Cost-neutral reduction of infection risk in picker-to-parts warehousing systems

Löffler, Maximilian; Schneider, Michael; Žulj, Ivan Article — Published Version Cost-neutral reduction of infection risk in picker-to-parts warehousing systems OR Spectrum Provided in Cooperation with: Springer Nature Suggested Citation: Löffler, Maximilian; Schneider, Michael; Žulj, Ivan (2022) : Cost-neutral reduction of infection risk in picker-to-parts warehousing systems, OR Spectrum, ISSN 1436-6304, Springer, Berlin, Heidelberg, Vol. 45, Iss. 1, pp. 151-179, https://doi.org/10.1007/s00291-022-00695-8 This Version is available at: https://hdl.handle.net/10419/306655 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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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. https://creativecommons.org/licenses/by/4.0/ Vol.:(0123456789) OR Spectrum (2023) 45:151–179 https://doi.org/10.1007/s00291-022-00695-8 1 3 ORIGINAL ARTICLE Cost‑neutral reduction ofinfection risk inpicker‑to‑parts warehousing systems MaximilianLöffler1· MichaelSchneider1· IvanŽulj2 Received: 22 September 2021 / Accepted: 8 August 2022 / Published online: 23 November 2022 © The Author(s) 2022 Abstract The rapid and severe outbreak of COVID-19 caused by SARS-CoV-2 has heavily impacted warehouse operations around the world. In particular, picker-to-parts warehousing systems, in which human pickers collect requested items by moving from picking location to picking location, are very susceptible to the spread of infection among pickers because the latter generally work close to each other. This paper aims to mitigate the risk of infection in manual order picking. Given multiple pickers, each associated with a given sequence of picking tours for collecting the items specified by a picking order, we aim to execute the tours in a way that minimizes the time pickers simultaneously spend in the same picking aisles, but without changing the distance traveled by the pickers. To achieve this, we exploit the degrees of freedom induced by the fact that picking tours contain cycles which can be traversed in both directions, i.e., at the entry to each of these cycles, the decision makers can decide between the two possible directions. We formulate the resulting picking tour execution problem as a mixed integer program and propose an efficient iterated local search heuristic to solve it. In extensive numerical studies, we show that an average reduction of 50% of the total temporal overlap between pickers can be achieved compared to randomly executing the picking tours. Moreover, we compare our approach to a zone picking approach, in which infection risk between pickers can be almost eliminated. However, compared to our approach, the results show that the zone picking approach increases the makespan by up to 1066%. Keywords Routing· Iterated local search· Picker routing· COVID-19· Pandemics * Ivan Žulj [email protected] Maximilian Löffler [email protected]h-aachen.de Michael Schneider [email protected] 1 Deutsche Post Chair – Optimization ofDistribution Networks, RWTH Aachen University, Aachen, Germany 2 Department ofProcurement andProduction, University ofHohenheim, Stuttgart, Germany 152 M.Löffler et al. 1 3 1 Introduction Warehousing is an essential component of many supply chains (see, e.g., Boysen etal. 2021) and has a strong influence on on-time deliveries and customer satisfaction. High cost pressure, steadily increasing global retail sales volumes (Statista 2019), and next- or same-day deliveries force warehouse managers to improve the performance of their warehousing processes to meet the customer expectations for fast delivery and to gain advantages over competitors (see, e.g., Weidinger 2018; Boysen etal. 2019). The most labor-intensive warehousing activity is order picking, which is the process of retrieving items from their storage locations to fulfill customer orders. Studies indicate that a large share of all order picking systems in Western Europe follow the traditional picker-to-parts warehousing setup, in which pickers move through the warehouse to retrieve the requested items from their storage locations (Napolitano 2012). The major drawback of such systems is the large fraction of unproductive picker walking time (deKoster etal. 2007; Tompkins etal. 2010). While there are technologies to automate order picking (see, e.g., Azadeh etal. 2019), warehouse managers rely on manual order picking because human pickers are flexible and can adapt to changes in real time compared to automated systems (Grosse etal. 2014). The rapid spread of SARS-CoV-2 forces warehouse managers to reduce the risk of spreading the virus to avoid a complete stop of the workflow due to infected pickers. Because SARS-CoV-2 is mainly spread person-to-person through close contact, and pickers work close to each other, the risk of virus transmission in manual order picking is very high. As a response to COVID-19, warehouse managers initially aimed at maintaining physical distance between pickers by restricting the number of pickers allowed in a picking area at one time or by implementing one-way picking aisles to avoid congestion or picker blocking (DC Velocity 2020). Because such safety measures make it hard to meet pre-pandemic picking performance due to longer picking routes, Amazon, for example, launched a social distancing monitor based on