A hypercube queueing model embedded into a genetic algorithm for

Ann Oper Res (2008) 157: 207–224 DOI 10.1007/s10479-007-0195-z

A hypercube queueing model embedded into a genetic algorithm for ambulance deployment on highways Ana Paula Iannoni · Reinaldo Morabito · Cem Saydam

Published online: 4 August 2007 © Springer Science+Business Media, LLC 2007

Abstract The hypercube is a well-known descriptive model for planning server-tocustomer systems. In the present study we adapt this model to analyze Emergency Medical Systems on highways involving partial backup and multiple dispatching of ambulances. The modified model is then embedded into a genetic algorithm to optimize the configuration and operation of the system. By embedding the hypercube into a genetic algorithm, we can support decisions, such as, determining the optimal districts for the system in order to optimize the mean performance measures. Computational results are analyzed applying the approach to the case study of an EMS operating on Brazilian highways. Keywords Emergency medical systems · Hypercube queueing model · Genetic algorithm · Ambulance deployment · Highways 1 Introduction When designing or modifying the configuration of Emergency Medical Systems (EMS), managers try to balance the benefits of improving user service (shorter mean response times, fewer lost calls, better quality of clinical, pre-hospital care) at the expense of higher capacity investments (more ambulances, stations, better training of personnel, improved screening of calls) in the system. In EMS on highways the mean response time to incidents is an important component of the user service. Lost calls and excessive delays in the response time are directly related to the conflict between randomly occurring service requests and occasional A.P. Iannoni · R. Morabito () Department of Production Engineering, Federal University of Sao Carlos, Sao Carlos, SP, Brazil e-mail: [email protected] A.P. Iannoni e-mail: [email protected] C. Saydam Department of Business Information Systems and Operations Management, University of North Carolina at Charlotte, Charlotte, NC, USA e-mail: [email protected]

208

Ann Oper Res (2008) 157: 207–224

unavailability of the closest ambulances. Comprehensive analysis of these systems requires that probabilistic factors related to the spatial (location) and temporal (time) distribution of both calls and servers should be modeled. Most models in the facility location literature take into account uncertainties only in the server availability and other important probabilistic variables of the system are not considered in the analysis. Swersey (1994), Owen and Daskin (1998) and Brotcorne et al. (2003) present surveys of the relevant location models to analyze emergency systems. The hypercube model (Larson 1974; Larson and Odoni 1981), which is based on spatially distributed queueing theory and Markovian analysis approximations, has been one of the most effective descriptive approaches to analyze EMS. The basic idea is to expand the state space description of a simple multi-server queueing system (e.g. M/M/N, where N is the number of servers) in order to represent each server individually and incorporate reasonably complex dispatching policies. The model requires the solution of linear systems of O(2N ) equations, where the variables involved are the equilibrium state probabilities of the system. With these probabilities, a number of interesting performance measures such as server workloads, mean server response times, fraction of dispatches of each server to each region, among others, can be evaluated. The hypercube model has been widely studied and applied to police and ambulance kinds of emergency response systems. For example, studies by Batta et al. (1989), Saydam and Aytug (2003), Chiyoshi et al. (2003), Galvão et al. (2005) and Rajagopalan et al. (2006) combined the hypercube model with optimization procedures for server deployment. Other studies have extended the original model to remove some of its limiting assumptions or improve its computational efficiency, such as Larson (1975), Jarvis (1985), Chelst and Barlach (1981), Swersey (1994), Mendonça and Morabito (2001), Atkinson et al. (2005) and Iannoni and Morabito (2007). In particular, Mendonça and Morabito (2001) adapted the single dispatch hypercube model to deal with partial backup on highways, and Chelst and Barlach (1981) extended the hypercube model by relaxing the assumption that only one server can be dispatched to a call, to analyze police patrol systems. Examples of applications of the hypercube model in urban EMS in the United States can be found in Larson and Odoni (1981), Chelst and Barlach (1981), Brandeau and Larson (1986), Burwell et al. (1993) and Sacks and Grief (1994). More recently, the hypercube has been considered as a deployment model for response to terrorism attacks and other major emergencies (Larson 2004). In Brazil, the hypercube model has been applied to analyze urban EMS (Takeda et al. 2007) and EMS on highways (Mendonça and Morabito 2001; Iannoni and Morabito 2007). Other models for ambulance location were considered in Eaton et al. (1985), Fujiwara et al. (1987), Goldberg and Paz (1991) and Harewood (2002). In the present study we extend the multiple dispatch hypercube model proposed by Chelst and Barlach (1981) to analyze EMS on highways with partial backup and zero-line capacity. The modified model is then integrated into a genetic algorithm to analyze and optimize such systems. Recent studies that combine the hypercube model in single objective optimization schemes were focused on determining optimal ambulance locations (location problem). On the other hand, the present approach is particularly concerned with the problem of determining the optimal primary and secondary response areas for the ambulances (districting problem), while considering different conflicting objectives such as the mean user response time, the imbalance of ambulances’ workloads and the fraction of calls with response times exceeding a predetermined threshold (e.g., 10 minutes). Computational results for a case study of an EMS on a Brazilian highway are provided. These results are validated using sample data and discrete simulation procedures. To the best of our knowledge, we are not aware of other studies dealing with this particular districting problem in EMS on highways.

Ann Oper Res (2008) 157: 207–224

209

The paper is organized as follows: Sect. 2 presents a brief description of the EMS under consideration and Sect. 3 describes the partial backup and multiple dispatch hypercube model to analyze this EMS. Section 4 analyzes the results of the main performance measures of the system; these results are validated using sample data and discrete simulation procedures. Section 5 presents a hypercube embedded genetic algorithm (GA/hypercube algorithm) to prescribe the best configurations (sizes of operation areas) for the system according to different performance measures: mean region wide travel time, server workload imbalance and fraction of calls not serviced within 10 minutes. Section 6 analyzes the outcomes from the application of the algorithm to the case study. Finally, Sect. 7 discusses the conclusions of this study and perspectives for future research.

