Nonlinear Mathematical Programming Problem Research Articles

Simplified evaluation of the ultimate strength of RC circular members subjected to axial force, bending moment and shear force

The paper proposes a simple analytical tool for the evaluation of the shear strength of reinforced concrete circular, solid or hollow, slender columns endowed with shear reinforcement and subjected to axial force, bending moment and shear force. The proposed method considers the sole beam action by means of simplified stress fields and simultaneously satisfies the equilibrium under the above combined internal forces. The paper identifies basic points in the N-M-V ultimate interaction curve and describes the procedures for the evaluation of the internal forces of these points as well as the procedures for the evaluation of the internal forces of the points in between. The method is first applied to a set of circular solid columns with different geometric and mechanical properties and a comparison is drawn between the shear resistance resulting from the simplified method and that from a more complex non-linear mathematical programming problem proposed in the past by one of the authors. Then, to demonstrate the value of the method and define the field of its accurate application, the proposed method is applied to circular (solid or hollow) columns tested in laboratory by other researchers. Finally, the results of the proposed method are compared with those of some code formulations.

Total Problem of Constructing Linear Regression Using Matrix Correction Methods with Minimax Criterion

A linear problem of regression analysis is considered under the assumption of the presence of noise in the output and input variables. This approximation problem may be interpreted as an improper interpolation problem, for which it is required to correct optimally the positions of the original points in the data space so that they all lie on the same hyperplane. The use of the quadratic approximation criterion for such a problem led to the appearance of the total least squares method. In this paper, we use the minimax criterion to estimate the measure of correction of the initial data. It leads to a nonlinear mathematical programming problem. It is shown that this problem can be reduced to solving a finite number of linear programming problems. However, this number depends exponentially on the number of parameters. Some methods for overcoming this complexity of the problem are proposed.

OPTIMAL DESIGN OF OIL TANKS ACCORDING TO ECONOMIC CRITERIA WITH CONSIDERATION OF RELIABILITY

Problem statement. According to state regulations, the main method of ensuring the reliability of steel vertical above-ground tanks for long-term storage of oil and petroleum products is the limit states method. This method ensures the reliability of tanks under the conditions and modes of operation envisaged by the relevant standards. However, issues relating to the reliability of tanks whose service life exceeds the normative one, as well as the economic efficiency of project decisions, require further scientific study. In this article, in accordance with the normative documents regulating the oil tanks design, the general method of oil storage tanks optimal design according to the criterion of minimum total expected costs while ensuring the required reliability level during a specified operation period is considered. Within the framework of this methodology, a model of optimal tanks design is formulated and an algorithm for its numerical realization is proposed. The results of numerical experiments using this model are given. The dependence of the rational values of the structural elements thickness for the tank with a volume of 5 000 cubic meters on the required level of reliability is investigated. The purpose of the article is to develop a method of making economically justified design decisions that ensure the required reliability level of oil tanks during a specified operation period, taking into account the current design standards. Conclusions. This article proposes an optimal oil tank design model formulated in terms of a non-linear mathematical programming problem with integer variables. With this model, rational design decisions can be made based on the criterion of the minimum total expected costs while ensuring the required reliability level during a specified operation period. The components of this model are the calculation models for assessing the reliability of the tanks, the parameters of which are determined based on the available results of statistical data processing from field surveys of similar objects. A special “greedy” algorithm was developed for the numerical realization of the proposed optimal design model. The obtained results show, in particular, that ensuring the oil tanks reliability is mainly related to ensuring the strength of the two lower wall belts and the tightness of the bottom. At the same time, the strength of the 1st belt has a key role. All this is in good agreement with the available data of field surveys and statistics of oil tanks capital repairs, which confirms the reliability of the obtained results.

