Search results for: non-stationary stochastic optimization

Authors: Khaled M. El Naggar

Abstract:

This paper introduces a digital method for fault section identification in transmission lines. The method uses digital set of the measured short circuit current to locate faults in electrical power systems. The digitized current is used to construct a set of overdetermined system of equations. The problem is then constructed and solved using the proposed digital optimization technique to find the fault distance. The proposed optimization methodology is an application of simulated annealing optimization technique. The method is tested using practical case study to evaluate the proposed method. The accurate results obtained show that the algorithm can be used as a powerful tool in the area of power system protection.

Keywords: optimization, estimation, faults, measurement, high voltage, simulated annealing

Procedia PDF Downloads 372

Authors: Elham Kiyani, S. Mansour Vaezpour, Javad Tavakoli

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In this paper, we first introduce a new distance function in a linear space not necessarily endowed with a topology. The algebraic concepts of interior and closure are useful to study optimization problems without topology. So, we define Q-weak efficient solutions generated by the algebraic interior of a set Q, where Q is not necessarily convex. Studying nonconvex vector optimization is valuable since, for a convex cone K in topological spaces, we have int(K)=cor(K), which means that topological interior of a convex cone K is equal to the algebraic interior of K. Moreover, we used the scalarization technique including the distance function generated by the vectorial closure of a set to characterize these Q-weak efficient solutions. Scalarization is a useful approach for solving vector optimization problems. This technique reduces the optimization problem to a scalar problem which tends to be an optimization problem with a real-valued objective function. For instance, Q-weak efficient solutions of vector optimization problems can be characterized and computed as solutions of appropriate scalar optimization problems. In the convex case, linear functionals can be used as objective functionals of the scalar problems. But in the nonconvex case, we should present a suitable objective function. It is the aim of this paper to present a new distance function that be useful to obtain sufficient and necessary conditions for Q-weak efficient solutions of general optimization problems via scalarization.

Keywords: weak efficient, algebraic interior, vector closure, linear space

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Authors: Abdelhak Bouchakour, Mustafa Brahami, Layachi Zaghba

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The use of photovoltaic energy offers an inexhaustible supply of energy but also a clean and non-polluting energy, which is a definite advantage. The geographical location of Algeria promotes the development of the use of this energy. Indeed, given the importance of the intensity of the radiation received and the duration of sunshine. For this reason, the objective of our work is to develop a data-processing tool (software) of calculation and optimization of dimensioning of the photovoltaic installations. Our approach of optimization is basing on mathematical models, which amongst other things describe the operation of each part of the installation, the energy production, the storage and the consumption of energy.

Keywords: solar panel, solar radiation, inverter, optimization

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Authors: Dina K. Badawy, Dina M. Ibrahim, Alsayed A. Sallam

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Distributed database is a collection of logically related databases that cooperate in a transparent manner. Query processing uses a communication network for transmitting data between sites. It refers to one of the challenges in the database world. The development of sophisticated query optimization technology is the reason for the commercial success of database systems, which complexity and cost increase with increasing number of relations in the query. Mariposa, query trading and query trading with processing task-trading strategies developed for autonomous distributed database systems, but they cause high optimization cost because of involvement of all nodes in generating an optimal plan. In this paper, we proposed a modification on the autonomous strategy K-QTPT that make the seller’s nodes with the lowest cost have gradually high priorities to reduce the optimization time. We implement our proposed strategy and present the results and analysis based on those results.

Keywords: autonomous strategies, distributed database systems, high priority, query optimization

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Authors: Varun Singh, Mainak Bandyopadhyay, Maharana Pratap Singh

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The shortest path (SP) problem concerns with finding the shortest path from a specific origin to a specified destination in a given network while minimizing the total cost associated with the path. This problem has widespread applications. Important applications of the SP problem include vehicle routing in transportation systems particularly in the field of in-vehicle Route Guidance System (RGS) and traffic assignment problem (in transportation planning). Well known applications of evolutionary methods like Genetic Algorithms (GA), Ant Colony Optimization, Particle Swarm Optimization (PSO) have come up to solve complex optimization problems to overcome the shortcomings of existing shortest path analysis methods. It has been reported by various researchers that PSO performs better than other evolutionary optimization algorithms in terms of success rate and solution quality. Further Geographic Information Systems (GIS) have emerged as key information systems for geospatial data analysis and visualization. This research paper is focused towards the application of PSO for solving the shortest path problem between multiple points of interest (POI) based on spatial data of Allahabad City and traffic speed data collected using GPS. Geovisualization of results of analysis is carried out in GIS.

