A closed loop based facility layout design using a cuckoo search algorithm

Accepted Manuscript

A closed loop based facility layout design using a cuckoo search algorithm Sumin Kang , Minhee Kim , Junjae Chae PII: DOI: Reference:

S0957-4174(17)30719-4 10.1016/j.eswa.2017.10.038 ESWA 11621

To appear in:

Expert Systems With Applications

Received date: Revised date: Accepted date:

14 June 2017 14 October 2017 15 October 2017

Please cite this article as: Sumin Kang , Minhee Kim , Junjae Chae , A closed loop based facility layout design using a cuckoo search algorithm, Expert Systems With Applications (2017), doi: 10.1016/j.eswa.2017.10.038

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Highlights We propose a novel approach to solve the closed loop layout design problem. The material travel distance between cells is evaluated along a closed loop. A cuckoo search based algorithm is applied to layout problems. Random-key based encoding is used to represent a combinatorial solution

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A closed loop based facility layout design using a cuckoo search algorithm Sumin Kang1, Minhee Kim2 and Junjae Chae3*

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1 Research Assistant School of Air Transport, Transportation and Logistics Korea Aerospace University, 76, Hanggongdaehang-ro, Deogyang-gu, Goyang-si, Gyeonggi-do, 10540, Korea email: [email protected] 2 Research Assistant School of Air Transport, Transportation and Logistics Korea Aerospace University, 76, Hanggongdaehang-ro, Deogyang-gu, Goyang-si, Gyeonggi-do, 10540, Korea email: [email protected]

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3 Associate Professor School of Air Transport, Transportation and Logistics Korea Aerospace University, 76, Hanggongdaehang-ro, Deogyang-gu, Goyang-si, Gyeonggi-do, 10540, Korea email: [email protected] *Corresponding Author: [email protected] (Tel:+82-2-300-0372)

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Abstract

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Facility layout problems (FLPs) are design problems that involve determining the most favorable arrangement of facilities in a given space. The closed loop layout problem (CLLP) is an FLP that employs a closed loop guided configuration. This type of layout is commonly discussed when designing a flexible manufacturing system (FMS). The CLLP is concerned with determining the efficient arrangement of manufacturing cells on a central loop based material handling system. Because the material flow between cells must pass through the loop path, distance is not measured in the conventional manner (i.e. the rectilinear or Euclidean distance). The problem is more complicated than a generic FLP because there are additional constraints that restrict cell shape and orientation as well as the positions of the pick-up and drop-off points. In this study, we propose a random-key and cuckoo search (CS) based approach to solve the CLLP. CS is a rather recently developed algorithm, and it has not yet been applied to FLPs in the literature. To evaluate the present algorithm, computational experiments are conducted using benchmark problems from a previous study. The obtained results show the remarkable performance of the proposed approach. Keywords: Facility layout design · Closed loop layout · Cuckoo search · Flexible manufacturing system

1. Introduction Facility layout problems (FLPs) are optimization problems that involve determining an efficient layout design for departments within a given facility. The primary objective of an FLP is to minimize the total material handling cost, which is evaluated as the sum of the product between the material flow and travel distance. FLPs from previous literature can be classified into two main groups – une2

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qual area facility layout problems (UAFLPs) and machine layout problems (MLPs) – according to department shapes. The basic form of a department is considered to be a rectangular block representing cells or machines. A department’s shape is given in either flexible or specified dimensions, which are denoted as Types A and B by Chae and Regan (2016). UAFLPs only restrict the department area, which enables departments to flexibly change their aspect ratio. However, MLPs fix the dimensions of each machine or department.

(a) Spine layout

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(c) Ladder layout

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(b) Closed loop layout

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Figure 1. The four common FMS layout types

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Layout design problems in flexible manufacturing systems (FMSes) are concerned with cells with inflexible shapes. These cells include a set of machines. Layout design is an important factor in FMSes because the layout significantly influences the material handling cost and throughput of a system (Nearchou, 2006). In most FMS layout problems, it is assumed that cell shape and the positions of the pick-up and drop-off (P/D) points are specified, so the problem focuses on determining cell position and orientation. The restrictions that are considered for these problems vary according to the layout configurations. Luggen (1991) classified the commonly considered layout configurations for FMSes, as shown in Figure 1. In general, with an open-field layout problem, solving a practical large instance is considered too difficult because it does not have a predefined configuration. On the other hand, Chae and Peters (2006) addressed a closed loop layout problem to produce an alternative solution for the open-field layout and showed quality results. In this paper, we focus on the closed loop layout configuration. The closed loop layout problem (CLLP) is concerned with arranging manufacturing cells along a material handling path, which forms a closed loop. In the CLLP, the material must travel along the loop. This forces the cells’ P/D points to be attached to the loop. Furthermore, cells can be located both inside and outside of the loop. However, there have been a very limited number of studies on this type of problem. Chae and Peters (2006) introduced the CLLP and proposed a simulated annealing (SA) based approach. In their approach, the cells were placed along a rectangular loop. This minimized the dead space both inside and outside of the closed loop because the cells were also rectangular in shape. Another advantage of the rectangular closed loop was that the material handling path was 3

