Picker routing and storage-assignment strategies for precedence-constrained order picking

Transcription

1 Picker routing and storage-assignment strategies for precedence-constrained order picking Ivan šulj Christoph H. Glock Eric H. Grosse Michael Schneider Ÿ May 15, 2018 Abstract Order picking describes the process of retrieving items from their storage locations to satisfy customer orders. Because order picking is considered the most labor-intensive process in warehousing, eectively routing order pickers through a warehouse can result in considerable time and cost savings. In practice, picker routing is often inuenced by precedence constraints, i.e., the order-picking sequence is partially predetermined due to fragility restrictions, stackability, shape, size, and preferred unloading sequence. Although many warehouses face such precedence constraints for picking items (especially in the grocery sector), they are hardly considered in the scientic literature. This paper is inspired by a practical case observed in a warehouse of a German manufacturer of household products, where heavy items are not allowed to be stored on top of light items to prevent damage to the light items. Currently, the sequence for retrieving the items from their storage locations is determined by applying a picker-routing strategy that neglects this precedence constraint, and the order pickers pack the items respecting their weights after picking. To avoid this sorting eort at the end of the order-picking process, we propose a picker-routing strategy that incorporates the precedence constraint by picking heavy items before light items. We develop an exact solution method to evaluate this strategy. Furthermore, we examine the inuence of dierent problem parameters on the proposed picker-routing strategy, and we derive managerial insights for dealing with precedence constraints in order picking. Keywords: logistics, warehousing systems, order-picking methods, precedence constraint, picker-routing strategies, storage-assignment strategies. zulj@uni-hohenheim.de, Department of Procurement and Production, University of Hohenheim, Schwerzstr. 40, Stuttgart, Germany glock@pscm.tu-darmstadt.de, grosse@pscm.tu-darmstadt.de, Department of Production and Supply Chain Management, TU Darmstadt, Hochschulstr. 1, 64289, Darmstadt, Germany schneider@dpo.rwth-aachen.de, Deutsche Post Chair Optimization of Distribution Networks, RWTH Aachen University, Kackertstr. 7 B, Aachen, Germany Ÿ Corresponding author. Tel.: , Fax:

2 1 Introduction Retrieving items from their storage locations according to customer orders (order picking) is considered the most laborious task in warehouses accounting for up to 65% of the total operating costs (Drury 1988, Frazelle 2001, Coyle et al. 2002, Tompkins et al. 2010). Therefore, the optimization of order-picking operations has signicant impact on the overall performance of a warehouse and the overall supply chain. Although advances in technology have enabled the use of automated storage and retrieval systems, in which items are transported via band conveyors or automated guided vehicles to a central depot, it is estimated that about 80% of all order-picking systems in Western Europe are picker-to-parts order-picking systems and still operated manually (Wäscher 2004, de Koster et al. 2007). In such systems, order pickers walk or drive through the warehouse to retrieve the requested items from their storage locations. Warehouses often rely on humans for order picking because of their exibility and ability to adapt to changes in real-time in contrast to automated sorting systems (Grosse et al. 2015). Among all order-picking activities in picker-to-parts systems (setup for the routes, searching, traveling and picking), traveling is the most time-consuming activity with a share of up to 50% of total order-picking time (Tompkins et al. 2010). Travel time (or travel distance) mainly depends on item storage assignment (assignment of items to storage locations) and routing pickers through the warehouse (determining the orderpicking sequence). With respect to the routing of order pickers through the warehouse, our survey of the literature showed that constraints arising in real-world application have often been neglected in prior research. Recently, research has started to consider more realistic characteristics of real-world warehouse activities like, e.g., product specic characteristics such as fragility or weight and also human factors (see Chackelson et al. 2013, Grosse et al. 2013, 2015, Matusiak et al. 2014, 2017, Chabot et al. 2017). In practice, order picking is often subject to precedence constraints (Chabot et al. 2017). These constraints dene that certain items need to be collected before other items due to fragility, stackability, shape and size, and preferred unloading sequence. Such constraints can often be found in the grocery sector. This paper is inspired by a practical case observed by the authors in a warehouse of a German manufacturer of household products, where the items to be picked can be roughly distinguished into light (fragile) and heavy (robust) items. In the warehouse of our industry partner, items are categorized as 'light' if their weight does not exceed 0.75 kg, and otherwise as 'heavy'. To prevent damage to light items, order pickers are not permitted to put heavy items on top of light items. Currently, the sequence for retrieving items from their storage locations is determined by applying a simple s-shape routing strategy that does not consider this precedence constraint. As a result, the order pickers 2

