Nonlinear Programming Techniques Research Articles

OPTIMIZING ROYALTY PAYMENTS FOR MAXIMUM ECONOMIC BENEFIT: A CASE STUDY UTILIZING MODIFIED SHOOTING AND DISCRETIZATION METHODS

This research delves into the application of the modified shooting method for the numerical resolution of non-standard optimal control (OC) problems. More precisely, it concentrates on scenarios where the final state value component remains unknown and unconstrained, leading to a non-zero final shadow value or costate variable. Moreover, the objective function involved a piecewise royalty function, which poses a challenge due to its lack of differentiability within a specific time interval. Consequently, the novel modified shooting method was employed to ascertain the elusive final state value. The model’s differentiability is maintained throughout by adopting a continuous hyperbolic tangent (tanh) approximation. In addition, the construction of the Sufahani-Ahmad-Newton-Golden-Royalty Algorithm (SANGRA) and Sufahani-Ahmad-Powell-Golden-Royalty Algorithm (SAPGRA) was accomplished using the C++ programming language to formulate the problem. The outcomes of these algorithms, satisfying the criteria for optimality, were juxtaposed with non-linear programming (NLP) techniques such as Euler and Runge-Kutta, aside from Trapezoidal and Hermite-Simpson approximations. This groundbreaking discovery carries extensive practical implications as it propels the field forward and ensures the application of contemporary problem-solving methodologies. Moreover, the study underscores the significance of fundamental theory in effectively tackling current economic challenges.

The applicability of Zermelo's equation to indirect and direct optimization of commercial aircraft flights

This paper focuses on the applicability of Zermelo's equation within the context of 3D commercial aircraft trajectory optimization problems. The associated optimal control problem includes two singular controls, specifically aerodynamic path angle and throttle setting, and one regular control, specifically heading angle. Using Pontryagin's maximum principle and the direct adjoining method, we show that the optimal heading angle is defined by Zermelo's equation. The significance of the presented analysis is that Zermelo's equation holds for 3D commercial aircraft flights, even in the presence of standard state-inequality constraints and a more general objective function. With the help of Zermelo's equation, the control problem is initially analyzed for a 3D time-optimal climb-phase scenario, which is solved by an indirect approach. Through the analysis of the switching function, we demonstrate the dependency of the optimal aerodynamic path angle on the optimal heading angle. Next, by tackling a 3D time-fuel-optimal free-routing flight, solved by a direct approach, we show that Zermelo's equation can help reduce the overall dimension of the associated nonlinear programming. To successfully handle the initial guess in both indirect and direct problems, it is proposed a diminutive nonlinear programming technique as a fast and robust initializer. This simple-yet-effective initializer provides sufficient information about the optimal controls and state dynamics required for initializing a direct optimization, as well as the optimal switching times and co-states required for initializing an indirect optimization.

Controller design using PSO and WOA algorithm for enhanced performance of vector-controlled PMSM drive

ABSTRACT This study presents the tuning strategy of PI controllers gain parameters to improve the performance of vector-controlled permanent magnet synchronous motor drive. In the proposed work PI controller parameters are optimized using Particle Swarm Optimization (PSO) and Whale Optimisation Algorithm (WOA). Nonlinear programming techniques, namely PSO and WOA optimisation, are employed to enhance the convergence rate of speed, torque, and flux errors using an offline tuning method. The simulation results of speed, torque, and flux show that the transient and steady state responses have been greatly improved and verified for different operating conditions. It has been observed that the quality of stator current, developed output voltage of inverter, and ripples in electromagnetic torque are also improved, which shows the feasibility and accuracy of the proposed method.

