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Proceedings of the 2016 International Conference on Industrial Engineering and Operations Management
Detroit, Michigan, USA, September 23-25, 2016
An ILP Model for Healthcare Facility Location Problem
with Long Term Demand
Ruilin Ouyang
Department of Mathematics
Northeastern University
360 Huntington Ave, Boston, MA 02115, USA
ouyang.ru@husky.neu.edu
Tasnim Ibn Faiz and Md Noor-E-Alam
Department of Mechanical and Industrial Engineering
Northeastern University
360 Huntington Ave, Boston, MA 02115, USA
faiz.t@husky.neu.edu, mnalam@neu.edu
Abstract
Facility location decisions is one of the most crucial commitments that manufacturing and service
industries face to impacts their interaction with the end users or customers. This is a long term decision
and recourse option is very difficult once the decision has been implemented. By optimally placing the
facilities considering the probable future expansion in the market, a business entity can gain the
dominance over its competitors. In the face of rapid urbanization and increasing demand, making facility
location decision based only on existing demand is not optimal; taking into account possible realization of
future demand would result in more robust decisions. The present study focuses on developing a
mathematical model for making optimal decisions regarding healthcare facilities taking into account long
term demand. It utilizes the concept of grid-based location problems to divide the area of interest into
discrete cells. The model provides the optimal locations for facilities to be built at present time and the
potential location of facilities in the future. Finally, the model is programmed with a standard modeling
language AMPL and solved with the CPLEX solver. Results show that the model is efficient in solving
small to moderate sized problems. The developed approach can be used by government or relevant
agencies to make optimal location decisions for healthcare or other service facilities.
Keywords
Healthcare facility location, Grid-based location problem, Long term location decision.
1. Introduction
Numerous studies have addressed the facility location problem for both manufacturing and service sectors; many
algorithms have been developed for determining the optimal numbers and locations for facilities to be built. The
survey done by Brandeau el al. (1989) provides an overview of the studies focusing on location decision problems
conducted in the earlier part of the twentieth century. According to the survey, one of the most important inputs for
the location problem is the demand for the products or services that the facility will provide. But very few of these
studies have taken into account the future demand of customers or end users, in addition to current or existing
requirements. In one of these studies, done by Brancolini et al. (2006), asymptotical location problem was
considered; the authors compared long term and short-term strategies and their effects on location decisions. Their
study suggested that considering long term demand in making location decisions is beneficial. In a slightly similar
manner, Fernández at el. (2007) considered a location problem and the price setting in order to maximize profit, in
which the authors considered long term competition on prices and showed the effect of it on location decisions.
Chou (2009) proposed an integrated short-term and long term MCDM (multiple-criteria decision-making) Model for
location problems. It showed the importance of integrating the short and long term evaluation method with
examples. Kim and Kim (2010) studied long term healthcare facilities problem, which can balance the numbers of
© IEOM Society International
840
Proceedings of the 2016 International Conference on Industrial Engineering and Operations Management
Detroit, Michigan, USA, September 23-25, 2016
patients assigned to each facility; although cost was not considered in the model, which is a limitation of this study.
Marić et al. (2015) used Hybrid metaheuristic method for long term health-care facility problem that focused on
minimizing the maximum number of patients assigned to established facilities. Carlo et al. (2012) discussed and
proposed several approaches for the long term location problem with the objective of minimizing the total cost of
interacting with a set of existing facilities. Öhman and Lämås (2003) studied the effect of long term location
planning on harvest activities. In considering the long term demand, a few studies have been done that strived to
make the optimal location decisions in presence of demand uncertainty. Hosseini and MirHassani (2015) proposed a
2-stage stochastic location model for refueling station under uncertainty. But the model is too complicated to be
solved by available solvers. Albareda-Sambola et al. (2013) presented a compact binary formulation for the
deterministic equivalent model of the problem under uncertainty. Temur (2016) presented a multi-attribute decision
making approach for location decision under high uncertainty. It showed that location decision is very sensitive to
the uncertainty. Bai and Liu (2014) examined the influence of uncertain transportation costs and customers’
demands on the location decisions.
