Picker routing in the mixed-shelves warehouses of e-commerce retailers

Accepted Manuscript

Picker routing in the mixed-shelves warehouses of e-commerce retailers Felix Weidinger, Nils Boysen, Michael Schneider PII: DOI: Reference:

S0377-2217(18)30874-9 https://doi.org/10.1016/j.ejor.2018.10.021 EOR 15413

To appear in:

European Journal of Operational Research

Received date: Revised date: Accepted date:

29 August 2017 28 August 2018 11 October 2018

Please cite this article as: Felix Weidinger, Nils Boysen, Michael Schneider, Picker routing in the mixedshelves warehouses of e-commerce retailers, European Journal of Operational Research (2018), doi: https://doi.org/10.1016/j.ejor.2018.10.021

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Highlights • We study picker routing in mixed-shelves warehouses with multiple depots • We model the problem as MIP and prove computational complexity • Heuristic solution methods are proposed and evaluated

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• We investigate whether different storage assignment strategies are beneficial

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Working Paper

Picker routing in the mixed-shelves warehouses of e-commerce retailers

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Felix Weidinger1 , Nils Boysen1,∗ , Michael Schneider2

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June 2017 Revised: May and August 2018

: Friedrich-Schiller-Universität Jena Lehrstuhl für Operations Management Carl-Zeiss-Str. 3, 07743 Jena, Germany http://www.om.uni-jena.de/ {felix.weidinger,nils.boysen}@uni-jena.de 1

:RWTH Aachen Deutsche Post Lehrstuhl für Optimierung von Distributionsnetzwerken Kackertstr. 7, 52072 Aachen, Germany http://www.dpor.rwth-aachen.de/ [email protected] 2

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Corresponding author, phone +49 3641 9-43100

ACCEPTED MANUSCRIPT Abstract

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E-commerce retailers face the challenge to assemble large numbers of time-critical picking orders, of which each typically consists of just a few order lines and low order quantities. To efficiently solve this task, many warehouses in this segment are organized according to the mixed-shelves paradigm. Incoming unit loads are isolated into single units, which are randomly spread all over the shelves of the warehouse. In such a setting, the probability that a picker always finds a demanded stock keeping unit (SKU) close-by is high, irrespective of his/her current position in the warehouse. In spite of this organizational adaption, picker routing, i.e., the sequencing of shelf visits when retrieving a set of picking orders, is still an important optimization problem. In a mixed-shelves warehouse, picker routing is much more complex than in traditional environments: Multiple orders are concurrently assembled by each picker, many alternative depots are available, and items of the same SKU are available in multiple shelves. This paper defines the resulting picker-routing problem in mixed-shelves warehouses and provides efficient solution methods. Furthermore, we use the developed methods to explore important managerial aspects. Specifically, we benchmark mixed-shelves storage against traditional storage policies for scenarios with different ratios between small-sized and large-sized orders. In this way, we investigate whether mixed-shelves storage is also a suited policy if an omnichannel sales strategy is pursued, and large-sized orders of brick-and-mortar stores as well as small-sized online orders are to be jointly processed by the same warehouse.

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Introduction

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Keywords: Facility logistics; Warehousing; Mixed-shelves; Picker routing

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When setting up their warehouse operations, e-commerce retailers directly serving final customer demands in the business-to-consumer (B2C) segment typically face the following challenges:

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• Small orders: Private households tend to order rather few items in low quantities, so that the typical picking order in the B2C segment consists of just a few order lines, each demanding only very few items. According to personal communications with a German Amazon warehouse manager, their average order contains only about 1.6 items.

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• Large assortment: The term “long tail” describes the phenomenon where niche products account for a much larger proportion of sales in e-commerce than they do in brick-andmortar stores [4]. Consequently, most online retailers offer a large assortment. • Varying workloads: Most online retailers have dynamically grown over the past years [15]. Depending on the offered products, they face highly volatile demands, e.g., due to end-of-season sales. Thus, warehouses should be scalable to flexibly react to varying workloads. • Tight delivery schedules: Next-day or even same-day deliveries have become an elementary component of many business models and supply chains especially in the B2C segment (e.g. [22]). This narrows the time window for order picking considerably and puts increasing stress on warehouse operations. 1

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A warehousing setup specifically designed for these requirements is a mixed-shelves warehouse (e.g., see [20]). Here, unit loads are purposefully broken down into single items and randomly spread all over the racks of the warehouse. This storage policy is also denoted as scattered storage [21]. With stock keeping units (SKUs) being stored all around, there is a good chance to always have one of the demanded items close-by, irrespective of where the picker is currently positioned in a warehouse. In this way, the unproductive walking time of the pickers, which is the main drawback of traditional picker-to-parts systems (in which unit loads are kept together, see, e.g., [7]), is considerably reduced. This helps to realize the tight delivery schedules of e-commerce. Moreover, storing a large assortment seems unproblematic as long as there is enough space for adding shelves to the warehouse. The standardized and low-tech equipment also facilitates the scalability in case of varying capacity situations; additional pickers and racks can simply be added to or removed from a warehouse. Finally, the small order sizes reduce the chance that a picker has to visit multiple storage positions to accumulate enough items of a SKU that is ordered in a larger quantity. This would increase the walking time, so that care has to be taken that the SKU diversity of the racks fits the order sizes in a mixed-shelves warehouse.

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Mixed-shelves are applied in the European distribution centers of online retailer Amazon but also by many other warehouses in the B2C segment [21].

Operations in a mixed-shelves warehouse

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A typical warehouse setting of B2C online retailers operated under the mixed-shelves policy applies the basic elements depicted in Figure 1 and can be described as follows.

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Figure 1: Basic elements of a mixed-shelves warehouse

The racks for storing items are headhigh to be conveniently accessible by human pickers, who walk along the aisles while pushing small picking carts. A cart carries multiple bins in parallel, each dedicated to a specific picking order. Due to the small order sizes the picking carts – although being small and agile – are still able to transport up to a handful of bins concurrently. To use space more efficiently, the racks are often arranged on top of each other in multi-story mezzanine systems. The SKUs are stored according to the mixed-shelves policy, so that all items of a SKU are spread all over the warehouse. To route the workers during order picking, they carry a small handheld scanner. The integrated display lists which order line from which storage location is to be retrieved next. The optimization approach for picker routing that is developed in this paper can thus be applied to navigate the pickers via their handhelds. Furthermore, the scanner is used to acknowledge the retrieval of an item from a shelf and its 2

ACCEPTED MANUSCRIPT put-away in a bin. In this way, the background information system also receives information on empty storage positions where newly retrieved items can be restocked. All around the warehouse, there are multiple access points to a conveyor system, which moves completed bins towards the packing stations of the shipping area. Thus, there is not a single central depot but multiple ones spread all over the warehouse. Next to the access points, there are stacks of empty bins, which can directly be loaded onto the cart for initiating the retrieval of subsequent picking orders.

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In this way, the shelves are successively emptied by the pickers, so that refilling the shelves is another important task. Other than picking, which is executed under great time pressure due to the tight delivery schedules, refill operations are not (that) time-critical. At a distribution center we visited, refill operations have been used to get new employees acquainted with the warehousing processes and to employ redundant workforce during off-peak hours.

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In different real-world applications, the above processes may, of course, slightly vary. For instance, a pick-by-voice system can direct the pickers and picking carts and mezzanine systems are not a fixed requirement. Also, each bin filled by a picker is not necessarily used to collect only a single order but can be assigned to a batch of orders that are jointly picked. Traditionally, batching is particularly effective if the same SKU is needed for multiple orders (see [7]). In the packing and shipping area, however, these batches need to be separated into customer orders again, e.g., by an automated sorter [13, 8].

Decision problems and literature review

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The main decision problems to be solved under a mixed-shelves policy are basically the same as in any other warehouse. Among the most elementary decisions are (see, e.g., [3]): • the storage assignment problem, which assigns SKUs to storage locations, and

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• the picker-routing problem, which determines the routes of the pickers through the warehouse when retrieving picking orders.

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Instead of trying to summarize the vast body of literature which has accumulated on these topics, we merely refer to the recent in-depth review papers, e.g., provided by De Koster et al. [7] and Gu et al. [10, 11]. Our focus is on the peculiarities of mixed-shelves warehouses. In realworld mixed-shelves warehouses, storage assignments are typically not optimized but simply randomly determined (see [21]). The paper on hand concentrates on the latter problem and aims to optimize picker routes in mixed-shelves warehouses. Compared to traditional pickerrouting problems, where a single order is to be picked from a given set of storage positions, and each tour starts and ends at a central depot, the following peculiarities have to be considered: • Items of the same SKU are available in multiple shelves, so that an additional selection problem is to be solved. Moreover, order lines requesting multiple items may need to be collected by visiting multiple storage positions of the respective SKU. • The picking cart is able to carry multiple bins concurrently, so that multiple orders are processed in parallel. 3

ACCEPTED MANUSCRIPT • There exist multiple access points to the conveyor system, so that we have many alternative depots where finished orders can be handed over, and new orders can be initiated.

