From Single Orders to Batches: A Sensitivity Analysis of Warehouse Picking Efficiency

Abstract

Currently, companies are called to meet variable market demand whilst having to comply with tighter delivery times, also due to the growing spread of e-commerce systems in the last decade. As never before, it is therefore mandatory to increase the efficiency within distribution centers to minimize operating costs and increase environmental and economical sustainability. The picking process is the most expensive task in a warehouse, both for the required resources and time for completing all the operations, which is typically carried out manually. Several policies can be identified, such as discrete or batch picking. Many studies tend to optimize both policies, treating them distinctly and integrating them with other factors including, for instance, the logic of product allocation. This article stands on a higher decision-making level: starting from a database obtained with a simulative approach that contains the average distances covered by pickers in different warehouse configurations, the aim is to provide an analysis of which factors have the greatest impact in preferring a discrete order picking policy over the batch one. The factors in question are shape factor, input–output point, routing and storage location assignment policies. Results can be useful for industrial practitioners in defining the most efficient managerial strategies.

1. Introduction

In the last ten years, there has been a continuous increase towards the use of e-commerce by customers [1]. In this environment, the enterprises, in order to maximize customer satisfaction and not to risk incurring additional costs, are called upon to improve their processes to meet even very short delivery times [2,3]. Lately, there has also been a further reduction in the time the market is willing to wait from the time a purchase order is placed; for instance, in the case of quick commerce, companies may have to manage processes that cannot last more than a few hours. Therefore, it is evident that it is mandatory for companies to constantly streamline and evaluate their delivery and related processes, as demonstrated by [4], in order to remain competitive over time. For this purpose, within distribution centers (DCs), it is necessary to continuously increase the efficiency of processes for minimizing operating costs that would otherwise tend to become unsustainable, also taking into account the operators. In other words, it is essential to be both economically and socially (e.g., ergonomically) sustainable.

In this context, the scientific literature has attempted to provide innovative tools and solutions that could support the optimization of specific processes or seek integrated optimization of various processes within the DC. For example, in [5], the authors explored issues related to replenishment processes and, more specifically, they tried to minimize the total travel time required to carry out the operations while ensuring the availability of items in subsequent picking processes. In particular, this study deals with those contexts in which replenishment is a process that is separated in time from picking, and therefore seeks to determine rules that allow operators to replenish all the necessary items whilst being able to do that using the most time-efficient route. Similarly, in [6], the authors addressed the replenishment process when this, due to time constraints, must be carried out in parallel with the picking process; under these conditions, they developed and compared different policies to prioritize the items’ replenishment, aiming to minimize possible stock-outs during the picking process.

Regarding the storage processes, with reference to the storage location assignments, numerous research studies have been carried out over the last years. In [7], the topic of storage location assignments (SLAs) developed in warehouse contexts with non-traditional layouts was addressed, for instance, to reduce the traveled pathways to store and retrieve the loads with a consequent rise in the warehouse sustainability due to the resources saved (e.g., energy). In [8], instead, the authors provide an analytical model to support the design of non-traditional warehouses, in particular with V-shaped layouts, within which load units are stored according to the class-based storage (CBS) allocation strategy. In this context, assumptions and analytical formulations are provided to quantify the average normalized horizontal distance covered in different warehouse configurations, evaluating its performance also with different demand shapes. The research conducted by [9] presents another example of how a storage system impacts the order-picking time. The authors tried to evaluate the performance while simultaneously considering the picking and replenishment processes, and also assessed the economic impact of high-density flow-rack solutions. It is important to highlight that, typically, full-pallet storage systems and high-density flow-rack storage systems are used in parallel in the same warehouse. The research conducted by [10], on the other hand, addressed the impact that the three most important decision-making processes within a warehouse have on the travel made by pickers using a simulative approach; specifically, the impact of different picking, routing and warehousing policies on the total fulfillment time as order size varies was analyzed in the case of manual bin-shelving order-picking operation.

In all the abovementioned studies, the main metric for evaluating the proposed solution is the ability to optimize the picking process. The reason is that it corresponds to the most labor-intensive and time-consuming activity of warehouses; indeed, several studies quantify the impact of the picking process on total operating costs as being between 55% and 75% [9,10]. As shown in studies [11,12], four major factors directly impact the distances traveled by pickers to fulfill their missions in a traditional discrete order-picking system, where each mission corresponds to a single end-customer order. According to the authors, two different categories of factors can be found: hardware and software. Among the first factors, which are directly connected to the characteristics of the specific layout of the warehouse, we can find the shape factor (S_F) and the input–output point (I/O); among the second category, on the other hand, related to the management policies used in a specific warehouse, the routing policies (R_P) and the storage location assignment policies (ζ) can be identified.