2 EMS on highways The EMS on Brazilian highways are typically zero-line capacity systems, since if a call arrives in the system when its candidate ambulances are busy, the call is immediately transferred to another local agency which is usually unable to provide the same service quality. The EMS on highways operate within a particular dispatching policy, which considers that only certain ambulances in the system can travel to a given region due to the limitations of travel distance or time. Moreover, in some of these EMS two ambulances can be simultaneously dispatched to respond to the same call (multiple dispatches). The operation of many EMS on Brazilian highways is under the management of private organizations as part of a privatization contract with the Brazilian government. Private operators are responsible for all aspects of running an EMS, including maintenance, design and responding to calls on the highways. In this paper we study an EMS located in portions of highways in Sao Paulo State and operated by a private firm. This system operates similar to other private and public EMS on highways in Brazil. This EMS deployment system has five ambulances located in five fixed bases along the highways. The operations center, located in one of the bases, handles the calls, dispatches the ambulances and tracks the ambulance movements. The EMS’ ambulance provides the first medical treatments to the persons involved in the accident, transports them to the nearest hospital (if necessary), and then goes back to its base. Further, the ambulance can also provide medical services to users that stop at its base requiring for care (e.g., epileptics, cardiac patients, alcoholics, etc). When the operations center receives a call, the nearest available ambulance is immediately dispatched to the call location. If the nearest ambulance is busy, then the next nearest ambulance (called backup) is dispatched. If the two closest ambulances are busy, the call is transferred to another system (for example, the nearest local hospital) and the call is considered lost to the system. According to this particular dispatching policy, the call can be serviced by only two ambulances (the nearest unit and the backup). By operational policy, the third nearest ambulance is never dispatched (partial backup). Furthermore, in this system, should there be a need then two ambulances can be simultaneously dispatched to a single call. The resulting system can be conceptualized as a spatially distributed multi-server loss (zero-line capacity) queueing system, with a specific and limited dispatching policy, which violates two assumptions of the basic hypercube model of Larson (1974). These are: (i) any server can travel (respond) to any call location and (ii) only one server is dispatched to a call. Figure 1 illustrates the configuration of this EMS on the stretches of highways in Sao Paulo State.

210

Ann Oper Res (2008) 157: 207–224

Fig. 1 EMS on portions of highways in Sao Paulo State

3 The hypercube model for EMS on highways The name hypercube derives from the state space of the system. For the zero-line capacity system, it can be described as the vertices of the N -dimensional unit hypercube in the positive orthant and each vertex corresponds to a particular combination of serves’ status (Larson 1974). Given that each server has two possible statuses: idle (0) or busy (1) at any given instant, there are 2N possible states to the system. A particular state of the system is given by the entire listing of ambulances that are free and busy. For example, the state 101 corresponds to a 3-server-system, with servers 1 and 3 busy and server 2 idle, and the state space is given by the vertices of a cube. If the system has more than 3 ambulances, we have a hypercube. The main assumptions of the partial backup and multiple dispatch hypercube model for application on highways are: • The highways are divided into NA geographical areas (segments of highways) or atoms, which corresponds to independent sources of calls. In each atom j calls arrive according to a Poisson process, independently from the other atoms. The calls can be of two types: type 1 calls (with arrival rate λj[1] ) require the dispatch of only one ambulance, whereas type 2 calls (with arrival rate λj[2] ) require the simultaneous response of two ambulances. For each atom there is a server dispatch preference list. According to this list, the highest ranked ambulance is dispatched to the type 1 calls, and if this ambulance is busy, the backup ambulance is dispatched. In the case of type 2 calls, the first two ambulances on the list are dispatched, and if only one of them is available, it is assigned as a single dispatch. If both ambulances are busy, then regardless of its type the call is lost, since a third ambulance is never assigned. • There are N ambulances spatially distributed along the highways and, when idle, each ambulance is stationed in its base waiting for a call. The ambulance service time includes the set-up time, the travel times and the on-scene time. In general, the servers have distinct mean service times. The model assumes negative exponentially distributed service times, however reasonable deviations from this assumption have been found not to alter the accuracy of the model, as pointed out in Larson and Odoni (1981) and Mendonça and Morabito (2001). • Type 1 calls are serviced by a single ambulance i with mean service rate μi . Type 2 calls are serviced by two ambulances i and k, which operate independently with mean service rates μi and μk , respectively. Therefore, two ambulances servicing a type 2 call are treated

Ann Oper Res (2008) 157: 207–224

211

as the same as two ambulances servicing two separated type 1 calls. If the service of type 2 calls would be modeled differently, we would have to consider an additional state for each server to differentiate servers busy at type 1 and type 2 calls, and the state space would grow to 3N (Chelst and Barlach 1981). • The travel time between two regions is known or can be estimated by geometric probability (Larson and Odoni 1981). The variations in the service time that are due to variations in travel time are assumed to be of second order compared to variations in set-up and on-scene time. It is important to note that, given that the system does not allow queue, the mean user response time is simply composed of the mean set-up time and the mean travel time. In some cases an ambulance may be available for service during its travel time back to the base. For example, if a type 1 call occurs during this time and the ambulance is the first preference of the call, this ambulance already on the highway might travel to the locale of the accident in a shorter time than the backup. Moreover, if the backup is busy, or the current ambulance is actually the backup and the first preference ambulance is busy, the call would not be lost. Note, however, that, as pointed out in Mendonça and Morabito (2001), the consideration of this alternative increases the state space of the model to 3N . For simplicity, in the present study we will disregard this alternative, since only in some cases the ambulances are available during their return travel times, the ambulance workloads and the system loss probability are low, and the return travel times are relatively short in comparison to the service times. Among the most interesting steady-state performance measures described by the partial backup and multiple dispatch hypercube model are: workloads of each ambulance, mean region wide travel time (considering the two types of calls), mean travel times to type 1 and type 2 calls, mean travel time of the two ambulances to a type 2 call, mean travel time of the first and second ambulance arriving at a type 2 call location, mean travel time of the preferential and backup ambulance arriving at a type 2 call location (note that this measure may be different of the former, since the preferential ambulance may not be the first to arrive at the location), mean fraction of dispatches of each server to each region according to the type of call, mean interval of time that the first arriving ambulance wait until the arrival of the second ambulance in a type 2 call location, and others. Some of these and other additional measures obtained by the partial backup and multiple dispatch hypercube model are defined below. 3.1 Loss probability This is an important measure to the EMS on highways, given that a call can be lost to the system even when there are ambulances available (partial backup). As result of the multiple dispatching of ambulances, we have: loss probability of type 1 calls Pp[1] , loss probability of type 2 calls Pp[2] and loss probability of any call Pp . For example, the loss probability Pp (adapted from the definition in Mendonça and Morabito (2001) to take into account different types of dispatches) is formulated as:   j ∈N [B] λj  A PB , (1) Pp = λ B where B represents the state of the system (e.g., B = {11000} is a possible state for a 5server-system), NA[B] is the set of atoms with all servers of the atom dispatching preference

