Optimal Assignment of Emergency Response Service ...

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Optimal Assignment of Emergency Response Service Units With TimeDependent Service Demand and Travel Time Ruey Long Cheua; Hao Leia; Raed Aldourib a Department of Civil Engineering, The University of Texas at El Paso, El Paso, Texas, USA b Geospatial Information Service Center, The University of Texas at El Paso, El Paso, Texas, USA Online publication date: 05 November 2010

To cite this Article Cheu, Ruey Long , Lei, Hao and Aldouri, Raed(2010) 'Optimal Assignment of Emergency Response

Service Units With Time-Dependent Service Demand and Travel Time', Journal of Intelligent Transportation Systems, 14: 4, 220 — 231 To link to this Article: DOI: 10.1080/15472450.2010.516232 URL: http://dx.doi.org/10.1080/15472450.2010.516232

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Journal of Intelligent Transportation Systems, 14(4):220–231, 2010 C Taylor and Francis Group, LLC Copyright
ISSN: 1547-2450 print / 1547-2442 online DOI: 10.1080/15472450.2010.516232

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Optimal Assignment of Emergency Response Service Units With Time-Dependent Service Demand and Travel Time RUEY LONG CHEU,1 HAO LEI,1 and RAED ALDOURI2 1 2

Department of Civil Engineering, The University of Texas at El Paso, El Paso, Texas, USA Geospatial Information Service Center, The University of Texas at El Paso, El Paso, Texas, USA

This article formulates an improved integer-programming model to assign multiple types of emergency response service units among their candidate base stations to maximize the coverage to the critical infrastructures, subject to station capacity, service time, and reliability constraints. Unlike past models, this new model formulation accounts for the fluctuation of travel time and demand frequency at different time periods of a day. The potential applications of the model have been illustrated via a case study of assigning firefighting units and ambulances in the city of El Paso, Texas. This article also demonstrates how the model can be modified for the selection of a new base station, increase in fleet size, station capacity expansion, or a combination of these. Keywords

Emergency Service; Integer Programming; Service Programming; Service Demand; Travel Time

BACKGROUND Emergency Response Services (ERS) units, such as fire, law enforcement, and medical services, are important services in most cities. ERS units must cover the maximum possible area of a city and yet arrive at the incident scene at the shortest possible time. In their day-to-day operation, ERS units must respond to emergency calls from residents and businesses. In addition, they must respond quickly to any incident that occurs at critical transportation infrastructures (CTIs), schools, hospitals, and other important facilities that may affect a large population. One type of ERS that is unique to the freeway systems is the Freeway Service Patrol (FSP), which is an ERS that travels on the fixed routes on the freeways and responds to incidents such as accidents and vehicle breakdowns. However, the focus The authors thank the El Paso Fire Department for providing the data and sharing their operational requirements for the case study. The authors also thank the Texas Department of Transportation El Paso District and the El Paso Metropolitan Planning Organization for providing the transportation network model. The Texas Department of Transportation El Paso District has also helped in identifying the critical transportation infrastructures. Address correspondence to Ruey Long Cheu, Department of Civil Engineering, The University of Texas at El Paso, El Paso, TX 79968. E-mail: [email protected]

of this research is on the stationing of ERS resources at fixed locations to protect CTIs, schools, and hospitals while responding to competing demands from residents and businesses. Critical transportation infrastructures are important facilities—such as highway interchanges, tunnels, bridges, transit terminals, and so forth—in a city’s transportation network. The ERS units must be able to reach CTIs quickly to restore the capacity of the affected node at the earliest possible time to minimize the delay to the users. Schools are densely populated buildings with children and young adults who are vulnerable to any life-threatening event. Hospitals are the central providers of medical services and have high concentration of medically impaired persons. Moreover, hospitals must continue to function around the clock. In this article, we refer to the CTIs, schools, and hospitals collectively as critical infrastructures (CIs). Two major decisions made by the ERS planning departments are where to locate the base stations and how many units to assign to each of the base stations. Assigning ERS units to potential base stations to serve CIs may be viewed as a facility location problem. A comprehensive review on the various simplified problem formulations and solution algorithms on this topic can be found in Owen and Daskin’s (1998) study. The problem may be formulated to minimize the number of base stations with

