A regional biogeography-based optimization algorithm for the distributed assembly permutation flow-shop scheduling problem with fuzzy processing time

Abstract

For solving the distributed assembly flow shop scheduling problem with fuzzy processing time (FDAPFSP), a regional biogeography-based optimization algorithm (RBBO) is proposed to minimize the maximum fuzzy completion time. The mathematical model is provided. In RBBO, all habitats are divided into regions based on the habitat suitability index, and the habitats of each region are subject to cross-regional migration and replacement procedures. A critical factory optimization strategy is developed to enhance local search capability. Taguchi method is used to determine the parameters of RBBO. In ten FDAPFSP instances, comparative testing of RBBO algorithm with various heuristic and swarm intelligence algorithms are conducted. The computation results show that in ten FDAPFSP cases, the proposed RBBO outperforms other algorithms in nine out of ten FDAPFSP cases.

Keywords

Fuzzy scheduling distributed scheduling permutation flow-shop regional biogeography-based optimization algorithm

1 Introduction

Distributed permutation flow-shop scheduling problem (DPFSP) was first proposed by Naderi and Ruiz and had extensive research in operations research, economics, and other applications [1, 2]. DPFSP is a coupling between distributed manufacturing systems [3] and the permutation flow-shop scheduling problem (PFSP) [4], which has high practical research value.

In recent years, many scholars have also proposed many methods for DPFSP. Wang et al. [5] proposed an effective hybrid immune algorithm to solve DPFSP, designed feasible scheduling that converts job permutation sequence into factory dispatching and job sequencing, and proposed four local search methods and unique crossover factors. Gao et al. [6] proposed a new neighborhood generation method and an enhanced local search method applied to the Tabu search algorithm to solve the DPFSP.

The assembly flow-shop scheduling problem (AFSP) was first proposed by Lee et al. [7]. AFSP refers to the independent process of each job and sending the produced parts to the assembly shop for assembly. It is composed of a flow-shop scheduling problem (FSP) [8] and an assembly system [9]. Different job processing sequences can be generated during the assembly stage based on different product assembly tables, representing close relationships between enterprise managers and suppliers [10]. Many scholars have proposed valuable research on AFSP. Cai and Lei [11] proposed a new shuffled frog-learning algorithm with Q-learning to solve the distributed assembly hybrid flow shop scheduling problem. Niloofar et al. [12] considered a three-stage assembly flow shop scheduling problem with machine availability constraints and proposed two multi-objective meta-heuristics. Mohammad et al. [13] incorporates preventive maintenance into the two-stage assembly flow shop scheduling problem.

Distributed assembly permutation flow-shop scheduling problem (DAPFSP) is a combination of DPFSP and AFSP, the combination problem of DAPFSP was first proposed by Hatami et al. [14]. In the classical DAPFSP, some jobs are assigned to factories, and each factory has some machines to process each process of these jobs. FDAPFSP is the extension of DAPFSP. In real life, the processing time of jobs is often subject to a certain degree of uncertainty, which various errors may cause. Therefore, studying DAPFSP with fuzzy processing time constraints will solve real-world processing problems better. There are relatively few scholars studying FDAPFSP, but most scholars have considered other constraints. Hatami et al. [10] proposed a distributed assembly permutation flow shop scheduling problem with sequence-dependent setup times (DAPFSP-SDST). Yang and Xu [15] proposed a novel distributed assembly permutation flow shop scheduling problem with flexible assembly and batch delivery (DAPFSP-FABD). Wei et al. [16] proposed an energy-efficient distributed assembly blocking flow shop problem (EEDABFSP). Huang et al. [17] proposed an effective memetic algorithm to solve the distributed flow shop scheduling problem with an assembly machine. Zhang et al. [36] proposed a distributed estimation algorithm to solve DAPFSP and developed a matrix cube-based probability model with effective sampling mechanism to estimate the probability distribution of the optimal solution and conduct global exploration to find promising regions. Song et al. [37] used genetic programming (GP) as a high-level strategy set for generating heuristic sequences from predesigned low-level heuristics, thus proposing the GP-HH algorithm to solve DAPFSP- SDST.

Alfred Wallace and Charles Darwin introduced the idea of biogeography in the 19th century. Simon’s study of the theoretical underpinnings of biogeography was awarded the BBO in 2008 [18]. BBO performs well in comparison to other algorithms [19]. Based on the characteristics of the problem, Lin and Zhang [20] suggested an improved local search approach and a hybrid biogeography algorithm to solve DAPFSP. The efficacy of the BBO in DAPFSP is shown by this combined strategy. By dividing habitat regions based on BBO, RBBO speeds up the trend toward the dynamic balance of the habitat suitability index (HSI) by allowing better habitat migration. For maintain a dynamic equilibrium and ensure that there will never be a difference between regions, habitat replacement activities are conducted simultaneously within the region. However, many researchers added a wide range of constraints to DAPFSP and solved them; they did not impose limitations on fuzzy processing. In this paper, task processing and product assembly are given fuzzy constraints, and then 10 cases are tested by using RBBO. The results were compared using BBO and GA. The designed algorithm for FDAPFSP produced positive results.

The current research work of scholars is summarized, as shown in Table 1.

