Double Coverage Ambulance Location Modeling using Fuzzy Traveling Time B. Lahijanian, M.H. Fazel Zarandi, F. Vasheghani Farahani Department of Industrial Engineering and Management Systems Amirkabir University of Technology Tehran, Iran
[email protected],
[email protected],
[email protected]
Abstract— In this paper, one of the novel issues of the world regarding the location of ambulance stations within a given area to cover the maximum amount of demand is studied. In this study, the classic version of location problem is improved using the double coverage models so that two radii are considered for covering. Furthermore, the developed study contains the meaningful factors indicating the demand for each patient location covered by each station (vehicle location). In the proposed model, the uncertainty existed in the travel time between the patient locations and vehicle locations have been considered as triangular fuzzy numbers. To solve the proposed model, the goal programming approach is applied in the GAMS software and desired outputs have been achieved. The obtained results represent a significant improvement compared to the past models with uncertainty. Keywords— Double coverage model; Ambulance location problem; Fuzzy Traveling time; Goal Programming;
I. INTRODUCTION Facility location plays an important role in logistic decisions. Each day, many enterprises resort to quantitative methods to estimate the best or the more economical way to meet clients' demand for goods or services [1]. In the last few years, the health care industry has become one of the largest branches of the economy of developed countries. A more extensive list of applications can be found for facility location models. An important application is the location of emergency facilities in city [2]. The action of the ambulances may be placed in two categories: emergency assistance and programmed transportation. The latter, perfectly plannable, consists of transporting certain types of ill people who need periodic treatment in health centers, and who cannot make the journey on their own account; however, the former, due to its random character and the need to respond to the demands in less than the maximum time so as not to risk the life of the patient, complicates planning and conditions the design of the system. Ambulance services are attractive to model for a number of reasons. First, the start and end of each task is well defined, tasks are self-contained and take place over short periods of time. Moreover, the processes lend themselves to a logistic analysis well-suited to modelling. This is not the case for all 978-1-5090-4492-4/16/$31.00 ©2016 IEEE
medical services, many of which are open ended, run concurrently with other medical interventions and stretch over long periods of time. Second, as we shall see, although there are many dimensions of uncertainty to modelling an ambulance service, these dimensions can be partitioned with relative ease and changed when moving from one ambulance service to another [3]. Ambulance deployment is a challenging problem encountered in emergency medical services (EMS) [4]. The aim of ambulance location models is to provide adequate coverage. This can be interpreted and measured in many ways. Recently, another class of facilitylocation problems with uncertain parameters has been developed based on uncertain environment such as fuzzy set theory, Stochastic and possibility theory which aims at dealing with cases of imprecise or vague data [5]. In this paper, we focus on the facility-location problems with respect to uncertain environment. The rest of the paper is organized as follows: Section II provides some reviews on previous research. The problem is described in Section III. Section IV discusses the computational experiments comprising data description, solution procedure and results. Finally, conclusion and future work will be presented in section V. II. BACKGROUND In the past few decades, the Emergency Medical Services (EMS) systems have drawn a great deal of attention from researchers. Traditionally, emergency facility location problems deal with decisions from two aspects: which sites should be selected as depots for facilities and how many facilities should be placed in each depot, given demand points and potential facility sites. Plenty of models have been developed to solve facility location problems. Most of these models simplify the facility location problems by treating emergency calls generated from discrete demand points. These models can be divided into three broad groups: (1) covering models, which emphasize providing coverage for emergency calls within a predefined distance standard; (2) p-median models, which minimize the total or average service distance for all demand points; and (3) p-center models, which aim to minimize the
