multi-objective optimization
در نشریات گروه ریاضی-
Iranian Journal of Numerical Analysis and Optimization, Volume:15 Issue: 2, Spring 2025, PP 600 -624
Investors need to grasp how liquidity affects both risk and return in order to optimize their portfolio performance. There are three classes of stocks that accommodate those criteria: Liquid, high-yield, and less-risky. Classifying stocks help investors build portfolios that align with their risk profiles and investment goals, in which the model was constructed using the one-versus-one support vector machines method with a radial basis function kernel. This model was trained using a combination of the Kompas100 index and the Indonesian industrial sectors stocks data. Single optimal portfolios were created using the real coded genetic algorithm based on different sets of objectives: Maximizing short-term and long-term returns, maximizing liquidity, and minimizing risk. In conclusion, portfolios with a balance on all these four investment objectives yielded better results compared to those focused on partial objectives. Furthermore, our proposed method for selecting portfolios of top-performing stocks across all criteria outperformed the approach of choosing top stocks based on a single criterion.
Keywords: Genetic Algorithm, Liquidity, Multi-Objective Optimization, One-Versus-One Support Vector Machines, Radial Basis Functions -
In this paper, we introduce the new concept of r-efficiency and classify the weakly efficient solutions by this concept, where r = 1, 2, . . . , m and m is the number of objective functions. Afterward, we decompose the multi-objective optimization problem into the collection of subproblems with cardinality r, for two reasons. First, we apply the scalarization techniques for these subproblems to generate r-efficient solutions. Second, in order to present the new mean-equity models for solving the location problem, we employ the mean and inequality measures on these subproblems. Moreover, by introducing the consistency property for the inequality measures and stating the sufficient conditions to maintain this property, we investigate the relationship between r-efficient solutions of the new mean-equity models and efficient solutions of the location problem.
Keywords: R-Efficiency, Decomposition, Inequality Measure, Location, Multi-Objective Optimization -
Iranian Journal of Numerical Analysis and Optimization, Volume:14 Issue: 4, Autumn 2024, PP 1106 -1139Numerous optimization problems comprise uncertain data in practical circumstances and such uncertainty can be suitably addressed using the concept of fuzzy logic. This paper proposes a computationally efficient solution methodology to generate a set of fuzzy non-dominated solutions of a fully fuzzy multi-objective linear programming problem, which incorporates all its parameters and decision variables expressed in form of triangular fuzzy numbers. The fuzzy parameters associated with the objective functions are transformed into interval forms by utilizing the fuzzy-cuts, which subsequently generates the equivalent interval valued objective functions. The concept of centroid of triangular fuzzy numbers derives the deterministic form of the constraints. Furthermore, the scalarization process of weighting sum approach and certain concepts of interval analysis are used to generate the fuzzy non-dominated solutions from which the compromise solution can be determined based on the corresponding real valued expressions of fuzzy optimal objective values resulted due to the ranking function. Three numerical problems and one practical problem are solved for illustration and validation of the proposed approach. The computational results are also discussed as compared to some existing methods.Keywords: Multi-Objective Optimization, Fully Fuzzy Programming, Triangular Fuzzy Numbers, Interval Valued Functions, Weighting Sum Approach
