Search results for: Economic dispatch problem

Authors: Vahid Mohammadi Saffarzadeh, Pourya Jafarzadeh, Masoud Mazloom

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This paper presents a hybrid approach for solving nqueen problem by combination of PSO and SA. PSO is a population based heuristic method that sometimes traps in local maximum. To solve this problem we can use SA. Although SA suffer from many iterations and long time convergence for solving some problems, By good adjusting initial parameters such as temperature and the length of temperature stages SA guarantees convergence. In this article we use discrete PSO (due to nature of n-queen problem) to achieve a good local maximum. Then we use SA to escape from local maximum. The experimental results show that our hybrid method in comparison of SA method converges to result faster, especially for high dimensions n-queen problems.

Keywords: PSO, SA, N-queen, CSP

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Authors: Pankaj Dutta, Debjani Chakraborty

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This paper investigates the effect of product substitution in the single-period 'newsboy-type' problem in a fuzzy environment. It is supposed that the single-period problem operates under uncertainty in customer demand, which is described by imprecise terms and modelled by fuzzy sets. To perform this analysis, we consider the fuzzy model for two-item with upward substitution. This upward substitutability is reasonable when the products can be stored according to certain attribute levels such as quality, brand or package size. We show that the explicit consideration of this substitution opportunity increase the average expected profit. Computational study is performed to observe the benefits of product's substitution.

Keywords: Fuzzy demand, Newsboy, Single-period problem, Substitution.

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Authors: Majid Abdolrazzagh-Nezhad, Salwani Abdullah

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The job-shop scheduling problem (JSSP) is an important decision facing those involved in the fields of industry, economics and management. This problem is a class of combinational optimization problem known as the NP-hard problem. JSSPs deal with a set of machines and a set of jobs with various predetermined routes through the machines, where the objective is to assemble a schedule of jobs that minimizes certain criteria such as makespan, maximum lateness, and total weighted tardiness. Over the past several decades, interest in meta-heuristic approaches to address JSSPs has increased due to the ability of these approaches to generate solutions which are better than those generated from heuristics alone. This article provides the classification, constraints and objective functions imposed on JSSPs that are available in the literature.

Keywords: Job-shop scheduling, classification, constraints, objective functions.

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Authors: Rajeev, N. K. Raigar

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In this study, one dimensional phase change problem (a Stefan problem) is considered and a numerical solution of this problem is discussed. First, we use similarity transformation to convert the governing equations into ordinary differential equations with its boundary conditions. The solutions of ordinary differential equation with the associated boundary conditions and interface condition (Stefan condition) are obtained by using a numerical approach based on operational matrix of differentiation of shifted second kind Chebyshev wavelets. The obtained results are compared with existing exact solution which is sufficiently accurate.

Keywords: Operational matrix of differentiation, Similarity transformation, Shifted second kind Chebyshev wavelets, Stefan problem.

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Authors: Bita Tadayon, Nasser Salmasi

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This research deals with a flexible flowshop scheduling problem with arrival and delivery of jobs in groups and processing them individually. Due to the special characteristics of each job, only a subset of machines in each stage is eligible to process that job. The objective function deals with minimization of sum of the completion time of groups on one hand and minimization of sum of the differences between completion time of jobs and delivery time of the group containing that job (waiting period) on the other hand. The problem can be stated as FFc / rj , Mj / irreg which has many applications in production and service industries. A mathematical model is proposed, the problem is proved to be NPcomplete, and an effective heuristic method is presented to schedule the jobs efficiently. This algorithm can then be used within the body of any metaheuristic algorithm for solving the problem.

Keywords: flexible flowshop scheduling, group processing, machine eligibility constraint, mathematical modeling.

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Authors: S. Balaji, V. Swaminathan, K. Kannan

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The Maximum Weighted Independent Set (MWIS) problem is a classic graph optimization NP-hard problem. Given an undirected graph G = (V, E) and weighting function defined on the vertex set, the MWIS problem is to find a vertex set S V whose total weight is maximum subject to no two vertices in S are adjacent. This paper presents a novel approach to approximate the MWIS of a graph using minimum weighted vertex cover of the graph. Computational experiments are designed and conducted to study the performance of our proposed algorithm. Extensive simulation results show that the proposed algorithm can yield better solutions than other existing algorithms found in the literature for solving the MWIS.

