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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.17 No.3 pp.531-549
DOI : https://doi.org/10.7232/iems.2018.17.3.531

# Cellular Bucket Brigades with Worker Collaboration on U-Lines with Discrete Workstations

Tirta Pratama Aditya, Takahashi Katsuhiko*, Morikawa Katsumi, Nagasawa Nagasawa, Hirotani Daisuke
Department of System Cybernetics, Graduate School of Engineering Hiroshima University, Japan
Faculty of Management and Information Systems, Prefectural University of Hiroshima, Japan
Corresponding Author, E-mail: takahasi@hiroshima-u.ac.jp
February 23, 2018 June 5, 2018 August 3, 2018

## ABSTRACT

Cellular bucket brigades (CBB) are a new design of bucket brigade that can reduce unproductive travel by workers on U-shaped production lines. With CBB, each worker works on one side of an aisle when proceeding in one direction, and on the other side when proceeding in the reverse direction; workers then exchange their work when they meet each other. The throughput of CBB may be reduced due to blocking and/or halting at discrete workstations. A method for countering these blocking/halting conditions is proposed here; it utilizes worker collaboration such that a maximum of two workers can collaborate on the same task. The collaboration velocity is proportional to the sum of individual worker velocities, and is influenced by the worker collaboration coefficient. The existing model and its as-sumptions are utilized here to compare the performance of CBB against worker collaboration on a three-station, two-worker U-line for which the difference collaboration coefficient and work content of the product at each station are deterministic. The results show that worker collaboration almost always outperforms CBB, except for certain cases in which each worker has a different velocity at different stations.

## 1. INTRODUCTION

U-shaped production lines (U-lines) are popular in cellular manufacturing, due to the use of just-in-time and lean manufacturing. In a typical U-line, workers work inside the U-line, and one worker oversees the ma-terial at both the entrance and exit. U-lines are common in manufacturing, since they possess several ad-vantages over serial lines. Miltenburg (2001) has re-viewed the advantages and different variants of U-line organization; the most common reasons for using these are to reduce the WIP (work in progress), setup time, throughput time, material handling, and quality improvement. Bischak (1996) considers a U-line module with a number of workers that is lower than the number of stations. The author proposes rules for workers’ movements within the mod-ule that are suitable for a straight-line layout. The throughput and flow time of this moving-worker mod-ule is compared with a system with one dedicated work-er per station, using simulation studies.

The bucket brigade is a well-known and widely used production module that has self-balancing and self-organizing features. Under some production condi-tions, however, it has been shown that the production efficiency can be improved through better organization. Bartholdi III and Eisenstein (1996a) introduce serial bucket brigades to coordinate workers, for an assembly line with more stations than workers. In their paper, they show that if workers are sequenced from slowest to fastest, based on their velocities in the direction of pro-duction flow, then the hand-off item will converge to a fixed point, and each worker repeatedly works on a fixed portion of work content on each item produced. Lim and Yang (2009) analyze the dynamics of bucket brigades on discrete workstations; they identify the best policy to achieve maximum throughput, and show that although fully cross-training the workers and sequencing them from slowest to fastest is not always the best poli-cy, it outperforms other schemes for most work content distributions.

A real experiment involving order picking with a bucket brigade has been reported by Bartholdi III and Eisenstein (1996b). When using a bucket brigade, work-ers are not confined to specific zones, and any worker can choose from any location. The results show that pick rates increase due to the more effective work shar-ing. However, potential waste can occur if the bucket brigade is too large; the line can become congested by boxes, meaning that workers must walk a certain dis-tance to put items into the appropriate boxes. Bartholdi III and Eisenstein (2005) explained the utilization of bucket brigades to migrate from a craft manufacturing to an assembly line of tractors and showed that the adoption of bucket brigades led to narrowing the tasks for each worker and thus accelerated learning and in-creased productivity.

To improve the efficiency of a bucket brigade for a long assembly line, Lim (2011) has introduced the idea of cellular bucket brigades (CBB) to coordinate workers on U-lines; this eliminates the unproductive walk-back that is inherent in traditional bucket brigades. Lim (2011) shows that a CBB, even with fewer workers, can be sig-nificantly (30%) more productive than its traditional counterpart if the aisle is sufficiently narrow. A case study of CBB was carried out by Lim (2012), using a computer simulation based on secondary data from warehousing in North America. This author showed that order picking with CBB can be significantly more pro-ductive than with a traditional bucket brigade. Lim and Wu (2014) have analyzed the features of cellular buck-et brigades, particularly for U-lines with discrete work-stations, and have concluded that CBB may perform differently in different situations, due to the blocking and halting of workers. Their analysis, which is based on a deterministic model, shows that the system always converges to a fixed point or a period-2 orbit for a given work content distribution.

Many modern work environments, both with and without cross-training, make use of teams to accom-plish certain tasks. The fundamental justification is that teamwork can ultimately improve productivity. An ex-periment on worker collaboration was carried out by Compaq on one of its assembly lines within its Houston facilities (McGraw, 1996). In this study, one workstation could allow three workers to construct computers simul-taneously. This experiment showed that there was an improvement in productivity and quality, although the operating costs increased. Andradόttir et al. (2001) used the assumption that when multiple servers are assigned to the same task, their combined service rate is additive. Several servers can work together for a single customer, in which case the combined rate of the server team is proportional to the sum of the rates of the individual servers. These authors use α as a measure of the magni-tude of the servers’ collaboration/synergy. The magni-tude of the server’s collaboration/synergy (α) can be calculated from the ratio between the combined rate of the server team and the sum of rates of the individual servers. If the combined rate of the server team equal with the sum of the rates if the individual servers, then the servers are additive (α = 1). If the combined rate of the server team is less than the sum of the rates if the individual servers, then α < 1 and the servers collabora-tion/synergic slows the processing. Peltokorpi et al. (2015) compared four different worker coordination policies (no helping, floater, pairs and complete helping) to test collaborative coefficients of 0.7 and 0.9. They suggested the use of a complete helping policy with a collaborative coefficient of 0.7, and a pairs policy with a collaborative coefficient of 0.9.

