An Adaptive Cooperated Shuffled Frog-Leaping Algorithm for Parallel Batch Processing Machines Scheduling in Fabric Dyeing Processes

1  Introduction

Batch processing machine (BPM) has been widely applied in many real applications such as fabric dyeing [1], chemical production [2], tire manufacturing [3], steel smelting [4], semiconductor manufacturing [5]. BPMs are capable of simultaneously processing multiple jobs within a single batch, where all jobs in the batch share the same start and end times. Scheduling problems involving BPM can be broadly divided into single BPM scheduling [6,7], parallel BPM scheduling, and other BPM problems [8,9]. Parallel BPM scheduling, as a representative type, is an extension of the parallel machine scheduling problem and has attracted extensive research attention in recent years.

Table 1 summarizes relevant research. In Scenario 1, researchers use different algorithms to solve BPM-related scheduling problems in various industries, aiming at different objectives. Scenario 2 focuses on the shuffled frog-leaping algorithm (SFLA) and its applications in scheduling problems. Scenario 3 covers other multi-objective scheduling problems solved by different algorithms. However, parallel BPM scheduling in fabric dyeing still has many unsolved problems.

The rest of the paper is organized as follows. Section 2 describes the parallel BPM scheduling problem defined by the fabric dyeing process. Section 3 introduces the SFLA. Section 4 presents the ACSFLA for the considered problem in the fabric dyeing process. The computational experiments are shown in Section 5. In the last section, conclusions are drawn, and potential topics for future research are discussed.

2  Problem Description

Zhang et al. [1] developed a Mixed-Integer Linear Programming (MILP) model for parallel BPM scheduling based on a real fabric dyeing process, which is described as follows. There are n jobs J1, J2, ..., Jn waiting to be processed by m parallel BPMs M1, M2, ..., Mm. For each job Ji, it has a weight of wi, a due date of di, a set Θi of eligible machines where Θi⊆{M1,M2,...,Mm}, and belongs to the family fai. Meanwhile, each BPM Mi has a volume of vk, representing its weight capacity. The notations and descriptions for MILP are listed in Table 2.

All jobs are classified into F families according to the required dyeing color. Jobs within the same family share the same processing time, with pg denoting the processing time of each job in family fg. Families are not compatible with each other, meaning jobs from different families cannot be processed simultaneously in a batch. Each bach Bh is composed of jobs from the same family. When forming a batch Bh on Mk with jobs form fg, job Ji can be added to Bh if wi+∑j∈Bhwj≤vk. Jobs in the same batch share start and end times, with batch processing time equal to that of the job family. When a machine Mk switches from family fa to a different family fb, a pre-specified setup time sab is needed for cleaning.

There are some constraints on jobs and machines:

Each BPM can only process one batch at a time.

No job can be processed in distinct batches across multiple BPMs.

The processing cannot be interrupted.

All BPMs and jobs are accessible at all times.

Solving the BPMs scheduling problem requires addressing three sub-problems: (1) batch formation determines which jobs are grouped to form each batch; (2) BPM assignment selects an Mk for each batch; and (3) batch scheduling defines the processing sequence of all batches on each Mk.

Suppose that batches B1,B2,...,Buk are processed on Mk consecutively. The completion time Ci of job Ji within these batches can be calculated as Eq. (1).

Ci=pfB1+∑a=2hsfBa−1fBa+pfBa(1)

where fB1 represents the family of batch Ba, which equal to fai, i∈Ba.

The goal of the problem is to simultaneously minimize the maximum completion time and total tardiness of all jobs, whilst satisfying all constraints. The makespan Cmax is defined as Eq. (2). The total tardiness TT is defined as Eq. (3).

Minimize Cmax=max{Ci|i=1,2,...,n}(2)

Minimize TT=∑i=1nmax{Ci−di,0}(3)

The MILP model proposed by Zhang et al. [1] can be directly applied once the total finishing time of all machines is replaced by the maximum finishing time of all machines, which is equal to Cmax.

