Research ArticleDistribution-Free Model for Ambulance LocationProblem with Ambiguous Demand

Feng Chu1 Lu Wang2 Xin Liu 3 Chengbin Chu34 and Yang Sui5

1Management Engineering Research Center Xihua University Chengdu 610039 China2Department of Aviation Transportation Shanghai Civil Aviation College Shanghai 200232 China3School of Economics amp Management Tongji University Shanghai 200092 China4Laboratoire Genie Industriel Centrale Supelec Universite Paris-Saclay Grande Voie des Vignes Chatenay-Malabry France5Glorious Sun School of Business amp Management Donghua University Shanghai 200051 China

Correspondence should be addressed to Xin Liu liuxin9735126com

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

Academic Editor Oded Cats

Copyright copy 2018 FengChu et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Ambulance location problem is a key issue in Emergency Medical Service (EMS) system which is to determine where to locateambulances such that the emergency calls can be responded efficiently Most related researches focus on deterministic problems orassume that the probability distribution of demand can be estimated In practice however it is difficult to obtain perfect informationon probability distributionThis paper investigates the ambulance location problem with partial demand information ie only themean and covariance matrix of the demands are known The problem consists of determining base locations and the employmentof ambulances to minimize the total cost A new distribution-free chance constrained model is proposedThen two approximatedmixed integer programming (MIP) formulations are developed to solve it Finally numerical experiments on benchmarks (Nickel etal 2016) and 120 randomly generated instances are conducted and computational results show that our proposed two formulationscan ensure a high service level in a short time Specifically the second formulation takes less cost while guaranteeing an appropriateservice level

1 Introduction

Thedesign of EmergencyMedical Service (EMS) systems thataffects peoplersquos health and life focuses on how to respond toemergencies rapidly Ambulance location problem is one ofthe key problems in EMS system which mainly consists ofdetermining where to locate bases also named as emergencyservice facilities and the employment of ambulances in orderto serve emergencies efficiently and to guarantee patient sur-vivability There have been various researches investigatingambulance location problem (eg [1 2]) since it is introducedby Toregas et al (1974)

Early studies addressing ambulance location problemmainly focus on deterministic environments including setcovering ambulance location problem that minimizes thenumber of ambulances to cover all demand points [3]maximal covering location problem tomaximize the numberof covered demand points with given number of ambulances

[4] and double standard model (DSM) in which eachdemand point must be covered by one or more ambulances[5] However in practice stochastic ambulance locationproblem is more realistic due to the inevitable uncertaintiesof emergency events (eg [6 7] etc) Usually an emergencyevent has the following characteristics (i) it is difficult toforecast precisely where an emergency will occur and (ii) therequired number of ambulances depends on the severity ofthe situation which is also unforeseen [8]

Most existing works investigate stochastic ambulancelocation problem by assuming that the demand probabilitydistribution at each possible emergency is known (eg [910] Beraldi and Bruni 2008 etc) However as stated byWagner [11] and Delage and Ye [12] it is usually impossibleto obtain the perfect information on probability distributiondue to (i) the lack of historical data and the fact that (ii) thegiven historical data may not be represented by probabilitydistribution

HindawiJournal of Advanced TransportationVolume 2018 Article ID 9364129 12 pageshttpsdoiorg10115520189364129

2 Journal of Advanced Transportation

Moreover how to cover as many emergencies as possibleto guarantee a high patient service level (ie the portion ofthe satisfied emergency points among all emergency points)has always been a main goal of EMS operators Motivatedby the above observation this work focuses on a stochasticambulance location problem with only partial informationof demand points ie the mean and covariance matrix ofthe demands Besides we introduce a new individual chanceconstraint which implies a minimum probability that eachemergency demand has to be satisfied For the problema new distribution-free model is presented and then twoapproximated formulations are proposed The contributionof this work mainly includes the following

(1) For the studied stochastic ambulance location prob-lem we introduce a new individual chance con-straint (ie safety level) guaranteeing each emergencydemand satisfied with a least probability while Nickelet al [8] use a coverage constraint where some serviceof individual emergencies may suffer insufficiency

(2) A new distribution-free model based on individualchance constraints is proposed To our best knowl-edge it is the first distribution-freemodel for stochas-tic ambulance location problem

(3) To solve the distribution-free model two approx-imated mixed integer programming (MIP) formu-lations are developed Experimental results showthat our approximated formulations are efficient andeffective for large size instances compared to thatproposed by Nickel et al [8] in terms of both com-putational time and service level

The remainder of this paper is organized as followsSection 2 gives a brief literature review In Section 3 we givethe problem description and propose a new distribution-freemodel In Section 4 two approximated MIP formulationsare proposed Computational results on benchmarks and120 randomly generated instances are reported in Section 5Section 6 summarizes this work and states future researchdirections

2 Literature Review

The deterministic ambulance location problem has been wellstudied in literature (eg [2 5 13] etc) Besides there havebeen some works addressing ambulance location problemunder uncertainty (eg [1 6 14] etc) Since our study fallswithin the scope of stochastic ambulance location problem inthe following subsections we first review existing studies onambulance location problem with uncertain demand Thenwe review the literature studying general and specific stochas-tic optimization problems with distribution-free approaches

21 Ambulance Location Problem with Uncertain DemandAmbulance location problem under uncertain demand hasbeen investigated by many researchers Most existing worksaddress the uncertainty with given scenarios or knownprobability distribution

Chapman and White [15] first investigate the ambulancelocation problem with uncertain demand in which the

complete information on probability distributions is assumedto be known Beraldi et al [9] study the emergency medicalservice location problemwith uncertain demand tominimizethe total cost with amarginal probability distribution Beraldiand Bruni (2008) investigate the emergency medical servicefacilities location problem with stochastic demand based ongiven set of scenarios and known probability distributionA stochastic programming formulation with probabilisticconstraints is proposed Noyan [10] studies the ambulancelocation problem with uncertain demand on given scenariosand probability distribution Then two stochastic optimiza-tion models and a heuristic are proposed

Recently Nickel et al [8] first study the joint optimizationof ambulance base location and ambulance employmentTheobjective is to minimize the total constructing base costsand ambulance employment costs It is assumed that variousscenarios are given beforehand A coverage constraint isintroduced in the paper They propose a sampling approachthat solves a finite number of scenario samples to obtaina feasible solution of the original problem But with thecoverage constraint some emergency demands risk insuffi-cient individual service or cannot be served For the studywe propose a new distribution-free model for stochasticambulance location problem

22 Distribution-Free Approaches In data-driven settingsthe probability distributions of uncertain parameters maynot always be perfectly estimated [16] Therefore in the lastdecade there have beenmany solution approaches developedto address stochastic problems under partial distributionalknowledgeMost related researches focus on the distribution-free approach via considering chance constraintsWagner [11]studies a stochastic 0-1 linear programming under partialdistribution information ie min119883isin01119899c1015840x a1015840119895x le119887119895 119895 = 1 119898 where a119895 are random vectors withunknown distributions The only information on a119895 aretheir moments up to order 119896 A robust formulation asa function of 119896 is given Given the known second-ordermoment knowledge ie 119896 = 2 an approximated formulationis developed as min119883isin01119899c1015840x radicx1015840Γ119895x le radic119901119895(1 minus 119901119895)(119887119895 =E[a119895]1015840x) 119895 = 1 119898 where Γ119895

119894119896= E[(119886119894119895 minus E[119886119894119895])(119886119896119895 minus

E[119886119896119895])] forall119894 119896 = 1 119899 Delage and Ye [12] investigate thestochastic program with limited distribution informationand they propose a new moment-based ambiguity set whichis assumed to include the true probability distribution todescribe the uncertainties There have been various workssuccessfully applying the distribution-free approaches Ng[17] investigates a stochastic vessel deployment problem forliner shipping in which only the mean standard deviationand 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 theliner shipping industry where only the mean and varianceof the uncertain demands are known New models areproposed and the provided bounds are shown to be sharpunder uncertain environment The stochastic dependenciesbetween the shipping demands are considered

Jiang and Guan [19] develop approaches to solve stochas-tic programs with data-driven chance constraints Two types

Journal of Advanced Transportation 3

of confidence sets for the possible probability distributionsare proposed For more distribution-free formulation appli-cations 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 operatingrooms and Zheng et al [24] for stochastic disassembly linebalancing problem To the best of our knowledge there is noresearch for the stochastic ambulance location problem withonly partial information on the uncertain demand

3 Problem Description and Formulation

In this section we first describe the considered problem andthen propose a new distribution-free model

31 Problem Description There is a given set of candidatebase locations 119868 = 1 2 |119868| and a set of potentialemergency demand points 119869 = 1 2 |119869| The emergencypoints may refer to a road a part of the urban area and avillage Base 119894 isin 119868 is said to be covering an emergency point119895 if the driving time 119905119894119895 between 119894 and 119895 is no more than apredetermined value119879The set 119868119895 of candidate bases coveringemergency point 119895 is denoted as 119868119895 = 119894 isin 119868 | 119905119894119895 le 119879

The number of ambulances required by an emergencypoint 119895 isin 119869 is denoted as 119889119895 which depends on the severityof the practical situationWe consider uncertain emergencieswhich are estimated or predicted by partial distributionalinformation thus the demand is regarded to be ambiguousBesides we focus on the case where the historical data cannotbe represented by a precise probability distribution It isassumed that the customer demands are independent of eachother

The objective of the problem is to select a subset ofbase locations and determine the number of ambulances ateach constructed base in order to minimize the total baseconstruction cost and ambulance costThroughout this paperwe assume that only the mean and covariance matrix ofdemands are known Moreover a predefined safety level 120572119895is given for each potential emergency point 119895 isin 119869 That is thenumber of ambulances serving each emergency point 119895 isin 119869 islarger than or equal to its demand with a least probability of12057211989532 Chance Constraint Construction In this section weintroduce a chance constraint to guarantee the safety levelof each emergency demand point In the following 119910119894119895 is adecision variable denoting the number of ambulances servingpoint 119895 isin 119869 from base 119894 isin 119868119895 and 119889119895 is the uncertain demandat point 119895 isin 119869 The chance constrained inequality is presentedas follows

ProbP (sum119894isin119868119895

119910119894119895 ge 119889119895) ge 120572119895 forall119895 isin 119869 (1)

where ProbP(sdot) denotes the probability of the event inparentheses under any potential probability distribution PConstraint (1) ensures that the number of ambulances servingemergency point 119895 isin 119869 is no less than its demand with a leastprobability of 120572119895 (ie safety level)

33 Distribution-Free Formulation In the following we givebasic notations define decision variables and propose thedistribution-free formulationDF for the ambulance locationproblem with uncertain demand

Parameters

119894 index of base locations119895 index of emergency points119868 set of candidate base locations119869 set of possible emergency demand points119905119894119895 driving time between base 119894 isin 119868 and demand point119895 isin 119869119879 maximum driving time for serving any emergencycall119868119895 set of candidate bases that can cover emergencypoint 119895 isin 119869 ie 119868119895 = 119894 isin 119868 | 119905119894119895 le 119879119869119894 set of potential emergency points covered by base119894 isin 119868 ie 119869119894 = 119895 isin 119869 | 119905119894119895 le 119879119891119894 fixed construction cost for installing a base atlocation 119894 isin 119868119892119894 fixed cost associated with an ambulance to belocated at 119894 isin 119868119889119895 number of (stochastic) ambulances requested byemergency point 119895 isin 119869M a sufficiently large positive number

Decision Variables

119909119894 a binary variable equal to 1 if a base is constructedat 119894 isin 119868 0 otherwise119911119894 number of ambulances assigned to possible baselocation 119894 isin 119868119910119894119895 number of ambulances serving emergency point119895 isin 119869 from possible base location 119894 isin 119868119895

Distribution-Free Model [DF]

[DF]

min sum119894isin119868

(119891119894 sdot 119909119894 + 119892119894 sdot 119911119894)st Constraint (1)

(2)

119911119894 le M sdot 119909119894 forall119894 isin 119868 (3)

sum119895isin119869119894

119910119894119895 le 119911119894 forall119894 isin 119868 (4)

119909119894 isin 0 1 forall119894 isin 119868 (5)

119911119894 isin Z+ forall119894 isin 119868 (6)

119910119894119895 isin Z+ forall119894 isin 119868 119895 isin 119869119894 (7)

4 Journal of Advanced Transportation

The objective function denotes the goal to minimize thetotal cost consisting of two parts (i) the cost for constructingbases ie sum119894isin119868 119891119894 sdot 119909119894 and (ii) the cost for deploying ambu-lances ie sum119894isin119868 119892119894 sdot 119911119894

Constraint (3) ensures that ambulances can only belocated at the opened bases Constraint (4) ensures that thenumber of ambulances sent to serve emergency points frombase 119894 isin 119868 does not exceed the total number of ambulanceslocated at 119894 isin 119868 Constraints (5)-(7) are the restrictions ondecision variables

4 Solution Approaches

The proposed distribution-free model is difficult to solvewith the commercial software due to chance constraints Inthis section we propose two approximatedMIP formulationsbased on those in Wagner [11] and in Delage and Ye [12]respectively In the following subsections we present the twoapproximated MIP formulations

41 Approximated MIP Formulation MIP-DF1 In this partwe first describe a widely used ambiguity set to describe theuncertainty Then an approximated MIP formulation MIP-DF1 is proposed

Given a set of independent historical data samples 119889119904|119878|119904=1of random vectors of demands where 119878 is the set of sampleindexes the mean vector 120583 and the covariance matrix Σ ofdemands can be estimated as follows

120583 = 1|119878|sum119904isin119878119889119904Σ = 1|119878|sum119904isin119878 (119889119904 minus 120583) (119889119904 minus 120583)⊤

(8)

where (sdot)⊤ implies the transposition of the vector in paren-theses Then an ambiguity set of all probability distributionsof demandsP1(120583 Σ) can be described as the following

P1 (120583 Σ) = P EP [119889] = 120583EP [(119889 minus 120583) (119889 minus 120583)⊤] = Σ (9)

whereP denotes a possible probability distribution satisfyingthe given conditions and EP[sdot] denotes the expected valueof the number in parentheses Then the chance constraint (1)with the ambiguity setP1 can be presented as follows

ProbP (sum119894isin119868119895

119910119894119895 ge 119889119895) ge 120572119895 forall119895 isin 119869 P isin P1 (10)

According to Wagner (2010) and Ng [18] constraint (1) canbe conservatively approximated by the following

sum119894isin119868119895

119910119894119895 ge 120583119895 + 120590119895 sdot radic 120572119895(1 minus 120572119895) forall119895 isin 119869 (11)

where 120590119895 = radicΣ119895119895 and Σ119895119895 denotes the 119895-th element in the 119895-thcolumn of matrix Σ In terms of conservative approximation

all solutions satisfying constraint (11) must satisfy constraint(1) Then we describe the approximated MIP formulationMIP-DF1 toDF combined with constraint (11)

[MIP-DF1]

min sum119894isin119868

(119891119894 sdot 119909119894 + 119892119894 sdot 119911119894)st Constraints (4) - (8) (12)

(12)

MIP-DF1 can be exactly solved by the commercialsoftware such as CPLEX As we can observe from thecomputational results in Section 5 it can obtain a relativelyhigh service level However the system cost is also high Inorder to save the system cost another approximated MIPformulation of DF is then proposed in the next subsection

42 Approximated MIP Formulation MIP-DF2 AmbiguitysetP1 focuses on exactly matching the mean and covariancematrix of uncertain parameters However in practice theremay be considerable estimation errors in the mean andcovariance matrix To take the inevitable estimation errorsinto consideration Delage and Ye [12] introduce a newmoment-based ambiguity set Therefore we in the followingemploy the moment-based ambiguity set

P2 (120583 Σ 1205741 1205742)= P (EP [119889] minus 120583)⊤ Σminus1 (EP [119889] minus 120583) le 1205741

EP [(119889 minus 120583) (119889 minus 120583)⊤] ⪯ 1205742Σ (13)

where 1205741 ge 0 and 1205742 ge 0 are two parameters of ambiguityset P2(120583 Σ 1205741 1205742) with the following assumptions (i) thetrue mean vector of demands is within an ellipsoid of sizeproportion to 1205741 centered at 120583 and (ii) the true covariancematrix of demands is in a positive semidefinite cone boundedby amatrix inequality of 1205742ΣThe ambiguity set describes howlikely the uncertain parameters are to be close to the meanin terms of the correlation [12] Besides under the moment-based ambiguity set various stochastic programs with partialdistributional information have been successfully modeledand solved [22 23 25]

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

radic 11 minus 119886 minus 119887 sdot (1 + radic1 minus 120572119895120572119895 sdot 119887) sdot 120590119895le radic1 minus 120572119895120572119895 sdot (sum

119894isin119868119895

119910119894119895 minus 120583119895) 119895 isin 119869(14)

where

119886 = 1 minus 1205741 + 11205742 minus 1205741 119887 = 12057411205742 minus 1205741

(15)

Journal of Advanced Transportation 5

Table 1 120572 = 095|119868| = |119869| Sampling approach MIP-DF1 MIP-DF2

CT Obj Sel () CT Obj Sel () CT Obj Sel ()1 7 49 23 9626 01 36 10000 01 23 100002 7 51 22 9863 01 35 10000 01 24 99823 6 41 14 9692 01 30 10000 01 14 98504 6 43 13 9713 02 29 10000 01 12 97425 6 57 12 7996 01 30 10000 02 12 100006 8 58 21 9850 01 24 10000 02 27 10000

Average 50 175 9457 01 307 10000 01 187 9929

The second approximated MIP formulationMIP-DF2 toDF is presented as follows

[MIP-DF2]

min sum119894isin119868

(119891119894 sdot 119909119894 + 119892119894 sdot 119911119894)st Constraints (4) - (8) (15)

(16)

Notice that when 1205741 = 0 and 1205742 = 1 inequality (14)reduces to (11) in MIP-DF1 which implies that MIP-DF1 isa special case of MIP-DF2

5 Computational Experiments

In this section the performance of our proposed formula-tions first evaluated and compared to the sampling approachproposed in Nickel et al [8] Specifically given a requiredservice level 120572 we apply our MIP formulations by restrictingthe chance constraint for each emergency point to satisfya safety level 120572119895 = 120572 while we adopt the samplingapproach by restricting the coverage constraint with 120572 Thencomputational results on 120 randomly generated instancesbased on the information on emergencies in Shanghai arereported Our approximated MIP formulations and thesampling approach are solved by coding in MATLAB 2014bcalling CPLEX 126 solver All numerical experiments areconducted on a personal computer with Core I5 and 330GHzprocessor and 8GB RAMunder windows 7 operating systemThe computational time for sampling approach is limited to3600 seconds

51 Out-of-Sample Test FollowingZhang et alrsquos [22]method-ology we examine the out-of-sample performance of solu-tions obtained by our proposed approximated MIP formula-tions and the sampling approach [8] The out-of-sample testincludes three steps

(1) The base locations and the employment of ambu-lances at each base are determined with known meanvector and covariance matrix of demands which isnamed as in-sample test

(2) 1000 scenarios are sampled from the underlying truedistribution which represent realizing the uncertaindemand

(3) Based on the solution obtained in step (1) ambu-lances are assigned to serve emergency points inthe 1000 scenarios to maximize the (out-of-sample)service level

The (out-of-sample) service level is defined as the ratioof the number of emergency points with demand satisfiedto the total number of emergency points in 1000 scenariosIn the following subsections the benchmarks are tested andthen three common distributions are employed ie uniformdistribution Poisson distribution and normal distributionrespectively

52 Benchmark Instances We first test the benchmarks inNickel et al [8] to compare our proposed two approximatedMIP formulationswith the sampling approach InNickel et al[8] they assume that demand nodes are also potential baselocations and the number of ambulances required by eachemergency point is chosen from set 0 1 2with given proba-bility distribution Computational results on benchmarks arereported in Tables 1 and 2

Table 1 reports the computational results with servicelevel requirement120572 = 095 In Table 1 CTObj and Sel denotethe computational time objective value (ie the system cost)and the service level respectively The first column includesthe indexes of instances and the second indicates the numberof nodes It shows that the average service level obtainedby the sampling approach is 9457 which is smaller thanthe required service level ie 85 Specifically the servicelevel of instance 5 obtained by the sampling approach is only7996 However MIP-DF1 and MIP-DF2 can guarantee theservice levels for all instances higher than the requirementConcluding from Table 1 we can observe that (i) the averagecomputational time of sampling approach is about 50 timesas those of the two approximated MIP formulations and (ii)MIP-DF1 and MIP-DF2 improve the average service level byabout 574 and 499 compared with sampling approachhowever (iii) system costs obtained by MIP-DF1 and MIP-DF2 are respectively 7543 and 686 higher than that ofsampling approach

Table 2 reports the computational results with servicelevel requirement 120572 = 099 For instances 2-6 the ser-vice levels obtained by sampling approach cannot meet therequirement ie 99 Besides it shows that the averagecomputational time of sampling approach is 48 which isabout 48 times greater than those of MIP-DF1 andMIP-DF2

6 Journal of Advanced Transportation

Table 2 120572 = 099|119868| = |119869| Sampling approach MIP-DF1 MIP-DF2

CT Obj Sel () CT Obj Sel () CT Obj Sel ()1 7 49 24 9979 01 64 10000 01 25 100002 7 48 23 9856 01 63 10000 01 24 100003 6 51 15 9717 01 55 10000 01 16 100004 6 47 15 9275 01 52 10000 01 13 100005 6 40 14 9000 01 55 10000 01 14 100006 8 52 23 9785 01 45 10000 01 27 10000

Average 48 190 9343 01 557 10000 01 198 10000

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

Sampling approach MIP-DF1 MIP-DF2Emergency 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 0Base locations Ambulance employment Ambulance employment Ambulance employment1 0 0 02 1 1 0 3 1 23 1 1 26 1 2 2 04 0 0 05 0 6 06 2 2 0 4 2

The average service level is 9343 Although the averageservice levels obtained by MIP-DF1 and MIP-DF2 are both100 the average system cost obtained by MIP-DF1 is about3 times larger than that of sampling approach Besides theaverage system cost obtained by MIP-DF2 is 421 higherthan that of sampling approach

Concluding for the benchmarks described in Nickel etal [8] we can observe that (1) MIP-DF1 and MIP-DF2 canbe solved in far less computational time compared with thesampling approach (2) the sampling approach cannot meetthe requirement of service level for all benchmarks whilethe two MIP formulations can (3) the average system cost ofMIP-DF2 is very close to that of sampling approach (4) withthe increase of 120572 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 ofthe lower system cost obtained by the sampling approachthere may exist extreme situations that demands of someemergency points are not satisfied Take one benchmark inNickel et al [8] ie instance 4 where there are 6 nodesA scenario in which demand at each emergency point isgenerated from the given probability distribution is shown inFigure 1 We can observe from Figure 1 that under the givenscenario the demand of each possible emergency point is1 0 2 0 2 0The solutions obtained by sampling approachMIP-DF1 andMIP-DF2 are shown in Table 3 which includesbase 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 poten-tial base locations The second column represents the baselocations and the ambulance employment at each location

＞1 = 1

＞2 = 0

＞3 = 2

＞4 = 0

＞5 = 2

＞6 = 0

1

2 4

3 5

6

Figure 1 Instance 4 in Nickel et al [8] under given scenario

in which a location with 0 ambulance implies that it is notselected for base construction From Table 3 we can observethat the in-sample solutions obtained by sampling approachis to construct bases at locations 2 3 and 6 where the numberof ambulances is 1 1 and 2 respectively Besides undergiven scenario there is one ambulance serving emergencypoint 1 from base location 3 and one ambulance servingemergency point from base location 2 and 2 ambulancesserving emergency point 5 from base location 6 It shows thatthe requirement of ambulances at emergency point 3 is notfully satisfied The insufficient service may cause serious lossof life However we can find that the solutions obtained byMIP-DF1 andMIP-DF2 can provide sufficient ambulances forall emergency points

In sum to guarantee the required service level withless cost MIP-DF2 is recommended for the solutions with

Journal of Advanced Transportation 7

Nickel et al [8]rsquos data set Moreover in the benchmarks itis assumed that all nodes are possible emergency points aswell as potential base locations and the number of requiredambulances is small which is not always consistent withthe situation in Shanghai Therefore we further test the twoapproximated MIP formulations and the sampling approachon randomly generated instances by analysing the informa-tion on emergencies in Shanghai [26]

53 Computational Tests on Randomly Generated InstancesIn the following subsection numerical experiments on 120randomly generated instances are described

531 Test Instances By analysing the information on emer-gencies in Shanghai [26] the mean value 120583119895 of demand 119889119895 ineach emergency point are randomly generated fromadiscreteUniform distribution on interval [5 25] The standard devia-tion 120590119895 is set to be 5 Since the sampling approach proposed inNickel et al [8] requires a sample of scenarios in numericalexperiments scenarios are produced with demands randomlygenerated from Uniform distribution Poisson distributionand Normal distribution consistent with given mean andstandard deviation

According to the instance settings in Erkut and Ingolfsson(2008) response time from a base to an emergency pointis randomly generated from a discrete Uniform distributionon interval [3 30] in units of minutes A base covers anemergency point if the driving time between them is lessthan 10 minutes to ensure the survival probability (Erkut andIngolfsson 2006) For an emergency point in each instanceif there is no base with 10 minutes driving time to it then thebase covering this point is set to be the nearest one The costof constructing a base and that of locating one ambulance areboth set to be 1 as in Nickel et al [8] Besides the number ofemergency points is 3 times that of potential base locations

532 Results and Discussion Different values of requiredservice level 120572 are tested ie 085 09 095 Computationalresults on 120 randomly generated instances are reported inTables 4-6

Table 4 reports the numerical results of instances withdemand generated from Uniform distribution We observethat when 120572 = 085 09 095 the average computationaltimes of the sampling approach are 4482 4379 and 4295seconds respectively However both approximated MIP for-mulations can produce solutions in much faster speeds iewithin 13 seconds on average With the increase of 120572 (i)the out-of-sample average service level is getting higher and(ii) the system costs of solutions obtained by all method aregetting larger and (iii) the system costs of solutions yieldedbyMIP-DF2 is getting closer to those of solutions obtained bythe sampling approach In terms of the average system costthe sampling approach performs the best while MIP-DF1 isthe poorest When 120572 = 085 for example the average systemcost ofMIP-DF2 and the sampling approach are about 3032and 442 less than that of MIP-DF1 respectively When120572 = 095 the above gaps are even larger Concluding fortheUniformdistribution (1) the proposed two approximatedMIP formulations are far more time saving than the sampling

approach (2)MIP-DF1 andMIP-DF2 can improve the servicelevel by about 1907 compared with the sampling approachhowever (3) despite of the highest service levels obtainedby MIP-DF1 the system cost obtained by MIP-DF1 is alsohigh (4) MIP-DF2 can ensure an appropriate service levelwith much less cost than MIP-DF1 and (5) for the situationswith small-scale instances and high service level requirementMIP-DF1 is recommended

Computational results of instances with demand at eachemergency point generated from Poisson distribution arepresented in Table 5 For MIP-DF1 and MIP-DF2 onlymean and variance are involved Therefore with givenmean and standard deviation of the demand at each pointcomputational results are the same for different probabilitydistributions We can obtain from Table 5 that the objectivevalue obtained by the sampling approach increases with theincrease of 120572 Besides when the value of 120572 equals 085the average service level obtained by the sampling approachis about 8143 which is smaller than that of the twoapproximated MIP formulations The highest service level9934 is obtained by MIP-DF1 and that obtained by MIP-DF2 is 9708 When 120572 = 09 the average service levelof the sampling approach is 8437 and those obtained byMIP-DF1 and MIP-DF2 are 9985 and 9797 respectivelyWhen 120572 = 095 the average service level obtained by thesampling approach is 9356 and those yielded by MIP-DF1 and MIP-DF2 are 9997 and 9781 respectively ForPoisson distribution concluding MIP-DF1 and MIP-DF2can improve the service level compared with the samplingapproach especially MIP-DF1 can obtain the highest servicelevel

Similarly Table 6 reports the computational results forinstances in which the demand at each emergency point isgenerated following Normal distribution When 120572 = 085the average service levels obtained by the sampling approachMIP-DF1 and MIP-DF2 are 8174 9925 and 9742respectively When 120572 = 09 the three values are 84579981 and 9743 respectively Moreover for 120572 = 095 thevalues are equal to 8732 9999 and 9794 respectivelyIt can be observed that the performance of the samplingapproach is the poorest in terms of the service level

By observing the above computational results on bench-marks and randomly generated instances we conclude that(1)The proposed approximated MIP formulations ie MIP-DF1 and MIP-DF2 are very time saving compared with thesampling approach (2) MIP-DF1 an MIP-DF2 can achievethe requirement of service level and improve the service levelby about 1547 and 1229 on average compared with thatof the sampling approach (3) MIP-DF1 is more conservativein terms of overhigh emergency service level close to 100and the system cost is also high which is about 568 largerthan that of MIP-DF2 (4) Compared with MIP-DF1 MIP-DF2 is less conservative and much more cost saving whileensuring an appropriately service level on average which isabout 97

In sum for medium and large problem instances (egShanghai) which are common in practice to guarantee therequired service level with less cost we recommendMIP-DF2as solution method However for very small instances with

8 Journal of Advanced Transportation

Table4Com

putatio

nalresultson

insta

nces

with

datageneratedfro

mun

iform

distrib

ution

Set (|119868||119869|)

120572=085

120572=09

120572=095

Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2CT

Obj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

1(26)

27

897317

01

149

9906

01

107

9222

28

967373

01

155

9933

01

107

9167

27

106

7585

01

185

9987

01

107

9128

2(412

)47

169

7423

00

291

9919

00

207

9122

58

181

7438

01

303

9997

01

207

9363

50

199

7702

00

363

9982

00

207

9338

3(618

)72

258

7506

01

438

9976

01

312

9106

75279

8082

01

456

9996

01

312

9369

78304

8337

01

546

9997

01

312

9357

4(824)

99314

7956

01

553

9993

01

385

9206

111

338

8202

01

577

9999

01

385

9602

112

366

8492

01

697

9994

01

385

9620

5(1030)

132

415

8406

01

722

9896

01

512

9406

143

441

8322

02

752

9983

01

512

9692

142

476

8609

01

902

9993

01

512

9692

6(1236)

170

475

8249

02

843

9870

01

591

9367

181

504

8251

01

879

9982

01

591

9608

189

535

8726

01

1059

9997

01

591

9609

7(14

42)

215

573

8111

02

1006

9849

02

712

9606

243

609

8217

02

1048

9953

02

712

9589

233

644

8608

02

1258

9992

02

712

9582

8(1648)

248

686

7532

02

1185

9896

02

849

9725

284

727

8307

02

1233

9949

02

849

9613

281

768

9063

02

1473

9999

03

849

9622

9(1854)

313

691

8353

03

1236

9974

04

858

9709

351

731

8397

03

1290

9988

03

858

9765

352

760

8906

03

1560

9994

03

858

9758

10(2060)

372

794

8174

03

1409

9887

03

989

9657

428

843

8489

03

1469

9947

03

989

9773

432

892

9004

03

1769

9997

03

989

9738

11(2266

)451

830

8072

04

1493

9949

03

1031

9758

518

880

8328

04

1559

9982

03

1031

9723

516

930

8967

05

1889

9995

03

1031

9727

12(2472)

543

1025

7970

04

1771

9927

04

1267

9835

601

1083

8482

05

1843

9949

04

1267

9776

637

1146

9014

04

2203

9993

04

1267

9777

13(2678)

631

1026

8142

05

1819

9898

04

1274

9782

743

1087

8678

05

1897

9936

04

1274

9778

765

1146

9023

04

2287

9997

05

1273

9837

14(2884)

782

1101

7903

05

1949

9869

08

1361

9804

878

1165

8790

05

2033

9935

07

1361

9889

870

1227

9056

05

2453

9998

06

1361

9808

15(3090)

907

1091

8098

06

1997

9915

06

1367

9629

1018

1155

8329

06

2087

9981

06

1367

9807

1020

1225

8986

06

2537

9999

08

1367

9817

16(3296)

1062

1160

8798

07

2123

9944

07

1449

9787

1183

1229

8383

07

2219

9988

06

1449

9849

1193

1300

9105

06

2698

9998

06

1449

9815

17(34102)

1202

1298

7572

07

2328

9969

07

1614

9821

1377

1374

8860

08

2430

9997

07

1614

9824

1260

1452

9085

07

2940

9999

08

1614

9818

18(36108)

1394

1408

8121

08

2509

9947

08

1752

9809

1544

1492

8928

08

2617

9983

08

1752

9799

1548

1580

9173

08

3156

9999

08

1752

9809

19(38114

)1618

1508

8170

09

2676

9899

111877

9742

1760

1598

9130

09

2790

9934

08

1877

9908

1755

1688

9186

09

3360

9999

09

1877

9872

20(40120)

2206

1628

7719

102869

9989

102029

9873

2007

1732

8753

112990

9997

09

2029

9879

1904

1827

9138

103590

9998

09

2029

9875

21(42126)

2780

1622

8354

102900

9978

102017

9732

2609

1718

8854

103026

9876

102017

9818

2670

1812

9089

123655

9999

102017

9848

22(44132)

3189

1735

8494

113088

9985

152165

9857

2970

1840

8922

123220

9925

122165

9886

2873

1952

9206

113880

9999

112165

9850

23(46138)

3630

1937

8141

123372

9893

132406

9796

3222

2048

8937

123510

9934

142406

9906

3461

2170

8989

124199

10000

122406

9809

24(48144)

4226

1968

8226

133461

9932

132453

9814

3674

2085

8846

133605

9956

152453

9912

3556

2211

9300

134326

9999

132453

9819

25(50150)

4727

1919

8111

143456

9999

152407

9629

4019

2037

8699

143606

9999

142407

9857

4257

2161

9259

154356

9995

142407

9819

26(52156)

5300

1906

8096

153484

9878

172392

9658

4495

2021

8952

143640

9999

152392

9568

4486

2142

9182

1544

209999

152392

9854

27(54162)

5913

2096

8387

163751

9846

172617

9761

5117

2221

9045

163913

9998

172617

9544

4830

2355

9346

184723

9999

162617

9893

28(56168)

6310

2110

7758

173819

9933

192643

9829

5777

2234

9157

183987

9998

182643

9526

5550

2365

9045

184827

9999

172643

9831

29(58174)

6780

2175

7998

183937

9898

182720

9714

6303

2307

9022

20

4111

9969

192720

9623

6189

2444

9128

194981

10000

192720

9914

30(60180)

7593

2384

8321

194240

9967

20

2979

9841

6791

2527

8978

1944

199979

23

2979

9721

6878

2679

9143

21

5316

9999

20

2979

9869

31(62186)

8220

2281

7604

20

4157

9984

21

2856

9796

7734

2416

8934

21

4343

9997

21

2856

9679

7466

2557

9158

22

5273

9999

23

2856

9902

32(64192)

8766

2517

8378

21

4489

9932

25

3145

9729

8276

2671

8296

23

4680

9977

24

3145

9825

7897

2822

9089

24

5640

9999

22

3145

9891

33(66198)

10123

2532

7806

23

4555

9967

23

3169

9615

9010

2682

8947

24

4752

9989

24

3169

9703

9051

2835

9157

24

5742

9998

24

3169

9867

34(68204)

11010

2535

7934

24

4602

9999

29

3174

9631

9898

2685

8858

30

4806

9998

28

3174

9762

9863

2844

9236

26

5824

9999

25

3174

9889

35(70210)

11666

2714

8062

25

4862

9986

29

3391

9787

1066

62875

9137

26

5072

9999

29

3391

9716

10822

3043

9315

26

6122

9999

29

3391

9923

36(72216)

12011

2827

8104

26

5047

9991

28

3535

9798

1246

52984

9016

28

5263

9999

28

3535

9878

11667

3169

9253

30

6343

9999

29

3535

9896

37(74222)

11437

2826

8228

28

5096

9933

29

3543

9884

13582

3098

8895

29

5318

9999

32

3543

9888

12654

3178

9139

29

6427

10000

31

3543

9893

38(76228)

12435

2839

8152

29

5161

9897

29

3565

9788

13976

3103

8926

29

5388

9999

31

3565

9724

13837

3185

9255

33

6528

10000

31

3565

9895

39(78234)

1464

32945

8412

31

5331

9998

32

3695

9876

14794

3210

9005

31

5565

9999

34

3695

9713

14561

3313

9081

34

6735

10000

33

3695

9878

40(80240)

1604

22975

8510

32

5414

9957

36

3734

9796

16263

3346

8999

35

5654

9999

35

3734

9806

15869

3343

9248

35

6854

10000

35

3734

9873

Average

4482

1584

680

67

1228

397

9936

131978

796

82

4379

16913

8654

1329

626

9975

131978

79721

4295

1778

889

85

1335774

9997

131978

79775

Journal of Advanced Transportation 9

Table5Com

putatio

nalresultson

insta

nces

with

datageneratedfro

mPo

isson

distr

ibution

set ( |119868 |

|119869 |)120572=

085120572=

09120572=

095Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2CT

Obj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

1(26)