Healthcare facility location decision has a significant impact on the effectiveness of a facility to provide reliable and
safe services to the patients in the long run. Once such decisions have been made and implemented, it is extremely
difficult and costly to take recourse actions. Therefore, in deriving an optimal solution for the facility location
problem, considering future demand is required in addition to current demand. To the best of our knowledge, a little
effort has been made to incorporate future demand in determining optimal healthcare facility locations. Our
objective of this study is to develop a new integer linear programming (ILP) model for a long term healthcare
facility location problem. This model will consider both, the present and future demand, i.e. total number of patients
requires treatment at present and additional number of patients that would require treatment in the future. Therefore,
for both the present time and the future time horizon, two types of decision variables will be considered. The first
type of decision variables will provide the optimal locations of facilities to be built, while the second type will
determine the optimal allocation of patients to the facilities. The objective of the model is to minimize the total cost
of building and maintaining all the facilities in the considered time horizon, while meeting all the demands.
2. Problem description
In order to formulate a mathematical model for the long term healthcare location problem, a region is considered,
where the patients are known to be present. The total number of patients can be treated as the demand and their
locations as the demand location for services. The region of interest can be divided into two-dimensional grids to
consider the problem as grid-based location problems (Noor-E-Alam et al., 2012), where each cell with the same
dimension; patients located at a particular grid can be represented by the coordinates of a cell. Cell coordinates (,)
serve as the demand locations, and number of patients present at each cell determines the demand. For each cell
(,), demand is during the time interval beginning from time point to time point , and during the time
interval (beginning from time point to the end of planning time horizon), demand is . These demands are
assumed to be determined and will be considered as parameters in our model. To serve the patients, healthcare
facilities need to be built, for which fixed costs of building facilities () and maintenance cost per unit time () will
be incurred. Each facility is capacitated (maximum capacity is ), and patients from each cell can only go to the
facility that is nearest to them. For the facility locations, the same region is considered, divided into two-dimensional
grids, where cells are indicated by a different set of coordinates (,). If a facility is to be built at (,) at time
point , the binary variable associated with this decision takes a value of 1, and if a patient located at (,) goes
to facility located at (,) during the time interval beginning from time point to time point , then the binary
decision variable takes a value of 1. The corresponding binary decision variables for time point and time
interval , i.e.
and
follow the same logic. While making these decisions, capacity restrictions of facilities
and conditions for allocation of patients to the nearest facilities must be met. The problem is to decide optimal
numbers and locations for facilities to be built now and in the future, and allocate patients to their nearest facilities.
The objective is to keep the total cost of building and maintaining the facilities at the minimum level.
3. Location model
In the following subsections, the assumptions, parameters and variables included in the model are described. Figure
1 shows the time horizon considered in developing the model. Following that, objective function and the constraints
of the developed model are presented and explained in detail.
© IEOM Society International
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Proceedings of the 2016 International Conference on Industrial Engineering and Operations Management
Detroit, Michigan, USA, September 23-25, 2016
3.1 Assumptions
We have made the following assumptions to simplify our decision problem:
• All the current patients at each cell go to one facility in the entire planning horizon.
• All the future patients estimated at each cell at time point b will also go to a single facility during the time
interval beginning from time point to the end of planning time horizon.
• All the patients will go to the nearest facility for the entire planning time horizon.
• All the facilities have the same capacity.
• Cost of maintenance for a facility will remain same in the planning time horizon.
• Costs of building a facility at time point is the same as that at time point .
Figure 1: Planning horizon to develop optimization model.