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Picker routing with alternative pick locations has been considered by Daniels et al. [5]. They solve the resulting problem with facultative distance matrices by a tabu search procedure. Weidinger [20] addresses the same problem for rectangular warehouses. These warehouses have specially structured distance matrices, for which the (pure) routing problem is solvable in polynomial time (see [17]). Integrating alternative pick locations, however, makes the problem strongly NP-hard even for rectangular warehouses [20], so that simple myopic selection rules coupled with the efficient routing procedure of Ratliff and Rosenthal [17] are applied. However, in both papers the latter two peculiarities have not been considered. They assume a single central depot, exclude picking carts and a parallel processing of multiple orders.

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With a single central depot, a picking cart with capacity for multiple bins leads to a traditional batching problem (see [7] for an overview on the literature). In addition to the routing problem, it has to be decided which orders should be jointly retrieved in a tour. With multiple access points a picker may pass some depot anyway, so that some completed order can be handed over en passant although other orders of the current batch are not yet completed. Thus, a dynamic batching problem arises, which (to the best of the authors’ knowledge) has not been treated in the warehousing literature.

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Multiple depots are also rarely considered in the warehousing literature. In a related setting, conveyors run along the cross-aisles, so that each tour can start and end anywhere in a cross aisle. De Koster and van der Poort [6] extend the algorithm of Ratliff and Rosenthal [17] to derive optimal routes for rectangular warehouses in polynomial time for such a setting. We, however, consider a discrete set of depots spread arbitrarily around the warehouse.

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Routing pickers in warehouses shares many similarities with routing delivery tours of vehicles. Here, our three peculiarities may be relevant too, so that we briefly summarize related problem settings for extensions of the vehicle-routing problem (VRP) and its derivatives. A truck has capacity for multiple items anyway and multiple depots are a common in VRPs too (e.g., [16]). However, contrary to our setting, the depots generally mark the end point of a vehicle route at which all orders are completely fulfilled. A more related variant are VRPs with intermediate stops [19], in which a certain resource can be replenished or unloaded at so-called intraroute facilities, and thus the capacity of the vehicle with regard to that resource is increased. Examples include stops for good replenishment at certain intermediate depots or for waste disposal at intermediate collection sites. Finally, alternative delivery locations are considered in the generalized VRP [9]: given a partitioning of the customer set, this variant determines cost-minimal routes that serve exactly one customer of each subset. However, all these VRP variants share some similarities, but are not directly applicable to our picker routing problem. It can be concluded that optimization approaches suited for picker routing in mixed-shelves warehouses are not yet available. Note that this result seems especially unsatisfying because of the extreme time pressure (and the large potential of optimized picker routes to reduce this pressure) existent in the warehouses of online retailers.

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Contribution and paper structure

This paper treats a rich picker-routing problem, which is relevant in the mixed-shelves warehouses of online retailers. Specifically, the following peculiarities are considered: Multiple 4

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orders are concurrently assembled by a picker, several alternative depots are available, and items of the same SKU are available in multiple shelves. The resulting problem is defined (Section 2), and efficient solution procedures are introduced (Section 3) and tested in a comprehensive computational study (Section 4). Furthermore, we address important managerial aspects. A major drawback of mixed-shelves warehouses is that multiple storage locations have to be accessed whenever larger order quantities occur that have to be accumulated from multiple shelves. Therefore, we benchmark mixed-shelves storage against traditional storage policies for scenarios with different ratios between small-sized and large-sized orders. In this way, we investigate whether mixed-shelves storage is also a suited policy if an omni-channel sales strategy is pursued, and large-sized orders of brick-and-mortar stores as well as smallsized online orders are to be jointly processed by the same warehouse. This issue is addressed in Section 5 and, finally, Section 6 concludes the paper.

Problem definition

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Consider a mixed-shelves warehouse consisting of a set P of storage positions, each initially storing γp ≥ 0 items of one SKU s ∈ S, where S denotes the total SKU set. Without loss of generality, we assume that each storage position stores just a single type of SKU, which is defined by ηp . If a shelf carries items of multiple SKUs, then multiple storage positions having zero distance among each other can be introduced. Furthermore, we have a given set D of depots (also denoted as access points), where completed picking orders are handed over to the conveyor system, and empty bins for new orders are retrieved. In total, we have V = P ∪ D positions in the warehouse, and the picker’s walking distance among storage positions and/or depots v and v 0 is denoted by dvv0 . We consider a single order picker, whose picking cart can carry up to C bins concurrently for collecting picking orders. Initially, the picker is located at an arbitrary depot. We have a given set of picking orders O, and aos defines the number of items of SKU s demanded by order o ∈ O.

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We aim at a solution defining the picker route through the warehouse such that all picking orders are retrieved. Specifically, a solution is represented by a sequence π defining the successive visits of the picker at positions in V . Each sequence position πi , i = 1, . . . , m, is specified by a triple (v, δ, o). We have to distinguish three cases: If the picker is located at a storage position, i.e., v ∈ P holds, such a triple defines that during the ith stop the picker retrieves δ items of SKU ηv stored at storage position v for order o. Note that if multiple successive triples all refer to an unaltered storage position v ∈ P , this means that the picker retrieves SKU ηv stored at this position for multiple orders currently being active on his/her cart. Stops do not only refer to storage positions but also to depots where finished orders are handed over and new orders are started. At these positions, e.g. if v ∈ D holds, no items can be retrieved, such that we have δ = 0 and o referring to a virtual order -1. Finally, triple (v, δ, o) at sequence position π0 refers to the initial position of the picker, so that v is fixed to the given starting point of the picker. Given these definitions, we call a solution π feasible if the following conditions hold: • An order o is started (handed over) at the depot preceding (succeeding) the first (last) occurrence of o in the triples of π. Note that there must always exist one optimal solution for which this property holds. Starting (finishing) orders at a depot earlier (later) than required cannot improve the solution value because, in this case, cart capacity is blocked 5

ACCEPTED MANUSCRIPT longer than necessary. Therefore, in a feasible solution π, there has to be at least one depot visit prior and after the first and last occurrence of each order, respectively. During the time between these two depot visits, we call this order active. • In each sequence position i = 1, . . . , m of π where a storage position is visited, we have at most C active orders, so that the given cart capacity is not exceeded. • For each order o ∈ O and all SKUs s ∈ S, it must hold that the number of items aos requested by order o is retrieved while it is active. Thus, retrieved items δ summed over all triples (v, δ, o0 ) in π with o0 = o and ηv = s must equal aos .

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• The available stock in each storage position can be depleted but may not become negative. Thus, for each storage position v ∈ P retrieved items δ summed over all triples (v 0 , δ, o) in π with v 0 = v may not exceed the given initial inventory γv . Among all feasible solutions π, we seek one that minimizes the length of the picker route, i.e., F (π) =

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dvπi−1 ,vπi ,

where vπi refers to position v ∈ V being visited during the ith visit of π. Recall that vπ0 refers to the given start position of the picker. Our problem definition is based on some (simplifying) assumptions, which we elaborate in the following:

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• We only consider a single order picker and neglect all interdependencies to other pickers concurrently working in the warehouse. Thus, pickers are assumed to neither block each other in front of shelves, in front of depots, or within narrow aisles (see, e.g., [12]) nor to influence each other’s inventory levels. Large warehouses in the B2C segment use a large number of pickers that work concurrently, so that it will regularly occur that multiple pickers compete for items. As soon as a picker has retrieved some items, they are no longer available for other pickers, who have to sidestep to other shelves. The coordination of pickers is not part of our research, and we assume that some preceding planning stage has already distributed the storage positions and orders among the workers. It seems a valid task for future research to integrate our (single picker) approach in a larger framework where it can be applied to evaluate different distribution plans of shelves among multiple pickers.