Application of the mixed integer nonlinear programming technique for the economic planning of transmission networks in new manner

Application of the mixed integer nonlinear programming technique for the economic planning of transmission networks in new manner

Artificial intelligence driven smart operation of large industrial complexes supporting the net-zero goal: Coal power plants

The true potential of artificial intelligence (AI) is to contribute towards the performance enhancement and informed decision making for the operation of the large industrial complexes like coal power plants. In this paper, AI based modelling and optimization framework is developed and deployed for the smart and efficient operation of a 660 MW supercritical coal power plant. The industrial data under various power generation capacity of the plant is collected, visualized, processed and subsequently, utilized to train artificial neural network (ANN) model for predicting the power generation. The ANN model presents good predictability and generalization performance in external validation test with R2 = 0.99 and RMSE =2.69 MW. The partial derivative of the ANN model is taken with respect to the input variable to evaluate the variable’ sensitivity on the power generation. It is found that main steam flow rate is the most significant variable having percentage significance value of 75.3 %. Nonlinear programming (NLP) technique is applied to maximize the power generation. The NLP-simulated optimized values of the input variables are verified on the power generation operation. The plant-level performance indicators are improved under optimum operating mode of power generation: savings in fuel consumption (3 t/h), improvement in thermal efficiency (1.3 %) and reduction in emissions discharge (50.5 kt/y). It is also investigated that maximum power production capacity of the plant is reduced from 660 MW to 635 MW when the emissions discharge limit is changed from 510 t/h to 470 t/h. It is concluded that the improved plant-level performance indicators and informed decision making present the potential of AI based modelling and optimization analysis to reliably contribute to net-zero goal from the coal power plant.

A Proposed New Non-Linear Programming Technique for Solving a Mixed Strategy Problem in Game Theory

A Proposed New Non-Linear Programming Technique for Solving a Mixed Strategy Problem in Game Theory

Design and Development of Neutronics and Thermal Hydraulics Modeling Code for ACP1000 Nuclear Reactor Dynamics in LabVIEW

An advanced neutronics and thermal hydraulics nuclear code, called GNTHACP code, is designed and developed in LabVIEW as Graphical Neutronics and Thermal Hydraulics toolkit for 1100 MWe Advanced Chinese PWR (ACP-1000) nuclear power plant. The reactor neutronics model is developed using a nonlinear point reactor kinetics model, while, the reactor thermal hydraulics model is developed based on nonlinear fuel and coolant temperature dynamics. The heart of the GNTHACP code is the control rod reactivity model. Control rod reactivity banks are comprised of four power compensation banks G1, G2, N1, N2 and one temperature compensation bank R. The reactivity control configuration of these banks is highly nonlinear, complex and challenging in nature. The control rod reactivity model as a function of G1, G2, N1, N2 and R banks is optimized using two distinct techniques. The control rod reactivity model is optimized using Simplex Linear Programming (SLP) technique under constraints of reactor power as safety limit and control rod speed as maximum speed limit in LabVIEW. The control rod reactivity model is also optimized and investigated using nonlinear Sequential Quadratic Programming (SQP) technique under same constraints in LabVIEW. All the models are integrated and the state-of-the-art virtual instruments (VIs) are designed for cost function optimization, configuring models and calibration of model parameters in LabVIEW. The integrated model as graphical coupled neutronics and thermal hydraulics modeling code is optimized and validated against the Final Safety Analysis Report (FSAR) and different parameters of interest are investigated and proved within design limits as reported with CORCA and CORTH benchmark nuclear codes. The proposed code is stable, highly efficient and accurate as compared to other nuclear codes.

