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A planning model for determining the optimal location and size of traffic centers: The case of Dammam Metropolitan, Saudi Arabia
272
European Journal of Operational Research 66 (1993) 272-278 North-Holland
Theory and Methodology
A planning model for determining the optimal location and size of traffic centers: The case of Dammam Metropolitan, Saudi Arabia Taqi N. AI-Faraj, Abdulaziz S. Alidi and Abdulla A. Al-Ibrahim King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia Received April 1990; revised July 1991
Abstract: Fixed traffic centers (FTC) are usually the dispatching points for traffic patrol units (TPU). Therefore, location of the optimal regional sites of TPU is a major prerequisite for determining the optimal location of FTC. In this paper, a planning model for determining the location and size of FTC is developed through two stages. In the first stage, the optimal regional sites of TPU are determined using a goal integer linear programming approach. Based on the results of the model at the first stage and other relevant data, the second stage determines the location and size of the FTC using a binary integer linear programming technique. The model has been used to determine the regional sites of TPU and the FTC for the Dammam Metropolitan, Saudi Arabia, utilizing historical data. Keywords: Traffic services; Allocation; Integer goal programming; 0-1 integer programming
I. Introduction In most developing countries such as Saudi Arabia, traffic departments usually serve two main functions. In addition to being dispatching centers for traffic patrol units (TPU), they also provide administrative services related to traffic such as issueing and renewing of driving licenses and vehicles registration. These centers are also used as temporary apprehension centers for major traffic violators. Hence, the location of fixed traffic centers Correspondence to: Dr. Abdulaziz S. Alidi, King Fahd University of Petroleum and Minerals, KFUPM Box 1645, Dhahran 31261, Saudi Arabia.
(FTC) in a metropolitan area should be determined by taking into consideration several factors: First, since the FTC are usually dispatching points for TPU, location of the optimal regional sites of these units is a major prerequisite for determining the optimal location, among several potential sites, for the FFC. Second, since a FTC usually serves several communities throughout the metropolitan area, the location of the center should be at a convenient site that takes into consideration the population density in the various districts of the metropolitan area and the sites of other FTC's. In this paper a planning model for determining the optimal location and size of FTC is developed through two stages. The first stage is concerned
0377-2217/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
T.N. Al-Faraj et al. / A planning model for traffic centers
with determining the optimal regional sites of T P U using the methodology of integer goal programming which is an appropriate technique to handle multi-objective planning situations. Based on the results of the model at the first stage and other relevant data, the optimal location and size of the FTC are determined, using a binary integer linear programming technique. The developed model has been used to determine the optimal location and size of FTC in the Dammam Metropolitan area of Saudi Arabia. The use of the model will help planners and managers of traffic departments in utilizing the available human and physical resources effectively and efficiently.
2. Literature review
The problem of locating TPU's within a geographical area can be considered as a special case of locating emergency service facilities. Like ambulance and fire prevention systems, TPU's also have the common objective of reaching the site of an emergency as quickly as possible. Therefore, the time between a call announcing a traffic incident and the arrival of a T P U at the site, the response time, should be as short as possible. In considering an area in which a T P U system is to be operated efficiently, the problem of where to locate the T P U is a crucial one for determining the response time. Many researchers have developed mathematical models and algorithms to address such a location problem. Mandl (1979) formulated the location of Emergency Medical Service Facilities as a network problem. He pointed out that such emergency services are usually vehicles using the road network of the area, and vertices of such a road network are considered as subareas of the given area. He also considered this location problem as one of finding the optimal location at vertices in a given network. Church and ReVelle (1974) presented a complete review and mathematical development of many facility location models. The maximum covering location (MCL) model, originally proposed by Church and ReVelle (1974), is a network optimization model which selects sites for a given number of facilities to maximize the total demand that can be covered within a user specified response time. Cover-
273
ing is defined as follows: point i is covered if there is a facility located within s miles (minutes) of i, where s is the maximum service distance (time). By applying location covering techniques within a goal programming framework, Charnes and Storbeck (1980) developed a method for the siting of multilevel emergency medical services systems (EMS) so that (a) each service level maximizes coverage of its own demand population, and (b) 'back-up' coordination between levels is assured. The working of their multilevel covering model was demonstrated by reference to EMS planning scenarios and related numerical exampies. The problem of fire fighters deployment within a district was studied by Beltrami (1977) who formulated a binary integer linear programming model of the set covering type. The model was applied to the city of New York to solve the problem of how many fire fighter units to allocate citywide into different sectors. The optimal allocation of the TPU within a geographical area is an important factor for the traffic police departments to operate efficiently and effectively. Among those who studied the problem of TPU allocation within an area are Larson (1974) and