We propose a worker collaboration approach to counter the blocking and halting conditions for U-lines with fewer workers than the number of stations, where the work content of product at each station is determin-istic and using the prior assumption of CBB. A maxi-mum of two workers can work collaboratively on the same task (at the same station), and thus each worker’s idle time due to blocking and halting can be minimized.

Furthermore, the conditions of worker collabora-tion on the U-lines are defined and the performance of CBB against worker collaboration is compared for dif-ferent collaboration coefficients. By identifying these conditions, we extend the working conditions on the U-lines to achieve better throughput using worker collabo-ration. Based on the numerical results, we show that worker collaboration almost always outperforms CBB, except for some cases in which workers have different velocities at different stations.

In the next section, we review several works from the literature that are related to the state of the art in terms of the concepts of U-lines, CBB and worker col-laboration. Section 3 presents the model definition of a three-station, two-worker U-line, with worker collabora-tion movement rules and condition classification. Sec-tion 4 explains the numerical analysis by comparing the performance of CBB and worker collaboration for different collabora-tion coefficients and working velocities. Finally, in Sec-tion 5, we present the managerial implications and our conclusions regarding CBB and worker collaboration.

## 2. LITERATURE REVIEW

This section presents the theories and significant works in the literature related to U-line balancing, CBB and worker collaboration. This review is divided into three parts: the first deals with U-lines, the second ex-amines CBB, and the third explores worker collabora-tion.

### 2.1. U-Shaped Line Balancing

According to Miltenburg and Wijngaard (1994), the assembly line can be divided into two main groups based on the shape of the line: straight and U-shaped. U-lines are popular in cellular manufacturing, due to the use of just-in-time and lean manufacturing, and have particular characteristics; for example, the entrance and exit of the line are close to each other, and workers can perform tasks and handle workpieces from both the front and the back of the line (Kucukkoc and Zhang, 2017). This kind of flexibility often results in a better balancing of the line. The closeness of the stations also increases the visibility and communication among workers. The superiority of the U-shaped line over the traditional serial line has been demonstrated and docu-mented in various studies (e.g. Miltenburg, 2001; Aase et al., 2004).

U-shaped line balancing (ULB) was first studied by Miltenburg and Wijngaard (1994), who developed a dynamic programming (DP) formulation to minimize the number of workstations for a given cycle time. Most of the papers on ULB assume that each assembly task requires fixed equipment and a fixed number of work-ers. (e.g. Toklu and Özcan, 2008; Ağpak et al., 2012; Lian et al., 2012). In a U-line with multiple workers, two types of worker allocation can exist (Nakade, 2016): separate allocation and carousel allocation. In a sepa-rate allocation, each worker deals with a unique subset of machines, and thus all workers have their own tasks. The cycle time becomes the maximal total task time of the workers. In a carousel allocation, the system is op-erated in a ‘chase’ mode in which all workers deal with all tasks. Initially, the workers are placed in a sequence on the line; each worker operates an item at a machine, takes the operated item and then brings it to the next machine. Thus, workers deal with all machines along the same route. If a worker catches up to the worker ahead, he/she waits until the latter finishes the operation and leaves the machine, since no worker can pass an-other worker.

Bischak (1996) considers a U-line module with a number of workers that is lower than the number of stastations. Using a U-line with multiple workers who work along the line, the author considers a system with mov-ing workers who may move backwards as in the bucket brigade, and investigates the performance of the mov-ing-worker module using a simulation. Theoretical anal-ysis is limited in a system where the movement of work-ers has such a high degree of flexibility.

In conclusion, the U-shaped layout has demon-strated its superiority over the traditional straight layout. U-lines offer greater flexibility in terms of line balancing; the serial line only allows balancing using operations to the left and right of the station, while U-lines allow bal-ancing using not only right and left stations, but also to the left and right stations behind the worker.

One way to increase the productivity of U-line is by increasing number of workers in the line. The carousel allocation (Nakade, 2016) is a possible approach to organize workers in cellular manufacturing. The workers always move in a circle to handle all process in a se-quence where the faster worker will be chasing the slow-er worker. Since workers work forward with the item, then less need inventory in the line. Another advantage is the worker has full responsibility for the item from start to the end, then the quality of the item is probably better. However, this approach comes with a drawback that the system speed is limited to the speed of slower worker, then faster worker may have to wait for the slower worker. In addition, if more workers are added in the line, then more workers have to wait. This situation will end up with a manufacturing version of traffic jam.

One technique to solve the idling is by sharing the task where the U-line has great flexible assignment since multi-skilled workers are located inside the line. Most works of literature on flexible assembly lines utilize work-sharing on a serial line. In this line, the number of workers is smaller than the number of machines and each worker operates only in his respective zone (McClain et al., 2000; Anuar and Bukchin, 2006; Buk-chin and Cohen, 2013). Workers may share the tasks by assuming that it requires expensive special purpose equipment to process the item (Bartholdi III and Eisen-stein, 2005). The two most common work-sharing sys-tems are TSS and the bucket brigade system. Studies of TSS have already been done based on the work-sharing system, where workers move down the line carrying an item with them and working on it at each machine in an assigned zone until they are pre-empted by another worker who is coming back upstream (Bischak, 1996).

Most recent literature discusses the study of serial bucket brigades, in which workers exchange an item on a serial production line. Bartholdi III and Eisenstein (1996a) introduced the serial bucket brigade to coordi-nate workers on an assembly line with more stations than workers, and proved that slowest-to-fastest sequencing always attains the maximum possible throughput. Most studies of bucket brigades have been done in the same way as the extended study of Bartholdi III and Eisenstein (1996b), i.e. based on the assumption that the work content is distributed continuously and uniformly over the entire line. (e.g. Bartholdi III et al., 1999; Bartholdi III et al., 2001; Bartholdi III and Eisenstein, 2005; Bartholdi III et al., 2006; Armbruster and Gel, 2006; Hirotani et al., 2006). However, the work content is grouped in various portion into discrete workstation under real conditions. Lim and Yang (2009) studied the main problem of serial bucket brigades for discrete workstations, in which max-imum throughput cannot always be achieved due to blocking conditions.

