Article

Hospital layout design renovation as a Quadratic Assignment Problem with geodesic distances

Abstract and Figures

Hospital facilities are known as functionally complex buildings. There are usually configurational problems that lead to inefficient transportation processes for patients, medical staff, and/or logistics of materials. The Quadratic Assignment Problem (QAP) is a well-known problem in the field of Operations Research from the category of the facility's location/allocation problems. However, it has rarely been utilized in architectural design practice. This paper presents a formulation of such logistics issues as a QAP for space planning processes aimed at renovation of existing hospitals, a heuristic QAP solver developed in a CAD environment, and its implementation as a computational design tool designed to be used by architects. The tool is implemented in C# for Grasshopper (GH), a plugin of Rhinoceros CAD software. This tool minimizes the internal transportation processes between interrelated facilities where each facility is assigned to a location in an existing building. In our model, the problem of assignment is relaxed in that a single facility may be allowed to be allocated within multiple voxel locations, thus alleviating the complexity of the unequal area assignment problem. The QAP formulation takes into account both the flows between facilities and distances between locations. The distance matrix is obtained from the spatial network of the building by using graph traversal techniques. The developed tool also calculates spatial geodesic distances (walkable, easiest, and/or shortest paths for pedestrians) inside the building. The QAP is solved by a heuristic optimization algorithm, called Iterated Local Search. Using one exemplary real test case, we demonstrate the potential of this method in the context of hospital layout design/re-design tasks in 3D. Finally, we discuss the results and possible further developments concerning a generic computational space planning framework.
Content may be subject to copyright.
Journal of Building Engineering 44 (2021) 102952
Available online 13 July 2021
2352-7102/© 2021 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/).
Hospital layout design renovation as a Quadratic Assignment Problem with
geodesic distances
Cemre Cubukcuoglu
a
,
b
,
*
, Pirouz Nourian
a
, M. Fatih Tasgetiren
c
, I. Sevil Sariyildiz
a
, Shervin Azadi
a
a
Department of Architectural Engineering and Technology, Chair of Design Informatics, Faculty of Architecture and the Built Environment, Delft University of Technology,
Delft, the Netherlands
b
Department of Interior Architecture and Environmental Design, Faculty of Architecture, Yasar University, Izmir, Turkey
c
Department of International Logistics Management, Faculty of Business, Yasar University, Izmir, Turkey
ARTICLE INFO
Keywords:
Computational design
Architectural space planning
Hospital layout
Quadratic assignment problem
Layout optimization tool
ABSTRACT
Hospital facilities are known as functionally complex buildings. There are usually congurational problems that
lead to inefcient transportation processes for patients, medical staff, and/or logistics of materials. The Quadratic
Assignment Problem (QAP) is a well-known problem in the eld of Operations Research from the category of the
facilitys location/allocation problems. However, it has rarely been utilized in architectural design practice. This
paper presents a formulation of such logistics issues as a QAP for space planning processes aimed at renovation of
existing hospitals, a heuristic QAP solver developed in a CAD environment, and its implementation as a
computational design tool designed to be used by architects. The tool is implemented in C# for Grasshopper
(GH), a plugin of Rhinoceros CAD software. This tool minimizes the internal transportation processes between
interrelated facilities where each facility is assigned to a location in an existing building. In our model, the
problem of assignment is relaxed in that a single facility may be allowed to be allocated within multiple voxel
locations, thus alleviating the complexity of the unequal area assignment problem. The QAP formulation takes
into account both the ows between facilities and distances between locations. The distance matrix is obtained
from the spatial network of the building by using graph traversal techniques. The developed tool also calculates
spatial geodesic distances (walkable, easiest, and/or shortest paths for pedestrians) inside the building. The QAP
is solved by a heuristic optimization algorithm, called Iterated Local Search. Using one exemplary real test case,
we demonstrate the potential of this method in the context of hospital layout design/re-design tasks in 3D.
Finally, we discuss the results and possible further developments concerning a generic computational space
planning framework.
1. Introduction
The term Space Planning refers to the processes aimed at arranging a
spatial conguration having logistics-related objectives, ergonomics,
and intended user-experience of a building [1]. The spatial congura-
tion of a building is an abstract representation of the particular ways in
which the spaces inside a building are related to one another. Mathe-
matically, spatial congurations are represented as graphs, which are
typically labelled but may or may not be assumed as directed and
weighted. Conventionally, if the graph in question is weighted and
directed, it is called a network.
1.1. Problem denition
Hospitals consist of a wide range of functional units, each serving for
different activities such as clinical, nursing, administration, service
(food, laundry, etc.), research, and teaching. There are also numerous
types of user groups and materials, moving or being transported be-
tween those various functions within the hospital. The owrates of these
movements in between the functional spaces can be modelled as a
graph/matrix indexed by the indices of the functional units, we can
consider this as a graph describing the functional requirements. On the
other hand, in an existing spatial conguration, how the spaces are inter-
related can be modelled as a dense graph/matrix encoding the distances
* Corresponding author. Department of Architectural Engineering and Technology, Chair of Design Informatics, Faculty of Architecture and the Built Environment,
Delft University of Technology, Delft, the Netherlands.
E-mail addresses: c.cubukcuoglu@tudelft.nl, cemre.cubukcuoglu@yasar.edu.tr (C. Cubukcuoglu), p.nourian@tudelft.nl (P. Nourian), fatih.tasgetiren@yasar.edu.
tr (M.F. Tasgetiren), i.s.sariyildiz@tudelft.nl (I.S. Sariyildiz), s.azadi-1@tudelft.nl (S. Azadi).
Contents lists available at ScienceDirect
Journal of Building Engineering
journal homepage: www.elsevier.com/locate/jobe
https://doi.org/10.1016/j.jobe.2021.102952
Received 17 February 2021; Received in revised form 14 June 2021; Accepted 3 July 2021
Journal of Building Engineering 44 (2021) 102952
2
between spaces of interest, computed with respect to the [temporal]
length of geodesics or optimal paths; we may refer to this graph as the
spatial conguration graph. Therefore, one of the challenges of space
planning can be seen as matching these two graphs such that the dis-
tance between pairs is small when the ow rate between them is large
and vice versa.
According to the literature, around 67% of health and care em-
ployees are not able to perform their jobs efciently due to the unsuit-
able layout of the working spaces [2]. Especially nurses sometimes
spend more time walking than the activities related to patient care in a
day [3] because of the spatial connectivity problems of the interrelated
spaces. In one study, it has been observed that 28.9% of nursing staff
time wasted on walking [4]. Previous studies also show that layout type
of nursing unit has an impact on the walking time of nursing staff, e.g.
Ref. [5] established that nursing staff in the radial unit walked 4.7 steps
per minute while the other staff working in rectangular unit walked 7.9
steps per minute, which is signicantly more. The poor placement of the
clinics combined with the increasingly overwhelming volume of trafc
between them was causing delays and heavy congestion in hospitals [6].
All these examples show that spatial conguration has a great impact on
the efcient functioning of hospital buildings in terms of walking dis-
tances and transportation processes. Therefore, providing efcient
transportation processes by minimizing the walking distances between
interrelated spaces should be the major concern in hospital layout
planning.
The use of Quadratic Assignment Problem (QAP) is highly desirable
to deal with layout planning problems in hospitals. It is a well-known
problem in the eld of Operations Research. Most of the facility layout
problems (FLP) are formulated as QAP to minimize the transportation
cost. The requirements for design in the production industry are parallel
to those in hospital design. Therefore, facility-planning methodologies,
which are widely used in industrial engineering, are needed in dealing
with hospital layouts.
1.2. Related literature
QAP is developed by Koopmans and Beckmann (1957) [7]. It is one
of the most difcult computational problems in the NP-hard class. Hence
solving them optimally in a reasonable time is a very challenging task.
Although exact problem-solving might be intractable, there exist heu-
ristics algorithms capable of solving the QAP even in large sizes with
nearly optimal solutions in a reasonable time.
1.2.1. QAP in general
Several solution techniques for the QAP have been suggested in the
literature. As early works of QAP for FLPs, Kaku et al. (1988) [8] pro-
posed a heuristic approach (exchange-improvement routine) for
multi-story layout design. Kaku (1992) [9] proposed a procedure that
combined a constructive heuristic and exchange improvement for loop
conveyor and linear-track layout cases. Rosenblatt (1992) [10] devel-
oped a hybrid method that combined branch and bound framework with
heuristics for equal-sized departmental layout. A novel study by Li and
Smith (1995) [11] proposed a sample test-pairwise exchange heuristic
procedure (STEP) for dynamic facility layout. Urban (1998) [12] pre-
sented two heuristics (multi-greedy algorithm and GRASP) for dynamic
facility layout design. Ulutas and Sarac (2006) [12] addressed QAP with
relocation cost thus handling a dynamic facility layout problem by
developing a heuristic algorithm, called modied sub-gradient (MSG).
Ramkumar et al. (2009) [13] proposed a new heuristic (iterated fast
local search) for equal area layout.
