Journal of Advanced Transportation
Volume 2018 (2018), Article ID 9364129, 12 pages
https://doi.org/10.1155/2018/9364129

Research Article

Distribution-Free Model for Ambulance Location Problem with Ambiguous Demand

Feng Chu,¹ Lu Wang,² Xin Liu,³ Chengbin Chu,3,4 and Yang Sui⁵

¹Management Engineering Research Center, Xihua University, Chengdu 610039, China
²Department of Aviation Transportation, Shanghai Civil Aviation College, Shanghai 200232, China
³School of Economics & Management, Tongji University, Shanghai 200092, China
⁴Laboratoire Génie Industriel, Centrale Supélec, Université Paris-Saclay, Grande Voie des Vignes, Chatenay-Malabry, France
⁵Glorious Sun School of Business & Management, Donghua University, Shanghai 200051, China

Correspondence should be addressed to Xin Liu; liuxin9735@126.com

Received 11 June 2018; Accepted 28 August 2018; Published 13 September 2018

Academic Editor: Oded Cats

Copyright © 2018 Feng Chu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 by

where

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: .

Sampling approach
MIP-DF1
MIP-DF2

CT
Obj
Sel (%)
CT
Obj
Sel (%)
CT
Obj
Sel (%)

1
7
4.9
23
96.26
0.1
36
100.00
0.1
23
100.00

2
7
5.1
22
98.63
0.1
35
100.00
0.1
24
99.82

3
6
4.1
14
96.92
0.1
30
100.00
0.1
14
98.50

4
6
4.3
13
97.13
0.2
29
100.00
0.1
12
97.42

5
6
5.7
12
79.96
0.1
30
100.00
0.2
12
100.00

6
8
5.8
21
98.50
0.1
24
100.00
0.2
27
100.00

Average
5.0
17.5
94.57
0.1
30.7
100.00
0.1
18.7
99.29

Table 2: .

Sampling approach
MIP-DF1
MIP-DF2

CT
Obj
Sel (%)
CT
Obj
Sel (%)
CT
Obj
Sel (%)

1
7
4.9
24
99.79
0.1
64
100.00
0.1
25
100.00

2
7
4.8
23
98.56
0.1
63
100.00
0.1
24
100.00

3
6
5.1
15
97.17
0.1
55
100.00
0.1
16
100.00

4
6
4.7
15
92.75
0.1
52
100.00
0.1
13
100.00

5
6
4.0
14
90.00
0.1
55
100.00
0.1
14
100.00

6
8
5.2
23
97.85
0.1
45
100.00
0.1
27
100.00

Average
4.8
19.0
93.43
0.1
55.7
100.00
0.1
19.8
100.00

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.

Table 3: Solutions to instance 4 in Nickel et al. [8] for the given scenario.

Sampling approach
MIP-DF1
MIP-DF2

Emergency points
1
2
3
4
5
6
Emergency points
1
2
3
4
5
6
Emergency points
1
2
3
4
5
6

Demand
1
0
2
0
2
0
Demand
1
0
2
0
2
0
Demand
1
0
2
0
2
0

Base locations
Ambulance employment

Ambulance employment

Ambulance employment

1
0

0

0

2
1

1

0

3
1

2

3
1
1

26
1

2

2

0

4
0

0

0

5
0

6

0

6
2

2

0

4

2

Figure 1: Instance 4 in Nickel et al. [8] 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: Computational results on instances with data generated from uniform distribution.

Sampling approach
MIP-DF1
MIP-DF2
Sampling approach
MIP-DF1
MIP-DF2
Sampling approach
MIP-DF1

MIP-DF2

CT
Obj
Sel (%)
CT
Obj
Sel (%)
CT
Obj
Sel (%)
CT
Obj
Sel (%)
CT
Obj
Sel (%)
CT
Obj
Sel (%)
CT
Obj
Sel (%)
CT

Obj

Sel (%)

CT

Obj

Sel (%)

1
(2,6)
2.7
89
73.17
0.1
149
99.06
0.1
107
92.22
2.8
96
73.73
0.1
155
99.33
0.1
107
91.67
2.7
106
75.85
0.1
185
99.87
0.1
107
91.28

