A data-driven ensemble algorithm of black widow optimizer and simulated annealing algorithms for multi-objective buffer allocation in production lines

Abstract

The multi-objective buffer allocation problem of production lines is a non-deterministic-polynomial-hard problem. Many metaheuristic algorithms have been proposed to solve this problem. However, further investigation of new algorithms is still required because metaheuristic algorithms highly depend on the problem types. Furthermore, the balance between the solution quality and computational efficiency requires further improvement. Therefore, a data-driven algorithm consisting of the black widow optimizer and simulated annealing algorithm is proposed to maximize throughput and minimize energy consumption in production lines. Numerical examples demonstrate that the proposed algorithm achieves better solution quality than other state-of-the-art algorithms without losing computational efficiency. This study contributes to multi-objective optimization of resource scheduling in production lines.

Introduction

The multi-objective buffer allocation problem (MBAP) in production lines is a non-deterministic-polynomial-hard problem. A production line consists of machines and buffers between the machines, which are typically sequentially arranged in a line. Because buffers can decrease the blocking phenomenon and keep machines working smoothly, it is vital to allocate buffers reasonably to improve the system performance and optimize the system energy consumption.^1,2

The MBAP can be solved by integrating evaluative algorithms with optimization algorithms.^3,4 Optimization algorithms search for candidate solutions, whereas evaluative algorithms select optimal solutions among candidate solutions. This process continues until a predefined objective is satisfied.

Evaluative algorithms consist of exact, approximation, simulation, and data-driven methods. Exact methods and simulation methods typically obtain results close to actual values ⁵ ; however, long computational times limit their application in MBAP, in which many evaluations are required. Approximation methods are widely used because of their higher computational efficiency than exact methods and simulation methods.^6–8 However, in MBAPs that require a large number of evaluations, a certain amount of computational time is required for approximation methods. Data-driven methods utilize machine learning algorithms to analyze the data obtained by mathematical models or industrial data and build performance prediction models.^9,10 Despite their high computational efficiency, their predictive accuracy remains a major challenge. To achieve a balance between predictive accuracy and computational efficiency, approximation methods are being combined with data-driven methods.

Optimization algorithms include enumeration algorithms (EAs), dynamic programing, search algorithms, and metaheuristics. Both enumeration algorithms and dynamic programing are typically used to obtain global-optimal solutions for production lines on a small scale owing to their long computational time.^11,12 Search algorithms can obtain the near-optimal solution quickly by using guidance information to update the candidate solutions.^13,14 However, the effectiveness of the guidance information is difficult to determine, which directly affects solution quality. Metaheuristics are widely used to solve resource scheduling problems, such as MBAP.^15–17 Yelkenci Kose and Kilincci ¹⁸ proposed a simulated annealing (SA) based nondominated sorting genetic algorithm II (NSGA-II) to maximize the average system throughput and minimize the total buffer size. Xi et al. ¹⁹ developed a decomposition coordination method to minimize the total cost investment by allocating buffers and machines and selecting machine types. Despite the rich number of previous studies on metaheuristics, the performance of a meta-heuristic algorithm highly depends on the problem being handled. ²⁰ Therefore, it is necessary to investigate new optimization algorithms against classical algorithms. Furthermore, the balance between solution quality and computational efficiency requires further improvement to satisfy increasing industrial application requirements.

This study proposes a data-driven ensemble algorithm (DDEA) of the black widow optimizer algorithm (BWOA) and SA to maximize the throughput and minimize the energy consumption of production lines subject to the total buffer capacity. In the proposed algorithm, the BWOA is improved for global search, whereas SA is used to intensify local search and detect promising regions. The nondominated-sorting-II method is used to efficiently sort the Pareto optimal solutions. Furthermore, an evaluative algorithm is utilized to calculate the throughput and energy consumption. Simultaneously, a gradient boosting regression tree (GBRT) is used to build surrogate models using a database, improving the evaluative efficiency of the local search.

This study proposes a novel DDEA to solve the MBAP of production lines, significantly improving solution quality without losing computational efficiency. This can help engineers design and optimize resource scheduling in production and logistics systems.

The remainder of this study is organized as follows. Section 2 describes the problem. Section 3 presents the solution methodology. Section 4 compares the proposed algorithm with other state-of-the-art algorithms to demonstrate its effectiveness. Finally, conclusions and future research are summarized in Section 5.

Problem statement

Assumption

The assumptions in this study are as follows.

• There are $I$ machines and $I-1$ buffers in the production line, as shown in Figure 1.

