This section of the manuscript presents a case study that was used to evaluate the effectiveness of the developed optimization models and solution approaches for the existing highway–rail grade crossings in the State of Florida. Note that only public highway–rail grade crossings will be considered throughout the analysis.
6.1. Input Data
Both
RAP-1 and
RAP-2 optimization models require certain input data in order to perform resource allocation among the highway–rail grade crossings in the State of Florida. The input data related to the set of highway–rail grade crossings (
) were extracted from the publicly available crossing inventory database, which is maintained by the FRA [
42]. The inventory database includes detailed information regarding basic crossing characteristics (e.g., average daily traffic volume, average daily train volume, protection type, train speed, crossing surface). The overall hazard at highway–rail grade crossings (
) for the
RAP-1 mathematical model was calculated using the Florida priority index formula [
22]:
where
= the Florida priority index at highway–rail grade crossing
(no units);
= average daily traffic volume at highway–rail grade crossing
(vehicles per day);
= average daily train volume at highway–rail grade crossing
(trains per day);
= train speed at highway–rail grade crossing
(mph);
= protection factor at highway–rail grade crossing
(PF = 1.00 for passive; PF = 0.70 for flashing lights; PF = 0.10 for gates);
= accident history parameter at highway–rail grade crossing
(accidents), representing the total number of accidents in the last 5 years or since the year of last improvement (if there was an upgrade).
Note that many factors may influence the overall hazard of a given highway–rail grade crossing (i.e., average daily train volume, number of cars in a train, train type, train speed, number of tracks, average daily traffic volume, sight distance, number of traffic lanes, highway vehicular speed, approach gradient, crossing angle, location, existing protection, etc.). Dulebenets et al. [
22] evaluated a wide range of different accident and hazard prediction models for the highway–rail grade crossings in the State of Florida, including the Coleman–Stewart model, NCHRP Report 50 accident prediction formula, Peabody–Dimmick formula, U.S. DOT accident prediction formula, New Hampshire formula, California hazard rating formula, Connecticut hazard rating formula, Illinois hazard index formula, Michigan hazard index formula, Texas priority index formula, and the Florida priority index formula. The considered accident and hazard prediction models directly accounted for many of the aforementioned factors that influence the overall hazard of highway–rail grade crossings. The actual 2007–2016 accident data provided by the FRA were used to estimate the predicted number of accidents and the predicted overall hazard by applying the candidate models, while the actual 2017 accident data were used to validate the candidate models. It was found that over-represented accident and hazard prediction models (i.e., the ones that consider many factors at the same time) have low accident and hazard prediction accuracy for the highway–rail grade crossings in the State of Florida. On the other hand, the Florida priority index formula demonstrated the best performance; therefore, it was used throughout this study.
The overall hazard of a given highway–rail grade crossing was set equal to the Florida priority index (i.e.,
). The FRA highway–rail grade crossing accident database [
13] was used for estimation of the accident history parameter (
). Both
RAP-1 and
RAP-2 mathematical models relied on a set of countermeasures, denoted as
, to improve the level of safety at the highway–rail grade crossings. Each countermeasure has a specific effectiveness factor, which represents the percentage reduction of a potential hazard at a given highway–rail grade crossing [
39,
43]. However, not all the countermeasures can be implemented at a given highway–rail grade crossing, considering its technical characteristics. In this study, a total of 11 countermeasures were adopted for the
RAP-1 and
RAP-2 mathematical models from the GradeDec.NET reference manual [
39]. The basic information regarding the countermeasures was retrieved from the GradeDec.NET reference manual (see
Table 2), including the (1) effectiveness factors (
) and (2) installation costs (
). Note that the
RAP-1 and
RAP-2 mathematical models can be applied for other types of countermeasures as well (i.e., not just the ones that are suggested by the GradeDec.NET reference manual). In order to introduce new countermeasures in the developed mathematical models, the user will need to add new elements to the set of countermeasures (
) and specify the effectiveness factor (
), as well as the installation cost (
), for each new countermeasure.