Multidisciplinary Analysis and Optimization Method for Conceptually Designing of Electric Flying-Wing Unmanned Aerial Vehicles

Current unmanned aerial vehicles have been designed by applying the traditional approach to aircraft conceptual design which has drawbacks in terms of the individual analysis of each discipline involved in the conception of new aircraft, the reliance on the designer’s experience and intuition, and the inability of evaluating all possible design solutions. Multidisciplinary analysis and optimization focus on solving these problems, by synthesizing all the disciplines involved and accounting for their mutual interaction. This study presents a multidisciplinary analysis and optimization method for conceptually designing electrical flying-wing micro-unmanned aerial vehicles. The conceptual design task was formulated as a non-linear mathematical programming problem. The method considers the trimming of the UAV during each mission profile phase, consisting of the climb, cruise, and descent. We used two algorithms, one for design space exploration and another for optimization. Typical examples of solving conceptual design problems were considered in the work: the modernization of an existing UAV; the effect of the change of the payload and endurance change on the takeoff weight; and the influence of different static margins on aerodynamic characteristics. The advantages of using this design method are the remotion of additional internal cycles to solve the sizing equation at each optimization step, and the possibility of not only obtaining a unique optimal solution but also a vector of optimal solutions.

Optimal sensor placement method for wastewater treatment plants based on discrete multi-objective state transition algorithm

Parameters monitoring is essential to maintain the stability and efficiency of the wastewater treatment process, which has spurred ubiquitous installation of sensors in wastewater treatment plants (WWTPs). As the rich process data of WWTPs is not effectively transformed into actionable knowledge for system optimization due to improper sensor installation, the sensor placement scheme needs to be optimized. In this paper, a weighted sensor placement optimization model based on sensor cost, information richness and reliability is established to transform the sensor optimization problem to a nonlinear mathematical programming problem. Then a discrete multi-objective state transition algorithm is proposed to find the Pareto optimal solutions. Finally, an evaluation strategy is designed to select the most suitable solution for industrial application. The results of simulation experiments on three different WWTPs demonstrate the validity and superiority of the proposed method, increasing the degree of variable observability and measurement redundancy while keeping the sensor cost at a low level.

MATLAB Implementation of Direct and Indirect Shooting Methods to Solve an Optimal Control Problem With State Constraints

The paper presents a general procedure to solve numerically optimal control problems with state constraints. Itis used in the case, when the simple time discretization of the state equations and expressing the optimal control problem as a nonlinear mathematical programming problem is too coarse. It is based on using in turn two multiple shooting BVP approaches: direct and indirect.The paper is supplementary to the earlier author’s paper on direct and indirect shooting methods, presenting the theory underlying both approaches. The same example is considered here and brought to an end, that is two full listings of two Matlab codes are shown.

Profit maximization in an inventory system with time-varying demand, partial backordering and discrete inventory cycle

In this paper, an inventory problem where the inventory cycle must be an integer multiple of a known basic period is considered. Furthermore, the demand rate in each basic period is a power time-dependent function. Shortages are allowed but, taking necessities or interests of the customers into account, only a fixed proportion of the demand during the stock-out period is satisfied with the arrival of the next replenishment. The costs related to the management of the inventory system are the ordering cost, the purchasing cost, the holding cost, the backordering cost and the lost sale cost. The problem is to determine the best inventory policy that maximizes the profit per unit time, which is the difference between the income obtained from the sales of the product and the sum of the previous costs. The modeling of the inventory problem leads to an integer nonlinear mathematical programming problem. To solve this problem, a new and efficient algorithm to calculate the optimal inventory cycle and the economic order quantity is proposed. Numerical examples are presented to illustrate how the algorithm works to determine the best inventory policies. A sensitivity analysis of the optimal policy with respect to some parameters of the inventory system is developed. Finally, conclusions and suggestions for future research lines are given.

ავტომატური მართვის სისტემის წრფივი მაკორექტირებელი მოწყობილობის პარამეტრების ოპტიმალური მნიშვნელობების განსაზღვრა ინტეგრალური კვადრატული კრიტერიუმის მიხედვით

When designing an automatic control system (ACS), it is important to determine the optimal values of the parameters of the correcting device according to any of the criteria. Recently, more and more publications have appeared on the use of the method of nonlinear mathematical programming for solving problems of synthesis of an automated control system. The article discusses a method for determining the optimal parameters of linear correcting devices according to the quadratic integral criterion. The novelty is the presentation of the above mentioned problem in the form of a nonlinear mathematical programming problem. To find a multiparametric, multiextremal, i.e. complex, objective function, a random search method is used. Also, a class of automatic control systems having one extreme objective function is highlighted. For this case, the theorems are proved, and the formula for determining extreme points is given in an analytical form. Numerical examples are also considered. Programs have been developed for implementing the corresponding algorithms on a computer in VBA language. The graphs of the corresponding transients are given.

