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ORIGINAL ARTICLE
OR Spectrum
https://doi.org/10.1007/s00291-025-00833-y
Abstract
Sequential zone picking is a strategy to pick stock keeping units in a warehouse. It
is preferred for its exibility and scalability. Among others, it is applied in use cases
in which the overall number of stock keeping units is low, and the volume of orders
is high. If there are only very few dierent stock keeping units compared to the
order volume, the same stock keeping units are placed in multiple zones to improve
performance. This gives rise to the problem of order routing, which determines
where the required stock keeping units for every order should be picked. Although
there is a large body of literature on picker-to-parts order picking, this paper is the
rst to consider order routing for sequential zone picking. From a theoretical per-
spective, we analyze problem complexity and provide insights into the mechanics
of the problem. Based on these insights, we develop three dierent model-based
solution heuristics. To evaluate the routing approaches, we collaborated with Hel-
loFresh, one of the largest cook-and-eat meal-kit providers worldwide. An extensive
case study based on real-world data shows that the proposed routing approaches
are applicable in practice and outperform heuristics currently in use with an up to
30% higher throughput.
Keywords Pick-and-pass · Zone picking · Routing policies · Operations · Order
picking · Order batching · Zoning · Storage assignment · Order routing · Routing
Received: 13 May 2025 / Accepted: 14 July 2025
© The Author(s) 2025
Order routing in sequential zone picking systems
SebastianDebold1· JochenGönsch1· RobertDochow2
Sebastian Debold
sebastian.debold@uni-due.de
Jochen Gönsch
jochen.goensch@uni-due.de
Robert Dochow
rodo@hellofresh.com
1 Chair of Service Operations, University Duisburg-Essen, Lotharstraße 65, 47057 Duisburg,
NRW, Germany
2 Algorithmic Solutions, HelloFresh SE, Prinzenstraße 89, 10969 Berlin, Germany
1 3
S. Debold et al.
1 Introduction
Order picking is the process of retrieving stock keeping units (SKUs) from a ware-
house to fulll customer orders. Each order requires assembling one delivery box,
while ensuring the selection of the correct SKUs in the right quantities. Designing
and implementing an ecient system for rapid and accurate order fulllment is a
complex challenge. The system developed for this purpose is known as an order
picking system. This paper focuses on sequential zone picking systems (ZPSs), also
known as pick-and-pass. ZPSs are widely used in practice due to their ability to sup-
port high throughput, scalability, and exibility in handling varying order volumes
and product sizes (e.g., van der Gaast et al. (2020)). In this strategy, the picking area
is divided into multiple zones, each managed by a dedicated operator. By limiting
an operator’s work to a specic zone, travel time is minimized, leading to increased
productivity.
A unidirectional conveyor belt connects all zones, transporting boxes through the
system. As a box reaches a designated zone, the required SKUs are picked and the
box is passed along to the next zone for further processing. This structured approach
enhances eciency and reduces bottlenecks in the order fulllment process (e.g.,
de Koster et al. (2007)). A key challenge in any ZPS is operating within a given
design while minimizing the makespan of all orders or, respectively, maximizing
throughput. If all zones are similar in size and have the same number of operators,
the ZPS performance is largely driven by the ability to distribute workload evenly
across all zones. Unequal demand for SKUs in certain zones can lead to some opera-
tors being overwhelmed while others remain idle van der (e.g. Gaast et al. (2018)).
Especially companies from the e-food sector (e.g. HelloFresh and Gousto, Wig-
more (2020)), that oer a small variety of SKUs compared to order volume, started
placing the same SKUs in multiple – homogeneous sized– picking zones. This gives
rise to the order routing problem, which determines for every order in which zone
each required SKU should be picked. These order routing decisions are handled by
computer-based management systems. The order routing decisions are a valuable
instruments which helps to balance the workload across zones and the load on con-
veyor merges. Suboptimal decisions can lead to heavily occupied zones, resulting in
delayed orders missing their pick-up window.
While there exists a vast amount of research on operator routing in picker-to-parts
systems, routing policies for customer orders in ZPSs have been largely overlooked
in the operations research literature. This paper addresses this gap by analyzing the
problem and introducing two novel model-based routing heuristics capable of mak-
ing just-in-time order routing decisions. The proposed algorithms are scalable to vari-
ous warehouse layouts, easy to implement, and eectively balance workload across
zones. In addition, our methods can be used in more complex ZPS layouts (even
without SKU repetitions) to guide boxes through the system and de-bottleneck con-
veyor merges by distributing boxes more equally.
The remainder of this paper is organized as follows: Section 2 introduces the fun-
damental picking operations in ZPSs. A review of related literature is presented in
Section. 3. Section 4 formally denes the order routing problem (ORP), introduces
relevant notation, and presents a concise problem statement. Given that the ORP
1 3
Order routing in sequential zone picking systems
is computationally intractable, we propose a static approximation. This approxima-
tion is already NP-hard and, therefore, we argue that ORP is unlikely to be solvable
in polynomial time. Additionally, the approximation is used to analyze key aspects
of the problem. Building on this, Section 5 introduces three model-based routing
algorithms designed to solve the ORP heuristically. Section 6 presents a case study,
in which the performance of the presented algorithms is compared against several
benchmark methods, using real-world customer data from meal-kit provider Hello-
Fresh. Section 7 summarizes the ndings of this paper.
2 Sequential zone picking systems
In sequential zone picking (hereafter zone picking), each customer order is repre-
sented by a single box. The boxes travel on a conveyor belt that connects the picking
zones, each zone managed by an operator. Figure 1 provides a schematic illustration
of a zone picking system with three zones. These zones have a homogeneous zone
layout and each consists of two input buers, two pick-faces, one operator, and one
output buer. Four distinct SKUs are available for picking. The process begins at the
carton erector machines, where boxes are assembled and labeled with a QR code for
unique order identication. Once labeled, the boxes are placed on the circular con-
veyor at the system’s entrance, ready for processing.
When a box reaches a designated zone – determined by an order routing algorithm
(which are discussed later) – the zone's input buer is checked for available space. If
at least one input buer is free, the box automatically diverts from the conveyor onto
the buer. Operators process boxes in the order of arrival, starting by scanning the
QR code. A pick-to-light system indicates which SKUs to pick and in what quantities.
Fig. 1 Schematic Illustration of a zone picking system with three picking zones
1 3
S. Debold et al.
Once picking is complete, the box is placed in the output buer, where it waits for
sucient space on the circular conveyor before continuing to the next zone.
An operator can only resume work if there is at least one empty space in the out-
put buer. If a box does not require any SKU from an upcoming zone, or if the input
buer is full when the box arrives, it remains on the main conveyor and bypasses
that zone. After visiting all necessary zones and collecting the required SKUs, the
box exits the ZPS. However, if SKUs are missing when the box reaches the exit, it
remains on the main conveyor and is recirculated as a box waiting in front of a zone
would congest the main conveyor. Please note that a box only needs all necessary
SKUs in the required number, their picking sequence does not matter.
ZPSs can vary in zone type, pick-face design, buer length, storage system lay-
out, number of operators per zone, and conveyor conguration (van der Gaast et al.
(2018)). This paper focuses on the design outlined above, which is widely used and
consistent with our practical experience and the use-case considered. Besides, simi-
lar layouts have been studied by van der Gaast et al. (2018) and van der Gaast et al.
(2020). With minimal modications, dierent design variants can be incorporated
into the proposed heuristics.
When evaluating ZPS performance, throughput is the most important metric. The
throughput measures the number of completed orders in a given period of time (e.g.
van der Gaast et al. (2018)). Since ZPS throughput depends on balancing workload
across zones, optimizing workload distribution is essential. Disruptions such as
starving (zones running out of orders, leaving operators idle) and blocking (zones
with consistently full input or output buers, causing delays in other areas) can sig-
nicantly impact throughput (Vanheusden et al. (2022)). Besides the ZPS design
(e.g. conveyor ow, conveyor merges and the ZPS layout itself, van der Gaast et al.
(2018)), there are two major and interdependent optimization problems that directly
impact the ZPS throughput, namely (i) the storage assignment problem, and (ii) the
order routing problem.
2.1 Storage assignment
Each zone contains a limited number of pick-faces, with each pick-face serving
as storage for a specic SKU. The process of determining which SKU is assigned
to which zone and pick-face is known as storage assignment or rackplanning. To
balance the workload across zones, SKUs are allocated so that each zone handles
a similar number of picks. When SKU demand is uniform, storage assignment is
straightforward: an ecient rack plan simply distributes an equal number of SKUs–
and consequently, an equal number of picks–across all zones.
