Heuristic Solution Research Articles

A heuristic solution-based motion planning framework for redundant manipulators

Motion planning and optimization are the key and challenging problems for redundant manipulators operating in cluttered environments. This paper proposes a motion planning framework based on the heuristic solution that explores the optimal solutions for path planning and kinematic solutions by estimating the cost of target configurations via dynamic programming methods. A heuristic function model based on artificial neural networks (ANN) is constructed in the path planning structure and rapidly trained through the RRT* algorithm, leveraging value iteration concepts to search the state space. This structure can utilize previous experience to guide future exploration behavior with significant improvements in path quality and algorithm efficiency. The kinematic solving structure is unified with path planning by building a global energy optimal heuristic function. K-means is employed to determine the initial policy, avoid ineffective searches in non-critical spaces, and introduce gradient concepts to explore the optimal policy rapidly. The proposed method can obtain better energy optimization results while ensuring solving efficiency. The optimal joint angles of the manipulator are determined through collision detection and posture adjustment methods. Finally, the performance of the proposed framework is simulated and experimentally verified.

A stochastic-MILP dispatch optimization model for concentrated solar thermal under uncertainty

Concentrated Solar Thermal (CST) offers a promising solution for large-scale solar energy utilization as Thermal Energy Storage (TES) enables electricity generation independent of daily solar fluctuations, shifting to high-priced electricity intervals. The development of dispatch planning tools is mandatory to account for uncertainties associated with weather and electricity price forecasts. A Stochastic Mixed-Integer Linear Program (SMILP) is proposed to maximize Sample Average Approximation (SAA) of expected profit within a specified scenario space. The SMILP exhibits robust performance, yet its computational time poses a challenge. Three heuristic solutions are developed which run a set of deterministic optimizations on different historical weather profiles to generate candidate Dispatch Plans (DPs). Subsequently, the DP with the best average performance on all profiles is selected. The new methods are applied to a hypothetical 115 MW CST plant in South Australia. When the historical database has a limited set of historical weather profiles, the SMILP achieves 6–9 % higher profit than the closest heuristic when the DPs are applied to novel weather conditions. With a large historical weather dataset, the performance of the SMILP and closet heuristic becomes nearly identical since the SMILP can only utilize a limited number of trajectories for optimization without becoming computationally infeasible. In this case, the heuristic emerges a practical alternative, providing similar average profit in a reasonable time. Taken together, the results illustrate the importance of considering uncertainty in DP optimization and indicate that straightforward heuristics on a large database are a practical method for addressing uncertainty.

Heuristic solution to the problem of diffraction of a TE-polarized electromagnetic field on a half-plane with generalized two-sided impedance boundary conditions

Employing the method of fundamental components, heuristic formulas are constructed for solving the problem of diffraction on a half-plane with non-ideal boundary conditions. The problem of diffraction of a TE-polarized wave on a half-plane with generalized two-sided impedance boundary conditions was chosen as a verification solution. The accuracy was increased by two types of adjustment functions. The effectiveness of the applied technique is proven by quantification of accuracy. This approach represents a new way to obtain high-accuracy heuristic formulas and allows one to create solvers that simultaneously have both high accuracy and high performance. The main result is the development and description of adjustment methods for increasing the accuracy of heuristic formulas.

Location and routing of armed Unmanned Aerial Vehicles and carrier platforms against mobile targets

In this study, we consider a real-life combinatorial optimization problem related to deploying and routing Unmanned Aerial Vehicles (UAVs) and naval carrier platforms. In particular, we seek to determine the initial locations for carrier platforms and the optimal type and number of UAVs to be stationed on each carrier platform as well as their spatial/temporal routes for engaging hostile surface targets in the region. Our modeling framework incorporates a number of realistic but challenging ingredients and assumptions such as the mobility of surface targets and carrier platforms during the mission, capacitated multiple platforms and UAVs, UAV-carrier platform compatibility, and allowance for different takeoff/land on platforms for UAVs. In the effort to represent the problem mathematically, we first formulated an Integer Linear Program (ILP) model which seeks to maximize the total time-dependent weights of the targets engaged. Next, we proposed a heuristic solution algorithm based on the ant colony optimization framework. Our computational experiments performed on instances with different sizes showed that the heuristic approach achieves high-quality solutions even for large-size problem instances in short CPU times.

