mixed-integer programming
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Scientia Iranica, Volume:30 Issue: 4, Jul-Aug 2023, PP 1518 -1533This paper focuses on a closed-loop supply chain that deals with disruptions in the distribution centers, and optimizes the network in two dimensions of sustainability: economic and environmental. Economically, the proposed network maximizes the profits of the customers, manufacturers and distributors. Three avenues for cost minimization are designed for the customer by adding the warranty periods, the reworking options, and the incentives for returning the used items. Non-dominated solutions via the Reservation Level-driven Tchebycheff procedure are found by appropriate choice of facility establishment and suitable allocation links considering the disruption in the distribution centers.Environmentally, the model adopts a zero-waste strategy by embedding various return-segmentation policies and a secondary chain. The backward flow depends on the customers' choice of reworking, the validity of the warranty contract, and the quality of the returns. The test results indicate that due to various revenue options, the manufacturing and distribution centers prefer returns with medium-range quality, while due to the incentives offered for the recyclable items, the customers benefit the most from returning the items with the lowest quality. The tests on the probability of disruptions indicate that establishing a minimum number of the manufacturing and/or distribution sites without disruption leads to better overall performance.Keywords: Closed-loop supply chain networks, disruption, facility failure, return quality management, Mixed integer programming, reservation level-driven Tchebycheff procedure
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Data envelopment analysis (DEA) technique is widely applied for performance assessment of decision making units (DMUs). The revenue efficiency (RE) evaluation is one of the controversial subject matters that can be performed through DEA context. The amount of productions and its prices are crucial factors in the RE. The classical DEA models consider the prices to be fixed and known which are not the case in real world. Also, the classical DEA models considers linear pricing in revenue assessment. However, most of real world problems deal with nonlinear prices. This paper evaluates the RE given the piecewise linear theory in non-competitive situations. In doing so, a stepwise pricing function is introduced which lets the prices to be changed in relation to the amount of the production. As an innovative idea, a more accurate mathematical modeling for the RE is proposed. We define a dynamic weights’ function in maximum revenue optimization model which no longer considers fixed prices. A case study validates our proposed model.Keywords: Data envelopment analysis (DEA), Revenue efficiency, Stepwise pricing, Mixed integer programming, Big M, Malmquist productivity index (MPI), Piecewise linear functions
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The quality of public transportation service has major effects on people’s quality of life. During frequency and timetable setting, synchronization is a very important and complicated issue which can directly influence the utility and attractiveness of the system. In this paper, a mixed-integer nonlinear programming model is proposed that aims at setting timetables on a bus transit network with the maximum synchronization and the minimum number of fleet size. The proposed model is shown to be applicable for both small and large-scale transit networks by employing it for setting timetables on two samples of both sizes. As an illustrative example, a simple version of the model is coded and run in GAMS Software and a completely reasonable timetable is obtained. As the second example, the proposed model is used to set timetables on Tehran BRT networks through the genetic algorithm; then the NSGA-II is used to obtain the Pareto optimal solutions of the problem for five different scenarios. The Pareto optimal solutions are used to draw the Pareto optimal fronts which act as an essential decision making tool. The overall results show that the proposed model is efficient enough to be employed setting timetables on transit networks with different sizes.Keywords: Public transportation, Bus line timetable setting, mathematical modelling, mixed-integer programming, Genetic Algorithm
