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

Winkelmann, David; Tolkmitt, Frederik; Ulrich, Matthias; Römer, Michael Article — Published Version Integrated storage assignment for an E-grocery fulfilment centre: accounting for day-of-week demand patterns Flexible Services and Manufacturing Journal Provided in Cooperation with: Springer Nature Suggested Citation: Winkelmann, David; Tolkmitt, Frederik; Ulrich, Matthias; Römer, Michael (2024) : Integrated storage assignment for an E-grocery fulfilment centre: accounting for day-of-week demand patterns, Flexible Services and Manufacturing Journal, ISSN 1936-6590, Springer US, New York, NY, Vol. 37, Iss. 2, pp. 558-598, https://doi.org/10.1007/s10696-024-09549-7 This Version is available at: https://hdl.handle.net/10419/323317 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Vol:.(1234567890) Flexible Services and Manufacturing Journal (2025) 37:558–598 https://doi.org/10.1007/s10696-024-09549-7 Integrated storage assignment foranE‑grocery fulfilment centre: accounting forday‑of‑week demand patterns DavidWinkelmann1 · FrederikTolkmitt1· MatthiasUlrich1· MichaelRömer1 Accepted: 15 May 2024 / Published online: 20 June 2024 © The Author(s) 2024 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 pickerto-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. Keywords Retailing· E-grocery· Storage assignment· Demand variation· Simulation Extended author information available on the last page of the article 559 Integrated storage assignment foranE‑grocery fulfilment… 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 fastpick area. In this area, boxes sequentially move between stations within a so-called 1 The large number of order lines in grocery retailing is also emphasised by previous literature, see e.g. Fernie etal. (2010). Additionally, Ulrich etal. (2021) state that an average shopping basket in e-grocery retailing has a value of about 80 to 90€. 560 D.Winkelmann et al. 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 pickand-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-ofweek-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 2 In the literature, this type of picking system is referred to as progressive zoning see e.g. the review on warehouse order picking by De Koster etal. (2007). 561 Integrated storage assignment foranE‑grocery fulfilment… 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 562 D.Winkelmann et al. 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 563 Integrated storage assignment foranE‑grocery fulfilment… 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. 564 D.Winkelmann et al. Therefore, the decision on which SKUs are included in the picking loop is based on an importance score assigned to each SKU. 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 0 200 400 600 −15 −10 −5 0 Log Importance Score Frequency (a) Log importance score 0 250 500 750 3579 Log Picks per Year Frequency (b)Log number of picks 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 565 Integrated storage assignment foranE‑grocery fulfilment… 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 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) 23.0 23.5 24.0 24.5 25.0 25.5 123456 Day of Week Total Number of Picks in Thousands 572 D.Winkelmann et al. a higher weight should be placed on this score (i.e. choosing a larger value for 𝛼 ). 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 𝛼 ). We will provide a discussion of the value of 𝛼 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 xv,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 zk , which represent the workload in terms of the total number of picks assigned to station k. Additionally, we introduce the integer variable yo , 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: max 𝛼 𝛾1 ∑ v∈V ∑ r∈Rv svxv,r ⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟ I + 1−𝛼 𝛾2 ∑ v∈V ∑ r∈Rv 1 dr pvxv,r ⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟ II (1) ∑ r∈R v xv,r≤1∀v∈V (2) kr x v,r ≤y o∀ o ∈ O , v ∈ V o , r ∈ R v (3) krxv,r ≥ yo−1∀o∈O⧵{1},v∈Vo,r∈Rv (4) z k= ∑ v∈V ∑ r∈Rk pvxv,r∀k∈ K (5) z k≤(1+𝛿)⋅ 1 | K |∑ l∈K zl∀k∈K (6) z k≥(1−𝛿)⋅ 1 | K |∑ l∈K zl∀k∈K 6 For a table containing all notation used in the model, see Table6 in the Appendix. 