Multi-cyclic scheduling of identical parts in an N-machine ...scheduling over the traditional scheduling theories in flexible manufacturing. (Timkovsky, 2004), (Seo and Lee, 2002),

Proceedings of the International Conference on Industrial Engineering and Operations Management Dubai, UAE, March 10-12, 2020

© IEOM Society International

Multi-cyclic scheduling of identical parts in an N-machine no-wait robotic cell

Dhirendra Prajapati1, Arjun R Harish2, Saurabh Pratap5

Department of Mechanical Engineering, Indian Institute of Information Technology, Design & Manufacturing, Jabalpur, India

*[email protected], [email protected], [email protected]

Mengdi Zhang3

School of Internet of Things , Nanjing University of Posts and Telecommunications, Nanjing 210003, People’s Republic of China

[email protected],

Yash Daultani4 Department of Operations Management,

Indian Institute of Management, Lucknow, India [email protected],

Abstract

This paper proposes a mixed integer linear programming model to determine the best average cycle time for multi-degree cyclic scheduling in a no-wait robotic cell producing identical parts using N-machines and one robot to perform handling operations. We make a conjecture that the best average cycle time is always produced by a k-degree cyclic schedule. The problem is NP-hard in nature and is solved using a brute-force method to obtain the exact solution. A numerical example is used to analyze the model and the proposal is validated through computational results obtained. The sensitivity analysis performed based on different parameters is used to obtain insights and implications of the model.

Keywords

Average cycle time, Multi-degree cyclic scheduling, NP-hard, No-wait, Robotic cell

1. Introduction

The processing on the cyclic environment is gaining popularity due to its ease of handling and higher throughput rates in a flexible manufacturing environment. A cyclic schedule can be defined as a schedule that repeats itself at fixed intervals of time. These fixed intervals are called cycle times and it corresponds to each cycle of operations. The number of parts produced in each cycle depends on the degree of cyclic scheduling. 𝑘𝑘 parts enter and leave the robotic cell during a cycle in a k-degree cyclic schedule. The cycle time controls the throughput rate and hence we consider it as the object to be minimized in this paper. Minimum cycle time results in maximum throughput. For a k-degree cyclic schedule, 𝑇𝑇𝑇𝑇1,𝑇𝑇𝑇𝑇2, …𝑇𝑇𝑇𝑇𝑘𝑘 represents the inter-arrival time for the loading station. The average ofthese times gives the average cycle time of the schedule. It can be defined as the mean time required in producing apart in the robotic cell.

One of the important questions that arise in scheduling theory is about the existence of an optimal cyclic schedule. According to (Dawande et al, 2008), different infinitely-long schedules will have one optimal schedule if the input parameter values are integer based. The parameters such as the status of the robot, the position of the robot, and the status of each part being processed determine the state of a schedule. Since the number of machines and robots in a cell are finite, the number of states will be finite. A schedule can be generated by arranging these states sequentially. Similar states repeat itself during an infinite schedule. However, the occurrence of these repeated states can be segmented in case of an optimal schedule and a schedule can be obtained which can be repeated to obtain the same

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throughput rate. If an optimal schedule exists, then the repetition of such schedules would result in maximum throughput.

Most of the available literature in this research area determines the best average cycle time for robotic cells with two or three machines. There is not much available literature, which gives the solution for the best average cycle time that can be produced from a 𝑁𝑁 machine robotic cell. Most of the research in this field concentrates on cyclic scheduling in robotic cells with 1 or 2 machines. There is very little literature available for cyclic scheduling in robotic cells with more than two machines. (Agnetis, 2000) considered the cyclic scheduling of systems with 2 or 3 machines and came to the conclusion that the best average cycle time is obtained from a cyclic schedule with degree less than or equal to 2. This paper aims to prove that for a schedule with 𝑁𝑁 machines, the best average cycle time will be given by a k-degree cyclic schedule where 𝑘𝑘 ≤ (𝑁𝑁 − 1)! . The reduced cycle time will result in increased throughput rates and reduces work-in-process inventory, thereby helping the companies to maximize their profit.

