In actual production and life, a parallel disassembly line can be used to completely separate different EOL products during disassembly, reduce the disassembly difficulty, and improve the workstation’s utilization rate. This work puts forward MPDMW. Multi-product means that EOL products can be combined in pairs or other single products in various forms to form multi-products for disassembly simultaneously. The multi-skilled worker means that each worker has different skills. The multi-skilled worker is assigned to the workstation to disassemble EOL products. The different skills required for each task in disassembling each product lead to the allocation disassembly sequence becoming more diverse, and there are often many complex task allocation problems. Enterprises can reduce labor costs through the reasonable distribution of multi-skilled workers. Solve the problem of insufficient emergency production personnel. Flexible response to emergencies to ensure the normal operation of production operations. Improve the production line’s flexibility and the enterprise’s overall productivity level. A parallel disassembly line is placed in two or more parallel disassembly lines and can disassemble various products synchronously, effectively improving disassembly efficiency and production line flexibility. According to the mathematical model established by MPDMW, the following assumptions are made:

Each worker has multiple skills.

Through the AND/OR graph of the flashlight in Fig. 1, different disassembly sequences can be obtained, but these sequences must meet certain constraints to ensure that the disassembly sequence is correct and feasible. In the AND/OR graph, subassemblies are represented by nodes, and the index of each subassembly is indicated by an integer within angle brackets. Directed edges between linked subassemblies depict each disassembly task. It is easy to observe that this product consists of 15 subassemblies and 10 disassembly tasks.

In order to describe the problem more deeply and systematically, four relation matrices, S, D, A, and B, are introduced in this paper to describe relevant constraint relations. The detailed definition of the matrix is as follows.

The disassembly precedence relation matrix S=Sij is used to describe the precedence relation between disassembly tasks i and j, specifically defined as: Sij={1,If task i is performed before task j;−1,If task i is conflict with task j;0,Otherwise.

Disassembly incidence matrix D=Dqi is used to describe the relationship between subassembly q and disassembly task i, specifically defined as: Dqi={1,If subassembly q is performed before task i;−1,If subassembly q is obtained by disassembly in i;0,Otherwise.

The matrix of workers and skills A=Apk is as follows: The matrix of workers p and skills k is described. According to the actual situation, each worker has different skills. We put the workers who can master relatively complex disassembly skills into the workstation behind to achieve greater disassembly of EOL products. Apk={1,If worker p has skill k; 0,If worker p doesn't have skill k.

1) Disassemble the ballpoint pen and radio for the multi-skilled worker and skill matrix below. The abscissa represents the multi-skilled workers required for disassembly, and the ordinate represents the skills required for the disassembly task composed of 10 skills. A=[101110101011010101100110111111]

Disassembly task and disassembly task skill matrix B=Bjk, which is used to describe the skill k required by disassembly task j, specifically defined as: Bjk={1,If disassembly task j requires skill k; 0,If disassembly task j doesn'trequire skill k.

2) Disassemble the ballpoint pen and radio for the disassembly task and the disassembly task required the skill matrix below. The abscissa represents that the disassembly task consists of 43 disassembly tasks, and the ordinate represents that the skills required by the disassembly task consist of 10 skills. B=[100000000010000000000100000000001000000000100000000001000000…00001000000000010000000000100000000001000000000001]

In MPDMW, a feasible disassembly task sequence is obtained, and it is reasonably allocated to each workstation according to the required skills of the disassembly task and the workers’ skills. This work aims to reduce costs, improve workstation utilization, and meet the environmental requirements of disassembly enterprises.

Notations used in the model to be presented are summarized as follows:

Sets:

Parameters:

Decision variables: xgilm={1,If disassemble the execution of task i in product g assigned to workstation m on side l;0,Otherwise. um={1,If the m−th workstation is started;0,Otherwise. zplm={1,If worker p is assigned to workstation m on side l;0,Otherwise. ygipklm={1,If the disassembly task i in product g requires the k skills of worker p and is arranged to be performed at workstation;0,Otherwise.

