A framework for stochastic scheduling of two-machine robotic rework cells with in-process inspection system

Accepted Manuscript Stochastic scheduling of an automated two-machine robotic cell with in-process inspection system Mehdi Foumani, Kate Smith-Miles, Indra Gunawan, Asghar Moeini PII: DOI: Reference:

S0360-8352(17)30069-4 http://dx.doi.org/10.1016/j.cie.2017.02.009 CAIE 4639

To appear in:

Computers & Industrial Engineering

Received Date: Revised Date: Accepted Date:

4 March 2016 6 December 2016 8 February 2017

Please cite this article as: Foumani, M., Smith-Miles, K., Gunawan, I., Moeini, A., Stochastic scheduling of an automated two-machine robotic cell with in-process inspection system, Computers & Industrial Engineering (2017), doi: http://dx.doi.org/10.1016/j.cie.2017.02.009

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Stochastic scheduling of an automated two-machine robotic cell with in-process inspection system Mehdi Foumani1, 2*, Kate Smith-Miles1, Indra Gunawan3 and Asghar Moeini4 1

School of Mathematical Sciences, Monash University, Clayton, Victoria 3800, Australia Department of Econometrics and Business Statistics, Monash University, Caulfield East, VIC 3145, Australia 3 Entrepreneurship, Commercialisation and Innovation Centre, University of Adelaide, Adelaide, SA 5005, Australia 4 School of Computer Science, Engineering and Mathematics, Flinders University, Tonsley Park, SA 5042, Australia 2

Abstract This study is focused on the domain of a two-machine robotic cell scheduling problem for three various kinds of pickup scenarios: free, interval, and no-wait pickup scenarios. Particularly, we propose the first analytical method for minimizing the partial cycle time of such a cell with a PCbased automatic inspection system to make the problem more realistic. It is assumed that parts must be inspected in one of the production machines, and this may result in a rework process. The stochastic nature of the rework process prevents us from applying existing deterministic solution methods for the scheduling problem. This study aims to develop an in-line inspection of identical parts using multiple contact/non-contact sensors. Initially, we convert a multiple-sensor inspection system into a single-sensor inspection system. Then, the expected sequence times of two different cycles are derived based on a geometric distribution, and finally the maximum expected throughput is pursued for each individual case with free pickup scenario. Results are also extended for the interval and no-wait pick up scenarios as two well-solved classes of the scheduling problem. The waiting time of the part at each machine after finishing its operation is bounded within a fixed time interval in cells with interval pickup scenario, whereas the part is processed from the input conveyor to the output conveyor without any interruption on machines in cells with no-wait pickup scenario. We show a simple approach for solving these two scenarios of the problem which are common in practice. Keywords: Scheduling, Rework, Robotic cell, Performance 1

Introduction

Robotic cells are one of the complicated application areas of flow-shops that have received a considerable amount of attention in different areas such as robot path planning [1, 2], robot selection [3, 4], task allocation in robotic systems [5] and robot move sequencing [6, 7]. They are basically classified into two categories: the robotic cells without rework assumption and robotic rework cells. *

*

Corresponding author. Email: [email protected]

1

The term “rework” here means that a processed part may need reprocessing. Therefore, it is cycled between test and processing stations until deemed acceptable. It is straightforward to find a deterministic model for the robotic cells without rework assumption. Following that, there are many studies in the literature dealing with the scheduling of the robot activities, as widely addressed in [8] for two-machine cells. Nonetheless, inspection and rework stages in a robotic cell is one of the important issues in the field of robotic cell scheduling which reflects most real-life cases. This paper addresses the stochastic issues that arise when considering inspection and rework stages, laying some important analytical foundations for this under-studied problem. A robotic cell with an additional inspection process in one of the rework stages is called a robotic rework cell. A two-machine robotic rework cell which is the smallest possible robotic rework cell is commonly captured by the following framework: the cell is made up of two production machines M1 and M2, multiple contact/non-contact sensors installed into M1 or M2, a gantry robot that serves the entire production line, an input conveyor (I or the axillary machine M0) and an output conveyor (O or the axillary machine M3) with unlimited storage capacity. This framework makes it clear that typical robotic cells are a special case of robotic rework cells where there is no inspection sensor on production machines, and all produced parts are failure-free.

Inspection controller Gantry Robot

Robot Controller

Machine vision (camera) Production controller

Crankshaft M1

M2

Figure 1. A two-machine robotic rework cell with end of line inspection. Two-machine robotic rework cells are classified into two groups: two-machine robotic rework cells with start of line inspection (RRCSI) and two-machine robotic rework cells with end of line inspection (RRCEI). They are also called “Start of Line” testing and “End of Line” testing, respectively. An example of two-machine robotic rework cells with end of line inspection is shown in Fig. 1 for the 2

