A Novel Collaborative Evolutionary Algorithm with Two-Population for Multi-Objective Flexible Job Shop Scheduling

1  Introduction

Planning is a very crucial problem in modern industries [1]. Through the elimination of scheduling conflicts, the decrease of flow time, the improvement of production resource usage, and the adaptation to unpredictable shop floor disruptions, the optimization of production scheduling can lead to considerable improvements. Job shop scheduling (JS) is a challenging problem for production planning [2]. This problem is an NP-hard problem, which is difficult to solve [3]. However, traditional JS, which assumes no flexibility of the resources for each operation of every job, might not be able to fulfill the demand of modern industries, because manufacturing systems have become increasingly flexible, and a considerable amount of automation equipment has been used [4,5]. Therefore, flexible job shop scheduling (FJS) is increasingly interesting for both industry and research.

FJS is a kind of JS that enables any machine from a certain set to process an operation, and it was first introduced in the 1990s by Bruker et al. [6]. Compared with JS, FJS is more complex, which means that JSP is NP-hard as well. Recently, a considerable amount of research has been examined, most of which has focused on a single object. However, different departments of a company expect to maximize their objectives, such as cost and efficiency. Therefore, a single objective is not sufficient to meet the requirements of realistic production, and there is a need for further research on multi-objective FJS (MOFJS). Recently, MOFJS has attracted increasing attention from both the industrial and academic, and several methods, including evolutionary algorithms [7], and ant colony algorithms [8]. Traditionally, MOFJS contains 2 subproblems: operation sorting (OS) and machine selection (MS). Most of the related work employs integrated encoding to solve MOFJS [9]. However, OS and MS are quite different, thereby limiting the global searching ability of the current methods and seriously affecting the solving effect. Therefore, a novel collaborative evolutionary algorithm with two populations based on Pareto optimality is proposed for MOFJS. The proposed method uses Pareto optimality to address the Mutli-objectives problems, and develops a collaborative optimization strategy, which simulates the optimization process of OS and MS wherein the subsystems interact with each other. The interaction with each other in every generation can improve the solution of FJS. Thus, the proposed method takes the merits of collaborative optimization and evolutionary algorithms, and combines them to prompt the solving. The experimental results also show the availability of the proposed method.

The rest of this paper is organized as follows. Section 2 is the literature review. Section 3 is the problem formulation. Section 4 is the proposed method and Section 5 presents the experimental results. Section 6 is the conclusion.

2  Literature Review

FJS has been a research hotspot for many years, and a considerable amount of research has been conducted in recent years. For the single objective, the FJS can be categorized by exact approaches, such as mathematical programming, and approximation approaches, such as GA [10]. Meng et al. [11] presented a novel integer linear programming for FJS. Alvarez-Valdes et al. [12] developed a heuristic algorithm and applied it to glass production.

Currently, MOFJS has attracted an increasing amount of attention from both academia and industry, and several approaches have been proposed, involving evolutionary algorithms, local search methods, and swarm intelligence. Tay et al. [13] evolved the dispatching rules by genetic programming. Baykasoğlu et al. [14] performed a deep analysis of the effects of dispatching rules. Rajkumar et al. [15] introduced a greedy adaptive search, which considered the maintenance and limited resource constraints. Caldeira et al. [16] developed a multi-objective discrete Jaya by maximizing the makespan, and workload. Li et al. [17] introduced a rescheduling method with a Monte Carlo tree search, and it considered the MOFJS with dynamic events.

In summary, most of the existing approaches use an integrated encoding. However, the two subproblems in FJS (MS and OS) are quite different, and the integrated encoding might cause the solution space to be more complex, and lead to a worse solution. Thus, this paper aims to address this drawback by proposing a novel collaborative evolutionary algorithm with two populations.

3  Problem Formulation

The formulation of an n×m FJS is defined as follows.

Given a set with n jobs J={J1,J2,J3,…,Jn} and a set of m M={M1,M2,M3,…,Mm}, a job ji has an operation series {Oi1,Oi2,Oi3,…,OiOni}. And Oni is the total number of operations of job Ji. Oij(i=1,2,…,n;j=1,2,…,Oni) must be handled by one in the given set M. In other words, FJS can be regarded as deciding the sequence of assignment and operation under the criteria.