Computer-integrated manufacturing is gaining popularity and as a result, computer-controlled robots are widely being used in industries to handle various manufacturing processes. These robots are kept responsible for transferring parts from one machine to another to perform processing operations. Hence, the throughput from each machine cell depends upon the performance of the robots in it. Proper scheduling of the robotic activities can maximize the throughput from these robotic cells. This will help to reduce the manufacturing cost and the work-in-process inventory. The following research questions are addressed in this paper: (1) can multi-cyclic schedules produce better throughputs than monocyclic schedule? (2) Up-to what value of 𝑘𝑘 should the cyclic scheduling be performed to produce throughputs better than its monocyclic counterpart? This is one of the earliest studies focusing on developing a model for cyclic scheduling in a robotic cell with 𝑁𝑁 machines, where 𝑁𝑁 is any natural number. This model focuses on producing results which imply better chances of obtaining a better average cycle time from multi-cyclic schedules, resulting in a higher throughput rate of the robotic cell.

In this paper, we consider the problem of minimizing the cycle time in a robotic cell that satisfies the no-wait criterion. The parts are transferred to the next machine as soon as the processing is completed in one machine. The robot-centered cell concept is used to perform the handling operations. Hence, all the machines are arranged in a semicircle and a robotic arm fixed at the center of the semi-circle is used to rotate around and transfer parts from one machine to another. Figure 1 illustrates the functioning of the robotic cell taken into consideration.

Figure 1. Schematic diagram representing the functioning of a robotic cell

This paper aims at producing a solution for best average cycle time in a k-cyclic schedule by developing a mathematical model for a robotic cell with 𝑁𝑁 machines. Further, the mathematical model is solved using an exhaustive brute-force search method to obtain the exact solution for this problem of NP-hard category.

2. Literature Review

There is an increasing trend in the significance and popularity for cyclic scheduling problems in Industries as well as in scheduling literature. This can be characterized as a scenario produced due to the advantages posed by the cyclic scheduling over the traditional scheduling theories in flexible manufacturing. (Timkovsky, 2004), (Seo and Lee, 2002), (Dawande et al, 2007) and much other literature describes the significance of cyclic scheduling over the

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traditional methods used in flexible manufacturing. The robotic cyclic scheduling was first introduced in early 1960, when (Suprunenko et al, 1962), (Tanaev, 1964), (Aizenshtat, 1963) analyzed the applicability of cyclic processes in manufacturing lines with transporting devices. These studies came up with the method of forbidden intervals, which is a combinatorial method to be followed while performing cyclic scheduling. This method was further modified and improved based on different scheduling problems by (Che and Chu, 2005a, 2005b), (Kats et al, 1999), (Chu, 2006), (Che et al, 2002, 2003). A detailed study in this field was carried out by (Crama et al, 2000), (Manier and Bloch, 2003), and (Dawande et al, 2005, 2007), which can be used for further reference. Brauner, (2005) introduced some of the basic concepts and tools related to cyclic production. It defined the concepts related to multi-cyclic scheduling in a flow shop with a robot performing the handling operations. Mishra et al, 2019 introduced the job shop-scheduling problem and used meta-heuristic algorithm (i.e. Chemical reaction optimization and Cuckoo Search Algorithm) to minimize the make-span.

There are several studies being performed to optimize the cycle times related to robotic cyclic scheduling. (Agnetis, 2000) studied the scheduling in a no-wait robotic cell. Sequencing of parts through scheduling of robotic moves was performed to obtain the maximum productivity. (Kampmeyer, 2006) conducted a study on general cyclic scheduling problems and came up with various applications and solutions related to cyclic scheduling. (Brawner et al, 2005) performed a study on the complexity of high-multiplicity scheduling problems. They developed frameworks to classify different scheduling algorithms based on their complexities. (Che and Chu, 2005) developed a model to study the scheduling in a robotic cell. They tried to minimize the cycle time in a no-wait robotic cell with 2 robots operating in parallel lines. (Kats & Levner, 1998) studied the periodic scheduling of parts in a robotic environment. They tried to determine processing time of operations and the start time to optimize the cycle length. (Florence et al, 2016) performed the cyclic scheduling in a circular robotic cell to study the factors affecting the throughput rate. They developed a dominating family of 1-cyclic schedule, which, leads to a polynomial algorithm. (Pratap et al, 2015) developed an MINLP model to capture the berth and quay crane scheduling problem and used GA to determine the optimal sequence of ships at the port.