We formulate the optimization problem of MPDMW as: (1)maxf1=∑g=1G∑i=1I∑q=1Q∑l=12∑m=1MVgqdgiqxgilm−∑g=1G∑i=1I∑p=1P∑k=1K∑l=12∑m=1MTgikDcgikDygipklm−∑p=1P∑l=12∑m=1MCpzplm (2)maxf2=CT (3)∑p=1Pzplm=um,∀m,∀l (4)∑m=1M∑l=12zplm≤1,∀p (5)∑l=12∑m=1Mxgilm≤1,∀g,∀i (6)∑g=1G∑i=1I∑l=12xgilm≥um,∀m (7)∑g=1G∑i=1I∑l=12xgilm≤Ium,∀m (8)∑k=1Kβkiygipklm≥xgilm+zplm−1,∀g,∀i,∀p,∀l,∀m (9)∑p=1P∑k=1Kαpkβkizplm≥xgilm,∀g,∀i,∀l,∀m (10)ygipklm≤xgilm,∀g,∀i,∀p,∀k,∀l,∀m (11)ygipklm≤zplm,∀g,∀i,∀p,∀k,∀l,∀m (12)∑g=1G∑i=1I∑k=1K∑l=12∑m=1MTgikDxgilm≤CT (13)∑g=1G∑i=1I∑p=1P∑k=1K∑l=12TgikDygipklm≤m∗um (14)∑m=1M(mxgilm)≤∑m=1M(mxgjlm)+(N1+N2)(1−∑m=1Mxgjlm),∀g,∀i,∀l,∀j (15)∑m=1M∑l=12(xgilm+xgjlm)≤1,∀g,Sijg=−1 (16)xgjlm,um,zplm,ygipklm,∀g,∀i,∀p,∀k,∀l,∀m

The objective function (1) represents the maximum profit from the disassembly of EOL products, composed of three parts: the total recovery value of disassembly EOL products, the total cost of disassembly, and the total cost of workers. (2) represents the minimization cycle time. Constraint (3) guarantees that only one worker can be assigned to each workstation on each side. Constraint (4) indicates that a person can be assigned to at most one side of the workstation. Constraint (5) indicates that each task can only be assigned to one side of a workstation. Constraint (6) indicates that each enabled workstation must be assigned at least one task. Constraint (7) indicates that a workstation that is not enabled cannot be assigned tasks. Constraint (8) ensures that each assigned task must be executed. Constraint (9) represents the relationship between skills and workers’ tasks. Constraint (10) ensures that the task can only be executed if it is assigned to the workstation side. Constraint (11) ensures that the task can only be performed if the worker is assigned to the workstation and has this skill. Constraint (12) indicates that the processing time of each workstation does not exceed the cycle time. Constraint (13) indicates that tasks on the demolition line must meet the precedence relation. Constraints (14) and (15) indicate that the feasible disassembly sequence must satisfy the conflict relation. Constraint (16) represents the range of decision variables.

The three-segment idea π=π1,π2,π3 in the U-shaped is continued in the encoding of union type to solve an MPDMW. π1 is the sequence of disassembly tasks, π2 determines whether the task is executed, and π3 determines which disassembly line the current task takes. As shown in Fig. 3. It is the feasible task sequence of disassembling the ballpoint pen and radio, where in π1, orange is the task sequence of disassembling the radio, and yellow is the task sequence of disassembling the ballpoint pen. In π3, orange is placed below the parallel disassembly line, and yellow is placed above the parallel disassembly line.

After the coding of the solution is completed, it needs to be decoded. The idea of decoding is to assign tasks to each workstation based on the fixed number of workstations. The decoding process of the parallel disassembly line is to assign feasible task sequences to each workstation. The allocation principle allocates disassembly tasks to each workstation. In the allocation process, the cycle time of each workstation is minimized. Fig. 4 shows the specific assignment of the task sequence at the workstation. Task 2 takes 10 s, task 14 takes 6 s, and task 11 takes 8 s. Assigned to workstation 1, Workstation 1 has a cycle time of 24 s. Task 16 has a time of 5 s, task 6 has a time of 7 s, task 8 has a time of 9 s, and workstation 2 has a cycle time of 21 s. Task 43 time is 4 s, Task 17 time is 5 s, task 18 time is 7 s, task 29 time is 5 s, task 10 time is 4 s, and workstation 3 cycle time is 25 s.

According to the above allocation, the cycle of each workstation can be minimized. Three workstations are required to complete the disassembly to optimize the two objective functions.