• The first machine is never starved, and the final machine is never blocked.

• Each machine had an identical failure rate, repair rate, and energy consumption for different operations, and their values were predefined.

• First come first serve ²¹ is used to describe the processing sequences for the parts in front of the machines.

• Blocking after service ²¹ is used to describe the blocking behavior of the parts between the buffers and machines.

• The transfer time between machines is assumed to be zero.

Figure 1.

Model of a production line.

Notation

Table 1 presents the notations used in this study.

Table 1.

Notations in this study.

Items	Description
$B$	Total buffer capacity.
$b_i$	Buffer capacity in buffer $i$ . $b_i^{*}$ denotes a buffer capacity in the new individual.
$\overline{b}$	Solution vector of buffer allocation. $\overline{b}=(b_1,\cdots b_i\cdots ,b_{I-1})$ . ${\overline{b}}^{*}$ denotes a neighborhood individual bymove representation or genetic operator.
$\mathrm{EC}$	Total energy consumption of a production line. $E{C}_n$ is the energy consumption of the $n$ th individual.
$F_l$	The $l$ th Pareto optimal floor. $F_1$ is the Pareto optimal front.
$i\mathrm{or}h$	A machine or buffer in production lines. $I$ is the total number of machines.
$j_{\mathrm{DDEA}}$	An iteration number in the DDEA. $J_{\mathrm{DDEA}}$ is the maximum iteration number.
$j_{\mathrm{SA}}$	An iteration number in the SA. $J_{\mathrm{SA}}$ is the maximum iteration number.
$m\mathrm{or}n$	An individual or a solution of buffer allocation.
$N_{\mathrm{ini}}$	The number of initial individuals
$P{H}_n$	The pheromone of the $n$ th individual in the BWOA.
$\mathrm{TH}$	Throughput of a production line. $T{H}_n$ is the throughput of the $n$ th individual.
$\overline{\mu }$	Vector of service rate allocation. $\overline{\mu }=({\mu }_1,\cdots {\mu }_i\cdots ,{\mu }_I)$
$\mathrm{\Pi }$	An archive that stores individuals.
$\mathrm{random}(X,Y)$	A random number between $X$ and $Y$
$\mathrm{randint}(X,Y)$	A random integer between $X$ and $Y$ .
$\mathrm{round}(X)$	The round value of $X$ .

Multi-objective optimization problem of buffer allocation

The MBAP in this study is aimed at maximizing the throughput and minimizing the energy consumption of production lines subject to the total buffer capacity. This problem can be expressed as follows:

$$\mathrm{Find}\overline{b}=(b_1,\cdots b_i\cdots ,b_{I-1})$$

to maximize the $\mathrm{TH}$ and minimize the $\mathrm{EC}$ ,

$$\mathrm{subject}\mathrm{to}\sum _{i=1}^{I-1}{b}_i=B,$$

(1)

$$b_i\ge 1,b_i\mathrm{is}\mathrm{an}\mathrm{integer},$$

(2)

where equation (1) regulates the decision variables $\overline{b}$ and equation (2) represents the bounds of the $b_i$ .

In the case where the sub-objective of $\mathrm{TH}$ conflicts with the sub-objective of $\mathrm{EC}$ , it is necessary to define the partial order relation among solutions.

Pareto dominance

A solution $n$ is dominated by another solution $m$ if

$$T{H}_n\le T{H}_m\mathrm{and}E{C}_n>E{C}_m$$

(3)

or

$$T{H}_n<T{H}_m\mathrm{and}E{C}_n\ge E{C}_m$$

(4)

Pareto optimality and Pareto optimal front

A given solution $n$ is called a Pareto optimal solution if it is not dominated by other solutions. The set of all Pareto optimal solutions is the Pareto optimal front, which is denoted by $F_1$ . Excluding the solutions in $F_1$ , the other solutions are iteratively sorted into different floors $F_l$ based on Pareto dominance. For individuals on the same floor, crowding distances of solutions are compared to represent the priority.^22,23

Solution methodology

In this study, the DDEA was integrated with the economic and energetic performance evaluation method (EEPEM) ²⁴ to solve the MBAP. In the DDEA, the BOWA ²⁵ is improved for a global search to exploit candidate solutions, whereas the SA is used for a local search to explore more promising regions. Surrogate models are constructed based on a database that stores the historical data of candidate solutions to reduce the computational time of the local search.

