Unified Modeling and Multi-Objective Optimization for Disassembly Line Balancing with Distinct Station Configurations

Abstract

Disassembly line balancing (DLB) is a crucial optimization item in the recycling and remanufacturing of waste products. Considering the variations in the number of operators assigned to each station, this study investigates DLBs with six distinct station configurations: single-manned, multi-manned, single-robotic, multi-robotic, single-manned–robotic, and multi-manned–robotic setups. First, a unified mixed-integer programming (MIP) model is established for Type-I DLBs with each configuration to minimize four objectives: the number of stations, the number of operators, the total disassembly time, and the idle balancing index. To obtain more solutions, a novel bi-metric is proposed to replace the quadratic idle balancing index and is used in lexicographic optimization. Subsequently, based on the unified Type-I models, a unified MIP model for Type-II DLBs is established to minimize the cycle time, the number of operators, the total disassembly time, and the idle balancing index. Finally, the correctness of the established unified models and the effectiveness of the proposed bi-metric are verified by solving two disassembly cases of lighters and hairdryers, which further shows that the mathematical integration method of unified modeling has significant theoretical value for the multi-objective optimization of the DLBs with six distinct station configurations.

1. Introduction

Recycling, remanufacturing, and reusing waste products are effective strategies to mitigate resource shortages and reduce environmental pollution [1,2]. As the first step in remanufacturing, disassembly is a crucial means to realize the separation of waste parts from the main body [3]. Industrial disassembly usually adopts disassembly production lines to improve efficiency and ensure output [4]. The disassembly line is an integrated device that includes many technologies, such as mechanism design, electromechanical control, lot-sizing problems, and balancing problems. Disassembly line balancing (DLB) is crucial for effective production scheduling. It involves assigning disassembly tasks that adhere to the product precedence relationship to operators (workers and robots) at various stations along the disassembly line, optimizing indicators to ensure smooth and efficient operation [5].

Based on the different types of operators, disassembly modes are classified into manual disassembly, robotic disassembly, manned–robotic shared-station disassembly, and manned–robotic collaborative disassembly. These modes can be applied to various disassembly scenarios, according to their advantages. The manual disassembly mode has good flexibility and cognitive ability, which can handle waste products with complex structures and uncertain recycling states, as well as respond to emergencies during the disassembly process [6]. The robotic disassembly mode can achieve standardized operation and has the advantage of high production efficiency [7]. When handling waste products with complex structures and hazardous parts, the manned–robotic shared-station disassembly mode can distribute complex tasks to workers and hazardous tasks to robots, which can improve the disassembly efficiency and minimize the risk posed by hazardous parts to workers [8,9]. In this mode, the workers and robots do not interact with each other and just share the same station. However, they interact in the manned–robotic collaborative disassembly mode; more specifically, workers and robots collaboratively execute the same interactive task [10,11,12]. Considering the different number of operators configured in each station, this paper investigates six categories of DLBs: single-manned (SM-DLB), multi-manned (MM-DLB), single-robotic (SR-DLB), multi-robotic (MR-DLB), single-manned–robotic (SMR-DLB), and multi-manned–robotic (MMR-DLB). To reduce the complexity of unified modeling, this study sets the stations with multiple operators as shared-station disassembly mode, where no interaction occurs between operators. The schematic diagrams of the six types of DLBs are shown in Figure 1.

The optimization objective of DLBs usually includes the number of stations, cycle time, idle balancing index, hazardous index, cost, profit, etc. [13,14,15,16]. According to the different optimization objectives, the existing DLBs can be divided into three types: (1) Type-I DLBs minimize the number of stations under a given cycle time; (2) Type-II DLBs minimize the cycle time under a given number of stations; (3) Type-E DLBs maximize the line efficiency under an unknown cycle time and the number of stations [17,18]. As Type-I and Type-II DLBs are most widely used in realistic production-line scheduling optimization, this study attempts to establish mathematical programming models to solve Type-I and Type-II DLBs under six different operator configurations. The number of stations, cycle time, the number of operators, and idle balancing index are the basic production-line indicators, so they are subject to optimization in this study as multiple objectives. Additionally, due to varying levels of worker proficiency and differences in robot performance, their disassembly efficiency is also different for the same task. This results in different processing times for each task performed by different workers and robots. Therefore, the total performing time of all tasks is uncertain and needs to be taken as another optimization objective to reduce the total workload of the operators.

The mathematical programming model is a formulaic expression of the objective functions and constraints of a DLB. An accurate model can not only correctly describe the characteristics of the DLB but can also be used to solve the global optimal value of each objective. Many existing studies have employed swarm intelligence optimization algorithms [19] and reinforcement learning [20] to solve DLBs, but these works either neglect the mathematical formulation for the optimization objectives and the constraints for the DLBs, or the most established DLB models are conceptual and are only used to mathematically describe the DLB, which means that they cannot solve the global optimal values. Therefore, it is necessary to establish accurate mathematical programming models for DLBs. To solve the optimal solutions of the investigated DLBs, this paper employs the lexicographic approach [21] to jointly optimize the multiple objectives.

In the existing DLBs, five commonly used idle balancing indices are mentioned [22,23]. However, the most commonly used and comprehensive expression is the sum of squares of the idle time of each operator [13,24]. The operators in this study involve workers and robots, and each station contains a different number of operators. Therefore, this paper uses the sum of squares of the idle time of each worker and each robot as the idle balancing index. In Type-I DLBs, the quadratic nonlinearity of the idle balancing index cannot be applied to the lexicographic approach, so this study proposes a novel bi-metric (minimizing the maximum idle time of the assigned operators first and then minimizing the maximum workload of the assigned operators) to realize the hierarchical optimization of multiple objectives and obtain more non-inferior solutions. In Type-II DLBs, the idle balancing index simultaneously contains two types of nonlinearities: the quadratic terms and the mixed product of two decision variables. Therefore, it cannot be replaced by the proposed bi-metric to participate in lexicographic optimization, and only the other three linear objectives are hierarchically optimized. Finally, the established models are applied to the actual disassembly cases of lighters and hairdryers to verify the effectiveness of the proposed bi-metric and the correctness of the models.

The main contributions of this study are summarized as follows:

• Type-I and Type-II DLBs with six distinct operator configurations are unified, modeled, and solved by the mixed-integer programming approaches;
• Considering the efficiency difference between operators (workers and robots) makes the DLBs investigated more practical;
• In Type-I DLBs, a novel bi-metric is proposed to replace the idle balancing index to participate in lexicographic optimization, which can obtain more non-inferior solutions.

The rest of this paper is organized as follows: Section 2 reviews the literature on the main factors influencing DLBs. Section 3 establishes a unified mixed-integer programming model for the Type-I DLBs. The novel bi-metric is proposed to replace the idle balancing index to participate in lexicographic optimization. Then, two actual disassembly cases of the lighter and hairdryer are solved, their optimal solutions for each operator configuration are obtained by the models and the proposed bi-metric, and some discussions are presented. Section 4 establishes the unified mixed-integer programming models for the six Type-II DLBs based on the common objectives and constraints of Type-I DLBs and solves the two disassembly cases. Section 5 concludes this study and presents some future research directions.

2. Related Work

DLB was proposed by Gupta et al. in 2001 [25]. With the emphasis on resource recovery and remanufacturing industries, it has gradually become a research hotspot in academic and engineering fields. DLB involves the sequence constraint of disassembly tasks, the assignment of operators to stations, and the distribution of disassembly tasks to operators. Thus, this section mainly reviews the factors that affect these three parts.

During optimization, the disassembly sequence of tasks must satisfy the precedence relationships of products to ensure that the removal of each part does not violate its physical connection constraints [26]. The existing expressions of precedence relationships are mainly divided into four types: the part precedence diagram (PPD), the task precedence diagram (TPD), the transformed AND/OR graph (TAOG), and the Petri net (PN). The PPD defines the removal of a part or component as a disassembly task [27], whereas the TPD defines the break of the connection constraint between two adjacent parts as a task [28]. When describing the precedence relationship of a product with many same-connection constraints, the number of tasks in TPD is lower than that in the PPD, which comes at the cost of losing the removal status of the parts (removal status represents the part removal and remaining situation corresponding to the current task). Specifically, a task in the TPD cannot distinguish the removal status of multiple parts in a same connection constraint, but the task in the PPD can distinguish this situation. This inexhaustive task definition in the TPD may lead to the fact that the optimized scheme may not be the actual optimal. The TAOG and PN have been developed based on the AOG [29,30], and both contain all the tasks (disassembly operations) and removal status [5,31]. Furthermore, the TAOG and PN contain all the feasible disassembly trees of a product (performing tasks along any one disassembly tree can finish the complete disassembly of a product), so they are too complicated to describe waste products with large-scale parts [17]. Therefore, the PPD will be employed in this study.

Traditional DLBs focus on SM-DLB. For example, Gupta et al. designed a single-manned disassembly line for a personal computer and studied the corresponding SM-DLB [32]. Ren et al. developed a heuristic algorithm combining multi-criterion decision making and general variable neighborhood search to optimize the SM-DLB [33]. Wang et al. considered the differences in worker efficiency in SM-DLB and solved it using a discrete flower pollination algorithm [34]. To improve efficiency, Cevikcan et al. designed a multi-manned shared-station disassembly line and proposed an MM-DLB model [35]. Subsequently, Ibrahim et al. proved that the disassembly efficiency of the MM-DLB model is higher than that of the SM-DLB model [18]. Abidin ÇIL used the constraint programming approach and an iterative genetic algorithm to solve the MM-DLB model [36]. Yılmaz et al. explored the MM-DLB model by considering the heterogeneity of workers and optimized it using an improved augmented ϵ-constrained method [37]. To reduce the risk to workers when disassembling products with complex structures and hazardous parts, Liu et al. added a robot to each single-manned station and proposed the SMR-DLB model. They stipulated that complex tasks can only be performed by workers, hazardous tasks only by robots, and normal tasks by both operators [38]. Xu et al. considered task failure in the SMR-DLB model and developed a multi-objective artificial bee colony algorithm to optimize this problem [39]. With the development of robot technology, Liu et al. used robots to replace workers in a single-manned disassembly line and developed a space interference matrix method and an improved discrete bee algorithm to study the SR-DLB [40,41]. Abidin ÇIL et al. established a mathematical programming model of Type-II SR-DLB based on the TAOG and proposed an improved ant colony algorithm to solve the problem [42]. Fang et al. proposed a multi-robotic shared-station disassembly mode and optimized the MR-DLB model using various algorithms [43,44,45,46]. Later, Liu et al. further developed their research based on the work of Fang [47]. The above literature either did not consider operator efficiency differences or the established models cannot be used to solve the optimal values of the objectives, so this paper will develop mixed-integer programming models for DLBs with six distinct operator quantity configurations and consider the differences in operator efficiency.