artificial intelligence, which visually alerts pickers when physical distance is not maintained (Amazon 2020). Softeon, a global supply chain software vendor, offers a software that alerts a picker if the picker’s next pick is in a picking aisle which is already occupied by another picker (DC Velocity 2020). This study aims at mitigating the risk of becoming infected or spreading SARS-CoV-2 (or other infectious diseases) in manual order picking by minimizing the time pickers simultaneously spend in the same picking aisles, which supports maintaining social distancing practices. We assume that the risk of infection is proportional to the time that pickers spend in close proximity to each other, which is in accordance with the current state of knowledge (see, e.g., World Health Organization 2020). We consider a rectangular single-block warehouse layout (see Fig.1) with a central depot and with parallel picking aisles that are connected by a cross aisle at the front and at the rear of the picking aisles. Items are stored in storage locations arranged along both sides of the picking aisles. We assume that each picker is given a sequence, in which the picking orders are to be executed. A picking order comprises a single or multiple customer 153 1 3 Cost‑neutral reduction ofinfection risk inpicker‑to‑parts… orders. For each picking order, the picking tour along the storage locations is given. A common feature of picking tours is that they contain cycles which can be traversed in both directions, i.e., the decision maker can choose between the two possible directions at the entry to each of these cycles. Figure2 gives an example of different execution possibilities (see Fig.2(b)–(e)) of a picking tour (see Fig.2(a)). The double circle marked by d represents the central depot, the other double circles indicate the front/rear locations of each picking aisle, and the standard circles denote the picking positions. The picking tour is represented by the solid lines. The front/rear end locations of picking aisles at which a degree of freedom occurs are marked red. Here, the picker can decide whether to enter the respective picking aisle (if so, we indicate this by a red vertical arc) or to proceed along the front/rear cross aisle (if so, we indicate this by a red horizontal arc). For each of the four depicted execution possibilities, the time periods in which the picker occupies a certain picking aisle differ. Consequently, to reduce virus transmission among pickers, we seek to execute the tours such that the time that pickers spend simultaneously in the same picking aisles is minimized. Note that the decision on how the picking tours are executed does not affect the distance traveled by the pickers. The risk of infection in the cross aisles is not considered because cross aisles are generally significantly wider than the picking aisles, and thus, wide enough to allow pickers to pass each other while maintaining a social distance. Moreover, the risk of virus transmission via goods is assumed to be negligible for the following reason: when a picker moves to the storage location of a requested item, she1 does not touch the surrounding items. The risk of infection via goods can be further decreased by regularly disinfecting hands. picking aisles Legend: item storage location central depot front cross aisle rear cross aisle Fig. 1 An example of a rectangular single-block warehouse layout 1 For the sake of readability, we have decided to speak exclusively of female pickers. Of course, a picker can also be of any other gender. 154 M.Löffler et al. 1 3 In Fig.3, we show the impact of different executions of picking tours on the total time pickers spend together in the same picking aisles. To this end, we consider an instance with four picking tours, each associated with one of four pickers. In the upper part of Fig.3(a), the picking tours are illustrated by the sequence in which the picking aisles are visited by the respective picker. The number in a rectangle indicates the picking aisle, and the length of a rectangle represents the time period the respective picker spends in this picking aisle. For example, picker1 starts in picking aisle1 (from t = 0 to t = 6) and from there proceeds to picking aisles4 (from t = 12 to t = 16), 5 (from t = 18 to t = 20), 6 (from t = 22 to t = 28), and so on. In the lower part of Fig.3(a), we report the time that the pickers simultaneously spend in the same picking aisles resulting from the given execution of the picking tours. We assume that in the case in which n pickers occupy a picking aisle at a point in time, 0.5 ⋅ n ⋅  (n−1) overlaps occur. Executing the picking tours given in Fig.3(a) leads to a total temporal overlap of Z = 102. Modifying the execution of the picking tours d (a) d (b) d (c) d (d) d (e) Fig. 2 Example of different execution possibilities (b)–(e) of the picking tour given in(a) 155 1 3 Cost‑neutral reduction ofinfection risk inpicker‑to‑parts… of pickers1 and 4 as shown in the upper part of Fig.3(b) leads to a reduction in the temporal overlap by approximately 80%. The example also shows that the length of the rectangles in Fig.3(a) is the same as in Fig.3(b), i.e., the distance traveled (or time required) by each picker is not affected. To the best of our knowledge, the resulting picking tour execution problem (PTEP) constitutes a novel setting, which has not been studied in the