Keywords: particle swarm optimization, GIS, traffic data, outliers

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Authors: Samia Sadouki Chibani, Abdelkamel Tari

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Web service composition combines available services to provide new functionality. Given the number of available services with similar functionalities and different non functional aspects (QoS), the problem of finding a QoS-optimal web service composition is considered as an optimization problem belonging to NP-hard class. Thus, an optimal solution cannot be found by exact algorithms within a reasonable time. In this paper, a meta-heuristic bio-inspired is presented to address the QoS aware web service composition; it is based on Elephant Herding Optimization (EHO) algorithm, which is inspired by the herding behavior of elephant group. EHO is characterized by a process of dividing and combining the population to sub populations (clan); this process allows the exchange of information between local searches to move toward a global optimum. However, with Applying others evolutionary algorithms the problem of early stagnancy in a local optimum cannot be avoided. Compared with PSO, the results of experimental evaluation show that our proposition significantly outperforms the existing algorithm with better performance of the fitness value and a fast convergence.

Keywords: bio-inspired algorithms, elephant herding optimization, QoS optimization, web service composition

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Authors: Benmoussa Dennai, H. Benslimane, A. Helmaoui

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Photo voltaic conversion is the direct conversion of electromagnetic energy into electrical energy continuously. This electromagnetic energy is the most solar radiation. In this work we performed a computer modelling using AMPS 1D optimization of hetero-junction solar cells GaInP/GaAs configuration for p/ n. We studied the influence of the thickness the base layer in the cell offers on the open circuit voltage, the short circuit current and efficiency.

Keywords: optimization, photovoltaic cell, GaInP / GaAs AMPS-1D, hetetro-junction

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Authors: Ludovic Favre, Thibaut M. Schafer, Jean-Luc Robyr, Elena-Lavinia Niederhäuser

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This paper presents an optimization method based on genetic algorithm for the energy management inside buildings developed in the frame of the project Smart Living Lab (SLL) in Fribourg (Switzerland). This algorithm optimizes the interaction between renewable energy production, storage systems and energy consumers. In comparison with standard algorithms, the innovative aspect of this project is the extension of the smart regulation over three simultaneous criteria: the energy self-consumption, the decrease of greenhouse gas emissions and operating costs. The genetic algorithm approach was chosen due to the large quantity of optimization variables and the non-linearity of the optimization function. The optimization process includes also real time data of the building as well as weather forecast and users habits. This information is used by a physical model of the building energy resources to predict the future energy production and needs, to select the best energetic strategy, to combine production or storage of energy in order to guarantee the demand of electrical and thermal energy. The principle of operation of the algorithm as well as typical output example of the algorithm is presented.

Keywords: building's energy, control system, energy management, energy storage, genetic optimization algorithm, greenhouse gases, modelling, renewable energy

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Authors: Nozar Kishi, Babak Kamrani, Filmon Habte

