A Hybrid Algorithm for Multi-Objective Optimization—Combining a Biogeography-Based Optimization and Symbiotic Organisms Search

Abstract

To solve the multi-objective, flexible job-shop scheduling problem, the biogeography-based optimization (BBO) algorithm can easily fall into premature convergence, local optimum and destroy the optimal solution. Furthermore, the symbiotic organisms search (SOS) strategy can be introduced, which integrates the mutualism strategy and commensalism strategy to propose a new migration operator. To address the problem that the optimal solution is easily destroyed, a parasitic natural enemy insect mechanism is introduced, and predator mutation and parasitic mutation strategies with symmetry are defined, which can be guided according to the iterative characteristics of the population. By comparing with eight multi-objective benchmark test functions with four multi-objective algorithms, the results show that the algorithm outperforms other comparative algorithms in terms of the convergence of the solution set and the uniformity of distribution. Finally, the algorithm is applied to multi-objective, flexible job-shop scheduling (FJSP) to test its practical application value, and it is shown through experiments that the algorithm is effective in solving the multi-objective FJSP problem.

1. Introduction

The breakthrough of flexible job-shop scheduling [1] (FJSP) is limited by the uniqueness of resources; each process is coordinated and processed by multiple machines. Because of its flexible feature, the problem is closer to real life, and it has been widely used in steel manufacturing, the chemical industry, automobile manufacturing and other fields. Multi-objective, flexible job-shop scheduling (MOFJSP) has received less research attention than single-objective FJSP in recent years. MOFJSP has the ability to simultaneously optimize numerous performance metrics. Finding the quasi-optimal solution set that satisfies the criteria is a pressing issue to be resolved given the MOFJSP problem’s complexity, randomness, various constraints, and multiple aims. Consequently, it has very important practical significance for the research of MOFJSP.

In the realm of multi-objective optimization research, multi-objective evolutionary algorithms (MOEAs) are a popular tool due to their high universality and independent function model. The development of MOEAs can be divided into two phases [2]: The first generation algorithm uses dominated sorting and fitness-sharing mechanisms, while the second generation introduces an elite retention strategy on top of the first generation. As research into MOPs has progressed, more non-dominated mechanisms and evolutionary algorithms have been introduced to the multi-objective research field.

In 2004, Carlos introduced the multiple objectives with a particle swarm optimization algorithm (MOPSO) as a way to address the loss of non-dominated solutions [3]. This was achieved by introducing a secondary (external) repository of particles along with additional mutation operators and boundary restriction strategies. This method showed strong competitiveness and feasibility in solving multi-objective optimization problems. Chen and Man et al. [4] proposed a multi-objective new whale optimization algorithm (MONWOA) that aimed to balance algorithm exploration and development. The Gaussian mutation operator and the differential evolution operator were introduced, along with a new constraint handling method to enhance the distribution of Pareto solution sets. Compared to the traditional MOPSO method, the MONOWA method displayed stronger stability and diversity.

In 2012, Tan suggested a new algorithm called the uniform design multi-objective differential algorithm based on decomposition [5]. This algorithm introduced the differential evolution algorithm and re-designed the weight vector. Experiments have shown that the distribution of population individuals in the algorithm has improved significantly. However, the convergence speed of the algorithm still needs improvement. To address this issue, Chen and Han et al. proposed a multi-objective novel improved water cycle algorithm (MONIWCA) [6]. This algorithm introduced the evaporation rate to improve the evaporation process and used a normal distribution optimization mechanism to change the individual position. Additionally, a global sorting strategy was proposed to obtain a high-quality Pareto optimal solution set and select the best compromise solution. Upon analysis of the generational distance and interval, the MONIWCA algorithm was found to have the characteristics of uniform distribution, high convergence, and strong stability.

In 2016, Tran and Cheng developed a novel multi-objective symbiotic organisms search algorithm (MOSOS) [7]. MOSOS is a population-based meta-heuristic algorithm that sequentially simulates the interactions between organisms in an ecosystem with three strategies of mutualism, commensalism, and parasitism [8]. The experimental results show that MOSOS exhibits better diversity characteristics. In 2017, Chen and Cao et al. proposed three modified multi-objective imperialist competitive algorithms (MOICA-I, MOICA-II, and MOICA-III) [9] to tackle multi-objective optimal reactive power dispatch problems. The results proved that the enhanced methods were effective in solving the problem, with MOICA-III demonstrating excellent performance. That same year, Chen and Qian et al. proposed a new multi-objective improved bat algorithm [10] to address the basic bat algorithm’s tendency to fall into a local optimum. The new algorithm includes a nonlinear inertia weight, a global optimal guidance mechanism, and a monotonic random filling model based on extremum. An elite non-dominated sorting method based on crowding distance was used to obtain a uniformly distributed Pareto optimal set. In 2022, Wang et al. [11] presented a novel multi-objective cellular memetic optimization algorithm (MOCMOA). The proposed MOMOA combined the advantages of a cellular structure for global exploration and a variable neighborhood search (VNS) for local exploitation. At last, MOCMOA was compared against other multi-objective optimization approaches by performing experiments. In 2023, Zhang et al. [12] proposed a dual-population genetic algorithm with Q-learning to minimize the maximum completion time and the number of tardy jobs for distributed, hybrid flow-shop scheduling problems, which have some symmetries in machines.

