Branch Algorithm Research Articles

Modeling Lung Architecture in the XCAT Series of Phantoms: Physiologically Based Airways, Arteries and Veins.

The purpose of this paper was to extend the extended cardiac-torso (XCAT) series of computational phantoms to include a detailed lung architecture including airways and pulmonary vasculature. Eleven XCAT phantoms of varying anatomy were used in this paper. The lung lobes and initial branches of the airways, pulmonary arteries, and veins were previously defined in each XCAT model. These models were extended from the initial branches of the airways and vessels to the level of terminal branches using an anatomically-based volume-filling branching algorithm. This algorithm grew the airway and vasculature branches separately and iteratively without intersecting each other using cylindrical models with diameters estimated by order-based anatomical measurements. Geometrical features of the extended branches were compared with the literature anatomy values to quantitatively evaluate the models. These features include branching angle, length to diameter ratio, daughter to parent diameter ratio, asymmetrical branching pattern, diameter, and length ratios. The XCAT phantoms were then used to simulate CT images to qualitatively compare them with the original phantom images. The proposed growth model produced 46369 ± 12521 airways, 44737 ± 11773 arteries, and 39819 ± 9988 veins to the XCAT phantoms. Furthermore, the growth model was shown to produce asymmetrical airway, artery, and vein networks with geometrical attributes close to morphometry and model based studies. The simulated CT images of the phantoms were judged to be more realistic, including more airways and pulmonary vessels compared with the original phantoms. Future work will seek to add a heterogeneous parenchymal background into the XCAT lungs to make the phantoms even more representative of human anatomy, paving the way towards the use of XCAT models as a tool to virtually evaluate the current and emerging medical imaging technologies.

An exact approach for the balanced k-way partitioning problem with weight constraints and its application to sports team realignment

In this work a balanced k-way partitioning problem with weight constraints is defined to model the sports team realignment. Sports teams must be partitioned into a fixed number of groups according to some regulations, where the total distance of the road trips that all teams must travel to play a Double Round Robin Tournament in each group is minimized. Two integer programming formulations for this problem are introduced, and the validity of three families of inequalities associated to the polytope of these formulations is proved. The performance of a tabu search procedure and a Branch & Cut algorithm, which uses the valid inequalities as cuts, is evaluated over simulated and real-world instances. In particular, an optimal solution for the realignment of the Ecuadorian Football league is reported and the methodology can be suitable adapted for the realignment of other sports leagues.

New Valid Inequalities for the Two-Echelon Capacitated Vehicle Routing Problem

We introduce new valid inequalities for the two-echelon variant of the Capacitated Vehicle Routing Problem (CVRP)In particular, a first group of inequalities is obtained by extending to 2E-CVRP some of the most effective among the existing CVRP valid inequalities. A second group of inequalities is explicitly derived for the 2E-CVRP and concerns the flow feasibility at customer nodes and the satellite-customer route connectivity. The inequalities are then introduced in a Branch & Cut algorithm. Computational results show that the proposed algorithm is able both to solve to optimality many open literature instances and significantly reduce the optimality gap for the remaining instances.

A multiobjective optimization model and an orthogonal design–based hybrid heuristic algorithm for regional urban mining management problems

ABSTRACTIn this paper, a multiobjective mixed-integer piecewise nonlinear programming model (MOMIPNLP) is built to formulate the management problem of urban mining system, where the decision variables are associated with buy-back pricing, choices of sites, transportation planning, and adjustment of production capacity. Different from the existing approaches, the social negative effect, generated from structural optimization of the recycling system, is minimized in our model, as well as the total recycling profit and utility from environmental improvement are jointly maximized. For solving the problem, the MOMIPNLP model is first transformed into an ordinary mixed-integer nonlinear programming model by variable substitution such that the piecewise feature of the model is removed. Then, based on technique of orthogonal design, a hybrid heuristic algorithm is developed to find an approximate Pareto-optimal solution, where genetic algorithm is used to optimize the structure of search neighborhood, and both local branching algorithm and relaxation-induced neighborhood search algorithm are employed to cut the searching branches and reduce the number of variables in each branch. Numerical experiments indicate that this algorithm spends less CPU (central processing unit) time in solving large-scale regional urban mining management problems, especially in comparison with the similar ones available in literature. By case study and sensitivity analysis, a number of practical managerial implications are revealed from the model.Implications: Since the metal stocks in society are reliable overground mineral sources, urban mining has been paid great attention as emerging strategic resources in an era of resource shortage. By mathematical modeling and development of efficient algorithms, this paper provides decision makers with useful suggestions on the optimal design of recycling system in urban mining. For example, this paper can answer how to encourage enterprises to join the recycling activities by government’s support and subsidies, whether the existing recycling system can meet the developmental requirements or not, and what is a reasonable adjustment of production capacity.