3 collect items that have been retrieved from the shelves of the warehouse into plastic boxes without stacking items on top of each other. Upon completion of the order-picking process, the order pickers travel back to the depot of the warehouse, where they pack the collected items into cardboard boxes that are used for shipping the items respecting the precedence constraint. The paper at hand intends to improve the order-picking process observed in the warehouse under study in two respects. First, to avoid that items have to be sorted at the end of the order-picking process, the paper proposes a picker-routing strategy that incorporates the described precedence constraint and collects heavy items before light items. This enables the order pickers to place retrieved items directly in the cardboard boxes required for shipping the items and, thus, avoids the use of plastic boxes and the sorting of items upon return to the depot. Second, to shorten travel distances in the warehouse, the paper determines an optimal order-picking route, which leads to a quicker order-picking process compared to the s-shape routing strategy. The paper compares the proposed picker-routing strategy to the one observed in the warehouse and to an exact solution approach that neglects the precedence constraint with regard to the resulting total tour length and the sorting eort. The main contributions of this paper are as follows: We propose a picker-routing strategy for the case where order picking is precedence-constrained. To determine the picking sequence for the exact solution approach in which the order-picker collects items disregarding the precedence constraint, we use the graph-based algorithm proposed by Ratli and Rosenthal (1983). For a detailed description of the algorithm, we refer the reader to the original work. To evaluate the proposed picker-routing strategy, we introduce an exact algorithm based on the concept of dynamic programming. We investigate the inuence of dierent storage-assignment strategies (SASs) on the proposed picker-routing strategy. Moreover, we derive insights for warehouse managers regarding the cost impact of the precedence constraint in manual order picking. The remainder of this paper is organized as follows. Section 2 gives a brief review of the related literature. Section 3 introduces the picker-routing problem with the precedence constraint. The exact algorithm is described in Section 4. Section 5 presents a numerical analysis to evaluate the proposed picker-routing strategy. The paper concludes with a summary and an outlook on possible future research in Section 7. 2 Literature review The literature on designing order-picking processes can be distinguished into four main research areas: warehouse layout design, order batching, storage assignment, and picker routing. Because traveling is the most time-consuming activity, research in this area mainly focuses on reducing the average travel distance necessary to pick all items of a 3

4 given set of customer orders. The four research areas and picker routing with a special focus on precedence constraints will be discussed briey in the following: Warehouse layout: In the context of order picking, the design of the warehouse layout deals with (i) the characteristics of the order-picking system such as the mechanization level (manual, mechanized, semi-automated, automated), (ii) the question of where to locate receiving, picking, storage, sorting, and shipping areas, and (iii) the layout within an order-picking system, i.e., the location of the depot, the size of the picking area, racking (ow racks, pallet racks or shelves), and equipment usage (picking trucks, picking carts) (see Caron et al. 2000, Petersen 2002, Roodbergen and Vis 2006, de Koster et al. 2007, Roodbergen et al. 2008). Rectangular warehouse layouts with parallel aisles are prevalent both in the literature and in practice (see Ratli and Rosenthal 1983, Bozer and Kile 2008, Henn and Wäscher 2012). Here, the layout decision concerns the number of blocks, and the number and dimension of aisles and cross aisles in each block. Few approaches deal with non-standard warehouse layouts such as ying-v, shbone, and U-shaped layouts (see Glock and Grosse 2012, Gue and Meller 2009, Pohl et al. 2009). Order batching: If the number of items per customer order is small, the total travel distance can be reduced by consolidating a set of customer orders into a single picking tour. Order batching groups customer orders to picking orders (batches) such that the total length of all tours through a warehouse is minimized. Because order picking is considered the most labor-intensive process in warehousing, eectively batching customer orders can result in considerable cost savings (see de Koster et al. 1999, Gademann and van de Velde 2005, Chen and Wang 2017, šulj et al. 2018). Storage assignment: The literature proposes various strategies for assigning items to storage locations in the warehouse. Common strategies are random storage, dedicated storage, and class-based storage (see Gu et al. 2007, 2010, de Koster et al. 2007). A random storage strategy arbitrarily assigns items to a storage location. This strategy aims on maximizing storage-space utilization, but often results in long travel times (see de Koster et al. 1999, Tompkins et al. 2010, Grosse et al. 2013). Dedicated storage assigns items to xed storage locations based on common characteristics, such as demand frequency, weight or measurements (see Brynzér and Johansson 1996, Frazelle 2001) or the cube-per-order index, i.e., the ratio of the stock volume to the demand frequency (see Heskett 1963, Malmborg 1995). Dedicated storage leads to a lower degree of storage-space utilization, but often reduces travel time as compared to random storage. Class-based storage rst groups items into classes and then assigns classes to dedicated areas of the warehouse (see Jarvis and McDowell 1991, Petersen and Schmenner 1999). Storage assignment within an 4

