#### Abstract

Ambulance location problem is a key issue in Emergency Medical Service (EMS) system, which is to determine where to locate ambulances such that the emergency calls can be responded efficiently. Most related researches focus on deterministic problems or assume that the probability distribution of demand can be estimated. In practice, however, it is difficult to obtain perfect information on probability distribution. This paper investigates the ambulance location problem with partial demand information; i.e., only the mean and covariance matrix of the demands are known. The problem consists of determining base locations and the employment of ambulances, to minimize the total cost. A new distribution-free chance constrained model is proposed. Then two approximated mixed integer programming (MIP) formulations are developed to solve it. Finally, numerical experiments on benchmarks (Nickel et al., 2016) and 120 randomly generated instances are conducted, and computational results show that our proposed two formulations can ensure a high service level in a short time. Specifically, the second formulation takes less cost while guaranteeing an appropriate service level.

#### 1. Introduction

The design of Emergency Medical Service (EMS) systems that affects people’s health and life focuses on how to respond to emergencies rapidly. Ambulance location problem is one of the key problems in EMS system, which mainly consists of determining where to locate bases, also named as emergency service facilities, and the employment of ambulances in order to serve emergencies efficiently and to guarantee patient survivability. There have been various researches investigating ambulance location problem (e.g., [1, 2]) since it is introduced by Toregas et al. (1974).

Early studies addressing ambulance location problem mainly focus on deterministic environments, including set covering ambulance location problem that minimizes the number of ambulances to cover all demand points [3], maximal covering location problem to maximize the number of covered demand points with given number of ambulances [4], and double standard model (DSM) in which each demand point must be covered by one or more ambulances [5]. However, in practice, stochastic ambulance location problem is more realistic due to the inevitable uncertainties of emergency events (e.g., [6, 7], etc.). Usually, an emergency event has the following characteristics: (i) it is difficult to forecast precisely where an emergency will occur, and (ii) the required number of ambulances depends on the severity of the situation, which is also unforeseen [8].

Most existing works investigate stochastic ambulance location problem by assuming that the demand probability distribution at each possible emergency is known (e.g., [9, 10], Beraldi and Bruni, 2008; etc.). However, as stated by Wagner [11] and Delage and Ye [12], it is usually impossible to obtain the perfect information on probability distribution, due to (i) the lack of historical data and the fact that (ii) the given historical data may not be represented by probability distribution.

Moreover, how to cover as many emergencies as possible to guarantee a high patient service level (i.e., the portion of the satisfied emergency points among all emergency points) has always been a main goal of EMS operators. Motivated by the above observation, this work focuses on a stochastic ambulance location problem with only partial information of demand points, i.e., the mean and covariance matrix of the demands. Besides, we introduce a new individual chance constraint, which implies a minimum probability that each emergency demand has to be satisfied. For the problem, a new distribution-free model is presented, and then two approximated formulations are proposed. The contribution of this work mainly includes the following:(1)For the studied stochastic ambulance location problem, we introduce a new individual chance constraint (i.e., safety level) guaranteeing each emergency demand satisfied with a least probability, while Nickel et al. [8] use a coverage constraint, where some service of individual emergencies may suffer insufficiency.(2)A new distribution-free model based on individual chance constraints is proposed. To our best knowledge, it is the first distribution-free model for stochastic ambulance location problem.(3)To solve the distribution-free model, two approximated mixed integer programming (MIP) formulations are developed. Experimental results show that our approximated formulations are efficient and effective for large size instances, compared to that proposed by Nickel et al. [8] in terms of both computational time and service level.

The remainder of this paper is organized as follows. Section 2 gives a brief literature review. In Section 3, we give the problem description and propose a new distribution-free model. In Section 4, two approximated MIP formulations are proposed. Computational results on benchmarks and 120 randomly generated instances are reported in Section 5. Section 6 summarizes this work and states future research directions.

#### 2. Literature Review

The deterministic ambulance location problem has been well studied in literature (e.g., [2, 5, 13], etc.). Besides, there have been some works addressing ambulance location problem under uncertainty (e.g., [1, 6, 14], etc.). Since our study falls within the scope of stochastic ambulance location problem, in the following subsections, we first review existing studies on ambulance location problem with uncertain demand. Then we review the literature studying general and specific stochastic optimization problems with distribution-free approaches.

