- Email: [email protected]
A Hybrid model for locating new emergency facilities to improve the coverage of the road crashes
Accepted Manuscript A Hybrid model for locating new emergency facilities to improve the coverage of the road crashes Seyed Sina Mohri, Meisam Akbarzadeh, Seyed Hamed Sayed Matin PII:
S0038-0121(18)30201-5
DOI:
https://doi.org/10.1016/j.seps.2019.01.005
Reference:
SEPS 683
To appear in:
Socio-Economic Planning Sciences
Received Date: 26 June 2018 Revised Date:
15 December 2018
Accepted Date: 27 January 2019
Please cite this article as: Mohri SS, Akbarzadeh M, Sayed Matin SH, A Hybrid model for locating new emergency facilities to improve the coverage of the road crashes, Socio-Economic Planning Sciences (2019), doi: https://doi.org/10.1016/j.seps.2019.01.005. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
A Hybrid model for locating new emergency facilities to improve the coverage of the road crashes Seyed Sina Mohri1*, Meisam Akbarzadeh2, Seyed Hamed Sayed Matin 3 Department of Transportation Engineering, Isfahan University of Technology, Isfahan, Iran. email: [email protected]
Department of Transportation Engineering, Isfahan University of Technology, Isfahan, Iran, email: [email protected]
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Department of Civil Engineering, Imam Khomeini International University, Qazvin, Iran, email:
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[email protected]
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A Hybrid model for locating new emergency facilities to improve the coverage of road crashes ABSTRACT We propose an emergency facility-locating model aimed at increasing the coverage of
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emergency demand throughout the city. The proposed model takes into account the status and location of the emergency facilities in the network and identifies locations suitable for the construction of new facilities. Here, Data Envelopment Analysis (DEA) and Maximum Coverage Location Problem (MCLP) have been combined in a single model. To do so, design
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problem and evaluation problem are considered concurrently to maximize the efficiency of services provided by emergency facilities across the city in response to the demand. Moreover, the total emergency demand in each district was considered in relation to the
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population density, the fatal, injurious, and property damage only (PDO) crashes. The coverage area of each emergency facility was assumed to be proportional to the average ambulance speed in the surrounding road network during rush hours. The available budget was included in the model to let the model function under various fiscal conditions. Model input variables consisted of average number of mortalities, injuries and PDO crashes as well
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as the population density of each urban district. The output variables of the model included the coverage share of proposed emergency centers and hospitals equipped with ambulances. The model was tested on the network of Tehran (Iran). It is recommended to add the location of some emergency centers and hospitals to the network. Moreover, the results showed that
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ten urban districts had efficiency problem in provision of emergency services. Keywords: Emergency facility location, Ambulance station, Road crashes, Data envelopment
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analysis
1. INTRODUCTION
Road crashes represent a leading cause of death and disability throughout the world. According to World Health Organization (WHO), crashes are the ninth cause of death in the world and first cause of death among young people in the age range of 15 to 29 years [1]. Each year about 1.24 million lives are lost in road crashes and 20 to 50 million people sustain injuries that lead to disabilities [1, 2]. One of the effective measures to reduce the severity of road crashes and protecting human lives is to increase the speed and quality of emergency service provision after the crashes [3]. This goal is the effective resolution of an Emergency 1
ACCEPTED MANUSCRIPT Facility Location Problem (EFLP) at strategic level. Optimum positioning of emergency facilities would diminish their construction and functional cost and improve their efficiency. In this regard, an EFLP with efficiency-based objective has been proposed in this paper. Expansion of urban networks requires locating new facilities. Moreover, increasing coverage in all urban areas with respect to population density, fatal, injurious and property
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damage only (PDO) crashes, along with the coverage of medical facilities can improve the vertical equity in the distribution of emergency services throughout city. In this regard, a systemic and managerial perspective has been employed for reinforcing and balancing the efficiency of all urban areas in covering emergency demand. Moreover, contrary to iterative
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methods of finding an optimum global solution to ensure the maximum efficiency of the system, a combination of Data Envelopment Analysis (DEA) and Maximum Coverage
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Location Problem (MCLP) has been adopted in a single model. Accordingly, the proposed mathematical model evaluates the location of existing emergency centers and hospitals that respond to emergency demands (in terms of population and number of road crashes in regions) and building of new facilities to increase the efficiency of regions in meeting the demand for emergency services.
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The demand for emergency services is in relation to the population, rates of fatal, injurious and PDO crashes of each region. Unlike most of previous studies [4-7] considering a discrete demand through placing demand on nodes (i.e. regions), we assume a continuous demand spread on demand areas (i.e. region areas) in this paper. Road crashes, in general or
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as part of the network demand in EFLPs, should be considered in terms of crash severity as well as crash frequency due to high costs of fatal and injurious crashes. Therefore, it is necessary that the proposed model be able to distinguish crashes in terms of severity in each
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region. Accordingly, the network demand was divided based on four variables of population, fatal, injurious and PDO crashes in the regions. Each part serves as an input criterion in the DEA structure.
PDO crashes along with injurious and fatal crashes were considered as part of network demand in recognition of one of the major characteristic of crashes, i.e. randomness. Random feature of crashes has received little attention in previous studies on EFLPs for covering road crashes. This feature means that several uncontrollable and controllable factors have a bearing on the occurrence of crashes [8]. In addition to its effect on the frequency of crashes, 2
ACCEPTED MANUSCRIPT randomness also affects the severity of crashes. The severity of crashes is a variable of road, human and vehicle factors. Hence, if one of these factors changes, the severity will be modified as well. For example, a PDO crash in an area may result in an injurious or fatal crash if the driver’s age or vehicle’s type changes to more intensifying states.
•
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In summary, this research contributes to the literature in the following ways: The proposed model considers the status and location of the existing emergency facilities, emergency centers and hospitals with ambulance stations in the network and searches for new sites for constructing new facilities to maximize efficiencies •
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in the area.
Network demand is continuous and dependent both of the population and the frequency and severity of road/highway crashes of regions.
Given the random characteristic of road crashes, all three types of crashes,
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•
mortality, injury and PDO, are considered in the network demand. •
The coverage radius of facilities, as a variable of the type and the average speed of ambulances during rush hours, varies in different regions. Since the average speed of ambulances during rush hours is considered, the system is risk-averse
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with the prospect of the worst-case scenario. 2. LITERATURE REVIEW
EFLP is a subtopic of programming and management of Emergency Medical Services
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(EMS). From the modeling perspective, EFLP is also a subcategory of Healthcare Facility Location Problem (HCFLP). In our review of literature, we found ten review papers on EMS and HCFLP published in the period of 2000 to 2015, which shows the importance of this
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topic [9-18]. The goal of EFLPs is to locate new emergency facilities for covering emergency demands of a network. In 2017, Ahmdi-Javid et al. [18] divided emergency facilities into two groups of permanent and temporary facilities. Temporary facilities are utilized in the event of disaster, while permanent facility are concerned with medical services, intended to respond to service emergency calls under all circumstances. Permanent facilities are mainly deployed in ambulance stations, hospitals and emergency center [19]. An ambulance station is the site for dispatching ambulances to crash scenes, which is usually embedded in emergency centers or hospitals or operates independently. Emergency centers are units that provide emergency services, either as a part of hospitals or small-scale independent centers. In this paper, two 3
ACCEPTED MANUSCRIPT emergency center facilities and hospital, both with an ambulance station, have been considered to respond to medical demands of network emergency. The simultaneous inclusion of these two facilities in the model can be explained in terms of limited medical equipment in ambulances and the fact that the process of responding to an emergency call is completed when a patient is delivered to an emergency center or a hospital. The choice of
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emergency center or hospital is a variable of the severity of disease or crash, which is a random feature. Therefore, for the purpose of exhaustiveness, the emergency centers must be able to respond efficiently to demand points/areas and the points/areas need to be in the response range of the hospitals.
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The mathematical models in EFLPs cover either the location models or median location models. In the former, the objective is to expand the area covered by facilities. In the latter,
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the goal is to minimize the geodesic or Euclidean distance between network facilities and demand points. Covering location models embrace various categories including Set Covering Programming (SCP) problem [20], MCLP [21], maximum expected covering location problem [22], and maximum available covering location problem [23]. In SCP problems, the aim is to enable the coverage of total network demands for facilities, and the objective function is typically utilized to minimize the cost of building facilities. In three other models,
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the objective is to provide maximum coverage and the objective function is to maximize the respond range of facilities. The mathematical model proposed in this research falls in MCLP category, but with an efficiency-based objective, as indicated by the DEA algorithm.
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Network demand in previous EFLPs is estimated by (1) history of medical emergency calls [24, 25], (2) population-based measures [26-28] and historical road/highway crashes [4, 29-33] of demand points/areas or (3) a combination of (1) and (2) [34]. However, medical
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emergency calls encompass a variety of emergency demands such as diseases, heart attacks and road injuries, which this information are either unavailable in most cities or characterized by inaccuracy in collection, presentation, etc. Also, these historical calls are characterized with a high level of uncertainty so that some researches regard the network demand as uncertain [5]. On the other hand, among studies that estimate network demand by different measures, only a few incorporate historical crash data into estimation of network demand [4, 29-33].