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Natural hazards such as earthquakes and tropical storms, are very frequent and highly destructive in Japan. Japan experiences, every year on average, more than 10 tropical cyclones that come within damaging reach, and earthquakes of moment magnitude 6 or greater. We have developed stochastic catastrophe models to address the risk associated with the entire suite of damaging events in Japan, for use by insurance, reinsurance, NGOs and governmental institutions. KCC’s (Karen Clark and Company) catastrophe models are procedures constituted of four modular segments: 1) stochastic events sets that would represent the statistics of the past events, hazard attenuation functions that could model the local intensity, vulnerability functions that would address the repair need for local buildings exposed to the hazard, and financial module addressing policy conditions that could estimates the losses incurring as result of. The events module is comprised of events (faults or tracks) with different intensities with corresponding probabilities. They are based on the same statistics as observed through the historical catalog. The hazard module delivers the hazard intensity (ground motion or wind speed) at location of each building. The vulnerability module provides library of damage functions that would relate the hazard intensity to repair need as percentage of the replacement value. The financial module reports the expected loss, given the payoff policies and regulations. We have divided Japan into regions with similar typhoon climatology, and earthquake micro-zones, within each the characteristics of events are similar enough for stochastic modeling. For each region, then, a set of stochastic events is developed that results in events with intensities corresponding to annual occurrence probabilities that are of interest to financial communities; such as 0.01, 0.004, etc. The intensities, corresponding to these probabilities (called CE, Characteristics Events) are selected through a superstratified sampling approach that is based on the primary uncertainty. Region specific hazard intensity attenuation functions followed by vulnerability models leads to estimation of repair costs. Extensive economic exposure model addresses all local construction and occupancy types, such as post-linter Shinand Okabe wood, as well as concrete confined in steel, SRC (Steel-Reinforced Concrete), high-rise.

Keywords: typhoon, earthquake, Japan, catastrophe modelling, stochastic modeling, stratified sampling, loss model, ERM

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Authors: Bohli Afef

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The fifth-generation (5G) wireless systems is featured by extreme densities of cell stations to overcome the higher future demand. Hence, interference management is a crucial challenge in 5G ultra-dense cellular networks. In contrast to the classical inter-cell interference coordination approach, which is no longer fit for the high density of cell-tiers, this paper proposes a novel virtual coordination based on the dynamic common cognitive monitor channel protocol to deal with the inter-cell interference issue. A tractable and flexible model for the coverage probability of a typical user is developed through the use of the stochastic geometry model. The analyses of the performance of the suggested protocol are illustrated both analytically and numerically in terms of coverage probability.

Keywords: ultra dense heterogeneous networks, dynamic common channel protocol, cognitive radio, stochastic geometry, coverage probability

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Authors: Mohammad Reza Ramezani, Shahriyar Afandizadeh

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Since the intersections play a crucial role in traffic delay, it is significant to evaluate them precisely. In this paper, three critical intersections in Tehran (Capital of Iran) had been simulated. The main purpose of this paper was to optimize the public transit delay. The simulation had three different phase in three intersections of Tehran. The first phase was about the current condition of intersection; the second phase was about optimized signal timing and the last phase was about prioritized public transit access. The Aimsun software was used to simulate all phases, and the Synchro software was used to optimization of signals as well. The result showed that the implement of optimization and prioritizing system would reduce about 50% of delay for public transit.

Keywords: transit signal priority, intersection optimization, public transit, simulation

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Authors: Yossiri Ariyakul, Piyakiat Insom, Poonyawat Sangiamkulthavorn, Takamichi Nakamoto

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In this paper, the method of optimization algorithm for sensor network comprised of movable sensor nodes which can be used for odor source localization was proposed. A sensor node is composed of an odor sensor, an anemometer, and a wireless communication module. The odor intensity measured from the sensor nodes are sent to the processor to perform the localization based on optimization algorithm by which the odor source localization map is obtained as a result. The map can represent the exact position of the odor source or show the direction toward it remotely. The proposed method was experimentally validated by creating the odor source localization map using three, four, and five sensor nodes in which the accuracy to predict the position of the odor source can be observed.

Keywords: odor sensor, odor source localization, optimization, sensor network

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Authors: Ahmed Cherif Megri, HossamEldin ElSherif

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In a power plant, the header pipe plays a pivotal role in optimizing the performance of diverse systems by serving as a central conduit for the collection and distribution of steam within the plant. This paper investigates the significance of header pipes within power plant setups, highlighting their critical influence on reliability, efficiency, and the performance of the power plant as a whole. The concept of shape optimization emerges as a crucial factor in power plant design and operation, with the potential to maximize performance while minimizing the use of materials. Shape optimization not only enhances efficiency but also contributes to reducing the environmental footprint of power plant installations. In this paper, we initially developed a methodology designed for optimizing header shapes with the primary goal of reducing the usage of costly new alloy materials and lowering the overall maintenance operation expenses. Secondly, we conducted a case study based on an authentic header sourced from an operational power plant.

