The paper discusses several extensions of the recursive representation of the flow shop scheduling problem. It is shown that recursive functions make it possible to describe multiple extensions in a single problem. The paper considers altogether six extensions. The examples consider three types of recursive functions: functions associated with the machine, functions that adjust the procession time based on constraints, and functions that control the feasibility of the schedule. The structure of the superpositions of these functions is presented, and also descriptions of several objective functions by recursive functions are presented. Then the general requirements for a recursive function are formulated and its properties are described. Finally, a demonstration of the formulation of new problems is provided using examples of simple flow shop extensions and branch and bound optimization.

We study makespan minimisation on two identical parallel machines that share a single server for both loading and unloading. Each job must be loaded, processed without interruption on its assigned machine, and unloaded immediately after processing, with a common positive integer duration for all loading and unloading operations. We prove that the decision problem is NP-complete for every fixed server-operation duration and strongly NP-complete when this duration is part of the input. We then analyse ordinary list scheduling and the longest-processing-time rule in the non-unit setting. List scheduling has a tight supremum ratio of two. For the longest-processing-time rule, we obtain the exact worst-case ratio when all processing times are at least the server-operation duration, and derive new parameter-dependent lower and upper bounds for unrestricted instances. The results show that both processing-time granularity and blocking generated by short jobs shape the approximation behaviour of the common-server problem.

Since the publication of the first scheduling paper in 1954, a huge number of works dealing with different types of single machine problems have appeared. They addressed many heuristics and enumerative procedures, complexity results or structural properties of certain problems. Regarding surveys, often particular subjects like special objective functions were discussed or more general scheduling problems were surveyed, in which a substantial part was devoted to single machine problems. In this paper, we focus on standard settings, basic structural properties of these settings, polynomial algorithms and complexity and approximation issues, which have not been reviewed so far, and suggest some future work in this area.

This is the fourth edition of a Special Issue of Algorithms that is of a different nature than the other Special Issues in the journal, which are usually dedicated to a particular subject in the area of algorithms [...]

Online grocery (E-grocery) offers a convenient alternative to traditional shopping by emphasizing seamless browsing, purchasing, and quick delivery. Accurate stock-keeping unit information is crucial, as in-stock availability strongly influences consumer choices and supports service-level objectives. Moreover, precise demand forecasts guide strategic and tactical decisions, reducing costs and risks while increasing profits. This paper introduces a novel three-stage hybrid forecasting algorithm that addresses time series with non-stationarity, dynamic variability, and noise. First, an enhanced empirical wavelet transform decomposes time series data into multiple frequency bands, isolating both high-frequency fluctuations and lower-frequency trends. Next, High-order Fuzzy Cognitive Maps (HFCM) are calibrated with Grey Wolf Optimization (GWO) to capture complex temporal interdependencies. Residual errors, often exhibiting long-memory or rough volatility, are then modeled via an Improved Fractional Brownian Motion (IfBm) component, which adaptively estimates the Hurst exponent over rolling windows. Finally, a deep ensemble reinforcement learning stage selects and ensembles the best-performing HFCM-GWO-IfBm model. This pipeline leverages advanced signal decomposition, cognitive modeling, evolutionary computation, and fractal-based corrections, culminating in robust and accurate forecasts. Evaluations on eight publicly available datasets with varied statistical properties, as well as real E-grocery market demand data, confirm the superiority of this framework over state-of-the-art benchmarks. Our method improves upon prior work by as much as 7%-80% in RMSE, underscoring its consistently superior predictive accuracy. In theoretical terms, the study demonstrates how wavelet-based decomposition, fuzzy cognitive maps refined by grey wolf optimization, and fractional Brownian motion can be synergistically applied to address the complexities of non-stationary time series forecasting.

This paper investigates a parallel machine scheduling problem featuring a single common server responsible for both loading and unloading operations. Each job consists of a unit-time loading operation, non-preemptive processing on one of m identical machines, and a unit-time unloading operation executed by the same server. The objective is to minimize the makespan. Unlike classical loading-only common-server models, our setting requires the server to handle an unloading operation immediately after a job's processing phase concludes. We prove that the corresponding decision problem is strongly NP-complete when the number of machines is given as part of the input. Furthermore, we analyse the worst-case performance of the List Scheduling (LS) and Longest Processing Time (LPT) heuristics. For three machines, we establish that Algorithm LS achieves an approximation ratio of at most 5/2. For an arbitrary fixed m ≥ 3, we show that the general LS bound approaches 4-3/m as the number of jobs grows, whereas for Algorithm LPT, we prove the finite-instance ratio 3-2/m+(m-1)(m-2)/mn. Thus, for each fixed m, the LPT bound approaches 3-2/m as the number of jobs grows.

