Introducing a trapezoidal interval type-2 fuzzy regression model

Abstract

The uncertainty is an important attribute about data that can arise from different sources including randomness and fuzziness, therefore in uncertain environments, especially, in modeling, planning, decision-making, and control under uncertainty, most data available contain some degree of fuzziness, randomness, or both, and at the same time, some of this data may be anomalous (outliers). In this regard, the new fuzzy regression approaches by creating a functional relationship between response and explanatory variables can provide efficient tools to explanation, prediction and possibly control of randomness, fuzziness, and outliers in the data obtained from uncertain environments.

In the present study, we propose a new two-stage fuzzy linear regression model based on a new interval type-2 (IT2) fuzzy least absolute deviation (FLAD) method so that regression coefficients and dependent variables are trapezoidal IT2 fuzzy numbers and independent variables are crisp. In the first stage, to estimate the IT2 fuzzy regression coefficients and provide an initial model (by original dataset), we introduce two new distance measures for comparison of IT2 fuzzy numbers and propose a novel framework for solving fuzzy mathematical programming problems. In the second stage, we introduce a new procedure to determine the mild and extreme fuzzy outlier cutoffs and apply them to remove the outliers, and then provide the final model based on a clean dataset. Furthermore, to evaluate the performance of the proposed methodology, we introduce and employ suitable goodness of fit indices. Finally, to illustrate the theoretical results of the proposed method and explain how it can be used to derive the regression model with IT2 trapezoidal fuzzy data, as well as compare the performance of the proposed model with some well-known models using training data designed by Tanaka et al. [55], we provide two numerical examples.

Keywords

Fuzziness randomness IT2 fuzzy regression parameters outlier cutoffs IT2 fuzzy goal programming problems

1 Introduction

Fuzziness and randomness are two important sources of uncertainty. Randomness describes the uncertainty of event occurrence and is determined through experimentation or over time. Fuzziness describes event ambiguity and measures the degree that an event occurs and nothing to do with time or experimentation, which is mainly caused by human subjective judgment, incomplete knowledge, and noise [1 , 35]. In the many mathematical programming problems (LP, NLP, and GP, etc.) and decision-making environments, data available contain fuzziness, randomness, or both, and at the same time, some of this data is outliers (anomalous). On the other hand, analysis of such data is one of the most important challenges for decision-makers and researchers. For this reason, many techniques have been introduced for modeling and analyzing such data. In this regard, classical regression techniques have useful, efficient, and complete tools to analyzing the randomness and outliers in crisp data [11 , 32]. In addition, fuzzy regression approaches via combining the statistical techniques with fuzzy logic approaches provide us efficient tools to analyze and management of randomness, fuzziness, and outliers in data [3 , 11]. It is worth noting that in the fuzzy regression analysis, the deviations (difference between estimated and observed responses values) are caused by system fuzziness and considered as fuzzy errors while in conventional regression analysis the deviations are caused by observations inconsistency and considered as random errors [9 –11].

The fuzzy regression models depending on whether each of its main components (i.e., input-output and parameters) are fuzzy or crisp, fall into one of the three categories as following [12, 13]:

Crisp input and fuzzy output with fuzzy parameters,

Fuzzy input-output with crisp parameters,

Fuzzy input-output with fuzzy parameters

According to [12 , 54] the fuzzy regression models from the point of view the techniques used to estimate the parameters, can be broadly classified are classified into four main categories as follows:

Fuzzy least absolute deviations method,

Fuzzy least squares deviations method,

Machine learning technique,

Heuristic approaches

In this part of the study, to identify potentially fruitful future directions, we present a comprehensive literature review based on the categories mentioned above.

1.1 Fuzzy least absolute deviations approach

Tanaka et al. [55] based on the fuzzy least absolute deviations approach (FLAD) in 1982 first introduced the fuzzy regression method. In this method, the fuzzy regression problem is formulated as a mathematical programming problem in which minimized the entire fuzziness of the model via minimizing the total spread of fuzzy regression coefficients [11]. Simplicity in programming and computations is one of the main advantages of this approach while sensitivity to outliers, and provides too wide ranges in estimates is one of its main disadvantages. This technique has been investigated and improved by numerous authors [9–11 , 54]. For instance, Shakouri et al. [50] proposed a fuzzy linear regression model with absolute errors and optimum uncertainty. They expanded the model based on the idea of reducing the distance between observed and predicted response variables. In the same year, [choy and buckly] suggested two methods to obtain the fuzzy least absolute deviation estimators for two common fuzzy linear regression models. Taheri and Kelkinnama [54] introduced a new metric based on absolute errors to develop fuzzy regression analysis. They determined the crisp parameters and fuzzy errors at the same time by minimizing the total of distances between the estimated and observed fuzzy responses. Zeng et al. [59] applied the fuzzy least absolute deviations approach to estimate the fuzzy regression coefficients with fuzzy parameters, fuzzy dependent variable, and crisp independent variables. Li et al. [37] introduced a fuzzy linear regression model based on the fuzzy least absolute error estimator with the trapezoidal fuzzy numbers.

Some other fuzzy regression models have been reported based on the fuzzy mathematical programming approaches. For instance, Hojati et al. [24] proposed a goal programming approach to estimate the parameters of a fuzzy regression model via estimating the predicted band with the help of the endpoints of the input intervals. Hassanpour et al. [21] suggested a goal programming (GP) approach to estimate crisp regression coefficients with fuzzy input-output data that minimizes total absolute errors between central points of estimated and observed output. Rafiei and Ghoreyshi [47] introduced a bi-objective GP approach for a fuzzy regression model with fuzzy input-output and fuzzy parameters in two stages. Hosseinzade et al. [28] presented a weighted GP method to estimate the type-2 fuzzy regression coefficients via minimizes the total absolute errors between the estimated and observed outputs for fuzzy output and crisp inputs. To handle the outlier problem, they proposed an omission approach and examined the behavior of value changes in the objective function when observations are omitted. In another study [29], they introduced a GP approach to estimate the fuzzy regression coefficients when the parameters and input-output variables are quasi type-2 fuzzy numbers.

1.2 Fuzzy least-squares deviations approach

The fuzzy least-squares deviations method (FLSD), is based on the ordinary least squares procedure which was initially proposed by Diamond in 1988 [15]. In the FLSD approach, the purpose is to minimize some square error between the estimated and observed responses [12, 13]. From the robustness point of view, in the fuzzy linear/nonlinear regression model when there exist some abnormal values or outliers in the dataset, the fuzzy least square estimator method is not robust and very sensitive to outliers [51, 58] while the least absolute estimator method is robust and less sensitive than least square to outliers. It is worth noting that sometimes even a single observation may dramatically influence the value of the parameter estimates.

In recent years, the FLSD approach has been improved and extended by a large number of researchers. For instance, D’urso and Massari [13] proposed a weighted fuzzy least median square and fuzzy least squares estimation to estimate the fuzzy regression coefficients. They also introduced some theoretical properties and a suitable generalization of the determination coefficient to investigate the goodness of fit the proposed model. Chachi et al. [7] introduced a squared distance based on intervals to estimate the fuzzy regression coefficients when the data available of input-output variables are triangular fuzzy numbers. Rabie et al. [45, 46], proposed a fuzzy least-squares technique to fuzzy regression modeling with interval-valued fuzzy input-output and parameters and showed how to obtain the fuzzy coefficients of the regression model. Mashinchi et al. [39] used a FLSD approach and applied a two-step method to detect the outlier in a dataset first, and then fit a model on the clean dataset. Shafaei Bajestan et al. [49] introduced a fuzzy nonlinear regression problem with IT2 fuzzy input-output based on the fuzzy least-squares method.

1.3 Machine learning methods

The generalization capability of fuzzy regression models has been enhanced via combining machine-learning approachs, such as support vector machines, neural networks, and evolutionary algorithms. Support vector machines are supervised learning models with associated learning algorithms that are employed for function estimation problems and pattern recognition [12]. In recent years, this approach improved by several papers. For instance, Hong and Hwang [25] suggested the convex optimization technique of fuzzy multiple linear and non-linear regression approaches using support vector machines, and develop support vector fuzzy regression machines. In another study [26] they introduced an estimation of fuzzy multiple non-linear regression approaches with fuzzy input-output data using a least-squares support vector machine.

The evolutionary algorithms and genetic algorithms are mostly used in optimization problems [12]. Some other fuzzy regression models have been reported based on evolutionary algorithms. For example, Gkountakou and Papadopoulos [20] introduced a linear regression model via an adaptive neuro-fuzzy inference system to estimate the compressive cement strength. Buckley and Hayashi [6] proposed a fuzzy genetic algorithm to solve fuzzy optimization problems, such as fuzzy linear regression analysis. Chan et al. [8] introduced an intelligent fuzzy regression method via an evolutionary algorithm and estimate the parameter of the model based on fuzzy least-squares method. Ezadi and Allahviranloo [17] proposed a numerical solution to fuzzy regression modeling based on z-numbers by the improved neural network.

1.4 Heuristic methods

Heuristic methods include some novel approaches, which combine the fuzzy least absolutes; fuzzy least squares, or Machine learning techniques [12]. In this context, Wei and Watada [56] introduced an expected regression technique based on credibility theory with type-2 fuzzy input-output data. Yang and Yin [58] presented a fuzzy varying coefficient regression model with robust analysis after deleting the abnormal data. Shakouri and Nadimi [51] introduced a basic curve by conventional regression to apply the fuzzy number concept and recognize the outlier data, and used the linguistic variables with conventional regression to determining outliers data. Hesamian and Akbari [22] proposed a fuzzy semi-parametric quantile regression methodology with fuzzy predictors, crisp coefficients, and fuzzy responses. In another study [23], they introduced an extension for the classical partial univariate regression model with crisp inputs and fuzzy output. Arefi [3] investigated a new approach to the problem of quantile regression modeling based on the fuzzy response variable and fuzzy parameters. They introduced a loss function between fuzzy numbers and then fit a quantile regression model between the available data based on the proposed loss function. Additionally, there are numerous papers in the framework of robust fuzzy regression analysis, fuzzy cluster-wise regression method, fuzzy regression approach with time series, fuzzy regression based on entropy, fuzzy regression analysis with Monte Carlo methods, fuzzy regression analysis based on bootstrap techniques, etc.

As mention above, in the last three decades, many studies have been performed in the field of fuzzy regression models and these models in terms of fitting accuracy, efficiency, robustness, sensitivity to outliers, and simplicity of calculations have been compared with each other in different ways. However, most of these studies have been done based on type-1 fuzzy sets [12] while type-1 fuzzy sets, despite the acceptable performance, to model data with high linguistic uncertainty, aggregation the knowledge of several experts in decision-making scenarios have not favorable abilities. In this regard, trapezoidal interval type-2 fuzzy numbers using linguistic rating systems have a favorable ability to overcome most of the above-mentioned shortcomings [18 , 57].

With the review of the articles published in the context of the fuzzy linear regression and so far as the authors know, there has not been any research on fuzzy linear regression technique based on perfectly normal trapezoidal IT2 fuzzy data. Accordingly, the main objective of this paper is to introduce a fuzzy linear regression model when dependent variable and coefficients are perfectly normal trapezoidal IT2 fuzzy numbers and independent variables are crisp. The novelty and main contributions of this study relative to the other existing fuzzy linear regression approaches, summarized as follows:

In this study, we introduce a new method to compute the multiplication and division of trapezoidal IT2 FNs.

To provide a new framework for estimating the IT2 fuzzy regression coefficients, we propose a new technique for using the unrestricted in sign IT2 fuzzy variables in crisp/fuzzy mathematical programming problems, and then highlights the differences and similarities between crisp/fuzzy LP, crisp/fuzzy GP, and crisp/fuzzy NLP.

We propose a new interval type-2 fuzzy least absolute deviations approach to build an interval type-2 fuzzy regression model.

We propose some new distance and similarity measures based on a new metric on the space of trapezoidal IT2 FSs, which are easy to handle and interpret, and reflect the intuitive meaning of distance and similarity between two trapezoidal IT2 FSs. In addition, we prove that the suggested measures satisfy the properties of the axiomatic definition for distance measures.

In this study, we explain the process of identification of outliers in the fuzzy dataset, suggest a new procedure to determine the mild and extreme fuzzy outlier cutoffs, and then propose a novel procedure to the parameter estimation of the case-deletion linear regression model in interval type-2 fuzzy environment.

In the present study, to evaluate the trapezoidal IT2 fuzzy regression models, we introduce some new performance methods based on new similarity and distance measures. We also propose a new leave-one-out cross-validation approach to investigate the goodness of fit of the proposed model

To do the above, in Section 2, first, the concepts of IT2 fuzzy sets, trapezoidal IT2 fuzzy sets, arithmetic operations between these sets is introduced in detail and then a new class of distance measures to rank and computing the difference between two trapezoidal IT2 fuzzy sets is provided. Next, introduce a new methodology for using the unrestricted in sign IT2 fuzzy variables in fuzzy/crisp mathematical programming approaches, and then highlight the similarities and differences between fuzzy/crisp GP, fuzzy/crisp LP, fuzzy/crisp NL. In Section 3, introduce and investigate a new fuzzy least absolute deviations approach to build a new fuzzy regression model, for crisp input, IT2 fuzzy output, and parameters. To estimate the parameters of the model, the proposed methodology is carried out in five steps. In Section 4, we introduce a new procedure to determine the mild and extreme fuzzy outlier cutoffs and apply them to remove the outliers (through the initial model), then final the model by clean dataset is provided. In addition, to investigate the performance of the proposed model, we provide a new leave-one-out cross-validation methodology then to illustrate the theoretical results of the proposed approach and its performances, employ a numerical example. Finally, we present a comprehensive conclusion in Section 5.

2 Preliminaries and fundamental concepts

Throughout this paper, we use ⊗, ⊕ , ⊖ and ø to denote the product, addition, subtraction, and division between fuzzy numbers, respectively, and Ω to denote the universe of discourse. In addition, we use R to denote the real numbers, $R_{0}^{+}$ to denote the all nonnegative real numbers, $F_{1}$ stands for the family of all trapezoidal type-1 fuzzy number, $F_{2}$ stands for the family of all perfectly normal trapezoidal IT2 FNs, and the family of all nonnegative perfectly normal trapezoidal IT2 FNs is denoted by $F_{2}^{+}$ .

Definition 2.1. [53] The trapezoidal type-1 fuzzy number $\dot{x}$ based on left and right spread, defined by $\dot{x} = (x^{l}, x^{u}, α, β)$ where α, β ≥ 0, x^u ≥ x^l. [x^l, x^u] is the core, α is the left-hand spread and β is the right-hand spread of $\dot{x}$ (see Fig. 2.1).

Fig. 2.1

The left and right spread of trapezoidal T1 FN $\dot{x}$ .

Fig. 2.2

The FOU and secondary membership grade of $\tilde{x}$ .

2.1 Type-2 and interval type-2 fuzzy sets

According to [19 , 57] A type-2 fuzzy set (also called a general type-2 fuzzy set) is a collection of infinite type-1 fuzzy sets into a single fuzzy set. The membership grade of each element of a type-2 fuzzy set is a type-1 fuzzy set with support bounded by the interval [0, 1], which provides an additional degree of freedom for handling linguistic uncertainties in the modeling of uncertainty. A type-2 fuzzy set denoted as $\tilde{x}$ , over the universal set Ω, defined as $\tilde{x} = {(t, u), μ_{\tilde{x}} (t, u) | \forall t \in Ω, \forall u \in J_{t} \subseteq [0, 1]}$ , Here t ∈ Ω is the primary variable, J_t is the primary membership of t, u ∈ J_t ⊆ [0, 1] is the secondary variable and $μ_{\tilde{x}} (t, u) \in [0, 1]$ is referred to as a secondary membership degree of t. Because of computational complexity of T2 FSs, the authors are interested to apply a simplified form of T2 FSs, which is called interval type-2 fuzzy sets (IT2 FSs) [57]. IT2 FSs allow to model high-level linguistic uncertainty about a variable coming from human perceptions and have enough degrees of freedom to save individual data of subjects about a word (words mean different things to different people, and so are uncertain) [18, 36].

An IT2 FS such as $\tilde{x}$ , is a simplification of T2 FS in the sense that the secondary membership function is assumed to be 1 and defined as $\tilde{x} = {(t, u), 1 | \forall t \in Ω, \forall u \in J_{t} \subseteq [0, 1]}$ . Uncertainty in the primary membership of type-2 fuzzy set $\tilde{x}$ consists of a bounded region that called the footprint of uncertainty (FOU). It is the union of all primary membership i.e., $FOU (\tilde{x}) = ⋃_{t \in Ω} J_{t}$ . $FOU (\tilde{x})$ is characterized by upper membership function (UMF) and lower membership function (LMF). Both of the membership functions are type-1 fuzzy sets. Upper membership function and lower membership function of $\tilde{x}$ is denoted by ${\dot{x}}^{U}$ and ${\dot{x}}^{L}$ , respectively. Given that it is ${\dot{x}}^{U} = UMF (\tilde{x})$ and ${\dot{x}}^{L} = LMF (\tilde{x})$ , the FOU of $\tilde{x}$ can be also expressed in the form $FOU (\tilde{x}) = ⋃_{\forall t \in Ω} [{\dot{x}}^{L} (t), {\dot{x}}^{U} (t)]$ [36, 53].

2.2 Trapezoidal interval type-2 fuzzy numbers

Trapezoidal IT2 fuzzy numbers are a kind of important fuzzy number, which can express linguistic assessments by transforming into numerical variables objectively.

Definition 2.2. [18, 36] A trapezoidal IT2 fuzzy numbers $\tilde{x}$ is called trapezoidal IT2 fuzzy number which upper membership function ( ${\dot{x}}^{u}$ ) and lower membership function( ${\dot{x}}^{l}$ ) are both trapezoidal type-1 fuzzy numbers i.e. $\tilde{x} = ({\dot{x}}^{l}, {\dot{x}}^{u}) = [(x^{l_{1}}, x^{l_{2}}, x^{l_{3}}, x^{l_{4}}; h_{1}^{l}, h_{2}^{l}), (x^{u_{1}}, x^{u_{2}}, x^{u_{3}}, x^{u_{4}}; h_{1}^{u}, h_{2}^{u})]$ where $h_{j}^{u}$ denotes the membership values of the elements x^{u
_j+1} in the ${\dot{x}}^{u}$ , and $h_{j}^{l}$ denotes the membership values of the elements x^{l
_j+1} in the ${\dot{x}}^{l}$ , j = 1, 2.

Fig. 2.3

Trapezoidal IT2 FN $\tilde{x}$ .

Remark 2.1. [18 , 53] A trapezoidal IT2 fuzzy number $\tilde{x}$ , is said to be a perfectly normal trapezoidal IT2 number if both its upper membership functions ( ${\dot{x}}^{u}$ ) and lower membership functions ( ${\dot{x}}^{l}$ ) are normal, and denoted by $\tilde{x} = ({\dot{x}}^{l}, {\dot{x}}^{u}) = [(x^{l_{1}}, x^{l_{2}}, x^{l_{3}}, x^{l_{4}}), (x^{u_{1}}, x^{u_{2}}, x^{u_{3}}, x^{u_{4}})]$ . Where ${\dot{x}}^{l} \subseteq {\dot{x}}^{u}$ , ${\dot{x}}^{l} (t) \leq {\dot{x}}^{u} (t)$ and ${\dot{x}}^{l} (t), {\dot{x}}^{u} (t) \in [0, 1]$ ∀t∈. This means the grade of membership of t belongs to the interval $[{\dot{x}}^{l} (t), {\dot{x}}^{u} (t)]$ , the greatest grade of membership at t is ${\dot{x}}^{u} (t)$ and the least grade of membership at t is ${\dot{x}}^{l} (t)$ [5].