In this research, the following criteria are involved:

Makespan: the maximal finish time;

Maximal machine workload (MMW): the maximal time on a machine.

Total workload of machines (TWM): the total running time of all machines.

Based on these criteria, the MOFJS in this research has three objectives, and the descriptions of these objectives are shown below. n denotes the number of jobs. m presents the number of machines. oij means the j-th operation of the job i. Oni is the total number of the operations of job i. cijk denotes the earliest finish time. ci denotes the earliest finish time of job i. Wk denotes the workload of machine k. All the objectives are formulated below:

f1=Makespan=maxi=1{ci}i∈[1,n]ci=maxj=1{cijk}j∈{1,2,…,Oni}(1)

f2=MMW=maxk=1{Wk}k∈{1,2,…,m}(2)

f3=TWM=∑k=1mWkk∈{1,2,…,m}(3)

The model of FJS is followed to [18], and the summary of the notations is presented in Table 1.

4  Proposed Method for MOFJS

4.1 Basic Concepts

Generally, the formulation of a multi-objective optimization [19,20] (contains n variables, k objectives and m constraints) can be defined as:

Miny=f(x)={ f1(x),f2(x),…,fk(x) }s.t.e(x)={ e1(x),e2(x),…,em(x) }≤0x=(x1,x2,…,xn)∈Xy=(y1,y2,…,yn)∈Y(4)

x and X are the variable and variable space, respectively. y and Y denote the objective and objective space, respectively. e(x) means the constraints, and xf={x∈X|e(x)≤0} is the feasible space.

Non-dominated solution: if the objective function fi(a) is all better than fi(b), then solution b is dominated by solution a. For example, for a minimization problem, if fi(a)≤fi(b), the solution b is dominated by solution a. Otherwise, solution b is not dominated by solution a. This is a non-dominated solution. Fig. 1a shows a group of solutions for a two-objective problem.

Figure 1: NDS and Pareto front

Pareto-optimal: when the solution cannot be dominated, it is called Pareto-optimal (PO). Fig. 1b shows the Pareto front of a two objectives problem.

4.2 Proposed Method

4.2.1 The Optimization Strategy for Collaborative Evolution

The proposed method uses a collaborative strategy to search for a solution in both the OS and MS. The OS and MS are optimized alternatively, and evaluated together. Moreover, it should be noted that this strategy is generic, and it can be suitable to combine with various algorithms, such as GA. In this paper, the proposed method uses an evolutionary algorithm (EA) for both OS and MS, and the optimization strategy is shown in Fig. 2 and below:

1):   Parameter Initialization. Assum given n jobs. Generating two initial populations for OS and MS individually;

2):   Fitness Evaluation. Select the Popsize randomly, and calculate the solutions population. After that, evaluate each solution individually and choose the NDS (NDS);

3):   Condition judgement. If it meets the end condition, just terminate and output the solution. Otherwise, going to 4);

4):   Collaborative evolution. Generate some new individuals. Go back to 2).

Figure 2: Optimization strategy of the proposed method

4.2.2 Pareto Archive Set

For the diversity, a Pareto archive set (PAS) is employed in the proposed method, which can retain the quantity of the NDS with the user requirement [21]. When optimizing, the PAS is moving to the Pareto-optimality front by replenishing some new NDS and eliminating some dominant solutions. Once the number of NDSs is sufficient, a crowding similarity in Eq. (5) is employed (CD) to remove the redundant solutions and guarantee diversity. Individuals with higher CD are preferentially kept in the PAS.

CDp=∑i=1k(fi(p+1)−fi(p−1))(5)

where p denotes the number of the individuals and i is the i-th objective.

4.2.3 Pareto Sort Algorithm

In MO problems, one objective is not enough to evaluate the quality of the solution individually, and all of the objectives must be involved. Thus, a sorting algorithm is essential, and the non-dominated sorting method [22] is adopted. The sorting algorithm separates the solutions into different levels. For example, as shown in Fig. 3, the solutions have three levels, and the lower level has the better fitness. For the same levels, the solutions that have higher CDs are also a priority. With this separation, the good solutions can be retained as a priority by the PAS.

