Manual order consolidation with put walls: the batched order bin sequencing problem

Abstract

Batching and zoning are popular picking strategies to speed up picker-to-parts order fulfillment systems. On the negative side, these strategies require a subsequent order consolidation to separate the customer orders. In a widespread setup implemented by many online retailers put walls are applied to consolidate orders manually. A batch of orders picked under a batching and/or zoning strategy and, thus, distributed over multiple bins arrives via a conveyor system in the consolidation area. Here, a logistics worker retrieves the items from the successively arriving bins and—guided by a put-to-light mechanism—places them into the put wall. The wall is a simple reach-through rack separated into multiple shelves each temporarily dedicated to a specific customer order. On the other side of the wall reside packing workers, who empty shelves and pack completed orders into cardboard boxes. We aim to optimize the bin sequence in which a batch arrives at the consolidation area, such that the probability of starving packers waiting idle for completed orders is reduced. For this purpose, we formulate the batched order bin sequencing problem and derive suited optimization procedures. In our computational study, we investigate under which circumstances optimized bin sequences are especially valuable to reduce the packers’ idle times.

Full text

RESEARCH PAPER
Manual order consolidation with put walls: the batched
order bin sequencing problem
Nils Boysen
1
•Konrad Stephan
1
•Felix Weidinger
1
Received: 16 March 2017 / Accepted: 16 January 2018 / Published online: 29 January 2018
Springer-Verlag GmbH Germany, part of Springer Nature and EURO - The Association of European
Operational Research Societies 2018
Abstract Batching and zoning are popular picking strategies to speed up picker-to-
parts order fulfillment systems. On the negative side, these strategies require a subse-
quent order consolidation to separate the customer orders. In a widespread setup
implemented by many online retailers put walls are applied to consolidate orders
manually. A batch of orders picked under a batching and/or zoning strategy and, thus,
distributed over multiple bins arrives via a conveyor system in the consolidation area.
Here, a logistics worker retrieves the items from the successively arriving bins and—
guided by a put-to-light mechanism—places them into the put wall. The wall is a simple
reach-through rack separated into multiple shelves each temporarily dedicated to a
specific customer order. On the other side of the wall residepacking workers, who empty
shelves and pack completed orders into cardboard boxes. We aim to optimize the bin
sequence in which a batch arrives at the consolidation area, such that the probability of
starving packers waiting idle for completed orders is reduced. For this purpose, we
formulate the batched order bin sequencing problem and derive suited optimization
procedures. In our computational study, we investigate under which circumstances
optimized bin sequences are especially valuable to reduce the packers’ idle times.
This research has been supported by the German Science Foundation (DFG) through the grant ‘‘Planning
and operating sortation conveyor systems’’ (BO 3148/5-1).
&Nils Boysen
nils.boysen@uni-jena.de
Konrad Stephan
konrad.stephan@uni-jena.de
Felix Weidinger
felix.weidinger@uni-jena.de
1
Friedrich-Schiller-Universita
¨t Jena, Lehrstuhl fu
¨r Operations Management, Carl-Zeiss-Str. 3,
07743, Jena, Germany
123
EURO J Transp Logist (2019) 8:169–193
https://doi.org/10.1007/s13676-018-0116-0

Keywords Facility logistics Warehousing Order consolidation Scheduling
1 Introduction
Online retailers from the business-to-consumer (B2C) segment face the following
challenges when setting up their order fulfillment processes. Typically, they have to
process
•many orders (e.g., in 2016, global e-retail sales grew 23.7% compared to the
previous year, making up 8.7% of the total retail market worldwide, see Statista
2017),
•with few demanded items (e.g., the average number of demanded items at online
retailer Amazon is just 1.6 items per order, see Weidinger and Boysen 2017),
•from a large assortment (e.g., due to the lower costs many online retailers
have a much larger assortment than traditional brick-and-mortar stores; this
phenomenon is known as the long-tail of e-commerce, see Brynjolfsson et al.
2003),
•under great time pressure (e.g., many online retailers offer their customers next-
or even same-day deliveries, see Yaman et al. 2012).
Many online retailers, therefore, apply a batching and/or zoning strategy to speed up
their picker-to-parts based order fulfilment processes (see de Koster et al. 2007):
•Batching Instead of returning to the central depot each time a picking order is
completed, multiple orders are unified to a batch of orders jointly assembled on a
picker tour. This way, the pick density per tour is increased and a more efficient
picking process is enabled.
•Zoning A further reduction of the picking effort is enabled, if the warehouse is
partitioned into disjoint zones. Order pickers only pick the part of an order that is
stored in their assigned zone. In this way, parallel order picking is enabled and
each picker only traverses smaller areas of the warehouse.
On the negative side, batching and/or zoning require a subsequent consolidation
process, where the picking orders are unified. Batched orders need to be separated
and zoning requires the merging of multiple partial orders picked in different zones.
There exist fully automated solutions for the consolidation process based on
sortation conveyors (Gallien and Weber 2010; Boysen et al. 2016). Many online
retailers, however, aim to avoid the high investment costs for these automated
solutions, which are, furthermore, hardly scalable in times of varying workloads.
Therefore, many retailers apply manual consolidation processes based on put walls.
For instance, we are aware of an order consolidation with put walls in warehouses of
Amazon Europe (e.g., in Poznan
´, Poland, and Bad Hersfeld, Germany) and of
European fashion retailer Zalando (e.g., in Erfurt, Germany). However, since many
warehouses orientate their processes on these market leaders, put walls should be
applied in many more warehouses especially of online retailers.
170 N. Boysen et al.
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1.1 Order consolidation with put walls
The fulfillment process of online retailers applying put walls can be differentiated
into the following three basic steps:
(i) order picking in a batching and/or zoning environment,
(ii) intermediate storage of bins, and
(iii) order consolidation and packing of orders.
First, the items demanded by customer orders need to be picked [process step (i)].
Most online retailers apply a picker-to-parts order picking in a batching and zoning
environment where, additionally, a mixed-shelves policy (also denoted as scattered
storage, see Weidinger and Boysen 2017) is applied. Under this policy unit loads of
stock keeping units (SKUs) are purposefully broken down and single items are
scattered all over the shelves of a warehouse. In this way, there is always some item
of a demanded SKU close by irrespective of the current picker location. In such a
setting, large online retailers apply dozens of pickers, which are typically assigned
to specific zones of the warehouse. They pick batched orders in parallel into bins
each finally containing partial orders for multiple customers.
Completed bins are handed over to the central conveyor system. Typically, each
bin is not directly released into the consolidation area, but intermediately stored
somewhere in the conveyor system [process step (ii)]. Especially in large
warehouses, the time span between completion of the first and last bin of a batch
may become large. For instance, inventory differences or misplaced items can
occur, so that some parts of a batch arrive considerably delayed. If the bins of such a
batch would directly be released into the consolidation area, then, at least some
positions where orders are collected, i.e., some dead-end lane of a conveyor-based
sorting system or the shelves of a put wall, would be blocked for a prolonged time
span until the delayed bins arrive. To avoid the excessive consolidation capacity
required by a direct bin release, bins are typically collected in the central conveyor
system and only released into the consolidation area once the batch is completed.
Automated (i.e., conveyor-based) consolidation systems often apply a closed-loop
conveyor where items circulate until the complete batch has arrived in the loop
(Meller 1997; Johnson 1998). A loop conveyor for a huge number of orders,
however, requires excessive space on the shop floor. Therefore, especially large-
sized facilities rather apply automated storage and retrieval systems (ASRS), e.g., a
crane-operated high-bay rack (Boysen and Stephan 2016) or a carousel rack (Litvak
and Vlasiou 2010), to intermediately store bins in a more space-efficient manner.
The basic layout of a consolidation and packing area applying a put wall [process
step (iii)] is schematically depicted in Fig. 1. The bins of a batch released from
intermediate storage successively arrive in the consolidation area. At the end of a
conveyor segment, a logistics worker we call the putter resides. The putter
successively removes the items from the current bin and puts them into the put wall.
The put wall is a simple reach-trough rack separated into multiple shelves, which
are accessible from both sides. Each shelf is temporarily assigned to a separate order
and once the putter scans the current item a put-to-light mechanism indicates into
Manual order consolidation with put walls…171
123