literature so far but is of high relevance in real-world warehouses. The contributions of our paper are the following: • We formally describe the PTEP as a mixed integer program and propose an efficient iterated local search heuristic (ILS) to provide solutions for large-sized problem instances. t 0 5 10 15 20 25 30 35 40 45 50 55 60 65 1 2 3 4 5 6 #overlaps Z= 102 (a) Picker 1 1 2 3 45 6 7 Picker 2 1 2 3 3 45 6 7 Picker 3 12 3 4 56 7 Picker 4 12 3 4 56 7 t0 5 10 15 20 25 30 35 40 45 50 55 60 65 1 #overlaps Z=20 (b) Picker 1 12 34 56 7 Picker 2 1 2 3 3 45 6 7 Picker 3 12 3 4 56 7 Picker 4 1 2 3456 7 Fig. 3 Total temporal overlap between pickers (Z) for different picking tours 156 M.Löffler et al. 1 3 • In extensive numerical studies, we test the performance of our PTEP model using the optimization software Gurobi. Our formulation is able to solve small and medium instances but is not able to consistently solve large instances within a given time limit. Moreover, we compare the performance of ILS to those of Gurobi solving PTEP. The results clearly demonstrate the ability of ILS to find high-quality solutions within reasonable runtimes. • Additional experiments show that reductions between 17% and 100% of the total temporal overlap between pickers can be achieved compared to randomly executing the picking tours without increasing the distance traveled by pickers. • To provide managerial insights, we compare our PTEP to a zone picking approach, in which the risk of infection spread is almost eliminated. The results show that the zone picking approach strongly increases the makespan compared to our approach. The remainder of the paper is structured as follows: In Sect.2, we discuss the related literature. In Sect.3, we introduce the PTEP and present a mixed integer formulation. Our ILS is detailed in Sect.4. Section5 is devoted to the numerical studies. In Sect.6, we compare the PTEP to a zone picking approach, followed by a conclusion in Sect.7. 2 Literature review Because traveling is considered the most time-consuming warehousing activity, research mainly focuses on reducing the average distance traveled (or time required) by pickers for collecting the items of a given set of picking orders. Travel distance depends on the design of the following four planning problems: warehouse layout (configuration of the warehouse, including the number of blocks and the number, length, and width of the picking aisles and cross aisles in each block), picker routing (determining the picking tour through the warehouse and the retrieval sequence of the items), order batching (grouping or splitting of customer orders), and storage assignment (assignment of items to storage locations). The PTEP is closely related to picker routing problems (PRPs), which, in general, aim at determining a cost-minimal picking tour along the storage locations defined by a picking order (see, e.g., Ratliff and Rosenthal 1983). The most well-studied PRP in the literature is the standard single picker routing problem (standard SPRP), which can be defined as follows: Given a single-block parallel-aisle warehouse with a central depot and dedicated storage, i.e., each item is available from one storage location, the standard SPRP seeks the cost-minimal picking tour for collecting the items defined by a picking order. For an extensive review on PRPs, we refer the reader to Masae etal. (2019). Contrary to PRPs, in our PTEP, the picking tour is already given as input, and it can be generated by an arbitrary solution method for the PRP. Moreover, in the PTEP, picking tours are executed such that the temporal overlap between pickers is minimized, whereas previous research on PRPs has mainly concentrated on determining picking tours such that the distance traveled (or time required) by pickers is minimized. 157 1 3 Cost‑neutral reduction ofinfection risk inpicker‑to‑parts… The risk of infection spread in warehousing operations is only considered in Ardjmand etal. (2021). The authors study an order batching problem which aims at grouping customer orders into (larger) picking orders such that the total order picking time, the makespan, and the time in which pickers are picking closer than at a specified physical distance are minimized. To solve the resulting problem, they propose three multi-objective metaheuristics. Ardjmand et al. (2021) do not aim to reduce the risk of infection by optimizing the routing of pickers but assume that pickers are fixedly routed according to an S-shape routing policy. Another closely related problem concerns the blocking of pickers. The studies that exist on picker blocking consider additional travel distance (or time) that occurs when, for example, pickers meet and passing one another is not possible due to narrow picking aisles. Gue etal. (2006) investigate the effect of pick density on picker blocking in warehousing systems with high space utilization, in which pickers move in an S-shape fashion along unidirectional picking aisles, and passing by other pickers is not allowed. To avoid blocking, the authors discuss different routing methods, e.g., forcing a blocked picker to leave the picking aisle and proceed along a picker-free picking aisle to continue order picking. Pan and Shih (2008) and Parikh and Meller (2009) investigate a multiple-picker setting with congestion considerations from a queuing theory perspective: Pan and Shih (2008) present a throughput model for a picker-to-parts warehouse involving multiple pickers and narrow picking aisles, which simultaneously considers the total travel time and the congestion effect. A picker is routed according to the S-shape routing policy, and in case that an aisle is already occupied by another picker, she has to wait in a buffer zone until the other picker leaves the picking aisle. Parikh and Meller (2009) develop analytical models to estimate picker idle time in a wide-aisle distribution center, in which picker blocking occurs if a picker blocks access to a picking position of another picker. Zhang etal. (2009) consider workflow congestion in the context of material handling equipment interruptions in a manufacturing or warehousing facility. The authors combine a probabilistic and physics-based model to evaluate the expected link travel time when interruptions occur. Two heuristic algorithms are developed to solve the combined optimization model. Hong etal. (2012) introduce an integrated batching and batch sequencing problem for a narrow-aisle order picking system that aims at minimizing the sum of travel time, pick time, and congestion delays. To address realistically sized instances, the authors present a simulated annealing heuristic. Chen etal. (2013, 2014) consider a warehousing system with narrow picking aisles, in which picker congestion has to be avoided. Chen etal. (2013) propose a routing algorithm based on ant colony optimization for two pickers and Chen etal. (2014) for multiple pickers. Schrotenboer etal. (2017) develop a genetic algorithm to evaluate the tradeoff between delays caused by so-called picker interactions and the time required to avoid an interaction. An interaction delay occurs, for example, when a picker blocks access to a storage location from which another picker needs to collect an item. To conclude, methods on picker blocking cannot be used to solve the PTEP because picker blocking problems assume that passing one another is not possible. 158 M.Löffler et al. 1 3 3 The picking tour execution problem In this section, we provide a detailed description of the PTEP (see Sect.3.1) and present a mathematical formulation (see Sect.3.2). 3.1 Problem description The PTEP considers a rectangular single-block warehouse with parallel picking aisles (see Fig.1). Let A denote the ordered set of picking aisles, which are numbered in ascending order from the leftmost to the rightmost picking aisle. We use a weighted graph G=(V,E) to describe a picking order within a warehouse. Set V indicates the vertices including the front (rear) locations ea ( e′ a ) of each picking aisle a ∈ A, the depot d, and the picking positions vi of each requested item i ∈ I, where I denotes the set of requested items of the picking order. Set E consists of an unlimited number of parallel edges between every pair of adjacent vertices. The weight of each edge represents the travel distance (or time) between two adjacent vertices. An example of a warehouse graph with |A| = 7 picking aisles and |I| = 9 picking positions is given in Fig.4. For simplicity, the figure represents the parallel edges connecting adjacent vertices by a single dashed edge. The items of a given set of picking orders have to be collected. Let K denote the set of pickers, where each picker k ∈ K is associated with an ordered set Ok of so-called tour subgraphs. A tour subgraph o is a subgraph of G from which a directed picking tour can be constructed such that every arc of the directed picking tour corresponds to exactly one edge in o. Note that the PTEP works independent of the underlying warehouse layout and routing policy because any routing policy used in any warehouse layout can be described such that the output is a tour subgraph, which is used as input to the PTEP and ILS. In Fig.2(a), a tour subgraph returned by the optimal routing policy is depicted, and in Fig.2(b)–(e), we show the four directed picking tours that can be constructed from the tour 2 2 2 2 2 2 2 2 2 2 2 2 4 2 2 2 2 1 4 1 4 2 5 1 3 3 2 4 d e 1 e1 e 2 e2 e 3 e3 e 4 e4 e 5 e5 e 6 e6 e 7 e7 v1v2 v3 v4 v5 v6 v7 v8 v9 Fig. 4 An example of a warehouse graph with |A| = 7 picking aisles and |I| = 9 picking positions 165 1 3 Cost‑neutral reduction ofinfection risk inpicker‑to‑parts… whereas in the case of (ii) or (iii), a picking aisle is accessed from the rear (ii) or the front cross aisle (iii). The largest gap policy provides a single degree of freedom at the beginning of the tour. • S-shape: In the S-shape routing policy, a picker enters picking aisles in an alternating manner from the front and the rear cross aisle and traverses them completely (if picks are required in there) or not at all (if no pick is required in there). • S-shape+: As in the S-shape routing policy, in S-shape+, picking aisles are entered in an alternating manner from the front and the rear cross aisle and traversed completely (if picks are required). The difference between these two routing policies is the following: In the standard S-shape routing policy, the picker directly travels from the front location of the rightmost picking aisle to be visited to the depot without accessing the other front locations of the picking aisles (see Fig.7(c)). Thus, all picks are executed on the way to the rightmost picking aisle that contains a picking position. In the S-Shape+ routing policy, the picker