Research Article

An effective co-evolutionary quantum genetic algorithm for the no-wait flow shop scheduling problem

Advances in Mechanical Engineering 2015, Vol. 7(12) 1–10 Ó The Author(s) 2015 DOI: 10.1177/1687814015622900 aime.sagepub.com

Guanlong Deng1, Ming Wei2, Qingtang Su1 and Mei Zhao1

Abstract This article proposes a competitive co-evolutionary quantum genetic algorithm for the no-wait flow shop scheduling problem with the criterion to minimize makespan, which is a renowned NP-hard combinatorial optimization problem. An innovative coding and decoding mechanism is proposed. The mechanism uses square matrix to represent the quantum individual and adapts the quantum rotation gate to update the quantum individual. In the algorithm framework, the store-with-diversity is proposed to maintain the diversity of the population. Moreover, a competitive co-evolution strategy is introduced to enhance the evolutionary pressure and accelerate the convergence. The store-with-diversity and competitive co-evolution are designed to keep a balance between exploration and exploitation. Simulations based on a benchmark set and comparisons with several existing algorithms demonstrate the effectiveness and robustness of the proposed algorithm. Keywords Genetic algorithm, no-wait flow shop, scheduling, competitive co-evolution, evolutionary algorithm

Date received: 14 May 2015; accepted: 23 November 2015 Academic Editor: Duc T Pham

Introduction The flow shop scheduling problem (FSP) is a renowned complex combinatorial optimization problem with strong practical background and has gained growing research in the past decades. However, in real industrial environment, the no-wait constraint exists when the production requires that each job must be processed from the start to completion with no waiting between or on consecutive machines.1 The no-wait flow shop scheduling problem (NWFSP) has been found in a host of industrial applications.2,3 With respect to its computational complexity, the NWFSP has been proved to be NP-hard.4,5 On account of its complexity and significance in both theory and application, the NWFSP has aroused much concern of researchers. The relatively earlier work was the heuristics developed by Bonney and Gundry6 and King and

Spachis.7 In 1990s, Rajendran8 and Gangadharan and Rajendran9 proposed heuristics which outperformed the previous ones. As for metaheuristic method, Aldowaisan and Allahverdi10 presented a genetic algorithm (GA) to minimize makespan, and Schuster and Framinan11 proposed two metaheuristics, genetic algorithm hybridized with simulated annealing (GASA) and variable neighborhood search (VNS), for the same problem. Then a descending search (DS) method and a

1

School of Information and Electrical Engineering, Ludong University, Yantai, China 2 Shandong Space Electronic Technology Institute, Yantai, China Corresponding author: Guanlong Deng, School of Information and Electrical Engineering, Ludong University, Yantai 264025, China. Email: [email protected]

Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License (http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage).

2 tabu search (TS) algorithm were also employed.12 In recent years, the particle swarm optimization was introduced for the problem and yielded outstanding results.13,14 Besides, the problem was also solved by an improved iterated greedy algorithm,15 a discrete differential evolution (DDE),16 and a novel discrete selforganizing migrating algorithm17 for the makespan minimization. In 2015, Ding et al.18 proposed a tabumechanism improved iterated greedy algorithm for the problem, which is a modification of the iterated greedy algorithm using a tabu-based reconstruction strategy. To minimize total flowtime, Gao et al.19 put forward an effective discrete harmony search (DHS) algorithm as well as a hybrid harmony search algorithm (HHS)20 for the NWFSP, but they did not compare these two algorithms. For the same criterion, a new particle swarm optimization hybridizing with a local search method was presented by Akhshabi et al.,21 and several constructive heuristics were shown with impressive performance.22–24 Since Grover’s25 database search algorithm and Narayanan and Moore’s26 quantum GA, quantum algorithm has been a fresh research area attracting much attention. Han and Kim27 proposed quantuminspired evolutionary algorithm (QEA) which proved to be efficacious for the knapsack problem. Like other evolutionary algorithms, QEA is also characterized by the representation of the individual, the evaluation function, and the population dynamics. However, instead of binary, numeric, or symbolic representation, QEA uses a Q-bit, defined as the smallest unit of information, for the probabilistic representation and a Q-bit individual as a string of Q-bits. Meanwhile, a Q-gate (or quantum gate) is introduced as a variation operator to drive the individuals toward better solutions. Although the solution representation of the scheduling problem is thoroughly dissimilar to that of knapsack problem, the researchers have devoted great efforts to adapting the QEA for the scheduling problem. Li and Wang28 presented a hybrid quantuminspired genetic algorithm (HQGA) for multiobjective flow shop scheduling, and Gu et al.29 proposed a competitive co-evolutionary quantum genetic algorithm (CCQGA) for stochastic job shop scheduling problem. Some other related research work can be found in Gu et al.30–32 So far, there is little published research work on quantum algorithm solving the NWFSP. This article focuses on presenting an effective quantum-inspired hybrid metaheuristic algorithm for the NWFSP. The main innovation of the proposed algorithm lies in three aspects. First, an original coding and decoding mechanism is presented according to the characteristics of flow shop. Second, QEA and GA are combined concretely in a fresh way to form a new algorithm framework. In this framework, a store strategy, named store-with-

