Control and Optimization in Applied Mathematics
https://mathco.journals.pnu.ac.ir/
Control and Optimization in Applied Mathematicsendaily1Wed, 01 May 2024 00:00:00 +0330Wed, 01 May 2024 00:00:00 +0330A Fuzzy Sliding Mode Control for Nonlinear Leader-Follower Multi-Agent Systems
https://mathco.journals.pnu.ac.ir/article_9920.html
In this paper&lrm;, &lrm;we present a new approach for achieving leader-follower consensus in a network of nonlinear dynamic agents with an undirected graph topology&lrm;, &lrm;using a fuzzy sliding mode controller (FSMC) for Multi-Agent Systems (MASs)&lrm;. &lrm;Our proposed sliding mode controller is based on a separating hyperplane that effectively addresses the consensus problem in MASs&lrm;. &lrm;Additionally&lrm;, &lrm;we design a fuzzy controller to eliminate the chattering phenomenon&lrm;. &lrm;According to the communication graph topology and the Lyapunov stability condition&lrm;, &lrm;the proposed FSMC satisfies the consensus condition&lrm;. &lrm;One significant advantage of our approach is that the system states converge to the sliding surface quickly and remain on the surface&lrm;, &lrm;thereby ensuring better tracking performance&lrm;. &lrm;We validate the effectiveness of our proposed approach through simulation results&lrm;.Dynamical Behaviour of Fractional Order SEIR Mathematical Model for Infectious Disease Transmission
https://mathco.journals.pnu.ac.ir/article_10189.html
This paper presents an extension of the SEIR mathematical model for infectious disease&lrm; &lrm;transmission to a fractional-order model&lrm;. &lrm;The model is formulated using the Caputo derivative of order &alpha; &isin; (0, 1]&lrm;. &lrm;We study the stability of equilibrium points&lrm;, &lrm;including the disease-free equilibrium $(E_{f})$&lrm;, &lrm;and the&lrm; &lrm;infected steady-state equilibrium $(E_{e})$ using the&lrm; &lrm;stability theorem of Fractional Differential Equations&lrm;. &lrm;The model is also analyzed under certain conditions&lrm;, &lrm;and&lrm; &lrm;it is shown that the disease-free equilibrium is locally asymptotically&lrm; &lrm;stable&lrm;. &lrm;Additionally&lrm;, &lrm;the extended Barbalat&rsquo;s lemma is applied to the&lrm; &lrm;fractional-order system&lrm;, &lrm;and a suitable Lyapunov functional is constructed&lrm; &lrm;to demonstrate the global asymptotic stability of the infected&lrm; &lrm;steady-state equilibrium&lrm;. &lrm;To validate the theoretical results&lrm;, &lrm;a numerical simulation of the problem is conducted&lrm;.&nbsp;Efficient Solution of Nonlinear Unconstraint Optimization Problems using Quasi-Newton's Method: A Revised Approach
https://mathco.journals.pnu.ac.ir/article_10288.html
While many real-world optimization problems typically involve multiple constraints, unconstrained problems hold practical and fundamental significance. They can arise directly in specific applications or as transformed versions of constrained optimization problems.&lrm;&nbsp;&lrm;Newton's method&lrm;, &lrm;a notable numerical technique within the category of line search algorithms, is widely used for function optimization&lrm;. The search direction and step length play crucial roles in this algorithm. &lrm;This paper introduces an algorithm aimed at enhancing the step length within the Broyden quasi-Newton process&lrm;. &lrm;Additionally&lrm;, &lrm;numerical examples are provided to compare the effectiveness of this new method with another approach&lrm;.Hesitant Fuzzy Equation
https://mathco.journals.pnu.ac.ir/article_10360.html
This paper presents the introduction of two novel equation types: the partial hesitant fuzzy equation and the half hesitant fuzzy equation&lrm;. Additionally, &lrm; an efficient method is proposed to solve these equations by defining four solution categories: Controllable&lrm;, &lrm;Tolerable Solution Set (TSS)&lrm;, Controllable &lrm;Solution Set (CSS)&lrm;, &lrm;and Algebraic Solution Set (ASS)&lrm;. &lrm; Furthermore, &lrm; the paper establishes eight theorems that explore different types of solutions and lay out the conditions for the existence and non-existence of hesitant fuzzy solutions&lrm;. &lrm; The practicality of the proposed method is demonstrated through numerical examples.Big Data Analytics and Data Mining Optimization Techniques for Air Traffic Management
