Neighborhood Search Mechanism Research Articles

A discrete moth-flame optimization algorithm for multiple automated guided vehicles scheduling problem in a matrix manufacturing workshop

The growing need for customized and varied production has highlighted the significance of smart and automated factories in the manufacturing sector. In this particular context, the scheduling of multiple Automated Guided Vehicles (AGVs) plays a pivotal role in enhancing the efficiency of operations within an intelligent manufacturing shop. This study centers on the material handling process within a matrix manufacturing shop with the objective of identifying the most cost-effective transportation routes for materials. To accomplish the specified objective, this research aims to identify an optimal solution for reducing transportation costs. In particular, this study formulates a mixed-integer linear programming model and introduces a novel discrete variant of the moth-flame optimization (MFO) algorithm, named DMFO, to address the scheduling problem. The DMFO algorithm incorporates several significant enhancements. Firstly, a population initialization method is proposed, which combines a nearest-neighbor-based adaptive heuristic and a random sorting technique to ensure the formation of a well-structured population. Secondly, the flame generation mechanism and spiral flight search processes within the MFO have been redefined to achieve a more optimal balance between exploration and exploitation. A neighborhood search mechanism is subsequently devised, employing the concept of neighborhood relevance to accelerate the convergence process. Additionally, a heuristic approach is introduced to reduce the computational cost. Moreover, a population regeneration mechanism is proposed to avoid the algorithm falling into a local optimum. To validate the effectiveness of the DMFO, a comparative analysis is conducted using a dataset of 110 real-world factory instances. In this analysis, eight well-established optimization algorithms are employed. The simulation results consistently demonstrate that the relative percentage deviation (RPD) value of the DMFO tends to approach 0% more closely compared to other algorithms, thereby substantiating the effectiveness of the proposed algorithm.

A Genetic Algorithm with Lower Neighborhood Search for the Three-Dimensional Multiorder Open-Size Rectangular Packing Problem

This paper addresses the multiorder open-dimension three-dimensional rectangular packing problem (3D-MOSB-ODRPP), which involves packing rectangular items from multiple orders into a single, size-adjustable container. We propose a novel metaheuristic approach combining a genetic algorithm with the Gurobi solver. The algorithm incorporates a lower neighborhood search strategy and is underpinned by a mathematical model representing the multiorder open-dimension packing scenario. Extensive experiments validate the effectiveness of the proposed approach. The LNSGA algorithm outperforms Gurobi and the traditional genetic algorithm in solution quality and computational efficiency. For small-scale instances, LNSGA achieves optimal values in most cases. LNSGA demonstrates significant optimization improvements over Gurobi and the genetic algorithm for large-scale instances. The superior performance is attributed to the effective integration of the lower neighborhood search mechanism and the Gurobi solver. This study offers valuable insights for optimizing the packing process in e-commerce warehousing and logistics operations.

3D real-time dynamic path planning for UAV based on improved interfered fluid dynamical system and artificial neural network

In complex and volatile unknown flight environments, the limited environmental information obtained by sensors in the face of sudden dynamic and static obstacles makes it extremely challenging for unmanned aerial vehicles (UAVs) to obtain a safe and efficient path to avoid obstacles and reach a designated target point. Therefore, a real-time dynamic path planning method based on an improved interfered fluid dynamical system (IFDS) and artificial neural network (ANN) is proposed to enhance path quality and computational efficiency. Firstly, to address the issue of insufficient sample quality and quantity, IFDS is employed as the fundamental method for path planning to simulate and generate an adequate amount of sample data for the ANN training. Then, an enhanced sand cat swarm optimization algorithm (ESCSO) with an adaptive social neighborhood search mechanism and Lévy flight strategy is proposed to improve the sample quality. Secondly, the information between the UAV and the target points and obstacles is extracted from the sample data as the input for the network, the parameters of the IFDS are used as the feature extraction at the output of the network, and the ESCSO is applied to optimize the weights and biases of the ANN, enabling offline training of the neural network. Finally, the trained neural network is utilized to dynamically output IFDS parameters based on the real-time environmental information obtained from the sensors, enabling the generation of real-time obstacle avoidance paths. Experimental results in a series of complex simulated environments demonstrate that the proposed method outperforms other algorithms in terms of path quality and meets real-time requirements. It provides excellent obstacle avoidance characteristics for the UAV.

