Combinatorial Optimization Problems Research Articles

Research on Multi-Objective Flexible Job Shop Scheduling Problem with Setup and Handling Based on an Improved Shuffled Frog Leaping Algorithm

Flexible job shop scheduling problem (FJSP), widely prevalent in many intelligent manufacturing industries, is one of the most classic problems of production scheduling and combinatorial optimization. In actual manufacturing enterprises, the setup of machines and the handling of jobs have an important impact on the scheduling plan. Furthermore, there is a trend for a cluster of machines with similar functionalities to form a work center. Considering the above constraints, a new order-driven multi-equipment work center FJSP model with setup and handling including multiple objectives encompassing the minimization of the makespan, the number of machine shutdowns, and the number of handling batches is established. An improved shuffled frog leading algorithm is designed to solve it through the optimization of the initial solution population, the improvement of evolutionary operations, and the incorporation of Pareto sorting. The algorithm also combines the speed calculation method in the gravity search algorithm to enhance the stability of the solution search. Some standard FJSP data benchmarks have been selected to evaluate the effectiveness of the algorithm, and the experimental results confirm the satisfactory performance of the proposed algorithm. Finally, a problem example is designed to demonstrate the algorithm’s capability to generate an excellent scheduling plan.

Quantum-enhanced hybrid feature engineering in thoracic CT image analysis for state-of-the-art nodule classification: an advanced lung cancer assessment

The intricate nature of lung cancer treatment poses considerable challenges upon diagnosis. Early detection plays a pivotal role in mitigating its escalating global mortality rates. Consequently, there are pressing demands for robust and dependable early detection and diagnostic systems. However, the technological limitations and complexity of the disease make it challenging to implement an efficient lung cancer screening system. AI-based CT image analysis techniques are showing significant contributions to the development of computer-assisted detection (CAD) systems for lung cancer screening. Various existing research groups are working on implementing CT image analysis systems for assessing and classifying lung cancer. However, the complexity of different structures inside the CT image is high and comprehension of significant information inherited by them is more complex even after applying advanced feature extraction and feature selection techniques. Traditional and classical feature selection techniques may struggle to capture complex interdependencies between features. They may get stuck in local optima and sometimes require additional exploration strategies. Traditional techniques may also struggle with combinatorial optimization problems when applied to a prominent feature space. This paper proposed a methodology to overcome the existing challenges by applying feature extraction using Vision Transformer (FexViT) and Feature selection using the Quantum Computing based Quadratic unconstrained binary optimization (QC-FSelQUBO) technique. This algorithm shows better performance when compared with other existing techniques. The proposed methodology showed better performance as compared to other existing techniques when evaluated by applying necessary output measures, such as accuracy, Area under roc (receiver operating characteristics) curve, precision, sensitivity, and specificity, obtained as 94.28%, 99.10%, 96.17%, 90.16% and 97.46%. The further advancement of CAD systems is essential to meet the demand for more reliable detection and diagnosis of cancer, which can be addressed by leading the proposed quantum computation and growing AI-based technology ahead.

Time-reliability optimization for the stochastic traveling salesman problem

This paper presents a novel approach to addressing the Stochastic Traveling Salesman Problem (STSP), a classical problem in combinatorial optimization, by integrating travel time and reliability factors into the decision-making process. Traditional TSP models primarily focus on minimizing the total travel distance or cost without considering the reliability of each route. In real-world situations, especially in logistics and network design, it's just as important to have reliable routes. A reliable route means there's a good chance it will be completed successfully and on time. Our research extends the conventional STSP framework by incorporating a reliability metric for each route, alongside the standard travel time metric. A tri-objective optimization model is proposed to minimize the mean and standard deviation of travel time and maximize route reliability simultaneously. A new algorithm called Permutation Binary-Addition-Tree (BAT) is proposed to solve the problem more efficiently when there is uncertainty. Our approach marks a significant step towards more realistic and practical solutions for route optimization problems in dynamic and uncertain environments. We also present a complexity analysis of our model against traditional cost-only TSP solutions, demonstrating the efficacy of considering reliability in route planning.