Moreover, Huntley and Brown (1991) [14] combined a genetic al-
gorithm (GA) and simulated annealing (SA) for an equal-area layout. Yip
and Pao (1994) [15] proposed a hybrid technique that combined GA and
SA for equal-area layout design. Bland and Dawson (1994) [16]
addressed QAP for large-scale layout using a hybrid heuristic algorithm
that combined SA and TS. Chiang and Chiang (1998) [17] proposed TS
and SA for facility layout. Kochar et al. (1998) [18] proposed a
meta-heuristic using a genetic algorithm, called HOPE, for unequal area
single and multi-row layouts. Chiang (2001) [19] addressed a modied
version of QAP with binary variables by proposing a TS algorithm for
interdepartmental layout. Solimanpur et al. (2004) [20] used ant colony
optimization for inter-cell layout. Nourelfalth et al. (2007) [21] used
metaheuristic (Ant Colony Optimization with EGD local search) for
equal-area layout. Jaramillo and Kendall (2010) [22] proposed a TS
heuristic using different construction algorithms for machine layout.
Moslemipour and Lee (2012) [23] developed a SA approach for dynamic
layout. Pourvaziri and Pierreval (2017) [6] presented SA algorithm for
dynamic facility layout design based on a QAP formulation. For an
extensive review of solution techniques for the QAP, refer to recent
survey paper by Singh and Sharma (2010) [24].
1.2.2. QAP in hospital layout planning
A relatively limited body of research has been published with respect
to layout planning in hospitals.
Elshafei (1977) [25] rstly proposed QAP for locating the clinics
within a hospital department using an improvement heuristic in order to
optimize traveling distances of patients and delay in patient ows.
Murtagh et al. (1982) [26] used QAP formulation for assigning 19 clinics
to predened locations in order to minimize transport costs by devel-
oping a new heuristic. Butler et al. (1992) [27] formulated a QAP for bed
allocation in a general-purpose hospital to minimize the distance be-
tween services taken by nurses using a constructive heuristic (CRAFT).
Hahn et al. (2001) [28] proposed QAP for assigning the facilities into
locations in order to minimize travel distance taken by all pairs in a
hospital comparing the results of several heuristics and metaheuristics
solution methods. Yeh (2006) [29] focused on adjacency of objects, the
distance between objects, availability of space for object location, po-
sitions of objects in relation to others for a case study of a hospital with
28 facilities. Facility layout design formulated as QAP and solved by
simulated annealing with an annealed neural network. Chraibi et al.
(2015) [30] minimized total traveling cost and rearrangement cost in a
dynamic facility layout problem of the Operating Theatre (OT) depart-
ment of a hospital. Recently, Helber et al. (2016) [31] have proposed a
hierarchical modelling approach. The rst stage is formulated as QAP
for assigning elements to locations using a x-optimize heuristic by
considering transportation processes, locating some units on specic
locations and ensuring the direct adjacency of two specic units. The
second stage is detailed positioning within a location considering space
requirements in a large hospital facility. Zuo et al. (2019) [32] proposed
a QAP formulation for an emergency department of a hospital using a
multi-objective tabu search algorithm by focusing on a real-case study.
As a result, most works focus on individual departments2D layout
optimization in the hospital such as OT planning and nursing units.
Considering the whole set of hospital departments in a 3D layout opti-
mization is scarce in the literature. There is only one study in Ref. [31]
that focuses on whole hospital departments with a practical case study.
Most of the application papers considered rectilinear distances (a.k.a.
Manhattan distances), not geodesic distances. To the best of our
C. Cubukcuoglu et al.
Journal of Building Engineering 44 (2021) 102952
3
knowledge, none of the works considered the combination of the QAP
with the graph-theoretical aspect of computing spatial geodesic dis-
tances (walkable, easiest, and/or shortest paths for pedestrians) inside
the building in real-case scenarios.
1.3. Utilization of QAP in computational design
QAP is rst developed for facility layout planning but it is also useful
for spatial layout planning in architecture. There are different facility
layout programs like CRAFT [33], COFAD [34], CORELAP [35], and
BLOCPLAN [36]. However, these tools have rarely been utilized in
architectural design practice. Considering spatial layout planning as a
fundamental aspect of architectural design that affects the functional
performance of hospital buildings, we argue that a systematic approach
for ensuring the effectiveness and efciency of a layout schema should
be an integral part of any architectural design process, particularly in the
case of critically complex building such as hospitals. The idea of bridging
the gap between both the parametric CAD platforms and the layout
design is not entirely new. There is a limited number of tools available in
Parametric CAD platforms to facilitate the space planning processes,
namely DeCoding Spaces, SpiderWeb, and Syntactic [3739]. To the
best of our knowledge, none of the tools proposed the QAP approach in
layout planning and none of them utilize the shortest paths for pedes-
trians. These three toolkits provide methods for formulating space
planning problems and analysing spatial congurations within the
framework of Space Syntax theories. However, other than heuristic
force-directed graph drawing solvers, none of these tool suites proffer an
explicit formulation of the space planning problem as an optimization
problem. Our proposed methodology formulates the problem of layout
optimization as a Quadratic Assignment Problem, proposes a measure of
quality as the Logistic Cost Function, as to which a benchmark can be
created for the improvements on the inner walking/transportation costs
within a complex layout. Finally, the proposed solver reduces these costs
to a minimum and nds a new spatial conguration.
1.4. Contributions
The main goal of this paper is to formulate and solve a 3D space
planning problem methodology in the form of a Quadratic Assignment
Problem (QAP), in the context of a re-design/renovation task, by
considering the effect of geodesic distances through a network of cir-
culation spaces in 3D. The QAP is a well-known problem in the eld of
Operations Research from the category of the facilitys location/allo-
cation problems. The methodology is rst and foremost developed for
hospital space planning; however, it can also be used in design and
optimization for other types of complex buildings, especially in order to
study reuse scenarios. The methodology is implemented and tested
partly in C# language [40] for McNeels Grasshopper3D [41] and partly
in VEX language [42] for SideFXHoudini [43]. The following steps are
presented in this paper:
Formulating a layout problem in architectural design as a QAP based
on geodesic distances
Introducing a practical CAD workow for applying QAP solvers to 3D
spatial layout problems
Discretizing and modularizing the design space as a way of struc-
turing the geodesic computation problem in 3D as well as relaxing
and simplifying the unequal area QAP problem
Estimating ows and entering the ow matrix as an input of the tool
Calculating spatial geodesic distances and entering the distance
matrix as an input of the tool
Proposing a heuristic algorithm for solving the QAP
Implementation of the model and heuristic optimization algorithm
Running the solver, reecting on the results, and comparing them to
the existing state of an actual hospital
1.5. Gaps in the literature
The detailed literature review is given in section 1.2. We have
identied and summarised the following gaps in the literature con-
cerning the layout optimization of existing hospitals:
There is no complete methodology for a 3D layout problem with a
QAP formulation, most papers are focused on 2D layout and focused
on specic departments such as the operation theatre and the nursing
units. However, we focus on the entirety of the hospital design
problem.
There is no study on QAP that focuses on geodesic distances (espe-
cially in 3D). Most studies focus on rectilinear distances on a at 2D
plan.
The combination of QAP with the graph theoretical aspects (way-
nding) in real-world test-cases is unique.
Modularization/Discretization of space layout problem both in terms
of modularization of the departments and the walkable space in
between the departments in 3D for solving a QAP.
There is no implementation of QAP problem-solving in computa-
tional design methodologies.
The Iterative Local Search algorithm is not entirely new, but it has
not been implemented in the case of space layout of hospitals.
2. Problem formulation
As explained in the literature review the Quadratic Assignment
Problem is well known in the eld of Facilities Layout Planning but
relatively unknown in architectural layout design. Interestingly, when
the focus of the optimization task is on improving the conguration of an
existing building, the Quadratic Assignment Problem arises naturally
when we consider the efciency of the spatial conguration of the
building as a whole. QAP is a combinatorial optimization problem that
aims at allocating a set of facilities to a set of locations such that the total
transportation cost is minimized. Total transportation cost is a function
of the ows between facilities and the distances between locations. The
QAP has a discrete representation of areas since the facility departments
can only be assigned to predened network locations. Essentially, the
problem of assignment here is an unequal area layout problem because
each department has a different required amount of surface area Table 1
(Appendix-A). Our model makes this discretization regular by topolog-
ically abstracting the whole 3D walkable oor space of the building as a
voxelated domain [44,45]. Voxels or (volumetric pixels/picture cells)
are 3D regular units of space for partitioning a 3D volume into a Car-
tesian grid of cells. The design domain, in this case, is voxelated for two
main reasons: 1) to modularize the units of space so that the problem of
layout can be formulated as an assignment problem, and 2) to use the
explicit topological relations between adjacent voxels for constructing a
network model of space to compute the geodesic distances for the
computation of the logistic cost function (the objective function). In
accordance with this tessellation, at a higher resolution corresponding to
the structural grid of the building, we break down the departments in
C. Cubukcuoglu et al.
Journal of Building Engineering 44 (2021) 102952
4
terms of their surface area requirements into ‘multiple facilities, each of
which has exactly the area of one pixel. Such a regular tessellation
brings about two main advantages: on the one hand computation of
geodesic distance becomes straightforward on the network generated
from the voxels, and on the other hand the problem of assignment is
relaxed in that a single facility may be allowed to be allocated within
multiple voxel locations, thus alleviating the complexity of the unequal
area assignment problem. Each facility can be assigned to exactly one
location and no location is assigned to more than one facility. Therefore,
the number of facilities should be the same as the number of locations.