Advances in Mechanical Engineering diversity, is proposed, and QEA adopts Q-bit representation and GA adopts job permutation representation. Third, a competitive co-evolution is introduced in the algorithm to achieve a balance between exploration and exploitation. Generally, co-evolution can be classified into two categories: cooperative co-evolution and competitive co-evolution.29 Rosin and Belew33 conducted some research on competitive co-evolution adopting several interactional species. Due to the competition between individuals of different species, the individuals in each species evolve toward better solutions and the convergence is accelerated. The remainder of this article is organized as follows: Section ‘‘Formulation of the NWFSP’’ provides the formulation of the NWFSP with makespan criterion. In section ‘‘CQGA,’’ a CQGA for NWFSP is proposed. Section ‘‘Simulations and comparisons’’ presents the computational results and comparisons. Finally, we summarize the contribution and draw some conclusions of this article in section ‘‘Conclusion.’’

Formulation of the NWFSP The NWFSP can be described as follows: Given the processing times p(j, k) for job j and machine k, each of n jobs (j = 1, 2,., n) needs to be sequenced though m machines (k = 1, 2,., m). The sequence in which the jobs are to be processed is the same for each machine. Each job j has a sequence of m operations (oj1, oj2,., ojm). To satisfy the no-wait restriction, the completion time of the operation ojk must be equal to the earliest time to start of the operation oj,k + 1 for k = 1, 2,., m 2 1. In other words, there must be no waiting time between the processing of any consecutive operation of each of n jobs. In this article, the makespan criterion is considered, so the problem is then to find a schedule minimizing the maximum completion time, that is, makespan. Let the job permutation p = fp1 , p2 , . . . , pn g represent the schedule of jobs to be processed, and d(pj1 , pj ) be the minimum delay on the first machine between the start of job pj and pj1 restricted by the no-wait constraint when the job pj is directly processed after the job pj1 . The minimum delay can be calculated from the following expression d(pj1 , pj ) = p(pj1 , 1) " ( )# k k 1 X X p(pj1 , h)  p(pj , h) + max 0, max 2k m

h=2

h=1

ð1Þ And the makespan can be calculated in O(mn) time as Cmax (p) =

n X j=2

d(pj1 , pj ) +

m X k =1

p(pn , k)

ð2Þ

Deng et al.

3

Coding and decoding method

Figure 1. Gantt chart of the example.

Let P denote the set of all permutations, then the NWFSP with the makespan criterion is to find a permutation p in P such that Cmax (p )  Cmax (p)

8p 2 P

ð3Þ

To illustrate the scheduling of jobs under the no-wait constraint, a simple example with n = 4 and m = 3 is used. The processing times p(j, k) compose matrix P with list of machines in columns and list of jobs in lines. Suppose that 0

2 B5 P=B @4 1

3 1 5 3

In the previous research by Li and Wang28 and Gu et al.,29 they applied the qubit representation to the flow shop and job shop scheduling. They obtained binary strings from qubit individuals, decimal strings from binary strings, and job permutation from decimal strings, which was called random-key representation. For example, suppose a binary sting is [1 0 1 | 0 1 1 | 1 1 0], which is obtained from a Q-bit representation by observation, then the decimal string is [5 3 6]. From small to large, the job permutation is gained as [2 1 3]. More details could be found in their research articles. According to the characteristics of NWFSP with n jobs, we propose that the state of a qubit can be represented as jCi = a1 j1i + a2 j2i +    + an jni

ð4Þ

where ai is a complex number and jai j2 is the probability that job i is selected (i = 1, 2,., n). Correspondingly, normalization of the state to unity requires n X

jai j2 = 1

ð5Þ

i=1

1

5 3C C 2A 2

Thus, a qubit individual for NWFSP is defined as a square matrix 0

and the job permutation is p = f1, 2, 3, 4g, then the scheduling Gantt chart is shown in Figure 1. The makespan is calculated as follows

a11 B .. @ . an1

... .. . 

