Research Article Improved Quantum-Inspired Evolutionary Algorithm

13 Pages·2013·0.12 MB·English

HindawiPublishingCorporation
MathematicalProblemsinEngineering
Volume2012,ArticleID836597,27pages
doi:10.1155/2012/836597
Research Article
Improved Quantum-Inspired Evolutionary
Algorithm for Engineering Design Optimization
Jinn-Tsong Tsai,1 Jyh-Horng Chou,2,3 and Wen-Hsien Ho4
1Department of Computer Science, National Pingtung University of Education, 4-18 Min-Sheng Road,
Pingtung900,Taiwan
2InstituteofSystemInformationandControl,NationalKaohsiungFirstUniversityofScienceand
Technology,1UniversityRoad,Yenchao,Kaohsiung824,Taiwan
3DepartmentofElectricalEngineering,NationalKaohsiungUniversityofAppliedSciences,
415Chien-KungRoad,Kaohsiung807,Taiwan
4DepartmentofHealthcareAdministrationandMedicalInformatics,KaohsiungMedicalUniversity,
100Shi-Chuan1stRoad,Kaohsiung807,Taiwan
CorrespondenceshouldbeaddressedtoWen-HsienHo,[email protected]
Received31August2012;Revised26October2012;Accepted31October2012
AcademicEditor:Jung-FaTsai
Copyrightq2012Jinn-TsongTsaietal.ThisisanopenaccessarticledistributedundertheCreative
CommonsAttributionLicense,whichpermitsunrestricteduse,distribution,andreproductionin
anymedium,providedtheoriginalworkisproperlycited.
Animprovedquantum-inspiredevolutionaryalgorithmisproposedforsolvingmixeddiscrete-
continuous nonlinear problems in engineering design. The proposed Latin square quantum-
inspiredevolutionaryalgorithm(cid:2)LSQEA(cid:3)combinesLatinsquaresandquantum-inspiredgenetic
algorithm(cid:2)QGA(cid:3).ThenovelcontributionoftheproposedLSQEAistheuseofaQGAtoexplore
the optimal feasible region in macrospace and the use of a systematic reasoning mechanism of
the Latin square to exploit the better solution in microspace. By combining the advantages of
explorationandexploitation,theLSQEAprovideshighercomputationalefficiencyandrobustness
compared to QGA and real-coded GA when solving global numerical optimization problems
withcontinuousvariables.Additionally,theproposedLSQEAapproacheffectivelysolvesmixed
discrete-continuous nonlinear design optimization problems in which the design variables are
integers, discrete values, and continuous values. The computational experiments show that the
proposedLSQEAapproachobtainsbetterresultscomparedtoexistingmethodsreportedinthe
literature.
1. Introduction
Solving engineering design optimization problems usually requires consideration of many
different types of design variables and many constraints. Practical problems in engineering
design often involve a mix of integers, discrete variables, and continuous variables. These
constraints are often problematic during the engineering design optimization process.
2 MathematicalProblemsinEngineering
Since the 1960s, researchers have attempted to solve this problem, which is known as
the mixed discrete nonlinear programming (cid:2)MDNLP(cid:3) problem. One of the most effective
solutions reported so far is a nonlinear branch and bound method (cid:2)BBM(cid:3) for solving
nonlinear and discrete programming in mechanical design optimization (cid:6)1, 2(cid:7). In BBM,
however, subproblems result from portioning the feasible domain to obtain solutions by
ignoring discrete conditions, and the number of times the problem needs to be resolved
increases exponentially with the number of variables (cid:6)3(cid:7). The better method, such as the
sequential linear programming (cid:2)SLP(cid:3), was developed by Bremicker et al. (cid:6)4(cid:7) and by Loh
and Papalambros (cid:6)5(cid:7) to solve general MDNLP problems. The linearized discrete problem
is solved by the simplex method to obtain information at each node of the tree. Their SLP
approach is compared with the pure BBM, where the sequential quadratic programming
is used to solve the nonlinear problem to obtain information at each node. The study
shows the SLP method to be superior to the pure BBM. Other approaches to solving
MDNLPproblemsincludethepenaltyfunctionapproach(cid:6)6–8(cid:7)andtheLagrangianrelaxation
approach (cid:6)9(cid:7). The penalty function approach to treat the requirement of discreteness is to
define additional constraints and construct a penalty function for them. This term imposes
penalty for deviations from the discrete values. The difficulties with a penalty approach
are the introduction of additional local minima and repeated minimizations by adjusting
the penalty parameters (cid:6)3(cid:7). The Lagrangian relaxation method is similar to the penalty
function method. The main difference is that the additional terms due to discrete variables
areaddedtoaLagrangianfunctioninsteadofapenaltyfunction.TheLagrangianrelaxation
approachdoesnotguaranteefindingaglobalsolution,evenifitisaconvexproblembefore
discretevariablesareintroduced.Itisobservedthatsomeofthemethodsfordiscretevariable
optimizationusethestructureoftheproblemtospeedupthesearchforthediscretesolution.
