A discrete cross aisle design model for order-picking warehouses

A discrete cross aisle design model for order-picking warehouses

Accepted Manuscript A Discrete Cross Aisle Design Model for Order-Picking Warehouses ¨ Ozt ¨ urko O glu, ¨ ˘ D Hoser PII: DOI: Reference: S0377-2217...

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Accepted Manuscript

A Discrete Cross Aisle Design Model for Order-Picking Warehouses ¨ Ozt ¨ urko O glu, ¨ ˘ D Hoser PII: DOI: Reference:

S0377-2217(18)30978-0 https://doi.org/10.1016/j.ejor.2018.11.037 EOR 15486

To appear in:

European Journal of Operational Research

Received date: Revised date: Accepted date:

13 June 2017 7 November 2018 16 November 2018

¨ Ozt ¨ urko Please cite this article as: O glu, D Hoser, A Discrete Cross Aisle Design Model ¨ ˘ for Order-Picking Warehouses, European Journal of Operational Research (2018), doi: https://doi.org/10.1016/j.ejor.2018.11.037

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Highlights • A new warehouse layout problem and its model are introduced. • An efficient optimal order-picking tour algorithm is developed. • New inspirational aisle designs are proposed for order-picking warehouses. • New designs offer up to 7% improvement over equivalent two-block layouts.

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• Two-block layouts seem to be good and robust for low number of picks.

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• One-block layouts appear to be the best when the number of picks is very high.

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A Discrete Cross Aisle Design Model for Order-Picking Warehouses ¨ Ozt¨ ¨ urko˘glua,∗, D. Hoserb O. a Department

Business Administration, Yasar University, Bornova, Izmir 35100, Turkey b Bornova, Izmir 35100, Turkey

Abstract

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In this paper, we develop new warehouse designs that provide a reduction in travel distance for the order-picking operation, which is the most costly operation and the one most closely associated with order delivery time. For this purpose, we propose a new layout problem called “discrete cross aisle warehouse design”. In this problem, a linear middle cross aisle is divided into segments called tunnels on each picking aisle. In order to calculate average tour length for the proposed design problem, we develop

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an efficient algorithm that solves the order-picking problem optimally. A harmony search algorithm is used to find optimal tunnel positions that minimize the average tour length under a randomized storage policy by searching the space of all possible designs. A numerical study shows that, for small size order lists, the best-found solutions have similar layouts to the traditional two-block designs. As the number of locations to be visited increases, tunnel positions move away from the center of the warehouse and construct a segregated tunnels on the layout. Compared to the traditional two-block layouts, new,

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tunnel-based designs provide savings of up to 7% average savings in order-picking tour length.

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Keywords: facilities planning and design, warehouse design, order-picking, routing, randomized storage

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1. Introduction

Increasing e-commerce sales and changing customer expectations about delivery speed have been reshaping logistics networks, logistics facilities, and their operations. According to estimates for 2016

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US holiday season, the main courier delivery companies were expected to deliver around 1,835 million packages (Lindner, 2016). Similar developments have also occurred in Europe B2C (business-to-customer)

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e-commerce sales grew from 440 billion in 2015 to 500 billion euros in 2016 (Statista, 2017). Hence, these figures can be seen as a trigger to modify operations in both transportation and material handling in facilities in order to be able to fulfill many small, individual orders as opposed to simply handling product pallets.

Particularly in warehouses that handle many small units, workers visit many locations per trip to fulfill customer orders in a system called picker-to-part order-picking systems. Despite recent technological ∗ Corresponding

author ¨ Ozt¨ ¨ urko˘ Email addresses: [email protected] (O. glu ), [email protected] (D. Hoser)

Preprint submitted to Elsevier

November 22, 2018

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advances in warehouses, this is still the most widely used system with 80% of relevant Western European warehouses using manual picker-to-part systems (De Koster et al., 2007) because of their flexibility. Due to the system’s labor intensive operations, order-picking comprises around 50% of total warehouse operating costs while traveling accounts for almost half of order-picking costs (Frazelle, 2002). Additionally, order-picking time is the main determinant “internal order cycle time”, which is highlighted to be one of the most important metrics for warehouse managers in WERC’s (Warehousing Education and Research Council) survey (Manrodt, 2016). Thus, much order-picking process design focuses on reducing travel

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time as an unproductive and a non-value-added activity (Bartholdi and Hackman, 2011). Even though reducing order-pickers’ travels have been extensively studied in the literature and efficient optimal routing algorithms have been provided, these studies and algorithms are limited to traditional warehouse designs, which have been the most commonly applied and studied layouts in both warehousing industry and academia.

Figure 1 shows two common examples of traditional designs. The traditional design of Layout A in

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Figure 1a, known as a one-block warehouse, contains only two cross aisles, at the front and rear end of the warehouse. In this layout, only the front and rear cross aisles allow order-pickers to change aisles. This layout type is a better choice for unit-load warehouses that apply single-command cycles (Pohl et al., 2009a). Layout B in Figure 1b, which is also fairly common and known as a two-block warehouse, contains an additional cross aisle passing through the center of the warehouse. By dividing the picking space into two equal parts, this middle cross aisle increases the number of opportunities for workers to

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change aisles. When more than one location is visited in a single tour, having extra cross aisles in a warehouse can reduce travel distances between locations by increasing the number of routing options.

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However, adding a large number of cross aisles may increase average travel times because the space occupied by the cross aisles has to be traversed as well (Roodbergen and de Koster, 2001a). Pohl et al. (2009a) demonstrated that the optimal place for a central cross aisle for dual-command operation, where

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the number of picks is two, is between the center and the rear end of the warehouse. Berglund and Batta (2012) also studied the optimal placement of the middle cross aisle in a picker-to-part warehouse under a

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class-based storage policy. After implementing across-aisle storage they showed that placing the middle cross aisle close to the products with higher turnover rates reduces tour length about 6–9% compared to Layout B in Figure 1b. They also found that adding a second middle cross aisle reduces the savings

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in travel by as much as 3% while adding a third one brings almost no savings. Overall, these studies show that even slightly changing the position of a straight cross aisle may affect order-picking length. However, two supposed design constraints, used by both researchers and industry for many years, were discussed by Gue and Meller (2009) as unspoken design rules in these traditional designs: (1) picking aisles are straight and arranged in parallel; (2) cross aisles are straight and positioned perpendicular to picking aisles. Several researchers have relaxed one or two of the supposed design rules in traditional designs and studied aisle designs in unit-load warehouses. In these studies, the aisles are angled to reduce travel

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(a) Layout A

(b) Layout B

Figure 1: Traditional rectangular warehouses. We adopt the terminology of Pohl et al. (2009a).

distance between a pick-up-and-deposit point and a pallet location. Gue and Meller (2009), Pohl et al. (2011), Gue et al. (2012), Ozturkoglu et al. (2012) and Ozturkoglu et al. (2014) significantly reduced

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single-command distances by using new aisle designs. Pohl et al. (2009b) and C ¸ elik and S¨ ural (2014) also considered the Fishbone design, a non-traditional aisle design proposed by Gue and Meller (2009), for dual-command and order-picking operations, respectively. It is shown that the Fishbone design may be a good candidate layout for order-picking operations where more than one location must be visited. Additionally, Henn et al. (2013) presented an inverted U-shaped layout that can reduce order-picking tour length for a wide range of medium-sizes pick numbers compared to traditional designs. However, it is

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seen that new aisle arrangements require larger storage areas than traditional designs, which also means higher investment cost, while they can potentially reduce travel distance in order-picking warehouses.

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Thus, the main objective of this study is to explore new warehouse layouts that reduces order-picking length by relaxing the second of the supposed design rules in traditional designs, namely straight cross aisles without causing any loss in storage area.

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There have been many routing algorithms proposed for traditional designs. Ratliff and Rosenthal (1983)’s pioneering study showed that an optimal order-picking route can be efficiently solved in running

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time linear in the number of aisles for one-block warehouses. De Koster and van der Poort (1998) solved the same problem using Ratliff and Rosenthal (1983)’s algorithm assuming that a picker can deposit the picked items at the front of any picking aisle. Roodbergen and de Koster (2001a) extended this algorithm

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for two-block warehouses, and developed an efficient algorithm to solve the optimal order-picker route in running time linear in the number of aisles and the number of pick locations. Gelders and Heeremans (1994) and Roodbergen and de Koster (2001b) solved the optimal order-picking route as a traveling salesman problem (TSP) by using Little et al. (1963)’s branch-and-bound algorithm. Recently, Scholz et al. (2016) developed an efficient mathematical programing formulation to solve the order-picking routing problem as a TSP problem, taking into account the specific structure of traditional warehouses. In addition to these exact algorithms, special heuristics, such as s-shape, aisle-by-aisle, largest-gap, mid-point, and return, have been proposed for generating reasonable routes in traditional warehouse

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designs (Kunder and Gudehus, 1975; Hall, 1993; Petersen, 1997; Roodbergen and de Koster, 2001b). These heuristics provide good, straightforward and easily memorable routes for workers, whereas they mostly generate solutions far from being optimal, especially in complex layouts. On the basis of these heuristics, Kunder and Gudehus (1975), Hall (1993), Le Duc and de Koster (2005) and Roodbergen et al. (2008) developed expected route length equations for one-, two- and multi-block warehouses. In most of these studies, the researchers focused on calculating optimal or close-optimal order-picking length in predefined traditional one-block or multi-block layouts under different storage policies, such

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as randomized, dedicated and class-based. Because the main order-picking routing algorithms rely on a specific layout, they lack the flexibility to be used if the layout changes. This makes it necessary to develop a new algorithm whenever a new layout structure is introduced. Thus, the second objective of this study is to develop a new and an efficient routing algorithm for order-picking operation in our proposed layout problem.