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International Journal Of Nonlinear Analysis And Applications, Volume:16 Issue: 4, Apr 2025, PP 41 -55This article solves the mathematical model of supply, planning, storage, and distribution in a supply chain using an agile supply chain. In modern supply chains, addressing financial risks in purchases and distribution, transportation for supply and distribution of products and lost policies or the storage of products in perishable products supply chains is critical. This article investigates a direct supply chain model for the executive policies of manufacturing companies and a four-stage supply chain problem using the operational financial and marketing approach under an uncertainty state, Lagrangian relaxation and Benders’ decomposition evaluation. This study introduced and provided the concepts of supply chains in perishable products. Considering supply chain issues and uncertainty in this environment, a fuzzy mathematical model was provided. In the end, a literature review has shown that most research has used heuristic and meta-heuristic algorithms due to supply chain models being NP-HARD., which are inefficient due to the approximation of optimal solutions in this domain. Meanwhile, the study used a Benders decomposition and Lagrangian relaxation algorithm for mathematical solutions, which would reduce the model’s solving time and provide accurate answers.Keywords: Lagrangian Relaxation, Benders Relaxation, Perishable Supply Chain, Multi-Objective Optimization
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International Journal Of Nonlinear Analysis And Applications, Volume:16 Issue: 3, Mar 2025, PP 63 -76The two requirements of impartiality and equitability expressed with the principle of transfers are fulfilled by all objective functions in equitable multi-objective optimization. However, in some practical situations, the decision-maker believes these requirements should only be satisfied by a subset of objective functions. To solve the problem in this paper, we first divide the set of objective functions into two subsets, the subset given by the decision maker and its complement. Then, we apply the concepts of equitable efficiency and efficiency for these two subsets, respectively. Furthermore, we apply the mean and inequality measures for these subsets of objective functions and present the new mean-equity models for solving the location problem. We investigate the relationship between $2$-efficient solutions of the new mean-equity models and efficient solutions of the location problem.Keywords: Efficiency, Equitable Efficiency, Inequality Measure, Location, Multi-Objective Optimization
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International Journal Of Nonlinear Analysis And Applications, Volume:16 Issue: 3, Mar 2025, PP 77 -88
In the contemporary supply chain management landscape, the intricacies of managing a single vendor-multi-buyer network amidst stochastic demand pose significant challenges. This paper delves into optimizing such supply chains, emphasizing resilience in the face of uncertain demand scenarios. Leveraging the NSGA-II (Non-dominated Sorting Genetic Algorithm II), a powerful evolutionary optimization technique, we explore the multifaceted dimensions of supply chain optimization. The proposed framework aims to enhance the robustness and adaptability of supply chain networks by simultaneously addressing two key objectives minimizing costs and maximizing service levels. By considering stochastic demand patterns, inherent uncertainties are meticulously accounted for, ensuring that the optimized solutions are efficient and resilient to unforeseen fluctuations in demand. This study comprehensively evaluates the single vendor-multi buyer supply chain model and highlights the efficacy of the NSGA-II algorithm in navigating the complex trade-offs inherent in supply chain optimization. By generating diverse Pareto-optimal solutions, the algorithm empowers decision-makers with actionable insights, enabling them to make informed choices that balance cost-effectiveness with service quality. Furthermore, this paper contributes to the evolving discourse on supply chain resilience by integrating advanced optimization methodologies with real-world supply chain dynamics. The findings underscore the importance of proactive optimization strategies in building resilient supply chain networks capable of withstanding the volatility of today's global marketplace. In conclusion, this research illuminates the path towards catalyzing resilience in single vendor-multi buyer supply chains, offering a nuanced understanding of the interplay between optimization algorithms, stochastic demand, and supply chain performance. Organizations can fortify their supply chain architectures through continuous refinement and adaptation, fostering agility and competitiveness in an ever-evolving business landscape.