The assignment process of tasks to operators is influenced by many factors, such as task attributes, disassembly level, and line layout. In terms of the task attribute, except for the above-mentioned complex and hazardous attributes, the tasks in waste products may have uncertain end-of-life states owing to wear, corrosion, looseness, and other reasons [48]. This uncertainty further affects the task assignment process. In terms of the disassembly level, the existing DLBs can be divided into complete and partial disassembly. Complete disassembly requires all parts of a product to be removed, whereas partial disassembly only involves removing the parts of interest [49]. Complete disassembly accounts for 82% of the existing literature, whereas the remaining studies focus on partial disassembly [13]. Thus, the complete disassembly mode is adopted in this study. In terms of line layout, there are four main types: straight, U-shaped, two-sided, and parallel lines. The straight line is the original type and is the basis for the other three layouts. Agrawal et al. bent the straight line to make the inlet and outlet on the same side to form the U-shaped line and used a collaborative ant colony algorithm to solve the U-shaped DLB [50]. Wang et al. arranged stations on the two sides of a straight line to form a two-sided line and studied the stochastic two-sided partial DLB [51]. Typically, a straight line can handle only one type of product. To improve the capacity of product types, Hezer et al. designed two straight lines in parallel, placed the shared stations in the middle of the two lines to form a parallel disassembly line, and then proposed a parallel DLB [52]. It can be observed from the listed instructions that the same disassembly task sequence corresponds to different disassembly assignment schemes, in different layouts. To date, the straight disassembly line is the most widely used in the actual disassembly scene. Therefore, this study adopts this layout for subsequent studies.

3. Type-I DLB: Modelling and Optimization

As mentioned in Section 1, Type-I DLBs optimize the number of stations under a given cycle time, so they are suitable for the design stage of a new disassembly line and can provide a reference for purchasing stations. To establish the models for the six distinct DLBs, the following notations are defined for tasks, workers, robots, and stations.

Indices:
i, j	Task index.
k	Station index.
w	Worker index.
r	Robot index.
Sets and parameters:
T	Task index set, i, j ∈ T.
K	Station index set, k ∈ K.
W	Worker index set, w ∈ W.
R	Robot index set, r ∈ R.
$N_{max}^{\mathrm{h}}$	The maximum number of workers per station (h means human).
$N_{max}^{\mathrm{m}}$	The maximum number of robots per station (m means machine).
CT	Cycle time.
KN	The number of the open station.
ON	The number of assigned operators.
TT	Total time to perform all tasks.
IB	Idle balancing index for assigned operators.
ST	The maximum idle time of assigned operators.
LT	The maximum workload of assigned operators.
P(i)	Immediate predecessor set of task i.
ψ	A large positive number.
tiw	Time for worker w performing task i. Here, the processing time of each task performed by different workers is different due to the influence of the skill differences of workers.
tir	Time for robot r to perform task i. Here, the processing time of each task performed by different robots is different due to the influence of the performance differences.
ci	Complex attribute: 1, task i is too complex to be performed by robots; 0, otherwise.
hi	Hazardous attribute: 1, task i is harmful to workers; 0, otherwise.
Decision Variables:
$t_i^s$	Start time of task i.
$x_{ikw}^{\mathrm{h}}$	Assignment of tasks to workers: 1, task i is assigned to worker w in station k; 0, otherwise.
$x_{ikr}^{\mathrm{m}}$	Assignment of tasks to robots: 1, task i is assigned to robot r in station k; 0, otherwise.
$y_{kw}^{\mathrm{h}}$	Employment of workers: 1, worker w is assigned to station k; 0, otherwise.
$y_{kr}^{\mathrm{m}}$	Employment of robots: 1, robot r is assigned to station k; 0, otherwise.
$z_{ijkw}^{\mathrm{h}}$	Task ranking variable in workers: 1, task i and j are assigned to worker w in station k, and i is ranked before j; 0, otherwise.
$z_{ijkr}^{\mathrm{m}}$	Task ranking variable in robots: 1, task i and j are assigned to robot r in station k, and i is ranked before j; 0, otherwise.
sk	Station usage variable: 1, station k is used; 0, otherwise.
Mathematical operator:
| |	Cardinal number of a set.

3.1. Models of the Type-I DLBs with the Six Operator Configurations

Because both the SMR-DLB and MMR-DLB adopt two types of operators, that is, workers and robots, whereas SM-DLB, MM-DLB, SR-DLB, and MR-DLB only adopt one type of operator, a unified model of SMR-DLB and MMR-DLB can be established first. Afterwards, the unified model can be simplified for the other four types of DLBs. In addition, in the SMR-DLB and MMR-DLB models, the tasks are divided into complex, hazardous, and normal tasks, only performed by workers, only by robots, and by both operators, respectively [8]. On the other hand, the task classification method is not valid in the SM-DLB, MM-DLB, SR-DLB, and MR-DLB models.

3.1.1. Unified Model of the Type-I SMR-DLB and MMR-DLB

(1)

Optimization objectives:

In Type-I DLBs, in addition to the number of stations, the number of operators and the idle balancing index are the basic objectives to be optimized. Minimizing the number of stations and operators can effectively reduce the purchase and operational costs of the disassembly line. Minimizing the idle balancing index can balance the workload of each operator as much as possible, avoid situations where some workers are idle while others are busy, and further improve the overall disassembly efficiency. In addition, the processing time of each task performed by different workers and robots is different, so minimizing the total performing time can help avoid choosing the operator who takes a longer time to perform the current task during the optimization. The four optimization objectives f₁–f₄ are calculated as follows:

1)

The number of opened stations:

$$f_1:\min KN={\displaystyle \sum _{k\in K}{s}_k}$$

(1)

2)

The number of assigned workers and robots:

$$f_2:\min ON={\displaystyle \sum _{k\in K}({\displaystyle \sum _{w\in W}{y}_{kw}^{\mathrm{h}}}+{\displaystyle \sum _{r\in R}{y}_{kr}^{\mathrm{m}}})}$$

(2)

3)

The total performing time of all tasks by the assigned operators:

$$f_3:\min TT={\displaystyle \sum _{k\in K}{\displaystyle \sum _{i\in T}({\displaystyle \sum _{w\in W}{x}_{ikw}^{\mathrm{h}}\cdot t_{iw}}+{\displaystyle \sum _{r\in R}{x}_{ikr}^{\mathrm{m}}\cdot t_{ir}})}}$$

(3)

4)

The idle balancing index for operators (the sum of squares of the idle time of each assigned worker and robot):

$$f_4:\min IB={\displaystyle \sum _{k\in K}({\displaystyle \sum _{w\in W}{(y_{kw}^{\mathrm{h}}\cdot CT-{\displaystyle \sum _{i\in T}{x}_{ikw}^{\mathrm{h}}\cdot t_{iw}})}^2}}+{\displaystyle \sum _{r\in R}{(y_{kr}^{\mathrm{m}}\cdot CT-{\displaystyle \sum _{i\in T}{x}_{ikr}^{\mathrm{m}}\cdot t_{ir}})}^2})$$

(4)

(2)

Constraints:

In the SMR-DLB and MMR-DLB models, the distribution of disassembly tasks to operators involves the following constraints:

1)

Because of the complete disassembly mode, each task is either performed by robots or workers (C₁ represents the codename of the first constraint).

$$C_1:{\displaystyle \sum _{k\in K}({\displaystyle \sum _{w\in W}{x}_{ikw}^{\mathrm{h}}}+{\displaystyle \sum _{r\in R}{x}_{ikr}^{\mathrm{m}}})}=1\hspace{1em}\forall i\in T$$

(5)

2)

According to the task classification mentioned above, constraint C₂ ensures that the complex tasks are only performed by workers, and constraint C₃ ensures that the hazardous tasks are only performed by robots. Notably, when a task contains both complex and hazardous attributes, the task is still handed over to workers because robots cannot process the complex attribute; namely, the complex attribute of a task has a higher priority than its hazardous attribute, and this situation is included in constraint C₂. In addition, because normal tasks can be performed by both workers and robots, they are not constrained, to ensure that they can be randomly distributed between the two types of operators.

$$C_2:{\displaystyle \sum _{k\in K}{\displaystyle \sum _{w\in W}{x}_{ikw}^{\mathrm{h}}}}=1\hspace{1em}\forall i\in \{\left.i\right|i\in T,c_i=1\}$$

(6)

$$C_3:{\displaystyle \sum _{k\in K}{\displaystyle \sum _{r\in R}{x}_{ikr}^{\mathrm{m}}}}=1\hspace{1em}\forall i\in \{i\left|i\in T,h_i>c_i\}\right.$$

(7)

3)

In the disassembly process, the execution order of the tasks must meet their precedence relationships. Specifically, on the time axis with the start time of the first station as the zero point, the start time of the immediately following task should be greater than the end time of the immediately preceding task.

$$C_4:t_j^s\ge t_i^s+{\displaystyle \sum _{k\in K}({\displaystyle \sum _{w\in W}{x}_{ikw}^{\mathrm{h}}\cdot t_{iw}}+{\displaystyle \sum _{r\in R}{x}_{ikr}^{\mathrm{m}}\cdot t_{ir}})}\hspace{1em}\forall j\in T,\forall i\in P(j)$$

(8)

4)

Each assigned worker or robot can only perform one task at a time; that is, the tasks assigned to one worker or robot are ordered. On the time axis, the start time of the tasks assigned to an operator later should be greater than the end time of the tasks first assigned to the operator. C₅ constrains the execution order of tasks assigned to workers, and C₆ constrains the execution order of tasks assigned to robots.

$$C_5:\psi \cdot (1-z_{ijkw}^{\mathrm{h}})+t_j^s\ge t_i^s+t_{iw}\forall i,j\in T,i\ne j,\forall k\in K,\forall w\in W$$

(9)

$$C_6:\psi \cdot (1-z_{ijkr}^{\mathrm{m}})+t_j^s\ge t_i^s+t_{ir}\forall i,j\in T,i\ne j,\forall k\in K,\forall r\in R$$

(10)