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• We neglect the interdependency to replenishment operations (see Section 1.1). If stow workers replenish shelves in parallel to the picking operations, it may occur that additional items become available after a while. It seems risky to integrate items (and their exact replenishment times) ahead of their actual placement in a shelf; any delay would result in an invalid picker tour. Therefore, we assume that items are only considered once they are finally placed, and the picker is about to start the next tour. • All items of a SKU are assumed to be equally valid to satisfy a demand. In warehouses storing perishable goods, dates of expiry may need to be considered, so that, for instance, the first-in-first-out rule has to be applied [2]. This can be easily considered by preselecting these preferred (urgent) items. Our picker routing approach is then only allowed to retrieve inventory from these preselected storage positions. 6

ACCEPTED MANUSCRIPT • We presuppose standardized bins for collecting orders, which all have equal size. This allows us to measure the picking cart capacity C in number of bins. Typically, standardized bins are a prerequisite for a fail-safe transport on picking carts and conveyors, so that this should rarely be a shortcoming.

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• We restrict ourselves to rectangular warehouses, i.e., single-deep parallel racks along both sides of an aisle with cross aisles at the top and bottom. This most basic setup is very common in the real world. For instance, at a German Amazon warehouse we visited, the pickers were not allowed to change levels in the multi-story mezzanine system, and in each level the setup was rectangular. Moreover, the restriction to rectangular warehouses allows us to apply the polynomial time routing algorithm of Ratliff and Rosenthal [17] for quickly solving an important subproblem to optimality. If more complex warehouse settings occur, this algorithm needs to be replaced either by one of the extensions (see [6, 18]) still solving the subproblem to optimality or by a heuristic (see [7]). However, we leave an evaluation of these adaptions to future research.

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• We follow the most common approach in the warehousing literature and presuppose that distance minimization is the right performance criterion for measuring picking performance. As most other parts of the picker’s time, e.g., the time to retrieve items from a shelf, are fixed once the pick list is determined, picker walking is the main factor influencing picker time. Note that in e-commerce often different priorities and varying deadlines have to be considered, which, e.g., vary whether or not privileged delivery programs have been booked by a customer. However, assigning each order a specific weight and deadline implies an unrealistically long planning horizon for picker routing, e.g., of several hours. We, however, assume a shorter planning horizon where the (urgent) orders to be processed next have already been determined on a preceding planning stage. In such a setting, if a given order set is assembled faster by reducing the travel distances, then it becomes more likely that the orders’ tight delivery schedules are met. A more detailed discussion on this matter is provided by De Koster et al. [7].

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Given such a rich routing problem, it is anything but astounding that (already a quite restricted version of) our mixed-shelves picker routing (MSPR) turns out to be a complex matter (see, e.g., [20]).

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Theorem 1. MSPR is strongly NP-hard even if only rectangular warehouses with a single depot are considered and only a single order is picked.

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The reduction is from the set cover problem, which belongs to Karp’s famous collection of 21 NP-complete problems [14]: Set cover: Given a set U of n elements and a collection S of subsets from U , is it possible to select exactly k subsets from S, such that every element in U is contained in at least one of the selected subsets. Proof. In our transformation, we introduce n SKUs, |S| aisles, and a single picking order. An aisle is added to our warehouse for each subset in S, and we store a single item of those SKUs corresponding to the elements contained in the respective subset exactly in the middle of the respective aisle. Note that items can be stored in different levels of a rack, such that the storage positions in the middle of an aisle can host multiple SKUs. The single order demands 7

ACCEPTED MANUSCRIPT a single unit of each of the n SKUs. Our single depot is located right under the leftmost aisle, and if the length of the cross aisle to enter the aisles has length m, then all our aisles have length l > 2m. The question we ask is whether a solution with a tour length smaller than (k + 1) · l exists.

As the movement along the cross aisle is smaller than reaching the middle of an aisle, we only have to choose aisles (subsets) that altogether cover our order (set U ). Thus, the one-to-one mapping between the k subsets of set cover to be selected and the k aisles to be visited in MSPR is readily available, and we abstain from a more detailed proof.

Solving the mixed-shelves picker routing problem with multiple depots

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In this section, we elaborate on how to solve MSPR. In Section 3.1, we formulate a mixedinteger program, which allows to solve MSPR with a standard solver. Then, we present heuristic solution procedures extending former work on picker routing with multiple storage locations per SKU. Daniels et al. [5] were the first treating picker routing in a mixed-shelves warehouse. Their solution approach, however, presupposes a single depot and assumes that only a single order is picked at a time. We adapt their nearest neighbor approach to our more complex problem setting in Section 3.2. Afterward (see Section 3.3), we develop a more complex pool-based solution procedure extending the work of Weidinger [20], in which picker routing with multiple storage locations per SKU in a rectangular warehouse is addressed, but, again, only a single depot and a single active picking order is assumed. In Section 3.4, a local search heuristic is presented, which can be used to further improve solutions of our construction heuristics.

Mixed-integer program

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Applying the notation summarized in Table 1 MSPR can be formulated as a mixed-integer problem (called MSPR-MIP) consisting of objective function (1) and constraints (2) to (17).

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Objective function (1) minimizes the total length of the picker tour. Constraints (2) and (3) ensure that position visits in the warehouse are properly sequenced. Inequalities (4) take care that the available stock of items in a shelf is not exceeded during order picking, and formulas (5) make sure that customer orders are satisfied. Since preemption is not allowed, constraints (6) and (7) determine that each order has only one start and one end point. Additionally, picking a specific order can only start and end at a depot (8), and SKUs can only be picked at corresponding storage positions (9). Moreover, picking items of an order is limited to the time span during which the order is active (10). The capacity of the picker’s cart is considered by constraints (11) and the auxiliary variable used to linearize the objective function is set in formulas (12). Constraints (13) and (14) state that the same position cannot be visited twice in a row and storage positions are only visited if demand is satisfied during the visit. The start of the picker tour is set by (15), and, finally, the domains of the variables are given by (16) and (17). MSPR-MIP Minimize F =

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ACCEPTED MANUSCRIPT set of SKUs (index: s) set of orders (index: o) set of storage positions set of storage positions holding SKU s set of depots set of positions, with V = P ∪ D (indices: v, v 0 ) maximum sequence position (index: i, i0 = 0, . . . , m) capacity of picking cart number of items stored at storage position v ∈ P distance between positions v and v 0 quantity of items of SKU s demanded by order o initial start position of the picker (with v 0 ∈ V ) big integer (e.g., M = maxo∈O,s∈S {aos , |O|}) binary variables: 1, if v is visited at sequence position i; 0 otherwise integer variables: quantity of items picked from storage position v at sequence position i for order o binary variables: 1, if picking order o starts at sequence position i; 0 otherwise binary variables: 1, if picking order o ends at sequence position i; 0 otherwise (auxiliary) binary variables: 1, if v is direct predecessor of v 0 at sequence position i; 0 otherwise

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S O P Ps D V m C γv dvv0 aos v0 M xvi δvio

Table 1: Notation used in the MSPR-MIP

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δvio = aos

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s kio = 1 ∀o ∈ O

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e = 1 ∀o ∈ O kio

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δvio ≤ γv

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xv0 ≤ 1

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∀v ∈ V ; i = 1, . . . , m

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v∈D

δvio ≤ M · xvi

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(kis0 o − kie0 o ) ≤ C

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∀i = 1, . . . , m

(11)

φvv0 i + 1 ≥ xv(i−1) + xv0 i ∀v, v 0 ∈ V, i = 1, . . . , m xv(i−1) + xvi ≤ 1 ∀v ∈ V, i = 1, . . . , m X xvi ≤ δvio ∀v ∈ P, i = 0, . . . , m

(12) (13)

i0 =0 o∈O

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xv 0 0 = 1 δvio ≥ 0 ∀v ∈ V ; o ∈ O; i = 0, . . . , m s e xvi , yvo , kio , kio , φvv0 i ∈ {0, 1} ∀v, v 0 ∈ V ; o ∈ O; i = 0, . . . , m

(15) (16) (17)

Nearest neighbor heuristic

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Note that our model deviates from our problem definition in Section 2, where picking at a single storage position for multiple orders is modeled by multiple successive (virtual) visits at the same position. In the model, picks for several orders during the same visit are possible due to the δ-variables. Consequently, successive visits of the same position are prohibited by constraints (13) of our model. Even if constraints (13) and (14) are optional and the same solutions are obtained without them, they reduce the search space considerably. During preliminary tests employing our small dataset (see Section 4.1), these additional constraints increased the fraction of instances solved to proven optimality from 25% to 99% when applying Gurobi with a time limit of 30 minutes. Further note that the binary x-variables can easily be eliminated and replaced by the φ-variables. However, the resulting model, although being much more handy, increases the computational time of the presented model by a factor of more than seven for our small dataset. Thus, we abstain from eliminating the x-variables.

