Volume 2012, Article ID 836597,27pages doi:10.1155/2012/836597
Research Article
Improved Quantum-Inspired Evolutionary
Algorithm for Engineering Design Optimization
Jinn-Tsong Tsai,
1Jyh-Horng Chou,
2, 3and Wen-Hsien Ho
41Department of Computer Science, National Pingtung University of Education, 4-18 Min-Sheng Road, Pingtung 900, Taiwan
2Institute of System Information and Control, National Kaohsiung First University of Science and Technology, 1 University Road, Yenchao, Kaohsiung 824, Taiwan
3Department of Electrical Engineering, National Kaohsiung University of Applied Sciences, 415 Chien-Kung Road, Kaohsiung 807, Taiwan
4Department of Healthcare Administration and Medical Informatics, Kaohsiung Medical University, 100 Shi-Chuan 1st Road, Kaohsiung 807, Taiwan
Correspondence should be addressed to Wen-Hsien Ho,[email protected]
Received 31 August 2012; Revised 26 October 2012; Accepted 31 October 2012 Academic Editor: Jung-Fa Tsai
Copyrightq 2012 Jinn-Tsong Tsai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
An improved quantum-inspired evolutionary algorithm is proposed for solving mixed discrete-continuous nonlinear problems in engineering design. The proposed Latin square quantum-inspired evolutionary algorithmLSQEA combines Latin squares and quantum-inspired genetic algorithmQGA. The novel contribution of the proposed LSQEA is the use of a QGA to explore 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 exploration and exploitation, the LSQEA provides higher computational efficiency and robustness compared to QGA and real-coded GA when solving global numerical optimization problems with continuous variables. Additionally, the proposed LSQEA approach effectively solves mixed discrete-continuous nonlinear design optimization problems in which the design variables are integers, discrete values, and continuous values. The computational experiments show that the proposed LSQEA approach obtains better results compared to existing methods reported in the 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.
Since the 1960s, researchers have attempted to solve this problem, which is known as the mixed discrete nonlinear programming MDNLP problem. One of the most effective solutions reported so far is a nonlinear branch and bound method BBM for solving nonlinear and discrete programming in mechanical design optimization 1, 2. 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 3. The better method, such as the sequential linear programming SLP, was developed by Bremicker et al. 4 and by Loh and Papalambros 5 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 MDNLP problems include the penalty function approach6–8 and the Lagrangian relaxation approach9. 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 3. The Lagrangian relaxation method is similar to the penalty function method. The main difference is that the additional terms due to discrete variables are added to a Lagrangian function instead of a penalty function. The Lagrangian relaxation approach does not guarantee finding a global solution, even if it is a convex problem before discrete variables are introduced. It is observed that some of the methods for discrete variable optimization use the structure of the problem to speed up the search for the discrete solution. These methods are not suitable for implementation into general purpose applications. The BBM is the most general methods; however, it is time consuming. In recent years, the focus has shifted to applications of soft-computing optimization techniques that naturally use mixed-discrete and continuous variables for solving practical engineering problems. These approaches include genetic algorithmsGAs 10–18, simulated annealing 19, differential evolution20,21, and evolutionary programming approach 22. The major challenge when solving MDNLP problems is that numerous local optima can result in the methods becoming trapped in the local optima of the objective functions 12. Therefore, an efficient and robust algorithm is needed to solve mixed discrete-continuous nonlinear design optimization problems in the engineering design field.
