Paul Schonfeld

Multiple period optimization of bus transit systems

Analytic models are developed for optimizing bus services with time dependence and elasticity in their demand characteristics. Some supply parameters, i.e. vehicle operating costs and speeds are also allowed to vary over time. The multiple period models presented here allow some of the optimized system characteristics (e.g. route structure) to be fized at values representing the best compromise over different time periods, while other characteristics (e.g. service headways) may be optimized within each period. In a numerical example the demand is assumed to fluctuate over a daily cycle (e.g. peak, offpeak and night), although the same models can also be used for other cyclical or noncyclical demand variations over any number of periods. Models are formulated and compared for four types of conditions, which include steady fixed demand, cyclical fixed demand, steady equilibrium demand and cyclical equilibrium demand. When fixed demand is assumed, the optimization objective is minimum total system cost, including operator cost and user cost, while operator profit and social welfare are the objective functions maximized for equilibrium demand. The major results consist of closed form solutions for the route spacings, headways, fares and costs for optimized feeder bus services under various demand conditions. A comparison of the optimization results for the four cases is also presented. When demand and bus operating characteristics are allowed to vary over time, the optimal functions are quite similar to those for steady demand and supply conditions. The optimality of a constant ratio between the headway and route spacing, which is found at all demand densities if demand is steady, is also maintained with a multi-period adjustment factor in cyclical demand cases, either exactly or with a relatively negligible approximation. These models may be used to analyze and optimize fairly complex feeder or radial bus systems whose demand and supply characteristics may vary arbitrarily over time.

An evolutionary model for simultaneously optimizing three-dimensional highway alignments

Abstract Optimizing highway alignments is a very complex engineering problem. The factors that should be considered in the design process are complex and interrelated. Although several mathematical models have been developed to solve the alignment optimization problem, most of them emphasize either horizontal or vertical alignments, and only yield a suboptimal solution to the problem. Models for simultaneously optimizing three-dimensional alignments are rare in the literature and their capabilities are quite limited. In this paper, an evolutionary model (a search algorithm that imitates the natural evolution process) for solving three-dimensional alignment optimization problems is developed. It overcomes some drawbacks in existing models. The cost components and design constraints embedded in it can be comprehensive. The proposed algorithm can optimize complex, comprehensive, and non-differentiable objective function. The model can also exploit detailed geographical information for highway analysis. The resulting alignments are smooth everywhere and can have backward bends (i.e., “backtracking”) to better fit terrain and land-use patterns. A numerical example is presented to illustrate the proposed model and the performance of the solution algorithm.

PRELIMINARY HIGHWAY DESIGN WITH GENETIC ALGORITHMS AND GEOGRAPHIC INFORMATION SYSTEMS

This paper presents a method that integrates geographic information systems (GIS) with genetic algorithms (GAs) for optimizing horizontal highway alignments between 2 given end points. The proposed approach can be used to optimize alignments in highly irregular geographic spaces. The resulting alignments are smooth and satisfy minimum-radius constraints, as required by highway design standards. The objective function in the proposed model considers land-acqusition cost, environmental impacts such as wetlands and flood plains, length-dependent costs, and user costs. A numerical example based on a real map is employed to demonstrate application of the proposed model to the preliminary design of horizontal alignments.

METAMODELS FOR ESTIMATING WATERWAY DELAYS THROUGH SERIES OF QUEUES

A numerical method has been developed for estimating delays on congested waterways. Analytic and numerical results are presented for series of G/G/1 queues, i.e. with generally distributed arrivals and service times and single chambers at each lock. One or two-way traffic operations are modelled. A metamodelling approach which develops simple formulas to approximate the results of simulation models is presented. The structure of the metamodels is developed from queueing theory while their coefficients are statistically estimated from simulation results. The numerical method consists of three modules: (1) delays, (2) arrivals and (3) departures. The first estimates the average waiting time for each lock when the arrival and service time distributions at this lock are known. The second identifies the relations between the arrival distributions at one lock and the departure distributions from the upstream and downstream locks. The third estimates the mean and variance of inter-departure times when the inter-arrival and service time distributions are known. The method can be applied to systems with two-way traffic through common bi-directional servers as well as to one-way traffic systems. Algorithms for both cases are presented. This numerical method is shown to produce results that are close to the simulation results. The metamodels developed for estimating delays and variances of inter-departure times may be applied to waterways and other series of G/G/1 queues. These metamodels for G/G/1 queues may provide key components of algorithms for analyzing networks of queues.

USING GIS, GENETIC ALGORITHMS, AND VISUALIZATION IN HIGHWAY DEVELOPMENT

A model for highway development is presented, which uses geographic information systems (GIS), genetic algorithms (GA), and computer visualization (CV). GIS serves as a repository of geographic information and enables spatial manipulations and database management. GAs are used to optimize highway alignments in a complex search space. CV is a technique used to convey the characteristics of alternative solutions, which can be the basis of decisions. The proposed model implements GIS and GA to find an optimized alignment based on the minimization of highway costs. CV is implemented to investigate the effects of intangible parameters, such as unusual land and environmental characteristics not considered in optimization. Constrained optimization using GAs may be performed at subsequent stages if necessary using feedback received from CVs. Implementation of the model in a real highway project from Maryland indicates that integration of GIS, GAs, and CV greatly enhances the highway development process.