Algorithm for maintaining the energy balance of the power supply system of the earth remote sensing spacecraft under critical operating conditions

In this paper, we consider the problem of maintaining the energy balance of the power supply system of the Earth remote sensing spacecraft in critical operating conditions with the most intense cyclogram of the spacecraft operation, the maximum state of degradation of energy sources and storage devices, as well as with the minimum time to replenish useful energy.The problem of maintaining balance is formulated as a problem of nonlinear mathematical programming with functional constraints. Goal of the work is improving the efficiency and survivability of the remote sensing spacecraft and maintaining the energy balance of the power system in critical operating conditions by transferring the storage devices of the accumulating subsystem of the spacecraft to a given level of charge in the minimum charge time. An algorithm for controlling the accelerated charge of the rechargeable battery with the analysis of the Nickel-hydrogen battery temperature and its time derivative is developed.in some cases, the rechargeable battery replenishment time was reduced by 12−20%. The proposed algorithm for maintaining the energy balance can be used to select the stages of the accumulating subsystem charge current and transition from the current stage to the next one in a minimum time, providing flexibility in managing the modes of electricity storage with the adaptation of the charging current profiles to changing critical operating conditions.

OPTIMAL DESIGN OF RESTRAINED SHEET PILE WALLS

Restrained shoring wall represents a commonly used economic solution for vertical deep excavation, when open cuts with side slopes are not allowed. It is mainly used to avoid failure that may be accompanied by considerable settlements, tilting or by bearing capacity failure of nearby foundations. The cost of these systems mainly depended on soil type and excavation depth. In this research, strutted shoring systems are analyzed and designed for sandy soil conditions and excavation depth 15m. The system is optimized using Genetic Algorithm. Finite Element Method is used for the analysis. The designed problem is formulated as a non-linear mathematical programming problem using FORTRAN 95. The developed model is used for parametric study to investigate the influence of different design parameters on the system cost. Genetic Algorithm, is used to perform the optimization study based on the minimum cost. The optimization process aims to minimize the system cost considering both deformation and stress constraints for the ground soil and construction material.

USING PSEUDOSPECTRAL METHOD TO SEARCH FOR PROPELLANT-OPTIMAL TRAJECTORIES OF THE INTERNATIONAL SPACE STATION ATTITUDE CONTROL MANEUVERS

The paper discusses the problem of searching for propellant-optimal trajectories of the International Space Station attitude control maneuvers involving spatial turns through large angles using attitude control jet thrusters. Development of such algorithms for controlling the ISS angular motion is currently a crucial task for Russian developers of the onboard software. In order to generate the optimal trajectory for the attitude control maneuver, the paper proposes to use the Lobatto pseudospectral method. This method allows stating the optimal control problem as a nonlinear mathematical programming problem which can be solved using the method of sequential quadratic programming. Simulation results demonstrated significant savings of attitude control thrusters propellant and life during station attitude control maneuvers in comparison with the algorithms that are currently used in the motion control system of the ISS Russian Segment. Key words: International Space Station, optimal control problem, angular motion control, pseudospectral method, nonlinear programming

Geometrical synthesis of four-bar gear train with related gears

The method of optimal geometric synthesis of four-bar cylindrical gear with related gears with one external and one internal gearing is proposed. The method is based on a new calculation system that increases the number of synthesis output parameters to six, which makes it possible to considerably expand the domain of existence of the synthesis problem solution with the simultaneous synthesis of two gearing. The software implementing the proposed technique is described. To solve the problem of nonlinear mathematical programming the library of Optimization Toolbox functions included in Matlab is used. Examples of calculations are given.

