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ScienceDirect Procedia - Social and Behavioral Sciences 96 (2013) 671 – 682

13th COTA International Conference of Transportation Professionals (CICTP 2013)

A Chance Constrained Programming Model for Reliable Emergency Vehicles Relocation Problem Yang Liua,*,Yun Yuana,Yu-hui Lia,Hao Panga a

College of Engineering, Nanjing Agricultural University, No.40,Dianjiangtai Rd., Pukou Dist., Nanjing 210032,P.R.China

Abstract Emergency vehicles relocation is one mechanism of increasing preparedness for potential emergencies. This paper addresses the problem of designing reliable emergency vehicles relocation system. Under this respect, we extend the DYNACO model with chance-constrained programming framework for the optimal redeployment of emergency vehicles. The model deals with the availability of emergency vehicles by approximate hypercube. In addition, other random elements including travel time and emergency demand are taken into account in the model. Solution procedure based on genetic algorithm and Monte-Carlo simulation is developed to solve the stochastic model. Computational experiences are reported to illustrate the performance and the effectiveness of the proposed solution. © TheAuthors. Authors. Published by Elsevier B.V.access under CC BY-NC-ND license. © 2013 2013 The Published by Elsevier Ltd. Open Selection and/or peer-review under responsibility ChineseTransportation Overseas Transportation Association (COTA). Selection and peer-review under responsibility of ChineseofOverseas Association (COTA). Keywords: Location; Emergency Response; Reliability; Stochastic programming; Queueing

1. Introduction Relocation problem of emergency response vehicles (ambulances, fire trucks, and police cars etc.), also known as redeployment or dynamic repositioning, calls for the optimal redistribution of existing idle resources or units to serve future emergencies to ensure adequate coverage throughout the area of responsibility in a short time (hours or real time) (Sajjadian, 2009).It's a flexible and adaptable measure in emergency response resources deployment. However, the technical hurdles, including electronic maps, order-handling systems, vehicle tracking systems, and other communication systems, have limited the research in this area until the past two decades. Besides, most of relocation models are NP-hard and suffer from excessive computational burdens. To date, the development of powerful local search algorithms, particularly meta-heuristic search, coupled with the increasing affordability of

* Corresponding author. Tel.: +86-25-58606570; fax: +86-25-58606570. E-mail address: [email protected]

1877-0428 © 2013 The Authors. Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and peer-review under responsibility of Chinese Overseas Transportation Association (COTA). doi:10.1016/j.sbspro.2013.08.078

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Yang Liu et al. / Procedia - Social and Behavioral Sciences 96 (2013) 671 – 682

