3.1 Problem description
There are different types of EOL products for remanufacturing with uncertain quantities. To achieve a reasonable allocation of disassembly tasks for different types of products, this paper proposes a parallel mixedflow disassembly line layout, as shown in Fig. 1.
If there were two kinds of EOL products to be disassembled and the number of components was uncertain, two disassembly lines were required. Parallel stations were arranged on each disassembly line, such as stations S1 and S3 in Fig. 1. The two adjacent disassembly lines had mixedflow disassembly stations, such as stations S2 and Sm. All disassembly tasks were assigned to N workstations according to the determined beat time, CT.
The parallel mixedflow remanufacturing disassembly line balancing problem focused on attaining a reasonable allocation of disassembly tasks in the layout shown in Fig. 1 to minimize the number of disassembly stations, prioritize the disassembly of components with high remanufacturing values and hazardous material properties, and rationally utilize the factory space of the enterprise.
To simplify the problem, three assumptions were made:
(1) The disassembly time and remanufacturing value of each component were known, and all disassembly tasks were independent.
(2) A disassembly task could not be interrupted.
(3) The same disassembly task could not be assigned to multiple stations at the same time.
3.2 Judgment conditions for the mixedflow disassembly of multivariety products
Similarities and differences exist in the physical, material, and geometrical structures of various types of EOL products. Only products with certain similarities can be disassembled using a parallel mixedflow disassembly line [12]. Therefore, it was necessary to determine the degree of product similarity.
It was assumed that the two disassembly task sets for the EOL products were
\({\mathbf{P}}_{1}=\left\{{a}_{1}^{1},{a}_{2}^{1},{a}_{3}^{1},\cdots ,{a}_{m1}^{1}\right\}\)and \({\mathbf{P}}_{2}=\left\{{a}_{1}^{2},{a}_{2}^{2},{a}_{3}^{2},\cdots ,{a}_{m2}^{2}\right\}\). The similar components set was
\(\mathbf{S}=\bigcap \left\{{a}_{1}^{1},{a}_{1}^{2},{a}_{2}^{1},{a}_{2}^{2},{a}_{3}^{1},{a}_{3}^{2}\cdots ,{a}_{m1}^{1},{a}_{m2}^{2}\right\}\), and the total components set was \({\text{P}}_{1}{\text{P}}_{2}=\bigcup \left\{{a}_{1}^{1},{a}_{1}^{2},{a}_{2}^{1},{a}_{2}^{2},{a}_{3}^{1},{a}_{3}^{2}\cdots ,{a}_{m1}^{1},{a}_{m2}^{2}\right\}\). Thus, the similarity degree between the two products could be defined as follows:
\({\lambda }_{pro}=\frac{\bigcap \left\{{a}_{1}^{1},{a}_{1}^{2},{a}_{2}^{1},{a}_{2}^{2},{a}_{3}^{1},{a}_{3}^{2}\cdots ,{a}_{m1}^{1},{a}_{m2}^{2}\right\}}{\bigcup \left\{{a}_{1}^{1},{a}_{1}^{2},{a}_{2}^{1},{a}_{2}^{2},{a}_{3}^{1},{a}_{3}^{2}\cdots ,{a}_{m1}^{1},{a}_{m2}^{2}\right\}}=\frac{\mathbf{S}}{{\mathbf{P}}_{1}{\mathbf{P}}_{2}}\),\(0\le {\lambda }_{pro}\le 1.\) (1)
where m1 and m2 are the numbers of components in the two EOL products to be disassembled. The larger the similarity degree was, the greater was the similarity between the components in geometrical, physical, and material aspects, among others.
3.3 Mathematical model for the PMRDLB problem
A mathematical model for the PMRDLB problem was developed based on the parallel mixedflow remanufacturing disassembly line layout shown in Fig. 1. For clarification, the symbols utilized in the mathematical model are defined in Table 1.
Table 1
Symbol

Illustration

mk 
Number of the kth EOL product’s disassembly tasks

N 
Number of disassembly stations

CT 
Disassembly line beat time

til 
The disassembly time of the task i on the lth disassembly line

xilr 
Station task allocation coefficient, when the disassembly task i on the lth disassembly line is assigned to the rth station, it equals 1, and otherwise, it equals 0.

Pil 
Remanufacturing value of the components disassembled in the disassembly task i of the lth disassembly line, if the component has no remanufacturing value, it equals 0, and otherwise, it equals 1.

