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Open Loop Layout Optimization: Feasible Path
Planning and Exact Door-to-Door Distance
Calculation
Seyed Mahdi Shavarani1*, B´ela Vizv´ari2and Kov´acs Gergely3
1*Centre for Logistics and Sustainability Analytics, Kent Business
School, University of Kent, UK.
2Department of Industrial Engineering, Eastern Mediterranean
University, Turkey.
3Edutus University, Hungary.
*Corresponding author(s). E-mail(s): m.shavarani@kent.ac.uk;
Contributing authors: bela.vizvari@emu.edu.tr;
kovacs.gergely@edutus.hu;
Abstract
The Open Loop Layout Problem (OLLP) seeks to position rectangular cells
of varying dimensions on a plane without overlap, minimizing transportation
costs computed as the flow-weighted sum of pairwise distances between cells.
A key challenge in OLLP is to compute accurate inter-cell distances along fea-
sible paths that avoid rectangle intersections. Existing approaches approximate
inter-cell distances using centroids, a simplification that can ignore physical con-
straints, resulting in infeasible layouts or underestimated distances. This study
proposes the first mathematical model that incorporates exact door-to-door dis-
tances and feasible paths under the Euclidean metric, with cell doors acting
as pickup and delivery points. Feasible paths between doors must either follow
rectangle edges as corridors or take direct, unobstructed routes. To address the
NP-hardness of the problem, we present a metaheuristic framework with a novel
encoding scheme that embeds exact path calculations. Experiments on standard
benchmark instances confirm that our approach consistently outperforms existing
methods, delivering superior solution quality and practical applicability.
Keywords: Open Loop Layout Optimization, Exact Distances, Evolutionary
Algorithm, Door-to-Door Path Planning
1
arXiv:2504.12792v1 [math.OC] 17 Apr 2025
1 Introduction
Layout optimization is a fundamental problem in operational research, focusing on
the arrangement of spatial entities within a bounded region to optimize specific objec-
tives, such as minimizing transportation costs or maximizing space utilization, while
adhering to constraints like non-overlapping placement and spatial relationships (Wan
et al. 2022). The spatial entities may represent machines in manufacturing, locations
in logistics, infrastructure in urban planning, chips in semiconductor design, and assets
in facility management, where optimized layouts enhance productivity, streamline
resource utilization, and improve process flows (Pourvaziri et al. 2021).
Layout problems are typically categorised into four main types: Spine,Closed
Loop,Ladder, and Open Loop (Drira et al. 2006;Kang et al. 2018), with examples
of each illustrated in Figure 1. Each layout type addresses specific constraints and
operational needs. Spine layouts streamline movement paths along a single linear
track, while Closed Loop layouts organize flow within a rectangular loop, often used in
manufacturing. Ladder layouts divide spaces into grid-based zones for efficient access,
whereas Open Loop layouts - the most flexible - accommodate diverse configurations
and unrestricted flow paths without predefined constraints (Yang et al. 2005).
Rectangular Cell
Transportation Track
(a) Spine (b) Closed Loop (c) Ladder (d) Open Loop
Fig. 1: Examples of different layout configurations.
In contrast to Spine and Closed Loop layouts, where distances can be calcu-
lated precisely, Open Loop layouts—even those featuring a single loop—incorporate
irregular pathways, branching tracks, and intersecting zones that complicate the com-
putation of shortest paths (De Guzman et al. 1997;Niroomand and Vizv´ari 2013).
Incorporating the concept of ”doors” enhances the realism of the model but also
significantly increases its complexity (Carlo and Bozer 2011). In realistic scenarios,
doors represent designated entry and exit points for inter-cell flow within rectangu-
lar entities, restricting movement to specific pathways (Gu et al. 2010). This realistic
yet complex feature, combined with the need for exact distance calculations, signals
the limitations of traditional approximation and centroid-based methods, which often
oversimplify spatial relationships and lead to sub-optimal solutions.
An inferior solution disrupts flow and reduces operational efficiency (Niroomand
and Vizv´ari 2013;Drira et al. 2006). In Open Loop Layout Problems (OLLP), these
inefficiencies arise from the challenge of accurately determining the shortest feasible
paths between objects (Niroomand and Vizv´ari 2013). We address this challenge by
introducing a mathematical model that employs exact feasible path calculations based
2
on the l2metric. We further introduce a scalable solution method that efficiently
solves benchmark problems in the field, demonstrating significant improvements over
existing results. The model incorporates the ”door-to-door” concept to enhance real-
ism and employs advanced metaheuristic algorithms for efficient problem-solving.
Through extensive experimentation on benchmark problems, we demonstrate signifi-
cant improvements over existing approaches, providing a robust framework for tackling
real-world layout optimization challenges.
The remainder of this paper is structured as follows: An overview of past effort on
OLLPs is prvided in Section 2. Section 3delineates the specific attributes of the open
layout problem tackled in this study. In Section 4, we introduce the mathematical
model developed to address this problem. Section 5describes the innovative solution
approach we devised, detailing the techniques and strategies employed. Section 6out-
lines the experimental design, including the benchmark problems and methodologies
utilized. The results are presented and critically compared with existing literature in
Section 7. Section 8offers additional insights and reflections derived from the study’s
findings and lessons learned. Finally, Section 9concludes the paper and discusses
potential directions for future inquiry.
2 OLLP: An Overview
Strategic layout design has shown remarkable potential for cost savings and efficiency
gains. For instance, optimal machinery placement in manufacturing can reduce pro-
duction lead times, material handling costs, and overall operational expenses by up
to 35% (Klausnitzer and Lasch 2019;Alduaij and Hassan 2020). Similarly, efficient
warehouse layouts in logistics and supply chain management improve inventory han-
dling, accelerate order processing, and boost overall performance (D´ıaz et al. 2024).
Beyond industrial applications, well-optimized layouts in healthcare and retail improve
service delivery by enhancing flow and space utilization, as seen in hospitals and
shopping malls (Meng et al. 2021). In office environments, optimized layouts promote
increased employee productivity and collaboration (Bukchin and Tzur 2014;Dorrah
and Marzouk 2021).