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Our nearest neighbor (NN) heuristic is a straightforward approach to find a first feasible solution for MSPR. First, picking orders are sequenced according to increasing index, and the first C orders according to this sequence are activated. As long as not all active demands are satisfied, the nearest storage position providing items of a demanded SKU is visited. When having collected all active demands, the nearest depot is visited and processing of the next batch of C successive orders starts, until all orders are completed. A more formal definition of NN is given in Algorithm 1. Algorithm 1: Nearest neighbor heuristic 1 for (i = 1; i ≤ |O|; i += C) do 2 determine demand of active orders i, i + 1, . . . , min{i + C − 1, |O|}; 3 while (not all demand satisfied) do 4 visit nearest storage position containing a not yet satisfied demand; 5 visit nearest depot to activate new picking orders; 6 return the generated tour;

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ACCEPTED MANUSCRIPT Note that batching, i.e., deciding which orders should be concurrently processed on the cart, is a complex matter especially in a multi-depot warehouse. Picking two orders jointly in a batch may be advantageous when starting from a specific depot but may turn out as a pretty bad choice if started from another access point. We tested several simple decision rules for order batching, but none of them turned out significantly better than our most basic index-based approach.

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Pool-based construction heuristic

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Real-world warehouses of online retailers contain thousands of SKUs (see Section 1), so that we aim at a fast procedure being able to handle a vast assortment of SKUs. Consequently, we apply a construction heuristic, which extends partial solutions in a stepwise manner. To gain promising solution values with small optimality gaps, we generate many partial solutions in parallel in each step and only add the most promising among them to a central pool of solutions. Furthermore, we apply the efficient algorithm of Ratliff and Rosenthal [17] as a subprocedure to determine the optimal circular picker tour in a rectangular warehouse once the orders to be processed and the storage positions to be visited in between two successive visits at a depot are selected. In the following, we sketch the basic idea of our heuristic, which mainly consists of two hierarchical steps, before we elaborate each step in detail:

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• On the upper level, a pool of multiple partial solutions is managed. Starting with an empty tour where the picker is located in his/her initial position, partial solutions are selected from the pool and extended in a stepwise manner. In each step of the lower level, we start at the picker’s current depot, select a small subset of yet unscheduled orders, schedule them, and end at the succeeding depot where finished orders are handed over. This is done λ · µ times by randomly selecting and extending a partial solution from the pool, where λ and µ are two parameters defining the search effort. Among all existing and new partial solutions, the λ most promising ones are kept in the pool. Once all partial solutions are completed and all orders are scheduled, the best complete solution of the pool is returned.

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• The extension of partial solutions is handled on the lower level of our heuristic. We receive a partial solution of positions already visited and orders already processed. This partial solution is extended up to a specific number of subsequent orders, which is also set by the upper level. To do so, we first select the (small) subset of orders to be scheduled next. Note that some orders may not be completed, so that the picking cart need not be completely empty when extending a partial solution. For the active orders, we select the respective storage positions to satisfy the demanded SKUs and determine the best picker tour along these selected positions by applying the dynamic programming procedure of Ratliff and Rosenthal [17]. Note that this procedure derives an optimal circular tour and, thus, always returns the picker to the initial depot. In our problem, however, this needs not be the best choice, so that we do not necessarily take over the complete solution but check whether a premature shortcut to another depot seems advisable. Based on the work of Weidinger [20], we apply three different priority rules for selecting the storage positions to be visited, so that multiple partial solutions are generated among which one is returned to the upper level, chosen with a probability based on the solution quality. 11

ACCEPTED MANUSCRIPT Note that by varying the steering parameters λ and µ, our pool-based heuristic can easily be adjusted to problem instances of varying size. In warehouses of the largest size, the dynamic programming approach of Ratliff and Rosenthal [17] applied as a subprocedure can be substituted by a faster heuristic, e.g., a nearest neighbor approach or some other routing heuristic (see [7]), to still get acceptable computational times. Further note that detailed tests on tuning the steering parameters λ and µ are reported in Section 4.2.

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In the following, we describe both levels in more detail and start with Algorithm 2 of the upper level. In this step, a pool of partial solutions is managed. By randomly selecting partial solutions from the pool and handing them over to the lower level, λ · µ new partial solutions are generated and among all existing and new partial solutions, the λ most promising ones are kept in the pool. Pool management is defined in detail by Algorithm 2.

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Algorithm 2: Pool management on the upper level of the pool-based procedure 1 determine UB F by solving the nearest neighbor heuristic; 0 2 define pool, pool ; // used to save (new) partial solutions 3 pool.Add(new PartialSolution(null)); // picker at start position 4 for (i = C; i ≤ |O|; i += max{1, min{C, |O| − i}}) do 5 pool0 .Add(NearestNeighbor(i)); // partial nearest neighbor solution 6 for (j = 1; j < λ · µ; j++) do 7 select partial solution π ˜ from pool randomly; 8 while (˜ π .OrdersP rocessed < i) do 9 π ˜ += new PartialSolution(˜ π ); // generate new partial solution 10 if (˜ π .T ourLength < UB F ) then 11 pool0 .Add(˜ π ); 12 foreach (PartialSolution π ˜ in pool ∪ pool0 ) do π ˜ .T ourLength .T ourLength 13 rate π ˜ by π˜ .OrdersCompleted (tie-breaker: π˜π˜.ItemsP ); icked 14 store exclusively the λ partial solutions with lowest rates from pool ∪ pool0 in pool; 15 return best solution in pool;

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At first, an upper bound UB F is determined by executing the nearest neighbor heuristic (see Section 3.2; line 1). Afterward, the pool is initialized with an empty partial solution, where the picker has not picked any orders yet and is located at the starting position of the tour. Additionally, pool0 , storing newly generated partial solutions, is defined (lines 2-3).

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In each of the following iterations, λ · µ new partial solutions are generated, which differ from the partial solutions generated in the previous iteration by C less orders not yet processed. One of these partial solutions is generated by partially executing the nearest neighbor approach. The remaining (λ · µ) − 1 solutions are constructed by extending solutions from the pool (lines 5-9). A new partial solution obtained from the lower level is added to the temporary pool pool0 , if the length of the partial tour is lower than upper bound UB F (lines 10-11). In the second step of each iteration, all partial solutions (pool ∪ pool0 ) are ranked by the length of the partial solution divided by the number of orders already fulfilled. As orders can be processed partially, the distance divided by the number of items picked acts as a tie-breaker. The λ partial solutions with the lowest ratios are kept in the pool and provide the basis for the next iterations (lines 12-14). In the last iteration, partial solutions are completed. Among all complete and feasible solutions the one with the lowest objective value is returned (line 15). 12

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Algorithm 3: Extending a partial solution on the lower level of the pool-based procedure ˜ C − |O|}; ˇ 1 c = randomly selected integer between 1 and min{|O|, ˆ = select c orders of O; ˜ 2 O ˆ and O; ˇ 3 determine (remaining) demand of O 4 maxLength = 0; List l = new List(); // the new subtours are stored in l 5 foreach (depot vd ) do 6 foreach (penalty rule Φ) do 7 r = randomly drawn double value between 0.0 and 1.0; 2 ) then 8 if (vd is picker position ∨ r < |D| 9 π ˘ = GenerateSubtour(demand, vd , Φ); 10 maxLength = max{maxLength, π ˘ .Length}; 11 l.Add(˘ π ); 12 foreach (˘ π in l) do π .Length+1 13 compute probability for selecting subtour π ˘ by P (˘ π ) = P 0 maxLength−˘ ; (maxLength−˘ π 0 .Length+1) π ˘ ∈l

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˜ and O; ˇ update π ˘ ∗ , P˜ , O, return π ˘∗;

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select subtour π ˘ ∗ based on probabilities; determine lowest position v ∗ of π ˘ ∗ at which an order is completely picked; // consider both possible directions of the tour and select based on distances r = randomly drawn double value between 0.0 and 1.0; if (r < 0.5) then stop subtour at storage position where order is completed (v ∗ unchanged); else proceed subtour until the picker comes along a depot (update v ∗ );

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Next, we detail how partial solutions are extended on the lower level by Algorithm 3. The ˜ of orders that are not picked yet and the set P˜ of storage positions, algorithm receives the set O ˇ whose initial inventory is reduced according to the already picked items. Additionally, set O contains orders, which are still active but not yet completed. Their demands are partly satisfied.