1 a1n .. C . A

ð6Þ

ann

d(1, 2) = p(1, 1) + max½0, maxfp(1, 2)  p(2, 1), p(1, 2) + p(1, 3)  (p(2, 1) + p(2, 2))g = 2 + 2 = 4 d(2, 3) = p(2, 1) + max½0, maxfp(2, 2)  p(3, 1), p(2, 2) + p(2, 3)  (p(3, 1) + p(3, 2))g = 5 + 0 = 5 d(3, 4) = p(3, 1) + max½0, maxfp(3, 2)  p(4, 1), p(3, 2) + p(3, 3)  (p(4, 1) + p(4, 2))g = 4 + 4 = 8 Cmax (p) = d(1, 2) + d(2, 3) + d(3, 4) +

3 X

p(4, k) = 23

k =1

CQGA Since it was proposed recently, QEA has been applied to both functional and combinatorial problems and attracted lots of attention, but there is little published research work about quantum algorithm solving the NWFSP. This section proposes co-evolutionary quantum genetic algorithm (CQGA) and describes each part of it in detail. Before putting forward the framework of CQGA, we introduce an innovative coding and decoding mechanism.

Pn  2 j = 1, 2, . . . , n, n is the numwhere i = 1 aij  =21, ber of jobs, and aij  denotes the probability of job i being scheduled in the jth position of permutation. When it comes to the operation of observation, a job is got from each column of the square matrix and the job permutation can be obtained by rouletee wheel selection from the first to the last. Note that if a job has been selected in the permutation from the hth column of the matrix, then when selecting the (h + 1)th job in the permutation from the (h + 1)th column of the matrix, we can only select the unselected jobs yet. Suppose these jobs are job k1, job k2,., job kn 2 h, and therefore, the probabilities corresponding to these jobs to be selected are not simply jak1 , h + 1 j2 , jak2 , h + 1 j2 ,.,

4

Advances in Mechanical Engineering

. . jaknh , h + 1 j2 , but jak1 , h + 1 j2 D, jak2 , h + 1 j2 D,., . nP h jaknh , h + 1 j2 D, where D = jaki , h + 1 j2 . i=1

For instance, consider a NWFSP with three jobs. Suppose a qubit individual is represented pffiffiffi as a square matrix with each elements equal to 1= 3. At first, a job is selected by rouletee wheel selection, and from the first column, we know the probabilities corresponding to job 1, job 2, and job 3 are 1/3, 1/3, and 1/3, respectively. According to rouletee wheel selection, a random number rnd is generated from the uniform distribution [0, 1]. If rnd  1=3, job 1 is selected, if 1=3\rnd  2=3, job 2 is selected, and if 2=3\rnd  1, job 3 is selected. Suppose job 2 has been selected in the permutation, and then when selecting the second job from the second column, there are two jobs to select, job 1 and job 3, and the corresponding probabilities are 1/2 and 1/2, respectively. Finally a job permutation will be generated. Such a mechanism which is an innovation in this article can make the permutation solution directly and will be used in the proposed algorithm.

CQGA for the NWFSP The procedure of CQGA for the NWFSP is described in the following: Step 1: let t = 0 and initialize all the parameters and two quantum populations Q1(t), Q2(t). Step 2: make P1(t), P2(t) by observation operation of Q1(t), Q2(t), respectively. Step 3: evaluate P1(t), P2(t). Step 4: store the solutions from P1(t), P2(t) to B1(t), B2(t), respectively, store b and let hbt = bt. Step 5: if a stopping condition is satisfied, then output the results; otherwise, continue the following steps. Step 6: let t = t + 1 and make P1(t), P2(t) by observation operation of Q1(t 2 1), Q2(t 2 1), respectively. Step 7: evaluate P1(t), P2(t). Step 8: update Q1(t 2 1), Q2(t 2 1) using Q-gate. Step 9: store-with-diversity in B1(t), B2(t), and store bt, hbt. Step 10: P1(t), P2(t) are replaced with B1(t), B2(t), respectively. Step 11: compete between P1(t) and P2(t). Step 12: selection, crossover and mutation for P1(t), P2(t), respectively. Step 13: evaluate P1(t), P2(t). Step 14: store-with-diversity in B1(t), B2(t), and store bt, hbt. Step 15: perform the migration and go to Step 5.

