Improved firefly algorithm with courtship learning for unrelated parallel machine scheduling problem with sequence-dependent setup times

Abstract

The Unrelated Parallel Machines Scheduling Problem (UPMSP) with sequence-dependent setup times has been widely applied to cloud computing, edge computing and so on. When the setup times are ignored, UPMSP will be a NP problem. Moreover, when considering the sequence related setup times, UPMSP is difficult to solve, and this situation will be more serious in the case of high-dimensional. This work firstly select the maximum completion time as the optimization objective, which establishes a mathematical model of UPMSP with sequence-dependent setup times. In addition, an improved firefly algorithm with courtship learning is proposed. Finally, in order to provide an approximate solution in an acceptable time, the proposed algorithm is applied to solve the UPMSP with sequence-dependent setup times. The experimental results show that the proposed algorithm has competitive performance when dealing with UPMSP with sequence-dependent setup times.

Introduction

The classical parallel machine scheduling problem (PMSP) widely exists in scientific and industrial production [1], including printed circuit board manufacturing [2], semiconductor wafer manufacturing dicing operations [3], computer multiprocessors task scheduling [4] and heterogeneous clusters scheduling [5], which has received extensive attention from researchers. However, in some applications, such as cloud computing [6] and edge computing [7], setup times such as waiting time for job sorting and job data transmission time are usually involved. In addition, due to the heterogeneity of devices, quite a few types of jobs also have sequence dependencies (such as Spark). This kind of problem is called the unrelated parallel machines scheduling problem (UPMSP) with sequence-dependent setup times [1].

In recent years, the research on UPMSP and its variants has received extensive attention. In UPMSP, the objective of optimization is usually to minimize the maximum completion time. Most of the UPMSP problems deal with the parallel scheduling, which requires the use of two or more machines. The UPMSP whose setup times depend on the sequence has the nature of non-deterministic polynomial, which makes it difficult to solve. Some precise algorithms have been proposed to solve the UPMSPs when the setup times are ignored, such as the branch and bound algorithm [8, 9], the cutting plane algorithm [9], etc. Nevertheless, due to the time-consuming characteristics, precise algorithms cannot be well applied to high-dimensional situations. For the case of high-dimensional, meta-heuristic algorithms are more considered to be adopted as they can find feasible solutions more efficiently and stably within an acceptable time range. However, the performance of these algorithms will decline in high-dimensional cases.

In addition, Firefly Algorithm with Courtship Learning (FACL) is a variant of Firefly Algorithm (FA) [10], which shows efficient performance in continuous numerical optimization problems, including high-dimensional cases [11]. This algorithm has the advantages of fewer hyper parameters and easy implementation. Therefore, we try to apply the FACL to solve the UPMSP with sequence-dependent setup times. However, the male and female attraction factors used by FACL decrease rapidly during search process, which makes it easy to terminate the search early and fall into the local optimal problem.

Aiming at the above problems, this work proposes an Improved Firefly Algorithm with Courtship Learning (IFACL) to solve the UPMSP with sequence dependent setup times. Firstly, we propose an effective technique for solving UPMSP with sequence-dependent setup times, that is, the proposed IFACL. Secondly, a new attraction factor based on Cauchy distribution is used to improve FACL’s search ability. Moreover, we use a two-stage solution representation method in combination with the UPMSP with sequence-dependent setup time. Without introducing additional parameters, the search performance of male firefly in courtship learning stage is enhanced, so that the algorithm can make full use of the social information of population. Therefore, the ability of the algorithm to jump out of the local optimum is enhanced, and the global search performance is improved. Finally, a lot of simulation experiments have been carried out to evaluate the effectiveness of the IFACL in solving UPMSP with sequence-dependent setup times.

The remaining sections of this work are constructed as follows. “Related work” section presents the related work. The UPMSP mathematical model is given in “Mathematical modeling” section. “UPMSP based on firefly algorithm with courtship learning” section presents the proposed IFACL for UPMSP in detail. “Simulation experiment” section shows the experimental results and discussion. Finally, a conclusion is provided in “Conclusion” section.

Related work

Recent research on UPMSPs

In the past few years, some progress has been made in solving this challenging problem. Few exact search algorithms have been applied to UPMSP with sequence-dependent setup times. The model proposed by Guinet [12] is the basis of other Mixed Integer Linear Programming (MILP) methods, although optimality can only be guaranteed in small instances. In Vallada and Ruiz [13], an improved method was proposed for the weighted earliness-tardiness minimization model proposed by Balakrishnan et al. [14]. As far as we know, this model greatly reduces the number of binary variables, and has not been used in the makespan target before. Rocha et al. [15] developed a branch and bound algorithm to solve the UPMSP with the sequence-dependent setup times, which achieved better results in the cases of low dimension. It is only in recent years that optimal solutions for larger UPMS instances have been obtained. Avalos Rosales et al. [16] proposed a MILP that can effectively solve some instances of up to 60 jobs and 8 machines. In some iterative algorithms proposed by Tran and Beck [17] and Tran et al. [18], a similar MILP was once the main problem.