The recent studies on cyclic scheduling of robotic cells can be summarized as follows. (Lei, Che, & Chu, 2014) proposed a branch and bound algorithm to solve a robotic flow shop scheduling problem with multiple parts where part processing time varies with the given time windows. They considered both the robot movement sequencing and the part input sequencing while performing the cyclic scheduling. A similar study on the scheduling of multiple parts in a single robot served hybrid robotic flow shop was performed by (Elmi & Topaloglu, 2014). They considered different part types and machines operating in parallel with different processing speeds and eligibility constraints while developing a mixed integer linear programming model to address the problem. (Elmi & Topaloglu, 2016) performs robot assignment to machines and robot movement sequencing in a robotic flow shop served by multiple robots. (Kats & Levner, 2018) studied a four machine robotic cell served by a single robot and proved that the optimal schedule will be 𝑘𝑘-cyclic with 𝑘𝑘 >= 6. This can be considered as one of the studies from the related literature that is closest to the problem being addressed by this paper.

This paper comprises different sections as follows: Section 3 gives the detailed problem description followed by the mathematical model and the constraints associated with the scenario. Section 4 describes the solution methodology. Section 6 gives the results and analysis of the problem taken into consideration. The conclusion and future research possibilities in this field are given in section 7.

3. Problem Description

The complete scheduling is performed over a robotic cell consisting of 𝑁𝑁 machines and 1 robot for transferring the parts. Machines 𝑁𝑁1, 𝑁𝑁2, …𝑁𝑁𝑁𝑁 are associated with the processing of parts and dummy stations 𝑁𝑁0 and 𝑁𝑁𝑁𝑁+1act as loading and unloading stations. Scheduling in this paper is based on the identical parts production scenario. The parts are loaded into the station 𝑁𝑁0 . The robot is programmed to carry the parts from 𝑁𝑁0 through 𝑁𝑁1, 𝑁𝑁2, …𝑁𝑁𝑁𝑁 for processing and finally to 𝑁𝑁𝑁𝑁+1 for unloading. The robot follows the same sequence repetitively for all the cycles. Each machine can process one part at a time. This avoids overlapping of operations taking place in each machine 𝑁𝑁𝑖𝑖 and provides the machine enough time to remove the part being processed before the machine part enters from the machine𝑁𝑁𝑖𝑖−1. Part processing time 𝐷𝐷𝑛𝑛 , part delivery time of robot from the machine 𝑁𝑁𝑖𝑖 → 𝑁𝑁𝑖𝑖+1, 𝑡𝑡𝑛𝑛, and the empty robot travel time from machine𝑁𝑁𝑖𝑖 → 𝑁𝑁𝑗𝑗 , 𝑒𝑒𝑖𝑖𝑗𝑗, constitute the different input parameters of the system.

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Here the parts are assumed to move satisfies the following criterion (triangular inequality): 𝐿𝐿𝑖𝑖𝑘𝑘 ≤ 𝐿𝐿𝑖𝑖𝑗𝑗 + 𝐿𝐿𝑗𝑗𝑘𝑘 , and 𝐿𝐿𝑖𝑖𝑗𝑗 = 𝑡𝑡𝑖𝑖 + 𝑒𝑒𝑖𝑖+1,𝑗𝑗 ; where 𝑖𝑖, 𝑗𝑗, 𝑘𝑘 ∈ 1, 2, 3 …𝑁𝑁.We consider that the robot moves instantaneously to the next machine as it becomes empty. Hence, 𝑒𝑒𝑖𝑖+1,𝑗𝑗 becomes zero and the criterion reduces to𝑡𝑡𝑖𝑖 ≤ 𝑡𝑡𝑖𝑖 + 𝑡𝑡𝑗𝑗. As the no-wait criterion needs to be satisfied, each part process completion time is calculated based on the start time 𝑆𝑆𝑇𝑇. It represents the time at which parts start moving initially from the loading station 𝑁𝑁0.Forbidden region and hence, the feasible region of cycle times can be calculated using the method of forbidden interval used in (Agnetis, 2000) and (Kats & Levner, 1997). For the reader’s understanding, the method of forbidden interval used in this paper is given in the Appendix.