Definition 2.3. [18, 38] The trapezoidal IT2 fuzzy number $\tilde{x}$ based on left and right spreads denoted by $\tilde{x} = [{\dot{x}}^{l}, {\dot{x}}^{u}] = [(x^{l_{2}}, x^{l_{3}}, α_{l}, β_{l}), (x^{u_{2}}, x^{u_{3}}, α_{u}, β_{u})]$ , where x^{u
₂} ≤ x^{u
₃}, x^{l
₂} ≤ x^{l
₃}, α_l, β_l, α_u, β_u≥ ∘. [x^{u
₂}, x^{u
₃}] is the core of ${\dot{x}}^{u}$ and α_u, β_u are the left and right spreads of ${\dot{x}}^{u}$ respectively. In addition, [x^{l
₂}, x^{l
₃}] is the core of ${\dot{x}}^{l}$ and α_l, β_l are the left and right spreads of ${\dot{x}}^{l}$ respectively.

Fig. 2.4

Trapezoidal IT2 FN $\tilde{x}$ based on left and right spreads.

Definition 2.4. [22, 57] Let $\tilde{x} = [{\dot{x}}^{l}, {\dot{x}}^{u}] = [(x^{l_{1}}, x^{l_{2}}, x^{l_{3}}, x^{l_{4}}), (x^{u_{1}}, x^{u_{2}}, x^{u_{3}}, x^{u_{4}})]$ and $\tilde{y} = [{\dot{y}}^{l}, {\dot{y}}^{u}] = [(y^{l_{1}}, y^{l_{2}}, y^{l_{3}}, y^{l_{4}}), (y^{u_{1}}, y^{u_{2}}, y^{u_{3}}, y^{u_{4}})]$ be two trapezoidal IT2 FNs, then

${\dot{x}}^{u} = {\dot{y}}^{u} \Leftrightarrow x^{u_{i}} = y^{u_{i}} i = 1, . . ., 4$

${\dot{x}}^{l} = {\dot{y}}^{l} \Leftrightarrow x^{l_{i}} = y^{l_{i}} i = 1, . . ., 4$

$\tilde{x} = \tilde{y}$ $\Leftrightarrow {\dot{x}}^{l} = {\dot{y}}^{l} and {\dot{x}}^{u} = {\dot{y}}^{u}$

$\tilde{x} \subseteq \tilde{y} \Leftrightarrow {\dot{x}}^{l} \subseteq {\dot{y}}^{l} and {\dot{x}}^{u} \subseteq {\dot{y}}^{u}$

For symmetric normalized fuzzy number $\tilde{x}$ : ${\tilde{x}}^{c} = [(1 - x^{l_{4}}, 1 - x^{l_{3}}, 1 - x^{l_{2}}, 1 - x^{l_{1}}), (1 - x^{u_{4}}, 1 - x^{u_{3}}, 1 - x^{u_{2}}, 1 - x^{u_{1}})]$

Definition 2.5. [18, 57] Let $\tilde{x}$ and $\tilde{y}$ be two trapezoidal IT2 FNs, where $\tilde{x} = [(x^{l_{1}}, x^{l_{2}}, x^{l_{3}}, x^{l_{4}}), (x^{u_{1}}, x^{u_{2}}, x^{u_{3}}, x^{u_{4}})]$ and $\tilde{y} = [(y^{l_{1}}, y^{l_{2}}, y^{l_{3}}, y^{l_{4}}), (y^{u_{1}}, y^{u_{2}}, y^{u_{3}}, y^{u_{4}})]$ . The arithmetic operations between $\tilde{x}$ and $\tilde{y}$ are defined as follows.

The addition operation: $\begin{matrix} [({\dot{x}}^{l}, {\dot{x}}^{u}) \oplus ({\dot{y}}^{l}, {\dot{y}}^{u})] = [({\dot{x}}^{l} \oplus {\dot{y}}^{l}), ({\dot{x}}^{u} \oplus {\dot{y}}^{u})] \\ = [(x^{l_{1}} + y^{l_{1}}, x^{l_{2}} + y^{l_{2}}, x^{l_{3}} + y^{l_{3}}, x^{l_{4}} + y^{l_{4}}), \\ (x^{u_{1}} + y^{u_{1}}, x^{u_{2}} + y^{u_{2}}, x^{u_{3}} + y^{u_{3}}, x^{u_{4}} + y^{u_{4}})] \end{matrix}$ (1)

$⊖ \tilde{x} = ⊖ [⊖ {\dot{x}}^{l}, ⊖ {\dot{x}}^{u}] = [{\dot{x}}^{l}, {\dot{x}}^{u}]$ $\begin{matrix} = [(- x^{l_{4}}, - x^{l_{3}}, - x^{l_{2}}, - x^{l_{1}}) \\ , (- x^{u_{4}}, - x^{u_{3}}, - x^{u_{2}}, - x^{u_{1}})] \end{matrix}$

The subtraction operation: $\begin{matrix} \tilde{x} ⊖ \tilde{y} = ({\dot{x}}^{l}, {\dot{x}}^{u}) ⊖ ({\dot{y}}^{l}, {\dot{y}}^{u})] = [({\dot{x}}^{l} ⊖ {\dot{y}}^{l}), \\ ({\dot{x}}^{u} ⊖ {\dot{y}}^{u})] = [(x^{l_{1}} - y^{l_{4}}, x^{l_{2}} - y^{l_{3}}, x^{l_{3}} - y^{l_{2}}, x^{l_{4}} \\ - y^{l_{1}}), (x^{u_{1}} - y^{u_{4}}, x^{u_{2}} - y^{u_{3}}, x^{u_{3}} - y^{u_{2}}, \\ x^{u_{4}} - y^{u_{1}})] \end{matrix}$ (2)

The multiplication with a scalar w: $\begin{matrix} w \tilde{x} = \\ {\begin{matrix} [({wx}^{l_{1}}, {wx}^{l_{2}}, {wx}^{l_{3}}, {wx}^{l_{4}}), ({wx}^{u_{1}}, {wx}^{u_{2}}, {wx}^{u_{3}}, {wx}^{u_{4}})], & w \geq 0 \\ [({wx}^{l_{4}}, {wx}^{l_{3}}, {wx}^{l_{2}}, {wx}^{l_{1}}), ({wx}^{u_{4}}, {wx}^{u_{3}}, {wx}^{u_{2}}, {wx}^{u_{1}})], & w < 0 \end{matrix} \end{matrix}$ (3)

$\frac{1}{w} \tilde{x} =$ ${\begin{matrix} [(\frac{x^{l_{1}}}{w}, \frac{x^{l_{2}}}{w}, \frac{x^{l_{3}}}{w}, \frac{x^{l_{4}}}{w}), (\frac{x^{u_{1}}}{w}, \frac{x^{u_{2}}}{w}, \frac{x^{u_{3}}}{w}, \frac{x^{u_{4}}}{w})], & w > 0 \\ [(\frac{x^{l_{4}}}{w}, \frac{x^{l_{3}}}{w}, \frac{x^{l_{2}}}{w}, \frac{x^{l_{1}}}{w}), (\frac{x^{u_{4}}}{w}, \frac{x^{u_{3}}}{w}, \frac{x^{u_{2}}}{w}, \frac{x^{u_{1}}}{w})], & w < 0 \end{matrix}$

The multiplication operation: $\begin{matrix} \tilde{x} \otimes \tilde{y} = [({\dot{x}}^{l} \otimes {\dot{y}}^{l}), ({\dot{x}}^{u} \otimes {\dot{y}}^{u})] = [(x^{l_{1}} y^{l_{1}}, x^{l_{2}} y^{l_{2}}, \\ x^{l_{3}} y^{l_{3}}, x^{l_{4}} y^{l_{4}}), (x^{u_{1}} y^{u_{1}}, x^{u_{2}} y^{u_{2}}, x^{u_{3}} y^{u_{3}}, x^{u_{4}} y^{u_{4}})] \\ if x^{u_{1}}, y^{u_{1}} \geq 0 \end{matrix}$ (4) $\begin{matrix} \tilde{x} \otimes \tilde{y} = [(z^{l_{1}}, z^{l_{2}}, z^{l_{3}}, z^{l_{4}}), (z^{u_{1}}, z^{u_{2}}, z^{u_{3}}, z^{u_{4}})] \\ where z^{m_{i}} = \min {x^{m_{i}} y^{m_{i}}, x^{m_{i}} y^{m_{(5 - i)}}, x^{m_{(5 - i)}} y^{m_{i}}, \\ x^{m_{(5 - i)}} y^{m_{(5 - i)}}}, i = 1, 2; m \in {l, u} z^{m_{j}} = \\ max {x^{m_{(5 - j)}} y^{m_{(5 - j)}}, x^{m_{(5 - j)}} y^{m_{j}}, x^{m_{j}} y^{m_{(5 - j)}}, \\ x^{m_{j}} y^{m_{j}}}, j = 3, 4 m \in {l, u} \end{matrix}$

The division operation: $\begin{matrix} \tilde{x} ø \tilde{y} = [(\frac{x^{l_{1}}}{y^{l_{1}}}, \frac{x^{l_{2}}}{y^{l_{2}}}, \frac{x^{l_{3}}}{y^{l_{3}}}, \frac{x^{l_{4}}}{y^{l_{4}}}), (\frac{x^{u_{1}}}{y^{u_{1}}}, \frac{x^{u_{2}}}{y^{u_{2}}}, \frac{x^{u_{3}}}{y^{u_{3}}}, \\ \frac{x^{u_{4}}}{y^{u_{4}}})] if x^{u_{1}} \geq 0, y^{u_{1}} > 0 \tilde{x} ø \tilde{y} = [(w^{l_{1}}, w^{l_{2}}, \\ w^{l_{3}}, w^{l_{4}}), (w^{u_{1}}, w^{u_{2}}, w^{u_{3}}, w^{u_{4}})] where w^{m_{i}} \\ = \min {\frac{x^{m_{i}}}{y^{m_{i}}}, \frac{x^{m_{i}}}{y^{m_{(5 - i)}}}, \frac{x^{m_{(5 - i)}}}{y^{m_{i}}}, \frac{x^{m_{(5 - i)}}}{y^{m_{(5 - i)}}}}, i = 1, 2; \\ m \in {l, u} w^{m_{j}} = \max {\frac{x^{m_{(5 - j)}}}{y^{m_{(5 - j)}}}, \frac{x^{m_{(5 - j)}}}{y^{m_{j}}}, \\ \frac{x^{m_{j}}}{y^{m_{(5 - j)}}}, \frac{x^{m_{j}}}{y^{m_{(5 - j)}}}}, j = 3, 4; m \in {l, u} \end{matrix}$

The n th root: $\begin{matrix} \sqrt[n]{\tilde{x}} = [(\sqrt[n]{x^{u_{1}}}, \sqrt[n]{x^{u_{2}}}, \sqrt[n]{x^{u_{3}}}, \sqrt[n]{x^{u_{4}}}), \\ (\sqrt{x^{l_{1}} n}, \sqrt[n]{x^{l_{2}}}, \sqrt[n]{x^{l_{3}}}, \sqrt[n]{x^{l_{4}}})], if x^{u_{1}} \geq 0 \end{matrix}$

The power operation: $\begin{matrix} (\tilde{x})^{n} = [((x^{l_{1}})^{n}, (x^{l_{2}})^{n}, (x^{l_{3}})^{n}, (x^{l_{4}})^{n}), (x^{u_{1}})^{n}, \\ (x^{u_{2}})^{n}, (x^{u_{3}})^{n}, (x^{u_{4}})^{n})], if x^{u_{1}} \geq 0 \end{matrix}$

2.2.1 A class of distance measures for trapezoidal IT2 fuzzy numbers

With no doubt, fuzzy distance measures play a vital role to differentiate between two fuzzy sets or objects and so far, many fuzzy distance measures have been introduced including the Hamming distance, the Euclidean distance, and the Minkowski distance. Based on the existing literature, fuzzy Hamming distance measures have been extensively studied due to their applications in many fields, e.g. regression analysis, risk analysis, data mining, medical diagnosis, signal processing, pattern recognition, decision-making, network comparison, and the like. On the other hand, in the trapezoidal IT2 fuzzy regression analysis, one of the most important aspects of the analysis of randomness and fuzziness is the usage of a convenient distance on the family of trapezoidal IT2 fuzzy sets. In this regard, the Hamming distance is a convenient distance for the family of trapezoidal IT2 fuzzy sets, which can easily provide the intuitive meaning of the difference between the estimated and observed trapezoidal IT2 fuzzy responses in IT2 fuzzy regression analysis [4 , 52].

Definition 2.6. [2 , 52] Suppose that $\dot{x} = [x_{1}, x_{2}, x_{3}, x_{4}], \dot{y} = [y_{1}, y_{2}, y_{3}, y_{4}] \in F_{1}$ . The Hamming distance (d_H) between $\dot{x}$ and $\dot{y}$ is defined by $d_{H} (\tilde{x}, \tilde{y}) = \frac{1}{4} [(| x_{1} - y_{1} |) + (| x_{2} - y_{2} |) + (| x_{3} - y_{3} |) + (| x_{4} - y_{4} |)] = \frac{1}{4} \sum_{i = 1}^{4} (| x_{i} - y_{i} |)$ .

Definition 2.7. [2 , 52] Let $\tilde{x} = [{\dot{x}}^{l}, {\dot{x}}^{u}] = [(x^{l_{1}}, x^{l_{2}}, x^{l_{3}}, x^{l_{4}}), (x^{u_{1}}, x^{u_{2}}, x^{u_{3}}, x^{u_{4}})]$ and $\tilde{y} = [{\dot{y}}^{l}, {\dot{y}}^{u}] = [(y^{l_{1}}, y^{l_{2}}, y^{l_{3}}, y^{l_{4}}), (y^{u_{1}}, y^{u_{2}}, y^{u_{3}}, y^{u_{4}})]$ be two trapezoidal IT2 FNs. The Hamming distance between $\tilde{x}$ and $\tilde{y}$ is defined by $d_{H} (\tilde{x}, \tilde{y}) = λ d_{H} ({\dot{x}}^{u}, {\dot{y}}^{u}) + (1 - λ) d_{H} ({\dot{x}}^{l}, {\dot{y}}^{l})$ where λ ∈ (0, 1), which is determined by the decision maker. The value $λ \in (0, \frac{1}{2})$ means that LMFs of given IT2FNs have more importance and $λ \in (\frac{1}{2}, 1)$ mentions that UMFs are more important.

It is obvious that if $λ = \frac{1}{2}$ , the LMFs and UMFs do have equal significance and $d_{H} (\tilde{x}, \tilde{y}) = \frac{1}{2} [d_{H} ({\dot{x}}^{u}, {\dot{y}}^{u}) + d_{H} ({\dot{x}}^{l}, {\dot{y}}^{l})]$ which in, ${\dot{x}}^{u}, {\dot{y}}^{u}, {\dot{x}}^{l}$ and ${\dot{y}}^{l}$ are trapezoidal type-1 fuzzy numbers. $d_{H} ({\dot{x}}^{u}, {\dot{y}}^{u})$ , $d_{H} ({\dot{x}}^{l}, {\dot{y}}^{l})$ and $d_{H} (\tilde{x}, \tilde{y})$ with $λ = \frac{1}{2}$ can be defined as $\begin{matrix} d_{H} (\tilde{x}, \tilde{y}) = \frac{1}{2} [d_{H} ({\dot{x}}^{u}, {\dot{y}}^{u}) + d_{H} ({\dot{x}}^{l}, {\dot{y}}^{l})] \\ = \frac{1}{8} [\sum_{m = 1}^{4} (| x^{u_{m}} - y^{u_{m}} | + | x^{l_{m}} - y^{l_{m}} |)] \\ d_{H} (\tilde{x}, \tilde{y}) = \frac{1}{2} [d_{H} ({\dot{x}}^{u}, {\dot{y}}^{u}) + d_{H} ({\dot{x}}^{l}, {\dot{y}}^{l})] \\ = \frac{1}{8} [\sum_{m = 1}^{4} (| x^{u_{m}} - y^{u_{m}} | + | x^{l_{m}} - y^{l_{m}} |)] \end{matrix}$ (5)

Definition 2.8. [2, 52] The Hamming distance between $\tilde{x}$ , $\tilde{y}$ $\in F_{2}$ namely $d_{H} (\tilde{x}, \tilde{y})$ is defined as the mapping $d_{H} : F_{2} \otimes F_{2} \to R^{+} \cup {0} .$

Theorem 2.1. $(F_{2}, d_{H})$ is a metric space.

Proof. Let $\tilde{x}$ , $\tilde{y}$ , $\tilde{z} \in F_{2}$ be three trapezoidal IT2 FNs. It suffices to prove the following items.

$d_{H} (\tilde{x}, \tilde{y}) = 0$ iff $\tilde{x} \approx \tilde{y}$ (i.e., ${\dot{x}}^{u} \approx {\dot{y}}^{u}$ and ${\dot{x}}^{l} \approx {\dot{y}}^{l}$ )

$d_{H} (\tilde{x}, \tilde{y}) = d_{H} (\tilde{y}, \tilde{x})$

$d_{H} (\tilde{x}, \tilde{y}) \geq 0$

$d_{H} (\tilde{x}, \tilde{z}) \leq d_{H} (\tilde{x}, \tilde{y}) + d_{H} (\tilde{y}, \tilde{z})$

Proof. Properties (i), (ii) and (iii) are obvious and we only prove (iv): $\begin{matrix} d_{H} (\tilde{x}, \tilde{z}) = \frac{1}{8} [\sum_{i = 1}^{4} (| x^{u_{i}} - z^{u_{i}} | + | x^{l_{i}} - z^{l_{i}} |)] \\ = \frac{1}{8} [\sum_{i = 1}^{4} (| x^{u_{i}} - z^{u_{i}} \pm y^{u_{i}} | + | x^{l_{i}} - z^{l_{i}} \pm y^{l_{i}} |)] \\ \leq \frac{1}{8} [\sum_{i = 1}^{4} (| x^{u_{i}} - y^{u_{i}} | + | y^{u_{i}} - z^{u_{i}} | \\ + | x^{l_{i}} - y^{l_{i}} | + | y^{l_{i}} - z^{l_{i}} |)] \\ = \frac{1}{8} [\sum_{i = 1}^{4} (| x^{u_{i}} - y^{u_{i}} | + | x^{l_{i}} - y^{l_{i}} |)] \\ + \frac{1}{8} [\sum_{i = 1}^{4} (| y^{u_{i}} - z^{u_{i}} | + | y^{l_{i}} - z^{l_{i}} |)] \\ = d_{H} (\tilde{x}, \tilde{y}) + d_{H} (\tilde{y}, \tilde{z}) \end{matrix}$

2.2.2 Fuzzy sign distance measure for trapezoidal IT2 FNs

To deal with the maximization or minimization of the mathematical programming approaches and compare with fuzzy numbers, some authors propose a ranking method to compute the signed distance from fuzzy number to y-axis [4, 16]. In this part of study, a new signed distance measure to rank the trapezoidal IT2 FNs is defined. The signed distance between $\tilde{x}$ , $\tilde{O} \in F_{2}$ namely $d_{S} (\tilde{x}, \tilde{O})$ is defined as $d_{S} : F_{2} \otimes F_{2} \to R$ .

