A Bi-Objective Location-Allocation Problem of Temporary Emergency Stations and Ambulance Routing in a Disaster Situation Reza Tavakkoli-Moghaddam School of Industrial Engineering University of Tehran Tehran, Iran LCFC, Arts et Métiers ParisTech Metz, France
[email protected]
Pedram Memari School of Industrial Engineering University of Tehran Tehran, Iran
[email protected]
Abstract—The aim of this study is to introduce a new temporary emergency stations distribution for emergency medical services (EMSs) simultaneously in order to find the optimal ambulance routing in a disaster response. Planning and preparedness are a set of activities in pre-disaster situations. When a disaster may not occur for a long time in a specific city, we cannot predict the time of its occurrence. However, in disaster situations, an EMS is faced with a large number of injured people, most of them with critical injuries, in which normal facilities and planning cannot respond. Last disasters teach us that we should be ready to respond correctly to decrease morbidity and mortality. This study seeks a better plan for patients in a post-disaster by categorizing them into very high emergency and normal patients. The main aim of the presented model is to find the best ambulance routing by minimizing the total response time, meanwhile finding the optimal number of a temporary emergency station and the best location setup by taking costs into consideration. The model is exactly solved by the ε-constraint method. Then, we can propose the optimal planning for the decision makers. Keywords—location-allocation, emergency medical service, ambulance routing, disaster response.
Ehsan Talebi School of Industrial Engineering University of Tehran Tehran, Iran
[email protected]
decision making at shortest possible time. In other words, response is to allocate the optimal number, location of temporary emergency stations and ambulance routings. Accident level
Collect patient data(symptoms and their locations)
Clustering patient according to their injured priority Finding the optimum number of temporary relief stations, type of vehicle to assign and allocating it to potential location simultaneously rout ambulance Sending the ambulances and Construct temporary stations Accident Level Occurrance Time Mitigation
Preparedness
Response
Recovery
I. INTRODUCTION
Fig. 1. Disaster phases
Recent researches in this subject includes: (1) locating of facilities, (2) relocation of ambulances, (3) dispatching and routing policies, (4) ambulance routing problems, and (5) cooperation with other emergency health care delivery systems. The ambulance routing is divided into two areas, namely static and dynamic routing problems[1]. Dynamic routing models can manage and update new coming patients by reassigning the routs, and static routing models cannot update itself for new incoming patients. As this study is done in pre-disaster and plans for post-disaster situation facing a large number of patients at the same moment, this model is categorized as the static one [2]. Generally, disaster management is divided into four phases: (1) mitigation, (2) preparedness, (3) response, and (4) recovery as shown in Fig. 1. This study covers preparedness and response phases [3]. Preparedness is to make the necessary equipment, such as first aids, tents, and ambulance ready to be deployed and in proper positions. Response is the process of most logical
A. Disaster Disaster is a situation, which suddenly happens and causes a huge damage in a short period of time. Infrastructures are damaged and a large number of suffered people are in need of medical services and shelter, which are by far more than the capacity of an emergency medical service (EMS). Every year, a disaster takes lives of thousands of people, brings economic problems, destruction, and ecological disruption. The time and magnitude of a disaster cannot be anticipated; however, we can be prepared for. Preparing for these situations is vital to save people’s lives and reduce the rate of mortality and morbidity. A disaster is divided in two sections, namely natural disaster (e.g., earthquake, floods, hurricanes, volcanic eruptions and tsunamis) [4], and man-made (e.g., war, terrorist attack and human error in hazardous jobs) [5]. In this study, a disaster is dealt with its
