New formulations for the setup assembly line balancing and scheduling problem

Abstract

We present three new formulations for the setup assembly line balancing and scheduling problem (SUALBSP). Unlike the simple assembly line balancing problem, sequence-dependent setup times are considered between the tasks in the SUALBSP. These setup times may significantly influence the station times. Thus, there is a need for scheduling the list of tasks within each station so as to optimize the overall performance of the assembly line. In this study, we first scrutinize the previous formulation of the problem, which is a station-based model. Then, three new formulations are developed by the use of new sets of decision variables. In one of these formulations, the schedule-based formulation, SUALBSP is completely formulated as a scheduling problem. That is, no decision variable in the model directly denotes a station. All the proposed formulations will be improved by the use of several enhancement techniques such as preprocessing and valid inequalities. These improved formulations can be applied to establishing lower bounds on the problem. To assess the performance of new formulations, results of an extensive computational study on the benchmark data sets are also reported.

Appendix

1.1 The secondary objective

Scholl et al. (2013) consider the objective function (76) and constraint sets (77) and (78) to embed a secondary objective into their formulation.

$$\begin{aligned}&\min \ z_n+\varepsilon \cdot \sum _{k\in K} {T_{k}} \end{aligned}$$

(76)

$$\begin{aligned}&(f_i-c\cdot (z_i-1))+t_i+\sum _{j\in F_{i}^{B}}\mu _{ij}\cdot w_{ij}\le T_k+M\cdot (1-x_{ik})&\forall i\in V\,\text {and}\, k \in FS_{i} \end{aligned}$$

(77)

$$\begin{aligned}&T_k\ge 0&\forall k\in K \end{aligned}$$

(78)

In this model, $T_k$ is a continuous variable representing the station time for station k. In addition, $\varepsilon$ is a sufficiently small number, i.e., $\varepsilon =1/(c\cdot (n+1))$. The secondary objective minimizes the total setup time by minimizing the sum of the station times. Constraint set (77) in conjunction with the objective function (76) captures the station time of station k. Constraint set (78) states non-negativity of the variables $T_k$. As is seen, including the secondary objective (76) requires addition of $O(n^2)$ disjunctive constraints.

One other way to incorporate the secondary objective into the formulation is to use the following objective function and constraints:

$$\begin{aligned}&\min \ z_n+\varepsilon \cdot \sum _{i\in V} {T'_{i}} \end{aligned}$$

(79)

$$\begin{aligned}&(f_i-c\cdot (z_i-1))+t_i+\sum _{j\in F_{i}^{B}}(c+\mu _{ij})\cdot w_{ij}\le T'_i+c&\forall i\in V \end{aligned}$$

(80)

$$\begin{aligned}&T'_i\ge 0&\forall i\in V \end{aligned}$$

(81)

According to this model, if task i is the last task in its station, then variable $T'_i$ gets the value of the station time of that station to which task i is assigned, and zero otherwise. This method requires only n additional constraints.

Still, the secondary objective can be included into the formulation more effectively. Let idle time be defined as the difference between the cycle time and the station time. If I be the total idle time, and S be the total setup time, the relation $S+I+t_\mathrm{sum}=m \cdot c$ holds. So, for a given value of the line capacity, maximization of the total idle time is equivalent to minimization of the total setup time. Thus, the following scheme is proposed.

$$\begin{aligned}&\min \ (\bar{m}\cdot c-t_\mathrm{sum}+1)\cdot z_n-I \end{aligned}$$

(82)

$$\begin{aligned}&\sum _{i\in V}\sum _{j\in F_{i}^{F}}\tau _{ij}\cdot y_{ij}+\sum _{i\in V}\sum _{j\in F_{i}^{B}}\mu _{ij}\cdot w_{ij}+t_\mathrm{sum}+I\le c\cdot z_n \end{aligned}$$

(83)

$$\begin{aligned}&I\ge 0 \end{aligned}$$

(84)

The objective function (82) minimizes the number of stations as the primary objective and the total setup time as the secondary objective. Because $z_n$ takes an integer value and $I \in [0,\bar{m}\cdot c-t_\mathrm{sum}]$, maximization of the total idle time (minimization of the total setup time) has no effect on optimization of the primary objective. This approach just needs one additional constraint.