Chaiken (1975). Larson (1974, 1985) developed a model and computer software, called Hypercube Queuing Model (HQM). Larson's work is a very useful planning tool for planners seeking to evaluate alternative plans for allocating police vehicles and redistricting in urban emergency services. The H Q M was used by Larson and Rich (1987), in the New York City Police Department, to predict the consequences of the effect on travel times to calls for service of alternative radio-monitored patrol cars. Chaiken (1975, 1985) developed another model and computer software for patrol car allocation named Patrol Car Allocation Model (PCAM) to assist police departments in evaluating alternative manpower schedules. Chu and Abdulghani (1985) applied Larson and Chaiken's work to allocate TPU's and determine expected performance levels for the traffic department of Khobar city, Saudi Arabia, under such an allocation scheme. Both H Q M and PCAM are considered to be effective planning tools for traffic police departments in developing manpower schedules and operations of police patrol vehicles. In another
274
T.N. ALFaraj et aL / A planning model for traffic centers
approach for deploying patrol officers, Taylor and Huxley (1989) developed an optimizationbased decision support system to forecast hourly needs of officers, schedule them to maximize coverage, and to allow fine-tuning to meet human needs. Their system was recently implemented by the San Francisco Police Department for deploying patrol officers and evaluating policy options for strategic deployment. A review of the literature indicates an abundance of research and models for determining the optimal location of various public service facilities. However, not much research has been conducted to locate FTC's in conjunction with siting TPU's along with other factors, such as the population density and the sites of other FTC's, that influence the location of these centers. This paper is an attempt to bridge the gap between planning models for mobile traffic service facilities and the stationary traffic centers.
3. Model development As previously mentioned, the first stage of the model is concerned with determining the optimal regional sites of TPU's. Usually TPU's are engaged in two types of activities. The first is a reactive activity in which the TPU reacts to random incident calls sent by the central traffic dispatcher through radio communications. The second is a preventive activity where the T P U usually stands by or drives along a route to deter potential violators. Generally speaking, it is desirable to locate TPU's so that a maximum number of calls for service are covered within some time standard, say T minutes, assuming a specific number of TPU's are available. In such a scenario, coverage of calls occurring at any location is defined in terms of that location's proximity to a TPU site. Specifically, if some location i is within T minutes of a T P U at site j, the call for service at location i is considered covered by site j. If, on the other hand, location i is beyond T minutes of site j, demand for service at that location remains uncovered. The primary focus of the model, then, is to site all available TPU's in such a way that a maximum number of calls are covered within T minutes. Planning for the traffic services requires the achievement, as much as possible, of several goals.
Some of these goals may be conflicting. By applying the location covering technique within the goal programming framework, a method for the siting of TPU's so that the demand for traffic services in each sub-area is satisfied to the maximum extent using the minimum number of TPU's. Additionally the coordination between the minimum number of FTC's is assured. Therefore, an integer goal programming model is developed to incorporate the multi-objective criteria in the traffic services maximal coverage problem. The parameters of the model are defined as follows: Indices: i = a sub-area of the metropolitan area, N = number of sub-areas in the metropolitan area, J = a group of sub-areas which can be served by a TPU within a pre-determined time, i.e. a cluster. Priorities: Pa = the first priority given to avoid uncovering the sub-areas with traffic services, P2 = the second priority given to avoid exceeding the maximum number of available TPU's. Constants: C = available number of TPU's, W/ = weight given to the various sub-areas according to number of fatal incidents, nonfatal incidents, population, the size of the sub-area, type of land use and number of major cross intersections. Variables: d 7 = the under-attainment of covering sub-area i with traffic services, d p = additional number of TPU's that should be available to achieve maximal coverage, dn = number of unused TPU's, 1 if a T P U is sited at sub-area i, xi = 0 otherwise, [ 1 if sub-area i is covered ~ with traffic services, 0 otherwise. The objective function at the first stage of the model aims at the maximal coverage of sub-areas with traffic patrol service using the least number of TPU's. The function is composed of two parts. The first part, which is given the highest priority, is concerned with the coverage of sub-areas according to the weight assigned to each sub-area. This weight is obtained using six factors: (a) num-
Yi =
T.N. AI-Faraj et al. / A planning model for traffic centers
ber of fatal incidents, (b) number of non-fatal incidents, (c) population, (d) size, (e) number of major cross intersections, and (f) an index for the type of land use in each sub-area. The weight is computed by dividing the sub-area factor value over the sum of all factor values. To show the importance of the ratio of the fatal incidents factor over the non-fatal incidents factor, the ratio is magnified ten-fold. The magnifying value is determined subjectively based on the perception of the traffic department management. Then, each sub-area weight is obtained by adding the values of the six factors. The second part is concerned with using the least possible number of TPU's for the maximal coverage. Mathematically, this is expressed as follows: N
Minimize
~ e ~ W i d 7 + P2 d p . i=l
10
N
(2)
i-I
Constraint set (3) seeks the coverage of every sub-area with traffic patrol services, or yi + d i - = 1,
i=1,2 .....
