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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 congurational problems that

lead to inefcient 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

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 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 conguration having logistics-related objectives, ergonomics,

and intended user-experience of a building [1]. The spatial congura-

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 congurations 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 denition

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 conguration, 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 conguration 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 efciently 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 signicantly more. The poor placement of the

clinics combined with the increasingly overwhelming volume of trafc

between them was causing delays and heavy congestion in hospitals [6].

All these examples show that spatial conguration has a great impact on

the efcient functioning of hospital buildings in terms of walking dis-

tances and transportation processes. Therefore, providing efcient

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 difcult 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 modied 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 modied

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 predened 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 specic

locations and ensuring the direct adjacency of two specic 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 departments’ 2D 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 efciency 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 [37–39]. 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 congurations 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 conguration.

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 facility’s 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 McNeel’s Grasshopper3D [41] and partly

in VEX language [42] for SideFX′Houdini [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 workow 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, reecting 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

identied 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 specic 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 conguration of an

existing building, the Quadratic Assignment Problem arises naturally

when we consider the efciency of the spatial conguration 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 ﬂows between facilities and the distances between locations. The

QAP has a discrete representation of areas since the facility departments

can only be assigned to predened 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

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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):N→N, 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 difculty 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) [49–52] 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.)

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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 “perturbation” of the current solution, we apply a “local

search” based 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

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Journal of Building Engineering 44 (2021) 102952

7

The general framework of the ILS algorithm is given in Algorithm 2.

Briey, 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 satised. If

we need to implement a xed-department constraint, we can dene a set

dubbed f and we can add a condition to the Swap procedure and change

it to:

if (a∕∈ f∧b∕∈ f)then Swap

π

aand

π

b

Algorithm 2.The General Framework of the Iterated Local Search

Algorithm

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8

4. Test & implementation

The purpose of our implementation at this point was to test the al-

gorithms. More specically, 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

McNeel’s Grasshopper3D [41] software application and partly as a VEX

add-on for SideFX′Houdini [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

“True” mode; 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

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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

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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 specic 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 specic locations due to a

specic 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 dene 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 dened

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 benecial in light of a higher level of satis-

faction with the logistics requirements. Based on the structural system of

the building, we dene 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 modied. 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 difculty 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 scientic 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. Denition 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

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(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 denition 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 network” of 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.)

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Journal of Building Engineering 44 (2021) 102952

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Fig. 5. Existing layout.

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Journal of Building Engineering 44 (2021) 102952

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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.

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Journal of Building Engineering 44 (2021) 102952

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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

efcient 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 efcient. 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 conguration 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 conguration is deemed

infeasible, the seed of the heuristic solver can be changed to nd

another conguration. The single conguration 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 reconguration 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 workows 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 recongu-

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

reect on the programme of requirements and consider revising it, e.g.

considering two Cardiology departments if a signicant 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 workow 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

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Journal of Building Engineering 44 (2021) 102952

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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 conict 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

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Appendix-B

The table below is a RELChart produced by collating expert interviews, site visits, design guidelines/standards, and recommendations from the

scientic literature.

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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

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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 Modied 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

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