Table 1
Literature comparison. Summary of literature, where the symbol “√” indicates that the corresponding assumptions has been considered

Authors Algorithm Constraint Heuristic algorithm Swarm intelligence algorithm Local search algorithm

Wang et al. [5] HIA √

Gao et al. [6] TS √

Lee et al. [7] HA √

Hatami et al. [10] IG √ √

Cai and Lei [11] Q-SFLA √

Niloofar et al. [12] HA √ √

Mohammad et al. [13] VNS-SA, VNS-TS √ √

Hatami et al. [14] VNS √

Yang and Xu [15] IG √ √

Wei et al. [16] NSGA-II √ √

Huang et al. [17] EM √

Lin and Zhang [20] HBBO √

Zhang et al. [36] MCEDA √

Song et al. [37] GP-HH √ √

Authors	Algorithm	Constraint	Heuristic algorithm	Swarm intelligence algorithm	Local search algorithm
Wang et al. [5]	HIA				√
Gao et al. [6]	TS				√
Lee et al. [7]	HA		√
Hatami et al. [10]	IG	√			√
Cai and Lei [11]	Q-SFLA			√
Niloofar et al. [12]	HA	√	√
Mohammad et al. [13]	VNS-SA, VNS-TS	√			√
Hatami et al. [14]	VNS		√
Yang and Xu [15]	IG	√	√
Wei et al. [16]	NSGA-II	√		√
Huang et al. [17]	EM			√
Lin and Zhang [20]	HBBO			√
Zhang et al. [36]	MCEDA				√
Song et al. [37]	GP-HH	√		√

From Table 1, adding constraints is an increasingly popular research direction in studying distributed scheduling problems, and swarm intelligence algorithms are also gradually applied in distributed scheduling problems.

The current distributed assembly permutation flow shop scheduling problem belongs to the distributed basic theory problem, and there are often practical constraints such as order delays, machine failures, and job loading and unloading in the actual production and processing process. These constraints will make problem more complex, and there is a certain error between the actual processing time and the ideal processing time in enterprise production, which disrupts the production rhythm. Therefore, it is necessary to add problem constraints to the distributed assembly permutation flow shop scheduling problem to make it more suitable for the actual production scheduling type.

However, although many scholars have proposed improved swarm intelligence algorithms to solve the distributed assembly permutation flow shop scheduling problem with constraints, there are still shortcomings in the solving process, such as falling into local optima, slow convergence speed, and long running time. Among them, BBO algorithms stand out among swarm intelligence algorithms. Compared to other swarm intelligence algorithms, BBO algorithms can achieve fast local convergence and rapid habitat evolution when solving distributed permutation flow shop scheduling problems. However, biogeographical algorithms are prone to falling into local optima, so improving the local search of biogeographical algorithms and enhancing their ability to jump out of local optima is a key research direction that needs to be focused on.

In Section 2, the FDAPFSP model and the concept of triangular fuzzy numbers (TFN) are provided. Section 3 provides the basic BBO operations as well as the migration rate formula and population mutation rate formula for the algorithm. Section 4 will introduce the encoding and decoding method for this problem as well as the migration and mutation methods used in RBBO, giving a complete algorithm model for RBBO. The performance comparison experiments between the RBBO and other algorithms in the design scenario, as well as the RBBO parameter design experiments and basic algorithm iteration speed comparison studies, are all included in Section 5. Section 6 summarizes the entire text and offers suggestions for further design work.

2 Problem description

2.1 Distributed assembly permutation flow-shop scheduling problem with fuzzy processing time

As seen in Fig. 1, FDAPFSP has fuzzy production stages and fuzzy assembly stages. In the stages of fuzzy production, n jobs are distributed among f factories, each of which contains m machines, each of which can only complete one procedure of a given job. Every job on the machine is processed in the same order, and every factory has the same machinery. The job cannot be stopped once it has begun processing. When the group of jobs that comprise a product p (i) is completed during the manufacturing phase and submitted to the assembly shop, product p starts the assembly process. Table 2 provides the definitions of the model’s parameters.

Fig. 1

FDAPFSP introduction.

Table 2

Symbols, parameters, and variables used in FDAPFSP

Symbol
i	the name of job i = 1, 2, . . . , n
j	the name of machine j = 1, 2, . . . , m
h	the name of factory h = 1, 2, . . . , f
k	the name of product k = 1, 2, . . . , p
Parameter
n	the number of jobs
m	the number of machines
f	the number of factories
p	the number of products
Variable
v	Encoding string
C _o,j	The length of processing time of operation o on machine j
S _o,j	Fuzzy start time of operation o on machine j
E _o,j	Fuzzy finish time of operation o on machine j
MFAT	Maximum fuzzy assembly time

In the fuzzy production stage, it is necessary to obtain the time for each product to prepare for assembly from zero and save this time for use in the fuzzy assembly stage. The method is to divide the fuzzy production problem into sub problems that solve the fuzzy start processing time of each product. v_k,i represents the encoding sequence of product k in the current encoding, which includes the encoding sequence before job i, that is, the encoding string of product k. Taking the code [3 10 9 6 7 2 11 8 5 12 4 1] as an example (In the fourth section, the specific encoding and decoding method will be described), this code consists of 12 jobs, 2 machines, 2 factories, and 2 products. [3 10 9 6 7] belongs to product 1, [2 11 8 5 12 4 1] belongs to product 2. v_1,i = [3 10 9 6 7], v_2,i = [2 11 8 5 12 4 1]. v_1,9 = [3 10 9], v_2,4 = [2 11 8 5 12 4]. E (v) represents the fuzzy finish processing time obtained by decoding the encoding string v during the fuzzy production stage. E (v) _j represents the fuzzy finish processing time of the machine j on the v encoding string. C_i,j represents the fuzzy production time of job i on machine j. E_k,i represents the maximum fuzzy finish time distributed across various factories. Taking [3 10 9 6 7 2 11 8 5 12 4 1] encoding as an example, assume their factory distribution is [2 1 1 2 1 1 1 2 1 2 2 2]. Encoding the job for two-dimensional expansion can obtain [10 9 7 2 11 5] [3 6 8 12 4 1]. Job 3 is the last job distributed by product 1 on factory 1, therefore E_k,i=E_1,3=max {E_3,1, E_3,2}. Using the same steps to calculate E_2,6 and comparing it with E_1,3, the start fuzzy processing time for product 1 can be obtained, represented by S₁. $S_{k, i} = E (v_{k, i})$ (1) $E_{i, 1} = E (v_{k, i})_{1} + C_{i, 1} if j = 1$ (2)