Since precise algorithms have not been proven effective in real-life examples of solving this problem until recently, some meta-heuristic algorithms have been developed to solve the UPMSP with sequence-dependent setup times. One of the meta-heuristic algorithms used to deal with the UPMSP with sequence-dependent setup times is the Variable Neighborhood Search (VNS) [19]. De Paula et al. [20] firstly used VNS to solve high-dimensional UPMSP with sequence-dependent setup times, which is to minimize task completion time and weighted delay. Vallada and Ruiz [21] proposed a genetic algorithm (GA) based on the local search enhanced crossover operator to solve the UPMSP, which is to minimize the task completion time. And they also considered the influence of the sequence dependent setup time between machines and jobs. Behnamian et al. [22] proposed a hybrid meta-heuristic algorithm that combines Ant Colony Optimization (ACO), Simulated Annealing (SA), and VNS algorithms to solve the UPMSP that also considers the sequence-dependent setup times, so as to minimize the task completion time. ACO is another meta heuristic algorithm that successfully applied to UPMSP with sequence-dependent set times. Arnaout et al. [23] proposed an enhanced ACO algorithm to solve the UPMSP with sequence-dependent setup times in the case of 10 machines and 120 jobs. Ezugwu and Akutsah [24] implemented an improved Salp Swarm Algorithm (SSA), and obtained the new complexity result of the sequence-dependent UPMSP algorithm. Niu Qun et al. [25] realized the parallel machine scheduling with adjustment time based on the improved clone selection algorithm, and it has a better performance compared with GA and the basic clone selection algorithm (CSA).

Standard firefly algorithm

Firefly Algorithm is a simple and efficient swarm intelligence algorithm for solving complex optimization problems in continuous search space. The flashing and attracting behavior of fireflies is crucial to their evolution. A firefly with a high brightness will attract a firefly with a darker brightness, and the brightness of each firefly depends on its light intensity, that is, the fitness of the objective function. The flashing brightness (the degree of mutual attraction) of fireflies can be expressed as follows:

$$ \beta_{i,j}(r_{i,j})=\beta_{0}\times e^{-\gamma r_{i,j}^{2}}. $$

(1)

where r_i,j is the Euclidian distance between two fireflies x_i and x_j, and β₀ represents the initial attraction factor, which is usually 1. Parameter γ is the fixed value of the light absorption coefficient, which is usually also 1. For two fireflies x_i and x_j randomly selected in the same search space, the distance r_i,j between them is calculated as follows:

$$ r_{i,j}=\left \|x_{i}-x_{j} \right \| =\sqrt{\sum\limits_{d=1}^{D}(x_{i,d}-x_{j,d})^{2}}. $$

(2)

where D represents the dimension of the problem, and d represents the d-th dimension of the position vector of the firefly.

During the search, firefly x_j is attracted to the brighter firefly x_i and moves towards x_i. Its position is updated as follows:

$$ x_{j}(t+1)=x_{j}(t)+\beta_{i,j}\times (x_{i}(t)-x_{j}(t))+\alpha\times \epsilon. $$

(3)

where, t represents the number of iterations of the algorithm, α is the step size parameter, which is usually a random number with the value of [0,1], and ε is a random number between [−0.5,0.5].

In the optimization process of FA, all fireflies will move to the brighter firefly, and finally the firefly individuals will gather around the firefly with the highest brightness to complete the optimization process.

Mathematical modeling

Problem description

The UPMSP with sequence-dependent setup times, which is considered and proposed based on the improved firefly algorithm of courtship learning, is described as follows:

• The scheduling problem assumes that there are N available jobs to be allocated to M unrelated machines for processing at time zero, and the machines are independent of each other.

• There is no preemptive execution.

• The data set, including the time required for machine k to process assigned job j represented by P_j,k and the sequence-dependent setup times of processing job j after job i on machine k given by S_i,j,k, are priori and determined. Among them, i,j={1,2,,N},k={1,2,,M}, usually S_i,j,k≠S_j,i,k.