Figure 2 illustrates the demand distribution of 183 SKUs in an OPS operated by
meal-kit provider HelloFresh. While the weekly demand for 50% of SKUs is fewer
than 10,000 picks, some SKUs are picked up to 450,000 times. When a small number
of SKUs with widely varying demand levels are assigned to a large number of zones,
achieving workload balance becomes impossible if each SKU is assigned to one zone
exclusively. To address this, high-demand SKUs are assigned to multiple zones. In
practice, the average repetition factor – the average number of zones to which an
SKU is assigned – is typically around 3. Our contribution to literature lies in explic-
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Order routing in sequential zone picking systems
itly considering the order routing decisions arising from SKU repetition in ZPSs.
Finding an optimal rackplan is a complex challenge itself. In this paper we assume
that a feasible rackplan exists and focus on the routing problem.
2.2 Order routing
Assigning the same SKUs to multiple zones helps balance the workload, but it intro-
duces a new challenge: when a customer order requires SKUs available in multiple
zones, a decision must be made on which zone each required SKU will be picked
from.
Figure 3 continues the scenario depicted in Fig. 1. Please note that this example
neglects travel time. For simplicity, each customer order requires one unit each of
SKU 1 and SKU 2. SKU 1 is stored in zones 1 and 2. SKU 2 is available in zones 1
and 3, allowing for four possible routes to fulll an order. Each route diers in terms
of workload distribution across zones and the total number of zone visits required.
In an empty ZPS, route 1 would be optimal, as it involves a single zone visit, mini-
mizing initial processing time and reducing congestion at that zone. This lowers the
likelihood of order recirculation and ensures a faster exit from the ZPS. Now assume
that two orders are picked. Obviously, the rst one takes route 1. If, however, the
Fig. 3 Example for routing decisions for one customer order
Fig. 2 Weekly demand for 183 SKUs in an OPS operated by meal-kit provider HelloFresh
1 3
S. Debold et al.
second one would also take route 1, zone 1 would process two boxes, while zones 2
and 3 remain idle.
Alternatively, consider that only the rst order takes route 1. Then, since zone 1 is
occupied, route 4 becomes the optimal choice for the new order. By bypassing zone
1, route 4 utilizes the idle capacity of zones 2 and 3, balancing the workload and
reducing idle time – even though it requires one additional zone visit.
3 Literature overview
The order routing problem is related to job shop scheduling with partially parallel
machines, yet there are two key dierences:
(1) In job shop scheduling, precedence relationships between tasks dictate the
sequence in which jobs are processed. In zone picking there is usually no specic
order for picking SKUs, but the sequence in which zones are visited is implicitly
given by the ZPS design.
(2) Large ZPSs are designed to accommodate several thousand orders, with indi-
vidual pick times measured in seconds. In most job shop systems, fewer than a
hundred jobs are processed with task durations spanning several minutes. This,
combined with the inherent stochasticity, makes it intractable to control the
sequence in which orders arrive at a zone.
Rather than determining a processing sequence for each zone as in job shop sched-
uling, our objective is to decide for each order which zones should be visited and
from which zones each SKU should be picked. For an extensive overview of the job
scheduling literature, see Xiong et al. (2022).
Further this paper falls in the warehousing literature. While there is a substantial
body of warehousing literature on operator routing in traditional picker-to-parts OPSs
(e.g., Baardman et al. (2021); for an overview, see Masae et al. (2020)), we are not
aware of any work on order routing in zone picking. This gap exists because previous
zone picking studies have assumed that SKUs are exclusively assigned to a single
picking zone, eliminating the need for explicit order routing. Given the fundamental
dierences between operator routing in unzoned order picking systems and order
routing in ZPS, existing operator routing approaches cannot be applied. Therefore,
rather than attempting to provide a comprehensive overview of warehousing litera-
ture, our focus is on summarizing previous research topics that have been explored in
the context of zone picking. For a detailed review of zone picking literature, see van
Gils et al. (2018); Vanheusden et al. (2022), and Boysen et al. (2017).
The literature on zone picking is limited, but gained attention in the last decade.
The earliest work is in publications by de Koster (1994), Malmborg (1996), and Jane
(2000). de Koster (1994) presents a framework to evaluate the eects of early-stage
layout decisions using a Jackson queuing network disregarding box recirculation.
Malmborg (1996) and Jane (2000) develop models for storage assignment to inves-
tigate the trade-o in space requirement and retrieval cost under randomized storage
policies. Petersen (2002) uses simulation to compare the eects of multiple picking
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Order routing in sequential zone picking systems
strategies and shows that the shapes of picking zones, the number of SKUs per cus-
tomer order, and the storage policy can signicantly inuence the average walking
time. Jewkes et al. (2004) extend the work of Malmborg (1996) and Jane (2000)
by considering the picker home base and the allocation of products to each picker.
Eisenstein (2008) studies a similar problem extension. Jane and Laih (2005) propose
an heuristic algorithm to balance workload among all pickers in a synchronized zone
order picking system. Yu and de Koster (2008) develop an approximation method for
a zone picking system without recirculation using G/G/m queuing networks. Pan and
Wu (2009) develop an analytic Markov chain model for a sequential zone picking
system, to study the storage assignment problem for equal and unequal sized picking
zones. Melacini et al. (2010) develop a framework for sequential zone picking system
design using queuing networks. Pan et al. (2015) introduce a genetic algorithm to
solve the order batching problem in a sequential zone picking system by simulating
dierent batching policies.
In van der Gaast et al. (2020), a block-and-recirculation protocol for a multi-seg-
ment zone picking system is introduced. They developed a product-form queuing
network with a jump-over-protocol to approximate the system. Earlier van der Gaast
et al. (2018) introduced a similar approach, that takes the performance of the con-
veyor merges into account. They developed an approximation method, which pre-
dicts the loss in throughput due to congestion and blocking of the conveyor merges.
Ostermeier et al. (2021) optimize time windows in a zone picking systems while con-
sidering the occurring vehicle routing problem. Tu et al. (2020) introduced a genetic
algorithm to solve the storage assignment problem optimizing the workload balance
for each zone and the emergency replenishment during picking operations using
simulation to validate their approach. Zhang et al. (2021) investigated the eects of
quantity and quality feedback on speed and quality performance metrics.
4 The order routing problem
This section precisely describes the problem this paper considers. Subsection 4.1
describes the used notation and shares a concise problem statement of the order rout-
ing problem (ORP). As the ORP is prohibitively large and intractable, Subsection 4.2
presents a static approximation of ORP as a linear program (ORP-apr).
4.1 Notation and problem statement
A set of SKUs J is oered. The set of customer orders O is known before production
starts. The demand of order
o∈O
is denoted
Djo
and takes on the value n, if SKU
j∈J
is included n-times in the order
o∈O
and zero if SKU j is not part of order o.
The ZPS consists of a set of I homogeneous zones. Each zone has one order picker
and P pick-faces. The processing times are deterministic, identical for each zone, and
given by a linear function. For each order, the operator initially needs BT seconds
to start processing it (e.g. scanning the barcode). Per pick the operator requires an
additional PT seconds. In the production horizon there is a rackplan
Rij
, which maps
each SKU
j∈J
to zones
i∈I
. The rackplan satises the following conditions:
1 3
S. Debold et al.
∑
i∈I
Rij ≥
1
∀j∈J (1)
∑
j
∈
J
Rij ≤P∀i∈I (2)
Rij ∈{0,1} ∀ i∈I, j ∈J
(3)
Every SKU
j∈J
is assigned to at least one zone (1). Every zone hosts at most P
dierent SKUs (2) and every SKU is, at most, assigned to a zone once (3). In the
decision process there are two types of routing decisions: The decision variable
aio
takes on the value one if customer order
o∈O
visits zone
i∈I
, and otherwise zero.
Decision variable
bijo
takes on the value one if SKU
j∈J
in customer order
o∈O
is picked at zone
i∈I
, and otherwise zero. By declaring
bijo
binary, it is assumed
that all units of one SKU are picked in the same zone (as done by HelloFresh and
Gousto, Wigmore (2020)).
Problem statement
Orders are released in an exogenously given sequence into the ZPS described in
Sect. 2. The Order Routing Problem (ORP) seeks a routing for each order (repre-
sented by decision variables
bijo
) that minimizes the makespan of the set of orders
O. We deliberately refrain from sharing a mathematical formulation for ORP since,
rst, it would be cumbersome to read as it needs to model the ZPS in great detail. For
example, travel times between zones, the sequence of orders on the conveyor belts,
etc. must be modeled. Second, it would quickly become intractable because, as we
show in the next subsection, a static approximation is already NP-hard.