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general meaning, in which the EMS faces a large number of injuries, blocked routes, destructed infrastructure, and hospitals. On the other hand, demand for help is more than a normal system capacity. Therefore, setting up temporary emergency stations is vital. B. Recognition severity and priority of injuries High emergency patients, who are close to death and have a little time for treatment, should be given priority for reducing the morbidity and mortality. In the medical research, scientists have developed an intelligent algorithm, called triage systems, for assessment of a patient by their symptoms [6] [7] [8]. In the first section of this study patients are categorized in two types: Red-code-patient: patients with serious injuries that if EMS does not respond immediately, their lives are in danger and needs to be taken to emergency station or hospital. Green-code-patient: patients with slight injuries. Dissatisfaction is the results if they do not get medical treatment. These patients are usually get standing treatment. C. Ambulance routing Each EMS situation needs to respond quickly. To find best route is seriously important to reduce a response time. Some studies by simulating from a patient receive call until treatment in hospitals [9]. Routing can describe in each phase of a disaster. Sheu [10] analyzed relief demands in a post-disaster and used a resilience survival concept. Chang et al. [11] considered a dynamic traffic assignment. Some research focused approached to a reliable emergency response [12]. Rath and Gutjahr [13] proposed a model of a location-routing problem in a disaster relief. This study simultaneously finding the optimal quantity of ambulances, ambulance routing and priority of dispatching. D. Location and allocation of ambulances and temporary emergency stations In a disaster, hospitals may block and we have a resource constraint. A seek EMS should have ambulances, first aid and tens stock for response [14]. Leknes et al. [15] extended the location of heterogeneous ambulances. Vahdani et al. [16] solved a model of location-routing problem for distributing the relief after disaster. Rezaei-Malek et al. [17] considered a tradeoff in reliable pre-positioning in disaster management. This study divides locations to areas and find out the quantity and their best areas for implementation of temporary emergency stations. II. AMBULANCE ROUTING PROBLEM The purpose of this study of routing for the ambulance fleet in terms of natural disasters is to help and relief patients while minimizing the time service of patient and cost of implementation of the temporary emergency stations. In this paper, patients are categorized in two types, namely red and green code patients. Red code patients should be transported by an ambulance to hospital h; however, green code patients are at
the scene treated patients and are not taken to the hospital. The set of all patients show P (where P = R G). The sets, parameters and decision variables are as follows: Sets: R G P H K J I
Set of red code patients Set of green code patients Set of all patients (P = R G) Set of hospitals Set of all ambulances Set of all patients and hospitals Set of green code patients and hospitals
Parameters: Binary parameter 1 if ambulance k is initially 𝑓 located at hospital h Travel time from i to p 𝑡 Travel time from r to h with ambulance k 𝑡 Service time of patient j 𝑑 Capacity of hospital h 𝑐 Transfer time to drop off a red code patient at 𝑑 hospital h Priority given to red code patients 𝑤 Priority given to green code patients 𝑤 The cost establish hospital in the region h 𝑠 Decision variables: Binary, 1 if ambulance k serves patient j directly 𝑥 before patient m visiting time of patient j 𝑏 Latest service completion time among all red code 𝐸 patients Latest service completion time among all green 𝐸 code patients Binary if the hospital be established in the region h 𝑢 𝑐 shows the hospital capacity. The total capacity of hospitals h should be more than of red code patients (∑ 𝑐 ≥ 𝑅 ). Also, it is assumed that any ambulance can only transfer a red code patient to the hospital h and every ambulance after serving patients should come back to temporary emergency stations. The goal of this study is to locate and allocate temporary emergency stations by minimizing the total response time of patients. The setup cost of temporary emergency stations are more than serving patients. The mathematical model for this problem is as follows: A. Bi-objective Mathematical Model Min 𝑍 = 𝑤 . 𝐸 + 𝑤 . 𝐸
(1)
Min 𝑍 =
(2)
𝑠 .𝑢
s.t. .𝑢 ≤ 𝑓
∑ ∑ 𝑥 𝑥 ∑ 𝑥
.𝑢 = 1
.𝑢 + 𝑥
.𝑢 ≤ 1
.𝑢 = ∑ 𝑥
.𝑢
∀𝑟
(19)
≤𝑐
∀ℎ
(20)
∀ 𝑖, 𝑝
(21)
∀ 𝑟, 𝑝, ℎ, 𝑘
(22)
∀𝑔
(23)
∀ 𝑟, ℎ
(24)
∀𝑘
(3)
∑ ∑ 𝑦
∀𝑝
(4)
𝑏 +𝑑 +𝑡
∀ ℎ, 𝑔, 𝑘
(5)
∀ 𝑝, 𝑘
(6)
∑ ∑ 𝑥
.𝑢 = 1
∀𝑟
(7)
∑ ∑ 𝑥
.𝑢 ≤ 𝑐
∀ℎ
(8)
∀ 𝑖, 𝑝
(9)
∀ 𝑟, 𝑝, ℎ, 𝑘
(10)
𝐸𝑔 ≥ 𝑏 + 𝑑
∀𝑔
(11)
𝐸𝑟 ≥ 𝑏 + 𝑑 + 𝑑 + (∑ 𝑢 . 𝑥 .𝑡 )
∀ 𝑟, ℎ
(12)
𝑏 +𝑑 +𝑡 ≤ 𝑏 + 1− ∑ 𝑥 .𝑢 .𝑀 𝑏 +𝑑 +𝑡 +𝑑 +𝑡 ≤𝑏 + 2 − 𝑥 .𝑢 + 𝑥 .𝑢 .𝑀
=1