These methods are not suitable for implementation into general purpose applications. The
BBMisthemostgeneralmethods;however, itistimeconsuming. Inrecentyears,thefocus
has shifted to applications of soft-computing optimization techniques that naturally use
mixed-discrete and continuous variables for solving practical engineering problems. These
approachesincludegeneticalgorithms(cid:2)GAs(cid:3)(cid:6)10–18(cid:7),simulatedannealing(cid:6)19(cid:7),differential
evolution(cid:6)20,21(cid:7),andevolutionaryprogrammingapproach(cid:6)22(cid:7).Themajorchallengewhen
solvingMDNLPproblemsisthatnumerouslocaloptimacanresultinthemethodsbecoming
trapped in the local optima of the objective functions (cid:6)12(cid:7). Therefore, an efficient and
robustalgorithmisneededtosolvemixeddiscrete-continuousnonlineardesignoptimization
problemsintheengineeringdesignfield.
Inthepastdecade,theemergingfieldofquantum-inspiredcomputinghasmotivated
intensive studies of algorithms such as the Shor factorizing algorithm (cid:6)23(cid:7) and the Grover
quantum search algorithm (cid:6)24, 25(cid:7). By applying quantum mechanical principles such as
quantum-bit representation and superposition of states, quantum-inspired computing can
simultaneouslyprocesshugenumbersofquantumstatessimultaneouslyandinparallel.To
introduceastrongparallelismintheevolutionaryalgorithm,Hanetal.(cid:6)26(cid:7)andHanandKim
(cid:6)27–29(cid:7)proposedthequantum-inspiredgeneticalgorithm(cid:2)QGA(cid:3).Forsolvingcombinatorial
optimizationproblems,QGAhasprovensuperiortoconventionalGAs.Malossinietal.(cid:6)30(cid:7)
showed that, by taking advantage of quantum phenomena, QGA improves the speed and
efficiencyofgeneticprocedures.Quantum-inspiredevolutionaryalgorithmshavealsobeen
used to solve optimization problems such as partition function calculation (cid:6)31(cid:7), nonlinear
blindsourceseparation(cid:6)32(cid:7),filterdesign(cid:6)33(cid:7),numericaloptimization(cid:6)34–36(cid:7),hyperspectral
anomaly detection (cid:6)37(cid:7), multiple sequence alignment (cid:6)38(cid:7), thermal process identification
(cid:6)39(cid:7), and multiobjective optimization (cid:6)40(cid:7). However, the performance of the simple QGA
MathematicalProblemsinEngineering 3
isoftenunsatisfactory,anditiseasilytrappedinthelocaloptima,whichresultsinpremature
convergence.Thatis,thequantum-inspiredbit(cid:2)Q-bit(cid:3)searchwithquantummechanismmust
bewellcoordinatedwiththegeneticsearchwithevolutionmechanism,andtheexploration
and exploitation behaviors must also be well balanced (cid:6)35(cid:7). Therefore, a big challenge is to
improveQGAcapabilityofexplorationandexploitationanddevelopanefficientandrobust
algorithm.
The efficient and robust Latin square quantum-inspired evolutionary algorithm
(cid:2)LSQEA(cid:3) proposed in this study solves global numerical optimization problems with con-
tinuous variables and mixed discrete-continuous nonlinear design optimization problems.
The LSQEA approach integrates Latin squares (cid:6)41–43(cid:7) and QGA (cid:2)i.e., quantum-inspired
individualandmechanismwithGAs(cid:3).TheconceptoftheuseofQGAcamefromtheworksof
Hanetal.(cid:6)26(cid:7)andHanandKim(cid:6)27–29(cid:7),whilethedevelopmentstepswereimplementedby
authorsandshowninSection3.TheroleoftheLatinsquareistogeneratebetterindividuals
byimplementingtheLatinsquare-basedrecombinationsincethesystematicreasoningability
ofLatinsquareoftheTaguchimethodis,induecourse,incorporatedintotherecombination
operation to select better Q-bits. This role is important for improving the efficiency of the
crossover operation in generating representative individuals and better-fit trial individuals.