crankshaft production lines [9]. A particular crankshaft being processed goes through I, the lathe machine M1, the lathe machine M2 and O under this part processing route. After loading the crankshaft to any one of the lathe machines, the robot either waits for the crankshaft to finish its operation or immediately moves to another occupied lathe machine or I for unloading a new crankshaft. The difference between two machines is that the crankshaft is failure-free when it is processing on M1, whereas the crankshaft may fail and need rework when it is processing on M2. Another example of robotic rework cells with inspection is extracted from cluster tools which are employed in processes such as deposition and inspection. The reason why we give this example is that cluster tools actually act as closed mini-environment robotic cells [8]. For fabrication of wafer in cluster tools, atomic layer deposition (ALD) is a process that controls the wafer thickness by repeating the deposition processes with mono-atomic layer precision as needed. The quality of the wafer is often inspected by Spectroscopic Ellipsometry (SE) inspection method in order to check whether a conformal layer is fabricated. Therefore, the thickness of the wafer is inspected by SE method during madding the depositions of each layer on wafer [10]. We should mention that a wide variety of real-life studies of production environments have been conducted on noncyclic production aiming at minimization of the maximum completion time (also referred to as makespan) [11, 12]. However, this study is limited to cyclic scheduling of the robotic rework cell due to its popularity in mass production environments in which the robot is applied for material handling. A cyclic schedule is based on a repeating pattern of part processing [13]. Robotic cells under consideration in this study can do rework processes, and consequently the stochastic nature of the rework process prevents us from applying existing deterministic solution methods for the cyclic scheduling problem. Therefore, the study of robotic cells without rework assumption, which has a deterministic processing route, will be briefly reviewed in this paper. We refer readers to the rigorous analysis of robotic cells with deterministic data elaborated in the book by [14]. It is interesting to address the following recent studies for three kinds of pickup scenarios: free, interval, and no-wait pickup scenarios. For the free pickup scenario with deterministic data, all corresponding studies assumed that the waiting time of the part on the machine is unbounded. The approach in [15] jointly analysed design and scheduling problems when the type of the robot is dualarm. An analytical approach for optimization of the cycle time of two-machine reentrant robotic cells was presented in [16]. The distinguishing feature of the robot in [16] is that it has a one-unit portable 3

buffer on its end-effector that can increase productivity. Adopting a genetic algorithm, the impact of setup time consideration on the optimal cycle was analysed in [17]. Following that, in [18], an ant colony algorithm was adopted for the multi-degree scheduling problem with multiple robots. The metaheuristic algorithm concurrently find the degree of the schedule, a robot task assignment matrix, and cycles of robots that minimize the cycle time. Finally, robotic cells with parallel machines was studied in [19] where an ant colony algorithm and a genetic algorithm were adopted. The computational results of this study have shown that the genetic algorithm has a higher performance in comparison with the ant colony algorithm. Additionally, a large number of works can be found in the literature for problems with interval and nowait pickup scenarios [20]. The reason behind this interest in interval and no-wait pickup scenarios is that they are more suitable for real-world problems than free pickup scenario which is a simplified version of them. Let us initially give a brief literature review on the recent problems with the interval pickup scenario. The interval pickup scenario is commonly seen in chemical industry where the waiting time of the part on the machine is bounded within a pre-defined time interval. More precisely, an applications of such cells can be found in the hoist scheduling problem on an electroplating line. For this problem, a set of electroplating baths with various chemicals are serially employed to layer a part. It should be noted that the automatic hoist, which can be a robot, transfer parts between chemical baths . Each particular electroplating bath has a processing window constraint for which a part is allowed to stay in a bath. Amraoui et al. [21] proposed an exact solving method to minimize the cycle time of r different part-jobs. A comparative analysis in [21] enabled insightful evaluation of the throughput improvement of electroplating by considering r-cyclic scheduling. In a similar study, mixed integer linear programming algorithm were formulated by Amraoui and Elhafsi [22] to minimize the makespan. Major goals in their study were to achieve higher productivity and product quality, and therefore a heuristic was developed to obtain the optimal hoist move sequence. Che et al. [23] considered optimization of the robustness in a cyclic hoist scheduling problem. They developed a bicriteria mixed integer linear programming algorithm to optimize cycle time and robustness of the problem concurrently. The result of their study shown that the problem has unlimited Pareto efficient solutions and also there is a direct relationship between the cycle time and the robustness. Yan et al. [24] minimized the cycle time and the transportation cost concurrently in a reentrant electroplating line in which parts revisit some particular baths. It should be emphasized that a reentrant electroplating line has a deterministic processing route although the processing route can be complex. 4

As our best knowledge, there is no earlier research in stochastic hoist scheduling problem due to its complexity. Only, a stochastic hoist scheduling problem with stochastic transportation times was analysed by Fleury et al. [25]. In particular, a metaheuristic-based method is suggested in their research to find schedules for which the stochasticity negative effects are low. The no-wait pickup scenario is a special case of interval pickup scenario in which the waiting time of the part on the machine is zero. For the no-wait pickup scenario with deterministic data, Paula et al. [26] established an approach to instruct a robot specifically employed in the process of manufacturing wings of transport aircrafts. As a case study, Brauner et al. [27] verified that surface-to-surface radar applications should be simulated by a no-wait robotic cell because their first operation is a signal transmission and their second operation is reflected signal receiving with no gap. In another study, a robotic cell with multiple robots was studied by Shabtay et al. [28] to minimize cycle time and the robot selection cost simultaneously. We refer readers to a recent survey of scheduling problems with no-wait assumption elaborated in [29]. The survey classified no-wait scheduling problems according to shop environments, and then it described how advanced manufacturing systems such as robotic cells should be modelled under a no-wait condition. Although there are many studies on deterministic robotic cells with interval and no-wait pickup scenarios, the problem of determining the optimal robot move cycle of such robotic cells under prescience of stochasticity is an open problem. Under this condition, many issues remain open for research. One may consider stochastic machine failures, stochastic processing times, or stochastic processing route under interval and no-wait pickup scenarios as they represent all stochasticity issues typically encountered in practice. The study of stochastic machine failures, stochastic processing times is outside the scope of this research since we specifically focus on stochasticity resulted from inspection processes. The analysis of stochastic robotized cells has a fragmented history of development in the literature. Some studies of stochastic robotic cells concentrated on two-machine cells which operate under a production system with machine failures and repairs [30,31], whereas some others touched on the robotic cells with stochastic processing times [32,33]. To our best knowledge, there is a lack of research on the stochastic part processing route although it is applicable to many practice-oriented scenarios. Note that the stochastic part processing route, which is named closed loop manufacturing in some industries, is actually possible when a set of sensors is integrated into at least one of the 5

machines. We will provide more details of the concept of robotic rework cells later. Let us now summarize contributions of our study as follows:



We present a proof of dynamicity of the problem of determining the optimal one-unit cycle for robotic rework cells with more than two machines. In contrast, we also prove that the problem is not dynamic for two-machine robotic rework cells, and the pickup scenario has no impact on the result of this theorem.



