Water cycle algorithm for solving constrained multi-objective optimization problems

A dynamic utopia point updating strategy for multi-objective optimization

Multi-objective optimization problems widely exist in engineering practice, which are usually solved by repeatedly searching Pareto fronts. However, the search process faces the challenges of heavy computational burden and local optimality. To overcome these difficulties, a novel utopia optimization method is developed in the paper, where the multi-objective problems are converted into single-objective ones without repeatedly searching, and then the computational burden is reduced. In particular, different from the classical utopia optimization, where the utopia point is always fixed during the whole optimization process, a dynamical updating strategy is proposed by comparing the current optimization solution and the original utopia point, and the local optimality problem is also solved. Moreover, an improved particle swarm optimization method with the Logistic map and adaptive weights is developed to further enhance the computing ability of the optimization solution. Finally, three examples are taken for simulations, and the results indicate that the relative errors between the global optimums and the optimal values are respectively reduced by 1.45, 1.06, and 0.14, compared with the existing approaches.

Structural optimization of different truss designs using two archive mult objective crystal structure optimization algorithm

Optimizing a multi-objective structure is a challenging design problem that requires handling several competing goals and constraints. Despite their success in resolving such issues, metaheuristics can be difficult to apply due to their stochastic nature and restrictions. This work proposes the multi-objective crystal structure optimizer (MOCRY), a potent and effective optimizer, to address this problem. The MOCRY algorithm, also known as MOCRY2arc, is built on a two-archive idea centered on diversity and convergence, respectively. The efficacy of MOCRY2arc in solving five typical planar and spatial real-world structure optimization issues was assessed. Because of these problems, safety and size limits were put on discrete cross-sectional regions and component stress. At the same time, different goals were being pursued, such as making nodal points bend more and reducing the mass of trusses. Four recognized standard evaluators-Hypervolume (HV), Generational-Inverted Generational Distance (GD, IGD), Spacing to Extent Metrics (STE), convergence, and diversity plots-were utilized to compare the results with those of nine sophisticated optimization techniques, including MOCRY and NSGA-II. Moreover, the Friedman rank test and comparison analysis showed that MOCRY2arc performed better at resolving big structure optimization issues in a shorter amount of computing time. In addition to identifying and realizing effective Pareto-optimal sets, the recommended method produced strong variety and convergence in the objective and choice spaces. As a result, MOCRY2arc may be a useful tool for handling challenging multi-objective structure optimization issues.

Advanced modeling of pharmaceutical solubility in solvents using artificial intelligence techniques: assessment of drug candidate for nanonization processing

This research is an analysis of multiple regression models developed for predicting ketoprofen solubility in supercritical carbon dioxide under different levels of T(K) and P(bar) as input features. Solubility of the drug was correlated to pressure and temperature as major operational variables. Selected models for this study are Piecewise Polynomial Regression (PPR), Kernel Ridge Regression (KRR), and Tweedie Regression (TDR). In order to improve the performance of the models, hyperparameter tuning is executed utilizing the Water Cycle Algorithm (WCA). Among, the PPR model obtained the best performance, with an R2 score of 0.97111, alongside an MSE of 1.6867E-09 and an MAE of 3.01040E-05. Following closely, the KRR model demonstrated a good performance with an R2 score of 0.95044, an MSE of 2.5499E-09, and an MAE of 3.49707E-05. In contrast, the TDR model produces a lower R2 score of 0.84413 together with an MSE of 7.4249E-09 and an MAE of 5.69159E-05.