27

907294

9967

9001

27

987770

9979

9013

24

103

9273

9999

9468

2(412

)46

170

7961

9921

9117

51

183

8305

9946

9128

49

201

9392

9998

9546

3(618

)68

258

7578

9938

8911

74280

8028

9968

9004

69

307

9182

9999

9658

4(824)

93315

8001

9928

9196

102

338

8438

9967

9172

92368

9468

9999

9613

5(1030)

120

413

8041

9939

9449

133

443

8447

9976

9272

121

479

9544

9999

9752

6(1236)

155

476

8084

9877

9554

167

508

8863

9991

9569

159

536

9456

9999

9734

7(14

42)

195

573

7876

9892

9663

213

609

9124

9998

9729

205

643

9314

10000

9768

8(1648)

232

687

7964

9997

9788

267

724

8523

9988

9687

238

772

9422

9998

9689

9(1854)

293

692

8267

9893

9809

327

730

8386

9999

9750

298

757

9228

9999

9755

10(2060)

356

796

7946

9899

9801

399

840

8299

9976

9686

360

885

9668

9999

9705

11(2266)

423

829

8201

9895

9756

489

877

8378

9987

9788

448

927

9476

9999

9812

12(2472)

512

1023

8338

9913

9853

554

1085

8501

9999

9881

514

1148

9398

9999

9888

13(2678)

592

1045

8416

9907

9873

687

1088

7998

9996

9802

607

1152

9504

9999

9821

14(2884)

723

1099

7946

9915

9788

838

1166

8056

9992

9829

776

1230

9546

9999

9839

15(3090)

855

1089

7915

9964

9703

942

1156

8432

9984

9778

903

1225

9223

9998

9785

16(3296)

1009

1158

8024

9897

9769

1163

1231

8500

9996

9824

1045

1222

9199

9999

9835

17(34102)

1165

1299

8034

9919

9779

1285

1375

8495

9937

9808

1190

1457

9258

10000

9846

18(36108)

1254

1411

8176

9943

9885

1479

1492

8561

9968

9868

1345

1579

9286

9999

9875

19(38114

)1504

1506

7970

9923

9842

1651

1594

8167

9972

9846

1564

1694

9376

9999

9786

20(40120)

1706

1635

7819

9916

9783

1854

1734

8607

9968

9776

1778

1825

9285

9904

9718

21(42126)

2054

1622

8354

9918

9834

2357

1720

8297

9978

9833

2226

1856

9501

9999

9834

22(44132)

2400

1736

8494

9922

9846

2852

1844

8269

9999

9816

2680

1946

9486

9999

9820

23(46138)

2651

1934

8141

9953

9793

3159

2049

8492

9969

9799

3072

2172

9358

10000

9813

24(48144)

3004

1961

8226

9947

9681

3522

2086

8513

9998

9706

3102

2209

9488

9999

9724

25(50150)

3308

1924

8111

9965

9746

3898

2038

8264

9999

9788

3639

2161

9546

9999

9813

26(52156)

3669

1911

8096

9939

9855

4305

2027

8722

9967

9874

4453

2142

9185

9999

9876

27(54162)

4418

2095

8387

9927

9754

4771

2219

8341

9989

9749

4806

2353

9289

9999

9755

28(56168)

5367

2113

8458

9968

9848

5356

2237

8452

9978

9855

5221

2369

9228

10000

9868

29(58174)

6217

2186

7998

9989

9694

5942

2315

8316

9996

9699

5853

2371

9167

9999

9705

30(60180)

7263

2382

8321

9991

9837

6527

2543

8359

9998

9886

6214

2678

9369

9997

9886

31(62186)

8214

2346

8404

9936

9888

7813

2430

8321

9999

9846

6876

2560

9358

9999

9847

32(64192)

8863

2518

8378

9938

9807

8106

2677

8356

9999

9833

7877

2822

9408

10000

9835

33(66198)

9985

2532

8406

9917

9709

9684

2689

8497

9996

9746

8088

2838

9507

9999

9749

34(68204)

11264

2534

8543

9908

9897

10986

2685

8809

9996

9782

8469

2843

8844

9998

9780

35(70210)

12035

2715

8258

9937

9799

12098

2886

8569

9991

9809

9946

3041

9459

9999

9813

36(72216)

12803

2825

8104

9929

9827

13076

3010

8651

9994

9834

11013

3170

9468

9999

9865

37(74222)

13021

2830

8228

9949

9826

13261

3009

8376

9999

9826

12014

3172

9346

9999

9888

38(76228)

13862

2843

8152

9933

9807

13995

3028

8760

9998

9841

12133

3187

9706

9999

9876

39(78234)

15387

2953

8312

9964

9799

14827

3133

9001

9995

9783

13381

3304

8913

9999

9799

40(80240)

16974

2975

8510

9985

9767

15256

3249

8217

9993

9781

15867

3347

9111

9999

9791

Average

4352

15875

8143

9934

9708

4362

16856

8437

9985

9707

3968

17763

9356

9997

9781

10 Journal of Advanced Transportation

Table6Com

putatio

nalresultson

insta

nces

with

datageneratedfro

mno

rmaldistr

ibution

set ( |119868 |

|119869 |)120572=

085120572=

09120572=

095Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2CT

Obj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

1(26)

27

907294

9964

9117

27

947844

9970

9202

2558

106

8422

10000

9208

2(412

)46

170

7961

9921

9283

48

183

8328

9982

9283

4508

183

8823

10000

9370

3(618

)68

258

7578

9916

9228

68

280

7798

9971

9417

6803

307

7658

9998

9447

4(824)

93315

8000

9899

9367

96337

8379

9979

9467

9373

368

8438

9999

9579

5(1030)

120

413

7841

9927

9437

124

443

8763

9963

9476

1262

499

8850

9998

9708

6(1236)

155

476

7921

9919

9519

162

504

8124

9982

9582

16021

536

8954

10000

9756

7(14

42)

195

573

7976

9933

9598

196

609

8287

9975

9619

2011

646

8467

9999

9767

8(1648)

232

687

8031

9918

9705

235

727

8115

9965

9650

2404

772

8278

9999

9765

9(1854)

293

692

8086

9898

9759

291

734

8464

9974

9755

29815

796

8617

9998

9835

10(2060)

356

796

7984

9937

9770

358

842

8223

9988

9765

37973

921

8846

10000

9838

11(2266

)421

829

8101

9940

9755

436

877

8211

9986

9786

4514

8959

8497

9999

9789

12(2472)

514

1023

8033

9951

9785

501

1085

8162

9983

9788

52073

1147

8321

9999

9824

13(2678)

592

1023

8138

9916

9885

599

1087

8411

9970

9845

62937

1194

8609

9998

9795

14(2884)

728

1099

8235

9913

9915

722

1168

8435

9987

9806

75053

1270

8567

9997

9834

15(3090)

855

1089

8187

9922

9861

880

1157

8376

9976

9796

89222

1300

8611

9996

9798

16(3296)

1009

1158

8770

9897

9788

966

1231

8845

9984

9823

106288

1359

8997

9999

9848

17(34102)

1154

1299

8434

9925

9849

1137

1377

8601

9980

9798

123869

1493

8798

9996

9877

18(36108)

1407

1411

8321

9927

9866

1339

1495

8301

9965

9822

133807

1601

8567

9999

9834

19(38114

)1702

1506

8206

9915

9805

1672

1598

8977

9978

9845

152999

1694

8998

9999

9847

20(40120)

2091

1635

8528

9918

9889

1916

1737

8805

9976

9837

1890

021833

9012

9999

9791

21(42126)

2843

1622

8550

9934

9913

2432

1718

8886

9974

9813

2399

131815

9008

10000

9815

22(44132)

3300

1736

8449

9940

9859

2795

1843

8786

9977

9756

253106

1948

8997

9999

9826

23(46138)

3741

1934

8133

9911

9876

3143

2050

8876

9981

9785

274583

2173

9014

9999

9858

24(48144)

4531

1961

8245

9909

9846

3515

2090

8344

9993

9819

308089

2211

8661

9999

9883

25(50150)

4715

1924

8111

9929

9888

3959

2041

8279

9996

9837

3510

932158

8527

9999

9827

26(52156)

5223

1911

8096

9898

9857

4301

2025

8251

9997

9757

421747

2146

8497

9999

9914

27(54162)

5994

2095

8387

9908

9775

4835

2226

8497

9974

9780

4614

432359

8695

9999

9837

28(56168)

6406

2111

8411

9906

9798

5336

2240

8558

9991

9820

494113

2371

9249

9999

9816

29(58174)

6897

2176

8012

9916

9791

5948

2311

8211

9984

9765

624835

2445

8531

10000

9851

30(60180)

7436

2391

8111

9913

9766

6558

2533

8305

9968

9795

696116

2682

8575

9999

9799

31(62186)

8118

2282

8119

9928

9806

7069

2422

8417

9973

9873

797768

2564

8667

9999

9846

32(64192)

8895

2519

8223

9919

9757

7857

2675

8631

9982

9769

864813

2827

8976

10000

9905

33(66198)

11235

2533

8563

9928

9811

9162

2688

8547

9979

9828

973089

2843

8657

9999

9846

34(68204)

11587

2534

8249

9926

9847

9549

2686

8798

9938

9856

10523

3041

8999

10000

9877

35(70210)

1164

92715

8098

9895

9872

10345

2880

8602

9998

9869

11272

3052

9021

9999

9897

36(72216)

12224

2825

8403

9958

9770

11165

3002

8657

9999

9839

12456

3175

9113

9999

9864

37(74222)

12978

2830

8357

9937

9795

11850

3005

8496

9998

9894

13458

3176

8827

9999

9905

38(76228)

13426

2844

8267

9966

9759

12795

3019

8604

9999

9797

15155

3121

8978

9999

9853

39(78234)

15003

2955

8303

9956

9867

13816

3132

8588

9996

9885

15041

3310

9016

9999

9912

40(80240)

16495

2978

8257

9954

9826

14896

3160

8495

9997

9837

16563

3349

8957

9999

9933

Average

4619

15855

8174

9925

9742

4077

16828

8457

9981

9743

4350

17938

8732

9999

9794

Journal of Advanced Transportation 11

high service level requirement we recommend MIP-DF1 forsolutions

6 Conclusion

This paper investigates an ambulance location problem withambiguity set of demand in which the probability distribu-tion of the uncertain demand is unknown In such a data-driven environment only the mean and covariance demandsat emergency points are knownThe problem is to determinethe base locations and the number of ambulances at eachbase aiming at minimizing the total cost associated withlocating bases and assigning ambulances We propose adistribution-free model with chance constraints Then twoapproximated MIP formulations with different approxima-tion methods of chance constraints are proposed which arebased on different ambiguity sets of the unknown probabilitydistribution Numerical experiments on benchmarks and120 randomly generated instances are conducted and thecomputational results show that the first approximated MIPformulation is much conservative with overhigh system costwhile the second approximatedMIP formulation ismore costsaving with service level appropriately ensured

In future research wemay consider the following relevantissues First the uncertainty of driving time from a base toan emergency point shall be analysed Besides unexpected orsudden disasters may be taken into consideration Moreoverother uncertainties should be considered such as travel timethe number of available vehicles and so on The stochasticdependence should also further considered

Data Availability

The data of the benchmark instances in our manuscript canbe obtained in Nickel et al [8] For the test instances thedata is randomly generated and the way for generating data isstated in our manuscript The datasets generated during thecurrent study are available from the corresponding author onreasonable request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China (NSFC) under Grants 7142800271531011 71571134 and 71771048 This work was also sup-ported by the Fundamental Research Funds for the CentralUniversities

References

[1] L Brotcorne G Laporte and F Semet ldquoAmbulance loca-tion and relocation modelsrdquo European Journal of OperationalResearch vol 147 no 3 pp 451ndash463 2003

[2] X Li Z Zhao X Zhu and T Wyatt ldquoCovering modelsand optimization techniques for emergency response facilitylocation and planning a reviewrdquo Mathematical Methods ofOperations Research vol 74 no 3 pp 281ndash310 2011

[3] C Toregas R Swain C ReVelle and L Bergman ldquoThe locationof emergency service facilitiesrdquoOperations Research vol 19 no6 pp 1363ndash1373 1971

[4] R Church and C ReVelle ldquoThe maximal covering locationproblemrdquo Papers in Regional Science vol 32 no 1 pp 101ndash1181974

[5] M Gendreau G Laporte and F Semet ldquoSolving an ambulancelocation model by tabu searchrdquo Location Science vol 5 no 2pp 75ndash88 1997

[6] A Ingolfsson S Budge and E Erkut ldquoOptimal ambulancelocation with random delays and travel timesrdquo Health CareManagement Science vol 11 no 3 pp 262ndash274 2008

[7] M Gendreau G Laporte and F Semet ldquoThemaximal expectedcoverage relocation problem for emergency vehiclesrdquo Journal ofthe Operational Research Society vol 57 no 1 pp 22ndash28 2006

[8] S Nickel M Reuter-Oppermann and F Saldanha-da-GamaldquoAmbulance location under stochastic demand A samplingapproachrdquo Operations Research for Health Care vol 8 pp 24ndash32 2016

[9] P Beraldi M E Bruni and D Conforti ldquoDesigning robustemergency medical service via stochastic programmingrdquo Euro-pean Journal of Operational Research vol 158 no 1 pp 183ndash1932004

[10] N Noyan ldquoAlternate risk measures for emergency medicalservice system designrdquo Annals of Operations Research vol 181pp 559ndash589 2010

[11] M R Wagner ldquoStochastic 0-1 linear programming underlimited distributional informationrdquoOperations Research Lettersvol 36 no 2 pp 150ndash156 2008

[12] EDelage andYYe ldquoDistributionally robust optimization undermoment uncertainty with application to data-driven problemsrdquoOperations Research vol 58 no 3 pp 595ndash612 2010

[13] J T Essen J L Hurink S Nickel and M Reuter ldquoModels forambulance planning on the strategic and the tactical levelrdquo BetaResearch School for Operations Management and Logistics p 272013

[14] E Erkut A Ingolfsson and G Erdogan ldquoAmbulance locationfor maximum survivalrdquoNaval Research Logistics (NRL) vol 55no 1 pp 42ndash58 2008

[15] S C Chapman and J A White ldquoProbabilistic formulations ofemergency service facilities location problemsrdquo in Proceedingsof ORSATIMS Conference San Juan Puerto Rico 1974

[16] A Ben-Tal D Den Hertog A De Waegenaere B Melenbergand G Rennen ldquoRobust solutions of optimization problemsaffected by uncertain probabilitiesrdquo Management Science vol59 no 2 pp 341ndash357 2013

[17] MNg ldquoDistribution-free vessel deployment for liner shippingrdquoEuropean Journal of Operational Research vol 238 no 3 pp858ndash862 2014

[18] M Ng ldquoContainer vessel fleet deployment for liner shippingwith stochastic dependencies in shipping demandrdquo Transporta-tion Research Part B Methodological vol 74 pp 79ndash87 2015

[19] R Jiang and Y Guan ldquoData-driven chance constrained stochas-tic programrdquo Mathematical Programming vol 158 no 1-2 SerA pp 291ndash327 2016

[20] C-M Lee and S-L Hsu ldquoThe effect of advertising on thedistribution-free newsboy problemrdquo International Journal ofProduction Economics vol 129 no 1 pp 217ndash224 2011

[21] K Kwon and T Cheong ldquoA minimax distribution-free proce-dure for a newsvendor problem with free shippingrdquo EuropeanJournal of Operational Research vol 232 no 1 pp 234ndash2402014

12 Journal of Advanced Transportation

[22] Y Zhang S Shen and S A Erdogan ldquoDistributionally robustappointment scheduling with moment-based ambiguity setrdquoOperations Research Letters vol 45 no 2 pp 139ndash144 2017

[23] Y Zhang S Shen and S A Erdogan ldquoSolving 0-1semidefinite programs for distributionally robust allocationof surgery blocksrdquo Social Science Electronic Publishing 2017httpsssrncomabstract=2997524

[24] F Zheng J He F Chu and M Liu ldquoA new distribution-freemodel for disassembly line balancing problem with stochastictask processing timesrdquo International Journal of ProductionResearch pp 1ndash13 2018

[25] J Cheng E Delage and A Lisser ldquoDistributionally robuststochastic knapsack problemrdquo SIAM Journal on Optimizationvol 24 no 3 pp 1485ndash1506 2014

[26] M Liu D Yang and F Hao ldquoOptimization for the locationsof ambulances under two-stage life rescue in the emergencymedical service a case study in Shanghai ChinardquoMathematicalProblems in Engineering vol 2017 Article ID 1830480 14 pages2017

Journal of Advanced Transportation 3

of confidence sets for the possible probability distributionsare proposed For more distribution-free formulation appli-cations 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 operatingrooms and Zheng et al [24] for stochastic disassembly linebalancing problem To the best of our knowledge there is noresearch for the stochastic ambulance location problem withonly partial information on the uncertain demand

3 Problem Description and Formulation

In this section we first describe the considered problem andthen propose a new distribution-free model

31 Problem Description There is a given set of candidatebase locations 119868 = 1 2 |119868| and a set of potentialemergency demand points 119869 = 1 2 |119869| The emergencypoints may refer to a road a part of the urban area and avillage Base 119894 isin 119868 is said to be covering an emergency point119895 if the driving time 119905119894119895 between 119894 and 119895 is no more than apredetermined value119879The set 119868119895 of candidate bases coveringemergency point 119895 is denoted as 119868119895 = 119894 isin 119868 | 119905119894119895 le 119879

The number of ambulances required by an emergencypoint 119895 isin 119869 is denoted as 119889119895 which depends on the severityof the practical situationWe consider uncertain emergencieswhich are estimated or predicted by partial distributionalinformation thus the demand is regarded to be ambiguousBesides we focus on the case where the historical data cannotbe represented by a precise probability distribution It isassumed that the customer demands are independent of eachother

The objective of the problem is to select a subset ofbase locations and determine the number of ambulances ateach constructed base in order to minimize the total baseconstruction cost and ambulance costThroughout this paperwe assume that only the mean and covariance matrix ofdemands are known Moreover a predefined safety level 120572119895is given for each potential emergency point 119895 isin 119869 That is thenumber of ambulances serving each emergency point 119895 isin 119869 islarger than or equal to its demand with a least probability of12057211989532 Chance Constraint Construction In this section weintroduce a chance constraint to guarantee the safety levelof each emergency demand point In the following 119910119894119895 is adecision variable denoting the number of ambulances servingpoint 119895 isin 119869 from base 119894 isin 119868119895 and 119889119895 is the uncertain demandat point 119895 isin 119869 The chance constrained inequality is presentedas follows

ProbP (sum119894isin119868119895

119910119894119895 ge 119889119895) ge 120572119895 forall119895 isin 119869 (1)

where ProbP(sdot) denotes the probability of the event inparentheses under any potential probability distribution PConstraint (1) ensures that the number of ambulances servingemergency point 119895 isin 119869 is no less than its demand with a leastprobability of 120572119895 (ie safety level)

33 Distribution-Free Formulation In the following we givebasic notations define decision variables and propose thedistribution-free formulationDF for the ambulance locationproblem with uncertain demand

Parameters

119894 index of base locations119895 index of emergency points119868 set of candidate base locations119869 set of possible emergency demand points119905119894119895 driving time between base 119894 isin 119868 and demand point119895 isin 119869119879 maximum driving time for serving any emergencycall119868119895 set of candidate bases that can cover emergencypoint 119895 isin 119869 ie 119868119895 = 119894 isin 119868 | 119905119894119895 le 119879119869119894 set of potential emergency points covered by base119894 isin 119868 ie 119869119894 = 119895 isin 119869 | 119905119894119895 le 119879119891119894 fixed construction cost for installing a base atlocation 119894 isin 119868119892119894 fixed cost associated with an ambulance to belocated at 119894 isin 119868119889119895 number of (stochastic) ambulances requested byemergency point 119895 isin 119869M a sufficiently large positive number

Decision Variables

119909119894 a binary variable equal to 1 if a base is constructedat 119894 isin 119868 0 otherwise119911119894 number of ambulances assigned to possible baselocation 119894 isin 119868119910119894119895 number of ambulances serving emergency point119895 isin 119869 from possible base location 119894 isin 119868119895

Distribution-Free Model [DF]

[DF]

min sum119894isin119868

(119891119894 sdot 119909119894 + 119892119894 sdot 119911119894)st Constraint (1)

(2)

119911119894 le M sdot 119909119894 forall119894 isin 119868 (3)

sum119895isin119869119894

119910119894119895 le 119911119894 forall119894 isin 119868 (4)

119909119894 isin 0 1 forall119894 isin 119868 (5)

119911119894 isin Z+ forall119894 isin 119868 (6)

119910119894119895 isin Z+ forall119894 isin 119868 119895 isin 119869119894 (7)

4 Journal of Advanced Transportation

The objective function denotes the goal to minimize thetotal cost consisting of two parts (i) the cost for constructingbases ie sum119894isin119868 119891119894 sdot 119909119894 and (ii) the cost for deploying ambu-lances ie sum119894isin119868 119892119894 sdot 119911119894

Constraint (3) ensures that ambulances can only belocated at the opened bases Constraint (4) ensures that thenumber of ambulances sent to serve emergency points frombase 119894 isin 119868 does not exceed the total number of ambulanceslocated at 119894 isin 119868 Constraints (5)-(7) are the restrictions ondecision variables

4 Solution Approaches

The proposed distribution-free model is difficult to solvewith the commercial software due to chance constraints Inthis section we propose two approximatedMIP formulationsbased on those in Wagner [11] and in Delage and Ye [12]respectively In the following subsections we present the twoapproximated MIP formulations

41 Approximated MIP Formulation MIP-DF1 In this partwe first describe a widely used ambiguity set to describe theuncertainty Then an approximated MIP formulation MIP-DF1 is proposed

Given a set of independent historical data samples 119889119904|119878|119904=1of random vectors of demands where 119878 is the set of sampleindexes the mean vector 120583 and the covariance matrix Σ ofdemands can be estimated as follows

120583 = 1|119878|sum119904isin119878119889119904Σ = 1|119878|sum119904isin119878 (119889119904 minus 120583) (119889119904 minus 120583)⊤

(8)

where (sdot)⊤ implies the transposition of the vector in paren-theses Then an ambiguity set of all probability distributionsof demandsP1(120583 Σ) can be described as the following

P1 (120583 Σ) = P EP [119889] = 120583EP [(119889 minus 120583) (119889 minus 120583)⊤] = Σ (9)

whereP denotes a possible probability distribution satisfyingthe given conditions and EP[sdot] denotes the expected valueof the number in parentheses Then the chance constraint (1)with the ambiguity setP1 can be presented as follows

ProbP (sum119894isin119868119895

119910119894119895 ge 119889119895) ge 120572119895 forall119895 isin 119869 P isin P1 (10)

According to Wagner (2010) and Ng [18] constraint (1) canbe conservatively approximated by the following

sum119894isin119868119895

119910119894119895 ge 120583119895 + 120590119895 sdot radic 120572119895(1 minus 120572119895) forall119895 isin 119869 (11)

where 120590119895 = radicΣ119895119895 and Σ119895119895 denotes the 119895-th element in the 119895-thcolumn of matrix Σ In terms of conservative approximation

all solutions satisfying constraint (11) must satisfy constraint(1) Then we describe the approximated MIP formulationMIP-DF1 toDF combined with constraint (11)

[MIP-DF1]

min sum119894isin119868

(119891119894 sdot 119909119894 + 119892119894 sdot 119911119894)st Constraints (4) - (8) (12)

(12)

MIP-DF1 can be exactly solved by the commercialsoftware such as CPLEX As we can observe from thecomputational results in Section 5 it can obtain a relativelyhigh service level However the system cost is also high Inorder to save the system cost another approximated MIPformulation of DF is then proposed in the next subsection

42 Approximated MIP Formulation MIP-DF2 AmbiguitysetP1 focuses on exactly matching the mean and covariancematrix of uncertain parameters However in practice theremay be considerable estimation errors in the mean andcovariance matrix To take the inevitable estimation errorsinto consideration Delage and Ye [12] introduce a newmoment-based ambiguity set Therefore we in the followingemploy the moment-based ambiguity set

P2 (120583 Σ 1205741 1205742)= P (EP [119889] minus 120583)⊤ Σminus1 (EP [119889] minus 120583) le 1205741

EP [(119889 minus 120583) (119889 minus 120583)⊤] ⪯ 1205742Σ (13)

where 1205741 ge 0 and 1205742 ge 0 are two parameters of ambiguityset P2(120583 Σ 1205741 1205742) with the following assumptions (i) thetrue mean vector of demands is within an ellipsoid of sizeproportion to 1205741 centered at 120583 and (ii) the true covariancematrix of demands is in a positive semidefinite cone boundedby amatrix inequality of 1205742ΣThe ambiguity set describes howlikely the uncertain parameters are to be close to the meanin terms of the correlation [12] Besides under the moment-based ambiguity set various stochastic programs with partialdistributional information have been successfully modeledand solved [22 23 25]

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

radic 11 minus 119886 minus 119887 sdot (1 + radic1 minus 120572119895120572119895 sdot 119887) sdot 120590119895le radic1 minus 120572119895120572119895 sdot (sum

119894isin119868119895

119910119894119895 minus 120583119895) 119895 isin 119869(14)

where

119886 = 1 minus 1205741 + 11205742 minus 1205741 119887 = 12057411205742 minus 1205741

(15)

Journal of Advanced Transportation 5

Table 1 120572 = 095|119868| = |119869| Sampling approach MIP-DF1 MIP-DF2

CT Obj Sel () CT Obj Sel () CT Obj Sel ()1 7 49 23 9626 01 36 10000 01 23 100002 7 51 22 9863 01 35 10000 01 24 99823 6 41 14 9692 01 30 10000 01 14 98504 6 43 13 9713 02 29 10000 01 12 97425 6 57 12 7996 01 30 10000 02 12 100006 8 58 21 9850 01 24 10000 02 27 10000

Average 50 175 9457 01 307 10000 01 187 9929

The second approximated MIP formulationMIP-DF2 toDF is presented as follows

[MIP-DF2]

min sum119894isin119868

(119891119894 sdot 119909119894 + 119892119894 sdot 119911119894)st Constraints (4) - (8) (15)

(16)

Notice that when 1205741 = 0 and 1205742 = 1 inequality (14)reduces to (11) in MIP-DF1 which implies that MIP-DF1 isa special case of MIP-DF2

5 Computational Experiments

In this section the performance of our proposed formula-tions first evaluated and compared to the sampling approachproposed in Nickel et al [8] Specifically given a requiredservice level 120572 we apply our MIP formulations by restrictingthe chance constraint for each emergency point to satisfya safety level 120572119895 = 120572 while we adopt the samplingapproach by restricting the coverage constraint with 120572 Thencomputational results on 120 randomly generated instancesbased on the information on emergencies in Shanghai arereported Our approximated MIP formulations and thesampling approach are solved by coding in MATLAB 2014bcalling CPLEX 126 solver All numerical experiments areconducted on a personal computer with Core I5 and 330GHzprocessor and 8GB RAMunder windows 7 operating systemThe computational time for sampling approach is limited to3600 seconds

51 Out-of-Sample Test FollowingZhang et alrsquos [22]method-ology we examine the out-of-sample performance of solu-tions obtained by our proposed approximated MIP formula-tions and the sampling approach [8] The out-of-sample testincludes three steps

(1) The base locations and the employment of ambu-lances at each base are determined with known meanvector and covariance matrix of demands which isnamed as in-sample test

(2) 1000 scenarios are sampled from the underlying truedistribution which represent realizing the uncertaindemand

(3) Based on the solution obtained in step (1) ambu-lances are assigned to serve emergency points inthe 1000 scenarios to maximize the (out-of-sample)service level

The (out-of-sample) service level is defined as the ratioof the number of emergency points with demand satisfiedto the total number of emergency points in 1000 scenariosIn the following subsections the benchmarks are tested andthen three common distributions are employed ie uniformdistribution Poisson distribution and normal distributionrespectively

52 Benchmark Instances We first test the benchmarks inNickel et al [8] to compare our proposed two approximatedMIP formulationswith the sampling approach InNickel et al[8] they assume that demand nodes are also potential baselocations and the number of ambulances required by eachemergency point is chosen from set 0 1 2with given proba-bility distribution Computational results on benchmarks arereported in Tables 1 and 2

Table 1 reports the computational results with servicelevel requirement120572 = 095 In Table 1 CTObj and Sel denotethe computational time objective value (ie the system cost)and the service level respectively The first column includesthe indexes of instances and the second indicates the numberof nodes It shows that the average service level obtainedby the sampling approach is 9457 which is smaller thanthe required service level ie 85 Specifically the servicelevel of instance 5 obtained by the sampling approach is only7996 However MIP-DF1 and MIP-DF2 can guarantee theservice levels for all instances higher than the requirementConcluding from Table 1 we can observe that (i) the averagecomputational time of sampling approach is about 50 timesas those of the two approximated MIP formulations and (ii)MIP-DF1 and MIP-DF2 improve the average service level byabout 574 and 499 compared with sampling approachhowever (iii) system costs obtained by MIP-DF1 and MIP-DF2 are respectively 7543 and 686 higher than that ofsampling approach

Table 2 reports the computational results with servicelevel requirement 120572 = 099 For instances 2-6 the ser-vice levels obtained by sampling approach cannot meet therequirement ie 99 Besides it shows that the averagecomputational time of sampling approach is 48 which isabout 48 times greater than those of MIP-DF1 andMIP-DF2

6 Journal of Advanced Transportation

Table 2 120572 = 099|119868| = |119869| Sampling approach MIP-DF1 MIP-DF2

CT Obj Sel () CT Obj Sel () CT Obj Sel ()1 7 49 24 9979 01 64 10000 01 25 100002 7 48 23 9856 01 63 10000 01 24 100003 6 51 15 9717 01 55 10000 01 16 100004 6 47 15 9275 01 52 10000 01 13 100005 6 40 14 9000 01 55 10000 01 14 100006 8 52 23 9785 01 45 10000 01 27 10000

Average 48 190 9343 01 557 10000 01 198 10000

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

Sampling approach MIP-DF1 MIP-DF2Emergency 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 0Base locations Ambulance employment Ambulance employment Ambulance employment1 0 0 02 1 1 0 3 1 23 1 1 26 1 2 2 04 0 0 05 0 6 06 2 2 0 4 2

The average service level is 9343 Although the averageservice levels obtained by MIP-DF1 and MIP-DF2 are both100 the average system cost obtained by MIP-DF1 is about3 times larger than that of sampling approach Besides theaverage system cost obtained by MIP-DF2 is 421 higherthan that of sampling approach

Concluding for the benchmarks described in Nickel etal [8] we can observe that (1) MIP-DF1 and MIP-DF2 canbe solved in far less computational time compared with thesampling approach (2) the sampling approach cannot meetthe requirement of service level for all benchmarks whilethe two MIP formulations can (3) the average system cost ofMIP-DF2 is very close to that of sampling approach (4) withthe increase of 120572 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 ofthe lower system cost obtained by the sampling approachthere may exist extreme situations that demands of someemergency points are not satisfied Take one benchmark inNickel et al [8] ie instance 4 where there are 6 nodesA scenario in which demand at each emergency point isgenerated from the given probability distribution is shown inFigure 1 We can observe from Figure 1 that under the givenscenario the demand of each possible emergency point is1 0 2 0 2 0The solutions obtained by sampling approachMIP-DF1 andMIP-DF2 are shown in Table 3 which includesbase 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 poten-tial base locations The second column represents the baselocations and the ambulance employment at each location

＞1 = 1

＞2 = 0

＞3 = 2

＞4 = 0

＞5 = 2

＞6 = 0

1

2 4

3 5

6

Figure 1 Instance 4 in Nickel et al [8] under given scenario

in which a location with 0 ambulance implies that it is notselected for base construction From Table 3 we can observethat the in-sample solutions obtained by sampling approachis to construct bases at locations 2 3 and 6 where the numberof ambulances is 1 1 and 2 respectively Besides undergiven scenario there is one ambulance serving emergencypoint 1 from base location 3 and one ambulance servingemergency point from base location 2 and 2 ambulancesserving emergency point 5 from base location 6 It shows thatthe requirement of ambulances at emergency point 3 is notfully satisfied The insufficient service may cause serious lossof life However we can find that the solutions obtained byMIP-DF1 andMIP-DF2 can provide sufficient ambulances forall emergency points

In sum to guarantee the required service level withless cost MIP-DF2 is recommended for the solutions with

Journal of Advanced Transportation 7

Nickel et al [8]rsquos data set Moreover in the benchmarks itis assumed that all nodes are possible emergency points aswell as potential base locations and the number of requiredambulances is small which is not always consistent withthe situation in Shanghai Therefore we further test the twoapproximated MIP formulations and the sampling approachon randomly generated instances by analysing the informa-tion on emergencies in Shanghai [26]

53 Computational Tests on Randomly Generated InstancesIn the following subsection numerical experiments on 120randomly generated instances are described

531 Test Instances By analysing the information on emer-gencies in Shanghai [26] the mean value 120583119895 of demand 119889119895 ineach emergency point are randomly generated fromadiscreteUniform distribution on interval [5 25] The standard devia-tion 120590119895 is set to be 5 Since the sampling approach proposed inNickel et al [8] requires a sample of scenarios in numericalexperiments scenarios are produced with demands randomlygenerated from Uniform distribution Poisson distributionand Normal distribution consistent with given mean andstandard deviation

According to the instance settings in Erkut and Ingolfsson(2008) response time from a base to an emergency pointis randomly generated from a discrete Uniform distributionon interval [3 30] in units of minutes A base covers anemergency point if the driving time between them is lessthan 10 minutes to ensure the survival probability (Erkut andIngolfsson 2006) For an emergency point in each instanceif there is no base with 10 minutes driving time to it then thebase covering this point is set to be the nearest one The costof constructing a base and that of locating one ambulance areboth set to be 1 as in Nickel et al [8] Besides the number ofemergency points is 3 times that of potential base locations

532 Results and Discussion Different values of requiredservice level 120572 are tested ie 085 09 095 Computationalresults on 120 randomly generated instances are reported inTables 4-6

Table 4 reports the numerical results of instances withdemand generated from Uniform distribution We observethat when 120572 = 085 09 095 the average computationaltimes of the sampling approach are 4482 4379 and 4295seconds respectively However both approximated MIP for-mulations can produce solutions in much faster speeds iewithin 13 seconds on average With the increase of 120572 (i)the out-of-sample average service level is getting higher and(ii) the system costs of solutions obtained by all method aregetting larger and (iii) the system costs of solutions yieldedbyMIP-DF2 is getting closer to those of solutions obtained bythe sampling approach In terms of the average system costthe sampling approach performs the best while MIP-DF1 isthe poorest When 120572 = 085 for example the average systemcost ofMIP-DF2 and the sampling approach are about 3032and 442 less than that of MIP-DF1 respectively When120572 = 095 the above gaps are even larger Concluding fortheUniformdistribution (1) the proposed two approximatedMIP formulations are far more time saving than the sampling

approach (2)MIP-DF1 andMIP-DF2 can improve the servicelevel by about 1907 compared with the sampling approachhowever (3) despite of the highest service levels obtainedby MIP-DF1 the system cost obtained by MIP-DF1 is alsohigh (4) MIP-DF2 can ensure an appropriate service levelwith much less cost than MIP-DF1 and (5) for the situationswith small-scale instances and high service level requirementMIP-DF1 is recommended

Computational results of instances with demand at eachemergency point generated from Poisson distribution arepresented in Table 5 For MIP-DF1 and MIP-DF2 onlymean and variance are involved Therefore with givenmean and standard deviation of the demand at each pointcomputational results are the same for different probabilitydistributions We can obtain from Table 5 that the objectivevalue obtained by the sampling approach increases with theincrease of 120572 Besides when the value of 120572 equals 085the average service level obtained by the sampling approachis about 8143 which is smaller than that of the twoapproximated MIP formulations The highest service level9934 is obtained by MIP-DF1 and that obtained by MIP-DF2 is 9708 When 120572 = 09 the average service levelof the sampling approach is 8437 and those obtained byMIP-DF1 and MIP-DF2 are 9985 and 9797 respectivelyWhen 120572 = 095 the average service level obtained by thesampling approach is 9356 and those yielded by MIP-DF1 and MIP-DF2 are 9997 and 9781 respectively ForPoisson distribution concluding MIP-DF1 and MIP-DF2can improve the service level compared with the samplingapproach especially MIP-DF1 can obtain the highest servicelevel

Similarly Table 6 reports the computational results forinstances in which the demand at each emergency point isgenerated following Normal distribution When 120572 = 085the average service levels obtained by the sampling approachMIP-DF1 and MIP-DF2 are 8174 9925 and 9742respectively When 120572 = 09 the three values are 84579981 and 9743 respectively Moreover for 120572 = 095 thevalues are equal to 8732 9999 and 9794 respectivelyIt can be observed that the performance of the samplingapproach is the poorest in terms of the service level