For the problem with Cmax and TT, x≻y means that x dominates y if Cmax(x)≤Cmax(y) and TT(x)≤TT(y), with either Cmax(x)<Cmax(y) or TT(x)<TT(y). If neither x≻y nor y≻x holds, then x and y are non-dominated.

3  Introduction to SFLA

The SFLA [24] is a meta-heuristic based on the behavior of frogs. In SFLA, each solution represents the position of a frog, and the population of possible solutions is modeled as a set of virtual frogs. After generating the initial population P, the algorithm iteratively performs population division, memeplex search, and population shuffling until the stopping criterion is satisfied.

Population division is shown below. All solutions are sorted in the descending order of fitness, suppose that Fit1≥Fit2≥...≥FitN, then assign xk, k=1,2,...,N into memeplex k(mod s)+1, s memeplexes ℳ1,ℳ2,...,ℳs are finally obtained, where k(mod s) means the remainder of k/s and Fiti is the fitness of solution xi.

The search process in memeplex ℳi is described below. xw is used as an optimization object, repeat the following steps μ times: a new solution xw′ is produced by Eq. (4) with xw and xb, if xw′ is better than xw, then replace xw with xw′; otherwise, xw and gbest are used to generate a solution xw′ becomes the new xw′ by Eq. (5); otherwise, replace xw with a random solution directly, where xw, xb and gbest are the worst solution and best solution and best solution in memeplex ℳi and the best solution of P:

xw′=xw+rand×(xb−xw),(4)

xw′=xw+rand×(gbest−xw),(5)

where rand is a random real number following uniform distribution in [0,1].

A new population P is constructed by shuffling all evolved memeplexes.

4  The Proposed ACSFLA

In ACSFLA, both the quality of the memeplex and its evolution quality are considered to determine the shuffling pool for partial population shuffling and the diverse search process of the memeplex. Meanwhile cooperated search is conducted within the diversified search processes, which is determined by the degree of dominance between the two memeplexes. The detailed steps of ACSFLA are shown below.

4.1 Population Initialization and Adaptive Division

The solution representation and decoding process follow the approach [25], which is described as follows. For the BPM scheduling problem with n jobs, m BPMs, and F families, use two-string representation. A scheduling string [π1,π2,…,πn] and a machine assignment string [θ1,θ2,…,θn] for batches, where πi∈{1,2,…,n} and θi∈{1,2,…,m}. The decoding process, based on the scheduling string sequence and machine capacity constraints, repeatedly assigns jobs to batches on machines until all jobs are assigned. Finally, the formed batches are processed on each machine in the first formed first processing rule.

The population initialization process generates an initial population P consists of N solutions, as follows: N/2 solutions are randomly generated to ensure diversity, while the remaining N/2 solutions are produced using a heuristic method [25].

ACSFLA introduces an adaptive population division strategy. This strategy can adjust the population division targets based on P¯ generated in Section 4.3. The adaptive division process is as follows:

(1)   If gen=1, set P¯=P, Γ=1,2,...,s

(2)   Let i=1

(3)   If i∈Γ, set ℳi be empty, then repeat the following steps N/s times: randomly select x,y∈P¯, if x≻y, add x to ℳi; if y≻x, add y to ℳi; if both x⊁y and y⊁x, randomly select one of x,y to add into ℳi

(4)   i=i+1, if i<s, then go to step(3).

The quality Mq of a memeplex quantifies its overall performance and is calculated as follows:

Mqi=∑x∈ℳi|y∈P∣x≻y|(6)

After calculating the quality Mq for each memeplex, the s memeplexes are sorted in descending order based on their Mq values, such that ℳ1,ℳ2,…,ℳs satisfy the condition Mq1>Mq2>⋯>Mqs.