https://mathco.journals.pnu.ac.ir/article_10476.html
With the advancements in science and technology&lrm;, &lrm;the industrial and aviation sectors have witnessed a significant increase in data&lrm;. &lrm;A vast amount of data is generated and utilized continuously&lrm;. &lrm;It is imperative to employ data mining techniques to extract and uncover knowledge from this data&lrm;. &lrm;Data mining is a method that enables the extraction of valuable information and hidden relationships from datasets&lrm;. &lrm;However&lrm;, &lrm;the current aviation data presents challenges in effectively extracting knowledge due to its large volume and diverse structures&lrm;. &lrm;Air Traffic Management (ATM) involves handling Big data&lrm;, &lrm;which exceeds the capacity of conventional acquisition&lrm;, &lrm;matching&lrm;, &lrm;management&lrm;, &lrm;and processing within a reasonable timeframe&lrm;. &lrm;Aviation Big data exists in batch forms and streaming formats&lrm;, &lrm;necessitating the utilization of parallel hardware and software&lrm;, &lrm;as well as stream processing&lrm;, &lrm;to extract meaningful insights&lrm;. &lrm;Currently&lrm;, &lrm;the map-reduce method is the prevailing model for processing Big data in the aviation industry&lrm;. &lrm;This paper aims to analyze the evolving trends in aviation Big data processing methods&lrm;, &lrm;followed by a comprehensive investigation and discussion of data analysis techniques&lrm;. &lrm;We implement the map-reduce optimization of the K-Means algorithm in the Hadoop and Spark environments&lrm;. &lrm;The K-Means map-reduce is a crucial and widely applied clustering method&lrm;. &lrm;Finally&lrm;, &lrm;we conduct a case study to analyze and compare aviation Big data related to air traffic management in the USA using the K-Means map-reduce approach in the Hadoop and Spark environments&lrm;. &lrm;The analyzed dataset includes flight records&lrm;. &lrm;The results demonstrate the suitability of this platform for aviation Big data&lrm;, &lrm;considering the characteristics of the aviation dataset&lrm;. &lrm;Furthermore&lrm;, &lrm;this study presents the first application of the designed program for air traffic management&lrm;.Optimizing Supply Chain Design for Sustainability and Reliability: A Comparative Study of Augmented Epsilon and Normalized Normal Constraint Methods
https://mathco.journals.pnu.ac.ir/article_10361.html
Integrating sustainability and reliability represents a synergistic approach that can be explored through the problem of a closed-loop supply chain network design (SCND)&lrm;. &lrm;This study is conducted in three stages&lrm;: &lrm;mathematical modeling&lrm;, &lrm;model solution using exact methods&lrm;, &lrm;and evaluation of the solution methods&lrm;. &lrm;In the first stage&lrm;, &lrm;a mixed-integer linear programming (MILP) model is developed in a multi-objective&lrm;, &lrm;multi-product&lrm;, &lrm;and multi-period framework&lrm;. &lrm;The objectives of the proposed model aim to maximize profitability&lrm;, &lrm;social responsibility&lrm;, &lrm;and reliability&lrm;. &lrm;In the second stage&lrm;, &lrm;two methods&lrm;, &lrm;namely Augmented &lrm;$\varepsilon&lrm;&lrm;$&lrm;-Constraint (AEC) and Normalized Normal Constraint (NNC)&lrm;, &lrm;are implemented in the GAMS software to solve the model and identify the optimal Pareto solutions&lrm;. &lrm;In the third stage&lrm;, &lrm;the Shannon Entropy technique is employed to determine the criteria weights&lrm;, &lrm;and the VIKOR technique is utilized to select the superior solution method&lrm;. &lrm;The overall performance accuracy of the proposed model is measured using four samples from a numerical example with randomly generated data based on the objective function coefficients&lrm;. &lrm;The