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ACCEPTED MANUSCRIPT Castañeda and Villegas (2017) [29] proposed a facility location model to extend the coverage of the existing emergency medical services in Medellín (Colombia) for people injured in traffic accidents. The proposed model was designed to locate some new emergency facilities and to compute the optimal number of ambulances in current and new locations. Kepaptsoglou et al. (2011) [30] proposed a double standard location model for identifying the
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optimal location of emergency response vehicles in a city to alleviate the severity of road crashes. The presented location model considered frequency and severity of road crashes in the network demand. However, this study had failed to consider the current site for emergency response vehicles and the average speed of emergency response vehicles in all
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parts of the city had been assumed constant. Increasing the coverage of emergency facilities does not ensure an equitable distribution of facilities in a city. Amorim et al. (2017) [4] proposed a two-step methodology: (1) assessing emergency demands in a city and finding
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high priority areas that need dispatching an ambulance with a medical team, and (2) applying a double standard covering model to accurately identify the location of emergency medical services with respect to road crashes. The results indicated the areas with heavy traffic and high travel speed had more road crash emergencies. Thus, new emergency facilities should be constructed near these areas.
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Navazi et al. (2018) [31] proposed a multi period location-allocation ambulance station problem to cover accident-prone spots with an uncertain demand. Cheng and Liang (2014) [32] developed a multi-objective EFLP to cover both urban and railway emergency demands. The presented model was designed to maximize the coverage of ambulance stations for a
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population within a predetermined time threshold and to maximize the risk coverage in various railway areas. Also, fairness was considered in their research. Hsia et al. (2009) [33]
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proposed a bi-objective ambulance location problem which minimized the maximum weighted sum of distances between an ambulance station or a hospital from a crash scene and maximized the preference function of ambulance stations. Ferrari et al. (2018) [34] proposed a multi-objective ambulance station location problem intended to maximize the number of patients served by ambulances and the coverage of demand points and to minimize the number of receptive facilities and the distance between the covered areas and facilities. Moreover, the demand network was integrated by emergency calls and population of demand points. In this paper, both population and crash data of different urban districts have been integrated to increase the coverage of current and new emergency centers and hospitals. The 5
ACCEPTED MANUSCRIPT crash data are divided into three categories based on severity (fatal, injurious and PDO) to see the random feature of crashes. In MCLP, different objective functions are considered, including enhanced system efficiency, reduced average response time (RT), distance, cost and the expected coverage of network facilities, and horizontal and/or vertical equity [35]. The multiplicity of objective
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functions has brought to the fore the multi-objective emergency facility location problems [36]. One of the emergency facility location indices is efficiency index based on data envelopment analysis (DEA). In these problems, a DEA mathematical model is used for evaluating the proposed location, and combining it with EFLP [37]. Thomas et al. (2002) [38]
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studied the facility location problem and used a DEA and location model iteratively to identify near-optimum values for proximity and efficiency scores. Sahin et al. (2007) [39]
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applied DEA to evaluate the efficiency of blood centers. Mitropoulos et al. (2013) [37] proposed a multi-objective location model by considering the effectiveness of proposed locations for building medical centers based on a DEA model. In their study, the efficiency of proposed locations was calculated with DEA algorithm before solving location model by incorporating a maximum objective in the facilities location model.
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A challenge of combining DEA and EFLP in an iterative structure is the absence of any guarantee to reach a global optimum solution. Therefore, it is more appropriate to integrate both models in a single mathematical model. For this reason, Khodaprasti et al. (2016) [35] studied the medical emergency facilities location problem with the goal of increasing social
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equity via Gini index and enhancing efficiency of proposed locations based on the efficiency score obtained from a DEA model. In this study, the DEA model and EFLP are integrated in a single mathematical model. This research is inspired by the following criticism leveled
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against Khodaprasti et al. (2016) [35] by prominent academics and practitioners: •
The efficiency objective function of Khodaprasti et al. (2016) was designed to
maximize the efficiency of emergency facilities, but in this paper, we maximized the sum of efficiency in all urban regions, with the efficiency of each region depending on its specific set of emergency facilities.
•
When an efficient service is offered to patients, the time between dispatching an ambulance from a station to a crash scene and from the crash scene to the hospital or emergency center is less than a threshold. In this regard, this paper 6
ACCEPTED MANUSCRIPT maximizes the efficiency of both emergency centers and hospitals for the coverage of network demand. •
Network demand in this paper is comprised of four parts called population, number of fatal, injurious and PDO crashes in the regions. Also, network demand has been assumed to be continuous. Given the random feature of crashes, we have considered fatal and PDO crash as a part of network demand for the first time.
•
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•
In order to guarantee the model against the economic variations, the cost of building facilities and the available budget were expressed in a mathematical
•
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model.
The coverage area of each emergency facility in the network depends on
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average ambulance speed during rush hours in the road network. In this regard, we proposed an EFLP based on MCLP with an efficiency-based objective and a modified continuous network demand. To convert nonlinear constraints into linear ones in the proposed model, the typical linearization methods were used, and finally a mixed integer programming model was developed. The proposed model was tested on a real
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network in capital and the largest city of Iran, Tehran.
The rest of the paper is organized as follows; Section 3 defines the problem and presents a mixed integer programming model. Section 4 tests the proposed model in Tehran
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and in Section 5 discusses the final results
3. PROBLEM DEFINITION AND FORMULATION In this section, the proposed model is explained in details. First, the existing urban network is
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divided into several regions to increase the efficiency (coverage performance) in all regions. The efficiencies of regions is computed by a DEA model embedded in the location model based on some input and output criteria. Population density (the ratio of a region’s population to its effective area) and the number of fatal, injurious and PDO crashes in each region represent the inputs of the model. Also, the coverage contribution of emergency centers and hospitals in each region divided by its effective area constitute the outputs of the model. The effective area of a region is the area occupied by residential, official and business land uses. The inputs of DEA model serve as parameters while outputs represent decision variable. Accordingly, the decision maker seeking to solve the model, usually the 7
ACCEPTED MANUSCRIPT government, can change the efficiency of a region by increasing the outputs of DEA model, which are the coverage of emergency centers and hospitals in the emergency demand of the region. The coverage of facilities (emergency centers and hospitals) can be changed by locating new facilities in each region. For instance, if the budget constraint allows constructing three emergency centers, the model determines the appropriate locations to
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maximize the overall efficiency of all regions. As such, it is possible that the optimal solution proposes three locations for the opening of facilities in a region, each one in a specific area or with other possible scenarios. Fig. (1) illustrates the inputs and outputs of the DEA model embedded in the location model for locating emergency centers and hospitals.
Outputs
Number of fatal, injury, and PDO crashes in each region
Coverage contribution of emergency centers divided by effective area in region k
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Decision unit k (region k)
Population density of each region
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Inputs
Coverage contribution of hospitals divided by effective area in region k
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Fig. 1. Inputs and outputs for evaluating the efficiency of units
It is possible to change inputs and outputs of the model or add new ones. For instance, instead of considering the total population of a region as the input, the population of different age groups or the frequency of emergency calls in the region could be assumed as the inputs.
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Before presenting the mathematical model, the notations for sets/indices, input parameters and decision variables used for problem modeling are introduced. ,
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Sets/indices
: Index representing decision units (regions)
: Index representing new proposed facilities for each decision unit (regions) : Index representing inputs for DEA : Index representing outputs for DEA
Input parameters
: Number of problem decision units (regions) : Number of inputs for DEA : Number of outputs for DEA 8
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: Number of proposed facilities of type b for decision unit j : Effective area of decision unit j (region j) : Total budget for building facilities
: Costs of building facilities of type b
: Added coverage of kth facilities of type b for decision unit j
: Value of input a for decision unit j : Value of output a for decision unit j
Decision variables
: Inefficiency of decision units j (region j)
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: Weight of input a for decision unit j
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: An upper bound for the value of
: Weight of output b for decision unit j
: As a binary decision variable, it is equal to one if the kth facilities of type b for decision unit j is built otherwise is equal to zero
= " Subject to:
%$#'
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#
(2)
" × = 1 ∀ = 1,2, … ,
%$./ #,
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#,
" × + " " × #,
$%
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%$#, ./
" 0 × + " " 0 ×
$% #
./ #,
" " " $% $% $%
$%
%$×
(1)
+ = 1 ∀ = 1,2, … , #'
× − " 0 × ≤ 0 ∀, = 1,2, … ,
(3)
(4)
%$(5)
≤
∈ 50,16 ∀ = 1,2, … , & ∀ = 1,2, … , & ∀ = 1,2, … ,
≥ ϵ ∀ = 1,2, … , & ∀ = 1,2, … , 9
(6) (7)
ACCEPTED MANUSCRIPT ≥ : ∀ = 1,2, … , & ∀ = 1,2, … ,
(8)
Equation (1) is the objective function of the problem. It minimizes the sum of inefficiency values in all decision units ( ), which results in Eq. (3). Equations (2)-(4), which are based on the DEA mathematical model, compute the efficiency or inefficiency of decision units. Equation (2) is a common constraint used in DEA models, which sets the value of
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virtual inputs for each decision unit to one. Equation (3) calculates the inefficiency of a decision unit based on allocated weights of output variables. In Equation (4), the subtraction of virtual output of each decision unit from virtual input is greater than or equal to zero. Equation (5) sets a constraint on the number of possible constructions in the network, which
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is designed to observe the budget constraint. Finally, Equations (6)-(8) show the range and type of decision variables including Input and output weights in a DEA method. These
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weights are the most favorable ones from the point of view of the target unit. Accordingly, units might value inputs and outputs differently and therefore adopt different weights in order to maximize their efficiency in comparison to the other units.