N,
i=1,2 .....
N.
(4)
i~j
Constraint set (5) guarantees that every cluster is served by at least one TPU.
ZXi>__l,
i=1,2 .....
N.
(5)
i~j
Finally, the non-negativity and integrality requirements are ensured by the following constraints:
Xi, Yi 1 or 0. =
if a TPU, located in sub-area i, reports to the FTC at sub-area j, otherwise. Constants: dii = average distance between sub-areas i and j, K = number of clusters of sub-areas where TPU's are located, M = the maximum possible number of FTC's to serve the metropolitan area. The objective function of the model at the second stage is stated as:
xij =
K
Minimize
(6)
The results of the solution of the above model will divide the sub-areas into clusters where every cluster will be served by a TPU.
K
~_, ~'~ dijxiy,
(7)
i=1 j=l
(3)
The constraints stated in (4) ensure the coverage of every sub-area with traffic patrol services by a T P U located in the sub-area itself or its cluster:
Exi--Yi>_O,
The second stage of the model is concerned with determining the optimal location and size of a given number of FTC's. The number of these FTC's is determined by the budget allocated for this purpose. The size of each FTC is proportional to the sub-areas it serves; that is to say, the human and physical resources available to the regional traffic department should be allocated to the various determined FTC's based on the number of subareas assigned to it. The results of the model at the first stage are utilized as feed data for solving the second stage of the model. The parameters used in the second stage of the model are defined as follows: Variables: 1 if the FTC is located at sub-area j, Yi = 0 otherwise;
(1)
Constraint (2) ensures that the total number of TPU's to cover the metropolitan area with traffic patrol services plus (minus) the unused (needed) units must be equal to the total number of the available TPU's. This is expressed mathematically as follows: ~_, x i + d n - d p = C.
275
which seeks to minimize the sum of the distances from each cluster center to its respective FTC. The first set of constraints (8) indicates that, unless a cluster center is a site for a FTC, no sub-areas can be assigned to it; that is, yj = 0 implies that xij = 0 for all j: K
Y', x # - Kyj < 0
for all j.
(S)
i-I
The second set of constraints (9) indicates that if cluster center j is a FTC, at least one other cluster must be assigned to it; that is, Yi = 1 implies that xij = 1 for at least one j. Mathematically, this is represented as follows: K
~ , xii - yj > 0 i=1
for all j.
(9)
T.N. Al-Faraj et al. / A planning model for traffic centers
276
The third set of constraints (10) indicates that either cluster center i is a FTC, or it is assigned to a F T C at another cluster center; that is, either Yi = 1 or xij = 1 for some j. This can be represented mathematically as follows:
4. Model application Siting of T P U ' s and location of FTC's in urban areas of Saudi Arabia, under the existing system, are based on subjective judgment and past incident records. The proposed two-stage model will be used to determine the optimal regional sites of T P U ' s and the location of FTC's within the metropolitan area of D a m m a m , Saudi Arabia. D a m m a m metropolitan area is located in the eastern province of Saudi Arabia and includes three cities: D a m m a m , Khobar and Dhahran (Figure 1). The population of this metropolitan area is estimated at 300000 people living in an area of approximately 160 square kilometers. To determine the optimal regional sites for the T P U ' s in the D a m m a m Metropolitan area, the area was partitioned into 73 sub-areas where each sub-area is surrounded by main streets and a T P U can circle the surroundings within 5 minutes. The seventy three sub-areas are then grouped such that a T P U sited in a given sub-area
K
Yi + E
Xij
=
1
for all i.
(10)
j=l Constraint (11) ensures that no more than M, the maximum permissible number of FTC's, can be established: K
Y'. yj < M .
(11)
j=l
Finally, the binary and the non-negativity requirements are expressed as xq = 0 or 1 and yj = 0 or 1.
