An Improved Variable Neighborhood Search for the Reconfigurable Assembly Line Reconfiguring Problem

Abstract

Throughout the past several decades, the manufacturing industry has been confronted with rapidly evolving market demands. The Reconfigurable Assembly Line (RAL) changes the quantity and variety of products produced through reconfiguration at different stages. The reconfiguration stages of RAL include the reallocation of operations and resources. However, existing research has neglected the resources. In the paper, a mixed integer linear program for the Reconfigurable Assembly Line Reconfiguring Problem is first formulated. Subsequently, an efficient data structure is developed to model the process precedence constraint and alternative resources. And an improved Variable Neighborhood Search which includes new problem-specific neighborhood structures and local search for resource selection is proposed to solve the problem. Computational results demonstrate the contribution of the proposed data structure and the improved procedure. Furthermore, the experiments verify the superiority and stability of the proposed method compared with other heuristic algorithms.

1. Introduction

Throughout the past several decades, the manufacturing industry has been confronted with an increasing variety of products and the consequent reduction in production volumes, together with the continuous shortening of products’ life cycle [1]. In this context, the design of manufacturing systems becomes a complex engineering task that entails manufacturing strategy decisions, has long-term impacts, and involves a major commitment of financial resources [2]. Hence, Koren et al. [3] proposed the solution of the Reconfigurable Manufacturing System (RMS). In terms of response functionality, the RMS can change the variety and quantity of manufactured products through phased reconfiguration; in terms of response time, since the RMS is initially designed and developed for a product family, when there is a need to manufacture a new product within the family, the RMS can undergo less reconfiguration, which greatly reduces the time needed to reconfigure the manufacturing system.

Assembly accounts for 40–60% of the total manufacturing time and over 30% of the total manufacturing cost of a product, making the Reconfigurable Assembly System (RAS) an important component of the RMS. Since the majority of RASs in the discrete assembly industry adopt the line layout, the design of the Reconfigurable Assembly Line (RAL) is the key [4].

Past researchers have conducted extensive studies on the initial design of the RAL. However, research on the reconfiguration phase of the RAL is limited, with most literature focusing on the Assembly Line Rebalancing Problem (ALRBP), which reallocates operations without considering the selection of specific manufacturing resources. These findings are challenging to apply in engineering practice because they do not account for actual resource selection. Moreover, the existing methods use matrix encoding, which is not suitable for modeling complex process information that includes resources. In fact, in the design of assembly line in the discrete assembly industry, most operations require selecting a resource from a specific set of optional resources, while fewer operations can select flexible resource such as robots. Therefore, there is a need to optimize the utilization of both the resources from old assembly line and new resources based on the allocation of new product operations to stations. This challenge is defined as the Reconfigurable Assembly Line Reconfiguring Problem (RALRP).

This paper stems from the research project “Configuration Design and Reconfiguration Optimization of RAL”, which divides the design of RAL into three stages: (1) product family formation for RALs, (2) configuration design of RALs, and (3) reconfiguration optimization of RALs. Firstly, various products are classified into product families based on co-line production, allowing a RAL to offer production flexibility for each category of product family. Secondly, the initial design of a RAL is based on the existing product varieties within a product family, ensuring capacity and a certain level of reconfigurability. Finally, when products within the product family are updated, the RAL is reconfigured and optimized to meet new operation requirements. This study aimed to solve the reconfiguration optimization of RAL.

The innovation of this paper is as follows.

• A mixed integer linear program (MILP) is developed to formulate the RALRP for product update.
• An efficient data structure is proposed to model product and assembly line, which includes more detailed process information.
• An improved Variable Neighborhood Search (VNS) is proposed to reduce reconfiguration cost for Reconfigurable Assembly Line.

The rest of this paper is organized as follows. Section 2 gives a literature review about the Reconfigurable Assembly Line Reconfiguring Problem and solution methods. Section 3 describes the problem and proposes a mathematical model. The data structure of the problem is detailed in Section 4. The proposed algorithm is introduced in Section 5. Extensive experimental studies are performed, and the results presented in Section 6. Section 7 is a discussion. Finally, Section 8 concludes this paper.

2. Literature Review

In this section, we review the Reconfigurable Assembly Line Reconfiguring Problem and the existing state-of-the-art work relating to the solution method.

2.1. Reconfigurable Assembly Line Reconfiguring Problem

Factors such as changes in market demand, product updates, the use of new assembly tools, and changes in worker operating skills, assembly line cycle time, assembly processes, and assembly operation time may change [5]. These changes make the existing assembly balance unfeasible. Schofield defined the problem of reallocating partial operations of products on old assembly line to restore efficient and stable production balancing as ALRBP [6]. RALRP considers the resource selection from both existing old assembly line and resource library, building upon ALRBP. However, most studies focus on ALRBP without considering resources. Gamberini et al. [7,8] mainly studied ALRBP caused by changes in product features and yield, considering changes in process precedence constraint. Makssoud et al. [9,10] researched ALRBP caused by the introduction of new products, considering changes in operations and process precedence constraint. Karas et al. [11] considered ALRBP caused by changes in worker task execution time. Tremblet et al. [12] comprehensively considered the reconfiguration involving changes in operations, process precedence constraint, and process time. R. Cajzek and U. Klansek [13] proposed a model for cost optimization of project schedules under constrained resources and alternative production processes. The model comprehensively considers practical constraints, including generalized precedence relationship constraints, activity duration and start time constraints, lag/lead time constraints, execution mode constraints, project duration constraints, working time unit assignment constraints, and resource constraints.

Research on RALRP considering resource selection mostly focuses on flexible resources. Karas [11] proposed the worker assignment and rebalancing problem. Perez-Wheelock et al. [14] reallocated workers to stations to achieve rebalancing of assembly line over multiple periods. Zhang et al. [15] addressed the balancing problem of robot resources across multiple periods of assembly line. In conclusion, currently no scholars have comprehensively studied the reconfiguration considering changes in process elements such as operations, process precedence constraint, process time, and the corresponding specific manufacturing resources for RAL.

