Multiobjective Model for Locating Automatic Vehicle Identification ...

Multiobjective Model for Locating Automatic Vehicle Identification Readers Anthony Chen, Piya Chootinan, and Surachet Pravinvongvuth one new sensor technology designed to measure travel times and to facilitate toll operations. In addition, it can provide other types of information unavailable from loop detectors. [See Dixon (5) for a list of potential data that can be collected by an AVI system]. An AVI system consists of fixed AVI reader stations placed at various locations in a transportation network and transponders (or AVI tags) placed in individual vehicles. Various traffic data can be collected by reading unique identification numbers from the AVI tags of individual vehicles as they pass through the AVI reader stations. A central problem in the deployment of the emerging AVI surveillance technology is to determine the number and locations of reading stations that would best cover the network. Sherali et al. (6) defined best in terms of providing a maximum degree of information about traffic variability in the network subject to certain resource constraints. On the other hand, Teodorovic et al. (7) proposed a twoobjective model by defining best in terms of the total number of readings along the shortest path for each origin–destination (O-D) pair and the total number of O-D pairs covered. A reading here means that the same vehicle is registered at two different AVI stations in the network. Both studies considered the best coverage in terms of one or more criteria for a given number of AVI readers (or a fixed cost). No consideration is given to the trade-off between the quality of AVI coverage and the total cost (density of AVI readers) of the AVI system. In this paper, we extend the AVI reader-location problem of Teodorovic et al. (7) to explicitly include cost as another objective in addition to the total number of O-D pairs covered and the total number of trips registered. The cost of an AVI system can be specified as the number of AVI readers installed. A multiobjective model is formulated for locating AVI readers in a network to catch a maximum number of trips and cover a maximum number of O-D pairs using a minimum number of AVI readers. When solving a multiobjective optimization problem, there may not exist a single best solution that satisfies all objectives. It is necessary to develop a solution procedure that explicitly generates and retains the nondominated solutions. Instead of specifying the priority (or weight) for each objective as in the study by Teodorovic et al. (7), the distance-based genetic algorithm (GA) is applied in this study to generate a set of nondominated solutions. The remainder of this paper is organized as follows. In the next section, we formulate the multiobjective AVI location problem. The distance-based GA is presented in the next section, which is followed by a discussion of numerical results and finally by a section containing the conclusion and suggested future research.

The problem of locating automatic vehicle identification (AVI) readers on a transportation network is one worth considering. AVI readers are strategically located to catch a maximum number of trips and cover a maximum number of origin–destination (O-D) pairs using a minimum number of AVI readers. There are three possible objectives when deciding locations for AVI readers: (a) a minimum number of AVI readers, (b) maximum O-D coverage, and (c) a maximum number of trips (or AVI readings). To satisfy all three objectives as much as possible, the problem is formulated as a multiobjective integer-optimization problem. A distance-based genetic algorithm is applied to solve this multiobjective AVI reader-location problem by explicitly generating the nondominated solutions. Numerical results are presented to demonstrate the feasibility of the proposed multiobjective model. The procedure proposed holds great promise for the development of a well-configured AVI system that can achieve a balance between quality and cost of coverage (i.e., trade-off between cost and coverage requirements).

In many cities, traffic congestion is a serious problem because of the limited road space and the ever-increasing travel demands. Intelligent transportation systems (ITS) are considered useful for dealing with traffic congestion, protecting the environment, and improving transportation safety. ITS take advantage of the latest sensor, communication, and traffic-control technologies to combat traffic congestion. These technologies hold promise in assisting state, county, and local governments in meeting the increasing travel demands on the surface transportation system. These methods of traffic surveillance, which are integral parts of ITS, have been described as the eyes of ITS to provide knowledge of existing networkwide traffic conditions (1). Traffic management and information systems usually rely on a system of sensors for traffic surveillance. Currently, the dominant technology for this purpose are inductive-loop detectors, which are buried underneath road pavement to count vehicles passing over them. The reliability and accuracy of traffic data collected by loop detectors are critical factors influencing the performance of trafficcontrol and -management systems. Because of the harsh environment in which loop detectors operate, malfunctioning is a common problem. Recent research (2) has showed that the percentage of good samples can range from 20% to 80% in California. Other studies (3, 4) also reported similar results, in which more than 50% of the loops can be malfunctioning or producing erroneous traffic data at any one time. With advances in traffic-surveillance technologies, new types of sensors have become available to provide new data with more reliability and accuracy. The automatic vehicle-identification (AVI) system is

FORMULATION OF MULTIOBJECTIVE AVI LOCATION PROBLEM

Department of Civil and Environmental Engineering, Utah State University, Logan, UT 84322-4110.