Keywords: shape optimization, header, power plant, inconel alloy, CFD, structural optimization

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Authors: Yaping Zhao

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In the present study, the exact solutions for the steady response of quasi-linear systems under non-white wide-band random excitation are considered by means of the stochastic averaging method. The non linearity of the systems contains the power-law damping and the cross-product term of the power-law damping and displacement. The drift and diffusion coefficients of the Fokker-Planck-Kolmogorov (FPK) equation after averaging are obtained by a succinct approach. After solving the averaged FPK equation, the joint probability density function and the marginal probability density function in steady state are attained. In the process of resolving, the eigenvalue problem of ordinary differential equation is handled by integral equation method. Some new results are acquired and the novel method to deal with the problems in nonlinear random vibration is proposed.

Keywords: random vibration, stochastic averaging method, FPK equation, transition probability density

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Authors: Faisel Elramalli, Ibrahim Althomali Amjad Sabbagh, Dhananjay Tambe

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SPEC CPU benchmarks are used to evaluate the performance of CPUs on computer systems. In our project we are going to use SPEC CPU suite that contains several benchmarks running on two different compilers gcc and icc in different optimizations levels to evaluate the performance of a CPU. The motivation of this project is to find out which compiler and in which optimization level makes the CPU reaches the best performance. The results of that evaluation will help users of these compilers to choose the best compiler and optimization level that perform efficiently for their work. In other words, it will give users the best performance of the CPU while doing their works. This project is interesting since it will provide the method used to measure the performance of CPU and how different optimization levels of compilers can help achieve a higher performance. Moreover, it will give a good understanding of how benchmarks are used to evaluate a CPU performance. For the reader, in reality SPEC CPU benchmarks are used to measure the performance of new released CPUs to be compared to other CPUs.

Keywords: SPEC, CPU, GCC, ICC, copilers

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Authors: Ali Dorostkar

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In this paper, a basic schematic of fractional dimensional optimization problem is presented. As will be shown, a method is performed based on a relation between roots and tangent lines of function in fractional dimensions for an arbitrary initial point. It is shown that for each polynomial function with order N at least N tangent lines must be existed in fractional dimensions of 0 < α < N+1 which pass exactly through the all roots of the proposed function. Geometrical analysis of tangent lines in fractional dimensions is also presented to clarify more intuitively the proposed method. Results show that with an appropriate selection of fractional dimensions, we can directly find the roots. Method is presented for giving a different direction of optimization problems by the use of fractional dimensions.

Keywords: tangent line, fractional dimension, root, optimization problem

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Authors: Seyedmahdi Mousavihashemi

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One of the most important subjects of interest in researches is 'improving' which result in various algorithms. In so many geometrical problems we are faced with target functions which should be optimized. In group practices, all the functions’ cooperation lead to convergence. In the study, the optimization algorithm of dense particles is used. Usage of the algorithm improves the given performance norms. The results reveal that usage of swarm algorithm for reinforced particles in designing state feedback improves the given performance norm and in optimized designing of multi-target state feedback controlling, the network will maintain its bearing structure. The results also show that PSO is usable for optimization of state feedback controllers.

Keywords: multi-objective, enhanced, feedback, optimization, algorithm, particle, design

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Authors: Citra Pertiwi, Muhammad Fikruzzaman Rahawarin

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Government of Indonesia had committed to increasing its national defense the budget up to 1,5 percent of GDP. However, the budget increase does not necessarily allocate efficiently and effectively. Using Data Envelopment Analysis (DEA), the operational units of Indonesian Armed Forces are considered as a proxy to measure those two aspects. The bootstrap technique is being used as well to reduce uncertainty in the estimation. Additionally, technological change is being measured as a nonstationary component. Nearly half of the units are being estimated as fully efficient, with less than a third is considered as effective. Longer and larger sets of data might increase the robustness of the estimation in the future.