computing power, have finally created ideal conditions for bringing the emergency response vehicles redeployment research to fruitful implementation. Relocation problem derives from the location one. The location problem, one of the most important, and studied, combinatorial optimization problems, has been studied in various forms over the past 40 years. Early models are drawn from static and deterministic formulations, which can be categorized as median and coverage models. Most public facility location models use one of these approaches or a combination of both to set the foundations of the formulation (Marianov and Serra, 2002). Detailed review of emergency service location problems are referred to Goldberg (2004), Marianov and Revelle (1995), Brotcorne et al. (2003), and Li et al. (2011). While the location problems are usually solved at the strategic level, redeployment problems are mainly operational and solved dynamically in a short time. An early relocation model was proposed by Kolesar and Walker (1974) for fire companies. Gendreau et al. (2001) dynamically maximized the total demand covered at least twice, minus a penalty term to reflect the change from the current state of the system. The model was applied in real-time through the use of tabu search algorithm and parallel-computing. A similar model for physician cars is presented by Gendreau et al. (2006). Sathe and Miller-hooks (2005) studied the relocation of first-response units (e.g., military units, police forces) in order to maintain protection coverage to critical facilities under disaster conditions. The problem was formulated with two objectives (maximize secondary coverage and minimize cost). Schmid and Doerner (2010) developed a multi-period model taking into account time-varying travel time. Haghani and Yang (2007) proposed a model integrated with dispatching, routing and relocation. Schmid (2012) presented a model to solve the ambulance dispatching and relocation decisions using appropriate dynamic programming (ADP). However, to the authors' best knowledge, reliable emergency vehicles relocation model has not been addressed in the literature. Availability of vehicles is a key component in emergency vehicle relocation problem. Broadly speaking, five alternatives for evaluating the availability of emergency vehicles (Budge et al.,2009): (1) exact hypercube model, such as Larson (1974); (2) discrete-event simulation, e.g., Goldberg et al.(1990); (3) approximate hypercube (AH) model, such as Larson (1975); (4) queuing model, e.g., Restrepo et al.(2009), and (5) busy possibility, e.g., Sorensen and Church (2010). The hypercube model, proposed by Larson (1974) and studied by several authors, Geroliminis et al. (2009), Iannoni et al. (2008), and Takeda et al. (2007), is a notable milestone for evaluating emergency service systems. Compared to exact approaches, the advantages of the various versions of the AH models (Jarvis, 1985; Budge et al., 2009; Baptista and Oliveira, 2012) are that they are fast, with computation times that are relatively insensitive to system characteristics, and they are sufficiently accurate for many practical purposes. The paper aims to fill this gap by developing a reliable emergency vehicle relocation design framework. We are here using chance-constrained programming method, a branch of stochastic programming method, with probabilistic constraints aimed to achieve a reliable level of service. The model developed here is an extension to the dynamic ambulance relocation model (DYNACO), which was originally proposed by Andersson and Värbrand (2007). The character of this model is the preparedness measure for emergency vehicle relocation problem. Lee (2011) proposed a dispatching algorithm for ambulances based on the preparedness measure. Approximate hypercube model is utilized to accurately compute emergency vehicles dispatching probabilities. The model also concerns the randomness elements of the system including travel time and emergency rescue demands. Solution method integrated genetic algorithm with stochastic simulation is used to solve the relocation model with increased accuracy and efficiency. The case study in Shanghai, the biggest city in China, demonstrates the validity of the proposed method.


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Our contribution is placed in two respects. Firstly, we propose a stochastic programming model under probabilistic constraints aimed at solving the reliable emergency response vehicles relocation problem. The stochastic programming methodology is particularly effective to deal with uncertainties. Secondly, the preparedness measure presented by Andersson and Värbrand (2007) are modified with approximate hypercube model. It brings a new quantitative measure for emergency management system, especially convenient for priority dispatching rule. The remainder of this paper is organized as follows: in the next section, the model is presented and analyzed. In Section 3, the algorithm is described. This is followed by a computational experiment in Section 4 and by some conclusions in Section 5. 2. Model formulation 2.1. Notation We use the following notation.

J -the set of emergency rescue zones j -the index of emergency rescue zones, j 1,..., m

K -the set of emergency vehicles k -the index of emergency vehicles, k 1,..., n -the arrival rate of emergency generated from zone

j

j

N j -the set of closest emergency vehicles contributing to the preparedness for zone j L j -the number of closest emergency vehicles contributing to the preparedness in zone j t jk -the travel time for emergency vehicles k to reach zone j k -the contribution factor of emergency vehicles k and the following properties hold: t j1 t j 2 t jL j 1

2

Lj

-the average system busy probability k l j

-the utilization of vehicle

k

-the busy fraction for the l

th

preferred vehicle for node

j

Q(n, , L j 1) -correction factor to relax the assumption that servers are independent f jk -the probability that emergency from zone j is responded by vehicle k p j -the preparedness of zone j pmin -the lowest level of preparedness , -the predetermined confidence levels -the coefficient of target

P -the number limitation of relocation allowed

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P0 -probability that all emergency vehicles are idle Pn -probability that all servers are busy 0,1 x jk -the decision variable, x jk -the vector form of x jk