Recently, there have been some studies concerned with the importance of the sensor-location problem to some transportation applications. Yang and Miller-Hooks (8) proposed a model to select the

Transportation Research Record: Journal of the Transportation Research Board, No. 1886, TRB, National Research Council, Washington, D.C., 2004, pp. 49–58.

49

50

Transportation Research Record 1886

information critical arcs such that the greatest benefits (e.g., relieving traffic congestion, providing real-time traveler information) could be derived. A modified maximum-covering formulation was presented along with a heuristic solution procedure. Chiu et al. (9) developed a method for strategically locating a number of variable message signs to maximize their expected benefit under a variety of trafficincident situations. A formulation for bilevel stochastic integer programming was proposed. A tabu search heuristic accompanied by a simulation-based dynamic traffic assignment was employed to solve the problem. Sherali et al. (6) proposed a quadratic 0–1 optimization model to determine the number and locations of AVI readers that would best cover the network by providing a maximum degree of information on travel time variability subject to certain resource constraints. Teodorovic et al. (7) provided a preference-based GA (e.g., preference of decision makers) for determining the best locations of AVI readers on the basis of two criteria. These recent research studies highlight the potential and importance of the traffic-sensorlocation problem to surface-transportation applications. The following list gives the symbols used in the remainder of this paper and their definitions. N A W Rw L δ war h wr xa yw z wr

Set of nodes in the network and N is size of set N. Set of links in the network and A is size of set A. Set of O-D pairs and W  is size of set W. Set of paths between O-D pair w. Number of AVI readers available. A path-link indicator denoting 1 if link a is on path r between O-D pair w, and 0 otherwise. Flow on path r between O-D pair w. An integer decision variable indicating the number of AVI readers to be installed on link a {0, 1, . . . , ρ}. A binary decision variable indicating whether O-D pair w is covered (or intercepted) or not. An integer decision variable indicating the number of readings along path r between O-D pair w {0, 1, . . . , A × ρ − 1}.

Definition of Coverage A vehicle is registered when it passes a pair of AVI readers. If there are multiple AVI readers along a vehicle’s path, the number of readings is simply the number of AVI readers minus 1. Hence, for a path to be covered (or intercepted), there must be at least two AVI readers along the path. Intercepting a vehicle multiple times along the path is considered beneficial because better and more accurate travel-time information can be obtained. For an O-D pair to be covered, all used paths of the O-D pair must have at least two AVI readers.

to identify the minimum number of AVI readers required and their corresponding locations to provide the maximum coverage (or other benefits). Hence, the first objective can be achieved by formulating the AVI location problem as a variant of the set-covering problem (10) as follows: A

∑x

minimize f1 ( x ) =

(1)

a

a =1

subject to A

∑δ

w ra

xa > 1

∀r ∈ R w , w ∈ W

and x a ∈ {0, 1, . . . , ρ}

∀a ∈ A

The AVI readers are strategically located to catch a maximum number of trips and cover a maximum number of O-D pairs using a minimum number of AVI readers. This statement leads to three possible objectives when deciding on locations for AVI readers: (a) to minimize the number of AVI readers, (b) to maximize the O-D coverage, and (c) to maximize the number of AVI readings. Although installing AVI readers at every link or movement can provide the most traffic information, it is not cost-effective and may not be feasible with a constrained budget. Therefore, it is essential

(3)

The objective of the set-covering problem, Equation 1, is to determine the minimum number and locations of AVI readers such that all O-D pairs are covered (or intercepted) as defined by Equation 2. Equation 3 constrains the solution to be an integer. There are three main differences between this formulation and the classical setcovering problem: (a) the location variable (xa) allows multiple AVI readers to be located on the same link; (b) demand in a network is expressed between an O-D pair, not at a demand point; and (c) the definition of AVI coverage requires a minimum of two readers. This variant of the set-covering formulation for the AVI reader-location problem is also similar to the formulation for the traffic-counting location problem recently proposed by Yang et al. (11) for determining the minimum number of traffic counting stations needed to separate all O-D pairs. The main differences are (a) and (c) earlier in this paragraph. In the traffic-counting location problem, the location variable is binary (i.e., 1 if a counting station is located on a link, and 0 otherwise) and only one counting station is sufficient to intercept (or cover) an O-D pair. In practice, a maximum number of AVI readers to be installed on any link could be limited by the spacing requirement between a pair of AVI readers. The second objective can be determined by formulating the AVI location problem as a variant of the maximal covering problem (12) as follows: maximize f2 ( x ) =