The basic QAP model can be formulated as:
minimize
π
Pn
n
i=1
n
j=1
Di,jT
π
i,
π
j
Where,
π
is a vector of integers denoting a permutation of facilities at a
moment in time, T
π
i,
π
j denotes the [transportation] ow between a
permuted facility
π
i=k, and another permuted facility
π
j=l and Di,j is
the distance between location i and j. Pn denotes the set of all permu-
tations
π
(t):NN, where the superscript denotes the iteration time.
Note that the cardinality of Pn will be n!, and so a brute-force search
becomes intractable as soon as the problem gets large, i.e. an order of
complexity of O(n!); which in the case of our example would be about
searching within 64!=1.2688693e+89 possible permutations. Note
that the objective function is measuring the expected travelled distance
for a typical building user; this is because the transition probabilities (e.
g. those given as percentage values in Table 2, Appendix-B) are
dimensionless/unitless and that the travelled distance between every
two nodes is multiplied with the probability of that transition.
3. Methodology
We have formulated the problem as a matter of reducing the logistic
cost function by choosing the right permutation of facilities within a set
of existing locations. Due to the physical nature of the costs (distances)
and the dimensionless (unitless) meaning of the ow rates, the physical
unit of the objective function is the same as the distances. This means
that the distances in between the departments in a 3D space must be
computed. In order to do so, we propose the following methodology:
1. Estimate the ow-rates as the transition probability between the
departments for a pedestrian (the ow-rates, in this case, are
considered as given, see Appendix B);
2. Discretise the walkable space in between the departments and model
the topology of the connectivity between the discrete spaces as a
graph/network;
3. Compute the geodesic distance between the department locations in
the discretized space and attribute the distances to the pairs of de-
partments using the Floyd-Warshall algorithm;
4. Heuristically improve (minimize) the objective cost function by try-
tting various permutations of facilities over the locations using the
Iterative Local Search algorithm.
3.1. Flows
As stated above, the objective of the QAP is a function of ows be-
tween facilities and distances between locations as constant parameters
and the only variable parameter is the permutation, hence it is called a
combinatorial optimization problem. The ow information refers to any
quantitative relationship score between any pair of items, typically the
estimated volume of transporting materials or probability of transition
of medical staff in between facilities. We have considered the latter
probabilistic interpretation in our formulation. This score can also be
interpreted from given adjacency requirements of a hospital, which are
typically recommended for ensuring effective logistics according to
medical procedures, providing privacy or community, security, safety,
hygiene, congruence of noise levels, etcetera. The so-called RELChart
table, e.g. the one used in this paper (Appendix B) is a matrix whose
entries indicate the relative importance of closeness between two de-
partments. Considering the central importance of this matrix in the
formulation of the QAP problem or even only in assessing the quality of a
particular assignment, it is important to compose this table with
objective information. However, in practice, such tables are often
composed following discussions of the board of directors of a hospital.
Nevertheless, for a building that does not exist yet, guring out such
importance ratings is a daunting task. For an existing building, however,
these relative importance ratings can be replaced with the measured or
estimated probabilities of transition between pairs of departments
objectively. Given the technical difculty of measuring such probabili-
ties in practice, estimating the probabilities according to the foreseen
procedures is an alternative that has been shown to be feasible using a
Discrete Event Simulation (q.v. [46,47]).
The number of spatial units must remain the same throughout the
QAP solving, and so, we rstly split the departments into spatial units
considering the size of the designated modules and their area re-
quirements. Then we divide the predicted ow rates mentioned in the
RELChart equally between the dividend units. We added this description
to the methodology section. The suggestion to set high intra-closeness
ratings for encouraging closeness between the split parts is very
logical and it is already implemented by setting high ow rates between
the divided units (100% closeness). The numbers written in our REL
chart are percentages [0,100] which are interpreted at the end as
numbers in the range of [0,1] as transition probabilities. This makes it
possible to have a physical interpretation of the objective cost function
as an expected travelled time as explained in the section Application.
However, if we raise these numbers to values higher than 100 (or 1) this
would disrupt the probabilistic interpretation and make it hard to
explain and justify the results.
3.2. Spatial geodesic distance
In order to compute walking distances within a building, spatial
geodesic (optimal paths on a network) are used in this method. In the
mathematical eld of graph theory, the geodesic distance between two
vertices in a graph is the total sum of the costs attributed to the edges
connecting them through an optimal path. For constructing such paths
and computing geodesic distances, rstly, the set of all spaces (locations,
corridors, and stairs) are discretized in a surface geometric model [45].
The spatial network of the indoor walkable space of the building is then
extracted from this mesh, a multi-source graph-traversal search is run,
and the distance matrix is obtained. The graph is constructed based on
6-neighbourhoods of voxels (i.e. voxels connected to their top, bottom,
left, right, back and front neighbours) [48]. Then multiple A* searches
are run within the constructed graph to nd the geodesic distance from
every location to every other location. Note that the location points (as
marked in red and shown in Fig. 1) are exactly 64 voxels corresponding
to the larger sets of voxels whose areas are equal to 100 m
2
. These large
C. Cubukcuoglu et al.
Journal of Building Engineering 44 (2021) 102952
5
areas are not included in the graph generated from the voxels because
the model only needs to have the distance from their access points to
other access points.
3.3. Heuristic problem solving
As stated before, QAP is an NP-hard problem. Therefore, heuristic
optimization algorithms are seen as remedies for tackling this complex
problem in large instances. In this tool, we selected Iterated Local Search
algorithm (ILS) [4952] for problem solving. Recently, the performance
of an ILS algorithm in Ref. [53] has been tested on QAP instances arising
from real-life problems as well as on several benchmark instances from
the QAPLIB [54]. Inspired by Refs. [53,55], we utilize the ILS algorithm
for the space planning tool presented in this paper. Details of the algo-
rithm that is considered in this paper are given in sub-sections below.
3.3.1. Solution encoding
The encoding scheme in our algorithm corresponds to a sequence of
integers that represents facilities in a feasible solution (permutation).
Fig. 1. The walkable space as a mesh (top) and its discretized voxel model (bottom); blue voxels: circulation areas & red voxels: location areas. (For interpretation of
the references to colour in this gure legend, the reader is referred to the web version of this article.)
C. Cubukcuoglu et al.
Journal of Building Engineering 44 (2021) 102952
6
3.3.2. Initial solution
In the Iterated Local Search (ILS) algorithm, the initial solution
π
(0)=
[
π
1,
π
2, ..,
π
n]T is constructed randomly, which is a permutation of the
integers between 1 and n, where n is the number of facilities.
3.3.3. Perturbation scheme
In the ILS procedure, the initial solution is perturbed with swap and
insertion neighbourhoods to escape from local minima. In this paper,
random swap and insertion neighbourhoods are employed. The swap
operator exchanges two facilities in a solution, whereas the insertion
operator removes a single facility from a solution and inserts it into a
random position in the solution. As an example, to a swap operator,
suppose that we are given a current solution
π
(0)= [5,4,2,1,3]T. Two
facilities are randomly selected and they are exchanged. As an example,
we randomly choose the facility
π
(0)
4=1 and
π
(0)
2=4 in order to swap
them. Thus, we end up with a solution as
π
(1)= [5,1,2,4,3]T. In addi-
tion, as an example of an insertion operator, we apply forward or
backward insertion with an equal probability. Suppose that we are given
a current solution
π
(0)= [5,4,2,1,3]T. Assume that we randomly choose
π
(0)
3=2. Then, we remove it from the solution and insert it into the
fourth position as a forward insertion to generate a new solution
π
(1)=
[5,4,1,2,3]T whereas in the backward insertion, we remove
π
(0)
3=2
from the current solution and insert into the second position as
π
(1)=
[5,2,4,1,3]T.
3.3.4. Local search
After the perturbationof the current solution, we apply a local
searchbased on swap neighbourhood. In the swap local search, the
perturbed solution
π
(1)goes under a swap local search procedure. The
iteration counter is xed at 1 at the beginning, we select two facilities
randomly and simply swap them. If the new solution obtained after the
swap neighbourhood is better than the current solution, it is replaced
with the current solution and the iteration counter is again xed at 1,
otherwise, we keep the current solution as it is. And the iteration counter
is increased by 1. The swap local search is repeated until the iteration
counter is reached at the number of facilities n. The pseudo-code of the
swap local search is given in Algorithm 1.
Algorithm 1.Swap Local Search
C. Cubukcuoglu et al.
Journal of Building Engineering 44 (2021) 102952
7
The general framework of the ILS algorithm is given in Algorithm 2.
Briey, the initial solution is constructed randomly. Then, a swap local
search is applied to the initial solution. A loop-based on the termination
criterion is started. Repeatedly, perturbation and swap local searches are
applied to the current solution until a termination criterion is satised. If
we need to implement a xed-department constraint, we can dene a set
dubbed f and we can add a condition to the Swap procedure and change
it to:
if (afbf)then Swap
π
aand
π
b
Algorithm 2.The General Framework of the Iterated Local Search
Algorithm
C. Cubukcuoglu et al.
Journal of Building Engineering 44 (2021) 102952
8
4. Test & implementation
The purpose of our implementation at this point was to test the al-
gorithms. More specically, the purpose was to verify whether the al-
gorithms work as expected in terms of the correctness of the results and
to validate whether the results are improved. We have implemented the
method presented above partly using the C# programming language and
developed a space planning tool, called QAP Solver as an add-on for
McNeels Grasshopper3D [41] software application and partly as a VEX
add-on for SideFXHoudini [43] for computing network geodesics and
the corresponding distance matrix.