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2.2. The preparedness measure Definition One of the most important features of any relocation model is the measure of service quality pursued by the decision makers. Especially in a geographic information system, it is useful of checking the measure in the zones automatically, and suggests ways to relocate emergency vehicles ensuring a sufficient service level for potential emergencies. In order to find a quantifiable measure for emergency response ability, Andersson and Värbrand (2007) Geographical area is partitioned into sets of zones, the preparedness for zone is calculated as Eq.(1):

pj

1

Lj

j l 1

l

t jl

(1)

The preparedness function weakness lies in the fact that they fail to take into consideration the probabilistic nature of the emergency vehicles. Here we present new expression which embedded with approximate hypercube, describing the probabilities of servers being busy, and the stochastic nature of emergency vehicles arriving at the emergency rescue demand zones. The dispatch probabilities may be defined as Eq.(2):

f jk

Pr vehicle k responds|emergency from zone j

(2)

The probability that a particular emergency will find busy the first L j 1 servers chosen, and the Lthj server idle may be defined as Eq.(3): Lj X

f jk

1

Q n, , L j X

l j

1

(3)

k

l 1

Q n, , L j X 1 is the correction factor relaxing the assumption that servers are independent. We adopt the formulation presented by Budge et al. (2009), Galvão et al (2005), Rajagopalan and Saydam (2009). Considering a general M/M/N/loss system, the correction factor can be expressed as Eq.(4) (Budge et al,2009):

Q ( n, , L j )

n! 1

P0 1 Pn

n Lj ! 1 Pn

Lj 1

The preparedness measure can be described as Eq.(5):

n 1 u Lj 1

u Lj 1

n u nu u Lj 1 !

(4)

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pj

l

Lj

1

f jl

(5)

t jl

j l 1

Thus, if the dispatching probability, that is, f jk , increases, the preparedness increases. On the contrary, the preparedness is inverse of travel time to the zone t jk and the emergency rescue demand

j

.

2.3. Formulation With the requirement of considering randomness, three types of stochastic programming, expected value model, chance-constrained programming, and dependent-chance programming, have been developed to suit the different purposes (Liu, 2009). Chance-constrained programming (CCP) was pioneered by Charnes and Cooper (1959) as a means of handling uncertainty by specifying a confidence level at which it is desired that the stochastic constraint holds. The model here is formulated by chance-constrained programming framework.

max f x

(6)

Subject to:

Pr{

pj

t jk x jk

j J

x jk

f}

j J k K

1

k

(7)

K

j J

(8)

x jk

P

(9)

k K j J

Pr

1

Lj

j l 1

X

0,1

l

f jl

t jl ( )

pmin

j

J (10) (11)

The twin challenge facing emergency vehicles relocation decision is to maximize service quality while minimizing operating costs. How to ensure a trade-off solution between effect and costs is the main concern in establishing a relocation model. In this paper, the quantification of the service can be expressed in terms of the preparedness measurement, and the cost challenge may be written as transportation costs as a proxy. The objective (6) of the model is to maximize the optimistic return with a given confidence level subject to some subtract the cost, the cost is described as the sum of travel times t jk ,which is the time required for emergency vehicle k to reach zone j .The variable x jk equals 1 if emergency vehicle k is relocated to zone j , namely, the time it will take until the preparedness is at least pmin in all zones, which is required in Eq.(10).Constraint (8) shows that each of the emergency vehicle can be relocated to at most one zone. Constraint (9) ensures that the relocation amount less than the limitation allowed. Constraint (10) ensures the probability of the acceptable


