scholarly journals Problems on Shortest k-Node Cycles and Paths

2021 ◽  
pp. 15-33
Author(s):  
Petro Stetsyuk ◽  
Dumitru Solomon ◽  
Maria Grygorak
Keyword(s):  
Linear Programming ◽  
Mathematical Models ◽  
Programming Problem ◽  
Linear Programming Problem ◽  
Optimization Problems ◽  
Travelling Salesman Problem ◽  
Modeling Language ◽  
Boolean Variable ◽  
Travelling Salesman ◽  
Wine Making

The paper is devoted to the construction of mathematical models for problems on the shortest cycles and paths, that pass through a given number of nodes of a directed graph. Such cycles and paths are called k-node, where 1<k <n, n is the number of nodes in the graph. Section 1 formulates two problems for finding the shortest k-node cycle – a mixed Boolean and linear programming problem and a discrete programming problem. Both problems include constraints from the classical assignment problem, describing a one-time entry into a node and a one-time exit from a node for those nodes through which the cycle passes. The cycle connectivity in the first problem is ensured by modeling the flow problem, and in the second problem, it is ensured by using the A. Tucker constraints for the travelling salesman problem. Section 2 establishes a connection between the formulations of both problems from Section 1 and the travelling salesman problem and investigates the efficiency of their solution using modern versions of gurobi and cplex programs and the AMPL modeling language. Section 3 contains the formulation of the shortest k-node path problem, which is represented by a mixed Boolean and linear programming problem. With its help, the optimal routes were found for visiting the wine-making points of the Malopolskie Wine Route in the direction Lviv-Wroclaw-Lviv (Section 4). Here a map for the 20 most visited wine-making points of the Malopolskie Wine Route and a table of the distances between them and the distances from them to Lviv and Wroclaw, calculated using the Google Maps web service, are presented. The developed mathematical models of the problems of finding the shortest k-node paths and cycles and the developed software in the AMPL modeling language can be used for the design and arrangement of technical objects, optimization of the transportation of products, analysis and forecasting of economic processes, determination of optimal routes when planning passenger and freight traffic, optimal organization of the process of managing a set of transactions and queries during their implementation in network databases and other classes of applied optimization problems. Keywords: digraph, shortest path, Boolean variable, linear programming, Hamiltonian cycle, Hamiltonian path, travelling salesman problem, AMPL, gurobi, cplex.

scholarly journals Mathematical modeling in economics for selection of optimum investment solution

2021 ◽  
Vol 273 ◽  
pp. 08003
Author(s):  
Arthur Alukhanyan ◽  
Olga Panfilova
Keyword(s):  
Mathematical Modeling ◽  
Mathematical Model ◽  
Linear Programming ◽  
Mathematical Models ◽  
Programming Problem ◽  
Linear Programming Problem ◽  
Target Function ◽  
Investment Portfolio ◽  
Monetary Relations ◽  
Selection Of

This work is devoted to development of economic and mathematical models for selection of the optimum investment solution. Moreover, it states the basis for development of model examples and correction of the model considering the results obtained in the examples. In the work the problem is set for selection of the investment sources and objects, which is limited to the linear programming problem. The controlled variable and basic limitations simulating real credit and monetary relations are distinguished in the provided model. The discounted profit obtained from implementation of the optimum investment portfolio is considered as a target function. The economic and mathematical model presented in the article allows finding the optimum investment solution within the limits of the credit and monetary relations taking place both at the micro- and macroeconomic level.

scholarly journals A METHOD FOR SOLVING LINEAR PROGRAMMING PROBLEMS WITH FUZZY PARAMETERS BASED ON MULTIOBJECTIVE LINEAR PROGRAMMING TECHNIQUE

2007 ◽  
Vol 24 (04) ◽  
pp. 557-573 ◽  
Author(s):  
M. ZANGIABADI ◽  
H. R. MALEKI
Keyword(s):  
Linear Programming ◽  
Programming Problem ◽  
Linear Programming Problem ◽  
Fuzzy Numbers ◽  
Optimization Problems ◽  
Multiobjective Linear Programming ◽  
Programming Technique ◽  
Linear Programming Problems ◽  
Linear Programming Technique ◽  
Fuzzy Parameters

In the real-world optimization problems, coefficients of the objective function are not known precisely and can be interpreted as fuzzy numbers. In this paper we define the concepts of optimality for linear programming problems with fuzzy parameters based on those for multiobjective linear programming problems. Then by using the concept of comparison of fuzzy numbers, we transform a linear programming problem with fuzzy parameters to a multiobjective linear programming problem. To this end, we propose several theorems which are used to obtain optimal solutions of linear programming problems with fuzzy parameters. Finally some examples are given for illustrating the proposed method of solving linear programming problem with fuzzy parameters.

scholarly journals On Linear and Quadratic Two-Stage Transportation Problem

2020 ◽  
pp. 5-14
Author(s):  
P. Stetsyuk ◽  
O. Lykhovyd ◽  
A. Suprun
Keyword(s):  
Linear Programming ◽  
Mathematical Programming ◽  
Quadratic Programming ◽  
Programming Problem ◽  
Linear Programming Problem ◽  
Transportation Problem ◽  
Quadratic Programming Problem ◽  
Modeling Language ◽  
Two Stage ◽  
Linear Programming Problems