By observing the above computational results on bench-marks and randomly generated instances we conclude that(1)The proposed approximated MIP formulations ie MIP-DF1 and MIP-DF2 are very time saving compared with thesampling approach (2) MIP-DF1 an MIP-DF2 can achievethe requirement of service level and improve the service levelby about 1547 and 1229 on average compared with thatof the sampling approach (3) MIP-DF1 is more conservativein terms of overhigh emergency service level close to 100and the system cost is also high which is about 568 largerthan that of MIP-DF2 (4) Compared with MIP-DF1 MIP-DF2 is less conservative and much more cost saving whileensuring an appropriately service level on average which isabout 97

In sum for medium and large problem instances (egShanghai) which are common in practice to guarantee therequired service level with less cost we recommendMIP-DF2as solution method However for very small instances with

8 Journal of Advanced Transportation

Table4Com

putatio

nalresultson

insta

nces

with

datageneratedfro

mun

iform

distrib

ution

Set (|119868||119869|)

120572=085

120572=09

120572=095

Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2CT

Obj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

1(26)

27

897317

01

149

9906

01

107

9222

28

967373

01

155

9933

01

107

9167

27

106

7585

01

185

9987

01

107

9128

2(412

)47

169

7423

00

291

9919

00

207

9122

58

181

7438

01

303

9997

01

207

9363

50

199

7702

00

363

9982

00

207

9338

3(618

)72

258

7506

01

438

9976

01

312

9106

75279

8082

01

456

9996

01

312

9369

78304

8337

01

546

9997

01

312

9357

4(824)

99314

7956

01

553

9993

01

385

9206

111

338

8202

01

577

9999

01

385

9602

112

366

8492

01

697

9994

01

385

9620

5(1030)

132

415

8406

01

722

9896

01

512

9406

143

441

8322

02

752

9983

01

512

9692

142

476

8609

01

902

9993

01

512

9692

6(1236)

170

475

8249

02

843

9870

01

591

9367

181

504

8251

01

879

9982

01

591

9608

189

535

8726

01

1059

9997

01

591

9609

7(14

42)

215

573

8111

02

1006

9849

02

712

9606

243

609

8217

02

1048

9953

02

712

9589

233

644

8608

02

1258

9992

02

712

9582

8(1648)

248

686

7532

02

1185

9896

02

849

9725

284

727

8307

02

1233

9949

02

849

9613

281

768

9063

02

1473

9999

03

849

9622

9(1854)

313

691

8353

03

1236

9974

04

858

9709

351

731

8397

03

1290

9988

03

858

9765

352

760

8906

03

1560

9994

03

858

9758

10(2060)

372

794

8174

03

1409

9887

03

989

9657

428

843

8489

03

1469

9947

03

989

9773

432

892

9004

03

1769

9997

03

989

9738

11(2266

)451

830

8072

04

1493

9949

03

1031

9758

518

880

8328

04

1559

9982

03

1031

9723

516

930

8967

05

1889

9995

03

1031

9727

12(2472)

543

1025

7970

04

1771

9927

04

1267

9835

601

1083

8482

05

1843

9949

04

1267

9776

637

1146

9014

04

2203

9993

04

1267

9777

13(2678)

631

1026

8142

05

1819

9898

04

1274

9782

743

1087

8678

05

1897

9936

04

1274

9778

765

1146

9023

04

2287

9997

05

1273

9837

14(2884)

782

1101

7903

05

1949

9869

08

1361

9804

878

1165

8790

05

2033

9935

07

1361

9889

870

1227

9056

05

2453

9998

06

1361

9808

15(3090)

907

1091

8098

06

1997

9915

06

1367

9629

1018

1155

8329

06

2087

9981

06

1367

9807

1020

1225

8986

06

2537

9999

08

1367

9817

16(3296)

1062

1160

8798

07

2123

9944

07

1449

9787

1183

1229

8383

07

2219

9988

06

1449

9849

1193

1300

9105

06

2698

9998

06

1449

9815

17(34102)

1202

1298

7572

07

2328

9969

07

1614

9821

1377

1374

8860

08

2430

9997

07

1614

9824

1260

1452

9085

07

2940

9999

08

1614

9818

18(36108)

1394

1408

8121

08

2509

9947

08

1752

9809

1544

1492

8928

08

2617

9983

08

1752

9799

1548

1580

9173

08

3156

9999

08

1752

9809

19(38114

)1618

1508

8170

09

2676

9899

111877

9742

1760

1598

9130

09

2790

9934

08

1877

9908

1755

1688

9186

09

3360

9999

09

1877

9872

20(40120)

2206

1628

7719

102869

9989

102029

9873

2007

1732

8753

112990

9997

09

2029

9879

1904

1827

9138

103590

9998

09

2029

9875

21(42126)

2780

1622

8354

102900

9978

102017

9732

2609

1718

8854

103026

9876

102017

9818

2670

1812

9089

123655

9999

102017

9848

22(44132)

3189

1735

8494

113088

9985

152165

9857

2970

1840

8922

123220

9925

122165

9886

2873

1952

9206

113880

9999

112165

9850

23(46138)

3630

1937

8141

123372

9893

132406

9796

3222

2048

8937

123510

9934

142406

9906

3461

2170

8989

124199

10000

122406

9809

24(48144)

4226

1968

8226

133461

9932

132453

9814

3674

2085

8846

133605

9956

152453

9912

3556

2211

9300

134326

9999

132453

9819

25(50150)

4727

1919

8111

143456

9999

152407

9629

4019

2037

8699

143606

9999

142407

9857

4257

2161

9259

154356

9995

142407

9819

26(52156)

5300

1906

8096

153484

9878

172392

9658

4495

2021

8952

143640

9999

152392

9568

4486

2142

9182

1544

209999

152392

9854

27(54162)

5913

2096

8387

163751

9846

172617

9761

5117

2221

9045

163913

9998

172617

9544

4830

2355

9346

184723

9999

162617

9893

28(56168)

6310

2110

7758

173819

9933

192643

9829

5777

2234

9157

183987

9998

182643

9526

5550

2365

9045

184827

9999

172643

9831

29(58174)

6780

2175

7998

183937

9898

182720

9714

6303

2307

9022

20

4111

9969

192720

9623

6189

2444

9128

194981

10000

192720

9914

30(60180)

7593

2384

8321

194240

9967

20

2979

9841

6791

2527

8978

1944

199979

23

2979

9721

6878

2679

9143

21

5316

9999

20

2979

9869

31(62186)

8220

2281

7604

20

4157

9984

21

2856

9796

7734

2416

8934

21

4343

9997

21

2856

9679

7466

2557

9158

22

5273

9999

23

2856

9902

32(64192)

8766

2517

8378

21

4489

9932

25

3145

9729

8276

2671

8296

23

4680

9977

24

3145

9825

7897

2822

9089

24

5640

9999

22

3145

9891

33(66198)

10123

2532

7806

23

4555

9967

23

3169

9615

9010

2682

8947

24

4752

9989

24

3169

9703

9051

2835

9157

24

5742

9998

24

3169

9867

34(68204)

11010

2535

7934

24

4602

9999

29

3174

9631

9898

2685

8858

30

4806

9998

28

3174

9762

9863

2844

9236

26

5824

9999

25

3174

9889

35(70210)

11666

2714

8062

25

4862

9986

29

3391

9787

1066

62875

9137

26

5072

9999

29

3391

9716

10822

3043

9315

26

6122

9999

29

3391

9923

36(72216)

12011

2827

8104

26

5047

9991

28

3535

9798

1246

52984

9016

28

5263

9999

28

3535

9878

11667

3169

9253

30

6343

9999

29

3535

9896

37(74222)

11437

2826

8228

28

5096

9933

29

3543

9884

13582

3098

8895

29

5318

9999

32

3543

9888

12654

3178

9139

29

6427

10000

31

3543

9893

38(76228)

12435

2839

8152

29

5161

9897

29

3565

9788

13976

3103

8926

29

5388

9999

31

3565

9724

13837

3185

9255

33

6528

10000

31

3565

9895

39(78234)

1464

32945

8412

31

5331

9998

32

3695

9876

14794

3210

9005

31

5565

9999

34

3695

9713

14561

3313

9081

34

6735

10000

33

3695

9878

40(80240)

1604

22975

8510

32

5414

9957

36

3734

9796

16263

3346

8999

35

5654

9999

35

3734

9806

15869

3343

9248

35

6854

10000

35

3734

9873

Average

4482

1584

680

67

1228

397

9936

131978

796

82

4379

16913

8654

1329

626

9975

131978

79721

4295

1778

889

85

1335774

9997

131978

79775

Journal of Advanced Transportation 9

Table5Com

putatio

nalresultson

insta

nces

with

datageneratedfro

mPo

isson

distr

ibution

set ( |119868 |

|119869 |)120572=

085120572=

09120572=

095Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2CT

Obj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

1(26)

27

907294

9967

9001

27

987770

9979

9013

24

103

9273

9999

9468

2(412

)46

170

7961

9921

9117

51

183

8305

9946

9128

49

201

9392

9998

9546

3(618

)68

258

7578

9938

8911

74280

8028

9968

9004

69

307

9182

9999

9658

4(824)

93315

8001

9928

9196

102

338

8438

9967

9172

92368

9468

9999

9613

5(1030)

120

413

8041

9939

9449

133

443

8447

9976

9272

121

479

9544

9999

9752

6(1236)

155

476

8084

9877

9554

167

508

8863

9991

9569

159

536

9456

9999

9734

7(14

42)

195

573

7876

9892

9663

213

609

9124

9998

9729

205

643

9314

10000

9768

8(1648)

232

687

7964

9997

9788

267

724

8523

9988

9687

238

772

9422

9998

9689

9(1854)

293

692

8267

9893

9809

327

730

8386

9999

9750

298

757

9228

9999

9755

10(2060)

356

796

7946

9899

9801

399

840

8299

9976

9686

360

885

9668

9999

9705

11(2266)

423

829

8201

9895

9756

489

877

8378

9987

9788

448

927

9476

9999

9812

12(2472)

512

1023

8338

9913

9853

554

1085

8501

9999

9881

514

1148

9398

9999

9888

13(2678)

592

1045

8416

9907

9873

687

1088

7998

9996

9802

607

1152

9504

9999

9821

14(2884)

723

1099

7946

9915

9788

838

1166

8056

9992

9829

776

1230

9546

9999

9839

15(3090)

855

1089

7915

9964

9703

942

1156

8432

9984

9778

903

1225

9223

9998

9785

16(3296)

1009

1158

8024

9897

9769

1163

1231

8500

9996

9824

1045

1222

9199

9999

9835

17(34102)

1165

1299

8034

9919

9779

1285

1375

8495

9937

9808

1190

1457

9258

10000

9846

18(36108)

1254

1411

8176

9943

9885

1479

1492

8561

9968

9868

1345

1579

9286

9999

9875

19(38114

)1504

1506

7970

9923

9842

1651

1594

8167

9972

9846

1564

1694

9376

9999

9786

20(40120)

1706

1635

7819

9916

9783

1854

1734

8607

9968

9776

1778

1825

9285

9904

9718

21(42126)

2054

1622

8354

9918

9834

2357

1720

8297

9978

9833

2226

1856

9501

9999

9834

22(44132)

2400

1736

8494

9922

9846

2852

1844

8269

9999

9816

2680

1946

9486

9999

9820

23(46138)

2651

1934

8141

9953

9793

3159

2049

8492

9969

9799

3072

2172

9358

10000

9813

24(48144)

3004

1961

8226

9947

9681

3522

2086

8513

9998

9706

3102

2209

9488

9999

9724

25(50150)

3308

1924

8111

9965

9746

3898

2038

8264

9999

9788

3639

2161

9546

9999

9813

26(52156)

3669

1911

8096

9939

9855

4305

2027

8722

9967

9874

4453

2142

9185

9999

9876

27(54162)

4418

2095

8387

9927

9754

4771

2219

8341

9989

9749

4806

2353

9289

9999

9755

28(56168)

5367

2113

8458

9968

9848

5356

2237

8452

9978

9855

5221

2369

9228

10000

9868

29(58174)

6217

2186

7998

9989

9694

5942

2315

8316

9996

9699

5853

2371

9167

9999

9705

30(60180)

7263

2382

8321

9991

9837

6527

2543

8359

9998

9886

6214

2678

9369

9997

9886

31(62186)

8214

2346

8404

9936

9888

7813

2430

8321

9999

9846

6876

2560

9358

9999

9847

32(64192)

8863

2518

8378

9938

9807

8106

2677

8356

9999

9833

7877

2822

9408

10000

9835

33(66198)

9985

2532

8406

9917

9709

9684

2689

8497

9996

9746

8088

2838

9507

9999

9749

34(68204)

11264

2534

8543

9908

9897

10986

2685

8809

9996

9782

8469

2843

8844

9998

9780

35(70210)

12035

2715

8258

9937

9799

12098

2886

8569

9991

9809

9946

3041

9459

9999

9813

36(72216)

12803

2825

8104

9929

9827

13076

3010

8651

9994

9834

11013

3170

9468

9999

9865

37(74222)

13021

2830

8228

9949

9826

13261

3009

8376

9999

9826

12014

3172

9346

9999

9888

38(76228)

13862

2843

8152

9933

9807

13995

3028

8760

9998

9841

12133

3187

9706

9999

9876

39(78234)

15387

2953

8312

9964

9799

14827

3133

9001

9995

9783

13381

3304

8913

9999

9799

40(80240)

16974

2975

8510

9985

9767

15256

3249

8217

9993

9781

15867

3347

9111

9999

9791

Average

4352

15875

8143

9934

9708

4362

16856

8437

9985

9707

3968

17763

9356

9997

9781

10 Journal of Advanced Transportation

Table6Com

putatio

nalresultson

insta

nces

with

datageneratedfro

mno

rmaldistr

ibution

set ( |119868 |

|119869 |)120572=

085120572=

09120572=

095Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2CT

Obj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

1(26)

27

907294

9964

9117

27

947844

9970

9202

2558

106

8422

10000

9208

2(412

)46

170

7961

9921

9283

48

183

8328

9982

9283

4508

183

8823

10000

9370

3(618

)68

258

7578

9916

9228

68

280

7798

9971

9417

6803

307

7658

9998

9447

4(824)

93315

8000

9899

9367

96337

8379

9979

9467

9373

368

8438

9999

9579

5(1030)

120

413

7841

9927

9437

124

443

8763

9963

9476

1262

499

8850

9998

9708

6(1236)

155

476

7921

9919

9519

162

504

8124

9982

9582

16021

536

8954

10000

9756

7(14

42)

195

573

7976

9933

9598

196

609

8287

9975

9619

2011

646

8467

9999

9767

8(1648)

232

687

8031

9918

9705

235

727

8115

9965

9650

2404

772

8278

9999

9765

9(1854)

293

692

8086

9898

9759

291

734

8464

9974

9755

29815

796

8617

9998

9835

10(2060)

356

796

7984

9937

9770

358

842

8223

9988

9765

37973

921

8846

10000

9838

11(2266

)421

829

8101

9940

9755

436

877

8211

9986

9786

4514

8959

8497

9999

9789

12(2472)

514

1023

8033

9951

9785

501

1085

8162

9983

9788

52073

1147

8321

9999

9824

13(2678)

592

1023

8138

9916

9885

599

1087

8411

9970

9845

62937

1194

8609

9998

9795

14(2884)

728

1099

8235

9913

9915

722

1168

8435

9987

9806

75053

1270

8567

9997

9834

15(3090)

855

1089

8187

9922

9861

880

1157

8376

9976

9796

89222

1300

8611

9996

9798

16(3296)

1009

1158

8770

9897

9788

966

1231

8845

9984

9823

106288

1359

8997

9999

9848

17(34102)

1154

1299

8434

9925

9849

1137

1377

8601

9980

9798

123869

1493

8798

9996

9877

18(36108)

1407

1411

8321

9927

9866

1339

1495

8301

9965

9822

133807

1601

8567

9999

9834

19(38114

)1702

1506

8206

9915

9805

1672

1598

8977

9978

9845

152999

1694

8998

9999

9847

20(40120)

2091

1635

8528

9918

9889

1916

1737

8805

9976

9837

1890

021833

9012

9999

9791

21(42126)

2843

1622

8550

9934

9913

2432

1718

8886

9974

9813

2399

131815

9008

10000

9815

22(44132)

3300

1736

8449

9940

9859

2795

1843

8786

9977

9756

253106

1948

8997

9999

9826

23(46138)

3741

1934

8133

9911

9876

3143

2050

8876

9981

9785

274583

2173

9014

9999

9858

24(48144)

4531

1961

8245

9909

9846

3515

2090

8344

9993

9819

308089

2211

8661

9999

9883

25(50150)

4715

1924

8111

9929

9888

3959

2041

8279

9996

9837

3510

932158

8527

9999

9827

26(52156)

5223

1911

8096

9898

9857

4301

2025

8251

9997

9757

421747

2146

8497

9999

9914

27(54162)

5994

2095

8387

9908

9775

4835

2226

8497

9974

9780

4614

432359

8695

9999

9837

28(56168)

6406

2111

8411

9906

9798

5336

2240

8558

9991

9820

494113

2371

9249

9999

9816

29(58174)

6897

2176

8012

9916

9791

5948

2311

8211

9984

9765

624835

2445

8531

10000

9851

30(60180)

7436

2391

8111

9913

9766

6558

2533

8305

9968

9795

696116

2682

8575

9999

9799

31(62186)

8118

2282

8119

9928

9806

7069

2422

8417

9973

9873

797768

2564

8667

9999

9846

32(64192)

8895

2519

8223

9919

9757

7857

2675

8631

9982

9769

864813

2827

8976

10000

9905

33(66198)

11235

2533

8563

9928

9811

9162

2688

8547

9979

9828

973089

2843

8657

9999

9846

34(68204)

11587

2534

8249

9926

9847

9549

2686

8798

9938

9856

10523

3041

8999

10000

9877

35(70210)

1164

92715

8098

9895

9872

10345

2880

8602

9998

9869

11272

3052

9021

9999

9897

36(72216)

12224

2825

8403

9958

9770

11165

3002

8657

9999

9839

12456

3175

9113

9999

9864

37(74222)

12978

2830

8357

9937

9795

11850

3005

8496

9998

9894

13458

3176

8827

9999

9905

38(76228)

13426

2844

8267

9966

9759

12795

3019

8604

9999

9797

15155

3121

8978

9999

9853

39(78234)

15003

2955

8303

9956

9867

13816

3132

8588

9996

9885

15041

3310

9016

9999

9912

40(80240)

16495

2978

8257

9954

9826

14896

3160

8495

9997

9837

16563

3349

8957

9999

9933

Average

4619

15855

8174

9925

9742

4077

16828

8457

9981

9743

4350

17938

8732

9999

9794

Journal of Advanced Transportation 11

high service level requirement we recommend MIP-DF1 forsolutions

6 Conclusion

This paper investigates an ambulance location problem withambiguity set of demand in which the probability distribu-tion of the uncertain demand is unknown In such a data-driven environment only the mean and covariance demandsat emergency points are knownThe problem is to determinethe base locations and the number of ambulances at eachbase aiming at minimizing the total cost associated withlocating bases and assigning ambulances We propose adistribution-free model with chance constraints Then twoapproximated MIP formulations with different approxima-tion methods of chance constraints are proposed which arebased on different ambiguity sets of the unknown probabilitydistribution Numerical experiments on benchmarks and120 randomly generated instances are conducted and thecomputational results show that the first approximated MIPformulation is much conservative with overhigh system costwhile the second approximatedMIP formulation ismore costsaving with service level appropriately ensured

In future research wemay consider the following relevantissues First the uncertainty of driving time from a base toan emergency point shall be analysed Besides unexpected orsudden disasters may be taken into consideration Moreoverother uncertainties should be considered such as travel timethe number of available vehicles and so on The stochasticdependence should also further considered

Data Availability

The data of the benchmark instances in our manuscript canbe obtained in Nickel et al [8] For the test instances thedata is randomly generated and the way for generating data isstated in our manuscript The datasets generated during thecurrent study are available from the corresponding author onreasonable request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China (NSFC) under Grants 7142800271531011 71571134 and 71771048 This work was also sup-ported by the Fundamental Research Funds for the CentralUniversities

References

[1] L Brotcorne G Laporte and F Semet ldquoAmbulance loca-tion and relocation modelsrdquo European Journal of OperationalResearch vol 147 no 3 pp 451ndash463 2003

[2] X Li Z Zhao X Zhu and T Wyatt ldquoCovering modelsand optimization techniques for emergency response facilitylocation and planning a reviewrdquo Mathematical Methods ofOperations Research vol 74 no 3 pp 281ndash310 2011

[3] C Toregas R Swain C ReVelle and L Bergman ldquoThe locationof emergency service facilitiesrdquoOperations Research vol 19 no6 pp 1363ndash1373 1971

[4] R Church and C ReVelle ldquoThe maximal covering locationproblemrdquo Papers in Regional Science vol 32 no 1 pp 101ndash1181974

[5] M Gendreau G Laporte and F Semet ldquoSolving an ambulancelocation model by tabu searchrdquo Location Science vol 5 no 2pp 75ndash88 1997

[6] A Ingolfsson S Budge and E Erkut ldquoOptimal ambulancelocation with random delays and travel timesrdquo Health CareManagement Science vol 11 no 3 pp 262ndash274 2008

[7] M Gendreau G Laporte and F Semet ldquoThemaximal expectedcoverage relocation problem for emergency vehiclesrdquo Journal ofthe Operational Research Society vol 57 no 1 pp 22ndash28 2006

[8] S Nickel M Reuter-Oppermann and F Saldanha-da-GamaldquoAmbulance location under stochastic demand A samplingapproachrdquo Operations Research for Health Care vol 8 pp 24ndash32 2016

[9] P Beraldi M E Bruni and D Conforti ldquoDesigning robustemergency medical service via stochastic programmingrdquo Euro-pean Journal of Operational Research vol 158 no 1 pp 183ndash1932004

[10] N Noyan ldquoAlternate risk measures for emergency medicalservice system designrdquo Annals of Operations Research vol 181pp 559ndash589 2010

[11] M R Wagner ldquoStochastic 0-1 linear programming underlimited distributional informationrdquoOperations Research Lettersvol 36 no 2 pp 150ndash156 2008

[12] EDelage andYYe ldquoDistributionally robust optimization undermoment uncertainty with application to data-driven problemsrdquoOperations Research vol 58 no 3 pp 595ndash612 2010

[13] J T Essen J L Hurink S Nickel and M Reuter ldquoModels forambulance planning on the strategic and the tactical levelrdquo BetaResearch School for Operations Management and Logistics p 272013

[14] E Erkut A Ingolfsson and G Erdogan ldquoAmbulance locationfor maximum survivalrdquoNaval Research Logistics (NRL) vol 55no 1 pp 42ndash58 2008

[15] S C Chapman and J A White ldquoProbabilistic formulations ofemergency service facilities location problemsrdquo in Proceedingsof ORSATIMS Conference San Juan Puerto Rico 1974

[16] A Ben-Tal D Den Hertog A De Waegenaere B Melenbergand G Rennen ldquoRobust solutions of optimization problemsaffected by uncertain probabilitiesrdquo Management Science vol59 no 2 pp 341ndash357 2013

[17] MNg ldquoDistribution-free vessel deployment for liner shippingrdquoEuropean Journal of Operational Research vol 238 no 3 pp858ndash862 2014

[18] M Ng ldquoContainer vessel fleet deployment for liner shippingwith stochastic dependencies in shipping demandrdquo Transporta-tion Research Part B Methodological vol 74 pp 79ndash87 2015

[19] R Jiang and Y Guan ldquoData-driven chance constrained stochas-tic programrdquo Mathematical Programming vol 158 no 1-2 SerA pp 291ndash327 2016

[20] C-M Lee and S-L Hsu ldquoThe effect of advertising on thedistribution-free newsboy problemrdquo International Journal ofProduction Economics vol 129 no 1 pp 217ndash224 2011

[21] K Kwon and T Cheong ldquoA minimax distribution-free proce-dure for a newsvendor problem with free shippingrdquo EuropeanJournal of Operational Research vol 232 no 1 pp 234ndash2402014

12 Journal of Advanced Transportation

[22] Y Zhang S Shen and S A Erdogan ldquoDistributionally robustappointment scheduling with moment-based ambiguity setrdquoOperations Research Letters vol 45 no 2 pp 139ndash144 2017

[23] Y Zhang S Shen and S A Erdogan ldquoSolving 0-1semidefinite programs for distributionally robust allocationof surgery blocksrdquo Social Science Electronic Publishing 2017httpsssrncomabstract=2997524

[24] F Zheng J He F Chu and M Liu ldquoA new distribution-freemodel for disassembly line balancing problem with stochastictask processing timesrdquo International Journal of ProductionResearch pp 1ndash13 2018

[25] J Cheng E Delage and A Lisser ldquoDistributionally robuststochastic knapsack problemrdquo SIAM Journal on Optimizationvol 24 no 3 pp 1485ndash1506 2014

[26] M Liu D Yang and F Hao ldquoOptimization for the locationsof ambulances under two-stage life rescue in the emergencymedical service a case study in Shanghai ChinardquoMathematicalProblems in Engineering vol 2017 Article ID 1830480 14 pages2017

4 Journal of Advanced Transportation

The objective function denotes the goal to minimize thetotal cost consisting of two parts (i) the cost for constructingbases ie sum119894isin119868 119891119894 sdot 119909119894 and (ii) the cost for deploying ambu-lances ie sum119894isin119868 119892119894 sdot 119911119894

Constraint (3) ensures that ambulances can only belocated at the opened bases Constraint (4) ensures that thenumber of ambulances sent to serve emergency points frombase 119894 isin 119868 does not exceed the total number of ambulanceslocated at 119894 isin 119868 Constraints (5)-(7) are the restrictions ondecision variables

4 Solution Approaches

The proposed distribution-free model is difficult to solvewith the commercial software due to chance constraints Inthis section we propose two approximatedMIP formulationsbased on those in Wagner [11] and in Delage and Ye [12]respectively In the following subsections we present the twoapproximated MIP formulations

41 Approximated MIP Formulation MIP-DF1 In this partwe first describe a widely used ambiguity set to describe theuncertainty Then an approximated MIP formulation MIP-DF1 is proposed

Given a set of independent historical data samples 119889119904|119878|119904=1of random vectors of demands where 119878 is the set of sampleindexes the mean vector 120583 and the covariance matrix Σ ofdemands can be estimated as follows

120583 = 1|119878|sum119904isin119878119889119904Σ = 1|119878|sum119904isin119878 (119889119904 minus 120583) (119889119904 minus 120583)⊤

(8)

where (sdot)⊤ implies the transposition of the vector in paren-theses Then an ambiguity set of all probability distributionsof demandsP1(120583 Σ) can be described as the following

P1 (120583 Σ) = P EP [119889] = 120583EP [(119889 minus 120583) (119889 minus 120583)⊤] = Σ (9)

whereP denotes a possible probability distribution satisfyingthe given conditions and EP[sdot] denotes the expected valueof the number in parentheses Then the chance constraint (1)with the ambiguity setP1 can be presented as follows

ProbP (sum119894isin119868119895

119910119894119895 ge 119889119895) ge 120572119895 forall119895 isin 119869 P isin P1 (10)

According to Wagner (2010) and Ng [18] constraint (1) canbe conservatively approximated by the following

sum119894isin119868119895

119910119894119895 ge 120583119895 + 120590119895 sdot radic 120572119895(1 minus 120572119895) forall119895 isin 119869 (11)

where 120590119895 = radicΣ119895119895 and Σ119895119895 denotes the 119895-th element in the 119895-thcolumn of matrix Σ In terms of conservative approximation

all solutions satisfying constraint (11) must satisfy constraint(1) Then we describe the approximated MIP formulationMIP-DF1 toDF combined with constraint (11)

[MIP-DF1]

min sum119894isin119868

(119891119894 sdot 119909119894 + 119892119894 sdot 119911119894)st Constraints (4) - (8) (12)

(12)

MIP-DF1 can be exactly solved by the commercialsoftware such as CPLEX As we can observe from thecomputational results in Section 5 it can obtain a relativelyhigh service level However the system cost is also high Inorder to save the system cost another approximated MIPformulation of DF is then proposed in the next subsection

42 Approximated MIP Formulation MIP-DF2 AmbiguitysetP1 focuses on exactly matching the mean and covariancematrix of uncertain parameters However in practice theremay be considerable estimation errors in the mean andcovariance matrix To take the inevitable estimation errorsinto consideration Delage and Ye [12] introduce a newmoment-based ambiguity set Therefore we in the followingemploy the moment-based ambiguity set

P2 (120583 Σ 1205741 1205742)= P (EP [119889] minus 120583)⊤ Σminus1 (EP [119889] minus 120583) le 1205741

EP [(119889 minus 120583) (119889 minus 120583)⊤] ⪯ 1205742Σ (13)

where 1205741 ge 0 and 1205742 ge 0 are two parameters of ambiguityset P2(120583 Σ 1205741 1205742) with the following assumptions (i) thetrue mean vector of demands is within an ellipsoid of sizeproportion to 1205741 centered at 120583 and (ii) the true covariancematrix of demands is in a positive semidefinite cone boundedby amatrix inequality of 1205742ΣThe ambiguity set describes howlikely the uncertain parameters are to be close to the meanin terms of the correlation [12] Besides under the moment-based ambiguity set various stochastic programs with partialdistributional information have been successfully modeledand solved [22 23 25]

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

radic 11 minus 119886 minus 119887 sdot (1 + radic1 minus 120572119895120572119895 sdot 119887) sdot 120590119895le radic1 minus 120572119895120572119895 sdot (sum

119894isin119868119895

119910119894119895 minus 120583119895) 119895 isin 119869(14)

where

119886 = 1 minus 1205741 + 11205742 minus 1205741 119887 = 12057411205742 minus 1205741

(15)

Journal of Advanced Transportation 5

Table 1 120572 = 095|119868| = |119869| Sampling approach MIP-DF1 MIP-DF2

CT Obj Sel () CT Obj Sel () CT Obj Sel ()1 7 49 23 9626 01 36 10000 01 23 100002 7 51 22 9863 01 35 10000 01 24 99823 6 41 14 9692 01 30 10000 01 14 98504 6 43 13 9713 02 29 10000 01 12 97425 6 57 12 7996 01 30 10000 02 12 100006 8 58 21 9850 01 24 10000 02 27 10000

Average 50 175 9457 01 307 10000 01 187 9929

The second approximated MIP formulationMIP-DF2 toDF is presented as follows

[MIP-DF2]

min sum119894isin119868

(119891119894 sdot 119909119894 + 119892119894 sdot 119911119894)st Constraints (4) - (8) (15)

(16)

Notice that when 1205741 = 0 and 1205742 = 1 inequality (14)reduces to (11) in MIP-DF1 which implies that MIP-DF1 isa special case of MIP-DF2

5 Computational Experiments

In this section the performance of our proposed formula-tions first evaluated and compared to the sampling approachproposed in Nickel et al [8] Specifically given a requiredservice level 120572 we apply our MIP formulations by restrictingthe chance constraint for each emergency point to satisfya safety level 120572119895 = 120572 while we adopt the samplingapproach by restricting the coverage constraint with 120572 Thencomputational results on 120 randomly generated instancesbased on the information on emergencies in Shanghai arereported Our approximated MIP formulations and thesampling approach are solved by coding in MATLAB 2014bcalling CPLEX 126 solver All numerical experiments areconducted on a personal computer with Core I5 and 330GHzprocessor and 8GB RAMunder windows 7 operating systemThe computational time for sampling approach is limited to3600 seconds

51 Out-of-Sample Test FollowingZhang et alrsquos [22]method-ology we examine the out-of-sample performance of solu-tions obtained by our proposed approximated MIP formula-tions and the sampling approach [8] The out-of-sample testincludes three steps

(1) The base locations and the employment of ambu-lances at each base are determined with known meanvector and covariance matrix of demands which isnamed as in-sample test

(2) 1000 scenarios are sampled from the underlying truedistribution which represent realizing the uncertaindemand

(3) Based on the solution obtained in step (1) ambu-lances are assigned to serve emergency points inthe 1000 scenarios to maximize the (out-of-sample)service level

The (out-of-sample) service level is defined as the ratioof the number of emergency points with demand satisfiedto the total number of emergency points in 1000 scenariosIn the following subsections the benchmarks are tested andthen three common distributions are employed ie uniformdistribution Poisson distribution and normal distributionrespectively

52 Benchmark Instances We first test the benchmarks inNickel et al [8] to compare our proposed two approximatedMIP formulationswith the sampling approach InNickel et al[8] they assume that demand nodes are also potential baselocations and the number of ambulances required by eachemergency point is chosen from set 0 1 2with given proba-bility distribution Computational results on benchmarks arereported in Tables 1 and 2

Table 1 reports the computational results with servicelevel requirement120572 = 095 In Table 1 CTObj and Sel denotethe computational time objective value (ie the system cost)and the service level respectively The first column includesthe indexes of instances and the second indicates the numberof nodes It shows that the average service level obtainedby the sampling approach is 9457 which is smaller thanthe required service level ie 85 Specifically the servicelevel of instance 5 obtained by the sampling approach is only7996 However MIP-DF1 and MIP-DF2 can guarantee theservice levels for all instances higher than the requirementConcluding from Table 1 we can observe that (i) the averagecomputational time of sampling approach is about 50 timesas those of the two approximated MIP formulations and (ii)MIP-DF1 and MIP-DF2 improve the average service level byabout 574 and 499 compared with sampling approachhowever (iii) system costs obtained by MIP-DF1 and MIP-DF2 are respectively 7543 and 686 higher than that ofsampling approach

Table 2 reports the computational results with servicelevel requirement 120572 = 099 For instances 2-6 the ser-vice levels obtained by sampling approach cannot meet therequirement ie 99 Besides it shows that the averagecomputational time of sampling approach is 48 which isabout 48 times greater than those of MIP-DF1 andMIP-DF2

6 Journal of Advanced Transportation

Table 2 120572 = 099|119868| = |119869| Sampling approach MIP-DF1 MIP-DF2

CT Obj Sel () CT Obj Sel () CT Obj Sel ()1 7 49 24 9979 01 64 10000 01 25 100002 7 48 23 9856 01 63 10000 01 24 100003 6 51 15 9717 01 55 10000 01 16 100004 6 47 15 9275 01 52 10000 01 13 100005 6 40 14 9000 01 55 10000 01 14 100006 8 52 23 9785 01 45 10000 01 27 10000

Average 48 190 9343 01 557 10000 01 198 10000

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

Sampling approach MIP-DF1 MIP-DF2Emergency 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 0Base locations Ambulance employment Ambulance employment Ambulance employment1 0 0 02 1 1 0 3 1 23 1 1 26 1 2 2 04 0 0 05 0 6 06 2 2 0 4 2

The average service level is 9343 Although the averageservice levels obtained by MIP-DF1 and MIP-DF2 are both100 the average system cost obtained by MIP-DF1 is about3 times larger than that of sampling approach Besides theaverage system cost obtained by MIP-DF2 is 421 higherthan that of sampling approach

Concluding for the benchmarks described in Nickel etal [8] we can observe that (1) MIP-DF1 and MIP-DF2 canbe solved in far less computational time compared with thesampling approach (2) the sampling approach cannot meetthe requirement of service level for all benchmarks whilethe two MIP formulations can (3) the average system cost ofMIP-DF2 is very close to that of sampling approach (4) withthe increase of 120572 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 ofthe lower system cost obtained by the sampling approachthere may exist extreme situations that demands of someemergency points are not satisfied Take one benchmark inNickel et al [8] ie instance 4 where there are 6 nodesA scenario in which demand at each emergency point isgenerated from the given probability distribution is shown inFigure 1 We can observe from Figure 1 that under the givenscenario the demand of each possible emergency point is1 0 2 0 2 0The solutions obtained by sampling approachMIP-DF1 andMIP-DF2 are shown in Table 3 which includesbase 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 poten-tial base locations The second column represents the baselocations and the ambulance employment at each location

＞1 = 1

＞2 = 0

＞3 = 2

＞4 = 0

＞5 = 2

＞6 = 0

1

2 4

3 5

6

Figure 1 Instance 4 in Nickel et al [8] under given scenario

in which a location with 0 ambulance implies that it is notselected for base construction From Table 3 we can observethat the in-sample solutions obtained by sampling approachis to construct bases at locations 2 3 and 6 where the numberof ambulances is 1 1 and 2 respectively Besides undergiven scenario there is one ambulance serving emergencypoint 1 from base location 3 and one ambulance servingemergency point from base location 2 and 2 ambulancesserving emergency point 5 from base location 6 It shows thatthe requirement of ambulances at emergency point 3 is notfully satisfied The insufficient service may cause serious lossof life However we can find that the solutions obtained byMIP-DF1 andMIP-DF2 can provide sufficient ambulances forall emergency points

In sum to guarantee the required service level withless cost MIP-DF2 is recommended for the solutions with

Journal of Advanced Transportation 7

Nickel et al [8]rsquos data set Moreover in the benchmarks itis assumed that all nodes are possible emergency points aswell as potential base locations and the number of requiredambulances is small which is not always consistent withthe situation in Shanghai Therefore we further test the twoapproximated MIP formulations and the sampling approachon randomly generated instances by analysing the informa-tion on emergencies in Shanghai [26]