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very simple; the material handling vehicle only needed two turns at most to access any desirable point on loop. This differentiated the approach from other loop based layout designs. The mathematical model for the problem was proposed by Niroomand and Vizvári (2013). In this model, a modified distance measure was proposed because the previous study used the rectilinear distance between cells, which is not appropriate for certain layouts. The exact travel distance along the loop path is measured in this model. In terms of the solution methodology, we employ CS, a recently developed meta-heuristic algorithm inspired by the brood parasitism behavior of cuckoos (Yang & Deb, 2009). Some birds in the cuckoo family lay their eggs in the nest of another species to increase their probability of reproduction. CS has been applied to numerous optimization problems and has shown remarkable performance. However, very few studies have utilized CS with FLPs. This is because its searching process was designed for continuous spaces, and increasing complexity occurs when the algorithm is applied to a discrete optimization problem. Several previous studies have converted the configuration of the solution space from continuous to discrete (Burnwal & Deb;2013, Li & Yin; 2013,Ouaarab et al.;2014). However, their encoding methods were tailored to their proposed heuristics and could not be verified for prevalent use. In that respect, random-key encoding is the most applicable conventional method of converting the configuration of the solution space because it has been successfully adopted to the CS process (Bean,1994). Therefore, we apply random-key encoding to CS, which enables the algorithm to perform on continuous spaces and convert a continuous solution into a combinatorial solution. Further discussion on random-key encoding is provided in Section 2 (the literature review) and Section 4 (which describes the design procedure for the closed loop layout problem with the loop distance metric). In this study, we propose a CS based approach to solve the CLLP. The loop distance measure is used as a distance metric in the CLLP. This measure represents the travel distance along the central loop path. Because there have been very few studies using the loop distance measure on the CLLP, sufficient comparative experiments are not available. This has motivated the authors to develop a quality approach to solve the CLLP. As mentioned, very few studies have applied CS to an FLP. In this study, we adopt CS to determine the solution for a CLLP. To appropriately apply the algorithm to the problem, we use the random-key-based encoding scheme. The outline of this paper is as follows. Section 2 provides a review on FLPs, CLLPs and CS. Section 3 offers a detailed description of the proposed problem. In Section 4, the proposed solving procedure is described. In Section 5, the computational results for the previously used problem sets are provided. Finally, conclusions are drawn and considerations for future research are listed in Section 6.

2. Literature review

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As mentioned, FLPs can be classified into two groups according to their dimension restrictions (Chae & Regan, 2016). First, UAFLPs restrict the department area while maintaining shape flexibility, and second, MLPs have specific department shapes and pick-up / drop-off (P/D) points. In UAFLPs, rectangular-shaped departments are located on a floor space to minimize the total material handling cost. Because department shape is only restricted by the area constraint, it can be transformed flexibly to fit in the given floor space. Since these problems are NP-hard (Castillo & Peters, 2003; TavakkoliMoghaddam & Panahi, 2007), numerous approaches based on meta-heuristic algorithms have been proposed such as genetic algorithms (GAs) (Tate & Smith, 1995), SA (Tam, 1992), Tabu Search (TS) (Scholz, Petrick & Domschke, 2009), artificial immune systems (Ulutas & Kulturel-Konak, 2012), particle swarm optimization (Kulturel-Konak & Konak, 2011), the island model genetic algorithm (Palomo-Romero et al., 2017) and harmony search (Kang & Chae, 2017). More specific information on the use of meta-heuristic approaches with FLPs can be found in (Drira, Pierreval, & Hajri-Gabouj, 4