maximum service distance for all demand points. Covering models are concern with covering demands, and in most covering models, demand is said to be covered when it can be reached within a predefined distance standard by at least one facility [6]. According to the literature, the location set covering problem (LSCP) was one of the earliest models introduced for seeking a minimum number of ambulances to cover all demands. The strategic location sites that provide full coverage with minimum ambulances can be identified from a given set of potential ambulance location sites. In this model, the number of ambulances is unlimited, and a demand node is assumed to be covered if it can be reached within a time threshold. LSCM can be used to determine the right number of ambulances and strategic location sites to cover all demands [7]. The later model was MCLP. The Maximal Covering Location Problem (MCLP) is a facility location problem which aims to select some location candidates to install facilities, in order to maximize the total demand of clients that are located within a covering distance from an existing facility [8]. Recently, another class of facility location problems with uncertain parameters has been developed based on fuzzy set theory, stochastic and possibility theory, which aims at dealing with cases of imprecise or vague data. Laporte et al. formulated a class of capacitated facility-location problems with random demands by using stochastic integer linear programming and proposed a branch and cut-solution approach [9]. Schutz et al. considered a stochastic facilitylocation problem with general long-run costs and convex short-run costs and solved the problem through a Lagrange relaxation-based method [10]. Bhattacharya et al. considered facilities located under multiple fuzzy criteria and proposed a fuzzy-goal programming to cope with the problem [11]. Ishii et al. developed a location model by considering the satisfaction degree expressed with respect to the distance from the facility to each customer [12]. Both LSCP and MCLP suffer from one main disadvantage though: If a vehicle is sent away from its location and a new emergency appears, no vehicle remains to provide coverage. The DSM introduced by Gendreau and solved by using Tabu search [13]. The DSM seeks to maximize demand covered by at least two vehicles (double coverage), and it implicitly recognizes that vehicles could become unavailable, to avoid the risk of under coverage. Doerner et al.extend the model to a capacitated version, in which a single vehicle can cover a specific amount of demand per period [14].
minimum facilities that can cover all demand locations. In other words, the fundamental requirement of this problem is that all demand nodes must be covered at least once. While in a maximal covering location problem (MCLP), the aim is to maximize the total demand covered with the determined number of facilities and stations. In the problems such as emergency services and fire station services which are very sensitive, allocation it's better to be in such a way that if a vehicle is sent away from its location to the far place and a new emergency situation appears, a vehicle remains in order to provide coverage. Therefore, one possible solution is to provide multiple coverage or double coverage approach as in the Double Standard Model (DSM) introduced by Gendreau et al. [13]. In this paper, the double coverage problem is considered by two coverage radii which show the necessity and preferences, respectively. Moreover, the question of how much of the demand for each location is covered by each vehicle location is being considered by taking into account more restrictions. Besides, the uncertainty of the travel time between the patient location and the vehicle locations is solved by fuzzy logic. An overview of the double coverage problem is demonstrated in the Fig. 1. In this figure, is the small radius which indicates the preference and is the large radius which denotes the necessity. The demand points are in the shape of circles and the vehicle locations are shown through triangles.
Fig. 1. A view of double coverage problem
In this article, the assumptions intended to cover the issue of ambulance double coverage location problem are as follows: 1. 2. 3. 4.
III. PROBLEM DESCRIPTION AND DCLP WITH FUZZY TRAVEL TIME: As it has been discussed in the literature, the covering problem could be divided into two separate categories (set covering and maximal covering)[15]. In a location set covering problem (LSCP), the objective is to find the
5. 6.
Objective Function: Maximizing the demand covered by at least two vehicles within the small radius. Each patient location must be covered within the large radius which indicates the necessity. Coverage within the small radius would not be possible until the patient location is not covered within the large radius. Each of the patient location has its own demand, therefore, the amount of demand in each place is different. The capacity of each vehicles (ambulances) that are at the station is determined. Travel time: Since the travel time between the patient location and the vehicle location is a parameter
depending on many factors such as speed of the vehicle or the person who is driving, it's possible not to have a definite or fixed value. Hence, in this article, the travel time between the patient location and the vehicle location are triangular fuzzy numbers. Also, all operations such as required comparisons are made based on the fuzzy operation. The general formula for creating the triangular fuzzy numbers and a specific sample produced for this case can be seen in the Fig. 2. x−a b−a μ x = x−c b−c 0
: The largest number of vehicles at the vehicle location : The capacity of each vehicle when it is placed in vehicle location . =
1 ≤ 0 .
