(RWIS) Network Considering the Needs of Winter ...

Computer-Aided Civil and Infrastructure Engineering 00 (2016) 1–15

Location Optimization of Road Weather Information System (RWIS) Network Considering the Needs of Winter Road Maintenance and the Traveling Public Tae J. Kwon Department of Civil and Environmental Engineering, University of Waterloo, Waterloo, ON, Canada

Liping Fu* Department of Civil and Environmental Engineering, University of Waterloo, Waterloo, ON, Canada and School of Transportation and Logistics, Southwest Jiaotong University, Chengdu, China

& Stephanie J. Melles Department of Chemistry and Biology, Ryerson University, Toronto, ON, Canada

Abstract: This study presents an innovative approach to the planning of a critical highway sensor infrastructure - road weather information system (RWIS). The problem is formulated to minimize the spatially averaged kriging variance of hazardous road surface conditions while maximizing the coverage of accident-prone areas. This optimization framework takes explicit account of the value of information from an RWIS network, providing the potential to enhance the overall efficacy of winter maintenance operations and the safety of the travelers. Spatial simulated annealing is used to solve the resulting optimization problem and its performance is demonstrated using a real-world case study from Minnesota, United States. The case study illustrates the distinct features of the proposed model, assesses the effectiveness of the current location setting, and recommends additional stations locations. The findings of our study suggest that the proposed model could become a valuable decisionsupport tool for planning a new RWIS network and evaluating the performance of alternative RWIS expansion plans.

∗ To

whom correspondence should be addressed. E-mail: lfu@ uwaterloo.ca.

C 2016 Computer-Aided Civil and Infrastructure Engineering. DOI: 10.1111/mice.12222

1 INTRODUCTION Wintery countries often experience a high frequency of inclement weather events, which can have a detrimental impact on the safety and mobility of motorists. Generally, road collision rates increase dramatically during inclement weather conditions due to degradation of visibility and traction on the roadway (Goodwin, 2002; Wallman, 2004; Qiu and Nixon, 2008). There is also extensive evidence showing that inclement winter events can significantly affect traffic mobility in terms of reduction in vehicular operating speed and capacity (Kyte et al., 2001; Agarwal et al., 2005; Datla and Sharma, 2008; Kwon and Fu, 2011; Kwon et al., 2013). To minimize the safety and mobility impacts of winter weather events, it is crucial to enforce systematic snow and ice control, which can be realized by integrating various winter road maintenance operations including snow plowing and salting. Although winter road maintenance is indispensable, it entails substantial financial costs and environmental damage. North American transportation authorities, for instance, expend more than US $3 billion annually on winter road maintenance activities such as executing snow removal and applying

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salt and other chemicals for ice control (Ye et al., 2009; Highway Statistics Publications, 2005). Use of these chemicals has become an increasing environmental concern because they contaminate the ground and the surface water, and damage roadside vegetation and infrastructures. Likewise, motor vehicles have shown to suffer greatly from severe corrosion by use of road salt, and the cost of vehicle damage caused by such is simply immeasurable (Kuemmel, 1994). To reduce the costs of winter road maintenance and the use of salt, many transportation agencies are seeking ways to optimize their winter maintenance operations and improve the safety and mobility of the traveling public. One approach to improving the decision-making process for road maintenance is to make use of road weather and surface condition information through utilization of innovative technologies such as road weather information systems (RWIS). RWIS disseminates road weather and surface condition information to help winter road operation personnel make timely and proactive winter maintenance decisions. As a result, RWIS have been considered a primary tool supporting decision making of winter road maintenance operations by most road authorities for better winter maintenance decision support and traveler information provision. Although effective in providing valuable information, RWIS stations are expensive to install and operate and, therefore, can only be deployed at a restricted number of locations. Considering the immense road network that often needs to be monitored and the diverse road conditions that could develop any time during winter, RWIS stations must be placed strategically to ensure they are most informative in providing the inputs required for stimulating competent winter maintenance operations and provision of timely information to travelers. Currently, however, there are significant gaps in knowledge and methodology for effective planning of RWIS stations over a statewide road network. A few RWIS siting guidelines are available but they are limited to providing very general information and/or local site recommendations such as the availability of power and communication utilities (Garrett et al., 2008; Manfredi et al., 2008). Mackinnon and Lo (2009) proposed an approach to prioritizing locations based on a weighted measure of multiple factors such as traffic loads, accidents rates, and climatic information. A more recent study by Kwon et al. (2014) and Zhao et al. (2015) used a cost-benefit-based approach to quantify the monetary benefits of individual RWIS stations, and delineated the sites with the highest projected benefits. Another study by Jin et al. (2014) used weather-related crash data and converted to a Safety Concern Index, using which the locations providing a good spatial coverage

