1. INTRODUCTION
In traditional assembly lines, each worker usually carries out their assigned job at a fixed work station. The methods of balancing traditional lines have been studied previously (e.g., Scholl, 1995). However, when an imbalance between worker's speeds exists, the slowest worker delays the production rate of the line. Thus, the “selfbalancing line” was introduced by Aisin Seiki Co., Ltd., a subsidiary of Toyota (Bartholdi and Eisenstein, 1996), for Toyota Production Systems (TPS), and has since been utilized in commercial environments such as apparel manufacturing and distribution warehousing (Bartholdi et al., 1999), as well as luxury assembly lines (De Carlo et al., 2013). Selfbalancing lines assign work dynamically; when the last worker in the line completes an item, he/she walks back to and takes over from the previous worker, who then walks back to and takes over from the worker before, and so onuntil the first worker, who instead begins work on a new item. Since faster workers therefore carry out more work per item, and slower workers less, a good balance is maintained. Moreover, to achieve the maximum production rate, the workers should be sequenced from slowest to fastest (Bartholdi and Eisenstein, 1996). Selfbalancing lines with three workers have been examined numerically by simulation (Bartholdi et al., 1999), and those with n workers have been analyzed analytically (Hirotani et al., 2006).
Furthermore, many recent papers have focused on orderpicking lines, as detailed by Lim (2012). Lim (2011) analyzed a cellular line, which divides into two to accommodate workers of different speeds, which was expanded on by Zhou et al. (2015) to consider the constraints of worker processing. Lim and Wu (2014) analyzed a cellular Ushaped line with discrete work stations. Hong et al. (2015) focused on the blocking for order picking systems with serial lines that were considered by Koo (2009). Webster et al. (2012) considered the probability distribution of the workload with serial lines.
Although typically assumed to be serial, recent studies have assumed a Yshaped type line (with two sublines that then combine). Yshaped lines are also suitable for orderpicking systems, because a worker can access the picked item easily. Xu et al. (2014) analyzed a treeshaped line, with two sublines that then combine like a Y; There are three points located according to the Y shape, and each truck picks an item according to the Y’s route. Bartholdi et al. (2006) found that, if walkback and travel times are ignored, the result of a traditional serial line can be applied. Xu et al. (2014) considered walkback and travel time for only two workers. In this paper, the authors consider a Yshaped line with more than two workers. The authors derive the balance condition for three workers, with consideration to walkback time, while comparing the result to that of serial lines (Hirotani et al., 2005). The authors then try to find the condition that balances the line for n workers.
This paper is organized as follows: In Section 2, assumptions and characteristics of this selfbalancing line with walkback time are explained. In Sections 3 and 4, the authors analyze the line for three and n workers, respectively. Finally, the authors present concluding remarks in Section 5.
2. THE LINE
2.1. Assumptions
In this paper, a line with the following assumptions is considered. These assumptions are almost the same as those in Xu et al. (2014), except that there are n workers.

1. Each worker processes only one identical item sequentially.

2. The line is shaped like a “Y” (see Figure 1). There are three edge points (L, R, E) and one center point (D) in this line. There are three sublines (LD, RD, DE). The length of each line is l, r, d, respectively. Note that l < r and l + r + d = 1. The authors call the sublines the L section, R section, and D section, respectively.

3. Workers are sequenced from 1 to n on the line, under the condition that these sublines combine into one line from point L to point E. Each worker processes according to the line from point L to point E. If any worker arrives at point D, he/she has to travel to point R, and then he/she continues to process from point R. The process flow is also shown in Figure 1. No worker can pass over the upstream or downstream workers.

4. Worker i processes with working speed vi regardless of the line (v_{i} > 0). Furthermore, each worker has the same walkback and travel speed v_{r} as all other workers (v_{r} > 0).

5. When the last worker (worker n) finishes processing an item, he/she walks back to and takes over from worker n − 1, who then walks back to and takes over from worker n − 2; similarly, each workers walks back to and takes over from their preceding worker— except worker 1, who introduces a new item into the system from point L. The takeover time is ignored.
The position of worker i when he/she starts to process is given by x_{i}. Then, the position at iteration t is defined as ${x}_{i}^{\left(t\right)}$. Note that ${x}_{1}{}^{\left(t\right)}$ = 0 for any iteration t, because the first worker always starts to process a new item.
2.2. SelfBalancing and Convergence
In the serial line, it has been proven that the line can maintain balance when workers are sequenced from slowest to fastest, with a constant working speed for each worker. Subsequently, the position of workers will converge to a unique fixed point. Under this condition, the production rate can be calculated as the sum of each worker's speed, as derived mathematically by Bartholdi and Eisenstein (1996). The condition for convergence, that the line can be balanced for n workers, has been found in a previous paper (Hirotani et al., 2006).