._...r
NORTH
AL-~AWF
A
tDUBA
~
L-WAJH
AL-KHAFJI JUBAII
fINN l . ~ 3 J
~W~ ~ANB~ MEDINA RIYADH0
\
0
AL-HUFU
,M~CCA --T~]F
SAUDI
ARABIA ,J
x) 8
SOUTH YEMEN
Figure 1. Map of Saudi Arabia
T.N. Al-Faraj et aL / A planning model for traffic centers
can respond to calls from nearby sub-areas within 15 minutes. The model objective at this stage is to have maximal coverage of the metropolitan area using a minimum number of TPU's. Maximal coverage is given the first and highest priority and the importance of a sub-area to the maximal coverage is indicated by a specific weight, which is computed based on the number of fatal and non-fatal incidents, size, population, type of land use and number of main intersections. The necessary parameters for the first stage of the model are the available number of TPU's and the weights which have to be computed for each sub-area using historical data. Based on the service level indicated by the response time of 5 and 15 minutes, the seventy three sub-areas of the Dammam Metropolitan area can be fully covered with traffic patrol services using only fourteen TPU's.The optimal regional sites of these TPU's, the sub-areas that could be served from each site and the number of sub-areas served from each site are shown in Table 1. Each cluster code in Table 1 is a recommended site for a TPU which serves the designated subareas and its own sub-area. It is noted that some of the sub-areas are s e r v e d from more than one T P U site.
277
Table 2 The results of the second stage of the model Traffic center location name
Clusters served by the center
Mubarakiah Adama Television station vicinity Agrabieah
x s, Xlo
N u m b e r of sub-areas served by the clusters 9
x17, x38 , x43
20
X l , X25 , X32
15 32
X49 , X54 , X57 , X66 , X69 , X73
As illustrated before, the model at the second stage is used to determine the location and size of FTC's in the Dammam Metropolitan area. The budget allocated to the Dammam Traffic Directorate permits the establishment of four traffic centers. The average distances between the centers of the fourteen clusters of the sub-areas obtained at the first stage are determined and used as a parameter in the objective function, as is the number of clusters. The results of this stage are presented in Table 2, showing an optimum of four FFC's. The optimal location of FTC's and the number of clusters and sub-areas to be serviced by each FTC are also presented in Table 2.
5. Summary and conclusion Table 1 Results of the first stage of the model Cluster code
Sub-areas served from the cluster
N u m b e r of sub-areas served from the cluster
xI x8 xi0 x17 x 25 x32 x38 x43 x49
2,3,4 5,6,7,9,11,12 7,11,13 15,16,18,20,21,22,29,30,31 4,6,24,26,27,28 23,24,26,33,34,39 35,36,37,39,40 14,19,42,44,45,46 41,46,47,48,50,51 52,53,55,59 52,56,58,59,64 59,60,63,64,65,67 33,39,40,41,68,70,72 61,62,71,72
3 6 3 9 6 6 5 6 6 4 5 6 7 4
x54 x57 x66 X69 X73
In this paper, a multi-objective planning model for determining the location and size of fixed traffic centers (FTC) is developed, consisting of two stages. The first stage is concerned with determining the optimal regional sites of traffic patrol units (TPU) while the second stage is concerned with determining the optimal location and size of a FTC. The model has been used for determining the optimal regional sites of TPU's and the location of FTC's in Dammam Metropolitan, Saudi Arabia, using historical data. The problem at the first stage of the model includes 221 decision variables and 220 constraints, while the problem at the second stage includes 210 decision variables and 44 constraints. All computations were done on an IBM-3090 using the Statistical Analysis System/
278
T.N. AI-Farajet al. / A planning model for traffic centers
operations Research (SAS/OR) computer package, using minimal computing time. computation time, however, is not as great a factor since the model is strategic, rather than operational in nature. The results obtained indicate that there is a need to deploy TPU's at fourteen regional sites per shift in the Dammam Metropolitan area. The solution of the second stage of the model resulted in determining the optimal location and size of four FTC's. It is anticipated that the implementation of the model will result in a significant improvement over the existing situation which is based on past experience and subjective judgment. The use of the model by planners will lead to an effective and efficient utilization of the traffic departments' human and physical resources. The model will also be of great assistance as the expected future population growth and urban expansion take place. The developed model does not consider time of the shift as a factor in determining the optimal regional sites of traffic patrol units. Further extension of the model is needed if time is to be considered.