2
(4,12)
4.7
169
74.23
0.0
291
99.19
0.0
207
91.22
5.8
181
74.38
0.1
303
99.97
0.1
207
93.63
5.0
199
77.02
0.0
363
99.82
0.0
207
93.38

3
(6,18)
7.2
258
75.06
0.1
438
99.76
0.1
312
91.06
7.5
279
80.82
0.1
456
99.96
0.1
312
93.69
7.8
304
83.37
0.1
546
99.97
0.1
312
93.57

4
(8,24)
9.9
314
79.56
0.1
553
99.93
0.1
385
92.06
11.1
338
82.02
0.1
577
99.99
0.1
385
96.02
11.2
366
84.92
0.1
697
99.94
0.1
385
96.20

5
(10,30)
13.2
415
84.06
0.1
722
98.96
0.1
512
94.06
14.3
441
83.22
0.2
752
99.83
0.1
512
96.92
14.2
476
86.09
0.1
902
99.93
0.1
512
96.92

6
(12,36)
17.0
475
82.49
0.2
843
98.70
0.1
591
93.67
18.1
504
82.51
0.1
879
99.82
0.1
591
96.08
18.9
535
87.26
0.1
1059
99.97
0.1
591
96.09

7
(14,42)
21.5
573
81.11
0.2
1006
98.49
0.2
712
96.06
24.3
609
82.17
0.2
1048
99.53
0.2
712
95.89
23.3
644
86.08
0.2
1258
99.92
0.2
712
95.82

8
(16,48)
24.8
686
75.32
0.2
1185
98.96
0.2
849
97.25
28.4
727
83.07
0.2
1233
99.49
0.2
849
96.13
28.1
768
90.63
0.2
1473
99.99
0.3
849
96.22

9
(18,54)
31.3
691
83.53
0.3
1236
99.74
0.4
858
97.09
35.1
731
83.97
0.3
1290
99.88
0.3
858
97.65
35.2
760
89.06
0.3
1560
99.94
0.3
858
97.58

10
(20,60)
37.2
794
81.74
0.3
1409
98.87
0.3
989
96.57
42.8
843
84.89
0.3
1469
99.47
0.3
989
97.73
43.2
892
90.04
0.3
1769
99.97
0.3
989
97.38

11
(22,66)
45.1
830
80.72
0.4
1493
99.49
0.3
1031
97.58
51.8
880
83.28
0.4
1559
99.82
0.3
1031
97.23
51.6
930
89.67
0.5
1889
99.95
0.3
1031
97.27

12
(24,72)
54.3
1025
79.70
0.4
1771
99.27
0.4
1267
98.35
60.1
1083
84.82
0.5
1843
99.49
0.4
1267
97.76
63.7
1146
90.14
0.4
2203
99.93
0.4
1267
97.77

13
(26,78)
63.1
1026
81.42
0.5
1819
98.98
0.4
1274
97.82
74.3
1087
86.78
0.5
1897
99.36
0.4
1274
97.78
76.5
1146
90.23
0.4
2287
99.97
0.5
1273
98.37

14
(28,84)
78.2
1101
79.03
0.5
1949
98.69
0.8
1361
98.04
87.8
1165
87.90
0.5
2033
99.35
0.7
1361
98.89
87.0
1227
90.56
0.5
2453
99.98
0.6
1361
98.08

15
(30,90)
90.7
1091
80.98
0.6
1997
99.15
0.6
1367
96.29
101.8
1155
83.29
0.6
2087
99.81
0.6
1367
98.07
102.0
1225
89.86
0.6
2537
99.99
0.8
1367
98.17

16
(32,96)
106.2
1160
87.98
0.7
2123
99.44
0.7
1449
97.87
118.3
1229
83.83
0.7
2219
99.88
0.6
1449
98.49
119.3
1300
91.05
0.6
2698
99.98
0.6
1449
98.15

17
(34,102)
120.2
1298
75.72
0.7
2328
99.69
0.7
1614
98.21
137.7
1374
88.60
0.8
2430
99.97
0.7
1614
98.24
126.0
1452
90.85
0.7
2940
99.99
0.8
1614
98.18