In this study, we first propose a new layout problem that differs from any of the warehouse layout

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problems in current research. We then investigate whether better layouts than traditional layouts exist for order-picking operations. We introduce this problem and its model in Section 2. We then present the developed algorithm for calculating order-picking tour length in a new warehouse layout problem in Section 3 before discussing how to find the best design variables for minimizing average order-picking tour length in Section 4. Here, we discuss our implementation of our Harmony Search algorithm. In Section 5, we compare our new aisle configurations to equivalent traditional designs before ending with

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our conclusions regarding the model’s potential contribution, its limitations, and future directions of the

2. Problem Description

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study.

In a picker-to-part order-picking system, the order-pickers move through the picking area to collect

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the required products in an appropriate order from the storage locations to make the order ready for shipping. The order-picker then returns to the pick-up and deposit (P&D) point to deposit the picked

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items and start picking the next batch. A pick list given to the order-pickers contains the order lines that should be processed in each tour. An order line includes information about the requested item, its quantity and storage location. The total distance an order-picker travels to process the pick list, starting

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and ending at the P&D point, is called the order-picking tour length. Because order-picking time is proportional to the tour length, the items requested in the pick list should be retrieved in such a way as to minimize the total tour length. As discussed above, previous studies have shown that the positions, number, and angles of straight cross aisles can reduce expected travel distances. Although having more than one middle-cross aisle perpendicular to the picking aisles reduces travel distance for order-picking, it also reduces total storage capacity due to including devoted aisle spaces, which may sometimes increase tour length due to additional cross aisles (Roodbergen and de Koster, 2001a). New non-traditional aisle designs also show that 5

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Figure 2: Discrete cross aisle design and its parameters.

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there is no good reason to adhere to the traditional design constraints noted by Gue and Meller (2009) so the way we think about the aisle design problem should change. That is, we can simply decide to arrange an existing middle cross aisle in such a way that makes order-picking efficient without reducing storage capacity in a traditional two-block layout. Figure 2b depicts the new layout problem for order-picking warehouses that we call the “discrete cross aisle design problem”.

In this design problem, the middle straight cross aisle in a two-block layout is divided into aisle

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segments in every aisle, such that every segment, called “tunnels” hereafter, can be arranged along the picking aisles independently in order to facilitate travel between picking aisles. By preserving the total

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space occupied by the aisles as in a two-block layout, the new layout problem focuses on finding the best positions for the tunnels in order to make the order-picking tour length less than the equivalent two-block layout. In defining the model for the discrete cross-aisle design problem, we made the following design

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assumptions. Additionally, the design parameters and their units used to depict the layout problem in Figure 2b are given in Table 1.

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1. There is only one tunnel on a picking aisle. 2. The warehouse uses a single-deep rack system, so the left- and right-most racks are attached to the

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walls. There are no tunnels on the left-most and right-most racks because there is nothing beyond them to access.

3. There is only one P&D point, where picked items are deposited or pickers receive order lists, which can be placed at the front of any picking aisle.

4. The aisles and tunnels are wide enough to be traversed in both directions simultaneously. 5. All picking aisles are the same width and run parallel to each other. 6. A tunnel is not allowed to overlap with its respective front or rear cross aisles. Additionally, there is always at least one pallet location both below and above each tunnel.

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Table 1: Design parameters and variables in the discrete cross aisle design problem.

depth of warehouse defined by number of pallets width of warehouse determined by either number of picking aisles or number of pallets width of tunnels defined by number of pallets width of picking aisles defined by number of pallets width of picking face of a storage location defined by unit length depth of storage location defined by unit length width of front and rear cross aisles defined by number of pallets number of picking aisles number of storage locations available on the left- or right-most aisle total number of storage locations (number of pallets) position of the tunnel on the ith picking aisle defined by number of pallets.

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H W a b c d e m q n hi

For the given design parameters, the main goal in our problem is to find the best tunnel positions. Let P be the set of tunnel positions P = {h1 , h2 , ..., hm−1 } where hi ∈ [2, q − a], ∀i ∈ 1, 2, .., m − 1 due to design assumption #6. With this representation, all possible layout configurations under our design assumptions can be easily generated. In order to evaluate the fitness of a given design, we consider

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“average order-picking tour length” which is the arithmetic average of the total distances a picker travels to fulfill a given number of orders. Because there is no existing quick and exact algorithm, to the best of our knowledge, for the proposed layout problem, we develop a new algorithm to calculate optimal tour length for any given any tunnel location under our design assumptions. The next section explains the

3. Order Picking Routing Model

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details of this algorithm.

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In spite of the similarity between the layout in the proposed design problem and the two-block layout, existing optimal tour algorithms cannot be used as they are due to varying tunnel locations in the new problem. Hence, we adopt the original approach of Ratliff and Rosenthal (1983), and modify

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the algorithm of Roodbergen and de Koster (2001a) to solve the optimal tour length in our problem. Because our algorithm relies on the aforementioned algorithms, we briefly explain these to make our

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proposed algorithm clear.

Ratliff and Rosenthal (1983) modeled the order-picking problem as a variant of the well-known traveling salesman problem (TSP) and presented an algorithm for optimally solving the problem in one-block

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warehouses (see Figure 1a) which is linear in the number of aisles. They also argued that the number of items has little effect on the solution time of the algorithm. By representing the storage locations, and the aisles’ entrance and exit points as vertices in a graph with appropriate arcs, they used the theorem on Eulerian graphs to generate subgraphs. They then characterized these subgraphs as “partial tour subgraphs (PTS)” in seven equivalence classes. Starting from the left-most aisle and moving along aisles at every stage, PTSs are extended sequentially by adding all possible transitions. Finally, they showed that the optimal order-picking tour and its length are found among the specific equivalence classes at the last stage because of the characteristics of the Eulerian circuit.

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Figure 3: Representation of a discrete cross-aisle design and its network. White squares represent storage locations. Black squares are the picking locations. White circles are transit vertices to facilitate travel among aisles. Black circles are the access vertices to the locations that must be visited.

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Roodbergen and de Koster (2001a) extended the algorithm of Ratliff and Rosenthal (1983) for twoblock layouts. Since a middle cross aisle brings additional access vertices to the aisles, they showed that number of equivalence classes increases to 25 in two-block layouts. The algorithm considers all aisles and items, and a constant number of operations, which are evaluation of equivalent classes and transitions, has to be performed for each aisle and item. Although the size of the problem increases due to increasing number of equivalence classes, they presented that problem complexity still grows linearly depending on

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the number of picking aisles and the number of items.

In order to solve the optimal tour length for a given order in our problem, we also first represent the

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given warehouse layout as a network of vertices and edges. Let G be a graph of a set of vertices V and a set of edges E; G = (V, E). Vertices consist of a single P&D point, access areas to storage locations of items to be picked, transit points at the front and rear of the warehouse, and tunnel access/exit points.

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Edges are determined by connecting two consecutive vertices, and the edge weight is the distance between them. For a given order that includes p locations where requested items are placed, let S be the set of

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vertices of picking locations and the P&D point which are called non-Steiner vertices; S ⊆ G. That is, vertices in G \ S are defined as Steiner vertices so the problem is a Steiner Traveling Salesman Problem, and described as finding the minimum length of the Steiner-tour that visits non-Steiner vertices at least

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once (Cornu´ejols et al., 1985; De Koster et al., 2007; Theys et al., 2010). Figure 3 presents an example representation of a warehouse and its graph with the locations for an example order. For our model, we adopt the same terminology of Ratliff and Rosenthal (1983), and Roodbergen

and de Koster (2001a). As seen in Figure 3, the single P&D point that each tour starts and ends at is represented by V0 and can be located at the front of any picking aisle. aj and cj vertices for all j = 1, ..m are used to access picking aisle j on the rear and front cross aisles, respectively (note that picking aisles are numbered from left to right). bj , j = 1, ..., m − 1 represents the tunnel inlet access point on the j th picking aisle. In contrast to Roodbergen and de Koster (2001a), because a tunnel can be a dead-end, its

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Figure 4: Six possible vertical transitions within an aisle between adjacent vertices, i.e. cj and bj . Adapted from Roodbergen and de Koster (2001a). If there is no item to be picked between two adjacent transit vertices, the only possible transitions are (1), (2) and (6). If there is exactly one item, transitions (1), (2), (3) and (4) are allowed. Transition (5) is only possible if there are two or more picks in this part of the aisle. Finally, all transitions except for (6) are possible for more than one pick.

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outlet point on the following aisle is defined by βj−1 , j = 2, .., m. Hence, vertex βj−1 may not provide access to aisle j + 1. Of course, this does not restrict continuous middle cross aisle that can occur when bj and βj−1 overlap. Hence, aj , bj , cj and βj are called transit vertices, hereafter. In graph G, an order-picking tour is a special subgraph T (tour subgraph) that contains the P&D point and each of the picking locations at least once to form a cycle (see Theorem A.1 in Appendix). Additionally, each vertex in T must have even or zero degree. Hence, the length of the tour is the sum of

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the weights of the edges included in the cycle or T . It is also noted that a minimum length tour subgraph contains no more than two arcs between any pair of vertices (see Corollary A.1 in Appendix). Last, while

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each tour subgraph provides a possible route for order-picking, the one with the minimum length gives the optimal order-picking tour. The details of the algorithm developed to construct the tour subgraphs

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and find the optimal tour are presented in sections 3.1 – 3.3. 3.1. Constructing partial tour subgraphs In Roodbergen and de Koster (2001a)’s algorithm for two-block warehouses, L− j is defined as a set

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of subgraphs of G consisting of aj , bj , cj and all the vertices and (some) edges located to the left of picking aisle j. In their algorithm, L+y and L+x sets of PTSs are constructed consecutively for aisle j j

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j. To construct PTSs in L+y j , vertices cj and bj together with all vertices between them are added to PTSs in L− j using the six possible vertical transitions described in Figure 4. These six transitions are the only possible edge combinations between any two adjacent transit vertices (i.e. c1 and b1 ) within a picking aisle because any pair of vertices does not need more than two edges (see Corollary A.1). For L+x PTSs, vertices bj and aj together with all vertices between them are added to L+y PTSs using the j j +x same six vertical transitions. In the next step, L− PTSs with j+1 PTSs are determined by extending Lj

edges between aisles j and j + 1 using the horizontal transitions described in Figure 5. Continuing this way, after obtaining L+x m PTSs for the last aisle, the algorithm terminates.