were identified as optimal locations. Although there are many nice properties and features in these previous studies, the models presented do not account for the trade-off between multiple location optimization criteria, and more importantly the ultimate use of RWIS information for spatial inference. In this research, we introduce a systematic framework to optimize the spatial design of a regional RWIS network, and demonstrate the value of the model using a real-world case study. The work has made both methodological and practical contributions to the field of interest. Methodologically, the formulation of the RWIS location optimization problem is foundational with several unique features, including explicit consideration of the enhanced monitoring capability of winter road weather conditions and road collision risk. Furthermore, the optimization framework proposed herein is the first in the literature targeted at simulating and optimizing RWIS station locations under a variety of settings, providing decision makers with the freedom to balance the needs of the traveling public, winter road maintenance requirements, and their respective priorities in locating RWIS stations. In particular, the objective is to maximize monitoring capabilities by minimizing the sum of the mean estimation errors (i.e., the kriging variance), of given RWIS measurements, and the coverage of winter collision-prone areas on the underlying highway network. The model allows development of a balanced solution considering maintenance needs as well as the traveling public. The practical value of the proposed model is demonstrated using a real-world case study from Minnesota Department of Transportation (Mn/DOT), with the solutions generated by our model being considered by the Mn/DOT RWIS manager for future deployment. The remainder of this article proceeds as follows. Section 2 describes the materials and methods used in this study whereas numerical results on a case study are presented in Section 3. Section 4 is followed to provide the conclusions and recommendations for future research.

2 METHODOLOGY To locate a monitoring network such as RWIS over a road network, it is critical to select a suitable criterion such that the fitness of any given location plan can be evaluated and quantified. An RWIS station typically consists of atmospheric, pavement, and/or water-level monitoring sensors that constantly (every 10–15 min) disseminate measurements including air and pavement temperatures; wind speed and direction, (sub)surface temperature and moisture, precipitation type, intensity

Location optimization of RWIS network

and accumulation, visibility, dew point, and relative humidity (Manfredi et al., 2008). Furthermore, each RWIS station reports road surface condition status based on current observations: areas that experience hazardous road surface conditions (HRSC) are flagged for a prompt remedial winter maintenance action. These socalled HRSCs include snow/ice warning, ice warning, wet below freezing, and frost. RWIS information makes it possible to perform proactive winter maintenance operations such as anti-icing (i.e., applying salt, mostly in liquid form, in advance of an event), which reduces the amount of time required to restore the roads to a clear and dry state at lower costs. As a large portion of RWIS benefits lies in the use of information, it is critical to locate stations in such a way that produces the most accurate information on various hazardous events over the whole road network. This argument remains valid under the assumption that an increase in estimation or monitoring capability of hazardous conditions will contribute to improving the overall quality of winter road maintenance operations. It is important to state that some regions may not have RWIS stations in place, and hence getting hazardous surface condition information directly from existing RWIS stations may not be possible. In a situation where there are no RWIS data available, data from local/regional weather stations and/or a record of winter maintenance activities can be acquired instead to best infer road weather and surface conditions and construct an underlying spatial relationship. To model the monitoring capability of a given RWIS network, this article proposes to apply a popular geostatistical approach called kriging (described further below) to estimate HRSCs for the entire road network based on the observations from a given RWIS network. The monitoring capability of the RWIS network can therefore be captured by summarizing the expected estimation errors – also called kriging errors, at all locations of interest. A nice property of the kriging errors is that they can be determined as part of the estimation process on the basis of the spatial correlation structure over the domain, which can be obtained as a function of distance (and perhaps direction) as a prior (van Groenigen and Stein, 1998). This indicates that kriging errors can be used as a criterion to optimize and evaluate an RWIS location solution. In addition, another optimization criterion, namely vehicular collision frequency, is introduced to reflect the needs of installing an RWIS station for reducing collisions in its vicinity. Selection of these criteria has been decided based primarily on the findings from a survey dedicated to reviewing and examining the current best practices for locating an RWIS station in North America (Kwon and Fu, 2012). The following section provides a formulation of the problem, followed by the detailed description of

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Fig. 1. An example of discretized network and the notations used (10 × 10 cells).

the kriging method for spatial inference as well as the optimization algorithm applied in this study. 2.1 Formulation of the problem Facility location problems have been well studied by operations researchers and engineers. Many innovative modeling techniques and solution algorithms have been developed, varying widely in terms of fundamental assumptions, mathematical complexity, and computational performance (Klose and Drexl, 2005). Discrete facility location problems (DFLP) are of particular interest in this research. DFLP problems are typically formulated as integer or mixed integer programming problems, for which, three broad types of discrete location models including covering- and median-based models, and others such as dispersion models are widely used (Daskin, 2008). Considering the nature of the proposed problem using the estimation errors, the RWIS station location problem cannot be simply classified as one definite type of facility location problem because the underlying spatial structure is defined by making a spatial inference on the location to be estimated using known data points. Perhaps the closest model can be a p-median in which the demand is defined based on the demand-weighted distance to all other available RWIS sites (i.e., one location is interrelated with all other locations). For more detailed information on facility location modeling, readers are advised to refer to comprehensive reviews documented by Daskin (2008) and Revelle et al. (2008). In this study, we also consider a discrete network, which is common in geostatistical analysis. Figure 1 shows an example of a discretized network with the notations defined herein. As shown in this figure, letus consider a region of interest, which is discretized into

