Integrated storage assignment for an E-grocery fulfilment centre: accounting for day-of-week demand patterns

Abstract

In this paper, we address a storage assignment problem arising in a fulfilment centre of a major European e-grocery retailer. The centre can be characterised as a hybrid warehouse, consisting of a highly efficient and partially automated fast-picking area designed as a pick-and-pass system with multiple stations, and a picker-to-parts area. The storage assignment problem involves the decisions of selecting products to be allocated to the fast-picking area, assigning these products to picking stations, and determining the specific shelves within the designated station. The objective is to achieve high picking efficiency while maintaining balanced workloads across stations and respecting precedence order constraints. We formulate this three-level problem using an integrated mixed-integer linear programming (MILP) model. Computational experiments with real-world data demonstrate that our integrated approach yields significantly better results than a sequential approach, where the selection of products to be included in the fast-picking area is performed before assigning stations and shelves. To enhance computational efficiency, we propose a heuristic solution approach that fixes SKUs to shelves, allowing us to find better solutions in shorter runtimes compared to directly solving the MILP model. Additionally, we extend the integrated storage assignment model to explicitly account for within-week demand variation. In a set of experiments with day-of-week-dependent demands, we show that while a storage assignment based on average demand figures can lead to highly imbalanced workloads on certain days, the augmented model provides well-balanced storage assignments for each day-of-week without compromising the solution quality in terms of picking efficiency. The benefits of accounting for demand variation are further demonstrated through a simulation-based analysis using sampled weekly data.

1 Introduction

In e-grocery retailing, grocery products are ordered online and delivered directly at a date and time chosen by the customer. In recent years, the e-grocery business has experienced a growth rate in sales of 18.4% in the US in 2023 and is expected to become the largest category within e-commerce until 2026 (Droesch 2024).

Many key players in the e-grocery business are omnichannel grocers, having their roots in traditional brick-and-mortar retailing. Wollenburg et al. (2018) provide a review of the transition from brick-and-mortar to a brick-and-clicks grocery retailing and the implications for underlying logistics networks. Initially, grocers started their e-grocery business using in-store picking, where online orders are picked in existing brick-and-mortar stores close to the customer and then delivered. Although this pick strategy is still used in rural regions, it is not suitable to handle the increasing volume of e-grocery purchases in large cities or metropolitan areas. Consequently, to increase the efficiency of the picking process, which, according to our business partner, accounts for a substantial share of total warehousing costs, major e-grocery retailers have established dedicated warehouses, so-called fulfilment centres or dark stores, solely for picking e-grocery orders. While Hübner and Kuhn (2023) develop a model for shelf space management in light of real-world replenishment processes in grocery retailing, research on e-grocery warehousing is still limited.

Warehousing is a key challenge for almost all retailers (Gu et al. 2007). A suitable warehouse configuration depends on the assortment of the retailer, the characteristics of stock keeping units (SKUs), as well as customer expectations, such as very high service levels of 97–99% (Ulrich et al. 2021) and short delivery times, with some retailers even offering same-day delivery. In e-grocery, most retailers offer an assortment of about 12,000 to 15,000 SKUs, some of which require special storage conditions like refrigeration. For our business partner, an average order includes about 30 to 40 different SKUs (order lines).¹ While the need for short delivery times also arises in classical (non-grocery) e-commerce, the number of order lines in a customer order is quite small—for example, Boysen et al. (2019) note that each order at Amazon Germany comprises 1.6 lines. This difference in lines per order significantly impacts warehouse design and operation, as highlighted by the fact that the recent review on warehousing for e-commerce by Boysen et al. (2019) explicitly excludes the e-grocery business.

In this paper, we address scientific decision support for a storage assignment problem arising in a fulfilment centre of a major European e-grocery retailer. This retailer operates various fulfilment centres with different designs and degrees of automation. The most recently established centre can be characterised as a hybrid or parallel warehouse, where a part of the assortment is allocated to a traditional picker-to-parts area, and the other SKUs are allocated to a partially automated fast-pick area. In this area, boxes sequentially move between stations within a so-called picking loop². At each station, a picker pulls the SKUs for a given customer from a shelf and places them into the box. This configuration can be classified as a pick-and-pass system (Chia Jane 2000; Pan and Wu 2009).

For this fulfilment centre, we consider an integrated storage assignment problem. The retailer’s goal is to assign SKUs to specific shelves within designated stations of the picking loop. Specifically, this decision can be divided into three (hierarchically related) sub-decisions: The first decision is to determine the subset of SKUs to be handled in the picking loop, considering the limited storage capacity. The second decision is to assign each SKU to a station within the picking loop to balance the workload. The third decision is to assign SKUs to specific shelves within the corresponding station, with the aim of placing SKUs with a high number of picks close to the picker. The overall objective associated with these decisions is to achieve a high level of operational efficiency within the warehouse. Additionally, as is typical for the retail business (see e.g. Trindade et al. 2022), the storage assignment must respect requirements such as maintaining space between SKUs located next to each other and adhering to precedence order constraints to ensure that heavy SKUs do not damage fragile ones in an order box.

As observed by Boysen et al. (2019) and other authors, demand variation is a key challenge in high-performance retail warehouses. If the demand for SKUs changes due to seasonal impacts or long-term trends, maintaining a high level of picking efficiency typically requires adapting the storage assignment by rearranging the storage locations of the SKUs. However, for short-term demand variations, such as day-of-week-dependent demand for certain SKUs, rearranging SKUs is often not possible or practical—as an example, this is the case in the e-grocery fulfilment centre considered in this paper. To mitigate the negative impact of such short-term demand variations, we propose determining a variation-aware storage assignment, that is, a storage assignment that performs well across multiple demand scenarios, particularly for day-of-week-dependent SKU demands. The contributions of this paper can be stated as follows:

We address a new three-level storage assignment problem arising in an e-grocery fulfilment centre with a pick-and-pass system for fast order picking. By doing so, we contribute to the literature on e-grocery warehouse logistics, which is relatively limited compared to the extensive body of research dealing with non-grocery e-commerce and brick-and-mortar warehousing (see Boysen et al. 2019, 2021).

We formulate this three-level problem as an integrated Mixed-Integer Linear Programming (MILP) model and provide a heuristic solution approach based on iterative variable fixing. In a set of experiments using real-world data provided by a leading European e-grocery retailer, we demonstrate that solving this integrated model is clearly superior to a standard sequential approach, where the selection of SKUs to be included in the fast-pick area is made before taking zone/station assignment decisions. Furthermore, we report results from a sensitivity analysis using simulated data to demonstrate the efficiency and applicability of our approach to a broader range of business cases.

Finally, to cope with day-of-week-dependent demand fluctuations, we propose solving an augmented MILP model that explicitly aims to find a storage assignment for the pick-and-pass system that performs well for each day of week. Using real-world data, we show that while an assignment based on average demands leads to substantially imbalanced station workloads on certain days, the variation-aware solution maintains balance on each day of week, almost without compromising the storage assignment objective. We further demonstrate the usefulness of the variation-aware approach through a simulation-based analysis with randomly sampled demand data for each week of the year.

The remainder of the paper is structured as follows: In the next section, we introduce the business case and the real-world data set considered in our study. Section 3 provides a review of related literature, followed by Section 4, which covers our integrated storage assignment model, computational experiments, and the heuristic solution approach. In Section 5, we develop a model extension that accounts for varying demand patterns with respect to days of week of the SKUs, which is evaluated using both real-world data and a simulation study. Finally, we summarise our major findings in the Conclusion.

2 Description of the logistic processes and available data

The e-grocery retailer analysed in this paper primarily operates with a two-stage distribution process. In the first step, SKUs are supplied from national distribution warehouses to local fulfilment centres. These supplies typically occur on each working day from Monday to Saturday. Upon arrival at a fulfilment centre, all SKUs are stored on their allocated shelves. In the second step, customer purchases are fulfilled by these local centres based on the customer’s location. Most orders are placed by customers at least one day in advance. Although the retailer allows same-day orders to a limited extent, there is a time lag before the delivery from the warehouse occurs. Additionally, the number of customer orders is restricted by the availability of delivery time slots. This allows the retailer to synchronise most orders for a certain day, leading to the reasonable assumption of a constant pick rate within our model. In the following, we describe the logistic processes within the fulfilment centre and detail the SKU data. Finally, we present historical picking data from the retailer, highlighting recurring patterns depending on the day of week.

2.1 The process of order picking

In most fulfilment centres, the retailer operates with a traditional picker-to-parts system. To improve operational efficiency, reduce operation times, and increase the number of purchases served within a day, the retailer has introduced higher levels of automation in certain fulfilment centres. While a fully automated picking process is cost-intensive, this paper considers a partially automated picking loop within a hybrid warehousing system established in one of the retailer’s fulfilment centres. This hybrid system consists of two storage areas: (1) a partially automated picking loop and (2) a traditional picker-to-parts area. Although the operational efficiency is higher in the first area, its available storage space is limited. Consequently, the retailer must decide which SKUs should be included in the picking loop and which should remain in the picker-to-parts area. Given that an average customer order comprises about 30 different SKUs, in general, no order can be completed by only one of the storage areas. Instead, assembling all SKUs for a single purchase usually requires two independent picking processes in both areas. While there is comprehensive literature on traditional picker-to-parts areas (see e.g. Caron et al. 1998; Franzke et al. 2017), the majority of picks in the warehouse of the business partner under consideration are performed within the picking loop. At the same time, optimising the picking loop is more complex due to the existence of different stations. Therefore, this paper focuses on optimising the picking process within the more crucial pick-and-pass area.

