Search results for: integer factorization problem

Authors: Amin Karimi, Lele Zhang, Mark Fackrell

Abstract:

Rural-to-urban population migrations are a global phenomenon, with projections indicating that by 2050, 68% of the world's population will inhabit densely populated urban centers. Concurrently, the popularity of e-commerce shopping has surged, evidenced by a 51% increase in total e-commerce sales from 2017 to 2021. Consequently, distribution and logistics systems, integral to effective supply chain management, confront escalating hurdles in efficiently delivering and distributing products within bustling urban environments. Additionally, events like environmental challenges and the COVID-19 pandemic have indicated that decision-makers are facing numerous sources of uncertainty. Therefore, to design an efficient and reliable logistics system, uncertainty must be considered. In this study, it examine fleet sizing and routing while considering uncertainty in demand rate. Fleet sizing is typically a strategic-level decision, while routing is an operational-level one. In this study, a carrier must make two types of decisions: strategic-level decisions regarding the number and types of drones to be purchased, and operational-level decisions regarding planning routes based on available fleet and realized demand. If the available fleets are insufficient to serve some customers, the carrier must outsource that delivery at a relatively high cost, calculated per order. With this hierarchy of decisions, it can model the problem using two-stage stochastic programming. The first-stage decisions involve planning the number and type of drones to be purchased, while the second-stage decisions involve planning routes. To solve this model, it employ logic-based benders decomposition, which decomposes the problem into a master problem and a set of sub-problems. The master problem becomes a mixed integer programming model to find the best fleet sizing decisions, and the sub-problems become capacitated vehicle routing problems considering battery status. Additionally, it assume a heterogeneous fleet based on load and battery capacity, and it consider that battery health deteriorates over time as it plan for multiple periods.

Keywords: drone-delivery, stochastic demand, VRP, fleet sizing

Procedia PDF Downloads 34

Authors: Ching-Shoei Chiang

Abstract:

The Malfatti’s Problem solves the problem of fitting 3 circles into a right triangle such that these 3 circles are tangent to each other, and each circle is also tangent to a pair of the triangle’s sides. This problem has been extended to any triangle (called general Malfatti’s Problem). Furthermore, the problem has been extended to have 1+2+…+n circles inside the triangle with special tangency properties among circles and triangle sides; we call it extended general Malfatti’s problem. In the extended general Malfatti’s problem, call it Tri(Tn), where Tn is the triangle number, there are closed-form solutions for Tri(T₁) (inscribed circle) problem and Tri(T₂) (3 Malfatti’s circles) problem. These problems become more complex when n is greater than 2. In solving Tri(Tn) problem, n>2, algorithms have been proposed to solve these problems numerically. With a similar idea, this paper proposed an algorithm to find the radii of circles with the same tangency properties. Instead of the boundary of the triangle being a straight line, we use a convex circular arc as the boundary and try to find Tn circles inside this convex circular triangle with the same tangency properties among circles and boundary Carc. We call these problems the Carc(Tn) problems. The CPU time it takes for Carc(T16) problem, which finds 136 circles inside a convex circular triangle with specified tangency properties, is less than one second.

Keywords: circle packing, computer-aided geometric design, geometric constraint solver, Malfatti’s problem

Procedia PDF Downloads 85

Authors: A. Chillali, M. Sahmoudi

Abstract:

In this paper we will give a method for encoding the elements of the ring of integers of sextic extension, namely L = Q(a,b) which is a rational quadratic over cubic field K =Q(a) where a^{2} is a rational square free integer and b is a root of irreducible polynomiale of degree 3.

