A decomposition approach for the integrated vehicle-crew-roster problem with days-off pattern

The integrated vehicle-crew-roster problem with days-off pattern aims to simultaneously determine minimum cost vehicle and daily crew schedules that cover all timetabled trips and a minimum cost roster covering all daily crew duties according to a pre-defined days-off pattern. This problem is formulated as a new integer linear programming model and is solved by a heuristic approach based on Benders decomposition that iterates between the solution of an integrated vehicle-crew scheduling problem and the solution of a rostering problem. Computational experience with data from two bus companies in Portugal and data from benchmark vehicle scheduling instances shows the ability of the approach for producing a variety of solutions within reasonable computing times as well as the advantages of integrating the three problems.     

A Lagrangian decomposition approach for the pump scheduling problem in water networks

Dynamic pricing has become a common form of electricity tariff, where the price of electricity varies in real time based on the realized electricity supply and demand. Hence, optimizing industrial operations to benefit from periods with low electricity prices is vital to maximizing the benefits of dynamic pricing. In the case of water networks, energy consumed by pumping is a substantial cost for water utilities, and optimizing pump schedules to accommodate for the changing price of energy while ensuring a continuous supply of water is essential. In this paper, a Mixed-Integer Non-linear Programming (MINLP) formulation of the optimal pump scheduling problem is presented. Due to the non-linearities, the typical size of water networks, and the discretization of the planning horizon, the problem is not solvable within reasonable time using standard optimization software. We present a Lagrangian decomposition approach that exploits the structure of the problem leading to smaller problems that are solved independently. The Lagrangian decomposition is coupled with a simulation-based, improved limited discrepancy search algorithm that is capable of finding high quality feasible solutions. The proposed approach finds solutions with guaranteed upper and lower bounds. These solutions are compared to those found by a mixed-integer linear programming approach, which uses a piecewise-linearization of the non-linear constraints to find a global optimal solution of the relaxation. Numerical testing is conducted on two real water networks and the results illustrate the significant costs savings due to optimizing pump schedules.     

A new decomposition approach for the single machine total tardiness scheduling problem

This paper deals with the single machine total tardiness problem. From Emmons’ basic dominance conditions a new partition theorem is derived which generalises Lawler’s decomposition rule and leads to a new double decomposition procedure. This procedure is embedded into a branch and bound method which applies a new lower bound based on due dates reassignment. The branch and bound method is tested on problems with size up to 150 jobs.     

A decomposition approach to the two-stage stochastic unit commitment problem

The unit commitment problem has been a very important problem in the power system operations, because it is aimed at reducing the power production cost by optimally scheduling the commitments of generation units. Meanwhile, it is a challenging problem because it involves a large amount of integer variables. With the increasing penetration of renewable energy sources in power systems, power system operations and control have been more affected by uncertainties than before. This paper discusses a stochastic unit commitment model which takes into account various uncertainties affecting thermal energy demand and two types of power generators, i.e., quick-start and non-quick-start generators. This problem is a stochastic mixed integer program with discrete decision variables in both first and second stages. In order to solve this difficult problem, a method based on Benders decomposition is applied. Numerical experiments show that the proposed algorithm can solve the stochastic unit commitment problem efficiently, especially those with large numbers of scenarios.     

A decomposition approach for solving a broadcast domination network design problem

We consider an optimization problem that integrates network design and broadcast domination decisions. Given an undirected graph, a feasible broadcast domination is a set of nonnegative integer powers f _i assigned to each node i, such that for any node j in the graph, there exists some node k having a positive f _k -value whose shortest distance to node j is no more than f _k . The cost of a broadcast domination solution is the sum of all node power values. The network design problem constructs edges that decrease the minimum broadcast domination cost on the graph. The overall problem we consider minimizes the sum of edge construction costs and broadcast domination costs. We show that this problem is NP-hard in the strong sense, even on unweighted graphs. We then propose a decomposition strategy, which iteratively adds valid inequalities based on optimal broadcast domination solutions corresponding to the first-stage network design solutions. We demonstrate that our decomposition approach is computationally far superior to the solution of a single large-scale mixed-integer programming formulation.     