The picking loop consists of eight picking stations, with boxes sequentially visiting the stations. Each box corresponds to one customer purchase, and at each station, the picker retrieves the SKUs for that purchase from the shelves and places them into the box. Once all SKUs for a specific customer purchase are placed into the box, it exits the loop and the purchase is loaded into a vehicle for delivery. Figure 1 provides a schematic sketch of the picking loop.

Fig. 1

Representation of the picking loop

Fig. 2

Representation of the structure of picking stations

Figure 2 illustrates the structure of a typical station within the picking loop. Each station consists of six racks: four in front of the picker (two outer and two inner) and two behind the picker. The racks in front of the picker contain four shelves, each with a height of 250 mm (type 1), while the racks behind the picker contain four shelves, each with a height of 450 mm (type 2). In total, there are 192 shelves within the entire picking loop: 128 of type 1 and 64 of type 2. The structure of shelves is represented in Fig. 13 in the Appendix. Note that some SKUs can only be allocated to type 2 shelves due to their individual height.

To avoid congestion within the picking loop and idle times at some stations, the retailer aims to balance the processing time and workload for pickers across all stations. For a given order, the time spent at a station depends on the number of picks and the shelf locations of the picked SKUs within the racks. While it is easy to pull SKUs from a shelf at face level and in front of the picker, the picking process is more time-intensive for SKUs located on the top or bottom shelves of a rack, as well as on shelves in the racks behind the picker. Therefore, in addition to deciding which SKUs to allocate to the picking loop, the retailer needs to assign each SKU to a specific station and shelf, considering the goals mentioned above. In contrast to brick-and-mortar retailing, where SKU allocation to shelves also takes into account marketing aspects (cf. Sigurdsson et al. 2009), the online retailing setting in dark stores allows the company the flexibility to decide on SKU placement based solely on efficiency-related objectives. However, the retailer must consider additional constraints implicitly taken into account by customers in brick-and-mortar retailing, such as placing large and heavy SKUs into the box first to mitigate the risk of damaging fragile items. To avoid the ergonomic burden of picking heavy items being allocated to just a few pickers, the pickers rotate between stations throughout the day. This rotation also reduces the variance in picking efficiency between stations induced by human touch and contributes to the plausibility of the assumption of a constant picking efficiency across stations.

2.2 SKU data

The data set provided by the e-grocery retailer covers a total of 4693 different SKUs. It includes information on the dimensions of each SKU, determining whether the SKU can be allocated to type 2 shelves only or also to type 1 shelves. Additionally, each SKU is associated with a precedence order rank, taking one of three values: 1, 2, or 3. Rank 1 corresponds to heavy items which need to be allocated to an early station, while rank 3 is used for fragile items. All other SKUs are associated with rank 2. Furthermore, each SKU has a target stock based on expected customer demand. This target, along with the size of the SKU, determines the amount of shelf space that needs to be allocated to the SKU. In fact, the target level is affected by the replenishment cycle. Shelf allocation must also consider handling-related aspects, such as the need to reserve space for a separator if two different SKUs are placed next to each other on a single shelf. As mentioned previously, the available space within the loop is insufficient to store all SKUs in the retailer’s assortment. Therefore, the decision on which SKUs are included in the picking loop is based on an importance score assigned to each SKU.

Fig. 3

Histograms of the log importance score and the log number of picks for the SKUs in the assortment of the retailer appropriate for the picking loop

Relying on an importance score allows retailers to include both quantitative and qualitative variables into the optimisation process. While some factors are obvious, such as the number of picks or space requirements determined by the volume of an SKU, retailers might also wish to attribute higher importance to certain SKUs based on historical data or qualitative human expert knowledge. Conversely, SKUs with a high value and facing a higher risk for larceny within the picking loop compared to a secured area receive a lower score. The importance score enables a generalisation of our approach in terms of a flexible, case-specific, or even warehouse-specific optimisation of SKU allocation. In particular, this approach can be generalised to other retailers, allowing them to include additional information based on factors such as the shape of the warehouse (i.e. the picking loop and the picker-to-parts area), the level of variation in customer demand (e.g. due to seasonality), or the weight of an SKU (e.g. it might be more convenient to carry heavy SKUs within a box in the picking loop rather than picking them from the picker-to-parts area). These considerations, while not directly part of the optimisation problem, can be implicitly included through the importance score, enhancing the overall efficiency and effectiveness of the allocation process.

In this paper, we consider the importance score as given and rely on the data provided by our business partner. The retailer mainly bases the score values on two dimensions: the space required on the shelves, determined by the SKU volume, and the target stock level. The volume of SKUs is normalised on a scale ranging from 0 to 1. Additionally, the number of order lines over recent years that include this specific SKU is considered, with values again normalised between 0 and 1. Multiplying both dimensions provides an importance score ranging from 0 to 1, where higher values correspond to a higher importance of including this SKU in the picking loop. There might be also some adjustments to the importance score based on considerations by human experts that are not directly quantifiable, as discussed in the previous paragraph. Notably, we find a high correlation between the importance score and the number of picks for a specific SKU, with a correlation coefficient of 0.718. The distribution of the importance score is strongly positively skewed, so we illustrate the frequency of the logarithm of importance scores for all SKUs in Fig. 3a. The log importance score is approximately symmetric around its mean of $-$7.81 with a standard deviation of 2.34. This implies that only a small number of SKUs have an importance score exceeding 0.1, while the score is fairly small and nearly equal for the majority of SKUs. Due to the positive skewness of the total number of picks for the SKUs in the retailer’s assortment appropriate for inclusion in the picking loop, we also show a histogram of the logarithm of the number of picks per SKU in Fig. 3b. This distribution is again roughly symmetric with a mean log number of picks of about 6.84. For more than 80% of the SKUs, the average number of units per order line is at most 2 (mean 1.70). This confirms prior statements by Boysen et al. (2021) on the characteristics of e-grocery purchases. For some SKUs, however, the average number of units per order line is larger, with up to 13.92 units (details can be seen in the boxplot in Fig. 14 in the Appendix). The target stock for SKUs varies across the assortment. More than 90% of the SKUs have a target stock of fewer than 20 units, with an average of 9.10 units, implying some flexibility in the assignment due to the limited space needed for individual SKUs. However, the 1% of SKUs with the highest target stock have an average of 100.35 units, with a maximum of 252 units. Additionally, the dimensions of 17.4% of the SKU require allocation to type 2 shelves.

2.3 Historical picking data

In addition to the characteristics of SKUs introduced above, the data set of the e-grocery retailer includes historical picking data. This provides information on the average number of picks per month for a specific day of week for SKUs within the retailer’s assortment that are suitable for the picking loop, for the year 2020. The data set includes the SKU ID, the day of week (with 1 corresponding to Monday and 6 to Saturday), the month, and the corresponding average number of picks for each month and day of week.³ Positive values on Sundays correspond to picks in the early Sunday morning hours if the purchases from the preceding Saturday could not be fully completed by midnight. Given 4693 SKUs with data for 12 months each, there are 2348 out of these 56,316 combinations with 0 picks for all days of the week. Furthermore, more than 30% of the combinations have 0 picks for at least one day of week, indicating a significant variation in demand on different days of the week and months. Considering the average number of picks per day of week, shown in Fig. 4, we find peaks on Tuesday, driven by demand from business companies, and on Friday, when leisure goods are primarily in demand. These findings are supported by Table 1, which displays the relative distribution of picks per day of week and also covers the number of SKUs with the highest demand on each specific day of week. Again, the highest values are observed on Tuesday, followed by Friday.

Fig. 4

Total number of picks in thousands in the assortment of the retailer appropriate for the picking loop depending on the day of week (1 equals Monday, 6 Saturday)

Table 1 Average relative number of picks per day of week, number of SKUs where the highest demand occurs on this day of week and number of SKUs with exceptional high demand on a specific day of week

It should be noted that a constant proportional change in demand across all SKUs included in the picking loop would not significantly affect the balancing between stations. However, if there is high demand for multiple SKUs at the same station compared to other stations on a specific day of week, this would cause congestion at that station and thus impact the retailer’s operational efficiency. For example, SKUs with high demand at the beginning of the week should be paired with those predominantly demanded right before the weekend to balance the workload across stations. Table 1 confirms the results of Fig. 4 and suggests that SKUs can be categorised into two main groups: one with the highest demand at the beginning of the week and one with the highest demand at the end of the week. The relative number of picks per day of week over all SKUs fluctuates between 15.6% and 17.3%. Therefore, we define an exceptionally high demand on a certain day of week if more than 25% of the picks per week for a specific SKU are accomplished on this day of week. Again, we find high values for Tuesday and Friday, but also for Monday. Figure 5a presents boxplots of the relative number of picks for each SKU on specific days of week. While there is a higher variance for Monday and Saturday, for the other days, 50% of the SKUs have a relative number of weekly picks between 15% and 19%. Notably, on Wednesday there is one SKU with a relative number of picks exceeding 40%. In more detail, Fig. 5b covers the six SKUs corresponding to those having the maximum relative amount of weekly picks on a specific day of week.⁴ We display their relative distribution of picks for each day of week, indicating strong variation in the number of picks for these SKUs across different days of week.