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کنترل پیش بین سیستم های هایبرید با دو چالش اساسی تضمین پایداری حلقه-بسته و همچنین کاهش پیچیدگی محاسباتی روبه رو می باشد. در این مقاله، پایداری نمایی حلقه-بسته سیستم های هایبرید توصیف شده توسط مدل مرکب منطقی دینامیکی توسط کنترل پیش بین مدل تحلیل می شود. برای این منظور، استفاده از شرط نزولی بودن یک تابع لیاپانوف مبتنی بر نرم بی نهایت متغیرهای حالت سیستم، به جای تحمیل قید مساوی نهایی در مسئله کنترل پیش بین مدل سیستم های مرکب منطقی دینامیکی پیشنهاد می شود. شرایط تضمین پایداری نمایی حلقه-بسته با استفاده از روش پیشنهادی دارای عملکرد بهتری هم از نظر پیاده سازی کنترل کننده و هم از نظر پیچیدگی محاسباتی است . علاوه بر این، با استفاده از این روش، شرایط پایداری نمایی حلقه-بسته نقطه تعادل به مقدار افق پیش بینی سیستم وابسته نمی باشد و همین امر می تواند یکی از مهم ترین مزایای این روش در نظر گرفته شود. با استفاده از شرط نزولی بودن تابع لیاپانوف در فرمول بندی کنترل پیش بین مدل برای سیستم های مرکب منطقی دینامیکی، نسخه زیربهینه سیگنال کنترل با حجم محاسباتی بسیار کمتر به دست می آید. به منظور بررسی عملکرد روش پیشنهادشده، مسئله پایدارسازی در سیستم تعلیق خودرو مورد مطالعه و شبیه سازی قرار می گیرد.کلید واژگان: کنترل کننده پیش بین مدل، سیستم های هایبرید مرکب منطقی دینامیکی، پایداری، برنامه ریزی صحیح-مرکبThere are two main challenges in control of hybrid systems which are to guarantee the closed-loop stability and reduce computational complexity. In this paper, we propose the exponential stability conditions of hybrid systems which are described in the Mixed Logical Dynamical (MLD) form in closed-loop with Model Predictive Control (MPC). To do this, it is proposed to use the decreasing condition of infinity norm based Lyapunov function instead of imposing the terminal equality constraint in the MPC formulation of MLD system. The exponential stability conditions have a better performance from both implementation and computational points of view. In addition, the exponential stability conditions of the equilibrium point of the MLD system do not depend on the prediction horizon of MPC problem which is the main advantage of the proposed method. On the other hand, by using the decreasing condition of the Lyapunov function in the MPC setup, the suboptimal version of the control signal with reduced complexity is obtained. In order to show the capabilities of the proposed method, the stabilization problem of the car suspension system is studied.Keywords: model predictive control, Mixed logical dynamical system, Mixed integer programming
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Scientia Iranica, Volume:27 Issue: 5, Sep Oct 2020, PP 2514 -2528Under conditions of consumer panic buying, satisfying demand with the available products is a complex problem. In reality, most retailers accept alternative products during panic situations. This study considers the case of firm-driven substitution of products (differing in weight) based on retailer preferences over two periods. In the proposed model, panic behavior emerged in the first period and supply disruption occurred in the second period. Under this model, retail stores were segmented into high index (valuable) and low index (less valuable) customers. Before meeting the demand of low-index customers, wholesalers attempt to satiate high-index customer’s panic buying behavior. We determined the optimal number of units to be substituted, order quantities, and leftover units that generated maximum total profits for the wholesaler. The performance of the model was analyzed both with and without customer-segmented substitution. To gain managerial insights, we also examined the influence of both the degree of supply disruption and substitution costs on decisions and profits. The results can assist business managers to improve the decision-making process.Keywords: panic buying, product substitution, service level, customer segmentation, mixed-integer programming
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Due to the ever-growing load, especially peak load, the increase in the capacity of plants is inevitable for the response to this growth. Peak load causes increases in customer costs and vast investments in generating and transmission parts. Therefore, restructuring in the electrical industry, competition in the electrical market and Demand Response Programs (DRPs) are of special importance in power systems. In DRPs, customers in certain periods, such as peak or times when the price is high, decrease self-consumption. It means profit for costumers and prevention of expensive production in peak time for a genera-tion source. Moreover, to decrease the operation cost of network and ever-growing technology significantly, the power sys-tems operators have employed new sources of energy production as well as thermal units, and it has led to the emergence of Electric Vehicles (EVs) technology as a new source of energy production. This paper studies the simultaneous presence of DRPs and EVs to minimize the total operation cost of a network from one hand and from the other to improve the level of system reserve in Unit Commitment (UC) problem with considering the security constraint. Here, the proposed framework is structured as a Mixed Integer Programming (MIP) and solved using CPLEX solver.