5 area is random. The goal of this strategy is to simultaneously achieve a high space utilization and short travel times. Picker routing: The goal of picker routing is to determine a sequence for collecting required items such that the travel time of the order picker is minimized. For rectangular warehouses with parallel aisles of equal length and width, this so-called picker-routing problem can be formulated as a special case of the traveling salesman problem. Solution approaches for picker routing can be distinguished into exact algorithms (see Ratli and Rosenthal 1983, Goetschalckx and Ratli 1998, de Koster and van der Poort 1998, Roodbergen and de Koster 2001) and heuristics (see Hall 1993, Petersen 1997). Ratli and Rosenthal (1983) present an exact and polynomialtime tour-construction algorithm based on dynamic programming for order picking in a single-block warehouse with a central depot. The time complexity of their algorithm is linear in the number of aisles and the number of items. Goetschalckx and Ratli (1998) present an algorithm for optimally routing order pickers in wide aisles, where the order picker cannot retrieve items from both sides of the aisle without additional eort. The algorithm of Ratli and Rosenthal (1983) is extended in de Koster and van der Poort (1998) by allowing decentralized depositing, i.e., dropping o items is allowed at the end of every aisle. Moreover, Löer et al. (2017) extend the algorithm to handle arbitrary start and end points for the order-picking tour. Besides exact solution approaches, several heuristics have been proposed in the literature for routing order pickers: s-shape (or traversal) by Goetschalckx and Ratli (1998), return, midpoint, and largest gap by Hall (1993), and composite by Petersen and Schmenner (1999). Picker routing with precedence constraints: In the order-picking literature, precedence constraints in picker routing have only rarely been studied. Dekker et al. (2004) examine combinations of SASs and routing heuristics for a real-world application arising in a warehouse of a wholesaler of tools and garden equipment. The warehouse is characterized by multiple cross aisles, dead-end aisles, two oors, and dierent start and end locations of a tour. Furthermore, a guideline requiring that fragile products have to be picked last has to be considered. To address this requirement, fragile items are positioned in the right-most aisle (with the start location being at the left-most aisle), so that this requirement is automatically met. Matusiak et al. (2014) present a simulated annealing method to address the joint order batching and precedence-constrained picker-routing problem in a warehouse with multiple depots. The shortest path through the warehouse is determined by using the exact A*-algorithm proposed by Hart et al. (1968). Arcs represent possible state transitions for moving to a location and indicate the reachability of states. This ensures that the pre-specied picking sequence is met. Chabot et al. (2017) 5

6 introduce the order-picking routing problem under weight, fragility and category constraints (OPP-WFCC). They propose a capacity-indexed mathematical model formulation and a two-index vehicle-ow formulation as well as four heuristics (sshape, largest gap, mid-point and adaptive large neighborhood search) to solve the OPP-WFCC. Furthermore, a branch-and-cut algorithm is applied to solve the two formulations of the OPP-WFCC considering the precedence constraints. Precedence constraints in related contexts: Precedence constraints have been considered in other applications as well. Junqueira et al. (2012), for example, introduce the container loading problem that considers the vertical and horizontal stability and fragility of the cargo. Fragility is ensured by limiting the number of boxes that can be loaded above each other. Precedence constraints also appear in the literature for vehicle-routing problems, in which one customer must be served before another. Practical applications include the dial-a-ride problem (Psaraftis 1983, Jaw et al. 1986, Cordeau and Laporte 2007), bus routing (Wren and Holiday 1972, Stein 1978, Park and Kim 2010), and pickup and delivery (Parragh et al. 2008a,b). For more details on vehicle-routing problems with precedence constraints, we refer to Lahyani et al. (2013). As the review of the literature shows, works considering precedence constraints in picker routing are rare. Thus, our work contributes to the literature by proposing a new exact algorithm for precedence-constrained order picking based on the concept of dynamic programming. In addition, the inuence of dierent SASs on the proposed picker-routing strategy is studied. 3 Problem description The order-picking system considered in this paper is a rectangular single-block warehouse with parallel aisles of equal length and width connected by crossing aisles at the front and rear of each vertical aisle (see Figure 1). The depot is the start and end point of all picking tours, and it is located at the front of the leftmost aisle. Here, the order picker receives a pick list for collecting the items required by a customer order, and a picking device. The picker then walks through the aisles and retrieves the required items from the storage locations until the customer order is completed. A customer order is picked in a single tour. Storage locations are arranged on both sides of the vertical picking aisles. A customer order may consist of two types of items, namely heavy (robust) items and light (fragile) items. We assume that the order picker handles one customer order per picking tour. Furthermore, we consider a one-dimensional stacking system. When collecting the items, the order picker is not allowed to put heavy items on top of light items to prevent damage to light items, i.e., heavy items can only be placed above other 6

