Linear Programming Research Articles

Approximate and Exact Optimization Algorithms for the Beltway and Turnpike Problems with Duplicated, Missing, Partially Labeled, and Uncertain Measurements.

The Turnpike problem aims to reconstruct a set of one-dimensional points from their unordered pairwise distances. Turnpike arises in biological applications such as molecular structure determination, genomic sequencing, tandem mass spectrometry, and molecular error-correcting codes. Under noisy observation of the distances, the Turnpike problem is NP-hard and can take exponential time and space to solve when using traditional algorithms. To address this, we reframe the noisy Turnpike problem through the lens of optimization, seeking to simultaneously find the unknown point set and a permutation that maximizes similarity to the input distances. Our core contribution is a suite of algorithms that robustly solve this new objective. This includes a bilevel optimization framework that can efficiently solve Turnpike instances with up to 100,000 points. We show that this framework can be extended to scenarios with domain-specific constraints that include duplicated, missing, and partially labeled distances. Using these, we also extend our algorithms to work for points distributed on a circle (the Beltway problem). For small-scale applications that require global optimality, we formulate an integer linear program (ILP) that (i) accepts an objective from a generic family of convex functions and (ii) uses an extended formulation to reduce the number of binary variables. On synthetic and real partial digest data, our bilevel algorithms achieved state-of-the-art scalability across challenging scenarios with performance that matches or exceeds competing baselines. On small-scale instances, our ILP efficiently recovered ground-truth assignments and produced reconstructions that match or exceed our alternating algorithms. Our implementations are available at https://github.com/Kingsford-Group/turnpikesolvermm.

The Frank-Wulf method in modeling information systems

Subject of research: the process of choosing the best option for the structure of an information system under given conditions at the design stage. Purpose of research: to increase the efficiency and quality of the validity of decision-making, to reduce time and financial costs when choosing the structure of an information system at the design stage. Methods and objects of research: the object of research is an information system. The problem of nonlinear programming is formulated. A mathematical model of choosing the optimal variant of an information system is presented. By the method of expert assessments, information system projects with point values are determined. Information system projects with maximum values of scores are selected to determine the optimal option. The method of the conditional Frank-Wolf gradient determines the optimal solution of a nonlinear mathematical model. The essence of gradient methods is a sequential change in the value of the objective function when moving from the starting point of the domain of acceptable solutions to the optimal value of the function. The algorithm of the Frank-Wulff method includes the following steps: determining the initial value of the point of the domain of acceptable solutions, calculating the gradient, determining the optimal solution to a linear programming problem, moving to a new point, checking the condition for the end of the iterative process. The iterative process of determining the optimal solution ends when the calculated value of the calculation accuracy is less than the given value. Main results of research: preliminary projects of the information system were determined using the method of expert assessments, the optimal solution of the mathematical model of nonlinear programming by the Frank-Wolf method was found. A mathematical model for determining the optimal design of an information system will improve the efficiency and quality of the validity of decisions made, reduce financial costs and design time of information systems. The results obtained can be used in further research on this topic.

Optimisation of Ammonia Production and Supply Chain from Sugarcane Ethanol and Biomethane: A Robust Mixed-Integer Linear Programming Approach

Optimisation of Ammonia Production and Supply Chain from Sugarcane Ethanol and Biomethane: A Robust Mixed-Integer Linear Programming Approach

EVALUATING PERFORMANCE, COMBUSTION, AND EMISSION CHARACTERISTICS OF WASTE PLASTIC OIL BLENDS IN CRDI DIESEL ENGINES USING DATA ENVELOPMENT ANALYSIS