53 Computational Tests on Randomly Generated InstancesIn the following subsection numerical experiments on 120randomly generated instances are described

531 Test Instances By analysing the information on emer-gencies in Shanghai [26] the mean value 120583119895 of demand 119889119895 ineach emergency point are randomly generated fromadiscreteUniform distribution on interval [5 25] The standard devia-tion 120590119895 is set to be 5 Since the sampling approach proposed inNickel et al [8] requires a sample of scenarios in numericalexperiments scenarios are produced with demands randomlygenerated from Uniform distribution Poisson distributionand Normal distribution consistent with given mean andstandard deviation

According to the instance settings in Erkut and Ingolfsson(2008) response time from a base to an emergency pointis randomly generated from a discrete Uniform distributionon interval [3 30] in units of minutes A base covers anemergency point if the driving time between them is lessthan 10 minutes to ensure the survival probability (Erkut andIngolfsson 2006) For an emergency point in each instanceif there is no base with 10 minutes driving time to it then thebase covering this point is set to be the nearest one The costof constructing a base and that of locating one ambulance areboth set to be 1 as in Nickel et al [8] Besides the number ofemergency points is 3 times that of potential base locations

532 Results and Discussion Different values of requiredservice level 120572 are tested ie 085 09 095 Computationalresults on 120 randomly generated instances are reported inTables 4-6

Table 4 reports the numerical results of instances withdemand generated from Uniform distribution We observethat when 120572 = 085 09 095 the average computationaltimes of the sampling approach are 4482 4379 and 4295seconds respectively However both approximated MIP for-mulations can produce solutions in much faster speeds iewithin 13 seconds on average With the increase of 120572 (i)the out-of-sample average service level is getting higher and(ii) the system costs of solutions obtained by all method aregetting larger and (iii) the system costs of solutions yieldedbyMIP-DF2 is getting closer to those of solutions obtained bythe sampling approach In terms of the average system costthe sampling approach performs the best while MIP-DF1 isthe poorest When 120572 = 085 for example the average systemcost ofMIP-DF2 and the sampling approach are about 3032and 442 less than that of MIP-DF1 respectively When120572 = 095 the above gaps are even larger Concluding fortheUniformdistribution (1) the proposed two approximatedMIP formulations are far more time saving than the sampling

approach (2)MIP-DF1 andMIP-DF2 can improve the servicelevel by about 1907 compared with the sampling approachhowever (3) despite of the highest service levels obtainedby MIP-DF1 the system cost obtained by MIP-DF1 is alsohigh (4) MIP-DF2 can ensure an appropriate service levelwith much less cost than MIP-DF1 and (5) for the situationswith small-scale instances and high service level requirementMIP-DF1 is recommended

Computational results of instances with demand at eachemergency point generated from Poisson distribution arepresented in Table 5 For MIP-DF1 and MIP-DF2 onlymean and variance are involved Therefore with givenmean and standard deviation of the demand at each pointcomputational results are the same for different probabilitydistributions We can obtain from Table 5 that the objectivevalue obtained by the sampling approach increases with theincrease of 120572 Besides when the value of 120572 equals 085the average service level obtained by the sampling approachis about 8143 which is smaller than that of the twoapproximated MIP formulations The highest service level9934 is obtained by MIP-DF1 and that obtained by MIP-DF2 is 9708 When 120572 = 09 the average service levelof the sampling approach is 8437 and those obtained byMIP-DF1 and MIP-DF2 are 9985 and 9797 respectivelyWhen 120572 = 095 the average service level obtained by thesampling approach is 9356 and those yielded by MIP-DF1 and MIP-DF2 are 9997 and 9781 respectively ForPoisson distribution concluding MIP-DF1 and MIP-DF2can improve the service level compared with the samplingapproach especially MIP-DF1 can obtain the highest servicelevel

Similarly Table 6 reports the computational results forinstances in which the demand at each emergency point isgenerated following Normal distribution When 120572 = 085the average service levels obtained by the sampling approachMIP-DF1 and MIP-DF2 are 8174 9925 and 9742respectively When 120572 = 09 the three values are 84579981 and 9743 respectively Moreover for 120572 = 095 thevalues are equal to 8732 9999 and 9794 respectivelyIt can be observed that the performance of the samplingapproach is the poorest in terms of the service level

By observing the above computational results on bench-marks and randomly generated instances we conclude that(1)The proposed approximated MIP formulations ie MIP-DF1 and MIP-DF2 are very time saving compared with thesampling approach (2) MIP-DF1 an MIP-DF2 can achievethe requirement of service level and improve the service levelby about 1547 and 1229 on average compared with thatof the sampling approach (3) MIP-DF1 is more conservativein terms of overhigh emergency service level close to 100and the system cost is also high which is about 568 largerthan that of MIP-DF2 (4) Compared with MIP-DF1 MIP-DF2 is less conservative and much more cost saving whileensuring an appropriately service level on average which isabout 97

In sum for medium and large problem instances (egShanghai) which are common in practice to guarantee therequired service level with less cost we recommendMIP-DF2as solution method However for very small instances with

8 Journal of Advanced Transportation

Table4Com

putatio

nalresultson

insta

nces

with

datageneratedfro

mun

iform

distrib

ution

Set (|119868||119869|)

120572=085

120572=09

120572=095

Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2CT

Obj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

1(26)

27

897317

01

149

9906

01

107

9222

28

967373

01

155

9933

01

107

9167

27

106

7585

01

185

9987

01

107

9128

2(412

)47

169

7423

00

291

9919

00

207

9122

58

181

7438

01

303

9997

01

207

9363

50

199

7702

00

363

9982

00

207

9338

3(618

)72

258

7506

01

438

9976

01

312

9106

75279

8082

01

456

9996

01

312

9369

78304

8337

01

546

9997

01

312

9357

4(824)

99314

7956

01

553

9993

01

385

9206

111

338

8202

01

577

9999

01

385

9602

112

366

8492

01

697

9994

01

385

9620

5(1030)

132

415

8406

01

722

9896

01

512

9406

143

441

8322

02

752

9983

01

512

9692

142

476

8609

01

902

9993

01

512

9692

6(1236)

170

475

8249

02

843

9870

01

591

9367

181

504

8251

01

879

9982

01

591

9608

189

535

8726

01

1059

9997

01

591

9609

7(14

42)

215

573

8111

02

1006

9849

02

712

9606

243

609

8217

02

1048

9953

02

712

9589

233

644

8608

02

1258

9992

02

712

9582

8(1648)

248

686

7532

02

1185

9896

02

849

9725

284

727

8307

02

1233

9949

02

849

9613

281

768

9063

02

1473

9999

03

849

9622

9(1854)

313

691

8353

03

1236

9974

04

858

9709

351

731

8397

03

1290

9988

03

858

9765

352

760

8906

03

1560

9994

03

858

9758

10(2060)

372

794

8174

03

1409

9887

03

989

9657

428

843

8489

03

1469

9947

03

989

9773

432

892

9004

03

1769

9997

03

989

9738

11(2266

)451

830

8072

04

1493

9949

03

1031

9758

518

880

8328

04

1559

9982

03

1031

9723

516

930

8967

05

1889

9995

03

1031

9727

12(2472)

543

1025

7970

04

1771

9927

04

1267

9835

601

1083

8482

05

1843

9949

04

1267

9776

637

1146

9014

04

2203

9993

04

1267

9777

13(2678)

631

1026

8142

05

1819

9898

04

1274

9782

743

1087

8678

05

1897

9936

04

1274

9778

765

1146

9023

04

2287

9997

05

1273

9837

14(2884)

782

1101

7903

05

1949

9869

08

1361

9804

878

1165

8790

05

2033

9935

07

1361

9889

870

1227

9056

05

2453

9998

06

1361

9808

15(3090)

907

1091

8098

06

1997

9915

06

1367

9629

1018

1155

8329

06

2087

9981

06

1367

9807

1020

1225

8986

06

2537

9999

08

1367

9817

16(3296)

1062

1160

8798

07

2123

9944

07

1449

9787

1183

1229

8383

07

2219

9988

06

1449

9849

1193

1300

9105

06

2698

9998

06

1449

9815

17(34102)

1202

1298

7572

07

2328

9969

07

1614

9821

1377

1374

8860

08

2430

9997

07

1614

9824

1260

1452

9085

07

2940

9999

08

1614

9818

18(36108)

1394

1408

8121

08

2509

9947

08

1752

9809

1544

1492

8928

08

2617

9983

08

1752

9799

1548

1580

9173

08

3156

9999

08

1752

9809

19(38114

)1618

1508

8170

09

2676

9899

111877

9742

1760

1598

9130

09

2790

9934

08

1877

9908

1755

1688

9186

09

3360

9999

09

1877

9872

20(40120)

2206

1628

7719

102869

9989

102029

9873

2007

1732

8753

112990

9997

09

2029

9879

1904

1827

9138

103590

9998

09

2029

9875

21(42126)

2780

1622

8354

102900

9978

102017

9732

2609

1718

8854

103026

9876

102017

9818

2670

1812

9089

123655

9999

102017

9848

22(44132)

3189

1735

8494

113088

9985

152165

9857

2970

1840

8922

123220

9925

122165

9886

2873

1952

9206

113880

9999

112165

9850

23(46138)

3630

1937

8141

123372

9893

132406

9796

3222

2048

8937

123510

9934

142406

9906

3461

2170

8989

124199

10000

122406

9809

24(48144)

4226

1968

8226

133461

9932

132453

9814

3674

2085

8846

133605

9956

152453

9912

3556

2211

9300

134326

9999

132453

9819

25(50150)

4727

1919

8111

143456

9999

152407

9629

4019

2037

8699

143606

9999

142407

9857

4257

2161

9259

154356

9995

142407

9819

26(52156)

5300

1906

8096

153484

9878

172392

9658

4495

2021

8952

143640

9999

152392

9568

4486

2142

9182

1544

209999

152392

9854

27(54162)

5913

2096

8387

163751

9846

172617

9761

5117

2221

9045

163913

9998

172617

9544

4830

2355

9346

184723

9999

162617

9893

28(56168)

6310

2110

7758

173819

9933

192643

9829

5777

2234

9157

183987

9998

182643

9526

5550

2365

9045

184827

9999

172643

9831

29(58174)

6780

2175

7998

183937

9898

182720

9714

6303

2307

9022

20

4111

9969

192720

9623

6189

2444

9128

194981

10000

192720

9914

30(60180)

7593

2384

8321

194240

9967

20

2979

9841

6791

2527

8978

1944

199979

23

2979

9721

6878

2679

9143

21

5316

9999

20

2979

9869

31(62186)

8220

2281

7604

20

4157

9984

21

2856

9796

7734

2416

8934

21

4343

9997

21

2856

9679

7466

2557

9158

22

5273

9999

23

2856

9902

32(64192)

8766

2517

8378

21

4489

9932

25

3145

9729

8276

2671

8296

23

4680

9977

24

3145

9825

7897

2822

9089

24

5640

9999

22

3145

9891

33(66198)

10123

2532

7806

23

4555

9967

23

3169

9615

9010

2682

8947

24

4752

9989

24

3169

9703

9051

2835

9157

24

5742

9998

24

3169

9867

34(68204)

11010

2535

7934

24

4602

9999

29

3174

9631

9898

2685

8858

30

4806

9998

28

3174

9762

9863

2844

9236

26

5824

9999

25

3174

9889

35(70210)

11666

2714

8062

25

4862

9986

29

3391

9787

1066

62875

9137

26

5072

9999

29

3391

9716

10822

3043

9315

26

6122

9999

29

3391

9923

36(72216)

12011

2827

8104

26

5047

9991

28

3535

9798

1246

52984

9016

28

5263

9999

28

3535

9878

11667

3169

9253

30

6343

9999

29

3535

9896

37(74222)

11437

2826

8228

28

5096

9933

29

3543

9884

13582

3098

8895

29

5318

9999

32

3543

9888

12654

3178

9139

29

6427

10000

31

3543

9893

38(76228)

12435

2839

8152

29

5161

9897

29

3565

9788

13976

3103

8926

29

5388

9999

31

3565

9724

13837

3185

9255

33

6528

10000

31

3565

9895

39(78234)

1464

32945

8412

31

5331

9998

32

3695

9876

14794

3210

9005

31

5565

9999

34

3695

9713

14561

3313

9081

34

6735

10000

33

3695

9878

40(80240)

1604

22975

8510

32

5414

9957

36

3734

9796

16263

3346

8999

35

5654

9999

35

3734

9806

15869

3343

9248

35

6854

10000

35

3734

9873

Average

4482

1584

680

67

1228

397

9936

131978

796

82

4379

16913

8654

1329

626

9975

131978

79721

4295

1778

889

85

1335774

9997

131978

79775

Journal of Advanced Transportation 9

Table5Com

putatio

nalresultson

insta

nces

with

datageneratedfro

mPo

isson

distr

ibution

set ( |119868 |

|119869 |)120572=

085120572=

09120572=

095Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2CT

Obj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

1(26)

27

907294

9967

9001

27

987770

9979

9013

24

103

9273

9999

9468

2(412

)46

170

7961

9921

9117

51

183

8305

9946

9128

49

201

9392

9998

9546

3(618

)68

258

7578

9938

8911

74280

8028

9968

9004

69

307

9182

9999

9658

4(824)

93315

8001

9928

9196

102

338

8438

9967

9172

92368

9468

9999

9613

5(1030)

120

413

8041

9939

9449

133

443

8447

9976

9272

121

479

9544

9999

9752

6(1236)

155

476

8084

9877

9554

167

508

8863

9991

9569

159

536

9456

9999

9734

7(14

42)

195

573

7876

9892

9663

213

609

9124

9998

9729

205

643

9314

10000

9768

8(1648)

232

687

7964

9997

9788

267

724

8523

9988

9687

238

772

9422

9998

9689

9(1854)

293

692

8267

9893

9809

327

730

8386

9999

9750

298

757

9228

9999

9755

10(2060)

356

796

7946

9899

9801

399

840

8299

9976

9686

360

885

9668

9999

9705

11(2266)

423

829

8201

9895

9756

489

877

8378

9987

9788

448

927

9476

9999

9812

12(2472)

512

1023

8338

9913

9853

554

1085

8501

9999

9881

514

1148

9398

9999

9888

13(2678)

592

1045

8416

9907

9873

687

1088

7998

9996

9802

607

1152

9504

9999

9821

14(2884)

723

1099

7946

9915

9788

838

1166

8056

9992

9829

776

1230

9546

9999

9839

15(3090)

855

1089

7915

9964

9703

942

1156

8432

9984

9778

903

1225

9223

9998

9785

16(3296)

1009

1158

8024

9897

9769

1163

1231

8500

9996

9824

1045

1222

9199

9999

9835

17(34102)

1165

1299

8034

9919

9779

1285

1375

8495

9937

9808

1190

1457

9258

10000

9846

18(36108)

1254

1411

8176

9943

9885

1479

1492

8561

9968

9868

1345

1579

9286

9999

9875

19(38114

)1504

1506

7970

9923

9842

1651

1594

8167

9972

9846

1564

1694

9376

9999

9786

20(40120)

1706

1635

7819

9916

9783

1854

1734

8607

9968

9776

1778

1825

9285

9904

9718

21(42126)

2054

1622

8354

9918

9834

2357

1720

8297

9978

9833

2226

1856

9501

9999

9834

22(44132)

2400

1736

8494

9922

9846

2852

1844

8269

9999

9816

2680

1946

9486

9999

9820

23(46138)

2651

1934

8141

9953

9793

3159

2049

8492

9969

9799

3072

2172

9358

10000

9813

24(48144)

3004

1961

8226

9947

9681

3522

2086

8513

9998

9706

3102

2209

9488

9999

9724

25(50150)

3308

1924

8111

9965

9746

3898

2038

8264

9999

9788

3639

2161

9546

9999

9813

26(52156)

3669

1911

8096

9939

9855

4305

2027

8722

9967

9874

4453

2142

9185

9999

9876

27(54162)

4418

2095

8387

9927

9754

4771

2219

8341

9989

9749

4806

2353

9289

9999

9755

28(56168)

5367

2113

8458

9968

9848

5356

2237

8452

9978

9855

5221

2369

9228

10000

9868

29(58174)

6217

2186

7998

9989

9694

5942

2315

8316

9996

9699

5853

2371

9167

9999

9705

30(60180)

7263

2382

8321

9991

9837

6527

2543

8359

9998

9886

6214

2678

9369

9997

9886

31(62186)

8214

2346

8404

9936

9888

7813

2430

8321

9999

9846

6876

2560

9358

9999

9847

32(64192)

8863

2518

8378

9938

9807

8106

2677

8356

9999

9833

7877

2822

9408

10000

9835

33(66198)

9985

2532

8406

9917

9709

9684

2689

8497

9996

9746

8088

2838

9507

9999

9749

34(68204)

11264

2534

8543

9908

9897

10986

2685

8809

9996

9782

8469

2843

8844

9998

9780

35(70210)

12035

2715

8258

9937

9799

12098

2886

8569

9991

9809

9946

3041

9459

9999

9813

36(72216)

12803

2825

8104

9929

9827

13076

3010

8651

9994

9834

11013

3170

9468

9999

9865

37(74222)

13021

2830

8228

9949

9826

13261

3009

8376

9999

9826

12014

3172

9346

9999

9888

38(76228)

13862

2843

8152

9933

9807

13995

3028

8760

9998

9841

12133

3187

9706

9999

9876

39(78234)

15387

2953

8312

9964

9799

14827

3133

9001

9995

9783

13381

3304

8913

9999

9799

40(80240)

16974

2975

8510

9985

9767

15256

3249

8217

9993

9781

15867

3347

9111

9999

9791

Average

4352

15875

8143

9934

9708

4362

16856

8437

9985

9707

3968

17763

9356

9997

9781

10 Journal of Advanced Transportation

Table6Com

putatio

nalresultson

insta

nces

with

datageneratedfro

mno

rmaldistr

ibution

set ( |119868 |

|119869 |)120572=

085120572=

09120572=

095Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2CT

Obj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

1(26)

27

907294

9964

9117

27

947844

9970

9202

2558

106

8422

10000

9208

2(412

)46

170

7961

9921

9283

48

183

8328

9982

9283

4508

183

8823

10000

9370

3(618

)68

258

7578

9916

9228

68

280

7798

9971

9417

6803

307

7658

9998

9447

4(824)

93315

8000

9899

9367

96337

8379

9979

9467

9373

368

8438

9999

9579

5(1030)

120

413

7841

9927

9437

124

443

8763

9963

9476

1262

499

8850

9998

9708

6(1236)

155

476

7921

9919

9519

162

504

8124

9982

9582

16021

536

8954

10000

9756

7(14

42)

195

573

7976

9933

9598

196

609

8287

9975

9619

2011

646

8467

9999

9767

8(1648)

232

687

8031

9918

9705

235

727

8115

9965

9650

2404

772

8278

9999

9765

9(1854)

293

692

8086

9898

9759

291

734

8464

9974

9755

29815

796

8617

9998

9835

10(2060)

356

796

7984

9937

9770

358

842

8223

9988

9765

37973

921

8846

10000

9838

11(2266

)421

829

8101

9940

9755

436

877

8211

9986

9786

4514

8959

8497

9999

9789

12(2472)

514

1023

8033

9951

9785

501

1085

8162

9983

9788

52073

1147

8321

9999

9824

13(2678)

592

1023

8138

9916

9885

599

1087

8411

9970

9845

62937

1194

8609

9998

9795

14(2884)

728

1099

8235

9913

9915

722

1168

8435

9987

9806

75053

1270

8567

9997

9834

15(3090)

855

1089

8187

9922

9861

880

1157

8376

9976

9796

89222

1300

8611

9996

9798

16(3296)

1009

1158

8770

9897

9788

966

1231

8845

9984

9823

106288

1359

8997

9999

9848

17(34102)

1154

1299

8434

9925

9849

1137

1377

8601

9980

9798

123869

1493

8798

9996

9877

18(36108)

1407

1411

8321

9927

9866

1339

1495

8301

9965

9822

133807

1601

8567

9999

9834

19(38114

)1702

1506

8206

9915

9805

1672

1598

8977

9978

9845

152999

1694

8998

9999

9847

20(40120)

2091

1635

8528

9918

9889

1916

1737

8805

9976

9837

1890

021833

9012

9999

9791

21(42126)

2843

1622

8550

9934

9913

2432

1718

8886

9974

9813

2399

131815

9008

10000

9815

22(44132)

3300

1736

8449

9940

9859

2795

1843

8786

9977

9756

253106

1948

8997

9999

9826

23(46138)

3741

1934

8133

9911

9876

3143

2050

8876

9981

9785

274583

2173

9014

9999

9858

24(48144)

4531

1961

8245

9909

9846

3515

2090

8344

9993

9819

308089

2211

8661

9999

9883

25(50150)

4715

1924

8111

9929

9888

3959

2041

8279

9996

9837

3510

932158

8527

9999

9827

26(52156)

5223

1911

8096

9898

9857

4301

2025

8251

9997

9757

421747

2146

8497

9999

9914

27(54162)

5994

2095

8387

9908

9775

4835

2226

8497

9974

9780

4614

432359

8695

9999

9837

28(56168)

6406

2111

8411

9906

9798

5336

2240

8558

9991

9820

494113

2371

9249

9999

9816

29(58174)

6897

2176

8012

9916

9791

5948

2311

8211

9984

9765

624835

2445

8531

10000

9851

30(60180)

7436

2391

8111

9913

9766

6558

2533

8305

9968

9795

696116

2682

8575

9999

9799

31(62186)

8118

2282

8119

9928

9806

7069

2422

8417

9973

9873

797768

2564

8667

9999

9846

32(64192)

8895

2519

8223

9919

9757

7857

2675

8631

9982

9769

864813

2827

8976

10000

9905

33(66198)

11235

2533

8563

9928

9811

9162

2688

8547

9979

9828

973089

2843

8657

9999

9846

34(68204)

11587

2534

8249

9926

9847

9549

2686

8798

9938

9856

10523

3041

8999

10000

9877

35(70210)

1164

92715

8098

9895

9872

10345

2880

8602

9998

9869

11272

3052

9021

9999

9897

36(72216)

12224

2825

8403

9958

9770

11165

3002

8657

9999

9839

12456

3175

9113

9999

9864

37(74222)

12978

2830

8357

9937

9795

11850

3005

8496

9998

9894

13458

3176

8827

9999

9905

38(76228)

13426

2844

8267

9966

9759

12795

3019

8604

9999

9797

15155

3121

8978

9999

9853

39(78234)

15003

2955

8303

9956

9867

13816

3132

8588

9996

9885

15041

3310

9016

9999

9912

40(80240)

16495

2978

8257

9954

9826

14896

3160

8495

9997

9837

16563

3349

8957

9999

9933

Average

4619

15855

8174

9925

9742

4077

16828

8457

9981

9743

4350

17938

8732

9999

9794

Journal of Advanced Transportation 11

high service level requirement we recommend MIP-DF1 forsolutions

6 Conclusion

This paper investigates an ambulance location problem withambiguity set of demand in which the probability distribu-tion of the uncertain demand is unknown In such a data-driven environment only the mean and covariance demandsat emergency points are knownThe problem is to determinethe base locations and the number of ambulances at eachbase aiming at minimizing the total cost associated withlocating bases and assigning ambulances We propose adistribution-free model with chance constraints Then twoapproximated MIP formulations with different approxima-tion methods of chance constraints are proposed which arebased on different ambiguity sets of the unknown probabilitydistribution Numerical experiments on benchmarks and120 randomly generated instances are conducted and thecomputational results show that the first approximated MIPformulation is much conservative with overhigh system costwhile the second approximatedMIP formulation ismore costsaving with service level appropriately ensured

In future research wemay consider the following relevantissues First the uncertainty of driving time from a base toan emergency point shall be analysed Besides unexpected orsudden disasters may be taken into consideration Moreoverother uncertainties should be considered such as travel timethe number of available vehicles and so on The stochasticdependence should also further considered

Data Availability

The data of the benchmark instances in our manuscript canbe obtained in Nickel et al [8] For the test instances thedata is randomly generated and the way for generating data isstated in our manuscript The datasets generated during thecurrent study are available from the corresponding author onreasonable request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China (NSFC) under Grants 7142800271531011 71571134 and 71771048 This work was also sup-ported by the Fundamental Research Funds for the CentralUniversities

References

[1] L Brotcorne G Laporte and F Semet ldquoAmbulance loca-tion and relocation modelsrdquo European Journal of OperationalResearch vol 147 no 3 pp 451ndash463 2003

[2] X Li Z Zhao X Zhu and T Wyatt ldquoCovering modelsand optimization techniques for emergency response facilitylocation and planning a reviewrdquo Mathematical Methods ofOperations Research vol 74 no 3 pp 281ndash310 2011

[3] C Toregas R Swain C ReVelle and L Bergman ldquoThe locationof emergency service facilitiesrdquoOperations Research vol 19 no6 pp 1363ndash1373 1971

[4] R Church and C ReVelle ldquoThe maximal covering locationproblemrdquo Papers in Regional Science vol 32 no 1 pp 101ndash1181974

[5] M Gendreau G Laporte and F Semet ldquoSolving an ambulancelocation model by tabu searchrdquo Location Science vol 5 no 2pp 75ndash88 1997

[6] A Ingolfsson S Budge and E Erkut ldquoOptimal ambulancelocation with random delays and travel timesrdquo Health CareManagement Science vol 11 no 3 pp 262ndash274 2008

[7] M Gendreau G Laporte and F Semet ldquoThemaximal expectedcoverage relocation problem for emergency vehiclesrdquo Journal ofthe Operational Research Society vol 57 no 1 pp 22ndash28 2006

[8] S Nickel M Reuter-Oppermann and F Saldanha-da-GamaldquoAmbulance location under stochastic demand A samplingapproachrdquo Operations Research for Health Care vol 8 pp 24ndash32 2016

[9] P Beraldi M E Bruni and D Conforti ldquoDesigning robustemergency medical service via stochastic programmingrdquo Euro-pean Journal of Operational Research vol 158 no 1 pp 183ndash1932004

[10] N Noyan ldquoAlternate risk measures for emergency medicalservice system designrdquo Annals of Operations Research vol 181pp 559ndash589 2010

[11] M R Wagner ldquoStochastic 0-1 linear programming underlimited distributional informationrdquoOperations Research Lettersvol 36 no 2 pp 150ndash156 2008

[12] EDelage andYYe ldquoDistributionally robust optimization undermoment uncertainty with application to data-driven problemsrdquoOperations Research vol 58 no 3 pp 595ndash612 2010

[13] J T Essen J L Hurink S Nickel and M Reuter ldquoModels forambulance planning on the strategic and the tactical levelrdquo BetaResearch School for Operations Management and Logistics p 272013

[14] E Erkut A Ingolfsson and G Erdogan ldquoAmbulance locationfor maximum survivalrdquoNaval Research Logistics (NRL) vol 55no 1 pp 42ndash58 2008

[15] S C Chapman and J A White ldquoProbabilistic formulations ofemergency service facilities location problemsrdquo in Proceedingsof ORSATIMS Conference San Juan Puerto Rico 1974

[16] A Ben-Tal D Den Hertog A De Waegenaere B Melenbergand G Rennen ldquoRobust solutions of optimization problemsaffected by uncertain probabilitiesrdquo Management Science vol59 no 2 pp 341ndash357 2013

[17] MNg ldquoDistribution-free vessel deployment for liner shippingrdquoEuropean Journal of Operational Research vol 238 no 3 pp858ndash862 2014

[18] M Ng ldquoContainer vessel fleet deployment for liner shippingwith stochastic dependencies in shipping demandrdquo Transporta-tion Research Part B Methodological vol 74 pp 79ndash87 2015

[19] R Jiang and Y Guan ldquoData-driven chance constrained stochas-tic programrdquo Mathematical Programming vol 158 no 1-2 SerA pp 291ndash327 2016

[20] C-M Lee and S-L Hsu ldquoThe effect of advertising on thedistribution-free newsboy problemrdquo International Journal ofProduction Economics vol 129 no 1 pp 217ndash224 2011

[21] K Kwon and T Cheong ldquoA minimax distribution-free proce-dure for a newsvendor problem with free shippingrdquo EuropeanJournal of Operational Research vol 232 no 1 pp 234ndash2402014

12 Journal of Advanced Transportation

[22] Y Zhang S Shen and S A Erdogan ldquoDistributionally robustappointment scheduling with moment-based ambiguity setrdquoOperations Research Letters vol 45 no 2 pp 139ndash144 2017

[23] Y Zhang S Shen and S A Erdogan ldquoSolving 0-1semidefinite programs for distributionally robust allocationof surgery blocksrdquo Social Science Electronic Publishing 2017httpsssrncomabstract=2997524

[24] F Zheng J He F Chu and M Liu ldquoA new distribution-freemodel for disassembly line balancing problem with stochastictask processing timesrdquo International Journal of ProductionResearch pp 1ndash13 2018

[25] J Cheng E Delage and A Lisser ldquoDistributionally robuststochastic knapsack problemrdquo SIAM Journal on Optimizationvol 24 no 3 pp 1485ndash1506 2014

[26] M Liu D Yang and F Hao ldquoOptimization for the locationsof ambulances under two-stage life rescue in the emergencymedical service a case study in Shanghai ChinardquoMathematicalProblems in Engineering vol 2017 Article ID 1830480 14 pages2017

Journal of Advanced Transportation 5

Table 1 120572 = 095|119868| = |119869| Sampling approach MIP-DF1 MIP-DF2

CT Obj Sel () CT Obj Sel () CT Obj Sel ()1 7 49 23 9626 01 36 10000 01 23 100002 7 51 22 9863 01 35 10000 01 24 99823 6 41 14 9692 01 30 10000 01 14 98504 6 43 13 9713 02 29 10000 01 12 97425 6 57 12 7996 01 30 10000 02 12 100006 8 58 21 9850 01 24 10000 02 27 10000

Average 50 175 9457 01 307 10000 01 187 9929

The second approximated MIP formulationMIP-DF2 toDF is presented as follows

[MIP-DF2]

min sum119894isin119868

(119891119894 sdot 119909119894 + 119892119894 sdot 119911119894)st Constraints (4) - (8) (15)

(16)

Notice that when 1205741 = 0 and 1205742 = 1 inequality (14)reduces to (11) in MIP-DF1 which implies that MIP-DF1 isa special case of MIP-DF2

5 Computational Experiments

In this section the performance of our proposed formula-tions first evaluated and compared to the sampling approachproposed in Nickel et al [8] Specifically given a requiredservice level 120572 we apply our MIP formulations by restrictingthe chance constraint for each emergency point to satisfya safety level 120572119895 = 120572 while we adopt the samplingapproach by restricting the coverage constraint with 120572 Thencomputational results on 120 randomly generated instancesbased on the information on emergencies in Shanghai arereported Our approximated MIP formulations and thesampling approach are solved by coding in MATLAB 2014bcalling CPLEX 126 solver All numerical experiments areconducted on a personal computer with Core I5 and 330GHzprocessor and 8GB RAMunder windows 7 operating systemThe computational time for sampling approach is limited to3600 seconds

51 Out-of-Sample Test FollowingZhang et alrsquos [22]method-ology we examine the out-of-sample performance of solu-tions obtained by our proposed approximated MIP formula-tions and the sampling approach [8] The out-of-sample testincludes three steps

(1) The base locations and the employment of ambu-lances at each base are determined with known meanvector and covariance matrix of demands which isnamed as in-sample test

(2) 1000 scenarios are sampled from the underlying truedistribution which represent realizing the uncertaindemand

(3) Based on the solution obtained in step (1) ambu-lances are assigned to serve emergency points inthe 1000 scenarios to maximize the (out-of-sample)service level

The (out-of-sample) service level is defined as the ratioof the number of emergency points with demand satisfiedto the total number of emergency points in 1000 scenariosIn the following subsections the benchmarks are tested andthen three common distributions are employed ie uniformdistribution Poisson distribution and normal distributionrespectively

52 Benchmark Instances We first test the benchmarks inNickel et al [8] to compare our proposed two approximatedMIP formulationswith the sampling approach InNickel et al[8] they assume that demand nodes are also potential baselocations and the number of ambulances required by eachemergency point is chosen from set 0 1 2with given proba-bility distribution Computational results on benchmarks arereported in Tables 1 and 2

Table 1 reports the computational results with servicelevel requirement120572 = 095 In Table 1 CTObj and Sel denotethe computational time objective value (ie the system cost)and the service level respectively The first column includesthe indexes of instances and the second indicates the numberof nodes It shows that the average service level obtainedby the sampling approach is 9457 which is smaller thanthe required service level ie 85 Specifically the servicelevel of instance 5 obtained by the sampling approach is only7996 However MIP-DF1 and MIP-DF2 can guarantee theservice levels for all instances higher than the requirementConcluding from Table 1 we can observe that (i) the averagecomputational time of sampling approach is about 50 timesas those of the two approximated MIP formulations and (ii)MIP-DF1 and MIP-DF2 improve the average service level byabout 574 and 499 compared with sampling approachhowever (iii) system costs obtained by MIP-DF1 and MIP-DF2 are respectively 7543 and 686 higher than that ofsampling approach

Table 2 reports the computational results with servicelevel requirement 120572 = 099 For instances 2-6 the ser-vice levels obtained by sampling approach cannot meet therequirement ie 99 Besides it shows that the averagecomputational time of sampling approach is 48 which isabout 48 times greater than those of MIP-DF1 andMIP-DF2

6 Journal of Advanced Transportation

Table 2 120572 = 099|119868| = |119869| Sampling approach MIP-DF1 MIP-DF2

CT Obj Sel () CT Obj Sel () CT Obj Sel ()1 7 49 24 9979 01 64 10000 01 25 100002 7 48 23 9856 01 63 10000 01 24 100003 6 51 15 9717 01 55 10000 01 16 100004 6 47 15 9275 01 52 10000 01 13 100005 6 40 14 9000 01 55 10000 01 14 100006 8 52 23 9785 01 45 10000 01 27 10000

Average 48 190 9343 01 557 10000 01 198 10000

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

Sampling approach MIP-DF1 MIP-DF2Emergency 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 0Base locations Ambulance employment Ambulance employment Ambulance employment1 0 0 02 1 1 0 3 1 23 1 1 26 1 2 2 04 0 0 05 0 6 06 2 2 0 4 2

The average service level is 9343 Although the averageservice levels obtained by MIP-DF1 and MIP-DF2 are both100 the average system cost obtained by MIP-DF1 is about3 times larger than that of sampling approach Besides theaverage system cost obtained by MIP-DF2 is 421 higherthan that of sampling approach

Concluding for the benchmarks described in Nickel etal [8] we can observe that (1) MIP-DF1 and MIP-DF2 canbe solved in far less computational time compared with thesampling approach (2) the sampling approach cannot meetthe requirement of service level for all benchmarks whilethe two MIP formulations can (3) the average system cost ofMIP-DF2 is very close to that of sampling approach (4) withthe increase of 120572 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 ofthe lower system cost obtained by the sampling approachthere may exist extreme situations that demands of someemergency points are not satisfied Take one benchmark inNickel et al [8] ie instance 4 where there are 6 nodesA scenario in which demand at each emergency point isgenerated from the given probability distribution is shown inFigure 1 We can observe from Figure 1 that under the givenscenario the demand of each possible emergency point is1 0 2 0 2 0The solutions obtained by sampling approachMIP-DF1 andMIP-DF2 are shown in Table 3 which includesbase 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 poten-tial base locations The second column represents the baselocations and the ambulance employment at each location

＞1 = 1

＞2 = 0

＞3 = 2

＞4 = 0

＞5 = 2

＞6 = 0

1

2 4

3 5

6

Figure 1 Instance 4 in Nickel et al [8] under given scenario

in which a location with 0 ambulance implies that it is notselected for base construction From Table 3 we can observethat the in-sample solutions obtained by sampling approachis to construct bases at locations 2 3 and 6 where the numberof ambulances is 1 1 and 2 respectively Besides undergiven scenario there is one ambulance serving emergencypoint 1 from base location 3 and one ambulance servingemergency point from base location 2 and 2 ambulancesserving emergency point 5 from base location 6 It shows thatthe requirement of ambulances at emergency point 3 is notfully satisfied The insufficient service may cause serious lossof life However we can find that the solutions obtained byMIP-DF1 andMIP-DF2 can provide sufficient ambulances forall emergency points

In sum to guarantee the required service level withless cost MIP-DF2 is recommended for the solutions with

Journal of Advanced Transportation 7

Nickel et al [8]rsquos data set Moreover in the benchmarks itis assumed that all nodes are possible emergency points aswell as potential base locations and the number of requiredambulances is small which is not always consistent withthe situation in Shanghai Therefore we further test the twoapproximated MIP formulations and the sampling approachon randomly generated instances by analysing the informa-tion on emergencies in Shanghai [26]

53 Computational Tests on Randomly Generated InstancesIn the following subsection numerical experiments on 120randomly generated instances are described