Keywords: Demand Response, Mixed Integer Programming, security constrained unit commitment, electric vehicles -
Scientia Iranica, Volume:26 Issue: 3, May-Jun 2019, PP 1824 -1841In this paper, we address the surgical case scheduling problem in multi operating theater environment with uncertain service times in order to minimize makespan. In surgical case scheduling, not only the hospital resources are allocated to surgical cases but also the start time of performing surgeries is determined based on sequence of cases in a short-term time horizon. We consider fuzzy numbers for duration times of all stages and hereafter the problem called fuzzy surgical case scheduling. Since the operational environment in the problem is similar to no-wait multi-resource fuzzy flexible job shop problem, we consider constraints of that for formulating and solving problem. This problem is strongly an NP-hard optimization problem, hence we employ ant system algorithm to tackle problem. The proposed approach is illustrated by detailed examples of three test cases, and numerical computational experiments. Therefore, the performance of proposed algorithm is compared with a schedule constructed by first-come-first-service rule on all test instances. Also, a real case is provided from Isfahan’s hospital to evaluate proposed algorithm. Consequently, computational experiments state that algorithm outperforms results obtained by hospital planning as well as fuzzy rule, and these indicate efficiency and capability of our algorithm for optimizing the makespan.Keywords: Surgical case scheduling, Ant System, Operating theater, Fuzzy duration time, Makespan, Mixed integer programming
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Scientia Iranica, Volume:25 Issue: 3, May - June 2018, PP 1750 -1767This study considers a multi-objective combined budget constrained facility location/network design problem (FL/NDP) in which the system uncertainty is considered. The most obvious practical examples of the problem are territorial designing and locating of academies, airline networks, and medical service centers. In order to assure the network reliability versus uncertainty, an efficient robust optimization approach is applied to model the proposed problem. The formulation is minimizing the total expected costs, including, transshipment costs, facility location (FL) costs, fixed cost of road/link utilization as well as minimizing the total penalties of uncovered demand nodes. Then, in order to consider of several system uncertainty, the proposed model is changed to a fuzzy robust model by suitable approaches. An efficient Sub-gradient based Lagrangian relaxation algorithm is applied. In addition, a practical example is studied. At the following, a series of experiments, including several test problems, is designed and solved to evaluate of the performance of the algorithm. The obtained results emphasize that considering of practical factors (e.g., several uncertainties, system disruptions, and customer satisfaction) in modelling of the problem can lead to significant improvement of the system yield and subsequently more efficient utilization of the established network.Keywords: Facility location, Network design, Robust optimization, Mixed integer programming, Fuzzy, Multi-objective, Sub- gradient based Lagrangian relaxation
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برنامه ریزی گیت یکی از فعالیت های کلیدی در فرودگاه هاست که به عنوان یک مساله بهینه سازی تعریف می شود. هدف اصلی این پژوهش پیدا کردن یک تخصیص مناسب برای پروازهای ورودی و خروجی با درنظر گرفتن مجموعه ایی از محدودیت های کاربردی است. یکی از اهدافی که کمتر مورد توجه قرار گرفته است، بالانس نمودن بار کاری گیت ها با استفاده از تعداد مسافران می باشد. در این مقاله، این هدف به همراه دو هدف کمینه کردن تاخیرهای بوجود آمده در زمان تخصیص گیت به هواپیما و بیشینه کردن امتیاز اولویت تخصیص گیت (کنترل ازدحادم مسافران) که تاکنون باهم در نظر گرفته نشده اند، به عنوان اهداف این مساله در نظر گرفته شده است. مساله به شکل برنامه ریزی عدد صحیح مختلط مدل سازی شده است. همچنین این مدل با استفاده از داده های واقعی فرودگاه بین المللی مهرآباد در ابعاد کوچک و متوسط حل شده است. به منظور یافتن مجموعه جواب های پارتو، الگوریتم NSGA-II پیشنهاد و برای نشان دادن کارآیی الگوریتم جواب های بدست آمده در ابعاد کوچک با جواب های بدست آمده از روش محدودیت اپسیلون مقایسه شده است. نتایج نشان می دهد که درصد خطای توابع هدف نسبت به روش محدودیت اپسیلون در تمامی مسایل حل شده کمتر از 1. 5% است که کارآیی الگوریتم پیشنهادی را نشان می دهد. افزایش نمایی زمان حل با استفاده از روش محدودیت اپسیلون در مقابل افزایش خطی توسط NSGA-II نشان دهنده کارآیی روش حل توسعه داده شده، برای حل مساله در ابعاد واقعی و بزرگ است.