We define three kinds of stochastic order relations and sort them through strength. Based on these relations, we find the dominancy regions of two feasible one-unit cycles by an analytical method. Then, we extend the result to a mass production environment under free pickup scenario.



We extend results for the interval and no-wait pick up scenarios as two well-solved classes of the problem. For the sake of generalization, we graphically represent the linkage between optimality regions of feasible cycles of different pickup scenarios to provide an integrated framework.

As the most similar study to this paper, Geismar and Pinedo [33] analysed the effect of stochastic processing times on the throughput and the scheduling of the robotic cells. However, aforementioned contributions make it clear that the current paper considerably extends their work. Firstly, Geismar and Pinedo [33] limited their search to one kind of stochastic order relations which is the weakest and only compare expected throughput of different cycles with each other. In other words, from a comparative point of view, two other kinds of stochastic order relations are ignored in [33] although these two kinds of stochastic order relations are stronger. The current paper not only focuses on stochasticity in robotic cells, but also it provides details of the concept of inspection in robotic cells for the first time in the literature. Consequently, in addition to comparing cycles with each other, it is important to understand the strength of the comparison. Secondly, the free pickup scenario is the only pickup scenario considered in [33] to optimize a robotic cell, whereas we provides an integrated framework summarizing both feasibility and optimality results for all pickup scenarios mentioned in the literature: free, interval, and no-wait pickup scenarios. Therefore, this paper presents a full package for solving the problem. Finally, in comparison with [33], it is more straightforward to extend results of this paper 6

for specific work related to inspection process. For instance, we can extend results of this paper for the situation in which a limited number of rework processes, say three times, is permitted for a part. Then, the part will be counted as a scrap if the inspection result shows the part’s failure occurrence three times. The overall structure of this paper is as follows. Section 2 contains fundamental concepts related to stochastic robotic cells. Sections 3 and 4 are dedicated to RRCSI and RRCEI scheduling problems to cover “Start of Line” and “End of Line” testing approaches for a cell with free pickup criterion, respectively. Following that, results are extended for interval and no-wait pickup scenarios in Section 5. Finally, Section 6 concludes the paper with perspectives.

2

Definitions, notations, and basics

We begin this section by first giving a precise definition of each problem class, and then summarizing the notations used in robotic rework cells to find the expected partial cycle times. The concept of the robot activity is one of the best tools to consider a cyclic formulation of the robot movement. In a twomachine robotic rework cell, the robot activity noted Ai, i ={0, 1, 2}, corresponds to the following sequence of actions: 1) The robot takes a part from Mi if the inspection process discovers no error. 2) The robot immediately carries the part to Mi+1. 3) The robot immediately completes the activity Ai by loading the part onto Mi+1. Consistent with [7], a particular n-unit cycle can be characterised by a permutation of activities in which any activity is repeated exactly n times. Since A2 is one of the activities with n repetition, we can conclude that an n-unit cycle is able to produce n final products. Therefore, the cycle time is per unit cycle time or the time to complete one part in a cyclic behavior. An important point about the permutation is that the rework cell must return to the initial state at the starting of the permutation after completing it. In addition, scheduling of cells must be constrained to the deadlock-free subregion of permutations. The deadlock-free schedule can be guaranteed only if the following activities are executed:



The receiving server and sending server must be empty and loaded before the load/unload process, respectively [34]. 7



The robot can load a part to Mi only if the last process of the part was performed by Mi-1, i ={1, 2, 3} [34].



Since the part’s failure occurrence is possible, additionally, it is forbidden to unload the machine when it’s current part needs rework.

It should be noted that the number of n-unit cycles of a particular cell depends on the first two deadlock prevention instructions, whereas the last instruction only increases the cycle time. The simplest case of n-unit cycles is one-unit cycles where exactly one completed part leaves from the cell after the cycle’s execution. Let us fix A1 in the last position, so the order of other activities can alternatively be A2, A0 or A0, A2. This results in two cycles, namely S1= A2, A0, A1 and S2= A0, A2, A1. It is necessary to mention that we choose A1 as the last activity of both cycles since it is the only activity that is able to act as switching point from one cycle to another. In other words, the only common state between S1 and S2 is exactly the moment when the robot loaded a part on M2 and waits at M2 until it receives the next order by the robot controller. The key note here is that any n-unit cycle is actually a combined sequence repeating S1 exactly q times and also S2 exactly n-q times in each iteration of the cycle. Therefore, it is expected that per unit partial cycle time is a convex combination of the expected partial cycle time of S1 and S2 as two given corner points. This means that the expected partial cycle time of one of cycles S1 or S2 always dominates the expected per unit partial cycle time of any given n-unit cycle. As a result, it is enough to limit our search to S1 and S2 [35]. Now, let us use the following parameters derived from [33] throughout the text:

ε

The load/unload time of machines (or buffers) by the robot

δ The required time for traveling between adjacent location pairs (I, M1),(M1, M2), and (M2, O) mi The number of sensors installed into Mi, i ={1, 2} a

The processing time of the first machine

b

The processing time of the second machine

Also, there are two variables, namely

and

, defined as follows: 8

The partial cycle time of Sj for the kth part fed to the rework cell, j ={1, 2} and The waiting time of the robot at Mi for the kth part fed to the rework cell, i ={1, 2} and

The notation

is useful for definition of three different pickup scenarios. This waiting time means

that the robot has to wait for machine i to finish processing if it reaches the machine before finishing the operation. However, instead, it is possible that the robot reaches the machine i after the machine finishes the part processing. Under this condition, the part has to wait for the robot to reach and remove it. This waiting time of the part can provide a framework for classification of pickup scenarios as follows:



The waiting time of the part on the machine is unbounded: this case is named free pickup scenario in which the completed part can stay indefinitely on the machine before removing by the robot. Hence, the waiting time of the part can be shown by interval [0, + ), i={1, 2}.