In the past decade, the emerging field of quantum-inspired computing has motivated intensive studies of algorithms such as the Shor factorizing algorithm23 and the Grover quantum search algorithm 24, 25. By applying quantum mechanical principles such as quantum-bit representation and superposition of states, quantum-inspired computing can simultaneously process huge numbers of quantum states simultaneously and in parallel. To introduce a strong parallelism in the evolutionary algorithm, Han et al.26 and Han and Kim 27–29 proposed the quantum-inspired genetic algorithm QGA. For solving combinatorial optimization problems, QGA has proven superior to conventional GAs. Malossini et al.30 showed that, by taking advantage of quantum phenomena, QGA improves the speed and efficiency of genetic procedures. Quantum-inspired evolutionary algorithms have also been used to solve optimization problems such as partition function calculation31, nonlinear blind source separation32, filter design 33, numerical optimization 34–36, hyperspectral anomaly detection 37, multiple sequence alignment 38, thermal process identification 39, and multiobjective optimization 40. However, the performance of the simple QGA
is often unsatisfactory, and it is easily trapped in the local optima, which results in premature convergence. That is, the quantum-inspired bitQ-bit search with quantum mechanism must be well coordinated with the genetic search with evolution mechanism, and the exploration and exploitation behaviors must also be well balanced35. Therefore, a big challenge is to improve QGA capability of exploration and exploitation and develop an efficient and robust algorithm.
The efficient and robust Latin square quantum-inspired evolutionary algorithm LSQEA proposed in this study solves global numerical optimization problems with con-tinuous variables and mixed discrete-concon-tinuous nonlinear design optimization problems. The LSQEA approach integrates Latin squares 41–43 and QGA i.e., quantum-inspired individual and mechanism with GAs. The concept of the use of QGA came from the works of Han et al.26 and Han and Kim 27–29, while the development steps were implemented by authors and shown inSection 3. The role of the Latin square is to generate better individuals by implementing the Latin square-based recombination since the systematic reasoning ability of Latin square of the Taguchi method is, in due course, incorporated into the recombination 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 microspace can be exploited. The QGA is used to explore the optimal feasible region in macro-space. Therefore, the LSQEA approach is highly robust and achieves quick convergence.
The paper is organized as follows. Section 2 gives the problem statements. The LSQEA for solving the mixed discrete-continuous nonlinear design optimization problems is described in Section 3. In Section 4, the proposed LSQEA approach is compared with QGA and real-coded GARGA 44–47 in terms of performance in solving global numerical optimization problems with continuous variables. The LSQEA approach is then used to solve mixed discrete-continuous nonlinear design problems encountered in the engineering design field, and the results obtained by LSQEA are compared with those obtained by existing methods reported in the literature. Finally,Section 5concludes the study.
2. Problem Statements
This section states the considered problems, which include a global numerical optimization problem with continuous variables and a mixed discrete-continuous nonlinear programming problem.
The following global numerical optimization problem with continuous variables is considered:
minimize fX
subject to XL≤ X ≤ XU,
gjX ≤ 0 j 1, 2, . . . , t,
2.1
where X x1, x2, . . . , xi, . . . , xn is a continuous variable vector, fX is an objective function, XL xL
1, . . . , xiL, . . . , xLn, and XU xU1, . . . , xUi , . . . , xUn define the feasible
solution vector spaces. The domain of xi is denoted by xiL, xUi , and the feasible solution
space is defined byXL, XU. For this problem, g
jX and j 1, 2, . . . , t are the constraint
efficiently obtaining optimal solution is difficult because the problem involves designs that are high-dimensional, nondifferentiable, and multimodal 35.
The mixed discrete-continuous nonlinear programming problem is expressed as follows12: minimize fX subject to gjX ≤ 0 j 1, 2, . . . , t, xLi ≤ xi ≤ xiU i 1, 2, . . . , n, X Xd, XcT, Xdx1, x2, . . . , xnq, xnq 1, . . . , xndT, Xc xnd 1, xnd 2, . . . , xnT, 2.2
where x1, x2, . . . , xnq are nonnegative discrete variables with permissible values equally spaced; xnq 1, . . . , xnd are nonnegative discrete variables with permissible values unequally spaced; xnd 1, xnd 2, . . . , xn are nonnegative continuous variables. Practical optimization
problems encountered in the engineering design field often have many constraints and require consideration of different types of design variables as shown in 2.2. Again, because these problems involve high-dimensional, nondifferentiable, and multimodal properties, an effective algorithm is needed to solve them optimally and efficiently. According to the above statements, the problem in2.2 is a special case of the problem in 2.1.
