a b s t r a c t

Shortening the travel time of patient transfer has clinical implications for many conditions such as cardiac arrest, trauma, stroke and STEMI. As resources are often limited precise calculations are needed. In this paper we con- sider the location problem for both ground and aerial emergency medical services. Given the uncertainty of when patients are in need of prompt medical attention we consider these demand points to be uncertain. We consider various ways in which ground and helicopter ambulances can work together to make the whole process go faster. We develop a Mathematical model that minimizes travel time and maximizes service level. We use a compromising programming method to solve this bi-objective mathematical model. For numerical experiments we apply our model to a case study in Lorestan, Iran, using geographical and population data, and the location of the actual hospital based in the capital of the province. Results show that low-accessibility locations are the main focus of the proposed problem and with mathematical modeling access to a hospital is vastly improved. We also found out that once the budget reaches a certain point which suffices for building certain ambulance bases more investments does not necessarily result in less travel time.

(C) 2017

1. Introduction

In the emergency medicine practice, physicians interact with emer- gency medical services (EMS) repeatedly. To provide the best possible care in prehospital settings, emergency physicians should be up to date on the recent researches which may impact the care of patients be- fore arriving at the hospital. All around the world people’s lives can face all sorts of crises such as natural disasters and unforeseeable diseases. These crises usually need emergency relief because they deal with peo- ple’s lives directly. Managing emergency relief requires relief logistics which is the process of preparation, controlling and supervising the flow of resources to casualties and sick people. EMS systems are specif- ically prearranged systems which provide emergency medical services to patients whether they are injured or sick. The area which EMS pro- vides service to, van be rural, urban or a combination of the two. Type of services is based on the specifications of the region such as population distribution and geographical location. Rural zones sometimes come second to urban areas in case of adequate services [1]. Evaluation of EMS systems is often based the response time and treatment of cardiac arrest patients outside the hospital. More advanced EMS systems which

* Corresponding author.

E-mail addresses: milad.shahriari@ut.ac.ir (M. Shahriari), alibozorgi@ut.ac.ir (A. Bozorgi-Amiri), sh.tavakoli@ut.ac.ir (S. Tavakoli), a.yousefibabadi@ut.ac.ir (A. Yousefi-Babadi).

are based in urban areas have a substantial higher survival rate in cardi- ac arrest patients.

A well-functioning EMS system involves the integration of air and ground EMS resources and has been shown to improve outcomes in a variety of conditions including cardiac arrest, STEMI, stroke, trauma and burns. The design of most ground EMS systems has been centered around the reduction in EMS response time, particularly for cardiac arrest. Urban system design is typically more complex and better resourced. Air medical is an important component of an EMS system that extends the benefits of care to rural, less-populated regions. There have been few sophisticated analyses to determine optimum placement of air and ground resources as well as remote helicopter landing zones. The purpose of the current investigation is use an improved mathe- matical modeling to develop optimal solutions for placing ground and aerial base and helipad locations. This model will then be applied to a representative geographical location to demonstrate the efficiency of the model. There are three possible cases that the combination of ambu- lance and helicopter can transfer the casualties. These cases are further discussed in the following sections. Because of this complex structure, designing an efficient HEMS requires more complex decisions than am-

bulance transfer systems.

This research is an extended of Bozorg-Amiri et al. [2]. An important analysis was performed by Bozorg-Amiri et al. [2]. The main contribu- tions and distinctive points of this paper as adding time window, ambu- lances bases, optimal capacity of ambulance bases, minimax time objective, multiple ambulance requests, bi-objective optimization and

http://dx.doi.org/10.1016/j.ajem.2017.06.026

0735-6757/(C) 2017

satisfaction objective by maximizing the service level. While this pro- vided important insights into planning an integrated EMS system, their model did not consider the following: (1) the impact of ground de- mand on the placement of air medical bases and remote landing zones;

(2) the capacity of ground ambulances; (3) the impact of ground trans- portation times and geography of routes traveled; (4) the capacity of ambulances at each base and (5) the potential need for multiple ambu- lance responses.

The remainder of this paper is organized in such a way that in the next section we review the papers existing in the literature of emergen- cy location. In Section 3 we describe the problem which we study in this paper, and present the mathematical model. Solution approach and nu- merical analysis of the case study are given in Sections 4 and 5. In the end conclusions and future research directions are stated in Section 6.