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2007; Mavridou & Pardalos, 1997; Meller & Gau, 1996; Singh & Sharma, 2006; García-Hernández et al., 2015). On the other hand, MLPs are concerned with the specific machine or cell shapes. The FMS layout problem is one type of MLP. It involves determining the position and orientation of the manufacturing cells with their fixed shape and P/D points. In the FMS layout problem, the machine arrangement in the cells and the positions of the P/D points are assumed to be given. The problem asks to arrange the FMS cells on an open-field space to minimize the total material handling cost. In terms of the openfield layout, which is difficult to solve because it has no floor space restrictions, several approaches have been proposed. Das (1993) presented a mixed integer programming (MIP) formulation and fourstep heuristic method. Rajasekharan, Peters, & Yang (1998) developed a GA based approach to solve large-sized problems with up to 18 cells. Kim and Kim (2000) also proposed an MIP formulation and a two-phase heuristic method for the open-field layout problem. Kim and Kim (2000) relaxed the restriction on P/D points as they could be separated into input and output. As another layout configuration, the closed loop layout arranges cells on a circular loop material handling path. There are many studies on the loop based layout design, but most studies only locate cells along the outside of the loop, or they arrange cells to form the loop themselves (Afentakis, 1989; Cheng & Gen, 1998; Cheng et al, 1996; Nearchou, 2006; Saravanan & Kumar, 2013, 2015; Kumar et al, 2008; Tavakkoli-Moghaddam & Panahi, 2007). Most of these studies have focused on the traffic congestion that occurs because of the unidirectional movement on the loop. On the other hand, there are very few studies that solve CLLPs, which arrange cells on both the inside and outside of the loop and allow bidirectional movement. This is because when cells are located inside the loop just as they are on the outside, the problem becomes more complicated to solve (Asef-Vaziri & Laporte, 2005). The additional complication is derived from the loop size selection. In a CLLP, the scale of the loop should be determined in the design procedure because it causes the cell arrangement to differ and avoid overlapping, even with the same solution. By relaxing the problem as mentioned above, the problem becomes similar to the open-field layout problem. Chae and Peters (2006) proposed an SA based procedure to solve a CLLP. This procedure searched for the best arrangement while varying the loop size. It generated competitive solutions to the open-field layout solution in small-sized problems. Most of the FMS layout problems mentioned above approximate distance between P/D points as rectilinear distance (or Manhattan distance). Rectilinear distance may significantly differ from the actual distance, especially in a CLLP, leading the approach to unsuitable solutions in practice. For instance, when two cells are placed on the left and right side of the loop, the material handling equipment must travel along the loop path, which causes increasing distance compared to that of rectilinear movement. Despite the importance of the distance measure, there are only two papers considering it with the CLLP. Niroomand and Vizvári (2013) applied the exact distance metric with the closed loop based layout to Das (1993)’s MIP formulation. Their work was extended to a meta-heuristic based approach. In that study, the migrating birds optimization (MBO) algorithm was used to find quality solutions for large-sized problems (Niroomand, Hadi-Vencheh, Şahin, & Vizvári, 2015). They forced the P/D points of the cells to be placed on one edge of the cells with no clearance, so the distance between P/D points only measured the distance on the loop (so-called loop distance). They reproduced Chae and Peters (2006)’s algorithm with the loop distance measure and used it to compare the results from their approaches. To achieve an improvement in problem efficiency, Wang et al. (2012) demonstrated CS’s global convergence capabilities via an analysis model. This has been applied in a wide range of optimization problems. According to the aforementioned studies, CS easily and simply deals with multi-criteria optimization problems. Also, CS’s computational time is relatively fast compared to other metaheuristic methodologies. In particular, in engineering design optimization problems, CS has shown remarkable efficiency compared to other algorithms such as GAs, particle swarm optimization and the 5

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firefly algorithm (Durgun & Yildiz, 2012; Gandomi, Yang, & Alavi, 2013; Yang & Deb, 2010, 2013). CS also shows remarkable performance in the machining parameter optimization problem (Yildiz, 2013) and the reliability optimization problem (Kanagaraj et al., 2013; Valian et al., 2013). These studies reported CS-produced results that outperformed other meta-heuristic algorithms, including artificial bee colony, harmony search and ant colony optimization. An overall review of CS applications was provided by Yang and Deb (2014). As mentioned, a number of approaches based on meta-heuristic algorithms have been adopted to solve UAFLPs and MLPs. However, despite the extensive range of studies, only a few studies have applied CS to solve an FLP. In terms of applying CS to an FLP, only one study, conducted by Maadi et al. (2016), has taken this approach. In addition, two studies related to quadratic assignment problems (QAPs) have been proposed (Dejam et al., 2012; Kazemi & Dejam, 2014). Maadi et al. (2016) used CS to solve a single-row FLP, which conceives the department as a linearly arranged space. Thus, the approach is very different from the CLLP in terms of how the candidate space is used. The sheet nesting problem, which is similar to an FLP, was solved by Elkeran (2013). In this study, a set of pattern polygons needs to be placed on a rectangular sheet. The objective of the problem is to minimize the length of the sheet, but the problem is quite similar to an FLP as the arrangement layout must be determined. Furthermore, there are no known CS approaches for discrete optimization problems because CS was originally designed to be applied to continuous optimization problems. The PSO is similar in this regard; it is a continuous-domain-based algorithm. However, the PSO has been modified using the mapping technique, bringing combinatorial solution spaces into continuous ones, and there have been numerous applications to FLPs (Hosseini-Nasab & Emami, 2013; Kulturel-Konak & Konak, 2011; Samarghandi et al., 2010). The implementation of CS to discrete optimization problems is difficult because of its unique random walk, levy flight. To perform levy flight on a discrete solution, which is represented as a sequence of integer or binary digits, a modification is required. Burnwal and Deb (2013) proposed a CS based approach to solve the scheduling optimization of an FMS. They modified the solution movement caused by the levy flight to be generated as a set of integer numbers so that it could be added to the integer based solution. In flow shop scheduling problems, the solution is a permutation of jobs. With this encoding process, the permutation is arbitrarily changed into an integer number and works with the levy random walk. Li and Yin (2013) proposed a random-key rule to represent this permutation, which was similar to the process used in the present study. The difference is that the random numbers were not restricted by the upper bound of 1.0. Marichelvam et al. (2014) also solved a hybrid flow shop scheduling problem with an improved version of CS. A binary based CS approach was proposed to solve a knapsack problem (Gherboudj et al., 2012). As another binary based approach, quantum inspired CS algorithms were proposed by Layeb (2011) and then again by Layeb and Boussalia (2012). Ouaarab et al. (2014) proposed a discrete version of CS to solve a traveling salesman problem (TSP). In the encoding process in their study, they adopted a 2-opt and double-bridge move in accordance with the value of the levy random walk. This work was extended in Ouaarab et al. (2015) to use a random-key encoding scheme. The study was the first to use the name random-key cuckoo search. The present random-key encoding concept is based on the concept originally introduced by Bean (1994). A set of real numbers is randomly generated between 0 and 1 and used to order the visiting nodes. This has been considered as the representative conventional method for encoding work. The random-key-based encoding scheme is basically operated on continuous spaces, and it projects a solution onto combinatorial spaces without any arbitrary settings such as 2-opt and integer conversions. This allows the random walk to be generated by levy flight to be unconditionally and easily applied to a solution. In addition, the feasibility can be maintained when levy flight is performed (Li & Yin, 2013; Ouaarab et al., 2015). Therefore, in the present study, we propose a randomkey CS (RKCS) based approach to solve the CLLP with the loop distance measure. 6

