scholarly journals Finding algorithm of optimal subset structure based on the Pareto layers in the knapsack problem

2020 ◽  
pp. 97-104
Author(s):  
Sergey V. Chebakov ◽  
Liya V. Serebryanaya
Keyword(s):  
Initial Data ◽  
Knapsack Problem ◽  
Optimization Model ◽  
Multicriteria Optimization ◽  
Combinatorial Problem ◽  
Optimal Subset ◽  
Preference Space

An algorithm is developed for finding the structure of the optimal subset in the knapsack problem based on the proposed multicriteria optimization model. A two-criteria relation of preference between elements of the set of initial data is introduced. This set has been split into separate Pareto layers. The depth concept of the elements dominance of an individual Pareto layer is formulated. Based on it, conditions are determined under which the solution to the knapsack problem includes the first Pareto layers. They are defined on a given set of initial data. The structure of the optimal subset is presented, which includes individual Pareto layers. Pareto layers are built in the introduced preference space. This does not require algorithms for enumerating the elements of the initial set. Such algorithms are used when finding only some part of the optimal subset. This reduces the number of operations required to solve the considered combinatorial problem.  The method for determining the found Pareto layers shows that the number of operations depends on the volume of the knapsack and the structure of the Pareto layers, into which the set of initial data in the entered two-criteria space is divided.

scholarly journals FINDING OF OPTIMAL SUBSET STRUCTURE IN THE KNAPSACK PROBLEM

Doklady BGUIR ◽  
2019 ◽  
pp. 72-79
Author(s):  
S. V. Chebakov ◽  
L. V. Serebryanaya
Keyword(s):  
Initial Data ◽  
Knapsack Problem ◽  
Combinatorial Problem ◽  
Pareto Set ◽  
Knapsack Problems ◽  
Data Set ◽  
Optimal Subset ◽  
Specific Subset ◽  
Large Numbers

An algorithm for solving the knapsack problem based on the proposed multi-criteria model is considered. The implementation of this algorithm allows to define the structure of the optimal subset as a union of certain elements of a Pareto layers group into which a initial data set is divided. The first such layer is the Pareto set. The optimal subset allows to find a specific subset of the initial data. Its elements as a result of belonging to the Pareto layers with large numbers cannot enter the optimal subset. The most expensive in terms of the number of operations required are knapsack problems, in which the number of elements in the set of initial data is quite large. The article shows that with a relatively small value of the knapsack volume, the number of elements required to find the optimal subset is significantly less than their total number in the original set. It can lead to a significant decrease in the total time to solve the combinatorial problem.  

scholarly journals Optimization Method to Address Psychosocial Risks through Adaptation of the Multidimensional Knapsack Problem

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1126
Author(s):  
Marta Lilia Eraña-Díaz ◽  
Marco Antonio Cruz-Chávez ◽  
Fredy Juárez-Pérez ◽  
Juana Enriquez-Urbano ◽  
Rafael Rivera-López ◽  
...  
Keyword(s):  
Risk Factors ◽  
Simulated Annealing ◽  
Knapsack Problem ◽  
Simulated Annealing Algorithm ◽  
Psychosocial Risk Factors ◽  
Psychosocial Risk ◽  
Multidimensional Knapsack Problem ◽  
Optimal Subset ◽  
Multidimensional Knapsack ◽  
A Company

This paper presents a methodological scheme to obtain the maximum benefit in occupational health by attending to psychosocial risk factors in a company. This scheme is based on selecting an optimal subset of psychosocial risk factors, considering the departments’ budget in a company as problem constraints. This methodology can be summarized in three steps: First, psychosocial risk factors in the company are identified and weighted, applying several instruments recommended by business regulations. Next, a mathematical model is built using the identified psychosocial risk factors information and the company budget for risk factors attention. This model represents the psychosocial risk optimization problem as a Multidimensional Knapsack Problem (MKP). Finally, since Multidimensional Knapsack Problem is NP-hard, one simulated annealing algorithm is applied to find a near-optimal subset of factors maximizing the psychosocial risk care level. This subset is according to the budgets assigned for each of the company’s departments. The proposed methodology is detailed using a case of study, and thirty instances of the Multidimensional Knapsack Problem are tested, and the results are interpreted under psychosocial risk problems to evaluate the simulated annealing algorithm’s performance (efficiency and efficacy) in solving these optimization problems. This evaluation shows that the proposed methodology can be used for the attention of psychosocial risk factors in real companies’ cases.

