Branching Strategy Research Articles

Post-disaster recovery sequencing strategy for road networks

Natural disasters cause significant disruption in road networks, rendering many crucial links unusable. We investigate how to identify a sequence for repairing these links that minimizes total travel time over the repair horizon, given that at each repair stage road traffic distributes according to the principle of user equilibrium. We derive an analogue of Bellman’s optimality principle, allowing us to solve the problem using methods of dynamic programming. We specifically develop a bidirectional search heuristic with customized pruning and branching strategies that exploit specific properties of traffic assignment. Our experiments show that our method is scalable and performs well even on networks involving thousands of links.

A column generation approach for the driver scheduling problem with staff cars

Given a set of timetabled bus trips, transport companies are faced with the challenge of finding a feasible driver schedule that covers all trips and abides by various labor union regulations. The regulations are primarily concerned with providing sufficient breaks for the drivers during the day. Practical limitations in the city network enforce drivers to travel by cars between bus stops to have breaks. Transport companies have a limited number of cars, known as staff cars, which have to be returned to their respective depots at the end of the day. The simultaneous scheduling of drivers and staff cars for the drivers is known as the driver scheduling problem with staff cars (DSPSC). It is estimated that the DSPSC accounts for 60% of a bus company’s operational expense, and this paper proposes a column generation approach that attempts to minimize operational expense. The column-generation framework iterates between a master problem, a subproblem for generating driver variables and a subproblem for generating staff car variables. The subproblem related to the drivers is formulated as a resource constrained shortest path problem, which is solved by a dynamic programming approach. Several heuristic branching strategies are explored to find integer solutions. The proposed methodology is tested on eight real-life instances from seven Northern European bus companies. A comparison with a state-of-the-art mixed integer programming (MIP) solver and an adaptive large neighborhood search (ALNS) heuristic indicate that the column generation approach provides improved solutions for six instances and the average improvement is 1.45%.

Non-Model-Based Search Guidance for Set Partitioning Problems

We present a dynamic branching scheme for set partitioning problems. The idea is to trace features of the underlying MIP model and to base search decisions on the features of the current subproblem to be solved. We show how such a system can be trained efficiently by introducing minimal learning bias that traditional model-based machine learning approaches rely on. Experiments on a highly heterogeneous collection of set partitioning instances show significant gains over dynamic search guidance in Cplex as well as instance-specifically tuned pure search heuristics.

A new branch-and-filter exact algorithm for binary constraint satisfaction problems

A binary constraint satisfaction problem (BCSP) consists in determining an assignment of values to variables that is compatible with a set of constraints. The problem is called binary because the constraints involve only pairs of variables. The BCSP is a cornerstone problem in Constraint Programming (CP), appearing in a very wide range of real-world applications. In this work, we develop a new exact algorithm which effectively solves the BCSP by reformulating it as a k-clique problem on the underlying microstructure graph representation. Our new algorithm exploits the cutting-edge branching scheme of the state-of-the-art maximum clique algorithms combined with two filtering phases in which the domains of the variables are reduced. Our filtering phases are based on colouring techniques and on heuristically solving an associated boolean satisfiability (SAT) problem. In addition, the algorithm initialization phase performs a reordering of the microstructure graph vertices that produces an often easier reformulation to solve. We carry out an extensive computational campaign on a benchmark of almost 2000 instances, encompassing numerous real and synthetic problems from the literature. The performance of the new algorithm is compared against four SAT-based solvers and three general purpose CP solvers. Our tests reveal that the new algorithm significantly outperforms all the others in several classes of BCSP instances.