For the mixed discrete-continuous design problem in2.2, the discrete variables with equal spacing i.e., individuals with arithmetical progression are xi xLi Ni − 1Δxi,
where i 1, 2, . . . , nq; xLi is the lower bound of xi; Ni is the natural number corresponding
to xi; Δxiis the discrete increment of the ith discrete variable; nq is the number of discrete
variables with equal spacing. Let xi eMi,1for the discrete variables with unequal spacing, where nq < i≤ nd, Midenotes the natural number corresponding to xi, Miis generated from
1, wi, eMi,1denotes the Mith element of the wi× 1 vector Ei, Eirepresents the wi× 1 vector of values of discrete variables with unequal spacing, and wi is the maximum permissible
number of discrete values for the ith discrete variable with unequal spacing.
3. The LSQEA Approach for Solving Mixed Discrete-Continuous
Design Problems
This section describes the details of the LSQEA approach for solving mixed discrete-continuous nonlinear programming problems.
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 superposition of the two. The state of a quantum bit can be represented as
where α and β are complex numbers that describe the probability amplitudes of the corresponding states. The|α|2 and|β|2are probabilities of the quantum bit being in the “0” state and “1” state, respectively, such that|α|2 |β|2 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 represented by a string of n Q-bits such as
α1 β1 α2· · · β2· · · αn βn , 3.2
where|αi|2 |βi|2 1, i 1, 2, . . . , n. Since Q-bits represent a linear superposition of states, a
Q-bit representation provides better population diversity compared to other representations used in evolutionary computing. For example, for following three Q-bits system with three pairs of amplitudes ⎡ ⎢ ⎢ ⎢ ⎣ 1 √ 2 1 √ 2 1 √ 2 −√1 2 1 2 √ 3 2 ⎤ ⎥ ⎥ ⎥ ⎦, 3.3
the states of the system can be represented as
1 4|000 √ 3 4 |001 − 1 4|010 − √ 3 4 |011 1 4|100 √ 3 4 |101 − 1 4|110 − √ 3 4 |111. 3.4
The above result means that the probabilities to represent the states |000, |001, |010, |011, |100, |101, |110, and |111 are 1/16, 3/16, 1/16,3/16, 1/16, 3/16, 1/16, and 3/16, respectively. By consequence, the above three Q-bits system contains the information of eight states.
3.2. Initial Population
The initialization procedure produces psQ-bit individuals where ps denotes the population
size.
3.3. Crossover Operation
The crossover operators are the one point operator, which randomly determines one cut-point and exchanges the cut-cut-point right parts of the Q-bits of two parents to generate new
offspring. If the ith position is selected as one cut-point, the one cut-point crossover operator is used for Q-bits as shown in3.5
α1,1 · · · α1,i α1,i 1 · · · α1,n β1,1 · · · β1,i β1,i 1 · · · β1,n α1,1 · · · α1,i α2,i 1 · · · α2,n β1,1 · · · β1,i β2,i 1 · · · β2,n ⇒ α2,1 · · · α2,i α2,i 1 · · · α2,n β2,1 · · · β2,i β2,i 1 · · · β2,n α2,1 · · · α2,i α1,i 1 · · · α1,n β2,1 · · · β2,i β1,i 1 · · · β1,n . 3.5
For example, for following four Q-bits system, the 2nd position is selected as one cut-point, the crossover operation is shown below
⎡ ⎢ ⎢ ⎣ 1 √ 2 1 √ 2 1 2 √ 3 2 1 √ 2 − 1 √ 2 √ 3 2 − 1 2 ⎤ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎣ 1 √ 2 1 √ 2 − 1 2 1 √ 2 1 √ 2 − 1 √ 2 √ 3 2 − 1 √ 2 ⎤ ⎥ ⎥ ⎦ ⇒ ⎡ ⎢ ⎢ ⎣ √ 3 2 − 1 √ 2 − 1 2 1 √ 2 −1 2 1 √ 2 √ 3 2 − 1 √ 2 ⎤ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎣ √ 3 2 − 1 √ 2 1 2 √ 3 2 −1 2 1 √ 2 √ 3 2 − 1 2 ⎤ ⎥ ⎥ ⎦. 3.6
3.4. Mutation Operation
Mutation of Q-bits is performed by randomly determining one positione.g., position i and then exchanging the corresponding αi and βi
α1 β1 · · ·· · ·αi βi · · ·· · ·αn βn ⇒ α1 β1 · · ·· · ·βi αi · · ·· · · αn βn . 3.7
For example, for following four Q-bits system, the 2nd position is selected for mutation, the mutation operation is shown below.