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3. Closed loop layout problem with the loop distance measure

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The CLLP was introduced by Chae and Peters (2006) and modified by Niroomand & Vizvári (2013) and Niroomand et al. (2015) to measure a distance along the loop. In the present study, the loop distance based CLLP is solved. In the CLLP, cell position and orientation, as well as the size of the material handling path, must be determined. As shown in Figure 2, the problem has a single material handling path, which is a rectangular shaped closed loop configuration. The dimensions of the loop are denoted by for the horizontal measurement and for the vertical measurement. Cells are represented as rectangular shaped blocks, and they are arranged along this loop path. For each cell , the coordinate of its centroid is denoted by , and the dimensions are denoted by and . Cells can be placed either inside or outside of the loop, and in either case, overlap between cells is not allowed. Notice that the material flow between cells must go through the material handling system, so the P/D point of a cell must be attached to the loop path. The P/D point is located at the central point of the pre-defined edge (either the longer or shorter side) and denoted by .

Pick-up/drop-off point

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Figure 2. Illustration of closed loop layout problem

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The objective of the CLLP is to minimize the total material handling cost, which is formulated as follows. The material handling cost can be evaluated as the sum of the weighted distance in this formulation. ∑ ∑

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Loop distance Cell j Rectilinear distance

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Cell i

Figure 3. Comparison of distance measures

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Case 2: Cells and are on two adjacent edges.

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Case 1: Cells and same edge.

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Case 3: Cells and are on two parallel edges.

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The distance metric used in this study is the loop distance, which is measured as the distance the material moves along the loop. This was proposed by Niroomand & Vizvári (2013) and Niroomand et al. (2015). This distance measure is equivalent to the exact travel distance when departing from one P/D point and arriving at another P/D point. Because the vehicle is forced to travel along the loop path, the loop distance can be longer than the rectilinear distance in certain cases. In the case shown in Figure 3, the material flow must go through the loop path, and it is expressed as the bold dash-dotted line. This loop based travel path is longer than the rectilinear distance shown by the thin dashed line. This is because the rectilinear distance is the shortest path that can be achieved via rectilinear movement. To evaluate the loop distance, Niroomand and Vizvári (2013) introduced three cases that identified the relative spatial relationships of the cells. The distance measurement according to these cases is defined as follows.

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For the case in which both cells are on the same edge, the distance is measured as the difference between their P/D points on an axis. The horizontal ( -axis) or vertical ( -axis) difference can be used, and it varies according to the edge where cells are located. For Case 2, where the cells are on two adjacent edges, the flow occurs along the loop path. For example, if Cell 1 is on the upper edge and Cell 2 is on the right edge of the loop, the exact travel distance from Cells 1 to 2 is measured as ( . This is equivalent to the rectilinear distance between cells because is equal to ) and is equal to . Therefore, Case 2 can be expressed as the rectilinear distance, as shown in Equation (2). The distance for Case 1 is also obtained with Equation (2) because it can represent a difference in a single axis. Case 3 is divided into two sub-cases: i) the two cells are on the upper and lower edges, and ii) the two cells are on the left and right edges. There are two candidate travel paths for both sub-cases. Fig8

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ures 4 (a) and (b) indicate these sub-cases, respectively. The model defines candidates and selects the shorter one. This distance evaluation is expressed in Equation (3). i i

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Figure 4. Illustration of Case 3

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The detailed formulations for the problem are given in the previous literature (Das, 1993; Niroomand & Vizvári, 2013). In the next section, we propose the CS based approach to solve the CLLP.