For a robot to operate without delay, the time interval between any two parts being loaded should not fall in the forbidden intervals. The feasible schedule consists of an infinite sequence 𝑆𝑆𝑆𝑆 = (𝑊𝑊1, ,𝑊𝑊2 … . ) where, each element is selected from the feasible region of cycle times. For a robotic cell with N machines, there exist 𝑁𝑁(𝑁𝑁 − 1)/2 + 1 forbidden regions. The feasible region is obtained after removing all the forbidden regions from 𝑅𝑅. This feasible region is further used to determine the elements of the cyclic schedule and finally, the best average cycle time is calculated. Since, integer-based input parameters are given, the elements in the schedule can be found to repeat after a certain fixed interval. Hence, these schedules can be segmented and repeated to obtain the same throughput rate and, the number of elements in these segments determine the number of parts going out per cycle and hence, the degree of the cyclic schedule. All the constraints associated with the determination of the best average cycle time are given below:

Mathematical Model

Sets N Machines to perform operations n ∈ 0, 1, .......N K Number of parts coming out per cycle k ∈ 1, 2, .......K R Set of real numbers

Parameters

ST Starting time of schedule nD Processing time associated with machine n nt Transfer time required for each loaded move from machine n

ije Time associated with empty move from machine i to machine j, where i >j

Variables

jM Completion time for machine j i∂ Time restriction from machine i for zero overlapping of operations

Ω Base time restriction ijL Robotic moves taking place between machine i and machine j ijB Restricted time intervals for operations between machine i and j

RT Forbidden cycle time region

AL Feasible cycle time region

Decision Variables

kyTC Time interval between loading of part y and part (y-1) in a k-cyclic schedule f (k) Average cycle time for k cyclic schedules

Objective Function:

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Determine: k

=

=

∑1(k) M in

k

kyy

TC

fk

(1)

Constraints:

( )j

j n-1 nn=1

M ST + t + D= ∑ ∀ ∈j N (2)

i i i,j= M M + L∂ − j ∀ = −∈, j N; 1 j ii (3) iMax( )Ω = ∂ ∀ ∈Ni (4)

ij i i+1,jL = + t e ∀ ∈, j Ni (5)

( )ij j i ij j i jiB = M M L M M +L− − − , ∀ ∈ ≥ +, j N; j 2i i (6)

ijRT B= −∞ Ω( , ) ∀ ∈ ≥ +, j N; j 2i i (7)

AL = R RT− (8)

TC AL∈ ky ∀ ∈ ∈,y k k K (9)

pTC AL∈11 ∀ = −1,2.....,( 1)p N (10) k(y+1) k(y+z)+TCTC + TC AL∈ ........... ky −∀ ∈ ∈ ≥ = 1,2....,( 2), ; k 2; Ny k k K z (11)

kyTC +∈ Z y k k K∀ ∈ ∈ , (12)

The objective function (1) determines the value of 𝑘𝑘 with minimum average cycle time. Constraint (2) to Constraint (7) are used to find the forbidden intervals. Constraint (2) calculates the completion time of operation in machine 𝑛𝑛. Base time restriction for each machine to avoid overlapping of operations is given by Constraint (3). Constraint (4) selects the maximum value from amongst the base time restrictions. Constraint (5) gives the net robot movement around a machine. The completion time associated with each machine generates a corresponding forbidden region, represented by constraint (6). Constraint (7) gives the complete forbidden region for the cyclic schedule under consideration. Constraint (8) represents the values of cycle time that are feasible for the schedule. Constraint (9) imposes a restriction on each cycle time in the schedule to be an element of 𝐴𝐴𝐿𝐿. Constraint (10) puts restriction on the cycle time of monocyclic schedules. For calculation of 𝑇𝑇𝑇𝑇11, the least value form 𝐴𝐴𝐿𝐿 which satisfies constraint (10) is selected. Constraint (11) is associated with selection of elements for a k-degree cyclic schedule where 𝑘𝑘 ≥ 2. Each element in the cyclic schedule must satisfy the constraint (11). For example, in a 3-degree cyclic schedule of a robotic cell with 3 machines; 𝑇𝑇𝑇𝑇31 + 𝑇𝑇𝑇𝑇32, 𝑇𝑇𝑇𝑇32 + 𝑇𝑇𝑇𝑇33, 𝑇𝑇𝑇𝑇33 + 𝑇𝑇𝑇𝑇31 ∈ 𝐴𝐴𝐿𝐿. Constraint (12) is related to non-negativity and integrity of the variable.