The Latin square is applied to recombine the better Q-bits so that potential individuals in
microspacecanbeexploited.TheQGAisusedtoexploretheoptimalfeasibleregioninmacro-
space.Therefore,theLSQEAapproachishighlyrobustandachievesquickconvergence.
The paper is organized as follows. Section2 gives the problem statements. The
LSQEA for solving the mixed discrete-continuous nonlinear design optimization problems
is described in Section3. In Section4, the proposed LSQEA approach is compared with
QGAandreal-codedGA(cid:2)RGA(cid:3)(cid:6)44–47(cid:7)intermsofperformanceinsolvingglobalnumerical
optimizationproblemswithcontinuousvariables.TheLSQEAapproachisthenusedtosolve
mixeddiscrete-continuousnonlineardesignproblemsencounteredintheengineeringdesign
field, and the results obtained by LSQEA are compared with those obtained by existing
methodsreportedintheliterature.Finally,Section5concludesthestudy.
2. Problem Statements
Thissectionstatestheconsideredproblems,whichincludeaglobalnumericaloptimization
problemwithcontinuousvariablesandamixeddiscrete-continuousnonlinearprogramming
problem.
The following global numerical optimization problem with continuous variables is
considered:
minimize f(cid:2)X(cid:3)
subject to XL ≤X ≤ XU, (cid:2)2.1(cid:3)
(cid:2) (cid:3)
g (cid:2)X(cid:3) ≤ 0 j (cid:8)1,2,...,t ,
j
where X (cid:8) (cid:6)x ,x ,...,x,...,x (cid:7) is a continuous variable vector, f(cid:2)X(cid:3) is an objective
1 2 i n
function, XL (cid:8) (cid:6)xL,...,xL,...,xL(cid:7), and XU (cid:8) (cid:6)xU,...,xU,...,xU(cid:7) define the feasible
1 i n 1 i n
solution vector spaces. The domain of x is denoted by (cid:6)xL, xU(cid:7), and the feasible solution
i i i
space is defined by (cid:6)XL, XU(cid:7). For this problem, g (cid:2)X(cid:3) and j (cid:8) 1,2,...,t are the constraint
j
functions. Although many design problems can be cast as the above optimization problem,
4 MathematicalProblemsinEngineering
efficiently obtaining optimal solution is difficult because the problem involves designs that
arehigh-dimensional,nondifferentiable,andmultimodal(cid:6)35(cid:7).
The mixed discrete-continuous nonlinear programming problem is expressed as
follows(cid:6)12(cid:7):
minimize f(cid:2)X(cid:3)
(cid:2) (cid:3)
subject to g (cid:2)X(cid:3)≤ 0 j (cid:8)1,2,...,t ,
j
xL ≤x ≤xU (cid:2)i (cid:8) 1,2,...,n(cid:3),
i i i
(cid:2)2.2(cid:3)
X (cid:8)(cid:6)Xd,Xc(cid:7)T,
(cid:4) (cid:5)
Xd (cid:8) x1,x2,...,xnq,xnq(cid:9)1,...,xnd T,
Xc (cid:8)(cid:6)xnd(cid:9)1,xnd(cid:9)2,...,xn(cid:7)T,
where x ,x ,...,x are nonnegative discrete variables with permissible values equally
1 2 nq
spaced; xnq(cid:9)1,...,xnd are nonnegative discrete variables with permissible values unequally
spaced; xnd(cid:9)1,xnd(cid:9)2,...,xn are nonnegative continuous variables. Practical optimization
problems encountered in the engineering design field often have many constraints and
requireconsiderationofdifferenttypesofdesignvariablesasshownin(cid:2)2.2(cid:3).Again,because
theseproblemsinvolvehigh-dimensional,nondifferentiable,andmultimodalproperties,an
effectivealgorithmisneededtosolvethemoptimallyandefficiently.Accordingtotheabove
statements,theproblemin(cid:2)2.2(cid:3)isaspecialcaseoftheproblemin(cid:2)2.1(cid:3).