A novel chaotic and neighborhood search-based artificial bee colony algorithm for solving optimization problems

With the development of artificial intelligence, numerous researchers are attracted to study new heuristic algorithms and improve traditional algorithms. Artificial bee colony (ABC) algorithm is a swarm intelligence optimization algorithm inspired by the foraging behavior of honeybees, which is one of the most widely applied methods to solve optimization problems. However, the traditional ABC has some shortcomings such as under-exploitation and slow convergence, etc. In this study, a novel variant of ABC named chaotic and neighborhood search-based ABC algorithm (CNSABC) is proposed. The CNSABC contains three improved mechanisms, including Bernoulli chaotic mapping with mutual exclusion mechanism, neighborhood search mechanism with compression factor, and sustained bees. In detail, Bernoulli chaotic mapping with mutual exclusion mechanism is introduced to enhance the diversity and the exploration ability. To enhance the convergence efficiency and exploitation capability of the algorithm, the neighborhood search mechanism with compression factor and sustained bees are presented. Subsequently, a series of experiments are conducted to verify the effectiveness of the three presented mechanisms and the superiority of the proposed CNSABC, the results demonstrate that the proposed CNSABC has better convergence efficiency and search ability. Finally, the CNSABC is applied to solve two engineering optimization problems, experimental results show that CNSABC can produce satisfactory solutions.

A neighborhood comprehensive learning particle swarm optimization for the vehicle routing problem with time windows

Vehicle routing problem with time windows (VRPTW), which is a typical NP-hard combinatorial optimization problem, plays an important role in modern logistics and transportation systems. Recent years, heuristic and meta-heuristic algorithms have attracted many researchers’ attentions to solve the VRPTW problems. As an outstanding meta-heuristic algorithm, particle swarm optimization (PSO) algorithm exhibits very promising performance on continuous problems. However, how to adapt PSO to efficiently deal with VRPTW is still challenging work. In this paper, we propose a neighborhood comprehensive learning particle swarm optimization (N-CLPSO) to solve VRPTW. To improve the exploitation capability of N-CLPSO, we introduce a new remove-reinsert neighborhood search mechanism, which consists of the removed operator and the reinsert operator. When performing the removed operator, the probability of adjacency between two customers is calculated by an information matrix (IM), which is constructed based on the customers’ time-space information and elite individuals’ local information. When executing the reinsert operator, the IM and a cost matrix (CM), which is introduced to record the cost of customer insertion, are used to find an optimal insert position. Moreover, to enhance the exploration of N-CLPSO, a semi-random disturbance strategy is proposed, in which elites’ longest common sequences (LCS) are saved, aiming to prevent population degradation. The N-CLPSO algorithm is tested on 56 Solomon benchmark instances, and it attains the optimal solutions on 29 instances. The simulation results and comparison results illustrate that the proposed algorithm outperforms or can compete with the majority of other 3 PSO variants as well as other 12 state-of-the-art algorithms.

Cost optimization model design of fresh food cold chain system in the context of big data

The assessment of cold chain logistics for fresh products can be more precise with high-dimensional information data, providing valuable insights for the optimization of associated costs. Nonetheless, traditional data processing techniques fail to meet the processing efficiency required for such high-dimensional cold chain logistics data. Therefore, this paper proposes a spectral clustering algorithm based on the local standard deviation and optimized initial center, which comprehensively analyzes the fixed, transportation, refrigeration, and cargo damage costs of cold chain logistics. Additionally, this algorithm includes a variation operator based on clustering and introduces a large neighborhood search mechanism for optimizing the individual connectivity gene layer after selecting the gene layer site for variation. Simulation results demonstrate that the proposed algorithm exhibits better convergence in 15 iterations, reduces error rates, and significantly cuts down on the clustering process time. This ultimately leads to a reduction in the total cost of cold chain calculation.

