are considered short (S), around 5.5 feet are considered
longestpathofthecircuit,takingintoconsiderationthe moderate (M), and heights around 6.0 feet are
estimated interconnect length, is used as a measure of considered tall (T). Thus, short, moderate, and tall are
the timing performance of the floorplan (Youssef et al., not crisply defined. Fuzzy sets for short, moderate, and
1995). tall may be expressed as sets of ordered pairs
The cost function should reflect all objectives. fðx;m ðxÞÞjall x2Xg, where the first element of the
A
Traditionally, multiple objectives have been combined pair is the height and the second is its membership in
intoascalarobjectivefunction,usuallythroughalinear thatset.Fromourpreviousknowledgewecandefinethe
combination (weighted sum) of the multiple attributes, fuzzy sets S, M and T as follows:
or by turning objectives into constraints. One way is to
assign a constant weight to each of the multiple S ¼fð3:5;1:0Þ; ð4:0;1:0Þ; ð4:5;1:0Þ; ð5:0;0:3Þ;
objective functions whose value will depend on the
importance of that objective. Assuming that all objec- ð5:5;0:3Þ; ð6:0;0:0Þ; ð6:5;0:0Þg
tives are to be maximized, the cost of a floorplan
M ¼fð3:5;0:0Þ; ð4:0;0:0Þ; ð4:5;0:1Þ; ð5:0;0:5Þ;
solution x can be expressed as follows:
ð5:5;1:0Þ; ð6:0;0:5Þ; ð6:5;0:1Þg
costðxÞ¼w (cid:9)f ðxÞþw (cid:9)f ðxÞþ(cid:9)(cid:9)(cid:9)þw (cid:9)f ðxÞ; ð1Þ
1 1 2 2 n n
where n is the number of objective functions, and w is T ¼fð3:5;0:0Þ; ð4:0;0:0Þ; ð4:5;0:0Þ; ð5:0;0:1Þ;
i
the weight of the ith objective f. The problem with this
i ð5:5;0:4Þ; ð6:0;1:0Þ; ð6:5;1:0Þg
weighted sum approach is the difficulty in determining
suitable weights. Adhoc methods are sometimes em-
ployed. Fig. 1 illustrates thethree membershipfunctions.For
Fuzzy logic. The logic used to infer a crisp outcome simplicity we have used piecewise linear membership
from fuzzy input values is called fuzzy logic (Zadeh, functions.Membershipfunctionscanalsobecontinuous
1965; Zimmermann, 1996; Kartalopoulos, 1996). Using curves of many different shapes. For example, Fig. 2
fuzzy logic approach, optimization of a vector-valued illustrates a continuous membership function of a fuzzy
function is replaced by the optimization of a scalar set: An individual’s weight around 50kg.
function, which is constructed from the levels of Fuzzy operators: As seen above, in fuzzy logic, the
satisfaction of decision-makers by values of the compo- values are not crisp, and their fuzziness exhibits a
nents of a vector function. Below, we present a brief distribution described by the membership function. In
introductiontofuzzylogic,andthenshowhowitcanbe ordinary set theory, operations such as union ([),
applied in optimization problems with multiple objec- intersection (\), and complementation (:) are used.
tives. Whatistheresultoftheseoperationsonfuzzysets?This
Fuzzy set theory: Unlike in ordinary set theory where question has been addressed by various fuzzy logics.
an element is either in a set or not in a set, in fuzzy set These are logics that have been defined for operations
theory, an element may partially belong to a set. Lotfi on fuzzy sets. One such logic defined by Zadeh is
Zadeh defined a fuzzy set as a class of objects with a calledthemin–maxlogic(Zadeh,1965).Thereareother
continuum of grades of membership. Formally, a fuzzy fuzzy logics, one of which we used is discussed later. In
set A of a universe of discourse X ¼fxg is defined as min–max logic, the ‘‘union’’, ‘‘intersection’’ and
170 H.Youssefetal./EngineeringApplicationsofArtificialIntelligence14(2001)167–181
For floorplanning, the designer seeks to find floor-
plansoptimizedwithrespecttotheareaoccupiedbythe
floorplan (area occupied by the cells and wires), wire
length required to interconnet all the cells, and delay
along the longest path of the corresponding circuit
(delays due to the cells and wires). Therefore, the
objective function is not a scalar, but a vector function
F(cid:2)ðxÞ¼ðAreaðxÞ;LengthðxÞ;DelayðxÞÞ;
where x is a particular floorplan solution with area
AreaðxÞ, wire length LengthðxÞ, and delay performance
Fig.1. Membershipfunctionsforshort,moderate,andtall.
