An Objective Approach for Constructing a Membership Function Based on Fuzzy Harvda-Charvat Entropy and Mathematical Programming

2016 ◽  
Vol 20 (4) ◽  
pp. 535-542
Author(s):  
Takashi Hasuike ◽  
◽  
Hideki Katagiri ◽  
Keyword(s):  
Mathematical Programming ◽  
Decision Maker ◽  
Membership Function ◽  
Piecewise Linear ◽  
Natural Extension ◽  
Interval Estimation ◽  
Optimal Solution ◽  
Mathematical Programming Problem ◽  
Partial Linear ◽  
Intermediate Value

This paper proposes an objective approach to the construction of an appropriate membership function that extends to our previous studies. It is important to set a membership function with subjectivity and objectivity to obtain a reasonable optimal solution that complies with the decision maker’s feelings in real-world decision making. To ensure objectivity and subjectivity of the obtained membership function, an entropy-based approach based on mathematical programming is integrated into the interval estimation considered by the decision maker. Fuzzy Harvda-Charvat entropy, which is a natural extension of fuzzy Shannon entropy, is introduced as general entropy with fuzziness. The main steps of our proposed approach are to set intervals with membership values 0 and 1 to enable a decision maker to judge confidently, and to solve the proposed mathematical programming problem strictly using nonlinear programming. In this paper, the given membership function is assumed to be a piecewise linear membership function as an approximation of nonlinear functions, and each intermediate value of partial linear function is optimally obtained.

scholarly journals Effect of graphical method for solving mathematical programming problem

1970 ◽  
Vol 5 (1) ◽  
pp. 29-36 ◽  
Author(s):  
Bimal Chandra Das
Keyword(s):  
Mathematical Programming ◽  
Programming Problem ◽  
Graphical Method ◽  
Optimal Solution ◽  
Mathematical Programming Problem ◽  
Feasible Region ◽  
Computer Implementation ◽  
International University ◽  
Short Time ◽  
Matlab Programming

In this paper, a computer implementation on the effect of graphical method for solving mathematical programming problem using MATLAB programming has been developed. To take any decision, for programming problems we use most modern scientific method based on computer implementation. Here it has been shown that by graphical method using MATLAB programming from all kinds of programming problem, we can determine a particular plan of action from amongst several alternatives in very short time. Keywords: Mathematical programming, objective function, feasible-region, constraints, optimal solution. DOI: 10.3329/diujst.v5i1.4379 Daffodil International University Journal of Science and Technology Vol.5(1) 2010 pp.29-36

scholarly journals Universal method for solving optimization problems under the conditions of uncertainty in the initial data

2021 ◽  
Vol 1 (4 (109)) ◽  
pp. 46-53
Author(s):  
Lev Raskin ◽  
Oksana Sira ◽  
Larysa Sukhomlyn ◽  
Yurii Parfeniuk
Keyword(s):  
Mathematical Programming ◽  
Objective Function ◽  
Membership Function ◽  
Fuzzy Numbers ◽  
Optimization Problems ◽  
Original Data ◽  
Structural Basis ◽  
Mathematical Programming Problem ◽  
Universal Method ◽  
Analytical Expression

This paper proposes a method to solve a mathematical programming problem under the conditions of uncertainty in the original data. The structural basis of the proposed method for solving optimization problems under the conditions of uncertainty is the function of criterion value distribution, which depends on the type of uncertainty and the values of the problem’s uncertain variables. In the case where independent variables are random values, this function then is the conventional theoretical-probabilistic density of the distribution of the random criterion value; if the variables are fuzzy numbers, it is then a membership function of the fuzzy criterion value. The proposed method, for the case where uncertainty is described in the terms of a fuzzy set theory, is implemented using the following two-step procedure. In the first stage, using the membership functions of the fuzzy values of criterion parameters, the values for these parameters are set to be equal to the modal, which are fitted in the analytical expression for the objective function. The resulting deterministic problem is solved. The second stage implies solving the problem by minimizing the comprehensive criterion, which is built as follows. By using an analytical expression for the objective function, as well as the membership function of the problem’s fuzzy parameters, applying the rules for operations over fuzzy numbers, one finds a membership function of the criterion’s fuzzy value. Next, one calculates a measure of the compactness of the resulting membership function of the fuzzy value of the problem’s objective function whose numerical value defines the first component of the integrated criterion. The second component is the rate of deviation of the desired solution to the problem from the previously received modal one. Absolutely similarly designed is the computational procedure for the case where uncertainty is described in the terms of a probability theory. Thus, the proposed method for solving optimization problems is universal in relation to the nature of the uncertainty in the original data. An important advantage of the proposed method is the ability to use it when solving any problem of mathematical programming under the conditions of fuzzily assigned original data, regardless of its nature, structure, and type

scholarly journals An Approach of Trustworthy Measurement Allocation Based on Sub-Attributes of Software