Definition 2.9. [18] Let $\tilde{x}, \tilde{O} \in F_{2}$ . Then the signed distance from $\tilde{x}$ to $\tilde{O}$ (y-axis) is defined as follows. $\begin{matrix} d_{S} (\tilde{x}, \tilde{O}) = \frac{1}{8} (x^{u_{1}} + x^{u_{2}} + x^{u_{3}} + x^{u_{4}} + x^{l_{1}} + x^{l_{2}} \\ + x^{l_{3}} + x^{l_{4}}) = \frac{1}{8} \sum_{i = 1}^{4} (x^{u_{i}} + x^{l_{i}}) . \end{matrix}$

Definition 2.10. [2 , 38] Let $\tilde{x}, \tilde{y} \in F_{2}$ , then based on signed distance $d_{S} (., \tilde{O})$ , we define the ranking order as:

$\tilde{x} ⪯ \tilde{y} iff d_{S} (\tilde{x}, \tilde{O}) \leq d_{S} (\tilde{y}, \tilde{O})$

$\tilde{x} ≻ \tilde{y} iff d_{S} (\tilde{x}, \tilde{O}) ≻ d_{S} (\tilde{y}, \tilde{O})$

$\tilde{x} \approx \tilde{y} iff d_{S} (\tilde{x}, \tilde{O}) = d_{S} (\tilde{y}, \tilde{O})$

Theorem 2.2. Let $\tilde{x} = [(x^{l_{1}}, x^{l_{2}}, x^{l_{3}}, x^{l_{4}})$ , (x^u
₁, x^u
₂, x^u
₃, x^u
₄)] and $\tilde{y} = [(y^{l_{1}}, y^{l_{2}}, y^{l_{3}}, y^{l_{4}})$ , $(y^{u_{1}}, y^{u_{2}}, y^{u_{3}}, y^{u_{4}})] \in F_{2}$ , then $d_{S} (w \tilde{x} + \tilde{y}, \tilde{O}) = {wd}_{S} (\tilde{x}, \tilde{O}) + d_{S} (\tilde{y}, \tilde{O})$ for all k ∈ R.

Proof.

For case w ≥ 0: $w {\tilde{x}}_{i} + {\tilde{y}}_{i} = [({wx}_{i}^{l_{1}} + y_{i}^{l_{1}}, {wx}_{i}^{l_{2}} + y_{i}^{l_{2}}, {wx}_{i}^{l_{3}} + y_{i}^{l_{3}}, {wx}_{i}^{l_{4}} + y_{i}^{l_{4}}), ({wx}_{i}^{u_{1}} + y_{i}^{u_{1}}, {wx}_{i}^{u_{2}} + y_{i}^{u_{2}}, {wx}_{i}^{u_{3}} + y_{i}^{u_{3}}, {wx}_{i}^{u_{4}} + x_{i}^{u_{4}})] d_{S} (w {\tilde{x}}_{i} + {\tilde{y}}_{i}, \tilde{O}) = \frac{1}{8} ({wx}_{i}^{l_{1}} + y_{i}^{l_{1}} + {wx}_{i}^{l_{2}} + y_{i}^{l_{2}} + {wx}_{i}^{l_{3}} + y_{i}^{l_{3}} + {wx}_{i}^{l_{4}} + y_{i}^{l_{4}} + {wx}_{i}^{u_{1}} + y_{i}^{u_{1}} + {wx}_{i}^{u_{2}} + y_{i}^{u_{2}} + {wx}_{i}^{u_{3}} + y_{i}^{u_{3}} + {wx}_{i}^{u_{4}} + x_{i}^{u_{4}}) d_{S} (w {\tilde{x}}_{i} + {\tilde{y}}_{i}, \tilde{O}) = \frac{1}{8} \sum_{m = 1}^{4} ({wx}_{i}^{l_{m}} + {wx}_{i}^{u_{m}}) + \frac{1}{8} \sum_{m = 1}^{4} (y_{i}^{l_{m}} + y_{i}^{u_{m}}) = w \frac{1}{8} \sum_{m = 1}^{4} (x_{i}^{l_{m}} + x_{i}^{u_{m}}) + \frac{1}{8} \sum_{m = 1}^{4} (y_{i}^{l_{m}} + y_{i}^{u_{m}}) = {kd}_{S} (\tilde{x}, \tilde{O}) + d_{S} (\tilde{y}, \tilde{O})$

For case w < 0 {% } $w {\tilde{x}}_{i} + {\tilde{y}}_{i} = [({wx}_{i}^{l_{4}} + y_{i}^{l_{1}}, {wx}_{i}^{l_{3}} + y_{i}^{l_{2}}, {wx}_{i}^{l_{2}} + y_{i}^{l_{3}}, {wx}_{i}^{l_{1}} + y_{i}^{l_{1}}), ({wx}_{i}^{u_{4}} + y_{i}^{u_{1}}, {wx}_{i}^{u_{3}} + y_{i}^{u_{2}}, {wx}_{i}^{u_{2}} + y_{i}^{u_{3}}, {wx}_{i}^{u_{1}} + y_{i}^{u_{4}})] d_{S} (w {\tilde{x}}_{i} + {\tilde{y}}_{i}, \tilde{O} = \frac{1}{8} ({wx}_{i}^{l_{4}} + y_{i}^{l_{1}} + {wx}_{i}^{l_{3}} + y_{i}^{l_{2}} + {wx}_{i}^{l_{2}} + y_{i}^{l_{3}} + {wx}_{i}^{l_{1}} + y_{i}^{l_{1}} + {wx}_{i}^{u_{4}} + y_{i}^{u_{1}} + {wx}_{i}^{u_{3}} + y_{i}^{u_{2}} + {wx}_{i}^{u_{2}} + y_{i}^{u_{3}} + {wx}_{i}^{u_{1}} + y_{i}^{u_{4}}) d_{S} (w {\tilde{x}}_{i} + {\tilde{y}}_{i}, \tilde{O}) = \frac{1}{8} \sum_{m = 1}^{4} ({wx}_{i}^{l_{m}} + {wx}_{i}^{u_{m}}) + \frac{1}{8} \sum_{m = 1}^{4} (y_{i}^{l_{m}} + y_{i}^{u_{m}}) = {wd}_{S} (\tilde{x}, \tilde{O}) + d_{S} (\tilde{y}, \tilde{O}) .$ □

2.3 Free variables in mathematical programming problems

The linear programming formulation restricts the sign of unknown variables to be nonnegative, and the nonnegative condition is a constraint (requirement) in solution procedures of these problems. Free (unrestricted in sign) variables are variables that have no upper or lower bound [41]. For each free variable ɛ_i, we use the decomposition into the negative and the positive parts as follows: $\begin{matrix} ɛ_{i}^{+} = max {0, ɛ_{i}} = \frac{ɛ_{i}}{2} + \frac{| ɛ_{i} |}{2}, ɛ_{i}^{-} = max {0, - ɛ_{i}} \\ = \frac{- ɛ_{i}}{2} + \frac{| - ɛ_{i} |}{2} \end{matrix}$ $ɛ_{i}^{+} + ɛ_{i}^{-} = (\frac{ɛ_{i}}{2} + \frac{| ɛ_{i} |}{2}) + (\frac{- ɛ_{i}}{2} + \frac{| ɛ_{i} |}{2}) = | ɛ_{i} |$ (6.1) $ɛ_{i}^{+} - ɛ_{i}^{-} = (\frac{ɛ_{i}}{2} + \frac{| ɛ_{i} |}{2}) - (\frac{- ɛ_{i}}{2} - \frac{| ɛ_{i} |}{2}) = ɛ_{i}$ (6.2)

Remark 2.2. In the GP problems, to remove absolute values from the achievement function, we replace |ɛ_i| with $ɛ_{i}^{+} + ɛ_{i}^{-}$ , ɛ_i with $ɛ_{i}^{+} - ɛ_{i}^{-}$ and add the non-negative constraint $ɛ_{i}^{+} \geq 0, ɛ_{i}^{-} \geq 0$ , $ɛ_{i}^{+} \times ɛ_{i}^{-} = 0$ and then solve the modified GP problem as $\begin{matrix} if ɛ_{i}^{+} = ɛ_{i}^{-} = 0 then ɛ_{i} = 0, \\ if ɛ_{i}^{+} > 0 then ɛ_{i} = ɛ_{i}^{+}, \\ if ɛ_{i}^{-} > 0 then ɛ_{i} = - ɛ_{i}^{-} . \end{matrix}$

2.4 Trapezoidal IT2 fuzzy LP and trapezoidal IT2 fuzzy GP problems

In the many fuzzy mathematical programming problems, the constraints and/or parameters are included non-probabilistic uncertainty and defined by the knowledge of several experts. To aggregation and convert the different subjective opinions and knowledge of several experts in decision-making scenarios into crisp, the trapezoidal IT2 FSs have favorable abilities. In this regard, several useful linguistic rating systems have been introduced [19, 53]. Figure 2.5 depicted aggregation of the different opinions of several experts in decision-making scenarios.

Fig. 2.5

Aggregation the different opinions of several experts’ in decision-making scenarios.

The general standard form of crisp LP problem is defined as $\begin{matrix} Min z \approx \sum_{j = 1}^{m} c_{j} x_{j} \\ s . t . \sum_{j = 1}^{m} a_{ij} x_{j} = b_{i}, i = 1, . . ., n, \\ x_{j} \geq 0, \\ c_{j}, a_{ij}, b_{i} \in R . \end{matrix}$ (8) where x_j is the decision variable, c_j is the coefficient of the x_j in the objective function, a_ij is the coefficient of the constraint matrix and b_i is the fuzzy right-hand side of the constraint [5]. In addition, the general standard form of IT2 fuzzy LP problem with ${\tilde{x}}_{i} \in F_{2}^{+}$ and ${\tilde{c}}_{j}, {\tilde{a}}_{ij}, {\tilde{b}}_{i} \in F_{2}$ is defined as [2 , 18] $\begin{matrix} Min \tilde{z} \approx \sum_{j = 1}^{m} {\tilde{c}}_{j} \otimes {\tilde{x}}_{j} \\ s . t . \sum_{j = 1}^{m} {\tilde{a}}_{ij} \otimes {\tilde{x}}_{j} \approx {\tilde{b}}_{i}, i = 1, . . ., n, \\ {\tilde{x}}_{j} ≽ \tilde{0}, j = 1, . . ., m . \end{matrix}$ (9)

Remark 2.3. [43] By taking the addition and multiplication operations between trapezoidal IT2 FNs given in Definition 2.5, $\sum_{j = 1}^{m} {\tilde{a}}_{ij} \otimes {\tilde{x}}_{j} \approx {\tilde{b}}_{i}$ can be written as $\begin{matrix} \sum_{j = 1}^{m} [(a_{ij}^{l_{1}} x_{j}^{l_{1}}, a_{ij}^{l_{2}} x_{j}^{l_{2}}, a_{ij}^{l_{3}} x_{j}^{l_{3}}, a_{ij}^{l_{4}} x_{j}^{l_{4}}), \\ (a_{ij}^{u_{1}} x_{j}^{u_{1}}, a_{ij}^{u_{2}} x_{j}^{u_{2}}, a_{ij}^{u_{3}} x_{j}^{u_{3}}, a_{ij}^{u_{4}} x_{j}^{u_{4}})] \\ \approx [(b_{i}^{l_{1}}, b_{i}^{l_{2}}, b_{i}^{l_{3}}, b_{i}^{l_{4}}), (b_{i}^{u_{1}}, b_{i}^{u_{2}}, b_{i}^{u_{3}}, b_{i}^{u_{4}})] \end{matrix}$ (8.1)

In addition, according to the equality between trapezoidal IT2 FNs given in Definition 2.4, the fuzzy equality (8.1) $\sum_{j = 1}^{m} {\tilde{a}}_{ij} \otimes {\tilde{x}}_{j} \approx {\tilde{b}}_{i}$ can be expressed as $\sum_{j = 1}^{m} (a_{ij} x_{j})^{l_{t}} = b_{i}^{l_{t}}, \sum_{j = 1}^{m} (a_{ij} x_{j})^{u_{t}} = b_{i}^{u_{t}} fort = 1, . . ., 4$ . So, the IT2 fuzzy LP problem (8) can be expressed as $\begin{matrix} Min \tilde{z} \approx \sum_{j = 1}^{m} {\tilde{c}}_{j} \otimes {\tilde{x}}_{j} \\ s . t \\ \sum_{j = 1}^{m} [(a_{ij} x_{j})^{t} = b_{i}^{t}, t \in {l_{1}, l_{2}, l_{3}, l_{4}, u_{1}, u_{2}, u_{3}, u_{4}}, \\ {\tilde{\tilde{x}}}_{j} ≽ \tilde{\tilde{0}}, j = 1, . . ., m . \end{matrix}$ (8.2)

Definition 2.11. [18, 43] A vector $\tilde{x} = ({\tilde{x}}_{1}, {\tilde{x}}_{2}, . . ., {\tilde{x}}_{m}) \in (F_{2})^{m}$ is named a feasible solution of problem (8) if and only if it satisfies all the constraints of the problem (8). We denote the set of all feasible solutions of problem (8) by $\tilde{S}$ , that is $\tilde{S} = {\tilde{x} \in (F_{2})^{m} : \sum_{j = 1}^{m} {\tilde{a}}_{ij} \otimes {\tilde{x}}_{j} \approx {\tilde{b}}_{i}, {\tilde{x}}_{j} ≽ \tilde{0}, i = 1, . . ., n, j = 1, . . ., m}$ .

Definition 2.12. [43] A vector ${\tilde{x}}^{*} = ({\tilde{x}}_{1}^{*}, {\tilde{x}}_{2}^{*}, . . ., {\tilde{x}}_{m}^{*}) \in (F_{2})^{m}$ is called an exact optimal solution of problem (8) if:

${\tilde{x}}_{j}^{*} ≽ \tilde{0}, j = 1, . . ., m$ and $\sum_{j = 1}^{m} {\tilde{a}}_{ij} \otimes {\tilde{x}}_{j}^{*} \approx {\tilde{b}}_{i} i = 1, . . ., n,$

If there exist any ${\tilde{x}}_{j}^{'} ≽ \tilde{0}, j = 1, . . ., m$ such that $\sum_{j = 1}^{m} {\tilde{a}}_{ij} \otimes {\tilde{x}}_{j}^{'} \approx {\tilde{b}}_{i} i = 1, . . ., n$ ; then $d_{S} (\sum_{j = 1}^{m} {\tilde{c}}_{j} \otimes {\tilde{x}}_{j}^{*}, \tilde{O}) < d_{S} (\sum_{j = 1}^{m} {\tilde{c}}_{j} \otimes {\tilde{x}}_{j}^{'}, \tilde{O})$ (in the case of Min problem) and $d_{S} (\sum_{j = 1}^{m} {\tilde{c}}_{j} \otimes {\tilde{x}}_{j}^{*}, \tilde{O}) > d_{S} (\sum_{j = 1}^{m} {\tilde{c}}_{j} \otimes {\tilde{x}}_{j}^{'}, \tilde{O})$ (in the case of Max problem).

Theorem 2.3. The following programming problem can be converted to LP form. $\begin{matrix} Min d_{S} (\tilde{z}, \tilde{O}) = \sum_{j = 1}^{m} d_{S} ({\tilde{c}}_{j} \otimes {\tilde{x}}_{j}, \tilde{O}) \\ s . t . \sum_{j = 1}^{m} [(a_{ij} x_{j})^{t} = b_{i}^{t}, \\ t \in {l_{1}, l_{2}, l_{3}, l_{4}, u_{1}, u_{2}, u_{3}, u_{4}}, i = 1, . . ., n, \\ d_{S} ({\tilde{x}}_{j}, \tilde{O}) \geq 0, j = 1, . . ., m, \\ {\tilde{c}}_{j}, {\tilde{a}}_{ij}, {\tilde{b}}_{i} \in_{2} . \end{matrix}$ (9)

Proof. For ${\tilde{c}}_{j}, {\tilde{x}}_{j}, {\tilde{a}}_{ij}, {\tilde{b}}_{i} \in F_{2}$ and ${\tilde{x}}_{j} = [(x_{j}^{l_{1}}, x_{j}^{l_{2}}, x_{j}^{l_{3}}, x_{j}^{l_{4}}), (x_{j}^{u_{1}}, x_{j}^{u_{2}}, x_{j}^{u_{3}}, x_{j}^{u_{4}})] \in F_{2}^{+}$ , from Remark 2.3 we know that $\sum_{j = 1}^{m} {\tilde{a}}_{ij} \otimes {\tilde{x}}_{j} = \sum_{j = 1}^{m} [(a_{ij}^{l_{1}} x_{j}^{l_{1}}, a_{ij}^{l_{2}} x_{j}^{l_{2}}, a_{ij}^{l_{3}} x_{j}^{l_{3}}, a_{ij}^{l_{4}} x_{j}^{l_{4}}), (a_{ij}^{u_{1}} x_{j}^{u_{1}}, a_{ij}^{u_{2}} x_{j}^{u_{2}}, a_{ij}^{u_{3}} x_{j}^{u_{3}}, a_{ij}^{u_{4}} x_{j}^{u_{4}})]$ can be written $\sum_{j = 1}^{m} a_{ij}^{t} x_{j}^{t} = b_{i}^{t}, t \in {l_{1}, l_{2}, l_{3}, l_{4}, u_{1}, u_{2}, u_{3}, u_{4}}$ where the functions $\sum_{j = 1}^{m} {\tilde{c}}_{j} \otimes {\tilde{x}}_{j} = \sum_{j = 1}^{m} [(c_{j}^{l_{1}} x_{j}^{l_{1}}, c_{j}^{l_{2}} x_{j}^{l_{2}}, c_{j}^{l_{3}} x_{j}^{l_{3}}, c_{j}^{l_{4}} x_{j}^{l_{4}}), (c_{j}^{u_{1}} x_{j}^{u_{1}}, c_{j}^{u_{2}} x_{j}^{u_{2}}, c_{j}^{u_{3}} x_{j}^{u_{3}}, c_{j}^{u_{4}} x_{j}^{u_{4}})]$ and $\sum_{j = 1}^{m} a_{ij}^{t} x_{j}^{t} = b_{i}^{t}, t \in {l_{1}, l_{2}, l_{3}, l_{4}, u_{1}, u_{2}, u_{3}, u_{4}}$ are linear functions with variable vector $(x_{j}^{u_{1}}, x_{j}^{l_{1}}, x_{j}^{u_{2}}, x_{j}^{l_{2}}, x_{j}^{l_{3}}, x_{j}^{u_{3}}, x_{j}^{l_{4}}, x_{j}^{u_{4}}) \in (^{+})^{8}, j = 1, . . ., m$ . Using Theorem 2.2, we get $d_{S} (\sum_{j = 1}^{m} {\tilde{c}}_{j} \otimes {\tilde{x}}_{j}, \tilde{O}) = \sum_{j = 1}^{m} d_{S} ({\tilde{c}}_{j} \otimes {\tilde{x}}_{j}, \tilde{O}) = f (x) = \frac{1}{8} \sum_{j = 1}^{m} [c_{j}^{l_{1}} x_{j}^{l_{1}} + c_{j}^{l_{2}} x_{j}^{l_{2}} + c_{j}^{l_{3}} x_{j}^{l_{3}} + c_{j}^{l_{4}} x_{j}^{l_{4}} + c_{j}^{u_{1}} x_{j}^{u_{1}} + c_{j}^{u_{2}} x_{j}^{u_{2}} + c_{j}^{u_{3}} x_{j}^{u_{3}} + c_{j}^{u_{4}} x_{j}^{u_{4}}]$ and $d_{S} ({\tilde{x}}_{j}, \tilde{O}) \geq 0, j = 1, . . ., m$ . So the model (9) can be written as: $\begin{matrix} min f (x) = \frac{1}{8} \sum_{j = 1}^{m} [c_{j}^{l_{1}} x_{j}^{l_{1}} + c_{j}^{l_{2}} x_{j}^{l_{2}} + c_{j}^{l_{3}} x_{j}^{l_{3}} + c_{j}^{l_{4}} x_{j}^{l_{4}} \\ + c_{j}^{u_{1}} x_{j}^{u_{1}} + c_{j}^{u_{2}} x_{j}^{u_{2}} + c_{j}^{u_{3}} x_{j}^{u_{3}} + c_{j}^{u_{4}} x_{j}^{u_{4}}] \end{matrix}$ (9.1) $\begin{matrix} s . t \\ \sum_{j = 1}^{m} a_{ij}^{t} x_{j}^{t} = b_{i}^{t}, t \in {l_{1}, l_{2}, l_{3}, l_{4}, u_{1}, u_{2}, u_{3}, u_{4}}, \\ x^{u_{1}} \leq x^{l_{1}} \leq x^{l_{4}} \leq x^{u_{4}} \\ x^{u_{1}} \leq x^{u_{1}} \leq x^{u_{3}} \leq x^{u_{4}} \end{matrix}$ $\begin{matrix} x^{l_{1}} \leq x^{l_{1}} \leq x^{l_{3}} \leq x^{l_{4}} \\ x_{j}^{u_{1}} \geq 0, j = 1, . . ., m \end{matrix}$ which is a crisp linear programming with decision variable $x_{j} = (x_{j}^{u_{1}}, x_{j}^{l_{1}}, x_{j}^{u_{2}}$ , $x_{j}^{l_{2}}, x_{j}^{l_{3}}, x_{j}^{u_{3}}$ , $x_{j}^{l_{4}}, x_{j}^{u_{4}}) \in (R^{+})^{8}$ , j = 1, . . . , m.□

Theorem 2.4. The programming problems (9) and (9.1) are equivalent.