Keywords: coding, integral bases, sextic, quadratic

Procedia PDF Downloads 273

Authors: Noppadon Phumeechanya, Panita Wannapiroon

Abstract:

The purpose of this research is to design the ubiquitous scaffold learning environment using problem-based learning activities that enhance problem-solving skills and context awareness, and to evaluate the suitability of the ubiquitous scaffold learning environment using problem-based learning activities. We divide the research procedures into two phases. The first phase is to design the ubiquitous scaffold learning environment using problem-based learning activities, and the second is to evaluate the ubiquitous scaffold learning environment using problem-based learning activities. The sample group in this study consists of five experts selected using the purposive sampling method. We analyse data by arithmetic mean and standard deviation. The research findings are as follows; the ubiquitous scaffold learning environment using problem-based learning activities consists of three major steps, the first is preparation before learning. This prepares learners to acknowledge details and learn through u-LMS. The second is the learning process, where learning activities happen in the ubiquitous learning environment and learners learn online with scaffold systems for each step of problem solving. The third step is measurement and evaluation. The experts agree that the ubiquitous scaffold learning environment using problem-based learning activities is highly appropriate.

Keywords: ubiquitous learning environment scaffolding, learning activities, problem-based learning, problem-solving skills, context awareness

Procedia PDF Downloads 480

Authors: Birome Holo Ba, Christian Prins, Caroline Prodhon

Abstract:

Optimization is an important aspect of logistics management. It can reduce significantly logistics costs and also be a good tool for decision support. In this paper, we address a planning problem specific to biomass supply chain. We propose a new mixed integer linear programming (MILP) model dealing with different feed stock production operations such as harvesting, packing, storage, pre-processing and transportation, with the objective of minimizing the total logistic cost of the system on a regional basis. It determines the optimal number of harvesting machine, the fleet size of trucks for transportation and the amount of each type of biomass harvested, stored and pre-processed in each period to satisfy demands of refineries in each period. We illustrate the effectiveness of the proposal model with a numerical example, a case study in Aube (France department), which gives preliminary and interesting, results on a small test case.

Keywords: biomass logistics, supply chain, modelling, optimization, bioenergy, biofuels

Procedia PDF Downloads 492

Authors: Kamariah Abu Bakar, Jennifer Way

Abstract:

This study investigated how young children (six years old) constructed and used representations in mathematics classroom; particularly in problem solving. The purpose of this study is to explore the ways children used representations in solving addition problems and to determine whether their representations can play a supportive role in understanding the problem situation and solving them correctly. Data collection includes observations, children’s artifact, photographs and conversation with children during task completion. The results revealed that children were able to construct and use various representations in solving problems. However, they have certain preferences in generating representations to support their problem solving.

Keywords: young children, representations, addition, problem solving

Procedia PDF Downloads 434

Authors: Muhammad Fiaz

Abstract:

The article in hand is the study of complex features like Zero Hopf Bifurcation, Chaos and Synchronization of integer and fractional order version of a new 3D finance system. Trusted tools of averaging theory and active control method are utilized for investigation of Zero Hopf bifurcation and synchronization for both versions respectively. Inventiveness of the paper is to find the answer of a question that is it possible to find a chaotic system which can be synchronized with any other of the same dimension? Based on different examples we categorically develop a theory that if a couple of master and slave chaotic dynamical system is synchronized by selecting a suitable gain matrix with special conditions then the master system is synchronized with any chaotic dynamical system of the same dimension. With the help of this study we developed generalized theorems for synchronization of n-dimension dynamical systems for integer as well as fractional versions. it proposed that this investigation will contribute a lot to control dynamical systems and only a suitable gain matrix with special conditions is enough to synchronize the system under consideration with any other chaotic system of the same dimension. Chaotic properties of fractional version of the new finance system are also analyzed at fractional order q=0.87. Simulations results, where required, also provided for authenticity of analytical study.

Keywords: complex analysis, chaos, generalized synchronization, control dynamics, fractional order analysis

Procedia PDF Downloads 40

Authors: Liudmyla Koliechkina, Olena Dvirna

Abstract:

The statement of the multi-objective optimization problem on combinatorial configurations is formulated, and the approach to its solution is proposed. The problem is of interest as a combinatorial optimization one with many criteria, which is a model of many applied tasks. The approach to solving the multi-objective optimization problem on combinatorial configurations consists of two stages; the first is the reduction of the multi-objective problem to the single criterion based on existing multi-objective optimization methods, the second stage solves the directly replaced single criterion combinatorial optimization problem by the horizontal combinatorial method. This approach provides the optimal solution to the multi-objective optimization problem on combinatorial configurations, taking into account additional restrictions for a finite number of steps.

