On the integration of dispatching and covering for emergency vehicles management system

Sarah Ibri Département d’informatique, Université Hassiba Benbouali de

Chlef, UHBC Chlef, Algérie.

sarah.ibri.1@ulaval.ca

Mustapha Nourelfath Centre Interuniversitaire de Recherche sur les Réseaux

d’Entreprise, la Logistique et le Transport, CIRRELT Québec, Canada.

Mustapha.nourelfath@gmc.ulaval.ca

Habiba Drias Département d’informatique, Université des Sciences et de la Technologie Houari Boumediene,USTHB,

Alger, Algérie. hdrias@usthb.dz

Abstract—In the vehicle management systems, two important issues have to be considered by managers: the dispatching problem with the aim of minimizing the response time for current emergency calls; and the covering problem with the objective of keeping proper coverage to satisfy future calls in best times. The purpose of this paper is to investigate the value, in term of service quality, of integrating the dispatching and covering problems in the same model. A heuristic algorithm, combining Ant optimization and Tabu search, is used as a solution approach. Several numerical examples are used to compare the integrated approach and the non-integrated one.

Keywords-emergency vehicles management; dispatching and covering problems; ant colony optimization; tabu search; real-time simulation.

I. INTRODUCTION The most important objective of the decision makers in

emergency response services (ERS) is to save human lives by making quick decisions to provide emergency vehicles in the shortest times to the emergency arriving calls. An obvious solution may consist in assigning the nearest available vehicles to calls, but it is far to be the best solution, because in long term and with a limited fleet of emergency vehicles, the quality of service becomes unsatisfactory, especially when inter-arrival time of calls is short. For these cases, another objective as important as minimizing the response time is introduced by the emergency calls management centres for public; it is the covering that consists of relocating available vehicles to allow for adequate coverage of the region under surveillance. To keep proper coverage, strategies used in practice include diversion, relocation and rerouting. Diversion means that dispatched vehicles on route are allowed to switch to a new emergency call if it is more severe. Relocation means that the idle vehicles may be relocated in order to maintain proper coverage for future demands. Rerouting means that vehicles are allowed to change the route to destinations based on real traffic information.

The emergency vehicle dispatching problem is solved in some studies either by static dispatching strategies like the first come first served, nearest origin or highest priority first served [9]. In [8] it has been formulated by a model that minimizes the total travel time in the system. In [1] the solver decides on the vehicle to dispatch according to the priority of the call: for the highest priority calls, the vehicle with the shortest travel time is dispatched, otherwise the vehicle that affects at least the covering issue is chosen. To avoid making calls with less priority waiting for long time, they use pseudo priorities that are incremented when a call is not served in some time limit. This problem can also be modelled as a Generalized Assignment Problem (GAP) when considering different types of vehicles and urgency priorities. It is solved by exact solvers of GAP [5][12] that have the inconvenient to be time-consuming, or by heuristic and meta-heuristic methods, such as genetic algorithms and tabu search [2].

For the covering problem many models were proposed. The most important static models are the “LSCM: Location Set Covering Model” proposed in [14] which minimizes the number of vehicles to cover all demand points. In the MCLP: Maximal Covering Location Problem [3] the aim is to maximize population coverage subject to limited vehicle availability. The static model in [4, 10] maximizes the number of points covered more than once using hierarchical objective. Other static models that maximize coverage with several types of vehicles at the same time are proposed in [13]. In [7] two coverage standards r1<r2 are used, all the demand points must be covered within r2 time units and a proportion α of population must be covered within r1 time units.

When vehicles are dispatched to calls, some points become uncovered; this issue is ignored in the static models. In dynamic models, relocation decisions must be periodically made to keep proper areas coverage. In [11] a relocation system was proposed for fire companies.

To meet the real time requirements of the relocation problem and accelerate the decisions making, authors in [7], proposed a

198978-1-4244-8611-3/10/$26.00 ©2010 IEEE

dynamic model solved in parallel for real time ambulance relocation.

In [7] the two problems are not properly integrated, but a solution methodology is developed by taking advantage of the available time between consecutive calls by anticipating future decisions on the redeployment of the fleet. More precisely, for each available ambulance, their approach consists in pre-computing the relocation decisions associated with the possible assignment of this ambulance to the next incoming call. Thus, when this call actually occurs, an ambulance is assigned according to the rules described previously and the pre-computed redeployment scenario is applied.

