
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12349-3
10.1016/j.heliyon.2024.e36318
e36318
Research Article
A multi-objective brain storm optimization for integrated distributed flexible job shop and distribution problems
Jia Yanhe yhejia@bistu.edu.cn
a
Zhou Yaoyao yaoyao_zhou2001@163.com
a
Fu Yaping fuyaping0432@163.com
b⁎
a School of Economics and Management, Beijing Information Science & Technology University, Beijing, 100192, China
b School of Business, Qingdao University, Qingdao, 266071, China
⁎ Corresponding author. fuyaping0432@163.com
14 8 2024
30 8 2024
14 8 2024
10 16 e3631824 5 2024
23 7 2024
13 8 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
Production and distribution are critical components of the furniture supply chain, and achieving optimal performance through their integration has become a vital focus for both the academic and business communities. Moreover, as economic globalization progresses, distributed manufacturing has become a pioneering production technique. Via leveraging a distributed flexible manufacturing system, mass flexible production at lower costs can be achieved. To this end, this study presents an integrated distributed flexible job shop and distribution problem to minimize makespan and total tardiness. In our research, a set of custom furniture orders from different customers are processed among flexible job shops and then delivered by vehicles to customers as the due date. To distinctly show the presented problem, a mixed integer mathematical programming model is created, and a multi-objective brain storm optimization method is introduced considering the problem's features. In comparison to the other three advanced methods, the superiority of the algorithm created is showcased. The findings of the experiments demonstrate that the constructed model and the introduced algorithm have remarkable competitiveness in addressing the problem being examined.

Keywords

Integrated production and distribution scheduling
Distributed flexible job shop
Vehicle routing problem
Brain storm optimization
==== Body
pmc1 Introduction

The furniture manufacturing sector is essential in numerous regions and countries, contributing significantly to their economic growth [1]. In China, the furniture market, comprising 6647 enterprises, produces approximately 1.12 billion pieces of products and generates total revenues of CNY 800.46 billion in 2022.1 Along with the rapid advancements in economy and technology, the custom furniture segment has expanded swiftly, reaching a market size of nearly CNY 527.10 billion in 2023,2 as depicted in Fig. 1. To achieve a competitive advantage, a growing number of custom furniture manufacturers are adopting flexible production systems to satisfy the varied demands of customers [2,3]. Driven by the ongoing expansion of globalization, distributed manufacturing across multiple facilities has emerged as a pioneering production paradigm [4,5]. Consequently, the distributed flexible job shop (DFJS), a crucial component of distributed manufacturing systems, has received wide attention in the custom furniture industry for its remarkable ability to facilitate mass flexible production in a rapid and cost-efficient manner [6].Fig. 1 The scale of the custom furniture market in China. *Note: 2023E = estimate.

Fig. 1

Delivery time is a crucial factor in determining customer satisfaction [7]. However, production and distribution activities are usually scheduled separately and sequentially in many customized furniture enterprises [2]. The former is first conducted by the manufacturing department, and then the latter is performed by a logistics enterprise. In this scenario, very little interaction exists between the two activities, posing a challenge to fulfilling customer's expectations for prompt delivery. Hence, integrated production and distribution scheduling (IPDS) has seen a rise in attention from both enterprises and researchers [8]. The existing work has demonstrated that implementing IPDS with a flexible job shop (FJS) is an available approach for furniture enterprises to decrease operating costs, adapt to dynamic market conditions, and keep superior service levels [2]. However, much research interest in IPDS focuses on a manufacturing system within a single factory, while distributed manufacturing systems are rarely studied owing to the intricate characteristics of integrated problems [9]. As a matter of fact, an integration of distributed manufacturing and distribution better suits the operational necessities of the custom furniture industry, providing superior production and delivery services. To this end, an integrated DFJS and distribution scheduling model is extended in our work, tailored to the requirements of customized furniture businesses.

Given the NP-hard nature of IPDS, meta-heuristics become commonly employed solution methods in the literature [10]. They possess good abilities to obtain near-optimal solutions within limited computational resources. Therefore, much effort has been given to develop various meta-heuristics to handle IPDS problems in the past decades, e.g., particle swarm optimization algorithms [2,11], and large neighborhood search methods [12,13]. The brain storm optimization (BSO) algorithm, proposed by Shi [14], is a meta-heuristic technique designed for continuous and single-objective problems. It has many advantages, such as being easy to code and implement, strong search abilities, and fast convergence. Presently, BSO techniques have been extensively conducted across a diverse range of optimization tasks, such as production scheduling, traveling salesman, and image segmentation problems [15]. A series of numerical experiments have validated its outstanding abilities [16]. Hence, this work adopts it as a fundamental optimizer. The presented problem aims to minimize makespan and total tardiness in production and distribution phases, respectively. To achieve this, a multi-objective brain storm optimization algorithm (MBSO) is designed by considering the inherent nature of the problem. Twofold contributions of this study include:(a) Different from existing studies that most of them are devoted to solving IPDSs with a single-factory setup, this work addresses such problems by integrating distributed manufacturing across multiple factories. In the proposed problem, a DFJS manufacturing system is managed in the production stage and a multi-depot vehicle routing problem is handled in the distribution stage. To clearly define the problem under study, a multi-objective mixed integer mathematical programming model is provided to minimize makespan and total tardiness.

(b) The considered problem is very complex since it needs to make five decisions simultaneously. Production decisions consist of assigning jobs among factories, determining the operation sequence, and allocating machines for operations. Distribution decisions include assigning jobs to vehicles, and deciding the job delivery routes. The proposed problem has a wide search space, posing a huge challenge to effectively handle it. To this end, this work designs an MBSO by considering some problem features. In MBSO, the solution representation, population initialization, clustering, generating, and selecting strategies are constructed. By comparing MBSO with three advanced algorithms, the results reveal that MBSO demonstrates superior performance in tackling this problem.

The organization of this work is below: The subsequent section surveys the related literature. Section 3 outlines and models the examined problem. The details of MBSO are discussed in Section 4. Section 5 carries out numerical experiments. Lastly, Section 6 summarizes this research and proposes future exploration ideas.

2 Literature review

This section is organized into three key parts. The initial part reviews the literature related to IPDS. The subsequent part reviews the relevant studies associated with DFJS. Finally, the third part examines the recent investigations of BSO algorithms.

