Branching Strategy Research Articles

A dedicated branch-price-and-cut algorithm for advance patient planning and surgeon scheduling

In this paper, we study the patient planning and surgeon scheduling in the operating room theatre. The problem considers the simultaneous planning of patients and the assignment of time blocks to surgeons so that they can perform the surgery of their patients. The timing and length of the allotted time blocks depend on the patient characteristics on the surgeons’ waiting lists. Solving this problem in an exact manner is challenging due to the large number of rooms, surgeons, and patients involved. To overcome this challenge, we propose an efficient branch-price-and-cut algorithm to find an optimal solution in an acceptable time span. For that purpose, we include different dedicated mechanisms to accelerate the solution-finding process. In this regard, the branch-price-and-cut tree is set up using an intelligent branching scheme, the nodes are searched in order of the lowest number of fractional variables, and improved bounds are computed to prune nodes earlier. To tighten the convex hull of the linear programming relaxation in each node, the algorithm relies on a row generation mechanism for adding valid inequalities. We conducted various computational experiments to demonstrate the performance of our algorithm and validate for each component the contribution of the implemented optimisation principles. Additionally, we show the superior performance of the proposed algorithm to alternative optimisation procedures.

Synergistic enhancement of poly(lactide) melt strength via copolymerization and multi-arm branching strategy

The lower melt strength of poly(lactide) (PLA) limits its broader applications. Here, a strategy combining copolymerization with multi-arm branching was propose to enhance the melt strength of PLA. Initially, stereoisomeric cyclic ester monomers (CEM) synthesized via zeolite catalysis were copolymerized into PLA chains. Subsequently, rheological testing revealed that the zero-shear viscosity (η0) of linear PLA increased by 467 % with only 1 mol% of CEM units. Our study further systematically explored the relationship between the side group structure and chirality of the comonomers and the rheological properties of the copolymers. CEMs with long-chain branched structures and opposite chirality had the best enhancement effect. In order to further enhance the melt strength, we successfully achieved alterations in polymer topology by employing trimethylolpropane as an initiator, corresponding three-arm copolymers achieve up to a 67-fold increase in η0 (1.0 kPa∙s to 68.1 kPa∙s). Tensile tests indicated that the mechanical properties of the copolymers were comparable to those of PLA, with a tensile strength of approximately 65 MPa. Additionally, due to the high melt strength, we successfully produced closed-cell PLA-based foam materials with uniform pore sizes. In summary, this study furnishes a feasible method for designing polymer materials possessing the desired melt strength.

Branching evolution for unknown objective optimization in biclustering

Biclusters hold significant importance in microarray analysis. Given the EA algorithm’s efficacy in tackling nonlinear problems, it has become a prevalent choice for evolutionary biclustering in microarray analysis. However, in conventional approaches, the objective of bicluster volume remains elusive, as it heavily relies on the yet-to-be-discovered real bicluster. This discrepancy introduces a novel research problem termed “unknown objective optimization” in our study. To address this issue, our paper introduces an innovative branching evolution strategy within a multi-objective framework. This strategy aims to resolve the challenge of unknown objectives. Throughout the biclustering search process, we meticulously observe the evolution of optimal bicluster individuals. Stability in both mean squared residue (MSR) and volume suggests a high likelihood of reaching an optimal solution, whether local or global. If a global optimal solution is attained at the end of the final evolution, our initial assumption is validated; otherwise, it necessitates an update. The proposed branching strategy is subsequently implemented to bifurcate the original evolution into two branches. One continues the original evolution with an unknown objective of bicluster volume, while the other pursues a new evolution with an estimated objective of bicluster volume. Our algorithm’s performance is assessed through comparisons with various traditional and evolutionary biclustering algorithms. The experimental results affirm its enhanced efficacy on both synthetic datasets and real gene datasets.

