Harmony search for the layout design of an unequal area facility
Introduction
Facility layout problems (FLPs) are typical optimization problems. They require an efficient non-overlapping arrangement of facilities and are considered as a rectangular block in a given space. They have been extensively studied, and many variants have appeared in areas such as manufacturing facility layout and distribution center layout design. Meller and Gau (1996) and Singh and Sharma (2006) reviewed these variants and solution approaches. In most variant cases, determining the optimal arrangement of n departments on a floor space is a primary concern. The variants known as unequal area facility layout problems (UA-FLPs) attempt to determine the optimal arrangement by minimizing the total material handling cost between departments while satisfying the unequal area and aspect ratio restrictions of the departments. Minimizing material movement is very important in facility planning since material handling costs account for 20–50% of total operating expenses in manufacturing (Francis, McGinnis, & White, 1992).
UA-FLPs are NP-hard (Gonçalves and Resende, 2015, Paes et al., 2017). This is because the quadratic assignment problem (QAP), which determines a layout within a finite set of department locations, is already NP-hard (Castillo & Westerlund, 2005). The UA-FLP has unequal area requirements with continuous department positions that have varying dimensions and can be anywhere in a rectangular floor space, and it increases its complexity more than QAP (Castillo & Westerlund, 2005). Because the complexity exponentially increases as the size of the problem increases, exact methods such as the branch and bound method run into difficulties when attempting to solve a large-sized problem optimally.
To achieve favorable solutions in large-sized problems, significant efforts have been made to develop meta-heuristic algorithm-based approaches. In terms of developing meta-heuristic algorithms for UA-FLPs, two views exist. First, encoding can be used to represent an FLP layout configuration, and second, a meta-heuristic algorithm can be used to search a layout configuration region.
The slicing tree structure (STS) is an encoding representation that organizes a layout into a tree structure. The STS consecutively divides the floor space either in a horizontal or vertical direction with a given block space (called a guillotine cut). The encoding vector of the STS is relatively long because it contains a variety of information about the consecutive operations of the slicing tree. The STS requires a (3n − 2)-sized encoding vector to represent a layout with n departments, whereas the flexible bay structure (FBS), another well-known layout representation, only requires a (n + α)-sized vector (where α is a decision variable related to the number of bays). The STS, however, can represent a layout that cannot be constructed in the flexible bay structure. Notwithstanding this advantage, the STS has not received much attention because of its unwieldiness (due to the long and complicated encoding vector).
As a search algorithm, HS was developed by Geem, Kim, and Loganathan (2001), and was inspired by a music improvisation process that attempted to determine superior states of harmony. The HS algorithm has been applied to various optimization problems, and it has thus far compared favorably with other traditional meta-heuristic approaches such as GAs (Lee, Geem, Lee, & Bae, 2005), PSOs and SAs (Al-Betar et al., 2015, Gao et al., 2015). The basic concept of the HS algorithm is very similar with that of a GA. However, the HS algorithm generates a new solution after considering all of the existing solutions, while the GA only considers two parent solutions. The remarkable advantages of HS are that its concept and steps are relatively simple, and it does not require initial solution settings (Geem et al., 2001). These advantages make the HS flexible for implementation and combination with other meta-heuristic approaches (Fesanghary, Asadi, & Geem, 2012). Until recently, however, only one paper has utilized an HS algorithm to solve a UA-FLP, and the approach proposed in that paper did not directly use the original HS algorithm; rather, it modified and hybridized the algorithm (Chang & Ku, 2013). This hybrid approach incorporated quadratic constraint programming (QCP) and HS, and it caused the algorithm to have a slower searching process. It was only used to test problems with fewer than twenty departments. This motivated the current authors to develop an entirely new HS-on-UA-FLP approach.