2.1 Relevant literature on IPDS

Integrating production and distribution operations can be seen as a valid strategy to enhance overall performance. Recently, the concept of IPDS has garnered significant attention across various industries and the academic community. Hilali et al. [17] tackled a multi-site IPDS problem for a phosphoric supply chain, where co-production and alternative routings were considered. They created an integer linear programming model to minimize the cost objective and applied an Xpress solver to conduct numerical experiments on a case study. Nogueira et al. [18] investigated an IPDS problem with parallel batching machines, aiming to gain an effective integration between production and delivery activities for maximizing profits. To achieve this goal, two fast polynomial algorithms were improved, namely RKP and CMFFDA. Chevroton et al. [19] studied an IPDS problem within flow shop and vehicle routing environments. To handle it, they constructed a mixed integer linear programming model (MILP) and applied a two-step method to effectively reduce the combined expenses of inventory, routing, and penalties. Yagmur and Kesen [20] addressed an IPDS problem incorporating flow shop and multi-trip vehicle routing sub-problems. They used memetic and simulated annealing algorithms to optimize the sum of total tour time and total tardiness. Luo et al. [21] presented an IPDS problem with an inventory phase, aiming at minimizing total energy consumption and total earliness/tardiness penalty costs. In this problem, the production and delivery tasks were respectively carried out by homogeneous flow shops and third-party logistics providers. To address it, they formulated an MILP model and developed a modified NSGA-II as a way. Mohammadi et al. [3] put forward an IPDS problem for a furniture manufacturing enterprise, where both an FJS environment and a vehicle routing problem were considered simultaneously. To settle it, they provided an MILP model and introduced a hybrid particle swarm optimization solver to minimize total cost and total weighted delivery earliness and tardiness. Su et al. [22] presented an IPDS problem within an open shop system, in which an MILP model was developed to minimize makespan and a group teaching optimization was introduced as an optimizer.

In addition to the above studies involving a single-factory manufacturing system, several studies focusing on distributed manufacturing systems were also investigated. Fu et al. [9] addressed an IPDS with distributed flow shops to minimize makespan. They modeled the problem as an MILP and designed a black widow optimization approach as a basic solver. Furthermore, they extended a multi-objective variant, in which an MBSO was given to optimize total weighted earliness/tardiness as well as total quality cost [23]. Qin et al. [24] discussed an IPDS problem within a distributed hybrid flow shop, aiming to reduce total delivery, earliness and tardiness costs. To handle it, they developed an adaptive human-learning based genetic algorithm. Moreover, an integrated DFJS and distribution problem was initially investigated by Zhang et al. [10], in which they modeled an MILP and introduced a cooperative evolutionary algorithm to minimize makespan.

From the above discussions, it is evident that most studies on IPDS mainly focus on single-factory manufacturing environments such as parallel machines, flow shops, job shops, and open shops, with limited work on distributed manufacturing systems and none on the DFJS with multi-objectives.

2.2 Relevant literature on DFJS

The DFJS problems integrate the characteristics of both distributed job shops and FJSs [25]. In the past years, a wide range of studies have addressed these problems with various constraints, employing numerous efficient algorithms to solve them. Meng et al. [26] studied a DFJS problem considering energy consumption. To cope with this challenge, an MILP model was formulated to minimize total energy consumption and a novel hybrid shuffled frog leaping algorithm was introduced via analyzing the NP-hardness features of the problem. Yu et al. [27] also explored an energy-efficient DFJS problem, focusing on reducing both makespan and total energy consumption. To tackle it, a knowledge-guided bi-population evolutionary algorithm was constructed. Tang et al. [28] handled a DFJS problem considering both the serial, discrete, and hybrid operation constraints. They used CPLEX for an MILP model and designed a memetic algorithm to reach a minimization of total energy consumption and makespan. Xu et al. [29] investigated a DFJS problem with operation outsourcing restraints, where four objectives were concurrently optimized, consisting of makespan, quality, total carbon emission, and total costs. To address this problem effectively, they developed a hybrid algorithm with genetic and tabu search features. Sang et al. [30] concentrated on a multi-objective DFJS problem while optimizing economic and green indicators. A high-dimensional memetic algorithm was given as a solution. Wei et al. [31] considered a shared manufacturing-based DFJS problem involving the supply-demand matching, and they proposed a hybrid algorithm to minimize total costs and makespan, including estimation of distribution and tabu search approaches. An analysis of the above work shows that prior studies concentrated on optimizing time, cost, and energy consumption objectives, while considering production and assembly constraints. However, studies on the integration of the DFJS and a distribution stage are limited. As a result, this paper seeks to explore these under-researched areas.

2.3 Relevant literature on BSO

BSO is a swarm intelligence algorithm that has proven its efficiency in settling various optimization problems, especially in addressing shop scheduling and vehicle routing problems [[32], [33], [34]]. Alzaqebah et al. [35] created a hybrid BSO framework to solve an FJS problem, where a new updating strategy and three novel neighborhood operators were implemented to enhance BSO's performance to explore the solution space. Hao et al. [36] extended a hybrid BSO approach for a distributed hybrid flow shop problem, in which a novel technique of computing the distance during the clustering phase was provided. Zhao et al. [37] presented a reinforcement learning-BSO algorithm to handle a distributed assembly no-wait flow shop problem, where a learning mechanism based on the clustering strategy was introduced. Li et al. [38] raised an enhanced BSO algorithm to figure out a fuzzy distributed hybrid flow shop problem, in which a novel constructive heuristic approach and several local search heuristics were considered. Ke [39] proposed an enhanced BSO method to address a cumulative capacitated vehicle routing problem. According to the features of the studied problem, new convergent and divergent strategies were designed. Ma et al. [34] applied an MBSO method to a home health care scheduling and routing problem. Besides, Hou et al. [23] provided a multi-objective framework of BSO to tackle an integrated problem of distributed flow shops and vehicle routings.

Overall, many studies have demonstrated the effectiveness of BSO in settling shop scheduling and vehicle routing optimization problems. Nevertheless, BSO has not been considered to address integrated DFJS and vehicle routing optimization problems. Thus, this work introduces an MBSO to fill this gap.

