Reduction In Solution Time Research Articles

Multi-stage planning of clean resources and energy storage assets with hybrid uncertainty modeling for low-carbon resilient distribution systems

Climate change drives the urgent need for low-carbon and resilient energy system transitions. However, current planning methods ignore the inherent conflicts between carbon emission reduction and resilience enhancement, failing to optimally balance asset allocation for both aspects. They also neglect the long-term dynamic and stochastic nature of transitions, exposing systems to hybrid uncertainties. This paper presents a multi-stage dynamic planning method for clean resources and energy storage assets in power distribution networks. First, to facilitate low-carbon and resilient transitions, adaptive, stage-wise planning decisions are optimally determined under various planning strategies to mitigate risks stemming from hybrid uncertainties. Second, to precisely quantify the impact of hybrid uncertainties on decision-making, a systematic hybrid-uncertainty modeling approach is designed to capture short- and long-term uncertainties arising from both exogenous and endogenous factors. Third, to alleviate computational complexity, a decomposition-based stochastic dynamic dual integer programming method with augmented cut generation is introduced, which decomposes the original problem into stage-wise subproblems coordinated through forward and backward steps. Numerical results show that the proposed method generates cost-effective planning schemes, yielding $60 K in savings and reducing carbon emissions by 3853 tons. The method also demonstrates strong scalability, with a 52% reduction in solution time for large-scale planning problems.

Structured pruning of neural networks for constraints learning

In recent years, the integration of Machine Learning (ML) models with Operation Research (OR) tools has gained popularity in applications such as cancer treatment, algorithmic configuration, and chemical process optimization. This integration often uses Mixed Integer Programming (MIP) formulations to represent the chosen ML model, that is often an Artificial Neural Networks (ANNs) due to their widespread use. However, ANNs frequently contain a large number of parameters, resulting in MIP formulations impractical to solve. In this paper we showcase the effectiveness of a ANN pruning, when applied to models prior to their integration into MIPs. We discuss why pruning is more suitable in this context than other ML compression techniques, and we highlight the potential of appropriate pruning strategies via experiments on MIPs used to construct adversarial examples to ANNs. Our results demonstrate that pruning offers remarkable reductions in solution times without hindering the quality of the final decision, enabling the resolution of previously unsolvable instances.

ReLU surrogates in mixed-integer MPC for irrigation scheduling

Efficient water management in agriculture is important for mitigating the growing freshwater scarcity crisis. Mixed-integer Model Predictive Control (MPC) has emerged as an effective approach for addressing the complex scheduling problems in agricultural irrigation. However, the computational complexity of mixed-integer MPC still poses a significant challenge, particularly in large-scale applications. This study proposes an approach to enhance the computational efficiency of mixed-integer MPC-based irrigation schedulers by employing Rectified Linear Unit (ReLU) surrogate models to describe the soil moisture dynamics of the agricultural field. By leveraging the mixed-integer linear representation of the ReLU operator, the proposed approach transforms the mixed-integer MPC-based scheduler with a quadratic cost function into a mixed-integer quadratic program, which is the simplest class of mixed-integer nonlinear programming problems that can be efficiently solved using global optimization solvers. The effectiveness of this approach is demonstrated through comparative studies conducted on a large-scale agricultural field across two growing seasons, involving other machine learning surrogate models, specifically Long Short-Term Memory (LSTM) networks, and the triggered irrigation scheduling method. The ReLU-based approach significantly reduces solution times—by up to 99.5%—while achieving comparable performance to the LSTM approach in terms of water savings and Irrigation Water Use Efficiency (IWUE). Moreover, the ReLU-based approach achieves enhanced performance in terms of irrigation water savings and IWUE compared to the triggered approach.