Keywords: discrete set, linear combinatorial optimization, multi-objective optimization, Pareto solutions, partial permutation set, structural graph

Procedia PDF Downloads 147

Authors: Jiayuan Wu. Lu Hu

Abstract:

With the development of the sharing economy and mobile technology, carsharing becomes more popular. In this paper, we focus on the joint optimization of one-way station-based carsharing systems. We model the problem as an integer linear program with six elements: station locations, station capacity, fleet size, initial vehicle allocation, vehicle relocation, and demand selection. A greedy-based heuristic is proposed to address the model. Firstly, initialization based on the location variables relaxation using Gurobi solver is conducted. Then, according to the profit margin and demand satisfaction of each station, the number of stations is downsized iteratively. This method is applied to real data from Chengdu, Sichuan taxi data, and it’s efficient when dealing with a large scale of candidate stations. The result shows that with vehicle relocation and demand selection, the profit and demand satisfaction of carsharing systems are increased.

Keywords: one-way carsharing, location, vehicle relocation, demand selection, greedy algorithm

Procedia PDF Downloads 110

Authors: Andrés I. Ávila, Patricia Aros, César San Martín, Elizabeth Kehr, Yovana Leal

Abstract:

Planning models for fresh products is a very useful tool for improving the net profits. To get an efficient supply chain model, several functions should be considered to get a complete simulation of several operational units. We consider a linear programming model to help farmers to decide if it is convenient to choose what area should be planted for three kinds of export fruits considering their future investment. We consider area, investment, water, productivity minimal unit, and harvest restrictions to develop a monthly based model to compute the average income in five years. Also, conditions on the field as area, water availability, and initial investment are required. Using the Chilean costs and dollar-peso exchange rate, we can simulate several scenarios to understand the possible risks associated to this market. Also, this tool help to support decisions for government and individual farmers.

Keywords: mixed integer problem, fresh fruit production, support decision model, agricultural and biosystems engineering

Procedia PDF Downloads 415

Authors: Shih Yu Pan, Pao Chen Hung, Chuan Yao Lin, Charles C.-K. Chou, Yu Chi Lin, Kai Hsien Chi

Abstract:

This study analyzed 16 atmospheric PAHs species which were controlled by USEPA and IARC. To measure the concentration of PAHs, four rural sampling sites and two urban sampling sites were selected in Central Taiwan during spring and summer. In central Taiwan, the rural sampling stations were located in the downstream of Da-An River, Da-Jang River, Wu River and Chuo-shui River. On the other hand, the urban sampling sites were located in Taichung district and close to the roadside. Ambient air samples of both vapor phase and particle phase of PAHs compounds were collected using high volume sampling trains (Analitica). The sampling media were polyurethane foam (PUF) with XAD2 and quartz fiber filters. Diagnostic ratio, Principal component analysis (PCA), Positive Matrix Factorization (PMF) models were used to evaluate the apportionment of PAHs in the atmosphere and speculate the relative contribution of various emission sources. Because of the high temperature and low wind speed, high PAHs concentration in the atmosphere was observed. The total PAHs concentration, especially in vapor phase, had significant change during summer. During the sampling periods the total PAHs concentration of atmospheric at four rural and two urban sampling sites in spring and summer were 3.70±0.40 ng/m3,3.40±0.63 ng/m3,5.22±1.24 ng/m3,7.23±0.37 ng/m3,7.46±2.36 ng/m3,6.21±0.55 ng/m3　; 15.0± 0.14 ng/m3,18.8±8.05 ng/m3,20.2±8.58 ng/m3,16.1±3.75 ng/m3,29.8±10.4 ng/m3,35.3±11.8 ng/m3, respectively. In order to identify PAHs sources, we used diagnostic ratio to classify the emission sources. The potential sources were diesel combustion and gasoline combustion in spring and summer, respectively. According to the principal component analysis (PCA), the PC1 and PC2 had 23.8%, 20.4% variance and 21.3%, 17.1% variance in spring and summer, respectively. Especially high molecular weight PAHs (BaP, IND, BghiP, Flu, Phe, Flt, Pyr) were dominated in spring when low molecular weight PAHs (AcPy, Ant, Acp, Flu) because of the dominating high temperatures were dominated in the summer. Analysis by using PMF model found the sources of PAHs in spring were stationary sources (34%), vehicle emissions (24%), coal combustion (23%) and petrochemical fuel gas (19%), while in summer the emission sources were petrochemical fuel gas (34%), the natural environment of volatile organic compounds (29%), coal combustion (19%) and stationary sources (18%).