results indicate the presence of a conflict among the three objective functions&lrm;. &lrm;Consequently&lrm;, &lrm;decision-makers should consider sacrificing some profitability to enhance environmental protection and improve reliability&lrm;. &lrm;In terms of three criteria&lrm;, &lrm;run time&lrm;, &lrm;diversification metric&lrm;, &lrm;and general distance&lrm;, &lrm;the NNC method is given priority over the AEC method&lrm;. &lrm;Even when the criteria are given equal weight&lrm;, &lrm;the superiority of the NNC method remains unchanged&lrm;. &lrm;The application of the proposed model across different industries represents a significant research direction for future research&lrm;.An Efficient Variable Neighborhood Search for Solving Multi-Criteria Project Portfolio Selection
https://mathco.journals.pnu.ac.ir/article_9818.html
Project portfolio selection is a critical challenge for many organizations as they often face budget constraints that limit their ability to support all available projects&lrm;. &lrm;To address this issue&lrm;, &lrm;organizations seek to select a feasible subset of projects that maximizes utility&lrm;. &lrm;While several models for project portfolio selection based on multiple criteria have been proposed&lrm;, &lrm;they are typically NP-hard problems&lrm;. &lrm;In this study&lrm;, &lrm;we propose an efficient Variable Neighborhood Search (VNS) algorithm to solve these problems&lrm;. &lrm;Our algorithm includes a formula for computing the difference value of the objective function&lrm;, &lrm;which enhances its accuracy and ensures that selected projects meet desired criteria&lrm;. &lrm;We demonstrate the effectiveness of our algorithm through rigorous testing and comparison with a genetic algorithm (GA) and CPLEX&lrm;. &lrm;The results of the Wilcoxon non-parametric test confirm that our algorithm outperforms both GA and CPLEX in terms of speed and accuracy&lrm;. &lrm;Moreover&lrm;, &lrm;the variance of the relative error of our algorithm is less than that of GA&lrm;.Solving Fractional Optimal Control-Affine Problems via Fractional-Order Hybrid Jacobi Functions
https://mathco.journals.pnu.ac.ir/article_10414.html
This paper proposes and analyzes an applicable approach for numerically computing the solution of fractional optimal control-affine problems. The fractional derivative in the problem is considered in the sense of Caputo. The approach is based on a fractional-order hybrid of block-pulse functions and Jacobi polynomials. &lrm;First&lrm;, &lrm;the corresponding Riemann-Liouville fractional integral operator of the introduced basis functions is calculated&lrm;. &lrm; Then, an approximation of the fractional derivative of the unknown state function is obtained by considering an approximation in terms of these basis functions&lrm;. &lrm; Next, &lrm;using the dynamical system and applying the fractional integral operator&lrm;, &lrm;an approximation of the unknown control function is obtained based on the given approximations of the state function and its derivatives&lrm;. &lrm; Subsequently&lrm;, &lrm;all the given approximations are substituted into the performance index&lrm;. &lrm;Finally&lrm;, &lrm;the optimality conditions transform the problem into a system of algebraic equations&lrm;. &lrm;An error upper bound of the approximation of a function based on the fractional hybrid functions is provided&lrm;. &lrm;The method is applied to several numerical examples&lrm;, and &lrm;the experimental results confirm the efficiency and capability of the method. &nbsp;Furthermore, they demonstrate a good agreement between the approximate and exact solutions&lrm;. &lrm;Solving Linear Fractional Programming Problems in Uncertain Environments: A Novel Approach with Grey Parameters
https://mathco.journals.pnu.ac.ir/article_10592.html
