## 1 Introduction
In real-life semiconductor fab applications, for cluster tools shown in Figure 2, two fingers on one of the robot arms are used as clean ones, while the other two fingers on the other robot arm are treated as dirty ones. The cluster tools shown in Figure 2 are normally used for chemical vapor deposition processes. The cluster tool illustrated in Figure 1 is a cluster tool with single-space PM and single-finger-arm robot (S\({}^{3}\)F-CT), while the cluster tool illustrated in Figure 2 is a cluster tool with four-space PM and four-finger-arm robot (SF\({}^{3}\)-CT). Previous studies provide the optimal scheduling strategy for S\({}^{3}\)F-CT. However, existing studies do not deal with the practical situations that a robot arm is constrained to dedicate a part of robot activities to as stated in this paper. To the best of the authors' knowledge, there is no research report on the scheduling of SF\({}^{3}\)-CTs, as the existing studies related to cluster tool scheduling mainly focus on S\({}^{3}\)F-CTs. Two or more LLs in a cluster tool can ensure that there are enough raw wafers entering the tool for processing, such that the tool can operate under a steady state most of the time. Practically, the wafer-processing time in a PM is quite longer than the robot task time [10; 11]. Nevertheless, the robot task time in a cycle might be longer than the wafer completion time at a step. Under a steady state, if the robot task time in a cycle is longer than the wafer completion time at the bottleneck step, the cluster tool is said to be transport-bound. In this situation, the robot is always busy. If the robot task time in a cycle is shorter than the wafer completion time at the bottleneck step, then the cluster tool is said to be process-bound. In this situation, the robot has idle time. For process-bound cluster tools with a single-arm robot, a backward strategy is optimal [11], while for process-bound cluster tools with a dual-arm robot, a swap strategy is efficient [12]. Significant advancements have been made in the area of modeling, performance assessment, control, and scheduling of cluster tools under the steady state [7; 13; 14; 15; 16; 17; 18]. However, RTCs are not considered in those studies. With RTCs, [PERSON] et al. [19], [PERSON] and [PERSON] [20], and [PERSON] et al. [8] propose several methods to find an optimal and feasible cyclic schedule for cluster tools with a dual-arm robot. Furthermore, [PERSON] et al. [6; 21] conduct studies on scheduling cluster tools equipped with a single-arm robot/dual-arm robot from a perspective of control theory so as to obtain an optimal and feasible schedule efficiently. [PERSON] et al. [22] introduce a novel category of robot task sequences that are more effective in meeting RTCs for situations with unbalanced processing parameters among steps. [PERSON] et al. [23] propose an approach for cluster tools with a dual-arm robot handling multiple wafer types, aiming to satisfy RTCs and achieve the minimum cycle time by identifying an optimal release sequence for different wafer types. For cluster tools with processing-time variation and RTCs, [PERSON] et al. [24] suggest a flexible strategy to improve the schedule robustness. For the scheduling problem of multi-cluster tools with RTCs, [PERSON] et al. [25] find a feasible steady-state schedule in dealing with two wafer types concurrently. For re-entrant processes with RTCs, efforts have also been made in the modeling, control, and scheduling of cluster tools with single-arm and dual-arm robots [26; 27]. Meanwhile, the studies in [28; 29] examine how activity time variation affects the wafer delay in PMs for cluster tools with single-arm and dual-arm robots. [PERSON] et al. [30; 31; 32] suggest partially loading wafers in parallel PMs to improve the productivity of cluster tools. However, they do not consider multi-space PMs, nor address the issue of cross-contamination between dirty and clean wafers.