In 2008, Professor Dan Simon introduced the biogeography-based optimization (BBO) algorithm in IEEE Transactions on Evolutionary Computation [13]. Due to its simplicity and minimal parameters, BBO has gained the interest of both domestic and international scholars and has been applied across various fields. In 2021, Li et al. [14] proposed an enhanced version of BBO by adjusting the migration rate using the habitat suitability index of a normalized individual and introducing a differential perturbation to the migration operator. This improved algorithm was utilized in the optimization dispatching of microgrids. In 2022, Zhang et al. [15] addressed the limitations of the Worst Opposition Learning and Random-Scaled Differential Mutation Biogeography-Based Optimization (WRBBO) algorithms which had an insufficient search ability and low efficiency in solving complex optimization problems. They proposed a Multi-Population BBO (MPBBO) which showed significant improvements. In 2023, Zhang et al. [16] introduced a hybrid biogeography-based optimization algorithm that solves high-dimensional optimization problems and real-world engineering challenges.

The BBO algorithm has been utilized for single-objective optimization problems since its inception. However, with the advancement of multi-objective evolutionary algorithms, the BBO algorithm has been extended to multi-objective optimization problems as well. Xu and Mo have proposed a multi-objective disturbance biogeography-based optimization algorithm [17] that incorporates the SPEA-II model and a perturbation migration model to enhance population diversity. The algorithm effectively solves MOPs as demonstrated by its comparison with standard function tests and classical algorithms. Bi and Wang have introduced a new MOBBO algorithm [18] that employs a self-adaptive method to determine immigration and emigration rates, a dynamic migration strategy, and a piecewise logistic chaos mutation strategy for better convergence performance in multi-objective optimization. Experimental results confirm that MOBBO significantly improves the convergence of the solution set and distribution uniformity. Wang and Li have proposed an adaptive constrained multi-objective biogeography-based optimization algorithm (ACMBBO) [19] based on a two-stage elite selection to address constrained multi-objective optimization problems. They establish a constrained multi-objective optimization model suitable for BBO by considering the characteristics of constrained multi-objective optimization and BBO’s evolution mechanism. Numerical experiments using a dynamic migration strategy show that the ACMBBO algorithm is competitive in terms of convergence and distribution.

When compared to traditional MOEAs, the MOEAs mentioned above have demonstrated an improvement in the convergence and distribution uniformity of the approximate Pareto optimal solution set. However, the algorithm still tends to become trapped in a local optimal front, making it challenging to balance both the convergence and uniformity of distribution. This problem can be attributed to the low efficiency of the evolution process of MOEAs when the current solution set is nearing the global optimum, there is a lack of an effective optimization mechanism to broaden the search area, and the search mechanism is incompatible with the population. The current evolution state is limited by randomness and a certain degree of blindness.

In order to solve the above-mentioned problems in MOEAs, this paper introduces the SOS symbiosis strategy into the BBO algorithm. The enhanced algorithm is integrated into the PESA algorithm model, resulting in the proposal of MOHBS—a multi-objective optimization approach based on a hybrid biogeography-based optimization and symbiotic organisms search. The algorithm utilizes a hyperbolic tangent mobility model that aligns with natural migration patterns. The migration operator in the BBO algorithm incorporates the mutualism and commensalism strategies from the symbiotic organism search algorithm to avoid premature convergence and cater to diversity. The “predation” mutation strategy is defined and used alongside the “parasitic” strategy of SOS, whereas the mutation operator based on symbiosis strategy is proposed to counter the damage caused by the mutation operator in finding a better solution.

2. Method

In biogeography-based optimization algorithms, the migration operator and mutation operator are the primary operators utilized.

2.1. Migration Operator

The BBO algorithm relies on migration as its core operation. Habitats exchange information with each other through species migration, with immigration and emigration rates denoted by ${\lambda }_k$ and ${\mu }_k$. As shown in Figure 1, this paper utilizes the hyperbolic tangent migration rate model suggested by Wang et al. [20] as the migration operator. This model imitates the migration pattern of natural species, resulting in a superior solution compared to the BBO algorithm’s cosine migration rate model and other migration models.

Hyperbolic tangent migration rate mode formula:

$$\begin{array}{l}{\lambda }_k=\frac{I}{2}\left(-\frac{e^{(k-n/2)}-e^{(-k+n/2)}}{e^{(k-n/2)}+e^{(-k+n/2)}}+1\right)\\ {\mu }_k=\frac{E}{2}\left(\frac{e^{(k-n/2)}-e^{(-k+n/2)}}{e^{(k-n/2)}+e^{(-k+n/2)}}+1\right)\end{array}$$

(1)

where $I=E=1$, $k$ is the number of species in the current habitat, $n$ is the maximum number of species, and $e$ is the base parameter. Thus, Equations (1) and (2) are expressed as:

$$\begin{array}{l}{\lambda }_k=\frac{1}{2}\left(-\frac{e^{(k-n/2)}-e^{(-k+n/2)}}{e^{(k-n/2)}+e^{(-k+n/2)}}+1\right)\\ {\mu }_k=\frac{1}{2}\left(\frac{e^{(k-n/2)}-e^{(-k+n/2)}}{e^{(k-n/2)}+e^{(-k+n/2)}}+1\right)\end{array}$$

(2)

2.2. Mutation Operator

The BBO algorithm’s species variation mechanism is highly efficient in enhancing the diversity of individuals in the population. There is a probability that habitat individuals will mutate, which provides them with more opportunities to locate the best solution in the search space. However, this random variation can increase the likelihood of better habitat individuals mutating into weaker ones. The mutation probability function of a certain habitat is in an inverse ratio to the number probability of the habitat; therefore, the probability $m_s$ of the variation in a habitat with several species $s$ is:

$$m_s=m_{max}\left(1-P_s/P_{max}\right)$$

(3)

where $P_S$ represents the probability that the number of species in the habitat is $s$, $m_{\mathit{max}}$ is the maximum value of the user-defined mutation rate, and $P_{\mathit{max}}$ is the maximum value of all $P_S$, which is $Max\left(P_S\right)$.