Proof.

Let ${\tilde{c}}_{j}, {\tilde{a}}_{ij}, {\tilde{b}}_{i} \in F_{2}$ , ${\tilde{x}}_{j} = [(x_{j}^{l_{1}}, x_{j}^{l_{2}}, x_{j}^{l_{3}}, x_{j}^{l_{4}})$ , $(x_{j}^{u_{1}}, x_{j}^{u_{2}}, x_{j}^{u_{3}}, x_{j}^{u_{4}})] \in F_{2}^{+}$ and $x_{j} = (x_{j}^{u_{1}}, x_{j}^{l_{1}}, x_{j}^{u_{2}}, x_{j}^{l_{2}}, x_{j}^{l_{3}}, x_{j}^{u_{3}}, x_{j}^{l_{4}}, x_{j}^{u_{4}}) \in (R_{0}^{+})^{8}, j = 1, . . ., m$ .

Suppose that $\tilde{S} = {\tilde{x} \in (F_{2}^{+})^{m} : \sum_{j = 1}^{m} {\tilde{a}}_{ij} \otimes {\tilde{x}}_{j} \approx {\tilde{b}}_{i}, {\tilde{x}}_{j} ≽ \tilde{0}, i = 1, . . ., n, j = 1, . . ., m}$ and $S = {x \in (R_{0}^{+})^{8 m} : \sum_{j = 1}^{m} a_{ij}^{t} x_{j}^{t} = b_{i}^{t}, t \in {l_{1}, l_{2}, l_{3}, l_{4}, u_{1}, u_{2}, u_{3}, u_{4}}, x_{j}^{t} \geq 0, i = 1, . . ., n, j = 1, . . ., m}$ are the sets of all feasible solutions of problems (9) and (9.1) respectively. Then, $\begin{matrix} ({\tilde{x}}_{1}, . . ., {\tilde{x}}_{m}) \in \tilde{S} \Leftrightarrow \sum_{j = 1}^{m} {\tilde{a}}_{ij} \otimes {\tilde{x}}_{j} = {\tilde{b}}_{i}, i = 1, . . ., n \\ \Leftrightarrow \sum_{j = 1}^{m} a_{ij}^{t} x_{j}^{t} = b_{i}^{t}, t \in {l_{1}, l_{2}, l_{3}, l_{4}, u_{1}, u_{2}, u_{3}, u_{4}} \\ \Leftrightarrow x_{j} = (x_{j}^{u_{1}}, x_{j}^{l_{1}}, x_{j}^{u_{2}}, x_{j}^{l_{2}}, x_{j}^{l_{3}}, x_{j}^{u_{3}}, x_{j}^{l_{4}}, x_{j}^{u_{4}}) \in S \end{matrix}$

Now, suppose that ${\tilde{x}}^{*} = ({\tilde{x}}_{1}^{*}, . . ., {\tilde{x}}_{1}^{*}) \in \tilde{S}$ is an optimal solution of (9). That is, for all $\tilde{x} = ({\tilde{x}}_{1}, . . ., {\tilde{x}}_{n}) \in \tilde{S}$ , then $\sum_{j = 1}^{m} {\tilde{c}}_{j} \otimes {\tilde{x}}_{j} ≽ \sum_{j = 1}^{m} {\tilde{c}}_{j} \otimes {\tilde{x}}_{j}^{*}$ for all ${\tilde{x}}_{j}$ in $\tilde{S}$ . According to the Theorem 2.2, we may obtain that $d_{S} (\sum_{j = 1}^{m} {\tilde{c}}_{j} \otimes {\tilde{x}}_{j}, \tilde{O}) \geq d_{S} (\sum_{j = 1}^{m} {\tilde{c}}_{j} \otimes {\tilde{x}}_{j}^{*}, \tilde{)}$ $\Leftrightarrow \frac{1}{8} \sum_{j = 1}^{m} \sum_{k = 1}^{4} (c_{j}^{l_{k}} x_{j}^{l_{k}} + c_{j}^{u_{k}} x_{j}^{u_{k}}) \geq \frac{1}{8} \sum_{j = 1}^{m} \sum_{k = 1}^{4} (c_{j}^{l_{k} *} x_{j}^{l_{k} *} + c_{j}^{u_{k} *} x_{j}^{u_{k} *})$ . This shows that $x_{j}^{*} = (x_{j}^{u_{1 *}}, x_{j}^{l_{1 *}}, x_{j}^{u_{2 *}}, x_{j}^{l_{2} *}, x_{j}^{l_{3 *}}, x_{j}^{u_{3 *}}, x_{j}^{l_{4 *}}, x_{j}^{u_{4 *}}) \in (R_{0}^{+})^{8}, j = 1, . . ., m$ is an optimal solution to the problem (9.1). Similarity, we may prove that any optimal solution of (9.1) is also that of (9). Hence, model (9) is equivalent to (9.1).□

3 The proposed model

In this section, based on the metric suggested in section 2, we will introduce and investigate a new fuzzy least absolute deviations approach to build a new fuzzy regression model, for crisp input and IT2 fuzzy output and parameters. Consider ${\tilde{a}}_{j}, {\hat{\tilde{a}}}_{j}, {\tilde{y}}_{i}, {\hat{\tilde{y}}}_{i} \in F_{2}$ and $x_{ij} \in R_{0}^{+}$ (x_ij ≥ 0), then the general form of the IT2 fuzzy linear regression model can be considered as $\begin{matrix} {\tilde{y}}_{i} = \oplus_{j = 0}^{p} {\tilde{a}}_{j} x_{ij} = {\tilde{a}}_{0} x_{i 0} \oplus {\tilde{a}}_{1} x_{i 1} \\ \oplus . . . \oplus {\tilde{a}}_{p} x_{ip}, i = 1, . . ., n \end{matrix}$ (10)

In the model (10), ${\tilde{y}}_{i}$ is the ith fuzzy observed output variable, ${\tilde{a}}_{j}$ is the jth fuzzy regression coefficient, and x_ij (i = 1, . . . , n ; j = 0, . . . , p) is the jth crisp input variable for the ith sample. Without loss of generality, we assume that all of x_ij are positive (if x_ji < 0, it can be transformed to the situation x_ji > 0). The general predicted IT2 fuzzy regression model for the aforementioned data set can be expressed as $\begin{matrix} {\hat{\tilde{y}}}_{i} \approx \oplus_{j = 0}^{p} {\hat{\tilde{a}}}_{j} x_{ij} \approx {\hat{\tilde{a}}}_{0} x_{i 0} \oplus {\hat{\tilde{a}}}_{1} x_{i 1} \\ \oplus . . . \oplus {\hat{\tilde{a}}}_{p} x_{ip}, i = 1, . . ., n \end{matrix}$ (11) $\begin{matrix} [({\hat{y}}_{i}^{l_{1}}, {\hat{y}}_{i}^{l_{2}}, {\hat{y}}_{i}^{l_{3}}, {\hat{y}}_{i}^{l_{4}}), ({\hat{y}}_{i}^{u_{1}}, {\hat{y}}_{i}^{u_{2}}, {\hat{y}}_{i}^{u_{3}}, {\hat{y}}_{i}^{u_{4}})] = \\ [(\sum_{j = 0}^{p} {\hat{a}}_{j}^{l_{1}} x_{ij}, \sum_{j = 0}^{p} {\hat{a}}_{j}^{l_{2}} x_{ij}, \sum_{j = 0}^{p} {\hat{a}}_{j}^{l_{3}} x_{ij}, \sum_{j = 0}^{p} {\hat{a}}_{j}^{l_{4}} x_{ij}), \\ (\sum_{j = 0}^{p} {\hat{a}}_{j}^{u_{1}} x_{ij}, \sum_{j = 0}^{p} {\hat{a}}_{j}^{u_{2}} x_{ij}, \sum_{j = 0}^{p} {\hat{a}}_{j}^{u_{3}} x_{ij}, \sum_{j = 0}^{p} {\hat{a}}_{j}^{u_{4}} x_{ij})], \\ i = 1, . . ., n \end{matrix}$

In model (11), ${\hat{\tilde{y}}}_{i}$ is the ith fuzzy estimated output variable and ${\hat{\tilde{a}}}_{j}$ is the jth estimated fuzzy regression coefficient.

3.1 Estimations of the parameters of model

To estimate the parameters of proposed model (in an optimal way), the proposed methodology is carried out in the six steps:

Step 1: Using (6.1), (6.2) and, Remark 2.2, we put ${\hat{\tilde{a}}}_{j} = {\hat{\tilde{a}}}_{j}^{+} ⊖ {\hat{\tilde{a}}}_{j}^{-}$ for j = 1, . . . , p, so that $if {\hat{a}}_{j}^{u_{1} +} > 0 then {\hat{\tilde{a}}}_{j} \approx {\hat{\tilde{a}}}_{j}^{+} else$ $if {\hat{a}}_{j}^{u_{1} -} > 0 then {\hat{\tilde{a}}}_{j} \approx {\hat{\tilde{a}}}_{j}^{-}$ $otherwise {\hat{\tilde{a}}}_{j} \approx \tilde{0}$ . Given that ${\hat{\tilde{a}}}_{j}^{+} = [({\hat{a}}_{j}^{l_{1} +}, {\hat{a}}_{j}^{l_{2} +}, {\hat{a}}_{j}^{l_{3} +}, {\hat{a}}_{j}^{l_{4} +}), ({\hat{a}}_{j}^{u_{1} +}, {\hat{a}}_{j}^{u_{2} +}, {\hat{a}}_{j}^{u_{3} +}, {\hat{a}}_{j}^{u_{4} +})]$ (12) ${\hat{\tilde{a}}}_{j}^{-} = [({\hat{a}}_{j}^{l_{1} -}, {\hat{a}}_{j}^{l_{2} -}, {\hat{a}}_{j}^{l_{3} -}, {\hat{a}}_{j}^{l_{4} -}), ({\hat{a}}_{j}^{u_{1} -}, {\hat{a}}_{j}^{u_{2} -}, {\hat{a}}_{j}^{u_{3} -}, {\hat{a}}_{j}^{u_{4} -})]$ (13)

By using (1), (2) and, substituting ${\hat{\tilde{a}}}_{j}^{+}$ , ${\hat{\tilde{a}}}_{j}^{-}$ from (12), (13) we have: $\begin{matrix} {\hat{\tilde{a}}}_{j} = {\hat{\tilde{a}}}_{j}^{+} ⊖ {\hat{\tilde{a}}}_{j}^{-} = [({\hat{a}}_{j}^{l_{1} +} - {\hat{a}}_{j}^{l_{4} -}, {\hat{a}}_{j}^{l_{2} +} - {\hat{a}}_{j}^{l_{3} -}, {\hat{a}}_{j}^{l_{3} +} \\ - {\hat{a}}_{j}^{l_{2} -}, {\hat{a}}_{j}^{l_{4} +} - {\hat{a}}_{j}^{l_{1} -}), ({\hat{a}}_{j}^{u_{1} +} - {\hat{a}}_{j}^{u_{4} -}, {\hat{a}}_{j}^{u_{2} +} - {\hat{a}}_{j}^{u_{3} -}, \\ {\hat{a}}_{j}^{u_{3} +} - {\hat{a}}_{j}^{u_{2} -}, {\hat{a}}_{j}^{u_{4} +} - {\hat{a}}_{j}^{u_{1} -})] \end{matrix}$ (14)

Using Definition 2.5, and substituting ${\hat{\tilde{a}}}_{j}$ from (14), we have: $\begin{matrix} {\hat{\tilde{a}}}_{j} x_{ji} = ({\hat{\tilde{a}}}_{j}^{+} ⊖ {\hat{\tilde{a}}}_{j}^{-}) x_{ji} = [({\hat{a}}_{j}^{l_{1} +} x_{ji} - {\hat{a}}_{j}^{l_{4} -} x_{ji}, {\hat{a}}_{j}^{l_{2} +} \\ x_{ji} - {\hat{a}}_{j}^{l_{3} -} x_{ji},, {\hat{a}}_{j}^{l_{3} +} x_{ji} - {\hat{a}}_{j}^{l_{2} -} x_{ji}, {\hat{a}}_{j}^{l_{4} +} x_{ji} - {\hat{a}}_{j}^{l_{1} -} x_{ji}), \\ ({\hat{a}}_{j}^{u_{1} +} x_{ji} - {\hat{a}}_{j}^{u_{4} -} x_{ji}, {\hat{a}}_{j}^{u_{2} +} x_{ji} - {\hat{a}}_{j}^{u_{3} -} x_{ji}, {\hat{a}}_{j}^{u_{3} +} \\ x_{ji} - {\hat{a}}_{j}^{u_{3} -} x_{ji}, {\hat{a}}_{j}^{u_{4} +} x_{ji} - {\hat{a}}_{j}^{u_{4} -} x_{ji})] \end{matrix}$ (15) $\begin{matrix} {\hat{\tilde{a}}}_{0} \oplus {\hat{\tilde{a}}}_{j} x_{ji} = {\hat{\tilde{a}}}_{0} \oplus ({\hat{\tilde{a}}}_{j}^{+} {\hat{\tilde{a}}}_{j}^{-}) x_{ji} = [({\hat{a}}_{0}^{l_{1}} + {\hat{a}}_{j}^{l_{1} +} x_{ji} \\ - {\hat{a}}_{j}^{l_{4} -} x_{ji}, {\hat{a}}_{0}^{l_{2}} + {\hat{a}}_{j}^{l_{2} +} x_{ji} - {\hat{a}}_{j}^{l_{3} -} x_{ji}, {\hat{a}}_{0}^{l_{3}} + {\hat{a}}_{j}^{l_{3} +} x_{ji} \\ - {\hat{a}}_{j}^{l_{2} -} x_{ji}, {\hat{a}}_{0}^{l_{4}} + {\hat{a}}_{j}^{l_{4} +} x_{ji} - {\hat{a}}_{j}^{l_{1} -} x_{ji}), ({\hat{a}}_{0}^{u_{1}} + {\hat{a}}_{j}^{u_{1} +} x_{ji} \\ - {\hat{a}}_{j}^{u_{4} -} x_{ji}, {\hat{a}}_{0}^{u_{2}} + {\hat{a}}_{j}^{u_{2} +} x_{ji} - {\hat{a}}_{j}^{u_{3} -} x_{ji}, {\hat{a}}_{0}^{u_{3}} + {\hat{a}}_{j}^{u_{3} +} x_{ji} \\ - {\hat{a}}_{j}^{u_{2} -} x_{ji}, {\hat{a}}_{0}^{u_{4}} + {\hat{a}}_{j}^{u_{4} +} x_{ji} - {\hat{a}}_{j}^{u_{1} -} x_{ji})] \end{matrix}$ (16)

Using (16), we have: ${\hat{y}}_{i}^{u_{m}} = {\hat{a}}_{0}^{u_{m}} + \sum_{j = 1}^{p} ({\hat{a}}_{j}^{u_{m} +} x_{ji} - {\hat{a}}_{j}^{u_{5 - m} -} x_{ji}), m = 1, . . ., 4$ (17) ${\hat{y}}_{i}^{l_{m}} = {\hat{a}}_{0}^{l_{m}} + \sum_{j = 1}^{p} ({\hat{a}}_{j}^{l_{m} +} x_{ji} - {\hat{a}}_{j}^{l_{5 - m}^{-}} x_{ji}), m = 1, . . ., 4$ (18)

Using (11) and substituting (17) and (18) in ${\hat{\tilde{y}}}_{i} = [({\hat{y}}_{i}^{l_{1}}, {\hat{y}}_{i}^{l_{2}}, {\hat{y}}_{i}^{l_{3}}, {\hat{y}}_{i}^{l_{4}}), ({\hat{y}}_{i}^{u_{1}}, {\hat{y}}_{i}^{u_{2}}, {\hat{y}}_{i}^{u_{3}}, {\hat{y}}_{i}^{u_{4}})]$ , we have: $\begin{matrix} {\hat{\tilde{y}}}_{i} = {[({\hat{a}}_{0}^{u_{1}} + (\sum_{j = 1}^{p} {\hat{a}}_{j}^{u_{1} +} x_{ji} - \sum_{j = 1}^{p} {\hat{a}}_{j}^{u_{4} -} x_{ji}), {\hat{a}}_{0}^{u_{2}} \\ + (\sum_{j = 1}^{p} {\hat{a}}_{j}^{u_{2} +} x_{ji} - \sum_{j = 1}^{p} {\hat{a}}_{j}^{u_{3} -} x_{ji}), \end{matrix}$ $\begin{matrix} {\hat{a}}_{0}^{u_{3}} + \sum_{j = 1}^{p} ({\hat{a}}_{j}^{u_{3} +} x_{ji} - {\hat{a}}_{j}^{u_{2} -} x_{ji}), {\hat{a}}_{0}^{u_{4}} \\ + \sum_{j = 1}^{p} ({\hat{a}}_{j}^{u_{4} +} x_{ji} - {\hat{a}}_{j}^{u_{1} -} x_{ji})], \\ [({\hat{a}}_{0}^{l_{1}} + \sum_{j = 1}^{p} ({\hat{a}}_{j}^{l_{1} +} x_{ji} - {\hat{a}}_{j}^{l_{4}^{-}} x_{ji}), {\hat{a}}_{0}^{l_{2}} \\ + \sum_{j = 1}^{p} ({\hat{a}}_{j}^{l_{2} +} x_{ji} - {\hat{a}}_{j}^{l_{3} -} x_{ji}), \\ {\hat{a}}_{0}^{l_{3}} + \sum_{j = 1}^{p} ({\hat{a}}_{j}^{l_{3} +} x_{ji} - {\hat{a}}_{j}^{l_{2} -} x_{ji}), {\hat{a}}_{0}^{l_{4}} \\ + \sum_{j = 1}^{p} ({\hat{a}}_{j}^{l_{4} +} x_{ji} - {\hat{a}}_{j}^{l_{1} -} x_{ji})]} \end{matrix}$