A penalty-free hybrid algorithm framework based on feasible stream matching principle for large-scale heat exchanger networks synthesis

In this work, a penalty-free hybrid stochastic-deterministic algorithm framework is proposed for large-scale heat exchanger networks (HENs) synthesis (HENS), formulated as a computationally-hard mixed-integer nonlinear programming (MINLP) problem. In the outer level, an improved genetic algorithm (GA) is developed to optimize process stream matches represented by integer variables whose values are generated by a unique heat exchanger vector. Unlike previous researches, the improved GA does not rely on any penalty terms, because we propose a feasible stream matching principle to exclude all infeasible process stream matches and only feasible matches are considered in optimization process. In the inner level, a reduced-size MINLP model is solved using deterministic methods to minimize total annualized costs (TACs), which are then used to evaluate the fitness of candidate HENs. Through this way, the proposed framework combines deterministic and stochastic methods to enhance optimization efficiency and global search capability. Illustrative tests on six benchmark cases demonstrate that the framework can efficiently achieve lower-cost solutions compared to deterministic, stochastic, or hybrid methods. The results show a decrease in TAC for all six cases and a reduction in solution time ranging from 11.1% to 97.2%. Importantly, the proposed framework can be extended to solve MINLP problems in other process networks.

Deep reinforcement learning algorithms for dynamic pricing and inventory management of perishable products

A perishable product has a limited shelf life, and inefficient management often leads to waste. This paper focuses on dynamic pricing and inventory management strategies for perishable products. By implementing effective inventory control and the right pricing policy, it is possible to maximize expected revenue. However, the exponential growth of the problem size due to the shelf life of the products makes it impractical to use methods that guarantee optimal solutions, such as Dynamic Programming (DP). Therefore, approximate solution algorithms become necessary. We use Deep Reinforcement Learning (DRL) algorithms to address the dynamic pricing and ordering problem for perishable products, considering price and age-dependent stochastic demand. We investigate Deep Q Learning (DQL) solutions for discrete action spaces and Soft Actor-Critic (SAC) solutions for continuous action spaces. To mitigate the negative impact of the stochastic environment inherent in the problem, we propose two different DQL approaches. Our results show that the proposed DQL and SAC algorithms effectively address inventory control and dynamic pricing for perishable products, even when products of different ages are offered simultaneously. Compared to dynamic programming, our proposed DQL approaches achieve an average approximation of 95.5 % and 96.6 %, and reduce solution times by 71.5 % and 79.9 %, respectively, for the largest problem. In addition, the SAC algorithm achieves on average 4.6 % and 1.7 % better results and completes the task 56.1 % and 48.2 % faster than the proposed DQL algorithms.

Efficiency enhancement techniques in finite element analysis: navigating complexity for agile design exploration

PurposeThis paper aims to comprehensively explore techniques for reducing solution time in finite element analysis (FEA), addressing the critical need for expediting computations to facilitate agile design exploration within project timelines.Design/methodology/approachDrawing from a wide array of literature sources, this paper synthesizes and analyzes various methodologies used to enhance the efficiency of FEA. Techniques are scrutinized in terms of their applicability, effectiveness and potential limitations.FindingsThe review signifies application of linear assumptions across multiple facets of analysis and delves into matrix order reduction strategies, geometry simplification, symmetry exploitation, submodeling and mesh attribute control. It reveals how these techniques can effectively reduce computational burdens while maintaining acceptable levels of accuracy.Research limitations/implicationsWhile this review provides a comprehensive overview of existing efficiency enhancement techniques in FEA, it acknowledges inherent limitations of any synthesis-based study. Future research should focus on refining these methodologies.Practical implicationsThe insights provided in this paper offer practical guidance for structural engineers and researchers seeking to optimize FEA workflows. By implementing these techniques, practitioners can expedite solution times and enhance their ability to explore design alternatives efficiently ultimately leading to cost savings and more robust structures.Originality/valueThis review contributes to the existing literature by offering a comprehensive synthesis of efficiency enhancement techniques in FEA. By highlighting the originality and value of each discussed methodology, this paper provides a roadmap for future research and practical implementation in the field of structural engineering.

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.

Study on Location Selection of Urban Two-Level Joint Express Delivery Stations Considering Fair Cost Allocation among Enterprises

The rapid growth of e-commerce has heightened the importance for express delivery companies to ensure timely deliveries. Consequently, it is essential to explore ways to deliver more packages to customers while simultaneously reducing costs through the adoption of a joint distribution mode. This study presents a two-level delivery location selection model within the joint distribution mode, considering factors such as delivery station capacity and the number of transport vehicles, with the objective of minimizing the total cost associated with selecting delivery station locations. The proposed model is addressed using a combination of the k-means algorithm and the improved discrete firefly algorithm. In addition, to facilitate equitable cost allocation among enterprises, the Shapley value method is introduced in this study. A case study based on real data from an urban distribution network in the city of Hebei Province, China, is adopted to perform the experiments. The results of this study indicate that the improved algorithm not only improves solution accuracy but also reduces solution time when compared to both the particle swarm optimization and artificial bee colony methods. Furthermore, the application of the Shapley value method demonstrates the efficacy of a rational allocation of costs.