4. Design procedure for closed loop layout problem with loop distance metric

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4.1. Placement strategy

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In this section, the detailed description of the design procedure for the CLLP is presented. The experimental results from (Niroomand & Vizvári, 2013) indicate that the mathematical model for the CLLP cannot find an optimal solution when the problem size increases beyond eight cells. To solve large-sized problems, we propose a meta-heuristic algorithm based approach. In a meta-heuristic algorithm, the representation method that is used to change a layout configuration into a numeric solution is an important issue. In the following sub-section, we describe a placement strategy that is used as the representation method, and then the random-key CS algorithm is proposed as the solution procedure to be used with the proposed placement method.

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The solution representation used in this study is a sequence of integer numbers, which indicates the number of cells. With the solution sequence, the placement strategy is performed to obtain a corresponding layout configuration. The cells ordered by the solution sequence are arranged along the loop path one by one according to placement priority. The placement priority is an arrangement configuration to be considered when placing a cell on the loop. We define the possible layout space of the loop path as eight spaces, as shown in Figure 5 (a). Cells can be located in these spaces according to the priority given in Figure 5 (b).

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Figure 6. Example of priority based placement

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The placement illustration is shown in Figure 6. The first cell is the base cell, which is placed at the outside left corner of the loop. The following cells are subsequently placed along the clockwise direction of the loop. The placement priority considers both the inside and outside of the loop. First, cells are placed on the inside of the upper side. If there is not enough remaining space to assign the next cell according to the current priority, the cell considers the following priority. In the case shown in Figure 6 (a), Cell 5 is located at the outside of the upper side (Priority 2) because this cell cannot be placed in the remaining space according to Priority 1. For Cases 2, 4, 6 and 8, the placement limit is not structurally defined because cells are placed outside of the loop, so they are not restricted by the loop. Cells for these outside cases are passed to the next priority if the P/D point of the cell cannot be attached in the current priority. As shown in Figure 6 (b), Cell 7 is assigned to the next priority because the P/D point cannot be attached to the loop when it is assigned to Priority 2. If all cells are placed on the loop, the solution is represented as a feasible layout configuration. The placement procedure is performed while the loop size is fixed, so varying the loop size may cause the overall layout configuration to change. This is important because the dimensions of the loop must also be decided for the CLLP. Even if the given solution sequence is the same, the cell arrangements generated for different loop shapes can differ significantly, as shown in Figure 7. To define the loop size for the best layout solution, a simple heuristic algorithm is performed.

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(a) Initial placement with initial

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Figure 7. Illustration of the produced layouts in different loop shapes

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This algorithm is based on the defining procedure presented by Chae and Peters (2006). First, the dimensions of the loop are initialized to be half of the sum of the cell’s length ∑ ( ). The cells are arranged on this initial loop by performing the proposed placement procedure. The algorithm evaluates the total material handling cost of this layout configuration and records it as the current best layout. Afterward, the algorithm produces the new cell arrangement, decreasing and by unit size ( ). We determine the unit size of 0.5 to be appropriate. The unit size influences the performance of the search procedure. A feasible solution cannot be found with a unit size over 1.0, and the process time critically increases when the unit size is less than 0.5. This new layout is compared to the current best layout solution. The current best layout is updated if the new layout is better than the previous one. This procedure is repeated until both and reach their minimum size, the minimum cell length. In summary, this placement strategy produces the best layout configuration considering every possible loop shape with a given solution sequence. The overall procedure for the placement strategy is shown in Figure 8. This strategy is performed within a CS algorithm when every new solution is produced and the objective value must be evaluated. According to Figure 8, the loop size does not decrease continuously at each iteration. However, at the end of the entire iteration, the loop size decreases successfully.