QAP Solver component implements Algorithm 2 for solving a QAP
instance based on the given input data (ow and distance matrices).
Inside the component, the rst initial solution is generated randomly,
and then the optimization algorithm is run after Boolean toggle is set to
Truemode; this will trigger the generation of new permutations, for
each of which the objective function of the QAP is evaluated and re-
ported. In each generation, the component is capable of showing the
change of decision variables (permutation) on a collection of number
sliders by realizing a slider update procedure inside the component. In
this way, users of the tool can see how well the layout is being improved
over the generations in terms of the value of the tness function and at
the same time see the generations in real-time, as the number sliders that
encode the permutation are used to pick and change the colours of
rooms in the 3D model. In addition, the tool allows users to set the
maximum number of trial times (tmax in Algorithm 2). The user is
expected to connect as many number sliders as the number of facilities
(functional units). All sliders should be connected to the QAP Solver
component. The output of the component presents the result of the
optimization as well as the amount of improvement in objective value.
The permutation results are also shown on the number sliders, e.g. if the
rst slider has resulted as 2, then the second facility is placed to the rst
location and so on. The QAP Solver component is shown in Fig. 2 when
working on a toy problem with three facilities to be assigned to three
locations.
For computing the geodesic distances, we rst extract a set of meshes
representing the connective spaces such as corridors, stairs, and ramps
[if any]; then we voxelate these spaces using openVDB [56]; then
construct a network out of the voxels, and then calculate shortest paths
from all locations to each other location, i.e. the same set of locations
will be used as both origins and as destinations. Technically, the loca-
tions are rst mapped onto their closest points/voxels on the network.
The output of the process will be the matrix of distances [Di,j]n×n where n
is the number of locations. This output is directly used in the QAP solver.
The working principle of the QAP solver can be shown in the video
available online (https://www.youtube.com/watch?v=Lv52qy1OjSw).
5. Application
In this section, we articulate an outlook for using the QAP tool in
space planning and design of existing hospitals in a larger context.
Due to the nature of the QAP method, it can be used in case of the
following scenarios in redesigning a hospital building, i.e. a building
with a set of facilities is to be moved to another or the same building
with the same number of locations:
1. When we know the transportation ows [logistics of pedestrians or
materials] between the facilities of a building;
2. When we can estimate the ows between the facilities of a building,
e.g. by utilizing a simulation procedure such as Discrete Event
Simulation [46].
C. Cubukcuoglu et al.
Journal of Building Engineering 44 (2021) 102952
9
Fig. 2. QAP_Solver component in Grasshopper3D (left) and the geodesic/network distance computing in houdini (right).
C. Cubukcuoglu et al.
Journal of Building Engineering 44 (2021) 102952
10
We illustrate the application of the proposed QAP method can be
used in computational space planning in the context of an exemplary
case study hospital (corresponding to the rst scenario described above).
All hospital departments are considered in the layout optimization
problem; however, it would be also possible to perform a similar pro-
cedure on the inner spatial layout of only one department like Operation
Theatre layout or Intensive Care Unit layout, i. e QAP at a higher level of
detail. The chosen hospital is a state hospital with a capacity of 250-beds
for the in-patient wards. It is a 9-story building including basement and
ground oors. In this model, we excluded the locations placed on the
basement and the 7th level of the building in the existing situation;
because, the 7th level has only a terrace area, which is not suitable for
placing any facility; and that the spaces located in the basement have
some specic features and their locations cannot be changed. These
spaces that are excluded from the model can be listed as a mortuary, a
worship-space, a bunker, parking lots, and storage. Furthermore, some
of the functional units may require some specic locations due to a
specic feature in hospitals, e.g. the emergency department and main
entrance should be at the ground level of the hospital. The locations of
these facilities are excluded in the model by making their locations
constant in proper locations. Based on this, there are 34 facilities
considered in this case model for renovating the hospital layout.
Utilizing the QAP method is limited, theoretically, to equal-area
layout problems. However, each functional unit differs with respect to
its space requirements in this case. Although this is arguably an inherent
limitation of the methodology, we can relax the requirements such that
this is no longer a limitation. Since each location is represented with
[modular] discrete spaces (boxes in 3D), the number of needed boxes for
each facility can also differ. As a new approach to adapt QAP to unequal
departments, we can repeat each facility according to the area
requirement in the ow matrix and distribute/divide the ows accord-
ingly. For instance, assuming that each box has a capacity of 100 m
2
and
that the cardiology department needs 300 m
2
then we can dene this
department 3 times in the ow matrix and then divide the row and the
column corresponding to this space in the ow matrix by 3 and use the
results in the new rows and columns. By this repetition, the number of
facilities becomes the same as the number of locations, which is dened
as 64 in the model; while this also entails that a single facility may not
necessarily stay at a single location. This is obviously a limitation but at
the same time, it might be benecial in light of a higher level of satis-
faction with the logistics requirements. Based on the structural system of
the building, we dene the locations as rectangular spaces, in line with
the structural axes. This ensures that during the renovation the struc-
tural system does not have to be modied. Afterward, we tessellate each
rectilinear oor space into four quadrangular faces. Each quad face re-
fers to a square-like space surrounded by vertical columns axes and
horizontal beam axes of the existing building (as shown in Fig. 3). A list
of spaces with the number of needed rectangular boxes is given in
Appendix-A. The ow matrix is given in Appendix-B. For calculating the
distance matrix, the spatial network of the building is given in Fig. 4.
The RELChart table in Appendix-B is a matrix whose entries indicate
the relative importance of closeness between two departments.
Considering the central importance of this matrix in the formulation of
the QAP problem or even only in assessing the quality of a particular
assignment, it is important to compose this table with objective infor-
mation. However, in practice, such tables are often composed following
discussions of the board of directors of a hospital. Nevertheless, for a
building that does not exist yet, guring out such importance ratings is a
daunting task. For an existing building, however, these relative impor-
tance ratings can be replaced with the measured or estimated proba-
bilities of transition between pairs of departments objectively. Given the
technical difculty of measuring such probabilities in practice, esti-
mating the probabilities according to the foreseen procedures is an
alternative that has been shown to be feasible using a Discrete Event
Simulation in Ref. [47]. The particular table added in the appendices of
this paper, however, is a RELChart produced by collating expert in-
terviews, site visits, design guidelines/standards, and recommendations
from the scientic literature.
Regarding the computational results, the proposed heuristic algo-
rithm for the QAP is tested on an Intel Core-i7 computer, with 2 GB of
RAM. Maximum trial time is taken as 50 000 iterations with a seed
number 5. Permutations of the existing and proposed layout are given
detailly in Appendix-C. Based on this table, existing tness is 483751000
expected travelled steps (roughly equal to 60 cm, i.e. the small voxel-size
in the model). After the optimization by QAP Solver, the new tness is
411578400. The improvement in the objective value is 72172600 steps
Fig. 3. Denition of rectangular location boxesoor surfaces (consisting of 4 structural grid pixels based on building axes).
C. Cubukcuoglu et al.
Journal of Building Engineering 44 (2021) 102952
11
(adjusted to 721726.00 after dividing by 100 for converting ow per-
centages to probability fractions), which equals about 469121.90 m of
travelled distance. To put this result in a more concrete context, let us
assume that an average person can walk 5 km per hour; then this number
means that we have reduced the time spent for walking in between the
facilities by 469121.90/5000 =93.82438 person hours for a typical day.
Note that as we explained in the denition of the unit of the objective
function, this is the ‘expected travelled distance(or the time spent on
walking) for a typical building user on a typical day of operation. This
means that the improvement can be attributed to the building as a whole
Fig. 4. Extracting spatial networkof the building (top), continuous version of paths (middle), a discrete version of paths (bottom), (red dots are location points).
(For interpretation of the references to colour in this gure legend, the reader is referred to the web version of this article.)
C. Cubukcuoglu et al.
Journal of Building Engineering 44 (2021) 102952
12
Fig. 5. Existing layout.
C. Cubukcuoglu et al.
Journal of Building Engineering 44 (2021) 102952
13
rather than an individual person. The existing layout and the new layout
result for the case study hospital model are visualized in Figs. 5 and 6.
Regarding the design results, the placement of the inpatient areas is
still located at the top levels of the building as expected due to the
daylight requirements of patient wards. However, in the proposed
layout, interrelated spaces with inpatient have moved to various loca-
tions. In the proposed layout, operation rooms are located closer to the
patient wards and there exists ease of access between these facilities
with a vertical short connection. Critical patient ow processes and the
shared equipment entail better access between intensive care units and
Fig. 6. Proposed layout.