which shelf the current item is to be put. In this way, bin after bin is sorted into the
wall. On the other side of the wall reside the packers. Here, another put-to-light
mechanism indicates completed orders, so that a packer can empty an indicated
shelf and pack the respective items into a cardboard box. Packed orders are, finally,
handed over to another conveyor system bringing them towards the shipping area.
Manual order consolidation with put walls is, for instance, applied in the Poznan
(Poland) facility of Amazon and we have also seen their application at online
retailer Zalando in Erfurt (Germany). Low-tech solutions like put walls do not
require large investment costs and are easily adaptable to varying capacity
situations. Therefore, put walls are applied by many other online retailers too.
In the following section, the basic decision problems to be solved when operating
such a fulfilment process and the related literature treating these problems are
described.
1.2 Decision problems and literature review
A vast body of literature has accumulated on warehousing in the recent decades.
Instead of trying to summarize all these approaches we refer to the in-depth review
papers provided by Gu et al. (2007), Gu et al. (2010) and de Koster et al. (2007).
We concentrate our literature survey on the three process steps elaborated in Sect.
1.1.
Existing research on batching and zoning [process step (i)] mainly focuses the
impact of both policies on the travel distances of pickers. In-depth surveys on these
research efforts are provided by de Koster et al. (2007) and Henn et al. (2012). In
our research, we presuppose that all batching and zoning decisions have already
been made. A picked batch of bins is assumed to wait in intermediate storage to be
released from the ASRS towards the consolidation area.
Not much work is dedicated to process step (ii), the intermediate storage of bins,
which is in the focus of this paper. Once a batch is completed and all bins have
arrived in the ASRS, the bin sequence in which they are released into the
consolidation area has to be determined. To the best of our knowledge in business
practice, bins are mostly released in random sequence. In this paper, we aim at
Fig. 1 Schematic layout of the consolidation and packing area
172 N. Boysen et al.
123

optimized release sequences. Specifically, we aim to minimize the sum of order
completion times. An order is completed once all belonging items have been sorted
into the respective shelf of the put wall. This way, we aim to quickly sort orders into
the put wall, so that the probability of starving packers waiting idle for completed
orders is minimized. This problem has not been treated in the literature yet. The
only other paper dealing with release sequences of bins into the consolidation area is
provided by Boysen et al. (2016). They, however, treat a setting where an
automated conveyor-based sorting system is applied instead of a put wall. To reduce
the probability of deadlocks where no lane for collecting items is available, they aim
to minimize the order spread within the release sequence. Thus, a completely
different consolidation technology and another objective function is applied
compared to the paper on hand.
Existing research on order consolidation [process step (iii)] exclusively treats
automated conveyor-based sorting systems (see Meller 1997; Johnson 1998;
Petersen 2000; Johnson and Meller 2002; Russell and Meller 2003; Briskorn et al.
2017). Manual order consolidation, however, has (to the best of the authors’
knowledge) not been treated in the literature yet. We simulate the consolidation and
packing process if a put wall is applied for manual order consolidation. In this way,
the impact of different bin release sequences from the ASRS can be evaluated with
regard to their impact on consolidation performance.
Sequencing of item retrievals from an ASRS is a vividly researched topic. Survey
papers are provided by Roodbergen and Vis (2009) as well as Boysen and Stephan
(2016). This stream of literature, however, focuses reducing the effort for the
storage and retrieval machine that moves the items between their storage positions
and the input–output point. Our sequencing approach presupposes that the ASRS is
not the bottleneck. Instead, we aim at an efficient manual consolidation process and
optimize the release sequence of bins according to this aim. From a methodological
point of view, our problem is closely related to single machine scheduling. The jobs
processed by a machine equal the bins to be released from the ASRS. Our objective
is to minimize the completion times of orders, such that the packers’ idle times are
reduced. The single machine scheduling problem minimizing the sum of completion
times is well known to be solvable in polynomial time (Smith 1956), whereas our
problem where each job (bin) may contribute to completing multiple orders is
shown to be strongly NP-hard. The main difference between machine scheduling
and our problem is that the former either has a 1:1 relation between each job and the
order the respective job refers to or a 1 : nrelation where multiple jobs, e.g., in a
job-shop, open-shop or flow-shop environment, are to be processed to complete an
order. In our problem, we have a m:nrelation instead. Each job (bin) contains
items for many orders and each order requires items from multiple bins. An
additional major distinction to open-shop and flow-shop scheduling problems is that
we have just a single machine (i.e., the ASRS). Thus, we treat a very basic, yet (to
the best of the authors’ knowledge) novel extension of traditional machine
scheduling.
It can be concluded that this paper treats a novel optimization problem, which has
not been treated in the scientific literature yet.
Manual order consolidation with put walls…173
123

