1. INTRODUCTION
An assembly line is a floworiented production system in which the productive units performing the operations, referred to as stations, are aligned in a serial manner. On an assembly line, a worker can become starved (no part is available) or blocked (there is no room to place a complete part), so buffers help to smooth and balance the flow of materials. Two common methods of balancing the line are by distributing the workload among the different stations (Becker and Scholl, 2006) and by assigning a storage space to different buffers (Gershwin and Schor, 2000). In real assembly lines, task variability occurs due to human labor, the product mix, and machine breakdowns. On these lines, different issues arise that are not considered in many assembly line balancing problem algorithms that assume deterministic task times.
In a market environment where product variety demands are high and product life cycles are short, assembly systems must achieve flexibility as an essential requirement in order to respond to shorter product life cycles, low to medium production volumes, changing demand patterns, and a higher variety of product models and options (Bukchin et al., 2002). One approach that can maintain flexibility while improving the line balance and productivity is the worksharing system. Based on McClain et al. (2000), there are fewer workers than machines and each worker operates within a zone in the worksharing system. The machine has unique operations that require one worker to be present. The Toyota Sewn System (TSS) and bucket brigades are the most common worksharing systems. According to Bartholdi III and Eisenstein (1996a), a bucket brigade differs from the TSS in two ways. First, although the TSS and bucket brigades allow preemption, the bucket brigade does not restrict workers to the zones of the machines. Preemption occurs when a worker returning (emptyhanded) toward the beginning of his zone encounters a shared machine that is being used and then takes over the task without stopping the process (McClain et al., 2000). Second, when a bucket brigade worker has finished an operation and finds that the next machine is being used, he must wait until the next worker finishes. In the TSS, a worker may leave and commence preemption or return to the beginning of his zone. Bartholdi III and Eisenstein (1996a) showed that a bucket brigade is selfbalancing, with the points where the item is transferred from one worker to the next worker determined by the worker’s speed and tending to be stable.
A real experiment on bucket brigades on an assembly line has been reported by Bartholdi III and Eisenstein (2005). Their study showed that productivity increased and the length of the training period for new workers decreased. However, idling is a potential source of waste even when the workers are properly sequenced. Another real experiment considered order picking at a warehouse (Bartholdi III and Eisenstein, 1996b). By using bucket brigades, workers are not restricted by zones, so any worker can choose from any location. The results showed that the pick rates increased due to more effective work sharing. However, potential waste could occur if the bucket brigade was too large. The line can become congested by boxes, and then workers must walk to put items in the appropriate boxes.
Most studies of bucket brigades have been carried out based on the assumption that the work content is distributed continuously and uniformly over the entire line (e.g., Bartholdi III and Eisenstein, 1996a, 2005; Bartholdi III et al., 1999, 2001, 2006; Armbruster and Gel, 2006; Hirotani et al., 2006). In real conditions, the work content is grouped in various proportions at discrete workstations. Lim and Yang (2009) studied the main problem of discrete workstations: that the maximum possible throughput is not always achieved due to blocking conditions, although preemption or simply handing over task is allowed. They found that workers may block each other, as each workstation can only accommodate one worker at one time.
To offset the decrease in performance as a result of the blocking condition, worker collaboration can be utilized. An experiment on worker collaboration was carried out by Compaq at one of its assembly lines at its Houston facility (McGraw, 1996). One workstation was designed to accommodate three workers to construct computers simultaneously. The experiment showed that there was an improvement in productivity and quality, although the operating costs increased.
In this paper, we propose, based on prior studies and adopting the assumptions of Lim and Yang (2009), to integrate bucket brigades and worker collaboration to counter the blocking condition. If the successor has finished with his item, he then walks to the predecessor’s position to work collaboratively. A maximum of two workers can collaborate on the same task (or at the same workstation), so the worker’s idle time due to blocking can be minimized. We assume that the work content of the product at each station is deterministic and that complicated results will be obtained. Analysis of the deterministic system shows that the working velocity and work content distribution along the line are the most important parameters related to throughput.
Another aim is to identify the conditions that produce higher throughput and to compare the performance of bucket brigades with and without worker collaboration. These conditions are used to extend the bucket brigade condition to achieve maximum throughput using worker collaboration. A procedure has been developed for deriving throughput for the case including worker collaboration.
In the next section, several studies in the literature related to the state of the art in the concept of bucket brigades and worker collaboration are reviewed. The production line model developed along with its major rules and assumptions is presented in Section 3. Section 4 explains the behavior analysis for the larger production line, followed by the procedures used to define the throughput of bucket brigades with worker collaboration and numerical calculations for a small production line and an mworkstation and nworker production line. Finally, Section 5 concludes the paper, presenting the managerial implications and explaining the conclusions regarding bucket brigades with and without worker collaboration.