212

Ann Oper Res (2008) 157: 207–224

list busy at state B, PB is the equilibrium probability of state B, λj = λj[1] + λj[2] is the arrival  rate of type 1 and type 2 calls at atom j , and λ = Na j =1 λj is the total arrival rate of the system. 3.2 Fraction of dispatches The model can produce additional measures related to fraction of server dispatches. We adapted some of these statistics from the studies in Larson (1974), Chelst and Barlach (1981) and Mendonça and Morabito (2001). For example: • Fraction of all type 1 dispatches that send ambulance i to atom j :  (λj[1] /λ[1] ) B∈Eij pB [1] , fij = (1 − Pp[1] )

(2)

where Eij corresponds to the set of states that ambulance i is the first available server in NA [1] the preference list of atom j, Pp[1] is defined as before, and λ[1] = j =1 λj is the total arrival rate of type 1 calls in the system; • Fraction of all type 2 dispatches that send ambulances i and k to atom j :  (λj[2] /λ[2] ) B∈E(i,k)j pB [2] , (3) f(i,k)j = (1 − Pp[2] ) where E(i,k)j corresponds to the set of states that ambulances i and k are the two first available servers in the preference list of atom j , Pp[2] is defined as before and λ[2] = NA [2] j =1 λj is the total arrival rate of type 2 calls in the system; • Fraction of all type 2 dispatches that ambulance i is the single ambulance sent to atom j :  (λi[2] /λ[2] ) B∈Fij pB [2] , (4) fij = (1 − Pp[2] ) where Fij corresponds to the set of states that only ambulance i can respond calls from atom j . More details related to the numerator of these formulas can be found in Chelst and Barlach (1981). Since in the present study we consider a loss queueing system, the denominator of these formulas include the terms (1 − Pp[1] ) or (1 − Pp[2] ), corresponding to the loss described in the aforementioned reference, we found that NA [1] Note that, N probabilities. NAas  N N−1 N [2] [2] f = 1 and [ f + f ] j =1 ij j =1 i=1 i=1 ij i=1 k=i+1 (i,k)j = 1. Other statistics can be calculated such as: fraction of all dispatches that send ambulance i to atom j (fij ), fraction of all dispatches that send ambulance i to atom j to respond a type 1 call (fij [1] ), fraction of all dispatches that send ambulance i to atom j to respond a type 2 call (fij [2] ), and fraction of all dispatches that send ambulances i and k to a type  Natom jN to respond  [2]  [2]  [1]  [2] A ), where fij = (fij [1] + fij [2] ) + N f ) and [ (f + f 2 call f(i,k)j ij ) + j =1 k=i (i,k)j i=1 ij N−1 N  [2] i=1 k=i+1 f(i,k)j ] = 1. 3.3 Aggregated measures of travel time Using the fractions of dispatches defined above, we can obtain interesting travel time measures. Some of these statistics are defined as follows (Chelst and Barlach 1981):

Ann Oper Res (2008) 157: 207–224

213

• Mean travel time to type 1 calls: T¯ [1] =

NA N  

fij[1] t ij ,

(5)

j =1 i=1

where t ij corresponds to the mean travel time from ambulance i base to atom j . • Mean travel time to type 2 calls for the first arriving ambulance for a type 2 dispatch: T

[2]

=

N−1 N NA    j =1

[2] f(i,k)j

min(t ij , t kj ) +

i=1 k=i+1

N 

 fij[2] t ij

.

(6)

i=1

• Mean travel time for the two ambulances for a type 2 dispatch: Tt

[2]

=

N−1 N NA    j =1

[2] f(i,k)j (t ij

i=1 k=i+1

+ t kj ) +

N 

 fij[2] t ij

.

(7)

i=1

Note that in expressions (6) and (7) we are considering either type 2 dispatches that send ambulances i and k to atom j , and type 2 dispatches that send the single ambulance i to atom j . • Mean travel time for the first arriving ambulance and mean travel time for the second arriving ambulance for type 2 dispatches, when ambulances i and k are sent: NA N−1 N

[2] i=1 k=i+1 f(i,k)j min(t ij , t kj ) , NA N−1 N [2] j =1 i=1 k=i+1 f(i,k)j NA N−1 N [2] j =1 i=1 k=i+1 f(i,k)j max(t ij , t kj ) Ts = . NA N−1 N [2] j =1 i=1 k=i+1 f(i,k)j

Tf =

j =1

(8)

In this case, we are considering only the dispatches that sent two ambulances (i and k) to respond a type 2 call in atom j . • Mean travel time for ambulance i (type 1 dispatch) and mean travel time for ambulance i (type 2 dispatch): [1] T Ui

NA =

[1] j =1 fij t ij NA [1] j =1 fij

,

[2] T Ui

NA N [2] [2] j =1 [ k=i f(i,k)j t ij + fij t ij ] = N N . [2] [2] A j =1 [ k=i f(i,k)j + fij ]

(9)

These two expressions were adapted from the ones in Larson and Odoni (1981), considering type 1 and type 2 dispatches.