3 Proposed problem and model

3.1 Problem statement

The examined problem encompasses two phases, i.e., production and distribution. In the production phase, multiple factories participate in performing the production task, each functioning as an FJS equipped with a collection of versatile machines. A group of personalized furniture orders from different customers must be assigned among these factories for manufacturing. Each order includes multiple operations, e.g., board cutting, edge banding, and drilling, and has its own production path. Each operation is carried out by selecting a machine from a predefined subset of available machines. Hence, the production phase can be abstracted as a DFJS scheduling problem, which involves three determinations: assigning jobs among factories, deciding the operation sequence, and allocating machines for operations. After the orders are completed, they are promptly delivered by a fleet of vehicles to their customers in various locations. Since each order has a due date limit, tardiness will be incurred if the delivery is late. Consequently, the distribution phase can be simplified to a multi-depot vehicle routing problem, including two crucial decisions: assigning jobs to vehicles and determining the job delivery route. In an effort to enhance production and distribution efficiency as well as boost customer satisfaction, the presented problem seeks to optimize makespan and total tardiness in the production and distribution phases, respectively.

To clarify the studied problem, a demonstrative example involving nine jobs and two factories is displayed in Fig. 2. Each factory incorporates three machines, and every job involves three operations. The notation “Oip” indicates the p-th operation of job i.Fig. 2 A demonstrative example of the investigated problem.

Fig. 2

To address our research, the following constraints are relevant: (a) All production factories, machines, and vehicles are available from the start; (b) Only one factory is assigned to every job; (c) A single operation is executed concurrently on each machine; (d) Only one machine is assigned to each operation; (e) No interruption is allowed; (f) Each customer is provided with the service only once; and (g) Each vehicle's capacity remains within acceptable limits.

3.2 Model establishment

To model the presented problem under investigation, we define the symbols used as follows, as displayed in Table 1.Table 1 The symbols for the investigated problem.

Table 1Symbol	Description	
N={0,1,2,…,n}	Set of jobs indexed by i,j, where 0 is a virtual job.	
F={n+1,n+2,…,n+f}	Set of factories indexed by g,l.	
M={1,2,…,m}	Set of machines indexed by k.	
O={1,2,…,o}	Set of operations indexed by p,q.	
V={1,2,…,v}	Set of vehicles indexed by h.	
pgkp	Production time of operation p on machine k at factory g.	
αgi	Transport time between factory g and customer i.	
βij	Transport time between customers i and j.	
ui	Unloading time at customer i.	
υ	Vehicle capacity.	
ωi	Weight of job i.	
ϑi	Due date for job i.	
εp	Job index denoting that operation p is an operation of job εp.	
λ	A sufficiently large number.	
χpq	= 1 if operation p is processed before q, 0 otherwise.	
ψgkp	= 1 if operation p is processed on machine k at factory g, 0 otherwise.	
xpqgk	= 1 if operation q is directly processed after p on machine k at factory g, 0 otherwise.	
ypg	= 1 if operation p is processed at factory g, 0 otherwise.	
zijgh	= 1 if job j is directly delivered after i by vehicle h at factory g, 0 otherwise.	
bp	Production beginning time of operation p.	
cp	Production completion time of operation p.	
ρi	Production completion time of job i.	
ηigh	Loading time of job i into vehicle h at factory g.	
dgh	Departure time of vehicle h from factory g.	
ai	Arrival time at customer i.	

Based on the above symbols, this work formulates a mathematical model below.(1) minf1=maxi∈N/{0}ρi.

(2) minf2=∑i∈N/{0}max{0,(ai−ϑi)}.

s.t.(3) ypg=∑q∈O∑k∈Mxpqgk,∀p∈O,p≠q,∀g∈F.

(4) ypg=yqg,∀p,q∈O′⊆O,O′={p′|p′∈O,εp′=i},∀i∈N,i≠0,∀g∈F.

(5) xpqgk≤ψgkp,∀p,q∈O,p≠q,∀g∈F,∀k∈M.

(6) bp+pgkp∙xpqgk–bq≤0,∀p,q∈O,p≠q,∀g∈F,∀k∈M.

(7) ∑p∈O∑k∈M∑g∈Fxpqgk=1,∀q∈O,p≠q.

(8) ∑q∈O∑k∈M∑g∈Fxpqgk=1,∀p∈O,p≠q.

(9) bp+pgkp∙xpq′gk–(1–χpq)∙λ–bq≤0,∀p,q,q′∈O,p≠q≠q′,∀g∈F,∀k∈M.

(10) bp+pgkp∙xpqgk≤cp,∀p,q∈O,p≠q,∀g∈F,∀k∈M.

(11) ρi≥cp,∀i∈N,i≠0,∀p∈O′⊆O,O′={p′|p′∈O,εp′=i}.

(12) ypg=∑j∈N∑h∈Vzijgh,∀i∈N,i≠0,∀p∈O′⊆O,O′={p′|p′∈O,εp′=i},∀g∈F.

(13) ∑j∈N/{0}z0jgh≤1,∀g∈F,∀h∈V.

(14) ∑j∈N∑g∈F∑h∈Vzijgh=1,∀i∈N,i≠j≠0.

(15) ηigh≥ρi–(1–∑j∈Nzijgh)∙λ,∀i∈N,i≠0,∀g∈F,∀h∈V.

(16) dgh≥ηigh,∀i∈N,i≠0,∀g∈F,∀h∈V.

(17) ai≥dgh+αgi–(1–z0igh)∙λ,∀i∈N,i≠0,∀g∈F,∀h∈V.

(18) aj≥ai+ui+βij–(1–∑g∈F∑h∈Vzijgh)∙λ,∀i,j∈N,i≠j≠0.

(19) ∑i∈N∑j∈N,i≠jzijgh∙ωi≤υ,∀g∈F,∀h∈V.

(20) xpqgk∈{0,1},∀p,q∈O,∀g∈F,∀k∈M.

(21) ypg∈{0,1},∀p∈O,∀g∈F.

(22) zijgh∈{0,1},∀i,j∈N,∀g∈F,∀h∈V.

(23) bp≥0,cp≥0,∀p∈O.

