Mathematics

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17 pages, 3549 KiB

Open AccessArticle

Optimal Design of Water Distribution Systems Considering Topological Characteristics and Residual Chlorine Concentration

by Mun Jin Ko and Young Hwan Choi

Mathematics 2022, 10(24), 4721; https://doi.org/10.3390/math10244721 - 12 Dec 2022

Cited by 2 | Viewed by 1630

Abstract

Water distribution systems (WDSs) are designed for supplying safe water under abnormal conditions. Therefore, the optimal design of WDSs should present a plan that satisfies the hydraulic constraint, pressure at the node, and flow rate of the pipe. The water quality constraint, that [...] Read more.

Water distribution systems (WDSs) are designed for supplying safe water under abnormal conditions. Therefore, the optimal design of WDSs should present a plan that satisfies the hydraulic constraint, pressure at the node, and flow rate of the pipe. The water quality constraint, that is, the residual chlorine standard, should be also satisfied. However, there is a problem of insufficient pressure or absence of water for the rapid increase in demand and abnormal situations caused by the destruction of pipes resulting from growing urbanization. This problem differs in node pressure and residence time, depending on the type of WDSs (i.e., loop, hybrid, and branch). Therefore, in this study, the optimal design of WDSs was determined by considering the form of the WDS and the residual chlorine concentration. To construct the layout of WDSs, the type was constructed and classified using the branch index, classification index, and hydraulic water-quality characteristics, which were analyzed accordingly. In addition, the objectives of the WDSs in terms of hydraulic (i.e., nodal pressure) and water-quality (i.e., reference values of residual chlorine concentrations) constraints were established to derive optimal designs that simultaneously stabilize and satisfy water. To stably supply water to the customer even in abnormal situations, an optimal multipurpose design was carried out by setting the sum of the surplus head and design cost as an objective function. These analyses can improve the water quality by simultaneously considering the residual chlorine concentration. They improved the hydraulic characteristics by considering only pressure in the existing design stage. In addition, by deriving an optimal design plan in terms of hydraulic quality according to topological features, we can derive an optimal design that assists the designer in decision making while improving the economic aspect and usability for the consumer. Full article

(This article belongs to the Special Issue Mathematical Modeling and Optimization)

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21 pages, 4346 KiB

Open AccessArticle

Model Predictive Paradigm with Low Computational Burden Based on Dandelion Optimizer for Autonomous Vehicle Considering Vision System Uncertainty

by Shimaa Bergies, Shun-Feng Su and Mahmoud Elsisi

Mathematics 2022, 10(23), 4539; https://doi.org/10.3390/math10234539 - 1 Dec 2022

Cited by 16 | Viewed by 2086

Abstract

The uncertainty due to road fluctuations and vision system dynamics represents a big challenge to adjusting the steering angle of autonomous vehicles (AVs). Furthermore, AVs require fast action to follow the target lane to overcome lateral deviation with minor errors. In this regard, [...] Read more.

The uncertainty due to road fluctuations and vision system dynamics represents a big challenge to adjusting the steering angle of autonomous vehicles (AVs). Furthermore, AVs require fast action to follow the target lane to overcome lateral deviation with minor errors. In this regard, this paper introduces a fast model predictive controller formulated based on the discrete-time Laguerre function (DTLF) to overcome the high computational burden of the traditional MPC. To improve the hybrid DTLF-MPC performance, a modern and effective dandelion optimizer (DO) strategy is used in this work, which resulted in obtaining the optimal DTLF-MPC parameters and achieving satisfactory results. Furthermore, the proposed hybrid DTLF-MPC is designed based on different algorithms in the literature to evaluate the performance of the DO. Several scenarios are discussed in this paper to confirm the effectiveness and efficiency of the proposed control strategy system against the vision system uncertainty and road fluctuations. The results emphasize that the proposed DTLF-MPC based on the DO can achieve the best damping performance for the lateral deviations with less overshoot; around 0.4533, and a settling time of around 0.01979 s compared with other algorithms. Full article

(This article belongs to the Special Issue Mathematical Modeling and Optimization)

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34 pages, 686 KiB

Open AccessArticle

A Non-Archimedean Interior Point Method and Its Application to the Lexicographic Multi-Objective Quadratic Programming

by Lorenzo Fiaschi and Marco Cococcioni

Mathematics 2022, 10(23), 4536; https://doi.org/10.3390/math10234536 - 30 Nov 2022

Cited by 4 | Viewed by 1574

Abstract

This work presents a generalized implementation of the infeasible primal-dual interior point method (IPM) achieved by the use of non-Archimedean values, i.e., infinite and infinitesimal numbers. The extended version, called here the non-Archimedean IPM (NA-IPM), is proved to converge in polynomial time to [...] Read more.

