Fuzzy Linear Regression II

Introduction

This chapter is a continuation of Chapter 11 and is based on [1]. We wish to use our Monte Carlo method to get approximate solutions for crisp numbers a _i , 0 ≤ i ≤ m, to the fuzzy linear regression model

$$ \overline{Y}=a_0+ a_1\overline{X}_1+...+a_m \overline{X}_m, $$

for $\overline{X}_i$, 1 ≤ i ≤ m, triangular fuzzy numbers and $\overline{Y}$ a triangular fuzzy number. The fuzzy linear regression model in equation (14.1) has been previously studied in [2]-[6]. In this model the independent variables $\overline{X}_i$ will be given triangular fuzzy numbers, the dependent variable $\overline{Y}$ will be a given triangular fuzzy number, so the best way to fit the model to the data is to use real numbers for the a _i . If a _i is also a triangular fuzzy number, then $a_i\overline{X}_i$ will be a triangular shaped fuzzy number and the right side of equation (14.1) is a triangular shaped fuzzy number which is used to approximate $\overline{Y}$ a triangular fuzzy number. If the a _i are real numbers the right side of equation (14.1) will be a triangular fuzzy number which is better to use to approximate a triangular fuzzy number $\overline{Y}$. The data will be $((\overline{X}_{1l},...,\overline{X}_{ml}),\overline{Y}_l)$, 1 ≤ l ≤ n, for the $\overline{X}_{il}=(x_{il1}/x_{il2}/x_{il3})$ triangular fuzzy numbers and $\overline{Y}_l=(y_{l1}/y_{l2}/y_{l3})$ triangular fuzzy numbers. Given the data the objective is to find the “best” a _j , 0 ≤ j ≤ m. We propose to employ our Monte Carlo methods to approximate the “best” values for the a _j , j = 0,1,...,m.