A mixing algorithm of ACO and ABC for solving path planning of mobile robot

Path planning is involved in many applications such as trajectory planning, mobile robotics, pipeline layout, etc. Researchers use artificial intelligence algorithms to solve path planning efficiently, among which the ant colony algorithm (ACO) is one of the common intelligent algorithms to solve path planning problems. However, the traditional ACO has defects such as low early search efficiency and easy to fall into local optimum, while the artificial bee colony algorithm (ABC) has high search efficiency. Therefore, an improved ant colony optimization-artificial bee colony algorithm (IACO-IABC) is proposed in this study. IACO-IABC contains three mechanisms. First, the heuristic mechanism with directional information for the ACO is improved to enhance the efficiency of steering towards the target direction. Secondly, the novel neighborhood search mechanism of the employed bee and the onlooker bee in the ABC is presented to enhance the exploitation of optimal solutions. Then, the path optimization mechanism is introduced further to reduce the number of turn times in the planned path. To verify the performance of the IACO-IABC, a series of experiments are conducted with 10 different maps. The experiments compare nine variants of ACO and eight commonly used intelligent search algorithms, and the results show the advantages of the IACO-IABC in reducing the number of turn times and path lengths and enhancing the convergence speed of the algorithm. Compared to the best results of other algorithms, the average improvement percentages of the proposed algorithm in terms of the path turn times are 375%, 258.33%, 483.33%, 186.67%, 166.77% and 255.33%, further demonstrating the ability of IACO-IABC to obtain high-quality path planning result.

Dynamic cyclic kitting part-feeding scheduling for mixed-model assembly line by a hybrid quantum-behaved particle swarm optimization

PurposeThe purpose of this paper is to cut down energy consumption and eliminate production waste on mixed-model assembly lines. Therefore, a supermarket integrated dynamic cyclic kitting system with the application of electric vehicles (EVs) is introduced. The system resorts to just-in-time (JIT) and segmented sub-line assignment strategies, with the objectives of minimizing line-side inventory and energy consumption.Design/methodology/approachHybrid opposition-based learning and variable neighborhood search (HOVMQPSO), a multi-objective meta-heuristics algorithm based on quantum particle swarm optimization is proposed, which hybridizes opposition-based learning methodology as well as a variable neighborhood search mechanism. Such algorithm extends the search space and is capable of obtaining more high-quality solutions.FindingsComputational experiments demonstrated the outstanding performance of HOVQMPSO in solving the proposed part-feeding problem over the two benchmark algorithms non-dominated sorting genetic algorithm-II and quantum-behaved multi-objective particle swarm optimization. Additionally, using modified real-life assembly data, case studies are carried out, which imply HOVQMPSO of having good stability and great competitiveness in scheduling problems.Research limitations/implicationsThe feeding problem is based on static settings in a stable manufacturing system with determined material requirements, without considering the occurrence of uncertain incidents. Current study contributes to assembly line feeding with EV assignment and could be modified to allow cooperation between EVs.Originality/valueThe dynamic cyclic kitting problem with sub-line assignment applying EVs and supermarkets is solved by an innovative HOVMQPSO, providing both novel part-feeding strategy and effective intelligent algorithm for industrial engineering.

An Accurate UAV 3-D Path Planning Method for Disaster Emergency Response Based on an Improved Multiobjective Swarm Intelligence Algorithm.

Planning a practical three-dimensional (3-D) flight path for unmanned aerial vehicles (UAVs) is a key challenge for the follow-up management and decision making in disaster emergency response. The ideal flight path is expected to balance the total flight path length and the terrain threat, to shorten the flight time and reduce the possibility of collision. However, in the traditional methods, the tradeoff between these concerns is difficult to achieve, and practical constraints are lacking in the optimized objective functions, which leads to inaccurate modeling. In addition, the traditional methods based on gradient optimization lack an accurate optimization capability in the complex multimodal objective space, resulting in a nonoptimal path. Thus, in this article, an accurate UAV 3-D path planning approach in accordance with an enhanced multiobjective swarm intelligence algorithm is proposed (APPMS). In the APPMS method, the path planning mission is converted into a multiobjective optimization task with multiple constraints, and the objectives based on the total flight path length and degree of terrain threat are simultaneously optimized. In addition, to obtain the optimal UAV 3-D flight path, an accurate swarm intelligence search approach based on improved ant colony optimization is introduced, which can improve the global and local search capabilities by using the preferred search direction and random neighborhood search mechanism. The effectiveness of the proposed APPMS method was demonstrated in three groups of simulated experiments with different degrees of terrain threat, and a real-data experiment with 3-D terrain data from an actual emergency situation.