This work presents a generalized implementation of the infeasible primal-dual interior point method (IPM) achieved by the use of non-Archimedean values, i.e., infinite and infinitesimal numbers. The extended version, called here the non-Archimedean IPM (NA-IPM), is proved to converge in polynomial time to a global optimum and to be able to manage infeasibility and unboundedness transparently, i.e., without considering them as corner cases: by means of a mild embedding (addition of two variables and one constraint), the NA-IPM implicitly and transparently manages their possible presence. Moreover, the new algorithm is able to solve a wider variety of linear and quadratic optimization problems than its standard counterpart. Among them, the lexicographic multi-objective one deserves particular attention, since the NA-IPM overcomes the issues that standard techniques (such as scalarization or preemptive approach) have. To support the theoretical properties of the NA-IPM, the manuscript also shows four linear and quadratic non-Archimedean programming test cases where the effectiveness of the algorithm is verified. This also stresses that the NA-IPM is not just a mere symbolic or theoretical algorithm but actually a concrete numerical tool, paving the way for its use in real-world problems in the near future. Full article

(This article belongs to the Special Issue Mathematical Modeling and Optimization)

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11 pages, 3466 KiB

Open AccessArticle

Research on an Accuracy Optimization Algorithm of Kriging Model Based on a Multipoint Filling Criterion

by Shande Li, Shuai Yuan, Shaowei Liu, Jian Wen and Qibai Huang

Mathematics 2022, 10(9), 1548; https://doi.org/10.3390/math10091548 - 5 May 2022

Cited by 1 | Viewed by 1838

Abstract

The optimization method based on the surrogate model has been widely used in the simulation and calculation process of complex engineering models. However, in this process, the low accuracy and computational efficiency of the surrogate model has always been an urgent problem that [...] Read more.

The optimization method based on the surrogate model has been widely used in the simulation and calculation process of complex engineering models. However, in this process, the low accuracy and computational efficiency of the surrogate model has always been an urgent problem that needs to be solved. Aimed at this problem, combined with the two characteristics of global search and local detection, a filling criterion with multiple points is firstly proposed named maximum of expected improvement & minimizing the predicted objective function & maximum of root mean squared error (EI&MP&RMSE) in this paper. Furthermore, the optimization procedure of the surrogate model based on EI&MP&RMSE is concluded. Meanwhile, the classical one-dimensional and two-dimensional functions are applied to verify the accuracy of the proposed method. The difference in the accuracy and mean square error of the surrogate model under different infill points criteria are analyzed. As expected, it shows that this method can effectively improve the accuracy of the surrogate model and reduce the number of iterations. It has extensive practicability and serviceability for the optimization of complex engineering structures. Full article

(This article belongs to the Special Issue Mathematical Modeling and Optimization)

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23 pages, 700 KiB

Open AccessArticle

An Evolutionary Justification of the Emergence of Leadership Using Mathematical Models

by Daniel Solow, Joseph Szmerekovsky and Sukumarakurup Krishnakumar

Mathematics 2021, 9(18), 2271; https://doi.org/10.3390/math9182271 - 15 Sep 2021

Viewed by 1957

Abstract

The value and importance of leadership is evident by its prevalence throughout human societies and organizations. Based on an evolutionary argument, models are presented here that provide a mathematical justification as to how and why leadership arose in the first place and then [...] Read more.

