A modified mathematical design for the Facility Layout Problem (FLP) was introduced in 2009, which led to a groundbreaking transformation of the plant layout optimisation field, providing an improved and innovative approach [
3]. One of the primary challenges in plant layout optimisation is overcoming the constraints of plant size, and to address this obstacle, The Origin (TO) algorithm was developed. The algorithm generates workable designs for a wide range of configurations with minimal design modifications to accommodate various layouts. To achieve this, the algorithm assumes all plants are rectangular, making for smooth implementation. The success of the TO algorithm highlights the importance of customised and inventive mathematical designs in tackling complex problems, such as plant layout optimisation.
Figure 1 illustrates the research methodology followed in this study, which involved identifying the research problem, reviewing literature, developing a theoretical framework, coding, collecting and testing data, evaluating algorithm effectiveness, analysing results, and drawing conclusions. These steps ensured high-quality research outcomes.
2.2. Input Information (Constant)
The following notations are used in the development of the mathematical model:
Facility length i
Facility breadth i
Sum between two facilities (flow per unit distance) i and j
Interval between the facility i center to facility j
A large number,
1, 2, …, N
1, 2, …, N
Facility orientation i
Facility area i
Output Data (Decision Variables)
coordinates of the facility centroid i
2.3. Objective Role and Limitations
Plant layout optimisation is a crucial aspect of production management that seeks to minimise the total material handling costs (MHC) between different facilities, which is calculated by multiplying the flow rate between facilities and the path length between them. The MHC is a quantitative factor that is pivotal to the success of any facility layout planning endeavour. The algorithmic solution of this equation can provide a layout that minimises the total MHC, assuming the cost per unit distance is one, which ultimately reduces the overall cost of production. The main objective of Facility Layout Planning (FLP) is to minimise material handling costs (MHCs) between facilities; an example of this is shown in Equation (1). The function to minimise the sum of total product between the costs, flows, and distance travelled from one facility to another from their centroids is by using the Manhattan (rectilinear) distance formulation, where
and
represents the centroid for facility
i to be placed in the layout. It is assumed that the cost per unit distance is one. This approach guarantees a cost-efficient, time-saving, and hassle-free process for manufacturers.
It is worth noting that the model has constraints that perform essential tasks, such as preventing facilities from overlapping, restricting facility extremities to the layout interior, and defining the domains of variables. This makes it a reliable and efficient model for plant layout optimisation, and it can be easily adapted to suit the needs of any facility. In conclusion, the application of the FLP model has proven to be effective in reducing material handling costs, and its adoption can lead to a significant reduction in overall production costs while simultaneously increasing efficiency and productivity.
The development of mathematical models for facility layout planning necessitates the inclusion of a set of constraints in order to provide a robust and efficient solution. Among the constraints utilised in the model, Equations (2) and (3) stand out as being particularly important. These constraints are designed to ensure that the first facility is located at the origin, (0,0), which serves as a reference point.
By guaranteeing that the first facility is placed at the origin, Equations (2) and (3) offer a precise starting point for the optimisation algorithm. This starting point, in turn, is vital for generating an efficient and accurate solution. The TO algorithm, when combined with these equations, produces a facility layout that minimises the MHC between facilities, resulting in a more efficient and cost-effective production process. The significance of these constraints to the overall success of the study cannot be overstated. They serve as the basis for the optimisation algorithm, and their inclusion in the model ensures that the facility layout planning problem is solved with precision and accuracy. Their contribution to the study is, therefore, significant, and their effectiveness in generating an optimal solution is essential to reducing the overall cost of production. The use of Equations (2) and (3) highlights the importance of precise and well-defined constraints in the development of effective mathematical models for facility layout planning.
The plant layout optimisation problem is further defined by a set of constraints, which ensure that the facilities are placed in a manner that satisfies the objective function and optimises the total material handling costs (MHCs) between the facilities. Constraints (4) to (7) are the disjunctive constraints that prevent every pair of facilities from overlapping or intercepting. These constraints specify that a facility, i, can be either to the left of, right of, above or below a facility, j, and there is a constraint for each case based on the shortest distance from the origin. In other words, based on the location of the facility and the shortest distance to the origin, the appropriate binary variable is activated, enforcing the appropriate constraints. This ensures that the facilities are placed in a manner that does not violate any spatial requirements, and prevents them from overlapping, which would result in increased material handling costs.
Moreover, Constraint (8) is used to ensure that only one of the four Constraints (4) to (7) is activated at any given time. This is because these constraints are mutually exclusive, and activating one constraint renders the other three redundant. By enforcing Constraint (8), the optimisation model ensures that only the relevant constraint is activated, depending on the location of the facility and the shortest distance to the origin. In this way, the optimisation model ensures that the layout satisfies all spatial requirements, while minimising the total MHCs between the facilities, resulting in a layout that is both efficient and cost-effective.
Constraints (9) and (10) specify the restrictions on each variable. As for Constraint (10), it serves as the non-negative constraint. indicating that in this situation, only the positive quadrant is taken into consideration.
2.4. Heuristic Methods
The process of facility layout planning can be challenging, as it requires careful consideration of numerous factors, including the flow rate between facilities, path length, and material handling costs. In order to address these challenges, the proposed method for facility layout planning involves a two-stage heuristic algorithm. This algorithm provides a systematic and efficient approach to facility layout planning, with the ultimate goal of improving the overall efficiency and effectiveness of facility layouts.