As far as we know, few studies have been dedicated to integrate the two problems in the same model. In [9], the authors propose a simulation model to the management of emergency vehicle fleet. They emphasized the integration and exchange of information across public safety and transportation agencies; and coordinate the activities of three public safety services namely; fire protection agencies, police and paramedics services. They deal with the covering problem by considering nodes one by one; a node is considered covered if there is at least one vehicle that can reach it in critical time.

The real-time emergency vehicle dispatching problem studied in this paper has a dynamic aspect which is present by the continual changing of the vehicle’s states (available, on route, in service), their positions and the asynchronous arrival of emergency calls. The solving method should adapt to this dynamism: the destination of vehicles for example can change at any time if their diversion improves the quality of service, especially the response time.

In this paper we investigate the effect and the value of integrating dispatching and covering problems on the quality of service. For this purpose we use a simulator and we compare the approach of solving dispatching followed by covering to the approach of solving them simultaneously.

Unlike in [9] in this work we consider that each zone contains a set of predefined nodes that represent potential calls and has a quantitative measure called “preparedness” that represent the covering rate of all the nodes that belong to this zone. This measure is function of the demand of the zone and its distance from the vehicles. A zone is considered covered if its preparedness is greater than a predefined rate. Regulating this rate has to be done in a tactical or strategic level based on some prognostic data as zone’s requirements, available vehicles and its population density.

The proposed solving method designed according to the above purposes is an efficient heuristic algorithm that couples two meta-heuristics, namely ant colony optimization and tabu search that we call “Ant-Tabu”.

The remainder of this paper is organized as follows. In Section 2, the mathematical model of the problem is presented. In Section 3 the proposed heuristic approach is detailed. In Section 4 we present the numerical results. Conclusions are drawn in Section 5.

II. MATHEMATICAL MODEL

A. Problem S tatement The problem under study is characterized by the

following practical constraints and assumptions:

• We consider a centre responsible of receiving all the emergency calls and controlling the movement of all the vehicles in the system. The region managed by this centre contains a set of stations geographically dispersed; each one has some emergency vehicles.

• When an emergency call arrives, the centre should make decisions about the vehicle to send to assist this call.

• The region under control is composed of zones from where emergency calls arrive. To respond to calls in best times; the centre has to control the assignment of vehicles to sites in order to guarantee a good covering rate of all the zones. The covering rate of a zone is given by its preparedness; it indicates how efficient the system is prepared to respond to an emergency call coming from this zone.

• A vehicle at station is considered idle, when it is on route toward a call/station it can be deviated; when it arrives to a call it should finish service to become available.

• To avoid reassigning (deviating) the same vehicle many times or deviating too much vehicles (because this can perturb the emergency crews and the caller), a limit is imposed by accepting only reassignments that bring in an important saving of time or covering to the system.

To describe the problem mathematically, the following notations are introduced.

B. Notations

Sets V set of emergency vehicles V1 set of idle vehicles (that are at stations) V2 set of vehicles that are moving to an

emergency call V3 set of vehicles that are servicing an

emergency V4 set of vehicles that are moving to station S set of stations W set of emergencies waiting for service Z set of zones

Indices j index of vehicles s index of stations I index of emergencies z index of zones

Parameters

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Prioi priority of emergency i Tij(t) predicted time to vehicle j to reach

emergency i while departing at time t Tjs(t) predicted time to vehicle j to reach

station s while departing at time t Tjz(t) predicted time to vehicle j to reach zone

z while departing at time t Tsz(t) predicted travel time from station s to

zone z at time t Cz weight representing the need of zone z ϕ required coverage rate τ rate of allowed deviations M a large number

Penal1 penalty applied when an emergency call is not satisfied

Penal2 penalty applied when a zone is not covered

Penal3 penalty applied when a vehicle is deviated from its destination

{ steplast at iemergency toassigned wasj vehicleif 1

otherwise 0 0 =jiX

{ steplast at sstation toassigned wasj vehicleif 1otherwise 0 0 =jsX

{ steplast at satisfiednot wasiemergency if 1otherwise 0 0 =iunsat

⎩⎨⎧ <= steplast at pr if 1

otherwise 0cov z0 ϕzun

Obj1 total travel time toward emergency calls at time t Obj2 total travel time toward stations at time t Obj3 number of unsatisfied emergencies Obj4 number of uncovered zones Obj5 number of deviated vehicles