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N grid cells with each cell represented by a single point with geographical coordinates labeled by i with i ࢠ 1, 2, . . . , N. There are a total of M monitoring stations (RWIS) labeled by k with k ࢠ 1, 2, . . . , M, and their locations are known and denoted by a vector X, where X = [x1 , . . . , xM ] and xk represents the location of RWIS station k. Let z be a variable of interest, which is observable at the M locations. Based on the observations from the M number of RWIS stations, we are interested in estimating the condition at any given geographic location i, denoted by zˆ (i|X ), which is an estimate of the true value z(i) given observations at X. In this study, average kriging variance is calculated to reflect the needs for installing RWIS stations for improved winter road maintenance operations (i.e., locations with higher errors require more attention than others with lower errors), and sum of average kriging variance should therefore be minimized. The traffic criterion pertaining to a vehicular collision frequency, on the other hand, should be maximized because an RWIS station should be located at high-risk areas. Therefore, to combine these two criteria, collision frequency measurements must be inverted such that the problem can be solved as a minimization problem. It is worthwhile noting that the inversion of the second criterion inevitably rescales the frequencies, hence, those values that are high are likely to be less penalized when compared to the lower values. To formulate the problem as an integer programming problem, we introduce a decision variable yki (i ࢠ 1, . . . , N, and k ࢠ 1, . . . , M) with yki = 1 if an RWIS station k is assigned to cell i, 0 otherwise. The decision variable yki is related to a geographic location, xk in X as follows: ∀i , ∀k (1) xk = (yki · i), i

The fitness function (objective function) combining the two location criteria is expressed in the following discrete formula: Min ϕ(X ) X ⊂

⎤ 2 σ [ˆz (i|X )] · ω ⎥ ⎢ i

=⎣ ⎦ , ∀i, ∀k (2) −1 1 +M · μi · yk,i · (1 − ω) ⎡

1 N

·

i

k

Subject to: i

ck,i · yk,i ≤ B,

∀i , ∀k

(3)

k

yk,i ∈ {0 , 1}, 0≤ω≤1

∀i , ∀k

(4) (5)

where is the index set that defines all of the candidate RWIS station locations in the study area, X is the subset of and a solution set, X = [x1 , . . . , xM ], N is the total number of all highway grid cells, M is the total number of available observations (i.e., RWIS stations), cki is the total cost of an RWIS station k at site i, B is the total available budget, σ 2 [ˆz (i)|X )] is the square root of the kriging variance at site i given X (to be defined in Section 2.2), μi−1 is the inverse of mean collision frequency at site i, and ω is the weighting factor. In Equation (2), the objective function represents the sum of average kriging standard deviation of estimating the HRSC frequency and average collision frequency, given X. The kriging variance term is root-squared, as appeared in the first part of the objective function so that estimation errors can be expressed in the same magnitude as the observations themselves (see Section 2.2 for more details). The second term of the objective function represents the sum of average collision frequency. Inclusion of this criterion is based on a number of prior studies, which have suggested that an RWIS station should be located at an accident-prone area to maximize the benefit of the RWIS, and to design a well-balanced RWIS network (Garrett et al, 2008; Buchanan and Gwartz, 2005; Mackinnon and Lo, 2009; Kwon et al., 2014). The binary decision variable yki is there to take account for those measured only when an RWIS station, k is allocated to site i. Average collision frequency is calculated using the minimum gridded cell, within each of which, all collision events are aggregated. As the two optimization criteria presented in Equation (2) have different units, they have been converted into a dimensionless term by normalizing their values into an equal scale. In addition, the candidate cells are predetermined by filtering out those cells that do not contain any segment of the highway network under investigation. This reduces the solution space of the optimization model significantly and thus the computational time. The constraint provided in Equation (3) represents the cost limit of installing RWIS stations in the study region. During installation, the stations may be equipped with different sensors based on various requirements. Furthermore, the annual maintenance costs for individual sites may also vary depending on the proximity to maintenance facilities. Hence, cki is added to take account for all supplementary costs in addition to the cost of installing a single RWIS station k at site i. The weighting term, ω is added so that an RWIS planning department can adjust and/or apply different weighting schemes according to their importance. It is


worthwhile noting that some sites may not have access to power and/or communication utilities; another important factor that must be considered to ensure that the data can be obtained and processed in real time (Manfredi et al., 2008). The optimization framework introduced in this article, however, can be easily extended to take additional factors into account by introducing another binary decision variable (i.e., 1 if a potential RWIS site has power/communication network in its vicinity, and 0 otherwise). Alternatively, the cells that do not satisfy the local requirements can be filtered out first such that only candidate locations are considered.