DelayðxÞ (Fig. 3).
To obtain a fuzzy logic definition of the above
multicriteriaobjectivefunction,threelinguisticvariables
area, length, and delay are defined Shragowitz et al.,
1997). In this implementation, only one linguistic value
is defined for each variable. That is, small area, short
length,and low delay. Theselinguistic values character-
ize the degree of satisfaction of the designer with the
values of objectives Area, Length, and Delay of the
floorplan.Thesedegreesofsatisfactionaredescribedby
membership functions m ; m , and m on fuzzy sets of
A L D
linguisticvaluessmallarea,shortlength,andlowdelay.
Membership functions for small area, short length
Fig.2. Continuousmembershipfunctionforthefuzzyset:individual’s
andlowdelayareeasytobuild.Theyarenon-increasing
weightaround50Kg.
functions,since thesmaller thearea,thelength,andthe
delay,thehigheristhedegreeofsatisfactionm ,m ,and
A L
‘‘complementation’’ are defined as follows:
m of the expert and vice versa (see Fig. 4).
D
m ðxÞ¼minðm ðxÞ; m ðxÞÞ; To make the membership functions applicable to
A\B A B
different designs, the base variables floorplan area,
m ðxÞ¼maxðm ðxÞ; m ðxÞÞ; length and delay are normalized to the interval [0,1]
A[B A B
(see Fig. 4). The values A ;L , and D are lower
min min min
m:AðxÞ¼1:0(cid:6)ðmAðxÞÞ: boundsonthearea,totallengthandclockperiodofthe
Fuzzy rules: Approximate reasoning can be made circuit. Amin is the sum of areas of all cells; Lmin is the
basedon linguisticvariablesandtheir values.Rulescan smallest possible length, computed assuming all con-
be generated based on previous experience. These rules nectedcellsareplacedadjacenttoeachother;Dministhe
can then be expressed with If ... Then statements. smallest possible delay of the circuit, which is the delay
Connectives such as AND and OR can be used in of the longest path due to cells and assuming minimum
approximate reasoning to join two or more linguistic interconnect delay for the nets. The values of Amax,
values. Lmax, and Dmax are values corresponding to the initial
As mentioned above, in min–max logic, the fuzzy solution.
AND is realized by the function min. If more than one The most desirable floorplan is the one with the
rule is used to perform decision-making, each rule can highest membership in the fuzzy subsets small area,
be evaluated to generate a numerical value. Then, these
numerical values from various evaluations of different
rules can be combined to generate a crisp value on a
higher level of hierarchy.
Fuzzification. In this work, we rely on the expressive
power of fuzzy logic (Zadeh, 1965, 1975, 1973) to
expressthedesignerobjectivesintheformoffuzzylogic
rules. The evaluation of a floorplanning solution
consists of the evaluation of the fuzzy rules which
returnavaluerepresentingthedegreeofmembershipof
thatsolutioninthefuzzysubsetofgoodfloorplans.The
Fig. 3. (a) Slicing floorplan. (b) Slicing tree with Polish expression
fuzzy logic rules are expressed on linguistic variables of
VH21HHV67V453. (c) Another possible slicing tree with Polish
the problem domain. expressionVH21HV67HV453.