2.2. Solution Methods

RALRP involves complex process information, such as operations and resources, so it is necessary to establish an efficient data structure modeling this information to accelerate the algorithm. Most studies [16,17] use a two-vector solution representation, including an Operations Sequence and Machines Sequence. Caldeira et al. [18] further construct the operation sequence based on precedence constraint in the encoding scheme. However, when the problem involves more complex process information, this low-level data structure consumes a significant amount of time during encoding and decoding, thereby affecting the efficiency of the algorithm.

As a combinatorial optimization problem, ALRBP and its variants are tackled using exact and approximate methods, as illustrated in Figure 1. Exact methods include mathematical programming and enumerative algorithms [4,19,20]. The main advantage of exact methods is guaranteeing global optimal results. However, they are suitable for small-scale problems, and cannot solve large instances within a reasonable time. Approximate methods include heuristic algorithms, meta-heuristic algorithms and hyper-heuristic algorithms. Research mainly focuses on heuristic algorithms and meta-heuristic algorithms. Heuristic algorithms employ rules derived from problem characteristics for efficient solving, such as tabu search [21], heuristic rules [5] and VNS. The basic component of the algorithm is basic VNS proposed by Mladenović and Hansen in 1997 [22,23]. VNS is a local search based meta-heuristic, which explores increasingly several neighborhoods of the current incumbent solution and jumps from this solution to a new one if and only if an improvement has been made. The selection of neighborhood structures can greatly influence the performance of the VNS method [22]. Yang et al. [24] propose four different neighborhood structures for the assembly line balancing problem. Lei [25] demonstrates the promising advantages of designing effective neighborhood structures for the Assembly Line Rebalancing Problem. Polat et al. [26] introduce a two-phase VNS algorithm. In the first phase, a VNS approach is applied to assign tasks to workstations with the aim of minimizing the cycle time, while in the second phase, a variable neighborhood descent method is applied to assign workers to workstations. Genetic Algorithm (GA) [27], Simulated Annealing Algorithm (SA), and Particle Swarm Optimization (PSO) are typical meta-heuristic algorithms. There is no research on applying SA and PSO to solve ALRBP. Paper [28] proposes a GA hybridized with a heuristic priority rule-based procedure. This hybridization is used to add more rich seeds to the initial population and, consequently, to improve the convergence capability and performance of the GA. Zhang et al. [29] propose a modified non-dominated sorting genetic algorithm, which employs some problem-specific designs for encoding and decoding, initial population, crossover operator, mutation operator, and selection operator. In addition, they also use experiments to determine the parameters for GA to solve ALRBP.

At present, there is a lack of data structures that represent the complex information of the RALRP, as well as a lack of efficient algorithms for the simultaneous allocation of operations and manufacturing resources.

3. Problem Description and Formulation

In the section, a clear definition of the studied problem is given. Subsequently, a corresponding MILP formulation is proposed.

3.1. Problem Statement

We consider a RAL that produces several products from the same product family at different stages. Each product requires a specific set of operations to complete its production. Each operation is performed by a resource selected from a set of optional resources, each with varying processing times, and must be executed according to process precedence constraint. Therefore, an acceptable RAL configuration for a product must include a resource for each operation and satisfy the process precedence constraint. Product updates lead to changes in operations, process precedence constraints, and optional resources. Thus, RALRP allocates new operations to stations while optimally utilizing the resources from the existing old assembly line and new resources. For example, in an assembly line, a resource reassignment can be seen as a relocation of a robot from one station to another. Without loss of generality, we suppose that the logistics time of both stations is less than the process time of the station. Moreover, we do not consider the inclusion and exclusion constraints that force certain tasks to be assigned, or not, to the same station.

In order to ensure high efficiency and quality reconfiguration, it is important to optimize the number of resource reassignments performed when product update. Since different types of resource reassignments have different costs. In this case, it is relevant to RALRP so that the resource reassignments cost is minimized.

To better illustrate the above-described RALRP, Figure 2 shows an example where previously Assembly Line A produced Product A. After some time, Product A is upgraded to Product B, prompting Assembly Line A to be reconfigured to Assembly Line B to accommodate the new operations and process precedence constraint. Thus, the input to the optimization problem consists of the old assembly line and the new product, and the output is the new assembly line configuration. In the figure, product information is represented by corresponding process precedence graphs with resources. Circles denote operation number, arrows denote the process precedence constraint, and rectangles represent optional manufacturing resource names and their process time. For example, in the process precedence graph with resources for Product A, circle 2 represents operation number 2, with the corresponding rectangle indicating optional manufacturing resources D2 and R1, with process time of 100 s and 80 s, respectively. An arrow from circle 2 to circle 5 signifies that operation 2 must be completed before operation 5. Assembly Line A, corresponding to Product A, consists of five stations, where the first station S1 includes resource D1, processing operation 1 with a total process time of 160 s. In Product B, sections highlighted in yellow denote the parts of the product that have changed. Assembly Line B represents a reconfigured configuration. Compared to Assembly Line A, resources D2 and D3 in station S2 have been relocated, resource D4 has been removed, and new resource D11 has been added. Additionally, new resource D12 has been introduced into station S2. A new station S4 has been added, equipped with resource R6.

3.2. Mathematical Model

The RALRP minimizing the reconfiguration cost is formulated in this paper as a mixed integer programming model, using the notations in

Table 1. Notations.