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Figure 5: Fourteen possible horizontal transitions between two neighboring aisles. Adapted from Roodbergen and de Koster (2001a) by defining tunnel vertex βj .

Because of the varying positions of βj in our model, we develop a new algorithm for constructing

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PTSs in a discrete cross aisle design. Let L− j be a set of subgraphs of G consisting of aj , bj , cj , βj−1 , and all the vertices and (some) edges located to the left of picking aisle j (see examples in Figures 6a

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+z and 7a). We then develop three sets of PTSs within every aisle j, which are L+y and L+x j , Lj j . These

PTSs are constructed based on the position of βj−1 in aisle j according to following procedure. If bj is +z located closer to the front cross aisle than βj−1 (bj is below βj−1 ) L+y and L+x are constructed j , Lj j

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are constructed next, respectively (L+z → L+y → L+x j j j ). The procedure that defines the sequences for

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constructing PTSs with respect to the position of βj−1 are explained in details as followings. a) L+y PTSs are obtained by adding the vertical transitions shown in Figure 4; j

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• between cj and bj in subgraphs L− j if bj is below βj−1 , (see example in Figure 7b);

• between βj−1 and bj in subgraphs L+z if bj is above βj−1 , (see example in Figure 6c). j

b) L+z partial subgraphs are obtained by adding the vertical transitions shown in Figure 4; j • between bj and βj−1 in subgraphs L+y if bj is below βj−1 , (see example in Figure 7c); j • between cj and βj−1 in subgraphs L− j if bj is above βj−1 , (see example in Figure 6b). c) L+x partial subgraphs are obtained by adding the vertical transitions shown in Figure 4; j • between βj−1 and aj in subgraphs L+z if bj is below βj−1 , (see example in Figure 7d); j 10

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Figure 7: Examples of L3 PTSs for the graph in Figure 3.

• between bj and aj in subgraphs L+y if bj is above βj−1 , (see example in Figure 6d). j

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+y +z +x The denotation Lj is used to indicate that a result holds for Lj = L− j , Lj = Lj , Lj = Lj or Lj = Lj .

As in Roodbergen and de Koster (2001a), the procedure is followed such a way that L− j+1 PTSs are

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constructed by extending L+x PTSs with the horizontal transitions given in Figure 5. The algorithm is j continued to construct L+x m PTSs in the last aisle. The first aisle is a special case where the PTSs do not include β0 . Hence, the algorithm starts by

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ˆ ˆ constructing an L− 1 PTSs with a set of vertices V = {a1 , b1 , c1 } and edges E = {∅}. The valid sets of

+x the PTSs in the first aisle are L+y 1 and L1 . The algorithm ends after completing the PTSs in the last +x aisle which is also special because bm does not exist. Thus, there L+z m and Lm PTSs exist. We design

the problem such that we can use dynamic programming in order to determine the optimal order-picking tour and its length. For this, we define all feasible states as equivalence classes. 3.2. Equivalence Classes Equivalence classes are described by the unique PTSs in the set of Lj PTSs. In addition to Roodbergen and de Koster (2001a)’s five-fold representation of equivalence classes, we use six-fold representation to 11

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characterize the classes of Lj PTSs due to the existence of the new variable βj . The first four elements in the six-fold representation indicate the degree parities of vertices aj , bj , cj , and βj−1 , respectively. Degree parity shows if a vertex has odd (u for uneven), even (e), or zero (0) number of edges incident to it. Since there is no vertex β in the first aisle and no vertex b in the last aisle, their degree parities are taken as zero in the equivalence classes of L1 and Lm PTSs, respectively. The fifth element shows the connectivity in terms of the number of connected components of the PTS. The number of connected components in a PTS varies between 0 and 4. The last element explains the distribution of aj , bj ,

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cj , and βj−1 vertices within the connected components. For simplicity in notation of some equivalence classes, we omit the last element if there is only one possible distribution of aj , bj , cj , and βj−1 in the components. The details of this with examples are explained in the following discussions.

For example, Figure 6 depicts four instances of L2 of the graph given in Figure 3. The equivalence class of the example PTS in Figure 6a is represented by (e, 0, u, u, 2). This representation shows that the degree parities of vertices c2 and β1 are odd (degree 1), and vertex a2 is even (degree 2). Since there is no

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incident edge to vertex b2 , its degree is zero. Excluding the vertices with zero degree (b2 ), the example PTS has 2 different connected components: c2 and β1 are in one component while a2 is in the other one. Similar to this, the equivalence classes of the PTSs given in Figure 6b, 6c and 6d are (e, 0, u, u, 2), (e, u, u, e, 2, a − bcβ) and (u, e, u, e, 1), respectively. Additionally, as the PTS in Figure 7a is represented by (u, 0, u, 0, 1), the others in Figure 7b, 7c and 7d are represented by (u, u, e, 0, 1). As seen in some of these examples, the sixth element was not included in the notations. The cases that generate a single

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possible distribution are explained in detail in the following discussion.

(i) A PTS that has zero connected component can only occur if the degrees of all vertices are zero.

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(ii) If a PTS has only one connected component, all vertices, excluding those with zero degree, are located in a single connected PTS.

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(iii) For the two-component Lj PTSs, the distribution of the transit vertices (aj , bj , cj , and βj−1 ) should be clearly specified in case one of the following four conditions is met (see Theorem A.3 in

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Appendix).

• Three of the transit vertices have even degree parity while one has zero degree parity,

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• Two have even degree parity while the other two have odd degree parity, • Each has uneven degree parity, • Each has even degree parity.

Otherwise, the sixth element in the encoding can be omitted because there is only one valid possible distribution. Any other distribution possibility different from that would violate condition (b) of Theorem A.2. To make this case clear, let us consider the equivalence class (0, e, e, e, 2). This class has even degree parity in three vertices (bj , cj and βj−1 ), zero degree parity in one vertex (aj ), and consists of two

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Table 2: Potential distribution of components.

a − bc ac − bβ b − βc c − bβ

a − βc a − β − bc b − aβc c − abβ

a − bβ a − b − βc b − β − ac β − bc

a − bβc a − c − bβ b − c − aβ β − ac

aβ − bc b − ac c − ab β − abc

ab − βc b − aβ c − aβ

connected components. This class is an example of the first above-mentioned condition. Since bj has even degree parity, it can be in one component, either by itself or with one other vertex.

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This leads to three different valid possibilities in the distribution of vertices between components: b − cβ, c − bβ or β − bc. Hence, these distributions must be stated clearly in the sixth element of (0, e, e, e, 2). Additionally, let us consider the previously explained class (e, 0, u, u, 2), depicted in Figure 6a. None of the above-mentioned conditions is applicable to this class; therefore, there is only one valid distribution in this class: (a − cβ). In detail, because cj has odd degree parity, at least one other vertex in the same component must have odd degree parity as well. Theorem A.2(b)

must be in the same component as cj .

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implies that the other vertex must be either aj , bj , or βj−1 . Only βj−1 has odd degree parity so it

(iv) For the three-component Lj PTS, we only need to clearly identify the distributions of the vertices if all vertices have even degree parity (see Theorem A.3 in Appendix). The valid distributions are (b−c−aβ), (a−β −bc), (a−b−βc), (a−c−bβ), and (b−β −ac). Otherwise, the other distributions do not need to be mentioned because they have only one possibility.

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(v) A PTS that has four connected components can only occur if the degrees of all vertices are even.

different possibilities.

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Thus, Table 2 shows the distributions of vertices among connected components in case there are

Using the theorems and corollaries in Appendix A, we developed 111 equivalence classes for our

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problem, which is more than Roodbergen and de Koster (2001a) because we introduce a new transit vertex (βj ) and the graph varies due to the position of βj . Table 3 presents the 111 equivalence classes

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used to characterize any Lj .