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preparedness is more than .Although the demand prediction over short time is more accurate than over long times, it can be predicted under some degree of certainty, but not precisely. So the demand is described as a stochastic variable j .In constraint (10), t jl ( ) is a function of the variable , which is the vector form of x jk . Naturally, the travel time for the l th closest ambulance to zone j , that is, t jl , depends on where the ambulances are located, which is decided by the values on the variable . Besides, the travel time is influenced by traffic conditions, so it should be described as stochastic variable. 3. Solution approach 3.1. Algorithm Framework Knowing that location problems are typically NP-completed problem, relocation problem is easily proven to be NP-completed (Saydam et al, 1994). This means that one is unable to solve relocation problem to optimality to moderate-sized or large sized problems. The DYNAROC model formulated by Andersson and Värbrand (2007) is solved by tree search algorithm, that is, starting with the current situation, the heuristic iteratively tries to raise the preparedness in the zone with the lowest preparedness. By setting the search circle low, the set of feasible solutions will be small. This will however also decrease the number of feasible solutions to the model, with the risk that there exists no solution on some instances. Therefore, one should resort to heuristics and meta-heuristics to solve the problem to near-optimality. A good way is to design some hybrid intelligent algorithms for solving them. Genetic algorithm approach is developed to solve the proposed formulation. Genetic Algorithms(GA) were first introduced by Holland (1975) and described as stochastic search methods based on the process of natural evolution (Goldberg, 1989). Over the past three decades, GAs exhibit inherent advantages to many complex optimization problems (Gen and Cheng, 2000). In this section, we integrate the Monte-Carlo simulation and genetic algorithm to produce a hybrid intelligent algorithm to obtain an approximate optimal solution. In this hybrid algorithm, the genetic algorithm handles the optimal solution; the stochastic simulation addresses uncertain functions

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Figure 1 Framework of the Hybrid Genetic Algorithm


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3.2. Stochastic simulation In order to solve the chance constrained programming model, we must handle the following two types of uncertain function. The first type of uncertain function is

U (1) : Pr{

pj

t jk x jk

j J

f}

j J k K

In order to compute f ,we describe the algorithm as the following procedure: Step1: Generate 1 , 2 , , s from the probability space S is a sufficiently large number; Step2: For each

s

, we can obtain the value Fs

according to the probability measure Pr , where

t jk x jk , s 1,2,..., S respectively;

pj j J

j J k K

Step3: Set S * as the integer part of S ; Step4: Return the S *th larger element in F1 , F2 ,

, FS as U (1) .

The second type of uncertain function is

U (2) : Pr

1

Lj

j l 1

l

f jl

t jl ( )

pmin

The simulation for calculating U (2) can be summarized as follows: Step1: Set N ' 0 ; according to the probability measure Pr ,where S Step2: Generate 1 , 2 , , s from the probability space is a sufficiently large number; l L f jl 1 j pmin 0 , s 1, 2, , S ,then N ' Step3: For each s ,if H s N' 1; ( ) t l 1 j jl Step4: Repeat Step2 and Step3 N times; Step5: U (2) N ' / N . 3.3. Genetic algorithm Here we propose the major steps of the simulation-based genetic algorithm procedure for solving the model. Step 1: Define input parameters: population size, crossover and mutation rates, maximum number of generations, and maximum number of simulations.

680


Step 2: Generate an initial population pool and initialize the generation index. Step 3: Evaluate the fitness of all chromosomes in the population pool using stochastic simulation procedures Step 4: Check whether the predefined maximum generation number is reached. If yes, go to Step 7; otherwise, go to Step 5. Step 5: Rank the chromosomes based on their fitness values and use the tournament selection scheme to select parent chromosomes for reproduction. Step 6: Update the chromosomes using the crossover and mutation operations, increment the generation index, and go to Step 3. Step 7: Report the best chromosomes as the optimal result. 4. Case study To evaluate the performance of proposed model and algorithm, this section will utilize the case study of Shanghai front of the Yangtze Delta, is a super mega-city with vast hinterland and a favorable geographical position. The city has 17 districts and one island county, covering an area of 6,340 km 2 in total and 23 million inhabitants in 2011.The population density of the whole metropolitan wide area is 3,632 person/ km 2 , while the central districts up to 34, 625 person/ km 2 . An example about fire stations at downtown of Shanghai is used to show the application of the model and algorithm. The network of 11 nodes, as depicted in Figure (2), is tested. In Figure (2), the points labelled with {A, B, C, D, E, F, G} are current locations of emergency vehicles, and the points {H, I, J, K} are potential relocation sites.