Notations	Description
Index:
i	Operation. $\left(i=1,2,\dots ,m\right)$.
j	Station. $\left(j=1,2,\dots ,n\right)$.
$(i,r)$	Resource. $\left((i,r)=\left(1,1\right),\dots ,\left(1,r_1^{max}\right),\dots ,\left(m,r_m^{max}\right)\right)$.
l	Operation index of the station. $\left(l\in L=\left\{1,2,\dots ,l_{max}\right\}\right)$.
k	Resource index of the station. $\left(k\in K=\left\{1,2,\dots ,k_{max}\right\}\right)$.
Parameter:
D	Set of dedicated resource.
R	Set of reconfigurable resource.
F	Set of flexible resource.
$AL$	Set of resources from old assembly line.
$z_1$	Number of resources in old assembly line.
m	Number of operations.
$ct$	Cycle time.
$\phi$	Cost of additional dedicated resource.
$\kappa$	Cost of additional reconfigurable resource.
$\lambda$	Cost of additional flexible resource.
$\mu$	Cost of moving a resource. ($\mu \ll \phi ,\kappa ,\lambda$).
$S{R^{\prime }}_{jk}$	Resource k in station j. ($S{R^{\prime }}_{jk}=(i,r)$).
$r_i^{max}$	Maximum number of resources of operation i.
$t{o}_{ir}$	Process time of resource r of operation i.
$P_i$	Set of precursor operation for operation i.
$S_i$	Set of successor operation for operation i.
$t{s}_j$	Process time of station j.
$do{m}_{i,r}$	Operation index of resource. $do{m}_{i,r}=i$.
Decision variable:
n	Number of stations.
$S{O}_{jl}$	Operation l in station j. ($S{O}_{jl}=1,2,\dots ,m$).
$S{R}_{jk}$	Resource k in station j. ($S{R}_{jk}=(i,r)$).
$SR{N}_{jk}$	Whether resource k in station j is a new resource.
	($SR{N}_{jk}=0,1$).
$SR{O}_{jk}$	Whether resource k in station j is from old assembly line.
	($SR{O}_{jk}=0,1$).
$SR{Q}_{jk}$	Number of resource k in station j. ($SR{Q}_{jk}\in N$).
$SR{O^{\prime }}_{jk}$	Resource k in station j is a move of the resource from
	other station in old assembly line. ($SR{O^{\prime }}_{jk}=0,1$).
$SR{D}_{jk}$	Resource k in station j is a new dedicated resource.
	($SR{D}_{jk}=0,1$).
$SR{R}_{jk}$	Resource k in station j is a new reconfigurable resource.
	($SR{R}_{jk}=0,1$).
$SR{F}_{jk}$	Resource k in station j is a new flexible resource.
	($SR{F}_{jk}=0,1$).

.

$$\begin{array}{cc}\hfill minrc& =\phi \sum _{j=1}^m\sum _{k=1}^{k_{max}}SR{O^{\prime }}_{jk}+\kappa \sum _{j=1}^m\sum _{k=1}^{k_{max}}SR{D}_{jk}\hfill \\ \hfill & +\lambda \sum _{j=1}^m\sum _{k=1}^{k_{max}}SR{R}_{jk}+\mu \sum _{j=1}^m\sum _{k=1}^{k_{max}}SR{F}_{jk}\hfill \end{array}$$

(1)

Subject to

$$\begin{array}{cc}\hfill & SR{N}_{jk}+SR{O}_{jk}=1,\forall j\in S,k\in K\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & SR{N}_{jk}\in \left\{0,1\right\},\forall j\in S,k\in K\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & SR{O}_{jk}\in \left\{0,1\right\},\forall j\in S,k\in K\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & S{R}_{jk}\ne S{R^{\prime }}_{j^{\prime }{k}^{\prime }},\exists S{R}_{j^{″}{k}^{″}}=S{R^{\prime }}_{j^{\prime }{k}^{\prime }},\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \forall j,j^{\prime },j^{″}\in S,\forall k,k^{\prime },k^{″}\in K\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & SR{O^{\prime }}_{jk}=0,\exists S{R}_{jk}=S{R^{\prime }}_{j^{\prime }k},j^{\prime }\in S,\forall j\in S,k\in K\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & SR{O^{\prime }}_{jk}\in \left\{0,1\right\},\forall j\in S,k\in K\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & SR{D}_{jk}=1,\exists S{R}_{jk}\in D,\exists SR{N}_{jk}=1,\forall j\in S,k\in K\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & SR{D}_{jk}\in \left\{0,1\right\},\forall j\in S,k\in K\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & SR{R}_{jk}=1,\exists S{R}_{jk}\in R,\exists SR{N}_{jk}=1,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \forall j\in S,k\in K\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & SR{R}_{jk}\in \left\{0,1\right\},\forall j\in S,k\in K\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & SR{F}_{jk}=1,\exists S{R}_{jk}\in F,\exists SR{N}_{jk}=1,\forall j\in S,k\in K\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & SR{F}_{jk}\in \left\{0,1\right\},\forall j\in S,k\in K\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & SR{D}_{jk}+SR{R}_{jk}+SR{F}_{jk}\le 1,\forall j\in S,k\in K\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & \sum _{j=1}^n\sum _{l=1}^{l_{max}}S{O}_{jl}/S{O}_{jl}=m\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & S{O}_{jl}\ne S{O}_{j^{\prime }{l}^{\prime }},\forall j\ne j^{\prime }\vee l\ne l^{\prime }\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & S{O}_{jl}=i,\forall j\in S,l\in L\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & S{O}_{jl}\in P_{S{O}_{j{l}^{\prime }}},\forall j\in S,l\le l^{\prime }\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & S{O}_{j{l}^{\prime }}\in P_{S{O}_{j^{\prime }{l}^{\prime \prime }}},\forall j<j^{\prime },l^{\prime }\in L,l^{\prime \prime }\in L\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & t{s}_j=\sum _{k=1}^{k_{max}}t{o}_{S{R}_{jk}}SR{Q}_{jk}\le ct,\forall j\in S\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & do{m}_{S{R}_{jk}}=S{O}_{jl},\forall j\in S,k\in K,\exists l\in L\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & n\in N\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & SR{\mathrm{Q}}_{jk}\in N,\forall j\in S,k\in K\hfill \end{array}$$

(23)