4 Results of the hypercube model application For the application of the hypercube model of Sect. 3, we divided the region under study into 8 atoms (segments of highways), according to the primary area of each base established by the system operators. The sample data was collected at the EMS’ operations center between December 2001 and June 2002, and 945 events were reported during this period. The configuration of the system did not change during this period. For each atom we conducted

214

Ann Oper Res (2008) 157: 207–224

Table 1 Statistics of the arrival and service processes and the server dispatch preferences Atom

Arrival rates (calls/h)

Dispatch preference list

Server

Service

Workload

j

[1] Type 1 λj

[2] Type 2 λj

First server

Backup server

i

rate (calls/h)

ρi

1

0.06181

0.00340

1

2

1

1.3085

0.0587

2

0.04819

0.00476

2

1

2

1.0642

0.0532

3

0.00693

0.00040

1

2

3

0.8247

0.0182

4

0.00996

0.00042

3

1

4

1.0852

0.0252

5

0.00494

–

3

2

5

1.1495

0.0185

6

0.01139

–

4

3

7

0.01440

–

4

5

8

0.01937

0.00196

5

4

graphical and statistical analyses to determine if there were significant variations in the call data by month and/or by day of the week. These analyses indicated that the monthly and daily user arrival rates remained approximately constant during this period; in particular the corresponding confidence intervals are small. To test the validity of the steady-state assumption in this study, we simulated the system starting with an empty system (all ambulances assumed idle). The simulation results showed that the warm-up period is relatively short (in order of hours), therefore, the steady-state assumption is acceptable for this study. Table 1 presents statistics of the atom arrival processes and the ambulance service processes. This table also shows the dispatchpreference list for each atom. Note that  λ = 8j =1 λj[1] + λj[2] = 0.1878 calls/h and μ = 5i=1 μi = 5.4321 calls/h. We applied goodness-of-fit tests (Kolmogorov-Smirnov, Andersen- Darling, χ -square) using the Best-Fit software to verify the assumption of Poisson arrival processes. For all atoms, the tests were unable to reject this assumption with 5% of significance. On the other hand, goodness-of-fit tests were able to reject the assumption that the service processes are exponentially distributed. However, as discussed in Sect. 3, since the system does not allow queue, reasonable deviations from this assumption do not alter the accuracy of the model. As shown below, this result is validated using sample data and discrete simulation. The data related to the travel time of each server to each atom was also obtained from the sample and used to determine the mean travel times of the system. When the system is in steady state, the equilibrium probability for each state is formulated taking into account that the in flow is equal to the out flow. Given that the EMS analyzed has N = 5 servers, there are 25 = 32 possible states. Following the preference list in Table 1, one can formulate the equilibrium equations for each state of the system. We illustrate this using the two examples below (for states B = {11001} and {00110}): Example 1 B = 11001: p11001 (λ4[1] + λ5[1] + λ6[1] + λ7[1] + λ8[1] + λ4[2] + λ5[2] + λ6[2] + λ7[2] + λ8[2] + μ1 + μ2 + μ5 ) = p00001 (λ1[2] + λ2[2] + λ3[2] ) + p10001 (λ1[1] + λ2[1] + λ3[1] + λ1[2] + λ2[2] + λ3[2] ) + p01001 (λ1[1] + λ2[1] + λ3[1] + λ1[2] + λ2[2] + λ3[2] ) + p11000 (λ8[1] ) + p11101 (μ3 ) + p11011 (μ4 ).

(10)

Ann Oper Res (2008) 157: 207–224

215

Note on the left side of this expression (outflow rate of state {11001}), that type 1 and type 2 calls arriving in the system from atoms 1, 2 or 3 on the state 11001 are lost since the two ambulances of their list are busy (Table 1). On the right side of the equation (inflow rate of state {1110}), we verify the following transitions into state {11001}: • {00001} → {11001}: servers 1 and 2 go simultaneously into service when a type 2 call arrives at atoms 1, 2 or 3. Note in Table 1 that servers 1 and 2 are the two first servers in the preference dispatching list of atoms 1, 2 and 3; • {01001}→ {11001}: server 1 goes into service when a type 1 call arrives at atoms 1, 2 or 3, or a type 2 call arrives at atoms 1, 2 or 3 (i.e., as server 2 is busy, server 1 is assigned as single server to a type 2 call); • {10001} → {11001}: server 2 goes into service when a type 1 or a type 2 call arrives at atoms 1, 2 or 3; • {11000} → {11001}: server 5 goes into service when a type 1 call arrives at atom 8 (note in Table 1 that server 5 is the preferential server of atom 8); • {11101} → {11001}: server 3 finishes service; • {11011} → {11001}: server 4 finishes service. Example 2 B = 10001: p10001 (λ + μ1 + μ5 ) = p00001 (λ1[1] + λ3[1] ) + p10000 (λ8[1] ) + p11001 (μ2 ) + p10101 (μ3 ) + p10011 (μ4 ).

(11)

Note on the left side of expression (11) (outflow rate of state {10001}) that any call can enter in the system in this state (i.e, there are no lost calls), since servers 1 and 5 have idle backup servers (servers 2 and 4, respectively). The transitions on the right side of the expression above are: • {00001}→ {10001}: server 1 goes into service when a type 1 call arrives at atoms 1 or 3 (note in Table 1 that server 1 is the preferential server); • {10000}→ {10001}: server 5 goes into service when a type 1 call arrives at atom 8; • {11001}→ {10001}: server 2 finishes service; • {10101}→ {10001}: server 3 finishes service; • {10011}→ {10001}: server 4 finishes service. The results of the state probabilities showed that the system stays most of the time with all servers available, given that p00000 = 0.8434 and p11111 = 0.00000034. The system loss probability for all calls Pp is only 0.6%, meaning that the user arrival and input rates are almost the same. Calculating the workload ρi for each ambulance i using the equilibrium state probabilities, we obtain: ρ1 = 0.0578, ρ2 = 0.0537, ρ3 = 0.0186, ρ4 = 0.0253 and ρ5 = 0.0185. Observe that the busiest (highest workload) server, ρ1 , is more than twice times busier than the least (lowest workload) busy server, ρ5 . These results present very small deviations to the results from the sample analysis on Table 1 (mean deviation of 1%). Calculating the frequencies statistics described in Sect. 4, we found that dispatches sending ambulance 1 to atom 1 and ambulance 2 to atom 2 present the high[1] [2] [2] [1] = 33.10%, f(1,2)1 + f11 = 29.48%, f11 = 32.89% and f22 = 25.92%, est frequencies: f11 [2] [2] f(2,1)2 + f22 = 41.46%, f22 = 24.73%. For illustration, Table 2 presents the complete results   for fij[1] ; note that 5i=1 8j =1 fij[1] = 1. We used the matrix of travel times from each base to each atom obtained via sample data analysis to calculate travel time measures. Table 3 presents some aggregate travel time

216

Ann Oper Res (2008) 157: 207–224

Table 2 Fraction of all type 1 dispatches that send ambulance i to atom j (fij[1] ) Server