Sr 
Disassemble task set in station r

Lil 
The position of component in the disassembly sequence in the ith disassembly task of the lth disassembly line

Sij 
Task i takes precedence over task j


Disassembly priority mapping matrix of the EOL product k


Disassembly task hazard mapping matrix of the EOL product k

k 
Number of the disassembly lines


The comprehensive priority relation matrix of the EOL product k

Gk 
The comprehensive priority relation matrix of the EOL product k

G 
The comprehensive disassembly tasks hierarchical matrix of EOL products

One clear difference between the PMRDLB problem and the traditional DLBP is the constraint conditions. All of the products in the parallel disassembly lines should not only meet the component disassembly priority relationship requirements but should also prioritize the disassembly of toxic and harmful components to reduce secondary pollution. This type of disassembly is more complex than singleproduct disassembly.
The disassembly priority relationship mapping matrix for the EOL product k is given by
$${\mathbf{P}}_{mk}^{k}=\begin{array}{c}1\\ \begin{array}{c}2\\ \begin{array}{c}⋮\\ mk\end{array}\end{array}\end{array}{\left\{\begin{array}{cc}\begin{array}{cc}{p}_{11}& {p}_{12}\\ {p}_{21}& {p}_{22}\end{array}& \begin{array}{cc}\cdots & {p}_{1mk}\\ \cdots & {p}_{2mk}\end{array}\\ \begin{array}{cc}⋮& ⋮\\ {p}_{mk1}& {p}_{mk2}\end{array}& \begin{array}{cc}{p}_{ij}& ⋮\\ \cdots & {p}_{mkmk}\end{array}\end{array}\right\}}_{mk\times mk}.$$
2
In Eq. (2), if task i is performed before task j, then\({p}_{ij}\) = 1; otherwise, \({p}_{ij}\) = 0.
The component hazard mapping matrix for the EOL product k is defined as follows:
$${\mathbf{B}}_{mk}^{k}=\begin{array}{c}1\\ \begin{array}{c}2\\ \begin{array}{c}⋮\\ mk\end{array}\end{array}\end{array}{\left\{\begin{array}{cc}\begin{array}{cc}{b}_{11}& {b}_{12}\\ {b}_{21}& {b}_{22}\end{array}& \begin{array}{cc}\cdots & {b}_{1mk}\\ \cdots & {b}_{2mk}\end{array}\\ \begin{array}{cc}⋮& ⋮\\ {b}_{mk1}& {b}_{mk2}\end{array}& \begin{array}{cc}{b}_{ij}& ⋮\\ \cdots & {b}_{mkmk}\end{array}\end{array}\right\}}_{mk\times mk}.$$
3
In Eq. (3), if disassembly task i is more hazardous than task j, then\({ b}_{ij}\) = 1; otherwise, \({b}_{ij}\) = 0.
The disassembly priority relationship for the EOL product k was deduced from \({\mathbf{P}}_{mk}^{k}, {\mathbf{B}}_{mk}^{k}\), and the comprehensive matrix \({\mathbf{S}}_{mk}^{k}, \text{a}\text{s} \text{f}\text{o}\text{l}\text{l}\text{o}\text{w}\text{s}\):
$${\mathbf{S}}_{mk}^{k}={\mathbf{P}}_{mk}^{k}\bigvee {\mathbf{B}}_{mk}^{k}$$
$$=\begin{array}{c}1\\ \begin{array}{c}2\\ \begin{array}{c}M\\ mk\end{array}\end{array}\end{array}{\left\{\begin{array}{cc}\begin{array}{cc}{p}_{11}\bigvee {b}_{11}& {p}_{12}\bigvee {b}_{12}\\ {p}_{21}\bigvee {b}_{21}& {p}_{22}\bigvee {b}_{22}\end{array}& \begin{array}{cc}L& {p}_{1mk}\bigvee {b}_{1mk}\\ L& {p}_{2mk}\bigvee {b}_{2mk}\end{array}\\ \begin{array}{cc}M& M\\ {p}_{mk1}{\bigvee b}_{mk1}& {p}_{mk2}{\bigvee b}_{mk2}\end{array}& \begin{array}{cc}{p}_{ij}\bigvee {b}_{ij}& M\\ L& {p}_{mkmk}\bigvee {b}_{mkmk}\end{array}\end{array}\right\}}_{mk\times mk}=\begin{array}{c}1\\ \begin{array}{c}2\\ \begin{array}{c}M\\ mk\end{array}\end{array}\end{array}{\left\{\begin{array}{cc}\begin{array}{cc}{S}_{11}& {S}_{12}\\ {S}_{21}& {S}_{22}\end{array}& \begin{array}{cc}L& {S}_{1mk}\\ L& {S}_{2mk}\end{array}\\ \begin{array}{cc}M& M\\ {S}_{mk1}& {S}_{mk2}\end{array}& \begin{array}{cc}{s}_{ij}& M\\ L& {S}_{mkmk}\end{array}\end{array}\right\}}_{mk\times mk}.$$