The ideas of CBB were first introduced by Lim (2011) to distribute work content on both sides of an aisle. The author proposed a novel design alternative that could provide a significant improvement in the performance of a bucket brigade. Under the new design, each worker works on one side of an aisle when pro-ceeding in one direction, and on the other side of the aisle when proceeding in the reverse direction. Based on this new design, the author proposed a self-balancing system for certain conditions. The numerical example used suggests that this new design can be 30% more productive than serial bucket brigades. Lim (2012) pro-vides a case study of order picking by CBB, using data from a distributor of service parts in North America, and compares the average throughput of cellular and serial bucket brigades using a computer simulation. This pa-per suggests that CBB can not only boost productivity but also save costs in terms of labor and wireless tech-nology. The improvement in throughput due to a CBB can be as high as 25% over a serial bucket brigade for a team of 20 workers. Even with fewer and slower work-ers, CBB can be more productive than serial bucket brigades that are equipped with wireless technology to reduce travel. Lim and Wu (2014) analyzed the features of CBB on U-lines with discrete workstations, and adapted the basic idea of CBB as introduced by Lim (2011) to coordinate workers on the U-lines. They showed that throughput is significantly improved when the number of stations at each stage increases from one to two, but that there are diminishing returns if each stage is divided into more stations. They concluded that the performance of CBB may vary according to the situation.

In conclusion, CBB can eliminate the unproductive walk-back inherent in traditional bucket brigades. How-ever, blocking and/or halting may occur if the work con-tent on each station is not balanced and workers are not sequenced properly.

### 2.3. Worker Collaboration

Teamwork or collaborative work can be used to improve the performance of assembly lines through workforce flexibility. Salas et al. (1992) have defined a team as a group of two or more people with specific or non-specific roles or functions, who interact dynamical-ly, interdependently and adaptively to target a common objective or mission. In collaborative systems where at least some of the tasks involved in a job can be per-formed simultaneously, teamwork may help increase the average task speed or labor productivity. There are several studies in the literature that have discussed worker collaboration in the manufacturing field. McGraw (1996) describes a scenario at Compaq where teams of three workers built, tested and shipped com-puters at a single workstation that showed improve-ments in productivity and quality of as much as 25%, although the operating costs were reported to be nearly four times higher than those of a conventional assem-bly line. With regard to multi-product scenarios with various product options, the teamworking experiments at Volvo’s Uddevalla and Kalmar plants are probably the most well-known (Engström and Medbo, 1994; Ellegärd et al., 1992). Workers could collaborate on the same task, and each team of workers was assigned to do all or most of the assembly tasks on a particular ve-hicle. This plant showed an improvement in quality and a decrease in the total lead time, and it was also easier to handle requests for customization from customers.

Buzacott (1996) used queuing models to analyze the performance of collaborative teams, and found that the mean job completion time was shorter for teams than for individuals. Oyen et al. (2001) assumed that all workers were identical and could collaborate on the same job without interfering with each other, and fur-thermore that sufficient machinery, tooling, and so on existed to allow all workers to work at the same station simultaneously. They showed that collaborative teams are beneficial for systems with high variability. Under some circumstances, such as operational environments with low utilization, low variability and a lack of bal-ance, cooperative teams may not improve system per-formance unless collaborative efficiency is very high.

Most works in the literature have studied the im-pact of collaboration on the tandem queuing system, where workers can work at the same task with an addi-tive collaboration coefficient or α = 1.0 (Andradόttir et al., 2001; Andradόttir and Ayhan, 2005; Andradόttir et al., 2011; Wang et al., 2015). According to Hopp and Oyen (2004), a basic measure of collaboration efficien-cy is the relative percentage increase in average task speed (or labor productivity) that results from assigning multiple workers to the same task. Sengupta and Jacobs (2004) compared an assembly line design without col-laboration to a parallel, cell-based design of two single tasks with collaboration. The paper introduced an inef-ficiency factor representing the loss of efficiency as a result of worker transfers, tool constraints, space in which to work or simply working in teams. The values of this factor represent the inefficien-cy arising from zero impact of collaboration, a negative impact in which the productivity of each worker in a collaboration decreases, and a situation where collabo-ration even increases the completion time for the cur-rent task. Peltokorpi et al. (2015) compared four differ-ent worker coordination policies (no helping, floater, pairs and complete helping) by studying the parallel assembly in terms of a continuous-time Markov process. This paper tests collaborative efficiency factors that are equivalent to the factors proposed by Sengupta and Jacobs (2004). They define minor collaborative ineffi-ciency in pair-working as 10% (with a collaborative co-efficient of 0.9) and major inefficiency as 30% (with a collaborative coefficient of 0.7). They also assume that when one worker helps another, their collaborative inef-ficiency reduces productivity, and suggest the use of a complete helping policy in conditions of minor collabo-rative inefficiency, especially for a specific set of jobs, and a pair policy as a reasonable alternative in condi-tions of major inefficiency with the continuous arrival of jobs.

The use of worker collaboration may help to in-crease the task speed. The integration of worker collab-oration may decrease the idling of workers in some cas-es, and increase the performance of the production line. Based on theoretical explanations, the current paper will test the performance of worker collaboration in counter-ing blocking or halting in CBB on a three-station, two-worker U-line. The model and assumptions used for the U-line are adopted from Lim and Wu (2014), while the concept of worker collaboration is adopted from Oyen et al. (2001) and Peltokorpi et al. (2015). We study col-laboration under additive conditions (α = 1.0) and inef-ficiency collaborative conditions (α = 0.7 and α = 0.9).

## 3. U-SHAPED PRODUCTION LINE

This section is divided into two parts: the first pre-sents the definition of the model and movement rules for collaborative work, and the second describes the classification and throughput formulation of worker collaboration.