1. Literature review

There are various areas in medical services which EMS systems can be beneficial. Nable et al. reviewed recent researches that studied emer- gency medicine and the use of EMS systems. Most important areas of these medical practices are cardiac arrest, trauma, sepsis and acute myocardial infarction. Our study is most applicable in using EMS in case of trauma [3]. Corrado et al. studied Transportation modes in which Pediatric trauma patients are transported to hospitals or trauma facilities. Based on their statistics two third of relatively or severely in- jured children in the US were transported by EMS systems and triage experts to reach to a trauma center. Results indicate that there is a great need for EMS transportation systems to be used more and the awareness on this issue should be raised in the public [4]. Study on the location problem of Emergency Medical Services (EMS) was started from 1970 in the literature. These studies drew more attention and began to grow especially from 1990?s [5]. By the look of papers in the lit- erature most of researchers’ focus has been on implementing emergen- cy medical services and ambulance locations. A comprehensive study on this subject has been done by Goldberg et al. [6] and Brotcorne et al. [7]. Aerial emergency medical services have been developed less than ambulance locations.

Schuurman et al. proposed a method which spots the location where a helicopter have to be placed in the EMS system for covering currently underserved regions with more population among their proximity, to analyze their model they applied it in a case where two EMS equipped hospitals switch their helicopters. They used population and transport time data from British Columbia Trauma Registry in a five year span [8]. Branas et al. study the location problem of a trauma facility and ae- rial and medical stations simultaneously [9]. Later they proposed a heu- ristic algorithm to solve the problem and applied it to several areas in the United States [10]. Fulton et al. studied the relocation of medical resources for evacuation which includes aerial and ground medical evacuation. They proposed a stochastic optimization approach. The model locates aerial evacuation sites, hospitalization sites, and routes the evacuation resources should take in order to minimize transport time in different scenarios [11]. Erdemir et al. studied the aerial medical base location problem. They assumed that demands such as car acci- dents occur both on nodes such as intersections and also on paths such as roads. They applied their findings in a study in New Mexico [12]. They also extended their model for service facilities to find loca- tions of aerial bases to maximize the amount of requests that can be attended to [13]. Erdemir et al. considered the problem where the aim is to locate ambulance and helicopters and also transfer points. They proposed covering models and assumed that demands are covered if a helicopter directly covers it, two ambulances cover it, or a combination of helicopter and ambulance does it. They considered a set-covering ob- jective that search for minimizing the cost needed to locate the facilities while all demand points are covered, in another model they considered a maximum-covering objective that search for maximizing the coverage of demands under a certain cost [14]. Furuta and Tanaka proposed a

model for a problem where instead of patients doctors are being transported to patients by helicopter. Their model intends to maximize the survival rate by locating the optimal helicopters and transfer points [15]. Hosseinijou and Bashiri developed the TPLP model with the as- sumption of uncertain demand points following a Uniform distribution [16]. Kalantari et al. also assumed the demand points to be uncertain but developed the model under the assumption that they are fuzzy in nature. They solved the model by a heuristic fuzzy logic controller [17]. Schneeberger et al. proposed a three phased model to locate and re- locate ambulances in a crisis. They implemented experimental compu- tations of their model in Salzburg, Austria [18]. Kim et al. used hypercube modeling to present an iterative optimization algorithm with parameter estimation for the ambulance location problem. To check the validity of the model they compared the results of the math- ematical model with simulation-optimization approach [19]. Nickel et al. used a scenario sampling approach to simulate a mathematical model optimizing location of ambulance bases. They assumed demands to be stochastic such that the number of ambulances requested by an emergency call is uncertain [20]. As it was mentioned in the previous section, Bozorgi-Amiri et al. proposed a model to locate helipads and ae- rial bases in the event of a crisis and need of transportation of casualties. They also considered the demand points to be uncertain. However they failed to include the location of ambulance bases in their model. They also did not consider the capacity of ambulance bases [2]. Jo Roislien et al. explored the optimal location of Air ambulances in Norway using maximal covering location problem to cover the increases population. They found out that by using mathematical modeling fewer bases with minor adjustments to existing bases can improve the coverAge SIg- nificantly [21]. Clark et al. used C programming language to develop a discrete-event Computer simulation in order to determine the optimal location for Trauma system helicopter base. They applied this simulation to Maine, USA which is a rural area with unevenly distributed popula- tion [22]. Summary of the most related papers in the literature are

shown in Table 1.

