Push and Pull Strategies in Automated Agile

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Push and Pull Strategies in Automated Agile Workforce Environments Wallace J. Hopp, Seyed M.R. Iravani and Biying Shou Department of Industrial Engineering and Management Sciences Northwestern University, Evanston, IL 60208, USA

Abstract Spearman and Zazanis (1992) demonstrated the superior performance of pull over push control of traditional production lines consisting of manual workstations staffed by dedicated operators. Specifically, they showed that pull systems have less congestion and are inherently easier to control. In this paper we show that the superior performance of pull over push carries over to manufacturing cells with a mix of manual and automated equipment staffed by a single agile (cross-trained) worker. We first derive an optimal control policy and evaluate practical heuristics for a cell operating under a CONWIP protocol. Then we compare the performance of CONWIP (pull) lines with equivalent push lines. Finally, we examine the impact of automation configuration on cell performance and conclude that CONWIP also offers greater design flexibility than push. Keywords: Workforce agility, CONWIP, Markov decision processes, Automation.

1

Introduction

Spearman and Zazanis (1992) described three advantages of CONWIP (Constant Work In Process) control of job releases relative to traditional push control: (a) observability, (b) efficiency and (c) robustness. The first of these is obvious; WIP, the control in a CONWIP system can be counted directly, while release rate, the control in a push system, must be set relative to capacity, which can only be estimated. Efficiency (the ability to achieve the same throughput with a lower average WIP level) and robustness (insensitivity to errors in control) are more subtle and have since been accepted as important principles underlying the effectiveness of pull systems (see Hopp and Spearman (2000), Chapter 10). To demonstrate these advantages, Spearman and Zazanis relied on comparisons of conventional open and closed queueing networks. This modeling assumption effectively restricted their conclusions to production lines in which the operators can be neglected (e.g., fully manual lines with operators dedicated to stations). However, recent years have seen a shift in manufacturing away from traditional manual lines. This was spurred by pressure along two competitive dimensions: (i) Efficiency: as pursued through 1

lean manufacturing, business process reengineering and other methods for making more with less, and (ii) Responsiveness: as emphasized in time based competition, agile manufacturing, and other methods for meeting diverse customer requirements in a prompt and personalized manner. To achieve these conflicting objectives, firms have increasingly adopted automated machinery (for efficiency) and workforce agility (for responsiveness). In cells with automated machinery, processing a job at a machine may not require the presence of a worker during the entire operation, and thus the agile (cross-trained) worker can operate another machine while the automated machine is running. We refer to systems with these features as Agile Automated Production (AAP) environments. In this note, we reconsider the relative effectiveness of push and pull (CONWIP) in the context of an AAP line staffed by a single cross-trained worker. Although the models required are very different from the simple queueing systems used by Spearman and Zazanis, our results show that their main conclusions of CONWIP efficiency and robustness extend to the more complex AAP environment. Furthermore, because automation and workforce agility introduce new considerations of design (e.g., where to place automation) and control (e.g., how to allocate cross-trained workers), we also consider these issues and their impact on the relative performance of push and pull.

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Literature Review

A significant amount of literature exists on production systems with cross-trained workers. Askin et al. (1999) and Hopp and Van Oyen (2004) provide a survey of these studies in the context of manufacturing cells and workforce agility, respectively. Ostalaza et al. (1990), McClain et al. (1992, 2000), Zavadlav et al. (1996), and Gel et al. (2002) examined worksharing in a variety of situations. Bartholdi, Eisenstein et al. (1996, 1999, 2001) studied bucket brigade lines. Farar (1993), Iravani et al. (1997), Duenyas et al. (1998), Ahn et al. (1999), Andradottir et al. (2001) and Van Oyen et al. (2001) investigated the optimal assignment of flexible labor in tandem lines. While these papers yield many useful insights into the subject of workforce agility, all of them have focused on production environments without automated machinery. Some other work, however, has explicitly modeled automation in agile production systems. For example, Nakade et al. (1997), and Ohno and Nakade (1997) analyzed a serial AAP system. These papers considered AAP lines in which cross-trained workers visit their assigned workstations according to a cyclic policy. 2