3.2 Parameters
total time interval under consideration (from time point to the end of planning time
horizon)
time interval from time point to the end of planning time horizon
current demand of cell (,) (demand during time interval beginning from time point
to time point )
future demand of cell (,) during time interval , beginning from time point to the
end of planning time horizon
fixed cost of building a facility
cost of maintain one facility per unit time
distance between points (,) and (,)
maximum capacity of one facility
large cost parameter
3.3 Variables
if patients from (,) go to facility (,) during time interval beginning from time point
to time point
, then
= 1
, else
= 0
if a facility is built at (,) at time point , then = 1, else = 0
if patients from (,) go to facility (,) during time interval ,then
= 1, else
=
0
if a facility is built at (,) at time point , then
= 1, else
= 0
3.4 ILP model
With the above assumptions, we have proposed the following ILP model to solve our long term healthcare facility
location problems:
(+)
+
(+)
(1)
© IEOM Society International
842
Proceedings of the 2016 International Conference on Industrial Engineering and Operations Management
Detroit, Michigan, USA, September 23-25, 2016
(2)
+(1 )
(3)
(4)
= 1
(5)
+
1
(6)
+
(7)
+(1
)
(8)
(9)
= 1
(10)
The objective function (1) is a cost minimization function that aims to minimize the total cost of building and
maintaining the facilities. It is assumed that the building cost is the same at time points and time . The term
indicates the maintenance cost of one facilities built at time point for time length , whereas the term
indicates the maintenance cost of one facilities setting at time point for time length . Constraint (2) describe the
relation between the variables and , that is a patient from (,) can go to (,) during time interval
beginning from time point to time point for service only if a facility is located at (,). Constraint (7) specify
the same relation during time point to the end of planning horizon between the variables
and
. Constraint
(3) restricts the patients to go only to the nearest facility built at time point . As Figure 2 indicates if there is a
facility built at (,) at time point , then for any other point (,), =1 if and only if distance .
Constraint (8) represents the same relation for facilities and patients for time point . Constraint (4) restricts the total
number of patients that can go to a facility to the maximum facility capacity for the current time period and
constraint (9) does the same for future time period. Constraints (5) and (10) restrict the number of facilities that
patients at each grid cell can visit to one. Constraint (6) specify that at any location there can be at most one facility,
i.e. if a facility is built at (,) at time point , then no facility can be built at the same location at time point .
Figure 2: Geometry of distance for three points.
4. Computational results
The model is solved on a computer with Intel(R) Core(TM) i5 CPU running at 3.19 GHz with 4 GB memory and a
64-bit operating system. For implementation, AMPL programming language is used, and CPLEX 12.6.3 solver is
utilized to solve. Three different instances are solved using the model, e.g. 5 × 5 Grid (Shown in Fig. 3), 5 × 8 Grid
(,)
(,)
(,)
© IEOM Society International
843
Proceedings of the 2016 International Conference on Industrial Engineering and Operations Management
Detroit, Michigan, USA, September 23-25, 2016
(Shown in Fig. 4) and 10 × 10 (Shown in Fig. 5) Grid. In these instances, other parameter are assumed as follows:
==10, =20, =10, =10 and =( )+ ( ). For , values were randomly generated
ranging from 0 to 5 and for , random values range from 0 to 8. In Figures 3-5, the numbers in each cell before the
sign ‘/’ specify the demand at time , and the number after the sign ‘/’ indicate the demand at time . The grey grid
cells represent facilities are built at those locations at time , while the black grid cells represent building of
facilities at time . The white gird cells indicate that no facility is required at those locations.
Figure 3: Demand distribution and result for 5×5 grid.
Figure 4: Demand distribution and result for 5×8 grid.
4/6
0/6
1/7
1/4
3/4
5/6
1/0
5/6
2/3
0/4
1/3
3/2
5/3
5/6
4/5
5/5
5/0
2/8
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3/1
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4/0
5/1
3/6
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1/4
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5/1
0/8
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2/5
3/0
1/3
1/3
1/2
5/2
4/0
1/1
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3/7
3/2
4/7
5/4
2/7
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1/8
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1/0
3/1
4/1
3/7
0/1
1/3
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4/1
5/0
© IEOM Society International
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Proceedings of the 2016 International Conference on Industrial Engineering and Operations Management
Detroit, Michigan, USA, September 23-25, 2016
Figure 5: Demand distribution and result for 10×10 grid.