In order to estimate the overall hazard severity at each highway–rail grade crossing for the
RAP-2 mathematical model, the overall hazard estimated by the Florida priority index should be divided into different severity categories. In this study, the hazard severity was categorized based on the GradeDec.NET reference manual as follows [
39]: (1) fatal accidents—accidents with at least one fatality; (2) injury accidents—accidents with at least one injury, but no fatality; and (3) property damage only accidents. The severity weights attributed to different severity categories in the
RAP-2 mathematical model were adopted based on the project conducted by the Iowa DOT [
18]. Specifically, the base values for the weights of fatality hazard (
), injury hazard (
), and property damage hazard (
) were set to
,
, and
, respectively.
6.2. Evaluation of the Solution Algorithms
All the solution algorithms developed for the
RAP-1 and
RAP-2 mathematical models were evaluated for all the public highway–rail grade crossings in the State of Florida. There were a total of 6089 public highway–rail grade crossings in the State of Florida based on the FRA crossing inventory database as of November 2018. A total of 121 scenarios were developed to evaluate the performance of the exact optimization algorithms and the heuristic algorithms. The scenarios were developed by varying the number of highway–rail grade crossings and the number of available countermeasures. The following values were considered for the number of highway–rail grade crossings:
. The number of countermeasures was altered from 1 to 11 by an increment of 1. A total of 12 problem instances were developed by changing the values for the total available budget as follows:
. Note that the adopted values for the total available budget are in line with the ones reported by the FDOT [
14].
All the developed heuristic algorithms (see
Section 5.2 of the manuscript) were encoded in the MATLAB environment. Moreover, the GAMS environment was used to encode the
RAP-1 and
RAP-2 mathematical models and to solve them to the global optimality with CPLEX. A CPU with Dell Intel(R) Core™ i7 Processor, 32 GB of RAM, and Windows 10 operating system was utilized to perform all the numerical experiments throughout this study. Note that the
RAP-1 and
RAP-2 mathematical models were solved using the function “intlinprog” available within the MATLAB optimization toolbox. However, the initial numerical experiments showed that “intlinprog” required a significant computational time as compared to the other solution methodologies adopted in this study. Furthermore, in some cases, “intlinprog” violated certain constraint sets of the
RAP-1 and
RAP-2 mathematical models. Therefore, the MATLAB “intlinprog” function had to be withdrawn from the analysis.
Section 6.2.1 and
Section 6.2.2 provide more details regarding evaluation of the candidate solution algorithms for the
RAP-1 and
RAP-2 mathematical models, respectively.
6.2.1. Solution Quality and Computational Efforts for RAP-1
The
RAP-1 mathematical model was solved utilizing CPLEX, MPHR, MEHR, PHR, and EHR solution algorithms for all the developed scenarios of each problem instance considered. Note that the same objective function values were returned by each algorithm after each iteration, as they are deterministic in nature. However, in order to estimate the average computational time, a total of 5 replications were performed for each scenario and each problem instance.
Table 3 and
Figure 5 present the average over the developed scenarios overall hazard values returned by CPLEX, MPHR, MEHR, PHR, and EHR for each of the considered problem instances of the
RAP-1 mathematical model. Furthermore, the average over the considered problem instances overall hazard values returned by CPLEX, MPHR, MEHR, PHR, and EHR for each of the developed scenarios of the
RAP-1 mathematical model are presented in
Figure 6.
Based on the information presented in
Table 3, the averages of the overall hazard values for the developed scenarios were 1,755,123.6, 1,769,527.6, 2,061,902.0, 1,769,495.1, and 2,034,204.7 for CPLEX, MPHR, MEHR, PHR, and EHR, respectively. Hence, the MPHR and PHR heuristics returned the solutions that were close to the optimal solutions identified by CPLEX for all the developed scenarios of each problem instance of the
RAP-1 mathematical model. However, PHR was found to be superior to MPHR for certain scenarios, as it consistently assigns countermeasures to the highway–rail grade crossings from the priority list using the hazard reduction-to-cost ratios (unlike MPHR, which assigns countermeasures with the highest hazard reduction-to-cost ratios to the highway–rail grade crossings from the priority list, while the remaining budget is allocated for implementation of eligible countermeasures at certain highway–rail grade crossings that were not selected for upgrading). The numerical experiments show that the MEHR and EHR heuristics were outperformed by CPLEX in terms of the overall hazard values on average by 17.48% and 15.90%, respectively, over the considered problem instances. Hence, prioritization of the highway–rail grade crossings based on the hazard reduction-to-cost ratios (implemented within MPHR and PHR) was found to be a more effective strategy throughout resource allocation as compared to prioritization of the highway–rail grade crossings based on either combination of the hazard reduction-to-cost ratios and the hazard reduction (implemented within MEHR) or based on the hazard reduction only (implemented within EHR).