In the notation, $z_{ijkw}^h$ defines the assignment of tasks to workers and the execution order of tasks assigned to the workers, whereas $x_{ikw}^h$ also defines the assignment of tasks to workers. Thus, from the closed-loop perspective of modelling, these two variables should be related. The relationship between the two variables can reflect three physical cases: (1) The different tasks i and j assigned to worker w must have execution order constraints; that is, when $x_{ikw}^h$ = 1 and $x_{jkw}^h$ = 1, one of $z_{ijkw}^h$ and $z_{jikw}^h$ must be equal to 1 (the definitions of $z_{ijkw}^h$ and $z_{jikw}^h$ guarantee that neither of them can be equal to 1 at the same time). (2) The different tasks i and j assigned to the different workers must have no execution order constraints, except for their precedence relationships; that is, when $x_{ikw}^h$ = 1 and $x_{jkw}^h$ = 0 or $x_{ikw}^h$ = 0 and $x_{jkw}^h$ = 1, both $z_{ijkw}^h$ and $z_{jikw}^h$ must be equal to 0. (3) The different tasks i and j that are not assigned to worker w have no execution order constraints for worker w; that is, when $x_{ikw}^h$= 0 and $x_{jkw}^h$ = 0, both $z_{ijkw}^h$ and $z_{jikw}^h$ must be equal to 0. These three physical cases can be represented by constraints C₇ and C₈. Similarly, constraints C₉ and C₁₀ define the three robot cases.

$$C_7:x_{ikw}^{\mathrm{h}}+x_{jkw}^{\mathrm{h}}\le 1+(z_{ijkw}^{\mathrm{h}}+z_{jikw}^{\mathrm{h}})\hspace{1em}\forall i,j\in T,i<j,\forall k\in K,\forall w\in W$$

(11)

$$C_8:{\displaystyle \frac{1}{2}}(x_{ikw}^{\mathrm{h}}+x_{jkw}^{\mathrm{h}})\ge z_{ijkw}^{\mathrm{h}}+z_{jikw}^{\mathrm{h}}\hspace{1em}\forall i,j\in T,i<j,\forall k\in K,\forall w\in W$$

(12)

$$C_9:x_{ikr}^{\mathrm{m}}+x_{jkr}^{\mathrm{m}}\le 1+(z_{ijkr}^{\mathrm{m}}+z_{jikr}^{\mathrm{m}})\hspace{1em}\forall i,j\in T,i<j,\forall k\in K,\forall r\in R$$

(13)

$$C_{10}:{\displaystyle \frac{1}{2}}(x_{ikr}^{\mathrm{m}}+x_{jkr}^{\mathrm{m}})\ge z_{ijkr}^{\mathrm{m}}+z_{jikr}^{\mathrm{m}}\hspace{1em}\forall i,j\in T,i<j,\forall k\in K,\forall r\in R$$

(14)

5)

Owing to the limitation of cycle time, the starting point of the time at which each operator can start executing tasks is the end time of the previous station. Any task assigned to an operator should wait until all tasks assigned to the operator before that task are completed. Therefore, the start time of a task assigned to an operator should be greater than the sum of the time start point of the operator and the total performing time of all tasks assigned to the operator before the task, which is expressed by constraint C₁₁. Evidently, the end time of a task cannot exceed the end time of the station where it is located, and this is expressed by constraint C₁₂.

$$C_{11}:t_j^s\ge CT\cdot ({\displaystyle \sum _{k\in K}({\displaystyle \sum _{w\in W}{x}_{jkw}^{\mathrm{h}}}+{\displaystyle \sum _{r\in R}{x}_{jkr}^{\mathrm{m}}})}\cdot k-1)+{\displaystyle \sum _{k\in K}{\displaystyle \sum _{i\in T,i\ne j}({\displaystyle \sum _{w\in W}{z}_{ijkw}^{\mathrm{h}}\cdot t_{iw}}+{\displaystyle \sum _{r\in R}{z}_{ijkr}^{\mathrm{m}}\cdot t_{ir}})}}\hspace{1em}\forall j\in T$$

(15)

$$C_{12}:t_i^s+{\displaystyle \sum _{k\in K}({\displaystyle \sum _{w\in W}{x}_{ikw}^{\mathrm{h}}\cdot t_{iw}}+{\displaystyle \sum _{r\in R}{x}_{ikr}^{\mathrm{m}}\cdot t_{ir}})}\le CT\cdot {\displaystyle \sum _{k\in K}({\displaystyle \sum _{w\in W}{x}_{ikw}^{\mathrm{h}}}+{\displaystyle \sum _{r\in R}{x}_{ikr}^{\mathrm{m}}})}\cdot k\hspace{1em}\forall i\in T$$

(16)

6)

Because every task in the waste product must be performed, the first station must be opened. Starting from the second station, each subsequent station can be opened only when at least one task and at least one operator are assigned to the station; otherwise, it will not be opened. Thus, the total number of opened stations is at least one and does not exceed the minimum value of the total number of tasks and the total number of given operators, which can be ensured by constraint C₁₃.

$$C_{13}:1\le {\displaystyle \sum _{k\in K}{s}_k}\le \min (\left|T\right|,\left|W\right|+\left|R\right|)$$

(17)

If only the first station is opened, all tasks must be assigned to this station, and no tasks are assigned to subsequent stations. If multiple stations are opened, the number of tasks assigned to each opened station should be less than the total number of tasks, while no tasks should be assigned to the unopened stations. These two situations can be constrained by C₁₄.

$$C_{14}:s_k\le {\displaystyle \sum _{i\in T}({\displaystyle \sum _{w\in W}{x}_{ikw}^{\mathrm{h}}+{\displaystyle \sum _{r\in R}{x}_{ikr}^{\mathrm{m}}}})}\le s_k\cdot \left|T\right|\forall k\in K$$

(18)

In addition, because the stations are arranged in sequence along the disassembly line, they must be turned on in sequence to reduce the transportation time of products on the line. Constraint C₁₅ can describe this case:

$$C_{15}:s_k\le s_{k-1}\hspace{1em}\hspace{1em}\forall k\in K,k\ne 1$$

(19)

7)

The SMR-DLB requires each station to employ at most one worker and one robot, whereas the MMR-DLB allows each station to employ multiple workers and robots. Thus, the variables $N_{max}^{\mathrm{h}}$ and $N_{max}^{\mathrm{m}}$ are used to limit the maximum number of operators assigned to each station. When $N_{max}^{\mathrm{h}}$ = 1 and $N_{max}^{\mathrm{m}}$ = 1, this model can be used to solve the SMR-DLB, and when $N_{max}^{\mathrm{h}}$ > 1 or $N_{max}^{\mathrm{m}}$ > 1, this model can be used to solve the MMR-DLB. Such limitations can be imposed by constraints C₁₆ and C₁₇.

$$C_{16}:0\le {\displaystyle \sum _{w\in W}{y}_{kw}^{\mathrm{h}}}\le N_{max}^{\mathrm{h}}\hspace{1em}\hspace{1em}\forall k\in K$$

(20)

$$C_{17}:0\le {\displaystyle \sum _{r\in R}{y}_{kr}^{\mathrm{m}}}\le N_{max}^{\mathrm{m}}\hspace{1em}\hspace{1em}\forall k\in K$$

(21)

Similar to constraint C₁₄, if no tasks are assigned to an operator, the operator does not need to be assigned to the stations; otherwise, the number of tasks assigned to the operator in the station cannot exceed the total number of tasks. Here, constraints C₁₈ and C₁₉ are used to consider the situations of workers and robots, respectively.

$$C_{18}:y_{kw}^{\mathrm{h}}\le {\displaystyle \sum _{i\in T}{x}_{ikw}^{\mathrm{h}}}\le y_{kw}^{\mathrm{h}}\cdot \left|T\right|\hspace{1em}\forall k\in K,\forall w\in W$$

(22)

$$C_{19}:y_{kr}^{\mathrm{m}}\le {\displaystyle \sum _{i\in T}{x}_{ikr}^{\mathrm{m}}}\le y_{kr}^{\mathrm{m}}\cdot \left|T\right|\hspace{1em}\forall k\in K,\forall r\in R$$

(23)

Considering that some workers or robots among the candidates may not be assigned, it is necessary to constrain the state of each candidate operator, as expressed by constraints C₂₀ and C₂₁. When the left side of the formula is 0, the operator is not assigned, and when it is 1, the operator is assigned.

$$C_{20}:{\displaystyle \sum _{k\in K}{y}_{kw}^{\mathrm{h}}}\le 1\hspace{1em}\forall w\in W$$

(24)

$$C_{21}:{\displaystyle \sum _{k\in K}{y}_{kr}^{\mathrm{m}}}\le 1\hspace{1em}\forall r\in R$$

(25)

8)

Constraint C₂₂ lists all the binary variables used in the model.

$$C_{22}:x_{ikw}^{\mathrm{h}},x_{ikr}^{\mathrm{m}},y_{kw}^{\mathrm{h}},y_{kr}^{\mathrm{m}},z_{ijkw}^{\mathrm{h}},z_{ijkr}^{\mathrm{m}},s_k,c_i,h_i\in \left\{0,1\right\}\hspace{1em}\forall i,j\in T,\forall k\in K,\forall w\in W,\forall r\in R$$

(26)

3.1.2. Unified Model of the Type-I SM-DLB, MM-DLB, SR-DLB, and MR-DLB

Because the SM-DLB, MM-DLB, SR-DLB, and MR-DLB models adopt a single type of operator, it is necessary to simplify the unified model in Section 3.1.1 to solve these four types of problems. To solve the SM-DLB and MM-DLB, except for the common objective functions and constraints, the constraints and variables related to robots in the unified model must be removed, and the simplification is as follows:

• Kept: f₁, C₅, C₇–C₈, C₁₅–C₁₆, C₁₈, C₂₀;
• Removed: C₂–C₃, C₆, C₉–C₁₀, C₁₇, C₁₉, C₂₁;
• Modified: f₂–f₄, C₁, C₄, C₁₁–C₁₄, C₂₂.
• The modified parts are executed in detail below:

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Optimization objective:

To distinguish these four types of DLBs, the superscript “*” is used to mark the stations with only workers, and the superscript “#” is used to mark the stations with only robots; for example, $f_1^{*}$ = f₁, and $f_1^{\#}$ = f₁.

$$f_2^{*}:\min ON={\displaystyle \sum _{k\in K}{\displaystyle \sum _{w\in W}{y}_{kw}^{\mathrm{h}}}}$$

(27)

$$f_3^{*}:\min TT={\displaystyle \sum _{k\in K}{\displaystyle \sum _{w\in W}{\displaystyle \sum _{i\in T}{x}_{ikw}^{\mathrm{h}}\cdot t_{iw}}}}$$