The overall framework of CQGA is illustrated in Figure 2. It can be seen that CQGA contains two spaces and each space consists of three populations: Q, P, and B, like QEA. The proposed CQGA involving a competitive co-evolution strategy is a hybridization of QEA with GA, and it adopts a coding and decoding mechanism introduced above. Besides, we put forward the store-with-diversity in CQGA. The details of each step are described as follows: Step 1: initialize: the Gen (iterative generation), Pc (crossover probability), Pm (mutation probability), N (popsize), u (rotation angle), Q1(0) and Q2(0) (quantum population). Q1(t) (or Q2(t)) is made up of N qubit individuals represented as in the proposed mechanism above. All elements in the matrix corresponding to every qubit individual are initiapffiffiffi lized with 1= n, where n is the number of jobs in FSP, which means that all possible jobs are selected in the permutation with the same probability originally. Note: Q1(0) = fq011 , q012 , . . . , q01N g and Q2(0) = fq021 , q022 , . . . , q02N g at generation t = 0. Step 2: this step makes permutation solutions in P1(0), P2(0) by observing the states of Q1(0), Q2(0), respectively, and the observation is performed as in the proposed mechanism above. Note: P1(0) = fp011 , p012 , . . . , p01N g and P2(0) = fp021 , p022 , . . . , p02N g at generation t = 0. For notational simplicity, q is used instead of any individual in Q1(t) or Q2(t), and p in P1(t) or P2(t) is used instead of the corresponding individual of q. Let p = (e1, e2,., en) and q = (aij )n 3 n , then ej (j = 1, 2,., n) is obtained by observing the jth column of q. Step 3: each solution in P1(0), P2(0) is evaluated by calculating the objective value. Given that we are solving the NWFSP, the objective value is obtained according to the formulation in section ‘‘Formulation of the NWFSP.’’ Step 4: the solutions in P1(0), P2(0) are stored into B1(0), B2(0), respectively, where B1(0) = fb011 , b012 , . . . , b01N g and B2(0) = fb021 , b022 , . . . , b02N g. B1(t) and B2(t) are used to store relatively better solutions. At the initial generation, we have b0ij = p0ij , i = 1, 2; j = 1, 2, . . . , N . The best solution in B1(t), B2(t) is stored as bt, and hbt represents the history best solution. Thus hb0 is the same as b0 at t = 0. Step 5: the stopping condition is t = Gen. The best solution for output is hbt. Step 6: permutation solutions in P1(t), P2(t) are formed by observing Q1(t 2 1), Q2(t 2 1), respectively, as in Step 2. Step 7: evaluate each solution as in Step 3.

Deng et al.

5

Figure 2. Overall framework of CQGA.

Step 8: this step updates qubit individuals by operation of a quantum gate which is modified in accordance with the novel coding and decoding mechanism. A Q-gate is an important updating method to drive the individuals toward better solutions in quantum evolutionary algorithm.28 A quantum gate is a reversible gate and can be represented as a unitary operator U satisfying U + U = UU + , where U + is the hermitian adjoint of U. There are several quantum gates, such as the NOT gate, controlled NOT gate, rotation gate, and Hadamard gate. Here, a rotation gate is used. To comply with the proposed coding and decoding mechanism, only two numbers may need to be updated in the jth Q-bit (a1j , a2j , . . . , anj ) as follows 

    cos (uj ) a0 xj axj = U (u = ) j a0 yj ayj sin (uj )

 sin (uj ) cos (uj )



axj ayj

 ð7Þ

where uj is a rotation angle determining the rotation direction and magnitude and its lookup table is presented in Table 1, and x, y are jobs scheduled in the jth position of p, b, respectively, assuming b in B1(t) or B2(t) is the corresponding individual of p.

Table 1. Lookup table of uj . f(p) . f(b)

x 6¼ y

Quadrant (axj , ayj )

uj

True True True True False False False False

True True False False True True False False

1 or 3 2 or 4 1 or 3 2 or 4 1 or 3 2 or 4 1 or 3 2 or 4

u u 0 0 0 0 0 0

Figure 3 depicts the polar plot of the rotation gate for Q-bit individuals. As illustrated in Table 1, the rotation occurs only when the condition f(p) . f(b) is true and x is not the same as y: if (axj , ayj ) is located in the first or the third quadrant in Figure 3, the angle value is set to a positive value u to increase the probability of job y being selected; if (axj , ayj ) is located in the second or the fourth quadrant, u is used to increase the probability of job y being selected. For all the other situations, even when (axj , ayj ) is located in the coordinate axes, uj is set to 0, which means no rotation. It is clear that unitary operator U (uj ) can still meet the demand of normalization. In fact, after the unitary operator U, we have

6

Advances in Mechanical Engineering Step 11: the improved competition is held between P1(t) and P2(t). For each individual p in P1(t), all the individuals in P2(t) are its competitors. The value of competition is defined as c(p) =

X 1 0 t

p2j 2P2(t)

if p is better than pt2j , else

j = 1, 2, . . . , N

ð10Þ It can be seen that c(p) is an integer in the range of [0, N], then the fitness of p is computed as fit(p) = r 3

ð8Þ

which still makes n  X  aij 2 = 1,

j = 1, 2, . . . , n

ð11Þ

where r affects the effects of competition. For each individual in P2(t), all the individuals in P1(t) are its competitors. The competition is performed as above.