Atom

Atom

Atom

Atom

Atom

Atom

Atom

Atom

i

1

2

3

4

5

6

7

8

1

0.3310

0.0124

0.0371

0.0010

–

–

–

–

2

0.0174

0.2592

0.0019

–

–

–

–

–

3

–

–

–

0.0556

0.0276

0.0016

–

–

4

–

–

–

–

0.0005

0.0631

0.0798

0.0019

5

–

–

–

–

–

–

0.0019

0.1081

Table 3 Aggregate travel time measures (in minutes) Mean travel times

T T

[1] [2]  [2]

Hypercube model

6.277

Sample data

Deviation

5.962

5.30%

8.186

Simulation model

6.237

Simulation Confidence Interval 6.176–6.298

8.118

7.906–8.330

7.807

7.613–8.001

7.776

7.201

7.90%

Tt

25.130

25.144

−0.06%

Tf

7.776

7.771

0.06%

7.807

7.613–8.001

Ts

17.373

17.944

−3.30%

17.344

17.112–17.576

T

 [2]

Average deviation

1.98%

Maximum deviation

7.90%

measures calculated by the hypercube model and compare them to the results from the sample data. As the sample data analysis could not identify and consider the dispatches that  [2]  [2] and T t (given by sent a single ambulance to respond a type 2 call, we also calculated T [2] [2] the expressions below) instead of T and T t (as defined in Sect. 3), respectively, in order to take into account only the dispatches that sent two ambulances to respond a type 2 call and compare the results to the sample data. Some of the model measures are also compared to the results from the simulation model. NA N−1 N [2]  [2] j =1 i=1 k=i+1 f(i,k)j min(tij , tkj ) T = , NA N−1 N [2] j =1 i=1 k=i+1 f(i,k)j (12) NA N−1 N [2]  [2] j =1 i=1 k=i+1 f(i,k)j (tij + tkj ) Tt = . NA N−1 N [2] j =1 i=1 k=i+1 f(i,k)j Note in Table 3 that the estimates of the travel time measures obtained by the hypercube model are close to the sample data, since the mean deviation is around 2%. The maximum deviation is 7.9%, which is related to the mean travel time of the first unit arriving at a type 2 call. The reason for this difference is that the hypercube model calculates the mean travel time for the first ambulance using the mean travel time data between atoms. Consequently, the first ambulance arriving in a call location corresponds to the first preferred ambulance to that call. The sample data analysis does not use the mean travel time between atoms and, as a

Ann Oper Res (2008) 157: 207–224

217

result, the mean travel time is calculated considering events that the preferential ambulance is the second arriving. Note that the mean interval that the first ambulance waits for the arrival of the second is T s − T f = 9.6 minutes. We also calculated the fraction of calls requiring more than 10 minutes to be reached by the ambulance, using the mean travel time from the ambulance’s base to the atom, and it can be defined as follows: Ptv>10 (x) =

NA N    [2] (fij [1] + fij [2] + f(i,k)j )p(tij > 10),

(13)

i=1 j =1  [2] )p(tij > 10) is the fraction of all dispatches of server where the term (fij [1] + fij [2] + f(i,k)j i to atom j to which the travel time exceeds 10 minutes, and p(tij > 10) is the probability that the travel time of server i to atom j is greater than 10 minutes. When the travel time data is not available, we can estimate p(tij > 10) by determining the portion of each atom j (call location) that server i cannot reach under 10 minutes (using geometric probability concepts). The model found that the mean travel time exceeded 10 minutes for 26.18% of all calls, 25.72% of type 1 calls and 33.53% of type 2 calls.

4.1 Simulation model In order to verify if the assumptions of the hypercube model affects the analysis of the main performance measures, we built a simulation model of the system using the Arena software. As the assumption of exponential distribution for the service times was rejected (as discussed above), statistical analysis was performed using the Best-Fit software in order to obtain the best-adjusted statistical distributions representing the data (using goodness-of-fit tests). The following distributions for the service time (in minutes), not including the travel time from the server’s base to the call location, were found to each ambulance in the system: ambulance 1—Lognormal (41.10, 57.24); ambulance 2—Erlang (16.51, 3); ambulance 3— Lognormal (69.46, 66.59); ambulance 4—Erlang (9.59, 5) and ambulance 5—Lognormal (49.51, 39.13). The transient period (warm up) was determined by the graphical analysis of the moving average of the results obtained during a long simulation run. Considering one of the main performance measures, the mean number of calls in the system, the transient period was close to 1,000 min (or around 17 hrs). The batch means method (Pegden et al. 1995; Banks 1998; Kelton et al. 2002) was applied to determine the simulation run-length period. The simulation software performed the statistical procedures determining the number of observations in each batch in order to assure a correlation close to zero between different batches (more details of this procedure can also be found in Iannoni and Morabito 2006). Following this procedure and considering the measure with the smallest number of observations, the simulation run length was calculated to be 5.18 ×106 minutes (e.g., including the transient period). The last two columns of Table 3 compare the results of the aggregate travel time measures obtained by the hypercube and the simulation models, including the 95% confidence intervals of the simulations results. Table 4 compares the results of travel times for each ambulance obtained by the hypercube and the simulation models. Note in Tables 3 and 4 that the average values obtained by the hypercube are reasonably close to the ones of simulation; note also that they are within the simulation confidence intervals, validating the model outputs.

218

Ann Oper Res (2008) 157: 207–224

Table 4 Travel time (in minutes) and workload statistics for each ambulance Server

Model

i 1

2

3

4

5

[1]

[2]

T Ui

T Ui

TU

type 1

type 2

overall

ρi

Hypercube

5.993

10.460

6.232

0.0578

Simulation

5.982

10.356

6.202

0.0601

(Conf. Interval)

5.888–6.075

10.444–10.567

6.108–6.296

0.0576–0.0624

Hypercube

7.342

6.862

7.300

0.0537

Simulation

7.287

6.855

7.250

0.0538

(Conf. Interval)

7.192–7.382

3.594–3.729

7.154–7.346

0.0521–0.0555

Hypercube

6.686

18.450

7.003

0.0186

Simulation

6.671

18.450

7.030

0.0189

(Conf. Interval)

6.533–6.809

–

6.865–7.195

0.0176–0.0202

Hypercube

6.705

15.250

6.716

0.0253

Simulation

6.681

15.250

6.689

0.0255

(Conf. Interval)