Keywords: bootstrap, effectiveness, efficiency, DEA, military, Malmquist, technological change

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Authors: Eva Queheille, Franck Taillandier, Nadia Saiyouri

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Waste management has become a major issue in demolition works, because of its environmental impact (energy consumption, resource consumption, pollution…). However, improving waste management requires to take also into account the overall demolition process and to consider demolition main objectives (e.g. cost, delay). Establishing a strategy with these conflicting objectives (economic and environment) remains complex. In order to provide a decision-support for demolition companies, a multi-objective optimization model was developed. In this model, a demolition strategy is computed from a set of 80 decision variables (worker team composition, machines, treatment for each type of waste, choice of treatment platform…), which impacts the demolition objectives. The model has experimented on a real-case study (demolition of several buildings in France). To process the optimization, different optimization algorithms (NSGA2, MOPSO, DBEA…) were tested. Results allow the engineer in charge of this case, to build a sustainable demolition strategy without affecting cost or delay.

Keywords: deconstruction, life cycle assessment, multi-objective optimization, waste management

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Authors: Ken Hampshire, Thomas Mazzuchi, Shahram Sarkani

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As a subset of metaheuristics, nature-inspired optimization algorithms such as particle swarm optimization (PSO) have shown promise both in solving intractable problems and in their extensibility to novel problem formulations due to their general approach requiring few assumptions. Unfortunately, single instantiations of algorithms require detailed tuning of parameters and cannot be proven to be best suited to a particular illustrative problem on account of the “no free lunch” (NFL) theorem. Using these algorithms in real-world problems requires exquisite knowledge of the many techniques and is not conducive to reconciling the various approaches to given classes of problems. This research aims to present a unified view of PSO-based approaches from the perspective of relevant systems engineering problems, with the express purpose of then eliciting the best solution for any problem formulation in an ensemble learning bucket of models approach. The central hypothesis of the research is that extending the PSO algorithms found in the literature to real-world optimization problems requires a general ensemble-based method for all problem formulations but a specific implementation and solution for any instance. The main results are a problem-based literature survey and a general method to find more globally optimal solutions for any systems engineering optimization problem.

Keywords: particle swarm optimization, nature-inspired optimization, metaheuristics, systems engineering, ensemble learning

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Authors: Benmoussa Dennai, H. Benslimane, A. Helmaoui

Abstract:

Photovoltaic conversion is the direct conversion of electromagnetic energy into electrical energy continuously. This electromagnetic energy is the most solar radiation. In this work we performed a computer modelling using AMPS 1D optimization of hetero-junction solar cells GaInP / GaAs configuration for p / n. We studied the influence of the thickness the base layer in the cell offers on the open circuit voltage, the short circuit current and efficiency.

Keywords: optimization, photovoltaic cell, GaInP / GaAs AMPS-1D, hetetro-junction

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Authors: Benalia Nadia, Zerzouri Nora, Ben Si Ali Nadia

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Abstract: Among the many different modern heuristic optimization methods, genetic algorithms (GA) and the particle swarm optimization (PSO) technique have been attracting a lot of interest. The GA has gained popularity in academia and business mostly because to its simplicity, ability to solve highly nonlinear mixed integer optimization problems that are typical of complex engineering systems, and intuitiveness. The mechanics of the PSO methodology, a relatively recent heuristic search tool, are modeled after the swarming or cooperative behavior of biological groups. It is suitable to compare the performance of the two techniques since they both aim to solve a particular objective function but make use of distinct computing methods. In this article, PSO and GA optimization approaches are used for the parameter tuning of the power system stabilizer and Proportional integral derivative regulator. Load angle and rotor speed variations in the single machine infinite bus bar system is used to measure the performance of the suggested solution.