The job shop scheduling problem (JSSP) is a paradigmatic and strongly NP-hard combinatorial optimisation problem that underpins production planning in modern manufacturing systems, and constraint programming (CP) has become one of the leading methodologies for tackling it. However, comparative studies of CP solvers for the JSSP have so far been restricted to a single benchmark family, a single instance-size range, or a single hardware setting, which limits the practical guidance they offer to both researchers and practitioners. This paper presents a controlled empirical evaluation of four state-of-the-art CP solvers—IBM ILOG CP Optimizer, Google OR-Tools (CP-SAT), Hexaly, and OptalCP—on the makespan-minimisation JSSP. The four engines are run with default parameters and a uniform 600 s wall-clock time budget on 332 instances drawn from nine canonical benchmark families (Fisher–Thompson, Lawrence, Adams–Balas–Zawack, Applegate–Cook, Yamada–Nakano, Storer–Wu–Vaccari, Taillard, Demirkol–Mehta–Uzsoy, and Da Col–Teppan), spanning sizes from 6×6 to 1000×1000 operations. OptalCP emerges as the most robust engine overall, certifying optimality on 191 of the 332 instances (57.5%) with the smallest average optimality gap (3.55%), followed by CP Optimizer (166 optima), OR-Tools (144), and Hexaly (116), while Hexaly dominates on industrial-scale problems and produces the bulk of the 22 new best-known upper bounds and one new best-known lower bound reported here. A Friedman test followed by Nemenyi post hoc comparisons confirms that OptalCP attains significantly smaller optimality gaps than the three other engines (p<0.001). Solver competitiveness depends sharply on instance size and the n/m ratio, with square instances confirmed as the hardest case. In practical terms, these findings support an instance-aware approach to CP solver selection: OptalCP is the default choice for small to large instances of moderate aspect ratio, whereas Hexaly is preferable for industrial-scale problems with tens of thousands of operations or extreme n/m ratios, where it is the only engine that reliably returns high-quality feasible schedules within the time budget.

Shop scheduling problems, where a set of jobs has to be processed on a set of machines, have wide practical applications in manufacturing, logistics and many other fields. In this mini review, we describe the main types of problems and survey the major approaches for solving such problems exactly or heuristically.

In this study, a completely different approach to optimization is introduced through the development of a novel metaheuristic algorithm called the Barber Optimization Algorithm (BaOA). Inspired by the human interactions between barbers and customers, BaOA captures two key processes: the customer’s selection of a hairstyle and the detailed refinement during the haircut. These processes are translated into a mathematical framework that forms the foundation of BaOA, consisting of two critical phases: exploration, representing the creative selection process, and exploitation, which focuses on refining details for optimization. The performance of BaOA is evaluated using 52 standard benchmark functions, including unimodal, high-dimensional multimodal, fixed-dimensional multimodal, and the Congress on Evolutionary Computation (CEC) 2017 test suite. This comprehensive assessment highlights BaOA’s ability to balance exploration and exploitation effectively, resulting in high-quality solutions. A comparative analysis against twelve widely known metaheuristic algorithms further demonstrates BaOA’s superior performance, as it consistently delivers better results across most benchmark functions. To validate its real-world applicability, BaOA is tested on four engineering design problems, illustrating its capability to address practical challenges with remarkable efficiency. The results confirm BaOA’s versatility and reliability as an optimization tool. This study not only introduces an innovative algorithm but also establishes its effectiveness in solving complex problems, providing a foundation for future research and applications in diverse scientific and engineering domains.

