scholarly journals On Chamfer Distances on the Square and Body-Centered Cubic Grids: An Operational Research Approach

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Gergely Kovács ◽  
Benedek Nagy ◽  
Gergely Stomfai ◽  
Neşet Deni̇z Turgay ◽  
Béla Vizvári
Keyword(s):  
Linear Programming ◽  
Operational Research ◽  
Optimization Problems ◽  
Shortest Paths ◽  
Optimal Solution ◽  
The Body ◽  
Research Approach ◽  
Body Centered Cubic ◽  
Solid State Physics ◽  
Chamfer Distances

Linear programming is used to solve optimization problems. Thus, finding a shortest path in a grid is a good target to apply linear programming. In this paper, specific bipartite grids, the square and the body-centered cubic grids are studied. The former is represented as a “diagonal square grid” having points with pairs of either even or pairs of odd coordinates (highlighting the bipartite feature). Therefore, a straightforward generalization of the representation describes the body-centered cubic grid in 3D. We use chamfer paths and chamfer distances in these grids; therefore, weights for the steps between the closest neighbors and steps between the closest same type points are fixed, and depending on the weights, various paths could be the shortest one. The vectors of the various neighbors form a basis if they are independent, and their number is the same as the dimension of the space studied. Depending on the relation of the weights, various bases could give the optimal solution and various steps are used in the shortest paths. This operational research approach determines the optimal paths as basic feasible solutions of a linear programming problem. A directed graph is given containing the feasible bases as nodes and arcs with conditions on the used weights such that the simplex method may step from one feasible basis to another one. Thus, the optimal bases can be determined, and they are summarized in two theorems. If the optimal solution is not integer, then the Gomory cut is applied and the integer optimal solution is reached after only one Gomory iteration. Chamfer distances are frequently used in image processing and analysis as well as graphics-related subjects. The body-centered cubic grid, which is well-known in solid state physics, material science, and crystallography, has various applications in imaging and graphics since less samples are needed to represent the signal in the same quality than on the cubic grid. Moreover, the body-centered cubic grid has also a topological advantage over the cubic grid, namely, the neighbor Voronoi cells always share a full face.

scholarly journals Applying Simulation Modelling in Quantifying Optimization Results

2021 ◽  
Vol 15 (4) ◽  
pp. 518-523
Author(s):  
Ratko Stanković ◽  
Diana Božić
Keyword(s):  
Linear Programming ◽  
Simulation Model ◽  
Input Data ◽  
Optimization Problems ◽  
Programming Model ◽  
Optimal Solution ◽  
Simulation Modelling ◽  
Simulation Software ◽  
Programming Models ◽  
Freight Forwarding

Improvements achieved by applying linear programming models in solving optimization problems in logistics cannot always be expressed by physically measurable values (dimensions), but in non-dimensional values. Therefore, it may be difficult to present the actual benefits of the improvements to the stake holders of the system being optimized. In this article, a possibility of applying simulation modelling in quantifying results of optimizing cross dock terminal gates allocation is outlined. Optimal solution is obtained on the linear programming model by using MS Excel spreadsheet optimizer, while the results are quantified on the simulation model, by using Rockwell Automation simulation software. Input data are collected from a freight forwarding company in Zagreb, specialized in groupage transport (Less Than Truckload - LTL).

scholarly journals Discrete Optimization: The Case of Generalized BCC Lattice

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 208
Author(s):  
Gergely Kovács ◽  
Benedek Nagy ◽  
Gergely Stomfai ◽  
Neşet Deniz Turgay ◽  
Béla Vizvári
Keyword(s):  
Integer Programming ◽  
Optimal Path ◽  
Optimal Solution ◽  
Linear Programming Relaxation ◽  
Integer Programming Problem ◽  
Body Centered Cubic ◽  
Solid State Physics ◽  
Negative Part ◽  
Integer Points ◽  
Bcc Lattice

