Fuzzy Random Regression-Based Modeling in Uncertain Environment

Introduction

Real-world decision-making is often referred to rely on human judgment and consideration. Human contributes directly to the decision making process as many evaluations depend on human judgment, which is usually based on experience and intuition. The expression of accurate values in mathematical models has been a complicated problem yet the conventional theory of decision making and problem solving methodologies are developed within the crisp interpretation to obtain the best solution (Zeleny, 1981; Jones and Tamiz, 2010; Jubril, 2012, Vasant, 2012). However, research in decision making methodologies is influential to address the existence of uncertainties that are occurring naturally from the real problem or during the model setting. For example, the information available to a decision maker has been often imprecise because of inaccurate attribute measurements and inconsistency in priorities. Until recently, the decision-making process still utilized subjective judgments when considering human evaluations for certain cases, such as resource planning problems. Hence, it is crucial to embark upon uncertainty to obtain a feasible solution. For this reason, fuzzy sets (Zadeh, 1965) are useful for representing uncertain and imprecise information in mathematical programming, as fuzzy sets reflect these uncertainties and can therefore play a significant role in dealing with such circumstances. Thus, fuzzy programming is valuable for dealing with uncertainties for cases in which the problem model’s parameters cannot be estimated precisely from the real situation.

Interpreting uncertainties or incomplete information to the precise numerical values requires an appropriate approach that is difficult and stills a challenge (Zeleny, 1981; Inuiguchi et al., 1990; Bouyssou, 1989; Hasuike and Ishii, 2009). In practical systems, coefficient values should typically be considered to be uncertain values. The uncertainty arises in probability and/or vague situations, such as in cases of predictions of future profits, incomplete historical data, and/or replacement of decision makers, all of which result in uncertain information. Various approaches can be utilized to accommodate the uncertainty. For example, probability distribution, fuzzy numbers, and thresholds types (Bouyssou, 1989). Fuzzy sets (Zadeh, 1965) have made remarkable significant role and are useful for representing uncertain and imprecise information. Probability and possibility theories are also introduced to treat the random and fuzzy information, respectively. Such theories are significant to tackle uncertain information which exists apparently in real-world situation. Additionally, stochastic programming model (i.e., Dantzig, 1955; Sengupta, 1979) and the fuzzy programming model (i.e., Zimmermann, 1976; Zimmermann, 1978; Inuiguchi et al., 1990; Sakawa, 1993; Vasant, 2003; Vasant, 2010) has been introduced to address various real-world problems. The application of fuzzy set and possibility theories to decision-making allows decisions based on imprecise information. In linear programming, possibility theory can be used to include imprecise parameters in the problem formulation (Inuiguchi and Sakawa, 1996; Julien, 1994). Apparently, fuzzy theories and other related theory have made successful to work on the uncertainty problem in modeling and dealing with real world problem.