Formula (1) minimizes the reconfiguration cost of the assembly line. This study subdivides the reconfiguration cost into “cost of moving a station”, “cost of moving a resource”, “cost of additional dedicated resource”, “cost of additional reconfigurable resource” and “cost of additional flexible resource”. The cost of moving a station is set to 0. Other costs are assigned as $\phi$, $\kappa$, $\lambda$, and $\mu$, respectively. The specific values of these costs can be obtained through on-site surveys of enterprises or determined based on the production preferences of the enterprises. Constraint (2) restricts the resources of new assembly line are from either old assembly line or resource library. Constraint (3) and (4) restrict the decision variable to the values $\left\{0,1\right\}$. Constraints (5)–(7) express all constraints regarding resources from the old assembly line. Constraint (5) restricts the usage of resources from the old assembly line to only once in the new assembly line. Constraint (6) determines whether resources within stations of the old assembly line are relocated as a whole. Constraint (7) restricts the decision variable to the values $\left\{0,1\right\}$. Constraints (8)–(14) express all constraints regarding resources from resource library. Constraint (8) specifies whether a resource is dedicated. Constraint (9) restricts the decision variable to the values $\left\{0,1\right\}$. Constraint (10) specifies whether a resource is reconfigurable. Constraint (11) restricts the decision variable to the values $\left\{0,1\right\}$. Constraint (12) specifies whether a resource is flexible. Constraint (13) restricts the decision variable to the values $\left\{0,1\right\}$. Constraint (14) ensures that if a resource from the resource library is selected, it must be one of “dedicated, reconfigurable, or flexible”. Constraints (15)–(23) express the fundamental constraints of assembly line balancing problems involving resources. Constraint (15) ensures that the number of operations in the assembly line equals the number of operations of the product. Constraint (16) prohibits duplicate operations in the assembly line. Constraint (17) specifies that the operations in the assembly line correspond to the operation number of the product. Together, these three constraints ensure that all operations of the product are uniquely allocated to stations in the assembly line. Constraint (18) ensures that the sequence of operations within the same station satisfies the process precedence constraint of the product. Constraint (19) ensures that the sequence of operations among different stations satisfies the process precedence constraint of the product. Together, these two constraints ensure that the operations on the assembly line adhere to the process precedence constraint of the product. Constraint (20) ensures that the process time of operations at stations is less than the cycle time, thus meeting cycle time constraints. Constraint (21) models that each operation in the assembly line must be assigned a resource. Together, these two constraints ensure that each operation in the assembly line is assigned an available resource, thereby satisfying resource constraint. Constraint (22) expresses that the number of stations in the assembly line must be an integer. Constraint (23) specifies that the number of the required the kth resource in station j of the assembly line must be an integer.

4. Data Structure

Compared to ALRBP, which does not account for resources, RALRP involves an expanded set of resources, thereby requiring maintenance of additional process information. This paper introduces a data structure for products, defined as Data Structure 1, including operation nodes and precedence nodes. Each operation node consists of a data field and a pointer field. The data field stores the operation number and operation data, along with the names of optional resources and their process time. The pointer field points to the first precedence relation of that operation. The precedence node also includes a data field and a pointer field, where the data field stores the operation number of the successor node, and the pointer field points to the next precedence relation of that operation. To illustrate the encoding scheme, Figure 3 displays partial information of Product a from Figure 2. For example, the information of the row numbered 6 is represented as follows: the operation number is 6, the operation information is $Dat{a}_6$, and the process precedence constraint are $\langle 6,9\rangle ,\langle 6,10\rangle ,\langle 6,11\rangle$.

An effective solution representation for the problem must comprehensively encompass all aspects necessary to fully define the problem. This paper introduces a data structure for assembly line composed of station nodes, defined as Data Structure 2, which is used to represent both the old and new assembly lines. Each station node includes a station number, station data, station process time, resources, and the operations processed. To illustrate the solution representation, Figure 4 presents partial information of Assembly Line A from Figure 2. For example, the information for the first station is represented as follows: the station number is $S1$, the station data are $Dat{a}_1$, the process time of the station is 160 s, and the resource in the station is D1 processing operation 1.

5. Proposed IVNS-RALRP

In this section, we present an Improved Variable Neighborhood Search for the Reconfigurable Assembly Line Reconfiguring Problem (IVNS-RALRP). First, the algorithm framework is described. The subsequent five sections provide a detailed explanation of the algorithm.

5.1. Algorithm Framework

VNS systematically explores different neighborhood structures, allowing it to escape local optima effectively. By switching between different neighborhoods, VNS can explore a broader search space than algorithms that rely on a single neighborhood structure. For certain specific combinatorial optimization problems, we can design specialized neighborhood structures based on the problem characteristics for VNS, thereby further enhancing its performance. So, we choose VNS as the algorithm framework. The basic component of the algorithm is basic VNS proposed by Mladenović and Hansen in 1997 [22,23]. Firstly, it is necessary to select the set of neighborhood structures, generate the initial solution x, choose the stopping condition, and complete the initialization of the algorithm. Subsequently, the main iterative process of VNS includes three parts: shaking, local search, and move or not. The current solution x in a certain neighborhood structure undergoes shaking to produce a temporary solution $x^{\prime }$. A local search is performed on the temporary solution $x^{\prime }$ to obtain a locally optimal solution $x^{″}$. Then, move or not is executed: if the locally optimal solution $x^{″}$ is better than the current solution x, $x^{″}$ is taken as the current solution x to initiate a new search; otherwise, shaking is continued on x within other neighborhood structures. The entire iterative process continues until a stopping condition is met. The stopping condition may be, e.g., maximum CPU time allowed, maximum number of iterations, or maximum number of iterations between two improvements. Steps of the VNS are presented as follows:

• Initialization. Select the set of neighborhood structures ${\mathcal{N}}_k,k=1,\dots ,k_{max}$, that will be used in the search; find an initial solution x; choose a stopping condition;
• Repeat the following until the stopping condition is met:

  (1)

  Set $k\leftarrow 1$;

  (2)

  Until $k=k_{max}$, repeat the following steps:

  (a)

  Shaking. Generate a point $x^{\prime }$ at random from the kth neighborhood of x ($x^{\prime }\in {\mathcal{N}}_k\left(x\right)$);

  (b)

  Local search. Apply some local search method with $x^{\prime }$ as initial solution; denote with $x^{″}$ the so obtained local optimum;

  (c)

  Move or not. If this local optimum is better than the incumbent, move there ($x\leftarrow x^{″}$), and continue the search with ${\mathcal{N}}_1\left(k\leftarrow 1\right)$; otherwise, set $k\leftarrow k+1$.