W

∑y

( 4)

w

w =1

subject to   yw ≤ max 0, ∑ δ raw x a − 1  a =1  A

Measures of Effectiveness

(2)

a =1

A

∑x

a

≤ L

∀r ∈ R w , w ∈ W

(5)

(6)

a =1

x a ∈ {0, 1, . . . , ρ}

∀a ∈ A

( 7)

and yw ∈ {0, 1}

∀w ∈ W

(8)

Chen, Chootinan, and Pravinvongvuth

51

For a given number of AVI readers (presumably less than the minimum number of AVI readers required to cover all O-D pairs), the objective of the maximal covering problem, Equation 4, is to determine the locations of AVI readers that maximize the O-D coverage. The O-D coverage is determined by Equation 5, which forces yw to be 0 unless there are at least two AVI readers located on every used path of O-D pair w. Here all O-D pairs are considered equal. However, if there is a priority for different O-D pairs, the objective function could be weighted by the importance of the O-D pair. Equation 6 constrains the total number of AVI readers to be installed to less than or equal to L, the number of readers available. Equations 7 and 8 constrain the solution to be an integer and a binary integer, respectively. As with the variant of the set-covering problem, the same three differences—location variable allowing multiple AVI readers on the same link, demand in a network expressed between an O-D pair, and definition of AVI coverage requiring at least two readers— also apply here. The third objective can be formulated as a variant of the flow-capturing problem (13) as follows: maximize f3 ( x ) =

W

road network, the obtained path set is considered an acceptable estimate of the used paths.

Multiobjective Approach for Locating AVI Readers In principle, multiobjective optimization problems are very different from single-objective optimization problems. This is because solving multiobjective problems often requires a set of nondominated solutions, not just a single best solution as in single-objective optimization problems. When the objectives are conflicting, there may not exist a single best solution that simultaneously optimizes all objectives. A solution may be best in one objective, but worst in the others. This also applies for the AVI reader-location problem in which several objectives are usually taken into account for the development of a well-configured AVI system. By incorporating the three singleobjective problems described above, the multiobjective model for locating AVI readers is formulated as follows:

Rw

∑∑h

w w r r

z

( 9)

w =1 r =1

maximize [ − f1 ( x ), f2 ( x ), f3 ( x )]

(14)

subject to

subject to A

∀r ∈ R w , w ∈ W

(10)

yw ≤ zrw

A

∑ xa ≤ L

(11)

a =1

xa ∈ {0, 1, . . . , ρ}

  zrw ≤ max 0, ∑ δ raw xa − 1  a =1  A

  zrw ≤ max 0, ∑ δ raw xa − 1  a =1 

∀a ∈ A

(12)

and zrw ∈ {0, 1, . . . , A − 1}

∀r ∈ R w , w ∈ W

xa ∈ {0, 1, . . . , ρ} yw ∈ {0, 1}

∀r ∈ R w , w ∈ W

(15) (16)

∀a ∈ A

(17)

∀w ∈ W

(18)

and ∀r ∈ R w , w ∈ W

(13)

For a given number of AVI readers, the objective of the flowcapturing problem, Equation 9, is to determine the locations of AVI readers to maximize the amount of captured flows measured by the product of flows and the number of readings on all used paths of all O-D pairs. Here double counting is preferable because the collected information (e.g., travel time) becomes more accurate when vehicles are registered many times along their traveled paths. This is different from the original flow-capturing problem (13) and its variants used for locating the vehicle-inspection stations (14, 15). The path coverage is determined by Equation 10. Equation 11 constrains the number of AVI readers to be installed to less than or equal to L. Equations 12 and 13 constrain the solutions to be integers. The variable z wr counts the number of AVI readings on each used path r between O-D pair w. In addition to the three differences mentioned earlier, these three models for the AVI location problem do not assume that all flows from a particular O-D pair must take the shortest path. All feasible paths without loops are explicitly considered through Equations 2, 5, and 10. Particularly for the third formulation, path-flow information is required. In practice, it may be difficult to collect such information. In this paper, we use the path-flow estimator (PFE), which is based on the stochastic user equilibrium principle (16 ), to derive path-flow information from traffic counts. Because the PFE estimates unique path flows based on the actual traffic condition of the

zrw ∈ {0, 1, . . . , A × ρ − 1}

∀r ∈ R w , w ∈ W

(19)