Algorithm: the proposed DDEA to solve the MBAP
Initialize: Generate $N_{\mathrm{DB}}$ initial solutions and their corresponding throughput and energy consumption in the database $\mathrm{DB};$ Randomly select $N_{\mathrm{ini}}$ initial individuals in $\mathrm{DB}$ to build the archive ${\mathrm{\Pi }}_{\mathrm{arc}}.$
1 Set $j_{\mathrm{DDAE}}=0$
2 While $j_{\mathrm{DDEA}}<J_{\mathrm{DDEA}}$ do
3   $j_{\mathrm{DDAE}}=j_{\mathrm{DDEA}}+1$
4   Set $j_{\mathrm{SA}}=0$
5   Build surrogate models $\hat{f}$ based on $\mathrm{DB}$ .
6   Set ${\mathrm{\Pi }}_{\mathrm{SA}}={\mathrm{\Pi }}_{\mathrm{arc}}$
7   Set $T_0$
8   While $j_{\mathrm{SA}}<J_{\mathrm{SA}}$ do
9   $j_{\mathrm{SA}}=j_{\mathrm{SA}}+1$
10   Generate neighborhood individuals of the individuals in ${\mathrm{\Pi }}_{\mathrm{SA}}$ using equations (8) and (9).
11   Accept the neighborhood individuals based on the acceptance criterion.
12   Predict $\mathrm{TH}$ and $\mathrm{EC}$ using $\hat{f}$ .
13   Give floor rank and calculate the crowding distance of the accepted neighborhood individuals.
14   Renew ${\mathrm{\Pi }}_{\mathrm{SA}}$ by the individuals in $F_1$ .
15   $T=T_0\times \lambda$
16   Generate neighborhood individuals of the individuals in ${\mathrm{\Pi }}_{\mathrm{arc}}$ and ${\mathrm{\Pi }}_{\mathrm{SA}}$ using equation (11), and store them in ${\mathrm{\Pi }}_{\mathrm{BWOA}}$ .
17   Revise buffer capacities in the neighborhood individuals by equation (14).
18   Normalize the neighborhood individuals by equation (15).
19   Select individuals and generate new individuals by equations (12) and (13), and store them in ${\mathrm{\Pi }}_{\mathrm{BWOA}}$ .
20   Revise buffer capacities in the new individuals by equation (14).
21   Normalize the new individuals by equation (15).
22   Give floor rank and calculate the crowding distance of the individuals in ${\mathrm{\Pi }}_{\mathrm{arc}}$ , ${\mathrm{\Pi }}_{\mathrm{SA}},$ and ${\mathrm{\Pi }}_{\mathrm{BWOA}}$ .
23   Renew ${\mathrm{\Pi }}_{\mathrm{bes}}$ by the individuals in $F_1$ .
24   Renew ${\mathrm{\Pi }}_{\mathrm{arc}}$ by the individuals in $F_1$ (If the number of individuals in $F_1$ is less than $N_{\mathrm{ini}}$ , the least crowed individuals in $F_2$ is added to ${\mathrm{\Pi }}_{\mathrm{arc}};$ otherwise, $N_{\mathrm{ini}}$ least crowed individuals in $F_2$ is added to ${\mathrm{\Pi }}_{\mathrm{arc}}$ ).
25   Update $\mathrm{DB}$ by adding the individuals in ${\mathrm{\Pi }}_{\mathrm{bes}}$ into $\mathrm{DB}$ and delete the same amount of individuals that were added to $\mathrm{DB}$ before.
26   Use the diversification strategy if it is necessary.
27 Output ${\mathrm{\Pi }}_{\mathrm{bes}}$

The EEPEM

EEPEM calculates the throughput and energy consumption of a production line based on discrete Markov chain formulations. ²⁴ First, it calculates the steady-state probabilities of the buffers and approximates the effective service rate of each machine based on these steady-state probabilities. The throughput of the production line is the lowest effective service rate among the machines. Furthermore, EEPEM computes the energy consumption of each machine based on failure rates, machine repair rates, steady-state probabilities of buffers, and energy consumption of various machine operations. The sum of all the machine energy consumption is the system energy consumption. Readers can refer to the literature ²⁴ for details on EEPEM.