3.3. Constructing the Minimum-Length Tour Subgraph In order to obtain the minimum-length tour subgraph, similar to Roodbergen and de Koster (2001a),

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the algorithm uses the concept of dynamic programming where potential states are defined by equivalence classes, transitions are possible horizontal and vertical transitions, and the cost is order-picking tour − lengths. The algorithm starts by constructing a L− 1 PTS. By definition, L1 PTS is a subgraph that

includes vertices a1 , b1 and c1 . It also does not include any edge. Hence, L− 1 is described by the equivalence class (0, 0, 0, 0, 0). Next, because there is no tunnel outlet in the first aisle, we directly add

+y the vertical transitions in Figure 4 between c1 and b1 in L− 1 PTS to construct L1 PTSs. Hence, adding +y each vertical transition to an L− 1 PTS generates different L1 PTSs. The cost of each transition is equal

to the sum of the lengths of the added edges. In the next step, L+y 1 PTSs are extended by adding each 13

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Table 3: Possible equivalence classes for Lj

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Equivalence Class (e, e, e, e, 4) (e, u, u, e, 2, a − bβc) (e, e, e, e, 2, a − bβc) (e, e, e, e, 3, a − b − βc) (e, u, e, u, 2, a − bβc) (e, e, u, u, 2, a − bβc) (e, e, u, u, 3) (0, u, 0, u, 1) (0, e, 0, e, 1) (e, u, 0, u, 1) (e, e, 0, e, 1) (u, u, 0, e, 1) (u, e, 0, u, 1) (e, u, 0, u, 2) (e, e, 0, e, 2a − bβ) (0, u, e, u, 2) (0, e, e, e, 2, c − bβ) (e, u, e, u, 2, c − abβ) (e, e, e, e, 2, c − abβ) (u, u, e, e, 2, c − abβ) (u, e, e, u, 2, c − abβ) (e, u, e, u, 3) (e, e, e, e, 3, a − c − bβ) (e, u, e, u, 2, ac − bβ) (e, e, e, e, 2, ac − bβ) (e, 0, e, e, 2, β − ac) (e, e, e, e, 3, b − β − ac) (e, e, e, e, 2, β − abc) (u, u, u, u, 2, ac − bβ) (u, e, u, e, 2, ac − bβ) (u, 0, u, e, 2) (u, e, u, e, 3) (e, u, u, e, 2, β − abc) (u, e, u, e, 2, β − abc) (u, u, e, e, 2, β − abc) (u, u, e, e, 2, ab − βc) (e, e, e, e, 2, ab − βc)

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Class No. 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111

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Equivalence Class (e, 0, e, e, 2, c − aβ) (e, 0, u, u, 2) (u, 0, e, u, 2) (e, 0, e, e, 3) (0, u, u, e, 2) (0, e, e, e, 2, β − bc) (0, e, 0, e, 2) (0, e, e, e, 3) (e, u, u, e, 2, aβ − bc) (e, e, e, e, 2, aβ − bc) (e, e, 0, e, 2, b − aβ) (e, e, e, e, 3, b − c − aβ) (0, u, u, e, 1) (0, e, e, e, 1) (0, e, e, e, 2, b − βc) (e, u, u, e, 1) (e, e, e, e, 1) (e, e, e, e, 2, b − aβc) (u, u, u, u, 2, aβ − bc) (u, e, e, u, 2, aβ − bc) (u, e, 0, u, 2) (u, e, e, u, 3) (0, u, e, u, 1) (0, e, u, u, 1) (0, e, u, u, 2) (e, u, e, u, 1) (e, e, u, u, 1) (e, e, u, u, 2, b − aβc) (u, u, e, e, 1) (u, e, u, e, 1) (u, e, u, e, 2, b − aβc) (u, u, u, u, 1) (u, e, e, u, 1) (u, e, e, u, 2, b − aβc) (e, u, u, e, 3) (e, e, e, e, 3, a − β − bc) (e, e, 0, e, 3)

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Class No. 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74

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Equivalence Class (0, 0, 0, 0, 0) (0, 0, 0, 0, 1) (e, 0, 0, 0, 1) (0, e, 0, 0, 1) (0, 0, e, 0, 1) (e, e, 0, 0, 1) (e, 0, e, 0, 1) (0, e, e, 0, 1) (e, e, e, 0, 1) (u, u, 0, 0, 1) (u, 0, u, 0, 1) (0, u, u, 0, 1) (e, u, u, 0, 1) (u, e, u, 0, 1) (u, u, e, 0, 1) (e, e, 0, 0, 2) (e, 0, e, 0, 2) (0, e, e, 0, 2) (e, e, e, 0, 2, a − bc) (e, e, e, 0, 2, b − ac) (e, e, e, 0, 2, c − ab) (e, u, u, 0, 2) (u, e, u, 0, 2) (u, u, e, 0, 2) (e, e, e, 0, 3) (0, 0, 0, e, 1) (e, 0, 0, e, 1) (0, 0, e, e, 1) (e, 0, e, e, 1) (u, 0, 0, u, 1) (0, 0, u, u, 1) (e, 0, u, u, 1) (u, 0, u, e, 1) (u, 0, e, u, 1) (e, 0, 0, e, 2) (0, 0, e, e, 2) (e, 0, e, e, 2, a − βc)

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Class No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

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vertical transition between b1 and a1 in order to derive L+x 1 PTSs. At the end of each step, if more than one subgraph is represented by the same equivalence class, we select the PTS with the smallest sum of edge weights for the corresponding equivalence class. After constructing L+x PTSs in the first (left-most) aisle, we move to the second aisle from the left. 1 Then, L− 2 PTSs are determined by sequentially adding all the possible horizontal transitions in Figure 5 that represent travel from aisle j to j + 1. As seen in the figure, only aj , bj , and cj transit vertices can be used to move from aisle j to j + 1. Meanwhile, βj−1 cannot be used to change aisle since they

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only represent the outlet of a tunnel. Even if bj and βj−1 overlap, for algorithmic purposes, βj−1 is not allowed for changing aisles.

We then determine the best L− 2 PTS for each equivalence class. Next, considering the relative positions +z +x of bj to βj−1 and the procedure discussed in Section 3.1, we sequentially construct L+y 2 → L2 → L2

or L+z → L+y → L+x PTSs within the second aisle by adding appropriate vertical transitions (1) 2 2 2

through (6). At every stage the best PTS is determined for each equivalence class. Moreover, L− 3 PTSs

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are determined by adding appropriate horizontal transitions to L+x 2 . Following the same procedure, we obtain L+x m PTSs in the last step.

+y +z The resulting equivalence classes we obtain by transitions L− and L− are given in j → Lj j → Lj

+y +z Tables D.2 and D.3, respectively. The equivalence classes in transitions L+y and L+z are j → Lj j → Lj

given in Tables D.4 and D.5, respectively. The last vertical transitions, L+z → L+x and L+y → L+x j j j j , generate the equivalence classes shown in Tables D.6 and D.7, respectively. The horizontal transitions

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between aisles j and j + 1 construct the equivalence classes given in Table D.8. In constructing these tables, we adhered to the approach of Roodbergen and de Koster (2001a) to make them easily comparable

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and understandable. For the sake of smooth flow in the manuscript, these tables are presented in the appendix. The detailed pseudo-code of the algorithm that forms PTSs through transitions is also given in Appendix E.

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Similar to Roodbergen and de Koster (2001a), using Theorem A.1, it can be shown that the optimal order-picking tour is one of the following L+x m PTSs that has the minimum tour length: (e, 0, e, e, 1),

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(0, 0, e, e, 1), (e, 0, 0, e, 1), (0, 0, 0, e, 1), (e, 0, e, 0, 1), (0, 0, e, 0, 1), (e, 0, 0, 0, 1), (0, 0, 0, 0, 1) or (0, 0, 0, 0, 0). The optimal tour can be determined going back from the equivalence class that results in minimum tour length.

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3.4. Computational performance of the algorithm The algorithm performs constant number of operations to evaluate all possible equivalence classes,

which are explicitly demonstrated in the aforementioned tables in section 3.3, obtained by adding appropriate transitions in each picking aisle. The algorithm terminates after considering the right-most picking aisle that includes a picking location. Thus, the computational time to obtain the optimal order-picking tour increases linearly especially as the number of picking aisles and the number of items increase. In order to evaluate the computational performance of the proposed algorithm, we generated an example warehouse layout with 11 aisles, and a width-to-depth (shape) ratio of 2:1. We located the P&D 15

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point in the middle of the front cross aisle. The layout includes 512 storage locations in the bottom racks (forward picking area). We then randomly generated 1,000 orders for varying pick list sizes from 3 to 50. Thus, an order with a pick list size of 50 requires 51 visits due to the single P&D point. We then calculated the tour lengths of all orders and took their average. We also used the improved Held-Karp algorithm proposed by Volgenant and Jonker (1982) as the basis for comparison. This algorithm uses 1-tree relaxation in a branch-and-bound algorithm that guarantees an optimal solution. It also uses both a depth-first and a breadth-first search algorithm to speed up the solution. Hence, it provides faster

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solutions than the original Held-Karp algorithm proposed by Held and Karp (1970). Both algorithms and the other algorithms used in this study were coded in the JAVA environment and run on 16GB RAM and a 3.60 GHz Intel® CORE i7 processor. The average length of optimal tours and total computational time requirements for each pick list size are given in Table 4. As seen in the table, the proposed algorithm is quite effective and requires much less time than the Held-Karp algorithm, especially for large pick lists. Because of the nature of the Steiner-TSP problem, we could not obtain optimal solutions for large pick

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lists and 1,000 orders using the Held-Karp algorithm due to memory problems. The proposed algorithm presented optimal solutions as expected. Besides, the computational time requirement does not seem to be very sensitive to pick list sizes. For example, the computational requirement only doubled as the pick list size increased from 10 to 40.

Table 4: Computational comparisons of the proposed algorithm and the Held-Karp algorithm. NA indicates that optimal solutions were not obtained due to memory problems.

CPU Time (sec) 1.16 2.05 6.64 29.12 13783.16 154285.70 NA NA

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Average Tour Length 1089.86 1372.72 1865.30 2245.86 2549.66 2998.60 NA NA

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Average Tour Length 1089.86 1372.72 1865.30 2245.86 2549.66 2998.60 3325.16 3561.66

CPU Time (sec) 0.63 0.81 0.97 1.19 1.28 1.67 2.03 2.17

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Pick List Size 3 5 10 15 20 30 40 50

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Held-Karp

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4. Solution Methodology: Harmony Search Optimization Our main focus in this study is to find the optimum discrete cross-aisle layouts that make the order-

picking process efficient rather than to study on a given layout. Thus, we search for the best tunnel positions (hi ) on each picking aisle in such a way that average order-picking tour length is minimized.