4.2 A static approximation
By ignoring the details of the ZPS and especially the conveyors connecting the pick-
ing zones, it is easy to formulate a compact approximation of the ORP. Thus, in the
following, we present ORP-apr, a static approximation to the ORP. Particularly, we
ignore time and place and only take picking times into account. For example, we dis-
regard idle times at zones, boxes’ traveling times on the conveyors, and the fact that
a box cannot simultaneously be in multiple zones. The advantage of this approach is
that the ORP-apr can be formulated with quite a simple linear program:
min Mmax
(4)
s.t.
bijo ≤
R
ij ∀
i
∈
I, j
∈
J, o
∈
O (5)
bijo ≤aio ∀i∈I, j ∈J, o ∈O
(6)
∑
i∈I
bijo
=1
[Djo>0] ∀j∈J, o ∈O (7)
1 3
Order routing in sequential zone picking systems
∑
o∈O
BT ·aio
+∑
o∈O∑
j∈J
PT ·Djo ·bijo ≤Mmax ∀i∈I (8)
aio,b
ijo ∈{0,1}∀i∈I, j ∈J, o ∈O
(9)
The objective aims to minimize the makespan for the given set of customer orders.
The makespan is denoted as
Mmax
and is given by the zone with the highest work-
load (8). Constraints (5) implement the rackplan and determine that SKUs have to
be picked from zones where they are available. Constraints (6) ensure that SKUs are
only picked in a zone if the zone is visited. Constraints (7) ensure for each order that
all SKUs are picked.
1[Djo>0]
denotes the indicator function that takes on the value
one, if SKU
j∈J
is part of order
o∈O
. Constraints (8) add up the picking times for
each zone for all orders that are processed in that zone. Constraints (9) set the domain
for our decision variables.
Sucient conditions for an optimal solution of the ORP-apr
There are two sucient conditions that, if met, ensure the optimality of a solution to
the ORP-apr. Both conditions serve as the foundation for the lower bounds presented
in Appendix A and motivate our solution approaches in Sect. 5. Consider a feasible
solution of the ORP-apr dened by the binary variables
aio ∈{0,1}∀i∈I,j ∈J
and
bijo ∈{0,1}∀i∈I,j ∈J, o ∈O
. This solution is optimal, if it satises two
conditions:
(1) The rst sucient condition requires that the total number of zone
visits – and, therefore, the overall required pick time – given by
∑i∈I∑o∈OBT ·a
io
+∑o∈O∑j∈JPT ·D
jo
·bijo
is minimized. We refer to
this as the minimal zone visit property. This condition is particularly relevant
when
BT > 0
, which is a reasonable assumption in practical applications.
(2) The same picking time is assigned to each zone, meaning that
∑o∈OBT ·aio
+∑o∈O∑j∈JPT
·
Djo
·
bijo
=∑o∈O
BT
·
ai
′
o
+∑o∈O∑j∈JPT
·Djo ·bi′jo ∀i, i′∈I:i=i′
. We refer to this condition as the workload balancing
property. Based on this, we dene a balanced rackplan as one that meets a relaxed
version of this property. Specically, a rackplan is considered balanced if there
exists a set
{˙
b
ijo |˙
b
ijo ≤
R
ij
,
˙
b
ijo ≥0
,
∑i∈I˙
b
ijo =1
[Djo>0]
,j
∈
J, o
∈
O
}
such that, for every picking zone
i∈I
, the following equation holds:
j
∈
J
o
∈
O
PT ·Djo ·˙
bijo =
i∈I
j∈J
o∈OPT ·Djo ·
˙
bijo
|
I
|
(10)
In other words, the total picking time allocated to each zone is equal to the aver-
age picking time over all zones.
Proof of Complexity
ORP is NP-hard and thus unlikely to be solved by polynomial time algorithms.
The proof is conducted by a reduction of ORP-apr’s decision version ORP-apr-dec to
the PARTITION problem. ORP-apr-dec asks if ORP-apr has a solution with a makes-
1 3
S. Debold et al.
pan no larger than
M′
max
. Remember that PARTITION asks whether a given set of
integers can be partitioned into two subsets with equal sums. More formally, PARTI-
TION is: Given a set X of n positive integers
xm
, does there exist a subset
X1⊂X
such that
∑xm∈X1x
m
=∑xm∈X\X1xm
?
For a given instance of PARTITION an instance of ORP-apr-dec with two picking
zones, one SKU for each element of X (i.e.,
J=X
) and without a rackplan restric-
tion (
Rij =1∀i∈I=1,2,j ∈J
),
PT =1
and
BT =0
is constructed. Only one
order (
|O|=1
), which needs exactly j units of SKU j (i.e.
Dj,1=j∀j∈J
) is
considered. Then ORP-apr-dec with
M′
max =1
/
2∑xm∈X
xm is satisable if and
only if the instance of PARTITION is satisable.
5 Solution approaches for the order routing problem
Since ORP-apr is NP-hard, three heuristics are proposed. All heuristics account for
the passage of time indirectly by decomposing ORP-apr into smaller sub-problems.
These sub-problems are solved sequentially, incorporating the most recent informa-
tion available at each step. To account for orders currently being processed in the
ZPS, a careful aggregation of the current ZPS state is introduced.
More precisely, a workload approximation
Wi
is dened for each zone
i∈I
, rep-
resenting the workload currently assigned to that zone (for simplicity, a time index is
omitted). To reect system dynamics, these workload approximations are updated on
the basis of events, increasing or decreasing as necessary. At the start of production,
the approximations are initialized as
Wi:= 1 ∀i∈I
. Each time a newly released
box
o∈O
passes the decision point illustrated in Fig. 4, order routing decisions –
represented by
aio
and
bijo
– are determined using one of the solution approaches
Fig. 4 Illustration of the workload approximation and decision points
1 3
Order routing in sequential zone picking systems
described in Sects. 5.1 and 5.2. The workload approximations are then updated
accordingly:
W
i
:=
Wi
+
BT ·aio
+∑
j∈J
PT ·Djo ·bijo ∀i∈I (11)
When order
o∈O
passes the ZPS exit – regardless of whether it is dispatched
or recirculated – the workload approximations are updated. This ensures that the
approximations accurately reect the evolving ZPS state and dynamically adjusts to
real-time order ow:
W
i
:=
Wi−BT ·aio −
∑
j∈J
PT ·Djo ·bijo ∀i∈I (12)
The following sections discuss three approaches for determining order routing deci-
sions at the decision point:
5.1 Rolling horizon order routing problem (RHORP)
The rst approach, called the Rolling Horizon Order Routing Problem (RHORP),
is a rolling horizon strategy based on ORP-apr. Each instance of the RHORP is an
instance of ORP-apr only considering a subset of future orders
¨
O
and the current sys-
tem state. The number of orders in each subset is denoted N. To consider the current
system state Constraint (8) of ORP-apr is modied:
W
i
+∑
o∈
¨
O
BT ·aio
+∑
o∈
¨
O
∑
j
∈
J
PT ·Djo ·bijo ≤
¨
Mmax ∀i∈I (13)
The constraint is modied so that, when calculating the route for the next N boxes, all
boxes that are currently in the ZPS (represented by the latest workload approximations)
are considered. ORP-apr’s objective function and other constraints remain unchanged.
If the n-th order enters the decision point and satises condition
n mod N !
=1
, the
RHORP is solved for the subset of orders
¨
O={n, ..., min{(n+N−1),|O|}}
. The
obtained routing decisions remain valid for the next N orders. This strategy requires
to solve
⌈|
O
|
N⌉
instances.
If a box reaches the ZPS exit, the workload approximations are decreased as shown
in Eq. (12). If the box is recirculated the routing decision for all remaining SKUs
remain the same and the workload approximation are immediately increased again
for the remaining SKUs using Eq. (12) (eectively only leading to a decrease for the
SKUs that have been picked). Decomposing ORP-apr enhances computational trac-
tability. By incorporating the current system state, the model implicitly accounts for
the passage of time. However, this approach has two major drawbacks. First, when
the set of orders
¨
O
considered in each instance is small, the end-of-horizon eect
leads to neglecting the minimal zone visit property of ORP-apr, shifting the focus
1 3
S. Debold et al.
solely to balancing workload. Second, temporary eects such as stock-outs, cannot
eectively be handled as the routing decisions stay xed until the next resolve.
5.2 The adaptive order routing algorithm (AORA)
The second proposed solution approach is called the Adaptive Order Routing Algo-
rithm (AORA). In this approach, all orders are routed sequentially one at a time. More
precisely, for each order
o∈O
that enters the decision point AORA’s routing model
is solved (just considering this particular order):
min ∑
i
∈
I
Wi·
A·aio +
∑
j
∈
J
Djo ·bijo
(14)
s.t.
bijo ≤
R
ij ∀
i
∈
I, j
∈
J (15)
bijo ≤aio ∀i∈I, j ∈J
(16)
∑
i∈I
bijo
=1
[Djo>0] ∀j∈J (17)
aio,b
ijo ∈{0,1}∀i∈I, j ∈J
(18)
In AORA’s objective the accumulated weighted picking time is minimized. Visiting
zones with a high workload is punished, while zones with a lower workload are pref-
erably selected. Constraints (15, 16, 17 and 18) are similar to ORP-apr, ensuring that
all SKUs of an order are picked, the decision variables are linked, that the rackplan is
adhered to, and that the decision variables are binary. The parameter A plays a critical
role in determining order routing decisions by inuencing the structure of the rout-
ing decisions. When A is set close to zero, AORA prioritizes balancing the workload
across zones, with less emphasis on minimizing the zone visits per box.