##### 2.1. Ambulance Location Problem with Uncertain Demand

Ambulance location problem under uncertain demand has been investigated by many researchers. Most existing works address the uncertainty with given scenarios or known probability distribution.

Chapman and White [15] first investigate the ambulance location problem with uncertain demand, in which the complete information on probability distributions is assumed to be known. Beraldi et al. [9] study the emergency medical service location problem with uncertain demand to minimize the total cost with a marginal probability distribution. Beraldi and Bruni (2008) investigate the emergency medical service facilities location problem with stochastic demand based on given set of scenarios and known probability distribution. A stochastic programming formulation with probabilistic constraints is proposed. Noyan [10] studies the ambulance location problem with uncertain demand on given scenarios and probability distribution. Then, two stochastic optimization models and a heuristic are proposed.

Recently, Nickel et al. [8] first study the joint optimization of ambulance base location and ambulance employment. The objective is to minimize the total constructing base costs and ambulance employment costs. It is assumed that various scenarios are given beforehand. A coverage constraint is introduced in the paper. They propose a sampling approach that solves a finite number of scenario samples to obtain a feasible solution of the original problem. But with the coverage constraint, some emergency demands risk insufficient individual service or cannot be served. For the study, we propose a new distribution-free model for stochastic ambulance location problem.

##### 2.2. Distribution-Free Approaches

In data-driven settings, the probability distributions of uncertain parameters may not always be perfectly estimated [16]. Therefore in the last decade, there have been many solution approaches developed to address stochastic problems under partial distributional knowledge. Most related researches focus on the distribution-free approach via considering chance constraints. Wagner [11] studies a stochastic 0-1 linear programming under partial distribution information, i.e., , where are random vectors with unknown distributions. The only information on are their moments, up to order . A robust formulation, as a function of , is given. Given the known second-order moment knowledge, i.e., , an approximated formulation is developed as , where . Delage and Ye [12] investigate the stochastic program with limited distribution information, and they propose a new moment-based ambiguity set, which is assumed to include the true probability distribution, to describe the uncertainties. There have been various works successfully applying the distribution-free approaches. Ng [17] investigates a stochastic vessel deployment problem for liner shipping, in which only the mean, standard deviation, and an upper bound of demand are known. A distribution-free optimization formulation is proposed. Based on that, Ng [18] studies a stochastic vessel deployment problem for the liner shipping industry, where only the mean and variance of the uncertain demands are known. New models are proposed, and the provided bounds are shown to be sharp under uncertain environment. The stochastic dependencies between the shipping demands are considered.

Jiang and Guan [19] develop approaches to solve stochastic programs with data-driven chance constraints. Two types of confidence sets for the possible probability distributions are proposed. For more distribution-free formulation applications, please see Lee and Hsu [20], Kwon and Cheong [21], etc., for the stochastic inventory problem, and Zhang et al. [23] for stochastic allocating surgeries problem in operating rooms, and Zheng et al. [24] for stochastic disassembly line balancing problem. To the best of our knowledge, there is no research for the stochastic ambulance location problem with only partial information on the uncertain demand.

#### 3. Problem Description and Formulation

In this section, we first describe the considered problem and then propose a new distribution-free model.

##### 3.1. Problem Description

There is a given set of candidate base locations and a set of potential emergency demand points . The emergency points may refer to a road, a part of the urban area and a village. Base is said to be covering an emergency point if the driving time between and is no more than a predetermined value . The set of candidate bases covering emergency point is denoted as .

The number of ambulances required by an emergency point is denoted as , which depends on the severity of the practical situation. We consider uncertain emergencies which are estimated or predicted by partial distributional information; thus the demand is regarded to be ambiguous. Besides, we focus on the case where the historical data cannot be represented by a precise probability distribution. It is assumed that the customer demands are independent of each other.

The objective of the problem is to select a subset of base locations and determine the number of ambulances at each constructed base in order to minimize the total base construction cost and ambulance cost. Throughout this paper we assume that only the mean and covariance matrix of demands are known. Moreover, a predefined* safety level * is given for each potential emergency point . That is, the number of ambulances serving each emergency point is larger than or equal to its demand with a least probability of .