2.2 Kriging for spatial inference The main idea behind kriging is that the predicted outputs are weighted averages of sample data, and the weights are determined in such a way that they are unique to each predicted point and a function of the separation distance between the observed location, and the location to be predicted. The method primarily deals with the characterization of spatial attributes or regionalized variables, for which deterministic models may harbor significant uncertainty for drawing the complex underlying relationships due to the intricacy of the natural processes and variations (Olea, 1999). Previous studies in a variety of different fields, mainly in hydrology and ecology, support the applicability and usefulness of geostatistics as a tool for an optimum selection of sites for monitoring environmental and meteorological variables (van Groenigen et al., 1999; Cameron and Hunter, 2002; Nunes et al., 2007; Yeh et al., 2006; Brus and Heuvelink, 2007; Amorim et al., 2012). In the field of transportation, Chen et al. (2013) used different types of kriging methods for the purpose of optimizing the network-wide average travel time in transportation network. Another study by Lajnef et al. (2011) utilized the kriging method, primarily as an interpolation technique for field mapping and sediment loads estimation. However, use of kriging and its application to transportation research is still new and has been less studied. Likewise, incorporation of kriging variance as an optimization criterion has never been explored for the purpose of optimizing transportation facility locations. There are several different types of kriging models, among which three main kriging variants are called simple kriging (SK), ordinary kriging (OK), and regression kriging (RK) or universal kriging (UK). The most widely utilized kriging approach is OK, which assumes constant but unknown mean, whereas SK assumes a constant and known mean over the entire study area. Such a strong assumption of SK often limits its application to real-world problems (Olea, 2006). RK/UK

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are similar hybrid methods that use point observations and various covariates to model the trend component of the regionalized random variable of interest (Hengl et al., 2004). With UK, the trend component is typically modelled as a function of spatial covariates (i.e., in the X and Y coordinates); whereas with RK, the covariates may be any spatial regressors. In this study, we adopted the OK model considering its ability to take local fluctuations into account by limiting the domain of mean stationarity to the local neighborhood (Olea, 1999). This model is capable of dealing with environmental or meteorological variables, that is, RWIS measurements that typically show fluctuations over a certain spatial interval (Goovaerts, 1997). Equally important, OK is relatively easy to implement and requires substantially less amount of data, thus making it a more favorable option. The expression of OK model is provided in Equation (6) (Goovaerts, 1997). m m λk z(xk ) + 1 − λk m(i) (6) zˆ [i|X ] = k=1

k=1

where m(i) is expected value of the random variable z(i); i is a prediction location having geographic coordinates (lat and long); λk is a kriging weight assigned to datum z(xk ) and its value is to be determined based on spatial variation pattern. The unknown local mean is filtered from the linear estimator by forcing the weights to sum to 1. The OK estimator can then be written as: m m λk z(xk ) subject to λk = 1 (7) zˆ [i|X ] = k=1

k=1

Kriging weights are determined such that the estimate has the following two properties: 1. The estimate is unbiased: E[ˆz (i) − z(i)] = 0

(8)

2. The estimate has minimum variance: E[ˆz (i) − z(i)]2 = min

(9)

where z and zˆ are observed and estimated values, respectively. The above constrained minimization problem can be transformed into an unconstrained problem by means of a new objective function using a Lagrangian function (Hillier and Lieberman, 1995). The Lagrangian function for OK can be expressed as (Olea, 1999): L(λ1 , λ2 , · · · λm ; ) = 2

m

λk γ (xk , i)

k=1

−

m m k=1 j=1

λk λ j γ (xk , x j ) + 2

m k=1

λk − 1

(10)

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where γ () is the semivariogram, and is a Lagrange parameter. The semivariogram measures the degree of dissimilarity between pairs of observations in terms of their separations distances and orientations. This degree of dissimilarity is represented by what is known as semivariances, thus higher the semivariances, more dissimilar the measurements are. The kriging weights that produce the minimum estimation variance are the solution to (Olea, 1999): ⎧ m ⎪ ⎪ λk γ (xk , x1 ) − = γ (x1 , i) ⎪ ⎪ ⎪ k=1 ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ ⎪ k=1 λk γ (xk , x2 ) − = γ (x2 , i) ⎨ · · · (11) ⎪ m ⎪ ⎪ ⎪ λk γ (xk , xm ) − = γ (xm , i) ⎪ ⎪ ⎪ k=1 ⎪ ⎪ m ⎪ ⎪ ⎪ λk = 1 ⎩

Fig. 2. A typical semivariogram (adapted and modified from Flatman et al., 1987).

semivariogram estimators but the predominant form of describing this functional relationship is to use the following unbiased estimator (Olea, 2006):

k=1

σ [ˆz (i|X )] = 2

m

1 [z(xk + h) − z(xk )]2 2n(h) n(h)

The weights and Lagrange parameter can be determined by solving the system of linear equations, then the error variance can be determined by Equation (12) (Olea, 1999): λk γ (xk , i) −

(12)

k=1

2.3 Semivariogram The OK estimator discussed in the previous section requires an estimate on the covariance/semivariance of a given variable at any two locations within the region of interest, that is, γ (xk , xj ). This problem is addressed in geostatistics by assuming that the correlation between any two locations is a function of separation and orientation delineated by the two locations. The underlying functional relationship is called semivariogram and can be calibrated in advance using available data. Under the more general conditions of second-order stationarity, it is assumed that the spatial dependence between observations depends only on the distance between points and not on direction; that is, the spatial covariance function is isotropic (Webster and Oliver, 1992). Under this assumption, the semivariogram and the more commonly used statistical property, the covariance are related as follows (Goovaerts, 1997): C(h) = C(0) − γ (h) = Sill − γ (h)