### 3.1. Model Definition and Movement Rules

Lim and Wu (2014) have defined the general as-sumptions of the U-line with discrete workstations. They assume that a U-line may consist of multiple stations at each stage and work by two workers. They have shown that on a U-line with one station at each stage, blocking and/or halting conditions still occur even if workers are sequenced properly; these are caused by imbalances arising when one stage has more work content than the other stages. They also analyzed the impact of multiple stations at each stage, and showed that increasing the number of stations can reduce the time for which workers are blocked, and thus in-crease the average throughput. However, if each stage is further divided into more stations, this gives rise to a continuous line, and throughput soon becomes constant.

In this paper, we adopt the assumptions of the U-shaped production line made by Lim and Wu (2014), and limit these assumptions only for a three-station system where the highest rates of blocking and/or halt-ing occur. Consider a U-shaped production line that consists of three stages. Stages 1 and 3 are separated by an aisle, and Stage 2 spans across the aisle. That is, Stage 1 consists of a station located on one side of the aisle; stage 2 has a station located across the aisle; and Stage 3 consists of a station located on the other side of the aisle. Since each stage in the U-line consists of only one station, then the work content on each stage j is deterministic; it is denoted as sj and the total work con-tent is normalized to 1. The U-line is worked by W1 and W2. Each worker is cross-trained to work on any stage of the U-line, and workers continuously move along a sta-tion as they progressively work at the station. Wi works with a constant and deterministic velocity vij at stage j, for i = 1, 2 and j = 1, 2, 3. The travel time between sta-tions is short compared to the time required to process an item; we therefore assume that the time to walk from one stage to another is instantaneous, and hand-off time is neglected.

With CBB, each item in the U-line is initiated at the start of stage 1, specifically by W1. The item is passed to W2 at some point during Stage 1. W2 then finishes the remaining work of Stage 1 and continues to assemble the item at Stage 3 before passing it back to W1 at Stage 3. W1 then completes the item at the end of Stage 3. When W1, who is working on Stage 1, meets with W2, who is working on Stage 3, their horizontal position co-incides, and a hand-off takes place between the two workers; each worker relinquishes the item, walks across the aisle and takes over the other’s item. After the hand-off, W1 works on Stage 3, while W2 proceeds with Stage 1. At most, one worker is allowed to work at a station at any one time. As a result, a worker may be idle at the end of the station while a colleague is still working at the next station.

When the hand-off occurs, we assume that the arriving worker takes over the task without stopping the process, then the work content is preemptible without any loss of work. Figure 1 shows a conceptualized U-line of length 1. The start and end of the path are represented by points 0 and 1, respectively; the intervals [0, s1], [s1, s1+s2], and [s1+s2, 1] correspond to the work content at Stages 1, 2, and 3, respectively; and the horizontal line segments [0, s1] and [s1+s2, 1] are parallel to each other, while the line segment [s1, s1+s2] is perpendicular to them.

For worker collaboration, we extend several of the assumptions made in CBB. One item at most can be processed by a maximum of two workers at a station at any given time, in the case of collaborative work. As a result, a worker may be idle at the end of the station while another colleague is still working at the next sta-tion, rather than being able to continue the individual task. The station can accommodate multiple workers working together simultaneously by sharing the space and tools required to complete an item. There is a buffer at the end of each stage; a worker can put an item into this buffer and then move to support another worker. A worker can also pick up an item from a buffer and then continue to process the item.

The collaborative velocity of a team is the sum of the velocities of the individual work, and is influenced by the worker collaboration coefficient. Since there are only two workers in the U-line, the collaboration velocity is equal to $v c o l l ( j ) = α ( v 1 j + v 2 j )$ and must be higher than the minimum worker velocity at each stage $( v c o l l ( j ) > min { v 1 j , v 2 j } ) .$ is a measure or coefficient of the collaboration or synergy between Worker 1 and Worker 2, and can be defined by 0<α≤1 We assume an additive scheme (α = 1.0) and an inefficient (α < 1.0) worker collaboration coefficient. At a station, a worker will process an item in individual work and/or partial work of the item will be processed by collaboration. We assume that collaborative work can start at any point at a station, and finish at the end of the station without any loss of work. Hand-off occurs only at the end of the station after finishing collaborative work. Table 1 summarizes the movement rules that must be followed by workers under the two types of worker collaboration.

### 3.2. Classification of Worker Collaboration

Before workers implementing the CBB or worker collaboration at the production line, we assume an ini-tial situation before the production line achieving the steady state as shown in Figure 2 and Figure 3. For ini-tial situation, Worker 1 initiates the new item and con-tinue to process the item along the production line, while Worker 2 will wait on the designated station. When they meet, Worker 1 hands off the item to Worker 2 and re-turn to initiate new item. Since both workers have their own item to be processed, they can continue to process the item based on CBB or worker collaboration move-ment rules.

Figure 2 shows an example of CBB behavior, in which the line has a “long” Stage 1 and a “short” Stage 3. Worker 1 starts at Stage 1, while Worker 2 starts at Stage 2. When Worker 2 reaches the end of Stage 3, then work will be halted, since Worker 1 is still working on Stage 1, where h1h2. If x* denotes the hand-off position between Worker 1 and Worker 2, then the halting condition before exchanging an item can be defined as $x * v 11 > s 2 v 22 + 1 − ( s 1 + s 2 ) v 23 .$ After exchanging the item, Worker 2 continues to work at Stage 1, while Worker 1 will be blocked in front of Stage 1. The blocking condition after exchanging the item can be defined as $( 1 − ( s 1 + s 2 ) ) − ( s 1 − x * ) v 11 < ( s 1 − x * ) v 21$. According to Lim and Wu (2014), this condition can be classified as Region 1, where Worker 1 is always blocked at point 0, and Worker 2 is always halted at point 1. The system will converge to a single fixed point at $x * = s 1 − s 3 ,$ and the cycle time of the system is $C T = x * v 11 + ( 1 − ( s 1 + s 2 ) ) v 21 .$