Unlined Length Effect on the Tunnel Face Stability and Collapse Mechanisms in c-ϕ Soils: A Numerical Study with Advanced Mesh Adaptive Strategies

This paper presents a stability study on the collapse mechanisms of a plane-strain tunnel face in c-ϕ soils using the upper bound finite element method with rigid translatory moving elements (UBFELA-RTME) and nonlinear programming technique. Practical considerations are given to the unlined length influence behind the tunnel face. An advanced mesh adaptive updating strategy is adopted, aiming to improve the computational efficiency, the accuracy of upper-bound solutions, as well as the produced collapse mechanisms. The unlined length influence on the face stability and collapse mechanism of the tunnel face are determined with various combinations of tunnel depth ratios, soil friction angles, and dilatancy angles. Using the UBFELA-RTME with the Davis’s approach and a mesh adapting strategy, the non-associated plasticity flow rule can be well approximated. The developed technique was validated against different numerical methods, and it is concluded that the tunnel face stability can be improved by increasing soil friction and dilatancy angles, and yet weakens as the unlined length increases where a mesh-liked collapse zone gradually appears on the tunnel vault top. It gradually evolves to a global collapse failure till the ground surface. The findings contribute to a better understanding of the ground surface failure under the unlined support length influence in tunnel construction.

Fuzzy modeling and cost optimization for machine repair problem with retrial under admission control F-policy and feedback

In the present paper, the machine repair problem (MRP) with real-life scenarios is investigated, which plays a significant role in the manufacturing process, industries, production lines, etc. An M∖M∖1∖K machine repair problem with retrial attempts under admission control policy F-policy and machine’s feedback is considered in which a single repairman is available to repair the failed machines. The recursive technique is applied to solve the mathematical model and establish the probability distributions for different states, which are further used to develop the performance indices. The parametric non-linear programming (P-NLP) technique is employed to develop the MRP with F-policy in a fuzzy environment using the trapezoidal and sigmoidal fuzzy numbers. Zadeh’s extension principle and parametric non-linear programming technique help us to find the α-cuts of the developed performance predictors of MRP. We frame and minimize the total cost of the system. The harmony search method (HSM) and quasi-Newton method are implemented to achieve the total minimum cost of the system corresponding to the optimal repair rate, which helps the system managers in the formulation of business policies in industries, including the automobile sector. A numerical example is also given, which produces several tables and graphs that show the validity of the model.

Nonlinear Programming Techniques in Optimization: An exhaustive comparative study understanding Quasi – Newton and Gauss – Newton methods

Non – linear Programming Applications are becoming increasingly important as managers and operations researchers become more sophisticated in implementing decision – oriented mathematical models, as well as computer routines capable of solving large – scale nonlinear problems become more widely available. By means of this paper, we investigate two of the very widely and varied NLP techniques, namely the Gauss – Newton method and the Quasi – Newton method. When computation or iteration is expensive, Quasi –Newton methods are an effective method for function optimization. Even if their precise approaches differ, when the issues are complicated, they can all determine the optimum more quickly and effectively than Newton's Method. Gauss – Newton can be used to locate a single point or, as it is most frequently use, to evaluate how well a theoretical model matches a collection of experimental data points. We get the most accurate estimates of the unknown variables in a theoretical model by solving the system of nonlinear equations. In this review we present an overview of the methods mentioned earlier, discuss the scope of them, and advocate a comparison between the two.

Unified AC Transmission Expansion Planning Formulation incorporating VSC-MTDC, FACTS devices, and Reactive Power compensation

The main aim of the static Transmission Network Expansion Planning (TNEP) is to determine which and where new transmission equipment must be installed. The complexity added by the non-linearities leads to simplifications, which include the DC model. However, most non-linearities are solvable nowadays. Thus, the new scenario of large non-dispatchable power sources penetration and the several developments in technologies, e.g, Flexible AC Transmission Systems (FACTS) devices and High Voltage Direct Current (HVDC) interconnections, motivate the use of the AC model with its non-linearities. Some research works address the use of some of those technologies for TNEP in an independent fashion, which can lead to sub-optimal solutions. In this work, Voltage Source Controlled-Multiterminal HVDC (VSC-MTDC) systems, FACTS devices, and Reactive Power Planning (RPP) are integrated into the same planning optimization process, so that a unified AC TNEP formulation is proposed. A non-linear mathematical programming technique and a differential evolution based metaheuristics are chosen to achieve an optimal transmission configuration. To evaluate the benefits of the proposed approach, two IEEE modified test systems (9 and 118 buses) are used. Results suggest that more economical solutions can be obtained if different types of reinforcement strategies are taken into account in a unified approach.