4.2 Adaptive Cooperated and Diversified Search

Unlike existing SFLA [26,27] and MGFLA [19], which apply the same process to all memeplexes or uniformly conduct cooperated searches, ACSFLA dynamically determines whether memeplexes cooperate based on their dominance relationship and adjust search strategies accordingly. This method emphasizes the balance of exploration and exploitation, executing cooperation when there is a significant advantage, to enhance solution quality and maintain population diversity. The diversified search processes are listed as follows:

(1)   For the best memeplex ℳ1:

If C(ℳ1, ℳs)≥γ2, execute the following steps:

Let l=1, τ=0

(ii)   If τ=0, randomly choose a solution x∈ℳ1 and a solution y∈ℳs

(iii)   Perform [10−δigen] times global search between x and y

(iv)   Perform δigen times local search on x

(v)   l=l+1, if l<μ1/μ, go to step(ii).

Else if C(ℳ1, ℳs)<γ2, execute the following steps:

(i)   Let j=1 and decide a non-dominated solution xb∈ℳ1

(ii)   Randomly choose a solution y∈ℳ1

(iii)   Perform [10−δigen] times global search between xb and y

(iv)   Perform δigen times local search on xb

(v)   Perform δigen times local search on y

(vi)   j=j+1, if j<μ1, go to step(ii).

For the worst memeplex ℳs:

If C(ℳ1, ℳs)≥γ2, execute the following steps:

(i)   Let l=1, τ=0 and decide a non-dominated solution xb∈ℳs

(ii)   If τ=0, randomly choose a solution y∈ℳ1

(iii)   Perform [10−δigen] times global search between xb and y

(iv)   Perform δigen times local search on xb

(v)   l=l+1, if l<μs/μ, go to step(ii).

Else if C(ℳ1, ℳs)<γ2, execute the following steps:

(i)   Let j=1 and decide a non-dominated solution xb∈ℳs

(ii)   Randomly choose a solution y∈ℳ1

(iii)   Perform [10−δigen] times global search between xb and y

(iv)   Perform δigen times local search on xb

(v)   j=j+1, if j<μs, go to step(ii).

For each memeplex ℳr,1<r<s, repeat the following steps μr times:

(i)   Decide a non-dominated solution xb∈ℳr. Randomly choose a solution y1∈ℳ1 and a solution y2∈ℳs

(ii)   Perform [10−δigen] times global search between xb and y1

(iii)   Perform [10−δigen] times global search between xb and y2

(iv)   Perform δigen times local search on xb.

Metric C [28] is employed to describe the dominance relationship between two solution sets. Specifically, C(A,B) represents the proportion of solutions in set B that are dominated by at least one solution in set A. When C(A,B)=0, it means that none of the solutions in set B are dominated by any solution in set A. Conversely, C(A,B)=1 indicates that set A completely dominates set B. μi(1≤i≤s) indicates the search times of the memeplex ℳi, which is decided on the quality Mqi. δigen is a dynamic factor that changes with iterations. It determines whether the memeplex search in generation gen is inclined towards global search or local search, based on the quality Mqi and evolution quality Moigen.

C(A,B), μi and δigen are computed as follows:

C(A,B)=|{b∈B:∃a∈A,a≻b}||B|(7)

μi=[5⋅Mqi−MqminMqmax−Mqmin+5]⋅μ(8)

δigen=[5⋅(Mqi−MqminMqmax−Mqmin−Moigen−1−Momingen−1Momaxgen−1−Momingen−1)]+5(9)

where μ is the basic search times.

There are 3 global search operators and 6 local search operators used in the search process. GS1 replaces the machines between positions k1 and k2 in the machine assignment string of x with those from y. GS2 applies the order crossover to the scheduling strings of x and y. GS3 executes GS1 and GS2 sequentially. NS1 and NS2 are insertion operators for the scheduling and machine assignment strings. NS3 and NS4 are swap operators for the scheduling and machine assignment strings. NS5 and NS6 are inversion operators for the scheduling and machine assignment strings.