(24) ηigh≥0,dgh≥0,∀i∈N,i≠0,∀g∈F,∀h∈V.

(25) ρi≥0,ai≥0,∀i∈N,i≠0.

where (1) and (2) aim at minimizing makespan and total tardiness, respectively. (3) and (4) guarantee that all operations related to a particular job are processed within a single factory. (5) defines that no more than one machine can be allocated to an operation. (6) implies that a machine is engaged in only one operation at a time. (7) and (8) impose that each operation is limited to having an immediate predecessor and an immediate successor, respectively. (9) shows that all operations of a job follow a preset production route. (10) requires the operation's production completion time. (11) provides the job's production completion time. (12) ensures the continuity of production and distribution phases. (13) signifies that a vehicle is applied at most once. (14) imposes that a customer is served only once. (15) stipulates the job's loading time. (16) regulates the vehicle's departure time. (17) and (18) calculate the arrival time of vehicles at each customer. (19) restricts the vehicle's capacity. (20)–(25) provide the ranges of variables.

4 Presented algorithm

BSO has gained popularity due to its capacity to effectively work out optimization problems by emulating brainstorming processes of human beings [14]. It begins with an initial population of size rs, and then proceeds iteratively and sequentially through three operations, namely, clustering, generating, and selecting. Finally, the best solution found is output when the stop condition is reached. BSO was initially applied to tackle continuous optimization problems with one objective. Considering the discrete and multi-objective characteristics of the problem, an MBSO is designed by employing certain specialized strategies. The components of MBSO are detailed below.

4.1 Solution representation

To express a solution, a three-dimensional vector [FA;OS;MA] is adopted, where symbols FA, OS, and MA mean factory assignment, operation sequence, and machine allocation vectors, respectively. The FA vector is represented by an integer string, i.e., π=(π1,π2,…,πn), where each integer πi is a factory index indicating the factory allocation of the job i, i∈{1,2,‥,n}. The OS vector gives a scheduling sequence of all operations, indicated by an integer string ϖ=(ϖ1,ϖ2,…,ϖo), where each integer ϖp is a job index, p∈{1,2,‥,o}, and the p-th occurrence of a job index indicates its p-th operation. The MA vector comprises a sequence of integers, and each integer is a machine index assigned to an operation, identified by its position. To clarify this method, an example including three jobs with nine operations that need to be processed on two factories and four machines is illustrated in Fig. 3. It is evident that factory 1 takes on jobs 1 and 3, while factory 2 handles job 2. The scheduling sequence of operations is O21→O22→O11→O31→O12→O23→O32→O13→O33. The machine allocations of the three operations for job 1 are machines 1, 2, and 1, respectively, and so on.Fig. 3 Solution representation.

Fig. 3

To optimize the objectives of minimizing makespan and total tardiness, a left-shift strategy [40] is applied to decode a solution, which only obtains production decisions. Accordingly, this work proposes two heuristic rules for obtaining distribution decisions, which are detailed as follows:(a) Priority is given to jobs that complete processing first, and they are assigned in sequence to a vehicle. The vehicle's maximum capacity must not be exceeded by the total weight of assigned jobs.

(b) The earlier the due date of the job, the higher its priority for delivery.

4.2 Population generation

To enhance the search performance of meta-heuristics, it is necessary to create an initial population that exhibits higher quality and better diversity. To this end, we first construct two solutions using a hybrid initialization method that combines randomness and heuristics. Then, a randomized approach is applied to produce the rest solutions. To be specific, FA and OS vectors of the first two initial solutions are formed randomly, whereas their MA vectors are created by using the global minimum processing time and the operation minimum processing time heuristics [41], respectively. These two heuristics aim to pick the machine with minimum processing time and the machine with the lowest overall workload for each operation, respectively, from their respective available machine sets. For the rest initial solutions, their factory, operation, and machine vectors are randomly produced.

4.3 Clustering

BSO typically segments the population into a predefined cluster quantity using the k-means clustering technique [42]. Nevertheless, the selection of algorithm parameters plays a significant role in search performance, and identifying an ideal number of clusters for a particular problem can be a labor-intensive process. Thus, this work employs an adaptive clustering strategy inspired by non-dominated sorting ideas [43]. The population is sorted following the dominance relationships of solutions, in which each solution's sort value corresponds to its non-dominated level. In this research, all solutions possessing the same sort value are grouped into a cluster in MBSO. Therefore, the population is successfully partitioned into multiple clusters. To comprehensively display the clustering strategy, Fig. 4 provides a diagram where 18 solutions are grouped into three clusters.Fig. 4 A demonstrative example of the clustering strategy.

Fig. 4

4.4 Generating

In the generation process, one or two center or normal solution(s) are adopted for creating new solutions, involving three parameters, i.e., rg, ro, and rt. The symbol rg figures out if the new solution is constructed by one solution or two solutions. Symbols ro and rt identify if the new solution is created by the center or normal solution(s). Given that all solutions in a cluster are non-dominated, we randomly select one solution from each cluster as its center solution, while the rest are considered as its normal ones.

When employing a randomly chosen center or normal solution Xold to create a new solution Xnew, the generation process of Xnew is described as follows:

In the FA vector of Xold, two randomly selected different factory assignments are interchanged. In the OS vector of Xold, the sequence of elements is inverted between a pair of randomly picked indices. In the MA vector of Xold, each operation's machine is reallocated at random. Therefore, Xnew is successfully achieved. In order to vividly present this generation process, Fig. 5 shows a schematic diagram.Fig. 5 An example of the generation process when using one solution.

Fig. 5

When employing two randomly selected center or normal solutions Xold1, Xold2 to create new solutions, the generation process is introduced below.(a) A two-point crossover operator is applied to FA vectors of Xold1 and Xold2. Firstly, two different points p1 and p2 are selected at random. Secondly, the segments between these two points of Xold1 and Xold2 are swapped. Finally, the FA vectors of two new solutions Xnew1 and Xnew2 are obtained.

(b) An order-based crossover operator is executed on OS vectors of Xold1 and Xold2. Firstly, two distinct points p3 and p4 are randomly given. Secondly, the segments between the two points of Xold1 and Xold2 are replicated and inserted into the same positions in Xnew1 and Xnew2. Thirdly, the absent elements in Xnew1 and Xnew2 are appended according to the order in which they appear in Xold2 and Xold1, respectively. Finally, the OS vectors of Xnew1 and Xnew2 can be derived.

