2.1. Problem Statement
The DLBP is considered when assigning disassembly tasks to workstations under various constraints. This paper studies the complete disassembly of linear disassembly lines. The disassembly task is assigned to the workstation under a certain workstation cycle time, and the impact of the degradation rate of the disassembly tool on the disassembly time is considered, and the workstation cycle time is minimized [
32,
33,
34,
35]. A complete disassembly line includes a disassembly workstation, a disassembly location, a disassembly task, and disassembly tools. The disassembly tool considered in this paper is a vulnerable tool. When the tool does the same thing again, the disassembly time will be longer. We assume that different disassembly tools disassemble different parts during the product disassembly process and use the tool’s degradation coefficient to represent the impact on disassembly time. Through the continuous use of tools during the dismantling process, the dismantling time has been improved, and the losses suffered by dismantling tools vary depending on the dismantling task.
The total disassembly time of a workstation is influenced by the degree of degradation in the disassembly tool due to the fixed original disassembly time for each component. As illustrated in
Figure 3, tool 1 in workstation 1 is capable of disassembling both tasks 1 and 5. However, after completing task 1, tool 1 undergoes deterioration. Consequently, the dismantling time of task 5 will be affected if tool 1 is used for dismantling. Therefore, when considering the allocation of workstations, we also need to consider the degradation of dismantling tools to minimize the dismantling time cost of our workstations.
An AND/OR graph depicts the disassembly process, incorporating information on disassembly tasks, disassembly subassemblies, and disassembly parts. The AND/OR diagram not only signifies the prioritization of disassembly tasks but also indicates the dependency relationships among subassemblies and parts. It comprises rectangles and directed angles, where a rectangle represents the subassembly number and part information. More precisely, the numbers within the angle brackets represent the subassembly number, while the letters outside the angle brackets represent the part information within the subassembly. Each directional angle corresponds to a decomposition task, with the number within the angle denoting the task number and the arrow symbolizing the relationship between subassemblies. The arrow tail points to the detachable subassembly, whereas the arrow points to the subassembly obtainable after disassembling the current subassembly. There are many directional angles before and after the rectangle, but the same subassembly can only perform one disassembly task; that is, it can only perform one directional angle. Utilizing the AND/OR graph, we can transform the disassembly problem into a graph-solving procedure, subsequently introducing various constraints to obtain optimal solutions. In the context of employing reinforcement learning for DLBP problems, the AND/OR graph can be viewed as an environment for agent interaction. To effectively combine disassembly line problems, we convert the task precedence relationship and conflicting task relationship represented in the AND/OR graph into matrix form, upon which algorithms can be applied.
Figure 4 illustrates the composition of the kettle, while
Figure 5 represents the disassembly AND/OR graph for the basic kettle subassembly. The AND/OR graph entails various constraints that include conflicted relationships, task priorities, and task-subassembly relationships. As demonstrated in
Figure 5, the disassembly process involves 19 subassemblies, 8 parts, and 14 disassembly tasks. The conflict and priority relationship of tasks needs to be met in the disassembly process. For instance, subassembly 1 can disassemble subassembly 2 and subassembly 10 through task 1, and subassembly 2 can subsequently proceed with downward disassembly tasks. Specifically, subassembly 2 can undertake disassembly tasks 3 and 4. However, a conflict arises when attempting to perform these two tasks simultaneously, allowing only one task to be executed. Consequently, disassembly task 1 cannot be repeated.
The conflicted relationship, precedence relationship of tasks, and correlation between tasks and subassemblies are discernible from the AND/OR graph. For the algorithm to effectively utilize these relationships, we employ matrix representation to express them. The following describes these relationships.
(1) To describe the conflicted relationship of each task, we added the conflict matrix
, where
(2) To describe the relationships between individual tasks and components, we built the task-component relationship matrix
, where
(3) To describe the relationships between individual tasks and tools, we built the task-tool relationship matrix
, where
According to the provided formula, the conflict matrix
C of the kettle can be defined as follows. From
Figure 5, we can see that task 1 is executed before task 3, so
is set to 1. Furthermore, since task 1 and task 2 both disassemble the same subassembly and cannot be performed simultaneously, values of -1 are assigned to
and
, with 0 being assigned in other cases.
Similarly, the relationship matrix
B for the kettle can be defined as follows. As can be seen from
Figure 5, component 9 can be disassembled by task 10 to obtain parts 16 and 17 from it, so
and
are set to 1. It is set to 0 for other task-component combinations for which parts cannot be obtained. For example, component 6 can be disassembled by task 8, from which part 12 and component 8 are obtained. Therefore,
is set to 1, while
is set to 0.
The following is the definition of the relationship between individual tasks and tools in matrix D for the kettle. Based on the complexity of the DLBP and the DLBP addressed in this work, we make the following assumptions for the mathematical model.
- 1.
We adopt linear single-target disassembly.
- 2.
One subassembly can only be removed by one disassembly tool.
- 3.
All disassembled parts are to be removed.
- 4.
The degradation rate of disassembly tools has been determined.
- 5.
The disassembly time of tools used on each workstation for each task is known.
- 6.
Workstations are started in sequence.
- 7.
Each task is disassembled only once during the disassembly process.
2.2. Notations
I Number of tasks.
W Number of workstations.
N Number of subassemblies.
R Number of tools.
K Number of task locations.
Collection of workstations .
Mission collection .
Collection of components .
Collection of task locations on the workstation .
Collection of tools .
Matrix of tasks j that conflict with task i.
Precedence task j matrix for task i.
w Workstation index, .
n Subassembly index, .
Task indexes, i, .
k Location index, .
Normal disassembly time of task i on workstation w.
Deterioration coefficient of processing task i on the workstation w.
An element in the n-th row and i-th column of B.
An element in the i-th row and r-th column of D.
Decision variables:
(1)
(2)
(3) The time that tool r on the w-th workstation has been used before task i is executed.
(4) Actual disassembly time of task i on the w-th workstation, which is the dismantling time considering the impact of tool deterioration.
(5) Maximum working time of a workstation.
2.3. Mathematical Model
The objective function (14) of this study aims to minimize the maximum working time of workstations. Constraint (2) ensures that each task is executed at most once. Constraint (3) guarantees that workstations are activated sequentially. Constraint (4) restricts each location on a workstation to execute only one task. Constraint (5) ensures that if there is a task in the last position of a workstation, there must be a task in the previous position as well. Constraint (6) allows for the execution of only one conflicting task at most. Constraint (7) imposes a limitation on the number of open workstations. Constraint (8) mandates that an open workstation must be assigned tasks. Constraint (9) determines whether a workstation is enabled. M represents an infinite number. Constraint (10) ensures that the disassembly task sequence adheres to the task priority relation. Constraint (11) calculates the usage time of disassembly tool r on workstation w prior to the execution of task i. Constraint (12) calculates the actual disassembly time of task i using disassembly tool r on workstation w. Lastly, constraint (13) guarantees the disassembly of all parts.