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Control and Optimization in Applied Mathematics - Volume:9 Issue: 1, Winter-Spring 2024

Control and Optimization in Applied Mathematics
Volume:9 Issue: 1, Winter-Spring 2024

  • تاریخ انتشار: 1403/02/12
  • تعداد عناوین: 12
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  • Negar Izadi *, Mohammad Taghi Dastjerdi Pages 1-34
    In this paper‎, ‎we present a new approach for achieving leader-follower consensus in a network of nonlinear dynamic agents with an undirected graph topology‎, ‎using a fuzzy sliding mode controller (FSMC) for Multi-Agent Systems (MASs)‎. ‎Our proposed sliding mode controller is based on a separating hyperplane that effectively addresses the consensus problem in MASs‎. ‎Additionally‎, ‎we design a fuzzy controller to eliminate the chattering phenomenon‎. ‎According to the communication graph topology and the Lyapunov stability condition‎, ‎the proposed FSMC satisfies the consensus condition‎. ‎One significant advantage of our approach is that the system states converge to the sliding surface quickly and remain on the surface‎, ‎thereby ensuring better tracking performance‎. ‎We validate the effectiveness of our proposed approach through simulation results‎.
    Keywords: Consensus‎, ‎Fuzzy Controller‎, ‎Multi-Agent System‎, ‎Sliding Mode Control
  • Reza Akbari *, Leader Navaei, Mohammad Shahriari Pages 35-48
    This paper presents an extension of the SEIR mathematical model for infectious disease‎ ‎transmission to a fractional-order model‎. ‎The model is formulated using the Caputo derivative of order α ∈ (0, 1]‎. ‎We study the stability of equilibrium points‎, ‎including the disease-free equilibrium $(E_{f})$‎, ‎and the‎ ‎infected steady-state equilibrium $(E_{e})$ using the‎ ‎stability theorem of Fractional Differential Equations‎. ‎The model is also analyzed under certain conditions‎, ‎and‎ ‎it is shown that the disease-free equilibrium is locally asymptotically‎ ‎stable‎. ‎Additionally‎, ‎the extended Barbalat’s lemma is applied to the‎ ‎fractional-order system‎, ‎and a suitable Lyapunov functional is constructed‎ ‎to demonstrate the global asymptotic stability of the infected‎ ‎steady-state equilibrium‎. ‎To validate the theoretical results‎, ‎a numerical simulation of the problem is conducted‎.
    Keywords: Fractional Calculus‎, ‎Caputo Derivatives‎, ‎SEIR ‎Model‎, ‎Lyapunov Function‎, ‎Stability
  • Hajar Alimorad * Pages 49-65
    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.‎ ‎Newton's method‎, ‎a notable numerical technique within the category of line search algorithms, is widely used for function optimization‎. The search direction and step length play crucial roles in this algorithm. ‎This paper introduces an algorithm aimed at enhancing the step length within the Broyden quasi-Newton process‎. ‎Additionally‎, ‎numerical examples are provided to compare the effectiveness of this new method with another approach‎.
    Keywords: Optimization, ‎Hessian Matrix‎, Quasi-Newton Method‎, Constrained, Unconstrained Problems
  • Fatemeh Babakordi *, Nemat Allah ‎Taghi-Nezhad Pages 67-79
    This paper presents the introduction of two novel equation types: the partial hesitant fuzzy equation and the half hesitant fuzzy equation‎. Additionally, ‎ an efficient method is proposed to solve these equations by defining four solution categories: Controllable‎, ‎Tolerable Solution Set (TSS)‎, Controllable ‎Solution Set (CSS)‎, ‎and Algebraic Solution Set (ASS)‎. ‎ Furthermore, ‎ 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. ‎ The practicality of the proposed method is demonstrated through numerical examples.