In this paper, we propose a formulation of the permutation flow and job shop scheduling problems using special recursive functions and show its equivalence to the existing classical formulation. Equivalence is understood in the sense that both ways of defining the problem describe the same set of feasible schedules for each pair of jobs and machine numbers. In this paper, the apparatus of recursive functions is used to describe and solve three problems: permutation flow shop; permutation flow shop with the addition of the ‘and’ predicate extending the machine chain to an acyclic graph; and permutation job shop. The predicate ‘and’ allows the description of the flow shop with assembly operation tasks. Recursive functions have a common domain and range. To calculate an optimal schedule for each of these three problems, a branch and bound method is considered based on a recursive function that implements a job swapping algorithm. The complexity of the optimization algorithm does not increase compared to the non-recursive description of the PFSP. This article presents some results for the calculation of optimal schedules on several test instances. It is expected that the new method, based on the description of recursive functions and their superposition, will be productive for formulating and solving some extensions of scheduling problems that have practical significance.

The problem of scheduling non-simultaneously released jobs with due dates on a single machine with the objective to minimize the maximum job lateness is known to be strongly NP-hard. Here we consider an extended model in which the compression of the job processing times is allowed. The compression is accomplished at the cost of involving additional emerging resources, whose use, however, yields some cost. With a given upper limit $U$ on the total allowable cost, one wishes to minimize the maximum job lateness. It is clear that, by using the available resources, some jobs may complete earlier and the objective function value may respectively be decreased. As we show here, for minimizing the maximum job lateness, by shortening the processing time of some specially determined jobs, the objective value can be decreased. Although the generalized problem is harder than the generic non-compressible version, given a ``sufficient amount'' of additional resources, we can solve the problem optimally. We determine the compression rate for some specific jobs and develop an algorithm that obtains an optimal solution. Such an approach can be beneficial in practice since the manufacturer can be provided with an information about the required amount of additional resources in order to solve the problem optimally. In case the amount of the available additional resources is less than used in the above solution, i.e., it is not feasible, it is transformed to a tight minimal feasible solution.

This Special Issue of the journal Mathematics was dedicated to compiling new results in the area of optimization algorithms, and both theoretical works and practical applications have been searched [...]

Multi-objective optimization is critical for problem-solving in engineering, economics, and AI. This study introduces the Multi-Objective Chef-Based Optimization Algorithm (MOCBOA), an upgraded version of the Chef-Based Optimization Algorithm (CBOA) that addresses distinct objectives. Our approach is unique in systematically examining four dominance relations—Pareto, Epsilon, Cone-epsilon, and Strengthened dominance—to evaluate their influence on sustaining solution variety and driving convergence toward the Pareto front. Our comparison investigation, which was conducted on fifty test problems from the CEC 2021 benchmark and applied to areas such as chemical engineering, mechanical design, and power systems, reveals that the dominance approach used has a considerable impact on the key optimization measures such as the hypervolume metric. This paper provides a solid foundation for determining the most effective dominance approach and significant insights for both theoretical research and practical applications in multi-objective optimization.

Reconfiguration, as well as optimal utilization of distributed generation sources and capacitor banks, are highly effective methods for reducing losses and improving the voltage profile, or in other words, the power quality in the power distribution system. Researchers have considered the use of distributed generation resources in recent years. There are numerous advantages to utilizing these resources, the most significant of which are the reduction of network losses and enhancement of voltage stability. Non-dominated Sorting Genetic Algorithm II (NSGA-II), Multi-Objective Particle Swarm Optimization (MOPSO), and Intersect Mutation Differential Evolution (IMDE) algorithms are used in this paper to perform optimal reconfiguration, simultaneous location, and capacity determination of distributed generation resources and capacitor banks. Three scenarios were used to replicate the studies. The reconfiguration of the switches, as well as the location and determination of the capacitor bank’s optimal capacity, were investigated in this scenario. However, in the third scenario, reconfiguration, and determining the location and capacity of the Distributed Generation (DG) resources and capacitor banks have been carried out simultaneously. Finally, the simulation results of these three algorithms are compared. The results indicate that the proposed NSGAII algorithm outperformed the other two multi-objective algorithms and was capable of maintaining smaller objective functions in all scenarios. Specifically, the energy losses were reduced from 211 to 51.35 kW (a 75.66% reduction), 119.13 kW (a 43.54% reduction), and 23.13 kW (an 89.04% reduction), while the voltage stability index (VSI) decreased from 6.96 to 2.105, 1.239, and 1.257, respectively, demonstrating significant improvement in the voltage profile.

