1. INTRODUCTION
Order fulfillment centers in ecommerce employ different picking strategies to respond quickly to customer demand. For example, when fulfilling a large number of smallsized orders with varying weights, sizes, and fragile parts, manual order picking combined with a zone picking strategy facilitates operational performance by allowing cooperation between pickers across the zones. Manual pickers can adjust their operational scope when neighboring zones experience a heavy workload. In a pickandpass strategy, any picker can pick at any pick face without violating the sequence of upstream and downstream pickers. A pickandpass strategy combined with dynamic resizing of the entire work area eliminates the need for work zone load balancing and improves order picking throughput.
Bucket brigade order picking (Bartholdi III and Eisenstein, 1996a), hereafter bucket brigade OPS, is well known for its flexible work range. During collaboration between neighboring pickers in a bucket brigade order picking, however, productivity losses are incurred when downstream pickers have to wait for new totes from the upstream pickers. According to the pickandpass characteristic, the upstream picker has priority and the downstream picker waits until the upstream picker passes the tote. The productivity loss caused by the waiting delay occurs at every handoff and across all pickers except for the most upstream pickers (Hong, 2018).
Solving handoff delay problems require synchronization between two neighboring pickers. To shorten handoff delay, Koo (2009) suggests a combined zone and bucket brigade picking while Bartholdi III and Eisenstein (1996b) suggest a batch order picking. The zone picking can relax the tight coupling of handoff delay between pickers while the batch order picking can reduce the frequency of handoffs. A smoother handoff relies on the pickers’ flexible adjustment of picking time and walking speed. This study approaches the handoff delay problem similarly to the smooth handoff strategy of Bartholdi III and Eisenstein (1996b), but from the viewpoint of the upstream picker.
We introduce a loosely coupled handoff operation, which is frequently preferred by upstream pickers to mitigate the expected handoff delay of downstream pickers, and develop an analytical model for less handoff delay in a bucket brigade OPS. During a handoff operation, early release of totes could improve order picking synchronization and quantify the impacts of the early release as looseness. We use the model and its numerical analysis to observe the effects of the early release of totes and then develop a practical control framework of the handoff delay that considers the expected operational time, the workinprocess level, and the control position based on the looseness parameters. A metaheuristic search obtains a combination of parameters for less handoff delays and higher throughput. We conduct a simulation study to clarify the effects of the control parameters in diverse situations.
The paper is organized as follows. Section 2 details the handoff delay in bucket brigade order picking systems, reviews the relevant literature, and identifies new opportunities. Section 3 introduces the loosely coupled handoff operation, develops an analytical model using a stochastic process, and conducts numerical and simulation analyses. Section 4 describes a mitigation framework of handoff delay, identifies an optimal operational environment with metaheuristics, and validates the framework with simulation studies. Section 5 concludes and suggests future research.
2. BUCKET BRIGADE ORDER PICKING
2.1. Handoff Delay in Bucket Brigade Order Picking
We consider a bucket brigade order picking system (OPS) with flowrack shelving (Figure 1) and a pickandpass strategy (Bartholdi III and Eisenstein, 1996a). Pickers make a trip for parts (items) and retrieve them from the flowrack shelves. Pickers, sequenced from upstream to downstream, collaborate to fill totes. After completing the picks assigned, the upstream picker passes the tote to the downstream picker, who is ready for the next picks. The pickers dynamically and independently resize the work area to balance the workload between them.
In a bucket brigade OPS, throughput relies on the flexibility of the handoff (pick face) location between neighboring pickers. If an upstream picker is picking when a downstream picker encounters the upstream picker, the downstream picker has to wait until the upstream picker completes the pick. This is termed handoff delay (Figure 2). If an upstream picker is not allowed to pass the downstream picker, because the pickers have to maintain the sequence in the work zone, this is termed picker blocking delay.
2.2. Literature Review
The synchronization “gap” between two pickers triggers a handoff delay. Hong (2018) proves that a handoff delay is comprised of both pickingdependent delays and walkingdependent delays, and that the former is responsible for most of the delay. The pickingdependent delay also includes the magnitude and variation of the picking time. Even if there is zero variation, however, the randomness of the upstream picker’s arrival causes a handoff delay.