Pixelation of time matrices for solving permutation flowshop scheduling problem

Over the last decades, scheduling theory has been put into practice so as to tackle production process as a combinatorial optimisation problem and specifically as a PFSP (Permutation Flowshop Scheduling Problem). Often, NP-completeness of PFSP leads numerous research towards suggesting heuristic algorithms. The objective is to propose an optimal or near-to-optimal order of n jobs processing on m machines with a minimum completion time of all jobs. In this paper, a two-phase heuristic algorithm will be presented, named IFRS (Improved FRS). FRS, within the first phase, is going to be used to find a superior order of jobs out of Taillard’s instances; then the order of jobs will be improved again in phase 2. While running IFRS, a completely new idea, named Pixelation of time matrices, will be used for the very first time and provide a final pattern, which can be used in any heuristic algorithm with makespan criterion. After using the hard benchmark instances of Taillard, the superiority of IFRS over 20 algorithms is going to be shown by tables and statistical graphics. Due to its low makespan values, IFRS benefits flow production, as does NEH.

Explanations for combinatorial optimization problems

We introduce a representation for generating explanations for the outcomes of combinatorial optimization algorithms. The two key ideas are (A) to maintain fine-grained representations of the values manipulated by these algorithms and (B) to derive explanations from these representations through merge, filter, and aggregation operations. An explanation in our model presents essentially a high-level comparison of the solution to a problem with a hypothesized alternative, illuminating why the solution is better than the alternative. Our value representation results in explanations smaller than other dynamic program representations, such as traces. Based on a measure for the conciseness of explanations we demonstrate through a number of experiments that the explanations produced by our approach are small and scale well with problem size across a number of different applications.

A deep reinforcement learning algorithm framework for solving multi-objective traveling salesman problem based on feature transformation

As a special type of multi-objective combinatorial optimization problems (MOCOPs), the multi-objective traveling salesman problem (MOTSP) plays an important role in practical fields such as transportation and robot control. However, due to the complexity of its solution space and the conflicts between different objectives, it is difficult to obtain satisfactory solutions in a short time. This paper proposes an end-to-end algorithm framework for solving MOTSP based on deep reinforcement learning (DRL). By decomposing strategies, solving MOTSP is transformed into solving multiple single-objective optimization subproblems. Through linear transformation, the features of the MOTSP are combined with the weights of the objective function. Subsequently, a modified graph pointer network (GPN) model is used to solve the decomposed subproblems. Compared with the previous DRL model, the proposed algorithm can solve all the subproblems using only one model without adding weight information as input features. Furthermore, our algorithm can output a corresponding solution for each weight, which increases the diversity of solutions. In order to verify the performance of our proposed algorithm, it is compared with four classical evolutionary algorithms and two DRL algorithms on several MOTSP instances. The comparison shows that our proposed algorithm outperforms the compared algorithms both in terms of training time and the quality of the resulting solutions.

Capacitated Clustering via Majorization-Minimization and Collaborative Neurodynamic Optimization.

This paper addresses capacitated clustering based on majorization-minimization and collaborative neurodynamic optimization (CNO). Capacitated clustering is formulated as a combinatorial optimization problem. Its objective function consists of fractional terms with intra-cluster similarities in their numerators and cluster cardinalities in their denominators as normalized cluster compactness measures. To obviate the difficulty in optimizing the objective function with factional terms, the combinatorial optimization problem is reformulated as an iteratively reweighted quadratic unconstrained binary optimization problem with a surrogate function and a penalty function in a majorization-minimization framework. A clustering algorithm is developed based on CNO for solving the reformulated problem. It employs multiple Boltzmann machines operating concurrently for local searches and a particle swarm optimization rule for repositioning neuronal states upon their local convergence. Experimental results on ten benchmark datasets are elaborated to demonstrate the superior clustering performance of the proposed approaches against seven baseline algorithms in terms of 21 internal cluster validity criteria.

Binary matrix factorization via collaborative neurodynamic optimization

Binary matrix factorization is an important tool for dimension reduction for high-dimensional datasets with binary attributes and has been successfully applied in numerous areas. This paper presents a collaborative neurodynamic optimization approach to binary matrix factorization based on the original combinatorial optimization problem formulation and quadratic unconstrained binary optimization problem reformulations. The proposed approach employs multiple discrete Hopfield networks operating concurrently in search of local optima. In addition, a particle swarm optimization rule is used to reinitialize neuronal states iteratively to escape from local minima toward better ones. Experimental results on eight benchmark datasets are elaborated to demonstrate the superior performance of the proposed approach against six baseline algorithms in terms of factorization error. Additionally, the viability of the proposed approach is demonstrated for pattern discovery on three datasets.