Carbon efficiency analysis in the provision of drinking water: Estimation of optimal greenhouse gas emissions

Assessing carbon efficiency (CE) in the provision of drinking water services is essential to achieve a net-zero greenhouse gas (GHG) urban water cycle. Previous studies evaluating the CE of water companies are very scarce and employed parametric and non-parametric. Both methodological approaches present limitations such as overfitting issues or require assumptions about the production technology which could lead to less reliable efficiency scores. To overcome these limitations, in this study, and for the first time, we estimated CE of English and Welsh water companies using the Efficiency Analysis Trees (EAT) approach. This technique brings together machine learning and non-linear programming techniques to estimate production frontier and efficiency scores. It also allowed us to quantify the optimal level of GHG emissions in the provision of water services and estimate potential GHG savings. Bootstrap truncated regression methods were employed to quantify the impact of operating characteristics on CE of water companies. The optimal level of GHG emissions was estimated to be between 0.062 and 133.03 tons of CO2 equivalent (CO2eq) per year and per connected property. The average CE was at the level of 0.632. This means that GHG emissions could reduce by 36.8% to maintain the same level of water services. Equivalently, this corresponds to a reduction of 488,321 tons of CO2eq per year. Water only companies exhibited a better performance than water and sewerage companies with an average CE of 0.785 and 0.540, respectively. The performance of the English and Welsh water companies decreased over time. In 2011 the average CE was 0.772 whereas it went down to 0.534 in 2020. It was also estimated that on average water companies could reduce 0.034 tons of CO2eq per cubic meter of drinking water supplied and 16.16 tons of CO2eq/connected property per year. The regression results showed that topography and water treatment complexity had a significant impact on CE. The conclusions of this study are relevant for policy makers to define policies toward a low-carbon urban water cycle.

Optimization of Planar Phased Arrays for Vehicles

Radar sensors are used for the detection of objects surrounding a vehicle to enable advanced driver assistance systems. Wide angle azimuth field-of-view (FOV) sensor having high angular resolution is beneficial, but presents challenges for low-cost implementation where element-to-element spacing may be larger than one-half wavelength. Irregular placement of array elements in a two-dimensional (2-D) array can achieve low-cost targets, but may generate large sidelobes if done without care. We introduce an optimization method to determine placement of array elements in a vehicular volumetric beam-scanning radar to minimize sidelobe power for typical automotive beam steering angles. Optimization with array constraints is realized using a gradient-based nonlinear programming technique. Optimization is accelerated by extending <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">uv</i> -projection planes beyond the unit circle. For a grating lobe-free scan area based on vehicular applications, the azimuth scanning FOV is increased by 11° bidirectionally to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\pm$</tex-math></inline-formula> 54° compared to a reference triangular grid array with 24 elements. The validity of an antenna design generated using the optimization method is established by a three-dimensional (3-D) electromagnetic wave simulation of a representative optimized array configuration.

Investigating the Utility of Water Cycle Optimization and Non-linear Programming Techniques in Electric Vehicle Stations Smart Planning

Transportation electrification shows enormous prospects regarding the global environmental pollution issue. However, the penetration of electric vehicles (EVs) is not massive in the global market. We attribute this to the long AC charging time and the limitations of the charging topologies at home. Accordingly, the need for a centralized electrical vehicle station (EVS) becomes a must to enlarge the contribution of the EVs industry market. Intelligent decision-making is essential in enhancing the EVS performance by optimizing the charging time and minimizing the queuing delay, especially during peak hours. In the current study, we propose an entire scenario for an EVS with a maximum of 14 EVs at peak hours. Multiple charging methodologies have been provided, including DC constant current constant voltage, DC multistage constant currents, AC 110/120 V, and AC 220/240 V. In addition, four various photovoltaic/grid integration scenarios have been introduced seeking the optimum EVS operation concerning; Levelized cost of energy, charging time and capacity. Finally, two charging schemes have been demonstrated, targeting a minimum total charging time and queuing delay, where the water cycle optimization technique has been enrolled in parallel with non-linear programming. A significant reduction in the total charging time, nearly 90%, has been reached concerning the overnight in-home AC charging time.