Starting from these brief considerations, this study is based on a higher decision-making profile, aiming to understand which specific hardware and software factors impact the convenience of a “Pick-by-Order” (PBO) strategy compared to a “Pick-by-Article” (PBA) strategy. Based on the database used in [12], it was possible to derive the average time spent to complete one picking mission in PBO and PBA contexts, with orders of different sizes. The conditions of indifference were obtained for all the configurations and, by implementing a design of experiments (DoE), the parameters having the greatest impact on making one strategy more appropriate than the other were identified. In addition, the interactions between the four factors were investigated to further understand the conditions in which PBO is preferable to PBA, and vice versa.

The results obtained can be leveraged by industrial practitioners and managers in making strategic decisions when dealing with picking policies in their DCs, to make reasoned choices that consider sustainability from an economic and social perspective. In fact, the selected strategy is supposed to have the minimum cost and the least impact on the pickers, since its convenience is evaluated in terms of minimum covered distance. In addition, the optimization of the picking policy increases the efficiency of warehouse management by reducing processing times. As a consequence, it is also possible to provide shorter delivery times than competitors, in accordance with customer requirements. Finally, reducing the traveled distances and times would lead to a decrease in the resources employed for picking missions, coherently with environmental sustainability goals.

The article is structured as follows. Section 2 presents the state-of-the-art of the analyzed topic, deepening the issues related to order picking and batch picking. Section 3 describes the nomenclature used, the experimental campaign, the mathematical model implemented, as well as the methods used for the data analysis. Section 4 presents an overview of the obtained results, discussed in Section 5. Finally, in Section 6, the main conclusions are illustrated, together with insights for future research developments.

2. Literature Review

This section provides the reader with an overview of the state-of-the-art concerning the order-picking problem (OPP) and batch-picking problem (BPP). Firstly, an overview of the OPP is presented; then, reflecting the context of the present study, two different systems are addressed: the abovementioned PBO and PBA policies.

2.1. Order-Picking Problem

The OPP is a topic that has seen increasing attention in terms of scientific research over time. As underlined in [13], the focus has shifted towards issues related to picker routing within warehouses with conventional layouts and, lately, the impact of the presence of cross-aisles.

The OPP has been widely studied and analyzed from many different perspectives. An exhaustive classification of the various categories pertaining to the subject is presented in [14], and will be referred to in the use of terminology. In that study, two distinct classifications can be observed: the first one considers in general the main picking systems, and the second one classifies the picking systems according to the different optimization policies and methods. Under these preliminary assumptions, it is possible to distinguish picker-to-parts systems from parts-to-picker; specifically, in the first one, the PBO systems and the PBA systems can be treated in a different manner. In the second classification, on the other hand, PBO and the PBA systems are collapsed in the order batching category, characterized by the single-order policy and order batching policy. In this study, therefore, these two classification methods were merged and, based on them, a summary classification was made, as illustrated in the following Figure 1.

2.1.1. Single Order

In this system, each order is treated separately from the others, and the complete fulfillment of an order is achieved when all the products on the picking list have been picked by the operator. It is a simple form of picking, with a low risk of error, which does not require further subsequent processes. Once all the products have been collected, they are immediately sent to the following phases of packing and shipping. Despite these positive aspects, it is a low-productivity system because only one order is processed at a time.

To increase the productivity of this picking system, it is necessary to identify the routing policy that reduces the total traveled time. However, if the picking list consists of many products stored in very distant locations, it becomes difficult to significantly reduce the total distances covered and the processing times. Several studies in the literature have tried to overcome these problems. In particular, [15] underlined how the optimization of the traveled distance is strictly correlated to the storage location of the product and, therefore, the authors proposed an optimization approach integrating the choice of the storage location with the optimization of the traveled distances. The efficiency of routing policies in the optimization of the entire process is a subject that is treated at a transversal level, even in warehouses with unconventional layouts [16].

2.1.2. Order Batches

In this picking system, a stock keeping unit (SKU), in case it belongs to different orders, is collected during the same mission by the picker. In this case, the main constraints concern the maximum capacities that can be taken on a mission by pickers. Overall, one can obtain greater performances if compared to the previous system because more orders are elaborated in parallel from the same pickers in the same missions, accordingly limiting the number of times in which the same location is visited.