##### 3.2. Chance Constraint Construction

In this section, we introduce a chance constraint to guarantee the safety level of each emergency demand point. In the following, is a decision variable denoting the number of ambulances serving point from base and is the uncertain demand at point . The chance constrained inequality is presented as follows: where denotes the probability of the event in parentheses under any potential probability distribution . Constraint (1) ensures that the number of ambulances serving emergency point is no less than its demand with a least probability of (i.e., safety level).

##### 3.3. Distribution-Free Formulation

In the following, we give basic notations, define decision variables and propose the distribution-free formulation** DF** for the ambulance location problem with uncertain demand.

*Parameters* : index of base locations : index of emergency points : set of candidate base locations : set of possible emergency demand points : driving time between base and demand point : maximum driving time for serving any emergency call : set of candidate bases that can cover emergency point , i.e., : set of potential emergency points covered by base , i.e., : fixed construction cost for installing a base at location : fixed cost associated with an ambulance to be located at : number of (stochastic) ambulances requested by emergency point : a sufficiently large positive number

*Decision Variables* : a binary variable equal to 1 if a base is constructed at ; 0 otherwise : number of ambulances assigned to possible base location : number of ambulances serving emergency point from possible base location

*Distribution-Free Model [DF]* **[DF]:**

The objective function denotes the goal to minimize the total cost consisting of two parts: (i) the cost for constructing bases, i.e., , and (ii) the cost for deploying ambulances, i.e., .

Constraint (3) ensures that ambulances can only be located at the opened bases. Constraint (4) ensures that the number of ambulances sent to serve emergency points from base does not exceed the total number of ambulances located at . Constraints (5)-(7) are the restrictions on decision variables.

#### 4. Solution Approaches

The proposed distribution-free model is difficult to solve with the commercial software due to chance constraints. In this section, we propose two approximated MIP formulations based on those in Wagner [11] and in Delage and Ye [12], respectively. In the following subsections, we present the two approximated MIP formulations.

##### 4.1. Approximated MIP Formulation:** MIP-DF1**

In this part, we first describe a widely used ambiguity set to describe the uncertainty. Then an approximated MIP formulation** MIP-DF1** is proposed.

Given a set of independent historical data samples of random vectors of demands, where is the set of sample indexes, the mean vector and the covariance matrix of demands can be estimated as follows:where implies the transposition of the vector in parentheses. Then, an ambiguity set of all probability distributions of demands can be described as the following: where denotes a possible probability distribution satisfying the given conditions, and denotes the expected value of the number in parentheses. Then the chance constraint (1) with the ambiguity set can be presented as follows:According to Wagner (2010) and Ng [18], constraint (1) can be conservatively approximated by the following: where , and denotes the -th element in the -th column of matrix . In terms of conservative approximation, all solutions satisfying constraint (11) must satisfy constraint (1). Then we describe the approximated MIP formulation** MIP-DF1** to** DF** combined with constraint (11): **[MIP-DF1]:**

**MIP-DF1** can be exactly solved by the commercial software, such as CPLEX. As we can observe from the computational results in Section 5, it can obtain a relatively high service level. However, the system cost is also high. In order to save the system cost, another approximated MIP formulation of** DF** is then proposed in the next subsection.

##### 4.2. Approximated MIP Formulation:** MIP-DF2**

Ambiguity set focuses on exactly matching the mean and covariance matrix of uncertain parameters. However, in practice, there may be considerable estimation errors in the mean and covariance matrix. To take the inevitable estimation errors into consideration, Delage and Ye [12] introduce a new moment-based ambiguity set. Therefore, we in the following employ the moment-based ambiguity set:where and are two parameters of ambiguity set with the following assumptions: (i) the true mean vector of demands is within an ellipsoid of size proportion to centered at , and (ii) the true covariance matrix of demands is in a positive semidefinite cone bounded by a matrix inequality of . The ambiguity set describes how likely the uncertain parameters are to be close to the mean in terms of the correlation [12]. Besides, under the moment-based ambiguity set, various stochastic programs with partial distributional information have been successfully modeled and solved [22, 23, 25].