A constraint programming approach for resource-constrained flexible assembly flow shop scheduling problem with batch direct delivery

Production and distribution are both crucial components of supply chains. Integrated production and distribution scheduling (IPDS) in the context of flexible assembly flow shop scheduling and batch delivery problems is often overlooked. A realistic problem inspired by the production and distribution processes of a dishwasher factory can be modeled as a resource-constrained flexible assembly flow shop scheduling problem with batch direct delivery (RCFAFSP-BDD). In this problem, order requirements are decomposed into several production tasks processed at different stages of the workshop, and are then delivered in batches via a third-party logistics provider to regional distributors at various locations. Auxiliary resource restrictions, hierarchical coupling constraints, machine eligibility restrictions and sequence-dependent setup times, are incorporated into the problem as operational constraints. To the best of our knowledge, this is the first attempt to solve this problem. This work formulates a mixed-integer linear programming (MIP) model to minimize total costs, including tardiness, inventory, and delivery costs. Given the problem’s strong NP-hard nature, the focus is on developing an efficient solution approach using constraint programming (CP). A CP model is proposed and enhanced with multiple redundant constraints. To reduce runtime, two branching strategies are designed for the CP model. Numerical experiments with varying instance scales reveal that the proposed CP model outperforms the MIP model in accuracy and efficiency within the given time limit. The redundant constraints and search strategy can reduce CP model runtime by up to 263.83%. Compared to manual scheduling at the studied factory, the CP model can cut costs by up to 26.59% for real data, offering viable alternatives for factory planners.

Parallel branch-and-price algorithms for the single machine total weighted tardiness scheduling problem with sequence-dependent setup times

Scheduling problems occur in a broad range of real-world application fields and have attracted a huge set of research articles. However, there is only little research on exact algorithms for scheduling problems, many of which are NP-hard in the strong sense. We investigate the problem on a single machine with a total weighted tardiness objective function and sequence-dependent setup times. First, we adopt a serial branch-and-price algorithm from the literature and present a modified branching strategy and a primal heuristic. Second, we use the potential of parallel computing architectures by presenting two parallel versions of the branch-and-price algorithm. Third, we conduct extensive computational experiments to show that our parallelization approaches provide substantial parallel speedups on well-known benchmark instances from the literature. We further observe that the parallel speedups achieved by our parallel algorithms are very robust among all tested instances.

Government-Led digital transformation in FinTech ecosystems

In this paper, we report on a qualitative exploratory case study of a national-level government-led digital transformation. We depart from most studies on government digital transformation that largely focus on improving existing services, bureaucratic processes, or adopting emerging digital technologies. Instead, we analyze the process of a government-led digital transformation aimed at addressing significant institutional voids within a resource-constrained context. Drawing from 60 interviews with stakeholders in the Ghanaian FinTech ecosystem, we theorize the concept of digital branching strategy as an alternative lens to envisage government-led digital transformation that considers the resource-constrained context and characteristics of governments. Our findings show that governments, especially those in resource-constrained contexts pursue digital transformation through exploring frugal innovations and leveraging established resources, structures, and relationships within an ecosystem. We subsequently develop a process model to explain the mechanisms of a national-level government-led digital transformation. Based on the findings and the model, our study offers critical insights to re-imagine government-led digital transformation in resource-constrained contexts by demonstrating how pursuing a digital branching strategy leads to planned and emergent outcomes because of the generative nature of the transformation.