Tunnel positions, as described in Section 2, can take discrete values between 2 and (q − a). In order to explore the search space efficiently, we use the Harmony Search (HS) algorithm because it is one of the most widely preferred meta-heuristic algorithms for design problems (Saka et al., 2011). The algorithm was initially proposed by Geem et al. (2001) to solve discrete combinatorial optimization problems. It imitates the behavior of musicians when composing their music and relates this to 16

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optimization. In the process of searching for a perfect state of harmony, musicians usually try various combinations of pitches by playing notes based on experience, adjusting pitches slightly to find a good harmony, and playing notes randomly. This process is simulated in each variable selection of the solution vector in the standard HS algorithm by the following three components: usage of harmony memory, pitch adjustment, and random selection. Harmony memory indicates a list of individual solutions that builds the population. Pitch adjustment refers to a mutation operation that randomly selects a value in a predetermined bandwidth (bw). Using this value, the variable in the solution is changed. Harmony

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memory considering rate (hr) is used to generate a value for a variable from the harmony memory. Pitch adjustment rate (pr) is used to slightly change (mutate) the value of a variable. In the random selection operation, a variable takes a randomly selected value within its range with a probability of (1 − hr). The population size of the algorithm is defined by harmony memory size (hs). For initialization, individuals in the population are randomly generated within their bounds.

The structure of the algorithm that we used for our optimization problem resembles the standard

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form of the HS algorithm proposed by Geem et al. (2001). The basic elements of our algorithm are harmony memory, number of harmonies in the memory, the best harmony, harmony memory considering rate, and pitch adjustment rate.

Harmony: It represents the solution vector Hit that denotes the ith harmony in the memory at iteration t. In our problem, it is represented by m − 1 dimensions because there are tunnels on m − 1 aisles. Hit = {hti1 , hti2 , ..., hti(m−1) }, where hti1 is the tunnel position on aisle 1 for the ith solution at

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t t iteration t. Additionally, Hbest is the best solution at iteration t; Hbest = {htbest,1 , htbest,2 , ..., htbest,m−1 }.

Harmony Memory: It is the set of hs number of harmonies at iteration t, and represented by HM t =

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t {H1t , H2t , ..., Hhs }.

Coefficients: Each variable in a new harmony solution is randomly chosen from the solutions in harmony memory with probability hr. If a variable does not take a value from harmony memory, it

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is chosen from the best harmony with a probability of (1 − hr). Next, in the pitch adjustment step, these variables are adjusted and moved to neighboring values with a probability of pr in order to gain

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the capability to move from a local optimum. These parameters thus aim to give the algorithm both exploration and convergence capability. The pseudo-code of the proposed HS algorithm is presented in detail in Algorithm 1. We also

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implement a local search to improve the quality of solutions in every specific number of iterations through the optimization. The pseudo-code of the local search algorithm is also explained in detail in Algorithm 2.

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Algorithm 1 Proposed Harmony Memory Search Algorithm

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Require: m as the number of aisles, q as the number of pallet locations on the left-most column rack, a as the tunnel width, o as the order size, hr as the harmony memory considering rate, pr as the pitch adjustment rate, bw as the bandwidth, hs as the population size, and N I as the number of iterations. for individuals i = 1 to hs in the population do for tunnel positions j = 1 to (m − 1) do Generate a random initial solution i; hij = randInteger(2, q − a), Hi0 ∪ {h0ij }. end for for order k = 1 to o do Call Algorithm 3 in Appendix E to calculate the optimal order-picking tour length of order k for Hi0 . end for Calculate average of o orders’ tour lengths and assign it as the fitness value of Hi0 . end for Return Initial harmony memory HM 0 . Determine the best solution vector H best that has the lowest average order-picking tour length in HM 0 t←1 while t ≤ N I do Let X be a new solution that includes tunnel positions, X = {x1 , x2 , ..., xm−1 }. for tunnel positions j = 1 to (m − 1) do if rand ∈ (0, 1) ≤ hr then Randomly select an individual from HM t−1 . i = randInteger(1, hs). xj = htij . if rand ∈ (0, 1) ≤ pr then xj := xj ± randInteger(1, bw) such that xj ∈ [2, q − a]. end if else xj := htbest,j if rand ∈ (0, 1) ≤ pr then xj := xj ± randInteger(1, bw) such that xj ∈ [2, q − a]. end if end if end for if t mod 10 = 0 (every 10 iterations.) then Apply local search algorithm 2 to X. end if for order k = 1 to o do Call Algorithm 3 to calculate the optimal order-picking tour length of order k for X. end for Calculate average of o orders’ tour lengths and assign it as the fitness value of X. t if F itness(X) < F itness(Hbest ) then t Hbest := X. else Determine the solution that has the lowest fitness value in the memory; Hwt ∈ HM t . if F itness(X) < F itness(Hwt ) then Hwt := X. end if end if t←t+1 end while NI Return The best-found solution. Hbest .

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Algorithm 2 Proposed Local Search Algorithm

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Require: X as the candidate solution vector that is generated in Algorithm 1, min and max as the minimum and maximum number of variables that will be changed, bw as the bandwidth, m as the number of aisles, N I as the number of iterations, and CI as the current iteration number. Let s be the step size to determine the number of variables that will be changed. s = b CI·(max−min+1) c. NI ˆ Let X be the copy of solution X. j ← 1. while # of variables j ≤ (min + s) do r = randInteger(1, m − 1) for counter i = −bw to bw do ˆ xˆr := xr + i, then update X. for order k = 1 to o do Call Algorithm 3 in Appendix E to calculate the optimal order-picking tour length of order k ˆ for X. end for ˆ Calculate average of o orders’ tour lengths and assign it as the fitness value X. ˆ if F itness(X) < F itness(X) then ˆ X := X. end if end for j ←j+1 end while To implement the HS algorithm in our study, we select constant coefficients which do not change over iterations. Because Yang (2009) and Lee and Geem (2005) recommended that hr ∈ [0.7, 0.95] and

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pr ∈ [0.2, 0.5] for an efficient HS algorithm, we assigned their central values 0.8 to hr and 0.35 to pr. For pitch adjustment (mutation) operation, we take bw as d0.3(q − a)e, which determines the size of

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the neighborhood. In our implementation, a pitch is randomly chosen from the range (−bw, bw). After performing pitch adjustment operation, if the variable goes out of its limits it is repaired by the so-called absorbing constraint handling scheme (Gandomi and Yang, 2012), which corrects the value of the variable

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to its closest bound. Additionally, we took 30 for harmony memory size (hs) as a population size because Geem (2012) showed that the performance of the HS algorithm does not vary much with hs between 30

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and 100 when efficient hr and pr are used. We applied our local search algorithm at every 10 iterations. During the local search, the algorithm gradually changes the value of the determined variable in a given solution starting from (−bw) to (bw).

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The number of variables that will be locally searched is dynamically increased as the number of iterations increases. For this implementation, max and min values in Algorithm 2 were taken as d0.7(m − 1)e and

b0.2(m − 1)c, respectively. Thus, the algorithm searches for max of 70% of the variables in a solution at the final iterations while it searches minimum of 20% of the variables at the beginning iterations. As a termination criterion, we use both the total number of iterations and the number of iterations for which the global best found was not improved. The HS algorithm runs until both limits are exceeded. In our implementation, we terminate the search after 5000 iterations if there is no improved solution in the last 100 iterations. In order to see the reproducibility of the solutions by the algorithm, we performed ten

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Table 5: Characteristics of the four warehouse instances.

replications using ten different random number generation seeds. 5. Numerical Experiments: New Designs

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Warehouse Instances (Layout B) parameters WH1 WH2 WH3 WH4 m 5 7 9 11 W 28 38 48 58 H 14 19 24 29 q 11 16 21 26 n 86 188 330 512 h (in Layout B) 5 8 10 13

This section describes the search for the best locations for tunnels that minimize the average tour length in a discrete cross aisle design problem. For the numerical study, we consider a rectangular

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warehouse with storage racks placed in a single-deep rack system. Picking operations are assumed to be performed from the lowest level of the racking system, called the forward-picking area. For ease of representation, storage locations have a square-shaped footprint, and one pallet unit (1 PU = 10 pixels) is the adopted measure of distance. That is, c and d are taken as one PU. The widths of picking aisles (b), tunnels (a), and front and rear cross aisles (e) are all assumed to be three PUs. We assume a randomized storage policy because of its simplicity and popularity in the industry, and its higher utilization of storage

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locations (Petersen, 1999). We therefore assume uniform picking. We also locate the single P&D point in the middle of the front cross aisle because Roodbergen and Vis (2006) showed that this is the optimal

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location for a single P&D point that minimizes the order-picking tour length under randomized storage in one-block warehouses. Additionally, we make the warehouse twice as wide as its deep, giving a shape ratio (width/depth) of 2:1, which is not only common in industry but also taken as optimal by several

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researchers especially when there is a central P&D point and single-command operation (Francis, 1967; Bassan et al., 1980).