${\begin{matrix} E_{i, j + 1} = E_{i, j} + C_{i, j + 1}, E_{i, j} > E {(v_{k, i})}_{j + 1} \\ E_{i, j + 1} = E {(v_{k, i})}_{1} + C_{i, j + 1}, E_{i, j} \leq E {(v_{k, i})}_{j + 1} \end{matrix} if j > 1$ (3)

$E_{k, i} = max {E_{i, 1}, E_{i, 2}, . . ., E_{i, j}}$ (4) $S_{k} = max {E_{k, 1}, E_{k, 2}, . . .}$ (5)

Equation (1) represents the fuzzy start processing time for solving the encoding string v_k,i. Equation (2) represents the fuzzy finish processing time on machine 1 when processing on the first machine. Equation (3) represents the fuzzy finish processing time formula on machine j + 1 when processing on machine j + 1. When the fuzzy start processing time of the encoding string v_k,i on machine j + 1 is less than the fuzzy start processing time of job i on machine j, the fuzzy processing time of job i on machine j + 1 is equal to the processing time of job i on machine j plus the fuzzy processing time required for job i on machine j + 1. On the contrary, it is equal to the fuzzy start processing time of the v_k,i encoding string on machine j + 1 plus the fuzzy processing time required for job i on machine j + 1. Equation (4) represents the maximum fuzzy finish time for solving the job i in the product k set. Equation (5) represents the maximum start fuzzy time for solving assembly product k.

In the fuzzy assembly stage, it is necessary to obtain the maximum fuzzy assembly time (MFAT). The method is to obtain the fuzzy product assembly sequence by processing the fuzzy start assembly schedule of the product, and then accumulate it in sequence to obtain the maximum fuzzy assembly time. C_k represents the fuzzy assembly time required to assemble product k. E_k represents the fuzzy finish time after assembling product k. $E_{1} = S_{1} + C_{1} if k = 1$ (6) ${\begin{matrix} E_{k} = E_{k - 1} + C_{k}, E_{k - 1} > S_{k} \\ E_{k} = S_{k} + C_{k}, E_{k - 1} \leq S_{k} \end{matrix} if k > 1$ (7) $MFAT = max {E_{1}, E_{2}, . . ., E_{p}}$ (8)

Equation (6) represents the calculation formula for the finfish fuzzy assembly time of the first product obtained when the assembly shop assembles the first product. Equation (7) represents the calculation formula for the finish fuzzy assembly time (FFAT) of the non-first product obtained when assembling it. When the FFAT of the previous product is greater than the start fuzzy assembly time (SFAT) of the current product, the FFAT of the current product is equal to the FFAT of the previous product plus the fuzzy processing time (FPT) required for the current product assembly. On the contrary, the FFAT of the current product is equal to the SFAT of the current product plus the required FPT for the current product assembly. By comparing the SFAT of each product, MFAT can be obtained. Equation (8) represents the maximum fuzzy assembly time of the target.

2.2 Operation on TFN

Processing fuzzy numbers is crucial while solving DAPFSP under fuzzy settings. In this study, work processing times and product assembly times are fuzzily processed using TFN [21].

When TFN is used to represent processing data of scheduling problem, three operators are often used to build a schedule, which are addition operator, max operator and ranking operator [22].

For two TFNs $\tilde{s} = (s_{1}, s_{2}, s_{3})$ and $\tilde{t} = (t_{1}, t_{2}, t_{3})$ , the addition operation is performed as follows: $\tilde{s} + \tilde{t} = (s_{1} + t_{1}, s_{2} + t_{2}, s_{3} + t_{3})$ (9)

To compare $\tilde{s}$ and $\tilde{t}$ , a calculation criterion is proposed, which includes three comparison formulas c₁, c₂ and c₃. For TFN $\tilde{s}$ , $c_{1} (\tilde{s}) = (s_{1} + 2 s_{2} + s_{3}) / 4$ , $c_{2} (\tilde{s}) = s_{2}$ and $c_{3} (\tilde{s}) = s_{3} - s_{1}$ . TFN $\tilde{t}$ , $c_{1} (\tilde{t}) = (t_{1} + 2 t_{2} + t_{3}) / 4$ , $c_{2} (\tilde{t}) = t_{2}$ and $c_{3} (\tilde{t}) = t_{3} - t_{1}$ . If $c_{1} (\tilde{s}) > (<) c_{1} (\tilde{t})$ , then $\tilde{s} > (<) \tilde{t}$ . If $c_{1} (\tilde{s}) = c_{1} (\tilde{t})$ and $c_{2} (\tilde{s}) > (<) c_{2} (\tilde{t})$ , then $\tilde{s} > (<) \tilde{t}$ . If $c_{1} (\tilde{s}) = c_{1} (\tilde{t})$ , $c_{2} (\tilde{s}) = c_{2} (\tilde{t})$ and $c_{3} (\tilde{s}) > (<) c_{3} (\tilde{t})$ , then $\tilde{s} > (<) \tilde{t}$ .