Decision-focused neural adaptive search and diving for optimizing mining complexes

Optimizing industrial mining complexes, from extraction to end-product delivery, presents a significant challenge due to non-linear aspects and uncertainties inherent in mining operations. The two-stage stochastic integer program for optimizing mining complexes under joint supply and demand uncertainties leads to a formulation with tens of millions of variables and non-linear constraints, thereby challenging the computational limits of state-of-the-art solvers. To address this complexity, a novel solution methodology is proposed, integrating context-aware machine learning and optimization for decision-making under uncertainty. This methodology comprises three components: (i) a hyper-heuristic that optimizes the dynamics of mining complexes, modeled as a graph structure, (ii) a neural diving policy that efficiently performs dives into the primal heuristic selection tree, and (iii) a neural adaptive search policy that learns a block sampling function to guide low-level heuristics and restrict the search space. The proposed neural adaptive search policy introduces the first soft (heuristic) branching strategy in mining literature, adapting the learning-to-branch framework to an industrial context. Deployed in an online fashion, the proposed hybrid methodology is shown to optimize some of the most complex case studies, accounting for varying degrees of uncertainty modeling complexity. Theoretical analyses and computational experiments validate the components’ efficacy, adaptability, and robustness, showing substantial reductions in primal suboptimality and decreased execution times, with improved and more robust solutions that yield higher net present values of up to 40%. While primarily grounded in mining, the methodology shows potential for enabling smart, robust decision-making under uncertainty.

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.

A Branch-and-Cut-and-Price Algorithm for Cutting Stock and Related Problems

In this project, we introduce a branch-and-cut-and-price framework to solve the Cutting Stock Problems with strong relaxations using the Set Covering (Packing) Formulations, which are solved through column generation. We propose an extended Ryan-Foster branching scheme tailored to non-binary models, a pricing algorithm that produces convergence in a few iterations, and a variable selection technique based on branching history. These strategies are combined with subset-row cuts and custom primal heuristics to create a framework that overcomes the current state-of-the-art of Cutting Stock Problem, Skiving Stock Problem, and other related problems, being at least twice faster in the first problem and at least 60% faster in the second one.

ERoots: A three-dimensional dynamic growth model of rice roots coupled with soil

Root architecture systems (RAS) reflect the spatial structure of roots in soil. To clarify the structure and distribution of rice roots and investigate the coupling between roots and soil, wetland rice was selected as the experimental object, and a three-dimensional (3D) growth model of rice root environment-roots (ERoots) based on the parameter Lindenmayer system (L-system) was proposed. ERoots combines a root morphological structure model with a growth model and defines L-system grammar iteration rules with the unit time and unit step length as parameters. At the same time, the basic growth parameters of rice roots were obtained via destructive detection, and 3D growth visualisation of roots was realised via MATLAB. In the soil coupling process, a soil nutrient simulation map was constructed based on the spatial soil characteristics per unit volume, and an adjustment strategy for roots reaching the growth boundary was designed. The flexibility of the model coupled with soil was reflected in the tropisms of root growth, growth rate and root branching strategy. Finally, combined with soil spatial characteristic simulation, geometric growth boundary and 3D root growth model, the ability of 3D growth visualisation of rice roots was verified under three soil conditions: (1) unconfined root growth, (2) confined spatial root growth, and (3) root growth with tropisms. The results indicated that the ERoots root model basically realised coupling with soil and achieved a satisfactory simulation effect in regard to the rice morphological structure. This study provides a reference for 3D growth modelling and visualisation of other crop roots.

Efficient verification of neural networks based on neuron branching and LP abstraction

With the rapid development and wide application of neural networks in various domains including safety-critical systems, it is more and more important to investigate formal methods to provide strict guarantees on their behavior. As formal verification of a neural network has high computational complexity, caused by the nonlinear nature of activation functions such as ReLU, it is vital to improve the efficiency to solve verification problems. In this work, we propose a complete verification method for neural networks by means of neuron branching and linear programming (LP) abstraction. Specifically, the approach of branch and bound is adopted and a branching strategy is developed to divide a complex verification problem into sub-problems with smaller search space. And for each sub-problem, a part of ReLU neurons are abstracted with LP constraints and the others are accurately encoded with mixed integer linear programming (MILP) constraints, and an abstraction strategy is deployed to guide such mixed LP/MILP encoding so that the problem can be solved with reduced complexity. The method is implemented as a tool named BAVerify (Branching and Abstraction based Verification of neural networks), which is evaluated through experiments on benchmark network models and their verification problems. Experimental results show that BAVerify achieves a 43.13% advantage in verification efficiency compared with state-of-the-art complete methods for formal verification of neural networks.