Keywords: Supply Chain Optimization, Stochastic Demand, NSGA-II Algorithm, Resilience, Multi-Objective Optimization, Single Vendor-Multi Buyer -
International Journal Of Nonlinear Analysis And Applications, Volume:15 Issue: 11, Nov 2024, PP 109 -120Some of the important and effective issues of HRM and senior managers of knowledge-based organizations are maintaining and improving the commitment level of employees who are the most valuable resources of these organizations. So, this study was conducted to determine and rank organizational commitment factors in knowledge-based organizations using the multi-objective gray wolf optimizer (MOGWO). The study was conducted using a mixed method (qualitative and quantitative) and is applied in terms of objectives and descriptive. The statistical population was selected using exploratory factor analysis (EFA) and included 200 senior managers of knowledge-based companies in Tehran and Alborz provinces, and the sample size was calculated as 132 using Cochran's formula. The best employee commitment (optimal) factors, which were 42 factors, were determined using the MOGWO. Finally, the SWARA method was used to weigh and rank the principal and sub-components from the opinions of 16 experts who were selected by purposive sampling, and it was found that corporate culture is the most important aspect of optimizing employee commitment in knowledge-based organizations. In today's changing and highly competitive environment, organizations must focus more on employee commitment to survive. Besides, knowledge-based organizations should use process re-engineering in human resource processes and working conditions to improve employee commitment levels.Keywords: Organizational Commitment, Multi-Objective Optimization, Gray Wolf Algorithm, Knowledge-Based Organizations
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Iranian Journal of Numerical Analysis and Optimization, Volume:14 Issue: 2, Spring 2024, PP 522 -544There are various techniques for separating natural gas liquid (NGL) from natural gas, one of which is refrigeration. In this method, the temper-ature is reduced in the dew point adjustment stage to condense the NGLs. The purpose of this paper is to introduce a methodology for optimizing the NGLs production process by determining the optimal values for specific set-points such as temperature and pressure in various vessels and equip-ment. The methodology also focuses on minimizing energy consumption during the NGL production process. To do this, this research defines a multi-objective problem and presents a hybrid algorithm, including a ge-netic algorithm (NSGA II) and artificial neural network (ANN) system. We solve the defined multi-objective problem using NSGA II. In order to de-sign a tool that is a decision-helper for selecting the appropriate set-points, the ability of the ANN algorithm along with multi-objective optimization is evaluated. We implement our proposed algorithm in an Iranian chemical factory, specifically the NGL plant, which separates NGL from natural gas, as a case study for this article. Finally, we demonstrate the effectiveness of our proposed algorithm using the nonparametric statistical Kruskal–Wallis test.Keywords: Natural Gas Liquid (NGL), Multi-Objective Optimization, Artifi-Cial Neural Network, NSGA II
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Feature selection is one of the most important tasks in machine learning. Traditional feature selection methods are inadequate for reducing the dimensionality of online data streams because they assume that the feature space is fixed and every time a feature is added, the algorithm must be executed from the beginning, which in addition to not performing real-time processing, causes many unnecessary calculations and resource consumption. In many real-world applications such as weather forecasting, stock markets, clinical research, natural disasters, and vital-sign monitoring, the feature space changes dynamically, and feature streams are added to the data over time. Existing online streaming feature selection (OSFS) methods suffer from problems such as high computational complexity, long processing time, sensitivity to parameters, and failure to account for redundancy between features. In this paper, the process of OSFS is modeled as a multi-objective optimization problem for the first time. When a feature stream arrives, it is evaluated in the multi-objective space using fuzzy Pareto dominance, where three feature selection methods are considered as our objectives. Features are ranked according to their degree of dominance in the multi-objective space over other features. We proposed an effective method to select a minimum subset of features in a short time. Experiments were conducted using two classifiers and eight OSFS algorithms with real-world datasets. The results show that the proposed method selects a minimal subset of features in a reasonable time for all datasets.