Applying the new multi-objective algorithms for the operation of a multi-reservoir system in hydropower plants

The optimal operation of the multi-purpose reservoir system is a difficult, and, sometimes, non-linear problem in multi-objective optimization. By simulating biological behavior, meta-heuristic algorithms scan the decision space and can offer a set of points as a group of solutions to a problem. Because it is essential to simultaneously optimize several competing objectives and consider relevant constraints as the main problem in many optimization problems, researchers have improved their ability to solve multi-objective problems by developing complementary multi-objective algorithms. Because the AHA algorithm is new, its multi-objective version, MOAHA (multi-objective artificial hummingbird algorithm), was used in this study and compared with two novel multi-objective algorithms, MOMSA and MOMGA. Schaffer and MMF1 were used as two standard multi-objective benchmark functions to gauge the effectiveness of the proposed method. Then, for 180 months, the best way to operate the reservoir system of the Karun River basin, which includes Karun 4, Karun 3, Karun 1, Masjed-e-Soleyman, and Gotvand Olia dams to generate hydropower energy, supply downstream demands (drinking, agriculture, industry, environmental), and control flooding was examined from September 2000 to August 2015. Four performance appraisal criteria (GD, S, Δ, and MS) and four evaluation indices (reliability, resiliency, vulnerability, and sustainability) were used in Karun's multi-objective multi-reservoir problem to evaluate the performance of the multi-objective algorithm. All three algorithms demonstrated strong capability in criterion problems by using multi-objective algorithms' criteria and performance indicators. The large-scale (1800 dimensions) of the multi-objective operation of the Karun Basin reservoir system was another problem. With a minimum of 1441.71 objectives and an average annual hydropower energy manufacturing of 17,166.47 GW, the MOAHA algorithm demonstrated considerable ability compared to the other two. The final results demonstrated the MOAHA algorithm's excellent performance, particularly in difficult and significant problems such as multi-reservoir systems' optimal operation under various objectives.

Computational intelligence modeling of hyoscine drug solubility and solvent density in supercritical processing: gradient boosting, extra trees, and random forest models

This work presents the results of using tree-based models, including Gradient Boosting, Extra Trees, and Random Forest, to model the solubility of hyoscine drug and solvent density based on pressure and temperature as inputs. The models were trained on a dataset of hyoscine drug with known solubility and density values, optimized with WCA algorithm, and their accuracy was evaluated using R², MSE, MAPE, and Max Error metrics. The results showed that Gradient Boosting and Extra Trees models had high accuracy, with R² values above 0.96 and low MAPE and Max Error values for both solubility and density output. The Random Forest model was less accurate than the other two models. These findings demonstrate the effectiveness of tree-based models for predicting the solubility and density of chemical compounds and have potential applications in determination of drug solubility prior to process design by correlation of solubility and density to input parameters including pressure and temperature.

Many‑objective meta-heuristic methods for solving constrained truss optimisation problems: A comparative analysis

Many-objective truss structure problems from small to large-scale problems with low to high design variables are investigated in this study. Mass, compliance, first natural frequency, and buckling factor are assigned as objective functions. Since there are limited optimization methods that have been developed for solving many-objective truss optimization issues, it is important to assess modern algorithms performance on these issues to develop more effective techniques in the future. Therefore, this study contributes by investigating the comparative performance of eighteen well-established algorithms, in various dimensions, using four metrics for solving challenging truss problems with many objectives. The statistical analysis is performed based on the objective function best mean and standard deviation outcomes, and Friedman's rank test. MMIPDE is the best algorithm as per the overall comparison, while SHAMODE with whale optimisation approach and SHAMODE are the runners-up.•A comparative test to measure the efficiency of eighteen state-of-the-practice methods is performed.•Small to large-scale truss design challenges are proposed for the validation.•The performance is measured using four metrics and Friedman's rank test.