successively accesses each front location of the picking aisles on the way back to the depot. As shown in Fig.7(d), this leads to a tour subgraph which allows the picker to skip required picking aisles on the way from the depot to the rightmost picking aisle to be visited and then to access the skipped picking aisles on the way back to the depot. Both routing policies lead to the same distance traveled by the picker but differ in the number of degrees of freedom they offer. The S-shape routing policy contains only one degree of freedom at the beginning of the tour, whereas the S-shape+ routing policy offers multiple degrees of freedom and therefore provides greater flexibility in designing the picker routes. Combining the above described parameter values leads to 160 instance groups, which are identified by the demand scenario (UDD or CBD), the number of pickers |K|, the number of tour subgraphs per picker |Ok| , and the routing policy used to generate the respective tour subgraphs. For each instance group, we generate five instances, i.e., 160⋅5=800 instances in total. All experiments are conducted on a computing cluster running CentOS7 with 2×Intel Xeon E5-2430v2 Processors at 2.50GHz and 64GB of memory per compute node. Gurobi is used to solve the mixed integer program presented in Sect.3.2, and each process runs multithreaded on 6CPU cores. We restrict the solution time of Gurobi to 3600s for all experiments. ILS is implemented in single-threaded C++ and compiled using GCC8.2 with full optimizations enabled. 5.2 Computational results In this section, we assess the performance of the PTEP model and of our ILS. Moreover, we provide managerial insights with respect to the potential avoidance of temporal overlaps between pickers in the warehousing system under study. Tables2 and 3 present aggregate results for different routing policies on the UDD instances and Tables4 and 5 on the CBD instances. Each table reports averages for groups of instances defined by the number of pickers (column K) and the number of tour subgraphs per picker (column |Ok| ). The remaining columns are 166 M.Löffler et al. 1 3 divided into two blocks, where each block reports the results for one of the underlying routing policies. In column frand , we give the average temporal overlap of 1000 randomly generated solutions, i.e., we randomly construct directed picking tours from the respective tour subgraphs. In addition, we report the average of the total travel distance of pickers (column td ). All other values that are reported are provided as percentage deviation from the objective function value reported in column frand . Column ub (%) ( lb (%)) denotes the percentage deviation of the best objective function (lower bound) values Table 2 Results on the UDD instances grouped by number of pickers |K| and number of tour subgraphs per picker | O k| using the optimal and largest gap routing policy Table 3 Results on the UDD instances grouped by number of pickers |K| and number of tour subgraphs per picker | O k| using the S-shape and S-shape+ routing policy 167 1 3 Cost‑neutral reduction ofinfection risk inpicker‑to‑parts… given by Gurobi. Column #opt indicates the number of instances solved to proven optimality. For ILS, the reported results are based on five runs on each instance. In column best (%) ( avg (%)), we show the percentage deviation of the best (average) solution value over the five runs. Column ta (s) gives the average runtimes in seconds. In the following, we discuss the results of our experiments: Comparison of ILS to Gurobi The results show that the number of pickers, the number of tour subgraphs per picker, and the routing policy significantly influence the performance of Gurobi. The main findings are as follows: Table 4 Results on the CBD instances grouped by number of pickers |K| and number of tour subgraphs per picker | O k| using the optimal and largest gap routing policy Table 5 Results on the CBD instances grouped by number of pickers |K| and number of tour subgraphs per picker | O k| using the S-shape and S-shape+ routing policy 168 M.Löffler et al. 1 3 • Optimal: Using the optimal routing policy, Gurobi is able to solve 55 UDD instances and 74 CBD instances to optimality (UDD instances with |K| =  {2, 5} and those with |K| =  {10} assuming |Ok|  =  {2, 5} as well as all CBD instances with |K| =  {2, 5, 10} ). On the UDD and CBD instances with |K| = 20, all instances with |Ok|  = 2 are solved, and on the CBD instances with |Ok|  = 5, four out of five instances are solved. On the instances with |K| = 40, Gurobi is not able to solve any of the UDD and CBD instances within the given runtime limit. • Largest gap and S-shape: Most of the instances can be solved to optimality using the largest gap policy (148 out of 200 instances), i.e., all UDD and CBD instances with |K| =  {2, 5, 10} , except a single CBD instance with |K| = 10 and |Ok|  = 20. On the CBD instances with |K| = 20 and |Ok|  =  {2, 5} , Gurobi provides optimal solutions for all tested instances and on the UDD instances, for all except for a single instance. Even on the larger UDD and CBD instances with |K| = 40 and |Ok|  = 2, Gurobi finds optimal solutions for all tested instances. Similar results can be observed for the case of the S-shape routing policy (146 out of 200 instances are solved to optimality). • S-shape+: The fewest of the instances can be solved using