This work uses the AND/OR graph of two products, ballpoint pen and radio, to illustrate the encoding and decoding of parallel disassembly lines. Fig. 5 describes the disassembly process of these two products on the parallel disassembly line. For illustration purposes, the feasible disassembly sequence it forms is {2, −14, −16, −43, −17, −18, 11, 6, −29, 8, 10}. The numbers in the disassembly sequence represent the corresponding disassembly tasks. The numbers prefixed with “−” indicate that the current task is disassembling the radio and assigned to the lower disassembly line. The numbers without prefix symbols indicate that the current task disassembles the ballpoint pen and is assigned to the upper disassembly line. For instance, −14 denotes that task 14 for the radio is allocated to the lower disassembly line, while 2 signifies that task 2 for the ballpoint pen is assigned to the upper disassembly line. The required skills refer to the skills needed to perform each disassembly task, and only workers possessing those specific skills can execute the disassembly. For example, task −14 corresponds to skill 4, meaning skill level 4 is necessary to accomplish task −14. At this moment, Worker 1 happens to have skill level 4, thus enabling the allocation of task −14 to Worker 1’s lower disassembly line station for disassembly.

Synthesis reaction is the process of two or more molecules colliding to form a single molecule. As shown in Fig. 6. The steps are as follows:

Step 1: Randomly select two molecules from the population and judge whether the kinetic energy of each molecule of the two molecules is greater than the threshold for the decomposition reaction. If the conditions are met, a synthesis reaction will occur. Currently, the number of molecular collisions at each position (POS1, POS2) is +1.

Step 2: Give the molecule at POS1 position to the Father and the molecule at POS2 position to the Mother, and calculate the length of each molecule, respectively. The beginning value of the current index is 0, and the corresponding current index is increased by 1 for each operation.

Step 3: Give StartChildent1 the {−11, −13, 3, 6, −17, −18, 7, −19} at positions {0, 2, 4, 6, 8, 10, 12, 14} in the Father.

Step 4: Give {2, −21, −40, 9, 6, −17, 10} at positions {1, 3, 5, 7, 9, 11, 13} in the Mother to the corresponding {1, 3, 5, 7, 9, 11, 13} in StartChildren1.

Step 5: When the sequence is added to StartChildren1 by the Father and the Mother individuals, respectively, the two are conducted by polling. If the sequence added does not exist in StartChildren1, it will be added; otherwise, it will not be added. For example, when the Mother individual adds sequence {6, −17}, If this sequence is found to already exist in StartChildren1, these two sequences cannot be added. The corresponding current index is also added.

Step 6: If the size of the current index is smaller than the length of the Father, then add the sequence of the corresponding index in the Father and the sequence does not exist in StartChildren1, then add the sequence to StartChildren1 and then give the fully allocated sequence to StartChildren2.

Step 7: If the size of the current index is smaller than the length of the Mother, add the sequence of the corresponding index in the Mother, and it does not exist in StartChildren1, then add the sequence to StartChildren1.

Step 8: Because it is so multi-objective here in calculating the potential energy, we introduced the thought of weight, combined with the actual reasonable weight distribution, due to the first goal being profit, the second being the workstation cycle, are essential for decision making, so we are here to set their weights are 0.5, calculation, draw the corresponding potential energy. To see if this reaction can happen.

Step 9: If the sum of the kinetic energy plus the potential energy of the Father and the Mother is greater than the potential energy of the StartChildren2 solution, this reaction can occur. The synthesis reaction is complete, and the next iteration is performed.

Step 10: Check and adjust the newly generated sequence of StartChildren2 to meet the constraints of precedence and conflict relationships.

In this reaction, two or more molecules collide with each other without changing their structure. Here, two molecules are taken as examples, as shown in Figs. 7 and 8, and the steps are as follows:

Step 1: Randomly select two molecules from the population and judge whether these two molecules meet the threshold for synthesis. An ineffective internal molecular collision reaction will occur if they do not meet the conditions. At this time, the molecular collision times of each position (POS1, POS2) + 1.

Step 2: Give the molecule at POS1 position to the Father and the molecule at POS2 position to the Mother, and calculate the length of each molecule, respectively. The beginning value of the current index is 0, and the corresponding current index is increased by 1 for each operation.

Step 3: Generate two random numbers p and q for a long time. If p>q, exchange values between them; otherwise, no exchange is required.