2.3. Behavior of Workers in YShaped Lines
In this line, the difference to the traditional serial line with walkback time is travel time. Figure 2 shows a time chart for three workers. In Figure 2, the dashed line represents instantaneous movement at point D. This phenomenon occurs because point D has two different points (i.e., 0.2 and 0.5 in Figure 2). If worker 1 arrives at point D, he/she has to move to point R. The travel takes time r × v_{r}. In addition, as circled in Figure 2, worker 2 takes an item to worker 1 while worker 1 is traveling. Thus, neither worker can process immediately when taking over an item, and have to travel or walkback more compared to a normal takeover.
3. ANALYSIS FOR THREE WORKERS
The authors compare the result of the serial line with walkback time. This is because if travel time is ignored, the line is the same as the serial line with walkback time. Therefore, the authors utilize the result of Hirotani et al. (2005). Based on the results, the authors analyze the line with walkback and travel time, expect with a Yshaped, rather than serial, line. The condition of convergence is that the line can balance for n workers for all i (i = 2, 3, …, n), given for the serial line as follows (Hirotani et al., 2005):
At convergence, the aforementioned fixed point at which worker i (i = 2, 3, …, n) starts to process can be defined for a serial line as follows (Hirotani et al., 2005):
where V_{n,k} is the productsum of all combinations of the velocities of k workers chosen from v_{1} to v_{n}, and C_{i} is the coefficient in V_{n,k}. For instance, C_{2}V_{3,1} = 1×v_{1} + 1×v_{2} + 0× v_{3} = v_{1}+v_{2}. Note that V_{n}_{,0} = 1 and C_{0}V_{n,k} = 0 for all n and k, and that ${x}_{1}^{*}$ = 0 since worker 1 always starts to process a new item.
By developing Equation (2), the fixed point of each worker for the three workers with walkback time is as follows:
The authors now divide the six areas according to the positions of ${x}_{2}{}^{*}$ and ${x}_{3}{}^{*}$. Further analysis is carried out according to area.
In the previous study by Xu et al. (2014), with two workers, and the assumption that v_{1} < v_{2} < v_{r}. Under these assumptions, they analyze according to s(3):
This is a relative processing time and special case of Equation (2) under n = 2. Using this condition, they divide into four areas: (a) s < l, (b) l < s < r, (c) r < s < (l + r), and (d) s > (l + r). With fewer than two workers, all workers are affected by the condition for balancing. However, if more than two workers are assumed, there is a case that some, but not all, of the workers affect the condition for balancing. Therefore, this previous result cannot be utilized easily. In this section, the authors assume three workers and analyze according to the positions of ${x}_{2}{}^{*}$ and ${x}_{3}{}^{*}$. Where these positions are located along the section is important. There are three sections in this line; R, L, and D. Thus, the authors can divide into six areas: (1) ${x}_{2}{}^{*}<{x}_{3}{}^{*}\le l,\hspace{0.17em}(2)\hspace{0.17em}{x}_{2}{}^{*}\le l<{x}_{3}{}^{*}\le \left(l+r\right),\hspace{0.17em}(3){x}_{2}{}^{*}\le l,\hspace{0.17em}\left(l+r\right)\le {x}_{2}{}^{*},$ $(4)l\le {x}_{2}{}^{*}<{x}_{3}{}^{*}\le \left(l+r\right),\hspace{0.17em}(5)\hspace{0.17em}l\le {x}_{3}{}^{*}<\left(l+r\right)\le {x}_{3}{}^{*},\hspace{0.17em}\text{and}\hspace{0.17em}(6)\left(l+r\right)\le {x}_{2}{}^{*}<{x}_{3}{}^{*}$.
3.1. Case of ${x}_{2}{}^{*}<{x}_{3}{}^{*}\le l$
In this case, worker 1 processes only the L section, while workers 2 and 3 process all sections (L, R, D) (see Figure 3). Therefore, worker 3 has to travel when moving from point D to R, with time r × v_{r}. If the line is balanced, the following equation can be made, since the time of one iteration is the same for all workers:
For worker 1, the processing time is ${x}_{2}{}^{*}$/v_{1}; the walkback time is ${x}_{2}{}^{*}$/v_{r}. Similarly, for worker 2, the processing time is (${x}_{3}{}^{*}$− ${x}_{2}{}^{*}$)/v_{2}; the walkback time is (${x}_{3}{}^{*}$− ${x}_{2}{}^{*}$)/v_{r}. For worker 3, the processing time is (1 − ${x}_{3}{}^{*}$)/v_{3}; the walkback time is (l − ${x}_{3}{}^{*}$+ d)/v_{r}; and the travel time is r/v_{r}.
As a result, this relationship is the same as that of Hirotani et al. (2005). Therefore, the same result (Equations (1) and (2)) can be used.