(13)

where Sill is the semivariance value for large lag distances (h), where spatial autocorrelation between separated data points appear to be negligible, C(h) is the covariance function, C(0) is the variance, and γ (h) is the semivariogram model (see Figure 2). There are several

γˆ (h) =

(14)

k=1

where γˆ (h) is the sample semivariogram (i.e., half the average squared difference between points separated by a distance h), z(xk ) is a measurement taken at location xk , and n(h) is the number of pairs of observations separated by h. A typical layout of the sample semivariogram is depicted in Figure 2, where sill, range, and nugget represent the level of the plateau (if it exists), the lag distance where the semivariogram reaches the sill, and the nugget effect which accounts for micro scale variation and measurement errors or any spatial variability that exists at a distance smaller than the shortest distance of two measurements. Note that the sample data as represented by dots in Figure 2 are the output of Equation (14), and it is customary to use mathematical models (e.g., exponential, Gaussian, spherical) to fit the sample data owing to the fact that true spatial structure of a region is never known (Oliver and Webster, 1990). For more information on how to construct a good semivariogram, readers are referred to a comprehensive guide prepared by Olea (2006). 2.4 Optimization with spatial simulated annealing (SSA) The problem formulated previously is a nonlinear integer programming (NIP) problem which is computationally intractable; heuristic techniques are often required to solve these types of problems of realistic sizes. In this research, we applied a variant of one of the most successful techniques called SSA, which is a spatial counterpart to simulated annealing (SA,


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Fig. 3. Workflow of SSA algorithm.

Kirkpatrick et al., 1983). SSA is an iterative combinatorial optimization algorithm in which a sequence of combinations is produced by deriving a new combination from slightly and randomly modifying the previous combination (van Groenigen et al., 1999). In the search process, SSA not only accepts improving solutions, but also worsening solutions, based on a certain probability that is defined to minimize the risk of falling prematurely into local minima (van Groeningen and Stein, 1998). More importantly, the method has a unique generation mechanism for transforming a randomly chosen sampling point over a vector h—the direction chosen at random, and the length also drawn randomly in the interval [0 and hmax ], thus giving the sampling scheme the chance to “freeze” in its optimal sampling design by terminating with very small perturbations (van Groeningen et al., 1997). This method is proven to produce dramatic improvements compared to its nonspatial counterparts (van Groeningen et al., 1997; van Groenigen and Stein, 1998). The workflow of the SSA algorithm is depicted in Figure 3. For more detailed information on the SSA algorithm (e.g., selection of optimization parameters), the reader is encouraged to review the papers by Kirkpatrick, 1984; van Groeningen et al., 1997; and van Groenigen and Stein, 1998).

3 CASE STUDY—MINNESOTA, UNITED STATES 3.1 Study area and RWIS network The Mn/DOT installed their first RWIS station in the Minneapolis area in 1988. Soon after, a task force was established to investigate implementation of a statewide integrated RWIS. Over the last decade, the benefits of RWIS have been widely recognized by the DOT, and thus the number of installations has increased dramatically. Currently, there are a total of 97 RWIS stations in Minnesota, and the number is expected to increase in the future. The current Mn/DOT’s RWIS network covering a state highway system of approximately 13,400 km is depicted in Figure 4. Individual RWIS stations and the Minnesota highway network are delineated by blue circles and grey lines, respectively. As discussed earlier, the study area was first discretized into a uniform grid of 3 km × 3 km. The grid cells that do not cover any highways are first filtered out as they are not the valid candidate sites for any RWIS stations. This sub-filtering step is beneficial computationally as it reduces the size of the candidate pool significantly. As a result, a total of 27,246 prediction locations, and

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Fig. 4. Minnesota RWIS (triangles) and highway network (solid lines).

8,124 candidate locations have been delineated and used in the optimization process. 3.2 Data description Mn/DOT provided their regional RWIS data, which were collected at 15-min intervals over three consecutive winters (i.e., October to March) between 2010 and 2013. The data came stratified by individual stations each containing nearly one million rows of measurements including the variable of interest—surface condition status. There are a total of 15 surface status codes describing the current surface conditions as shown in Table 1. These status descriptions are listed in order of severity and further classified into several categories with the most critical category listed first. In case of Mn/DOT, a total of 11 different categories are used to illustrate the severity of each condition, providing a prioritized list of hazardous surface conditions. In this study, we define the top four categories as HRSC, which include snow/ice warning, ice warning, wet below freezing, and frost, as they pose a great danger to the public

and thus require fast remedial actions. Each data entry was checked and counted if it reported anything that belonged to the category under consideration. VBA scripts were written to efficiently process over 58 million rows of data, returning a yearly (seasonal) average HRSC frequency for each of the existing RWIS stations. As pointed out in previous sections, a fundamental assumption of implementing geostatistics is the existence of spatially correlated data, which describes the commonly held notion that near-by measurements will have similar values and the degree of similarity decreases as the distance of separation increases (Olea, 2006). As the assessment of such spatial correlation is a prerequisite to further pursue and apply kriging, it is critical to model the semivariogram based on the information provided by the attribute of interest—the HRSC. The processed data were subsequently used to construct a semivariogram—a model characterizing the spatial autocorrelation of HRSC measurements to be used in estimating the kriging variance in the objective function. Note that our investigation of spatial structure in different cardinal directions revealed no significant