573 Integrated storage assignment foranE‑grocery fulfilment… 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 𝛾1 and 𝛾2 , respectively. Here, 𝛾1 corresponds to the objective value when solely optimising the total importance score (Part I of the objective function), while 𝛾2 represents the situation where only the picking efficiency is optimised. By adjusting the parameter 𝛼 , 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 yo is at least as large as the maximum station index kr 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 ≥ yo−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 zk , 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 wr of the shelf. Finally, Constraints (8) and (9) enforce the domains of the variables xv , r and yo . 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 𝛼 on the values of the two parts of the objective function, considering a fixed relative deviation 𝛿 in the number of picks between stations. This enables us to determine a range of reasonable values for the weighting factor 𝛼 . Furthermore, we analyse the effect of the allowed deviation 𝛿 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 (7) w r≥ ∑ v∈V (wv+g)⋅xv,r−g∀r∈ R (8) xv , r∈{0, 1}∀v∈V,r∈Rv (9) yo∈{1, …,|K|}∀o∈O 574 D.Winkelmann et al. 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 𝛼=0.5 and allow for a deviation of picks between stations of 𝛿=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. 4.2.2 Effect oftheobjective weight ˛ 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 𝛼 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 𝛿=1% . Figure7 provides an overview of the values for both parts of the (normalised) objective function across different values for 𝛼 . For clarity, we exclude the results for 𝛼=0 (score 0.912; efficiency 0.997) and 𝛼=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 98.6 98.8 99.0 99.2 99.4 99.6 99.8 0100 200 300 400 500 600 700 Runtime in Minutes Objective Value (a)Runtime vs. objectivevalue 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0100 200300 400500 600700 Runtime in Minutes Gap in Percent (b)Runtime vs. gap 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 𝛼=0.5 and 𝛿=1% . The red dotted line corresponds to a runtime of 1h 575 Integrated storage assignment foranE‑grocery fulfilment… 0.990 0.993 0.996 0.999 0.00 0.25 0.50 0.75 1.00 Alpha Relative Score (a) PartI(importance score) 0.984 0.988 0.992 0.996 0.00 0.25 0.50 0.75 1.00 Alpha Relative Distance (b)Part II (pickingefficiency) Fig. 7 Normalised values of Part I (importance score) and Part II (picking efficiency) of the objective function depending on the weighting factor 𝛼 for 𝛿=1% 0 500 1000 1500 2000 2500 3000 010000 20000 Picks Distance alpha = 1 0 500 1000 1500 2000 2500 3000 010000 20000 Picks Distance alpha = 0.75 0 500 1000 1500 2000 2500 3000 010000 20000 Picks Distance alpha = 0.5 0 500 1000 1500 2000 2500 3000 010000 20000 Picks Distance alpha = 0.25 0 500 1000 1500 2000 2500 3000 0 10000 20000 Picks Distance alpha = 0 Fig. 8 Allocation of SKUs and corresponding picks to shelves with given distance to the picker for different values of 𝛼 and 𝛿=1% . Note that the figure is limited to SKUs with a height of up to 250mm 576 D.Winkelmann et al. picking loop achieves its peak for 𝛼≥0.35 . Meanwhile, Part II of the objective function remains relatively stable for 𝛼≤0.6 and declines for larger values of 𝛼 .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 𝛼∈{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 𝛼=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 𝛼≤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 𝛼 . However, there remain some outliers in each scenario. For instance, in the allocation for 𝛼=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 𝛼=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 𝛼 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 7 Note: We encounter an outlier for the score when using 𝛼=0.45 , primarily due to the runtime limitation, which would not typically occur in practice with larger runtimes. 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 𝛿 and a fixed weighting factor 𝛼=0.5 𝛿 # SKUs Space utilisation (%) Norm. dev. of obj Part I (%) Norm. dev. obj Part II (%) Dev. of obj value (%) Maximum rel. dev. (%) Gap (%) 0.1% 1491 98.00 0.18 1.00 0.59 0.097 0.63 1.0% 1496 98.51 0.10 0.54 0.32 0.908 0.35 5.0% 1505 98.52 0.10 0.56 0.33 4.961 0.37 10.0% 1502 98.64 0.14 0.45 0.29 9.450 0.33 unrestricted 1531 99.41 0.07 0.11 0.09 16.572 0.12 577 Integrated storage assignment foranE‑grocery fulfilment… and the importance of a combination within the objective function. Consequently, 𝛼 should fall within the interval [0.35, 0.6]. For our ongoing analyses, we fix 𝛼 at 0.5. 