The maximum procedures for TFNs are as follows: $If \tilde{s} > \tilde{t}, then \tilde{s} \lor \tilde{t} = \tilde{s}, else \tilde{s} \lor \tilde{t} = \tilde{t}$ (10)

3 Biogeography-based optimization algorithm

BBO has strong integration and stability compared to other swarm intelligence optimization algorithms, which has attracted the attention of numerous researchers and has promising research possibility. BBO mainly creates a dynamic balancing effect in habitats through species’ migration and mutation, which could contribute to a balanced total HSI value for all habitats.

BBO divides ecosystems into groups, each of which has a unique starting population of species. Describe the state of each habitat using the HSI. The HSI is affected by a wide range of factors, such as surface temperature, humidity, rainfall, and others. All these qualities are together referred to as the habitat suitability index variables (SIVs). The HSI will change because of species migration and habitat mutation in each habitat in BBO. A high HSI indicates the presence of many species in the environment; at this point, the resources the habitat supplies for each species will drop, and it will become more likely that species will move out from the habitat. Contrarily, a low HSI indicates that there are fewer species present, and that each species can be allocated more resources, which increases the risk of invasive species invading the environment. Habitat mutation is the process through which, because of some irrevocable and unexpected natural disasters, the number of species and habitats can both vary considerably. Habitat enhances the ability for species to share information through migration and mutation, eventually boosting species variety. It also forced the algorithm to abandon local optima.

The migration operator in BBO is used to change the HSI of an existing environment by exchanging information between two habitats [23, 24]. The migration process includes both the choice of habitat for immigration and the choice of habitat for emigration. Their immigration rate (λ_r) and emigration rate (μ_r) are tied to the environments they choose. In Equations (11) and (12), the migration formula for BBO is displayed. $λ_{r} = I (1 - \frac{N (r)}{N_{max}})$ (11) $μ_{r} = E \frac{N (r)}{N_{max}}$ (12)

Along with the quadratic, exponential, and cosine mobility models [25], BBO also includes a linear migration model in (11–12). These models’ λ_r and μ_r maintain a constant rise or drop when the number of species in the environment changes, but when the number of species is too few or too many, the iteration rate of λ_r and μ_r changes. varied migration models will have varied impacts when applied to various problem, and to make conclusions, particular effects must be compared through experiments. This paper uses a straightforward, computationally efficient linear mobility model that is ideal for BBO to handle distributed job shop scheduling problem.

Among them, I and E represent the maximum immigration rate and maximum emigration rate, usually constants. Both the I and E values are treated as 1 in this paper. N (r) represents the current number of species contained in the habitat H_r, N_max is the maximum possible species that can be supported by all habitats [26]. In this paper, the habitat HSI is arranged in decreasing order, the number of species (N (1)) in the first habitat (H₁) is equal to 1, the number of species (N (2)) in the second habitat (H₂) is equal to 2, and so on. Where N_max = P (P is the maximum number of habitats).

When the habitat immigration rate and emigration rate attain dynamic equilibrium in BBO, mutation operation is often required to cause the algorithm to exit local optimization [27 –29]. Through unexpected circumstances, the mutation operation significantly alters the HSI of the present habitat H_r. In Equations (13) and (14), the formulae for mutation operations are shown.

${\dot{P}}_{r} = {\begin{matrix} - (λ_{r} + μ_{r}) P_{r} + μ_{r + 1} P_{r + 1} \\ - (λ_{r} + μ_{r}) P_{r} + λ_{r - 1} P_{r - 1} + μ_{r + 1} P_{r + 1} \\ - (λ_{r} + μ_{r}) P_{r} + λ_{r - 1} P_{r - 1} \end{matrix} \begin{matrix} r = 0 \\ 1 \leq r \leq P - 1 \\ r = P \end{matrix}$ (13)

$m_{n} = m_{max} (1 - \frac{P (n)}{P_{max}})$ (14)

Equation (13) is used to solve the probability formula for the number of species in each current habitat. Equation (14) is used to calculate the mutation probability of the current habitat, where m_max is the maximum mutation rate and is a constant [30, 31].

From (13–14), it may be inferred that the habit is most likely to alter as λ_r and μ_r become closer to one another. The fitness of all habitats approaches an ideal value as the number of habitat iterations rises, with λ_r nearing μ_r and progressively reaching the local optimum. The habitat will go through mutations at this stage, which will force it to leave its current local optimal state to search for a more ideal solution. This is the main concept of BBO’s “Nature’s Renewal Iteration”.

4 Regional biogeography-based optimization algorithm

4.1 The framework of RBBO

The RBBO algorithm divides three different sorts of areas to redistribute the original habitats. Three different sorts of areas must be divided in the early stages of the algorithm: region A (R_A), region B (R_B), and region C (R_C). All habitats must also be sorted and assigned to R_A, R_B, and R_C. The technique of allocation entails sorting by HSI level and sequentially filling in R_A, R_B, and R_C. Each generation’s iteration process involves two rounds of region migration and replacement, which are activities carried out between adjacent regions. In Fig. 2, the RBBO model is displayed. In Equations (15)–(17), the RBBO iteration formula is displayed.

Fig. 2

RBBO model.