Covered within the large radius.
=
≤ 1 0 .
Covered within the small radius.
if a ≤ x ≤ b if b ≤ x ≤ c
(1)
Otherwise
Variables: : The number of vehicles located at the vehicle location . = 1 If the place is covered by one vehicle through the small radius. = 1 If the place is covered by two or more vehicles through the small radius. : Current demand at the patient location which is covered by one vehicle at the vehicle location .
=
(2)
Subject to: ≥1
∀ ∈
(3)
Fig. 2. Travel time membership function
≥
In this part, the double coverage problem for ambulances location is discussed. The aim is to maximize the demand covered by at least two ambulances with a small radius. The patient location set is represented by and the potential ambulance stations set is displayed by . Parameters:
(4)
≥
+
≤
∈ : Patient location
∀ ∈
∀ ∈
(5) (6)
=
(7)
∈ : Potential vehicle location : Small radius
≤
: Large radius
∀ ∈ ≤
(8) ∀ ∈
(9)
: Total available vehicles =
: Travel time from node to : Percentage of total demand which must be covered by an ambulance located within radius. : Total demand at patient location
∀ ∈
(10)
≥0
∀ ∈
(11)
≥0
∀ ∈ ,∀ ∈
(12)
0,1 ∀ ∈ ,
∈ 1,2
(13)
∀ ∈
(14)
∈ ∀ ∈ , ∀ ∈
(15)
In this model, the objective function (1) is seeking to maximize the demand covered by at least two vehicles within the small radius. Constraints (2) ensures that each demand in the place must be covered within the large radius at least through one vehicle. In fact, all demands must be covered within the large radius. Constraint (3) ensures that a fraction of total demand has to be covered within the small radius. The left-hand side of the constraint (4) counts the number of vehicles covering node within and the right-hand side is equal to 1 if the node is covered exactly once within the small radius, and equal to 2 if it is covered at least twice within the small radius. Constraint (5) states that if the node is not covered at least once, it cannot be covered at least twice. Constraints (4) and (5) both guarantee that if two or more facilities are within , the node will be covered twice. By constraint (6), vehicle must be assigned to the vehicle locations. Constraint (7) controls the maximum capacity of each vehicle location . Constraint (8) ensures that the demand of vehicle location can be controlled by the positioned vehicles. Constraint (9) states that the total demand for node must be covered within the large radius . Constraints (10) and (11) express the non-negativity of some variables and constraint (12) presenting the coverage index as a binary variable. Constraint (13) shows that the total available vehicles for allocation should be an integer. Finally, constraints (14) states that must be an integer.
IV.
and m/10 stations are generated in each of the eight remaining zones. In other words, more weight is placed in the center of the square. Again, these points are generated according to a continuous uniform distribution. To convert distances into times, the side of the square region is assumed to be equal to 30 km and the ambulance speed to be 40 km/h. Thus, a function is designed to obtain the distance between any two points inside the square and then by another function, the points are converted to time. Each ambulance which is located in the corresponding station has a specific capacity. The demand points and the ambulance stations of an example are depicted in the Fig. 3.
COMPUTATIONAL EXPERIMENTS: 1.