4.2.3 Effect oftheworkload balancing parameter ı In the following analysis, we delve into the effect of the permitted deviation of picks between stations 𝛿 on the resulting objective value obtained after a runtime of one hour. Using 𝛼=0.5 , we vary the permitted relative deviation between stations across 𝛿∈{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. 𝛾1 and 𝛾2 ), 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 𝛿∈{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 𝛿 in each case. Remarkably, we are even able to balance the assignment on the level 𝛿=0.1 . As this setting is more complex, the objective value deviation from 1 is nearly twice as high as for 1%≤𝛿≤10% (0.59% compared to 0.32% for 𝛿=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 𝛿≤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. 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 578 D.Winkelmann et al. 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 𝛿=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 𝛼=0.5 again. As this objective value is clearly inferior compared to the integrated approach (deviation of the objective value 0.3% for 𝛿=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 𝛿 , 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 𝛿=1% , are achieved when limiting the runtime to 5 min and fixing 100 SKUs in each iteration. After 15 579 Integrated storage assignment foranE‑grocery fulfilment… 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 Σ2 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 0 100 200 300 0100 200300 Height in Basic Set Height in Simulated Set (a)Height 0 100 200 300 400 500 0 100 200 300 40 05 00 Width in Basic Set Width in Simulated Set (b)Width 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) 580 D.Winkelmann et al. 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. 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 𝛾1 and 𝛾2 . 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 8 Note: There is a small probability of generating SKUs with a negative height and/or width. In these cases, we exclude the corresponding SKU and draw again from the underlying distribution. 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 Set Space utilisation (%) Deviation of objective value (%) Maximum relative deviation (%) Gap (%) 1 99.73 0.07 0.91 0.06 2 99.54 0.15 0.41 0.14 3 99.74 0.06 0.92 0.05 4 99.79 0.05 0.98 0.04 5 99.79 0.05 0.99 0.04 6 99.59 0.08 0.99 0.07 7 99.66 0.09 0.72 0.08 8 99.71 0.14 0.80 0.13 9 99.70 0.06 0.96 0.05 10 99.70 0.07 0.99 0.06 Basic 99.60 0.07 0.94 0.11 581 Integrated storage assignment foranE‑grocery fulfilment… 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. 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. 588 D.Winkelmann et al. 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 ⋅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 variationaware 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 𝛿=1% (red solid line) to those obtained under the variation-aware model with 𝛿=1% (blue dotted line), both solved by the heuristic. While we limit the deviation in the number of picks between different stations to 𝛿=1% (for the basic model) and 𝛿t=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 589 Integrated storage assignment foranE‑grocery fulfilment… 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 biobjective 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 dayof-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. 590 D.Winkelmann et al. 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 𝛿t with respect to the day of week t∈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 simulationbased 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. Appendix1: Representation ofshelves See Fig.13. Fig. 13 Representation of the structure of shelves 591 Integrated storage assignment foranE‑grocery fulfilment… Appendix2: Boxplot ofthenumber ofunits perorder line See Fig.14. Appendix3: MILP formulation fortheintegrated three‑level storage assignment See Table6. 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 246810 12 14 Average Number of Units per SKU Within an Order 592 D.Winkelmann et al. max 𝛼 𝛾1 ∑ v∈V ∑ r∈Rv svxv,r ⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟ I + 1− 𝛼 𝛾2 ∑ v∈V ∑ r∈Rv 1 dr pvxv,r ⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟ II Table 6 Table of set, parameters, and decision variables for the MILP Sets VSet of SKUs KSet of stations in the picking loop, ordered and indexed by integers ( K ={1, …, |K|} RSet of shelves in the picking loop with Rk denoting the subset of shelves at station k∈K and Rv corresponding to the subset of shelves R that can fit SKU v∈V OSet of precedence order ranks with O={1, …,|O|} , where ov≤ov′ iff v has to be assigned to an earlier station than v′ Parameters sv Importance score of SKU v∈V pv Number of picks for SKU v∈V hv Height taken in the shelf SKU v∈V wv Width taken in the shelf by the target stock of SKU v∈V kr Station k∈K shelf r∈R is located in zk Workload at station k∈K . The average over all stations is denoted by z. 