(See Section 3.1 for a detailed description of a cyclic policy). They obtained performance measures such as cycle time and worker waiting time under this policy. Desruelle and Steudel (1996) investigated a similar system in the context of work cell design. By modeling the work cell as two interacting queuing networks: an open part/machine network, and a closed machine/operator network, they evaluated machine utilization and waiting times for the operator. In Hopp et al. (2003), we considered AAP cells operating under a push protocol and showed that the capacity of production lines with automatic machines can be significantly lower than the rate of the bottleneck. We also showed that automation is more effective when placed toward the end of a push line rather than toward the front. Finally, we showed that the automation level increases the priority workers should give to a station when selecting a work location. However, because many cellular manufacturing environments make use of pull mechanisms to promote efficiency, it is important to understand AAP cells in pull environments. The literature on pull is ample (for reviews, see Uzsoy and Martin-Vega (1990), and Huang and Kusiak 1996).A key part of the pull literature has focused on the operational benefits of pull relative to push. Again, these papers are all for traditional production lines (i.e., with neither automated machinery nor workforce agility). A natural question to ask, therefore, is how the relative performance of pull and push strategies is affected by the introduction of automated equipment staffed by agile workers.

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Control of AAP Push and CONWIP Lines

In CONWIP lines the control is the WIP level, while in push systems the control is throughput (i.e., job release rate). In other words, in CONWIP the WIP level is set and throughput is observed, while in push, the throughput is set and WIP is observed. However, due to the use of cross-trained workers, AAP lines have one additional control parameter, namely, the worker control policy. When a worker is in charge of more than one station, the control question is how to allocate his/her effort among those stations.

3.1

Worker Control in AAP Push and CONWIP Lines

In Hopp et al. (2003), we proved for a two-station, one-worker push line with one automatic machine that, when the automatic machine is placed upstream, the optimal worker control policy

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(to minimize average WIP per unit time) is of a threshold type; and that the optimal threshold changes with the number of jobs at stations 1 and 2. However, when the automatic machine is placed downstream, the optimal worker control policy is a strict-priority policy, i.e., to load the automatic machine first whenever possible. Threshold-type policies are not easy to implement, especially when the thresholds change with the state of the system. The structure of the optimal policy becomes even more complex in a three-station push line in which an automated machine is located at station 1 or station 2. In order to study the structure of the optimal worker control policy in CONWIP lines, we consider a simple three-station line with one automated machine and one cross-trained worker as our base for comparison. We call the station with the automatic machine, the automated station, and the other stations, the manual stations. The line operates under a CONWIP protocol, according to which a new job is released to the beginning of the line each time a job departs from the end of the line (Hopp and Spearman, 2001). We assume that the automated station is the first station, while the second and the third stations are manual stations.1 For simplicity, we assume that the operation at an automated station includes a manual loading time and an automatic processing time. The machines at the manual stations require the presence of the worker during their entire operations, while the automatic machine requires the presence of the worker only during the manual loading operations. After a job is loaded on an automatic machine, the machine can complete the job without worker supervision. We assume that loading the automatic machine requires an exponential amount of time with mean 1/l1 . Although in practice the actual automated process times themselves may be close to deterministic, occasional interruptions (e.g., failures, adjustments, cleanings, material outages, etc.) will sometimes inflate the effective process time. We approximate this behavior by representing the (effective) automatic processing times as exponential random variables with mean 1/µ1 . We also model the manual process times at stations i = 2, 3 as exponential with meane 1/li . The above assumptions allow us to formulate the problem of finding the operating policy that maximizes the throughput of the line as a Markov decision process (MDP). We define: • System State is (n1 , n2 , s), where n1 and n2 are the WIP levels (including jobs in process) at the first 1 Later we will show that this assumption is without loss of generality because of the closed-loop property of a CONWIP line.

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and the second stations, respectively, while s refers to the status of the automatic machine at station 1. Specifically, s = 1 implies that the automatic machine is performing automatic processing, while s = 0 implies it is not processing a job. • Decision Epochs consist of machine loading completion epochs at station 1, machine processing completion epochs at station 1, job processing completion epochs at station 2, and job processing completion epochs at station 3. • Action Space includes: (i) Idling, (ii) Loading the automatic machine at station 1 (if the machine is idle and the station is non-empty), (iii) Processing a job at station 2 (if there is a job at station 2), and (iv) Processing a job at station 3 (if there is a job at station 3).