0.5 , In this example, we set 0.5 , Pmin 0.0002 , working as preset constant variables. A run of the hybrid intelligent algorithm (300 generations in GA, 10000 cycles in each fitness calculation progress) shows that the optimal solution. Crossover rate of 0.2 and mutation rate of 0.02 are used for the following analysis. The test is conducted on a 2.00GHz Intel(R) Core(TM)2 Duo T6400 PC with 2GB of RAM and the algorithms are coded in C# language (Visual Studio 2010). The evolution progress of the proposed algorithm is depicted in Figure3. Note that the proposed algorithm converges after 300 generations progressed. The optimal solution is {A->J, D->I}. The CPU searching time is 27380 seconds (approximately 456 minutes).The Monte Carlo simulation of each fitness calculation progress cost about 10 seconds, which enlarges the calculation time.


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Figure 2 Sample network for proposed example

Figure 3 Total Fitness of Each Generation

5. Conclusion As a tactical decision of emergency vehicles management, the relocation problem is important to ensure an adequate coverage and a quick response time to potential emergencies. However, finding reliable emergency vehicles relocation schemes is a difficult task. The complexity of this problem has limited much of the relocation literature to simplified static and deterministic models. In this paper, we explicitly address the stochastic nature of emergency vehicles relocation problems by considering the demand, travel time, and vehicles availability congestion character. A chance-constrained programming approach is described to find the best set of relocations based on the state of emergency vehicles and potential travel time and demand fluctuation throughout the area at the time of the relocation. Compared with static location problem, the relocation problem is needed to worry about computing time. Acknowledge This work was partly supported by the National Science Foundation of China under Grants 51008160, and by the Faculty Research Support Fund from the College of Engineering, Nanjing Agricultural University through project GXZ09005.These supports are gratefully acknowledged. References Andersson, T., & Värbrand, P. (2007). Decision support tools for ambulance dispatch and relocation. Journal of the Operational Research Society, 58(2), 195-201. Baptista, S., & Oliveira, R. (2012). A case study on the application of an approximated hypercube model to emergency medical systems management. Central European Journal of Operations Research, 20(4), 559-581. Brotcorne, L., Laporte, G., & Semet., F. (2003). Ambulance location and relocation models. European Journal of Operational Research, 147(3), 451-463. Budge, S., Ingolfsson, A., & Erkut, E. (2009). Approximating vehicle dispatch probabilities for emergency service systems with locationspecific service times and multiple units per location. Operations Research, 57(1), 251-255. Charnes, A., & Cooper, W. (1959). Chance-constrained programming. Management Science, 6(1), 73-79. Daskin, M. S. (1995). Network and Discrete Location: Models, Algorithms, and Applications. New York: John Wiley & Sons Inc.