Recently, operations research, especially linear integer-programming, is used in various grids to find optimal paths and, based on that, digital distance. The 4 and higher-dimensional body-centered-cubic grids is the nD (n≥4) equivalent of the 3D body-centered cubic grid, a well-known grid from solid state physics. These grids consist of integer points such that the parity of all coordinates are the same: either all coordinates are odd or even. A popular type digital distance, the chamfer distance, is used which is based on chamfer paths. There are two types of neighbors (closest same parity and closest different parity point-pairs), and the two weights for the steps between the neighbors are fixed. Finding the minimal path between two points is equivalent to an integer-programming problem. First, we solve its linear programming relaxation. The optimal path is found if this solution is integer-valued. Otherwise, the Gomory-cut is applied to obtain the integer-programming optimum. Using the special properties of the optimization problem, an optimal solution is determined for all cases of positive weights. The geometry of the paths are described by the Hilbert basis of the non-negative part of the kernel space of matrix of steps.

scholarly journals Use of Linear Programming Model to Determine the Optimum Cropping Pattern for an Irrigation Scheme in Masvingo, Zimbabwe

2013 ◽  
Vol 1 (4) ◽  
pp. 450-452
Author(s):  
Majeke F ◽  
Mubvuma S M T ◽  
J. Chirima, K. Makaza ◽  
T. Hungwe R. Gwazan ◽  
Nyoni ◽  
...  
Keyword(s):  
Linear Programming ◽  
Optimization Problems ◽  
Programming Model ◽  
Optimal Solution ◽  
Linear Programming Model ◽  
No Production ◽  
Cropping Pattern ◽  
Irrigation Scheme ◽  
Irrigation Planning ◽  
Optimal Cropping

Agricultural systems are often faced by challenges such as crop selection and irrigation planning which can be formulated as optimization problems. Decisions have to be made on the proper set of crops to be cultivated and a proper irrigation scheme. The objectives of such decisions are to maximize net profit or to minimize water waste. In this study, a linear programming model was developed that helped to determine the optimal cropping pattern for an irrigation scheme in Masvingo, Zimbabwe. Crops which considered were wheat, sugar beans for winter and cotton and maize for summer for the 2012/13 agricultural season. The linear programming model was solved by using Microsoft Excel (2007). The model recommended no production of wheat and cotton. Sugar beans and maize gained acreage by 50 percent and 88 percent respectively. On the whole, the optimal cropped acreage did not change as compared to the existing cropping plan. As a result of the optimal solution, a farmer‘s income could be increased by $1,668.60. The optimal income increased from existing level of $1,919.40 to $3,588.00 showing an improvement of 87 percent. The results show that LP models solutions are worthy implementing.

SIMULATION OBJECTIVES OF FUZZY LINEAR PROGRAMMING WITH AN α-LEVEL METHOD OF λ-CONTINUE

2019 ◽  
Vol 6 (2) ◽  
pp. 71-76
Author(s):  
Alevtina Jur'evna Shatalova ◽  
Konstantin Andreevich Lebedev Konstantin Andreevich
Keyword(s):  
Linear Programming ◽  
Objective Function ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Optimal Solution ◽  
Investment Project ◽  
Fuzzy Linear Programming ◽  
Distribution Law ◽  
Linear Optimization Problems ◽  
Level Method

The article describes an approach that allows to formally describe the arising uncertainties in linear optimization problems. The generalized parametric alpha-level method of lambda-continuation of the fuzzy linear programming problem is considered. The model offers two methods that take into account the expansion of the binary fuzzy ratio (“strong” and “weak”). After the condition is formed taking into account the incoming quantities in the form of fuzzy numbers (the objective function and the system of constraints), the optimal solution (the value of the objective function) for each alpha and lambda is calculated using the simplex method implemented in Mathcad. On its basis, a mathematical model is built that will take into account the random values of alpha and lambda with a uniform distribution law. The paper presents a description of the simulation study, which confirms the conclusions about the possibilities of the method. Using the proposed theory, the decision-maker receives more information showing the behavior of the system with small changes in the input parameters to make more informed conclusions about the choice of financing of an investment project. The developed method of simulation of fuzzy estimation can be applied to other economic models with the appropriate necessary modification, for example, to assess the creditworthiness of the enterprise.