6.589–6.772

–

6.595–6.783

0.0244–0.0282

Hypercube

3.692

4.600

3.765

0.0185

Simulation

3.662

4.600

3.748

0.0193

(Conf. Interval)

3.594–3.729

–

3.692–3.804

0.0177–0.0213

5 A hypercube embedded genetic algorithm In this section we present a genetic algorithm embedded with the hypercube model as outlined in Sect. 3 (called GA/hypercube algorithm) to search for near-optimal atom sizes for the system. Our choice of GA approach is motivated by recent successful implementations of hypercube embedded metaheuristic search methods applied to ambulance location problems, such as in Chiyoshi et al. (2003), Saydam and Aytug (2003) and Galvão et al. (2005). It should be pointed out that while these studies dealt with single objective location problems, the present algorithm is concerned with a districting problem involving conflicting objectives. We use a standard genetic algorithm as described in Holland (1975), Goldberg (1989) and Michalewicz (1996). The main components considered in the implementation of the GA/hypercube algorithm are briefly presented in the following sections. 5.1 Chromosome representation and generation We defined a procedure that modifies the sizes of the atoms to produce different feasible configurations (chromosomes) for the system. Figure 2 illustrates a size variation of the atoms 1 and 2 between the two adjacent bases 1 and 2. Note in the figure that the sizes of atoms 1 and 2 change from z10 and z20 to z1 = x1 d1 and z2 = d1 − z1 , respectively, where x1 is a percentage of d1 , the distance between bases 1 and 2. Each chromosome is represented as a vector x = (x1 , x2 , . . . , xN−1 ), where each xi is the fraction of the distance between bases i and i + 1. The size of the preferential atom for server i is given by xi di . The remaining distance between the bases i and i + 1 becomes the first preferred atom for base i + 1. As suggested by the system managers, we impose that 0.2 ≤ xi ≤ 0.8, limiting the preferential area of each server i from 20 to 80 percent of the

Ann Oper Res (2008) 157: 207–224

219

Fig. 2 Example of the size variation of atoms 1 and 2 between bases 1 and 2

distance di . To generate possible system configurations (chromosomes) for initial and subsequent populations we conducted computational experiments which favored a finite discrete approach over a continuous approach where each chromosome is populated by a sorted set of continuous random numbers between 0.2 and 0.8. The finite approach simply adds an increment k to 0.2 (the lower limit of the interval), where  is fixed and k is an integer randomly sorted in the range [0, M = (0.8 − 0.2)/]. Therefore, there are M + 1 possible values for each xi . This simple representation worked reasonably well for the purposes of this study; we are not aware of other alternative chromosome representations for districting problems on highways in the literature. Note that there are M + 1 possible values for each yi . The enumerative algorithm was applied with  = 0.03, that is, to (M + 1)N−1 = 214 possible combinations. We define a procedure that modifies the input data to the hypercube model according to the different configurations of the system, which are generated by the variation of the atom’s sizes. In this way, for each configuration the algorithm calculates the arrival rates in each atom in order to preserve the arrival rate distribution along the entire highway. After a change in the sizes of the atoms j and j + 1 between the adjacent servers i and i + 1, the new arrival rates λj and λj +1 are calculated considering the initial arrival rates λ0j and λ0j +1 as follows:  λj = zj (λ0j /zj0 ), 0 If zj < zj then λj +1 = λ0j +1 + (zj +1 − zj0+1 )(λ0j /zj0 ), (14)  λj = λ0j + (zj − zj0 )(λ0j +1 /zj0+1 ), If zj > zj0 then λj +1 = zj +1 (λ0j +1 /zj0+1 ), where, as in Fig. 2, zj0 is the initial size of atom j , zj is the new size of atom j , and di corresponds to the distance between the two adjacent servers i and i + 1. As mentioned before, the travel time statistics presented in Tables 3 and 4 were calculated using the sample data travel times, which consider the distribution of the call location along the highways. However, using the proposed procedure to generate new atom sizes, we recalculated the matrix of travel times based on the distance from each base to the atom’s centroid (as proposed in Larson and Odoni 1981). Therefore, each new configuration generates a new matrix of travel times as a consequence of the new position of the atom’s centroid. We also use the average speed for each ambulance obtained from the sample data to calculate the travel times. 5.2 Evaluation and fitness function The evaluation procedure of the GA/hypercube utilizes the multiple dispatch hypercube model outlined in the previous sections to compute the performance measures for each configuration (represented by a chromosome). Some performance measures described by

220

Ann Oper Res (2008) 157: 207–224

the hypercube model may be conflicting in terms of the different interests of the parties involved. For example, the balance of the server workloads is an internal performance measure of the system, which is of interest to operations managers. On the other hand, the response time is an external performance measure, which is important to the users of the system, the public. As shown in this section, when we reduce mean response time, we may worsen the workload imbalances, and vice-versa. In this study, we conducted three different experiments using three different measures (fitness function f (x) of chromosome x) in order to evaluate the generated configurations and prescribe the best one. In one of the experiments, the objective is to minimize the mean region wide travel time, that is, min f (x) = T¯ (x):

N NA N−1 N    [1]    [2]  [2] T (x) = (fij + fij )tij + f(i,k)j min(tij , tkj ) , j =1

i=1

(15)

i=1 k=i+1

 [2] where fij [1] , fij [2] , f(i,k)j and tij were defined in Sect. 3. In other experiments, the objective is minimizing the fraction of calls not serviced within 10 minutes, that is, min f (x) = Pt>10 (x) (as described in Sect. 4), or minimizing the unbalance of server workloads evaluated by the standard deviation, min f (x) = σρ (x), as follows:

 σρ (x) =

N

i=1 (ρi

− ρ)2

N

,

(16)

where ρi is the workload of server i (i.e.,the sum of the equilibrium probabilities of the states for which server i is busy) and ρ¯ = N i=1 ρi /N is the mean server workload. 5.3 Selection of chromosomes, crossover and mutation The selection of parents follows the roulette wheel method, with probabilities based on the fitness function value. After selecting two parents, with probability pc we applied the wellknown single-point crossover (and with probability 1−pc , the selected parents are preserved in the next generation). The mutation procedure is applied to each gene in the chromosome with a predefined probability pm (Goldberg 1989; Michalewicz 1996). In order to randomly replace the gene, we use the same initialization discrete procedure described before. For example, for each gene xi (with probability pm ) this value is mutated to xi = k, where k is uniformly sorted in the interval [0, . . . , M]. There is no possibility of invalid chromosomes through crossover and mutation. 5.4 Parameter tuning In setting the critical parameters of the GA/hypercube algorithm such as crossover and mutation probabilities (pc and pm ), number of generations (G) and population size (Pop), we first ran extensive tests varying the values of these parameters within certain ranges. The set of values: pc = 0.5, pm = 0.06, G = 1000 generations and Pop = 100 chromosomes yielded the best results in most of the tests. An additional parameter is the precision  required for the finite method we devised to generate the chromosomes. The intervals  = 0.03 and  = 0.01 (i.e., M = 20 and M = 60, respectively) were tested in each of the three fitness functions.