The waiting time of the part on the machine is bounded: this case is named interval pickup scenario in which the waiting time of the part at Mi, i={1, 2}, is bounded within a pre-defined time interval. The interval is [0, ) for M1 and [0, ) for M2. Therefore, the maximum wait times of the part on M1 and M2 are



and , respectively.

The waiting time of the part on the machine is zero: it is no-wait pickup scenario in which the part must be removed from Mi, i={1, 2}, without delay and carried to the next machine, Mi+1.

It is noticeable that different pickup scenarios do not affect the number of n-unit cycles although they may affect the cycle times. More precisely, the waiting times of cycles change according to the above description of pickup scenarios and this may also change the total cycle times. We can simply extend this result for one-unit cycles S1 and S2 in robotic rework cells with two machines, and say that the permutations of S1 and S2 are independent from pickup scenarios. We conduct our analysis of the two-machine robotic rework scheduling problem under the assumption that the inspection process is error-free (perfect). Nonetheless, it is sometimes possible that the 9

inspection processes is recognized as an error-prone (imperfect) one. The importance of an error-prone inspection varies according to the impact it may have on the industry for which the robotic cell is applied. Although this impact can be ignore in many industries, there is a clear need for a high accuracy of the inspection in high-tech industries such as aircraft and nuclear industries. Consequently, it is necessary to give some general notes on the effect of accuracy of inspection process on the robotic rework cells before determining the optimal cycle. As mentioned in [36], in terms of hypothesis testing, two type of errors namely type I and type II errors may occur when the inspection process is errorprone. Type I error occurs if inspection process wrongly rejects a part that should be accepted, and type I error occurs if inspection process wrongly accepts a part that should be rejected.

Remark 1. For a robotic rework cell, the accuracy of the inspection process has no effect on the analytical method in order to find the optimal robot move cycle.

Obviously, Remark 1 is true in that the analytical method here is indeed a comparative analysis between partial cycle times of of S1 and S2. The accuracy of the inspection process can change the partial cycle time of each of S1 and S2. However, the ammount of this change is same for both of these cycles, and therefore it has no effect on the process of optimal cycle selection when we compare them. This is true even when the number of machines in the cell is more that two since, likewise, we should compare a set of cycles in orther to find the optimal one. So, it is not a matter how much their cycle time change when the amount of this change is same for all of them. In other words, we specifically provide a sequence-oriented analysis in this paper and inspection-oriented analysis of robotic rework cells is out of is outside the scope of this research. However, the following remark is borrowed from a general study in [37] to provide a specific insight into how an inspection-oriented analysis of robotic rework cell can be performed, especially in practice. Although it assume a single-sensor inspection sysmtem, it results can be esily extended for a multi-sensor system.

Remark 2. Inspection policy can be classified according to the accuracy of inspection which is needed for the implemented industry as follows:

10



Multi-test rejects policy: the second (or more depending on the demanded accuracy level) inspection is performed on the part rejected in the first inspection process. It needs two (or more depending on the demanded accuracy level) successive rejections before the part be reprocessed by the current machine. However, the part can be uploaded by the robot immediately after receiving the first acceptance.



Multi-test accepts policy: the inspection process of the part needs two (or more depending on the demanded accuracy level) successive acceptances before transferring the part to the next machine by the robot. Nonetheless, a rejection by each of the multiple inspection repetitions can be cause of a rework process on the current machine.



Single-test policy: here, a part is inspected once. When the sensor identifies no defect the part is transferred to the next machine whereas the part must be reworked if the sensor identifies a defect.

Remark 2 gives us an insight into how we can justify the accuracy level of inspection for each indusrty by changine the number of required acceptance (or rejection) befoe passing a machine (reworking). The inspection polices in Remark 2 represent accuracy level in industries that go in opposite directions. The goal of a multi-test rejects policy is to decrease type I error and equivalently internal failure, but the goal of a multi-test accepts policy is to decrease type II error and equivalently external failure. The internal failure (or cost) here can be interpreted as the energy consumed by the machine and the availability time of the machine for the next part. Likewise, the external failure (or cost) can be interpreted as the quality of outgoing parts. Thus, we conclude that implementation of the multi-test rejects policy is appropriate for industries such clothing industry where there is no obligation to have a completely accurate inspection process and quick manufacturing plays a more important role. Also, the implementation of the multi-test accepts policy leads to a high accuracy of inspection in industries such as aircraft and nuclear industries where occurrence of a failures has fatal consequences. On the contrary to these two policies, the single-test policy is something in between that covers most real-world robotic cells and it can decrease inspection costs. Accordingly, considering Remarks 1 and 2, we assume throughout this paper that inspection processes are error-free, and it is required to perform the inspection process once to recognize whether a rework process should be performed on the part or not. At the end of this section, we should emphasized that we cannot be assured that the robotic rework cell operates in steady state as a result of its dynamic nature. In this regard, we define the variable C(A1 , k) 11

as the completion time of the kth implementation of A1 to explain why there is no guarantee of cyclic behaviour of the cell [38]. Obviously, C(A1, k)-C(A1, k-1) does not remain constant for

, and it

is vital to carry out a separate analysis for any C(A1, k)-C(A1, k-1), which is named the partial cycle time

3

for the kth implementation of Sj.