Reconfigurable stochastic neurons based on strain engineered low barrier nanomagnets.

Stochastic neurons are efficient hardware accelerators for solving a large variety of combinatorial optimization problems. "Binary" stochastic neurons (BSN) are those whose states fluctuate randomly between two levels +1 and -1, with the probability of being in either level determined by an external bias. "Analog" stochastic neurons (ASNs), in contrast, can assume any state between the two levels randomly (hence "analog") and can perform analog signal processing. They may be leveraged for such tasks as temporal sequence learning, processing and prediction. Both BSNs and ASNs can be used to build efficient and scalable neural networks. Both can be implemented with low (potential energy) barrier nanomagnets (LBMs) whose random magnetization orientations encode the binary or analog state variables. The difference between them is that the potential energy barrier in a BSN LBM, albeit low, is much higher than that in an ASN LBM. As a result, a BSN LBM has a clear double well potential profile, which makes its magnetization orientation assume one of two orientations at any time, resulting in the binary behavior. ASN nanomagnets, on the other hand, hardly have any energy barrier at all and hence lack the double well feature. That makes their magnetizations fluctuate in an analog fashion. Hence, one can reconfigure an ASN to a BSN, and vice-versa, by simply raising and lowering the energy barrier. If the LBM is magnetostrictive, then this can be done with local (electrically generated) strain. Such a reconfiguration capability heralds a powerful field programmable architecture for a p-computer whereby hardware for very different functionalities such as combinatorial optimization and temporal sequence learning can be integrated in the same substrate in the same processing run. This is somewhat reminiscent of heterogeneous integration, except this is integration of functionalities or computational fabrics rather than components. The energy cost of reconfiguration is miniscule. There are also other applications of strain mediated barrier control that do not involve reconfiguring a BSN to an ASN or vice versa, e.g., adaptive annealing in energy minimization computing (Boltzmann or Ising machines), emulating memory hierarchy in a dynamically reconfigurable fashion, and control over belief uncertainty in analog stochastic neurons. Here, we present a study of strain engineered barrier control in unconventional computing.

Extension of CMSA with a Learning Mechanism: Application to the Far from Most String Problem

One of the problems with exact techniques for solving combinatorial optimization problems is that they tend to run into problems with growing problem instance size. Nevertheless, they might still be very usefully employed, even in the context of large problem instances, as a sub-ordinate method within so-called hybrid metaheuristics. “Construct, Merge, Solve and Adapt” (Cmsa) is a hybrid metaheuristic technique that allows the application of exact methods to large-scale problem instances through intelligent instance reduction. However, Cmsa does not make use of an explicit learning mechanism. In this work, an algorithm called LEARN_CMSA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsc {Learn}\\_\ extsc {Cmsa}$$\\end{document} is presented for the application to the far from most string problem (FFMSP), which is an NP-hard combinatorial optimization problem from the field of string consensus problems. LEARN_CMSA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsc {Learn}\\_\ extsc {Cmsa}$$\\end{document} results from hybridization between Cmsa and a population-based algorithm. By means of this hybridization, explicit learning is introduced to Cmsa. Even though the FFMSP is a well-studied problem, LEARN_CMSA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsc {Learn}\\_\ extsc {Cmsa}$$\\end{document} achieves superior performance when compared to current state-of-the-art solvers.

Restricted global optimization for QAOA

The Quantum Approximate Optimization Algorithm (QAOA) has emerged as a promising variational quantum algorithm for addressing NP-hard combinatorial optimization problems. However, a significant limitation lies in optimizing its classical parameters, which is in itself an NP-hard problem. To circumvent this obstacle, initialization heuristics, enhanced problem encodings and beneficial problem scalings have been proposed. While such strategies further improve QAOA’s performance, their remaining problem is the sole utilization of local optimizers. We show that local optimization methods are inherently inadequate within the complex cost landscape of QAOA. Instead, global optimization techniques greatly improve QAOA’s performance across diverse problem instances. While global optimization generally requires high numbers of function evaluations, we demonstrate how restricted global optimizers still show better performance without requiring an exceeding amount of function evaluations.