531 Test Instances By analysing the information on emer-gencies in Shanghai [26] the mean value 120583119895 of demand 119889119895 ineach emergency point are randomly generated fromadiscreteUniform distribution on interval [5 25] The standard devia-tion 120590119895 is set to be 5 Since the sampling approach proposed inNickel et al [8] requires a sample of scenarios in numericalexperiments scenarios are produced with demands randomlygenerated from Uniform distribution Poisson distributionand Normal distribution consistent with given mean andstandard deviation

According to the instance settings in Erkut and Ingolfsson(2008) response time from a base to an emergency pointis randomly generated from a discrete Uniform distributionon interval [3 30] in units of minutes A base covers anemergency point if the driving time between them is lessthan 10 minutes to ensure the survival probability (Erkut andIngolfsson 2006) For an emergency point in each instanceif there is no base with 10 minutes driving time to it then thebase covering this point is set to be the nearest one The costof constructing a base and that of locating one ambulance areboth set to be 1 as in Nickel et al [8] Besides the number ofemergency points is 3 times that of potential base locations

532 Results and Discussion Different values of requiredservice level 120572 are tested ie 085 09 095 Computationalresults on 120 randomly generated instances are reported inTables 4-6

Table 4 reports the numerical results of instances withdemand generated from Uniform distribution We observethat when 120572 = 085 09 095 the average computationaltimes of the sampling approach are 4482 4379 and 4295seconds respectively However both approximated MIP for-mulations can produce solutions in much faster speeds iewithin 13 seconds on average With the increase of 120572 (i)the out-of-sample average service level is getting higher and(ii) the system costs of solutions obtained by all method aregetting larger and (iii) the system costs of solutions yieldedbyMIP-DF2 is getting closer to those of solutions obtained bythe sampling approach In terms of the average system costthe sampling approach performs the best while MIP-DF1 isthe poorest When 120572 = 085 for example the average systemcost ofMIP-DF2 and the sampling approach are about 3032and 442 less than that of MIP-DF1 respectively When120572 = 095 the above gaps are even larger Concluding fortheUniformdistribution (1) the proposed two approximatedMIP formulations are far more time saving than the sampling

approach (2)MIP-DF1 andMIP-DF2 can improve the servicelevel by about 1907 compared with the sampling approachhowever (3) despite of the highest service levels obtainedby MIP-DF1 the system cost obtained by MIP-DF1 is alsohigh (4) MIP-DF2 can ensure an appropriate service levelwith much less cost than MIP-DF1 and (5) for the situationswith small-scale instances and high service level requirementMIP-DF1 is recommended

Computational results of instances with demand at eachemergency point generated from Poisson distribution arepresented in Table 5 For MIP-DF1 and MIP-DF2 onlymean and variance are involved Therefore with givenmean and standard deviation of the demand at each pointcomputational results are the same for different probabilitydistributions We can obtain from Table 5 that the objectivevalue obtained by the sampling approach increases with theincrease of 120572 Besides when the value of 120572 equals 085the average service level obtained by the sampling approachis about 8143 which is smaller than that of the twoapproximated MIP formulations The highest service level9934 is obtained by MIP-DF1 and that obtained by MIP-DF2 is 9708 When 120572 = 09 the average service levelof the sampling approach is 8437 and those obtained byMIP-DF1 and MIP-DF2 are 9985 and 9797 respectivelyWhen 120572 = 095 the average service level obtained by thesampling approach is 9356 and those yielded by MIP-DF1 and MIP-DF2 are 9997 and 9781 respectively ForPoisson distribution concluding MIP-DF1 and MIP-DF2can improve the service level compared with the samplingapproach especially MIP-DF1 can obtain the highest servicelevel

Similarly Table 6 reports the computational results forinstances in which the demand at each emergency point isgenerated following Normal distribution When 120572 = 085the average service levels obtained by the sampling approachMIP-DF1 and MIP-DF2 are 8174 9925 and 9742respectively When 120572 = 09 the three values are 84579981 and 9743 respectively Moreover for 120572 = 095 thevalues are equal to 8732 9999 and 9794 respectivelyIt can be observed that the performance of the samplingapproach is the poorest in terms of the service level

By observing the above computational results on bench-marks and randomly generated instances we conclude that(1)The proposed approximated MIP formulations ie MIP-DF1 and MIP-DF2 are very time saving compared with thesampling approach (2) MIP-DF1 an MIP-DF2 can achievethe requirement of service level and improve the service levelby about 1547 and 1229 on average compared with thatof the sampling approach (3) MIP-DF1 is more conservativein terms of overhigh emergency service level close to 100and the system cost is also high which is about 568 largerthan that of MIP-DF2 (4) Compared with MIP-DF1 MIP-DF2 is less conservative and much more cost saving whileensuring an appropriately service level on average which isabout 97

In sum for medium and large problem instances (egShanghai) which are common in practice to guarantee therequired service level with less cost we recommendMIP-DF2as solution method However for very small instances with

8 Journal of Advanced Transportation

Table4Com

putatio

nalresultson

insta

nces

with

datageneratedfro

mun

iform

distrib

ution

Set (|119868||119869|)

120572=085

120572=09

120572=095

Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2CT

Obj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

1(26)

27

897317

01

149

9906

01

107

9222

28

967373

01

155

9933

01

107

9167

27

106

7585

01

185

9987

01

107

9128

2(412

)47

169

7423

00

291

9919

00

207

9122

58

181

7438

01

303

9997

01

207

9363

50

199

7702

00

363

9982

00

207

9338

3(618

)72

258

7506

01

438

9976

01

312

9106

75279

8082

01

456

9996

01

312

9369

78304

8337

01

546

9997

01

312

9357

4(824)

99314

7956

01

553

9993

01

385

9206

111

338

8202

01

577

9999

01

385

9602

112

366

8492

01

697

9994

01

385

9620

5(1030)

132

415

8406

01

722

9896

01

512

9406

143

441

8322

02

752

9983

01

512

9692

142

476

8609

01

902

9993

01

512

9692

6(1236)

170

475

8249

02

843

9870

01

591

9367

181

504

8251

01

879

9982

01

591

9608

189

535

8726

01

1059

9997

01

591

9609

7(14

42)

215

573

8111

02

1006

9849

02

712

9606

243

609

8217

02

1048

9953

02

712

9589

233

644

8608

02

1258

9992

02

712

9582

8(1648)

248

686

7532

02

1185

9896

02

849

9725

284

727

8307

02

1233

9949

02

849

9613

281

768

9063

02

1473

9999

03

849

9622

9(1854)

313

691

8353

03

1236

9974

04

858

9709

351

731

8397

03

1290

9988

03

858

9765

352

760

8906

03

1560

9994

03

858

9758

10(2060)

372

794

8174

03

1409

9887

03

989

9657

428

843

8489

03

1469

9947

03

989

9773

432

892

9004

03

1769

9997

03

989

9738

11(2266

)451

830

8072

04

1493

9949

03

1031

9758

518

880

8328

04

1559

9982

03

1031

9723

516

930

8967

05

1889

9995

03

1031

9727

12(2472)

543

1025

7970

04

1771

9927

04

1267

9835

601

1083

8482

05

1843

9949

04

1267

9776

637

1146

9014

04

2203

9993

04

1267

9777

13(2678)

631

1026

8142

05

1819

9898

04

1274

9782

743

1087

8678

05

1897

9936

04

1274

9778

765

1146

9023

04

2287

9997

05

1273

9837

14(2884)

782

1101

7903

05

1949

9869

08

1361

9804

878

1165

8790

05

2033

9935

07

1361

9889

870

1227

9056

05

2453

9998

06

1361

9808

15(3090)

907

1091

8098

06

1997

9915

06

1367

9629

1018

1155

8329

06

2087

9981

06

1367

9807

1020

1225

8986

06

2537

9999

08

1367

9817

16(3296)

1062

1160

8798

07

2123

9944

07

1449

9787

1183

1229

8383

07

2219

9988

06

1449

9849

1193

1300

9105

06

2698

9998

06

1449

9815

17(34102)

1202

1298

7572

07

2328

9969

07

1614

9821

1377

1374

8860

08

2430

9997

07

1614

9824

1260

1452

9085

07

2940

9999

08

1614

9818

18(36108)

1394

1408

8121

08

2509

9947

08

1752

9809

1544

1492

8928

08

2617

9983

08

1752

9799

1548

1580

9173

08

3156

9999

08

1752

9809

19(38114

)1618

1508

8170

09

2676

9899

111877

9742

1760

1598

9130

09

2790

9934

08

1877

9908

1755

1688

9186

09

3360

9999

09

1877

9872

20(40120)

2206

1628

7719

102869

9989

102029

9873

2007

1732

8753

112990

9997

09

2029

9879

1904

1827

9138

103590

9998

09

2029

9875

21(42126)

2780

1622

8354

102900

9978

102017

9732

2609

1718

8854

103026

9876

102017

9818

2670

1812

9089

123655

9999

102017

9848

22(44132)

3189

1735

8494

113088

9985

152165

9857

2970

1840

8922

123220

9925

122165

9886

2873

1952

9206

113880

9999

112165

9850

23(46138)

3630

1937

8141

123372

9893

132406

9796

3222

2048

8937

123510

9934

142406

9906

3461

2170

8989

124199

10000

122406

9809

24(48144)

4226

1968

8226

133461

9932

132453

9814

3674

2085

8846

133605

9956

152453

9912

3556

2211

9300

134326

9999

132453

9819

25(50150)

4727

1919

8111

143456

9999

152407

9629

4019

2037

8699

143606

9999

142407

9857

4257

2161

9259

154356

9995

142407

9819

26(52156)

5300

1906

8096

153484

9878

172392

9658

4495

2021

8952

143640

9999

152392

9568

4486

2142

9182

1544

209999

152392

9854

27(54162)

5913

2096

8387

163751

9846

172617

9761

5117

2221

9045

163913

9998

172617

9544

4830

2355

9346

184723

9999

162617

9893

28(56168)

6310

2110

7758

173819

9933

192643

9829

5777

2234

9157

183987

9998

182643

9526

5550

2365

9045

184827

9999

172643

9831

29(58174)

6780

2175

7998

183937

9898

182720

9714

6303

2307

9022

20

4111

9969

192720

9623

6189

2444

9128

194981

10000

192720

9914

30(60180)

7593

2384

8321

194240

9967

20

2979

9841

6791

2527

8978

1944

199979

23

2979

9721

6878

2679

9143

21

5316

9999

20

2979

9869

31(62186)

8220

2281

7604

20

4157

9984

21

2856

9796

7734

2416

8934

21

4343

9997

21

2856

9679

7466

2557

9158

22

5273

9999

23

2856

9902

32(64192)

8766

2517

8378

21

4489

9932

25

3145

9729

8276

2671

8296

23

4680

9977

24

3145

9825

7897

2822

9089

24

5640

9999

22

3145

9891

33(66198)

10123

2532

7806

23

4555

9967

23

3169

9615

9010

2682

8947

24

4752

9989

24

3169

9703

9051

2835

9157

24

5742

9998

24

3169

9867

34(68204)

11010

2535

7934

24

4602

9999

29

3174

9631

9898

2685

8858

30

4806

9998

28

3174

9762

9863

2844

9236

26

5824

9999

25

3174

9889

35(70210)

11666

2714

8062

25

4862

9986

29

3391

9787

1066

62875

9137

26

5072

9999

29

3391

9716

10822

3043

9315

26

6122

9999

29

3391

9923

36(72216)

12011

2827

8104

26

5047

9991

28

3535

9798

1246

52984

9016

28

5263

9999

28

3535

9878

11667

3169

9253

30

6343

9999

29

3535

9896

37(74222)

11437

2826

8228

28

5096

9933

29

3543

9884

13582

3098

8895

29

5318

9999

32

3543

9888

12654

3178

9139

29

6427

10000

31

3543

9893

38(76228)

12435

2839

8152

29

5161

9897

29

3565

9788

13976

3103

8926

29

5388

9999

31

3565

9724

13837

3185

9255

33

6528

10000

31

3565

9895

39(78234)

1464

32945

8412

31

5331

9998

32

3695

9876

14794

3210

9005

31

5565

9999

34

3695

9713

14561

3313

9081

34

6735

10000

33

3695

9878

40(80240)

1604

22975

8510

32

5414

9957

36

3734

9796

16263

3346

8999

35

5654

9999

35

3734

9806

15869

3343

9248

35

6854

10000

35

3734

9873

Average

4482

1584

680

67

1228

397

9936

131978

796

82

4379

16913

8654

1329

626

9975

131978

79721

4295

1778

889

85

1335774

9997

131978

79775

Journal of Advanced Transportation 9

Table5Com

putatio

nalresultson

insta

nces

with

datageneratedfro

mPo

isson

distr

ibution

set ( |119868 |

|119869 |)120572=

085120572=

09120572=

095Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2CT

Obj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

1(26)

27

907294

9967

9001

27

987770

9979

9013

24

103

9273

9999

9468

2(412

)46

170

7961

9921

9117

51

183

8305

9946

9128

49

201

9392

9998

9546

3(618

)68

258

7578

9938

8911

74280

8028

9968

9004

69

307

9182

9999

9658

4(824)

93315

8001

9928

9196

102

338

8438

9967

9172

92368

9468

9999

9613

5(1030)

120

413

8041

9939

9449

133

443

8447

9976

9272

121

479

9544

9999

9752

6(1236)

155

476

8084

9877

9554

167

508

8863

9991

9569

159

536

9456

9999

9734

7(14

42)

195

573

7876

9892

9663

213

609

9124

9998

9729

205

643

9314

10000

9768

8(1648)

232

687

7964

9997

9788

267

724

8523

9988

9687

238

772

9422

9998

9689

9(1854)

293

692

8267

9893

9809

327

730

8386

9999

9750

298

757

9228

9999

9755

10(2060)

356

796

7946

9899

9801

399

840

8299

9976

9686

360

885

9668

9999

9705

11(2266)

423

829

8201

9895

9756

489

877

8378

9987

9788

448

927

9476

9999

9812

12(2472)

512

1023

8338

9913

9853

554

1085

8501

9999

9881

514

1148

9398

9999

9888

13(2678)

592

1045

8416

9907

9873

687

1088

7998

9996

9802

607

1152

9504

9999

9821

14(2884)

723

1099

7946

9915

9788

838

1166

8056

9992

9829

776

1230

9546

9999

9839

15(3090)

855

1089

7915

9964

9703

942

1156

8432

9984

9778

903

1225

9223

9998

9785

16(3296)

1009

1158

8024

9897

9769

1163

1231

8500

9996

9824

1045

1222

9199

9999

9835

17(34102)

1165

1299

8034

9919

9779

1285

1375

8495

9937

9808

1190

1457

9258

10000

9846

18(36108)

1254

1411

8176

9943

9885

1479

1492

8561

9968

9868

1345

1579

9286

9999

9875

19(38114

)1504

1506

7970

9923

9842

1651

1594

8167

9972

9846

1564

1694

9376

9999

9786

20(40120)

1706

1635

7819

9916

9783

1854

1734

8607

9968

9776

1778

1825

9285

9904

9718

21(42126)

2054

1622

8354

9918

9834

2357

1720

8297

9978

9833

2226

1856

9501

9999

9834

22(44132)

2400

1736

8494

9922

9846

2852

1844

8269

9999

9816

2680

1946

9486

9999

9820

23(46138)

2651

1934

8141

9953

9793

3159

2049

8492

9969

9799

3072

2172

9358

10000

9813

24(48144)

3004

1961

8226

9947

9681

3522

2086

8513

9998

9706

3102

2209

9488

9999

9724

25(50150)

3308

1924

8111

9965

9746

3898

2038

8264

9999

9788

3639

2161

9546

9999

9813

26(52156)

3669

1911

8096

9939

9855

4305

2027

8722

9967

9874

4453

2142

9185

9999

9876

27(54162)

4418

2095

8387

9927

9754

4771

2219

8341

9989

9749

4806

2353

9289

9999

9755

28(56168)

5367

2113

8458

9968

9848

5356

2237

8452

9978

9855

5221

2369

9228

10000

9868

29(58174)

6217

2186

7998

9989

9694

5942

2315

8316

9996

9699

5853

2371

9167

9999

9705

30(60180)

7263

2382

8321

9991

9837

6527

2543

8359

9998

9886

6214

2678

9369

9997

9886

31(62186)

8214

2346

8404

9936

9888

7813

2430

8321

9999

9846

6876

2560

9358

9999

9847

32(64192)

8863

2518

8378

9938

9807

8106

2677

8356

9999

9833

7877

2822

9408

10000

9835

33(66198)

9985

2532

8406

9917

9709

9684

2689

8497

9996

9746

8088

2838

9507

9999

9749

34(68204)

11264

2534

8543

9908

9897

10986

2685

8809

9996

9782

8469

2843

8844

9998

9780

35(70210)

12035

2715

8258

9937

9799

12098

2886

8569

9991

9809

9946

3041

9459

9999

9813

36(72216)

12803

2825

8104

9929

9827

13076

3010

8651

9994

9834

11013

3170

9468

9999

9865

37(74222)

13021

2830

8228

9949

9826

13261

3009

8376

9999

9826

12014

3172

9346

9999

9888

38(76228)

13862

2843

8152

9933

9807

13995

3028

8760

9998

9841

12133

3187

9706

9999

9876

39(78234)

15387

2953

8312

9964

9799

14827

3133

9001

9995

9783

13381

3304

8913

9999

9799

40(80240)

16974

2975

8510

9985

9767

15256

3249

8217

9993

9781

15867

3347

9111

9999

9791

Average

4352

15875

8143

9934

9708

4362

16856

8437

9985

9707

3968

17763

9356

9997

9781

10 Journal of Advanced Transportation

Table6Com

putatio

nalresultson

insta

nces

with

datageneratedfro

mno

rmaldistr

ibution

set ( |119868 |

|119869 |)120572=

085120572=

09120572=

095Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2CT

Obj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

1(26)

27

907294

9964

9117

27

947844

9970

9202

2558

106

8422

10000

9208

2(412

)46

170

7961

9921

9283

48

183

8328

9982

9283

4508

183

8823

10000

9370

3(618

)68

258

7578

9916

9228

68

280

7798

9971

9417

6803

307

7658

9998

9447

4(824)

93315

8000

9899

9367

96337

8379

9979

9467

9373

368

8438

9999

9579

5(1030)

120

413

7841

9927

9437

124

443

8763

9963

9476

1262

499

8850

9998

9708

6(1236)

155

476

7921

9919

9519

162

504

8124

9982

9582

16021

536

8954

10000

9756

7(14

42)

195

573

7976

9933

9598

196

609

8287

9975

9619

2011

646

8467

9999

9767

8(1648)

232

687

8031

9918

9705

235

727

8115

9965

9650

2404

772

8278

9999

9765

9(1854)

293

692

8086

9898

9759

291

734

8464

9974

9755

29815

796

8617

9998

9835

10(2060)

356

796

7984

9937

9770

358

842

8223

9988

9765

37973

921

8846

10000

9838

11(2266

)421

829

8101

9940

9755

436

877

8211

9986

9786

4514

8959

8497

9999

9789

12(2472)

514

1023

8033

9951

9785

501

1085

8162

9983

9788

52073

1147

8321

9999

9824

13(2678)

592

1023

8138

9916

9885

599

1087

8411

9970

9845

62937

1194

8609

9998

9795

14(2884)

728

1099

8235

9913

9915

722

1168

8435

9987

9806

75053

1270

8567

9997

9834

15(3090)

855

1089

8187

9922

9861

880

1157

8376

9976

9796

89222

1300

8611

9996

9798

16(3296)

1009

1158

8770

9897

9788

966

1231

8845

9984

9823

106288

1359

8997

9999

9848

17(34102)

1154

1299

8434

9925

9849

1137

1377

8601

9980

9798

123869

1493

8798

9996

9877

18(36108)

1407

1411

8321

9927

9866

1339

1495

8301

9965

9822

133807

1601

8567

9999

9834

19(38114

)1702

1506

8206

9915

9805

1672

1598

8977

9978

9845

152999

1694

8998

9999

9847

20(40120)

2091

1635

8528

9918

9889

1916

1737

8805

9976

9837

1890

021833

9012

9999

9791

21(42126)

2843

1622

8550

9934

9913

2432

1718

8886

9974

9813

2399

131815

9008

10000

9815

22(44132)

3300

1736

8449

9940

9859

2795

1843

8786

9977

9756

253106

1948

8997

9999

9826

23(46138)

3741

1934

8133

9911

9876

3143

2050

8876

9981

9785

274583

2173

9014

9999

9858

24(48144)

4531

1961

8245

9909

9846

3515

2090

8344

9993

9819

308089

2211

8661

9999

9883

25(50150)

4715

1924

8111

9929

9888

3959

2041

8279

9996

9837

3510

932158

8527

9999

9827

26(52156)

5223

1911

8096

9898

9857

4301

2025

8251

9997

9757

421747

2146

8497

9999

9914

27(54162)

5994

2095

8387

9908

9775

4835

2226

8497

9974

9780

4614

432359

8695

9999

9837

28(56168)

6406

2111

8411

9906

9798

5336

2240

8558

9991

9820

494113

2371

9249

9999

9816

29(58174)

6897

2176

8012

9916

9791

5948

2311

8211

9984

9765

624835

2445

8531

10000

9851

30(60180)

7436

2391

8111

9913

9766

6558

2533

8305

9968

9795

696116

2682

8575

9999

9799

31(62186)

8118

2282

8119

9928

9806

7069

2422

8417

9973

9873

797768

2564

8667

9999

9846

32(64192)

8895

2519

8223

9919

9757

7857

2675

8631

9982

9769

864813

2827

8976

10000

9905

33(66198)

11235

2533

8563

9928

9811

9162

2688

8547

9979

9828

973089

2843

8657

9999

9846

34(68204)

11587

2534

8249

9926

9847

9549

2686

8798

9938

9856

10523

3041

8999

10000

9877

35(70210)

1164

92715

8098

9895

9872

10345

2880

8602

9998

9869

11272

3052

9021

9999

9897

36(72216)

12224

2825

8403

9958

9770

11165

3002

8657

9999

9839

12456

3175

9113

9999

9864

37(74222)

12978

2830

8357

9937

9795

11850

3005

8496

9998

9894

13458

3176

8827

9999

9905

38(76228)

13426

2844

8267

9966

9759

12795

3019

8604

9999

9797

15155

3121

8978

9999

9853

39(78234)

15003

2955

8303

9956

9867

13816

3132

8588

9996

9885

15041

3310

9016

9999

9912

40(80240)

16495

2978

8257

9954

9826

14896

3160

8495

9997

9837

16563

3349

8957

9999

9933

Average

4619

15855

8174

9925

9742

4077

16828

8457

9981

9743

4350

17938

8732

9999

9794

Journal of Advanced Transportation 11

high service level requirement we recommend MIP-DF1 forsolutions

6 Conclusion

This paper investigates an ambulance location problem withambiguity set of demand in which the probability distribu-tion of the uncertain demand is unknown In such a data-driven environment only the mean and covariance demandsat emergency points are knownThe problem is to determinethe base locations and the number of ambulances at eachbase aiming at minimizing the total cost associated withlocating bases and assigning ambulances We propose adistribution-free model with chance constraints Then twoapproximated MIP formulations with different approxima-tion methods of chance constraints are proposed which arebased on different ambiguity sets of the unknown probabilitydistribution Numerical experiments on benchmarks and120 randomly generated instances are conducted and thecomputational results show that the first approximated MIPformulation is much conservative with overhigh system costwhile the second approximatedMIP formulation ismore costsaving with service level appropriately ensured

In future research wemay consider the following relevantissues First the uncertainty of driving time from a base toan emergency point shall be analysed Besides unexpected orsudden disasters may be taken into consideration Moreoverother uncertainties should be considered such as travel timethe number of available vehicles and so on The stochasticdependence should also further considered

Data Availability

The data of the benchmark instances in our manuscript canbe obtained in Nickel et al [8] For the test instances thedata is randomly generated and the way for generating data isstated in our manuscript The datasets generated during thecurrent study are available from the corresponding author onreasonable request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China (NSFC) under Grants 7142800271531011 71571134 and 71771048 This work was also sup-ported by the Fundamental Research Funds for the CentralUniversities

References

[1] L Brotcorne G Laporte and F Semet ldquoAmbulance loca-tion and relocation modelsrdquo European Journal of OperationalResearch vol 147 no 3 pp 451ndash463 2003

[2] X Li Z Zhao X Zhu and T Wyatt ldquoCovering modelsand optimization techniques for emergency response facilitylocation and planning a reviewrdquo Mathematical Methods ofOperations Research vol 74 no 3 pp 281ndash310 2011

[3] C Toregas R Swain C ReVelle and L Bergman ldquoThe locationof emergency service facilitiesrdquoOperations Research vol 19 no6 pp 1363ndash1373 1971

[4] R Church and C ReVelle ldquoThe maximal covering locationproblemrdquo Papers in Regional Science vol 32 no 1 pp 101ndash1181974

[5] M Gendreau G Laporte and F Semet ldquoSolving an ambulancelocation model by tabu searchrdquo Location Science vol 5 no 2pp 75ndash88 1997

[6] A Ingolfsson S Budge and E Erkut ldquoOptimal ambulancelocation with random delays and travel timesrdquo Health CareManagement Science vol 11 no 3 pp 262ndash274 2008

[7] M Gendreau G Laporte and F Semet ldquoThemaximal expectedcoverage relocation problem for emergency vehiclesrdquo Journal ofthe Operational Research Society vol 57 no 1 pp 22ndash28 2006

[8] S Nickel M Reuter-Oppermann and F Saldanha-da-GamaldquoAmbulance location under stochastic demand A samplingapproachrdquo Operations Research for Health Care vol 8 pp 24ndash32 2016

[9] P Beraldi M E Bruni and D Conforti ldquoDesigning robustemergency medical service via stochastic programmingrdquo Euro-pean Journal of Operational Research vol 158 no 1 pp 183ndash1932004

[10] N Noyan ldquoAlternate risk measures for emergency medicalservice system designrdquo Annals of Operations Research vol 181pp 559ndash589 2010

[11] M R Wagner ldquoStochastic 0-1 linear programming underlimited distributional informationrdquoOperations Research Lettersvol 36 no 2 pp 150ndash156 2008

[12] EDelage andYYe ldquoDistributionally robust optimization undermoment uncertainty with application to data-driven problemsrdquoOperations Research vol 58 no 3 pp 595ndash612 2010

[13] J T Essen J L Hurink S Nickel and M Reuter ldquoModels forambulance planning on the strategic and the tactical levelrdquo BetaResearch School for Operations Management and Logistics p 272013

[14] E Erkut A Ingolfsson and G Erdogan ldquoAmbulance locationfor maximum survivalrdquoNaval Research Logistics (NRL) vol 55no 1 pp 42ndash58 2008

[15] S C Chapman and J A White ldquoProbabilistic formulations ofemergency service facilities location problemsrdquo in Proceedingsof ORSATIMS Conference San Juan Puerto Rico 1974

[16] A Ben-Tal D Den Hertog A De Waegenaere B Melenbergand G Rennen ldquoRobust solutions of optimization problemsaffected by uncertain probabilitiesrdquo Management Science vol59 no 2 pp 341ndash357 2013

[17] MNg ldquoDistribution-free vessel deployment for liner shippingrdquoEuropean Journal of Operational Research vol 238 no 3 pp858ndash862 2014

[18] M Ng ldquoContainer vessel fleet deployment for liner shippingwith stochastic dependencies in shipping demandrdquo Transporta-tion Research Part B Methodological vol 74 pp 79ndash87 2015

[19] R Jiang and Y Guan ldquoData-driven chance constrained stochas-tic programrdquo Mathematical Programming vol 158 no 1-2 SerA pp 291ndash327 2016

[20] C-M Lee and S-L Hsu ldquoThe effect of advertising on thedistribution-free newsboy problemrdquo International Journal ofProduction Economics vol 129 no 1 pp 217ndash224 2011

[21] K Kwon and T Cheong ldquoA minimax distribution-free proce-dure for a newsvendor problem with free shippingrdquo EuropeanJournal of Operational Research vol 232 no 1 pp 234ndash2402014

12 Journal of Advanced Transportation

[22] Y Zhang S Shen and S A Erdogan ldquoDistributionally robustappointment scheduling with moment-based ambiguity setrdquoOperations Research Letters vol 45 no 2 pp 139ndash144 2017

[23] Y Zhang S Shen and S A Erdogan ldquoSolving 0-1semidefinite programs for distributionally robust allocationof surgery blocksrdquo Social Science Electronic Publishing 2017httpsssrncomabstract=2997524

[24] F Zheng J He F Chu and M Liu ldquoA new distribution-freemodel for disassembly line balancing problem with stochastictask processing timesrdquo International Journal of ProductionResearch pp 1ndash13 2018

[25] J Cheng E Delage and A Lisser ldquoDistributionally robuststochastic knapsack problemrdquo SIAM Journal on Optimizationvol 24 no 3 pp 1485ndash1506 2014

[26] M Liu D Yang and F Hao ldquoOptimization for the locationsof ambulances under two-stage life rescue in the emergencymedical service a case study in Shanghai ChinardquoMathematicalProblems in Engineering vol 2017 Article ID 1830480 14 pages2017

6 Journal of Advanced Transportation

Table 2 120572 = 099|119868| = |119869| Sampling approach MIP-DF1 MIP-DF2

CT Obj Sel () CT Obj Sel () CT Obj Sel ()1 7 49 24 9979 01 64 10000 01 25 100002 7 48 23 9856 01 63 10000 01 24 100003 6 51 15 9717 01 55 10000 01 16 100004 6 47 15 9275 01 52 10000 01 13 100005 6 40 14 9000 01 55 10000 01 14 100006 8 52 23 9785 01 45 10000 01 27 10000

Average 48 190 9343 01 557 10000 01 198 10000

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

Sampling approach MIP-DF1 MIP-DF2Emergency 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 0Base locations Ambulance employment Ambulance employment Ambulance employment1 0 0 02 1 1 0 3 1 23 1 1 26 1 2 2 04 0 0 05 0 6 06 2 2 0 4 2

The average service level is 9343 Although the averageservice levels obtained by MIP-DF1 and MIP-DF2 are both100 the average system cost obtained by MIP-DF1 is about3 times larger than that of sampling approach Besides theaverage system cost obtained by MIP-DF2 is 421 higherthan that of sampling approach

Concluding for the benchmarks described in Nickel etal [8] we can observe that (1) MIP-DF1 and MIP-DF2 canbe solved in far less computational time compared with thesampling approach (2) the sampling approach cannot meetthe requirement of service level for all benchmarks whilethe two MIP formulations can (3) the average system cost ofMIP-DF2 is very close to that of sampling approach (4) withthe increase of 120572 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 ofthe lower system cost obtained by the sampling approachthere may exist extreme situations that demands of someemergency points are not satisfied Take one benchmark inNickel et al [8] ie instance 4 where there are 6 nodesA scenario in which demand at each emergency point isgenerated from the given probability distribution is shown inFigure 1 We can observe from Figure 1 that under the givenscenario the demand of each possible emergency point is1 0 2 0 2 0The solutions obtained by sampling approachMIP-DF1 andMIP-DF2 are shown in Table 3 which includesbase 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 poten-tial base locations The second column represents the baselocations and the ambulance employment at each location

＞1 = 1

＞2 = 0

＞3 = 2

＞4 = 0

＞5 = 2

＞6 = 0

1

2 4

3 5

6

Figure 1 Instance 4 in Nickel et al [8] under given scenario

in which a location with 0 ambulance implies that it is notselected for base construction From Table 3 we can observethat the in-sample solutions obtained by sampling approachis to construct bases at locations 2 3 and 6 where the numberof ambulances is 1 1 and 2 respectively Besides undergiven scenario there is one ambulance serving emergencypoint 1 from base location 3 and one ambulance servingemergency point from base location 2 and 2 ambulancesserving emergency point 5 from base location 6 It shows thatthe requirement of ambulances at emergency point 3 is notfully satisfied The insufficient service may cause serious lossof life However we can find that the solutions obtained byMIP-DF1 andMIP-DF2 can provide sufficient ambulances forall emergency points

In sum to guarantee the required service level withless cost MIP-DF2 is recommended for the solutions with

Journal of Advanced Transportation 7

Nickel et al [8]rsquos data set Moreover in the benchmarks itis assumed that all nodes are possible emergency points aswell as potential base locations and the number of requiredambulances is small which is not always consistent withthe situation in Shanghai Therefore we further test the twoapproximated MIP formulations and the sampling approachon randomly generated instances by analysing the informa-tion on emergencies in Shanghai [26]

53 Computational Tests on Randomly Generated InstancesIn the following subsection numerical experiments on 120randomly generated instances are described

531 Test Instances By analysing the information on emer-gencies in Shanghai [26] the mean value 120583119895 of demand 119889119895 ineach emergency point are randomly generated fromadiscreteUniform distribution on interval [5 25] The standard devia-tion 120590119895 is set to be 5 Since the sampling approach proposed inNickel et al [8] requires a sample of scenarios in numericalexperiments scenarios are produced with demands randomlygenerated from Uniform distribution Poisson distributionand Normal distribution consistent with given mean andstandard deviation

According to the instance settings in Erkut and Ingolfsson(2008) response time from a base to an emergency pointis randomly generated from a discrete Uniform distributionon interval [3 30] in units of minutes A base covers anemergency point if the driving time between them is lessthan 10 minutes to ensure the survival probability (Erkut andIngolfsson 2006) For an emergency point in each instanceif there is no base with 10 minutes driving time to it then thebase covering this point is set to be the nearest one The costof constructing a base and that of locating one ambulance areboth set to be 1 as in Nickel et al [8] Besides the number ofemergency points is 3 times that of potential base locations

532 Results and Discussion Different values of requiredservice level 120572 are tested ie 085 09 095 Computationalresults on 120 randomly generated instances are reported inTables 4-6

Table 4 reports the numerical results of instances withdemand generated from Uniform distribution We observethat when 120572 = 085 09 095 the average computationaltimes of the sampling approach are 4482 4379 and 4295seconds respectively However both approximated MIP for-mulations can produce solutions in much faster speeds iewithin 13 seconds on average With the increase of 120572 (i)the out-of-sample average service level is getting higher and(ii) the system costs of solutions obtained by all method aregetting larger and (iii) the system costs of solutions yieldedbyMIP-DF2 is getting closer to those of solutions obtained bythe sampling approach In terms of the average system costthe sampling approach performs the best while MIP-DF1 isthe poorest When 120572 = 085 for example the average systemcost ofMIP-DF2 and the sampling approach are about 3032and 442 less than that of MIP-DF1 respectively When120572 = 095 the above gaps are even larger Concluding fortheUniformdistribution (1) the proposed two approximatedMIP formulations are far more time saving than the sampling

approach (2)MIP-DF1 andMIP-DF2 can improve the servicelevel by about 1907 compared with the sampling approachhowever (3) despite of the highest service levels obtainedby MIP-DF1 the system cost obtained by MIP-DF1 is alsohigh (4) MIP-DF2 can ensure an appropriate service levelwith much less cost than MIP-DF1 and (5) for the situationswith small-scale instances and high service level requirementMIP-DF1 is recommended

Computational results of instances with demand at eachemergency point generated from Poisson distribution arepresented in Table 5 For MIP-DF1 and MIP-DF2 onlymean and variance are involved Therefore with givenmean and standard deviation of the demand at each pointcomputational results are the same for different probabilitydistributions We can obtain from Table 5 that the objectivevalue obtained by the sampling approach increases with theincrease of 120572 Besides when the value of 120572 equals 085the average service level obtained by the sampling approachis about 8143 which is smaller than that of the twoapproximated MIP formulations The highest service level9934 is obtained by MIP-DF1 and that obtained by MIP-DF2 is 9708 When 120572 = 09 the average service levelof the sampling approach is 8437 and those obtained byMIP-DF1 and MIP-DF2 are 9985 and 9797 respectivelyWhen 120572 = 095 the average service level obtained by thesampling approach is 9356 and those yielded by MIP-DF1 and MIP-DF2 are 9997 and 9781 respectively ForPoisson distribution concluding MIP-DF1 and MIP-DF2can improve the service level compared with the samplingapproach especially MIP-DF1 can obtain the highest servicelevel

Similarly Table 6 reports the computational results forinstances in which the demand at each emergency point isgenerated following Normal distribution When 120572 = 085the average service levels obtained by the sampling approachMIP-DF1 and MIP-DF2 are 8174 9925 and 9742respectively When 120572 = 09 the three values are 84579981 and 9743 respectively Moreover for 120572 = 095 thevalues are equal to 8732 9999 and 9794 respectivelyIt can be observed that the performance of the samplingapproach is the poorest in terms of the service level

By observing the above computational results on bench-marks and randomly generated instances we conclude that(1)The proposed approximated MIP formulations ie MIP-DF1 and MIP-DF2 are very time saving compared with thesampling approach (2) MIP-DF1 an MIP-DF2 can achievethe requirement of service level and improve the service levelby about 1547 and 1229 on average compared with thatof the sampling approach (3) MIP-DF1 is more conservativein terms of overhigh emergency service level close to 100and the system cost is also high which is about 568 largerthan that of MIP-DF2 (4) Compared with MIP-DF1 MIP-DF2 is less conservative and much more cost saving whileensuring an appropriately service level on average which isabout 97