Sequencing of robot activities in RRCSIs with free pickup scenario

We start this section by reducing the scheduling problem to a problem where pickup scenario is always free. Under this condition for the RRCSI case, a part processing on M1 is monitored to detect the presence of different types of defects using a multi-sensor system. Each sensor i, i

{1, 2,…,m1},

detects the part’s failure with specified probability q1i each time the part is processing on M1, and therefore this individual sensor identifies no defect in each time inspection with probability p1i=1-q1i. This means that the inspection result of each sensor i after each time processing of the part k on M1 is a random variable

which follows a Bernoulli probability distribution of parameter p1i. Needless to say

that here we have a sub-inspection set of m1 different sensors where there is no dependency between them. This means that the random variables

,

, …,

that represent the inspection results of m1

sensors are independent Bernoulli trials that are not necessarily identically distributed. We know the part cannot pass the multi-sensor inspection process even if one of the sensors detects the part’s failure. Following that, the generalized Bernoulli distributed variable

supporting the success of the multi-

sensor inspection system of the part k is:

P(

=1)=

P(

=0)=1-

(1)

The analysis of M1 with multiple sensors integrated into it can be converted to the analysis of M1 with a single sensor using the above equations. The important random variable for us here to determine is the number of inspections performed by the newly established single-sensor inspection system before passing all sub-inspections. Clearly, for the kth parts interred to the cell, this number can be represented by a random variable

which is associated with a geometric distribution with parameter p1= 12

and the time between two inspections equals a. The reason for this intuition is that the geometric distribution is defined as a discrete distribution counting the number of Bernoulli trials until the first success. The memoryless property implies that the current sub-inspection process is performed independent of the number of completed sub-inspections before.

Lemma 1. Having a RRCSI with free pickup scenario, the partial cycle time

for kth

implementation of S1 is:

=6ε+6δ+a

+b

(2)

Proof: The kth implementation of S1 includes the tasks below: the robot unloads (k-1)th part from M2 after a full waiting at this machine to finish the processing of the (k-1)th part (b+ε). The robot carries this part from M2 to O, and then moves backward to I to pick up the kth part and load it on M1 (3ε+5δ). In this stage, the robot undergoes a random waiting depending on the number of rework processes performed by M1, and eventually it picks up the part k and loads it on M2 (a

+2ε+δ) □.

Lemma 2. Having a RRCSI with free pickup scenario, the partial cycle time for kth implementation of S2 is the random variable TkS2 as:

=6ε+8δ+max{0, a

-(2ε+4δ), b-(2ε+4δ)}

(3)

Proof: Here, the robot visits I, M2, and M1 in the kth implementation of S2, respectively. Our goal is to find the time required for all intervening activities which are implemented between two consecutive loadings of M2. The robot initially moves backward to I to unload the kth part and load it on M1 (2ε+3δ). In the second phase, it returns to M2 to unload the (k-1)th part and transfer it to O after a partial waiting at M2 (2ε+2δ+

). Likewise, the empty robot returns to M1 to unload the kth part and

transfer it to M2 after a partial waiting at M1 (2ε+3δ+ 13

). Thus,

=6ε+8δ+

, and we have:

Recall that

and

are waiting times of the robot, not parts k-1 and k. Accordingly, these two

variables can be equal to zero since the pickup scenario is free and the completed part can stay indefinitely on the machine to be removed by the robot as soon as it reaches the machine □.

Our task now is to find an optimal long-term production strategy considering Lemmas 1 and 2. Before proceeding with the following theorem, let us notice that it is pickup scenario-independent, and therefore it is applicable for free, interval, and no-wait pickup scenarios. The reason behind this intuition is that the dynamic nature of a cell only can change cycle times, not the number of cycles.

Theorem 1. For two-machine robotic rework cells, there is no dynamic state change from S1 to S2, and vice versa. In contrast, robotic rework cells with over two machines always have dynamic behaviour.

Proof: in regard with the first segment of this theorem, as mentioned before, the starting point of each one of S1 and S2 is the moment when the robot loaded a part on M2 and waits at M2 until it receives the next order by the robot controller. If M1 was loaded at this given moment, the stochastic modelling was essential due to the fact that the extent to which each part has been processed on M1 at the starting point of each partial cycle is random. Fortunately, M1 which has a random number of rework on the part is empty at this common state. Therefore, there is no relationship between the optimal partial cycle in the current state and the number of rework performed on the part in the previous partial cycle. Regarding the second segment of this theorem, let us initially consider three-machine case. For this case, there in an extra machine M3 is comparison with two-machine case. Therefore, activity A3 should be added to the list of activities so that the following tree scheme shows all possible cycles.

14

Three-machine robotic rework cells

Two-machine robotic rework cells S1= A2, A0, A1

S1= A2, A3, A0, A1 S3= A3, A2, A0, A1 S4= A2, A0, A3, A1 A3

S2= A0, A2, A1

S2= A3, A0, A2, A1 S5= A0, A2, A3, A1 S6= A0, A3, A2, A1 A3

Figure 2. A tree scheme for cycle generation of three-machine robotic rework cells. In Fig. 2, it is clear that six possible cycles are originated from S1 and S2. More specifically cycles S1, S3 and S4 rise from S1 in two-machine case. Also, cycles S2, S5 and S6 rise from S2 in two-machine case. It should be emphasized that all S1, S3, S4 and S5 can reach a state where the robot has just completed loading a part on a machine and the rest of machines are unoccupied. This means that these cycles get a chance to exhibit non-dynamic behaviour. The machine is either M1, M2 or M3 for S1; the machine is M3 for S3; the machine is either M2 or M3 for S4; and finally the machine is M2 for S5. However, S2 and S6 always have dynamic behaviour since their permutations shows that cell never reach an state in which there is only one part is processing. Therefore, three-machine robotic rework cells always have dynamic behaviour. An indirect result from the aforementioned tree scheme is that possible cycles of an m-machine robotic rework cell originate from cycles of a rework robotic cell with m-1 machines. We only need to take into account activity Am. Therefore, the m-machine robotic rework cell will certainly has dynamic behaviour if the robotic rework cell with m-1 machines has it. This means that all robotic rework cell with over three machine also have dynamic behaviour □.

This results in two cycles, namely S1= A2, A0, A1 and S2= A0, A2, A1. It is necessary to mention that we choose A1 as the last activity of both cycles since it is the only activity that is able to act as switching point from one cycle to another.

