In this paper, we introduce a new class of fuzzy linear regression model with fuzzy input – fuzzy output data using shape preserving arithmetic operations on L–R fuzzy numbers for least-squares fitting. A mixed nonlinear programming approach is used to derive the satisfying solution. This answers one of Diamond's questions [Inform. Sci. 46 (1988) 141].
Since Tanaka et al. in 1982 [11] proposed a study in linear regression analysis with fuzzy model, the fuzzy regression analysis has been widely studied and applied in a variety of substantive areas. A collection of recent papers dealing with several approaches to fuzzy regression analysis can be found in [7]. In this paper, we concentrate on the model of Diamond [2]. Diamond [2] proposed the so-called fuzzy least-squares. Recently, Diamond's method has been revisited in [3].
Let X=(m,α,β) be a triangular fuzzy number where m is the modal value of X and α and β are the left and right spreads, respectively. Diamond [2] gave a metric d on the space of all triangular fuzzy numbers by
where X=(mX,αX,βX) and Y=(mY,αY,βY) are any two triangular fuzzy numbers in . A linear structure is defined on by if t⩾0,t(m,α,β)=(tm,|t|β,|t|α) if t<0.
There are three simple fuzzy regression models considered in [2]:
The corresponding least-squares optimization problems are:
The models are rigorously justified by a projection-type theorem for cones on a Banach space containing the cone of triangular fuzzy numbers. In conclusion of [2], Diamond mentioned about the following model:
where A,B, and X are fuzzy numbers and BX is fuzzy multiplication. If (Xi,Yi) are received in the form of triangular fuzzy numbers, such a fit is neither possible, nor would it make good sense. This is because multiplication BX results in membership functions with drumlike sides.
In 1992, Bardossy et al. [1] defined a new class of distance on L–R fuzzy numbers and considered the model (F4). Recently, Hong and Do [5] introduced a shape preserving operations based on sup-t-norm convolutions in both addition and multiplication and Hong et al. [6] presented a new method to evaluate fuzzy linear regression models based on Tanaka's approach, where both input data and output data are fuzzy numbers, using TW-based fuzzy arithmetic operations. Actually, TW (the weakest t-norm) based fuzzy arithmetic operations is the only shape preserving operations in both addition and multiplication [4].
In this paper, we apply shape preserving arithmetic operations based on the TW (the weakest t-norm) convolution in both addition and multiplication and consider the least-squares optimization problem in association with the model (F4). Then a mixed nonlinear programming approach to derive the satisfying solution is developed.
When the data consist of pairs of fuzzy numbers (Xi,Yi), if we use TM-based operations, they cannot be fitted well for least-square fitting with models of the form Y=A+BX, where A,B, and X are L–R fuzzy numbers. This is because BX is not in general L–R fuzzy numbers. To overcome this computational difficulty, we use TW-based operation which is the unique shape preserving operation in both addition and multiplication. A mixed nonlinear programming approach is used to derive the satisfying solution. This answers one of Diamond's questions.