2. LITERATURE REVIEW
This section presents the theories and significant studies related to bucket brigades and worker collaboration. The review has been divided into two parts: first, studies dealing with bucket brigades and second, studies dealing with worker collaboration.
2.1. Bucket Brigades
The main challenge in the management of a production line is to balance the work assignments so there are no bottlenecks in order to achieve low cost and high productivity. In operation management, this important decision problem is known as the “Assembly Line Balancing Problem” (ALBP) (Becker and Scholl, 2006). The ALBP is Nondeterministic Polynomialtime (NP)hard (Bratcu and Dolgui, 2005), meaning that the optimal line structure can be obtained, but it is sensitive to changes in the environment (i.e., short product life cycles, high demand for product variety, many models of products in small lot sizes, and short leadtimes to the customer), and then the assembly line has to be configured or reconfigured. The assembly system must have flexibility in order to respond to the changes.
Most studies on flexible assembly lines utilize dynamic assignment or worksharing on a serial line. In this type of line, the number of workers is smaller than the number of machines and each worker operates only in his respective zone (McClain et al., 1992, 2000; Gel et al., 2002; Askin and Chen, 2006; Sennott et al., 2006; Anuar and Bukchin, 2006; Bukchin and Cohen, 2013). The two most common worksharing systems are the TSS and the bucket brigade system. Studies of the TSS have already been done based on the worksharing 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 preempted by another worker who is coming back upstream (Bischak, 1996; Zavadlav et al., 1996).
Bartholdi III and Eisenstein (1996a) introduced theoretical studies of serial bucket brigades to dynamically coordinate workers along one assembly line. Workers are not restricted by zones but must maintain their sequence with no passing allowed, so a worker may be blocked by a successor. They proved that slowesttofastest sequencing always attains the maximum possible throughput. Then, they enriched the initial model (Bartholdi III and Eisenstein, 1996b, 1998; Bartholdi III et al., 1999). Bucket brigades were in use in at least two commercial environments: apparel manufacturing/assembly lines (Bartholdi III and Eisenstein, 1996a, 2005) and distribution warehouses (Bartholdi III and Eisenstein, 1996b; Bartholdi III et al., 1999; Koo, 2009; Lim, 2012). A deterministic model of the work is appropriate for apparel manufacturing, where successive units of work are identical, whereas in warehousing, the location and amount of work varies from order to order.
An example of applying bucket brigades for assembling complex products is provided by TUG Manufacturing (Bartholdi III and Eisenstein, 2005). Bucket brigades were chosen to replace a nonstandard procedure for constructing tractors where the turnover rate was particularly high. The new solution was to set up the bucket brigades in a cluster of four workstations around a shared crane. As a result, the productivity increased and the training period for new workers decreased considerably.
In a case study of order picking in a warehouse, the bucket brigade replaced the sequential zonepicking method. Sequential zonepicking restricts each picker to work only within his zone and requires constant supervision to ameliorate unavoidable imbalance (Bartholdi III and Eisenstein, 1996b). The results showed more effective worksharing and increased pick rates using bucket brigades due to the absence of zone restrictions. However, if the bucket brigade was too large, the line could become congested by boxes and then pickers must walk to put items in the appropriate boxes. This walking reduced the pick rates and thus the throughput.
Other studies have expanded the condition of order picking, especially in the narrowaisle picking system. This system is typically characterized by a nopassing restriction in an aisle, and the congestion created by the nopassing condition is termed picker blocking. Several researchers have investigated the effect of picker blocking on order picking performance (Hong et al., 2013, 2015, 2016; Hong, 2015). In order picking situations, bucket brigades provide selforganizing and selfbalancing characteristics. Hong et al. (2015) showed that the circularaisle congestion model is applicable for a bucket brigade order picking situation. They proved that when the backward walk time is instantaneous and the handoff time is zero, the congestion model of bucket brigade order picking is equivalent to the congestion model of the circular aisle abstraction.
Several extended studies have been done by relaxing the simplifying assumptions that the work content is distributed continuously and uniformly over the entire line. Bartholdi III et al. (2001) considered the stochastic work content at workstations and proved a similarity between the deterministic and stochastic systems as the number of stations approaches infinity. Hirotani et al. (2006) considered that blocking happens when the slower worker slows the process by preventing faster workers from continuing the process. They concluded that the positions of workers will not converge to a fixed point, and then the production rate will decrease.
