Hybrid algorithm based scheduling optimization in robotic cell with ...

Proceeding of the IEEE International Conference on Information and Automation Hailar, China, July 2014

Hybrid algorithm based scheduling optimization in robotic cell with dual-gripper Na Li and Jie Cheng*

Xinyu Fang and Jiafan Zhang

Shandong University, Weihai Weihai,264209, China [email protected], [email protected]

ABB Corporate Research Center Shanghai,201319, China [email protected]

Abstract - With the maturity of the industrial robotic technology, robotic cells are gradually regarded as a kind of stand equipment to replace human work in every walk of life. How to obtain the maximum or approximate maximum throughput in a robotic cell is always the highlighted goal, especially in the rapid growing 3C industry market. In this paper, the objective is to get a 1-unit cycle sequence of robot actions that approximately minimizes the cycle time to produce a part and maximizes the throughput by using a new hybrid algorithm in the robotic cell with a dual-gripper robot. In this algorithm, different constrains are considered during computing the cycle time, including free/non-free process, allowed time window. The resulting diagrams provide very intuitive insights into the accuracy of the hybrid algorithm compared with the exact algorithm. Additional 100 simulation results prove the effectiveness of the hybrid algorithm, with a solid performance to achieve the maximum productivity of robotic cell.

with the identical part to be processed. A robot transports the processed part from one workstation to next one. In cyclic robotic operations, a k-unit cycle means that a sequence of robot actions that loads and unloads each workstation exactly k times. It is the simplest case when k is 1. Fig. 2 shows an example of a 1-unit cycle. From the classical robotic cell model 𝛼|𝛽|𝛾 [1], 𝛾 is the optimizing objective. For a given processing requirement, the objective of manufacturers is almost to minimum the part produced time for the maximization of the throughput [3], so the objective of this paper is also to get minimum time to produce a part in 1-unit cycle.

Index Terms - robotic cell; optimal scheduling; hybrid algorithm; dual-gripper;

About the robot actions scheduling problem, it previously has been described in some classical scheduling literatures. Reference [4] initially developed the necessary framework for scheduling problem. And most of prior studies were implemented based on the constant or addictive travelling time [5][6]. However, the travelling time is always general in practical engineering robotic cell. This paper is based on the general travelling time, loading time, unloading time and processing time. To improve the throughput, the efficient use of dual-gripper robot has been proved to increase the rate of production and provides a competitive advantage in productivity [9]. But due to the dual-gripper flexibility, and the increase in combinational possibilities severely complicates its theoretical analysis, and this problem belongs to scheduling optimization problem which is proved as NP-hard problem [2]. As a result, there are only few papers that contribute to the scheduling in dual-gripper robotic cells. For robotic cell scheduling optimization with dual-gripper robot, reference [11] is the earliest research, but it only studies five cases in a 2-workstation robotic cell except the feed-in and feed-out workstation. Then a new suitable frame to express the actions of the dual-gripper robot is proposed by [12]. Reference [13] classified robotic cell into three types by allowed time window constrains to study the scheduling problem. But all of the researches above are under the situation of assuming the travelling time, unloading time and loading time are constant, and they are not practical for real engineering. Reference [9] adopted branch and bound technique based on enumerated to solve the problem in general robotic cell, and the algorithm to compute the cycle time is very practical and also taken by this paper.

Fig. 2 A general 1-unit cycle

I. INTRODUCTION Industrial manufacturers have been compelled to incorporate automation and repetitive processing for improving productivity in terms of flexibility, repeatability, and with new functions that offer improved accuracy. Therefore many modern manufacturing systems use robotic cells to reduce cost and improve productivity.

Fig. 1 A general robotic cell

Generally, a robotic cell consists of a series of processing workstations, a feed-in workstation, a feed-out workstation and some robots to transport parts. Fig. 1 shows a general robotic cell diagram. In this robotic cell, there are m robots and n workstations. In practical engineering of industry, according to the part processed order, two types of robotic cell can be classified. One is flow-shop type, and the other is job-shop type [1]. Nowadays most schedule problem research are about the flow-shop. In this paper the robotic cell is about flow-shop type *

Corresponding author: School of Mechanical, Electrical and Information Engineering, Shandong University, Weihai, Shandong 264209, China.