Equation 14 is the multiobjective function of the AVI readerlocation problem, which is to configure the AVI readers into a trafficsensor network that is capable of capturing as much flows as possible (i.e., flow-capturing objective) while covering a maximum number of O-D pairs (i.e., maximal covering objective) with a minimum number of AVI readers (i.e., set covering objective). Equation 15 ensures that, if a used path is covered, there must be a minimum of two readers along the path; otherwise, it is considered as not covered. Furthermore, for O-D pair w to be covered, Equation 16 requires z wr (all used paths r serving this O-D pair w) to be positive. Equations 17 to 19 constrain the solutions to be either binary integers or integers.

GA FOR MULTIOBJECTIVE OPTIMIZATION PROBLEM GA is one of the evolutionary computing techniques that have been widely used to solve complex problems known to be difficult for traditional optimization techniques. The classical calculus-based optimization techniques generally require the problem to possess certain mathematical properties, such as continuity, differentiability, convexity, and others, which may not be satisfied in many real-world problems. As a result, GA, which is not bothered by these requirements (17), has been considered and adopted as a practical tool for

52

Transportation Research Record 1886

problem solving in many disciplines. Another advantage of GA is that it works with a population of solutions as opposed to a single solution in the classical techniques. Because the population of solutions is processed in each generation, the outcome of GA is also a population of solutions. If the optimization problem has only a single optimum, all solutions in the population can be expected to converge to that optimal solution. However, if the problem has multiple optimal solutions (is multimodal), GA can be used to capture this multiplicity. The ability of finding multiple optima within a single simulation run makes GA unique for generating nondominated solutions in multiobjective optimization problems (18, 19). A detailed description of GA may be found in Goldberg (17 ) and in Gen and Cheng (19).

Chromosome Representation Although there are three sets of decision variables (xa, yw, and z wr) in the multiobjective formulation, only one set is required to determine the objective values. We just need to represent xa (an integer decision variable indicating the number of AVI readers to be installed on link a) as a chromosome of length A, while the other two sets of decision variables can be implicitly referred to by xa using the relationships defined in Equations 20 and 21. Since the number of AVI readers to be installed on any link can be 0 to L, the problem is combinatorial with the size of the solution space equal to ρ + 1 A .     yw = min max 0, min ∑ δ raw xa − 1,   ∀r ∈R w  a =1  A

  zrw = max 0, ∑ δ raw xa − 1  a =1  A

 1 

∀w ∈ W

∀r ∈ R w , w ∈ W

tance from the existing set of nondominated solutions. The relative distance dk from the existing nondominated solution xk defines the fitness of new solution x as follows. dk =

m

 fi ( x k ) − fi ( x )    fi ( x k )

∑  i =1

2

(22)

where m = the number of objectives, fi (xk) = the value of the ith objective of the nondominated solution xk, and fi (x) = the value of the ith objective of the new solution x. There are two sets of solutions involved: the population set and the nondominated set. The population set is used to perform the genetic search (i.e., GA operators) while the nondominated set is the solutions to the problem. The newly generated solutions can be classified as either superior (Type I), nondominated (Type II), or inferior (Type III) solutions. They will be added to the nondominated set if they are superior or nondominated. The inferior solutions will not be introduced into the nondominated set because they are worse than the existing ones. In the case that some existing nondominated solutions are dominated by a newly generated solution, they will be discarded from the nondominated set. The solution with a distance farther away from the existing nondominated solutions will receive a higher fitness value. The fitness evaluation according to the distance-based method is summarized as follows.

(20)

(21)

Fitness of Chromosome by Distance-Based Method One of the difficulties of a multiobjective genetic algorithm is the determination of the fitness value of individual chromosomes when considering multiple objectives simultaneously. According to Gen and Cheng (19), there are generally two approaches: the preferencebased approach and the generating approach. If the decision maker knows the preference structure (priority) of all objectives, each of them can be weighted and combined into a single measure. However, doing this requires a good knowledge of the problem. In addition, selecting improper weight factors may lead to the domination of one objective over the others (18). For the generating approach, the concept of Pareto optimality is adopted. The solution can be classified either as the superior, nondominated, or inferior solutions. Intuitively, the superior solution should have the highest fitness value, while the inferior solution receives the lowest fitness. Each nondominated solution is considered equally good and has the same fitness. In this study, we use the distance-based method, one of the generating methods proposed by Osyczka and Kundu (20), to determine the fitness of solutions. The basic concept of this method is to evolve the genetic search toward the ideal Pareto solutions such that all individual objectives are simultaneously achieved at the highest possible level. In practice, such ideal solutions are rarely known; therefore, an alternative benchmark, which is the current set of nondominated solutions, is considered. The solutions are improved based on the dis-