the S-shape+ routing policy (106 out of 200 instances), i.e., all UDD and CBD instances with |K| =  {2, 5} and |Ok|  =  {2, 5, 10, 20} . On the larger UDD instances, Gurobi finds optimal solutions for all instances with |K| = 10 and |Ok|  = 2, and for 4 out of 5 instances with |K| = 20 and |Ok|  = 2. In the case of larger CBD instances, Gurobi provides optimal solutions for all instances with |K| =  {10, 20} and |Ok|  = 2, and for 4 out of 5 instances with |K| = 10 and |Ok|  = 5, and 3 out of 5 instances with |K| = 40 and |Ok|  = 2. The results show that Gurobi succeeds in solving small and medium instances within reasonable runtimes but is not able to consistently solve larger instances to optimality within the given time limit. The tables show that ILS finds the same or slightly worse solutions as Gurobi on the smaller and medium instances but clearly outperforms Gurobi on the larger instances: It is worth noting that the solution quality of ILS on the largest instances is close to the lower bound of Gurobi (in the cases in which the lower bound is not equal to zero). In addition, the results indicate a very good robustness of our ILS, i.e., the best solutions found by ILS deviate from the average solutions by only 0.5% for the UDD case and 0.6% for the CBD case. The runtimes of ILS are reasonable and stay below 10minutes for all but the largest instances with |K| = 40 and |Ok|  = 20. The results of ILS clearly demonstrate that ILS is able to solve the largest instances within reasonable runtimes. Performance of ILS for different routing policies In the following, we investigate the performance of ILS for different routing policies. To this end, we compare the results of the arbitrary solution (see column “ frand ”) with those obtained by our ILS (see column “ best ”) for the same picker routing policy. We use the reduction in temporal overlap between pickers to compare the results of an arbitrary solution to the PTEP with that obtained by our ILS for the same picker routing policy. This can be an insightful measure for managers if a fixed routing policy is used in their warehouse. 169 1 3 Cost‑neutral reduction ofinfection risk inpicker‑to‑parts… The results show that ILS is able to significantly reduce the temporal overlap between pickers independent of the picker routing policy at hand. For example, on the smallest UDD and CBD instances with |K| = 2 and S-shape+ routing policy, ILS generates solutions that are completely free of temporal overlaps. Even on the largest UDD (CBD) instances with |K| =  {20, 40} , a reduction of 39% (35%) and 26% (23%) is achieved although the instances assume a warehouse with only 10 picking aisles. ILS achieves the smallest reduction in the temporal overlap between pickers in the case of S-shape and largest gap routing policies. For instance, even on the smallest UDD instances with |K| = 2, temporal overlaps between pickers cannot be completely avoided. This is because both routing policies provide only one degree of freedom in the routing: the picker performs the S-shape routing either via the leftmost or via the rightmost picking aisle to be visited, and the largest gap routing starts either along the front or the rear cross aisle. In the following, we investigate the effect of (i) the number of pickers, (ii) the number of tour subgraphs per picker, and (iii) the demand scenario on the total temporal overlap between pickers. Moreover, we briefly discuss the results with respect to the total travel distance of the pickers. • For all routing policies, we observe that the larger the number of pickers |K|, the smaller the average reduction in the total temporal overlap between pickers. Obviously, in the case of a large number of pickers, there is less potential for reduction in comparison with a small number of pickers because the probability of temporal overlaps increases with the number of pickers. For instance, on the UDD instances using the S-shape+ routing policy, the average reduction ranges between 26% in the case of |K| = 40 and 100% for |K| = 2. • The influence of the number of tour subgraphs per picker is hardly visible. For example, on the UDD instances with |K| = 20 using largest gap routing policy, the reductions achieved for |Ok|  =  {2, 5, 10, 20} are 24%, 23%, 23%, and 23%, respectively. • For CBD, ILS generates solutions that are slightly worse compared to those for UDD. For example, for CBD, |K| = 40, and the S-shape+ routing policy, ILS yields a reduction of 23% on average compared to 26% for UDD. • As expected, the optimal routing leads to the shortest picking tours, and the S-shape and S-shape+ routing policies result in the longest routes. This is because all picking aisles containing even a single item to be picked must be completely traversed. Our numerical results show that large reductions in the total temporal overlap between pickers are achieved independent of the underlying routing policy. The smallest temporal overlap is observed for the optimal routing policy (see Tables7, 8, 9, and 10). With respect to picking performance, the optimal routing policy is obviously superior to the other routing policies because it leads to shorter picking tours. However, a shortcoming of using the optimal routing policy is that pickers may get confused by the complex routes, and therefore, tend to deviate from optimal routing patterns (see, e.g., Elbert etal. 2017). Deviating