An improved GRASP method for the multiple row equal facility layout problem

Highlights

• Successful application of a metaheuristic to solve a challenging NP-Problem.

• Probabilistic approach to improve only promising solutions.

• Efficient evaluation of move operators.

• Thorough experimentation over a set of benchmark instances.

• Comparison with the state-of-the-art methods over a set of 552 instances.

Abstract

As it is well documented in the literature, an effective facility layout design of a company significantly increases throughput, overall productivity, and efficiency. Symmetrically, a poor facility layout results in increased work-in process and manufacturing lead time. In this paper we focus on the Multiple Row Equal Facility Layout Problem (MREFLP) which consists in locating a given set of facilities in a layout where a maximum number of rows is fixed. We propose a Greedy Randomized Adaptive Search Procedure (GRASP), with an improved local search that relies on an efficient calculation of the objective function, and a probabilistic strategy to select those solutions that will be improved. We conduct a through preliminary experimentation to investigate the influence of the proposed strategies and to tune the corresponding search parameters. Finally, we compare our best variant with current state-of-the-art algorithms over a set of 552 diverse instances. Experimental results show that the proposed GRASP finds better results spending much less execution time.

Introduction

Facility Layout Problems (FLP) are a well-known family of optimization problems where the objective is to find optimal locations of facilities in a given area. See Ahmadi et al. (2017) and Hosseini-Nasab et al. (2017) for recent surveys. FLP are considered a challenge for both, exact and heuristic procedures. Therefore, the layout is usually restricted to specific shapes. The very first paper tackling this family of problems dates from 1969 (Simmons, 1969), and was motivated for the necessity of arranging a set of rooms in a linear way. Hence, it is stated as the Single-Row Facility Layout Problem (SRFLP). Specifically, the authors provided a branch-and-bound algorithm to solve it and a first set of instances for this problem. Later, some other studies were made about the SRFLP like Anjos and Vannelli, 2008, Anjos and Yen, 2009 and Hungerländer and Rendl (2013). A new approach combining a Greedy Randomized Adaptive Search Procedure (GRASP) (Feo & Resende, 1989) with Path Relinking (Glover et al., 1998) was proposed in Rubio-Sánchez et al. (2016), obtaining better results than the previous papers. Recently, these results were further improved in Palubeckis (2015), using a Variable Neighborhood Search (VNS) algorithm (Mladenović & Hansen, 1997), and then improved again in Palubeckis (2017) with a Simulated Annealing method (Kirkpatrick et al., 1983).

Many variants of the FLP can be also found in the literature. For instance, in Hungerländer (2014), the Single-Row Equidistant Facility Layout Problem (SREFLP) is introduced, where the main difference is that facilities have the same width. Considering two rows instead of one, we find the Double-Row Facility Layout Problem (DRFLP) (Amaral, 2019, Chung and Tanchoco, 2010), and its particular case, called the Corridor Allocation Problem (CAP) (Ahonen et al., 2014, Amaral, 2012) which has been also studied in a double corridor version (Guan et al., 2019). As the interested reader can find, either exact and heuristic approaches are applied to these problems.

When considering two or more rows to allocate the facilities, we find the Multiple-Row Facility Layout Problem (MRFLP) (Hungerländer & Anjos, 2015), where a new mathematical formulation and an exact algorithm are proposed. This problem is an extension of the SRFLP, where optimal layouts may include spaces between neighboring facilities. For further details about this family of problems, we refer the reader to Ahmadi et al. (2017).

In this paper we focus on the Multiple Row Equal Facility Layout Problem (MREFLP) (Anjos et al., 2018), which consists in locating a given set of $n$ facilities in a layout where a maximum number of rows ($m$) is fixed. This $\mathcal{NP}$-hard problem (Anjos et al., 2018) is also known as Multi-Row Facility layout with departments of Equal Length (Fischer et al., 2019), or Equidistant Multiple Row Facility Layout Problem (Anjos et al., 2018). Additionally, this problem can also be considered as an extension of the Linear Arrangement Problem (Díaz et al., 2002) with an additional constraint that ensures the allocation of, at most, $m$ nodes to one position.

The MREFLP has several real-life applications. Among them, the most common ones are those related with industrial applications like assembly lines (Gane, 1978), chip-set design (Hong et al., 2014, Yang and Peters, 1997), data memory layout (Wess & Zeitlhofer, 2004), single machine scheduling (Bracht et al., 2018) or even computational biology (Karp, 1993). More recently, this problem has been related also to museum, shopping centers, or hospital arrangement (Lin et al., 2015, Rubio-Sánchez et al., 2016). For more real-life applications, we suggest the reader the review of Cravo and Amaral, 2019, Dahlbeck et al., 2020 and Fischer et al. (2019). In brief, this problem is useful in those circumstances where a number of elements with variable relationships among them have to be placed in a given environment.

From an algorithmic perspective, MREFLP was originally tackled in Anjos et al. (2018). The authors proposed a new integer linear and semidefinite optimization models that reach optimal solutions in all instances up to $25$ facilities. Considering this paper as the state of the art, we propose a new metaheuristic approach able to reach the same optimal results, but spending much less execution time. Our proposal is based on a Greedy Randomized Adaptive Search Procedure (GRASP) (Feo and Resende, 1989, Feo et al., 1994), improved with a hybrid local search method, and a probabilistic strategy to select those solutions that will be improved. Besides, we propose an efficient computation of the cost of the solutions that is able to deal with larger instances (up to $100$ facilities) in affordable execution times. Our approach improves upon the results of Anjos et al. (2018), spending a small amount of time compared with the state-of-the-art algorithms.

The rest of the paper is structured as follows. In Section 2, we formally define the problem and illustrate the computation of the cost. Then, we describe the GRASP approach of the constructive methods and the proposed neighborhoods to explore in Section 3. We detail the improved GRASP proposal in Section 4, which incorporates the shaking procedure to deal with large plateaus and the probabilistic quality threshold. We perform an exhaustive computational experience in Section 5, drawing the conclusions and future work in Section 6. Finally, we have included two Appendices, where the reader can find the mathematical details of the efficient exploration of the neighborhoods, and the detailed results for each instance of the problem.