A Branch and Bound Algorithm for Cyclic Scheduling of Timed Petri Nets

Abstract

A timed Petri net (TPN) has been widely used for modeling, scheduling, and analyzing discrete event dynamic systems. This study examines cyclic scheduling problems of a TPN to minimize the cycle time especially for automated manufacturing systems. Appropriate token routing at each conflict place can make a TPN repeat an identical firing sequence. We propose a systematic procedure to transform a TPN with such cyclic token routing into an equivalent timed event graph (TEG) for which the cycle time and firing schedules can be evaluated by a linear programming (LP). Based on the transformation procedure, we develop an efficient branch and bound algorithm to solve the scheduling problem. A partial solution is defined as a partial token route that has only a subset of token routes for determining the complete schedule. The lower bound of a partial solution is determined by the cycle time of a TEG that has the partial token route. The cycle time of a TEG with an additional token route for a new partial solution is computed by a dual-simplex algorithm which avoids solving the LP completely again. A dynamic branching strategy that prevents unnecessary branching for the scheduling decision is also proposed. We demonstrate the computational efficiency through intensive experiments of cluster tools and robotic flow shops.Note to Practitioners-There are many systems which repeat an identical task sequence such as manufacturing systems, transportation systems, and robotic systems. Maximizing the throughput of such a system by optimizing the cyclic task sequence, which is called a cyclic scheduling problem, has been an important problem. In order to deal with cyclic scheduling problems, this paper uses a timed Petri net (TPN) which is a graphical modeling tool for discrete event dynamic systems. As the size of a TPN model increases, the computational complexity of the cyclic scheduling problem exponentially increases due to the combinatorial nature of sequencing problems. Therefore, an efficient algorithm for optimal cyclic scheduling of TPNs is needed. This study proposes an efficient branch and bound algorithm which is a kind of a tree search algorithm. Several important techniques are developed. First, a transformation procedure from a TPN to a timed event graph is developed. Second, an efficient branching rule which reduces the size of the search tree is proposed. Third, the lower bound which evaluates the cycle time of each node in the search tree is suggested. We verify the efficiency of the algorithm Manuscript through the experiments on manufacturing systems such as a cluster tool and a robotic flow shop.Index Terms-Branch and bound algorithm, cluster tool, cyclic scheduling, timed event graph (TEG), timed Petri net (TPN). 1545-5955 © 2013 IEEE

Scheduling In-Line Multiple Cluster Tools

“…Since each PM can process one wafer at a time and a wafer should visit one PM in a tool, constraints (8) and (9) are also needed. Constraints (10) and (11) are for synchronizing variables x ijk and z ijkl by introducing s jkl .…”

Noncyclic Scheduling of Cluster Tools With a Branch and Bound Algorithm

“…Then a TPN is transformed into a decision-free timed event graph (TEG), and a lower bound is computed with the makespan of each TEG. The new TEG transformation procedures we develop further improve the B&B procedure compared to the previous works [21], [22], [26]. We also propose a new lower bound evaluation method based on a resource workload and verify the efficiency of the B&B procedure with various cluster tool scheduling problems.…”

Scheduling Lot Switching Operations for Cluster Tools

“…More detailed properties, analyses and applications of Petri nets can be found in [43]- [44]. There have been works on modeling, analyzing and scheduling of cluster tools by Petri nets [10]- [11], [16]- [18], [20]- [30], [40]. We note that although a mutual exclusion place is needed to prevent to perform two loading (or unloading) tasks simultaneously for a dual-armed robot [16], we do not depict it in our Petri net model in Fig.…”

“…The firing order of the output transitions of each conflict place corresponds to a scheduling sequence of jobs or tasks processed at the resource. There are several methods of finding an optimal solution for such Petri net-based scheduling problems, including an MIP and a B&B algorithm [17]- [18], [40]. We briefly introduce an MIP model based on the proposed Petri net model for an optimal schedule [17], [40].…”