In the global search between x and y, randomly select a global search operator GSi (i∈1,2,3) to generate a new solution xnew. If xnew≻x, then set x=xnew and τ=1; if xnew≻y, set y=xnew; if both xnew⊁x and xnew⊁y, update Ω with xnew.

In the local search on x, randomly choose a local search operators NSi (i∈1,2,3,4,5,6) to generate a new solution xnew. If xnew≻x, x=xnew, τ=1, otherwise update Ω with xnew.

Initially, Ω consists of N/s randomly chosen solutions from the initial population Pop, and it is updated with the solution x in the following way: non-dominated sorting is performed on Ω after adding x into it. Then, the solution with the largest rankx and smallest crowding distance is removed from Ω.

4.3 Partial Population Shuffling

Partial Population Shuffling is a key step in ACSFLA, aimed at dynamically constructing the division pool P¯. The process evaluates the evolutionary quality Mo of each memeplex. Memeplexes with low evolutionary potential are selected to P¯ for reconstruction in the next iteration. Memeplexes with high evolutionary potential are retained to continue their evolution. The partial population shuffling process is as follows:

(1)   Let i=1, Γ and P¯ be empty

(2)   Compute Moigen

(3)   If Moigen<γ1 and Moigen−Moigen−1<0, add i to Γ and add ℳi to P¯

(4)   i=i+1, if i<s, then go to step(2).

where Moigen−1 indicates the evolution quality of the memeplex ℳi in the latest generation. The set Γ stores the indices of memeplexes added to P¯, which are used in the next iteration to decide if memeplex need to be reconstructed. Moigen is calculated as follows:

Moigen=λigenμi(10)

where λigen represents the updated times of ℳi during the memeplex search. Each time the optimization object of ℳi is replaced with a new solution, λigen=λigen+1. μi is the search times of ℳi.

4.4 Algorithm Description

ACSFLA is described in Algorithm 1.

Unlike previous SFLA, ACSFLA incorporates several features that improve optimization performance. First, it determines the search times for each memeplex based on its quality, efficiently focusing on high-quality memeplexes and minimizing effort on low-quality ones. Second, it decides whether to perform a cooperated search by evaluating dominance relationships, maintaining population diversity, and avoiding premature convergence. Third, during the cooperated search, the algorithm uses a segmented and dynamic targeted search strategy, while in non-cooperated scenarios, it focuses on local search around higher-quality solutions to improve efficiency and explore promising regions more effectively. Finally, adaptive population division and partial shuffling select low-quality memeplexes for reconstruction in the next generation. These features balance exploration and exploitation, sustain diversity, and improve search efficiency. Fig. 1 shows the flowchart of ACSFLA.

Figure 1: Flowchart of ACSFLA

5  Computational Experiments

Extensive experiments are conducted to test the performances of ACSFLA for the BPM scheduling problems in the fabric dyeing process. Experiments are implemented by using Matlab R2021a and run on 16 G RAM 3.1 GHz CPU PC.

5.1 Test Instances, Metrics and Comparative Algorithms

A total of 90 test instances are utilized, which is provided by Zhang et al. [1]. The related data are shown below. (n,F)∈{(100,6),(100,9),(200,9),(200,12),(300,12),(300,15)}, m∈{5,7,9},Pg∈[10,30],sab∈[2,5],sab=sba−1,a>b,wi∈[5,60],di=ε×5nm,ε∈[0.5,1.5],vk=30+10k. There are 18 instance combinations, each of which is presented as n×F×m. Then randomly produces five instances for each combination.

Metric 𝒞 [28] is used to compare the approximate Pareto optimal sets generated by different algorithms. The calculation formula for metric 𝒞 has already been provided in Eq. (7) above.

The metric ρ [29] presents the ratio of |{x∈Ωl|x∈Ω∗}| to |Ω∗|, where Ωl denotes the non-dominated set obtained by Algorithm l, and Ω∗ refers to the reference set, which consists of all non-dominated solutions in the union of non-dominated sets produced by all algorithms.