(28)

$$f_4^{*}:\min BI={\displaystyle \sum _{k\in K}{\displaystyle \sum _{w\in W}{(y_{kw}^{\mathrm{h}}\cdot CT-{\displaystyle \sum _{i\in T}{x}_{ikw}^{\mathrm{h}}\cdot t_{iw}})}^2}}$$

(29)

(2)

Constraints:

$$C_1^{*}:{\displaystyle \sum _{k\in K}{\displaystyle \sum _{w\in W}{x}_{ikw}^{\mathrm{h}}}}=1\hspace{1em}\forall i\in T$$

(30)

$$C_4^{*}:t_j^s\ge t_i^s+{\displaystyle \sum _{k\in K}{\displaystyle \sum _{w\in W}{x}_{ikw}^{\mathrm{h}}\cdot t_{iw}}}\hspace{1em}\forall j\in T,\forall i\in P(j)$$

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$$C_{11}^{*}:t_j^s\ge CT\cdot ({\displaystyle \sum _{k\in K}{\displaystyle \sum _{w\in W}{x}_{jkw}^{\mathrm{h}}}}\cdot k-1)+{\displaystyle \sum _{k\in K}{\displaystyle \sum _{w\in W}{\displaystyle \sum _{i\in T,i\ne j}{z}_{ijkw}^{\mathrm{h}}\cdot t_{iw}}}}\hspace{1em}\forall j\in T$$

(32)

$$C_{12}^{*}:t_i^s+{\displaystyle \sum _{k\in K}{\displaystyle \sum _{w\in W}{x}_{ikw}^{\mathrm{h}}\cdot t_{iw}}}\le CT\cdot {\displaystyle \sum _{k\in K}{\displaystyle \sum _{w\in W}{x}_{ikw}^{\mathrm{h}}}\cdot k}\hspace{1em}\forall i\in T$$

(33)

$$C_{13}^{*}:1\le {\displaystyle \sum _{k\in K}{s}_k}\le \min (\left|T\right|,\left|W\right|)$$

(34)

$$C_{14}^{*}:s_k\le {\displaystyle \sum _{i\in T}{\displaystyle \sum _{w\in W}{x}_{ikw}^{\mathrm{h}}}}\le s_k\cdot \left|T\right|\hspace{1em}\forall k\in K$$

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$$C_{22}^{*}:x_{ikw}^{\mathrm{h}},y_{kw}^{\mathrm{h}},z_{ijkw}^{\mathrm{h}},s_k,c_i,h_i\in \left\{0,1\right\}\hspace{1em}\forall i,j\in T,\forall k\in K,\forall w\in W$$

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The mathematical model formed by the above kept and modified parts is the unified form of the SM-DLB and MM-DLB models, and it can be used to solve these two types of DLBs by adjusting $N_{max}^{\mathrm{h}}$ = 1 and $N_{max}^{\mathrm{h}}$ > 1, respectively. In addition, from the optimization point of view, there is no essential difference between the workers in the SM-DLB and MM-DLB and the robots in the SR-DLB and MR-DLB, and only the number of operators in each station differs. Therefore, the above-simplified model can also be used to solve the SR-DLB and MR-DLB models by replacing all the worker variables with robot variables.

3.1.3. A Bi-Metric to Replace Quadratic Idle Balancing Index

The expressions of the four objectives above show that the first three objectives (the number of stations, number of operators, and total disassembly time) are linear, and the fourth objective (the idle balancing index) is quadratic nonlinear. The existing exact optimizers can solve their single-objective global optimal values individually, but the multi-objective solution corresponding to the single-objective optimal value (namely, the single-objective optimal solution) is not necessarily optimal for the other three objectives. Thus, this study employs the lexicographic approach to achieve multi-objective hierarchical joint optimization and explore the real Pareto frontier.

The lexicographic approach in exact optimizers cannot address quadratic nonlinear terms, so the fourth objective cannot directly participate in lexicographic optimization. To address this issue, the nature of the idle balancing index needs to be explored: each operator has as little idle time as possible, and their workloads are distributed as equally as possible. Figure 2 illustrates the workload distribution of nine operators (five workers and four robots) in three stations, under a given cycle time. The symbols k1–k3 are the three station numbers; w1–w5 and r1–r4 are the numbers of the five workers and four robots, respectively; and the rectangles represent the workloads of the assigned operators. To make the idle time of each assigned operator as small as possible, the maximum idle time of the assigned operators ST (corresponding to the operator with the worst workload) should be focused on. As can be seen in Figure 2, minimizing ST causes station 3 to be closed, workers 4–5 and robot 4 are no longer assigned, and the workloads of workers 1–3 and robots 1–3 increase (correspondingly, the idle time of the assigned operators is relatively minimal and does not exceed the minimized ST value). After minimizing the ST, the workload of each operator may still be unbalanced. Therefore, the maximum workload of the assigned operators LT should also be minimized to suppress the maximum workload and balance the workloads of assigned operators as much as possible. The essence of minimizing ST first and then LT is to first obtain the upper limits of the minimum workload of assigned operators and then the lower limit of the maximum workload of assigned operators to make their workloads relatively balanced. While the old idle balancing index focuses on the idle balancing state of each assigned operator, the hierarchical alliance of the bi-metric belongs to the slack objective of the idle balancing index.

After minimizing the ST, the minimized SL is definitely equal to or less than the ST value in the single-objective global optimal scheme of the idle balancing index. However, the idle time of other operators whose idle time is not equal to the minimized SL may not be less than that of these operators in the single-objective optimal scheme. This would lead to the bi-metric method being unable to solve the single-objective optimal value of the idle balancing index. Then, minimizing the LT will result in choosing the operators with less time to perform the tasks and further make the total disassembly time of all tasks (the third optimization objective) less than or equal to that in the single-objective optimal solution of the idle balancing index. Therefore, the bi-metric method offers the possibility of obtaining new additional non-inferior solutions that are different from the single-objective optimal solution.

The expressions of the idle balancing index in the MMR-DLB and MM-DLB (SMR-DLB and SM-DLB) are different, so their bi-metrics are expressed differently, as shown below. Although the expressions of ST and LT are also nonlinear, the existing exact optimizers provide the maximum functions to internally linearize these two alternative indices. For example, the CPLEX optimizer provides the max() keyword to calculate the maxima of a linear expression and the maxl() function to obtain the maxima of multiple linear expressions. Thus, the bi-metric objectives f₅ and f₆ ($f_5^{*}$ and $f_6^{*}$) can replace the idle balancing index to participate in lexicographic optimization. Notably, objective f₅ must precede objective f₆, which is the case for objectives $f_5^{*}$ and $f_6^{*}$.

$$f_5:\min ST=\underset{\forall k\in K}{\max }(\underset{\forall w\in W,\forall r\in R}{max}(y_{kw}^{\mathrm{h}}\cdot CT-{\displaystyle \sum _{i\in T}{x}_{ikw}^{\mathrm{h}}\cdot t_{iw}},y_{kr}^{\mathrm{m}}\cdot CT-{\displaystyle \sum _{i\in T}{x}_{ikr}^{\mathrm{m}}\cdot t_{ir}}))$$

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$$f_6:\min LT=\underset{\forall k\in K}{\max }(\underset{\forall w\in W,\forall r\in R}{\max }({\displaystyle \sum _{i\in T}{x}_{ikw}^{\mathrm{h}}\cdot t_{iw}},{\displaystyle \sum _{i\in T}{x}_{ikr}^{\mathrm{m}}\cdot t_{ir}}))$$

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$$f_5^{*}:\min ST=\underset{\forall k\in K,\forall w\in W}{\max }(y_{kw}^{\mathrm{h}}\cdot CT-{\displaystyle \sum _{i\in T}{x}_{ikw}^{\mathrm{h}}\cdot t_{iw}})$$

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$$f_6^{*}:\min LT=\underset{\forall k\in K,\forall w\in W}{\max }({\displaystyle \sum _{i\in T}{x}_{ikw}^{\mathrm{h}}\cdot t_{iw}})$$

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3.2. Actual Case Calculation and Verification

Two actual disassembly cases of waste lighters and hairdryers are solved using the above-mentioned mixed-integer programming models, with six distinct operator configurations, to verify the effectiveness of the bi-metric and the correctness of these models.

3.2.1. Case I: Lighter

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Basic disassembly information:

The correspondence between the parts and tasks and the PPD of the lighter are shown in Figure 3a,b, respectively. Because the cap and wick are not easily removed, tasks 1 and 5 are considered complex tasks and marked by the red symbol “c”. Because of the electrical potential in the igniter and the flammable liquid in the plastic tank, tasks 6 and 8 are considered hazardous tasks and marked by “h”. The other tasks (2—switch, 3—lever, 4—adjuster, and 7—electronic igniter padding) are considered normal tasks. The PPD can be converted into a binary matrix TP, as shown in Figure 3c, to optimize the DLBs [53]. Moreover, the given cycle time is CT = 30, the candidate worker numbers are 1 to 4, the candidate robot numbers are 1 to 3, and the processing times of all tasks performed by the candidate operators are listed in

Table 1. Disassembly time of a lighter by candidate workers and robots.

Tasks	Workers (tiw)				Robots (tir)
	1	2	3	4	1	2	3
1	12	12	14	13	14	12	13
2	6	7	5	6	3	5	3
3	8	5	6	6	7	9	8
4	11	12	10	12	7	9	11
5	16	15	17	18	18	17	16
6	6	9	5	8	4	5	3
7	5	4	6	7	3	5	4
8	7	4	5	6	5	3	5

.

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Optimization results and analysis:

The programming models of the six types of DLBs were programmed using the OPL language in IBM ILOG CPLEX Studio IDE 20.1.0, and solved using the CPLEX optimizer in Win10 system with an Intel (R) Core (TM) i5-9400 2.9 GHz and 8 GB RAM. Because the idle balancing index cannot participate in lexicographic optimization, its single-objective global optimal solution can be obtained first. Lexicographic optimization is then performed for the other three linear objectives. Because the total number of permutations of the three objectives is six, the lexicographic optimization should be calculated six times for each type of DLB. The solutions obtained through these two steps are used as benchmarks to verify the effectiveness of the bi-metric method. When the bi-metric participates in lexicographic optimization together with the other three objectives, the number of optimizations is 60. This study focuses on whether the bi-metric method can obtain new solutions, so the bi-metrics are bundled together, and only 24 permutations need to be calculated for each type of DLB. The results of solving the six DLBs using the original model and bi-metric method are given in

Table 2. Results of Type-I MMR-DLB and SMR-DLB for the lighter case.