Keywords: Online streaming feature selection, Fuzzy Pareto dominance, High-dimensional data, multi-objective optimization -
ارائه یک مدل بهینه سازی چندهدفه جهت زمانبندی و مسیریابی پرستاران در ارائه خدمات پزشکی در منزل
امروزه با رشد روزافزون جمعیت و همچنین عواملی مانند افزایش افراد سالمند، افزایش تعداد بیمارانی که ازبیماریهای مزمن رنج می برند، تقاضا برای دریافت مراقبتهای پزشکی در منزل (HHC) در حال افزایش است. مراکز ارائه دهنده خدمات مراقبتی پزشکی همواره به دنبال راهکارهایی جهت برنامه ریزی و زمانبندی دقیق ارائه خدمات مراقبتی - درمانی هستند تا علاوه برکمینه کردن هزینه های خود، میزان رضایت بیماران و پرستاران را بیشینه نمایند. در این راستا، ازیک سو بیمار تمایل دارد تا با در نظر گرفتن مهارت پرستار اختصاص داده شده، در پنجره زمانی مورد ترجیحش ملاقات شود. از سوی دیگر، پرستار نیز ترجیح می دهد که در پنجره زمانی مورد مطلوب خود به ارائه خدمات بپردازد. علاوه بر این موارد، حفظ قوانین ساعت کاری در قرارداد، رعایت بازه های زمانی نرم و سخت ارائه خدمات و استراحت های الزامی از محدودیت هایی هستند که ضروری است در این نوع مسائل در نظر گرفته شوند. اهداف این مقاله کمینه کردن زمان رفت و آمد پرستاران و بیشینه نمودن همزمان سطح رضایت بیماران و پرستاران و همچنین کاهش ساعات اضافه کاری پرستاران می باشد که با در نظر گرفتن ترجیحات پرستاران و بیماران و اختصاص استراحت های اجباری به پرستاران بعد از مدت زمان کاری مشخص،در قالب یک مدل برنامه ریزی ریاضی چندهدفه ارائه می گردد. در ادامه به منظوربررسی و تحلیل عملکرد مدل پیشنهادی، با در نظر گرفتن معیار توقف بر روی زمان حل مسئله، مدل پیشنهادی را بر روی مجموعه ای از مسائل تصادفی متفاوت مورد آزمون قرار می دهیم.
کلید واژگان: بهینه سازی چند هدفه، زمانبندی، برنامه ریزی ریاضی، مراقبت پزشکی در منزل، مسیریابیA multi-objective optimization model for scheduling nurses and routing them in home health care servicesRecently with the growing population and the consequences of factors such as the increase in the number of elderly and patients with chronic diseases, the demand for receiving Home Health Care (HHC) is increasing. HHC services providers are looking for the optimal solutions in planning and scheduling HHC service delivery to maximize the satisfaction of patients and nurses in addition to minimizing the costs to patients. Accordingly on the one hand, the patients prefer to be visited at some specific periods based on their nursing skills. On the other hand, nurses are willing to provide services during their desired time windows. Following the rules corresponding to the working times in the contract, observing the soft and hard time windows, and taking required breaks are some of the restrictions that must be considered.The main objectives of this paper are to minimize the total traveling times and overtime of all nurses and to maximize the satisfaction level of patients as well as nurses, which are achieved through a multi-objective mathematical programming model.The proposed model considers the preferences of the nurses, as well as their patience. Moreover it establishes mandatory breaks for nurses after a certain period of work to assign qualified nurses to patients, optimize schedules, and determine the route of nurses, and provides high-quality services. Finally by applying the proposed model to a set of different random test problems, and by considering the stopping criterion on the problem solving time, we analyze the numerical results corresponding to optimal scheduling and allocation.
Keywords: Scheduling, Routing, Home Health Care, Mathematical Programming, Multi-Objective Optimization -
This paper is motivated by high dose rate brachytherapy treatment planning problems which involve the specification of the movement schedule of a radiation source so that the target volumes are adequately covered with sufficient doses and organs at risk are not radiated beyond the clinical acceptance threshold. It utilizes four powerful multi-objective evolutionary algorithms (MOEA), which create a set of equally-weighted Pareto optimal solutions instead of only one and produce better results compared to other optimization methods. These algorithms include non-dominated sorting genetic algorithms, Pareto envelope-based selection algorithm, non-dominated ranking genetic algorithm, and strength Pareto evolutionary algorithm. The results indicate that the last algorithm uses the dependency between decision variables to solve them efficiently and is the best type of MOEA both in terms of convergence criteria and solution diversity maintenance for the brachytherapy problems.Keywords: Multi-objective optimization, Fuzzy logic, Evolutionary algorithms, Brachytherapy
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در این مقاله با استفاده از یک رویکرد تلفیقی فازی به حل مساله فروشندگان دوره گرد چند ایستگاهی به عنوان یک مساله بهینه سازی دوهدفه می پردازیم. این رویکرد، با تعریف مفهوم جدید چیره گی فازی، به هر یک از بردارهای توابع هدف مساله، یک درجه نزدیکی گوسی متناظر می کند که بر اساس آن امکان رتبه بندی و در نتیجه مقایسه جواب های پارتو در یک مساله بهینه سازی چندهدفه فراهم می شود. به عبارت دقیق تر، با استفاده از این رویکرد، مساله بهینه سازی چندهدفه را می توان به صورت یک مساله بهینه سازی تک هدفه در نظر گرفت. در این مقاله با تلفیق مفهوم چیره گی فازی و یک الگوریتم فراابتکاری مانند شبیه سازی تبریدی حل مساله فروشندگان دوره گرد چند ایستگاهی را مورد مطالعه قرار می دهیم. برای این منظور، با انجام شبیه سازی های مختلف، عملکرد این رویکرد پیشنهادی را ارزیابی می کنیم. نتایج عددی حاکی از تاثیر این رویکرد در بهبود کیفیت جواب ها و همچنین کاهش زمان محاسباتی حل مساله می باشد.