Step 1. If there exist solutions in the nondominated set, go to Step 2. Otherwise, add the qth solution into the set. Arbitrarily assign the fitness value vq to the qth solution (e.g., vq = Pmax). Set the number of nondominated solutions, k = 1. Step 2. Determine the minimum distance (dmin) and the corresponding nondominated solution  by Equation 23, where  is the kth solution in the nondominated set with the minimum distance dmin and  dmin = min  k 

 fi ( x k ) − fi ( x q )    ∑   fi ( x k ) i =1 m

2

  

(23)

Step 3. Determine the type of solution. If the qth solution is Type I, add this solution to the nondominated set and remove all dominated solutions (r solutions). Set k = k − r + 1, compute the fitness of new nondominated solution pk using Equation 24, and go to Step 4. pk = pmax + dmin

(24)

If the qth solution is Type II, add this solution to the nondominated set. Set k = k + 1, compute the fitness of new nondominated solution pk using Equation 25, and go to Step 4. pk = pl + dmin

(25)

If the qth solution is Type III, compute the fitness of the solution using Equation 26. vq = max{0.0, pl − dmin }

(26)

Step 4. Compute the fitness of the qth solution using Equation 27. Update the fitness of the nondominated set according to Equation 28.

Chen, Chootinan, and Pravinvongvuth

53

vq = pk

(27)

Pmax = max{Pmax , pk }

(28)

54, 57, 58, 62, and 63, and the destination nodes are 7, 21, 38, 55, and 61.

Estimation of Path Flows NUMERICAL EXPERIMENTS Description of Test Network In this section, we present the application of the proposed multiobjective model for locating AVI readers to a simplified Irvine network in Orange County, California, as shown in Figure 1. It consists of three major freeways (I-5, I-405, and SR-133) in the city of Irvine. The Irvine network and associated demand data were extracted from the Orange County Transportation Analysis Model (OCTAM), which contains data for the whole county. OCTAM incorporates the best state-of-the-practice modeling components that are consistent with the new Southern California Regional Transportation Model recently released by the Southern California Association of Governments. OCTAM 3.0 was developed and validated for Base Year 1991 condition and revalidated for Year 1998 to reflect more on the current highway and transit data. OCTAM 3.0 has 2,940 traffic analysis zones within Orange County and 1,282 external stations (including cordon stations). The extracted network is composed of 63 nodes, 75 links, and 34 O-D pairs. The simplified network includes only freeway links, on ramps, and off ramps. The demands originating at the onramp nodes represent the internal-to-external travel demands, while those terminating at the off-ramp nodes represent the external-tointernal travel demands. The origin nodes are 1, 15, 24, 40, 49, 53,

FIGURE 1

The logit-based PFE, originally developed by Bell and Shield (16) as a one-stage network observer to estimate path flows (thus O-D flows) from traffic counts, is adopted in this study. The core component of PFE is a logit-based path-choice model, which interacts with link cost functions to produce a stochastic-user-equilibrium (SUE) traffic pattern. We use the PFE to estimate path flows and O-D flows for the simplified network using synthetic traffic counts. These counts were obtained by assigning the known trip table of the original network to the full network based on the SUE principle. The partial O-D trip table for the simplified network is provided in Table 1. Because of the simplified topology of the network, there is only one path per O-D pair for a total of 34 paths. Hence, the O-D flows are the same as path flows. A dashed line indicates that such an O-D pair does not exist in this simplified network.

Numerical Results The distance-based GA is used to solve the multiobjective AVI location problem. The parameters of GA are set as follows: the population size is 128, the probability of crossover is .50, the mutation rate is 0.05, and the maximum number of generation is 30,000. For the genetic search, it is assumed that the maximum number of AVI

Simplified test network in city of Irvine, Orange County, California.