3. Migration Operator Based on Symbiosis Strategy

3.1. Improved Mutualism Strategy

In the migration phase, for the immigration habitat $X_i$, choose one of the species $X_{i,k}$, and choose the emigration habitat $X_j$. The two habitat species, the emigration habitat, and the immigration habitat evolve to each other through a win-win symbiosis relationship to improve their habitat fitness. The improved mutualism strategies formula is:

$$\begin{array}{l}{X}_{i,k}=X_{i,k}+\omega \times \left(X_{best,k}-X_{mutual}\times B{F}_1\right)\\ X_{j,k}=X_{j,k}+\omega \times \left(X_{best,k}-X_{mutual}\times B{F}_2\right)\end{array}$$

(4)

$$X_{Mutual\_Vector}=\frac{X_{i,j}+X_{j,k}}{2}$$

(5)

where $X_{i,k}$ and $X_{j,k}$ are the habitat of the species that immigrate and the habitat of the species that emigrate, respectively. $X_{best,k}$ represents the species in the optimal habitat, and under the guidance of the optimal habitat, it gradually approaches the optimal solution. $X_{Mutual\_Vector}$ is a “mutually beneficial vector”, which represents the reciprocal characteristics between two organisms. $B{F}_1$ and $B{F}_2$ take a value of 1 or 2, indicating the degree of profit from the interaction between organisms.

$$\omega =1-\frac{1}{e^{{(1-\frac{G}{G\max })}^2}}$$

(6)

where it can be seen from Equation (6) that in the early phase of the algorithm’s iteration, $\omega \approx 1$, the algorithm has a strong global exploration ability. As the algorithm progresses iteratively and the value of $\omega$ gradually decreases, the algorithm enhances its refined search in a specific area, leading to an improvement in the convergence rate and precision of the algorithm. Incorporating a mutualism strategy during this operational phase can effectively balance the exploration and development capabilities of the algorithm.

3.2. Improved Commensalism Strategy

In the migration phase, for the immigration habitat $X_i$, choose one of the species $X_{i,k}$, and choose the emigration habitat $X_j$, the emigration habitat $X_{j,k}$ and the immigration habitat $X_{i,k}$ interact with the two habitat species, and the species $X_{i,k}$ benefits from this interaction. However, the species $X_{j,k}$ neither benefits nor suffers from this interaction, only $X_{i,k}$ benefits from the influence of the relationship. According to the symbiotic relationship between $X_i$ and $X_j$ in the habitat, a new candidate solution is calculated. The improved commensalism strategy formula is:

$$X_{i,k}=X_{i,k}+\phi \times \left(X_{best,k}-X_{j,k}\right)$$

(7)

where $\phi$ represents a random number between [−1,1]. $X_{best,k}-X_{j,k}$ reflects the beneficial advantages provided, and $X_i$ helps to maximize the survival advantage of the current ecosystem and improves the convergence rate of the algorithm under the guidance of the optimal habitat species.

3.3. Improved Migration Operator

In view of the shortage of the original migration operator, the improved mutualism strategy and the improved commensalism strategy in the SOS algorithm are introduced. The pseudocode of the improved operator is shown in Algorithm 1.

The use of the migration operator to improve the quality of the habitat individuals is limited due to its heavy reliance on individuals with superior habitats during the evolution process. This results in the inability of individuals in need of growth to learn from their peers, ultimately leading to a single model of individual evolution. Additionally, the algorithm lacks the ability to effectively mine new solutions and its search ability is inadequate. During the later stages of evolution, there is a high risk of minimal differentiation between habitat individuals which can lead to a lack of evolutionary motivation, resulting in premature evolution and the algorithm becoming trapped in a local optimal solution.

Algorithm 1: Migration operator based on symbiosis strategy

for $i=1$ to $n$   select the emigration habitat based on the hyperbolic tangent migration rate model   for $j=1$ to $D$   if $rand<{\lambda }_i$   improved the mutualism strategy (Formula (4))   else                       improved the commensalism strategy (Formula (6))   end if   end for end for

In this paper, we implement a strategy of mutualism and commensalism in the SOS algorithm’s migration operator. The mutualism strategy addresses the issue of algorithm diversity insufficiency. In the early stages of algorithm iteration, the global search equation is primarily used in the mutualism strategy, providing a strong global exploration capability. As the algorithm progresses, the guided local search is employed to reinforce the refined search in specific regions, effectively enhancing the algorithm’s convergence speed. This operation phase allows for the full utilization of population information in exploring the entire search space in the early stages, maintaining population diversity, and performing local refined searches in the later phases. This strategy helps to balance the algorithm’s exploration and development capabilities. The commensalism strategy prevents the algorithm from becoming stuck in the local optimal solution and improves the search process’s convergence rate.

4. Mutation Operator Based on Symbiosis Strategy

4.1. Parasitic Strategy

In a symbiotic relationship, parasitism occurs when one party benefits while the other suffers. The most familiar example of this phenomenon in nature is the relationship between anopheles mosquitos and humans. Anopheles mosquitos reap the benefits of this parasitic relationship, while humans bear the burden of their bites.

At this phase, $X_i$ randomly selects certain dimensions and replaces them with random values in the search space, forming an artificial parasite $Parasite\_Vector$. After that, an individual $X_j\left(j\ne i\right)$ is randomly selected in the population, and the fitness values of the two are compared, keeping the optimal one as the new $X_j$.

The SOS algorithm parasitic strategy formula is:

$$Parasite\_Vector(pick)=(UB(pick)-LB(pick))\times rand+LB(pick)$$

(8)

where $pick$ is the variation bit, and $UB$ and $LB$ represent upper and lower bounds, respectively.