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OPTIMAL ASSIGNMENT OF EMERGENCY RESPONSE SERVICE UNITS

the requirement that every CI in the network should be covered within a certain service time (Toregas and ReVelle, 1973; Toregas, Swain, ReVelle, and Bergman, 1971). When the number of base stations or units is limited, the objective can be set to maximize the coverage among the CIs (Church and ReVelle, 1974; Schilling, Elzinga, Cohon, Church, and ReVelle, 1979). A model that simultaneously assigns different types of ERS units (e.g., fire engines, ambulances) among their respective base stations has also been proposed (Schilling et al.). The aforementioned models, known collectively as deterministic covering models, do not consider the demand frequency or the probability that an ERS unit may be busy serving another incident when a new emergency call is received. Daskin (1983) and ReVelle and Marianov (1991) first formulated probabilistic covering models to account for the competing demand when computing coverage. Huang, Cheu, and Fan (2007) and Cheu, Huang, and Huang (2008) expanded the formulations of the probabilistic covering models from one type of ERS to three. An important input in the determination of coverage is the travel time from the stations to the CIs. Most of the earlier studies estimated the travel times on the basis of either the Euclidean distance or distance measured along the links in a road network. Both Huang et al. (2007) and Cheu et al. (2008) computed the travel times according to the estimated driving speed and other traffic regulations in the road network. To date, all the models assume that the travel times from the stations to the CIs are the same throughout the day, ignoring the traffic congestion in the morning and evening peak periods. Furthermore, perhaps because of the lack of location-specific demand data, they assumed that the competing demand was the same spatially throughout the city and remained the same temporally throughout the day. The objective of this article is to formulate and solve an improved version of the probabilistic covering integerprogramming model for the optimal assignment of multiple types of ERS units to their respective base stations in order to maximize the coverage of the CIs. During the computation of coverage of the CIs, the improved model takes the following into account: (a) the spatial and temporal distributions of competing demand for ERSs and (b) the temporal distribution of travel time as a result of traffic congestion during different time periods of a day. The case of the El Paso Fire Department (EPFD) allocating its fire fighting units and ambulances among its fire stations has been used to demonstrate the applications of this model to difference scenarios. After this introduction, we review the most recent probabilistic covering model. In the subsequent section of this article, we present the improved model, then discuss the case study and results.

LITERATURE REVIEW This section presents the integer-programming probabilistic covering model (from Huang et al., 2007) that simultaneously assigns multiple types of ERS units among the candidate base

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stations. The objective of the optimization is to cover as many CIs as possible, taking into consideration the competing demands from residents and businesses. This is the version of the model from which improvements are made. Two steps were involved in developing the probabilistic covering model. In the first step, the historical demand frequency and ERS response data were used to derive the minimum numbers of units that will meet the service reliability requirement (in the different service areas and different time of a day). In the second step, the calculated minimum numbers of units were entered as constraint values in the integer-programming model. The two-step process may be viewed as similar to the approach that Edara and Dougald (2007) used in their planning model for FSP coverage. In their model, historical incident data along existing FSP routes were used to develop a model that predicts the number of incidents assisted by a FSP vehicle as a function of several explanatory variables such as route length, traffic volume, and so forth. In the second step, the number of incidents is used as one of the input into a planning tool that ranks the routes competing for FSP deployment. This probabilistic covering model assumes the following: 1. The network has a set of critical infrastructure nodes I ; 2. The network has a set of candidate base stations J , each has a uniform station capacity b for each type of ERS unit; 3. The types of ERS units are denoted by set V and each type of ERS has a fleet size pv , ∀v ∈ V ; 4. Incident occurrence (emergency call) rates for the different types of unit are known from the historical data; 5. The travel time between a base station j and CI node i, tj i , is predetermined,∀j ∈ J, ∀i ∈ I ; 6. For each type of ERS unit v, ∀v ∈ V , the service standard (maximum time to reach the CIs) is given as s v and the percentage of responses that must meet this standard (also known as reliability) is given as α. When there is a request for an ERS unit of type v (v ∈ V ) at CI node i (i ∈ I ), the call center should dispatch a vehicle from the station j (j ∈ J ) that is nearest to i. However, the ERS unit of type v at station j may be busy serving a competing demand (which may be another CI node, resident, or business) at that time. ReVelle and Hogan (1989) and ReVelle and Marianov (1991) introduced the concept of local busy fraction (around the CI node i for a particular ERS unit of type v) which is essentially the probability that an ERS unit is busy. It is defined as the total service time spent (by all the units of type v) in the area (within s v around the CI node i) divided by the available service time (of all the units of type v in the same area). In mathematical terms, qiv , the local busy fraction of a unit of type v, at CI node i is expressed as follows:
t¯v l∈MEi fl ρv v
i
(1) qi = v = v 24 j ∈NEi xj j ∈NEi xj

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where t¯v = average service time of a unit of type v, in hours/call; = frequency of emergency calls at competing defl mand node l, in calls per day; = number of units of type v assigned to station j ; xjv = set of competing demand nodes within s v of CI MEi node i; = set of stations within s v of CI node i; NEi = utilization ratio of a unit of type v around CI node ρiv i. Most ERS departments recognize that it is impossible for their units to reach the scenes within s v for all emergency calls. Therefore, they normally set a target that s v must be fulfilled at 100α% of the time (0 < α < 1). Typical values of
are 0.90, 0.95, and 0.99. In another words, when CI node i requests for an emergency service, the probability of having at least one ERS unit of type v available within s v of node i must be larger than or equal to α. The aforementioned probability is equivalent to the following expression: 
j ∈NE xjv  i ρiv 1−
≥ α. (2) v x j ∈NEi

j

which may be replaced by the following:  v eiv ρ 1 − vi ≥ α, ei

(3)

where eiv is the smallest integer that satisfies the inequality in (3). The derivation of this expression can be found in ReVelle and Hogan (1989). Typically ρiv , ∀v ∈ V , ∀i ∈ I are estimated from historical data using (1). Once ρiv is known, (3) is used to obtain eiv , ∀v ∈ V , ∀i ∈ I . The eiv values will be entered as constants in the following linear programming problem. The complete probabilistic covering integer-programming model is as follows: Maximize  yi (4) i∈I