Bartholdi III and Eisenstein (1996b) improve the smoothness of picking to reduce blocking delay. Less handoff delay also results from batch picking introduced by Bartholdi III and Eisenstein (1996b) and the model optimized by Hong et al. (2016). Koo (2009) adopts a zoning strategy where an upstream picker limits the boundary of the downstream move. When the upstream picker reaches the boundary, the tote is left there, and then the upstream picker walks backward. Because some handoffs occur in a zone boundary, there is less blocking delay.
Several studies quantify handoff delay to understand bucket brigades. To analyze an assembly line, Bartholdi III and Eisenstein (2005) define handoff delay as a constant value and explain that handing off a workload requires significant time. To analyze order picking throughput, they quantify the amount of delay as a preliminary value. Hong et al. (2015) adjust a blocking delay model by using the expected handoff delay. Bartholdi III and Eisenstein (1996a) suggest a smooth handoff operation, but do not describe the operational implementation. In addition, simultaneous and synchronous handoff (Bartholdi III et al., 2001) does not apply when both pick time and walk time are finite.
Lim (2017) asserts that the celluar bucket brigade is preferable compared to the serial bucket brigade because the cellular bucket brigade experiences half the number of handoff occurrences. Hong (2018) quantifies the loss by the handoff delay as a closedform expression of pick time distribution over walk time in a discrete bucket brigade order picking system and observes less handoff delay with the small variance of the pick time.
3. PROBLEM DEFINITION
3.1. A Loosely Coupled Handoff Operation
In a tightly coupled operation, the upstream picker hands off the tote to the downstream picker when the downstream picker arrives at the adjacent next pick face. Our proposed policy relaxes this tight restriction. As depicted in Figure 3, the upstream picker decides to: 1) move forward to the next pick face and retrieve the next pick; or 2) move backward after releasing a tote at the current location. In the latter case, which we term a loosely coupled handoff, the downstream picker starts the next pick upon taking over the tote.
The upstream picker’s decision to release a current tote is based on the synchronization gap between the expected completion time of the upcoming pick face and the expected arrival time of the downstream picker at the same pick face. Without loss of generality, the upstream picker decides to release the tote on the boundary if the expected returning time of the downstream picker is less than or equal to threshold τ. Below, we describe a renewal process model to quantify the effects of threshold τ on handoffs.
3.2. The Expected Handoff Delay Over τ
We assume instantaneous walk time, i.e., infinite walk speed, random pick time, and constant WIP and consider a handoff between two identical pickers. In Figure 4 (a), the upstream picker conducts picks and walks repeatedly. Because of the infinite walk speed assumption, the downstream picker observes only a series of picks occurring to the upstream picker. During the inter completion times between stops, the downstream picker observes the arrival process of the upstream picker. We define the sequence, A_{1}, A_{2}, ..., as the specific times when the upstream picker completes the retrieval operation at the first stop, the retrieval operation at the second stop, …. and define the sequence, S_{1}, S_{2}, S_{3}, ... as the arrival time for the downstream picker. The waiting time of the downstream picker for arrival j is expressed as HD_{j}, which equals to A_{i} – S_{j}. In Figure 4 (a), S_{1} is later than the time that the upstream picker starts the next pick. At time S_{1}, the downstream picker arrives, but the upstream picker is still picking. Until A_{1}, the downstream picker idles. In the case of S_{2}, the arrival time of the downstream picker is earlier than S_{1}, and HD_{2} is the expected delay of the downstream picker.
We define τ as a threshold period of time. If the expected arrival time of the downstream picker is longer than τ, the upstream picker does not start the next pick. In Figure 4 (b), S_{1} is later than τ and the upstream picker starts the next pick. When S_{2} is earlier than τ, the upstream picker does not start the next pick, but leaves from the current tote and moves backward. In Figure 4 (b), the new handoff time, zero, replaces HD_{2} in Figure 4 (a). The remaining timeline of the loosely coupled handoff bucket brigade differs from the timeline of the regular bucket brigade, because the upstream picker does not pick at the picking cycle in S_{2}; instead, the downstream picker retrieves the item(s). Thus, the remaining timeline uses A'_{3}, A'_{4}, and S'_{2}. Note that at S'_{2}, the downstream picker does not wait, but picks an item. The second handoff occurs at S'_{2} without a tightly coupled handoff between the two pickers.