4.2. Predator Strategy

Predation: Predation is also a symbiosis strategy, a phenomenon in which one species preys on another species to obtain nutrients and maintain life. The former is called the predator and the latter is called the prey. One species improves its fitness by hunting another species. Only two species are involved in this phase. The migration rate of the habitat represents the food chain of a species; the higher the migration rate, the higher the food chain. Species with higher food chains prey on species with lower food chains to improve their fitness. If the fitness improves after hunting, it means the hunting is successful; otherwise, the hunting fails.

The hunting phase is:

$$Predator(new)=Predator+\alpha \times (Predator-prey)$$

(9)

where $\alpha$ is the hunting intensity, and $\alpha ={\mu }_{predator}-{\mu }_{prey}$, $\alpha$ is in proportion to the difference between the emigration rate of the habitat predator and the habitat prey. The smaller the $\alpha$, the closer the food chain of the two species, and the higher the hunting intensity; predator–prey is called the species difference.

4.3. Improved Mutation Operator

The SOS algorithm’s parasitic strategy is similar to the biogeography-based optimization algorithm’s mutation strategy, both using a single random mutation approach. However, this method has limitations in considering the global and local search, resulting in certain drawbacks. While it may create better habitats for poor environments, it may not be effective for better habitats, especially in later iterations of the algorithm. Random mutations can produce habitats with no use value, resulting in a low efficiency for the evolution of the population towards the optimal solution.

Parasitic natural enemy insects are those that parasitize the body or surface of other insects or animals during a certain developmental phase or throughout their life. They feed on the nutrients of the host, and most parasitize during the larval stage. These larvae cannot survive independently from the host and grow inside or on a single host. After the parasitic natural enemy insect larvae complete their development, the host gradually dies or is destroyed. The adults of most parasitic natural enemy insects are free-living and hunt for survival. The parasitic strategy is employed in the early phase of the algorithm iteration, while the predation strategy is used in the later phase. Algorithm 2 shows the pseudo code of the improved operator.

Algorithm 2: Mutation operator based on symbiosis strategy

for $i=1$ to $n$   for $j=1$ to $D$   if $rand<m_i$   if $rand\ge kg/G$   parasitic strategy   else   predation strategy   end if   end if   end for end for

The dual-strategy cooperative mutation operator combines the parasitic and predation strategies to promote species diversity and the optimal evolution of the population. In the initial iteration, the parasitic strategy is used to introduce random mutations that may lead to new and improved habitats. Meanwhile, the predation strategy is employed in the later phase to conduct local searches for better solutions.

4.4. Low-Discrepancy Sequences Initialization Population Strategy

Low-discrepancy sequence [21] is also called quasi-random sequence, and its super-uniform distribution allows sequences of any length to be uniformly filled in the entire function space. A uniformly distributed random number means a better sample distribution. To ensure a more even distribution in the sample, it is ideal to use uniformly distributed random numbers. In order to avoid repetitive patterns during the initialization process, a low-discrepancy sequence with super-uniform distribution characteristics is used to initialize the population. This results in better algorithm performance, reduced errors, and faster convergence. The Halton sequence is the low-discrepancy sequence utilized in this study.

4.5. PESA Multi-Objective Evolutionary Algorithm Model

PESA (proposed by Corne) [22] is an algorithm that selects based on the Pareto envelope. Its distinctive feature is the use of a hyper-grid scheme to manage selection and ensure population diversity. This algorithm employs a small population size and stores non-dominated individuals in an external archive collection.

To ensure a varied and widespread distribution of solutions, the hyper-grid approach divides the individual space into several hyper-boxes, each corresponding to a specific individual’s correlation. Figure 2 illustrates the optimization problem, where the objective is to minimize two goals. Individual A is located in a hyper-box with two other individuals, resulting in a squeezing coefficient of 2, while Individual B has only one other individual in their hyper-box, resulting in a squeezing coefficient of 1. Non-dominated individuals are represented by circles, while dominated individuals are represented by squares.

When a new individual enters the external Pareto population, an individual needs to be deleted from the external Pareto population, and the specific method is to find the individual with the largest squeezing coefficient in the external population of Pareto and remove it. If there are multiple individuals with the same squeezing coefficient, one is chosen randomly for removal.

This paper uses the objective function combination method in literature [2] to calculate the habitat fitness. The objective function combination method is: Suppose the optimization objective function has $r$ sub-objectives, and the fitness of the sub-objective is: $fit\left(i\right)\left(j=1,2,\dots ,r\right)$. The individual $i$ is defined as: The $fitness\left(i\right)$ of individual $i$ is defined as:

$$fitness(i)=\underset{j=1}{\overset{r}{\Pi }}{(fi{t}_j(i))}^2$$

(10)

This method is characterized by simplicity, low computational complexity, and it is not easy to lose boundary points.

5. Mutation Operator Based on Symbiosis Strategy

This paper proposes a novel approach for multi-objective optimization based on a combination of the hyperbolic tangent migration rate model, improved migration and mutation operators, and low-discrepancy sequence initialization population strategy. The algorithm, called MOHBS, integrates hybrid biogeography-based optimization and symbiotic organisms search. The proposed method is designed to enhance the efficiency and effectiveness of multi-objective optimization. The steps of the algorithm are as follows:

Step 1: Set the relevant parameters, the number of species $N$, the current number of iterations is $t$, and the maximum number of iterations $g_{\max }$;

Step 2: Low-discrepancy sequence initial internal population $IP$, and evaluate it; at the same time, initialize the external population $EP$ to make it empty;

Step 3: Incorporate the non-dominated individuals in the $IP$ into the $EP$; if the number of habitats in the $EP$ is less than $N$, select the individuals with the smallest fitness to join the $EP$, until the number of individuals is equal to $N$; if the number of non-dominated solutions in the $EP$ is greater than $N$, after that, look for the individual with the largest squeezing coefficient and remove it until the number of individuals is equal to $N$;

Step 4: If the termination conditions are met, it ends, and takes $EP$ as the return result. Otherwise, execute step 5;

Step 5: Clear all individuals in the $IP$, and sequentially execute Algorithm 1 to improve the migration operator and Algorithm 2 to improve the dual-strategy cooperative mutation operator; obtain the population $P\left(t+1\right)$, $t=t+1$;

Step 6: Go to Step 3.