Finding small feedback arc sets on large graphs

The minimum feedback arc set problem (FASP), which seeks to remove a minimum set of arcs from a directed graph to make the remaining graph acyclic, is fundamental in graph algorithms with many applications in social network analysis, circuit testing, deadlock resolution, algorithmic games, and other fields. However, in theory, FASP is NP-Hard and cannot be approximated within any constant under the Unique-Games Conjecture. In practice, the performance of existing algorithms on large graphs is still unsatisfactory. To solve FASP on large graphs, we propose two novel methods in this paper. First, we systematically study the reduction rules for FASP that can quickly reduce the scale of input graphs without losing optimality. We integrate the reduction rules into an algorithm that can be used as a preprocessor for all existing FASP algorithms to simplify the input graphs. Second, we provide a scalable heuristic algorithm based on the divide-and-conquer method for better solution quality. Extensive experiments on a wide range of real-world graphs show that the reduction algorithm effectively reduces the size of input instances. Furthermore, for half of the circuit benchmark graphs, our final algorithm improves the quality of the existing heuristic solution by 24% to 40% within an acceptable running time.

Capacity design in school choice

We study a new variant of the school choice problem in which capacities can be altered by distributing additional seats across schools in response to students' reported preferences. We show that heuristic solutions to this capacity design problem can be inefficient, even if they focus on allocating seats to the most demanded schools. We introduce a simple class of algorithms that, in the problem where additional seats can be distributed, characterizes the set of efficient matchings among those that respect priorities. We also investigate the incentive properties of this class of efficient algorithms.

Optimisation of process plan generation for reconfigurable manufacturing systems: efficient heuristics and lower bounds

ABSTRACT We address the process plan generation problem in a reconfigurable manufacturing environment. The objective is to minimise the total production time of a given part which includes the processing times of operations, as well as the changeover times for machines, configurations, and tools. A new efficient 0–1 mathematical programming formulation is proposed, together with a heuristic and two lower bounds. The formulation is based on one main decision variable (1MDV). It outperforms a well known two-main decision variable formulation (2MDV) from the literature. Using a commercial solver, the 1MDV formulation is on average almost five times faster than the 2MDV formulation across all instances where optimality was achieved. The proposed heuristic is a decomposition mathematical programming-based heuristic. For small benchmark instances solved to optimality using 1MDV formulation, the heuristic obtains solutions at less than 1% distance on average from optimum in few seconds. The best of the two proposed lower bounds, LB2, is obtained in less than 0.56 seconds, across all tested instances, while the 1MDV model needs about 30 seconds on average to reach the same result. These lower bounds are used to measure the performance of the heuristics on large size instances

Hierarchical planning-scheduling-control — Optimality surrogates and derivative-free optimization

Planning, scheduling, and control typically constitute separate decision-making units within chemical companies. Traditionally, their integration is modelled sequentially, but recent efforts prioritize lower-level feasibility and optimality, leading to large-scale, potentially multi-level, hierarchical formulations. Data-driven techniques, like optimality surrogates or derivative-free optimization, become essential in addressing ensuing tractability challenges. We demonstrate a step-by-step workflow to find a tractable solution to a tri-level formulation of a multi-site, multi-product planning-scheduling-control case study. We discuss solution tractability-accuracy trade-offs and scaling properties for both methods. Despite individual improvements over conventional heuristics, both approaches present drawbacks. Consequently, we synthesize our findings into a methodology combining their strengths. Our approach remains agnostic to the level-specific formulations when the linking variables are identified and retains the heuristic sequential solution as fallback option. We advance the field by leveraging parallelization, hyperparameter tuning, and a combination of off- and on-line computation, to find tractable solutions to more accurate multi-level formulations.

Model and heuristics for the multi-manned assembly line worker integration and balancing problem

This paper examines the balancing of assembly lines with multi-manned stations and a heterogeneous workforce. Both topics received considerable attention in the literature, but not in an integrated fashion. Combining these two characteristics gives rise to a highly combinatorial Multi-manned Assembly Line Worker Integration and Balancing Problem. When considering multi-manned stations, the already coupled decisions on assigning tasks to heterogeneous workers and workers to stations must be further linked with task scheduling assessments. We propose a Mixed-Integer Linear Programming model and develop two heuristic solution procedures, which tackle the problem with a hierarchical decomposition approach. Computational tests on a large dataset indicate that the proposed method can obtain good primal bounds in short computational times. We demonstrate that these results can be applied to the monolithic model either as a warm start or in a proximity search procedure to obtain synergistic gains with statistically significant differences. From a managerial perspective, we show that multi-manned stations can reduce the assembly line's length even in the presence of a heterogeneous workforce, which is crucial for many industries manufacturing large-size products.