In RALRP, the sequence of operations in each solution must adhere to process precedence constraints. This method enhances the basic VNS by incorporating a legal check for these constraints. Each solution generated by the shaking process requires this legal check for process precedence constraint. If the constraints are satisfied, the next process proceeds; otherwise, the shaking process is repeated. Since the solution for RALRP involves both operation and resource dimensions, where resources are constrained by operations, the method first generates a partial solution through shaking that includes only operations. Subsequently, it employs local search to explore the solutions with resource.

The detailed procedure of IVNS-RALRP is presented in Algorithm 1.

• The first three lines initialize the problem. Line 1 initializes iteration for the algorithm. Line 2 computes the indegree of all operations of the product to facilitate legal check for process precedence constraint. Line 3 obtains an initial solution $NAL$ as the current solution through the initialization algorithm.
• Line 4 calculates the reconfiguration cost of the current solution $NAL$ using the cost computation method.
• Line 5 to 21 represent the iteration process of IVNS-RALRP, with the stopping condition being the maximum iteration. Line 6 counts the iteration.
• Line 7 to 20 sequentially search kth neighborhood of the current solution $NAL$.
• Line 8 to 10 generate a shaking in the kth neighborhood to create a partial solution, i.e., operations of new assembly line $NALO$ containing only operations. Then, the isLegal algorithm is used to check whether the partial solution $NALO$ satisfies the process precedence constraint. If not, shaking is performed again.
• Line 11 performs local search on the partial solution $NALO$ to generate a locally optimal solution $NA{L}^{\prime }$. Line 12 calculates the reconfiguration cost of the locally optimal solution $NA{L}^{\prime }$ using the cost computation method in Section 5.3.
• Line 13 to 19 determine move or not. If the reconfiguration cost $r{c}^{\prime }$ of the locally optimal solution $NA{L}^{\prime }$ is better than the reconfiguration cost $rc$ of current solution $NAL$, then the locally optimal solution $NA{L}^{\prime }$ replaces the current solution $NAL$ and the search restarts. Otherwise, it moves to the $k+1$th neighborhood to repeat the search.

5.2. Initialization

VNS iteratively optimizes based on an initial solution until a final solution is obtained. If the objective value of initial solution is poor, the entire algorithm converges slowly. Additionally, the initial solution must satisfy both cycle time constraint and process precedence constraint. This paper proposes a two-stage initialization method. Since the cost of moving individual resources in the old assembly line is much lower than adding new resources ($\lambda \ll \phi ,\kappa ,\lambda$), the optimization objective is to minimize the sum of moving old resources and adding new resource costs. Therefore, in the first stage of the initialization method, resources that can be reused, and their corresponding operations for new products, are searched for. In the second stage, operations are sequentially assigned to stations according to process precedence constraint, prioritizing the use of resources identified in the first stage from the old assembly line. This process continues, generating subsequent stations according to the cycle time constraint until all operations are allocated, thereby forming a new assembly line as the initial solution. The pseudocode in Algorithm 2 illustrates this procedure.

Algorithm 1: IVNS-RALRP

First Stage:

• The first two lines focus on searching for resources that can be maximally utilized by operations, where each operation corresponds uniquely to a resource. In Line 1, if operation i of product G can be processed by resource j of the old assembly line $OAL$, then $r_{ij}=1$; otherwise, $r_{ij}=0$. The matrix $R_{m\times z}$ represents the many-to-many relationship between operations and resources. To maximize the number of reusable resources, this problem is transformed into an assignment problem. Line 2 utilizes the Hungarian algorithm [30] to obtain an optimal assignment scheme $M_{m\times z}$ for operations and resources.
• Due to the ability of reconfigurable and flexible resources to process multiple operations, Line 3 to 9 allocate feasible operations to the identified reconfigurable and flexible resources. And the result is stored in $M_{m\times z}$.
  Second Stage:
• Line 10 to 11 perform initialization. Line 12 identifies operations without process precedence constraint.
• Line 13 to 28 establish the new assembly line. Line 14 randomly selects a feasible operation. Line 15 to 19 match resources for the selected operation. If a resource corresponding to the operation selected in the first stage is available, a random resource from those corresponding resources is chosen; otherwise, a random resource is selected from the resource library.
• Line 20 to 24 indicate that if placing the resource at the current station meets the cycle time constraint, it is placed; otherwise, the next station is opened and the resource is placed there.
• Line 25 records the results of operations and resources into the solution $NAL$.
• Line 26 and Line 27 obtain the optional operations for the next iteration.

Algorithm 2: initialization

5.3. Solution Evaluation

The overall reconfiguration cost $rc$ is not only related to the types and quantities of three new resources added from the resource library, but also to whether resources in the old assembly line $OAL$ are moved individually or relocated as a whole within stations. To identify a new solution $NAL$ using Data Structure 2, a fast method with $O\left(z_1{z}_2\right)$ (where $z_1$ is the number of resources in the old assembly line and $z_2$ is the number of resources in the new assembly line) can be presented as Algorithm 3.

• Line 1 initializes variables necessary to compute the reconfiguration cost. Here, $SR{O}^{\prime }$ records the number of individual resource movements in the old assembly line, $SRD$ records the number of newly added dedicated resource, $SRR$ records the number of newly added reconfigurable resource, and $SRF$ records the number of newly added flexible resource. Since the cost of relocating entire station in the assembly line is set to 0, there is no need to track the number of resources involved.
• Line 2 to 28 search each station of the new assembly line. Line 3 initializes $allOld\leftarrow 1$, indicating that the resources of the entire station in the new assembly line are relocated from the stations in the old assembly line.
• Line 4 to 19 search each resource of each station. Line 5 checks whether the resource is a new resource. If it is a new resource, Line 7 to 15 determine its type and update the respective counters; otherwise, in Line 17, it is assumed to be an individual resource moved from the old assembly line and counted in $SR{O}^{\prime }$.
• Line 14, judge $allOld\leftarrow 1$, which means that the resources of the entire station of the new assembly line are the overall relocation of the station of the old assembly line. Further, Line 20 to 27 check if it involves relocating the entire station. If so, the number of resources counted in $SR{O}^{\prime }$ in Line 17 is adjusted by subtracting the number of resources involved in relocating the entire station $SR{O}^{\prime }\leftarrow SR{O}^{\prime }-k_{max}$.
• Line 29 calculates the total reconfiguration cost.