The study by Lim and Yang (2009) is the most relevant for determining solutions for balancing the serial line by considering the real conditions in which the work content is grouped into discrete workstations in various proportions. Lim and Yang (2009) analyzed the throughput of a bucket brigade, focusing on an mworkstation and nworker serial production line. They considered fully and partially crosstrained workers and assumed constant worker speeds for all tasks. They showed that using fully crosstrained teams with slowesttofastest sequencing is not always the best policy for the system. Lim and Wu (2014) analyzed the features of cellular bucket brigades (CBBs), focusing on an mworkstation and twoworker Uline with discrete workstations. They adapted the basic idea of CBBs introduced by Lim (2011) to coordinate workers on the Uline. In the CBB design, the work content is distributed on both sides of an aisle. Two workers work on the production line, with the working velocity depending on the worker and station. Each worker works on one side when he proceeds in one direction and on the other side when he proceeds in the reverse direction; then the two workers exchange their work when they meet. They showed that the throughput is significantly improved when the number of stations in a stage increases from one to two, but there are diminishing returns when each stage is divided into more stations. They concluded that the performance of CBBs will differ according to the situation.
In conclusion, bucket brigades are able to achieve a better balance because they redistribute the work based on the actual time taken by a particular worker to perform a particular task. However, they have the weakness of idling even if the workers are sequenced from slowesttofastest, and the chances of idling increase when the number of workers approaches the number of stations and the workstations are discrete.
2.2. Worker Collaboration
Another way to improve the performance of assembly lines through workforce flexibility has been studied by introducing teamwork or collaborative work on the assembly line. Hackman (2002) mentioned several reports on the advantage of teamwork for manufacturing firms; for example, Procter & Gamble achieves 30 to 40% higher productivity at its 18 teambased plants, Tektronix reports that one selfdirected work team now turns out as many products in three days as it once took an entire assembly line to produce in 14 days, and Shenandoah Life processes 50% more applications and customer service requests using work teams with 10% fewer people.
McGraw (1996) described a team scenario at Compaq where a team of three workers built, tested, and shipped computers at one workstation showing productivity and quality improvements of as much as 25%, although the operating costs were reported to be nearly four times higher than those of a conventional assembly line. With regard to multiproduct scenarios with various product options, the teamworking experiments at Volvo’s Uddevalla and Kalmar plants are probably the most wellknown (Engström and Medbo, 1994; Ellegärd et al., 1992). Workers can collaborate on the same task where a team of workers is assigned to do all or most of the assembly tasks on a particular vehicle. This plant showed an improvement in quality and a decrease in the total lead time, and it was also easy to resolve a request for customization from a customer.
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. Van Oyen et al. (2001) assumed that all workers are identical and can collaborate on the same job without interfering with each other, and furthermore that sufficient machinery, tooling, and so on exist for all workers to be working 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 balance, cooperative teams may not improve system performance unless collaborative efficiency is very high. Most works in the literature have studied the impact of collaboration on the tandem queuing system where workers can work at the same task and have included the additive magnitude of the worker’s collaboration/synergy, or α = 1.0 (Andradόttir et al., 2001; Andradόttir and Ayhan, 2005; Andradόttir et al., 2011; Wang et al., 2015).
Hopp and Oyen (2004) defined a basic measure of collaboration efficiency as the relative percentage increase in average task speed that results from assigning multiple workers to the same task. Sengupta and Jacobs (2004) compared assembly line design without collaboration to the parallel cellbased design of two single tasks with collaboration. They introduced an inefficiency factor that rates the efficiency due to collaboration from no impact to a negative impact in which the productivity of each worker in the collaboration decreases. Peltokorpi et al. (2015) compared four different worker coordination policies (no helping, floater, pairs, and complete helping) on a parallel assembly line under the assumption that their collaborative inefficiency reduces the productivity. Their paper tests collaborative efficiency factors equivalent to the factors of Sengupta and Jacobs (2004), defining minor collaborative inefficiency in pairworking as 10% (with a collaborative coefficient of 0.9) and major inefficiency as 30% (with a collaborative coefficient of 0.7). They suggested the use of a complete helping policy in conditions of minor collaborative inefficiency and a pairs policy in conditions of major collaborative inefficiency.
Utilizing worker collaboration may help to increase the task speed. The integration of bucket brigades and worker collaboration might decrease blocking in some cases and increase the performance of the production line. Based on the theoretical explanations, this paper will test the performance of the integration of a bucket brigade with worker collaboration to counter blocking. The model and some assumptions are adopted from Lim and Yang (2009), and the worker collaboration concept is adopted from Van Oyen et al. (2001) and Peltokorpi et al. (2015). We studied collaboration with an additive condition (α = 1.0) and with inefficient collaborative conditions (α = 0.7 and α = 0.9).