INTEGRATING GENETIC ALGORITHMS AND GEOGRAPHIC INFORMATION SYSTEM TO OPTIMIZE HIGHWAY ALIGNMENTS

A comprehensive highway cost model can be used for optimizing highway alignments subject to a number of design constraints. Because a geographic information system (GIS) can spatially represent the locations of properties, floodplains, streams, and other geographical characteristics of significance in a highway cost model, it can provide valuable input to a highway design optimization model. Additionally, a GIS-based model can be developed to compute geographically sensitive costs to be used with an iterative optimization scheme. However, connecting a GIS to a highway optimization model requires full automation of an entire search process in which there is a continuous exchange of inputs and outputs until the optimized solution is obtained. An integrated model is developed by linking a GIS model with an optimization model employing genetic algorithms (GAs). The GIS model provides accurate geographical features, computes location-dependent costs, and transmits these costs to an external program. That program computes length-dependent costs and user costs and then, using GAs, optimizes the highway alignment to minimize the sum of all costs. An example study using real land use and environmental features is presented for a part of Talbot County, Maryland. The computational performance of the integrated model is assessed.

Geographic Information System-Based Analysis of Right-of-Way Cost for Highway Optimization

At the planning stages of a highway project, various location alternatives must be explored, subject to a set of design constraints. A computerized tool with which to compare alignment alternatives would significantly reduce the time and resources spent as well as help find a minimum cost (or maximum net benefit) solution. Highway design optimization (HDO) is a computerized process that minimizes an objective function composed of significant highway costs, subject to a set of design constraints, including curvature, gradient, and sight distance. Several costs of alignments, such as right-of-way, earthwork, and environment costs, are sensitive to geography. A geographic information system (GIS) may be exploited to compute such costs for use in HDO models (HDOMs). Most known HDOMs focus only on refining the optimization approach and do not provide a comprehensive formulation for all costs sensitive to alignment. Provided is a comprehensive formulation for right-of-way cost computation. A GIS-based algorithm is developed to compute the right-of-way cost, which is integrated with an HDOM based on genetic algorithms. Two examples are used to demonstrate the effectiveness of the proposed approach.

Cost Functions for Optimizing Highway Alignments

Optimizing highway alignments through complex environments is a difficult problem. Optimal alignment is influenced by many factors and should be based on trade-offs among various supplier and user cost components. Previous works on alignment optimization usually have neglected some important components of the total cost function that should influence the optimal solution. The formulations of different cost components for two-lane rural highways are presented. The proposed cost functions are based on the geometric features of both horizontal and vertical alignments. The cost formulations can be employed in models for simultaneously optimizing three-dimensional alignments. Possible methods for solving such optimization problems are provided. A numerical example also is presented to illustrate the optimization of three-dimensional alignments with the proposed cost functions.

Optimization Through Simulation of Waterway Transportation Investments

The cost of tow delays is a serious problem in a waterway network. One way to reduce the delay cost is to increase capacity at waterway locks. Planners must determine how much additional capacity to provide at particular lock sites and when to implement the capacity expansion projects. Answers for such project sizing and timing problems are difficult to obtain analytically. The use of a new approach for optimizing through simulation, called simultaneous perturbation stochastic approximation (SPSA), is investigated. This approach, which seeks optimal values for all decision variables after each pair of simulation runs, is quite promising for optimizing large problems relatively fast. A small numerical example tests how this simulation and optimization algorithm may be used to optimize lock capacities and implementation times.

Queuing network analysis for waterways with artificial neural networks

An Artificial Neural Network (ANN) model has been developed for analyzing traffic in an inland waterway network. The main purpose of this paper is to determine how well such a relatively fast model for analyzing a queuing network could substitute for far more expensive simulation. Its substitutability for simulation is judged by relative discrepancies in predicting tow delays between the ANN and simulation models. This model is developed by integrating five distinct ANN submodels that predict tow headway variances at (1) merge points, (2) branching (i.e., diverging) points, (3) lock exits, and (4) link outflow points (e.g., at ports, junctions, or lock entrances), plus (5) tow queuing delays at locks. Preliminary results are shown for those five submodels and for the integrated network analysis model. Eventually, such a network analyzer should be useful for designing, selecting, sequencing, and scheduling lock improvement projects, for controlling lock operations, for system maintenance planning, and for other applications where many combinations of network characteristics must be evaluated. More generally, this method of decomposing complex queuing networks into elements that can be analyzed with ANNs and then recombined provides a promising approach for analyzing other queuing networks (e.g., in transportation, communication, computing, and production systems).