(c) A uniform crossover operator is conducted on the MA vector of Xold1 and Xold2. Firstly, a binary string is uniformly produced with the same length as the MA vector. Secondly, the elements at the positions with bit 1 in Xold1 and the elements at the positions with bit 0 in Xold2 are inherited to Xnew1. Thirdly, the elements at the positions with bit 0 in Xold1 and the elements at the positions with bit 1 in Xold2 are added to Xnew2. Finally, MA vectors of Xnew1 and Xnew2 can be achieved.

Note that the better one between Xnew1 and Xnew2 is regarded as the Xnew, and a random one is chosen when they are non-dominated. To clearly display the above generation process, Fig. 6 presents a schematic diagram.Fig. 6 An example of the generation process when using two solutions.

Fig. 6

4.5 Selecting

Upon the conclusion of the generation phase, a collection of rs new solutions are merged with solutions in the existing population. To identify the best solutions among this merged pool, the sorting and crowding distance techniques [43] are applied. rs better solutions, with higher sort and crowding distance values, are selected as the next population, enabling the algorithm to progress towards better solutions in a more efficient way and ultimately converge on the optimal solution.

4.6 Framework of MBSO

Based on the above design, an MBSO framework is given in Algorithm 1, and its relevant flowchart is presented in Fig. 7.Algorithm 1: MBSO.	
Input: Parameters rs, rg, ro, rt, and a stop criterion.	
Output: All nondominated solutions.	
Begin	
 Initialize rs solutions, and evaluate them.	
 while (a stop criterion has not been met) do	
 Carry out the clustering operation.
 the clustering operation.	
 for (each solution in the population)//Execute the generating operation.	
 if (rand(0,1)<rg) then	
 if (rand(0,1)<ro) then	
 Create a new solution through applying a center solution.	
 else	
 Create a new solution by employing a normal solution	
 end if	
 else	
 if (rand(0,1)<rt) then	
 Construct a new solution via adopting two center solutions.	
 else	
 Construct a new solution through using two normal solutions.	
 end if	
 end if	
 end for	
 Perform the selection strategy.	
 end while	
End.	

Fig. 7 The flowchart of MBSO.

Fig. 7

4.7 Computational complexity of MBSO

The complexity of the computational process in our algorithm is influenced by initialization, sorting, crossover, mutation, and crowding distance operations. For the convenience of computational complexity analysis, this work utilizes L to indicate the size of a solution. A hybrid initialization method is applied to create an initial population, and the computational complexity of this process is O(0.1∙rs∙o∙m). To cluster the population, a non-dominated sorting method is used and its computational complexity is O(rs2). To create new solutions, crossover and mutation methods are applied and their computational complexity is O(rs∙L2) and O(rs∙L), respectively. Furthermore, the computational complexity of the selecting operation is O(2∙rs2) by conducting sorting and crowding distance methods. Combined with the above analysis, the overall computational complexity of MBSO under the worst condition is O(rs∙(L2+L+0.1∙o∙m)+3∙rs2).

5 Numerical experiments

In this section, numerical experiments with a variety of test instances are conducted to assess the performance of MBSO in addressing the problem. Three cutting-edge peers comprise the comparison pool, i.e., the memetic algorithm (MA) [44], nondominated sorting genetic algorithm II (NSGA-II) [43], and discrete Jaya algorithm (DJA) [41]. All algorithms are coded in C++ and executed on a computer outfitted with an Intel Core i5-7500 CPU (3.40 GHz/12 GB RAM).

5.1 Test instances construction

This research tackles an integrated DFJS and distribution scheduling problem with minimizing makespan and total tardiness. As a matter of fact, this work was inspired by a real-world application at a furniture company, and we have not found a similar research problem in the current body of literature. Therefore, no available test instances are directly suitable for experiments. To this end, this work develops new test instances by conducting field investigation and inspection at an actual company (i.e., Guangzhou Yuanfang Computer Software Engineering Co., Ltd.), which provided us with real data for experiments. To ensure data protection and compliance with laws and regulations, we have collaborated with company workers to carefully desensitize the first-hand data. Therefore, we have randomly generated 18 test instances based on the actual operating data, containing 18 combinations of f×m×n, i.e., f∈{2,3,4}, m∈{5,10}, and n∈{20,40,60}. For each test instance, the parameter data is randomly generated from the specified intervals as follows: pgkp∈[75,95], αgi∈[10,100], βij∈[10,100], ui∈[10,50], ωi∈[10,60], and δi∈[200,1800]. In addition, the vehicle capacity υ is fixed at 100 and each job involves three operations.

5.2 Performance criteria

This work adopts three performance criteria, namely C-, IGD-, and HV-metrics [[45], [46], [47], [48]], to assess the excellence of the non-dominated solution sets obtained.(a) The formula of the C-metric is defined as follows:

(26) C(X,Y)=|{y∈Y|∃x∈X:x≺y}||Y|.

where X and Y indicate non-dominated solution sets of two algorithms, respectively. C(X,Y) represents the proportion of solutions in Y that are dominated by at least one solution from X, and |Y| implies the size of Y. C(X,Y)=1 reveals that all solutions within Y are dominated by those in X, while C(X,Y)=0 suggests that no solution in Y is outperformed by solutions in X. Thus, the smaller C(X,Y), the better Y.(b) The formula of the IGD-metric is provided as follows:

(27) IGD(U*,U)=1|U*|∑u∈U*d(u,U).

where U* means the true pareto front and U indicates a solution set obtained by a specific algorithm. Due to U* being unknown in real life, this work considers it as the nondominated subset of all acquired nondominated solution sets. d(u,U) means the Euclidean distance from a solution u in U* to the nearest solution in U. In this study, all the objective function values are scaled to fit within [0,1]. Typically, the algorithm performs better as the IGD-metric value decreases.(c) The formula of the HV-metric is given as follows:

(28) HV=δ(∪i=1|T|ti).

where δ is the Lebesgue measure, and ti expresses the super volume formed by the i-th solution in a solution set T and the reference point (1,1). The objective function values of all algorithms are normalized within the range of [0,1]. Generally, higher HV-metric values indicate better performance.

