CONTENTS

1. Knowledge Integration: The Future of Multidisciplinary Engineering Education - Solving the Highway Merging Problem using a Non-linear Regression Model Atan, I.B.1, Samat, N.A.1, Adnan, M.A.1, Shamsudin, M.B.2, Ashaari, Y.1, Jaafar, J.1, Mohamed, S.N.3 and Baki, A.2* 1. Faculty of Civil Engineering, Universiti Teknologi MARA, Shah Alam, Selangor, Malaysia 2. Envirab Services, Bandar Baru Bangi, Selangor, Malaysia 3. Faculty of Civil Engnineering, Universiti Teknologi MARA, Cawangan Johor Kampus Pasir Gudang, Johor, Malaysia

2. CONTENTS INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSIONS CONCLUSIONS REFERENCES

3. INTRODUCTION

4. INTRODUCTION Merging is a special case of lane changing, where vehicles must move into the target lane because the lane they are currently travelling ends. On major roads the merging is provided with an acceleration lane, which enables drivers to enter the priority stream to synchronise their speeds and make use of available gap. Heavy merging on acceleration lane at entrance ramp operation can cause either congestion or accident or both. Therefore there is a need to explore into the conflicts on entrance ramp operation. The objective of this study is to discover more complete models to describe the operation of ramps on highway with the presence of acceleration lane in order to solve the highway merging problem. As a result, a higher model to describe the behaviour of merging time and to predict the merging time of acceleration lane can be developed.

5. The result of this study will assist local authorities to allow for future implementation and plan better strategies in order to reduce or prevent the merging problems. This will lead to improve assessment of our highway and techniques to control merging problem especially congestion at entrance ramp. A ramp is a length of roadway providing an exclusive connection between two highway facilities. HCM [11] explained that ramp consist of three geometric elements, namely, the ramp-freeway junction, the ramp roadway and the ramp-street junction. Illustration is shown in Figure 1. As the analysis of merging area focuses on influence area including the two right-most lanes of the freeway, a critical step in the methodology is to estimate lane distribution of traffic immediately upstream of the merging. Specifically, a determination of the approaching flow in the remaining lane 1 and 2 immediately upstream of the merging is required [11].

6. The interaction of vehicles within a single stream or the intersections of two separate traffic streams are two of the most important traffic operation aspects [3]. The theory behind the traffic intersection associated with merging issues is gap acceptance concept. Abdulhai and Kattan [1] has described that gap can be measured as the time that elapses between the departure of the first vehicle and the arrival of the second at the designated test point. Gap also can be defined as the time between the rear bumper of the first vehicle and the front bumper of the second vehicle. Headway is similar to gap except that it is a measure of the temporal space between two vehicles, or, more specifically, the time that elapses between the arrival of the leading vehicle and the following vehicle at the designated test point along the lane.

7. 450 m 450 m VF VF VFO VFO V12 V12 VR12 DR,SR VR DR,SR VR Figure 1: Critical varable in Merge and Diverge Analysis. (HCM, 2000)

8. Headway between two vehicles is measured by starting a chronograph when the front bumper of the first vehicle crosses the selected point and subsequently recording the time that the second vehicle’s front bumper crosses over the designated point. Figure 2 illustrates the difference between gap and headway. A few mathematical formulation of merging vehicles, which incorporate the length of acceleration lane are originated from a series of gap-acceptance models. Gap can be measured as the time that elapses between the departure of the first vehicle and the arrival of the second at the designated test point [1]. The gap-acceptance model is a probabilistic treatment of such random variables as headways or time-gaps created by successive vehicles in the right lane of the freeway [7].

9. Gap (sec) Clearance (ft or m) Headway, h (sec) Spacing, S (ft or m) Figure 2: Illustration of gap and headway definition

10. The accuracy of capacity estimation is mainly determined by the accuracy of the critical gap. The maximum likelihood method for estimating critical gap is based on the fact that a driver is between the range of his largest rejected gap [10]. Pollatschek [5] stipulated that not all gaps presented to the driver should be considered in the process while waiting at an intersection. Kittelson and Vandehey [2] showed that nearly all gaps longer than 12s are acceptable and, therefore, should not be considered when determining the critical gap. Teply et al. [9] similarly suggested that drivers facing gap greater than 13s had no option situation because all greater gaps were accepted. Polus and Shmueli [6], who studied gap acceptance at roundabout intersections, suggested a threshold gap value above 8-9s would not be relevant for the determination of the critical gap and, consequently, for the capacity analysis.

11. MATERIALS AND METHODS

12. MATERIALS AND METHODS Data Data from four observation sites are analysed using the multiple regression and non-linear regression model. The first task is to separate the variables which could feasibly be used as either dependent or independent variables in the modelling process later. These parameters fall into two general categories namely volume or flow rate and geometric variables. According to HCM [11], all volume and flow rate data for a five minute study period are converted to passenger car unit (PCU). The length of acceleration lane is measured from the merging gore area up to a point where the acceleration lane ends (Figure 3).

13. Ramp La Lane 1 Lane 2 Lane 3 Figure 3: Typical acceleration lane length

14. . The dependent variable is time merging. This can be explained as the average merging time by ramp vehicles to merge into freeway lane 1 for a five minute study period. Average time merging is measured in second. The parameters that involve in analysis will be estimated using method of least squares. In least squares estimation, the unknown values of the parameters, in the regression function, are estimated by finding numerical values for the parameters that minimise the sum of the squared deviations between the observed responses and the functional portion of the model are treated as the variables in the optimization and the predictor variable values, are treated as coefficients. To emphasize the fact that the estimates of the parameter values are not the same as the true values of the parameters, the estimates are denoted by .