References Beltrami, E.J. (1977), Models for Public System Analysis, Academic Press, New York. Chaiken, J. (1975), Patrol Allocation Methodology for Police Departments, Rand Corporation, Santa Monica, CA. Chaiken, J. (1985) "Patrol car allocation model", (software program), Abt Associates, Cambridge, MA. Charnes, A., and Storbeck, J. (1980), "A goal programming model for the siting of multilevel EMS systems", Journal of Socio-Economic Planning Sciences 14/4, 155-161. Chu, C., and Abdulghani, K. (1985), "Enhancing the traffic patrol operation in the kingdom of Saudi Arabia", Second Saudi Engineers Conference, 16-19 November 1985, Dhahran, Saudi Arabia. Church, R., and ReVelle, C. (1974), "The maximal covering location problems", Papers of the Regional ScienceAssociation 32, 101-118. Larson, R.C. (1974), "A hypercube queuing model for facility location and redistricting in urban emergency services", Computers and Operations Research 1/1, 67-95. Larson, R.C. (1985), "Hypercube" Software, Enforth, Cambridge, MA. Larson, R.C., and Rich, T.F. (1987), "Travel-time analysis of New York city police patrol cars", Interfaces 17/2, 15-20. Mahdi, C.E. (1979), Interactive Network optimization System, Institute for Advanced Studies, Vienna. Taylor, P.E., and Huxley, S.J. (1989), "A break from tradition for the San Francisco police: Patrol officer scheduling using an optimization-based decision support system", Interfaces 19/1, 4-24.
European Journal of Operational Research 66 (1993) 272-278 North-Holland
Theory and Methodology
A planning model for determining the optimal location and size of traffic centers: The case of Dammam Metropolitan, Saudi Arabia Taqi N. AI-Faraj, Abdulaziz S. Alidi and Abdulla A. Al-Ibrahim King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia Received April 1990; revised July 1991
Abstract: Fixed traffic centers (FTC) are usually the dispatching points for traffic patrol units (TPU). Therefore, location of the optimal regional sites of TPU is a major prerequisite for determining the optimal location of FTC. In this paper, a planning model for determining the location and size of FTC is developed through two stages. In the first stage, the optimal regional sites of TPU are determined using a goal integer linear programming approach. Based on the results of the model at the first stage and other relevant data, the second stage determines the location and size of the FTC using a binary integer linear programming technique. The model has been used to determine the regional sites of TPU and the FTC for the Dammam Metropolitan, Saudi Arabia, utilizing historical data. Keywords: Traffic services; Allocation; Integer goal programming; 0-1 integer programming
I. Introduction In most developing countries such as Saudi Arabia, traffic departments usually serve two main functions. In addition to being dispatching centers for traffic patrol units (TPU), they also provide administrative services related to traffic such as issueing and renewing of driving licenses and vehicles registration. These centers are also used as temporary apprehension centers for major traffic violators. Hence, the location of fixed traffic centers Correspondence to: Dr. Abdulaziz S. Alidi, King Fahd University of Petroleum and Minerals, KFUPM Box 1645, Dhahran 31261, Saudi Arabia.
(FTC) in a metropolitan area should be determined by taking into consideration several factors: First, since the FTC are usually dispatching points for TPU, location of the optimal regional sites of these units is a major prerequisite for determining the optimal location, among several potential sites, for the FFC. Second, since a FTC usually serves several communities throughout the metropolitan area, the location of the center should be at a convenient site that takes into consideration the population density in the various districts of the metropolitan area and the sites of other FTC's. In this paper a planning model for determining the optimal location and size of FTC is developed through two stages. The first stage is concerned
0377-2217/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
T.N. Al-Faraj et al. / A planning model for traffic centers
with determining the optimal regional sites of T P U using the methodology of integer goal programming which is an appropriate technique to handle multi-objective planning situations. Based on the results of the model at the first stage and other relevant data, the optimal location and size of the FTC are determined, using a binary integer linear programming technique. The developed model has been used to determine the optimal location and size of FTC in the Dammam Metropolitan area of Saudi Arabia. The use of the model will help planners and managers of traffic departments in utilizing the available human and physical resources effectively and efficiently.
