4.1. Free-Flow Speeds Based on Vehicle Classes
Initially, assessments of the mean free-flow speed according to vehicle classification were conducted. In this study, free-flow speed was estimated based on the average the speed of the vehicle with headway to the nearest vehicle in front of more than 8 s and no recent or immediate meeting with a vehicle in the opposing direction. Thus, the data that do not represent free flow conditions were eliminated from the analysis, and free-flow speed data for a total of 41,154 vehicles were collected; the highest number of vehicles was recorded in Class 1 at 28,795 vehicles (or 70.0%), followed by Class 2 with 4943 vehicles (12.0%), Class 5 with 4265 vehicles (10.4%), Class 3 with 2835 vehicles (6.9%), and, lastly, Class 4 with only 316 vehicles (0.8%).
The descriptive statistics of the average free-flow speeds observed for vehicles in Classes 1, 2, 3, 4, and 5 based on individual lanes are as shown in
Table 3.
Table 2 shows that the mean free-flow speed for vehicles in Class 1 yields the highest value (82.5 km/h), while the lowest mean free-flow speed of 71.8 km/h was observed for vehicles in Class 5.
However, in order to compare the mean free-flow speed of each vehicle class, a one-way ANOVA test was conducted. The results of the one-way ANOVA are shown in
Table 2. Based on the results in
Table 4, the
p-value is less than 0.05, which means that the null hypothesis was rejected. Therefore, it can be concluded that the mean free-flow speeds for different classes of vehicles were significantly different. Subsequently, post hoc range tests and pairwise multiple comparisons were conducted to determine which means differed from each other. Range tests were used to identify the homogeneous subsets of means that are not different from each other, while pairwise multiple comparisons tested the difference between each pair of means.
However, in order to select an appropriate post hoc test, a Levene test for equality of variances was used to test whether the two samples had statistically equivalent variances.
Table 5 shows the results. Based on the results from the Levene test of homogeneity of variance, the Levene statistic is 3.060 with a
p-value of 0.017. Thus, at a 95% confidence interval, the null hypothesis of equal variances across the five classes of vehicles was rejected.
Upon determining that the means of free-flow speed were significantly different for different classes of vehicles with unequal variances, post hoc multiple comparison procedures were conducted to determine exactly which groups were different from the others, either because the means of free-flow speed were different for all five classes of vehicles or because only one class differed from the rest. The results from the post hoc test are shown in
Table 6. Based on the results, at a 95% confidence interval, the mean free-flow speeds for vehicles in Class 1, 2, 3, and 4 were not significantly different from each other, but they were significantly different from the mean free-flow speed of the vehicles in Class 5.
4.2. Free-Flow Speed Models Using Multiple Linear Regression
In this study, only three groups of vehicle classifications were used to develop free-flow speed models for several reasons. Firstly, as traffic in Malaysia consist of five vehicle classes, there is a need to ascertain whether the free-flow speed of all vehicle classes should be used in the formulation of a free-flow speed equation. Therefore, the free-flow speeds of all vehicles were initially grouped in Group 1, which is the base group. Subsequently, because the mean free-flow speed of motorcycles (Class 5) is the lowest at only 71.77 km/h, motorcycles deviate the most from the free-flow speed of other vehicle classifications. Hence, there is a need to investigate the impact of this vehicle class on the overall free-flow speed by comparing the results from Group 2 with the results from Group 1. Lastly, because the average free-flow speed of light vehicles or cars in Class 1 is only slightly higher than the mean free-flow speed of heavy vehicles in Class 2, Class 3, and Class 4, there is a need to investigate the impact of heavy vehicles on the overall free-flow speed, which can be done by comparing the results from Group 2 with the results from Group 3.