Immediately after the routing decision for box
o∈O
is obtained, the workload
approximation are updated using Eq. (11). If a box is dispatched or recirculated the
workload approximation are updated using (12). In case of a recirculation – deviating
from the RHORP strategy – AORA’s routing model is solved again, only considering
the remaining SKUs and the current ZPS state. By solving the routing for each order
at a time, the most recent information are available. Eects like stock-outs, absent
operators, etc. can be considered by adapting the rackplan Constraint (15) (changing
the relevant bit from 1 to 0).
Illustrative example
In Fig. 5 AORA’s mechanism is illustrated using the example from Sect. 2. The
following parameter setup is used:
A=4,BT =4
and
PT =2
.
Three boxes are released in the displayed sequence. At production start, zone
weights are initialized. Once box 1 is released, the AORA’s routing model for this
1 3
Order routing in sequential zone picking systems
box is solved. There are two optimal solutions: pick both quantities of SKU 2 in
either zone 1 or zone 3. Here zone 1 is selected. Next, the zone weights are updated
and the process is repeated for the remaining boxes. The obtained routing decisions
are displayed in Table 1. It is assumed that no box had reached the system’s exit by
the time a following box was routed and therefore no zone weights are decreased.
AORA’s routing decisions are clearly not optimal in the sense of the above formu-
lated ORP-apr. In an optimal solution both picks of box 1 would be allocated to zone
3 and the picks of boxes 2 and 3 to zone 1. However, for a large number of boxes
the AORA works eciently. In the above example a static setup for parameter A was
used. In practice a static setup works quite well (as show in Sect. 6.3); however, the
used parameter should be optimized. As a rule of thumb, the more ecient the rack-
plan, the stronger zone visits can be punished.
5.3 The adaptive order routing algorithm with instant updates (AORA IU)
A second variant based on AORA is proposed which considers instant updates of
the workload approximations. This approach is called the Adaptive Order Routing
Algorithm with Instant Updates (AORA IU). Similar to the base variant of AORA,
AORA’s routing model is solved once a box travels by the decision point and the
workload approximations are immediately increased as proposed in Eq. (11).
Diering from the base approach, the workload approximation for a zone
i∈I
is instantly updated when a box is processed at zone
i∈I
, using Eq. (12). If a box
misses a zone due to a full input buer, the workload approximation for this zone
remains unchanged until the box travels by the decrease point, where the workload
approximation for the remaining zones is decreased and afterwards a new route is
recalculated.
Table 1 Example for the adaptive order routing algorithm
Allocated picks Updated weights
Zone 1 Zone 2 Zone 3
W1
W2
W3
Initial – – – 1 1 1
Box 1 Item 2 (2) – – 9 1 1
Box 2 – Item 1 Item 2 9 7 7
Box 3 Item 1, Item 2 – – 17 7 7
Fig. 5 Example for the adaptive order routing algorithm
1 3
S. Debold et al.
6 Computational study
This computational study focuses on the meal-kit industry. In 2022 the worldwide
market size of the meal-kit industry was valued at 21 billion USD, with an expected
annual growth rate of 15% from 2022 to 2030 (GrandViewResearch (2023)). The
industry is divided in two dierent segments—the “heat-and-eat” and the “cook-
and-eat” segment. Companies in the “heat-and-eat” segment sell “precooked” meals,
while companies in the “cook-and-eat” segment deliver ingredients to the customer,
including a recipe for each dish. Customers use the ingredients to freshly cook the
ordered dishes at home, with limited preparation time and eort. In 2022 the cook-
and-eat segment held a revenue share of 60% (GrandViewResearch (2023)).
In our research we cooperate with the world’s largest cook-and-eat meal-kit pro-
vider HelloFresh. We selected HelloFresh for our case study because it exhibits
several traits that make it ideal for this setting. First, HelloFresh handles very high
weekly order volumes while maintaining a relatively small, agile SKU portfolio. Sec-
ond, because its SKU assortment changes on a weekly basis, HelloFresh fully lever-
ages the exibility of a sequential ZPS. Most importantly, they use the same ZPS
as considered in this paper and already apply order routing strategies in production.
In this simulation study, two dierent HelloFresh ZPSs are modeled and real-world
customer data from ten dierent production weeks is used. In Subsect. 6.1, both mod-
eled ZPSs and the used customer data are described in more detail. In Subsection 6.2,
four dierent benchmark approaches are proposed. The computational results are
presented in Subsect. 6.3. Finally, we recommend an approach based on the compu-
tational results in Subsect. 6.4.
6.1 Instances
HelloFresh oers a weekly rotating menu of meals to its customers. Thus, each week,
the existing SKU portfolio is replaced with a new portfolio, resulting in changes to
both the SKUs themselves and their total count. Thus, a completely new rackplan is
calculated and implemented each week. Customers have to place orders several days
before production begins, ensuring that all orders are known in advance. This study
considers ten HelloFresh production weeks from dierent markets. Table 2 provides
a summary of the customer orders and the SKU portfolio. Each week,
|O|= 20.000
customer orders are considered. Columns 1 and 2 display the average number of picks
per customer order and their standard deviation. Columns 3 to 5 show the total num-
ber of SKUs
|I|
, the average number of picks per SKU and their standard deviation.
6.1.1 Zone picking system at HelloFresh
Two ZPS were implemented in MatLAB SimEvents 2022b. The schematic layouts
of both ZPSs are similar to Fig. 1. ZPS 1 comprises 21 picking zones. Each zone is
equipped with 15 pick-faces, ve input buers, one output buer, and one opera-
tor. ZPS 2 consists of 11 picking zones, where each zone has four input buers, one
output buer and 15 pick-faces. ZPS 1 is designed to handle weeks 1–4 and ZPS 2
weeks 6–10.
1 3
Order routing in sequential zone picking systems
The picking process in both systems is identically to the description in Sect. 2. For
an order
o∈O
requiring
P
io
=∑j∈Jb
ijo
·Djo
picks from picking zone
i∈I
, the
processing time is calculated by:
BT ·
aio
+∑P
io
p=1
PT
p, where BT and
PT
p
follow
a normal distribution. The normal distributions are truncated for negative values.
The picking times in ZPS 1 are
BT ∼N(2,4)
,
PT
p∼N(5,5)
and
BT ∼N(1,4)
,
PT
p∼N(1,4)
in ZPS 2.
6.1.2 Storage assignment and order sequencing
In this section, we detail the storage assignment algorithm and the order sequenc-
ing used in our experiments. For each instance described above, 30 simulation runs
are conducted, with each run employing a randomly generated order sequences and
rackplans. The same 30 order sequences and rackplans are used for all approaches
ensuring consistency. The order sequences were generated entirely at random, the
rackplans were created using xed repetition factors – remember a SKU’s repetition
factor indicates to how many zones this SKU is assigned to – to create practically
relevant rackplans.
The calculation of the repetition factors (the repetition factor for SKU
j∈J
is
denoted
RFj
) is done with a non-linear, integer program called the Repetition Factor
Calculation Problem (RFCP) proposed by Debold (2025):
min max
j
∈
J
∑
o∈ODjo
RFj
(19)
s.t.
∑
j∈J
RFj≤P·|I| (20)
RFj≥1∀j∈J
(21)
Customer data SKU portfolio
Average
picks per
order
Standard
deviation
picks per
order
Num-
ber of
SKUs
Average
picks per
SKU
Standard
deviation
picks per
SKU
Week 1 13.2 4.8 183 1437 3842
Week 2 10.7 4.0 178 1202 1624
Week 3 10.8 4.1 168 1290 1999
Week 4 10.2 4.1 167 1219 1933
Week 5 10.1 4.2 65 3119 6462
Week 6 10.1 5.1 62 3246 4870
Week 7 7.8 3.5 53 2916 3362
Week 8 9.9 4.1 90 2211 5349
Week 9 7.6 3.4 75 2038 2718
Week 10 7.0 3.4 74 1890 2674
Table 2 Summary of the cus-
tomer data and SKU portfolio
1 3
S. Debold et al.
RFj≤|
I
|∀
j
∈
J (22)
RFj∈N∀j∈J
(23)
In the RFCP it is assumed that the picks for each SKU are uniformly distributed
across all the pick faces where it is available. The objective of the RFCP is to mini-
mize the number of picks from the pick-face with the highest number of picks (19).