The DDEA

Coding and initialization operator

Individuals denote candidate solutions, and coding represents the solution expression. The coding of an individual is defined as:

$$\overline{b}=(b_1,\cdots ,b_i,\cdots b_{I-1}).$$

(5)

Furthermore, $N_{\mathrm{ini}}$ initial individuals were randomly generated with the constraints of equations (1) and (2), respectively. The initialization operator is expressed as follows:

$$b_i=\mathrm{randint}(1,B-I+1),i=1,\cdots I-1.$$

(6)

$$\sum _{i=1}^{I-1}{b}_i=B.$$

(7)

Surrogate model and $\mathit{DB}$

The GBRT, which is effective in prediction research, was used to construct surrogate models. ²⁶ The surrogate models were trained using the sklearn library (version 0.24.2) in Python (version 3.6). For more details on GBRT, readers can refer to the literature. ²⁶ $\mathrm{DB}$ stores the historical data of individuals and their corresponding $\mathrm{TH}$ and $\mathrm{EC}$ . In each iteration of the DDEA, the obtained new individuals are used to update $\mathrm{DB}$ to make all data closer to the current Pareto optimal front and improve the predictive accuracy of the surrogate model. Considering the fitting efficiency and accuracy, the number of data in $\mathrm{DB}$ was set to 20 $B$ .

The SA in the DDEA

Move representation and neighborhood structure

The SA has satisfactory local search ability, ¹⁸ and its move representation determines the neighborhood individuals of the current optimal individuals. The move representation is defined in this study as changing a certain number of capacities between two buffers and a certain service rate between the two machines of each individual in ${\Pi }_{\mathrm{arc}}$ . The move representation is expressed as follows:

$$\begin{array}{c}\multicolumn{1}{c}{\overline{b}=(\cdots ,b_i,\cdots ,b_h,\cdots )\to {\overline{b}}^{*}}\\ \multicolumn{1}{c}{=(\cdots ,b_i-\mathrm{\Delta }b,\cdots ,b_h+\mathrm{\Delta }b,\cdots )}\end{array}$$

(8)

$$\Delta b=\mathrm{randint}(1,2)$$

(9)

subject to $b_i-\mathrm{\Delta }b\ge 1$ .

The neighborhood structure is a subset of all feasible neighborhood individuals considering the computational efficiency. Each individual in ${\Pi }_{\mathrm{arc}}$ generates a new neighborhood individual.

Temperature settings

$T_0$ is related to optimal solution quality. ²⁷ A high $T_0$ typically generates a high-quality solution; however, it requires a longer computational time. $\lambda$ is related to convergence velocity. $T_0$ and $\lambda$ are problem-dependent.

Acceptance and stopping criteria for SA

The acceptance criterion is defined using the Metropolis criterion and Pareto optimality. If an individual is dominated by its neighborhood individual, the neighborhood individual is accepted; otherwise, if equation (10) is satisfied, the neighborhood individual is also accepted.

$$\mathrm{rand}(0,1)<\exp (\mathrm{abs}(l-l^{*})/T),$$

(10)

where $l$ and $l^{*}$ denote the sum of the $\mathrm{TH}$ and $\mathrm{EC}$ corresponding to the selected individual and its neighborhood individual, respectively.

The stopping criterion for SA is defined as the maximum number of iterations ( $J_{\mathrm{SA}}$ ).

Genetic operator

The BWOA has a satisfactory global search ability ²⁵ inspired by the movement and pheromones of black widows. In this study, the BWOA was extended to solve the MBAP. By imitating the linear and spiral movements of black widows, each individual’s buffer capacity can be expressed as follows:

$$b_i\left(j+1\right)=\{\begin{array}{c}{b}_i^{*}\left(j\right)-random\left(0.4,0.9\right)\times b_i^{r1}\left(j\right),ifrandom\left(0,1\right)<0.3\\ b_i^{*}\left(j\right)-cos\left(2\pi \times random\left(-1,1\right)\right)\times b_i\left(j\right),inothercases\end{array}$$

(11)

where $b_i^{*}\left(j\right)$ denotes the buffer capacity of the best individual in the current iteration, and $b_i^{r1}\left(j\right)$ denotes the buffer capacity of a randomly selected individual in the current iteration. Here, the best individual was obtained by randomly selecting one from the highest 10% of individuals with large crowding distances.