3. THE PRODUCTION LINE
The production line model and assumptions for bucket brigades without worker collaboration are adopted from Lim and Yang (2009). They considered a production line in which each instance of a product is progressively assembled in the same sequence of m stations. The work content of the product at station j is deterministic and is denoted as s_{j}, and the total work content is normalized to 1 so that ${\sum}_{j=1}^{m}{s}_{j}=1}.$ Workers are indexed from 1 to n and remain in this sequence along the production line in the direction of production flow and each worker works with constant and deterministic velocity v_{i}. Workers i  1 and i + 1 are the predecessor and successor, respectively, of worker i. Each worker i is crosstrained to work in zone Z_{i}, a set of adjacent stations along the line. Worker i is fully crosstrained if Z_{i} contains all stations on the line. A worker i<n will be blocked if he finishes his work at station j while worker i + 1 is still working at the next station $j+1\in {Z}_{i}.$ The blocked worker remains idle until the next station becomes available. A worker i<n will be halted if he finishes his work at all stations in Z_{i} before he can hand off his item to worker i+1. The halted worker remains idle until the successor takes over his item. A worker i>1 will be starved if he reaches the beginning of his zone before worker i1 can hand over the item to him. The starved worker remains idle until his predecessor hands over an item. The production line can be conceptually represented by a line with length equal to 1 as shown in Figure 1.
In a bucket brigade without worker collaboration, when the last worker (worker n) finishes work on his item, he then walks back to get the item from worker n1, who in turn walks back to get the item from worker n2 and so on until worker 1 initiates a new item. Each worker i works along the line with constant velocity v_{i} within zone Z_{i}, except when the workers are blocked, halted, starved, and/or working collaboratively.
Worker collaboration at a collaborative station begins when the successor and predecessor can work at any place in the station; they work collaboratively until the task is complete at the end of the station. A collaborative station is defined as a station that allows multiple workers to work together by sharing spaces and tools to complete an item. At a station, a worker will process an item as individual work and/or partial work will be performed by collaboration without any loss of work. At most two workers can work together on the same task, in which case the combined velocity of a team is proportional to the sum of the velocities of the individual workers. Worker collaboration cannot be established at the last station because only the last worker can work at the last station. The collaborative velocity is ${v}_{coll}(i)={\alpha}_{i}({v}_{i}+{v}_{i+1})$ and must be higher than the minimum worker velocity $({v}_{coll}(i)>\text{min}\{{v}_{i},\hspace{0.17em}{v}_{i+1}\}).$ α_{i} is a measure or coefficient of the collaboration or synergy between worker i and worker i + 1, defined as $0<{\alpha}_{i}\le 1.$
We assume that the work content is preemptible when an item is handed off without any loss of work. In the production line, the time taken to process an item is significantly longer than the time taken to hand over the item and the time taken to walk back along the entire length of the line. Therefore, each worker spends negligible time handing over the item and walking back to the predecessor or the beginning of his zone. Table 1 explains the forward and backward rules that must be followed by each worker.
4. BEHAVIOR ANALYSIS
This section explains the behavior of bucket brigades with worker collaboration on a general production line with discrete workstations. The condition will be focused on fully crosstrained workers utilizing slowesttofastest sequencing, which is based on the most reasonable condition and most of the works in the literature.
4.1. Behavior of Worker Collaboration
Figure 2 gives four examples of time charts for a fourstation, threeworker line with similar velocity conditions and a collaboration coefficient α = 1.0. We indicate the collaboration meeting point between worker 1 and worker 2 by a solid black circle and the collaboration meeting point between worker 2 and worker 3 by a solid black triangle. The vertical line in which time elapses without any work progress indicates the worker’s idle time. The behavior of worker collaboration can be divided into singleorbit and multiorbit cyclic behavior. Singleorbit cyclic behavior means that each collaboration meeting point is always at the same station in every iteration. Multiorbit cyclic behavior means that the collaboration meeting points move between the downstream and upstream stations alternately in every iteration.
Maximum throughput can be defined as the throughput of the bucket brigade’s line without idle time. The necessary condition for achieving the maximum throughput with worker collaboration is no idleness occurring and α = 1.0. The throughput decreases when idling occurs on the bucket brigade line, whereas by using worker collaboration with α < 1.0 and elimination of idling, the throughput may be higher than that of the bucket brigade but still lower than the maximum possible throughput.