C. Cubukcuoglu et al.
Journal of Building Engineering 44 (2021) 102952
14
operating rooms. In the new layout, intensive care units and operation
rooms have a better connection since Intensive Care Units are the
neighbour of the Operating Theatre at the upper levels of the building. In
addition, delivery rooms have to get closer to Operation Theatre, which
is an advantage for shared staff and facilities. The kitchen and dining
hall were located at the last level of the building whereas they are
located at lower levels in the proposed layout. This provides quicker
food transportation from the kitchen to the medical spaces e.g. dining &
kitchen area has horizontal access to outpatient and vertical access to
inpatient. In the new layout, all departments related to maternity like
Paediatrics Intensive Care Unit, Paediatrics Outpatient, Delivery Rooms
and Obstetrics departments are located close to each other.
Laboratories and diagnostic units are located adjacent to each other
on the ground oor and become closer to the outpatient departments as
expected. In addition, inpatient departments were placed in the areas
that have one single corridor in the old layout. Whereas, these de-
partments are mostly located in areas with a radial layout structure.
Surgical outpatients like Obstetrics, Orthopedy, and Urology de-
partments are located closer to the operating rooms in the new layout.
Transportation processes between interrelated spaces became more
efcient with the proposed approach.
6. Conclusions
This paper introduces a new computational space planning meth-
odology based on the well known Quadratic Assignment Problem, pre-
sents a heuristic solver for it and presents the test results on a hospital re-
design case study. One of the novelties of the presented methodology is
that it utilizes the spatial network of the existing building for computing
geodesic distances, which are then used as inputs of the QAP model.
Results show that objective value is reasonably minimized, and design
results seem more logistically efcient. We have estimated an aspect of
the operational cost of the building as expected travel time of em-
ployees/users of the building using our objective cost function. It must
be noted that this is a matter of ex-ante assessment and not a mea-
surement, as measuring the actual travel time would require tracking the
personnel inside the building and fall out of the scope of this paper. The
time saving achieved with our methodology based on Operations
Research has achieved an estimated reduction of around 90 person-
hours for a typical operational day of the hospital. Due to its aggre-
gate nature, such a reduction should be of interest for the management
of the hospital as it implies not only a reduction of costs but also
implicitly an increased comfort for the employees, users, and thus a
higher-quality service. The contributions of the paper can be recapitu-
lated as below:
The obtained results, i.e. the new conguration and its correspond-
ing logistic cost function, reveal a major difference made by reas-
signing the departments to alternative locations, hence validating
the major contribution of the paper on improving existing layouts.
The reduction of the logistic cost function in this case corresponds to
a total reduction of around 93 person-hours of expected travel time
between departments of the hospital for a typical day.
We have considered the physical constraints pertaining to the size of
the departments and assumed that the departments can be accom-
modated into modular/rectilinear spatial units (colored in the pic-
tures). The newly found assignment can be visually inspected from
the point of view of an architect/manager and it seems to be a
feasible/logical assignment in terms of other constraints that are not
taken into account in this formulation. If the conguration is deemed
infeasible, the seed of the heuristic solver can be changed to nd
another conguration. The single conguration found as an example
in this paper seems to be feasible. However, in practice, more ex-
periments are needed to list layout alternatives and choose the one
with the least transformation costs and/or the best suitability with
respect to other architectural criteria.
The proposed methodology bases the reconguration problems on a
completely discretized and modularized design space, and the pro-
posed algorithm is reasonably fast, it would be theoretically feasible
to dissect the spatial units into smaller units and generalize the
method to broaden the application areas and the versatility of the
method for incorporating more diverse validity constraints.
Integrating QAP into computational design workows can, to say the
least, provides awareness of the logistics performance of the building in
terms of the expected walking time for personnel, and in that sense, it
can even be used as an informative tool for conceptual design of new
buildings as well as re-designing existing buildings.
7. Limitations & future work
The method presented in this paper is only suitable for recongu-
ration of existing buildings, especially because it requires computing the
distance between available locations for computing the main objective
function. Even though most hospital buildings have a regular structural
grid, it must be noted that our proposed way of dissecting departments
into modular areal units is only feasible on such highly modular and
regularly structured buildings. It must be noted that in our problem
formulation we consider all facilities to be accommodatable in all
available locations, while in reality there might be facilities that can
only be accommodated in certain locations due to particular technical
requirements. While this constraint is handled by excluding a list of xed
facilities, we have disregarded the exchangeability of other de-
partments. Our methodology does not take the contiguity constraints
into account explicitly as hard constraints, e.g. in cases where we split a
facility into 2 or 3 facilities to t it into our modularized spaces. We
cannot enforce the new units to stay contiguous/adjacent to each other
during the optimization process; however, by adding extra closeness
ratings in between the split parts, we relax such constraints and add
them to the objective function effectively. Moreover, this limitation can
also be considered in another way: that the obtained results, which may
not strictly entail the initially conceived contiguity, can be used to
reect on the programme of requirements and consider revising it, e.g.
considering two Cardiology departments if a signicant expected travel-
time saving can be made by splitting it into two departments. This can be
observed from the objective function. As future work, limitations of the
proposed workow to the existing buildings can be addressed by
modifying the problem-formulation, for instance by a Mixed-Integer
Programming formulation of the hospital layout problem. More
advanced optimization algorithms for solving the QAP can be proposed
e.g. populated version of the proposed algorithm. The geodesic distances
computed on the voxelated corridors are currently more accurate than
Euclidean distance but still quite simplistic in that they do not consider
the cost of waiting times for the elevators, nor do they differentiate
between going downstairs and upstairs with distances on the same level.
In general, instead of measuring distance in meters, it would be more
general to measure distance in travel time/effort, to also account for
path complexity for the visitors, for instance by using the Easiest Paths
weighting [57] Corridor Allocation Problem (CAP) [58] can be added,
which has the same tness structure with the QAP but with an extra
C. Cubukcuoglu et al.
Journal of Building Engineering 44 (2021) 102952
15
decision on locating the facilities on either sides of a corridor. Finally,
the proposed layout optimization tool can be potentially extended to
solve a Multi-Objective QAP [59,60] in further versions.
CRediT authorship contribution statement
Cemre Cubukcuoglu: Conceptualization, Methodology, Software,
QAP solver, Writing original draft. Pirouz Nourian: Conceptualiza-
tion, Methodology, Software, Geodesic Distance, Slider Update in QAP
solver, Writing review & editing, Supervision. M. Fatih Tasgetiren:
Conceptualization, Methodology, Software, QAP solver, Writing re-
view & editing, Supervision. I. Sevil Sariyildiz: Conceptualization,
Methodology, Supervision. Shervin Azadi: Software, Geodesic
Distance.
Declaration of competing interest
The authors declare that there is no conict of interest in this paper.
Appendix
Appendix-A
Table 1
A List of Spaces with a needed number of boxes (modular units of roughly 100 m
2
)
Facility Name Number of Boxes Needed
NEUROLOGY 1
OBSTETRICS 1
INTERNAL MEDICINE 1
CARDIOLOGY 1
PEDIATRICS 1
GASTROENTEROLOGY 2
DIAGNOSTIC UNITS 2
LABORATORIES 2
ADMINISTRATION-1 1
ORTHOPEDY 1
PHYSIOTHERAPY 2
HEMODIALYSIS 2
INFECTION DISEASES 2
SURGERY ICU 2
GENERAL ICU 1
PEDIATRY ICU 1
ICU WAITING AREA 1
ENT 1
UROLOGY 1
EYE DISEASES 2
BRAIN&GENERAL SURGERY 1
DERMATOLOGY 1
INPATIENT-1 2
DELIVERY ROOMS 3
OPERATION THEATRE 5
ADMINISTRATION-2 4
CONFERENCE HALL 3
LIBRARY 1
INPATIENT-2 4
INPATIENT-3 4
INPATIENT-4 4
KITCHEN 2
DINING HALL 2
Total number of locations (modular units) 64
C. Cubukcuoglu et al.
Journal of Building Engineering 44 (2021) 102952
16
Appendix-B
The table below is a RELChart produced by collating expert interviews, site visits, design guidelines/standards, and recommendations from the
scientic literature.