Comparison of multi-objective evolutionary algorithms applied to watershed management problem

Simulation-based optimization (S-O) frameworks are effective in developing cost-effective watershed management strategies, where optimization algorithms have substantial effect on the quality of strategies. Despite the development and improvement of multi-objective evolutionary algorithms (MOEAs) provide more robust alternatives for optimization, they typically have limited applications in real-world decision contexts. In this study, three advanced MOEAs, including NSGA-II, MOEA/D and NSGA-III, were introduced into the S-O framework and applied to a real-world watershed management problem, and their performance and characteristics were quantified through performance metrics. Results show that a higher crossover or mutation probability do not necessarily promote convergence and diversity of solutions, while a larger generation and population size is helpful for MOEAs to find high-quality solutions. Compared to the other two MOEAs, NSGA-II consistently exhibits robust performance in finding solutions with good convergence and high diversity, and provides more options at the same computational cost, while the degenerate Pareto front of the proposed watershed management problem may account for the poor performance of MOEA/D and NSGA-III in terms of diversity. For a 10% TN or TP reduction target, the average cost of the NSGA-II optimized strategies is 32.22% or 47.83% of the commonly used strategies. In addition, this study also discussed the development of resilient watershed management to buffer the impacts of climate change on aquatic system, the incorporation of fuzzy programming into the S-O framework to develop robust watershed management strategies under uncertainty, and the application of machine learning-based surrogate models to reduce computational cost of the S-O framework. These results can contribute to the understanding of MOEAs and provide useful guidance to decision makers.

A Cooperative Evolutionary Framework Based on an Improved Version of Directed Weight Vectors for Constrained Multiobjective Optimization With Deceptive Constraints

When solving constrained multiobjective optimization problems (CMOPs), the most commonly used way of measuring constraint violation is to calculate the sum of all constraint violations of a solution as its distance to feasibility. However, this kind of constraint violation measure may not reflect the distance of an infeasible solution from feasibility for some problems, for example, when an infeasible solution closer to a feasible region does not have a smaller constraint violation than the one farther away from a feasible region. Unfortunately, no set of artificial benchmark problems focusing on this area exists. To remedy this issue, a set of CMOPs with deceptive constraints is introduced in this article. It is the first attempt to consider CMOPs with deceptive constraints (DCMOPs). Based on our previous work, which designed a set of directed weight vectors to solve CMOPs, this article proposes a cooperative framework with an improved version of directed weight vectors to solve DCMOPs. Specifically, the cooperative framework consists of two switchable phases. The first phase uses two subpopulations-one to explore feasible regions and the other to explore the entire space. The two subpopulations provide useful information about the optimal direction of objective improvement to each other. The second phase aims mainly at finding Pareto-optimal solutions. Then an infeasibility utilization strategy is used to improve the objective function values. The two phases are switchable based on the information found to date at any time in the evolutionary process. The experimental results show that this method significantly outperforms the algorithms with which it is compared on most of the DCMOPs, in terms of reliability and stability in finding a set of well-distributed optimal solutions.

A new optimization algorithm to solve multi-objective problems

Simultaneous optimization of several competing objectives requires increasing the capability of optimization algorithms. This paper proposes the multi-objective moth swarm algorithm, for the first time, to solve various multi-objective problems. In the proposed algorithm, a new definition for pathfinder moths and moonlight was proposed to enhance the synchronization capability as well as to maintain a good spread of non-dominated solutions. In addition, the crowding-distance mechanism was employed to select the most efficient solutions within the population. This mechanism indicates the distribution of non-dominated solutions around a particular non-dominated solution. Accordingly, a set of non-dominated solutions obtained by the proposed multi-objective algorithm is kept in an archive to be used later for improving its exploratory capability. The capability of the proposed MOMSA was investigated by a set of multi-objective benchmark problems having 7 to 30 dimensions. The results were compared with three well-known meta-heuristics of multi-objective evolutionary algorithm based on decomposition (MOEA/D), Pareto envelope-based selection algorithm II (PESA-II), and multi-objective ant lion optimizer (MOALO). Four metrics of generational distance (GD), spacing (S), spread (Δ), and maximum spread (MS) were employed for comparison purposes. The qualitative and quantitative results indicated the superior performance and the higher capability of the proposed MOMSA algorithm over the other algorithms. The MOMSA algorithm with the average values of CPU time = 2771 s, GD = 0.138, S = 0.063, Δ = 1.053, and MS = 0.878 proved to be a robust and reliable model for multi-objective optimization.