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We also consider four different warehouse sizes to see their effects on the allocation of tunnels. Table 5 shows the characteristics of the four layout instances that we study in this paper. Because we assume that there is only one tunnel at each picking aisle, the tunnel designs consume the same amount of aisle

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space as an equivalent Layout B. Therefore, Layout B is primarily taken as the basis for comparisons in the design problems. h in Table 5 shows the position of the middle cross aisle in Layout B. To make this

clear, there are 4 pallet locations between the front and the middle cross aisles for WH1. Since there are 11 pallet locations available on the left-most side of the warehouse in this setting, there are also 4 (i.e. q − h − a + 1) pallet locations available between the middle and the rear cross aisles. Additionally, there are eighty six (n = 2q + 2(m − 1)(q − a)) locations in the forward picking area in this setting. As mentioned before, the performance measure is average tour length, which is the arithmetical

average of the optimal tours of all orders. As noted in section 3.4, pick list sizes of 3, 5, 10, 15, 20, 30, 40,

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Add location v to order i and move to the next pick j = j + 1.

Start

No Specify pick list size (p) Yes Specify total numbers of orders (o) being generated

Selected location v is already located in order i ?

Start with order i = 1

Yes

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Select a random location v = rand(1,n)

No

j≤p Finish

No

i≤o Yes

Move to next order i = i + 1.

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Start with pick location j = 1

Figure 8: Generating random orders under a randomized storage policy.

and 50 units are taken into account in addition to the P&D point to evaluate the performance measure. For each pick list size and warehouse setting, we then randomly generate 1,000 orders to estimate the

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mean tour length within a 1% relative error with probability 95%. We are confident that 1,000 orders are enough after conducting an experiment on a sample with a size of 1,000, a sample mean of 2220.4,

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and a standard deviation of 167.5. Finally, at the beginning of the optimization process, after setting the pick list size and warehouse design parameters, orders were randomly generated using the procedure depicted in Figure 8.

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For a given warehouse setting, the solution space includes (q − a − 1)m−1 different solutions where tunnels can be located on (m − 1) aisles and there are only (q − a − 1) feasible positions for locating a

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tunnel on an aisle because of the design assumptions in section 2. For the WH1 setting, we prefer to use brute-force search method for determining the optimal tunnel positions because there are only 2,401 ((11−3−1)5−1 ) possible different solutions which is much less than the computational requirement of HS

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algorithm. Because the number of possible solutions increases dramatically, about 3 million for the WH2 setting, as the warehouse size increases, we used the developed HS algorithm for the other warehouse instances. Table 6 shows, for each pick list size, the best solutions for the WH1 setting and the best solutions found so far for the other warehouse settings after running HS algorithm for 10 replications. Table 7 also shows the worst solutions, the average and the standard deviations of the obtained solutions within the 10 replications. The representations of the best solutions found so far in WH1 are shown in Figure 10. For the other warehouse settings, they are shown in Figures B.1, B.2 and B.3 in Appendix B for the sake of smooth flow. 21

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The representations of the solutions show that, for small pick list sizes, such as 3 and 5, the best layouts found so far and Layout B look alike. The only difference between them is the position of the middle cross aisle where it is placed between the center and the rear of the warehouse in the best found Layout B-like designs. This result of placing middle cross aisle beyond the midpoint of the warehouse in the best-found solutions also agrees with the observations of Pohl et al. (2009a) and Berglund and Batta (2012), in which they presented that the best position of the single middle cross aisle is located between the center and rear of the warehouse for dual-command and order-picking operations, respectively. Additionally,

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our results show that middle cross aisle moves closer to the center of the warehouse as the pick list size increases. For instance, whereas the cross aisle is placed at the 13th pallet position for 3 picks in WH3, it is located at the center when the pick list size is 15. Besides, as the pick list size increases, the best found solutions generate segregated tunnels. In these solutions, as a tunnel moves closer to the front of the warehouse in a picking aisle, the tunnel in the adjacent aisle moves in the opposite direction towards the rear of the warehouse, and vice versa. In other words, while a tunnel is located below the center of

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the warehouse, the tunnel in the adjacent aisle is located above the center, so this pattern repeats till the edge of the warehouse. For simplicity, we call these type of layouts “the designs with up-and-down tunnels”, hereafter. Our analysis of optimal tours in those designs yields the following general insights. For an example optimal tour, see Figure 9.

• Because the probability of having picks in every aisle increases as the number of picks increases, it

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becomes more important to travel to neighboring aisles than more distant aisles. • The designs with up-and-down tunnels create one short and one long rack segment along an aisle

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in contrast to two equal rack segments in Layout B. Hence, a picker does not have to traverse 50% of the locations in an aisle to access the neighboring aisle in the new layouts.

aisle.

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• Up-and-down tunnels with front and rear cross aisles provide four alternative access areas along an

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• When the number of picks is very high, the model locates tunnels at the limiting positions so it no longer seems to need a tunnel (or a cross aisle) at all (e.g. the best found design in WH2 for 50 picks). This is an expected result because a picker tends to traverse every picking aisle so a middle

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cross aisle only increases travel distance in an aisle when the number of picks is large with respect to warehouse size. This becomes obvious when every location must be visited in the extreme case. Therefore, Layout A would seem to be the best choice for relatively large number of picks.

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P&D

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Figure 9: An optimal route for a single order of 40 picks in WH3.

Table 6: The best tunnel positions found so far in each warehouse and pick list setting. WH2 Tunnels 10 10 10 10 10 10 999999 999999 5 10 5 10 5 10 5 11 5 11 5 11 3 13 3 13 3 13 2 13 2 13 2 13 2 13 2 13 2 13

WH3

Fitness 686.6 858.9 1156.6 1344.2 1457.0 1605.0 1680.0 1737.2

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Fitness 493.2 609.8 787.7 876.2 931.0 999.7 1040.2 1064.3

Tunnels 13 13 13 13 13 13 13 13 12 12 12 12 12 12 12 12 11 11 11 11 11 11 11 11 10 10 10 10 10 10 10 10 8 8 13 8 13 8 13 8 6 14 6 14 6 14 6 14 5 16 5 16 5 16 5 16 3 18 3 18 3 18 3 18

Fitness 876.4 1105.1 1505.8 1791.3 1989.0 2253.6 2432.2 2539.8

WH4

Tunnels 15 15 15 15 15 15 15 15 15 15 14 14 14 14 14 14 14 14 14 14 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 17 9 17 9 17 9 17 9 17 9 18 8 18 8 18 8 18 8 18 8 19 7 19 7 19 7 19 7 19 7

Fitness 1077.0 1355.0 1839.3 2219.6 2521.4 2915.4 3181.4 3392.3

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Tunnels 7777 7777 4747 3838 2828 2828 2428 2428

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WH1 # picks 3 5 10 15 20 30 40 50

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Table 7: The worst solution and the mean and the standard deviations of the obtained solutions within 10 replications.

Worst

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# picks 3 5 10 15 20 30 40 50

689.8 861.1 1158.0 1345.9 1462.4 1613.8 1696.6 1754.4

WH2 Average Std. Dev. 687.3 859.3 1157.1 1345.0 1458.6 1607.7 1688.9 1747.6

1.4 1.0 0.5 0.7 2.4 3.6 7.5 7.2

Worst 878.9 1109.5 1509.8 1791.9 1990.5 2262.8 2442.7 2548.9

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WH3 Average Std. Dev. 877.0 1107.2 1508.4 1791.5 1989.3 2259.4 2436.5 2543.4

1.0 1.6 1.6 0.3 0.7 4.1 4.3 4.8

Worst 1083.8 1365.3 1843.9 2227.8 2523.1 2548.9 3189.0 3410.4

WH4 Average Std. Dev. 1080.3 1359.5 1842.5 2224.1 2521.9 2543.4 3187.3 3401.7

2.8 4.9 1.7 2.8 0.8 4.8 3.4 7.1

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Table 8: Characteristics of Layout A instances.

Equivalent Layout A WH1 WH2 WH3 WH4 5 7 9 11 28 38 48 58 12 16 21 26 9 13 18 23 90 182 324 506

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parameters m W H q n

P&D

P&D

(b) 10 picks

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(a) 3 and 5 picks

P&D

(d) 20 and 30 picks

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P&D

(c) 15 picks

P&D

(d) 40 and 50 picks

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Figure 10: Representation of the best solutions found so far in WH1.

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Even though Layout A does not provide flexible travels between locations for the order-picking operation due to lack of the middle cross aisle, the last insight showed that Layout A may be a good choice for large number of picks. Additionally, one-block warehouse layouts are preferred over two-block warehouse layouts in small storage areas because of not consuming space for an additional cross aisle. Therefore, we also compare Layout A with the developed designs in order to gain more insights. The design parameters for an equivalent Layout A with very close storage capacity to Layout B are given in Table 8 for each warehouse instance. For an accurate comparison, we also use the same generated order lists and warehouse settings to evaluate the performance of the given instances of Layout A and Layout B. We used our tour algorithm 24

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Improvement over Layout B

Improvement over Layout A

%

% WH1 WH2 WH3 WH4

6

WH1 WH2 WH3 WH4

15

10

4 5 2

0

10

20

30

40

50

0

10

pick list size

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0

0

20

30

40

50

pick list size

Figure 11: Comparison of proposed layouts to Layouts B and A.

to calculate optimal tour lengths in Layout B, and we used Ratliff and Rosenthal (1983)’s algorithm for

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optimal tours in Layout A. Table 9 shows the calculated average tour lengths of the optimal tours for the given set of orders in the problem instances of Layout A and Layout B. We then analyze if the best found solutions provide any improvement on average tour length over Layouts A and B. Improvement is calculated by dividing the difference of the average tour lengths of the best found solution and Layout A (or B) by the respective tour length of Layout A (or B). As can be seen from Figure 11, which presents the percentage improvement of the best found solutions over traditional designs, the improvement over

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Layout B increases as the pick list size increases. On the contrary, the improvement over Layout A decreases as the pick list size increases. The reason for this finding is that the picker is forced to traverse

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every aisle due to the high number of visits. Thus, the maximum improvement over Layout A occurs for 10 picks. However, the best found solutions, except for the ones in WH1 because of its size, provide only a marginal improvement over Layout B for up to 10 picks because they resemble Layout B. Additionally,

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the best found solutions obtained for more than 30 picks in WH2, WH3, and WH4 provide 2% to 7% savings over Layout B and up to 12% over Layout A. As a result, whereas the improvement over Layout

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B decreases as the warehouse size increases, the improvement over Layout A increases.