Accelerated Benders decomposition and local branching for dynamic maximum covering location problems

The maximum covering location problem (MCLP) is a key problem in facility location, with many applications and variants. One such variant is the dynamic (or multi-period) MCLP, which considers the installation of facilities across multiple time periods. To the best of our knowledge, no exact solution method has been proposed to tackle large-scale instances of this problem. To that end, in this work, we expand upon the current state-of-the-art branch-and-Benders-cut solution method in the static case, by exploring several acceleration techniques. Additionally, we propose a specialised local branching scheme which exploits the separability of the problem by time period. This scheme uses a novel distance metric in its definition of subproblems and features a new method for efficient and exact solving of the subproblems. These methods are then compared through extensive computational experiments, highlighting the strengths of the proposed methodologies.

BiGNN: Bipartite graph neural network with attention mechanism for solving multiple traveling salesman problems in urban logistics

The multiple traveling salesman problems (MTSP), which arise from real world problems, are essential in urban logistics. Variations such as MinMax-MTSP and Bounded-MTSP aim to distribute workload evenly among salesmen and impose constraints on visited cities, respectively. Branch-and-bound (B&B) provides an exact algorithm solution for these problems. The Learn to Branch (L2B) approach guides branch node selection through deep learning. In this study, we utilize mathematical modeling of Bipartite Graph Neural Network (BiGNN) and an attention mechanism to support B&B in exploring solution spaces through imitation learning. The problems are framed to formulate mixed integer linear programming, which is different from conventional algorithms. It is proposed that a bipartite graph network approach makes a feature representation by setting a structure of constraints and variables. Experimental results showed that our model can generate more accurate solutions than three benchmark models. The BiGNN model can effectively learn the strong branch strategy, which reduces solution time by replacing complex calculations with fast approximations. Additionally, the small-scale instances model can be applied to larger-scale ones.

Computing Bipath Multicommodity Flows with Constraint Programming–Based Branch-and-Price-and-Cut

We propose a constraint programming (CP)–based branch-and-price-and-cut framework to exactly solve bipath multicommodity flow (MCF): an MCF problem with two paths for each demand. The goal is to route demands in a capacitated network under the minimum cost. The two paths must have disjoint arcs, and the delays accumulated along the two paths must be within a small deviation of each other. CP is used at multiple points in this framework: for solving pricing problems, for cut generation, and for primal and branching node heuristics. These modules use a CP solver designed for network routing problems and can be adapted to other combinatorial optimization problems. We also develop a novel, complete, two-level branching scheme. On a set of diverse bipath MCF instances, experimental results show that our algorithm significantly outperforms monolithic CP and mixed integer linear programming models and demonstrate the efficiency and flexibility brought by the tailored integration of linear programming and CP methodologies. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: This work was supported by the Natural Sciences and Engineering Research Council of Canada; Huawei Technologies. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0128 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0128 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

A chance-constrained optimization approach integrating project scheduling and material ordering to manage the uncertain material supply