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Table 1 Surface status definition (adopted from Mn/DOT Scan Web, 2015) Surface condition status Snow/Ice Warning Ice Warning Wet Below Freezing Frost Ice Watch Snow/Ice Watch Chemical Wet Wet Damp Trace Moisture Absorption at Dew Point Dry Other No Report Error

Description Continuous film of ice and water mixture at or below freezing (32 °F/0 °C) with insufficient chemical to keep the mixture from freezing Continuous film of ice and water mixture at or below freezing (32 °F/0 °C) with insufficient chemical to keep the mixture from freezing Moisture on pavement sensor with a surface temperature below freezing (32 °F/0 °C) Moisture on pavement at or below freezing (32 °F/0 °C) with a pavement temperature at or below the dew point temperature Thin or spotty film of moisture at or below freezing (32 °F/0 °C) Thin or spotty film of moisture at or below freezing (32 °F/0 °C) Continuous film of water and ice mixture at or below freezing (32 °F/0 °C) with enough chemical to keep the mixture from freezing Continuous film of moisture on the pavement sensor with a surface temperature above freezing (32 °F/0 °C) Thin or spotty film of moisture above freezing (32 °F/0 °C) Thin or spotty film of moisture above freezing (32 °F/0 °C). Surface moisture occurred without precipitation being detected Currently not detected Absence of moisture on the surface sensor Conditions not explicitly included in this table The surface sensor is not operating properly and requires maintenance The surface sensor is not operating properly and requires maintenance

difference among all experimental semivariograms tested. The semivariogram was fit with the exponential model given an estimated nugget variance of 0, a sill of 0.94, and range of 95,000 m. To assure the accuracy of the semivariogram, a quantitative assessment namely cross-validation is employed as a verification process, where each observation is removed with replacement to produce an estimate at the same site of the removal (Olea, 1999). A resulting correlation rate between the predicted and observed values is found to be 0.88 (ideally 1) indicating that the model is able to produce accurate estimates. Ordinary kriging was performed using R (R Development Core Team, 2014) with package gstat (Pebesma, 2004). The square root of kriging variance discussed in Section 2 was implemented as the first criterion in the objective function. Another set of data provided by Mn/DOT was collision data collected over a 5-year period (i.e., 2008 to 2013). The data contained individual crash records with detailed information. Each record represented a single incident and listed day, month, year, and locational attributes (latitude and longitude). As the focus of the intended study is a winter season, weather-related collisions that occurred only during winter months were considered. As such, a total of 44,674 winter vehicle collision records were extracted and geocoded on a GIS platform for efficient processing. As the minimum spatial unit was 3 km, the collision data were aggregated

and averaged over a 9-km2 gridded surface so that the traffic portion of the optimization criterion could be evaluated. 3.3 The results The proposed model was implemented in designing an optimal RWIS network using the dual criteria discussed earlier. First, as there are currently a total of 97 RWIS stations in Minnesota, the optimization algorithm was run to obtain an all-new 97 RWIS station design using different weights to better appreciate different network design scenarios. The optimized and existing RWIS networks were then compared by determining the objective function value of each network design to demonstrate the superiority of the optimized network. For simplicity and convenience herein, a uniform cost of RWIS stations was used and a total available budget was assumed to be equal to the proposed number of RWIS stations to be deployed. RWIS network expansion scenarios were proposed by locating a new set of 5 and 15 additional RWIS stations in the existing RWIS network of Minnesota. When running the optimization with the SSA algorithm, the optimization parameters were selected based on the existing literature (Kirkpatrick, 1984; Brus and Heuvelink, 2007; Heuvelink et al., 2010; Melles et al., 2011). The optimization was run over a total of 10,000 iterations in

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Fig. 5. All-new 97 optimized RWIS station locations using the combined criterion optimized with (a) ω = 1, (b) ω = 0.5, (c) ω = 0.3, and (d) ω = 0.1.

generating each optimal network design solution. Analyses were performed on a Windows operating desktop computer equipped with a 3.39 GHz processor and 8.00 GB of RAM, and a series of customized functions coded in R was used in this study. 3.3.1 All-new optimal RWIS network. The first analysis was to evaluate the location quality of the current RWIS network in relation to the optimal solution obtained from our model under different weightings between the two location criteria. As discussed earlier, the greatest benefit of the proposed approach is its ability to simulate and optimize RWIS station locations under any given settings that users define. It also

provides decision makers with the freedom to choose different weights depending on the needs of the traveling public, winter road maintenance requirements, and their respective priorities in locating RWIS stations. Table 2 Comparison of objective function values of the optimized networks with respect to the current network Scenarios Figure 5(a) ω = 1.0 Figure 5(b) ω = 0.5 Figure 5(c) ω = 0.3 Figure 5(d) ω = 0.1

Obj. function

% Improvement

0.1424 0.1517 0.1569 0.1624

21.12% 19.60% 17.31% 16.55%


Fig. 6. Decrease of the combined objective function as a function of iterations (ω = 1).