Taking the human out of decomposition-based optimization via artificial intelligence, Part II: Learning to initialize

The repeated solution of large-scale optimization problems arises frequently in process systems engineering tasks. Decomposition-based solution methods have been widely used to reduce the corresponding computational time, yet their implementation has multiple steps that are difficult to configure. We propose a machine learning approach to learn the optimal initialization of such algorithms which minimizes the computational time. Active and supervised learning is used to learn a surrogate model that predicts the computational performance for a given initialization. We apply this approach to the initialization of Generalized Benders Decomposition for the solution of mixed-integer model predictive control problems. The surrogate models are used to find the optimal initialization, which corresponds to the number of initial cuts that should be added in the master problem. The results show that the proposed approach can lead to a significant reduction in solution time, and active learning can reduce the data required for learning.

Computationally efficient solution of mixed integer model predictive control problems via machine learning aided Benders Decomposition

Mixed integer Model Predictive Control (MPC) problems arise in the operation of systems where discrete and continuous decisions must be taken simultaneously to compensate for disturbances. The efficient solution of mixed integer MPC problems requires the computationally efficient online solution of mixed integer optimization problems, which are generally difficult to solve. In this paper, we propose a machine learning-based branch and check Generalized Benders Decomposition algorithm for the efficient solution of such problems. We use machine learning to approximate the effect of the complicating variables on the subproblem by approximating the Benders cuts without solving the subproblem, therefore, alleviating the need to solve the subproblem multiple times. The proposed approach is applied to a mixed integer economic MPC case study on the operation of chemical processes. We show that the proposed algorithm always finds feasible solutions to the optimization problem, given that the mixed integer MPC problem is feasible, and leads to a significant reduction in solution time (up to 97% or 50×) while incurring small error (in the order of 1%) compared to the application of standard and accelerated Generalized Benders Decomposition.

FGeo-TP: A Language Model-Enhanced Solver for Euclidean Geometry Problems

The application of contemporary artificial intelligence techniques to address geometric problems and automated deductive proofs has always been a grand challenge to the interdisciplinary field of mathematics and artificial intelligence. This is the fourth article in a series of our works, in our previous work, we established a geometric formalized system known as FormalGeo. Moreover, we annotated approximately 7000 geometric problems, forming the FormalGeo7k dataset. Despite the fact that FGPS (Formal Geometry Problem Solver) can achieve interpretable algebraic equation solving and human-like deductive reasoning, it often experiences timeouts due to the complexity of the search strategy. In this paper, we introduced FGeo-TP (theorem predictor), which utilizes the language model to predict the theorem sequences for solving geometry problems. The encoder and decoder components in the transformer architecture naturally establish a mapping between the sequences and embedding vectors, exhibiting inherent symmetry. We compare the effectiveness of various transformer architectures, such as BART or T5, in theorem prediction, and implement pruning in the search process of FGPS, thereby improving its performance when solving geometry problems. Our results demonstrate a significant increase in the problem-solving rate of the language model-enhanced FGeo-TP on the FormalGeo7k dataset, rising from 39.7% to 80.86%. Furthermore, FGeo-TP exhibits notable reductions in solution times and search steps across problems of varying difficulty levels.

New optimization models for optimal classification trees

The most efficient state-of-the-art methods to build classification trees are greedy heuristics (e.g. CART) that may fail to find underlying characteristics in datasets. Recently, exact linear formulations that have shown better accuracy were introduced. However they do not scale up to datasets with more than a few thousands data points. In this paper, we introduce four new formulations for building optimal trees. The first one is a quadratic model based on the well-known formulation of Bertsimas et al. We then propose two different linearizations of this new quadratic model. The last model is an extension to real-valued datasets of a flow-formulation limited to binary datasets (Aghaei et al.). Each model is introduced for both axis-aligned and oblique splits. We further prove that our new formulations have stronger continuous relaxations than existing models. Finally, we present computational results on 22 standard datasets with up to thousands of data points. Our exact models have reduced solution times with learning performances as strong or significantly better than state of the art exact approaches.