OLLPs represents a significant extension of layout optimization, focusing on
arranging rectangular entities on a plane without overlap to minimize transporta-
tion costs (P´erez-Gosende et al. 2021). In OLLPs, entities can be placed anywhere
on the plane as long as they meet the non-overlapping constraints. Unlike closed-
loop systems with predefined and structured flow paths, open-loop configurations are
characterised by unrestricted and flexible pathways. However, this flexibility intro-
duces unique optimisation challenges, making OLLPs more complex to model and
solve effectively (Niroomand and Vizv´ari 2013;Chae and Peters 2006). The inherent
complexity of OLLPs comes from the need to model obstacles, distance metrics, and
spatial constraints when planning the paths between entities. These factors contribute
to the classification of OLLP as NP-hard, where the exponential growth in the number
of possible configurations renders the problem computationally intractable for exact
solutions. Consequently, approximation algorithms and heuristic methods are essen-
tial for addressing real-world applications of OLLP (Drira et al. 2006). The evolution
3
of layout optimisation techniques reflects a gradual shift from simplistic models to
sophisticated, real-world applications. (Kundu and Dan 2012).
Early approaches that leveraged the concept of weighted centroids—as introduced
in (Muther 1973) through analyses of material flow and departmental placement—laid
the foundation for later centroid-based distance models, offering computationally effi-
cient mechanisms for solving layout problems. However, these models have limited
applicability as they overlook practical constraints such as obstacles and non-linear
flow paths (Meller and Bozer 1996). These limitations were particularly evident
in scenarios involving irregularly shaped entities or dynamic flow interactions. The
introduction of rectangular layout problems in the late 1970s marked a significant
advancement in addressing challenges posed by irregular shapes (Thornton et al. 1979).
Since then, a range of mathematical models, along with heuristic and metaheuristic
approaches, have been proposed. However, these methods are predominantly based
on distance approximations, limiting their accuracy and practical applicability due to
inherent suboptimality (see Figure 2for an example).
Rectangle
Flow Path
Door
Optimal Solution Suboptimal Solution
Fig. 2: Suboptimality in centroid-based methods
For instance, Niroomand and Vizv´ari (2013) introduced mixed-integer program-
ming models, incorporating rigorous constraints to account for feasible flow paths.
However, their reliance on approximated l1distance metrics limited efficiency in
scenarios where optimal paths deviated from linear assumptions. Similarly, Ma
et al. (2025) employed a centroid-based approximation to improve urban emergency
response, raising concerns about the accuracy of the model. While using l1distance
approximations may be justified in large-scale problems where computing exact dis-
tances is computationally prohibitive, this limitation becomes particularly critical
in facility layout contexts. Metaheuristic methods are no exception, and their solu-
tion quality is influenced by the reliance on approximate distance calculations. For
example, Hauser and Chung (2006) and ¨
On¨ut et al. (2008) demonstrated the effec-
tiveness of GA and PSO, respectively, in tackling high-dimensional and non-linear
optimization challenges. To enhance efficiency and adaptability, many studies have
explored the integration of machine learning techniques and hybrid approaches within
metaheuristic frameworks. Powell (2010) demonstrated the potential of combining arti-
ficial intelligence with traditional optimization strategies to address complexity and
dynamic constraints. Hybrid approaches, such as simulation-based optimization (Yang
et al. 2023) and surrogate models like Kriging (Zhang et al. 2017), capture complex
4
interactions and enhance computational efficiency while maintaining solution quality.
However, the reliance on approximate distance metrics remains a critical limitation,
often resulting in suboptimal outcomes. Precise distance calculations become even
more complex when the concept of doors or input/output (IO) points is introduced,
as it necessitates door-to-door path planning (Khalilabadi et al. 2024).
In layout optimization, door-to-door distance refers to the distance between entry
points (doors) of spatial entities such as rooms, departments, or workstations. This
metric is critical for accurately modeling flow dynamics within a facility, as it cap-
tures the true spatial relationships and movement patterns between different areas
(Khalilabadi et al. 2024). By considering the distance between doors, along with traf-
fic volume and directionality, realistic simulations of flow efficiency can be developed,
accounting for congestion, bottlenecks, and pathway effectiveness (Benson and Foote
1997). In warehouses, for example, reducing door-to-door distances improves efficiency
by cutting travel time and reducing congestion along transportation routes (Kim and
Kim 2015).
In the following sections, we present a concise description of the rectangular
OLLP with door-to-door distance requirements. For the first time, an exact math-
ematical model is formulated to address this problem, complemented by a novel
solution approach that integrates exact door-to-door path planning with advanced
metaheuristic algorithms.
3 Problem Definition
We consider a specific type of layout problem that involves placing a number of rect-
angles of different dimensions on a plane without overlapping (Figure 3a). These
rectangles represent different objects across different domains. A typical example is
the arrangement of machines or cells in a factory, where the objective is to minimise
the total transportation cost of semi-finished products (Shi et al. 2022).
To provide a more realistic representation of the problem, we assume each rect-
angle (cell) is equipped with a pick-up point positioned at the midpoint of one of its
edges. Transportation occurs only between these pick-up points, here called ”doors”.
A general constraint is that the rectangles must have vertical and horizontal edges and
can be rotated by 90, 180, or 270 degrees (→Section 4→Figure 5), only. Transporta-
tion paths cannot cross the rectangles, and any intersection (Figure 3b) is infeasible;
they must move either outside or along the edges of the rectangles.
4 Problem Formulation
An exact mathematical model is provided for the case where distances are measured
using the Euclidean metric. It should be noted that in mathematics, there are infinitely
many other ways to measure distances, commonly represented as lpnorms, where
1≤p≤ ∞. The Euclidean distance corresponds to the l2norm. An example of a fea-
sible l2path is illustrated in Figure 4. In engineering, two other distance measures are
particularly significant: the l1and l∞norms. The l1distance, also known as the Man-
hattan distance, is the sum of the absolute differences between the coordinates. The
l∞distance, on the other hand, is defined as the maximum of the absolute differences
5
Rectangle
Overlapping Area
Flow Path
Intersection
Door
Rectangle 1
Rectangle 2
(a) Overlapping
Rectangle 1
Rectangle 2
(b) Intersection
Fig. 3: Illustrations of (a)Overlapping and, (b)Intersection of a path and the edges.
