A green-oriented bi-objective disassembly line balancing problem with stochastic task processing times

Abstract

Remanufacturing and recycling industry has developed rapidly in recent years due to its benefits in reducing waste and protecting the environment. However, the uncertain environment and excessive emission during production become two main obstacles for its further development. In this paper, a green-oriented bi-objective disassembly line balancing problem with stochastic task processing times is studied. The objectives are to minimize the total line configuration cost respecting the given budget, and minimize the total contaminant emission, respectively. To depict stochastic processing times, their mean, standard deviation and change-rate upper bound are assumed to be known since it may be difficult to obtain the complete historical data. For the problem, a bi-objective model with chance constraints is first formulated, which is further approximated into a linear distribution-free one. To solve the second model, an efficient $\varepsilon$-constraint method is proposed based on problem analysis. Finally, a fuzzy-logic-based approach is applied to recommend preferred solutions for managers according to their perspectives. The solution methods are first examined by a case study, then by 247 benchmark-based instances and randomly generated instances. Experimental results indicate the efficiency and effectiveness of the proposed methods for solving the green-oriented bi-objective problem.

Appendices

Appendix 1

The proof of Lemma 1

Proof

Since $t_j={\mathbb {E}}[p_j]$ and $p_j=t_j(1+d_j)$, then we have ${\mathbb {E}}[d_j]=0$. Using the well-known Taylor series expansion $e^x=\sum \nolimits _{n=1}^{\infty } \frac{x^n}{n!}$, we have ${\mathbb {E}}[e^{\lambda d_j}]=1+\sum \nolimits _{n=2}^{\infty } \frac{\lambda ^n {\mathbb {E}}[d_j^2d_j^{n-2}]}{n!}$. Because that $Pr (d_i \le b_i)=1$, the following formulas satisfy:

$$\begin{aligned} {\mathbb {E}}[e^{\lambda d_j}]\le & {} 1+ {\mathbb {E}}[d_j^2] \sum _{n=2}^{\infty } \frac{\lambda ^n b_j^{n-2}}{n!} \\= & {} 1+ \frac{{\mathbb {E}}[d_j^2]}{b_j^2} \sum _{n=2}^{\infty } \frac{\lambda ^n b_j^n}{n!} \\= & {} 1+\frac{{\mathbb {E}}[d_j^2]}{b_j^2}(e^{\lambda b_j}-\lambda b_j-1) \end{aligned}$$

It completes the proof. $\square$

Appendix 2

The proof of Proposition 1

Proof

Knowing that $\Pr \left( \sum \nolimits _{j \in J} p_j \cdot x_{j,m}> CT \right) = \Pr \left( \sum \nolimits _{j \in J} t_j(1+d_j) \cdot x_{j,m} > CT \right)$. If $\sum \nolimits _{j \in J} t_j(1+d_j) \cdot x_{j,m} > CT$, then it must follow that $t_j d_j > \mu _j$ since $x_{j,m}$ is a feasible solution to model P2. Hence, the following formula can be established:

$$\begin{aligned} \Pr \big ( t_j(1+d_j) \cdot x_{j,m}> CT \big )\le & {} \Pr \left( t_j d_j> \mu _j \right) \\= & {} \Pr \left( e^{\lambda _j d_j}> e^{\lambda _j} \frac{\mu _j}{t_j} \right) \\\le & {} \min _{\lambda _j>0} e^{\lambda _j \mu _j / t_j} {\mathbb {E}}[e^{\lambda _j d_j}] \quad \%\,\textit{Markov}\hbox {'}{} \textit{s inequality}\\= & {} \min _{\lambda _j>0} e^{\lambda _j \mu _j / t_j} \left( 1+\frac{{\mathbb {E}}[d_j^2]}{b_j^2}(e^{\lambda _j b_j}-\lambda _j b_j-1) \right) \end{aligned}$$

Let the right side of the above inequality equal $\beta _m$, it completes the proof. $\square$

Appendix 3

The related information of the used instance.

The DCG of the instance is shown in Fig. 6, and the input information is collected in Table 6.

Fig. 6

The DCG of this instance (Zheng et al. 2018)

Table 6 Input information of the instance

Appendix 4

The proof of Theorem 1

Proof

For model $\mathbf{P} _E(\varepsilon _i)$, the minimum unit of the total line cost C depends on $\sum \nolimits _{m \in M} c_m \cdot y_m$ and $h \cdot \sum \nolimits _{j \in H} \sum \nolimits _{m \in M} x_{j,m}$ in the $\varepsilon$-constraint. Since the latter is a constant independent of decision variables due to assumption (viii), therefore, it can be removed when deciding the step length. Then, the minimum unit of C is determined by the former part. Consider that at least one workstation should be opened and each candidate workstation has a cost $c_m$, thus the minimum unit of C should be the Greatest Common Divisor among the cost of candidate workstations, i.e., $\bigtriangleup _c = GCD(c_m)$. Finally, according to the first theorem in Wu et al. (2015), the Pareto front for the studied problem can be obtained by exactly solving model $\mathbf{P} _E(\varepsilon _i)$ with step length $GCD(c_m)$. $\square$

Appendix 5

The complementary input information for the used instance

The complementary input of the instance introduced in Appendix 3 is shown as follows:

• The regular budget RB: 20

• The invest cost of each workstation $c_m$: [6, 5, 4, 6, 5]

• The contaminant emission of task j generated by machine m, i.e., $e_{j,m}$:

  $$\begin{aligned} \left[ \begin{matrix} 6 &{}\quad 8 &{}\quad 10 &{}\quad 6 &{}\quad 8 \\ 9 &{}\quad 11 &{}\quad 13 &{}\quad 9 &{}\quad 11 \\ 4 &{}\quad 6 &{}\quad 8 &{} \quad 4 &{} \quad 6 \\ 6 &{}\quad 8 &{}\quad 10 &{}\quad 6 &{}\quad 8 \\ 5 &{}\quad 7 &{} \quad 9 &{} \quad 5 &{} \quad 7 \\ 4 &{}\quad 6 &{} \quad 8 &{}\quad 4 &{} \quad 6 \\ 5 &{}\quad 7 &{} \quad 9 &{}\quad 5 &{} \quad 7 \\ 6 &{} \quad 8 &{} \quad 10 &{}\quad 6 &{} \quad 8 \\ 6 &{}\quad 8 &{} \quad 10 &{}\quad 6 &{} \quad 8 \\ 7 &{} \quad 9 &{} \quad 11 &{} \quad 7 &{} \quad 9 \end{matrix} \right] \end{aligned}$$

Appendix 6

The detailed computational results in Sect. 5.3.

See Tables 7, 8, 9, and 10.

Table 7 Computational results on 40-disassembly-task instances

Table 8 Computational results on 60-disassembly-task instances

Table 9 Computational results on 80-disassembly-task instances

Table 10 Computational results on 100-disassembly-task instances