Forthemixeddiscrete-continuousdesignproblemin(cid:2)2.2(cid:3),thediscretevariableswith
equal spacing (cid:2)i.e., individuals with arithmetical progression(cid:3) are x (cid:8) xL (cid:9) (cid:2)N − 1(cid:3)Δx,
i i i i
where i (cid:8) 1,2,...,nq; xL is the lower bound of x; N is the natural number corresponding
i i i
to x; Δx is the discrete increment of the ith discrete variable; nq is the number of discrete
i i
variables with equal spacing. Let x (cid:8) e for the discrete variables with unequal spacing,
i Mi,1
wherenq<i≤nd,M denotesthenaturalnumbercorrespondingtox,M isgeneratedfrom
i i i
(cid:6)1,w(cid:7),e denotestheMthelementofthew ×1 vectorE,E representsthew ×1 vector
i Mi,1 i i i i i
of values of discrete variables with unequal spacing, and w is the maximum permissible
i
numberofdiscretevaluesfortheithdiscretevariablewithunequalspacing.
3. The LSQEA Approach for Solving Mixed Discrete-Continuous
Design Problems
This section describes the details of the LSQEA approach for solving mixed discrete-
continuousnonlinearprogrammingproblems.
3.1. Q-Bit Representation
In quantum-inspired computing, the smallest unit of information stored in a two-state
quantum computer is called a quantum bit. It may be in a “0” state, a “1” state, or any
superpositionofthetwo.Thestateofaquantumbitcanberepresentedas
|Ψ(cid:3)(cid:8)α|0(cid:3)(cid:9)β|1(cid:3), (cid:2)3.1(cid:3)
MathematicalProblemsinEngineering 5
where α and β are complex numbers that describe the probability amplitudes of the
corresponding states. The |α|2 and |β|2 are probabilities of the quantum bit being in the “0”
stateand“1”state,respectively,suchthat|α|2(cid:9)|β|2 (cid:8)1.
The use of a Q-bit to represent an individual in this study was inspired by
quantum computing concepts. The advantage of the representation is the capability to use
linear superposition method to generate any possible solution. A Q-bit individual can be
representedbyastringofnQ-bitssuchas
(cid:6) (cid:7) (cid:7) (cid:8)
α1 (cid:7)(cid:7)(cid:7)α2···(cid:7)(cid:7)(cid:7)αn , (cid:2)3.2(cid:3)
β β ··· β
1 2 n
where|α|2(cid:9)|β|2 (cid:8) 1, i (cid:8) 1,2,...,n.SinceQ-bitsrepresentalinearsuperpositionofstates,a
i i
Q-bitrepresentationprovidesbetterpopulationdiversitycomparedtootherrepresentations
used in evolutionary computing. For example, for following three Q-bits system with three
pairsofamplitudes
⎡ (cid:7) (cid:7) ⎤
(cid:7) (cid:7)
√1 (cid:7) √1 (cid:7) 1
⎢ (cid:7) (cid:7) ⎥
⎢⎢ 2 (cid:7)(cid:7) 2 (cid:7)(cid:7) √2 ⎥⎥, (cid:2)3.3(cid:3)
⎣ (cid:7) (cid:7) ⎦
√1 (cid:7)−√1 (cid:7) 3
(cid:7) (cid:7)
2 2 2
thestatesofthesystemcanberepresentedas
√ √ √ √
1|000(cid:3)(cid:9) 3|001(cid:3)− 1|010 (cid:3)− 3|011(cid:3)(cid:9) 1|100(cid:3)(cid:9) 3|101(cid:3)− 1|110 (cid:3)− 3|111(cid:3). (cid:2)3.4(cid:3)
4 4 4 4 4 4 4 4
The above result means that the probabilities to represent the states |000(cid:3),|001(cid:3),|010(cid:3),
|011(cid:3),|100(cid:3),|101(cid:3),|110(cid:3), and |111(cid:3) are 1/16,3/16,1/16,3/16, 1/16, 3/16, 1/16, and 3/16,
respectively.Byconsequence,theabovethreeQ-bitssystemcontainstheinformationofeight
states.
3.2. Initial Population
Theinitializationprocedureproducesp Q-bitindividualswherep denotesthepopulation
s s
size.
