IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION,VOL. 18, NO. 1, FEBRUARY 2002 69 Multicyclic Hoist Scheduling With Constant Processing Times Ada Che, Chengbin Chu, Member, IEEE, and Feng Chu Abstract—This paper proposes an exact algorithm for the mul- ticyclic schedules of hoist moves in a printed circuit board (PCB) electroplating facility, where exactly parts enter and parts leave the production line during each cycle, and the pro- cessing time at each production stage is a given constant. The mul- ticyclic scheduling problem is transformed into enumeration of in- tervals for linear functions of decision variables. This enumeration is accomplished with a branch and bound procedure. At each node of the search tree, by solving a linear programming problem (LPP), either the corresponding partial solution is proved to be unable to lead to a feasible solution, or a lower bound is computed. Due to its particular structure, this LPP is equivalent to a cycle time evalu- ation problem in a bivalued graph which can be solved efficiently. The proposed algorithm is polynomial in the number of tanks for a fixed , but exponential if is arbitrary. Computational experi- ence with both benchmark and randomly generated test instances is presented. Index Terms—Algorithms, production systems, scheduling. I. INTRODUCTION T HIS PAPER addresses cyclic scheduling problem in a no-wait production system involving material handling facilities such as hoists or robots which are widely used in industrial processes for material handling. In such systems, the hoist or robot is in charge of transporting parts from one machine to another for the next operation. The productivity of these systems largely depends on the schedule of the robot activities, especially when the intermediate storage area is limited in order to limit the work-in-process. Due to the importance of the problem, the number of articles addressing production systems involving robots or hoists has increased rapidly in recent years [1]–[6], [8]–[19] both in cyclic [1]–[3], [5], [8]–[19]) and noncyclic environments [4], [6]. The model considered in this paper is particularly relevant in printed circuit board (PCB) electroplating processes and other galvanization processes with a hoist for material handling. Such a production line is composed of a sequence of chemical tanks. Each tank contains chemicals required for a specific electro- plating step in the processing of parts, such as acid cleaning, acid activating, copper plating, rinsing, etc. During the process, a part must be successively soaked in each chemical tank for a Manuscript received March 19, 2000; revised November 20, 2000 and May 23, 2000. This paper was receommended for publication by Associate Editor R. Kumar and Editor N. Viswanadham upon evaluation of the reviewers’ com- ments. This work was supported by the French–Israeli co-operation grant “Fac- tory of the Future.” The authors are with the Laboratoire d’optimization des systèmes industriels (LOSI), Université de Technologie de Troyes, BP 2060, 10010 Troyes Cedex, France (e-mail: ada.che@utt.fr; chengbin.chu@utt.fr; feng.chu@utt.fr). Publisher Item Identifier S 1042-296X(02)01778-0. specified period of time. This problem is commonly known as hoist scheduling problem [1], [3], [10]–[15], [17]–[19]. When the time that parts can stay on machines is not limited, the sys- tems are called robotic cells [2], [5], [8], [9], [16]. This paper considers a cyclic production environment. A cyclic production system periodically repeats the same state. The length of the period is called cycle time. The cycle time measures the throughput rate of a production system. Therefore, the criterion considered in this paper is cycle time minimization which is equivalent to maximizing the throughput rate, as in almost all works addressing cyclic schedules. Cyclic production is very important for its simplicity of implemen- tation and ease of management. It is particularly relevant in mass production, such as PCB electroplating and many other galvanization processes. In such a system, production lines are specially configured for a quite long period to produce very few specific types of parts with very large lot size and therefore can be approximated by cyclic production model. This explains why cyclic production environment has received very much attention from researchers and practitioners [1]–[3], [5], [7]–[19]. Cyclic schedules can be distinguished between simple cycle schedules and multicyclic schedules. In a general cyclic schedule, parts of type 1, parts of type 2, etc. are introduced into the system every period (cycle). If is the largest common divider of these ’s, this schedule is called -cyclic schedule or -degree cyclic schedule. In simple cycle schedules, . In this paper, we consider a single part type multicyclic schedule. In such a schedule, identical parts are introduced during each cycle, that is, parts are introduced in and removed from each tank during a cycle. The mean cycle time of an -cyclic schedule is defined as the whole cycle time divided by . The throughput rate is the inverse of the mean cycle time. As far as hoist scheduling is concerned, many researchers have studied simple cycle schedules [3], [10], [14], [15], [17], [19]; i.e., 1-cyclic schedules. In practice, however, simple cycle schedules are not necessarily optimal. In general, -cyclic schedules would have larger throughput. Lei and Wang [13], Song et al. [18] have noticed that a 2-cyclic schedule may be better than the optimal simple cycle schedule. Our experiments have also confirmed this fact. Little research work has been done on multicyclic scheduling problems. Due to the characteristics of chemical process in PCB elec- troplating or galvanization considered in this paper, we address the no-wait environment where as soon as the operation of a part is completed in a tank, it must be immediately removed from that tank and transported to the next one. In the litera- ture, some works [3], [13]–[15], [17], [19] deal with models 1042–296X/02$17.00 © 2002 IEEE