./ , ∑# $% In Equations (3) and (4), the multiplicative terms ( ∑$% × × = <./
>/
>/
and
) render the model nonlinear. These terms are concerned with
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./ , ∑# $% ∑$% 0 × ×
= <./
making decision about construction of new facilities. These nonlinear multiplications were composed of continuous ( or 0 ) and binary variables ( ). To linearize these terms,
the method used in [35] and [40] was employed. Based on this method first the nonlinear
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term (e.g. × must be replaced with a virtual variable ? . Then, four new
constraints are added to the problem. (9)
? ≥ 0 ∀ = 1,2, … , & ∀ = 1,2, … ,
(10)
? ≤ × ∀ = 1,2, … , & ∀ = 1,2, … ,
(11)
? ≤ ∀ = 1,2, … , & ∀ = 1,2, … ,
(12)
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? ≥ × + − ∀ = 1,2, … , & ∀ = 1,2, … ,
= 1, then ? must be equal to , which According to Equations (9) and (12), if
is the expected result. On the other hand, if = 0, then Equations (10) and (11) ensure that
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ACCEPTED MANUSCRIPT ? is equal to zero and the expected result is achieved. Similar to the abovementioned
linearization, another nonlinear term, 0 × , must be swapped with another virtual variable ? 0 and Equations (13)-(16) should be added to the problem constraints.
? × + 0 − 0 ∀, = 1,2, … , &∀ 0 ≥ 0 = 1,2, … , & ∀ = 1,2, … ,
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(13)
? 0 ≥ 0 ∀, = 1,2, … , &∀ = 1,2, … , & ∀ = 1,2, … , ? × ∀, = 1,2, … , &∀ = 1,2, … , & ∀ = 1,2, … , 0 ≤ 0
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? 0 ≤ 0 ∀, = 1,2, … , &∀ = 1,2, … , & ∀ = 1,2, … ,
(14) (15) (16)
One of the inputs of the proposed model is , which is the amount of added coverage
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of kth facilities of type b for decision unit j, and was calculated using the concept of catchment area. An example describing the catchment area concept is illustrated in Fig. (2).
A
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@A
C
@%
Facility location
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Region boundary
Catchment boundary of new stations Catchment boundary of existing stations
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Catchment area of new stations
B
Fig. 2. A hypothetical network to explain the concept of catchment area
In Fig. (2), facilities of type A and type B are added to the set of facilities already available in the network. The hatched areas are the coverage area generated as a result of adding new facilities. The double coverage is disregarded as one of the extra output variables of the problem. For example, by adding facility B, a part of the region is covered with facilities C and B (the overlapping area between B and C), which improves the emergency services in this area. Therefore, the coverage of new stations can be calculated from
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ACCEPTED MANUSCRIPT uncovered areas in the regions. Another important point is that decision variables and
Region boundary
B
ΙΙ: With Overlap
A
B
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A
E: Without Overlap
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their possible effect is a function of . Consider the example shown in Fig. (3).
Facility location
Catchment boundary of new stations Catchment boundary of existing stations
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Catchment area of new stations
Fig. 3: Instance of presence/absence of conflicts between decision variables
In Fig. (3), in situation E (on the left), there is no overlap between covered areas in the
new stations. However, in situation EE (on the right), the covered areas of new stations A and
B (both are of the same type) overlap. If there is an overlap, value of new stations would
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be dependent on both areas, and this would increase the complexity of the problem.
Therefore, in this research, situation E has been considered, and the proposed model is based on the assumption of non-overlap of new stations of the same type. In the next section, the
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model is presented and tested on a real network, and the results are discussed. 4. NUMERICAL RESULTS
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The proposed model was tested in the city of Tehran. With an area of 730 km2, Tehran has over 12 million population, which are dispersed in 22 Metropolitan Statistical Areas (MSA). In this paper, the same fragmentation was used. Values of frequency variables such as damage, injurious and fatal crashes were calculated based on accident statistics of Tehran in the period of 2013 to 2016. The average rate of crashes was taken during this time interval. The coverage contribution of current emergency centers and hospitals were calculated from the effective area in each region. Table (1) shows the numerical values of input and output variables in 22 regions of Tehran. The details of calculating the coverage share of new stations in each region is described in the following section. Fig. (4) shows the location of existing emergency centers and hospitals of Tehran. 12
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Table 1: Values of existing input and output variables for different city regions Fatal
Effective area (1000 square meters)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
3664 2960 2891 4035 3512 2647 1283 1144 933 1143 882 491 639 1122 1770 1080 535 1446 1906 1079 2360 956
762 956 780 1075 1326 1507 728 459 649 748 1020 1118 355 743 1065 712 343 909 995 719 867 356
9 5 5 10 11 5 3 2 5 3 2 2 1 4 9 8 4 7 12 12 13 11
36497 42871 23517 36902 42109 20335 14612 13224 7804 8074 12076 16018 11384 15957 21337 14833 7932 34154 17735 22769 38869 43717
Emergency Population Density (1000 Hospitals centers coverage Year 2016 people per area) coverage area area
462323 633905 325193 848433 865467 223240 309844 377419 158112 293734 296179 234370 287943 393640 621197 272113 248816 438919 233608 351781 161054 140567
12.67 14.79 13.83 22.99 20.55 10.98 21.2 28.54 20.26 36.38 24.53 14.63 25.29 24.67 29.11 18.35 31.37 12.85 13.17 15.45 4.14 3.22
0.65 0.8 0.85 0.5 0.9 0.94 0.93 0.85 0.9 1 0.85 0.7 0.83 0.8 0.55 0.62 0.9 0.48 0.1 0.56 0.35 0.4
0.9 1 1 0.83 0.8 1 1 1 0.9 1 1 1 0.94 0.94 0.3 0.94 1 0.82 0.4 0.78 0.23 0.4
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Injury
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Damage
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Region
Fig. 4: Location of existing and proposed emergency centers and hospitals in Tehran network
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ACCEPTED MANUSCRIPT Based on Fig. (4), 50 locations were proposed for building emergency centers and 16 for building hospitals. Average travel speed during rush hours for all 22 regions of Tehran was derived from 2016 statistics book of Tehran [41], and used as the basis for calculating coverage radius of each facility. Eq. (17) shows the method of calculating coverage radius of each emergency facility. (17)
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@ = UV × W X × Y × Z ∀ = 1,2, … , & ∀ = 1,2, … , [
Where @ is the coverage radius of facilities of type in region j, W is the average
travel speed during rush hours in region j, Z is a suitable response time for facilities of type
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b, V is a constant larger than one used to calculate the average ambulance speed during rush
hours in the network based on mean vehicle travel speed. Because of usually specialized paths assigned to ambulances, the average speed of ambulance is faster than that of traffic
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flow in a route. Intuitively, in Tehran network, the value of α was set equal to 1.2. Due to topographic features of the roads, emergency services, dispatching from center of an emergency facility to the boundary of its coverage area, pass more distance than the coverage
radius. Therefore, Y is a decreasing coefficient, which reduces the real coverage radius of the
facility. For Tehran network, Y was set equal to 0.8. In cases where the proposed station is located on the border of two or more regions (near the boundaries of regions), the minimum
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speed of travel time between regions was considered as W . Then, the response time of each
facility was determined. Based on national EMS code enacted in 1973, the maximum allowable time for response time in urban areas is 10 min. Therefore, the response time of
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emergency centers equal to 5 min was considered. Once primary emergency services are provided to the injured people, it can take a longer time for ambulances to arrive at the
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hospital. Therefore, the coverage radius of hospitals was assumed to be larger than that of emergency centers. Accordingly, the response time of hospitals (maximum allowable time for an ambulance travel from the accident scene to the hospital) was considered as 10 min. After determining the input values in Eq. (17), the coverage radius of all existing and proposed centers in the network were calculated. For example, if the average travel speed of vehicles during rush hours is 20 kph in a region, the coverage area of emergency centers and hospital could be calculated as follows:
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ACCEPTED MANUSCRIPT @%\]^_`ab = c1.2 ×
20 e f
10e × g0.7 × j = 2.8 60
e f
@%lmnom#pq = c1.2 × 20
× g0.7 ×
(18)
5e j = 1.4 60
(19)
Table (2) shows the speed and coverage radius of emergency centers and hospitals in
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all regions of Tehran. By determining the coverage radius of each emergency center and hospital, the added coverage to each region could be computed by adding stations. Fig. (5) shows the coverage areas (catchment areas) of the existing facilities.
Table 2: Coverage radius of emergency centers and hospitals in 22 regions of Tehran 1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22
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Region Average speed
24 24 21 27 31 14 16 22 18 18 13 11 19 21 22 21 18 31 32 29 31 39
Hospital radius
3.3 3.3 3.0 3.8 4.3 1.9 2.3 3.0 2.6 2.5 1.9 1.5 2.7 2.9 3.1 2.9 2.5 4.3 4.5 4.1 4.3 5.5
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Emergency center 1.7 1.7 1.5 1.9 2.2 1.0 1.1 1.5 1.3 1.2 0.9 0.8 1.3 1.4 1.5 1.4 1.3 2.2 2.2 2.0 2.2 2.7 radius
Fig. 5: Coverage area of emergency centers and hospitals in Tehran network
To guarantee the model against economic variations, the costs of building facilities and the available budget were expressed in terms of a hypothetical unit. That is, the cost of building an emergency center was considered as one unit and the costs of building a hospital with conventional equipment was considered as three units. The proposed model was 15
ACCEPTED MANUSCRIPT executed based on different values of the budget (B), and the outcomes of each scenario were compared. The model was resolved in a commercial software Cplex 12.6 with a Java interface in a system with these specifications (4 GB RAM, 4-core CPU and 4 MB cache). Table (3) shows the results of each scenario.
