The Extended Theory Of Soft Set And Its Application In Decision Making

A number of real life problems in economics, engineering, environment, socialscience which are not always all crisp. There are theories such as theory of probabilityand theory of interval mathematics can be considered as the classical mathematical toolsfor dealing with uncertainties. But theory of probabilities can deal only withstochastically stable phenomena. To verify the law of stochastically phenomena, wemust perform a large number of trials. Interval mathematics cannot appropriatelydescribe a smooth changing of information. The fuzzy set and rough set are newmathematical tools to deal with uncertainties. The membership function of fuzzy set isdifficult to determine. The upper and lower limits of equivalence relation in rough setare difficult to determine. In1999, D.Molodtsov pointed out all these theories had theirinherent difficulties. The reason for these difficulties is, possibly, the inaccuracy of theparameterization tool of the theories. Consequently, he initiated the concept of soft settheory as a mathematical tool for dealing with uncertainties which is free from theabove difficulties. The soft set theory differs from the way in which we use classicalmathematics. In classical mathematics, we construct a mathematical model of an objectand define the notion of the exact solution of this model. Usually the mathematicalmodel is too complicated to find the exact solution. So we use the approximate solutionto instead of the exact solution. In the soft set theory, we have the opposite approach tothis problem. The initial description of the object has an approximate nature, and we donot need to introduce the notion of exact solution. Maji et al. presented the concept ofthe fuzzy soft set which is based on a combination of the fuzzy set and soft set models.Maji presented some results on an application of soft sets and fuzzy soft sets in singlestage decision making problem. In the paper, the soft set was expanded considering theapplied range of the extended theory and the multistage decision problem in real lifebased on the existing research results. The theory of composite soft set and generalisedinterval-valued fuzzy soft sets were discussed in decision making problems. The mainresearch contents and results are as follows:(1)In this paper, the concept of composite soft set was proposed. Operations likeAND, OR and also the operations of union, intersection were studied. The properties ofthese operations were analysed.(2)The application of composite soft set such as arithmetic in multistage decision problems was studied.(3)The concept of generalized interval-valued fuzzy soft sets was proposed basedon the theory of interval-valued fuzzy soft sets. Operations like AND, OR and also theoperations of union, intersection were studied. The properties of these operations wereanalysed.(4)The application of generalized interval-valued fuzzy soft sets in decisionmaking problems was studied. In the last, the general interval-valued fuzzy soft set waschanged into fuzzy soft set and standard soft set to reduce the calculation.