Smart economic development patterns in Europe: interaction with competitiveness

PurposeThis paper aims to investigate the smart economic development (SED) patterns in Europe in relation to competitiveness. Motivational focus corresponds to global events: the fourth industrial revolution, transition to a low-carbon economy, economic shocks (such as the 2008 financial crisis, Brexit or the coronavirus pandemic), which requires rethinking development policies, targeting competitiveness increase and reducing imbalances in economic development.Design/methodology/approachThe analysis includes self-organising neural networks cluster analysis and correlations, comparative analysis of SED indicators structure and cumulative index estimation with World Economic Forum (WEF) global competitiveness index. The panel data set of 19 years from 2000 to 2018 for 30 European countries.FindingsOverall, cross-country examination suggests that European countries of higher competitiveness illustrate higher estimates in SED. The key determinants are juridical fairness, social responsibility, competence building, intelligence and welfare employment to develop smart patterns for reaching higher competitiveness.Research limitations/implicationsThe limitations relate to the particular sample of European countries and gathering statistical data and a methodology of the SED index calculation. In addition, the paper contains a macroeconomic environment focus on competitiveness estimation. Further research may be improved with micro and mezzo environment incorporation at a cross-country analysis level.Practical implicationsBy linking well-known terms of competitiveness and economic development with a concept of smartness, new approaches to policymaking emerged. The methodology presented in this paper has implications for territorial cohesion policies, competitiveness and branching strategies. The combination of SED sub-indexes and WEF GCI might aid a more accurate ex ante measurement.Social implicationsThe findings are essential for fostering a smart approach in economic development for long-term competitiveness.Originality/valueThis paper provides original empirical evidence about the relationship between SED and competitiveness and adds new knowledge that smartness becomes a way for building countries’ competitiveness by identified two profiles of SED patterns by development stages, namely, integrated to economic development and institutional-based which is divided to focus and balanced.

Dynamic branching in qualitative constraint-based reasoning via counting local models

We introduce and evaluate dynamic branching strategies for solving Qualitative Constraint Networks (QCNs), which are networks for representing and reasoning about spatial and temporal information in a natural manner, e.g., a constraint can be “Task A is scheduled after or during Task C”. Specifically, we propose heuristics that dynamically associate a weight with a relation in the branching decisions that occur during backtracking search, based on the count of local models that the relation is involved with in a given QCN. Experimental results with a random and a structured dataset of QCNs of Interval Algebra show that it is possible to achieve up to 5 times better performance for structured instances, whilst maintaining non-negligible gains of around 20% for random ones. Finally, we show that these results may be notably improved via a selection protocol algorithm that synthesizes the involved heuristics into an overall better performing meta-heuristic in the phase transition.

A stand-alone branch-and-price algorithm for identical parallel machine scheduling with conflicts

We consider a scheduling problem where a set of n jobs has to be processed in a non-preemptive way on a set of m identical parallel machines. Each job j is associated with a positive integer processing time pj. The problem is also characterized by a conflict graph where adjacent nodes in the graph represent conflicting jobs that cannot be processed on the same machine. A schedule is an assignment of a time interval of pj time units on m machines for each job j. The schedule is feasible if intervals on the same machine do not overlap, and jobs to be processed on the same machine are pairwise not conflicting. The aim is to find a feasible schedule that minimizes the maximum completion time of the jobs. The problem is NP-hard as it generalizes both the problem of scheduling jobs on identical parallel machines to the aim of minimizing the makespan (P||Cmax) and the vertex coloring problem (VCP), two well-known NP-hard problems. We present the first stand-alone branch-and-price (BP) algorithm that works directly on the problem. In comprehensive computational experiments with benchmark instances, we prove that the new BP algorithm, even without using ad hoc primal heuristics and feasibility check algorithms, is competitive with the best exact solution algorithm proposed so far in the literature. These results are mainly achieved thanks to the branching scheme we propose.

Computational study of a branching algorithm for the maximum [formula omitted]-cut problem

This work considers the graph partitioning problem known as maximum k-cut. It focuses on investigating features of a branch-and-bound method to obtain global solutions. An exhaustive experimental study is carried out for the two main components of a branch-and-bound algorithm: Computing bounds and branching strategies. In particular, we propose the use of a variable neighborhood search metaheuristic to compute good feasible solutions, the k-chotomic strategy to split the problem, and a branching rule based on edge weights to select variables. Moreover, we analyze a linear relaxation strengthened by semidefinite-based constraints, a cutting plane algorithm, and node selection strategies. Computational results show that the resulting method outperforms the state-of-the-art approach and discovers the solution of several instances, especially for problems with k≥5.