Rectangle 1
Rectangle 2
Rectangle 3
Rectangle
Flow Path
Door
Fig. 4: Exact feasible l2distances.
between coordinates. Each of these distance metrics requires a distinct mathematical
model to determine the path that is optimal (i.e., minimal) with respect to the given
distance measure and thus are not discussed here.
4.1 Mathematical Model for OLLPs with Exact Euclidean
Distances
The exact description of the objective function requires certain data from the con-
straint set to ensure the non-overlapping property of the layout. Montreuil (1990) was
the first in discussing this type of constraint. More complicated models can be found
in (Heragu and Kusiak 1991;Tompkins et al. 1996;Meller et al. 1999;Sherali et al.
2003), and (Das 1993). Any of the aforementioned sets of non-overlapping constraints
can be used in an exact model, although significant differences may occur in the effec-
tiveness of the model. Here we use the latter. In what follows, we will outline these
basic constraints along with specific constraints required for open layout problems for
l2distances considering the positions of the doors which are unique to this paper. The
nomenclature is defined in Table 1.
Each cell contains five key points: its four corner points and a pick-up point. When
the cell is rotated, the locations and indices of these points are illustrated in Figure
5. The upper-left corner is always assigned index 1, while the upper-right corner is
consistently indexed as 2, and so on. However, the position of the pick-up point varies
depending on the cell’s rotation. In its default position (0◦rotation), the length of
the vertical (horizontal) edges of cell iis si(ti). Without loss of generality, we assume
that the door (or the edge containing the door) is located below the centroid in the
6
Table 1: Sets, Parameters and Variables
Symbol Description
ISet of cell (rectangle) indices. Alias: i, j, p
K={1,2,3,4,5}Set of node (vertices + door) indices for each cell
R={1,2,3,4}Edge index
Q={1,2,3,4}Orientation index
si(ti) Length of the vertical (horizontal) edge of cell i
Fi,j The flow between rectangles iand j
ANode–arc incidence matrix of the network connecting
door (pick–up) points.
ai(bi) Horizontal (vertical) coordinate of the center of cell i
eij , fij (gij , hij ) Horizontal (vertical) distances between the centers of cells
iand j
ziBinary variable; 1 if cell iis in a vertical position, 0 if
horizontal
xik(yik) Coordinate of node k∈Kof cell i
λiqBinary variable describing the door position for cell i
dikjlDistance between nodes ikand jl
MikjlprPenalty if the line connecting ikto jlcrosses the edge r
of cell p
γikjlpr, δikjlprAuxiliary variables for penalty calculation
uikjlpr, vikjlpr, wikjlpr, zikjlpr, mikjlprBinary auxiliary variables for penalty calculation
αij , βij , µij Binary auxiliary variables for non-overlapping conditions
between cells iand j
ei5Unit vector associated with cell i(at its pick-up point)
gij Path vector between the pick-up points i5and j5
λi4= 1
P
j=1,2,3
(λij)=0
si
ti
i1i2
i4i3
λi3= 1
P
j=1,2,4
(λij)=0
si
ti
i1i2
i4i3
λi2= 1
P
j=1,3,4
(λij)=0
si
ti
λi1= 1
P
j=2,3,4
(λij)=0
ti
si
i1i2
i4i3
Fig. 5: Orientation of the ith rectangular box (si,ti), described by λij.
default position. The coordinates of the doors of the ith rectangle depend only on
its orientation/rotation on the plane. The orientation of the ith cell is described by
four binary variables, λij;j∈ {1,2,3,4}. When λi1= 1 the door is located below
the centroid of the rectangle i, when λi2= 1 the door is located to the right of the
centroid, when λi3= 1 the door is located above the centroid, and finally λi4= 1
means the door is located to the left of the centroid. λij= 1 when the pick-up point
is in the specified position dictated by j, otherwise 0 such that Σ4
j=1λij = 1 ∀i. The
7
complete mathematical model will be described as follows:
min X
i,j
Fij dT
ij gij (1)
A gij =ei5−ej5∀i, j ∈I(2)
eij +fij −1−zi
2ti−zi
2si−1−zj
2tj−zj
2sj≥ −Mµij ∀i, j ∈I(3)
gij +hij −1−zi
2si−zi
2ti−1−zj
2sj−zj
2tj≥−M(1−µij )∀i, j ∈I(4)
ai−aj=eij −fij ∀i, j ∈I(5)
bi−bj=gij −hij ∀i, j ∈I(6)
eij ≤Mαij ∀i, j ∈I(7)
fij ≤M(1 −αij )∀i, j ∈I(8)
gij ≤Mβij ∀i, j ∈I(9)
hij ≤M(1 −βij )∀i, j ∈I(10)
xi1=xi4=ai−1−zi
2ti−zi
2si∀i∈I(11)
xi2=xi3=ai+1−zi
2ti+zi
2si∀i∈I(12)
yi1=yi2=bi+1−zi
2si+zi
2ti∀i∈I(13)
yi3=yi4=bi−1−zi
2si−zi
2ti∀i∈I(14)
zi=λi2+λi4∀i∈I(15)
1−zi=λi1+λi3∀i∈I(16)
xi5=ai+ (λi2−λi4)si
2∀i∈I(17)
yi5=bi+ (λi3−λi1)si
2∀i∈I(18)
di1i2=di3i4=zisi+ (1 −zi)ti∀i∈I(19)
di2i3=di1i4=ziti+ (1 −zi)si∀i∈I(20)
di1i3=di2i4=M∀i∈I(21)
di5i1= (λi1+λi2)M+ (λi3+λi4)ti
2∀i∈I(22)
di5i2= (λi1+λi4)M+ (λi2+λi3)ti
2∀i∈I(23)
di5i3= (λi3+λi4)M+ (λi1+λi2)ti
2∀i∈I(24)
di5i4= (λi2+λi3)M+ (λi1+λi4)ti
2∀i∈I(25)
dikjl=q(xik−xjl)2+ (yik−yjl)2+
n
X
p=1
4
X
r=1
Mikjlpr∀i, j ∈I;k, l ∈K(26)
xp2=xp3=γikjlp1xik+ (1 −γikjlp1)xjl∀i, j, p ∈I;k, l ∈K(27)
γikjlp1yik+(1−γikjlp1)yjl=δikjlp1yp2+(1 −δikjlp1)yp3∀i, j, p ∈I;k, l ∈K(28)
8
xp1=xp4=γikjlp3xik+ (1 −γikjlp3)xjl∀i, j, p ∈I;k, l ∈K(29)
γikjlp3yik+(1−γikjlp3)yjl=δikjlp3yp1+(1 −δikjlp3)yp4∀i, j, p ∈I;k, l ∈K(30)
yp4=yp3=δikjlp2yik+ (1 −δikjlp2)yjl∀i, j, p ∈I;k, l ∈K(31)
δikjlp2xik+(1−δikjlp2)xjl=γikjlp2xp4+(1 −γikjlp2)xp3∀i, j, p ∈I;k, l ∈K(32)
yp1=yp2=δikjlp4yik+ (1 −δikjlp4)yjl∀i, j, p ∈I;k, l ∈K(33)