A decomposition approach for a new test-scenario in complex problem solving

Over the last years, psychological research has increasingly used computer-supported tests, especially in the analysis of complex human decision making and problem solving. The approach is to use computer-based test scenarios and to evaluate the performance of participants and correlate it to certain attributes, such as the participant's capacity to regulate emotions. However, two important questions can only be answered with the help of modern optimization methodology. The first one considers an analysis of the exact situations and decisions that led to a bad or good overall performance of test persons. The second important question concerns performance, as the choices made by humans can only be compared to one another, but not to the optimal solution, as it is unknown in general.Additionally, these test-scenarios have usually been defined on a trial-and-error basis, until certain characteristics became apparent. The more complex models become, the more likely it is that unforeseen and unwanted characteristics emerge in studies. To overcome this important problem, we propose to use mathematical optimization methodology not only as an analysis and training tool, but also in the design stage of the complex problem scenario.We present a novel test scenario, the IWR Tailorshop, with functional relations and model parameters that have been formulated based on optimization results. We also present a tailored decomposition approach to solve the resulting mixed-integer nonlinear programs with nonconvex relaxations and show some promising results of this approach.     

A heuristic for the container loading problem: A tertiary-tree-based dynamic space decomposition approach

Increasing fuel costs, post-911 security concerns, and economic globalization provide a strong incentive for container carriers to use available container space more efficiently, thereby minimizing the number of container trips and reducing socio-economic vulnerability. A heuristic algorithm based on a tertiary tree model is proposed to handle the container loading problem (CLP) with weakly heterogeneous boxes. A dynamic space decomposition method based on the tertiary tree structure is developed to partition the remaining container space after a block of homogeneous rectangular boxes is loaded into a container. This decomposition approach, together with an optimal-fitting sequencing and an inner-right-corner-occupying placement rule, permits a holistic loading strategy to pack a container. Comparative studies with existing algorithms and an illustrative example demonstrate the efficiency of this algorithm.     

A Benders decomposition approach for the robust spanning tree problem with interval data

The robust spanning tree problem is a variation, motivated by telecommunications applications, of the classic minimum spanning tree problem. In the robust spanning tree problem edge costs lie in an interval instead of having a fixed value.Interval numbers model uncertainty about the exact cost values. A robust spanning tree is a spanning tree whose total cost minimizes the maximum deviation from the optimal spanning tree over all realizations of the edge costs. This robustness concept is formalized in mathematical terms and is used to drive optimization.This paper describes a new exact method, based on Benders decomposition, for the robust spanning tree problem with interval data. Computational results highlight the efficiency of the new method, which is shown to be very fast on all the benchmarks considered, and in particular on those that were harder to solve for the methods previously known.     

A decomposition approach for a very large scale optimal diversity management problem

This paper focuses on the solution of the optimal diversity management problem formulated as a p-Median problem. The problem is solved for very large scale real instances arising in the car industry and defined on a graph with several tens of thousands of nodes and with several millions of arcs. The particularity is that the graph can consist of several non connected components. This property is used to decompose the problem into a series of p-Median subproblems of a smaller dimension. We use a greedy heuristic and a Lagrangian heuristic for each subproblem. The solution of the whole problem is obtained by solving a suitable assignment problem using a Branch-and-Bound algorithm.Received: June 2004 / Revised version: December 2004MSC classification: 49M29, 90C06, 90C27, 90C60All correspondence to: Antonio SforzaIgor Vasilev: Support for this author was provided by NATO grant CBP.NR.RIG.911258.     

A phase decomposition approach and the Riemann problem for a model of two-phase flows

We present a phase decomposition approach to deal with the generalized Rankine–Hugoniot relations and then the Riemann problem for a model of two-phase flows. By investigating separately the jump relations for equations in conservative form in the solid phase, we show that the volume fractions can change only across contact discontinuities. Then, we prove that the generalized Rankine–Hugoniot relations are reduced to the usual form. It turns out that shock waves and rarefaction waves remain on one phase only, and the contact waves serve as a bridge between the two phases. By decomposing Riemann solutions into each phase, we show that Riemann solutions can be constructed for large initial data. Furthermore, the Riemann problem admits a unique solution for an appropriate choice of initial data.     

A dyadic decomposition approach to a finitely degenerate hyperbolic problem

Abstract We use the Littlewood-Paley decomposition technique to obtain a C^∞-well-posedness result for a weakly hyperbolic equation with a finite order of degeneration. Keywords: Littlewood-Paley decomposition, Hyperbolic equations, C^∞-well-posedness, Approximate energy method     