Initially, a descriptive statistical analysis was conducted to investigate the trends of the grouped free-flow speeds, and the results are shown in
Table 7. The calculated mean values shown in
Table 7 indicate that the mean free-flow speed of all vehicles yields the lowest value of 80.8 km/h, while the mean free-flow speed of cars is the highest at 82.5 km/h. Additionally, for free-flow speeds, Group 1 yields the highest standard deviation value, while the free-flow speeds for Group 3 yield the lowest standard deviation value. This is because, in Group 1, the free-flow speeds of motorcycles are included in the analysis, and, due to their lower free-flow speed values, grouping them together with other higher free-flow speed values causes the data points to become more spread out over a wider range of values. On the other hand, due to the proximity of the free-flow speed values obtained for the vehicles in Classes 1 to 4, when grouped together (as in Groups 2 and 3), they yield smaller standard deviation values. However, due to different vehicle classes in Group 2 and the different operating conditions of cars in Group 3, there is still variation in the observed free-flow speeds. Considering the differences between the minimum and maximum free-flow speeds observed for the vehicles in Groups 2 and 3 (around 40 km/h), the standard deviation of around 9.7 km/h is still acceptable as the standard deviation value, which demonstrates how spread out the data are. Further analyses to determine the spread of the data points in Groups 1, 2, and 3 indicated that, respectively, 66%, 98%, and 100% of the data points in Group 1 are within ± 1SD, ± 2SD, and ± 3SD of the mean. Similarly, 66%, 95%, and 100% of the data points in Group 2, respectively, fall within ± 1SD, ± 2SD and ± 3SD of the mean. Lastly, 64%, 97%, and 100% of the data points in Group 3 are, respectively, within ± 1SD, ± 2SD, and ± 3SD of the mean.
Subsequently, multiple linear regression analyses were conducted to develop the equation to estimate free-flow speed. A similar approach was adopted by Balakrishnan and Sivanandan [
18] to develop free-flow speed models for urban roads under heterogeneous traffic conditions in India. This approach was also used in the Highway Capacity Manual [
1] to estimate free-flow speed based on the physical characteristics of any multilane segment in the United States of America. The independent variable used for each model is as follows:
Model 1—free-flow speed of all vehicles (Class 1, 2, 3, 4, and 5);
Model 2—free-flow speed of vehicles excluding motorcycles (Class 1, 2, 3, and 4);
Model 3—free-flow speed of cars only (Class 1).
In order to develop free-flow speed models under prevailing conditions, the factors affecting free-flow speed, such as lane width, shoulder width, median clearance, access point density, and lane position, were investigated and are included in the model. In this study, lateral clearance is divided into two types: shoulder width and median clearance. Shoulder width refers to the width of the paved shoulder, which is measured from the left edge of the travel lane, while median clearance is measured from the right edge of the travel lane to the median. For divided highways, shoulder width is considered to be the lateral clearance for the outer lane, and median clearance is considered to be lateral clearance for the inner lane. However, for undivided highways, only shoulder width is considered as the lateral clearance for the outer lane, while the lateral clearance for the inner lane is zero. Additionally, a dummy variable indicating lane position was also included in the regression model. The ideal conditions for multilane highways based on the Malaysian road geometry design guidelines [
1] and Highway Capacity Manual [
2] are as follows:
Lane width greater than or equal to 3.65 m;
Lateral clearance wider than or equal to 1.8 m;
Divided highways;
No impediment to traffic due to traffic control or turning vehicles;
Level terrain.
Initially, the obtained values of free-flow speed were investigated based on lane position.
Figure 3 shows that all the values of free-flow speed obtained for the inner lane or fast lane were higher than those obtained for outer lane. As such, a dummy variable indicating lane position was included in all three regression models. The general equation adopted in this study to estimate the free-flow speed of multilane highways in Malaysia is shown as Equation (2):
where
BFFS | = | Base free flow speed; |
fLW | = | Adjustment for lane width; |
fLC | = | Adjustment for total lateral clearance; |
fM | = | Adjustment for median type; |
fA | = | Adjustment for access-point density. |
The developed free-flow speed models are as shown in Equations (3)–(5).