H.Youssefetal./EngineeringApplicationsofArtificialIntelligence14(2001)167–181 171
Fig.4. Normalizedmembershipfunctionsforarea,lengthanddelay.
short length and low delay. Such a solution most likely aggregation technique. According to this orlike opera-
does not exist since some of the criteria conflict with tion, the fuzzy logic rule R.0 evaluates to the following:
each other. Therefore, one has to tradeoff these
individual criteria against each other. This tradeoff is 1X3
m ðxÞ¼b(cid:5)maxðm ;m ;m Þþð1(cid:6)bÞ(cid:5) mðxÞ;
conveniently specified in linguistic terms in the form of ðSÞ 1 2 3 3 i
i¼1
one or several fuzzy logic rules. ð2Þ
As mentioned earlier, a fuzzy logic rule is an If–Then
where m is the membership function of the fuzzy
rule.TheIfpart(antecedent)isafuzzypredicatedefined ðSÞ
subsetofgoodsolutions,andbisaparameterbetween0
in terms of linguistic values and fuzzy operators (AND
and 1 indicating the degree of nearness of this orlike
andOR).TheThenpartiscalledtheconsequent.Inour
operator to the strict meaning of the logical OR
case, the linguistic value used in the consequent part
operator. When b¼1, the orlike operator behaves like
identifies the fuzzy subset of good solutions. Therefore,
a regular max operator, and for b¼0 it behaves like a
the result of evaluation of the antecedent part identifies
weighted averaging operator.
the degree of membership in the fuzzy subset of good
Individual objective criteria can be given relative
floorplan solutions according to the fuzzy rule in
rankingtoreflecttheirimportance.Thiscanbeachieved
question (Shragowitz et al., 1997).
by assigning weights as exponents to individual mem-
The fuzzy subset of good floorplan solutions is
bership functions. When the membership function of a
characterized by the following fuzzy rule:
particular criterion is assigned a positive exponent less
R:0 If ðsmall areaÞ OR ðshort lengthÞ OR ðlow delayÞ than 1, we say that the corresponding fuzzy subset is
dilated. This corresponds to giving more empha-
Then good solution:
sis=importance to that criterion. Concentration corre-
According to the traditional min–max fuzzy logic, the spondstothecaseofassigninganexponentgreaterthan
above rule R.0 evaluates to the following: 1 (Mendel, 1995). For example, in the case of floor-
planning problem, Area is a more significant criterion,
m ðxÞ¼maxðm ;m ;m Þ;
ðSÞ A L D whereas Length and Delay are of equal but less
importance. Such preferences can be conveniently
where m is the membership function of the fuzzy
ðSÞ expressedbyusingdilationandconcentrationoperators.
subset of good solutions, and m , m , and m are the
A L D
With these preferences, Eq. (2) becomes
membershipfunctionsinthefuzzysubsetsofsmall-area
solutions, short wire-length solutions, and small delay m ðxÞ¼b(cid:5)maxðm1=2;m2;m2Þþð1(cid:6)bÞ
solutions, respectively. The min and max operators are ðSÞ A L D
non-compensatory, i.e., the weaker elements cannot (cid:5)1ðm1=2þm2 þm2Þ: ð3Þ
3 A L D
compensate for the stronger element. Such property is
In the above equation, the fuzzy subset representing
undesirable in the case of multi-criteria optimiza-
solutions with small area is dilated, whereas the fuzzy
tion=decision problems. For this category of problems
subset of solutions with short length and the fuzzy
it is important to make use of all the information
subset of solutions with small delay are concentrated.
sources. In this case it is recommended to use a
compensatory interpretation of the fuzzy OR operator,
which allows all the criteria elements to influence the
outcome of the operator. One such interpretation is the 4. GA, SA, AND TS algorithms
orlike ordered weighted averaging (OWA) operator
suggested by Yager in Yager (1994); Yager and Filev Anyiterativeapproximationalgorithm,beitGA,SA,
(1994) in the context of the softening of the rule or TS, requires decisions with respect to the following
172 H.Youssefetal./EngineeringApplicationsofArtificialIntelligence14(2001)167–181
elements: expressions with n operands and n(cid:6)1 operators. The
fitness function is a measure of how good each
* a solution encoding,
individual of current generation is with respect to the
* amechanismtogenerateinitialsolutionsfrom where
desired features. For each chromosome x; fitness
the iterative search will proceed,
ðxÞ¼m ðxÞ, where m is as defined in Eq. (3). The
* an evaluation function to rate solutions encountered ðSÞ ðSÞ
choice of parents from the set of individuals that
during the search,
comprise the population (also known as the mating
* perturbationoperatorstocreatenewsolutionsfroma
pool) is probabilistic. In keeping with the ideas of
given current solution,
natural selection, we assume that stronger individuals,
* assignment to the parameters of the algorithm, and
that is, those with higher fitness values, are more likely
* stopping criteria.
to mate than the weaker ones. One way to simulate this
Below, we provide a brief description of each of GA, is to select parents with a probability that is directly
SA, and TS, and illustrate how each algorithm is proportional to their fitness values. This method is
adapted to the floorplanning problem. called the roulette-wheel method (Goldberg, 1989).