scholarly journals Hierarchical Decomposition Synthesis in Optimal Systems Design

1997 ◽  
Vol 119 (4) ◽  
pp. 448-457 ◽  
Author(s):  
R. S. Krishnamachari ◽  
P. Y. Papalambros
Keyword(s):  
Optimal Design ◽  
Optimization Model ◽  
Multicriteria Optimization ◽  
Systems Design ◽  
System Optimization ◽  
Hierarchical Decomposition ◽  
Objective Functions ◽  
Formal Process ◽  
Large Systems ◽  
System Objective

Optimal design of large systems is easier if the optimization model can be decomposed and solved as a set of smaller, coordinated subproblems. Casting a given design problem into a particular optimization model by selecting objectives and constraints is generally a subjective task. In system models where hierarchical decomposition is possible, a formal process for selecting objective functions can be made, so that the resulting optimal design model has an appropriate decomposed form and also possesses desirable properties for the scalar substitute functions used in multicriteria optimization. Such a process is often followed intuitively during the development of a system optimization model by summing selected objectives from each subsystem into a single overall system objective. The more formal process presented in this article is simple to implement and amenable to automation.

A strong integer linear optimization model to the compartmentalized knapsack problem

2019 ◽  
Vol 26 (5) ◽  
pp. 1633-1654
Author(s):  
John J. Quiroga‐Orozco ◽  
J.M. Valério de Carvalho ◽  
Robinson S. V. Hoto
Keyword(s):  
Knapsack Problem ◽  
Optimization Model ◽  
Linear Optimization ◽  
Linear Optimization Model ◽  
Integer Linear Optimization

scholarly journals Formation of an optimal portfolio of venture projects

2021 ◽  
pp. 128-132
Author(s):  
I.A Korkhina ◽  
V.O Petrenko ◽  
V.L Khomenko ◽  
V.O Kulyk
Keyword(s):  
Mathematical Programming ◽  
Initial Data ◽  
Stochastic Optimization ◽  
Development Strategy ◽  
Optimization Model ◽  
Programming Model ◽  
Optimal Portfolio ◽  
Financial Resources ◽  
Proposed Model ◽  
The Face

Purpose. Development of a method for forming an optimal portfolio of venture projects taking into account risks, uncertainty in initial data and limited financial resources. Methodology. To calculate the accuracy of forecasting prices, which are necessary for calculating the parameters of the stochastic optimization model for the formation of a portfolio of projects, we used the theory of random variables and regression analysis. The problem of choosing the optimal portfolio of venture projects was solved using stochastic mathematical programming. Findings. A model for creating an optimal portfolio of venture projects has been developed. It is a stochastic mathematical programming model that can be used to solve problems of investing in venture projects in the extractive industry. This model takes into account the risks associated with obtaining the expected income from the implementation of each venture project, the uncertainty in the initial data for calculating the income from the projects selected in the portfolio, as well as the limited funds required to finance the project portfolio. Originality. The stochastic optimization model for the formation of an optimal portfolio of projects, taking into account the peculiarities of venture projects, in particular, their high riskiness, has been significantly improved and adapted. Practical value. The proposed model for the formation of an optimal portfolio of venture projects can be used at mining enterprises, whose development strategy involves the implementation of innovative, high-risk projects. The use of this model in strategic planning will allow an enterprise to receive the maximum income from venture projects in the face of a lack of financial resources, as well as instability of the innovation market.

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