As fossil fuels run out quickly, researchers studying engines are increasingly interested in alternative fuels. This paper covers the use of plastic paralysis oil as biodiesel in diesel engines. This was done using a four-stroke, multi-cylinder CRDI diesel engine running between 2000 and 2500 rpm. From zero loads to fifty percent load, the performance, combustion, and emission characteristics were computed for a variety of load scenarios. The analysis revealed that the blend D80WPO20 at 2000 rpm reduced hydrocarbon emissions from 0.059 g/kW to 0.045 g/kW, which is lower than that of 25% diesel at 26 kW. At high speed, the blend D80WPO20 at 2000 rpm gave the maximum BTE of 34.53%. The nitric oxide (NOx) emissions from D50WPO50 and diesel at a 26 kW load are 6.48 g/kWh and 5.37 g/kWh, respectively, according to reports. Data envelopment analysis, a multi-response linear programming optimization tool, was used to evaluate the output and emissions of DI diesel engines using waste plastic oil mixtures.

Robust day-ahead scheduling of cooperative energy communities considering multiple aggregators

Future cities must play a vital role in reducing energy consumption and decarbonizing the electricity sector, thus evolving from passive structures towards more efficient smart cities. This transition can be facilitated by energy communities. This emerging paradigm consists of collectivizing a set of residential installations equipped with onsite renewable generators and storage assets (i.e., prosumers), which can eventually share resources to pursue collective welfare. This paper focuses on cooperative communities, where prosumers share resources without seeking selfish monetary counterparts. Despite their apparent advantages, energy management and scheduling of energy communities suppose a challenge for conventional tools due to the high level of uncertainty (especially due to intermittent renewable generation and random demand), and privacy concerns among prosumers. This paper addresses these issues. Specifically, a novel management structure based on multiple aggregators is proposed. This paradigm preserves users' confidential features while allowing them to extract the full potential of their assets. To efficiently manage the variety of assets available under uncertainty, an adaptive robust day-ahead scheduling model is developed, which casts as a solvable and portable Mixed Integer Linear Programming framework, which eases its implementation in real-world cases. The new proposal concerns uncertain generation and demand using a polyhedral representation of the uncertainty set. A case study is conducted to validate the developed model, showing promising results. Moreover, different results are obtained and analysed. Finally, it is worth remarking on how the level of robustness impacts the collective bill, incrementing it by 75 % when risk-averse conditions are assumed. In addition, the role of storage assets under pessimistic conditions is remarked, pointing out that these assets rule the scheduling plan of the community instead of renewable generators.

Linear programming based heuristic algorithms for tunnel assignment and reconfiguration in MPLS networks

Linear programming based heuristic algorithms for tunnel assignment and reconfiguration in MPLS networks

WAOA: A hybrid whale-ant optimization algorithm for energy-efficient routing in wireless sensor networks

Wireless Sensor Networks (WSNs) are vital for collecting data from remote environments. Nevertheless, the limited energy resources of sensor nodes render energy-efficient routing a critical concern for the successful operation of WSNs. To address these concerns, clustering, and routing are essential tasks in WSNs; clustering aims to organize sensor nodes into groups or clusters to minimize energy usage and prolong the network's lifespan. On the other hand, routing involves determining the optimum paths for transmitting data from the source nodes to the destination nodes. Nonetheless, it has been established that the current energy-efficient routing problem is an NP-hard, requiring a trade-off between energy and overall network performance. In this paper, we proposed a Hybrid Whale-Ant Optimization Algorithm (WAOA) for energy-efficient routing in WSNs. The proposed WAOA utilizes the Whale Optimization Algorithm (WOA) to find the suitable cluster head in the predefined search space, while the Ant Colony Optimization (ACO) searches the optimal route from the source cluster sensors to the cluster head within its predefined space. Linear programming construction is employed to formulate optimization problems for cluster head selection and search for the optimal route. The performance analysis demonstrates that the proposed WAOA performs better than MOORP, MMABC, and AZEBR by 5.78 %,16.11 %, and 18.52 %, respectively, in terms of network lifetime.