The waiting time by the downstream picker (E[HD]) is conditioned on the expected picking time of the upstream picker X. To obtain the new expected handoff delay, we derive the waiting time under the new policy (0 ≤ τ).
Theorem 1. The expected handoff delay (E[HD]) under the loosely coupled handoff operations becomes
Proof.
We use the renewal reward process to derive the expected waiting time (Ross, 2010). We assume that the new waiting time is a reward during a renewal cycle. Then,(2)
The waiting occurs only when the expected waiting time is longer than τ. In addition, over the given cycle length X, we obtain the following expected value,(3)
Because X is always greater than τ, the expectation for the overall X ranges from τ to infinite,(4)
The time of a pick cycle is a renewal process independent of τ, and the cycle time is E[time of a pick cycle] = E[X].
Finally,(5)
End of proof.
We obtain the numerical models of the uniform distribution and exponential distribution cases. First, we consider a uniform distribution function [min, max] = [a, b]. When τ = 0, E[HD_{unif(a,b)}] = (a^{2} + a·b + b^{2})/(3·(b^{2} – a^{2})). As the threshold becomes large, the expected handoff delay decreases. The derivative with respective to τ shows (2τ – (a + b))/(b^{2} – a^{2}) < 0 for τ < a and – (τ – b)^{2}/(b^{2} – a^{2}) < 0 when a ≤ τ < b. In addition, the values show zero handoff delay at τ ≥ b.(6)
Where $\begin{array}{l}{\displaystyle {\int}_{\tau}^{\infty}{\left(X\tau \right)}^{2}}\cdot dF\left(X\right)=\\ \{\begin{array}{ll}\left[{\tau}^{2}(a+b)\tau +\frac{{a}^{2}+ab+{b}^{2}}{3}\right]\frac{1}{(ba)}\hfill & \tau <a\hfill \\ \begin{array}{l}\\ \left[\frac{{\tau}^{3}}{3}+b{\tau}^{2}{b}^{2}\tau +\frac{{b}^{3}}{3}\right]\frac{1}{(ba)}\end{array}\hfill & \begin{array}{l}\\ a\le \tau <b\end{array}\hfill \\ \begin{array}{l}\\ 0\end{array}\hfill & \begin{array}{l}\\ b\le \tau \end{array}\hfill \end{array}\\ \text{and}E[X]=(a+b)/2.\end{array}$
The exponential distribution shows a relatively high handoff delay, because of the high variation of the pick time, and it decreases slowly over τ. The exponential distribution experiences a high handoff delay despite τ = 2, because $E[H{D}_{\mathrm{exp}(a)}]={e}^{a\tau}/a$ decreases very slowly. The derivative with respect to τ is $\tau \cdot {e}^{\tau}<0$. Thus, the expected handoff delay also decreases as τ increases.(7)
$\begin{array}{l}\text{where}{\displaystyle {\int}_{\tau}^{\infty}{\left(X\tau \right)}^{2}}\cdot {e}^{ax}dx={e}^{a\tau}{\displaystyle {\int}_{0}^{\infty}{\left(Y\right)}^{2}}\cdot {e}^{ay}dy=\frac{2{e}^{a\tau}}{{a}^{2}}\\ \text{and}E[X]=1/a.\end{array}$
We validate the analytical models compared to the simulation models with OPS size = 200 pick faces, forward and backward walk times = 0.01 unit time per pick face, and pick time exp (1). Compared to the simulation models developed by Tecnomatix Plant Simulation 12 (Bangsow, 2016), the analytical models shows a stable error gap, where gap % = (the measures by the analytical model – the measures by the simulation model)/(the measures by the simulation model) × 100. The error gaps show 0.37 to 8.90% across τ = 0.0, 0.1,…, 1.9, 2.0. We limit the range of τ as 0.0 to 2.0. When τ is greater than 1, the loosely coupled handoff operation needs to consider the picking time over one or more pick faces.