978-1-4799-4100-1/14/$31.00 ©2014 IEEE

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In previous research, exact method is employed as the main method to solve scheduling problems. Generally some exact methods are suitable to fix problems with little workstations in the robotic cell, including polynomial algorithm [6], mixedinteger linear program [7][8] and based on enumerated branch and bound technique [9]. Though an accurate result can be given by using exact method, it will take more computing time and space when the robotic cell has many workstations with large scale or in robotic cell with dual-gripper robot. So the approximate methods, for example, genetic algorithm, ant colony algorithm are alterative to large scale scheduling problem. In this paper, a hybrid algorithm is proposed and used for scheduling optimization in large scale robotic cell with dualgripper, by considering the constrains in real engineering practice mentioned in prior research [9]. The constraints are free/non-free pick, allowed time window and free /not-free process. In the next section the model is formulated, and the proposed method is outlined in section 3. Section 4 introduces the result of the method through 100 practical cases in 6worksation, 7-workstation and 8-workstation robotic cell with dual-gripper robot. The resulting diagrams provide very intuitive insights into the accuracy of the hybrid algorithm, compared with the exact algorithm. The conclusions are given in section 5.

Since each of the workstations performs different function, in general, every workstation needs different processing time to produce the given part. The processing time in workstation 𝑊𝑠𝑖 is denoted as 𝛿𝑇𝑝,𝑖 . Similarly, the time of loading and unloading at each workstation 𝑊𝑠𝑖 can be denoted as 𝛿𝑇𝑙,𝑖 and 𝛿𝑇𝑢,𝑖 respectively. A natural and widely used measure of productivity is throughput, the number of finished parts produced per unit time slot. Normally, a robotic cell refers to the production of finished parts by repeating a fixed sequence of robot moves, until the required production is completed. Thus, each element of the cycle time matrix can be expressed as: 𝛿𝑇𝑖,𝑗 = {

𝑛

𝑇=∑

𝑖=1

⋯ ⋱ ⋯

𝛿𝑇𝑡,1𝑛 ⋮ ] 0

∆𝑇𝑡,𝑔 = [

𝛿𝑇𝑡,𝑔,𝑛1

⋯ ⋱ ⋯

𝛿𝑇𝑡,𝑔,1𝑛 ⋮ ] 0

𝑛

∑

𝑗=1

(4)

𝛿𝑇𝑖,𝑗

B. Some notations of robotic cell with dual-gripper Because the dual-gripper robot is used in the robotic cell, the robot can pick up two parts simultaneously. To show the states and actions of two grippers respectively, we uniquely define the travelling of the robot between different workstations in terms of loading and unloading activities. The robot action sequence S is described by the notion 𝑖 + and 𝑖 − , 𝑖 ∈ {1,2, ⋯ , 𝑛 − 1, 𝑛}. 𝐴𝑗 is a popular notation that is based on the concept of activity, according to the similar definition given by prior paper [9]. The activity 𝐴𝑗 can be presented by the two notions above. TABLE I shows the corresponding relationship between them. Here a developed definition of activity 𝐴𝑗 is proposed considering the practical situations in engineering. For example, the sequence 𝑆 = (𝐴1 , 𝐴2 , ⋯ , 𝐴2𝑛−1 , 𝐴2𝑛−2 ) represents a standard processing order. It can be translated into the sequence of robot actions as follows: picking up a part from the feed-in 𝑊𝑠1 ; loading 𝑊𝑠2 , unloading 𝑊𝑠2 till unloading 𝑊𝑠𝑛−1 , and then dropping the established part to the feed-out 𝑊𝑠𝑛 . Obviously, such a sequence of “loading” and “unloading” activities uniquely defines the robot travelling between workstations.