⎡ ⎢ ⎢ ⎣ 1 √ 2 1 √ 2 1 2 √ 3 2 1 √ 2 − 1 √ 2 √ 3 2 − 1 2 ⎤ ⎥ ⎥ ⎦ ⇒ ⎡ ⎢ ⎢ ⎣ 1 √ 2 − 1 √ 2 − 1 2 1 √ 2 1 √ 2 1 √ 2 √ 3 2 − 1 √ 2 ⎤ ⎥ ⎥ ⎦. 3.8
3.5. Q-Bit Rotation Operation
The purpose of a rotation gate Uθ is to update a Q-bit individual by rotating the Q-bit toward the direction of the corresponding Q-bit to obtain a better value. Theαi, βi of the ith
Q-bit is updated as follows: αi βi Uθi αi βi cos θi − sin θi sin θi cos θi αi βi , 3.9
where θiis a rotation angle by0, 0.05π.
For example, if|Ψ 2/√5|0 1/√5|1 1/√52 1 and U 1/√21 1 1−1 , the result obtained by Q-bit rotation operation is|Ψ 3/√10|0 1/√10|1. The probability of |0 becomes larger, and the probability of|1 becomes smaller.
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 1, R where R is the number of design constraints. The higher this index of violation is, the larger the penalty should be. Given these considerations, the penalty value P is defined as follows: P wp R j1 wL Lj− yj wU yj− Uj, 3.10
where yj is a value computed from the constraint function when the values of design
variables are determined; Uj and Lj are the upper and lower bounds, respectively, of the
constraint function; wpis a value distinguishing the feasible from the unfeasible individuals;
and wLand wU denote the weights. Additionally, if yj < Lj, then wL 1 and wU 0; if yj > Uj, then wL 0 and wU 1; if Lj ≤ yj ≤ Uj, then wL 0 and wU 0. Equation
3.10 generally requires that each value calculated for yj of the constraint function should
be limited to its upper and lower bounds. If the value is located within the feasible region, the value is not punished. Otherwise, the value is punished by being multiplied with a large number wp. The penalty value equals 0 when the optimization process is complete since the
values of the design variables no longer violate the design constraints.
3.7. Latin Square
The Latin square experimental design method screens for the important factors that impact product performance. Therefore, it can be used to study a large number of decision variables with a small number of experiments. The design variablesparameters are called factors, and parameter settings are called levels. The name Latin square originates from Leonhard Euler,
who used Latin characters as symbols. The details regarding the experimental design method can be found in texts by Phadke41, Montgomery 42, and Park 43. For an orthogonal array, matrix experiments are conducted by randomly choosing two individuals from the Q-bit population pool. Each factor of one Q-Q-bit individual is designated level 1, and each factor of the other Q-bit individual is designated level 2. The two-level orthogonal array of Latin squares applied here is Lm2m−1. Additionally, each of Z number of design factors has two
levels. To establish a two-level orthogonal array of Z factors, let Lm2m−1 represent m − 1
columns and m individual experiments corresponding to the m rows, where m 2k, k is a positive integerk > 1 and Z ≤ m − 1. If Z < m − 1, only the first Z columns are used while the other m− 1 − Z columns are ignored. For example, if each of six factors has two levels, only six columns are needed to allocate these factors. In this case, L827 is sufficient for this purpose because it has seven columns.