18
(36,108)
139.4
1408
81.21
0.8
2509
99.47
0.8
1752
98.09
154.4
1492
89.28
0.8
2617
99.83
0.8
1752
97.99
154.8
1580
91.73
0.8
3156
99.99
0.8
1752
98.09

19
(38,114)
161.8
1508
81.70
0.9
2676
98.99
1.1
1877
97.42
176.0
1598
91.30
0.9
2790
99.34
0.8
1877
99.08
175.5
1688
91.86
0.9
3360
99.99
0.9
1877
98.72

20
(40,120)
220.6
1628
77.19
1.0
2869
99.89
1.0
2029
98.73
200.7
1732
87.53
1.1
2990
99.97
0.9
2029
98.79
190.4
1827
91.38
1.0
3590
99.98
0.9
2029
98.75

21
(42,126)
278.0
1622
83.54
1.0
2900
99.78
1.0
2017
97.32
260.9
1718
88.54
1.0
3026
98.76
1.0
2017
98.18
267.0
1812
90.89
1.2
3655
99.99
1.0
2017
98.48

22
(44,132)
318.9
1735
84.94
1.1
3088
99.85
1.5
2165
98.57
297.0
1840
89.22
1.2
3220
99.25
1.2
2165
98.86
287.3
1952
92.06
1.1
3880
99.99
1.1
2165
98.50

23
(46,138)
363.0
1937
81.41
1.2
3372
98.93
1.3
2406
97.96
322.2
2048
89.37
1.2
3510
99.34
1.4
2406
99.06
346.1
2170
89.89
1.2
4199
100.00
1.2
2406
98.09

24
(48,144)
422.6
1968
82.26
1.3
3461
99.32
1.3
2453
98.14
367.4
2085
88.46
1.3
3605
99.56
1.5
2453
99.12
355.6
2211
93.00
1.3
4326
99.99
1.3
2453
98.19

25
(50,150)
472.7
1919
81.11
1.4
3456
99.99
1.5
2407
96.29
401.9
2037
86.99
1.4
3606
99.99
1.4
2407
98.57
425.7
2161
92.59
1.5
4356
99.95
1.4
2407
98.19

26
(52,156)
530.0
1906
80.96
1.5
3484
98.78
1.7
2392
96.58
449.5
2021
89.52
1.4
3640
99.99
1.5
2392
95.68
448.6
2142
91.82
1.5
4420
99.99
1.5
2392
98.54

27
(54,162)
591.3
2096
83.87
1.6
3751
98.46
1.7
2617
97.61
511.7
2221
90.45
1.6
3913
99.98
1.7
2617
95.44
483.0
2355
93.46
1.8
4723
99.99
1.6
2617
98.93

28
(56,168)
631.0
2110
77.58
1.7
3819
99.33
1.9
2643
98.29
577.7
2234
91.57
1.8
3987
99.98
1.8
2643
95.26
555.0
2365
90.45
1.8
4827
99.99
1.7
2643
98.31

29
(58,174)
678.0
2175
79.98
1.8
3937
98.98
1.8
2720
97.14
630.3
2307
90.22
2.0
4111
99.69
1.9
2720
96.23
618.9
2444
91.28
1.9
4981
100.00
1.9
2720
99.14

30
(60,180)
759.3
2384
83.21
1.9
4240
99.67
2.0
2979
98.41
679.1
2527
89.78
1.9
4419
99.79
2.3
2979
97.21
687.8
2679
91.43
2.1
5316
99.99
2.0
2979
98.69

31
(62,186)
822.0
2281
76.04
2.0
4157
99.84
2.1
2856
97.96
773.4
2416
89.34
2.1
4343
99.97
2.1
2856
96.79
746.6
2557
91.58
2.2
5273
99.99
2.3
2856
99.02

32
(64,192)
876.6
2517
83.78
2.1
4489
99.32
2.5
3145
97.29
827.6
2671
82.96
2.3
4680
99.77
2.4
3145
98.25
789.7
2822
90.89
2.4
5640
99.99
2.2
3145
98.91

33
(66,198)
1012.3
2532
78.06
2.3
4555
99.67
2.3
3169
96.15
901.0
2682
89.47
2.4
4752
99.89
2.4
3169
97.03
905.1
2835
91.57
2.4
5742
99.98
2.4
3169
98.67