(1)

A new variable ε is introduced as the grippers switching time which is the time required for the dual-gripper robot to reposition its two grippers. Normally, the travelling time required by the robot to reposition its gripper while travelling from one workstation to another is different to that without gripper repositioning. Another travelling time matrix with gripper reposition would be employed with another notation ∆𝑇𝑡,𝑔 : 0 ⋮

(3)

Where, c and σ represent the number of the processing times on 𝑊𝑠𝑖 and travelling times between 𝑊𝑠𝑖 and 𝑊𝑠𝑗 . For a 1-unit cycle, each process occurs only once during the part whole producing process, namely ci =1. In general, the shorter cycle time, the larger throughput. Thus the scheduling optimization objective can be considered to achieve the minimum cycle time 𝑇.

A. Time matrix and cycle time Assumed that the robotic cell is composed of an nworkstation working serially with a dual-gripper robot. Then n workstations can be denoted as 𝑊𝑠1 , 𝑊𝑠2 , ⋯ 𝑊𝑠𝑛−1 , 𝑊𝑠𝑛 , where 𝑤𝑠1 is denoted as the feed-in workstation and 𝑤𝑠𝑛 is denoted as the feed-out workstation. . In a complete production process for an identified part in such an n-workstation robotic cell with dual-gripper robot, comes four main sub-tasks: travelling, processing, loading and unloading. The exact definition can be found in [9]. Let 𝛿𝑇𝑡,𝑖𝑗 denote the robot travelling time between any two workstations 𝑊𝑠𝑖 and 𝑊𝑠𝑗 , where 1 ≤ 𝑖, 𝑗 ≤ 𝑛 . A general travel time matrix ∆𝑇𝑡 can be expressed as follows: 0 ⋮ 𝛿𝑇𝑡,𝑛1

𝑖𝑓 𝑖 = 𝑗 𝑖𝑓 𝑖 ≠ 𝑗

In this kind of cyclic production, cycle time is referred to the duration during which the sequence of operations is completed in a normal iteration to produce the part. The straightforward approach for computing the cycle time is to sum all of the times cost in a normal iteration of the part production. It can be calculated as:

II. PRELIMINARIES AND PROBLEM FORMULATION

∆𝑇𝑡 = [

𝑐𝑖 (𝛿𝑇𝑙,𝑖 + 𝛿𝑇𝑢,𝑖 + 𝛿𝑇𝑝,𝑖 ), 𝜎𝑖,𝑗 𝛿𝑇𝑡,𝑖𝑗 ,

(2)

TABLE I ACTIVITY LIST FOR AN N-WORKSTATION ROBOTIC CELL Activity 𝐴1 𝐴2 𝐴3 𝐴4 ⋯ 𝐴2𝑛−3 𝐴2𝑛−2 Action

148

1−

2+

2−

3+

⋯

(𝑛 − 1)−

𝑛+

sequence of the robot activity is 𝑠 = {𝐴1 , 𝐴2 , ⋯ , 𝐴5 , 𝐴6 }, with the resulting chromosome notated as [1,2,3,4,5,6]. Because the scheduling is cyclic, it does not matter which number as the first gene. As a result, by default we set number one as the first gene in every chromosome. The initial population consists of 𝑁 randomly generated robot activity sequences. 𝑁 is referred to as the population size. From TABLE II, we can see that the number of feasible cyclic sequences grows exponentially based on (2𝑛 − 2)! with the increasing of the number of workstations n. Hence, 𝑁 is set according to a given rule in consideration of the number of workstations. Here we set 3𝑛−2 as the population size based on n. During the generation of 𝑁 individuals, heuristic search algorithm is used to eliminate the infeasible individuals according to the criteria in [9].

In this paper, we only focus on the active cycles that are defined in [12]. For an active cycle, the transportation of a part from workstation Wsi to workstation Wsi+1 always takes less than the cycle time. To be feasible, some necessary criteria should not be violated, these criteria are listed in [9] in detail. TABLE II summarizes the number of feasible 1-unit cycles for both single- and dual-gripper robotic cell. Although a dualgripper robotic cell has productivity advantage over a singlegripper robotic cell, yet as we can summarize from the TABLE II, with the increase in the number of workstations, the number of feasible 1-unit cycles becomes huge. It is difficult to find the minimum cycle time using the exact way to enumerate all the possible solutions as the prior paper [9]. To solve such a problem, the intelligent algorithms would be employed to reach approximate optimal solutions. Some previous researches proved that hybrid genetic algorithm, such as genetic local search and genetic simulated annealing have higher performance than purely using intelligent algorithm [14].

