Mathematical Foundations of Fuzzy Sets

6Fuzzification of Crisp Functions

Using fuzzy set theory to study engineering and economics problems has received much attention for a long time. The main reason is that we may have problems collecting the actual data from the environment because of the uncertainty. When the uncertainty can be regarded as fuzziness, we can consider fuzzy data in engineering and economics problems. In this case, we may need to fuzzify real‐valued functions into fuzzy functions.

For example, the cost functions and benefit functions should be considered as fuzzy functions when fuzzy data are involved in the formulated problems. In this chapter, we are going to use the extension principle and the form of mathematical expression in the decomposition theorem to fuzzify crisp functions into fuzzy functions. The fuzzification of real‐valued functions and vector‐valued functions will be studied separately.

There are two ways to fuzzify a vector‐valued function , where and denotes the ‐th component function of for . We can directly fuzzify the vector‐valued function without considering its component functions ...