Constraint (20) limits the total number of available pick-faces, while Constraints (21)
and (22) ensure that each SKU is assigned at least once overall and at most once to
the same zone. Finally, Constraint (23) denes the decision variables’ domain.
The RFCP is solved using a commercial solver. The assignment of SKUs to zones
is done entirely at random so that Constraints (1, 2 and 3) are satised, while ensuring
that each SKU is exactly assigned to as many zones as calculated in the RFCP. Note,
that at HelloFresh also all SKUs can theocratically be placed at any zone.
6.2 Benchmark order routing approaches
Given the scarcity of routing approaches in the literature, ve benchmark strategies
are considered.
6.2.1 Greedy algorithm (GA)
The rst algorithm represents the simplest conceivable approach. Each time a box
o∈O
reaches a zone that contains at least one missing item and the zone’s buer
is not occupied, the box is routed to that zone. The operator will pick all missing,
available SKUs. This method results in the ZPS being front-loaded, with zones at the
beginning experiencing a higher workload than those toward the end.
6.2.2 Greedy algorithm including a round robin strategy (GA+)
To mitigate the front-loading issue, practitioners enhanced the GA by incorporating a
round-robin strategy. When order
o∈O
is launched, boxes are prohibited from stop-
ping at the rst
(o−1) mod |I|
zones. If an order is recirculated, it is permitted to
visit all zones to ensure that the order can eventually be fullled.
6.2.3 Minimum zone visits (MZV)
Another common approach is a shortest path algorithm, that selects the minimal
amount of zone visits possible for each order. If order
o∈O
is launched, a special
case of AORA’s routing model is solved, in which A is a large number and neglecting
the workload approximations in the objective funtion.
6.2.4 Gousto routing algorithm (GOUSTO)
The GOUSTO approach was developed concurrently with AORA by meal-kit pro-
vider Gousto and is a special case of AORA. GOUSTO is a two-stage optimization
1 3
Order routing in sequential zone picking systems
process. When an order
o∈O
enters the ZPS, GOUSTO’s routing model is solved
determining which zones to visit (Wigmore (2020)):
min ∑
i∈I
...
Wi·
...
aio (24)
s.t.
∑
i∈I
Rij ·...
aio ≥1[Djo>0] ∀j∈J (25)
...
aio ∈{0,1} ∀ i∈I,j ∈J
(26)
In the objective the weighted sum of all selected zones is minimized, under the con-
straint that all SKUs are picked. The workload approximations for all zones
i∈I
are
initialized with
...
Wi:= 1
. After the zones have been selected, the following algorithm
is used to decide for each order o, where the required SKUs
Dio
are picked. For each
required SKU
j∈J
, the number of possible zones within the chosen path is calcu-
lated as
∑i∈IR
ij
·...
aio
and sorted in descending order by their number of alterna-
tives. The SKU with the lowest number of alternatives is assigned to the picking zone
with the minimal approximated workload
...
i:= argmin
{...
Wi
|
i
∈
I,
...
aio =1
}
.
Next, the decision
...
bijo
is used to update the workload approximations using:
...
W
i
:=
...
Wi
+∑
j
∈
J
...
bijo ∀i∈I (27)
The algorithm iteratively repeats, using the updated workload approximations, until
all required SKUs are assigned. In contrast to AORA these approximations display
how many picks are assigned to each picking zone, decreasing every time a box is
processed at a zone
i∈I
using:
...
W
i
:=
...
Wi−
∑
j∈J
...
bijo ∀i∈I (28)
If an order recirculates new routing decisions are obtained.
6.2.5 ORP static approximation (ORP-apr)
As the nal benchmark, we employ the ORP-apr as described in Sect. 4.2. We solve
the ORP-apr model using a commercial solver. While ORP-apr’s objective value is
neglected, the routing decisions from ORP-apr are integrated into our simulation.
Precomputing the order routing decisions results in a static policy. Nonetheless, this
setup enables the evaluation of the real ORP’s objective value.
1 3
S. Debold et al.
6.3 Computational results
All optimization models are solved using Gurobi 11. In Sect. 6.3.1 the performance
of all approaches is evaluated under perfect conditions. In Sect. 6.3.2 all approaches
are tested for robustness. For AORA and AORA IU parameter
A= 10
is used, which
was determined by simulating dierent values for week 1 and choosing the value
that minimized the makespan. The value is used in all instances. For the RHORP
|¨
O|= 300
and a time limit of 120 s for the solver is used.
6.3.1 Performance evaluation
This section provides an in-depth comparison of all routing approaches. As described
earlier, each instance undergoes 30 simulation runs, each run using a randomly gen-
erated order sequence and rackplan, which remain consistent across all approaches.
Figure 6 shows the makespan of all approaches and a lower bound for the makes-
pan (LB3 as proposed in Appendix A). Deviating from the previous sections, the
makespan is dened as the time when the last customer order is dispatched. Further
key metrics are outlined in Table 3, comparing the Average Zone Visits per Box (AZV),
the Standard Deviation of Planned Zone Visits (StD PZV), the Standard Deviation of
Zone Utilization (StD Util), and the Average Recirculation Rate (ARR): The AZV
metric represents the average number of zone visits per box across all simulation
runs, providing insight into the eectiveness of the zone visit minimization. The StD
PZV and StD Util metrics assess the workload balancing. Both metrics are calculated
as follows: For each zone, the number of boxes sent to that zone – including those
processed at the zone and those bypassed due to full input buers – is counted. Simi-
larly, the zone utilizations are determined. The standard deviation of these numbers
across all zones is then calculated. Then, the average across simulation runs is com-
puted. A StD PVZ or StD Util value of zero indicates a perfectly balanced workload,
where each zone processes the same number of boxes and maintains equal utilization.
Finally, the recirculation rate is the ratio of total recirculations (a box with two recir-
Fig. 6 Comparison of the average makespan without operational disturbances in hours