B. Mutation and crossover operator In general, compared with mutation operator, crossover operator is regarded as the main genetic operator to generate some new individuals, mutation operator is only regarded as the ancillary method when generating some new individuals. But in this paper we set mutation operator as the main way to generate new individuals. The individuals will get more diversity and the optimal solution will be generated earlier. Because of practical meaning of the gene and the encoding method, the order between different genes has close relationship with the feasibility of the chromosome sequence. Traditional genetic operators may lead to large quantity of infeasible chromosome sequences. Hence, to make the new generated individuals feasible, we choose translation change mutation and precedence preservation crossover. Probability (Pm) and crossover probability (Pc) are used to determine whether individuals are mutated or crossed. The detailed operation process of mutation operator is described in [15]. For example, in a 4-workstation robotic cell with dual-gripper, the picked chromosome is [1,2,3,4,5,6]. We choose 5 as the gene point and 2 as the insertion point randomly. The resulting child is [1,5,2,3,4,6]. Similar to mutation operator, there is an example about crossover operator. In a 4-workstation robotic cell with dualgripper, the picked chromosomes are: 𝑃1 = [1,2,3,4,5,6] 𝑃2 = [1,3,2,5,4,6] A group of randomly generated numbers are: 𝑝𝑖𝑐𝑘 = [2,3,5,10,13,40] When the random number above is odd, the gene at the corresponding position is chosen from the first parent. Otherwise one gene from the second parent is filled in. Then the generated individual is: 𝑐ℎ𝑖𝑙𝑑 = [1,2,3,5,4,6]

TABLE II NUMBER OF FEASIBLE 1-UNIT CYCLES FOR BOTH SINGLE- AND DUAL-GRIPPER ROBOTIC CELL Number of 1-unit cycle Number of workstations Single-gripper cell Dual-gripper cell 3 1 6 4 2 46 5 6 456 6 24 5688 7 120 86640 8 720 1568880 9 5040 33022080 10 40320 793215360 11 362880 21423709440

III. PROCEDURE OF THE HYBIRD ALGORITHM As an intelligent global search algorithm, genetic algorithms (GAs) are search techniques based on mechanics of natural selection and natural evolutions. However, many practical application of the GAs showed that the traditional GAs always have no steadily high performance and better approximate optimal solutions can’t be found. Here a hybrid intelligent algorithm that combines heuristic search algorithm and improved GA is proposed in this paper. A chromosome coding based on the robot activity sequence is adopted, but this coding method may lead to many infeasible cycles. In order to promise the quality of initial cyclic sequences, some unreasonable sequences are eliminated by using heuristic search algorithm. And we take precedence preservation crossover (PPC) [16] and translation change mutation (TCM) [15] to avoid the generation of unreasonable sequences efficiently. During the genetic operations, local neighborhood search is used to improve the quality of GA optimization and keep the search efficiency.

C. Fitness function and select operator The fitness function 𝑓𝑖 of the 𝑖𝑡ℎ cyclic sequence is set according the equitation as follows: (5) (𝑇 − 𝑇𝑖 ) 𝑓𝑖 = 𝑚𝑎𝑥 ⁄𝑇 𝑚𝑎𝑥

A. Encoding and initial population In this paper, we set one feasible cyclic sequence based on robot activity as one chromosome. For example, for a 4workstation robotic cell with dual-gripper, the robot acts according to the order as follows: picking up a part from the feed-in 𝑊𝑠1 , loading 𝑊𝑠2 , unloading 𝑊𝑠2 till unloading 𝑊𝑠3 , and then dropping the established part to the feed-out 𝑊𝑠4 . The

According to the equitation above, 𝑇𝑖 is expressed as the cycle time of the 𝑖𝑡ℎ robot activity sequence. The cycle time is smaller, the fitness value is larger. During the evolutionary

