A Generalized Fuzzy Integer Programming Approach for Environmental Management under Uncertainty

In this study, a generalized fuzzy integer programming (GFIP) method is developed for planning waste allocation and facility expansion under uncertainty. The developed method can (i) deal with uncertainties expressed as fuzzy sets with known membership functions regardless of the shapes (linear or nonlinear) of these membership functions, (ii) allow uncertainties to be directly communicated into the optimization process and the resulting solutions, and (iii) reflect dynamics in terms of waste-flow allocation and facility-capacity expansion. A stepwise interactive algorithm (SIA) is proposed to solve the GFIP problem and generate solutions expressed as fuzzy sets. The procedures of the SIA method include (i) discretizing the membership function grade of fuzzy parameters into a set ofα-cutlevels; (ii) converting the GFIP problem into an inexact mixed-integer linear programming (IMILP) problem under eachα-cut level; (iii) solving the IMILP problem through an interactive algorithm; and (iv) approximating the membership function for decision variables through statistical regression methods. The developed GFIP method is applied to a municipal solid waste (MSW) management problem to facilitate decision making on waste flow allocation and waste-treatment facilities expansion. The results, which are expressed as discrete or continuous fuzzy sets, can help identify desired alternatives for managing MSW under uncertainty.

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A Stochastic Mixed-Integer Programming approach to the energy-technology management problem

Computers & Industrial Engineering ◽

10.1016/j.cie.2011.07.012 ◽

2012 ◽

Vol 63 (3) ◽

pp. 594-606 ◽

Cited By ~ 12

Author(s):

Stephen J. Stoyan ◽

Maged M. Dessouky

Keyword(s):

Integer Programming ◽

Technology Management ◽

Mixed Integer Programming ◽

Mixed Integer ◽

Programming Approach ◽

Management Problem ◽

Energy Technology ◽

Stochastic Mixed Integer Programming

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Approximation of real data by fuzzy sets for the classification problem

Ukrainian journal of remote sensing ◽

10.36023/ujrs.2019.22.154 ◽

2019 ◽

pp. 22-26

Author(s):

Kostiantyn Sukhanov

Keyword(s):

Fuzzy Sets ◽

Membership Function ◽

Fuzzy Set ◽

Real Data ◽

Training Data ◽

Learning Object ◽

Membership Functions ◽

One Dimensional ◽

Set Membership ◽

The Membership Function

The article deals with the method of classification of real data using the apparatus of fuzzy sets and fuzzy logic as a flexible tool for learning and recognition of natural objects on the example of oil and gas prospecting sections of the Dnieper-Donetsk basin. The real data in this approach are the values for the membership function that are obtained not through subjective expert judgment but from objective measurements. It is suggested to approximate the fuzzy set membership functions by using training data to use the approximation results obtained during the learning phase at the stage of identifying unknown objects. In the first step of learning, each traditional future of a learning data is matched by a primary traditional one-dimensional set whose membership function can only take values from a binary set — 0 if the learning object does not belong to the set, and 1 if the learning object belongs to the set. In the second step, the primary set is mapped to a fuzzy set, and the parameters of the membership function of this fuzzy set are determined by approximating this function of the traditional set membership. In the third step, the set of one-dimensional fuzzy sets that correspond to a single feature of the object is mapped to a fuzzy set that corresponds to all the features of the object in the training data set. Such a set is the intersection of fuzzy sets of individual features, to which the blurring and concentration operations of fuzzy set theory are applied in the last step. Thus, the function of belonging to a fuzzy set of a class is the operation of choosing a minimum value from the functions of fuzzy sets of individual features of objects, which are reduced to a certain degree corresponding to the operation of blurring or concentration. The task of assigning the object under study to a particular class is to compare the values of the membership functions of a multidimensional fuzzy set and to select the class in which the membership function takes the highest value. Additionally, after the training stage, it is possible to determine the degree of significance of an object future, which is an indistinctness index, to remove non-essential data (object futures) from the analysis.

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