Fig. 5

Boxplots of the relative number of picks and detailed description of SKUs corresponding to those with the highest relative number of picks at a specific day of week in the assortment of the retailer appropriate for the picking loop

3 Related work

As discussed by Hübner et al. (2019), establishing efficiency in distribution logistics is one of the most challenging and success-critical tasks for e-grocery retailers. For recent overviews on the specific challenges faced by omnichannel and e-grocery retailers, and the resulting implications on the design and operation of their logistics networks, see e.g. Wollenburg et al. (2018), Hübner et al. (2019), Rodriguez Garcia et al. (2022). Given these challenges and the increasing relevance of e-grocery retailing, it is not surprising that the Operations Research community has recently begun developing optimisation-based approaches to support decision-making in distribution logistics.

3.1 Decision support for e-grocery warehousing

To provide a few examples of a fulfilment process based on in-store picking, Vazquez-Noguerol et al. (2022) propose an optimisation model that allocates customer orders to stores where these orders are picked, and schedules both the picking as well as the delivery of the orders to customers. Meanwhile, Dethlefs et al. (2022) consider a setting where the e-grocer operates with both in-store picking and picking in distribution centres, offering an approach that integrates the assignment of orders to stores or distribution centres with the scheduling and routing of deliveries. A recent overview of the new role of brick-and-mortar stores in omnichannel retailing is provided by Hübner et al. (2022). However, we are not aware of any paper addressing optimisation-based decision support for tactical problems such as storage assignments in dedicated e-grocery fulfilment centres. This research gap is particularly relevant because, as detailed by Hübner et al. (2019), e-grocery fulfilment differs substantially from fulfilment for non-grocery e-commerce and warehousing in distribution centres supplying grocery stores. These differences are also discussed in recent reviews on warehousing for e-commerce by Boysen et al. (2019) and for brick-and-mortar retailing by Boysen et al. (2021), both of which explicitly exclude the case of e-grocery fulfilment warehouses. To illustrate the differences between e-grocery and other e-commerce characteristics and their impact on warehousing, consider the usefulness of a fast-pick area based on a pick-and-pass system: In classical e-commerce settings, each customer purchase typically involves only a few order lines, leading Boysen et al. (2019) to consider the use of a pick-and-pass system as inappropriate for e-commerce warehouses. However, an e-grocery order typically involves dozens of order lines, completely changing this assessment. In fact, a leading European e-grocery retailer has opted to use a variant of such a system as part of the fulfilment centre we consider in this paper.

3.2 Storage assignment

Using this warehouse as the underlying business case, we propose an integrated approach for making the three main storage assignment decisions identified in the review paper by De Koster et al. (2007): allocating SKUs to a specific area of the warehouse (in our case, either to a standard picker-to-parts-based area or to the fast-picking area), assigning SKUs to zones within a given area (in our case, stations in the pick-and-pass-based picking loop), and the assignment of SKUs to shelves within a given zone (in our case, the picking station). While the exact problem considered here, which involves specific aspects such as order constraints, has not been discussed in the literature thus far, we are also not aware of any work that addresses all three storage assignment decisions in an integrated manner in other warehousing settings. The problem addressed in this paper is an extension of the generic storage assignment problem considered in Abdel-Hamid and Borndörfer (1994), for which the authors show that it is NP-hard in the strong sense. Next, we will briefly review existing research dealing with storage assignment decisions, with an emphasis on pick-and-pass systems considered in this paper.

The highest-level storage assignment decision considered in this paper is determining which SKUs to assign to the highly efficient picking loop system and which to assign to the picker-to-parts area of the warehouse. A similar decision arises in the so-called forward reserve allocation problem (see e.g. Hackman et al. 1990; Walter et al. 2013), which involves selecting the SKUs to be allocated to the fast-picking area along with the number of units to be allocated for each selected SKU. In that problem, it is assumed that the fast-picking area is refilled from the reserve area, whereas in our setting, each SKU is assigned to a single location in one of the two parts of the warehouse. In contrast to the integrated problem considered in our paper, the forward-reserve allocation problem does not consider the assignment of SKUs to storage locations but only aims at ensuring that the SKUs allocated to the fast-picking area can be assigned to the shelves.

The next decision to be considered in our problem is the assignment of SKUs to zones or stations in the picking loop. As discussed in the review by De Koster et al. (2007), when it comes to assigning SKUs to zones or stations in a pick-and-pass system (also referred to as progressive zoning), the most important goal is to balance the workload among the zones. In fact, workload balancing is either part of the objective or the constraint set in most of the works dealing with the optimisation of zone assignment decisions (see e.g. Chia Jane 2000; Jewkes et al. 2004; De Koster and Yu 2008; Hong et al. 2016). Actually, the positive effect of workload balancing on the performance of pick-and-pass systems has been verified in several studies using simulation (Chia Jane 2000; De Koster and Yu 2008; Pan et al. 2015) and approximate models based on queuing theory (Yu and De Koster 2008; Pan et al. 2015; Van Der Gaast et al. 2020). Except for (Chia Jane 2000), the majority of the papers dealing with storage assignment in pick-and-pass systems combine the allocation to zones with storage location assignment on shelves. In general, the main goal of this shelf assignment is to determine the shelf locations in a way that SKUs with a high picking frequency have a short picking time. Instead of focusing solely on picking efficiency, Otto et al. (2017) propose focusing on ergonomic aspects, and, similar to Jewkes et al. (2004), consider an order line system in which the configuration of the zones in terms of zone borders/allocation of rack columns is part of the decision problem (which is not the case in our setting).

3.3 Dealing with demand variation

Irrespective of the type of picking system, the majority of articles dealing with storage assignment assume a given demand scenario for which the assignment is optimised. However, in practice, demand is subject to variation. For example, in case of the e-grocery retailer considered in this paper, there are multiple demand variation patterns depending on the day of week and the season, and there are long-term trends leading to structural changes in customer buying behaviour (e.g. increased demand for vegan and vegetarian products). As noted by Pazour and Carlo (2015), a storage assignment that is optimal for a given demand scenario may be suboptimal for another scenario. Consequently, there are several papers, such as those by Christofides and Colloff (1973), Chen et al. (2011), or Pazour and Carlo (2015), dealing with rearranging the storage location configuration of warehouses. While such adaptation of the storage assignment is useful in case of long-term demand fluctuations, when it comes to short-term demand variations, such an adaptation is typically not feasible or meaningful from an economic perspective.

Research on variation-aware storage assignment is relatively scarce and mostly considers warehouse designs that substantially differ from the setting of the business partner considered in this paper. For the case of a unit-load warehouse, Ang et al. (2012) deal with finding a storage allocation policy in the presence of varying demands. They demonstrate that their policy, obtained using robust optimisation, significantly outperforms variation-agnostic policies from the literature in terms of expected performance. For a warehouse storing pallets from a car parts manufacturer, Kofler et al. (2015) discuss a robust storage reallocation strategy, that is, one that is robust to small demand variations, thereby reducing the need for storage reallocations. However, when it comes to pick-and-pass warehouses, we are not aware of any article dealing with finding a storage assignment that performs well in the presence of short-term demand variations.

4 Integrated three-level storage assignment

In this section, we first propose a MILP model formulation that simultaneously considers the three decisions outlined above: selection of SKUs to be included in the picking loop, assignment of SKUs to picking stations, and assignment of SKUs to the shelves within each station. In Subsection 4.2, we present the results from a series of experiments with this model using real-world data. This is followed by the introduction of a heuristic solution approach in Subsection 4.3, and a simulation-based analysis on the sensitivity of our results in Subsection 4.4.

4.1 Problem description and model formulation

As described in the Introduction, the assignment problem of SKUs to shelves consists of three sub-decisions. First, given a set V of SKUs and a hybrid warehouse consisting of a picker-to-parts area with relatively low picking efficiency and a picking loop with high picking efficiency, we have to decide which SKUs to allocate to the picking loop.⁵ This allocation is based on an importance score $s_v$ associated with each SKU v, which, as described in Sect. 2, is provided by the retailer in our case study. If we consider only this first decision and aim to maximise the sum of the importance scores, the resulting problem is very similar to the so-called forward-reserve problem reviewed in Sect. 3. Note, however, that while in the forward-reserve problem it is assumed that the reserve area serves both as a picking area for the SKUs not assigned to the fast-picking zone and as a reserve area from which the fast-picking system is restocked, in the problem considered here, each SKU is either assigned to the picking loop or the picker-to-parts area. Consequently, it is assumed that the (fixed) shelf space taken by an SKU v (characterised by its height $h_v$ and the width $w_v$) is sufficiently large to store all units of the SKU until the next re-supply of the SKU v. In this context, observe that $w_v$ is not the width of a unit of an SKU but the width required for the target stock of v if it is included in the picking loop; for the problem considered here, $w_v$ is assumed to be a given and fixed parameter.