Assuming that the WIP level in the CONWIP line is W and the worker can preempt a task to switch between stations, the optimality equation for the MDP with the objective of maximizing the average throughput rate can be expressed as: g + V (n1 , n2 , 0)

=

g + V (n1 , n2 , 1) =

µ1 V (n1 , n2 , 0) Λ   (l1 + l2 + l3 )V (n1 , n2 , 0)   1 I1 [l1 V (n1 , n2 , 1) + (l2 +l3 )V (n1 , n2 , 0)] + max  I2 [l2 V (n1 , n2 −1, 0) + (l1 +l3 )V (n1 , n2 , 0)] Λ   I3 [l3 V (n1 +1, n2 , 0) + (l1 +l2 )V (n1 , n2 , 0)]

µ1 l1 [1 + V (n1 −1, n2 + 1, 0)] + V (n1 , n2 , 1) Λ Λ  (l2 +l3 )V (n1 , n2 , 1)  1 I2 [l2 V (n1 , n2 − 1, 1) + l3 V (n1 , n2 , 1)] + max  Λ I3 [l3 V (n1 + 1, n2 , 1) + l2 V (n1 , n2 , 1)]

where Λ = l1 +l2 +l3 + µ1 , and for i = 1, 2,  0 ; if ni = 0 Ii = and 1 ; otherwise

I3 =



; ; ; ;

Idling Loading at station 1 (1) Processing at station 2 Processing at station 3

; Idling ; Processing at station 2 ; Processing at station 3

(2)

0 ; if n1 + n2 = W 1 ; otherwise.

Note that assuming exponential distribution for operation times helps us formulate the worker control problem as an MDP model and gain insights into simple lines. However, in this note, we will also study more general lines in which operation times are not exponentially distributed. Theorem 1 characterizes the structure of the optimal worker control policy in our three-station AAP CONWIP line. The proof is presented in the On-Line Appendix. Theorem 1: (i) The optimal policy for the three-station line with an automated station 1 is non-idling.2 2

Note that the worker can be forced to idle when all jobs are at station 1, and the automated machine at that station is processing a job.

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(ii) When the automatic machine in non-empty station 1 is not processing a job, the optimal policy is to always load that machine. (iii) When the automatic machine in station 1 is either starved or processing a job, the optimal policy is to process a job, if there is any, at station 3 which directly feeds the automated machine. As Theorem 1 describes, the principle of optimal worker control is very simple in our AAP CONWIP line. First check station 1 (the automated station); load it if there is a job and the machine is not automatically processing. Then check station 3 which directly feeds station 1 and work there if a job is available; otherwise, process a job, if there is any, at station 2. Idle only when none of the above conditions are satisfied. It is clear that this optimal control policy is easier to implement than a threshold-type policy which is optimal in push systems. In other words, from the perspective of optimal control, single-worker AAP CONWIP lines are inherently easier to control than their AAP push counterparts. In larger lines, in which several stations are automated and operation times may not be exponentially distributed, the optimal worker control policy has a complex structure. Therefore, in practice, two simple heuristic policies, namely cyclic and fixed-priority policies are often used. (i) Under a cyclic policy, the worker attends workstations in a cyclic fashion. When the worker arrives at an automated station, she waits for the machine to finish processing the preceding job (if it is not finished), then unloads the processed job, sends it to the next machine, loads the new job on the machine, switches the machine on, and then goes to the next machine. If the worker arrives at a manual station, she processes the job at that station. If the worker arrives at a station with no jobs, she immediately switches to the next station in her cycle. (ii) Under a fixed-priority policy, the worker attends stations in a fixed order. The order specifies stations from the highest priority to the lowest. When the worker arrives at a station, she works at that station until she becomes idle, whereupon she switches to the next lower-priority station. If a job becomes available in a higher-priority station (e.g., an automatic machine finishes processing and requires unloading), the worker interrupts her work at a station and immediately switches to the higher-priority station.

In the rest of the paper, in addition to the optimal worker control policy, we also study CONWIP and push lines under both cyclic and fixed-priority policies.