In the next section the problem is defined in detailed, the nomencla- tures and proposed model are described.

1. Methods

The aim of this paper is to find the locations of ambulance bases, aerial bases and helipads under uncertainty in the order that the total time of transferring casualties from demand point to the hospital is minimized. We also aim to assign ambulances to accidents based on the demand of the emergency call and also based on the capacity of ambulances on hand. Ambulance capacity is referred to the optimal number of ambulances needed in each ambulance base. Due to the un- known exact location of the accident in the real world, in this paper we assume demand points to be in a square shaped area where length and width follow a uniform distribution. The demands are assumed to be stochastic. We assume that by the use of previous data, we know the probability distribution of the number of ambulances associated with an emergency call made at each demand point. We also assume that probability of demands at each demand point is uniformly distributed. We assume that a demand point originates an emergency call in which they may request more than one ambulance. We also consider a time window in our model which means that there is a penalty if a pa- tient transfer takes more than a specific amount of time.

The hub and spoke models are the most relevant models to transfer point location problem (TPLP). The hub location problem (HLP) studies the location of hub facilities and allocation of nodes demands to hubs to find the best route from origins to destinations [12]. Goldman presented the first paper on center location in network which addressed the con- ditions for optimum center nodes in the Hakimi problem [23]. Berman et al. first presented the TPLP. The problem is concerned with locating a transfer point in a situation where a facility already exists and its loca- tion is known [24]. Although TPLP can be considered a in the same

Table 1

Summary of papers in literature and research gap

Author

Year

Ground Base

Aerial Base

EMS

Transfer Point

Uncertainty

Capacity

Objective

Covering

Time

Cost

Satisfaction

Branas et al.

2000

*

*

*

Berman et al.

2004

*

*

Berman et al.

2005

*

*

*

Berman et al.

2007

*

*

*

Berman et al.

2007

*

*

*

Berman and Drezner

2008

*

*

*

Sasaki et al.

2008

*

*

*

Erdemir et al.

2008

*

*

*

Schuurman et al.

2009

*

*

*

Erdemir et al.

2010

*

*

*

*

*

*

Fulton et al.

2010

*

*

*

*

*

*

Hosseinijou and Bashiri

2012

*

*

*

*

Furuta and Tanaka

2013

*

*

*

*

Kalantari et al.

2014

*

*

*

Schneeberger et al.

2014

*

*

*

Kim et al.

2015

*

*

*

Nickel et al.

2016

*

*

*

*

Bozorgi-Amiri

2016

*

*

*

*

*

This Paper

*

*

*

*

*

*

*

*

group as HLP two major differences distinct these categories. First, HLPs are usually considered on network but TPLPs are not bound by this. Berman et al. also studied the problem of multiple locations of transfer points (MLTP) and the facility and TPLP (FTPLP). These models are used to locate multiple facilities. They presented multiple heuristic methods to solve the MLTP and network FTPLP [25]. Berman et al. ex- tended the TPLP and proposed the multiple TPLP (MTPLP), where more than one transfer point is allowed to be established [26]. Sasaki et al. proposed a new approach to model the MLTP by considering a problem like the p-median location problem. By formulation of their mathematical programming by use of a solver exact optimal solutions can be obtained; they also proposed another approach to model FTPLP and a new method to optimize single facility problems [27].

Once our additional assumptions are added we intend to apply the new model to the same region previously studied by Bozorgi-Amiri et al. [2]. Such differences in the mathematical modeling from the previ- ous study by Bozorgi-Amiri et al. [2] is as follows;

Adding a time window for the transfer process and also a penalty if time exceeds the limit,

Adding ambulance bases to the transfer process,

Calculating the optimal capacity of ambulances in each base

Optimizing the bi-objective model with the objectives of minimum time and maximum service level

Considering requests of multiple ambulances due to the magnitude of the medical emergency at the demand point We apply our model to Lorestan in Iran. Lorestan is one of the west-

ern provinces in Iran with coordinates of 33.4871?N 48.3538?E,

28,294 km² area and around 1.7 million population. Due to being in the transportation pathway of Tehran-South and mountainous area, lots of accidents happen in the roads of this province. And therefore it is necessary to deploy a suitable emergency medical system.