𝛿 Threshold denoting the maximum permitted relative deviation of workload zk of a station k∈K from the average z over all stations hr Height of shelve r∈R wr Width of shelve r∈R dr Distance between the picker and shelve r∈R with picking efficiency 1 dr gMinimum distance between each two SKUs stored next to each other 𝛼 Weighting factor for the objective function 𝛾1 Objective value when optimising the total importance score only 𝛾2 Objective value when optimising the picking efficiency only Decision variables xv,r∈{0, 1} 1 iff SKU v∈V is assigned to shelve r∈R yo Last station to which an SKU with order rank o∈O can be assigned 593 Integrated storage assignment foranE‑grocery fulfilment… Appendix4: Augmented MILP formulation accounting fordemand variation inthestorage assignment See Table7. ∑ r ∈Rv xv,r≤1∀v∈ V krxv,r≤yo∀o∈O,v∈Vo,r∈R v krxv,r≥yo−1∀o∈O⧵{1},v∈Vo,r∈R v zk=∑ v∈V ∑ r∈Rk pvxv,r∀k∈ K zk≤(1+𝛿)⋅ 1 |K|∑ l∈K zl∀k∈ K zk≥(1−𝛿)⋅ 1 |K|∑ l∈K zl∀k∈ K wr≥∑ v∈V (wv+g)⋅xv,r−g∀r∈ R xv,r∈{0, 1 }∀ v∈V,r∈R v y o ∈{1, …, | K |}∀ o∈ O max 𝛼 𝛾1 ∑ v∈V ∑ r∈Rv svxv,r ⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟ I + 1−𝛼 𝛾2 ∑ v∈V ∑ r∈Rv 1 dr pvxv,r ⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟ II ∑ r∈Rv x v,r ≤1∀v∈V krxv,r≤yo∀o∈O,v∈Vo,r∈R v krxv,r≥yo−1∀o∈O⧵{1},v∈Vo,r∈R v zt k=∑ v∈V ∑ r∈Rk pt vxv,r∀k∈K,t∈ T zt k≤(1+𝛿t)⋅ 1 |K|∑ k∈K zt k∀k∈K,t∈ T zt k≥(1−𝛿t)⋅ 1 |K|∑ k∈K zt k∀k∈K,t∈ T wr≥∑ v∈V (wv+g)⋅xv,r−g∀r∈ R xv,r∈{0, 1 }∀ v∈V,r∈R v y o ∈{ 1, … ,| K |}∀ o∈ O 594 D.Winkelmann et al. Table 7 Table of set, parameters, and decision variables for the MILP Sets VSet of SKUs KSet of stations in the picking loop, ordered and indexed by integers ( K ={1, …, |K|} RSet of shelves in the picking loop with Rk denoting the subset of shelves at station k∈K and Rv corresponding to the subset of shelves R that can fit SKU v∈V OSet of precedence order ranks with O={1, …,|O|} , where ov≤ov′ iff v has to be assigned to an earlier station than v′ TSet of days of week Parameters sv Importance score of SKU v∈V pv Number of picks for SKU v∈V hv Height taken in the shelf SKU v∈V wv Width taken in the shelf by the target stock of SKU v∈V kr Station k∈K shelf r∈R is located in. zk Workload at station k∈K . The average over all stations is denoted by z. The workload at station k∈K at day of week t∈T is denoted by zt k 𝛿 Threshold denoting the maximum permitted relative deviation of workload zk of a station k∈K from the average z over all stations. The threshold at day of week t∈T is denoted by 𝛿t hr Height of shelve r∈R wr Width of shelve r∈R dr Distance between the picker and shelve r∈R with picking efficiency 1 dr gMinimum distance between each two SKUs stored next to each other 𝛼 Weighting factor for the objective function 𝛾1 Objective value when optimising the total importance score only 𝛾2 Objective value when optimising the picking efficiency only Decision variables xv,r∈{0, 1} 1 iff SKU v∈V is assigned to shelve r∈R yo Last station to which an SKU with order rank o∈O can be assigned Appendix5: Shelf distance ofSKUs withheight exceeding 250mm See Figs.15 and 16. 595 Integrated storage assignment foranE‑grocery fulfilment… 2000 2500 0 500010000 15000 Picks Distance alpha = 1 2000 2500 0500010000 15000 Picks Distance alpha = 0.75 2000 2500 05000 10000 15000 Picks Distance alpha = 0.5 2000 2500 0 500010000 15000 Picks Distance alpha = 0.25 2000 2500 0500010000 15000 Picks Distance alpha = 0 Fig. 15 Allocation of SKUs and corresponding picks to shelves with given distance to the picker for different values of 𝛼 and 𝛿=1% . Note that the figure is limited to SKUs with a height exceeding 250mm 596 D.Winkelmann et al. Funding Open Access funding enabled and organized by Projekt DEAL. Declarations Conflict of interest The authors declare that they have no conflict of interest. Ethical approval This article does not contain any studies with human participants or animals performed by any of the authors. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References Abdel-Hamid AA-A, Borndörfer R (1994) On the complexity of storage assignment problems. (ZIB Report SC-94-14). 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