کلید واژگان: مساله فروشنده دوره گرد، بهینه سازی چندهدفه، جواب پارتو، پیچیدگی الگوریتم، الگوریتم فراابتکاریIn this paper, the multi-traveling salesman problem as a two-objective optimization problem, are solved by using an integrated fuzzy approach. This approach, by defining the new concept of fuzzy domination, corresponds to each of the vectors of the objective functions of the problem a degree of Gaussian proximity, so it can be ranked and hence, we can compare the obtained Pareto solutions in a multi- objective optimization problem. More precisely, using this approach, the multi-objective optimization problem was considered as a single optimization problem. In this paper, by combining the concept of fuzzy domination and a meta-heuristic algorithm such as simulating annealing we study the multi-traveling salesman problem. To this end, by performing some different simulation, the performance of this proposed approach is evaluated. Numerical results indicate the effect of this approach in improving the quality of results and also reducing the computational time of problems.
Keywords: Traveling salesman problem, Multi-objective optimization, Pareto solution, Complexity algorithm, Heuristic algorithm -
The literature of linear programming problem with trapezoidal intuitionistic parameters is full of solution approaches which are mainly ranking function based. Use of ranking function in the solution approaches could be a weakness as different ranking functions mag give different solutions. This paper, proposes a new solution approach without any ranking function for linear programming problem with trapezoidal intuitionistic parameters. For this aim, the trapezoidal intuitionistic fuzzy objective function is converted to a multi-objective function, and consequently, the problem is converted to a multi-objective crisp problem. As another contribution, in order to solve the obtained multi-objective problem for its efficient solutions, a new multi-objective optimization approach was developed and suited to the obtained multi-objective problem. The computational experiments of the study show the superiority of the proposed multi-objective optimization approach over the multi-objective optimization approaches of the literature.Keywords: linear programming, Trapezoidal intuitionistic fuzzy number, Multi-Objective Optimization, Fuzzy programming approach
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The existence of different solution approaches that generate approximations to the optimal Pareto frontiers of a multi-objective optimization problem lead to different sets of non-dominated solutions. To evaluate the quality of these solution sets, one requires a comprehensive evaluation measure to consider the features of the solutions. Despite various valuation measures, the deficiency caused by the lack of such a comprehensive measure is visible. For this reason, in this paper, by considering some evaluation measures, first we evaluate the quality of the approximations to the optimal Pareto front resulting from the decomposition-based multi-objective evolutionary algorithm equipped with four decomposition approaches and investigate the related drawbacks. In the second step, we use the concept of Gaussian degree of closeness to combine the evaluation measures, and hence, we propose a new evaluation measure called the quasi-Gaussian integration measure. The numerical results obtained from applying the proposed measure to the standard test functions confirm the effectiveness of this measure in examining the quality of the non-dominated solution set in a more accurate manner.Keywords: Multi-objective optimization, Evolutionary algorithm, Evaluation measure, Pareto frontier, Decomposition
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افزایش آلودگی زیست محیطی که موجب گرم شدن کره زمین شده است و برای سلامت انسان و تخریب محیط زیست خطرناک است، باعث نگرانی بسیاری از طراحان و مدیران زنجیره تامین شده است. هدف این پژوهش ارایه یک مدل ریاضی برای طراحی خرید، تولید و توزیع در یک شبکه زنجیره تامین چند سطحی و چند محصولی است که تاثیرات زیست محیطی و هزینه های کلی زنجیره تامین به حداقل برساند و سطح رضایت مشتری به بالاترین سطح برسد. عدم اطمینان تقاضا به خاطر نامشخص بودن سطح تقاضا به نظر مشکل ساز است. با توجه به پیچیدگی مدل ریاضی پیشنهادی و سختی های حل مسئله با روش های دقیق در اندازه بزرگ، یک NSGA II پیشنهادشده است. برای ارزیابی NSGA II پیشنهادی، 5 نمونه در اندازه های مختلف ساخته می شود و به وسیله روش محدودیت اپسیلون و NSGAII حل می شود. بر اساس نتایج به دست آمده، NSGA II پیشنهادی یک روش قابل اطمینان برای یافتن مرزهای پارتویی کارآمد در زمان قابل قبول محسوب می شود.