3.3. Simulation Analysis
The loosely coupled handoff could reduce the handoff delay (Figure 5) and partially contribute to throughput improvement (Figure 6). Over different order picking environments, i.e., number of pick faces PF, the allowed WIP increment, and the forward walk time, the simulation results in Figure 5 (a) and (b) show the gaps compared with the analytical models, where PF200 is the number of pick faces = 200, W2 is the maximal WIP increment = 2, and F is the forward walk time. We set the backward walk time to half of the forward walk time. With the small value of τ, the handoff delay follows the analytical models. Under the restriction with a tight WIP limitation, there is a limited reduction of the handoff delay. Even though the upstream picker releases a tote earlier, the upstream picker cannot take a new tote from the loading station because of the constant WIP rule. In the case of an increase in WIP level, however, there is a consistent reduction of handoff delay and a similar value to Eq. (1). Note that Eq. (1) assumes instantaneous walk time, i.e., very fast walk speed. Since τ ranges from 0 to 2, the handoff delay becomes e^{−2} ≈ 0.13 at τ = 2.
The reduction of the handoff delay, however, does not guarantee improved throughput (Figure 6). Less handoff delay could shorten a portion of the lead time, but offsets the order picking throughput. One reason is the increase in the blocking delay (Hong, 2014;Hong et al., 2015). Initially, throughput increases since there is some throughput gain of pickers and a decrease in the lead time from the handoff delay reduction, but the throughput gain gradually decreases as both lead time and blocking delay lengthen. The cases with W3 show relatively high throughput and consistent improvement compared to the cases with no WIP relaxation. The TH improvement decreases after a peak when the previous tote is released near the loading station, because the next trip is blocked at the location if the downstream picker does not take over the tote prior to the upstream picker’s arrival.
4. A LOOSELY COUPLED HANDOFF FRAMEWORK AND SIMULATION STUDIES
While the loosely coupled handoff operation can reduce handoff delay, the model requires additional adjustment since the reduction affects both blocking delay and cycle time. The WIP level also needs adjustment to obtain more handoff reduction and throughput improvement.
In practice, pickers tend to make collaborative decisions based on their remaining operational times. For instance, the upstream picker considers both the downstream picker’s expected completion time and the upstream picker’s expected remaining time. If the upstream picker’s expected time is less than the downstream picker’s completion time, the upstream picker can release the tote at the next pick face.
Additional WIP can also create a blocking delay, i.e., when the upstream picker releases a tote in the loading station area, it blocks the release of the next tote. In this case, management should implement a loosely coupled handoff zone.
4.1. The Bucket Brigade Framework with Loosely Coupled Handoffs
We propose the following framework for the δbased approach to reduce the handoff delay considering operational throughput. The decision variables include the value of relative looseness (δ), the increment of WIP (ω), and the release allowance zone of the loosely coupled handoff (ψ). First, we reformat τ as δ = τ minus the expected completion time of the upstream pickers at the next pick face (E[CT_{up}]). The relative looseness δ makes the loosely coupled handoff operation compensate for both the upstream picker’s pick and the downstream picker’s pick. The upstream picker makes a decision based on the expected completion time of his/her operation at the next pick face, i.e., E[CT_{up}]. The relative looseness δ expresses the confidence adjustment over the expected handoff delay. The range of δ has three insights: 1 denotes that an upstream picker yields the current tote to the downstream picker approaching when the upstream picker starts a pick at the next pick face; 0 denotes that the upstream picker yields the current tote when the downstream picker approaches within a one pick distance if the upstream picker has a pick, and if there is no pick at the next pick, the upstream picker yields only when the downstream picker is at the next pick faces; and 1 denotes that the upstream picker yields the current tote if the downstream picker may locate within two picks away when the upstream picker has a pick at the next pick face, and if there is no pick, it becomes a one pick distance. While determining the gap allowance (δ), we assume that δ is one of [1.0, 0.9, …, 0.9, 1.0]. Our analysis in Section 3 shows that the handoff delays have a smooth curve over δ. Since the delicate control of δ is relatively difficult in practice, we decode the gap allowance into [1.0,  0.9, …, 0.9, 1.0].
Second, as we relax the tight handoff operation in order to exploit the looseness, we also need to relax the WIP increment. We introduce ω as the level parameter. If the current WIP level is below ω + the number of pickers, the most upstream picker can take a new tote from the loading station. This relaxation is the systemlevel looseness of the handoff delay such that the system obtains additional WIP against the idleness of the most upstream picker.