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ˇ are to be picked in parallel during the At first, we determine how many orders (c + |O|) extension of the partial solution. We choose c with equal probabilities from integers 1 to ˜ C − |O|}. ˇ min{|O|, In doing so, the number of partly satisfied orders has to be considered, ˆ of orders because they reduce the remaining cart capacity. Based on this first decision, a set O ˆ ˆ ˜ ˜ with to be picked with |O| = c and O ⊆ O is generated by randomly selecting orders of O ˇ∪O ˆ can be equal probabilities. Having these two sets, the (remaining) demand of orders O determined (lines 1-3). This demand is addressed partly or fully during the next subtour. In this context a subtour is defined as a sequence of stopovers starting and ending at a depot and visiting pick positions in between solely. Making use of the notation employed in Section 2, a partial solution π ˘ = ((v0 , δ0 , o0 ), (v1 , δ1 , o1 ), . . . , (vm , δm , om )) is a subtour if v0 , vm ∈ D and vi ∈ P ∀i = 1, . . . , (m − 1) holds. To determine a set of subtours, then, storage positions to be visited are selected. Afterwards, the shortest circular tour between these storage positions is determined by using the efficient procedure of Ratliff and Rosenthal [17]. Note that this procedure derives the optimal circular 13

ACCEPTED MANUSCRIPT tour and, thus, always returns the picker to the initial depot. In our problem, however, this may not the best choice, so that we do not necessarily take over the complete solution but check whether a premature shortcut to a depot seems advisable (see below). We apply three different penalty values to select the storage position, which base on the approximated distances to be additionally covered if the respective position is chosen:

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For all three rules, SKUs are processed in non-descending order by the quantity of storage positions available per SKU. Among all storage positions not depleted yet and belonging to the currently considered SKU, the one with the lowest penalty value is selected until the demand of the considered SKU is satisfied. To avoid visiting more positions than required, the selected positions are processed in non-ascending order of their penalty values, in a second step, and deselected if the demand can still be satisfied afterwards. The three approaches differ only by the computation of the penalty values. The SinglePosition rule uses the distance starting from a given depot vd to the considered storage position v to approximate the additional distance to be covered when visiting v. The two other rules MinMax and MinMin employ the maximum/minimum distance to all storage positions selected until now. The set of selected storage positions is initialized containing the predefined depot vd in both rules.

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All three rules depend on a given start depot. In a first step, we create subtours based on the depot the picker is currently located at by applying all three rules. All resulting subtours are added to set l of potential subtours. However, it can be advantageous to change the depot of the picker first before continuing the picking process. To enable our procedure to do so, each of the remaining depots is used as the start point of an additional subtour. Each of these 2 } for each rule. Since we additional subtours is added to l with a probability of min{1, |D| 2 have three rules, the mean value of subtours generated is 3 · (1 + (|D| − 1) · |D| ) ≈ 9. Among these subtours, we choose one with a probability based on the tour length needed to satisfy the total demand. The longer a subtour, the lower the chances to pick it (line 13).

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Having selected a subtour, we check in both directions at which point of the tour at least one order is completely picked and choose the direction with a shorter distance to this point. In a second step, we cut the subtour at that very position with a probability of 50%. In the other half of the cases, we cut the tour when the picker comes along a depot anyway. Now that we have the pick positions of the subtour, we choose the depot to be visited afterward with probabilities based on the distance to be covered from the last storage position visited to reach the respective depot in the same way used to select the subtour (line 21). Finally the ˜ and O ˇ are updated and the generated subtour is returned (lines 22-23). parameters P˜ , O,

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Improvement heuristic

Once a feasible solution is generated, whether by the nearest neighbor heuristic or our poolbased approach, an iterative search procedure can be employed to further improve the solution. The procedure repeatedly unfixes parts of the tour and tries to connect both loose ends of the tour by a shorter subtour still satisfying the predefined demand. In doing so, the 14

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Algorithm 4: Heuristic improvement procedure 1 for (i = 0; i < κ; i++) do 2 reduce inventory γ according to tour π; 3 π 0 = π; 4 foreach subtour π ˘ in π 0 in increasing order do 5 increase inventory γ according to subtour π ˘; 0 6 initialize new subtour π ˘; 7 foreach SKU s demanded in π ˘ in randomized sequence do 8 while demand of s not satisfied do 9 determine additional distance for each storage position v ∈ Ps at each potential position within π ˘0; 10 insert best storage position at best position into π ˘ 0 and reduce inventory and demand; 11 while capacity C not fully utilized during subtour π ˘ 0 do 12 foreach order o in randomized sequence do 13 if any demand of o satisfiable without additional effort then 14 plan new picking job substituting last possible job of order o; 15 replace π ˘ by π ˘ 0 in π 0 ; 16 if (π 0 .Length < π.Length) then 17 π = π0; 18 return improved solution;

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subtour is generated making use of a greedy insertion algorithm based on a randomly generated sequence of demanded SKUs. Additionally, the tour is tried to be shortened by preponing order processing intervals. Note that previous and subsequent pick positions have to be considered when generating a new subtour to obtain a feasible solution. In the following, we will detail the single steps of the iterative improvement heuristic (see Algorithm 4).

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First, the inventory of each storage position is reduced by the total number of items picked from that very position during the initial tour (line 2). In this way, improving a subtour does not result in a shortage of items during a downstream picking stop. The new solution π 0 is then generated on the basis of the current solution π (line 3). During the improvement phase, subtours are unfixed and improved in a subsequent manner. Note that the definition of a subtour in Section 3.3 still holds in this context. The improvement procedure considers subtours in increasing order (line 4). The inventory of the warehouse is increased according to the picking jobs during the unfixed subtour π ˘ (line 5) and a greedy insertion logic is applied to replan the unfixed subtour in a successive manner. Starting with a subtour π ˘ 0 consisting only of the fixed start point vstart and end point vend (˘ π 0 = {(vstart , 0, −1), (vend , 0, −1)}), all currently demanded SKUs are processed in a randomly determined sequence during the repair phase (lines 6-7). For each SKU all combinations of storage positions and sequence positions within the newly generated subtour π ˘ 0 are compared and the combination leading to the shortest additional distance is added to the subtour as long as the demand of the current SKU is not satisfied, before the next SKU is processed (lines 8-10). Afterwards, it is checked if full picker capacity is utilized during the currently unfixed subtour (line 11). If not, the orders starting right after the currently unfixed subtour are evaluated in a randomized sequence and new picking jobs are added to subtour π ˘ 0 if the corresponding position could be visited without additional effort (lines 12-14). Newly planned picking jobs of 15

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preponed orders substitute the latest picking job of the same SKU of that very order. In this way, there is a high probability that the order can both be started and completed earlier. After the preponement step, subtour π ˘ is substituted by π ˘ 0 in the new tour π 0 (line 15). Note that even if the newly generated subtour is longer than π ˘ , a better solution can be obtained (e.g., 0 if an order was preponed). For this reason, π ˘ is accepted for sure, whereas an evaluation is executed once all subtours are processed. The new tour π 0 , obtained by the single improvement runs, is accepted as the current solution, if it is the new best known solution (lines 16-17). This procedure is repeated for κ iterations, such that κ is a parameter determining the search effort of the greedy improvement heuristic (line 1). Finally, the best solution found is returned (line 18).

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Performance of solution procedures

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In this section, the performance of our solution procedures is tested. As no established testbed is available for our routing problem, instance generation is detailed in Section 4.1. Afterward, parameter tuning (Section 4.2) and performance results of our solution procedures are presented (Section 4.3).

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All computations are executed on a 64-bit PC with an Intel Core i7-6700K CPU (4 x 4.0 GHz), 64 GB main memory, and Windows 7 Enterprise. The procedures are implemented using C# (Visual Studio 2017), and the off-the-shelf solver Gurobi (version 7.5.1) is applied for solving the MIP models.

Instance generation

description number of aisles length of aisles [in #storage positions] number of SKUs number of depots number of orders number of items per order max. number of items stored per storage position capacity of picking cart

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symbol w h |S| |D| |O| omax γmax C

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We generate two different sizes of test instances. The ones in the small dataset are still solvable to optimality by a standard solver, while the ones labeled large represent instances of real-world size. The parameters handed over to our data generator are presented in Table 2. The procedure of instance generation is summarized in the following. small 2 5 {3, 6, 9} {2, 4} 3 {1, 2} {1, 2} 2

large 40 150 {2000, 3500, 5000} {26, 80} 10 {1, 2} {1, 4} 4

Table 2: Parameter values for instance generation Warehouse layout: The warehouse is equipped with parallel racks arranged along w aisles, which lead into the cross aisles at the front and rear of the warehouse. Storage positions within a rack are quadratic and the depth of a rack is assumed to equal the width of an aisle (see Figure 4). Therefore, we can measure distances via the resulting grid of squares. Two 16

ACCEPTED MANUSCRIPT depots are located at both cross aisles at the end of each (normal) aisle. Whenever |D| < 2·w, not any aisle can house depots at both ends. They are skipped such that depots are located in an equidistant manner along the cross aisles.