Elite Multi-Criteria Decision Making—Pareto Front Optimization in Multi-Objective Optimization

Optimization is a process of minimizing or maximizing a given objective function under specified constraints. In multi-objective optimization (MOO), multiple conflicting functions are optimized within defined criteria. Numerous MOO techniques have been developed utilizing various meta-heuristic methods such as Evolutionary Algorithms (EAs), Genetic Algorithms (GAs), and other biologically inspired processes. In a cooperative environment, a Pareto front is generated, and an MOO technique is applied to solve for the solution set. On other hand, Multi-Criteria Decision Making (MCDM) is often used to select a single best solution from a set of provided solution candidates. The Multi-Criteria Decision Making–Pareto Front (M-PF) optimizer combines both of these techniques to find a quality set of heuristic solutions. This paper provides an improved version of the M-PF optimizer, which is called the elite Multi-Criteria Decision Making–Pareto Front (eMPF) optimizer. The eMPF method uses an evolutionary algorithm for the meta-heuristic process and then generates a Pareto front and applies MCDM to the Pareto front to rank the solutions in the set. The main objective of the new optimizer is to exploit the Pareto front while also exploring the solution area. The performance of the developed method is tested against M-PF, Non-Dominated Sorting Genetic Algorithm-II (NSGA-II), and Non-Dominated Sorting Genetic Algorithm-III (NSGA-III). The test results demonstrate the performance of the new eMPF optimizer over M-PF, NSGA-II, and NSGA-III. eMPF was not only able to exploit the search domain but also was able to find better heuristic solutions for most of the test functions used.

ESKEMAP: exact sketch-based read mapping

BackgroundGiven a sequencing read, the broad goal of read mapping is to find the location(s) in the reference genome that have a “similar sequence”. Traditionally, “similar sequence” was defined as having a high alignment score and read mappers were viewed as heuristic solutions to this well-defined problem. For sketch-based mappers, however, there has not been a problem formulation to capture what problem an exact sketch-based mapping algorithm should solve. Moreover, there is no sketch-based method that can find all possible mapping positions for a read above a certain score threshold.ResultsIn this paper, we formulate the problem of read mapping at the level of sequence sketches. We give an exact dynamic programming algorithm that finds all hits above a given similarity threshold. It runs in O(|t|+|p|+ℓ2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O} (|t| + |p| + \\ell ^2)$$\\end{document} time and O(ℓlogℓ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O} (\\ell \\log \\ell )$$\\end{document} space, where |t| is the number of k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k$$\\end{document}-mers inside the sketch of the reference, |p| is the number of k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k$$\\end{document}-mers inside the read’s sketch and ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell$$\\end{document} is the number of times that k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k$$\\end{document}-mers from the pattern sketch occur in the sketch of the text. We evaluate our algorithm’s performance in mapping long reads to the T2T assembly of human chromosome Y, where ampliconic regions make it desirable to find all good mapping positions. For an equivalent level of precision as minimap2, the recall of our algorithm is 0.88, compared to only 0.76 of minimap2.

Long-term upgrade strategies in multiband and multifiber optical transport networks

Flexible-grid optical transport networks (OTNs) are currently undergoing a massive scale upgrade to bandwidth variable transceivers (BVTs) to cater to the exponential Internet traffic growth. Network operators seeking to plan future capacity in a scalable manner can also add additional bands of operations or additional fibers in the OTN. We propose two strategies for planning multiband upgrades in OTNs, which use an integer linear programming (ILP) solution and a heuristic solution with a reinforcement learning (RL)-based extension. Network simulation studies on three differently sized OTNs show that the heuristic and its RL-based extension perform extremely close to the optimal ILP-based solution while reducing execution times drastically. Finally, a techno-economic study based on what we believe to be a novel cost model highlights the cost-per-bit-per-second savings through carefully planned upgrades.

Obtaining the Grundy chromatic number: How bad can my greedy heuristic coloring be?