1.3 Contribution and paper structure
This paper aims to optimize the bin sequence in which a batch of orders is released
from intermediate storage into the consolidation area.The batch of orders is assumed
to be already completely assembled in the ASRS after being picked under a
batching and/or zoning policy. We aim at a release sequence of all bins from the
ASRS, such that orders are quickly sorted into the put wall and unproductive idle
time of packers is avoided. Our optimization objective for the bin sequencing
problem is to minimize the sum of completion times at which orders are readily
assembled in the put wall. The impact of optimized bin sequences on the subsequent
packing process on the other side of the wall is evaluated by a simulation study. This
way, we quantify potential performance gains of optimized release sequences
(compared to random sequences) in different warehouse settings. The contribution
of this paper is, therefore, twofold. From a methodological point of view, we treat a
basic extension of single machine scheduling where we have a m:nrelation among
the job processed and the orders these jobs belong to. We settle computational
complexity for the sum of completion times objective and provide exact and
heuristic solution procedures. From an application perspective, we show that an
optimized release of bins from intermediate storage has the potential to considerably
improve consolidation performance compared to the random release sequences that
are typically applied in real-world warehouses.
The remainder of the paper is structured as follows. Section. 2defines our
batched order bin sequencing problem and investigates computational complexity.
Suited exact and heuristic solution procedures are developed in Sect. 3, whose
computational performance is investigated in Sect. 4. Then, Sect. 5introduces the
setup of our simulation study and explores the impact of optimized release
sequences on the packers’ idle time. Finally, Sect. 6concludes the paper.
2 The batched order bin sequencing problem
2.1 Problem definition
This section defines the batched order bin sequencing (BOBS) problem, which
decides on the release sequence of bins from intermediate storage into the
consolidation area. Our batch of orders O¼f1;...;mghas been picked under a
batching and/or zoning policy, so that the items of this batch are spread over a set
I¼f1;...;ngof bins. Depending on the specific content of the bins sets IoI
define the subset of bins, which contain at least one item for order o. Processing a
bin at the put wall requires the putter to withdraw item after item from the bin. The
putter scans the current item, so that the put-to-light mechanism can indicate the
shelf where the respective order is collected, and puts the item into the shelf.
Depending on the number of items contained in a bin, processing bin itakes
processing time pi. A specific bin sequence is represented by a sequence /, i.e., a
permutation of bins i¼1;...;n, with /ðkÞreturning the bin at sequence position
k¼1;...;n. Let jð/;oÞ¼maxfk¼1;...;n:/ðkÞ2Iogbe the sequence position
174 N. Boysen et al.
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of the last bin containing items for order o. Among all sequences /, BOBS seeks
one which minimizes
Zð/Þ¼X
o2OX
jð/;oÞ
k¼1
p/ðkÞ:
Thus, we aim to optimize the sum of completion times at which all items of each
order are readily placed in the respective shelf of the put wall. This way, the orders
of the batch are quickly assembled, so that unproductive idle time of packers on the
other side of the wall is reduced.
BOBS is based on some (simplifying) assumptions, which we elaborate in the
following:
•The bins are intermediately stored in an ASRS until all bins of the batch have
arrived. With regard to the retrieval process we assume that the ASRS is able to
release the bins in arbitrary sequence. Thus, assignment restrictions where
specific sequence positions are forbidden for some bins are a non-issue.
•All data is assumed to be deterministic. Thus, there are no inventory differences
of items missing or being picked into wrong bins. We also assume that the
number of items per bin is a reliable predictor for the processing time, so that
modelling it as deterministic is not too far away from real-world operations.
•Typically, no information on the stacking plan of items within a bin is available.
Thus, the retrieval sequence of items from a bin is unknown. Therefore, we take
some worst-case perspective and assume that an order is completely stored in the
wall not before the complete processing time of its final bin has elapsed.
•To simplify the optimization problem we do not explicitly model the packing
process behind the wall. The total completion time is just a surrogate objective
for an efficient packing process. It will be part of our simulation study in Sect. 5
to explore whether our proxy is indeed well suited to improve packing
productivity.
Example 1 Consider the example of Fig. 2where a batch of m¼4 orders is spread
over n¼5 bins. Items are represented by circles within the bins and the orders these
items are dedicated to are referenced by the numbers within the circles. The
processing times of the bins correspond to the number of items contained, so that a
bin iwith two items inside is supposed to have a processing time of pi¼2. Figure 2
depicts two alternative solutions. Solution (a) has a total completion time of
Zð/Þ¼39, whereas solution (b) only amounts to Zð/Þ¼34.
2.2 A mixed-integer program
Based on the notation presented in Table 1, BOBS can be formulated as a mixed-
integer programming (MIP) model, which consists of objective function (1) and
constraints (2)to(7).
Manual order consolidation with put walls…175
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BOBS MIP:MinimizeZðC;X;YÞ¼X
o2O
yoð1Þ
subject to
X
i2I[f0g
xi;j¼18j2I[fnþ1gð2Þ
X
j2I[fnþ1g
xi;j¼18i2I[f0gð3Þ
Cið1xi;jÞMCjpj8i2I[f0g;j2I[fnþ1gð4Þ
C0¼0ð5Þ
Ciyo8o2O;i2Ioð6Þ
xi;j2f0;1g8i2I[f0g;j2I[fnþ1gð7Þ
Objective function (1) minimizes the sum of completion times of all orders. Con-
straints (2) and (3) make sure that the sequence of bins is well defined. Constraints
Fig. 2 Two alternative bin sequences for the example
Table 1 Notation
OSet of orders (index: o¼1;...;m)
ISet of bins (indices: i;j¼1;...;n)
IoSet of bins containing items for order o
piProcessing time of bin i
MBig integer (e.g., M¼Pi2Ipi)
yoContinuous variable: completion time of order o
CiContinuous variable: completion time of bin i
xi;jBinary variable: 1, if bin jis direct successor of bin i; 0, otherwise
x0;iBinary variable: 1, if bin iis the first one to be processed; 0, otherwise
xi;nþ1Binary variable: 1, if bin iis the last one to be processed; 0, otherwise
176 N. Boysen et al.
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(4) set the completion times of bins and avoids overlap of their processing intervals.
The first (virtual) bin with index 0 has completion time 0 (see constraint 5) and the
orders’ completion times are set by (6). Finally, constraints (7) define the domain of
the binary variables.
2.3 Computational complexity
In this section, we investigate the complexity status of BOBS and prove NP-
hardness in the strong sense. The transformation is from the linear arrangement
problem (LAP), which is well-known to be NP-complete in the strong sense (see
Garey and Johnson 1979).
LAP: Given a graph G¼ðV;EÞand a positive integer K. Is there a one-to-one-
function #:V!f1;2;...;jVjg, i.e., a numbering of nodes Vwith integer values
from 1 to |V|, such that Pðu;vÞ2Ej#ðuÞ#ðvÞj  K?
Theorem 1 BOBS is strongly NP-hard even if all bins have unit processing time,
i.e., pi¼1 for all i¼1;...;n.
Proof Within our transformation of LAP to BOBS we introduce a bin for each
node, so that n¼jVj. The integer value #ðuÞassigned to a node uwithin LAP
corresponds to the sequence position /1ðiÞassigned to bin iwithin BOBS. Given
the maximum degree dðGÞ¼maxu2Vv2V:ðu;vÞ2E
fgjjfg
of the LAP graph, we
introduce dðGÞorders for each node u2V: First, an order fu;vgis generated for
each adjacent node v, so that for each edge ðu;vÞ2Etwo orders fu;vgand fv;ug
are generated. Then, for each node having a degree less than dðGÞwe extend the
order set by additional single-bin-orders fuguntil dðGÞorders per node are
generated. In total, dðGÞjVjsingle- and two-bin-orders are introduced. The
question we ask is whether we can find a solution for BOBS with objective value
Z¼dðGÞjVjðjVjþ1Þ
2þK:
Obviously, this transformation can be done in polynomial time. The dðGÞorders
associated with each bin uare either single-bin-orders, which have completion time
/1ðuÞ, or two-bin-orders. Each order fu;vgof the latter kind always exists twice,
because in the name of each edge (u,v) two identical orders are introduced, i.e., one
when generating the dðGÞorders for node uand the other when generating orders
for v. The unit processing times allow us to measure completion time in sequence
positions of the bin sequence. The sum of completion times for both of these orders
is, thus, twice the sequence position of the later of both bins uand v.If
/1ðuÞ\/1ðvÞ, this amounts to 2/1ðvÞ. Due to the inequality of /1ðuÞand
/1ðvÞ, we can rearrange 2/1ðvÞto /1ðvÞþ/1ðuÞþð/1ðvÞ/1ðuÞÞ.Ifwe
assign the former two time spans /1ðvÞand /1ðuÞto bins vand u, respectively,
then it becomes obvious that to each sequence position i¼1;...;nexactly dðGÞ
time spans are assigned. Thus, we have an inevitable amount of completion time,
i.e., independent of the sequence positions of bins, of
Manual order consolidation with put walls…177
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dðGÞX
u2V
/1ðuÞ¼dðGÞX
jVj
i¼1
i¼dðGÞjVjðjVjþ1Þ
2:
The remaining time spans /1ðvÞ/1ðuÞwithin BOBS, which are dependent of
the sequence positions of bins, exactly equal the difference in the node numbers
assigned to each edge within LAP, so that both problems are directly transferable
from each other and the theorem holds. h
Example 2 For a more intuitive understanding consider the LAP instance given in
Fig. 3a and its belonging optimal linear ordering (b) with an objective value of
K¼6. The orders to be generated are illustrated in the put wall on the left side of
Fig. 3c. The corresponding BOBS bin sequence with
Z¼3jVjðjVjþ1Þ
2þK
¼2þ5þ2þ3þ5
|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
orders 11 to 15
þ1þ5þ3þ5þ4
|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
orders 21 to 25
þ1þ5þ2þ4þ4
|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
orders 31 to 35
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
completion times
¼51
is depicted in (d).
Theorem 1settles the complexity status for the rather theoretical case of unit
processing times. In business practice, however, the processing time varies from bin
to bin and depends on the number of items that are contained in each bin. Therefore,
the following corollary transfers our previous result to the case where processing
times are proportional to the number of items contained in each bin.
Fig. 3 Example for the transformation of LAP to BOBS
178 N. Boysen et al.
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Corollary 1 BOBS is strongly NP-hard even if all processing times piequal the
number of items contained in bin ifor all i¼1;...;n.
Proof This result immediately follows from the proof of Theorem 1. All we have
to do is to make sure that all bins contain the same number cof items. Extending the
previous transformation scheme, we fill up each bin, so that each of them contains
exactly c¼2dðGÞþ1 items. Additionally, we introduce one extra order requiring
all these fill-up items (e.g., at least one item from every bin). This additional order
will be completed right after processing the last bin and has, therefore, no further
influence on the optimal bin sequence. Although all bins have processing times
depending on the number of items contained, the transformation from LAP is still
valid, as we seek now for a solution with objective value
Z¼2dðGÞþ1ðÞdðGÞjVjðjVjþ1Þ
2þKþjVj