Table 8 shows a summary of the regression analyses for Models 1, 2, and 3. In the regression equation, the coefficient of the lane width factor will only determine the rate at which the speed will decrease for every 0.1 m of lane width less than 3.65 m, with no increment in free-flow speed for lane widths of more than 3.65 m. Similarly, the coefficient of lateral clearance will indicate the rate at which the speed will decrease for every meter of the shoulder width or median clearance less than 1.8 m. For the factor of access point density, the coefficient provided by the regression equation directly estimates the speed reduction penalty for every access point found within 1 km. Lastly, since all surveyed sites have level terrain, the effect of the gradient is not considered in the regression analyses.
where
BFFS | = | Base free flow speed (100 km/h for multilane highways); |
LW | = | Lane width (ideal lane width = 3.65 m); |
LC | = | Lateral clearance, adjustment of shoulder width for the outer lane, and adjustment of median clearance for the inner lane (ideal lateral clearance = 1.8 m); |
APD | = | Access point density; |
LD | = | Lane dummy (0 if inner lane, 1 if outer lane). |
However, regression analysis requires the assumption that the residual (error) is normally distributed with a zero mean and constant variance to obtain the best model. Residual analyses were then conducted to check these assumptions. A normal probability plot can be used to test whether the error terms are normally distributed or otherwise and presents a plot of a variable’s cumulative proportions (the proportion of the distribution that is less than the specified value) against the cumulative proportions of a theoretical normal distribution. If the error terms are normally distributed, the points cluster around a 45° straight line. The normal probability plots generated for Models 1, 2, and 3 using statistical software are shown in
Figure 4. Based on the results, the data points are distributed roughly around a 45º straight line, thus indicating that the normality assumption for each model is satisfied.
Additionally, residuals are plotted against the fitted values determine to determine whether non-constant variance exists. If the points in the residual plot show no pattern, then the variance is constant. Based on the residual plots shown in
Figure 5, which were generated using statistics, the data points appear to be randomly scattered around the horizontal axis; thus, the variance is constant. Therefore, the developed regression equations are valid and acceptable.
Subsequently, analyses using performance indicators (PIs) were conducted to select the best free-flow speed model. These analyses involved three error measures (root mean square error (RMSE), normalized absolute error (NAE), and Mean Absolute Percentage Error (MAPE)) and three accuracy measures (index of agreement (IA), prediction accuracy (PA), and coefficient of determination (R2)). To produce a good estimator, the error measures must have values close to zero, as small values of RMSE, NAE, and MAPE indicate that the developed models have smaller errors. For the measures of accuracy, the obtained values should be nearer to 1, as higher values of PA, IA, and R2 suggest that the developed models can predict well.
A score system was then formulated to select the best model. For the error measure, the models are ranked from the highest, with a score of 1, to the smallest, with a score of 3. The accuracy measures are ranked from the smallest, with a score of 1, to the highest, with a score of 3. The total score for each model is calculated by adding the score obtained in each PI, which can be between a minimum score of 5 (i.e., each PI only obtained a minimum score of 1) and a maximum score of 18 (each PI obtained a maximum score of 3). The results are summarized in
Table 9. Based on the results shown in
Table 9, Model 2 has the lowest score of 9, which indicates that Model 2 is the least favorable model. Conversely, Model 1 which has the highest score of 15, making it the most favorable model. This suggests that the mean free-flow speed of all vehicles, including heavy vehicles and motorcycles, should be used to measure free-flow speed.
