Branch Algorithm Research Articles

Designing a resilient supply chain network: A multi-objective data-driven distributionally robust optimization method

Uncertainty of supply chain disruption poses a significant challenge to decision-making in many real-life problems. In this paper, a new multi-objective data-driven distributionally robust optimization (MDDRO) model for the resilient supply chain network (RSCN) design problem under disruption scenario is developed, which includes minimizing the total cost, carbon emissions, and delivery time. In the context of supply chain disruption, it is important self-evidently that products can be supplied normally and the practical demand can be met. In handling the partial probability distribution information of the considered uncertain demand, this paper uses a data-driven Wasserstein-Moment ambiguity set (WMAS), which incorporate the Wasserstein metric and Moment information, and a robust counterpart to transform the developed model into a tractable approximation form. The finite-sample performance guarantee of the optimal solution of MDDRO model with Wasserstein-Moment ambiguous chance constraint is given. An accelerated Branch and cut algorithm (BCA) is constructed to solve the MDDRO model. Finally, through the investigation of a real-life case and sensitivity analysis of key parameters, some managerial insights are put forward.

A branch-and-price approach for the nurse rostering problem with multiple units

In this paper, we study the nurse rostering problem that considers multiple units and many soft time-related constraints. An efficient branch and price solution approach that relies on a fast algorithm to solve the pricing subproblem of the column generation process is presented. For the nurse rostering problem, its pricing subproblem can be formulated as a shortest path problem with resource constraints, which has been the backbone of several solutions for several classical problems like vehicle routing problems. However, approaches that perform well on these problems cannot be used since most constraints in the nurse rostering problem are soft. Based on ideas borrowed from global constraints in constraint programming to model rostering problems, an efficient dynamic programming algorithm with novel label definitions and dominating rules specific to soft time-related constraints is proposed. In addition, several acceleration strategies are employed to improve the branch and price algorithm. Computational results on two sets of instances ranging from 2 to 4 units over a horizon of 2 or 4 weeks demonstrate the effectiveness of the proposed algorithms. The accelerated dynamic programming algorithm is able to solve pricing subproblems to optimality with the fewest extended labels. When contrasted with Cplex, the branch and price approach yields solutions that are either equivalent or superior across all instances.

Quadratic p-Median Problem: A Bender’s Decomposition and a Meta-Heuristic Local-Based Approach

In this paper, the quadratic p-median optimization problem is discussed, where the goal is to connect users to a selected group of facilities (emergency services, telecommunications servers, healthcare facilities) at the lowest possible cost. The problem is aimed at minimizing the cost of connecting these selected facilities. The costs are symmetric, meaning connecting two different points is the same in both directions. This problem extends the traditional p-median problem, a combinatorial problem used in various fields like facility location, network design, transportation, supply chain networks, emergency services, healthcare, and education planning. Surprisingly, the quadratic version has not been thoroughly considered in the literature. The paper highlights the formulation of two mixed-integer quadratic programming models to find optimal solutions to this problem. One model is a classic formulation, and the other is based on set cover theory. Linear versions and Bender’s decomposition formulations for each model are also derived. A Bender’s decomposition is solved using an algorithm that adds constraints during each iteration to improve the solution. Lazy constraints in the Gurobi solver’s branch and cut algorithm are dynamically added whenever a mixed-integer programming solution is found. Additionally, an efficient local search meta-heuristic is proposed that usually finds optimal solutions for tested instances. Challenging instances with up to 60 facilities and 2000 users are successfully solved. Our results show that Bender’s models with lazy constraints are the most effective for Euclidean and random test cases, achieving optimal solutions in less CPU time. The meta-heuristic also finds near-optimal solutions rapidly for these cases.