Table 4 presents the average over the developed scenarios computational time recorded for CPLEX, MPHR, MEHR, PHR, and EHR for each of the considered problem instances of the
RAP-1 mathematical model. The averages of computational time values for the developed scenarios were 81.19 sec, 29.89 sec, 10.25 sec, 16.78 sec, and 11.68 sec for CPLEX, MPHR, MEHR, PHR, and EHR, respectively. Although all the candidate solution algorithms were able to solve all the developed scenarios of each problem instance of the
RAP-1 mathematical model within a reasonable computational time, it is expected that the CPLEX computational time may significantly increase for certain problem instances due to the computational complexity of the
RAP-1 mathematical model. Therefore, based on the results from the conducted analysis, the PHR heuristic algorithm is recommended for resource allocation among the highway–rail grade crossings in the State of Florida, aiming to minimize the overall crossing hazard.
6.2.2. Solution Quality and Computational Efforts for RAP-2
The
RAP-2 mathematical model was solved utilizing CPLEX, MPSR, MESR, PSR, and ESR solution algorithms for all the developed scenarios of each problem instance considered. Note that the same objective function values were returned by each algorithm after each iteration, as they are deterministic in nature. However, in order to estimate the average computational time, a total of 5 replications were performed for each scenario and each problem instance.
Table 5 and
Figure 7 present the average over the developed scenarios overall hazard severity values returned by CPLEX, MPSR, MESR, PSR, and ESR for each of the considered problem instances of the
RAP-2 mathematical model. Furthermore, the average over the considered problem instances overall hazard severity values returned by CPLEX, MPSR, MESR, PSR, and ESR for each of the developed scenarios of the
RAP-2 mathematical model are presented in
Figure 8.
Based on the information presented in
Table 5, the average overall hazard severity values for the developed scenarios were 338,700.3, 341,696.5, 401,785.0, 341,687.4, and 398,079.7 for CPLEX, MPSR, MESR, PSR, and ESR, respectively. Hence, the MPSR and PSR heuristics returned solutions that were close to the optimal solutions identified by CPLEX for all of the developed scenarios of each problem instance of the
RAP-2 mathematical model. However, PSR was found to be superior to MPSR for certain scenarios, as it consistently assigns countermeasures to the highway–rail grade crossings from the priority list using the hazard severity reduction-to-cost ratios (unlike MPSR, which assigns countermeasures with the highest hazard severity reduction-to-cost ratios to the highway–rail grade crossings from the priority list, while the remaining budget is allocated for implementation of eligible countermeasures at certain highway–rail grade crossings that were not selected for upgrading). The numerical experiments show that the MESR and ESR heuristics were outperformed by CPLEX in terms of the overall hazard severity values on average by 18.63% and 17.53%, respectively, over the considered problem instances. Hence, prioritization of the highway–rail grade crossings based on the hazard severity reduction-to-cost ratios (implemented within MPSR and PSR) was found to be a more effective strategy throughout resource allocation as compared to prioritization of the highway–rail grade crossings based on either combination of the hazard severity reduction-to-cost ratios and the hazard severity reduction (implemented within MESR) or based on the hazard severity reduction only (implemented within ESR).
Table 6 presents the average over the developed scenarios computational time recorded for CPLEX, MPSR, MESR, PSR, and ESR for each of the considered problem instances of the
RAP-2 mathematical model. The average computational time values for the developed scenarios were 79.64 sec, 14.52 sec, 13.94 sec, 15.01 sec, and 15.24 sec for CPLEX, MPSR, MESR, PSR, and ESR, respectively. Although all the candidate solution algorithms were able to solve all the developed scenarios of each problem instance of the
RAP-2 mathematical model within a reasonable computational time, it is expected that the CPLEX computational time may significantly increase for certain problem instances due to the computational complexity of the
RAP-2 mathematical model. Therefore, based on the results from the conducted analysis, the PSR heuristic algorithm is recommended for resource allocation among the highway–rail grade crossings in the State of Florida, aiming to minimize the overall crossing hazard severity.