In order to visualize and compare the estimated free-flow speeds based on Model 1 with the observed free-flow speeds, a bar chart showing the percentages of differences was plotted, and the results are shown in
Figure 6. Based on the plot in
Figure 6, the highest difference existed between Lane 60 and Lane 15. For Lane 15, Model 1 overestimated the free-flow speed by 22.5%, while for Lane 60, Model 1 underestimated the free-flow speed by −22.4%. However, on average, the percentages of overestimation and underestimation are only 6.1% and −7.4%, respectively, with 30 lanes recording an overestimation of free-flow speeds and 34 lanes recording an underestimation of free-flow speeds. Consequently, the few lanes with high percentages of differences might be due to other external factors, such as a high volume of heavy vehicles and motorcycles, as heterogeneous traffic streams usually contain a mix of different vehicles, such as trucks, trailers, passenger cars, buses, and motorcycles. The high percentage of heavy vehicles in this traffic stream will cause a significant decrease in the free-flow speed. Moreover, heavy vehicles tend to impede passenger cars due to their slower speeds, which will create difficulties for passenger cars to overtake them. Most motorcycles travel on the road shoulder. However, if the road has a smaller shoulder width, motorcyclists will tend to ride in the same lane, either beside, in front of, or following other vehicles, albeit with a much slower speed. As a motorcycle is much smaller than other vehicles, motorcycles can easily be overtaken, usually without needing to move into an adjacent lane to overtake them. Therefore, if the volume of motorcycles is high, the tendency to overtake them will also be higher, which will affect the free-flow speed of the other vehicles, especially if the motorcycles are traveling beside other vehicles in the same lane.
Figure 7 shows the effects of heavy vehicles and motorcycles on the free-flow speed of passenger cars. It can be observed from the trend of the scatterplot that a higher percentage of heavy vehicles and motorcycles causes a reduction in the free-flow speed of passenger cars.
The findings of this study are consistent to some extent with the findings of Balakrishnan and Sivandan [
18], who mentioned that, for heterogeneous traffic, a combined class-wise model is more efficient in predicting the overall free-flow speed compared to the base model, which is predominantly focused on passenger cars. However, in this study, the proportions of different vehicle classes were not considered in the free-flow speed estimation model as a traffic composition factor, and, together with the peak hour factor, is applied during the calculation of the demand flow rate to determine the level-of-service of multilane highways. Therefore, by adopting the level-of-service criteria given in the Malaysian Highway Capacity Manual [
24] (as shown in
Table 10), the level-of-service of each lane determined based on the estimated free-flow speeds using Model 1 were compared with the level-of-service determined based on the observed free-flow speeds. The results are shown in
Table 11.
Based on the results shown in
Table 11, only five lanes (Lanes 19, 41, and 58 to 60) recorded different levels-of-service between the estimated and observed free-flow speeds, while the other 59 lanes recorded the same level-of-service. Lanes 19 and 41 yielded a better level-of-service based on the estimated free-flow speeds, while Lanes 58 to 60 produced a slightly worse level-of-service due to their considerably lower estimated free-flow speed values, as this site recorded a much higher access point density of 7.43 access points/km eastbound and 11.43 access points per km westbound compared to the other sites, which only recorded between 0.26 to 3.14 access points/km. This suggests that access point density is a more sensitive parameter and has the greatest impact on the estimation of free-flow speeds, as the coefficient given in the regression equation can directly estimate the reduction in free-flow speed for every access point found over 1 km. Nevertheless, based on the 92% correctly projected level-of-service, the estimated free-flow speeds can be applied to determine the level-of-service. However, the developed model has yet to be validated. As sites suitable for data collection are limited, data recorded for all 64 lanes were used to develop the free-flow speed model. Further research will be needed to validate the developed model. As free-flow speed is the fundamental parameter needed to estimate level-of-service, an inaccurate estimation of free-flow speed will result in the incorrect determination of level-of-service. Therefore, the developed model can be used as the basis for assessing the design consistency of new and existing multilane highways to facilitate the sustainable development of road networks, particularly the inter urban multilane highways in Malaysia, as free-flow speed is the most fundamental and important parameter in determining the capacity and level-of-service of any uninterrupted road facility.