A branch and cut algorithm to optimize a weighted sum-of-ratios in multiobjective mixed-integer fractional programming

AbstractMultiobjective linear fractional programming is useful to model multiobjective problems where all or some of the objective functions are a ratio or proportion of one linear/affine function to another linear/affine function. In practice, many of such problems include integer variables. If the weighted-sum scalarization is used to compute efficient solutions to the multiobjective problem, then the scalar problem to be solved for each weight vector turns out to be a weighted sum-of-ratios. There are several algorithms reported in the literature to optimize weighted sum-of-ratios, but almost all of them cannot deal with integer variables. In this paper we propose a Branch & Cut algorithm to optimize weighted-sums of the objective functions in multiobjective mixed integer fractional programming (MOMIFP). Several theoretical properties that support the algorithm are presented and proved. Computational experiments with randomly generated general problems are presented and discussed, which show that the algorithm is able to deal with practical MOMIFP problems.

A mathematical programming approach to export pathway planning of distributed hydrogen production in offshore wind farm

Hydrogen production is a vital development trend to improve the economics and penetration rate of offshore wind farms (OWFs), and distributed hydrogen production (DHP) is a preferred solution for deep and far sea OWFs without expensive submarine cables and offshore substations. The export pathway of OWF-DHP consists of collection pipelines, transmission pipelines, and offshore compressor stations. In this paper, mathematical programming is introduced to explore the export pathway planning problem of OWF-DHP. Firstly, the area of OWF is discretized into multiple grids with the center of the grids being the candidate locations of the offshore compressor station. Then, a binary integer quadratic programming problem is established to optimize both the offshore compressor station location and pipeline construction with different topologies. Further, the mathematical model is solved by the branch and cut algorithm integrated with the proposed dimension reduction methods. Finally, the effectiveness of the proposed export pathway planning approach is verified by the actual data of Baltic Eagle OWF in Germany, which would support the construction of the OWF-DHP project.

An exact algorithm for simultaneous pickup and delivery problem with split demand and time windows

This study introduces a new variant of the vehicle routing problem (VRP) called the simultaneous pickup and delivery problem with split demand and time windows (SPDP-SDTW). The motivation behind this study stems from real-life urban and rural delivery scenarios, encompassing features such as split demand, simultaneous pickup and delivery, many-to-many pickup and delivery, and time windows. The study thoroughly investigates the properties of the optimal solution for the SPDP-SDTW. Based on these properties, an arc flow model is developed for the SPDP-SDTW. Dantzig Wolfe (DW) decomposition techniques are employed to obtain the master problem and the pricing subproblem. In order to effectively address the SPDP-SDTW, an improved branch and price (I-BP) algorithm is proposed, incorporating a tailored column generation (CG) algorithm, branching strategies, and dual stabilization strategies. The proposed CG algorithm provides a framework that combines the improved adaptive degree heuristic (I-AGH) algorithm and the solver Gurobi. This integration substantially mitigates the computational burden involved in solving the subproblem. Extensive computational experiments conducted on datasets of varying sizes, including small, medium, and large instances, consistently demonstrate that the I-BP algorithm performs the best in both solution quality and computational efficiency when compared to existing exact and heuristic algorithms.

Mapping and quantifying unique branching structures in lentil (Lens culinaris Medik.)