Let us define ${x}_{i}^{\text{*}}$ as the fixed point of a bucket brigade where workers maintain balance by repeating the respective portion of the work content for each item produced, ${x}_{i}^{\text{**}}$ as the collaboration meeting point at which a pair of workers will start collaboration, and ${j}^{\text{*}}(i)$ as the smallest index of a station within which the fixed point of the bucket brigade falls.
Figure 2a shows singleorbit cyclic behavior without blocking where the collaboration meeting points are unique and are at the same station as the fixed points of the bucket brigade. In singleorbit cyclic behavior, each pair of workers always carries out collaborative work at the same station repeatedly. Multiorbit cyclic behavior without blocking is shown in Figure 2b where the collaboration meeting points move between the upstream and downstream stations in every iteration.
Figure 2c shows singleorbit cyclic behavior with blocking. Since all fixed points of the bucket brigade are located at a single station, the upstream worker is blocked when he finishes his work earlier, while others are still working at the next station. Two workers always meet and start collaboration at the same point in every iteration. Multiorbit cyclic behavior with blocking is shown in Figure 2d. The collaboration meeting points move between the upstream and downstream stations. The upstream worker is blocked when he has finished his work because the next station is still being operated by other workers.
In this paper, we neglect the analysis of multiorbit cyclic behavior for several reasons. Multiorbit cyclic behavior is hard to identify because the system does not converge into singleorbit cyclic behavior. We expect a similar behavior to occur on larger production lines and the movement of collaboration meeting points will be hard to identify. Furthermore, the number of configurations with multiorbit cyclic collaboration behavior will be smaller than the number of configurations with singleorbit cyclic collaboration behavior.
4.2. SingleOrbit Cyclic Behavior
The complexity of the analysis to achieve higher throughput on a line with m stations and n workers increases rapidly with m and n. Based on Figure 2a, a production line can obtain higher throughput with singleorbit cyclic behavior by utilizing worker collaboration if the meeting point for collaborative work is always at the same point in the next iteration. Furthermore, there is no idleness due to blocking, halting, or starvation when collaboration occurs.
Based on Figure 3, with the same definition of ${x}_{i}^{*},{x}_{i}^{**},$ and ${j}^{\text{*}}(i),$ we demonstrate that singleorbit cyclic behavior without blocking will be achieved if all workers start the collaborative work at each collaboration meeting point $({x}_{i}^{**}),$ obtained without any loss or blocking condition. The obtained collaboration meeting points for each pair of workers can be derived simultaneously based on the mathematical formulation developed in this section.
Let us assume that all expected collaboration stations are the stations where the fixed points of bucket brigades are located and that the collaboration meeting points for different pairs of workers occur at different collaboration stations. If all collaboration meeting points $({x}_{i}^{**})$ are obtained and all workers start collaborative work at the obtained points, then singleorbit cyclic behavior can be achieved. Below, we derive the singleorbit cyclic behavior based on those assumptions.
When worker n and worker n1 have just finished the collaborative work at the intended collaboration station, worker n starts the individual work while worker n1 will collaborate with worker n2 at the upstream collaboration station as shown in Figure 3a. The inequality $\frac{\left({{\displaystyle \sum}}_{j=1}^{{j}^{\text{*}}(\text{n}2)}{s}_{j}{x}_{n2}^{**}\right)}{{\alpha}_{n2}{{\displaystyle \sum}}_{j=n2}^{n1}{v}_{j}}<\frac{{{\displaystyle \sum}}_{j=1}^{{j}^{\text{*}}(n)}{s}_{j}{{\displaystyle \sum}}_{j=1}^{{j}^{\text{*}}(n1)}{s}_{j}}{{v}_{n}}$ expresses that the collaboration time for worker n2 and worker n1 at the expected collaboration station is shorter than the individual processing time by worker n. After finishing collaboration, worker n1 starts individual work while worker n has already started his work. If this condition is not satisfied, worker n will finish earlier at the end of the line and then interrupt the collaboration process and continue the collaborative work with worker n1 at collaboration station ${j}^{\text{*}}(n2),$ while worker n2 will go upstream to collaborate with worker n3 or introduce a new item.
Based on the inequality condition, we can determine the position of worker $n({P}_{n})$ after worker n1 and n2 worker have finished collaborating.(1)(2)
When worker n2 and worker n1 have just finished collaborating, worker n1 starts individual work while worker n has already begun his work as shown in Figure 3b. The inequality $\frac{\left({{\displaystyle \sum}}_{j=1}^{{j}^{\text{*}}(n1)}{s}_{j}{{\displaystyle \sum}}_{j=1}^{{j}^{\text{*}}(n2)}{s}_{j}\right)}{{v}_{n1}}\ge \frac{\left(1{{\displaystyle \sum}}_{j=1}^{{j}^{\text{*}}(n1)}{s}_{j}\right){P}_{n}}{{v}_{n}}$ expresses that the processing time by worker n1 must be longer than or equal to the processing time by worker n. Worker n will go to support worker n1 at a collaboration station. If this condition is not satisfied, then worker n1 will be halted until worker n finishes with his item at the end of the line and takes over the item from worker n1.