5. Solution Approach

The model taken into consideration is a variant of Multiple Subset Sum Problem (MSSP), which makes it NP-hard in nature. There is no existing literature on algorithms that can give the best possible solution of this model with a time less than polynomial time complexity. In this paper, the solution to the model is given prominence over the computational time. Hence, we use a brute-force method to extract the exact solution.

The values of 𝑀𝑀𝑗𝑗, for j= 0, 1…. N, are calculated. All the values of 𝑀𝑀𝑗𝑗 are used to calculate the values corresponding to 𝜕𝜕𝑖𝑖. The maximum value obtained from 𝜕𝜕𝑖𝑖 is assigned to Ω, which forms the base restriction or the first restriction to be imposed on the solution space. For determining other time restrictions, 𝐿𝐿𝑖𝑖𝑗𝑗 needs to be calculated. The remaining restricted regions,𝐵𝐵𝑖𝑖𝑗𝑗 , are determined using the values of 𝑀𝑀𝑗𝑗 and 𝐵𝐵𝑖𝑖𝑗𝑗 calculated earlier. The values obtained from 𝐵𝐵𝑖𝑖𝑗𝑗 are combined with the base restriction Ω to obtain the complete forbidden cycle time region. All the values coming outside the forbidden cycle time region are given by 𝐴𝐴𝐿𝐿.

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The cycle time for monocyclic schedule, 𝑇𝑇𝑇𝑇11, is calculated. This is taken as a reference for higher degrees of scheduling. It is assumed that the optimal cyclic schedule will have average cycle time less than or equal to the monocyclic schedule. Hence, all the elements within the feasible region with their values less than or equal to two times 𝑇𝑇𝑇𝑇11 are considered for determining the best cycle time for a k-cyclic schedule as the average cycle time should be less than or equal to 𝑇𝑇𝑇𝑇11. This part of the feasible region can be termed as the selection region.

All the elements of a cyclic schedule are selected in such a way that the values always satisfy the constraint (11). Hence, every possible permutation of the set of elements in each schedule is generated using the elements in the selected region. For a k-degree cyclic schedule, the permutations containing 𝑘𝑘 elements are developed. This helps to ensure that all the possible combinations are analyzed and helps to find the best solution out of all the possible solutions. Figure 2 gives a sample representation for the formation of the permutations.

Figure 2. Formation of all possible elements through permutation

The permutations are analyzed to check whether they satisfy constraint (11) and the element with least average value is selected as the best average cycle time for the cyclic schedule. Similarly, sets of permutation for 𝑘𝑘 = 1,2, …𝐾𝐾 are created and the value of 𝑘𝑘 that gives the best average cycle time is determined. A flowchart representing the flow of operations while calculating the solution is given in Figure 3.

Figure 3. Flowchart representing the steps involved in determining the best solution

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6. Results and Discussion

The proposed model focuses on determining the minimum value of 𝑘𝑘 associated with a k-degree cyclic schedule, which would lead to maximum throughput and minimum average cycle time. The model is solved using brute-force search method, which solves the model by analyzing every possible permutation and gives the best possible solution. The algorithm was coded into a personal computer with an Intel core i5, 2.9 GHz processor and 8 GB RAM, in a Windows 10 environment. This section illustrates the values assigned to different input parameters. The number of machines in the work cell is varied to analyze the results produced from the problem for 3 different cases of work cell design. For all the calculations,𝑒𝑒𝑖𝑖+1,𝑗𝑗 is assumed to be zero to represent instantaneous movement of robot from one machine to another as it becomes empty.

Case-1 (N = 2)

Here, the machine cell is considered to have 2 machines and 1 robot serving it. Dummy station 𝑁𝑁0 will be acting as the loading station and other dummy station 𝑁𝑁3 acts as the unloading station. Table 1 gives the input parameters corresponding to this cell design. Here, the average cycle times for multi-cyclic schedules with degree up to 𝐾𝐾 is calculated. We consider two cell configurations with different processing and handling times as shown in Table 1.