from given routes can have a negative impact on picking efficiency and also increase the risk of 170 M.Löffler et al. 1 3 infection spread. This effect is particularly strong if no modern technologies such as tablets, pick-by-light, or pick-by-voice are used to assist pickers. To reduce infection risk in picker-to-parts warehousing systems through our PTEP, it is paramount that pickers follow prescribed routes. While pickers will not always follow prescribed routes in practice in non-pandemic periods for several reasons (e.g., social interactions), in pandemic periods, pickers are more likely to do so to avoid infection. Moreover, although our PTEP is not able to completely prevent possibilities for social interaction, it significantly reduces them through exploiting the degrees of freedom that the routing policies offer. 6 Zone picking In this section, we compare our approach to a zone picking approach, which is another possible way to reduce infection risk in picker-to-parts warehouses. To make a fair comparison possible, we choose the following setup for the zone picking approach: • Warehouse layout: The picking area follows the single-block parallel-aisle warehouse layout described in Sect.5.1, but is divided into smaller zones, each with a single picker assigned to it. A zone comprises a fixed number of picking aisles, and each zone is associated with a handover location located in the front cross aisle. Figure 8 illustrates the zone picking system with different numbers of zones. Obviously, physical distancing practices are easy to implement, and infection risk between pickers can be almost eliminated in such systems. • Parallel zone picking: We assume parallel zoning, i.e., the items of a picking order are simultaneously retrieved by multiple zone pickers in multiple zones as described in the following: Each zone picker is initially positioned at her handover location. Starting from the handover location, the zone pickers collect the picking order items from their zone according to an optimal routing pattern (see Ratliff and Rosenthal 1983). Once a zone picker has collected all picking order items from her zone, she returns to her handover location to deposit the items. Subsequently, she executes the next picking order from her picking list, starting from her handover location. We assume that each zone picker uses a bin with sufficient capacity for temporarily storing the picking order items and that each handover location has infinite storage capacity (buffer) for depositing items. • Sequence for picking orders: The sequence according to which the picking orders have to be collected by the zone pickers is given. • Consolidator: A consolidator collects the items of a single picking order from the relevant handover locations (i.e., those from which the picking order items are to be collected) on a single tour as follows: The consolidator starts at the depot, where she is equipped with a trolley with sufficient capacity for the items of a single picking order, proceeds along the front cross aisle to the relevant handover locations, and from the last visited handover location to the depot. The sequence according to which a consolidator visits the relevant handover loca- 171 1 3 Cost‑neutral reduction ofinfection risk inpicker‑to‑parts… tions is determined in optimal fashion (concerning the objective of minimizing the completion time of a picking order). We assume that no effort for consolidating the items occurs at the depot. The consolidator collects the picking orders in the given picking sequence. zone 1 zone2 front crossaisle (a) Picking areadivided into2zones each with5picking aisles. zone 1zone 2zone 3 zone4 zone5 front crossaisle (b) Picking areadivided into 5zones each with 2picking aisles. zone 1zone 2zone 3zone 4zone 5 zone6 zone7 zone8 zone9 zone10 front crossaisle (c) Picking areadivided into10 zones each with asinglepicking aisle. Legend: item storagelocation handover location central depot Fig. 8 Warehouse layout assumed in the zone picking system 172 M.Löffler et al. 1 3 • Test instances: Our experiment is based on the instances described in Sect.5.1. For both approaches, we use the exact algorithm of Ratliff and Rosenthal (1983) to determine a picking tour. For our PTEP, we consider |K| =  {3, 6, 11} pickers and |Ok|  =  {2, 5, 10, 20} tour subgraphs per picker k ∈ K. For the zoning approach, the picking area is divided into 2, 5, and 10 zones comprising 5, 2, and 1 picking aisles, respectively. We assume |K| =  {3, 6, 11} pickers, where the 3rd, 6th, and 11th picker represents the consolidator. The number of picking orders in the zone picking system is defined as |O| =  |Ok|  ⋅ |K|. Combing the above described parameter values leads to 12 instance groups, which are identified by the number of pickers |K| and the number of picking orders in the system |O|. For each instance group, we generate 5 instances, i.e., 12 ⋅ 5 = 60 instances in total. In the following, we use the makespan, i.e., the time when all picking orders are completed, as performance measure for comparing the PTEP and the zone picking approach. In Table6, we present aggregate results obtained by our ILS for the PTEP and by the zone picking approach on the UDD instances and the CBD instances. The table reports averages for groups of instances defined by the number of pickers (column |K|) and the number of picking orders in the