http://cepac.cheme.cmu.edu/pasilectures/sahinidis/NVSpaperthree.pdf

Optimization under uncertainty: state-of-the-art and opportunities

http://geocities.ws/smhurtado/2000-111-1-3-283.pdf

Possibilistic linear programming: a brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem

A fuzzy restriction may be visualized as an elastic constraint on the values that may be assigned to a variable In terms of such restrictions, the meaning of a proposition of the form “x is P,” where x is the name of an object and P is a fuzzy set, may be expressed as a relational assignment equation of the form R(A(x)) = P, where A(x) is an implied attribute of x, R is a fuzzy restriction on x, and P is the unary fuzzy relation which is assigned to R For example, “Stella is young ,” where young is a fuzzy subset of the real line, translates into R(Age(Stella))= young  The calculus of fuzzy restrictions is concerned, in the main, with (a) translation of propositions of various types into relational assignment equations, and (b) the study of transformations of fuzzy restrictions which are induced by linguistic modifiers, truth-functional modifiers, compositions, projections and other operations An important application of the calculus of fuzzy restrictions relates to what might be called approximate reasoning , that is, a type of reasoning which is neither very exact nor very inexact The main ideas behind this application are outlined and illustrated by examples

/pdf/calculus-of-fuzzy-restrictions-3y42lczg56.pdf

Calculus of fuzzy restrictions

https://link.springer.com/content/pdf/10.1007%2F978-1-4615-6103-3_15.pdf

Fuzzy Mathematical Programming

In optimization analysis by linear programming, uncertainties may exist in model coefficients and stipulations (right-hand side constraints). These uncertainties can propagate through the analysis and generate uncertainties in the results. However, among the previous methods dealing with uncertainty, some were too complicated to be applied to actual problems, and some were unable to reflect completely the uncertainties of the input and output information. In this paper, a grey linear programming (GLP) model is introduced to the civil engineering area. This method allows uncertainties in the model inputs to be communicated into the optimization process, and thereby solutions reflecting the inherent uncertainties can be derived. A grey linear programming problem can be solved easily by running a simplex program several times. The modelling approach is applied to a hypothetical problem of waste flow allocation planning within a municipal solid waste management system. The results indicate that reaso...

A grey linear programming approach for municipal solid waste management planning under uncertainty

In problems of system analysis, it is customary to treat imprecision by the use of probability theory. It is becoming increasingly clear, however, that in the case of many real world problems involving large scale systems such as economic systems, social systems, mass service systems, etc., the major source of imprecision should more properly be labeled ‘fuzziness’ rather than ‘randomness.’ By fuzziness, we mean the type of imprecision which is associated with the lack of sharp transition from membership to nonmembership, as in tall men, small numbers, likely events, etc. In this paper our main concern is with the application of the theory of fuzzy sets to decision problems involving fuzzy goals and strategies, etc., as defined by R. E. Bellman and L. A. Zadeh [1]. However, in our approach, the emphasis is on mathematical programming and the use of the concept of a level set to extend some of the classical results to problems involving fuzzy constraints and objective functions.

On Fuzzy-Mathematical Programming

Although the decision‐making problem at the lower level is generally well‐defined, the decision‐making problem at the higher level would not contain the detail. Much of decision‐making at the higher level might take place in a fuzzy environment, so that it is only necessary to decide roughly what actions, what states and what parameters should be considered. This paper deals with the higher level problem in which we can regard the elements‐states of nature, feasible actions and available information‐as fuzzy objects. Since the uncertainty of meaning of objects is represented by the fuzzy sets and the uncertainty of occurrence of objects is defined by the probability, a specific formulation of the higher level decision problem can be defined by the probability of fuzzy events. From the same aspect, the definitions of worth, entropy and quantity concerning fuzzy information are given in this paper, and we have tried to extend some of the statistical decision theory to the fuzzy decision problem. To explain our formulation, an investment problem is presented as an example.

A formulation of fuzzy decision problems and its application to an investment problem

Publisher Summary There are many problems in a fuzzy environment where it is necessary to decide a present estimated goal. Therefore, it becomes necessary to formulate decision problems in such a sense that an estimated goal can be decided. This chapter defines N-decision problems from that point of view and discusses the properties of optimal decision and goal with a view to solve N-decision problems. It discusses some properties of 1-decision problems and shows that 1-decision problems can be reduced to simply 0-decision problems. As it seems that almost real world problems involving economic systems and public systems satisfy a pseudo complement in the domain under consideration, almost N-decision problems can be solved by the method for solving 0-decision problems. 0-decision problems may be regarded as optimization problems of logical functions. The chapter discusses the properties of optimization problems including logical functions.

Tetsuji Okuda

Papers

On Fuzzy-Mathematical Programming

A formulation of fuzzy decision problems and its application to an investment problem

Fuzzy Mathematical Programming

Decision-making and its goal in a fuzzy environment

Statistical Estimation by Fuzzy Observation Data