149

process, we efficiently evaluate the fitness of every individual in each generation. After doing all the genetic operators, the new generation was selected by synthesizing roulette wheel and optimization selecting. We always choose the optimal individual from last generation to next generation. Then the rest of the individuals are chosen according to the probability of the individual. The probability of the 𝑖𝑡ℎ individual is computed according to (6), from this equitation we can conclude that the probability of the individual may be chosen is bigger for the bigger fitness value. 𝑓 𝑃𝑖 = 𝑖⁄ 𝑁 ∑𝑖=1 𝑓𝑖

(6)

D. The terminal criterion In this paper, we terminate the algorithm after 𝐺 iterations, and then choose the sequence with the minimum cycle time as the optimal solution. Similar is the method to set population size of every generation, the generation number 𝐺 is set according to a given rule in consideration of the number of workstations. Here we set the population size equal to 2𝑛2 based on the practical experiments, where n is the number of workstations. The algorithm for scheduling the optimal cycle time with dual-gripper cell is in TABLE III.

Fig. 3 Pictorial diagram of a 6-workstation robotic cell

Here some workstation parameters in the robotic cell, workstation names, workstation roles, workstation processing time are showed in TABLE IV and TABLE V. The travel time without gripper exchange between any two workstations is listed in TABLE VI, where the “-”means that the path is not feasible. And TABLE VII shows the travel time between any pair of workstations while switching its gripper. The time required for the dual-gripper robot to reposition its grippers is 0.2s. In TABLE VIII, we set some parameters as the best combination to apply the hybrid algorithm.

IV. IMPLEMENTATION SCENARIO AND ANALYSIS To validate that the hybrid algorithm is more suitable for large scale schedule problems, the optimal solution which is obtained from the exact method [9] is compared with the result obtained by using the hybrid algorithm. In this section, we take two ways to demonstrate the performance of the hybrid algorithm, which are respectively described in part A and Part B.

TABLE IV WORKSTATION PARAMETERS Workstation Workstation Process Robot Allow Name Role Stage Free Timeslot [s] Feed-in 1 Yes 𝑾𝒔𝟏 Workstation 2 Yes 4.00 𝑾𝒔𝟐 Workstation 3 Yes 𝑾𝒔𝟑 Workstation 4 Yes 6.00 𝑾𝒔𝟒 Workstation 5 Yes 𝑾𝒔𝟓 Feed-out 6 Yes 𝑾𝒔𝟔

TABLE III SEARCHIING ALGORITHM Input: Data for a robotic cell with dual-gripper robot; Set the factors of GA ; Get initial generation by heuristic search algorithm; Loop1: IF the terminal criterion is not satisfied, THEN Loop2: IF the loop number is less than the number of initial population, THEN IF the mutation condition is satisfied, THEN mutation operation and local search the child solution to keep a solution with the biggest fitness value; END IF the crossover condition is satisfied, THEN crossover operation and local search the child solution to keep a solution with the biggest fitness value; END END Select the next generation and keep the optimal solution; END Loop2 END END Loop1 Output: An optimal cycle time and corresponding solution;

TABLE V TIME MATRIX OF PROCESS, LOADING, UNLOADING Workstation Processing Loading Unloading Name Time [s] Time [s] Time [s] 0.00 0 0.10 𝑾𝒔𝟏 1.20 0.5 0.35 𝑾𝒔𝟐 4.00 0.7 0.12 𝑾𝒔𝟑 1.50 0.45 0.30 𝑾𝒔𝟒 1.40 0.80 0.13 𝑾𝒔𝟓 0.00 0.30 0.00 𝑾𝒔𝟔 TABLE VI TRAVEL TIME MATRIX WITHOUT GRIPPER REPOSITION, UNIT [SEC] 𝑾𝒔𝟏 𝑾𝒔𝟐 𝑾𝒔𝟑 𝑾𝒔𝟒 𝑾𝒔𝟓 𝑾𝒔𝟔

A. A practical case Fig. 3 is a pictorial description of a general 6-workstation robotic cell in real life engineering for semiconductor fabrication. We only take the robot-center operations into count except the external equipment on the product line.