Based on the inequality condition, the collaboration meeting point for worker n1$({x}_{n1}^{\text{**}})$ and worker can be defined as the individual processing time for worker n multiplied by the velocity of worker n1 added to the length of work contents at ${j}^{\text{*}}(n2).$.
By defining ${x}_{n1}^{**}$ as the collaboration meeting point for worker n and worker n1, if all necessary conditions are satisfied without any loss, then the equilibrium condition will be obtained:(3)
Based on Equation 3, the total time on the left side must be equal to the total time on the right side. The left side indicates the collaboration time for worker n2 and worker n1, the individual processing time for worker n1, and the collaboration time for worker n1 and worker n. The right side indicates the collaboration time for worker n1 and worker n and the processing time for worker n. Equation 3 ensures that the cycle time will be the same for all workers without any loss, and then worker n and worker n1 will always start collaboration at the same position of ${x}_{n1}^{**}.$ in every iteration.
The noncyclic condition may occur if the conditions for cyclic behavior are not satisfied. Consider the situation in which the expected collaboration of worker n and worker n1 at station ${j}^{*}(n1)$ is obtained. If $\frac{\left({\displaystyle {\sum}_{j=1}^{{j}^{\text{*}}(n2)}{s}_{j}}{\displaystyle {\sum}_{j=1}^{{j}^{\text{*}}(n3)}{s}_{j}}\right)}{{v}_{n2}}\ge \frac{\left({{\displaystyle \sum}}_{j=1}^{{j}^{\text{*}}(n1)}{s}_{j}{x}_{n1}^{**}\right)}{{\alpha}_{n1}{{\displaystyle \sum}}_{j=n1}^{n}{v}_{j}}+\frac{\left({x}_{n1}^{**}{{\displaystyle \sum}}_{j=1}^{{j}^{\text{*}}(n1)}{s}_{j}\right)}{{v}_{n2}},$ then the next expected collaboration will take place at station ${j}^{*}(n2).$ If this condition is not satisfied, worker n2 will be blocked at the end of station ${j}^{*}(n2)$ or each pair of collaborative workers will shift to the next downstream station to prevent the blocking condition. Due to noncyclic movement, the next expected meeting point will be hard to determine.
4.3. Procedure to Determine Throughput
We propose a procedure to determine the worker collaboration throughput with m stations and n workers in accordance with Section 4.2. Figure 4 shows the proposed procedure to determine the throughput of bucket brigades with worker collaboration by considering the collaboration coefficient. As seen in Figure 4, with the same definitions of ${x}_{i}^{*},\hspace{0.17em}{x}_{i}^{**},\hspace{0.17em}\text{and}\hspace{0.17em}{j}^{\text{*}}(i),$ we can determine the expected collaboration station and propose condition 1 as follows:(4)
Equation 4 ensures that:

No two neighboring collaboration locations can be at the same station. If there are two or more neighboring collaboration locations at the same station, then the predecessor will always be blocked by successor workers.

No collaboration occurs at station m. Only the last worker can work at station m.
In the next step of Figure 4, we determine the collaboration meeting point for each pair of workers. By substituting Equations 1 and 2, we determine ${x}_{n2}^{**}$ as the collaboration meeting point for worker n2 and worker n1.(5)
Then, by substituting Equations 3 and 5, we can determine ${x}_{n1}^{**}$ as the meeting point for collaboration between worker n and worker n1.(6)
Assume that all collaboration coefficients are additive such that ${\alpha}_{i}=1.0;$ then the relationship between the fixed point of the bucket brigade and the collaboration meeting point is as follows:(7)
Based on singleorbit cyclic behavior and the equilibrium condition, the collaboration meeting point of the next upstream pair can be determined for $i=\text{3},\text{}\dots ,\text{(}n\text{1)}$ as follows:(8)
Halting can occur if the processing time for the last worker is longer than the collaboration time for worker n1 and worker n2 and the processing time for worker n1. Thus, the no halting condition can be derived as follows:(9)(10)
On a larger production line, blocking may occur when the collaboration time for a downstream pair is longer than the collaboration time for the next upstream pair. Under the blocking condition, after the collaboration between worker n3 and worker n4 has finished, worker n3 will do individual work at the downstream station and will be blocked at the end of the downstream station, because worker n1 and worker n2 are still performing collaboration. Condition 2 ensures that the noblocking condition exists for $i=\text{1},\hspace{0.17em}\dots ,\hspace{0.17em}n\text{1}.$.