$R_{A}^{t + 1} = R_{A}^{t} + (R_{B}^{t + 1} (h_{best}) - R_{A}^{t} (h_{worst}))$ (15)

$\begin{matrix} R_{B}^{t + 1} = & R_{B}^{t} + (R_{B}^{t} (h_{A . M}) - R_{B}^{t} (h_{m})) \\ + (R_{C}^{t + 1} (h_{best}) - R_{B}^{t} (h_{worst})) \end{matrix}$ (16) $R_{C}^{t + 1} = R_{C}^{t} + (R_{C}^{t} (h_{B, M}) - R_{C}^{t} (h_{m}))$ (17)

t represents the current number of iterations, and t + 1 represents the number of iterations that the next generation is about to iterate. R_k represents the habitat aggregation of region k. h_best represents the optimal habitat in the region, while h_worst represents the worst habitat in the region. h_k,M is the habitat selected for emigration in region k. h_m is the habitat selected for immigration.

Equation (15) represents the habitat set of R_A in t + 1 iteration is equal to the original habitat set of the region plus the optimal habitat of R_B in t + 1 iteration minus the worst habitat of R_A in t iteration. Equation (16) represents the habitat set of R_B in iteration t + 1 is equal to the original habitat set plus the optimal habitat of R_C in iteration t + 1 minus the worst habitat of R_B in iteration t, plus the habitat of R_B after migration through R_A in iteration t minus the selected immigration habitat of R_B in iteration t. Equation (17) represents the habitat set representing R_C in iteration t + 1 is equal to the original habitat set plus the habitat of R_C after migration through R_B in iteration t minus the selected immigration habitat of R_C in iteration t.

4.2 Encode and decode method

This paper uses the dual-code method. The allocation order encoding of the jobs is on the first line of encoding. The factory-assigned encoding of the jobs is on the second line of encoding.

In this paper the problem does not have fuzzy restrictions for the sake of the readers’ convenience and better understanding. Taking the example from Chapter 2 as an example, the encoding is [3 10 9 6 7 2 11 8 5 12 4 1] [2 1 1 2 1 1 1 2 1 2 2 2], after being assigned to the factory is [3 10 9 7 2 11 5] [3 6 8 12 4 1]. The specific cases are shown in Table 3. The first layer encoding for this paper is distributed to the factory in accordance with the second layer encoding, going from left to right. The Gantt chart of the case is shown in Fig. 3.

Table 3
Case in DAPFSP

Job Processing time Product number Product Assembly time

1 1 6 2 1 8

2 4 3 1

3 4 2 2

4 4 1 2

5 5 2 1

6 2 4 2

7 3 1 1 2 4

8 6 3 2

9 3 7 1

10 2 9 1

11 1 3 1

12 6 1 2

Job	Processing time	Product number	Product	Assembly time
1	1 6	2	1	8
2	4 3	1
3	4 2	2
4	4 1	2
5	5 2	1
6	2 4	2
7	3 1	1	2	4
8	6 3	2
9	3 7	1
10	2 9	1
11	1 3	1
12	6 1	2

Fig. 3

Case gantt chart.

Considering that swarm intelligence algorithms require multiple individuals as initial solutions, the initialization method adopted by RBBO is to obtain a batch of initial habitats through random coding and NR₂ [1] factory allocation. These initial populations are then two-dimensional encoded to obtain the initial solution of the algorithm. NR₂ factory allocation rule: Assign the jobito the factory that contains the minimum and maximum completion time after the job i.The specific initialization method is as follows: Firstly, obtain a one-dimensional random solution based on the number of jobs in the instance. Secondly, obtain a scheduling solution for the instance by using the NR₂ factory allocation method for this one-dimensional random solution. Finally, convert this scheduling scheme into dual encoding. In this way, the initial solution of the algorithm can be obtained.

4.3 Migration operation

Biogeography-based optimization techniques heavily rely on migration. Migration, a phase in the algorithm’s global search, can successfully increase the suitability of habitats overall and more efficiently enhance the appropriateness of habitats with lower suitability. The migration strategy used in this paper aims to locate and enhance habitat HSI by trading product assembly sets. $ω_{k} (h_{e}) \to ω_{k} (h_{i})$ (18)

As shown in Equation (18), the set of k product ω_k in the emigration habitath_eis represented by ω_k (h_e), while the ω_k in the immigration habitat h_iis represented by ω_k (h_i). It is possible to improve the destination’s HSI by substituting the set of products k from the immigration habitat with the set of products k from the emigration habitat.

For example, emigration habitat h_e=[3 10 9 6 7 2 11 8 5 12 4 1] [2 1 1 2 1 1 1 2 1 2 2 2], immigration habitat h_i=[2 5 11 4 1 8 12 6 3 10 9 7][1 2 2 1 2 1 2 2 1 1 2 1]. Choose to replace the job set and factory allocation set of product 1 in the relocation location h_e with the job set and factory allocation set of product 1 in h_i, and obtain $h_{i}^{'}$ =[2 5 11 4 1 8 12 3 10 9 6 7][1 2 2 1 2 1 2 2 1 1 2 1]. If $HSI (h_{i}^{'}) > HSI (h_{i})$ , stop migration, otherwise continue to replace the job set and factory allocation set for product 2. The Pseudocode for migration is shown in Fig. 4.

Fig. 4

Pseudocode for migration.

4.4 Mutation operation

The local search procedure is crucial to forcing the algorithm out of local optima. The mutation process used in this paper disrupts a specific set of jobs and associated factory allocation sets of the product, changing the chosen habitat’s appropriateness significantly. Instead of performing individual mutation procedures inside the region, the RBBO mutation operation integrates the habitats of all regions into a single region for mutation.