In this case, the production rate (PR) is as follows:(4)
3.2. Case of ${x}_{2}^{*}\text{\hspace{0.17em}}\le \text{\hspace{0.17em}}l\text{\hspace{0.17em}}<\text{\hspace{0.17em}}{x}_{\text{3}}{}^{*}\text{\hspace{0.17em}}\le \text{\hspace{0.17em}}(l\text{\hspace{0.17em}}+\text{\hspace{0.17em}}r)$
In this case, worker 1 processes the L section, worker 2 processes the L and R sections, and worker 3 processes the R and D sections (see Figure 4). Therefore, worker 2 has to travel from point D to R with time r × v_{r}. If the line is balanced, the following equation can be made, since the time of one iteration is the same for all workers:
As a result, the processing time for each worker is the same as in the serial line (Hirotani et al., 2005). However, walkback and travel time are not the same for worker 2. The distance of walkback and travel is $2\left(r+l\right){x}_{3}{}^{*}{x}_{2}{}^{*}$ Therefore, increasing distance from serial line is $2\left(r+1{x}_{3}{}^{*}\right)$. As a result, the result of Hirotani et al. (2005) cannot be utilized here.
Deriving the above equations, ${x}_{2}{}^{*}$ and ${x}_{3}{}^{*}$ can be given as follows:
Compared to the equations for ${x}_{2}{}^{*}$ and ${x}_{3}{}^{*}$ in section 3, this solution is not feasible according to the value of d. Therefore, there is a region in which the fixed point does not exist.
In this case, the production rate (PR) is as follows:
Of course, this PR is smaller than that in the case of the previous subsection.
3.3. Case of ${x}_{\text{2}}{}^{*}\le \text{\hspace{0.17em}}l,\text{}(l\text{\hspace{0.17em}}+\text{\hspace{0.17em}}r)\text{\hspace{0.17em}}\le \text{\hspace{0.17em}}{x}_{\text{3}}{}^{*}$
In this case, worker 1 processes only the L section, worker 2 processes all sections, and worker 3 processes only the D section (see Figure 5). Therefore, worker 2 has to travel when moving from point D to R with time r × v_{r}. If the line is balanced, the following equation can be made, since the time of one iteration is the same for all workers:
As a result, this relationship is the same as that of Hirotani et al. (2005). Therefore, the same result (Equations (1) and (2)) can be used.
3.4. Case of $l<{x}_{2}{}^{*}<{x}_{3}{}^{*}\le \left(l+r\right)$
In this case, worker 1 processes the L and R sections, worker 2 processes only the R section, and worker 3 processes the R and D sections (see Figure 6). Therefore, worker 1 has to travel from point D to R with time r × v_{r}. If the line is balanced, the following equation can be made, since the time of one iteration is the same for all workers:
As a result, the processing time for each worker is the same in the serial line (Hirotani et al., 2005). However, walkback and travel time are not the same for worker 1. The distance of walkback and travel is $2\left(r+l\right){x}_{3}{}^{*}{x}_{2}{}^{*}$; an increasing of 2(r + l − ${x}_{2}{}^{*}$). Thus, the result of Hirotani et al. (2005) cannot be utilized.
Deriving the above equations, ${x}_{2}{}^{*}$ and ${x}_{3}{}^{*}$ can be derived as follows:
Compared to the equations for ${x}_{2}{}^{*}$ and ${x}_{3}{}^{*}$ in section 3, this solution is not feasible according to the values of l and r. Therefore, there is a region in which the fixed point does not exist.
In this case, the production rate (PR) is as follows:
Of course, this PR is smaller than that in the case of the previous subsection.
3.5. Case of $l\text{\hspace{0.17em}}<\text{\hspace{0.17em}}{x}_{\text{2}}{}^{*}<\text{\hspace{0.17em}}(l\text{\hspace{0.17em}}+\text{\hspace{0.17em}}r)\text{\hspace{0.17em}}\le \text{\hspace{0.17em}}{x}_{\text{3}}{}^{*}$
In this case, worker 1 processes the L and R sections, worker 2 processes the R and D sections, and worker 3 processes only the D section (see Figure 7). Therefore, worker 1 has to travel from point D to R with time r × v_{r}. If the line is balanced, the following equation can be made, since the time of one iteration is the same for all workers:
This result is the same as in the previous subsection.
As a result, the result of Hirotani et al. (2005) cannot be utilized and an additional condition is needed to balance the line. The production rate is the same as in the previous subsection.
3.6. Case of $\left(l+r\right)\le {x}_{2}{}^{*}<{x}_{3}{}^{*}$
In this case, worker 1 processes all sections, and workers 2 and 3 process only the D section (see Figure 8). Therefore, worker 1 has to travel from point D to R with time r × v_{r}. If the line is balanced, the following equation can be made, since the time of one iteration is the same for all workers:
As a result, this relationship is the same as that of Hirotani et al. (2005). Therefore, the same result (Equations (1) and (2)) can be used.