BackgroundLentil (Lens culinaris Medik.) is a globally-significant agricultural crop used to feed millions of people. Lentils have been cultivated in the Australian states of Victoria and South Australia for several decades, but efforts are now being made to expand their cultivation into Western Australia and New South Wales. Plant architecture plays a pivotal role in adaptation, leading to improved and stable yields especially in new expansion regions. Image-based high-throughput phenomics technologies provide opportunities for an improved understanding of plant development, architecture, and trait genetics. This paper describes a novel method for mapping and quantifying individual branch structures on immature glasshouse-grown lentil plants grown using a LemnaTec Scanalyser 3D high-throughput phenomics platform, which collected side-view RGB images at regular intervals under controlled photographic conditions throughout the experiment. A queue and distance-based algorithm that analysed morphological skeletons generated from images of lentil plants was developed in Python. This code was incorporated into an image analysis pipeline using open-source software (PlantCV) to measure the number, angle, and length of individual branches on lentil plants.ResultsBranching structures could be accurately identified and quantified in immature plants, which is sufficient for calculating early vigour traits, however the accuracy declined as the plants matured. Absolute accuracy for branch counts was 77.9% for plants at 22 days after sowing (DAS), 57.9% at 29 DAS and 51.9% at 36 DAS. Allowing for an error of ± 1 branch, the associated accuracies for the same time periods were 97.6%, 90.8% and 79.2% respectively. Occlusion in more mature plants made the mapping of branches less accurate, but the information collected could still be useful for trait estimation. For branch length calculations, the amount of variance explained by linear mixed-effects models was 82% for geodesic length and 87% for Euclidean branch lengths. Within these models, both the mean geodesic and Euclidean distance measurements of branches were found to be significantly affected by genotype, DAS and their interaction. Two informative metrices were derived from the calculations of branch angle; ‘splay’ is a measure of how far a branch angle deviates from being fully upright whilst ‘angle-difference’ is the difference between the smallest and largest recorded branch angle on each plant. The amount of variance explained by linear mixed-effects models was 38% for splay and 50% for angle difference. These lower R2 values are likely due to the inherent difficulties in measuring these parameters, nevertheless both splay and angle difference were found to be significantly affected by cultivar, DAS and their interaction. When 276 diverse lentil genotypes with varying degrees of salt tolerance were grown in a glasshouse-based experiment where a portion were subjected to a salt treatment, the branching algorithm was able to distinguish between salt-treated and untreated lentil lines based on differences in branch counts. Likewise, the mean geodesic and Euclidean distance measurements of branches were both found to be significantly affected by cultivar, DAS and salt treatment. The amount of variance explained by the linear mixed-effects models was 57.8% for geodesic branch length and 46.5% for Euclidean branch length.ConclusionThe methodology enabled the accurate quantification of the number, angle, and length of individual branches on glasshouse-grown lentil plants. This methodology could be applied to other dicotyledonous species.

Current clinical practice and needs assessment in inherited eye diseases from the perspective of ophthalmologists

ABSTRACT Background The clinical approach to inherited eye diseases has evolved due to advances in genetic testing methods and treatment opportunities. However, no data are available on the current practices of ophthalmologists in countries, such as Turkey, with higher rates of consanguinity and inherited eye diseases. The aim of this study was to evaluate the current practices, knowledge, and needs of ophthalmologists in Turkey regarding inherited eye diseases. Methods A 29-item self-administered survey with a branching algorithm was developed through Google Forms. The survey link was sent to 2983 ophthalmologists in Turkey. The survey assessed respondents’ occupational characteristics, current practices, knowledge about available diagnostic and therapeutic options, and opinions on improving continuing education and healthcare services. Results Responses from 414 ophthalmologists (20.8%) were analyzed. The responses suggested that ophthalmologists mainly collaborate with medical geneticists in respect of inherited eye diseases. The majority of ophthalmologists reported a lack of knowledge about genetic diagnostic tests, and approximately 90% of the ophthalmologists thought training after residency was inadequate for inherited eye diseases. Conclusion This is the most extensive survey exploring ophthalmologists’ practice patterns and needs in a setting without specialists or specialized centers in ophthalmic genetics. The results emphasize the need for continued education on updated approaches to inherited eye diseases.

The cage effect of electron beam irradiation damage in cryo-electron microscopy

Electron beam irradiation can cause damage to biological and organic samples, as determined via transmission electron microscopy (TEM). Cryo-electron microscopy (cryo-EM) significantly reduces such damage by quickly freezing the environmental water around organic molecules. However, there are multiple hypotheses about the mechanism of cryo-protection in cryo-EM. A lower temperature can cause less molecular dissociation in the first stage, or frozen water can have a “cage” effect by preventing the dissociated fragments from flying away. In this work, we use real-time time-dependent density functional theory molecular dynamics(rt-TDDFT-MD) simulations to study the related dynamics. We use our recently developed natural orbital branching (NOB) algorithm to describe the molecular dissociation process after the molecule is ionized. We find that despite the difference in surrounding water molecules at different temperatures, the initial dissociation process is similar. On the other hand, the dissociated fragments fly away at room temperature, while they remain in the same cage when frozen water is used. Our results provide direct support for the cage effect mechanism.