Subject to

 j ∈NEi

 j ∈J

xjv ≥ eiv yi , ∀i ∈ I, ∀v ∈ V

(5)

xjv ≤ pv , ∀v ∈ V

(6)

0 ≤ xjv ≤ b, ∀j ∈ J, ∀v ∈ V yi = 0, 1, ∀i ∈ I ; xjv = 0, 1, 2, ..., ∀j ∈ J, ∀v ∈ V

(7) (8)

where yi is a binary integer that represents the coverage of CI node i. A CI node i is covered when there are eiv or more ERS units of type v within s v of node i, ∀v ∈ V simultaneously. yi = 1 if CI node i is covered; otherwise, yi = 0. The objective presented in (4) maximizes the total number of covered CIs. Constraint (5) states that at each CI node i,

the number of units of type v assigned to stations within s v of node i must be greater than or equal to the number of units of the same type needed within s v of node i (service reliability constraint). Constraint (6) restricts the total number of vehicles to be assigned (fleet size constraint). Inequality (7) imposes the capacity limitation at each station (station capacity constraint). The aforementioned formulation has several shortcomings, including the following: 1. Although eiv is node specific, researchers including Huang et al. (2007) and Cheu et al. (1008) have calculated eiv using citywide average value of ρiv . That is, an average value of ρiv was calculated from a city’s ERS data, ignoring the spatial distribution of demand frequency in different areas (represented by different i) in the city. 2. The model considers only one time period in a day. It is obvious that the emergency call frequency varies by time of day. The travel time tj i between station j and CI node i also varies by time of the day because of traffic congestion that happens during the peak hours. Consequently, the sets of MEi and NEi may change at different hours of the day. Because of this, a CI node that is covered during the off-peak call demand and traffic hours may not be covered during the peak hours. 3. The station capacity b in (7) may vary between stations and unused capacity for one type of ERS unit may be used by another type of ERS unit. In the next section, we present a model that takes into account the aforementioned considerations: (a) spatial and temporal distributions of call frequency and travel time and (b) different station capacities.

THE IMPROVED MODEL Assume that a typical 24-hr day is divided into T discrete time periods τ = 1,. . . , T . The time period may be as short as 1 hr or as long as one crew shift. In addition, we assume that, in the historical data set, for every emergency call, the location of the call, the vehicle dispatch time and service time are known. On the basis of this information, the location and time specific busy fraction may be computed by the following equation:
v ρv l∈ME tl v
i,τ v =
i,τ v (9) qi,τ = dτ j ∈NEi,τ xj j ∈NEi,τ xj v = local busy fraction of a unit of type v at CI node where qi,τ i in time period τ ; = service time of a unit of type v in response to an tlv emergency call from node l; = set of all the nodes (including CIs, residents and MEi,τ businesses) within s v of CI node i that generated historical emergency calls in time period τ ;

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= set of fire stations within s v of CI node i in time period τ ; = duration of time period τ : = utilization ratio of a unit of type v at CI node i in time period τ .

NEi,τ dτ v ρi,τ

The time and location dependent utilization ratio is calculated as follows:
v l∈MEi,τ tl v (10) ρi,τ = dτ Equation (2) can therefore be rewritten as follows: 
j ∈NE xjv  v i,τ ρi,τ ≥ α. 1−
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j ∈NEi,τ

(11)

j

Equation (2), in turn, may be replaced by the following: v  v ei,τ ρi,τ 1− v ≥ α, ei,τ

Subject to  j ∈NEi ,τ

i∈I

v xjv ≥ ei,τ yi,τ , ∀i ∈ I, ∀v ∈ V , τ = 1, . . . , T

(14)  j ∈J

 v∈V

xjv ≤ pv , ∀v ∈ V

(15)

xjv ≤ bj , ∀j ∈ J

(16)

yi,τ = 0, 1, ∀i ∈ I, τ = 1, . . . , T ; xjv = 0, 1, 2, . . . , ∀j ∈ J, ∀v ∈ V

bj = capacity of station j . The station capacity constraint in (15) has been modified such that all the units of different types that are assigned to station j cannot exceed bj . In this new model formulation, xjv is independent of time period τ . that is, once an ERS unit is assigned to a station, it will remain in service for the entire day. this is to due to practical consideration to avoid moving one set of vehicle and equipment between stations for the different time periods of the day. comparing (14) with (5), the number of constraints in this improved model is multiplied by the factor T . another point worth noting is that this formulation is able to handle different service standards or reliability requirements at different v CI nodes in different time periods, resulting in different ei,τ v values in (14). in this case, the notation s that leads to MEi,τ and N Ei,τ in (9) to (11), and α in (11) and (12) must be modified to include the subscripts i and/or τ .

