Symmetry is a widespread phenomenon in nature. Recognizing symmetry can minimize redundancy to improve computing efficiency. In this paper, we take permutation-related combinatorial optimization problems as a starting point and explore the symmetric structure of its solution space through group theory. From a new perspective of group action, we discover that the meaningful symmetric feature within the solution space is subject to two conditions regarding the form of objective function and the number of objects to be permuted. To exploit the symmetric features, we design a half-solution-space search strategy for various search operators, which are commonly used for permutation-related combinatorial optimization problems. The half-solution-space search strategy can make these operators explore more promising regions without additional computational effort. When the condition of object number for symmetry is unsatisfied, we propose two dimension mapping approaches to construct the symmetric feature, making the half-solution-space search strategy applicable. We evaluate the proposed strategy on three classes of popular 68 benchmark instances, including the single row facility layout problem (SRFLP), traveling salesman problem (TSP), and multi-objective traveling salesman problem (MOTSP). Experimental results show that algorithms embedded with the half-solution-space search strategy can achieve a more competitive performance than those not exploiting the symmetric features.

Reinforcement learning is a promising method for solving decision problems, and its potential has been increasingly recognized for large-scale combinatorial optimization problems in recent years. However, the existing studies on reinforcement learning for cutting stock problems mostly rely on sequence-to-sequence or graph neural network approaches that use the learned experience to make decisions while neglecting the combination features of cutting stock problems. In this paper, we propose a novel reinforcement learning framework for cutting stock problems that integrate integer programming and monotone comparative statics to construct a Markov decision process with a high-quality action space. We start by constructing a new Markov decision process that considers the diagonal structure of the integer programming model for combinatorial optimization problems, and then use column generation to obtain each action by combining multiple decision variables. Furthermore, we design a bipartite graph and related bipartite graph convolutional network to find the solutions. The results show that the proposed reinforcement learning framework provides a high-quality action space, and the designed bipartite graph convolutional network can effectively select the best actions from the action set.Note to Practitioners—This article was motivated by the cutting stock problems that exist in various industrial scenarios such as the wood, steel, paper, and glass industries. We improve the reinforcement learning for the cutting stock problem that can be adopted in industrial scenarios, which can increase the profile and reduce the production cost of industrial enterprises. Our improvement can also be referred to when solving other combinatorial optimization problems that can promote making decisions in industrial production.

Density functional theory (DFT) forms the foundation for computing the electronic structure of many-electron systems through the Kohn-Sham (KS) equations, with electron densityn(r) serving as the fundamental variable. This approach allows for the understanding and manipulation of material properties at the atomic and electronic levels. However, the exchange-correlation (XC) functional in DFT is not known in an exact form and is typically approximated using analytical models, which limits the accuracy of DFT calculations. The recent advancements in data analytics techniques, such as machine learning, which excel in pattern recognition, offer new opportunities for more effective approximations of the XC functional. In this study, we introduce a novel approach for XC functional modeling by fusing shared representation learning. This method leverages information sharing across tasks to improve the fused XC (FXC) functional model generalization. To improve the robustness of the FXC functional model, we further develop a multi-objective constrained multitask adjustment mechanism. Additionally, to expedite the determination of the optimal FXC model, we propose a group-theory-based multi-objective optimization (MOO) algorithm, which enables faster exploration of the solution space. Our experimental results demonstrate that the KS equations incorporating the FXC functional substantially improve the accuracy of molecular atomization energies, ionization potentials, and electron density calculations. Moreover, the method exhibits applicability to the computation of isomerization reaction energies in large molecules. These findings validate the effectiveness of our data analytics and optimization-based approach in enhancing the precision of the XC functional approximation.

This article studies an operation optimization problem in a steelmaking process. Shortly before the tapping of molten steel from the basic oxygen furnace (BOF), end-point control measures are applied to achieve the required final molten steel quality. While it is difficult to build an exact mathematical model for this process, the control inputs and the corresponding outputs are available by collecting production data. We build a data-driven model for the process. To optimize the control parameters, an improved estimation of distribution algorithm (EDA) is developed using a probabilistic model comprising different distributions. A resampling mechanism is incorporated into the EDA to guide the new population to a broader and more promising area when the search becomes ineffective. To further enhance the solution quality, we add a local improvement to update the current best individual through simplified gravitational search and information learning. Experiments are conducted using real data from a BOF steelmaking process. The results show that the algorithm can help to achieve the specified molten steel quality. To evaluate the proposed algorithm as a general optimization algorithm, we test it on some complex benchmark functions. The results illustrate that it outperforms other state-of-the-art algorithms across a wide range of problems.