Crossover.Crossoverisamechanismforprobabilistic
4.1. Genetic algorithms inheritance of useful information from two fit indi-
viduals to offsprings. The main idea is that the genetic
Genetic algorithms (GAs) are robust and effective information of a good solution is spread over the entire
optimization techniques inspired by the mechanism of population. Thus, the best solution can be obtained by
evolutionandnaturalgeneticsGoldberg,1989;Holland, thoroughly combining the chromosomes in the popula-
1975. They are characterized by a parallel search of the tion.Crossoveroperationachievesrecombinationofthe
state space as against a point-by-point search by the genetic material. The recombination process includes
conventional optimization techniques. The parallel domain specific knowledge to enforce the inheritance of
search is achieved by keeping a set of possible solutions desirable features from individuals of current popula-
to the optimization problem, called population. An tion. The followingthree crossoveroperators havebeen
individual in the population is a string of symbols and used in this work (Cohoon et al., 1991).
is an abstract representation of the solution. The In the first crossover scheme, the operands from
symbols are called genes and each string of genes is parent P are copied into the offspring O. Then, the
1
termedachromosome.Theindividualsinthepopulation operatorsfromparentP arecopiedbymakingaleft-to-
2
are evaluated by some fitness measure. The population right scan to complete the chromosome. Hence, each
of chromosomes evolves from generation to the next offspring inherits the block structure of one of the
through the use of two types of genetic operators: (1) parents. For example, if P ¼VH21HHV67V453 and
1
unary operators such as mutation and inversion which P ¼VH12HV47HV653, then, according to this cross-
2
alter the genetic structure of a single chromosome, and over, the offspring would be VH21HVH67V453.
(2)higher-orderoperator,referredtoascrossoverwhich The second crossover operates by copying the
consists of obtaining new individual by combining operators from parent P into the corresponding
1
geneticmaterialfromtwoselectedparentchromosomes. positions in offspring O. Then, it completes the
Then the new population is selected out of the indi- construction of O by copying the operands from parent
viduals of the current population and its offsprings. P by making a left-to-right scan. By propagating
2
Based on the fitness value, two individuals (parents) are the groups of operators from P to O, this cros-
1
selected at a time from the population. The genetic sover produces an offspring having the same overall
operators (crossover and mutation) are applied on the slicing structure as that of P . We call this crossover
1
selectedparentstogeneratenewpossiblesolutionscalled ‘‘slicing inheritance crossover’’. For example, if
offsprings. The structure of basic GA is given in Fig. 5. P ¼VH21HHV67V453 and P ¼VH12HV47HV653,
1 2
Theapplication ofgeneticalgorithm foranyproblem then,accordingtothiscrossover,theoffspringwouldbe
requires a representation of the solution to the problem VH12HHV47V653. Both the above crossoversproduce
as a string of symbols, a choice of genetic operators, an valid Polish expressions.
evaluation function, a selection mechanism, and deter- ThethirdcrossoveremployedinthisworkisthePMX
mination of genetic parameters (population size and crossover (partially mapped crossover) which has been
probabilities of crossover, mutation, and inversion). widely used in other genetic implementations for
Each of these greatly influences the performance of the combinatorial optimization problems (Shahookar and
genetic algorithm. Mazumder,1991).PMX isapplied onlyonthemodules
In this work, a chromosome in the population (operands), the operatorsand thepositionsfrom one of
(solution)isaPolishexpressionrepresentingaparticular the parents are used in the offspring (see Figs. 6 and 7).
slicing floorplan (see Section 3.1, Fig. 3). The initial Mutation. Mutation is a means of introducing new
population consists of randomly generated Polish informationintothepopulation.Threesimplemutation
H.Youssefetal./EngineeringApplicationsofArtificialIntelligence14(2001)167–181 173
Fig.5. Structureofgeneticalgorithmforfloorplanning.