scholarly journals Optimisasi Jumlah Produksi dalam Memperoleh Keuntungan Maksimal Pada Penjualan Dompet dan Tas Jimshoney “NAYA Online Shopping”

2020 ◽  
Vol 1 (1) ◽  
pp. 45-52
Author(s):  
Rochmat Umar
Keyword(s):  
Linear Programming ◽  
Operational Research ◽  
Online Shopping ◽  
Graphical Method ◽  
Optimal Solution ◽  
Sensitivity Test ◽  
Graphical Methods ◽  
Maximum Capacity ◽  
Maximum Profit ◽  
Research Techniques

ABSTRACT Linear programming is one of the most widely used operational research techniques in practice and is known for being easy to understand. The method in solving linear programming in "Naya Online Shopping" is by the graphical method and compared with LINGO software. The purpose of the completion of linear programming is to optimize the amount of production in obtaining maximum profits. And an optimal solution is obtained, namely a maximum profit of IDR 1,750,000. with 50 bags of "ting-ting bags" to be purchased from agents and many "crown wallets" to be purchased from agents of 100 pieces. From the maximum capacity of the shelter 150 pieces can be maximized by purchasing as many as 50 bags and purchasing 100 pieces of wallet. From the sensitivity test, it can be concluded that each capital increase or reduction of IDR 750,000 will result in a gain or loss of IDR 12,500. Completion of linear programming with research into the operation of graphical methods and LINGO software obtained the same value. Keywords: Linear programming, Operations of graphical method operations, LINGO software, sensitivity test ABSTRAK Linear progamming adalah salah satu teknik riset operasi yang paling banyak dipergunakan dalam praktik dan dikenal karena mudah dipahami. Metode dalam penyelesaian linear progamming pada “Naya Online Shopping” yaitu dengan metode grafik dan dibandingkan dengan software LINGO. Tujuan dari penyelesaian linear progamming ini adalah untuk mengoptimalkan jumlah produksi dalam memperoleh keuntungan maksimal. Dan didapatkan solusi optimal yaitu keuntungan maksimum sebesar Rp1.750.000, dengan banyaknya tas “ting-ting bag” yang akan dibeli dari agen sebanyak 50 buah dan banyaknya dompet “crown wallet” yang akan dibeli dari agen sebanyak 100 buah. Dari kapasitas maksimal penampungan 150 buah dapat dimaksimalkan dengan pembelian tas sebanyak 50 buah dan pembelian dompet 100 buah. Dari uji sensitivitas dihasilkan bahwa setiap penambahan atau pengurangan modal sebesar Rp750.000, maka akan didapatkan keuntungan atau kerugian sebesar Rp12.500,. Penyelesaian linear progamming dengan riset operasi metode grafik dan software LINGO didapatkan hasil yang sama nilainya. Kata Kunci: Linear progamming, Riset operasi, Software LINGO, Optimasi, Pemodelan Matematik.

scholarly journals Tuning Successive Linear Programming to Solve AC Optimal Power Flow Problem for Large Networks

2021 ◽  
Author(s):  
Sayed Abdullah Sadat ◽  
mostafa Sahraei-Ardakani
Keyword(s):  
Linear Programming ◽  
Power Flow ◽  
Optimal Power Flow ◽  
Optimization Problems ◽  
Optimal Solution ◽  
Computational Time ◽  
Test Cases ◽  
Successive Linear Programming ◽  
Optimal Power ◽  
Number Of Iterations