Step 2: We employed the fuzzy distance measure introduced in Definition 2.7 to calculate the absolute deviations between ${\tilde{y}}_{i}$ and ${\hat{\tilde{y}}}_{i}$ as follows: $d_{H} ({\tilde{y}}_{i}, {\hat{\tilde{y}}}_{i}) = λ d_{H} ({\dot{y}}_{i}^{U}, {\hat{\dot{y}}}_{i}^{U}) + (1 - λ) d_{H} ({\dot{y}}_{i}^{L}, {\hat{\dot{y}}}_{i}^{L})]$ (19) we consider lower membership functions ( ${\dot{y}}_{i}^{l}, {\hat{\dot{y}}}_{i}^{l}$ ) and upper membership functions ( ${\dot{y}}_{i}^{u}, {\hat{\dot{y}}}_{i}^{u}$ ) are of equal significance, then $λ = \frac{1}{2}$ and the relation (19) will be as follows: $\begin{matrix} d_{H} ({\dot{y}}_{i}^{U}, {\hat{\dot{y}}}_{i}^{U}) = \frac{1}{4} \sum_{t = 1}^{4} (| y_{i}^{u_{t}} - {\hat{y}}_{i}^{u_{t}} |), d_{H} ({\dot{y}}_{i}^{L}, {\hat{\dot{y}}}_{i}^{L}) \\ = \frac{1}{4} \sum_{t = 1}^{4} (| y_{i}^{l_{t}} - {\hat{y}}_{i}^{l_{t}} |), d_{H} ({\tilde{y}}_{i}, {\hat{\tilde{y}}}_{i}) \\ = \frac{1}{2} [d_{H} ({\dot{y}}_{i}^{U}, {\hat{\dot{y}}}_{i}^{U}) + d_{H} ({\dot{y}}_{i}^{L}, {\hat{\dot{y}}}_{i}^{L})] \\ d_{H} ({\tilde{y}}_{i}, {\hat{\tilde{y}}}_{i}) = \frac{1}{8} {\sum_{t = 1}^{4} (| y_{i}^{u_{t}} - {\hat{y}}_{i}^{u_{t}} | + | y_{i}^{l_{t}} - {\hat{y}}_{i}^{l_{t}} |)} \end{matrix}$ (20)

Step 3: We applied the new IT2 fuzzy mathematical programming problem to minimizing the overall absolute error between ${\tilde{y}}_{i}$ and ${\hat{\tilde{y}}}_{i}$ as follows $\begin{matrix} min \sum_{i = 1}^{n} d_{H} ({\hat{\tilde{y}}}_{i}, {\tilde{y}}_{i}) = min \sum_{i = 1}^{n} \\ {[\frac{1}{8} \sum_{t = 1}^{4} (| {\hat{y}}_{i}^{u_{t}} - y_{i}^{u_{t}} | + | {\hat{y}}_{i}^{l_{t}} - y_{i}^{l_{t}} |)]} \end{matrix}$ (21) $\begin{matrix} = min \frac{1}{8} \sum_{i = 1}^{n} {\sum_{t = 1}^{4} [| {\hat{a}}_{0}^{u_{t}} + \sum_{j = 1}^{p} ({\hat{a}}_{j}^{u_{t} +} x_{ji} - {\hat{a}}_{j}^{u_{5 - t}^{-}} x_{ji}) \\ - y_{i}^{u_{t}} | + | {\hat{a}}_{0}^{l_{t}} + \sum_{j = 1}^{p} ({\hat{a}}_{j}^{l_{t} +} x_{ji} - {\hat{a}}_{j}^{l_{5 - m}^{-}} x_{ji}) - y_{i}^{l_{t}} |]} \\ s . t . \\ y_{i}^{u_{t}}, y_{i}^{l_{t}}, {\hat{a}}_{j}^{u_{t}}, {\hat{a}}_{j}^{l_{t}} \in R, t = 1, . . ., 4; j = 0, . . ., p, \\ i = 1, . . ., n \\ {\hat{a}}_{j}^{l_{1}} \leq {\hat{a}}_{j}^{l_{2}} \leq {\hat{a}}_{j}^{l_{3}} \leq {\hat{a}}_{j}^{l_{4}}; j = 0, . . ., p \\ {\hat{a}}_{j}^{u_{1}} \leq {\hat{a}}_{j}^{u_{2}} \leq {\hat{a}}_{j}^{u_{3}} \leq {\hat{a}}_{j}^{u_{4}}; j = 0, . . ., p \\ y_{i}^{l_{1}} \leq y_{i}^{l_{2}} \leq y_{i}^{l_{3}} \leq y_{i}^{l_{4}}, i = 1, . . ., n \\ y_{i}^{u_{1}} \leq y_{i}^{u_{2}} \leq y_{i}^{u_{3}} \leq y_{i}^{u_{4}}, i = 1, . . ., n \\ x_{ij} \geq 0; i = 1, . . ., n, j = 0, . . ., p \end{matrix}$

Step 4: To remove absolute values from the achievement function, we employed positive and negative deviational variables (i.e., $d_{i}^{λ_{t} +}$ and $d_{i}^{λ_{t} -}$ ) as $\begin{matrix} d_{i}^{λ_{t} +} = max (0, {\hat{y}}_{i}^{λ_{t}} - y_{i}^{λ_{t}}) \\ = \frac{({\hat{y}}_{i}^{λ_{t}} - y_{i}^{λ_{t}})}{2} + \frac{| {\hat{y}}_{i}^{λ_{t}} - y_{i}^{λ_{t}} |}{2} \\ d_{i}^{λ_{t} -} = max (0, y_{i}^{λ_{t}} - {\hat{y}}_{i}^{λ_{t}}) \\ = \frac{(y_{i}^{λ_{t}} - {\hat{y}}_{i}^{λ_{t}})}{2} + \frac{| y_{i}^{λ_{t}} - {\hat{y}}_{i}^{λ_{t}} |}{2} \end{matrix}$ $\begin{matrix} d_{i}^{λ_{t} +} + d_{i}^{λ_{t} -} = | {\hat{y}}_{i}^{λ_{t}} - y_{i}^{λ_{t}} | \end{matrix}$ (22) $\begin{matrix} d_{i}^{λ_{t} +} - d_{i}^{λ_{t} -} = {\hat{y}}_{i}^{λ_{t}} - y_{i}^{λ_{t}} \\ {\hat{y}}_{i}^{λ_{t}} + d_{i}^{λ_{t} -} - d_{i}^{λ_{t} +} = y_{i}^{λ_{t}} \\ i = 1, . . ., n; t = 1, . . ., 4; λ = l, u \end{matrix}$ (23)

Minimizing the $| y_{i}^{λ_{t}} - {\hat{y}}_{i}^{λ_{t}} |$ , means minimizing either $d_{i}^{λ_{t} +}$ or $d_{i}^{λ_{t} -}$ or $d_{i}^{λ_{t} +} + d_{i}^{λ_{t} -}$ . We minimize $d_{i}^{λ_{t} -}$ when we need $y_{i}^{λ_{t}} \leq {\hat{y}}_{i}^{λ_{t}}$ (Max problem), minimize $d_{i}^{λ_{t} +}$ , when we need ${\hat{y}}_{i}^{λ_{t}} \leq y_{i}^{λ_{t}}$ (Min problem) and minimize $d_{i}^{λ_{t} +} + d_{i}^{λ_{t} -}$ when we need ${\hat{y}}_{i}^{λ_{t}} = y_{i}^{λ_{t}}$ . In fact, $d_{i}^{λ_{t} -}$ and $d_{i}^{λ_{t} +}$ are the negative and positive deviations between the parameters of the ith estimated and observed response, respectively. It can be easily seen that: ${\hat{y}}_{i}^{l_{t}} - d_{i}^{l_{t} +} + d_{i}^{l_{t} -} = y_{i}^{l_{t}} i = 1, . . ., n; t = 1, . . ., 4$ (24) ${\hat{y}}_{i}^{u_{t}} - d_{i}^{u_{t} +} + d_{i}^{u_{t} -} = y_{i}^{u_{t}} i = 1, . . ., n; t = 1, . . ., 4$ (25)

By substitution (22–25) in problem (21), the IT2 fuzzy mathematical programming problem (21) can be transformed into the GP problem as following: $\begin{matrix} min \frac{1}{8} \sum_{i = 1}^{n} \sum_{t = 1}^{4} [(d_{i}^{u_{t} +} + d_{i}^{u_{t} -}) + (d_{i}^{l_{t} +} + d_{i}^{l_{t} -})] \\ s . t . \end{matrix}$ (26) $y_{i}^{l_{t}} = {\hat{a}}_{0}^{l_{t}} + \sum_{j = 1}^{p} ({\hat{a}}_{j}^{l_{t} +} x_{ji} - {\hat{a}}_{j}^{l_{5 - t}^{-}} x_{ji}) + d_{i}^{l_{t} +} - d_{i}^{l_{t} -}$ (26.1) $y_{i}^{u_{t}} = {\hat{a}}_{0}^{u_{t}} + \sum_{j = 1}^{p} ({\hat{a}}_{j}^{u_{t} +} x_{ji} - {\hat{a}}_{j}^{u_{5 - t} -} x_{ji}) + d_{i}^{u_{t} +} - d_{i}^{u_{t} -}$ (26.2) $d_{i}^{l_{t} +} \times d_{i}^{l_{t} -} = 0$ (26.3) $d_{i}^{u_{t} +} \times d_{i}^{u_{t} -} = 0$ (26.4) $d_{i}^{l_{t} +}, d_{i}^{l_{t} -} \geq 0$ (26.5) $d_{i}^{u_{t} +}, d_{im}^{u_{t} -} \geq 0$ (26.6) $y_{i}^{u_{1}} \leq y_{i}^{l_{1}} \leq y_{i}^{l_{4}} \leq y_{i}^{u_{4}}$ (26.7) ${\hat{y}}_{i}^{u_{1}} \leq {\hat{y}}_{i}^{l_{1}} \leq {\hat{y}}_{i}^{l_{4}} \leq {\hat{y}}_{i}^{u_{4}}$ (26.8) ${\hat{a}}_{j}^{u_{1}} \leq {\hat{a}}_{j}^{l_{1}} \leq {\hat{a}}_{j}^{l_{4}} \leq {\hat{a}}_{j}^{u_{4}}$ (26.9) $\begin{matrix} y_{i}^{l_{t}}, y_{i}^{u_{t}}, {\hat{y}}_{i}^{l_{t}}, {\hat{y}}_{i}^{u_{t}}, {\hat{a}}_{j}^{l_{t}}, {\hat{a}}_{j}^{u_{t}} \in R; x_{ij} \geq 0 \\ t = 1, . . ., 4; j = 0, . . ., p; n = 1, . . ., n \end{matrix}$ (26.10)

Step 5: We remove the constraint (26.1) and (26.2) (see Theorem 3.1) then convert the GP problem (26) into traditional LP as $\begin{matrix} min \frac{1}{8} \sum_{i = 1}^{n} \sum_{t = 1}^{4} [(d_{i}^{u_{t} +} + d_{i}^{u_{t} -}) + (d_{i}^{l_{t} +} + d_{i}^{l_{t} -})] \\ s . t . \end{matrix}$ (27) $y_{i}^{l_{t}} = {\hat{a}}_{0}^{l_{t}} + \sum_{j = 1}^{p} ({\hat{a}}_{j}^{l_{t} +} x_{ji} - {\hat{a}}_{j}^{l_{5 - t}^{-}} x_{ji}) + d_{i}^{l_{t} +} - d_{i}^{l_{t} -}$ (27.1) $y_{i}^{u_{t}} = {\hat{a}}_{0}^{u_{t}} + \sum_{j = 1}^{p} ({\hat{a}}_{j}^{u_{t} +} x_{ji} - {\hat{a}}_{j}^{u^{5 - t} -} x_{ji}) + d_{i}^{u_{t} +} - d_{i}^{u_{t} -}$ (27.2) $d_{i}^{l_{t} +}, d_{i}^{l_{t} -} \geq 0$ (27.3) $d_{i}^{u_{t} +}, d_{i}^{u_{t} -} \geq 0$ (27.4) $y_{i}^{u_{1}} \leq y_{i}^{l_{1}} \leq y_{i}^{l_{4}} \leq y_{i}^{u_{4}}$ (27.5) ${\hat{y}}_{i}^{u_{1}} \leq {\hat{y}}_{i}^{l_{1}} \leq {\hat{y}}_{i}^{l_{4}} \leq {\hat{y}}_{i}^{u_{4}}$ (27.6) ${\hat{a}}_{j}^{u_{1}} \leq {\hat{a}}_{j}^{l_{1}} \leq {\hat{a}}_{j}^{l_{4}} \leq {\hat{a}}_{j}^{u_{4}}$ (27.7) $\begin{matrix} y_{i}^{l_{t}}, y_{i}^{u_{t}}, {\hat{y}}_{i}^{l_{t}}, {\hat{y}}_{i}^{u_{t}}, {\hat{a}}_{j}^{l_{t}}, {\hat{a}}_{j}^{u_{t}} \in R; x_{ij} \geq 0 \\ t = 1, . . ., 4; j = 0, . . ., p; n = 1, . . ., n \end{matrix}$ (27.8)

Step 6: By solving the problem in (27) with the help of MATLAB, the all trapezoidal IT2 fuzzy parameters ( ${\tilde{\tilde{b}}}_{j}; j = 0, . . ., n$ ) are estimated and the proposed model is constructed as ${\hat{\tilde{y}}}_{i} \approx \oplus_{j = 0}^{p} {\hat{\tilde{a}}}_{j} x_{ij} \approx {\hat{\tilde{a}}}_{0} x_{i 0} \oplus {\hat{\tilde{a}}}_{1} x_{i 1} \oplus . . . \oplus {\hat{\tilde{a}}}_{p} x_{ip}, i = 1, . . ., n$ .

Theorem 3.1. The constraints $d_{i}^{l_{t} +} \times d_{i}^{l_{t} -} = 0$ t = 1, . . . , 4 ; i = 1, . . . , n and $d_{i}^{u_{t} +} \times d_{i}^{u_{t} -} = 0$ t = 1, . . . , 4 ; i = 1, . . . , n in (26.3) and (26.4) of the problem (26) can be deleted.

Proof. Suppose $d_{i}^{u_{t}^{\circ} +}$ and $d_{i}^{u_{t}^{\circ} -}$ be an optimal solution of problem (26), i.e. their does not exist another $d_{i}^{u_{t}^{⌣} +}$ or $d_{i}^{u_{t}^{⌣} -}$ such that $d_{i}^{u_{t}^{⌣} +} + d_{i}^{u_{t}^{⌣} -} \leq d_{i}^{u_{t}^{\circ} +} + d_{i}^{u_{t}^{\circ} -}$ with a strict inequality holding for at least one. Let $d_{i}^{u_{t}^{\circ} +} \times d_{i}^{u_{t}^{\circ} -} \neq 0$ . Suppose that $d_{v}^{u_{t} +} \geq 0$ and $d_{v}^{u_{m} -} \geq 0$ such that $d_{v}^{u_{t} +} \times d_{v}^{u_{t} -} = 0$ and $d_{i}^{u_{t}^{\circ} -} - d_{i}^{u_{t}^{\circ} +} = d_{v}^{u_{t} -} - d_{v}^{u_{t} +}$ . For $d_{v}^{u_{t} +} \times d_{v}^{u_{t} -} = 0$ , two cases are possible:

If $d_{v}^{u_{t} +} = 0$ and $d_{v}^{u_{t} -} > 0$ then $d_{i}^{u_{t}^{\circ} -} - d_{i}^{u_{t}^{\circ} +} = d_{v}^{u_{m} -} - 0 = d_{v}^{u_{m} -}$ , and because $d_{i}^{u_{t}^{\circ} +} \times d_{i}^{u_{t}^{\circ} -} \neq 0$ we have $d_{i}^{u_{t}^{\circ} +} > 0$ and $d_{i}^{u_{t}^{\circ} -} > 0$ , Hence, relation $d_{i}^{u_{t}^{\circ} -} > d_{v}^{u_{m} -}$ is satisfied.

If $d_{v}^{u_{t} -} = 0$ and $d_{v}^{u_{m} +} > 0$ then $d_{i}^{u_{t}^{\circ} -} - d_{i}^{u_{t}^{\circ} +} = 0 - d_{v}^{u_{t} +}$ , and we have $d_{i}^{u_{t}^{\circ} +} > 0$ and $d_{i}^{u_{t}^{\circ} -} > 0$ , hence relation $d_{i}^{u_{t}^{\circ} +} > d_{v}^{u_{t} +}$ is satisfied.

According to above relations, $d_{i}^{u_{t}^{\circ} -} - d_{i}^{u_{t}^{\circ} +} > d_{v}^{u_{t} +} - d_{v}^{u_{t} -}$ and this contradicts the optimality of $d_{i}^{u_{t}^{\circ} +}$ and $d_{i}^{u_{t}^{\circ} -}$ . Hence, $d_{i}^{u_{t}^{\circ} +} \times d_{i}^{u_{t}^{\circ} -} = 0$ . The case of $d_{i}^{l_{t}^{\circ} +} \times d_{i}^{l_{t}^{\circ} -} = 0$ can prove in the same manner.

The Theorem 3.2 shows that problem (27) yields the exact regression parameters and estimated outputs if the given trapezoidal IT2 fuzzy input-output data satisfy in a trapezoidal IT2 fuzzy linear model.

Theorem 3.2. Suppose that ${\tilde{a}}_{j}, {\tilde{y}}_{i} \in F_{2}$ , $x_{ij} \in R_{0}^{+}$ (x_ij ≥ 0). If $(x_{ji}, {\tilde{y}}_{i})$ satisfy in the model ${\tilde{y}}_{i} = {\tilde{a}}_{0} \oplus {\tilde{a}}_{1} x_{1 i} \oplus . . . \oplus {\tilde{a}}_{p} x_{pi}, i = 1, . . ., n$ , then:

The optimal solution of problem (27) contains $\tilde{a} = ({\tilde{a}}_{0}, {\tilde{a}}_{1}, . . ., {\tilde{a}}_{p})$ .

The observed responses ( ${\tilde{\tilde{y}}}_{i}$ ) and the estimated responses ( ${\tilde{\tilde{y}}}_{i}$ ) from problem (27) are the same (i.e., ${\hat{\tilde{y}}}_{i} \approx {\tilde{y}}_{i}$ )

Proof. (i) (see also [32] p.9-10). First, we show that $\tilde{a} = ({\tilde{a}}_{0}, {\tilde{a}}_{1}, . . ., {\tilde{a}}_{p})$ yield a feasible solution for problem (27).