Figure 3. Polar plot of rotation gate for Q-bit individuals.

 0 2  0 2  2  2 a xj  + a yj  = axj  + ayj 

c(p) + 1 N

ð9Þ

i=1

Step 9: in population B1(t) and B2(t), there may be some individuals with the same permutations. Therefore, store-with-diversity is proposed in the framework of CQGA to maintain the diversity of populations, which is different from the initial store in Step 4. After store-with-diversity, if the best solution in B1(t) and B2(t) is better than the stored bt21, bt is replaced with the new one, then if bt is better than the stored hbt21, hbt is replaced with bt. In store-with-diversity, the generation t is omitted for simplicity. For population P1, we try to store p11 , p12 , . . . , p1N into b11 , b12 , . . . , b1N , respectively, in sequence. b11 , b12 , . . . , b1N are randomly rearranged to get ready to accept the individuals in P1 before storing. At first, if p11 is better than or equal to b11 , b11 is replaced with p11 , otherwise compare p11 with b12 , if p11 is better than or equal to b12 , b12 is replaced with p11 , otherwise go on till p11 and b1N are compared, when it is worse than all individuals in P1, p11 is not stored. Then we consider p12 , . . . , p1N in sequence. The same operation is performed for population P2. It is clear that store-with-diversity can keep the diversity of populations. Step 10: P1(t), P2(t) are replaced with B1(t), B2(t), respectively. This step prepares for the ensuing competition and the genetic operations.

Step 12: this step executes the genetic evolution on P1(t) and P2(t), respectively. Here, Rouletee wheel selection based on fitness value, partially mapped crossover (PMX),34 and mutation operator INSERT28 is adopted. Step 13: the same as in Step 7. Step 14: the same as in Step 9. Step 15: the migration26 is performed in each generation. The best in B1(t) is swapped with the best in B2(t), so is the worst.

Simulations and comparisons All the algorithms involved in this article were programmed in Visual C++ and conducted on an Intel Pentium IV 3.06 GHz PC with 1024 MB memory. To compare the performance of different algorithms, the experiments were conducted using the well-known Taillard35 benchmark set, which is composed of 120 problem instances, ranging from 20 jobs and five machines to 500 jobs and 20 machines. In our simulations, only the former 90 instances were treated by the compared algorithms for simplicity. The satisfied parameter setting of the CQGA was calibrated, and then we employed the CQGA to solve each instance with replications. The solution quality is evaluated according to the reference makespans. As in Ding et al.,18 the indicator percentage relative deviation (PRD) was adopted to measure the amount of improvement over the reference makespans. Obviously, the smaller PRD is, the better the corresponding solution is. To be specific, PRD is calculated as PRD =

(M  Mref ) 3 100 Mref

ð12Þ

Deng et al.

7

Table 2. Best known solutions for Taillard benchmark set. Instance 20 3 5 Ta01 Ta02 Ta03 Ta04 Ta05 Ta06 Ta07 Ta08 Ta09 Ta10 20 3 10 Ta11 Ta12 Ta13 Ta14 Ta15 Ta16 Ta17 Ta18 Ta19 Ta20 20 3 20 Ta21 Ta22 Ta23 Ta24 Ta25 Ta26 Ta27 Ta28 Ta29 Ta30

Best known solution 1486 1528 1460 1588 1449 1481 1483 1482 1469 1377 2044 2166 1940 1811 1933 1892 1963 2057 1973 2051 2973 2852 3013 3001 3003 2998 3052 2839 3009 2979

Instance 50 3 5 Ta31 Ta32 Ta33 Ta34 Ta35 Ta36 Ta37 Ta38 Ta39 Ta40 50 3 10 Ta41 Ta42 Ta43 Ta44 Ta45 Ta46 Ta47 Ta48 Ta49 Ta50 50 3 20 Ta51 Ta52 Ta53 Ta54 Ta55 Ta56 Ta57 Ta58 Ta59 Ta60

Best known solution 3161 3432 3211 3339 3356 3347 3231 3235 3072 3317 4274 4177 4099 4399 4322 4289 4420 4318 4155 4283 6129 5725 5862 5788 5886 5863 5962 5926 5876 5958

where M, Mref are makespan obtained by algorithms in each run and the reference makespan, respectively. The best known solutions provided in Ding et al.18 are used as reference makespan, as shown in Table 2.