								MMR-DLB (${\mathit{N}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathbf{h}}$ = 2 and ${\mathit{N}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathbf{m}}$ = 2)
	No.	f1	f2	f3	f4	f5	f6	Priority Order of Objectives f (Time/s)	Notes
Original model	1	2	3	66	224	12	26	4(1.72).	-
	2	2	3	59	353	-	-	123(0.47), 231(1.08).	-
	3	2	3	59	371	-	-	213(0.59).	Inferior to No.2
	4	2	5	52	2112	-	-	132(0.48).	-
	5	3	5	51	2129	-	-	312(0.33), 321(0.34).	-
Bi-metric method	6	2	3	64	244	12	24	56(1.72).	Non-inferior to No.1
	7	2	3	59	353	14	24	12,356(1.38), 21,356(1.78), 23,561(2.36), 23,156(1.78).	Same as No.2
	8	2	3	61	301	12	24	56,123(1.98), 56,132(1.95), 56,213(2.08), 56,231(2.30), 56,312(2.08), 56,321(2.24), 15,623(1.26), 15,632(1.23), 12,563(1.25), 25,613(1.89), 25,631(1.95), 21,563(1.56).	New solution
	9	2	5	52	2070	27	17	13,256(1.03).	Dominates No.4
	10	2	6	52	2830	27	15	13,562(1.05).	Inferior to No.9
	11	3	5	51	2129	27	20	35,612(1.13), 35,621(1.09).	Same as No.5
	12	3	5	51	2171	27	20	31,562(0.94), 31,256(0.94), 32,561(0.91), 32,156(0.92).	Inferior to No.11
								SMR-DLB ($N_{\mathrm{m}ax}^{\mathrm{h}}$ = 1 and $N_{\mathrm{m}ax}^{\mathrm{m}}$ = 1)
Original model	1	2	3	66	224	12	26	4(1.20).	-
	2	2	3	59	353	-	-	231(0.86).	-
	3	2	3	59	371	-	-	123(0.47), 213(0.47).	Inferior to No.2
	4	2	4	54	1166	-	-	132(0.44).	-
	5	3	4	52	1322	-	-	312(0.51), 321(0.48).	-
Bi-metric method	6	2	3	61	301	12	24	56(1.25)	Non-inferior to No.1
	7	2	3	59	353	14	24	12,356(1.34), 21,356(1.27), 23,561(2.44), 23,156(1.77).	Same as No.2
	8	2	3	61	301	12	24	56,123(2.19), 56,132(2.26), 56,213(2.25), 56,231(2.53), 56,312(2.20), 56,321(2.42), 15,623(1.33), 15,632(1.39), 12,563(1.53), 25,613(1.58), 25,631(1.72), 21,563(1.22).	Same as No.6
	9	2	4	54	1166	24	17	13,562(1.31), 13,256(1.30).	Same as No.4
	10	3	4	52	1322	27	20	31,256(1.64), 32,561(1.73), 32,156(1.42).	Same as No.5
	11	3	5	52	2070	27	17	31,562(1.03).	Inferior to No.10
	12	4	6	52	2830	27	15	35,612(2.05), 35,621(2.16).	Inferior to No.10, 11

and

Table 3. Results of the Type-I SM-DLB, MM-DLB, SR-DLB, and MR-DLB for the lighter case.

								MM-DLB (${\mathit{N}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathbf{h}}$ = 2)
	No.	${\mathit{f}}_1^{*}$	${\mathit{f}}_2^{*}$	${\mathit{f}}_3^{*}$	${\mathit{f}}_4^{*}$	${\mathit{f}}_5^{*}$	$f_6^{*}$	Priority Order of Objectives f (Time/s)	Notes
Original model	1	3	3	75	77	6	26	4(0.41).	-
	2	2	3	64	410	-	-	123(0.25), 213(0.25).	-
	3	2	4	63	1061	-	-	132(0.27).	-
	4	3	3	61	389	-	-	231(0.26), 312(0.17), 321(0.27).	-
Bi-metric method	5	3	3	74	86	6	25	56(0.23).	Non-inferior to No.1
	6	2	3	64	410	19	30	12,356(0.34), 21,356(0.42).	Same as No.2
	7	2	3	70	142	9	25	15,623(0.42), 15,632(0.38), 12,563(0.33), 21,563(0.39).	New solution
	8	2	4	63	895	19	23	13,562(0.33), 13,256(0.41).	Dominates No.3
	9	3	3	61	389	18	26	23,561(0.52), 23,156(0.49), 35,612(0.39), 35,621(0.45), 31,562(0.44), 31,256(0.41), 3,2561(0.38), 32,156(0.45).	Same as No.4
	10	3	3	74	86	6	25	56,123(0.52), 56,132(0.44), 56,213(0.49), 56,231(0.53), 56,312(0.45), 56,321(0.48), 25,613(0.48), 25,631(0.50).	Same as No.5
								SM-DLB ($N_{\mathrm{m}ax}^{\mathrm{h}}$ = 1)
Original model	1	3	3	75	77	6	26	4(0.30).	-
	2	3	3	61	389	-	-	123(0.20), 132(0.22), 213(0.23), 231(0.25), 312(0.14), 321(0.14).	-
Bi-metric method	3	3	3	74	86	6	25	56(0.20)	Non-inferior to No.1
	4	3	3	61	389	18	26	12,356(0.31), 13,562(0.35), 13,256(0.28), 21,356(0.28), 23,561(0.42), 23,156(0.39), 35,612(0.26), 35,621(0.28), 31,562(0.26), 31,256(0.20), 32,561(0.39), 32,156(0.30).	Same as No.2
	5	3	3	74	86	6	25	56,123(0.33), 56,132(0.41), 56,213(0.44), 56,231(0.34), 56,312(0.33), 56,321(0.39), 15,623(0.33), 15,632(0.38), 12,563(0.30), 25,613(0.38), 25,631(0.39), 21,563(0.45).	Same as No.3
								MR-DLB ($N_{\mathrm{m}ax}^{\mathrm{m}}$ = 2)
Original model	1	3	3	66	206	11	24	4(0.13).	-
	2	2	3	57	369	-	-	123(0.09), 213(0.20).	-
	3	2	3	57	405	-	-	312(0.13), 321(0.13).	Inferior to No.2
	4	2	3	57	497	-	-	132(0.14), 231(0.20).	Inferior to No.2
Bi-metric method	5	3	3	64	238	11	24	56(0.14).	Non-inferior to No.1
	6	2	3	57	369	13	20	12,356(0.24), 13,562(0.23), 13,256(0.28), 21,356(0.33), 23,561(0.22), 23,156(0.34), 35,612(0.22), 35,621(0.28), 31,562(0.27), 31,256(0.30), 32,561(0.27), 32,156(0.34).	Same as No.2
	7	2	3	64	238	11	24	56,123(0.28), 56,132(0.28), 56,213(0.26), 56,231(0.30), 56,312(0.31), 56,321(0.36), 15,623(0.23), 15,632(0.33), 12,563(0.24), 25,613(0.25), 25,631(0.45), 21,563(0.23).	Same as No.5
								SR-DLB ($N_{\mathrm{m}ax}^{\mathrm{m}}$ = 1)
Original model	1	3	3	66	206	11	24	4(0.11).	-
	2	3	3	57	377	-	-	312(0.11), 321(0.17).	-
	3	3	3	57	497	-	-	123(0.11), 132(0.09), 213(0.19), 231(0.13).	Inferior to No.2
Bi-metric method	4	3	3	66	206	11	24	56(0.14).	Same as No.1
	5	3	3	57	377	14	21	12,356(0.20), 13,562(0.14), 13,256(0.13), 21,356(0.25), 23,561(0.27), 23,156(0.22), 35,612(0.16), 35,621(0.13), 31,562(0.19), 31,256(0.16), 32,561(0.11), 32,156(0.22).	Same as No.2
	6	3	3	64	238	11	24	56,123(0.28), 56,132(0.25), 56,213(0.30), 56,231(0.20), 56,312(0.23), 56,321(0.23), 15,623(0.27), 15,632(0.34), 12,563(0.22), 25,613(0.28), 25,631(0.28), 21,563(0.22).	New solution

.

Table 2 lists the optimization results of MMR-DLB and SMR-DLB for the lighter case. The penultimate column shows the optimization priority order of the objectives and the corresponding solution time. For instance, 4(1.72) indicates that the time to solve the single-objective optimal value (224) of objective f₄ individually is 1.72 s. The other objective values (f₁–f₃ and f₅–f₆) are then calculated by the single-objective optimal scheme corresponding to the single-objective optimal value. The values of objectives f₁–f₄ consist of a multi-objective solution (No.1 solution) of the MMR-DLB, while 123 (0.47) means that the lexicographic approach is used to optimize f₁ first, then f₂, and finally f₃, and the optimization time is 0.47 s. The value of the objective f₄ is then calculated from the scheme optimized by the lexicographic approach, and another multi-objective solution (No. 2 solution) is obtained.

In the results of the MMR-DLB, the first five solutions were obtained by the original model, and the solutions numbered 6 to 12 were obtained through the lexicographic optimization of the bi-metric and the other three linear objectives. The comparison reveals that solutions 1 and 6 do not dominate each other. Although the objective f₅ solved by the bi-metric method reaches f₅ = 12 of the single-objective optimal scheme of the idle balancing index, the solved f₆ (24) is smaller than that (f₆ = 26) of the single-objective optimal scheme. To explore the reasons for this, Figure 4 shows the disassembly schemes corresponding to solutions 1 and 6, where the virescent and pink rectangles represent the complex and hazardous tasks, respectively. Except for the workloads of worker 4 and robot 2, which are the same in these two schemes, respectively, the workload of worker 2 (f₆ = 24) in the scheme obtained by the bi-metric method differs from that of worker 1 (f₆ = 26) in the single-objective optimal scheme. The difference is caused by the fact that objective f₆ (min LT) requires selecting the worker with the lowest time to perform tasks 1–3. The difference further occurs in the total disassembly time of all tasks (f₃), and results in the f₄ (244) value obtained by the bi-metric method being 20 units higher than the single-objective optimal value (224) of the idle balancing index, 20 = (30−24)²−(30−26)². This is the detailed reason why solutions 1 and 6 do not dominate each other, which is also consistent with the explanation in Section 3.1.3. In addition, solutions 3, 10, and 12 are inferior solutions and are dominated by solutions 2, 9, and 11, respectively. Solutions 2 and 5 are identical to solutions 7 and 11, respectively. Solution 9 dominates solution 4, and solutions 6 and 8 are new solutions obtained additionally by the bi-metric method. These Pareto dominance comparisons are noted in the last column and indicate that the bi-metric method not only obtains solutions that are not worse than those of the original model but also obtains additional new solutions. This conclusion is also verified by the results of the SMR-DLB.