Table 1: Values of Parameters

Parameters Configuration 1 (C1) Configuration 2 (C2) nD [14, 13] [28, 16]

nt [3, 2, 1] [3, 1, 2] K 4 4 ST 0 0

Case-2 (N = 3)



Parameters Configuration 1 (C1) Configuration 2 (C2) nD [14, 13, 12] [28, 8, 30]

nt [3, 2, 1, 2] [3, 1, 2, 4] K 4 4 ST 0 0

Case-3 (N = 4)



Parameters Configuration 1 (C1) Configuration 2 (C2) nD [14, 13, 12, 15] [28, 26, 24, 30]

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nt [3, 2, 1, 2, 3] [6, 4, 2, 4, 6] K 6 6 ST 0 0

Table 4: Computation Experiment

Case Average Cycle time for k=1, 2, K Minimum f (k) kyTC Solution

k 1 C1 19, 19, 19, 19 19 [19] 1

C2 32, 32, 32, 32 32 [32] 1 2 C1 19, 19, 19, 19 19 [19] 1

C2 45, 40.5, 42, 40.5 40.5 [36, 45] 2 3 C1 24, 24, 24, 24, 24, 23.833 23.833 [20, 22, 26, 22, 20, 33] 6

C2 48, 48, 48, 48, 48, 47.667 47.667 [40, 44, 52, 44, 40, 66] 6

From table 4, it is observed that for all the problem cases the best solution is always obtained for 𝑘𝑘 ≤ (𝑁𝑁 − 1)!. This gives the answer to the research questions (1) and (2) propounded in this paper. Therefore, it can be clearly stated that a cyclic schedule with 𝑘𝑘 ≤ (𝑁𝑁 − 1)! always produces the maximum throughput. Figure 4 represents the cyclic scheduling of parts in the robotic cell for the solution obtained from case 3. In this figure, Machine 0 represents the loading station and Machine 5 represents the unloading station.

Figure 4. Gantt chart representing the movement of parts for solution obtained from case-3

6.1 Sensitivity Analysis

The effect of input parameters on the solution generated is analyzed by performing a sensitivity analysis (Pratap et al, 2019). The parameters such as 𝐷𝐷𝑛𝑛 and 𝑡𝑡𝑛𝑛 vary from their original value to observe the impact of the solution.

6.1.1 The effect of processing time

The processing time associated with each machine is varied by -60%, -40%, -20%, 20%, 40% and 60% and the corresponding changes in average cycle time and the degree of cyclic schedule is observed. Figure 5 depicts the changes observed during the analysis. The average cycle time is found to vary by -24.47% when the processing time was changed to -60% of its current value. Similarly, the average cycle time was observed to vary by +21.68% when the processing time was changed to +60% of its initial value.

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Figure 5. Graphical representation of change in average cycle time with change in processing time

6.1.2 The influence of Transfer time of robot

The transfer time associated with robot transferring parts between the machines is varied by -60%, -40%, -20%, 20%, 40% and 60% and the corresponding change in average cycle time and the degree of cyclic schedule is observed. Figure 6 depicts the changes observed during the analysis performed. The average cycle time is found to vary by -24.47% when the transfer time was changed to -60% of its current value. Similarly, the average cycle time was observed to vary by +30.07% when the transfer time was changed to +60% of its initial value.

Figure 6. Graphical representation of change in average cycle time with change in transfer time

6.2 Managerial Insights

The results obtained from the sensitivity analysis are very crucial when it comes to deciding the degree of cyclic schedule 𝑘𝑘. From Fig. 5 and Fig. 6 it is observed that the average cycle time for the schedule shows a steady increase with increase in processing and transfer times of parts present inside the robotic cell. Similarly, it can also be observed that at certain points in these graphs where the value of k increases with respect to the point before it, the average cycle time shows a slight decrease or a more gradual increase. It can be observed in Fig. 5 that when the transition takes place from -60% to -40%, the value of 𝑓𝑓(𝑘𝑘) decreases and the value of 𝑘𝑘 increases from 1 to 2. Similarly, in the Fig. 4 it can be observed that when the transition takes place between 20% and 40%, the increase in 𝑓𝑓(𝑘𝑘) becomes more gradual or less steep, indicating an increase in value of 𝑘𝑘. This further enhances the significance of research question (1).

k=1

k=1

k=6

k=6

k=1k=2

k=2

0

10

20

30

40

50

60

70

-60% -40% -20% 0% 20% 40% 60%

f(k)

Percentage change in part transfer time

f(k)

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From these observations it can be inferred that the degree of a cycle plays an important role in determining the average cycle time. Hence, this analysis concludes that, for a schedule with suitable input parameters, a multi degree cyclic schedule always gives relatively lower cycle time and higher production rate.