Model construction

This work uses a model based on Mixed Integer Programming (MIP) [1, 3, 21] to represent the (P_j,k|S_i,j,k|C_max) model, which is convenient to minimize the maximum completion time (C_max). In order to describe the mathematical model of this problem, Table 1 gives some symbol definitions.

Table 1 Symbol definition

Based on the symbol definition in Table 1 above, this article establishes a mathematical model of (P_j,k|S_i,j,k|C_max), as shown below.

$$\begin{array}{*{20}l} & \text{minimize}~C_{max} \end{array} $$

(4)

$$\begin{array}{*{20}l} &{\mathrm{~s.t.~}}\sum\limits_{k}^{M}\sum\limits_{\substack{i=0 \\ i\neq j}}^{N}x_{i,j,k}=1,~\forall~j=1, \cdots, N; \end{array} $$

(5)

$$\begin{array}{*{20}l} &~~~~~~~~\sum\limits_{j}^{N}x_{0,j,k}=1,~\forall~k=1,\cdots,M; \end{array} $$

(6)

$$\begin{array}{*{20}l} &~~~~~~~~\sum\limits_{\substack{i=0 \\ i\neq h}}^{N}x_{i,j,k}-\sum\limits_{\substack{j=0 \\ j\neq h}}^{N}x_{h,j,k}=0,~\forall~h=1, \cdots, N; \end{array} $$

(7)

$$\begin{array}{*{20}l} &~~~~~~~~C_{j}-\left[C_{i}+\sum\limits_{k=1}^{M}x_{i,j,k}(S_{i,j,k}+P_{j,k})+\right.\notag\\ &~~~~~~~~~~~~~~\left. V\left(\sum\limits_{k=1}^{M}x_{i,j,k}-1\right)\right ]\geq 0,~\forall~i=0, \cdots, N; \end{array} $$

(8)

$$\begin{array}{*{20}l} &~~~~~~~~C_{j}\leq C_{max},~\forall~j=1,\cdots,N; \end{array} $$

(9)

$$\begin{array}{*{20}l} &~~~~~~~~C_{0}=0; \end{array} $$

(10)

$$\begin{array}{*{20}l} &~~~~~~~~C_{j}\geq 0,~\forall~j=1,\cdots, N; \end{array} $$

(11)

$$\begin{array}{*{20}l} &~~~~~~~~x_{i,j,k}\in \{0,1\},~\forall~i=0,\cdots,N;~\forall~j=0,\cdots,N;\notag\\ &~~~~~~~~~~~~~~~~~~~~~~~~~~~~\forall~k=1,\cdots,M. \end{array} $$

(12)

where, the goal of formula (4) is to minimize the maximum completion time, the constraint in formula (5) ensures that the job will be executed only once, the constraint (6) indicates that the number of jobs to be processed first by each machine is 1, the constraint in formula (7) ensures that each job is processed first or constructed into a tight post-processing job of other jobs, the formula (8) is used to calculate the completion time of each job, and the formula (9) defines C_max as having to be greater than the completion time of any other job. The constraint (10) ensures that the completion time of virtual job 0 is 0, the constraint (11) ensures that the completion time of the job is not negative, and the constraint (12) defines the value range of decision variable x.

UPMSP based on firefly algorithm with courtship learning

Firefly algorithm with courtship learning

Given that fireflies attract mates in nature by glowing, interaction and information sharing between males and females is important. However, the standard FA algorithm does not distinguish the gender of individuals in the population and cannot effectively use the gender information of fireflies, which restricts the global search of FA in the presence of a large number of local extreme points.

FACL is based on the original FA algorithm, by adding a courtship learning mechanism to promote population information interaction and communication, and guide fireflies to fly, thereby enhancing the algorithm’s ability to jump out of the local optimum. The courtship learning process is: the male fireflies in the population select their mates from the female firefly’s external archive (A) and generate better solutions to improve the performance of the algorithm. The four key aspects of this mechanism are described as follows:

(1) Scaling mechanism. In order to make full use of the female firefly information in external archive A, individuals with lower fitness are more likely to be selected from external archive A. In this section, each female firefly in external archive A is scaled according to its fitness. The proportion transformation process of each female firefly is as follows:

$$ R_{i}=\frac{1}{f(X_{i})},~\forall~i=1,2,\cdots,Np. $$

(13)

where Np represents the archive size, which can be the same size as the male firefly population; f(X_i) represents the fitness of the i-th female firefly. Therefore, the female firefly with the lowest fitness in the external archive A has the largest estimation standard.