Deep reinforcement learning driven trajectory-based meta-heuristic for distributed heterogeneous flexible job shop scheduling problem

As the production environment evolves, distributed manufacturing exhibits heterogeneous characteristics, including diverse machines, workers, and production processes. This paper examines a distributed heterogeneous flexible job shop scheduling problem (DHFJSP) with varying processing times. A mixed integer linear programming (MILP) model of the DHFJSP is formulated with the objective of minimizing the makespan. To solve the DHFJSP, we propose a deep Q network-aided automatic design of a variable neighborhood search algorithm (DQN-VNS). By analyzing schedules, sixty-one types of scheduling features are extracted. These features, along with six shaking strategies, are used as states and actions. A DHFJSP environment simulator is developed to train the deep Q network. The well-trained DQN then generates the shaking procedure for VNS. Additionally, a greedy initialization method is proposed to enhance the quality of the initial solution. Seven efficient critical path-based neighborhood structures with three-vector encoding scheme are introduced to improve local search. Numerical experiments on various scales of instances validate the effectiveness of the MILP model and the DQN-VNS algorithm. The results show that the DQN-VNS algorithm achieves an average relative percentage deviation (ARPD) of 3.2%, which represents an approximately 88.45% reduction compared to the best-performing algorithm among the six compared, with an ARPD of 27.7%. This significant reduction in ARPD highlights the superior stability and performance of the proposed DQN-VNS algorithm.

Stochastic Risk-driven Bidding of a Solar and Storage Aggregator in Primary Frequency and Energy Markets: A Performance-based Capacity Allocation Approach

The ever-increasing impact of deploying renewable energy resources reducing power system inertia, requires distributed energy resource (DER) aggregators to secure high-performance, fast-responding primary frequency reserve (PFR) in ancillary service markets. However, enabling immediate and local regulation of the operating point upon detecting frequency deviations imposes the necessity of allocating specific headroom for DERs. The individual headroom susceptibility to uncertainty and response speed of a certain DER not only compromises aggregated headroom performance in the PFR market but also leads to incurring unnecessary opportunity costs in the energy market and inadequate frequency control in the power system. To address this challenge, this paper introduces an integrated probabilistic performance index for DERs that considers their individual interactive heterogeneous response speed and uncertainty. Using this index, aggregators overseeing photovoltaic and smart buildings equipped with energy storage can evaluate, allocate, and remunerate optimal headroom capacity for individual DERs based on their performance. This approach ensures maximum aggregated profits and minimum opportunity costs in the PFR and energy markets while improving the performance of the provided PFR capacity, respectively. These tasks are aimed by a novel optimal capacity allocation strategy for an aggregator in both markets. This strategy employs a mixed-integer linear programming model to find the optimal solution and the conditional value-at-risk measure to tackle the uncertainties faced with the problem. The efficiency of the proposed model is assessed within a system reflecting the ERCOT market structure, demonstrating improved aggregated headroom performance, reduced energy opportunity costs, diminished risk of capacity shortage, and improved load frequency control (LFC) from the power system perspective.

Augmented ɛ‐constraint‐based matheuristic methodology for Bi‐objective production scheduling problems

AbstractMatheuristic is an optimisation methodology that integrates mathematical approaches and heuristics to address intractable combinatorial optimisation problems, where a common framework is to insert mixed integer linear programming (MILP) models as local search functions for evolutionary algorithms. However, since a mathematical programming formulation only tries to find the solution with the best objective value, matheuristics are rarely adopted to multi‐objective scenarios asking for a set of Pareto optimal solutions, for example, vehicle routing problems and production scheduling problems. In this situation, the ɛ‐constraint, which transforms multi‐objective problems into single‐objective formulations by considering selected objectives as constraints, seems to be a promising approach. First, an augmented ɛ‐constraint‐based matheuristic methodology (ɛ‐MH) is proposed to apply the idea of ɛ‐constraint to embedded MILP models, so that Pareto fronts obtained by meta‐heuristics can be further improved by solving a set of MILP models. Afterwards, four speed‐up strategies are developed to alleviate the computational burden resulting from repeatedly solving mathematical formulations, which also imply preferable scenarios for taking advantages of the ɛ‐MH. Finally, several real‐world bi‐objective scheduling problems are discussed to present potential applications for the proposed methodology.