Metaheuristics with variable diversity control and neighborhood search for the Heterogeneous Site-Dependent Multi-depot Multi-trip Periodic Vehicle Routing Problem

The planning of vehicle routes is a major issue involved in supply chains. In real environment, we can find situations involving a very large number of clients or constraints which indicate that exact methods should be avoided. In this context, this paper presents two metaheuristcs which are used to solve a complex problem named the Heterogeneous Site-Dependent Multi-depot Multi-trip Periodic Vehicle Routing Problem (HSDMDMTPVRP). The HSDMDMTPVRP is a real problem found in the automotive industry and considers several well-known Vehicle Routing Problems (VRP). The first metaheuristic is an adaptation of the Unified Hybrid Genetic Search (UHGS) which considers an advanced diversity control, feasibility control and a restart mechanism. The second one is a new metaheuristic named Adaptive Variable Neighborhood Race (AVNR) which combines variable neighborhood search and adaptive mechanisms integrated with a shrinking population managed with a diversity mechanism. Both approaches are also used for solving some variants of the VRP: Heterogeneous VRP, Site dependent VRP, Periodic VRP and Multi-trip VRP. Our computational experiments used 398 available instances in the literature with generic code path and also present 20 new instances for the HSDMDMTPVRP. The metaheuristics solved all instances with only one set of parameters and the results outperform or present the same solutions found by several state-of-the-art algorithms, showing the good performance of the approaches. Out of the 398 previously tested literature instances, the proposed metaheuristics found 140 new best-known solutions and 209 of the best-known ones. For the remaining instances, both approaches found results very close to best ones known.

Neighborhood Search Based Improved Bat Algorithm for Web Service Composition

Web services are provided as reusable software components in the services-oriented architecture. More complicated composite services can be combined from these components to satisfy the user requirements represented as a workflow with specified Quality of Service (QoS) limitations. The workflow consists of tasks where many services can be considered for each task. Searching for optimal services combination and optimizing the overall QoS limitations is a Non-deterministic Polynomial (NP)-hard problem. This work focuses on the Web Service Composition (WSC) problem and proposes a new service composition algorithm based on the micro-bats behavior while hunting the prey. The proposed algorithm determines the optimal combination of the web services to satisfy the complex user needs. It also addresses the Bat Algorithm (BA) shortcomings, such as the tradeoff among exploration and exploitation searching mechanisms, local optima, and convergence rate. The proposed enhancement includes a developed cooperative and adaptive population initialization mechanism. An elitist mechanism is utilized to address the BA convergence rate. The tradeoff between exploration and exploitation is handled through a neighborhood search mechanism. Several benchmark datasets are selected to evaluate the proposed bat algorithm’s performance. The simulation results are estimated using the average fitness value, the standard deviation of the fitness value, and an average of the execution time and compared with four bat-inspired algorithms. It is observed from the simulation results that introduced enhancement obtains significant results.

Research on Multi-Objective Scheduling Algorithm of Job Shop Considering Limited Storage and Transportation Capacity

As with the continuous improvement of the workshop automation rate and the importance in energy consumption, more and more enterprises not only need to make scheduling decision on production equipment, but also need to consider whether the scheduling of transportation equipment supports scheduling decisions on workshop production. At the same time, because both workshop production scheduling decision and transportation scheduling decision are NP-hard problems, it is necessary to design an efficient algorithm to improve productivity of the workshop. In order to solve this problem, firstly, based on the analysis of the problem structure, production environment and optimization objectives, a “manufacturing-transportation” multi-objective joint scheduling optimization mathematical model is established. By converting the energy consumption into the total transportation time objective of the transportation equipment, both total transportation time and makespan are taken as the optimization objectives. Secondly, based on the design idea of memetic algorithm (MA), non-dominated sorting genetic algorithm-Ⅱ(NSGA-II) is employed as the basis framework of our new developed algorithm. An effective discrete encoding scheme of MO-MA, a new initialization method for initial population and a neighborhood search mechanism based on critical path are incorporated into our new proposed algorithm. Then the parameter design of the algorithm is completed through variance analysis. Finally, the proposed algorithm is compared and analyzed with other algorithms in the dimension of hypervolume and Set Coverage (SC), and advantages of the algorithm in solving this problem are verified.