According to the approximation method in Zhang et al. [23], chance constraint (1) can be approximated bywhere

The second approximated MIP formulation** MIP-DF2** to** DF** is presented as follows: **[MIP-DF2]:**

Notice that when and , inequality (14) reduces to (11) in** MIP-DF1**, which implies that** MIP-DF1** is a special case of** MIP-DF2**.

#### 5. Computational Experiments

In this section, the performance of our proposed formulations first evaluated and compared to the sampling approach proposed in Nickel et al. [8]. Specifically, given a required service level , we apply our MIP formulations by restricting the chance constraint for each emergency point to satisfy a safety level , while we adopt the sampling approach by restricting the coverage constraint with . Then computational results on 120 randomly generated instances based on the information on emergencies in Shanghai are reported. Our approximated MIP formulations and the sampling approach are solved by coding in MATLAB_2014b calling CPLEX 12.6 solver. All numerical experiments are conducted on a personal computer with Core I5 and 3.30GHz processor and 8GB RAM under windows 7 operating system. The computational time for sampling approach is limited to 3600 seconds.

##### 5.1. Out-of-Sample Test

Following Zhang et al.’s [22] methodology, we examine the out-of-sample performance of solutions obtained by our proposed approximated MIP formulations and the sampling approach [8]. The out-of-sample test includes three steps:(1)The base locations and the employment of ambulances at each base are determined with known mean vector and covariance matrix of demands, which is named as in-sample test.(2)1000 scenarios are sampled from the underlying true distribution, which represent realizing the uncertain demand.(3)Based on the solution obtained in step , ambulances are assigned to serve emergency points in the 1000 scenarios to maximize the (out-of-sample) service level.

The (out-of-sample) service level is defined as the ratio of the number of emergency points with demand satisfied to the total number of emergency points in 1000 scenarios. In the following subsections, the benchmarks are tested and then three common distributions are employed, i.e., uniform distribution, Poisson distribution, and normal distribution, respectively.

##### 5.2. Benchmark Instances

We first test the benchmarks in Nickel et al. [8] to compare our proposed two approximated MIP formulations with the sampling approach. In Nickel et al. [8], they assume that demand nodes are also potential base locations, and the number of ambulances required by each emergency point is chosen from set with given probability distribution. Computational results on benchmarks are reported in Tables 1 and 2.

Table 1 reports the computational results with service level requirement . In Table 1, CT, Obj, and Sel denote the computational time, objective value (i.e., the system cost), and the service level, respectively. The first column includes the indexes of instances, and the second indicates the number of nodes. It shows that the average service level obtained by the sampling approach is 94.57%, which is smaller than the required service level, i.e., 85%. Specifically, the service level of instance 5 obtained by the sampling approach is only 79.96%. However, MIP-DF1 and MIP-DF2 can guarantee the service levels for all instances higher than the requirement. Concluding, from Table 1, we can observe that (i) the average computational time of sampling approach is about 50 times as those of the two approximated MIP formulations and (ii) MIP-DF1 and MIP-DF2 improve the average service level by about 5.74% and 4.99% compared with sampling approach, however, (iii) system costs obtained by MIP-DF1 and MIP-DF2 are, respectively, 75.43% and 6.86% higher than that of sampling approach.

Table 2 reports the computational results with service level requirement . For instances 2-6, the service levels obtained by sampling approach cannot meet the requirement, i.e., 99%. Besides, it shows that the average computational time of sampling approach is 4.8, which is about 48 times greater than those of MIP-DF1 and MIP-DF2. The average service level is 93.43%. Although the average service levels obtained by MIP-DF1 and MIP-DF2 are both 100%, the average system cost obtained by MIP-DF1 is about 3 times larger than that of sampling approach. Besides, the average system cost obtained by MIP-DF2 is 4.21% higher than that of sampling approach.

Concluding, for the benchmarks described in Nickel et al. [8], we can observe that MIP-DF1 and MIP-DF2 can be solved in far less computational time compared with the sampling approach; the sampling approach cannot meet the requirement of service level for all benchmarks, while the two MIP formulations can; the average system cost of MIP-DF2 is very close to that of sampling approach; with the increase of , the average system costs obtained by MIP-DF2 is getting closer to that obtained by sampling approach.