2. Literature review
The problem of locating TPU's within a geographical area can be considered as a special case of locating emergency service facilities. Like ambulance and fire prevention systems, TPU's also have the common objective of reaching the site of an emergency as quickly as possible. Therefore, the time between a call announcing a traffic incident and the arrival of a T P U at the site, the response time, should be as short as possible. In considering an area in which a T P U system is to be operated efficiently, the problem of where to locate the T P U is a crucial one for determining the response time. Many researchers have developed mathematical models and algorithms to address such a location problem. Mandl (1979) formulated the location of Emergency Medical Service Facilities as a network problem. He pointed out that such emergency services are usually vehicles using the road network of the area, and vertices of such a road network are considered as subareas of the given area. He also considered this location problem as one of finding the optimal location at vertices in a given network. Church and ReVelle (1974) presented a complete review and mathematical development of many facility location models. The maximum covering location (MCL) model, originally proposed by Church and ReVelle (1974), is a network optimization model which selects sites for a given number of facilities to maximize the total demand that can be covered within a user specified response time. Cover-
273
ing is defined as follows: point i is covered if there is a facility located within s miles (minutes) of i, where s is the maximum service distance (time). By applying location covering techniques within a goal programming framework, Charnes and Storbeck (1980) developed a method for the siting of multilevel emergency medical services systems (EMS) so that (a) each service level maximizes coverage of its own demand population, and (b) 'back-up' coordination between levels is assured. The working of their multilevel covering model was demonstrated by reference to EMS planning scenarios and related numerical exampies. The problem of fire fighters deployment within a district was studied by Beltrami (1977) who formulated a binary integer linear programming model of the set covering type. The model was applied to the city of New York to solve the problem of how many fire fighter units to allocate citywide into different sectors. The optimal allocation of the TPU within a geographical area is an important factor for the traffic police departments to operate efficiently and effectively. Among those who studied the problem of TPU allocation within an area are Larson (1974) and Chaiken (1975). Larson (1974, 1985) developed a model and computer software, called Hypercube Queuing Model (HQM). Larson's work is a very useful planning tool for planners seeking to evaluate alternative plans for allocating police vehicles and redistricting in urban emergency services. The H Q M was used by Larson and Rich (1987), in the New York City Police Department, to predict the consequences of the effect on travel times to calls for service of alternative radio-monitored patrol cars. Chaiken (1975, 1985) developed another model and computer software for patrol car allocation named Patrol Car Allocation Model (PCAM) to assist police departments in evaluating alternative manpower schedules. Chu and Abdulghani (1985) applied Larson and Chaiken's work to allocate TPU's and determine expected performance levels for the traffic department of Khobar city, Saudi Arabia, under such an allocation scheme. Both H Q M and PCAM are considered to be effective planning tools for traffic police departments in developing manpower schedules and operations of police patrol vehicles. In another
274
T.N. ALFaraj et aL / A planning model for traffic centers
approach for deploying patrol officers, Taylor and Huxley (1989) developed an optimizationbased decision support system to forecast hourly needs of officers, schedule them to maximize coverage, and to allow fine-tuning to meet human needs. Their system was recently implemented by the San Francisco Police Department for deploying patrol officers and evaluating policy options for strategic deployment. A review of the literature indicates an abundance of research and models for determining the optimal location of various public service facilities. However, not much research has been conducted to locate FTC's in conjunction with siting TPU's along with other factors, such as the population density and the sites of other FTC's, that influence the location of these centers. This paper is an attempt to bridge the gap between planning models for mobile traffic service facilities and the stationary traffic centers.
3. Model development As previously mentioned, the first stage of the model is concerned with determining the optimal regional sites of TPU's. Usually TPU's are engaged in two types of activities. The first is a reactive activity in which the TPU reacts to random incident calls sent by the central traffic dispatcher through radio communications. The second is a preventive activity where the T P U usually stands by or drives along a route to deter potential violators. Generally speaking, it is desirable to locate TPU's so that a maximum number of calls for service are covered within some time standard, say T minutes, assuming a specific number of TPU's are available. In such a scenario, coverage of calls occurring at any location is defined in terms of that location's proximity to a TPU site. Specifically, if some location i is within T minutes of a T P U at site j, the call for service at location i is considered covered by site j. If, on the other hand, location i is beyond T minutes of site j, demand for service at that location remains uncovered. The primary focus of the model, then, is to site all available TPU's in such a way that a maximum number of calls are covered within T minutes. Planning for the traffic services requires the achievement, as much as possible, of several goals.
Some of these goals may be conflicting. By applying the location covering technique within the goal programming framework, a method for the siting of TPU's so that the demand for traffic services in each sub-area is satisfied to the maximum extent using the minimum number of TPU's. Additionally the coordination between the minimum number of FTC's is assured. Therefore, an integer goal programming model is developed to incorporate the multi-objective criteria in the traffic services maximal coverage problem. The parameters of the model are defined as follows: Indices: i = a sub-area of the metropolitan area, N = number of sub-areas in the metropolitan area, J = a group of sub-areas which can be served by a TPU within a pre-determined time, i.e. a cluster. Priorities: Pa = the first priority given to avoid uncovering the sub-areas with traffic services, P2 = the second priority given to avoid exceeding the maximum number of available TPU's. Constants: C = available number of TPU's, W/ = weight given to the various sub-areas according to number of fatal incidents, nonfatal incidents, population, the size of the sub-area, type of land use and number of major cross intersections. Variables: d 7 = the under-attainment of covering sub-area i with traffic services, d p = additional number of TPU's that should be available to achieve maximal coverage, dn = number of unused TPU's, 1 if a T P U is sited at sub-area i, xi = 0 otherwise, [ 1 if sub-area i is covered ~ with traffic services, 0 otherwise. The objective function at the first stage of the model aims at the maximal coverage of sub-areas with traffic patrol service using the least number of TPU's. The function is composed of two parts. The first part, which is given the highest priority, is concerned with the coverage of sub-areas according to the weight assigned to each sub-area. This weight is obtained using six factors: (a) num-
Yi =
T.N. AI-Faraj et al. / A planning model for traffic centers
ber of fatal incidents, (b) number of non-fatal incidents, (c) population, (d) size, (e) number of major cross intersections, and (f) an index for the type of land use in each sub-area. The weight is computed by dividing the sub-area factor value over the sum of all factor values. To show the importance of the ratio of the fatal incidents factor over the non-fatal incidents factor, the ratio is magnified ten-fold. The magnifying value is determined subjectively based on the perception of the traffic department management. Then, each sub-area weight is obtained by adding the values of the six factors. The second part is concerned with using the least possible number of TPU's for the maximal coverage. Mathematically, this is expressed as follows: N
Minimize
~ e ~ W i d 7 + P2 d p . i=l
10
N
(2)
i-I
Constraint set (3) seeks the coverage of every sub-area with traffic patrol services, or yi + d i - = 1,
i=1,2 .....