δikjlp4xik+(1−δikjlp4)xjl=γikjlp4xp1+(1 −γikjlp4)xp2∀i, j, p ∈I;k, l ∈K(34)
1−γikjlpr−uikjlprM≤0∀i, j, p ∈I;k, l ∈K;r∈R(35)
γikjlpr−vikjlprM≤0∀i, j, p ∈I;k, l ∈K;r∈R(36)
1−δikjlpr−wikjlprM≤0∀i, j, p ∈I;k, l ∈K;r∈R(37)
δikjlpr−zikjlprM≤0∀i, j, p ∈I;k, l ∈K;r∈R(38)
uikjlpr+vikjlpr+wikjlpr+zikjlpr≤3 + mikjlpr∀i, j, p ∈I;k, l ∈K;r∈R(39)
Mikjlpr=mikjlprM∀i, j, p ∈I;k, l ∈K;r∈R(40)
eij , fij , gij , hij ≥0∀i, j ∈I(41)
dikjl, gikjl≥0∀i, j ∈I, ∀k , l ∈K(42)
uikjlpr, vikjlpr, wikjlpr, zikjlpr, mikjlpr∈ {0,1} ∀i, j, p ∈I;k, l ∈K;r∈R(43)
µij , αij , βij , zi, λiq∈ {0,1} ∀ i, j ∈I;q∈Q(44)
The objective function (1) seeks to minimize the total flow between cells, weighted
by the distances between their respective pick-up points. These pick-up points rep-
resent the vertices of a graph, with edges formed between vertices whenever a direct
movement is feasible. Consequently, the edge set dynamically depends on the spatial
arrangement of the rectangles. The lengths of these edges correspond to the l2distances
between vertices, which are inherently influenced by the locations of the rectangles.
The constraint 2establishes a path from pick-up point i5to pick-up point j5
for all non-negative flow values fij, where Ais the node-arc adjacency matrix of
the complete graph, consisting of the critical points of all cells. Constraints 3-10
guarantee the non-overlapping of cells iand j. Two cells are overlapping if and only
if their centers are too close to each other. The minimal required horizontal (vertical)
distance such that two cells are not overlapping is half the sum of the length of their
edges in the horizontal (vertical) position. The sum depends on the rotation of the
cells. Let eij = max{0, xi−xj},fij = max{0, xj−xi},gij = max{0, yi−yj}and
hij = max{0, yj−yi}. Notice that eij +fij (gij +hij ) is the horizontal(vertical) distance
of the centers of the cells iand j. If there is no horizontal (vertical) overlap, then the
distance must be at least as long as the sum of the two horizontal (vertical) half edges.
This requirement is described by 3and 4. If the binary variable µij is 1, then cells i
and jare not overlapping vertically, and if it is 0, then cells iand jare not overlapping
horizontally. It is difficult to use the formulae of eij,fij ,gij , and hij explicitly in an
optimization problem; therefore, they are described implicitly by 5-10. Here αij is a
binary variable: it is 1 if xi≥xj; similarly, βij is 1 if yi≥yj. Constraints 11 and 12
describe the horizontal coordinates of the vertices of cell i. Constraints 13-14 indicate
the vertical coordinates of the vertices of cell i. Constraints 15 and 16 determine if
the cell is in horizontal or vertical position. It should be remembered that a cell is in
a horizontal position by definition if the pick-up point is below or above the center,
9
and it is in a vertical position if the pick-up point is beside the center. Thus, the λ
and zvariables are not independent. Constraints 17 and 18 identify the coordinates of
the pick-up point of cell i. In our model we use the Euclidean distance for the ikand
jlpoints if there are no cells between the points. Otherwise the distance will be M.
Based on equations 19-20 the distance between two adjacent vertices of cell iis equal
to the length of the edge. Constraint 21 prohibits direct movement between opposite
vertices. The constraints 22 -25 determines the distance between the pick-up point
of cell iand its adjacent vertices: it is half of the length of the edge, i.e., ti
2. Direct
movement between the pick-up point and the opposite vertices of the same cell is not
feasible; hence, the distance in this case is M.
Constraints 26 -40 determine the distance between points ikand jl: it is the
Euclidean distance if there are no cells between them. Otherwise, the distance is
increased by M. Here Mikjlpris a penalty variable; it is Mif the line ikjlcrosses
the r-th edge of cell pand it is 0 otherwise. If r= 1, then the r-th edge of cell pis
the vertical edge between points p2and p3. Then there exists the unique value γikjlp1
for which 27 holds. Moreover there exists the value δikjlp1which satisfies 28. If the
segment between ikand jlcrosses the 1st edge of cell p, then both of the values γikjlp1
and δikjlp1are between 0 and 1, and 27 determines the horizontal, and 28 determines
the vertical coordinate of the cross-point. Similarly, if r= 3, then there exist the same
γikjlp3and δikjlp3for 29 and 30. If r= 2 or r= 4, then the edge of cell pis horizontal,
thus the vertical equation is described first. If r= 2, then there exist the only δikjlp2
and γikjlp2which satisfy 31 and 32. Similarly, if r= 4, then there exist the same δikjlp4
and γikjlp4for 33 and 34. (For example if r= 4, then the segment between ikand jl
crosses the 4th edge of cell pif and only if, then both of the values δikjlp4and γikjlp4
are between 0 and 1.) If 0 < γikjlpr<1, then for feasibility of 35 and 36 the values
of the two binary variables need to be equal to 1. Similarly, in 37 and 38 wand zare
both equal to 1 if 0 < δikjlpr<1. If the segment between ikand jlcrosses the r-th
edge of cell p, then all four binary variables are 1. Therefore in 39 the value of mis 1
if the segment crosses the edge, which leads to the realization of the penalty in 40.