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Table 3: Model results for 22 districts of Tehran
45
50
55
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efficiency of region 1 0.42 0.42 0.42 0.42 0.42 0.42 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 efficiency of region 2 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.41 0.41 0.41 0.41 0.41 0.41 efficiency of region 3 0.46 0.46 0.46 0.46 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50
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efficiency of region 4 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.32 0.32 0.32 0.32 0.32 0.32 0.32 efficiency of region 5 0.28 0.28 0.28 0.28 0.28 0.28 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 efficiency of region 6 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 efficiency of region 7 0.56 0.56 0.60 0.60 0.60 0.60 0.60 0.59 0.59 0.59 0.59 0.59 0.59 0.59 0.59
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efficiency of region 8 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.83 0.84 0.84 0.84 0.84 0.84 0.84 efficiency of region 9 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 efficiency of region 10 0.67 0.67 0.67 0.67 0.67 0.67 0.67 0.65 0.65 0.65 0.65 0.65 0.65 0.65 0.65 efficiency of region 11 0.72 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 efficiency of region 12 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 efficiency of region 13 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 efficiency of region 14 0.52 0.59 0.59 0.59 0.59 0.59 0.59 0.59 0.59 0.59 0.59 0.59 0.59 0.59 0.59 efficiency of region 15 0.22 0.22 0.26 0.33 0.33 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35
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efficiency of region 16 0.52 0.52 0.52 0.52 0.52 0.52 0.52 0.56 0.56 0.56 0.56 0.56 0.56 0.56 0.56 efficiency of region 17 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 efficiency of region 18 0.35 0.35 0.35 0.35 0.35 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42 efficiency of region 19 0.15 0.15 0.15 0.15 0.21 0.21 0.21 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 efficiency of region 20 0.39 0.39 0.39 0.46 0.46 0.46 0.46 0.46 0.46 0.46 0.46 0.46 0.46 0.46 0.46 efficiency of region 21 0.16 0.16 0.30 0.34 0.37 0.37 0.37 0.39 0.39 0.39 0.39 0.39 0.39 0.39 0.39 No emergency centers No hospital stations
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efficiency of region 22 0.43 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 5
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10.80 10.17 9.94 9.77 9.63 9.53 9.46 9.39 9.34 9.31 9.31 9.31 9.31 9.31 9.31
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0.05 0.84 2.79 4.55 4.93 7.47 10.87 9.47 8.08 6.71 7.40 6.80 6.96 8.30 8.08
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Objective function
According to Table (3), under the scenario of existing condition (B=0), regions 12, 13 and 17 are efficient in covering emergency crashes. The important point about this scenario is that region 22 was selected as the most suitable candidate for improving the total network efficiency. This is despite the fact that Regions 19, 21 and eight other regions have an efficiency score critically lower than Region 22. This result implies that combining DEA and EFLP in an iterative method does not ensure an efficient solution. Therefore, constructing emergency centers in regions with the lowest efficiency score, which are in a critical 16
ACCEPTED MANUSCRIPT condition, is not always the most efficient strategy. Fig. (6) shows variation in the efficiency of each region for budget values below 50 units. No efficiency change was observed for budget levels higher than 50. 1.6
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Fig. 6. Changes of units and system efficiency at different budget levels
According to Fig. (6), Regions 11, 14, and 22 are the most suitable candidates for increasing the coverage of emergency crashes. Assigning facilities to these regions would require five
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budget units, which decreases the objective function (sum of a units’ inefficiency) by 0.6. The information of Table (3) can be used for scheduling the construction of network
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emergency facilities and prioritizing projects with the goal of increasing system efficiency. Based on Table (3), the construction of a new hospital in Tehran occurs at the budget level of 55 and above with a maximum of one hospital unit. However, this investment does not improve the sum of efficiency. Therefore, budget levels of 45 to 70 will not cause any further improvement in the efficiency of decision units. Table (5) shows the ID number of selected emergency centers under each effective scenario. Each ID number has two parts. The first shows the region ID and the second indicates the center ID in the corresponded region. Also, Fig. A1 in the appendix illustrates the location of emergency centers allocated to each region in Tehran. 17
ACCEPTED MANUSCRIPT Table 4. Region number and number of selected emergency centers for effective scenarios The emergency centers selected (region ID- center ID)
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5
11-1,14-1,22-1,22-2,22-3
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7-1,11-1,14-1,15-1,20-1,20-2,21-1,21-3,21-6,22-1,22-2,22-3
15
7-1,11-1,14-1,15-1,15-3,15-4,20-1,20-2,21-1,21-3,21-4-21-5-22-1-22-2-22-3
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3-1,3-2,7-1,11-1,13-1,14-1,15-1,15-3,15-4,19-1,19-2,20-1,20-2,21-1,21-2,21-3, 21-4,215,22-1,22-2,22-3
25
3-1,3-2,7-1,11-1,13-1,14-1,15-1,15-2,15-3,15-4,18-1,18-2,18-3,18-4,19-1,19-2, 20-1,202,21-1,21-2,21-3,21-4,21-6,22-1,22-2,22-3
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1-1,1-2,1-3,1-4,3-1,3-2,5-1,7-1,11-1,13-1,14-1,15-1,15-2,15-3,15-4,17-1,18-1, 18-2,183,18-4,19-2,19-3,20-1,20-2,21-1,21-2,21-3,21-4,21-5,22-1,22-2,22-3 1-1,1-2,1-3,1-4,3-1,3-2,5-1,7-1,11-1,13-1,14-1,15-1,15-2,15-3,15-4,16-1,16-2, 16-3,171,18-1,18-2,18-3,18-4,19-1,19-3,19-4,20-1,20-2,21-1,21-2,21-3,21-4,21-5,21-6,22-1,222,22-3 1-1,1-2,1-3,1-4,3-1,3-2,4-1,4-2,4-3,4-4,5-1,7-1,8-1,11-1,13-1,14-1,15-1,15-2,15-3,15-4,161,16-2,16-3,17-1,18-1,18-2,18-3,18-4,19-1,19-3,19-4,20-1,20-2,21-1, 21-2,21-3,21-4,215,21-6,22-1,22-2,22-3 1-1,1-2,1-3,1-4,2-1,2-2,2-3,3-1,3-2,4-1,4-2,4-3,4-4,5-1,7-1,8-1,8-2,11-1,13-1,14-1,15-1,152,15-3,15-4,16-1,16-2,16-3,17-1,18-1,18-2,18-3,18-4,19-1,19-3,19-4, 20-1,20-2,21-1,212,21-3,21-4,21-5,21-6,22-1,22-2,22-3
5. CONCLUSION
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In this study, by combining DEA and MCLP, the problem of locating new emergency
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facilities in urban scale was addressed. The goal was to increase the efficiency of all urban districts in covering emergency demand. In this study, particular attention was given to emergency service demands for road crashes. The network demand considers not only the
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population of regions but also the fatal, injurious and PDO crashes. The proposed model considers the current condition of emergency facilities in the network and seeks to identify locations for building new emergency facilities. The coverage area of each emergency facility was calculated proportional to the average ambulance speed during rush hours in the road network of that region. In this study, site locations in the network were carried out by adding two types of facilities, emergency centers and hospitals. Also four input variables, including the average fatal, injurious and PDO crashes and the population density of each region as well as two output variables, the coverage contribution of emergency centers and hospitals in each region, were used to calculate the efficiency of regions in the mathematical model. The 18
ACCEPTED MANUSCRIPT share of new stations in the coverage of each region was presented as the decision variable of the model. The initial model was a nonlinear mathematical model, which was then converted to a mixed integer programming model using a linearization technique. We tested the proposed model in of 22 regions of Tehran network. The results suggested that inefficiency of Tehran regions was related to lack of emergency centers facilities. Based on the existing
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facilities, ten regions were in a critically less efficient condition than Region 22, but establishing new emergency centers in Region 22 would provide the largest network coverage.
This paper had some limitations in terms of data collection and statistics. For instance,
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the number of ambulances in the network and their positions in facilities over different time periods were unknown. If we had access to the number of in-service ambulances, we could consider allocation problem along with the location problem and therefore measure the
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efficiency of regions more accurately. In addition, it was possible to propose a multi-period model and consider ambulance deployment decision along other decisions. Future studies can focus on determining the number of ambulances in facilities to maximize the efficiency in the regions. Also, if the model is solved for multi periods, the ambulance deployment decisions could be integrated with location and allocation decisions.
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Moreover, besides locating facilities, it is recommended to determine the level of necessary equipment with respect to the budget constraint.
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[31] Navazi F, Tavakkoli-Moghaddam R, Sazvar Z. A Multi-Period Location-AllocationInventory Problem for Ambulance and Helicopter Ambulance Stations: Robust Possibilistic Approach. IFAC-PapersOnLine. 2018;51:322-7. [32] Cheng Y-H, Liang Z-X. A strategic planning model for the railway system accident rescue problem. Transportation research part E: logistics and transportation review. 2014;69:75-96. [33] Hsia H-C, Ishii H, Yeh K-Y. Ambulance service facility location problem. Journal of the Operations Research Society of Japan. 2009;52:339-54. [34] Ferrari T, Camara MVO, Nassi CD, Ribeiro GM, Costa Junior RR, Ribeiro Júnior C, et al. Analysis of the Location of Rescue Ambulance Dispatch Bases: A Case Study in Rio de Janeiro, Brazil. Geographical Analysis. 2018. [35] Khodaparasti S, Maleki H, Bruni ME, Jahedi S, Beraldi P, Conforti D. Balancing efficiency and equity in location-allocation models with an application to strategic EMS design. Optimization Letters. 2016;10:1053-70. [36] Araz C, Selim H, Ozkarahan I. A fuzzy multi-objective covering-based vehicle location model for emergency services. Computers & Operations Research. 2007;34:705-26. [37] Mitropoulos P, Mitropoulos I, Giannikos I. Combining DEA with location analysis for the effective consolidation of services in the health sector. Computers & Operations Research. 2013;40:2241-50. [38] Thomas P, Chan Y, Lehmkuhl L, Nixon W. Obnoxious-facility location and dataenvelopment analysis: A combined distance-based formulation. European Journal of Operational Research. 2002;141:495-514. [39] Şahin G, Süral H, Meral S. Locational analysis for regionalization of Turkish Red Crescent blood services. Computers & Operations Research. 2007;34:692-704. [40] McCormick GP. Computability of global solutions to factorable nonconvex programs: Part I—Convex underestimating problems. Mathematical programming. 1976;10:14775. [41] Traffic Department Of Tehran Municipality. Tehran Statistical Yearbook. 2016.