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# picks 3 5 10 15 20 30 40 50

Table 9: The average tour lengths in Layout B and Layout A.

WH1 502.8 618.2 798.8 909.3 986.6 1063.0 1101.1 1123.1

Layout B WH2 WH3 689.6 883.0 861.1 1110.7 1157.0 1509.0 1356.7 1791.3 1500.2 2008.5 1696.8 2332.5 1801.8 2551.5 1869.6 2694.6

WH4 1082.4 1358.3 1839.3 2219.6 2521.4 2978.9 3313.2 3558.0

WH1 572.0 691.7 866.6 944.5 979.6 1019.0 1039.5 1053.2

Layout A WH2 WH3 814.5 1084.2 996.9 1323.3 1293.3 1748.8 1464.3 2043.7 1565.2 2243.2 1672.1 2481.3 1717.2 2595.1 1743.8 2656.8

WH4 1333.2 1644.7 2208.7 2614.6 2928.6 3333.5 3569.5 3704.6

Because some solutions look like Layout B and some seem to converge on Layout A, we determine 25

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which layout, Layouts A, B or best founds, is superior to others in terms of average tour length. These superior layouts are presented in Table 10, which shows that layouts similar to Layout B are best when pick number is relatively small compared to warehouse size whereas Layout A is best when pick number is relatively very large compared to warehouse size. Therefore, based on the data from our numerical analysis, we investigate the effects of the following measures on determining the superior layouts. 1. The effect of the percentage ratio number of picks to total number of locations in the warehouse (100 · p/n).

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p 2. The effect of the expected number of picks in an aisle ( 2(q−a) ) for uniform picking. This number is

only an approximation because there are q pallet locations, not q − a, on the left- and right-most columns.

Table 11 presents the values of these two measures and their effects. As can be seen from Table 11(a), Layout B-like designs dominate others when a picker must visit up to 6% of the storage locations whereas

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Layout A seems to be the superior design if a picker visits more than 35% of all storage locations. That is, tunnel based layouts only provide savings when 6% to 35% of locations must be visited. Similarly, Table 11(b) shows that Layout B-like designs dominate others if the average expected number of picks per aisle is less than about 0.6. If it is more than 0.6 and less than 2, tunnel-based solutions outperform both Layouts A and B. Finally, Layout A is preferred when the expected number of picks per aisle is more than 2.5.

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Table 10: Superior designs with respect to pick list and warehouse sizes. A indicates one-block Layout A. B indicates traditional two-block layout with middle cross aisle or Layout B-like designs where the cross aisle is located between the center and the rear of the warehouse. T indicates the new, tunnel-based discrete-aisle designs.

WH2

WH3

WH4

B B T T T T A A

B B B T T T T T

B B B B T T T T

B B B B B T T T

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WH1

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3 5 10 15 20 30 40 50

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Up to this point, we only considered travel distance in our analysis. However, we also would like

provide insights about travel time on the proposed designs. Because the new tunnel-based solutions propose segmented cross aisles instead of a straight middle-cross aisle for some cases, this may impose complex paths including more U-turns and turns than those in Layout B. As known that these turns reduce the speed of the pickers especially if they use vehicles in comparison to straight travels. Therefore, we also compared the numbers of turns and U-turns that pickers make during a tour within the bestfound and Layout B designs (for an example path see Figure 9) in order to provide deeper insights to the practitioners.

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Table 11: Effects of relative pick number on determining the superior layout. Dark and light gray shaded cells represent Layouts A and B in Table 10, respectively. The other cells represent tunnel-based solutions.

WH1

WH2

WH3

WH4

# picks

WH1

WH2

WH3

WH4

3 5 10 15 20 30 40 50

3.5% 5.8% 11.6% 17.4% 23.3% 34.9% 46.5% 58.1% (a)

1.6% 2.7% 5.3% 8.0% 10.6% 16.0% 21.3% 26.6% 100 · p/n

0.9% 1.5% 3.0% 4.5% 6.1% 9.1% 12.1% 15.2%

0.6% 1.0% 2.0% 2.9% 3.9% 5.9% 7.8% 9.8%

3 5 10 15 20 30 40 50

0.2 0.3 0.6 0.9 1.3 1.9 2.5 3.1

0.1 0.2 0.4 0.6 0.8 1.2 1.5 1.9

0.1 0.1 0.3 0.4 0.6 0.8 1.1 1.4

0.1 0.1 0.2 0.3 0.4 0.7 0.9 1.1

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# picks

(b)

p 2(q−a)

Let turnB and turnt be the average numbers of turns, and U turnB and U turnt be the average numbers of U-turns in an optimal tour within Layout B and the best-found solutions, respectively. For the clarification, the average number of turns consists of U-turns, if exist. We considered the aforementioned

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order set and the obtained optimal paths in our previous analysis. Table 12 shows the difference of average numbers of turns (∆t = turnt − turnB ) and U-turns (∆ut = U turnt − U turnB ) in the solutions. As seen in the light gray shaded cells in the table, both Layout B and the best-found solutions impose similar number of turns when the number of picks is relatively small because the-best found solutions look like Layout B. As the number of picks increases, tunnel based layouts cause less number of turns and U-turns than Layout B. Thus, this shows that tunnel based designs seem not to provide more complex

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tours than Layout B for moderate number of picks. However, when pick number is relatively large the tunnel based layouts impose more turns than Layout B. The reason of this finding may be that the picker

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seems to visit neighboring aisle through a tunnel instead of bottom or top cross aisles, hence the picker makes U-turns to visit the locations above or below the tunnel before traversing whole aisle. However, the above-mentioned observation already revealed that Layout A becomes preferred when the number of

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picks is relatively large. The detailed numbers of turns and U-turns are given in Tables C.1 and C.2.

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Table 12: The difference of average numbers of turns and U-turns during the optimal routes within Layout B and the bestfound solutions. Light gray shaded cells indicate the cases where the associated Layout B is superior than tunnel-based solutions (see Table 10) in terms of travel distance.

# picks 3 5 10 15 20 30 40 50

WH1 ∆t 0.2 0.9 -1.4 -1.3 -0.7 2.7 2.4 3.5

∆ut 0.0 0.3 -0.5 -0.3 0.2 1.8 0.9 1.2

WH3 ∆t 0.2 0.4 0.7 -2.8 -2.1 -2.2 -1.1 2.2

∆ut 0.0 0.1 0.2 -0.9 -0.6 0.1 1.2 2.6

WH3 ∆t 0.4 0.6 0.4 0.0 -3.0 -2.6 -0.6 -2.6

∆ut 0.0 0.1 0.1 0.0 -1.0 -0.5 0.9 0.8

WH4 ∆t 0.0 0.3 0.0 0.0 0.0 -4.5 -2.4 0.2

∆ut 0.0 0.0 0.0 0.0 0.0 -1.4 -0.3 1.2

In addition to the insights about layouts, we also present the computational effort of the algorithm in

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optimization. Table 13 shows how long it took in seconds to obtain our solutions in a single replication. As seen in Table 13, computational time requirement to search the best design for a given pick number doubles while total number of storage locations in a warehouse, except for WH1, approximately doubles (see Table 5 for the warehouse capacities). Table 13: Computational time (seconds) of the optimization algorithm for different warehouse sizes.

WH1∗

WH3

WH4

405 2492 504 2862 577 4019 613 4730 645 5401 681 5540 741 5850 882 6088 force search.

6243 7262 9546 10191 12126 13140 12988 12066

12358 14288 16083 17492 19409 21707 25887 26755

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3 5 10 15 20 30 40 50 * Using brute

WH2

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# picks

6. Conclusion

The main contribution of this study is to present a new layout problem for warehouses that we call the discrete cross-aisle design problem. In doing so, we relax one of the unspoken design rules in traditional warehouse layouts by assuming that a linearly placed middle cross aisle can be segmented

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into tunnels along the aisles. We then focus on the order-picking operation in order to investigate the optimal locations of tunnels in a layout to minimize order-picking tour length.