The pheromone of the $n$ th individual can be calculated by

$$P{H}_n(j)=\frac{1}{2}\times (\frac{T{H}_{\max }-T{H}_n\left(j\right)}{T{H}_{\max }-T{H}_{\min }}+\frac{E{C}_{\max }-E{C}_n\left(j\right)}{E{C}_{\max }-E{C}_{\min }})$$

(12)

where $T{H}_{\max }$ and $T{H}_{\min }$ represent the maximum and minimum throughputs in the current individuals, respectively; $E{C}_{\max }$ and $E{C}_{\min }$ represent the maximum and minimum throughputs in the current individuals, respectively; and $T{H}_n(j+1)$ and $E{C}_n(j+1)$ represent the throughput and energy consumption corresponding to individual $n$ , respectively. Subsequently, more new individuals can be generated by

$$b_i(j+1)=b_i^{*}\left(j\right)+\frac{1}{2}(b_i^{r1}\left(j\right)-{(-1)}^{\mathrm{random}(0,1)}\times b_i^{r2}\left(j\right))$$

(13)

where $b_i^{r1}\left(j\right)$ and $b_i^{r2}\left(j\right)$ represent the buffer capacity in two different individuals randomly selected in ${\Pi }_{\mathrm{SA}}$ and ${\Pi }_{\mathrm{arc}}$ .

To avoid the buffer capacity from being lower than the lower limit, the reflection operator is expressed as

$$b_i(j+1)=b_{\min },$$

(14)

where $b_{\min }$ is the lower limit of the buffer capacity. Furthermore, to maintain the total buffer capacity of the new individuals, the normalization operator is expressed as

$$b_i(j+1)=\mathrm{round}(B\times b_i(j+1)/\sum _{i=1}^{I-1}{b}_i(j+1))$$

(15)

Stopping criterion

The stopping criterion for either DDEA was defined as the maximum number of iterations ( $J_{\mathrm{DDEA}}$ ).

Numerical example

The proposed DDEA was tested in production lines involving 5, 10, 15, 20, and 25 machines. To test its sensitivity, the effects of the total buffer capacity, initial individual size, and service rate allocation were considered in the numerical examples. The experimental settings are presented in Table 2.

Table 2.

Input parameters of different patterns in the experiment.

Patterns	$N_{\mathrm{ini}}$	$B$	$I$	$\overline{\mu }$
1	10	20	5	(1,1,1,1,1)
2	20	20	5	(1,1,1,1,1)
3	10	20	5	(1,0.8,1.2,0.9,1.1)
4	10	40	5	(1,1,1,1,1)
5	10	40	10	(1,1,1,1,1,1,1,1,1,1)
6	20	40	10	(1,1,1,1,1,1,1,1,1,1)
7	10	40	10	(1,0.9,1,0.8,1.2,1.1,0.8,1,1.2,1)
8	10	80	10	(1,1,1,1,1,1,1,1,1,1)
9	10	60	15	(1,⋯,1)
10	20	60	15	(1,⋯,1)
11	10	60	15	(1,1,[1.1*3],[0.9*3],[0.8*3],[1.2*3],1)
12	10	120	15	(1,⋯,1)
13	10	80	20	(1,⋯,1)
14	10	100	25	(1,⋯,1)

The proposed DDEA was compared with the NSGA-II, SA, and BWOA. The settings of the four algorithms are listed in Table 3. In addition, an EA was used to demonstrate the absolute solution quality of small-scale problems, including Patterns 1–4.

The proposed DDEA, NSGA-II, SA, and BOWA were written in Python 3.6, and executed for all experiments on a laptop computer with a 2.5 GHz Intel Core 2 Duo CPU.

Comparison criteria

The number of Pareto optimal solutions ( $N_{\mathrm{sol}}$ ), ²³ number of Pareto solutions ( $\widetilde{N_{\mathrm{sol}}}$ ), ²³ and average distance ( $h_{\mathrm{ave}}$ ) ²³ were used to evaluate the algorithm convergence performance for small-scale problems, including Patterns 1–4. $N_{\mathrm{sol}}$ , Riise distance ( $u$ ), ¹⁸ and Zitzler and Thiele measure ( $C_{V,W}$ ) ³⁰ were used to evaluate the algorithm convergence performance for medium- and large-scale problems, including Patterns 5–12. The diversity metric ( $\mathrm{\Delta }$ ) ²³ and limit distance ( $\tau$ ) ²³ were used to assess the diversity performance of the algorithm. Here, higher $N_{\mathrm{sol}}$ and $\widetilde{N_{\mathrm{sol}}}$ , and lower $h_{\mathrm{ave}}$ and $u$ indicate better convergence. $C_{V,W}$ denotes the percentage of solutions obtained by algorithm $V$ dominated by or equal to at least one solution obtained by algorithm $W$ . Algorithm $V$ is better than algorithm $W$ if $C_{V,W}$ is lower than $C_{W,V}$ . A lower $\mathrm{\Delta }$ indicates better diversity, and a higher $\tau$ represents a larger scope of the Pareto optimal solution. For more details on the calculation of $h_{\mathrm{ave}}$ , $u$ , $\mathrm{\Delta }$ , and $\tau$ readers can refer to the literature. ²³

Experiment

Setting

Production lines with different total buffer capacities ( $B$ ), total machine size ( $I$ ), and service rate allocation ( $\overline{\mu }$ ) were tested. The parameter settings of NSGA-II, SA, BWOA, and DDEA are listed in Table 3.