150

𝑾𝒔𝟏

𝑾𝒔𝟐

𝑾𝒔𝟑

𝑾𝒔𝟒

𝑾𝒔𝟓

𝑾𝒔𝟔

0 0.9 1.5 1.02 1.5 0.8

1 0 0.3 0.35 0.85

1.5 0.3 0 0.45 1.1

0.83 0.5 1.5 0 1.6 2.3

1.52 0.5 0.6 0 1.9

0.9 0.8 1.1 2.2 1.7 0

algorithm in 3 different types of robotic cells. In order to do comparison easily, all the results from propose hybrid algorithm is normalized by the corresponding results from exact method introduced in [9]. The box chart in Fig. 6 indicates that the proposed hybrid algorithm have a solid performance. Though with the increasing of the number of workstations, the probability to get the global optimal solution is smaller, yet in most of cases the global optimal or near global optimal solution can be obtained. From the box-chart, we can see that above 90% cases can obtained the optimal or approximate optimal solution and the median of three type robotic cell are close to 100%.

TABLE VII TRAVEL TIME MATRIX WITH GRIPPER REPOSITION, UNIT [SEC] 𝑾𝒔𝟏 𝑾𝒔𝟐 𝑾𝒔𝟑 𝑾𝒔𝟒 𝑾𝒔𝟓 𝑾𝒔𝟔 0 0.95 1.57 0.91 1.46 0.81 𝑾𝒔𝟏 0.81 0.00 0.38 0.57 0.57 0.71 𝑾𝒔𝟐 𝑾𝒔𝟑 𝑾𝒔𝟒

1.50 1.01

𝑾𝒔𝟓

1.46

𝑾𝒔𝟔

0.78

0.34 0.33

0.00 0.49

1.54 0.00

0.60

1.14 2.28

-

-

1.59

0.00

1.75

0.78

1.18

2.40

1.87

0.00

TABLE VIII PARAMETERS USED IN HYBRID ALGORITHM IN 6WORKSTATION ROBOTIC CELL Workstation Generation Population Pc Pm number number Size 6 72 81 0.9 0.9

TABLE IX PARAMETERS USED IN HYBRID ALGORITHM Workstation Generation Population No. Pc Pm number number Size 1 6 72 81 0.9 0.9 2 7 98 243 0.9 0.9 3 8 128 729 0.9 0.9 Relative Cycle Time (%)

Here the result of the hybrid algorithm is given by using C# language program. The optimal solution is obtained when the generation number is 14. And Fig. 4 shows the relative minimum cycle time in each generation. Here the minimum cycle time obtained by the hybrid algorithm is normalized by the final optimal result calculated by the exact method. Fig. 5 gives the Gantt char of the robot actions in a series of the sequence 𝑆 = (1− , 6+ , 2− , 2+ , 3− , 3+ , 4− , 4+ , 5− , 5+ ) over time. And the minimum cycle time is 10.11s. Compared to the exact method, the hybrid algorithm can obtain the optimal solution and the hybrid algorithm is even faster than to search all the possible solutions. Relative Cycle Time (%)

The Number of Workstations Fig. 6 The cycle time comparison between existing exact method and proposed approximate method

Computing Consumption

On the other hand, the computing consumption is also evaluated. As show in Fig. 7, it can be found that the computing consumption of proposed hybrid method is much less than existing exact method.

The Number of Generation Fig. 4 Convergent process of minimum cycle time

The Number of Workstations Fig. 7 The comparison of computing consumption between two methods

V. CONCLUSIONS

Fig. 5 Gantt chart of a free-process dual-gripper robotic cell

In this paper, a hybrid algorithm is proposed and used for scheduling optimization in large scale robotic cell with dualgripper by considering the constrains in real engineering practice. The resulting diagrams provide very intuitive insights into the accuracy of the hybrid algorithm by comparing the exact algorithm. Experiments proved that hybrid algorithm has a nice

B. 100 random cases In order to analysis the performance of the hybrid algorithm further, 100 virtual robotic cell cases with 6 up-to 8 workstations are generated. TABLE IX shows the parameters of hybrid

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performance to achieve the minimum cycle in large scale robotic cells with dual-gripper robot. With the increasing of the number of workstations, the number of possible solutions grows exponentially, the probability to get the optimal solution has close links with the number of the initial population. We can choose more initial population to get better performance when the number of workstations is very big. In the future work, the hybrid algorithm also can be used to fix scheduling optimization problem in robotic cell with parallel workstations.