(2) Selection probability. After scaling the females in external archive A, the selection probability is designed to select the females. The selection probability is calculated as follows:

$$ p_{i}=\frac{R_{i}}{(\sum\limits_{j=1}^{Np}R_{j})},~\forall~i=1,2,\cdots,Np. $$

(14)

where p_i represents the probability of selecting the i-th female firefly. The selection mechanism ensures that female firefly individuals with lower fitness have higher selection probability.

(3) Female individual selection. If the selection is always based on the value order without the probability selection, FA is easy to fall into the local optimum, and it is difficult to achieve the global optimum search. Therefore, after calculating the selection probability of each female firefly by formula (14), the roulette strategy is used to select the female firefly individual to avoid the local optimum.

(4) New movement formula. According to formula (1), the attraction factor β_i,j decreases with the increase of distance. In FA, when the brightness of the selected male firefly x_j is higher than the brightness of the current male firefly x_i, there will be no movement operation in this iteration; When the brightness of the selected male firefly x_j is lower than the brightness of the current male firefly x_i, the current male firefly x_j will perform a movement operation according to Eq. (15). However, when the distance between the two fireflies is large, the attractive force is extremely low (as shown in Fig. 1a, which can easily lead to the premature termination of the movement process, and an ideal solution may not be obtained. For this reason, a new movement operation is adopted in FACL to handle this situation, which is defined as follows:

$$ x_{j}(t+1)=x_{j}(t)+v_{k,j}\times (x_{k}(t)-x_{j}(t))+\alpha\times \epsilon. $$

(15)

Fig. 1

The relationship between the three different attracting factors and distance r. For Fig. 1a, β₀ = 1,γ = 1; for Fig. 1b, β₀ = 1,γ = 1,t_max = 1000

where x_k is the currently selected female firefly individual, and x_j is the currently selected male firefly individual with higher brightness. v is the newly defined male and female attractiveness factor, which is defined as follows:

$$ v_{k,j}=\left(\frac{r_{k,j}}{800}\right)\times \text{logsig}\left((-\beta_{k,j})^{\frac{t}{600}}\right)\times e^{-\gamma r_{k,j}^{2}}. $$

(16)

In the above formula, logsig(.) represents a logistic regression function with a range of [0,1], and t represents the number of iterations. In formula (15), v is used to replace the parameter β in formula (3) to maintain the attractiveness between male and female firefly individuals within a certain distance r, which improves the probability of finding a better solution. The male and female attractiveness factor v defined by formula (16) is shown in Fig. 1b, where, based on the consideration of control variables, the maximum number of iterations of 1000 is taken as the value of t.

Improved firefly algorithm with courtship learning

A careful analysis of the variation curve of male and female individual attraction factor v introduced in FACL algorithm with the distance r between male and female fireflies in Fig. 1b shows that the substitution of v to some extent improves the situation that β decreases rapidly to nearly 0 with the increase of r and the attraction between fireflies decreases rapidly. However, when r increases to a certain extent (near r=1), at this time, the attraction factor of male and female will decrease again rapidly, and the information interaction between male and female individuals is no longer possible. What’s more, compared with the random term (α×ε) in formula (15), v is actually a very small value, which makes it difficult for male and female fireflies to interact and exchange information. Hence, the random term α×ε has a greater impact on the position update process, which degenerates into a random walk operation and is not conducive to the fast search of the optimal solution, especially in the discrete case. Therefore, although FACL still has a certain ability to jump out of the local optimum, it cannot search for the optima quickly.

Based on the above analysis, this work proposes a firefly algorithm based on FACL framework to improve male and female attraction factors. The algorithm introduces the Cauchy distribution function as the new attractiveness factor f to ensure that a certain attractiveness can be guaranteed when the distance r is far, so that the male firefly individual can still maintain a certain information interaction ability with the female firefly individual, continue to move in courtship behavior to ensure a certain ability to jump out of the local optimum, thereby improving the global search ability. This algorithm not only maintains the original simple structure of FACL, but also improves the accuracy of optimization.

As a new attraction factor, Cauchy distribution function has the characteristics similar to Gaussian distribution, but the peak value at the origin of Cauchy distribution is smaller and the distribution at both ends is longer. This characteristic is easy to generate random numbers with a large distance from the origin, so as to avoid the situation that the attraction rapidly decreases to nearly 0 when r is too large [26]. In this way, the individual male firefly can keep the interaction and movement of population information towards the female when the distance is large, and can jump out when trapped in the local optimum. The new attractiveness factor f is defined as follows:

$$ f=\text{Cauchy}(0,1). $$

(17)

where Cauchy(0,1) represents a random variable subject to the standard Cauchy distribution, defined as follows:

$$ f_{k,j}=\frac{1}{\pi(r_{k,j}^{2}+1)}. $$

(18)

According to the above definition, the curve of the attraction factor f of male and female fireflies in the IFACL with the distance r between the male and female fireflies will be a shape with a wide distribution, a gentle falling speed, and a non-zero end. Both the FA attraction factor β and the male and female attraction factor v in FACL have a better chance of searching for a better solution. The change curve of f is shown in Fig. 1c.