3.2

Efficiency and Robustness of Push and Pull

We now compare the relative performance of CONWIP and push AAP systems. Our analysis parallels that of Spearman and Zazanis (1992) for traditional CONWIP and push lines (i.e., lines 6

with no automated machinery or workforce agility). In particular, we consider a simple static optimization model that balances the cost of lost production with the cost of added WIP, via a simple profit function of the form: P rof it = p T H − h W IP, where p is the marginal profit per job, T H is the throughput of the line, and h is the cost for each unit of WIP (including costs for increased cycle time, decreased quality, etc.). As in Spearman and Zazanis (1992), we assume that the relative holding cost h/p equals 0.01. For a CONWIP line, throughput is a function of WIP, so we seek a value of WIP that maximizes profit. In contrast, for a push line the average WIP is a function of the release rate, so we find the value of T H (i.e., the job release rate) that maximizes profit. The question of efficiency is concerned with the relative performance of the two release protocols when their controls are optimized, while robustness is concerned with how fast profit degrades when WIP or TH are set at suboptimal levels. Figure 1(left) compares the profit curves of a two-station CONWIP system and a push system in which the second station is automated with an automation level of 80% (i.e., 80% of the total operation in the second station is automated). The manual operation time at station 1, and the total operation time (i.e., loading and automatic processing time) are set to one unit to represent a balanced line. We used our MDP models to obtain the throughput in CONWIP line and the MDP model from Hopp et al. (2003) to compute the average WIP in the push line. We chose to automate the second station since the optimal worker control policy in both lines assigns strict priority to the second station. Because WIP and throughput are measured in different units, we measure suboptimality in terms of percent error. For example, 70% on the horizontal axis indicates that the current control parameter (i.e., WIP in the CONWIP system or T H in the push system) is set at 70% of the optimal level. Figure 1(left) shows that CONWIP is more efficient than push, since it generates higher profit at the optimum. Furthermore, the profit function of the CONWIP system is very flat near the optimal WIP level, while in contrast, the profit function of the push system declines sharply when the release rate is set above or below the optimum level. Hence, Figure 1(left) also indicates that CONWIP is more robust than push. 7

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0.7 25

Efficiency gap

0.6

Profit

0.5 0.4

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low automation (20%) high automation (80%) mild automation (50%)

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0.3 10

CONWIP High automation (80%) Push High automation (80%)

0.2

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0.1 0 0

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Figure 1: Comparisons for a Balanced Line Operating Under Optimal Push and Pull Policies: (Left) Efficiency and Robustness, (Right) Impact of Automation on Robustness.

To characterize the relative efficiency of push and pull in greater detail, we compared a range of CONWIP and push lines operating under both the optimal and heuristic control policies, with their parameters set so that throughput was the same for each pair of CONWIP/push lines. Hence, the question of efficiency hinges on which line achieves this throughput with less WIP. For a given throughput rate, we define the efficiency ratio between the CONWIP and the push systems as: Efficiency ratio =

Average WIP of the push system . WIP of the equivalent CONWIP system

In order to analyze the efficiency ratio between the CONWIP and push systems under the optimal worker control policy, we focused on two-station lines in which the second station is automated. We assume that the operation at an automated station consists of loading and processing while that on a manual station includes only processing. We also assume (for now) that the line is balanced. Without loss of generality, we assume that the total operation time in each station is 1 unit. Furthermore, in order to investigate how the magnitude of automation in the line affects the relative performance of CONWIP and push strategies, we considered three scenarios: (i) high automation, where the automated machine requires 0.2 units of manual loading time and 0.8 units of automatic processing time, which represents an 80% automation level; (ii) medium automation, where the automation level is 50%; and (iii) low automation, where the automation level is 20%. In our experiments, we first increased the WIP level in the CONWIP line from 1, 2, ... , until the line saturated and throughput reached its maximum. Then we computed the average WIP level in 8