C. Cubukcuoglu et al.
Journal of Building Engineering 44 (2021) 102952
17
Appendix-C
Table 3
The Existing Assignment of the Case Study Hospital
Location Index
The Existing Assignment The Proposed Assignment
Facility Index Facility Name Facility Index Facility Name
0 44 CONFERENCE HALL-1 56 INPATIENT-4-1
1 35 OPERATION THEATRE-1 19 SURGERY ICU-1
2 22 PEDIATRY ICU 29 DERMATOLOGY
3 7 DIAGNOSTIC UNITS-1 30 INPATIENT-1-1
4 8 DIAGNOSTIC UNITS-2 61 KITCHEN-2
5 3 CARDIOLOGY 15 HEMODIALYSIS-1
6 4 PEDIATRICS 25 UROLOGY
7 2 INTERNAL MEDICINE 50 INPATIENT-2-3
8 1 OBSTETRICS 48 INPATIENT-2-1
9 0 NEUROLOGY 55 INPATIENT-3-4
10 5 GASTROENTOLOGY-1 49 INPATIENT-2-2
11 9 LABORATORIES-1 32 DELIVERY ROOMS-1
12 6 GASTROENTEROLOGY-2 2 INTERNAL MEDICINE
13 10 LABORATORIES-2 54 INPATIENT-3-3
14 15 HEMODIALYSIS-1 60 KITCHEN-1
15 16 HEMODIALYSIS-2 47 LIBRARY
16 11 ADMINISTRATION-1 13 PHYSIOTHERAPHY-1
17 12 ORTHOPEDY 63 DINING HALL-2
18 13 PHYSIOTHERAPHY-1 57 INPATIENT-4-2
19 14 PHYSIOTHERAPY-2 6 GASTROENTEROLOGY-2
20 19 SURGERY ICU-1 4 PEDIATRICS
21 20 SURGERY ICU-2 17 INFECTION DISEASES-1
22 21 GENERAL ICU 3 CARDIOLOGY
23 17 INFECTION DISEASES-1 31 INPATIENT-1-2
24 18 INFECTION DISEASES-2 44 CONFERENCE HALL-1
25 23 ICU WAITING AREA 52 INPATIENT-3-1
26 30 INPATIENT-1-1 23 ICU WAITING AREA
27 31 INPATIENT-1-2 59 INPATIENT-4-4
28 26 EYE DISEASES-1 1 OBSTETRICS
29 28 BRAIN&GENERAL SURGERY 0 NEUROLOGY
30 27 EYE DISEASES-2 41 ADMINISTRATION-2-2
31 25 UROLOGY 9 LABORATORIES-1
32 24 ENT 26 EYE DISEASES-1
33 36 OPERATION THEATRE-2 45 CONFERENCE HALL-2
34 37 OPERATION THEATRE-3 7 DIAGNOSTIC UNITS-1
35 38 OPERATION THEATRE-4 12 ORTHOPEDY
36 39 OPERATION THEATRE-5 46 CONFERENCE HALL-3
37 32 DELIVERY ROOMS-1 11 ADMINISTRATION-1
38 29 DERMATOLOGY 40 ADMINISTRATION-2-1
39 33 DELIVERY ROOMS-2 24 ENT
40 34 DELIVERY ROOMS-3 62 DINING HALL-1
41 48 INPATIENT-2-1 28 BRAIN& GENERAL SURGERY
42 49 INPATIENT-2-2 51 INPATIENT-2-4
43 45 CONFERENCE HALL-2 35 OPERATION THEATRE-1
44 50 INPATIENT-2-3 10 LABORATORIES-2
45 47 LIBRARY 14 PHYSIOTHERAPY-2
46 51 INPATIENT-2-4 39 OPERATION THEATRE-5
47 46 CONFERENCE HALL-3 16 HEMODIALYSIS-2
48 52 INPATIENT-3-1 37 OPERATION THEATRE-3
49 53 INPATIENT-3-2 22 PEDIATRY ICU
50 54 INPATIENT-3-3 34 DELIVERY ROOMS-3
51 55 INPATIENT-3-4 36 OPERATION THEATRE-2
52 56 INPATIENT-4-1 38 OPERATION THEATRE-4
53 57 INPATIENT-4-2 20 SURGERY ICU-2
54 58 INPATIENT-4-3 43 ADMINISTRATION-2-4
55 59 INPATIENT-4-4 53 INPATIENT-3-2
56 60 KITCHEN-1 8 DIAGNOSTIC UNITS-2
57 61 KITCHEN-2 21 GENERAL ICU
58 62 DINING HALL-1 33 DELIVERY ROOMS-2
59 63 DINING HALL-2 5 GASTROENTEROLOGY-1
60 40 ADMINISTRATION-2-1 42 ADMINISTRATION-2-3
61 41 ADMINISTRATION-2-2 27 EYE DISEASES-2
62 42 ADMINISTRATION-2-3 58 INPATIENT-4-3
63 43 ADMINISTRATION-2-4 18 INFECTION DISEASES-2
C. Cubukcuoglu et al.
Journal of Building Engineering 44 (2021) 102952
18
Appendix-D
Table 4
Nomenclature
Notation Explanation
QAP Quadratic Assignment Problem
C# C sharp programming language
GH Grasshopper
CAD Computer Aided Design
ILS Iterated Local Search
FLP Facility Layout Planning
STEP A sample test-pairwise exchange heuristic procedure
GRASP Greedy randomized adaptive search procedure
MSG Modied sub-gradient
GA Genetic algorithm
SA Simulated annealing
TS Tabu search
EGD Extended great deluge
OT Operating Theatre
2D Two dimensional
3D Three dimensional
OR Operations Research
QAPLIB A Quadratic Assignment Problem Library
CAP Corridor Allocation Problem
A* A-star
π
A vector of integers denoting a permutation of facilities at a moment in time
T
π
i,
π
j The [transportation] ow between a permuted facility
π
i =k, and another permuted facility
π
j =l
Pn The set of all permutations
Di,j The distance between location i and j
t_max Maximum number of trial times
n Number of facilities & locations
CRAFT Computerized Relative Allocation of Facilities Technique
COFAD COmputerized FAcilities Design
CORELAP Computerized Relationship Layout Planning
References
[1] Space Planning Basics, fourth ed., Wiley, (n.d.), https://www.wiley.com/en-gb/
Space+Planning+Basics%2C+4th+Edition-p-9781118882009. (Accessed 13
December 2019). accessed.
[2] A. Chraibi, S. Kharraja, I.H. Osman, O. Elbeqqali, A mixed integer programming
formulation for solving operating theatre layout problem: a multi-goal approach,
in: Proc. - IESM Conf. Oct. 2013, Rabat, Morocco, 2013.
[3] R. Ulrich, X. Quan, H. Systems, C. Architecture, A. Texas, The role of the physical
environment in the hospital of the 21 st Century : a once-in-a-lifetime opportunity,
Environment 439 (2004) 69.
[4] L.D. Burgio, B.T. Engel, A. Hawkins, K. McCormick, A. Scheve, A descriptive
analysis of nursing staff behaviors in a teaching nursing home: differences among
NAs, LPNs, and RNs, Gerontol. 30 (1990) 107112.
[5] M.M. Shepley, K. Davies, Nursing unit conguration and its relationship to noise
and nurse walking behavior: an AIDS/HIV unit case study, AIA Acad. J. 6 (2003)
1214.
[6] H. Pourvaziri, H. Pierreval, Dynamic facility layout problem based on open
queuing network theory, Eur. J. Oper. Res. 259 (2017) 538553, https://doi.org/
10.1016/j.ejor.2016.11.011.
[7] T.C. Koopmans, M. Beckmann, Assignment problems and the location of economic
activities, Econom. J. Econom. Soc. (1957) 5376.
[8] B.K. Kaku, G.L. Thompson, I. Baybars, A heuristic method for the multi-story layout
problem, Eur. J. Oper. Res. 37 (1988) 384397, https://doi.org/10.1016/0377-
2217(88)90202-0.
[9] B.K. Kaku, R. Rachamadugu, Layout design for exible manufacturing systems,
Eur. J. Oper. Res. 57 (1992) 224230, https://doi.org/10.1016/0377-2217(92)
90044-A.
[10] M.J. Rosenblatt, B. Golany, A distance assignment approach to the facility layout
problem, Eur. J. Oper. Res. 57 (1992) 253270, https://doi.org/10.1016/0377-
2217(92)90047-D.
[11] W.J. Li, J.M. Smith, An algorithm for quadratic assignment problems, Eur. J. Oper.
Res. 81 (1995) 205216, https://doi.org/10.1016/0377-2217(93)E0162-Q.
[12] T.L. Urban, Solution procedures for the dynamic facility layout problem, Ann.
Oper. Res. 76 (1998) 323342.
[13] A.S. Ramkumar, S.G. Ponnambalam, N. Jawahar, A new iterated fast local search
heuristic for solving QAP formulation in facility layout design, Robot. Comput.
Integrated Manuf. 25 (2009) 620629, https://doi.org/10.1016/j.
rcim.2008.03.022.
[14] C.L. Huntley, D.E. Brown, A parallel heuristic for quadratic assignment problems,
Comput. Oper. Res. 18 (1991) 275289, https://doi.org/10.1016/0305-0548(91)
90029-Q.
[15] P.P.C. Yip, Y.-H. Pao, A guided evolutionary simulated annealing approach to the
quadratic assignment problem, IEEE Trans. Syst. Man. Cybern. 24 (1994)
13831387, https://doi.org/10.1109/21.310514.
[16] J.A. Bland, G.P. Dawson, Large-scale layout of facilities using a heuristic hybrid
algorithm, Appl. Math. Model. 18 (1994) 500503, https://doi.org/10.1016/0307-
904X(94)90328-X.
[17] C. Wen-Chyuan, C. Chi, Intelligent local search strategies for solving facility layout
problems with the quadratic assignment problem formulation, Eur. J. Oper. Res.
2217 (1998) 457488, https://doi.org/10.1016/S0377-2217(97)00285-3.
[18] J.S. Kochhar, B.T. Foster, S.S. Heragu, Hope, A genetic algorithm for the unequal
area facility layout problem, Comput. Oper. Res. 25 (1998) 583594, https://doi.
org/10.1016/S0305-0548(97)00100-7.
[19] W.-C. Chiang, Visual facility layout design system, Int. J. Prod. Res. 39 (2001)
18111836, https://doi.org/10.1080/00207540110035192.
[20] M. Solimanpur, P. Vrat, R. Shankar, Ant colony optimization algorithm to the inter-
cell layout problem in cellular manufacturing, Eur. J. Oper. (2004).