3.3. Crossover Operation
Thecrossoveroperatorsaretheonecut-pointoperator,whichrandomlydeterminesonecut-
point and exchanges the cut-point right parts of the Q-bits of two parents to generate new
6 MathematicalProblemsinEngineering
offspring.Iftheithpositionisselectedasonecut-point,theonecut-pointcrossoveroperator
isusedforQ-bitsasshownin(cid:2)3.5(cid:3)
(cid:6) (cid:8) (cid:6) (cid:8)
α1,1 ··· α1,i α1,i(cid:9)1 ··· α1,n α1,1 ··· α1,i α2,i(cid:9)1 ··· α2,n
β1,1 ··· β1,i β1,i(cid:9)1 ··· β1,n β1,1 ··· β1,i β2,i(cid:9)1 ··· β2,n
(cid:8)⇒ (cid:2)3.5(cid:3)
(cid:6) (cid:8) (cid:6) (cid:8)
α2,1 ··· α2,i α2,i(cid:9)1 ··· α2,n α2,1 ··· α2,i α1,i(cid:9)1 ··· α1,n .
β2,1 ··· β2,i β2,i(cid:9)1 ··· β2,n β2,1 ··· β2,i β1,i(cid:9)1 ··· β1,n
Forexample,forfollowingfourQ-bitssystem,the2ndpositionisselectedasonecut-point,
thecrossoveroperationisshownbelow
⎡ √ ⎤ ⎡ ⎤
√1 √1 1 3 √1 √1 −1 √1
⎢ ⎥ ⎢ ⎥
⎢ 2 2 √2 2 ⎥ ⎢ 2 2 √2 2 ⎥
⎣ ⎦ ⎣ ⎦
√1 −√1 3 −1 √1 −√1 3 −√1
2 2 2 2 2 2 2 2
(cid:8)⇒ (cid:2)3.6(cid:3)
⎡ √ ⎤ ⎡ √ √ ⎤
3 1 1 1 3 1 1 3
−√ − √ −√
⎢ ⎥ ⎢ ⎥
⎢ 2 2 √2 2 ⎥ ⎢ 2 2 √2 2 ⎥
⎣ ⎦ ⎣ ⎦.
1 1 3 1 1 1 3 1
− √ −√ − √ −
2 2 2 2 2 2 2 2
3.4. Mutation Operation
MutationofQ-bitsisperformedbyrandomlydeterminingoneposition(cid:2)e.g.,positioni(cid:3)and
thenexchangingthecorrespondingα andβ
i i
(cid:6) (cid:7) (cid:7) (cid:7) (cid:7) (cid:8) (cid:6) (cid:7) (cid:7) (cid:7) (cid:7) (cid:8)
α1 (cid:7)(cid:7)(cid:7)···(cid:7)(cid:7)(cid:7)αi (cid:7)(cid:7)(cid:7)···(cid:7)(cid:7)(cid:7)αn (cid:8)⇒ α1 (cid:7)(cid:7)(cid:7)···(cid:7)(cid:7)(cid:7)βi (cid:7)(cid:7)(cid:7)···(cid:7)(cid:7)(cid:7) αn . (cid:2)3.7(cid:3)
β ··· β ··· β β ··· α ··· β
1 i n 1 i n
Forexample,forfollowingfourQ-bitssystem,the2ndpositionisselectedformutation,the
mutationoperationisshownbelow.
⎡ √ ⎤ ⎡ ⎤
√1 √1 1 3 √1 −√1 −1 √1
⎢ ⎥ ⎢ ⎥
⎢⎣ 2 2 √2 2 ⎥⎦(cid:8)⇒⎢⎣ 2 2 √2 2 ⎥⎦. (cid:2)3.8(cid:3)
√1 −√1 3 −1 √1 √1 3 −√1
2 2 2 2 2 2 2 2
MathematicalProblemsinEngineering 7
3.5. Q-Bit Rotation Operation
The purpose of a rotation gate U(cid:2)θ(cid:3) is to update a Q-bit individual by rotating the Q-bit
towardthedirectionofthecorrespondingQ-bittoobtainabettervalue.The(cid:2)α,β(cid:3)oftheith
i i
Q-bitisupdatedasfollows:
(cid:6) (cid:8) (cid:6) (cid:8) (cid:6) (cid:8)(cid:6) (cid:8)
α(cid:6) α cos θ −sin θ α
β(cid:6)i (cid:8) U(cid:2)θi(cid:3) βi (cid:8) sin θi cos θ i βi , (cid:2)3.9(cid:3)
i i i i i
whereθiisarotationangleby(cid:6)0√,0.05π(cid:7). √ √ (cid:2) (cid:3) √ (cid:2) (cid:3)
For example, if |Ψ(cid:3) (cid:8) 2/ 5|0(cid:3) (cid:9) 1/ 5|1(cid:3)(cid:8)√1/ 5 21 and√U (cid:8) 1/ 2 11−11 , the result
obtained by Q-bit rotation operation is |Ψ(cid:6)(cid:3) (cid:8) 3/ 10|0(cid:3)(cid:9) 1/ 10|1(cid:3). The probability of |0
becomeslarger,andtheprobabilityof|1becomessmaller.