The first stage of the proposed method involves determining the choosing sequence of the facilities. This is a crucial step in the process, as the order of facility placement can significantly impact the efficiency of the layout plan. In order to determine the optimal choosing sequence, various factors such as flow rate and path length between the facilities are considered. Once the optimal choosing sequence has been established, the second stage of the algorithm involves the placement of the facilities. This stage aims to determine the most effective location for each facility, while adhering to the constraints of the layout plan.
By dividing the facility layout planning process into two distinct stages, the proposed method provides an efficient and effective framework for optimising facility placement. This two-stage heuristic algorithm offers numerous benefits, including increased accuracy, reduced computational time, and improved efficiency. Additionally, the algorithm has the potential to enhance the overall efficiency and effectiveness of facility layouts, resulting in increased productivity and cost savings. Overall, the proposed method represents a significant contribution to the field of facility layout planning, providing a comprehensive and effective approach to optimising facility placement.
The Origin Heuristic
Stage 1: Pre-analysis
The first stage of the study involved exploring four different types of pre-processing techniques: decreasing length (DL), decreasing breadth (DB), decreasing area (DA), and no fixed arrangement (none). These techniques were crucial in arranging the study’s facilities before they were placed in the manufacturing plant. By arranging the facilities in accordance with the pre-processing stages, the MHC between facilities was minimised, and optimal positions for each facility within the plant were identified.
Figure 1 illustrated the different pre-processing stages with the exception of “none”, which had no fixed pre-processing. In situations where multiple facilities shared the same DL or DB values, a tiebreaker was used to determine their order. Similarly, in cases where some facilities shared the same area values, the breadth or length was used to break the tie. If the breadths or lengths were equal, the tie was broken arbitrarily.
Overall, the purpose of the first stage was to establish a foundation for the subsequent phases of the study, which aimed to further optimise the placement of the study’s facilities to achieve the study’s objectives. This process allowed for a systematic and efficient approach to facility layout planning, enabling the study to minimise the total MHC between facilities and ultimately reduce the overall cost of production.
Stage 2: Operating
The processing stage of the study played an essential role in optimising the position of the study’s facilities within the manufacturing plant to achieve the study’s objectives. The primary procedures for developing the Origin heuristic were executed during this stage with the aim of minimising the MHC.
Figure 2 presents all the procedures involved in the processing stage in a diagrammatic form, providing a clear visual representation of the steps taken. In
Figure 2a illustrates the first facility is placed at the origin (0,0). Then, the shortest distance is calculated at each corner point of the first facility as shown in
Figure 2b.
Figure 2c is where the facility is tested at each corner point before placement, for example, reference point = (0,0), distance
a = 5, distance
b = 6.4, distance
c = 5, shortest distance = Min {5, 6.4, 5} = 5, break tie arbitrarily. Then, choose the minimum distance and place the second facility as presented in
Figure 2d. All the other facilities are placed by filling up the plane based on the shortest distance from the origin at the bottom-left corner as shown in
Figure 2e–g. The finalised layout is shown in
Figure 2h. Then, the calculation of the overall cost from their centroid after placement of all facilities. Algorithm 1 illustrates the step-by-step approach of the heuristic method, outlining the specific actions taken at each stage.
The heuristic method employed during the processing stage was designed to determine the optimal position of each study facility within the manufacturing plant to reduce the overall MHC. This approach was critical to the study as it provided a means of achieving its objectives, specifically reducing costs and improving efficiency. The results obtained from this stage were used to inform subsequent phases of the study, which aimed to further optimise the position of the study’s facilities. Overall, the processing stage of the study was a critical component in achieving the study’s objectives, with the heuristic approach providing a systematic and efficient means of determining the optimal placement of each facility within the manufacturing plant. This stage served as a foundation for subsequent phases, which aimed to build on the results obtained to further optimise the study’s outcomes. Through this approach, the study was able to achieve its objectives and contribute to the field of facility layout planning.
Algorithm 1: Facility Layout Planning Algorithm
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Input: Set of facilities F, manufacturing plant boundaries B |
Output: Optimal facility layout with minimised MHC |
Initialise F by arranging facilities according to decreasing length (DL), decreasing breadth (DB), decreasing area (DA), or none.
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- 2.
Place the first facility at the origin (0,0).
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- 3.
Calculate all possible intervals from the origin for the remaining facilities.
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- 4.
Choose the smallest interval and allocate the corresponding facility Fi at the determined location.
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- 5.
Check the viability of the allocation based on the constraints.
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- 6.
If Fi violates any constraints, return to Step 3 and choose the next smallest interval.
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- 7.
Repeat Steps 3 to 6 for all remaining facilities.
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- 8.
Determine the objective function by calculating the MHC for the allocated layout.
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- 9.
If the objective function is satisfactory, terminate the algorithm and output the optimal facility layout.
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- 10.
Otherwise, return to Step 2 and explore alternative facility arrangements until an optimal layout is achieved.
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This algorithm outlines the proposed method for facility layout planning, which involves a two-stage heuristic approach. In the first stage, facilities are arranged according to decreasing length (DL), decreasing breadth (DB), decreasing area (DA), and no fixed arrangement (none). The second stage involves the placement of facilities based on the calculated intervals and the objective function, while adhering to the constraints of the layout plan. By dividing the facility layout planning process into two distinct stages, the proposed method offers a systematic and efficient approach to facility layout planning.