6. Numerical Experiment and Result Analysis

The ZDT series offers several test functions, namely ZDT1, ZDT2, ZDT3, ZDT6, Schaffer, and Kursawe. Additionally, Viennet2 and Viennet3 are chosen as three objective test functions in literature [2]. The specific description of the test function is shown in

Table 1. Eight test functions used in the experiment.

Problem	Objective Function	Constraint Condition
ZDT1 PFtrue convex	$\begin{array}{l}F=(f_1(x),f_2(x)),where\\ f_1=(x_1)=x_1\\ f_2(x)=g(1-\sqrt{(f_1/g)})\\ g(x)=1+g{\displaystyle \sum _{i=2}^m{x}_i/(m-1)}\end{array}$	$m=30;0\le x_i\le 1$
ZDT2 PFtrue convex	$\begin{array}{l}F=(f_1(x),f_2(x)),where\\ f_1=(x_1)=x_1\\ f_2(x)=g(1-{(f_1/g)}^2)\\ g(x)=1+g{\displaystyle \sum _{i=2}^m{x}_i/(m-1)}\end{array}$	$m=30;0\le x_i\le 1$
ZDT3 PFtrue discontinuous	$\begin{array}{l}F=(f_1(x),f_2(x)),where\\ f_1=(x_1)=x_1\\ f_2(x)=g(1-\sqrt{(f_1/g)}-(f_1/g)\sin (10\pi f_1))\\ g(x)=1+9{\displaystyle \sum _{i=2}^m{x}_i/(m-1)}\end{array}$	$m=10;0\le x_i\le 1$
ZDT6 PFtrue concave	$\begin{array}{l}F=(f_1(x),f_2(x)),where\\ f_1=(x_1)=1-\exp (-4x_i){\sin }^6(6\pi x_1)\\ f_2(x)=g(1-{(f_1/g)}^2)\\ g(x)=1+9(({\displaystyle \sum _{i=2}^m{x}_i/{(m-1)}^{0.25}}))\end{array}$	$m=10;0\le x_i\le 1$
Schaffer PFtrue convex	$\begin{array}{l}F=(f_1(x),f_2(x)),where\\ f_1(x)=x^2\\ f_2(x)={(x-2)}^2\end{array}$	$-{10}^5\le x\le {10}^5$
Kurwase Ptrue discontinuous PFtrue discontinuous	$\begin{array}{l}F=(f_1(x),f_2(x)),where\\ f_1(x)={\displaystyle \sum _{i=1}^{n-1}\left(-10e^{\left(-0.2\right)\times \sqrt{x_i^2+x_{i+1}^2}}\right)}\\ f_2(x)={\displaystyle \sum _{i=1}^{n-1}\left({\left|x_i\right|}^{0.8}+5\sin {(x_i)}^3\right)}\end{array}$	$\begin{array}{l}-5\le x_i\le 5(i=1,2,3)\\ \end{array}$
Viennet2 Ptrue continuous PFtrue discontinuous	$\begin{array}{l}F=(f_1(x,y),f_2(x,y),f_3(x,y)),where\\ f_1(x,y)=0.5\times {(x-2)}^2+\frac{{(y+1)}^2}{13}+3\\ f_2(x,y)=\frac{{(x+y-3)}^2}{36}+\frac{{(-x+y+2)}^2}{8}-17\\ f_3(x,y)=\frac{{(x+2y-1)}^2}{175}+\frac{{(2y-x)}^2}{17}-13\end{array}$	$m=2;-4\le x,y\le 4$
Viennet3 Ptrue discontinuous asymmetry PFtrue continuous (Pareto curve in 3-dimensional space)	$\begin{array}{l}F=(f_1(x,y),f_2(x,y),f_3(x,y)),where\\ f_1(x,y)=0.5\times (x^2+y^2)+\sin (x^2+y^2)\\ f_2(x,y)=\frac{{(3x-2y+4)}^2}{8}+\frac{{(x-y+1)}^2}{27}+15\\ f_3(x,y)=\frac{1}{(x^2+y^2+1)}-1.1\exp (-x^2-y^2)\end{array}$	$m=2;-30\le x,y\le 30$

. These test functions are useful in assessing the effectiveness of multi-objective optimization algorithms in achieving convergence towards the true Pareto front and maintaining group diversity. To ensure thorough testing, each algorithm is executed 30 times on each test function.

All algorithms are re-implemented using the MatlabR2014a programming language and run on a PC with 2.5 GHZ Intel(R)Core (TM)i5-7300 CPU 8 GB RAM 64-bit OS.

6.1. Test Problems and Comparison Algorithms

In order to verify the performance of the MOHBS algorithm, it is compared with the advanced algorithms of MVCMOPSO [23], NSGA-II [24], SPEA2 [25], MOPSO [3], and GWASF-GA [26]. The initial population size of MOHBS and the other seven algorithms is $n=100$, $EP=100$, $I=E=1$, $P_{\mathit{max}}=0.05$ in MOHBS, and for other algorithm parameter settings, please refer to the performance metrics in literature [23].