54

Transportation Research Record 1886

The result from the distance-based GA suggests that, to purely satisfy Objective 1 (regardless of the value of Objectives 2 and 3), it does not require any AVI readers (i.e., do nothing). Similarly, the maximum number of O-D pairs covered is 34 O-D pairs (all O-D pairs are covered), and the maximum number of readings is 729,216 (using ρ = 5 on all links). It is important to note that these three extreme nondominated solutions cannot be obtained simultaneously. In other words, if one wants to choose a plan that purely satisfies Objective 1 (minimum number of AVI readers), the number of O-D pairs covered and the number of readings will definitely be less than 34 and 729,216. Figure 3 presents all nondominated solutions found at the final generation. As mentioned earlier, these objectives are, in general, conflicting. There is no single best solution that simultaneously optimizes all three objectives. Each solution shown in Figure 3a is nondominated. The relationship between each pair of objectives is projected on each associated plane as shown in Figures 3b, 3c, and 3d. From Figure 3c, it is clear that the number of AVI readers and the number of readings exhibit a strong positive correlation; the more AVI readers that are used, the larger is the portion of traffic that will be captured. Clearly, there is a trade-off between these two objectives. If one wants to minimize the number of readers (Z1), one will get fewer readings (Z3). Similarly, there is a trade-off between the number of readers (Z1) and the number of O-D pairs covered (Z2) in Figure 3d but with a weaker positive correlation. Figure 3b shows a positive but not a strong correlation between the number of readings

TABLE 1 Estimation of Partial O-D Trip Table for the Simplified Network From/To 1 15 24 30 40 49 53 54 57 58 62 63 Total

7 966 2,989 115 3,185 84 24 384 2,551 10,298

21 350 140 433 1,888 26 290 3,127

38 559 314 184 1,936 590 1,044 4,627

55 35 930 1,075 95 972 6 48 3,161

61 131 794 53 1,447 27 11 658 3,121

Total 2,006 35 4,713 168 5,847 425 747 4,796 384 622 1,382 3,209 24,334

readers allowed to be installed on each link is five; therefore, an integer value varied from 0 to 5 is used to represent the decision variables. Figure 2 displays the evolution of Pareto (nondominated) solutions starting from the initial generation to the final generation considering all three objectives: Objective 1: minimum number of AVI readers (Z1), Objective 2: maximum number of O-D pairs covered (Z2), and Objective 3: maximum number of readings (Z3).

35

Number of OD Pairs Covered

30 25 20 15

Initialization 10th Generation 5000th Generation 9000th Generation

10 5 0 8 6

x 105 Number of Readings

4 2 0

0

50

100

150

200

250

Number of AVI Readers FIGURE 2

Convergence curve of the distance-based method.

300

350

Chen, Chootinan, and Pravinvongvuth

55

(a)

(b)

(c)

(d)

FIGURE 3 Nondominated solutions of the multiobjective AVI reader-location problem: (a) three-dimensional representation of nondominated solutions among numbers of O-D pairs covered, readings, and AVI readers; (b) number of readings versus number of O-D pairs covered; (c) number of AVI readers versus number of readings; and (d ) number of AVI readers versus number of O-D pairs covered.

(Z3) and the number of O-D pairs covered (Z2). Because both Z2 and Z3 are to be maximized, Figure 3b shows a complementary relationship between these two objectives. In general, more AVI readers will result in more readings as well as O-D pairs covered. However, the most important factor affecting the performance of the AVI system is the locations of AVI readers. The quality and quantity of information obtained from the same number of AVI readers with different locations may be quite different. Because each of 1,023 nondominated solutions at the final generation in Figure 3a is considered equally good, one might select any of those solutions for implementation based on some additional criteria. For a given number of AVI readers and their optimal locations, it might be impossible to increase the number of readings by relocating the AVI readers unless one is willing to accept a reduction in the number of O-D pairs covered. The set of nondominated solutions generated is quite useful because it provides a complete picture of all solutions and their trade-offs. This is an advantage over the preference-based approach for multiobjective problems because all the nondominated solutions are generated in one single run. The preference-based method requires solving the problem many times with different weight combinations to generate the nondominated

solutions. The selection of weight combinations will influence the nondominated solutions. In addition, some of the nondominated solutions may not be attainable because there exists no proper weight combination able to generate such solutions, the so-called duality-gap points (21). For this simplified network, three nondominated solutions representing Plans A, B, and C (Figures 4 and 5) were selected to illustrate the trade-off among them. The criteria used for choosing these three points are based on the same number of AVI readers required to cover all 34 O-D pairs (solution to the set-covering problem), which is 14 AVI readers, as shown in Figure 3d. These three nondominated solutions are also depicted in Figures 3a to 3c. The characteristics of these three plans are shown in Figure 4. The numbers shown in the network are the number of AVI readers installed on the link. With 14 AVI readers, Plan A is the best in terms of the O-D coverage; however, it has the least number of AVI readings. On the other hand, Plan C has the highest number of AVI readings but with the least number of O-D pairs covered. Plan B is a compromise solution between plans A and C. These solutions clearly demonstrate that, with the same number of AVI readers—though differently located—the quality and quantity of information obtained are significantly different.