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Appendix A
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Fig. A1: Proposed emergency centers in each region of Tehran
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The proposed model considers the status and location of the existing emergency facilities in the network and searches for identifying new locations to construct new facilities.
•
Presented model is an integrated model of data envelopment analysis and maximum coverage location problem considering both population and accident demands in various regions of a city Applying the model on a real network with 22 regions and two different facilities
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Considering an specific ambulance speed for each region based on the observed average
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traffic flow in the region’s links
S0038-0121(18)30201-5
DOI:
https://doi.org/10.1016/j.seps.2019.01.005
Reference:
SEPS 683
To appear in:
Socio-Economic Planning Sciences
Received Date: 26 June 2018 Revised Date:
15 December 2018
Accepted Date: 27 January 2019
Please cite this article as: Mohri SS, Akbarzadeh M, Sayed Matin SH, A Hybrid model for locating new emergency facilities to improve the coverage of the road crashes, Socio-Economic Planning Sciences (2019), doi: https://doi.org/10.1016/j.seps.2019.01.005. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
A Hybrid model for locating new emergency facilities to improve the coverage of the road crashes Seyed Sina Mohri1*, Meisam Akbarzadeh2, Seyed Hamed Sayed Matin 3 Department of Transportation Engineering, Isfahan University of Technology, Isfahan, Iran. email: [email protected]
Department of Transportation Engineering, Isfahan University of Technology, Isfahan, Iran, email: [email protected]
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Department of Civil Engineering, Imam Khomeini International University, Qazvin, Iran, email:
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[email protected]
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A Hybrid model for locating new emergency facilities to improve the coverage of road crashes ABSTRACT We propose an emergency facility-locating model aimed at increasing the coverage of
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emergency demand throughout the city. The proposed model takes into account the status and location of the emergency facilities in the network and identifies locations suitable for the construction of new facilities. Here, Data Envelopment Analysis (DEA) and Maximum Coverage Location Problem (MCLP) have been combined in a single model. To do so, design
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problem and evaluation problem are considered concurrently to maximize the efficiency of services provided by emergency facilities across the city in response to the demand. Moreover, the total emergency demand in each district was considered in relation to the
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population density, the fatal, injurious, and property damage only (PDO) crashes. The coverage area of each emergency facility was assumed to be proportional to the average ambulance speed in the surrounding road network during rush hours. The available budget was included in the model to let the model function under various fiscal conditions. Model input variables consisted of average number of mortalities, injuries and PDO crashes as well
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as the population density of each urban district. The output variables of the model included the coverage share of proposed emergency centers and hospitals equipped with ambulances. The model was tested on the network of Tehran (Iran). It is recommended to add the location of some emergency centers and hospitals to the network. Moreover, the results showed that
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ten urban districts had efficiency problem in provision of emergency services. Keywords: Emergency facility location, Ambulance station, Road crashes, Data envelopment
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analysis
1. INTRODUCTION
Road crashes represent a leading cause of death and disability throughout the world. According to World Health Organization (WHO), crashes are the ninth cause of death in the world and first cause of death among young people in the age range of 15 to 29 years [1]. Each year about 1.24 million lives are lost in road crashes and 20 to 50 million people sustain injuries that lead to disabilities [1, 2]. One of the effective measures to reduce the severity of road crashes and protecting human lives is to increase the speed and quality of emergency service provision after the crashes [3]. This goal is the effective resolution of an Emergency 1
ACCEPTED MANUSCRIPT Facility Location Problem (EFLP) at strategic level. Optimum positioning of emergency facilities would diminish their construction and functional cost and improve their efficiency. In this regard, an EFLP with efficiency-based objective has been proposed in this paper. Expansion of urban networks requires locating new facilities. Moreover, increasing coverage in all urban areas with respect to population density, fatal, injurious and property
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damage only (PDO) crashes, along with the coverage of medical facilities can improve the vertical equity in the distribution of emergency services throughout city. In this regard, a systemic and managerial perspective has been employed for reinforcing and balancing the efficiency of all urban areas in covering emergency demand. Moreover, contrary to iterative
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methods of finding an optimum global solution to ensure the maximum efficiency of the system, a combination of Data Envelopment Analysis (DEA) and Maximum Coverage
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Location Problem (MCLP) has been adopted in a single model. Accordingly, the proposed mathematical model evaluates the location of existing emergency centers and hospitals that respond to emergency demands (in terms of population and number of road crashes in regions) and building of new facilities to increase the efficiency of regions in meeting the demand for emergency services.
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The demand for emergency services is in relation to the population, rates of fatal, injurious and PDO crashes of each region. Unlike most of previous studies [4-7] considering a discrete demand through placing demand on nodes (i.e. regions), we assume a continuous demand spread on demand areas (i.e. region areas) in this paper. Road crashes, in general or
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as part of the network demand in EFLPs, should be considered in terms of crash severity as well as crash frequency due to high costs of fatal and injurious crashes. Therefore, it is necessary that the proposed model be able to distinguish crashes in terms of severity in each
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region. Accordingly, the network demand was divided based on four variables of population, fatal, injurious and PDO crashes in the regions. Each part serves as an input criterion in the DEA structure.
PDO crashes along with injurious and fatal crashes were considered as part of network demand in recognition of one of the major characteristic of crashes, i.e. randomness. Random feature of crashes has received little attention in previous studies on EFLPs for covering road crashes. This feature means that several uncontrollable and controllable factors have a bearing on the occurrence of crashes [8]. In addition to its effect on the frequency of crashes, 2
ACCEPTED MANUSCRIPT randomness also affects the severity of crashes. The severity of crashes is a variable of road, human and vehicle factors. Hence, if one of these factors changes, the severity will be modified as well. For example, a PDO crash in an area may result in an injurious or fatal crash if the driver’s age or vehicle’s type changes to more intensifying states.
•
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In summary, this research contributes to the literature in the following ways: The proposed model considers the status and location of the existing emergency facilities, emergency centers and hospitals with ambulance stations in the network and searches for new sites for constructing new facilities to maximize efficiencies •
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Network demand is continuous and dependent both of the population and the frequency and severity of road/highway crashes of regions.
Given the random characteristic of road crashes, all three types of crashes,
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mortality, injury and PDO, are considered in the network demand. •
The coverage radius of facilities, as a variable of the type and the average speed of ambulances during rush hours, varies in different regions. Since the average speed of ambulances during rush hours is considered, the system is risk-averse
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with the prospect of the worst-case scenario. 2. LITERATURE REVIEW
EFLP is a subtopic of programming and management of Emergency Medical Services
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(EMS). From the modeling perspective, EFLP is also a subcategory of Healthcare Facility Location Problem (HCFLP). In our review of literature, we found ten review papers on EMS and HCFLP published in the period of 2000 to 2015, which shows the importance of this
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topic [9-18]. The goal of EFLPs is to locate new emergency facilities for covering emergency demands of a network. In 2017, Ahmdi-Javid et al. [18] divided emergency facilities into two groups of permanent and temporary facilities. Temporary facilities are utilized in the event of disaster, while permanent facility are concerned with medical services, intended to respond to service emergency calls under all circumstances. Permanent facilities are mainly deployed in ambulance stations, hospitals and emergency center [19]. An ambulance station is the site for dispatching ambulances to crash scenes, which is usually embedded in emergency centers or hospitals or operates independently. Emergency centers are units that provide emergency services, either as a part of hospitals or small-scale independent centers. In this paper, two 3
ACCEPTED MANUSCRIPT emergency center facilities and hospital, both with an ambulance station, have been considered to respond to medical demands of network emergency. The simultaneous inclusion of these two facilities in the model can be explained in terms of limited medical equipment in ambulances and the fact that the process of responding to an emergency call is completed when a patient is delivered to an emergency center or a hospital. The choice of
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emergency center or hospital is a variable of the severity of disease or crash, which is a random feature. Therefore, for the purpose of exhaustiveness, the emergency centers must be able to respond efficiently to demand points/areas and the points/areas need to be in the response range of the hospitals.
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The mathematical models in EFLPs cover either the location models or median location models. In the former, the objective is to expand the area covered by facilities. In the latter,
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the goal is to minimize the geodesic or Euclidean distance between network facilities and demand points. Covering location models embrace various categories including Set Covering Programming (SCP) problem [20], MCLP [21], maximum expected covering location problem [22], and maximum available covering location problem [23]. In SCP problems, the aim is to enable the coverage of total network demands for facilities, and the objective function is typically utilized to minimize the cost of building facilities. In three other models,
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the objective is to provide maximum coverage and the objective function is to maximize the respond range of facilities. The mathematical model proposed in this research falls in MCLP category, but with an efficiency-based objective, as indicated by the DEA algorithm.
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Network demand in previous EFLPs is estimated by (1) history of medical emergency calls [24, 25], (2) population-based measures [26-28] and historical road/highway crashes [4, 29-33] of demand points/areas or (3) a combination of (1) and (2) [34]. However, medical
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emergency calls encompass a variety of emergency demands such as diseases, heart attacks and road injuries, which this information are either unavailable in most cities or characterized by inaccuracy in collection, presentation, etc. Also, these historical calls are characterized with a high level of uncertainty so that some researches regard the network demand as uncertain [5]. On the other hand, among studies that estimate network demand by different measures, only a few incorporate historical crash data into estimation of network demand [4, 29-33].
