Multi-commodity flow estimation with partial counts on selected links

The purpose of this research is to formulate a multi-commodity network flow model for vehicular traffic in a geographic area and develop a procedure for estimating traffic counts based on available partial traffic data for a selected subset of highway links. Due to the restriction of time and cost, traffic counts are not always observed for every highway link. Typically, about 50% of the links have traffic counts in urban highway networks. Also, it should be noted that the observed traffic counts are not free from random errors during the data collection process. As a result, an incoming flow into a highway node and an outgoing flow from the node do not usually match. They need to be adjusted to satisfy a flow conservation condition, which is one of the fundamental concepts in network flow analysis. In this dissertation, the multi-commodity link flows are estimated in a two-stage process. First, traffic flows of "empty" links, which have no observation data, are filled with deterministic user equilibrium traffic assignments. This user equilibrium assignment scheme assumes that travelers select their routes by their own interests without considering total cost of the system. The assignment also considers congestion effects by taking a link travel cost as a function of traffic volume on the link. As a result, the assignment problem has a nonlinear objective function and linear network constraints. The modified Frank-Wolfe algorithm, which is a type of conditional gradient method, is used to solve the assignment problem. The next step is to consider both of the observed traffic counts on selected links and the deterministic user equilibrium assignments on the group of remaining links to produce the final traffic count estimates by the generalized least squares optimization procedure. The generalized least squares optimization is conducted under a set of relevant constraints, including the flow conservation condition for all highway intersections

Models and Algorithms for Stochastic Network Design and Flow Problems: Applications in Truckload Procurement Auctions and Renewable Energy.

This dissertation presents novel mathematical models and algorithms for stochastic network design and flow (SNDF) problems: the optimal design and flow of a network under uncertainty to meet specific requirements while minimizing expected total cost. The focus of this dissertation is SNDF problems characterized by uncertainties in node supplies and/or demands and in arc capacities and/or costs. SNDF problems often have characteristics that render them difficult to model and computationally challenging to solve, including nonlinearities, probabilistic constraints, and stochastic parameters, all of which lead to large-scale, nonlinear, and discrete models. The work in this dissertation is motivated by problems in combinatorial truckload procurement auctions (CTPA) and wind farm network design (WFND). We use these two applications both for their own sake, as they present important and computationally challenging practical problems, and as a basis for the development of more general SNDF models and algorithmic approaches. In studying CTPA, we develop a novel bidding framework, the Implicit Bidding Approach (IBA), that permits the solution of fully-enumerated combinatorial auctions in a single round. Using IBA, we can circumvent the computational challenges of CTPAs by reposing the problem as a polynomially-sized integer multicommodity flow problem. We then extend our CTPA models to consider network uncertainties and show that the resulting model is a special case of a two-stage multicommodity flow problem (TS-MFP). We develop an efficient decomposition algorithm for solving problems in this class and provide extensive computational results to demonstrate its efficacy. In WFND, we present the integrated generation- and transmission- expansion planning problem for a network of interconnected wind farms. We develop an efficient decomposition algorithm for solving WFND problems and present computational results to demonstrate its efficacy. We then extend this model to include a probabilistic constraint on loss-of-load-expectation. We demonstrate that this model is extremely challenging and that direct applications of mathematical programming approaches are not viable. We present a hybrid algorithm, which we called Iterative Test-and-Prune (I-T&P), that leverages mathematical programming to solve a series of easy feasibility problems within a larger meta-search algorithm. Computational results for several test systems demonstrate the efficacy of I-T&P.Ph.D.Industrial & Operations EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/76005/1/richchen_1.pd

Clemson Newsletter, 1989-1991

Information for the faculty and staff of Clemson Universityhttps://tigerprints.clemson.edu/clemson_newsletter/1021/thumbnail.jp