Low-thrust tour of the main belt asteroids

This work presents some initial results on a possible low-thrust tour of the main asteroid belt. The asteroids are visited through a series of fly-by’s to be completed within a given time-frame and limit on the mass of the spacecraft at launch. The asteroids to be visited are automatically selected out of a large database of possible candidates. The initial shortlist of targets is based on the Minimum Orbit Intersection Distance (MOID) between the orbit of the asteroids in the database and the initial orbit of the spacecraft traversing the main belt. The final sequence is then obtained with an efficient deterministic branch and prune algorithm. The transfers between asteroids are designed using a low-thrust analytical model that provides a good estimation of the propellant consumption and transfer time. The mission analysis is completed with a study of the cost of the launch. In this paper two databases will be analysed: one containing only targets with a particular scientific relevance and one containing all available asteroids.

Multi-Objective Optimal Sizing Design of a Diesel- PV-Wind-Battery Hybrid Power System in Colombia

This paper proposes a design of an isolated hybrid microgrid using an iterative optimization method: the branch and cut algorithm. The Colombian community of Unguía, found in the state of Chocó, has been proposed for the microgrid design. The optimization objectives are the minimization of the grid operational costs and CO2 emissions, and ensuring 80 % of load demand coverage. The elements considered in the microgrid design are: Two Diesel generators that already exists in community, a battery bank, photovoltaic panels and wind turbines. In addition a comparison between the designs obtained with the branch and cut algorithm and the software HOMER is performed for the Unguía community.

Generalized local branching heuristics and the capacitated ring tree problem

In this paper we present a heuristic framework that is based on mathematical programming to solve network design problems. Our techniques combine local branching with locally exact refinements. In an iterative strategy an existing solution is refined by sequentially solving restricted mixed integer programs (MIPs) to optimality. These are obtained from the master problem MIP by limiting the number of variable flips for structured subsets of the binary edge variables which are selected based on the underlying network cost structure. We introduce generalized local branching cuts which enforce the latter using two parameters at the same time: the number of considered variables and the number of allowed variable flips.Using this concept we develop an efficient algorithm for the capacitated ring tree problem (CRTP), a recent network design model for partially reliable capacitated networks that combines cycle and tree structures. Our implementation operates on top of an efficient branch and cut algorithm for the CRTP. The sets of refinement variables are deduced from single-ball network node clusters. We provide computational results and an extensive analysis of the algorithm for a set of literature instances. We show that the approach is capable of improving existing best results for the CRTP and outperforms the pure refinement or local branching approaches.

Fréchet Distance Between a Line and Avatar Point Set

Frechet distance is an important geometric measure that captures the distance between two curves or more generally point sets. In this paper, we consider a natural variant of Frechet distance problem with multiple choice, provide an approximation algorithm and address its parameterized and kernelization complexity. A multiple choice problem consists of a set of color classes $$\mathcal {Q}=\{Q_1,Q_2,\ldots ,Q_n\}$$ , where each class $$Q_i$$ consists of a pair of points $$Q_i = \{q_i, \bar{q_i}\}$$ . We call a subset $$A\subset \{q_i , \bar{q_i}:1\le i\le n\}$$ conflict-free if A contains at most one point from each color class. The standard objective in multiple choice problem is to select a conflict-free subset that optimizes a given function. Given a line-segment $$\ell $$ and a set $$\mathcal {Q}$$ of a pair of points in $$\mathbb {R}^2$$ , our objective is to find a conflict-free subset that minimizes the Frechet distance between $$\ell $$ and the point set, where the minimum is taken over all possible conflict-free subsets. We first show that this problem is NP-hard, and provide a 3-approximation algorithm. Then we develop a simple randomized FPT algorithm for the problem when parametrized by the solution size, which is later derandomized using universal family of sets. We believe that our derandomization technique can be of independent interest, and can be used to solve other parameterized multiple choice problems. The randomized algorithm runs in $$\mathcal {O}(2^k n \log ^2 n)$$ time, and the derandomized deterministic algorithm runs in $$2^k k^{\mathcal {O}(\log k)} n \log ^2 n$$ time, where k, the parameter, is the number of elements in the conflict-free subset solution. Finally we present a simple branching algorithm for the problem running in $$\mathcal {O}(2^k n\log n)$$ time. We also show that the problem does not have a polynomial sized kernel under standard complexity theoretic assumptions.

A branch and price algorithm for a Stackelberg Security Game

Mixed integer optimization formulations are an attractive alternative to solve Stackelberg Game problems thanks to the efficiency of state of the art mixed integer algorithms. In particular, decomposition algorithms, such as branch and price methods, make it possible to tackle instances large enough to represent games inspired in real world domians.In this work we focus on Stackelberg Games that arise from a security application and investigate the use of a new branch and price method to solve its mixed integer optimization formulation. We prove that the algorithm provides upper and lower bounds on the optimal solution at every iteration and investigate the use of stabilization heuristics. Our preliminary computational results compare this solution approach with previous decomposition methods obtained from alternative integer programming formulations of Stackelberg games.