operators are used in this work (Wong and Liu, 1986): Inversion. This operator consists of randomly select-
(1) inversion of a randomly selected chain of operators, ing two positions within the string (a chromosome) and
(2)swappingoftworandomlyselectedoperands,and(3) then laterally inverting the string between them. In this
swapping an operand with an adjacent operator work,itisimplementedsoastomakeafullright-to-left
(provided it results in a valid Polish expression). or left-to-right lateral inversion on the entire string of
174 H.Youssefetal./EngineeringApplicationsofArtificialIntelligence14(2001)167–181
Fig.6. Crossoverthatpropagatesslicingstructuretotheoffspring.
Fig.7. IllustrationofPMXcrossover.
operands(only).Forexample,ifthechromosomebefore (higher fitness translates to higher survival). The
inversion is 268HH7H5V4H13VV, then after inversion probability of mutation is set to 5%, and the prob-
thenewsolutiongeneratedwillbe:314HH5H7V8H62VV. ability of crossover to 80%. Inversion is applied
A number of methods have been proposed to select with a probability of 30%. Population size chosen is
individuals that can survive and be used in the next 10, and in each generation the number of offsprings
generation (Goldberg, 1989). In this work, among the producedisthesameasthenumberofparents.Stopping
parents and their offsprings, the chromosomes for the criterion in this case was set to a fixed number of
next generation are chosen based on their fitness values generations.
H.Youssefetal./EngineeringApplicationsofArtificialIntelligence14(2001)167–181 175
4.2. Simulated annealing peratures.TemperatureisinitializedtoavalueT atthe
0
beginning of the procedure (explained below), and is
Simulated annealing (SA) is an iterative search slowly reduced (in a geometric progression); the par-
methodinspiredbytheannealingofmetals(Kirkpatrick ameter a is used to achieve this cooling. The amount of
et al., 1983; Cerny, 1985). Starting with an initial time spent in annealing at a temperature is gradually
solution and armed with adequate perturbation and increased as temperature is lowered. This is done using
evaluation functions, the algorithm performs a stochas- the parameter b>1. The variable Time keeps track of
tic partial search of the state space. Uphill moves are the time being expended in each call to the Metropolis.
occasionallyacceptedwithaprobabilitycontrolledbya The annealing procedure halts when Time exceeds the
parameter called temperature ðTÞ. The probability of allowed time.
acceptanceofuphillmoves decreasesasT decreases.At The Metropolis procedure is shown in Fig. 9. It uses
hightemperature,the search isalmostrandom,whileat the procedure Neighbor to generate a local neighbor
low temperature the search becomes almost greedy. At NewS of any given solution S. The function Fitness
zerotemperature,thesearchbecomestotallygreedy,i.e., returns the fitness of a given solution S. If the fitness of
only good moves are accepted (Kirkpatrick et al., 1983; the new solution NewS is better than the fitness of the
Cerny, 1985). current solution S , then the new solution is accepted,
c
ThebasicSAalgorithmisshowninFig.8.Thecoreof and we do so by setting CurS ¼NewS. If the fitness of
the algorithm is the Metropolis procedure, which the new solution is better than the best solution (BestS)
simulates the annealing process at a given temperature seen thus far, then we also replace BestS by NewsS. If
T (Fig. 9) Metropolis et al., 1953. This procedure is thenewsolutionhasalowerfitnessincomparisontothe
named after the scientist who devised a similar scheme original solution CurS, Metropolis will accept the new
to simulate a collection of atoms in equilibrium at a solution on a probabilistic basis. A random number is
giventemperature.TheMetropolisprocedurereceivesas generated in the range 0 to 1. If this random number is
input the current temperature T, and the current smaller than eDfitness=T, where Dfitness¼fitness
solution CurS which it improves through local search. ðNewSÞ(cid:6)FitnessðcurSÞ, and T is the current tempera-
Metropolis must also be provided with the value M, ture, the inferior solution is accepted. This criterion for
whichistheamountoftimeforwhichannealingmustbe accepting the new solution is known as the Metropolis
applied at temperature T. The procedure Simulated_an- criterion. The Metropolis procedure generates and
nealing simply invokes Metropolis at decreasing tem- examines M solutions.