    Keywords: Categories Of Hesitant Fuzzy Equations‎, ‎Partial Hesitant Fuzzy Equation‎, ‎Half Hesitant Fuzzy Equation‎, ‎Algebraic Solution Set
  • Abbas Ali Rezaee *, Hadis Ahmadian Yazdi, Mahdi Yousefzadeh Aghdam, Sahar Ghareii Pages 81-96
    With the advancements in science and technology‎, ‎the industrial and aviation sectors have witnessed a significant increase in data‎. ‎A vast amount of data is generated and utilized continuously‎. ‎It is imperative to employ data mining techniques to extract and uncover knowledge from this data‎. ‎Data mining is a method that enables the extraction of valuable information and hidden relationships from datasets‎. ‎However‎, ‎the current aviation data presents challenges in effectively extracting knowledge due to its large volume and diverse structures‎. ‎Air Traffic Management (ATM) involves handling Big data‎, ‎which exceeds the capacity of conventional acquisition‎, ‎matching‎, ‎management‎, ‎and processing within a reasonable timeframe‎. ‎Aviation Big data exists in batch forms and streaming formats‎, ‎necessitating the utilization of parallel hardware and software‎, ‎as well as stream processing‎, ‎to extract meaningful insights‎. ‎Currently‎, ‎the map-reduce method is the prevailing model for processing Big data in the aviation industry‎. ‎This paper aims to analyze the evolving trends in aviation Big data processing methods‎, ‎followed by a comprehensive investigation and discussion of data analysis techniques‎. ‎We implement the map-reduce optimization of the K-Means algorithm in the Hadoop and Spark environments‎. ‎The K-Means map-reduce is a crucial and widely applied clustering method‎. ‎Finally‎, ‎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. ‎The analyzed dataset includes flight records‎. ‎The results demonstrate the suitability of this platform for aviation Big data‎, ‎considering the characteristics of the aviation dataset‎. ‎Furthermore‎, ‎this study presents the first application of the designed program for air traffic management‎.
    Keywords: Data Mining‎, ‎Air Traffic Management‎, ‎Clustering‎, ‎K-Means Algorithm‎, ‎Hadoop Platform‎, ‎Spark Platform Optimization
  • Sajad Amirian, Maghsoud Amiri *, Mohammad Taghi Taghavifard Pages 97-130
    Integrating sustainability and reliability represents a synergistic approach that can be explored through the problem of a closed-loop supply chain network design (SCND)‎. ‎This study is conducted in three stages‎: ‎mathematical modeling‎, ‎model solution using exact methods‎, ‎and evaluation of the solution methods‎. ‎In the first stage‎, ‎a mixed-integer linear programming (MILP) model is developed in a multi-objective‎, ‎multi-product‎, ‎and multi-period framework‎. ‎The objectives of the proposed model aim to maximize profitability‎, ‎social responsibility‎, ‎and reliability‎. ‎In the second stage‎, ‎two methods‎, ‎namely Augmented ‎$\varepsilon‎‎$‎-Constraint (AEC) and Normalized Normal Constraint (NNC)‎, ‎are implemented in the GAMS software to solve the model and identify the optimal Pareto solutions‎. ‎In the third stage‎, ‎the Shannon Entropy technique is employed to determine the criteria weights‎, ‎and the VIKOR technique is utilized to select the superior solution method‎. ‎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. ‎The results indicate the presence of a conflict among the three objective functions‎. ‎Consequently‎, ‎decision-makers should consider sacrificing some profitability to enhance environmental protection and improve reliability‎. ‎In terms of three criteria‎, ‎run time‎, ‎diversification metric‎, ‎and general distance‎, ‎the NNC method is given priority over the AEC method‎. ‎Even when the criteria are given equal weight‎, ‎the superiority of the NNC method remains unchanged‎. ‎The application of the proposed model across different industries represents a significant research direction for future research‎.
    Keywords: Supply Chain Network Design (SCND)‎, ‎Sustainability‎, ‎Reliability‎, ‎Augmented Ε-Constraint (AEC)‎, ‎Normalized Normal Constraint (NNC)
  • Ali Akbar Sohrabi, Reza Ghanbari *, Khatere Ghorbani-Moghadam Pages 131-147
    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. ‎To address this issue‎, ‎organizations seek to select a feasible subset of projects that maximizes utility‎. ‎While several models for project portfolio selection based on multiple criteria have been proposed‎, ‎they are typically NP-hard problems‎. ‎In this study‎, ‎we propose an efficient Variable Neighborhood Search (VNS) algorithm to solve these problems‎. ‎Our algorithm includes a formula for computing the difference value of the objective function‎, ‎which enhances its accuracy and ensures that selected projects meet desired criteria‎. ‎We demonstrate the effectiveness of our algorithm through rigorous testing and comparison with a genetic algorithm (GA) and CPLEX‎. ‎The results of the Wilcoxon non-parametric test confirm that our algorithm outperforms both GA and CPLEX in terms of speed and accuracy‎. ‎Moreover‎, ‎the variance of the relative error of our algorithm is less than that of GA‎.