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. The signal-to-noise ratioη refers to the mean-square deviation in the objective function. For cases of the larger-the-better characteristic and the smaller-the-better characteristic, Taguchi defined η, which is expressed in decibels, as η −10 log1/nnt11/y2
t and η
−10 log1/nn
t1y2t, respectively, where {y1, y2, . . . , yn} denotes a set of characteristics.
Further details can be found in works by Phadke41, Montgomery 42, and Park 43. If only the degree of η in the orthogonal array experiments is being described, the previous equations can be modified as ηi yi if the objective function is to be maximized
larger-the-better and as 1/yi if the objective function is to be minimized
smaller-the-better. Let yi denote the evaluation value of the objective function of experiment i, where i 1, 2, . . . , m, and m is the number of orthogonal array experiments. The effects of the various
factorsvariables or individuals can be defined as follows:
Efl sum of ηi for factor f at level l, 3.11
where i is the number of experiments, f is the factor name or number, and l is the level number. The main objective of the matrix experiments is to choose a new Q-bit individual from the two Q-bit individuals at each locusfactor. At each locus factor, a Q-bit is chosen if the
Efl has the highest value in the experimental region. That is, the objective is to determine the best level for each factor. The best level for a factor is the level that maximizes the value of
Eflin the experimental region. For the two-level problem, if Ef1> Ef2, the better level is level
1 for factor f ∈ 1, Z. Otherwise, level 2 is better. After the best level is determined for each 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-noise ratio of the Taguchi method ensures that the new Q-bit individual has the best or close-to-best evaluation value of the objective function among the 2Zcombinations of factor levels,
where 2Zis the total number of experiments needed for all combinations of factor levels.
For the matrix experiments of an orthogonal array of Latin squares, generation of better individuals requires random selection of two 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
algorithm46. The two individuals recombine the better Q-bits to be a better-fit individual, so that potential individuals in microspace can be exploited. The detailed steps for each matrix experiment are described as follows.
Algorithm
Step 1. Set j 1. Generate two sets U1and U2, each of which has Z design factorsvariables. From the first Z columns of the orthogonal array Lm2m−1, allocate Z design factors, where m≥ Z 1.
Step 2. Designate sets U1 and U2 as level 1 and level 2, respectively, by using a uniformly distributed random method to choose two Q-bit individuals from the Q-bit population pool generated by the crossover operation.
Step 3. Assign the level 1 values obtained from U1and the level 2 values obtained from U2to level cells of the j experiment in the orthogonal array.
Step 4. Calculate the fitness value and the signal-to-noise ratio for the new individual.
Step 5. If j > m, then go toStep 6. Otherwise, j j 1, and repeat Steps3–5.
Step 6. Calculate the effects of the various factors Ef1and Ef2, where f 1, 2, . . . , Z. Step 7. The Q-bit of locus f of the new Q-bit individual is obtained from U1 if Ef1 > Ef2.
Otherwise, it is obtained from U2, where f 1, 2, . . . , Z. Implementing the process for each Q-bit at each locus then obtains the new Q-bit individual.
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. The penalty function is considered for a constrained problem, as the fitness value is calculated. The steps of the LSQEA approach are described as follows.
Step 1. Set parameters, including population size ps, crossover rate pc, mutation rate pm, and
number of generations.
Step 2. Generate an initial Q-bit population, and calculate the fitness values for the
population.
Step 3. Perform selection operation by roulette wheel approach44.
Step 4. Perform the one-cut-point crossover operation for each Q-bit. Select Q-bit individuals
for crossover according to crossover rate pc.
Step 5. Perform matrix experiments for Latin squares method, and use signal-to-noise ratios
to generate the better offspring.
Step 7. Generate the Q-bit population via Latin squares method.
Step 8. Perform the mutation operation in the Q-bit population. Select Q-bits for mutation
according to mutation rate pm.
Step 9. Except for the best individual, select ps Q-bit individuals for the Q-bit rotation operation.
Step 10. Generate the new Q-bit population.