Therefore, by introducing the new male and female attraction factor f into Eq. (15) to replace v, the movement formula of male firefly courtship learning from female firefly in IFACL algorithm is as follows:

$$ x_{j}(t+1)=x_{j}(t)+f_{k,j}\times (x_{k}(t)-x_{j}(t))+\alpha\times \epsilon. $$

(19)

Based on the above definition, the complete search process of the IFACL is illustrated as Algorithm 1.

The implementation of this additional female archiving mechanism in this article is simple. When it is time to update the female archive, choose the brighter firefly as the new female firefly to replace the original female firefly.

IFACL is applied to UPMSP with sequence-dependent setup times

Based on the literature [1], this work designs a two-stage individual representation scheme to deal with this problem. This work applies the proposed IFACL algorithm to solve the UPMSP with sequence-dependent setup times.

a)Machine assignment

The first stage is the machine assignment process. This process describes the feasible solution of assigning N tasks to M unrelated parallel machines so as to minimize the maximum completion time (C_max) of all machines, which can be represented by an integer vector Seq₁ whose dimension is equal to the number of jobs. Taking the assignment of 10 jobs and 3 parallel machines as an example, if the vector Seq₁=[3,1,3,2,1,2,2,3,1,1], it means that the jobs {2, 5, 9, 10} will be performed on machine 1, the jobs {4, 6, 7} will be performed on machine 2, and the jobs {1, 3, 8} will be performed on machine 3. This requires the IFACL algorithm to adopt floor(.) in the iterative optimization process, where floor(.) is a function used to convert a real value to an integer value (i.e., floor(2.4) is 2). Hence, the position information of fireflies in FACL is discretized by the floor(.) function.

b)Sequence scheduling

The second stage is to determine the sequence of jobs on each machine. The sequence can be represented as a matrix of the same length as the machine-assigned vector. Therefore, the job sequence represented by Seq₂ can be expressed as a M×N matrix, which illustrates the sequence of operations on each machine. For the same previous example, the following instance of Seq₂ matrix, where extending on the Seq₁ vector, denotes that the sequence of operations on machine 1 is: job 9, job 5, job 10 and job 2; the sequence of operations on machine 2 is: job 4, job 7 and job 6; and the sequence of operations on machine 3 is: job 3, job 1 and job 8. The zero after job 2 in the first row denotes that job 2 is to be the last one to be performed by machine 1. Using the same convention, jobs 6 and 8 are the last jobs performed by machines 2 and 3, respectively.

$$ Seq_{2} = { \left[\begin{array}{cccccccccc} 9 & 5 & 10& 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 4 & 7 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 3 & 1 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right]}. $$

(20)

From the definition of Seq₂ and x_i,j,k, the value of x_i,j,k can be easily obtained. Then, we can calculate the completion time of the jobs at the machines C_j according to formula (8). This is performed by using a large positive number (V=∞), where $\sum \limits _{k=1}^{M}x_{i,j,k}=1$ when the jth job is processed after the ith job, so $V\left (\sum \limits _{k=1}^{M}x_{i,j,k}-1\right)=0$ and C_j=C_i+S_i,j,k+P_j,k.

In this work, for simplicity, the corrected processing times AP_i,j,k is used to replace the times of the two independent stages (machine processing times P_j,k and sequence-dependent setup times S_i,j,k) to determine the maximum completion time. For the same example above, in Tables 2, 3, 4, 5, the input parameters related to the example are illustrated as an instance of the problem under study.

Table 2 Example of processing times for an instance of 10 jobs and 3 machines

Table 3 Example of sequence-dependent setup times on machine 1

Table 4 Example of sequence-dependent setup times on machine 2

Table 5 Example of sequence-dependent setup times on machine 3

AP_i,j,k can be expressed mathematically as follows:

$$ \begin{aligned} \forall~j=1,2,\cdots,N, \forall~k=1,2,\cdots,M:\ \ AP_{i,j,k}=S_{i,j,k}+P_{j,k}, \\~~~~~~\forall~i=1,2,\cdots,N. \end{aligned} $$

(21)

Therefore, by introducing the concept of AP_i,j,k into Eq. (4), the optimization objective function of UPMSP with sequence-dependent setup times can be given by:

$$ fb(.)=\ C_{max}=\max_{1\leq j\leq N}\{C_{j}\} =\max_{\substack{k=1,\cdots,M;\\i=1,\cdots,N}}\sum\limits_{j=1}^{N} AP_{i,j,k}. $$

(22)

The target for solving UPMSP with sequence-dependent setup times problem is to minimize this objective function fb(.).