For the SKUs to be allocated to the picking loop, the second decision to be considered is the assignment of each SKU v to a station k from the set K of stations, which we assume is ordered and indexed by integers ($K = \{1,\ldots , |K|\}$). This station assignment, which can be viewed as a zone assignment in a pick-and-pass system, needs to consider two main aspects. First, the assignment must respect a set of precedence order constraints, to avoid damaging SKUs. Each SKU is associated with a precedence rank $o_v \in O$, with $O = \{1,\ldots , |O|\}$. The station assignment decision needs to ensure that for each pair of two SKUs v and $v'$ with $o_v \le o_{v'}$, v is assigned to the same or an earlier station as $v'$. In addition to respecting these precedence order constraints, the workload among the stations in terms of the number of picks should be balanced. According to the requirements of our business partner, we consider a given threshold $\delta$ denoting the maximum permitted relative deviation of the workload $z_k$ of a station k from the average value over all stations in terms of picking operations per day. This average workload can be calculated as $z = \frac{1}{|K|} \sum \limits _{k \in K} z_k$.

The third storage assignment decision is the assignment of SKUs to shelves within the stations, ensuring hat SKUs with a high number of picks are stored on shelves that are fast and easy for the picker to reach. Each SKU is assigned to a single shelf. We denote the set of shelves at a given station k with $R_k$ and the set of all shelves in the picking loop with R, that is, $R= \bigcup \limits _{k \in K} R_k$. Each shelf $r \in R$ has a height $h_r$ and a width $w_r$. An SKU v can only be assigned to a shelf r if $h_v \le h_r$ and $w_v \le w_r$, leading to the definition of a set $R^v \subseteq R$ of shelves that can fit SKU v. In addition, the set $R^v$ can be further restricted based on the precedence rank $o_v$. For example, if all SKUs with rank 1 fit into the first two stations, this implies that an SKU v with $o_v =1$ cannot be assigned to a shelf in a station $k > 2$ without either leaving shelf space empty in the first two stations or violating the precedence constraints. If more than one SKU is stored on a shelf, there must be a minimum distance g between each two SKUs stored next to each other. Each shelf r is associated with a distance $d_r$ from the picker at the corresponding station $k_r$. We simplify by considering the same distance for all SKUs within a shelf of one rack. However, we explicitly differentiate between the different shelves based on their height within one rack. We assume that every pick is carried out separately. Following the definition used by the management of the e-grocery retailer, we define the picking efficiency of a shelf r as the inverse of the distance $d_r$, that is, as $\frac{1}{d_r}$. Using this definition, the goal in this subproblem is to allocate SKUs with a large number of picks to shelves exhibiting high picking efficiency. Accordingly, in this subproblem, we aim to maximise the average efficiency per pick.

Since the three storage assignment decisions described above are strongly related, we aim to consider them simultaneously in an integrated problem. Among these three decisions, only the first and the third are associated with an objective function: namely maximising the sum of the importance scores of the SKUs selected for the picking loop and maximising the picking efficiency of the shelf allocation. In the integrated problem, we combine these two objectives in the form of a convex combination by introducing weights $\alpha$ and $1-\alpha$, where the first part of the objective function is multiplied by $\alpha$ and the second part by $1-\alpha$. In practice, the choice of the weighting factor $\alpha$ depends on the thoroughness of the importance score elaborated by the retailer. If this score covers critical information and is well-conceived, a higher weight should be placed on this score (i.e. choosing a larger value for $\alpha$). Conversely, if it is only some rule of thumb or, e.g., highly correlated with the number of picks, the importance score should not be overvalued (i.e. choosing a smaller value for $\alpha$). We will provide a discussion of the value of $\alpha$ in our computational experiments.

Retailers typically aim to maximise profits. These profits are influenced by revenues and costs, such as those for order picking. While it is difficult to precisely quantify the consequences of a certain storage assignment regarding associated picking costs, our model aims to reduce those costs by enhancing operational efficiency.

Next, we present a MILP formulation for the integrated problem. The primary decision variables in this formulation are the binary variables $x_{v,r}$, which take value 1 if SKU v is assigned to shelf r and 0 otherwise. The values of these variables determine the values of the second set of variables considered in the model: the variables $z_k$, which represent the workload in terms of the total number of picks assigned to station k. Additionally, we introduce the integer variable $y_o$, which represents the last station (i.e. the station with the highest index k) to which an SKU with precedence rank o is assigned. Given these variables and the parameters introduced above, we can now present the MILP formulation of the integrated problem⁶:

$$\begin{aligned}{} & {} \max \frac{\alpha }{\gamma _1} \underbrace{\sum \limits _{v \in V}\sum \limits _{r \in R^v} s_v x_{v,r}}_{\begin{array}{c} I \end{array}} + \frac{1-\alpha }{\gamma _2} \underbrace{ \sum \limits _{v \in V}\sum \limits _{r \in R^v} \frac{1}{d_r} p_v x_{v,r}}_{\begin{array}{c} II \end{array}} \end{aligned}$$

$$\begin{aligned} \sum \limits _{r\in R^v} x_{v,r}&\le 1 \,\,\,{} & {} \forall \,\,\, v\in V \end{aligned}$$

(1)

$$\begin{aligned} k_r x_{v,r}&\le y_{o} \,\,\,{} & {} \forall \,\,\, o \in O, v \in V^{o}, r \in R^{v} \end{aligned}$$

(2)

$$\begin{aligned} k_r x_{v,r}&\ge y_{o-1} \,\,\,{} & {} \forall \,\,\, o \in O \setminus \{ 1 \}, v \in V^{o}, r \in R^{v} \end{aligned}$$

(3)

$$\begin{aligned} z_k&= \sum \limits _{v \in V}\sum \limits _{r \in R_k} p_v x_{v,r} \,\,\,{} & {} \forall \,\,\, k \in K \end{aligned}$$

(4)

$$\begin{aligned} z_k&\le (1+\delta ) \cdot \frac{1}{|K|} \sum \limits _{l \in K} z_l \,\,\,{} & {} \forall \,\,\, k \in K \end{aligned}$$

(5)

$$\begin{aligned} z_k&\ge (1- \delta ) \cdot \frac{1}{|K|} \sum \limits _{l \in K} z_l \,\,\,{} & {} \forall \,\,\, k \in K \end{aligned}$$

(6)

$$\begin{aligned} w_r&\ge \sum \limits _{v \in V} (w_v + g) \cdot x_{v,r} - g \,\,\,{} & {} \forall \,\,\, r\in R \end{aligned}$$

(7)

$$\begin{aligned} x_{v,r}&\in \{0, 1\} \,\,\,{} & {} \forall \,\,\, v \in V, r \in R^v \end{aligned}$$

(8)

$$\begin{aligned} y_{o}&\in \{1,\ldots ,|K|\} \,\,\,{} & {} \forall \,\,\, o \in O \end{aligned}$$

(9)

The objective function is a weighted combination of two parts: Part I corresponds to the maximisation of the total importance score, while Part II represents the maximisation of the average efficiency per pick. To ensure that both parts of the objective function, and consequently the total objective value, fall within the interval [0,1], we normalise the objective function by dividing through $\gamma _1$ and $\gamma _2$, respectively. Here, $\gamma _1$ corresponds to the objective value when solely optimising the total importance score (Part I of the objective function), while $\gamma _2$ represents the situation where only the picking efficiency is optimised. By adjusting the parameter $\alpha$, the decision maker can control the relative importance of the two objectives.

Constraint set (1) ensures that each SKU is assigned to at most one shelf in the picking loop. Constraints (2) and (3) enforce the precedence order constraints: Constraint set (2) requires that $y_o$ is at least as large as the maximum station index $k_r$ of a shelf r to which an SKU with order rank o is assigned, and (3) ensures that all SKUs with a precedence rank o other than 1 are assigned to a station $k \ge y_{o-1}$. This means they are either assigned to the last station containing an SKU with the next smaller rank or to a station later in the loop. Constraints (4)–(6) enforce balanced workload among the stations. Constraint set (4) determines the value of the auxiliary variables $z_k$, representing the total number of picking operations allocated to station k. Using this variable, Constraints (5) and (6) ensure that the workload allocated to each station respects the maximum permitted relative deviation from the average workload among all stations. Constraints (7) ensure that the total width of the SKUs assigned to a shelf r plus the required gaps between each pair of SKUs in a shelf does not exceed the width $w_r$ of the shelf. Finally, Constraints (8) and (9) enforce the domains of the variables $x_{v,r}$ and $y_o$.

4.2 Computational experiments

In this section, we present the results of several experiments conducted with the model described above, using real-world data from the e-grocery retailer considered in this paper. In a first set of experiments, we explore the solution behaviour concerning the convergence of the duality gap, i.e. the relative difference between a solution found by the optimiser and an upper bound, over time. In addition, we discuss the impact of the weighting factor $\alpha$ on the values of the two parts of the objective function, considering a fixed relative deviation $\delta$ in the number of picks between stations. This enables us to determine a range of reasonable values for the weighting factor $\alpha$. Furthermore, we analyse the effect of the allowed deviation $\delta$ between stations on the structure of the solutions. Finally, we compare our integrated three-level storage assignment approach to a sequential approach, where we solve the area allocation problem (similar to the forward reserve problem), the assignment to stations, and the assignment of selected SKUs to shelves consecutively. All experiments were conducted with the Gurobi optimiser version 9.0.2 on a computer with 16 GB RAM and a AMD Ryzen™  5 1600 3.2 GHz CPU.