An Improved Variable Neighborhood Search for the Reconfigurable Assembly Line Reconfiguring Problem

Throughout the past several decades, the manufacturing industry has been confronted with rapidly evolving market demands. The Reconfigurable Assembly Line (RAL) changes the quantity and variety of products produced through reconfiguration at different stages. The reconfiguration stages of RAL include the reallocation of operations and resources. However, existing research has neglected the resources. In the paper, a mixed integer linear program for the Reconfigurable Assembly Line Reconfiguring Problem is first formulated. Subsequently, an efficient data structure is developed to model the process precedence constraint and alternative resources. And an improved Variable Neighborhood Search which includes new problem-specific neighborhood structures and local search for resource selection is proposed to solve the problem. Computational results demonstrate the contribution of the proposed data structure and the improved procedure. Furthermore, the experiments verify the superiority and stability of the proposed method compared with other heuristic algorithms.

Game theoretic operation optimization of photovoltaic storage charging station considering uncertainty and carbon trading

With the advancement of energy conservation and emission reduction efforts, the orderly charging of electric vehicles and the operation of photovoltaic-storage-charging stations associated with electric vehicles have become increasingly important topics. This study constructs an optimization model for the operation of stations under the synergy of electricity and carbon markets from a game theory perspective. Firstly, Latin hypercube sampling and Monte Carlo sampling are employed to handle the uncertainties in photovoltaic output and the stochastic nature of electric vehicles charging. Secondly, a ladder-type carbon trading mechanism is introduced, and a charging optimization model based on Stackelberg game theory is developed to describe the benefit interaction between charging stations and electric vehicles users. In this model, the upper level represents the charging station operator aiming to maximize joint electricity and carbon revenue while minimizing load fluctuations, whereas the lower level represents electric vehicles users aiming to maximize consumer surplus. Finally, a genetic algorithm nested with mixed-integer linear programming is used to solve the optimization model. The simulation results validate the model's effectiveness and superiority. The results indicate that, compared to centralized optimization methods, the Stackelberg game mechanism can increase consumer surplus by 119.40 %, reduce carbon emissions by 217.92 %, and achieve a win-win situation for both parties. Compared to a fixed carbon trading value, the ladder-type carbon trading mechanism can reduce carbon emissions by 23.84 % and smooth load fluctuations.

Practical solutions for microgrid energy management: Integrating solar forecasting and correction algorithms

Nowadays, the shift towards renewable energies is happening at an exponential pace. Microgrids are an integral part in integrating renewable energies into the global energy mix. Due to the volatility of renewables, such as solar or wind, proper energy management is needed to avoid generation or load curtailment. This issue is addressed throughout the literature using a wide variety of strategies, ranging from classical linear to nature inspired metaheuristic optimization algorithms. This paper goes beyond the simulation phase of different optimization algorithms and addresses the optimal energy management problem by proposing a novel framework to integrate optimization strategies into the daily operation of microgrids. The proposed framework showcases 3 practical methods for controlling the distributed generators, as well as a communication scheme which facilitates the data transfer between the machine running the optimization strategy and the energy management system of the microgrid. The framework is demonstrated on a small-scale islanded microgrid setup, located at the Technical University of Cluj-Napoca. The microgrid setup consists of a photovoltaic array, a battery energy storage system and two dispatchable generators. A two-stage optimization strategy is used, consisting of a day-ahead stage, which uses the solar forecast provided by the Global Forecast System, prior to each day, to issue a working plan for the dispatchable generators, and an intra-day stage, which uses hourly solar forecasts, issued during the day, to adjust the operation of the microgrid and minimize the error between the day-ahead solar forecast and the actual weather conditions. Mixed integer linear programming is used to solve the optimization problem and validate the correct operation of the proposed framework over two case studies. The results show that the microgrid works as intended, in both scenarios, as well as the benefits of using an intra-day correction stage.