Before presenting the mathematical model we discuss the different cases for transportation of casualties to the hospital which is expanded in this paper. First of all in all the cases as an accident happens based on the call to the hospitals and the number of casualties one or more

ambulances come from the ambulance base to the location of accident which we call demand point. The number of ambulances is based on the number of casualties and the capacity of ambulances on hand. The first case would be that the ambulance takes the casualties to the hospi- tal directly (Fig. 1). In the second case casualties are taken to an aerial base and then are brought to the hospital by a helicopter (Fig. 2). The third case is that the casualties are taken to a helipad and a helicopter came from the aerial base brings the casualties to the hospital (Fig. 3).

The decision of which case to be used relies on the total time of transportation. Whichever one is the lowest of all is going to be used for patient transfer. This means the transportation time of the whole process is equal to minimum time of the three cases.

T_D = min(T₁; T₂; T₃) (1)

As discussed above, time of patient transfer from demand points to hospitals rely much on the location of helipads, aerial bases and ambu- lance bases. In this section we present a mixed integer mathematical programming to solve the problem. The location of demand points and hospital are known therefore location of ambulance bases, aerial stations and helipads are obtained in such way that the maximum of transportation time is minimized; the second objective is maximizing the service level. For modeling of the problem the following parameters and decision variables are defined.

Indices and parameters

I: set of demand points, specified by index i.

J: set candidate locations for helipad, specified by index j.

K: set of candidate locations for aerial base, specified by index k.

M: set of candidate locations for ambulance base, specified by index m.

C_t: cost of building a helipad.

C_a: cost of building and equipping an aerial base.

C_b: cost of building an ambulance base.

B: whole budget available.

Fig. 1. Case 1 for delivering patients to hospital.

Fig. 2. Case 2 for delivering patients to hospital.

D_i: weighted demand of demand point i. V: The speed of helicopter.

W: The speed of ambulance.

L: Limit of time window.

P: Penalty of exceeding time window per time unit.

n_i: Number of ambulances requested by demand point i.

with the goal to minimize the sum over the normalized difference be- tween each of the objectives and their corresponding optimum values. For the presented model, we call the objectives obj₁ and obj₂. According to the Lp-metrics method, this model must be solved for both objectives separately. Now we call the optimum values for these two single-

objective problems as obj^* and obj^* , Eq. (40) demonstrates the formula-

1 2

f_m: Fixed setup cost for installing a base at location m.

g_m: Fixed cost for adding an ambulance to ambulance base m.

tion of Lp-metrics objective function.

Decision Variables

obj1-obj^*

obj2-obj^*

X_j: Binary decision variable, equals 1 if a helipad is built in point j;

Min obj₃ = ?.

obj1

1 + (1-?).

2

obj2

(40)

otherwise 0.

Y_k: Binary decision variable, equals 1 if an aerial base is built in point k; otherwise 0.

S_m: Binary decision variable, equals 1 if an ambulance base is built in point m; otherwise 0.

?_i: Binary decision variable, equals 1 if case one happens; otherwise

0.

?_ik: Binary decision variable, equals 1 if case two happens; otherwise

0.

?_ijk: Binary decision variable, equals 1 if case three happens; other- wise 0.

R_m: Integer variable, required ambulances at ambulance base m ? M

(Ambulance Capacity).

q_im = Integer variable, Number of ambulances from base m to allo- cate to an emergency call from demand point i.

T_ijk: auxiliary variable which represents the maximum time of trav- eling the distance of demand point to helipad by ambulance and the time which helicopter reaches to the helipad.

The mathematical model is presented in Appendix 2.

There are different approaches to solve the bi-objective problem. Most famous methods of decision making in a bi-objective problem are the Goal Programming methods, Compromise Programming and the Reference Point Method. The one which is still often used is Com- promise Programming [28-31]. We also try to solve our bi-objective op- timization problem by Compromise Programming. For implementing this method we solve the bi-objective problem concerning each objec- tive functions separately. Then reformulate a single objective model

The decision maker specifies the weight of the influence of each ob- jective which is indicated by 0 <= ? <= 1, also called Lp-metrics coefficient. By the use of this Lp-metrics objective function and taking the model constraints into consideration, we reach at a single-objective, model, that we will solve easily by a suitable solver.

To analyze the results of the model in the most efficient way we apply the model to the Lorestan province as a case study that Bozorgi- Amiri et al. (2016) also used in their research. In Table 2 most populated cities of Lorestan and their population and their population weight are mentioned. Shohadaye Ashayer Hospital is based in Khoram Abad the capital of Lorestan. This hospital has the capability of attending to trau- ma patients. Target people in this study are the ones which have to be transferred to the hospital after the occurrence of an accident in a timely manner and as fast as possible.