Successive linear programming (SLP) is a practical approach for solving large-scale nonlinear optimization problems. Alternating current optimal power flow (ACOPF) is no exception, particularly the large size of real-world networks. However, in order to achieve tractability, it is essential to tune the SLP algorithm presented in the literature. This paper presents a modified SLP algorithm to solve the ACOPF problem, specified by the U.S. Department of Energy's (DOE) Grid Optimization (GO) Competition Challenge 1, within strict time limits. The algorithm first finds a near-optimal solution for the relaxed problem (i.e., Stage 1). Then, it finds a feasible solution in the proximity of the near-optimal solution (i.e., Stage 2 and Stage 3). The numerical experiments on test cases ranging from 500-bus to 30,000-bus systems show that the algorithm is tractable. The results show that our proposed algorithm is tractable and can solve more than 80\% of test cases faster than the well-known Interior Point Method while significantly reduce the number of iterations required to solve ACOPF. The number of iterations is considered an important factor in the examination of tractability which can drastically reduce the computational time required within each iteration.

scholarly journals Tuning Successive Linear Programming to Solve AC Optimal Power Flow Problem for Large Networks

2021 ◽  
Author(s):  
Sayed Abdullah Sadat ◽  
mostafa Sahraei-Ardakani
Keyword(s):  
Linear Programming ◽  
Power Flow ◽  
Optimal Power Flow ◽  
Optimization Problems ◽  
Optimal Solution ◽  
Computational Time ◽  
Test Cases ◽  
Successive Linear Programming ◽  
Optimal Power ◽  
Number Of Iterations

Successive linear programming (SLP) is a practical approach for solving large-scale nonlinear optimization problems. Alternating current optimal power flow (ACOPF) is no exception, particularly the large size of real-world networks. However, in order to achieve tractability, it is essential to tune the SLP algorithm presented in the literature. This paper presents a modified SLP algorithm to solve the ACOPF problem, specified by the U.S. Department of Energy's (DOE) Grid Optimization (GO) Competition Challenge 1, within strict time limits. The algorithm first finds a near-optimal solution for the relaxed problem (i.e., Stage 1). Then, it finds a feasible solution in the proximity of the near-optimal solution (i.e., Stage 2 and Stage 3). The numerical experiments on test cases ranging from 500-bus to 30,000-bus systems show that the algorithm is tractable. The results show that our proposed algorithm is tractable and can solve more than 80\% of test cases faster than the well-known Interior Point Method while significantly reduce the number of iterations required to solve ACOPF. The number of iterations is considered an important factor in the examination of tractability which can drastically reduce the computational time required within each iteration.

Voids in Irradiated Tungsten and Molybdenum

1969 ◽  
Vol 27 ◽  
pp. 190-191
Author(s):  
Robert C. Rau ◽  
Robert L. Ladd
Keyword(s):  
Melting Point ◽  
Fast Neutron ◽  
Neutron Irradiation ◽  
Elevated Temperatures ◽  
Irradiation Temperature ◽  
The Body ◽  
Face Centered Cubic ◽  
Body Centered Cubic ◽  
Cubic Metals ◽  
Face Centered

Recent studies have shown the presence of voids in several face-centered cubic metals after neutron irradiation at elevated temperatures. These voids were found when the irradiation temperature was above 0.3 Tm where Tm is the absolute melting point, and were ascribed to the agglomeration of lattice vacancies resulting from fast neutron generated displacement cascades. The present paper reports the existence of similar voids in the body-centered cubic metals tungsten and molybdenum.

MERITS OF USING THE NON - LINEAR PROGRAMMING IN SOLVING THE OPTIMIZATION PROBLEMS OF GEODETIC NETWORKS

2005 ◽  
Vol 28 (3) ◽  
pp. 293-300
Author(s):  
Abo El Hassan Rahil ◽  
Ahmed El Gohary ◽  
Mohamed Ismail
Keyword(s):  
Linear Programming ◽  
Optimization Problems ◽  
Non Linear Programming ◽  
Geodetic Networks ◽  
Non Linear
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