Table 3 shows the optimization results of SM-DLB, MM-DLB, SR-DLB, and MR-DLB for the lighter case. The notes in the last column show that the solutions solved by the bi-metric method in the four DLBs are either non-inferior to, identical to, or dominate the solutions obtained by the original model, and that the bi-metric method can obtain additional new solutions. Therefore, the effectiveness of the bi-metric method is verified.

3.2.2. Case II: Hairdryer

To further verify the effectiveness of the bi-metric method, a larger-scale actual case of a hairdryer was applied to optimize the corresponding six DLBs.

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Basic disassembly information:

Figure 5 presents the parts and corresponding PPD of the hairdryer. Because the air-inlet grille is fixed by multiple snaps and cannot be easily removed, and the resistance wire and power cord are wire-mounted parts that are not conducive to robotic disassembly, tasks 6, 11, and 12 are classified as complex tasks. The edges of the air-inlet grille and mica support are very sharp and can easily hurt the hands of the operators, so tasks 6 and 14 are classified as hazardous tasks. Moreover, the cycle time is set to 89, five candidate workers and four robots are assigned, and the processing times of all tasks performed by these candidates are listed in

Table 4. Disassembly time of a hairdryer by candidate workers and robots.

Tasks	Workers (tiw)					Robots (tir)
	1	2	3	4	5	1	2	3	4
1	22	20	25	19	23	19	18	20	21
2	8	9	12	8	11	8	6	7	9
3	14	11	12	13	15	13	11	12	14
4	5	4	6	4	5	4	6	5	4
5	10	9	7	8	10	9	6	7	8
6	40	36	37	39	38	39	35	33	36
7	15	18	14	16	17	14	13	12	15
8	12	10	13	12	11	13	12	11	14
9	30	28	33	32	29	29	28	31	26
10	9	8	10	11	11	11	10	12	9
11	28	28	25	26	27	25	26	27	24
12	35	31	34	33	32	34	36	33	35
13	24	23	21	22	25	24	20	26	22
14	18	16	19	17	15	18	16	14	20
15	32	30	33	31	29	30	33	34	36
16	7	6	8	7	9	6	4	5	6

.

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Optimization results and analysis:

Table A1. Calculation results for the six Type-I DLBs for the hairdryer case.

								MMR-DLB (${\mathit{N}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathbf{h}}$ = 3 and ${\mathit{N}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathbf{m}}$ = 2)
	No.	f1	f2	f3	f4	f5	f6	Priority Order of Objectives f (Time/s)	Notes
Original model	1	3	4	321	373	12	87	4(5257.78).	-
	2	2	4	284	1794	-	-	123(11.00), 213(27.03).	-
	3	2	6	263	13,231	-	-	132(10.68).	-
	4	3	4	273	2609	-	-	231(46.50).	-
	5	3	7	262	22,327	-	-	312(19.98), 321(32.11).	-
Bi-metric method	6	3	4	309	553	12	78	56(323.75).	Non-inferior to No.1
	7	2	4	284	1794	28	88	15,623(26.75), 15,632(26.83), 12,563(23.31), 12,356(23.28), 21,563(90.20), 21,356(100.38).	Same as No.2
	8	2	6	263	13,231	63	61	13,562(22.13), 13,256(21.56).	Same as No.3
	9	3	4	273	2609	42	85	23,561(260.98), 23,156(286.91).	Same as No.4
	10	3	4	309	553	12	78	56,123(149.73), 56,132(150.13), 56,213(159.53), 56,231(174.19), 56,312(164.20), 56,321(174.38), 25,613(213.39), 25,631(225.34).	Same as No.6
	11	3	7	262	22,327	85	85	35,612(118.24), 35,621(136.42), 31,562(42.22), 31,256(50.06), 32,561(125.14), 32,156(56.48).	Same as No.5
								SMR-DLB ($N_{max}^{\mathrm{h}}$ = 1 and $N_{max}^{\mathrm{m}}$ = 1)
Original model	1	3	4	321	373	12	87	4(2449.86)	-
	2	3	4	273	2609	-	-	231(38.17).	-
	3	3	4	273	2613	-	-	123(15.41).	Inferior to No.2
	4	3	4	273	3157	-	-	213(33.58).	Inferior to No.2,3
	5	3	6	265	14,279	-	-	132(13.88).	-
	6	4	7	262	22,327	-	-	312(22.45), 321(29.11).	-
Bi-metric method	7	3	4	309	553	12	78	56(111.86).	Non-inferior to No.1
	8	3	4	273	2609	42	85	12,356(40.94), 21,356(71.70), 23,561(195.31), 23,156(81.92).	Same as No.2
	9	3	4	309	553	12	78	56,123(159.56), 56,132(153.38), 56,213(164.63), 56,231(182.97), 56,312(168.44), 56,321(184.13), 15,623(27.22), 15,632(36.13), 12,563(41.89), 25,613(142.53), 25,631(164.25), 21,563(100.95).	Same as No.6
	10	3	6	265	14,279	63	80	13,562(31.86), 13,256(35.27).	Same as No.5
	11	4	7	262	22,327	85	85	35,612(141.06), 35,621(156.56), 31,562(52.61), 31,256(57.64), 32,561(123.53), 32,156(80.09).	Same as No.6
								MM-DLB ($N_{max}^{\mathrm{h}}$ = 3)
Original model	1	3	4	317	473	15	87	4(794.23).	-
	2	2	5	280	7651	-	-	123(2.22), 132(2.23).	-
	3	3	4	277	2799	-	-	213(4.94).	-
	4	3	4	277	2853	-	-	231(6.98).	Inferior to No.3
	5	3	5	276	9731	-	-	321(6.36).	-
	6	3	5	276	9785	-	-	312(4.61).	Inferior to No.5
Bi-metric method	7	3	4	304	692	15	78	56(7.53).	Non-inferior to No.1
	8	2	5	280	7651	65	84	12,356(3.41), 13,562(3.47), 13,256(3.44).	Same as No.2
	9	2	5	289	5076	39	68	15,623(3.42), 15,632(3.45), 12,563(3.40).	New solution
	10	3	4	277	2799	50	82	21,356(7.52), 23,561(12.80), 23,156(9.33).	Same as No.3
	11	3	4	300	796	15	78	56,123(13.53), 56,132(13.90), 56,213(13.97), 56,231(16.10), 56,312(15.89), 56,321(16.38), 25,613(12.31), 25,631(13.76), 21,563(10.06).	New solution
	12	3	5	276	9731	82	82	35,612(11.97), 35,621(13.40), 31,562(7.50), 31,256(7.14), 32,561(13.45), 32,156(8.74)	Same as No.5
								SM-DLB ($N_{\mathrm{m}ax}^{\mathrm{h}}$ = 1)
Original model	1	4	4	316	494	15	87	4(156.94).	-
	2	4	4	278	2854	-	-	123(4.46), 132(4.09), 213(4.50), 231(5.64).	-
	3	5	5	276	9785	-	-	312(5.12), 321(5.42).	-
Bi-metric method	4	4	4	307	611	15	78	56(3.81).	Non-inferior to No.1
	5	4	4	278	2854	50	85	12,356(5.44), 13,562(5.66), 13,256(5.87), 21,356(5.92), 23,561(7.46), 23,156(6.81).	Same as No.2
	6	4	4	304	686	15	78	56,123(6.51), 56,132(6.46), 56,213(6.75), 56,312(7.07), 56,321(7.33), 25,613(6.57), 25,631(6.73).	New solution
	7	4	4	304	692	15	78	56,231(7.01), 15,623(5.86), 15,632(6.08), 12,563(6.18), 21,563(7.10).	Inferior to No.6
	8	5	5	276	9785	82	85	35,612(9.88), 35,621(9.78), 31,562(8.02), 31,256(8.81), 32,561(9.46), 32,156(9.71).	Same as No.3
								MR-DLB ($N_{max}^{\mathrm{m}}$ = 2)
Original model	1	3	4	311	533	15	81	4(94.94).	-
	2	2	4	275	3049	-	-	123(1.14), 132(1.17), 213(1.90).	-
	3	3	4	270	2186	-	-	231(2.96), 312(2.11), 321(3.01).	-
Bi-metric method	4	3	4	309	557	13	79	56(3.05).	Non-inferior to No.1
	5	2	4	275	3049	40	88	15,623(1.84),15,632(1.86), 12,563(1.81), 12,356(1.79), 13,562(1.80), 13,256(1.80), 21,563(3.06), 21,356(2.69).	Same as No.2
	6	3	4	270	2186	30	83	23,561(4.70), 23,156(4.05), 35,612(3.92), 35,621(3.91), 31,562(3.78), 31,256(3.70), 32,561(4.06), 32,156(4.09).	Same as No.3
	7	3	4	308	582	13	79	56,123(4.96), 56,132(5.32), 56,213(5.47), 56,231(5.79), 56,312(5.61), 56,321(5.77), 25,613(5.51), 25,631(6.11).	New solution
								SR-DLB ($N_{\mathrm{m}ax}^{\mathrm{m}}$ = 1)
Original model	1	4	4	310	618	18	84	4(4.88).	-
	2	4	4	272	2666	-	-	123(2.52), 132(2.41), 213(2.53), 231(2.42), 312(2.30), 321(2.03).	-
Bi-metric method	3	4	4	301	777	16	79	56(2.11).	Non-inferior to No.1
	4	4	4	301	777	16	79	56,123(3.38), 56,132(3.36), 56,213(3.42), 56,231(3.39), 56,312(3.45), 56,321(3.55), 15,623(4.00), 15,632(3.88), 12,563(4.45), 25,613(3.42), 25,631(3.44), 21,563(4.09).	Same as No.3
	5	4	4	272	2666	41	83	12,356(4.42), 13,562(3.64), 13,256(3.55), 21,356(4.67), 23,561(4.33), 23,156(3.94), 35,612(3.75), 35,621(3.72), 31,562(4.00), 31,256(3.72), 32,561(3.66), 32,156(3.84).	Same as No.2

presents the optimization results of the six DLBs for the hairdryer case. The Pareto dominance comparison results presented in the “Notes” column verify that the proposed bi-metric method is not only able to obtain solutions no worse than those of the original model, but also to find additional new solutions. Moreover, in the six DLBs, the time to solve the idle balancing index ($f_4$, $f_4^{*}$, $f_4^{\#}$) alone is much longer than that of the lexicographic optimization of the bi-metric ($f_5$ and $f_6$, $f_5^{*}$ and $f_6^{*}$, $f_5^{\#}$ and $f_6^{\#}$), which also demonstrates the high efficiency of the bi-metric method in solving problems. This high efficiency is realized by restricting only the upper and lower limits of the workloads of the assigned operators but does not guarantee that the workloads of the other operators are maximized.