7 D Figure 1: Rectangular single-block warehouse layout. heavy items, while light items can be placed above heavy items or other light items. Thus, the retrieval sequence is precedence-constrained. In this paper, we compare the following picker-routing strategies: BC In the case company under study, the sequence for retrieving items from the storage locations is determined by applying a s-shape routing strategy without considering the precedence constraint. The order pickers collect the items into plastic boxes without stacking them on top of each other. The sorting of items takes place at the end of the picking process, and it is integrated into our model by means of a sorting penalty that is added to the total traveled distance of the order picker. We refer to this picker-routing strategy as the base case (BC). PRS-1 According to the exact solution approach (PRS-1), items are collected without considering the precedence constraint. In this case, order picking is carried out in optimal fashion with respect to the routing using the algorithm of Ratli and Rosenthal (1983). Again, the order pickers sort the items at the end of the picking process, and the sorting eort is added to the total traveled distance of the order picker. PRS-2 According to the newly proposed picker routing strategy (PRS-2), heavy items are collected before light items to avoid the sorting eort. Note that it would be possible to consider a hybrid of these two extreme picker-routing strategies, i.e., a picker-routing strategy that determines the optimal retrieval sequence when sorting is carried out while picking. However, such a solution approach is likely to be less useful for practical applications due to the complexity of the resulting order-picking process and the high potential for errors: the order picker would have to implement a predened sorting scheme (due to the one-dimensional stacking system) in addition to traveling on a given route through the warehouse. Furthermore, Elbert et al. (2017) nd that order pickers deviate from complex routes (e.g., due to confusion) and recommend more straightforward and non-confusing routing methods. In light of these limitations, the paper refrains from studying such a hybrid strategy. 7

8 4 Solution algorithm In this section, we present an exact algorithm to evaluate PRS-2 that is based on a modication of Ratli and Rosenthal (1983). In order to incorporate the precedence constraint into the algorithm, we dene dierent types of subtours. A heavy subtour t H i denes an optimal route through the warehouse for collecting all heavy items i = 1,..., h, starting at the depot and ending at a predetermined (heavy) item storage location. A light subtour t L i denes an optimal route through the warehouse for collecting all light items j = 1,..., l, starting at a predetermined (heavy) item storage location and ending at the depot. Obviously, the start and end locations cannot be determined a priori. Each location that contains a heavy item to be picked may be the end location of a heavy tour and the start location of a light tour that leads to the minimum tour length for collecting both heavy items and light items in sequence. The algorithm of Ratli and Rosenthal (1983) does not consider arbitrary start and end locations. Löer et al. (2017) extend the algorithm proposed by Ratli and Rosenthal (1983) by allowing arbitrary start and end locations. We use their algorithm to determine the optimal picking sequence for the heavy and light subtours. The optimal sequence for retrieving the required items m = h + l from their storage locations is determined by nding a combination of a heavy subtour t H i and a light subtour that leads to a minimum total tour length c(t ). Note that the optimality of the picking t L i sequence is guaranteed by evaluating all possible combinations of heavy and light subtours: c(t ) = min i { c(t H i ) + c(t L i ) }. (1) Figure 2 illustrates the resulting picking tours for dierent end locations of a heavy subtour and start locations of a light subtour in two examples. We assume that a customer order consists of heavy items (h 1, h 2 ) and light items (l 1, l 2 ). There are two possible end/start locations for the heavy/light subtour. In (a), (heavy) item location h 1 is the end location of the heavy subtour t h 1 and the start location of the light subtour t l 1. In (b), (heavy) item location h 2 is the end location of the heavy subtour t h 2 and the start location of the light subtour t l 2. The minimum total tour length is realized in (b) by picking sequence h 1, h 2, l 2, and l 1. Moreover, the gure shows that start and end locations of the subtours cannot be determined a priori. For rectangular single-block warehouse layouts, the picker-routing problem with the precedence constraint can be solved in polynomial time. The algorithm has a run-time complexity of O((m 3 + p m 2 ) h l) O((m 3 + p m 2 ) m) O(m 5 ), where p denotes the number of picking aisles. h l calculates all possible combinations for linking a heavy subtour with a light subtour. 8

9 h 1 l 2 h 1 l 2 l 1 h 2 l 1 h 2 D (a) Picking tour for the case where h 1 is the end location of the heavy subtour (solid line) and the start location of the light subtour (dashed line). D (b) Picking tour for the case where h 2 is the end location of the heavy subtour (solid line) and the start location of the light subtour (dashed line). Figure 2: Resulting picking tours for dierent end locations of a heavy subtour and start locations of a light subtour. 5 Practical case study This section is devoted to the assessment of the performance of the picker-routing strategies within the practical case study. Section 5.1 introduces a practical case that motivated the study at hand. Section 5.2 evaluates the described picker-routing strategies and investigates the inuence of dierent SASs on the performance of the order picker. 5.1 Case description The proposed method was applied to a scenario motivated by a practical case to investigate the inuence of dierent item weight classes and dierent SASs on the routing of order pickers through the warehouse. The case company considered here produces household products (e.g., uid bath additives and natural cosmetics) and operates a distribution warehouse that stores a large variety of items and ships orders to customers worldwide. Products stored in the warehouse range from very small glass phials weighing 50 grams up to big wreaths of plastic vessels weighing up to 10 kg. Order picking in the warehouse is completely manual, and technical equipment for supporting the order picking process, such as pick-by-light or pick-by-vision, is not available. The existing order-picking process in the case company can be described as follows: For each customer order, an order picker receives a paper-based pick list at the depot of the warehouse that contains the items that need to be picked and then travels along the shelves of the warehouse to retrieve the requested items. The order picker uses a standard hand trolley for transporting items and places collected items next to each other on the 9