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Figure 8. Overall procedure of the proposed placement strategy 11

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4.2. Random-key encoding scheme

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Because the generic formulation of the CS algorithm was designed for continuous optimization problems, modifications must be carried out to allow the combinatorial solution to be converted into a continuous solution region. The present study applies a random-key encoding scheme from a randomkey genetic algorithm (Andrade et al., 2015). In the random-key encoding scheme, a sequence of random real numbers is generated, and this initializes the cell arrangement order. Each real number ( ) is uniformly generated in a range of [0, 1], as denoted in Equation (4). In the case in which the six-cell problem is solved, six corresponding numbers are generated, as shown in Figure 9. Each number represents the key that is used to determine the order of corresponding cells. In Figure 9, because Cell 5 has the smallest key, it comes first in the produced solution sequence. Afterward, Cell 6 comes second because it has the second largest key. As a result of this ordering procedure, the solution sequence is obtained as [5, 6, 4, 2, 1, 3]. (4)

Produced solution sequence

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Figure 9. Illustration of random-key encoding scheme

4.3. Random-key cuckoo search based methodology

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To solve the problem, we present the CS based algorithm with the random-key encoding scheme. In the general CS, this behavior is mimicked with three principle rules (Yang & Deb, 2014). First, each cuckoo can lay one egg (solution) at a time and bring it into a randomly selected nest. Second, the best nest (with the best egg) will remain in the next generation. And finally, the number of nests is fixed during the whole procedure. The nest will be abandoned and reestablished if the host bird discovers the cuckoo eggs. The host bird can find the eggs as per the probability . In other words, the last rule indicates that a fraction ( ) of the nests will be reestablished as random nests for each iteration. Based on these principle rules, the general CS is conducted in a similar manner to that used by other population-based algorithms. However, the main difference is that CS uses a unique random walk (levy flight) to derive a new solution. Levy flight is more efficient than a random walk based on Gaussian random steps. This is because levy flight has an infinite mean and variance, and this enables the algorithm to explore the search space more efficiently on a global scale (Yang & Deb, 2014). 4.3.1. Levy flight

Levy flight is a type of random walk in which the step-lengths have a heavy-tailed probability distribution. When a new solution for the generation ( ) is generated, a levy random walk is performed, as follows (Yang & Deb, 2014). (5)

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where is a scaling factor that should be related to the scales of the problem of interest, is the step size, and denotes the Gamma function, . To generate a random ∫ walk based on a levy stable distribution, we use an index ( ) of 1.5 and Mantegna’s algorithm (Mantegna, 1994). This algorithm is one of the most efficient ways to obtain the step size (Yang, 2010). In the present algorithm, a random levy walk is generated and added to a random key. Once a set of random levy walks are obtained, it is added to a sequence of random-keys, and it makes adjustments to these random-keys. This may cause the corresponding solution to be rearranged, and the results are similar to those of the swap and insert operation, as shown in Figure 10. The integer number on the top at left hand side is the cell identifying number, and the candidate solution is represented as the group of square boxes in the upper right of the figure. The number sequence in these boxes indicates the ascending order of which represents the current configuration, . The sequence changes after it is combined with the levy numbers, as shown in the lower left of the figure, and this changes the order and the configuration. In the event that more than one has the same value after its change, the first has priority to be in the first sequence in the group. This is shown in the lower right of the figure where the solution changes from 5-6-4- to 6-4-5-. In most cases, levy walks result in the swapping of two cells so that the neighboring solution can be obtained. However, a global searching walk can occasionally be achieved in the form of either the insert or the shuffle operation. This enables the algorithm to achieve balanced searching capabilities with a single random walk via levy flight. 4

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Figure 10. Illustration of a new solution representation generated via levy flight

4.3.2. Solution procedure

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The proposed solution procedure is based on the three rules mentioned above. These rules are implemented in the algorithm as follows. For each iteration, i) the best solution is maintained for the next generation, ii) a new solution is generated via levy flight, and iii) a fraction ( ) of solutions is discarded according to their objective value and replaced with new solutions. In the present study, we assume that one nest can hold one egg, which means that both a nest and an egg can be used to represent a solution. To produce a new solution, the algorithm randomly selects a solution ( ) from the existing solutions, and it then searches a neighboring region via levy flight. The new solution (( ) ) is compared to another randomly selected solution ( ), and it replaces that solution if its objective value is superior. In iii), the present algorithm replaces the abandoned nests with the nests that are neighbors with the best nests. In the case where is 0.25 and the nest size is 8, as shown in Figure

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11, two of the worst solutions are abandoned with every iteration. According to these operations, the overall procedure of the present algorithm is summarized in Figure 11.

(t)th Generation

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Figure 11. Overall procedure of the proposed algorithm

5. Computational results

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5.1. Parameter tuning

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We conduct computational experiments to evaluate the present procedure for a CLLP. The produced results are compared to those from previous studies. The algorithm is implemented in C++ and conducted on a single core system with an Intel Core i5 CPU (3.2 GHz) processor, 8 GB of memory and Windows 10. The experiments use problem datasets with up to 30 cells. The datasets proposed by Chae and Peters (2006) are taken from Das (1993) for problems with four to 12 cells and Rajasekharan et al. (1998) for problems with 14 to 18 cells. We generate more problems by extending the previous problems to include problems with 20, 25, 30, 40 and 50 cells, which are named K20, K25, K30, K40 and K50, respectively. Note that we create problem names by using a combination of the first letter of the author’s name and problem size.