Ann Oper Res (2008) 157: 207–224

221

Fig. 3 General structure of the GA/hypercube algorithm

Figure 3 presents the general scheme of the GA/hypercube algorithm. Several studies, such as Hertz and Kobler (2000), Jaszkiewicz (2002), Beasley (2002) and Arroyo and Armentano (2005) have pointed out the superior performance of the hybrid genetic algorithms, which includes a local search procedure after crossover and mutation operators. However, in the present study, this alternative was not adopted because of the high computational costs due to the hypercube model involved in the evaluation of the local neighborhood (i.e., solution of a large number of linear systems of size 2N ).

6 Computational results for the case study The GA/hypercube algorithm proposed was coded in Pascal language and run on a 1.66 GHz Athlon microcomputer. To verify the quality of the solutions produced by the GA/hypercube algorithm, we developed a simple procedure that incorporates the hypercube model of Sect. 3 into a simple enumerative algorithm. This exhaustive procedure determines the optimal atom sizes (under a precision) testing all possible configurations to the system (using

222

Ann Oper Res (2008) 157: 207–224

Table 5 Three performance measures of the best solution Measures

Original

First

Percent

Second

Percent

Third

Percent

Configuration

objective

Improved

objective

Improved

objective

Improved

T (x)

5.9584

3.4054

42.85%

4.8047

19.36%

5.9654

−0.12%

Ptv>10 (x)

0.3119

0.3472

−11.32%

0.2779

10.90%

0.2970

4.78%

σρ (x)

0.01733

0.0207

−19.73%

0.01732

0.05%

0.0163

6.00%

the discrete method described in Sect. 5). Given that there are M + 1 possible values for each xi , the enumerative algorithm considers (M + 1)N−1 possible combinations. It is computationally treatable only for problems of moderate size, as for the EMS under consideration with only N = 5 ambulances and a precision of  = 0.03 or  = 0.01 (yielding 214 or 614 combinations, respectively). The original configuration of the system based on the atoms’ size is represented by the chromosome x = (x1 , x2 , x3 , x4 ) as discussed above, where: x1 = 0.61, x2 = 0.28, x3 = 0.47 and x4 = 0.61. The results of the three performance measures evaluated by the hypercube model in the original configuration are: T¯ = 5.958 minutes, σρ = 0.0173 and Ptv>10 = 0.3119. Using  = 0.03, the enumerative algorithm found a different configuration (represented by vector x) for each objective, that is: first objective: x1 = 0.23, x2 = 0.32, x3 = 0.62 and x4 = 0.53; second objective: x1 = 0.44, x2 = 0.20, x3 = 0.74 and x4 = 0.50 and third objective: x1 = 0.62, x2 = 0.20, x3 = 0.59 and x4 = 0.53. Table 5 shows the results for the best solution (in bold) in terms of each of the three objectives, as well as the relative deviation (percent improved) to the original configuration. We then solved each experiment using the GA/hypercube algorithm, which consistently found the optimal solutions. We used the parameter values defined in Sect. 5 and performed 10 runs using different seeds for G = 1000 generations (using  = 0.03). Besides the optimal values (in bold) for each objective, Table 5 also shows the values of the other two measures obtained by the GA/hypercube algorithm in each experiment, as well as the relative deviation (percent improved) to the original configuration. Regarding the computational time, a single run of the GA/hypercube for these experiments (where, N = 5 servers and G = 1000 generations) took an average of 70 seconds to be completed. Note in Table 5 that, in the first experiment, T¯ (to be minimized) reduces 42.85% at the expenses of increases of 19.73% and 11.32% in σρ and Ptv>10 , respectively, when compared to the initial configuration of the system. In the second experiment, Ptv>10 (to be minimized) reduces 10.9% at no expenses in σρ and T¯ . Finally, in the third experiment, σρ (to be minimized) reduces 6% at the expense of an increase of 0.12% in T¯ . Therefore, as expected, the results of these analyses (applying each objective separately) show that the evaluated measures may be conflicting. For example, the best solution in terms of T¯ results in the worst solution in terms of σρ and Ptv>10 . The GA/hypercube algorithm can be used to generate the trade-off curves (Pareto’s curves) among different performance measures of the system. It should be emphasized that the results above shown that T¯ , Pt>10 and σρ could be improved by simply modifying the atom sizes of the system, without relocating ambulances and without requiring additional capacity investments. 7 Conclusions In this study we adapted the hypercube model to analyze EMS on highways considering particular dispatching policies with zero-line capacity, partial backup and multiple dispatch-

Ann Oper Res (2008) 157: 207–224

223

ing of ambulances. We developed a method that embeds the extended hypercube model into a genetic algorithm, The GA/hypercube algorithm is effective to support design and operational decisions, for example, to determine the best operation areas for the ambulance’s bases of the system, minimize the mean user response time and the ambulances’ workload imbalances. Computational results were analyzed applying the GA/hypercube algorithm to the case study of an EMS on a Brazilian highway. We verified the optimality of the solutions generated by the algorithm using an exhaustive enumeration procedure for systems of moderate size. In particular, it was shown that different system performance measures or objectives, such as the mean user response time, fraction of calls not serviced within 10 minutes and workload imbalance among the ambulances could be improved by simply modifying the atom sizes of the system, without relocating ambulances and without requiring additional capacity investments. An interesting line of research is the combination of metaheuristics, such as tabu search and simulated annealing, with the hypercube model, instead of a genetic algorithm, in order to treat the districting problem in EMS on highways. In particular, one encouraging extension of the present study would be to combine location and districting decisions (i.e., location of ambulances and their response areas) in the same optimization approach. Acknowledgements The authors thank the anonymous referees for the useful comments and suggestions, Centrovias for providing the case study data, and the financial support from CNPq (grants 140178/01-5, 522973/95-4), CAPES (grant BEX2468/02-6), FAPESP (grant 05/53126-5) and from the Belk College of Business, UNC Charlotte.