56

Transportation Research Record 1886

Number of AVI Readers

Number of Readings

Number of O-D Pairs Covered

(a)

14

31,399

34

(b)

14

56,221

27

(c)

14

86,703

18

Location of AVI Readers

FIGURE 4 AVI reader locations: (a) minimum number of AVI readers to cover all O-D pairs; (b) one of the nondominated solutions at L  14; (c) maximum number of trips (or readings) covered at L  14.

Flow-Capturing Analysis To assess the quality of AVI readers recommended by the proposed multiobjective model, the authors use the path flows estimated by PFE and check how many times a path between an O-D pair is intercepted by the AVI readers. The three nondominated solutions (Plans A, B, and C) are assessed based on their capability of capturing traffic flows and intercepting O-D pairs. The percentage of O-D pairs covered (from 34 O-D pairs) and flow captured (from 24,334 trips) are depicted in Figures 5a and 5b, respectively. On the x-axis, k refers to the number of times that a vehicle is intercepted (or registered) by a pair of AVI readers. For illustration, in Figure 5a, the percentage of O-D pairs intercepted by Plan B at k = 1 is 79.41%. This means that there are 27 O-D pairs covered by the 14 AVI readers in Plan B, compared with 18 O-D pairs in Plan C and 34 O-D pairs in Plan A. Though Plan A can cover all 34 O-D pairs, the number of readings is only 31,399 trips as shown in Figure 4 compared with 56,221 trips in Plan B and 86,703 trips in Plan C. Similarly, Figure 5b shows that, in Plan B, there are 84.45% of trips (20,550 trips) intercepted at

least once, 46.61% of trips (11,342 trips) intercepted at least twice, and so on. The trade-offs among the three plans are clearly depicted in Figures 4 and 5. In general, for the purpose of travel-time collection, the plan that has a curve farther to the right is generally more preferred. Being farther to the right implies that the selected plan can intercept vehicles more times along the paths (higher number of vehicles registered). From the flow-capturing analysis, Plan B represents a compromise solution among the three plans.

CONCLUSION AND FUTURE RESEARCH The multiobjective optimization model for locating AVI readers in a transportation network is proposed in this study. The objectives are designed to explicitly consider the trade-off between the quality and the cost of coverage. The objectives for quality of coverage are chosen to maximize the O-D coverage and the number of trips intercepted by the AVI readers. For the cost of coverage, the objective is simply to minimize the number of AVI readers. Because there may

Chen, Chootinan, and Pravinvongvuth

57

100.00

Percentage of O-D Covered (%)

Plan A

Plan B

Plan C

80.00

60.00

40.00

20.00

0.00 1

2

3

4 5 6 7 Number of Vehicles Registered (k) (a)

8

10

9

100.00

Percentage of Flow Captured (%)

Plan A

Plan B

Plan C

80.00

60.00

40.00

20.00

0.00 1

2

3

4 5 6 7 Number of Vehicles Registered (k) (b)

8

9

10

FIGURE 5 Comparison among three alternatives by percentages of (a) O-D covered and (b) flow captured.

not exist a single best solution that simultaneously optimizes all objectives, a distance-based GA is applied to explicitly generate nondominated solutions. The multiobjective AVI location formulation proposed in this paper holds great promise, because a wellconfigured AVI traffic-sensor network can provide reliable and accurate traffic data that are crucial not only for effective traffic operations and control, traveler information, and transportation planning but also for monitoring critical infrastructures. In this study, the O-D demand matrix was assumed known with certainty. The nondominated solutions, however, are highly dependent on the accuracy and reliability of the O-D demand matrix, which

are likely to be uncertain. Thus, future research effort should account for the uncertainty of the O-D demand matrix. In addition, other metaheuristics should be explored for solving the multiobjective AVI location formulation. REFERENCES 1. Conroy, P. Surveillance R&D: Directions. www.path.berkeley.edu/ PATH/Research/1999_progwide/41.ppt. Accessed May 2003. 2. Chen, C., J. Y. Kwon, and P. Varaiya. The Quality of Loop Data and the Health of California’s Freeway Loop Detectors (Draft). 2002. paleale.