4.2.1 Experiments on the runtime

In a first analysis, we set an exemplary weighting factor of $\alpha =0.5$ and allow for a deviation of picks between stations of $\delta =1\%$. Figure 6 depicts (a) the objective value and (b) the gap to the lower bound over a given runtime of up to 12 h. We highlight the resulting values after one hour by the red dotted lines. In this exemplary setting, after the intended runtime, a gap of 0.35% remains. Since we are addressing a tactical problem of the retailer, that is not regularly solved, even longer runtimes could be permissible. However, our findings indicate slow progress in further reducing the gap. For instance, even after an additional four hours of runtime, the gap only diminishes by another 0.04 percentage points.

Fig. 6

Representation of the objective value and gap to the lower bound in per cent depending on the runtime in minutes of up to 12 h using $\alpha =0.5$ and $\delta =1\%$. The red dotted line corresponds to a runtime of 1 h

4.2.2 Effect of the objective weight $\alpha$

Fig. 7

Normalised values of Part I (importance score) and Part II (picking efficiency) of the objective function depending on the weighting factor $\alpha$ for $\delta = 1\%$

As previously introduced, the objective function comprises two parts: the first (I) involves the sum of importance scores of SKUs allocated to the picking loop area, while the second (II) relates to the picking efficiency in the picking loop. We explore the impact of different values for the weighting factor $\alpha$ through a series of experiments. Due to computational constraints, we terminate the optimisation when either a gap of 0.5% or a predefined time limit of 30 min is reached, while limiting the relative deviation of picks between stations to $\delta =1\%$. Figure 7 provides an overview of the values for both parts of the (normalised) objective function across different values for $\alpha$. For clarity, we exclude the results for $\alpha =0$ (score 0.912; efficiency 0.997) and $\alpha =1$ (score 0.999; efficiency 0.381). The left part of the figure illustrates that the normalised sum of importance scores of SKUs assigned to the picking loop achieves its peak for $\alpha \ge 0.35$. Meanwhile, Part II of the objective function remains relatively stable for $\alpha \le 0.6$ and declines for larger values of $\alpha$.⁷

Figure 8 provides further insights into the structure of the solutions by displaying the number of picks for a given distance between the picker and the shelf across different values $\alpha \in \{0, 0.25, 0.5, 0.75, 1\}$. SKUs with a height exceeding 250 mm are excluded from these plots as they can only be allocated to type II shelves (see Fig. 15 in the Appendix for their allocation). For $\alpha =1$ (i.e. a scenario where the distance between the picker and the corresponding shelf does not influence the objective value), Fig. 8 reveals a non-systematic pattern in the allocation of SKUs to shelves. Conversely, for $\alpha \le 0.75$ SKUs with a high number of picks tend to be allocated to shelves closer to the picker, with only marginal changes observed for smaller values of $\alpha$. However, there remain some outliers in each scenario. For instance, in the allocation for $\alpha =0.5$, certain SKUs with a high number of picks are still allocated to shelves with distances of 1900 mm and 2850 mm, respectively. These SKUs typically have a larger width, leading the model to prioritise the allocation of more but smaller SKUs with a high number of picks over these SKUs to shelves closer to the picker.

For $\alpha =0$, where the focus is solely on picking efficiency without considering the importance score, there is a decrease in the number of SKUs allocated to shelves close to the picker, particularly for SKUs with a small number of picks. Meanwhile, the total number of SKUs allocated to the picking loop increases by approximately 20%, and the total number of picks rises by 5–6%. However, the total importance score decreases by nearly 10% compared to other values for $\alpha$ considered. The average width taken on the shelf by SKUs allocated to the picking loop is around 20% smaller in this case. This confirms that the importance score considers additional factors, such as the volume of the SKU (correlation coefficient of 0.52 between the required width and the ratio of the score to the number of picks for an SKU). Overall, the analyses underline the contrasting behaviour of both parts and the importance of a combination within the objective function. Consequently, $\alpha$ should fall within the interval [0.35, 0.6]. For our ongoing analyses, we fix $\alpha$ at 0.5.

Fig. 8

Allocation of SKUs and corresponding picks to shelves with given distance to the picker for different values of $\alpha$ and $\delta =1\%$. Note that the figure is limited to SKUs with a height of up to 250 mm

4.2.3 Effect of the workload balancing parameter $\delta$

In the following analysis, we delve into the effect of the permitted deviation of picks between stations $\delta$ on the resulting objective value obtained after a runtime of one hour. Using $\alpha =0.5$, we vary the permitted relative deviation between stations across $\delta \in \{0.1\%, 1.0\%, 5.0\%, 10.0\%\}$, while also examining the results when balancing constraints are disregarded. Table 2 presents the number of SKUs included in the picking loop, along with the corresponding deviation of the (normalised) total importance score of these SKUs (Part I of the objective function), picking efficiency (Part II of the objective function), total objective value relative to the values obtained by optimising both parts individually without respecting balancing constraints (i.e. $\gamma _1$ and $\gamma _2$), and the remaining gap after one hour of runtime. Specifically, the numbers presented for both parts, as well as the total objective, correspond to one minus the actual value of the (part of the) objective function. When balancing constraints are not respected, the highest objective value is achieved with a remaining gap of 0.12%. While this model is easy to solve, the suggested assignment is notably imbalanced, with a deviation between stations of up to 16.57%. For $\delta \in \{1\%, 5\%, 10\%\}$, the objective values and gaps exhibit similarities, while the actual maximum relative deviations vary considerably, albeit remaining slightly below the corresponding permitted deviation $\delta$ in each case. Remarkably, we are even able to balance the assignment on the level $\delta =0.1$. As this setting is more complex, the objective value deviation from 1 is nearly twice as high as for $1\% \le \delta \le 10\%$ (0.59% compared to 0.32% for $\delta = 1\%$) driven by a relatively higher remaining gap which is also about twice as high after a runtime of one hour. Across all cases, we observe a very high utilisation of the picking loop of at least 98%, with approximately one-third of the suitable SKUs allocated to the picking loop. Notably, reducing the workload deviation marginally decreases the space utilisation as it becomes more challenging to find an allocation meeting this limit. From a managerial point of view, these findings suggest focusing on limiting the allowed deviation to $\delta \le 1\%$. This approach ensures workload balance between stations, minimising the risk of congestion, while maintaining a high objective value accounting for the importance of SKUs allocated to the picking loop as well as picking efficiency.

Table 2 Summary statistics on the number of SKUs included in the picking loop, the space utilisation in the picking loop, the objective value, the maximum relative deviation in picks between stations, and the resulting gap after a runtime of one hour for different values of allowed deviation $\delta$ and a fixed weighting factor $\alpha =0.5$

4.2.4 Integrated vs sequential storage assignment

Finally, we compare our integrated model to a sequential three-stage approach, where we first solve the subproblem akin to the forward reserve allocation problem, i.e. the selection of SKUs to be allocated to the efficient picking loop area (Part I of our objective function), without considering picking efficiency (Part II of our objective function) or respecting balancing constraints. Remarkably, this model can be solved with a gap of 0.11% after a runtime of only 24 s. It suggests including 1533 SKUs into the picking loop, leading to a deviation of the total (normalised) score from 1 of 0.1%. Comparing these results to Table 2, we find that the score improves only slightly, while we allocate 37 SKUs more to the picking loop than when also accounting for picking efficiency and limiting the deviation in picks between stations to $\delta =1\%$. However, the runtime reduces comprehensively.

We then assume the set of these 1533 SKUs as given and allocate them to stations with the aim of minimising the deviation in the number of picks between stations. Within a runtime of one hour, which is sufficient to solve the integrated problem efficiently, we are not able to find a feasible solution for an assignment that satisfies a maximum deviation of 1%. Instead, we obtain an objective value for the deviation between stations of more than 21%. Thus, we remove those SKUs with the smallest importance score until we are able to solve the problem within the 1% deviation constraint. This holds for the 1516 SKUs with the highest importance score in the set determined before, while the corresponding total importance score decreases only slightly.

Finally, assuming the assignment of SKUs to stations as given, we aim to maximise the picking efficiency within the stations by determining the location on the shelves for each SKU. Since each station can be optimised independently, this decomposed problem can be solved to optimality within less than 4 min for an individual station. We obtain a deviation of 2.2% for the (normalised) objective Part II and a total deviation of the objective value of 1.1% when using a weighting factor $\alpha =0.5$ again. As this objective value is clearly inferior compared to the integrated approach (deviation of the objective value 0.3% for $\delta =1\%$), the results underscore the importance of an integrated model compared to a sequential approach for the problem under consideration.