:
h
3 Solution procedures
This section is dedicated to exact and heuristic solution procedures for BOBS. At
first, a simple greedy heuristic is presented in Sect. 3.1. Afterwards, BOBS is
tackled by dynamic programming in Sect. 3.2.
3.1 Greedy heuristic
BOBS is an extension of the single machine scheduling problem minimizing the
sum of completion times. This problem is well known to be solvable to optimality in
polynomial time by applying the shortest-processing-time (SPT) rule (Smith 1956).
To transfer the basic logic of the SPT rule to BOBS, where multiple jobs may be
required to complete each order, we prioritize all not yet finished (active) orders by
their remaining processing time if all missing bins of the respective order are
scheduled in direct succession. Among all active orders we select one with the
shortest remaining processing time. As a tiebreaker the lower order index is applied.
All missing bins of the selected order are, then, added to the sequence. This greedy
approach cannot guarantee optimal solutions, but we, nonetheless, hope that our
simple adaption of the SPT logic yields solutions with small optimality gaps very
quickly. It will be part of our computational study in Sect. 4to explore whether this
hope is justified.
Manual order consolidation with put walls…179
123

Algorithm 1 details the greedy procedure in a more formal way using pseudo
code. First, the applied data structures are initialized (lines 1–4). Then, we calculate
the remaining processing times of all not yet finished (active) orders by adding the
processing times of the respective (active) bins not yet processed (lines 6–9). In line
10, we select the active order with the shortest remaining processing time and add
all active bins still required by the selected order to the bin sequence (lines 11–14).
Then, the selected order is removed from the set of active orders (line 15). We
proceed until all orders are processed and, finally, the bin sequence is returned.
Example 1 (cont.): For the example of Fig.2our greedy heuristic leads to solution
(b) and an objective value of Zð/Þ¼34.
3.2 A dynamic programming procedure
For determining optimal solutions we introduce a dynamic programming (DP)
procedure, which applies the traditional DP scheme for sequencing problems
defined by Held and Karp (1962). The decision process of DP is subdivided into
nþ1 stages, where each stage k¼1;...;nrepresents a sequence position of the bin
sequence (plus a virtual start stage 0). We plan the sequence in reverted order and
start in final sequence position n. It is the last bin that determines the order’s
completion time, so that the reverted order allows gaining at least some completion
times quickly. As we extend the basic DP scheme with upper and lower bounds and
also apply it in a heuristic beam search procedure, backwards planning showed
considerably superior compared to forward planning in preliminary computational
tests. Any stage kcontains a set of states, where each state represents a possible
subset I0I,jI0j¼k, of bins already assigned to sequence positions
ðjIjkþ1Þ;...;jIj. The optimum schedule for subset I0is found according to
recursive equations
180 N. Boysen et al.
123