In sum for medium and large problem instances (egShanghai) which are common in practice to guarantee therequired service level with less cost we recommendMIP-DF2as solution method However for very small instances with

8 Journal of Advanced Transportation

Table4Com

putatio

nalresultson

insta

nces

with

datageneratedfro

mun

iform

distrib

ution

Set (|119868||119869|)

120572=085

120572=09

120572=095

Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2CT

Obj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

1(26)

27

897317

01

149

9906

01

107

9222

28

967373

01

155

9933

01

107

9167

27

106

7585

01

185

9987

01

107

9128

2(412

)47

169

7423

00

291

9919

00

207

9122

58

181

7438

01

303

9997

01

207

9363

50

199

7702

00

363

9982

00

207

9338

3(618

)72

258

7506

01

438

9976

01

312

9106

75279

8082

01

456

9996

01

312

9369

78304

8337

01

546

9997

01

312

9357

4(824)

99314

7956

01

553

9993

01

385

9206

111

338

8202

01

577

9999

01

385

9602

112

366

8492

01

697

9994

01

385

9620

5(1030)

132

415

8406

01

722

9896

01

512

9406

143

441

8322

02

752

9983

01

512

9692

142

476

8609

01

902

9993

01

512

9692

6(1236)

170

475

8249

02

843

9870

01

591

9367

181

504

8251

01

879

9982

01

591

9608

189

535

8726

01

1059

9997

01

591

9609

7(14

42)

215

573

8111

02

1006

9849

02

712

9606

243

609

8217

02

1048

9953

02

712

9589

233

644

8608

02

1258

9992

02

712

9582

8(1648)

248

686

7532

02

1185

9896

02

849

9725

284

727

8307

02

1233

9949

02

849

9613

281

768

9063

02

1473

9999

03

849

9622

9(1854)

313

691

8353

03

1236

9974

04

858

9709

351

731

8397

03

1290

9988

03

858

9765

352

760

8906

03

1560

9994

03

858

9758

10(2060)

372

794

8174

03

1409

9887

03

989

9657

428

843

8489

03

1469

9947

03

989

9773

432

892

9004

03

1769

9997

03

989

9738

11(2266

)451

830

8072

04

1493

9949

03

1031

9758

518

880

8328

04

1559

9982

03

1031

9723

516

930

8967

05

1889

9995

03

1031

9727

12(2472)

543

1025

7970

04

1771

9927

04

1267

9835

601

1083

8482

05

1843

9949

04

1267

9776

637

1146

9014

04

2203

9993

04

1267

9777

13(2678)

631

1026

8142

05

1819

9898

04

1274

9782

743

1087

8678

05

1897

9936

04

1274

9778

765

1146

9023

04

2287

9997

05

1273

9837

14(2884)

782

1101

7903

05

1949

9869

08

1361

9804

878

1165

8790

05

2033

9935

07

1361

9889

870

1227

9056

05

2453

9998

06

1361

9808

15(3090)

907

1091

8098

06

1997

9915

06

1367

9629

1018

1155

8329

06

2087

9981

06

1367

9807

1020

1225

8986

06

2537

9999

08

1367

9817

16(3296)

1062

1160

8798

07

2123

9944

07

1449

9787

1183

1229

8383

07

2219

9988

06

1449

9849

1193

1300

9105

06

2698

9998

06

1449

9815

17(34102)

1202

1298

7572

07

2328

9969

07

1614

9821

1377

1374

8860

08

2430

9997

07

1614

9824

1260

1452

9085

07

2940

9999

08

1614

9818

18(36108)

1394

1408

8121

08

2509

9947

08

1752

9809

1544

1492

8928

08

2617

9983

08

1752

9799

1548

1580

9173

08

3156

9999

08

1752

9809

19(38114

)1618

1508

8170

09

2676

9899

111877

9742

1760

1598

9130

09

2790

9934

08

1877

9908

1755

1688

9186

09

3360

9999

09

1877

9872

20(40120)

2206

1628

7719

102869

9989

102029

9873

2007

1732

8753

112990

9997

09

2029

9879

1904

1827

9138

103590

9998

09

2029

9875

21(42126)

2780

1622

8354

102900

9978

102017

9732

2609

1718

8854

103026

9876

102017

9818

2670

1812

9089

123655

9999

102017

9848

22(44132)

3189

1735

8494

113088

9985

152165

9857

2970

1840

8922

123220

9925

122165

9886

2873

1952

9206

113880

9999

112165

9850

23(46138)

3630

1937

8141

123372

9893

132406

9796

3222

2048

8937

123510

9934

142406

9906

3461

2170

8989

124199

10000

122406

9809

24(48144)

4226

1968

8226

133461

9932

132453

9814

3674

2085

8846

133605

9956

152453

9912

3556

2211

9300

134326

9999

132453

9819

25(50150)

4727

1919

8111

143456

9999

152407

9629

4019

2037

8699

143606

9999

142407

9857

4257

2161

9259

154356

9995

142407

9819

26(52156)

5300

1906

8096

153484

9878

172392

9658

4495

2021

8952

143640

9999

152392

9568

4486

2142

9182

1544

209999

152392

9854

27(54162)

5913

2096

8387

163751

9846

172617

9761

5117

2221

9045

163913

9998

172617

9544

4830

2355

9346

184723

9999

162617

9893

28(56168)

6310

2110

7758

173819

9933

192643

9829

5777

2234

9157

183987

9998

182643

9526

5550

2365

9045

184827

9999

172643

9831

29(58174)

6780

2175

7998

183937

9898

182720

9714

6303

2307

9022

20

4111

9969

192720

9623

6189

2444

9128

194981

10000

192720

9914

30(60180)

7593

2384

8321

194240

9967

20

2979

9841

6791

2527

8978

1944

199979

23

2979

9721

6878

2679

9143

21

5316

9999

20

2979

9869

31(62186)

8220

2281

7604

20

4157

9984

21

2856

9796

7734

2416

8934

21

4343

9997

21

2856

9679

7466

2557

9158

22

5273

9999

23

2856

9902

32(64192)

8766

2517

8378

21

4489

9932

25

3145

9729

8276

2671

8296

23

4680

9977

24

3145

9825

7897

2822

9089

24

5640

9999

22

3145

9891

33(66198)

10123

2532

7806

23

4555

9967

23

3169

9615

9010

2682

8947

24

4752

9989

24

3169

9703

9051

2835

9157

24

5742

9998

24

3169

9867

34(68204)

11010

2535

7934

24

4602

9999

29

3174

9631

9898

2685

8858

30

4806

9998

28

3174

9762

9863

2844

9236

26

5824

9999

25

3174

9889

35(70210)

11666

2714

8062

25

4862

9986

29

3391

9787

1066

62875

9137

26

5072

9999

29

3391

9716

10822

3043

9315

26

6122

9999

29

3391

9923

36(72216)

12011

2827

8104

26

5047

9991

28

3535

9798

1246

52984

9016

28

5263

9999

28

3535

9878

11667

3169

9253

30

6343

9999

29

3535

9896

37(74222)

11437

2826

8228

28

5096

9933

29

3543

9884

13582

3098

8895

29

5318

9999

32

3543

9888

12654

3178

9139

29

6427

10000

31

3543

9893

38(76228)

12435

2839

8152

29

5161

9897

29

3565

9788

13976

3103

8926

29

5388

9999

31

3565

9724

13837

3185

9255

33

6528

10000

31

3565

9895

39(78234)

1464

32945

8412

31

5331

9998

32

3695

9876

14794

3210

9005

31

5565

9999

34

3695

9713

14561

3313

9081

34

6735

10000

33

3695

9878

40(80240)

1604

22975

8510

32

5414

9957

36

3734

9796

16263

3346

8999

35

5654

9999

35

3734

9806

15869

3343

9248

35

6854

10000

35

3734

9873

Average

4482

1584

680

67

1228

397

9936

131978

796

82

4379

16913

8654

1329

626

9975

131978

79721

4295

1778

889

85

1335774

9997

131978

79775

Journal of Advanced Transportation 9

Table5Com

putatio

nalresultson

insta

nces

with

datageneratedfro

mPo

isson

distr

ibution

set ( |119868 |

|119869 |)120572=

085120572=

09120572=

095Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2CT

Obj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

1(26)

27

907294

9967

9001

27

987770

9979

9013

24

103

9273

9999

9468

2(412

)46

170

7961

9921

9117

51

183

8305

9946

9128

49

201

9392

9998

9546

3(618

)68

258

7578

9938

8911

74280

8028

9968

9004

69

307

9182

9999

9658

4(824)

93315

8001

9928

9196

102

338

8438

9967

9172

92368

9468

9999

9613

5(1030)

120

413

8041

9939

9449

133

443

8447

9976

9272

121

479

9544

9999

9752

6(1236)

155

476

8084

9877

9554

167

508

8863

9991

9569

159

536

9456

9999

9734

7(14

42)

195

573

7876

9892

9663

213

609

9124

9998

9729

205

643

9314

10000

9768

8(1648)

232

687

7964

9997

9788

267

724

8523

9988

9687

238

772

9422

9998

9689

9(1854)

293

692

8267

9893

9809

327

730

8386

9999

9750

298

757

9228

9999

9755

10(2060)

356

796

7946

9899

9801

399

840

8299

9976

9686

360

885

9668

9999

9705

11(2266)

423

829

8201

9895

9756

489

877

8378

9987

9788

448

927

9476

9999

9812

12(2472)

512

1023

8338

9913

9853

554

1085

8501

9999

9881

514

1148

9398

9999

9888

13(2678)

592

1045

8416

9907

9873

687

1088

7998

9996

9802

607

1152

9504

9999

9821

14(2884)

723

1099

7946

9915

9788

838

1166

8056

9992

9829

776

1230

9546

9999

9839

15(3090)

855

1089

7915

9964

9703

942

1156

8432

9984

9778

903

1225

9223

9998

9785

16(3296)

1009

1158

8024

9897

9769

1163

1231

8500

9996

9824

1045

1222

9199

9999

9835

17(34102)

1165

1299

8034

9919

9779

1285

1375

8495

9937

9808

1190

1457

9258

10000

9846

18(36108)

1254

1411

8176

9943

9885

1479

1492

8561

9968

9868

1345

1579

9286

9999

9875

19(38114

)1504

1506

7970

9923

9842

1651

1594

8167

9972

9846

1564

1694

9376

9999

9786

20(40120)

1706

1635

7819

9916

9783

1854

1734

8607

9968

9776

1778

1825

9285

9904

9718

21(42126)

2054

1622

8354

9918

9834

2357

1720

8297

9978

9833

2226

1856

9501

9999

9834

22(44132)

2400

1736

8494

9922

9846

2852

1844

8269

9999

9816

2680

1946

9486

9999

9820

23(46138)

2651

1934

8141

9953

9793

3159

2049

8492

9969

9799

3072

2172

9358

10000

9813

24(48144)

3004

1961

8226

9947

9681

3522

2086

8513

9998

9706

3102

2209

9488

9999

9724

25(50150)

3308

1924

8111

9965

9746

3898

2038

8264

9999

9788

3639

2161

9546

9999

9813

26(52156)

3669

1911

8096

9939

9855

4305

2027

8722

9967

9874

4453

2142

9185

9999

9876

27(54162)

4418

2095

8387

9927

9754

4771

2219

8341

9989

9749

4806

2353

9289

9999

9755

28(56168)

5367

2113

8458

9968

9848

5356

2237

8452

9978

9855

5221

2369

9228

10000

9868

29(58174)

6217

2186

7998

9989

9694

5942

2315

8316

9996

9699

5853

2371

9167

9999

9705

30(60180)

7263

2382

8321

9991

9837

6527

2543

8359

9998

9886

6214

2678

9369

9997

9886

31(62186)

8214

2346

8404

9936

9888

7813

2430

8321

9999

9846

6876

2560

9358

9999

9847

32(64192)

8863

2518

8378

9938

9807

8106

2677

8356

9999

9833

7877

2822

9408

10000

9835

33(66198)

9985

2532

8406

9917

9709

9684

2689

8497

9996

9746

8088

2838

9507

9999

9749

34(68204)

11264

2534

8543

9908

9897

10986

2685

8809

9996

9782

8469

2843

8844

9998

9780

35(70210)

12035

2715

8258

9937

9799

12098

2886

8569

9991

9809

9946

3041

9459

9999

9813

36(72216)

12803

2825

8104

9929

9827

13076

3010

8651

9994

9834

11013

3170

9468

9999

9865

37(74222)

13021

2830

8228

9949

9826

13261

3009

8376

9999

9826

12014

3172

9346

9999

9888

38(76228)

13862

2843

8152

9933

9807

13995

3028

8760

9998

9841

12133

3187

9706

9999

9876

39(78234)

15387

2953

8312

9964

9799

14827

3133

9001

9995

9783

13381

3304

8913

9999

9799

40(80240)

16974

2975

8510

9985

9767

15256

3249

8217

9993

9781

15867

3347

9111

9999

9791

Average

4352

15875

8143

9934

9708

4362

16856

8437

9985

9707

3968

17763

9356

9997

9781

10 Journal of Advanced Transportation

Table6Com

putatio

nalresultson

insta

nces

with

datageneratedfro

mno

rmaldistr

ibution

set ( |119868 |

|119869 |)120572=

085120572=

09120572=

095Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2CT

Obj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

1(26)

27

907294

9964

9117

27

947844

9970

9202

2558

106

8422

10000

9208

2(412

)46

170

7961

9921

9283

48

183

8328

9982

9283

4508

183

8823

10000

9370

3(618

)68

258

7578

9916

9228

68

280

7798

9971

9417

6803

307

7658

9998

9447

4(824)

93315

8000

9899

9367

96337

8379

9979

9467

9373

368

8438

9999

9579

5(1030)

120

413

7841

9927

9437

124

443

8763

9963

9476

1262

499

8850

9998

9708

6(1236)

155

476

7921

9919

9519

162

504

8124

9982

9582

16021

536

8954

10000

9756

7(14

42)

195

573

7976

9933

9598

196

609

8287

9975

9619

2011

646

8467

9999

9767

8(1648)

232

687

8031

9918

9705

235

727

8115

9965

9650

2404

772

8278

9999

9765

9(1854)

293

692

8086

9898

9759

291

734

8464

9974

9755

29815

796

8617

9998

9835

10(2060)

356

796

7984

9937

9770

358

842

8223

9988

9765

37973

921

8846

10000

9838

11(2266

)421

829

8101

9940

9755

436

877

8211

9986

9786

4514

8959

8497

9999

9789

12(2472)

514

1023

8033

9951

9785

501

1085

8162

9983

9788

52073

1147

8321

9999

9824

13(2678)

592

1023

8138

9916

9885

599

1087

8411

9970

9845

62937

1194

8609

9998

9795

14(2884)

728

1099

8235

9913

9915

722

1168

8435

9987

9806

75053

1270

8567

9997

9834

15(3090)

855

1089

8187

9922

9861

880

1157

8376

9976

9796

89222

1300

8611

9996

9798

16(3296)

1009

1158

8770

9897

9788

966

1231

8845

9984

9823

106288

1359

8997

9999

9848

17(34102)

1154

1299

8434

9925

9849

1137

1377

8601

9980

9798

123869

1493

8798

9996

9877

18(36108)

1407

1411

8321

9927

9866

1339

1495

8301

9965

9822

133807

1601

8567

9999

9834

19(38114

)1702

1506

8206

9915

9805

1672

1598

8977

9978

9845

152999

1694

8998

9999

9847

20(40120)

2091

1635

8528

9918

9889

1916

1737

8805

9976

9837

1890

021833

9012

9999

9791

21(42126)

2843

1622

8550

9934

9913

2432

1718

8886

9974

9813

2399

131815

9008

10000

9815

22(44132)

3300

1736

8449

9940

9859

2795

1843

8786

9977

9756

253106

1948

8997

9999

9826

23(46138)

3741

1934

8133

9911

9876

3143

2050

8876

9981

9785

274583

2173

9014

9999

9858

24(48144)

4531

1961

8245

9909

9846

3515

2090

8344

9993

9819

308089

2211

8661

9999

9883

25(50150)

4715

1924

8111

9929

9888

3959

2041

8279

9996

9837

3510

932158

8527

9999

9827

26(52156)

5223

1911

8096

9898

9857

4301

2025

8251

9997

9757

421747

2146

8497

9999

9914

27(54162)

5994

2095

8387

9908

9775

4835

2226

8497

9974

9780

4614

432359

8695

9999

9837

28(56168)

6406

2111

8411

9906

9798

5336

2240

8558

9991

9820

494113

2371

9249

9999

9816

29(58174)

6897

2176

8012

9916

9791

5948

2311

8211

9984

9765

624835

2445

8531

10000

9851

30(60180)

7436

2391

8111

9913

9766

6558

2533

8305

9968

9795

696116

2682

8575

9999

9799

31(62186)

8118

2282

8119

9928

9806

7069

2422

8417

9973

9873

797768

2564

8667

9999

9846

32(64192)

8895

2519

8223

9919

9757

7857

2675

8631

9982

9769

864813

2827

8976

10000

9905

33(66198)

11235

2533

8563

9928

9811

9162

2688

8547

9979

9828

973089

2843

8657

9999

9846

34(68204)

11587

2534

8249

9926

9847

9549

2686

8798

9938

9856

10523

3041

8999

10000

9877

35(70210)

1164

92715

8098

9895

9872

10345

2880

8602

9998

9869

11272

3052

9021

9999

9897

36(72216)

12224

2825

8403

9958

9770

11165

3002

8657

9999

9839

12456

3175

9113

9999

9864

37(74222)

12978

2830

8357

9937

9795

11850

3005

8496

9998

9894

13458

3176

8827

9999

9905

38(76228)

13426

2844

8267

9966

9759

12795

3019

8604

9999

9797

15155

3121

8978

9999

9853

39(78234)

15003

2955

8303

9956

9867

13816

3132

8588

9996

9885

15041

3310

9016

9999

9912

40(80240)

16495

2978

8257

9954

9826

14896

3160

8495

9997

9837

16563

3349

8957

9999

9933

Average

4619

15855

8174

9925

9742

4077

16828

8457

9981

9743

4350

17938

8732

9999

9794

Journal of Advanced Transportation 11

high service level requirement we recommend MIP-DF1 forsolutions

6 Conclusion

This paper investigates an ambulance location problem withambiguity set of demand in which the probability distribu-tion of the uncertain demand is unknown In such a data-driven environment only the mean and covariance demandsat emergency points are knownThe problem is to determinethe base locations and the number of ambulances at eachbase aiming at minimizing the total cost associated withlocating bases and assigning ambulances We propose adistribution-free model with chance constraints Then twoapproximated MIP formulations with different approxima-tion methods of chance constraints are proposed which arebased on different ambiguity sets of the unknown probabilitydistribution Numerical experiments on benchmarks and120 randomly generated instances are conducted and thecomputational results show that the first approximated MIPformulation is much conservative with overhigh system costwhile the second approximatedMIP formulation ismore costsaving with service level appropriately ensured

In future research wemay consider the following relevantissues First the uncertainty of driving time from a base toan emergency point shall be analysed Besides unexpected orsudden disasters may be taken into consideration Moreoverother uncertainties should be considered such as travel timethe number of available vehicles and so on The stochasticdependence should also further considered

Data Availability

The data of the benchmark instances in our manuscript canbe obtained in Nickel et al [8] For the test instances thedata is randomly generated and the way for generating data isstated in our manuscript The datasets generated during thecurrent study are available from the corresponding author onreasonable request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China (NSFC) under Grants 7142800271531011 71571134 and 71771048 This work was also sup-ported by the Fundamental Research Funds for the CentralUniversities

References

[1] L Brotcorne G Laporte and F Semet ldquoAmbulance loca-tion and relocation modelsrdquo European Journal of OperationalResearch vol 147 no 3 pp 451ndash463 2003

[2] X Li Z Zhao X Zhu and T Wyatt ldquoCovering modelsand optimization techniques for emergency response facilitylocation and planning a reviewrdquo Mathematical Methods ofOperations Research vol 74 no 3 pp 281ndash310 2011

[3] C Toregas R Swain C ReVelle and L Bergman ldquoThe locationof emergency service facilitiesrdquoOperations Research vol 19 no6 pp 1363ndash1373 1971

[4] R Church and C ReVelle ldquoThe maximal covering locationproblemrdquo Papers in Regional Science vol 32 no 1 pp 101ndash1181974

[5] M Gendreau G Laporte and F Semet ldquoSolving an ambulancelocation model by tabu searchrdquo Location Science vol 5 no 2pp 75ndash88 1997

[6] A Ingolfsson S Budge and E Erkut ldquoOptimal ambulancelocation with random delays and travel timesrdquo Health CareManagement Science vol 11 no 3 pp 262ndash274 2008

[7] M Gendreau G Laporte and F Semet ldquoThemaximal expectedcoverage relocation problem for emergency vehiclesrdquo Journal ofthe Operational Research Society vol 57 no 1 pp 22ndash28 2006

[8] S Nickel M Reuter-Oppermann and F Saldanha-da-GamaldquoAmbulance location under stochastic demand A samplingapproachrdquo Operations Research for Health Care vol 8 pp 24ndash32 2016

[9] P Beraldi M E Bruni and D Conforti ldquoDesigning robustemergency medical service via stochastic programmingrdquo Euro-pean Journal of Operational Research vol 158 no 1 pp 183ndash1932004

[10] N Noyan ldquoAlternate risk measures for emergency medicalservice system designrdquo Annals of Operations Research vol 181pp 559ndash589 2010

[11] M R Wagner ldquoStochastic 0-1 linear programming underlimited distributional informationrdquoOperations Research Lettersvol 36 no 2 pp 150ndash156 2008

[12] EDelage andYYe ldquoDistributionally robust optimization undermoment uncertainty with application to data-driven problemsrdquoOperations Research vol 58 no 3 pp 595ndash612 2010

[13] J T Essen J L Hurink S Nickel and M Reuter ldquoModels forambulance planning on the strategic and the tactical levelrdquo BetaResearch School for Operations Management and Logistics p 272013

[14] E Erkut A Ingolfsson and G Erdogan ldquoAmbulance locationfor maximum survivalrdquoNaval Research Logistics (NRL) vol 55no 1 pp 42ndash58 2008

[15] S C Chapman and J A White ldquoProbabilistic formulations ofemergency service facilities location problemsrdquo in Proceedingsof ORSATIMS Conference San Juan Puerto Rico 1974

[16] A Ben-Tal D Den Hertog A De Waegenaere B Melenbergand G Rennen ldquoRobust solutions of optimization problemsaffected by uncertain probabilitiesrdquo Management Science vol59 no 2 pp 341ndash357 2013

[17] MNg ldquoDistribution-free vessel deployment for liner shippingrdquoEuropean Journal of Operational Research vol 238 no 3 pp858ndash862 2014

[18] M Ng ldquoContainer vessel fleet deployment for liner shippingwith stochastic dependencies in shipping demandrdquo Transporta-tion Research Part B Methodological vol 74 pp 79ndash87 2015

[19] R Jiang and Y Guan ldquoData-driven chance constrained stochas-tic programrdquo Mathematical Programming vol 158 no 1-2 SerA pp 291ndash327 2016

[20] C-M Lee and S-L Hsu ldquoThe effect of advertising on thedistribution-free newsboy problemrdquo International Journal ofProduction Economics vol 129 no 1 pp 217ndash224 2011

[21] K Kwon and T Cheong ldquoA minimax distribution-free proce-dure for a newsvendor problem with free shippingrdquo EuropeanJournal of Operational Research vol 232 no 1 pp 234ndash2402014

12 Journal of Advanced Transportation

[22] Y Zhang S Shen and S A Erdogan ldquoDistributionally robustappointment scheduling with moment-based ambiguity setrdquoOperations Research Letters vol 45 no 2 pp 139ndash144 2017

[23] Y Zhang S Shen and S A Erdogan ldquoSolving 0-1semidefinite programs for distributionally robust allocationof surgery blocksrdquo Social Science Electronic Publishing 2017httpsssrncomabstract=2997524

[24] F Zheng J He F Chu and M Liu ldquoA new distribution-freemodel for disassembly line balancing problem with stochastictask processing timesrdquo International Journal of ProductionResearch pp 1ndash13 2018

[25] J Cheng E Delage and A Lisser ldquoDistributionally robuststochastic knapsack problemrdquo SIAM Journal on Optimizationvol 24 no 3 pp 1485ndash1506 2014

[26] M Liu D Yang and F Hao ldquoOptimization for the locationsof ambulances under two-stage life rescue in the emergencymedical service a case study in Shanghai ChinardquoMathematicalProblems in Engineering vol 2017 Article ID 1830480 14 pages2017

Journal of Advanced Transportation 7

Nickel et al [8]rsquos data set Moreover in the benchmarks itis assumed that all nodes are possible emergency points aswell as potential base locations and the number of requiredambulances is small which is not always consistent withthe situation in Shanghai Therefore we further test the twoapproximated MIP formulations and the sampling approachon randomly generated instances by analysing the informa-tion on emergencies in Shanghai [26]

53 Computational Tests on Randomly Generated InstancesIn the following subsection numerical experiments on 120randomly generated instances are described

531 Test Instances By analysing the information on emer-gencies in Shanghai [26] the mean value 120583119895 of demand 119889119895 ineach emergency point are randomly generated fromadiscreteUniform distribution on interval [5 25] The standard devia-tion 120590119895 is set to be 5 Since the sampling approach proposed inNickel et al [8] requires a sample of scenarios in numericalexperiments scenarios are produced with demands randomlygenerated from Uniform distribution Poisson distributionand Normal distribution consistent with given mean andstandard deviation

According to the instance settings in Erkut and Ingolfsson(2008) response time from a base to an emergency pointis randomly generated from a discrete Uniform distributionon interval [3 30] in units of minutes A base covers anemergency point if the driving time between them is lessthan 10 minutes to ensure the survival probability (Erkut andIngolfsson 2006) For an emergency point in each instanceif there is no base with 10 minutes driving time to it then thebase covering this point is set to be the nearest one The costof constructing a base and that of locating one ambulance areboth set to be 1 as in Nickel et al [8] Besides the number ofemergency points is 3 times that of potential base locations

532 Results and Discussion Different values of requiredservice level 120572 are tested ie 085 09 095 Computationalresults on 120 randomly generated instances are reported inTables 4-6

Table 4 reports the numerical results of instances withdemand generated from Uniform distribution We observethat when 120572 = 085 09 095 the average computationaltimes of the sampling approach are 4482 4379 and 4295seconds respectively However both approximated MIP for-mulations can produce solutions in much faster speeds iewithin 13 seconds on average With the increase of 120572 (i)the out-of-sample average service level is getting higher and(ii) the system costs of solutions obtained by all method aregetting larger and (iii) the system costs of solutions yieldedbyMIP-DF2 is getting closer to those of solutions obtained bythe sampling approach In terms of the average system costthe sampling approach performs the best while MIP-DF1 isthe poorest When 120572 = 085 for example the average systemcost ofMIP-DF2 and the sampling approach are about 3032and 442 less than that of MIP-DF1 respectively When120572 = 095 the above gaps are even larger Concluding fortheUniformdistribution (1) the proposed two approximatedMIP formulations are far more time saving than the sampling

approach (2)MIP-DF1 andMIP-DF2 can improve the servicelevel by about 1907 compared with the sampling approachhowever (3) despite of the highest service levels obtainedby MIP-DF1 the system cost obtained by MIP-DF1 is alsohigh (4) MIP-DF2 can ensure an appropriate service levelwith much less cost than MIP-DF1 and (5) for the situationswith small-scale instances and high service level requirementMIP-DF1 is recommended

Computational results of instances with demand at eachemergency point generated from Poisson distribution arepresented in Table 5 For MIP-DF1 and MIP-DF2 onlymean and variance are involved Therefore with givenmean and standard deviation of the demand at each pointcomputational results are the same for different probabilitydistributions We can obtain from Table 5 that the objectivevalue obtained by the sampling approach increases with theincrease of 120572 Besides when the value of 120572 equals 085the average service level obtained by the sampling approachis about 8143 which is smaller than that of the twoapproximated MIP formulations The highest service level9934 is obtained by MIP-DF1 and that obtained by MIP-DF2 is 9708 When 120572 = 09 the average service levelof the sampling approach is 8437 and those obtained byMIP-DF1 and MIP-DF2 are 9985 and 9797 respectivelyWhen 120572 = 095 the average service level obtained by thesampling approach is 9356 and those yielded by MIP-DF1 and MIP-DF2 are 9997 and 9781 respectively ForPoisson distribution concluding MIP-DF1 and MIP-DF2can improve the service level compared with the samplingapproach especially MIP-DF1 can obtain the highest servicelevel

Similarly Table 6 reports the computational results forinstances in which the demand at each emergency point isgenerated following Normal distribution When 120572 = 085the average service levels obtained by the sampling approachMIP-DF1 and MIP-DF2 are 8174 9925 and 9742respectively When 120572 = 09 the three values are 84579981 and 9743 respectively Moreover for 120572 = 095 thevalues are equal to 8732 9999 and 9794 respectivelyIt can be observed that the performance of the samplingapproach is the poorest in terms of the service level

By observing the above computational results on bench-marks and randomly generated instances we conclude that(1)The proposed approximated MIP formulations ie MIP-DF1 and MIP-DF2 are very time saving compared with thesampling approach (2) MIP-DF1 an MIP-DF2 can achievethe requirement of service level and improve the service levelby about 1547 and 1229 on average compared with thatof the sampling approach (3) MIP-DF1 is more conservativein terms of overhigh emergency service level close to 100and the system cost is also high which is about 568 largerthan that of MIP-DF2 (4) Compared with MIP-DF1 MIP-DF2 is less conservative and much more cost saving whileensuring an appropriately service level on average which isabout 97

In sum for medium and large problem instances (egShanghai) which are common in practice to guarantee therequired service level with less cost we recommendMIP-DF2as solution method However for very small instances with

8 Journal of Advanced Transportation

Table4Com

putatio

nalresultson

insta

nces

with

datageneratedfro

mun

iform

distrib

ution

Set (|119868||119869|)

120572=085

120572=09

120572=095

Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2CT

Obj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

1(26)

27

897317

01

149

9906

01

107

9222

28

967373

01

155

9933

01

107

9167

27

106

7585

01

185

9987

01

107

9128

2(412

)47

169

7423

00

291

9919

00

207

9122

58

181

7438

01

303

9997

01

207

9363

50

199

7702

00

363

9982

00

207

9338

3(618

)72

258

7506

01

438

9976

01

312

9106

75279

8082

01

456

9996

01

312

9369

78304

8337

01

546

9997

01

312

9357

4(824)

99314

7956

01

553

9993

01

385

9206

111

338

8202

01

577

9999

01

385

9602

112

366

8492

01

697

9994

01

385

9620

5(1030)

132

415

8406

01

722

9896

01

512

9406

143

441

8322

02

752

9983

01

512

9692

142

476

8609

01

902

9993

01

512

9692

6(1236)

170

475

8249

02

843

9870

01

591

9367

181

504

8251

01

879

9982

01

591

9608

189

535

8726

01

1059

9997

01

591

9609

7(14

42)

215

573

8111

02

1006

9849

02

712

9606

243

609

8217

02

1048

9953

02

712

9589

233

644

8608

02

1258

9992

02

712

9582

8(1648)

248

686

7532

02

1185

9896

02

849

9725

284

727

8307

02

1233

9949

02

849

9613

281

768

9063

02

1473

9999

03

849

9622

9(1854)

313

691

8353

03

1236

9974

04

858

9709

351

731

8397

03

1290

9988

03

858

9765

352

760

8906

03

1560

9994

03

858

9758

10(2060)

372

794

8174

03

1409

9887

03

989

9657

428

843

8489

03

1469

9947

03

989

9773

432

892

9004

03

1769

9997

03

989

9738

11(2266

)451

830

8072

04

1493

9949

03

1031

9758

518

880

8328

04

1559

9982

03

1031

9723

516

930

8967

05

1889

9995

03

1031

9727

12(2472)

543

1025

7970

04

1771

9927

04

1267

9835

601

1083

8482

05

1843

9949

04

1267

9776

637

1146

9014

04

2203

9993

04

1267

9777

13(2678)

631

1026

8142

05

1819

9898

04

1274

9782

743

1087

8678

05

1897

9936

04

1274

9778

765

1146

9023

04

2287

9997

05

1273

9837

14(2884)

782

1101

7903

05

1949

9869

08

1361

9804

878

1165

8790

05

2033

9935

07

1361

9889

870

1227

9056

05

2453

9998

06

1361

9808

15(3090)

907

1091

8098

06

1997

9915

06

1367

9629

1018

1155

8329

06

2087

9981

06

1367

9807

1020

1225

8986

06

2537

9999

08

1367

9817

16(3296)

1062

1160

8798

07

2123

9944

07

1449

9787

1183

1229

8383

07

2219

9988

06

1449

9849

1193

1300

9105

06

2698

9998

06

1449

9815

17(34102)

1202

1298

7572

07

2328

9969

07

1614

9821

1377

1374

8860

08

2430

9997

07

1614

9824

1260

1452

9085

07

2940

9999

08

1614

9818

18(36108)

1394

1408

8121

08

2509

9947

08

1752

9809

1544

1492

8928

08

2617

9983

08

1752

9799

1548

1580

9173

08

3156

9999

08

1752

9809

19(38114

)1618

1508

8170

09

2676

9899

111877

9742

1760

1598

9130

09

2790

9934

08

1877

9908

1755

1688

9186

09

3360

9999

09

1877

9872

20(40120)

2206

1628

7719

102869

9989

102029

9873

2007

1732

8753

112990

9997

09

2029

9879

1904

1827

9138

103590

9998

09

2029

9875

21(42126)

2780

1622

8354

102900

9978

102017

9732

2609

1718

8854

103026

9876

102017

9818

2670

1812

9089

123655

9999

102017

9848

22(44132)

3189

1735

8494

113088

9985

152165

9857

2970

1840

8922

123220

9925

122165

9886

2873

1952

9206

113880

9999

112165

9850

23(46138)

3630

1937

8141

123372

9893

132406

9796

3222

2048

8937

123510

9934

142406

9906

3461

2170

8989

124199

10000

122406

9809

24(48144)

4226

1968

8226

133461

9932

132453

9814

3674

2085

8846

133605

9956

152453

9912

3556

2211

9300

134326

9999

132453

9819

25(50150)

4727

1919

8111

143456

9999

152407

9629

4019

2037

8699

143606

9999

142407

9857

4257

2161

9259

154356

9995

142407

9819

26(52156)

5300

1906

8096

153484

9878

172392

9658

4495

2021

8952

143640

9999

152392

9568

4486

2142

9182

1544

209999

152392

9854

27(54162)

5913

2096

8387

163751

9846

172617

9761

5117

2221

9045

163913

9998

172617

9544

4830

2355

9346

184723

9999

162617

9893

28(56168)

6310

2110

7758

173819

9933

192643

9829

5777

2234

9157

183987

9998

182643

9526

5550

2365

9045

184827

9999

172643

9831

29(58174)

6780

2175

7998

183937

9898

182720

9714

6303

2307

9022

20

4111

9969

192720

9623

6189

2444

9128

194981

10000

192720

9914

30(60180)

7593

2384

8321

194240

9967

20

2979

9841

6791

2527

8978

1944

199979

23

2979

9721

6878

2679

9143

21

5316

9999

20

2979

9869

31(62186)

8220

2281

7604

20

4157

9984

21

2856

9796

7734

2416

8934

21

4343

9997

21

2856

9679

7466

2557

9158

22

5273

9999

23

2856

9902

32(64192)

8766

2517

8378

21

4489

9932

25

3145

9729

8276

2671

8296

23

4680

9977

24

3145

9825

7897

2822

9089

24

5640

9999

22

3145

9891

33(66198)

10123

2532

7806

23

4555

9967

23

3169

9615

9010

2682

8947

24

4752

9989

24

3169

9703

9051

2835

9157

24

5742

9998

24

3169

9867

34(68204)

11010

2535

7934

24

4602

9999

29

3174

9631

9898

2685

8858

30

4806

9998

28

3174

9762

9863

2844

9236

26

5824

9999

25

3174

9889

35(70210)

11666

2714

8062

25

4862

9986

29

3391

9787

1066

62875

9137

26

5072

9999

29

3391

9716

10822

3043

9315

26

6122

9999

29

3391

9923

36(72216)

12011

2827

8104

26

5047

9991

28

3535

9798

1246

52984

9016

28

5263

9999

28

3535

9878

11667

3169

9253

30

6343

9999

29

3535

9896

37(74222)

11437

2826

8228

28

5096

9933

29

3543

9884

13582

3098

8895

29

5318

9999

32

3543

9888

12654

3178

9139

29

6427

10000

31

3543

9893

38(76228)

12435

2839

8152

29

5161

9897

29

3565

9788

13976

3103

8926

29

5388

9999

31

3565

9724

13837

3185

9255

33

6528

10000

31

3565

9895

39(78234)