4.3 Heuristic solution approach

Our computational experiments conducted in the previous section reveal that after 12 h of runtime, an optimal solution is not attained; a small gap of less than 1% remains (with the exact magnitude depending on the allowed workload deviation $\delta$, see Table 2). Figure 6 particularly illustrates that improvements in the objective value diminish as runtimes increase. Consequently, we introduce a heuristic solution approach to tackle the model, aiming to find satisfactory solutions within shorter runtimes. Our heuristic focuses on reducing the solution space to enhance search efficiency. For this purpose, we fix the storage locations of SKUs in an iterative scheme. Starting with the basic model without any fixations, the solver is endowed with a predefined runtime to find a solution. Subsequently, we fix the shelf locations of a certain number of SKUs, based on the solution values obtained in the best solution found. The selection of SKUs to be fixed is accomplished according to their importance score (higher importance score first). Subsequent iterations are conducted on the model with an increasing number of fixed SKUs, utilising the solution from the previous iteration.

Based on a set of initial experiments, we test different values for the runtime, the number of fixed SKUs in each iteration, and criteria for selecting the SKUs to fix. We find that the best results for our data set, with $\delta = 1\%$, are achieved when limiting the runtime to 5 min and fixing 100 SKUs in each iteration. After 15 iterations, taking about 75 min in total, we find a solution covering 1526 SKUs with a deviation of the (normalised) objective of 0.07% (with an optimality gap of 0.11%). Comparing these results to those obtained in Sect. 4.2, we observe that the solution obtained with the heuristic approach is superior even to the scenario with unrestricted workload deviation (see Table 2). The superiority also holds compared to the deviation of the objective value from the basic model after a runtime of 12 h (deviation of 0.11%).

4.4 Sensitivity analysis with artificial SKU data

Given that our computational experiments rely solely on the data set provided by our business partner, it raises questions about the sensitivity of the results with respect to the set of SKUs. At the same time, the heuristic solution approach proposed in the previous section offers us the opportunity to efficiently solve the model within reasonable runtimes. To assess the impact of the data set on the solution behaviour, we proceed as follows: (1) we generate simulated SKU data sets based on the structure of the data set provided by the retailer, and (2) we solve the model for these sets. In the following, we will describe the data-generating process before presenting the solution results.

4.4.1 Generating SKU data

The data set provided by the retailer includes five variables for each SKU relevant for optimising the storage assignment: the importance score, the number of picks accomplished, the width, the height, and the order rank of the SKU. As depicted in Fig. 3, we can approximate both the log number of picks and the log importance score with normal distributions. Additionally, we can simplify by assuming that the height and width of SKUs also follow normal distributions. Calculating the covariance matrix $\Sigma ^2$ between the logarithm of importance scores and picks accomplished, as well as height and width, enables us to randomly generate SKU data based on a multivariate normal distribution with the same means of the marginal distributions and dependence structure as in the basic data set provided by the retailer. To ensure the generated data set aligns with the business case, we maintain the same number of SKUs in each set. Figure 9, displaying the sorted heights (a) and widths (b) for the basic data set (x-axis) and the first generated data set (y-axis), confirms the relatively good fit of normal distributions in this case.⁸ Finally, we determine the order rank for each SKU. Since the basic data set covers very few SKUs with rank 1, we simplify by considering the binary case with rank 2 and 3 only. Given the positive correlation between the height and the order rank, as well as between the width and the order rank, we estimate a logistic regression model attempting to explain the order rank by the height and width of the corresponding SKU. Utilising the estimated coefficients of this regression model allows us to randomly select the order rank for generated SKUs based on the underlying probability determined by their height and width. In total, we generate ten different sets of SKUs with information on the five variables stated above.

Fig. 9

Relation between the sorted height (a) and width (b) of the SKUs for the basic set (x-axis) and the generated set 1 (y-axis)

4.4.2 Results

Table 3 summarises the results obtained from the heuristic solution approach applied to ten different data sets generated according to the data structure of the SKU set provided by the retailer. We compare our findings to those of the basic set (see Sect. 4.3 for detailed results) in the bottom line of the table. Each set requires its own $\gamma _1$ and $\gamma _2$. However, computing these the same way as before for all sets would be too time-consuming, so we simplify by not using the solutions but the upper bounds of the optimisations mentioned in Sect. 4.1. Across all summary statistics, i.e. the space utilisation, the deviation of the objective value, the maximum relative deviation between stations, and the gap, we observe similar results, that are also close to those obtained in the basic set. In each case, the space utilisation is close to 100%, indicating that selecting the most important SKUs to be allocated to the picking loop is a crucial task in each set. Furthermore, we demonstrate that we are able to solve the model with the heuristic solution approach in reasonable runtimes close to optimality. This is evident from the deviations of the objective value being close to 0, as well as very small gaps. Only regarding the maximum relative workload deviation between different stations, we do find some variation among the different sets. While in seven out of ten sets there is a deviation of more than 0.9%, which also holds for the basic set, for set 2, this variation is only 0.41%. However, in conclusion, we can state that even for different sets of SKUs generated according to the same structure as present in the SKU set provided by the retailer, our model allows for efficient solutions. This sensitivity analysis generalises the results obtained before and also suggests considering the business case of other retailers by adjusting the covariance matrix, e.g. to allow for a different composition of the importance score with less correlation to the number of picks in future work.

Table 3 Results on the simulation-based analysis for 10 different sets of SKUs stating the space utilisation, the deviation of the objective value from 1, the maximum relative deviation in picks between stations, and the remaining gap for each set

5 Coping with short-term demand variation

The general model developed in the previous section enables the retailer to address the three-level storage assignment problem: deciding which SKUs from the assortment should be allocated to the picking loop, as well as determining the assignment of SKUs to stations and shelves within the warehouse. However, as evidenced by the data described in Sect. 2.3, the demand for SKUs (and hence the number of picks) varies significantly across days, weeks, or even months of the year. This underscores the importance of balancing workload for each day of week individually, enabling the retailer to optimise the assignment of SKUs to stations and shelves. In this section, we delve into the significance of accounting for variation in demand when assigning SKUs to shelves. We start by analysing the efficacy of the storage assignment generated by the heuristic introduced in the previous section, particularly regarding the potential imbalance between stations across different days of week. Subsequently, we enhance our model formulation by constraining the deviation of picks between stations on the level of days of week, and then compare both approaches. Again, we employ the heuristic approach introduced in Sect. 4.3 to solve the model. This comparative analysis allows us to quantify the benefits of explicitly considering variation in demand when making storage assignment decisions. However, it is important to acknowledge that the retailer needs to spend effort on data collection, data processing and computational power for the detailed analysis. Therefore, this analysis serves as the foundation for determining whether the benefits of the detailed solution outweigh the associated efforts of the retailer.

5.1 Evaluating variation-agnostic storage assignments

The descriptive data analysis reveals distinct demand patterns based on the day of week. Consequently, we proceed to examine the extent of imbalance in the number of picks across different stations on the day-of-week level. The variation in demand can lead to two different issues: (1) imbalance resulting from the assignment of SKUs to stations, and (2) congestion arising from the assignment of SKUs to shelves within each station.

In the basic model formulation as proposed in Sect. 4.1, Constraints (5) and (6) restrict the maximum deviation in picks between stations. Specifically, the results of the heuristic for the basic model outlined in Sect. 4.3 indicate that we can efficiently limit the workload deviation at each station. However, since this model solely considers the average number of picks across all days of week, the deviation may be significantly higher for individual days of week due to demand flactuations. Thus, we now assess this allocation using the day-of-week data detailed in Sect. 2.3. This analysis enables us to evaluate the effectiveness of the basic (variation-agnostic) solution for each day of week.

Table 4 Relative deviation in the number of picks between single stations and the average number of picks over all stations as well as corresponding minimum and maximum values for all days of week

Table 4 provides an overview of the deviation in picks between each station and the average value across all stations for individual days of week. Additionally, the two bottom rows indicate the maximum positive and negative deviations. Our findings reveal that the intended maximum deviation level of $\delta = 1\%$ is breached every day of week, even if this constraint is met when averaging across all days of week. While minor deviations may have a negligible impact on the overall efficiency, we focus on deviations exceeding 1%, highlighted in bold font. Falling below the intended deviation level will not affect the completion time of the entire picking process within a day but may result in workforce dissatisfaction due to workload imbalances. Moreover, there is potential to reduce completion time compared to the solution proposed by the basic model for specific days of week. However, exceeding the intended deviation directly leads to congestion at some stations and, consequently, inefficiencies in the retailer’s operational processes, which should be avoided. Given the deviations from the intended level are also significant (exceeding 1%) on days with a high total number of picks, such as Tuesdays and Fridays (see Fig. 4), our results strongly advocate for considering variation in demand when determining SKU assignments to stations.

Fig. 10

Allocation of SKUs and corresponding average picks per day of week to shelves with given distance to the picker for SKUs not exceeding a height of 250 mm when applying the basic model formulation with $\alpha =0.5$ and $\delta = 1\%$

As SKUs are assigned to shelves within these stations based on the average number of picks over all days of week, any variation and thus a potentially high relative demand on specific days are disregarded in the assignment. This could result in further increased congestion at certain stations if an SKU is assigned to an outer shelf despite relatively high demand on a particular day of week. We extend the analysis depicted in Fig. 8 to the day-of-week level by presenting the number of picks relative to the distance between the picker and the shelves in Fig. 10.⁹ While there is variation in the figures for different days of week, only a few outliers are evident. As mentioned in Sect. 4.2, these outliers can be attributed to their large width and also occur in the solution proposed by the basic model on the level of averages. Consequently, it can be inferred that the assignment of SKUs to shelves within stations does not lead to operational inefficiencies for specific days of week. Nonetheless, it remains crucial to address the substantial deviation in the number of picks between different stations on the level of days of week.