Parameter calibration This part considers the values of the parameters to make CQGA obtain satisfying results for the NWFSP. For fair comparison, we set CQGA with N = 10 n and the stopping criterion as elapsed CPU time not less than 3 n2m milliseconds. For parameter r, our pilot experiment showed that a value between 600 and 1200 is acceptable, and here we set it to 1000 to get a proper competition pressure. Besides, there are three essential parameters u, Pc, and Pm to be determined. To get a convincing parameter setting, a design of experiments was performed. The parameters were tested at the following levels: (a) u = 0.0005p, 0.002p, 0.01p, 0.05p; (b) Pc = 0.6, 0.7, 0.8, 0.9; and (c) Pm = 0.005, 0.025, 0.1, 0.2. Therefore, 43 parameter configurations were examined. We selected nine instances, Ta01, Ta11,.,

Instance 100 3 5 Ta61 Ta62 Ta63 Ta64 Ta65 Ta66 Ta67 Ta68 Ta69 Ta70 100 3 10 Ta71 Ta72 Ta73 Ta74 Ta75 Ta76 Ta77 Ta78 Ta79 Ta80 100 3 20 Ta81 Ta82 Ta83 Ta84 Ta85 Ta86 Ta87 Ta88 Ta89 Ta90

Best known solution 6397 6234 6121 6026 6200 6074 6247 6130 6370 6381 8077 7880 8028 8348 7958 7801 7866 7913 8161 8114 10,700 10,594 10,611 10,607 10,539 10,690 10,825 10,839 10,723 10,798

Ta81, from each problem group to avoid bias of the results, and the algorithm is run five replications for each configuration for each considered instance. In total, we have 43 3 9 3 5 = 2880 runs. Based on such a large data set, the analysis of variance (ANOVA) results are shown in Table 3. It can be seen from Table 3 that since its p-value is less than 0.0001, the parameters u and Pc are both significant, while that parameters Pm with p-value equal to 0.1586 is less significant. To show the performance differences of parameter levels more clearly, the one factor means plots with 95% least significant difference (LSD) confidence intervals of the factors u and Pc are shown in Figure 4. Note that if the LSD intervals for two means are not overlapping, then the means are considered significantly different. Therefore, Figure 4 illustrates that for factor u, the value 0.01p is significantly better than the others, while for factor Pc, the value 0.8 is a relatively better choice. As regards parameter Pm, the differences are small and its means plot is omitted for simplicity, and in the performance test, it is set as 0.1.

8

Advances in Mechanical Engineering

Table 3. ANOVA results for the experiment on the parameter calibration. Source Main effects A: theta B: pc C: pm Interactions AB AC BC

Sum of squares

Df

F-value

p-Value

6.38 4.85 0.72

3 3 3

57.43 22.66 2.82

\0.0001 \0.0001 0.1586

2.174e2004 3.509e2004 4.21

9 9 9

4.346e2003 2.105e2003 2.64

1.0000 1.0000 \0.0001

Figure 4. Means plot with 95% LSD intervals for the parameters of the algorithm.

Comparisons of VNS, GASA, hybrid particle swarm optimization, and CQGA For the NWFSP, the GA10 and GASA11 were impressive because they successfully solved the problem in a considerably minor time. Besides, the hybrid particle swarm optimization (HPSO) proposed by Liu et al.13 also showed its excellent performance. As far as we know, no literature could be found about quantum algorithms for the NWFSP, and in this part, we will compare CQGA with GA, GASA, and HPSO. The computational experiments were based on the 90 Taillard instances, and each algorithm was run for each instance with 30 replications. The stopping criterion was 3 n2m milliseconds CPU time to get an unbiased statistical result. The average percentage relative

deviation (APRD) values grouped in subsets of different sizes are summarized in Table 4. It should be noted that the compared algorithms were executed in the same running environment and with the stopping criterion, so the results are comparable. We can see from Table 4 that the overall mean PRD value obtained by the CQGA is 2.76, which is substantially lower than that of the HPSO (3.20), that of the GASA (4.13), and that of the HPSO (5.55). The similar priority relation holds when the APRD values of each subset of different sizes are examined. In order to observe whether the performance differences in the PRD values are indeed statistically significant, we employed the original PRD data points and conducted an ANOVA. To be brief, we only provide

Deng et al.