Keywords: PAHs, source identification, diagnostic ratio, principal component analysis, positive matrix factorization

Procedia PDF Downloads 249

Authors: M. G. M. Khan, V. D. Prasad, D. K. Rao

Abstract:

In this manuscript, we discuss the problem of determining the optimum stratification of a study (or main) variable based on the auxiliary variable that follows a uniform distribution. If the stratification of survey variable is made using the auxiliary variable it may lead to substantial gains in precision of the estimates. This problem is formulated as a Nonlinear Programming Problem (NLPP), which turn out to multistage decision problem and is solved using dynamic programming technique.

Keywords: auxiliary variable, dynamic programming technique, nonlinear programming problem, optimum stratification, uniform distribution

Procedia PDF Downloads 306

Authors: Guilherme Baldo Carlos

Abstract:

The flexible job-shop scheduling problem (FJSP) is an NP-hard combinatorial optimization problem, which can be applied to model several applications in a wide array of industries. This problem will have its importance increase due to the shift in the production mode that modern society is going through. The demands are increasing and for products personalized and customized. This work aims to apply a meta-heuristic called a genetic algorithm (GA) to solve this problem. A GA is a meta-heuristic inspired by the natural selection of Charles Darwin; it produces a population of individuals (solutions) and selects, mutates, and mates the individuals through generations in order to find a good solution for the problem. The results found indicate that the GA is suitable for FJSP solving.

Keywords: genetic algorithm, evolutionary algorithm, scheduling, flexible job-shop scheduling

Procedia PDF Downloads 122

Authors: Laissa Mae Francisco, John Rolex Ingreso, Anna Krizel Menguito, Criselda Robrigado, Rej Maegan Tuazon

Abstract:

This study aims to determine and describe the existing relationship between number sense proficiency and problem-solving performance of grade seven students from Victorino Mapa High School, Manila. A paper pencil exam containing of 50-item number sense test and 5-item problem-solving test which measures their number sense proficiency and problem-solving performance adapted from McIntosh, Reys, and Bana were used as the research instruments. The data obtained from this study were interpreted and analyzed using the Pearson – Product Moment Coefficient of Correlation to determine the relationship between the two variables. It was found out that students who were low in number sense proficiency tend to be the students with poor problem-solving performance and students with medium number sense proficiency are most likely to have an average problem-solving performance. Likewise, students with high number sense proficiency are those who do excellently in problem-solving performance.

Keywords: number sense, performance, problem solving, proficiency

Procedia PDF Downloads 401

Authors: Pei Chin Lim

Abstract:

School of Mathematics and Science of Singapore Polytechnic offers a Basic Mathematics module to students who did not pass GCE O-Level Additional Mathematics. These students are weaker in Mathematics. In particular, they struggle with word problems and tend to leave them blank in tests and examinations. In order to improve students’ problem-solving skills, the school redesigned the Basic Mathematics module to incorporate Polya’s problem-solving methodology. During tutorial lessons, students have to work through learning activities designed to raise their metacognitive awareness by following Polya’s problem-solving process. To assess the effectiveness of the redesign, students’ working for a challenging word problem in the mid-semester test were analyzed. Sixty-five percent of students attempted to understand the problem by making sketches. Twenty-eight percent of students went on to devise a plan and implement it. Only five percent of the students still left the question blank. These preliminary results suggest that with regular exposure to an explicit and systematic problem-solving approach, weak students’ problem-solving skills can potentially be improved.