15

This theorem in particular implies that all k, where k→+ , have a similar production behaviour. As a direct result of this, it can be concluded that it is enough to compare a particular implementation of S1 and S2, for example the kth implementation of both of them, and then extend the result to a mass production environment with an infinite period of time. To point out this subject more clearly, let us find the dominancy regions of any one of S1 and S2 for kth implementation. We must initially recall some concepts related to stochastic dominancy. There are three kinds of stochastic order relations that are sorted through strength [39]: 1) Absolute dominance (AD): we say that over

≥

, written

, only if P(

≥

)=1 and

order stochastic dominance (FSD): we say that only if P(

>θ)≥P(

>

is absolute dominant

be satisfied for at least one

is first-order dominant over

, written

>θ) for all θ. 3) Second-order stochastic dominance (SSD): if

order dominant over

, then E(

determine the regions of AD of

. 2) First≥st

,

is second-

)≥E(

). Since AD has priority over other dominance relations, we

and

in the first stage.

Theorem 2. Under a RRCSI, it is absolutely better to instruct the robot for implementation of S2 in an infinite period of time if

.

Proof: we start with considering P( (2ε+4δ)})= P(a

+b≥2δ+max{0, a

≥

≥

+b≥6ε+8δ+max{0, a

-(2ε+4δ), b-(2ε+4δ)})=P(a

necessary to recall that the discrete variable equals 1. Therefore, we have P(

)= P(6ε+6δ+a

)=P(

+b≥2δ) =P(

-(2ε+4δ), b≥

). It is

is associated with a geometric distribution and at least ≥

)=1 when

□.

Due to the fact that FSD and SSD are our second and third priorities, we determine the regions of the FSD and SSD of

and

, respectively, in the Theorems 3 and 4.

Theorem 3. There is no FSD relationship between

and

16

executed for a RRCSI.

Proof: It is enough to prove that the inequality P( the intersection of

>θ) is not satisfied for all θ. Obviously,

>θ)≥ P(

is 6ε+6δ. Therefore, θ can be replaced with λ=θ-(6ε+6δ), and this yields

and

the following results where the cumulative distribution function equal to P( P(

of the geometric variable

and also = max{0, b-(2ε+4δ)} for the sake of simplicity.

>θ)=P(a

+b>λ)=P(

)=1-F(

>θ)=P(2δ+max{0, a

)=

-(2ε+4δ), b-(2ε+4δ)}>λ)= P(

)+ P(

) P(

) P(

)

None of α, λ, and δ is a random variable. So, any one of two following cases may occur for P( 1. α>λ-2δ → P(

>θ)=P(

2. α≤λ-2δ → P(

)+P(

) P( ) =1-F(

)=P(

)=

If the case 1 is taken into consideration, P(

>θ)≤P(

>θ), whereas P(

second case. Thus, it is impossible to satisfy inequality P( >θ)≤P(

>θ):

)=1

>θ)=P(

)= P(

P(

is

>θ)≥P(

>θ)≥P(

>θ) (or inequality

>θ)) for any θ, and this means that there is no FSD relationship between

As mentioned before, SSD order relations get through the expected values of

and

that, before proceeding with the next theorem, we must calculate E(

).

) and E(

>θ) for the

and

□.

. As a result of

Lemma 3. For the case of a RRCSI, the expected values of cycle time of S1 and S2 are given by:

= 6ε+6δ+

+b

(4)

=max{6ε+8δ, b+4ε+4δ}+

(5)

17

Proof: The fact that

is associated with a geometric distribution with parameter p1=

that

implies . Regarding

, it contains a triple-sided max term where 0 and b-(2ε+4δ) are fixed and a

-(2ε+4δ) is

variable. For simplicity, hereinafter β=4ε+4δ and γ=max{2ε+4δ, b}, and therefore we have the calculation of the conditional expected value below: =E(6ε+8δ+max{0, a )P(

)+E(max{γ, a

(

)+a(

-(2ε+4δ), b-(2ε+4δ)})=β+ E(max{γ, a

})=β+E(max{γ, a

} |

) +a(

) P(

)

)=β+γF(

) =β+γ+(a

+

}|

)(1- F(

-γ) □.

=max{6ε+8δ, b+4ε+4δ}+

≤2δ in a RRCSI, then

Theorem 4. if b+ b+

>2δ, then

))=β+γ

is second-order larger than

is second-order smaller than

; else if

.

Proof: The proof will be presented in a structure similar to that of Theorem 3: we use Equations (4) and (5) and initially consider the case in which E( which E(

)
1. E( 2. E(

)≤E( )>E(

) ≥E(

). If we relax the first bracket in E(

) → 2ε+2δ+b+

), and then extend the proof for the case in ), then we have:

≤max{2ε+4δ, b}+

) → 2ε+2δ+b+

>max{2ε+4δ, b}+

→ b+

)≤2δ → b+

)>2δ□.

Let us consider a couple of examples to shed light on application of stochastic order relations for optimization of different robotic rework cells. 18

Example 1: As the first data set, we assume that we have two operations to be performed by M1 and M2. In the cell, three sensors are installed onto M1, and they identify no defect in each times inspection of the part with the probability p11=0.93, p12=0.96, and p13=0.98, respectively. This means that we can convert the three-sensor inspection system into a single-sensor inspection system with p1=0.93*0.96*0.98=0.875 using Equation 1. For any particular part, assume the processing times are fixed as a=10 and b=11, the load/unload time of the machines is ε=1, and the robot takes δ=10 to move between machines. Considering these parameters for any particular part k, we can use Equations (2) and (3) to calculate cycle times for S1 and S2 as follows: =77+10 =86+max{0, 10 , where

-42}

can be rewritten as:

We know that the integer random variable

has a geometric distribution (meaning

), and it is

the number of inspection performed by the single-sensor inspection system before transferring the part k to M2. Obviously,

>

107, and 117 when

is 1, 2, 3, and 4, respectively. Note that

= 77+10

and

since the smallest value that

=

for

can take is 87. In other words,

is 87, 97,

equals 86 for these cases. Likewise,

, respectively. So, the result here is similar to Theorem

2 where it is absolutely better to implement S2 since

.