We know that ${\hat{\tilde{a}}}_{j} = {\hat{\tilde{a}}}_{j}^{+} ⊖ {\hat{\tilde{a}}}_{j}^{-} = [({\hat{a}}_{j}^{l_{1} +} - {\hat{a}}_{j}^{l_{4} -}, {\hat{a}}_{j}^{l_{2} +} - {\hat{a}}_{j}^{l_{3} -}, {\hat{a}}_{j}^{l_{3} +} - {\hat{a}}_{j}^{l_{2} -}, {\hat{a}}_{j}^{l_{4} +} - {\hat{a}}_{j}^{l_{1} -}), ({\hat{a}}_{j}^{u_{1} +} - {\hat{a}}_{j}^{u_{4} -}, {\hat{a}}_{j}^{u_{2} +} - {\hat{a}}_{j}^{u_{3} -}, {\hat{a}}_{j}^{u_{3} +} - {\hat{a}}_{j}^{u_{2} -}, {\hat{a}}_{j}^{u_{4} +} - {\hat{a}}_{j}^{u_{1} -})]$ and ${\tilde{\tilde{a}}}_{j} = [(a_{j}^{l_{1}}, a_{j}^{l_{2}}, a_{j}^{l_{3}}, a_{j}^{l_{4}}), (a_{j}^{u_{1}}, a_{j}^{u_{2}}, a_{j}^{u_{3}}, a_{j}^{u_{4}})]$ . $\begin{matrix} Set {\hat{a}}_{0}^{u_{m}} = a_{0}^{u_{m}}, {\hat{a}}_{0}^{l_{m}} = a_{0}^{l_{m}}, {\hat{a}}_{j}^{l_{m} +} = {\begin{matrix} a_{j}^{l_{m}} {ifa}_{j}^{l_{m}} \geq 0 \\ 0 if a_{j}^{l_{m}} < 0 \end{matrix}, \\ {\hat{a}}_{j}^{l_{m} -} = {\begin{matrix} 0 & if a_{j}^{l_{m}} \geq 0 \\ - b_{j}^{l_{m}} & if a_{j}^{l_{m}} < 0 \end{matrix}, {\hat{a}}_{j}^{u_{m} +} = \\ {\begin{matrix} a_{j}^{u_{m}} & if a_{j}^{u_{m}} \geq 0 \\ 0 & if a_{j}^{u_{m}} < 0 \end{matrix}, {\hat{a}}_{j}^{u_{m} -} = {\begin{matrix} 0 & if a_{j}^{u_{m}} \geq 0 \\ - b_{j}^{u_{m}} & if a_{j}^{u_{m}} < 0 \end{matrix}, \end{matrix}$ j = 1, . . . , p ; m = 1, . . . , 4 and $e_{i}^{u_{m} +} = e_{i}^{u_{m} -} = e_{i}^{l_{m} +} = e_{i}^{l_{m} -} = 0;$ j = 1, . . . , p ; m = 1, . . . , 4 ; i = 1, . . . , n.

Clearly, they satisfy in the constraints (27.3)–(27.6). Without loss of generality, for simplicity, we assume that ${\tilde{a}}_{j} ≽ \tilde{0}$ i.e., $a_{j}^{l_{m}}, a_{j}^{u_{m}} \geq 0, j = 1, . . ., p; m = 1, . . ., 4$ Hence ${\hat{a}}_{j}^{u_{m} +} = a_{j}^{u_{m}}, {\hat{a}}_{j}^{l_{m} +} = a_{j}^{l_{m}} j = 1, . . ., p; m = 1, . . ., 4$ (28) ${\hat{a}}_{j}^{u_{m} -} = a_{j}^{l_{m} -} = 0 j = 1, . . ., p; m = 1, . . ., 4$ (29)

By substituting (28) and (29) in (27.1) and (27.2), we have: $\begin{matrix} {\hat{y}}_{i}^{u_{m}} + e_{i}^{u_{m} -} - e_{i}^{u_{m} +} = {\hat{a}}_{0}^{u_{m}} + \sum_{j = 1}^{p} ({\hat{a}}_{j}^{u_{m} +} \times x_{ji}^{u_{m}} \\ - {\hat{a}}_{j}^{u_{5 - m}^{-}} \times x_{ji}^{u_{m}}) + e_{i}^{u_{m} +} - e_{i}^{u_{m} -} = a_{0}^{u_{m}} \\ + \sum_{j = 1}^{p} a_{j}^{u_{m}} \times x_{ji}^{u_{m}} = y_{i}^{u_{m}} \end{matrix}$ (30) $\begin{matrix} {\hat{y}}_{i}^{l_{m}} + e_{i}^{l_{m} -} - e_{i}^{l_{m} +} = {\hat{a}}_{0}^{l_{m}} + \sum_{j = 1}^{p} ({\hat{a}}_{j}^{l_{m} +} \times x_{ji}^{l_{m}} \\ - {\hat{a}}_{j}^{l_{5 - m}^{-}} \times x_{ji}^{l_{m}}) + e_{i}^{l_{m} -} - e_{i}^{l_{m} +} = a_{0}^{l_{m}} \\ + \sum_{j = 1}^{p} a_{j}^{l_{m}} \times x_{ji}^{l_{m}} = y_{i}^{l_{m}} \end{matrix}$ (31)

The Equations (30) and (31) show that $\tilde{a} = ({\tilde{a}}_{0}, {\tilde{a}}_{1}, . . ., {\tilde{a}}_{p})$ satisfies in the constraints (27.1) and (27.2) thus $\tilde{a} = ({\tilde{a}}_{0}, {\tilde{a}}_{1}, . . ., {\tilde{a}}_{p})$ is a feasible solution for problem (27). To show the optimality, note that the achievement function value of (27) for this feasible solution is 0, which is the least possible value for a non-negative function. Therefore, the suggested values yield the optimal solution of problem (27) and we have ${\hat{\tilde{a}}}_{j}^{+} ⊖ {\hat{\tilde{a}}}_{j}^{-} \approx {\tilde{a}}_{j}$ i.e., the optimal solution of problem (27) contains $\tilde{a} = ({\tilde{a}}_{0}, {\tilde{a}}_{1}, . . ., {\tilde{a}}_{p})$ .

Proof. (ii). The second part of the theorem immediately follows from (27.1) and (27.2). In fact, from (30) and (31) for m = 1, . . . , 4, j = 1, . . . , p, i = 1, . . . , n we have: $\begin{matrix} {\hat{y}}_{i}^{u_{m}} = {\hat{a}}_{0}^{u_{m}} + \sum_{j = 1}^{p} ({\hat{a}}_{j}^{u_{m} +} \times x_{ji}^{u_{m}} - {\hat{a}}_{j}^{u_{5 - m}^{-}} \times x_{ji}^{u_{m}}) \\ = a_{0}^{u_{m}} + \sum_{j = 1}^{p} a_{j}^{u_{m}} \times x_{ji}^{u_{m}} = y_{i}^{u_{m}} m = 1, . . ., 4, \\ j = 1, . . ., p, i = 1, . . ., n \end{matrix}$ (32) $\begin{matrix} {\hat{y}}_{i}^{l_{m}} = {\hat{a}}_{0}^{l_{m}} + \sum_{j = 1}^{p} ({\hat{a}}_{j}^{l_{m} +} \times x_{ji}^{l_{m}} - {\hat{a}}_{j}^{l_{5 - m}^{-}} \times x_{ji}^{l_{m}}) \\ = a_{0}^{l_{m}} + \sum_{j = 1}^{p} a_{j}^{l_{m}} \times x_{ji}^{l_{m}} = y_{i}^{l_{m}} m = 1, . . ., 4, \\ j = 1, . . ., p, i = 1, . . ., n \end{matrix}$ (33)

From (32) and (33) we conclude that ${\tilde{\tilde{Y}}}_{i} \approx {\tilde{\tilde{y}}}_{i}$ and the proof is completed.

3.2 Computational complexity of proposed model

Consider that there are n sample data in the Ω (the universe of discourse), denoted by $({\tilde{x}}_{ij}, {\tilde{y}}_{i})$ , i = 1, . . . , n ; j = 0, . . . , p where all of the explanatory variables $({\tilde{x}}_{ij})$ , the response variable $({\tilde{y}}_{i})$ and the regression coefficients $({\tilde{a}}_{j})$ are trapezoidal IT2 FNs (let ${\tilde{x}}_{i 0} = \tilde{1}$ ). Then in the model (27), the number of constraints (denoted by N_C) and variable (denoted by N_V) as follows: $N_{V} = 16 \times n + 5 (p + 1)$ (34.1) $N_{C} = 8 \times n$ (34.2)

From (34.1) and (34.2), it can be concluded that in a fuzzy linear regression based on FLAD, when increase the number of data and explanatory variables, the number of constraints increases rapidly and this leads to computational complexity in fuzzy regression. Also with removing or adding an explanatory variable, all constraints must be readjusted. Therefore, the number of data and explanatory variables might be limited in fuzzy regression analysis based on FLAD based on LP. Of course, with available linear programming solvers (e.g. MATLAB, LINGO, MPL, Maximal Software), the size of linear programming problems is not important.

3.3 Outliers

Frequently in fuzzy/crisp mathematical programming problems, the data set contains some elements that are outliers [26]. Among the data set, an observation that has a bigger residual value than the majority of data is called an outlier or abnormal data [39 , 51]. Outliers may occur due to unwanted errors during the collection, recording, or transcribing of the data. [13 , 33] In the dataset may have various types of outliers including

Outlier is in the spreads and/or centers of the output variables,

Outlier is in the spreads and/or centers of the input variables,

Outlier is in the spreads and/or centers of the input-output variables,

Outlier is a very important aspect of data analysis and the existence of outliers in a set of experimental observations may dramatically influence the value of the parameter estimates, therefore detection and elimination of outliers have a significant effect on overall results. Of course, the presence of the outliers in the dataset is not always a disadvantage and drawback. For example, in data mining, the detection of outliers in the dataset is more interesting than the detection of inliers; while in clustering problems, outliers supposed as noise observations that should be deleted to performing a more reliable clustering [39, 51].

The first and third quartile are insensitive to outlying data values and the difference between them (i.e., third quartile first quartile=Q₃ - Q₁) is known as the interquartile range. The interquartile range (IQR) is often used to find outliers in dataset. In this study we use the interquartile range (IQR) to define the mild and extreme outlier cutoffs as [Q₁ - 1.5 × IQR, Q₃ + 1.5 × IQR] and [Q₁ - 3 × IQR, Q₃ + 3 × IQR]], respectively. Therefore

the data values that lie below Q₁ - 1.5 × IQR or above Q₃ + 1.5 × IQR are called mild outliers and marked with a circle “o”

the data values that lie below Q₁ - 3 × IQR or above Q₃ + 3 × IQR are called extreme outliers and marked with an asterisk “∗”

3.4 Performance measures

To evaluate the fuzzy regression models, several performance measures have been proposed by authors [3 , 59]. In this section, we introduce some new performance measures to evaluate the IT2 fuzzy regression models.

3.4.1 Fuzzy similarity measure

The fuzzy similarity measure proposed by [48] is used to determine the similarity of two fuzzy sets [10, 54]. In a fuzzy regression model, a larger value of fuzzy similarity measure indicates better fuzzy model performance.

To determine the similarity degree between the estimated and observed IT2 fuzzy responses, we introduce new mean similarity measures as

$MS = \frac{1}{n} \sum_{t = 1}^{3} S ({\hat{\dot{y}}}_{i}, {\dot{y}}_{i}) = \frac{1}{n} [\frac{1}{3} \sum_{t = 1}^{3} \frac{min ({\hat{y}}_{i}^{t}, y_{i}^{t})}{max ({\hat{y}}_{i}^{t}, y_{i}^{t})}]$ where, $\dot{y}$ and $\hat{\dot{y}}$ are the observed and estimated values of output variable respectively.<

$MS = \frac{1}{n} \sum_{i = 1}^{n} S ({\hat{\tilde{y}}}_{i}, {\tilde{y}}_{i}) = \frac{1}{n} [\frac{1}{8} \sum_{t = 1}^{4} (\frac{min ({\hat{y}}_{i}^{u_{t}}, y_{i}^{u_{t}})}{max ({\hat{y}}_{i}^{u_{t}}, y_{i}^{u_{t}})} + \frac{min ({\hat{y}}_{i}^{l_{t}}, y_{i}^{l_{t}})}{max ({\hat{y}}_{i}^{l_{t}}, y_{i}^{l_{t}})})]$

where,

\tilde{y}

and

\hat{\tilde{y}}

are the observed and estimated values of trapezoidal IT2F response variable respectively.

3.4.2 Fuzzy distance measure

This measure is used to determine the absolute differences between the estimated and observed fuzzy responses. A smaller mean of fuzzy distances indicates smaller estimation errors, and thus higher prediction performance [10 , 48]. A good fuzzy regression model will produce a smaller mean fuzzy absolute error (mean fuzzy distance). To determine the distance between the estimated and observed fuzzy responses, we introduce new mean absolute error (ME) and mean relative erro (MRD) measures as

$ME = \frac{1}{n} \sum_{i = 1}^{n} d_{H} (\hat{\dot{y}}, \dot{y}) = \frac{1}{n} \sum_{i = 1}^{n} [\frac{1}{3} \sum_{t = 1}^{3} | {\hat{y}}_{i}^{t} - y_{i}^{t} |]$ , $MRD = \frac{1}{n} \sum_{i = 1}^{n} [\frac{1}{3} \sum_{t = 1}^{3} \frac{| {\hat{y}}_{i}^{t} - y_{i}^{t} |}{y_{i}^{t}}]$

where

\dot{y}

and

\hat{\dot{y}}

are the observed and estimated values of triangular T1F response variable respectively.

$ME = \frac{1}{n} \sum_{i = 1}^{n} d_{H} ({\hat{\tilde{y}}}_{i}, {\tilde{y}}_{i}) = \frac{1}{n} \sum_{i = 1}^{n} [\frac{1}{8} \sum_{t = 1}^{4} (| {\hat{y}}_{i}^{u_{t}} - y_{i}^{u_{t}} | + | {\hat{y}}_{i}^{l_{t}} - y_{i}^{l_{t}} |)]$ , $MRE = \frac{1}{n} \sum_{i = 1}^{n} [\frac{1}{8} \sum_{t = 1}^{4} (\frac{| {\hat{y}}_{i}^{u_{t}} - y_{i}^{u_{t}} |}{y_{i}^{u_{t}}} + \frac{| {\hat{y}}_{i}^{l_{t}} - y_{i}^{l_{t}} |}{y_{i}^{l_{t}}})]$

where

\tilde{y}

and

\hat{\tilde{y}}

are the observed and estimated values of trapezoidal IT2F response variable respectively.

3.5 Goodness-of-fit criteria

To investigate the goodness of fit of the proposed model, a leave-one-out cross-validation (LOOCV) approach is employed, which uses a single observation from the whole data sets as the validation data set, and the remaining observations as the training data sets. The validation process is repeated until each observation of the whole data sets is used as follows steps [31]:

Step 1: The 1th observation (i.e., $(x_{1}, {\tilde{y}}_{1})$ ) left out from the data set and we develop the new fuzzy regression model based on the n - 1 remaining observations and then based on the new obtained model, predict the response value of the 1th observation (denoted by ${\hat{\tilde{y}}}_{(- 1)}$ ).

Step 2: Repeat procedure step 1 to calculate ${\hat{\tilde{y}}}_{(- 2)}, . . ., {\hat{\tilde{y}}}_{(- n)}$ , $d_{(- i)} = | {\tilde{y}}_{i} - {\hat{\tilde{y}}}_{(- i)} |$ for i = 1, . . . , n and obtain ${Md}_{(- i)} = \frac{1}{n} \sum_{i = 1}^{n} d_{(- i)} = \frac{1}{n} \sum_{i = 1}^{n} | {\tilde{y}}_{i} - {\hat{\tilde{y}}}_{(- i)} |$ , $Md = \frac{1}{n} \sum_{i = 1}^{n} d_{i} = \frac{1}{n} \sum_{i = 1}^{n} | {\tilde{y}}_{i} - {\hat{\tilde{y}}}_{i} |$ , and then calculate the relative error of the estimated responses as $R_{E} = \frac{1}{Md} | Md - {Md}_{(-)} |$

Step 3: If the value of Md_(-i) is near to the value of Md and R_E become smaller, then the predictive ability of the proposed model is convenient and it is an optimal model.

3.6 Numerical examples

In this section, we provide two numerical examples. The first example illustrates the theoretical results of the proposed approach and explain how the proposed method is applicable to derive the regression model for trapezoidal interval type-2 fuzzy data. In the second example, we compare the performance of proposed model with some existing well-known models using the training data designed by Tanaka et al. [55]

Example 1. Since examples containing trapezoidal IT2F data did not exist in previous studies, in this example, we transformed the fuzzy output data given in [29, 42] to trapezoidal IT2F data with the same support. The dataset used in this example (listed in Table 1) includes 22 observations with one crisp explanatory variable and one fuzzy response variable (the input observations are crisp numbers and the output observations are trapezoidal IT2 fuzzy numbers).