Keywords: SMIB, genetic algorithm, PSO, transient stability, power system stabilizer, PID

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Authors: Mahmoud Nadir, Adel Ghenaiet

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The main purpose of this study is the thermodynamic modeling, the parametric analysis, and the optimization of three pressure level reheat HRSG (Heat Recovery Steam Generator) using PSO method (Particle Swarm Optimization). In this paper, a parametric analysis followed by a thermodynamic optimization is presented. The chosen objective function is the specific work of the steam cycle that may be, in the case of combined cycle (CC), a good criterion of thermodynamic performance analysis, contrary to the conventional steam turbines in which the thermal efficiency could be also an important criterion. The technologic constraints such as maximal steam cycle temperature, minimal steam fraction at steam turbine outlet, maximal steam pressure, minimal stack temperature, minimal pinch point, and maximal superheater effectiveness are also considered. The parametric analyses permitted to understand the effect of design parameters and the constraints on steam cycle specific work variation. PSO algorithm was used successfully in HRSG optimization, knowing that the achieved results are in accordance with those of the previous studies in which genetic algorithms were used. Moreover, this method is easy to implement comparing with the other methods.

Keywords: combined cycle, HRSG thermodynamic modeling, optimization, PSO, steam cycle specific work

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Authors: Kenza Aida Amara, Bachir Djebbar

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We treat the two-dimensional bin packing problem which involves packing a given set of rectangles into a minimum number of larger identical rectangles called bins. This combinatorial problem is NP-hard. We propose a pretreatment for the oriented version of the problem that allows the valorization of the lost areas in the bins and the reduction of the size problem. A heuristic method based on the strategy first-fit adapted to this problem is presented. We present an approach of resolution by bee colony optimization. Computational results express a comparison of the number of bins used with and without pretreatment.

Keywords: bee colony optimization, bin packing, heuristic algorithm, pretreatment

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Authors: Vincenzo Capalbo, Gianpaolo Ghiani, Emanuele Manni

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Global e-commerce sales have reached unprecedented levels in the past few years. As this trend is only predicted to go up as we continue into the ’20s, new challenges will be faced by companies when planning and controlling e-commerce logistics. In this paper, we survey the related literature on Optimization and Machine Learning as well as on combined methodologies. We also identify the distinctive features of next-generation planning algorithms - namely scalability, model-and-run features and learning capabilities - that will be fundamental to cope with the scale and complexity of logistics in the next decade.

Keywords: e-commerce, hardware acceleration, logistics, machine learning, mixed integer programming, optimization

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Authors: Amit Ghosh, Chanchal Kundu

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Recently, the notion of past cumulative inaccuracy (CPI) measure has been proposed in the literature as a generalization of cumulative past entropy (CPE) in univariate as well as bivariate setup. In this paper, we introduce the notion of CPI of order α (alpha) and study the proposed measure for conditionally specified models of two components failed at different time instants called generalized conditional CPI (GCCPI). We provide some bounds using usual stochastic order and investigate several properties of GCCPI. The effect of monotone transformation on this proposed measure has also been examined. Furthermore, we characterize some bivariate distributions under the assumption of conditional proportional reversed hazard rate model. Moreover, the role of GCCPI in reliability modeling has also been investigated for a real-life problem.

Keywords: cumulative past inaccuracy, marginal and conditional past lifetimes, conditional proportional reversed hazard rate model, usual stochastic order

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Authors: Liping Yang, Curran Crawford, Jr. Ren, Zhengyi Ren

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Considering the cost evaluation and the stress analysis, a fuzzy satisfactory optimization (FSO) method has been developed for a hybrid composite flywheel. To evaluate the cost, the cost coefficients of the flywheel components are obtained through calculating the weighted sum of the scores of the material manufacturability, the structure character, and the material price. To express the satisfactory degree of the energy, the cost, and the mass, the satisfactory functions are proposed by using the decline function and introducing a satisfactory coefficient. To imply the different significance of the objectives, the object weight coefficients are defined. Based on the stress analysis of composite material, the circumferential and radial stresses are considered into the optimization formulation. The simulations of the FSO method with different weight coefficients and storage energy density optimization (SEDO) method of a flywheel are contrasted. The analysis results show that the FSO method can satisfy different requirements of the designer and the FSO method with suitable weight coefficients can replace the SEDO method.