However, monitoring some critical issues is also required in this system. Surely, it is a more complex system that includes an additional phase prior to packing. Precisely, because in the same mission the same reference can belong to different orders, a sorting phase is typically inserted (between picking and packing activities) in which the products are separated according to the order they belong to. Only at the end of this phase are orders consolidated and ready to be prepared for packing and shipping. Under these assumptions, it becomes important to right-size the batches of the products to be picked for each mission so as to not have too many incomplete orders waiting during the sorting processes. In addition, the correct batch assignment sequencing must also be considered to avoid the same risk. An example of a case study that demonstrates how to optimize the order batching problem is presented by [17], where they use an approach based on the use of genetic algorithms.

Moreover, the efficiency objectives must be tailored to the routing policies, as demonstrates study [18], whose aim is the joint optimization of the order batching and picker routing problem. Another critical issue, then, is characterized by the storage allocation policy that impacts in the same general problem. Several studies deal with these issues in an integrated manner, such as the research carried out by [19] that analyses the possibility of grouping customer orders in batches so that the items belonging to a batch are collected during a single picking tour. They want to treat three challenges with a collective and balanced resolution. More specifically, the main issues to consider are (i) how to create the batches starting from various customers’ orders, (ii) how assign to the picker the batches, (iii) determining the sequence and finally (iv) determining the correct routing to collect the items of each picking mission.

At the same time, [20] developed an innovative three-stage model for considering the clustering, storage and joint online order batching and picker routing problems in the online shopping process. They note that, in this specific area, the performance of routing policies is affected by product allocation, but at the same time the correct allocation depends on clustering into the most homogeneous product classes possible. Under these assumptions, in their model, the first phase aims to classify the products to maximize the similarity. The second phase, on the other hand, assigns product classes to carriers and storage locations in a multi-block 3D warehouse model to minimize total transport costs and energy costs of vehicles. Finally, the third step provides a routing model that minimizes completion times. All of this considers the dissimilarity of the search times and withdrawal as well as the horizontal/vertical speeds of the collectors. In this way, different learning effects are also considered due to the presence of a variety of operators, each with a specific “way of working”.

Despite the large amount of research in the literature about picking systems, there is a gap in studies aiming to support the management in their operational decisions. More specifically, there were no insights into which parameters, both hardware and software, can have the greatest impact on making a warehouse more suitable for the use of a single-order system rather than an order batching system. With the intent to fill this gap, this research aims to analyze and identify which typical parameters of a warehouse (such as the shape factor, the routing policy adopted, the entry and exit points of pickers and the scope of a dedicated product allocation policy) have a greater impact on the picking system to be adopted through the evaluation of the time necessary for the fulfillment of orders in the two different systems analyzed.

3. Materials and Methods

3.1. Nomenclature

The nomenclature and acronyms used in this paper are listed and presented in

Table 1. Nomenclature and acronyms involved in the study.

Symbol	Description	Measurement Unit
DC	Distribution Center	-
SLA	Storage Location Assignments	-
PBO	Pick-by-Order	-
PBA	Pick-by-Article	-
DoE	Design of Experiments	-
OPP	Order-Picking Problem	-
BPP	Batch-Picking Problem	-
OS	Order Single	-
OB	Order Batches	-
SKU	Stock Keeping Unit	-
RP	Routing Policy	-
RS	Return Simple	-
RAD	Return Advanced	-
SSS	S-Shaped Simple	-
SSAD	S-Shaped Advanced	-
SF	Warehouse Shape Factor	-
CA	Number of Cross-Aisles	-
a	Width of the storage location	m
b	Depth of the storage location	m
I/O	Input–Output picker’s position	-
SCP	Single Central Picking	-
SLP	Single Lateral Picking	-
OLPSS	Opposite Lateral Picking, Same Side	-
OCP	Opposite Central Picking	-
OSP	Opposite Side Picking	-
ζ	Shape factor of the demand function	-
N	Warehouse storage capacity	-
Vt	Picking Transpallet speed	$\mathrm{k}\mathrm{m}/\mathrm{h}$
PL	Number of Items Picked up each mission	-
Conf	Configuration	-
T	Time required for a picking mission	min
t	Picking time	min
$t_s$	Sorting time	min
$\widehat{t_s}$	Limit sorting time	min
i	Iteration number	-
ATD	Average Traveled Distance	m

below.

3.2. Experimental Campaign

The analysis started from the database developed by [12], where the detailed description of the design parameters that will be proposed below is reported.

Specifically, this study focuses on a warehouse with a traditional double-side rack layout (Figure 2), with a fixed receptivity (N) of 1200 items. Each location measures 1 m in width (a) and 1.25 m in depth (b), suitable to store a standard-sized EPAL pallet (0.80 m × 1.20 m). The warehouse has also a fixed number of 2 cross aisles (CA), in addition to the top and bottom aisles.