The mutation operation used in this paper is to identify critical factory optimization, lower the critical factory fuzzy finish time (FFT), and subsequently lower MFAT. Taking the example in 4.2 as an example. Assuming a mutation operation is required on the habitat h_m. Find the factory with the largest FFT by decoding the h_m first. Figure 5 demonstrates that factory 2 is the factory with the largest FFT. From right to left, will select products from factory 2, and that product set is ω₂ = [81241]. Choose the first job from this product set, then choose a factory at random to place it in the associated product set. As the optimization target in this instance, select job 4 from product 2 in factory 2 and add it to product 2 in factory 1. In comparison to the original scheme, the sorting results based on [10 9 7 2 11 5 4] [3 6 8 12 1] are better as shown in Fig. 6. The Pseudocode for mutation operation is shown in Fig. 7.

Fig. 5

Gantt chart of cases before mutation operations.

Fig. 6

Gantt chart of cases after mutation operations.

Fig. 7

Pseudocode for mutation.

4.5 Description of RBBO

The flowchart of the RBBO algorithm is shown in Fig. 8.

Fig. 8

Flowchart of RBBO.

A batch of initial habitats must first be initialized, and their HSI must be determined. Based on HSI, divide the area and provide habitat to each section. It is essential to merge the habitat into one region and carry out habitat transformation activities after the migration and replacement of the current habitat region is complete. When the goal maximum number of iterations has been reached, continue the iterative process one more time, then halt the algorithm’s operation and output the HSI value of the ideal habitat.

5 Computational experiments

All trials were carried out on a 16.0 RAM, 2.8 GHz CPU PC utilizing Microsoft Visual C++2022 programming. First, as shown in Table 2, created RBBO parameters and provided four sets of algorithm parameters with different accuracies. As indicated in Table 4, the orthogonal test table and average HSI (AVG) statistical results were produced using parameters with various degrees of accuracy. The Taguchi experiment was the methodology employed, with cases J = 16, M = 5, F = 4, and P = 4. Based on the information in Tables 2 and 4, the response values for each parameter in Table 5 and the response trends for each parameter in Fig. 9 were derived. Through experiments, the most acceptable parameter accuracy for RBBO can be accurately defined. The effectiveness of the algorithm must be confirmed after setting the parameters. The GA, BBO, VND, SPT, LPT, NEH and RBBO solutions that were used in the case are displayed in Tables 6 and 7. Figures 10 and 22 illustrates the algorithm’s analysis of variance, which enables us to see the algorithm’s benefits and drawbacks more easily. To verify our hypothesis about the RBBO algorithm, Figs. 11–20 shows the evolutionary trend of the FDAPFSP case under BBO and RBBO. Figure 23 compares the CPU runtime among different algorithms.

Table 4
Combinations of parameter values

Parameter Factory Level

1 2 3 4

Number of habitats P 10 30 50 100

Maximum mutation rate M_max 0.05 0.10 0.20 0.40

Maximum iteration number Iter_max 30 50 100 150

Parameter	Factory Level
Number of habitats P	10	30	50	100
Maximum mutation rate M_max	0.05	0.10	0.20	0.40
Maximum iteration number Iter_max	30	50	100	150

Table 5

TFN corresponding ranking

Rank	TFN
1	(776 1036 1266)
2	(777 1029 1278)
3	(778 1037 1252)
4	(781 1036 1252)
5	(786 1034 1274)
6	(795 1033 1257)
7	(798 1062 1271)
8	(808 1060 1279)
9	(813 1054 1280)
10	(815 1051 1285)

Table 6

Orthogonal table of parameters settings

Parameter combination	Parameter level			AR
	P	M _max	Iter _max
1	1	1	1	5.3
2	1	2	2	6.7
3	1	3	3	4.3
4	1	4	4	4.7
5	2	1	2	5.1
6	2	2	1	3.4
7	2	3	4	4.1
8	2	4	3	5
9	3	1	3	3.9
10	3	2	4	3.4
11	3	3	1	4
12	3	4	2	4
13	4	1	4	3.3
14	4	2	3	3.3
15	4	3	2	3.5
16	4	4	1	3.2

Table 7

Average response values of each parameter

Level	Parameter
	P	M _max	Iter _max
1	5.3	4.4	4.0
2	4.4	4.2	4.8
3	3.8	4.0	4.1
4	3.3	4.3	3.9
range	2.0	0.4	0.7
Ranking of influence	1	3	2

Fig. 9

The influence trend of parameters.

Fig. 10

Analysis of variance chart between VND, GA, BBO and RBBO.

Fig. 11

Case 1.

Fig. 12

Case 2.

Fig. 13

Case 3.

Fig. 14

Case 4.

Fig. 15

Case 5.

Fig. 16

Case 6.

Fig. 17

Case 7.

Fig. 18

Case 8.

Fig. 19

Case 9.

Fig. 20

Case 10.

Fig. 21

Analysis of variance chart between SPT, LPT, NEH and RBBO.

Fig. 22

Line chart of algorithm CPU running time in FDAPFSP case.

5.1 RBBO parameter design

The ten cases provided in this paper are all randomly generated based on DAPFSP basic cases through random numbers. The parameters of RBBO include the number of habitats P, the maximum mutation rate M_max, and the maximum iteration number Iter_max. This paper will use four different parameter values for the Taguchi experiments [32] with a test case. Four sets of parameter combinations will generate 16 test cases. Test each case 10 times and record the data for each test. After the test is completed, perform TFNcomparison on the data and generate a set of sorting tables from small to large. Each test TFN has its corresponding sorting sequence number, and the sorting sequence number is used to replace the TFN to generate a parameter Run chart. Calculate the average ranking (AR) of each test case. Table 4 shows the four sets of parameters provided in this paper. Table 3 shows the TFN corresponding ranking from 16 test cases. Table 4 shows the AR generated by 16 parameter combinations. Figure 6 is the parameter trend chart.