5.3 Parameters setting

This study uses an orthogonal experimental method [49,50] to determine the parameters of MBSO, MA, and NSGA-II. A medium-scale instance 3×10×40 including 3 factories, 10 machines, and 40 jobs is employed. The parameter settings of these three algorithms refer to Table 2 for details. In this study, 16 parameter combinations are devised, and each algorithm is applied to handle the selected instance 20 times based on these combinations. The fitness evaluation number reaching 80×n×m is established as the termination criterion. This work employs the IGD-metric to examine results, and computes the average IGD value (AIGD) over 20 runs. Table 3 lists the experimental results, along with the recommended parameters for each algorithm, which are given below. For the MBSO, its parameters are displayed as: rs=80, rg=0.40, ro=0.20, and rt=0.80. For the MA, its parameters are given as: pa=80, pb=2, pc=7, and pd=0.15. For the NSGA-II, the population size, the crossover probability, and the mutation probability are 100, 0.80, and 0.30, respectively. Furthermore, this work sets the population size of DJA as 50 by referring to the results proposed by Gao et al. (2018) because it only requires one parameter.Table 2 Parameter settings of the three algorithms.

Table 2Algorithm	Parameter	Level	
MBSO	rs	60, 80, 100, 120	
rg	0.20, 0.40, 0.60, 0.80	
ro	0.20, 0.40, 0.60, 0.80	
rt	0.20, 0.40, 0.60, 0.80	
MA	population size pa	60, 80, 100, 120	
the number of interval iterations pb	2, 4, 6, 8	
the number of jobs for destruction pc	1, 3, 5, 7	
mutation probability pd	0.10, 0.15, 0.20, 0.25	
NSGA-II	population size pe	60, 80, 100, 120	
crossover probability pf	0.80, 0.85, 0.90, 0.95	
mutation probability pg	0.10, 0.20, 0.30, 0.40	

Table 3 Orthogonal matrix and experimental outcomes from three algorithms.

Table 3No.	MBSO	MA	NSGA-II	
rs	rg	ro	rt	AIGD	pa	pb	pc	pd	AIGD	pe	pf	pg	AIGD	
1	60	0.20	0.20	0.20	0.7852	60	2	1	0.05	0.7618	60	0.80	0.10	0.8594	
2	60	0.40	0.40	0.40	0.9552	60	4	3	0.10	0.9320	60	0.85	0.20	0.8454	
3	60	0.60	0.60	0.60	0.6811	60	6	5	0.15	0.7780	60	0.90	0.30	0.9406	
4	60	0.80	0.80	0.80	0.7359	60	8	7	0.20	0.8055	60	0.95	0.40	0.9844	
5	80	0.20	0.40	0.60	0.7297	80	6	3	0.05	0.9513	80	0.80	0.20	0.8428	
6	80	0.40	0.20	0.80	0.6245	80	8	1	0.10	0.9106	80	0.85	0.10	0.9673	
7	80	0.60	0.80	0.20	0.8783	80	2	7	0.15	0.7374	80	0.90	0.40	0.9069	
8	80	0.80	0.60	0.40	0.7296	80	4	5	0.20	0.8746	80	0.95	0.30	0.8401	
9	100	0.20	0.60	0.80	0.7134	100	8	5	0.05	0.9590	100	0.80	0.30	0.7375	
10	100	0.40	0.80	0.60	0.7221	100	6	7	0.10	0.8499	100	0.85	0.40	0.7871	
11	100	0.60	0.20	0.40	0.8808	100	4	1	0.15	0.9446	100	0.90	0.10	0.9440	
12	100	0.80	0.40	0.20	0.6878	100	2	3	0.20	0.9193	100	0.95	0.20	0.8358	
13	120	0.20	0.80	0.40	0.7990	120	4	7	0.05	0.9744	120	0.80	0.40	0.8861	
14	120	0.40	0.60	0.20	0.7765	120	2	5	0.10	0.8789	120	0.85	0.30	0.9908	
15	120	0.60	0.40	0.80	0.9327	120	8	3	0.15	0.8385	120	0.90	0.20	0.9947	
16	120	0.80	0.20	0.60	0.7508	120	6	1	0.20	0.7616	120	0.95	0.10	0.9310	

5.4 Results and discussions

In this segment, comparison experiments among MBSO, MA, NSGA-II, and DJA are executed. MBSO and its rivals terminate once 80×n×m fitness evaluations are conducted. Table 4, Table 5, Table 6 list the mean values of C-, IGD-, and HV-metrics over 20 times.Table 4 Comparison outcomes of the four algorithms concerning the C-metric.

Table 4Instance	C (BSO,MA)	C (MA,BSO)	C (BSO,GA)	C (GA,BSO)	C (BSO,DJA)	C (DJA,BSO)	
2 × 5 × 20	0.9500	0.0000	1.0000	0.0000	1.0000	0.0000	
2 × 5 × 40	1.0000	0.0000	1.0000	0.0000	1.0000	0.0000	
2 × 5 × 60	1.0000	0.0000	1.0000	0.0000	1.0000	0.0000	
2 × 10 × 20	0.9500	0.0500	0.9500	0.0250	1.0000	0.0000	
2 × 10 × 40	0.9917	0.0000	1.0000	0.0000	1.0000	0.0000	
2 × 10 × 60	1.0000	0.0000	1.0000	0.0000	1.0000	0.0000	
3 × 5 × 20	0.9750	0.0000	1.0000	0.0000	1.0000	0.0000	
3 × 5 × 40	1.0000	0.0000	1.0000	0.0000	1.0000	0.0000	
3 × 5 × 60	1.0000	0.0000	1.0000	0.0000	1.0000	0.0000	
3 × 10 × 20	0.9500	0.0500	0.9417	0.0167	1.0000	0.0000	
3 × 10 × 40	1.0000	0.0000	0.9500	0.0125	1.0000	0.0000	
3 × 10 × 60	0.9500	0.0500	1.0000	0.0000	1.0000	0.0000	
4 × 5 × 20	1.0000	0.0000	1.0000	0.0000	1.0000	0.0000	
4 × 5 × 40	1.0000	0.0000	1.0000	0.0000	1.0000	0.0000	
4 × 5 × 60	1.0000	0.0000	1.0000	0.0000	1.0000	0.0000	
4 × 10 × 20	1.0000	0.0000	0.7750	0.1083	1.0000	0.0000	
4 × 10 × 40	0.9750	0.0167	1.0000	0.0000	1.0000	0.0000	
4 × 10 × 60	1.0000	0.0000	1.0000	0.0000	1.0000	0.0000	
Average	0.9857	0.0093	0.9787	0.0090	1.0000	0.0000	

Table 5 Comparison results of the four algorithms about the IGD-metric.