hðI0Þ¼mini2I0hðI0nfigÞ þ fi;I0nfigðÞ
fg
ð8Þ
with hð;Þ ¼ 0. The objective value of the optimal bin sequence for bins I0nfigis
added to the contribution fði;I0nfigÞ of assigning bin ito sequence position ðjIj
kþ1Þand having bin set I0nfigassigned to subsequent sequence positions
ðjIjkþ2Þ;...;jIj, where fði;e
IÞis calculated according to
fði;e
IÞ¼ X
j2IneI
pj
0
B
@1
C
Ajfo2O:i2Io^Io\~
I¼ ;gj:
The first term in brackets calculates the completion time of the currently assigned
bin iby summing up the processing times of the set of not yet fixed bins, which then
is multiplied with the number of orders just being completed by bin i. The overall
objective of DP is to find optimal solution value hðIÞfor the set of all bins. We can
determine hðIÞby a stage-wise recursion using (8). Finally, when stage 0 is reached
and optimal solution value hðIÞis determined, a simple recursion in reverted
direction can be applied to determine the optimal bin sequence.
There are 2nstates and n2n1transitions to be evaluated. Each transition
requires the inspection of all morders, which in the worst case refer to all nbins.
Thus, the computational complexity of DP amounts to Oðmn22nÞ, so that (in line
with the previous complexity result) there is an exponential increase of runtime in
the number of bins n.
To improve the runtime of DP (without altering the worst-case runtime stated
above), we additionally consider a global upper bound as well as local lower bounds
in each state. The initial upper bound is determined by solving our greedy heuristic
(see Sect. 3.1). In each state, we try to improve the upper bound by completing the
partial solution represented by the current state and schedule all bins not sequenced
yet by applying our greedy heuristic. Note that the set of active orders not yet fixed
at the beginning of the iterative greedy procedure is fo2O:Io\~
I¼;g, with ~
I
being the set of bins already scheduled in the partial solution.
The basic idea of our local lower bound procedure is to relax the fact that only
one bin can be processed at a time. We assume that each active order is processed as
soon as possible, while we only consider pairwise disjunct orders. To determine the
local lower bound of a state, we at first determine sets 
Oo¼fo02O:Io\Io0¼;g
of orders being disjunct to order o. Based on these sets of disjunct orders 
Ooand the
set of orders already planned e
O, e.g., orders o2Owith at least one bin i2Io
sequenced, we define parameter 
toin each state by

to¼Pi2Iopiif 
Oone
O¼;
maxo02
Oone
Ofdoo0gotherwise
(8o2One
O;ð9Þ
with
Manual order consolidation with put walls…181
123

doo0¼Pi2Io0piþPi2Iopiif Pi2Io0pi\Pi2Iopi
Pi2Iopiotherwise.
(ð10Þ
This means that for all orders still active o2One
Othe completion time 
tois
assumed to be the earliest possible point in time, which is at least the sum of
processing times of the single bins of this order Pi2Iopi(see Eq. 9). To sharpen the
bound, we consider that pairwise disjunct orders cannot be processed in parallel. If
two active orders are pairwise disjunct, the one with longer processing times cannot
be scheduled before the other one is completed (see Eq. 10). Based on the values

to8o2One
O, we can define a local lower bound LB depending on the set of already
planned bins ~
Ias
LBð~
IÞ¼hð~
IÞþ X
o2One
O

to:
Using these bounds, we can extend our DP scheme to a bounded DP, where states
are only further developed if their lower bounds LBð~
IÞare lower than the current
upper bound value. If the state space becomes too large our bounded DP scheme can
also be heuristically applied in a bounded iterative beam search approach. Using
beam search, only the !best states (dubbed beam width) according to their partial
costs hðe
IÞare further considered in each stage. The intensity of the beam search,
thus, depends on three parameters ðX;!0;WÞ.Xrepresents the number of beam
search iterations performed, while !0is the initial beam width in the first iteration
of the DP scheme. To intensify the search with every iteration, the beam width is
multiplied with Win each of the following iterations, such that !n¼!n1W¼
!0Wnin iteration n. In this way, the upper bound is updated in each iteration, so
that a more exhaustive search with a larger beam width is affordable. Finally, after
Xiterations the best solution, which equals the current upper bound, is returned.
Example 1 (cont.): The resulting DP graph for our example of Fig. 2is depicted
in Fig. 4. Starting with an initial upper bound of 34 provided by the greedy heuristic,
the bounded DP approach is able to prove optimality of this solution in stage 3 of
the graph. In this stage, the states f1;2;4gand f1;2;5gboth represent partial
solutions where the completion times of all orders are already known. Therefore, the
objective value of the represented set of bin sequences can be computed by
hðf1;2;4gÞ¼hðf1;2;5gÞ ¼ 111 þ110 þ28¼37. As this value is larger
than the current upper bound, both states can be fathomed. The already fixed
completion times of state f1;2;3gamount to 29, while the completion time of order
2 is not yet known. Since no other order is still active (i.e., 
O2ne
O¼;), we can
determine the local lower bound by LBðf1;2;3gÞ ¼ hðf1;2;3gÞ þ ðp4þp5Þ¼34.
In consequence, this state can be fathomed, too, and optimality of the greedy upper
bound solution is proven.
182 N. Boysen et al.
123