Keywords: mathematics education, metacognition, problem solving, weak students

Procedia PDF Downloads 138

Authors: Hashem Sazegar

Abstract:

Legendre’s conjecture states that there is a prime number between n^2 and (n + 1)^2 for every positive integer n. In this paper we prove that every composite number between n2 and (n + 1)2 can be written u^2 − v^2 or u^2 − v^2 + u − v that u > 0 and v ≥ 0. Using these result as well as induction and residues (modq) we prove Legendre’s conjecture.

Keywords: bertrand-chebyshev theorem, landau’s problems, goldbach’s conjecture, twin prime, ramanujan proof

Procedia PDF Downloads 336

Authors: Wennan Long, Yuhao Nie, Yunan Li, Adam Brandt

Abstract:

To fight against climate change, California government issued the Senate Bill No. 100 (SB-100) in 2018 September, which aims at achieving a target of 100% renewable electricity by the end of 2045. A capacity expansion problem is solved in this case study using a binary quadratic programming model. The optimal locations and capacities of the potential renewable power plants (i.e., solar, wind, biomass, geothermal and hydropower), the phase-out schedule of existing fossil-based (nature gas) power plants and the transmission of electricity across the entire network are determined with the minimal total annualized cost measured by net present value (NPV). The results show that the renewable electricity contribution could increase to 85.9% by 2030 and reach 100% by 2035. Fossil-based power plants will be totally phased out around 2035 and solar and wind will finally become the most dominant renewable energy resource in California electricity mix.

Keywords: 100% renewable electricity, California, capacity expansion, mixed integer non-linear programming

Procedia PDF Downloads 152

Authors: Derkaoui Orkia, Lehireche Ahmed

Abstract:

The topic of this article is to exploring the potentialities of a powerful optimization technique, namely Semidefinite Programming, for solving NP-hard problems. This approach provides tight relaxations of combinatorial and quadratic problems. In this work, we solve the maximum clique problem using this relaxation. The clique problem is the computational problem of finding cliques in a graph. It is widely acknowledged for its many applications in real-world problems. The numerical results show that it is possible to find a maximum clique in polynomial time, using an algorithm based on semidefinite programming. We implement a primal-dual interior points algorithm to solve this problem based on semidefinite programming. The semidefinite relaxation of this problem can be solved in polynomial time.

Keywords: semidefinite programming, maximum clique problem, primal-dual interior point method, relaxation

Procedia PDF Downloads 197

Authors: Mehmet Hakan Karaata

Abstract:

In this paper, we first coin a new graph theocratic problem with numerous applications. Second, we provide two algorithms for the problem. The first solution is using a brute-force techniques, whereas the second solution is based on an initial identification of the cycles in the given graph. We then provide a correctness proof of the algorithm. The applications of the problem include graph analysis, graph drawing and network structuring.

Keywords: algorithm, cycle, graph algorithm, graph theory, network structuring

Procedia PDF Downloads 364

Authors: Hashini E. Mohottala

Abstract:

We report the combine use of Wikispaces (WikiS) and Group Problem Solving (GPS) sessions conducted in the introductory level physics classes. As a part of this new teaching tool, some essay type problems were posted on the WikiS in weekly basis and students were encouraged to participate in problem solving without providing numerical final answers but the steps. Wikispace is used as a platform for students to meet online and create discussions. Each week students were further evaluated on problem solving skills opening up more opportunity for peer interaction through GPS. Each group was given a different problem to solve and the answers were graded. Students developed a set of skills in decision-making, problem solving, communication, negotiation, critical and independent thinking and teamwork through the combination of WikiS and GPS.