Figure 3: NDS levels

4.2.4 Encoding and Decoding Method

The encoding method is essential for an EA, and the proposed method adopts the method in [23] for encoding. Moreover, since the OS and MS are quite different, the details of the encoding method for each subproblem are also different. The encoding for OS is based on the operation representation. In addition, the representation employs an undivided arrangement. With this encoding, each job can be selected by Oni times. The f-th occurrence of a work number refers to the f-th operation of that work number, and thus, any arrangement of chromosomes can convert into a viable solution.

The MS chromosome refers to the M, and the length of this chromosome is ∑Oni. The i-th part means the chosen set of machines for the operation corresponding to job i. Assuming that Oh of Ji can be handled by a machine set sih={mih1,mih2,…,mihcih}, the i-th part is denoted as {gi1gi2…gih…giOni}. This denotes that Oh is assigned to the gih-th machine.

An encoding solution contains an OS chromosome and an MS chromosome, and the decoding manner involves semi-active, active, non-delay, and hybrid. In the proposed method, to achieve a better solution, an active schedule is used. The decoding process is shown below:

m: number of the machines.

oij: the j-th operation of the i-th job.

asij: the valid start time of oij.

sij: the earliest start time of oij.

k: the alternative machine corresponding to oij.

tijk: the processing time of operation oijon machine k.

cij: the earliest finish time of operation oij, i.e., cij=sij+tijk.

The process of decoding in the proposed method:

1):   Generate the machine of each operation by the MS chromosome.

2):   Choose a set of the operations for each machine: ma={oij}1≤a≤m.

3):   Choose a set of machines for each job: Jmd={machine}1≤d≤n.

4):   The allowable begin time for every operation: asij=ci(j−1)(oij∈ma), ci(j−1) is the finish time of the pre-operation of oij for the same job.

5):   Checking the unused time of the machine of oij, and getting the unused areas [ts,te], checking the areas in turn (if: max(asit,ts)+tijk≤te, the earliest begin time is sij=ts, else: check the next area), if the area cannot meet the condition: sij=max(asij,c(oij−1)), c(oij−1) is the finish time of the pre-operation of oij for the same machine.

6):   Calculate the finish time of each operation by cij=sij+tijk.

7):   Generate the sets of the begin time and finish time for each operation of each job by Td(sij,cij)1≤d≤n.

In the decoding process, it can obtain the set of start time and finish time for each operation of each job, and it is a schedule solution for the workshop. Fig. 4 gives an example of the encoding and decoding in the proposed method. Fig. 4a is the background of this case, which involves 3 jobs and 3 machines. Fig. 4b presents an OS chromosome with repetitions of job numbers, and Fig. 4c is an MS chromosome. The OS and MS chromosomes in Fig. 4 form a solution of FJS. And this solution can be converted into a schedule by decoding, as shown in Fig. 4d. The decoding steps are as follows:

1):   Generate the machine of each operation, O11 on M2, O12 on M1, O21 on M3, O22 on M1, O31 on M3, O32 on M2;

2):   Choose the set of operations for each machine: M1={O12,O22}, M2={O11,O32}, M3={O21,O31};

3):   Choose the set of machines for each job: J1={M2,M1}, J2={M3,M1}, J3={M3,M2};

4):   The allowable begin time for each operation is calculated as follows: asij=ci(j−1)(oij∈ma). ci(j−1) is the finish time of the pre-operation of oij for the same job. For example, the allowable start time of O12 is the finish time of O11. This can guarantee that the obtained schedule is feasible. However, this cannot ensure that the schedule is active. Therefore, on the premise of avoiding destroying the feasibility of the schedule, Step 5 tries to insert the following operations into the earlier hole in schedule;

5):   Check the unused time of the machine of oij, and give the idle areas [ts,te], check these areas in turn (if: max(asit,ts)+tijk≤te, the earliest begin time is sij=ts, else: checking the another area), if there is no area that can meet this condition: sij=max(asij,c(oij−1)), c(oij−1) is the finish time of the pre-operation of oij for the same machine. For example, for arranging O22 on M1, its allowable stating time is 6 (as22=6). However, before time 6 in M1, there is a hole (from time 1 to 3). Now, we need to consider whether this operation can be inserted into the hole or not. Because this hole is smaller than the processing time of O22, this operation cannot be inserted into this hole. Its earliest begin time is 6 (s22=6);

Figure 4: Encoding and decoding example

Step 6: The finish time of each operation is calculated as follows: cij=sij+tijk. For example, for the O22, its finish time is 13 (c22=s22+t221=6+7=13);

Step 7: Generate the sets of begin time and finish time: Td(sij,cij)1≤d≤n.

