A spring-embedding approach for the facility layout problem

The facility layout problem is concerned with finding the most efficient arrangement of a given number of departments with unequal area requirements within a facility. The facility layout problem is a hard problem, and therefore, exact solution methods are only feasible for small or greatly restricted problems. In this paper, we propose a spring-embedding approach that unlike previous approaches results in a model that is convex. Numerical results demonstrating the potential of our model and the efficiency of our solution procedure are presented.     

The facility layout problem

In this paper, the facility layout problem is surveyed. Various formulations of the facility layout problem and the algorithms for solving this problem are presented. Twelve heuristic algorithms are compared on the basis of their performance with respect to eight test problems commonly used in the literature. Certain issues related to the facility layout problem and some aspects of the machine layout problem in flexible manufacturing systems are also presented.     

A convex optimisation framework for the unequal-areas facility layout problem

The unequal-areas facility layout problem is concerned with finding the optimal arrangement of a given number of non-overlapping indivisible departments with unequal area requirements within a facility. We present a convex-optimisation-based framework for efficiently finding competitive solutions for this problem. The framework is based on the combination of two mathematical programming models. The first model is a convex relaxation of the layout problem that establishes the relative position of the departments within the facility, and the second model uses semidefinite optimisation to determine the final layout. Aspect ratio constraints, frequently used in facility layout methods to restrict the occurrence of overly long and narrow departments in the computed layouts, are taken into account by both models. We present computational results showing that the proposed framework consistently produces competitive, and often improved, layouts for well-known large instances when compared with other approaches in the literature.     

An improved two-stage optimization-based framework for unequal-areas facility layout

The unequal-areas facility layout problem is concerned with finding the optimal arrangement of a given number of non-overlapping indivisible departments with unequal area requirements within a facility. We present an improved optimization-based framework for efficiently finding competitive solutions for this problem. The framework is based on the combination of two mathematical optimization models. The first model is a nonlinear approximation of the problem that establishes the relative position of the departments within the facility, and the second model is an exact convex optimization formulation of the problem that determines the final layout. Aspect ratio constraints on the departments are taken into account by both models. Our computational results show that the proposed framework is computationally efficient and consistently produces competitive, and often improved, layouts for well-known instances from the literature as well as for new large-scale instances with up to 100 departments.     

A scatter search algorithm for the single row facility layout problem

The single row facility layout problem (SRFLP) is the problem of arranging facilities with given lengths on a line, with the objective of minimizing the weighted sum of the distances between all pairs of facilities. The problem is NP-hard and research has focused on heuristics to solve large instances of the problem. In this paper we present a scatter search algorithm to solve large size SRFLP instances. Our computational experiments show that the scatter search algorithm is an algorithm of choice when solving large size SRFLP instances within limited time.     

Lower bounds for the two-stage uncapacitated facility location problem

In the two-stage uncapacitated facility location problem, a set of customers is served from a set of depots which receives the product from a set of plants. If a plant or depot serves a product, a fixed cost must be paid, and there are different transportation costs between plants and depots, and depots and customers. The objective is to locate plants and depots, given both sets of potential locations, such that each customer is served and the total cost is as minimal as possible. In this paper, we present a mixed integer formulation based on twice-indexed transportation variables, and perform an analysis of several Lagrangian relaxations which are obtained from it, trying to determine good lower bounds on its optimal value. Computational results are also presented which support the theoretical potential of one of the relaxations.     

A heuristic method for the multi-story layout problem

In this paper we present a heuristic procedure designed expressly for solving a large layout problem in a multi-story setting, where the objective is to minimize total fixed and interaction costs. This is achieved by decomposing the original facilities layout problem into several similar but smaller problems, thus enabling solution of problems with as many as 150 facilities in reasonable time. Some of the novel features of the procedure described are the use of a heuristic K-median subroutine to obtain groupings of facilities, and a simple and fast exchange-improvement method. Computational results for randomly generated problems compare the effectiveness of this method with the space planning heuristic method of Liggett and Mitchell.     

A graph-pair representation and MIP-model-based heuristic for the unequal-area facility layout problem

Owing to its theoretical as well as practical significance, the facility layout problem with unequal-area departments has been studied for several decades, with a wide range of heuristic and a few exact solution procedures developed by numerous researchers. In one of the exact procedures, the facility layout problem is formulated as a mixed-integer programming (MIP) model in which binary (0/1) variables are used to prevent departments from overlapping with one another. Obtaining an optimal solution to the MIP model is difficult, and currently only problems with a limited number of departments can be solved to optimality. Motivated by this situation, we developed a heuristic procedure which uses a “graph pair” to determine and manipulate the relative location of the departments in the layout. The graph-pair representation technique essentially eliminates the binary variables in the MIP model, which allows the heuristic to solve a large number of linear programming models to construct and improve the layout in a comparatively short period of time. The search procedure to improve the layout is driven by a simulated annealing algorithm. The effectiveness of the proposed graph-pair heuristic is demonstrated by comparing the results with those reported in recent papers. Possible extensions to the graph-pair representation technique are discussed at the end of the paper.     

A polyhedral approach to the single row facility layout problem

The single row facility layout problem (SRFLP) is the NP-hard problem of arranging facilities on a line, while minimizing a weighted sum of the distances between facility pairs. In this paper, a detailed polyhedral study of the SRFLP is performed, and several huge classes of valid and facet-inducing inequalities are derived. Some separation heuristics are presented, along with a primal heuristic based on multi-dimensional scaling. Finally, a branch-and-cut algorithm is described and some encouraging computational results are given.     