[6] [7] [8] [9]

ACKNOWLEDGMENT [10]

This work is supported by the Science and Technology Development Plan of Shandong Province of China (No. 2011GGH22203).

[11]

REFERENCES [1] [2] [3]

[4]

[5]

[12]

M. W. Dawande, H. N. Geismar, S. P. Sethi, C. Sriskandarajah, Throughput optimization in robotic cells. New York: Springer. 2007. N. Brauner, G. Finke, W. Kubiak, “Complexity of one-cycle robotic flow-shops,” Journal of Scheduling, Vol.6, No.4, pp.355-372, 2003. J. F. Zhang, X. Y. Fang. “LCM cycle based optimal scheduling in robotic cell with parallel workstations,” IEEE International conference on Robotics and Automation, Hongkong, China, May 31-4 June, 2014, accepted. S. P. Sethi, C. Sriskandarajah, G. Sorger, J. Blazewicz, W. Kubiak, “Sequencing of parts and robot moves in a robotic cell,” International Journal of Flexible Manufacturing Systems, Vol.4, No.3-4, pp.331-358, 1992. V. Kats, E. Levner, “A polynomial algorithm for 2-cyclic robotic scheduling: A non-Euclidean case,” Discrete Applied Mathematics, Vol.157, No.2, pp.339-355, 2009.

[13] [14] [15] [16]

152

M. W. Dawande, C. Sriskandarajah, S. Sethi, “On throughput maximization in constant travel-time robotic cells,” Manufacturing & Service Operations Management, Vol.4, No.4, pp.296-312, 2002. P. Brucker, T. Kampmeyer, “A general model for cyclic machine scheduling problems,” Discrete Applied Mathematics, Vol.156, No.13, pp.2561-2572, 2008. M. S. Akturk, H. Gultekin, O. E. Karasan, “Robotic cell scheduling with operational flexibility,” Discrete Applied Mathematics, Vol.145, No.3, pp.334-348, 2005. J. F. Zhang, X. Y. Fang, “Robot move scheduling optimization for maximizing cell throughput with constraints in real-life engineering,” IEEE International conference on Robotics and Biomimetics, Shenzhen, China, Dec 12-16, 2013, pp. 221-227. A. Kampa, “Planning and scheduling of work in robotic manufacturing systems with flexible production,” Journal of Machine Engineering, Vol.12, pp. 34-44, 2012. Q. Su, and F. Frank Chen, “Optimal sequencing of double-gripper gantry robot moves in tightly-coupled serial production systems,” Robotics and Automation, IEEE Transactions on, Vol.12, No.1,pp. 22-30, 1996. S. Sethi, J. B. Sidney, C. Sriskandarajah, “Scheduling in dual gripper robotic cells for productivity gains,” Robotics and Automation, IEEE Transactions, Vol.17, No.3, pp.324-341, 2001. M. Dawande, H. N. Geismar, M. Pinedo, C. Sriskandarajah, “Throughput optimization in dual-gripper interval robotic cells,” IIE Transactions, Vol.42, No.1, pp. 1-15, 2009. T. Murata and H. Ishibuchi, “Performance evaluation of genetic algorithms for flowshop scheduling problems,” Proc. 1st IEEE Conf. Evolutionary Computation, Orlando, Jun 27-29, 1994, Vol.2, pp. 812-817. T. Murata, H. Ishibuchi, and H. Tanaka, “Genetic algorithms for flowshop scheduling problems,” Computer & Industrial Engineering, Vol.30, No.4, pp.1061-1071, 1996. C. Bierwirth, D. C. Mattfeld, and H. Kopfer, “On permutation representations for scheduling problems,” in Proc. 4th Int. Conf. PPS from Nature, Voigt, Ebeling, Rechenberg, and Schwefel, Eds. Berlin, Germany: Springer-Verlag, 1996, pp. 310–318.