Minimizing the total travel distance for the locker-based drone delivery: A branch-and-cut-based method

This article investigates the problem of minimizing the total travel distance in locker-based drone delivery, where the roofs of lockers are reused as parking platforms for drones. It is a drone routing and parking hybrid problem with modeling challenges. We find the sufficient and necessary conditions for feasible solutions and transform the original problem into a scale-tractable one. Subsequently, we propose a compact lower-bound formulation for the transformed problem and prove the total unimodality of the coefficient matrix. Furthermore, we develop a two-stage method in which a branch and cut algorithm solves the transformed problem and a heuristic constructs practical schedules for the original problem. Simulated tests demonstrate that the method can solve each simulated instance within one second. Random tests reveal that the method can efficiently solve instances with 1000 sites and 1500 tasks within an acceptable CPU time. A sensitivity analysis indicates that the complexity arising from routing flexibility is greater than that arising from parking flexibility.

Innovative supply chain network design with two-step authentication and environmentally-friendly blockchain technology

AbstractBlockchain Technology (BT) has the potential to revolutionize supply chain management by providing transparency, but it also poses significant environmental and security challenges. BT consumes energy and emits carbon gases, affecting its adoption in Supply Chains (SCs). The substantial energy demand of blockchain networks contributes to carbon emissions and sustainability risks. Moreover, for secure and reliable transactions, mutual authentication needs to be established to address security concerns raised by SC managers. This paper proposes a tri-objective optimization model for the simultaneous design of the SC-BT network, considering a two-step authentication process. The model considers transparency caused by BT members, emissions of BT, and costs related to BT and SC design. It also takes into account uncertainty conditions for participating BT members in the SC and the range of transparency, cost, and emission targets. To solve the model, a Branch and Efficiency (B&E) algorithm equipped with BT-related criteria is developed. The algorithm is implemented in a three-level SC and produces cost-effective and environmentally friendly outcomes. However, the adoption of BT in the SC can be costly and harmful to the environment under uncertain conditions. It is worth mentioning that implementing the proposed algorithm from our article in a three-level SC case study can result in a significant cost reduction of over 16% and an emission reduction of over 13%. The iterative nature of this algorithm plays a vital role in achieving these positive outcomes.

Identifying and minimizing critical driving range thresholds for electric vehicles in intercity networks

It is well known that range anxiety faced by electric vehicle (EV) drivers nowadays can be effectively mitigated by either increasing the driving range of vehicles or increasing the distribution density of charging stations. To enact an effective research and development plan, vehicle manufacturers should understand well the relationship between the EV adoption and usage rates and the performance level of battery and energy management technologies. On the other hand, the underlying relationship between EV adoption and usage rates and the number and distribution of charging stations is also critical for charging infrastructure investors in determining their construction plans. This paper is dedicated to analytically identifying two critical driving range thresholds and further reducing these driving range thresholds as much as possible by optimally deploying charging station locations in intercity highway networks. These critical thresholds represent the turning points of individual routing convenience conditions or satisfaction levels for long-distance trips, implying that the inconvenient travel restrictions may be largely overcome if the driving range of EVs goes beyond these thresholds. For this purpose, we develop two integer programming models and dynamic programming algorithms for the driving range threshold identification problems, and extend the modeling framework to accommodate two relevant driving range threshold minimization problems as well, which are solved by a branching algorithm encapsulating the aforementioned dynamic programming procedures. The output of this research offers a set of new analytical tools for EV manufacturers to make strategic decisions on extending the driving range of their products and for charging infrastructure investors to make planning decisions on selecting charging station locations.