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The present algorithm operates with three parameters, the number of nests (𝑚), a fraction of the worst nests to be abandoned ( ) and the step scaling factor ( ). To tune the parameters to an appropriate setting, we use the iterated racing procedure (López-Ibáñez et al., 2016). This procedure samples parameter configurations according to a particular distribution and evaluates them by using either Friedman’s test or the -test. The sampling distribution is subsequently refined to bias the sampling toward the elite configurations, and this procedure is iterated to derive the best configuration. The ‘irace’ software package (version 2.1.1662) is used to perform the iterated racing procedure and obtain the best configuration for the entire problem set. With the irace package, the improved work performance with the representative fixed parameter set can be achieved. In the tuning process, the parameter sets are sampled for the selected distribution. When finding the better parameter set, the sampling technique differs from other simple methods such as factorial design. Also, running the package is comparable to carrying out a full factorial experimental design or other parameter tuning method in that it finds the parameter combination that provides a quality solution for a specific problem. The irace package draws several elite parameter configurations and specifies the best one for every iteration. For each tuning procedure, we set a budget of 1000 experiments and consider the follow] [ ] ing range of parameters: 𝑚 [ [ ], where is the number of 14

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cells. We obtain the parameters of the training data set modified from the experiment problem set, which are summarized in Table 1. Table 1. The best individual parameter configurations obtained from the irace package

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Table 2. The fixed parameter set obtained from the irace package 𝑚

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As shown in Table 1, the selected value of 𝑚 increases as the number of cells ( ) increases with regularity. As such, the number of nests is set as equal to the number of cells (𝑚 . The individual parameter sets for each problem are also tuned, but for the fairness of the experiment and to determine the representative parameter set, all of the training datasets are tuned via the irace package as a single big problem set. The obtained fixed parameter set is shown in Table 2. In terms of , most configurations are set near 0.3. This is because is closely related to the balance between diversification and intensification strategies in the solution procedure. If has a value that is too large, the local search using the current best solution is very intensively performed, while the global search via levy flight takes a relatively small amount of time. Therefore, most of the studies in the literature have used and recommended a value near 0.3 for (Yang & Deb, 2009, 2014).

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5.2. Experiments for smaller instances The algorithm is replicated 30 times for each problem with the parameter settings. The results obtained in this study are summarized in Table 3. The experiment with the individual parameter set is also conducted, and a better OFV for R18 is found, as shown in Table 4. However, the average OFV and average processing time are not better than those of R18 with the fixed parameters. The results obtained in this study are compared with the computational results obtained by Niroomand & Vizvári (2013) and Niroomand et al. (2015), as shown in Table 5. Niroomand et al. (2015) updated the exact solutions from the MIP model proposed by Niroomand and Vizvári (2013), so we refer to the proposed results in that paper. Note that the SA-based approach was introduced by Chae and Peters (2006), but the proposed results were reproduced by Niroomand et al. (2015) to modify its distance metric. 15

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For the smallest problem, D4, the present algorithm achieves the optimal solution in all replications in less than 0.02 sec. For D8, the best solution found by Niroomand & Vizvári (2013) is verified to be the optimal value with further MIP work in this study. For D6 and D8, the algorithm cannot find the optimal solutions, but it improves upon the previous algorithm-based solutions in both cases. We confirm that the obtained solutions for D6 and D8 are the best solutions that can be achieved within the proposed placement strategy. The difference between these solutions and the optimal solutions is derived from the position of dead spaces. As shown in Figure 12, the evaluated cost varies according to the position of Cell 5, although the arrangements of these solutions are identical. The difference in D8 can be explained in the same manner as for D6. We conduct additional tests for D8, adding one dummy cell, and we confirm that the present approach is able to determine the reported optimal solution, as shown in Figure 13. The originally obtained solution for D8 is given in Figure 14.

Table 3. Summary of computational results with the fixed parameter set Problem

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75991.2

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Table 5. Comparison of test results

(Niroomand et al., 2015)

This study

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SA (Chae & MBO MMBO RKCS Improvement Peters, 2006) a (%) b c D4 547.5 547.5 547.5 547.5 547.5 0.00 D6 1601.5 c 1659.0 1659.0 1659.0 1651.5 0.45 D8 5943.5 d 6354.0 6354.0 6354.0 6176.5 2.79 D10 13417.0 12747.0 12747.0 12747.0 12168.0 4.54 D12 37281.5 34333.0 34333.0 34333.0 32371.5 5.71 R14 45402.5 44407.0 44407.0 44407.0 43091.5 2.96 R16 69337.0 57903.0 57913.0 57675.0 55610.5 3.58 R18 88807.5 75933.0 75933.0 75933.0 71963.5 5.23 a The reproduced results of Chae and Peters (2006) with the loop distance from (Niroomand et al., 2015) b Improvement = 100 (the previous best heuristic-derived solution – the best solution obtained in this study) (the previous best heuristic-derived solution) c The previously reported optimal solutions d The optimal MIP solution is verified with CPLEX in this study.