Algorithm 3: reCalculate

5.4. Legal Check

Considering the process precedence constraint in RALRP, each operation in the partial solution $NALO$ must satisfy it. Therefore, this paper proposes a legal check method for the operation precedence constraint about partial solution $NALO$, as shown in Algorithm 4. Line 1 to 8 sequentially check whether each operation in the sequence represented by $NALO$ satisfies the process precedence constraint denoted by G. The indegree table $inDegree$ records the indegree of each operation node, indicating how many operations precede each operation. Line 2 to 4 use the indegree table $inDegree$ to determine if the operation has any predecessor nodes. If predecessor nodes exit, the process precedence constraint are not satisfied, and the partial solution $NALO$ is deemed invalid. Line 5 to 7 remove the checked operation nodes (decrementing the indegree of successor operation nodes). If all operations pass the legal check, then every operation in the sequence represented by $NALO$ satisfies the process precedence constraint denoted by G, and Line 9 outputs ‘True’.

Algorithm 4: isLegal

5.5. Shaking

VNS method is applied to find more promising solutions in the solution space through changing neighborhood structures during the search process. Since a neighborhood is usually defined based on moves of operation, the search process can benefit much from suitably selected moves. This paper proposes new problem-specific neighborhood structures ${\mathcal{N}}_k\left(k=1,2,3,4,5\right)$, as illustrated in Figure 5. The neighborhood structures include: Swap operation between the ith operation and the 1th operation following it ($swa{p}_1\left(i\right)$). Swap operation between the ith operation and the 2th operation following it ($swa{p}_2\left(i\right)$). Swap operation between the ith operation and the 3th operation following it ($swa{p}_3\left(i\right)$). Insert operation from the ith operation to the 3th operation following it ($inser{t}_3\left(i\right)$). Insert operation from the ith operation to the 4th operation following it ($inser{t}_4\left(i\right)$). Neighborhood change about swap operation and insert operation can transform the partial solution $NALO$ into all possible partial solutions.

The changes in neighborhoods are as follows:

• ${\mathcal{N}}_1:swa{p}_1\left(i\right)$: randomly select an integer i ($1\le i\le m-1$) and swap the positions of the ith operation and the $(i+1)$th operation in the partial solution $NALO$ of operations.
• ${\mathcal{N}}_2:swa{p}_2\left(i\right)$: randomly select an integer i ($1\le i\le m-2$) and swap the positions of the ith operation and the $(i+2)$th operation in the partial solution $NALO$ of operations.
• ${\mathcal{N}}_3:swa{p}_3\left(i\right)$: randomly select an integer i ($1\le i\le m-3$) and swap the positions of the ith operation and the $(i+3)$th operation in the partial solution $NALO$ of operations.
• ${\mathcal{N}}_4:inser{t}_3\left(i\right)$: randomly select an integer i ($1\le i\le m-3$) and insert the ith operation into the position of the $(i+3)$th operation in the partial solution $NALO$ of operations.
• ${\mathcal{N}}_5:inser{t}_4\left(i\right)$: randomly select an integer i ($1\le i\le m-4$) and insert the ith operation into the position of the $(i+4)$th operation in the partial solution $NALO$ of operations.

5.6. Local Search

Local search is an important component to enhance the optimization performance of IVNS-RALRP. In each iteration, a new partial solution is generated from the current partial solution $NALO$ by applying the aforementioned neighborhood changes. Then, local search is employed to complete resource selection and obtain the final locally optimal solution $NAL$. The description of local search is shown in Algorithm 5.

• The first two lines initialize the parameters.
• Line 3 to 26 conduct iterative local search. The stopping condition, as indicated in Line 26, is to exceed the maximum iteration and have consistent results in two consecutive searches. Line 4 counts the iteration.
• Line 5 to 20 allocate resources for the partial solution $NALO$ and design a new assembly line $NA{L}^{\prime }$. Line 6 to 13 allocate resources for each operation in the partial solution $NALO$. Line 7 determines whether the operation corresponds to the resource, which is selected in the matrix $M_{m\times z}$ in Algorithm 2. If satisfied, a corresponding resource is randomly selected; otherwise, from Line 10 to 12, a resource is randomly selected from the resource library. Line 14 to 18 indicate that if placing the resource in the current station meets the cycle time $ct$, it is placed; otherwise, the next station is opened and the resource is placed.
• Line 21 calculates the reconfiguration cost of the new assembly line $NA{L}^{\prime }$ using the cost calculation method.
• Line 22 to 25 update the solution. If the reconfiguration cost $r{c}^{\prime }$ of the locally optimal solution $NA{L}^{\prime }$ is better than the reconfiguration cost $rc$ of the current solution $NAL$, then the locally optimal solution $NA{L}^{\prime }$ is adopted as the current optimal solution $NAL$.

Algorithm 5: search

6. Experimental Results

This section includes the experimental analysis to evaluate the performance of the proposed IVNS-RALRP. The experiments are coded in C++, based on Microsoft Visual Studio Professional 2019, run on a computer with an Intel Core i7-11700F 2.50 GHz processor and 16 GB RAM. The experimental analysis includes the following aspects:

• Assess the effectiveness of the data structure for IVNS-RALRP.
• Assess the effectiveness of the shaking and local search procedure in IVNS-RALRP.
• Examine the performance of IVNS-RALRP with other heuristic algorithms.