1 3
Order routing in sequential zone picking systems
Weeks
1 2 3 4 5 6 7 8 9 10 ø
ORP-apr AZV 5.0 5.0 4.5 4.6 2.4 2.4 2.0 2.8 2.4 2.5 3.4
StD PZV 982 766 652 622 909 887 873 1,109 626 827 825
StD Util 1% 1% 0.9% 0.9% 1% 1% 1% 1% 0.9% 1% 1%
ARR 51% 54% 52% 49% 85% 82% 58% 88% 59% 55% 63%
RHORP AZV 5.0 4.9 4.6 4.6 2.4 2.4 2.0 2.8 2.5 2.6 3.4
StD PZV 974 741 635 604 620 754 620 810 364 362 648
StD Util 1% 1% 1% 1% 1% 2% 2% 2% 1% 1% 1%
ARR 39% 43% 40% 38% 61% 59% 38% 66% 44% 42% 47%
AORA AZV 5.0 4.9 4.6 4.6 2.4 2.4 2.0 2.9 2.5 2.5 3.4
StD PZV 4,918 1,146 774 632 1,101 1,128 521 7,085 445 492 1,824
StD Util 10% 6% 6% 5% 5% 5% 4% 10% 3% 2% 6%
ARR 120% 54% 48% 41% 66% 63% 37% 143% 42% 38% 65%
AORA IU AZV 5.1 4.9 4.6 4.6 2.5 2.4 2.0 2.9 2.5 2.6 3.4
StD PZV 4,571 1,486 1,197 1,114 1,136 1,201 1,004 6,990 912 1,162 2,077
StD Util 15% 11% 10% 11% 4% 7% 4% 12% 5% 5% 8%
ARR 116% 59% 51% 48% 62% 63% 35% 141% 43% 41% 66%
GOUSTO AZV 5.2 5.0 4.7 4.7 2.6 2.5 2.1 3.0 2.5 2.6 3.5
StD PZV 3,145 2,294 1,610 1,726 2,209 2,894 1,344 8,680 1,377 1,717 2,700
StD Util 18% 10% 10% 11% 15% 17% 5% 20% 5% 5% 12%
ARR 134% 73% 57% 58% 108% 121% 36% 204% 45% 42% 88%
GA+ AZV 6.8 6.7 6.5 6.4 3.5 3.6 3.2 3.9 3.6 3.6 4.8
StD PZV 728 570 603 479 750 717 698 771 744 504 657
StD Util 13% 11% 10% 10% 3% 3% 5% 4% 6% 5% 7%
ARR 94% 88% 85% 80% 82% 79% 57% 93% 68% 68% 80%
Table 3 Key performance indicators
1 3
S. Debold et al.
Weeks
1 2 3 4 5 6 7 8 9 10 ø
GA AZV 7.0 6.8 6.6 6.4 3.6 3.6 3.3 4.0 3.7 3.7 4.9
StD PZV 7,843 4,172 5,265 4,681 9,409 8,137 7,495 8,886 5,977 6,344 6,821
StD Util 34% 28% 32% 29% 35% 29% 31% 29% 27% 27% 30%
ARR 160% 94% 112% 90% 137% 116% 81% 150% 79% 79% 110%
MZV AZV 5.0 4.9 4.5 4.6 2.4 2.4 2.0 2.8 2.4 2.5 3.4
StD PZV 7,547 5,483 6,855 4,803 15,003 11,585 10,319 24,398 7,185 3,444 9,662
StD Util 17% 15% 17% 16% 22% 21% 19% 21% 17% 12% 18%
ARR 196% 138% 169% 126% 257% 212% 178% 393% 138% 86% 189%
AZV Average Zone Visits per box across all simulation runs, StD PZV Standard Deviation of Planned Zone Visits per zone. For each zone, the number of planned visits
is calculated. Next, the standard deviation of planned zone visits is calculated across all zones. Finally, the average standard deviation across all simulation runs is
calculated. StD Util Standard Deviation of Zone Utilization. For each zone, the zone utilization is calculated. Then, the standard deviation of these averages across all
zones is determined. Finally, the average standard deviation across all simulation runs is calculated. ARR Average Recirculation Rate. The recirculation rate is calculated
as the total count of recirculations divided by the total number of orders. The average recirculation rate is then determined across all simulation runs
Table 3 (continued)
1 3
Order routing in sequential zone picking systems
culations is counted twice) to the total number of boxes. The ARR is the averaged
recirculation rate across all simulation runs.
Across all instances, RHORP signicantly outperforms all approaches at the 95%
condence level (condence intervals of the dierences are shown in Table 4 in
Appendix B). AORA ranks second, outperforming all approaches except RHORP and
showing no statistically signicant dierence from ORP-apr. On average, AORA’s
makespan is 6.4% longer than RHORP’s and 0.5% shorter than ORP-apr’s. AORA
IU performs similarly to AORA, with an average makespan increase of 0.6%, which
is statistically signicant. On average, GOUSTO performs 2% worse than GA+,
though this dierence is non-signicant. Both GOUSTO and GA+ require approxi-
mately 17% more time than RHORP on average, but still outperform GA and MZV.
The performance dierences can be explained by the metrics in Table 3. In terms
of zone visit minimization, ORP-apr, RHORP, AORA, AORA IU, GOUSTO, and
MZV perform similarly, with most approaches closely matching or even achieving
the theoretical minimum number of zone visits – dened by the MZV baseline. GA
and GA+, on the other hand, lack a zone visit minimization mechanism and therefore
perform noticeably worse. On average, they require 38% more zone visits in ZPS 1
and 47% more in ZPS 2, which translates to an additional 8.5 h of pick time in ZPS
1 and 9.5 h in ZPS 2.
Signicant performance dierences also stem from the workload balancing capa-
bilities. RHORP demonstrates the most eective balancing, closely followed by
ORP-apr. Both show low standard deviations in PZV and zone utilization, indicating
that zones handle a similar number of boxes and maintain balanced utilization (indi-
cating that zones handle a similar number of picks). Compared to ORP-apr, RHORP
achieves an even better balance in the number of boxes processed per zone. This
is attributed to RHORP’s dynamic routing mechanism, which considers the current
system state by sequentially routing batches of boxes. This indirect consideration of
time allows RHORP to make more informed routing decisions, resulting in improved
balance.
AORA, AORA IU, and GOUSTO perform at a similar level, with AORA achieving
the best workload balance among the three. Both AORA IU and GOUSTO introduce
biases in their workload approximation. These biases stem from the way workload
is updated: Instantly decreasing the zone workload for boxes which are processed
at a zone causes zones at the beginning of the ZPS to appear less busy and attract
more boxes, while zones at the end appear more loaded and receive fewer boxes
due to their greater distance from the decision point. This imbalance grows with
the size of the ZPS, as the distance-related bias becomes more pronounced (shown
in Fig. 9 in Appendix B). As a result, AORA consistently outperforms AORA IU in
ZPS 1, while in ZPS 2 the performance gap narrows, and AORA IU even slightly
outperforms AORA in half of the instances. Finally, GOUSTO’s workload balancing
is weaker due to its less granular workload approximation –by only considering the
number of boxes sent to each zone, without including the number of picks allocated.
This simplication leads to less accurate load distribution and, consequently, a worse
workload balancing.
A similar imbalance is introduced by the GA approach. As each box is routed to
the rst available zone where at least one required SKU is present, a front-loading
1 3
S. Debold et al.
eect is created, in which zones at the beginning of the ZPS process signicantly
more picks than those at the end. As a result, zones at the front of the ZPS have a high
utilization, which drops o towards zones at the end (shown in Fig. 10 in Appendix
B). Again, this imbalance becomes more pronounced as the ZPS size increases.
This issue is eectively addressed in GA+ through the introduction of a round-
robin strategy. As a result, the zone visit balancing in GA+ becomes comparable to
that of RHORP. However, its utilization balancing is still slightly worse than that of
AORA. GA+ benets from indirect awareness of the system state, since a box can
only be routed to a zone if the zone’s input buer is not full. This helps distribute
boxes evenly. Compared to RHORP which routes multiple boxes simultaneously, the
approach falls short in utilization balancing because the assignment of picks to zones
is made greedily for each box at a time. Finally, MZV exhibits the worst workload
balancing of all approaches. This is due to its routing decisions being made without
any direct or indirect consideration of the system state, leading to highly uneven dis-
tribution of workload across zones.
The combined eect of zone visit minimization and workload balancing is reected
in the ARR metric, which indicates the number of recirculated boxes. Thanks to its
superior performance, RHORP achieves the lowest average recirculation rate at 47%,
followed by ORP-apr, AORA, and AORA IU, with rates between 63% and 66%.
GA+ records a higher rate of 80%, while GOUSTO reaches 88%. GA shows a signi-
cantly higher rate of 110%, and MZV performs the worst, with 189% recirculations.
On average, RHORP performs best and is only 3% above the proposed lower
bound, while AORA lags by 9.5%. This indicates that the lower bound is relatively
tight, largely due to the characteristics of the customer data used. In most ZPS con-
gurations, two primary bottlenecks typically arise: the operator performance and
the conveyor merges. In the instances considered, customers order between 7 and 13
SKUs –signicantly higher than in many other ZPS applications. This higher SKU
count necessitates a lower carton limit (fewer boxes in the system at any given time),
which reduces the required throughput of the conveyor merges. Consequently, the
operators become the dominant bottleneck. This explains why the proposed lower
bounds are particularly eective in these scenarios.
6.3.2 Operational robustness
This section investigates the impact of operational disturbances on the performance
of the routing approaches. Specically, the two – from our experience – most impor-
tant operational disturbances are considered: analysis of the eect of unexpected
increases in picking times and the impact of out-of-stock events. Due to their com-
paratively poor performance, GA and MZV are excluded from these experiments.
Systematic increase of pick time
To assess the impact of unexpected changes in pick times, scenarios where a sin-
gle picking zone operates at half its normal speed—representing disruptions such
as untrained workers—are simulated. Again, 30 simulation runs per instance are
performed. In each run, one picking zone is randomly drawn and its pick time is
increased by 100%. This change is not reected in the solution process, meaning it is
neither accounted for in the decision-making nor in the workload approximations. To
1 3
Order routing in sequential zone picking systems
ensure comparability, the same zones are used across all approaches in each simula-
tion instance.
Figure 7 shows the makespans for the scenarios with systematically increased
pick times and compares it to the makespans from the base scenarios. The condence
intervals of the dierences for the makespans of this experiment are shown in Table
5 in Appendix B).
The observed makespan increases under a single-zone pick-time disruption are as
follows: ORP-apr +75.7 %, RHORP +16 %, AORA +11.7 %, AORA IU +10.6 %,
GOUSTO +18 %, and GA+ +6 %. ORP-apr exhibits the largest increase because it
remains completely uninformed – it neither directly nor indirectly accounts for higher
pick times in any zone. Although RHORP, AORA, AORA IU, and GOUSTO also
don’t explicitly model pick-time increases, they “feel” the impact via their dynamic
workload approximations: as the disrupted zone’s input buer lls up, an increasing
number of boxes misses that zone caused by the full buer. When these boxes are
recirculated and re-routed, many must revisit the same congested zone to collect
SKUs only placed in this particular zone, creating a backlog that gradually raises
the estimated workload for that zone. This leads to less future boxes and picks being
assigned to this zone and mitigates, to some extent, the makespan increase.