As such, the RWIS network optimized under different weights is presented in Figure 5. The aggregated collision frequencies and kriged HRSC measurements are superimposed on the same map to help better appreciate and recognize how assignment of varying weights could contribute to deciding the optimal location for individual RWIS stations. In this figure, the all-new optimized RWIS stations are represented by circles. It is worthwhile to note that for each weighting scheme, the optimization was run three times and the outputs were visually compared to confirm that the optimization outputs were very similar and comparable to each other. The intent of multiple tryouts was to ensure that the SSA algorithm had reached an (near-) optimal solution without being trapped in local minima, which is an inherent problem of the SSA algorithm and all other metaheuristic algorithms currently available today. Figure 5a represents a case when OK variance is solely used in the objective function to minimize the spatially averaged kriging variance (i.e., ω = 1). In this figure, it is evident that RWIS stations are well spread over the entire study area without offering much consideration to the traveling public or in our case the collision frequency. In Figure 5b, the traffic criterion, namely collision frequency, has been added to the first criterion with equal weights. As can be seen clearly, incorporation of the traffic criterion enabled capturing the coverage of accident-prone areas, providing an improved balance in terms of monitoring HRSC and collision frequencies. Figures 5c and d illustrate the optimized RWIS networks based on different weights using ω = 0.3, and ω = 0.1, respectively. Such a difference in its pattern is well manifested; by decreasing the weight on the first criterion of the objective function, a higher number of RWIS stations are allocated to areas exposed to high collision risk areas, for instance, the City of Minneapolis and several highway corridors branching out from the city. The findings reveal that

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Fig. 7. Similarity index using different weights.

the proposed method is able to provide transportation agencies with the unique solutions dedicated to serving their respective priorities in terms of the monitoring capability of dangerous road surface conditions, and the coverage of accident-prone areas. As for the iteration schedule of the performed optimizations, Figure 6 illustrates the decrease of the dual objective function as the number of iterations increases for three runs. It is apparent after around 5,000 iterations, the objective function for all runs starts to level off and slowly reaches its minimum value as evidenced by the lower value obtained at the 10,000th iteration. As explained previously, the SSA algorithm has a mechanism that reduces the risk of falling prematurely into local minima, provided that there is a certain probability and a decaying function controlling how fast the objective function converges. Such behavior is well presented by the continuous fluctuations and peaks observed until it stabilizes at around 5,000 iterations. In terms of computational efficiency, optimization took an average running time of approximately 11 hours for optimizing each given network. To evaluate the overall efficacy of each optimized network (Figure 5) with respect to the existing network (Figure 4), the objective function was used to calculate its corresponding numerical value for all individual outputs as well as the current RWIS network. This evaluation metric is simply the lowest value obtained at the end of each optimization. For the existing network, a comparable yet equivalent approach is exercised by adding the averaged OK variance and collision frequency given the current RWIS station locations. Table 2 compares the lowest objective function value (three runs) associated with each optimal solution and the current network, along with percentage of improvement. It is worthwhile noting that the objective function values from all three runs for all tests were very similar and comparable to each other (less than 0.5% in difference) indicating that the solutions generated are deemed (sub-) optimal. The percentage of

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Kwon, Fu & Melles

Fig. 8. Expansion of current RWIS network with (a) 5 and (b) 15 additional stations.

improvement, which can also be interpreted as perceived benefits, was found to vary between 17% and 21%, signifying that the optimized networks are “better” in terms of monitoring capabilities of various HRSC and vehicular collisions, as defined in the objective function. Although the key benefit of the proposed method lies with its unique ability to provide RWIS planning agencies with the freedom to choose different weights for each optimization criterion, depending on their presumed relative importance, a sensitivity analysis, which examines how model results vary with respect to different weighting schemes, may be able to offer important insights on the implication of choosing different weights. To examine such effects, spatial overlay analysis was used as an evaluation method that quantifies the degree of similarities between the optimized and existing network. First, the existing RWIS network as well as all other optimal solutions were buffered with a reasonable size and dissolved to obtain the total spatial area. The optimal solutions generated using 11 different weighting schemes (i.e., 0 indicating only traffic to 1 indicating only weather) were used to determine the corresponding spatial overlapping area with respect to the existing network. Lastly, a similarity index was calculated by using the overlapping area such that the degree of similarity between one network and another can be effectively examined. Figure 7 depicts how different weights affect the optimization outputs with respect to the spatial distribution of the existing RWIS network. These findings indicate that the optimal solutions generated are sensitive to weights applied in the objective function, having a maximum difference of 16% (i.e., the difference of the values between ω = 0.4 and ω = 0.9). Likewise, visual inspection of the results also revealed that the current Minnesota RWIS network is able to provide a balanced coverage in terms