The multidimensional knapsack problem (MKP) is a well-known combinatorial optimization problem with many real-life applications. In this article, a memetic algorithm based on probability learning (MA/PL) is proposed to solve MKP. The main highlights of this article are two-fold: 1) problem-dependent heuristics for MKP and 2) a novel framework of MA/PL. For the problem-dependent heuristics, we first propose two kinds of logarithmic utility functions (LUFs) based on the special structure of MKP, in which the profit value and weight vector of each item are considered simultaneously. Then, LUFs are applied to effectively guide the repair operator for infeasible solutions and the local search operator. For the framework of MA/PL, we propose two problem-dependent probability distributions to extract the special knowledge of MKP, that is, the marginal probability distribution (MPD) of each item and the joint probability distribution (JPD) of two conjoint items. Next, learning rules for MPD and JPD, which borrow ideas from competitive learning and binary Markov chain, are proposed. Thereafter, we generate MA/PL's offspring by integrating MPD and JPD, such that the univariate probability information of each item as well as the dependency of conjoint items can be sufficiently used. Results of experiments on 179 benchmark instances and a real-life case study demonstrate the effectiveness and practical values of the proposed MKP.

Estimation of distribution algorithm (EDA) is an efficient population-based stochastic search technique. Since it was proposed, many attempts have been made to improve its performance in the context of nonlinear continuous optimization. However, the success of EDA depends on the accuracy of modeling, the effectiveness of sampling, and the ability of exploration. An effective EDA often needs to take some measures to adjust the model and to guide sampling. In this article, we propose a novel EDA which applies the idea of Kalman filtering to revise the modeling data and a learning strategy to improve sampling. The filtering scheme modifies the modeling data set using an estimation error matrix based on historic solution data. During the sampling process, the learning strategy determines the region to sample next based on the sampling outcomes so far, instead of completely random sampling. The proposed EDA also employs a multivariate probabilistic model based on copula function and can quickly reach the promising area in which the optimal solution is likely to be located. A collection of general benchmark functions are used to test the performance of the proposed algorithm. Computational experiments show that the EDA is effective.Note to Practitioners—In many process industries, there exist black-box operation optimization problems and large-scale nonlinear optimization problems with variable coupling. For these problems, it is difficult to establish mechanism models between input and output. However, real-time data can be measured from the system through sensors. We can utilize this process information to optimize the system so as to attain the desired objective. In this article, we propose a novel estimation of distribution algorithm (EDA) which applies a filtering scheme to revise the modeling data and a learning strategy to improve sampling, which can solve the problems with the characteristics of nonlinearity, variable coupling, and large scale. Computational experiments show that the EDA is effective. In the future, the proposed algorithm can be applied to some practical optimization problems such as operation optimization in blast furnace, which is considered as a continuous production process with variable coupling. The algorithm has the potential to help optimizing the process control parameters.

The steelmaking process in the iron and steel industry involves complicated physicochemical reactions. The main aim of steelmaking is to adjust the quality of molten steel. During the steel-tapping process, the temperature and carbon content are the most essential quality indices for end-point prediction. This article presents a novel machine learning framework for the endpoint prediction problems in the smelting process. Considering the importance of data representation in modeling, the original data are inputted to a stacked autoencoder (SAE) to extract the essential features in an unsupervised manner. The top layer is then designed as a sparse Bayesian regression (SBR) layer to obtain the predicted mean values and error bars that measure the uncertainty in the prediction. To improve the generalization ability of the prediction model, an intelligent optimization algorithm based on improved differential evolution (DE) is used to optimize the hyperparameters of the model. The main advantage of this model is that the underlying characteristics of the samples can be learned automatically layer by layer, instead of designing them manually. Finally, the effectiveness of the proposed method is verified using real data collected from two steel plants. The experimental results show that the proposed model gives a more precise prediction than other existing models and can provide error bars for the end-point prediction.