34
(68,204)
1101.0
2535
79.34
2.4
4602
99.99
2.9
3174
96.31
989.8
2685
88.58
3.0
4806
99.98
2.8
3174
97.62
986.3
2844
92.36
2.6
5824
99.99
2.5
3174
98.89

35
(70,210)
1166.6
2714
80.62
2.5
4862
99.86
2.9
3391
97.87
1066.6
2875
91.37
2.6
5072
99.99
2.9
3391
97.16
1082.2
3043
93.15
2.6
6122
99.99
2.9
3391
99.23

36
(72,216)
1201.1
2827
81.04
2.6
5047
99.91
2.8
3535
97.98
1246.5
2984
90.16
2.8
5263
99.99
2.8
3535
98.78
1166.7
3169
92.53
3.0
6343
99.99
2.9
3535
98.96

37
(74,222)
1143.7
2826
82.28
2.8
5096
99.33
2.9
3543
98.84
1358.2
3098
88.95
2.9
5318
99.99
3.2
3543
98.88
1265.4
3178
91.39
2.9
6427
100.00
3.1
3543
98.93

38
(76,228)
1243.5
2839
81.52
2.9
5161
98.97
2.9
3565
97.88
1397.6
3103
89.26
2.9
5388
99.99
3.1
3565
97.24
1383.7
3185
92.55
3.3
6528
100.00
3.1
3565
98.95

39
(78,234)
1464.3
2945
84.12
3.1
5331
99.98
3.2
3695
98.76
1479.4
3210
90.05
3.1
5565
99.99
3.4
3695
97.13
1456.1
3313
90.81
3.4
6735
100.00
3.3
3695
98.78

40
(80,240)
1604.2
2975
85.10
3.2
5414
99.57
3.6
3734
97.96
1626.3
3346
89.99
3.5
5654
99.99
3.5
3734
98.06
1586.9
3343
92.48
3.5
6854
100.00
3.5
3734
98.73

Average
448.2
1584.6
80.67
1.2
2839.7
99.36
1.3
1978.7
96.82
437.9
1691.3
86.54
1.3
2962.6
99.75
1.3
1978.7
97.21
429.5
1778.8
89.85
1.3
3577.4
99.97
1.3
1978.7
97.75

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.

Table 5: Computational results on instances with data generated from Poisson distribution.

Sampling approach
MIP-DF1
MIP-DF2
Sampling approach
MIP-DF1
MIP-DF2
Sampling approach
MIP-DF1
MIP-DF2

CT
Obj
Sel (%)
Sel (%)
Sel (%)
CT
Obj
Sel (%)
Sel (%)
Sel (%)
CT
Obj
Sel (%)
Sel (%)
Sel (%)

1
(2,6)
2.7
90
72.94
99.67
90.01
2.7
98
77.70
99.79
90.13
2.4
103
92.73
99.99
94.68

2
(4,12)
4.6
170
79.61
99.21
91.17
5.1
183
83.05
99.46
91.28
4.9
201
93.92
99.98
95.46

3
(6,18)
6.8
258
75.78
99.38
89.11
7.4
280
80.28
99.68
90.04
6.9
307
91.82
99.99
96.58

4
(8,24)
9.3
315
80.01
99.28
91.96
10.2
338
84.38
99.67
91.72
9.2
368
94.68
99.99
96.13

5
(10,30)
12.0
413
80.41
99.39
94.49
13.3
443
84.47
99.76
92.72
12.1
479
95.44
99.99
97.52

6
(12,36)
15.5
476
80.84
98.77
95.54
16.7
508
88.63
99.91
95.69
15.9
536
94.56
99.99
97.34

7
(14,42)
19.5
573
78.76
98.92
96.63
21.3
609
91.24
99.98
97.29
20.5
643
93.14
100.00
97.68

8
(16,48)
23.2
687
79.64
99.97
97.88
26.7
724
85.23
99.88
96.87
23.8
772
94.22
99.98
96.89

9
(18,54)
29.3
692
82.67
98.93
98.09
32.7
730
83.86
99.99
97.50
29.8
757
92.28
99.99
97.55