The objective function (1) minimizes the service completion time among all red code and green code patients. The objective function (2) minimizes the setup cost of temporary emergency stations. Constraint (3) guarantees that each ambulance that initially in temporary emergency stations, can be served to the patients. Constraints (4) and (5) guarantee each patient is visited exactly once by one of the ambulances. Constraints (6) guarantee that each ambulance after visit the patient, at the end goes to one of the temporary emergency stations. In Constraints (7), the red code patient must go to temporary emergency stations. In Constraints (8), the number of patient cannot exceed the capacity of temporary emergency stations. Constraints (9) and (10) propagate the arrival times of ambulances at the patient locations. Constraints (11) and (12) determine the latest service completion time red code patients and green code patients. B. Linearization of the Model Due to the multiplication of two binary variables (i.e.,𝑥 and 𝑢 ) in some of the constraints of the model are nonlinear. For converting model to linear, the following constraints should be added to the model. Constraints (25) - (27) are used for the linearization of model. The linear model is presented below:
.𝑀
𝐸𝑔 ≥ 𝑏 + 𝑑 𝐸𝑟 ≥ 𝑏 + 𝑑 + 𝑑 + (∑ 𝑦
.𝑡
)
C. Computational Results This study considers the Tehran city for assessing and planning a disaster situation. Tehran is divided to 22 districts. As a pre-disaster situation, a random generation is used for each code patient that assumed three red-patients and seven greenpatients. There are two objective functions in this model, the first objective proposed is to minimize the total response time to green-patients and red-patients. The second objective function is to minimize the setup cost of temporary emergency stations. The model is measured and analyzed based on collecting the patient requests for aid. Also, locating and allocating the optimum quantity of temporary emergency stations. Each temporary emergency station has two ambulances and should find the best ambulance routing. As example, the results of the sixth optimal solutions are shown in Fig. 2. The total response time is 38.4 minutes and the cost is 400. Table I shows the allocation of temporary emergency stations. One of ambulances is in used. Fig. 3 shows ambulance routes by considering the best location of temporary emergency stations. For example, a redpatient by number 8 is served by an ambulance in District 4 and then comes back to the temporary emergency station. 1000 800 600 400 200 0 0
20
40
60
80
Total response time
(13)
Min 𝑍 = 𝑤 . 𝐸 + 𝑤 . 𝐸
≤𝑏 + 1−∑ 𝑦
𝑏 +𝑑 +𝑡 +𝑑 +𝑡 ≤𝑏 + 2−𝑦 +𝑦 .𝑀
Total cost
∑ ∑ 𝑥
∑ ∑ 𝑦
Fig. 2. Optimal solutions for the decision makers
Min 𝑍 =
𝑠 .𝑢
(14) TABLE I.
s.t. ∑ ∑ 𝑦
≤𝑓
∑ ∑ 𝑦 𝑦 ∑ 𝑦
=1
+𝑦
≤1
=∑ 𝑦
∀𝑘
(15)
∀𝑝
(16)
∀ ℎ, 𝑔, 𝑘
(17)
∀ p, k
(18)
TEMPORARY EMERGENCY STATION IN THE SIXTH SOLUTION
Station
Allocation area
1
4
2
6
3
7
4
13
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9
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N N
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7 5
2
[6] 6
4
Red-code-patient rout
Red-patient
Green-code-patient rout
Green-patient
Temporary emergency station
Fig. 3. Location-allocation temporary emergency stations and ambulance routes for the sixth solution in the Tehran city
III. CONCLUSION This study developed a bi-objective model for locating temporary emergency stations and ambulances routes in disasters. In addition, it determined the optimal quantity, setup costs and minimized the total response time to patients. This model was in a pre-disaster planning section and was prepared for reducing the morbidity and mortality in disaster and postdisaster phases. This paper demonstrated the usefulness of the developed model on disaster situations. A case study was considered that might happen in an old city. The most predictable disaster for this city was earthquake. The old and densely populated areas of this city were prone to a catastrophe. The computational results in small-sized problems demonstrated the quantity and location of temporary emergency stations and ambulance routing preparation planning. The model was validated by the ε-constraint method. Furthermore, this paper considered different scenarios to develop an applicable plan for more realistic situations. For future research, we may consider the mortality rate as an objective. Additionally, we can consider the time windows for survival time of high-risk injuries patients and solve this model by a meta-heuristic algorithm in order to find out the best solution for a quick response. REFERENCES [1]
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