All cities have been inscribed to a square based on their space. (Fig. 4) the weight of each city’s population is calculated by dividing the population to the whole population of all cities. From conducted surveys seven points have been chosen for the candidate locations of ambulance and aerial bases. In Aleshtar, Nurabad, Borujerd, Dorud, Aligudarz, Kuhdasht, Pol Dokhtar also seven other points have been chosen for helipad candidate locations. In Firuzabad, Sarab Dowreh, Beyranvande Junubi, Razan, Kuhdasht, Azna, Pol Dokhtar.

1. Results

The model has been run several times with different values for the compromise programming weight of components, by GAMS software

Fig. 3. Case 3 for delivering patients to hospital.

Table 2

Population and weights of population of cities

Table 3

Demonstration of how casualties are being transferred

Row	City	Population	Population Weight		Row	City	Case of	Helipad	Ambulance	Aerial
1	Khoram Abad	354,855	0.393				Transport		Base	Base
2	Boroojerd	245,737	0.272		1	Khoram Abad	1	–	Borujerd	–
3	Dorood	100,977	0.124		2	Borujerd	2	Firuzabad	Borujerd	Dorud
4	Koohdasht	111,736	0.112		3	Dorud	2	Beyranvande Jonubi	Borujerd	Kuhdasht
5	Aligoodarz	89,520	0.099		4	Koohdasht	2	Pol Dokhtar	Borujerd	Kuhdasht
Total		902,825	1		5	Aligudarz	2	Razan	Borujerd	Kuhdasht

v24.1 by GAMS Development Corporation, USA, Washington. Average runtime for solving the model was 1.5 min.

We used compromise programming method to solve the model. The results which are shown in Table 3 are such that ambulance base is opened in Borujerd, with the capacity of seventeen ambulances. The ambulance base which is essential in computing the travel time of the whole patient transfer is one of the items that Bozorgi-Amiri et al. 2016 failed to consider. Helipads are built in Beyranvande Junubi, Razan, Firuzabad and Pol Dokhtar. Aerial Bases are opened in Dorud and Kuhdasht. Table 3 shows how casualties are being transferred, and through which base and helipads.

To analyze the sensitivity of the parameters we ran the model sever- al times with different sets of parameters. Figs. 5-8 demonstrate these analyses. As it is shown in Fig. 5, the optimal service level is at most 90%. As opposed to Bozorgi-Amiri et al. 2016, in our model we aimed to maximize the service level. The emergency problems usually assume a very high service level. But our model is also useful when there is an essential need for emergency medical service but minimizing costs is also very important to the people in charge. Although if a manager feels that the maximum service level is needed he/she could simply change ? in our model to the desired service level and solve the single-objective problem.

Also from Fig. 5, we understand that when the budget spent is higher the service level increases and the maximum travel time decreases. The reason which causes the lines to collide in the chart is that one objective is minimizing the time and the other is maximizing the service level.so when we are increasing or decreasing a parameter, the objectives move in different directions. 50% decrease in budget resulted in the model to have no feasible solution, and the reason is that the bases could not be built. By 50% increase in budget the results do not tend to change so

we omit it in the graph. From this we obtain that once the budget is as much as it is needed to build certain bases then because demands do not tend to change greatly over time more budget does not lead to a bet- ter solution.

Corollary 1. In the ground and aerial ambulance base location problem, where both objectives of service level and cost are being optimized; once the budget reaches to a certain amount where it is enough to build sufficient bases which are able to cover the demand of casualties and have an optimum service level, spending more money does not necessarily result in less travel time.

The managerial insight we obtain from corollary 1 is that when a manager is trying to decide on the number of emergency facilities such as ground and aerial ambulance bases in our model, or the number of ambulances and helicopters to buy, there should be a comprehensive analysis as to what decisions are optimal. And assigning more budgets to the problem does not always lead to a better solution. Fig. 6 illustrates the impact of the exceeding time window penalty on patient Transfer time. It can be seen that the more penalty set for times when transport time crosses the time window, the more positive effect it has on the time to be lowered which is better for the whole EMS system.

Corollary 2. Increasing the penalty of crossing time window leads to less transport time.

Although in our computations we observed that assigning higher than usual amounts for the penalty leads to the system to be unable to perform and it will not be possible to have an optimal system.