On Mathematical Modelling of Automated Coverage Optimization in Wireless 5G and beyond Deployments

The need to optimize the deployment and maintenance costs for service delivery in wireless networks is an essential task for each service provider. The goal of this paper was to optimize the number of service centres (gNodeB) to cover selected customer locations based on the given requirements. This optimization need is especially emerging in emerging 5G and beyond cellular systems that are characterized by a large number of simultaneously connected devices, which is typically difficult to handle by the existing wireless systems. Currently, the network infrastructure planning tools used in the industry include Atoll Radio Planning Tool, RadioPlanner and others. These tools do not provide an automatic selection of a deployment position for specific gNodeB nodes in a given area with defined requirements. To design a network with those tools, a great deal of manual tasks that could be reduced by more sophisticated solutions are required. For that reason, our goal here and our main contribution of this paper were the development of new mathematical models that fit the currently emerging scenarios of wireless network deployment and maintenance. Next, we also provide the design and implementation of a verification methodology for these models through provided simulations. For the performance evaluation of the models, we utilize test datasets and discuss a case study scenario from a selected district in Central Europe.

TOPSIS Decision on Approximate Pareto Fronts by Using Evolutionary Algorithms: Application to an Engineering Design Problem

A common technique used to solve multi-objective optimization problems consists of first generating the set of all Pareto-optimal solutions and then ranking and/or choosing the most interesting solution for a human decision maker (DM). Sometimes this technique is referred to as generate first–choose later. In this context, this paper proposes a two-stage methodology: a first stage using a multi-objective evolutionary algorithm (MOEA) to generate an approximate Pareto-optimal front of non-dominated solutions and a second stage, which uses the Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) devoted to rank the potential solutions to be proposed to the DM. The novelty of this paper lies in the fact that it is not necessary to know the ideal and nadir solutions of the problem in the TOPSIS method in order to determine the ranking of solutions. To show the utility of the proposed methodology, several original experiments and comparisons between different recognized MOEAs were carried out on a welded beam engineering design benchmark problem. The problem was solved with two and three objectives and it is characterized by a lack of knowledge about ideal and nadir values.

Water Cycle Algorithm for Probabilistic Planning of Renewable Energy Resource, Considering Different Load Models

This work introduces multi-objective water cycle algorithm (MOWCA) to find the accurate location and size of distributed energy resource (DERs) considering different load models for two seasons (winter, and summer). The impact of uncertainties produced from load and renewable energy resource (RES) such as wind turbine (WT) and photovoltaic (PV) on the performance of the radial distribution system (RDS) are covered as this is closer to the real operation condition. The point estimate method (PEM) is applied for modeling the RES uncertainties. An optimization technique is implemented to find the multi-objective optimal allocation of RESs in RDSs considering uncertainty effect. The main objectives of the work are to maximize the technical, economic and environmental benefits by minimizing different objective functions such as the dissipated power, the voltage deviation, DG cost and total emissions. The proposed multi-objective model is solved by using multi-objective water cycle algorithm (MOWCA), considering the Pareto criterion with nonlinear sorting based on fuzzy mechanism. The proposed algorithm is carried out on different IEEE power systems with various cases.

Energy optimization of Multiple Stage Evaporator system using Water Cycle Algorithm