Lim and Wu (2014) have classified the other dynamic behaviors and the throughput of the system according to CBB rules. Based on the deterministic model, if the CBB’s convergence condition $( 1 / v 11 − 1 / v 13 ≥ 1 / v 21 − 1 / v 23 ) ,$ is satisfied, then the system can be divided into five regions, where Regions 1 through 4 experience CBB behavior with blocking/halting, and neither blocking nor halting occurs in Region 5, since the work content of the three stages is balanced, allowing the system to avoid blocking and halting. Meanwhile, if the convergence condition for CBB is not satisfied $( 1 / v 11 − 1 / v 13 < 1 / v 21 − 1 / v 23 ) ,$ the behavior of all regions remain the same except for Region 5, which now can be partitioned into seven regions (Regions 5a through 5g). All of the regions (Regions 1 through 5g) are dominated by blocking/halting or combinations of blocking and halting. The term $1 / v i 1 − 1 / v i 3$ represents the extra work time needed by worker i to complete a unit of work at Stage 1 compared with Stage 3. Even though there is no idleness in Region 5, the throughput may be lower than for other regions. When each worker has a different work velocity at different stations, a worker may repeatedly work at a station where s/he is slow although there is no idleness in Region 5. Instead, a worker may repeatedly work on a station where s/he is fast, although a worker may be blocked or halted in the other regions. For details of the behavior and classification of CBB with three stations and two workers, see Lim and Wu (2014).

Figure 3 shows the principle of worker collabora-tion for types 1 and 2. Figure 3a shows that Worker 1 starts at Stage 1, while Worker 2 starts at Stage 3. Work-er 1 has finished at the end of Stage 1, while Worker 2 is still working on Stage 3; this can be expressed as $s 1 v 11 < ( 1 − ( s 1 + s 2 ) ) v 23 .$. Figures 3b-d show the next movement of the worker for type 1 worker collaboration, while Figure 3e shows the next movement for type 2.

In Figure 3b, if Worker 1 reaches the end of Stage 1 earlier than Worker 2 finishes Stage 3, Worker 1 will put the item into the buffer and go to Worker 2’s position for collaboration at Stage 3. Let $x * *$ be defined as the meeting point for Workers 1 and 2 to start collaborative work at a station. The collaboration meeting point is located at Stage 3, and can be defined as $x 1 * * = s 1 + s 2 + s 1 v 23 v 11 .$ Figure 3c shows that after finishing the collaboration, Worker 1 will introduce a new item to Stage 1, while Worker 2 will take an item from the buffer and continue to work at Stage 2. Worker 1 reaches the end of Stage 1 earlier than Worker 2 on Stage 2, which can be expressed as $s 1 v 11 < s 2 v 22 .$ Worker 1 puts the item into the buffer and goes to the position of Worker 2 for collaboration at Stage 2. The collabora-tion meeting point is located at Stage 2, and can be defined as $x 2 * * = s 1 + s 1 v 22 v 11 .$ Figure 3d shows that after finishing the collaboration, Worker 1 will continue to work at Stage 3, while Worker 2 takes the item from the buffer and works at Stage 2. Worker 1 reaches the end of Stage 3 earlier than Worker 2 at Stage 2, which can be expressed as $( 1 − ( s 1 + s 2 ) ) v 13 < s 2 v 22 .$ Worker 1 goes to the position of Worker 2 for collaboration at Stage 2. The collaboration meet-ing point is located at Stage 2, and can be defined as $x 3 * * = s 1 + ( 1 − ( s 1 + s 2 ) ) v 22 v 13 .$. After finishing the collaboration, Worker 1 introduces a new item, while Worker 2 works at Stage 3. The behaviors of this condition are shown as Class 1.1 in Table 2, and the cycle time of the system can be defined as $C T = 2 s 1 v 11 + ( ( s 1 + s 2 ) − x 2 * * ) α ( v 12 + v 22 ) + ( ( s 1 + s 2 ) − x 3 * * ) α ( v 12 + v 22 ) + ( 1 − ( s 1 + s 2 ) ) v 13 + ( 1 − x 1 * * ) α ( v 13 + v 23 ) .$

Figure 3e shows that when Worker 1 reaches the end of Stage 1, s/he can continue to work at the next stage, while Worker 2 still works at Stage 3. When Work-er 2 has finished at the end of the line, and Worker 1 is still working on an item at Stage 2, this condition can be defined as Worker 2 then goes to the position of Worker 1 for collaborative work at Stage 2. The collaboration meeting point is located at Stage 2, and can be defined as $x * * = s 1 − s 1 v 12 v 11 + ( 1 − ( s 1 + s 2 ) ) v 12 v 13 .$ After finish-ing the collaboration, Worker 1 introduces a new item, while Worker 2 continues to work on an item at Stage 3. The behaviors of this condition are shown as Class 2.2 in Table 3, and the cycle time of the system can be de-fined as $C T = s 1 v 11 + ( 1 − ( s 1 + s 2 ) ) v 23 + ( ( s 1 + s 2 ) − x * * ) α ( v 12 + v 22 ) .$. Tables 2 and 3 show the classification conditions for worker collabora-tion of types 1 and 2, respectively. Both tables are based on the assumptions and worker collaboration movement rules for a U-line.

## 4. ANALYSIS AND DISCUSSION

Consider a U-line with a narrow aisle consisting of m = 3 stations and n = 2 workers, where each worker has different work velocities at different stages. Since each worker has a different work velocity at different stages, the throughput for one region might be lower than for others. A worker may be slow in working at a particular station, even though there is no idling, or may work quickly at a station, giving rise to idling; these two situa-tions therefore produce different throughputs. These situations cannot be found on a U-line where each worker has constant working velocity at all stations; here, possible maximum throughput can always be ob-tained if there is no idling.

In this section, we adopt the same concept of col-laboration coefficient at additive condition (α = 1.0) and inefficient collaborative condition (α = 0.7 and α = 0.9) which refers to Oyen et al. (2001) and Peltokor-pi et al. (2015), respectively. By analyzing those condi-tions, we can determine the relation and tendency of α to impact of worker collaboration on CBB. If the α in-creases and close to the additive condition, then the advantage of worker collaboration on CBB increases. Meanwhile if the α decreases and close to 0, then the advantage of worker collaboration on CBB decreases since the collaboration velocity becomes smaller than the velocity of the individual workers. Our preliminary analysis at α = 0.5 shows that worker collaboration im-provement is limited only in Region 1, while if the α is less than 0.5, the worse performance will be obtained, and no area can be improved by worker collaboration.

