Abstract: The goal of this study is to examine a stochastic approximation technique for simulation of highway vertical curve elements by using Microsoft Excel 2010 program. The program is general purpose application software and in graphical user interface. Stochastic approximation simulation method established here is approached thus for reducing modeling-development time and towards getting inexpensive design methodology. The vertical curve elements of highway for investigation in this study are parabolic curves. The basic concept presented here is how to move progressively toward realisation of the best-value stochastic approximation algorithm for the development of vertical curves while providing adequate sight distance leading to the ability to see ahead on highway for safe, comfort and efficiency operation of a vehicle. Stochastic approximation simulation work is described in this study as a result of greater emphasis on automation modeling via Information Communication Technology (ICT) to improve productivity and quality of analysis and design of complex system such as highway. It is pertinent to note that alternative proposed vertical curves during design stage can be compared through stochastic approximation simulation modeling analysis methodology based upon an authentic policy for geometric design of highways and streets. Hence, a successful optimal system to meet specified requirements conforming to able policy on geometric design of highways and street has been established in this study via Microsoft Excel 2010 similarly like conventional hand method.

INTRODUCTION

Background: Stochastic approximation is a method of stochastic-optimization including technique such as gradient search (Weisstein, 2010) that is useful for designing a vertical curve for highway geometric design. Stochastic-optimization refers to the minimization (or maximization) of a function (a vertical curve) in the presence of stochastic approximation in the optimization process.

A parabolic curve is used exclusively in connecting profile grade tangents according to Wright and Dixon (2003) and the paradigm is useful in this study to automate crest and sag vertical curves by using Microsoft Excel 2010 which is general purpose application software and it is in graphical user interface. When a vertical curve connects a positive grade with a negative grade, it is referred to as a crest curve and for a vertical curve to connect a negative grade with positive grade is termed a sag curve.

Processing and evaluation of highway vertical curve elements by simulation approach is currently an active area of research (Garber and Hoel, 2010). Since, the early days of simulation people have constantly looked for new and better ways to model a system as well as novel ways to use existing computer hardware and software simulation. A simulation model (gradient or curve) can be stochastic if contains probabilistic or random (subjective) parameter according to Law and Kelton (2007) and Institute of Transportation Engineers (ITE, 2008). Highway vertical curve elements are usually modelled as having at least some arbitrary input components as asserted by AASHTO (2004) (America Association of State Highway and Transportation Officials), Amekudi and Meyer (2006) and FHWA (2006) (Federal Highway Authority) together with Wooddrige et al. (2003). This paradigm has paved way to stochastic approximation simulation modeling for highway vertical curve that is being examined in this study.

Stochastic is often used as counterpart of the word deterministic which means that random phenomena are not involve (Vincenzo, 2010). The subjective input components for vertical curve design include design speed, gradients, vertical curvature, length of curve, station locations for vertical point of curvature, intersection and tangent lengths.

Most real-world systems are not amenable to compact analytical tools and such systems must be studied by means of simulation (Law and Kelton, 2007; Akiije, 2009). In a simulation practice, the use of Microsoft (MS) Excel package is unique to evaluate mathematical modeling numerically and graphically when data are gathered in order to investigate the desired true characteristics of the model. To understand and cope with change in a stochastic environment, the frontiers of knowledge are being rapidly expanded as a result of new development in computers and information technology (Ibidapo-Obe, 1996).

The aim of this study, therefore is to advance stochastic approximation modeling approach to simulate vertical curves elements in highway geometric design that are dependable and consistent to help reduce accidents on roads and other problems related to driver expectance through the use of Microsoft Excel 2010. Specific objectives of this study include:

Justification for this study is that in the course of the stochastic approximation modeling while contemplating through analysis and design subjectively, a careful simulation of vertical curve elements could shed vivid light on proposed highway safety and cost-effectiveness.

The significance of this study is that the simulated elemental curves with adequate design parameters are genuine parts of a highway that are to serve as guides for the processing of the highway geometry alignments during construction and as they would be when the road work is completed.

Stochastic approximation modeling methodology of highway vertical curve elements is quite independent of the software and hardware used for it is the rudiment of the conceptual framework of geometric design of highways that matters most. This is because geometric design of highways could be approached by any one out of the conventional hand method, general purpose application program or specific purpose application program.

Conceptual framework: The design of the parabolic vertical curves of highways depends primarily on the design speed selected for the highway (Garber and Hoel, 2010). Design standards may vary with the type of terrain (level, rolling and hilly), anticipated traffic to be served and whether the highway is to be in an urban or rural area (Wright and Dixon, 2003). However, mechanism to the solution of indeterminacy to vertical curve length is made in relationship to each of the parabolic curve. In other words, for a parabolic curve in highway geometric design the length is the same whether measured along the tangents, the chord, the horizontal or the curve itself (Uren and Price, 2010).

The cognizance of this mechanism and the use ICT approach leads to stochastic approximation process of modeling to analysis the development of standard vertical curves in this study in MS Excel program environment for optimal achievement.

Meanwhile, available specific purpose programs go obsolete quickly despite being under licence, costly and they are yet to realise the target of the parabolic curve design mechanism optimally.

