A quantum ‐ inspired cuckoo co ‐ evolutionary algorithm for no ‐ wait flow shop scheduling

No ‐ wait flow shop scheduling problems (NWFSPs) are widespread in practical applications. The authors propose a quantum ‐ inspired cuckoo co ‐ evolutionary algorithm for the NWFSP to minimize the makespan. There are three algorithm components: quantum solution construction, quantum population evolution, and an improved neighbourhood local search. They generate initial solutions, search solutions, and improve solution qualities, respectively. Parameters of the proposed algorithm are calibrated statistically. The proposal with calibrated parameters is compared with three existing algorithms on Reeves and Taillard benchmark instances with middle scales. Experimental results show that the proposal outperforms the compared algorithms.


| INTRODUCTION
The no-wait flow shop scheduling problem (NWFSP) is a significant combinatorial optimization problem, which has been studied widely because of its applications in the industrial field, for example, steel, food processing, and chemical engineering [1][2][3]. The NWFSP has attracted significant attention from researchers since the 1950s [4][5][6][7]. A comprehensive survey on the research and application of the NWFSP can be found in Hall and Sriskandarayah's review paper [8]. In the NWFSP, waiting is not permitted, that is, once the processing of a job begins, subsequent processing must be continuously carried out on all machines with no interruption until its completion because of the constraints on processing jobs by equipment and technique. The common optimal objectives of the NWFSP include minimizing the makespan [9], total tardiness [10], and total flow time [11]. Among them, minimizing the makespan is an essential objective of the actual application.
NWFSP has been proved to be an NP-hard problem when the number of machines is more than two [12]. The heuristic and meta-heuristic algorithms are the most usual methods to solve the NWFSP. For decades, heuristics and meta-heuristics have been mainly developed for solving the NWFSP. Regarding heuristics, Allahverdi and Aldowaisan [13,14] researched the NWFSP with a separate setup of three machines and sequence-dependent additive changeover time. Aldowaisan and Allahverdi [15] presented several heuristics for the NWFSP with setup times and m-machine constraints. Li et al. [16] introduced an objective increment method to judge whether a new schedule is better than its parent. The time complexity of the proposed heuristic algorithm can be decreased by one order. Gao et al. [17] presented two algorithms: ISDH (Improved Standard Deviation Heuristic) and IBH (Improved Bertolissi Heuristic). Computational results show that the composite heuristics perform significantly better than the existing ones. Sapkal and Laha [18] presented an efficient heuristic method in which the initial sequence of jobs is generated based on bottleneck machines.
Various noteworthy meta-heuristics have been developed for solving the NWFSP. A hybrid differential evolution (HDE) algorithm is proposed by Qian et al. [19]. Simulation results demonstrated its effectiveness. Pan et al. [20] presented a discrete particle swarm optimization (DPSO) algorithm with variable neighbourhood descent (VND) local search for minimizing both makespan and total flow time. It was hybridized with the VND algorithm to improve the quality of solutions further. Tseng and Lin [21] proposed a hybrid genetic algorithm (HGA), which integrated the genetic algorithm (GA) into a novel local search scheme, and the results showed the superiority of the HGA. Jarboui et al. [22] designed a block order crossover operator for GA, regarded as GA-VNS. The experiments showed that GA-VNS provides competitive results and better upper bounds. Samarghandi and ElMekkawy [23] presented a hybrid tabu search (TS) and particle swarm optimization (PSO) algorithm, in which a new encoding scheme based on the natural relation between permutations and factoradics was developed. Akrout et al. [24] presented a hybrid greedy randomized adaptive search procedure (GRASP) of which the parameters are adjusted by differential evolution. Ding et al. [25] presented a tabu-mechanism improved iterated greedy algorithm (TMIIG) which adopts a tabu-based reconstruction strategy in the iterated greedy (IG) algorithm to improve the searching ability and the neighbourhood methods to obtain better solutions. Engin and Guclu [26] designed a hybrid ant colony optimization algorithm for solving the NWFSP with the makespan and the flowtime, which hybridized ant colony optimization (ACO) and simulated annealing (SA) algorithm. Zhao et al. [27] proposed a factorial based particle swarm optimization with a population adaptation (PA) mechanism to minimize the makespan in which the PA mechanism was designed to control the diversity of the population. Lin and Ying [28] proposed two efficient meta-heuristics to minimize the makespan for an asymmetric travelling salesman problem. Li et al. [29]researched the NWFSP with sequence-dependent setup times, learning and forgetting effects, and proposed an iterated greedy heuristic algorithm to minimize total flow time. Wang et al. [30] proposed an iterated greedy heuristic algorithm for solving mixed NWFSP with both wait and no-wait constraints. Shao et al. [31] considered a multiobjective distributed NWFSP with sequence-dependent setup time (MDNWFSP-SDST). In MDNWFSP-SDST, a Pareto-based estimation of distribution algorithm is presented for the makespan and the total weight tardiness objectives.
In addition, quantum computing has been drawn much attention and has widespread applications in combinatorial optimization. Since the late 1990s, a variety of quantuminspired evolutionary algorithms such as the geneticquantum algorithm [32], quantum-inspired gravitational search algorithm [33], and quantum differential evolution algorithm [34] have been successfully utilized in solving different engineering and combinatorial optimization problem. It is found that there is a growing interest in emerging quantum computing with evolutionary algorithms. One of the developed evolutionary algorithms is the cuckoo search (CS) algorithm [35] with global convergence, convenient to be coupled with other algorithms. Furthermore, the CS algorithm has been proved to be a useful optimization tool in practice. It is desirable to combine them for global optimization problems. As NWFSPs have unique characteristics, it is challenging to develop quantum-inspired evolutionary algorithms for them [36]. To the best of our knowledge, there are quite a few co-evolutionary algorithms for NWFSPs.
The authors propose an effective Quantum-inspired Cuckoo Co-evolutionary Algorithm (QCCA) with an improved local neighbourhood search method for the considered NWFSP to minimize makespan. The proposed algorithm combines quantum evolution and the cuckoo search mechanism. Furthermore, the QCCA with improved local neighbourhood search, which is called QCCA-INLS, enhances the quality of the solution. The main contributions of this paper are summarized as follows: � A novel quantum solution construction (QSC) method is developed for the NWFSP to encode the population and constructs diversiform initial solutions. � A quantum population evolutionary (QPE) method is designed with the cuckoo search mechanism and quantum NOT, which can effectively find the global solution. � To efficiently improve the solution's quality, an improved local neighbourhood search (INLS) is presented for minimizing the makespan.
The remainder of the paper is organized as follows. Section 2 describes and formalizes the problem of minimizing the makespan in NWFSP. The proposed algorithm is presented in Section 3. Experimental evaluations of parameter calibration and algorithm comparison are shown in Section 4, followed by the conclusions and future research in Section 5.