10 trolley. After picking all items on the pick list, the order picker returns to the depot, where the items are packed in a cardboard box for shipping (this process is often referred to as pick-and-sort in the literature, see, e.g., de Koster et al. (2007)). As the products signicantly dier in size, weight and physical features, it is necessary to pack light items on top of heavy items to avoid damage during shipping. In the warehouse under study, items are categorized as 'light' if their weight does not exceed 0.75 kg, and otherwise as 'heavy'. Light and heavy items each account for about 50% of the total number of items in the warehouse. The packing and sorting of items at the end of an order-picking process is a time-consuming process step in the considered warehouse. During on-site visits, the warehouse manager informed us that the company has tested a sort-while-pick process in the past in which the order pickers already sort items during the order-picking process; however, due to the frequent (re-)handling of items, this process proved to be too error-prone in the warehouse at hand. With respect to the storing of products, the case company does not use a specic dedicated SAS (such as demand- or turnover-based assignment) but instead assigns items randomly to the storage locations in the warehouse. To retrieve items, order pickers travel through the warehouse using the so-called s-shape or traversal strategy, i.e., the order picker starts at the leftmost aisle, enters aisles alternately from the front cross aisle or the rear cross aisle if they contain at least one picking location, and then traverses them completely. If the order picker enters the rightmost aisle from the front cross aisle, she travels the aisle to the last item to be picked, returns to the front cross aisle and from there to the depot. For the numerical analysis, a simplied model of the real case warehouse was built which consists of a rectangular picking area composed of 10 aisles with 100 storage locations per aisle (50 on each side), with the depot being located at the front of the leftmost aisle (see Figure 3). The length of the warehouse is set to L = 39 meters, the width to W = 60 meters, the length of an aisle to l = 25 meters, the width of an aisle to w = 1.5 meters and the width of the front and back aisles to w c = 2 meters. The aisle number is denoted as a (from 1 to 10), and the storage locations sl are numbered consecutively from 1 to Based on a dataset provided to us by the case company, a case study instance was generated that consists of 2089 customer orders. Each customer order is given by a single pick list including item numbers, the storage location of each item, and the classication of the item as 'light' or 'heavy'. Possible correlations between the items are not considered. 5.2 Results of the case study The aim of the numerical analysis is to compare the current order-picking performance in the case company (which induces high sorting eort) to the performance obtained using 10

11 Figure 3: Layout of the case warehouse. the proposed picker-routing strategy that integrates the precedence constraint and enables the order picker to pack items directly after retrieving them without additional sorting eort. For a fair comparison, sorting eort has to be considered by means of penalties when comparing the strategies. According to the BC in the warehouse under study and the proposed PRS-1, sorting takes place at the end of the order-picking process, i.e., all items need to be sorted into cardboard boxes used for shipping the items. We dene dierent sorting-penalty scenarios based on experimental tests that were conducted in the case company. We observed that resorting of items ranges approximately between 3 seconds and 4 seconds on average per item. This resorting also includes the time for searching an item in the plastic boxes and the time for identifying an item as 'light' or 'heavy' in order to determine the stacking sequence to avoid damage to light items. Assuming that the picker's travel velocity is constant, the total travel time is equivalent to the total length of all picker tours (Jarvis and McDowell 1991). Therefore, the resorting time can be added as a sorting penalty measured in length units (LUs) to the objective function value. The width of a storage location is set to 0.5 meters and equals 1 LU in our study. To evaluate the picker-routing strategies, we assume the travel velocity of an order picker to be 1.45 meter per second and dene the following scenarios for the sorting eort: 3 LUs (approximately 1 second), 6 LUs (approximately 2 seconds), 9 LUs (approximately 3 seconds), and 12 LUs (approximately 4 seconds). Comparison to the BC In the following, we use the average total tour length that includes the sorting penalty as a performance measure for comparing the picker-routing strategies. In Table 1, we compare the performance of the picker-routing strategies applying a random SAS. For all comparison strategies, we report the average of the gaps of the objective function values over the single customer orders obtained with the respective strategy f (%) to the best-known solution (BKS) for dierent sorting-eort scenarios. 11