The goal of solving multi-objective optimization problems is usually that the obtained Pareto optimal solution set can approach the real Pareto front, and the more uniform the Pareto front is, the better. In order to quantitatively analyze the convergence and uniform distribution of the algorithm, this subsection selects the following performance indicators, which are defined as follows:

Generational distance (GD): The distance between the evaluated Pareto front ($P{F}_{know}$) and the optimal Pareto front ($P{F}_{true}$), which is the index of the convergence evaluation.

$$GD=\frac{({\displaystyle {\sum }_{i=1}^n{d}_i{}^2{)}^{\frac{1}{2}}}}{n}$$

(11)

where $n$ is the number of solutions in $P{F}_{know}$. $d_i$ is the Euclidean distance between the $i$th solution in the target space and the nearest solution in $P{F}_{true}$. The smaller the GD value, the closer the found non-dominated solution set is to the real non-dominated solution set.

Spread (SP): It can be used to evaluate the distribution of the obtained solution set; that is, the characteristic of being uniformly distributed near the real Pareto front. The SP indicator definition formula is:

$$SP=\sqrt{\frac{1}{n-1}{\displaystyle \sum _{i=1}^n{(\overline{d}-d_i)}^2}}$$

(12)

where $d_i=\underset{j}{\min }\left|f_m{}^i(x)-f_m{}^j(x)\right|i,j=1,\cdots ,n,i\ne j$, $n$ is the number of solutions in $P{F}_{know}$, which represents the Euclidean distance between the particles closest to itself in the same dimensional objective function. $\overline{d}$ is the average value of $d_i$. $n$ is the number of particles in the external file. $m$ is the target dimension. The smaller the SP value, the better the distribution of the obtained solution set.

Inverse generational distance (IGD): It is used to comprehensively evaluate the convergence and diversity of the algorithm. The definition formula of IGD is as follows:

$$IGD=\frac{({\displaystyle {\sum }_{i=1}^n{d}_i)}}{n}$$

(13)

where $d_i$ is the shortest Euclidean distance between the real front $P{F}_{true}$ and the obtained Pareto optimal solution. $n$ is the number of particles in the external file. The smaller the IGD value, the better the convergence and diversity of the obtained solution set.

6.2. Experimental Results and Analysis of Algorithm Performance Testing

The experimental results and analysis of algorithm performance testing are shown in

Table 2. Mean results of six performance GD performance indicators.

Test Function	MVCMOPSO	NSGA-II	SPEA2	MOPSO	GWASF-GA	MOHBS
ZDTI	3.8503 × 10−4	7.6045 × 10−4	2.7854 × 10−4	8.8984 × 10−4	2.7307 × 10−4	7.9000 × 10−5
ZDT2	7.8843 × 10−4	4.5078 × 10−4	5.2788 × 10−4	9.0701 × 10−4	6.1593 × 10−4	4.3000 × 10−5
ZDT3	1.0618	1.1099	1.3646	1.1553	1.4029	1.021345
ZDT6	0.1186	4.4517 × 10−4	0.0318	0.0407	8.0986 × 10−4	0.037538
Schaffer	2.1211 × 10−4	0.0482	9.9408 × 10−4	2.3225 × 10−4	9.5901 × 10−4	3.4181 × 10−4
Kursawe	1.8259 × 10−4	2.0158 × 10−4	1.7407 × 10−4	1.4351 × 10−4	1.5173 × 10−4	1.1047 × 10−4
Viennet2	3.1272 × 10−4	5.0305 × 10−4	9.6306 × 10−4	3.6359 × 10−4	1.4321 × 10−4	4.6080 × 10−4
Viennet3	1.3427 × 10−4	8.0194 × 10−4	4.3650 × 10−4	1.4416 × 10−4	9.3362 × 10−5	6.4399 × 10−4

,

Table 3. Mean results of six performance SP performance indicators.

Test Function	MVCMOPSO	NSGA-II	SPEA2	MOPSO	GWASF-GA	MOHBS
ZDT1	0.2340	0.3709	0.3027	0.2400	0.5404	0.307352
ZDT2	0.3288	0.4300	0.3042	0.2625	0.3095	0.323597
ZDT3	1.0881	0.7763	0.7733	1.1010	1.0380	1.089784
ZDT6	1.1689	0.3543	1.1614	0.7876	0.3613	0.866827
Schaffer	0.3364	0.6316	0.9984	0.4017	0.8273	0.2759
Kursawe	0.5150	0.5837	0.5260	0.5277	0.7453	0.5526
Viennet2	0.8992	1.1094	0.9706	0.9003	1.0743	0.849299
Viennet3	0.8238	0.9199	0.7310	0.9282	1.0496	0.859002

and

Table 4. Mean results of six performance IGD performance indicators.

Test Function	MVCMOPSO	NSGA-II	SPEA2	MOPSO	GWASF-GA	MOHBS
ZDT1	20173 × 10−4	1.8189 × 10−4	1.8186 × 10−4	3.0572 × 10−4	2.5985 × 10−4	1.7680 × 10−4
ZDT2	1.1176 × 10−4	1.9840 × 10−4	1.7650 × 10−4	3.360 × 10−4	2.8607 × 10−4	1.7779 × 10−4
ZDT3	0.3485	1.4118 × 10−4	1.5941 × 10−4	0.3490	5.0567 × 10−4	0.301239
ZDT6	8.7958 × 10−4	2.2133 × 10−4	5.6466 × 10−4	3.6115 × 10−4	3.4561 × 10−4	1.1072 × 10−4
Schaffer	3.4222 × 10−4	0.0216	0.0104	4.9902 × 10−4	0.0103	4.6461 × 10−4
Kursawe	1.8822 × 10−4	1.7837 × 10−4	1.5668 × 10−4	2.1107 × 10−4	3.1832 × 10−4	1.1047 × 10−4
Viennet2	2.0231 × 10−4	1.9383 × 10−4	1.7501 × 10−4	2.0878 × 10−4	3.2433 × 10−4	2.7127 × 10−4
Viennet3	1.0130 × 10−4	9.9469 × 10−5	1.2881 × 10−4	1.0242 × 10−4	0.0023	1.0827 × 10−4