Through experiments, ten different TFNsolutions can be created. These ten solutions were ordered using the TFNcomparison approach, but rank values rather than TFNvalues were employed. These ten sets of solutions are shown in Table 5. Table 6 shows the orthogonal table of parameters settings.

It can be seen from the simulation results in Fig. 9 and the simulation results of parameter combinations in Table 7 that when P increases, the algorithm’s search performance improves and M_max is too large or too small, it will cause poor changes in the algorithm’s performance, and so will Iter_max. Therefore, choose P = 100, M_max = 0.40 and Iter_max = 30 are used to solve the problem, the RBBO algorithm has the best result. Therefore, this parameter is selected as the algorithm parameter for RBBO.

5.2 Algorithm comparison

BBO and GA were chosen for comparison testing to assess the efficacy of RBBO in FDAPFSP. Except for various algorithm models, the experiment is conducted fairly by employing the same search methodology across the board. Table 8 displays the comparison table of the outcomes. Using c₂ in TFN as the comparison technique because there is presently no way that can calculate the average value of TFN. Three metrics are offered for each algorithm. The worst-case solution (WST) among the five runs of the algorithm, the best-case solution (BST), and the average output result (AVG) of the five runs of the method are all represented by the acronyms. The cases designed in this article are all based on the DAPFSP case. A total of ten cases were provided, with a size of J = {8, 12, 16, 20, 24}, M = {2, 5}, F = {2, 4}, P = {2, 4}.

Table 8
RBBO’s effectiveness verification

Problem scale VND GA BBO RBBO

WST BST AVG WST BST AVG WST BST AVG WST BST AVG

8×2×2×2 501 501 501 501 501 501 501 501 501 501 501 501

8×5×4×4 876 876 876 876 876 876 876 876 876 876 876 876

12×2×2×2 969 955 961 956 949 952 949 945 942 962 940 950

12×5×4×4 763 763 763 763 763 763 763 763 763 763 763 763

16×2×2×2 926 917 920 921 910 911 915 901 909 906 894 899

16×5×4×4 1077 1062 1071 1062 1029 1051 1062 1020 1045 1025 1020 1024

20×2×2×2 1444 1439 1442 1436 1430 1432 1438 1435 1436 1434 1430 1432

20×5×4×4 1685 1685 1685 1685 1685 1685 1685 1685 1685 1685 1685 1685

24×2×2×2 1903 1886 1894 1880 1861 1867 1878 1856 1869 1875 1849 1863

24×5×4×4 2451 2451 2451 2451 2451 2451 2451 2451 2451 2451 2451 2451

Average 1259 1253 1256 1253 1246 1249 1251 1243 1247 1247 1240 1244

Problem scale	VND	GA	BBO	RBBO
8×2×2×2	501	501	501	501	501	501	501	501	501	501	501	501
8×5×4×4	876	876	876	876	876	876	876	876	876	876	876	876
12×2×2×2	969	955	961	956	949	952	949	945	942	962	940	950
12×5×4×4	763	763	763	763	763	763	763	763	763	763	763	763
16×2×2×2	926	917	920	921	910	911	915	901	909	906	894	899
16×5×4×4	1077	1062	1071	1062	1029	1051	1062	1020	1045	1025	1020	1024
20×2×2×2	1444	1439	1442	1436	1430	1432	1438	1435	1436	1434	1430	1432
20×5×4×4	1685	1685	1685	1685	1685	1685	1685	1685	1685	1685	1685	1685
24×2×2×2	1903	1886	1894	1880	1861	1867	1878	1856	1869	1875	1849	1863
24×5×4×4	2451	2451	2451	2451	2451	2451	2451	2451	2451	2451	2451	2451
Average	1259	1253	1256	1253	1246	1249	1251	1243	1247	1247	1240	1244

The variance analysis on the test results of RBBO, BBO, GA and VND [1] in 10 separate cases allows for a more in-depth examination of the distinctions between these three algorithms. The VND termination condition is CPU runtime=F × C, where C = 0.1. GA, BBO and RBBO use the same number of iterations and population and set the termination condition to the maximum number of iterations. Figure 10 displays the Tukey’s HSD test confidence intervals at a 95% level of assurance together with the mean change curves of the three algorithms.

Table 8 and Fig. 10 show that RBBO significantly outperforms GA and VND, with RBBO’s WST, BST, and AVG values being far greater than GA’s. This shows RBBO’s advantages for handling distributed problems in an informal way. Due to its greater iteration up than BBO, RBBO provides more new solutions than BBO. It may speed up the process of mutation by quickly achieving local optima in the overall habitat. Ten sets of iterative speed images of BBO and RBBO cases were provided, and tracking and observing the overall evolution of habitats based on the optimal population of each habitat generation. In the same way, compare each TFN value using the TFN comparison method to get a Rank table. To track a trend of image changes, c₂ values are used instead of TFN values.