3.3. Discussion for Type-I DLBs

In both the lighter and hairdryer cases, the bi-metric method obtained solutions that are not worse than the corresponding original models and also obtained new solutions. However, minimizing the bi-metric belongs to the minimizing maximum problems, that is, only the outstanding operator in the current optimization phase is suppressed, and the states of other operators are ignored. Further, the bi-metrics are the slack objectives of the idle balancing index, so using these bi-metrics together with the other three linear objectives to participate in the lexicographic optimization might still fail to achieve the solutions of the true Pareto frontier. Therefore, to obtain more and better solutions, this study suggests combining the idle balancing index with the bi-metric method to solve the DLBs.

Because the models of the MM-DLB, SM-DLB, MR-DLB, and SR-DLB are derived from the simplified models of the MMR-DLB and SMR-DLB in Section 3.1.2, their solution times for solving the same case are less than those of the MMR-DLB and SMR-DLB models, which is confirmed by the comparison of the computation time in Table 2, Table 3 and Table 4. This indicates that the MMR-DLB and SMR-DLB models are more difficult to optimize than the other four DLBs.

In addition, the model in Section 3.1.1 can solve the MMR-DLB and SMR-DLB by adjusting the parameters $N_{max}^{\mathrm{h}}$ and $N_{max}^{\mathrm{m}}$ in the constraints (20, 21), so the SMR-DLB belongs to the reduction problem of the MMR-DLB, and the solutions of SMR-DLB are also the solutions of MMR-DLB. The same is true for the MM-DLB, SM-DLB, MR-DLB, and SR-DLB models. For instance, solutions 4 and 5 of the SMR-DLB model in Table 2 are also the new non-inferior solutions of the MMR-DLB model; solution 4 of the SM-DLB model and solution 3 of the SR-DLB model in Table A1 is also the new non-inferior solutions of the MM-DLB and MR-DLB models, respectively. Thus, to obtain more non-inferior solutions of the MMR-DLB, MM-DLB, and MR-DLB models, except for their original models and bi-metric methods, these three problems can also be reduced to SMR-DLB, SM-DLB, and SR-DLB to obtain new solutions.

Finally, owing to the abundance of solutions, all the solutions and their disassembly schemes in Table 2, Table 3 and Table 4 are given in the Supplementary Files, and their Gantt charts are drawn. The correctness and reasonableness of these Gantt charts validate the correctness of the established models for the six types of DLBs. Further analysis reveals that the same multi-objective solution may correspond to multiple disassembly schemes because the four optimization objectives in the models do not consider the starting time of each task, and different operators may have the same time to perform the same task. For example, the solution 10 of the MMR-DLB model in Table A1 corresponds to two different schemes obtained by the lexicographic optimization of f₅-f₆-f₁-f₂-f₃ and f₂-f₅-f₆-f₁-f₃, respectively, as shown in Figure 6. The orange rectangle represents task 6, which has both complex and harmful attributes.

4. Type-II DLB: Modelling and Optimization

Type-II DLBs aim to optimize cycle time under a given number of stations, so they are suitable for the scheduling adjustment of existing disassembly lines. To model such problems, the given number of stations is denoted by the symbol “N_k”, and the other notations follow the definitions in Section 3. Here, the cycle time is unknown, so the CT becomes a decision variable.

4.1. Models of the Type-II DLB with the Six Operator Configurations

Similar to the Type-I DLBs, Type-II DLBs still comply with the sequence constraints of disassembly tasks, the assignment constraints of operators to stations, and the distribution constraints of tasks to operators. Therefore, the programming models in Section 3 are modified to solve the six types of Type-II DLBs. To distinguish the Type-I DLB, codenames of the objectives and constraints in the Type-II DLB are designated by “ff” and “CC”, respectively.

4.1.1. Unified Model of the Type-II SMR-DLB and MMR-DLB

(1)

Optimization objectives:

The first objective is to minimize the cycle time, and it is expressed in Equation (41). The other three objectives are the same as those in Equations (2)–(4), namely ff₂ = f₂, ff₃ = f₃, ff₄ = f₄.

$$f{f}_1:\min CT$$

(41)

(2)

Constraints:

After the analysis, constraints C₁–C₁₀ and C₁₄–C₂₂ can be used directly for the Type-II MMR-DLB and SMR-DLB. The unknown CT renders constraints C₁₁ and C₁₂ nonlinear, so they are split into four separate linear constraints to represent the relationships between the cycle time and the start and end times of the tasks performed by each operator. The four modified constraints are shown in Equations (42)–(45).

$$C{C}_{11}:\psi \cdot (1-x_{jkw}^{\mathrm{h}})+t_j^s\ge (k-1)\cdot CT+{\displaystyle \sum _{k\in K}{\displaystyle \sum _{w\in W}{\displaystyle \sum _{i\in T,i\ne j}{z}_{ijkw}^{\mathrm{h}}\cdot t_{iw}}}}\hspace{1em}\forall j\in T,\forall k\in K,\forall w\in W$$

(42)

$$C{C}_{11}^{\prime }:\psi \cdot (1-x_{jkr}^{\mathrm{m}})+t_j^s\ge (k-1)\cdot CT+{\displaystyle \sum _{k\in K}{\displaystyle \sum _{r\in R}{\displaystyle \sum _{i\in T,i\ne j}{z}_{ijkr}^{\mathrm{m}}\cdot t_{ir}}}}\hspace{1em}\forall j\in T,\forall k\in K,\forall r\in R$$

(43)

$$C{C}_{12}:t_i^s+x_{ikw}^{\mathrm{h}}\cdot t_{iw}\le \psi \cdot (1-x_{jkw}^{\mathrm{h}})+k\cdot CT\hspace{1em}\forall i\in T,\forall k\in K,\forall w\in W$$

(44)

$$C{C}_{12}^{\prime }:t_i^s+x_{ikr}^{\mathrm{m}}\cdot t_{ir}\le \psi \cdot (1-x_{ikr}^{\mathrm{m}})+k\cdot CT\hspace{1em}\forall i\in I,\forall k\in K,\forall r\in R$$

(45)

Owing to the given number of stations, the maximum number of opened stations should be less than N_k, and constraint C₁₃ is modified as follows:

$$C{C}_{13}:1\le {\displaystyle \sum _{k\in K}{s}_k}\le \min (N_k,|T|,|W|+|R|)$$

(46)

In addition, the cycle time of the stations should be no smaller than the workload of each assigned operator. To avoid increasing the idle balance index, the maximum workload among all assigned operators is defined as the cycle time. This is a newly added constraint and is shown in Equation (47).

$$C{C}_{+23}:CT=\underset{\forall k\in K}{\max }(\underset{\forall w\in W,\forall r\in R}{\max }({\displaystyle \sum _{i\in T}{x}_{ikw}^{\mathrm{h}}\cdot t_{iw}},{\displaystyle \sum _{i\in T}{x}_{ikr}^{\mathrm{m}}\cdot t_{ir}}))$$

(47)

In summary, the unified model of the Type-II MMR-DLB and SMR-DLB can be composed of the objective functions ff₁–ff₄ and constraints C₁–C₁₀, CC₁₁–CC₁₂, CC’₁₁–CC’₁₂, CC₁₃, C₁₄–C₂₂, and CC₊₂₃.

4.1.2. Unified Model of the Type-II SM-DLB, MM-DLB, SR-DLB, and MR-DLB

Some objective functions and constraints in Section 3 and Section 4.1.1 are selected or modified to establish a unified model for the Type-II SM-DLB and MM-DLB.

(1)

Optimization objectives: ${\mathit{ff}}_1^{*}$ = ff₁, ${\mathit{ff}}_2^{*}$ = $f_2^{*}$, ${\mathit{ff}}_3^{*}$ = $f_3^{*}$, ${\mathit{ff}}_4^{*}$ = $f_4^{*}$.

(2)

Constraints:

Selected: $C_1^{*}$, $C_4^{*}$, C₅, C₇–C₈, CC₁₁–CC₁₂, $C_{14}^{*}$, C₁₅–C₁₆, C₁₈, C₂₀, $C_{22}^{*}$;

Modified: the variables related to robots in constraints CC₁₃ and CC₊₂₃ need to be removed as follows:

$$C{C}_{13}^{*}:1\le {\displaystyle \sum _{k\in K}{s}_k}\le \min (N_k,|T|,|W|)$$

(48)

$$C{C}_{+23}^{*}:CT=\underset{\forall k\in K,\forall w\in W}{\max }({\displaystyle \sum _{i\in T}{x}_{ikw}^{\mathrm{h}}\cdot t_{iw}})$$

(49)

Furthermore, this model can be used to solve the Type-II SR-DLB and MR-DLB after replacing the worker variables with robot variables.