To deal with the impact of uncertain material supply on the implementation of projects, we investigate a problem of concurrently finding a robust baseline schedule and a material ordering plan under different levels of uncertainty. Assuming the material warehouses are restricted, a chance-constrained model of the integrated resource-constrained project scheduling and material ordering problem with a confidence level (1−θ) is thus formulated. To tackle the difficult joint chance constraints, we firstly transform the chance-constrained model into a scenario-based integer programming model by the sample average approximation approach. Based on dominance relationship among scenarios, we then derive an exact branch-and-bound algorithm and a sampling-based genetic algorithm (SGA) to solve the model. For the branch-and-bound algorithm, a tailor-made branching scheme and four pruning rules are developed to accelerate the search process. For the SGA, two sampling methods and a local search procedure are presented to boost solution quality. Furthermore, extensive numerical experiments are carried out to test the performance of the proposed algorithms. Specifically, 4608 instances with four different confidence levels and six different sample sizes are generated. Then the Taguchi method is employed to calibrate the parameters of the SGA. Besides, the numerical results demonstrate that the proposed branch-and-bound algorithm can solve more instances with shorter CPU times than the CPLEX solver, and the solution quality of the proposed SGA is far better than that of two existing meta-heuristics and two variants of SGA. Finally, a sensitivity analysis for the key parameters of the model is conducted. By considering the uncertain material supply, limited warehouse capacity and various confidence levels, our research contributes to the effective decision on project scheduling and material ordering in real-world contexts.

The effect of instructional design according to Neural Branching Strategies for Acquisition of biological concepts and analytical thinking skills for the third intermediate grade students

هدف البحث الى التعرف على أثر تصميم تعليمي وفقا لأستراتيجيات التفكير المتشعب في اكتساب المفاهيم الاحيائية ومهارات التفكير التحليلي لطالبات الصف الثالث المتوسط، ولتحقيق هدف البحث اعتمدت الباحثتان التصميم التجريبي ذا الضبط الجزئي، اذ تألفت عينة البحث من (62) طالبة من طالبات الصف الثالث المتوسط، بواقع (32) طالبة في المجموعة التجريبية التي درست على وفق التصميم التعليمي لاستراتيجيات التفكير المتشعب, و(30) طالبة في المجموعة الضابطة التي درست على وفق الطريقة الاعتيادية، تم إجراء التكافؤ بين المجموعتين في بعض المتغيرات (العمر، والذكاء، والمعلومات السابقة، والتفكير التحليلي). قامت الباحثتان بإعداد أداتي البحث وهي (اختبار اكتساب المفاهيم الاحيائية، واختبار مهارات التفكير التحليلي)، وتم التأكد من الخصائص السايكومترية لهما. طبقت الباحثتين الاختبارين على مجموعتي البحث، وقد اظهرت النتائج تفوق طالبات المجموعة التجريبية على طالبات المجموعة الضابطة في اكتساب المفاهيم الاحيائية ومهارات التفكير التحليلي

Efficient k-Clique Listing: An Edge-Oriented Branching Strategy

k-clique listing is a vital graph mining operator with diverse applications in various networks. The state-of-the-art algorithms all adopt a branch-and-bound (BB) framework with a vertex-oriented branching strategy (called VBBkC), which forms a sub-branch by expanding a partial k-clique with a vertex. These algorithms have the time complexity of O(k · m · (δ/2)k-2 ), where m is the number of edges in the graph and δ is the degeneracy of the graph. In this paper, we propose a BB framework with a new edge-oriented branching (called EBBkC), which forms a sub-branch by expanding a partial k-clique with two vertices that connect each other (which correspond to an edge ). We explore various edge orderings for EBBkC such that it achieves a time complexity of O( m · δ + k · m · (τ/2)k-2 ), where τ is an integer related to the maximum truss number of the graph and we have τ < δ. The time complexity of EBBkC is better than that of VBBkC algorithms for k>3 since both O(m · δ) and O(k · m · (τ/2)k-2 ) are bounded by O(k · m · (δ/2)k-2 ). Furthermore, we develop specialized algorithms for sub-branches on dense graphs so that we can early-terminate them and apply the specialized algorithms. We conduct extensive experiments on 19 real graphs, and the results show that our newly developed EBBkC based algorithms with the early termination technique consistently and largely outperform the state-of-the-art (VBBkC based) algorithms.