3.7. Case Where a Takeover Occurs When a Worker is Traveling
This is a very special case. As shown in Figure 2, worker 2 takes over an item when worker 1 is traveling. In this case, both workers have to travel more after taking over an item. Worker 1 has to walkback to position L, and worker 2 has to travel to position R. The worker’s behavior in this case is shown in Figure 9.
If the line is balanced and ${x}_{3}{}^{*}$ is located in the D section, the following equation can be made, since the time of one iteration is the same for all workers:
Where ${x}_{2}{}^{*}$′ is a fixed point when worker 1 is traveling. The positions of ${x}_{2}{}^{*}$ and ${x}_{2}{}^{*}$′ are the same; however, the behavior after item takeover is different. In this equation, for worker 1, processing time is l/v_{1}; walkback time is $\left[l+\left(l+r{x}_{2}{{}^{*}}^{\prime}\right)\right]/{v}_{r}$; and travel time is $\left(l+r{x}_{2}{{}^{*}}^{\prime}\right)/{v}_{r}$. Similarly, for worker 2, processing time is (${x}_{3}{}^{*}$− l)/v_{2}; walkback time is $\left[\left({x}_{3}{}^{*}l\right)+\left({x}_{2}{}^{*}l\right)\right]/{v}_{r}$; and travel time is $\left({x}_{2}{{}^{*}}^{\prime}l\right)/{v}_{r}$. For worker 3, processing time is (1− ${x}_{3}{}^{*}$)/v_{3}; and walkback time is $\left(1{x}_{3}{}^{*}\right)/{v}_{r}$. In this result, both workers 1 and 2 have to travel or walkback more. Therefore, the previous result of Hirotani et al. (2005) cannot be utilized.
3.8. Summary
The authors now summarize the results of the previous sections. Table 1 is a production rate (PR) comparicomparison between the results. In this table, three cases have the same PR. Therefore, these three cases should be arranged. Considering these results, authors can say that only one worker should process in the R section. Even if this cannot be achieved, ${x}_{2}{}^{*}$ and ${x}_{3}{}^{*}$ should be near the D point to prevent decreasing the production rate.
4. ANALYSIS FOR N WORKERS
In this section, the authors try to expand the result of the previous section to n workers. The authors found similarity with Hirotani et al. (2005). Furthermore, the condition for n workers for a serial line with walkback time has been found previously. Therefore, the authors can easily expand this to generalize to n workers. As shown in the previous sections, only one worker should process in the D section to prevent decreasing the production rate. Applied to n workers, this should be considered by using Equations (1) and (2). The flowchart for balancing the line and achieving the maximum production rate is shown in Figure 10. If the maximum production rate cannot be achieved, the production rate can be calculated using Equations (5) and (6), and any sequence can be set according to the desired production rate.
Note that the slowest to fastest sequence is the best in the serial line (Bartholdi and Eisenstein, 1996). Therefore, the authors used this sequence first.
The authors show an example for four workers (A, B, C, D). Assume v_{A} = 1, v_{B} = 2, v_{C} = 3, v_{D} = 4, v_{r} = 10, l = 0.15, r = 0.3, and d = 0.55. According to Figure 10, the authors sequenced workers as ABCD. Next, by Equation (2), the authors calculate fixed points. By calculation, ${x}_{2}{}^{*}$ = 0.117, ${x}_{3}{}^{*}$ = 0.323, and ${x}_{4}{}^{*}$ = 0.631. Then, authors judge the condition that only one worker processes in the R section. By calculation, this condition is not satisfied. Therefore, sequence must be changed. According to Equation (1), the authors sequenced workers as ADBC. Then, by Equation (2), the authors calculate fixed points. By calculation, ${x}_{2}{}^{*}$ = 0.117, ${x}_{3}{}^{*}$ = 0.487, and ${x}_{4}{}^{*}$ = 0.702. Then, the authors judge the condition that only one worker processes in the R section. By calculation, this condition is satisfied, giving the solution ADBC. In general, the fastest worker should process in the R section. However, in the slowest–fastest sequence, this is not satisfied. In fact, if the fastest worker works in R section.
5. CONCLUDING REMARKS
In this paper, the authors consider a Yshaped selfbalancing line. By analysis and comparison to the traditional serial line with walkback time, the authors derive the production rate for each case of fixed worker points. To achieve the maximum production rate, only one worker should process in the R section. Accordingly, the authors propose an algorithm for balancing the line and achieving the maximum production rate for generalized n workers.
However, the derived condition for balancing the line is very limited; future research is needed to proposing an algorithm for increasing the production rate.