Step 4: Assign {−15, 1} and {10, −39, −41, −43, −42, 11, −22} in the Father to the corresponding position of the Child 1. The middle blue part of Child 1 is filled with 0.

Step 5: Traverse the Mother individual and add the elements not in Child 1 to the 0 part of Child 1. In Fig. 7, fill {−14, 3, −16, −17, −18} into the corresponding position of Child 1, and {−19, −30, 6} in Mother individual into the opposite end of Child 1.

Step 6: If the Mother does not complete the 0 in Child 1 in Step 5, the Father will be randomly selected from the middle of p and q and filled into Child 1.

Step 7: Give {−14, 1, 3} and {11, −22} in the Mother to Child 2, and fill the middle blue part of Child 2 with 0.

Step 8: Traverse the individual Father and add the elements not found in Child 2 to the 0 part of Child 2. As shown in Fig. 8, fill {−15, 6, −24, −27, −32, 10, −39, −41, −43} into the corresponding position of Child 2, and {−42} in the individual Father is added to the opposite end of Child 2.

Step 9: If the Father does not complete the 0 in Child 2 in Step 8, the Mother will be randomly selected from the middle of p and q and filled into Child 2.

Step 10: Check and adjust the newly generated sequence of Child 1 and Child 2 to meet the constraints of precedence and conflict relationships.

The molecules hit the wall, and the structure of the molecule doesn’t change very much. This reaction is an example of a reaction in which one molecule forms another.

Step 1: As shown in Fig. 9, generate two random numbers p and q, p and q belong to [1, the length of the father sequence].

Step 2: If p = q, create a new q until the two are different.

Step 3: Swap the disassembly tasks at positions p and q to generate a new sequence.

Step 4: Check and adjust the disassembly sequence, turn it into a feasible solution, and calculate the objective function.

Decomposition reaction refers to the decomposition of one molecule into two or more molecules. The specific steps are shown in Fig. 10.

Step 1: The operation of new molecule 1 is: the loop starts from 0 and ends with the length/2 of the parent task sequence.

Step 2: Once a cycle, a random number will be generated. If the number is greater than 0.5, Step 3 will be repeated three times.

Step 3: Randomly generate a task. If the task is larger than 13, multiply by −1. If the task is not in the sequence, add it to the sequence.

Step 4: Perform Step 1 until the loop ends. The new individual is Children1.

Step 5: The operation of new molecule 2 is: the loop starts at length/2 of the parent task sequence and stops at the length of the parent task sequence.

Step 6: Once a cycle, a random number will be generated. If the number is greater than 0.5, Step 7 will be repeated three times.

Step 7: Randomly generate a task. If the task is larger than 13, multiply by −1. If the task is not in the sequence, add it to the sequence.

Step 8: Go to Step 5 until the loop ends. The new individual is Children2.

Step 9: Check and adjust the disassembly sequence, turn it into a feasible solution, and calculate the objective function.

In this paper, multi-product composed of a Ballpoint pen and radio (BR), flashlight and radio (FR), washing machine and radio (WR) are used as experimental cases to verify the feasibility of MDCRO in solving MPDMW, and NSGA-II, MOEA/D, NSGA-III and MOEAD-CRA are selected to solve MPDMW together to study the advantages and disadvantages of the five algorithms in solving this problem. The reason for choosing these metaheuristic methods lies in their unique strengths in multi-objective optimization. For instance, MOEA/D exhibits problem decomposition capabilities, MOEAD-CRA incorporates a control variable adjustment mechanism, NSGA-II utilizes non-dominated sorting and crowding distance calculation, and NSGA-III introduces layered non-dominated sorting. These characteristics contribute to effectively addressing the complex objectives and constraints of the problem. To ensure the accuracy and correctness of the experiment, all the algorithms are compared under the condition of complete fairness.

Among the experimental parameters, the molecular collision rate is set as 0.5, the synthesis threshold is 10, the energy loss rate is 0.3, the crossover rate of NSGA-II, MOEA/D, NSGA-III, and MOEAD-CRA Px = 0.7, the mutation probability Pm = 0.1/number of subassemblies, the number of workstations is 3, the total number of initialization groups of all algorithms is the same. And the population is 100, 200, and 300, three comparisons, respectively. The maximum iteration count equals three times the product of the subassemblies’ quantity and the tasks.

