# An Application of an Unequal-Area Facilities Layout Problem with Fixed-Shape Facilities
## 1 Introduction
### Related Research  
[PERSON] [3] presented a mixed integer programming (MIP) model for the FLP based on the continuous representation of the floor space, but the areas of the facilities were equal. A similar model was developed by [PERSON] and [PERSON] [4]. However, in this cases the facility areas are unequal with fixed dimensions. Since only small-size problems can be solved optimally using exact methods, heuristic methods were developed to solve the UA-FLP. For reviews of the FLP literature see [PERSON] and [PERSON] [5], [PERSON] and [PERSON] [6], [PERSON] and [PERSON] [7], and [PERSON] et al. [8]. Based on the assumptions defined above for the proposed problem, there are only a few papers in the literature which consider the UA-FLP with fixed facilities dimensions. For example, [PERSON] et al. [9] introduced a zone concept to reduce the solution space and used a zone algorithm and simulated annealing algorithm to solve an UA-FLP where input/output points are not restricted to the center of facilities. They can be located within or on the boundary of the facilities. The same problem was considered in [PERSON] and [PERSON] [10]. The authors presented a construction algorithm and a median method for the problem. Also, [PERSON] et al. [11] presented a mixed integer linear program (MILP) and a GA for the UA-FLP where facilities may have one, two, or more input/output points. They decomposed the problem by forming groups of facilities with relatively high flows between them. The layout for each group of facilities is obtained by the GA. After layouts for groups are obtained, rectangles are drawn around facilities in each group, and the arrangement of these rectangles is found. The chromosomes store information on the relative locations of facilities in each group, which is used to fix corresponding binary variables in the MILP formulation, and the corresponding LP is solved to get the layout of facilities on the plant floor. [PERSON] and [PERSON] [12] used a modified particle swarm optimization to solve the UA-FLP. The authors generalized their solution technique to solve the dynamic UA-FLP. [PERSON] and [PERSON] [13] presented a hybrid GA (HGA), which employs both tabu search and simulated annealing, for the UA-FLP. When a population of solutions is obtained in the HGA, which give the order in which facilities are placed on the plant floor, a shape-based block layout (SBL) approach, based on hay structure, is used to place facilities on the plant floor. This produces the layout for each solution obtained from the HGA. Other researchers used the SBL approach for placing facilities on the plant floor. For example, [PERSON] and [PERSON] [14] used a firefly algorithm and [PERSON] and [PERSON] [15] used a biogeography-based optimization algorithm to solve the UA-FLP. [PERSON] and [PERSON] [16] presented a multi-start simulated annealing algorithm for the UA-FLP, which considered aisle, but the facilities are not orientation-free. [PERSON] et al. [17] presented a particle swarm optimization algorithm for a multi-objective UA-FLP. The objectives of the problem are to minimize material handling cost while maximizing the sum of the total adjacency value and utilization ratio of the plant floor. A few papers in the literature consider the dynamic UA-FLP with fixed facilities. This is the problem of finding positions of fixed-dimension facilities on the plant floor for a multi-period planning horizon such that facilities do not overlap, and the sum of the material handling and rearrangement costs is minimized. As stated above, [PERSON] and [PERSON] [12] used a modified particle swarm optimization to solve the UA-FLP and generalized their solution technique to solve a dynamic UA-FLP. [PERSON] and [PERSON] [18] presented a heuristic which consisted of a boundary search heuristic (BSH) with a tabu search (TS) heuristic for the UA-FLP and the dynamic UA-FLP. BSH is used in this paper to construct layouts for the proposed genetic algorithm (GA). [PERSON] and [PERSON] [19] presented a hybrid heuristic for the dynamic UA-FLP.