The Implementation of Geogebra-Assisted Discovery Learning Approach Towards Mathematical Communication Ability of Vocational Students

Mathematicals communication ability is one of essential mathematical abilities that every person, who learns mathematics, needs to have. However, many facts found in many studies has shown us that students’ achievements on this ability are still low. Factors that may cause it spread from teacher factors, tools, approaches, learning media to students themselves. To overcome these issues, many efforts are made to select and implement learning approaches that can improve students' mathematical communication ability as well as their activities in instructional process. Thus, the purpose of this study is to examine whether the implementation of GeoGebra-assisted discovery learning approach is effective in improving vocational school students' mathematicals communication ability or not. This quasi experimental research design involved 60 students of Gema Karya Bahana Vocational School as samples. The samples were divided into two groups in which 30 students of Accounting-1 class as the experimental group and 30 students of Accounting-2 as the control group. The data collected through a description test consisted of 3 questions on a linear program. The data collected through the test, both in pre-test and post-test, then analyzed using inferential statistics. The results of this analysis showed that the implementation of the discovery Learning approach assisted by GeoGebra was effective in improving vocational school students' mathematicals communication ability. Thus, the discovery learning approach assisted by GeoGebra can be implemented on instructional processes as one of instructional approach alternatives.

A sustainable network design for municipal solid waste management considering waste-to-energy conversion under uncertainty

This research proposes a sustainable municipal solid waste management (MSWM) system to address population growth, consumerism, and resource scarcity. It introduces a two-phase decision structure combining multi-attribute decision-making (MADM) tools. Phase one develops a hybrid risk-oriented fuzzy MADM tool for selecting optimal waste-to-energy technologies, considering environmental and technological factors. Phase two uses a fuzzy bi-objective multi-period mixed-integer linear programming model to design the MSWM supply chain efficiently under uncertainty. A case study in North Tehran demonstrates the framework’s practical applicability and effectiveness in real-world scenarios, highlighting that effective source separation boosts recycling, reduces costs, and provides social benefits. Sensitivity analyses offer insights to enhance waste segregation practices. The study emphasizes integrating economic, environmental, and social sustainability in waste management decision-making. By offering a novel, holistic approach, this research addresses existing gaps in systematic decision-making processes and provides a robust tool for municipalities to optimize strategies under uncertainty.

AUTOMAP: Inferring Rank-Polymorphic Function Applications with Integer Linear Programming

Dynamically typed array languages such as Python, APL, and Matlab lift scalar operations to arrays and replicate scalars to fit applications. We present a mechanism for automatically inferring map and replicate operations in a statically-typed language in a way that resembles the programming experience of a dynamically-typed language while preserving the static typing guarantees. Our type system---which supports parametric polymorphism, higher-order functions, and top-level let-generalization---makes use of integer linear programming in order to find the minimum number of operations needed to elaborate to a well-typed program. We argue that the inference system provides useful and unsurprising guarantees to the programmer. We demonstrate important theoretical properties of the mechanism and report on the implementation of the mechanism in the statically-typed array programming language Futhark.

CMSA based on set covering models for packing and routing problems

AbstractMany packing, routing, and knapsack problems can be expressed in terms of integer linear programming models based on set covering. These models have been exploited in a range of successful heuristics and exact techniques for tackling such problems. In this paper, we show that integer linear programming models based on set covering can be very useful for their use within an algorithm called “Construct, Merge, Solve & Adapt”(CMSA), which is a recent hybrid metaheuristic for solving combinatorial optimization problems. This is because most existing applications of CMSA are characterized by the use of an integer programming solver for solving reduced problem instances at each iteration. We present applications of CMSA to the variable-sized bin packing problem and to the electric vehicle routing problem with time windows and simultaneous pickups and deliveries. In both applications, CMSA based on a set covering model strongly outperforms CMSA when using an assignment-type model. Moreover, state-of-the-art results are obtained for both considered optimization problems.