15. RESULTS AND DISCUSSIONS

16. RESULTS AND DISCUSSIONS Model Development Using the four independent variables namely, lane 1 flow rate, ramp flow rate, freeway flow rate and length of acceleration lane, multiple regression analyses were performed. These types of analyses were chosen because most probability functions cannot analyse more than two variables. The following regression model for the merging times was obtained. Tm = 9.08499-0.00151V1-0.00061Vf+ 0.001404Vr-0.01323La(1) R- Squared = 0.159338

17. Table 4.1: Regression Equation

18. Table 4.2: Analysis of Variance

19. The multiple regression model for the merging time has an R-squared of 0.159. This means that only 15.9% of the variation can be explained by the model, which the merging time is due to lane 1 flow, freeway flow, ramp flow and also length of the acceleration lane, whereas the 84.1% of the variation cannot be explained. Since the multiple regressions give a small value of R-squared, the data were then analysed using the non-linear regression model. In order to produce a good model with a decent R-squared, a number of iteration was considered for the variables. By using the same variable as the multiple regression namely, lane 1 flow rate, freeway flow rate, ramp flow rate, and length of acceleration lane as dependent variable while merging time as the independent variable, the following regression model was obtained:

20. Tm = 64.0 + 0.0310(V1) - 0.0448 (Vr) - 0.301 (La) - 0.000171 (V1)(La) + 0.000003 (Vf)(Vr) + 0.000237 (Vr)(La) - 0.000001 (Vf)2 – 0.000004 (Vr)2 (2) R-Square = 0.678

21. Table 4.3: Regression Equations

22. Table 4.4: Analysis of Variance

23. The non-linear regression model shows that the R-squared is 0.678. This means that only 67.8% of the variations can be explained by the model in which the merging time is due to lane 1 flow, ramp flow, and also length of the acceleration lane, whereas only 32.2% of the variation cannot be explained. The p-value also shows highly statistically significant and the model should be able to predict the merging time as a function of a set predictor variables namely V1, Vr, La, and Vf. The calibrated model for predictive lane 1 volume is shown in equation (3).

24. V1 = 37552-9.17(Vf)+322(Vol/Du)-338(La)-0.0560(Vf)(Vol/Du)+0.0441 (Vf)(La)-1.55(Vol/Du)(La)+0.000766(Vf)2+1.05(Vol/Du)2+0.767(La)2 (3) R-Squared = 0.804947 The statistical analysis of the model is as shown in table 4.4. The non-linear regression model showed that the R-squared value closed to 1. This means that 80.5% of the variation in the lane 1 flow rate can be explained by the model whereas only 19.5% of the variation cannot be explained. So, generally the model is acceptable to solve the merging problem.

25. Table 4.5: Regression Equation

26. Table 4.6: Analysis of Variance

27. CONCLUSIONS

28. CONCLUSIONS The differences observed in predicting lane 1 volume and merging time showed that the non-linear regression is better predictive model than multiple regression. Based on the data analysis, the impact of an acceleration lane length and merging time were significantly high due to the R-squared and p-value for each of the variables involved and also p-value of the model itself. Therefore, it is suggested to use the acceleration lane length as the geometric variable in a model. This non-linear regression model clearly explained the impact due to the length of the acceleration lane since the drivers in lane 1 tend to stay on that lane if they drive along a long acceleration lane. Long acceleration lane also provides longer merging time and helps ramp driver to merge comfortably from the ramp lane to the lane 1 of the freeway. It can be concluded that application of mathematical techniques (regression) in solving traffic engineering problems is the way to go in the future, where integration of knowledge (in this case mathematics and traffic engineering) is the way forward in engineering education.

29. REFERENCES

30. [1] Abdulhai, B. and L. Kattan. (2003), Reinforcement Learning Crystallized: Introduction To Theory And Potential For Transport Applications, CSCE Journal, 2002. [2] Kittelson, W.K and Vandehey, M.A. (1991), Delay Effects On Driver Gap Acceptance Characteristics At Two-Way Stop-Controlled Intersection, Transportation research record, 1320, 155-159. [3] Kou, C.C. (1997), Development Of A Methodology For Modeling Ramp Driver Behavior During Freeway Merge Maneuvers, Ph.D dissertation, University of Texas. [4] Mandenhall, W et all. (2006), Probability and Statistics, Thomson Books/Cole, United State of America, Edition 12. [5] Pollatscheck, M.A. (2002), A Decision Model for Gap Acceptance and Capacity at Intersections, Transportation Research Part B, 36, 649-663. [6] Polus, A. and Shmueli, S. (1999), Entry Capacity at Roundabouts and Impact of Waiting Times, Road and transport research, 8, 43-54. [7] Road Engineering Association of Malaysia – REAM, A Guide On Geometric Design Of Road, Malaysia. [8] Roger, M. (2003), Highway Engineering, Blackwell, Oxford, Edition 1. [9] Teply, S. et all. (1997), Gap Acceptance Behavior: Aggregate and Logit Lerspective, Traffic engineering and control, 38, 474-482. [10] Tian, Z. et al. (1999), Implementing The Maximum Likelihood Methodology To Measure A Driver’s Critical Gap, Transportaion research part A, 33, 187-197. [11] Transportation Research Board (2000), Highway Capacity Manual 2000, Washington D.C.

31. THANK YOU