Location Planning of Emergency Medical Facilities Using the p-Dispersed-Median Modeling Approach

This research employs a spatial optimization approach customized for addressing equitable emergency medical facility location problems through the p-dispersed-median problem (p-DIME). The p-DIME integrates two conflicting classes of spatial optimization problems, dispersion and median problems, aiming to identify the optimal locations for emergency medical facilities to achieve an equitable spatial distribution of emergency medical services (EMS) while effectively serving demand. To demonstrate the utility of the p-DIME model, we selected Gyeongsangbuk-do in South Korea, recognized as one of the most challenging areas for providing EMS to the elderly population (aged 65 and over). This challenge arises from the significant spatial disparity in the distribution of emergency medical facilities. The results of the model assessment gauge the spatial disparity of EMS, provide significantly enhanced solutions for a more equitable EMS distribution in terms of service coverage, and offer policy implications for future EMS location planning. In addition, to address the computational challenges posed by p-DIME’s inherent complexity, involving mixed-integer programming, this study introduces a solution technique through constraint formulations aimed at tightening the lower bounds of the problem’s solution space. The computational results confirm the effectiveness of this approach in ensuring reliable computational performance, with significant reductions in solution times, while still producing optimal solutions.

A Lagrangian relaxation approach to an electricity system investment model with a high temporal resolution

AbstractThe global production of electricity contributes significantly to the release of carbon dioxide emissions. Therefore, a transformation of the electricity system is of vital importance in order to restrict global warming. This paper proposes a modelling methodology for electricity systems with a large share of variable renewable electricity generation, such as wind and solar power. The model developed addresses the capacity expansion problem, i.e. identifying optimal long-term investments in the electricity system. Optimal investments are defined by minimum investment and production costs under electricity production constraints—having different spatial resolutions and technical detail—while meeting the electricity demand. Our model is able to capture a range of strategies to manage variations and to facilitate the integration of variable renewable electricity; it is very large due to the high temporal resolution required to capture the variations in wind and solar power production and the chronological time representation needed to model energy storage. Moreover, the model can be further extended—making it even larger—to capture a large geographical scope, accounting for the trade of electricity between regions with different conditions for wind and solar power. Models of this nature thus typically need to be solved using some decomposition method to reduce solution times. In this paper, we develop a decomposition method using so-called variable splitting and Lagrangian relaxation; the dual problem is solved by a deflected subgradient algorithm. Our decomposition regards the temporal resolution by defining 2-week periods throughout the year and relaxing the overlapping constraints. The method is tested and evaluated on some real-world cases containing regions with different energy mixes and conditions for wind power. Numerical results show shorter computation times as compared with the non-decomposed model and capacity investment options similar to the optimal solution provided by the latter model.

A fast-response mathematical programming approach for delivering disaster relief goods: an earthquake case study

ABSTRACT This paper tackles a complex logistics challenge of disaster management, encompassing warehouse location, pre-disaster inventory planning, routing, and post-disaster relief supply delivery. We establish an iterative process for optimizing relief distribution to shelters. Adaptable warehouse inventory reallocation responds to fluctuating demands, guided by a two-phase mathematical programming approach. In the first phase, a two-stage stochastic programming (TSSP) model determines optimal warehouse and shelter locations and inventory levels. In the subsequent phase, we introduce a mixed-integer programming (MIP) model to minimize the overall delivery time by making routing decisions. To streamline the process, we introduce a novel enumeration algorithm that trims down route options by considering unavailable links, effectively transforming the MIP model into an assignment-based model. This innovation results in a noticeable 74% reduction in solution time. Further efficiency is achieved by developing a branch-and-cut algorithm for swift MIP resolution. A real-world case study confirms the practicality of our approach.

Modeling and IAHA Solution for Task Scheduling Problem of Processing Crowdsourcing in the Context of Social Manufacturing

The paper addresses the discrete characteristics of the processing crowdsourcing task scheduling problem in the context of social manufacturing, divides it into two subproblems of social manufacturing unit selecting and subtask sorting, establishes its mixed-integer programming with the objective of minimizing the maximum completion time, and proposes an improved artificial hummingbird algorithm (IAHA) for solving it. The IAHA uses initialization rules of global selection, local selection, and random selection to improve the quality of the initial population, the Levy flight to improve guided foraging and territorial foraging, the simplex search strategy to improve migration foraging to enhance the merit-seeking ability, and the greedy decoding method to improve the quality of the solution and reduce solution time. For the IAHA, orthogonal tests are designed to obtain the optimal combination of parameters, and comparative tests are made with variants of the AHA and other algorithms on the benchmark case and a simulated crowdsourcing case. The experimental results show that the IAHA can obtain superior solutions in many cases with economy and effectiveness.