the equivalent push system for each throughput level achieved in the CONWIP line. The results are shown in Figure 1(right).3 Since the efficiency ratios are always above one, these experiments confirm that CONWIP is more efficient than push. Moreover, this efficiency advantage increases in the system utilization. However, it decreases in the level of automation. The reason for this behavior can be understood by looking at the two extreme cases of 100% and 0% automation. When the automation level is 100%, then the one-worker, two-station AAP CONWIP and push lines are both equivalent to a one-worker, two-station manual production line (because jobs flow through the automated station automatically without any help from the worker), and so the efficiency ratio is one. However, when the automation level of the automatic machine is 0% (i.e., both the CONWIP and push lines are manual), then it is easy to see that a WIP level of 1 is sufficient for the CONWIP line to reach its capacity, while it takes infinite WIP for the push line to reach its capacity; hence, the efficiency ratio is infinite. We also conducted experiments for various line imbalanced scenarios, by fixing the operation time in one station to be 1 unit and adjusting that in the other station to be 0.1, 0.2, ... , 1. For each of these scenarios, we placed high (80%), medium (50%), and low (20%) automation at one station and used our MDP model to calculate the efficiency ratio, in a manner similar to our approach for balanced lines. We conducted similar experiment for our 3-station general lines with one and three automated stations. In our general lines, operations in automated stations include loading, automatic processing, and unloading. We assumed that automatic processing times are deterministic while the manual operation times (i.e., loading and unloading in automated stations, and the entire operation in a manual station) are stochastic. The manual operation times follow Erlang-4 distributions4 and the automatic processing times are deterministic. We considered each case under both the best fixed-priority policy and the best cyclic policy, and we developed a simulation model to calculate the WIP and throughput in push and CONWIP lines under those two 3

Note that systems with different automation levels have different capacities. This would make the comparison between scenarios unfair if we were directly compare the efficiency ratios with regard to throughput rates directly. In order to avoid this, we plot the efficiency ratio vs. throughput as percent of capacity (which can be considered as system utilization). Note that there are fewer data points for the lower automation-level scenarios, because the line saturates at lower WIP levels. For example, in the 20% automation level scenario, throughput is essentially constant for all WIP levels of two or higher. Hence, the CONWIP line produces only two values of throughput to feed into the push line. 4 Erlang-4 distribution models operations with lower variability (i.e., CV < 1) than those in our optimal (MDP) model (i.e., operation times in our MDP model have CV = 1).

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policies. With respect to efficiency, we observed behavior similar to that shown in Figure 1(right). Hence, again, CONWIP outperforms push with regard to efficiency. We also broadened our study to our general lines with various automation levels (ranging from 30% to 90%) and various automation scenarios (i.e., one automated station or three automated stations). The results of these extensions consistently supported our earlier insight from the optimal model that CONWIP is both more efficient and more robust than push.

4

The Design of AAP CONWIP Lines

We now turn to two questions that did not arise in the manual line scenarios of Spearman and Zazanis (1992) but which are important in the design of an AAP CONWIP line: (i) Where in the line is automation most effective? (ii) Should automation be concentrated or distributed? By comparing our observations for CONWIP lines with those made for push lines in Hopp et al. (2003) we can determine the impact of these design considerations on the relative performance of push and pull.

4.1

The Impact of Automation Position

In the previous section, we characterized the optimal operating policy for a CONWIP line as one which places the automatic machine at the first station in the line. However, what if the automatic machine were placed at the middle or the downstream station instead? Hopp et al. (2003) showed that if an AAP line is run as a push system, automating downstream stations is generally more effective than automating upstream stations. In this section we address the question of automation position in an AAP line when the line is run under a CONWIP rule. We focus on our general line, and we begin with the following theorem: Theorem 2: In CONWIP production lines with only one automated station, if job processing times in all manual stations follow the same probability distribution, the position of automation does not matter. The proof is simple and follows from the fact that CONWIP lines are closed, cyclic queueing systems, so that the throughput of the line can be measured at any station in the line. Theorem 2 applies only to balanced systems, such as those where operations have been evenly

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divided to facilitate flow when stations are staffed simultaneously by separate workers. In general, however, the question regarding the position of automation in an unbalanced line is not easy to answer, since it often depends on the worker control policy. Theorems 3 and 4, however, address this question in systems in which a cyclic policy is implemented. Proofs are presented in the On-Line Appendix. Theorem 3: For CONWIP lines with k automated stations, m manual stations, and 1 server, the maximum throughput under a cyclic policy in which the worker visits stations in the direction of the material flow is obtained with a WIP level of k + 1. In order to study the position of automation in CONWIP lines, consider a manual two-station line with one worker in which ti , the operation time at station i, follows probability density function fi (t), i = 1, 2. Suppose that the worker employs a cyclic policy in which the worker visits the stations in the direction of the material flow. Consider the case where we have a certain amount of automation time, a, to allocate to any one of the two manual stations. (For example, if we choose to place the automation at station i, then the automatic processing time at station i is a while its total manual operation time (i.e., loading and unloading time) is ti − a, where a < ti , for i = 1, 2, ..., n). An intuitive choice for automation is the bottleneck station, since it has the longest total operation time and might well seem to be the constraint resource in the line. However, we show that this is not always true. On the contrary, it turns out that automating the bottleneck station may be the worst choice. In other words, in our two-station line, when we choose to place a time units of automation at station i, then we will have the following theorem: Theorem 4: If job processing time at station 1 (station 2) is stochastically larger than that at station 2 (station 1), then under a cyclic policy automating station 2 (station 1) results in a larger throughput. Note that the station with the stochastically larger operation time has a larger average job processing time, and is therefore the bottleneck. Under these circumstances, Theorem 4 implies that when a fixed amount of automation must be placed in a one-worker unbalanced CONWIP line, automating the bottleneck station is less effective than automating a non-bottleneck station. To further investigate the impact of automation position in larger CONWIP lines when either 11