4 Performance of solution procedures
In this section, we investigate the performance of our solution procedures. Since no
established testbed is available for our BOBS problem, instance generation is
detailed in Sect. 4.1. Then, the performance results of our solution procedures are
presented in Sect. 4.2.
All computations have been executed on a 64-bit PC with an Intel Core i7-6700K
CPU (4 4.0 GHz), 64 GB main memory, and Windows 7 Enterprise. The
procedures have been implemented using C# (Visual Studio 2015) and off-the-shelf
solver Gurobi (version 7.0.2) has been applied for solving the MIP model.
4.1 Instance generation
We generate two differently sized datasets for our computational study. The
instances in the small dataset are still solvable to optimality using our bounded DP
approach, while the large instances are of real-world size, so that only heuristic
solutions can be obtained by our procedures. The parameters handed over to our
instance generator are listed in Table 2. The parameter values are chosen based on
Fig. 4 Bounded dynamic programming graph for Example 1
Table 2 Parameter values for
instance generation Symbol Description Small Large
|O| Number of order 12 20
|I| Number of bins 24 40
lMean number of bins per order 2.3
rStandard deviation of bins per order {0.5,1.5,2.5}
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personal information from practitioners. We generate 25 instances per parameter
setting, so that we receive 75 small and 75 large instances. In the following, we
detail how each single instance is obtained.
In a first step, we draw the quantity of bins per order from a standard normal
distribution having mean value land standard deviation r. Note that, for obvious
reason, each order is at least contained in one bin, so that we cut the lower end of the
standard normal distribution. We also check if the total number of bins is less than
or equal to the sum of bins over all orders. If not, we return to the first step,
otherwise we assign one randomly chosen order to each bin to ensure that each bin
contains at least one order. Then, we draw the remaining bins of each order with
equal probabilities. Having a mapping between bins and orders, we set the number
of items of each order in a bin by drawing an integer from interval [1; 3] using a
uniform distribution. Based on the assumption that the scanning and putting process
for each item lasts 4 s, the processing time of each bin is finally computed.
4.2 Computational performance
In this section, we compare the solution quality of our two exact procedures, i.e.,
Gurobi solving the MIP model presented in Sect. 2.2 and our bounded DP of
Section 3.2. Furthermore, we benchmark the two heuristic approaches, i.e., our
greedy heuristic of Sect. 3.1 and the iterative beam search (IBS) procedure of Sect.
3.2. Table 3summarizes the results for the small dataset. For all four solution
methods, we report the average relative gap to the optimal solution (gap), the
number of optimal solutions obtained (#opt) by the respective approach, and the
average CPU seconds (s). IBS is executed with steering parameter values
ðX;!0;WÞ¼ð10;1:5;8Þ, which in preliminary computational tests showed a
promising performance. The following conclusions can be drawn from these results:
•When comparing the exact solution procedures, it can be concluded that
bounded DP clearly outperforms standard solver Gurobi. Although Gurobi uses
all its allowed solution time (i.e., the timeout is set to 600 s) it only solves 15 out
of 75 instances (20%) to optimality. For none of these instances the optimum is
proven. On average, the gap to the optimal solutions is more than 1.2%.
Bounded DP, instead, is able to find all optimal solutions in only 5 s on average.
Table 3 Computational results for the small dataset [gap to optimal solution (gap)/number of optimal
solutions found (#opt)/computation time in seconds (s)]
rbounded DP Gurobi IBS (10,1.5,8) Greedy
gap/#opt/s gap/#opt/s gap/#opt/s gap/#opt/s
0.5 0.00%/25/8.77 1.43%/5/600.00 0.00%/25/0.49 1.98%/7/\0.00001
1.5 0.00%/25/4.12 1.16%/6/600.00 0.00%/25/0.49 0.81%/13/\0.00001
2.5 0.00%/25/2.28 1.09%/4/600.00 0.00%/25/0.47 0.26%/15/\0.00001
Total 0.00%/75/5.06 1.23%/15/600.00 0.00%/75/0.48 1.02%/35/\0.00001
184 N. Boysen et al.
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•Our heuristic methods also lead to promising results. IBS also finds all optimal
solutions, while the average time consumption of this approach is only half a
second. The greedy heuristic is even faster. Its time consumption is barely
measurable and the average gap is only 1.0%, which is even better than the
optimality gap produced by Gurobi.
When solving the large dataset both bounded DP and Gurobi ran out of memory,
so that optimal solutions are not available. We, thus, only benchmark our two
heuristic approaches and report the gap to the best solution found by both
competitors (gap), the number of best solutions (#best), and the average
computational time (s). IBS is executed with steering parameter values
ðX;!0;WÞ¼ð25;2:0;15Þ. As can be seen in Table 4, the time consumption of
the greedy approach is still barely measurable and its average gap to the best
solutions found by IBS after more than an hour of computational time is less than
1.5% on average. Given the long runtime of IBS and a selection of steering
parameters, which increases the number of iterations and states generated per stage
compared to the small dataset, we conjecture that IBS comes pretty close to the
optimal solutions. If this is true, our simple greedy heuristic leads to an
astonishingly good performance and solves even large-sized instances with very
good results. Therefore, we apply the greedy heuristic for all following tests, where
we explore the impact of optimized bin release sequences on the performance of the
consolidation process.
5 Managerial aspects
In this section, we address some managerial aspects. To explore the impact of
different bin release sequences we apply a simulation study, where work of the
putter and the packer in the consolidation is emulated. The setup of the simulation is
described in Sect. 5.1. With the help of the simulation we can explore whether our
surrogate objective for deriving the bin release sequence is indeed a good choice. In
Sect. 5.2, we compare the performance of optimized and random bin release
sequences. In Sect. 5.3, we investigate how our performance measures are
Table 4 Computational results for the large dataset [gap to best solution (gap)/number of best solutions
found (#best)/computation time in seconds (s)]
rIBS(25,2.0,15) Greedy
gap/#best/s gap/#best/s
0.5 0.00%/25/3823.12 1.95%/0/\0.00001
1.5 0.00%/25/3843.56 1.08%/5/\0.00001
2.5 0.00%/25/3722.46 1.25%/3/\0.00001
Total 0.00%/75/3796.38 1.43%/8/\0.00001
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influenced by important system characteristics, i.e., packing speed, the number of
packers, and differently sized batches. In this way, we gain insight into optimal
process design and identify scenarios in which optimization considerably speeds up
order consolidation and others where not.
Note that considering a deterministic offline scheduling problem is no
shortcoming from the practitioner’s point of view. We presuppose that the complete
batch of orders is first assembled in intermediate storage, only once all bins
belonging to the same batch have arrived they are released into the consolidation
area. This is exactly the mode of operation we observed in business practice. The
next batch of orders to be picked is selected among all those orders whose cutoff
date is approaching. According to the zones the requested items of these orders are
stored in, the set of bins to be picked next is derived. Identified by the barcodes on
the bins the system controls the arrival of all bins belonging to a batch and marks it
as completed finally. Completed batches, which are pairwise disjunct, are
successively released towards the put wall. In this way, an excessive blocking of
sortation capacity due to some required but not yet picked items is avoided (see
Sect. 1.1). Our approach is, thus, easily implementable in real-world operations. A
completed batch of bins has to be retrieved in a specific sequence that is to be
announced to the IT system of the ASRS anyway. Instead of announcing just some
random bin sequence, which is the current approach at least in the warehouses we
visited, it is easily possible to hand over an optimized bin sequence without any
further investment into hardware.