[21] M. Nourelfath, N. Nahas, B. Montreuil, Coupling ant colony optimization and the
extended great deluge algorithm for the discrete facility layout problem, Eng.
Optim. 39 (2007) 953968, https://doi.org/10.1080/03052150701551461.
[22] J.R. Jaramillo, A.R. McKendall Jr., The generalised machine layout problem, Int. J.
Prod. Res. 48 (2010) 48454859, https://doi.org/10.1080/00207540903117840.
[23] G. Moslemipour, T.S. Lee, Intelligent design of a dynamic machine layout in
uncertain environment of exible manufacturing systems, J. Intell. Manuf. 23
(2012) 18491860, https://doi.org/10.1007/s10845-010-0499-8.
[24] S.P. Singh, R.R.K. Sharma, A review of different approaches to the facility layout
problems, Int. J. Adv. Manuf. Technol. 30 (2006) 425433, https://doi.org/
10.1007/s00170-005-0087-9.
[25] A.N. Elshafei, Hospital layout as a quadratic assignment problem, Oper. Res. Q. 28
(1977) 167179.
[26] B.A. Murtagh, T.R. Jefferson, V. Sornprasit, A heuristic procedure for solving the
quadratic assignment problem, Eur. J. Oper. Res. 9 (1982) 7176, https://doi.org/
10.1016/0377-2217(82)90013-3.
[27] T.W. Butler, K.R. Karwan, J.R. Sweigart, G.R. Reeves, An integrative model-based
approach to hospital layout, IIE Trans. 24 (1992) 144152, https://doi.org/
10.1080/07408179208964211.
[28] P.M. Hahn, J. Krarup, A hospital facility layout problem nally solved, J. Intell.
Manuf. 12 (2001) 487496, https://doi.org/10.1023/A:1012252420779.
[29] I.C. Yeh, Architectural layout optimization using annealed neural network, Autom.
ConStruct. 15 (2006) 531539, https://doi.org/10.1016/j.autcon.2005.07.002.
[30] A. Chraibi, S. Kharraja, I.H. Osman, O. Elbeqqali, Multi-agent system for solving
Dynamic Operating Theater facility layout problem, IFAC-PapersOnLine. 28 (2015)
11461151, https://doi.org/10.1016/j.ifacol.2015.06.238.
C. Cubukcuoglu et al.
Journal of Building Engineering 44 (2021) 102952
19
[31] S. Helber, D. B¨
ohme, F. Oucherif, S. Lagershausen, S. Kasper, A hierarchical facility
layout planning approach for large and complex hospitals, Flex. Serv. Manuf. J. 2
(2014) 529, https://doi.org/10.1007/s10696-015-9214-6.
[32] X. Zuo, B. Li, X. Huang, M. Zhou, C. Cheng, X. Zhao, Z. Liu, Optimizing hospital
emergency department layout via multiobjective tabu search, IEEE Trans. Autom.
Sci. Eng. 16 (2019) 11371147.
[33] G.C. Armour, E.S. Buffa, A heuristic algorithm and simulation approach to relative
location of facilities, Manag. Sci. 9 (1963) 294309.
[34] J.A. Tompkins, R. Reed Jr., An applied model for the facilities design problem, Int.
J. Prod. Res. 14 (1976) 583595.
[35] R.C. Lee, CORELAP-computerized relationship layout planning, Jour. Ind. Engg. 8
(1967) 195200.
[36] C.E. Donaghey, V.F. Pire, Facilities layout with BLOCPLAN, Univ. Houst. TX Work.
Pap., 1991.
[37] A. Abdulmawla, M. Bielik, P. Buˇ
s, M.-C. Chang, M. Dennemark, E. Fuchkina,
Y. Miao, K. Knecht, R. Koenig, S. Schneider, DeCodingSpaces toolbox for
grasshopper. Computational analysis and generation of street network, Plots and
Buildings (2017).
[38] R. Schaffranek, SpiderWeb. https://www.food4rhino.com/app/spiderweb, 2012.
[39] P. Nourian, S. Rezvani, S. Sariyildiz, A syntactic architectural design methodology,
in: Meet. Ninth Int. Sp. Syntax Symp, 2013. Seoul, Korea.
[40] Introduction to the C# Language and the, NET Framework | Microsoft Docs, (n.d.),
https://docs.microsoft.com/en-us/dotnet/csharp/getting-started/introduction-t
o-the-csharp-language-and-the-net-framework. (Accessed 22 January 2020).
accessed.
[41] D. Rutten, Grasshopper3D, (2007).
[42] VEX language reference (n.d.), https://www.sidefx.com/docs/houdini/vex/lang.
html. (Accessed 22 January 2020). accessed.
[43] Houdini - 3D modeling, animation, VFX, look development, lighting and rendering
| SideFX (n.d.), https://www.sidefx.com/. (Accessed 22 January 2020). accessed.
[44] P. Nourian, R. Gonçalves, S. Zlatanova, K.A. Ohori, A. Vu Vo, Voxelization
algorithms for geospatial applications: computational methods for voxelating
spatial datasets of 3D city models containing 3D surface, curve and point data
models, Methods (Duluth) 3 (2016) 6986, https://doi.org/10.1016/j.
mex.2016.01.001.
[45] S. Azadi, P. Nourian, topoGenesis: an open-source python library providing
topological structures and functions for Generative Systems and Sciences. https://d
oi.org/10.5281/ZENODO.4006514, 2020.
[46] M.M. Gunal, A guide for building hospital simulation models, Heal. Syst. 1 (2012)
1725.
[47] C. Cubukcuoglu, P. Nourian, I.S. Sariyildiz, M.F. Tasgetiren, A discrete event
simulation procedure for validating programs of requirements: the case of hospital
space planning, Software 12 (2020) 100539, https://doi.org/10.1016/j.
softx.2020.100539.
[48] S. Laine, A topological approach to voxelization, Comput. Graph. Forum 32 (2013)
7786, https://doi.org/10.1111/cgf.12153.
[49] A.S. Ramkumar, S.G. Ponnambalam, N. Jawahar, A new iterated fast local search
heuristic for solving QAP formulation in facility layout design, Robot. Comput.
Integrated Manuf. 25 (2009) 620629.
[50] M.S. Hussin, T. Stützle, Hierarchical iterated local search for the quadratic
assignment problem, in: Int. Work. Hybrid Metaheuristics, 2009, pp. 115129.
[51] H.R. Lourenço, O.C. Martin, T. Stützle, Iterated local search: framework and
applications, in: Handb. Metaheuristics, Springer, 2019, pp. 129168.
[52] T. Stützle, Iterated local search for the quadratic assignment problem, Eur. J. Oper.
Res. 174 (2006) 15191539.
[53] M.F. Tasgetiren, Q.-K. Pan, Y. Ozturkoglu, O.K. Cotur, Variable block insertion
heuristic for the quadratic assignment problem, 2017 IEEE Congr. Evol. Comput.
(2017) 17651770.
[54] R.E. Burkard, S.E. Karisch, F. Rendl, QAPLIB–a quadratic assignment problem
library, J. Global Optim. 10 (1997) 391403.
[55] M.F. Tasgetiren, Q.-K. Pan, P.N. Suganthan, I.E. Dizbay, Metaheuristic algorithms
for the quadratic assignment problem, 2013 IEEE Symp. Comput. Intell. Prod.
Logist. Syst. (2013) 131137.
[56] OpenVDB (n.d.), https://www.openvdb.org/. (Accessed 22 January 2020).
accessed.
[57] P. Nourian, F. van der Hoeven, S. Rezvani, S. Sariylidiz, Easiest paths for walking
and cycling : combining syntactic and geographic analyses in studying walking and
cycling mobility, Proc. 10th Int. Sp. Syntax Symp. (2015) 115.
[58] A.R.S. Amaral, The corridor allocation problem, Comput. Oper. Res. 39 (2012)
33253330, https://doi.org/10.1016/j.cor.2012.04.016.
[59] C. Sanhueza, F. Jim´
enez, R. Berretta, P. Moscato, PasMoQAP: a parallel
asynchronous memetic algorithm for solving the Multi-Objective Quadratic
Assignment Problem, in: Evol. Comput. (CEC), 2017 IEEE Congr, ., 2017,
pp. 11031110.
[60] C. Cubukcuoglu, M.F. Tasgetiren, I.S. Sariyildiz, L. Gao, M. Kucukvar, A memetic
algorithm for the bi-objective quadratic assignment problem, Procedia Manuf 39
(2019) 12151222.
C. Cubukcuoglu et al.
... Especially in the early phase of design process, their functionality should not be disregarded and their spatial configuration has a great impact on their functional performance. Otherwise, unsuitable layout of the working spaces affected medical staff performance and can cause difficulties such as medical error, stress levels or work concentration [5]. Hospital designs typically require systematic design methodologies and computational design tools in order to satisfy sophisticated adjacency/closeness requirements in response to programmatic necessities, project site specifications, surrounding buildings, in addition to environmental requirements such as daylight. ...