10
(20,60)
35.6
796
79.46
98.99
98.01
39.9
840
82.99
99.76
96.86
36.0
885
96.68
99.99
97.05

11
(22,66)
42.3
829
82.01
98.95
97.56
48.9
877
83.78
99.87
97.88
44.8
927
94.76
99.99
98.12

12
(24,72)
51.2
1023
83.38
99.13
98.53
55.4
1085
85.01
99.99
98.81
51.4
1148
93.98
99.99
98.88

13
(26,78)
59.2
1045
84.16
99.07
98.73
68.7
1088
79.98
99.96
98.02
60.7
1152
95.04
99.99
98.21

14
(28,84)
72.3
1099
79.46
99.15
97.88
83.8
1166
80.56
99.92
98.29
77.6
1230
95.46
99.99
98.39

15
(30,90)
85.5
1089
79.15
99.64
97.03
94.2
1156
84.32
99.84
97.78
90.3
1225
92.23
99.98
97.85

16
(32,96)
100.9
1158
80.24
98.97
97.69
116.3
1231
85.00
99.96
98.24
104.5
1222
91.99
99.99
98.35

17
(34,102)
116.5
1299
80.34
99.19
97.79
128.5
1375
84.95
99.37
98.08
119.0
1457
92.58
100.00
98.46

18
(36,108)
125.4
1411
81.76
99.43
98.85
147.9
1492
85.61
99.68
98.68
134.5
1579
92.86
99.99
98.75

19
(38,114)
150.4
1506
79.70
99.23
98.42
165.1
1594
81.67
99.72
98.46
156.4
1694
93.76
99.99
97.86

20
(40,120)
170.6
1635
78.19
99.16
97.83
185.4
1734
86.07
99.68
97.76
177.8
1825
92.85
99.04
97.18

21
(42,126)
205.4
1622
83.54
99.18
98.34
235.7
1720
82.97
99.78
98.33
222.6
1856
95.01
99.99
98.34

22
(44,132)
240.0
1736
84.94
99.22
98.46
285.2
1844
82.69
99.99
98.16
268.0
1946
94.86
99.99
98.20

23
(46,138)
265.1
1934
81.41
99.53
97.93
315.9
2049
84.92
99.69
97.99
307.2
2172
93.58
100.00
98.13

24
(48,144)
300.4
1961
82.26
99.47
96.81
352.2
2086
85.13
99.98
97.06
310.2
2209
94.88
99.99
97.24

25
(50,150)
330.8
1924
81.11
99.65
97.46
389.8
2038
82.64
99.99
97.88
363.9
2161
95.46
99.99
98.13

26
(52,156)
366.9
1911
80.96
99.39
98.55
430.5
2027
87.22
99.67
98.74
445.3
2142
91.85
99.99
98.76

27
(54,162)
441.8
2095
83.87
99.27
97.54
477.1
2219
83.41
99.89
97.49
480.6
2353
92.89
99.99
97.55

28
(56,168)
536.7
2113
84.58
99.68
98.48
535.6
2237
84.52
99.78
98.55
522.1
2369
92.28
100.00
98.68

29
(58,174)
621.7
2186
79.98
99.89
96.94
594.2
2315
83.16
99.96
96.99
585.3
2371
91.67
99.99
97.05

30
(60,180)
726.3
2382
83.21
99.91
98.37
652.7
2543
83.59
99.98
98.86
621.4
2678
93.69
99.97
98.86

31
(62,186)
821.4
2346
84.04
99.36
98.88
781.3
2430
83.21
99.99
98.46
687.6
2560
93.58
99.99
98.47

32
(64,192)
886.3
2518
83.78
99.38
98.07
810.6
2677
83.56
99.99
98.33
787.7
2822
94.08
100.00
98.35

33
(66,198)
998.5
2532
84.06
99.17
97.09
968.4
2689
84.97
99.96
97.46
808.8
2838
95.07
99.99
97.49

34
(68,204)
1126.4
2534
85.43
99.08
98.97
1098.6
2685
88.09
99.96
97.82
846.9
2843
88.44
99.98
97.80