Next we consider ambulance capacities of each ambulance base and what effects the penalty has on capacity. For example, when in the

Fig. 4. Demand points and candidate points on the map.

Fig. 5. Pareto optimal and sensitivity analysis of the budget.

assumption where the penalty is set for 5 units ambulance the optimal capacity of ambulances is shown in Table 4.

The effects of penalty of exceeding time window on ambulance capacities in each base are shown in Fig. 7.

It may seem strange that as the penalty increases, the capacity of am- bulances in ambulance bases decrease. There are two reasons for this kind of behavior. One is that the more fine set for ambulances that arrive late to the point of crisis, the less budget will be on hand to spend for buying more ambulances. The other reason is that as it was mentioned when a higher penalty is set, the model performs in such a way that travel time is decreasing. Less travel time equals more free time for ambulances overall. And also when demands are in such a way that a certain amount of ambulances will be able to answer them, more ambulances will not arrive at the demand point, which is a reason why less fine equals more ambulance capacity.

Due to the fact that our model is a Np-hard Problem, the solve runtime is exponential relative to the number of nodes. As it is present- ed in Fig. 8, as the number of demand points increases it takes much lon- ger for the solver to reach to the optimal solution. Our case study is relatively small in size therefore it does not take much time to be solved. Although for bigger areas with demand points, the model should be solved by exact methods such as decompositions or non-exact methods such as heuristic and meta-heuristic algorithms.

1. Discussion

As it was outlined in Literature review section, there are deficiencies in previous studies of aerial emergency medical services. Few of these studies considered the satisfaction in terms of service in their model or simulation. Also ambulance capacity was rarely discussed before. We considered these aspects among other novel approaches in HEMS such as limited time windows and multiple ambulance requests. In

Fig. 6. Sensitivity analysis of the penalty of exceeding time window on transport time.

Fig. 7. Sensitivity analysis of the penalty of exceeding time window on required ambulance capacity.

this paper we extended the model previously studied by Bozorgi- Amiri et al. 2016 in several ways. In this section we compare the results and offer managerial insights gained from our research.

One the weaknesses of Bozorgi-Amiri et al. 2016 were that they as- sumed that an ambulance is present at the place of crisis which is rarely the case and causes the travel time not to be accurate. Even in low- traffic areas it takes a while for the ambulance to reach its destination. To overcome this matter we assumed that there are ambulance bases that can respond to the demands of patient transfer. We also considered the capacity of these bases, which allows the manager of such emergen- cy transportation system to control their budget more wisely.

Another extension to their model in our work was setting a limit for the time window in which the whole system should have completed the transfer of casualties to the hospital. In the category of emergency medical transportation time is of the essence therefore we have set a time window for the transfer and if transfer time exceeds the limit, the manager is obligated to pay a fine. By setting this penalty we try to minimize circumstances in which transfer time crosses the time window.

Bozorgi-Amiri et al. 2016 considered the minimum objective for time optimization. They minimized the total time of the transportation in all of the nodes. We on the other hand minimize the maximum travel time. By doing so we guarantee that the maximum time spent for transporting patients is minimized. In an emergency medical situation, it is more suitable to have a backup system. By minimizing the maxi- mum time, we allow the optimal location of the ambulance and aerial bases to be as close as possible to the demand points. It is beneficial if the system faces disruption and is in need of a backup.

The managerial insight we obtain from Corollary 1 is that when a manager is trying to decide on the number of emergency facilities such as ground and aerial ambulance bases in our model, or the number of ambulances and helicopters to buy, there should be a comprehensive

Fig. 8. Exponential growth of the solving runtime by increasing the number of demand points.

Table 4

d2 = q(ffiffihffiffiffixffiffi-ffiffiffiffiffirffiffixffiffiffi2ffiffiffiffiffiffiffiffiffi ffiffihffiffiyffiffi-ffiffiffiffiffirffiffiyffiffi ffiffi2ffiffi

(3)

Ambulance capacity of ambulance bases in a 500 $ penalty setting ) +

Ambulance Base

Aligoudarz Borujerd Aleshtar Nurabad

Ambulance Capacity 10 4 2 1

d3 = q(ffiffisffiffixffiffi-ffiffiffiffiffirffiffixffiffi)ffiffi2ffiffiffi+ffiffiffiffiffi ffiffisffiffiyffiffi-ffiffiffiffiffirffiffiyffiffi ffiffi2ffiffi

(4)

analysis as to what decisions are optimal. And assigning more budget to the problem does not always lead to a better solution.