The throughput and region classification in the CBB system will be explained in the following sub-section; we then utilize the region classification to demonstrate the impact of worker collaboration on CBB, by assuming an additive worker collaboration coefficient (α = 1.0) and an inefficiency collaboration coefficient (α = 0.7 and α = 0.9). In the second sub-section, using the same assumption of a collaboration coefficient, we compare the performance of CBB and both types of worker collaboration with random work-ing velocities, corresponding to a numerical calculation of the relative difference in throughput be-tween CBB and both types of worker collaboration.

### 4.1. Impact of Worker Collaboration in Each CBB Region

Based on Section 3 and in accordance with Lim and Wu (2014), the CBB regions will depend on the convergence condition. Figure 4 and Figure 5 show the throughput and region of CBB when the convergence condition $( 1 / v 11 − 1 / v 13 ≥ 1 / v 21 − 1 / v 23 )$ is satisfied and the convergence condition is not satisfied, respectively. In Figure 4a, we set the working velocities as v11 = 0.8, v13 = 1.2, v21 = 1.2, v23 = 0.8, v12 = v22 = 1, while in Figure 5a we set the working velocities as v11 = 1.2, v13 = 0.8, v21 = 0.8, v23 = 1.2, and v12 = v22 = 1. Since there are block-ing and/or halting conditions within a region, then the throughput in each region will have a different expres-sion. The red scale indicates the highest throughput in a region, while the blue scale indicates the lowest.

If the convergence condition is satisfied, the sys-tem can be partitioned into five regions as designated by Lim and Wu (2014), shown in Figure 4b and the be-havior of the system can be described as follows:

Region 1: At the fixed point, Worker 1 is always blocked at point 0 and Worker 2 is always halted at point 1 at each iteration; Region 2: At the fixed point, Worker 1 is blocked at point 0 at each iteration; Region 3: At the fixed point, Worker 2 is blocked at point s1+s2 at each iteration; Region 4: At the fixed point, Worker 1 is halted at point s1 at each iteration; Region 5: Neither blocking nor halting occurs.

If the convergence condition is not satisfied, the system can be divided into eleven regions as designated by Lim and Wu (2014), shown in Figure 5b. The behav-ior remains the same for Region 1 to 4, while Region 5 can be partitioned into seven sub-regions as follows:

Region 5a: The system converges to a period-2 or-bit, where Worker 1 is blocked at point 0 at every other iteration; Region 5b: The system converges to a period-2 orbit, where Worker 2 is blocked at point s1+s2 at every other iteration; Region 5c: The system converges to a period-2 orbit, where Worker 1 is halted at point s11 at every other iteration; Region 5d: The system converges to a period-2 orbit, where Worker 1 is blocked at point 0 at one iteration and Worker 2 is halted at point 1 at the next iteration; Region 5e: The system converges to a period-2 orbit, where Worker 1 is blocked at point 0 at one iteration and Worker 2 is blocked at point s1+s2 at the next iteration; Region 5f: The system converges to a period-2 orbit, where Worker 1 is halted at point s1 at one iteration and Worker 2 is blocked at point s1+s2 at the next iteration; Region 5g: The system converges to a period-2 orbit, where Worker 1 is first blocked at point 0 and then halted at point s1 at one iteration, and Work-er 2 is halted at point 1 at the next iteration.

Using the obtained CBB regions, the impact of worker collaboration on each CBB region can be ana-lyzed based on the relative difference in throughput between CBB and both types of worker collaboration. Figure 6 shows this performance comparison when CBB’s convergence condition is satisfied. For this condi-tion, the CBB system converges to a fixed point with blocking and/or halting, and can be partitioned into Regions 1 through 5, as shown in Figure 6. Regions 1 through 4 have blocking/halting conditions, while Re-gion 5 has a balance condition. Figures 6a-c show the performance comparison based on the relative differ-ence in throughput between CBB and worker collabora-tion of type 1, for α = 0.7, α = 0.9, and α = 1.0, respec-tively. Figure 6d-f show the performance comparison based on the relative difference in throughput between CBB and worker collaboration of type 2, for α = 0.7, α = 0.9, and α = 1.0, respectively.

Comparing Figures 6a and 6d, it can be seen that both types of worker collaboration are dominated by an area with decreasing performance, almost entirely with-in the CBB regions. Both types of worker collaboration can only improve the throughput of CBB, especially in Regions 1 to 4 when α = 0.7.

Increasing the collaboration coefficient can im-prove the performance of worker collaboration signifi-cantly in certain regions. When the collaboration coeffi-cient increases, the area with decreased performance shrinks, while the area with equal or increased perfor-mance expands, as shown in Figures 6b and 6e for types 1 and 2 with α = 0.9, and Figures 6c and 6f for types 1 and 2 with α = 1.0, respectively.

Based on Figures 6c and 6f, and by calculating the number of configurations with a positive relative differ-ence, it can be shown that worker collaboration type 1 covers 70.17% of all CBB regions, while type 2 covers 57.74% of all CBB regions. Both worker collaborations achieve the maximum relative difference in Region 1. Type 1 achieves a maximum relative difference of 143.52%, while type 2 reaches 142.47%. Based on Fig-ures 6c and 6f, the performance of type 1 is superior to type 2, especially in Regions 1 through 4, while CBB is superior to both types in Region 5.

In the case where each worker has a constant working velocity at all stations, then the variable of working velocity will satisfy the convergence condition for CBB. The halting/blocking condition or combina-tions of blocking and halting can still occur only in Re-gions 1 through 4, while the maximum possible throughput can be obtained in Region 5, where neither blocking nor halting occurs. Since worker collaboration is utilized to eliminate the blocking/halting condition, then the maximum throughput in Regions 1 through 5 can be obtained for both types of worker collaboration with an additive collaboration coefficient.