Two-Stage Probe-Based Search Optimization Algorithm for the Traveling Salesman Problems

As a classical combinatorial optimization problem, the traveling salesman problem (TSP) has been extensively investigated in the fields of Artificial Intelligence and Operations Research. Due to being NP-complete, it is still rather challenging to solve both effectively and efficiently. Because of its high theoretical significance and wide practical applications, great effort has been undertaken to solve it from the point of view of intelligent search. In this paper, we propose a two-stage probe-based search optimization algorithm for solving both symmetric and asymmetric TSPs through the stages of route development and a self-escape mechanism. Specifically, in the first stage, a reasonable proportion threshold filter of potential basis probes or partial routes is set up at each step during the complete route development process. In this way, the poor basis probes with longer routes are filtered out automatically. Moreover, four local augmentation operators are further employed to improve these potential basis probes at each step. In the second stage, a self-escape mechanism or operation is further implemented on the obtained complete routes to prevent the probe-based search from being trapped in a locally optimal solution. The experimental results on a collection of benchmark TSP datasets demonstrate that our proposed algorithm is more effective than other state-of-the-art optimization algorithms. In fact, it achieves the best-known TSP benchmark solutions in many datasets, while, in certain cases, it even generates solutions that are better than the best-known TSP benchmark solutions.

A spiral modeling based on manta ray foraging optimization for wireless networks resource allocation

SummaryThe wireless network resource allocation is a NP‐hard combinatorial optimization problem. This paper proposes a new optimization method using the manta ray foraging optimization (MRFO) based on spiral modeling to solve the wireless network resource problem. In order to reduce the computational complexity and ensure the optimal performance of the allocation scheme, a MRFO algorithm based on spiral modeling and mutation strategy is proposed. In the first stage, spiral modeling is introduced to narrow the exploration area, while the mutation strategy of the genetic algorithm enhances the ability of the algorithm to jump out of the local optimal. In the second stage, a binary MRFO algorithm for using different transfer functions is proposed to solve the mixed integer programming problem. The experimental results show that IMRFO and BIMRFO have high comprehensive performance advantages, achieve better effects than the other optimization algorithms, which lead to faster convergence, faster accuracy, and lower complexity.

Research on the trajectory planning and global optimization strategy of cold spraying technique for complex products coating preparation

PurposeThis paper aims to present a set of processes for obtaining the global spraying trajectory of a cold spraying robot on a complex surface.Design/methodology/approachThe complex workpiece surfaces in the project are first divided by triangular meshing. Then, the geodesic curve method is applied for local path planning. Finally, the subsurface trajectory combination optimization problem is modeled as a GTSP problem and solved by the ant colony algorithm, where the evaluation scores and the uniform design method are used to determine the optimal parameter combination of the algorithm. A global optimized spraying trajectory is thus obtained.FindingsThe simulation results show that the proposed processes can achieve the shortest global spraying trajectory. Moreover, the cold spraying experiment on the IRB4600 six-joint robot verifies that the spraying trajectory obtained by the processes can ensure a uniform coating thickness.Originality/valueThe proposed processes address the issue of different parameter combinations, leading to different results when using the ant colony algorithm. The two methods for obtaining the optimal parameter combinations can solve this problem quickly and effectively, and guarantee that the processes obtain the optimal global spraying trajectory.

Application of Lexisearch Algorithm to Vehicle Routing Problem with Time Windows

Vehicle Routing Problem with Time Windows (VRPTW) is an NP hard combinatorial scheduling optimization problem in which a minimum number of routes have to be determined so as to serve each of the destinations within their specified time windows. In this paper, the mail delivery of the Nigerian Postal Services (NIPOST) is modelled as a VRPTW in order to address the problem of delay in mail delivery occurring regularly in NIPOST. The Abuja Post Office is used as a case study and the Model of a related literature is applied with modifications to solve the problem. The problem is solved by applying Lexisearch algorithm using data that was obtained from Abuja Post Office and computational results on Solomon’s 100 instances were used to validate the algorithm.