5.2 Accounting for short-term demand variation

Our findings from the previous section suggest that explicitly considering day-of-week-related demand variations when determining storage allocation for the picking loop could be beneficial. Indeed, we can enhance the basic model outlined in Sect. 4.1 to restrict the deviation in the number of picks between different stations in the picking loop for each day of week. To achieve this, we introduce a set of days of week $t\in T$, where the parameters $p_v^t$ represent the number of picks for SKU $v\in V$ and day of week t, and decision variables $z_k^t$ represent the workload assigned to station k on day of week t. Similarly, we utilise $\delta _t$ to denote the allowed relative deviation for each day of week t. Although considering day-of-week-dependent deviations, in general, enables the retailer to incorporate day-of-week-specific deviation limits, in our analysis, we assume that $\delta _t$ remains identical for each day of week. To enforce these maximal allowed deviations per day of week, we adjust the basic model formulation from Sect. 4.1 by replacing Constraints (4)–(6) with the following constraints, which limit the deviation at the level of days of week¹⁰:

$$\begin{aligned} z_k^t&= \sum \limits _{v \in V}\sum \limits _{r \in R_k} p_v^t x_{v,r} \,\,\,{} & {} \forall k \in K \,\,\, \forall t \in T \end{aligned}$$

(10)

$$\begin{aligned} z_k^t&\le (1+\delta _t) \cdot \frac{1}{|K|} \sum \limits _{k \in K} z_k^t \,\,\,{} & {} \forall k \in K \,\,\, \forall t \in T \end{aligned}$$

(11)

$$\begin{aligned} z_k^t&\ge (1- \delta _t) \cdot \frac{1}{|K|} \sum \limits _{k \in K} z_k^t \,\,\,{} & {} \forall k \in K \,\,\, \forall t \in T \end{aligned}$$

(12)

5.3 Computational experiments with the variation-aware model

The analysis in Table 4 exposes a maximum deviation between the number of picks within one station and the average over all stations of more than 3%. To mitigate this day-of-week variation, we employ the variation-aware model with the heuristic introduced in Sect. 4.3. By constraining the deviation at the day-of-week level to $\delta _t = 1\%$, we achieve a remaining gap of 0.22% after approximately one and a half hours, with a deviation of the objective value of 0.18%. While the deviation of the objective value increases, indicating that the solution is inferior compared to the objective value under the basic model of 0.07%, we notably balance the workload deviation between stations. However, for $\delta _t = 0.1\%$, we do not find a feasible solution. This suggests that there are no promising solutions in terms of the objective value under such strong restrictions.

Table 5 Minimum and maximum relative deviation in the number of picks between single stations and the average number of picks over all stations for the basic model (see Table 4) and the extended model

We now assess the deviation in the number of picks from the average over all 8 stations within the picking loop for each station and day of week individually. Conducting this analysis for both the basic model and the extended model, both solved using the heuristic approach, enables a comparison between the two models. Table 5 presents the minimum and maximum relative deviation for the stations with the highest absolute values. We are able to decrease the absolute values of the highest negative deviations for each day of week. Simultaneously, the highest positive deviations decrease for four out of six days, while we observe an increase of up to 20% for the remaining two days. However, the positive deviations do not exceed 1%. Compared to the highest positive value under the basic model (3.34% on Mondays), this represents a reduction of more than 70%.

In summary, we are able to decrease the deviation between stations at the day-of-week level from over 3% to less than 1% by implementing the variation-aware model with additional constraints. However, we observe that the deviation of the objective value increases by 0.11 percentage points, with approximately the same runtime provided as for the basic model formulation with $\delta = 1\%$. In most scenarios, retailers will benefit from the comprehensive reduction in the deviation for most days of week, which outweighs the slight increase in the deviation of the objective value.

5.4 Out-of-sample simulation: individual working days

Our previous analysis is based on the number of picks for each SKU within a year, utilising a data set that offers insight into the distribution over days of week, again on the level of averages. However, in practice, the actual number of picks for a particular working day may differ from the average for that day of week throughout the entire year. To assess the impact of this variation, in this section, we generate simulated data on the number of picks for a certain working day based on the data provided by the retailer.

To achieve this, we divide the year into $k = 1,\ldots , 52$ weeks, each consisting of operation on six days of week $t = 1, \ldots , 6$ (Monday to Saturday). Leveraging the available data, we parameterise a multivariate normal distribution with mean $\mu _v = (\mu _v^1, \ldots , \mu _v^6)$ equal to the average number of picks accomplished on day of week t, denoted by $\bar{p}^{t}_v$, for SKU $v \in V$ over the entire year. To accommodate different scenarios regarding the variation in the number of picks between different weeks, we define the standard deviation $\sigma _v = (\sigma _v^1, \ldots , \sigma _v^6)$ as the product of the mean and the coefficient of variation $CV \in \{0.05, 0.1, 0.15, 0.2, 0.25, 0.3\}$: $\sigma _v = CV \cdot \mu _v$. Additionally, we aim to incorporate the correlation structure between the different days of week. For each SKU, we consider the average number of picks for a specific day of week within each month provided in the available data set. Subsequently, for each month, we calculate the deviation in the number of picks for each day of week from the average across all six days of week. Considering the vectors with deviation for each day of week over the entire year allows us to compute the matrix of correlation coefficients $C_v$ for SKU v, where $c_v[i,j], i, j = 1,\ldots , 6$ represents the correlation between day of week i and day of week j. To determine the covariance matrix $\Sigma ^2_v$, we multiply the matrix of correlation coefficients by the standard deviation for both days of week, that is, $\sigma ^2_v[i,j] = c_v[i,j] \cdot \sigma _v^i \cdot \sigma _v^j$. Assuming independence between weeks and SKUs enables us to randomly generate pick data $p_v^{tk}$ from a multivariate normal distribution.

Fig. 11

Density functions for the number of picks on an individual working day for the SKU with the highest importance score on Mondays for different values of the coefficient of variation CV

To illustrate the impact of the coefficient of variation CV on the marginal distributions of the number of picks, Fig. 11 provides an exemplary density function for picks of the SKU with the highest importance score on Mondays. The average number of picks is 68.80. For the lowest values of the coefficient of variation, $CV = 0.05$, we observe most realisations between 62 and 78, while for $CV = 0.3$, the variation is significantly larger, with the 95% interval for the number of picks ranging from 28 to 110. Thus, there is a higher risk of an imbalanced allocation of SKUs to stations in this scenario, even if we limit the deviation at the level of averages per month and day of week. Considering different values for the coefficient of variation enables us to compare various settings and generalise the results obtained in this section to other retailers, for example.

Using the generated values for the number of picks accomplished on individual working days, we compute the workload for each station and working day over the entire year. This enables us to determine the deviation from the average over all eight stations for a total of $52 \cdot 6 = 312$ days. Repeating the data generation process according to the underlying distribution for each SKU and calculating the workload deviation 100 times ensures the reliability of the analysis. Specifically, we calculate both the average absolute deviation between a single station and the average across all stations for each working day and the maximum absolute deviation. Comparing the average results across all simulation runs for the solution based on the allocation under the basic model (see Section 4) to those obtained when using the variation-aware model (see Section 5), Fig. 12a gives the average over 100 simulation runs for the mean absolute deviation between a single station and all stations over 52 weeks and six days of week within each week. Figure 12b shows the maximum absolute deviation over all individual working days and stations, again averaged over the same 100 simulation runs. In both figures, we compare results obtained under the allocation determined by the basic model with $\delta = 1\%$ (red solid line) to those obtained under the variation-aware model with $\delta = 1\%$ (blue dotted line), both solved by the heuristic.

Fig. 12

Mean absolute a and maximum absolute b deviation between a single station and the average over all stations over 52 weeks with six days of week in each week averaged over 100 simulation runs according to the allocation determined by the basic model (red solid line) and the variation-aware model (blue dotted lines) (Color figure online)

While we limit the deviation in the number of picks between different stations to $\delta = 1\%$ (for the basic model) and $\delta _t = 1\%$ (for the variation-aware model, accounting for day-of-week variation) in the optimisation model, these deviations are based on the average number of picks over a whole year. However, in practice, the actual number of picks might vary from week to week for the same day of week. Specifically, our results demonstrate that even with a small coefficient of variation ($CV = 0.05$), the maximum absolute deviation for a single station from the average over all stations results in a value of more than 4%, which is four times higher than the intended level of 1%. When relying on the variation-aware model, which takes into account variation between different days of week, again on the level of averages over the whole year, we find an actual deviation of 2.46% (compared to 1% intended by the model). Even though the actual deviation is again larger than intended by the model, we can show that day-of-week-specific constraints allow for a considerable reduction in deviation, in this case by about 43%. The same result holds for the mean absolute deviation, which reduces from 1.01% to 0.57%. For increasing values of the coefficient of variation, we find higher mean absolute deviations (increasing to 2.33% for the basic model and 2.16% for the variation-aware model, respectively, for $CV = 0.3$), as well as higher maximum absolute deviations in this case (11.18% for the basic model and 10.43% for the variation-aware model). At the same time, the benefit of the variation-aware model reduces to 7.48% for the mean absolute deviation and 6.75% for the maximum absolute deviation. In summary, this analysis demonstrates the benefit of the variation-aware model introduced in this section in a practical setting where the number of picks varies between different weeks, with the highest benefit observed when the variation is relatively small.