12 The BKS corresponds to the average of the best objective function values obtained for each of the single customer order by one of the tested picker-routing strategies. We compute the percentage gap as 100 (BKS f K )/(BKS), where f K denotes the average of the objective values over the individual instances for picker-routing strategy k K. The smallest average gap found by any of the strategies is indicated in bold. The BKS for each individual instance is available for download at sh/zlxtg5cya8xnk54/aacqmqedbqyaucbdys7zigmya?dl=0/. On all tested instances, the run-time of the proposed algorithm is below one second. BC PRS-1 PRS-2 SE f (%) f (%) f (%) s = s = s = s = s = Table 1: Performance of the picker-routing strategies for a random SAS. For all comparison strategies, we report the average of the gaps of the objective function values over the customer orders obtained with the respective strategy f (%) to the best-known solution (BKS) for dierent sorting-eort scenarios. The BKS corresponds to the average of the best objective function values obtained for each of the single customer order by one of the tested picker-routing strategies. PRS-1 and PRS-2 outperform the BC for all sorting-eort scenarios with respect to the average total tour length. The BC deviates by 32.4% to 47.2% from the BKS for dierent sorting-eort scenarios. Even if no sorting eort is considered ( s = 0), PRS-2 shows a signicantly smaller deviation from the BKS compared to the BC. If the sorting eort is 6 LUs or higher, PRS-2 outperforms PRS-1. Weight-based storage-assignment strategies Besides routing, the allocation of items to storage locations in the warehouse inuences the resulting tour length of order pickers through the warehouse when collecting items. Obviously, the separation of heavy items and light items in the warehouse, and the allocation of heavy items close to the depot are in favour of PRS-2 because heavy items are collected before light items. Therefore, dierent weight-based storage-assignment strategies (W-SASs) are proposed in the following, and their performance in combination with the presented picker-routing strategies is evaluated. Figure 4 depicts four dierent W-SASs that can be described as follows: W-SAS (a) assigns heavy items to the rst half of the warehouse, light items are assigned to the second half of the considered warehouse. In W-SAS (b), heavy and light items are alternately 12

13 assigned to aisles starting with heavy items in the rst aisle. In W-SAS (c), heavy items are stored at the respective entrances of the aisles, whereas light items are stored within aisles. W-SAS (d) stores heavy items below the midpoint of the aisle, and light items are stored above. D (a) Heavy items are stored in the rst half of the warehouse, light items are stored in the second half of the warehouse. D (b) Heavy and light items are alternately stored in the aisles, starting with heavy items in the rst aisle. D (c) Heavy items are stored at the entrances of the aisles, light items are stored within the aisles. D (d) Heavy items are stored below the midpoint of the aisle, light items are stored above. Figure 4: Weight-based storage-assignment strategies. Table 2 shows the performance of the BC, PRS-1, and PRS-2 for dierent W-SASs and sorting-eort scenarios. Figure 5 depicts the average tour lengths of the investigated picker-routing strategies for s = {0, 3, 6, 9, 12} and dierent W-SASs. Comparison of the picker-routing strategies without sorting eort If sorting eort is neglected, PRS-1 and PRS-2 clearly outperform the BC in the case company on all tested instances with respect to the average total tour length. The average gap to the BKS of BC is approximately 35%. The comparison of PRS-1 and PRS-2 shows that PRS-2 deviates between 3.5 and 19.8% from the optimal solutions that are obtained with PRS-1. Obviously, PRS-1 is the best performing picker-routing strategy. This can be explained by the fact that the sorting of the items takes place after the order-picking process and is not considered in the objective function value for s = 0. Interestingly, for W-SAS (a), PRS-2 is able to nd a near-optimal solution with a deviation of only 3.5% from PRS-1 although for PRS-1 sorting is not considered yet. 13

14 Comparison of the picker-routing strategies with increasing sorting eort Again, PRS-1 and PRS-2 beat the solution quality of the BC on all instances. When comparing the performance of PRS-1 and PRS-2, we observe that the superiority of PRS- 2 in comparison to PRS-1 increases with the sorting eort. For s = 3 and W-SAS (a), (b), and (d), PRS-2 matches or outperforms PRS-1. Recall that a sorting eort of 3 LUs corresponds to 1 second and includes the time for searching an item in the plastic boxes and the time for identifying an item as 'light' or 'heavy'. For the practically more realistic sorting-eort scenarios (s = 9, 12), PRS-1 deviates between 7.9 and 33.6% from the BKS that is obtained by PRS-2. This indicates a convincing performance of PRS-2. Eect of dierent W-SASs The results that are reported in Table 1 and Table 2 show that the SASs signicantly aect the performance of PRS-2. In particular, a strong reduction of the average tour length can be achieved by assigning heavy items to the rst half of the warehouse and light items to the second half of the warehouse (W-SAS (a)). Comparing the results that assume a random SAS to those obtained for W-SAS (a) and s = 0, the deviation of PRS-2 from the BKS is signicantly smaller (17.7% versus 3.5%). W-SAS (c) seems not to be appropriate for precedence-constrained order picking because PRS-2 deviates by 19.8% from the BKS. PRS-2 benets from a SAS where heavy items are clearly separated from light items in the warehouse. To summarize, both the picker-routing strategy and the SAS have a signicant in- uence on the resulting total tour length when addressing the picker-routing problem with the studied precedence constraint. As can be seen from the numerical example, the combination of PRS-2 and W-SAS (a) is recommendable for warehouse managers dealing with similar problem settings. Note that it is quite likely that the superiority of PRS-2 would increase with further item categories because of the increasing complexity of the sorting process. 14