Table 3.

Parameter settings of comparison algorithms.

NSGA-II	Initial individual size	Same with the DDEA
	Crossover operation	Simulated binary crossover (SBX) operator 28
	Crossover rate	0.8
	Mutation operation	Mutation operator based on Gauss perturbation, ${\epsilon }_i~N(0,1)$ 29
	Mutation rate	0.1
	Sorting method	Same with Sections 2.3.1 and 2.3.2
	Maximum iteration	400
SA	Initial individual size	Same with the DDEA
	Initial temperature	$T_0=500$ ( $B\le 20$ ), $T_0=1000$ ( $B>20$ )19,27
	Cooling rate	$\lambda =0.95$ ( $B\le 20$ ), $\lambda =0.9$ ( $B>20$ )19,27
	Neighborhood generation	Move representation in the literature.19,27
	Sorting method	Same with Sections 2.3.1 and 2.3.2
	Maximum iteration	400
BWOA	Initial individual size	Same with the DDEA
	Neighborhood generation	Same with the Section 3.2.3.
	Best solution selection	Same with the $\mathrm{DDEA}.$
	Sorting method	Same with Sections 2.3.1 and 2.3.2
	Maximum iteration	400
DDEA	Initial individual size	10 or 20
	Maximum iteration in the SA	$I+5$
	Initial temperature	$T_0=500$ ( $B\le 20$ ), $T_0=1000$ ( $B>20$ )
	Cooling rate	$\lambda =0.95$ ( $B\le 20$ ), $\lambda =0.9$ ( $B>20$ )
	Maximum iteration	40

Results and discussion

Figures 2 to 8 show the performance comparison of the NSGA-II, SA, BWOA, and DDEA. In Patterns 1–4, the Pareto optimal front obtained by the DDEA is closer to the Pareto front obtained by the EA than the NSGA-II, SA, and BWOA, indicating better performance of the DDEA. In Patterns 5–14, the Pareto optimal front obtained by the DDEA is under those obtained by the NSGA-II, SA, and BWOA. Consequently, DDEA still performs better, although the total buffer capacity and machine size increase. The following conclusions were drawn based on Tables 4 to 7.

Figure 2.

Performance comparison of the NSGA-II, SA, BWOA, and DDEA in Patterns (a) 1 and (b) 2 for the production line involving five machines.

Figure 3.

Performance comparison of the NSGA-II, SA, BWOA, and DDEA in Patterns (a) 3 and (b) 4 for the production line involving five machines.

Figure 4.

Performance comparison of the NSGA-II, SA, BWOA, and DDEA in Patterns (a) 5 and (b) 6 for the production line involving 10 machines.

Figure 5.

Performance comparison of the NSGA-II, SA, BWOA, and DDEA in Patterns (a) 7 and (b) 8 for the production line involving 10 machines.

Figure 6.

Performance comparison of the NSGA-II, SA, BWOA, and DDEA in Patterns (a) 9 and (b) 10 for the production line involving 15 machines.

Figure 7.

Performance comparison of the NSGA-II, SA, BWOA, and DDEA in Patterns (a) 11 and (b) 12 for the production line involving 15 machines.

Figure 8.

Performance comparison of the NSGA-II, SA, BWOA, and DDEA in Patterns (a) 13 and (b) 14 for the production line involving 20 and 25 machines, respectively.

Table 4.

Performance comparison of the NSGA-II, SA, BWOA, and DDEA for the production line involving five machines.