Table 5Instance	MBSO	MA	NSGA-II	DJA	
2 × 5 × 20	0.1905	0.6184	0.7447	0.9516	
2 × 5 × 40	0.1551	0.7695	0.8824	1.0497	
2 × 5 × 60	0.1055	0.7576	0.6637	1.1504	
2 × 10 × 20	0.2831	0.7237	0.7706	1.1927	
2 × 10 × 40	0.2222	0.8561	0.7216	1.3201	
2 × 10 × 60	0.1289	0.8102	0.5688	1.1544	
3 × 5 × 20	0.2402	0.6985	1.0227	1.0727	
3 × 5 × 40	0.1408	0.8337	0.9138	1.1159	
3 × 5 × 60	0.1308	0.7490	0.7618	1.1382	
3 × 10 × 20	0.2997	0.6525	0.8028	1.1545	
3 × 10 × 40	0.1558	0.7500	0.5458	1.1593	
3 × 10 × 60	0.0606	0.5777	0.4940	1.1084	
4 × 5 × 20	0.2727	0.9444	1.0593	1.1573	
4 × 5 × 40	0.1268	0.7159	0.9414	1.0689	
4 × 5 × 60	0.1347	0.8363	0.6389	1.1563	
4 × 10 × 20	0.2854	1.0406	0.5119	1.1090	
4 × 10 × 40	0.1425	0.6874	0.5217	1.2752	
4 × 10 × 60	0.1956	0.7124	0.5255	1.0997	
Average	0.1817	0.7630	0.7273	1.1352	

Table 6 Comparison results of the four algorithms about the HV-metric.

Table 6Instance	MBSO	MA	NSGA-II	DJA	
2 × 5 × 20	0.9615	0.3193	0.1888	0.0650	
2 × 5 × 40	0.9788	0.1915	0.1253	0.0186	
2 × 5 × 60	0.9909	0.2101	0.2663	0.0067	
2 × 10 × 20	0.9710	0.4140	0.3728	0.0354	
2 × 10 × 40	0.9844	0.2539	0.3501	0.0007	
2 × 10 × 60	0.9970	0.2277	0.4374	0.0234	
3 × 5 × 20	0.9731	0.3450	0.0842	0.0409	
3 × 5 × 40	0.9898	0.1637	0.1174	0.0212	
3 × 5 × 60	0.9815	0.2054	0.1786	0.0048	
3 × 10 × 20	0.9826	0.4560	0.3087	0.0263	
3 × 10 × 40	0.9872	0.2175	0.3649	0.0015	
3 × 10 × 60	0.9991	0.4635	0.5432	0.0593	
4 × 5 × 20	0.9786	0.1994	0.0913	0.0328	
4 × 5 × 40	0.9936	0.2495	0.1017	0.0370	
4 × 5 × 60	0.9964	0.2477	0.4027	0.0335	
4 × 10 × 20	0.8800	0.0763	0.5365	0.0365	
4 × 10 × 40	0.9908	0.3578	0.4900	0.0025	
4 × 10 × 60	0.9993	0.3895	0.5684	0.0576	
Average	0.9798	0.2771	0.3071	0.0280	

The comparison results of four algorithms concerning the C-metric are provided in Table 4, where the symbols ‘BSO’ and ‘GA’ are respectively the MBSO and NSGA-II. When comparing MBSO with its peers, it is seen that for all instances, C(BSO,W)>C(W,BSO), where W∈{MA,GA,DJA}. The average mean values of C(BSO,W) for W∈{MA,GA,DJA} are 0.9857, 0.9787, and 1.000, respectively, which are larger than those of C(W,BSO), i.e., 0.0093, 0.0090, and 0.0000. Hence, we declare that MBSO is superior to its peer algorithms.

Table 5 showcases the comparative outcomes of four algorithms about the IGD-metric are given. It can be observed that MBSO outperforms MA, NSGA-II, and DJA on all instances utilized. Moreover, the average mean values of MBSO, MA, NSGA-II, and DJA are 0.1817, 0.7630, 0.7273, and 1.1352, respectively. It's clear that MBSO has a higher average mean value than the other algorithms. In light of the acquired findings and analysis, we can deduce that MBSO is a superior solver for the examined problem.

Table 6 reports the comparison results of four algorithms about the HV-metric. We observe that MBSO wins MA, NSGA-II, and DJA on all the employed instances. In addition, the average mean values of MBSO, MA, NSGA-II, and DJA are 0.9798, 0.2771, 0.3071, and 0.0280, respectively. Clearly, MBSO has a bigger average mean value than those of its peers. Accordingly, based on the previous results and analysis, our assertion is that MBSO proves to be an exceptional optimizer.

Furthermore, six box plots concerning the IGD- and HV-metrics are respectively drawn in Fig. 8, Fig. 9 to visually show the performance of all employed optimizers. As depicted in Fig. 8, MBSO exhibits a superior median value compared to other techniques. Similarly, the median value of MBSO in Fig. 9 is significantly better than that of the opponents. Based on the above findings, it is evident that MBSO stands out as superior in providing stable and centralized results compared to the other methods. Consequently, we acknowledge that MBSO can achieve superior performance over its rivals when addressing the given problem.Fig. 8 Box plots about the IGD-metric for the four algorithms, (a) 2×5×20, (b) 3×10×40, (c) 4×5×60, (d) 2×10×60, (e) 3×5×40, and (f) 4×10×20.

Fig. 8

Fig. 9 Box plots about the HV-metric for the four algorithms, (a) 2×5×20, (b) 3×10×40, (c) 4×5×60, (d) 2×10×60, (e) 3×5×40, and (f) 4×10×20.