Step 11. Has the stopping criterion been met? If so, then go to Step 12. Otherwise, repeat Step 3toStep 11.
Step 12. Display the best individual and fitness value.
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, the LSQEA approach is used to solve mixed discrete-continuous nonlinear design problems in the engineering design field, and its solutions are compared with those of other methods reported in the literature.
4.1. Solving Nonlinear Programming Optimization Problems with
Continuous Variables
For performance evaluation, the proposed LSQEA approach was used to solve the nonlinear programming optimization problems shown inTable 148–50. The test functions included quadratic, linear, polynomial, and nonlinear forms. The constraints of these functionsf1,
f2, f3, and f4 were linear and nonlinear inequalities, and their dimensions were 13, 8, 7, and 10, respectively. The penalty function of 3.10 was used to handle constrains of linear and nonlinear inequalities for optimization. Therefore, the test functions had sufficient 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 the test functions.
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 parameters51,52. For example, Chou et al. 53 applied an experimental design method to improve the performance of a GA by optimizing its evolutionary parameters. Therefore, this study adjusted evolutionary parameters by using the same experimental design method applied in Chou et al.53. The evolutionary environments used for experimental computation by LSQEA, QGA, and RGA approaches were as follows. For f1, f2, f3, and f4, the population size pswas 300, the crossover rate pcwas
0.9, and the mutation rate pmwas 0.1. For f1, f2, and f4, the stopping criterion for all methods and test functions was 540000 function calls. For f3, however, the stopping criterion was set to only 300000 function calls because it approached the optimal value fastest. Additionally,
Table 1: Test functions. f1X 5x1 5x2 5x3 5x4− 54i1x2i− 13 i5xi, subject to 2x1 2x2 x10 x11≤ 10, 2x1 2x3 x10 x12≤ 10, −8x1 x10≤ 0, 2x2 2x3 x11 x12≤ 10, −8x2 x11≤ 0, −8x3 x12≤ 0, −2x4− x5 x10≤ 0, −2x6− x7 x11≤ 0, −2x8− x9 x12≤ 0, 0≤ xi≤ 1, i 1, . . . , 9, 0 ≤ xi≤ 100, i 10, 11, 12, 0 ≤ x13≤ 1. f2X x1 x2 x3, subject to 1− 0.0025x4 x6 ≥ 0, 1 − 0.0025x5 x7− x4 ≥ 0, 1− 0.01x8− x5 ≥ 0, x1x6− 833.33252x4− 100x1 83333.333 ≥ 0, x2x7− 1250x5− x2x4 1250x4≥ 0, x3x8− 1250000 − x3x5 2500x5≥ 0, 100≤ x1≤ 10000, 1000 ≤ xi≤ 10000, i 2, 3, 10 ≤ xi≤ 1000, i 4, . . . , 8. f3X x1− 102 5x2− 122 x34 3x4− 112 10x65 7x26 x74− 4x6x7− 10x6− 8x7, subject to 127− 2x2 1− 3x42− x3− 4x24− 5x5≥ 0, 282 − 7x1− 3x2− 10x23− x4 x5≥ 0, 196− 23x1− x22− 6x26 8x7≥ 0, −4x12− x22 3x1x2− 2x23− 5x6 11x7≥ 0, −10 ≤ xi≤ 10, i 1, . . . , 7. f4X x12 x22 x1x2− 14x1− 16x2 x3− 102 4x4− 52 x5− 32 2x6− 12 5x72 7x8− 112 2x9− 102 x10− 72 45, subject to 105− 4x1− 5x2 3x7− 9x8≥ 0, −10x1 8x2 17x7− 2x8≥ 0, 8x1− 2x2− 5x9 2x10 12 ≥ 0, −5x12− 8x2− x3− 62 2x4 40 ≥ 0, −3x1− 22− 4x2− 32− 2x23 7x4 120 ≥ 0, −x2 1− 2x2− 22 2x1x2− 14x5 6x6≥ 0, −0.5x1− 82− 2x2− 42− 3x52 x6 30 ≥ 0, 3x1− 6x2− 12x9− 82 7x10≥ 0, −10 ≤ xi≤ 10, i 1, . . . , 10.
each test function was performed in 30 independent runs, and data collection included1 the best value,2 the mean function value, and 3 the standard deviation of the function values.