1464

32945

8412

31

5331

9998

32

3695

9876

14794

3210

9005

31

5565

9999

34

3695

9713

14561

3313

9081

34

6735

10000

33

3695

9878

40(80240)

1604

22975

8510

32

5414

9957

36

3734

9796

16263

3346

8999

35

5654

9999

35

3734

9806

15869

3343

9248

35

6854

10000

35

3734

9873

Average

4482

1584

680

67

1228

397

9936

131978

796

82

4379

16913

8654

1329

626

9975

131978

79721

4295

1778

889

85

1335774

9997

131978

79775

Journal of Advanced Transportation 9

Table5Com

putatio

nalresultson

insta

nces

with

datageneratedfro

mPo

isson

distr

ibution

set ( |119868 |

|119869 |)120572=

085120572=

09120572=

095Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2CT

Obj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

1(26)

27

907294

9967

9001

27

987770

9979

9013

24

103

9273

9999

9468

2(412

)46

170

7961

9921

9117

51

183

8305

9946

9128

49

201

9392

9998

9546

3(618

)68

258

7578

9938

8911

74280

8028

9968

9004

69

307

9182

9999

9658

4(824)

93315

8001

9928

9196

102

338

8438

9967

9172

92368

9468

9999

9613

5(1030)

120

413

8041

9939

9449

133

443

8447

9976

9272

121

479

9544

9999

9752

6(1236)

155

476

8084

9877

9554

167

508

8863

9991

9569

159

536

9456

9999

9734

7(14

42)

195

573

7876

9892

9663

213

609

9124

9998

9729

205

643

9314

10000

9768

8(1648)

232

687

7964

9997

9788

267

724

8523

9988

9687

238

772

9422

9998

9689

9(1854)

293

692

8267

9893

9809

327

730

8386

9999

9750

298

757

9228

9999

9755

10(2060)

356

796

7946

9899

9801

399

840

8299

9976

9686

360

885

9668

9999

9705

11(2266)

423

829

8201

9895

9756

489

877

8378

9987

9788

448

927

9476

9999

9812

12(2472)

512

1023

8338

9913

9853

554

1085

8501

9999

9881

514

1148

9398

9999

9888

13(2678)

592

1045

8416

9907

9873

687

1088

7998

9996

9802

607

1152

9504

9999

9821

14(2884)

723

1099

7946

9915

9788

838

1166

8056

9992

9829

776

1230

9546

9999

9839

15(3090)

855

1089

7915

9964

9703

942

1156

8432

9984

9778

903

1225

9223

9998

9785

16(3296)

1009

1158

8024

9897

9769

1163

1231

8500

9996

9824

1045

1222

9199

9999

9835

17(34102)

1165

1299

8034

9919

9779

1285

1375

8495

9937

9808

1190

1457

9258

10000

9846

18(36108)

1254

1411

8176

9943

9885

1479

1492

8561

9968

9868

1345

1579

9286

9999

9875

19(38114

)1504

1506

7970

9923

9842

1651

1594

8167

9972

9846

1564

1694

9376

9999

9786

20(40120)

1706

1635

7819

9916

9783

1854

1734

8607

9968

9776

1778

1825

9285

9904

9718

21(42126)

2054

1622

8354

9918

9834

2357

1720

8297

9978

9833

2226

1856

9501

9999

9834

22(44132)

2400

1736

8494

9922

9846

2852

1844

8269

9999

9816

2680

1946

9486

9999

9820

23(46138)

2651

1934

8141

9953

9793

3159

2049

8492

9969

9799

3072

2172

9358

10000

9813

24(48144)

3004

1961

8226

9947

9681

3522

2086

8513

9998

9706

3102

2209

9488

9999

9724

25(50150)

3308

1924

8111

9965

9746

3898

2038

8264

9999

9788

3639

2161

9546

9999

9813

26(52156)

3669

1911

8096

9939

9855

4305

2027

8722

9967

9874

4453

2142

9185

9999

9876

27(54162)

4418

2095

8387

9927

9754

4771

2219

8341

9989

9749

4806

2353

9289

9999

9755

28(56168)

5367

2113

8458

9968

9848

5356

2237

8452

9978

9855

5221

2369

9228

10000

9868

29(58174)

6217

2186

7998

9989

9694

5942

2315

8316

9996

9699

5853

2371

9167

9999

9705

30(60180)

7263

2382

8321

9991

9837

6527

2543

8359

9998

9886

6214

2678

9369

9997

9886

31(62186)

8214

2346

8404

9936

9888

7813

2430

8321

9999

9846

6876

2560

9358

9999

9847

32(64192)

8863

2518

8378

9938

9807

8106

2677

8356

9999

9833

7877

2822

9408

10000

9835

33(66198)

9985

2532

8406

9917

9709

9684

2689

8497

9996

9746

8088

2838

9507

9999

9749

34(68204)

11264

2534

8543

9908

9897

10986

2685

8809

9996

9782

8469

2843

8844

9998

9780

35(70210)

12035

2715

8258

9937

9799

12098

2886

8569

9991

9809

9946

3041

9459

9999

9813

36(72216)

12803

2825

8104

9929

9827

13076

3010

8651

9994

9834

11013

3170

9468

9999

9865

37(74222)

13021

2830

8228

9949

9826

13261

3009

8376

9999

9826

12014

3172

9346

9999

9888

38(76228)

13862

2843

8152

9933

9807

13995

3028

8760

9998

9841

12133

3187

9706

9999

9876

39(78234)

15387

2953

8312

9964

9799

14827

3133

9001

9995

9783

13381

3304

8913

9999

9799

40(80240)

16974

2975

8510

9985

9767

15256

3249

8217

9993

9781

15867

3347

9111

9999

9791

Average

4352

15875

8143

9934

9708

4362

16856

8437

9985

9707

3968

17763

9356

9997

9781

10 Journal of Advanced Transportation

Table6Com

putatio

nalresultson

insta

nces

with

datageneratedfro

mno

rmaldistr

ibution

set ( |119868 |

|119869 |)120572=

085120572=

09120572=

095Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2CT

Obj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

1(26)

27

907294

9964

9117

27

947844

9970

9202

2558

106

8422

10000

9208

2(412

)46

170

7961

9921

9283

48

183

8328

9982

9283

4508

183

8823

10000

9370

3(618

)68

258

7578

9916

9228

68

280

7798

9971

9417

6803

307

7658

9998

9447

4(824)

93315

8000

9899

9367

96337

8379

9979

9467

9373

368

8438

9999

9579

5(1030)

120

413

7841

9927

9437

124

443

8763

9963

9476

1262

499

8850

9998

9708

6(1236)

155

476

7921

9919

9519

162

504

8124

9982

9582

16021

536

8954

10000

9756

7(14

42)

195

573

7976

9933

9598

196

609

8287

9975

9619

2011

646

8467

9999

9767

8(1648)

232

687

8031

9918

9705

235

727

8115

9965

9650

2404

772

8278

9999

9765

9(1854)

293

692

8086

9898

9759

291

734

8464

9974

9755

29815

796

8617

9998

9835

10(2060)

356

796

7984

9937

9770

358

842

8223

9988

9765

37973

921

8846

10000

9838

11(2266

)421

829

8101

9940

9755

436

877

8211

9986

9786

4514

8959

8497

9999

9789

12(2472)

514

1023

8033

9951

9785

501

1085

8162

9983

9788

52073

1147

8321

9999

9824

13(2678)

592

1023

8138

9916

9885

599

1087

8411

9970

9845

62937

1194

8609

9998

9795

14(2884)

728

1099

8235

9913

9915

722

1168

8435

9987

9806

75053

1270

8567

9997

9834

15(3090)

855

1089

8187

9922

9861

880

1157

8376

9976

9796

89222

1300

8611

9996

9798

16(3296)

1009

1158

8770

9897

9788

966

1231

8845

9984

9823

106288

1359

8997

9999

9848

17(34102)

1154

1299

8434

9925

9849

1137

1377

8601

9980

9798

123869

1493

8798

9996

9877

18(36108)

1407

1411

8321

9927

9866

1339

1495

8301

9965

9822

133807

1601

8567

9999

9834

19(38114

)1702

1506

8206

9915

9805

1672

1598

8977

9978

9845

152999

1694

8998

9999

9847

20(40120)

2091

1635

8528

9918

9889

1916

1737

8805

9976

9837

1890

021833

9012

9999

9791

21(42126)

2843

1622

8550

9934

9913

2432

1718

8886

9974

9813

2399

131815

9008

10000

9815

22(44132)

3300

1736

8449

9940

9859

2795

1843

8786

9977

9756

253106

1948

8997

9999

9826

23(46138)

3741

1934

8133

9911

9876

3143

2050

8876

9981

9785

274583

2173

9014

9999

9858

24(48144)

4531

1961

8245

9909

9846

3515

2090

8344

9993

9819

308089

2211

8661

9999

9883

25(50150)

4715

1924

8111

9929

9888

3959

2041

8279

9996

9837

3510

932158

8527

9999

9827

26(52156)

5223

1911

8096

9898

9857

4301

2025

8251

9997

9757

421747

2146

8497

9999

9914

27(54162)

5994

2095

8387

9908

9775

4835

2226

8497

9974

9780

4614

432359

8695

9999

9837

28(56168)

6406

2111

8411

9906

9798

5336

2240

8558

9991

9820

494113

2371

9249

9999

9816

29(58174)

6897

2176

8012

9916

9791

5948

2311

8211

9984

9765

624835

2445

8531

10000

9851

30(60180)

7436

2391

8111

9913

9766

6558

2533

8305

9968

9795

696116

2682

8575

9999

9799

31(62186)

8118

2282

8119

9928

9806

7069

2422

8417

9973

9873

797768

2564

8667

9999

9846

32(64192)

8895

2519

8223

9919

9757

7857

2675

8631

9982

9769

864813

2827

8976

10000

9905

33(66198)

11235

2533

8563

9928

9811

9162

2688

8547

9979

9828

973089

2843

8657

9999

9846

34(68204)

11587

2534

8249

9926

9847

9549

2686

8798

9938

9856

10523

3041

8999

10000

9877

35(70210)

1164

92715

8098

9895

9872

10345

2880

8602

9998

9869

11272

3052

9021

9999

9897

36(72216)

12224

2825

8403

9958

9770

11165

3002

8657

9999

9839

12456

3175

9113

9999

9864

37(74222)

12978

2830

8357

9937

9795

11850

3005

8496

9998

9894

13458

3176

8827

9999

9905

38(76228)

13426

2844

8267

9966

9759

12795

3019

8604

9999

9797

15155

3121

8978

9999

9853

39(78234)

15003

2955

8303

9956

9867

13816

3132

8588

9996

9885

15041

3310

9016

9999

9912

40(80240)

16495

2978

8257

9954

9826

14896

3160

8495

9997

9837

16563

3349

8957

9999

9933

Average

4619

15855

8174

9925

9742

4077

16828

8457

9981

9743

4350

17938

8732

9999

9794

Journal of Advanced Transportation 11

high service level requirement we recommend MIP-DF1 forsolutions

6 Conclusion

This paper investigates an ambulance location problem withambiguity set of demand in which the probability distribu-tion of the uncertain demand is unknown In such a data-driven environment only the mean and covariance demandsat emergency points are knownThe problem is to determinethe base locations and the number of ambulances at eachbase aiming at minimizing the total cost associated withlocating bases and assigning ambulances We propose adistribution-free model with chance constraints Then twoapproximated MIP formulations with different approxima-tion methods of chance constraints are proposed which arebased on different ambiguity sets of the unknown probabilitydistribution Numerical experiments on benchmarks and120 randomly generated instances are conducted and thecomputational results show that the first approximated MIPformulation is much conservative with overhigh system costwhile the second approximatedMIP formulation ismore costsaving with service level appropriately ensured

In future research wemay consider the following relevantissues First the uncertainty of driving time from a base toan emergency point shall be analysed Besides unexpected orsudden disasters may be taken into consideration Moreoverother uncertainties should be considered such as travel timethe number of available vehicles and so on The stochasticdependence should also further considered

Data Availability

The data of the benchmark instances in our manuscript canbe obtained in Nickel et al [8] For the test instances thedata is randomly generated and the way for generating data isstated in our manuscript The datasets generated during thecurrent study are available from the corresponding author onreasonable request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China (NSFC) under Grants 7142800271531011 71571134 and 71771048 This work was also sup-ported by the Fundamental Research Funds for the CentralUniversities

References

[1] L Brotcorne G Laporte and F Semet ldquoAmbulance loca-tion and relocation modelsrdquo European Journal of OperationalResearch vol 147 no 3 pp 451ndash463 2003

[2] X Li Z Zhao X Zhu and T Wyatt ldquoCovering modelsand optimization techniques for emergency response facilitylocation and planning a reviewrdquo Mathematical Methods ofOperations Research vol 74 no 3 pp 281ndash310 2011

[3] C Toregas R Swain C ReVelle and L Bergman ldquoThe locationof emergency service facilitiesrdquoOperations Research vol 19 no6 pp 1363ndash1373 1971

[4] R Church and C ReVelle ldquoThe maximal covering locationproblemrdquo Papers in Regional Science vol 32 no 1 pp 101ndash1181974

[5] M Gendreau G Laporte and F Semet ldquoSolving an ambulancelocation model by tabu searchrdquo Location Science vol 5 no 2pp 75ndash88 1997

[6] A Ingolfsson S Budge and E Erkut ldquoOptimal ambulancelocation with random delays and travel timesrdquo Health CareManagement Science vol 11 no 3 pp 262ndash274 2008

[7] M Gendreau G Laporte and F Semet ldquoThemaximal expectedcoverage relocation problem for emergency vehiclesrdquo Journal ofthe Operational Research Society vol 57 no 1 pp 22ndash28 2006

[8] S Nickel M Reuter-Oppermann and F Saldanha-da-GamaldquoAmbulance location under stochastic demand A samplingapproachrdquo Operations Research for Health Care vol 8 pp 24ndash32 2016

[9] P Beraldi M E Bruni and D Conforti ldquoDesigning robustemergency medical service via stochastic programmingrdquo Euro-pean Journal of Operational Research vol 158 no 1 pp 183ndash1932004

[10] N Noyan ldquoAlternate risk measures for emergency medicalservice system designrdquo Annals of Operations Research vol 181pp 559ndash589 2010

[11] M R Wagner ldquoStochastic 0-1 linear programming underlimited distributional informationrdquoOperations Research Lettersvol 36 no 2 pp 150ndash156 2008

[12] EDelage andYYe ldquoDistributionally robust optimization undermoment uncertainty with application to data-driven problemsrdquoOperations Research vol 58 no 3 pp 595ndash612 2010

[13] J T Essen J L Hurink S Nickel and M Reuter ldquoModels forambulance planning on the strategic and the tactical levelrdquo BetaResearch School for Operations Management and Logistics p 272013

[14] E Erkut A Ingolfsson and G Erdogan ldquoAmbulance locationfor maximum survivalrdquoNaval Research Logistics (NRL) vol 55no 1 pp 42ndash58 2008

[15] S C Chapman and J A White ldquoProbabilistic formulations ofemergency service facilities location problemsrdquo in Proceedingsof ORSATIMS Conference San Juan Puerto Rico 1974

[16] A Ben-Tal D Den Hertog A De Waegenaere B Melenbergand G Rennen ldquoRobust solutions of optimization problemsaffected by uncertain probabilitiesrdquo Management Science vol59 no 2 pp 341ndash357 2013

[17] MNg ldquoDistribution-free vessel deployment for liner shippingrdquoEuropean Journal of Operational Research vol 238 no 3 pp858ndash862 2014

[18] M Ng ldquoContainer vessel fleet deployment for liner shippingwith stochastic dependencies in shipping demandrdquo Transporta-tion Research Part B Methodological vol 74 pp 79ndash87 2015

[19] R Jiang and Y Guan ldquoData-driven chance constrained stochas-tic programrdquo Mathematical Programming vol 158 no 1-2 SerA pp 291ndash327 2016

[20] C-M Lee and S-L Hsu ldquoThe effect of advertising on thedistribution-free newsboy problemrdquo International Journal ofProduction Economics vol 129 no 1 pp 217ndash224 2011

[21] K Kwon and T Cheong ldquoA minimax distribution-free proce-dure for a newsvendor problem with free shippingrdquo EuropeanJournal of Operational Research vol 232 no 1 pp 234ndash2402014

12 Journal of Advanced Transportation

[22] Y Zhang S Shen and S A Erdogan ldquoDistributionally robustappointment scheduling with moment-based ambiguity setrdquoOperations Research Letters vol 45 no 2 pp 139ndash144 2017

[23] Y Zhang S Shen and S A Erdogan ldquoSolving 0-1semidefinite programs for distributionally robust allocationof surgery blocksrdquo Social Science Electronic Publishing 2017httpsssrncomabstract=2997524

[24] F Zheng J He F Chu and M Liu ldquoA new distribution-freemodel for disassembly line balancing problem with stochastictask processing timesrdquo International Journal of ProductionResearch pp 1ndash13 2018

[25] J Cheng E Delage and A Lisser ldquoDistributionally robuststochastic knapsack problemrdquo SIAM Journal on Optimizationvol 24 no 3 pp 1485ndash1506 2014

[26] M Liu D Yang and F Hao ldquoOptimization for the locationsof ambulances under two-stage life rescue in the emergencymedical service a case study in Shanghai ChinardquoMathematicalProblems in Engineering vol 2017 Article ID 1830480 14 pages2017

8 Journal of Advanced Transportation

Table4Com

putatio

nalresultson

insta

nces

with

datageneratedfro

mun

iform

distrib

ution

Set (|119868||119869|)

120572=085

120572=09

120572=095

Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2CT

Obj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

CTObj

Sel()

1(26)

27

897317

01

149

9906

01

107

9222

28

967373

01

155

9933

01

107

9167

27

106

7585

01

185

9987

01

107

9128

2(412

)47

169

7423

00

291

9919

00

207

9122

58

181

7438

01

303

9997

01

207

9363

50

199

7702

00

363

9982

00

207

9338

3(618

)72

258

7506

01

438

9976

01

312

9106

75279

8082

01

456

9996

01

312

9369

78304

8337

01

546

9997

01

312

9357

4(824)

99314

7956

01

553

9993

01

385

9206

111

338

8202

01

577

9999

01

385

9602

112

366

8492

01

697

9994

01

385

9620

5(1030)

132

415

8406

01

722

9896

01

512

9406

143

441

8322

02

752

9983

01

512

9692

142

476

8609

01

902

9993

01

512

9692

6(1236)

170

475

8249

02

843

9870

01

591

9367

181

504

8251

01

879

9982

01

591

9608

189

535

8726

01

1059

9997

01

591

9609

7(14

42)

215

573

8111

02

1006

9849

02

712

9606

243

609

8217

02

1048

9953

02

712

9589

233

644

8608

02

1258

9992

02

712

9582

8(1648)

248

686

7532

02

1185

9896

02

849

9725

284

727

8307

02

1233

9949

02

849

9613

281

768

9063

02

1473

9999

03

849

9622

9(1854)

313

691

8353

03

1236

9974

04

858

9709

351

731

8397

03

1290

9988

03

858

9765

352

760

8906

03

1560

9994

03

858

9758

10(2060)

372

794

8174

03

1409

9887

03

989

9657

428

843

8489

03

1469

9947

03

989

9773

432

892

9004

03

1769

9997

03

989

9738

11(2266

)451

830

8072

04

1493

9949

03

1031

9758

518

880

8328

04

1559

9982

03

1031

9723

516

930

8967

05

1889

9995

03

1031

9727

12(2472)

543

1025

7970

04

1771

9927

04

1267

9835

601

1083

8482

05

1843

9949

04

1267

9776

637

1146

9014

04

2203

9993

04

1267

9777

13(2678)

631

1026

8142

05

1819

9898

04

1274

9782

743

1087

8678

05

1897

9936

04

1274

9778

765

1146

9023

04

2287

9997

05

1273

9837

14(2884)

782

1101

7903

05

1949

9869

08

1361

9804

878

1165

8790

05

2033

9935

07

1361

9889

870

1227

9056

05

2453

9998

06

1361

9808

15(3090)

907

1091

8098

06

1997

9915

06

1367

9629

1018

1155

8329

06

2087

9981

06

1367

9807

1020

1225

8986

06

2537

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08

1367

9817

16(3296)

1062

1160

8798

07

2123

9944

07

1449

9787

1183

1229

8383

07

2219

9988

06

1449

9849

1193

1300

9105

06

2698

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06

1449

9815

17(34102)

1202

1298

7572

07

2328

9969

07

1614

9821

1377

1374

8860

08

2430

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07

1614

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1260

1452

9085

07

2940

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08

1614

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18(36108)

1394

1408

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08

2509

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08

1752

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1544

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8928

08

2617

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1752

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08

3156

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1752

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19(38114

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09

2676

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1760

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9130

09

2790

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08

1877

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09

3360

9999

09

1877

9872

20(40120)

2206

1628

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2007

1732

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112990

9997

09

2029

9879

1904

1827

9138

103590

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09

2029

9875

21(42126)

2780

1622

8354

102900

9978

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9732

2609

1718

8854

103026

9876

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9818

2670

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123655

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9848

22(44132)

3189

1735

8494

113088

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152165

9857

2970

1840

8922

123220

9925

122165

9886

2873

1952

9206

113880

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112165

9850

23(46138)

3630

1937

8141

123372

9893

132406

9796

3222

2048

8937

123510

9934

142406

9906

3461

2170

8989

124199

10000

122406

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24(48144)

4226

1968

8226

133461

9932

132453

9814

3674

2085

8846

133605

9956

152453

9912

3556

2211

9300

134326

9999

132453

9819

25(50150)

4727

1919

8111

143456

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152407

9629

4019

2037

8699

143606

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142407

9857

4257

2161

9259

154356

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142407

9819

26(52156)

5300

1906

8096

153484

9878

172392

9658

4495

2021

8952

143640

9999

152392

9568

4486

2142

9182

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209999

152392

9854

27(54162)

5913

2096

8387

163751

9846

172617

9761

5117

2221

9045

163913

9998

172617

9544

4830

2355

9346

184723

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162617

9893

28(56168)

6310

2110

7758

173819

9933

192643

9829

5777

2234

9157

183987

9998

182643

9526

5550

2365

9045

184827

9999

172643

9831

29(58174)

6780

2175

7998

183937

9898

182720

9714

6303

2307

9022

20

4111

9969

192720

9623

6189

2444

9128

194981

10000

192720

9914

30(60180)

7593

2384

8321

194240

9967

20

2979

9841

6791

2527

8978

1944

199979

23

2979

9721

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9143

21

5316

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20

2979

9869

31(62186)

8220

2281

7604

20

4157

9984

21

2856

9796

7734

2416

8934

21

4343

9997

21

2856

9679

7466

2557

9158

22

5273

9999

23

2856

9902

32(64192)

8766

2517

8378

21

4489

9932

25

3145

9729

8276

2671

8296

23

4680

9977

24

3145

9825

7897

2822

9089

24

5640

9999

22

3145

9891

33(66198)

10123

2532

7806

23

4555

9967

23

3169

9615

9010

2682

8947

24

4752

9989

24

3169

9703

9051

2835

9157

24

5742

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24

3169

9867

34(68204)

11010

2535

7934

24

4602

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29

3174

9631

9898

2685

8858

30

4806

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28

3174

9762

9863

2844

9236

26

5824

9999

25

3174

9889

35(70210)

11666

2714

8062

25

4862

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29

3391

9787

1066

62875

9137

26

5072

9999

29

3391

9716

10822

3043

9315

26

6122

9999

29

3391

9923

36(72216)

12011

2827

8104

26

5047

9991

28

3535

9798

1246

52984

9016

28

5263

9999

28

3535

9878

11667

3169

9253

30

6343

9999

29

3535

9896

37(74222)

11437

2826

8228

28

5096

9933

29

3543

9884

13582

3098

8895

29

5318

9999

32

3543

9888

12654

3178

9139

29

6427

10000

31

3543

9893

38(76228)

12435

2839

8152

29

5161

9897

29

3565

9788

13976

3103

8926

29

5388

9999

31

3565

9724

13837

3185

9255

33

6528

10000

31

3565

9895

39(78234)

1464

32945

8412

31

5331

9998

32

3695

9876

14794

3210

9005

31

5565

9999

34

3695

9713

14561

3313

9081

34

6735

10000

33

3695

9878

40(80240)

1604

22975

8510

32

5414

9957

36

3734

9796

16263

3346

8999

35

5654

9999

35

3734

9806

15869

3343

9248

35

6854

10000

35

3734

9873

Average

4482

1584

680

67

1228

397

9936

131978

796

82

4379

16913

8654

1329

626

9975

131978

79721

4295

1778

889

85

1335774

9997

131978

79775

Journal of Advanced Transportation 9

Table5Com

putatio

nalresultson

insta

nces

with

datageneratedfro

mPo

isson

distr

ibution

set ( |119868 |

|119869 |)120572=

085120572=

09120572=

095Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2CT

Obj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

1(26)

27

907294

9967

9001

27

987770

9979

9013

24

103

9273

9999

9468

2(412

)46

170

7961

9921

9117

51

183

8305

9946

9128

49

201

9392

9998

9546

3(618

)68

258

7578

9938

8911

74280

8028

9968

9004

69

307

9182

9999

9658

4(824)

93315

8001

9928

9196

102

338

8438

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9172

92368

9468

9999

9613

5(1030)

120

413

8041

9939

9449

133

443

8447

9976

9272

121

479

9544

9999

9752

6(1236)

155

476

8084

9877

9554

167

508

8863

9991

9569

159

536

9456

9999

9734

7(14

42)

195

573

7876

9892

9663

213

609

9124

9998

9729

205

643

9314

10000

9768

8(1648)

232

687

7964

9997

9788

267

724

8523

9988

9687

238

772

9422

9998

9689

9(1854)

293

692

8267

9893

9809

327

730

8386

9999

9750

298

757

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9999

9755

10(2060)

356

796

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9899

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399

840

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360

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9668

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9705

11(2266)

423

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489

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9812

12(2472)

512

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13(2678)

592

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9802

607

1152

9504

9999

9821

14(2884)

723

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838

1166

8056

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9829

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1230

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9839

15(3090)

855

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8432

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9778

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1225

9223

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16(3296)

1009

1158

8024

9897

9769

1163

1231

8500

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9824

1045

1222

9199

9999

9835

17(34102)

1165

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8034

9919

9779

1285

1375

8495

9937

9808

1190

1457

9258

10000

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18(36108)

1254

1411

8176

9943

9885

1479

1492

8561

9968

9868

1345

1579

9286

9999

9875

19(38114

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1506

7970

9923

9842

1651

1594

8167

9972

9846

1564

1694

9376

9999

9786

20(40120)

1706

1635

7819

9916

9783

1854

1734

8607

9968

9776

1778

1825

9285

9904

9718

21(42126)

2054

1622

8354

9918

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2357

1720

8297

9978

9833

2226

1856

9501

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9834

22(44132)

2400

1736

8494

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2852

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8269

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9816

2680

1946

9486

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9820

23(46138)

2651

1934

8141

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9969

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3072

2172

9358

10000

9813

24(48144)

3004

1961

8226

9947

9681

3522

2086

8513

9998

9706

3102

2209

9488

9999

9724

25(50150)

3308

1924

8111

9965

9746

3898

2038

8264

9999

9788

3639

2161

9546

9999

9813

26(52156)

3669

1911

8096

9939

9855

4305

2027

8722

9967

9874

4453

2142

9185

9999

9876

27(54162)

4418

2095

8387

9927

9754

4771

2219

8341

9989

9749

4806

2353

9289

9999

9755

28(56168)

5367

2113

8458

9968

9848

5356

2237

8452

9978

9855

5221

2369

9228

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9868

29(58174)

6217

2186

7998

9989

9694

5942

2315

8316

9996

9699

5853

2371

9167

9999

9705

30(60180)

7263

2382

8321

9991

9837

6527

2543

8359

9998

9886

6214

2678

9369

9997

9886

31(62186)

8214

2346

8404

9936

9888

7813

2430

8321

9999

9846

6876

2560

9358

9999

9847

32(64192)

8863

2518

8378

9938

9807

8106

2677

8356

9999

9833

7877

2822

9408

10000

9835

33(66198)

9985

2532

8406

9917

9709

9684

2689

8497

9996

9746

8088

2838

9507

9999

9749

34(68204)

11264

2534

8543

9908

9897

10986

2685

8809

9996

9782

8469

2843

8844

9998

9780

35(70210)

12035

2715

8258

9937

9799

12098

2886

8569

9991

9809

9946

3041

9459

9999

9813

36(72216)

12803

2825

8104

9929

9827

13076

3010

8651

9994

9834

11013

3170

9468

9999

9865

37(74222)

13021

2830

8228

9949

9826

13261

3009

8376

9999

9826

12014

3172

9346

9999

9888

38(76228)

13862

2843

8152

9933

9807

13995

3028

8760

9998

9841

12133

3187

9706

9999

9876

39(78234)

15387

2953

8312

9964

9799

14827

3133

9001

9995

9783

13381

3304

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9999

9799

40(80240)

16974

2975

8510

9985

9767

15256

3249

8217

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9781

15867

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9111

9999

9791

Average

4352

15875

8143

9934

9708

4362

16856

8437

9985

9707

3968

17763

9356

9997

9781

10 Journal of Advanced Transportation

Table6Com

putatio

nalresultson

insta

nces

with

datageneratedfro

mno

rmaldistr

ibution

set ( |119868 |

|119869 |)120572=

085120572=

09120572=

095Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2CT

Obj

Sel()

Sel()

Sel()

CTObj

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CTObj

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1(26)

27

907294

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9117

27

947844

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9202

2558

106

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10000

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2(412

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170

7961

9921

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48

183

8328

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4508

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8823

10000

9370

3(618

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258

7578

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68

280

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9417

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307

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4(824)

93315

8000

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9467

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368

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5(1030)

120

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124

443

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1262

499

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9708

6(1236)

155

476

7921

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162

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8124

9982

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16021

536

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10000

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7(14

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195

573

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9933

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196

609

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9619

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646

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8(1648)

232

687

8031

9918

9705

235

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8115

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9650

2404

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8278

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9765

9(1854)

293

692

8086

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9759

291

734

8464

9974

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29815

796

8617

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9835

10(2060)

356

796

7984

9937

9770

358

842

8223

9988

9765

37973

921

8846

10000

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11(2266

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829

8101

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436

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9786

4514

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12(2472)

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8162

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13(2678)

592

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14(2884)

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15(3090)

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1300

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16(3296)

1009

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9823

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17(34102)

1154

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1137

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8601

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123869

1493

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9877

18(36108)

1407

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9927

9866

1339

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8301

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9822

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19(38114

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20(40120)

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21(42126)

2843

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22(44132)

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23(46138)

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24(48144)

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25(50150)

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26(52156)

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27(54162)

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28(56168)

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29(58174)

6897

2176

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9765

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30(60180)

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31(62186)

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32(64192)

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33(66198)

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34(68204)

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10523

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35(70210)

1164

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10345

2880

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11272

3052

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36(72216)

12224

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37(74222)

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38(76228)

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9999

9853

39(78234)

15003

2955

8303

9956

9867

13816

3132

8588

9996

9885

15041

3310

9016

9999

9912

40(80240)

16495

2978

8257

9954

9826

14896

3160

8495

9997

9837

16563

3349

8957

9999

9933

Average

4619

15855

8174

9925

9742

4077

16828

8457

9981

9743

4350

17938

8732

9999

9794

Journal of Advanced Transportation 11

high service level requirement we recommend MIP-DF1 forsolutions

6 Conclusion

This paper investigates an ambulance location problem withambiguity set of demand in which the probability distribu-tion of the uncertain demand is unknown In such a data-driven environment only the mean and covariance demandsat emergency points are knownThe problem is to determinethe base locations and the number of ambulances at eachbase aiming at minimizing the total cost associated withlocating bases and assigning ambulances We propose adistribution-free model with chance constraints Then twoapproximated MIP formulations with different approxima-tion methods of chance constraints are proposed which arebased on different ambiguity sets of the unknown probabilitydistribution Numerical experiments on benchmarks and120 randomly generated instances are conducted and thecomputational results show that the first approximated MIPformulation is much conservative with overhigh system costwhile the second approximatedMIP formulation ismore costsaving with service level appropriately ensured

In future research wemay consider the following relevantissues First the uncertainty of driving time from a base toan emergency point shall be analysed Besides unexpected orsudden disasters may be taken into consideration Moreoverother uncertainties should be considered such as travel timethe number of available vehicles and so on The stochasticdependence should also further considered

Data Availability

The data of the benchmark instances in our manuscript canbe obtained in Nickel et al [8] For the test instances thedata is randomly generated and the way for generating data isstated in our manuscript The datasets generated during thecurrent study are available from the corresponding author onreasonable request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China (NSFC) under Grants 7142800271531011 71571134 and 71771048 This work was also sup-ported by the Fundamental Research Funds for the CentralUniversities

References

[1] L Brotcorne G Laporte and F Semet ldquoAmbulance loca-tion and relocation modelsrdquo European Journal of OperationalResearch vol 147 no 3 pp 451ndash463 2003

[2] X Li Z Zhao X Zhu and T Wyatt ldquoCovering modelsand optimization techniques for emergency response facilitylocation and planning a reviewrdquo Mathematical Methods ofOperations Research vol 74 no 3 pp 281ndash310 2011

[3] C Toregas R Swain C ReVelle and L Bergman ldquoThe locationof emergency service facilitiesrdquoOperations Research vol 19 no6 pp 1363ndash1373 1971

[4] R Church and C ReVelle ldquoThe maximal covering locationproblemrdquo Papers in Regional Science vol 32 no 1 pp 101ndash1181974

[5] M Gendreau G Laporte and F Semet ldquoSolving an ambulancelocation model by tabu searchrdquo Location Science vol 5 no 2pp 75ndash88 1997

[6] A Ingolfsson S Budge and E Erkut ldquoOptimal ambulancelocation with random delays and travel timesrdquo Health CareManagement Science vol 11 no 3 pp 262ndash274 2008

[7] M Gendreau G Laporte and F Semet ldquoThemaximal expectedcoverage relocation problem for emergency vehiclesrdquo Journal ofthe Operational Research Society vol 57 no 1 pp 22ndash28 2006

[8] S Nickel M Reuter-Oppermann and F Saldanha-da-GamaldquoAmbulance location under stochastic demand A samplingapproachrdquo Operations Research for Health Care vol 8 pp 24ndash32 2016

[9] P Beraldi M E Bruni and D Conforti ldquoDesigning robustemergency medical service via stochastic programmingrdquo Euro-pean Journal of Operational Research vol 158 no 1 pp 183ndash1932004

[10] N Noyan ldquoAlternate risk measures for emergency medicalservice system designrdquo Annals of Operations Research vol 181pp 559ndash589 2010

[11] M R Wagner ldquoStochastic 0-1 linear programming underlimited distributional informationrdquoOperations Research Lettersvol 36 no 2 pp 150ndash156 2008

[12] EDelage andYYe ldquoDistributionally robust optimization undermoment uncertainty with application to data-driven problemsrdquoOperations Research vol 58 no 3 pp 595ndash612 2010

[13] J T Essen J L Hurink S Nickel and M Reuter ldquoModels forambulance planning on the strategic and the tactical levelrdquo BetaResearch School for Operations Management and Logistics p 272013

[14] E Erkut A Ingolfsson and G Erdogan ldquoAmbulance locationfor maximum survivalrdquoNaval Research Logistics (NRL) vol 55no 1 pp 42ndash58 2008

[15] S C Chapman and J A White ldquoProbabilistic formulations ofemergency service facilities location problemsrdquo in Proceedingsof ORSATIMS Conference San Juan Puerto Rico 1974

[16] A Ben-Tal D Den Hertog A De Waegenaere B Melenbergand G Rennen ldquoRobust solutions of optimization problemsaffected by uncertain probabilitiesrdquo Management Science vol59 no 2 pp 341ndash357 2013

[17] MNg ldquoDistribution-free vessel deployment for liner shippingrdquoEuropean Journal of Operational Research vol 238 no 3 pp858ndash862 2014

[18] M Ng ldquoContainer vessel fleet deployment for liner shippingwith stochastic dependencies in shipping demandrdquo Transporta-tion Research Part B Methodological vol 74 pp 79ndash87 2015

[19] R Jiang and Y Guan ldquoData-driven chance constrained stochas-tic programrdquo Mathematical Programming vol 158 no 1-2 SerA pp 291ndash327 2016

[20] C-M Lee and S-L Hsu ldquoThe effect of advertising on thedistribution-free newsboy problemrdquo International Journal ofProduction Economics vol 129 no 1 pp 217ndash224 2011

[21] K Kwon and T Cheong ldquoA minimax distribution-free proce-dure for a newsvendor problem with free shippingrdquo EuropeanJournal of Operational Research vol 232 no 1 pp 234ndash2402014

12 Journal of Advanced Transportation

[22] Y Zhang S Shen and S A Erdogan ldquoDistributionally robustappointment scheduling with moment-based ambiguity setrdquoOperations Research Letters vol 45 no 2 pp 139ndash144 2017