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Storage assignment: As storage turnover often follows the Pareto principle (see, e.g., [7]), we subdivide SKUs into 20% A-, 30% B-, and 50% C-products. First, between 1 and γmax items of each SKU are assigned randomly to one storage position per SKU to assure that each product can be found in the warehouse at least once. The remaining storage positions are assigned with a chance of 80% to an A-product, with a chance of 15% to a B-product, and with a chance of 5% to a C-product. Once the class is selected, the SKU to be assigned is drawn from the set of SKUs belonging to that class with equal probabilities. Finally, between 1 and γmax items of the chosen SKU are assigned to the respective storage position.

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Generation of pick lists: Pick lists are generated by randomly selecting |O| · omax items stored in the warehouse. In this way, we can ensure feasible problem instances because the quantity of demanded items cannot exceed the number of those available. Note that the cart capacity C of order bins within the mixed-shelves warehouses we visited varied between 2 and 4 bins, so that these values are realistic. Further note that the average number of items per order ranges between 1 and 2 items (see Section 1), so that parameter omax is realistic for all mixed-shelves warehouses where no batching is applied. We repeat this procedure 20 times for each parameter setting, so that we receive 480 small instances and another 480 large instances.

Parameter tuning for the heuristic procedures

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This section explores the impact of the steering parameters of both, the pool-based procedure (Section 3.3) as well as the improvement heuristic (Section 3.4). Recall that parameters λ and µ of the pool-based procedure define the number of partial solutions held in the pool and the relative number of new solutions generated, respectively. Specifically, we combine parameter values λ ∈ {10, 15, 20, 30} and µ ∈ {1, 3, 5, 10} in a full-factorial manner and solve all small and large data instances to evaluate the impact of these parameters. The results are summarized in Figure 2, where we relate objective value (’F ’) and solution time (’sec’) to the varying values of the steering parameters.

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The results reveal that an increasing λ value leads to a more extensive search and, therefore, lower objective values and a higher computation time. This holds true for both datasets. The analogous effect is recorded for µ in most cases. However, an interesting effect can be observed for the small dataset. Choosing a µ value of 10, up to 300 new partial solutions are generated in each iteration. As the size of the instances is restricted, however, many of these partial solutions are similar. In consequence, the pool becomes degenerated and further search is mainly based on the same partial solutions. This leads to a nearly constant solution quality irrespective of the λ value. The same effect, although at a much larger µ value, is assumed for the large dataset.

The search effort of the improvement heuristic introduced in Section 3.4 is specified by steering parameter κ. To get insight into the impact of this parameter, we vary parameter values κ ∈ {0, 1, 2, 3, 5, 10, 15, 20, 30, 40, 50, 100, 150}, where the start solution is determined making use of the nearest neighbor procedure (NN) and the pool-based procedure (using steering parameters µ = 5, λ = 20 and µ = 10, λ = 30), respectively. Results are presented in Figure 3. 17

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Figure 2: Influence of parameters µ and λ on solution quality (a, b) and time consumption (c, d) for small (a, c) and large (b, d) instances.

Computational performance

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The local search procedure is able to improve average solution values of both procedures and all parameter combinations investigated. Naturally, a higher number of iterations leads to a better solution quality. The potential of the improvement procedure, however, is depending on the solution quality of the start solution. The solutions of the pool-based procedure can only be improved marginally for the small dataset, while the nearest neighbor heuristic provides plenty potential for further improvement. The time consumption of the improvement procedure is hardly measurable. Only for the large dataset a slight linear increase exists. Even when executing κ = 150 iterations, the total computation time rises only by about 0.15 seconds, irrespective of the start solution.

This section is dedicated to the performance of our solution methods. Specifically, we compare the performance of standard solver Gurobi solving MSPR-MIP, nearest neighbor heuristic (NN), pool-based procedure (pool), and both of the latter two heuristics executed with the additional improvement procedure via local search (NN+ , pool+ ). Note that Gurobi solving MSPR-MIP is given a maximum solution time of 30 minutes. First, we report the solution performance for the small dataset. The results are summarized in Table 3. We report the number of optimal solution found (’#opt’), the average relative gap to the optimal solution (’gap’), and the solution time (’sec’) in CPU-seconds for each procedure. Our small instances have only 20 storage positions and between 3 and 6 items to be picked in total. Nonetheless, standard solver Gurobi is not able to find all optimal solutions, i.e., 18

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Figure 3: Influence of parameter κ on solution quality (a, b) and time consumption (c, d) for small (a, c) and large (b, d) instances. pool (λ = 30, µ = 3) #opt/gap/sec 20/0.00%/0.01 20/0.00%/0.01 20/0.00%/0.01 20/0.00%/0.01 20/0.00%/0.01 20/0.00%/0.01 19/1.25%/0.01 20/0.00%/0.01 19/0.91%/0.01 18/1.96%/0.01 19/1.25%/0.01 20/0.00%/0.01 20/0.00%/0.01 20/0.00%/0.01 20/0.00%/0.01 20/0.00%/0.01 17/2.83%/0.01 18/1.34%/0.01 18/1.68%/0.01 20/0.00%/0.01 16/1.68%/0.01 18/1.18%/0.01 17/1.11%/0.01 19/0.50%/0.01 458/0.65%/0.01

NN+ (κ = 15) #opt/gap/sec 14/9.05%/<0.001 14/8.75%/<0.001 16/7.62%/<0.001 14/8.75%/<0.001 8/18.93%/<0.001 13/8.51%/<0.001 12/11.22%/<0.001 14/14.40%/<0.001 7/19.15%/<0.001 8/20.63%/<0.001 9/14.56%/<0.001 10/19.10%/<0.001 0/34.46%/<0.001 3/33.86%/<0.001 5/23.69%/<0.001 3/27.74%/<0.001 1/34.23%/<0.001 2/31.84%/<0.001 3/31.13%/<0.001 6/23.20%/<0.001 4/34.88%/<0.001 4/28.87%/<0.001 1/32.50%/<0.001 4/32.68%/<0.001 175/22.00%/<0.001

NN #opt/gap/sec 14/9.05%/<0.001 13/9.75%/<0.001 15/8.53%/<0.001 13/9.46%/<0.001 6/23.74%/<0.001 11/11.49%/<0.001 10/13.04%/<0.001 13/16.55%/<0.001 2/35.73%/<0.001 6/23.31%/<0.001 9/16.14%/<0.001 8/24.93%/<0.001 0/38.10%/<0.001 2/36.63%/<0.001 5/23.69%/<0.001 2/29.17%/<0.001 1/39.27%/<0.001 2/34.70%/<0.001 3/39.52%/<0.001 4/27.64%/<0.001 4/42.76%/<0.001 3/35.21%/<0.001 1/42.60%/<0.001 2/38.75%/<0.001 149/26.14%/<0.001

Table 3: Solution performance for the small dataset

477 of 480 instances are solved to proven optimality. The main impact factor on Gurobi’s performance is the length of the pick list. If each order demands just omax = 1 item, Gurobi finds optimal solutions for all instances in about 1.92 seconds on average. However, if the order size increases and omax = 2 items are demanded, the average computation time increases to 193.62 seconds and 98.8% of the instances are solved to proven optimality in the given time frame. Our improved pool-based heuristic pool+ (executed with parameter values λ = 30, 19

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NN+ (κ = 150) #best/gap to best/sec 8/9.42%/0.08 1/21.68%/0.08 9/9.37%/0.08 2/19.57%/0.07 7/8.27%/0.11 2/21.77%/0.11 3/13.53%/0.11 2/22.60%/0.11 6/10.03%/0.15 0/20.12%/0.15 3/14.45%/0.15 0/26.74%/0.15 13/3.14%/0.07 3/10.34%/0.07 7/7.33%/0.08 6/9.95%/0.08 10/4.35%/0.12 6/5.81%/0.12 10/5.90%/0.12 3/11.91%/0.12 8/5.41%/0.15 4/5.84%/0.15 11/4.23%/0.15 4/10.36%/0.15 128/11.76%/0.11