4.2.5 Genetic Operators

The genetic operator (GO) is important for good individuals and solutions. Generally, GO is classified into three categories: selection, crossover and mutation. In the proposed method, for a better solution, OS and MS use different genetic operators. The details of the genetic operations are shown below:

(1) Selection

OS and MS randomly select the individuals.

(2) Crossover

A precedence operation (PO) is used for the crossover in the OS. The PO crossover can retain the good of the parents and propagate it to the offspring. The flowchart of the PO is shown below (P1 and P2 are parents, and O1 and O2 are offspring).

1):   Dividing the job set J={J1,J2,J3,…,Jn} into two parts Jobset1 and Jobset2.

2):   For an element belonging to Jobset1 in P1, it is retained in the same position in O1 and removed in P1. For an element belonging to Jobset2 in P1, it is retained in the same position in O2 and removed in P2.

3):   Remains in P2 are moved to the empty positions in O1, while the remainder in P1 are moved to O1.

The MS population uses a two-point crossover, which selects two positions randomly first, and the two strings swap all elements.

Fig. 5a is an example of the PO for OS and Fig. 5b is for MS.

Figure 5: The crossover operations for OS and MS

(3) Mutation

The OS uses a neighborhood mutation operator.

1):   Select three elements in a parent (each element has different value), and generate the neighborhood chromosomes.

2):   Choose a chromosome randomly for the chromosome.

MS uses the mutation operator below.

1):Select r positions in a parent (r is the 1/2 of the chromosome);

2):   For each position, change the value of the chosen position to the corresponding operations.

4.2.6 Stop Condition

If the iteration has reached the maximum epoch, the proposed method stops running.

4.2.7 Framework of the Proposed Method

Fig. 6 presents the flowchart, and the description of the proposed method is shown below:

1):   Setting the parameters, including Popsize1, Popsize2, PopsizeSP, PopsizeAS, maxGen, crossover rate of OS (pc1), crossover rate of MS population (pc2), mutation probability of OS population (pm1), mutation probability of MS population (pm2).

2):   Initialization. For n jobs, the encoding methods are used to generate two populations for OS with Popsize1 individuals and MS with Popsize2 individuals.

3):   Evaluation. Randomly select PopsizeSP individuals from every population separately to form an FJSP solution population with PopsizeSP FJSP solutions. The considered objectives of each solution are calculated, and the Pareto sort algorithm is used to sequence the population and divide these solutions into several levels. If Gen=1, generate the PAS, copy all the NDS to the PAS. If the number of solutions SizeSL≥PopsizeAS, selecting the solutions with bigger CD to copy into the PAS until the PAS is full; if SizeSL<PopsizeAS, select the NDS to the PAS and set the other positions in PAS is NULL; If Gen>1, update the PAS: compare each new non-dominant solution with the PAS in turn. If the new solution dominates the PAS, it can be retained in the PAS.

4):   If the model meets the stop condition, go to 6), otherwise, go to Step 5).

5):   Collaboration evaluating. Upgrade the populations: generate new individuals for each population;

6):   Output the NDS in PAS.

Figure 6: Work flow of the proposed method

5  Experimental Results of the Proposed Method

5.1 Experimental Setting and Results

The proposed method codes in C++ on a laptop. To evaluate the proposed method, six experiments were selected. The five experiments are adopted from the related papers, and the last experiment is employed by the author themselves. For the parameters, the sizes of the OS and MS populations are Popsize1=400 and Popsize2=400. The PopsizeSP is 400 and PopsizeAS is 5, and the maxGEN is set to 200. The crossover rates of OS and MS are 0.9 and 0.9, and the crossover rate of OS and MS are 0.9. The mutation rate of OS and MS are 0.1.