58

3.

4. 5. 6.

7.

8.

9.

10. 11.

Transportation Research Record 1886

eecs.berkeley.edu/∼varaiya/papers_ps.dir/QualityOfLoopData.pdf. Accessed May 2003. Ygnace, J. L., C. Drane, Y. B. Yim, and R. de Lacvivier. Travel Time Estimation on the San Francisco Bay Area Network Using Cellular Phones as Probes. Technical Report UCB-ITS-PWP-2000-18. California PATH Program, University of California, Berkeley, 2000. SRI International. Data Coverage Plan: Data Source Status and Recommendations. Presented at TravInfo Contractor Procurement Workshop and Pre-Proposal Meeting, Dec. 2, 1999. Dixon, M. P. Incorporation of Automatic Vehicle Identification Data into the Synthetic OD Estimation Process. Ph.D. dissertation. Texas A&M University, College Station, 2000. Sherali, H. D., J. Desai, H. Rakha, and I. El-Shawarby. A Discrete Optimization Approach for Locating Automatic Vehicle Identification Readers for the Provision of Roadway Travel Times. Presented at 82nd Annual Meeting of the Transportation Research Board, Washington, D.C., 2003. Teodorovic, D., M. Van Aerde, F. Zhu, and F. Dion. Genetic Algorithms Approach to the Problem of Automated Vehicle Identification Equipment Locations. Journal of Advanced Transportation, Vol. 36, No. 1, 2002, pp. 1–21. Yang, B., and E. Miller-Hooks. Determining Critical Arcs for Collecting Real-Time Travel Information. Transportation Research Record: Journal of the Transportation Research Board, No. 1783, TRB, National Research Council, Washington, D.C., 2002, pp. 34–41. Chiu, Y.-C., N. H. Huynh, and H. S. Mahmassani. Determining Optimal Locations for Variable Message Signs Under Stochastic Incident Scenarios. Presented at 80th Annual Meeting of the Transportation Research Board, Washington, D.C., 2001. Toregas, C., and C. ReVelle. Binary Logic Solutions to a Class of Location Problems. Geographical Analysis, Vol. 5, 1973, pp. 145–155. Yang, H., L. Gan, and W. H. Tang. Determining Cordons and Screen Lines for Origin–Destination Trip Studies. Proc., 3rd Eastern Asia Soci-

12. 13. 14. 15. 16.

17. 18. 19. 20. 21.

ety for Transportation Studies, Vol. 3, No. 2, Hanoi, Vietnam, Oct. 2001, pp. 85–99. Church, R., and C. ReVelle. The Maximal Covering Location Problem. Papers of the Regional Science Association, Vol. 32, 1974, pp. 101–118. Hodgson, M. J. A Flow-Capturing Location-Allocation Model. Geographical Analysis, Vol. 22, 1990, pp. 270–279. Mirchandani, P. B., R. Rebello, and A. Agnetis. The Inspection Station Location Problem in Hazardous Materials Transportation: Some Heuristics and Bounds. INFOR, Vol. 33, 1995, pp. 100–113. Hodgson, M. J., K. E. Rosing, and J. Zhang. Locating Vehicle Inspection Stations to Protect a Transportation Network. Geographical Analysis, Vol. 28, 1996, pp. 299–314. Bell, M. G. H., and C. M. Shield. A Log-Linear Model for Path Flow Estimation. In Proceedings of the 4th International Conference on the Application of Advanced Technologies in Transportation Engineering, Capri, Italy, 1995, pp. 695–699. Goldberg, D. Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley Publishing, Reading, Mass., 1989. Coello Coello, C. A. A Comprehensive Survey of Evolutionary-Based Multiobjective Optimization Techniques. Knowledge and Information System, Vol. 1, No. 3, 1999, pp. 269–308. Gen, M., and R. Cheng. Genetic Algorithms and Engineering Optimization. John Wiley and Sons, Inc., New York, 2000. Osyczka, A., and S. Kundu. A New Method to Solve Generalized Multicriteria Optimization Problems Using the Simple Genetic Algorithm. Structural Optimization, Vol.10, 1995, pp. 94–99. Daskin, M. S. Network and Discrete Location: Models, Algorithms, and Applications. John Wiley and Sons, Inc., New York, 1995.

Publication of this paper sponsored by Intelligent Transportation Systems Committee.