6 Conclusion

In this paper, we present an integrated approach to tackle a three-level storage assignment problem encountered in a fulfilment centre operated by a leading European e-grocery retailer. The fulfilment centre is characterised as a hybrid warehouse, combining a highly efficient, partially automated picking loop with a less efficient picker-to-parts area. While the demand for e-groceries has increased in recent years, the market has become more competitive, necessitating e-grocery retailers to enhance their operational efficiency. A key challenge lies in the assignment of SKUs to shelves within a fulfilment centre. We optimise a bi-objective value function of the retailer, considering the importance of SKUs allocated to the highly efficient picking loop, while also addressing picking efficiency dependent on the distance between a picker and the shelves. To prevent congestion within the picking loop, we additionally impose constraints in our proposed optimisation model, limiting the permitted relative deviation in the number of picks between different stations.

Our results indicate that we can efficiently solve the model with a remaining gap of less than 0.7% within one hour in most scenarios. Since we address a tactical problem of the retailer, which has not to be solved regularly but only in cases of significant changes in the assortment of the retailer or customer preferences, such runtimes are reasonable and can even be extended. Additionally, we propose a heuristic solution approach that takes less than two hours to obtain solutions surpassing those found by a standard solver after 12 h. The obtained results clearly demonstrate the superiority of our integrated approach compared to solving the allocation to the picking loop and the assignment to stations and shelves sequentially. The findings remain consistent across different sets of SKUs generated within a simulation-based analysis.

This paper also addresses the challenge of day-of-week-dependent demand variation for specific SKUs. In our business case, demand variation is notably high at the beginning of a week and just before the weekend. Through a series of experiments, we demonstrate that a storage assignment based solely on day-of-week-agnostic average demand figures tends to exhibit a highly imbalanced workload on certain days of week. To mitigate this issue, we extend the aforementioned storage assignment model to consider day-of-week-dependent demand variation. Our findings reveal that this extended model produces storage assignments that meet the workload balance requirements imposed for each day of week without compromising the quality of the solutions in terms of the (efficiency-oriented) objective value. Furthermore, by generating simulated data based on different coefficients of variations to account for variation between different weeks, we underscore the benefits of the extended model formulation. This approach also offers managerial insights into the actual deviation, going beyond reliance on the average number of picks over the entire year.

Future work could include refinements such as individual levels of permitted deviation between stations based on the total number of picks on a particular day of week, that is, varying $\delta _t$ with respect to the day of week $t\in T$. Since congestion is more critical on days of week with high workload, this model extension could further reduce operational inefficiencies for e-grocery retailers. The simulation-based analysis on the variation in the number of picks across weeks also offers the potential to develop advanced models. For instance, incorporating methods to learn from weeks with a high level of deviation could further reduce workload imbalance. While random demand variation might be addressed by employing robust or stochastic optimisation approaches, the availability of data spanning multiple years would additionally enable the detection of structural long-term demand changes, thereby justifying a rearrangement of the storage assignment.

Furthermore, given that our paper focuses solely on the detailed storage assignment in the fast-picking area, a natural extension would be to include the storage assignment for the picker-to-parts area in an integrated approach. Furthermore, setting up a detailed simulation of daily operations could generate insights into processing times or the number of orders that could be accepted on a single day, translating into a measure of operational costs. Finally, while we address the optimisation of an existing fulfilment centre, future research could also consider the strategic problem of designing warehouses. This would involve the decision on the size of the picking loop, the number of stations, and the configuration of shelves.

Appendices

Appendix 1: Representation of shelves

See Fig. 13.

Fig. 13

Representation of the structure of shelves

Appendix 2: Boxplot of the number of units per order line

See Fig. 14.

Fig. 14

Boxplot of the average number of units within one order line for the SKUs in the assortment of the retailer appropriate for the picking loop

Appendix 3: MILP formulation for the integrated three-level storage assignment

See Table 6.

Table 6 Table of set, parameters, and decision variables for the MILP

$$\begin{aligned} \max \frac{\alpha }{\gamma _1} \underbrace{\sum \limits _{v \in V}\sum \limits _{r \in R^v} s_v x_{v,r}}_{\begin{array}{c} I \end{array}} + \frac{1-\alpha }{\gamma _2} \underbrace{ \sum \limits _{v \in V}\sum \limits _{r \in R^v} \frac{1}{d_r} p_v x_{v,r}}_{\begin{array}{c} II \end{array}} \end{aligned}$$

$$\begin{aligned} \sum \limits _{r\in R^v} x_{v,r}&\le 1 \,\,\,{} & {} \forall \,\,\, v\in V \\ k_r x_{v,r}&\le y_{o} \,\,\,{} & {} \forall \,\,\, o \in O, v \in V^{o}, r \in R^{v} \\ k_r x_{v,r}&\ge y_{o-1} \,\,\,{} & {} \forall \,\,\, o \in O \setminus \{ 1 \}, v \in V^{o}, r \in R^{v} \\ z_k&= \sum \limits _{v \in V}\sum \limits _{r \in R_k} p_v x_{v,r} \,\,\,{} & {} \forall \,\,\, k \in K \\ z_k&\le (1+\delta ) \cdot \frac{1}{|K|} \sum \limits _{l \in K} z_l \,\,\,{} & {} \forall \,\,\, k \in K \\ z_k&\ge (1- \delta ) \cdot \frac{1}{|K|} \sum \limits _{l \in K} z_l \,\,\,{} & {} \forall \,\,\, k \in K \\ w_r&\ge \sum \limits _{v \in V} (w_v + g) \cdot x_{v,r} - g \,\,\,{} & {} \forall \,\,\, r\in R\\ x_{v,r}&\in \{0, 1\} \,\,\,{} & {} \forall \,\,\, v \in V, r \in R^v \\ y_{o}&\in \{1,\ldots ,|K|\} \,\,\,{} & {} \forall \,\,\, o \in O \end{aligned}$$

Appendix 4: Augmented MILP formulation accounting for demand variation in the storage assignment

See Table 7.

Table 7 Table of set, parameters, and decision variables for the MILP

$$\begin{aligned} \max \frac{\alpha }{\gamma _1} \underbrace{\sum \limits _{v \in V}\sum \limits _{r \in R^v} s_v x_{v,r}}_{\begin{array}{c} I \end{array}} + \frac{1-\alpha }{\gamma _2} \underbrace{ \sum \limits _{v \in V}\sum \limits _{r \in R^v} \frac{1}{d_r} p_v x_{v,r}}_{\begin{array}{c} II \end{array}} \end{aligned}$$

$$\begin{aligned} \sum \limits _{r\in R^v} x_{v,r}&\le 1 \,\,\,{} & {} \forall \,\,\, v\in V \\ k_r x_{v,r}&\le y_{o} \,\,\,{} & {} \forall \,\,\, o \in O, v \in V^{o}, r \in R^{v} \\ k_r x_{v,r}&\ge y_{o-1} \,\,\,{} & {} \forall \,\,\, o \in O \setminus \{ 1 \}, v \in V^{o}, r \in R^{v} \\ z_k^t&= \sum \limits _{v \in V}\sum \limits _{r \in R_k} p_v^t x_{v,r} \,\,\,{} & {} \forall \,\,\, k \in K , t \in T \\ z_k^t&\le (1+\delta _t) \cdot \frac{1}{|K|} \sum \limits _{k \in K} z_k^t \,\,\,{} & {} \forall \,\,\, k \in K , t \in T \\ z_k^t&\ge (1- \delta _t) \cdot \frac{1}{|K|} \sum \limits _{k \in K} z_k^t \,\,\,{} & {} \forall \,\,\, k \in K , t \in T\\ w_r&\ge \sum \limits _{v \in V} (w_v + g) \cdot x_{v,r} - g \,\,\,{} & {} \forall \,\,\, r\in R\\ x_{v,r}&\in \{0, 1\} \,\,\,{} & {} \forall \,\,\, v \in V, r \in R^v \\ y_{o}&\in \{1,\ldots ,|K|\} \,\,\,{} & {} \forall \,\,\, o \in O \end{aligned}$$

Appendix 5: Shelf distance of SKUs with height exceeding 250 mm

See Figs. 15 and 16.

Fig. 15

Allocation of SKUs and corresponding picks to shelves with given distance to the picker for different values of $\alpha$ and $\delta =1\%$. Note that the figure is limited to SKUs with a height exceeding 250 mm

Fig. 16

Allocation of SKUs and corresponding average picks per day of week to shelves with given distance to the picker for SKUs exceeding a height of 250 mm when applying the basic model formulation with $\alpha =0.5$ and $\delta =0.1\%$