5.1 Setup of simulation study
In the simulation study, we emulate the consolidation process of a single putter and
one or multiple packers in the consolidation area. Each simulation run covers 21
batches of orders spread over multiple bins, which successively arrive in the
consolidation area to be processed. The first batch is used as a tuning phase, whereas
the remaining ones are used to collect performance data.
Each batch is equivalent to a problem instance of our BOBS problem.
Consequently, we generate each batch in the same way as we generate our large
problem instances using a rvalue of 1.5 (see Sect. 4.1). Furthermore, we have to
define some additional simulation parameters, which are summarized in Table 5.
Note that the bold parameter values are the default values, which are applied if not
explicitly stated otherwise. Once again, we chose these values based on personal
Table 5 Parameter values for simulation study
Description Value(s)
Seconds required to scan an item and put it into the put wall 4
Seconds required to pack an order {8, 12, 16, 20,..., 60}
Number of packers {1,2,..., 10}
Number of bins containing all items of a batch {1, 5, 10, 15,...,40, 45}
186 N. Boysen et al.
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information of practitioners or averaged data collected during warehouse tours. We
generate 100 basic simulation instances, which are adapted to the different
parameter values of Table 5, such that each data point in the figures of Sect. 5.3
represent the average result of 100 simulation runs.
The basic setup of our simulation study is based on a put wall configuration,
which we observed at an Amazon facility and is schematically depicted in Fig. 1.
On one side of the put wall, the putter receives the bins via a conveyor belt. He/she
successively removes items from the bins, scans them, and puts them into the rack.
In business practice, these steps consume about 4 s of working time per item. On the
other side of the put wall one or multiple packers withdraw completed orders, pack
them into cardboard boxes, and hand them over to a conveyor system. Depending on
the size and weight of products and the technical support equipment time
consumption of the packing process can vary within a broad range. As a default
value we assume 20 s for packing each completed order, but we vary this value to
explore the impact of differing packing speeds.
Putting and packing on both sides of the wall impacts each other. Once a
predefined fill level of the put wall is reached, the background IT system initiates the
release of the next batch of orders from intermediate storage, so that the respective
bins timely reach the put wall once the previous batch is completed and the put wall
is empty again. Only then, the putter starts filling the shelves of the put wall with the
next batch of orders. Thus, the putter is idle once the last bin of the previous batch is
placed in the wall until the next batch starts again, so that the length of the idle time
depends on how fast the packers complete the orders. On the other side of the wall,
packers may have to wait for the putter if right after completing their previous order
the next order is not yet available. These interdependent waiting times on both sides
of the wall are recorded by our simulation, which leads us to the three performance
indicators of our simulation:
•average putter idle time,
•average packer idle time, and
•average makespan of a batch.
With these performance indicators on hand, we can test whether optimized bin
release sequences can indeed speed up the consolidation process.
5.2 Appropriateness of surrogate objective
Minimizing the sum of completion times within BOBS is just a surrogate objective
for the efficiency of the consolidation process. Therefore, we have to test whether
optimized bin release sequences positively impact our three performance indicators.
To evaluate this, we derive an optimized BOBS sequence with our greedy heuristic
and a random bin sequence for each instance. Recall that in business practice,
typically, random release sequences are applied. Each simulation instance is run
twice, once based on an optimized bin release sequence and once more for a random
sequence. For both settings, the performance measures of the consolidation process
derived by the simulation are recorded and compared. The results of this test
Manual order consolidation with put walls…187
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averaged over our 100 basic simulation instances are summarized in Table 6. Here,
we report all three performance measures as well as the relative (percentage point)
reduction of the respective performance indicator of optimized bin sequences in
relation to random ones and the p-value of a paired two-sample t-test. Note that the
relative (percentage point) reduction is calculated by
ðZsimðrndÞZsimð/ÞÞ=ZsimðrndÞ, where ZsimðrndÞand Zsim ð/Þdenote the value
of the respective performance measure obtained by simulation for the random and
the optimized bin sequence, respectively.
These results clearly indicate that both resources, i.e., the putter and the packer,
are more efficiently utilized if optimized bins release sequences are applied. The
idle time of the putter (packer) is significantly reduced by 68.77% (73.8%).
Additionally, our optimization approach reduces the average makespan of a batch
by more than a minute, which results in a reduction of 14.38%. Note that the p-
values of the paired two-sample t-tests indicate that the advantages of optimized
over randomized bin sequences are statistically significant. Further note that the
(theoretical) lower bound of the average makespan per batch, determined by the
sum of processing times of all bins plus the packing time of the last order, amounts
to 416.2 s. Our optimized bin release sequences have a gap to this bound of only
1.4%, while the randomized simulation runs have a gap of 18.5%. Thus, our
optimized bin sequences lead to a consolidation performance, which is pretty close
to the optimum.
It can be concluded that optimized bin release sequences considerably speed up
the consolidation process and our surrogate objective applied within BOBS seems
well suited. This conclusion holds for our default setting of simulation parameters.
In the following section, we explore the impact of different parameters, i.e., packing
time, the number of packers, and the number of bins over which each batch is
spread, on consolidation performance.
5.3 On the impact of varying system parameters
The packing time it takes a packer to pack an order into a cardboard box
considerably varies in business practice. It depends, for instance, on the size and
weight of the handled products, the type of packaging, and whether additional
protective packing material, advertising flyers, and invoices have to be added.
Furthermore, technical equipment to automatically erect, close, seal, and label
Table 6 Results of simulation study for optimized and random bin sequences
Performance measure Optimized Random Relative reduction
(%)
p-value
Average putter idle time [% of makespan] 6.42% 20.56% 68.77 \0.000001
Average packer idle time [% of makespan] 5.02% 18.65% 73.08 \0.000001
Average makespan of a batch [seconds] 422.1 493.0 14.38 \0.000001
188 N. Boysen et al.
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cardboard boxes can be applied to further reduce manual operations. To gain insight
into the effect of different packing times we run our simulation for optimized and
random bin release sequences for varying parameter values of the packing time (see
Sect. 5.1).
The results depicted in Fig. 5show that optimized bin release sequences
considerably improve consolidation performance only in a corridor where the
packing times range somewhere between 15 and 30 s. In this corridor, which (at
least to our experience) contains most packing times relevant in online retailing, the
ratio between putting and packing times is rather balanced. A quick assembly of
orders in the put wall (as is enabled by our BOBS objective) leads to a better
synchronization of putting and packing and, thus, to considerable performance
gains. If the system is rather unbalanced, however, no improvement over random
sequences can be realized. If the packing time is very short, e.g., only 8 s because all
process steps except for the retrieval of items from the wall and their placement in
an already erected cardboard box are automated, then the putter is the unique