The value and importance of leadership is evident by its prevalence throughout human societies and organizations. Based on an evolutionary argument, models are presented here that provide a mathematical justification as to how and why leadership arose in the first place and then persisted. In this setting, by a leader is meant a person whose overall actions are ultimately responsible for the well-being and survival of the group. The proposed models contain parameters whose values reflect group size, harshness of the environment, diversity of actions taken by individuals, and the amount of group cohesion. Mathematical analysis and computer simulations are used to identify conditions on these parameters under which leadership results in an increased survival probability for the community. Full article

(This article belongs to the Special Issue Mathematical Modeling and Optimization)

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19 pages, 314 KiB

Open AccessArticle

A Hybrid Genetic Algorithm for the Simple Assembly Line Balancing Problem with a Fixed Number of Workstations

by Eduardo Álvarez-Miranda, Jordi Pereira, Harold Torrez-Meruvia and Mariona Vilà

Mathematics 2021, 9(17), 2157; https://doi.org/10.3390/math9172157 - 4 Sep 2021

Cited by 10 | Viewed by 3245

Abstract

The assembly line balancing problem is a classical optimisation problem whose objective is to assign each production task to one of the stations on the assembly line so that the total efficiency of the line is maximized. This study proposes a novel hybrid [...] Read more.

The assembly line balancing problem is a classical optimisation problem whose objective is to assign each production task to one of the stations on the assembly line so that the total efficiency of the line is maximized. This study proposes a novel hybrid method to solve the simple version of the problem in which the number of stations is fixed, a problem known as SALBP-2. The hybrid differs from previous approaches by encoding individuals of a genetic algorithm as instances of a modified problem that contains only a subset of the solutions to the original formulation. These individuals are decoded to feasible solutions of the original problem during fitness evaluation in which the resolution of the modified problem is conducted using a dynamic programming based approach that uses new bounds to reduce its state space. Computational experiments show the efficiency of the method as it is able to obtain several new best-known solutions for some of the benchmark instances used in the literature for comparison purposes. Full article

(This article belongs to the Special Issue Mathematical Modeling and Optimization)

12 pages, 275 KiB

Open AccessArticle

Time-Optimal Control for Semilinear Stochastic Functional Differential Equations with Delays

by Yong Han Kang and Jin-Mun Jeong

Mathematics 2021, 9(16), 1956; https://doi.org/10.3390/math9161956 - 16 Aug 2021

Viewed by 1642

Abstract

The purpose of this paper is to find the time-optimal control to a target set for semilinear stochastic functional differential equations involving time delays or memories under general conditions on a target set and nonlinear terms even though the equations contain unbounded principal [...] Read more.

The purpose of this paper is to find the time-optimal control to a target set for semilinear stochastic functional differential equations involving time delays or memories under general conditions on a target set and nonlinear terms even though the equations contain unbounded principal operators. Our research approach is to construct a fundamental solution for corresponding linear systems and establish variations of a constant formula of solutions for given stochastic equations. The existence result of time-optimal controls for one point target set governed by the given semilinear stochastic equation is also established. Full article

(This article belongs to the Special Issue Mathematical Modeling and Optimization)

24 pages, 3018 KiB

Open AccessArticle

Optimal Software Feature-Limited Freemium Model Design: A New Consumer Learning Theoretical Framework

by Kang Li, Jingwei Zhang and Lunchuan Zhang

Mathematics 2021, 9(9), 944; https://doi.org/10.3390/math9090944 - 23 Apr 2021

Cited by 5 | Viewed by 2702

Abstract

The software industry is increasingly adopting a feature-limited freemium business model that combines “free” and “premium” contents in one product, to sell its products. How to determine the optimal product quality differences between the free and premium versions of software is a central [...] Read more.

The software industry is increasingly adopting a feature-limited freemium business model that combines “free” and “premium” contents in one product, to sell its products. How to determine the optimal product quality differences between the free and premium versions of software is a central business problem facing many software vendors. In this paper, we study the optimal feature-limited freemium software strategy design, as well as the associated pricing strategies based on consumer learning and network externality effects. We propose a new consumer learning framework induced by cross-module synergies that contains both direct and indirect learning processes. By employing a two-stage mathematical theoretical model and a numerical analysis method, we gained some insights regarding the feature-limited free trial strategy design and associated pricing strategies while considering the associated trade-off between the benefits and costs of the free trial strategy. In our modeling and numerical results, consumers’ prior beliefs about the quality of premium content before the free trial, network effect intensity, and indirect learning intensity were found to be three conditions that need to be studied to examine software vendors’ management decisions. For the software industry, the quality difference between free and premium functionality or the service and price strategy for a feature-limited free trial model can be designed while considering these factors, which will provide some useful guidelines for the industry. Full article

(This article belongs to the Special Issue Mathematical Modeling and Optimization)

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