The application of the algorithm

After the encoding process is completed, the UPMSP with sequence-dependent setup times is solved through the optimization steps of IFACL algorithm described in “IFACL is applied to UPMSP with sequence-dependent setup times” section. In the initial stage, the IFACL algorithm first randomly assigned N planned jobs to M available processing machines to generate an initial male firefly population (represented by matrix X) consisting of Np male firefly individuals. Each male firefly individual corresponds to a job sequence on a particular machine selected after encoding in the M×N matrix described in “IFACL is applied to UPMSP with sequence-dependent setup times” section. Female firefly populations in external archive A can be initialized by male firefly populations.

The specific steps of IFACL for the UPMSP with sequence dependent setup times are as follows:

Step 1: Initialization. According to the mathematical model of UPMSP and the given range of machine processing times and sequence-dependent setup times, the parameters of IFACL are initialized.

Step 2: Population evolution. The population evolution process of IFACL is described in “Improved firefly algorithm with courtship learning” section.

Step 3: When the evolution meets the stop condition, the relevant information of the optimal female firefly is output as the approximate optimal scheduling scheme.

The psuedocode of the IFACL for UPMSP with sequence-dependent setup times is illustrated in Algorithm 2. Figure 2a and 2b are respectively the scheduling scheme result diagram and corresponding objective function variation curve obtained by IFACL after 50 iterations when processing 10 planned jobs and assigning them to 3 available processing machines according to the above steps. As can be seen from Fig. 2b, IFACL has a high convergence rate in the case of small scale.

Fig. 2

IFACL solves the UPMSP problem through 50 iterations

Computational complexity

Literature [11] has demonstrated that the CL framework of FACL does not increase the method’s computational complexity significantly. The IFACL algorithm is directly derived from the FACL algorithm, but the attraction factor is different, so its complexity is similar to that of FACL. Here, we analyze the time complexity of IFACL according to Algorithm 2. The related symbols are defined as follows: the population size, the number of jobs (problem dimension) and the total number of iterations are denoted as Np, N and t_max, respectively. First, in the initialization stage, the main computation cost of IFACL is O(Np× N)+O(Np), which is obtained by step 1-4. Then, during the optimization process in IFACL, since the average number of attractions for each firefly is (Np−1)/2 [27], the computation cost of the while loop in IFACL is O(t_max × Np × Np × N). Therefore, the overall computation cost of IFACL is given as follow.

$$\begin{array}{@{}rcl@{}} O(\text{IFACL})&=&O(N\!p \!\times\! N)\,+\,O(N\!p)\,+\,O(t_{max}\!\times\!N\!p\times \!N\!p\! \times\! N)\!\\ &\approx&O(t_{max} \times \! N\!p \times \! N\!p \times N) \end{array} $$

(23)

Compared with other heuristic algorithms, such as PSO, whose computational complexity is O(t_max × N p×N), the IFACL and FA variants have higher computational complexity.

Simulation experiment

In order to verify the effectiveness of the IFACL algorithm in solving the UPMSP with sequence-dependent setup times, two benchmark test datasets were used to verify the results. The first benchmark test dataset (named Dataset1) is generated by ourselves based on the relevant literature [28], which has been applied in many related studies [3, 21, 24, 25, 28, 29]. On this dataset, the performance of the IFACL algorithm in the (P_j,k|S_i,j,k|C_max) problem is compared with some existing meta-heuristics algorithms in the literature, including GA [21], SA [28], basic FA and FACL algorithms. In order to avoid the randomness of the test results, each algorithm was repeated for 15 times. In consideration of the need to obtain a feasible solution within an acceptable time range, the maximum number of iterations was selected as MaxIt=1000,Np=50, and all other parameters were directly derived from literature [24]. The second benchmark test dataset (named Dataset2) is provided by Rabadi [23], which also has been applied in many existed studies [30–33]. Dataset2 is also available via the scheduling research website (http://schedulingresearch.com/). The performance of the proposed algorithm is further verified by running it on Dataset2 and comparing it with the state-of-the-art work. All experiments were implemented by MATLAB 2020a programming and run in Windows environment with 8.0G RAM and 3.6GHz CPU.