Black liquor, a residual stream from the Kraft recovery process of paper mills is an incipient biomass energy resource which finds prospective biofuel-based industrial applications to ensure process self-sufficiency and sustainability. Black liquor is concentrated using Multiple Stage Evaporator, the utmost energy intensive unit, before using it as biofuel. Pertaining to the contemporary global energy scenario, improvement in energy efficiency of Multiple Stage Evaporator becomes indispensable. The present work investigates the non-linear modeling and simulation-based optimization of Heptads' stage based Multiple Stage Evaporator in backward feed flow configuration integrated with various energy saving strategies. A novel metaheuristic approach, Water Cycle Algorithm has been employed to search the optimum estimates of unknown process variables and therefore, the optimum energy efficiency parameters. The optimization results demonstrate the efficiency of Water Cycle Algorithm in screening the most appropriate operating strategy, i.e., hybrid model of all energy saving strategies (steam-split, feed-split and feed-preheating) with optimum energy efficiency i.e. Steam Economy of 7.092 and Steam Consumption of 1.919 kg/s. Moreover, a comparative analysis of the results with previous literature and real-time plant estimates reveal that the hybrid model offers improvement of 52.84% in Steam Economy and reduction in Steam Consumption by 28.13% when compared to the real plant data.

Shape Design Optimization of a Robot Arm Using a Surrogate-Based Evolutionary Approach

In the design optimization of robot arms, the use of simulation technologies for modeling and optimizing the objective functions is still challenging. The difficulty is not only associated with the large computational cost of high-fidelity structural simulations but also linked to the reasonable compromise between the multiple conflicting objectives of robot arms. In this paper we propose a surrogate-based evolutionary optimization (SBEO) method via a global optimization approach, which incorporates the response surface method (RSM) and multi-objective evolutionary algorithm by decomposition (the differential evolution (DE ) variant) (MOEA/D-DE) to tackle the shape design optimization problem of robot arms for achieving high speed performance. The computer-aided engineering (CAE) tools such as CAE solvers, computer-aided design (CAD) Inventor, and finite element method (FEM) ANSYS are first used to produce the design and assess the performance of the robot arm. The surrogate model constructed on the basis of Box–Behnken design is then used in the MOEA/D-DE, which includes the process of selection, recombination, and mutation, to optimize the robot arm. The performance of the optimized robot arm is compared with the baseline one to validate the correctness and effectiveness of the proposed method. The results obtained for the adopted example show that the proposed method can not only significantly improve the robot arm performance and save computational cost but may also be deployed to solve other complex design optimization problems.

Optimal Allocation of Hybrid Renewable Energy System by Multi-Objective Water Cycle Algorithm

This article offers a multi-objective framework for an optimal mix of different types of distributed energy resources (DERs) under different load models. Many renewable and non-renewable energy resources like photovoltaic system (PV), micro-turbine (MT), fuel cell (FC), and wind turbine system (WT) are incorporated in a grid-connected hybrid power system to supply energy demand. The main aim of this article is to maximize environmental, technical, and economic benefits by minimizing various objective functions such as the annual cost, power loss and greenhouse gas emission subject to different power system constraints and uncertainty of renewable energy sources. For each load model, optimum DER size and its corresponding location are calculated. To test the feasibility and validation of the multi-objective water cycle algorithm (MOWCA) is conducted on the IEEE-33 bus and IEEE-69 bus network. The concept of Pareto-optimality is applied to generate trilateral surface of non-dominant Pareto-optimal set followed by a fuzzy decision-making mechanism to obtain the final compromise solution. Multi-objective non-dominated sorting genetic (NSGA-III) algorithm is also implemented and the simulation results between two algorithms are compared with each other. The achieved simulation results evidence the better performance of MOWCA comparing with the NSGA-III algorithm and at different load models, the determined DER locations and size are always righteous for enhancement of the distribution power system performance parameters.

Solving the Manufacturing Cell Design Problem through an Autonomous Water Cycle Algorithm

Metaheuristics are multi-purpose problem solvers devoted to particularly tackle large instances of complex optimization problems. However, in spite of the relevance of metaheuristics in the optimization world, their proper design and implementation to reach optimal solutions is not a simple task. Metaheuristics require an initial parameter configuration, which is dramatically relevant for the efficient exploration and exploitation of the search space, and therefore to the effective finding of high-quality solutions. In this paper, the authors propose a variation of the water cycle inspired metaheuristic capable of automatically adjusting its parameter by using the autonomous search paradigm. The goal of our proposal is to explore and to exploit promising regions of the search space to rapidly converge to optimal solutions. To validate the proposal, we tested 160 instances of the manufacturing cell design problem, which is a relevant problem for the industry, whose objective is to minimize the number of movements and exchanges of parts between organizational elements called cells. As a result of the experimental analysis, the authors checked that the proposal performs similarly to the default approach, but without being specifically configured for solving the problem.