It is also interesting to find out how much the throughput will be improved if we execute S2 here. Due to the fact that both

and

are random variable depending on

, we have to calculated their

expected value by Equations (4) and (5) to determine the throughput improvement.

and

are approximately 88.4285 and 86.0023. Clearly, these values lead to the throughput improvement equal to (2.4262/88.4285)×100=2.74% if we execute S2.

Example 2: Now, as the second data set, only assume that a is changed to 2 in Example 1. Under this condition, we can use Equations (2) and (3) in a similar manner to Example 1 to show that 19

<

when

, whereas

relationship between

> and

when

. Consequently, there is no absolute dominance

.

In regard with the FSD relationship between

and

, we need to show that P(

>θ)≥ P(

>θ) is

satisfied for all θ as mentioned earlier. Consider the breakpoint θ=86, let us present a counterexample: 1) We assume that θ=80. This leads to: P(

>80)= P(77+2

>84)= P(

P(

>80)= P(86+max{0, 2

>3.5)= 0.1253

-42}>80)= 1

2) We assume that θ=88. This leads to: P(

>88)= P(77+2

>88)= P(

P(

>88)= P(86+max{0, 2

This counterexample shows P(

>5.5)= 0.1255

-42}>88)= P(max{0, 2 >θ)≥ P(

-42}>2)= P(

>22)=0.12522

>θ) is not satisfied for all θ, as mentioned in Theorem 3.

Therefore, there is no FSD relationship between

and

are approximately 79.2857 and 86, and therefore

in this example. However, is second-order larger than

and (similar to

the result can be extracted from Theorem 4). Clearly, we should expect that the throughput improvement be equal to (6.7143/86)×100=7.8% if we execute S1.

At the end of this section, it should be emphasized that Theorems 1 to 4 give an appropriate structure to select the robot’s partial cycle with the maximum expected throughput of RRCSIs, and Lemma 3 helps us to calculate this maximum expected throughput. Clearly, this structure assists industry in both designing and developing basic rework cells.

4

Sequencing of robot activities in RRCEIs with free pickup scenario

For the RRCEI case, the final parts processed on M2 are monitored to detect the presence of different types of defects before delivering them to the customers. Similar to RRCSIs, there is no difficulty with converting the multi-sensor system into a single-sensor system in RRCEIs. Each sensor j, j

{1,

2,…,m2}, identifies no defect in each time inspection with probability p2j. This builds up a sub20

inspection set of m2 different sensors, and the sequence of random variables [

,

, …,

] that

represent the inspection results of m2 sensors with independent Bernoulli probability distributions. Following that, the generalized Bernoulli distributed variable

supporting the success of the multi-

sensor inspection system of the part k is expressed as:

P(

=1)=

P(

=0)=1-

(6)

According to Equation (6), the number of inspection of the kth parts interred to the rework cell performed by the single-sensor inspection system is the random variable geometric distribution with success parameter p2=

which is associated with a

and the time between two inspections

equals b. The reason for this intuition is that the geometric distribution is defined as a discrete distribution counting the number of Bernoulli trials until the first success. The partial cycle times for kth implementation of S1 and S2 are presented in the following lemma, respectively.

Lemma 4. Having a RRCEI, the partial cycle times for kth implementations of S1 and S2 are:

=6ε+6δ+a+b =6ε+8δ+max{0, a-(2ε+4δ), b

(7) -(2ε+4δ)}

(8)

Proof: if we follow the order of tasks performed in Lemmas 1 and 2, once again, we achieve the desired results. The only difference here is that the processing time of M2 is the random variable, not M1. The rest of this proof is easy and therefore omitted □.

Our task now is to find an optimal long-term production strategy considering Lemmas 4.

Corollary 1. Theorem 1 to 4 are also correct for RRCEIs if a and b be swapped with each other in all associated inequalities. 21

Proof: easy and omitted□.

Corollary 2. For the case of a RRCEI, the expected values of partial cycle time of S1 and S2 are:

=6ε+6δ+a+

(9)

=max{6ε+8δ, a+4ε+4δ}+

(10)

Proof: easy and omitted□.

It is worth noting that the result of this section along with the previous section create a parallel mechanism for analysing both RRCSIs and RRCEIs with free pickup scenarios. Actually, we must make it clear that the main purpose of inspection is to control the quality of parts at the early stage of production to decrease production complexity, or control the quality of parts through a final inspection at the last stage. According to the first or the second priorities, a RRCSI or RRCEI with free pickup scenarios can be designed, respectively, and the result of Sections 3 and 4 can be easily applied for optimizing the performance. However, we still need to extend results of these sections for cases in which the pickup scenario is either interval or no-wait.

5

Analysis of robotic rework cells interval and no-wait pickup scenarios

We begin with generalizing results to the case of interval pickup scenario and then provide a set of guidance notes to determine the optimality region of S1 and S2 for rework cells with no-wait pickup scenario. Let us first provide a more precise definition of the interval pickup scenario. A wide variety of robotic rework cells, in particular those used in steel, chemical and plastic industries, work under interval pickup scenario. For these cells, we keep the temperature of the part within a fixed range after completing any particular process, and hence, the waiting time of the part at the machine must be 22

bounded [40]. Otherwise, the part will be certainly scrapped because it overwaits for the robot to unload it. Under this condition, it seems more economical to impose waiting time’s restrictions in order to have no scrap in the production line, especially for expensive parts.