At the 95 % condence level, RHORP achieves on average the best makespan,
with AORA IU in second place and AORA in third. Compared to AORA, AORA IU
performs better in the ZPS 2 while AORA performs better in ZPS 1 (caused by the
same reasons as explained in Sect. (6.3.1). GA+ ranks fourth, followed by GOUSTO,
and nally ORP-apr. Interestingly, although GA+ is beaten by most methods in abso-
lute makespan, it exhibits the smallest relative increase over the baseline. This resil-
ience stems from GA+’s built-in feedback: by detecting when a zone’s input buer is
full, it dynamically adjusts routing to avoid further congestion.
Consideration of out-of-stock events
To evaluate the impact of out-of-stock events, the following ZPS protocol is
implemented. Each pick-face is stocked with 40 units per SKU. When a zone opera-
tor depletes all available units of a SKU, an emergency restock is automatically initi-
Fig. 7 Makespan in scenario with unexpected pick-time increase
1 3
S. Debold et al.
ated. In 3% of cases, the restock is not completed in time, causing a 5?minute delay
during which no picks for that SKU can be made. During this period, the operator
continues with all other assigned picks in the zone, and the box proceeds to the next
designated zone without the out?of?stock SKU.
In both the ORP-apr and RHORP strategies, current out-of-stock events do not
aect the routing decisions of upcoming boxes or those already in the ZPS. We inten-
tionally avoid incorporating such dynamic behavior, as it is unclear how to do so
within the framework of these strategies, particularly given the high computational
cost of repeatedly solving the ORP-apr and RHORP models. AORA, AORA IU, and
GOUSTO follow a similar logic: If a box misses a pick caused by a SKU being out-
of-stock, it is recirculated. When a newly launched or recirculated box reaches the
routing point, the rackplan is dynamically updated to reect current stock availabil-
ity –by setting the corresponding bit from 1 to 0– thereby preventing the box from
being routed toward unavailable SKUs. In contrast, GA+ handles out-of-stock events
dierently: If a box misses a pick, it will attempt to access the next available zone
where the out-of-stock SKU is still in stock. In all approaches, a box will only enter
a designated zone, if – by the time the box reaches the zone – at least one designated
in-stock SKU is available.
Figure 8 shows the makespans for the scenarios with out-of-stock events and com-
pares it to the makespans from the base scenarios. The condence intervals of the dif-
ferences for the makespans of this experiment are shown in Table 6 in Appendix B).
On average AORA outperforms all other approaches with the results being statisti-
cally signicant at the 95% level, followed by AORA IU (which on average performs
5% worse than AORA), which is signicantly better than the remaining approaches.
Even though RHORP is uniformed towards the out-of-stock events it ranks third, fol-
lowed by GOUSTO, ORP-apr and GA+.
Compared to the scenario without out-of-stock events (Sect. 6.3), AORA’s makes-
pan increases by only 0.5% on average. Except for GOUSTO – which actually
reduces its makespan by 2.2% – this is the smallest average increase among all meth-
ods (ORP-apr 8.5%, RHORP 12.2%, AORA IU 5%, GA+ 4%). Notably, AORA,
Fig. 8 Makespan in scenario with out-of-stock events
1 3
Order routing in sequential zone picking systems
AORA IU, and GOUSTO even outperform their baseline results in certain instances.
In these instances, the approaches tend to concentrate more boxes in a couple of
zones leading to higher SKU consumption rates in these zones, making stock-outs
more likely. When stock-outs occur, fewer boxes are routed to those congested zones
and instead are redirected to previously underutilized zones, inadvertently creating
an additional load-balancing eect. This is especially the case for instances in which
approaches reached a high StD PVZ metric (Table 3) in the Base Scenarios. For these
instances, the out-of-stock events de facto work as an additional balancing feature.
6.4 Recommendation for practical implementation
As shown in previous sections, RHORP delivers the best performance in terms
of makespan in scenarios without operational disturbances. However, for practi-
cal implementation, we recommend adopting AORA. Ecient order routing using
RHORP requires solving its routing model for relatively large batches of boxes. This
is essential to fully leverage the zone visit minimization feature – otherwise, solu-
tion quality deteriorates. In our experiments, RHORP was solved for batches of 300
boxes. Only 95% of all instances were solved by Gurobi within the 120-second time
limit (with an average runtime close to the 120 s). During these computations, the
simulation was paused, meaning the solution time had no impact on the ZPS perfor-
mance. These conditions are obviously very favoring for RHORP and almost surely
unacceptable in practice, leaving only the possibility to start the optimization earlier
with, thus, older information. In contrast, all instances of AORA’s routing model
were solved in under one second.
Moreover, RHORP’s centralized, batch-based nature makes it less exible in han-
dling real-time operational disturbances such as stock-outs or last-minute changes
in the order sequence, etc. While hybrid strategies (e.g., falling back to AORA in
such cases) are possible, they would diminish RHORP’s key strength – its ability
to plan large batches with foresight. Although AORA performs slightly worse than
RHORP (in scenarios without operational disturbances), its main advantage lies in
its decomposition of the routing problem into individual, box-level decisions. This
structure not only simplies implementation but allows for straightforward handling
of real-world challenges. AORA is highly scalable and adaptable to dierent ZPS
layouts, including complex, multi-segmented systems –as AORA is already used in
production environments like HelloFresh. Given its performance, speed, exibility,
and ease of integration, AORA emerges as the most practical choice for real-world
deployment.
7 Conclusion
Traditional research in sequential zone picking typically assumes that each SKU is
assigned to only one zone. Under this restriction, no routing decisions are necessary,
as the zones in which orders must be processed are implicitly dened. This paper
relaxes that assumption by allowing SKUs to appear in multiple zones. As a result,
routing policies become essential to determine the most ecient processing paths.
1 3
S. Debold et al.
The starting point of our formal routing considerations is a simplied linear
program that provides a theoretical lower bound on the makespan. However, this
model becomes computationally intractable for realistic problem sizes due to its
NP-hardness, limiting its practical applicability. To address this, we developed three
model-based decomposition heuristics suitable for just-in-time decision-making in
operational settings. These are benchmarked against four commonly used routing
strategies.
Our simulation study demonstrates that the choice of routing policy signicantly
inuences ZPS performance. The two best-performing approaches outperform one
of the most widely adopted industry baselines (GA) by up to 30%. While RHORP
achieves the best results under ideal conditions, we recommend the AORA approach
for real-world implementation. AORA oers a compelling balance between per-
formance and practicality – it is simple to implement, scalable across diverse ZPS
layouts, and adaptable to the unique operational requirements of dierent business
models. Variants of AORA have already been successfully deployed for various ZPSs
at HelloFresh.
Besides algorithmic performance and layout, conveyor merges can have a signi-
cant impact on the ZPS performance, as highlighted by van der Gaast et al. (2018).
Routing strategies can play a critical role in mitigating the impact caused by merging
operations. In our computational experiments, AORA reduces the total number of
merging operations by 48% compared to the GA approach. This reduction is primar-
ily achieved through fewer zone visits and a lower number of recirculated boxes.
Crucial conveyor merges—such as those combining recirculated and newly launched
boxes—experience trac reductions of up to 28%, while smaller merges, such as
those at zone entrances, see reductions of up to 60% caused by a reduction and more
ecient balancing of zone visits.
The use of routing policies proves especially advantageous for companies with
small to medium-sized product assortments and high demand variability. However,
there are no inherent limitations to applying routing policies in other sequential zone
picking system congurations. Currently there are two promising extensions for the
research developed in this paper. First, future research in the eld of order routing
algorithms can focus on increasing the robustness regarding stochastic eects, while
considering ZPS layouts with heterogeneous zone sizes. Secondly, the relationship
between order routing and storage assignment considering SKU repetitions should be
examined. Then, there are of course optimization problems regarding system design,
for example, the distance between the zones, conveyor specics, the number of zones
and buers, and trying to t everything in the available space.
Appendix A: Bounds on the order routing problem
In this appendix we derive three lower bounds (LB) for the ORP. Note that due to
the simplifying assumptions, the ORP-apr itself is a lower bound for the actually
required ZPS makespan. We focus on lower bounds that rely on the characteristics
of the ORP-apr. When calculating the makespan for a given set of orders, two kinds
of information are interdependent and unknown: The total duration of picking time
1 3
Order routing in sequential zone picking systems
(if
BT > 0
) and the allocation of the picking time to zones. The required amount of
picking time per order
o∈O
can be divided into two dierent components. One part
is taken up by the required SKUs of an order
o∈O
, as is given by
∑j∈JPT ·Djo
.
Although it is unclear to which zones this time is assigned, the amount remains con-
stant. A second part is taken up by time required for the number of zone visits. For
each order
o∈O
it is given
∑i∈IBT ·aio
. The total amount depends on the under-
lying routing decisions.