From Tables 1 and 2, we can observe that, in spite of the lower system cost obtained by the sampling approach, there may exist extreme situations that demands of some emergency points are not satisfied. Take one benchmark in Nickel et al. [8], i.e., instance 4, where there are 6 nodes. A scenario in which demand at each emergency point is generated from the given probability distribution is shown in Figure 1. We can observe from Figure 1 that under the given scenario, the demand of each possible emergency point is . The solutions obtained by sampling approach, MIP-DF1 and MIP-DF2, are shown in Table 3, which includes base locations and ambulance employment obtained by in-sample test and ambulance assignment under given scenario.

In Table 3, the first column includes the indexes of potential base locations. The second column represents the base locations and the ambulance employment at each location, in which a location with 0 ambulance implies that it is not selected for base construction. From Table 3, we can observe that the in-sample solutions obtained by sampling approach is to construct bases at locations 2, 3 and 6, where the number of ambulances is 1, 1, and 2, respectively. Besides, under given scenario, there is one ambulance serving emergency point 1 from base location 3, and one ambulance serving emergency point from base location 2, and 2 ambulances serving emergency point 5 from base location 6. It shows that the requirement of ambulances at emergency point 3 is not fully satisfied. The insufficient service may cause serious loss of life. However, we can find that the solutions obtained by MIP-DF1 and MIP-DF2 can provide sufficient ambulances for all emergency points.

In sum, to guarantee the required service level with less cost, MIP-DF2 is recommended for the solutions with Nickel et al. [8]’s data set. Moreover, in the benchmarks, it is assumed that all nodes are possible emergency points as well as potential base locations and the number of required ambulances is small, which is not always consistent with the situation in Shanghai. Therefore, we further test the two approximated MIP formulations and the sampling approach on randomly generated instances by analysing the information on emergencies in Shanghai [26].

##### 5.3. Computational Tests on Randomly Generated Instances

In the following subsection, numerical experiments on 120 randomly generated instances are described.

###### 5.3.1. Test Instances

By analysing the information on emergencies in Shanghai [26], the mean value of demand in each emergency point are randomly generated from a discrete Uniform distribution on interval . The standard deviation is set to be 5. Since the sampling approach proposed in Nickel et al. [8] requires a sample of scenarios, in numerical experiments scenarios are produced with demands randomly generated from Uniform distribution, Poisson distribution and Normal distribution consistent with given mean and standard deviation.

According to the instance settings in Erkut and Ingolfsson (2008), response time from a base to an emergency point is randomly generated from a discrete Uniform distribution on interval in units of minutes. A base covers an emergency point if the driving time between them is less than 10 minutes to ensure the survival probability (Erkut and Ingolfsson, 2006). For an emergency point in each instance, if there is no base with 10 minutes driving time to it, then the base covering this point is set to be the nearest one. The cost of constructing a base and that of locating one ambulance are both set to be 1, as in Nickel et al. [8]. Besides, the number of emergency points is 3 times that of potential base locations.

###### 5.3.2. Results and Discussion

Different values of required service level are tested, i.e., 0.85, 0.9, 0.95. Computational results on 120 randomly generated instances are reported in Tables 4-6.

Table 4 reports the numerical results of instances with demand generated from Uniform distribution. We observe that when , the average computational times of the sampling approach are 448.2, 437.9 and 429.5 seconds, respectively. However, both approximated MIP formulations can produce solutions in much faster speeds, i.e., within 1.3 seconds on average. With the increase of , (i) the out-of-sample average service level is getting higher, and (ii) the system costs of solutions obtained by all method are getting larger, and (iii) the system costs of solutions yielded by MIP-DF2 is getting closer to those of solutions obtained by the sampling approach. In terms of the average system cost, the sampling approach performs the best, while MIP-DF1 is the poorest. When , for example, the average system cost of MIP-DF2 and the sampling approach are about 30.32% and 44.2% less than that of MIP-DF1, respectively. When , the above gaps are even larger. Concluding, for the Uniform distribution, the proposed two approximated MIP formulations are far more time saving than the sampling approach, MIP-DF1 and MIP-DF2 can improve the service level by about 19.07% compared with the sampling approach, however, despite of the highest service levels obtained by MIP-DF1, the system cost obtained by MIP-DF1 is also high, MIP-DF2 can ensure an appropriate service level with much less cost than MIP-DF1, and for the situations with small-scale instances and high service level requirement, MIP-DF1 is recommended.