4
ACCEPTED MANUSCRIPT Castañeda and Villegas (2017) [29] proposed a facility location model to extend the coverage of the existing emergency medical services in Medellín (Colombia) for people injured in traffic accidents. The proposed model was designed to locate some new emergency facilities and to compute the optimal number of ambulances in current and new locations. Kepaptsoglou et al. (2011) [30] proposed a double standard location model for identifying the
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optimal location of emergency response vehicles in a city to alleviate the severity of road crashes. The presented location model considered frequency and severity of road crashes in the network demand. However, this study had failed to consider the current site for emergency response vehicles and the average speed of emergency response vehicles in all
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parts of the city had been assumed constant. Increasing the coverage of emergency facilities does not ensure an equitable distribution of facilities in a city. Amorim et al. (2017) [4] proposed a two-step methodology: (1) assessing emergency demands in a city and finding
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high priority areas that need dispatching an ambulance with a medical team, and (2) applying a double standard covering model to accurately identify the location of emergency medical services with respect to road crashes. The results indicated the areas with heavy traffic and high travel speed had more road crash emergencies. Thus, new emergency facilities should be constructed near these areas.
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Navazi et al. (2018) [31] proposed a multi period location-allocation ambulance station problem to cover accident-prone spots with an uncertain demand. Cheng and Liang (2014) [32] developed a multi-objective EFLP to cover both urban and railway emergency demands. The presented model was designed to maximize the coverage of ambulance stations for a
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population within a predetermined time threshold and to maximize the risk coverage in various railway areas. Also, fairness was considered in their research. Hsia et al. (2009) [33]
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proposed a bi-objective ambulance location problem which minimized the maximum weighted sum of distances between an ambulance station or a hospital from a crash scene and maximized the preference function of ambulance stations. Ferrari et al. (2018) [34] proposed a multi-objective ambulance station location problem intended to maximize the number of patients served by ambulances and the coverage of demand points and to minimize the number of receptive facilities and the distance between the covered areas and facilities. Moreover, the demand network was integrated by emergency calls and population of demand points. In this paper, both population and crash data of different urban districts have been integrated to increase the coverage of current and new emergency centers and hospitals. The 5
ACCEPTED MANUSCRIPT crash data are divided into three categories based on severity (fatal, injurious and PDO) to see the random feature of crashes. In MCLP, different objective functions are considered, including enhanced system efficiency, reduced average response time (RT), distance, cost and the expected coverage of network facilities, and horizontal and/or vertical equity [35]. The multiplicity of objective
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functions has brought to the fore the multi-objective emergency facility location problems [36]. One of the emergency facility location indices is efficiency index based on data envelopment analysis (DEA). In these problems, a DEA mathematical model is used for evaluating the proposed location, and combining it with EFLP [37]. Thomas et al. (2002) [38]
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studied the facility location problem and used a DEA and location model iteratively to identify near-optimum values for proximity and efficiency scores. Sahin et al. (2007) [39]
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applied DEA to evaluate the efficiency of blood centers. Mitropoulos et al. (2013) [37] proposed a multi-objective location model by considering the effectiveness of proposed locations for building medical centers based on a DEA model. In their study, the efficiency of proposed locations was calculated with DEA algorithm before solving location model by incorporating a maximum objective in the facilities location model.
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A challenge of combining DEA and EFLP in an iterative structure is the absence of any guarantee to reach a global optimum solution. Therefore, it is more appropriate to integrate both models in a single mathematical model. For this reason, Khodaprasti et al. (2016) [35] studied the medical emergency facilities location problem with the goal of increasing social
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equity via Gini index and enhancing efficiency of proposed locations based on the efficiency score obtained from a DEA model. In this study, the DEA model and EFLP are integrated in a single mathematical model. This research is inspired by the following criticism leveled
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against Khodaprasti et al. (2016) [35] by prominent academics and practitioners: •
The efficiency objective function of Khodaprasti et al. (2016) was designed to
maximize the efficiency of emergency facilities, but in this paper, we maximized the sum of efficiency in all urban regions, with the efficiency of each region depending on its specific set of emergency facilities.
•
When an efficient service is offered to patients, the time between dispatching an ambulance from a station to a crash scene and from the crash scene to the hospital or emergency center is less than a threshold. In this regard, this paper 6
ACCEPTED MANUSCRIPT maximizes the efficiency of both emergency centers and hospitals for the coverage of network demand. •
Network demand in this paper is comprised of four parts called population, number of fatal, injurious and PDO crashes in the regions. Also, network demand has been assumed to be continuous. Given the random feature of crashes, we have considered fatal and PDO crash as a part of network demand for the first time.
•
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•
In order to guarantee the model against the economic variations, the cost of building facilities and the available budget were expressed in a mathematical
•
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model.
The coverage area of each emergency facility in the network depends on
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average ambulance speed during rush hours in the road network. In this regard, we proposed an EFLP based on MCLP with an efficiency-based objective and a modified continuous network demand. To convert nonlinear constraints into linear ones in the proposed model, the typical linearization methods were used, and finally a mixed integer programming model was developed. The proposed model was tested on a real
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network in capital and the largest city of Iran, Tehran.
The rest of the paper is organized as follows; Section 3 defines the problem and presents a mixed integer programming model. Section 4 tests the proposed model in Tehran
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and in Section 5 discusses the final results
3. PROBLEM DEFINITION AND FORMULATION In this section, the proposed model is explained in details. First, the existing urban network is
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divided into several regions to increase the efficiency (coverage performance) in all regions. The efficiencies of regions is computed by a DEA model embedded in the location model based on some input and output criteria. Population density (the ratio of a region’s population to its effective area) and the number of fatal, injurious and PDO crashes in each region represent the inputs of the model. Also, the coverage contribution of emergency centers and hospitals in each region divided by its effective area constitute the outputs of the model. The effective area of a region is the area occupied by residential, official and business land uses. The inputs of DEA model serve as parameters while outputs represent decision variable. Accordingly, the decision maker seeking to solve the model, usually the 7
ACCEPTED MANUSCRIPT government, can change the efficiency of a region by increasing the outputs of DEA model, which are the coverage of emergency centers and hospitals in the emergency demand of the region. The coverage of facilities (emergency centers and hospitals) can be changed by locating new facilities in each region. For instance, if the budget constraint allows constructing three emergency centers, the model determines the appropriate locations to
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maximize the overall efficiency of all regions. As such, it is possible that the optimal solution proposes three locations for the opening of facilities in a region, each one in a specific area or with other possible scenarios. Fig. (1) illustrates the inputs and outputs of the DEA model embedded in the location model for locating emergency centers and hospitals.
Outputs
Number of fatal, injury, and PDO crashes in each region
Coverage contribution of emergency centers divided by effective area in region k
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Decision unit k (region k)
Population density of each region
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Inputs
Coverage contribution of hospitals divided by effective area in region k
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Fig. 1. Inputs and outputs for evaluating the efficiency of units
It is possible to change inputs and outputs of the model or add new ones. For instance, instead of considering the total population of a region as the input, the population of different age groups or the frequency of emergency calls in the region could be assumed as the inputs.
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Before presenting the mathematical model, the notations for sets/indices, input parameters and decision variables used for problem modeling are introduced. ,
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Sets/indices
: Index representing decision units (regions)
: Index representing new proposed facilities for each decision unit (regions) : Index representing inputs for DEA : Index representing outputs for DEA
Input parameters
: Number of problem decision units (regions) : Number of inputs for DEA : Number of outputs for DEA 8
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: Number of proposed facilities of type b for decision unit j : Effective area of decision unit j (region j) : Total budget for building facilities
: Costs of building facilities of type b
: Added coverage of kth facilities of type b for decision unit j
: Value of input a for decision unit j : Value of output a for decision unit j
Decision variables
: Inefficiency of decision units j (region j)
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: Weight of input a for decision unit j
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: An upper bound for the value of
: Weight of output b for decision unit j
: As a binary decision variable, it is equal to one if the kth facilities of type b for decision unit j is built otherwise is equal to zero
= " Subject to:
%$#'
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#
(2)
" × = 1 ∀ = 1,2, … ,
%$./ #,
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#,
" × + " " × #,
$%
%$AC C
%$#, ./
" 0 × + " " 0 ×
$% #
./ #,
" " " $% $% $%
$%
%$×
(1)
+ = 1 ∀ = 1,2, … , #'
× − " 0 × ≤ 0 ∀, = 1,2, … ,
(3)
(4)
%$(5)
≤
∈ 50,16 ∀ = 1,2, … , & ∀ = 1,2, … , & ∀ = 1,2, … ,
≥ ϵ ∀ = 1,2, … , & ∀ = 1,2, … , 9
(6) (7)
ACCEPTED MANUSCRIPT ≥ : ∀ = 1,2, … , & ∀ = 1,2, … ,
(8)
Equation (1) is the objective function of the problem. It minimizes the sum of inefficiency values in all decision units ( ), which results in Eq. (3). Equations (2)-(4), which are based on the DEA mathematical model, compute the efficiency or inefficiency of decision units. Equation (2) is a common constraint used in DEA models, which sets the value of
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virtual inputs for each decision unit to one. Equation (3) calculates the inefficiency of a decision unit based on allocated weights of output variables. In Equation (4), the subtraction of virtual output of each decision unit from virtual input is greater than or equal to zero. Equation (5) sets a constraint on the number of possible constructions in the network, which
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is designed to observe the budget constraint. Finally, Equations (6)-(8) show the range and type of decision variables including Input and output weights in a DEA method. These
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weights are the most favorable ones from the point of view of the target unit. Accordingly, units might value inputs and outputs differently and therefore adopt different weights in order to maximize their efficiency in comparison to the other units.