3.6. Penalty Function
When using evolutionary method to solve a constrained optimization problem, a penalty
function is used to relax the constraints by penalizing the unfeasible individuals in the
population. This method improves the probability of approaching a feasible region of the
search space by navigating through unfeasible regions and by reducing the penalty when
a feasible region is approached. To clarify this point, it is important to distinguish between
feasible and unfeasible individuals. The unfeasible individuals violate constraints included
in the range (cid:6)1,R(cid:7) where R is the number of design constraints. The higher this index of
violationis,thelargerthepenaltyshouldbe.Giventheseconsiderations,thepenaltyvalueP
isdefinedasfollows:
(cid:15)R (cid:7) (cid:2) (cid:3) (cid:2) (cid:3)(cid:7)
P (cid:8)wp (cid:7)wL Lj −yj (cid:9)wU yj −Uj (cid:7), (cid:2)3.10(cid:3)
j(cid:8)1
where y is a value computed from the constraint function when the values of design
j
variables are determined; U and L are the upper and lower bounds, respectively, of the
j j
constraintfunction;w isavaluedistinguishingthefeasiblefromtheunfeasibleindividuals;
p
and w and w denote the weights. Additionally, if y < L , then w (cid:8) 1 and w (cid:8) 0; if
L U j j L U
y > U , then w (cid:8) 0 and w (cid:8) 1; if L ≤ y ≤ U , then w (cid:8) 0 and w (cid:8) 0. Equation
j j L U j j j L U
(cid:2)3.10(cid:3) generally requires that each value calculated for y of the constraint function should
j
be limited to its upper and lower bounds. If the value is located within the feasible region,
thevalueisnotpunished.Otherwise,thevalueispunishedbybeingmultipliedwithalarge
numberw .Thepenaltyvalueequals0whentheoptimizationprocessiscompletesincethe
p
valuesofthedesignvariablesnolongerviolatethedesignconstraints.
3.7. Latin Square
TheLatin square experimental design method screens forthe important factors that impact
productperformance.Therefore,itcanbeusedtostudyalargenumberofdecisionvariables
withasmallnumberofexperiments.Thedesignvariables(cid:2)parameters(cid:3)arecalledfactors,and
parametersettingsarecalledlevels.ThenameLatinsquareoriginatesfromLeonhardEuler,
8 MathematicalProblemsinEngineering
whousedLatincharactersassymbols.Thedetailsregardingtheexperimentaldesignmethod
can be found in texts by Phadke (cid:6)41(cid:7), Montgomery (cid:6)42(cid:7), and Park (cid:6)43(cid:7). For an orthogonal
array,matrixexperimentsareconductedbyrandomlychoosingtwoindividualsfromtheQ-
bitpopulationpool.EachfactorofoneQ-bitindividualisdesignatedlevel1,andeachfactor
of the other Q-bit individual is designated level 2. The two-level orthogonal array of Latin
squares applied here is L (cid:2)2m−1(cid:3). Additionally, each of Z number of design factors has two
m
levels. To establish a two-level orthogonal array of Z factors, let L (cid:2)2m−1(cid:3) represent m − 1
m
columns and m individual experiments corresponding to the m rows, where m (cid:8) 2k, k is a
positiveinteger(cid:2)k > 1(cid:3)andZ ≤ m−1.IfZ < m−1,onlythefirstZcolumnsareusedwhile
the other m−1−Z columns are ignored. For example, if each of six factors has two levels,
onlysixcolumnsareneededtoallocatethesefactors.Inthiscase,L (cid:2)27(cid:3)issufficientforthis
8
purposebecauseithassevencolumns.
The better combinations of decision variables are also determined by integrating
the orthogonal array of the Latin square and the signal-to-noise ratio of the Taguchi
method. The concept of Taguchi method is to maximize signal-to-noise ratios used as
performance measures by using the orthogonal array to run a partial set of experiments.