In addition, the design variables were defined and grouped into software variables, i.e., related to the management of the warehouse, and hardware variables, i.e., physical constraints related to its layout. All the design parameters adopted in the development of the model are listed in

Table 2. Design values/parameters evaluated.

	Values/Parameters	Measurement Unit
Constraints
N	1200	-
CA	2	-
a	1	m
b	1.25	m
PL	10; 20; 30; 50	-
Software Variables
RP	RS; RAD; SSS; SSAD	-
ζ	~0; 0.001; 0.005; 0.01; 0.02; 0.03; 0.05	-
Hardware Variables
SF	0.2083; 0.2946; 0.5057; 0.7639; 1.0645; 1.5865; 2.6190; 3.5676; 3.8194	-
I/O	SCP; SLP; OPLSS; OCP; OSP	-

.

3.3. Mathematical Model

The configuration of a single warehouse model corresponds to the combination of these previously listed factors, as reported in Equation (1). A warehouse configuration, therefore, is defined by the combination of two software variables and two hardware variables. With the input data reported in Table 2, therefore, a total of 1260 warehouse configurations can be identified.

$$Conf=f(R_P,\mathsf{\zeta },S_F,\mathrm{I}/\mathrm{O})$$

(1)

For each warehouse configuration, it was then possible to define a simulation campaign of picking missions with different PL lengths to obtain a dataset that could be analyzed to perform a comparison between OS and OB picking policies. The number of items picked with OB or OS logic is equal to K and J, respectively ((2) and (3)).

$${PL}_{OB}=K,\hspace{1em}\hspace{1em}\hspace{1em}K\mathsf{\in }\mathbb{N}$$

(2)

$${PL}_{OS}=J,\hspace{1em}\hspace{1em}\hspace{1em}J<K,J\mathsf{\in }\mathbb{N}$$

(3)

The parameter n can be defined as the ratio of items picked with OB logic (2) to items picked with OS (3). This ratio, presented in (4a) and (4b), is defined as an integer value greater than zero and corresponds to the number of missions to be performed with OS logic to pick up the same number of items picked up with OB logic.

$$n={\displaystyle \frac{K}{J}},\hspace{1em}\hspace{1em}\hspace{1em}{\displaystyle \frac{K}{J}}\in \mathbb{N}$$

(4a)

$$n=\left({\displaystyle \frac{K}{J}}\right)-mant\left({\displaystyle \frac{K}{J}}\right)+1,\hspace{1em}\hspace{1em}\hspace{1em}{\displaystyle \frac{K}{J}}\notin \mathbb{N}$$

(4b)

$T_{OB}$ is defined as the time required to complete a mission in OB. Specifically, $T_{OB}$ includes two operations: $t_K^{OB}$, which is the time required for picking items from the warehouse, and $t_s$, which is the time required for sorting the items (5). $T_{OS}$ is defined as the time required to complete the task in OS. In particular, $T_{OS}$ is defined by the execution time of a single picking mission to pick J items ($t_J^{OS}$), multiplied by n (6). In Figure 3, the difference between the two picking strategies is presented, together with the representation of their specific timings introduced in Equations (5) and (6).

$$T_{OB}=t_K^{OB}+t_s,\hspace{1em}\hspace{1em}\hspace{1em}{T}_{OB},t_K^{OB},t_s\mathsf{\in }{\mathbb{R}}_0^{+}$$

(5)

$$T_{OS}=n·t_J^{OS},\hspace{1em}\hspace{1em}\hspace{1em}{T_{OS},t}_J^{OS}\mathsf{\in }{\mathbb{R}}_0^{+},n\in \mathbb{N}$$

(6)

The indifference condition, in which the time necessary for the two picking strategies is equivalent, is defined in (7). The substitution of (5) and (6) into (7) results in (8), where $\widehat{t_s}$ is the sorting time that makes $T_{OB}$ and $T_{OS}$equivalent—limit sorting time. By solving (8) with respect to $\widehat{t_s}$, it is possible to calculate the threshold sorting time (9).

$$T_{OB}=T_{OS}$$

(7)

$$t_K^{OB}+\widehat{t_s}=n·t_J^{OS}\hspace{1em}\hspace{1em}\hspace{1em}{t}_J^{OS},t_K^{OB},\widehat{t_s}\mathsf{\in }{\mathbb{R}}_0^{+},n\in \mathbb{N}$$

(8)

$$\widehat{t_s}=n·t_J^{OS}-t_K^{OB}\hspace{1em}\hspace{1em}\hspace{1em}{t}_J^{OS},t_K^{OB},\widehat{t_s}\mathsf{\in }{\mathbb{R}}_0^{+},n\in \mathbb{N}$$

(9)

In a specific context, if $t_s<\widehat{t_s}$, it is convenient to adopt an OB policy; on the other hand, if $t_s>\widehat{t_s}$, OS policy should be preferred. Cases with $t_s=\widehat{t_s}$ represent the indifference condition. Figure 4 below illustrates this concept, reporting on the x-axis all possible warehouse configurations.