| PROBLEM FORMULATION
In this section, we present a formal description of the problem. The major symbols and definitions used for the description are given in Table 1.
The NWFSP with makespan minimization is denoted as F m |nwt|C max , described as follows. There are n jobs (J 1 , J 2 , …, J n ) to be processed on a set of m machines (M 1 , M 2 , …, M m ). All jobs are processed in the same order without any pre-emption and interruption, that is, once a job is started on the first machine, it has to be continuously processed through m machines without interruption. At a time, job is being processed at most by one machine, and machine M k (1 ≤ k ≤ m) can execute no more than one job. The objective of the problem is to find a feasible schedule π = (π 1 , π 2 , …, π n ) for the n jobs such that the makespan denoted as C max (π) is minimized, where C max (π) is also equivalent to the finishing time of the last job on the last machine, which can be denoted as C max (π n , m) and is obtained using the following equation The no-wait constraints of the problem ensure that the completion time distance between two adjacent jobs is determined by the processing times of two jobs, regardless of the other jobs in the permutation. Therefore, to satisfy the no-wait constraint, the start of some jobs on the first machine needs to be delayed. A completion time distance is defined between each pair of jobs. The completion time distance between two adjacent jobs, π k−1 and π k , is calculated by As an example shown in Figure 1, three jobs are processed on three machines. It assumes that the processing sequence of three jobs is π = (π 1 , π 2 , π 3 ). As illustrated in Figure 1, it is clear that the C max (π) for processing three jobs can be easily obtained by summing up the completion time distances ðD π 1 ;π 2 ; D π 2 ;π 3 Þ and the total processing time ∑ 3 j¼1 P π 3 ;j of these three jobs, which can be formulated by

| QUANTUM-INSPIRED CUCKOO CO-EVOLUTIONARY ALGORITHM
In this section, we introduce an efficient quantum-inspired cuckoo co-evolutionary algorithm for F m |nwt|C max . QCCA is cooperative with quantum-inspired evolution and cuckoo search mechanism. Firstly, we adopt a QSC to initialize the job sequence, a better characteristic of population diversity. Then, QPE hybridized CS is used to iterate the population to drive them towards better solutions. However, in the generation, Best schedule involving a permutation of the given jobs Completion time distance between adjacent jobs in position k − 1 and k of a schedule π Makespan of a given schedule π ZHU ET AL. -107 there are some individuals eliminated with a certain probability. Thus, a variation operator is applied to the quantum encoding of these individuals. Finally, an INLS is presented to intensify the quality of the global solution of QCCA. Therefore, the QCCA with INLS is also called QCCA-INLS. The QCCA-INLS framework is detailed in Algorithm 1.