IGD [30] is used to calculate the distance of the non-dominated set Ωl relative to a reference set Ω∗.

IGD(Ωl,Ω∗)=1|Ω∗|∑xϵΩ∗minyϵΩ∗d(x,y)(11)

where d(x,y) is the distance between a solution x and a reference solution y in the normalized objective space.

Three comparative algorithms are chosen. Zhang et al. [1] presented MOABC for parallel BPM scheduling in the fabric dyeing process and MOABC is directly used in this study. Lei et al. [25] proposed ASFLA, an adaptive SFLA-based algorithm, that integrating dynamic population division and adaptive search processes. Li et al. [20] provided an NSGA-CC, which is formed by using NSGA-II [31] framework, incorporating a hierarchical clustering-based environmental selection strategy. An SFLA is also applied to show the effect of new strategies of ACSFLA, deleting the adaptive population and diversified cooperated search.

5.2 Parameter Settings

ACSFLA has following parameters: N, s, μ, γ1, γ2 and stopping condition. Regarding the stopping condition, ACSFLA demonstrates good convergence when the CPU time reaches 0.05×n×m seconds CPU time. Moreover, all comparative algorithms also converge effectively within this CPU time. Therefore, 0.05×n×m seconds CPU time is set as the stopping condition for all algorithms.

Taguchi method [32] is used to decide the settings for other parameters by using instance 25, which belongs to combination 100×9×7. Table 3 shows the levels of each parameter, in which five four levels are set for each parameter.

Fig. 2 shows the results of ρ and S/N ratio, which is defined as −10×log10⁡(ρ2). It can be found from Fig. 2 that ACSFLA with the following combination: N=120, s=6, μ=6, γ1=0.03 and γ2=0.7 can obtain better results than ACSFLA with other parameter combinations, so choose the above parameter settings.

Figure 2: Main effect plot for means and S/N ratios

SFLA has N=150, s=5, μ=60, and the above stopping condition. With respect to ASFLA, MOABC, NSGA-CC, Lei et al. [25], Zhang et al. [1] and Li et al. [20] provided their parameter settings. These settings except the stopping condition are directly used in this study. The experimental results show that these settings of each comparative algorithm are still effective, so they are kept.

5.3 Results and Discussion

ACSFLA, ASFLA, and three comparative algorithms are used. Each algorithm randomly runs 10 times for each instance. Tables 4–7 describe the corresponding results of five algorithms, in which AC, S, A, M, N indicate ACSFLA, SFLA, ASFLA, MOABC, NSGA-CC. Fig. 3 gives a mean plot with 95% confidence interval. Fig. 4 shows distributions of non-dominated solutions produced by each algorithm.

Figure 3: Mean plot with 95% confidence interval

Figurer 4: Distribution of non-dominated solutions

ACSFLA demonstrates clear superiority in terms of the ρ metric. As shown in Table 4, ACSFLA achieves higher ρ values than the other algorithms in 79 of 90 instances, achieving a ρ value of 1 in 60 instances. This indicates that all the non-dominated solutions produced by ACSFLA are also non-dominated in the combined reference set Ω, demonstrating that ACSFLA not only generates high-quality solutions but also outperforms the other algorithms. The average ρ value for ACSFLA is 0.82, with a standard deviation of 0.29, reflecting consistent high performance. Furthermore, when ACSFLA is compared with other algorithms using a t-test, the p-values are all far smaller than 0.05, highlighting its significant advantage.