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#best/gap to best/sec 2/18.98%/<0.001 0/34.61%/<0.001 0/22.27%/<0.001 0/30.03%/<0.001 1/14.59%/<0.001 0/30.62%/<0.001 0/24.11%/<0.001 0/30.56%/<0.001 0/21.37%/<0.001 0/25.37%/<0.001 1/18.77%/<0.001 0/33.87%/<0.001 0/22.86%/<0.001 0/32.24%/<0.001 0/35.83%/<0.001 0/38.14%/<0.001 0/20.64%/<0.001 0/24.99%/<0.001 0/24.71%/<0.001 0/34.64%/<0.001 0/20.79%/<0.001 0/24.32%/<0.001 0/16.36%/<0.001 0/26.58%/<0.001 4/26.14%/<0.001

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pool (λ = 30, µ = 10) #best/gap to best/sec 3/4.15%/19.53 4/5.52%/14.71 2/6.67%/19.32 1/5.02%/14.50 5/3.22%/21.54 5/3.18%/16.49 1/4.32%/21.77 7/3.60%/15.99 5/3.26%/22.98 5/2.88%/17.51 5/2.22%/24.12 5/3.05%/17.38 2/10.00%/20.64 0/8.71%/16.01 0/9.94%/20.50 0/11.32%/15.99 1/7.18%/23.51 1/7.58%/17.41 0/5.71%/23.39 1/5.80%/17.37 0/5.24%/25.06 2/5.19%/19.07 1/6.30%/25.91 3/4.05%/18.40 59/5.59%/19.54

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pool+ (λ = 30, µ = 10, κ = 150) #best/gap to best/sec 18/0.77%/19.60 19/0.53%/14.78 13/2.28%/19.39 19/0.17%/14.57 18/0.38%/21.68 18/0.10%/16.62 20/0.00%/21.91 19/0.07%/16.12 18/0.46%/23.18 20/0.00%/17.69 18/0.25%/24.31 20/0.00%/17.56 11/4.61%/20.74 17/1.03%/16.09 15/1.91%/20.60 14/1.58%/16.07 11/2.86%/23.70 15/1.31%/17.58 11/1.86%/23.59 17/0.53%/17.54 14/2.23%/25.35 16/1.94%/19.28 13/1.99%/26.17 17/0.53%/18.63 391/1.14%/19.70

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µ = 3, and κ = 150) finds the proven optimum in 96.6% of all instances for which the optimal objective value is available. The average gap to the optimal solutions is merely about 0.55%, and in spite of this very convincing performance, the average solution time ranges around onehundredth of a second only. Compared to the pool-based procedure without improvement, the time consumption does not vary in a measurable way, the solution quality, however, increases slightly due to the improvement heuristic applying local search. The nearest neighbor approach is even faster; its solution time is barely measurable. The solution quality, though, is not that good. The average gap to the optimum amounts to 26.14%. Even when the solution is further improved making use of the local search procedure, the average optimality gap still adds up to 22.0%.

Table 4: Solution performance for the large dataset

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The results for the large dataset are reported in Table 4. Unfortunately, standard solver Gurobi is not able to find feasible solutions for any of the instances. In spite of 64GB of main memory, it always returned a buffer overflow. Therefore, we can only compare the results of our four heuristics. Again, the pool-based(+ ) approach clearly outperforms the NN(+ ) heuristic. The pool+ procedure is able to find the best known solution for 391 instances (81.5%). However, in 89 cases, NN+ finds a solution of higher quality than pool+ . The solutions of NN+ , though, are only slightly better, resulting in a mean gap to the best solution of about 1% for the pool+ approach. The gap of NN+ amounts to more than 11%. Solving the large dataset, the additional improvement step shows a higher effect, so that both (unimproved) construction heuristics show significant gaps compared to their improved version. While the computational time of NN(+ ) is still well below one fifth of a second, the CPU time of our pool-based(+ ) approach amounts to approximately 20 seconds. This still seems acceptable in practical scenarios, so that we conclude that our pool-based heuristic with improvement (pool+ ) is well suited for solving MSPR of real-world size.

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Managerial aspects

Compared to traditional storage policies, where unit loads are kept together and are jointly stored in specific shelves, mixed-shelves warehouses have two main drawbacks, which give rise to the three research questions to be answered in this section: 20

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• To spread items of the same SKU all over the shelves, the unit loads that the SKUs arrive on need to be broken down and logistics workers have to bring the individual items to diverse storage positions. Thus, to harvest the advantages of mixed shelves during order picking, additional effort when replenishing the racks arises [21]. Therefore, we are interested in a performance comparison of two different warehouse settings: (a) a mixedshelves warehouse, and (b) a traditional warehouse applying dedicated storage, where all items of a SKU are assigned to a joint and unique storage position. We generate picking orders and pick the same set of orders from both settings. By comparing both results, we can quantify the gains of scattered storage in terms of reduced picker travel. Practitioners can then weigh up these gains against the additional effort to be spent during the put-away process of mixed-shelves storage.

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• Mixed-shelves storage is especially suited for small-sized orders demanding just a few items per order line. As soon as an occasional larger order with multiple items occurs, the stock in a single shelf may not be sufficient, and the picker has to visit multiple storage positions in order to gather enough items. Thus, large-sized orders increase the picking effort under mixed-shelves storage. Therefore, it is interesting to know at what apportionment of large- and small-sized orders the turning point between dedicated and mixed-shelves storage is reached.

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• Today, many retailers aim at an omni-channel sales strategy, so that often (large-sized) orders of physical brick-and-mortar stores and (small-sized) customer orders placed via a webshop need to be fulfilled from the same warehouse [1]. In this case, depending on the apportionment of large- and small-sized orders, either dedicated or mixed-shelves storage will show advantageous. However, there is also a third alternative, which is to subdivide the warehouse into a dedicated storage area, where items of the same SKU are kept together and batch orders of brick-and-mortar stores are processed, and a mixed-shelves section dedicated to small-sized online orders. Therefore, another aim of this section is to benchmark a split-warehouse strategy against the other two alternatives.

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To answer these research questions, our computational experiment is designed as follows. We generate large instances with 2000 SKUs. We assume that, in total, 120 items, i.e., 1% of the total stock, have to be picked in each instance. The items are demanded by two different kinds of orders. There are small-sized online orders each demanding two items of randomly determined SKUs and large-sized batch orders requiring 10 items of two SKUs each. These SKUs are randomly chosen among all those SKUs having a minimum stock level of 20 items, so that we are sure to generate feasible instances. Within our test instances, we vary the apportionment between online and batch orders. When picking a total of 120 items, up to 6 batch orders are possible. We vary the fraction of items belonging to online orders as follows: 0%, 17%, 33%, 50%, 67%, 83%, and 100%. Depot settings as shown in Table 2 are used and we generate 240 instances per parameter setting, so that in total 3360 instances, i.e., 480 instances per data point (item fraction of online orders), are generated. All resulting instances are solved with our improved pool-based heuristic (executed with parameter values λ = 30, µ = 10, and κ = 150). Irrespective of the storage assignment strategy, the setting of the warehouse is identical to the one described in Section 4.1. Depots are spread all over the warehouse, and two depots can always be found on both ends of the same aisle. However, we additionally have to process batch orders now. As those orders are often packed on separate stations in business practice, 21

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we introduce a special batch order depot on the upper left corner of the warehouse, where each batch order picker tour starts and ends. Due to the higher volume of these orders, picker capacity amounts to C = 1 for this kind of order. Picker capacity for online orders still amounts to C = 4. Having this basic setting, the orders are to be satisfied by three alternative warehouse settings, which are schematically depicted (in reduced size) by Figure 4 and explained in the following:

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Figure 4: Warehouse setup for (a) dedicated, (b) mixed-shelves, and (c) split storage

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(a) Dedicated storage: All items of the same SKU are stored at clustered positions within the warehouse. A-class SKUs are stored in the aisles closest to the depot; C-class SKUs farthest away.

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(b) Mixed-shelves storage: The items of the SKUs are randomly spread all over the warehouse.

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(c) Split storage: The warehouse is subdivided into two parts. On the left side, there is the batch area with the depot in the upper left corner and dedicated storage applied. In the remaining part of the warehouse, mixed-shelves storage is applied. The size of the scattered section as well as the relative quantity of items per SKU stored there corresponds to the relative amount of items picked for online orders. Note that anticipating the fraction of online orders is an aggregated forecast, which is assumed to be possible with the required precision.

