A New Dantzig-Wolfe Reformulation And Branch-And-Price Algorithm For The Capacitated Lot Sizing Problem With Set Up Times

The textbook Dantzig-Wolfe decomposition for the Capacitated LotSizing Problem (CLSP),as already proposed by Manne in 1958, has animportant structural deficiency. Imposingintegrality constraints onthe variables in the full blown master will not necessarily givetheoptimal IP solution as only production plans which satisfy theWagner-Whitin condition canbe selected. It is well known that theoptimal solution to a capacitated lot sizing problem willnotnecessarily have this Wagner-Whitin property. The columns of thetraditionaldecomposition model include both the integer set up andcontinuous production quantitydecisions. Choosing a specific set upschedule implies also taking the associated Wagner-Whitin productionquantities. We propose the correct Dantzig-Wolfedecompositionreformulation separating the set up and productiondecisions. This formulation gives the samelower bound as Manne'sreformulation and allows for branch-and-price. We use theCapacitatedLot Sizing Problem with Set Up Times to illustrate our approach.Computationalexperiments are presented on data sets available from theliterature. Column generation isspeeded up by a combination of simplexand subgradient optimization for finding the dualprices. The resultsshow that branch-and-price is computationally tractable andcompetitivewith other approaches. Finally, we briefly discuss how thisnew Dantzig-Wolfe reformulationcan be generalized to other mixedinteger programming problems, whereas in theliterature,branch-and-price algorithms are almost exclusivelydeveloped for pure integer programmingproblems.branch-and-price;Lagrange relaxation;Dantzig-Wolfe decomposition;lot sizing;mixed-integer programming

Reformulation and decomposition of integer programs

In this survey we examine ways to reformulate integer and mixed integer programs. Typically, but not exclusively, one reformulates so as to obtain stronger linear programming relaxations, and hence better bounds for use in a branch-and-bound based algorithm. First we cover in detail reformulations based on decomposition, such as Lagrangean relaxation, Dantzig-Wolfe column generation and the resulting branch-and-price algorithms. This is followed by an examination of Benders’ type algorithms based on projection. Finally we discuss in detail extended formulations involving additional variables that are based on problem structure. These can often be used to provide strengthened a priori formulations. Reformulations obtained by adding cutting planes in the original variables are not treated here.Integer program, Lagrangean relaxation, column generation, branch-and-price, extended formulation, Benders' algorithm

A New Dantzig-Wolfe Reformulation And Branch-And-Price Algorithm For The Capacitated Lot Sizing Problem With Set Up Times

The textbook Dantzig-Wolfe decomposition for the Capacitated Lot Sizing Problem (CLSP),as already proposed by Manne in 1958, has an important structural deficiency. Imposingintegrality constraints on the variables in the full blown master will not necessarily give theoptimal IP solution as only production plans which satisfy the Wagner-Whitin condition canbe selected. It is well known that the optimal solution to a capacitated lot sizing problem willnot necessarily have this Wagner-Whitin property. The columns of the traditionaldecomposition model include both the integer set up and continuous production quantitydecisions. Choosing a specific set up schedule implies also taking the associated Wagner-Whitin production quantities. We propose the correct Dantzig-Wolfe decompositionreformulation separating the set up and production decisions. This formulation gives the samelower bound as Manne's reformulation and allows for branch-and-price. We use theCapacitated Lot Sizing Problem with Set Up Times to illustrate our approach. Computationalexperiments are presented on data sets available from the literature. Column generation isspeeded up by a combination of simplex and subgradient optimization for finding the dualprices. The results show that branch-and-price is computationally tractable and competitivewith other approaches. Finally, we briefly discuss how this new Dantzig-Wolfe reformulationcan be generalized to other mixed integer programming problems, whereas in the literature,branch-and-price algorithms are almost exclusively developed for pure integer programmingproblems

Combining Column Generation and Lagrangian Relaxation

Although the possibility to combine column generation and Lagrangian relaxation has been known for quite some time, it has only recently been exploited in algorithms. In this paper, we discuss ways of combining these techniques. We focus on solving the LP relaxation of the Dantzig-Wolfe master problem. In a first approach we apply Lagrangian relaxation directly to this extended formulation, i.e. no simplex method is used. In a second one, we use Lagrangian relaxation to generate new columns, that is Lagrangian relaxation is applied to the compact for-mulation. We will illustrate the ideas behind these algorithms with an application in Lot-sizing. To show the wide applicability of these techniques, we also discuss applications in integrated vehicle and crew scheduling, plant location and cutting stock problems.column generation;Lagrangean relaxation;cutting stock problem;lotsizing;vehicle and crew scheduling

On the trade-off between staff-decomposed and activity-decomposed column generation for a staff scheduling problem.

In this paper a comparison is made between two decomposition techniques to solve a staff scheduling problem with column generation. In the first approach, decomposition takes place on the staff members, whereas in the second approach decomposition takes place on the activities that have to be performed by the staff members. The resulting master LP is respectively a set partitioning problem and a capacitated multi-commodity flow problem. Both approaches have been implemented in a branch-and-price algorithm. We show a trade-off between modeling power and computation times of both techniques.decomposition; staff scheduling; set partitioning; multi-commodity flow; branch-and-price; branch-and-price; programs;

Circulation of Railway Rolling Stock: A Branch-and-Price Approach

We describe an algorithmic approach to determine an efficient railway rolling stock circulation on a single line or on a set of interacting lines. Given the timetable and the passengers? seat demand, we develop a branch-and-price algorithm that results in an allocation of rolling stock material to the daily trips. In order to efficiently utilize the train units, they can be added to or removed from the trains at some stations along the line. These changes in train composition are subject to several constraints, for example corresponding to the order of the train units within a train. A solution is evaluated based on three criteria, i.e. the service to passengers, the robustness, and the cost of the circulation. The branch-and-price algorithm that we developed is tested on real-life instances from NS Reizigers, the main Dutch operator of passenger trains

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

Column generation approaches to ship scheduling with flexible cargo sizes

We present a Dantzig-Wolfe procedure for the ship scheduling problem with flexible cargo sizes. This problem is similar to the well-known pickup and delivery problem with time windows, but the cargo sizes are defined by an interval instead of a fixed value. We show that the introduction of flexible cargo sizes to the column generation framework is not straightforward, and we handle the flexible cargo sizes heuristically when solving the subproblems. This leads to convergence issues in the branch-and-price search tree, and the optimal solution cannot be guaranteed. Hence we have introduced a method that generates an upper bound on the optimal objective. We have compared our method with an a priori column generation approach, and our computational experiments on real world cases show that the Dantzig-Wolfe approach is faster than the a priori generation of columns, and we are able to deal with larger or more loosely constrained instances. By using the techniques introduced in this paper, a more extensive set of real world cases can be solved either to optimality or within a small deviation from optimalityTransportation; integer programming; dynamic programming