Third, we allow the loosely coupled handoff operation after pick face ψ. When the WIP level is greater than the number of pickers, the early released tote could block the upstream picker’s travel when the upstream picker takes a trip. Thus, we allow the loosely coupled handoff operation only when the picker is in [ψ + PK + ω  NTS, PF], where PK denotes the number of pickers, NTS denotes the total number of totes in the system, and PF denotes the number of pick faces in the bucket brigade OPS. Note that PK + ω – NTS compensates for the current totes in the bucket brigade OPS.
Based on the rules above, the loosely coupled handoff operation framework investigates the optimal combination of parameters (δ, ω, ψ) to optimize the bucket brigade OPS. Below, we summarize the decision variables, intermediate decision variables, objective functions, and rules and constraints.
Decision variables

δ: the gap allowance over the expected pick completion time, where E[CT_{up}] + δ = τ, δ in [1.0,  0.9, …, 0.9, 1.0]

ω: the increment of workinprocess (WIP) in the system; available maximum WIP level = PK + ω

ψ: the loosely coupled handoff is allowed in [ψ + PK + ω  NTS, PF], ψ in [0, 1, …, PF1, PF]
Intermediate decision variables

NTS = the total number of totes in the system

CT_{dn} = the estimated returning time of the down stream picker

CT_{up} = the estimated returning time of the upstream picker
We investigate two optimization situations. First, we consider the picker’s viewpoint, which is to minimize the expected handoff delay (Min HD). Second, we consider the OPS operation and maximize OPS throughput (Max TH).
Objective functions
Min Handoff delay (HD) or Max Throughput per hour (TH)
Rules and Constraints

CT_{dn} = the remaining walk time + the remaining pick time

CT_{up} = the remaining walk time + the pick time at the next pick face
If CT_{dn} ≤ CT_{up} + δ and the pick face pf of the upstream picker is in [ψ + PK + ω  NTS, PF], release a tote at pick face pf
If NTS < PK + ω, take a new tote from the loading station, NTS = NTS + 1
We search the solutions with full enumeration analysis and heuristically search an optimal combination of parameters with the Genetic Algorithm. Section 4.2 discusses the full enumeration analysis, and Section 4.3 characterizes the solutions and the objective values based on the heuristics solutions.
4.2. Full Enumeration Analysis
From the twopicker model, we investigate the handoff mitigation characteristics using a simulation on Tecnomatix Plant Simulation 12. We define the framework of the handoff delay regarding the blocking delay. Over the predefined solution spaces of δ, ω, and ψ, the simulation study conducts a full enumeration analysis. δ ranges [1.0,  0.9, …, 0.9, 1.0], ω considers one WIP increment in [0, 1, 2, 3], and ψ assumes one pick face in [0, 1, …, PF1, PF]. The elementary analysis with the full enumeration is conducted over WIP increment ects of WIP Increment(ω), and allowance position (ψ) across δ = [1.0, 0.9, …, 0.9, 1.0]. Each instance includes 20 runs with the same configurations of solution space. Performance measure of the handoff per occurrence is the average handoff delay of 20 runs. TH improvement compares the number of totes completed during 10·3600 unit times. Table 1 summarizes the other configurations.
Effects of WIP Increment (ω)
It is necessary that the loosely coupled handoff framework relaxes the WIP level in order to maximize the benefit from the loosely coupled handoff operation. When the WIP level is as tight as the number of pickers, the upstream picker, who has to release a tote early, confronts the limitation of additional WIP because of the constant WIP restriction. The most upstream picker stays idle at the loading station until the most downstream picker completes the tote. To obtain the additional benefit of more totes, management should allow additional WIP. Figure 7 (a) shows the additional reduction of the handoff delay from the additional WIP, i.e., ω > 0. Figure 7 (b) confirms the significant improvement of throughput as more totes are available, but the improvement is limited to a small range of δ. One reason is the added blocking delays. Even though more totes are allowed, the release of totes to the loading station affects the upstream picker’s travel scope. Thus, when δ is large and ω is greater than 0, congestion increases and throughput drops, i.e., more adjustment is necessary.