Corollary 2 suggests that from a managerial standpoint the optimal penalty should be computed in different cases and then it should be set in a way that the transport time would become lower while the whole system has an acceptable performance.

b–a

In this problem we assume that each demand point has the coordi- nate P_i = (A_i,B_i). A_i and B_i are independent random variables with a uni- form distribution in interval of [a,b]. In other words demand points are a square shaped area.

A; B ? uniform [a; b] (5)

The model studied in this research can be applied to other geograph- ical regions which do not already have a reliable HEMS. There is one ex- ception which our model may not actually help and that is if there are

f A(A) = 1

1

; f B (B) = b–a

(6)

reliable efficient roads and population of the region is low enough that the traffic of the ground does not interfere with ground ambulances to deliver patients in time. Other than that of we are faced with mountain- ous areas and crowded cities our model can guarantee the optimal med- ical services in a timely manner with the least budget.

Information needed for applying our model to other regions requires the knowledge of an expert in the region who can specify some basic pa- rameters needed for implementation. Parameters such as helicopter and ambulance speed is based on the type of vehicle best suited for transporting patients in the region. Penalty of the delayed patient deliv- ery is also something that requires insight from an expert in the region, as to how much penalty will have an impact on the medical service sat- isfaction based on aspects like people’s culture.

1. Conclusions and future research

This paper presented the bi-objective location model for helicopter emergency medical systems. While helicopters are much faster than ground ambulances, they require landing and takeoff areas, making transportation decisions complex. We designed a system that locates ambulance and aerial bases and helipads to take the casualties from the point where disasters e.g. car accidents to a hospital. There are 3

P = (A; B)?[a x b] x [a x b] (7)

By the above assumptions we can say that the demand points are ac- tually square shaped areas and the probability of an accident happening is constant all over the area. The distance from demand points to hospi- tal d_i (h_x,h_y), demand points to aerial bases d_i (s_x,s_y) demand points to ambulance base d_i (a_x,a_y) and demand points to helipad d_i (r_x,r_y) is cal- culated by Eqs. (8)-(11).

d_i h_x; h_y = |A–h_x| + B–h_y (8)

d_i s_x; s_y = |A–s_x| + B–s_y (9)

d_i r_x; r_y = |A–r_x| + B–r_y (10)

d_i a_x; a_y = | A–a_x| + B–a_y (11)

Mathematical Expectation of distance between demand points and hospital is as Eq. (12).

+?

E D E A–h B–h u–h f a dA

cases that a patient can be transferred to the hospital. We minimize maximum of the total time of whichever case that happens to be the

( ) = |

_x| +

b

= Z

_y = |

-?

Z

_x| A( )

minimum. We also maximize the service level in which a number of ambulances are assigned to the emergency call based on the capacity of ambulances in the ambulance bases. The demands are assumed to be stochastic and following a uniform distribution as well as the de-

1

b–a

a

| u–h_x|dx (12)

mand points. We applied the model to a case study in the province of Lorestan. The locations for building helipads, ambulance and aerial bases and capacity of ambulances are obtained. And the implementa-

tion of HEMS improves the objectives than when there is no system

Moreover we have Eqs. (13) and (14).

+? b

Z Z

f (h_x) = E[| A–h_x|] = | A–h_x| f (a)dA = 1 | A–h_x|dx (13)

applied. We also found out that spending more money does not neces- sarily improve the time which casualties are being transported.

Future research directions can be the following, considering disrup-

-?

8> –h + a + b

A

h <= a

b–a

a

tions in patient transfer, using scenario-based modeling to solve the x x

2

>

problem, considering a hard time window which means minimizing f h < (h_x–a ² h_x–b ² <= <= 14

( _x) =

)

+ ( )

a

h_x

b

(

)

the fatality based on when time exceeds the limit of time window.

Appendix 1

>>:

2(b–a)

hx– a + b

2

h_x >= b

The distance between aerial base and the hospital is d₁, the distance between helipad and the hospital is d₂ and the distance between aerial base and helipad is d₃.

f(h_y), f(r_x), f(r_y), f(s_x), f(s_y), f(a_x), f(a_y) are also obtained the same way.