Data description
The proposed model can be implemented on real-world data but here we express a way to generate numbers randomly and then the model has been implemented on this random data. It is assumed that we have a square-shaped province with the dimension of 0,30 (scale-kilometer) and the points are scattered in this square according to what will be explained. Random sample generated is as follows: First, we scatter n = 100 points randomly using a continuous uniform distribution in the 30 × 30 square which represent the demand nodes (or patient location). Then, we consider a demand for these points. The amount of demand has been generated according to a negative exponential distribution of mean 1 and is assigned to each of the points. Then, m = 30 points which represent the location of ambulance stations are placed in a square space using a continuous uniform distribution. The 0,30 square is divided into nine equal square zones. Due to the fact that it is better to have more stations in the center than other zones, 2m/10 potential stations are generated in the central zone,
Fig. 3. Proposed example
Some parameter values such as small and large radius coverage are in accordance with what has been announced by the industry standards. = 7 min; (small coverage radius) = 15 min; (large coverage radius) = 15 : Random numbers with exponential distribution of mean 1 are generated as the demand of each patient locations and assigned to them. : Random integer between 1 and 3 as the maximum number of ambulances location is allocated to each station. : Capacity of each vehicle when located at vehicle location . Here we use the value of 2 for . : The distance between the demand points and the vehicle locations is calculated using the written function. Next, the time is achieved using ambulance velocity. Afterward, we change the time into fuzzy numbers and compare with the and . radii coverage in order to define
2.
Solution procedure
According to the classical model, two main constraints as well as a parameter and a new variable as a result of these two constraints are defined for the proposed model. These two constraints are considered as hard constraints. Hence, to convert them into soft constraints, the goal programming approach is taken. In the goal programming, each goal is presented using a goal constraint. Therefore, different goals could be considered in a model. There is a possibility of violation by the decision-maker in the goal constraints unlike the constraints of the linear programming and integer programming problem. For this reason, the goal constraints are known as soft constraints. Both constraints (8) and (9), which are the hard ones, are considered as goal constraints and deviation values are reduced from the objective function by taking a ratio into account. Significant variables could be defined through the CPLEX solver in GAMS including the number of ambulances assigned to each vehicle location, the variables which represent the number of vehicles covering the demand within the small radius, and the demand is covered by a vehicle in a particular place. In the next section, we will analyze the results and numerical values obtained by solving the model through the GAMS. 3.
Results
A sensitivity analysis is performed to find the optimal value of which represents the percentage of the demand that must be covered within the small radius. The value of 0.65 is selected for this parameter. After running the model through GAMS, acceptable results have been achieved. The objective function that maximizes the demand by at least two vehicles covered within the small radius is equal to 43. This value has a significant improvement compared to the previous models that have overlooked the uncertainty. The number of ambulances located in each station must be lower than (or equal to) the capacity of each station which represents the
Fig. 4. Number
highest number of ambulances that can be positioned in each station. In the Fig. 4. The number of ambulances located in each vehicle station and the capacity of each vehicle station is shown. and state that how many vehicles are The variables able to cover the demand for the location within the small radius (indicating the preference). The patient location will gets a value be covered by one vehicle if the variable is equal to 1, it means that equal to 1. Also, if the variable the mentioned location will be covered by at least two vehicles. One of the most important policies and assumptions considered for the modeling of this problem is that each location needs to be covered within the large radius which represents the main requirement of our approach. In fact, there is not any possibility of coverage within the small radius until each location is covered within the large radius. 58% of the patient locations within the small radius have been covered by one vehicle and 26% of the patient locations within the small radius have been covered by at least two vehicles. It is obvious that the variable indicating the coverage by one vehicle gets the value equal to 1 in the areas that have been covered with more than two ambulances. Meanwhile, 32% of the patient locations are covered only by one vehicle within the small radius. have been Another substantial variable called presented as the contribution of the proposed model than previous models. This variable indicates the amount of demand in the patient location which is covered by a vehicle from station . The output of this variable could be illustrated in the form of a matrix in spreadsheets such as EXCEL. Moreover, this variable can be a useful signal for decisionmakers to select the best vehicle locations and stations. In this paper, we concentrate more on the amount of covered demand. Further studies could be done for the sake of improving the proposed model such as applying the travel cost.