Effects of allowance position (ψ)
The allowance position control plays a pivotal role in preventing throughput drop when 1 ≤ δ ≤ 1 and ω ≥ 0 as shown in Figure 8 and Figure 9. In Figure 8 (a) (ω = 0), as the allowance position is located closer to the loading station, i.e., W0P0 and ψ = 0, the handoff delay reduces because there is more chance of early tote release without any picks. In Figure 8 (b), when δ is smaller than 0, the early release improves throughput, but is relatively low. As δ increases, throughput improvement drops. When δ is large, the early release becomes available at nearby position 0, with a high priority of poor throughput.
As the WIP level is relaxed (ω > 0), the handoff delay reduces more effectively (Figure 9 (a)) and there is additional throughput gain (Figure 9 (b)). Thus, the combined WIP increment and allowance position adjustments are critical for reducing handoff delay while guaranteeing throughput gains.
Optimal solutions in twopicker experiments
Table 2, which summarizes the full enumeration results, ranks the decision values by handoff delay per occurrence and throughput. Consistently, the minimal handoff delay experiences a huge loss in throughput. However, enough WIP and handoff allowance position could prevent pickers from loss of production while reducing the handoff delay. Pickers need to release a tote when there is enough space for the next trip of the upstream picker. When there is enough space, additional tote release is also available and it results in a higher throughput.
4.3. Extended Analysis with Heuristic Search
To obtain a set of solutions parameters for three and fourpicker situations, we implement the metaheuristic optimization model with the Genetic Algorithm module (GA) of Tecnomatix Plant Simulation 12 (Bangsow, 2016) on a server system with Windows 10 Pro (Intel i74790 3.60 Ghz CPU, 16GB memory). We configure the GA module based on the one presented in Heragu (2008). We encode the solution space to represent all possible δ, ω, and ψ, where δ in [1.0, 0.9, …, 0.9, 1.0], ω in [0, 1, 2, 3], and ψ in [0, 1, …, PF1, PF]. We use the objective values to compare the fitness of a candidate solution. The GA module also sets the maximum number of individuals in the population as (P) = 10 and the maximum number of generations as (G) = 50. During the iteration, the GA module keeps 0.1P of the solutions with the best fitness in the previous population, generates 0.8P solutions via crossover, randomly selects 0.1P solutions from the previous population, and mutates them. We use the twopoint crossover operator which initially selects two randomly crossing points and then exchanges the ranges between the two points. The mutation operator uniformly and randomly determines a value chosen from the solutions space per the selected gene.
Table 3 summarizes our four simulation scenarios: slow walk time, small OPS size, normal configuration, and fast walk time, and reports the computation time in seconds (CPU). As performance measures, we report the handoff delay (HD), the number of totes per hour (TH), and the average life time in the OPS system per tote (LT). The three objectives are: no parameter configuration (Reg); minimization of handoff delay (Min HD); and maximization of throughput (Max TH). Note that the full enumeration requires 21·4·51 = 4284 experiment instances in a twopicker normal scenario. Due to the computation burden, we rely on the GA modules to find the best performance of the loosely coupled handoff cases. The solutions by the GA are validated on the 100 additional simulations runs with different random seeds.
Table 4 reports the heuristic solutions from the GA modules, and the related objective values of HD, TH, and LT. The heuristic search identifies an optimal solution over all scenarios and the Min HD in the Small scenario. The heuristic search requires about 3 hours, whereas the full enumeration requires 27 to 70 hours per scenario. The gap (%) represents the improvement of the objectives compared to the Reg in percentage. The Min HD approach shows the handdelay reduction of 92.7~98.5% of the handoff delays. A huge loss in TH is reported as 4.6~15.1%. Mostly, the Min HD has been achieved when allowance position (ψ) = 0. As shown in Figure 8 (a) (ω = 0), the handoff delay reduces because there is more chance of early tote release before any pick. However, throughput improvement is not guaranteed.