In this model the distances traveled by ambulance is assumed to be rectilinear. On the other hand distances traveled by helicopter are as- sumed to be Euclidean because the helicopter can move in a direct

line unlike the ambulance which most likely has to move in streets.

d1 = q(ffiffihffiffiffixffiffi-ffiffiffiffiffisffiffixffiffiffi2ffiffiffiffiffiffiffiffi ffiffiffihffiffiyffiffi-ffiffiffiffiffisffiffiyffiffi ffiffi2ffiffi

)

+

(2)

Coordinates of the hospital is represented by (h_x,h_y), ambulance base

by (a_x,a_y), aerial base by (s_x,s_y) and helipads by (r_x,r_y).

Appendix 2

s.t.

>8> f ax(i) + f ay(i) + f hx(i) + f hy(i) ? 9>

w

w

i

> >

5

f hx(i) + f hy(i)

+6

x(i)

x(k)

y(i)

y(k)

Max ? (15)

>>=

• 2

r ffiffiffihffiffiffiffiffiffiffiffiffi-ffiffiffiffisffiffiffiffiffiffiffiffi ffiffiffi2ffiffiffiffi+ffiffiffiffi ffiffiffihffiffiffiffiffiffiffiffiffi-ffiffiffiffisffiffiffiffiffiffiffiffiffi ffiffi2ffiffi3 >

8> f a

<>

+

7?

+ f a

f h + f h

9> z = ? ? ? 4 w

v ik

x(i)

>

y(i) +

w

x(i)

w

y(i) ?i

i?I j? J k?K

• > 2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 >

• 2

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffi2ffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffi2ffiffiffi3 >

• h –s

2 + h –s 2 >

• 6 f h_{x i}

+ f h_yi

hx(i) –sx(k)

+ hy(i) –sy(k)

>

4

7

w

>

x(i)

• 6

x(k)

y(i)

y(k)

7 >

5

Min ? ? ? D_i

>< +64 ( )

( ) +

v 75?ik >=

+ T_ijk + v

>:

?ijk

>>;

i?I j? J k?K

• 2

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffi3 >

s.t

>>>

+64T_ijk +

:

hx(i)

–sx(k)

2

+ hy(i) v

>

–sy(k)

2

75?ijk

>>

;

(16)

(35)

?>= D_iz ?i?I (36)

(17) to (32)

First objective function (33) maximizes the service level. Second ob- jective function (34) minimizes the maximum demand weighted trans-

?_ik <= yk ?i?I; k?K (17)

?_ijk <= x_j ?i?I; k?K; j? J (18)

?_ijk <= yk ?i?I; k?K; j? J (19)

?_i + ? ?_ik + ? ? ?_ijk = 1 ?i?I (20)

portation time. Constraint (35) is the total transportation time to the hospital, Constraint (36) is for minimizing the maximum time. Constraint (17) states that demand i can only be assigned to case 2 at ae- rial base k if a base is located at point k. Constraints (18) and (19) com- bined, state that demand can be met by case 3 if a helipad and an aerial base are located at points j and k respectively. Constraint (20) states that demand should be covered only by one of the three cases. Constraint

k?K

j? J k?K

(21) represents the constraint of budget for building helipads, aerial and ambulance bases and buying ambulances. In Constraints (22) and

? C_tx_j + ? C_syk + ? ( f ms_m + gmR_m) + PE <= B (21)

(23) the maximum time of arrival for ambulance and helicopter to heli-

j? J

k?K

m?M

pad is obtained for partly linearizing z and reducing computation time.

Constraints (24) ensure in each base, allocation of ambulances will not

f rx(i) + f ry(i) <= T

w

ijk

?i?I; k?K; j? J

(22)

exceed the available ambulance capacity. Constraints (25) does not let more ambulances to be allocated than the amount demanded. The ser- vice level for coverage is ensured by Constraint (26). By Constraint (27)

it is mandated that ambulances can only be located in a base if, in fact,

r ffiffiffirffiffiffiffiffiffiffiffi-ffiffiffiffisffiffiffiffiffiffiffiffiffi ffiffiffi2ffiffiffi+ffiffiffiffiffi ffiffirffiffiffiffiffiffiffiffi-ffiffiffiffiffisffiffiffiffiffiffiffiffi ffiffiffi2ffiffi

x(j) x(k) y(j) y(k)

v

<= Tijk

?i?I; k?K; j? J (23)

the base is opened. Constraint (38) computes the amount of time which exceeds the limit of time window.

References

? qim <= R_m ?m?M (24)

i?I

? qim <= n_i ?i?I (25)

m?M

? ? qim >= ? ? n_i (26)

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