Fig.8. Procedureforsimulatedannealingalgorithm.
176 H.Youssefetal./EngineeringApplicationsofArtificialIntelligence14(2001)167–181
anumberofstatetransitionsaremadesoastoreachthe
probabilistic steady state.
The perturb mechanisms employed are similar to the
mutation operators in GA. The stopping criterion is
whenthefinalT50:001orA=M50:05,whereA isthe
t t t
totalnumberofallacceptedmoves(bothgoodandbad)
and M is the number of all moves made at a given
t
temperature.
4.3. Tabu search
Tabu search is a higher-level method or meta strat-
egy for solving combinatorial optimization problems
(Glover, 1989, 1990). It is an iterative procedure that
startsfromsomeinitialfeasiblesolutionandattemptsto
determineabettersolution.TSmakesseveralneighbor-
hood moves and selects the move producing the best
solution among all candidate moves for current itera-
Fig.9. TheMetropolisprocedure. tion. This best candidate solution may not improve the
current solution.
Selecting the best move (which may or may not
Initial temperature. The SA algorithm needs to start improve the current solution) is based on the suppo-
from a high temperature ðTÞ. However, if this initial sition that good moves are more likely to reach the
value of T is too high, it causes a waste of processing optimalornear-optimalsolutions.Thesetofadmissible
time.Theinitialtemperaturevalueshouldbesuchthatit solutions attempted at a particular iteration forms a
allows virtually all proposed uphill or down hill moves candidate list. TS selects the best solution from the
to be accepted. The temperature parameter is initialized candidate list. Candidate list size is a trade-off between
using the procedure described in Wong and Liu (1986). quality and performance.
The idea is basically to use the Metropolis function To avoid move reversals, a device called tabu
(eDFitness=T) to determine the initial value of the restrictionisusedthatmakesselectedattributesofthese
temperature parameter. Before the start of actual SA moves tabu (forbidden). Tabu restrictions allow the
procedure, a constant number of moves, say M, in the search togobeyondthepointsoflocaloptimalitywhile
neighborhoodofthecurrent solutionaremade,andthe still making the best possible move in each iteration.
respective fitness values of these moves are determined. Taburestrictionsareenforcedbyatabulistwhichstores
Thefitnessdifferenceforeachmovei; DFitness isgiven the move attributes to avoid move reversals. Tabu list
i
by DFitness ¼Fitness (cid:6)Fitness .Let M and M be hasanassociatedsizeandcanbevisualizedasawindow
i i i(cid:6)1 u d
the number of uphill and downhill moves, respectively on accepted moves. The moves which tend to undo
(downhill refers to inferior moves) (M ¼M þM ). moves within this window are forbidden.
u d
The average DFitness is then given by Aspiration level component is used to temporarily
d
override the tabu status if the move is sufficiently good.
1 XMd If a move is made tabu in iteration i and its reversal
DFitness ¼ DFitness:
d Md i¼1 i comes in iteration j, where j5iþc, (where c is the size
ofthetabulist),thenitispossiblethatthereversemove
Since we wish to keep the probability, say P0, of may take the search into a new region because of the
acceptinguphillmoveshighintheinitialstageofSA,we effects of c intermediate moves. The simplest aspiration
estimate the value of the temperature parameter by criterion, also known as the ‘‘best cost aspiration
substituting the value of P0 in the following expression criterion’’, consists of overriding the tabu status if the
derived from the Metropolis function: reversal produces a solution better than the best
DFitness obtained thus far. Another approach is to use the same
T0 ¼ lnðP Þ d; attribute of the move which is used to identify the tabu
0
statusandassociateanaspirationlevelvaluewithit.The
where P (cid:17)1 ðP ¼0:999Þ. reversal has to do better than this historical aspiration
0 0
Parameters, stopping criterion. The parameter a for level. Fig. 10 illustrates the flow of the short-term tabu
updating the temperature is user specified. In our search algorithm.
implementation a takes on the value of 0.85, and b is Only short-term tabu search was implemented in this
setequalto1.Ateachupdatedvalueofthetemperature, work. The program is executed for 250 iterations. The