N,
i=1,2 .....
N.
(4)
i~j
Constraint set (5) guarantees that every cluster is served by at least one TPU.
ZXi>__l,
i=1,2 .....
N.
(5)
i~j
Finally, the non-negativity and integrality requirements are ensured by the following constraints:
Xi, Yi 1 or 0. =
if a TPU, located in sub-area i, reports to the FTC at sub-area j, otherwise. Constants: dii = average distance between sub-areas i and j, K = number of clusters of sub-areas where TPU's are located, M = the maximum possible number of FTC's to serve the metropolitan area. The objective function of the model at the second stage is stated as:
xij =
K
Minimize
(6)
The results of the solution of the above model will divide the sub-areas into clusters where every cluster will be served by a TPU.
K
~_, ~'~ dijxiy,
(7)
i=1 j=l
(3)
The constraints stated in (4) ensure the coverage of every sub-area with traffic patrol services by a T P U located in the sub-area itself or its cluster:
Exi--Yi>_O,
The second stage of the model is concerned with determining the optimal location and size of a given number of FTC's. The number of these FTC's is determined by the budget allocated for this purpose. The size of each FTC is proportional to the sub-areas it serves; that is to say, the human and physical resources available to the regional traffic department should be allocated to the various determined FTC's based on the number of subareas assigned to it. The results of the model at the first stage are utilized as feed data for solving the second stage of the model. The parameters used in the second stage of the model are defined as follows: Variables: 1 if the FTC is located at sub-area j, Yi = 0 otherwise;
(1)
Constraint (2) ensures that the total number of TPU's to cover the metropolitan area with traffic patrol services plus (minus) the unused (needed) units must be equal to the total number of the available TPU's. This is expressed mathematically as follows: ~_, x i + d n - d p = C.
275
which seeks to minimize the sum of the distances from each cluster center to its respective FTC. The first set of constraints (8) indicates that, unless a cluster center is a site for a FTC, no sub-areas can be assigned to it; that is, yj = 0 implies that xij = 0 for all j: K
Y', x # - Kyj < 0
for all j.
(S)
i-I
The second set of constraints (9) indicates that if cluster center j is a FTC, at least one other cluster must be assigned to it; that is, Yi = 1 implies that xij = 1 for at least one j. Mathematically, this is represented as follows: K
~ , xii - yj > 0 i=1
for all j.
(9)
T.N. Al-Faraj et al. / A planning model for traffic centers
276
The third set of constraints (10) indicates that either cluster center i is a FTC, or it is assigned to a F T C at another cluster center; that is, either Yi = 1 or xij = 1 for some j. This can be represented mathematically as follows:
4. Model application Siting of T P U ' s and location of FTC's in urban areas of Saudi Arabia, under the existing system, are based on subjective judgment and past incident records. The proposed two-stage model will be used to determine the optimal regional sites of T P U ' s and the location of FTC's within the metropolitan area of D a m m a m , Saudi Arabia. D a m m a m metropolitan area is located in the eastern province of Saudi Arabia and includes three cities: D a m m a m , Khobar and Dhahran (Figure 1). The population of this metropolitan area is estimated at 300000 people living in an area of approximately 160 square kilometers. To determine the optimal regional sites for the T P U ' s in the D a m m a m Metropolitan area, the area was partitioned into 73 sub-areas where each sub-area is surrounded by main streets and a T P U can circle the surroundings within 5 minutes. The seventy three sub-areas are then grouped such that a T P U sited in a given sub-area
K
Yi + E
Xij
=
1
for all i.