Fractional programming is a significant nonlinear planning tool within operation research&lrm;. &lrm;It finds applications in diverse domains such as resource allocation&lrm;, &lrm;transportation&lrm;, &lrm;production programming&lrm;, &lrm;performance evaluation&lrm;, &lrm;and finance&lrm;. &lrm;In practical scenarios&lrm;, &lrm;uncertainties often make it challenging to determine precise coefficients for mathematical models&lrm;. &lrm;Consequently&lrm;, &lrm;utilizing indefinite coefficients instead of definite ones is recommended in such cases&lrm;. &lrm;Grey systems theory&lrm;, &lrm;along with probability theory&lrm;, &lrm;randomness&lrm;, &lrm;fuzzy logic&lrm;, &lrm;and rough sets&lrm;, &lrm;is an approach that addresses uncertainty&lrm;. &lrm;In this study&lrm;, &lrm;we address the problem of linear fractional programming with grey coefficients in the objective function&lrm;. &lrm;To tackle this problem&lrm;, &lrm;a novel approach based on the variable change technique proposed by Charnes and Cooper&lrm;, &lrm;along with the convex combination of intervals&lrm;, &lrm;is employed&lrm;. &lrm;The article presents an algorithm that determines the solution to the grey fractional programming problem using grey numbers&lrm;, &lrm;thus capturing the uncertainty inherent in the objective function&lrm;. &lrm;To demonstrate the effectiveness of the proposed method&lrm;, &lrm;an example is solved using the suggested approach&lrm;. &lrm;The result is compared with solutions obtained using the whitening method&lrm;, &lrm;employing Hu and Wong's technique and the Center and Greyness Degree Ranking method&lrm;. &lrm;The comparison confirms the superiority of the proposed method over the whitening method&lrm;, &lrm;thus suggesting adopting the grey system approach in such situations&lrm;.A Hybrid Floyd-Warshall and Graph Coloring Algorithm for Finding the Smallest Number of Colors Needed for a Distance Coloring of Graphs
https://mathco.journals.pnu.ac.ir/article_10626.html
Graph coloring is a crucial area of research in graph theory, with numerous algorithms proposed for various types of graph coloring, particularly graph p-distance coloring&lrm;. In this study, we employ a recently introduced graph coloring algorithm to develop a hybrid algorithm approximating the chromatic number &lrm;p-distance, where $p$ represents a positive integer number. We apply our algorithm to molecular graphs as practical applications of our findings.Mathematical Modeling and Optimal Control of Carbon Dioxide Emissions
https://mathco.journals.pnu.ac.ir/article_10443.html
&lrm;This paper aims to demonstrate the flexibility of mathematical models in analyzing carbon dioxide emissions and account for memory effects. &lrm;The use of real data amplifies the importance of this study&lrm;. &lrm;This research focuses on developing a mathematical model utilizing fractional-order differential equations to represent carbon dioxide emissions stemming from the energy sector. By comparing simulation results with real-world data, it is determined that the fractional model exhibits superior accuracy when contrasted with the classical model&lrm;. &lrm;Additionally&lrm;, &lrm;an optimal control strategy is proposed to minimize the levels of carbon dioxide, CO2, and associated implementation costs&lrm;. &lrm;The fractional optimal control problem is addressed through the utilization of an iterative algorithm&lrm;, &lrm; and the effectiveness of the model is verified by presenting comparative results.Optimality Conditions for Properly Efficient Solutions of Nonsmooth Multiobjective GSIP
https://mathco.journals.pnu.ac.ir/article_10567.html
This paper aims to establish first-order necessary optimality conditions for non-smooth multi-objective generalized semi-infinite programming problems&lrm;. &lrm;These problems involve inequality constraints whose index set depends on the decision vector&lrm;, &lrm;and all emerging functions are assumed to be locally Lipschitz&lrm;. &lrm;We introduce a new constraint qualification for these problems&lrm;. &lrm;Building upon this qualification&lrm;, &lrm;we derive an upper estimate for the Clarke sub-differential of the value function of the problem&lrm;. &lrm;Furthermore&lrm;, &lrm;we demonstrate the necessary optimality conditions for properly efficient solutions to the problem&lrm;.Control and Synchronization of the Genesio-Tesi Chaotic System: A Contraction Analysis-Based Graphical Method