Figure 1 shows the 4 types of standard vertical parabolic curves models that are classified as Crest and Sag vertical curves (AASHTO, 2004). Vertical curves may be calculated in relationship to Fig. 2 which is an approximation of a circular arc to a simple parabola as follows:

(1)

(2)

(3)

For any point p on curve:

(4)

(5)

Figure 2 is showing typical vertical curve type I. In Fig. 1 and 2, together with Eq. 1-5: G₁ and G₂ are grades of initial and final tangents, respectively; A is the percentile algebraic difference; L is the length of vertical curve; PVC is the point of vertical curve; PVI is the point of vertical intersection; PVT is the point of vertical tangent; g₁ is the grade of initial tangent in percent; g₂ is the grade of final tangent in percent; K is the vertical curve length coefficient as determined for stopping sight distance; x is the horizontal distance to point on curve, measured from VPC; E_x is the elevation of point on curve located at distance x from VPC; X_m is the location of min/max point on curve, measured from VPC; E_m is the elevation of min/max point on curve at distance X_m from VPC; e = external distance = middle ordinate; y = offset of curve from initial grade line.

Fig. 1:	Type of vertical curves (AASHTO, 2004)

Fig. 2:	Typical vertical curve (Wright and Dixon, 2003)

According to AASHTO (2004) desirable maximum gradient is 3% for motorway; 4% for all-purpose dual carriageway; 6% for all-purpose single carriageway and for effective drainage a minimum gradient of 0.5% is to be maintained wherever possible.

Excel application program, as presented for use in this study is software that performs simulation by modeling calculations and analysing information in addition to visualising data and graphics as spreadsheet at optimal level. A spreadsheet is a document that stores data in a grid of horizontal rows and vertical columns.

The most commonly used spreadsheet application is Microsoft Excel but several other spreadsheet programs are available including Lotus 1-3 for Windows IBM, Apple Works and Numbers for Mac OS X., Tech Terms.com (2010).

Excel application program is amenable to stochastic approximation simulation for highway vertical curves elements.

Fig. 3:	Optimisation simulation flow chart

MATERIALS AND METHODS

The stochastic approximation simulation methodology introduced in this study is by using spreadsheet program for the verification of the validation of the presented four types of vertical curves in Fig. 1 by modeling approach (Adedimila and Akiije, 2006). This methodology generally involves simulating sequence of a selected vertical curve out of the six configurations as shown in Fig. 1.

In the stochastic approximation process VPI, VPC, VPT, the two intercepting gradients are specified, sight distance and K-values are defined and the reduced levels along the vertical curve are calculated. The process will result to simulated configurations that will suggest promising new directions to search through the space within the right of way of possible input-factor combinations subjectively that will lead to better configurations of a selected vertical curve. The subjective methodology can be made possible while utilising optimisation simulation flow chart setup of Fig. 3. This is indeed a convenient way to make the stochastic approximation process workable to simulate an optimum-seeking sequence of vertical curve configurations in the context of the spreadsheet Table 1.

Table 1:	Modeling module for the optimisation of type I crest vertical curve

Fig. 4:	Type 1 (positve and negative gardients) crest vertical curves simulation

Fig. 5:	Type 2 ( both gardients positve) crest vertical curves simulation

RESULTS AND DISCUSSION

Stochastic approximation simulation methodology has being successfully employed in this research which generated highway vertical curves elements of different configurations similarly like conventional hand approach while using Microsoft Excel 2010, a general purpose application program. Statistically, Eq. 1, 2 are linear regression logical models while Eq. 3-5 are nonlinear regression models to define the vertical parabolic curves subjectively before arriving at a logical conclusion. These equations have been used successfully to determine modeling module for the optimisation of Types I-IV as labelled by AASHTO (2004) while employing the vertical curve variables subjectively within the given boundary conditions of a right of way Table 2. Table 3 and 4 are showing the simulation run of a crest vertical curve Type I that is equally applicable to other Types II-IV in conformity to AASHTO (2004) and Federal Ministry of Works (FMW, 2007) Highway Standards.

The success of the stochastic approximation simulation methodology developed here is being reported upon as a prodigy approach to highway geometric design. With the use of the statistical equations that are related to vertical curves design, it has been made possible to produce stochastic approximation simulation models for the graphical expression of the four types with six representations of vertical curves as presented by AASHTO (2004) as in Fig. 1 completely while using the Microsoft Excel program. The six representations are in Fig. 4-9 slated for crest and sag vertical curves as carried out in this study that are stated as:

Table 2:	Final elevation on curve modeling module for the simulation of type I

Table 3:	Simulation module for the optimisation of type I crest vertical curve

Fig. 6:	Type 2 (both gardients negitive) crest vertical curves simulation

Table 4:	Final elevation on curve simulation module of type I crest vertical curve

Fig. 7:	Type3 (negitive and positive gardients) sag vertical curves simulation

Fig. 8:	Type 4 (both gardients negitive) sag vertical curves simulation

Fig. 9:	Type 4 (both gardients positive) sag vertical curves simulation

CONCLUSION

This study has vividly come out with the Optimisation Simulation Flow Chart and two Modeling Modules in Microsoft Excel 2010 office program for use in support of stochastic approximation simulation of highway geometric vertical curves. The objective-function values that include g₁, g₂, K, A, L and different station locations for VPT, VPC and VPT are variables that have been used subjectively via stochastic approximation to simulate reduced levels of vertical curves for highway vertical alignments easily and rapidly while utilising Microsoft Excel program.

Microsoft Excel is readily available and cheaper software than any other program for use in this facet. With this methodology, it is easily possible while employing stochastic approximation method advanced in this study to generate the simulation of the six vertical curves representations as presented by AASHTO (2004) in an electronics office similarly like that of the conventional hand method while carrying out geometric design of highway. It is pertinent to note that stochastic approximation process is sequel to subjective trials and it is a means to deterministic process to evolve the same output from the finally arrived vertical curve variables for the required solution to complex problems such as in defining highway vertical curves as shown in Fig. 4-9. This methodology is highly recommended to researchers, professional engineers and students as a cheaper and cost effective methodology of highway geometric design on comparison with the use of specific purpose application programs and conventional hand method.