system (column |O|). The remaining columns are divided into two blocks, where the first block represents the results on the UDD instances and the second block those on the CBD instances. In column ATO, we give the average temporal overlap. Column MPTEP ( MZONE ) reports the average makespan required for the PTEP (zone picking approach). Column Δ (%) denotes the average percentage deviation between MZONE and MPTEP . The results show that our PTEP approach strongly outperforms the zone picking approach with respect to makespan. For example, on the smallest instances with |K| = 3 pickers and |O| = 6 picking orders in the system, the average percentage deviation between the PTEP and the zone picking approach is approximately 40% in the UDD case (49% in the CBD case). Moreover, on these instances, our ILS finds solutions that are almost free of temporal overlaps between pickers. The larger the number of pickers |K|, the greater the deviation in makespan between the PTEP and the zone picking approach. For example, on the largest CBD instances with |K| = 11 pickers and |O| = 220 picking orders in the system, the makespan obtained by the zone picking approach is on average 1066% higher than that obtained by our ILS for the PTEP. To sum up, the zone picking approach presented above significantly increases the makespan although many favorable assumptions are made (e.g., no effort for consolidating the collected items at the depot, infinite storage capacity for depositing items at the handover locations). Clearly, the zone picking approach is able to prevent pickers from operating in close proximity to each other, and thus, to almost eliminate the infection risk between pickers. Consequently, the tradeoff between basically no temporal overlap but excessive makespan (zone picking approach) and some temporal overlap without raising costs (PTEP) must be evaluated by warehouse managers. Given medical tests and personal protective measures (e.g., vaccinations and/or mouth-nose protection masks), we believe that our PTEP is an interesting approach for warehouse managers to reduce infection risk between pickers without compromising order picking performance. 173 1 3 Cost‑neutral reduction ofinfection risk inpicker‑to‑parts… 7 Conclusion This paper aims to reduce the risk of infection in a picker-to-parts warehousing system in which multiple pickers operate in the same picking area. To this end, we introduce the PTEP that generates directed picking tours from given tour subgraphs such that the time period in which pickers simultaneously occupy the same picking aisles is minimized. We formally describe the PTEP as a mixed integer program and provide an efficient ILS. Our main finding is that significant reductions in the total temporal overlap between pickers can be achieved by exploiting the degrees of freedom that routing policies offer. On average, a reduction in approximately 52% is achieved in the UDD scenario, while the reduction in the CBD scenario is slightly lower with 49%. The smallest temporal overlap between pickers is achieved in the case of the optimal routing policy. The S-shape routing policy tends to perform worst. We believe that our results are quite interesting for warehouse managers in times of pandemics: Avoiding temporal overlaps between pickers does not only reduce the infection risk, but also increases the order picking efficiency (e.g., by reducing in-aisle congestion). Moreover, the reductions are achieved without compromising order picking performance, i.e., without changing the distance traveled (or time required) by the pickers. The comparison of our approach to a zone picking approach reveals that the zone picking approach results in excessive makespan: the deviation between the PTEP and the zone picking approach amounts to 396% on average. To further reduce temporal overlaps between pickers, our problem could be extended to incorporate modern warehousing concepts, such as scattered storage or multiple end depots. For example, with scattered storage, an item type can be available from several storage locations (see Goeke and Schneider 2018). Assigning the stock keeping units of a frequently requested item type to multiple storage locations Table 6 Results on the UDD and CBD instances 174 M.Löffler et al. 1 3 could reduce the probability of pickers working close to each other. The PTEP could be extended to incorporate congestion in picking aisles and picker blocking, which is particularly relevant for warehouses with narrow picking aisles. Moreover, future research could investigate alternative objective functions for reducing the infection risk, e.g., one could minimize infection risk that results from pickers operating within a given physical distance for more than a critical period of time. Pandemics or epidemics will certainly be given more focus in future research. Incorporating infection risks into well-known order picking problems will pose new challenges in the research on warehousing to ensure a safe order picking environment during pandemics. Appendix: Additional results Tables7 and 8 present aggregate results for different routing policies on the UDD instances and Tables9 and 10 on the CBD instances. Each table reports averages for Table 7 Results on the UDD instances grouped by number of pickers |K| and number of tour subgraphs per picker | O k| using the optimal and largest gap routing policy