Keywords: weighted independent set, vertex cover, vertex support, heuristic, NP - hard problem.

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Authors: T. Kumaresan, S. Sanjushree, C. Palanisamy

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Image spam is a kind of email spam where the spam text is embedded with an image. It is a new spamming technique being used by spammers to send their messages to bulk of internet users. Spam email has become a big problem in the lives of internet users, causing time consumption and economic losses. The main objective of this paper is to detect the image spam by using histogram properties of an image. Though there are many techniques to automatically detect and avoid this problem, spammers employing new tricks to bypass those techniques, as a result those techniques are inefficient to detect the spam mails. In this paper we have proposed a new method to detect the image spam. Here the image features are extracted by using RGB histogram, HSV histogram and combination of both RGB and HSV histogram. Based on the optimized image feature set classification is done by using k- Nearest Neighbor(k-NN) algorithm. Experimental result shows that our method has achieved better accuracy. From the result it is known that combination of RGB and HSV histogram with k-NN algorithm gives the best accuracy in spam detection.

Keywords: File Type, HSV Histogram, k-NN, RGB Histogram, Spam Detection.

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Authors: Joselito Medina Marin, Norberto Hernández Romero, Juan Carlos Seck Tuoh Mora, Erick S. Martinez Gomez

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The Flow Shop Scheduling Problem (FSSP) is a typical problem that is faced by production planning managers in Flexible Manufacturing Systems (FMS). This problem consists in finding the optimal scheduling to carry out a set of jobs, which are processed in a set of machines or shared resources. Moreover, all the jobs are processed in the same machine sequence. As in all the scheduling problems, the makespan can be obtained by drawing the Gantt chart according to the operations order, among other alternatives. On this way, an FMS presenting the FSSP can be modeled by Petri nets (PNs), which are a powerful tool that has been used to model and analyze discrete event systems. Then, the makespan can be obtained by simulating the PN through the token game animation and incidence matrix. In this work, we present an adaptive PN to obtain the makespan of FSSP by applying PN analytical tools.

Keywords: Flow-shop scheduling problem, makespan, Petri nets, state equation.

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Authors: Kamil Aida-zade, C. Ardil

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In this paper, some problem formulations of dynamic object parameters recovery described by non-autonomous system of ordinary differential equations with multipoint unshared edge conditions are investigated. Depending on the number of additional conditions the problem is reduced to an algebraic equations system or to a problem of quadratic programming. With this purpose the paper offers a new scheme of the edge conditions transfer method called by conditions shift. The method permits to get rid from differential links and multipoint unshared initially-edge conditions. The advantage of the proposed approach is concluded by capabilities of reduction of a parametric identification problem to essential simple problems of the solution of an algebraic system or quadratic programming.

Keywords: dynamic objects, ordinary differential equations, multipoint unshared edge conditions, quadratic programming, conditions shift

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Authors: K. -H. Yang

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Supplier selection problem is one of the important issues of supply chain problems. Two categories of methodologies include qualitative and quantitative approaches which can be applied to supplier selection problems. However, due to the complexities of the problem and lacking of reliable and quantitative data, qualitative approaches are more than quantitative approaches. This study considers operational cost and supplier’s reliability factor and solves the problem by using a quantitative approach. A mixed integer programming model is the primary analytic tool. Analyses of different scenarios with variable cost and reliability structures show that the effectiveness of this approach to the supplier selection problem.

Keywords: Mixed integer programming, quantitative approach, supplier’s reliability, supplier selection.

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Authors: S. Kavitha, Nirmala P. Ratchagar

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This paper presents the novel deterministic dynamic programming approach for solving optimization problem with quadratic objective function with linear equality and inequality constraints. The proposed method employs backward recursion in which computations proceeds from last stage to first stage in a multi-stage decision problem. A generalized recursive equation which gives the exact solution of an optimization problem is derived in this paper. The method is purely analytical and avoids the usage of initial solution. The feasibility of the proposed method is demonstrated with a practical example. The numerical results show that the proposed method provides global optimum solution with negligible computation time.

Keywords: Backward recursion, Dynamic programming, Multi-stage decision problem, Quadratic objective function.

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Authors: Flur R. Ismagilov, Vyacheslav E. Vavilov, Denis V. Gusakov

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This paper is devoted to solve the problem of increasing the electrification of aircraft engines by installing a synchronous generator at high pressure shaft. Technical solution of this problem by various research centers is discussed. A design solution of the problem was proposed. To evaluate the effectiveness of the proposed cooling system, thermal analysis was carried out in ANSYS software.