کلید واژگان: زنجیره تامین سبز، بهینه سازی چندهدفه، الگوریتم ژنتیک با مرتب سازی نا مغلوب، عدم اطمینان، محدودیت اپسیلونThe purpose of this research is to provide a mathematical model for designing the purchase, production, and distribution in a multi-level and multi-product supply chain network such that the environmental impact and total costs of supply chain is minimized and the customers' satisfaction level is maximized. According to the results, the proposed NSGAII is a reliable method to find efficient Pareto frontiers in a reasonable time.
Keywords: Green supply chain, Multi-objective optimization, Non-dominated Sorting Genetic Algorithm, Uncertainty, Epsilon Constraint -
بازآرایی و نصب منابع تولید پراکنده از جمله روش هایی است که برای کاهش تلفات، بهبود پایداری ولتاژ و افزایش قابلیت اطمینان درشبکه های توزیع برق بکار می روند. یافتن کلید هایی که به طور قطع باید در برنامه بازآرایی شرکت داشته باشند وتعیین وضعیت آنها در هر مرحله از باز آرایی از مهم ترین اهداف در بهره برداری بهینه شبکه است. این مقاله به بررسی جایابی بهینه کلیدهای شبکه توزیع ومنابع تولید پراکنده به منظور بهبود قابلیت اطمینان، کاهش تلفات و بهبود پایداری ولتاژ ودرنتیجه افزایش بارپذیری شبکه می پردازد. دراین مقاله جهت بهبود پایداری ولتاژ، درمقایسه با شاخص بارگزاری حداکثر معروف به (λmax)، شاخص دیگری بنام مقادیر منفرد ماتریس ژاکوبین نیزمعرفی وکارآرایی این دو باهم مقایسه می شود. همچنین جهت کاهش محاسبات سنگین مربوط به قابلیت اطمینان که درروش مونت کارلومشاهده می شود ازروش حداقل مجموعه انقطاع (کات ست) و از مدل احتمالاتی به منظور مدل سازی المان های سیستم توزیع در نقاط بار استفاده شده است. منابع تولید پراکنده باماهیت تصادفی و متغیر و بارهای سیستم بصورت ساعتی و با ماهیت سه گانه مسکونی، تجاری وصنعتی درنظرگرفته شده ودرنتیجه نتایج بازآرایی شبکه بصورت ساعتی بیان شده اند. باتوجه به توابع هدف متعدد آلگوریتم بهینه سازی چندهدفه NSGA2 جهت بهینه سازی توابع هدف بکارگرفته شده و از روش عضویت توابع فازی جهت تعیین جواب بهینه استفاده شده است. نتایج شبیه سازی برروی شبکه توزیع 33 باسه IEEE انجام وکارایی، دقت ونقاط ضعف احتمالی روش پیشنهادی نشان داده شده است.کلید واژگان: قابلیت اطمینان، پایداری ولتاژ، کلید های توزیع، تولید پراکنده، بهینه سازی چندهدفهReconfiguration and installation of distributed generation sources are some of the methods used to reduce losses, improve voltage stability and increase reliability in power distribution networks. Finding the switches that should definitely participate in the reconfiguration program and determining their status at each stage of the reconfiguration is one of the most important goals in the optimal operation of the network. This paper examines the optimal placement of distribution network switches and distributed generation sources in order to improve reliability, reduce losses and improve voltage stability and thus increase network load. In this paper, in order to improve the voltage stability, in comparison with the maximum load index known as (λmax), another index called singular values of Jacobin matrix is introduced and the efficiency of the two is compared. Also, in order to reduce the heavy reliability calculations observed in the Monte Carlo method, the minimum cut set method and the probabilistic model have been used to model the elements of the distribution system at load points. Distributed generation sources with random and variable nature and system loads hourly and with the triple nature of residential, commercial and industrial are considered. Due to the multiple objective functions, the NSGA2 multi-objective optimization algorithm is used to optimize the objective functions and the fuzzy function membership method is used to determine the optimal answer. The simulation results are performed on the 33-bus IEEE distribution network and the efficiency, accuracy and possible weaknesses of the proposed method are shown.Keywords: Reliability, Voltage stability, Distribution Switches, distributed generation, Multi objective Optimization