A decomposition heuristics for the container ship stowage problem

In this paper we face the problem of stowing a containership, referred to as the Master Bay Plan Problem (MBPP); this problem is difficult to solve due to its combinatorial nature and the constraints related to both the ship and the containers. We present a decomposition approach that allows us to assign a priori the bays of a containership to the set of containers to be loaded according to their final destination, such that different portions of the ship are independently considered for the stowage. Then, we find the optimal solution of each subset of bays by using a 0/1 Linear Programming model. Finally, we check the global ship stability of the overall stowage plan and look for its feasibility by using an exchange algorithm which is based on local search techniques. The validation of the proposed approach is performed with some real life test cases. This work has been developed within the research area: “The harbour as a logistic node” of the Italian Centre of Excellence on Integrated Logistics (CIELI) of the University of Genoa, Italy     

Using an interior point method for the master problem in a decomposition approach

We address some of the issues that arise when an interior point method is used to handle the master problem in a decomposition approach. The main points concem the efficient exploitation of the special structure of the master problem to reduce the cost of a single interior point iteration. The particular structure is the presence of GUB constraints and the natural partitioning of the constraint matrix into blocks built of cuts generated by different subproblems.The method can be used in a fairly general case, i.e., in any decomposition approach whenever the master is solved by an interior point method in which the normal equations are used to compute orthogonal projections.Computational results demonstrate its advantages for one particular decomposition approach: Analytic Center Cutting Plane Method (ACCPM) is applied to solve large scale nonlinear multicommodity network flow problems (up to 5000 arcs and 10000 commodities)     

A decomposition approach for facility location and relocation problem with uncertain number of future facilities

In this paper, we discuss two challenges of long term facility location problem that occur simultaneously; future demand change and uncertain number of future facilities. We introduce a mathematical model that minimizes the initial and expected future weighted travel distance of customers. Our model allows relocation for the future instances by closing some of the facilities that were located initially and opening new ones, without exceeding a given budget. We present an integer programming formulation of the problem and develop a decomposition algorithm that can produce near optimal solutions in a fast manner. We compare the performance of our mathematical model against another method adapted from the literature and perform sensitivity analysis. We present numerical results that compare the performance of the proposed decomposition algorithm against the exact algorithm for the problem.     

A Benders decomposition approach for a distribution network design problem with consolidation and capacity considerations

We develop a model for a strategic freight-forwarding network design problem in which the design decisions involve the locations and capacities of consolidation and deconsolidation centers, and capacities on linehaul linkages as well as the shipment routes from origins to destinations through centers. We devise a solution approach based on Benders decomposition and conduct a computational study that illustrates the efficiency and the effectiveness of the approach.     

A decomposition algorithm for linear relaxation of the weightedr-covering problem

In this paper the linear relaxation of the weightedr-covering problem (r-LCP) is considered. The dual problem (c-LMP) is the linear relaxation of the well-knownc-matching problem and hence can be solved in polynomial time. However, we describe a simple, but nonpolynomial algorithm in which ther-LCP is decomposed into a sequence of 1-LCP’s and its optimal solution is obtained by adding the optimal solutions of these 1-LCP’s. An 1-LCP can be solved in polynomial time by solving its dual as a max-flow problem on a bipartite graph. An accelerated algorithm based on this decomposition scheme to solve ar-LCP is also developed and its average case behaviour is studied.     

A decomposition algorithm for the single machine total tardiness problem

The problem of sequencing jobs on a single machine to minimize total tardiness is considered. An algorithm, which decomposes the problem into subproblems which are then solved by dynamic programming when they are sufficiently small, is presented and is tested on problems with up to 100 jobs.     

Multi-item single source ordering problem with transportation cost: A Lagrangian decomposition approach

As a part of supply chain management literature and practice, it has been recognized that there can be significant gains in integrating inventory and transportation decisions. The problem we tackle here is a common one both in retail and production sectors where several items have to be ordered from a single supplier. We assume that there is a finite planning horizon to make the ordering decisions for the items, and in this finite horizon the retailer or the producer knows the demand of each item in each period. In addition to the inventory holding cost, an item-base fixed cost associated with each item included in the order, and a piecewise linear transportation cost are incurred. We suggest a Lagrangean decomposition based solution procedure for the problem and carry out numerical experiments to analyze the value of integrating inventory and transportation decisions under different scenarios.     

A simplicial decomposition method for the transit equilibrium assignment problem

A transit equilibrium assignment problem assigns the passenger flows on to a congested transit (public transportation) network with asymmetric cost functions and a fixed origin-destination matrix. This problem which may be formulated in the space of hyperpath flows, is transformed into an equivalent problem in the space of total arc flows and an auxiliary variable. A simplicial decomposition algorithm is developed and its convergence is proved under the usual assumptions on the cost functions. The algorithm requires relatively little memory and its efficiency is demonstrated with computational results.