Influence of hydrogen import prices on hydropower systems in climate-neutral Europe

AbstractWhile climate and energy policy targets require fundamental changes and expansions in the energy infrastructure, hydropower systems across Europe remain essential for low-carbon energy systems. With renewable fuel import prices being subject to large uncertainties, this work aims to substantiate the relationship between these fuel import prices and multireservoir hydropower systems in a climate-neutral energy system. To that end, three green hydrogen import price scenarios are combined with two aggregated modelling approaches for pan-European hydropower assets. Using the integrated energy system model SCOPE SD, the analysis shows that import prices for green hydrogen have a significant impact on European electricity generation (+ 595 GW$$_\text {el}$$ el and + 650 TWh$$_\text {el}$$ el /yr), domestic hydrogen production (+ 396 TWh$$_\text {th}$$ th /yr), and water values of European hydropower assets (+ 33 % of average value in Norway). The results further indicate that the different aggregation methods only have a minor impact, suggesting that the computationally more efficient approach with up to 90% reductions in solution time provides suitable approximations of hydropower generation and flexibility in future analyses.

An efficient quantum partial differential equation solver with chebyshev points

Differential equations are the foundation of mathematical models representing the universe’s physics. Hence, it is significant to solve partial and ordinary differential equations, such as Navier–Stokes, heat transfer, convection–diffusion, and wave equations, to model, calculate and simulate the underlying complex physical processes. However, it is challenging to solve coupled nonlinear high dimensional partial differential equations in classical computers because of the vast amount of required resources and time. Quantum computation is one of the most promising methods that enable simulations of more complex problems. One solver developed for quantum computers is the quantum partial differential equation (PDE) solver, which uses the quantum amplitude estimation algorithm (QAEA). This paper proposes an efficient implementation of the QAEA by utilizing Chebyshev points for numerical integration to design robust quantum PDE solvers. A generic ordinary differential equation, a heat equation, and a convection–diffusion equation are solved. The solutions are compared with the available data to demonstrate the effectiveness of the proposed approach. We show that the proposed implementation provides a two-order accuracy increase with a significant reduction in solution time.

Tightening discretization-based MILP models for the pooling problem using upper bounds on bilinear terms

Discretization-based methods have been proposed for solving nonconvex optimization problems with bilinear terms such as the pooling problem. These methods convert the original nonconvex optimization problems into mixed-integer linear programs (MILPs). In this paper we study tightening methods for these MILP models for the pooling problem, and derive valid constraints using upper bounds on bilinear terms. Computational results demonstrate the effectiveness of our methods in terms of reducing solution time.

Learning-Assisted Variables Reduction Method for Large-Scale MILP Unit Commitment

The security-constrained unit commitment (SCUC) challenge is solved repeatedly several times every day, for operations in a limited time. Typical mixed-integer linear programming (MILP) formulations are intertemporal in nature and have complex and discrete solution spaces that exponentially increase with system size. Improvements in the SCUC formulation and/or solution method that yield a faster solution hold immense economic value, as less time can be spent finding the best-known solution. Most machine learning (ML) methods in the literature either provide a warm start or convert the MILP-SCUC formulation to a continuous formulation, possibly leading to sub-optimality and/or infeasibility. In this paper, we propose a novel ML-based variables reduction method that accurately determines the optimal schedule for a subset of trusted generators, shrinking the MILP-SCUC formulation and dramatically reducing the search space. ML indicators sets are created to shrink the MILP-SCUC model, leading to improvement in the solution quality. Test results on IEEE systems with 14, 118, and 300 busses, the Ontario system, and Polish systems with 2383 and 3012 busses report significant reductions in solution times in the range of 48% to 98%. This is a promising tool for system operators to solve the MILP-SCUC with a lower optimality gap in a limited-time operation, leading to economic benefits.