Three numerical approaches to find mutually unbiased bases using Bell inequalities

Mutually unbiased bases correspond to highly useful pairs of measurements in quantum information theory. In the smallest composite dimension, six, it is known that between three and seven mutually unbiased bases exist, with a decades-old conjecture, known as Zauner's conjecture, stating that there exist at most three. Here we tackle Zauner's conjecture numerically through the construction of Bell inequalities for every pair of integers n,d&amp;#x2265;2 that can be maximally violated in dimension d if and only if n MUBs exist in that dimension. Hence we turn Zauner's conjecture into an optimisation problem, which we address by means of three numerical methods: see-saw optimisation, non-linear semidefinite programming and Monte Carlo techniques. All three methods correctly identify the known cases in low dimensions and all suggest that there do not exist four mutually unbiased bases in dimension six, with all finding the same bases that numerically optimise the corresponding Bell inequality. Moreover, these numerical optimisers appear to coincide with the ``four most distant bases'' in dimension six, found through numerically optimising a distance measure in [P. Raynal, X. Lü, B.-G. Englert, {Phys. Rev. A}, { 83} 062303 (2011)]. Finally, the Monte Carlo results suggest that at most three MUBs exist in dimension ten.

Systematic Design of Solvent Recovery Pathways: Integrating Economics and Environmental Metrics

The increasing trend of solvent usage by chemical industries has caused an upsurge in hazardous waste solvents and, consequently, increased their toxicity potential within the environment. The increase in overall carbon and ecological footprints due to conventional waste handling techniques such as incineration and onsite and offsite disposal can hardly be overstated. Even though solvent recovery methods present a better alternative to these techniques, analysis is generally centered on economics. This paper presents a framework that evaluates the economic and sustainability metrics of solvent recovery processes. In the first approach, standalone software (SimaPro) was used to evaluate the impact of the recovery process. The framework was extended by incorporating the economics, Sustainable Process Index (SPI), and Emergy metrics. Thus, solvent recovery challenges have been transformed into a multiobjective optimization problem that is solved through superstructure and mixed-integer nonlinear programming techniques. The applicability of the framework is illustrated by two comprehensive case studies with varying complexities, and the results are compared with conventional methods. The results indicate that about 76–85% of the ecological and energy burden can be prevented if solvent recovery is the preferred option to incineration.

A cost analysis on Multi-item Inventory model for Factory Outlets with investment constraint under ranking Asteroid Fuzzy Set

Purpose of the study: A multi-item inventory model for factory outlets in crisp and fuzzy sense are formulated in the fuzzy environment with investment under one constraint has been considered. In this model, demand is constant and is related to the price per unit item. The asteroid fuzzy set is defined and is properties are given.&#x0D; Methodology: The parameters involved in this model represented by asteroid fuzzy set. The average total cost is defuzzify by ranking method.&#x0D; Main Findings: The analytical expressions for maximum inventory level and average total cost are derived for the proposed model by using nonlinear programming technique. A numerical example is presented to illustrate the results.&#x0D; Applications of this study: Ranking Asteroid fuzzy set is considered profitable in small businesses. Here we considered the factory outlet .Also its use is considered to help the small scale entrepreneurs during festival period and pandemic situation.&#x0D; Novelty/Originality of this study: In this paper, a novel approach to handle the asteroid fuzzy set is proposed. It uses ranking the cost parameters of the asteroid fuzzy set with the best approximation level. The parameters involved are asteroid fuzzy set and are all ill-defined in nature.