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The challenge in this problem is to calculate optimal tour length for any given number of pick locations because this problem was shown to resemble the very-well known, NP-Hard traveling salesman problem. Hence, our second contribution is to develop an optimal algorithm to calculate the best tour in a given

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discrete cross aisle warehouse. Although our algorithm is more complex than Roodbergen and de Koster (2001a)’s algorithm which is specifically developed for two-block warehouses, we demonstrated that our

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algorithm is very efficient and requires only a couple of seconds to calculate 1,000 optimal tours of 51 locations including the P&D point. Even our algorithm can be used for any tunnel configuration under the assumption of one tunnel for one aisle in warehouses. We then use one of the most common meta-

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heuristics algorithm Harmony Search to look for the best tunnel positions that minimize average tour length for the given number of orders. Our goal was to reduce labor costs for order-picking in labor intensive picker-to-part warehouses by

reducing traveling time between storage locations. However, we believe that the results presented in this study are also valid for automated warehouses where robots are responsible for visiting several locations to pick items. Therefore, we demonstrated that segmenting a linear cross aisle into tunnels or positioning the middle cross aisle closer to the rear of the warehouse can reduce average order-picking tour length. As Figure 11 shows, it is a potential to reduce average tour length by around 5% compared to equivalent

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traditional Layouts A and B by implementing segregated tunnels like up and down through adjacent aisles, especially where there is a pick every aisle (see Table 11(b) for these cases). In a large warehouse such as WH3 and WH4, approximately 5% to 9% savings in tour length can be obtained over equivalent Layouts B and A, respectively, if a picker needs to visit around 10% of locations (see Table 11(a) for these cases). In our detailed analysis, we also show that, for Layout B-like designs, the best option for small pick numbers (e.g. 3, 5 or 10) is to place the cross aisle linearly at the center or between the center and the rear of the warehouse. As expected, we also found that Layout A is the superior design when

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the number of picks is relatively very large. The proposed designs are developed based on several important assumptions: uniform picking, a centrally-located single P&D point, wide aisles that allow return in an aisle, and one tunnel along each picking aisle. First, even though the exact middle of the front cross aisle was shown to be the optimal position for a single P&D point for minimizing order-picking tour length by Roodbergen and Vis (2006), several studies also investigated the effect of its positions, such as the left corner of the front cross

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aisle, on tour length. For instances, Petersen (1997) presented that the average tour length in one-block warehouses with the left P&D point is 6% more than those with the central P&D point for low number of picks, while it is only 0.03% more for high number of picks. Similarly, Roodbergen and Vis (2006) demonstrated that the central P%D point in two-block warehouses results in lower tour length than the left P&D point even though the difference decreases as the number of picks increases: the difference between them decreases from 1.5% to 0.15% as the number of picks increases from 10 to 30. Thus, in

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the light of these discussions we can argue that the presented percentage improvements by new designs in this study may even increase when the position of the single P&D point is changed. However, this

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should be deeply analyzed in a further study because the proposed layouts might change when its position is changed. Additionally, Thomas and Meller (2014) and Ozturkoglu et al. (2014) demonstrated that order-picking tour lengths and warehouse layouts are affected by the number of multiple P&D points,

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respectively. Therefore, this research may be extended by relaxing both the position of a single P&D point or its number and positions.

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Second, because uniform picking may cause highly dispersed locations in the warehouse, one can consider volume-based or class-based storage policy in which some locations require more frequent visits than others with respect to their demand. Third, narrow-aisle designs that allow only one-way travel

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through picking aisles may also be considered because it certainly has effect on both travel distance and travel path in a warehouse. Similarly, a simulation study might be conducted to investigate the effect of multiple pickers or multiple turns within aisles on travel time instead of distance because they may cause congestion in aisles, especially in narrow aisles. Although our analysis of turns showed that tunnel based designs seem not to impose more turns or U-turns than Layout B, one can also consider the effect of speed of vehicles or pickers during a travel in a simulation study. Last but not the least, multiple tunnels per aisle that provide more access areas between aisles may be included in the model.

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Acknowledgment This research was supported by the TUBITAK (The Scientific and Technological Research Council of Turkey) under Grant 214M220. The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper. References

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https://www.statista.com/statistics/292928/

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domestic-and-cross-border-b2c-e-commerce-sales-europe/, accessed: 2017-02-16.

Theys, C., Br¨ aysy, O., Dullaert, W., Raa, B., 2010. Using a TSP heuristic for routing order pickers in warehouses. European Journal of Operational Research 200 (3), 755–763.

Thomas, L. M., Meller, R. D., 2014. Analytical models for warehouse configuration. IIE transactions 46 (9), 928–947. Volgenant, T., Jonker, R., 1982. A branch and bound algorithm for the symmetric traveling salesman problem based on the 1-tree relaxation. European Journal of Operational Research 9 (1), 83–89. 32

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Yang, X.-S., 2009. Harmony search as a metaheuristic algorithm. In: Music-Inspired Harmony Search Algorithm. Springer, pp. 1–14. Appendix A

Used theorems for constructing partial tour subgraphs and the optimal tour subgraph

The theorems and their proofs are largely similar to Ratliff and Rosenthal (1983) and Roodbergen and de Koster (2001a). We adopted their theorems and proofs for our problem if needed. We refer these studies for further details and readings.

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Theorem A.1. Ratliff and Rosenthal (1983). A subgraph T ⊂ G is a tour subgraph if and only if (a) all vertices including picking locations and the P&D point have positive degree in T, (b) excluding vertices with zero degree, T is connected, (c) every vertex in T has even or zero degree. Corollary A.1. Ratliff and Rosenthal (1983). A minimum length tour subgraph contains no more than two edges between any pair of vertices.

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Corollary A.2. Ratliff and Rosenthal (1983). If (P1 , P2 ) is any node partition of a tour subgraph, there is an even number of edges with one end in P1 and the other end in P2 .

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Theorem A.2 (Adapted from Ratliff and Rosenthal (1983)). Necessary and sufficient conditions for a tour subgraph in aisle j to be a Lj partial tour subgraph are (Tj ⊂ Lj ) (a) All vertices including picking locations and the P&D point in Lj (∀vi ∈ Lj ) PTSs should have positive degree in Tj . (b) Every vertex, except possibly aj , bj , cj and βj−1 , has even degree or zero degree, (c) excluding vertices with zero degree, Tj has either • no connected component, • a single connected component containing at least one of aj , bj , cj and βj−1 , • two or three connected components within each component at least one of aj , bj , cj and βj−1 , and each of aj , bj , cj and βj−1 is contained in at most one component, • four connected components with aj , bj , cj and βj−1 each in a different component. (d) βj−1 has either zero degree or even degree within a component with at least one of aj , bj and cj • in L+x if bj is below βj−1 j

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• in L+y if bj is above βj−1 . j

Proof. The proof is similar to that of Theorem 2 of Ratliff and Rosenthal (1983).

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Theorem A.3 (Adapted from Roodbergen and de Koster (2001a)). Two Lj partial tour subgraphs are equivalent if (a) aj , bj , cj and βj−1 each have the same degree parity (i.e. even, odd or zero) in both partial tour subgraphs. (b) excluding vertices with zero degree, both partial tour subgraphs have either • no connected component, • a single connected component containing at least one of aj , bj , cj and βj−1 , • two or three connected components with in each component at least one of aj , bj , cj and βj−1 , and each of aj , bj , cj and βj−1 contained in at most one component, • four connected components with aj , bj , cj and βj−1 each in a different component. (c) the distribution of aj , bj , cj and βj−1 over the various components is the same for both partial tour subgraphs. Proof. The theorem and therefore the proof is largely similar to Theorem A.3 of Roodbergen and de Koster (2001a) and Theorem 2 of Ratliff and Rosenthal (1983). Only vertex βj is added into the conditions. 33

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Representations of the best-found solutions

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Appendix B

P&D

P&D

(b) 5 and 10 picks

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(a) 3 picks

P&D

(d) 20 picks

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(c) 15 picks

P&D

P&D

P&D

(e) 30 picks

(f) 40 and 50 picks

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Figure B.1: Representation of the best solutions found so far in WH2

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P&D

P&D

(b) 5 picks

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(a) 3 picks

P&D

P&D

P&D

(f) 30 picks

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P&D

(e) 20 picks

(d) 15 picks

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(c) 10 picks

P&D

P&D

(g) 40 picks

(h) 50 picks

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Figure B.2: Representation of the best solutions found so far in WH3.

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P&D

P&D

(b) 5 picks

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(a) 3 picks

P&D

P&D

(d) 30 picks

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(c) 10, 15 and 20 picks

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P&D

(e) 40 picks

(f) 50 picks

Appendix C

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Figure B.3: Representation of the best solutions found so far in WH4.

Evaluation of turns and U-turns in the designs.

The following Tables C.1 and C.2 show the average numbers of turns and U-turns during a tour

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within the associated Layout B and the best designs found so far.

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Table C.1: The average number of turns in a tour.

WH1

Layout B 8.0 10.6 15.3 17.2 17.3 15.9 15.5 15.2

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# picks 3 5 10 15 20 30 40 50

Best-found 8.2 11.5 14.0 15.9 16.5 18.6 17.9 18.7

WH2 Layout B 10.2 14.0 20.5 24.2 25.7 26.2 24.7 23.0

WH3

Best-found 10.4 14.3 21.2 21.4 23.5 24.0 23.7 25.2

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Layout B 12.0 16.2 24.9 29.7 33.0 36.0 36.2 34.5

Best-found 12.4 16.8 25.3 29.7 30.0 33.4 35.6 31.9

WH4 Layout B 14.5 19.3 28.1 34.9 39.6 44.7 46.4 46.2

Best-found 14.5 19.6 28.1 34.9 39.6 40.1 44.0 46.5

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Table C.2: The average number of U-turns in a tour.

WH1

Layout B 1.0 1.7 3.6 4.7 5.0 4.6 3.6 2.8

Best-found 1.1 1.8 3.8 3.8 4.4 4.7 4.8 5.5

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Best-found 1.0 1.6 2.3 2.9 3.2 4.1 3.0 3.2

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Layout B 1.0 1.4 2.8 3.3 3.1 2.3 2.1 2.0

WH3

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# picks 3 5 10 15 20 30 40 50

WH2

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Layout B 1.1 1.8 4.1 5.6 6.5 7.0 6.6 5.6

Best-found 1.1 1.9 4.2 5.6 5.5 6.5 7.4 6.4

WH4 Layout B 1.1 1.9 4.5 6.5 7.8 9.1 9.1 8.6

Best-found 1.1 2.0 4.5 6.5 7.8 7.7 8.8 9.7