ACSFLA also demonstrates superior performance in terms of the IGD metric. Table 5 shows that ACSFLA achieves smaller IGD values than SFLA in 88 instances, ASFLA in 78 instances, MOABC in 83 instances, and NSGA-CC in 82 instances. These results highlight ACSFLA’s superior ability to approximate the ideal reference set Ω and its better convergence ability toward the Pareto front. The average IGD value for ACSFLA is 0.076, with a standard deviation of 0.14, indicating consistent performance. The maximum IGD value achieved is 0.724, demonstrating that ACSFLA maintains a competitive edge in most cases. Furthermore, when ACSFLA is compared with other algorithms using a t-test, the p-values are all far smaller than 0.05, confirming its significant advantage in terms of convergence and solution quality.

In terms of the 𝒞 metric, ACSFLA shows a significant advantage over other algorithms. As presented in Tables 6 and 7, ACSFLA achieves smaller 𝒞(S,AC) values than 𝒞(AC,S) in 89 of 90 instances, smaller 𝒞(A,AC) than 𝒞(AC,A) in 79 instances, smaller 𝒞(M,AC) than 𝒞(AC,M) in 86 instances, and smaller 𝒞(N,AC) than 𝒞(AC,N) in 85 instances. Specifically, ACSFLA reaches 𝒞(AC,S)=1 in 85 instances, 𝒞(A,AC)=1 in 69 instances, 𝒞(AC,M)=1 in 68 instances, and 𝒞(AC,N)=1 in 69 instances. The average values for 𝒞(AC,S), 𝒞(AC,A), 𝒞(AC,M), and 𝒞(AC,N) are 0.98, 0.79, 0.89, and 0.87, respectively, with standard deviations of 0.08, 0.38, 0.24, and 0.27. This indicates that the non-dominated solutions generated by ACSFLA are stronger than other solutions from other algorithms.

The above result analyses show that ACSFLA obtains better results than its three comparative algorithms. Initialization is done by the heuristic method, Mo and Mq are used for adaptive population division. Executing adaptive cooperated and diversified memeplex search based on the dominance relationship between memeplexes. As a result, the exploration capability is intensified and high diversity is maintained. Thus, ASFLA is a very competitive method for solving parallel BPM scheduling in the fabric dyeing process.

6  Conclusion and Future Topics

In this paper, we have effectively resolved the parallel BPM scheduling problem with machine eligibility in fabric dyeing processes using the newly developed adaptive cooperated shuffled frog-leaping algorithm (ACSFLA). ACSFLA incorporates innovative strategies based on the quality Mq and evolutionary quality Mo of memeplexes, along with the dominance relationships 𝒞 between them. By eliminating global population shuffling and conducting targeted shuffling within the division pool for selected memeplexes, ACSFLA significantly enhances algorithm efficiency. Extensive experiments have demonstrated the effectiveness of ACSFLA’s new strategies, with the algorithm outperforming comparative algorithms from the literature in minimizing makespan and total tardiness, thus highlighting its substantial advantages in solving this specific scheduling problem.

For future research directions in parallel BPM scheduling for fabric dyeing, several promising paths could be explored. Incorporating energy consumption as an additional objective might be a key step toward achieving more energy-efficient processes. This could involve devising algorithms considering the power consumption patterns of machines and job sequences to minimize energy use during dyeing. Developing new neighborhood search methods could be an effective way to enhance search efficiency, enabling the algorithm to find better solutions in a shorter time. Applying reinforcement learning techniques may also hold great potential in making the scheduling more intelligent and adaptable to dynamic production situations. The key features of the algorithm include adaptive population division, as well as an adaptive cooperated and diversified search strategy.

Acknowledgement: We sincerely appreciate the perceptive feedback from the anonymous reviewers. It has substantially contributed to improving the overall quality of this paper.

Funding Statement: The authors received no specific funding for this study.

Author Contributions: Study conception and design: Lianqiang Wu; analysis and interpretation of results: Yutong Cai; draft manuscript preparation: Lianqiang Wu; review and editing: Deming Lei. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: The data supporting this study are described in the first paragraph of Section 5.1.

Ethics Approval: Not applicable.

Conflicts of Interest: The authors declare no conflicts of interest to report regarding the present study.

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