Keywords: group problem solving (GPS), wikispace (WikiS), physics education, learning

Procedia PDF Downloads 391

Authors: Amauri Josafat Gomez Aguilar, Jean Pierre Kenné

Abstract:

This research presents the development of a self-sustainable manufacturing system from a circular economy perspective, structured by a multi-stage serial production system consisting of a series of machines under deterioration in charge of producing a single product and a reverse remanufacturing system constituted by the same productive systems of the first scheme and different tooling, fed by-products collected at the end of their life cycle, and non-conforming elements of the first productive scheme. Since the advanced production manufacturing system is unable to satisfy the customer's quality expectations completely, we propose the development of a mixed integer linear mathematical model focused on the optimal search and assignment of quality stations and preventive maintenance operation to the machines over a time horizon, intending to segregate the correct number of non-conforming parts for reuse in the remanufacturing system and thereby minimizing production, quality, maintenance, and customer non-conformance penalties. Numerical experiments are performed to analyze the solutions found by the model under different scenarios. The results showed that the correct implementation of a closed manufacturing system and allocation of quality inspection and preventive maintenance operations generate better levels of customer satisfaction and an efficient manufacturing system.

Keywords: closed loop, mixed integer linear programming, preventive maintenance, quality inspection

Procedia PDF Downloads 60

Authors: Sourabh Joshi, Tarun Sharma, Anurag Sharma

Abstract:

Ant Colony Optimization is heuristic Algorithm which has been proven a successful technique applied on number of combinatorial optimization problems. Two variants of Ant Colony Optimization algorithm named Ant System and Max-Min Ant System are implemented in MATLAB to solve travelling Salesman Problem and the results are compared. In, this paper both systems are analyzed by solving the some Travelling Salesman Problem and depict which system solve the problem better in term of cost and time.

Keywords: Ant Colony Optimization, Travelling Salesman Problem, Ant System, Max-Min Ant System

Procedia PDF Downloads 456

Authors: Boris Melnikov, Dmitrii Chaikovskii, Elena Melnikova

Abstract:

The traveling salesman problem (TSP) is a well-known optimization problem that seeks to find the shortest possible route that visits a set of points and returns to the starting point. In this paper, we apply some heuristics of the TSP for the task of restoring the DNA matrix. This restoration problem is often considered in biocybernetics. For it, we must recover the matrix of distances between DNA sequences if not all the elements of the matrix under consideration are known at the input. We consider the possibility of using this method in the testing of distance calculation algorithms between a pair of DNAs to restore the partially filled matrix.

Keywords: optimization problems, DNA matrix, partially filled matrix, traveling salesman problem, heuristic algorithms

Procedia PDF Downloads 129

Authors: Terence Soule, Tami Al Ghamdi

Abstract:

To study how to transfer knowledge from multiple source problems to the target problem, we modeled the Transfer Learning (TL) process using Genetic Algorithms as the model solver. TL is the process that aims to transfer learned data from one problem to another problem. The TL process aims to help Machine Learning (ML) algorithms find a solution to the problems. The Genetic Algorithms (GA) give researchers access to information that we have about how the old problem is solved. In this paper, we have five different source problems, and we transfer the knowledge to the target problem. We studied different scenarios of the target problem. The results showed combined knowledge from multiple source problems improves the GA performance. Also, the process of combining knowledge from several problems results in promoting diversity of the transferred population.

Keywords: transfer learning, genetic algorithm, evolutionary computation, source and target

Procedia PDF Downloads 117

Authors: Ocotlán Díaz-Parra, Jorge A. Ruiz-Vanoye, Alejandro Fuentes-Penna, Beatriz Bernabe-Loranca, Patricia Ambrocio-Cruz, José J. Hernández-Flores

Abstract:

The Maritime Platform Transport Problem of Solid, Special and Dangerous Waste consist of to minimize the monetary value of carry different types of waste from one location to another location using ships. We offer a novel mathematical, the characterization of the problem and the use CPLEX to find the optimal values to solve the Solid, Special and Hazardous Waste Transportation Problem of offshore platforms instances of Mexican state-owned petroleum company (PEMEX). The set of instances used are WTPLib real instances and the tool CPLEX solver to solve the MPTPSSDW problem.