15 BC PRS-1 PRS-2 W-SAS SE f (%) f (%) f (%) s = (a) s = s = s = s = s = (b) (c) s = s = s = s = s = s = s = s = s = s = (d) s = s = s = s = Table 2: Performance of the picker-routing strategies for alternative weight-based storage assignments (W-SAS). For all comparison strategies, we report the average of the gaps of the objective function values over the customer orders obtained with the respective strategy f (%) to the best-known solution (BKS) for dierent sorting-eort scenarios. The BKS corresponds to the average of the best objective function values obtained for each of the single customer order by one of the tested picker-routing strategies. 15

16 1,100 1,100 1,000 1, average tour length average tour length random W-SAS (a) W-SAS (b) W-SAS (c) W-SAS (d) 300 random W-SAS (a) W-SAS (b) W-SAS (c) W-SAS (d) storage-assignment strategy storage-assignment strategy BC PRS-1 PRS-2 BC PRS-1 PRS-2 (a) s=0 (b) s=3 1,100 1,100 1,000 1, average tour length average tour length random W-SAS (a) W-SAS (b) W-SAS (c) W-SAS (d) 300 random W-SAS (a) W-SAS (b) W-SAS (c) W-SAS (d) storage-assignment strategy storage-assignment strategy BC PRS-1 PRS-2 BC PRS-1 PRS-2 (c) s=6 (d) s=9 1,100 1, average tour length random W-SAS (a) W-SAS (b) W-SAS (c) W-SAS (d) storage-assignment strategy BC PRS-1 PRS-2 (e) s=12 Figure 5: Performance of the BC, PRS-1 and PRS-2 for dierent storage assignments and sorting eort scenarios. 16

17 6 Inuence of dierent problem parameters In this section, we present numerical studies to evaluate the inuence of dierent problem parameters. In Section 6.1, we describe the generation of new problem instances. Section 6.2 is devoted to investigate the inuence of dierent problem parameters on the quality of the picker-routing strategies. 6.1 Test instances The new instances assume a rectangular single-block warehouse with three dierent sizes 10, 25, and 50 parallel picking aisles. Each aisle contains 100 storage locations, 50 on the left-hand side and 50 on the right-hand side. The depot is located at the front of the leftmost aisle. We assume W-SAS (a) due to its superior performance in the practical case study. The physical dimensions of the warehouse are dened as follows: The distance between the depot and the rst storage location in the leftmost picking aisle amounts to 1 LU. A storage location has a length of 1 LU. When leaving an aisle, the order picker moves 1 LU in vertical direction. The distance between two vertical picking aisles amounts to 5 LUs. Furthermore, we consider 40 dierent customer orders per instance. The instances assume an uniformly distributed number of items per customer order that is randomly drawn from all of the three intervals [5, 35], [36, 70], and [71, 100]. Customer orders vary with respect to the share of heavy and light items. More precisely, we consider three dierent mixes with approximately 75% heavy items/25% light items, 50% heavy items/50% light items, and 25% heavy items/75% light items per customer order. The combination of the described parameter values results in 27 instance classes that are identied by the size of the warehouse, the mix of heavy and light items per customer order, and the number of items per customer order. Each instance class contains 20 instances. This leads to = 540 instances in total. 6.2 Results of the numerical study In Tables 3, 4, and 5, we aggregate the results of the numerical study according to the mix of heavy/light items, the number of items per customer order, and the sorting eort. Table 3 reports the results for a warehouse with 10 picking aisles, Table 4 for 25 picking aisles, and Table 5 for 50 picking aisles. Again, we use the average total tour length that includes the sorting penalty as a performance measure for comparing the picker-routing strategies. For all comparison strategies, we report the average of the gaps of the objective function values over the individual instances obtained with the respective strategy f (%) to the BKS for dierent problem parameters with respect to warehouse size, item mix, number of items per customer order, and sorting eort scenarios. The smallest gap found 17

18 by any of the picker-routing strategies is indicated in bold. Overall comparison of the performance of the picker-routing strategies The results reported in Tables 3, 4, and 5, show that in all problem settings, PRS-1 and PRS-2 clearly outperform the BC. For s = 0, BC deviates between 8.4% and 51.2% from PRS-1. For increasing sorting eorts, the BC has gaps of up to 113.2% from the BKS that is obtained with PRS-2. When comparing the performance of PRS-1 and PRS-2, we observe that PRS-1 slightly outperforms PRS-2 with respect to the average tour length if sorting eort is not considered. Similarly to the results of the case study, PRS-2 is able to nd near-optimal solutions with a deviation of between 0.6% and 5.4% from PRS-1 although for PRS-1 sorting is not considered yet. Interestingly, already for s = 3, PRS-1 has a deviation of up to 24.6% from PRS-2. When assuming a sorting eort of s = 12, this gap rises to 100.2%. In the following, we investigate the inuence of the dierent problem parameters on the solution quality of PRS-1 and PRS-2. Eect of warehouse size When comparing the performance of PRS-1 and PRS-2, we observe that PRS-1 performs slightly better with increasing warehouse size. Nevertheless, PRS-2 shows a more robust performance with a maximum gap to the BKS of 5.4%, whereas the gaps of PRS-1 uctuate between 0.0% and 100.2%. Eect of item mixes With a higher percentage of light items, the average tour length for the picker-routing strategies increases. This is due to the fact that light items are stored in the second half of the warehouse. Eect of number of items per customer order The results reported show that the number of items per customer order signicantly aects the performance of all pickerrouting strategies. PRS-2 performs signicantly better if larger customer orders are assumed. For example, PRS-2 shows a gap of only 0.6% to the BKS for the problem setting in which 10 picking aisles, 75% heavy items, 25% light items, [71-100] items per customer orders, and no sorting eort are considered. 7 Conclusions This paper is inspired by a practical case of a manual order-picking warehouse for household products where the item weight inuences the sorting sequence of items into cardboard boxes used for shipping the items. When dealing with the routing of order pickers through a warehouse in the literature and in practice, precedence constraints are often 18