(12)

v Using (12), given the values of α and ρi,τ , we can obtain v ei,τ , the minimum number of units of type v that must be assigned to stations within s v of CI node i in time period τ in order to cover CI node i with reliability α. It should be noted that the definition of tlv and MEi,τ is based on the assumption that none of the historical demand nodes generated more than one call. If a node has in fact generated multiple calls, the calls are treated as coming from separate nodes at the same locav will be calculated correctly by (9) and tion. In this way ρi,τ (10). The new model formulation is, then, the following: Maximize T  yi,τ wi,τ (13) τ =1

223

(17)

where yi,τ = binary variable that defines the coverage of CI v units node i in time period τ . yi,τ = 1 if node i is covered by ei,τ v within s , ∀v ∈ V simultaneously, otherwise, yi,τ = 0; wi,τ = weight for the coverage of CI node i in time period. τ . The weight may depend on the location of the node and the time period.

CASE STUDY El Paso: Fire Department and Critical Infrastructures The city of El Paso is located at the western end of Texas. It has a population of more than 600,000 people. The El Paso Fire Department (EPFD) is the primary agency in the city that is responsible for responding to fire, medical emergencies, incidents with hazardous materials, building explosions, and rescue events. The EPFD currently has 35 fire stations. Figure 1 shows a map of the fire districts (by fire battalions) and the locations of the fire stations. The EPFD operates the following ERS vehicles: 4 ladders, 33 pumpers, 8 quints, and 23 rescues. Ladders refer to fire trucks equipped with ladders to reach multistory buildings. Pumpers are fire trucks with high-pressure water pumps. Quints have the combined functions of ladders and pumpers. Last, rescues refer to ambulances. In the present article, we refer to ladders, pumpers, and quints collectively as firefighting units. The existing assignment of the EPFD units among the fire stations is listed in Table 1. Note that Fire Station 31 was under construction, and therefore it was not included in the initial model. The EPFD handled 70,472 calls in 2006, of which 24,992 required the dispatch of both firefighting units and rescues. The remaining 45,480 cases resulted in the dispatch of only rescues. The EPFD’s target is for all of its units to reach the scenes within 4.5 min for at least 90% of the cases. Therefore, s v = 4.5 min, ∀v ∈ V , and α = .90. The CIs to be covered by EPFD were CTIs, schools, and hospitals in its fire districts. With the assistance of the Texas Department of Transportation, we identified 20 CTIs. They included the bridges, tunnels, freeway interchanges, and major intersections along the interstate and Texas state highways. We found a total of 105 public elementary, middle, and high schools on the Web sites of the school districts. In addition, we identified 13 hospitals from online public sources. The locations of

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the CTIs, schools, and hospitals are shown in Figure 2. The total number of CIs was 138.

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Data Collection and Processing We obtained the transportation network data in El Paso from the Texas Department of Transportation and the El Paso Metropolitan Planning Organization (2005). The network was based on the model year 2005, provided in TransCAD (Caliper, 2006) files. It has 681 traffic analysis zones, 4,836 links, and 3,060 nodes, and an origin-destination matrix for a typical weekday in 2005. The network is shown in Figure 2. The ERS activity log between September 2006 and February 2007 (6 months) was provided by EPFD. There were 32,116 emergency calls during the 6-month period. Each record consisted of the date, time, location coordinate, ERS unit(s) dispatched, time of dispatch, time of arrival at the scene, and service time at the scene. The first task in data processing was to decide time periods of the day. After examining the hourly fluctuation of traffic flow in the city (El Paso Metropolitan Planning Organization, 2005)

and the hourly distribution of ERS activities, it was decided to divide a 24-hr day into three 8-hr periods (T = 3, τ = {1, 2, 3}). Each period was assumed to correspond to an 8-hr shift by the firefighting and rescue crews. The three periods were 3:00 a.m. to 11:00 a.m. (τ = 1, morning shift), 11:00 a.m. to 7:00 p.m. (τ = 2, afternoon shift), and 7:00 p.m. to 3:00 a.m. (τ = 3, night shift). The period from 11:00 a.m. to 7:00 p.m. (τ = 2) covered the hours with the highest emergency call frequencies. According to EPFD, a CI node was considered covered if one rescue and one fire fighting unit (ladder, pumper, or quint) could simultaneously reach the CI node within s v = 4.5 min with α = .90. An easier way to handle this requirement is to regard ladder, pumper, and quint as one type of ERS unit and rescue as another type. However, at the request of EPFD, it was decided to keep the four types of units in the model. Therefore, constraint (14) was modified to the following:   pumper quint  f ire + xj xjladder + xj ≥ ei,τ yi,τ , ∀i ∈ I, j ∈NEi

τ = 1, . . . , T

Figure 1 Fire districts and fire stations in El Paso, Texas.