Keywords: Flur R. Ismagilov, Vyacheslav E. Vavilov, Denis V. Gusakov

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Authors: Y. Egercioglu, S. Yalciner

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With the development of technology, the growing trend of fast and safe passenger transport, air pollution, traffic congestion, increase in problems such as the increasing population and the high cost of private vehicle usage made many cities around the world with a population of more or less, start to build rail systems as a means of urban transport in order to ensure the economic and environmental sustainability and more efficient use of land in the city. The implementation phase of rail systems costs much more than other public transport systems. However, social and economic returns in the long term made these systems the most popular investment tool for planned and developing cities. In our country, the purpose, goals and policies of transportation plans are away from integrity, and the problems are not clearly detected. Also, not defined and incomplete assessment of transportation systems and insufficient financial analysis are the most important cause of failure. Rail systems and other transportation systems to be addressed as a whole is seen as the main factor in increasing efficiency in applications that are not integrated yet in our country to come to this point has led to the problem.

Keywords: Urban Transportation Projects, Urban Light Rail Systems, Urbanization, Izmir.

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Authors: Y. Abdelrazig

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Evolutionary optimization methods such as genetic algorithms have been used extensively for the construction site layout problem. More recently, ant colony optimization algorithms, which are evolutionary methods based on the foraging behavior of ants, have been successfully applied to benchmark combinatorial optimization problems. This paper proposes a formulation of the site layout problem in terms of a sequencing problem that is suitable for solution using an ant colony optimization algorithm. In the construction industry, site layout is a very important planning problem. The objective of site layout is to position temporary facilities both geographically and at the correct time such that the construction work can be performed satisfactorily with minimal costs and improved safety and working environment. During the last decade, evolutionary methods such as genetic algorithms have been used extensively for the construction site layout problem. This paper proposes an ant colony optimization model for construction site layout. A simple case study for a highway project is utilized to illustrate the application of the model.

Keywords: Construction site layout, optimization, ant colony.

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Authors: Zohreh Bang Tavakoli, Shuktika Chatterjee

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This study investigates the fiscal policy impact on countries’ economic growth in developing countries with a different external debt level. The fiscal policy effectiveness has been re-emphasized in the global financial crisis of 2008 with the external debt as its new contemporary driver. Different theories have proposed the economic consequence of fiscal policy, specifically for developing countries. However, fiscal policy literature is lacking research regarding the fiscal policy’s effectiveness with the external debt’s contributions through comprehensive study. Also, high levels of external debt will influence economic growth. Through foreign resources and channel of investment in which high level of debt decreases the amount of foreign investment in the developing countries. The finding of this study suggests that only countries with a low external debt level and appropriate fiscal policies and good quality institutions can gain the proper quantity and quality of foreign investors in which will help the economic growth. For this, this research is examining the impact of fiscal policy on developing countries' economic growth in the situation of different external debt levels.

Keywords: fiscal policy, external debt, gross domestic product, developing countries

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Authors: A. Shirzadeh Chaleshtari, Sh. Shadrokh

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Renewable and non-renewable resource constraints have been vast studied in theoretical fields of project scheduling problems. However, although cumulative resources are widespread in practical cases, the literature on project scheduling problems subject to these resources is scant. So in order to study this type of resources more, in this paper we use the framework of a resource constrained project scheduling problem (RCPSP) with finish-start precedence relations between activities and subject to the cumulative resources in addition to the renewable resources. We develop a branch and bound algorithm for this problem customizing precedence tree algorithm of RCPSP. We perform extensive experimental analysis on the algorithm to check its effectiveness and performance for solving different instances of the problem in question.

Keywords: Resource constrained project scheduling problem, cumulative resources, branch and bound algorithm, precedence tree.