کلید واژگان: حمل و نقل هوایی، برنامه ریزی گیت، تصمیم گیری چند هدفه، برنامه ریزی عدد صحیح مختلط، الگوریتم NSGA-II، روش محدودیت اپسیلونGate scheduling is a key activity at airports that is proposed as an optimization problem. The main purpose of this problem is to find an assignment for the flights arriving and departing while satisfying a set of practical constraints. Studies show that the gate assignment tables have been used to minimize the gate flights delay and maximize the gate efficiency and productivity. Depending on the situation, different objectives become important. If the load balancing with number of passengers in the gates becomes a bottleneck one has to make sure that the flights are equally spread over the different gates. This load balancing objective function has to be balanced with other objectives, especially minimization total delay time and maximization of the total gate assignment preference score. The related problem is formulated as a mixed-integer programming (MIP). We address this problem using real life data from Mehrabad International Airport for both small and medium size problem. To find the set of Pareto solutions, NSGA-II algorithm is proposed to demonstrate the effectiveness of the solutions which is obtained in small dimensions compared with the results obtained by the method of epsilon constraint. The results show that the percentage of error of objective function compared to epsilon constraint method is less than 1.5% for all problems. Indeed, this shows the efficiency of proposed algorithm which is recommended for solving the medium and large size problem.Keywords: Air Transportation, Gate Scheduling, Multi, objective decision making, Mixed Integer Programming, NSGA, II, Epsilon constraint
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Scientia Iranica, Volume:24 Issue: 4, 2017, PP 2105 -2118This study is concerned with how the quality of perishable products can be improved by shortening the time interval between production and distribution. Since special types of food, such as dairy products, decay fast, the Integration of Production and Distribution Scheduling (IPDS), is investigated. This article deals with a variation of IPDS that contains a short shelf life product; hence, there is no inventory of the product in the process. Once a speci c amount of the product is produced, it must be transported with the least transportation time directly to various customer positions within its limited lifespans to minimize the delivery and tardy costs required to complete producing and distributing of the product to satisfy the demand of customers within the limited deadline. After developing a mixed-integer nonlinear programming model of the problem, because it is NP-hard, an Improved Particle Swarm Optimization (IPSO) is proposed. IPSO performance is compared with commercial optimization software for small-size and moderate-size problems. For large-size ones, it is compared with the genetic algorithm existing in the literature. Computational experiments show the eciency and e ectiveness of the proposed IPSO in terms of both the quality of the solution and the time of achieving
the best solution .Keywords: Production, distribution, Permutation flow shop scheduling, Vehicle routing problem, Integration, Mixed integer programming, particle swarm optimization -
این مقاله، به معرفی یک روش بهینه سازی هیبرید به منظور تحلیل مسائل بهینه سازی مختلط با عدد صحیح می پردازد. مدل هیبرید یاد شده به صورت یک برنامه بهینه سازی دو سطحی، شامل مسئله اصلی و تعدادی زیرمسئله پیاده سازی شده است. در روش پیشنهادی از یک پایگاه داده مجازی به منظور ذخیره سازی مجموعه حالات ارزیابی شده در حین اجرای الگوریتم بهینه سازی استفاده شده است. این پایگاه داده مجازی ضمن سرعت بخشیدن به روند شبیه سازی، امکان به اشتراک گذاشتن و استفاده از آن در برنامه های شبیه ساز با بیش از یک پروسسور را نیز فراهم می نماید. مسئله برنامه ریزی توسعه همزمان تولید و شبکه انتقال به عنوان یک مسئله بهینه سازی مختلط با عدد صحیح دینامیک جهت ارزیابی مدل بهینه سازی هیبرید پیشنهادی مورد توجه قرار گرفته است. نتایج شبیه سازی بر روی شبکه های تست نمونه نشان دهنده کارایی آن در حل مسئله برنامه ریزی بلندمدت در سیستم قدرت می باشد.
کلید واژگان: الگوریتم ژنتیک، برنامه ریزی توسعه، بهینه سازی مختلط با عدد صحیح، پایگاه داده مجازیThis paper presents a hybrid optimization technique in order to solve large scale mixed integer optimization problems. The aforementioned technique is modeled as a two-level optimization problem consists of a master problem and some slave sub-problems. In the proposed method، a virtual database has been considered in line with the master problem to store evaluated cases during simulation. The virtual database accelerates the simulation process and also could be incorporated in multi-processor simulators. Composite generation and transmission expansion planning problem is modeled as a dynamic mixed integer optimization problem has been considered here to evaluate the proposed technique. The simulation results show that the presented method is satisfactory and consistent with the expectation.Keywords: Genetic Algorithm, Expansion Planning, Mixed Integer Programming, Virtual Database, Hybrid Model
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