References Arroyo, J. C., & Armentano, V. A. (2005). Genetic local search for multi-objective flowshop scheduling. European Journal of Operational Research, 167, 717–738. Atkinson, J. B., Kovalenko, I. N., Kuznetsov, N., & Mykhalevych, K. V. (2005, to appear). Heuristic solution methods for a hypercube queueing model of the deployment of emergency systems. Cybernetics and Systems Analysis. Banks, J. (1998). Handbook of simulation (pp. 3–389). Atlanta: Wiley. Batta, R., Dolan, J. M., & Krishnamurthy, N. N. (1989). The maximal expected covering location problem: revisited. Transportation Science, 23, 277–287. Beasley, J. E. (2002). Population heuristics. In P. M. Pardalos & M. G. C. Resende (Eds.), Handbook of applied optimization (pp. 138–157). Oxford: University Press. Brandeau, M., & Larson, R. C. (1986). Extending and applying the hypercube queueing model to deploy ambulances in Boston. In A. J. Swersey & E. J. Ingnall (Eds.), TIMS studies in the management science: Vol. 22. Delivery of urban services (pp. 121–153). Amsterdam: Elsevier. Brotcorne, L., Laporte, G., & Semet, F. (2003). Ambulance location and relocation models. European Journal of Operational research, 147, 451–463. Burwell, T. H., Jarvis, J. P., & Mcknew, M. A. (1993). Modeling co-located servers and dispatch ties in the hypercube model. Computers & Operations Research, 20(2), 113–119. Chelst, K., & Barlach, Z. (1981). Multiple unit dispatches in emergency services: models to estimate system performance. Management Science, 27(12), 1390–1409. Chiyoshi, F., Galvão, R. D., & Morabito, R. (2003). A note on solutions to the maximal expected covering location problem. Computers & Operations Research, 30(1), 87–96. Eaton, D. J., Daskin, M. S., Simmons, D., Bulloch, B., & Jansma, G. (1985). Determining emergency medical service vehicle deployment in Austin, Texas. Interfaces, 15, 96–108. Fujiwara, O., Makjamroen, T., & Gupta, K. K. (1987). Ambulance deployment analysis: a case study of Bangkok. European Journal of Operational Research, 31, 9–18. Galvão, R. D., Chiyoshi, F., & Morabito, R. (2005). Towards unified formulations and extensions of two classical probabilistic location models. Computers & Operations Research, 32, 15–33. Goldberg, D. E. (1989). Genetic algorithms in search, optimization and machine learning. New York: Addison–Wesley.

224

Ann Oper Res (2008) 157: 207–224

Goldberg, J., & Paz, L. (1991). Location of emergency vehicle bases when service times depends on call location. Transportation Science, 25, 264–280. Harewood, S. I. (2002). Emergency ambulance deployment in Barbados: a multi-objective approach. Journal of the Operational Research Society, 53, 185–192. Hertz, A., & Kobler, D. (2000). A framework for the description of evolutionary algorithms. European Journal of Operational Research, 126, 1–12. Holland, J. H. (1975). Adaptation in natural and artificial systems. Cambridge: MIT Press. Iannoni, A., & Morabito, R. (2006). A discrete simulation analysis of a logistics supply system. Transportation Research Part E, 42, 191–210. Iannoni, A. P., & Morabito, R. (2007) A multiple dispatch and partial backup hypercube queuing model to analyze emergency medical systems on highways. Transportation Research E, doi:10.1016/j.tre.2006.05.005 Jarvis, J. P. (1985). Approximating the equilibrium behavior of multi-server loss systems. Management Science, 31, 235–239. Jaszkiewicz, A. (2002). Genetic local search for multi-objective combinatorial optimization. European Journal of Operational Research, 137, 50–71. Kelton, W. D., Sadowski, R. P., & Sadowski, D. (2002). A simulation with arena (2nd ed.) New York: McGraw–Hill. Larson, R. C. (1974). Hypercube queueing model for facility location and redistricting in urban emergency services. Computers & Operations Research, 1, 67–95. Larson, R. C. (1975). Approximating the performance of urban emergency service systems. Operations Research, 23, 845–868. Larson, R. C. (2004). OR models for homeland security. OR/MS Today, 31, 22–29. Larson, R. C., & Odoni, A. R. (1981). Urban operations research. New Jersey: Prentice Hall. Mendonça, F. C., & Morabito, R. (2001). Analyzing emergency service ambulance deployment on a Brazilian highway using the hypercube model. Journal of the Operation Research Society, 52, 261–268. Michalewicz, Z. (1996). Genetic algorithms + data structures = evolution programs (3rd ed.) Berlin: Springer. Owen, S. H., & Daskin, M. S. (1998). Strategic facility location: a review. European Journal of Operational Research, 111, 423–447. Pegden, C. D., Shannon, R. E., & Sadowski, R. P. (1995). Introduction to simulation using SIMAN (2nd ed.) New York: McGraw–Hill. Rajagopalan, H. K., Saydam, C., & Xiao, J. (2006, to appear) A multiperiod set covering location model for a dynamic redeployment of ambulances. Computers & Operations Research. Sacks, S. R., & Grief, S. (1994) Orlando Police Department uses OR/MS methodology, new software to design patrol districts. OR/MS Today, Baltimore, 30–32. Saydam, C., & Aytug, H. (2003). Accurate estimation of expected coverage: revisited. Socio-Economic Planning Sciences, 37, 69–80. Swersey, A. J. (1994). Handbooks in OR/MS (Vol. 6, pp. 151–200). Amsterdam: Elsevier. Takeda, R. A., Widmer, J. A., & Morabito, R. (2007). Analysis of ambulance decentralization in an urban medical emergency service using the hypercube queueing model. Computers & Operations Research, 34(3), 727–741.