The minimum area spanning tree problem: Formulations, Benders decomposition and branch-and-cut algorithms

The Minimum Area Spanning Tree Problem (MASTP) is defined in terms of a complete undirected graph G, where every vertex represents a point in the two dimensional Euclidean plane. Associated to each edge, there is a disk placed right at its midpoint, with diameter matching the length of the edge. MASTP seeks a spanning tree of G that minimizes the area in the union of the disks associated to its edges. This paper presents Integer Programming (IP) approaches for MASTP, introduces pre-processing techniques to reduce the size of the formulations, and characterizes valid inequalities for reinforcing its Linear Programming Relaxation bounds. Several Branch-and-cut (BC) algorithms exploiting such ideas are introduced. Additionally, we also apply Benders Decomposition to one of these formulations. An accompanying BC method, that separates Benders optimality cuts, is also introduced. Aiming to save linear programming re-optimization times, that algorithm makes use of an early branching strategy. Given the set of valid inequalities used in the polyhedral representation of the problem and the best available upper bound for the optimal cost, it detects if the node cannot be pruned by bounds and then stops the cutting plane generation, in order to branch. Our algorithms manage to solve instances with up to only 15 vertices, suggesting thus that MASTP is hard to solve in practice, at least with the currently available methods. Thanks to the early branching strategy, the Benders based BC method obtained the best computational results, by far. It solved more instances to proven optimality than the other algorithms (31 out of 45 cases) and it is about three times faster than the second best performing algorithm.

SALSA: Combining branch-and-bound with dynamic programming to smoothen workloads in simple assembly line balancing

We consider a version of the well-known simple assembly line balancing problem (called SALBP-SX) where, given the cycle time and the number of stations, the workloads of the stations are to be leveled according to an adequately defined smoothness index SX. Our index SX involves for each station the quadratic deviation of its workload from the average (or ideal) workload and is therefore closely related to the variance, which is a common measure of dispersion in statistics. Contrary to the existing literature on workload smoothing in ALB, which often treats the optimization of a prespecified smoothness index as a secondary objective, we consider our SX-objective as the single one in order to account for the practical relevance of fair workload distributions and avoiding overloaded bottleneck stations.To optimally solve SALBP-SX, we develop a tailored branch-and-bound procedure. It contains a new station-oriented branching scheme, new lower bound arguments, logical tests and, in particular, a dynamic programming scheme for the pre-calculation of potential workloads, which accelerates the procedure greatly. In comprehensive computational experiments, we show that our method clearly outperforms a class of recently developed task-oriented branch-and-bound procedures and also the mathematical programming solver Gurobi.

The determinants of bank branch location in India: an empirical investigation

PurposeBank branching plays a significant role in a wide range of economic activities. Existing studies on determinants of bank branching activities largely focus on developed countries; studies devoted to developing countries are scant. The purpose of this paper is to examine the determinants of bank branching activities in one of the largest developing country India.Design/methodology/approachThe authors employ a unique longitudinal data to study the determinants of bank branch location in India. These data are collected at the state level covering 25 Indian states for the period 2006–2017. The authors employ Poisson regression that are better suited for modeling counted dependent variable.FindingsFirst, region and bank specific factors such as size of population and bank deposits influence location of bank branches. Second, the relationship between these factors and branch locations is heterogeneous across different types of banks and across states with different business environments.Practical implicationsFirst, from the view of banks, considering the factors of branch location are crucial in order to set out branching strategy. Irrespective of policy measures aimed at promoting financial inclusion in India, the authors show that banks consider economic activities in the region in locating their branches. Second, from the view of policy makers and regulators, such branching strategy could potentially contribute to financial exclusion. As a result, population in the less developed regions may be excluded from accessing financial services. Hence, policy makers and regulators should take into this account when formulating policies aimed at promoting financial inclusion.Originality/valueFirst, while existing studies largely focus on developed countries, studies devoted to developing countries are scant. To the best of our knowledge, the authors have not come across any study that investigates the determinants of bank branch location in India, so the authors reasonably believe that this study is a first-of-its-kind. Second, the study provides a new perspective concerning how regional and bank specific factors influence banks of different ownership in locating branches. Third, while traditional regression used to be a method of choice among early studies, the authors employ Poisson regression that is better suited for modeling counted dependent variable.