Computational results of instances with demand at each emergency point generated from Poisson distribution are presented in Table 5. For MIP-DF1 and MIP-DF2, only mean and variance are involved. Therefore, with given mean and standard deviation of the demand at each point, computational results are the same for different probability distributions. We can obtain from Table 5 that the objective value obtained by the sampling approach increases with the increase of . Besides, when the value of equals 0.85, the average service level obtained by the sampling approach is about 81.43%, which is smaller than that of the two approximated MIP formulations. The highest service level 99.34% is obtained by MIP-DF1, and that obtained by MIP-DF2 is 97.08%. When , the average service level of the sampling approach is 84.37%, and those obtained by MIP-DF1 and MIP-DF2 are 99.85% and 97.97%, respectively. When , the average service level obtained by the sampling approach is 93.56%, and those yielded by MIP-DF1 and MIP-DF2 are 99.97% and 97.81%, respectively. For Poisson distribution, concluding, MIP-DF1 and MIP-DF2 can improve the service level compared with the sampling approach, especially, MIP-DF1 can obtain the highest service level.

Similarly, Table 6 reports the computational results for instances in which the demand at each emergency point is generated following Normal distribution. When , the average service levels obtained by the sampling approach, MIP-DF1 and MIP-DF2 are 81.74%, 99.25%, and 97.42%, respectively. When , the three values are 84.57%, 99.81%, and 97.43% respectively. Moreover, for , the values are equal to 87.32%, 99.99%, and 97.94%, respectively. It can be observed that the performance of the sampling approach is the poorest in terms of the service level.

By observing the above computational results on benchmarks and randomly generated instances, we conclude that The proposed approximated MIP formulations, i.e., MIP-DF1 and MIP-DF2 are very time saving compared with the sampling approach; MIP-DF1 an MIP-DF2 can achieve the requirement of service level and improve the service level by about 15.47% and 12.29% on average, compared with that of the sampling approach; MIP-DF1 is more conservative in terms of overhigh emergency service level close to 100%, and the system cost is also high, which is about 56.8% larger than that of MIP-DF2; Compared with MIP-DF1, MIP-DF2 is less conservative and much more cost saving while ensuring an appropriately service level on average, which is about 97%.

In sum, for medium and large problem instances (e.g., Shanghai), which are common in practice, to guarantee the required service level with less cost, we recommend MIP-DF2 as solution method. However, for very small instances with high service level requirement, we recommend MIP-DF1 for solutions.

#### 6. Conclusion

This paper investigates an ambulance location problem with ambiguity set of demand, in which the probability distribution of the uncertain demand is unknown. In such a data-driven environment, only the mean and covariance demands at emergency points are known. The problem is to determine the base locations and the number of ambulances at each base, aiming at minimizing the total cost associated with locating bases and assigning ambulances. We propose a distribution-free model with chance constraints. Then two approximated MIP formulations with different approximation methods of chance constraints are proposed, which are based on different ambiguity sets of the unknown probability distribution. Numerical experiments on benchmarks and 120 randomly generated instances are conducted, and the computational results show that the first approximated MIP formulation is much conservative with overhigh system cost, while the second approximated MIP formulation is more cost saving with service level appropriately ensured.

In future research, we may consider the following relevant issues. First, the uncertainty of driving time from a base to an emergency point shall be analysed. Besides, unexpected or sudden disasters may be taken into consideration. Moreover, other uncertainties should be considered, such as travel time, the number of available vehicles and so on. The stochastic dependence should also further considered.

#### Data Availability

The data of the benchmark instances in our manuscript can be obtained in Nickel et al. [8]. For the test instances, the data is randomly generated and the way for generating data is stated in our manuscript. The datasets generated during the current study are available from the corresponding author on reasonable request.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (NSFC) under Grants 71428002, 71531011, 71571134, and 71771048. This work was also supported by the Fundamental Research Funds for the Central Universities.