7. Conclusion

This paper gives a few new results for scheduling in a no-wait robotic cell. The results obtained are convincing and it justifies the proposal. Multi-degree cyclic schedules are very useful when it comes to the scheduling of robotic cells that operates for very long periods of time. The results give the optimal value of 𝑘𝑘, which is of high significance for industrial applications and has immense theoretical value. This paper proves that the best optimal cyclic schedule is obtained from a cyclic schedule with 𝑘𝑘 ≤ (𝑁𝑁 − 1)!.

For future research, the mathematical model can be extended to analyze whether the same proposition stands in the case of a robotic cells working on a different parts scenario. Also, studies can be performed to analyze the validity of the results in the case of robotic cells having N>4. Algorithms with lower time complexity can be developed to solve the model, which will make the model more productive in an industrial environment.

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Biographies Dhirendra Prajapati is a PhD Scholar in the Department of Mechanical Engineering at Indian Institute of Information Technology, Design & Manufacturing, Jabalpur, India. He earned Bachelor of Engineering in the Department of Mechanical Engineering from Rajiv Gandhi Proudyogiki Vishwavidyalaya, Bhopal, India, and Master of Technology in Manufacturing from Indian Institute of Information Technology, Design & Manufacturing, Jabalpur, India. His area of interest in E-commerce, logistics and supply chain network design, process planning and scheduling, operations management, mathematical modeling and optimization, sustainability. Arjun R Harish is a Master student in the Department of Industrial and Manufacturing Systems Engineering at University of Hog Kong HK. He has completed his Bachelor of Technology in the Department of Mechanical Engineering at the Indian Institute of Information Technology, Jabalpur, India. His area of interest in logistics and supply chain network design, process planning and scheduling, nature inspired algorithm and mathematical modeling. Mengdi Zhang is serving as a Lecturer in School of Internet of Things, Nanjing University of Posts and Telecommunications, Nanjing, China. She has completed his Master and PhD degree in the Department of Industrial & Manufacturing Systems Engineering at University of Hong Kong, HK. She has authored in several journals papers. Her publications appeared in such journals as Journal of Intelligent Manufacturing, International Journal of Production Research and International Journal of Production Economy etc. He is working as a reviewer in various renowned International Journals. She has worked as a reviewer in some reputed journals. Dr. Yash Daultani is working as an Assistant Professor in the Operations Management, Indian Institute of Management, Lucknow, UP, India. Previously, He was working as an Assistant Professor in the Department of Operations Management, Indian Institute of Information Technology, Gwalior, M.P, India. He has completed his FPM (PhD) degree in Operations Management, Indian Institute of Management, Lucknow, UP in 2016. He has authored several technical papers. His publications appeared in such journals as International Journal of Production Research, Journal of Intelligent Manufacturing etc. He is working as a reviewer in various renowned International Journals.

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https://doi.org/10.1016/j.ifacol.2015.06.193



Dr. Saurabh Pratap is currently working as an Assistant Professor in the Department of Mechanical Engineering, Indian Institute of Information Technology, Jabalpur, M.P, India. Previously, He worked as a Post-Doctoral Fellow in the Department of Industrial & Manufacturing Systems Engineering at University of Hong Kong, Hong Kong. He received his B.Tech degree in Mechanical Engineering from Institute of Engineering & Technology, (India) and M.Tech degree in Production Engineering from National Institute of Technology, Durgapur (India) in the year 2008 and 2011, respectively. He has completed his Ph.D. degree in Industrial & Systems Engineering Department at IIT Kharagpur, India in 2016. He has authored several technical papers. His publications appeared in such journals as International Journal of Production Research, International Journal of Production Economics, Computers & Industrial Engineering, Annals of Operation Research, Journal of Intelligent Manufacturing and Maritime Economics & logistics. He carried out collaborative research with the Laboratory for Production Management and Process in EPFL, Switzerland and University of Hong Kong, HK. He is working as a reviewer for 16 renowned International Journals.

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Multi-cyclic scheduling of identical parts in an N-machine ...scheduling over the traditional scheduling theories in flexible manufacturing. (Timkovsky, 2004), (Seo and Lee, 2002),