A multiobjective robust controller synthesis approach aided by multicriteria decision analysis

This paper proposes a methodology for robust dynamic output-feedback control synthesis of uncertain linear systems represented by polytopic models. This control problem results in a semi-infinite optimization problem. The proposed synthesis procedure improves a previous two-step procedure, composed of synthesis and analysis, by employing both a multiobjective evolutionary algorithm (MOEA) and a multiple criteria decision making (MCDM) strategy in the synthesis step. The analysis stage is performed via a Branch and Bound (B&B) algorithm, enabling the validation of the former step. In the proposed multiobjective approach, the project aim is to meet the specifications of (i) reference signal tracking response, (ii) disturbance rejection and (iii) measurement noise attenuation. The project specifications are quantified in terms of H∞ and H2 norms of closed-loop transfer functions. Essentially, this technique combines the flexibility of a dedicated MOEA together with the minimax semi-infinite programming problem. Since a MOEA evolves a set of candidate solutions in parallel, the suggested strategy provides a diverse set of controller designs, which is very useful for an a posteriori decision-making process. A proposed multicriteria decision-aid strategy is employed, in addition, as a controller design tool, aiming to incorporate the decision-maker preferences throughout the process, which (i) enables a guided evolutionary search to a trade-off region (of solutions) of practical interest and (ii) assists the definition of an adequate final controller, characterized by a reasonable practical trade-off concerning the criteria. The application of the suggested framework is illustrated on three case studies in order to demonstrate the effectiveness of the method presented.

Edge Bipartization Faster than 2^k

In the Edge Bipartization problem one is given an undirected graph G and an integer k, and the question is whether k edges can be deleted from G so that it becomes bipartite. Guo et al. (J Comput Syst Sci 72(8):1386–1396, 2006) proposed an algorithm solving this problem in time mathcal {O}(2^kcdot {m}^2); today, this algorithm is a textbook example of an application of the iterative compression technique. Despite extensive progress in the understanding of the parameterized complexity of graph separation problems in the recent years, no significant improvement upon this result has been yet reported. We present an algorithm for Edge Bipartization that works in time mathcal {O}(1.977^kcdot {nm}), which is the first algorithm with the running time dependence on the parameter better than 2^k. To this end, we combine the general iterative compression strategy of Guo et al. (2006), the technique proposed by Wahlström (in: Proceedings of SODA’14, SIAM, 2014) of using a polynomial-time solvable relaxation in the form of a Valued Constraint Satisfaction Problem to guide a bounded-depth branching algorithm, and an involved Measure&Conquer analysis of the recursion tree.

The Delicate Analysis of Short–Term Load Forecasting

This paper proposes a new method for short-term load forecasting based on the similar day method, correlation coefficient and Fast Fourier Transform (FFT) to achieve the precision analysis of load variation from three aspects (typical day, correlation coefficient, spectral analysis) and three dimensions (time dimension, industry dimensions, the main factors influencing the load characteristic such as national policies, regional economic, holidays, electricity and so on). First, the branch algorithm one-class-SVM is adopted to selection the typical day. Second, correlation coefficient method is used to obtain the direction and strength of the linear relationship between two random variables, which can reflect the influence caused by the customer macro policy and the scale of production to the electricity price. Third, Fourier transform residual error correction model is proposed to reflect the nature of load extracting from the residual error. Finally, simulation result indicates the validity and engineering practicability of the proposed method.

A multiple ship routing and speed optimization problem under time, cost and environmental objectives

The purpose of this paper is to investigate a multiple ship routing and speed optimization problem under time, cost and environmental objectives. A branch and price algorithm as well as a constraint programming model are developed that consider (a) fuel consumption as a function of payload, (b) fuel price as an explicit input, (c) freight rate as an input, and (d) in-transit cargo inventory costs. The alternative objective functions are minimum total trip duration, minimum total cost and minimum emissions. Computational experience with the algorithm is reported on a variety of scenarios.

The demand weighted vehicle routing problem

In this paper, we present an exact method for the demand weighted vehicle routing problem. This problem arises when the objective is to minimize the distance traveled by the vehicles weighted by the number of passengers on the routes. The resulting model is a non-convex, mixed-integer quadratic program and we show how to reformulate this problem as a linear integer program with auxiliary variables. The reformulation provides very tight lower bounds on the optimal solution value and little enumeration is required. However, the reformulated model is very large and therefore, it is solved with a branch, price and cut algorithm. Using COIN-OR software, we provide computational results with actual data from a large urban university.