. In the table, the optimal values obtained by each algorithm are represented in bold font. Based on the experimental data provided in literature [27], MOHBS performed exceptionally well in the GD performance test by obtaining four optimal GD values out of eight test problems. In comparison, the algorithm proposed in literature [28] only obtained two optimal values for GWASF-GA and one for MVCMOPSO and NSGA-II. Additionally, MOHBS outperformed the MVCMOPSO and MOPSO algorithms in the ZDT6 test function in terms of GD value. However, while MOHBS did not perform excellently in the Viennet2 and Viennet3 test functions, it still outperformed the NSGA-II algorithm and the SPEA2 algorithm.

Based on the findings, it appears that the MOHBS algorithm demonstrates the highest level of convergence performance in the dual-objective test function. The utilization of mutualism and commensalism strategies can aid in the swift convergence of the BBO algorithm. In the SP performance test, the MOHBS algorithm yields two optimal SP values out of eight test questions, while the SPEA2 algorithm also yields two optimal SP values. The remaining algorithms each possess one optimal SP value, except for the GWASF-GA algorithm, which does not have an SP optimal value. However, there is not a significant difference in the mean SP value for each algorithm. The IGD evaluation indicators reveal that the algorithm in this paper obtains three optimal values out of eight test problems, while NSGA-II and MVCMOPSO have two IGD optimal values. On the other hand, SPEA-II has one IGD optimal value, and MOPSO and GWASF-GA do not have an optimal IGD value. In conclusion, MOHBS is highly competitive in terms of convergence, distribution, and diversity.

Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10 show the Pareto frontier diagram of MOHBS in the test function. As shown in the figure, in the dual objective test function, the Pareto front is uniformly distributed, while in the three objective test function, although the Pareto front is not uniform enough, there is no dense mixing.

7. Multi-Objective Flexible Job-Shop Scheduling

This paper has applied the improved algorithm to solve the multi-objective flexible job-shop scheduling problem in order to verify its practical application value. FJSP involves uncertain processing machines for each process, and multiple machines can be selected for each workpiece, each with varying processing times. Given its flexible nature, this problem is highly applicable to real-life situations and has been widely employed in various fields such as steel manufacturing, the chemical industry, and automobile manufacturing.

7.1. Problem Description

If a workshop has m processing machines and $n$ workpieces, each workpiece would have various processes to be processed in a certain order. These processes can be carried out on multiple machines, each with different required processing times. The aim of MOFJSP is to assign the suitable machines for each process and sort the process sequence on each machine to achieve optimal multiple scheduling objectives.

According to the needs of actual production, this paper aims to minimize the maximum completion time of the workshop, minimize the maximum machine compliance, and minimize the total machine load. The definition of the objective function is as follows:

The maximum completion time: It refers to the duration needed for all machines to complete all processes for all workpieces, from the first process to the last. This metric reflects an enterprise’s production efficiency and is the most basic and commonly used indicator for measuring the effectiveness of scheduling schemes. The expression for calculating it is as follows:

$$C_{\max }=\min \left(\underset{1\le i\le n}{\max }{C}_i\right)$$

(14)

The maximum load of the machine: After all the machines have finished all the processes of all the workpieces, there must be one or more machines with the longest processing time. The minimum maximum load of the machine can effectively improve the life of the machine, and the expression formula is as follows:

$$wl=\min ({\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^{h_j}{p}_{kij}{x}_{kij}}})$$

(15)

The total machine load: It refers to the combined processing time of all machines when all workpiece processes are completed. Decreasing the total load results in lower processing hours and costs, leading to improved machine utilization. The expression for calculating the total machine load is as follows:

$$wal=\min ({\displaystyle \sum _{k=1}^m{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^{h_j}{p}_{kij}{x}_{kij}}}})$$

(16)

where $n$: the total number of workpieces. $i$ is the workpiece serial number and $x_{kij}=1$. $m$: the total number of machines. $k$ is the machine serial number and $x_{kij}=1$. $C_{\max }$ represents the maximum completion time of the workpiece, and $C_i$ represents the completion time of the workpiece $i$. $j$: the process serial number of a workpiece. $P_{kij}$: processing time on the process $O_{ij}$ machine $k$. $x_{kij}$ is the judgment statement, indicating whether the process $O_{ij}$ selects a machine $k$ for processing; if so, $x_{kij}=1$, otherwise $x_{kij}=0$.

For the multi-objective flexible workshop scheduling problem, the following assumptions need to be considered:

• The workpiece and the machine are available at zero time.
• The same machine can only process one process at the same time.
• Process processing cannot be interrupted.
• The processing priority of each workpiece is the same.
• The processes of different workpieces are independent of each other, and the processes of the same workpiece have a definite relationship of sequential processing.

7.2. Encoding and Decoding

FJSP entails dividing an individual’s coding into two parts: machine coding and process coding. Together, these parts create a chromosome that represents a viable solution.

Coding method: Assuming that a workshop contains $m$ workpieces and each workpiece has $n$ processing processes, after that, the total length of the individual position vector is $2\times m\times n$, and the value of the position vector is between $\left[-m,m\right]$.