Table 5 reports the results by the Max TH and simulations studies over twopicker cases. The Max TH approach shows the advantage of 2.7~5.6% of throughput and the advantage of 57.8~88.7% of handoff delay. In the solution, the higher level of WIP takes advantage of more throughput and less handoff delay. The results are similar to the batch order picking results in Bartholdi III and Eisenstein (1996b). As the WIP level becomes larger than the number of pickers, the loosely coupled handoff allowance position (ψ) narrows in [7~14th pick face, PF]. Note that the starting pick face for early tote release is offset as much as PK + ω – NTS, based on ψ.
In a threepicker model, we need more information about the most upstream picker’s release to calculate the downstream picker’s expected return time. For the first picker, we set the following rule to obtain the current pick face of the downstream picker: When the downstream picker walks forward, the expected returning pick face = (the current pick face of the downstream picker + the current pick face of the downstream picker’s downstream picker)/2; if the downstream picker travels backward, the current pick face is adopted.
Table 6 reports that the GA solution reduces handoff reduction 66.4~87.5% compared with the original cases (Reg) in the Min HD cases. However, the throughput gains are tiny or negative. In addition, the Min HD consistently requires allowance position (ψ) to the nearby loading station as 07. Note that in the Fast scenario, the second picker allowance position precedes the first picker allowance position. The results are plausible because the bucket brigade zone picking is a dynamic zone picking system, the solution is heuristic, and the starting pick face adjustments for early tote release by ω equalize the starting pick face for early tote release, i.e., ω + ψ are identical.
Table 7 reports that the Max TH guarantees the throughput gain of 3.66.4% with the longer LT as much as 3.45.7%. The threepicker model has two chances for handoff reduction by the most upstream picker and the second most upstream picker. The allowance position of the loosely coupled handoff operation plays a pivotal role in guaranteeing a high throughput while showing a relatively short handoff delay. The most upstream picker should have space to release totes. The most upstream picker is suggested to release a tote after 717th pick face and the second picker is best after the 29th pick face in a 50 pick face system and 10~12th pick face in a 20 pick face system. Note that a 50 pick face system balances workload at [1, 1617th pick face], [1617th pick face, 32~34th pick face], and [3234th pick face, PF].
Table 8 reports the heuristics solutions from fourpicker cases with the Max TH objective. Because of the computation burden, we limit the number of runs per instance = 5 runs, which may search relatively poor solutions. The solution consistently shows less handoff delay and more throughput improvement. In addition, the most upstream picker needs to manage the tight WIP level mostly as high as 1, and adjust the work range depending on the work speed. The second and the third most upstream pickers have relatively stable management on the relative looseness of the handoff (δ), the level of workinprocess (WIP) (ω), and the work range of the loosely coupled handoff (ψ) allowed. The level of WIP is as high as 3 and the work range is evenly balanced into 24~25th pick face or 29th pick face similar to the threepicker cases.
5. CONCLUSIONS
This paper developed an analytical model of handoff delays in bucket brigade OPS with a looseness parameter τ. Based on analytical studies and simulation studies, a framework was developed to completely manage the handoff delay based on the additional parameters of the WIP level and the early release boundary. The analytical models quantified the expected delay related to handoffs over the handoff looseness parameter by extending the analogy of an excess time renewal model to bucket brigade OPS.
Our results in Max TH heuristic solutions show that the rule relies on the WIP level and the zone restriction of the early handoff for the allowance boundary of the early release of totes with δ = [0.5, 1]. The most upstream picker’s releasement requires a relatively lower level of WIP (ω) and a release close to the loading station with the walk speed adjustment. The second and other upstream pickers have the higher WIP level and the workload balance point, which is very close to the boundary of a zone picking system.
The Min HD results indicate that a very large reduction in handoff delay does not guarantee improvement in throughput and reduction in lead time. Our extended handoff framework and the simulation study showed that the benefit was only available when the WIP level limited the frequency of the early handoff, and the zone restriction of the early handoff controlled the allowance boundary of the early release of totes by the Max TH. A poorly intended release of a tote by the Min HD could worsen lead time and reduce throughput. Our results emphasize the need to carefully manage the loosely coupled handoff to obtain overall system throughput.
Based on our findings, future research should focus on generalizing the proposed approach for bucket brigades used in multiplepicker environments, and identifying applications in other manufacturing and service areas such as the assembly line described in Bartholdi III and Eisenstein (2005).