Benchmarking the Dual and Compound Techniques-Based Branching Design Strategy Used for Upgrading of Pressurized Hydraulic Systems

Abstract Prior research has recognized that the compound- and dual-technique-based branching redesign measures, used as alternatives to the conventional technique-based one, were effective in upgrading steel pipe-based pressurized hydraulic systems. Principally, the compound technique used two different plastic material types for the short-penstock instead of the single material type utilized in the conventional technique. However, the dual technique is based on splitting the single penstock installed in the conventional technique into a set of dual subpenstocks placed at each connection of the main-piping system to hydraulic parts. This handling aimed at improving the conventional technique efficiency with regard to the tradeoff between the magnitude attenuation and period expansion effects of the transient pressure-wave signal. Accordingly, this study proposed a comprehensive comparison between the compound- and dual-technique-based branching strategy with particular focus on the tradeoff between the two last parameters. The plastic material types demonstrated in this study included the high- or low-density polyethylene. The application addressed a waterhammer maneuver initiated into a reservoir-steel-pipe-valve system. Numerical computations used the method of characteristics for the discretization of the 1D extended pressurized-pipe flow model, embedding the Kelvin–Voigt and Vitkovsky formulations. The finding of this study suggested that the high- or low-density polyethylene (HDPE–LDPE) setup of the compound technique is the most prominent protected system setup, providing an acceptable tradeoff between the attenuation of magnitude and the expansion of the period of pressure-wave oscillation.

An efficient and accurate Mixed‐integer linear programming model for long‐term operations of large‐scale hydropower systems

Mixed-integer linear programming is widely used in hydropower reservoirs operation optimisation because of its modelling flexibility, solution stability, and global search capability. A major obstacle to applying mixed-integer linear programming to hydropower reservoirs operation optimisation, especially for long-term operations of large-scale hydropower systems, is the non-linear phenomenon while converting water energy into electricity, which is normally expressed as a non-linear bivariate function, that is, hydropower production function. To cross this technological barrier, a novel linearisation method for hydropower production function is proposed. The method uses rectangular meshing techniques to approximate hydropower production function. Then Special Ordered Sets of type 2 (SOS2)constraints are adopted for sub-rectangle selection, which are formulated based on the binary branching schemes generated by binary reflected grey code. The proposed method can linearise the hydropower production function with only logarithmic-sized binary variables of the traditional methods and maintain high accuracy. The method is then applied to optimising the operation of the Lancang River Basin hydropower system, which has 13 head-dependent reservoirs with a deterministic model and a two-stage stochastic model. The results show that the authors' method significantly outperforms conventional linearisation methods and can obtain accurate solutions within an acceptable time for deterministic and stochastic large-scale hydropower reservoirs operation optimisation problems.

Constraint programming approach for multi-resource-constrained unrelated parallel machine scheduling problem with sequence-dependent setup times

This paper studies the multi-resource-constrained unrelated parallel machine scheduling problem under various operational constraints with the objective of minimising maximum completion time among the scheduled jobs. Sequence-dependent setup times, precedence relations, machine eligibility restrictions and release dates are incorporated into the problem as operational constraints to reflect real-world manufacturing environments. The considered problem is in NP-hard class of problems, which cannot be solved in deterministic polynomial time. Our aim in this study is to develop an exact solution approach based on constraint programming (CP), which shows good performance in solving scheduling problems. In this regard, we propose a CP model and enrich this model by adding lower bound restrictions and redundant constraints. Moreover, to achieve a reduction in computation time, we propose two branching strategies for the proposed CP model. The performance of the CP model is tested using randomly generated and benchmark instances from the literature. The computational results indicate that the proposed CP model outperforms the best solutions with an average gap of 15.52%.