of monitoring capabilities of HRSC and accident-prone areas, as indicated by a high similarity index at ω = 0.4. 3.3.2 Expansion of current RWIS network. In the previous section, the proposed method was applied to find optimal locations for the entire existing set of RWIS stations. However, this may be not a realistic option from a practical standpoint because it is extremely costly to relocate the entire network. In this section, we develop solutions for several hypothetical expansion plans over the current RWIS network. Consultation with Mn/DOT revealed that there is an RWIS network expansion plan to deploy 5–15 additional stations over the next few years to increase coverage and enhance winter maintenance operation programs. The optimization problem has therefore been modified to reflect the changes in the base condition. The objective function is evaluated at each iteration considering that there are permanently fixed 97 RWIS stations (i.e., a current number of RWIS stations in Minnesota) throughout the entire optimization process. As mentioned, the total budget constraint was assumed to equal the number of RWIS stations added to the existing network. In addition, identical optimization parameters and an equal weighting scheme (ω = 0.5) were used to locate 5 and 15 additional RWIS stations (circles) as depicted in Figure 8. As can be seen in this figure, minimization of sum of averaged kriging variance effectively prevented new stations from being closely located in the vicinity of existing stations. From a visual inspection, it can be asserted that new stations fill gaps in the existing RWIS network where vehicular collision frequency was also relatively high. In addition, evaluation of objective function values show that the performance of the current network is improved by 8.4% and 12.6% with the placement of 5 and 15 additional stations,


respectively. It should be noted that the 15 optimized stations depicted in Figure 8b is the result of running the optimization on the existing 97 stations, not on the 102 stations (i.e., 97 + 5), hence there are no duplicates in their optimized locations. This way of delineating the optimal locations is considered more desirable, especially when there is a large number of RWIS stations to be deployed in the near future in the area under analysis. Finally, we have presented the results to the Mn/DOT and received overwhelming positive feedback regarding the reasonableness of the solutions identified by our model. They have also confirmed that these solutions will be considered as the primary input to their decision on the location of their future RWIS stations.

4 CONCLUSIONS AND RECOMMENDATIONS In this research, an innovative optimization model was introduced for the purpose of locating RWIS stations over a regional highway network. The method of determining the optimal RWIS sampling design is new—the first of its kind that provides transportation agencies with a tool that helps them determine the optimal location of one of the most important transportation sensor infrastructures—RWIS network. In addition, the RWIS location allocation optimization framework represents the first attempt to address the challenging problem with a formal mathematical programing approach. In the proposed method, the weighted sum of average kriging variance of HRSC and collision frequencies was used to determine the optimal RWIS network design. The basic assumption is that minimizing the total estimation error would, in due course, contribute to improving the global effectiveness and efficiency of winter road maintenance operations. Collision frequencies over the underlying road network were incorporated to provide a balanced solution that considers the information need of the traveling public. The SSA algorithm was employed to solve the combinatorial optimization problem. A case study based on Minnesota, United States exemplified two distinct scenarios—redesign and expansion of the existing RWIS network. Findings from the case study indicate that optimally redesigned RWIS networks are, on average, 18.6% better in capturing the dual criteria considered when compared to the existing RWIS network. The study further revealed that the deployment of 5 and 15 additional RWIS stations would improve the current network by 8.4% and 12.6%, respectively. The proposed model has recently been implemented in a web application that is currently being evaluated by the various transportation agencies for solving their real-world problems.

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The work can be extended in several directions. First, other variants of kriging, such as regression kriging or universal kriging (Hengl et al., 2004; Heuvelink et al., 2006; Amorim et al., 2012), can be used to take into account auxiliary variables to obtain more accurate and detailed results in modeling the trend component of the regionalized random variable (e.g., HRSC measurements). Second, although the SSA algorithm is known for its ease of implementation and requires fewer parameters to configure, its relatively longer processing time remains as its main drawback. Therefore, other heuristic algorithms including greedy algorithm (Cormen et al., 2001), genetic algorithm (Kim and Adeli, 2001; Sarma and Adeli, 2001; Paris et al., 2015), and tabu search (Glover and Laguna, 1997) should also be explored and tested. Finally, more case studies should be conducted to investigate the generality and sensitivity of the model results to external conditions including network size, size of grid, parameters used in the SSA, and input parameters including use of other traffic variables (accident rates, highway class), and weather variables (snow intensity, road surface temperature).

ACKNOWLEDGMENTS The authors wish to thank the Minnesota DOT for providing the data necessary to complete this study. We would also like to thank anonymous reviewers for their valuable comments and suggestions that have contributed tremendously to improving the quality of our manuscript. This research was funded by the Aurora Program, and was partially funded by National Sciences and Engineering Research Council of Canada (NSERC).

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