6.1. Performance Metrics

For each test experiment, each algorithm performs $y_{max}$ iterations. Then, the results $r\left(y\right)$ and the running time $t\left(y\right)$, where $r\left(y\right)$ denotes the objective value obtained in the yth computation, and $t\left(y\right)$ denotes the running time of the yth computation in seconds, are computed and recorded. This paper proposes four metrics to evaluate the effectiveness of the algorithm in Equations (24)–(27). R represents the optimal objective value among $y_{max}$ iterations, $\overline{r}$ is the average objective value for $y_{max}$ iterations, S reveals the computation stability across $y_{max}$ iterations, and $\overline{t}$ indicates the computation speed over $y_{max}$ iterations.

$$\begin{array}{cc}\hfill R& =min\left(r\left(1\right),r\left(2\right),\dots ,r\left(y_{max}\right)\right)\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill \overline{r}& ={\sum }_{y=1}^{y_{max}}r\left(y\right)/y_{max}\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill S& ={\sum }_{y=1}^{y_{max}}{\left(r\left(y\right)-\overline{r}\right)}^2/y_{max}\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill \overline{t}& ={\sum }_{y=1}^{y_{max}}t\left(y\right)/y_{max}\hfill \end{array}$$

(27)

6.2. Test Cases

To the best of our knowledge, there is currently no benchmark dataset available for RALRP. Therefore, we have collated and simplified RAL information from our partner OEM, categorizing the problems into two groups: small-scale and large-scale.

Table 2. Cases of different sizes.

Case Size	No.	Number of Operations	Cycle Time (s)
small-scale	1	20	200
	2	25	200
	3	30	200
large-scale	4	70	200
	5	75	200
	6	80	200

illustrates the different scale problems used to evaluate the performance of the algorithms studied in this section. The detailed benchmarking dataset is given in the in Appendix A.1.

6.3. Performance Analysis of Each Component

In this section, the contribution of the proposed data structure and the improvement of VNS are investigated. We employ all the four performance metrics described in Section 6.1 to evaluate the solutions obtained from all algorithms. All algorithms are applied the same maximum iteration ($maxIter=1000$) for a fair comparison.

6.3.1. Effect of the Data Structure

To assess the effectiveness of the proposed data structure, we consider the IVNS-RALRP without these data structure (IVNS-WDS) and compare it to IVNS-RALRP for all problems.

Table 3. The computational results of IVNS-RALRP and IVNS-WDS.

No.	IVNS-RALRP				IVNS-WDS
	$R$	$\overline{r}$	$S$	$\overline{t}$	$R$	$\overline{r}$	$S$	$\overline{t}$
1	31	31.3	0.21	6.52	31	31.3	0.41	7.80
2	38	39.0	0.80	6.85	38	39.0	1.60	8.38
3	43	43.8	0.56	7.06	43	43.9	0.69	8.60
4	108	109.5	1.45	12.99	108	109.3	1.21	16.70
5	113	114.1	1.69	13.40	113	114.5	2.45	17.32
6	121	121.7	1.01	13.70	121	121.8	0.96	17.89

shows four performance metrics across 10 independent runs for all test cases. As shown in Table 3, IVNS-RALRP outperforms IVNS-WDS for $\overline{t}$ metrics, while showing nearly identical performance for the other three metrics for all data sets. The competitive advantage of IVNS-RALRP in computation speed is attributed to the data structure, which enhances search efficiency while maintaining more process information.

To further evaluate computation speed, we compare the two algorithms for the same case of $\overline{t}$.

Table 4. The computation speed metric $\overline{t}$ comparison of IVNS-RALRP and IVNS-WDS.

No.	IVNS-RALRP	IVNS-WDS	Decrease *
1	6.52	7.80	−16%
2	6.85	8.38	−18%
3	7.06	8.60	−18%
4	12.99	16.70	−22%
5	13.40	17.32	−23%
6	13.70	17.89	−23%

(*): Decrease in the results of IVNS-RALRP compared with IVNS-WDS.

presents the comparative results, indicating a significant reduction in computation time for IVNS-RALRP compared to IVNS-WDS as the problem size increases. This improvement is due to the implementation of efficient data structure in the iterative search. On the whole, it can be concluded that the proposed data structure is a vital component of the approach, contributing significantly to the computation speed of IVNS-RALRP.

6.3.2. Effect of the Shaking and Local Search Procedure

To evaluate the influence of the shaking and local search procedure described in Section 5.5 and Section 5.6, we consider the proposed IVNS-RALRP and compare it to basic VNS. The results across 10 independent runs are shown in

Table 5. The computational results of IVNS-RALRP and VNS.

No.	IVNS-RALRP				VNS
	$R$	$\overline{r}$	$S$	$\overline{t}$	$R$	$\overline{r}$	$S$	$\overline{t}$
1	31	31.3	0.21	6.52	31	32.3	0.61	4.33
2	38	39.0	0.80	6.85	38	39.1	2.29	4.36
3	43	43.8	0.56	7.06	44	45.2	1.56	4.47
4	108	109.5	1.45	12.99	108	111.8	13.56	6.00
5	113	114.1	1.69	13.40	114	116.9	7.09	6.15
6	121	121.7	1.01	13.70	122	125.8	8.56	6.35

. Concerning the optimal objective value metrics R, IVNS-RALRP searches for lower reconfiguration cost than VNS in three cases. Concerning the average objective value metrics $\overline{r}$ and the computation stability metrics S in Table 5, IVNS-RALRP is superior to VNS across all data sets. The notable performance of the IVNS-RALRP for the three metrics can be attributed to the improvement of the shaking and local search procedure. Concerning $\overline{t}$ metrics, the improved algorithm consumes additional computation time. As the reconfiguration of assembly line is not real-time responsive, allowing for low reconfiguration cost at computation times of a few seconds.

To further assess the reconfiguration cost obtained by both algorithms, the reconfiguration cost from 10 computations are displayed as box plots in Figure 6. From these box plots, it is evident that IVNS-RALRP performs better than VNS in terms of optimal objective value and computation stability. Moreover, as the problem size increases, the computation stability becomes more pronounced. Hence, given the above analysis, it can be ascertained that the contribution of the shaking and local search procedure is substantial in improving the performance of proposed approach.