Patterns	Algorithms	$N_{\mathrm{sol}}$	$\widetilde{N_{\mathrm{sol}}}$	$h_{\mathrm{ave}}$	$\mathrm{\Delta }$	$\tau$	$t$ [s]
1	EA	36	36	0	0.6399	8.4037	3
	NSGA-II	12	3	0.0435	0.4924	8.1969	22
	SA	14	8	0.0153	0.5215	8.0697	28
	BWOA	12	4	0.0383	0.4492	7.9132	28
	DDEA	23	15	0.0061	0.5441	7.9132	13
2	EA	36	36	0	0.6399	8.4037	3
	NSGA-II	19	9	0.0234	0.3958	7.1920	48
	SA	22	14	0.0065	0.4519	7.9028	47
	BWOA	21	11	0.0362	0.3807	8.0802	74
	DDEA	29	26	0.0046	0.5941	7.9237	18
3	EA	48	48	0	0.6652	8.0917	2
	NSGA-II	12	3	0.0526	0.4924	7.6537	24
	SA	15	8	0.0441	0.5198	8.0917	33
	BWOA	12	5	0.0415	0.3887	7.4802	37
	DDEA	26	21	0.0116	0.5283	8.0917	16
4	EA	100	100	0	0.8223	11.3741	32
	NSGA-II	14	1	0.0514	0.5810	7.17026	26
	SA	15	4	0.0537	0.6240	10.7211	26
	BWOA	12	5	0.0244	0.4972	10.5633	28
	DDEA	42	19	0.0219	0.7273	10.7211	28

Table 5.

Performance comparison of the NSGA-II, SA, BWOA, and DDEA for the production line involving 10 machines.

Patterns	Algorithms	$N_{\mathrm{sol}}$	$u$	$C_{1,2}$	$C_{2,1}$	$\mathrm{\Delta }$	$\tau$	$t$ [s]
5	NSGA-II	13	−34.14	0.92	0.02	0.5608	11.9477	100
	SA	10	−15.67	0.43	0.10	0.3935	15.2955	88
	BWOA	13	−0.29	0.61	0.09	0.4029	18.4322	112
	DDEA	36	/	/	/	0.7712	17.1231	63
6	NSGA-II	23	−75.16	0.79	0.05	0.5953	12.4747	165
	SA	19	−36.13	0.41	0.24	0.4283	14.4767	135
	BWOA	20	−10.82	0.57	0.17	0.3460	17.9540	183
	DDEA	33	/	/	/	0.6539	17.8580	91
7	NSGA-II	13.	−29.93	0.90	0	0.6121	11.3826	97
	SA	10	−12.11	0.34	0.11	0.4473	13.9110	112
	BWOA	13	−5.81	0.76	0.05	0.4670	16.0569	115
	DDEA	35	/	/	/	0.6337	15.9739	79
8	NSGA-II	14	−68.57	0.73	0.04	0.7631	12.8043	65
	SA	9	−6.99	0.36	0.16	0.4736	10.0643	113
	BWOA	13	−7.71	0.58	0.07	0.4823	23.9795	98
	DDEA	40	/	/	/	0.8775	24.1684	149

$u$ denotes the Riise distance of the Pareto optimal front of the DDEA to the NSGA-II, SA, and BWOA; $C_{1,2}$ denotes the Zitzler measure of the NSGA-II SA and BWOA to the DDEA; $C_{2,1}$ denotes the Zitzler measure of the DDEA to the NSGA-II, SA, and BWOA.

Table 6.

Performance comparison of the NSGA-II, SA, BWOA, and DDEA for the production line involving 15 machines.

Patterns	Algorithms	$N_{\mathrm{sol}}$	$u$	$C_{1,2}$	$C_{2,1}$	$\mathrm{\Delta }$	$\tau$	$t$ [s]
9	NSGA-II	13	−50.96	0.82	0.06	0.5098	12.1164	127
	SA	9	−17.22	0.64	0.24	0.4153	15.3320	117
	BWOA	12	−11.87	0.59	0.10	0.3916	20.9466	136
	DDEA	23	/	/	/	0.6773	19.9594	151
10	NSGA-II	20	−51.72	0.67	0.04	0.5083	13.3673	222
	SA	17	−39.27	0.3	0.31	0.6046	17.0022	258
	BWOA	20	−0.97	0.48	0.23	0.3667	20.3782	294
	DDEA	24	/	/	/	0.5029	19.6232	202
11	NSGA-II	12	−54.94	0.75	0.01	0.5107	11.0780	240
	SA	8	−5.85	0.31	0.21	0.3945	12.9314	305
	BWOA	12	−1.99	0.46	0.20	0.4876	18.8622	124
	DDEA	26	/	/	/	0.6793	17.8220	216
12	NSGA-II	12	−35.00	0.64	0.08	0.7001	10.9401	320
	SA	8	−4.00	0.35	0.21	0.5884	11.9588	219
	BWOA	13	−7.21	0.48	0.16	0.5493	26.6050	209
	DDEA	34	/	/	/	0.8576	28.7578	532

Table 7.