To verify the correctness of the model, the CPLEX solver is used for the first time to solve the established model. Since CPLEX does not support multi-objective optimization, only one objective can be optimized. Therefore, when verifying the model, the maximum profit is taken as the goal to solve. As can be seen from Table 2, comparing the sequence obtained by CPLEX solving BR, FR, AND WR with the AND/OR graph composed of actual cases, it can be seen whether the sequence is a feasible solution, thus verifying the correctness of the established model. To verify the feasibility and superiority of MDCRO, the algorithm is compared with NSGA-II, MOEA/D, NSGA-III, and MOEAD-CRA. The experimental results show that MDCRO is superior to the other four algorithms in solving MPDMW.

To analyze the performance of MDCRO in solving MPDMW compared with other algorithms, IGD-metric [34], Hypervolume-metric [35], and Epsilon-metric are used for comparison. Verification is done using the t-test. The t-test is a statistical method used to compare whether the means of two samples are significantly different. It assesses whether the difference in means between two samples is likely due to random factors or if it reflects a true difference in population means. When calculating the objective function, we try to find the maximization of profit and the minimization of workstation cycle time, so the value of the first objective function is multiplied by −1 to facilitate the comparison of results.

Through experiments, MDCRO can provide a feasible solution for MPDMW. Table 3 shows the nine non-dominated solutions obtained by MDCRO solving the above three combination cases. Among them, numbers 1–3 refer to the disassembly sequence obtained when BR is disassembled, numbers 4–6 refer to the disassembly sequence obtained when FR is disassembled, and numbers 7–9 refer to the disassembly sequence obtained when WR is disassembled. The disassembly sequence represents the sequence of feasible tasks for disassembling the above multi-product, and the required skills represent the skills required for disassembling the corresponding tasks. Workstation allocation represents the allocation of disassembly tasks between workstations, with f1 and f2 representing profit maximization and minimization of workstation cycle time, respectively.

In order to conduct an in-depth study on the performance of MDCRO, this paper chooses NSGA-II, MOEA/D, NSGA-III, MOEAD-CRA, and MDCRO to conduct an in-depth study around MPDMW. For multi-product cases composed of BR, FR, and WR under the condition of the same implementation parameters, The performance gap of five algorithms under different population sizes is studied.

Table 4 shows the IGD index results obtained by the five algorithms when the population is 100, 200, and 300. The experimental results indicate that MDCRO’s IGD index significantly outperforms NSGA-II, MOEA/D, NSGA-III, and MOEAD-CRA in the three multi-product combination cases mentioned above. It’s only in the case of a population size of 200 that the disassembly of WR is not as effective as NSGA-III.

Table 5 shows the Hypervolume index obtained by five algorithms when the population is 100, 200, and 300 for analysis. The Hypervolume index can comprehensively evaluate the convergence and distribution of non-dominant solution sets. The experimental results show that the Hypervolume index of MDCRO is slightly lower than that of NSGA-II except when the product BR is disassembled and the population size is 100 and 200. The disassembly Hypervolume index of MDCRO of the above three combinations is significantly higher than that of NSGA-II, MOEA/D, NSGA-III, and MOEAD-CRA.

Table 6 provides the Epsilon index obtained by five algorithms for population sizes of 100, 200, and 300, along with the corresponding analysis. The Epsilon index can measure the approximation degree of the solution. According to the results of the Epsilon index, the disassembly values of the above three composite products by MDCRO are significantly lower than those of NSGA-II, MOEA/D, NSGA-III, and MOEAD-CRA.

In order to further verify the superiority of MDCRO in the processing of MPDMWS, the performance of MDCRO is shown by the box graph more intuitively. Fig. 11 shows MDCRO has a better Hypervolume-metric, IGD-metric, and Epsilon-metric when removing WR. Experimental results and analysis show that the MDCRO algorithm is superior to NSGA-II, MOEA/D, NSGA-III, and MOEAD-CRA in solving MPDMW.

As shown in Fig. 12, when the profit is less than 1500, there is little difference between the solutions of the five algorithms. However, when the profit reaches 1500, the feasible solution cannot be found by other algorithms at all. MDCRO can still find a feasible solution. The results show that MDCRO performs well in depth and breadth of solution. As a result, MDCRO has a broader distribution in solving MPDMW than NSGA-II, MOEA/D, NSGA-III, and MOEAD-CRA.