Table 1: Runtime statistics.
Grid
size
Number of
facilities at
time
Number of
facilities at
time
CPU time
(seconds)
Simplex
iterations
Branch and
bound nodes
MIP
Gap
5×5
9
4
5.834
38360
1388
0.001
5×8
11
5
281.005
865350
6091
0.001
10×10
27
16
>36338.577
21005880
27202
0.05
Table 1 summarizes the runtimes and other measures that were obtained through solving for these three instances
using the developed model. From our preliminary experiment we see that our proposed model is capable to identify
the optimum healthcare facility locations and assigned facilities for the patients located in each grid for a long-term
planning horizon within a reasonable runtime at least for 5x5 and 5X8 grids. However, when we try to solve larger
grids, it takes very long time to reach the optimality (after running a 10x10 grid for more than 10 hours, we saw that
the mipgap was 5%). Therefore, we future plan is to reformulate the problem such that it will be scalable and
capable to solve large-scale instances.
5. Conclusion
In this paper, a grid-based integer linear programming model is developed for identifying optimal location of
healthcare facilities to be built now and in the future. It takes into consideration the long term, i.e. future demand
realizations in addition to the existing demand. Although this study focuses on the healthcare facilities, this model
can also be used for a variety of service sectors, for example parks and recreational centers, as well as in placing
retail stores. The model is scalable up to a degree, but for solving the large instances it takes longer time than we
expect. To rectify this limitation, future extensions would consider relaxation of some constraints and addition of
some logical constraints. Furthermore, another extension of the model will consider multiple periods and demands
as functions of time periods, which will make the model more robust.
4/1
1/4
3/1
4/3
2/7
1/3
4/6
4/2
2/4
0/7
5/6
5/1
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2/5
3/0
1/3
1/3
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0/6
1/2
5/2
4/0
1/1
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1/0
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1/2
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3/4
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2/3
2/5
3/1
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1/8
1/6
4/5
2/4
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1/2
0/4
4/1
1/3
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2/5
3/6
0/7
5/6
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1/6
0/1
4/2
1/1
1/0
2/4
3/1
2/6
© IEOM Society International
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Proceedings of the 2016 International Conference on Industrial Engineering and Operations Management
Detroit, Michigan, USA, September 23-25, 2016
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© IEOM Society International
846
Proceedings of the 2016 International Conference on Industrial Engineering and Operations Management
Detroit, Michigan, USA, September 23-25, 2016
Biography
Ruilin Ouyang is a Master Student in the Department of Mathematics at Northeastern University, Massachusetts,
USA. He earned a Bachelor of Science degree in Mathematics from Zhejiang University, Hangzhou, China. He is
currently working under Professor Muhammad Noor E Alam.
Tasnim Ibn Faiz is pursuing his PhD in Industrial Engineering at Northeastern University. He earned his B.S. in
Industrial Engineering from Bangladesh University of Engineering and Technology, Bangladesh, and Masters in
Industrial Engineering from Louisiana State University, USA. He has published journal and conference papers,
worked in academic research projects as well as in industrial sector. His research interests include optimization,
scheduling, healthcare and renewable energy. He is a member of IIE and INFORMS.
Md Noor-E-Alam is an Assistant Professor in the Department of Mechanical & Industrial Engineering at the
Northeastern University. Prior to his current role, he was working as a Postdoctoral Research Fellow at
Massachusetts Institute of Technology. His current research interests lie in the intersection of operations research
and data analytics, particularly as applied to healthcare, manufacturing systems and supply chain. He has completed
his PhD in Engineering Management in the Department of Mechanical Engineering at the University of Alberta
(UofA) in 2013. Before coming to the UofA, he served as a faculty member (first as a Lecturer and then as an
Assistant Professor) in the Department of Industrial and Production Engineering at Bangladesh University of
Engineering & Technology (BUET). He also previously received a B.Sc. and M.Sc. in Industrial and Production
Engineering from BUET.
© IEOM Society International
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