19 BC PRS-1 PRS-2 Item mix Number of articles SE f (%) f (%) f (%) [5-35] s = s = s = s = s = % / 25% [36-70] s = s = s = s = s = [71-100] s = s = s = s = s = [5-35] s = s = s = s = s = % / 50% [36-70] s = s = s = s = s = [71-100] s = s = s = s = s = [5-35] s = s = s = s = s = % / 75% [36-70] s = s = s = s = s = [71-100] s = s = s = s = s = Table 3: Performance of the picker-routing strategies for dierent problem parameters in a warehouse with 10 picking aisles. For all comparison strategies, we report the average of the gaps of the objective function values over the individual instances obtained with the respective strategy f (%) to the best-known solution (BKS) for dierent sortingeort scenarios. The BKS corresponds to the average of the best objective function values obtained for each of the individual instances by one of the tested picker-routing strategies. 19

20 BC PRS-1 PRS-2 Item mix Number articles SE f (%) f (%) f (%) [5-35] s = s = s = s = s = % / 25% [36-70] s = s = s = s = s = [71-100] s = s = s = s = s = [5-35] s = s = s = s = s = % / 50% [36-70] s = s = s = s = s = [71-100] s = s = s = s = s = [5-35] s = s = s = s = s = % / 75% [36-70] s = s = s = s = s = [71-100] s = s = s = s = s = Table 4: Performance of the picker-routing strategies for dierent problem parameters in a warehouse with 25 picking aisles. For all comparison strategies, we report the average of the gaps of the objective function values over the individual instances obtained with the respective strategy f (%) to the best-known solution (BKS) for dierent sortingeort scenarios. The BKS corresponds to the average of the best objective function values obtained for each of the individual instances by one of the tested picker-routing strategies. 20

21 BC PRS-1 PRS-2 Item mix Number articles SE f (%) f (%) f (%) [5-35] s = s = s = s = s = % / 25% [36-70] s = s = s = s = s = [71-100] s = s = s = s = s = [5-35] s = s = s = s = s = % / 50% [36-70] s = s = s = s = s = [71-100] s = s = s = s = s = [5-35] s = s = s = s = s = % / 75% [36-70] s = s = s = s = s = [71-100] s = s = s = s = s = Table 5: Performance of the picker-routing strategies for dierent problem parameters in a warehouse with 50 picking aisles. For all comparison strategies, we report the average of the gaps of the objective function values over the individual instances obtained with the respective strategy f (%) to the best-known solution (BKS) for dierent sortingeort scenarios. The BKS corresponds to the average of the best objective function values obtained for each of the individual instances by one of the tested picker-routing strategies. 21

22 neglected and sorting often takes place at the end of the order-picking process. To avoid this sorting eort, we propose a picker-routing strategy that integrates the precedence constraint by collecting heavy items before light items in an optimal fashion. In numerical studies, we compare our proposed picker-routing strategy to the pickerrouting strategy applied in the case company and to an exact solution approach that neglects the precedence constraint. The analysis showed that we improved the current order-picking process in the following aspects: With the proposed picker-routing strategy, warehouse managers are able to completely avoid sorting eort and reduce the average travel tour length an order picker needs for completing customer orders. The intention of our picker-routing strategy was to develop an approach that is easy to understand and that can easily be implemented in practice. In practice, information systems such as warehouse management systems (WMSs) are often used for handling warehouse operations eciently. The implementation of our algorithm within a WMS can be easily done. The WMS can deliver all necessary orderpicking information directly to the order pickers' portable device, such as a radio frequency handheld scanner, a smartphone, or a tablet. Moreover, it is possible to extend the software to feature a graphical user interface where the warehouse layout, the orderpicking route, and the storage locations to be visited are graphically visualized. An interesting topic for future research could be the evaluation of the proposed pickerrouting strategy when integrating precedence constraints into order-batching processes. Grouping of customer orders to picking orders (batches) can reduce the total length of all tours through a warehouse. Moreover, the inuence of further weight categories on the proposed picker-routing strategy could be investigated. 22

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