5.1.1 Experiment 1

This experimental problem is from [24], which includes 5 kinds of problems. Tables 2 and 3 show the comparison results. From Table 2, the proposed method finds another new Pareto solution for problem 4 × 5, which is (13, 7, 33), as shown in Fig. 7a. For problem 10 × 7, compared with other algorithms, the proposed algorithm also finds two Pareto solutions. Based on Table 2, the proposed method finds two solutions that dominate the P-DABC. One solution (11, 11, 61) is shown in Fig. 7b.

Table 3 is the comparisons with the other methods for problems 8 × 8, 10 × 10 and 15 × 10. From Table 3, the proposed method finds several solutions that dominate the other methods. Fig. 8 is the Gantt chart with different solutions that are solved by the proposed method. From the experimental results, it can be seen that the proposed method achieves the best performance. Compared with the other methods, the results of the proposed method are improved for the makespan, MMW and TWM, which means that the proposed method can find new Pareto solutions.

Figure 8: Gantt charts solved by the proposed method

5.1.2 Experiment 2

This experimental problem is from [15], which includes 3 kinds of problems. Table 4 is the comparison with the other algorithms. From Table 4, the proposed method finds several solutions that dominate the other methods. Fig. 9 is the Gantt chart of a solution for problem 12 × 5. Similar to the experimental results in Section 5.1.1, the experimental results also suggest that the proposed method has improved performance. Compared with the other methods, the proposed method can find a better solution.

Figure 9: Gantt chart of one solution for problem 12 × 5 (33, 33, 137)

5.1.3 Experiment 3

This experimental problem is from [36], which is the famous benchmark instance for the single objective FJS. This data contains 10 problems. Table 5 is the comparison with the other method, which also shows that the proposed method finds several dominant solutions.

5.1.4 Experiment 4

This experimental problem is from [38], which is another famous benchmark instance for the single objective FJS and is very hard to solve. The data contain 18 problems, and the range of the problems is from 8 × 5 to 20 × 10. Table 6 shows the experimental results. The results indicate that the proposed method can address the multi-objective FJS well.

5.1.5 Experiment 5

This experimental problem is from [39], which is another famous benchmark instance for the single objective FJSP and is very hard to solve. The experiment contains 21 problems and Table 7 presents the experimental results. The experimental results indicate the proposed method has significant improvement.

5.1.6 Experiment 6

The experimental problems are built by the authors. The range of the problems involves 10 × 8 and 16 × 8, which are shown in Table 8. The data of problem 16 × 8 are shown in Table 9. Table 10 shows the experimental results. From Table 10, the proposed method provides several Pareto solutions for each problem.

5.2 Discussion

In the experimental results, the proposed method finds some dominant solutions of the other methods. These results suggest that the proposed method addresses the MOFJS well. The main reasons for the good performance are because the OS and MS are quite different, and the integrated encoding might cause solution space intricacy and complexity. The complex space might prevent the search ability of the existing methods. Moreover, the proposed method evolves OS and MS separately, which can improve the overall searching ability. With this improvement, the proposed method achieves the more effective results. Furthermore, the proposed method uses PAS to save the solutions, and uses crowd similarity for diversity, which is also effective for the improvement of MOFJS.

6  Conclusions

FJS is very important for production, and a novel collaborative evolutionary algorithm with two-population based on Pareto optimality is proposed for MOFJS. The experimental results suggest that the proposed method is feasible for improving the solution of MOFJS, and several dominant solutions are found by the proposed method, which achieved significant improvement. The main contributions of this paper are as follows:

•   With the problem features of MOFJS, a collaborative evolutionary method is designed. The proposed method reflects the essential feature of MOFJS, and the PAS is used to save the Pareto solutions. The comparison demonstrates that the proposed method can address MOFJS successfully with a promising result.

•   The proposed method combines the merits of collaborative optimization and EA. It uses crowd similarity to guarantee diversity. It provides a novel way to solve MO problems by containing several sub-problems. The experimental results of the multi-objective FJSP show that the proposed method may solve these problems effectively.

Funding Statement: This research work is the Key R&D Program of Hubei Province under Grant No. 2021AAB001, and National Natural Science Foundation of China under Grant No. U21B2029.

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

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