https://mathco.journals.pnu.ac.ir/article_9910.html
In this paper&lrm;, &lrm;we use a graphical algorithm to control and synchronization of a chaotic system&lrm;. &lrm;Most of the controllers designed for synchronizing chaotic systems are complex&lrm;, &lrm;but the controllers designed using contraction and graphical methods are often simple and linear&lrm;. &lrm;Therefore&lrm;, &lrm;we explain the relationship between contraction analysis and the graphical method for controlling and synchronizing chaotic systems&lrm;. &lrm;We apply this approach to control and synchronize the chaotic Genesio-Tesi system&lrm;. &lrm;The stability of the error system in synchronization is investigated using the contraction method&lrm;. &lrm;Finally&lrm;, &lrm;we provide numerical simulations to demonstrate the effectiveness of the proposed method&lrm;.Optimal Adaptive Sliding Mode Control for a Class of Nonlinear Affine Systems
https://mathco.journals.pnu.ac.ir/article_10333.html
This paper presents an optimal robust adaptive technique for controlling a certain class of uncertain nonlinear affine systems&lrm;. &lrm;The proposed approach combines sliding mode control&lrm;, &lrm;a linear quadratic regulator for optimality, and gradient descent as an adaptive controller&lrm;. &lrm; The convergence of the sliding mode control process is proven using two theorems based on the Lyapunov function. Simulation results for pendulum and inverted pendulum systems demonstrate that the proposed method outperforms both the linear quadratic regulator technique and &lrm;the&lrm; &lrm;sliding&lrm; &lrm;mode&lrm; &lrm;control regarding reduced chattering and improved reaching time&lrm;.An Adaptive Time-Stepping Algorithm to Solve a Stochastic Lotka-Volterra Competition System with Time-Variable Delays
https://mathco.journals.pnu.ac.ir/article_10841.html
&lrm;This paper introduces &lrm;a variable step size strategy for a stochastic time-delays Lotka-Volterra competition system&lrm;. &lrm;This adaptive strategy utilizes the Milstein method for numerical solutions. It employs two local error estimates, corresponding to the diffusion and drift components of the model, to select and control the step sizes&lrm;. &lrm;The algorithm is described in detail&lrm;, &lrm;and numerical experiments are conducted to demonstrate the efficiency of the proposed method&lrm;. &lrm;The primary objective of this research is to propose a dynamic strategy for generating and controlling the step sizes in the finite difference algorithm employed. &lrm;This adaptive approach accelerates the numerical procedure and improves efficiency compared to a constant-size scheme&lrm;. &lrm; As an analytical solution for the model is unavailable&lrm;, &lrm;a numerical estimation with a small fixed step size is considered a reference solution&lrm;. &lrm;The numerical results demonstrate the superior accuracy of the proposed strategy compared to a reference solution&lrm;.A Multi-Objective Model for Humanitarian Logistics Model During an Earthquake Crisis: A Case Study of Iran
https://mathco.journals.pnu.ac.ir/article_10880.html
Natural disasters&lrm;, &lrm;such as earthquakes&lrm;, &lrm;result in significant financial and human losses&lrm;. &lrm;Rescue operations play a crucial role in managing such crises&lrm;. &lrm;However&lrm;, &lrm;the lack of precise information and the damage or destruction of urban transportation routes following earthquakes introduces uncertainty into these operations&lrm;. &lrm;This study presents a multi-objective humanitarian logistics model that utilizes a mixed-integer nonlinear programming (MINLP) approach&lrm;. &lrm;The model considers the reliability of transportation routes after an earthquake&lrm;, &lrm;the standard response time for allocating personnel and relief equipment&lrm;, &lrm;and the coverage maximization&lrm;. &lrm;This model incorporates various uncertainties&lrm;, &lrm;including the reliability of the transportation network&lrm;. &lrm;Real data from the city of Gonabad&lrm;, &lrm;Iran&lrm;, &lrm;was used to evaluate the proposed model&lrm;. &lrm;The results and sensitivity analysis demonstrated that the model exhibits desirable performance&lrm;.On Constraint Qualifications and Optimality Conditions in Nonsmooth Semi-infinite Optimization