Fig. 9

In order to assess how the problem scale affects the achievement of objectives, this study categorizes all cases into three groups according to the job count, specifically 30, 60, and 90 jobs, respectively. For every group, the AIGD and average mean HV (AHV) values are computed, as demonstrated in Table 7. Based on these data, the percentage improvement achieved by MBSO (M) in comparison to its one rival (N) regarding the IGD- and HV-metrics is calculated separately in Eqs. (29), (30). Table 7 presents the improvement results. It is evident that MBSO obtains the minimum AIGD and the maximum AHV values in each group, and its IIGD and IHV values are greater than 0. Hence, we conclude that MBSO outperforms its opponents. Besides, the performance improvement of MBSO over other methods concerning the IGD-metric becomes more significant as the problem scale increases. Although the IHV results of MBSO increase faster on small-scale instances and slightly slower on large-scale instances, they are still greater than 0, indicating that MBSO has been improving on both objectives. Therefore, MBSO is a viable solution, particularly effective in tackling large-scale problems.(29) IIGD=((AIGD(N)−AIGD(M))/AIGD(M))×100%.

(30) IHV=((AHV(M)−AHV(N))/AHV(M))×100%.

Table 7 Results of MBSO and its three peers for all the groups.

Table 7Metric	n	Algorithms	
MBSO	MA	NSGA-II	DJA	
AIGD	20	0.2619	0.7797	0.8187	1.1063	
40	0.1572	0.7688	0.7545	1.1649	
60	0.1260	0.7405	0.6088	1.1346	
AHV	20	0.9578	0.3017	0.2637	0.0395	
40	0.9874	0.2390	0.2582	0.0136	
60	0.9940	0.2907	0.3994	0.0309	
IIGD	20	MBSO vs.	1.9766	2.1255	3.2236	
40	3.8904	3.7993	6.4100	
60	4.8765	3.8310	8.0033	
IHV	20	MBSO vs.	0.6850	0.7247	0.9588	
40	0.7580	0.7385	0.9862	
60	0.7076	0.5982	0.9689	

In order to analyze the convergence performance of MBSO, this work transforms the convergence of MBSO into the convergence of a convergence metric sequence. In MBSO, the archive A(t) at the t-th iteration stores the non-dominated solutions found in the evolutionary process. This work sets {A(t);t=1,2,…} as the archive sequence. Considering the random nature of MBSO, the archive sequence is seen as a stochastic process. Thus, its convergence to the Pareto optimal front A*, plays a crucial role in determining the convergence of MBSO. Since {A(t);t=1,2,…} and A* are sets, the convergence of MBSO relates to set sequence convergence, which is difficult to analyze directly. Therefore, this work proposes a convergence metric based on the IGD-metric value between A(t) and A*, as shown in Eq. (31). Based on the archive updating mechanism in MBSO, A(t) gradually approaches A* as the iteration process proceeds. Usually, a smaller IGD-metric value means that A(t) is closer to A*. Hence, the value of CM(A(t)) should be decreasing. In this context, MBSO's convergence is converted into the convergence of {CM(A(t));t=1,2,…}. Specifically, MBSO converges to A* if limt→∞CM(A(t))=0.(31) CM(A(t))=IGD(A*,A(t)).

To prove it, we randomly collected the archive of four test instances with different scales during their iteration processes, which are 2×5×20, 3×10×40, 4×5×60, and 2×10×60. Fig. 10 intuitively exhibits the convergence curves of the CM values on these instances. The x-axis denotes the iteration count, and the y-axis shows the CM value. Note that it is difficult to obtain A* in reality. We set it as A(max), where max is the maximum number of iterations. From Fig. 10, it can be observed that MBSO decreases during iteration until it converges to 0, illustrating that global exploration and local exploitation abilities are well balanced. According to the above analysis, we can state that MBSO is effective and efficient in addressing the investigated problem.Fig. 10 Convergence curves on four instances with different scales., (a) 2×5×20, (b) 3×10×40, (c) 4×5×60, and (d) 2×10×60.

Fig. 10

To identify significant differences among all algorithms and enhance the credibility of the results, Friedman-test [51] and Wilcoxon-test [52] are applied to the HV-metric results with a 95 % confidence level. In the Friedman-test, the average ranking values of MBSO, MA, NSGA-II, and DJA are 1.0000, 2.5000, 2.5000, and 4.0000, respectively. The calculated test statistic value TF equals 153 that exceeds the critical value of 2.76. Hence, there is a significant difference in performance among all algorithms. In the Wilcoxon-test, the R+ values are 171, which are larger than the R− values of 0 when comparing MBSO with another method. The p-values are equal to 0.0001, which are smaller than 0.05. Thus, MBSO significantly outperforms the other three methods.

6 Conclusion

This work focuses on how to schedule a DFJS and a vehicle routing problem in an integrated way while satisfying makespan and total tardiness minimization requirements. To tackle this problem, a mixed-integer mathematical programming model is established and an MBSO is provided. In MBSO, the solution representation method, clustering, generating, and selection approaches are developed by considering the characteristics of the problem. Through performing numerical experiments between MBSO and MA, NSGA-II, DJA, we verify that MBSO outperforms its peers when tackling the problem being studied. Additionally, the significance of MBSO on different scale problems and its convergence performance have been analyzed. The results reveal that MBSO can help managers tackle large-scale practical problems.

In terms of future research, there are several potential directions that can be pursued. Firstly, the problem definition can consider integrated production-inventory-distribution scheduling problems, stochastic models, and energy-efficient objectives [53,54]. Secondly, the algorithm design can concentrate on exploring solution mechanisms that incorporate machine learning [55]. Finally, new models can be formulated based on more practical scenarios.

Data availability statement

Data can be available upon reasonable request.

CRediT authorship contribution statement

Yanhe Jia: Writing – original draft, Formal analysis, Conceptualization. Yaoyao Zhou: Writing – original draft, Methodology, Formal analysis. Yaping Fu: Writing – review & editing, Supervision, Investigation.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work is supported by the 10.13039/501100012166 National Key Research and Development Program of China under Grant No. 2021YFF0901300 .

1 https://www.askci.com.

2 https://www.sohu.com/a/636825732_121641637.
==== Refs
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