Thesignal-to-noiseratio(cid:2)η(cid:3)referstothemean-squaredeviationintheobjectivefunction.For
cases of thelarger-the-better characteristic and thesmaller-the-better(cid:16)characteristic, Taguchi
defined η, wh(cid:16)ich is expressed in decibels, as η (cid:8) −10 log(cid:2)(cid:2)1/n(cid:3) nt(cid:8)1(cid:2)1/yt2(cid:3)(cid:3) and η (cid:8)
−10 log(cid:2)(cid:2)1/n(cid:3) nt(cid:8)1(cid:2)yt2(cid:3)(cid:3), respectively, where {y1,y2,...,yn} denotes a set of characteristics.
FurtherdetailscanbefoundinworksbyPhadke(cid:6)41(cid:7),Montgomery(cid:6)42(cid:7),andPark(cid:6)43(cid:7).
If only the degree of η in the orthogonal array experiments is being described, the
previous equations can be modified as η (cid:8) y if the objective function is to be maximized
i i
(cid:2)larger-the-better(cid:3) and as 1/y if the objective function is to be minimized (cid:2)smaller-the-
i
better(cid:3). Let y denote the evaluation value of the objective function of experiment i, where
i
i(cid:8)1,2,...,m,andmisthenumberoforthogonalarrayexperiments.Theeffectsofthevarious
factors(cid:2)variablesorindividuals(cid:3)canbedefinedasfollows:
E (cid:8) sum of η for factor f at level l, (cid:2)3.11(cid:3)
fl i
whereiisthenumberofexperiments,f isthefactornameornumber,andlisthelevelnumber.
ThemainobjectiveofthematrixexperimentsistochooseanewQ-bitindividualfrom
thetwoQ-bitindividualsateachlocus(cid:2)factor(cid:3).Ateachlocus(cid:2)factor(cid:3),aQ-bitischosenifthe
E hasthehighestvalueintheexperimentalregion.Thatis,theobjectiveistodeterminethe
fl
best level for each factor. The best level for a factor is the level that maximizes the value of
E intheexperimentalregion.Forthetwo-levelproblem,ifE >E ,thebetterlevelislevel
fl f1 f2
1forfactorf ∈ (cid:6)1,Z(cid:7).Otherwise,level2isbetter.Afterthebestlevelisdeterminedforeach
factor, the best levels can be combined to obtain the new individual. Therefore, systematic
reasoning ability of the orthogonal array of the Latin square combined with the signal-to-
noiseratiooftheTaguchimethodensuresthatthenewQ-bitindividualhasthebestorclose-
to-bestevaluationvalueoftheobjectivefunctionamongthe2Z combinationsoffactorlevels,
where2Z isthetotalnumberofexperimentsneededforallcombinationsoffactorlevels.
For the matrix experiments of an orthogonal array of Latin squares, generation of
better individuals requires random selection of two Q-bit individuals at a time from the Q-
bit population pool generated by the one cut-point crossover operation. A new individual
generated by each matrix experiment is superior to its parents by using the systematic
reasoning ability of an orthogonal array of Latin squares and by following the below
MathematicalProblemsinEngineering 9
algorithm(cid:6)46(cid:7).ThetwoindividualsrecombinethebetterQ-bitstobeabetter-fitindividual,
so that potential individuals in microspace can be exploited. The detailed steps for each
matrixexperimentaredescribedasfollows.
Algorithm
Step1. Set j (cid:8)1.GeneratetwosetsU andU ,eachofwhichhasZdesignfactors(cid:2)variables(cid:3).
1 2
FromthefirstZ columnsoftheorthogonalarrayL (cid:2)2m−1(cid:3),allocateZ designfactors,where
m
m≥Z(cid:9)1.
Step 2. Designate sets U and U as level 1 and level 2, respectively, by using a uniformly
1 2
distributedrandommethodtochoosetwoQ-bitindividualsfromtheQ-bitpopulationpool
generatedbythecrossoveroperation.
Step3. Assignthelevel1valuesobtainedfromU andthelevel2valuesobtainedfromU to
1 2
levelcellsofthej experimentintheorthogonalarray.
Step4. Calculatethefitnessvalueandthesignal-to-noiseratioforthenewindividual.
Step5. Ifj >m,thengotoStep6.Otherwise,j (cid:8)j(cid:9)1,andrepeatSteps3–5.