An Improved Whale Optimization Algorithm for Web Service Composition

In the current circumstance, the Web Service Composition (WSC) was introduced to address complex user needs concerning the Quality of Services (QoS). In the WSC problem, the user needs are divided into a set of tasks. The corresponding web services are retrieved from the web services discovery according to the functionality of each task, and have different non-functional constraints, such as QoS. The WSC problem is a multi-objective optimization problem and is classified as an NP-hard problem. The whale optimization algorithm (WOA) is proven to solve complex multi-objective optimization problems, and it has the advantage of easy implementation with few control parameters. In this work, we contribute to improving the WOA algorithm, where different strategies are introduced to enhance its performance and address its shortcomings, namely its slow convergence speed, which produces low solution accuracy for the WSC problem. The proposed algorithm is named Improved Whale Optimization Algorithm (IWOA) and has three different strategies to enhance the performance of the WOA. Firstly, the Sine chaos theory is proposed to initiate the WOA’s population and enhance the initialization diversity. Secondly, a Lévy flight mechanism is proposed to enhance the exploitation and exploration of WOA by maintaining the whales’ diversity. Further, a neighborhood search mechanism is introduced to address the trade-off between exploration and exploitation searching mechanisms. Different experiments are conducted with datasets on 12 different scales (small, medium, and large), and the proposed algorithm is compared with standard WOA and five state-of-the-art swarm-based algorithms on 30 different independent runs. Furthermore, four evaluation criteria are used to validate the comparison: the average fitness value, best fitness values, standard deviation, and average execution time. The results show that the IWOA enhanced the WOA algorithm’s performance, where it got the better average and best fitness values with a low variation on all datasets. However, it ranked second regarding average execution time after the WOA, and sometimes third after the WOA and OABC, which is reasonable because of the proposed strategies.

Blocking flowshop scheduling problems with release dates

In a flowshop, the current stage has to remain work-in-process and cannot release any task to the downstream stage once the buffer is stuffed. This scheduling phenomenon is called blocking, and it usually occurs in an industrial setting because of the buffer capacity or the product volume. This study addresses several blocking flowshop scheduling problems to individually optimize the following criteria, i.e. makespan, maximum lateness, or maximum delivery-completion time, in which the tasks are available at different release dates. Given that these scheduling problems are strongly NP-complete, exact and metaheuristic algorithms are designed for different scale instances. An effective branch and bound method is proposed to search exact solutions for small-scale instances within a given period, where well-designed branching rules and lower bounds prune considerable invalid nodes. Further, to seek high-quality feasible schedules in a short time, a discrete artificial bee colony algorithm is provided for medium-scale instances, where a dispatching-rule-based initialization, a series of efficient non-delay variable neighborhood search mechanisms, and a well-designed disturbing scheme enhance the optimization capability. Comprehensive computational tests are conducted to evaluate the effectiveness of the proposed algorithms.

An efficient iterated local search algorithm for the corridor allocation problem

The corridor allocation problem (CAP) seeks the optimal arrangement of departments along the two sides of a corridor with respect to two key considerations; there should be no gap between two consecutive departments, and both of the rows should start at the same level of the corridor. In this study, a new and comprehensive multi-start iterated local search (ILS) algorithm is developed for CAP to search the solution space more effectively. The proposed algorithm utilizes a variable neighborhood descent (VND) for local search. The VND algorithm employs five neighborhood search mechanisms (swap, insert, inverse, cross, and block insert) and a tabu list to avoid cycling. The performance of the proposed ILS is evaluated on well-known instances of various sizes. Objective function values of 34 out of 82 benchmark instances are improved by the proposed algorithm. The computational results also show that the algorithm is capable of dealing with larger-sized problems with up to eighty departments.

A Comparison between Memetic Algorithm and Genetic Algorithm for an Integrated Logistics Network with Flexible Delivery Path

The distribution/allocation problem is known as one of the most comprehensive strategic decision. In real-world cases, it is impossible to solve a distribution/allocation problem in traditional ways with acceptable time. Hence researchers develop efficient non-traditional techniques for the large-term operation of the whole supply chain. These techniques provide near optimal solutions particularly for large-scale test problems. This paper presents an integrated supply chain model which is flexible in the delivery path. As the solution methodology, we apply a memetic algorithm with a neighborhood search mechanism and novelty in population presentation method called “extended random path direct encoding method.” To illustrate the performance of the proposed memetic algorithm, LINGO optimization software serves as comparison basis for small size problems. In large-size cases that we are dealing with in real world, a classical genetic algorithm as the second metaheuristic algorithm is considered to compare the results and show the efficiency of the memetic algorithm.