Therefore, in this paper, a new approach is proposed that applies HS to solve a UA-FLP. This approach is intended to produce quality solutions while utilizing the STS in a reasonable amount of time. The HS approach is modified with respect to the procedure by which a new solution vector is generated. An additional adjustment occurs after the new solution vector is generated. This enhances the search process and enables the algorithm to generate multiple solutions. The approach's slicing tree encoding is represented in a (3n − 2)-sized vector. Note that the previous approach required a (4n − 2)-sized vector (Chang & Ku, 2013). The smaller vector allows the algorithm to achieve a relatively faster search process. Furthermore, the possible range of encoding values is also modified to diversify the search range of the layout configurations. Finally, a penalty scheme is implemented to improve the feasible region-searching capabilities. The results from the computational tests indicate that the approach is relatively stronger and faster than the results offered by previous studies.
The outline of this paper is as follows. In Section 2, the literature on the UA-FLP is reviewed. Section 3 defines the objective function and constraints of the UA-FLP. In Section 4, the proposed approach is described in detail. Subsequently, the computational results of the test problems from previous studies are provided in Section 5. Finally, the conclusions and future research considerations are provided.
Section snippets
Literature review
As mentioned, the UA-FLP is a block layout design problem concerned with the optimal arrangement of departments on a floor space within given unequal areas. Armour and Buffa (1963) formally introduced this problem, and they proposed heuristics to solve it. Heuristic algorithms such as CRAFT (Armour & Buffa, 1963), CORELAP (Moore, 1974) and MULTIPLE (Bozer, Meller, & Erlebacher, 1994) have been proposed to solve UA-FLPs, but large-sized unequal area problems were not considered because of the
Layout design of unequal area facility
In this section, the mathematical model of a generic block layout design problem with unequal area restrictions is addressed. Each department needs to be determined with respect to its position on a floor space and with respect to the width and height dimensions of the unequal area.
In Fig. 1, the dimensions of the floor space are denoted by Lx and Ly, and for each department i, the coordinate of its centroid is denoted by (), and the width and height dimensions are and ,
Harmony search approach
In this section, details regarding the solution approach are explained. First, a discussion on the STS as a solution representation is discussed with a focus on its encoding. Afterwards, an explanation is given regarding how the HS algorithm is implemented to solve UA-FLPs, and, in addition, the ways in which this study might contribute to the original HS- and STS-based approaches are listed.
Computational results
Computational experiments are implemented to evaluate the performance of the approach, and the results are compared to those from the previous literature. The algorithm is coded using JAVA, and the tests are conducted on a computer with an Intel Core i5 CPU (3.2 GHz) and 8GB of memory. Datasets of well-known problems with 7–62 departments are used. Table 1 shows the names and setups for the problems. Note that the number in the name of the problem indicates the number of departments required to
Conclusion
An unequal area facility layout problem is a typical optimization problem that occurs while constructing an efficient block layout within given unequal areas. In this research, a harmony search-based heuristic algorithm is presented to solve UA-FLPs. The block layout configuration is represented via a slicing tree representation. Modification methods are proposed to improve the performance of the algorithm. A more effective slicing representation is proposed for the block layout to diversify
Acknowledgment
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-2016-H8601-16-1010)) supervised by the IITP (Institute for Information & communications Technology Promotion).
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2020, Engineering Applications of Artificial IntelligenceCitation Excerpt :For that matter, these approaches utilized interactive evolutionary computation to solve the UA-FLP. For representing a plant layout, three main structures have been commonly used: Block Layout (Meller and Gau, 1996; Castillo et al., 2005; Gonçalves and Resende, 2015), Slicing Trees (Shayan and Chittilappilly, 2004; Scholz et al., 2009; Azadivar and Wang, 2000; Komarudin and Wong, 2010; Kang and Chae, 2017) and Flexible Bays (Tate and Smith, 1995; Wong and Komarudin, 2010; Kulturel-Konak and Konak, 2011; Kulturel-Konak, 2012; Palomo-Romero et al., 2017; Garcia-Hernandez et al., 2019; Liu and Liu, 2019). The last category has been the most used because it offers the advantage of simplifying the UA-FLP by means of reducing the problem complexity (Wong and Komarudin, 2010).
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