This research presents a novel nature-inspired metaheuristic algorithm called Frilled Lizard Optimization (FLO), which emulates the unique hunting behavior of frilled lizards in their natural habitat. FLO draws its inspiration from the sit-and-wait hunting strategy of these lizards. The algorithm’s core principles are meticulously detailed and mathematically structured into two distinct phases: (i) an exploration phase, which mimics the lizard’s sudden attack on its prey, and (ii) an exploitation phase, which simulates the lizard’s retreat to the treetops after feeding. To assess FLO’s efficacy in addressing optimization problems, its performance is rigorously tested on fifty-two standard benchmark functions. These functions include unimodal, high-dimensional multimodal, and fixed-dimensional multimodal functions, as well as the challenging CEC 2017 test suite. FLO’s performance is benchmarked against twelve established metaheuristic algorithms, providing a comprehensive comparative analysis. The simulation results demonstrate that FLO excels in both exploration and exploitation, effectively balancing these two critical aspects throughout the search process. This balanced approach enables FLO to outperform several competing algorithms in numerous test cases. Additionally, FLO is applied to twenty-two constrained optimization problems from the CEC 2011 test suite and four complex engineering design problems, further validating its robustness and versatility in solving real-world optimization challenges. Overall, the study highlights FLO’s superior performance and its potential as a powerful tool for tackling a wide range of optimization problems.

Scheduling and optimization have a central place in the research area of computing because it is increasingly important to achieve fully automated production processes to adjust manufacturing systems to the requirements of Industry 4.0. In this paper, we demonstrate how an automated wet-etch scheduling problem for the semiconductor industry can be solved by constraint answer set programming (CASP) and its solver called clingcon. A successful solution to this problem is achieved, and we found that for all tested problems, CASP is faster and obtains smaller makespan values for seven of the eight problems tested than the solutions based on mixed integer linear programming and constraint paradigms. The considered scheduling problem includes a robot for lot transfers between baths. CASP is a hybrid approach in automated reasoning that combines different research areas such as answer set programming, constraint processing, and Satisfiability Modulo Theories. For a long time, exact methods such as constraint programming have displayed difficulties in solving real large-scale problem instances. Currently, the performance of state-of-the-art constraint solvers is comparatively better than a decade ago, and some complex combinatorial problems can be better solved. These theoretical and technical achievements open new horizons for declarative programming applications such as logistics.

The Cherenkov Telescope Array Observatory (CTAO) is the next-generation atmospheric Cherenkov gammaray Observatory. CTAO will be constructed on two sites, one array in the Northern and the other in the Southern hemisphere, containing telescopes of three different sizes, for covering different energy domains. To combine and orchestrate the different telescopes and auxiliary instruments (array elements), the Array Control and Data Acquisition (ACADA) system is the central element for the Observatory on-site operations: it controls, supervises, and handles the data generated by the array elements. Considering the criticality of the ACADA system for future Observatory operations, corresponding quality assurance provisions have been made at the different steps of the software development lifecycle, with focus on continuous integration and testing at all levels. To enable higher-level tests of the software deployed on a distributed system, an ACADA test cluster has been set up to facilitate testing and debugging of issues in a more realistic environment. Furthermore, a separate software integration and test cluster has also been established that allows for the off-site testing of the integrated software packages of ACADA and of the corresponding array elements. Here the software integration can be prepared, interfaces and interactions can be tested, and on-site procedures that are required later in the process can be checked beforehand, only limited by the simulation capabilities that are delivered as part of the software packages. Once preparations and testing with the off-site test cluster are completed, the integrated software can be deployed at the target site. The software packages and setup parameters are kept under configuration control at all stages, and deployment steps are documented to ensure that installations are reproducible. This methodology has been applied for the first time in the context of the integration of ACADA with the first CTAO Large-sized Telescope (LST-1) in October 2023.

In this paper, we consider some problems that arise in connected and autonomous vehicle (CAV) systems. Their simplified variants can be formulated as scheduling problems. Therefore, scheduling solution algorithms can be used as a part of solution algorithms for real-world problems. For four variants of such problems, mathematical models and solution algorithms are presented. In particular, three polynomial algorithms and a branch and bound algorithm are developed. These CAV scheduling problems are considered in the literature for the first time. More complicated NP-hard scheduling problems related to CAVs can be considered in the future.