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 237 ◽  
Author(s):  
Hongwei Tao ◽  
Hengyang Wu ◽  
Yixiang Chen
Keyword(s):  
Mathematical Programming ◽  
Software Quality ◽  
Necessary Conditions ◽  
Optimal Solution ◽  
Research Field ◽  
Programming Approach ◽  
Mathematical Programming Problem ◽  
Important Research ◽  
Mathematical Programming Approach ◽  
Software Trustworthiness

Measurement of software trustworthiness is an important research field in the software engineering, which is very useful for analyzing the software quality. In this paper, we propose a mathematical programming approach to allocate the trustworthy degree to each sub-attribute of some software attribute appropriately and then to make the trustworthy degree of this attribute maximize under some constraint conditions. Some sufficient or necessary conditions for analyzing this mathematical programming problem are investigated. Moreover, a polynomial allocation algorithm is given for computing the optimal solution of this mathematical programming. Finally, an example is given in order to show the significance of this work. The results obtained here are useful for improving the software quality by adjusting the trustworthy degree of each sub-attribute under the same cost.

scholarly journals A Novel Approach for Solving Nonsmooth Optimization Problems with Application to Nonsmooth Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Hamid Reza Erfanian ◽  
M. H. Noori Skandari ◽  
A. V. Kamyad
Keyword(s):  
Nonsmooth Optimization ◽  
Optimization Problems ◽  
Piecewise Linear ◽  
Linear Optimization ◽  
Optimal Solution ◽  
Piecewise Linear Function ◽  
Nonsmooth Equations ◽  
Smooth Functions ◽  
Generalized Derivative ◽  
Novel Approach

We present a new approach for solving nonsmooth optimization problems and a system of nonsmooth equations which is based on generalized derivative. For this purpose, we introduce the first order of generalized Taylor expansion of nonsmooth functions and replace it with smooth functions. In other words, nonsmooth function is approximated by a piecewise linear function based on generalized derivative. In the next step, we solve smooth linear optimization problem whose optimal solution is an approximate solution of main problem. Then, we apply the results for solving system of nonsmooth equations. Finally, for efficiency of our approach some numerical examples have been presented.

Predictive Search for Capacitated Multi-Item Lot Sizing Problems

2021 ◽  
Author(s):  
Tao Wu
Keyword(s):  
Mathematical Programming ◽  
Heuristic Search ◽  
Lot Sizing ◽  
Optimal Solution ◽  
Solution Space ◽  
Search Method ◽  
Benchmark Problems ◽  
Solution Quality ◽  
Programming Technique ◽  
Advanced Analytics

For capacitated multi-item lot sizing problems, we propose a predictive search method that integrates machine learning/advanced analytics, mathematical programming, and heuristic search into a single framework. Advanced analytics can predict the probability that an event will happen and has been applied to pressing industry issues, such as credit scoring, risk management, and default management. Although little research has applied such technique for lot sizing problems, we observe that advanced analytics can uncover optimal patterns of setup variables given properties associated with the problems, such as problem attributes, and solution values yielded by linear programming relaxation, column generation, and Lagrangian relaxation. We, therefore, build advanced analytics models that yield information about how likely a solution pattern is the same as the optimum, which is insightful information used to partition the solution space into incumbent, superincumbent, and nonincumbent regions where an analytics-driven heuristic search procedure is applied to build restricted subproblems. These subproblems are solved by a combined mathematical programming technique to improve solution quality iteratively. We prove that the predictive search method can converge to the global optimal solution point. The discussion is followed by computational tests, where comparisons with other methods indicate that our approach can obtain better results for the benchmark problems than other state-of-the-art methods. Summary of Contribution: In this study, we propose a predictive search method that integrates machine learning/advanced analytics, mathematical programming, and heuristic search into a single framework for capacitated multi-item lot sizing problems. The advanced analytics models are used to yield information about how likely a solution pattern is the same as the optimum, which is insightful information used to divide the solution space into incumbent, superincumbent, and nonincumbent regions where an analytics-driven heuristic search procedure is applied to build restricted subproblems. These subproblems are solved by a combined mathematical programming technique to improve solution quality iteratively. We prove that the predictive search method can converge to the global optimal solution point. Through computational tests based on benchmark problems, we observe that the proposed approach can obtain better results than other state-of-the-art methods.

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