Table 1
Values of input (x_j) and output the response variable ( ${\tilde{y}}_{j}$ )

j x _j ${\tilde{y}}_{j}$

1 6.73 [(6.13,6.45,7.12,7.38),(5.92,6.85,7.59,8.02)]

2 7.15 [(5.08,5.35,6.45,6.97),(4.68,5.95,7.08,7.41)]

3 7.23 [(6.93,7.208.50,9.53,),(6.75,8.05,9.69,10.2)]

4 8.14 [(7.00,8.61,9.31,9.63),(6.90,9.05,9.71,10.1)]

5 7.91 [(9.70,9.70,10.1,10.2),(9.50,9.75,10.5,10.8)]

6 5.14 [(5.24,5.55,6.00,6.24),(4.33,5.91,6.43,6.92)]

7 5.64 [(5.40,5.71,6.01,6.20),(4.73,5.91,6.33,6.72)]

8 6.86 [(7.50,7.82,8.43,8.71),(6.55,8.56,8.72,9.53)]

9 6.43 [(5.48,6.857.94,8.31),(5.76,6.10,7.25,7.58)]

10 8.78 [(10.0,10.5,11.3,11.5),(9.00,10.8,11.7,12.0)]

11 4.83 [(5.35,5.62,6.00,6.50),(4.56,5.72,6.32,6.75)]

12 7.66 [(9.01,9.25,12.1,12.3),(8.50,10.8,13.7,14.1)]

13 5.74 [(5.10,5.35,6.00,6.14),(4.99,5.85,6.95,7.00)]

14 6.95 [(7.60,8.20,8.43,8.95),(6.20,8.76,9.00,9.53)]

15 8.59 [(9.72,10.6,11.4,11.6),(9.15,11.0,11.7,12.0)]

16 4.27 [(4.50,4.80,5.50,6.20),(4.20,5.30,6.50,7.20)]

17 5.31 [(5.83,5.94,6.41,6.48),(5.11,6.23,7.12,7.23)]

18 6.64 [(5.61,6.00,6.31,6.45),(4.93,6.21,6.54,7.02)]

19 7.08 [(5.93,6.12,7.00,7.13),(5.25,6.35,7.39,8.21)]

20 6.12 [(7.48,8.02,8.93,9.21),(7.05,8.76,9.22,10.1)]

21 6.23 [(5.89,5.93,6.31,6.75),(5.34,6.05,6.93,7.13)]

22 5.93 [(5.20,5.75,6.20,6.50),(4.64,5.95,6.65,7.20)]

j	x _j	${\tilde{y}}_{j}$
1	6.73	[(6.13,6.45,7.12,7.38),(5.92,6.85,7.59,8.02)]
2	7.15	[(5.08,5.35,6.45,6.97),(4.68,5.95,7.08,7.41)]
3	7.23	[(6.93,7.208.50,9.53,),(6.75,8.05,9.69,10.2)]
4	8.14	[(7.00,8.61,9.31,9.63),(6.90,9.05,9.71,10.1)]
5	7.91	[(9.70,9.70,10.1,10.2),(9.50,9.75,10.5,10.8)]
6	5.14	[(5.24,5.55,6.00,6.24),(4.33,5.91,6.43,6.92)]
7	5.64	[(5.40,5.71,6.01,6.20),(4.73,5.91,6.33,6.72)]
8	6.86	[(7.50,7.82,8.43,8.71),(6.55,8.56,8.72,9.53)]
9	6.43	[(5.48,6.857.94,8.31),(5.76,6.10,7.25,7.58)]
10	8.78	[(10.0,10.5,11.3,11.5),(9.00,10.8,11.7,12.0)]
11	4.83	[(5.35,5.62,6.00,6.50),(4.56,5.72,6.32,6.75)]
12	7.66	[(9.01,9.25,12.1,12.3),(8.50,10.8,13.7,14.1)]
13	5.74	[(5.10,5.35,6.00,6.14),(4.99,5.85,6.95,7.00)]
14	6.95	[(7.60,8.20,8.43,8.95),(6.20,8.76,9.00,9.53)]
15	8.59	[(9.72,10.6,11.4,11.6),(9.15,11.0,11.7,12.0)]
16	4.27	[(4.50,4.80,5.50,6.20),(4.20,5.30,6.50,7.20)]
17	5.31	[(5.83,5.94,6.41,6.48),(5.11,6.23,7.12,7.23)]
18	6.64	[(5.61,6.00,6.31,6.45),(4.93,6.21,6.54,7.02)]
19	7.08	[(5.93,6.12,7.00,7.13),(5.25,6.35,7.39,8.21)]
20	6.12	[(7.48,8.02,8.93,9.21),(7.05,8.76,9.22,10.1)]
21	6.23	[(5.89,5.93,6.31,6.75),(5.34,6.05,6.93,7.13)]
22	5.93	[(5.20,5.75,6.20,6.50),(4.64,5.95,6.65,7.20)]

Table 2

Values of observed and predicted response variable and the distance between them

i	x _i	$\tilde{y}$	$\hat{\tilde{y}}$	$\| {\tilde{y}}_{i} - {\hat{\tilde{y}}}_{i} \|$
1	6.73	[(6.13,6.45,7.12,7.38),(5.92,6.85,7.59,8.02)]	[(6.60,7.07,7.80,8.10),(6.09,7.41,8.45,8.82)]	0.610
2	7.15	[(5.08,5.35,6.45,6.97),(4.68,5.95,7.08,7.41)]	[(6.64,7.10,7.83,8.14),(6.12,7.45,8.49,8.87)]	1.482
3	7.23	[(6.93,7.208.50,9.53),(6.75,8.05,9.69,10.2)]	[(6.64,7.11,7.84,8.10),(6.13,7.46,8.49,8.87)]	0.771
4	8.14	[(7.00,8.61,9.31,9.63),(6.90,9.05,9.71,10.1)]	[(6.71,7.18,7.92,8.23),(6.19,7.54,8.58,8.97)]	1.21
5	7.91	[(9.70,9.70,10.1,10.2),(9.50,9.75,10.5,10.8)]	[(6.69,7.16,7.90,8.21),(6.17,7.52,8.56,8.95)]	2.386
6	5.14	[(5.24,5.55,6.00,6.24),(4.33,5.91,6.43,6.92)]	[(6.49,6.94,7.66,7.95),(5.99,7.27,8.30,8.65)]	1.577
7	5.64	[(5.40,5.71,6.01,6.20),(4.73,5.91,6.33,6.72)]	[(6.52,6.98,7.70,8.00),(6.02,7.32,8.34,8.70)]	1.572
8	6.86	[(7.50,7.82,8.43,8.71),(6.55,8.56,8.72,9.53)]	[(6.61,7.05,7.81,8.11),(6.10,7.42,8.46,8.83)]	0.673
9	6.43	[(5.76,6.10,7.25,7.58),(5.48,6.857.94,8.31)]	[(6.58,7.04,7.77,8.07),(6.07,7.39,8.42,8.79)]	0.608
10 °	8.78	[(10.0,10.5,11.3,11.5),(9.00,10.8,11.7,12.0)]	[(6.76,7.23,7.98,8.29),(6.23,7.56,8.64,9.04)]	3.123
11	4.83	[(5.35,5.62,6.00,6.50),(4.56,5.72,6.32,6.75)]	[(6.46,6.91,7.63,7.92),(5.97,7.25,8.27,8.62)]	1.525
12 °	7.66	[(9.01,9.25,12.1,12.3),(8.50,10.8,13.7,14.1)]	[(6.67,7.14,7.88,8.18),(6.16,7.49,8.53,8.92)]	3.595
13	5.74	[(5.10,5.35,6.00,6.14),(4.99,5.85,6.95,7.00)]	[(6.10,6.52,7.20,7.47),(6.03,7.33,8.35,8.71)]	1.291
14	6.95	[(7.60,8.20,8.43,8.95),(6.20,8.76,9.00,9.53)]	[(6.62,7.09,7.82,8.12),(6.11,7.43,8.47,8.84)]	0.772
15 °	8.59	[(9.72,10.6,11.4,11.6),(9.15,11.0,11.7,12.0)]	[(6.74,7.22,7.96,8.27),(6.22,7.58,8.62,9.02)]	3.182
16	4.27	[(4.50,4.80,5.50,6.20),(4.20,5.30,6.50,7.20)]	[(6.42,6.87,7.58,7.87),(5.93,7.20,8.22,8.56)]	1.804
17	5.31	[(5.83,5.94,6.41,6.48),(5.11,6.23,7.12,7.23)]	[(6.50,6.95,7.67,7.97),(6.00,7.29,8.31,8.67)]	1.126
18	6.64	[(5.61,6.00,6.31,6.45),(4.93,6.21,6.54,7.02)]	[(6.60,7.06,7.79,8.09),(6.09,7.41,8.44,8.81)]	1.401
19	7.08	[(5.93,6.12,7.00,7.13),(5.25,6.35,7.39,8.21)]	[(6.63,7.107.83,8.13),(6.12,7.44,8.48,8.86)]	0.901
20	6.12	[(7.48,8.02,8.93,9.21),(7.05,8.76,9.22,10.1)]	[(6.56,7.02,7.74,8.04),(6.05,7.36,8.39,8.75)]	1.110
21	6.23	[(5.89,5.93,6.31,6.75),(5.34,6.05,6.93,7.13)]	[(6.57,7.03,7.75,8.05),(6.06,7.35,8.40,8.77)]	1.208
22	5.93	[(5.20,5.75,6.20,6.50),(4.68,5.95,6.65,7.20)]	[(6.54,7.00,7.73,8.02),(6.04,7.34,8.37,8.73)]	1.462

To illustrate the theoretical results of the proposed two-stage method and explain how to use it to derive a model with such data, in the first stage, to provide an initial model (before removing outliers from the original dataset), we estimate the IT2 fuzzy regression coefficients ( ${\hat{\tilde{a}}}_{0}, {\hat{\tilde{a}}}_{1}$ ) as ${\hat{\tilde{a}}}_{0} = [(6.0999, 6.5213 - . 5 pt, 7.198 - . 5 pt, 7.46921) - . 5 pt, (5.64389 - . 5 pt, 6.8244 - . 5 pt, 7.8154, 8.1002)]$ and ${\hat{\tilde{a}}}_{1} = [(0.07499, 0.0812, 0.089, 0.0934), (0.067, 0.0875, 0.09385, 0.1071)]$ . By replacing ${\hat{\tilde{a}}}_{0}, {\hat{\tilde{a}}}_{1}$ , the derived initial regression model will be as: $\begin{matrix} \hat{\tilde{y}} = [({\hat{a}}_{0}^{l_{1}} + x {\hat{a}}_{1}^{l_{1}}, {\hat{a}}_{0}^{l_{2}} + x {\hat{a}}_{1}^{l_{2}}, {\hat{a}}_{0}^{l_{3}} + x {\hat{a}}_{1}^{l_{3}}, {\hat{a}}_{0}^{l_{4}} + x {\hat{a}}_{1}^{l_{4}}), \\ ({\hat{a}}_{0}^{u_{1}} + x {\hat{a}}_{1}^{u_{1}}, {\hat{a}}_{0}^{u_{2}} + x {\hat{a}}_{1}^{u_{2}}, {\hat{a}}_{0}^{u_{3}} + x {\hat{a}}_{1}^{u_{3}}, {\hat{a}}_{0}^{u_{4}} + x {\hat{a}}_{1}^{u_{4}})] \end{matrix}$ (35) $\begin{matrix} \hat{\tilde{y}} = [(6.099 + 0.074 x, 6.521 + 0.081 x, 7.198 \\ + 0.089 x, 7.469 + 0.093 x), (5.644 + 0.067 x, \\ 6.824 + 0.087 x, 7.815 + 0.093 x, 8.100 + 0.107 x)] \end{matrix}$

Figure 4.1 depicts the observed and fitted values of ${\dot{y}}^{u} = (y^{u_{1}}, y^{u_{2}}, y^{u_{3}}, y^{u_{4}})$ based on the results presented in the fourth and fifth columns of Table 2, in which the vertical axis represents the components of ${\dot{y}}^{u} = (y^{u_{1}}, y^{u_{2}}, y^{u_{3}}, y^{u_{4}})$ and the horizontal axis represents the values of x.

Fig. 4.1

Plot of the observed and fitted values of ${\dot{y}}^{u}$

Table 3

Results of LOOCV approach

i	x _i	${\tilde{y}}_{i}$	${\hat{\tilde{y}}}_{i}$	${\hat{\tilde{y}}}_{(- i)}$	d _i	d _(-i)
1	7.15	[(5.08,5.35,6.45,6.78),(4.68,5.95,7.08,7.41)]	[(6.52,7.01,7.79,8.10),(6.03,7.37,8.46,8.85)]	[(6.41,6.92,7.59,7.90),(5.94,7.22,8.44,8.60)]	1.419	1.281
2	7.23	[(6.93,7.20,8.50,9.53),(6.75,8.05,9.69,10.2)]	[(6.53,7.02,7.80,8.11),(6.04,7.38,8.47,8.86)]	[(6.41,6.89,7.64,7.49),(5.95,7.22,8.17,8.61)]	0.831	1.06
3	8.14	[(7.00,8.61,9.31,9.63),(6.90,9.05,9.71,10.1)]	[(6.11,7.46,8.55,8.69),(6.10,7.46,8.55,8.96)]	[(6.39,6.89,7.67,7.96),(5.69,7.25,7.86,8.70)]	1.016	1.482
4	6.73	[(6.13,6.45,7.12,7.38),(5.92,6.85,7.59,8.02)]	[(5.96,7.26,8.34,8.71),(6.01,7.34,8.42,8.80)]	[(6.44,6.91,7.69,8.01),(5.96,7.27,8.54,8.73)]	0.715	0.512
5	7.91	[(9.70,9.70,10.1,10.2),(9.50,9.75,10.5,10.8)]	[(6.69,7.16,7.90,8.21),(6.17,7.52,8.56,8.95)]	[(6.39,6.99,7.77,8.01),(5.79,7.45,7.96,8.75)]	2.387	2.643
6	5.14	[(5.24,5.55,6.00,6.24),(4.33,5.91,6.43,6.92)]	[(6.37,6.84,7.61,7.91),(5.90,7.20,8.27,8.63)]	[(6.50,6.98,7.76,8.07),(6.04,7.34,8.43,8.80)]	1.514	1.664
7	5.64	[(5.40,5.71,6.01,6.20),(4.73,5.91,6.33,6.72)]	[(6.41,8.88,7.65,7.96),(5.94,7.24,8.32,8.68)]	[(6.43,6.92,7.70,8.00),(6.00,7.26,8.41,8.72)]	1.759	1.557
8	6.86	[(7.50,7.82,8.43,8.71),(6.55,8.56,8.72,9.53)]	[(6.47,6.95,7.73,8.04),(6.02,7.13,8.39,8.77)]	[(6.46,6.95,7.74,8.06),(6.01,7.32,8.44,8.79)]	0.734	0.753
9	6.43	[(5.76,6.10,7.25,7.58),(5.48,6.85,7.94,8.31)]	[(6.47,6.95,7.73,8.04),(5.99,7.13,8.39,8.77)]	[(6.54,7.02,7.85,8.14),(6.07,7.39,8.53,8.89)]	0.525	0.648
10	4.83	[(5.35,5.62,6.00,6.50),(4.56,5.72,6.32,6.75)]	[(6.35,6.71,7.58,7.88),(5.88,7.16,8.24,8.59)]	[(6.38,6.86,7.62,7.93),(5.91,7.20,8.31,8.65)]	1.446	1.507
11	5.74	[(5.10,5.35,6.00,6.14),(4.99,5.85,6.95,7.00)]	[(6.41,6.89,7.58,7.88),(5.94,7.25,8.33,8.69)]	[(6.49,6.92,7.73,8.03),(6.02,7.27,8.41,8.78)]	1.448	1.537
12	6.95	[(7.60,8.20,8.43,8.95),(6.20,8.76,9.00,9.53)]	[(6.51,6.99,7.77,8.09),(6.02,7.36,8.44,8.83)]	[(6.39,6.91,7.72,8.05),(5.90,7.31,8.41,8.82)]	0.833	0.892
13	4.27	[(4.50,4.80,5.50,6.20),(4.20,5.30,6.50,7.20)]	[(6.30,6.77,7.53,7.83),(5.84,7.12,8.19,8.53)]	[(6.44,6.93,7.71,8.04),(5.96,7.30,8.40,8.83)]	1.739	1.929
14	5.31	[(5.83,5.94,6.41,6.48),(5.11,6.23,7.12,7.23)]	[(6.38,6.85,7.62,7.93),(5.91,7.21,8.28,8.65)]	[(6.57,7.02,7.79,8.11),(6.07,7.37,8.48,8.89)]	1.060	1.243
15	6.64	[(5.61,6.00,6.31,6.45),(4.93,6.21,6.54,7.02)]	[(6.48,6.96,7.75,8.06),(6.00,7.33,8.41,8.79)]	[(6.50,7.01,7.75,8.02),(6.01,7.38,8.39,8.78)]	1.339	1.347
16	7.08	[(5.93,6.12,7.00,7.13),(5.25,6.35,7.39,8.21)]	[(6.44,6.92,7.70,8.01),(6.03,7.28,8.36,8.74)]	[(6.55,7.05,7.83,8.43),(6.08,7.47,8.52,8.87)]	0.840	0.929
17	6.12	[(7.48,8.02,8.93,9.21),(7.05,8.76,9.22,10.1)]	[(6.44,6.92,7.70,8.01),(5.97,7.28,8.36,8.74)]	[(6.60,7.11,7.86,8.18),(6.12,7.49,8.55,8.93)]	1.146	0.993
18	6.23	[(5.89,5.93,6.31,6.75),(5.34,6.05,6.93,7.13)]	[(6.45,6.93,7.71,8.02),(5.98,7.29,8.37,8.75)]	[(6.50,6.99,7.75,8.06),(6.02,7.34,8.42,8.79)]	1.343	1.503
19	5.93	[(5.20,5.75,6.20,6.50),(4.68,5.95,6.65,7.20)]	[(6.43,6.90,7.68,7.99),(5.50,7.27,8.34,8.72)]	[(6.50,7.00,7.78,8.10),(6.02,7.37,8.47,8.85)]	1.160	1.224

In the second stage, in order to identify and remove the outliers, the predicted response values ( ${\hat{\tilde{y}}}_{i}$ ) and absolute residuals values ( $d_{i} = | {\tilde{y}}_{i} - {\hat{\tilde{y}}}_{i} |$ ) are obtained which are shown in the fourth and fifth columns of Table 2. Based on the results presented in the third and fourth columns of Table 2, the median, the first and third quartiles of d_i, denoted by Me, Q₁, Q₃, respectively, are obtained as Me = 1.346, Q₁ = 0.8365, Q₃ = 1.6906. Therefore, the IQR = 0.8541 and cutoffs are Q₁ - 1.5IQR = -0.4446, Q₃ + 1.5IQR = 2.9717, Q₁ - 3IQR = -1.7258, Q₃ + 3 × IQR = 4.2529. Hence, the data point of Nos. 10, 12, and 15 are considered mild outlier, which are marked with blue circles in the first and fifth columns of Table 2.

After identifying and eliminating the outliers, using clean dataset given in the second and third columns of Table 3, the trapezoidal IT2F regression coefficients are obtained as ${\hat{\tilde{a}}}_{0} = [(5.9793, 6.4223, 7.141, 7.41923), (5.5594, 6.7284, 7.7802, 8.064)]$ , ${\hat{\tilde{a}}}_{1} = [(0.0759, 0.0818, 0.091, 0.0956), (0.067, 0.0905, 0.09514, 0.1099)]$ . By replacing the estimated fuzzy regression coefficients, the final model will be as: $\hat{\tilde{y}}$ $\begin{matrix} \hat{\tilde{y}} \hat{\tilde{y}} = [(5.979 + 0.076 x, 6.422 + 0.082 x, 7.114 \\ + 0.091 x, 7.419 + 0.095 x), (5.559 + 0.067 x, 6.728 \\ + 0.090 x, 7.800 + 0.095 x, 8.064 + 0.109 x)] \end{matrix}$ (36)

The trapezoidal IT2 fuzzy regression model (36) can be applied to predictive of $\tilde{\tilde{y}}$ for a new case of x. For example, if x = 7.45 then we predict the $\tilde{y}$ as $\hat{\tilde{y}} = [(6.542, 7.032, 7.791, 8.126), (6.058, 7.398, 8.507, 8.876)]$ .