A branch, bound and remember algorithm for maximizing the production rate in the simple assembly line balancing problem

The simple assembly line balancing problem is a classical optimization problem. It aims at efficiently assigning elementary operations to the workstations of a paced assembly line to optimize some efficiency measure. In this paper, we deal with the type-2 version of the problem in which the efficiency measure is the maximization of throughput with a given number of workstations. This problem has been scarcely studied compared to its type-1 counterpart, in which the throughput is fixed and the objective is to minimize the number of workstations. This has led to the development of indirect methods using type-1 exact solution methods to tackle the type-2 case.In this paper, we provide lower bounds, logical tests, and dominance rules for the type-2 problem and adapt the state-of-the-art method for the type-1 problem to develop a direct approach for solving the type-2 problem with a direct approach. Computational results show the effectiveness of the direct method, which can outperform recent type-1-based indirect approaches. The proposed method is able to close all instances from the classical instance set and solves to optimality over 90% of the large (100 tasks) and 76.5% of the very large (1000 tasks) instances derived from the current reference set for its type-1 counterpart. The average gaps between the upper and lower bounds on the optimal cycle time are also tight and are less than 0.13% (0.4%) for large (very large) instances. Finally, our results show that direct and indirect methods are comparable in terms of solution quality, but the direct method outperforms its indirect counterpart in terms of running time and optimality gaps. Moreover, we identify that instances averaging two to three tasks per workstation and featuring large variability between task times pose significant challenges to solve to optimality.

An impossible combinatorial counting method in distance geometry

The Distance Geometry Problem asks for a geometric representation of a given weighted graph in RK so that vertices are points and edges are segments with lengths equal to the corresponding weights. Two problem variants are defined by a vertex order given as part of the input, which allows the use of a branching algorithm based on K-lateration: find two possible positions for the next vertex j using the positions of K predecessors and their distances to j, then explore each position recursively, pruning positions at need. Whereas the first variant only requires the K predecessors to exist, the second variant also requires them to be contiguous and immediately preceding j. For this variant, fixed-parameter tractability of the algorithm can be established by means of a solution counting method that only depends on the graph edges (rather than their weights). Only in the first variant, however, it is possible to efficiently construct a suitable vertex order directly from the graph. Since both fixed-parameter tractability and efficient vertex order construction are desirable properties, one would need an analogous solution counting method for the first variant. In this paper we prove that such a counting method cannot be devised for the first variant.

Demand-adapted train timetabling with coupling-decoupling operations on a bidirectional intercity railway line

Many factors need to be considered when designing the timetable of intercity railway lines, where the passenger demand is variable and unbalanced along time in both directions of the line. This paper introduces a new multi-objective nonlinear integrated optimization model for designing full-day train timetables incorporating coupling-decoupling operations and rolling stock assignment on a bidirectional railway line. The model explicitly considers train capacity constraints and oversaturation while minimizing the total passenger waiting time and operating costs. To solve the problem, a hybrid algorithm governed by a genetic algorithm is proposed. Before evaluating the fitness of each individual, to complete the definition of each chromosome, a truncated branch and cut algorithm is used to solve a capacity and train unit allocation subproblem. A good-quality initial population is partially generated using the spatiotemporal synchronous coupling algorithm. The proposed approach is applied to a real case using the Shanghai-Hangzhou intercity railway line. The results of different experiments show the effectiveness of the model and algorithm, both in terms of passenger convenience and cost savings.

An FPT Algorithm for Directed Co-Graph Edge Deletion

In the directed co-graph edge-deletion problem, we are given a directed graph and an integer k, and the question is whether we can delete, at most, k edges so that the resulting graph is a directed co-graph. In this paper, we make two minor contributions. Firstly, we show that the problem is NP-hard. Then, we show that directed co-graphs are fully characterized by eight forbidden structures, each having, at most, six edges. Based on the symmetry properties and several refined observations, we develop a branching algorithm with a running time of O(2.733k), which is significantly more efficient compared to the brute-force algorithm, which has a running time of O(6k).