9

Table 4. APRD of each algorithm on Taillard benchmark set. n3m

GA

GASA

HPSO

CQGA

20 3 5 20 3 10 20 3 20 50 3 5 50 3 10 50 3 20 100 3 5 100 3 10 100 3 20 Overall mean

4.39 4.65 4.77 4.94 5.30 6.62 5.85 6.52 6.94 5.55

2.97 3.19 3.36 3.45 3.83 5.23 4.29 5.14 5.67 4.13

1.97 2.25 2.46 2.57 2.96 4.48 3.32 4.17 4.62 3.20

1.62 1.84 1.98 2.17 2.49 3.91 2.86 3.75 4.17 2.76

GA: genetic algorithm; GASA: genetic algorithm hybridized with simulated annealing; HPSO: hybrid particle swarm optimization; CQGA: co-evolutionary quantum genetic algorithm.

individual is represented as a square matrix, and a job permutation can thus be simply gained without use of random-key representation. In the framework combining QEA and GA, we put forward the store-withdiversity to maintain the diversity of populations. A competitive co-evolution is introduced in the algorithm to achieve a balance between exploration and exploitation. Simulation results and comparisons based on a set of benchmarks demonstrate the effectiveness and robustness of the proposed algorithm. Our future work lies in two directions. One is to apply quantum-inspired hybrid algorithms to the FSP with no-idle constraint,36 as well as job shop scheduling problems. The other is to consider more powerful problem-dependent algorithms for the NWFSP and its multiobjective optimization. Declaration of conflicting interests

Figure 5. Means plot with 95% LSD intervals for the compared algorithms.

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding the one factor means plots with 95% least significant difference (LSD) confidence intervals of the algorithms in Figure 5. The statistical results indicate that the differences between these algorithms are significant. The GA yields the poorest results while the CQGA is superior to the others. It is worth noting that the HPSO is also a positive algorithm because it achieved excellent results whereas its structure is quite simple.

Conclusion This article presents an effective and robust quantuminspired hybrid metaheuristic combining QEA and GA for the NWFSP. According to the characteristics of the problem, an innovative coding and decoding mechanism is presented. In this mechanism, a quantum

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by National Natural Science Foundation of China (Grant No. 61403180), The Project for Introducing Talents of Ludong University (LY2013005), National Natural Science Foundation of China (Grant No. 51407088), National Natural Science Foundation of China (Grant No. 61573144), and The Project of Shandong Province Higher Educational Science and Technology Program (Grant No. J14LN20).

References 1. Graham RL, Lawler EL, Lenstra JK, et al. Optimization and approximation in deterministic sequencing and scheduling: a survey. Ann Discrete Math 1979; 5: 287–326. 2. Moradinasab N, Shafaei R and Rabiee M. No-wait two stage hybrid flow shop scheduling with genetic and

10

3.

4.

5.

6.

7. 8. 9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

Advances in Mechanical Engineering adaptive imperialist competitive algorithms. J Exp Theor Artif In 2013; 25: 207–225. Rabiee M, Rad R and Mazinani M. An intelligent hybrid meta-heuristic for solving a case of no-wait two-stage flexible flow shop scheduling problem with unrelated parallel machines. Int J Adv Manuf Tech 2014; 71: 1229–1245. Garey MR and Johnson DS. Computers and intractability: a guide to the theory of NP-completeness. San Francisco, CA: Freeman, 1979. Papadimitriou C and Kanellakis PC. Flowshop scheduling with limited temporary storage. J Assoc Comput Mach 1980; 27: 533–549. Bonney MC and Gundry SW. Solutions to the constrained flowshop sequencing problem. Oper Res Quart 1976; 24: 869–883. King JR and Spachis AS. Heuristics for flowshop scheduling. Int J Prod Res 1980; 18: 343–357. Rajendran C. A no-wait flowshop scheduling heuristic to minimize makespan. J Oper Res Soc 1994; 45: 472–478. Gangadharan R and Rajendran C. Heuristic algorithms for scheduling in the no-wait flowshop. Int J Prod Econ 1993; 32: 285–290. Aldowaisan T and Allahverdi A. A new heuristics for no-wait flowshops to minimize makespan. Comput Oper Res 2003; 30: 1219–1231. Schuster CJ and Framinan JM. Approximate procedure for no-wait job shop scheduling. Oper Res Lett 2003; 31: 308–318. Grabowski J and Pempera J. Some local search algorithms for no-wait flow-shop problem with makespan criterion. Comput Oper Res 2005; 32: 2197–2212. Liu B, Wang L and Jin YH. An effective hybrid particle swarm optimization for no-wait flow shop scheduling. Int J Adv Manuf Tech 2007; 31: 1001–1011. Pan QK, Tasgetiren MF and Liang YC. A discrete particle swarm optimization algorithm for the no-wait flow shop scheduling problem. Comput Oper Res 2008; 35: 2807–2839. Pan QK, Wang L and Zhao BH. An improved iterated greedy algorithm for the no-wait flow shop scheduling problem with makespan criterion. Int J Adv Manuf Tech 2008; 38: 778–786. Qian B, Wang L, Hu R, et al. A DE-based approach to no-wait flow-shop scheduling. Comput Ind Eng 2009; 57: 787–805. Davendra D, Zelinka I and Bialic-Davendra M. Discrete self-organising migrating algorithm for flow-shop scheduling with no-wait makespan. Math Comput Model 2013; 57: 100–110. Ding J, Song S, Gupta JND, et al. An improved iterated greedy algorithm with a Tabu-based reconstruction strategy for the no-wait flowshop scheduling problem. Appl Soft Comput 2015; 30: 604–613. Gao K, Pan Q and Li J. Discrete harmony search algorithm for the no-wait flow shop scheduling problem with total flow time criterion. Int J Adv Manuf Tech 2011; 56: 683–692. Gao K, Pan Q and Li J. A hybrid harmony search algorithm for the no-wait flow-shop scheduling problems. Asia Pac J Oper Res 2012; 29: 1250012.1–1250012.23.