Performance comparison of the NSGA-II, SA, BWOA, and DDEA for the production line involving 20 and 30 machines, respectively.

Patterns	Algorithms	$N_{\mathrm{sol}}$	$u$	$C_{1,2}$	$C_{2,1}$	$\mathrm{\Delta }$	$\tau$	$t$ [s]
13	NSGA-II	13	−70.02	0.77	0.01	0.4990	18.0593	156
	SA	9	−25.70	0.56	0.25	0.3160	18.1363	196
	BWOA	11	−16.80	0.55	0.10	0.3977	26.4299	143
	DDEA	20	/	/	/	0.6007	27.3400	292
14	NSGA-II	16	−64.24	0.69	0.11	0.4634	18.7345	253
	SA	9	−15.73	0.44	0.16	0.3676	20.3343	346
	BWOA	12	−12.59	0.75	0.21	0.4080	32.6902	259
	DDEA	19	/	/	/	0.5413	32.4775	447

Convergence analysis

• In Pattern 1–4, the DDEA obtained the highest $N_{\mathrm{sol}}$ , indicating the best search ability of the DDEA among the four comparison algorithms. $\widetilde{N_{\mathrm{sol}}}$ of the DDEA was the highest and $h_{\mathrm{ave}}$ of DDEA was the lowest, indicating its better convergence. This is because SA intensifies the local search and provides more promising solutions for the BWOA, whereas the efficacy of the BWOA maintains a satisfactory global search. With the increase in the total buffer capacity in Patterns 2, 3, and 4, the search scope and difficulty increased. Therefore, the convergence of the four comparison algorithms decreased, although the DDEA was still the best. In Patterns 5–14, the machine number and total buffer capacity increase, enlarging the MBAP scale. $u$ was lower than zero and $C_{2,1}$ was lower than $C_{1,2}$ , indicating that the convergence of DDEA was better than that of NSGA-II, SA, and BWOA.

Diversity analysis

• $\mathrm{\Delta }$ of the DDEA was higher than those of NSGA-II, SA, and BWOA. The diversity of the DDEA is unsatisfactory because the SA focuses on local search; however, the new solution size in each generation is limited, leading to new solutions around limited small regions in a new generation. However, as shown in Figures 2 to 8, the Pareto optimal solutions are relatively evenly distributed, although there are some concentrations in the small regions. Therefore, the diversity of the DDEA is still acceptable. In addition, the solution scopes of DDEA and BWOA were better than those of NSGA-II and SA because of their higher $\tau$ .

Computational time analysis

• In Patterns 1–7, the search scope was small because of the low total buffer capacity and small machine size. The computation time of the DDEA was the shortest among the four algorithms, indicating its high computational efficiency. With increasing problem scale and extending the search scope, the computational time of the DDEA increased more significantly than that of the other three comparison algorithms because the SA searched for more promising regions and the BWOA tested more candidate solutions. However, the maximum computational time is less than 10 min. Considering the convergence improvement of DDEA, its computational efficiency is still satisfactory.

Sensitivity analysis

• The changes in the total buffer capacity, machine size, and service rate allocation hardly affected the efficacy of the DDEA, demonstrating its stability. However, the unbalanced service rate allocation in Patterns 3, 7, and 11 increased the difficulty of MBAP in production lines. Consequently, the computational time of DDEA increased.

Overall, the proposed DDEA can balance exploration and exploitation, generating better convergence without losing diversity. Furthermore, the computational efficiency is satisfactory. This proves the feasibility of integrating different optimization and machine learning algorithms to solve the MBAP in production lines.

Conclusions and future work

This study presented a DDEA of the BWOA and SA to maximize throughput and minimize energy consumption subject to the total buffer capacity in production lines. The proposed algorithm was compared to NSGA-II, SA, and BWOA using numerical examples to test its performance. The experimental results showed that the proposed algorithm achieved better convergence and computational efficiency than the compared algorithms without losing much diversity for small- and medium-scale problems. In large-scale problems, the computational efficiency of DDEA decreases.

This study integrates a data-driven surrogate into optimization algorithms to decrease computational efficiency and extend the solution methods of the MBAP. This can assist engineers design and optimize the layout of production lines in the industry more efficiently.

The accuracy of the surrogate model in the DDEA may fluctuate in large-scale problems. Therefore, one future direction is to study surrogates with higher and more stable predictive accuracy. Furthermore, the MBAP sorting method requires further investigation in order to reduce the effects of local searches and improve the diversity of the DDEA.