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Figure 14. Best solution derived for D8 (OFV = 6176.5)

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Figure 16. Best solution found for D12, R14, R16 and R18

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For problems with more than 10 cells, there are no reported optimal solutions. For D10, the proposed algorithm finds the new best solution and outperforms MIP in terms of using limited computational resources. Figure 15 (a) illustrates this new best solution for D10. It indicates that the new solution (a) is better than the previous solution (b) because it has a much smaller and compact loop path shape. The present solution for D12 improves upon the best-known solution by 5.71 percent. In addition, in the 14-, 16- and 18-cell problems, the algorithm improves upon the best-known solutions by 2.96, 3.58 and 4.99 percent, respectively. These solutions are illustrated in Figure 16. From the aforementioned comparison, it is observed that the present algorithm outperforms the previous approaches in all problem sets. In addition, the required processing times for most cases are under 4 minutes. Compared to other methods like SA, which is a searching neighborhood solution, CS, which is a population based meta-heuristic algorithm, can search in a broad solution space, thereby creating more opportunities to produce better starts toward the global optima. Thus, CS is superior to SA in terms of diversification. In addition, unlike MBO and MMBO, which are based on GAs, this algorithm needs comparatively less computational effort to tune the parameter settings (Yang & Deb, 2009). This facilitates further intensification, which leads to higher quality solutions. 5.3. Experiments for larger instances To demonstrate the performance with large-sized problems, we present datasets with 20-50 cells (K20-K50). The computational results for these problems with the fixed parameter set are summarized in Table 6. For K20 and K25, better solutions are found by the individual parameter set in Table 18

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7. The OFV processing time for K30 takes less time than that of K25 because of the data structure of the specific problem combined with the random generation in the algorithm. The generated value in this algorithm randomly moves from one solution space to another when determining the global solution. However, the average processing time for K30 is generally longer than that of K25. Generally, as the number of cells increases, constructing an efficient loop shape becomes extremely difficult because the number of cells arranged inside of the loop increases, as does the size of the loop. As shown in Figure 17, the obtained solutions for these problems show that the present algorithm aptly reduces the empty space in the loop to minimize the loop size and material handling cost. Problems with more than 30 cells (K40, K50), which are obviously much more complex, are feasible. However, the dead space does not decrease sufficiently. The representative decreasing loop size graph for K20 is shown in Figure 18. The perimeter of the loop discretely decreases with the iterations, as shown. Table 6. Summarized results for the large problems with the fixed parameter set Problem

Best OFV

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101725.0

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461.08

423.51

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318423.3

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1383050.0

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Loop size ( = 2v+2h )

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6. Conclusion The closed loop layout problem (CLLP) is a type of FLP concerned with arranging manufacturing cells along a loop-shaped material handling path. We propose a cuckoo search (CS) based algorithm to solve the CLLP. In this problem, cells can be located both inside and outside of the loop. Because the problem considers a closed loop path, we measure the distance between cells in the loop distance metric. To apply CS to this CLLP, levy flight is used on discrete spaces by applying randomkey based encoding that converts the solution into a set of continuous keys. The computational results are derived by conducting experiments with previously used problem sets. These results indicate that the proposed approach leads to improvements in all problems sets. In the two smaller problems, D6 and D8, the present algorithm finds the best solutions that can be ob21

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tained with the present placement strategy. We confirm that the present CS algorithm can produce the optimal solutions for D6 and D8 with dummy cells. For the problems with more than eight cells, CS improves upon the best-known solutions in all cases. In the experiments for the large-sized problems, we demonstrate the performance of CS for problems with up to 30 cells. The illustrations show that CS can obtain reasonable solutions in practice by reducing the empty space in the loop even with large problems. Further research can be carried out to investigate placement strategies with certain goals such as the maximization of area compactness, including the proper use of artificial dead space and dummy cells. As discussed with the D6 and D8 results, this could facilitate better quality solutions. A distance measure for separate P/D points could also be considered. For practical cases, cell input and output can be arranged in different places. Also, it might be feasible to relax the form of the rectilinear material handling path to reduce the entire space and the dead space of the layout. The present research limits the loop (i.e. the material handling path) to a rectangular shape. The rectangular loop may be very efficient in terms of the operation of material handling equipment, but it could be inefficient in terms of managing the entire space in a facility. Thus, relaxing the loop shape could potentially lead to a more realistic layout configuration. Other configurations such as a general layout problem other than an FMS could be considered to adopt this proposed method. Finally, adopting a hybrid algorithm with cuckoo search could be helpful in further studies.

Acknowledgment

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This research was supported by the MSIP(Ministry of Science, ICT and Future Planning), Korea, under the ITRC(Information Technology Research Center) support program (IITP-2017-2014-0-00678) supervised by the IITP(Institute for Information & communications Technology Promotion).

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