Given a simple undirected graph G, its Grundy chromatic number Γ(G) (or Grundy number) defines the worst-case behavior for the well-known and widely-used greedy first-fit coloring heuristic. Specifically, Γ(G) is the largest k for which a k-coloring can be obtained with the first-fit heuristic. In this paper, we address the Grundy coloring problem, the optimization problem consisting of obtaining the Grundy number of a graph. First, we propose a new combinatorial upper bound for the problem. Second, we describe a standard integer programming formulation and a formulation by representatives. The latter attempts to overcome the symmetries in the problem and relies on the idea that a subset of the vertices in the graph can be represented by one of its vertices, denoted as a representative. Finally, as integer programming approaches can struggle to tackle instances of the problem, we devise a biased random-key genetic algorithm (BRKGA) to obtain heuristic solutions in low computational times. Computational experiments show that our new upper bound can improve over well-established combinatorial bounds available in the literature for several instances. The results also indicate that the formulation by representatives has an overall superior performance than the standard formulation, achieving better results for the denser instances while slightly underperforming on the sparser ones. Furthermore, the BRKGA can find high-quality solutions and can be used with confidence in large instances where the formulations fail.

Heuristic solutions to optimize the traffic routing in space division multiplexing networks

This paper underscores the symbiosis between industrial engineering techniques and optical communication technologies, bringing valuable insights to the optimization of traffic routing in communication networks using constructive heuristics. Exploring and realizing the potential of optical fiber networks has received more attention due to the popularity of various network services and the ensuing increase in overall internet traffic. Space Division Multiplexing (SDM) stands out among the technologies designed to increase the functionality of core and urban mesh/ring networks. SDM's primary goal is to significantly increase transmission system bandwidths by effectively using the spatial dimension. The routing of traffic within flex-grid SDM optical networks has been optimized in this paper using two novel heuristic algorithms. The experimental results show that the proposed algorithms outperform the solutions presented in previous research from the literature.

On the tropical two-sided discrete logarithm and a key exchange protocol based on the tropical algebra of pairs

Since the existing tropical cryptographic protocols are either susceptible to the Kotov-Ushakov attack and its generalization, or to attacks based on tropical matrix periodicity and predictive behavior, several attempts have been made to propose protocols that resist such attacks. Despite these attempts, many of the proposed protocols remain vulnerable to attacks targeting the underlying hidden problems, one of which we call the tropical two-sided discrete logarithm with shift. An illustrative case is the tropical Stickel protocol, which, when formulated with a single monomial instead of a polynomial, becomes susceptible to attacks based on solutions of the above mentioned tropical version of discrete logarithm. In this paper we will formally introduce the tropical two-sided discrete logarithm with shift, discuss how it is solved, and subsequently demonstrate an attack on a key exchange protocol based on the tropical semiring of pairs. This particular protocol is compromised due to the existence of efficient (albeit heuristic) solution of the tropical two-sided logarithm problem, and this highlights the ongoing challenges in search of a “good” key exchange protocol in tropical cryptography.

Formulating and heuristic solving of contact problems in hybrid data-driven computational mechanics

In this work we consider the hybrid Data-Driven Computational Mechanics (DDCM) approach, in which a smooth constitutive manifold is reconstructed to obtain a well-behaved nonlinear optimization problem (NLP) rather than the much harder discrete-continuous NLP (DCNLP) of the direct DDCM approach. The key focus is on the addition of geometric inequality constraints to the hybrid DDCM formulation. Therein, the required constraint force leads to a contact problem in the form of a mathematical program with complementarity constraints (MPCC), a problem class that is still less complex than the DCNLP. For this MPCC we propose a heuristic quick-shot solution approach, which can produce verifiable solutions by solving up to four NLPs. We perform various numerical experiments on three different contact problems of increasing difficulty to demonstrate the potential and limitations of this approach.

Naval Air Defense Planning problem: A novel formulation and heuristics

AbstractThis article focuses on air defense in maritime environment, which involves protecting friendly naval assets from aerial threats. Specifically, we define and address the Naval Air Defense Planning (NADP) problem, which consists of maneuvering decisions of the ships and scheduling weapons and sensors to the threats in order to maximize the total expected survival probability of friendly units. The NADP problem is more realistic and applicable than previous studies, as it considers features such as sensor assignment requirements, weapon and sensor blind sectors, sequence‐dependent setup times, and ship's infrared/radar signature. In this study, a mixed‐integer nonlinear programming model of the NADP problem is presented and heuristic solution approaches are developed. Computational results demonstrate that these heuristic approaches are both fast and efficient in solving the NADP problem.