bottleneck of the consolidation process and the packer is idle for more than half of
the makespan. On the other hand, if the packing time is large, e.g., 60 s because
large, heavy, and/or fragile items need to packed in a completely manual process,
the packer is the unique bottleneck and the putter is idle for more than half of the
makespan. In both cases, the system is in imbalance which cannot be countervailed
by merely optimizing bin release sequences. One lever to balance the system
whenever one side is considerably faster than the other is to increase the division of
labor on the other side. The impact of multiple packers is considered in the
following paragraph.
The performance results when varying the workforce of packers are depicted in
Fig. 6. More packers have a similar effect like shortening packing times. The
makepan is reduced, until the putter becomes the unique bottleneck and no further
improvement can be realized. In our default setting, even having just two packers
seems not advisable. The processing time of a batch is shortened by about 5% and
the putter is fully utilized, but the second packer is idle most of the time. The mean
value of the relative idle time per packer is 49.9%, such that this setting can only be
recommended if the average makespan per batch is to be reduced for any price.
Finally, we investigate the following question: given a batch of orders to be
collected, is it better to collect the items in fewer bins or is it advantageous to
10 20 30 40 50 60
0
10
20
30
40
50
60
packing time per order in seconds
average packer idle time [% of makespan]
optimized
random
10 20 30 40 50 60
0
20
40
60
packing time per order in seconds
average putter idle time [% of makespan]
10 20 30 40 50 60
400
600
800
1,000
1,200
packing time per order in seconds
average makespan per batch [sec]
Fig. 5 Performance benchmark for varying packing speed
Manual order consolidation with put walls…189
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distribute the items among more bins? With regard to the picking process more bins
means more pickers that collect items concurrently. A larger division of labor
accelerates the parallel assembly of the batch, which is an important effect in times
of tight delivery times. On the other hand, more pickers increase wage costs and
reduce the number of items to be collected per picker and tour. The latter increases
the frequency of returns to the depots, so that additional (unproductive) walking
results. With regard to order picking these effects are well researched in the
literature (see Sect. 1.2). We investigate the impact of the bin number on the
consolidation process. For this purpose, we spread the batches of our simulation
dataset over varying quantities of bins, while all other instance parameters
introduced in Table 2receive their default value.
Figure 7summarizes the results if batches are distributed over varying numbers
of bins. With just a few bins, no considerable improvement can be achieved by
optimized bin sequences. In the very extreme, if all orders are collected in a single
bin, we, naturally, have no difference between random and optimized sequences.
Already with 10 bins, however, over which our 20 orders are spread all three
performance measures considerably profit when employing optimized instead of
random sequences. The more bins we have, the shorter the processing time of each
single bin, which leads to less waiting times for putter and packer on both sides of
the wall. Thus, random sequences profit from more bins too. This effect, however, is
much more pronounced if optimized bin release sequences are applied. The more
bins we have, the more flexibility there is to reduce the completion times of orders,
246810
0
20
40
60
80
number of packers
average packer idle time [% of makespan]
optimized
random
246810
0
5
10
15
20
number of packers
average putter idle time [% of makespan]
246810
400
420
440
460
480
500
number of packers
average makespan per batch [sec]
Fig. 6 Performance benchmark for a differing number of packers
010203040
10
20
30
40
50
number of bins per batch
average packer idle time [% of makespan]
optimized
random
0 10203040
10
20
30
40
50
number of bins per batch
average putter idle time [% of makespan]
010203040
400
500
600
700
800
number of bins per batch
average makespan per batch [sec]
Fig. 7 Performance benchmark for a differing number of bins
190 N. Boysen et al.
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which, in turn, increases consolidation performance. Thus, more bins induce more
unproductive walking effort during order picking, but lead to a more efficient
consolidation process, and vice versa. Both effects need to be carefully traded off
against each other when setting up warehouse operations.
In summary, our computational tests suggest that optimizing the release sequence
of bins from intermediate storage is unhelpful for unbalanced systems, where the
packing time is very short or extraordinarily long or too many packers are applied,
so that a unique bottleneck on either side of the wall exists. Too few bins and, thus,
few sequencing flexibility has the same effect. In these cases, optimization cannot
improve over random bin sequences. However, if the working speed on both sides
of the wall is rather balanced, then bin release sequences optimized according to our
BOBS approach considerably outperform random sequences. Well-balanced
systems have no unique bottleneck, which in our simulation is avoided for a single
packer having packing times between 15 and 30 s. Furthermore, optimization leads
to considerable advantages if the number of bins a batch is contained in is large.
6 Conclusion
This paper considers manual order consolidation based on put walls. Batches of
orders are picked according to a batching and/or zoning policy and arrive in an
intermediate storage system. We show that optimized sequences in which these bins
are finally released from intermediate storage into the consolidation area can
considerably reduce the putter’s and packer’s idle times on both sides of the put
wall. This is good news for practitioners, because a better consolidation
performance can be achieved without any further organizational adjustments; only
the desired bin sequence has to be computed and communicated to the storage and
retrieval machine operating the intermediate storage system. Specifically, our bin
sequencing problem aims to minimize the sum of order completion times. We show
that this optimization problem is strongly NP-hard and provide suited exact and
heuristic solution procedures. In our computational tests, we show that a simple
priority rule based approach leads to an astonishingly good solution performance.
Furthermore, we quantify the performance gains of optimized versus random bin
release sequences for different warehouse settings.
Put walls need not be fixedly erected on the shop floor, but they also exist
equipped with small wheels in form of mobile put walls. With mobile walls, the
manual consolidation process is organized slightly different. The putter fills a wall
with a batch of orders and rolls it towards the packing workstations of packers. In
this setting, idle time of packers can be avoided if the putter timely delivers the new
wall towards each station before the current wall is completely depleted. On the
other hand, the delivery process increases the unproductive time of the putter. With
mobile put walls, optimizing the bin release sequence is not worthwhile, because the
makespan of completely filling each wall, such that it can be delivered, is not
impacted by the bin release sequence. All bins have to be processed anyway and the
sequence does not matter. However, future research should compare consolidation
processes based on mobile and immobile put walls, e.g., by a simulation study, such
Manual order consolidation with put walls…191
123

that practitioners having to decide between both alternatives receive some decision
support. From a theoretical point of view, our optimization problem is closely
related to traditional machine scheduling where, however, we have a m:nrelation
between jobs (bins) and orders (see Sect. 1.2). Given this fundamental extension of
machine scheduling, not only the sum of completion times is an interesting
optimization objective, but also all other traditional objectives, e.g., lateness-related
objectives, seem worth being investigated. Complexity results as well as exact and
heuristic solution procedures are required here.
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