Decoding method: To convert the individual position into the scheduling scheme, the machine allocation follows the method outlined in literature [27], using Formula (30) to convert the individual position vector into the number of the process optional allocation machine set.

$$u\left(j\right)=round\left(\frac{1}{2\lambda }\left(x\left(j\right)+\tau \right)\left(s\left(j\right)-1\right)+1\right)$$

(17)

where $u\left(j\right)$ represents the serial number $\left(1\le j\le l\right)$ of the machine set to be allocated by individual $j$ in the corresponding process. $\tau$ represents the number of workpieces. $x\left(j\right)$ represents the position vector value of individual $j$. $s\left(j\right)$ represents the number of machines that can be allocated to the process corresponding to individual $j$. $l$ represents the total number of processes.

To assign machines, the scheduling plan is transformed into an individual position vector using the inverse operation of Formula (17).

To convert the individual position vector into a scheduling scheme, the process ordering part utilizes the Ranked Order Value (ROV) technique discussed in literature [28]. Initially, the position vector values of each individual are sorted into ascending order, and the resulting order represents the ROV value. The process numbers of the vector values are then arranged in the order corresponding to the ROV value to obtain the process ranking.

7.3. Experimental Simulation and Analysis

To test the MOHBS algorithm’s practical performance, this paper applies it to MOFJSP and simulates four Kacem examples. The number of workpieces in the algorithm varies from 4 to 10, and the number of machines varies from 5 to 10. The experimental results were compared with hybrid particle swarm optimization and random-restart hill climbing (PSO-RRHC) [29], a hybrid imperial competition algorithm and genetic algorithm (ICA-GA) [30], an effective estimation of distribution algorithm (EDA) [31], and the hybrid artificial bee colony algorithm (HTABC) [32]. The comparison results are shown in

Table 5. Comparison results of Kacem example.

	PSO-RRHC			ICA-CA			EDA			HTABC			MOHBS
4 × 5	-	-	-	-	-	-	13	11	11	11	11	11	11	11	11
	-	-	-	-	-	-	7	10	9	10	9	10	10	10	9
	-	-	-	-	-	-	32	32	34	32	34	33	32	32	34
8 × 8	14	15	16	14	15	16	16	14	15	14	15	15	15	14	14
	12	12	13	12	12	13	13	12	12	12	15	12	12	12	12
	77	75	73	77	75	73	73	77	75	77	75	75	75	77	77
10 × 7	-	-	-	-	-	-	12	11	11	11	11	11	11	11	11
	-	-	-	-	-	-	12	11	10	11	11	11	10	10	11
	-	-	-	-	-	-	60	61	62	61	61	61	62	62	61
10 × 10	7	7	8	7	7	8	8	7	7	7	7	7	8	7	7
	5	6	5	5	6	5	5	6	5	6	7	6	5	5	5
	43	42	42	43	42	42	42	44	43	41	41	42	42	44	44

, where each instance consists of three rows; from top to bottom, maximum completion time, the maximum load of the machine, and total machine load. If an algorithm has no data about a particular instance, it is indicated by the symbol “-”.

In Table 5, the parameters are set as follows: The species size is 40, the number of iterations of the algorithm is 200, each calculation example is run 10 times, and 3 groups of Pareto solutions are arbitrarily selected for comparison. The experimental environment adopts the Windows10 system, MatlabR2014a programming platform, and the processor is Intel(R) Core (TM) i5-7300.

As shown in Table 5, for instance 4 × 5 and instance 8 × 8, by comparing with several other algorithms, it can be found that the three indicators including the maximum completion time have reached the best state of other algorithms, but not all indicators of the solution are the best. For the special example 10 × 7, the number of workpieces in this example is more than the number of machines, which is in line with the actual production scheduling situation. It can be found that the MOHBS algorithm and the HTABC algorithm and EDA algorithm have the same optimal value. For the example of 10 × 10, through comparison, it is found that MOHBS is superior to the PSO-RRHC algorithm and EDA algorithm, and it is comparable to the HTABC algorithm. Moreover, when $Wal$ is certain, the maximum load of this algorithm is the smallest.

This algorithm solves one of the Pareto optimal solutions obtained by the example 10 × 7: $C_{\max }=11$, $Wl=10$, $Wal=62$ and $C_{\max }=11$, $Wl=10$, $Wal=61$; its Gantt chart is shown in Figure 11 and Figure 12. Each of the squares represents a process. There will be a job number and a process number in the square. The ordinate corresponding to the square is the operating machine. The vertical line on the left of the square represents the start time of the workpiece, and the vertical line on the right represents the end time.

8. Conclusions and Future Work

This study addressed the problems of decreasing population diversity, easy to fall into local optimum, easy to destroy the optimal solution in the variation phase, and uneven solution distribution when solving multi-objective flexible job-shop scheduling problems by using biogeography optimization algorithms. By incorporating symbiotic biological algorithms, a new migration operator is proposed through an improved mutualism strategy and commensalism strategy, which can effectively balance exploration and exploitation capabilities, enhance population diversity, and avoid the algorithm from falling into localization. A dual-strategy variation mechanism of a parasitic mutation and predator mutation is proposed in the mutation phase. In the early stage of the algorithm iteration to satisfy the species diversity, the use of a parasitic mutation can find the new better solution, and the more use of a predator mutation in the late stage of the algorithm iteration can enhance the local search capability and improve the solution accuracy. Through experimentation, it has been demonstrated that the algorithm excels in achieving convergence and the uniform distribution in solving multi-objective optimization problems. Furthermore, its effectiveness has been confirmed in addressing the complex issue of multi-objective FJSP.

In future work, we plan to delve into more intricate shopfloor scheduling problems, such as flexible job-shop scheduling and a distributed shop scheduling environment. Moreover, extensive research will be conducted on the meta-heuristic algorithm to effectively tackle the workshop scheduling issue.