5 Methods
Exact methods struggle to scale as the problem size increases due to the inherent NP-
hardness and combinatorial complexity of OLLPs (J A Bennell and Whitehead 2002).
In the case of OLLP with door-to-door distances, even the smallest problem instances
require significant time to solve due to the nonlinearity introduced by l2distance
calculations and the large number of binary variables. Consequently, metaheuristic
algorithms have emerged as crucial tools for efficiently exploring the solution space
and identifying high-quality solutions. Metaheuristics, such as evolutionary algorithms
and swarm intelligence, offer a balance between exploration and exploitation, allowing
efficient navigation of the vast solution space (Tongur et al. 2020).
Metaheuristics rely on solution representation and efficient search operators to
explore and refine candidate layouts iteratively. The effectiveness of these algorithms
depends on how solutions are encoded and decoded, as these representations directly
impact the search process. A well-structured encoding ensures that the evolutionary
10
operators—such as crossover, mutation, and selection—can generate diverse and fea-
sible solutions while preserving spatial constraints. Decoding, in turn, translates these
encoded representations into real-world layouts, enabling the evaluation of objective
functions and guiding the optimization process. Thus, an effective encoding-decoding
mechanism is crucial for leveraging the full potential of metaheuristic algorithms in
solving OLLP.
5.1 Encoding
A feasible layout must be incrementally constructed, ensuring non-overlapping place-
ments, valid door-to-door paths, and optimized spatial arrangements. This process
involves determining the insertion sequence of rectangles, their rotation, and their
positioning while maintaining feasibility constraints. The proposed encoding comprises
three sections, each with a length of n, where ndenotes the total number of rectan-
gles. The required information for constructing a feasible solution is extracted from
the chromosome as described below. A visual example is provided in Figure 6.
1. Insertion Order: A floating-point representation defining the sequence of rect-
angle placements. The sequence is sorted based on numerical values, ensuring that
lower values correspond to earlier placement. Each rectangle is initially located at
the center of the coordinates.
2. Rotation Angle: The values in this section of the chromosome are mapped to one
of {0◦,90◦,180◦,270◦}by a function f(x) = ⌊4x⌋ · 90◦,where x∈[0,1], dictating
each rectangle’s orientation.
3. Shift Angle: Each rectangle is associated with a continuous value θ∈[0,1], which
dictates the initial displacement direction. The angle along which the rectangle will
be shifted is determined by multiplying the value of θby 360. The shifting process
continues iteratively in the given direction until no further overlapping with other
rectangles exists.
Fig. 6: Illustration of a chromosome for a problem with 3 rectangles.
5.2 Path Computation and Adjacancy Graph Construction
Once all rectangles are placed without overlaps, an adjacency graph is created to
represent accessible node pairs. Each rectangle contributes five key nodes: a door at
the center of one edge and one at each vertex.
11
Intra-rectangle connections: Feasible paths connect vertices on the same edge,
with the path length equal to the edge’s length. The door, located at the midpoint of
an edge, is connected to its adjacent vertices with paths of half the edge’s length.
Inter-rectangle connections: Feasible paths between nodes on different rectan-
gles are considered if they can be established without intersecting the edges of other
rectangles. To decide if the path intersects any edges of positioned rectangles, we use
an orientation-based approach (Preparata and Shamos 1985).
Definition 1 (The Orientation Function) For any three points P= (Px, Py), Q= (Qx, Qy),
and R= (Rx, Ry), we define the orientation as:
orient(P, Q, R) = (Qx−Px)(Ry−Py)−(Qy−Py)(Rx−Px).
For segments AB and CD to intersect in the general case (i.e., when the endpoints
are not all collinear), the following conditions must be satisfied:
orient(A, B, C )·orient(A, B, D)≤0,
and
orient(C, D, A)·orient(C, D, B)≤0.
When one or more of the orientation values is zero, the corresponding points are
collinear. In such instances, it is necessary to verify whether the collinear point lies on
the line segment formed by the other two endpoints. For example, if
orient(A, B, C )=0,
then point Cis on segment AB if and only if
min(x1, x2)≤x3≤max(x1, x2) and min(y1, y2)≤y3≤max(y1, y2).
If the path between two points is feasible, its length is computed using the
Euclidean metric. Additionally, paths traversing along shared edges of neighboring
rectangles are considered feasible. Two rectangles may share an edge or even a single
point (at their edges or vertices) as long as their boundaries do not cross. To enforce
this and to mitigate spurious intersections caused by floating-point precision issues, a
small buffer threshold is incorporated during the feasibility checks. This ensures com-
putational robustness and the accuracy of path evaluations. This structured approach
guarantees that all potential paths are evaluated effectively, minimizing the overall
transportation cost while adhering to spatial constraints.
With the adjacency matrix established, we can then compute the shortest distance
matrix between pairs of doors using the Floyd-Warshall algorithm. The objective func-
tion can then be calculated as Z=PN
i=1 PN
j=1 wij dij , where Zis the total weighted
door-to-door distance, Nis the total number of rectangles, wij : flow between door of
rectangle iand door of rectangle j, representing the frequency or cost of travel and dij
is the shortest path distance between door iand door j, calculated using the adjacency
graph and Euclidean distances.
12
5.3 Selected Algorithms
To address the Open Loop Layout Problem (OLLP), we apply four metaheuristic opti-
mization algorithms. These algorithms have been chosen for their ability to effectively
navigate the vast and complex solution space of combinatorial problems. Below, we
present a brief overview of each algorithm, with more detailed information available
in the original studies cited in the respective sections.
5.3.1 Simple Genetic Algorithm (SGA)
The Simple Genetic Algorithm (SGA) is a population-based evolutionary algorithm
introduced by Holland (1973). It mimics the process of natural selection through
the iterative application of selection (Chooses individuals based on fitness), crossover
(Combines parent solutions to generate offspring), and mutation (Introduces random
variations to maintain diversity) operators. SGA balances exploration and exploita-
tion to identify high-quality solutions. The pseudocode for the SGA is illustrated in
Algorithm 1.