Keywords: flywheel energy storage, fuzzy, optimization, stress analysis

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Authors: Fábio A. S. Mota, Mauro A. S. S. Ravagnani, E. P. Carvalho

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In the present paper the design of plate heat exchangers is formulated as an optimization problem considering two mathematical modeling. The number of plates is the objective function to be minimized, considering implicitly some parameters configuration. Screening is the optimization method used to solve the problem. Thermal and hydraulic constraints are verified, not viable solutions are discarded and the method searches for the convergence to the optimum, case it exists. A case study is presented to test the applicability of the developed algorithm. Results show coherency with the literature.

Keywords: plate heat exchanger, optimization, modeling, simulation

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Authors: Epaminondas G. Kyriakidis, Theodosis D. Dimitrakos, Constantinos C. Karamatsoukis

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The vehicle routing problem (VRP) is a well-known problem in Operations Research and has been widely studied during the last fifty-five years. The context of the VRP is that of delivering products located at a central depot to customers who are scattered in a geographical area and have placed orders for these products. A vehicle or a fleet of vehicles start their routes from the depot and visit the customers in order to satisfy their demands. Special attention has been given to the capacitated VRP in which the vehicles have limited carrying capacity of the goods that must be delivered. In the present work, we present a specific capacitated stochastic vehicle routing problem which has realistic applications to distributions of materials to shops or to healthcare facilities or to military units. A vehicle starts its route from a depot loaded with items of two similar but not identical products. We name these products, product 1 and product 2. The vehicle must deliver the products to N customers according to a predefined sequence. This means that first customer 1 must be serviced, then customer 2 must be serviced, then customer 3 must be serviced and so on. The vehicle has a finite capacity and after servicing all customers it returns to the depot. It is assumed that each customer prefers either product 1 or product 2 with known probabilities. The actual preference of each customer becomes known when the vehicle visits the customer. It is also assumed that the quantity that each customer demands is a random variable with known distribution. The actual demand is revealed upon the vehicle’s arrival at customer’s site. The demand of each customer cannot exceed the vehicle capacity and the vehicle is allowed during its route to return to the depot to restock with quantities of both products. The travel costs between consecutive customers and the travel costs between the customers and the depot are known. If there is shortage for the desired product, it is permitted to deliver the other product at a reduced price. The objective is to find the optimal routing strategy, i.e. the routing strategy that minimizes the expected total cost among all possible strategies. It is possible to find the optimal routing strategy using a suitable stochastic dynamic programming algorithm. It is also possible to prove that the optimal routing strategy has a specific threshold-type structure, i.e. it is characterized by critical numbers. This structural result enables us to construct an efficient special-purpose dynamic programming algorithm that operates only over those routing strategies having this structure. The findings of the present study lead us to the conclusion that the dynamic programming method may be a very useful tool for the solution of specific vehicle routing problems. A problem for future research could be the study of a similar stochastic vehicle routing problem in which the vehicle instead of delivering, it collects products from ordered customers.

Keywords: collection of similar products, dynamic programming, stochastic demands, stochastic preferences, vehicle routing problem

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Authors: Jyotsna Rosario, K. R. Shanmugam

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Since elementary education is a merit good, considerable public resources are allocated to universalise it. However, elementary education outcomes vary across the Indian States. Evidences indicate that while some states are lagging in elementary education outcome primarily due to lack of resources and poor schooling infrastructure, others are lagging despite resource abundance and well-developed schooling infrastructure. Addressing the issue of efficiency, the study employs Stochastic Frontier Analysis for panel data of 27 Indian states from 2012-13 to 2017-18 to estimate the technical efficiency of State governments in generating enrolment. The mean efficiency of states was estimated to be 58%. Punjab, Meghalaya, and West Bengal were found to be the most efficient states. Whereas Jammu and Kashmir, Nagaland, Madhya Pradesh, and Odisha are one of the most inefficient states. This study emphasizes the efficient utilisation of public resources and helps in the identification of best practices.

Keywords: technical efficiency, public expenditure, elementary education outcome, stochastic frontier analysis

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