Article
Full-text available
The paper proposes a generative design workflow for three major hospital layout planning steps to satisfy multiplex configurational requirements. The initial step is stacking through clustering functional spaces into floor plans, for which a spectral method is presented. Subsequently, a novel simultaneous process of zoning and routing is proposed as a Mixed-Integer Programming problem-solving task; performed on a quadrilateral mesh whose faces and edges are allocated respectively to the rooms and the corridors. The paper situates the workflow in the context of an Activity-Relations-Chart for a general hospital while demonstrating, explaining, and justifying the generated optimal floor plans. The conversion of the hospital layout problem to a Mixed-Integer Programming problem enables the use of existing Operations Research solvers, allowing for the generation of optimal solutions in a digital design environment. The comprehensive problem formulation for a real-world scenario opens a new avenue for utilization of mathematical programming/optimization in healthcare design.
Article
Construction site layout planning (CSLP) refers to the facilities' placement in the site boundaries by optimizing layout objectives and considering constraints. This study aims to provide a framework for optimizing the construction site layout dynamically, utilizing information from Building Information Modeling (BIM) and Geospatial Information System (GIS). The framework arranges temporary facilities within the site boundaries by considering a multi-objective optimization problem for minimizing the total on-site travel distance of personnel and equipment between facilities and enhancing the safety level of the construction site. Since a good site layout would increase the safety level of the construction site, facilities are placed based on the proximity relationships among them, which represent the planner preference in locating the facilities near or far from one another due to safety concerns. Then, the Guided Population Archive Whale Optimization Algorithm (GPAWOA) is applied for solving the problem. The contribution of this study includes developing a more realistic model using BIM and GIS together for acquiring the spatial data, navigating dynamically, and considering available indoor and outdoor space continuously. Accordingly, a plugin is developed to find the optimum construction site layout using the ArcGIS APIs and Microsoft C#. Finally, an illustrative case is considered to validate the practicality and functionality of the suggested framework. The results depict that employing the proposed method decreases about 20% of the total on-site traveling distance. Additionally, it results in a safer facilities arrangement regarding the proximity preference, compared with conventional methods.
Article
Full-text available
Purpose: This article aims to show the methods used to optimize layout and tools that have been applied since 2010 in the most diverse production environments. Theoretical Reference: Due to the intense competitiveness and uncertainties in the current market, improving processes and increasing production efficiency by managing the layout of a facility can be one of the methods to benefit organizations. In this context, knowledge of the main tools and methods is essential in defining the conduct of studies. Design/methodology/approach: Research was carried out through a systematic literature review. The databases used were Science Direct and Portal Capes. Using keywords, a defined scope and the relevance of the theme, international articles were selected for reading and discussion. Findings: Through the review, it was possible to select 51 articles which were relevant to the topic. Due to the complexity of the layout management study, it was found that increasingly more algorithms, mathematical and computational models are being used to solve these NP-hardness problems. Research, Practical & Social implications: adjusting to new methods and ways of solving problems laying out facilities, although scenarios can be extremely varied, they can improve business results by making the process more efficient. Originality/value: The article compiles and briefly explains the methods found to optimize the layout, which is of great importance considering the knowledge that these models spread on production management practices. Keywords: Layout optimization. Algorithm. Method. Facility layout planning
Article
Full-text available
This paper introduces a Discrete-Event Simulation (DES) tool developed as a parametric CAD program for validating a program of requirements (PoR) for hospital space planning. The DES model simulates the procedures of processing of patients treated by doctors, calculating patient throughput and patient waiting times, based on the number of doctors, patient arrivals, and treatment times. In addition, the tool is capable of defining space requirements by taking hospital design standards into account. Using this tool, what-if scenarios and assumptions on the PoR about space planning can be tested and/or validated. The tool is ultimately meant for reducing patient waiting times and/or increasing patient throughput by checking the match of the layout of a hospital with respect to its procedural operations. This tool is envisaged to grow into a toolkit providing a methodological framework for bringing Operations Research into Architectural Space Planning. The tool is implemented in Python for Grasshopper (GH), a plugin of Rhinoceros CAD software using the SimPy library.
Article
Full-text available
Multi-Objective Optimization Problems (MOPs) have attracted growing attention during the last decades. Multi-Objective Evolutionary Algorithms (MOEAs) have been extensively used to address MOPs because are able to approximate a set of non-dominated high-quality solutions. The Multi-Objective Quadratic Assignment Problem (mQAP) is a MOP. The mQAP is a generalization of the classical QAP which has been extensively studied, and used in several real-life applications. The mQAP is defined as having as input several flows between the facilities which generate multiple cost functions that must be optimized simultaneously. In this study, we propose PasMoQAP, a parallel asynchronous memetic algorithm to solve the Multi-Objective Quadratic Assignment Problem. PasMoQAP is based on an island model that structures the population by creating sub-populations. The memetic algorithm on each island individually evolve a reduced population of solutions, and they asynchronously cooperate by sending selected solutions to the neighboring islands. The experimental results show that our approach significatively outperforms all the island-based variants of the multi-objective evolutionary algorithm NSGA-II. We show that PasMoQAP is a suitable alternative to solve the Multi-Objective Quadratic Assignment Problem.
Article
Full-text available
Determining the location of machines for given periods of time, depending on changes in the material flow between the machines, is known in the literature as a dynamic facility layout problem (DFLP). While most approaches focus on reducing handling and rearrangement costs, this article also considers the amount of work-in-process (WIP) for a particular class of problems. WIP results from the queuing phenomenon and depends on the availability of transportation units. To solve these types of problems, we suggest the use of an analytical approach which uses open queuing network theory and is based on a quadratic assignment problem formulation. Since a queuing model approximates the studied system, the accuracy of the results is evaluated through a comparison with simulation results. Considering both the NP-hardness and the multi-objective nature of the problem, a meta-heuristic optimization approach is proposed. It aims at determining the Pareto front using cloud-based multi-objective simulated annealing (C-MOSA). The performance of C-MOSA is evaluated against other published multi-objective approaches. Computational experiments are performed and the results show that C-MOSA is capable of obtaining efficient results to solve the multi-objective dynamic facility layout problem.
Article
Recently, multi-objective evolutionary algorithms (MOEAs) have been extensively used to solve multi-objective optimization problems (MOPs) since they have the ability to approximate a set of non-dominated solutions in reasonable CPU times. In this paper, we consider the bi-objective quadratic assignment problem (bQAP), which is a variant of the classical QAP, which has been extensively investigated to solve several real-life problems. The bQAP can be defined as having many input flows with the same distances between the facilities, causing multiple cost functions that must be optimized simultaneously. In this study, we propose a memetic algorithm with effective local search and mutation operators to solve the bQAP. Local search is based on swap neighborhood structure whereas the mutation operator is based on ruin and recreate procedure. The experimental results show that our bi-objective memetic algorithm (BOMA) substantially outperforms all the island-based variants of the PASMOQAP algorithm proposed very recently in the literature.
Code
The DeCodingSpaces Toolbox for Grasshopper is a collection of analytical and generative components for algorithmic architectural and urban planning. The toolbox is free software released by the Computational Planning Group (CPlan) and is a result of long term collaboration between academic institutions and praxis partners across the globe with the common goal to increase the efficiency and quality of architecture and urban planning. Development of the toolbox is on-going. Please visit https://toolbox.decodingspaces.net/ for the latest tutorials and case studies as well as to download the latest version of the DeCodingSpaces Grasshopper library.
Article
Hospital department layout problems (HDLPs) are significant in enhancing service quality and reducing patients' travel distance and time. Their studies are scarce in comparison with those for facility layout problems in manufacturing systems. Existing approaches to HDLPs usually adopt simplified models and thus gain very limited applications in a real world. HDLPs typically involve multiple objectives that may conflict with each other. There have been no studies on their multiobjective heuristic approaches to our best knowledge. In this paper, we propose multiobjective tabu search (MTS) for a real-world HDLP. Beside the frequently used objective of flow cost, ensuring the closeness among certain departments is introduced as another one. A solution coding scheme is designed to represent a solution. A penalty function is devised to handle infeasible solutions. Local search is integrated into tabu search to optimize the assignment of departments. Experiment results show that MTS is able to produce Pareto solutions that outperform those of the comparative method. Compared to the actually implemented layout, solutions produced by MTS can save about 5%-15% patients' travel time (distance).
Conference Paper
Multi-Objective Optimization Problems (MOPs) have attracted growing attention during the last decades. Multi- Objective Evolutionary Algorithms (MOEAs) have been extensively used to address MOPs because are able to approximate a set of non-dominated high-quality solutions in a reasonable time. The Multi-Objective Quadratic Assignment Problem (mQAP) is a MOP. The mQAP is a generalization of the classical QAP which has been extensively studied, and used in several real-life applications. The mQAP is defined as having as input several flows between the facilities which generate multiple cost functions that must be optimized simultaneously. In this study, we propose PASMOQAP, a parallel asynchronous memetic algorithm to solve the Multi- Objective Quadratic Assignment Problem. PASMOQAP is based on an island model that structures the population by creating sub- populations. The memetic algorithm on each island individually evolve a reduced population of solutions, and they asynchronously cooperate by sending selected solutions to the neighboring islands. The experimental results show that our approach significatively outperforms all the island-based variants of the multi-objective evolutionary algorithm NSGA-II. We show that PASMOQAP is a suitable alternative to solve the Multi-Objective Quadratic Assignment Problem.