Algorithm 1 Simple Genetic Algorithm (SGA)
Require: Population size N, Crossover rate pc, Mutation rate pm, Maximum
generations Gmax, Selection method S
Ensure: Best solution found
Initialize population P0of size Nrandomly
Set generation counter g←0
while g < Gmax and stopping condition not met do
Evaluate fitness of individuals in Pg
Select parents from Pgusing selection method S
Apply crossover with probability pcto generate offspring
Apply mutation with probability pmto offspring
Replace old population Pgwith new population Pg+1
Update generation counter g←g+ 1
end while
Return the best solution in the final population
5.3.2 Particle Swarm Optimization (PSO)
Particle Swarm Optimization, introduced by Kennedy and Eberhart (1995), is a swarm
intelligence algorithm inspired by the social behavior of birds and fish. Each particle
updates its velocity (by combining the particle’s inertia, personal best, and global best
information) and position (by moving the particle based on its updated velocity) in
the solution space based on its own experience and that of its neighbors. A general
pseudocode of the PSO is provided in Algorithm 2.
13
Algorithm 2 Particle Swarm Optimization (PSO)
Require: Population size N, Max generations Gmax, Inertia weight w, Cognitive
coefficient c1, Social coefficient c2
Ensure: Best solution found
1: Initialize positions xiand velocities viof Nparticles randomly
2: Initialize personal best positions piand global best position g
3: Set generation counter g←0
4: while g < Gmax and stopping condition not met do
5: Evaluate fitness f(xi) for each particle i
6: Update personal best positions pibased on fitness
7: Update global best position gbased on the best fitness
8: for each particle ido
9: Update velocity: vi←wvi+c1r1(pi−xi) + c2r2(g−xi)
10: Update position: xi←xi+vi
11: end for
12: Increment generation counter g←g+ 1
13: end while
14: return the global best solution g
5.3.3 Differential Evolution (DE)
Differential Evolution, proposed by Storn and Price (1997), is a population-based opti-
mization algorithm that employs differential mutation to explore the search space. It is
simple yet highly effective for continuous optimization problems. Mutation operator
adds the weighted difference of two individuals to a third individual. The Crossover
operator combines the mutated vector with the target vector. And the Selection oper-
ator retains the individual with the best fitness in the next generation. The pseudocode
of DE is provided in Algorithm 3.
5.3.4 Self-Adaptive Differential Evolution (SADE)
Self-Adaptive Differential Evolution (SADE), introduced by Omran et al. (2005),
enhances the classic Differential Evolution by dynamically adjusting mutation and
crossover rates. This self-adaptive mechanism improves convergence performance and
reduces the need for parameter tuning. The Mutation operator creates new solutions
by adding weighted differences between individuals to a base vector. Crossover com-
bines mutated vectors with the current population. And the Self-adaptation tech-
nique updates parameters based on success rates of previous iterations. Algorithm 4
shows the pseudocode for the DE.
5.3.5 Hybrid Methods
In our pursuit to enhance the efficacy of EAs, we also consider experiments with
integrating exact methods into the evolutionary framework, exploring several hybrid
approaches:
14
Algorithm 3 Differential Evolution (DE)
1: Input: Population size N, maximum generations G, mutation factor F∈[0,2],
crossover rate CR ∈[0,1]
2: Output: Best solution found
3: Initialize population {x0
i|i= 1, . . . , N }randomly within the bounds of the search
space
4: Evaluate fitness f(x0
i) for all individuals in the initial population
5: for generation g= 1 to Gdo
6: for each individual xg
iin the population do
7: Select three distinct indices r1, r2, r3=i
8: Compute mutant vector as vg
i=xg
r1+F·(xg
r2−xg
r3)
9: Perform crossover:ug
ij =(vg
ij ,if rj≤CR or j=jrand
xg
ij ,otherwise
10: Selection: Compare trial and target vectors3:xg+1
i=
(ug
iif f(ug
i)≤f(xg
i)
xg
iotherwise
11: end for
12: end for
13: return the best solution x∗from the final population
•Optimal Placement of Rectangles: Here, we attempt to optimally position each
rectangle individually given a randomly generated entry order. This involved solv-
ing a mathematical model aimed at minimizing the weighted distance from already
placed rectangles. However, this approach consistently yielded inferior results com-
pared to standalone EAs in our preliminary experiments and thus is not included
in the results.
•Individual Rectangle Optimization: After positioning all rectangles, we inves-
tigate the potential improvement of the solution quality by changing the position
of individual rectangles.
Due to space constraints, a detailed discussion of hybrid methods is omit-
ted; however, supplementary results are available in the repository at www.
to-be-updated-after-peer-review.com for interested readers.
6 Experimental Setup
Benchmark Problems: We conducted our experiments on a set of existing bench-
mark problems with the number of rectangles ∈ {8,10,12,14,16,18,20,25,30}
acquired from (Das 1993;Rajasekharan et al. 1998). Each experiment was assigned
a maximum CPU time of 24 hours, which is a traditional limit in this field and for
the benchmark problems used. Each experiment was run 40 times with different ran-
dom seed values. All experiments were executed on a HPC server equipped with 16
cores, 32 threads, and 192 gigabyte of RAM. The experiments are designed to evaluate
algorithmic performance, scalability and feasibility.
15
Algorithm 4 Self-Adaptive Differential Evolution (SADE)
1: Input: Population size N, max generations G
2: Output: Best solution found
3: Initialize population {x0
1,x0
2,...,x0
N}randomly
4: Initialize mutation factor Fand crossover rate CR
5: while stopping condition not met do
6: for each individual i∈ {1, . . . , N}do
7: Select three distinct indices r1, r2, r3=i
8: Generate mutant vector: vg
i=xg
r1+F·(xg
r2−xg
r3)
9: Perform crossover: ug
ij =(vg
ij ,if rj≤CR or j=jrand
xg
ij ,otherwise
10: Evaluate fitness f(ug
i)
11: if f(ug
i)≤f(xg
i)then
12: Update population: xg+1
i=ug
i
13: else
14: Keep old solution: xg+1
i=xg
i
15: end if
16: end for
17: Adapt mutation factor Fand crossover rate CR based on success rates
18: end while
19: return best solution xbest
Parameter Tuning Selected Algorithms: To make a fair comparison and get the
best results, we tune the parameters of the selected algorithms to identify the opti-
mal parameters that consistently deliver superior performance across various problem
instances. Parameter tuning is a pivotal step in the optimization process, aimed at
enhancing the performance of metaheuristic algorithms.