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Authors: Irena Palić

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The main objective of the research in this paper is to empirically assess the causal relationship of private savings and economic growth in the Republic of Croatia. Households’ savings are approximated by household deposits in banks, while domestic income is approximated by industrial production volume indices. Vector Autoregression model and Granger causality tests are used to in order to analyse the relationship among private savings and economic growth. Since ADF unit root tests have shown that both mentioned series are non stationary at levels, series are first differenced in order to become stationary. Therefore, VAR model is estimated with percentage change in private savings and percentage change in domestic income, which can be interpreted as economic growth in case of positive percentage change in domestic income. The Granger causality test has shown that there is no causal relationship among private savings and economic growth in Croatia. The impulse response functions have shown that the impact of shock in domestic income on private savings change is stronger than the impact of private saving on growth. Variance decompositions show that both economic growth and private saving change explain the largest part of its own forecast variance. The research has shown that the link between private savings economic and growth in Croatia is weak, what is in line with relevant empirical research in small open economies.

Keywords: Economic growth, Granger causality, innovation analysis, private savings, Vector Autoregression model.

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Authors: Mevlüt Gül, Hilal Yılmaz, M. Göksel Akpınar, Ayse Karadağ Gürsoy, Özge Bayındır

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Milk is a very important nutrient. Low productivity is a problem of Turkish dairy farming. During recent years, Turkish government has supported cooperatives that assist milk producers and encouraged farmers to become cooperative members. Turkish government established several ways to support specially smallholders. For example Ministry of Agriculture and Rural Affairs (MARA) provided two to four cows to villagers on a grant or loan basis with a long repayment period at low interest rates by cooperatives. Social Support Project in Rural Areas (SSPRA) is another support program targeting only disadvantaged people, especially poor villager. Both programs have a very strong social support component and similar objectives. But there are minor differences between them in terms of target people, terms and conditions of the credit supplied Isparta province in Mediterranean region of Turkey is one of the supported regions. MARA distributed dairy cows to 1072 farmers through 16 agricultural cooperatives in Isparta province in the context of SSPRA. In this study, economic-social impacts on dairy cattle project implemented through cooperatives were examined in Isparta. Primary data were collected from 12 cooperatives- president. The data were obtained by personal interview through a questionnaire and to cooperatives and given to farms benefiting from the project in order to reveal the economic and social developments. Finding of the study revealed that project provided new job opportunities and improved quality of livestock. It was found that producers who benefited from the project were more willing to participate in cooperative or other producer organizations.

Keywords: ooperative, Dairy Cattle, Economic Impact, Livestock Support Project, Social Impact.

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Authors: Ilias A. Makris

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In this work, we attempt to analyze the contribution of innovative activities to firm performance and growth. We examine economic data from some of the economies that were heavily affected by current economic crisis: the countries of southern Europe (Portugal, Italy, Greece and Spain) and Ireland. Following literature, an appropriate econometric model is developed and several indicators are tested in order to disclose possible relation with innovative activity. Findings confirm the crucial effect of innovative process in economic activity, in firm and country level.

Keywords: Eurozone Periphery, Firm Performance, Innovative activity, R&D.

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Authors: Bin Cai, Shilong Wang, Haibo Hu

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The job shop scheduling problem (JSSP) is well known as one of the most difficult combinatorial optimization problems. This paper presents a hybrid genetic algorithm for the JSSP with the objective of minimizing makespan. The efficiency of the genetic algorithm is enhanced by integrating it with a local search method. The chromosome representation of the problem is based on operations. Schedules are constructed using a procedure that generates full active schedules. In each generation, a local search heuristic based on Nowicki and Smutnicki-s neighborhood is applied to improve the solutions. The approach is tested on a set of standard instances taken from the literature and compared with other approaches. The computation results validate the effectiveness of the proposed algorithm.

Keywords: Genetic algorithm, Job shop scheduling problem, Local search, Meta-heuristic algorithm

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Authors: Ludk Benada, Jindika Sedova

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The current situation in the eurozone raises a number of topics for discussion and to help in finding an answer to the question of whether a common currency is a more suitable means of coping with the impact of the financial crisis or whether national currencies are better suited to this. The economic situation in the EU is now considerably volatile and, due to problems with the fulfilment of the Maastricht convergence criteria, it is now being considered whether, in their further development, new member states will decide to distance themselves from the euro or will, in an attempt to overcome the crisis, speed up the adoption of the euro. The Czech Republic is one country with little interest in adopting the euro, justified by the fact that a better alternative to dealing with this crisis is an independent monetary policy and its ability to respond flexibly to the economic situation not only in Europe, but around the world. One attribute of the crisis in the Czech Republic and its mitigation is the freely floating exchange rate of the national currency. It is not only the Czech Republic that is attempting to alleviate the impact of the crisis, but also new EU member countries facing fresh questions to which theory have yet to provide wholly satisfactory answers. These questions undoubtedly include the problem of inflation targeting and the choice of appropriate instruments for achieving financial stability. The difficulty lies in the fact that these objectives may be contradictory and may require more than one means of achieving them. In this respect we may assume that membership of the euro zone might not in itself mitigate the development of the recession or protect the nation from future crises. We are of the opinion that the decisive factor in the development of any economy will continue to be the domestic economic policy and the operability of market economic mechanisms. We attempt to document this fact using selected countries as examples, these being the Czech Republic, Poland, Hungary, and Slovakia.