This article studies a multiple double-load crane scheduling problem in steel slab yards. Consideration of multiple cranes and their double-load capability makes the scheduling problem more complex. This problem has not been studied previously. We first formulate the problem as a mixed-integer linear programming (MILP) model. A two-phase model-based heuristic is then proposed. To solve large problems, a pointer-based discrete differential evolution (PDDE) algorithm was developed with a dynamic programming (DP) algorithm embedded to solve the one-crane subproblem for a fixed sequence of tasks. Instances of real problems are collected from a steel company to test the performance of the solution methods. The experiment results show that the model can solve small problems optimally, and the solution greatly improves the schedule currently used in practice. The two-phase heuristic generates near-optimal solutions, but it can still only solve comparatively modest problems within reasonable (4 h) computational timeframes. The PDDE algorithm can solve large practical problems relatively quickly and provides better results than the two-phase heuristic solution, demonstrating its effectiveness and efficiency and therefore its suitability for practical use. Note to Practitioners-Bridge cranes are commonly used to move heavy items in manufacturing and logistics systems. Generally, more than one crane runs on a common track. The latest versions of such cranes, such as those used in slab yards in the steel industry, can hold two items simultaneously. Operations scheduling of multiple double-load cranes involves the assignment of tasks to the cranes, the combination of tasks to double-load operations, and sequencing of the tasks, considering the noncrossing constraint between cranes. Effective solution of this complex problem can help to fully utilize the crane capability, increase productivity, and reduce energy consumption. This article models this problem and develops a heuristic solution that combines differential evolution (DE) and DP. Experiment results show that the algorithm is effective and efficient for practical use in slab yards. It may also be applicable to other systems using similar cranes.

This paper proposes a dynamic analytics method based on the least squares support vector machine with a hybrid kernel to address real-time prediction problems in the converter steelmaking process. The hybrid kernel function is used to enhance the performance of the existing kernels. To improve the model's accuracy, the internal parameters are optimized by a differential evolution algorithm. In light of the complex mechanisms of the converter steelmaking process, a multistage modeling strategy is designed instead of the traditional single-stage modeling method. Owing to the dynamic nature of the practical production process, great effort has been made to construct a dynamic model that uses the prediction error information based on the static model. The validity of the proposed method is verified through experiments on real-world data collected from a basic oxygen furnace steelmaking process. The results indicate that the proposed method can successfully solve dynamic prediction problems and outperforms other state-of-the-art methods in terms of prediction accuracy.

Differential evolution (DE) is a well-known optimization algorithm that utilizes the difference of positions between individuals to perturb base vectors and thus generate new mutant individuals. However, the difference between the fitness values of individuals, which may be helpful to improve the performance of the algorithm, has not been used to tune parameters and choose mutation strategies. In this paper, we propose a novel variant of DE with an individual-dependent mechanism that includes an individual-dependent parameter (IDP) setting and an individual-dependent mutation (IDM) strategy. In the IDP setting, control parameters are set for individuals according to the differences in their fitness values. In the IDM strategy, four mutation operators with different searching characteristics are assigned to the superior and inferior individuals, respectively, at different stages of the evolution process. The performance of the proposed algorithm is then extensively evaluated on a suite of the 28 latest benchmark functions developed for the 2013 Congress on Evolutionary Computation special session. Experimental results demonstrate the algorithm's outstanding performance.

This paper studies a challenging problem of dynamic scheduling in steelmaking-continuous casting (SCC) production. The problem is to re-optimize the assignment, sequencing, and timetable of a set of existing and new jobs among various production stages for the new environment when unforeseen changes occur in the production system. We model the problem considering the constraints of the practical technological requirements and the dynamic nature. To solve the SCC scheduling problem, we propose an improved differential evolution (DE) algorithm with a real-coded matrix representation for each individual of the population, a two-step method for generating the initial population, and a new mutation strategy. To further improve the efficiency and effectiveness of the solution process for dynamic use, an incremental mechanism is proposed to generate a new initial population for the DE whenever a real-time event arises, based on the final population in the last DE solution process. Computational experiments on randomly generated instances and the practical production data show that the proposed improved algorithm can obtain better solutions compared to other algorithms.

This paper studies a large-scale scheduling problem in iron and steel industry, called color-coating production scheduling for coils in inventory (CCPSCI). The problem is to select steel coils from those in the coil yard and to create a production schedule so that the productivity and product quality are maximized, while the production cost and other penalties are minimized. A tabu search (TS) algorithm is proposed for this problem. Results on real production instances show that the proposed method is much more effective and efficient than manual scheduling.

In this paper, we study the no-wait multihoist cyclic scheduling problem, in which the processing times in the tanks and the transfer times between tanks are constant parameters, and develop a polynomial optimal solution to minimize the production cycle length. We first analyze the problem with a fixed cycle length and identify a group of hoist assignment constraints based on the positions of and the relationships among the part moves in the cycle. We show that the feasibility of the hoist scheduling problem with fixed cycle length is consistent with the feasibility of this group of constraints which can be solved efficiently. We then identify all of the special values of the cycle length at which the feasibility property of the problem may change. Finally, the whole problem is solved optimally by considering the fixed-cycle-length problems at these special values.