Multivariable Branching: A 0-1 Knapsack Problem Case Study

We explore the benefits of multivariable branching schemes for linear-programming-based branch-and-bound algorithms for the 0-1 knapsack problem—that is, the benefits of branching on sets of variables rather than on a single variable (the current default in integer-programming solvers). We present examples where multivariable branching has advantages over single-variable branching and partially characterize situations in which this happens. Chvátal shows that for a specific class of 0-1 knapsack instances, a linear-programming-based branch-and-bound algorithm (employing a single-variable branching scheme) must explore exponentially many nodes. We show that for this class of 0-1 knapsack instances, a linear-programming-based branch-and-bound algorithm employing an appropriately chosen multivariable branching scheme explores either three or seven nodes. Finally, we investigate the performance of various multivariable branching schemes for 0-1 knapsack instances computationally and demonstrate their potential; the multivariable branching schemes explored result in smaller search trees (some in search trees that are an order of magnitude smaller), and some also result in shorter solution times.Summary of Contribution: As a powerful modeling tool, mixed-integer programming (MIP) is ubiquitous in Operations Research and is usually solved via the branch-and-bound framework. However, solving MIPs is computationally challenging in general, where branching affects the performance of solvers dramatically. In this paper, we explore the benefits of branching on multiple variables, which can be viewed as a generalization of the standard single-variable branching. We analyze its theoretical behavior on a special instance introduced by Chvátal, which is proved to be hard for single-variable branching. We also partially characterize situations in which branching on multiple variables is superior to its single-variable counterpart. Lastly, we demonstrate its potential in reducing the overall computational time and possible memory usage for storing unexplored nodes through numerical experiments on 0-1 knapsack problems.

Effects of grazing intensities on functional traits of Cleistogenes squarrosa in a typical grassland of Inner Mongolia, China.

We examined the response of 11 functional traits of Cleistogenes squarrosa, including plant height and clump width, to the grazing of different intensity (no grazing; twice grazing in May and July; five grazing in the whole growing season). After five-year treatments, the number of reproductive branches, plant height, stem quality, specific leaf area and total leaf area were significantly reduced. Such reduction was significantly increased with the increases of grazing intensity. Under the treatments of grazing in May and July, the declines of each index were 61.7%, 21.5%, 33.3%, 21.6% and 26.7%, respectively; and the declines in the treatment of whole growing season grazing were 75.7%, 24.7%, 46.7%, 28.5% and 43.7%, respectively. The number of vegetative branches, number of leaf, total leaf area, stem quality, total leaf quality and clump width of the C. squarrosa had a synergistic relationship with aboveground biomass of the whole clump under different grazing intensities. The number of vegetative branches was significantly positively correlated with stem quality and total leaf quality. C. squarrosa had a dwarf-dense branching strategy to avoid grazing. The reduction of grazing intensity would be conducive to maintaining the stability of functional traits.

A machine learning-based branch and price algorithm for a sampled vehicle routing problem

Planning of operations, such as routing of vehicles, is often performed repetitively in rea-world settings, either by humans or algorithms solving mathematical problems. While humans build experience over multiple executions of such planning tasks and are able to recognize common patterns in different problem instances, classical optimization algorithms solve every instance independently. Machine learning (ML) can be seen as a computational counterpart to the human ability to recognize patterns based on experience. We consider variants of the classical Vehicle Routing Problem with Time Windows and Capacitated Vehicle Routing Problem, which are based on the assumption that problem instances follow specific common patterns. For this problem, we propose a ML-based branch and price framework which explicitly utilizes those patterns. In this context, the ML models are used in two ways: (a) to predict the value of binary decision variables in the optimal solution and (b) to predict branching scores for fractional variables based on full strong branching. The prediction of decision variables is then integrated in a node selection policy, while a predicted branching score is used within a variable selection policy. These ML-based approaches for node and variable selection are integrated in a reliability-based branching algorithm that assesses their quality and allows for replacing ML approaches by other (classical) better performing approaches at the level of specific variables in each specific instance. Computational results show that our algorithms outperform benchmark branching strategies. Further, we demonstrate that our approach is robust with respect to small changes in instance sizes.