6.3. Managerial Insights
This section of the manuscript provides the results of the sensitivity analyses, which were conducted to reveal how the developed optimization models and solution methodologies can assist the relevant stakeholders, including the smart city authorities, with important managerial insights. In particular, the sensitivity of the resource allocation decisions to the total available budget and the number of available countermeasures will be further investigated.
6.3.1. Sensitivity Analysis for the Total Available Budget
A sensitivity analysis was conducted to determine how the total available budget may impact the resource allocation decisions among the highway–rail grade crossings in the State of Florida. A total of 12 budget availability scenarios were developed by altering the total available budget from
$7.5 million to
$13.0 million, with increments of
$500,000. The impact of the total available budget on the number of upgraded highway–rail grade crossings by
RAP-1 and
RAP-2 is illustrated in
Figure 9. Note that PHR and PSR were used as solution approaches for
RAP-1 and
RAP-2, respectively. As expected, the total number of highway–rail grade crossings upgraded by
RAP-1 and
RAP-2 increased after increasing the total available budget. Specifically, a total of 1198 and 1723 highway–rail grade crossings were selected for upgrading by
RAP-1 for budget availability scenarios “1” and “12”, respectively. On the other hand, a total of 1212 and 1705 highway–rail grade crossings were selected for upgrading by
RAP-2 for budget availability scenarios “1” and “12”, respectively. Therefore, the total available budget may substantially influence the resource allocation decisions and directly affect the number of upgraded highway–rail grade crossings.
Figure 10 illustrates the overall hazard before and after implementation of countermeasures at the highway–rail grade crossings that were suggested for upgrading by
RAP-1 for all 12 developed budget availability scenarios. It can be noticed that neither the overall hazard before nor the overall hazard after implementation of countermeasures at the highway–rail grade crossings that were suggested for upgrading by
RAP-1 substantially changed from increasing the total available budget. Such a finding can be explained by the fact that
RAP-1 selected for upgrading the most hazardous highway–rail grade crossings, even with the total available budget of
$7.5 million (i.e., scenario “1”). Increasing the total available budget allowed upgrading of additional highway–rail grade crossings, however their overall hazard was not as significant as the one recorded for the highway–rail grade crossings upgraded under budget availability scenario “1”. Nevertheless, implementation of the countermeasures that were suggested by
RAP-1 significantly reduced the overall hazard at the highway–rail grade crossings. Specifically, the overall hazard was reduced by
for budget availability scenario “1”. Furthermore, the overall hazard was reduced by
for budget availability scenario “12”. Therefore, the developed
RAP-1 mathematical model can serve as an efficient methodology for reducing the overall hazard at the highway–rail grade crossings under various budget availability scenarios.
Figure 11 illustrates the overall hazard before and after implementation of countermeasures at the highway–rail grade crossings that were suggested for upgrading by
RAP-2 for all 12 developed budget availability scenarios. Similar to the resource allocation results that were observed for
RAP-1, neither the overall hazard before nor the overall hazard after implementation of countermeasures at the highway–rail grade crossings that were suggested for upgrading by
RAP-2 substantially changed by increasing the total available budget. Nevertheless, implementation of the countermeasures that were suggested by
RAP-2 significantly reduced the overall hazard at the highway–rail grade crossings. Specifically, the overall hazard was reduced by
for budget availability scenario “1”. Furthermore, the overall hazard was reduced by
for budget availability scenario “12”. Therefore, the developed
RAP-2 mathematical model can serve as an efficient methodology for reducing the overall hazard at the highway–rail grade crossings under various budget availability scenarios.
Note that similar percentages were observed in terms of the overall hazard reduction at the highway–rail grade crossings that were suggested for upgrading for both RAP-1 and RAP-2. However, RAP-1 suggested a total of 1198 and 1723 highway–rail grade crossings to be upgraded for budget availability scenarios “1” and “12”, respectively, while RAP-2 suggested a total of 1212 and 1705 highway–rail grade crossings to be upgraded for budget availability scenarios “1” and “12”. Such a variation can be explained by differences in the resource allocation objectives (i.e., RAP-1 aims to minimize the overall crossing hazard, while RAP-2 aims to minimize the overall crossing hazard severity).