In Table 9, by comparing the results of RBBO with six heuristic algorithms in ten FDAPFSP cases, RBBO has been found to have good local search ability and can find better solutions for each situation. meanwhile, the CPU runtime line charts of RBBO and BBO, GA, VND, and six heuristic algorithms were provided. Among them, the three heuristic algorithms SPT (short processing time), LPT (long processing time), and NEH were all taken from reference [1], and six heuristic algorithms were obtained through two different factory allocation methods in [1]. Although heuristic algorithms cannot provide the optimal solution, the quality of the solution is much better than that of the random solution, and heuristic algorithms usually have a shorter running time. Through heuristic algorithms, redundant solutions can be screened for the algorithm. As shown in Fig. 22, RBBO has an advantage in running time compared to GA and is longer than BBO, but it sacrifices some CPU running time in exchange for a better solution. This is a reasonable balancing strategy.

Table 9

RBBO’s effectiveness verification

Problem scale	SPT1	SPT2	LPT1	LPT2	NEH1	NEH2	RBBO
							BST
8×2×2×2	651	651	543	543	503	503	501
8×5×4×4	876	876	876	876	876	876	876
12×2×2×2	1150	1200	1294	1200	1275	1281	940
12×5×4×4	797	863	927	927	842	842	763
16×2×2×2	1667	1691	1850	1846	1869	1869	894
16×5×4×4	1447	1356	1485	1365	1489	1320	1020
20×2×2×2	2977	2977	2977	2977	2992	2991	1430
20×5×4×4	2291	2432	2291	2427	2485	2456	1685
24×2×2×2	1928	1928	2189	2226	1846	1850	1849
24×5×4×4	3719	3471	4059	4028	3828	3812	2451
Average	1750	1744	1849	1841	1800	1780	1240

Through experiments, it was found that under these ten cases, the habitat’s development speed of RBBO was much faster than that of BBO. Due to the increase in habitat evolution speed, it is expected that the algorithm will have sufficient time to search for better solutions through local optima. This is due to the faster development speed of RBBO compared to BBO. This habitat mechanism was developed during the migration phase.

6 Conclusions and future research

This paper proposes the problem of combining DAPFSP with fuzzy processing time constraints and designs a model for this problem. This article proposes the RBBO algorithm, designs four sets of parameters, and conducts Taguchi experiments to obtain the optimal parameter combination. Designed and solved 10 sets of examples. By comparing SPT, LPT, NEH, VND, GA, BBO, and RBBO algorithms, obtained interval graphs for the application of these three algorithms in FDAPFSP cases. To verify the hypothesis of the RBBO, iteration calculations were conducted and the iteration curves between the BBO algorithm and the RBBO were obtained.

Fuzzy processing time constraint is a common scheduling constraint in actual production and processing processes. Due to the uncertainty of processing time, it is necessary to consider reducing the difference between delayed processing time and early processing time in actual scheduling arrangements, so that this difference is close to 0. The ideal processing time value cannot be blindly regarded as the goal of fuzzy scheduling, and comprehensive consideration should be given, this scheduling arrangement is only feasible and practical.

The structural complexity of the RBBO algorithm will be reduced in future work by optimization based on the algorithm, and a new search strategy will be created to be added to it. To bring the DAPFSP problem into line with the real situation, add new constraints. To determine which method is most self-adaptive, an attempt will also be made to combine the new algorithm with the BBO. As an instance, consider the krill swarm algorithm [34, 35] and the grey wolf optimization method [33].

Footnotes

Acknowledgments

This work was supported by Anhui Provincial University Research Projects(2023AH050935), the Research Initiation Foundation of Anhui Polytechnic University (2022YQQ002), Anhui Polytechnic University Research Project (Xjky2022002), the Open Research Fund of Anhui Key Laboratory of Detection Technology and Energy Saving Devices (JCKJ2021A06, JCKJ2022B01), Key Natural Science Research Projects of Colleges and Universities in Anhui Province (2022AH050978, 2023AH052915), Anhui Province University Excellent Top Talent Training Project(gxbjZD2022023), Wuhu science and technology project (2022jc26), and Anhui Polytechnic University-Jiujiang District Industrial Collaborative Innovation Special Fund Project (2022cyxtb6).

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A regional biogeography-based optimization algorithm for the distributed assembly permutation flow-shop scheduling problem with fuzzy processing time

Abstract

Keywords

1 Introduction

2.1 Distributed assembly permutation flow-shop scheduling problem with fuzzy processing time

4.1 The framework of RBBO

Table 3 Case in DAPFSP Job Processing time Product number Product Assembly time 1 1 6 2 1 8 2 4 3 1 3 4 2 2 4 4 1 2 5 5 2 1 6 2 4 2 7 3 1 1 2 4 8 6 3 2 9 3 7 1 10 2 9 1 11 1 3 1 12 6 1 2

Table 4 Combinations of parameter values Parameter Factory Level 1 2 3 4 Number of habitats P 10 30 50 100 Maximum mutation rate Mmax 0.05 0.10 0.20 0.40 Maximum iteration number Itermax 30 50 100 150

5.2 Algorithm comparison

Footnotes

Acknowledgments

References

Table 3
Case in DAPFSP

Job Processing time Product number Product Assembly time

1 1 6 2 1 8

2 4 3 1

3 4 2 2

4 4 1 2

5 5 2 1

6 2 4 2

7 3 1 1 2 4

8 6 3 2

9 3 7 1

10 2 9 1

11 1 3 1

12 6 1 2

Table 4
Combinations of parameter values

Parameter Factory Level

1 2 3 4

Number of habitats P 10 30 50 100

Maximum mutation rate M_max 0.05 0.10 0.20 0.40

Maximum iteration number Iter_max 30 50 100 150