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(18)

OPTIMAL ASSIGNMENT OF EMERGENCY RESPONSE SERVICE UNITS

225

Table 1 Fire station capacity and vehicle assignments. After optimization with α = .90

Existing number of units

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Fire station 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Total



Station capacity

Ladder

Pumper

10 4 4 1 5 3 3 1 2 1 5 2 1 2 3 3 3 4 2 3 2 4 3 5 3 3 3 4 3 3 Under construction 2 3 3 3 106

1

1 1 1 1 1

1 1

–

1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 –

4

1 1 1 33

1

rescue xjrescue ≥ ei,τ yi,τ , ∀i ∈ I, τ = 1, . . . , T

(19)

j ∈NEi f ire

where ei,τ denotes the minimum total number of firefighting units (sum of ladders, pumpers, and quints) that must be assigned to fire stations within s v of CI node i to provide sufficient service reliability. f ire f ire rescue used in (18) and (19), ρi,τ , To estimate ei,τ and ei,τ rescue must first be computed using (10). The historical call ρi,τ f ire rescue . To make the comdata were used to compute ρi,τ and ρi,τ putation less complex and yet realistic enough, we assumed that, in time period τ , all the CI nodes i in each fire district shared f ire rescue values. Furthermore, some of the common ρi,τ and ρi,τ CTI nodes are at the boundary of two fire districts. We therefore decided to combine districts 1 and 3 as a joint district, and 2 f ire and 7 as another joint district. Hence, the calculations of ρi,τ , f ire rescue rescue and the corresponding ei,τ and ei,τ were performed ρi,τ by districts or joint districts (instead of citywide average in

Quint

Rescue

Ladder

Pumper

1

Quint

3 4

Rescue 3

1 1 1 1

1 1

1

2 1

1 1 1 1

1 1 1 1 1

1 1 1 1 1 1 1 1 1 1

2 1 1 1 1 1

1 2 1 1

1

1 –

– 1

2

3

–

2 2 2 1 3 1

2 2

3 1 3

3

–

1

–

1 8

23

4

2 3 1 1

33

–

1 4

23

past research). The ranges of calculated utilization ratios were f ire rescue ≤ 2.3864. 0.5749 ≤ ρi,τ ≤ 2.6867 and 0.4404 ≤ ρi,τ Details of computations are documented in (Lei, 2008). The f ire f ire rescue rescue , ei,τ and ei,τ values are summacomputed ρi,τ , ρi,τ rized in Table 2. As expected, because the time period between 11:00 a.m. to 7:00 p.m. (τ = 2) has the highest call frequency, f ire rescue , for the same district or combined district, the ρi,τ , ρi,τ f ire rescue values are equal or higher than the correspondei,τ , and ei,τ ing values obtained for other time periods (τ = 1 and 3). Constraint (14) requires N Ei,τ , the set of fire stations within the s v from CI node i. The travel times between all the fire stations ∀j ∈ J and CI nodes ∀i ∈ I during different time periods (τ = 1, 2, 3) must therefore be computed. The steps of computations were as follows: 1. The CI nodes were added to the existing transportation network in the TransCAD geographic file. Links were added to connect these CI nodes and the nearest nodes in the network.

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Figure 2 Locations of critical transportation infrastructures, schools, and hospitals in El Paso, Texas.

2. The daily 24-hr origin-destination matrix was scaled, by means of hourly factors reported in (El Paso Metropolitan Planning Organization, 2005), to three 8-hr origindestination matrices, each corresponds to a time period τ with the unit expressed in vehicles per hour. 3. The link capacities in the network were also factored from the 24-hr capacities to the hourly link capacities. 4. For each time period τ , a. Static user equilibrium traffic assignment (using the Frank-Wolfe algorithm) was performed in TransCAD (Caliper, 2005); Table 2 Utilization ratios and desired minimum units for time dependent model. f ire

rescue ρi,τ

ρi,τ Fire district

τ =1

τ =2

τ =3

τ =1

τ =2

τ =3

1&3 2&7 4 5 6

1.8409 0.6506 1.1424 0.8264 0.5749

(a) Utilization ratios 2.6867 1.8262 1.2251 0.9761 0.5957 0.4271 1.6298 1.2568 0.7374 1.1497 0.7790 0.6026 0.7945 0.6480 0.4404

2.3864 0.6599 1.2350 0.9319 0.8111

1.5523 0.4577 0.9486 0.7051 0.5956

1&3 2&7 4 5 6

4 3 3 3 2

5 3 3 3 3

4 2 3 3 2

(b) Desired minimum units 5 4 3 3 2 2 4 3 3 3 3 2 3 2 2

b. The shortest paths and the corresponding travel times between all the stations ∀j ∈ J and CI nodes ∀i ∈ I were computed; c. For each CI node i, the fire stations within s v = 4.5 min of travel time were identified to form N Ei,τ .