In particular, from (9), it is possible to obtain values of $\widehat{t_s}$ that are lower, equal or greater than zero. Cases with $\widehat{t_s}$ < 0, although possible mathematically, are not real, and their occurrence simply means that, independently of the context, OS policy is always to be preferred. For cases with $\widehat{t_s}$ ≥ 0, the indifference curve, in red in Figure 4, can be obtained. This curve allows identification of two distinct areas in which, for a specific real configuration, it is possible to assess the most convenient policy: if a point, identified by the intersection of a given configuration and value of $t_s$, lies above the indifference curve, then OS policy is more convenient; otherwise, OB policy must be adopted.

3.4. Data Pre-Processing

As mentioned in the previous paragraphs, the initial dataset used in this study was derived from [12], which presented the results of the simulations performed in several configurations resulting from a factorial design that included all the possible combinations of the input factors. In the dataset, the calculated values of average distance traveled by pickers (ATD) were reported for each configuration.

In this study, the dataset was pre-processed with MS Excel™ to (i) convert all the distances traveled by the pickers into time and (ii) fix the number of items on each PL retrieved under OS policy for the following analyses (${PL}_{OS})$. The second point was achieved by arbitrarily setting ${PL}_{OS}$ equal to the minimum PL length simulated in the initial dataset, i.e., 10 items. Consequently, ${PL}_{OB}$ are the remaining PLs simulated in [10], i.e., 20, 30 and 50 items. The corresponding n values, calculated with (4), are 2, 3 and 5, i.e., OB logic is applied to group 2, 3 and 5 orders, respectively (10). For the first point, a technical-commercial analysis was conducted to obtain an average value of transpallet velocity (Vt), which was equal to 4 km/h, and the ATDs were converted into time with (11) and (12). Then, by applying (9) with n values of 2, 3 and 5, $\widehat{t_{s,2}}$, $\widehat{t_{s,3}}$ and $\widehat{t_{s,5}}$ were computed for the 1260 configurations considered and included in the dataset, respectively.

$${PL}_{OB}=n·{PL}_{OS}\hspace{1em}\hspace{1em}{PL}_{OB},{PL}_{OS},n\in \mathbb{N}$$

(10)

$$t_J^{OS}=Vt·{\displaystyle \frac{ATD}{1000}}\hspace{1em}\hspace{1em}{t}_J^{OS},Vt,ATD\mathsf{\in }{\mathbb{R}}_0^{+}$$

(11)

$$t_K^{OB}+\widehat{t_s}=n·t_J^{OS}=n·Vt·ATD\hspace{1em}\hspace{1em}{t}_J^{OS},t_K^{OB},Vt,ATD\mathsf{\in }{\mathbb{R}}_0^{+},n\in \mathbb{N}$$

(12)

3.5. Data Analysis

The updated dataset was analyzed in two subsequent steps: (i) analysis in MS Excel™ and (ii) elaboration in Design Expert. For the first step, all the configurations were sorted by decreasing values of $\widehat{t_{s,2}}$, $\widehat{t_{s,3}}$ and $\widehat{t_{s,5}}$, and the indifference curve was generated for each case. Finally, the relation between each input variable and the $\widehat{t_s}$ values was investigated, highlighting all the configurations characterized by $\widehat{t_s}<0$, where the OS policy is always more convenient compared to grouping orders into batches.

As mentioned, OB is assumed to be more convenient than OS when $\widehat{t_s}\ge 0$ and the actual operating point; the function of the operating configuration and the actual $t_s$, lies below the indifference curve (Figure 4). This is also due to the fact that configurations with high values of $t_s$ increase the probability of OS being more convenient due to the long time required for sorting in the case of OB.

In the second step, the dataset was analyzed with Design Expert to determine if, and to what extent, the hardware and software parameters impacted $\widehat{t_s}$. During this analysis, the effects of the four input factors, as well as their combinations, on the output responses of interest were evaluated (

Table 3. Input factors and responses of the statistical analysis.