A memetic discrete differential evolution algorithm for the distributed permutation flow shop scheduling problem

The distributed manufacturing has become a prevail production mode under the economic globalization. In this article, a memetic discrete differential evolution (MDDE) algorithm is proposed to address the distributed permutation flow shop scheduling problem (DPFSP) with the minimization of the makespan. An enhanced NEH (Nawaz–Enscore–Ham) method is presented to produce potential candidate solutions and Taillard’s acceleration method is adopted to ameliorate the operational efficiency of the MDDE. A new discrete mutation strategy is introduced to promote the search efficiency of the MDDE. Four neighborhood structures, which are based on job sequence and factory assignment adjustment mechanisms, are introduced to prevent the candidates from falling the local optimum during the search process. A neighborhood search mechanism is selected adaptively through a knowledge-based strategy which focuses on the adaptive evaluation for the neighborhood selection. The optimal combinations of parameters in the MDDE algorithm are testified by the design of experiment. The computational results and comparisons demonstrated the effectiveness of the MDDE algorithm for solving the DPFSP.

Minimization of maximum lateness in a flowshop learning effect scheduling with release dates

In fiercely competitive industries, customer satisfaction is a significant evaluation for a modern enterprise. Reduction of delivery lateness is an effective way to improve satisfaction and increase revenues. This study addresses a flowshop scheduling problem to optimize maximum lateness, where learning effect is introduced for each task, i.e., more familiar is a worker with a particular task, shorter is the execution time. For simulating the dynamic setting in an enterprise, each task has an individual release date. For this strongly NP-hard problem, exact and approximate algorithms are presented to satisfy different production scenes. An earliest-due-date- (EDD)-based heuristic is provided to obtain an approximate solution in a short time. Asymptotic analysis indicates the convergence of the EDD-based heuristic under marginal cost condition, which suggests that it can substitute the optimal schedule in sense of probability limit. The optimal solution is achieved by an effective branch and bound (B&B) algorithm for small-scale instances, where well-designed branching rules and lower bound improve its search ability. For medium-scale instances, a discrete artificial bee colony algorithm with hybrid neighborhood search mechanism is introduced to capture high-quality solutions. The performance of the metaheuristic is enhanced by EDD-based initialization and key-interval-based operators. A series of numerical simulations is implemented to highlight the advantages of the proposed algorithms.

Feature Selection Algorithm for High-dimensional Biomedical Data Using Information Gain and Improved Chemical Reaction Optimization

Background: There have been rapid developments in various bioinformatics technologies, which have led to the accumulation of a large amount of biomedical data. However, these datasets usually involve thousands of features and include much irrelevant or redundant information, which leads to confusion during diagnosis. Feature selection is a solution that consists of finding the optimal subset, which is known to be an NP problem because of the large search space. Objective: For the issue, this paper proposes a hybrid feature selection method based on an improved chemical reaction optimization algorithm (ICRO) and an information gain (IG) approach, which called IGICRO. Methods: IG is adopted to obtain some important features. The neighborhood search mechanism is combined with ICRO to increase the diversity of the population and improve the capacity of local search. Results: Experimental results of eight public available data sets demonstrate that our proposed approach outperforms original CRO and other state-of-the-art approaches.

Improved Artificial Bee Colony Algorithm and Its Application to Fundus Retinal Blood Vessel Image Binarization

The content of this work is based on the characteristics of standard artificial bee colony(ABC) algorithm with weak local search ability and slow convergence speed. Then, an improved algorithm named KD-ABC is proposed. For improving the diversity and quality of the solution, it changes the generation method of honey source. In the initialization phase, it uses the cluster center generated by the K-MEANS method as the initial honey source instead of the initialization in the standard method. For improving the local optimization ability and the convergence speed without reducing the global search, we proposed a dynamic neighborhood search mechanism based on the number of iterations in terms of ABC search strategy and neighborhood selection stage. In order to find a suitable threshold to divide the grayscale image into blood vessels and background parts, we applied the characteristics of the KD-ABC algorithm to the binary processing stage of the fundus retinal blood vessel image, which lays the foundation for future image recognition.