Optimal Assignment of Firefighting Vehicles and Ambulances We constructed the improved integer-programming model with all the preprocessed data. The decision variables were pumper quint , xj , xjrescue , ∀j ∈ J , and yi,τ , ∀i ∈ I, xjladder , xj τ = {1, 2, 3}. For the objective function, all of the CIs were treated as equally important, but we assigned different weights for the coverage in the different time periods. The historical ERS calls in the respective time periods were used to calculate the relative weights. Therefore, wi,1 = 0.85, wi,2 = 1.40, wi,3 = 1.00, ∀i ∈ I (normalized such that wi,3 = 1.00). The model has 866 constraints (828 service reliability constraints, 4 fleet size constraints, and 34 capacity constraints). The optimization model was solved by ILOG CPLEX 11.1 Interactive Optimizer (ILOG, 2007) in a standard desktop personal computer. The optimized total coverage are 34, 21, and 36 nodes for τ = 1, 2, and 3, respectively. This is equivalent to 91 node shifts. The coverage at τ = 2 is lower because during this period there was f ire rescue a higher competing demand, resulting in higher ei,τ and ei,τ values. Figure 3 shows a map of the CIs with their coverage among the three shifts. Because the map will be too cluttered if three data points (one for each shift) are plotted for every CIs

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Figure 3 Coverage before and after optimization.

the coverage for each CI is color coded according to the number of shifts. The optimal distribution of ERS units is listed in the righthand side of Table 1. Because of the reliability requirement, f ire rescue values listed in Table 2 each CTI needs the ei,τ and ei,τ to be considered covered. Therefore, the vehicles tend to be clustered around fewer nearby stations, rather than spreading evenly across most of the stations. As noted in Table 1, after optimization, there is no vehicle assigned to stations 3, 7, 9, 11, 14, 26, 30, 32, 33, 35. The clustering of the vehicles around the remaining stations also reflect that, when there is a limited fleet size and station capacity, in order to maximize the coverage of the CIs, the coverage of residents and busi-

nesses has to be compromised. The last row of Table 1 indicates that, all ladders, pumpers, and rescues are assigned to stations, but only 4 of 8 quints are used. The remaining quints are not used because doing so will not increase the coverage of CIs f ire rescue requirements in Table 2). (see the simultaneous ei,τ and ei,τ However, they can be deployed to other stations to cover the background demands generated by residents and businesses. For the coverage of CIs and because of (18), the EPFD may swap ladders, pumpers, and quints between the assigned fire stations. This situation implies that there is more than one optimal solution to this problem. However, there are other operational considerations (e.g., the necessity to cover residents and businesses, trained crews, high-rise buildings) that may restrict the

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assignment of a particular type of vehicle to certain fire stations. This type of constraints can easily be handled by preassigning certain units of type v to specific stations before optimization and reducing the station capacities and fleet size constraints correspondingly.

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Comparison With Existing Coverage For comparison purpose, we also computed the coverage given by the current assignment of ERS units (shown in Table 1). We performed the calculations by replacing xjladder , pumper qu int xj , xj , ∀j ∈ J in the constraints by their actual values and then solved for yi,τ , ∀i ∈ I , for τ = {1, 2, 3} without optimization. The existing total coverage of CIs was found to be 9, 0, and 2 for τ = 1, 2, and 3 respectively, or the total coverage of only 11 node shifts. Our optimization model could improve the total coverage of CIs from 9 to 91 node shifts. Figure 3(b) plots the existing coverage of each of the CIs among the three shifts by color codes. Comparing Figures 3(a) and 3(b), there are obviously more CIs that are covered (either for the entire day or part of the day) after the optimization of the assignment of ERS units. The improvements in coverage occur mostly in fire districts 4 and 6, in the center and east side of the city.

f ire

f ire

rescue calculated ei,1 and ei,1 values, and the corresponding ρi,1 rescue and ρi,1 values are shown in Table 3. In the static model, the point-to-point travel times were assumed to be the same throughout the 24 hr. The travel times were estimated by the static traffic assignment in TransCAD (Caliper, 2005), using the 24-hr traffic demand and 24-hr link capacities in the 2005 El Paso network. This traffic assignment approach (including the time period, traffic demand, and link capacities) was used in the official El Paso 2030 Gateway Transportation Plan (El Paso Metropolitan Planning Organization, 2005). On the basis of this modeling approach, the optimized static coverage was 34 nodes. If there are three shifts in 24 hr, this coverage may be considered as 34 × 3 = 102 node shifts. The optimized time dependent coverage obtained with T = 3 was 34, 21, and 36 nodes for τ = 1, 2, and 3 respectively, or 91 node shifts. It is obvious that the CIs covered and the ERS vehicle assigned to the fire stations are different between T = 3 and T = 1. The differences are caused by the different travel times (as a result of different traffic demand and link capacities) and reliability requirements (as a result of the different demand frequencies). However, because the model with T = 1 does not have as good a time resolution as in T = 3, we have decided not to report the detailed differences in this article.