6.4. Comparison with Other Heuristic Algorithms

IVNS-RALRP is further compared to two other well-known heuristic algorithms, which are GA, TS, SA, and PSO. To make a fair comparison of the results, the maximum iteration is set to 1000 for all the compared algorithms. To determine the optimal parameters of GA, we carry out the Taguchi method of Design of Experiments (DOE). The detailed process of parameter calibration is given in Appendix A.2. The experimental results for the four performance metrics are tabulated in

Table 6. The computational results of IVNS-RALRP, GA, TS, SA and PSO.

No.	IVNS-RALRP				GA				TS				SA				PSO
	$R$	$\overline{r}$	$S$	$\overline{t}$	$R$	$\overline{r}$	$S$	$\overline{t}$	$R$	$\overline{r}$	$S$	$\overline{t}$	$R$	$\overline{r}$	$S$	$\overline{t}$	$R$	$\overline{r}$	$S$	$\overline{t}$
1	31	31.3	0.21	6.52	31	31.5	0.25	3.83	31	31.3	0.21	5.13	31	31.9	0.5	4.45	31	31.8	0.36	3.96
2	38	39.0	0.80	6.85	39	40.5	2.05	3.86	38	39.2	1.16	5.18	39	40.8	2.4	4.49	39	41.1	3.69	4.03
3	43	43.8	0.56	7.06	43	44.3	1.61	3.94	44	45.5	1.45	5.29	43	44.6	1.6	4.6	44	45.2	1.96	4.15
4	108	109.5	1.45	12.99	110	115.0	8.20	5.51	108	111.6	11.04	6.82	111	115.4	8.4	6.22	110	115.7	7.61	5.75
5	113	114.1	1.69	13.40	113	116.8	7.76	5.63	113	117.3	8.01	6.97	113	117.2	7.4	6.38	114	117.5	8.85	5.86
6	121	121.7	1.01	13.70	122	125.9	7.89	6.00	122	126.0	6.40	7.16	122	126.4	11.0	6.57	122	126.4	9.24	6.26

. It is evident that compared to GA, TS, SA and PSO, IVNS-RALRP achieves equivalent or superior performance in terms of the R, $\overline{r}$, and S metrics. In terms of solution time, IVNS-RALRP does not exhibit superiority; however, the time difference of seconds can be ignored in the reconfiguration stage of the assembly line.

In addition, the reconfiguration cost from 10 computations are illustrated as box plots in Figure 7. These box plots clearly demonstrate that IVNS-RALRP outperforms GA, TS, SA, and PSO in terms of optimal objective value and computation stability. Moreover, as the problem size increases, the computation stability becomes more pronounced. This underscores that IVNS-RALRP effectively maintains superior results while enhancing solution stability.

7. Discussion

From the experimental analysis in the previous section, IVNS-RALRP exhibits superior performance compared to IVNS-WDS, VNS, GA and TS. This enhanced performance is primarily attributed to the effective data structure and the improvement of the basic VNS which guides the search process aids in the convergence of the proposed approach. The data structure focuses on reducing algorithm runtime in solving problems that involve more process information. The shaking and local search procedure deals with reducing the algorithm solution results, specifically the reconfiguration cost. Both designs are intended to reduce algorithmic runtime to acceptable levels while improving solution quality to meet practical engineering requirements. The impact of the data structure on computation speed is evident from the comparison of average running time in Table 4. For the same algorithms, using the data structure proposed in this paper significantly reduces runtime. From the perspective of algorithm execution, when using low-level data structures to encode complex process information, the algorithm needs to perform decoding operation at each iteration, which greatly increases the computational complexity of the algorithm. As the scale of the case increases, the difference in computation speed becomes larger. The improvement of the shaking and local search procedure contributes to enhance computation quality and stability in the search process. These contributions are computationally proven in Section 6.3.1, with the results demonstrating the significantly superior performance of IVNS-RALRP as compared to VNS. From the perspective of the algorithm execution, the shaking and local search procedure we proposed encapsulates engineering heuristic experience, while the unimproved shaking and local search procedure is a general stochastic process. Therefore, IVNS-RALRP effectively guides each iteration towards a better direction, leading to a better result after multiple iterations. Furthermore, this paper compares IVNS-RALRP with four other well-known heuristic algorithms which are GA, TS, SA and PSO. The basic VNS provides a customizable neighborhood structure and local search procedure. The experimental results once again demonstrate that compared to other algorithms, the specialized neighborhood structures and local search procedure of IVNS-RALRP effectively improve the algorithm’s performance. Although this algorithm takes a few extra seconds to run, the reconfiguration phase of the assembly line does not require real-time responsiveness. The practical engineering requirement is to achieve higher quality results within an acceptable timeframe. Hence, the authors can conclude that the proposed IVNS-RALRP is effective in solving the RALRP.

8. Conclusions

This paper proposed a data structure about RAL and an IVNS-RALRP algorithm to solve RALRP. The major contributions of this paper are summarized as follows.

• A MILP is developed to formulate the RALRP for product update.
• An efficient data structure is proposed to model product and assembly line, which includes more detailed process information.
• An improved VNS is proposed to reduce reconfiguration cost for Reconfigurable Assembly Line.

The reconfiguration of RAL is an effective process for addressing product update within a product family. During the reconfiguration phase, it is necessary not only to allocate operations to the old assembly line but also to select corresponding resources. In practical cases, improvements in various aspects of IVNS-RALRP have proven effective. Furthermore, the solving performance of IVNS-RALRP surpasses that of other heuristic algorithms. Therefore, in theory, it is clear that IVNS-RALRP is an outstanding algorithm for solving RALRP, and it is effectively used in reconfiguration phase of RAL for product update. In practical engineering production, the reconfiguration phase of RAL not only involves reallocating operations, but also selecting specific manufacturing resources. For production management, excessive resource investment signifies needless waste. Moreover, reducing reconfiguration time means production can be brought forward, thereby increasing capacity. Therefore, what this paper contributes to the production management in a broader sense is an effective planning tool that reduces reconfiguration costs and increases capacity.

Since more information about manufacturing resources, such as the size of it and Mean Time Between Failure, is not considered, this method may be limited. Therefore, in the future research, more real practical information will be addressed in depth. In addition, there is still potential improvement in computation efficiency and optimality by introducing new operators and characteristics of other algorithms.