[23] Y Zhang S Shen and S A Erdogan ldquoSolving 0-1semidefinite programs for distributionally robust allocationof surgery blocksrdquo Social Science Electronic Publishing 2017httpsssrncomabstract=2997524

[24] F Zheng J He F Chu and M Liu ldquoA new distribution-freemodel for disassembly line balancing problem with stochastictask processing timesrdquo International Journal of ProductionResearch pp 1ndash13 2018

[25] J Cheng E Delage and A Lisser ldquoDistributionally robuststochastic knapsack problemrdquo SIAM Journal on Optimizationvol 24 no 3 pp 1485ndash1506 2014

[26] M Liu D Yang and F Hao ldquoOptimization for the locationsof ambulances under two-stage life rescue in the emergencymedical service a case study in Shanghai ChinardquoMathematicalProblems in Engineering vol 2017 Article ID 1830480 14 pages2017

Journal of Advanced Transportation 9

Table5Com

putatio

nalresultson

insta

nces

with

datageneratedfro

mPo

isson

distr

ibution

set ( |119868 |

|119869 |)120572=

085120572=

09120572=

095Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2CT

Obj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

1(26)

27

907294

9967

9001

27

987770

9979

9013

24

103

9273

9999

9468

2(412

)46

170

7961

9921

9117

51

183

8305

9946

9128

49

201

9392

9998

9546

3(618

)68

258

7578

9938

8911

74280

8028

9968

9004

69

307

9182

9999

9658

4(824)

93315

8001

9928

9196

102

338

8438

9967

9172

92368

9468

9999

9613

5(1030)

120

413

8041

9939

9449

133

443

8447

9976

9272

121

479

9544

9999

9752

6(1236)

155

476

8084

9877

9554

167

508

8863

9991

9569

159

536

9456

9999

9734

7(14

42)

195

573

7876

9892

9663

213

609

9124

9998

9729

205

643

9314

10000

9768

8(1648)

232

687

7964

9997

9788

267

724

8523

9988

9687

238

772

9422

9998

9689

9(1854)

293

692

8267

9893

9809

327

730

8386

9999

9750

298

757

9228

9999

9755

10(2060)

356

796

7946

9899

9801

399

840

8299

9976

9686

360

885

9668

9999

9705

11(2266)

423

829

8201

9895

9756

489

877

8378

9987

9788

448

927

9476

9999

9812

12(2472)

512

1023

8338

9913

9853

554

1085

8501

9999

9881

514

1148

9398

9999

9888

13(2678)

592

1045

8416

9907

9873

687

1088

7998

9996

9802

607

1152

9504

9999

9821

14(2884)

723

1099

7946

9915

9788

838

1166

8056

9992

9829

776

1230

9546

9999

9839

15(3090)

855

1089

7915

9964

9703

942

1156

8432

9984

9778

903

1225

9223

9998

9785

16(3296)

1009

1158

8024

9897

9769

1163

1231

8500

9996

9824

1045

1222

9199

9999

9835

17(34102)

1165

1299

8034

9919

9779

1285

1375

8495

9937

9808

1190

1457

9258

10000

9846

18(36108)

1254

1411

8176

9943

9885

1479

1492

8561

9968

9868

1345

1579

9286

9999

9875

19(38114

)1504

1506

7970

9923

9842

1651

1594

8167

9972

9846

1564

1694

9376

9999

9786

20(40120)

1706

1635

7819

9916

9783

1854

1734

8607

9968

9776

1778

1825

9285

9904

9718

21(42126)

2054

1622

8354

9918

9834

2357

1720

8297

9978

9833

2226

1856

9501

9999

9834

22(44132)

2400

1736

8494

9922

9846

2852

1844

8269

9999

9816

2680

1946

9486

9999

9820

23(46138)

2651

1934

8141

9953

9793

3159

2049

8492

9969

9799

3072

2172

9358

10000

9813

24(48144)

3004

1961

8226

9947

9681

3522

2086

8513

9998

9706

3102

2209

9488

9999

9724

25(50150)

3308

1924

8111

9965

9746

3898

2038

8264

9999

9788

3639

2161

9546

9999

9813

26(52156)

3669

1911

8096

9939

9855

4305

2027

8722

9967

9874

4453

2142

9185

9999

9876

27(54162)

4418

2095

8387

9927

9754

4771

2219

8341

9989

9749

4806

2353

9289

9999

9755

28(56168)

5367

2113

8458

9968

9848

5356

2237

8452

9978

9855

5221

2369

9228

10000

9868

29(58174)

6217

2186

7998

9989

9694

5942

2315

8316

9996

9699

5853

2371

9167

9999

9705

30(60180)

7263

2382

8321

9991

9837

6527

2543

8359

9998

9886

6214

2678

9369

9997

9886

31(62186)

8214

2346

8404

9936

9888

7813

2430

8321

9999

9846

6876

2560

9358

9999

9847

32(64192)

8863

2518

8378

9938

9807

8106

2677

8356

9999

9833

7877

2822

9408

10000

9835

33(66198)

9985

2532

8406

9917

9709

9684

2689

8497

9996

9746

8088

2838

9507

9999

9749

34(68204)

11264

2534

8543

9908

9897

10986

2685

8809

9996

9782

8469

2843

8844

9998

9780

35(70210)

12035

2715

8258

9937

9799

12098

2886

8569

9991

9809

9946

3041

9459

9999

9813

36(72216)

12803

2825

8104

9929

9827

13076

3010

8651

9994

9834

11013

3170

9468

9999

9865

37(74222)

13021

2830

8228

9949

9826

13261

3009

8376

9999

9826

12014

3172

9346

9999

9888

38(76228)

13862

2843

8152

9933

9807

13995

3028

8760

9998

9841

12133

3187

9706

9999

9876

39(78234)

15387

2953

8312

9964

9799

14827

3133

9001

9995

9783

13381

3304

8913

9999

9799

40(80240)

16974

2975

8510

9985

9767

15256

3249

8217

9993

9781

15867

3347

9111

9999

9791

Average

4352

15875

8143

9934

9708

4362

16856

8437

9985

9707

3968

17763

9356

9997

9781

10 Journal of Advanced Transportation

Table6Com

putatio

nalresultson

insta

nces

with

datageneratedfro

mno

rmaldistr

ibution

set ( |119868 |

|119869 |)120572=

085120572=

09120572=

095Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2CT

Obj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

1(26)

27

907294

9964

9117

27

947844

9970

9202

2558

106

8422

10000

9208

2(412

)46

170

7961

9921

9283

48

183

8328

9982

9283

4508

183

8823

10000

9370

3(618

)68

258

7578

9916

9228

68

280

7798

9971

9417

6803

307

7658

9998

9447

4(824)

93315

8000

9899

9367

96337

8379

9979

9467

9373

368

8438

9999

9579

5(1030)

120

413

7841

9927

9437

124

443

8763

9963

9476

1262

499

8850

9998

9708

6(1236)

155

476

7921

9919

9519

162

504

8124

9982

9582

16021

536

8954

10000

9756

7(14

42)

195

573

7976

9933

9598

196

609

8287

9975

9619

2011

646

8467

9999

9767

8(1648)

232

687

8031

9918

9705

235

727

8115

9965

9650

2404

772

8278

9999

9765

9(1854)

293

692

8086

9898

9759

291

734

8464

9974

9755

29815

796

8617

9998

9835

10(2060)

356

796

7984

9937

9770

358

842

8223

9988

9765

37973

921

8846

10000

9838

11(2266

)421

829

8101

9940

9755

436

877

8211

9986

9786

4514

8959

8497

9999

9789

12(2472)

514

1023

8033

9951

9785

501

1085

8162

9983

9788

52073

1147

8321

9999

9824

13(2678)

592

1023

8138

9916

9885

599

1087

8411

9970

9845

62937

1194

8609

9998

9795

14(2884)

728

1099

8235

9913

9915

722

1168

8435

9987

9806

75053

1270

8567

9997

9834

15(3090)

855

1089

8187

9922

9861

880

1157

8376

9976

9796

89222

1300

8611

9996

9798

16(3296)

1009

1158

8770

9897

9788

966

1231

8845

9984

9823

106288

1359

8997

9999

9848

17(34102)

1154

1299

8434

9925

9849

1137

1377

8601

9980

9798

123869

1493

8798

9996

9877

18(36108)

1407

1411

8321

9927

9866

1339

1495

8301

9965

9822

133807

1601

8567

9999

9834

19(38114

)1702

1506

8206

9915

9805

1672

1598

8977

9978

9845

152999

1694

8998

9999

9847

20(40120)

2091

1635

8528

9918

9889

1916

1737

8805

9976

9837

1890

021833

9012

9999

9791

21(42126)

2843

1622

8550

9934

9913

2432

1718

8886

9974

9813

2399

131815

9008

10000

9815

22(44132)

3300

1736

8449

9940

9859

2795

1843

8786

9977

9756

253106

1948

8997

9999

9826

23(46138)

3741

1934

8133

9911

9876

3143

2050

8876

9981

9785

274583

2173

9014

9999

9858

24(48144)

4531

1961

8245

9909

9846

3515

2090

8344

9993

9819

308089

2211

8661

9999

9883

25(50150)

4715

1924

8111

9929

9888

3959

2041

8279

9996

9837

3510

932158

8527

9999

9827

26(52156)

5223

1911

8096

9898

9857

4301

2025

8251

9997

9757

421747

2146

8497

9999

9914

27(54162)

5994

2095

8387

9908

9775

4835

2226

8497

9974

9780

4614

432359

8695

9999

9837

28(56168)

6406

2111

8411

9906

9798

5336

2240

8558

9991

9820

494113

2371

9249

9999

9816

29(58174)

6897

2176

8012

9916

9791

5948

2311

8211

9984

9765

624835

2445

8531

10000

9851

30(60180)

7436

2391

8111

9913

9766

6558

2533

8305

9968

9795

696116

2682

8575

9999

9799

31(62186)

8118

2282

8119

9928

9806

7069

2422

8417

9973

9873

797768

2564

8667

9999

9846

32(64192)

8895

2519

8223

9919

9757

7857

2675

8631

9982

9769

864813

2827

8976

10000

9905

33(66198)

11235

2533

8563

9928

9811

9162

2688

8547

9979

9828

973089

2843

8657

9999

9846

34(68204)

11587

2534

8249

9926

9847

9549

2686

8798

9938

9856

10523

3041

8999

10000

9877

35(70210)

1164

92715

8098

9895

9872

10345

2880

8602

9998

9869

11272

3052

9021

9999

9897

36(72216)

12224

2825

8403

9958

9770

11165

3002

8657

9999

9839

12456

3175

9113

9999

9864

37(74222)

12978

2830

8357

9937

9795

11850

3005

8496

9998

9894

13458

3176

8827

9999

9905

38(76228)

13426

2844

8267

9966

9759

12795

3019

8604

9999

9797

15155

3121

8978

9999

9853

39(78234)

15003

2955

8303

9956

9867

13816

3132

8588

9996

9885

15041

3310

9016

9999

9912

40(80240)

16495

2978

8257

9954

9826

14896

3160

8495

9997

9837

16563

3349

8957

9999

9933

Average

4619

15855

8174

9925

9742

4077

16828

8457

9981

9743

4350

17938

8732

9999

9794

Journal of Advanced Transportation 11

high service level requirement we recommend MIP-DF1 forsolutions

6 Conclusion

This paper investigates an ambulance location problem withambiguity set of demand in which the probability distribu-tion of the uncertain demand is unknown In such a data-driven environment only the mean and covariance demandsat emergency points are knownThe problem is to determinethe base locations and the number of ambulances at eachbase aiming at minimizing the total cost associated withlocating bases and assigning ambulances We propose adistribution-free model with chance constraints Then twoapproximated MIP formulations with different approxima-tion methods of chance constraints are proposed which arebased on different ambiguity sets of the unknown probabilitydistribution Numerical experiments on benchmarks and120 randomly generated instances are conducted and thecomputational results show that the first approximated MIPformulation is much conservative with overhigh system costwhile the second approximatedMIP formulation ismore costsaving with service level appropriately ensured

In future research wemay consider the following relevantissues First the uncertainty of driving time from a base toan emergency point shall be analysed Besides unexpected orsudden disasters may be taken into consideration Moreoverother uncertainties should be considered such as travel timethe number of available vehicles and so on The stochasticdependence should also further considered

Data Availability

The data of the benchmark instances in our manuscript canbe obtained in Nickel et al [8] For the test instances thedata is randomly generated and the way for generating data isstated in our manuscript The datasets generated during thecurrent study are available from the corresponding author onreasonable request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China (NSFC) under Grants 7142800271531011 71571134 and 71771048 This work was also sup-ported by the Fundamental Research Funds for the CentralUniversities

References

[1] L Brotcorne G Laporte and F Semet ldquoAmbulance loca-tion and relocation modelsrdquo European Journal of OperationalResearch vol 147 no 3 pp 451ndash463 2003

[2] X Li Z Zhao X Zhu and T Wyatt ldquoCovering modelsand optimization techniques for emergency response facilitylocation and planning a reviewrdquo Mathematical Methods ofOperations Research vol 74 no 3 pp 281ndash310 2011

[3] C Toregas R Swain C ReVelle and L Bergman ldquoThe locationof emergency service facilitiesrdquoOperations Research vol 19 no6 pp 1363ndash1373 1971

[4] R Church and C ReVelle ldquoThe maximal covering locationproblemrdquo Papers in Regional Science vol 32 no 1 pp 101ndash1181974

[5] M Gendreau G Laporte and F Semet ldquoSolving an ambulancelocation model by tabu searchrdquo Location Science vol 5 no 2pp 75ndash88 1997

[6] A Ingolfsson S Budge and E Erkut ldquoOptimal ambulancelocation with random delays and travel timesrdquo Health CareManagement Science vol 11 no 3 pp 262ndash274 2008

[7] M Gendreau G Laporte and F Semet ldquoThemaximal expectedcoverage relocation problem for emergency vehiclesrdquo Journal ofthe Operational Research Society vol 57 no 1 pp 22ndash28 2006

[8] S Nickel M Reuter-Oppermann and F Saldanha-da-GamaldquoAmbulance location under stochastic demand A samplingapproachrdquo Operations Research for Health Care vol 8 pp 24ndash32 2016

[9] P Beraldi M E Bruni and D Conforti ldquoDesigning robustemergency medical service via stochastic programmingrdquo Euro-pean Journal of Operational Research vol 158 no 1 pp 183ndash1932004

[10] N Noyan ldquoAlternate risk measures for emergency medicalservice system designrdquo Annals of Operations Research vol 181pp 559ndash589 2010

[11] M R Wagner ldquoStochastic 0-1 linear programming underlimited distributional informationrdquoOperations Research Lettersvol 36 no 2 pp 150ndash156 2008

[12] EDelage andYYe ldquoDistributionally robust optimization undermoment uncertainty with application to data-driven problemsrdquoOperations Research vol 58 no 3 pp 595ndash612 2010

[13] J T Essen J L Hurink S Nickel and M Reuter ldquoModels forambulance planning on the strategic and the tactical levelrdquo BetaResearch School for Operations Management and Logistics p 272013

[14] E Erkut A Ingolfsson and G Erdogan ldquoAmbulance locationfor maximum survivalrdquoNaval Research Logistics (NRL) vol 55no 1 pp 42ndash58 2008

[15] S C Chapman and J A White ldquoProbabilistic formulations ofemergency service facilities location problemsrdquo in Proceedingsof ORSATIMS Conference San Juan Puerto Rico 1974

[16] A Ben-Tal D Den Hertog A De Waegenaere B Melenbergand G Rennen ldquoRobust solutions of optimization problemsaffected by uncertain probabilitiesrdquo Management Science vol59 no 2 pp 341ndash357 2013

[17] MNg ldquoDistribution-free vessel deployment for liner shippingrdquoEuropean Journal of Operational Research vol 238 no 3 pp858ndash862 2014

[18] M Ng ldquoContainer vessel fleet deployment for liner shippingwith stochastic dependencies in shipping demandrdquo Transporta-tion Research Part B Methodological vol 74 pp 79ndash87 2015

[19] R Jiang and Y Guan ldquoData-driven chance constrained stochas-tic programrdquo Mathematical Programming vol 158 no 1-2 SerA pp 291ndash327 2016

[20] C-M Lee and S-L Hsu ldquoThe effect of advertising on thedistribution-free newsboy problemrdquo International Journal ofProduction Economics vol 129 no 1 pp 217ndash224 2011

[21] K Kwon and T Cheong ldquoA minimax distribution-free proce-dure for a newsvendor problem with free shippingrdquo EuropeanJournal of Operational Research vol 232 no 1 pp 234ndash2402014

12 Journal of Advanced Transportation

[22] Y Zhang S Shen and S A Erdogan ldquoDistributionally robustappointment scheduling with moment-based ambiguity setrdquoOperations Research Letters vol 45 no 2 pp 139ndash144 2017

[23] Y Zhang S Shen and S A Erdogan ldquoSolving 0-1semidefinite programs for distributionally robust allocationof surgery blocksrdquo Social Science Electronic Publishing 2017httpsssrncomabstract=2997524

[24] F Zheng J He F Chu and M Liu ldquoA new distribution-freemodel for disassembly line balancing problem with stochastictask processing timesrdquo International Journal of ProductionResearch pp 1ndash13 2018

[25] J Cheng E Delage and A Lisser ldquoDistributionally robuststochastic knapsack problemrdquo SIAM Journal on Optimizationvol 24 no 3 pp 1485ndash1506 2014

[26] M Liu D Yang and F Hao ldquoOptimization for the locationsof ambulances under two-stage life rescue in the emergencymedical service a case study in Shanghai ChinardquoMathematicalProblems in Engineering vol 2017 Article ID 1830480 14 pages2017

10 Journal of Advanced Transportation

Table6Com

putatio

nalresultson

insta

nces

with

datageneratedfro

mno

rmaldistr

ibution

set ( |119868 |

|119869 |)120572=

085120572=

09120572=

095Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2Samplingapproach

MIP-D

F1MIP-D

F2CT

Obj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

CTObj

Sel()

Sel()

Sel()

1(26)

27

907294

9964

9117

27

947844

9970

9202

2558

106

8422

10000

9208

2(412

)46

170

7961

9921

9283

48

183

8328

9982

9283

4508

183

8823

10000

9370

3(618

)68

258

7578

9916

9228

68

280

7798

9971

9417

6803

307

7658

9998

9447

4(824)

93315

8000

9899

9367

96337

8379

9979

9467

9373

368

8438

9999

9579

5(1030)

120

413

7841

9927

9437

124

443

8763

9963

9476

1262

499

8850

9998

9708

6(1236)

155

476

7921

9919

9519

162

504

8124

9982

9582

16021

536

8954

10000

9756

7(14

42)

195

573

7976

9933

9598

196

609

8287

9975

9619

2011

646

8467

9999

9767

8(1648)

232

687

8031

9918

9705

235

727

8115

9965

9650

2404

772

8278

9999

9765

9(1854)

293

692

8086

9898

9759

291

734

8464

9974

9755

29815

796

8617

9998

9835

10(2060)

356

796

7984

9937

9770

358

842

8223

9988

9765

37973

921

8846

10000

9838

11(2266

)421

829

8101

9940

9755

436

877

8211

9986

9786

4514

8959

8497

9999

9789

12(2472)

514

1023

8033

9951

9785

501

1085

8162

9983

9788

52073

1147

8321

9999

9824

13(2678)

592

1023

8138

9916

9885

599

1087

8411

9970

9845

62937

1194

8609

9998

9795

14(2884)

728

1099

8235

9913

9915

722

1168

8435

9987

9806

75053

1270

8567

9997

9834

15(3090)

855

1089

8187

9922

9861

880

1157

8376

9976

9796

89222

1300

8611

9996

9798

16(3296)

1009

1158

8770

9897

9788

966

1231

8845

9984

9823

106288

1359

8997

9999

9848

17(34102)

1154

1299

8434

9925

9849

1137

1377

8601

9980

9798

123869

1493

8798

9996

9877

18(36108)

1407

1411

8321

9927

9866

1339

1495

8301

9965

9822

133807

1601

8567

9999

9834

19(38114

)1702

1506

8206

9915

9805

1672

1598

8977

9978

9845

152999

1694

8998

9999

9847

20(40120)

2091

1635

8528

9918

9889

1916

1737

8805

9976

9837

1890

021833

9012

9999

9791

21(42126)

2843

1622

8550

9934

9913

2432

1718

8886

9974

9813

2399

131815

9008

10000

9815

22(44132)

3300

1736

8449

9940

9859

2795

1843

8786

9977

9756

253106

1948

8997

9999

9826

23(46138)

3741

1934

8133

9911

9876

3143

2050

8876

9981

9785

274583

2173

9014

9999

9858

24(48144)

4531

1961

8245

9909

9846

3515

2090

8344

9993

9819

308089

2211

8661

9999

9883

25(50150)

4715

1924

8111

9929

9888

3959

2041

8279

9996

9837

3510

932158

8527

9999

9827

26(52156)

5223

1911

8096

9898

9857

4301

2025

8251

9997

9757

421747

2146

8497

9999

9914

27(54162)

5994

2095

8387

9908

9775

4835

2226

8497

9974

9780

4614

432359

8695

9999

9837

28(56168)

6406

2111

8411

9906

9798

5336

2240

8558

9991

9820

494113

2371

9249

9999

9816

29(58174)

6897

2176

8012

9916

9791

5948

2311

8211

9984

9765

624835

2445

8531

10000

9851

30(60180)

7436

2391

8111

9913

9766

6558

2533

8305

9968

9795

696116

2682

8575

9999

9799

31(62186)

8118

2282

8119

9928

9806

7069

2422

8417

9973

9873

797768

2564

8667

9999

9846

32(64192)

8895

2519

8223

9919

9757

7857

2675

8631

9982

9769

864813

2827

8976

10000

9905

33(66198)

11235

2533

8563

9928

9811

9162

2688

8547

9979

9828

973089

2843

8657

9999

9846

34(68204)

11587

2534

8249

9926

9847

9549

2686

8798

9938

9856

10523

3041

8999

10000

9877

35(70210)

1164

92715

8098

9895

9872

10345

2880

8602

9998

9869

11272

3052

9021

9999

9897

36(72216)

12224

2825

8403

9958

9770

11165

3002

8657

9999

9839

12456

3175

9113

9999

9864

37(74222)

12978

2830

8357

9937

9795

11850

3005

8496

9998

9894

13458

3176

8827

9999

9905

38(76228)

13426

2844

8267

9966

9759

12795

3019

8604

9999

9797

15155

3121

8978

9999

9853

39(78234)

15003

2955

8303

9956

9867

13816

3132

8588

9996

9885

15041

3310

9016

9999

9912

40(80240)

16495

2978

8257

9954

9826

14896

3160

8495

9997

9837

16563

3349

8957

9999

9933

Average

4619

15855

8174

9925

9742

4077

16828

8457

9981

9743

4350

17938

8732

9999

9794

Journal of Advanced Transportation 11

high service level requirement we recommend MIP-DF1 forsolutions

6 Conclusion

This paper investigates an ambulance location problem withambiguity set of demand in which the probability distribu-tion of the uncertain demand is unknown In such a data-driven environment only the mean and covariance demandsat emergency points are knownThe problem is to determinethe base locations and the number of ambulances at eachbase aiming at minimizing the total cost associated withlocating bases and assigning ambulances We propose adistribution-free model with chance constraints Then twoapproximated MIP formulations with different approxima-tion methods of chance constraints are proposed which arebased on different ambiguity sets of the unknown probabilitydistribution Numerical experiments on benchmarks and120 randomly generated instances are conducted and thecomputational results show that the first approximated MIPformulation is much conservative with overhigh system costwhile the second approximatedMIP formulation ismore costsaving with service level appropriately ensured

In future research wemay consider the following relevantissues First the uncertainty of driving time from a base toan emergency point shall be analysed Besides unexpected orsudden disasters may be taken into consideration Moreoverother uncertainties should be considered such as travel timethe number of available vehicles and so on The stochasticdependence should also further considered

Data Availability

The data of the benchmark instances in our manuscript canbe obtained in Nickel et al [8] For the test instances thedata is randomly generated and the way for generating data isstated in our manuscript The datasets generated during thecurrent study are available from the corresponding author onreasonable request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China (NSFC) under Grants 7142800271531011 71571134 and 71771048 This work was also sup-ported by the Fundamental Research Funds for the CentralUniversities

References

[1] L Brotcorne G Laporte and F Semet ldquoAmbulance loca-tion and relocation modelsrdquo European Journal of OperationalResearch vol 147 no 3 pp 451ndash463 2003

[2] X Li Z Zhao X Zhu and T Wyatt ldquoCovering modelsand optimization techniques for emergency response facilitylocation and planning a reviewrdquo Mathematical Methods ofOperations Research vol 74 no 3 pp 281ndash310 2011

[3] C Toregas R Swain C ReVelle and L Bergman ldquoThe locationof emergency service facilitiesrdquoOperations Research vol 19 no6 pp 1363ndash1373 1971

[4] R Church and C ReVelle ldquoThe maximal covering locationproblemrdquo Papers in Regional Science vol 32 no 1 pp 101ndash1181974

[5] M Gendreau G Laporte and F Semet ldquoSolving an ambulancelocation model by tabu searchrdquo Location Science vol 5 no 2pp 75ndash88 1997

[6] A Ingolfsson S Budge and E Erkut ldquoOptimal ambulancelocation with random delays and travel timesrdquo Health CareManagement Science vol 11 no 3 pp 262ndash274 2008

[7] M Gendreau G Laporte and F Semet ldquoThemaximal expectedcoverage relocation problem for emergency vehiclesrdquo Journal ofthe Operational Research Society vol 57 no 1 pp 22ndash28 2006

[8] S Nickel M Reuter-Oppermann and F Saldanha-da-GamaldquoAmbulance location under stochastic demand A samplingapproachrdquo Operations Research for Health Care vol 8 pp 24ndash32 2016

[9] P Beraldi M E Bruni and D Conforti ldquoDesigning robustemergency medical service via stochastic programmingrdquo Euro-pean Journal of Operational Research vol 158 no 1 pp 183ndash1932004

[10] N Noyan ldquoAlternate risk measures for emergency medicalservice system designrdquo Annals of Operations Research vol 181pp 559ndash589 2010

[11] M R Wagner ldquoStochastic 0-1 linear programming underlimited distributional informationrdquoOperations Research Lettersvol 36 no 2 pp 150ndash156 2008

[12] EDelage andYYe ldquoDistributionally robust optimization undermoment uncertainty with application to data-driven problemsrdquoOperations Research vol 58 no 3 pp 595ndash612 2010

[13] J T Essen J L Hurink S Nickel and M Reuter ldquoModels forambulance planning on the strategic and the tactical levelrdquo BetaResearch School for Operations Management and Logistics p 272013

[14] E Erkut A Ingolfsson and G Erdogan ldquoAmbulance locationfor maximum survivalrdquoNaval Research Logistics (NRL) vol 55no 1 pp 42ndash58 2008

[15] S C Chapman and J A White ldquoProbabilistic formulations ofemergency service facilities location problemsrdquo in Proceedingsof ORSATIMS Conference San Juan Puerto Rico 1974

[16] A Ben-Tal D Den Hertog A De Waegenaere B Melenbergand G Rennen ldquoRobust solutions of optimization problemsaffected by uncertain probabilitiesrdquo Management Science vol59 no 2 pp 341ndash357 2013

[17] MNg ldquoDistribution-free vessel deployment for liner shippingrdquoEuropean Journal of Operational Research vol 238 no 3 pp858ndash862 2014

[18] M Ng ldquoContainer vessel fleet deployment for liner shippingwith stochastic dependencies in shipping demandrdquo Transporta-tion Research Part B Methodological vol 74 pp 79ndash87 2015

[19] R Jiang and Y Guan ldquoData-driven chance constrained stochas-tic programrdquo Mathematical Programming vol 158 no 1-2 SerA pp 291ndash327 2016

[20] C-M Lee and S-L Hsu ldquoThe effect of advertising on thedistribution-free newsboy problemrdquo International Journal ofProduction Economics vol 129 no 1 pp 217ndash224 2011

[21] K Kwon and T Cheong ldquoA minimax distribution-free proce-dure for a newsvendor problem with free shippingrdquo EuropeanJournal of Operational Research vol 232 no 1 pp 234ndash2402014

12 Journal of Advanced Transportation

[22] Y Zhang S Shen and S A Erdogan ldquoDistributionally robustappointment scheduling with moment-based ambiguity setrdquoOperations Research Letters vol 45 no 2 pp 139ndash144 2017

[23] Y Zhang S Shen and S A Erdogan ldquoSolving 0-1semidefinite programs for distributionally robust allocationof surgery blocksrdquo Social Science Electronic Publishing 2017httpsssrncomabstract=2997524

[24] F Zheng J He F Chu and M Liu ldquoA new distribution-freemodel for disassembly line balancing problem with stochastictask processing timesrdquo International Journal of ProductionResearch pp 1ndash13 2018

[25] J Cheng E Delage and A Lisser ldquoDistributionally robuststochastic knapsack problemrdquo SIAM Journal on Optimizationvol 24 no 3 pp 1485ndash1506 2014

[26] M Liu D Yang and F Hao ldquoOptimization for the locationsof ambulances under two-stage life rescue in the emergencymedical service a case study in Shanghai ChinardquoMathematicalProblems in Engineering vol 2017 Article ID 1830480 14 pages2017

Journal of Advanced Transportation 11

high service level requirement we recommend MIP-DF1 forsolutions

6 Conclusion

This paper investigates an ambulance location problem withambiguity set of demand in which the probability distribu-tion of the uncertain demand is unknown In such a data-driven environment only the mean and covariance demandsat emergency points are knownThe problem is to determinethe base locations and the number of ambulances at eachbase aiming at minimizing the total cost associated withlocating bases and assigning ambulances We propose adistribution-free model with chance constraints Then twoapproximated MIP formulations with different approxima-tion methods of chance constraints are proposed which arebased on different ambiguity sets of the unknown probabilitydistribution Numerical experiments on benchmarks and120 randomly generated instances are conducted and thecomputational results show that the first approximated MIPformulation is much conservative with overhigh system costwhile the second approximatedMIP formulation ismore costsaving with service level appropriately ensured

In future research wemay consider the following relevantissues First the uncertainty of driving time from a base toan emergency point shall be analysed Besides unexpected orsudden disasters may be taken into consideration Moreoverother uncertainties should be considered such as travel timethe number of available vehicles and so on The stochasticdependence should also further considered

Data Availability

The data of the benchmark instances in our manuscript canbe obtained in Nickel et al [8] For the test instances thedata is randomly generated and the way for generating data isstated in our manuscript The datasets generated during thecurrent study are available from the corresponding author onreasonable request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China (NSFC) under Grants 7142800271531011 71571134 and 71771048 This work was also sup-ported by the Fundamental Research Funds for the CentralUniversities

References

[1] L Brotcorne G Laporte and F Semet ldquoAmbulance loca-tion and relocation modelsrdquo European Journal of OperationalResearch vol 147 no 3 pp 451ndash463 2003

[2] X Li Z Zhao X Zhu and T Wyatt ldquoCovering modelsand optimization techniques for emergency response facilitylocation and planning a reviewrdquo Mathematical Methods ofOperations Research vol 74 no 3 pp 281ndash310 2011

[3] C Toregas R Swain C ReVelle and L Bergman ldquoThe locationof emergency service facilitiesrdquoOperations Research vol 19 no6 pp 1363ndash1373 1971

[4] R Church and C ReVelle ldquoThe maximal covering locationproblemrdquo Papers in Regional Science vol 32 no 1 pp 101ndash1181974

[5] M Gendreau G Laporte and F Semet ldquoSolving an ambulancelocation model by tabu searchrdquo Location Science vol 5 no 2pp 75ndash88 1997

[6] A Ingolfsson S Budge and E Erkut ldquoOptimal ambulancelocation with random delays and travel timesrdquo Health CareManagement Science vol 11 no 3 pp 262ndash274 2008

[7] M Gendreau G Laporte and F Semet ldquoThemaximal expectedcoverage relocation problem for emergency vehiclesrdquo Journal ofthe Operational Research Society vol 57 no 1 pp 22ndash28 2006

[8] S Nickel M Reuter-Oppermann and F Saldanha-da-GamaldquoAmbulance location under stochastic demand A samplingapproachrdquo Operations Research for Health Care vol 8 pp 24ndash32 2016

[9] P Beraldi M E Bruni and D Conforti ldquoDesigning robustemergency medical service via stochastic programmingrdquo Euro-pean Journal of Operational Research vol 158 no 1 pp 183ndash1932004

[10] N Noyan ldquoAlternate risk measures for emergency medicalservice system designrdquo Annals of Operations Research vol 181pp 559ndash589 2010

[11] M R Wagner ldquoStochastic 0-1 linear programming underlimited distributional informationrdquoOperations Research Lettersvol 36 no 2 pp 150ndash156 2008

[12] EDelage andYYe ldquoDistributionally robust optimization undermoment uncertainty with application to data-driven problemsrdquoOperations Research vol 58 no 3 pp 595ndash612 2010

[13] J T Essen J L Hurink S Nickel and M Reuter ldquoModels forambulance planning on the strategic and the tactical levelrdquo BetaResearch School for Operations Management and Logistics p 272013

[14] E Erkut A Ingolfsson and G Erdogan ldquoAmbulance locationfor maximum survivalrdquoNaval Research Logistics (NRL) vol 55no 1 pp 42ndash58 2008

[15] S C Chapman and J A White ldquoProbabilistic formulations ofemergency service facilities location problemsrdquo in Proceedingsof ORSATIMS Conference San Juan Puerto Rico 1974

[16] A Ben-Tal D Den Hertog A De Waegenaere B Melenbergand G Rennen ldquoRobust solutions of optimization problemsaffected by uncertain probabilitiesrdquo Management Science vol59 no 2 pp 341ndash357 2013

[17] MNg ldquoDistribution-free vessel deployment for liner shippingrdquoEuropean Journal of Operational Research vol 238 no 3 pp858ndash862 2014

[18] M Ng ldquoContainer vessel fleet deployment for liner shippingwith stochastic dependencies in shipping demandrdquo Transporta-tion Research Part B Methodological vol 74 pp 79ndash87 2015

[19] R Jiang and Y Guan ldquoData-driven chance constrained stochas-tic programrdquo Mathematical Programming vol 158 no 1-2 SerA pp 291ndash327 2016

[20] C-M Lee and S-L Hsu ldquoThe effect of advertising on thedistribution-free newsboy problemrdquo International Journal ofProduction Economics vol 129 no 1 pp 217ndash224 2011

[21] K Kwon and T Cheong ldquoA minimax distribution-free proce-dure for a newsvendor problem with free shippingrdquo EuropeanJournal of Operational Research vol 232 no 1 pp 234ndash2402014

12 Journal of Advanced Transportation

[22] Y Zhang S Shen and S A Erdogan ldquoDistributionally robustappointment scheduling with moment-based ambiguity setrdquoOperations Research Letters vol 45 no 2 pp 139ndash144 2017

[23] Y Zhang S Shen and S A Erdogan ldquoSolving 0-1semidefinite programs for distributionally robust allocationof surgery blocksrdquo Social Science Electronic Publishing 2017httpsssrncomabstract=2997524

[24] F Zheng J He F Chu and M Liu ldquoA new distribution-freemodel for disassembly line balancing problem with stochastictask processing timesrdquo International Journal of ProductionResearch pp 1ndash13 2018

[25] J Cheng E Delage and A Lisser ldquoDistributionally robuststochastic knapsack problemrdquo SIAM Journal on Optimizationvol 24 no 3 pp 1485ndash1506 2014

[26] M Liu D Yang and F Hao ldquoOptimization for the locationsof ambulances under two-stage life rescue in the emergencymedical service a case study in Shanghai ChinardquoMathematicalProblems in Engineering vol 2017 Article ID 1830480 14 pages2017

12 Journal of Advanced Transportation

[22] Y Zhang S Shen and S A Erdogan ldquoDistributionally robustappointment scheduling with moment-based ambiguity setrdquoOperations Research Letters vol 45 no 2 pp 139ndash144 2017

[23] Y Zhang S Shen and S A Erdogan ldquoSolving 0-1semidefinite programs for distributionally robust allocationof surgery blocksrdquo Social Science Electronic Publishing 2017httpsssrncomabstract=2997524

[24] F Zheng J He F Chu and M Liu ldquoA new distribution-freemodel for disassembly line balancing problem with stochastictask processing timesrdquo International Journal of ProductionResearch pp 1ndash13 2018

[25] J Cheng E Delage and A Lisser ldquoDistributionally robuststochastic knapsack problemrdquo SIAM Journal on Optimizationvol 24 no 3 pp 1485ndash1506 2014

[26] M Liu D Yang and F Hao ldquoOptimization for the locationsof ambulances under two-stage life rescue in the emergencymedical service a case study in Shanghai ChinardquoMathematicalProblems in Engineering vol 2017 Article ID 1830480 14 pages2017

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