With a New Station in Operation Comparison With Static Model This subsection compares the optimized coverage obtained with T = 3, against the optimized coverage obtained with the static case which assumed that T = 1. In the static case, there was only one time period for a 24-hr day, i.e., T = 1, τ = {1}. f ire rescue values for α = In this case, we recalculated the ei,1 and ei,1 .90, with the same ERS data set, using d1 = 24 hr in (9) for the districts or joint districts. Therefore, the spatial distribution of call frequency was still considered in the calculation. The

Table 3 Utilization ratios and desired minimum units for static model. Fire district

f ire

ρi,1

rescue ρi,1

1&3 2&7 4 5 6

(a) Utilization ratios 2.1179 0.7408 1.3430 0.9183 0.6725

1.7213 0.5149 0.9737 0.7465 0.6157

1&3 2&7 4 5 6

(b) Desired minimum units 4 3 3 3 3

4 2 3 3 2

This subsection applies the model to analyze the coverage and vehicle assignment when a new fire station is added to the system. The procedure describe in this section can be used to compare and select one among several candidate new fire stations to determine which one will lead to the maximum improvement in total coverage. Table 1 shows that fire station 31 (j = 31, with capacity b31 = 3) was under construction. There are six CIs within s v = 4.5 min from this new station. These CIs, which were not covered previously, may possibly be covered by ERS units assigned to station j = 31. The basic changes to the optimization model were the constraints related to CI nodes i that are within s v of station j = 31. That is, for (18) and (19) that were related to these CI nodes, NE i,τ now included the new fire stations j = 31. The optimization results show that the coverage of the CIs is improved from 91 to 106 node-shifts. Four previously noncovered CIs around fire station j = 31 have been covered. One previously covered CI was no longer covered. In other words, during the optimization, vehicles were redistributed. The model recommended 43 firefighting vehicles and 23 rescues. In comparing the total vehicles assigned in both optimization scenarios, two more firefighting vehicles have been deployed when station j = 31 is added to the system. In this case, one ladder and three rescues were assigned from to this new fire station j = 31. The three rescues were moved from other stations to j = 31.

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capacity constraints of the fire stations, the minimum capacity of each fire station (including the new fire station j = 31) was increased to five. The capacity constraint (16) was replaced by the following equations:    (20) xjv ≤ max 5,Bj ∀j ∈ J v∈V

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Figure 4 Total coverage with increasing number of rescues.

Increase in Fleet Size After analyzing the optimized results and vehicle distributions in the aforementioned scenarios, we observed that all rescues were assigned to the stations, but not all firefighting vehicles. It implied that the total number of the rescues was not sufficient to match the total number of the fire fighting vehicles. Increasing the fleet size of rescue prescue should increase the optimized coverage of the CIs. In this scenario, the total number of the rescues (prescue in constraint (14)) was increased to address this mismatch. The value of prescue was increased from 23 to 27, at increment of 1. For each prescue value, the optimized coverage was obtained and plotted in Figure 4. It was observed that, after a significant improvement in the coverage from 91 to 93 node shifts when the total number of the rescues was increased from 23 to 25, there was no further improvement in the coverage no matter how many more rescues are added. After a careful examination of the constraints, the reason causing this result was identified. Under the service reliability f ire rescue values are between 2 and 5 for of α = 90%, the ei,τ and ei,τ each CI. It implies that for a CI node i to be covered, there must be 2 to 5 firefighting vehicles and 2 to 5 rescues (depending on the fire district) assigned to the fire stations within s v of node i. The reliability constraint equations showed that, for some CIs, there was only one fire station within s v . To cover these CIs, those fire stations must have enough capacity to accommodate these firefighting vehicles and rescues.

Increase in Station Capacity The results in the previous sub-section indicated that the factors limiting the improvement of the coverage were not only the number of the ERS units (in this case the total number of rescues prescue ), but also the capacities of the fire stations (bj ). In this scenario, both factors were considered to investigate how the improvement in the coverage would be. To overcome the

In Figure 4, it can be seen that with the new minimum capacity, the coverage improved with the total number of the rescues. There was a significant increase in coverage when the capacities of all fire stations were expanded to a minimum of five even if the total number of rescues remained at 23 vehicles. In this case, the total coverage was increased from 91 to 151 node shifts. The reason is that all the fire stations now have the capacity to each accommodate the necessary number of fire fighting vehicles and rescues to cover the CIs within their s v . Therefore, one single fire station can accommodate enough ERS vehicles to cover as many CIs as possible within its s v . In the previous situation, the service reliability requirement may need at least two fire stations to provide the sufficient number of ERS units. With the increased capacity of five, one fire station may be sufficient to provide the coverage for all the CIs within its s v .