4
In Eq. (4), \({\mathbf{S}}_{ij}^{k}\) indicates that if disassembly task i has priority over task j, then \({\mathbf{S}}_{ij}=1\); otherwise, \({\mathbf{S}}_{ij}=0\).
According to Eq. (4), the feasibility conditions for disassembly task j were defined as follows:
$$\sum _{i=1}^{mk}{\mathbf{S}}_{ij}^{k}=0, \text{(5)}$$
The products’ disassembly tasks could be obtained from Eq. (5), and then, \({\mathbf{S}}_{mk}^{k}\) could be updated after disassembly. When \({\mathbf{S}}_{mk}^{k}={\left[0\right]}_{mk\times mk}\), all the disassembly tasks were finished, and the disassembly task hierarchical matrix, \({\mathbf{G}}^{k}\), for the EOL product k could be obtained.
The parallel mixedflow disassembly task allocation matrix, \(\mathbf{G}=\left\{{\mathbf{G}}^{1},{\mathbf{G}}^{2},{\mathbf{G}}^{3},\cdots ,{\mathbf{G}}^{k}\right\}\), shown in Fig. 2, could then be obtained from Eqs. (2)–(5).
Considering the uncertainty in the number of parts, during the construction of the disassembly sequence matrix for mixedflow products, the largest number of parts among k products should be taken as the matrix column standard, and the elements of the matrix with insufficient parts among the other products should be filled with 0.
The mathematical model for the PMRDLB problem was formulated utilizing Eqs. (6)–(13).
$$\text{m}\text{i}\text{n}F=\left({f}_{1},{f}_{2},{f}_{3}\right)$$
6
,
$${f}_{2}=\sqrt{\sum _{r=1}^{N}{\left(CT\sum _{l=1}^{k}\sum _{i=1}^{mk}{t}_{il}\times {x}_{ilr}\right)}^{2}/N} , \left(8\right)$$
$${f}_{3}=\frac{\sum _{l=1}^{k}\sum _{i=1}^{mk}\left({L}_{il}\times {P}_{il}\right)}{\sum _{l=1}^{k}\sum _{i=1}^{mk}{L}_{il}}$$
9
,
$$\sum _{i=1}^{mk}\sum _{l=1}^{k}\sum _{r=1}^{N}{x}_{ilr}=1,$$
10
$$\underset{r\in \{\text{1,2},\cdots ,N\}}{\text{max}}\left\{\sum _{l=1}^{k}\sum _{i=1}^{mk}({x}_{ilr}\times {t}_{il})\right\}\le CT,$$
11
$$\frac{\sum _{l=1}^{k}\sum _{i=1}^{mk}\left({x}_{ilr}\times {t}_{il}\right)}{CT}\le N\le \sum _{l=1}^{k}\sum _{i=1}^{mk}{x}_{ilr},$$
12
\(\sum _{r=1}^{N}\left({x}_{nlr}\times r\right)\le \sum _{r=1}^{N}\left({x}_{mlr}\times r\right)\) , \(n=(\text{1,2},3,L,mk1)\), \(m=\left\{mm=n+1\right\}.\) (13)
Eqs. (6)–(9) represent the optimization objects. In these equations, f1 is the number of parallel mixedflow disassembly line stations, f2 is the station equalization rate, and f3 is the average remanufacturing value index, which ensures disassembly of the higher value remanufacturing components first to avoid secondaryoperation damage to the remanufacturing cores. Eq. (10) ensures that each disassembly line and disassembly task are assigned only to one station. Eq. (11) guarantees that the maximum total disassembly time in each disassembly station does not exceed the beat time, CT. Eq. (12) represents the workstation number range in the parallel disassembly line. Eq. (13) ensures that the priority relationship constraint is met for all of the disassembly tasks during an EOL product’s disassembly.