21. Akhshabi M, Tavakkoli-Moghaddam R and RahnamayRoodposhti F. A hybrid particle swarm optimization algorithm for a no-wait flow shop scheduling problem with the total flow time. Int J Adv Manuf Tech 2014; 70: 1181–1188. 22. Gao K, Pan Q and Suganthan PN. Effective heuristics for the no-wait flow shop scheduling problem with total flow time minimization. Int J Adv Manuf Tech 2013; 66: 1563–1572. 23. Sapkal SU and Laha D. A heuristic for no-wait flow shop scheduling. Int J Adv Manuf Tech 2013; 68: 1327–1338. 24. Laha D and Sapkal SU. An improved heuristic to minimize total flow time for scheduling in the m-machine no-wait flow shop. Comput Ind Eng 2014; 67: 36–43. 25. Grover LK. A fast quantum mechanical algorithm for database search. In: Proceedings of the 28th ACM symposium theory of computing, Philadelphia, Pennsylvania, 26– 29 May 1996, pp.212–219. New York: ACM Press. 26. Narayanan A and Moore M. Quantum-inspired genetic algorithms. In: Proceedings of the 1996 IEEE international conference on evolutionary computation (ICEC96), Nagoya, Japan, 20–22 May 1996, pp.61–66. New York: IEEE Press. 27. Han KH and Kim JH. Quantum-inspired evolutionary algorithm for a class of combinatorial optimization. IEEE T Evolut Comput 2002; 6: 580–593. 28. Li B and Wang L. A hybrid quantum-inspired genetic algorithm for multiobjective flow shop scheduling. IEEE T Syst Man Cy B 2007; 37: 576–591. 29. Gu J, Gu M, Cao C, et al. A novel competitive co-evolutionary quantum genetic algorithm for stochastic job shop scheduling problem. Comput Oper Res 2010; 37: 927–937. 30. Gu J, Gu X and Jiao B. A quantum genetic based scheduling algorithm for stochastic flow shop scheduling problem with random breakdown. In: Proceeding of the 17th IFAC world congress, Seoul, South Korea, 6–9 July 2008, pp.63–68. COEX: IFAC. 31. Gu J, Gu X and Jiao B. Solving stochastic earliness and tardiness parallel machine scheduling using quantum genetic algorithm. In: Proceedings of the seventh world congress on intelligent control and automation, Chongqing, China, 25–27 June 2008, pp.4154–4159. New York: IEEE Press. 32. Gu J, Gu M and Gu X. A mutualism quantum genetic algorithm to optimize the flow shop scheduling with pickup and delivery considerations. Math Probl Eng 2015; 2015: 387082. 33. Rosin CD and Belew RK. Methods for competitive co-evolution: finding opponents worth beating. In: Proceedings of the sixth international conference on genetic algorithms, Pittsburgh, PA, 15–19 July 1995, pp.373–380. San Mateo, CA: Morgan Kaufman. 34. Chen CL, Neppalli RV and Aljaber N. Genetic algorithms applied to the continuous flow shop problem. Comput Ind Eng 1996; 30: 919–929. 35. Taillard E. Benchmarks for basic scheduling problems. Eur J Oper Res 1993; 64: 278–285. 36. Deng GL and Gu XS. A hybrid discrete differential evolution algorithm for the no-idle permutation flow shop scheduling problem with makespan criterion. Comput Oper Res 2012; 39: 2152–2160.