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Galvão, R., Chiyoshi, F., & Morabito, R. (2005). Towards unified formulation and extensions of two classical probabilistic location models. Computers & Operations Research, 32(1), 15-33. Gen M and Cheng R (2000). Genetic Algorithms and Engineering Optimization, New York: John Wiley & Sons Inc. Gendreau, M., Laporte, G., & Semet, F. (2001). A dynamic model and parallel tabu search heuristic for real-time ambulance relocation. Parallel Computing, 27(12), 1641-1653. Gendreau, M., Laporte, G. & Semet, F. (2006). The maximal expected coverage relocation problem for emergency vehicles. Journal of the Operational Research Society, 57(1), 22-28. Geroliminis, N., Karlaftis, M., & Skabardonis, A. (2009). A spatial queuing model for the emergency vehicle districting and location problem. Transportation Research Part B, 43(7), 798-811. Goldberg, E. (1989). Genetic Algorithms in Search, Optimization and machine Learning. New York: Addison-Wesley. Goldberg, J. (2004). Operations research models for the deployment of emergency services vehicles. EMS Management Journal, 1(1), 20-39. Goldberg, J., Dietrich, R., Chen, J., Mitwasi, M., Valenzuela, T., & Criss, E. (1990). A simulation model for evaluating a set of emergency vehicle base locations: Development, validation, and usage. Socio-Economic Planning Sciences, 24(2), 125-141. Haghani, A., & Yang, S. (2007). Real-time emergency response fleet deployment: Concepts, systems, simulation & case studies, In V. Zeimpekis, C. Tarantilis, G. Giaglis, & I. Minis(Eds), Dynamic Fleet Management: Concepts, Systems, Algorithms& Case Studies(pp.133162). New York: Springer Science+Business Media. Holland, J. (1992). Adaptation in natural and artificial systems. Cambridge MA: MIT Press. Iannoni, A., Morabito, R., & Saydam, C. (2008). A hypercube queueing model embedded into a genetic algorithm for ambulance deployment on highways. Annals of Operations Research, 157(1), 207-224. Jarvis, J. (1985). Approximating the equilibrium behavior of multi-server loss systems. Management Science, 31(2), 235-239. Kolesar, P., & Walker, W. (1974). An algorithm for the dynamic relocation of fire companies. Operations Research, 22(2), 249-274. Larson, R. (1974). A hypercube queuing model for facility location and redistricting in urban emergency services. Computers & Operations Research, 1(1), 67-95. Larson, R. (1975). Approximating the performance of urban emergency service systems. Operations Research, 23(5), 845-868. Lee S (2011).The role of preparedness in ambulance dispatching. Journal of the Operational Research Society, 62(10), 1888-1897. Li, X., Zhao, Z., Zhu, X., & Wyatt, T. (2011). Covering models and optimization techniques for emergency response facility location and planning: A review. Mathematical Methods of Operations Research, 74(3), 281-310. Liu, B. (2009). Theory and Practice of Uncertain Programming (2nd ed). Berlin: Springer-Verlag. Marianov, V. & Revelle, C. (1995). Siting emergency services. In Z. Drezner (Ed). Facility Lcoation: A Survey of Applications and Methods (pp.199-223).New York: Springer-Verlag. Marianov V and Serra D (2002). Location problems in the public sector. In: Z. Drezner, & H. Hamacher(Eds). Facility Location: Applications and Theory (pp. 119-150). Berlin: Springer-Verlag. Rajagopalan, H., & Saydam, C. (2009). A Minimum Expected Response Model: Formulation, Heuristic Solution and Application. SocioEconomic Planning Sciences, 43(4):253-262. Restrepo, M., Henderson, S., & Topaloglu, H. (2009). Erlang loss models for the static deployment of ambulances. Health Care Management science, 12(1), 67-79. Sajjadian, S. (2009). Dynamic Model for Real-Time Ambulance Relocations based on Coverage Variation. Master's thesis. Simon Fraser University. Sathe, A., & Miller-Hooks, E. (2005). Optimizing location and relocation of response units in guarding critical facilities. Transportation Research Record: Journal of the Transportation Research Board, 1923, 127-136. Saydam, C., Repede, J. & Burwell, T. (1994). Accurate estimation of expected coverage: A comparative study. Socio-Economic Planning Sciences, 28(2), 113-120. Schmid, V., & Doerner, K. (2010). Ambulance location and relocation problems with time-dependent travel times. European Journal of Operational Research, 207(3), 1293-1303. Schmid, V. (2012). Solving the dynamic ambulance relocation and dispatching problem using approximate dynamic programming. European Journal of Operational Research, 219(3), 611-621. Sorensen, P., & Church, R. (2010). Integrating expected coverage and local reliability for emergency medical services location problems. Socio-Economic Planning Sciences, 44(1), 8-18.