| Quantum solution construction
To encode the population, we use a qubit as the probabilistic representation of an individual, represented by the superposition of '1' state, '0' state, or any superposition of the two. The state of a qubit is given by where α and β are complex numbers that specify the probability amplitudes of the corresponding states. Normalization of the state to unity guarantees |α| 2 + |β| 2 = 1. Therefore, the probability amplitudes of a qubit can be expressed by a quantum angle, which is given by where θ is the quantum angle. Thus, a qubit can be expressed by |Φ〉 = [ cos θ sin θ] T . A qubit description is shown in Figure 2.The quantum encoding of the qubit is directly used to generate cuckoo populations initially. The cuckoo population can be formulated by Q(t) = {q i (t)|i = 1, 2, …, ps} at generation t. A quantum individual q i (t) can be given by where ps is the population size, n is the length of a qubit as well as the number of jobs, and θ is randomly generated in [0, 2π]. For solving F m |nwt|C max , a quantum individual solution construction (QISC) is used to transform individuals from quantum encoding to job sequences so that the quantum representation of solutions can be applied for solving NWFSPs.
The quantum individual solution construction is used for the QSC, formally described as Algorithm 2. We denote the maximum value MAX as the initial C max (π 0 ) and the initial solution π 0 is set to ∅. π i (0) is the corresponding solution of the ith quantum individual. Each quantum individual is generated by the procedure of QISC and evaluated by C max . The time complexity of QSC is O(n).

| Quantum population evolution
In quantum-inspired evolutionary algorithm (QEA), the following quantum rotation gate is adopted to update the iterative population.
Therefore, a qubit can be updated by where ▵θ is the rotation angle as Figure 4 illustrated. The quantum individual q i (t + 1) is updated by The cuckoo search simulates the process that cuckoos lay their eggs in the nests of other host birds. The cuckoo population searches new nests by Lévy flight at the generation. We consider generating quantum angle increment of the population by cuckoo search behaviour, which is performed by where α(α > 0) is the step factor which should be related to the scale of F m |nwt|C max . In most cases, we can use α = O(1) in Equation (10). The symbol ⊗ presents entry-wise multiplications. Lévy flights essentially provide a random walk while their random steps are drawn from a Lévy distribution for large steps where β ∈ [1, 2] is Lévy exponent, μ and ν are drawn from normal distributions with respectively, and Γ being the standard Gamma function.
During the procedure of quantum population evolution, there are some worse nests abandoned. Therefore, new nests can be discovered with a P a probability based on the cuckoo search mechanism. Moreover, a quantum mutation operation can modify the job sequence and the best solution is reserved for the next generation. Quantum NOT gate is used to generate some mutated individuals to avoid the premature convergence in quantum population evolution. The qubit of a mutated quantum individual is operated by -109 The procedure of QPE is described as Algorithm 3. The QPE contains two layers of the loop. The inner loop shows the procedure of quantum individual updates. The time complexity of this loop is O(n). The outer loop presents the number of iterations t and t is constant. The time complexity of QPE is O (cn).
For illustrating the mutation operation procedure distinctly, an example is shown in Figure 5, which is associated with Figure 3. From Figure 5, the job sequence is a solution of the NWFSP, obtained in the procedure of population evolution. It assumes that the quantum individuals of jobs J 2 and J 6 are mutational with a P a probability, which mutates according to quantum NOT gate. Then, the mutated quantum individuals are encoded to a new job sequence by the QISC method.

| Improved neighbourhood local search
Because of F m |nwt|C max is an NP-hard problem in the which solution space is very complicated, so a fast and efficient local search algorithm is essential [37].
The proposed INLS maintains the diversity of the population and avoids falling into a local optimum. The INLS is based on insert operation and swap operation. We give some job set definitions for searching optimized solutions.
There are two random jobs π i , π j (1 ≤ i ≤ n, 1 ≤ j ≤ n) and their neighbourhood jobs π i−1 , π i+1 , π j−1 , π j+1 . π local = {π i−1 , π i , π i+1 , π j−1 , π j , π j+1 } is denoted as a local neighbourhood job set, which is illustrated in Figure 6. The remaining jobs are denoted as π rem = π global −π local . If i and j is the insert position, we denote the insert point set I = {I point } = {(π i−1 , π i ), (π i , π i+1 ), (π j−1 , π j ), (π j , π j+1 )}. Any job from π rem can execute swap operation with jobs from π local . Two jobs π y from π rem and π x from π local can be swapped as shown in Figure 7. In Figure 7, π x can be selected from π i−1 , π i , π i+1 and π j−1 , π j , π j+1 which are also π local . π y can be exchanged with π x , which is anyone of π local . In the same way, π y can also be inserted to any point from I, as shown in Figure 8. After swap operation and insert operation, the new π global is re-

| EXPERIMENTAL EVALUATIONS
To evaluate the performance of QCCA-INLS for solving the F m |nwt|C max , the parameter calibration and performance evaluation are described in this section. All tested algorithms are coded in C++ and run on an Intel i7-3770 CPU and 3.10 GHz with 8 GB of RAM. The termination criterion is set to a maximum iteration.