35
(70,210)
1203.5
2715
82.58
99.37
97.99
1209.8
2886
85.69
99.91
98.09
994.6
3041
94.59
99.99
98.13

36
(72,216)
1280.3
2825
81.04
99.29
98.27
1307.6
3010
86.51
99.94
98.34
1101.3
3170
94.68
99.99
98.65

37
(74,222)
1302.1
2830
82.28
99.49
98.26
1326.1
3009
83.76
99.99
98.26
1201.4
3172
93.46
99.99
98.88

38
(76,228)
1386.2
2843
81.52
99.33
98.07
1399.5
3028
87.60
99.98
98.41
1213.3
3187
97.06
99.99
98.76

39
(78,234)
1538.7
2953
83.12
99.64
97.99
1482.7
3133
90.01
99.95
97.83
1338.1
3304
89.13
99.99
97.99

40
(80,240)
1697.4
2975
85.10
99.85
97.67
1525.6
3249
82.17
99.93
97.81
1586.7
3347
91.11
99.99
97.91

Average
435.2
1587.5
81.43
99.34
97.08
436.2
1685.6
84.37
99.85
97.07
396.8
1776.3
93.56
99.97
97.81

Table 6: Computational results on instances with data generated from normal distribution.

Sampling approach
MIP-DF1
MIP-DF2
Sampling approach
MIP-DF1
MIP-DF2
Sampling approach
MIP-DF1
MIP-DF2

CT
Obj
Sel (%)
Sel (%)
Sel (%)
CT
Obj
Sel (%)
Sel (%)
Sel (%)
CT
Obj
Sel (%)
Sel (%)
Sel (%)

1
(2,6)
2.7
90
72.94
99.64
91.17
2.7
94
78.44
99.70
92.02
2.558
106
84.22
100.00
92.08

2
(4,12)
4.6
170
79.61
99.21
92.83
4.8
183
83.28
99.82
92.83
4.508
183
88.23
100.00
93.70

3
(6,18)
6.8
258
75.78
99.16
92.28
6.8
280
77.98
99.71
94.17
6.803
307
76.58
99.98
94.47

4
(8,24)
9.3
315
80.00
98.99
93.67
9.6
337
83.79
99.79
94.67
9.373
368
84.38
99.99
95.79

5
(10,30)
12.0
413
78.41
99.27
94.37
12.4
443
87.63
99.63
94.76
12.62
499
88.50
99.98
97.08

6
(12,36)
15.5
476
79.21
99.19
95.19
16.2
504
81.24
99.82
95.82
16.021
536
89.54
100.00
97.56

7
(14,42)
19.5
573
79.76
99.33
95.98
19.6
609
82.87
99.75
96.19
20.11
646
84.67
99.99
97.67

8
(16,48)
23.2
687
80.31
99.18
97.05
23.5
727
81.15
99.65
96.50
24.04
772
82.78
99.99
97.65

9
(18,54)
29.3
692
80.86
98.98
97.59
29.1
734
84.64
99.74
97.55
29.815
796
86.17
99.98
98.35

10
(20,60)
35.6
796
79.84
99.37
97.70
35.8
842
82.23
99.88
97.65
37.973
921
88.46
100.00
98.38

11
(22,66)
42.1
829
81.01
99.40
97.55
43.6
877
82.11
99.86
97.86
45.148
959
84.97
99.99
97.89

12
(24,72)
51.4
1023
80.33
99.51
97.85
50.1
1085
81.62
99.83
97.88
52.073
1147
83.21
99.99
98.24

13
(26,78)
59.2
1023
81.38
99.16
98.85
59.9
1087
84.11
99.70
98.45
62.937
1194
86.09
99.98
97.95

14
(28,84)
72.8
1099
82.35
99.13
99.15
72.2
1168
84.35
99.87
98.06
75.053
1270
85.67
99.97
98.34

15
(30,90)
85.5
1089
81.87
99.22
98.61
88.0
1157
83.76
99.76
97.96
89.222
1300
86.11
99.96
97.98

16
(32,96)
100.9
1158
87.70
98.97
97.88
96.6
1231
88.45
99.84
98.23
106.288
1359
89.97
99.99
98.48