The cycle time (CT) of singleorbit cyclic behavior for worker collaboration on an mstation and nworker line without loss can be determined as follows:(11)
Based on Figure 4, if one of those conditions is not satisfied, then a line might sustain singleorbit cyclic behavior with the halting or blocking condition. In this condition, the movement of bucket brigades without worker collaboration will be utilized. We have developed a computational calculation to determine the throughput of bucket brigades without worker collaboration, not only under the condition in which the maximum throughput possible can be achieved but also under the blocking and halting conditions. The computational calculation is developed based on the iteration process by comparing the processing time among workers in accordance with the fixed points and work content at each station.
4.4. Numerical Calculation
In this section, we present two performance comparisons between bucket brigades with and without worker collaboration. First, we consider a threestation, twoworker production line with two conditions: slowesttofastest sequencing and fastesttoslowest sequencing. Most studies in the literature state that sequencing workers from slowesttofastest leads to superior performance compared to fastesttoslowest sequencing. We consider these two worker sequences to observe the usability of worker collaboration when there is variability in the work content. As a result, we show the characteristics of the regions with fully and partially crosstrained workers. Second, the performance analysis considers fully trained workers, slowesttofastest sequencing, and variation of the collaboration coefficient index. We focus on the behavior that achieves higher throughput according to Section 4.2.
4.4.1. ThreeStation, TwoWorker Production Line
Figures 5 and 6 show the regions of a production line in work content distribution with fully or partially crosstrained workers and slowesttofastest and fastesttoslowest sequencing, respectively. The horizontal and vertical axes correspond to s_{1} and s_{2}, respectively. Each point on the diagram represents the distribution of work content at the workstations. The feasible region is s_{2} < 1  s_{1}, and the behavior in each region can be summarized as follows:

Region 1:
The predecessor is always halted at point s_{1} + s_{2} because the work content at the first two stations is small, so the predecessor can finish his work at these stations before the successor arrives to collaborate.

Region 2:
The meeting point for collaboration is located at station 2. There is no blocking, so the maximum throughput possible may be achieved if α = 1.0.

Region 3:
The first and second collaborations are located at stations 1 and 2, respectively. After finishing collaboration at station 1, the predecessor will be blocked at point s_{1}.

Region 3a:
The collaboration meeting point is located at station 1. After finishing collaboration, the predecessor will be blocked at point s_{1} and halted at point s_{1} + s_{2}.

Region 4:
The first and second collaborations are located at stations 1 and 2, respectively. The maximum throughput possible may be achieved if α = 1.0.

Region 4a:
The collaboration is located at station 1. After finishing collaboration, the predecessor will be halted at point s_{1} + s_{2}.

Region 5:
The meeting point for collaboration is located at station 1. With fully crosstrained workers, there is no blocking, so the maximum throughput possible may be achieved if α = 1.0. Meanwhile, with partially crosstrained workers, the successor is always starved at point s_{1}.
Figure 7 shows the impact of worker collaboration with fully and partially crosstrained workers and slowesttofastest and fastesttoslowest sequencing. Figures 7a, 7b, and 7c show the percentage difference in throughput between bucket brigades with and without worker collaboration for α = 0.7, α = 0.9, and α = 1.0, respectively. Based on these figures, regions 2, 4, and 5 are most affected by the increase in α. The workers’ collaborative performance can be equal to or better than their performance without worker collaboration at α = 1.0.
Figure 7d shows the percentage difference in throughput between bucket brigades with and without worker collaboration with fully crosstrained workers, fastesttoslowest sequencing, and α = 1.0. Although region 1 is the largest compared to the other regions, worker collaboration still has a significant impact on increasing the performance when blocking occurs, which is represented in the parts of regions 2 and 5.
Figures 7e and 7f show the percentage difference in the throughput of bucket brigades with and without worker collaboration with partially crosstrained workers at α = 1.0 for slowesttofastest and fastesttoslowest sequencing, respectively. In region 5, the successor will be starved in front of s_{2} while the predecessor is still working on s_{1}. Worker collaboration can only increase the performance in the part of region 2.
Based on Figure 7, with fully crosstrained workers, better performance is achieved with worker collaboration than without it in regions 2 and 5. Meanwhile, with partially crosstrained workers, worker collaboration only performs better in region 2 due to the starvation condition in region 5.