Fast heuristics for the time‐constrained immobile server problem

AbstractWe propose easy‐to‐implement heuristics for time‐constrained applications of a problem referred to in the literature as the facility location problem with immobile servers, stochastic demand, and congestion, the service system design problem, or the immobile server problem (ISP). The problem is typically posed as one of allocating capacity to a set of M/M/1 queues to which customers with stochastic demand are assigned with the objective of minimizing a cost function composed of a fixed capacity‐acquisition cost, a variable customer‐assignment cost, and an expected‐waiting‐time cost. The expected‐waiting‐time cost results in a nonlinear term in the objective function of the standard binary programming formulation of the problem. Thus, the solution approaches proposed in the literature are either sophisticated linearization or relaxation schemes, or metaheuristics. In this study, we demonstrate that an ensemble of straightforward, greedy heuristics can rapidly find high‐quality solutions. In addition to filling a gap in the literature on ISP heuristics, new stopping criteria for an existing cutting plane algorithm are proposed and tested, and a new mixed‐integer linear model requiring no iterating algorithm is developed. In many cases, our heuristic approach finds solutions of the same or better quality than those found by exact methods implemented with expensive, state‐of‐the‐art mathematical programming software, in particular a commercial nonlinear mixed‐integer linear programming solver, given a five‐minute time limit.

Sustainable optimization of balancing valve settings in urban heating systems with an enhanced Jaya algorithm

With the continuous growth of global energy demand, energy conservation in heating has become a research hotspot for urban sustainable development. However, the research on energy conservation by adjusting the balance valve in the secondary network of heating system is relatively few. To address this gap, this paper solves the balancing valve opening adjustment (BVOA) problem with the goal of minimizing heat consumption and the energy usage of circulation pumps. A mixed-integer linear programming model is designed for this problem, and an effective EJaya algorithm is proposed. Initially, a random initialization method with three scrambling strategies is employed to generate high-quality solutions. Moreover, a segmented operator is proposed to enhance the algorithm’s search efficiency. Subsequently, Q-learning is utilized to select from three neighborhood operators, enhancing the algorithm’s local search capabilities. Additionally, an adaptive search balance strategy is designed to balance the methods of updating solutions, aiding in finding better solutions. Finally, extensive experiments have validated the effectiveness and efficiency of the proposed algorithm in solving the BVOA problem, contributing to the promotion of urban energy sustainability and social welfare.

Fast and Optimal Extraction for Sparse Equality Graphs

Equality graphs (e-graphs) are used to compactly represent equivalence classes of terms in symbolic reasoning systems. Beyond their original roots in automated theorem proving, e-graphs have been used in a variety of applications. They have become particularly important as the key ingredient in the popular technique of equality saturation, which has notable applications in compiler optimization, program synthesis, program verification, and symbolic execution, among others. In a typical equality saturation workflow, an e-graph is used to store a large number of equalities that are generated by local rewrites during a saturation phase, after which an optimal term is extracted from the e-graph as the output of the technique. However, despite its crucial role in equality saturation, e-graph extraction has received relatively little attention in the literature, which we seek to start addressing in this paper. Extraction is a challenging problem and is notably known to be NP-hard in general, so current equality saturation tools rely either on slow optimal extraction algorithms based on integer linear programming (ILP) or on heuristics that may not always produce the optimal result. In fact, in this paper, we show that e-graph extraction is hard to approximate within any constant ratio. Thus, any such heuristic will produce wildly suboptimal results in the worst case. Fortunately, we show that the problem becomes tractable when the e-graph is sparse, which is the case in many practical applications. We present a novel parameterized algorithm for extracting optimal terms from e-graphs with low treewidth, a measure of how “tree-like” a graph is, and prove its correctness. We also present an efficient Rust implementation of our algorithm and evaluate it against ILP on a number of benchmarks extracted from the Cranelift benchmark suite, a real-world compiler optimization library based on equality saturation. Our algorithm optimally extracts e-graphs with treewidths of up to 10 in a fraction of the time taken by ILP. These results suggest that our algorithm can be a valuable tool for equality saturation users who need to extract optimal terms from sparse e-graphs.