Figure 7 shows a performance comparison in terms of the relative difference in throughput between CBB and both types of worker collaboration, where the con-vergence condition for CBB is not satisfied. In this situ-ation, the CBB system converges to a fixed point or a period of 2-orbit, with blocking and/or halting, and can be partitioned into Regions 1 through 5g, as shown in Figure 7. Figures 7a-c show performance comparisons based on the relative difference between CBB and worker collaboration type 1, for α = 0.7, α = 0.9, and α = 1.0, respectively. Figure 7d-f show performance comparisons based on the relative difference between CBB and worker collaboration type 2, for α = 0.7, α = 0.9, and α = 1.0, respectively.

Comparing Figures 7a and 7d, it can be seen that type 2 worker collaboration gives a smaller region with decreased performance compared to type 1, when α = 0.7. Type 1 shows worse performance in terms of throughput for almost all of the CBB regions when α = 0.7, as shown in Figure 7a.

We can see the impact on the increase of collabo-ration coefficient has a significant relation on perfor-mance improvement of worker collaboration compared with CBB. When the collaboration coefficient for both types of worker collaboration increases, the area with decreasing performance shrinks, while the area with equal or increased performance expands, as shown in Figures 7b and 7e for types 1 and 2 for α = 0.9, and Fig-ures 7c and 7f for types 1 and type 2 for α = 1.0, respec-tively.

Based on Figures 7c and 7f, using the workers’ ve-locity profile with an additive collaboration coefficient and calculating the number of configurations which have a positive relative difference, type 2 worker col-laboration represents 99.09% of all CBB regions, while type 1 represents 98.31% of all CBB regions. Both types achieve a maximum relative difference in Region 3. Type 2 achieves a maximum relative difference of 145.89%, while type 1 achieves 144.95%. Based on Figures 7c and 7f, we can see that the performance of type 2 is superior to type 1.

### 4.2. Performance Comparison for Random Working Velocities

In this section, we compare the performance of CBB and both types of worker collaboration for ran-dom working velocities, by considering an additive worker collaboration coefficient (α = 1.0) and an ineffi-ciency collaboration coefficient (α = 0.7 and α = 0.9). Tables 4 and 5 show numerical results for the relative difference in throughput between CBB and worker col-laboration when CBB’s convergence condition is satis-fied and unsatisfied, respectively. In these tables, we apply work content multiplication factors of 0.01 for each station. For instance, the first possible configura-tion is s1 = 0.01, s2 = 0.01 and s3 = 0.98; the second is s1 = 0.01, s2 = 0.02 and s3 = 0.97; and so on. Since zero work content at a station is neglected, there are 4851 possible configurations. There are 10 sets of scenarios for working velocity, where we use a uniform random number generator to generate working velocities; each worker then has different velocities at different stations. Based on both tables, an increase in collaboration coef-ficient has a significant impact on the increase in per-formance of worker collaboration compared with CBB.

Based on Table 4, in which the CBB convergence condition is satisfied, the CBB system is partitioned into five regions, where Regions 1 through 4 are dominated by blocking and/or halting, while Region 5 experiences neither blocking nor halting. Regions 1 through 4 show the most significant improvement that can be achieved by both worker collaboration, as indicated by the max-imum relative difference and the average number of configurations with a positive relative difference. In Region 1, both types of worker collaboration achieve the highest number of configurations with a positive relative difference and maximum relative difference. For α = 0.7, type 1 worker collaboration achieves 890.9 out of 929 (or 95.90%) configurations with a positive relative difference and a maximum relative difference of 679.52%, while type 2 achieves 909.6 out of 929 (or 97.91%) configurations with a positive relative differ-ence and a maximum relative difference of 676.65%. For α = 0.9, type 1 achieves 924.5 of 929 (or 99.52%) configurations with a positive relative difference and a maximum relative difference of 893.81%, while type 2 achieves 928.5 of 929 (or 99.95%) configurations with a positive relative difference and a maximum relative difference of 884.36%. For α = 1.0, type 1 achieves 927.3 of 929 (or 99.82%) configurations with a positive relative difference and a maximum relative difference of 999.77%, while type 2 achieves 929 of 929 (or 100%) configurations with a positive relative difference and a maximum relative difference of 986.58%. In Region 1, the CBB system corresponds to “more work” at Station 1 and “less work” at Station 3, where Worker 1 is always blocked at point 0 and Worker 2 is always halted at point 1 at each iteration.

In Region 5, however, both types of worker collab-oration show worse performance than CBB based on the average number of configurations with a positive relative difference and the maximum relative differ-ence. For α = 0.7, type 1 achieves 143.4 of 923.9 (or 15.52%) configurations with a positive relative differ-ence and a maximum relative difference of 7.71%, while type 2 achieves 114.8 of 923.9 (or 12.43%) con-figurations with a positive relative difference and a maximum relative difference of 10.32%. For α = 0.9, type 1 achieves 251.5 of 923.9 (or 27.22%) configura-tions with a positive relative difference and a maximum relative difference of 16.79%, while type 2 achieves 134.8 of 923.9 (or 14.59%) configurations with a posi-tive relative difference and a maximum relative differ-ence of 17.43%. For α = 1.0, type 1 achieves 297.3 of 923.9 (or 32.18%) configurations with a positive relative difference and a maximum relative difference of 20.87%, while type 2 achieves 146 of 923.9 (or 15.80%) configurations with a positive relative difference and a maximum relative difference of 20.55%.

As shown in Table 4, type 1 almost always outper-forms type 2; this is indicated by the total of the aver-age number of configurations with a positive relative difference and maximum relative difference, especially for Regions 1 through 4. However, CBB has better per-formance in Region 5 than worker collaboration of both types. For an additive collaboration coefficient, type 1 can improve 78.33% of all CBB regions, with the max-imum relative difference of 999.77% in Region 1, while type 2 can improve 56.58% of all CBB regions, with a maximum relative difference of 986.58 in Region 1.

