IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 2, NO. 2, APRIL 2005 173 Multi-Degree Cyclic Scheduling of Two Robots in a No-Wait Flowshop Ada Che and Chengbin Chu, Member, IEEE Abstract—This paper addresses multi-degree cyclic scheduling of two robots in a no-wait flowshop, where exactly iden- tical parts with constant processing times enter and leave the pro- duction line during each cycle, and transportation of the parts be- tween machines is performed by two robots on parallel tracks. The objective is to minimize the cycle time. The problem is transformed into enumeration of pairs of overlapping moves that cannot be per- formed by the same robot. This enumeration is accomplished by enumerating intervals for some linear functions of decision vari- ables. The algorithm developed is polynomial in the number of ma- chines for a fixed , but exponential if is arbitrary. Computational results with benchmark instances are reported. Note to Practitioners—This paper was motivated by the problem of cyclic scheduling of a no-wait production line, where a part must be processed without any interruption either on or between machines due to characteristics of the processing technology itself or the absences of storage capacity between operations of a part. Multi-degree schedules, in which multiple parts enter and leave the line during a cycle, usually have larger throughput rate than simple ones. This paper proposes an algorithm for multi-degree cyclic scheduling of a no-wait flowshop with two robots. Com- putational results show that the throughput rate can be really improved by using multi-degree schedules with two robots. How- ever, we have not addressed the decision of the optimal value of the degree of the cycle. Furthermore, since we consider that the two robots travel along parallel tracks, the collision–avoidance constraints have been relaxed in the algorithm. In future research, we will address the two problems and generalize the algorithm to multi-robot cases. Index Terms—Algorithms, production systems, scheduling. I. INTRODUCTION C OMPUTER-CONTROLLED robots have been used widely in many industrial processes for material han- dling. A typical application is an automated electroplating line for processing printed circuit boards (PCBs). In these systems, the robots are in charge of transporting parts from one machine Manuscript received September 7, 2003; revised January 22, 2004. This paper was recommended for publication by Associate Editor F. Cheng and Editor N. Viswanadham upon evaluation of the reviewers’ comments. This work was supported by a post-doctoral grant from the French Ministry of National Education. A. Che was with the Laboratoire d’optimisation des systèmes industriels (LOSI), Université de Technologie de Troyes, BP 2060, 10010 Troyes Cedex, France. He is now with the Laboratoire d’informatique, de modélisation et d’optimisation des systèmes (LIMOS), Université Clermont-Ferrand II, Campus des Cézeaux, BP10125, 63173 Aubière Cedex, France (e-mail: ada.che@isima.fr). C. Chu is with the Laboratoire d’optimisation des systèmes industriels (LOSI), Université de Technologie de Troyes, BP 2060, 10010 Troyes Cedex, France, and also with Hefei University of Technology, Hefei 230009, China (e-mail: chengbin.chu@utt.fr). Digital Object Identifier 10.1109/TASE.2004.835600 to another for the next operation. The performance of these systems largely depends on the schedule of robot activities. A good schedule may increase the throughput rate, reduce work-in-process and production costs. That is why the number of articles addressing production systems involving robots for material handling has increased rapidly over the past decades. This paper considers a cyclic production environment. The cyclic production model is very important due to its simplicity of implementation and ease of management. A cyclic produc- tion system periodically repeats the same state. The length of the period is called the cycle time [1]–[4]. The cycle time mea- sures the throughput rate of a production system. The criterion considered in this paper is cycle time minimization, which is equivalent to maximizing the throughput rate, as is the case in almost all works addressing cyclic schedules. Cyclic schedules can be distinguished by the degree of the cycle [4]. In an -degree cyclic schedule, exactly parts enter and leave the production line during each cycle. In the literature, one-degree cyclic schedules are commonly known as simple- cycle schedules. In this paper, we consider a single part type multi-degree cyclic schedule. The mean cycle time of an -de- gree cyclic schedule is defined as the whole cycle time divided by . The throughput rate is the inverse of the mean cycle time. In this paper, we address the no-wait environment, where as soon as the processing of a part is completed on a machine, it must be immediately removed from that machine and trans- ported to the next one. In other words, the processing time of the part on each machine is a given constant. In the literature, some works [3]–[7] deal with models where the processing time of a part on each machine must fall into a time window. The time-window models are less constrained than the one consid- ered in this paper, and, therefore, can lead to better productivity with an optimal solution. The model considered in this paper is also relevant in practice for the following reasons. Firstly, with the no-wait model, the nominal processing times are strictly re- spected. This is necessary in some surface treatment processes, such as the PCB electroplating process and other galvanization processes, where the surface treatment quality of parts mainly depends on the processing time. Secondly, when a time window is allowed, the problem has been proved to be NP-hard, even with the simple-cycle and single-robot assumption [8]. Since a no-wait model can be solved to optimality in polynomial time as shown in this paper, it can be used as a heuristic to solve the problem with time windows. We expect this heuristic to perform particularly well when the time windows are narrow. One-degree (simple) cyclic scheduling problems with a single robot have been deeply studied over the past decades, both in 1545-5955/$20.00 © 2005 IEEE