of ambulances located in each vehicle station and the capacity of each vehicle station
V. CONCLUSION AND FUTURE WORK In this paper, the problem of ambulances location with double coverage concept is investigated. The main contributions of this paper than previous models are adding a new constraint which indicates the coverage of demand for each patient location by each vehicle station as well as considering the travel time between the patient locations and the vehicle stations through the fuzzy numbers. To solve the proposed model, the goal programming approach has been used. The performance of the presented model has been evaluated using the generated data. This article is a starting point for research in the field of location problem with regard to double coverage idea under the uncertainty situation. Therefore, we can propose interesting future studies such as: some other parameters have an uncertain nature like the demand in our study, so they can be examined under the uncertainty. Moreover, the issue can be studied in larger scale and efficient methods can be proposed for its solution. REFERENCES [1]
M. A. Pereira, L. C. Coelho, L. A. Lorena, and L. C. De Souza, “A hybrid method for the Probabilistic Maximal Covering Location–Allocation Problem,” Comput. Oper. Res., vol. 57, pp. 51–59, 2015.
[2]
M. Dzator and J. Dzator, “An effective heuristic for the P-median problem with application to ambulance location,” Opsearch, vol. 50, no. 1, pp. 60–74, 2013.
[3]
S. G. Henderson and A. J. Mason, “Ambulance service planning: simulation and data visualisation,” in Operations research and health care, Springer US, 2005, pp. 77–102.
[4]
J. Naoum-Sawaya and S. Elhedhli, “A stochastic optimization model for real-time ambulance redeployment,” Comput. Oper. Res., vol. 40, no. 8, pp. 1972–1978, 2013.
[5]
S. Wang, J. Watada, and W. Pedrycz, “Recourse-based facilitylocation problems in hybrid uncertain environment,” Syst. Man, Cybern. Part B Cybern. IEEE Trans., vol. 40, no. 4, pp. 1176– 1187, 2010.
[6]
X. Li, Z. Zhao, X. Zhu, and T. Wyatt, “Covering models and optimization techniques for emergency response facility location and planning: a review,” Math. Methods Oper. Res., vol. 74, no. 3, pp. 281–310, 2011.
[7]
C. Toregas, R. Swain, C. ReVelle, and L. Bergman, “The location of emergency service facilities,” Oper. Res., vol. 19, no. 6, pp. 1363–1373, 1971.
[8]
R. Church and C. R. Velle, “The maximal covering location problem,” Pap. Reg. Sci., vol. 32, no. 1, pp. 101–118, 1974.
[9]
G. Laporte, F. V Louveaux, and L. van Hamme, “Exact solution to a location problem with stochastic demands,” Transp. Sci., vol. 228, no. 2, pp. 95–103, 1994.
[10]
P. Schütz, L. Stougie, and A. Tomasgard, “Stochastic facility location with general long-run costs and convex short-run costs,” Comput. Oper. Res., vol. 35, no. 9, pp. 2988–3000, 2008.
[11]
U. Bhattacharya, J. Rao, and R. Tiwari, “Fuzzy multi-criteria facility location problem,” Fuzzy Sets Syst., vol. 51, no. 3, pp. 277–287, 1992.
[12]
H. Ishii, Y. L. Lee, and K. Y. Yeh, “Fuzzy facility location problem with preference of candidate sites,” Fuzzy Sets Syst., vol. 158, no. 17, pp. 1922–1930, 2007.
[13]
M. Gendreau, G. Laporte, and F. Semet, “Solving an ambulance location model by tabu search,” Locat. Sci., vol. 5, no. 2, pp. 75– 88, 1997.
[14]
K. F. Doerner, W. J. Gutjahr, R. F. Hartl, M. Karall, and M. Reimann, “Heuristic solution of an extended double-coverage ambulance location problem for Austria,” Cent. Eur. J. Oper. Res., vol. 13, no. 4, pp. 325–340, 2005.
[15]
S. Davari, M. H. F. Zarandi, and I. B. Turksen, “A greedy variable neighborhood search heuristic for the maximal covering location problem with fuzzy coverage radii,” Knowledge-Based Syst., vol. 41, pp. 68–76, 2013.