Keywords: oil platform, transport problem, waste, solid waste

Procedia PDF Downloads 448

Authors: Kalpana Dahiya

Abstract:

This paper discusses a two-level hierarchical time minimization transportation problem, which is an important class of transportation problems arising in industries. This problem has been studied by various researchers, and a number of polynomial time iterative algorithms are available to find its solution. All the existing algorithms, though efficient, have some shortcomings. The current study proposes an alternate solution algorithm for the problem that is more efficient in terms of computational time than the existing algorithms. The results justifying the underlying theory of the proposed algorithm are given. Further, a detailed comparison of the computational behaviour of all the algorithms for randomly generated instances of this problem of different sizes validates the efficiency of the proposed algorithm.

Keywords: global optimization, hierarchical optimization, transportation problem, concave minimization

Procedia PDF Downloads 137

Authors: Farzaneh Rajabighamchi, Stan van Hoesel, Christof Defryn

Abstract:

The generalized traveling salesman problem (GTSP) is an extension of the traveling salesman problem (TSP) where the set of nodes is partitioned into clusters, and the salesman must visit exactly one node per cluster. In this research, we apply the definition of the GTSP to an order picker routing problem with multiple locations per product. As such, each product represents a cluster and its corresponding nodes are the locations at which the product can be retrieved. To pick a certain product item from the warehouse, the picker needs to visit one of these locations during its pick tour. As all products are scattered throughout the warehouse, the product clusters not separated geographically. We propose an exact LP model as well as heuristic and meta-heuristic solution algorithms for the order picking problem with multiple product locations.

Keywords: warehouse optimization, order picking problem, generalised travelling salesman problem, heuristic algorithm

Procedia PDF Downloads 87

Authors: Ivan Baravy

Abstract:

Interpolation problem, as it was initially posed in terms of polynomials, is well researched. However, further mathematical developments extended it significantly. Trigonometric interpolation is widely used in Fourier analysis, while its generalized representation as exponential interpolation is applicable to such problem of mathematical physics as modelling of Ziegler-Biersack-Littmark repulsive interatomic potentials. Formulated for finite fields, this problem arises in decoding Reed--Solomon codes. This paper shows the relation between different interpretations of the problem through the class of matrices of special structure - Hankel matrices.

Keywords: Berlekamp-Massey algorithm, exponential interpolation, finite fields, Hankel matrices, Hankel polynomials

Procedia PDF Downloads 491

Authors: Mahmoud Lot

Abstract:

In this article, we will discuss the solution of elliptic optimal control problem. First, by using the nite element method, we obtain the discrete form of the problem. The obtained discrete problem is actually a large scale constrained optimization problem. Solving this optimization problem with traditional methods is difficult and requires a lot of CPU time and memory. But split Bergman method converts the constrained problem to an unconstrained, and hence it saves time and memory requirement. Then we use the split Bregman method for solving this problem, and examples show the speed and accuracy of split Bregman methods for solving these types of problems. We also use the SQP method for solving the examples and compare with the split Bregman method.

Keywords: Split Bregman Method, optimal control with elliptic partial differential equation constraint, finite element method

Procedia PDF Downloads 127

Authors: Cagri Memis, Muzaffer Kapanoglu

Abstract:

The school bus routing problem (SBRP) is a variant of the Vehicle Routing Problem (VRP) classified as a location-allocation-routing problem. In this study, the SBRP is decomposed into two sub-problems: (1) bus route generation and (2) bus stop selection to solve large instances of the SBRP in reasonable computational times. To solve the first sub-problem, we propose a genetic algorithm to generate bus routes. Once the routes have been fixed, a sub-problem remains of allocating students to stops considering the capacity of the buses and the walkability constraints of the students. While the exact method solves small-scale problems, treating large-scale problems with the exact method becomes complex due to computational problems, a deficiency that the genetic algorithm can overcome. Results obtained from the proposed approach on 150 instances up to 250 stops show that the matheuristic algorithm provides better solutions in reasonable computational times with respect to benchmark algorithms.

Keywords: genetic algorithm, matheuristic, school bus routing problem, vehicle routing problem

Procedia PDF Downloads 46