Table 1shows that the test functions involved 13, 8, 7, or 10 variablesfactors, which required 13, 8, 7, or 10 columns, respectively, to allocate them in the Latin square used in the LSQEA approach. The Latin square L827 was used for 7 variables because it had 7 columns. The Latin square L16215 was used for 13, 8, or 10 variables because it had 15 columns. In this case, the first 13, 8, or 10 columns were used, whereas the remaining 2, 7, or 5 columns, respectively, were ignored. The computational procedures and evolutionary environments used to solve the test functions by QGA and RGA approaches were the same as those used in the LSQEA approach. However, Latin square was not used in QGA and RGA, and quantum-inspired computing was not used in the RGA.
InTable 2, the comparison of results obtained by LSQEA, QGA, and RGA approaches reveals the following.
1 The proposed LSQEA finds optimal or near-optimal solutions.
2 For all test functions, the LSQEA solutions are closer to optimal compared to the QGA and RGA solutions.
Table 2: Results of performance comparisons of LSQEA, QGA, and RGA. Best value Mean function valuestandard deviation
Test function
Globally minimal function value
LSQEA QGA RGA LSQEA QGA RGA
f1X −15.000 −15.000 −15.000 −14.9980.005 −14.9970.007 −14.9880.055 −15.000 f2X 7117.961 7135.440 7178.168 197.3177390.259 322.8217614.255 480.058 7049.3317791.224 f3X 680.630 680.660 680.804 680.7720.104 681.4590.836 681.5150.891 680.630 f4X 24.306 24.462 24.694 0.47924.981 0.88625.980 1.69726.051 24.306
3 For all test functions, the deviations in function values are smaller in the proposed LSQEA than in the QGA and RGA. That is, the proposed LSQEA has a relatively more stable solution quality. Since the RGA is largely based on stochastic search techniques, the standard deviations in all evaluations of test functions are higher in the RGA than in the LSQEA and QGA.
Figure 1shows convergence results on test functions f1, f2, f3, and f4 by using the LSQEA, QGA, and RGA. The LSQEA requires fewer function calls to reach the best value and has the sharper decline than the QGA and GA. That is, the LSQEA has faster convergence speed than the QGA and GA.
In the computational experiment by using the systematic reasoning ability of Latin squares, it was confirmed that a new individual generated by each matrix experiment is superior to its parents, two Q-bit individuals. That is, potential individuals in microspace can be exploited. In micro Q-bit space, the systematic reasoning mechanism of the Latin square with signal-to-noise ratio enhanced the performance of the LSQEA by accelerating convergence to the global solution. In macro Q-bit space, quantum-inspired computing with the GA enhanced the performance of the LSQEA.Table 2shows that the QGA outperformed the RGA, which indicates that quantum-inspired computing with GA improves the performance of the QGA. Therefore, the LSQEA outperforms the QGA and RGA methods in both exploration and exploitation.
Garc´ıa et al. 54,55 confirmed the use of the most powerful nonparametric statistical tests to carry out multiple comparisons. Therefore, this study used the nonparametric Wilcoxon matched-pairs signed-rank test 56 to tackle a multiple-problem analysis to compare two algorithms over a set of problems simultaneously. Let di be the difference
between the performance scores of the two algorithms on ith out of n different runs. The differences are ranked according to their absolute values, and average ranks are assigned in case of ties. Let T be the sum of ranks for the different runs on which the second algorithm outperformed the first, and let T−be the sum of ranks for the opposite. Ranks of di 0 are
split evenly among the sums, and if there is an odd number of them, one is ignored
T di>0 rankdi 1 2 di0 rankdi, T− di<0 rankdi 1 2 di0 rankdi. 4.1
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