# Blocking Cyclic Job-Shop Scheduling Problems
## 1 Introduction
A sequence of the jobs, together with the starting times of all the operations, is referred to as a cyclic or periodical schedule. A regular time interval in which all the operations are repeated is referred to as the period or cycle time. There have been numerous studies on cyclic scheduling [15; 16; 17; 18; 19; 20]. However, there are far fewer studies on scheduling cyclic shops with blocking [21; 22]. [PERSON] [2] developed a branch-and-bound approach to deal with a generalised version of the cyclic job shop by incorporating precedence constraints. This approach was used in the context of a computer pipeline. [PERSON] and [PERSON] [23] investigated a scheduling problem for cyclic job shops with blocking where each machine had an input buffer of finite capacity. They developed Petri net models for the cyclic job shops with blocking. [PERSON] et al. [24] presented a general approach for solving the cyclic job-shop scheduling problem, based on the coupling of a genetic algorithm and a scheduler. This scheduler utilised a Petri net model of the linear precedence constraints between cyclic tasks. The goal of this genetic algorithm was to define an order of priority for jobs on the machines, to be used by the scheduler for solving  
Figure 1: Blocking conditions in job-shop scheduling. resource conflicts. [PERSON] and [PERSON] (2018) described a model for cyclic machine-scheduling problems with blocking. In that study, a tabu search algorithm solved the considered problem with different neighbourhood structures. [PERSON] et al. (2018) proposed a recurrent neural network approach for the cyclic JSSP that attempted to find the optimum solution by minimising the energy state of the network. They also extended the recurrent neural network technique by coupling it with the Lagrangian relaxation method. A more general cyclic JSSP with special restrictions has rarely been encountered in the literature. Nonetheless, a few studies have indeed investigated variants of this problem ([PERSON], 2017; [PERSON], 2018; [PERSON], 2019; [PERSON], 2019). The studies by [PERSON] ([PERSON], 2017) and [PERSON] and [PERSON] (2018) presented a mixed linear integer program for variants of the cyclic JSSP. Following this, [PERSON] et al. (2019) made use of the same model to solve a problem in transportation. [PERSON] et al. (2019) studied the cyclic JSSP with blocking and transportation and proposed a branch-and-bound method to solve the problem. The authors wish to point out that there have been a significant number of research reports on the scheduling of blocking job shops and flow shops. Furthermore, there are studies on these problems that take into account setup times and resources such as robotics and servers. However, this study examines the cyclic aspects of the scheduling problem in job shops. The literature review presents the previous studies in this area, indicating that there are not enough studies on cyclic job-shop scheduling. The following are a few studies on cyclic scheduling in order to clarify their differences from the problems investigated in this paper. [PERSON] et al. (2018) considered cyclic scheduling, which involves processing a single part type on machines arranged in a flow-line arrangement. Their study focused on the cyclic scheduling of transportation operations. Then, [PERSON] et al. (2018) studied a real-life example of the same problem with intermediate buffers and proposed a hybrid genetic algorithm for solving it. Furthermore, [PERSON] et al. (2018) studied stochastic optimisation of two machine robotic cells processing a single type of part. Specifically, our study focuses on the cyclic job-shop scheduling problem, in which multiple jobs are processed on a set of machines in a different order. The aim of our study was to bridge a gap in the literature, i.e., to investigate the line of research that concerns the cyclic JSSP with blocking and sequence-dependent setups. We propose a novel mathematical modelling approach for the cyclic job-shop scheduling.