Experimental results and performance comparison on dataset1

Dataset1 contains two categories, namely processing times and setup times, and its data are randomly generated by two discrete uniform distribution: U[50, 100] and U[125, 175]. The choice of the uniform distribution boundaries for processing times and setup times determines the level of dominance [28]. In other words, when the processing times and setup times are balanced (indicated by P, SBalanced), they are both generated by U[50, 100]. When the processing times dominate (indicated by PDominant), the processing times and the setup times are generated by U[125, 175] and U[50, 100], respectively. When the setup times are dominate (indicated as SDominant), the processing times and the setup times are extracted from [50, 100] and [125, 175], respectively.

The dataset generated from the above distribution consists of N jobs and M machines, covering 6 scenarios. The number of machines in each case is 2, 4, 6, 8, 10, and 12, and the number of jobs ranges from 20, 40, 60, 80, 100, and 120, respectively. Therefore, a total of 36 sets of calculation examples are constructed, and each set of calculation examples is run in each algorithm. Considering that the ratio of the computational time of algorithms to the execution time of the tasks is small, this work mainly evaluates the solving accuracy of the algorithms, which is measured by the Mean and variance SD.

Tables 6, 7, 8 illustrate the results on Dataset1 when the processing and setup times are balanced, when the processing times are dominant, and when the setup times are dominant, respectively. Among them, the optimal value of each case is marked in black and bold, and the result retains 2 decimal places.

Table 6 Comparison between IFACL algorithm and four comparison algorithms in P,SBalanced instances on Dataset1

Table 7 Comparison between IFACL algorithm and four comparison algorithms in PDominant instances on Dataset1

Table 8 Comparison between IFACL algorithm and four comparison algorithms in SDominant instances on Dataset1

When the processing times and setup times are balanced, according to Table 6, the results obtained by the IFACL algorithm on Dataset1 can obtain 24 optimal Mean values out of the 36 test cases in terms of accuracy. Therefore, in terms of accuracy, it is obvious that the IFACL is superior to the other four compared algorithms in terms of accuracy, followed by the other two FA variants (FA and FACL). In terms of stability, SA obtains 34 best SD values in the 36 cases, indicating that SA is of the highest stability. Besides that, IFACL also performs pretty good stability on Dataset1.

When the processing times dominate, as can be seen from Table 7, IFACL also shows the best performance in terms of accuracy as it achieves 19 optimal results out of the 36 test cases on Dataset1, followed by FA and FACL. Since the SA obtains 33 best SD values out of the 36 test cases, the SA still shows the best stability.

When the setup times dominate, according to Table 8, the IFACL achieves 23 best Mean values out of the 36 test cases on Dataset1, which shows that IFACL still performs best in terms of accuracy. Obviously, SA is the most stable, too.

To make the comparison of the five algorithms in terms of accuracy more intuitively and precisely, on the basis of Tables 6, 7, 8, this work also compares the rank of Mean values obtained by each comparison method in each of the 36 cases on three different categories of Dataset1, as is shown in Fig. 3. For each cases on different categories, the top-ranked approach will add 1 on the corresponding bar and the last-ranked approach will add 5, which denotes that the best method would correspond to the lowest bar. As can be seen from the Fig. 3, IFACL ranks first in three different categories, while SA ranks last. The result illustrates that when solving the UPMSP with sequence-dependent setup times on Dataset1, our proposed IFACL-based scheduling approach has best accuracy in all compared algorithms.

Fig. 3

Rank comparison of the five algorithms for Mean values over different categories on Dataset1

Figures 4, 5, 6, 7, 8 and 9 show the average convergence curves of these five algorithms on Dataset1 with different job numbers when the number of machines is 2 and 10, respectively, including the P,S,Balanced instances, PDominant instances and SDominant instances. These figures show that the convergence rate of IFACL is similar to that of FA and FACL, but better than that of GA and SA. In addition, these figures further demonstrate the high performance of IFACL in terms of accuracy.