4.4.2. An mStation and nWorker Production Line
The workstation configurations are derived from all possible work contents that can be accommodated by each station. The work content of the product at each station is deterministic and the total work content is normalized to 1. For each station, we apply a workcontent multiplication factor of 0.05; that is, the first configuration is s_{1} = 0.05, s_{2} = 0.05, s_{3} = 0.05, and s_{4} = 0.85, the second configuration is s_{1} = 0.05, s_{2} = 0.05, s_{3} = 0.1, and s_{4} = 0.8, and so on. Worker velocities are generated by a uniform random number with a range of 0.1 to 1.0, sorted from the lowest to the highest value to represent the sequence of workers on the production line.
Table 2 shows a performance comparison of bucket brigades with and without worker collaboration at an mworkstation, nworker line by considering the variation of the collaboration coefficient (α). For a fourworkstation, threeworker line, there are 969 work content configurations, comprising 37.34 configurations for bucket brigades with maximum throughput (no idling) and 931.66 configurations for bucket brigades with idling. Worker collaboration with α < 1.0 cannot achieve maximum throughput but can decrease the number of configurations with idling compared to the bucket brigade: 759.46 and 698.96 for α = 0.7 and α = 0.9, respectively. However, worker collaboration can increase the number of configurations with the maximum throughput at α = 1.0: 292.08 configurations compared to 37.34 configurations when using bucket brigades.
For a fiveworkstation, fourworker line, the number of combinations of work contents increases to 3876 configurations, but the number of bucket brigades with maximum throughput (no idling) decreases to 17.8 configurations, and the number of bucket brigades with idling increases to 3858.2 configurations. Utilizing worker collaboration can decrease the number of configurations with idling, and though it cannot achieve maximum throughput at α < 1.0, it can increase the number of configurations with the maximum throughput at α = 1.0. The application of worker collaboration with α < 1.0 cannot achieve maximum throughput but can decrease the number of configurations with idling compared to the bucket brigade: 3603.24 and 3492.26 for α = 0.7 and α = 0.9, respectively. At α = 1.0, however, worker collaboration can increase the number of configurations with the maximum throughput: 447.84 configurations compared to 17.8 configurations when using bucket brigades.
Based on Table 2, worker collaboration can effectively reduce the number of configurations with idling that may occur when using bucket brigades. If the number of stations and the number of workers increase, then without worker collaboration the number of configurations with the maximum throughput will decrease. However, by utilizing worker collaboration at α < 1.0, maximum throughput cannot be achieved, but the number of configurations without idling is higher than for the case without worker collaboration. The maximum throughput can be obtained by worker collaboration only at α = 1.0.
7. MANAGERIAL IMPLICATIONS AND CONCLUSIONS
We desire a condition where worker collaboration at discrete workstations can achieve its full production capacity. Cyclic behavior and the equilibrium condition for worker collaboration can be achieved if each pair of workers always starts the collaboration at the same meeting point in every iteration; then the characteristics of a selfbalancing line can still be preserved, and a performance improvement can be obtained.
To obtain the desired condition in a system with changeable size, workers need to have the flexibility to work at any station, and the work content needs to be adjustable across all stations. However, in practical cases, it is difficult and expensive to train workers to be flexible, and task allocation cannot be adjustable. Another limitation is that a station must be able to accommodate two or more workers simultaneously and the work content must be able to be divided among the workers at any time. Moreover, the condition where a downstream worker can join and assist the upstream worker with an additive collaboration coefficient is rare. If a single task is processed by multiple workers at the same time, the collaborative processing time might be faster or slower than the sum of the processing time by individual workers.
This paper focuses on how to improve the bucket brigade’s performance with discrete workstations by integrating worker collaboration. Possible extended conditions for improvement and a procedure for achieving a possibly higher throughput have been described. By analyzing the singleorbit cyclic behavior of worker collaboration, the characteristics and throughput formulation can be obtained.
On a threestation, twoworker line and an mstation, nworker line with fully crosstrained workers and slowesttofastest sequencing, the bucket brigade with worker collaboration almost always outperforms that without worker collaboration. When the collaboration coefficient is not additive, two workers can process the same task less than twice as fast as one worker, but worker collaboration can still effectively contribute some performance improvement.
Most of our results depend only on the assumptions that each worker works at a constant work velocity that is associated with the working zone, walking back is instantaneous, and handover time is neglected. Relaxing those assumptions would be an interesting topic for future research.
We also suggest a case study of worker collaboration in the order picking process at a warehouse where workers are flexible in that they may work at any shelf, task allocation can be adjusted, and the shelf (station) can accommodate two workers to evaluate the behavior and performance of worker collaboration in this type of setting.