Decision Variables PRz(t) = preparedness of zone z at time t

⎩⎨⎧= tat time iemergency toassigned is j vehicleif 1

otherwise. 0)(tjiX

⎩⎨⎧= tat time sstation toassigned is j vehicleif 1

otherwise. 0)(tjsX

⎩⎨⎧= tat time satisfiednot is iemergency if 1

otherwise. 0)(tiunsat

⎩⎨⎧ <

= )(zPR if 1

otherwise. 0)(cov

ϕttzun

⎩⎨⎧= tat time deviated is j vehicleif 1

otherwise. 0)(tjdev

C. The Model The aim of the proposed model is to find an assignment of

vehicles to emergencies and stations that minimizes the sum of the total travel time at time t to prioritized emergencies (Obj1), the total travel time at time t to stations (Obj2), the number of unsatisfied emergencies (Obj3), the number of uncovered zones (Obj4) and the number of deviated vehicles (Obj5). The

following equations express the terms of the objective function:

(1) )

iPrio )( )((Obj1

3∑ ∑∉ ∈

=Vj Wi

tji

Ttji

X

(2) ))( )((Obj23

∑ ∑∉ ∈

=Vj Ss

tjs

Ttjs

X

(3) )(1Obj3 ∑∈

=wi

ti

unsatPenal

(4) )(cov2Obj4 ∑∈

=Zz

tz

unPenal

(5) )(3Obj5 ∑∈

=Vj

tj

devPenal

The model minimizes the objective function (Obj1 + Obj2 + Obj3 + Obj4 + Obj5) subject to the constraints (C1 to C8) in Table1. The first constraint (C1) in Table 1 states that each vehicle should be assigned to one and only one location (emergency or station) at a time. Constraint (C2) state that at most one vehicle is assigned to each emergency call. Constraint (C3) serves to determine if an emergency is satisfied or not. (C4) and (C5) enable to determine if a zone is covered. (C6), (C7) and (C8) control the vehicles’ deviation. They allow only the deviations that bring in a gain higher than τ.

III. A HYBRID SOLVING APPROACH Hybrid methodologies have shown their efficiency for

many applications these last years. They are often designed to take into account the advantages of the combined methods and get rid from their drawbacks. The solution we propose for the problem in study is a hybrid algorithm based on two well recognized meta-heuristics that are Ant Colony Optimization and Tabu search. It is outlined as follows:

The following pseudo-random-proportional rule is used by ants to build solutions: If q<=q0 then:

{ })6(

else 0

],[],[argmax )( if 1)(

⎪⎩

⎪⎨⎧ =

=βα ljheurljpheroj,l

tljP

AnTabu()For nb_generations do For nb_ants do

s=build_solution(); Online_Delayed_Phero_Update(s);

Enddo s*=Select the best solution (); Tabu_search(s*) Offline_Phero_Update(s*); Enddo

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TABLE 1. THE MODEL CONSTRAINTS.

else

(7) ],[],[

],[],[)(

)(∑∈

=

tVv

lj lvheurlvphero

ljheurljpherotP βα

βα

q: a random variable uniformly distributed over [0,1]. q0: a tuneable parameter ∈[0,1]. phero[j,l]: pheromone quantity of the assignment (j,l). heur[j,l]: heuristic of assigning vehicle j to location l. α ,β: parameters witch control respectively the pheromone and the heuristic balance. V(t): set of available and deviable vehicles at time t. In the tabu search procedure two types of operators are used for neighbourhood evaluation: Swap(vi,vj) that exchanges the locations of vehicles vi and vj. Move(v,l) that moves vehicle v from its current location to location l. Swap operator is used to build the neighbourhood in the main procedure, whereas the second is used in the intensification function. The intensification phase starts when the number of iterations without improvement of the solution reaches some limit. After an intensification phase, a diversification is applied by choosing the less recently used movements and so directing the search to new regions of the space.

• best_s: the best solution found from the beginning of the algorithm

• current_s: the current position of the search process • neighbourhood(current_s): the procedure that return

the best neighbour of current_s which is not tabu or that satisfies the aspiration criterion

• no_improve: number of successive iterations without improvement of best_s

• update_t_length(): the procedure that updates the tabu list length

• max_no_improve: the maximum number of iterations without improvement before starting the intensification process.