The results of these tests are depicted in Figure 5. Specifically, we report the total picking distance (left) and the relative increase of picking distance over split storage (right) for all three warehouse settings and relate these results to the fraction of items demanded by online orders. The following answers to our research questions can be derived from these graphs: 22

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0 20 40 60 80 100 percentage of items demanded by online orders [%]

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Figure 5: Performance benchmark for the three warehouse settings depending on the fraction of items demanded by online orders

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• If we have only online orders, i.e., the percentage of items demanded by online orders is 100%, mixed-shelves storage leads to much better picking performance than dedicated storage; the total travel distance of the picker more than halves. This considerable advantage seems to outweigh the additional effort of the put-away process in order to realize mixed-shelves storage. This assessment is supported by the fact that only order picking is the time-critical process determining the customers’ waiting times, whereas replenishment has no direct impact on customer satisfaction and is, thus, not (that) time-critical (see Section 1.1).

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• The turning point between mixed-shelves storage and dedicated storage is already reached once 33% of all items are demanded by online orders. At a larger (smaller) fraction mixed-shelves (dedicated) storage becomes preferable. Evidently, dedicated storage considerably suffers when having to process small-sized online orders, so that the turning point is reached astonishingly early. Dedicated storage requires long walks between the storage positions of online orders, whereas mixed-shelves enlarge the probability of close-by storage locations of requested items. On the other hand, our results clearly indicate that applying mixed-shelves storage when less than 33% of items stem from online orders is not recommendable. If no online orders occur, dedicated storage reduces picker travel by a remarkable 189%.

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• Finally, we benchmark mixed-shelves storage and dedicated storage with the compromise strategy denoted as split-warehouse storage. The maximum advantage of split-warehouse storage, i.e., 39.7%, is reached at the turning point where 33% of all items are ordered via the online channel. We have to keep in mind, however, that the price for this advantage is organizational overhead for managing and operating two separate warehouse areas. Therefore, switching to split-warehouse storage will not pay if the gains in terms of reduced picker travel are too small, so that the corridor where split-warehouse storage should be considered as a serious alternative is comparatively small. For instance, the corridor where picker travel is reduced by about 25% just ranges somewhere between 25% and 50% of all items ordered online.

Note that another managerial aspect is considered in the appendix, where we explore the impact of an increasing capacity of the picking cart on the picker travel. The results presented 23

ACCEPTED MANUSCRIPT there show that the positive effect of additional cart capacity quickly diminishes. Small and agile carts with a capacity for up to a handful of orders lead to considerable reductions of picker travel. Larger and bulky carts that would considerably increase the ergonomic burden for the pickers only lead to small marginal reductions and, thus, need not be considered.

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Conclusion

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This paper defines the picker-routing problem for the mixed-shelves warehouses applied by many online retailers. In such warehouses, unit loads are purposefully broken down, and single items of the same SKU are spread all over the shelves. During picker routing, this gives the additional flexibility of having alternative shelves that a specific SKU can be retrieved from. Furthermore, our picker-routing problem considers (a) a cart pushed by the picker that allows to assemble multiple orders concurrently, and (b) multiple access points to the central conveyor system where completed orders are handed over. The resulting optimization problem is defined, computational complexity is proven, and a suited heuristic solution procedure is developed. This procedure is able to handle the thousands of SKUs relevant in real-world warehouses. Furthermore, we address managerial aspects in our computational studies. Specifically, we compare mixed-shelves storage with dedicated storage and a split warehouse where both policies are applied in parallel in separate areas. We benchmark these warehouse settings by varying the apportionment between small-sized online orders and large-sized orders, e.g., posed by brick-and-mortar stores, additionally serviced under an omni-channel sales strategy. Our results show which storage policy is superior under which circumstances.

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This paper considers only the single-picker case. Future research should extend our pickerrouting problem to a multiple-picker environment. In real-world warehouses, multiple pickers are employed in parallel, who influence each other with regard to the shelves they access and the inventory they retrieve from there. We assume that the coordination of pickers has already been accomplished on a superordinate planning level. How exactly our single-picker case can be applied in a larger solution framework that also includes picker coordination remains up to future research.

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References [1] Agatz, N.A.; Fleischmann, M.; Van Nunen, J.A. (2008): E-fulfillment and multi-channel distribution – A review. European Journal of Operational Research 187, 339-356. [2] Akkerman, R.; Farahani, P.; Grunow, M. (2010): Quality, safety and sustainability in food distribution: A review of quantitative operations management approaches and challenges. OR Spectrum 32, 863-904.

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[3] Bartholdi III, J.J.; Hackman, S.T. (2014): Warehouse & distribution science. Release 0.96. Supply Chain and Logistics Institute. [4] Brynjolfsson, E.; Hu, Y.J.; Smith, M.D. (2003): Consumer surplus in the digital economy: Estimating the value of increased product variety at online booksellers. Management Science 49, 1580-1596. [5] Daniels, R.L.; Rummel, J.L.; Schantz, R. (1998): A model for warehouse order picking. European Journal of Operational Research 105, 1-17.

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[6] de Koster, R.; van der Poort, E.S. (1998): Routing orderpickers in a warehouse: A comparison between optimal and heuristic solutions. IIE Transactions 30, 469-480. [7] de Koster, R.; Le-Duc, T.; Roodbergen, K.J. (2007): Design and control of warehouse order picking: A literature review. European Journal of Operational Research 182, 481-501.

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[8] Gallien, J.; Weber, T. (2010): To wave or not to wave? Order release policies for warehouses with an automated sorter. Manufacturing & Service Operations Management 12, 642-662.

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[9] Ghiani, G.; Improta, G. (2000): An efficient transformation of the generalized vehicle routing problem. European Journal of Operational Research 122, 11-17.

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[10] Gu, J.; Goetschalckx, M.; McGinnis, L.F. (2007): Research on warehouse operation: A comprehensive review. European Journal of Operational Research 177, 1-21.

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[11] Gu, J.; Goetschalckx, M.; McGinnis, L.F. (2010): Research on warehouse design and performance evaluation: A comprehensive review. European Journal of Operational Research 203, 539-549.

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[12] Hong, S.; Johnson, A.L.; Peters, B.A. (2012): Batch picking in narrow-aisle order picking systems with consideration for picker blocking. European Journal of Operational Research 221, 557-570. [13] Johnson, E.; Meller, R.D. (2002): Performance analysis of split-case sorting systems. Manufacturing & Service Operations Management 4, 258-274. [14] Karp, R.M. (1972): Reducibility among combinatorial problems, In: R.E. Miller; J.W. Thatcher (ed.): Complexity of Computer Computations. Plenum Press, New York, 1972, 85-103. [15] Laudon, K.C.; Traver, C.G. (2007): E-commerce. Pearson/Addison Wesley.

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ACCEPTED MANUSCRIPT [16] Montoya-Torres, J.R.; Franco, J.L.; Isaza, S.N.; Jiménez H.F.; Herazo-Padilla, N. (2015): A literature review on the vehicle routing problem with multiple depots. Computers & Industrial Engineering 79, 115-129. [17] Ratliff, H.D.; Rosenthal, A.S. (1983): Order-picking in a rectangular warehouse: A solvable case of the traveling salesman problem. Operations Research 31, 507-521. [18] Roodbergen, K.J.; de Koster, R. (2001): Routing order pickers in a warehouse with a middle aisle. European Journal of Operational Research 133, 32-43.

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[19] Schneider, M,; Stenger, A.; Hof, J. (2015): An adaptive VNS algorithm for vehicle routing problems with intermediate stops. OR Spectrum 37, 353-387. [20] Weidinger, F. (2018): Picker routing in rectangular mixed shelves warehouses. Computers & Operations Research 95, 139-150. [21] Weidinger, F.; Boysen, N. (2017): Scattered storage: How to distribute stock keeping units all around a mixed-shelves warehouse. Transportation Science (to appear).

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[22] Yaman, H.; Karasan, O.E.; Kara, B.Y. (2012): Release time scheduling and hub location for next-day delivery. Operations Research 60, 906-917.

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Appendix: Impact of cart capacity on picker travel

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total picking distance [distance units]

This appendix explores the impact of the capacity of the picking cart on the picking performance. Specifically, we vary parameter C ∈ {1, 2, 4, 6, 8}, which defines the number of orders that can concurrently be assembled by the picker. We solve each of the large instances (see Section 4.1) with any of the aforementioned capacities by applying our improved pool-based heuristic (with parameter values λ = 30, µ = 10, and κ = 150). The resulting average total picking distance for any of these capacities is plotted in Figure 6.

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Figure 6: Impact of cart capacity C on picking performance

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The results show that the positive effect of additional cart capacity quickly diminishes. Increasing the capacity from a single order to two and four concurrent orders leads to considerable reductions of the picker travel. From then on, only moderate additional gains can be realized. This is good news for the practitioner. Already relatively small and agile picking carts with a capacity for just a few orders lead to a good performance, so that larger carts that would considerably increase the ergonomic burden for the pickers need not be considered.

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