./ , ∑# $% In Equations (3) and (4), the multiplicative terms ( ∑$% × × = <./
>/
>/
and
) render the model nonlinear. These terms are concerned with
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./ , ∑# $% ∑$% 0 × ×
= <./
making decision about construction of new facilities. These nonlinear multiplications were composed of continuous ( or 0 ) and binary variables ( ). To linearize these terms,
the method used in [35] and [40] was employed. Based on this method first the nonlinear
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term (e.g. × must be replaced with a virtual variable ? . Then, four new
constraints are added to the problem. (9)
? ≥ 0 ∀ = 1,2, … , & ∀ = 1,2, … ,
(10)
? ≤ × ∀ = 1,2, … , & ∀ = 1,2, … ,
(11)
? ≤ ∀ = 1,2, … , & ∀ = 1,2, … ,
(12)
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? ≥ × + − ∀ = 1,2, … , & ∀ = 1,2, … ,
= 1, then ? must be equal to , which According to Equations (9) and (12), if
is the expected result. On the other hand, if = 0, then Equations (10) and (11) ensure that
10
ACCEPTED MANUSCRIPT ? is equal to zero and the expected result is achieved. Similar to the abovementioned
linearization, another nonlinear term, 0 × , must be swapped with another virtual variable ? 0 and Equations (13)-(16) should be added to the problem constraints.
? × + 0 − 0 ∀, = 1,2, … , &∀ 0 ≥ 0 = 1,2, … , & ∀ = 1,2, … ,
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(13)
? 0 ≥ 0 ∀, = 1,2, … , &∀ = 1,2, … , & ∀ = 1,2, … , ? × ∀, = 1,2, … , &∀ = 1,2, … , & ∀ = 1,2, … , 0 ≤ 0
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? 0 ≤ 0 ∀, = 1,2, … , &∀ = 1,2, … , & ∀ = 1,2, … ,
(14) (15) (16)
One of the inputs of the proposed model is , which is the amount of added coverage
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of kth facilities of type b for decision unit j, and was calculated using the concept of catchment area. An example describing the catchment area concept is illustrated in Fig. (2).
A
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@A
C
@%
Facility location
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Region boundary
Catchment boundary of new stations Catchment boundary of existing stations
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Catchment area of new stations
B
Fig. 2. A hypothetical network to explain the concept of catchment area
In Fig. (2), facilities of type A and type B are added to the set of facilities already available in the network. The hatched areas are the coverage area generated as a result of adding new facilities. The double coverage is disregarded as one of the extra output variables of the problem. For example, by adding facility B, a part of the region is covered with facilities C and B (the overlapping area between B and C), which improves the emergency services in this area. Therefore, the coverage of new stations can be calculated from
11
ACCEPTED MANUSCRIPT uncovered areas in the regions. Another important point is that decision variables and
Region boundary
B
ΙΙ: With Overlap
A
B
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A
E: Without Overlap
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their possible effect is a function of . Consider the example shown in Fig. (3).
Facility location
Catchment boundary of new stations Catchment boundary of existing stations
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Catchment area of new stations
Fig. 3: Instance of presence/absence of conflicts between decision variables
In Fig. (3), in situation E (on the left), there is no overlap between covered areas in the
new stations. However, in situation EE (on the right), the covered areas of new stations A and
B (both are of the same type) overlap. If there is an overlap, value of new stations would
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be dependent on both areas, and this would increase the complexity of the problem.
Therefore, in this research, situation E has been considered, and the proposed model is based on the assumption of non-overlap of new stations of the same type. In the next section, the
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model is presented and tested on a real network, and the results are discussed. 4. NUMERICAL RESULTS
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The proposed model was tested in the city of Tehran. With an area of 730 km2, Tehran has over 12 million population, which are dispersed in 22 Metropolitan Statistical Areas (MSA). In this paper, the same fragmentation was used. Values of frequency variables such as damage, injurious and fatal crashes were calculated based on accident statistics of Tehran in the period of 2013 to 2016. The average rate of crashes was taken during this time interval. The coverage contribution of current emergency centers and hospitals were calculated from the effective area in each region. Table (1) shows the numerical values of input and output variables in 22 regions of Tehran. The details of calculating the coverage share of new stations in each region is described in the following section. Fig. (4) shows the location of existing emergency centers and hospitals of Tehran. 12
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Table 1: Values of existing input and output variables for different city regions Fatal
Effective area (1000 square meters)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
3664 2960 2891 4035 3512 2647 1283 1144 933 1143 882 491 639 1122 1770 1080 535 1446 1906 1079 2360 956
762 956 780 1075 1326 1507 728 459 649 748 1020 1118 355 743 1065 712 343 909 995 719 867 356
9 5 5 10 11 5 3 2 5 3 2 2 1 4 9 8 4 7 12 12 13 11
36497 42871 23517 36902 42109 20335 14612 13224 7804 8074 12076 16018 11384 15957 21337 14833 7932 34154 17735 22769 38869 43717
Emergency Population Density (1000 Hospitals centers coverage Year 2016 people per area) coverage area area
462323 633905 325193 848433 865467 223240 309844 377419 158112 293734 296179 234370 287943 393640 621197 272113 248816 438919 233608 351781 161054 140567
12.67 14.79 13.83 22.99 20.55 10.98 21.2 28.54 20.26 36.38 24.53 14.63 25.29 24.67 29.11 18.35 31.37 12.85 13.17 15.45 4.14 3.22
0.65 0.8 0.85 0.5 0.9 0.94 0.93 0.85 0.9 1 0.85 0.7 0.83 0.8 0.55 0.62 0.9 0.48 0.1 0.56 0.35 0.4
0.9 1 1 0.83 0.8 1 1 1 0.9 1 1 1 0.94 0.94 0.3 0.94 1 0.82 0.4 0.78 0.23 0.4
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Injury
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Damage
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Region
Fig. 4: Location of existing and proposed emergency centers and hospitals in Tehran network
13
ACCEPTED MANUSCRIPT Based on Fig. (4), 50 locations were proposed for building emergency centers and 16 for building hospitals. Average travel speed during rush hours for all 22 regions of Tehran was derived from 2016 statistics book of Tehran [41], and used as the basis for calculating coverage radius of each facility. Eq. (17) shows the method of calculating coverage radius of each emergency facility. (17)
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@ = UV × W X × Y × Z ∀ = 1,2, … , & ∀ = 1,2, … , [
Where @ is the coverage radius of facilities of type in region j, W is the average
travel speed during rush hours in region j, Z is a suitable response time for facilities of type
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b, V is a constant larger than one used to calculate the average ambulance speed during rush
hours in the network based on mean vehicle travel speed. Because of usually specialized paths assigned to ambulances, the average speed of ambulance is faster than that of traffic
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flow in a route. Intuitively, in Tehran network, the value of α was set equal to 1.2. Due to topographic features of the roads, emergency services, dispatching from center of an emergency facility to the boundary of its coverage area, pass more distance than the coverage
radius. Therefore, Y is a decreasing coefficient, which reduces the real coverage radius of the
facility. For Tehran network, Y was set equal to 0.8. In cases where the proposed station is located on the border of two or more regions (near the boundaries of regions), the minimum
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speed of travel time between regions was considered as W . Then, the response time of each
facility was determined. Based on national EMS code enacted in 1973, the maximum allowable time for response time in urban areas is 10 min. Therefore, the response time of
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emergency centers equal to 5 min was considered. Once primary emergency services are provided to the injured people, it can take a longer time for ambulances to arrive at the
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hospital. Therefore, the coverage radius of hospitals was assumed to be larger than that of emergency centers. Accordingly, the response time of hospitals (maximum allowable time for an ambulance travel from the accident scene to the hospital) was considered as 10 min. After determining the input values in Eq. (17), the coverage radius of all existing and proposed centers in the network were calculated. For example, if the average travel speed of vehicles during rush hours is 20 kph in a region, the coverage area of emergency centers and hospital could be calculated as follows:
14
ACCEPTED MANUSCRIPT @%\]^_`ab = c1.2 ×
20 e f
10e × g0.7 × j = 2.8 60
e f
@%lmnom#pq = c1.2 × 20
× g0.7 ×
(18)
5e j = 1.4 60
(19)
Table (2) shows the speed and coverage radius of emergency centers and hospitals in
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all regions of Tehran. By determining the coverage radius of each emergency center and hospital, the added coverage to each region could be computed by adding stations. Fig. (5) shows the coverage areas (catchment areas) of the existing facilities.
Table 2: Coverage radius of emergency centers and hospitals in 22 regions of Tehran 1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22
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Region Average speed
24 24 21 27 31 14 16 22 18 18 13 11 19 21 22 21 18 31 32 29 31 39
Hospital radius
3.3 3.3 3.0 3.8 4.3 1.9 2.3 3.0 2.6 2.5 1.9 1.5 2.7 2.9 3.1 2.9 2.5 4.3 4.5 4.1 4.3 5.5
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Emergency center 1.7 1.7 1.5 1.9 2.2 1.0 1.1 1.5 1.3 1.2 0.9 0.8 1.3 1.4 1.5 1.4 1.3 2.2 2.2 2.0 2.2 2.7 radius
Fig. 5: Coverage area of emergency centers and hospitals in Tehran network
To guarantee the model against economic variations, the costs of building facilities and the available budget were expressed in terms of a hypothetical unit. That is, the cost of building an emergency center was considered as one unit and the costs of building a hospital with conventional equipment was considered as three units. The proposed model was 15
ACCEPTED MANUSCRIPT executed based on different values of the budget (B), and the outcomes of each scenario were compared. The model was resolved in a commercial software Cplex 12.6 with a Java interface in a system with these specifications (4 GB RAM, 4-core CPU and 4 MB cache). Table (3) shows the results of each scenario.