https://mathco.journals.pnu.ac.ir/article_10985.html
The primary objective of this paper is to enhance several well-known geometric constraint qualifications and necessary optimality conditions for nonsmooth semi-infinite optimization problems (SIPs). We focus on defining novel algebraic Mangasarian-Fromovitz type constraint qualifications, and on presenting two Karush-Kuhn-Tucker type necessary optimality conditions for nonsmooth SIPs defined by locally Lipschitz functions. Then, by employing a new type of generalized invex functions, we present sufficient conditions for the optimality of a feasible point of the considered problems. It is noteworthy that the new class of invex functions we considered encompasses several classes of invex functions introduced previously. Our results are based on the Michel-Penot subdifferential.Mordukhovich Normal Cone of Optimization Problems with Switching Constraints
https://mathco.journals.pnu.ac.ir/article_11075.html
This paper examines normal cones of the feasible set for mathematical programming problems with switching constraints (MPSC)&lrm;. &lrm;Functions involved are assumed to be continuously differentiable&lrm;. &lrm;The primary focus is on providing the upper estimate of the Mordukhovich normal cone for the feasible set of MPSCs&lrm;. &lrm;First&lrm;, &lrm;a constraint qualification&lrm;, &lrm;called the ``MPSC-No Nonzero Abnormal Multiplier Constraint Qualification''&lrm;, &lrm;is considered for the problem&lrm;. &lrm;Based on this qualification&lrm;, &lrm;the main result of the paper is presented&lrm;. &lrm;Finally&lrm;, &lrm;an optimality condition&lrm;, &lrm;called the ``necessary M-stationarity condition'' is proposed for optimal solutions of the considered problems&lrm;. &lrm;Since other optimization problems with multiplicative constraints can be rewritten in the form of MPSCs&lrm;, &lrm;results obtained in this paper can be extended to a wider class of problems involving multiplicative constraints&lrm;.Impact of Carrier Relaxation Time on the Performance of Quantum Dot Laser with Planar Cavities Using Artificial Neural Networks
https://mathco.journals.pnu.ac.ir/article_11086.html
This study presents a model of &lrm;a quantum dot laser with a planar cavity, employing numerical methods and artificial neural networks for simulation purposes. The investigation focuses on the influence of critical parameters, including the injection current into the active layer of the quantum dot laser and the carrier relaxation time to a lower energy state level. The model delves into the intricate carrier and photon dynamics within the laser, solving a system of coupled equations that describe these interactions. The fourth-order Runge-Kutta method is utilized to solve these equations numerically. &lrm;&lrm;The results indicate that increased pumping power enhances the stable power levels and the peak power output of the laser. Additionally, analysis of the power versus intensity of current ($P-I$) characteristic curve&lrm; &lrm; reveals that a longer carrier relaxation time to a lower energy state leads to a higher threshold current and a reduction in the quantum efficiency of the device&lrm;. &lrm;The study also examines the laser switch-on time against the injection current. Finally, the deterioration in the quality of quantum dots and quantum wells is scrutinized&lrm;. To gain deeper insights into the effect of increased pumping current on laser switch-on time&lrm;, &lrm;the study complements numerical findings with the application of artificial neural networks, yielding significant results.