Keywords: Currency exchange rate, Maastricht convergence criteria, monetary union, public finances.

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Authors: Sayyed R Mousavi

Abstract:

The Far From Most Strings Problem (FFMSP) is to obtain a string which is far from as many as possible of a given set of strings. All the input and the output strings are of the same length, and two strings are said to be far if their hamming distance is greater than or equal to a given positive integer. FFMSP belongs to the class of sequences consensus problems which have applications in molecular biology. The problem is NP-hard; it does not admit a constant-ratio approximation either, unless P = NP. Therefore, in addition to exact and approximate algorithms, (meta)heuristic algorithms have been proposed for the problem in recent years. On the other hand, in the recent years, hybrid algorithms have been proposed and successfully used for many hard problems in a variety of domains. In this paper, a new metaheuristic algorithm, called Constructive Beam and Local Search (CBLS), is investigated for the problem, which is a hybridization of constructive beam search and local search algorithms. More specifically, the proposed algorithm consists of two phases, the first phase is to obtain several candidate solutions via the constructive beam search and the second phase is to apply local search to the candidate solutions obtained by the first phase. The best solution found is returned as the final solution to the problem. The proposed algorithm is also similar to memetic algorithms in the sense that both use local search to further improve individual solutions. The CBLS algorithm is compared with the most recent published algorithm for the problem, GRASP, with significantly positive results; the improvement is by order of magnitudes in most cases.

Keywords: Bioinformatics, Far From Most Strings Problem, Hybrid metaheuristics, Matheuristics, Sequences consensus problems.

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Authors: Adam Roman

Abstract:

Finding synchronizing sequences for the finite automata is a very important problem in many practical applications (part orienters in industry, reset problem in biocomputing theory, network issues etc). Problem of finding the shortest synchronizing sequence is NP-hard, so polynomial algorithms probably can work only as heuristic ones. In this paper we propose two versions of polynomial algorithms which work better than well-known Eppstein-s Greedy and Cycle algorithms.

Keywords: Synchronizing words, reset sequences, Černý Conjecture

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Authors: Libena Cernohorska, Jan Cernohorsky

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This article aims to evaluate the impact of loans provided within the Czech banking sector on the growth of the Czech economy. The article is based on research of current scientific findings in respect to bank loans and economic development. The paper is based on data taken from the Czech Statistical Office on the development of the gross domestic product and data from the Czech National Bank on the development of loans from the period 2004-2015. Links between selected variables are tested using Granger causality tests. The results calculated confirm the hypothesis of the impact of the loans on economic growth, with a six-month delay. The results thus correspond to the standard economic findings and results of most previous studies.

Keywords: Bank, business cycle, economic growth, loans.

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Authors: G. Erkomaishvili

Abstract:

This paper discusses small entrepreneurship development strategy in Georgia and the tools and regulations that will encourage development of small entrepreneurship. The current situation in the small entrepreneurship sector, as well as factors affecting growth and decline in the sector and the priorities of state support, are studied and analyzed. The objective of this research is to assess the current situation of the sector to highlight opportunities and reveal the gaps. State support of small entrepreneurship should become a key priority in the country’s economic policy, as development of the sector will ensure social, economic and political stability. Based on the research, a small entrepreneurship development strategy is presented; corresponding conclusions are made and recommendations are developed.

Keywords: Economic policy for small entrepreneurship development, small entrepreneurship, regulations, small entrepreneurship development strategy.