A Branch & Cut algorithm for the prize-collecting capacitated location routing problem

The Capacitated location routing problem (CLRP) is the combination of two well-studied problems in Operations Research: the capacitated facility location problem (CFLP) and the multiple depots vehicle routing problem (MDVRP). Given a set of available locations and a fleet of vehicles, the aim is to determine a set of depots to open and routes of the vehicles to satisfy the customers demands. The objective of the CLRP is to minimize the total cost, that is the cost of the opened depots, the fixed cost of the vehicles and the cost of the routes while satisfying vehicle and depot capacity constraints. In this work the prize-collecting capacitated location routing problem (PC-CLRP), a new variant of the CLRP is presented. In this case it is possible to leave some customers unvisited and if a customer is visited it gives a gain. The objective is to maximize the overall benefit. A two-index formulation for the PC-CLRP and a Branch & Cut algorithm based on this model are proposed. Valid inequalities for the CLRP are adapted for the PC-CLRP. Also new valid inequalities and optimality cuts are proposed together with their corresponding separation algorithms. A hierarchical branching strategy is also included. The initial solution was provided by an Ant Colony Algorithm. The algorithm is tested on a set of instances specially designed for the new problem. Computational results showed very promising. We also compare the performance of the algorithm with recent work for CLRP on instances from the literature.

On tackling reverse convex constraints for non-overlapping of unequal circles

We study the unequal circle-circle non-overlapping constraints, a form of reverse convex constraints that often arise in optimization models for cutting and packing applications. The feasible region induced by the intersection of circle-circle non-overlapping constraints is highly non-convex, and standard approaches to construct convex relaxations for spatial branch-and-bound global optimization of such models typically yield unsatisfactory loose relaxations. Consequently, solving such non-convex models to guaranteed optimality remains extremely challenging even for the state-of-the-art codes. In this paper, we apply a purpose-built branching scheme on non-overlapping constraints and utilize strengthened intersection cuts and various feasibility-based tightening techniques to further tighten the model relaxation. We embed these techniques into a branch-and-bound code and test them on two variants of circle packing problems. Our computational studies on a suite of 75 benchmark instances yielded, for the first time in the open literature, a total of 54 provably optimal solutions, and it was demonstrated to be competitive over the use of the state-of-the-art general-purpose global optimization solvers.

A branch-and-bound approach for a Vehicle Routing Problem with Customer Costs

• A hard Vehicle Routing Problem with additional time-dependent customer costs. • A new partition and permutation model that allows a direct calculation of execution times. • Two appropriate branch-and-bound schemes based on the partition and permutation model. • New lower bounds for the time-dependent customer costs and application in branch-and-bound schemes. • Numerical comparison of lower bounds and branch-and-bound schemes. An important aspect in railway maintenance management is the scheduling of tamping actions in which two aspects need to be considered: first, the reduction of travel costs for crews and machinery; and second, the reduction of time-dependent costs caused by bad track condition. We model the corresponding planning problem as a Vehicle Routing Problem with additional customer costs. Due to the particular objective function, this kind of Vehicle Routing Problem is harder to solve with conventional methods. Therefore, we develop a branch-and-bound approach based on a partition and permutation model. We present two branching strategies, the first appends one job at the end of a route in each branching step and the second includes one job inside a route in each branching step; and analyze their pros and cons. Furthermore, different lower bounds for the customer costs and the travel costs are defined and compared. The performance of the branch-and-bound method is analyzed and compared with a commercial solver.