To investigate the performance of the proposed model, a leave-one-out cross-validation (LOOCV) approach is employed and the results presented in the fifth to seventh columns of Table 3. $\begin{matrix} Md = \frac{1}{19} \sum_{i = 1}^{19} d_{i} = \frac{1}{19} \sum_{i = 1}^{19} | {\tilde{y}}_{i} - {\hat{\tilde{y}}}_{i} | = 1.224 \\ {Md}_{(-)} = \frac{1}{19} \sum_{i = 1}^{19} d_{(- i)} = \frac{1}{19} \sum_{i = 1}^{19} | {\tilde{y}}_{i} - {\hat{\tilde{y}}}_{(- i)} | = 1.298 \\ R_{E} = \frac{1}{Md} | Md - {Md}_{(-)} | = \frac{1}{1.224} | 1.224 - 1.298 | \\ = 0.0604 \end{matrix}$

Table 4

Dataset for Example 2

Obs.	x	${\dot{y}}_{T}$	${\ddot{y}}_{H}$	${\tilde{y}}_{P}$
1	1	(6.2,8.0,9.8)	[(5.45,6.2,6.7),8.0,(9.3,9.8,10.55,)]	[(6.2,8.0,8.5,9.8),(5.45,6.7,9.3,10.55)]
2	2	(4.2,6.4,8.6)	[(3.45,4.2,4.7),6.4,(8.1,8.6,10.95)]	[(4.2,6.0,6.9,8.6),(3.45,4.7,8.1,10.95)]
3	3	(6.9,9.5,12.1)	[(5.95,6.9,7.4),9.5,(11.6,12.1,12.95)]	[(6.9,9.0,10.0,12.1),(5.95,7.4,11.6,12.58)]
4	4	(10.9,13.5,16.1)	[(9.65,10.4,10.9),13.0,(15.1,15.6,16.35)]	[(10.4,13.0,13.5,15.6),(9.65,10.9,15.1,16.35)]
5	5	(10.6,13.0,15.4)	[(10.4,10.6,11.6),13.5,(15.4,15.9,16.65)]	[(10.6,13.0,14.0,15.9)10.36,(10.36,11.6,15.4,16.65)]

Table 5

Comparison of fuzzy regression models and their capability indices in Example 2

Proposed by	Model	Method	Criteria’s
MRE	MS
Chen and Nien [10]	${\hat{\dot{y}}}_{CN} = (5.1, 6.75, 8.4) + (1.1, 1.25, 1.4) x$	T1 FLSD	0.328	0.789
Diamond [15]	$\hat{\dot{y}} = (3.11, 4.95, 6.79) + (1.55, 1.71, 1.87) x$	T1 FLSD	0.282	0.801
Hosseinzadeh et al. [27]	${\hat{\tilde{y}}}_{HO} = [(6.75; 2.55; 1.8; 1.3) + (1.25; 0.2; 0.2; 0.2) x$	Quasi T2 FLAD	0.152	0.884
Kim and Bishu [34]	${\hat{\dot{y}}}_{KB} = (3.11, 4.95, 6.84) + (1.55, 1.71, 1.82) x$	T1 FLAD	0.279	0.803
Li et al. [37]	${\hat{\dot{y}}}_{LI} = (5.527, 2.200) + (1.494, 0.0999) x$	T1 FLAD	0.173	0.478
Shafaei Bajestani et al. [49]	$\begin{matrix} {\hat{\tilde{y}}}_{SH} = [(1.5, 5.3), 5.52 (5.7, 9.06)] + \\ x \end{matrix}$	Triangular
T2 FLSD	0.22	0.792
Tanaka et al. [55]	${\hat{\dot{y}}}_{TA} = (0, 3.85, 7.7) + 2.1 x$	T1 FLAD	0.336	0.759
Zeng et al. [59]	${\hat{\dot{y}}}_{ZE} = (5.521, 2.200) + (1.498, 0.1000) x$	T1 FLAD	0.173	0.478
Proposed model	$\begin{matrix} {\hat{\tilde{y}}}_{PM} = [(4.95, 6.25, 7.35, 8.40), (4.30, 5.45, 7.95, 9.1) + \\ [(1.22, 1.25, 1.3, 1.38), (1.20, 1.24, 1.35, 1.45)] x \end{matrix}$	Trapezoidal IT2 FLAD	0.151	0.891

Since the value of Md_(-) = 1.298 is near to the value of Md = 1.224 and R_E = 0.0604 become smaller, the predictive ability of the proposed model is convenient and it is an optimal model. The obtained results of Table 3 are illustrated in Fig. 4.2. In view of the fact that the trapezoidal IT2 fuzzy number cannot be represented as a crisp in a coordinate system, in Fig. 4.2, we use the line segments to represent the values of ${\tilde{y}}_{i}$ , ${\hat{\tilde{y}}}_{i}$ and ${\hat{\tilde{y}}}_{(- i)}$ that the vertical axis represents the components of $\tilde{y}$ and the horizontal axis represents the values of input.

Fig. 4.2

The line segments to represent of ${\tilde{y}}_{i}$ , ${\hat{\tilde{y}}}_{i}$ and ${\hat{\tilde{y}}}_{(- i)}$ .

Example 2. (Comparative study) In this example, we intend the compare the performance of the proposed model with some existing well-known models, but none of these studies have been done with trapezoidal IT2 fuzzy data. On the other hand, in most of these studies, the data designed by Tanaka et al. [55] have been used as training data; furthermore, some researchers (including Hosseinzadeh et al. [27]) have transformed mentioned training data to T2F data. Therefore, we convert the quasi T2F output data of Example 1 given in Hosseinzadeh et al. [27] to trapezoidal IT2F output data with the same support. Then perform a comparison between proposed model (based on distance and similarity criterions proposed in this study) with some existing well-known models (including Chen-Nien [10], Diamond [15], Hosseinzadeh et al. [27], Kim-Bishu [34], Li et al [37], Shafaei Bajestani et al. [49], Tanaka et al. [55] and Zeng et al. [59]). In this example, the input- output data designed by [55] ( $x, {\dot{y}}_{T}$ ), the quasi T2F output data provided by [27] ( ${\ddot{y}}_{H}$ ) and the trapezoidal IT2F output data provided in this study ( ${\tilde{y}}_{P}$ ) are listed in Table 4.

The results of fitting the proposed model and aforementioned models as well as their performances based on the mean similarity degree (MS) between the estimated and observed fuzzy responses and men relative error (MRE) between the estimated and observed fuzzy responses are presented in Table 5. From the fourth column of Table 5, the mean relative error obtained by our model is 0.151, which is less than the mean relative error of Chen-Nien model [10], Diamond model [15], Kim-Bishu model [34], Li model [37], Shafaei Bajestani model [49], Tanaka model [55] and Zeng model [59]. The mean relative error obtained by Hosseinzadeh model [27] is 0.152, which is close to the amount obtained from the proposed model. On the other hand, by referring to the fifth column of Table 5, the mean similarity degree (MS) between the estimated and observed fuzzy responses for the proposed method is 0.891, which clearly larger than the MS obtained from the models mentioned in the second column of Table 5. The results presented in the fourth and fifth columns of Table 5 show that, while the indexes for some models (especially model Hosseinzadeh et al.) are close to that of the proposed model, the proposed model has better performance concerning MS and MRE.

4 Conclusion

The uncertainty can arise from different sources including randomness and fuzziness, which is related to chance, incomplete information, human subjective judgment, noise, etc. In this context, Fuzzy regression analysis (through creating a functional relationship between response and explanatory variables), provide efficient and effective tools for the explanation, prediction, and possibly control of randomness and fuzziness in the fuzzy dataset. In recent years, many studies in the field of fuzzy regression models have been performed and these models in terms of fitting accuracy, efficiency, robustness, sensitivity to outliers, and simplicity of calculations have been compared with each other in different ways, but most of these studies have been done based on type-1 fuzzy sets. Type-1 fuzzy sets, despite the acceptable performance, to handle the high-level linguistic uncertainty, aggregation the knowledge of several experts in decision-making scenarios have not favorable abilities. Given that trapezoidal IT2 fuzzy numbers using linguistic rating systems have a favorable ability to overcome the shortcomings of type-1 fuzzy sets, we introduced a new fuzzy linear regression model with trapezoidal IT2 fuzzy output, trapezoidal IT2 fuzzy parameters, and crisp inputs based on the fuzzy least absolute estimator. Some features and advantage of the present study are as

Some algebraic operators on type-1 fuzzy numbers are extended to perfectly normal trapezoidal IT2 fuzzy numbers then two new distance measures have been introduced.

For use of the unrestricted in sign IT2 fuzzy variables in fuzzy/crisp mathematical programming approaches a new approach have been proposed.

To estimate the IT2 fuzzy regression coefficients a novel framework is provided. In order to achieve this objective, the similarities and differences between fuzzy/crisp GP, fuzzy/crisp LP, and fuzzy/crisp NLP problems have been highlighted then answered the question how each one can lead to the other.

To determine the mild and extreme fuzzy outlier cutoffs, first, a new procedure has been introduced and then it is employed to detect and remove the fuzzy outliers in the original dataset.

To evaluate and compare the performance of trapezoidal IT2 fuzzy regression models, some new performance measures have been introduced.

Though the experimental results represent that the suggested algorithm has better efficiency and performance, but the complexity of computation is a potential problem. So, in the future study, we will further study the algorithm optimization of fuzzy least absolute deviation, and provide better solutions for the fuzzy least absolute deviation applications which contain a large amount of data. In the future, we will also introduce new IT2F distance measures based on extended Euclidean distance and a new IT2F least square deviation procedure to estimate the IT2 fuzzy regression coefficients.

Meanwhile, the extension of our suggested procedure to formulate the fuzzy linear regression approach with interval-valued Atanassov’s intuitionistic fuzzy sets based on fuzzy least absolute estimator and/or fuzzy least squares estimator can be potential topics for future work.

References

Allahviranloo

, Uncertain information and linear systems, Springer Nature Switzerland AG. (2020).

Allahviranloo

, Haghi

and Ghanbari

, The nearest symmetric fuzzy solution for a symmetric fuzzy linear system, An. St. Univ. Ovidius Constanta 20 (2012), 151–172.

Arefi

, Quantile fuzzy regression based on fuzzy outputs and fuzzy parameters, Soft Computing 24 (2020), 311–320.

Berkachy

and Donze

, Linguistic questionnaire evaluation an application of the signed distance defuzzification method on different fuzzy numbers, the impact on the skewness of the output distributions, International Journal of Fuzzy Systems and Advanced Applications 3 (2018), 12–19.

Bhartia

S.K.

and Singh

S.R.

, Interval-valued intuitionistic fuzzy linear programming problem, New Mathematics and Natural Computation 16 (2020), 53–71.

Buckley

J.J.

and Hayashi

, Fuzzy genetic algorithm and applications, Fuzzy Sets and Systems 61 (1994), 129–136.

Chachi

and Taheri

S.M.

, Multiple fuzzy regression model for fuzzy input-output data, Iranian Journal of Fuzzy Systems 13 (2016), 63–78.

Chan

K.Y.

, Kwong

C.K.

, Dillon

T.S.

and Fung

K.Y.

, An intelligent fuzzy regression approach for affective product design that captures nonlinearity and fuzziness, Journal of Engineering Design 225 (2011), 23–542.

Chen

L.H.

and Hsueh

, A mathematical programming method for formulating a fuzzy regression model based on distance criterion, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics) 37 (2007), 705–712.

10.

Chen

L.H.

and Nien

S.H.

, A new approach to formulate fuzzy regression models, Applied Soft Computing 86 (2020), 1–10.

11.

Choi

S.H.

and Buckly

, Fuzzy regression using least absolutedeviation estimators, Soft Compute 12 (2007), 257–263.

12.

Chukhrova

and Johannssen

, Fuzzy regression analysis, systematic review and bibliography, Applied Soft Computing Journal (2019), https://doi.org/10.1016/j.asoc.2019.105708.

13.

D’Urso

and Massari

, Weighted least squares and least median squares estimation for the fuzzy linear regression analysis, METRON 71 (2013), 279–306.

14.

Das

S.K.

, Mandal

and Edalatpanah

S.A.

, A mathematical model for solving fully fuzzy linear programming problem with trapezoidal fuzzy numbers, Appl Intell 46 (2017), 509–519.

15.

Diamond

, Fuzzy least squares, Information Sciences 46 (1988), 141–157.

16.

Ebrahimnejad

, An effective computational attempt for solving fully fuzzy linear programming using MOLP problem, Journal of Industrial and Production Engineering (2019), 1–11.

17.

Ezadi

and Allahviranloo

, Numerical solution of linear regression based on Z-numbers by improved neural network, Intelligent Automation & Soft Computing (2017), 1–11.

18.

Farhadinia

, Sensitivity analysis in interval valued trapezoidal fuzzy number linear programming problems, Applied Mathematical Modeling, 38 (2014), 50–62.

19.

Figueroa-Garcia

G.C.

and Hernandez

, Linear programming with interval type -2 fuzzy constraints, Constraint Programming and Decision Making 539 (2014), 19–34.

20.

Gkountakou

and Papadopoulos

, The use of fuzzy linear regression and ANFIS methods to predict the compressive strength of cement, Symmetry 12 (2020), 1–11.

21.

Hassanpour

, Maleki

and Yaghoobi

, A goal programming approach to fuzzy linear regression with fuzzy input-output data, Soft Computing 15 (2011), 1569–1580.

22.

Hesamian

and Akbari

M.G.

, Fuzzy quantile linear regression model adopted with a semi-parametric technique based on fuzzy predictors and fuzzy responses, Expert Systems with Applications 118 (2019), 585–597.

23.

Hesamian

, Akbari

G.M.

and Shams

, Parameter estimation in fuzzy partial univariate linear regression model with non fuzzy inputs and triangular fuzzy outputs, Iranian Journal of Fuzzy Systems (2021), 51–64.

24.

Hojati

, Bector

C.R.

and Smimou

, A simple method for computation of fuzzy linear regression, European Journal of Operational Research 166 (2005), 172–184.

25.

Hong

D.H.

and Hwang

, Support vector fuzzy regression machines, Fuzzy Sets and Systems 138 (2003), 271–281.

26.

Hong

D.H.

and Hwang

, Fuzzy nonlinear regression model based on LS-SV min features space, Fuzzy Systems and Knowledge Discovery, Lecture Notes in Computer Science 4223 (2006), 208–216.

27.

Hosseinzadeh

, Hassanpour

and Arefi

, A weighted goal programming approach to fuzzy linear regression with crisp inputs and type-2 fuzzy outputs, Soft Computing 19 (2015), 1143–1151.

28.

Hosseinzadeh

, Hassanpour

and Arefi

, A weighted goalprogramming approach to estimate the linear regression model in fullquasi type-2 fuzzy environment, Journal of Intelligent & FuzzySystems 30 (2016), 1319–1330.

29.

Hosseinzadeh

, Hassanpour

, Arefi

and Aman

, A weighted goal programming approach to fuzzy linear regression with quasi type-2 input-output data, TWMS Journal of Applied and Engineering Mathematics 6 (2016), 192–213.

30.

Hung

W.L.

and Yang

M.S.

, An omission approach for detecting outliers in fuzzy regression models, Fuzzy Sets and Systems 157 (2006), 3109–3122.

31.

Jiang

, Kwong

C.K.

, Chan

C.Y.

and Yung

K.L.

, A multi-objective evolutionary approach for fuzzy regression analysis, Expert Systems with Applications 130 (2019), 225–235.

32.

Kelkinnama

and Taheri

S.M.

, Fuzzy least-absolutes regression using shape-preserving operations, Information Sciences 214 (2012), 105–120.

33.

Khammar

A.H.

, Arefi

and Akbari

M.G.

, A robust least squares fuzzy regression model based on kernel function, Iranian Journal of Fuzzy Systems 17 (2020), 105–119.

34.

Kim

and Bishu

R.R.

, Evaluation of fuzzy linear regression models by comparing membership functions, Fuzzy Sets and Systems 100 (1998), 343–352.

35.

Kosko

, Fuzziness vs. probability, Int J General Systems 17 (1990), 211–240.

36.

, Zhang

, Yi

and Wang

, Uncertainty degree and modeling of interval type-2 fuzzy sets: Definition, method and application, Computers and Mathematics with Applications 66 (2013), 1822–1835.

37.

, Zeng

, Xie

and Yin

, A new fuzzy regression model based on least absolute deviation, Engineering Applications of Artificial Intelligence 52 (2016), 54–64.

38.

Manna

, Basu

and Mondal

S.k.

, Trapezoidal interval type-2 fuzzy soft stochastic set and its application in stochastic multi-criteria decision-making, Granular Computing, (2018), https://doi.org/10.1007/s41066-018-0119-0.

39.

Mashinchi

M.H.

, Orgun

M.A.

and Mashinchi

M.R.

, A least square approach for the detection and removal of outliers for fuzzy linear regressions, Second World Congress on Nature and Biologically Inspired Computing, Japan. (2010).

40.

Mishmast Nehi

and Keikha

, TOPSIS and Choquet integral hybrid technique for solving MAGDM problems with interval type 2 fuzzy numbers, Journal of Intelligent & Fuzzy Systems 30 (2016), 1301–1310.

41.

Mohamed

, The relationship between goal programming and fuzzy programming, Fuzzy Sets and Systems 89 (1997), 215–222.

42.

Mohammadi

and Taheri

S.M.

, Pedomodels fitting with fuzzy least squares regression, Iranian Journal of Fuzzy Systems 1 (2004), 45–61.

43.

Pérez-Cañedo

and Concepción-Morales

E.R.

, A method tofind the unique optimal fuzzy value of fully fuzzy linearprogramming problems with inequality constraints having unrestrictedL-R fuzzy parameters and decision variables, Expert Systems WithApplications, 123 (2019), 256–269.

44.

Pérez-Cañedo

, Rosete

and Verdegay

J.L.

, A fuzzygoal programming approach to fully fuzzy linear regression, CCIS 1238 (2020), 677–688.

45.

Rabiei

M.R.

, Arghami

N.R.

, Taheri

S.M.

and Sadeghpour Gildeh

, Least-squares approach to regression modeling in full, interval-valued fuzzy environment, Soft Computing 18 (2013), 2043–2059.

46.

Rabiei

M.R.

, Arghami

N.R.

, Taheri

and Sadeghpour

M.B.

, Fuzzy regression model with interval valued fuzzy input-output data, Soft Compute (2014). DOI 10.1007/s00500-013-1185-5

47.

Rafiei1

and Ghoreyshi

S.M.

, Symmetric fuzzy linear regression using multi-objective optimization, International Journal of Management Science and Engineering Management 6 (2012), 183–191.

48.

Rezaei

, Emoto

and Mukaidono

, New similarity measure between two fuzzy sets, Journal of Advanced Computational Intelligence and Intelligent Informatics 10 (2006), 946–953.

49.

Shafaei Bajestani

, Vahidian Kamyad

and Zare

, A piecewise type 2 fuzzy regression model, International Journal of Computational Intelligence Systems 10 (2017), 734–744.

50.

Shakouri

, Nadimi

and Ghaderi

R.F.

, Fuzzy linear regression models with absolute errors and optimum uncertainty, In: International Conference on Industrial Engineering and Engineering Management, pp. 917-912, IEEE, Singapore. (2007).

51.

Shakouri

and Nadimi

, Outlier detection in fuzzy linearregression with crisp input–output by linguistic variableview, Applied Soft Computing 13 (2013), 734–742.

52.

Sharaf

, An interval type-2 fuzzy TOPSIS using the extended vertex method for MAGDM, SN Applied Sciences 7 (2019), https://doi.org/10.1007/s42452-019-1825-1.

53.

Srinivasan

and Geetharamani

, Linear programming problem with interval type-2 fuzzy coefficients and an interpretation for its constraints, Journal of Applied Mathematics (2016), ID 8496812, 11.

54.

Taheri

S.M.

and Kelkinnama

, Fuzzy linear regression based on least absolute deviations, Iranian Journal of Fuzzy Systems 9 (2012), 121–140.

55.

Tanaka

, Uejima

and Asai

, Linear regression analysis with fuzzy model, IEEE Transactions on Systems, Man, and Cybernetics 12 (1982), 903–907.

56.

Wei

, Watada

and Pedrycz

, Design of a qualitative classification model through fuzzy support vector machine with type 2 fuzzy expected regression classifier preset, IEEJ Trans 11 (2016), 348–356.

57.

and Xu

, An approach for achieving consistency for symmetric trapezoidal interval type 2 fuzzy sets, Mathematical Problems in Engineering Article ID 4031485.

58.

Yang

and Yin

, Robust fuzzy varying coefficient regression analysis with crisp inputs and gaussian fuzzy output, Journal of Computing Science and Engineering 7 (2013), 263–227.

59.

Zeng

, Fengb

and Li

, Fuzzy least absolute linear regression, Applied Soft Computing 52 (2017), 1009–1019.

60.

Zhang

and Lu

, Robust regression analysis with L-R type fuzzy input variables and fuzzy output variable, Journal of Data Analysis and Information Processing 4 (2016), 64–80.