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Authors: Seyed Habib A. Rahmati

Abstract:

During last decades, developing multi-objective evolutionary algorithms for optimization problems has found considerable attention. Flexible job shop scheduling problem, as an important scheduling optimization problem, has found this attention too. However, most of the multi-objective algorithms that are developed for this problem use nonprofessional approaches. In another words, most of them combine their objectives and then solve multi-objective problem through single objective approaches. Of course, except some scarce researches that uses Pareto-based algorithms. Therefore, in this paper, a new Pareto-based algorithm called controlled elitism non-dominated sorting genetic algorithm (CENSGA) is proposed for the multi-objective FJSP (MOFJSP). Our considered objectives are makespan, critical machine work load, and total work load of machines. The proposed algorithm is also compared with one the best Pareto-based algorithms of the literature on some multi-objective criteria, statistically.

Keywords: Scheduling, Flexible job shop scheduling problem, controlled elitism non-dominated sorting genetic algorithm

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Authors: Priyadarsini P.L.K, Hemalatha T.

Abstract:

In the Top Right Access point Minimum Length Corridor (TRA-MLC) problem [1], a rectangular boundary partitioned into rectilinear polygons is given and the problem is to find a corridor of least total length and it must include the top right corner of the outer rectangular boundary. A corridor is a tree containing a set of line segments lying along the outer rectangular boundary and/or on the boundary of the rectilinear polygons. The corridor must contain at least one point from the boundaries of the outer rectangle and also the rectilinear polygons. Gutierrez and Gonzalez [1] proved that the MLC problem, along with some of its restricted versions and variants, are NP-complete. In this paper, we give a shorter proof of NP-Completeness of TRA-MLC by findig the reduction in the following way.

Keywords: NP-complete, 2-connected planar graph, Grid embedding of a plane graph.

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Authors: Peyman Sindareh Esfahani, Jeffery Kurt Pieper

Abstract:

In this paper, the problem of robust model predictive control (MPC) for discrete-time linear systems in linear fractional transformation form with structured uncertainty and norm-bounded disturbance is investigated. The problem of minimization of the cost function for MPC design is converted to minimization of the worst case of the cost function. Then, this problem is reduced to minimization of an upper bound of the cost function subject to a terminal inequality satisfying the l₂-norm of the closed loop system. The characteristic of the linear fractional transformation system is taken into account, and by using some mathematical tools, the robust predictive controller design problem is turned into a linear matrix inequality minimization problem. Afterwards, a formulation which includes an integrator to improve the performance of the proposed robust model predictive controller in steady state condition is studied. The validity of the approaches is illustrated through a robust control benchmark problem.

Keywords: Linear fractional transformation, linear matrix inequality, robust model predictive control, state feedback control.

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Authors: Hazlina Hamdan, Fatimah Yusof, Dasimah Omar, Faizul Abdullah, Naasah Nasrudin, Ishak Che Abullah

Abstract:

In the globalization context and competitiveness, the role of a university is further enhanced. University is no longer confined to traditional roles. Universities need to interact with others in order to be relevant and progressive. Symbiosis relationships between the university and industry are very significant because the relationship between those two can foster economic development of a nation. In a world of fast changing technology and competition, it is necessary for the university to collaborate with industry to combine efforts fostering the diffusion of knowledge, increasing research and development, patenting innovation and commercializing products. It has become increasingly accepted that the necessity of close university-industry interactions as a mean of national economic prosperity. Therefore, this paper is aim to examine the level of linkages in university-industry interactions to which promotes the regional economic growth and development. This paper will explore the formation of linkages between the Higher Education Institution (University Technology MARA) and industries located in the Klang Valley region of Malaysia. It will present the university-industry linkages with emphasis on the type of linkages existed, the benefits of having such linkages to promote regional economic development and finally the constraints that might impede the linkages and potentials to enhance the linkages towards economic growth and development.

Keywords: Industry, Interaction, Linkages, Regional development, University

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Authors: D.K. Rao, M.G.M. Khan, K.G. Reddy

Abstract:

The focus of this paper is to develop a technique of solving a combined problem of determining Optimum Strata Boundaries(OSB) and Optimum Sample Size (OSS) of each stratum, when the population understudy isskewed and the study variable has a Pareto frequency distribution. The problem of determining the OSB isformulated as a Mathematical Programming Problem (MPP) which is then solved by dynamic programming technique. A numerical example is presented to illustrate the computational details of the proposed method. The proposed technique is useful to obtain OSB and OSS for a Pareto type skewed population, which minimizes the variance of the estimate of population mean.

Keywords: Stratified sampling, Optimum strata boundaries, Optimum sample size, Pareto distribution, Mathematical programming problem, Dynamic programming technique.

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