Budget scenarios
0
5
10
15
20
25
30
35
40
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Table 3: Model results for 22 districts of Tehran
45
50
55
60
65
70
efficiency of region 1 0.42 0.42 0.42 0.42 0.42 0.42 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 efficiency of region 2 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.41 0.41 0.41 0.41 0.41 0.41 efficiency of region 3 0.46 0.46 0.46 0.46 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50
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efficiency of region 4 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.32 0.32 0.32 0.32 0.32 0.32 0.32 efficiency of region 5 0.28 0.28 0.28 0.28 0.28 0.28 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 efficiency of region 6 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 efficiency of region 7 0.56 0.56 0.60 0.60 0.60 0.60 0.60 0.59 0.59 0.59 0.59 0.59 0.59 0.59 0.59
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efficiency of region 8 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.83 0.84 0.84 0.84 0.84 0.84 0.84 efficiency of region 9 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 efficiency of region 10 0.67 0.67 0.67 0.67 0.67 0.67 0.67 0.65 0.65 0.65 0.65 0.65 0.65 0.65 0.65 efficiency of region 11 0.72 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 efficiency of region 12 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 efficiency of region 13 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 efficiency of region 14 0.52 0.59 0.59 0.59 0.59 0.59 0.59 0.59 0.59 0.59 0.59 0.59 0.59 0.59 0.59 efficiency of region 15 0.22 0.22 0.26 0.33 0.33 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35
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efficiency of region 16 0.52 0.52 0.52 0.52 0.52 0.52 0.52 0.56 0.56 0.56 0.56 0.56 0.56 0.56 0.56 efficiency of region 17 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 efficiency of region 18 0.35 0.35 0.35 0.35 0.35 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42 efficiency of region 19 0.15 0.15 0.15 0.15 0.21 0.21 0.21 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 efficiency of region 20 0.39 0.39 0.39 0.46 0.46 0.46 0.46 0.46 0.46 0.46 0.46 0.46 0.46 0.46 0.46 efficiency of region 21 0.16 0.16 0.30 0.34 0.37 0.37 0.37 0.39 0.39 0.39 0.39 0.39 0.39 0.39 0.39 No emergency centers No hospital stations
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efficiency of region 22 0.43 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 5
10
15
20
25
30
35
40
44
45
46
45
44
44
44
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
10.80 10.17 9.94 9.77 9.63 9.53 9.46 9.39 9.34 9.31 9.31 9.31 9.31 9.31 9.31
Run time
0.05 0.84 2.79 4.55 4.93 7.47 10.87 9.47 8.08 6.71 7.40 6.80 6.96 8.30 8.08
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Objective function
According to Table (3), under the scenario of existing condition (B=0), regions 12, 13 and 17 are efficient in covering emergency crashes. The important point about this scenario is that region 22 was selected as the most suitable candidate for improving the total network efficiency. This is despite the fact that Regions 19, 21 and eight other regions have an efficiency score critically lower than Region 22. This result implies that combining DEA and EFLP in an iterative method does not ensure an efficient solution. Therefore, constructing emergency centers in regions with the lowest efficiency score, which are in a critical 16
ACCEPTED MANUSCRIPT condition, is not always the most efficient strategy. Fig. (6) shows variation in the efficiency of each region for budget values below 50 units. No efficiency change was observed for budget levels higher than 50. 1.6
10
9
17
13
12
6
8
5
2
7
4
16
3
1
14
18
11
20
19
15
21
22
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1.4 1.2
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0.8 0.6 0.4 0.2 0 0
5
10
15
-0.2
20
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Efficiency Level
1
25
30
35
40
45
50
Budget
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Fig. 6. Changes of units and system efficiency at different budget levels
According to Fig. (6), Regions 11, 14, and 22 are the most suitable candidates for increasing the coverage of emergency crashes. Assigning facilities to these regions would require five
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budget units, which decreases the objective function (sum of a units’ inefficiency) by 0.6. The information of Table (3) can be used for scheduling the construction of network
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emergency facilities and prioritizing projects with the goal of increasing system efficiency. Based on Table (3), the construction of a new hospital in Tehran occurs at the budget level of 55 and above with a maximum of one hospital unit. However, this investment does not improve the sum of efficiency. Therefore, budget levels of 45 to 70 will not cause any further improvement in the efficiency of decision units. Table (5) shows the ID number of selected emergency centers under each effective scenario. Each ID number has two parts. The first shows the region ID and the second indicates the center ID in the corresponded region. Also, Fig. A1 in the appendix illustrates the location of emergency centers allocated to each region in Tehran. 17
ACCEPTED MANUSCRIPT Table 4. Region number and number of selected emergency centers for effective scenarios The emergency centers selected (region ID- center ID)
0
-
5
11-1,14-1,22-1,22-2,22-3
10
7-1,11-1,14-1,15-1,20-1,20-2,21-1,21-3,21-6,22-1,22-2,22-3
15
7-1,11-1,14-1,15-1,15-3,15-4,20-1,20-2,21-1,21-3,21-4-21-5-22-1-22-2-22-3
20
3-1,3-2,7-1,11-1,13-1,14-1,15-1,15-3,15-4,19-1,19-2,20-1,20-2,21-1,21-2,21-3, 21-4,215,22-1,22-2,22-3
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3-1,3-2,7-1,11-1,13-1,14-1,15-1,15-2,15-3,15-4,18-1,18-2,18-3,18-4,19-1,19-2, 20-1,202,21-1,21-2,21-3,21-4,21-6,22-1,22-2,22-3
40
45
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1-1,1-2,1-3,1-4,3-1,3-2,5-1,7-1,11-1,13-1,14-1,15-1,15-2,15-3,15-4,17-1,18-1, 18-2,183,18-4,19-2,19-3,20-1,20-2,21-1,21-2,21-3,21-4,21-5,22-1,22-2,22-3 1-1,1-2,1-3,1-4,3-1,3-2,5-1,7-1,11-1,13-1,14-1,15-1,15-2,15-3,15-4,16-1,16-2, 16-3,171,18-1,18-2,18-3,18-4,19-1,19-3,19-4,20-1,20-2,21-1,21-2,21-3,21-4,21-5,21-6,22-1,222,22-3 1-1,1-2,1-3,1-4,3-1,3-2,4-1,4-2,4-3,4-4,5-1,7-1,8-1,11-1,13-1,14-1,15-1,15-2,15-3,15-4,161,16-2,16-3,17-1,18-1,18-2,18-3,18-4,19-1,19-3,19-4,20-1,20-2,21-1, 21-2,21-3,21-4,215,21-6,22-1,22-2,22-3 1-1,1-2,1-3,1-4,2-1,2-2,2-3,3-1,3-2,4-1,4-2,4-3,4-4,5-1,7-1,8-1,8-2,11-1,13-1,14-1,15-1,152,15-3,15-4,16-1,16-2,16-3,17-1,18-1,18-2,18-3,18-4,19-1,19-3,19-4, 20-1,20-2,21-1,212,21-3,21-4,21-5,21-6,22-1,22-2,22-3
5. CONCLUSION
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In this study, by combining DEA and MCLP, the problem of locating new emergency
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facilities in urban scale was addressed. The goal was to increase the efficiency of all urban districts in covering emergency demand. In this study, particular attention was given to emergency service demands for road crashes. The network demand considers not only the
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population of regions but also the fatal, injurious and PDO crashes. The proposed model considers the current condition of emergency facilities in the network and seeks to identify locations for building new emergency facilities. The coverage area of each emergency facility was calculated proportional to the average ambulance speed during rush hours in the road network of that region. In this study, site locations in the network were carried out by adding two types of facilities, emergency centers and hospitals. Also four input variables, including the average fatal, injurious and PDO crashes and the population density of each region as well as two output variables, the coverage contribution of emergency centers and hospitals in each region, were used to calculate the efficiency of regions in the mathematical model. The 18
ACCEPTED MANUSCRIPT share of new stations in the coverage of each region was presented as the decision variable of the model. The initial model was a nonlinear mathematical model, which was then converted to a mixed integer programming model using a linearization technique. We tested the proposed model in of 22 regions of Tehran network. The results suggested that inefficiency of Tehran regions was related to lack of emergency centers facilities. Based on the existing
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facilities, ten regions were in a critically less efficient condition than Region 22, but establishing new emergency centers in Region 22 would provide the largest network coverage.
This paper had some limitations in terms of data collection and statistics. For instance,
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the number of ambulances in the network and their positions in facilities over different time periods were unknown. If we had access to the number of in-service ambulances, we could consider allocation problem along with the location problem and therefore measure the
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efficiency of regions more accurately. In addition, it was possible to propose a multi-period model and consider ambulance deployment decision along other decisions. Future studies can focus on determining the number of ambulances in facilities to maximize the efficiency in the regions. Also, if the model is solved for multi periods, the ambulance deployment decisions could be integrated with location and allocation decisions.
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Moreover, besides locating facilities, it is recommended to determine the level of necessary equipment with respect to the budget constraint.
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Appendix A
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Fig. A1: Proposed emergency centers in each region of Tehran
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The proposed model considers the status and location of the existing emergency facilities in the network and searches for identifying new locations to construct new facilities.
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Presented model is an integrated model of data envelopment analysis and maximum coverage location problem considering both population and accident demands in various regions of a city Applying the model on a real network with 22 regions and two different facilities
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Considering an specific ambulance speed for each region based on the observed average
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traffic flow in the region’s links