a cyclic or a fixed-priority policy is implemented, we used our simulation model and conducted a series of experiments with our general three-station lines. The total operation times of the three stations were 1.2, 1.1 and 1 units, respectively. The problem was to determine which station should receive 0.9 units of automation time. We assume the manual operation times follow an Erlang-4 distribution. Using our simulation for general lines under a cyclic policy, we found that automating either stations 1 or 2 results in a throughput rate of 0.413, while automating station 3 (the fastest machine) results in a throughput rate of 0.414. That is, automation position makes no practical difference in throughput. If we increase the amount of automation from 0.9 to 0.98 units, our simulation shows that the throughput rates from automating stations 1, 2 and 3 are 0.425, 0.426, and 0.427, respectively. Moreover, if we further change the distribution of the manual operation times to exponential, these numbers become 0.415, 0.416, and 0.417. These examples, along with several others that we studied for lines under different automation levels (e.g., 20%, 50%, and 80%), and different line imbalance scenarios (e.g., the bottleneck was twice slower than the fastest station in the line) showed that under both fixed-priority and cyclic policies the automation position has little impact on the throughput of the AAP CONWIP line. From our experiments, we also observed that the best fixed-priority policy always prioritizes the automated station, while the station before the automated station gets second priority, and the station after it gets the least priority. This is no surprise, since we have proven that under Markovian assumptions such a policy is indeed the optimal policy. In conclusion, our analytical and numerical analysis suggests that when considering which station to automate in an AAP CONWIP line, managers can concentrate on issues like financial and technological constraints without worrying too much about the impact of automation position on operational performance. This is in contrast with the results for AAP push lines, which led to the recommendation that automating downstream stations can have a significant effect on the operational performance of the line. This design flexibility under CONWIP is an additional benefit over push.

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4.2

The Impact of Automation Concentration

The question we consider in this section is: Should automation be concentrated or distributed? In more practical terms, under a fixed budget, should we install one highly automated machine or several partially automated machines? First we consider a simple 2-station line in which both stations are automated, and which operates under a cyclic policy. For each station i = 1, 2, assume that the manual loading rate is li and the automatic processing rate is µi , so that the total operation time is ti = 1/li + 1/µi (we assume unloading time is zero). In order to investigate the effect of automation concentration, we perform the following experiment. We start with the case in which both stations have an equal amount of automatic processing time, A. Then we transfer δ units of automatic processing time from station 1 to station 2, so that the total amount of automatic processing time in the line is kept constant but the distribution of automation changes from distributed to concentrated (as δ increases from zero to A, or decreases from zero −A). We are interested in the impact of shifting automation on the throughput of the line. By Theorem 3 we know that when two stations are automated, the maximum throughput can be obtained under a cyclic policy with a WIP level of 3. Assuming all manual operation times and automatic processing times are exponential, we can establish a simple Markov chain and obtain the throughput of the line (see the Appendix for details of the Markov model). For a balanced line where t1 = t2 = t, the throughput is as follows: TH =

−4A2 + 14A2 δ 2 + A4 + δ 4 . 2t(−4A2 + 10A2 δ 2 + A5 − 2A3 δ 2 + Aδ 4 + 4A3 − 3A4 + δ 4 )