On parallel Branch and Bound frameworks for Global Optimization

Branch and Bound (B&B) algorithms are known to exhibit an irregularity of the search tree. Therefore, developing a parallel approach for this kind of algorithms is a challenge. The efficiency of a B&B algorithm depends on the chosen Branching, Bounding, Selection, Rejection, and Termination rules. The question we investigate is how the chosen platform consisting of programming language, used libraries, or skeletons influences programming effort and algorithm performance. Selection rule and data management structures are usually hidden to programmers for frameworks with a high level of abstraction, as well as the load balancing strategy, when the algorithm is run in parallel. We investigate the question by implementing a multidimensional Global Optimization B&B algorithm with the help of three frameworks with a different level of abstraction (from more to less): Bobpp, Threading Building Blocks (TBB), and a customized Pthread implementation. The following has been found. The Bobpp implementation is easy to code, but exhibits the poorest scalability. On the contrast, the TBB and Pthread implementations scale almost linearly on the used platform. The TBB approach shows a slightly better productivity.

The Generalized Arc Routing Problem

This paper focuses on the generalized arc routing problem. This problem is stated on an undirected graph in which some clusters are defined as pairwise-disjoint connected subgraphs, and a route is sought that traverses at least one edge of each cluster. Broadly speaking, the generalized arc routing problem is the arc routing counterpart of the generalized traveling salesman problem, where the set of vertices of a given graph is partitioned into clusters and a route is sought that visits at least one vertex of each cluster. A mathematical programming formulation that exploits the structure of the problem and uses only binary variables is proposed. Facets and families of valid inequalities are presented for the polyhedron associated with the formulation and the corresponding separation problem studied. The numerical results of a series of computational experiments with an exact branch and cut algorithm are presented and analyzed.

Branch and price for covering shipments in a logistic distribution network with a fleet of aircraft

We consider the problem of covering a set of shipments in a logistic distribution network with a fleet of aircraft. The aim is to cover as many shipments as possible, while also minimizing the number of aircraft utilized for that purpose. We develop an integer programming model and a branch and price solution algorithm for this problem. The proposed methodology utilizes a master problem that covers the maximum possible number of shipments using a given set of aircraft-routes, and a column generation subproblem that generates cost-effective aircraft-routes which are fed into the master problem. We describe the proposed methodology, illustrating how it can be modified in order to accommodate several problem extensions. We also investigate how its efficiency is affected by various key design parameters. We conclude with extensive experimental results demonstrating its computational performance.

A power storage station placement algorithm for power distribution based on electric vehicle

Inspired by the concept of energy internet and energy problems including greenhouse gas emission and regional shortage, this article will propose an idea of electric power distribution using city bus lines running electric vehicles. The designed power distribution system includes renewable energy sources as the energy input, power storage stations placed at some bus stops as the fixed energy output, and bus lines running electric vehicles as the connections between them. As the basic and priority problem, choosing the suitable location to place power storage stations can not only reduce the total number of construction but also increase the utilization rate of each one. Main work involves the two branch algorithms for solving the placement problem; simulation while using real-world transportation data of two city bus maps and analysis about the advantages and disadvantages of algorithms.

Optimizing the train timetable with consideration of different kinds of headway time

The train timetable is a crucial document for organization of railway traffic. A high-quality train timetable is not only implementable in operational environment, but also efficient in terms of capacity utilization of railway infrastructure. In this paper, we try to solve the train timetabling problem of a double track high-speed railway line with heavy traffic and trains with different operational speeds are operated. In order to obtain a feasible timetable with high efficiency, different kinds of headway time are distinguished in the process of train timetabling. With a time expanded graph describing train movements, an integer linear programming is formulated. A branch and price algorithm is proposed to solve the model. The case study shows that more trains will be scheduled when different kinds of headway time considered.

An Optimization Method of Passenger Assignment for Customized Bus

This study proposes an optimization method of passenger assignment on customized buses (CB). Our proposed method guarantees benefits to passengers by balancing the elements of travel time, waiting time, delay, and economic cost. The optimization problem was solved using a Branch and Bound (B&B) algorithm based on the shortest path for the selected stations. A simulation-based evaluation of the proposed optimization method was conducted. We find that a CB service can save 38.33% in average travel time, 38.33% in average waiting time, and 27.78% in delay penalties, all at an acceptable ticket price. The variations are caused by different degrees of busload. A detailed investigation revealed that a CB can significantly reduce travel time, waiting time, and delays, with the degree of benefits showing a decreasing trend with corresponding increases in busload. This finding indicates that CBs with smaller busloads can more flexibly transport passengers and provide a greater level of benefits. At the same time, a small busload strategy can be implemented more easily and in the near future on a test basis in urban China areas.