Heuristics for the dynamic facility layout problem with unequal-area departments

The dynamic facility layout problem (DFLP) is the problem of finding positions of departments on the plant floor for multiple periods (material flows between departments change during the planning horizon) such that departments do not overlap, and the sum of the material handling and rearrangement costs is minimized. In this paper, the departments may have unequal-areas and free orientations, and the layout for each period is generated on the continuous plant floor. Because of the complexity of the problem, only small-size problems can be solved in reasonable time using exact techniques. As a result, a boundary search (construction) technique, which places departments along the boundaries of already placed departments, is developed for the DFLP. The solution is improved using a tabu search heuristic. The heuristics were tested on some instances from the DFLP and static facility layout problem (SFLP) literature. The results obtained demonstrate the effectiveness of the heuristics.     

A computational study and survey of methods for the single-row facility layout problem

The single-row facility layout problem (SRFLP) is an NP-hard combinatorial optimization problem that is concerned with the arrangement of n departments of given lengths on a line so as to minimize the weighted sum of the distances between department pairs. (SRFLP) is the one-dimensional version of the facility layout problem that seeks to arrange rectangular departments so as to minimize the overall interaction cost. This paper compares the different modelling approaches for (SRFLP) and applies a recent SDP approach for general quadratic ordering problems from Hungerländer and Rendl to (SRFLP). In particular, we report optimal solutions for several (SRFLP) instances from the literature with up to 42 departments that remained unsolved so far. Secondly we significantly reduce the best known gaps and running times for large instances with up to 110 departments.     

A polyhedral study of triplet formulation for single row facility layout problem

The single row facility layout problem (SRFLP) is the problem of arranging n departments with given lengths on a straight line so as to minimize the total weighted distance between all department pairs. We present a polyhedral study of the triplet formulation of the SRFLP introduced by Amaral [A.R.S. Amaral, A new lower bound for the single row facility layout problem, Discrete Applied Mathematics 157 (1) (2009) 183-190]. For any number of departments n, we prove that the dimension of the triplet polytope is n(n−1)(n−2)/3 (this is also true for the projections of this polytope presented by Amaral). We then prove that several valid inequalities presented by Amaral for this polytope are facet-defining. These results provide theoretical support for the fact that the linear program solved over these valid inequalities gives the optimal solution for all instances studied by Amaral.     

A projection method for the uncapacitated facility location problem

Several algorithms already exist for solving the uncapacitated facility location problem. The most efficient are based upon the solution of the strong linear programming relaxation. The dual of this relaxation has a condensed form which consists of minimizing a certain piecewise linear convex function. This paper presents a new method for solving the uncapacitated facility location problem based upon the exact solution of the condensed dual via orthogonal projections. The amount of work per iteration is of the same order as that of a simplex iteration for a linear program inm variables and constraints, wherem is the number of clients. For comparison, the underlying linear programming dual hasmn + m + n variables andmn +n constraints, wheren is the number of potential locations for the facilities. The method is flexible as it can handle side constraints. In particular, when there is a duality gap, the linear programming formulation can be strengthened by adding cuts. Numerical results for some classical test problems are included.     

An improved per-scenario bound for the two-stage stochastic facility location problem

We study the two-stage stochastic facility location problem(2-SFLP)by proposing an LP(location problem)-rounding approximation algorithm with 2.3613 per-scenario bound for this problem,improving the previously best per-scenario bound of 2.4957.     

An efficient tabu algorithm for the single row facility layout problem

The general goal of the facility layout problem is to arrange a given number of facilities to minimize the total cost associated with the known or projected interactions between them. One of the special classes of the facility layout problem is the Single Row Facility Layout Problem (SRFLP), which consists of finding an optimal linear placement of rectangular facilities with varying dimensions on a straight line. This paper first presents and proves a theorem to find the optimal solution of a special case of SRFLP. The results obtained by this theorem prove to be very useful in reducing the computational efforts when a new algorithm based on tabu search for the SRFLP is proposed in this paper. Computational results of the proposed algorithm on benchmark problems show the greater efficiency of the algorithm compared to the other heuristics for solving the SRFLP.     

A linear programming embedded probabilistic tabu search for the unequal-area facility layout problem with flexible bays

In this paper, a probabilistic tabu search (PTS) approach is proposed to solve the facility layout problem (FLP) with unequal area departments. For the representation, the flexible bay structure (FBS), which is a very common layout in many manufacturing and retail facilities, is used. In this paper, the FBS is relaxed by allowing empty spaces within bays, which results in more flexibility in assigning departments into bays. In addition, departments are allowed to be located more freely within the bays, and they can have different side lengths as long as they are within the bay boundaries and do not overlap. To achieve these goals, department shapes and their locations within bays are determined LP. A PTS approach is developed to search an overall layout structure that describes relative positions of departments for the relaxed-FBS (RFBS). The proposed LP embedded PTS–RFBS approach is used to solve thirteen FLP instances from the literature with varying sizes. The comparative results show that this approach is very promising and able to find new best solutions for several test problems.     

STaTS: A Slicing Tree and Tabu Search based heuristic for the unequal area facility layout problem

In this paper, a slicing tree based tabu search heuristic for the rectangular, continual plane facility layout problem (FLP) is presented. In addition to the incorporation of facilities with unequal areas we also integrate the possibility to specify various requirements regarding (rectangular) shape and dimensions of each individual facility by using bounding curves. Therefore, it is possible to solve problems containing facilities of fixed and facilities of flexible shapes at the same time. We present a procedure that calculates the layout corresponding to a given slicing tree on the basis of bounding curves. These layouts are slicing structures which are able to contain empty spaces to guarantee that stringent shape restrictions of facilities are kept. Due to these features this approach is better suited for practical use than so far existing ones. The effectiveness of our approach in terms of objective function value is shown by comparing our results to those found in the literature. Even a large problem instance comprised of 62 facilities has been solved.