∑ ∑ =+i s

jsji tXtX 1)()( 3 Vj ∉∀ (C1)

1)( ≤∑j

ji tX Wi ∈∀ (C2)

unsatMtX ijiVj

(t))(13

⋅≤− ∑∉

Wi ∈∀ (C3)

)(cov)( tunMtPR zz ⋅≤−ϕ Zz ∈∀ (C4)

∑∑⋅=1

)/)((/1)(j

szjss

zz ttXCtPR Zz ∈∀ (C5)

)()()( 0 tdevMXtXtX jjijiji ⋅≤⋅− WVj ∈∪∈∀ i ,V 42 (C6)

)()()( 0 tdevMXtXtX jjsjsjs ⋅≤⋅− SsVVj ∈∪∈∀ , 42 (C7)

))()(()( 00∑ ∑∪∈ ∪∈

+⋅−⋅SWl SWl

jljljljl tttXtX

))(( .1i

0 +−∑ ∑i

ii tunsatunsatpenal

)())(covcov.(2 0 tdevMtununpenal jz z

zz ⋅≤−−∑ ∑ τ

42 VVj ∪∈∀ (C8)

Tabu search procedure

Input: S.

Output: best_s.

(a) best_s=S; no_improve=0;

current_s=S;

(b) if (stopping criterion==true)

then return(best_s).

(c) current_s=neighbourh(current_s)

(f) update_t_length();

(g) if(cost(current_s)>cost(best_s))

then no_improve++;

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IV. EXPERIMENTS For validation purposes, we developed a set of benchmark

instances for tests since no benchmarks are available for the problem. All data: emergency calls, stations and zones correspond to real addresses and are geo-localized on real map (a Switzerland map) with real street network. Geo-localization, itinerary, distances and travel times are computed using MapPoint tool. Each instance is composed of a:

• Set of vehicles, each one has a station as initial position.

• Set of stations, a station is characterized by its position on the map.

• Set of vehicles, each one has a station as initial position.

• Set of stations, a station is characterized by its position on the map.

• Set of zones, each one has a position on the map (the centre of the zone) and a value representing its demand (number of calls possibly coming from this zone) that serves to compute the preparedness.

• Set of calls, each one is characterized by its position on the map, the zone to which it belongs, its priority, arrival time and service time.

Two data sets are tested. In the first one average arrival time is set to 30 and in the second to 15. In this section we study the impact of integrating dispatching and covering on the system performance. For this purpose we compare two approaches: 1. Solving dispatching and covering consecutively:

dispatching first (minimizes the sum: Obj1+ Obj3 + Obj5), the obtained solutions is used to solve the covering problem next (minimize the sum: Obj2 + Obj4 + Obj5).

2. Solving dispatching and covering simultaneously. Results in Figure1 and Figure 2 show that the first approach, namely approach1 gives better response time especially in the case of dataset2.

Fig. 1. Response Time Comparison for Dataset1

0

5

10

15

20

25

30

Ins1

Ins4

Ins7

Ins11

Ins14

Instance

Res

pons

e tim

e (M

inut

es)

Approach1Approach2

Fig. 2. Response Time Comparison for Dataset2.

In Table 2 we report the percentage of the zones that become uncovered during the simulation process (column1: %uncov) and the average of cumulative time that zones stay uncovered during the simulation even discontinuously (column 2: uncov_time) These results show that coverage is better maintained by the second approach called approach2 since for most instances either the percentage of uncovered zones (column1) is less than approach1 or the average time that zones stay uncovered (column2) is shorter. Another additional result shown in table3 is that the above results are in 50% of cases obtained using less vehicles by approach2 in the other cases the same number is used by the two approaches. We should note that the coverage criterion is as important as the response time because when zones are well covered, even if the solution of the model gives bad response time, human

0

5

10

15

20

25

30

Ins1

Ins5

Ins9

Ins13 Instance

Res

pons

e tim

e (M

inut

es)

Approach1

Approach2

(h) if(no_improve==max_no_improve)

then

no_improve=0;

current_s=Intensif(current_s);

if(cost(current_s)<cost(best_s))

then best_s=current_s;

diversif(current_s);

(i) go to (b)

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decision makers will still have choice to select another vehicle to respond to critical call if they decide so (they also can change the parameters values to favour response time on

coverage when needed), but with bad covering, they will not even find such vehicle.

TABLE 2. NUMERICAL RESULTS ON UNCOVERED ZONES AND TIME OF UNCOVERING TABLE 3. NUMERICAL RESULTS PERCENTAGE OF USED VEHICLES.

V. CONCLUSION In this paper, the value of integrating vehicle allocation

and covering issues in the same model was investigated. The model is solved using an algorithm that takes advantage of both ant colony optimization and tabu search meta-heuristics. The obtained results show that by solving the two problems simultaneously zone coverage is better controlled using less number of vehicles. However, by solving them consecutively, we give priority to the dispatching problem over the covering problem which, according to experimental results, produces better response times with less coverage. In the near future, we plan to study the problem with multi-objective optimization techniques hoping to increase the system performance.