| Parameter calibration
As previously discussed, the proposed algorithm has several parameters: (1) the population size (ps), (2) maximum iteration (max_iter), (3) the step factor (α), (4) the Lévy exponent (β), and (5) the mutation probability (P a ). The tested values of the five parameters to be calibrated are set as ps ∈ {50, 100, 150}, max_iter ∈ {30, 50, 70}, α ∈ {0.01, 0.05, 0.1}, β ∈ {1, 1.5, 2}, and P a ∈ {0.25, 0.5, 0.75}. There are 3 5 = 243 combinations of calibration instances with five parameters. Each combination runs 15 times on each NWFSP instance. Solutions are evaluated by the RPD (Relative Percentage Deviation) defined as follows: where S is the solution obtained by the corresponding algorithm on a given instance and S best is the best solution obtained on the same instance. Experimental results are analysed by the ANOVA (analysis of variance) technique. First, the three main hypotheses (normality, homoscedasticity, and independence of the residuals) are checked from the experiments' residuals. All three hypotheses are acceptable from this analysis. When the p-value is infinitesimal and close to zero, they are significant statistically. Figures 9-13 show the means plot and 95.0% confidence level Tukey HSD (honest significant difference) intervals for ps, max_iter, α, β, and P a , respectively.
Since the p-value in Figure 9 is close to 0, Figure 9 illustrates that the observed differences are statistically significant for ps. It can be observed that ps with 150 is significantly better than that with 100 and 50. It implies that a relatively big population size can get a good solution. Similar to   Figure 9, Figure 10 with infinitesimal p-value illustrates that the differences are statistically significant for max_iter. It follows that the proposed algorithm has minimal RPD when max_iter = 70. Thus, relatively numerous iterations can also get a good solution. Figures 11-13 indicate that the differences are statistically significant for different values of α, β, and P a . It implies that different values of α, β, and P a exert a great influence on the proposed algorithm's performance. Similarly, it can be observed that there is an increasing tendency of RPD with a decrease in α, β, and P a . As Figures 11-13 show, the proposed algorithm has the minimal RPD when α = 0.01, β = 1.0, and P a = 0.25. Therefore, we set α = 0.01, β = 1.0, and P a = 0.25 in the following experiments.
To make comparisons more intuitively, Figure 14 shows the interactions between the compared algorithms and Reeves benchmark instances with 95.0% confidence level Tukey HSD intervals. The p-value is close to 0, indicating that the RPD of QCCA-INLS is significantly smaller than those of the other four algorithms from Figure 14. Therefore, we can see that the QCCA-INLS is statistically better than HGA, TS-PSO, TMIIG, and QCCA in all the Reeves benchmark instances. Table 3 shows the RPDs and the average RPDs of HGA, TMIIG, and QCCA-INLS in Taillard benchmark instances. From Table 3 Figure 15 shows the interactions between the compared algorithms and Taillard benchmark instances with 95.0% confidence level Tukey HSD intervals. The RPDs of QCCA-ILNS are significantly smaller than those of HGA and TMIIG statistically with the number of jobs when n ≤ 100, which indicates that the QCCA-ILNS outperforms HGA and TMIIG statistically in small and middle scales of the Taillard benchmark. Therefore, the QCCA-ILNS is effective for solving NWFSP in small and middle scales.

| CONCLUSIONS AND FUTURE WORKS
The authors have studied NWFSP for minimizing the makespan. Based on the characteristics and objective of the NWFSP, we proposed the QCCA-INLS framework which consists of QSC, QPE, and INLS. QSC generates population by quantum encode to enhance the population diversity.
In the QPE, we developed a co-evolutionary procedure to search global solutions iteratively. To improve the intensification and efficiency of QCCA, quantum NOT is used to enhance the search to avoid falling into local optimization.
For improving the quality of the solution, the INLS method is presented for searching optimized solution. INLS is based on neighbourhood search and local search, which can speed up the search. By comparing with three existing algorithms, QCCA and QCCA-INLS show better performance for the problem under study. It is worthwhile to investigate mixed FSPs that consider the wait and no-wait constraints. It may commonly exist in practical industries and are promising topics to investigate in the future. Further refinements of the algorithm performance are also critical future avenues of research.