For 0 < A < 0.5t, we can show that the derivative of T H with respect to δ is negative when 0 < δ < A and positive when −A < δ < 0. This implies that distributed automation is more effective than concentrated automation when a cyclic policy is implemented in a balanced AAP CONWIP line. However, when the line is unbalanced, i.e., t1 6= t2 , this result may not hold. Experimenting with our Markov chain model, we observed that in unbalanced lines, the maximum throughput sometimes achieved with a mixed automation distribution (i.e., neither the most distributed nor the most concentrated). In addition, we observed that the optimal δ is nondecreasing in t2 − t1 , 13

0-9-0 1-7-1 2-5-2 3-3-3 4-1-4 4.5-0-4.5 max difference

(20,10,10) 0.03202 0.03223 0.03227 0.03227 0.03223 0.03225 0.8%

Cyclic policy (10,20,10) (10,10,20) 0.03220 0.03221 0.03226 0.03223 0.03226 0.03227 0.03226 0.03225 0.03225 0.03225 0.03225 0.03226 0.18%

0.18%

(10,10,10) 0.04706 0.04757 0.04761 0.04761 0.04761 0.04756

(20,10,10) 0.03231 0.03135 0.03067 0.03055 0.03136 0.03228

1.2%

5.8%

Fixed-Priority policy (10,20,10) (10,10,20) 0.03231 0.03229 0.03135 0.03135 0.03072 0.03069 0.03058 0.03055 0.03137 0.03132 0.03228 0.03228 5.7%

5.7%

(10,10,10) 0.04765 0.04566 0.04432 0.0439 0.04566 0.04763 8.5%

Table 1: Experiments on the effect of automation concentration and equals 0 when t2 = t1 . These examples indicate that clean general theoretical results regarding automation concentration in unbalanced lines are not possible. However, what really matters in practice is the magnitude of the impact. Therefore, we used simulation to study our general model with three-station lines, and evaluated the magnitude of the impact of automation distribution. We first considered a threestation balanced line where the total operation time on each station is 10 units, which we denote by (10,10,10). We also studied these unbalanced lines: (20,10,10), (10,20,10) and (10,10,20). We assumed that the total amount of automation (i.e., the sum of the automatic processing times on all stations) is 9 units. For each of the balanced and unbalanced lines, we started with the case in which station 2 is highly automated (having all 9 minutes of automation), while the other two stations are manual, denoted as 0-9-0. Then we distributed automation gradually and evenly to stations 1 and 3. Note that 0-9-0 is the most concentrated case, while 3-3-3 is the most distributed case. Moreover, since we observed in our numerical study that four jobs are sufficient to obtain maximum throughput when either the cyclic or fixed-priority policy is used, we set the constant WIP level at 4. The throughput for the various automation distributions, under both the cyclic and fixed-priority policies, are given in Table 1. These results suggest the following insights: 1. Under a cyclic policy, concentrated automation is not as effective as distributed automation, although the difference is quite small. For example, in the balanced line, the maximum difference between various automation distributions is no more than 1.2%; for the unbalanced lines, the maximum difference is no more than 0.8%. 2. Under a fixed-priority policy, concentrated automation is more effective than distributed automation. Table 1 shows that for the balanced line, the maximum difference between 14

various automation distributions is about 8.5%, while for the unbalanced lines, the maximum difference is around 5.8%. 3. For lines in which a single station is highly automated, the fixed-priority policy performs slightly better than the cyclic policy (i.e., around 1% difference in our examples in Table 1); however, if all stations in the line have the same level of automation, the cyclic policy is more effective (i.e., around 5% difference in our examples in Table 1). Therefore, in designing a CONWIP line with one cross-trained worker, one should consider either a concentrated automation configuration with a priority policy, or a distributed automation configuration with a cyclic policy. However, the performance difference between the two will be small. Since automation concentration has a larger impact on performance in push systems (Hopp et. al 2003), this observation further substantiates our conclusion that design flexibility is a benefit of CONWIP.

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Conclusion

This note confirms that the primary benefits of pull, namely observability, efficiency and control, extend to systems with equipment and cross-trained (agile) workers. Moreover, it identifies line design flexibility as a fourth advantage of CONWIP control in AAP environments. That is, while push systems are sensitive to the placement and concentration of automation, CONWIP systems are not, which means that automation decisions can be based on other issues, such as cost and quality. To see whether our insights are indeed robust in more general systems, further research is needed into systems with multiple-product types, multiple routings, multiple workers, machine failures, rework, batching, and partial cross-training. Given the growing importance of AAP systems in industry, such research would be of great practical significance.

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