6.3.2. Sensitivity Analysis for the Number of Available Countermeasures
A sensitivity analysis was conducted to determine how the number of available countermeasures may impact the resource allocation decisions among the highway–rail grade crossings in the State of Florida. A total of 11 countermeasure availability scenarios were developed by altering the number of available countermeasures from 1 to 11 by an increment of 1 countermeasure. The impact of the number of available countermeasures on the number of highway–rail grade crossings upgraded by
RAP-1 and
RAP-2 is illustrated in
Figure 12. Note that PHR and PSR were used as solution approaches for
RAP-1 and
RAP-2, respectively. As expected, the total number of highway–rail grade crossings upgraded by
RAP-1 and
RAP-2 increased after increasing the number of available countermeasures. Specifically, a total of 100 and 1198 highway–rail grade crossings were selected for upgrading by
RAP-1 for countermeasure availability scenarios “1” and “11”, respectively. On the other hand, a total of 100 and 1212 highway–rail grade crossings were selected for upgrading by
RAP-2 for countermeasure availability scenarios “1” and “11”, respectively. Such a significant increase in the number of upgraded highway–rail grade crossings can be explained by the introduction of low-cost countermeasures (i.e., countermeasures “7”, “8”, and “9”—see
Table 2). Therefore, the number of available countermeasures may substantially influence the resource allocation decisions and directly affect the number of upgraded highway–rail grade crossings.
Figure 13 illustrates the overall hazard before and after implementation of countermeasures at the highway–rail grade crossings that were suggested for upgrading by
RAP-1 for all 11 developed countermeasure availability scenarios. It can be noticed that the overall hazard before and after implementation of countermeasures at the highway–rail grade crossings that were suggested for upgrading by
RAP-1 substantially increased after increasing the number of available countermeasures. This finding can be explained by the fact that the number of upgraded highway–rail grade crossings significantly increased after introducing additional countermeasures. Moreover, implementation of the countermeasures that were suggested by
RAP-1 significantly reduced the overall hazard at the highway–rail grade crossings. Specifically, the overall hazard was reduced by
for countermeasure availability scenario “1”. Furthermore, the overall hazard was reduced by
for countermeasure availability scenario “11”. Therefore, the developed
RAP-1 mathematical model can serve as an efficient methodology for reducing the overall hazard at the highway–rail grade crossings under various countermeasure availability scenarios.
Figure 14 illustrates the overall hazard before and after implementation of countermeasures at the highway–rail grade crossings that were suggested for upgrading by
RAP-2 for all 11 developed countermeasure availability scenarios. Similar to the resource allocation results that were observed for
RAP-1, the overall hazard before and after implementation of countermeasures at the highway–rail grade crossings that were suggested for upgrading by
RAP-2 substantially increased after increasing the number of available countermeasures. Moreover, implementation of the countermeasures that were suggested by
RAP-2 significantly reduced the overall hazard at the highway–rail grade crossings. Specifically, the overall hazard was reduced by
for countermeasure availability scenario “1”. Furthermore, the overall hazard was reduced by
for countermeasure availability scenario “11”. Therefore, the developed
RAP-2 mathematical model can serve as an efficient methodology for reducing the overall hazard at the highway–rail grade crossings under various countermeasure availability scenarios.
Note that similar percentages were observed in terms of the overall hazard reduction at the highway–rail grade crossings that were suggested for upgrading for both RAP-1 and RAP-2. However, RAP-1 suggested a total of 100 and 1198 highway–rail grade crossings to be upgraded for countermeasure availability scenarios “1” and “11”, respectively, while RAP-2 suggested a total of 100 and 1212 highway–rail grade crossings to be upgraded for countermeasure availability scenarios “1” and “11”. Such a variation can be explained by differences in the resource allocation objectives (i.e., RAP-1 aims to minimize the overall crossing hazard, while RAP-2 aims to minimize the overall crossing hazard severity).