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International Journal Of Nonlinear Analysis And Applications, Volume:13 Issue: 1, Winter-Spring 2022, PP 1709 -1720
Our research includes studying the case 1//F(∑Ui,∑Ti,Tmax) minimized the cost of a three-criteria objective function on a single machine for scheduling n jobs. and divided this into several partial problems and found simple algorithms to find the solutions to these partial problems and compare them with the optimal solutions. This research focused on one of these partial problems to find minimize a function of sum cost of (∑Ui) sum number of late job and (∑Ti) sum Tardiness and (Tmax) the Maximum Tardiness for n job on the single machine, which is NP-hard problem, first found optimal solutions for it by two methods of Complete Enumeration technique(CEM) and Branch and Bounded ((BAB)). Then use some Local search methods(Descent technique(DM), Simulated Annealing (SA) and Genetic Algorithm (GA)), Develop algorithm called ((A)) to find a solution close to the optimal solution. Finally, compare these methods with each other.
Keywords: Descent Method(DM), Genetic Algorithm(GA), Maximum tardiness, Multi-objective optimization, Simulated annealing ((SA)), Total Number of Late job, Total Tardiness -
International Journal Of Nonlinear Analysis And Applications, Volume:12 Issue: 2, Summer-Autumn 2021, PP 2303 -2331
Nowadays, energy consumption is curtailed in an effort to further protect the environment as well as to avoid service level agreement (SLA) breach, as critical issues in task scheduling on heterogeneous computing centers. Any reliable task scheduling algorithm should minimize energy consumption, makespan, and cost for cloud users and maximize resource utilization. However, reduction of energy consumption leads to larger makespan and decreases load balancing and customer satisfaction. Therefore, it is essential to obtain a set of non-domination solutions for these multiple, conflicting objectives, as a non-linear, multi-objective, NP-hard problem. This paper formulates the energy efficient task scheduling in green data centers as a multi-objective optimization problem so that fuzzy Non-dominated Sorting Genetic Algorithm 2 (NSGA-II) has been applied using the concept of Dynamic Voltage Frequency Scaling (DVFS). In this procedure, we adopted fuzzy crossover and mutation for optimal convergence of initial solutions. For this purpose, the binary variance function of gene values and the mean variance function of objective values are proposed for fuzzy control of mutation rate, increasing the variation in the optimal Pareto front as well as the correct frequency variance function of the processors engaged in scheduling to control the crossover rate. This serves to add the objective of indirect load balancing to the optimization objectives, thereby to replace the three-objective optimization process with four-objective optimization. In the experiments, the proposed NSGA-II with fuzzy algorithm is compared against the NSGA-II algorithm, involving three scheduling strategies namely Green, Time and Cost Oriented Scheduling Strategy. The simulation results illustrate that the newly method finds better solutions than others considering these objectives and with less iteration. In fact, the optimal Pareto solutions obtained from the proposed method improved the objectives of makespan, cost, energy and load balance by 4%, 17%, 1% and 13%, respectively.
