Job-shop scheduling-joint consideration of production, transport, and storage/retrieval systems

Abstract

This paper proposes a new problem by integrating the job shop scheduling, the part feeding, and the automated storage and retrieval problems. These three problems are intertwined and the performance of each of these problems influences and is influenced by the performance of the other problems. We consider a manufacturing environment composed of a set of machines (production system) connected by a transport system and a storage/retrieval system. Jobs are retrieved from storage and delivered to a load/unload area (LU) by the automated storage retrieval system. Then they are transported to and between the machines where their operations are processed on by the transport system. Once all operations of a job are processed, the job is taken back to the LU and then returned to the storage cell. We propose a mixed-integer linear programming (MILP) model that can be solved to optimality for small-sized instances. We also propose a hybrid simulated annealing (HSA) algorithm to find good quality solutions for larger instances. The HSA incorporates a late acceptance hill-climbing algorithm and a multistart strategy to promote both intensification and exploration while decreasing computational requirements. To compute the optimality gap of the HSA solutions, we derive a very fast lower bounding procedure. Computational experiments are conducted on two sets of instances that we also propose. The computational results show the effectiveness of the MILP on small-sized instances as well as the effectiveness, efficiency, and robustness of the HSA on medium and large-sized instances. Furthermore, the computational experiments clearly shown that importance of optimizing the three problems simultaneous. Finally, the importance and relevance of including the storage/retrieval activities are empirically demonstrated as ignoring them leads to wrong and misleading results.

Appendix A: Automated storage/retrieval system and shuttle travel times calculation

The ASRS considered has four rectangular racks each of which with $m=10$ rows and $n=10$ columns. The storage cells are numbered sequentially from 1 to $nm=100$ in the first rack, from $nm+1=101$ to $2nm=200$ in the second rack, from $2nm+1=201$ to $3nm=300$ in the third rack, and from $3nm+1=301$ to $4nm=400$ in the fourth rack, as depicted in Fig. 9. Each pair of storage racks is serviced by a single shuttle that moves along the aisle between them.

Fig. 9

ASRS illustration, adapted from (Jawahar et al. 1998)

Each job i is allocated to a storage cell ${SC}_i$. The specific storage cell can be easily identified as its row ($x_i$) and column ($y_i$) numbers can be obtained through a simple algebraic calculation, as given by Equation (A1), where q is the result of the integer division of ${SC}_i$ by n and r its remainder.

$$ \begin{aligned}&{SC}_i=q\times n+r, \\&x_i=\left\{ \begin{matrix} r &{}r\ne 0,\\ n &{}r=0, \end{matrix} \right. \nonumber \\&y_i=\left\{ \begin{matrix} q+1 &{} q \le m \ \& \ r\ne 0, \\ q &{} q \le m \ \& \ r=0,\\ q-m+1 &{} q> m \ \& \ r\ne 0,\\ q-m &{} q > m \ \& \ r=0. \end{matrix} \right. \nonumber \end{aligned}$$

(A1)

The shuttles travel at a constant speed (i.e., the shuttle has no acceleration or deceleration). The travel time $TS_{i}^k$ between storage cells i and k is calculated by dividing the Chebyshev distance between the two storage cells by the shuttle speed, as shown in Equation (A2), where $D_r$ and $D_c$ are, respectively, row and column center distances between any two adjacent cells and the operator $|\bullet |$ returns the absolute value. Similarly, the travel time $TS_{i}$ between storage cell i and the LU, and vice versa, is obtained through Equation (A3), where C is a constant representing the time required to transfer a job between the shuttle and the LU.

$$\begin{aligned}&{TS}_i^k=\frac{\max \left( D_r \times |x_i- x_k |, \ D_c\times |y_i-y_k |\right) }{speed}, \end{aligned}$$

(A2)

$$\begin{aligned}&{TS}_i= \frac{\max \left( D_r\times x_i,\ D_c\times y_i \right) }{speed}+C. \end{aligned}$$

(A3)

The storage and retrieval cells allocated to each job as well as all other data can be found at https://fastmanufacturingproject.wordpress.com/problem-instances/.