    Keywords: Project Portfolio Selection‎, ‎Project Interaction‎, ‎Multi-Criteria‎, ‎Meta-Heuristic Algorithms
  • Zeinab Barary, Allahbakhsh Yazdani Cherati *, Somayeh Nemati Pages 149-168
    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. ‎First‎, ‎the corresponding Riemann-Liouville fractional integral operator of the introduced basis functions is calculated‎. ‎ Then, an approximation of the fractional derivative of the unknown state function is obtained by considering an approximation in terms of these basis functions‎. ‎ Next, ‎using the dynamical system and applying the fractional integral operator‎, ‎an approximation of the unknown control function is obtained based on the given approximations of the state function and its derivatives‎. ‎ Subsequently‎, ‎all the given approximations are substituted into the performance index‎. ‎Finally‎, ‎the optimality conditions transform the problem into a system of algebraic equations‎. ‎An error upper bound of the approximation of a function based on the fractional hybrid functions is provided‎. ‎The method is applied to several numerical examples‎, and ‎the experimental results confirm the efficiency and capability of the method.  Furthermore, they demonstrate a good agreement between the approximate and exact solutions‎. ‎
    Keywords: Fractional Optimal Control-Affine Problems‎, ‎Fractional-Order Hybrid Functions‎, ‎Caputo Derivative‎, ‎Riemann-Liouville Integral
  • Farid Pourofoghi *, Davood Darvishi Salokolaei Pages 169-183
    Fractional programming is a significant nonlinear planning tool within operation research‎. ‎It finds applications in diverse domains such as resource allocation‎, ‎transportation‎, ‎production programming‎, ‎performance evaluation‎, ‎and finance‎. ‎In practical scenarios‎, ‎uncertainties often make it challenging to determine precise coefficients for mathematical models‎. ‎Consequently‎, ‎utilizing indefinite coefficients instead of definite ones is recommended in such cases‎. ‎Grey systems theory‎, ‎along with probability theory‎, ‎randomness‎, ‎fuzzy logic‎, ‎and rough sets‎, ‎is an approach that addresses uncertainty‎. ‎In this study‎, ‎we address the problem of linear fractional programming with grey coefficients in the objective function‎. ‎To tackle this problem‎, ‎a novel approach based on the variable change technique proposed by Charnes and Cooper‎, ‎along with the convex combination of intervals‎, ‎is employed‎. ‎The article presents an algorithm that determines the solution to the grey fractional programming problem using grey numbers‎, ‎thus capturing the uncertainty inherent in the objective function‎. ‎To demonstrate the effectiveness of the proposed method‎, ‎an example is solved using the suggested approach‎. ‎The result is compared with solutions obtained using the whitening method‎, ‎employing Hu and Wong's technique and the Center and Greyness Degree Ranking method‎. ‎The comparison confirms the superiority of the proposed method over the whitening method‎, ‎thus suggesting adopting the grey system approach in such situations‎.
    Keywords: Uncertainty‎, ‎Optimization‎, ‎Fractional Programming‎, ‎Grey System‎, ‎Grey Interval Numbers
  • Hanifa Mosawi, Mostafa Tavakolli *, Khatere Ghorbani-Moghadam Pages 185-194
    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. In this study, we employ a recently introduced graph coloring algorithm to develop a hybrid algorithm approximating the chromatic number ‎p-distance, where $p$ represents a positive integer number. We apply our algorithm to molecular graphs as practical applications of our findings.
    Keywords: P-Distance Coloring‎, ‎P-Distance Chromatic Number‎, ‎Graph Adjacency Matrix‎, ‎Hybrid Algorithm
  • Fahimeh Akhavan Ghassabzade *, Mina Bagherpoorfard Pages 195-202
    ‎This paper aims to demonstrate the flexibility of mathematical models in analyzing carbon dioxide emissions and account for memory effects. ‎The use of real data amplifies the importance of this study‎. ‎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. ‎Additionally‎, ‎an optimal control strategy is proposed to minimize the levels of carbon dioxide, CO2, and associated implementation costs‎. ‎The fractional optimal control problem is addressed through the utilization of an iterative algorithm‎, ‎ and the effectiveness of the model is verified by presenting comparative results.
    Keywords: {Fractional‎, ‎Mathematical Model‎, ‎Optimal Control‎, ‎Carbon Dioxide
  • Ali Asghar Hojatifard, Nader Kanzi *, ‎Shahriar Farahmand Rad Pages 203-219
    This paper aims to establish first-order necessary optimality conditions for non-smooth multi-objective generalized semi-infinite programming problems‎. ‎These problems involve inequality constraints whose index set depends on the decision vector‎, ‎and all emerging functions are assumed to be locally Lipschitz‎. ‎We introduce a new constraint qualification for these problems‎. ‎Building upon this qualification‎, ‎we derive an upper estimate for the Clarke sub-differential of the value function of the problem‎. ‎Furthermore‎, ‎we demonstrate the necessary optimality conditions for properly efficient solutions to the problem‎.
    Keywords: Constraint Qualification, Generalized Semi-Infinite Optimization, Clarke Subdifferential, Marginal Function