Keywords: Green Computing, Multi Objective Optimization, Pareto solutions, DVFS, Taskscheduling -
In this paper, a new fuzzy multi-objective multi-product pipeline scheduling problem is introduced. The system consists of a single refinery, a unique distribution center, and a multi-product pipeline. Restrictions such as batchsizing, discharging rate, forbidden sequences, batch volumetric, etc. are considered. Due to the uncertain nature of real-world problems, some parameters of the system are considered as fuzzy values. Tardiness and earliness penalties are considered with a time dependent non-linear function. The basic aim of this scheduling problem is to achieve the optimal sequence for pumping batches of oil products to maximize the financial benefit and simultaneously satisfies the customers with on-time delivery as a multi-objective problem. To tackle this problem, a two-stage methodology is proposed. In the first stage, the fuzzy formulation is converted to its equivalent crisp form by a credibility-based chance-constrained programming approach. In the second stage, the multi-objective crisp formulation is solved by some well-known approaches in the literature. Some test problems are generated and solved by the proposed approaches and the obtained Pareto-optimal solutions are analyzed and compared using some distance-based comparison metrics.
Keywords: Credibility-based chance-constrained modeling, Fuzzy number, Multi-objective optimization, Multi-product pipeline scheduling -
توازن میان توابع هدف در بهینه سازی چندهدفه یکی از ابزارهای تفسیر و بررسی جواب های کارا است. جواب های کارای سره یکی از مفاهیم مهم از نظر تیوری و عملی می باشد که نشان دهنده رفتار توابع هدف طی یک فرایند تغییر می باشد؛ جواب-های کارای سره جواب های کارایی هستند که ناهنجاری های توابع هدف در بعضی از نقاط را فیلتر می کنند و این به تصمیم گیری برای به دست آوردن جواب های با اهمیت بیشتر توسط مدیریت کمک شایانی خواهد کرد. یکی از مهمترین ابزارهای به دست آوردن جواب با توازن کراندار در بهینه سازی چندهدفه، روش اسکالرسازی مجموع وزنی است که بسیاری از نویسندگان این نوع از اسکالرسازی را در بهینه سازی تعاملی بررسی کرده اند. این مقاله روشی برای به دست آوردن جواب-های کارای سره نزدیک به نقطه ایدآل با دیدگاه تیوری و تعاملی و با استفاده از اسکالرسازی وزنی ارایه می دهد. با توجه به اینکه نزدیکی به نقطه ایدآل می تواند یکی از ترجیحات تصمیم گیرنده باشد؛ این روش، ترجیحات تصمیم گیرنده را بدون از دست دادن تیوری در نظر می گیرد. بنابراین این مقاله رویکردی برای یافتن جواب های کارای سره نزدیک به نقطه ایدآل ارایه می دهد.
کلید واژگان: بهینه سازی چندهدفه، کارای سره، توازن، نقطه ایدآل، اسکالرسازی مجموع وزن دار شدهTrade-off between objective functions in multi-objective optimization is one of the tools for interpreting and studying efficient solutions. Properly efficient solutions are one of the most important theoretical and practical concepts that represent the behavior of the objective functions during a process change. Actually, these solutions are those efficient solutions that filter the anomalies of objective functions at some points, and this will help the manager to decision making to choose more important solutions. One of the most important tools for obtaining solutions with bounded trade-off in multi-objective optimization field is the Sum weighted scalarization method, which many authors have been studying it in interactive optimization field. This paper provides a method for obtaining properly efficient solutions near the ideal point with a theoretical and interactive view and using Sum weighted scalarization method. Since being near to ideal point will be abele to a preference of decision maker; this method examines the preferences of the decision maker without sacrifice the theory. Therefore, this paper presents an approach to finding properly efficient solutions near to the ideal point.
Keywords: Multi-objective optimization, Proper efficiency, trade-off, Ideal point, weighted sum scalarization
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