Chance-constrained stochastic assembly line balancing with branch, bound and remember algorithm

Assembly lines are widely used mass production techniques applied in various industries from electronics to automotive and aerospace. A branch, bound, and remember (BBR) algorithm is presented in this research to tackle the chance-constrained stochastic assembly line balancing problem (ALBP). In this problem variation, the processing times are stochastic, while the cycle time must be respected for a given probability. The proposed BBR method stores all the searched partial solutions in memory and utilizes the cyclic best-first search strategy to quickly achieve high-quality complete solutions. Meanwhile, this study also develops several new lower bounds and dominance rules by taking the stochastic task times into account. To evaluate the performance of the developed method, a large set of 1614 instances is generated and solved. The performance of the BBR algorithm is compared with two mixed-integer programming models and twenty re-implemented heuristics and metaheuristics, including the well-known genetic algorithm, ant colony optimization algorithm and simulated annealing algorithm. The comparative study demonstrates that the mathematical models cannot achieve high-quality solutions when solving large-size instances, for which the BBR algorithm shows clear superiority over the mathematical models. The developed BBR outperforms all the compared heuristic and metaheuristic methods and is the new state-of-the-art methodology for the stochastic ALBP.

A deep branching solver for fully nonlinear partial differential equations

We present a multidimensional deep learning implementation of a stochastic branching algorithm for the numerical solution of fully nonlinear PDEs. This approach is designed to tackle functional nonlinearities involving gradient terms of any orders, by combining the use of neural networks with a Monte Carlo branching algorithm. In comparison with other deep learning PDE solvers, it also allows us to check the consistency of the learned neural network function. Numerical experiments presented show that this algorithm can outperform deep learning approaches based on backward stochastic differential equations or the Galerkin method, and provide solution estimates that are not obtained by those methods in fully nonlinear examples.

An Underwater Distributed SLAM Approach Based on Improved GMRBnB Framework

Multi-vehicle collaborative mapping proves more efficient in constructing maps in unfamiliar underwater environments in comparison to single-vehicle methods. One of the pivotal hurdles of Simultaneous Localization and Mapping (SLAM) with multiple underwater vehicles is map registration. Due to the inadequate characteristics of the underwater grid maps, matching map features poses a challenge, and outliers between maps add to the complexity. We propose an algorithm to solve this problem. This approach employs the Gaussian Mixture Robust Branch and Bound (GMRBnB) algorithm with an interior point filtering technique. Feature point extraction, registration using the GMRBnB algorithm, inlier extraction based on density, and registration of the inlier are performed to obtain a more precise transformation matrix. The results of the simulation and experiments demonstrate that this technique heightens outlier tolerance and reinforces map registration accuracy. The proposed approach surpasses Iterative Closest Point (ICP) and Normal Distributions Transform (NDT) methods with respect to map registration quality.

An exact algorithm for Two-Echelon Location-Routing problem with simultaneous pickup and delivery

This study considers an extension of the two-Echelon Location-Routing Problem (2E-LRP) namely two-Echelon Location-Routing Problem with Simultaneous Pickup and Delivery (2E-LRPSPD). A Mixed Integer Programming (MIP) formulation is developed and strengthened by the use of valid inequalities to solve the 2E-LRPSPD, and then a Branch and Cut-based exact algorithm (B&C) is presented. To improve the upper bounds, Iterated Local Search and Variable Neighborhood Search-based metaheuristic algorithms, namely, ILS-VNS and VNS, are developed and embedded into the B&C. Computational results, conducted on instances produced from the literature, demonstrate that the B&C yields tight lower and upper bounds, and also is capable of solving some instances optimally with up to 50 customers and 10 satellites in a reasonable time. Furthermore, sensitivity analysis reveals that the heuristic approaches and an increase in the runtime limit assist the algorithm to reach tight the upper bounds, whereas the number of potential depots does not have any effect on the performance of the algorithm. Besides, further computational analysis on the 2E-LRPSPD with a single depot, which is studied in the literature, shows that the B&C improves some best-known solutions while proving the optimality of some instances. Finally, the B&C is applied to a two-echelon supermarket distribution system in Türkiye. The computational results show that simultaneous pickup and delivery operations, as well as depot location decisions are critical elements to consider in the design of cost-effective distribution networks.