Lower Bound 1: One zone visit—perfect balancing without considering
rackplan
The simplest bound for the ORP is denoted as
LB1
and is obtained by relaxing
Constraints (5) and (8) of the ORP-apr. If Constraint (5) is ignored, every SKU
j∈J
can be picked at every zone
i∈I
. Additionally, Constraint (8) can be replaced by the
average zone workload. This is permitted, since the average workload is always equal
to or less than the makespan. Thus, the lower bound is given by:
LB
1= min
∑
i∈I
∑
o∈OBT ·
ˆ
aio
+∑
i∈I
∑
o∈O
∑
j∈JPT ·Djo ·
ˆ
bijo
|I|
(A1)
s.t.
ˆ
bijo ≤
ˆa
io ∀
i
∈
I, j
∈
J, o
∈
O (A2)
∑
i∈I
ˆ
bijo
=1
[Dio>0] ∀j∈J (A3)
ˆaio
,
ˆ
b
ijo ∈{0
,
1} ∀
i
∈
I, j
∈
J, o
∈
O (A4)
In the optimal solution, every customer order
o∈O
visits one zone, where all SKUs
are picked in order to minimize the picking time taken up by zone visits. Equation
(A5) gives the simplest way of calculating
LB1
:
LB
1=BT
·|
O
|+∑
o∈O
∑
j∈J
(
PT
·
Djo
)
|I|
(A5)
In
LB1
, all information about the rackplan and the routing is dropped.
Lower Bound 2: Minimal zone visits—perfect balancing without considering
rackplan
To incorporate more information, the linear program (A1) - (A4) can be solved
including the rackplan constraint (5). The resulting lower bound is denoted as
LB2
.
In the optimal solution, considering the rackplan, every customer order
o∈O
visits
the minimal number of zones. This approach ensures that for each order
o∈O
the
accruing picking time due to zone visits
∑i∈I
BT
·
aio remains minimal. Analogous
to
LB1
,
LB2
can be calculated by solving the following shortest path problem, which
minimizes the number of visited zones separately for each order
o∈O
:
min ˜aio
(A6)
1 3
S. Debold et al.
s.t.
˜
bijo ≤
R
ij ∀
i
∈
I, j
∈
J (A7)
˜
bijo ≤˜
a
io ∀
i
∈
I, j
∈
J (A8)
∑
i∈I
˜
bijo
=1
[Dijo>0] ∀j∈J (A9)
˜aio
,
˜
b
ijo ∈{0
,
1} ∀
i
∈
I, j
∈
J (A10)
After having solved the shortest path problems for each box, lower bound
LB2
can
be calculated using the following equation:
LB
2=BT
·∑
i∈I
∑
o∈O
˜
aio
+∑
o∈O
∑
j∈JPT
·
Djo
|I|
(A11)
Compared to
LB1
the computational eort of calculating
LB2
is higher, but
LB2
delivers a tighter bound since
|
O
|≤∑i∈I∑o∈O˜
aio.
Lower Bound 3: Minimal zone visits–perfect balancing considering rackplan
In the last approach, we restate the ORP-apr so that the order structure is disre-
garded. Instead of deciding on the routing policy, picking times are directly assigned
to zones. First the total picking time for each SKU is calculated:
T
j
=∑
o∈O
PT ·Djo ∀j∈J (A12)
The minimal picking time, due to zone visits is given by
∑o∈O∑i∈I˜
aio
·
BT. If
SKU
j∈J
is placed in only one zone
i∈I
(
∑i∈I
Rij
=1
), all orders including
SKU j have to visit zone i. SKUs that satisfy this condition are called unique SKUs.
For all unique SKUs, the picking time due to zone visits can be assigned to the associ-
ated zone
i∈I
. This time is denoted as
MBTi
and is calculated using:
MBT
i
=
o∈O
BT ·
1
[
j′∈{j∈J:Rij′=1}(1[Djo>0] ·1[
i
′∈
IRi′j′=1])>0] ∀i∈I
(A13)
The indicator functions in Eq. (A13) ensure that for a given customer order
o∈O
BT is assigned to zone
i∈I
, if that unique SKU
j∈J
is assigned to zone i and is
part of the order o. If order o requires two unique SKUs in the same zone, Equation
(A13) ensures that BT is only added once. The remaining part of the picking time due
to zone visits is denoted with MBT and is calculated using:
MBT =∑
o∈O∑
i∈I
BT ·
˜
aio −
∑
i∈I
MBTi (A14)
1 3
Order routing in sequential zone picking systems
Lower bound
LB3
can be determined by solving the restated ORP-apr formulation
(A15) - (A20):
LB3= min Lmax
(A15)
s.t.
xij ≤
M
·
R
ij ∀
i
∈
I,j
∈
J (A16)
∑
i∈I
xij ≥Tj∀j∈J (A17)
∑
i∈I
yi≥MBT (A18)
MBT
i
+
yi
+∑
j
∈
J
xij ≤Lmax ∀i∈I (A19)
xij ,y
i≥0∀i∈I, j ∈J
(A20)
In the restated formulation the lower bound of the makespan
Lmax
is minimized
(A15). The decision variables
xij ,y
i
are used to assign the prior calculated picking
times to zones considering the rackplan (A16). Decision variable
xij
assigns a por-
tion of the picking time for SKU
j∈J
to zone
i∈I
and variable
yi
assigns a por-
tion of the picking time due to zone visits to each zone
i∈I
.
LB3
contains the most
information about the rackplan and the routing decisions.
LB3
is the tightest bound:
LB3≥LB2≥LB1
. In particular,
LB3
is a tighter bound than
LB2
, if the rackplan
creates bottlenecks in one or several zones.
1 3
S. Debold et al.
Appendix B: Extended computational results
Table 4 Makespan condence intervals of dierences at 95% condence level Sect. 6.3.1
Approach RHORP AORA AORA
IU
GOUSTO GA+ GA MZV
ORP-apr [1.26,
1.33]
[
−
0.43, 0.17] [
−
0.58,
0.01]
[
−
2.80,
−
1.88] [
−
2.08,
−
1.74]
[
−
7.78,
−
7.02]
[
−
11.75,
−
10.11]
RHORP [
−
1.74,
−
1.11] [
−
1.88,
−
1.28]
[
−
4.11,
−
3.17] [
−
3.36,
−
3.07]
[
−
9.08,
−
8.32]
[
−
13.06,
−
11.39]
AORA [
−
0.19,
−
0.11]
[
−
2.49,
−
1.92] [
−
2.06,
−
1.51]
[
−
7.51,
−
7.03]
[
−
11.51,
−
10.09]
AORA IU [
−
2.35,
−
1.76] [
−
1.89,
−
1.37]
[
−
7.34,
−
6.88]
[
−
11.37,
−
9.91]
GOUSTO [
−
0.03,
0.88]
[
−
5.41,
−
4.70]
[
−
9.21,
−
7.96]
GA+ [
−
5.78,
−
5.18]
[
−
9.86,
−
8.16]
GA [
−
4.28,
−
2.77]
The values represent a comparison across all approaches of the true dierence in average makespan
(expressed in hours) through the subtraction of approach “i” minus approach “ j” (i.e., row - column).
For each comparison, the rst value corresponds to the lower bound of the condence interval, while the
second corresponds to the upper bound
Approach RHORP AORA AORA
IU
GOUSTO GA+
ORP-apr [15.45,
15.90]
[14.48,
15.53]
[14.57,
15.62]
[9.7358,
11.50]
[14.23,
14.72]
RHORP [
−
1.21,
−
0.12]
[
−
1.12,
−
0.03]
[
−
6.02,
−
4.09]
[
−
1.41,
−
0.98]
AORA [0.06,
0.12]
[
−
5.01,
−
3.75]
[
−
1.08,
0.02]
AORA IU [
−
5.11,
−
3.84]
[
−
1.17,
−
0.06]
GOUSTO [2.89,
4.81]
Table 5 Makespan condence
intervals of dierences at 95%
condence level Sect. 6.3.2:
Pick Speed Decrease
1 3
Order routing in sequential zone picking systems
Fig. 9 Average zone utilization AORA versus AORA IU in ZPS 2 in %
Approach RHORP AORA AORA
IU
GOUSTO GA+
ORP-apr [0.58,
0.71]
[1.88,
2.14]
[0.71,
0.95]
[0.35, 0.81] [
−
1.11,
−
0.78]
RHORP [1.21,
1.51]
[0.03,
0.33]
[
−
0.30,
0.17]
[
−
1.79,
−
1.40]
AORA [
−
1.25,
−
1.10]
[
−
1.60,
−
1.25]
[
−
3.08,
−
2.83]
AORA IU [
−
0.41,
−
0.08]
[
−
1.92,
−
1.64]
GOUSTO [
−
1.78,
−
1.28]
Table 6 Makespan condence
intervals of dierences at 95%
condence level Sect. 6.3.2:
Out-Of-Stock
1 3
S. Debold et al.
Funding Open Access funding enabled and organized by Projekt DEAL.
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