Improved multiobjective salp swarm optimization for virtual machine placement in cloud computing

In data center companies, cloud computing can host multiple types of heterogeneous virtual machines (VMs) and provide many features, including flexibility, security, support, and even better maintenance than traditional centers. However, some issues need to be considered, such as the optimization of energy usage, utilization of resources, reduction of time consumption, and optimization of virtual machine placement. Therefore, this paper proposes an alternative multiobjective optimization (MOP) approach that combines the salp swarm and sine-cosine algorithms (MOSSASCA) to determine a suitable solution for virtual machine placement (VMP). The objectives of the proposed MOSSASCA are to maximize mean time before a host shutdown (MTBHS), to reduce power consumption, and to minimize service level agreement violations (SLAVs). The proposed method improves the salp swarm and the sine-cosine algorithms using an MOP technique. The SCA works by using a local search approach to improve the performance of traditional SSA by avoiding trapping in a local optimal solution and by increasing convergence speed. To evaluate the quality of MOSSASCA, we perform a series of experiments using different numbers of VMs and physical machines. The results of MOSSASCA are compared with well-known methods, including the nondominated sorting genetic algorithm (NSGA-II), multiobjective particle swarm optimization (MOPSO), a multiobjective evolutionary algorithm with decomposition (MOEAD), and a multiobjective sine-cosine algorithm (MOSCA). The results reveal that MOSSASCA outperforms the compared methods in terms of solving MOP problems and achieving the three objectives. Compared with the other methods, MOSSASCA exhibits a better ability to reduce power consumption and SLAVs while increasing MTBHS. The main differences in terms of power consumption between the MOSCA, MOPSO, MOEAD, and NSGA-II and the MOSSASCA are 0.53, 1.31, 1.36, and 1.44, respectively. Additionally, the MOSSASCA has higher MTBHS value than MOSCA, MOPSO, MOEAD, and NSGA-II by 362.49, 274.70, 585.73 and 672.94, respectively, and the proposed method has lower SLAV values than MOPSO, MOEAD, and NSGA-II by 0.41, 0.28, and 1.27, respectively.

A systematic approach for watershed ecological restoration strategy making: An application in the Taizi River Basin in northern China

Aiming to protect freshwater ecosystems, river ecological restoration has been brought into the research spotlight. However, it is challenging for decision makers to set appropriate objectives and select a combination of rehabilitation acts from numerous possible solutions to meet ecological, economic, and social demands. In this study, we developed a systematic approach to help make an optimal strategy for watershed restoration, which incorporated ecological security assessment and multi-objectives optimization (MOO) into the planning process to enhance restoration efficiency and effectiveness. The river ecological security status was evaluated by using a pressure-state-function-response (PSFR) assessment framework, and MOO was achieved by searching for the Pareto optimal solutions via Non-dominated Sorting Genetic Algorithm II (NSGA-II) to balance tradeoffs between different objectives. Further, we clustered the searched solutions into three types in terms of different optimized objective function values in order to provide insightful information for decision makers. The proposed method was applied in an example rehabilitation project in the Taizi River Basin in northern China. The MOO result in the Taizi River presented a set of Pareto optimal solutions that were classified into three types: I - high ecological improvement, high cost and high benefits solution; II - medial ecological improvement, medial cost and medial economic benefits solution; III - low ecological improvement, low cost and low economic benefits solution. The proposed systematic approach in our study can enhance the effectiveness of riverine ecological restoration project and could provide valuable reference for other ecological restoration planning.