A Review of Robust Machine Scheduling

Robust optimization (RO) has been recognized as an effective means to deal with unanticipated events in highly uncertain and risky environments. This paper systematically reviews two types of emerging RO machine scheduling approaches—robust machine scheduling (R-MS) and distributionally R-MS (DR-MS) methods—which usually offer tractable formulations and analytical results for machine scheduling problems under uncertainty. First, after highlighting the advantages of RO methods over the stochastic approach in terms of tractability and robustness, we use the bibliometric method to analyze the literature related to R-MS/DR-MS problems and classify them from the following aspects: (1) uncertain factors, (2) uncertainty descriptions, (3) robustness criteria, (4) machine environments and (5) solution methods. Second, we discuss the uncertainty descriptions, and the robust feasibility and robust optimality criteria. We further provide a state-of-the-art review of R-MS/DR-MS models in different machine environments and discuss the performance of the R-MS/DR-MS models. Third, we review and discuss the existing exact, approximation, online, and heuristic solution methods for solving R-MS/DR-MS models. Finally, we present future research opportunities in two promising areas: green machine scheduling problems and machine learning-enabled algorithms. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —Machine scheduling plays an essential role in industrial and service systems, such as manufacturing, power generation, transportation and medical systems. However, in practice, scheduling systems usually operate in highly uncertain environments due to noisy measurements, prediction errors, and implementation deviations. To ensure robust feasibility and robust optimality, robust machine scheduling (R-MS) and distributionally R-MS (DR-MS) approaches have been recently proposed to hedge against the uncertainties related to processing time, release time, due date, machine breakdown, etc. This paper provides a comprehensive review of the R-MS/DR-MS models and algorithms in different machine environments from the aspects of uncertainty descriptions, robustness criteria and solution methods. This paper further highlights the challenges of R-MS problems and provides promising and valuable research opportunities in terms of problem formulations and algorithm designs.

Multi-objective Optimization for Green Delivery Routing Problems with Flexible Time Windows

ABSTRACT This paper presents a model and heuristic solution algorithms for the Green Vehicle Routing Problem with Flexible Time Windows. A scenario of new vehicle routing is analyzed in which customers are asked to provide alternative time windows to offer flexibility to help route planners find more fuel-efficient routes (“green delivery”). Customers can rank their preferred time windows as first, second, and third. The optimization model aims to reduce tour costs, promote electromobility over fossil fuels, such as diesel, and meet customer preferences when possible and affordable. The study incorporates a multi-objective optimization model with three objectives, which are overall cost, use of fossil fuel, and customer satisfaction. For the new problem, a set of realistic benchmark problems is created and four mainstream solvers are applied for the Pareto front approximation: NSGA-II, NSGA-III, MOEA/D, and SMS-EMOA. These algorithms are compared in terms of their effectiveness in achieving the objectives of minimizing travel costs, promoting electromobility, and meeting customer preferences. The study uses five different problems of single-vehicle route planning. Two major findings are that the selection of the metaheuristic can make a big difference in terms of algorithm performance. The resulting 3-D Pareto fronts reveal the nature of this new class of problems: Interestingly, in the new model with flexible time windows, most users can still be delivered in their most preferred time windows with only small concessions to the other objectives. However, using only one time window per user can lead to an increasingly drastic cost and fossil fuel consumption.

SAT-Based Tree Decomposition with Iterative Cascading Policy Selection

Solvers for propositional satisfiability (SAT) effectively tackle hard optimization problems. However, translating to SAT can cause a significant size increase, restricting its use to smaller instances. To mitigate this, frameworks using multiple local SAT calls for gradually improving a heuristic solution have been proposed. The performance of such algorithmic frameworks heavily relies on critical parameters, including the size of selected local instances and the time allocated per SAT call. This paper examines the automated configuration of the treewidth SAT-based local improvement method (TW-SLIM) framework, which uses multiple SAT calls for computing tree decompositions of small width, a fundamental problem in combinatorial optimization. We explore various TW-SLIM configuration methods, including offline learning and real-time adjustments, significantly outperforming default settings in multi-SAT scenarios with changing problems. Building upon insights gained from offline training and real-time configurations for TW-SLIM, we propose the iterative cascading policy—a novel hybrid technique that uniquely combines both. The iterative cascading policy employs a pool of 30 configurations obtained through clustering-based offline methods, deploying them in dynamic cascades across multiple rounds. In each round, the 30 configurations are tested according to the cascading ordering, and the best tree decomposition is retained for further improvement, with the option to adjust the following ordering of cascades. This iterative approach significantly enhances the performance of TW-SLIM beyond baseline results, even within varying global timeouts. This highlights the effectiveness of the proposed iterative cascading policy in enhancing the efficiency and efficacy of complex algorithmic frameworks like TW-SLIM.