GRASP for sequencing mixed models in an assembly line with work overload, useless time and production regularity

Abstract

A GRASP algorithm is presented for solving a sequencing problem in a mixed-model assembly line. The problem is focused on obtaining a manufacturing sequence that completes the greatest possible amount of required work and fulfils the production regularity property. The implemented GRASP algorithm is compared with other resolution procedures by means of instances from a case study linked to the Nissan’s engine plant in Barcelona.

Appendix

Model for MAX–SAT production mix restriction problem:

Nomenclature:

I :

Set of product types $\left( {I\,{:}\,i=1,\ldots ,|I|} \right)$

$\vec {d},D$ :

Vector of demand $\vec {d}=(d_{1},\ldots ,d_{|I|})$, and total demand $D\equiv T=\sum \nolimits _{\forall i} d_i$

$\vec {\lambda }$ :

Vector of production mix $\vec {\lambda }=(\lambda _{1},\ldots ,\lambda _{|I|})$: $\vec {\lambda }=\vec {d}/D$

$\pi (T)$ :

Sequence of products $\pi (T)=\left( {\pi _1 ,\ldots ,\pi _T } \right)$

$x_{i,t}$ :

Binary variable equal to 1 if a product unit $i\in I$ is assigned to the position $t \quad \left( {t=1,\ldots ,T} \right) \,$of the sequence $\pi (T)$ and to 0 otherwise

$X_{i,\,t}$ :

Number of units of type $i\in I$ in the subsequence of products $\pi (t)\subseteq \pi (T)$. Obviously $X_{i,\,t} =\sum \nolimits _{\tau =1}^t x_{i,\tau } \quad \forall i\forall t$

$z_{i,t}^+$ :

Binary variable equal to 1 if $X_{i,\,t} \left( {i\in I,\,t=1,\ldots ,T} \right)$ is greater than the upper limit $\lfloor \lambda _i t\rfloor$ of production mix preservation and to 0 otherwise

$z_{i,t}^-$ :

Binary variable equal to 1 if $X_{i,\,t}\left( i\in I,\,t=1,\ldots ,T \right)$ is less than the lower limit $\lceil \lambda _i t\rceil$ of production mix preservation and to 0 otherwise

The objective is: $\lfloor \lambda _i t\rfloor \le X_{i,\,t} \le \lceil \lambda _i t\rceil ,\,\,X_{i,\,T} =d_i \,\,\forall i\in I\,\forall t=1,\ldots ,T$

MAX–SAT–PMR problem model:

$$\begin{aligned} \min Z=\mathop {\sum }\limits _{i=1}^{|I|} \mathop {\sum }\limits _{t=1}^T \left( {z_{i,t}^+ +z_{i,t}^- } \right) \end{aligned}$$

(18)

Subject to:

$$\begin{aligned}&\sum \limits _{i=1}^{|I|} x_{i,t} =1\quad \forall t=1,\ldots ,T \end{aligned}$$

(19)

$$\begin{aligned}&\sum \limits _{t=1}^T x_{i,t} =d_i \quad \forall i\in I \end{aligned}$$

(20)

$$\begin{aligned}&X_{i,t} -\sum \limits _{\tau =1}^t x_{i,\tau } =0\quad \forall i\in I\,\forall t=1,\ldots ,T \end{aligned}$$

(21)

$$\begin{aligned}&x_{i,t} =\left\{ {{\begin{array}{l} {1\Leftrightarrow \pi _t =i} \\ {0\Leftrightarrow \pi _t \ne i} \\ \end{array} }} \right\} \quad \forall i\in I\,\forall t=1,\ldots ,T \end{aligned}$$

(22)

$$\begin{aligned}&z_{i,t}^+ =\left\{ {{\begin{array}{l} {1\Leftrightarrow X_{i,\,t} > \lceil \lambda _i t}\rceil \\ {0\Leftrightarrow X_{i,\,t} \le \lceil \lambda _i t}\rceil \\ \end{array} }} \right\} \quad \forall i\in I\,\forall t=1,\ldots ,T \end{aligned}$$

(23)

$$\begin{aligned}&z_{i,t}^- =\left\{ {{\begin{array}{l} {1\Leftrightarrow X_{i,\,t} < \lfloor \lambda _i t}\rfloor \\ {0\Leftrightarrow X_{i,\,t} \ge \lfloor \lambda _i t\rfloor } \\ \end{array} }} \right\} \quad \forall i\in I\,\forall t=1,\ldots ,T \end{aligned}$$

(24)