17
(34,102)
115.4
1299
84.34
99.25
98.49
113.7
1377
86.01
99.80
97.98
123.869
1493
87.98
99.96
98.77

18
(36,108)
140.7
1411
83.21
99.27
98.66
133.9
1495
83.01
99.65
98.22
133.807
1601
85.67
99.99
98.34

19
(38,114)
170.2
1506
82.06
99.15
98.05
167.2
1598
89.77
99.78
98.45
152.999
1694
89.98
99.99
98.47

20
(40,120)
209.1
1635
85.28
99.18
98.89
191.6
1737
88.05
99.76
98.37
189.002
1833
90.12
99.99
97.91

21
(42,126)
284.3
1622
85.50
99.34
99.13
243.2
1718
88.86
99.74
98.13
239.913
1815
90.08
100.00
98.15

22
(44,132)
330.0
1736
84.49
99.40
98.59
279.5
1843
87.86
99.77
97.56
253.106
1948
89.97
99.99
98.26

23
(46,138)
374.1
1934
81.33
99.11
98.76
314.3
2050
88.76
99.81
97.85
274.583
2173
90.14
99.99
98.58

24
(48,144)
453.1
1961
82.45
99.09
98.46
351.5
2090
83.44
99.93
98.19
308.089
2211
86.61
99.99
98.83

25
(50,150)
471.5
1924
81.11
99.29
98.88
395.9
2041
82.79
99.96
98.37
351.093
2158
85.27
99.99
98.27

26
(52,156)
522.3
1911
80.96
98.98
98.57
430.1
2025
82.51
99.97
97.57
421.747
2146
84.97
99.99
99.14

27
(54,162)
599.4
2095
83.87
99.08
97.75
483.5
2226
84.97
99.74
97.80
461.443
2359
86.95
99.99
98.37

28
(56,168)
640.6
2111
84.11
99.06
97.98
533.6
2240
85.58
99.91
98.20
494.113
2371
92.49
99.99
98.16

29
(58,174)
689.7
2176
80.12
99.16
97.91
594.8
2311
82.11
99.84
97.65
624.835
2445
85.31
100.00
98.51

30
(60,180)
743.6
2391
81.11
99.13
97.66
655.8
2533
83.05
99.68
97.95
696.116
2682
85.75
99.99
97.99

31
(62,186)
811.8
2282
81.19
99.28
98.06
706.9
2422
84.17
99.73
98.73
797.768
2564
86.67
99.99
98.46

32
(64,192)
889.5
2519
82.23
99.19
97.57
785.7
2675
86.31
99.82
97.69
864.813
2827
89.76
100.00
99.05

33
(66,198)
1123.5
2533
85.63
99.28
98.11
916.2
2688
85.47
99.79
98.28
973.089
2843
86.57
99.99
98.46

34
(68,204)
1158.7
2534
82.49
99.26
98.47
954.9
2686
87.98
99.38
98.56
1052.3
3041
89.99
100.00
98.77

35
(70,210)
1164.9
2715
80.98
98.95
98.72
1034.5
2880
86.02
99.98
98.69
1127.2
3052
90.21
99.99
98.97

36
(72,216)
1222.4
2825
84.03
99.58
97.70
1116.5
3002
86.57
99.99
98.39
1245.6
3175
91.13
99.99
98.64

37
(74,222)
1297.8
2830
83.57
99.37
97.95
1185.0
3005
84.96
99.98
98.94
1345.8
3176
88.27
99.99
99.05

38
(76,228)
1342.6
2844
82.67
99.66
97.59
1279.5
3019
86.04
99.99
97.97
1515.5
3121
89.78
99.99
98.53

39
(78,234)
1500.3
2955
83.03
99.56
98.67
1381.6
3132
85.88
99.96
98.85
1504.1
3310
90.16
99.99
99.12

40
(80,240)
1649.5
2978
82.57
99.54
98.26
1489.6
3160
84.95
99.97
98.37
1656.3
3349
89.57
99.99
99.33

Average
461.9
1585.5
81.74
99.25
97.42
407.7
1682.8
84.57
99.81
97.43
435.0
1793.8
87.32
99.99
97.94

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.

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