Data-driven approach to optimal control of ACC systems and layout design in large rooms with thermal comfort consideration by using PSO

In recent years, there are increasing concerns in energy savings for buildings since they are responsible for a large proportion of energy use. A public room in buildings could hold a number of persons who may prefer dissimilar thermal environment. Furthermore, different areas in such rooms may have different temperatures. Also, facility layout in such a room has effect on the distribution of the people in the room. Thus, it may affect its thermal environment and energy consumption as well. It is meaningful and challenging to effectively operate an air-conditioning control (ACC) system by taking the above mentioned factors into account such that the thermal environment is improved and energy is saved. With the lack of research reports on this issue, this work aims at optimally and dynamically controlling the set-point temperature of an ACC system and designing the facility layout so as to maximize the total thermal satisfaction rate (TSR) as well as energy savings. To do so, a non-linear mathematical programming model is proposed to optimize TSR by determining the set-point of an ACC system and the room layout. Then, a particle swarm optimization (PSO) algorithm is constructed to find an optimal or near optimal solution since it is hard to solve a non-linear mathematical programming problem in a reasonable time. Besides, for further energy saving, two more mathematical programming models are proposed to find a set-point of an ACC system under a given outside temperature and room layout determined by the PSO algorithm. Finally, by using a large library room at Macau University of Science and Technology (MUST) as a case, investigations with a large number of experiments are conducted to collect necessary data. Based on the data, regression analysis is done to predict its indoor temperatures in different areas and TSR at a given temperature. Numerical results show that, by the proposed method, it can improve the thermal satisfaction rate by about 27% and save the daily power cost by about 24.3% in comparison with the currently used manual control method. • Study is conducted on energy efficiency in large public rooms. • Methods are derived to predict temperature distribution and occupants' satisfaction rate. • PSO algorithm is developed for optimal input temperature control and facility layout. • The proposed approach can significantly reduce energy consumption.

The Method of Shut-off Valves Optimal Placing on Trunk Oil Pipeline

The shut-off valves are an important element to the safe operation of a trunk oil pipeline. Actual problem of shut-off valves optimal placing on trunk oil pipeline is defined with respect to minimal oil spill during accidents and planning repair operations. The method of optimal placing is presented as a problem of nonlinear mathematical programming with limits and these limits are minimal distance between line valves. The solution of mathematical programming problem is given with the help of rigorous (alternance) parametric optimization method which is founded on special qualities ultimate solution. The approach of shut-off valves placing on three and four linear block valves taking into account typical cases. The results of optimal linear block valves placing calculation are shown at a site of trunk oil pipeline 77 km long. As a result of the application of the proposed technique, a reduction of the maximum volume of oil spill on the linear part of the oil trunk pipeline by 25 percent or a reduction in the number of shut-off valves from four to three for the initial value of the maximum oil spill can be achieve.

An optimization model for traceable closed-loop supply chain networks

• A closed-loop supply chain (CLSC) network comprises forward and reverse logistics. • A stochastic multi-product traceable CLSC using a radio frequency identification system is modelled. • Customer acquisition behavior when choosing internet over conventional sales is considered. • A greedy randomized adaptive search procedure (GRASP) is used to solve the problem. • A real-life case study of network marketing shows the applicability of the proposed model. In this paper, a new non-linear mixed-integer mathematical programming problem is proposed to model a stochastic multi-product closed-loop supply chain (CLSC). The radio frequency identification (RFID) system is implemented in the supply chain to decrease product losses and the overall lead time of transportation while computing the profit derived from internet and conventional sales. The resulting traceable CLSC improves upon the existing literature by allowing us to: (1) boost the incorporation of traceability assumptions in mathematical programming problems so as to enhance the efficiency and visibility of a supply chain, (2) analyze the strategic effects that different internet sale formats have on customers’ evaluations and acquisition choices, and (3) account for the environmental and socio-economical dimension by explicitly formalizing employment-based incomes as part of the profit function. Two meta-heuristic algorithms are introduced to solve the proposed optimization problem, namely, the greedy randomized adaptive search procedure (GRASP) and particle swarm optimization (PSO). Twelve test problems of different sizes are generated and solved using these algorithms. The computational results show that GRASP outperforms PSO in terms of both profit and CPU time values. Finally, a case study in the network marketing industry is presented and managerial implications outlined to show the validity of the proposed model and shed more light on its practical implications.