Introduction. When formulating the classical two-stage transportation problem, it is assumed that the product is transported from suppliers to consumers through intermediate points. Intermediary firms and various kinds of storage facilities (warehouses) can act as intermediate points. The article discusses two mathematical models for two-stage transportation problem (linear programming problem and quadratic programming problem) and a fairly universal way to solve them using modern software. It uses the description of the problem in the modeling language AMPL (A Mathematical Programming Language) and depends on which of the known programs is chosen to solve the problem of linear or quadratic programming. The purpose of the article is to propose the use of AMPL code for solving a linear programming two-stage transportation problem using modern software for linear programming problems, to formulate a mathematical model of a quadratic programming two-stage transportation problem and to investigate its properties. Results. The properties of two variants of a two-stage transportation problem are described: a linear programming problem and a quadratic programming problem. An AMPL code for solving a linear programming two-stage transportation problem using modern software for linear programming problems is given. The results of the calculation using Gurobi program for a linear programming two-stage transportation problem, which has many solutions, are presented and analyzed. A quadratic programming two-stage transportation problem was formulated and conditions were found under which it has unique solution. Conclusions. The developed AMPL-code for a linear programming two-stage transportation problem and its modification for a quadratic programming two-stage transportation problem can be used to solve various logistics transportation problems using modern software for solving mathematical programming problems. The developed AMPL code can be easily adapted to take into account the lower and upper bounds for the quantity of products transported from suppliers to intermediate points and from intermediate points to consumers. Keywords: transportation problem, linear programming problem, AMPL modeling language, Gurobi program, quadratic programming problem.

scholarly journals Privacy-Preserving Two-Party Computation of Parameters of a Fuzzy Linear Regression

2021 ◽  
pp. 11-19
Author(s):  
Elena Volkova ◽  
◽  
Vladimir Gisin ◽  
Keyword(s):  
Linear Programming ◽  
Linear Regression ◽  
Programming Problem ◽  
Linear Programming Problem ◽  
Fuzzy Numbers ◽  
Optimization Problems ◽  
Observation Data ◽  
Fuzzy Linear Regression ◽  
Explanatory Variables ◽  
Common Criterion

Purpose: describe two-party computation of fuzzy linear regression with horizontal partitioning of data, while maintaining data confidentiality. Methods: the computation is designed using a transformational approach. The optimization problems of each of the two participants are transformed and combined into a common problem. The solution to this problem can be found by one of the participants. Results: A protocol is proposed that allows two users to obtain a fuzzy linear regression model based on the combined data. Each of the users has a set of data about the results of observations, containing the values of the explanatory variables and the values of the response variable. The data structure is shared: both users use the same set of explanatory variables and a common criterion. Regression coefficients are searched for as symmetric triangular fuzzy numbers by solving the corresponding linear programming problem. It is assumed that both users are semihonest (honest but curious, or passive and curious), i.e. they execute the protocol, but can try to extract information about the source data of the partner by applying arbitrary processing methods to the received data that are not provided for by the protocol. The protocol describes the transformed linear programming problem. The solution of this problem can be found by one of the users. The number of observations of each user is known to both users. The observation data remains confidential. The correctness of the protocol is proved and its security is justified. Keywords: fuzzy numbers, collaborative solution of a linear programming problem, two-way computation, transformational approach, cloud computing, federated machine learning.

scholarly journals A comparative study on interval arithmetic operations with intuitionistic fuzzy numbers for solving an intuitionistic fuzzy multi–objective linear programming problem

2017 ◽  
Vol 27 (3) ◽  
pp. 563-573 ◽  
Author(s):  
Rajendran Vidhya ◽  
Rajkumar Irene Hepzibah
Keyword(s):  
Linear Programming ◽  
Programming Problem ◽  
Linear Programming Problem ◽  
Fuzzy Numbers ◽  
Optimization Method ◽  
Arithmetic Operations ◽  
Intuitionistic Fuzzy Numbers ◽  
Multi Objective ◽  
Intuitionistic Fuzzy ◽  
Interval Valued

AbstractIn a real world situation, whenever ambiguity exists in the modeling of intuitionistic fuzzy numbers (IFNs), interval valued intuitionistic fuzzy numbers (IVIFNs) are often used in order to represent a range of IFNs unstable from the most pessimistic evaluation to the most optimistic one. IVIFNs are a construction which helps us to avoid such a prohibitive complexity. This paper is focused on two types of arithmetic operations on interval valued intuitionistic fuzzy numbers (IVIFNs) to solve the interval valued intuitionistic fuzzy multi-objective linear programming problem with pentagonal intuitionistic fuzzy numbers (PIFNs) by assuming differentαandβcut values in a comparative manner. The objective functions involved in the problem are ranked by the ratio ranking method and the problem is solved by the preemptive optimization method. An illustrative example with MATLAB outputs is presented in order to clarify the potential approach.

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