Term	Parameter Type
RP	Categoric input factor
ζ	Discrete numeric input factor
SF	Discrete numeric input factor
I/O	Categoric input factor
$\widehat{t_{s,2}}$	Response
$\widehat{t_{s,3}}$	Response
$\widehat{t_{s,5}}$	Response

). Analysis of variance (ANOVA) was conducted to evaluate the significance of the input factors on the responses, up to a four-factor-interaction. Then, to make the impacts comparable, the standardized effects were calculated and plotted in a half-normal plot of effects. The standardized effects allow assessment of which contributions most influence the magnitude of $\widehat{t_s}$, thus the convenience of OB over OS, while the half-normal plot of effects provides an immediate representation of the impacts.

4. Results

4.1. Results of Indifference Curve Generation

Following the methodology described, for each configuration, $\widehat{t_s}$ values were sorted in descending order for the three batch sizes (n = 2, 3 and 5) (

Table 4. Number of configurations enabling assessment between OB and OS policies and their sample proportions.

$\widehat{{\mathit{t}}_{\mathit{s}}}$	Number of Configurations			Proportion
	<0	≥0	Total	<0	≥0	Total
$\widehat{t_{s,2}}$	527	733	1260	41.83%	58.17%	100.00%
$\widehat{t_{s,3}}$	378	882	1260	30.00%	70.00%	100.00%
$\widehat{t_{s,5}}$	152	1108	1260	12.06%	87.94%	100.00%

). A first consideration that can be made by looking at the results is that in moving from $\widehat{t_{s,2}}$ to $\widehat{t_{s,5}}$, there is a growing number of configurations with $\widehat{t_s}\ge 0$ for which a convenience of OB over OS policy could be found. Moreover, the number of configurations where OS policy is always more convenient independently of the context considered ($\widehat{t_s}<0$), decreased for bigger batches.

A more comprehensive representation is provided in Figure 5, where it can be seen that, by increasing the batch size, both the number of configurations with $\widehat{t_s}\ge 0$, and the $\widehat{t_s}$ values increased. In particular, it can be seen that values up to over 40 min can be achieved in the case of $\widehat{t_{s,5}}$.

As explained in Section 3.4, a single PL_OS consists of 10 items. Thus, under OS policy, if a total of 20, 30 or 50 items were ordered from the warehouse, 2, 3 and 5 missions would be required, respectively. Adopting an OB policy, on the other hand, a single mission would be executed but a sorting time $t_s$ would be required at the end of the mission. For example, assuming $t_s=2$ min for each PL_OS, the percentages of configurations among the 1260 analyzed for which OB policy is more convenient than OS are reported in

Table 5. Percentages of configurations where OB is more convenient than OS in case of t_s = 2 min/PL_OS.

n	${\mathit{t}}_{\mathit{s}}$ [min]	% Cases Where OB Is More Convenient
2	4	14.92%
3	6	29.60%
5	10	41.35%

.

Subsequently, all configurations sharing the same value of a given variable were evaluated; then, it was assessed how many of them were characterized by a negative $\widehat{t_s}$ value, and how many by a positive $\widehat{t_s}$ value. This activity was carried out for all three scenarios (n = 2, 3 and 5), and the results are presented in Figure 6, Figure 7 and Figure 8.

In general terms, it could be observed that, when the batch size increases, the percentage of configuration characterized by a $\widehat{t_s}<0$ strongly decreases. These results confirm the insights derived from Figure 5: the greater the batch size, the greater the convenience of the OB policy over OS.

The trends in $\widehat{t_s}$ as each of the hardware or software warehouse variables varied are discussed in detail in the following paragraphs.

First of all, as already discussed, the behavior of $\widehat{t_s}$ appears to be strongly impacted by the batch size.

Regarding the I/O location, it can be seen that the configuration characterized by an opposite side picking (OSP, i.e., with the entrance point on one side and the exit point on the opposite side) always generates $\widehat{t_s}\ge 0$, representing a possible convenience of OB policy over OS. As the batch size increases, the SCP configuration becomes interesting from the convenience-of-batching point of view; indeed, compared to the other I/O locations, in the case of n = 5, this configuration allows minimizing the area of $\widehat{t_s}<0$ to less than 10% of the cases evaluated.

The impact of the routing policy appears to be quite low: the percentages of configurations characterized by $\widehat{t_s}<0$ or $\widehat{t_s}\ge 0$ do not vary significantly when the routing policy changes.

Analyzing the behavior of ζ, batch picking is most convenient when all storage locations in the warehouse have approximately the same probability of being visited (i.e., ζ values between 0 and 0.005). When, on the other hand, some storage locations are visited more than others (e.g., allocation based on individual item turnover rates), the percentage of configurations in which OS policy is more convenient increases. In particular, when ζ is equal to 0.02, regardless of the batch size, in 40% of the configurations considered the OS policy is more convenient.