As shown in Table 5, when the CBB regions are dominated by blocking and/or halting conditions, then both types of worker collaboration can be applied to counter these conditions. When α < 1.0, type 1 has a lower performance than type 2 for several regions, as indicated by the average number of configurations with a positive relative difference and a maximum relative difference. For α = 0.7, type 1 can improve 82.31% of all CBB regions with a maximum relative difference of 271.02% in Region 3, while type 2 can improve 87.54% of all CBB regions with a maximum relative difference of 342.08% in Region 3. For α = 0.9, type 1 can improve 93.43% of all CBB regions with a maximum relative difference of 374.26% in Region 3, while type 2 can improve 94.32% of all CBB regions with a maximum relative difference of 383.23% in Region 3. In Region 3, the CBB system corresponds to “less work” at Station 1 and “more work” at Station 3, where Worker 2 is always blocked at point s1+s2 at each iteration.

Furthermore, type 1 outperforms type 2 for α = 1.0, as indicated by the total average number of configura-tions with a positive relative difference and the maxi-mum relative difference. Type 1 can improve 97.21% of all CBB regions with a maximum relative difference of 425.43% in Region 3, while type 2 can improve 96.64% of all CBB regions with a maximum relative difference of 433.26% in Region 3.

## 5. MANAGERIAL IMPLICATIONS AND CONCLU-SIONS

Lim and Wu (2014) have explained that CBB can maximize the productivity of the U-line with discrete workstations by choosing the worker sequence properly and preserve the productivity by allowing the workers dynamically to share the work. However, CBB shows a drawback where blocking and/or halting condition can occur in case of different amount work content distribu-tion. This paper puts forward another possibility for overcoming the impact of blocking/halting in CBB through the use of worker collaboration, where each worker has different work velocities at different stations. For a three-station, two-worker system, CBB always converges to a fixed point or period-2 orbit for a given work content distribution, when blocking and/or halting may occur. Numerical analysis shows that worker col-laboration can effectively improve the performance of CBB in a region where blocking and/or halting occurs.

Based on the analysis of a three-station, two-worker U-line where each worker has a different work-ing velocity for each stations, we tested the worker col-laboration condition with several sets of random worker velocities, with the CBB convergence condition satisfied and unsatisfied (as defined by Lim and Wu, 2014). When the CBB convergence condition is satisfied, both types of worker collaboration perform well in countering blocking and/or halting in Regions 1 through 4. In a region where the U-line is balanced, without blocking and/or halting, CBB shows higher performance than both types of worker collaboration. When the CBB convergence condition is not satisfied, then all regions of CBB are dominated by blocking and/or halting con-ditions. Both types of worker collaboration almost al-ways outperform CBB in this case. In addition, α is strongly related to the performance of both types of worker collaboration to counter the blocking and/or halting in certain regions. By increasing the α, there is a tendency that area with increased performance ex-pands and throughput relative difference also increases. Meanwhile, by decreasing α, then the performance worker collaboration on CBB becomes worse. The area with increased performance shrinks and throughput relative difference also decreases.

Based on the analysis of a three-station, two-worker U-line, a manager can directly use those results to boost the productivity in the real implementation. Moreover, workers can easily adopt and follow the rules, therefore it can be implemented in practical condition. If a balanced U-line can be obtained by CBB, then our approach can be neglected since CBB shows higher performance than worker collaboration. A manager should do a prelimi-nary check whether CBB or worker collaboration can give better performance since worker collaboration al-most always outperform than CBB in halting/blocking case. In addition, a manager can still utilize those results even though each stage consists of multiple stations. We expect that the characteristic of self-balancing line can still be preserved, and the performance improvement can be obtained with discrete workstation by integrating worker collaboration. Although increasing the number of stations can reduce the idling time and increase the average throughput, blocking/halting conditions may still occur, and the performance might be improved by using worker collaboration.

Ideally, we want a condition that worker collabora-tion can give better performance than cellular bucket brigade in a system with changeable size, then workers need to have flexibility to work at any station, and the work content needs to be adjustable across all stations. However, it is difficult and costly to train workers to be flexible, and task allocation is not easy to adjust. An-other constraint is the additive rate of the collaboration coefficient. In practice, if a single task is processed by multiple workers at the same time, the collaborative processing time may be slower or faster than the total processing time for the individual workers.

Most of our results depend only on the assump-tions that handover time is neglected and walking back is instantaneous. Relaxing those assumptions would be an interesting topic for future research. In addition, we recommend conducting a case study of worker collabo-ration in order picking process at a warehouse. Analyz-ing the behavior and performance of worker collabora-tion would be an interesting topic for future study. In order picking process, workers are flexible to work at any shelf, task allocation of each worker can be adjust-ed, and the shelf can accommodate two workers to work collaboratively to fulfill the order list.

## Figure

A conceptualized U-line with length 1. W1 works at Stage 1 and W2 at Stage 3. The horizontal posi-tion hi is determined by projecting the point where each worker i is located on the horizontal axis.

Operational principle of CBB.

Operational principle of worker collaboration for types 1 and 2.

Throughput and region of CBB when the convergence condition is satisfied (v11 = 0.8, v13 = 1.2, v21 = 1.2, v23 = 0.8, v12 = v22 = 1).

Throughput and region of CBB when the convergence condition is not satisfied (v11 = 1.2, v13 = 0.8, v21 = 0.8, v23 = 1.2, v12 = v22 = 1).

Relative difference in throughput for CBB and worker collaboration when the CBB convergence condition is satisfied (three-station, two-worker U-line).

Relative difference in throughput for CBB and worker collaboration when the CBB convergence condition is not satisfied (three-station, two-worker U-line).

## Table

Movement rules for worker collaboration on a U-line

Classification conditions for worker collaboration type 1

Classification conditions for worker collaboration type 2

Numerical result of relative difference in throughput between CBB and worker collaboration when the CBB convergence condition is satisfied (three-station, two-worker U-line)

Numerical results for relative difference in throughput between CBB and worker collaboration when the CBB convergence condition is not satisfied (three-station, two-worker U-line)

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