REFERENCES

[1] T.Andersson.,P.Varbrand “Decision support tools for ambulance dispatch and relocation” Journal of the operation research society, 2006 pp. 1-7

[2] P.C Chu, J.E Beasley “A genetic algorithms for the generalized assignment problem”, operations research 1997 24 (1), pp. 17-23.

[3] R.L Church, C.S ReVelle. “The maximal covering location problem”. Papers for regional science association. 1974:32 pp. 101-118.

[4] M.S Daskin, E.H Stern. “A hierarchical objective set covering model for emergency medical service vehicle deployment” Transportation Science 1981, 15 pp. 135-152

[5] M.L.Fisher, R.Jaikumar, L.N.Wassenhove. “A multiplier adjustment method for the generalized assignment

Dataset1 Dataset2 Approach1 Approach2 Approach1 Approach2

Inst %uncov uncov_time %uncov uncov_time %uncov uncov_time %uncov uncov_time 1 0% 0 0% 0 0% 0 0% 0 2 60% 270.66 60% 270.66 80% 106.74 80% 106.74 3 0% 0 0% 0 33% 41.0 33% 41.0 4 20% 17.0 20% 2.0 40% 85.5 40% 85.5 5 0% 0 0% 0 30% 42.33 20% 29.5 6 20% 576.99 20% 576.99 80% 124.99 60% 153.33 7 20% 81.0 20% 81.0 20% 264.0 20% 255.0 8 0% 0 0% 0 0% 0 0% 0 9 30% 126.0 20% 22.0 60% 124.45 60% 88.5

10 0% 0 0% 0 20% 22.0 10% 87.0 11 0% 0 0% 0 10% 84.0 10% 3.0 12 10% 628.99 10% 623.99 30% 170.66 30% 160.49 13 0% 0 0% 0 0% 0 0% 0 14 0% 0 0% 0 0% 0 0% 0 15 33% 617.19 20% 614.57 20% 511.47 20% 499.66 16 10% 878.99 10% 636.92 10% 525.995 10% 501.99

Data Set1 Data Set2 Inst Approach1 Approach2 Approach1 Approach2

1 50% 37% 62% 62% 2 80% 80% 90% 80% 3 66% 66% 66% 66% 4 53% 46% 60% 60% 5 53% 53% 60% 53% 6 86% 86% 93% 86% 7 40% 35% 55% 50% 8 36% 36% 44% 40% 9 52% 48% 60% 56% 10 40% 40% 46% 46% 11 63% 56% 80% 70% 12 37% 34% 40% 37% 13 34% 34% 45% 45% 14 40% 40% 42% 42% 15 60% 60% 55% 55% 16 40% 35% 48% 44%

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problems”. Management science, 1986 32, pp. 1095-1103 [6] M.Gendreau, G.Laporte, F.Semet. “The Maximal

expected coverage relocation problem for emergency vehicles”. Journal of Operation Reasearch Society, 2006 pp.22-28.

[7] M.Gendreau,G.Laporte,F.Semet. "A dynamic model and parallel tabu search heuristic for real time ambulance relocation”. Parallel computing 2001, 27, pp. 1641-1653.

[8] A. Haghani, H. Hu, and Q. Tian, “An Optimization Model for Real Time Emergency Vehicle Dispatching and Routing”, Presented at the 82nd annual meeting of the Transportation Research Board, Washington, D.C 2003.

[9] A.Haghani, S.Yang. “Real time emergency response fleet deployment concepts, systems, simulation and case studies”. Operations research/computer science interfaces series volume 2007, 38 pp.133-162.

[10] Hogan, C.S ReVelle. “Concepts and. applications of backup coverage” Management science 1986, 34. pp.1434-1444.

[11] P.Kolesar, W.E walker, “An algorithm for the dynamic relocation of fire companies” Operation research 1974,22 pp.249-274.

[12] G.T Ross, M.S Soland. “A branch and bound algorithm for the generalized assignment problem” Mathematical programming 1975,PP. 91-103

[13] D.A Schilling, D.J Elzinga, J.Cohon, RL Church, C.S ReVelle. ‘The TEAM/FLEET models for simultaneous facility and equipment siting’ Transportation Science. 1979, 13 pp.163-175.

[14] C.R Toregas, R.S Wain, C.S ReVelle, L.Bergman “The location of emergency service facilities” Operation Reseach 1971,19 pp. 1363-1373.

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