Branch-and-price and multicommodity flows

Tese de doutoramento em Engenharia de Produção e Sistemas, área de Investigação OperacionalIn this Thesis, we address column generation based methods for linear and integer programming and apply them to three multicommodity flow problems. For (mixed) integer programming problems, the approach taken consists in reformulating an original model, using the Dantzig-Wolfe decomposition principle, and then combining column generation with branch-and-bound (branch-and-price) in order to obtain optimal solutions. The main issue when developing a branch-and-price algorithm is the branching scheme. The approach explored in this work is to branch on the variables of the original model, keeping the structure of the subproblems of the column generation method unchanged. The incorporation of cuts (branch-and-price-and-cut), again without changing the structure of the subproblem, is also explored. Based on that general methodology, we developed a set of C++ classes (ADDing - Automatic Dantzig-Wolfe Decomposition for INteger column Generation), which implements abranch-and-price algorithm. Its main distinctive feature is that it can be used as a “black-box”: all the user is required to do is to provide the original model. ADDing can also be customised to meet a specific problem, if the user is willing to provide a subproblem solver and/or specific branching schemes. We developed column generation based algorithms for three multicommodity flow problems. In this type of problems, it is desired to route a set of commodities through a capacitated network at a minimum cost. In the linear problem, each unit of each commodity is divisible. By using a model with variables associated with paths and circuits, we obtained significant improvements on the solution times over the standard column generation approach, for instances defined in planar networks (in several instances the relative improvement was greater than 60%). In the integer problem, each unit of each commodity is indivisible; the flow of a commodity can be split between different paths, but the flow on each of those paths must be integer. In general, the proposed branch-and-price algorithm was more efficient than Cplex 6.6 in the sets of instances where each commodity is defined by an origin-destination pair; for some of the other sets of instances, Cplex 6.6 gave better time results. In the binary problem, all the flow of each commodity must be routed along a single path. We developed a branch-and-price algorithm based on a knapsack decomposition and modified (by using a different branching scheme) a previously described branch-and-price-and-cut algorithm based on a path decomposition. The outcome of the computational tests was surprising, given that it is usually assumed that specific methods are more efficient than general ones. For the instances tested, a state-of-the-art general-purpose (Cplex 8.1) gave, in general, much better results than both decomposition approaches.Nesta Tese, abordam-se métodos de geração de colunas para programação linear e inteira. A sua aplicação é feita em três problemas de fluxo multicomodidade. Para problemas de programação inteira (mista), a abordagem seguida é a de reformular um modelo original, utilizando o princípio de decomposição de Dantzig-Wolfe, e combinar geração de colunas com o método de partição e avaliação (partição e geração de colunas) para a obtenção de soluções óptimas. A questão essencial no desenvolvimento de um algoritmo deste tipo é a estratégia de partição. A abordagem seguida neste trabalho é a de realizar a partição nas variáveis do modelo original, mantendo a estrutura do subproblema do método de geração de colunas. A incorporação de cortes, ainda sem alteração da estrutura do subproblema, é também explorada. Com base nesta metodologia geral, foi desenvolvido um conjunto de classes em C++ (ADDing - Automatic Dantzig-Wolfe Decomposition for INteger column Generation), que implementa um algorithmo de partição e geração de colunas. A sua característica fundamental é apenas ser requerido ao utilizador a definição de um modelo original. Num modo mais avançado, o utilizador pode implementar algoritmos para resolver o subproblema e/ou esquemas de partição. Foram desenvolvidos algoritmos baseados em geração de colunas para três problemas de fluxo multicomodidade. Neste tipo de problemas, pretende-se encaminhar um conjunto de comodidades através de uma rede capacitada, minimizando o custo. No problema linear, cada unidade de cada comodidade é divisível. Utilizando um modelo com variáveis associadas a caminhos e a circuitos, obtiveram-se melhorias significativas nos tempos de resolução em relação ao método de geração de colunas usual, para instâncias definidas em redes planares (em várias instâncias a melhoria relativa foi superior a 60%). No problema inteiro, cada unidade de cada comodidade é indivisível; o fluxo de uma comodidade pode ser dividido por diferentes caminhos, mas o fluxo em cada um deles tem de ser inteiro. Em geral, o algoritmo de partição e geração de colunas foi mais eficiente do que o software Cplex 6.6 nos conjuntos de instâncias em que cada comodidade é definida por um par origem-destino; para alguns dos outros conjuntos de instâncias, o software Cplex 6.6 obteve melhores resultados. No problema binário, todo o fluxo de cada comodidade apenas pode utilizar um caminho. Foi desenvolvido um algoritmo de partição e geração de colunas baseado numa decomposição de mochila e modificado (através de um esquema de partição diferente) um algoritmo de partição e geração de colunas com cortes, previamente descrito, baseado numa decomposição por caminhos. Os resultados dos testes computacionais foram surpreendentes, dado que é usualmente assumido que métodos específicos são mais eficientes do que métodos gerais. Para as instâncias testadas, o software Cplex 8.1 obteve, em geral, resultados muito melhores do que as duas decomposições