For parameter tuning process we employed the Optuna framework, an efficient
hyperparameter optimization tool (Akiba et al. 2019). Optuna provides a flexible and
lightweight environment for defining and optimizing hyperparameters using a range of
search algorithms and pruning strategies. The tuning was performed across a selection
of benchmark problems with 8, 10, and 12 rectangles. This approach ensured that
the chosen parameters offered robust performance across different problem instances
rather than being tailored to a specific instance. The number of generations was capped
at 50 across all algorithms to accelerate the tuning process, with performance evaluated
based on the average objective function value across multiple trials. The parameters
to be tuned for each algorithm and the best parameter combinations found and used
in the experiments are illustrated in the Table 2.
Implementations: All the implementations, benchmark problems, and solution
methods were implemented in Python 3.7.6. For evolutionary algorithms we used the
Pygmo library version 2.16.0 (Biscani et al. 2010). We used Optuna library in python
for parameter tuning (Akiba et al. 2019). We also utilized the CPLEX optimization
library. To encourage further research, we have made our code publicly available at
www.to-be-updated-after-peer-review.com.
16
Table 2: Optimal parameters for PSO, SADE, SGA, and DE algorithms.
Algo Parameter Definition Value
PSO
ωInertia weight, controlling the influence of the previous
velocity
0.51
η1Cognitive parameter, influencing the particle’s ten-
dency to return to its best-known position
2.42
η2Social parameter, influencing the particle’s tendency
to move towards the swarm’s best-known position
2.37
Max vel Maximum velocity, limiting the particle’s speed 0.31
Neighb type Neighborhood type, determining the topology of par-
ticle interactions
2
Neighb param Neighborhood parameter, specifying the size of the
neighborhood
4
SADE Variant DE strategy variant 1
SGA
Crossover Type of crossover operator Binomial
Mutation Type of mutation operator Uniform
Selection Method used to select individuals for reproduction Tournament
DE
F Differential weight, controlling the amplification of dif-
ferential variations
0.11
CR Crossover probability, determining the probability of
recombination
0.86
Variant DE strategy variant 9
Table 3: Performance comparison of evolutionary algorithms. The format X(Y)
represents the Average(Best) objective value. Ndenotes the number of rectangles.
N Literature DE PSO SADE SGA
8 8906 6583 (5617) 6665 (5590) 6822 (5016) 6558 (5152)
10 12111 13946 (12008) 14600 (11937) 14585 (10712) 14343 (10934)
12 36677 36434 (31674) 36567 (31542) 35526 (31233) 36302 (30239)
14 41691 44290 (42393) 45478 (42131) 42855 (40689) 39642 (36693)
16 55064 58244 (53763) 61733 (55035) 57370 (55519) 51052 (48386)
18 66489 80770 (76474) 81147 (67949) 65923 (65179) 64451 (61330)
20 - 127275 (115777) 126114 (108435) 117537 (100901) 102672 (94828)
25 - 278156 (266444) 283498 (256110) 283330 (257964) 234835 (223442)
30 - 486497 (469408) 494729 (467387) 481058 (464196) 411587 (403351)
7 Experimental Results
The experimental results are benchmarked against the best-known solutions from the
literature and are presented in Table 3. For each problem size n, the table reports the
average objective function value over 40 runs, with the best solution found across those
runs enclosed in curly brackets. The best objective value for each problem instance is
highlighted in bold.
17
The results demonstrate that our optimizer consistently delivers competitive
outcomes, surpassing the best-known results documented in the literature for prob-
lem sizes where comparable data exists. GA proved to be more efficient in solving
most problems. This superiority is more prevalent in problems with larger sizes. It
is also noteworthy that the optimal objective value for the benchmark problem of
size 10 reported in the literature was achieved through an extensive computational
effort involving human-computer collaboration, exceeding a 24-hour time constraint
(Vizv´ari 2014). Despite this, our approach yields superior results. For larger prob-
lem sizes (n= 20,25,30), where literature results are unavailable, our optimizer
effectively achieves feasible solutions within the specified time frame. We invite the
research community to utilize the benchmark problems presented in this study as a
standard for evaluating and comparing their methods, thereby fostering clarity and
collaboration in the field.
Figure 7illustrates the impact of integrating exact solution methods into evolu-
tionary algorithms. In most cases, hybridization leads to a deterioration in solution
quality and increased variance, suggesting that the introduction of exact methods dis-
rupts the natural search dynamics of the algorithms. The only exception is PSO, where
the hybrid approach enhances performance, likely due to improved guidance in the
particle updates. These observations align with the results in Table 3, which highlight
the superior performance of SGA across various problem sizes. The findings suggest
that while hybridization with exact methods may provide benefits in some scenarios
and is capable of improving the quality of individuals, it generally introduces instabil-
ity and does not guarantee improved convergence, particularly for algorithms that rely
on iterative evolvement. This is inline with our preliminary observations where using
the exact methods for improving the quality of an individual withing initial iterations
showed promise, subsequent adjustments failed to produce significant improvements,
rendering the results inferior compared to pure EA outcomes.
The convergence analysis illustrated in Figure 8, highlights distinct performance
trends among the evolutionary algorithms across six problem instances (8, 12, 16, 20,
25, and 30 rectangles). DE consistently finds high-quality solutions early but stagnates
quickly, suggesting its suitability for time-constrained scenarios, particularly in small-
to-medium-sized problems. In contrast, SGA demonstrates a steady improvement over
iterations, benefiting from prolonged evolution and population refinement. While the
rate of improvement diminishes beyond a certain point, SGA’s exploratory capability
remains evident. PSO, on the other hand, exhibits fewer total iterations in some cases,
indicating a slower internal update mechanism, likely influenced by inertia, velocity
updates, and particle interactions. Overall, SGA outperforms the others in terms of
both convergence speed and final solution quality, making it a robust choice for layout
optimization.