Step6. Calculatetheeffectsofthevariousfactors(cid:2)E and E (cid:3),wheref (cid:8)1,2,...,Z.
f1 f2
Step 7. The Q-bit of locus f of the new Q-bit individual is obtained from U if E > E .
1 f1 f2
Otherwise, itis obtained fromU ,where f (cid:8) 1,2,...,Z. Implementing the process for each
2
Q-bitateachlocusthenobtainsthenewQ-bitindividual.
3.8. Steps of LSQEA
The LSQEA approach is a method of integrating Latin squares and QGA. The Latin square
method is performed between the one-cut-point crossover operation and the mutation
operation.Thepenaltyfunctionisconsideredforaconstrainedproblem,asthefitnessvalue
iscalculated.ThestepsoftheLSQEAapproacharedescribedasfollows.
Step1. Setparameters,includingpopulationsizep ,crossoverratep ,mutationratep ,and
s c m
numberofgenerations.
Step 2. Generate an initial Q-bit population, and calculate the fitness values for the
population.
Step3. Performselectionoperationbyroulettewheelapproach(cid:6)44(cid:7).
Step4. Performtheone-cut-pointcrossoveroperationforeachQ-bit.SelectQ-bitindividuals
forcrossoveraccordingtocrossoverratep .
c
Step5. PerformmatrixexperimentsforLatinsquaresmethod,andusesignal-to-noiseratios
togeneratethebetteroffspring.
Step6. RepeatStep5untiltheloopnumber(cid:2)(cid:2)1/4(cid:3)×p ×p (cid:3)hasbeenmet.
s c
10 MathematicalProblemsinEngineering
Step7. GeneratetheQ-bitpopulationviaLatinsquaresmethod.
Step 8. Perform the mutation operation in the Q-bit population. Select Q-bits for mutation
accordingtomutationratep .
m
Step 9. Except for the best individual, select p Q-bit individuals for the Q-bit rotation
s
operation.
Step10. GeneratethenewQ-bitpopulation.
Step 11. Has the stopping criterion been met? If so, then go to Step12. Otherwise, repeat
Step3toStep11.
Step12. Displaythebestindividualandfitnessvalue.
4. Design Examples and Comparisons
This section first describes the performance evaluation results for the proposed LSQEA
approach. The performance of the LSQEA is then compared with those of the QGA and
RGA methods in solving nonlinear programming optimization problems with continuous
variables.Finally,theLSQEAapproachisusedtosolvemixeddiscrete-continuousnonlinear
designproblemsintheengineeringdesignfield,anditssolutionsarecomparedwiththoseof
othermethodsreportedintheliterature.
4.1. Solving Nonlinear Programming Optimization Problems with
Continuous Variables
Forperformanceevaluation,theproposedLSQEAapproachwasusedtosolvethenonlinear
programming optimization problems shown in Table1 (cid:6)48–50(cid:7). The test functions included
quadratic, linear, polynomial, and nonlinear forms. The constraints of these functions (cid:2)f ,
1
f , f , and f (cid:3) were linear and nonlinear inequalities, and their dimensions were 13, 8,
2 3 4
7, and 10, respectively. The penalty function of (cid:2)3.10(cid:3) was used to handle constrains of
linearandnonlinearinequalitiesforoptimization.Therefore,thetestfunctionshadsufficient
local minima to provide a challenging problem for the purpose of performance evaluation.
To identify any performance improvements obtained by application of Latin square and
quantum computing-inspired concepts, the QGA and RGA approaches were used to solve
thetestfunctions.
Optimizing the main parameters in evolutionary environments continues to be a
area of active research in this field. Studies have shown how the performance of a GA
can be improved by modifying its main parameters (cid:6)51, 52(cid:7). For example, Chou et al. (cid:6)53(cid:7)
appliedanexperimentaldesignmethodtoimprovetheperformanceofaGAbyoptimizing
its evolutionary parameters. Therefore, this study adjusted evolutionary parameters by
using the same experimental design method applied in Chou et al. (cid:6)53(cid:7). The evolutionary
environments used for experimental computation by LSQEA, QGA, and RGA approaches
wereasfollows.Forf ,f ,f ,andf ,thepopulationsizep was300,thecrossoverratep was
1 2 3 4 s c
0.9,andthemutationratep was0.1.For f ,f ,andf ,thestoppingcriterionforallmethods
m 1 2 4
andtestfunctionswas540000functioncalls.Forf ,however,thestoppingcriterionwasset
3
to only 300000 function calls because it approached the optimal value fastest. Additionally,