4.2. Partial Linearization and Results Presentation

4.2.1. Partial Linearization for the Idle Balancing Index

The unknown CT makes the expressions inside the max functions of objectives f₅ and $f_5^{*}$ nonlinear, so the bi-metric method proposed in Section 3.1.3 cannot be used to solve Type-II DLBs. The objectives ff₄ and ${ff}_4^{*}$ (idle balancing index) contain two types of nonlinearities simultaneously: bilinear terms ($y_{kw}^{\mathrm{h}}\cdot CT$ and $y_{kr}^{\mathrm{m}}\cdot CT$) and quadratic terms, which cannot be addressed directly by the CPLEX. The CPLEX can only handle the nonlinearity of the quadratic terms, so the bilinear terms need to be linearized. Here, two continuous variables, $e_{kw}^{\mathrm{h}}$ and $e_{kr}^{\mathrm{m}}$, are introduced to represent the two bilinear terms, so the objectives ff₄ can be transformed into equivalent objectives ff₇ and linear constraints below:

$${\mathit{ff}}_7:\min IB={\displaystyle \sum _{k\in K}({\displaystyle \sum _{w\in W}{(e_{kw}^{\mathrm{h}}-{\displaystyle \sum _{i\in T}{x}_{ikw}^{\mathrm{h}}\cdot t_{iw}})}^2}}+{\displaystyle \sum _{r\in R}{(e_{kr}^{\mathrm{m}}-{\displaystyle \sum _{i\in T}{x}_{ikr}^{\mathrm{m}}\cdot t_{ir}})}^2})$$

(50)

$${\mathit{CC}}_{24}:0\le e_{kw}^{\mathrm{h}}\le \psi \cdot y_{kw}^{\mathrm{h}}$$

(51)

$${\mathit{CC}}_{25}:e_{kw}^{\mathrm{h}}\le CT$$

(52)

$${\mathit{CC}}_{26}:e_{kw}^{\mathrm{h}}\ge CT-\psi \cdot (1-y_{kw}^{\mathrm{h}})$$

(53)

$${\mathit{CC}}_{27}:0\le e_{kr}^{\mathrm{m}}\le \psi \cdot y_{kr}^{\mathrm{m}}$$

(54)

$${\mathit{CC}}_{28}:e_{kr}^{\mathrm{m}}\le CT$$

(55)

$${\mathit{CC}}_{29}:e_{kr}^{\mathrm{m}}\ge CT-\psi \cdot (1-y_{kr}^{\mathrm{m}})$$

(56)

In addition, the other three objectives of the models in Section 4.1 are still linear, so lexicographic optimization is performed for them in the CPLEX optimizer. The results of the six Type-II DLBs of the two actual cases are presented below.

4.2.2. Cases I and II: Lighter and Hairdryer

The basic disassembly data of the lighter in Section 3.2.1 are still used here, and N_k = 2.

Table 5. Calculation results of the six Type-II DLBs for the lighter case.

					MMR-DLB (${\mathit{N}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathbf{h}}$ = 2 and ${\mathit{N}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathbf{m}}$ = 2)							SMR-DLB (${\mathit{N}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathbf{h}}$ = 1 and ${\mathit{N}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathbf{m}}$ = 1)
No.	f1	f2	f3	f4	Priority Order of f (Time/s)	Notes	No.	f1	f2	f3	f4	Priority Order of f (Time/s)	Notes
1	25	4	60	568	4(2723.62).	-	1	35	3	66	773	4(864.82).	-
2	25	4	57	705	123(2.20), 132(1.30),	-	2	35	3	62	929	123(2.09).	-
					312(3897.56), 321(10,898.11).		3	35	4	60	2208	321(5377.33).	-
3	51	2	63	1521	213(1.25), 231(925.69).	-	4	35	4	60	2304	132(5.47), 312(517.34).	Inferior to No.3
							5	51	2	63	1521	213(0.55), 231(1858.75).	-
					MM-DLB ($N_{max}^{\mathrm{h}}$ = 2)							SM-DLB ($N_{\mathrm{m}ax}^{\mathrm{h}}$ = 1)
No.	$f_1^{*}$	$f_2^{*}$	$f_3^{*}$	$f_4^{*}$	Priority order of f * (Time/s)	Notes	No.	$f_1^{*}$	$f_2^{*}$	$f_3^{*}$	$f_4^{*}$	Priority order of f * (Time/s)	Notes
1	68	1	68	0	4(647.14).	-	1	68	1	68	0	4(130.93).	-
2	30	3	64	410	123(0.41), 132(0.44).	-	2	35	2	65	25	123(0.18), 132(0.19).	-
3	35	3	63	890	312(0.42).	-	3	40	2	63	289	312(0.20), 321(0.25).	-
4	40	2	63	289	321(0.41).	-	4	68	1	68	0	213(0.18), 231(0.17).	Same as No.1
5	68	1	68	0	213(0.19), 231(0.21).	Same as No.1
					MR-DLB ($N_{\mathrm{m}ax}^{\mathrm{m}}$ = 2)							SR-DLB ($N_{\mathrm{m}ax}^{\mathrm{m}}$ = 1)
No.	$f_1^{\#}$	$f_2^{\#}$	$f_3^{\#}$	$f_4^{\#}$	Priority order of f * (Time/s)	Notes	No.	$f_1^{\#}$	$f_2^{\#}$	$f_3^{\#}$	$f_4^{\#}$	Priority order of f * (Time/s)	Notes
1	65	1	65	0	4(15.53).	Inferior to No.3	1	61	1	61	0	4(1.21).	-
2	27	3	57	338	123(0.17), 132(0.12),	-	2	31	2	59	9	123(0.17), 132(0.14).	-
					312(0.16), 321(0.23).		3	38	2	58	324	312(0.13), 321(0.17).	-
3	61	1	61	0	213(0.13), 231(0.12).	-	4	61	1	61	0	213(0.10), 231(0.13).	Same as No.1

lists the optimization results of the six Type-II DLBs for the lighter. Similarly, the basic disassembly data of the hairdryer in Section 3.2.2 are used here, and N_k = 3.

Table 6. Calculation results for the six Type-II DLBs for the hairdryer case.

					MMR-DLB (${\mathit{N}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathbf{h}}$ = 3 and ${\mathit{N}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathbf{m}}$ = 2)							SMR-DLB (${\mathit{N}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathbf{h}}$ = 1 and ${\mathit{N}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathbf{m}}$ = 1)
No.	f1	f2	f3	f4	Priority Order of f (Time/s)	Notes	No.	f1	f2	f3	f4	Priority Order of f (Time/s)	Notes
1	99	3	297	0	4(149,319.52).	-	1	99	3	297	0	4(104,045.24).	-
2	69	9	284	14,959	123(5154.2).	-	2	75	6	270	6946	123(642.66), 132(231.66).	-
3	70	7	273	8211	132(65.07).	-	3	85	6	266	14,930	312(64,504.5), 321(121,497.74).	-
4	85	7	262	19,551	312(53,249.57), 321(72,964.44).	-	4	155	2	280	900	213(116.81).	-
5	155	2	280	900	213(65.29).	-	5	221	2	276	27,556	231(53,608.5).	-
6	221	2	276	27,556	231(3706.36).	-
					MM-DLB ($N_{max}^{\mathrm{h}}$ = 2)							SM-DLB ($N_{\mathrm{m}ax}^{\mathrm{h}}$ = 1)
No.	$f_1^{*}$	$f_2^{*}$	$f_3^{*}$	$f_4^{*}$	Priority order of f * (Time/s)	Notes	No.	$f_1^{*}$	$f_2^{*}$	$f_3^{*}$	$f_4^{*}$	Priority order of f * (Time/s)	Notes
1	298	1	298	0	4(14,268.43).	Inferior to No.4	1	287	1	287	0	4(4144.13).	-
2	75	5	287	2460	123(20.39), 132(22.07).	-	2	97	3	287	10	123(11.77), 132(15.52).	-
3	124	5	273	31,987	312(5.93), 321(6.19).	-	3	166	3	279	24,005	312(14.75), 321(28.08).	-
4	287	1	287	0	213(2.32), 231(1.71).	-	4	287	1	287	0	213(1.88), 231(1.72).	Same as No.1
					MR-DLB ($N_{\mathrm{m}ax}^{\mathrm{m}}$ = 2)							SR-DLB ($N_{\mathrm{m}ax}^{\mathrm{m}}$ = 1)
No.	$f_1^{\#}$	$f_2^{\#}$	$f_3^{\#}$	$f_4^{\#}$	Priority order of f * (Time/s)	Notes	No.	$f_1^{\#}$	$f_2^{\#}$	$f_3^{\#}$	$f_4^{\#}$	Priority order of f * (Time/s)	Notes
1	143	2	286	0	4(8671.79).	-	1	296	1	296	0	4(4249.63).	Inferior to No.4
2	93	3	279	0	123(11.32), 132(8.69).	-	2	94	3	278	8	123(5.75), 132(5.87).	-
3	142	3	269	12,365	312(15.95), 321(18.45).	-	3	95	3	273	122	312(14.92), 321(15.75).	-
4	280	1	280	0	213(1.02), 231(0.99).	-	4	280	1	280	0	213(1.10), 231(1.00).	-

presents the optimization results of the six Type-II DLBs.

4.3. Discussion for Type-II DLBs

A comparison of the computation times in Table 5 and Table 6 reveals that the time taken by the Type-II SM-DLB, MM-DLB, SR-DLB, and MR-DLB to solve the same case is less than that of the Type-II SMR-DLB and MMR-DLB. This is consistent with the Type-I problems, further indicating that the SMR-DLB and MMR-DLB are more difficult to optimize than the other four DLBs.

As discussed in Section 3.3, the non-inferior solutions of the Type-II SMR-DLB, SM-DLB, and SR-DLB may still be new solutions of the Type-II MMR-DLB, MM-DLB, and MR-DLB, respectively. For example, the solutions highlighted by the yellow shading in Table 5 and Table 6 also belong to the new non-inferior solutions of the corresponding multi-operator shared-station problems. Therefore, the Type-II multi-operator shared-station problems can also be reduced to Type-II SMR-DLBs, SM-DLB, and SR-DLB to obtain new possible solutions.

In addition, the disassembly schemes and Gantt charts of the solutions in Table 5 and Table 6, provided in the Supplementary Files, further verify the correctness of the established models for the six types of Type-II DLBs.

5. Conclusions and Future Work

The mixed-integer programming models of six types of Type-I and Type-II DLPBs, with distinct operator configurations, are established to minimize the basic indices of the disassembly line: the number of stations, cycle time, the number of operators, the total disassembly time, and the idle balancing time. The consideration of the influence of the efficiency difference of operators on the task disassembly time makes the established models more practical. In Type-I DLBs, the proposed bi-metric can be used to replace the quadratic idle balancing index to participate in lexicographic optimization, providing a new auxiliary method to obtain new and potentially better solutions and explore the real Pareto frontier. In addition, MMR-DLB, MM-DLB, and MR-DLB can be reduced to SMR-DLB, SM-DLB, and SR-DLB, respectively, to obtain new solutions. The optimization of SM-DLB and SR-DLB is essentially the same, as is the optimization of MM-DLB and MR-DLB. The comparison of the computational times reveals that the MMR-DLB and SMR-DLB are more difficult to optimize than the other four DLBs.

Future research directions: (1) DLPBs with six types of different operator configurations based on the U-shaped, two-sided, and parallel line layouts will be summarized and modelled to provide full-scenario solution models for DLBs. (2) The idle balancing index in Type-II DLBs cannot participate in lexicographic optimization because of its two types of nonlinear characteristics, which will be explored to obtain more true Pareto frontier solutions. (3) Heuristic algorithms or other methods will be developed or summarized to efficiently solve large-scale disassembly instances of DLBs with six distinct operator configurations.