Theorem 5. Theorems 2, 3, and 4 (Corollary 1) hold(s) for a RRCSI (a RRCEI) with interval pickup scenario if and only if a+ ≥2ε+4δ and b+ ≥2ε+4δ. Otherwise, S1 is the optimal cycle of rework cells under interval pickup scenario.

Proof: the definition of interval pickup scenario makes it clear that it is a special case of the free pickup scenario in which neither M1 nor M2 can keep the part for an infinite time (

and

).

This means that converting the pickup scenario into interval has no impact on the optimality region of S1 and S2. Nonetheless, we should check whether these cycles are always feasible. For S1, the robot loads the part to the machine (M1 or M2) and waits in front of it throughout that the part processing. Therefore, this cycle is always feasible (and also the optimal cycle if S2 be infeasible). In contrast, it is possible that S2 be an infeasible cycle if a+ <2ε+4δ or b+ <2ε+4δ. More particularly, the proof of Equations (3) shows that the total waiting time of the robot in a RRCSI is . In this max term,

and

are

waiting times of the part on M1 and M2 since they are the reciprocal of the waiting time of the robot on machines. This yields the result: ≥ ≥

→a

+ ≥2ε+4δ → a+ ≥2ε+4δ

→ b+ ≥2ε+4δ

Also, considering the proof of Equations (8) gives us same results for a RRCEI. Accordingly, Theorems 2, 3, 4 and Corollary 1, which are related to the optimality conditions of S1 and S2 hold □.

The results of analysing the rework cell with interval pickup scenario is even extendable for more specific pickup scenarios. One of these well-solved pickup scenarios originating from interval pickup scenario is no-wait pickup scenario. Within a manufacturing context, the no-wait means that the robot is instructed to unload the part immediately after completing its operation on the machine [41]. 23

Subsequently, we can consider it as a special case of the interval pickup scenario in which

and

.

Corollary 3. The cycle times of S2 for RRCSIs and RRCEIs with no-wait pickup scenario are 4ε+4δ+max{a

, b} and 4ε+4δ+max{a, b

}, respectively.

Proof: typically, the issues involved in no-wait pick up is that the waiting time of the robot in front of that machine is not zero, or equivalently the waiting time of the part at any particular machine is zero. This implies that (2ε+4δ), b

and

-(2ε+4δ)}= max{a-(2ε+4δ), b

for Equations (3), and therefore max{0, a-(2ε+4δ)}. This is enough to prove the first part of this

corollary. The proof of the second part of this corollary is completely analogous to above derivation and therefore it is omitted □.

Theorem 6. S2 is the optimal cycle for both two-machine RRCSIs and RRCEIs if it be feasible. Otherwise, S1 is the optimal cycle of rework cells under no-wait pickup scenario.

Proof: It is enough to prove that the inequality

≥

is always satisfied if S2 is feasible. Therefore,

considering Corollary 3, we have: 1. For RRCSIs: 6ε+6δ+a 2. For RRCEIs: 6ε+6δ+a+b

+b ≥ 4ε+4δ+max{a ≥ 4ε+4δ+max{a, b

, which are always satisfied □.

24

, b} → 2ε+2δ+a } → 2ε+2δ+a+b

+b ≥ max{a ≥ max{a, b

, b} }

Two-machine robotic rework cells

Two-machine RRCSI

Free pickup

Interval pickup

a+ ≥2ε+4δ a+ <2ε+4δ And

Or

b+ ≥2ε+4δ b+ <2ε+4δ S*=S1

Two-machine RRCEI

No-wait pickup

No-wait pickup

a≥2ε+4δ

a<2ε+4δ

A<2ε+4δ

a≥2ε+4δ

And

Or

Or

And

b≥2ε+4δ

b<2ε+4δ

b<2ε+4δ

b≥2ε+4δ

S*=S2

S*=S1

S*=S1

S*=S2

Free pickup

Interval pickup

a+ <2ε+4δ a+ ≥2ε+4δ Or

And

b+ <2ε+4δ b+ ≥2ε+4δ S*=S1 S*=S2

*

S =S2 b+

>2δ

S*=S2

b+

a+

≤2δ

≤2δ

S*=S1

S*=S1

a+

>2δ

S*=S2

Figure 3. A summary of the results of robotic rework cell scheduling problems

Fig. 3 also summarizes the feasibility and optimality results obtained in this research. This figure shows an integrated framework for both start of line and end of line inspections in small-scale robotic rework cells with three common pickup scenarios. This framework makes a significant contribution to real-life applications of industrial automation, and it assists manufacturers in designing automated inspection cells. In more detail, it helps companies to remain competitive since suggested cycles in this framework minimize the cycle time for the first time in the literature related to robotic cells with inspection process.

6

Concluding remarks

Some of the small-scale robotic cells, especially two-machine ones, are still used in automated manufacturing systems. The analysis of these cells is not an easy task if stochastic variables such as 25

inspection processes through a multiple-sensor inspection system are taken into account. Therefore, an analytical method for minimizing the partial cycle time of such cells has been developed in this study. Regardless of the pickup scenario, we have proven that it is possible to reach a steady state of these cells which have a dynamic behaviour, and then maximize the expected throughput of associated cells. Comparing the two-machine robotic rework cell under free pickup scenario with the same robotic cell without a rework assumption, it has been realized that the performance of the partial cycle S2 is improved due to the fact that the average time of producing a part is definitely increased. With regard to the interval pickup scenario, we conclude there is no guarantee that S2 (in comparison with S1) will be an optimal cycle when it is feasible. Nonetheless, it is enough to check whether S2 satisfies feasibility conditions for rework cells with interval and no-wait pickup scenarios to conclude that it is the optimal cycle. Further work should be done to consider the limited number of permitted rework processes for a particular part. If the part could not pass the inspection process even after this number of rework processes, it must be considered as scrap, not as a final product. Clearly, this assumption makes the analysis of the system more complex.

7

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29