Optimization of regeneration temperature for energy integrated water allocation networks

The conservation of energy and water resources is vital to tackle climate change and ensure sustainability. Energy integrated water allocation networks simultaneously conserve these resources using heat exchangers and regeneration units. A hybrid solution strategy is proposed in this paper to achieve the minimum total annualized cost of these networks and offer market competitiveness. The proposed algorithm solves the mixed-integer non-linear programming problem formulated in this study through a heuristic-based non-linear programming technique. The ideology of Pinch Analysis is implemented to identify heuristics that explore the non-isothermal mixing potential of streams and optimize the regeneration temperatures. These heuristics can drastically reduce energy requirements, while the mathematical optimization of water reuse, recycling, and regeneration decreases water consumption. Capturing the trade-offs between energy consumption and heat exchangers ensures a lower total annualized cost. The potency of the algorithm developed in this work is exhibited using demonstrative examples from the literature. The variation in the operating cost, investment expense, and thereby, the total annualized costs with the temperature of regeneration units and the regeneration throughput are demonstrated in these examples. Optimal regeneration temperature can significantly reduce the total annualized costs of the overall system, as observed through these examples. In the single contaminant water network example, the total annualized cost is 45.3% lower compared to the literature. • Proposed approach integrates conceptual Pinch Analysis with numerical algorithm. • Energy conservation is achieved by optimizing regeneration temperature. • The hybrid algorithm minimizes the total annualized cost prior to network design. • The proposed algorithm reduces the total annualized cost by 45.3%.

A Fuzzy Multi Objective Inventory Model with Production Cost and Set-up-Cost Dependent on Population

AbstractIn this paper, I have developed a multi item production inventory model for the non-deteriorating items with constant demand rate under the limitation on set up cost. The production price and set-up price are the most vital problem within the inventory system of the marketplace in international. Here the production cost is dependent on the demand as well as populations. Set up cost is dependent on the average inventory level. Holding cost is the most challenging issue in the business world. In order to reduce the holding cost, the holding cost function has been considered as on the number of peoples. Due to uncertainty all the cost parameters are taken as the generalized triangular fuzzy number. Multi objective fuzzy inventory model has been solved by various techniques like Fuzzy programming technique with hyperbolic membership function, Fuzzy non-linear programming technique and Fuzzy additive goal programming technique. Numerical example is given to illustrate the inventory model. Sensitivity analysis and the graphical representations have been shown to illustrate the reality of the inventory model.

Terahertz Meets Untrusted UAV-Relaying: Minimum Secrecy Energy Efficiency Maximization via Trajectory and Communication Co-Design

Unmanned aerial vehicles (UAVs) and Terahertz (THz) technology are envisioned to play paramount roles in next-generation wireless communications. In this paper, we present a novel secure UAV-assisted mobile relaying system operating at THz bands for data acquisition from multiple ground user equipments (UEs) towards a destination. We assume that the UAV-mounted relay may act, besides providing relaying services, as a potential eavesdropper called the untrusted UAV-relay (UUR). To safeguard end-to-end communications, we present a secure two-phase transmission strategy with cooperative jamming. Then, we devise an optimization framework in terms of a new measure $-$ secrecy energy efficiency (SEE), defined as the ratio of achievable average secrecy rate to average system power consumption, which enables us to obtain the best possible security level while taking UUR's inherent flight power limitation into account. For the sake of quality of service fairness amongst all the UEs, we aim to maximize the minimum SEE (MSEE) performance via the joint design of key system parameters, including UUR's trajectory and velocity, communication scheduling, and network power allocation. Since the formulated problem is a mixed-integer nonconvex optimization and computationally intractable, we decouple it into four subproblems and propose alternative algorithms to solve it efficiently via greedy/sequential block successive convex approximation and non-linear fractional programming techniques. Numerical results demonstrate significant MSEE performance improvement of our designs compared to other known benchmarks.