This may mean that in contexts characterized by products with similar turnover rates, there is a possible convenience of OB policy regardless of the quantity of products to be collected. Conversely, in contexts where the product turnover rates vary significantly, the OS policy appears to be more convenient in many configurations ($\widehat{t_s}<0)$.

Finally, analyzing the results as the S_F of the warehouse varies, it can be seen that even in the case of large batches, a percentage of at least 6% of configurations always requires an OS policy. In general, it can be observed that within wide and shallow warehouses, it is possible to have the convenience of an OB policy in more than 70% of contexts ($\widehat{t_s}$ > 0). On the other hand, with narrow and long warehouses, the percentage of configurations where OS policy is more convenient increases.

4.2. Statistical Analysis

The results of the statistical analysis, in terms of standardized effects, are shown in

Table 6. Standardized effect of the input factors and their interaction on the three responses.

Term	Standardized Effects on:
	$\widehat{{\mathit{t}}_{\mathit{s},2}}$	$\widehat{{\mathit{t}}_{\mathit{s},3}}$	$\widehat{{\mathit{t}}_{\mathit{s},5}}$
ζ	34.453	35.126	36.554
I/O	26.555	20.474	15.294
$\mathsf{\zeta }·$I/O	14.508	12.892	11.634
SF	11.600	12.173	12.204
$R_P·$I/O	8.875	8.882	9.701
$R_P·$ζ	8.127	7.846	8.115
$S_F·$I/O	6.530	5.093	4.174
$R_P·$SF	5.138	7.745	8.207
$R_P·$$\mathsf{\zeta }·$I/O	2.742	2.642	2.863
$\mathsf{\zeta }·$$S_F·$I/O	2.269	2.221	2.219
$R_P·$$S_F·$I/O	2.233	2.495	2.601
RP	2.184	4.522	11.527
$R_P·$$\mathsf{\zeta }·$SF	1.248	1.209	1.934
$\mathsf{\zeta }·$SF	1.206	1.091	3.239
$R_P·$$\mathsf{\zeta }·$$S_F·$I/O	0.134	0.0671	0.021

for the cases with n = 2, 3 and 5. The trends in the calculated effects, as the number of batched orders varies, can be observed in Figure 9. For greater detail, the trends in the standardized effects of the most significant input factors are presented in Figure 10.

The half-normal plot of effects for the response $\widehat{t_{s,5}}$ is reported for reference in Figure 11. In the plot, the squares farther away from the reference straight line represent larger and more influential effects. The straight line, indeed, represents the expected distribution of effects under the assumption that the evaluated factors have no impact on the response and the residuals follow a normal distribution.

From the results, it can clearly be seen that the most impactful variable is the one related to the allocation policy (ζ), i.e., how the allocation policy reflects the turnover rates of the items. Its impact also slightly increases with the dimension of the PL.

The same trend, although more pronounced, is followed by the impact of the routing policy (RP). As a consequence, when the size of the PL increases, the choice of the appropriate RP gains greater importance. On the other hand, the impact of the I/O point decreases when the items to be picked become numerous and potentially spread across the warehouse. Other factors, such as shape factor (S_F) and the two-factor combinations reported in Figure 9, although significant, maintain the same impact in all the scenarios analyzed; thus, they are not influenced by the PL length.

From Figure 9 and Figure 10, it can also be observed that the interaction among all four factors is not significant with regard to the convenience of batch picking over order picking, and neither are the interactions among the three variables.

In general, both the software and hardware variables’ results were significant, highlighting the importance of both the design and the management phase of the warehouse. In particular, when the number of items to be picked increases, the software variables (related to the management of the warehouse) become more significant in contrast to what happens to the hardware variables. Among the hardware variables, the I/O position resulted in being predominant over S_F. Moreover, with regard to the interaction between the two variable types, it was observed that the most significant ones were between software and hardware variables, with the hardware variable always being the I/O point.

5. Discussion

The analysis performed on published articles on picking operations revealed that there is a large body of literature that addresses the optimization of software aspects of warehouse management, i.e., picking policies and goods allocation, aiming to minimize the distances traveled by pickers, and thus reducing the time and the cost related to the picking process.

This study confirms the importance of these aspects and, in addition, with respect to the existing literature, it also highlights how they are related to the hardware aspects, i.e., warehouse shape factor and input/output position. Moreover, this study demonstrates that hardware and software aspects need to be considered together to assess the convenience of an OS policy versus an OB policy. This allows to better represent the specific scenario, without neglecting the synergy of the warehouse characteristics that obviously influence the final outcome.