Lagrangian-Relaxation-Based Self-Repairing Mechanism for Wi-Fi Networks

Wi-Fi was developed to support software-defined networks that are crucial for achieving high quality of service for next generation wireless networks. In this software-defined networking technology, network resources are utilized to reduce the effects of several influential factors such as the limited number of nonoverlapping channels, co-channel interference, rapidly changing distribution of customers, and access point (AP) failure. These factors cause irregular network traffic and service unavailability in some users. In this paper, we address the problems causing the AP failure and interference. The studied problems were modeled as linear and nonlinear mathematical programming problems. The Lagrangian relaxation approach, which is a type of divide-and-conquer method, was used to solve the load distribution problems near optimally. A self-repairing heuristic multiple-level load-balancing traffic adjustment was proposed to manage the problems pertaining to the AP failure. The delay difference metric, which is defined as the difference between tolerable delay and transmission delay, was used to evaluate the difference between the upper bound and lower bound of the delay. Moreover, the metric was used to estimate the improvement ratio between the existing methods and the proposed method. Modifiable transmission power ranges (which involve the cell breathing method) and association managements are adjusted to minimize the narrow gap and improve the self-repairing performance. Thus, resources are appropriately allocated to users, and adequate QoS is attained.

Modified Courant-Beltrami penalty function and a duality gap for invex optimization problem

In this paper, we modified a Courant-Beltrami penalty function method for constrained optimization problem to study a duality for convex nonlinear mathematical programming problems. Karush-Kuhn-Tucker (KKT) optimality conditions for the penalized problem has been used to derived KKT multiplier based on the imposed additional hypotheses on the constraint function g. A zero-duality gap between an optimization problem constituted by invex functions with respect to the same function η and their Lagrangian dual problems has also been established. The examples have been provided to illustrate and proved the result for the broader class of convex functions, termed invex functions.

A Multistage Feedback Control Strategy for Producing 1,3‐Propanediol in Microbial Continuous Fermentation

In this paper, we consider a multistage feedback control strategy for producing 1,3‐propanediol in microbial continuous fermentation. Both the dilution rate and the concentration of glycerol in the input feed are used as control variables, and these variables are further assumed to be in the form of a linear combination of biomass and glycerol concentrations. Unlike the general form of linear feedback control, the coefficients of linear combination are continuous functions with respect to time. Inspired by the control parameterization method, we use the piecewise‐constant functions to approximate the coefficient functions; then we get the multistage feedback control law by solving nonlinear mathematical programming problems. Numerical results indicate the flexibility and effectiveness of our strategy.

Reliability-based design optimization with frequency constraints using a new safest point approach

ABSTRACTIn the reliability-based design optimization (RBDO) model, the mean values of uncertain variables are usually applied as design variables, and the cost is optimized subject to prescribed probabilistic constraints as defined by a nonlinear mathematical programming problem. Therefore, an RBDO solution that reduces the structural weight in non-critical regions provides not only an improved design, but also a higher level of confidence in the design. Solving such nested optimization problems is extremely expensive for large-scale multidisciplinary systems that are likewise computationally intensive. This article focuses on the study of a particular problem representing the failure mode of structural vibration analysis. A new method is proposed, called safest point, that can efficiently give the reliability-based optimum solution under frequency constraints, and then several probability distributions are developed, which are mathematically nonlinear functions, for the proposed method. Finally, the efficiency of the extended approach is demonstrated for probability distributions such as log-normal and uniform distributions, and its applicability to the design of structures undergoing fluid–structure interaction phenomena, especially the design process of aeroelastic structures, is also demonstrated.

Stress-strain state analysis and optimization of rod system under periodic pulse load

The paper considers the problem of analysis and optimization of rod systems subjected to combined static and periodic pulse load. As a result of the study the analysis method was developed based on traditional approach to solving homogeneous matrix equations of state and a special algorithm for developing a particular solution. The influence of pulse parameters variations on stress-strain state of a rod system was analyzed. Algorithms for rod systems optimization were developed basing on strength recalculation and statement and solution of optimization problem as a problem of nonlinear mathematical programming. Recommendations are developed for efficient organization of process for optimization of rod systems under static and periodic pulse load.