A branch-and-price algorithm for unrelated parallel machine scheduling with machine usage costs

This paper considers unrelated parallel machine scheduling involving machine usage costs, in addition to classic job completion time-related costs. The usage cost of each machine is made up of a fixed usage cost and a variable usage cost proportional to the total processing time of the jobs assigned to it. These features model many practical situations where machine usage costs include, for example, rental fees when the machines are not owned but rented. To tackle this problem, four mathematical models based on the Shortest Weighted Processing Time (SWPT) rule are introduced. Additionally, the problem is formulated into a set-partitioning model, for which a branch-and-price algorithm is proposed with an appropriate branching strategy. This facilitates the development of an efficient pseudo-polynomial dynamic programming algorithm and a polynomial-time heuristic to solve the pricing problem. Extensive numerical experiments demonstrate the superior performance of the proposed branch-and-price algorithm over the four SWPT-based mathematical formulations and an existing branch-and-price algorithm designed for a special case. Notably, it can optimally solve instances involving up to 225 jobs and 15 machines within one hour. Moreover, statistical analyses reveal that the proposed polynomial-time heuristic significantly reduces the computation time, and the mathematical model based on the contribution of every job to the total weighted completion time exhibits the best overall performance.

An exact algorithm for unpaired pickup and delivery vehicle routing problem with multiple commodities and multiple visits

This paper addresses a new multi-commodity unpaired pickup and delivery vehicle routing problem in which each vehicle is allowed to visit each customer more than once for both pickup and delivery of each commodity. It is a complex variant of the classical capacitated vehicle routing problem and comes from resource transfer among different plants in a distributed manufacturing environment. Notably, the pairing decisions of pickup and delivery for each commodity and the routing decisions among customers are interdependent, and they need to be made simultaneously and collaboratively. Firstly, a mixed integer programming model is developed, and then new valid inequalities are derived to improve the linear programming relaxation. Afterwards, a branch-and-cut algorithm is proposed by exploring different branching strategies and tailored separation algorithms. Computational results based on real-case instances from an automobile manufacturing company demonstrate that the costs are considerably reduced by considering the multiple visits compared to the single visit of each customer. In addition, our proposed branch-and-cut algorithm outperforms a manual solution method adopted by the company in practice and two solution methods in the literature. Furthermore, two adaptations of our branch-and-cut algorithm also perform better than two state-of-the-art algorithms in solving two closely-related problems in terms of solution quality and computation time. Sensitivity analyses with respect to vehicle capacity, vehicle duration, and vehicle speed are conducted to derive several valuable managerial insights.

Identification and expression of strigolactone biosynthesis and signaling genes and the in vitro effects of strigolactones in olive (Olea europaea L.).

Strigolactones (SLs), synthesized in plant roots, play a dual role in modulating plant growth and development, and in inducing the germination of parasitic plant seeds and arbuscular mycorrhizal fungi in the rhizosphere. As phytohormones, SLs are crucial in regulating branching and shaping plant architecture. Despite the significant impact of branching strategies on the yield performance of fruit crops, limited research has been conducted on SLs in these crops. In our study, we identified the transcript sequences of SL biosynthesis and signaling genes in olive (Olea europaea L.) using rapid amplification of cDNA ends. We predicted the corresponding protein sequences, analyzed their characteristics, and conducted molecular docking with bioinformatics tools. Furthermore, we quantified the expression levels of these genes in various tissues using quantitative real-time PCR. Our findings demonstrate the predominant expression of SL biosynthesis and signaling genes (OeD27, OeMAX3, OeMAX4, OeMAX1, OeD14, and OeMAX2) in roots and lateral buds, highlighting their importance in branching. Treatment with rac-GR24, an SL analog, enhanced the germination frequency of olive seeds in vitro compared with untreated embryos. Conversely, inhibition of SL biosynthesis with TIS108 increased lateral bud formation in a hard-to-root cultivar, underscoring the role of SLs as phytohormones in olives. These results suggest that modifying SL biosynthesis and signaling pathways could offer novel approaches for olive breeding, with potential applicability to other fruit crops.