Even if the convenience, in absolute terms, can only be assessed on a case-by-case basis, as it depends on the time required to sort orders at the end of the picking mission in the batch-picking policy, which of course depends on the context, it is nevertheless possible to make a general quantitative analysis. In particular, by analyzing the results of an extensive simulation campaign where several software and hardware warehouse parameters were varied, this study presents a comprehensive sensitivity analysis of the impact of different warehousing scenarios on the convenience of different batching logics. This is performed both through the observation of the overall trends in the time available for sorting and through statistical analysis of the results of the full-factorial simulation campaign, under different management and structural conditions. In this way, this study provides both researchers and industrial decision makers with a preliminary screening tool that allows them to assess whether order batching could be convenient in a given context, thus allowing them to better and more efficiently focus the following analyses and evaluations and streamline the optimization process.

From the analysis performed, it emerged that the routing policy has little impact on the convenience of OB over OS. On the other hand, both the shape factor of the warehouse and the I/O position had a great impact. In particular, for OB, a wide and shallow warehouse is to be preferred over a narrow and long warehouse. Regarding the I/O position, when both points are located on the same side of the warehouse, but in opposite positions (Opposite Lateral Picking, Same Side), the OS policy is more convenient in a high percentage of configurations. The same happens when the items’ locations reflect their turnover rates (high values of ζ).

Another interesting aspect emerging from the present study is that the impact of both software and hardware parameters varies with the dimension of the picking list. In particular, in all scenarios considered, the aspect that was most significant in the performance of picking operations was the allocation policy, whose impact was found to be little affected by the size of the picking list. Going deeper into detail, for short picking lists (less than 30 items per mission), the routing policy adopted as well as the shape factor of the warehouse had a low impact on the performance of the picking process. In these scenarios, the most impactful parameters were the allocation policy and the input/output position of the picker within the warehouse. On the other hand, when the picking list dimension increases (more than 50 items per mission), the impact of the routing policy adopted strongly increased, while the impact of the input/output position of the picker significantly decreased, still remaining significant. As a consequence, in these contexts, routing policy also becomes a key variable in assessing the convenience of batch picking versus order picking.

6. Conclusions

In this study, a simulation tool was used to reproduce 1260 different warehouse configurations, considering four different routing policies, seven allocation policies based on the product turnover rate, nine shape factors and five different input/output positions. These parameters were classified into software parameters (allocation policy and routing policy), more connected to managerial aspects, and hardware parameters (shape factor and input/output position), more related to the structure and the layout of the warehouse. For each configuration, three different picking lists of different lengths (20, 30 and 50 items) were simulated, and, for each of them, the process time needed to complete the picking operations was assessed considering two different scenarios: order picking and batch picking. The data generated were used to assess the convenience of batch picking over order picking and evaluate the significance of each parameter considered, both software and hardware, on the system performance.

The results highlighted that all the parameters considered contribute to the performance of the process and must be taken into account when planning and managing picking operations; their impact, however, is not independent of the context but influenced by the size of the picking list. This result has important managerial implications since, depending on the context, it allows identification of the most significant factors to consider when facing such operations. For example, in contexts characterized by picking lists composed of numerous items, hardware parameters are less significant: this means that to optimize system performance, picking and allocation policies can be addressed without necessarily disrupting the warehouse configuration, which generally implies both time-consuming and costly activities. On the other hand, when the picking lists are short, the layout of the warehouse results in having a significant impact on the performance of the picking process: hence, in this scenario, it might be convenient to also address the hardware characteristics of the warehouse (shape factor, input/output position) to significantly improve the overall performance of the picking operations.

The results obtained significantly contribute to the streamlining and optimization of warehouse management, aiming to reduce the processing times in line with the market request strongly shifting towards e-commerce purchases and constantly shorter delivery times. Being able to determine when batching multiple orders allows optimization of the picking performance could be a significant added value that increases the competitiveness of a company against competitors. Furthermore, in case a company was interested in enhancing the performance of order batching policies, the results obtained could be of great support in revamping operations, providing industrial players with relevant insights for the adjustment of software (management) variables, or even for the alteration of hardware (structural) parameters.

Future research activities may include the investigation of non-traditional warehouse configurations, to assess the convenience of order batching in those cases. Moreover, the convenience of order batching over single orders in cases when $t_s\ge 0$ could be investigated in more detail, for example, starting from real data from industrial case studies. In this context, it will be interesting to evaluate if, and to what extent, specific product characteristics and the magnitude of the operating costs influence the identified trends in the convenience of order batching.