OTHER POTENTIAL APPLICATIONS In the previous section, we presented some applications of the improved probabilistic covering integer-programming model. These are just some illustrative applications. Using the El Paso Fire Department as a case study, the last section demonstrates how the model can be applied to optimize the assignment of ERS units, to analyze the coverage (and unit assignment) with a new base station, an increase in fleet size, and an increase in station capacity. This section presents some other potential applications of the model. For example, if an ERS department is granted a specific budget to expand its operations, how to allocate this money to achieve maximum improvement in the coverage? There are several possible alternatives. One alternative is to build one or more base stations. Another one is to purchase more units of type v. The third alternative is to expand the capacity of one or more specific base stations. The application of the improved integerprogramming model for the combined alternatives requires detailed economic information about the alternatives such as the following: (a) the cost of a new fire station; (b) the cost of a new ERS unit including the initial operation and maintenance costs of vehicle and crew members; and (c) the cost of expanding an existing fire station. If this information is known or can be estimated with a reasonable degree of accuracy, the model can be modified for each alternative to find out the respective improvement in coverage, and cost-benefit analysis performed to find out the best alternative. If the budget permits, the ERS department can even select a combination of the aforementioned alternatives. The model can be modified for a fixed budge M.

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The objective function of maximizing the total weighted coverage remains unchanged but the following constraints are added to the model:    um C m + gv C v + βj C j ≤ M (21) m∈NS

j ∈J

v∈V



xmv ≤ bm um

∀m ∈ N S

(22)

v∈V



xjv ≤ pv + g v

∀v ∈ V

be further extended as a probabilistic backup covering integerprogramming model. Another possible modification is to distinguish the weights of the CIs (wi,τ in Eq. (13)) at the different times of a day or months of a year. For example, during the summer and winter holidays, schools are usually closed. During these periods, the importance to cover the schools is not as great as that in the middle of a semester. However, the determination of weights between CIs, schools, and hospitals requires further research.

(23)

CONCLUSIONS

(24)

In this article, we have presented an improved integerprogramming model for the assignment of multiple types of ERS units to their base stations. Several improvements have been made in this model compared to past models:

j ∈(J +NS)



xjv ≤ bj + βj ∀j ∈ J

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v∈V

um = 0, 1, ∀m ∈ NS; g v ≥ 0, ∀v ∈ V ; βj ≥ 0,

∀j ∈ J (25)

where NS = the set of candidate new fire stations; um 1 = if the candidate fire station m is selected; otherwise 0; = the capacity of candidate new fire station m, a bm known nonnegative integer; = the number new type v units, an integer; gv = the additional capacity for existing fire station j , βj an integer; = the cost to of a new candidate fire station m; Cm = the total initial cost (vehicle and staff costs) of a Cv type v unit; = the cost to increase the capacity of existing fire Cj station j by 1 unit; The new decision variables are um , ∀m ∈ N S, g v , ∀v ∈ V, and βj , ∀j ∈ J . Constraint (21) guarantees that the total cost is not greater than the available budget. Constraint (22) is to guarantee that if the candidate fire station m is not selected, the capacity of that station is set to 0 so that no unit is assigned to that station. Constraint (23) caps that the total number of units including the possible new purchase. Constraint (24) updates capacity of existing fire stations. Another possible application is taking the resource redundancy into consideration. In the optimized results in the case study, the firefighting vehicles and rescues are not assigned to every fire station. Some fire stations have no assigned vehicles (see Table 1). How to deal with those fire stations? Note that, in the day-to-day operations, the EPFD still needs to assign the remaining vehicles to these stations to respond to the routine calls from residents and businesses. The remaining firefighting vehicles may be distributed among the remaining fire stations with this consideration. An optimization model may be formulated for this task. One possible consideration is to use those fire stations as backups to cover CIs. In this way, this model can

1. A typical day is divided into several time periods. Coverage of CIs is computed for the different time periods and the combined to form the total weighted coverage. 2. In each period, the requirement of service reliability for a CI is determined based on the frequency of emergency calls during that period and in the area surrounding the CI. This way of computation takes into account the temporal and spatial distribution of the competing demand for emergency services. 3. In each time period, the travel time between base stations and CIs is estimated by means of traffic assignment in the transportation network during the same time period. This method of estimation takes into account the traffic congestion during certain hours of a day. 4. The model has been modified to handle different capacities at the different stations. The improved model has been applied to the city of El Paso, Texas, to assign 45 firefighting units and 23 ambulances among 34 active fire stations to cover 138 CIs. Three time periods were used to represent three shifts. After performing optimization, the total coverage for the CI is improved from 11 to 91 node shifts. The flexibility of the model has been demonstrated with the applications to the following scenarios: (a) when a new station is added to the system, (b) when there is an increase in the fleet size of a certain type of unit, and (c) when there is an increase in station capacity. Further changes to the model may be made by modifying the objective function or adding/modifying the constraints. This article has suggested a set of changes for a combination of the aforementioned scenarios in ERS infrastructure planning. However, the full extension of the model requires detailed planning and economic data which warrant further research. The applications previously mentioned are intended to show the flexibility of this model. This model is not limited to the application to cover CIs. It can be easily modified for other services and/or to cover other facilities.

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