Perimeter efficiency is a critical metric in layout optimization, directly influencing
spatial utilization, design effectiveness, and energy performance (Maling et al. 1982).
Griffel et al. (2020) define perimeter efficiency as the ratio of the layout perimeter and
the occupied area. Figure 9demonstrates a clear negative correlation between perime-
ter efficiency and the objective function value across selected problem sizes (8, 12, 16,
18
Fig. 7: Comparison of algorithm performance with and without hybridization using
exact methods.
and 20 rectangles). This indicates that as the objective function improves, perimeter
efficiency also increases, emphasizing the trade-off between these metrics. SGA consis-
tently outperforms other algorithms, achieving the highest perimeter efficiency and the
lowest objective function values while maintaining robust performance across problem
sizes, particularly as complexity increases. In contrast, other algorithms perform com-
petitively in smaller problems but rapidly fall behind as problem size grows. SADE
exhibits higher variance, while SGA’s results are notably more consistent, with smaller
deviations, reinforcing its reliability for both small and large problem instances.
Finally, Figure 10 illustrates the relationship between hull efficiency and the objec-
tive function value across different problem sizes (8, 10, 20, and 30 rectangles). Hull
efficiency is defined as:
HE =Ahull
Amin bounding box
Where: HE is the Hull Efficiency, Ahull is the hull area and Amin bounding box is the
area of the smallest rectangle that can accomodate the layout.
Each subplot in Figure 10 represents a specific problem size, highlighting how
hull efficiency evolves as the objective function improves. A clear downward trend is
expected across all cases, indicating that as the objective function value decreases,
hull efficiency improves, suggesting a more compact and efficient layout. Interestingly,
this is not the case in the problem with 8 rectangles, where a slight downward trend
is observed. This highlights the complexity of open layout problems and the precision
required to effectively handle such configurations. The variation in trends among prob-
lem sizes also suggests that certain algorithms perform more effectively in maintaining
high hull efficiency as problem complexity increases.
19
Fig. 8: Convergence analysis of evolutionary algorithms for different problem sizes.
8 Discussion
Evolutionary algorithms (EAs) have proven effective in addressing a broad range of
optimization problems, from test suite generation (Dorigo and Gambardella 1997) to
mechanism synthesis (Huang et al. 2022). Various EAs, such as SGA, DE and PSO,
each demonstrate unique strengths in overcoming diverse optimization challenges, and
this study reaffirms their applicability to open loop layout problems. We also explored
a hierarchical optimization process involving three stages: optimizing the sequence of
cells, their orientation, and their positioning. Techniques ranged from sequential exact
methods to adjustments guided by EAs. In every instance, pure EAs outperformed
these hybrid strategies, underscoring their superior capability in configuring efficient
layouts.
The implications of this study are substantial, affecting cost efficiency across mul-
tiple domains. For instance, In manufacturing, combining Lean methods with facility
layout design can lead to substantial improvements in efficiency, productivity, and cost
reduction (Kov´acs 2020). Architecturally, optimizing the layout of facilities enhances
space utilization and workflow efficiency, applicable in floor planning and other spatial
arrangement challenges. Similar benefits extend to logistics through improved pack-
ing strategies and to project management via optimized resource allocation. Even in
robotic path planning and network design, efficient layouts can minimize operational
costs and enhance system performance.
Despite the successes reported here, potential improvements remain unexplored
and will be pursued in future studies. Particularly, in large-scale problems, each addi-
tional cell can substantially lengthen computation times, especially in calculating
20
Fig. 9: Perimeter efficiency versus objective function.
the shortest path matrices. Accelerating EA processes through surrogate modeling
or alternative techniques such as parallel computation could dramatically enhance
computational efficiency and solution quality within fixed time constraints.
9 Conclusion
In this paper, we introduced a comprehensive mathematical model for the open loop
layout problem, incorporating the novel concept of door-to-door paths. This con-
cept enhances the model’s capacity to simulate real-life scenarios more accurately,
albeit at the cost of increased computational complexity. We proposed an innova-
tive optimization methodology designed to efficiently arrange and position rectangles
within a defined plane, accounting for both geometric and adjacency constraints. The
core of our approach lies in a novel encoding scheme that effectively integrates with
evolutionary algorithms (EAs).
Extensive experimentation was conducted on a variety of benchmark problems,
demonstrating the superiority of our proposed approach over existing methods in
the literature. The experimental results indicate that the proposed optimizer consis-
tently yields superior solutions across different problem instances. Additionally, we
21
Fig. 10: Hull efficiency versus objective function.
explored the integration of exact methods at strategic phases of the optimization pro-
cess. Although these hybrid methods initially exhibited some improvements during the
early stages of optimization, their overall performance proved inferior compared to the
standalone evolutionary algorithms. Nevertheless, these observations provide valuable
insights into the role of exact methods in potentially accelerating initial convergence.
Potential applications of our approach span various fields, including VLSI design,
architectural floorplanning, packing problems, resource allocation, robotics, and net-
work design. The significant implications of our method include improved efficiency,
enhanced performance, cost reduction, scalability, and fostering innovation and
competitiveness.
Future work will focus on further refining the hybrid optimization approach, explor-
ing adaptive strategies to dynamically balance between heuristic and exact methods
based on problem characteristics. One limitation that could be further explored is the
rotation of the rectangles, which is currently restricted to only four options. Addi-
tionally, extending the approach to three-dimensional packing and exploring other
practical applications will be valuable directions for further research. Finally, applica-
tion of machine learning techniques to implement surrogate models for improving the
efficiency of the proposed method in large-scale problems will be interesting. To facili-
tate the contributions in the field we make our algorithms and data publicly available
at www.to-be-updated-after-peer-review.com.
22
Statements and Declarations
Funding
No funds, grants, or other support was received.
Conflict of Interest
The authors declare that they have no conflicts of interest. The authors have no
relevant financial or non-financial interests to disclose.
Ethics Approval and Consent to Participate
Not applicable.
Consent for Publication
All authors have given their consent for the publication of this study.
Code and Data Availability
The implemented code and datasets generated and analyzed during this study will be
made available after peer review. To comply with the journal’s double-blind review
process, links to these resources have been temporarily replaced with placeholders.
Materials Availability
Not applicable.
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