Fig. 4

Average convergence curve for each algorithm on P,SBalanced instances of Dataset1 when the number of machines M=2

Fig. 5

Average convergence curve for each algorithm on P,SBalanced instances of Dataset1 when the number of machines M=10

Fig. 6

Average convergence curve for each algorithm on PDominant instances of Dataset1 when the number of machines M=2

Fig. 7

Average convergence curve for each algorithm on PDominant instances of Dataset1 when the number of machines M=10

Fig. 8

Average convergence curve for each algorithm on SDominant instances of Dataset1 when the number of machines M=2

Fig. 9

Average convergence curve for each algorithm on SDominant instances of Dataset1 when the number of machines M=10

Experimental results and performance comparison on dataset2

In this subsection, another UPMSP benchmark dataset named Dataset2 is used to evaluate the proposed algorithm. Dataset2 also contains the two categories, including the processing times and the setup times. In this section, the IFACL is compared with the state-of-the-art from the literature such as FSS [30], WO [31], GADP2 [32], SADP [32], IFA [33]. These approaches are chosen since they established their quality to address the UPSMP with sequence-dependent setup times and all of them adopt the same benchmark instances (that is, Dataset2) to solve UPMSPs in previous studies.

Similar to the comparison approaches, such as IFA, only show results of P,SBalanced instances on Dataset2, we also conduct experiments in P,SBalanced cases here. According to literature [30], the maximum number of iterations is set to 10000 and the size of population is set to 100. Note that the results of the comparison algorithms are derived directly from the corresponding papers.

Table 9 presents the results of the methods on Dataset2 when the processing and setup times are balanced. Since the FSS only present the Mean results, we also just present the Mean results in Table 9. Furthermore, WO, GADP2 and SADP only present the results for problem instances with 2, 6 and 12 machines and 20, 40, 60, and 80 jobs, therefore, Table 9 also provides the corresponding results for the same problem instances. Among them, the optimal value of each case is marked in black and bold.

Table 9 Comparison between IFACL and the state-of-the-art in P,SBalanced instances on Dataset2

As can be seen from Table 9, both the IFACL and the FSS obtain the 5 best Mean values out of 12 instances, followed by the WO and the IFA. It can also be observed that when the number of jobs is 20, IFACL manages to outperform the other approaches. When the number of jobs is 40, IFACL also performs the best or second best. This demonstrates that IFACL is suitable for processing problem instances with a relatively small number of jobs. Besides, comparing with IFA, IFACL obtains 9 better results in 12 problem instances, which also verifies the effectiveness of the introduction of Cauchy distribution as a factor of female and male attraction on the basis of the FACL framework aforementioned in this work in improving FA performance to a certain extent. Although IFACL does not perform optimally when the number of jobs is 60 or 80, it is still a valid candidate algorithm overall.

In summary, the IFACL has great potential to achieve high-quality solutions over the Dataset1 and Dataset2 of the UPMSP problems. It outperforms the FA, FACL, GA and SA in most problem instances on Dataset1, and it performs quite well on Dataset2 compared to the state-of-the-art work, especially for the problem instances with a relatively small number of jobs. Whereas, the proposed IFACL benefits from the advantages of both the FACL framework and the new female and male attractiveness factor, thereby improving its search performance and raising its ability to avoid getting stuck in local optima. Furthermore, compared with the FA, FACL, GA and SA, the IFACL is also the fastest algorithm to reach the minimum makespan, as shown in the convergence curves. The comparison with the state-of-the-art work demonstrates the effectiveness of the IFACL in some of the problem instances on Dataset2, and the IFACL is competitive with other algorithms. However, the proposed approach has some limitations, such as performance degradation when the number of jobs is large, as shown in the Table 9. It stands for the reason that the new attractiveness factor in IFACL is also close to 0 at the later stage of the search, so the ability of IFACL to jump out of the local optimum in high dimensions is weaker than that of some state-of-the-art works. In addition, the processing times, the initial job sequences and the sequence-dependent setup times should be generated before starting the search process.

Conclusion

Aiming at the UPMSP with sequence-dependent setup times, this work proposes the IFACL algorithm, which takes minimizing the maximum completion time as the scheduling optimization goal to solve the UPMSP with sequence dependent setup times. For the IFACL algorithm, the Cauchy distribution function with wide distribution and gentle decline is introduced as a new attraction factor of male and female. This design can enhance the social information exchange between fireflies and enhance the ability of the algorithm to jump out of the local optimum in high-dimensional situations. The test results of two benchmark test datasets show that the scheduling algorithm proposed in this work is effective in dealing with sequence-dependent setup times UPMSP. Compared with GA and SA in terms of accuracy on Dataset1, the IFACL has obvious advantages, and it is also greatly improved compared with FA and FACL. Furthermore, based on the empirical results on Dataset2, the IFACL is shown to be competitive with other state-of-the-art algorithms.

The scheduling problem of application scenarios such as edge computing is a hot topic of scheduling research. However, there is no unified model standard yet. In the future, we will continue to use the UPMSP model for reference and study under the conditions of considering more constraints such as bandwidth and different network service providers’ charges, and propose efficient algorithm to solve the problem.