(10)
j=l Constraint (11) ensures that no more than M, the maximum permissible number of FTC's, can be established: K
Y'. yj < M .
(11)
j=l
Finally, the binary and the non-negativity requirements are expressed as xq = 0 or 1 and yj = 0 or 1.
._...r
NORTH
AL-~AWF
A
tDUBA
~
L-WAJH
AL-KHAFJI JUBAII
fINN l . ~ 3 J
~W~ ~ANB~ MEDINA RIYADH0
\
0
AL-HUFU
,M~CCA --T~]F
SAUDI
ARABIA ,J
x) 8
SOUTH YEMEN
Figure 1. Map of Saudi Arabia
T.N. Al-Faraj et aL / A planning model for traffic centers
can respond to calls from nearby sub-areas within 15 minutes. The model objective at this stage is to have maximal coverage of the metropolitan area using a minimum number of TPU's. Maximal coverage is given the first and highest priority and the importance of a sub-area to the maximal coverage is indicated by a specific weight, which is computed based on the number of fatal and non-fatal incidents, size, population, type of land use and number of main intersections. The necessary parameters for the first stage of the model are the available number of TPU's and the weights which have to be computed for each sub-area using historical data. Based on the service level indicated by the response time of 5 and 15 minutes, the seventy three sub-areas of the Dammam Metropolitan area can be fully covered with traffic patrol services using only fourteen TPU's.The optimal regional sites of these TPU's, the sub-areas that could be served from each site and the number of sub-areas served from each site are shown in Table 1. Each cluster code in Table 1 is a recommended site for a TPU which serves the designated subareas and its own sub-area. It is noted that some of the sub-areas are s e r v e d from more than one T P U site.
277
Table 2 The results of the second stage of the model Traffic center location name
Clusters served by the center
Mubarakiah Adama Television station vicinity Agrabieah
x s, Xlo
N u m b e r of sub-areas served by the clusters 9
x17, x38 , x43
20
X l , X25 , X32
15 32
X49 , X54 , X57 , X66 , X69 , X73
As illustrated before, the model at the second stage is used to determine the location and size of FTC's in the Dammam Metropolitan area. The budget allocated to the Dammam Traffic Directorate permits the establishment of four traffic centers. The average distances between the centers of the fourteen clusters of the sub-areas obtained at the first stage are determined and used as a parameter in the objective function, as is the number of clusters. The results of this stage are presented in Table 2, showing an optimum of four FFC's. The optimal location of FTC's and the number of clusters and sub-areas to be serviced by each FTC are also presented in Table 2.
5. Summary and conclusion Table 1 Results of the first stage of the model Cluster code
Sub-areas served from the cluster
N u m b e r of sub-areas served from the cluster
xI x8 xi0 x17 x 25 x32 x38 x43 x49
2,3,4 5,6,7,9,11,12 7,11,13 15,16,18,20,21,22,29,30,31 4,6,24,26,27,28 23,24,26,33,34,39 35,36,37,39,40 14,19,42,44,45,46 41,46,47,48,50,51 52,53,55,59 52,56,58,59,64 59,60,63,64,65,67 33,39,40,41,68,70,72 61,62,71,72
3 6 3 9 6 6 5 6 6 4 5 6 7 4
x54 x57 x66 X69 X73
In this paper, a multi-objective planning model for determining the location and size of fixed traffic centers (FTC) is developed, consisting of two stages. The first stage is concerned with determining the optimal regional sites of traffic patrol units (TPU) while the second stage is concerned with determining the optimal location and size of a FTC. The model has been used for determining the optimal regional sites of TPU's and the location of FTC's in Dammam Metropolitan, Saudi Arabia, using historical data. The problem at the first stage of the model includes 221 decision variables and 220 constraints, while the problem at the second stage includes 210 decision variables and 44 constraints. All computations were done on an IBM-3090 using the Statistical Analysis System/
278
T.N. AI-Farajet al. / A planning model for traffic centers
operations Research (SAS/OR) computer package, using minimal computing time. computation time, however, is not as great a factor since the model is strategic, rather than operational in nature. The results obtained indicate that there is a need to deploy TPU's at fourteen regional sites per shift in the Dammam Metropolitan area. The solution of the second stage of the model resulted in determining the optimal location and size of four FTC's. It is anticipated that the implementation of the model will result in a significant improvement over the existing situation which is based on past experience and subjective judgment. The use of the model by planners will lead to an effective and efficient utilization of the traffic departments' human and physical resources. The model will also be of great assistance as the expected future population growth and urban expansion take place. The developed model does not consider time of the shift as a factor in determining the optimal regional sites of traffic patrol units. Further extension of the model is needed if time is to be considered.
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