An efficient improved exponential distribution optimizer: application to the global, engineering and combinatorial optimization problems

An efficient improved exponential distribution optimizer: application to the global, engineering and combinatorial optimization problems

A Feasible Flow-based Algorithm for Solving Classic Assignment Problem with Bottleneck

Classic assignment problem with bottleneck is a fundamental combinatorial optimization problem with numerous practical applications in diverse domains. In order to comprehensively explore the classic assignment problem with bottleneck and to develop efficient algorithm, a feasible flow-based algorithm is proposed for optimizing classic assignment problem with bottleneck. The bottleneck constraints are addressed by updating the bottleneck of the problem, and a feasible-flow approach is leveraged to efficiently identify optimal solutions within the network framework associated with such problems. To assess the effectiveness of our proposed algorithm, a comprehensive set of numerical experiments is carried out across various instances of classic assignment problem with bottleneck. Our numerical results demonstrate the superior performance of our algorithm, particularly when applied to problems with smaller cost scales, making it particularly suitable for addressing medium-sized to smaller-scale problems.

A CMOS-compatible oscillation-based VO2 Ising machine solver

Phase-encoded oscillating neural networks offer compelling advantages over metal-oxide-semiconductor-based technology for tackling complex optimization problems, with promising potential for ultralow power consumption and exceptionally rapid computational performance. In this work, we investigate the ability of these networks to solve optimization problems belonging to the nondeterministic polynomial time complexity class using nanoscale vanadium-dioxide-based oscillators integrated onto a Silicon platform. Specifically, we demonstrate how the dynamic behavior of coupled vanadium dioxide devices can effectively solve combinatorial optimization problems, including Graph Coloring, Max-cut, and Max-3SAT problems. The electrical mappings of these problems are derived from the equivalent Ising Hamiltonian formulation to design circuits with up to nine crossbar vanadium dioxide oscillators. Using sub-harmonic injection locking techniques, we binarize the solution space provided by the oscillators and demonstrate that graphs with high connection density (η > 0.4) converge more easily towards the optimal solution due to the small spectral radius of the problem’s equivalent adjacency matrix. Our findings indicate that these systems achieve stability within 25 oscillation cycles and exhibit power efficiency and potential for scaling that surpasses available commercial options and other technologies under study. These results pave the way for accelerated parallel computing enabled by large-scale networks of interconnected oscillators.

A hybrid multi-objective evolutionary algorithm with high solving efficiency for UAV defense programming

Emerging Unmanned Aerial Vehicle (UAV) application patterns, including the Loyal Wingman and Unmanned Swarms, have significantly challenged the administration and defense against illegal or trespassing UAVs. The weapon-target assignment (WTA) problem, a famous combinatorial optimization problem in military operations research, is decisive for the success of UAV defense programming. Related investigations face two main challenges: 1) Constructing a multi-objective optimization model reflecting the complexities of new UAV application patterns, and 2) solving the time-consuming WTA problem with high-dimension decision variables as it is NP-complete. This paper introduces a modified differential evolution based on a new mapping method called Pareto-Optimal-Matching that solves multi-objective binary optimization problems and constructs a 3 objective Sensor-Weapon-Target Assignment model. The proposed algorithm overcomes the inherent limitation of traditional differential evolution, which only performs well in continuous problems. Promising results were obtained based on different simulation metrics under various problems (proposed model, MOKP, and ZDT5), validating the superior performance of the proposed algorithm.

Improved Quantum Approximate Optimization Algorithm for Low‐Density Parity‐Check Channel Decoding

AbstractQuantum computing shows promise for 6G networks due to its parallel computing capabilities. In the context of the Noisy Intermediate‐Scale Quantum era, the introduction of hybrid quantum‐classical algorithms like Quantum Approximate Optimization Algorithm (QAOA) offer powerful solutions to many combinatorial optimization problems in 6G. This paper focuses on Low‐Density Parity‐Check (LDPC) channel decoding and proposes an improved QAOA algorithm assisted by the learning‐to‐learn strategy. We also investigate the parameter concentration phenomenon in QAOA‐based LDPC decoding to assess the rationality. To evaluate effectiveness, a comprehensive numerical expression for the energy expectation of single‐layer QAOA and propose indicators for transfer performance evaluation is provided. Based on simulation results, the similarity in parameter distribution across specific LDPC configurations is investigated. This similarity facilitates the transfer of training outcomes from smaller to larger‐scale problems for optimization initialization, thereby avoiding the need for retraining. This approach offers insights and potential solutions for rapid, large‐scale channel decoding in 6G networks, despite the current limitations of quantum hardware.